From Polychords to Polya Adventures in Musical Combinatorics
Michael Keith
Vinculum Press • Princeton
Lines from "Sultans of Swing": Lyrics and music by Mark Knopfler © 1978 Straitjacket Songs, Ltd. All rights administered by Rondor Music (London) Ltd. Administered in the U.S. and Canada by Almo Music Corp. (ASCAP) All rights reserved. Used by permission.
© 1991 by Vinculum Press (P.O. Box 486, Princeton NJ 08542-0486). All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher, except for brief excerpts in connection with reviews and scholarly analysis.
Library of Congress Catalog Card Number: 91-75182 ISBN 0-9630097-0-2
"For, as you of course perceive, the literary artist plays: and the sole end of his endeavor is to divert himself." JAMES BRANCH CABELL,
Straws and Prayer-Books
Preface I have often admired the mystical way of Pythagoras, and the secret magic of numbers. - Sir Thomas Browne
Music has many resemblances to algebra. - Novalis
(Baron Friedrich Von Hardenberg)
From the time of Pythagoras, who was the first to realize that the difficulty in constructing musical scales is due to the fact that there are no integer solutions to the equation
until the present time, when computer algorithms are used to compose musical pieces, there has been a lively interaction between the musical arts and the mathematical sciences. In this book, we explore various connections. b'etween the basic musical building blocks - chords, scales, and rhythms - and the area of mathematics known-as combinatorics, which is primarily concerned with counting and classifying configuratiQns of objects. We consider questions such as the following: • How many essentially different chords are there? • How many different chords are there of certain types, such as containing exactly one pair of adjacent notes of the scale? • How many different scales are there? How many of a certain type, such as 7-note diatonic scales? • How many essentially different rhythms can be constructed? In addition to these counting problems, we also consider questions of a combined musical and mathematical nature, such as: • Which chords are the harmonically simplest or most complex? Or, loosely speaking, which are the simplest or the strangest chords? • Which are most ordinary or the most unusual musical scales? • Why are certain rhythms more popular than others?
ii
Preface
There are several motivations for the mathematical study of various musical objects: • There are many mathematical questions involved which are interesting in their own right. Their solutions involve the application of combinatorics, number theory, and other mathematical techniques to new and practically useful problems. • This leads to new musical insights into the structure of music. For example, the classification of musical chords by "number of adjacencies" (Chap. 3) leads to a better understanding of the subjective qualities of different chords. The concept of the idea/ness of a musical scale (Chap. 4) provides a measure for quantifying the musical characteristics of various scales. In Chapter 5 we suggest several mathematical reasons why certain meters and rhythms are more popular than others. • The classifications can be applied to algorithms for computer music composition. By mathematically quantifying (to some degree) the different musical qualities of chords, scales, and rhythms, we can improve the musicality of algorithms for music composition, by incorporating the results of our analyses into them. • They also find applications not only in computer music, but also in human composition and performance. For example, the 351 chords and 462 scales described in Chapters 3 and 4, and the enumeration of all possible n-beat rhythms given in Chapter 5, provide a rich source of musical ideas for jazz improvisation, or for music composition in general. By systematically studying the various ingredients, we extend the boundaries of musical possibility. Overview of topics. Chapter 1 begins by introducing the necessary musical terminology and developing precise mathematical definitions of the terms chord and scale. We also consider the question of isomorphisms; i.e., under what conditions two chords or scales are considered "essentially the same". Chapter 2 introduces three mathematical concepts- chains, necklaces, and partitions- which form the basis for deriving solutions to the various musical counting problems. It also describes P6lya's theorem, a powerful tool for counting combinatorial configurations, which is used to solve the basic necklace-counting problem. In Chapter 3 we solve the basic chord-counting problems, including classifying chords by the number of notes they contain, their minimum interval, the number of semitone intervals present, and other properties. In addition, we consider the problem alluded to in the title of this book,
Preface
iii
which involves counting and classifying polychords. This leads to a (probably) new mathematical concept: the polynecklace coverage problem. Chapter 4 discusses various problems in counting musical scales. Several useful measures for classifying scales are discussed, including a surprising connection between musical scales and least-squares linear regression. We also introduce the spelling problem for scales, which asks for a classification of the 7-note scales in each musical key according to the number of accidentals required to spell the scale using the 7 musical letters (A,B,C,D,E,F,G) once each. In Chapter 5 we conclude by discussing the combinatorics of musical rhythms. Among other things, we discover a surprising relationship between musical rhythms and the delayed Fibonacci sequence. Throughout, we attempt to keep the mathematics and musical applications equally in mind. Several musical pieces are offered as examples of the usefulness of the mathematical study of musical structures.
What's new. Chapters 1 and 2 are primarily expository. We believe that most of the combinatorial/musical applications in Chapters 3-5 are new. The author would be pleased to hear of any errors in what follows, or of improvements to tables such as Table 3.21 which represent unsolved problems. Acknowledgements. For providing inspiration, either wittingly or unwittingly, thanks to: Keith Jarrett, Martin Gardner, Thomas De Quincey, Pat and Rebecca Mercuri, F. J. Budden, George Crumb, Sandra Morris, Karlheinz Stockhausen, Dave Brubeck, and, of course, George P6lya. Special thanks to Chris Kocher and Rich Poulo for reviewing the entire manuscript and making numerous valuable suggestions for improvement.
iv
Preface
List of Symbols
(~)
binomial coefficient; value in Pascal's triangle
{k1 ,k2,n····km)
multinomial coefficient
A(L,n,a)
number of n-note chords in an L-note scale containing exactly a semitone intervals
Ac(L,n,k,a)
number of n-note scales in an L-note musical scale that require a accidentals to spell in key k
Ch(L,n)
number of n-note chords in an L-note scale
d
fraction denominator in rational representation of a musical frequency or interval
e
Mean-square error between intervals in a scale and the ideal set of all-equal intervals
E
Mean-square error between the notes of a scale and the ideal scale
cp(n)
Euler cp function; number of integers < n and relatively prime to n.
F(n)
nth Fibonacci number
Fs(n)
nth "s-step delayed Fibonacci number"
I(L,n,m)
number of n-note chords in an L-note scale with minimum interval ~ m
K(n,c)
number of n-beat metrical patterns with with complexity c
K(L,n,m)
number of n-note chords in an L-note scale with minimum interval equal to m
L
length of musical scale
m(n,k)
minimum number of subnecklaces of an (n,k) necklace
M(n,k)
maximum number of subnecklaces of an (n,k) necklace
Preface
N(n,k)
v
number of n-bead 2-color necklaces with k beads of one color number of n-bead m-color necklaces with lq beads of color i
Tt(n)
number of n's in a partition of an integer
p(n,k)
number of partitions of n into exactly k parts
P(L,n,p)
number of n-note chords in an L-note scale with period equal top
P(G; zt,Z2, ... ;zn)
cycle-index polynomial of a group G
rc(n,k)
number of elementary k-fold polychords in an n-note equal-tempered scale.
r(n)
number of metrical patterns on n beats
q(n,k)
number of partitions of n into parts ~ k (or into~ k parts)
S(L,n,s)
number of n-note chords in an L-note scale with span equal to s
Sc(L,n)
number of n-note scales in an L-note musical scale
Sp(L,n,p)
number of n-note scales with period p
Sy(p,n,s)
number of p-position, n-event rhythms with syncopation values
T
taxicab distance between two scales or chords
v
variety value of a chord: number of distinct intervals present
nlm
n is an even divisor of m ("n divides m")
nfm
n is not an even divisor of m (n does not divide m)
rnl
smallest integer~ n; e.g, r3.61
= 4.
[n]
greatest integer ~ n; e.g., [3.6]
= 3.
Preface
vi
Contents Preface ................................................................................... i Chapter 1. Notes, Chords, and Scales .............................................. 1 Notes ............................................................................ 1 Chords .......................................................................... 6 Scales ............................................................................ 10 Keys ............................................................................. 11 Chapter 2. Chains, Necklaces and Partitions ....................................... 15 Basic Definitions ............................................................... 15 Counting Chains .......· ........................................................ 17 Counting Necklaces ........................................................... 23 Counting Partitions ............................................................ 34 The Musical Connection ...................................................... 36 Chapter 3. Counting Chords .......................................................... 39 Basic Results ................................................................... 39 Interval Analysis ............................................................... 41 Adjacency Analysis ............................................................ 47 ' Monochromatic Chords ....................................................... 55 Period Classification: .......................................................... 58 Span (and Maximum Interval) Analysis ..................................... 63 Interval Sets .................................................................... 69 Graceful Chords ............................................................... 72 Harmonic Simplicity ........................................................... 76 Polychords ...................................................................... 80 Musical Applications .......................................................... 86 Chapter 4. Scale-Counting Problems ................................................ 89 Basic Results ................................................................... 89 The 7 -note Scales .............................................................. 92 Idealness of a Scale ............................................................ 97 Distance Measures ............................................................. 101 Chord Containment. .................•.......................................... 106 Spelling Analysis .............................................................. 109 Musical Applications .......................................................... 116 Chapter 5. Combinatorics of Rhythms .............................................. 121 Metrical Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Rhythmic Patterns ............................................................. 131 Musical Applications ................................................................... 140 Appendix A. The 351 Chords ........................................................ 142 Appendix B. The 462 Scales ......................................................... 151 Appendix C. Some Musical Numbers ............................................... 164
1
Notes, Chords, and Scales "/don't", she added, "know anything about music, really. But I know what /like." - Max Beerbohm, Zuleika Dobson.
Scientifically,/ could neverbe made to understand what a note in music is, or how one note differs from another. - Charles Lamb, Essays.
This chapter introduces and describes the three basic musical concepts notes, chords, and scales - which will be studied from a mathematical perspective in the following chapters.
Notes At the most basic level, a musical note is any sound with a well-defined pitch. The pitch of a musical note is determined by the frequency of the audio waveform of the sound, measured in units such as cycles per second(= Hertz, or Hz). Figure l.l(a) shows the waveform for a simple musical note with a frequency off Hz. Since there are f cycles per second, each cycle, or period, of the waveform is 1// seconds long. In this example, the waveform is a simple sine wave. Figure 1.1 (b) shows the waveform of a more complex-sounding musical note, also with frequency f. The distinction between (a) and (b) is a subtle one.
Both are
one cycle (period) Figure 1.1. Two notes with the same frequency if) and therefore the same pitch. Note (a) is a simple sine wave, whereas note (b) is a more complex waveform.
2
Chapter 1
waveforms for a note with frequency f, and thus have the same basic pitch. The difference between them lies in the complexity of the shape of each period of the waveform, which causes note (b) to, in effect, contain reduced-amplitude sounds of other (higher) frequencies. These other frequencies (which are all integral multiples of the basic frequency f) are referred to as harmonics or partials. The harmonic content of a particular musical sound is referred to as its timbre. Thus, notes (a) and (b) are said to have the same pitch but a different timbre. The different waveforms (a) and (b) might, for example, represent the same pitch played on two different musical instruments. In standard music notation, timbre is ignored, and all notes with a given pitch are notated the same. So, a note is completely defined by its pitch, which is in turn determined by the frequency, f, of its audio waveform.
The octave. Two notes- one with, say, frequency f and one with frequency g - are related to each other by the ratio of their frequencies if/g), which is referred to as the interval between the two notes. Iff= g, the notes are the same; this is the unison interval. The next simplest interval isf/g = 2; that is, one note is twice the frequency of the other. This interval is referred to as an octave. (The reason for the "oct-" prefix in this term, meaning 8, when in fact the interval equals 2, will be explained below!) Two notes that are an octave apart also sound rather alike. This is because, as illustrated in Figure 1.2, the periods of the two waveforms coincide exactly, with the only difference being that the higher-pitched note cycles an extra time in between each cycle of the lower note. This similarity of notes an octave apart is reflected in the fact that musicians call such notes by the same name. For example, the note "middle C" has a
Figure 1.2. The waveforms of one note (solid line) and another note (dotted line) exactly one octave higher. In timeT, the higher note contains two cycles compared to the lower note's one cycle.
Notes, Chords, and Scales
3
frequency of about 261.6 Hz. All the notes with frequencies 65.4, 130.8, 261.6, 523.2, 1046.4, ... Hz are also called C, since these are the notes with inteiVals equal to powers of 2 from middle C. Similarly, the notes with frequencies 110, 220, 440, 880, etc. are all called "A".l In mathematics, the term for two things that are "essentially the same" is isomorphic. Thus, we can say that two notes an octave apart are "isomorphic".
The fundamental scale. In most traditional music (i.e., ignoring certain 20th-century music and computer music) all the notes in a musical piece are selected from some set of notes with particular frequencies. We refer to this pool of notes from which compositions are created as the fundamental musical scale. In this book we will only be concerned with one type of fundamental scale, known as the L-tone equal-tempered scale. The equal-tempered scale is defined as follows. Take a single pitch P as a starting point, then add all pitches which are any number of octaves above or below P (i.e., whose frequency ratio toP is a power of 2). Finally, add L-1 notes, equally spaced in frequency, between each of the octave notes. "Equally-spaced" means that the inteiVal of 2:1 between each octave is subdivided into L equal inteiVals. If we denote this smaller inteiVal by e, then this is saying that
e · e · e · ... ·e
=2
where there are L e's on the left-hand side. This means therefore the value of e is
eL = 2,
and
e=T2. That is, the frequency interval between any two adjacent notes in the equal-tempered scale is the Lth root of 2. This inteiVal, between two adjacent notes, is referred to in music as a semitone. More generally, the inteiVal between two notes whose positions in the scale differ by i places is
1The
exact frequency corresponding to a particular musical note (such as middle C) is somewhat arbitrary. The frequency numbers given in this paragraph are based on an international standard in which the A below middle C has a frequency of 440 Hz.
Chapter 1
4
Lis referred to as the length of the scale, since there are L notes within each octave of the scale. The L-tone equal-tempered scale is illustrated schematically in Figure 1.3. The equal-tempered scale is the most commonly-used fundamental scale in music, and is also quite amenable to mathematical analysis due to its symmetry and simplicity.
=
The case L 12. Within the general framework of the equal-tempered fundamental scale, the case L = 12 is by far the most important, at least in Western music, since the twelve-tone scale is used almost exclusively. The entire system of music notation familiar to most musicians, and the construction of most musical instruments, are both based on the assumption that L = 12. Therefore, we will primarily consider the case L = 12, although many of the combinatorial equations in Chapters 3 and 4 will be expressed in terms of any L value. The 12 essentially different notes in the 12-tone equal-tempered scale are illustrated in Figure 1.4 using three different representations commonly encountered in music. At the bottom is a piano keyboard containing two octaves (24 notes) of the 12-tone scale. The twelve notes in each octave consist of 7 white keys interspersed with 5 black keys. The 7 white keys are named using the ?letters A, B, C, D, E, F, and G; these are known as the 7 natural notes of the scale. As shown in the middle of the figure, the other 5 notes (corresponding to the black keys) are named by taking an adjacent note and applying a suffix (known as an accidental) to its name, either# (sharp) orb (flat). The sharp(#) signifies an offset of one position forward in the equal-tempered scale, and a flat (b) denotes a shift down by one position. Thus, a C# (C sharp) is the 2ndoctave
L
U----------------+-
(!:,n)L:lf -------------1--1st octave (L notes)
(~3·f------~~------
frequency (log scale)
(~2·f-------------------
(~
•f---1------------f-.~--------------
Figure 1.3. The L-tone equal-tempered scale. The notes of the scale are equally-spaced (logarithmically) in frequency, and there are L notes within each octave of the scale.
Notes, Chords, and Scales
5
note one position higher than a C, which corresponds to the black key between the C and D keys.. Observe that this note can also be indicated as Db, since it is one position lower than a D. Finally, notes can be represented using the musical staff as shown across the top of Figure 1.4. The lines, and spaces between the lines, on the musical staff correspond to the white keys on the piano (or, equivalently, to the letter-named notes A, B, C, D, E, F, and G). Notes which require a sharp or flat to be notated have a # or b symbol placed to the left of the note. (In the figure we have used only sharps, corresponding to the top row of letter names.) We can now see the origin of the term "octave". If we pick a given natural note and call it "one", the note which is one octave higher is the eighth natural note in sequence. In very early music, only the natural notes were used, hence the designation "octave". Note that there are at least three interesting numbers connected with an octave: Its frequency ratio is 2; It is 7 natural notes higher; It is 12 notes of the scale higher. Since the 12 notes of the equal-tempered scale repeat in a cyclic octave
I~
.... *
octave
.. • • ....• •#•+U+.._ I.
.... *I* • *I* ••• ~
Figure 1.4. Two full octaves of the 12-tone equal-tempered scale, in three different musical representations: Notes on the musical staff, letter names, and keys on the piano keyboard.
6
Chapter 1
fashion, they can also be conveniently represented in a circular fashion as shown in Figure 1.5. In this figure, each position on the circle represents one of the 12 notes, and each note makes an interval of precisely
1~ with each adjacent note. In other words, the equal-tempered scale is completely symmetric: every note has precisely the same relationship with the other notes of the scale. Therefore, when considering the equaltempered scale mathematically, we can eschew musical designations (C, C#, D, etc.) entirely, and just denote the 12 notes by the 12 integers 0 through 11, as illustrated in the figure. We need not even define which note 0 corresponds to; since the scale is symmetric, any statements we make about the notes 0-11 will remain true regardless of exactly which musical tone (C, etc.) we identify with 0. (We will often identify 0 with C, as in Figure 1.5, since C is the simplest musical key, but this is merely conventional.) To summarize: In an L-tone equal-tempered musical scale, there are only L essentially different notes (since notes differing by an octave are considered essentially the same), and they repeat in a cyclic fashion as you step through the notes of the scale.
Chords A chord is simply a group of notes sounded together, to make a musical sound with more complexity than is possible with a single note. The
c
F#/Gb
-
Figure 1.5. The 12 musical notes repeat in a cyclic fashion, and therefore can be represented as 12 positions on a circle.
7
Notes, Chords, and Scales
resulting sound may be either more or less "pleasing" to the ear than a single note, depending on the notes chosen for use in the chord. Some sample musical chords, with their musical names, are shown in Figure 1.6. In music, chords are generally designated by their "root" or "tonic" note, which defines the harmonic center, or "key" of the chord, followed by a name (major, minor, augmented, etc.) specifying the type of chord. If no name is given, the major chord, which is perhaps the most ubiquitous chord in music, is assumed as the "default". As illustrated in Figure 1.6, a major chord consists of a root note combined with two notes which are 4 and 7 tones (in the equal-tempered scale) higher. Other types of chords are composed of a root note in combination with other notes at various intervals above the root. Observe that the root or tonic note can be in any position within the chord. More generally, the notes of a given type of chord can be arranged in any order. The third chord in Figure 1.6 illustrates this; it contains exactly the same notes as the first C major chord (C E G) but in a different order (E G C), and thus is also referred to as a C major chord. In musical terminology, different orderings of the same chord type are called inversions. That a chord is denoted by its root or tonic plus a "chord type" designation is highly significant. This is the musical consequence of the statement that two chords of the same type (regardless of tonic) are considered essentially the same, or isomorphic. For example, the first two chords in Figure 1.6 are both major chords (with different root notes), and thus are, at a certain level, essentially the same, as reflected in the fact that they are both called "major" chords. Two chords of the same type are considered isomorphic because they have a very similar sound. The reason for this is that the notes in the chord have a certain fixed relationship to the root note, within the context of the equal-tempered scale from which notes are selected to form the chord. Specifically, two chords of the same type have the same intervals between the notes of the chord. Because of this, for example, the Ab major chord shown in Figure 1.6 sounds "just like" the C major chord,
C
Ab
C
I
BbM7#11 Ddim
i
Gaug7
.£ ·I
Figure 1.6. Some sample chords, with their musical names.
8
Chapter I
only with a higher overall pitch. Alteration of a chord type from one tonic to another in this way is referred to in music as transposition. So, we can say that two chords are isomorphic if one can be transposed into the other. One of the main problems discussed in this book is the enumeration of distinct chord types; therefore, it is important to define precisely under what conditions two chords are considered isomorphic (i.e., of the same type). Based on the above discussion, we arrive at the following rule: ( 1) Mark all the positions in the circular scale of Figure 1.5 which correspond to notes contained in the chord. Repeated notes of the same name but in different octaves are counted as one, since, as discussed in the previous section, such notes are isomorphic. (2) Two chords are isomorphic if the set of marked notes for one chord can be rotated together around the diagram of Figure 1.5 so they exactly coincide with the notes of the other. This corresponds to transposing the chord to another root, which as we have discussed should be regarded as isomorphic. This rule can also be expressed numerically. Part (1) corresponds to writing down the notes of the chord as a list of numbers from 0-11, using the numbers-to-notes correspondence shown in Figure 1.5. The order of the numbers is not important, since all the notes are sounded together when playing a chord; therefore, we can without loss of generality always write the numbers in ascending order. Part (2) of the rule says that two chords are isomorphic if the set of note numbers for one chord equals the set of numbers for the other plus or minus a constant modulo 12. In comparing the two sets, the order of the numbers is not important. (Arithmetic modulo 12 means to add or subtract two numbers and then, if necessary, successively add or subtract 12 from the result until a number between 0 and 11 is obtained. Diagramatically, adding (subtracting) k modulo 12 simply means to step k steps clockwise (counterclockwise) along the diagram of Figure 1.5.) Figure 1.7 shows an example of the application of this rule for chord isomorphism. All the chords in this figure are isomorphic, as demonstrated by the calculations shown. The first line shows the notes of the chord listed in the order shown on the musical staff (from bottom to top). The second line shows the notes of the chord translated to numbers, using the numbering in Figure 1.5. In the third line, duplicate numbers arising from octave notes are eliminated, and the numbers are written in ascending order. In the fourth line, we have added or subtracted some value modulo 12. Finally, the last line shows the list of numbers rearranged into ascending order. In all cases, we arrive at the same set,
9
Notes, Chords, and Scales
{0,1,4,7,10}. Thus, all the chords are isomorphic, i.e., they are all of the same basic type; but just in different keys, in different orders, or having duplicate octave notes. In musical terminology, they are all 7th chords with a flatted 9th. If we subtract each number in the set {0,1,4,7,10} from the next one (in modulo 12 fashion), we arrive at the numbers
(1 3 3 3 2) which is a list of the intervals in the chord. These numbers say that a chord of this type is formed by starting on a given root note, taking the note 1 step higher, then taking the note 3 steps higher than that note, then 3 steps again, and finally 3 more steps. The last interval, 2, represents the interval from the last note of the chord back to the first note in the cyclic diagram of Figure 1.5. This number is actually redundant, since we know that the sum of the interval numbers must be 1+3+3+3+2::; 12, but it is convenient for the list of interval numbers to have the same number of elements as the list of note numbers; thus we will usually include the last interval. The point of the previous paragraph is this:· The fact that the list of intervals for all the chords in Figure 1.7 is the same- (1 3 3 3 2)- is precisely a reflection of the musical fact that it is really the intervals that . determine the "sound" of a particular chord. To summarize: Two chords are isomorphic if the set of notes in one chord (on a circular diagram such as Figure 1.5) can be rotated (musically speaking, transposed) so that they exactly coincide with the set of notes in the other chord. Equivalently, two chords are isomorphic if their set of intervals is (cyclically) the same. Given the vast array of chords that are typically encountered in music,
lt~1 CEGBbDb {047101) order: {0 1 4 7 10) rotate: {0 14 7 10) reorder: {0 1 4 7 10)
·tl Eb G Bb Db Fb(E)
{371014) {13 4710) -3={100147) {0 1 4 7 10)
~i CEbGbACF {036905) {0 3 56 9) -5={710014) {0 1 4 7 10)
·:!I C#GABbC#E
{1791014} {147910) +3={471001) {0 1 4 7 10)
Figure 1.7. Four isomorphic chords. Rotating (transposing) and reordering the notes of eachchordproducesthesamesetofnumbers: {0, 1,4, 7, 10).
Chapter 1
10
especially modern music such as 20th-century classical or jazz, it is somewhat surprising that, if we use the 12-tone equal-tempered scale as the fundamental basis for making chords, the number of essentially different (non-isomorphic, according to the above definition) chords is quite small. One of the main results of this book is that there are only 351 distinct chords - even fewer in certain natural special cases. These combinatorial facts are explored in detail in Chapters 2 and 3.
Scales All music which is based on a length-£ equal-tempered scale is, by definition, limited to the L essentially different notes available in the fundamental scale, for the purpose of creating chords and melodies. In addition to the fundamental scale constraints, however, the melodic element of a piece of music in the 12-tone scale is usually limited further, to some subset of the 12 tones available. This subset of the fundamental scale is referred to as the scale or mode in which the piece (or section of a piece) is written. Figure 1.8 shows a few common scales (or modes). The first twomajor and minor- are the most commonly heard scales, and use only 7 of the 12 notes available. Scale (c) is a slightly more unusual scale (the Dorian), also containing 7 notes. The fourth example, the pentatonic, only uses 5 notes. Observe that the notes of a scale are always listed in ascending order. As described so far, a scale is no different than a chord - it is simply a subset of the L different notes in the length-£ equal-tempered scale. There is one difference, however: in testing for isomorphism between scales, the order of the notes is important. That is, when we add or subtract some value modulo L in order to make two sets of note numbers coincide, we are not allowed to reorder the sets (as in the last line of Figure 1.7). The (a) CMajor
(b) F Minor
1& JJJJJJJ 1~ JJ .J ~rr ~n 1 1& J J ~rr rn 1 1~ J .J .J J •r (c) G Dorian
(d) E Pentatonic
Figure 1.8. Some sample scales, with their musical names.
11
Notes, Chords, and Scales
(o)
1& 1 JJ J J d J 1 1& JJ J~r
Notes:
(b)
0 2 4 5
(c)
7
8 11
5 7 -5 mod 12 = 0 2
r•r r 1
9 10 0 4 5 7
1 4 8 11
1& J~J .J~J J ~M I
0 +4 mod 12 = 4
1 3 5 7
4 7 8 11
8
10
0
2
Figure 1.9. Scales (a) and (b) are isomorphic, since (b) transposed down by 5 steps is identical to (a). Scale (c) transposed up by 4 steps yields the same notes as (a) and (b) but in a different order, so it is not isomorphic (but would be if these were chords).
reason for this is that the first note of a scale is special: it must always be the tonic or "key" note. For example, the first note in the C major scale in Figure 1.8 is a C. Figure 1.9 illustrates some scale isomorphisms. The first two scales are both isomorphic, since their note sets are identical after addition modulo 12 (which is equivalent to transposition, or rotation around the . diagram of Figure 1.5). The third scale is not isomorphic to these, since it cannot be made identical by addition or subtraction modulo 12. Notice, however, that it would be isomorphic if it were a chord! By adding 4 modulo 12 we arrive at the same set of numbers as the first two scales but in a different order. This satisfies the rule for chord isomorphism, since note order is irrelevant for a chord, but not for scale isomorphism, where note order is important. Scale isomorphism is reflected in musical terminology in the same way as chord isomorphism: isomorphic scales are called by the same name (Major, Dorian, Phrygian, etc.). In Chapter 4 we will explore the question of how many non-isomorphic scales there are of various kinds; for example, there are exactly 462 non-isomorphic scales with 7 notes, which is the number of notes typically used in an L = 12 scale.
Keys A piece of music that primarily uses the notes in a major scale starting on note X is said to be written "in the key of X". Since there are 12 different possible starting notes, one might expect that there are exactly 12 different musical keys; however, it turns out that there are 13 commonly-used musical keys!
Chapter 1
12
The reason for this surprising fact has to do with the ambiguity of notating the 12 notes of the equal-tempered scale using only 7 letters. As shown in Figure 1.5, many notes can be notated in two ways, such as C#, which also equals Db. In fact, even those notes which only have one name in Figure 1.5 really have two names- for example, the note E can also be written as Fb. Even less obvious is the fact that D can also be written as Ebb; this uses the standard musical notation for a double-flat, indicating a note two steps lower than the note E. (Also available is the double-sharp, which in musical notation is usually written as x (not##). So, for instance, G =Fx.) Since musical notation is based on the 7 letter names, each of the two notational possibilities for each tonic produces different descriptions for the notes in the major scale - both when written using the 7 letter names, and when written on the musical staff, which is also based on the 7 natural notes. Therefore, there are, strictly speaking, 24 notationallydistinct musical keys. These 24 keys are: B# C
C#
D
Db
Ebb
D# Eb
E Fb
E# F
F# Gb
G
G#
A
A#
B
Abb
Ab
Bbb
Bb
Cb
In the above listing, the two keys in each column are musically the same, since they are based on the same one of the 12 notes of the equal-tempered scale, but are notationally distinct. Even though there are 24 possible musical keys, only 13 are commonly used. The reason has to do with economy of notation, according to the following rule: For a given one of the 12 equal-tempered notes, write down all 7 notes of the major scale for each of the two notational possibilities. Count up the total amount of accidentals used to "spell" the 7 notes, and choose among the two the one which minimizes this number. Table 1.1 shows the result of doing this for all 24 major scales. (A major scale is formed by taking the notes {0,2,4,5,7,9,11}, where these numbers are relative to the first (tonic) note of the scale.) Note that each scale uses either all sharps (#,which counts as 1, or x, which counts as 2) or all flats (b = 1, or bb =2). In addition, the two possibilities for each tonic consist of one all-flats scale and one all-sharps scale. Each of these pairs satisfies the following equation: No. of sharps + No. of flats = 12. The number-of-sharps and number-of-flats values in Table 1.1 are all unique - for example, the A major scale is the only one with 3 sharps. Because of this fact, musicians often refer to scales by the number of
Notes, Chords, and Scales
13
sharps or flats. For instance, "This piece is in the key of 4 flats" is synonymous with "It is in the key of Ab". If we now apply the economy-of-notation rule, we see that for each of the twelve notes there is a unique scale with the fewest accidentals required, with the exception of F#!Gb! This tonic is completely ambiguous: it can be notated as either the key ofF#, which has 6 sharps, or as the key of Gb, which has 6 flats. Since there is no reason to prefer one of these over the other, both are in common use. Therefore, there are 13 common musical keys: the most economical one for each note, with the exception of F#/Gb, where both are used. Put another way, the 13 keys are: The key requiring 0 accidentals: C plus the keys requiring 1-6 sharps: G, D, A, E, B, F# plus the keys requiring 1-6 flats: F, Bb, Eb, Ab, Db, Gb. Key B#
c
C#
Db D
Ebb D#
Eb E
Fb E# F F# Gb G
Abb
Notes of the major scale B# Cx Dx E# Fx c D E F G C# D# E# F# G# Db Eb F Gb Ab D E F# G A Ebb Fb Gb Abb Bbb D# E# Fx G# A# Eb F G Ab Bb E F# G# A B Fb Gb Ab Bbb Cb E# Fx Gx A# B# F Bb A G F# G# A# B C# Gb Ab Bb Cb Db G A B c D
c
Abb Bbb Cb Dbb Ebb
G#
G#
Ab
Ab Bb
c
A
A
C#
Bbb
Bbb Cb Db Ebb Fb
A#
A#
. Bb
Bb
A# B
B#
C#
D#
Db Eb D
E
Gx A A#
Ax B B#
Bb
c
B
C#
#flats #sharps 12 0
7
5 2
Cb Db
10
Cx D
3
B#
c
C# D# Db Eb Cx Dx D E D# E# Eb F E F# Fb Gb E# Fx F G F# G# Gb Ab Fx Gx A G G# A#
9 4
8 11 1
6 6 1 11 8 4
3 9
10 E# 2 F B B F# 5 C# D# 7 Cb Cb Db Eb Fb Gb Ab Bb Table 1.1. The two ways of spellmg each of the 12 maJor scales, wh1ch produce the 24 notationally-different keys.
B#
c
Cx D
D#
Eb E
14
Chapter 1
These ideas will become important in Chapter 4, where we discuss some combinatorial problems related to spelling an arbitrary (not necessarily major) scale in each of the 13 keys.
2 Chains, Necklaces and Partitions There is music wherever there is harmony, order, or proportion. - Sir Thomas Browne
Among scientists are: collectors, classifiers, and compulsive tidiers-up. - Sir Peter Medawar, The Art of the Soluble
In this chapter we describe three kinds of mathematical objects which form the basis for solving most of the musical questions considered in the following chapters. These three key concepts are chains, necklaces, and partitions. In the process of counting chains, necklaces, and partitions, we will encounter P6lya's Theorem, a beautiful and powerful mathematical result that deserves to be more widely known.
Basic Definitions A chain is a linear string of beads of various colors, as shown in Figure 2.1 a. The total number of beads in the chain is n and there are m different colors; the example in Figure 2.1a has n = 7 and m = 3. Two n-bead chains are considered the same (isomorphic) if and only if each bead in corresponding positions of the two chains is the same color. So, reversing the two ends of the string, for example, yields a chain considered different from the original (unless the original chain was a palindrome- reading the same backwards as forwards). A necklace is simply a chain with the two ends of the string connected to form a circular chain. Two n-bead necklaces are considered the same if one can be rotated so that the colors of its beads correspond, one to one, with the colors of the other. Turning the necklace over is not permitted. Figure 2.1 b shows a necklace with n =9, m =2. A partition of an integer n is an expression of n as a sum of positive
(a)
0 (b)
Figure 2.1. (a) A chain with n =7, m =3. (b) A necklace with n =9, m =2.
Chapter 2
16
integers ~ n. For example, 12 = 4+3+2+1+1+1. The number of summands is denoted k:, in this example, n = 12 and k = 6. The basic counting problems, which are fundamentally important for analyzing musical combinatorics, are to count: (1) The number of n-bead chains with m different colors and a given number of beads of each color. (2) The number of n-bead necklaces with m different colors and a given number of beads of each color. (3) The number of partitions of n into exactly k parts. The musical applications of these three mathematical problems arises from the following connections, which will be explored in detail in Chapters 3 and 4: Counting chains
is equivalent to counting scales
chords interval sets
necklaces partitions
We now consider these three basic problems in detail. Before doing this, some basic facts about permutations are required. A permutation is an ordered arrangement of the elements of a set of n objects. For example, (15324) is a permutation of the elements in the set { 1,2,3,4,5}. A permutation can also be thought of as a "scrambling operation" in which a given ordering of a set is replaced by another ordering. Permutations are also written in the form
1 2 3 4 5) (1 5 3 2 4 to emphasize this fact. This notation indicates that the permutation operation replaces each element in the first row with the corresponding element in the second row. How many different permutations of a set of n objects are there? We can write down a permutation one element at a time, bearing in mind that each element in the set can only be used once. We have n choices for the first element, n-1 choices for the next, and so on. The total number of choices, and hence the total number of permutations, is therefore
n · (n-1) · (n-2) · ... · 1. This expression is denoted in mathematics by the symbol n! (n factorial). n! is the product of all integers between 1 and n. By convention, 0! = 1.
Chains, Necklaces and Partitions
17
Counting Chains To keep things simple, for the moment, consider first the case m = 2. In other words, there are only two different colors of beads - say black and white. (Of course, the case m = 1 is completely trivial, as there is exactly one chain for any value of n). How many n"'bead chains are there with exactly k black beads? The solution to this problem is quite elementary. Since there are k black beads, there must be n-k white beads. Suppose we label the k black beads with the labels bt, b2, ... , bk and the n-k white beads WI, w2, ... , Wn-k· Since there are n labels in all, there are n! possible arrangements of these labels. For the purpose of counting chains, we do not distinguish between the different b and w labels. In other words, all arrangements of labels with the b's in given positions and the w's in given positions are considered the same chain, regardless of the ordering of the labels within the b's and w's. So, how many have the b's and w's in given positions? It is simply the number of possible permutations of the b's times the number of permutations of thew's; that is, k!(n-k)!. The total number of chains is simply n! (the total number of arrangements of labels) divided by k!(n-k)! (the number of times each arrangement with the b's and w's in a given position is counted), and so is n! k!(n-k)t· This number, which is of great importance in mathematics, is usually written
(~). This expression, called a combination number, is read as "n choose k", since another interpretation of this number is as the number of different ways of choosing k distinct objects from a set of n distinguishable objects. In this case, we are choosing the positions of the k black beads
Figure 2.2. The 10 chains with n =5, k =2.
18
Chapter 2
from the set of all n positions in the chain. Combination numbers turn up in the solutions to many combinatorial counting problems.
As an example of chain counting, consider n = 5 and k = 2. Since ·
(~) = 21 (;~2 )! = ~:~ = 10, this means that there are 10 distinct 5-bead chains with 2 black beads. These chains are shown in Figure 2.2. It is convenient to be able to write down chains without drawing pictures as in Figure 2.2. To do this we can label the n positions in the chain with the n different numbers and then simply list the positions occupied by the black beads. For various reasons, we will find it convenient to label the positions 0 through n-1 (rather than 1 through n), so the 10 chains shown in Figure 2.2 can be concisely described as follows: {0,1}. {0,2}. {0,3}. {0,4}, {1,2}. {1,3}. {1,4}. {2,3}. {2,4}. {3,4}.
Another notation we will use for two-color chains or necklaces is (•.---)
in which • represents a black bead and - represents a white bead, listed in position order. This chain corresponds to the first chain in Figure 2.2. Now consider the more general case of m different colors (rather than just 2). How many m-color, n-bead chains are there with exactly ki beads of color i (for 1 :::; i:::; m)? By the same reasoning as above, it is simply the total number of permutations of a set of distinct labels (= n!) divided by the number of permutations of each set of labels corresponding to one color; i.e., n!
-o • • • -o • • • -o • • •
~
~
------.o-oFigure 2.3. The 12 4-bead, 3-color chains with color distribution (2,1,1)
19
Chains, Necklaces and Partitions
which is denoted by the symbol
(k 1, k2,n···• km ). Note that the ki sum to n; that is, they are a partition of n into m parts. The total number of m-color, n-bead chains (of all kinds) is the sum of these values over all possible partitions of n into m parts with parts ;::: 0. Figure 2.3 shows the 12 chains corresponding to (2,1,1)
= 21iiu = 12·
(k)
Table 2.1 shows the values of for n,k:::; 12. Notice the triangular arrangement of numbers in this table; the positions shown as "." are all
(k)
zero, and are due to the obvious fact that = 0 if k > n, since it is impossible to have more than n black beads when there are only n beads in all (or, equivalently, it is impossible to select more than n distinct things from a set of n things). This table is called Pascal's triangle, after its original constructor, Blaise Pascal (1623-1662). Observe that each row sums to a power of 2. In this book we will use the symbol "*" in an expression, in place of a parameter, to mean "the sum of the expression over all relevant values of the parameter." So, for example, the observation that each row of Pascal's triangle sums to a k
0
1
1 1 1 1 1 1 1 1 1 1 1 1
1 2 3 4 5 6 7 8 9 10 11 12
2
3
4
5
6
7
8
9
10
11
12
Total
n 1
2 3 4 5 6 7 8 9 10 11 12
1 3 1 6 1 4 10 10 5 1 1 15 20 15 6 7 21 35 35 21 1 1 28 56 70 56 28 8 9 1 36 84 126 126 84 36 45 120 210 252 210 120 45 10 55 165 330 462 462 330 165 55 66 220 495 792 924 792 495 220
l 11
66
1 12
Table 2.1. Values of(~), or Pascal's triangle.
1
2 4 8 16 32 64 128 256 512 1024 2048 4096
Chapter 2
20
power of 2 can also be written as:
The explanation for this equation is simple. The total number of chains, without regard to the value of k, is simply the total number of chains composed of white and black beads. Since either bead can be white or black independently, there are 2 choices for each bead, and there are n beads; thus, the total number of combinations is 2n. This is also the number of binary strings of length n. Similarly, the total number of mcolor chains is mn, which corresponds to the number of m-ary strings of length n. A more interesting observation is that each number is the sum of two numbers in the previous row, in the following way:
That is,
(~) = (~=D +(nil). The truth of this assertion can be seen by using the interpretation of these numbers as "n choose k". All possible selections of k objects from n can be divided into two sets: a set P containing all selections which include a particular object p , and a set Q containing all selections not using p. The number of selections in Pis {~=~). since we have already chosen the object p, and then choose k-1 objects from the remaining n-1 objects. The 1 number of selections in Q is ). because we select k objects from the set of n-1 objects (p not being allowed to be chosen). Summing the choices in P and the choices in Q yields the above formula.
(nk
Notice the symmetry in each row of Table 2.1. That is,
Chains, Necklaces and Partitions
21
An explanation for this is that the number of ways of selecting k objects from a set of n is equivalent to the number of ways of selecting which n-k objects are tWt selected. Another useful relation can be derived by selecting objects from the set of n objects, numbered 0 through n-1, in numerical order. If the first object we select is numbered r, then the remaining k-1 objects we select must be from the set {r+1, ... ,n-1}, which contains n-r-1 objects. The total number of possible selections is the sum of these numbers over all possible r, which is 0 ton-k. Therefore,
For example,
(~) =
I (ir) = (i) + (~) + (~) = 6+3+1 = 10.
0 S r S 2
This relation provides a simple recipe for explicitly enumerating all {~) combinations for a given n and k. We simply choose each value 0 through n-k for the first element, and then, for each choice, list the ( n-r-1) k- 1 combinations of {r+ 1,... ,n-1} after it. For example, let's enumerate all {~) selections of 3 objects from the set {0,1 ,2,3,4} using this procedure. First, we select {0} followed by all
(i) selections of 2 objects from the set {1,2,3,4}: {0}
+ { 1,2} { 1,3} { 1,4} {2,3} {2,4} {3,4}
and then {1 } followed by all
{~) selections of 2 objects from {2,3,4}: {1} + {2,3} {2,4} {3,4}
Chapter 2
22
and finally, {2} followed by the (~) = 1 selection of 2 objects from {3,4} (namely, {3,4 }). The final list of all 10 combinations is: {0,1,2} {0,1,3} {0,1,4} {0,2,3} {0,2,4} {0,3,4}.
{1,2,3} {1,2,4} {1,3,4}
{2,3,4}
Binomial coefficients. There is another interpretation for the (~) numbers which will be useful in the next section. A sum of two different symbols, such as x+y, is called a binomial. The binomial coefficients are the coefficients ai in the expansion of (x+y)n
= atxn + azxn-ly +... +
an-l.xyn-1
+ anYn·
We ask: what is the coefficient of xkyn-k in this expression? To answer this, note that (x+y)n
= (x+y)(x+y) ... (x+y).
Consider the multiplication as the sum of all products formed by selecting either x or y from each term on the right-hand side. How many of these products will equal xkyn·k? The answer is: the number of products in which there are exactly k x's; i.e., in which x is selected k times from among the n terms on the right-hand side. But this is simply (~). Therefore,
(~)is the coefficient of xkyn-k in the expansion of (x+y)n. For this reason, the
(~) are also called binomial coefficients. By exactly
n k ) is the coefficient of the same reasoning, ( k 1. k 2 •... , m xktyk2 ... wkm in the expansion of (x+y+ ... +w)n. These are called multinomial coefficients.
Chains, Necklaces and Partitions
23
m object types
•••• •••• •
••••••• ••
wl
w2
1
2
3
• ••••
4
n
n boxes Figure 2.4. The classic "configuration problem". Then boxes each contain one object chosen from among the m different object types with "weights" Wj.
Counting Necklaces There is another way of looking at the chain-counting problem which generalizes to a description of the necklace-counting problem, and which can then be solved by applying a powerful theorem developed by the Hungarian mathematician George P6lya (1887-1985). The power of P6lya's theorem is required because necklace counting is a much harder problem than chain counting. The concept which generalizes from chain-counting to necklacecounting is called a configuration. Suppose we have a collection of objects of m different types, and a set of n boxes, as shown in Figure 2.4. Each object type is said to have a certain weight, which we denote by Wi (1 :s; i :s; m), and there is an unlimited supply of objects of each type 1• A configuration is obtained by putting one object (selected in any manner from the m object types available) in each box. The weight of the configuration is defined as the sum of the weights of the individual objects in the boxes. The configuration problem asks for the number of distinct configurations having a given weight. This formulation includes the chain-counting problem. The objects are beads, and them different object types (or weights) are them colors. The n boxes are the n positions in the chain. Asking for the number of 1The weights w; should always be thought of as symbols, rather than, for example, as
numbers. The precise mathematical condition is that no weight should be a linear combination of the others.
Chapter 2
24
distinct m-color, n-bead chains with exactly ki beads of color i is equivalent to asking for the number of different configurations with weight ktWI + k2w2 +... + kmwm. The configuration problem can be generalized to necklaces by allowing different conditions under which two configurations are considered distinct. Specifically, we impose the restriction that two configurations are considered equivalent (isomorphic) if they are equal under any permutation (from a given set of permutations G) of the boxes. In other words, consider one of the two configurations, and apply a given permutation to the boxes. After the permutation, if the object in each box is exactly the same as in the second configuration, the two configurations are considered isomorphic, and are not to be counted as two distinct configurations. Repeat this procedure for all permutations in the set G. If any of the permutations cause the first configuration to be equal to the second, they are isomorphic. The revised problem now asks for the number of distinct configurations having a given weight and with a given set of permutations G under which configurations are considered isomorphic. This more involved problem now includes necklace counting as a special case. As shown in Figure 2.5, we simply arrange the boxes in a circle, and for the set G pick all cyclic permutations, which are all permutations of the form
1 2 3 .... n ) ( .. n 1 2 3 .. where the second row is the sequence 1,2,3 ... n shifted cyclically by some number of places. These permutations correspond to the condition that
3
1 Figure 2.5. Necklace counting becomes a problem in configurations by arranging the boxes in a circle and using G all cyclic permutations of the boxes.
=
25
Chains, Necklaces and Partitions
two necklaces are considered the same if they are rotations of each other. The power of P6lya's theorem, which we will now describe, is that it can solve the more general configuration-counting problem using any set of permutations G that satisfies a certain restriction: that of being a group. We will see that the set of all cyclic permutations is a group, and therefore P6lya's theorem can be used to solve the necklace-counting problem.
Pblya's Theorem. Before stating P6lya's theorem, we need a few preliminaries. A set of permutations G is a group if the following two conditions are met: (1) The set of permutations G is closed, which means that if we take any two permutations p and q in G, the permutation that results from applying p followed by q is also in G.
(2) The inverse of every permutation in G is also in G. The inverse of a permutation p is the permutation q that "undoes" the effect of p. In other words, if we apply p and then q, the resulting permutation . (1 2 3 ... . th .de . Is e z ntzty permutatwn, 1 = 1 2 3 . . . n ·
n)
P6lya's theorem applies to counting configurations with a given weight, where two configurations are considered the same under any permutation in a given permutation group G. The next concept we need is that of the cycles of a permutation. Consider a permutation, such as, for example
1 2 3 4 5 6 7 8 9) (2 4 8 7 3 6 1 5 9 . This permutation can be "decomposed" into a series of sub-permutations, which are called its cycles. Start, forinstance, with the element "1". The permutation maps "1" into "2". In turn, "2" is mapped into "4", "4" into "7", and "7" into "1 "; note that we have returned to the starting element
G j i).
"1 ". This portion of the permutation is the cyclic permutation ~ which is said to have length 4, since it contains 4 elements. Similarly, if
G
we start on the element "3" we obtain the cycle ~ ~). Finally, 6 and 9 map to themselves, thus each yielding a trivial cycle of length 1. Thus, we can write the above permutation as
12 4 7)(3853 8 5)(6)(9) (2471 6 9
Chapter 2
26
to illustrate its cycle structure. It consists of 4 cycles of lengths 4,3,1,1. In a similar way, every permutation on n elements can be decomposed (uniquely) into a set of cycles of various lengths where the sum of the · cycle lengths is n. The cycle-index monomial of a permutation on n elements is the expression Z
a1z a2 z an 1 2 ··· n
where z is an arbitrary symbol, and the ai are the number of cycles of length i in the permutation. For example, the above permutation has the cycle-index monomial 2 Zt Z3Z4
since there are 2 cycles of length 1 and one cycle each of lengths 3 and 4. The cycle-index polynomial of a permutation group G, which is denoted by P(G; Zt>Z2, ... ,Zn)
is the sum of the cycle-index monomials for each permutation in G. For instance, consider the permutation group 12345) ( 12345) ( 12345) ( 12345) ( 12345) } G = { ( 12345 • 23451 • 34512 • 45123 • 51234 · The first permutation has cycle-index monomial z15, since it has 5 cycles of length 1, and the other four permutations have cycle-index z5, since they have one cycle of length 5. Therefore, P(G; z1,z2,z 3,z4 ,z 5)
=z15+z5+z5+z5+z5 =z15+4z5.
We are now in a position to state Polya's Theorem: The number of configurations (using n boxes and m different object types) with weight ktwl + k2w2 +... + kmwm , where two configurations are considered equivalent under any permutation in a permutation group G, is equal to the coefficient of w 1klwl2 •.. wmkm in the polynomial
1 2 n IGI P(G; ~wi, ~wi , ... , ~wi )
Chains; Necklaces and Partitions
27
In other words, replace each Zr in the cycle-index polynomial with ~w{, and then divide by IGI, where IGI denotes the number of elements in G (i.e., the number of permutations in the permutation group )2 . An example is in order. How many 5-bead necklaces are there with exactly 2 black beads? Here, n = 5 and m = 2, and we assign the two types of beads (black and white) the weights WI and w2. We are asking for the number of configurations with weight= 2wt+3w2 (i.e., 2 black beads and 3 white beads). The permutation group G is the set of all 5 possible cyclic permutations of {1,2,3,4,5}, which is precisely the example group G given above:
G
= { (g~!~).(i~~~i).(~~;i~).(!~ii~).(~i~j~)} ·
which has cycle-index polynomial P(G; z1,z2 ,z3 ,z4,z5)
=z15+4z 5.
If we replace each zr with ~w{ = (wtr+w{), and divide by IGI = 5, we get
~ [ (wt+W2)5 + 4(wt 5+w25)] which can be expanded to Wt
+ W}W24 + 2wt 2W23 + 2w1 3w22 + Wt 4W2 + W2S·
5
P6lya's theorem says that the number of configurations with exactly 2 black beads and 3 white beads (i.e., with weight 2wt+3w2) is the coefficient of w1 2W23 in this polynomial; i.e. 2. These two necklaces are:
00
It is clear that all other necklaces with two black beads are cyclic permutations of one of these. Observe that the straightforward application of P6lya's theorem counts the number of configurations for all possible weights at once. Often we 2We will not give a proof of this theorem, since it is somewhat lengthy. See, for example, C. L. Liu, Introduction to Combinatorial Mathematics, McGraw-Hill, 1968.
Chapter2
28
are only interested in a specific weight, so it is not necessary to compute the whole polynomial that results from substituting the :Ew{ into the cycle-index polynomial- only the term(s) of interest. P6lya's theorem should, of course, give the correct result for counting chains, in which the permutation group G is the trivial group consisting of only the identity permutation (because in counting chains no permutation of the boxes is allowed). The cycle-index polynomial for the trivial group is simply z1n, since the identity permutation has n cycles of length 1. P6lya's theorem says to replace z1 with :Ewi> which yields (:Ewi)n
= (wt+w2+... +wm)n
and then to take the coefficient of w 1klwl2 ... wm krn in the resulting polynomial. But this expression is a sum of m terms raised to a power, and therefore the coefficient of w 1klwl2 ... wmkrn is simply the multinomial coefficient (k 1, k 2,n... , k m ). This agrees with the analysis of the previous section: the number of chains with a given number of beads of each type is equal to a multinomial coefficient. We are now in a position to solve the general necklace-counting problem. Denote by N(n,k) the number of two-color n-bead necklaces with k black beads, and let N(n,kt.k2, ... ,km) denote the number of mcolor necklaces with ki beads of color i (for 1 :5 i :5 m). These numbers are exactly analogous to the binomial and multinomial coefficients, except that they count necklaces rather than chains. Using P6lya's theorem, we can now derive a formula for both N(n,k) and the more general N(n,kt,k2, ... ,km). For simplicity, consider the two-color case first. A formula for N(n,k). In order to apply P6lya's theorem, we need to determine the permutation group G associated with ann-bead necklace, and then determine the cycle-index polynomial of G. The group G is simply all cyclic permutations of n objects; i.e., all permutations of the form
1 2 3 .... n ) ( . . n 1 2 3 .. There are n such permutations, with the bottom row of numbers being shifted 0, 1, ... ,or n-1 places relative to the top row, corresponding to a rotation of the necklace by 0, 1, ... , n-1 places. This group is called the cyclic group of order n, and is denoted by the symbol Cn.
Chains, Necklaces and Partitions
29
First, we must verify that the set of permutations Cn is in fact a group, since P6lya's theorem only applies when the set G forms a group (not for an arbitrary set of permutations). Recall the two group properties: to be a group, G must be closed, and the inverse of any permutation in G must also be in G. The set Cn is indeed closed, since the net effect of applying one cyclic permutation(= rotation) followed by another is also a cyclic permutation. The inverse property is also satisfied, since the inverse of a rotation by k places is a rotation by -k places (or, equivalently, n-k places), which is also in Cn since Cn contains all possible rotations. We now need to determine the cycle-index polynomial of C 0 • Consider the element of Cn corresponding to a shift by k places. In this permutation, the cycle length of a given element of the permutation is the smallest integer d such that d·k =m·n
for some integer m, since the operation of the permutation is to advance each element by k places, and if d such operations produce advancement by a multiple of n places then we will have returned to the starting point, which is the definition of the length of a cycle. The value d·k = m·n is also equal to the least common multiple of k and n, or lcm(k,n). If k is relatively prime ton, then lcm(k,n) = kn, which means that d = n. That is, if k is relatively prime to n, the cycle length of the permutation =n. Now suppose k and n are not relatively prime, but have a common factors. Then the above equation becomes
d·-sk = m·-sn = lcm<sk 's)n As above, if these values ~and ~) are relatively prime then d = ~· Note that in all cases d is a divisor of n. For a given d, if s =
a• the number of elements of Cn with cycle length
dis the number of integers less than and relatively prime to~= d. In a permutation with cycle length d, each cycle uses up d elements of the permutation, and therefore there are
asuch cycles.
In summary: The possible cycles in elements of Cn are all divisors d of n. For a given d, there are cj)(d) such elements, where cj)(d) is the
Chapter2
30
number of integers < d and relatively prime to d, and each such element n has d such cycles. Therefore: P(Cn; Zt•···•Zn)
:1: cp(d)·zdn/d
=
din
The function cp(d) is a famous mathematical function known as Euler's phi function. Defined as the number of integers less then d and relatively prime to d, it is also given by the following formula: cp(d)=d
II p~l· p =prime
pld
We can now apply P6lya's theorem, replacing each Zr in the cycleindex polynomial with ;Ew{. There are only two weights (wl and w2), which we denote by x andy for simplicity. Therefore, replacing each Zr by (r+y), we get
.!.
:1: cp(d)·(xd+yd)n/d
n dIn
The value of N(n,k), is, by P6lya's theorem, the coefficient of })n-k in this polynomial. Among the various values of d, there are two cases to consider: (1) Suppose d is not a divisor of k. Now all powers of x which will appear in the expanded polynomial will be xt, where tis a multiple of d. If d is not a divisor of k, then no multiple of d can equal k. Therefore, no terms of the form }cyn-k can be generated if d is not a divisor of k. (2) Suppose d is a divisor of k. · Instead of looking for the coefficient of _tcyn-k in the expansion of (xd+yd)n/d, we can look for the coefficient of ~<),
=
(x+y)nfd. But this is simply a binomial coefficient:
{~~).
This yields the final formula for N(n,k), the number of distinct 2-color, n-bead necklaces with exactly k beads of one color:
Chains, Necklaces and Partitions
31
00000 00000 Figure 2.6. The N(8,4)=10 8-bead necklaces with 4 beads of each color.
N(n,k)
=~
L (p(d)·(~f}
d I n,k
For example, let us compute the value of N(8,4): the number of 8bead necklaces with 4 black beads (and 4 white beads). The values of d which are divisors of both n = 8 and k =4 are 1,2, and 4; therefore:
N(8,4)
=t [(p(l)-(~)
+ (p(2)·(i) + (p(4)-(i)]
1 =g[ 1·70+ 1·6+2·2]
= 10. These 10 necklaces are shown in Figure 2.6. Table 2.2 gives the values of N(n,k) for n,k s; 12. Notice that, as with the values in Pascal's triangle, the numbers in each row are symmetrical; that is, N(n,n-k)
= N(n,k).
The reason for this is exactly the same as for chains: the set of necklaces with n black beads is clearly the same as the set with n white beads (and n-k black beads). The numbers in the right column of Table 2.2 give the total number of n-bead necklaces with all possible color distributions; i.e., the value of N(n,*). An explicit formula for N(n,*) can be derived by summing up the values of the N(n,k) formula:
32
Chapter2
N(n,*)
=
L N(n,k) OSk;;n
L L rp
-(tc;;)
=!
n 0Sk;;n d I n ,k
For a given d I n, the values of k for which d also divides k are the various multiples of d. So, k ranges from 0 to n in multiples of d, which means that kid ranges from 0 to nld. Therefore, we can rewrite this as
N(n, *) = !
L
L (tc;;)
(p(d)
n din
OSkldSnld
But the second summation is simply the sum of row n!d of Pascal's triangle. Therefore, N(n,*) =
L rp(d)·2nld
! n
dIn
For example, for n = 10 we have d = 1,2,5, and 10, and thus 1 N(lO,*) = 10 (1·210 + 1·25 +4·22 +4·21) = 108.
The m -color case. The same procedure that led to the formula for
N(n,k) generalizes to yield a formula for N(n,k1, ... ,km): the number of nk
0
1
2
1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1
1 1 2 2 3 3 4 4 5 5 6
3
5
6
7
8
9
10
11
12
Total
1
2 3 4 6 8 14 20 36 60 108 188 352
n 1 2 3 5 6 7
8 9 10 11
12
1 1 2 5 7
10 12 15 19
1 1 3 5 10 14 22 30 43
1 1 3
1 1
7
14 26 42 66
10 22 42 80
1 1 12 30 66
1 1 5 15 43
1 1 5 19
1 1 6
1 1
Table 2.2. N(n,k): number of n-bead necklaces with k beads of one color
Chains, Necklaces and Partitions
33
bead necklaces of m colors with exactly ki beads of color i (1
=
1 n
~
i
~ m):
~ cj>(d) ( nld ) £..J · ktld, k2/d, ... , kml d
din d I ki (lSiSm)
For instance, how many distinct 6-bead, 3-color necklaces are there with exactly 2 beads ofeach color? The above formula yields N(6, 2,2,2)
=i [cj>(1){2,~,2) 1 (
6!
=6 2!2!2! = 16.
+ cj>(2)·(1,i,1)]
3! )
+ 1!1!1!
These necklaces are illustrated in Figure 2. 7. Similar reasoning to the 2-color case also leads to the formula for the total number of m-color necklaces with n beads (including all possible color distributions):
N(n,*, ... ,*) =
!.
L cj>(d)·mnld
n dIn
For example, the total number of 6-bead, 3-color necklaces is N(6, *,*,*)
i[
=
cj>(1)·36 + cj>(2)·33 + cj>(3)·32 + cj>(6)·31]
=61 (729 + 27 + 18 + 6) = 130. Counting Partitions A partition of an integer n is an expression of n as a sum of k positive
00000000
00000000 Figure 2.7. The N(6, 2,2,2)= 16 3-color, 6-bead necklaces with 2 beads of each color.
34
Chapter2
integers ::;;; n; for example, 13 = 4+4+2+1+1+1, which has k = 6. Partitions will be encountered in several musical problems in Chapter 3 and 4; therefore, we desire the values of the function p(n,k), the number of distinct partitions of n into exactly k parts. The order of the summands is not important, so these are more precisely called unordered partitions. This problem is even harder than necklace counting; in fact, the determination of an explicit formula for p(n,k) (or even just p(n, *)) is a notorious mathematical problem, finally solved in 1918 by G. H. Hardy and the Indian genius, S. Ramanujan (1887-1920). A derivation of their explicit formula is well beyond the scope of this book, however, so we shall content ourselves with an easily-obtained recurrence formula. We begin by defining an auxiliary function, q(n,k), which is the number of distinct partitions of n into integers ::;;; k. A recurrence relation for q(n,k) follows from the observation that all such partitions fall into two categories: (1) Those which include k as a summand. There are q(n-k,k) of these, since k is included in each, and only the different partitions of n-k need to be counted. (2) Those which do not include k as a summand. Since k is not included, the largest possible summand is k-1, and so the number of partitions is q(n,k-1 ). Therefore, we have the following recurrence formula for q(n,k): q(n,k)
= q(n-k,k) + q(n,k-1)
(fork::;;; n).
Clearly, if k > n, q(n,k) simply equals q(n,n), since the maximum possible integer in a partition of n is n itself. In addition, we have the following values for small cases: q(O,k) = q(l,k) = q(n,1) = 1.
Armed with this recurrence formula, we can easily compute values of q(n,k). These values, for n,k ::;;; 12, are shown in Table 2.3.
There is another interpretation for the function q(n,k) which leads immediately to a formula for p(n,k). Suppose we write a partition with summands ::;;; k as an array of dots, with the rows sorted by decreasing size. So, for example, the partition 13 = 5 + 3 + 3 + 2 can be described · by the following diagram:
€hains, Necklaces and Partitions
35
••••• ••• ••• •• In this array, the horizontal dimension represents the sizes of the summands, and the vertical dimension represents the number of summands. Now, simply switch the interpretations of the two dimensions (that is, read the array by columns rather than by rows); this yields the partition 13 = 4 + 4 + 3 + 1 + 1, which, instead of a partition into summands ~ 5, is a partition into ~ 5 summands. In general, we see that the number of partitions into parts ~ k is equal to the number of partitions into~ k parts. Therefore, q(n,k) is also equal to the number of partitions of n into ~ k parts. But the number of partitions of n into exactly k parts (which is p(n,k)) is simply the number of partitions into ~ k parts minus the number of partitions into < k parts (or, into~ k-1 parts). Hence: p(n,k)
=q(n,k) - q(n,k-1 ).
But the above formula, q(n,k)
=q(n-k,k) + q(n,k-1)
can be rearranged to q(n-k,k) =q(n,k)- q(n,k-1) k
n 1 2 3 4 5 6 7 8 9 10 11
12
1
2
3
4
5
6
7
8
9
10
11
12
1 1 1 1 1 1 1 1 1 1 1 1
1 2 2 3 3 4 4 5 5 6 6 7
1 2 3 4 5 7 8 10 12 14 16 19
1 2 3 5 6 9
1 2 3 5 7 10 13 18 23 30 37 47
1 2 3 5 7
1 2 3 5 7
1 2 3 5 7
1 2 3 5 7 11 15 22 30 41 54 73
1 2 3 5 7
1 2 3 5 7
1 2 3 5 7
11
11
11
15 22 30 42 55 75
15 22 30 42 56 76
15 22 30 42 56 77
11
15 18 23 27 34
11
11
11
14 20 26 35 44 58
15 21 28 38 49 65
15 22 29 40 52 70
Table 2.3. Values of q(n,k): the number of partitions of n into parts s k, or into s k parts.
Chapter2
36
which means that p(n,k)
= q(n-k,k).
Thus, we can calculate values of p(n,k) by first calculating values of q(n,k) using the recurrence formula above. In fact, the rows of a p(n,k) table are equal to the diagonals of a q(n,k) table. Of course, p(n,k) = 0 if k > n. Values of p(n,k), computed in this way, are given in Table 2.4.
The Musical Connection Having introduced the mathematics of chains, necklaces, and partitions, the stage is now set for the exploration of the mathematical properties of musical chords and scales. As we saw in Chapter 1, the 12-tone equal-tempered scale is a circular arrangement of notes. A musical chord is a subset of these 12 notes, and two chords are considered isomorphic if one can be rotated into the other, which is the same condition for isomorphism between two necklaces. Therefore, there is an intimate connection between the number of distinct necklaces of various kinds and the number of distinct chords. A musical scale is likewise a subset of the 12 notes, but without rotational isomorphism. Thus, scales correspond to chains, and can be counted using the binomial coefficients. In music, the number of steps separating the notes of a chord or scale are referred to as its intervals. This means that the intervals of ann-note k
1
n 1 2 3 4 5 6 7 8 9 10 11 12
1 1 1 1 1 1 1 1 1 1 1 1
2
1 1 2 2 3 3 4 5 5 6
3
1 1 2 3 4 5 7 8
10 12
5
1 1 2 3 5 6 9 11 15
1 1 2 3 5 7 10 13
6
1 1 2 3 5 7 11
7
1 1 2 3 5 7
8
1 1 2 3 5
9
1 1 2 3
10
1 1 2
11
1 1
12
1
Total 1 2 3 5 7 11 15 22 30 42 56 77
Table 2.4. Values of p(n,k): the number of partitions of n into exactly k parts.
Chains, Necklaces and Partitions
37
chord or scale are simply a partition of 12 into n parts. Partitions are also involved in the combinatorics of musical rhythms, since a rhythm constructed from N beats per measure can be viewed as a partition of N into certain values, as will be discussed in Chapter 5. Figure 2.8 summarizes the various ways of representing chords and scales, both musically and mathematically, which will be used interchangeably throughout the remainder of this book: (1) Standard musical notation.
(2) A necklace (for a chord) or chain (for a scale). Black beads indicate which notes are a part of the chord or scale; white beads are unused notes. (3) A set of numbers (written {0,3,7,9}, for example) showing which notes are in the chord or scale, where the notes are numbered 0 to L-1, where Lis the length of the equal-tempered scale (or, the number of tones per octave). (4) A symbolic representation of the set in #3, where the L symbols represent the numbers 0 to L-1, and • and - represent notes used and unused, respectively, by the chord or scale. This is essentially the same representation as the 2-color necklace or chain, but is more convenient as it can be easily inserted in text (like so: • - - • - - - • - • - -). (5) A partition of L, written (3 4 2 3), for example, showing the successive differences between the numbers in the set of #3. Since the scale is circular, these should be thought of as distances between the successive notes in a circular necklace. For example, (3 4 2 3) corresponds to the set {0,3,7,9}, with the final3 representing the distance between the last note in the set, 9, and the first note, 0, when written in a circular arrangement of the 12-tone scale. Mathematically, we are taking differences modulo L.
Chapter 2
38
Musical notation
0
0 1 2 3 4 5 6 7 8 9 1011
Necklace
-e-oeeoeoeoeeo-
or
F F# G Ab A Bb B C Db DEb E
Chain
Subset of {O,... ,L-1}
{0,3,7,9)
{0,2,3,5,7,9,10}
Symbolic subset
·--·---·-·--
·-··-·-·-··-
(3 4 2 3)
(2 1 2 2 2 1 2)
Partition of L (intervals)
Figure 2.8. Five ways of representing chords and scales.
3 Counting Chords "Jobling, there are chords in the human mind." - Charles Dickens, Bleak House.
Check out Guitar George; He knows all the chords. -Mark Knopfler, "Sultans of Swing".
In this chapter we describe the enumeration and classification of chords based on various properties. Following the fundamental result that there are precisely 351 distinct chords with a given tonic, we then classify each chord based on its number of notes, the intervals of the chord, the number of semitone intervals, its period, span, variety, and simplicity. We also discuss the connections between these mathematical classifications and their musical implications.
Basic Results The fundamental theorem. As discussed in Chapter 1, the common 12-tone equal-tempered musical scale consists of a cyclically-repeating sequence of the 12 musical notes C, Db, D, Eb, E, F, Gb, G, Ab, A, Bb, B, which we also identify with the integers 0 through 11, as shown in Figure 3.1. More generally, an equal-tempered musical scale oflength L consists of L notes numbered 0 through L-1.
c
F#/Gb Figure 3.1. The 12 musical notes repeat in a cyclic fashion, and thus are equivalent to a necklace of 12 beads.
40
Chapter3
Two chords are considered the same (isomorphic) if one can be transposed into the other, which corresponds to rotation of the diagram in Figure 3.1. In other words, we are asking for the number of distinct chord types for a given tonic (first note), in an L-note scale. Such chords are equivalent to 2-color necklaces of L beads, with the 2 colors corresponding to (a) a note being selected for use in the chord (represented by a black bead), and (b) a note not being selected (represented by a white bead). From this correspondence between chords and necklaces, it follows that the number of distinct n-note chords, which we denote by Ch(L,n), is precisely the number of L-bead necklaces with n black beads. That is, C h(L,n)
=N(L,n).
where the values of N(L,n) can be calculated as described in Chapter 2. The values of Ch(12,n) are shown in Table 3.1. Notice the symmetry of the values of Ch(12,n) with respect to the number of notes n in the chord: the number of chords with n notes is the same as the number of chords with 12-n (more generally, L-n) notes. The reason for this is clear: there is a one-to-one correspondence between the chords with n notes and the chords with L-n notes obtained by simply interchanging all the notes and non-notes. This is equivalent to swapping the colors black and white in the corresponding necklaces. The total number of chords for a given value of L is: Ch(L,*)
= N(L,*)-
1.
The "minus 1" comes from the fact that among alllength-L necklaces is the one with all white beads, which corresponds to the "chord" consisting of no notes (i.e., n=O). Aside from its possible use as an object of Zen meditation, this chord is not particularly interesting, and therefore is not to be counted. For L=12, this notable number is Ch(12,*) = 352- 1 = 1 + 2(1 + 6 + 19 + 43 + 66) + 80 = 351. That is, There are precisely 351 essentially different chords.
Counting Chords
41
Number of notes (n) Number of chords 1 12 1 or 11 1 2 or 10 6 19 3 or9 4 or8 43 5 or 7 66 6 80 .. Table 3.1. Values of Ch(12,n): the number of dtstmct n-note chords in the 12-tone equal-tempered scale.
Also of interest are the cumulative sums in Table 3.1; that is, the number of chords having::;; n notes (rather than precisely n notes). These values are shown in table 3.2: These numbers tell us, for example, that there are precisely 135 different chords that can be played easily with one hand on a piano (i.e., using at most one' finger per note). The remainder of this chapter is concerned with further classifying the Ch(L,n) n-note chords according to other properties, such as their minimum interval or their period. That is, we seek the value of F(L,n,p) where F is some function and F(L,n,p) is the number of chords in an Lnote scale having exactly n notes and some additional parameter equal to p. As in the foregoing discussion, we will usually fix L = 12, thus yielding a two-dimensional table of counts based on the two variables n and p. Extensions to arbitrary L will be discussed where appropriate.
Interval Analysis Introduction. The first property of interest is the minimum interval between notes in the chord. The number of chords with n notes and minimum interval ~ m is denoted /(L,n,m). Note that, because of the circularity of the scale as shown in Figure 3.1, we must also count the interval from the last note of a chord (when listed linearly) back to the first note. For example, the chord C E G B has m = 1, since there is an interval of a single note from the final B back to the first note, C. There are several reasons why interval analysis is important The frrst n Number of chords
1
2
3
4
5
6
7
8
9
10
11
12
1
7
26
69
135
215
281
324
343
349
350
351
Table 3.2.
.. The number of distmct chords wtth :;:;; n notes.
Chapter 3
42
reason is that the sound of a chord is largely determined by the intervals it contains. The intervals between notes determine their relative frequencies, and hence the harmonic structure of the compound waveform. In addition to this general fact, there is a specific physical phenomenon related to the minimum interval between two notes of a chord. This phenomenon is known as a beat frequency. Consider, for simplicity, a chord with only two notes (n = 2), and further suppose that the waveform of each note is a simple cosine wave. If the first note has frequency f and the second note has frequency f+d, then the waveform resulting from sounding both notes together will be y(t)
= cos 27ift + cos 21C(f+d)t,
where t represents time andy is the waveform amplitude. Since cos a + cos b
b+a = 2 cos - b-a - cos - 2- , 2
this can be written as y(t)
= [2 cos ndt ] cos 21C(f+d/2)t,
or, y(t) =A cos 21C(f+d/2)t where A = 2 cos ndt is the amplitude of the composite waveform; that is, the amplitude of the composite waveform varies with frequency equal to d, the difference between the frequencies of the two notes. This amplitude variation is referred to as a beat, and d is referred to as the beat frequency. This is illustrated in Figure 3.2. Two notes that differ in frequency by d produce a secondary amplitude variation (beat) with frequency d. More generally, each pair of notes in a chord produces a separate beat frequency which is proportional to the interval between the two notes. The smaller the minimum interval in the chord is, the lower the beat frequency will be. For example, a chord consisting of notes around A=440Hz with a minimum interval of 1 (m = 1) will have a beat frequency of around 26Hz, which is far below the range of fundamental frequencies in the chord, and is very audible. A value of m = 2 will yield a beat frequency of 54Hz. For this reason, chords with m = 1 (i.e., chords containing two adjacent notes) are quite different in character from chords with m > 1. Even the difference between m = 2 and m > 2 can be discerned by many listeners.
Counting Chords
43
The second reason why the minimum interval in a chord is important is more psychological then physical: smaller values of m lead to chords that seem to have emotional qualities such as "tension", "suspense", or "terror". This is undoubtedly related to the physical phenomenon of beat frequencies, but is also probably reinforced by association with various musical cliches, such as music from mystery or horror films. For example, the well-known and highly effective music from the "shower scene" in Alfred Hitchcock's Psycho is rife with chords having m = 1 (the minimum value possible). Another classic example using m = 1 chords, which evokes the mysterious rather than the terrifying, is the theme from The Twilight Zone. Other less mundane examples abound in late 19th and 20th century music, SIJCh as the works of Stravinsky.
A formula for l(L,n,m). The values of I(L,n,m) can be determined easily by realizing that a chord with minimum interval m can be viewed as a necklace consisting of two kinds of "meta-beads": one meta-bead (b) comprised of a single black bead followed by m-1 white beads (representing a note included in the chord followed by m-1 unused notes), and one meta-bead (w) consisting of a single white bead (representing an unused note), as illustrated in Figure 3.3. The empty spaces in the first kind of meta-bead ensure that a chord created from these meta-beads Note 1: Frequency f
(\(\(\(\(\(\(\(\(\
v V VIU V'VIQ V V ururv v v urv v v v v
Figure 3.2. The beat frequency phenomenon. The sum of two waveforms whose frequencies differ by a small amount (d) is a waveform with frequency f + d(2 whose amplitude varies with frequency d.
Chapter 3
44
included note
unused notes
--o-
~···-om- 1 Type (b) meta-bead
Type (w) meta-bead
Figure 3.3. All chords with minimum interval m can be constructed as meta-necklaces with two kinds of meta-beads, where the b-type metabead contains m-1 unused spaces, to guarantee the minimum interval required in the chord (which is made of beads, not meta-beads).
(when considered as a string of beads) will have at least a minimum interval of m between adjacent notes. The meta-necklace will have exactly n b-type meta-beads, since each contains one note. Each of these meta-beads consists of m beads, for a total of nm beads. Since the complete necklace must contain L beads, there must beL - nm w-type meta-beads, for a total of L- mn + n =L- n(m- 1)
meta-beads. The key point is this: since no string of b-type meta-beads is isomorphic under bead rotation to any string of w-type meta-beads, the number of distinct meta-necklaces is exactly equal to the number of distinct necklaces (which equals the number of distinct chords). Therefore, I(L,n,m) = N(L- n(m - 1), n).
Of course, I(L,n,m)
=0
if nm > L,
since nm is the number of positions required to fit n notes with a minimum separation of m, and if this number is larger than the length of the scale L, no such chords are possible. The values of 1(12,n,m), computed using the above formulas, are shown in Table 3.3.
45
Counting Chords
m = Min interval 1 2 1 2 3 4 5 6
7 •• 12
1 1 1 1 1 1 1
6
5 4 3 2 1
= Number of notes
n 3
4
5
6
7
8
9
10
11
12
19 10 4 1
43 10 1
66
80 1
66
43
19
6
1
1
3
Total 351 30 10 5 3 2 1
Table 3.3. Values of /(12,n,m):number of n-note chords with minimum interval between notes
As can be seen from Table 3.3, /(12,*,2) =30; that is, there are only 30 distinct chords with a minimum interval of at least 2 between notes (or, equivalently, that contain no "1 ", or semitone, intervals). These 30 chords are the most commonly-used chords in music but are, in fact, only a very small percentage (30 out of 351, or about 9%) of all possible chords! This provides another nice example of a "90-10" rule: approximately 90% of all music probably uses only 10% of the available Partition n=1 12
Chord inC
c
n=2 4 8 5 7 102 3 9 6 6 n=3 435 462 345 525 336 723 732 444 228 426
Comments
Partition
Chord inC
Comments
~
single note
222222
CDEF#AbBb whole tones
n=5 CE CF CBb CEb C F# CEG CEBb CEbG CFG CEbF# CGA CGBb CEG# CDE CEF#
maj3 interval 22332 4th interval 22323 maj7 interval 22224 min3 interval dimS interval n=4 C (major) 2235 C7 (partial) 4332 Cm 4323 Csus 3333 Cdim (part.) 2523 C6 (partial) 3225 C7 (partial) 3423 C+ 2262 2244 2424
CDEGBb CDEGA CDEF#Ab
C9 C6+9
CDEG CEGBb CEGA CEbF#A CDGA CEbFG CEbGA CD EBb CDEAb CDF#Ab
C9 (partial) C7 C6 Cdim C6+9 (part.) Cm(sus) Cm6 C9 (partial)
Table 3.4. The 30 chords containing no semitone (=1) intervals.
Chapter 3
46
chords. Table 3.4 gives an explicit list of the 30 chords with no semitone intervals, with the notes of the chord listed for the key of C. If we e~iminate the trivial chords from this list, and those that are subsets of another, we arrive at an even smaller set: the set of "musically distinct" chords with minimum interval 2. Although this classification is more subjective that the strict mathematical formulation of I(L,n,m), it is even more instructive, as it reduces the list of Table 3.4 to a list of only twelve chords. This list is shown in Table 3.5. Further study of Table 3.5 shows that this list can be reduced even further. The frrst six chords are essentially variations on the major chord, and the next three are variations on the minor chord. The last three Diminished, Augmented, and a rare chord not often found in music, consisting entirely of 2-semitone intervals - are all quite different and cannot be combined further. Thus, the majority of all music can be reduced to five chords (or four, since one of these is very rarely used). These five chords are marked with a"*" in the table. These five chords all share a common property, that of being composed of equal or nearly-equal intervals. The diminished, augmented, and whole-tone chord are perfect, in that all intervals are the same, and the major and minor chord are nearly so, being composed of the two possible permutations of the intervals (3,4,5). (Why are only two permutations possible? Because inversions are isomorphic, and so the other four permutations are generated by the cyclic permutations of the two chords Partition 435 4332 4323 22332 22323 525 345 3423 3225 3333 444 222222
Chord inC CEG CEGBb CEGA CDEGBb CDEGA CFG CEbG CEbGA CEbFG CEbF#A CEG# CDEF#Ab.Bb
Symbol CorCM C7 C6 C9 C6+9 Csus Cm Cm6 Cmll co C+
-
Description Major* Seventh Sixth Ninth Sixth with added 9th Sustained chord Minor* Minor sixth Minor eleventh Diminished * Augmented* Whole-tone chord *
Table 3.5. The 12 "musically distinct" chords with no semitone intervals. All of these have standard musical names except for the final one, which is composed entirely of 2-semitone intervals.
47
Counting Chords
present in the list.) There are two senses, then, in which these chords are simple in structure: they contain only large intervals, and all their intervals are nearly equal. Notice that the whole-tone chord, which is rarely seen, is the only one with minimum interval 2 (the others all have m ~ 3), agreeing with the theory that smaller-m chords are more unusual (and, therefore, more interesting!). Another function of interest, closely related to I(L,n,m), is K(L,n,m) -the number of chords having minimum interval exactly equal tom (rather than ~ m). From the definition, it is easy to see that
K(L,n,m)
=I(L,n,m)- I(L,n,m+l)
A table of these values is given in Table 3.6. This table shows that the number of distinct chords with minimum interval equal to 1 (i.e., those containing a semitone interval) is K(12,*,1) = 321. As discussed earlier in this section, these 321 chords are particularly noteworthy because of the musical and ·psychological effects of the semitone interval(s) they contain. In the next section we study these 321 chords in more detail.
Adjacency Analysis Introduction. The minimum-interval analysis described in the previous section provides a first level of classification of the 351 chords into two major classes: the 30 with no semitone intervals, and the 321 with at least one semitone interval. (We also refer to a semitone interval as a clash, because of the harsh sound produced, or an adjacency, because it involves two adjacent notes on a piano keyboard). In order to further refine this m = Min Interval 1 2
= Number
n
3
4
5
6
7
8
9
10
11
12
Total
66
43
19
6
1
1
321
1
1
9
33
63
79
2 3
1 1
6 3
9 1
3
1
1
1
5 6
1
12 Total
of notes
20 5 2 1
1 1
1 1
1
6
19
43
66
80
66
43
19
6
1
1
351
Table 3.6. Values of K(l2,n,m): number of n-note chords with minimum interval equal to m.
Chapter'3
48
Figure 3.4. Graphs of a(t) for excerpts from several compositions: Handel, Hallelujah Chorus, from Messiah (1741) Beethoven, Piano Sonata Op. 79 (1808) Gershwin, Rhapsody in Blue Theme (1924) Debussy, L'Apres-Midi d'un Faune (1892) George Crumb, Makrocosmos Vol./, Section #12 (1972)
notion, we define a new function, the adjacency count, as follows: A(L,n,a) is the number of n-note chords in a scale of L notes having exactly a adjacencies. The purpose of A(L,n,a) is to classify the 351 chords by the exact number of adjacencies they contain (rather than just into two categories: no adjacencies and at least 1 adjacency, which is what K(L,n,m) does). Chords containing adjacencies sound "harsh", "peculiar", "mysterious", "modern", "strange", and so on; the result of enumerating the number of adjacencies is a classification of all chords by their degree of strangeness. The case a = 1 is of particular interest, since the chords with exactly one adjacency are just slightly more strange than the 30 a =0 chords. This idea has many applications in the analysis and composition of music. For instance, the parameter a (number of adjacencies in chords) could be used, in a computer music composition algorithm, as one of the variables that controls the overall "sound" of a piece (by affecting the chords and harmonies used at various points in time). Low values of a (especially a= 0) would produce more traditional harmonies, while larger values of a would produce a more 20th-century sound. The average value of a could be made to vary over the course of the composition, yielding a piece whose harmonic structure evolves over time. Is the number of adjacencies as a function of time really correlated to the "sound" of a piece of music? To see that there is at least some merit to this suggestion, Figure 3.4 shows a graph of the value of a versus time for some classical pieces of music dating from 1750 to the present (including one late-20th-century composition). As expected, the average
Counting Chords
49
value of a(t) is roughly correlated with the time period of the composition, ranging from the section of Handel's Messiah which has a constant value of a= 0 to Makrocosmos by the 20th-century composer George Crumb, for which a(t) averages about 5! This graph, and the idea of classifying chords by the number of adjacencies, suggests that all music can be loosely divided into several categories: • Pieces with a = 0. A very large body of music (most music from the Classical period and earlier, for example) satisfies this trivial condition. Indeed, one of the achievements of the late Classical, Romantic, and later periods can be viewed as simply "breaking the a =0 barrier". • Pieces with an occasional a > 0 chord, but with an average value of a well under 1. The late works of Beethoven, for example, have this property, as do early works of the Romantic period. • Pieces where a averages close to 1, such as the piece by Debussy shown in Figure 3.4. Traditional jazz also falls into this category. • Pieces where the average value of a is greater than 1 (perhaps much greater). Examples of this extreme are harder to find, but are still easily found among 20th-century (or "new") music and more experimental jazz. Because of our definition of a chord in which octaves are isomorphic, note that the maximum value possible for a is 12, which is achieved by a chord consisting of all 12 notes of the scale. Therefore, the average value of a = 5 for the Crumb piece shown in Figure 3.4 is quite amazing. In the middle of this piece a series of chords with a = 8 actually occurs! An example is shown in Figure 3.5. We now return to the enumeration question: how many distinct chords
II
Figure 3.5. A chord with a= 8, from Makrocosmos by G. Crumb
Chapter 3
50
• • oType/ (contains 1 adjacency)
-oType II (adjacency-free)
Type III
Figure 3.6. Meta-beads used in the derivation of the foonula for A(L,n,m).
containing exactly a adjacencies are there?
Computation of A(L,n,a). The computation of A(L,n,a) is somewhat more involved than the computation of I(L,n,m) given in the last section, since, as we shall see, it involves necklaces of 3 or more colors. For simplicity, consider first the case a= 1 (which is the case we are most interested in, as it the next most interesting class of chords after the 30 with a= 0). Since these chords have exactly 1 adjacency, they can be constructed from a meta-necklace having three kinds of meta-beads, as shown in Figure 3.6: one meta-bead containing the adjacency, one containing a note followed by a space, and one consisting of an unused note. The unused notes in the first two meta-beads guarantee that no additional adjacencies (other than the one present in the first type of metabead) will occur in the chord. The meta-necklace will contain one type-I meta-bead and n-2 type-11 meta-beads (to satisfy the requirement of having n notes present). Therefore, there will be L- 3(1) - 2(n-2) =L + 1 - 2n type-III meta-beads (the total length of the necklace minus the number of beads used up by type-1 and type-11 meta-beads). This means that the total size of the meta-necklace is 1 + (n-2) + (L+1-2n) =L-n. Since there are three different types of meta-beads, this is a meta-necklace of three colors; therefore, we have the following formula for A(L,n,1): A(L,n,l)
=N(L-n,
1, n-2, L+1-2n).
Now, recall the formula for computing 3-color necklaces, from Chapter 2. We have (k 1,k2,k 3) = (1, n-2, L+1-2n), which means that only d = 1 is possible, since d must divide all the ki· Thus, there will only be one term in the summation, and we have
51
Counting Chords
1
A(L,n,l)
=L-n
(L-n)! 1! (n-2)! (L+l-2n)!
(L-n-1)! (L+l-2n)!
= (n-2)!
But this final expression is simply a binomial coefficient, so we can state this more elegantly as:
A(L,n, 1)
= ( L-n-1) n-2 ,
2~n ~
[L+ 2 1]
(2)
The reason this reduces to a binomial coefficient is clear: since there is exactly one adjacency, we can without loss of generality fix its position at the beginning of the chord. Then we are free to place the remaining notes anywhere (subject to the restriction that each note be followed by an unused note). What is the value of A(L, *,1) (the total number of chords with exactly 1 adjacency)? The answer to this question reveals a surprising connection between musical chords and the famous sequence of numbers known as the Fibonacci sequence. The Fibonacci numbers F(n) are defined by the relations
F(n) = F(n-1) + F(n-2) =1 F(l) = 1.
F(O)
The first few terms of the sequence are: 1,1,2,3,5,8,13,21,34,55,89. Now, from equation (2) it follows that [(L+l)/2]
A(L,*,l) =
L (L~~il) L (
L-j-i).
=
n=2
i~O
From this last expression, we see that A(L,*,l) is equal to the sum of a 3 ) and diagonal of Pascal's triangle, since the first term of the sum is
(L0
(Z)
with n decremented by each successive term is a binomial coefficient 1 and k incremented by 1; i.e., the values along a diagonal of Pascal's triangle. Denote by D(n) the sum of the diagonal of Pascal's triangle starting at ( ~). Then, we claim that:
D(n) =F(n).
Chapter 3
52
The proof of this fact rests upon the identity
(Z) = (nk1) + (Z=D· which, as illustrated symbolically in Figure 3.7, implies that D(n) =D(n-1) +D(n-2)
which is precisely the relation defining the Fibonacci sequence. Since =1 and D(1) =1, we find thatD(n) =F(n) for all n.
D(O)
Combining these results, we conclude that A(L,*,1)
=F(L-3).
That is, the number of chords in a scale of L notes having exactly 1 adjacency is equal to the (L-3)th Fibonacci number. For the case L = 12, there are therefore F(9) = 55 such chords. These chords are listed in Table 3.7.
-k0 0
0
0
0
e~l) kl~
0
0
n
e-1) k
(;) 0 0
0 '-...... Pascal's Triangle
D(n-2)
0
D(n-1)
0
0 0
0 0
\ 0
D(n) Figure 3.7. Pictorial demonstration that
=
(Z) = ( nk1)
+(
z:D
implies D(n) D(n-1) + D(n-2), and therefore that the sums of the diagonals of Pascals' triangle equal the Fibonacci sequence.
Counting Chords
53
Only a few of these 55 chords ate commonly used in music; perhaps the most frequently heard is the major seventh chord (4,3,4,1) which is ubiquitous in jazz music as a more "colorful" substitute for the major chord (4,3,5). Note that, as illustrated by this example, many of the a= 1 chords can be viewed as an a = 0 chord with one of the intervals split into two intervals (one of which is an interval of 1).
The case a= 2 and beyond. The case a= 2 is even more tricky, Partition n=2 111 n=3 219 3 18 417 5 16 6 15 714 8 13 912 n=4 4341 3441 4431 2145 4215 5241 4125 2451 4251 3135 33 15 3 3 51 6321 2631 3621 6231 2361 3261 2127 2217 2271
Chord inC
Comments
CC#
single clash
CDEb CEbE CEF CFF# CF#G CGG# CG#A CABb CEGB CEbGB CEG#B CDEbG CEGbG CFGB CEFG CDF#B CEF#B CEbEG C EbF# G CEbF#B CF#AB CDG#B CEbAB CF#G#B CDFB CEbFB CDEbF CDEF CDEB
M7 m+M7 aug+M7 m9 M+b5 M7sus M+ll M/m m+b5 dim+M7
Partition
n::::6 223122 223212 223221 221232 322221 n=5 21432 43212 43122 41232 21423 42132 12423 22341 32241 22314 22134 42123 13332 13323 31323 3 3 3 12 22215 22125 21225 12225
Chord inC
Comments
CDEGAbBb CDEGABb CDEGAB CDEFGBb CEbFGAB
9th+b6 9th+6 M7+6+9 7th+9+11
CDEbGBb CEGABb CEGG#Bb CEFGBb CDEbGA CEGbGBb CDbEbGA CDEGB CEbFGB CDEGG# CDEFG# CEF#GA CDbEGBb CDbEGA CEbEGA CEbF# ABb CDEGbG CDEFG CDEbFG CDbEbFG
m7+9 7th+6 7th+aug5 7th(sus) m6+9 7th+b5 m6+b9 M7+9 m4+M7 9th+aug5 aug+4+9 6th+b5 7th+b9 6th+b9 M/m+6 dim+7 9th+b5 9th+4 m9+4 mll+b9
Table 3.7. The 55 chords containing exactly one adjacency.
54
Chapter 3
-e--o---
--o-
n-3
L+2-2n
--e-e-o--
-e--o---
--o-
2
n-4
L+2-2n
• • • o-
=
Figure 3.8. The two different ways to make a chord with a 2. The values below each type of meta-bead indicate how many meta-beads of that type are required.
since there are two different kinds of 3-color meta-necklaces that can generate chords with 2 adjacencies, as shown in Figure 3.8. These two types arise from the fact that the two adjacencies can themselves be adjacent (in other words, the two adjacencies arise from using three consecutive notes in the chord) or they can be separated. From Figure 3.8, we derive the following formula:
A(L,n,2)
=N(L-n, 1, n-3, L+2-2n) + N(L-n, 2, n-4, L+2-2n) n-3 + N(L-n, 2, n-4, L+2-2n) = (L-n-1) n = Number of notes
a =
Number of Adjacencies 1 0 1 2 3 4 5 6
1
2
3
4
5
6
5 1
10 8 1
10 21
3 20 30 12 1
1 5 26 34 13 1
11
1
7
8
9
10
11
12
30 55 3 20 30 12 1
7
71
1
67 54 34 22 9 6 1 1 0 1
1
351
10 21 11
1
8 9 10
10 8 1
5 1 1
11
12
Total
1
6
19
43
66
80
Total
66
43
19
6
1
Table 3.8. Values of A(l2,n,a): number of n-note chords with exactly a adjacencies.
Counting Chords
55
Proceeding in this fashion we can derive formulas for a = 3, a = 4, and so on (although they become increasingly complex). The results of these computations for L = 12 are shown in Table 3.8. We have also included the values of A(l2,*,0) in this table, which are simply equal to the values of /(12,*,2). This table provides the complete classification of all 351 chords by the number of adjacencies. Of course, there are no chords with a= 11, since the only way to have 11 adjacencies is to use all 12 notes, and in that case there is also a 12th adjacency (due to the circularity of the scale). Notice that the largest numbers of chords are found under a = 2 and a= 3. Indeed, if a chord is chosen at random from the 351, the expected
number of adjacencies is approximately 3. This is quite high - recall from our earlier discussion that a =1 is quite sufficient to provide a subjectively "strange" sound, and chords with a > 1 are relatively rare, except in very modem music. The fact that the "average" chord has a= 3 is one reason why it sounds "bad" when, for instance, a non-musician "plays" a piano by randomly striking keys. (Of course, we are walking a fine line here, between "bad" music with a= 3 and "good" music with a= 3. There are, naturally, other things beside the value of a that determine the quality of music; the point is that random a = 3 music has a high probability of sounding "bad".) This motivates the following realization: it is possible for someone who knows nothing about music to play "music" that contains only chords with a= 0 (and, therefore, sounds somewhat "reasonable") on a piano. The rule for doing this is simple: use only the black keys. The black keys are located at positions(- •- •-- •- •- •-) in the 12-note scale; that is, no two black keys are adjacent, and therefore chords with a > 0 will never occur. In the next section we consider the generalization of this idea, by asking which of the 351 chords can be played on keys of the same color on a piano.
Monochromatic Chords Define a monochromatic chord as one which can be played on a piano using either all white keys or all black keys. How many of the 351 chords are monochromatic, and which are of which type (all white, called type W, or all black, called type B)?
Chapter 3
56
Recall (from Fig. 1.4) that the white and black keys on a piano form the following pattern (starting on the note C):
-·-·--·-·-·where- is a white key and • is a black key. We immediately have the following theorem: if a chord is of type B, then it is also of type W. The proof of this is easy: if we shift one note down from the 5 black keys, we arrive at 5 white keys. Therefore, any chord which can be played on all black keys can also be played on all white keys - namely, using either set of 5 white keys itnmediately adjacent. Remember that in asking about monochromatic chords, we consider transpositions to be isomorphic. So, this theorem merely states that if a chord is playable using only black keys, a chord of the same type (but not in the same key) is playable using only white keys. In other words, we are asking which chord types are monochromatic, irrespective of key. Even monochromatic chords will require both white and black keys in some keys. Given this theorem, we can divide the 351 chords into 3 disjoint classes: (1) Monochromatic of type B (and, therefore, also of type W), (2) Monochromatic of type W, but not type B, (3) Polychromatic; i.e., requires both white and black keys to play (in every key). As discussed in the last section, all the chords in class ( 1) will have a = 0. All the chords in class (2) will have 0 ~ a ~ 2, and it is therefore of interest to divide this class into three sub-classes, for a = 0,1 ,2. In addition, we also seek to enumerate the chords in all of these classes based on n, the number of notes in the chord. The number of n-note chords in each of these categories is shown in Table 3.9. There are four manifestations of Pascal's triangle (i.e., binomial coefficients) in this table. First, observe that the first, second, and fourth rows of Table 3. 9 are rows of Pascal's triangle:
=1
(1)
n-note B (and W) chords
(2)
a=O chords =. 1 3 3 1 n-note W-only, a=2 chords = ... 1 3 3 1
(3)
n-note W-only,
4 6 4 1
- (
-
4
n-1 )
= (n~2). =
(n~4)
Counting Chords
n B (and W) W only, a=O W only, a=l W only, a=2
57
2
3
4
5
1
4 1 1
6 3 6 4
4 3 12 1 23
1 1 10 3 51
3 3 74
1 65
43
19
6
1
1
16 8 32 8 287
19
43
66
80
66
43
19
6
1
1
351
Polychromatic Total
1
6
6
7
8
12
l
9
10
11
To-tal
Table 3.9. The number of monochromatic and polychromatic chords with n notes.
Secondly, if we sum the first four rows of Table 3.9, to produce the total number of monochromatic chords (type B or type W), we get: (4)
n-note monochromatic chords= 1
6 15 21 15 6 1
=
(n~t)·
Equation (3) is the simplest to explain. The only way of achieving a=2 in a W-only chord is to choose the four notes BC (which are adjacent) and EF (which are adjacent); the other n-4 notes can then be chosen from the 3 remaining white keys. Since the pairs BC and EF are separated by 5 notes, which is relatively prime to 12, no rotation of one of these chords can be equal to one of the others; therefore all distinct, and hence equation (3).
(n~4 )
such chords are
Is there a simple combinatorial explanation for equations (1), (2), and (4)? This question is left as an exercise for the reader. As discussed in the previous section, a type-B chord (which is also type-W) always has a = 0. Therefore, the expected value of a for a randomly-chosen type-W (=monochromatic) chord is: a= (16·0 + 8·0 + 32·1 + 8·2)/64 = 0.75
This says that our hypothetical non-musician discussed in the last section could make more interesting, but still not too weird, music (with a=0.75, on average) by following another simple rule: use only the white keys. Partition 444 381 83 1 I 110
Chord inC
Comments
CEG# CEbB CG#B CC#D
C+; only one with a=O b3 + M7; a=1 #5+ M7; a=1 Onlv one with a=2
Table 3.10. The 4 polychromatic chords containing only 3 notes.
Chapter 3
58
There are polychromatic chords with as few as 3 notes. The four such chords are shown in Table 3.10.
Period Classification Definition. Another property of interest, especially when playing chords on a stringed instrument such as a guitar, is the period of a chord. By this we mean the period of a chord under transposition, or, equivalently, under rotation of the corresponding necklace, where the period P is defined as the smallest integer P 2: 1 for which a given chord, when transposed by P notes (equivalently, when rotated P positions as a necklace) comes into exact coincidence with itself. For example, consider the three necklaces shown in Figure 3.9. The first of these has period 1, since a rotation by only 1 position produces the same necklace. (Clearly, there are only two two-color necklaces with period 1 -the ones consisting of either all white or all black beads.) The second example has period 3, since a rotation of 3 positions brings it into coincidence with itself, and no lesser rotation will do. The final example has period 6; notice that P = n, the number of beads in the necklace. In other words, no rotation of less than n positions produces the same necklace. The connection with stringed instruments is the following. On a stringed instrument, a chord is played by placing fingers on the strings at various positions (usually demarcated by frets which occur at each semitone position), thus effectively transposing the notes sounded by each string, perhaps by a different amount for each string. Given a particular arrangement of the fingers on the strings, the resulting chord can be transposed by simply moving the entire hand up or down the neck of the instrument. Therefore, the period of a chord is also precisely the number of positions after which the chord repeats exacdy (i.e., in the same key). For this reason, chords with small period are especially interesting,
000 Period= 1
Period= 3
Period=6
Figure 3.9. Several6-bead necklaces with different periods.
Counting Chords
59
since they repeat within a short distance on a stringed instrument. Furthermore, when such a chord repeats, it is in a different inversion. Thus, this property can be used to play several different inversions of the same chord in very rapid succession, since the finger position does not change, and the distance between successive chords is small. The classic example of such a chord is the diminished chord as played on the guitar. Figure 3.10 shows the finger positions for a diminished chord; the numbers across the top of the figure indicate the note number (0-11) corresponding to each string of the guitar. (The x's indicate strings which are not used in this chord.) The four strings being sounded, from the right, are 0, 7, 3, 10; the finger positions increase the first and third notes by 2 and the second and fourth notes by 1, to yield {2,8,5,11 }, or {2,5,8,11}, or, transposing down by 2, {0,3,6,9}. This is precisely the diminished chord, with intervals (3,3,3,3). Since all the intervals are 3, this chord has period 3, and therefore repeats exactly every 3 frets on the guitar. The general question, then, is to find the number of n-note chords in a scale of L notes having period exactly equal to p. In necklace terminology, this is the number of necklaces with L beads and n black beads having period p. We denote this value by P(L,n,p), and seek a general formula for P(L,n,p ).
A formula for P(L,n,p). A simple recursive formula for P(L,n,p) can be derived by the following strategy. First, we count the number of necklaces with period d ~ p (where d I p ), including all distinct cyclic permutations. Then, we subtract out the number of necklaces having period strictly less than p (and their cyclic permutations). This leaves just those having period p plus all their distinct cyclic permutations. But since these all have period p, each necklace appears in exactly p incarnations; therefore, the number of distinct necklaces (with rotations considered isomorphic) is simply this number divided by p. A prototypical necklace with period pis shown in Figure 3.11. By 0510370
X X
1m
Figure 3.10. The diminished chord on guitar. Each vertical line represents one of the 6 strings, and each horizontal line a fret. The circles represent finger positions.
60
Chapter 3
c;;o+. ·••I I• o-e-• • ·# · · · -£€00+· ~·~;J '"-----.J" /
p beads
Llp identical pieces
Fig. 3.11. A generic necklace of n beads having period p (or, possibly, period equal to a divisor of p), consisting of Lip identical pieces, each containing p beads.
definition, a rotation by p places is an isomorphism; therefore, the necklace must consist of a series of identical segments of length p, where p is a divisor of L. There are Lip such segments. Since there are n black beads in all, there must be (n I (Lip)) black beads in each segment. Therefore, if we ignore rotational isomorphism, and just count the number of necklaces of the form shown in Figure 3.11, we see that there are
such necklaces. However, among these necklaces are also, possibly, those with period less than p (and their cyclic permutations). For each possible such period d (which must be a divisor of p ), there are P(L,n,d) distinct such necklaces, and each one occurs d times (since cyclic permutations are included), for a total of d·P(L,n,d) necklaces. After subtracting the necklaces with period < p, we are left with only those with period =p, plus their cyclic permutations. A final division by p produces the value of P(L,n,p). The resulting formula for P(L,n,p) is:
P(L,n,p)
=
t [ (L;J ·
L d · P(L,n,d)
1~
dip (Lid) In
with
P(L,n,Lin) = 1 P(L,n,p) = 0 if p
! L or (Lip) ! n.
]
Counting Chords
61
Note that if n is relatively prime to L then no period less than L is possible. (Why? Because Lip cannot divide n for any p, except for p=L.). In this case, the summation term is always 0, and so: P(L,n,L)
t
= (~)
P(L,n,p) = 0 (for p
L).
¢
For example, in a 12-note scale (L = 12), there are two integers less than 12 and relatively prime to 12: 5 and 7. This means, for example, that all 7-note chords have period 12, and the number of such chords is 1 (12) 792 12 1 =u= 66 · Table 3.11 shows the values of P(l2,n,p), calculated using the above formula. Notice that this table is symmetric about then= 6 column (then= 12 entry appears to violate this symmetry, but in fact does not, since it is symmetric with the trivial n = 0 chord which we do not show in these tables). This leads to the conjecture that P(L,L-n,p) = P(L,n,p) in general. The proof that this is indeed the case is by induction on p. The value of P(L,L-n,p) is, by the above formula,
=~ [
P(L,L-n,p)
(£;)- L
d·P(L,L-n,d)]
n = Number of notes
p =
Period 1
2
5
3
1 2 3
Total
7
8
9
10
11
12
1
1 1 2 3 9 335
1
351
1
1
1 5
18
40
66
1 3 75
1
6
19
43
66
80
2
1 2
66
40
18
1 5
1
66
43
19
6
1
Total
1 1 1
6 12
6
Table 3.11. Values of P{l2,n,p): number of n-note chords with period p.
62
Chapter 3
Period Comments Partition Chord inC Whole-tone scale 222222 C DEF# Ab Bb 2 Diminished C EbF# A 3 3333 444 4 Augmented CEG# C+ and C#+ combined! CC#EFG#A 4 13 13 13 dim5 interval C F# 6 66 2424 CDF#G# 6 a=O a=;2 1 51 5 C C#F# G 6 a=;2 CDFF#G#B 231231 6 a=;2 321321 CEbFF#AB 6 114114 CC#DF#GG# 6 a=4 Table 3.12. The 10 chords with p<12 and n~6.
=
.!. [ (
p
\_P
~.!L)L d·P(L,L-n,d)] Lip
which is, in fact, equal to the expression for P(L,n,p ); the binomial coefficient is equal, by the identity
and the summation is equal, by the induction hypothesis (since values of d in the summation are strictly less thanp). Another, somewhat more direct, explanation of this symmetry is that the period of a necklace is unaffected by exchanging the colors of the beads. A necklace with n black beads has L-n white beads, and therefore there is a direct isomorphism (including period) between the necklaces with n black beads and those with L-n black beads; therefore, P(L,L-n,p) =P(L,n,p). As Table 3.11 shows, the vast majority of the 351 chords have period
p =2 Fig. 3.12. Two interesting guitar fingerings: the unique chord with p=2, and the unique chord with a=5. The symbol "o" means that a string is played but not fingered.
Counting Chords
63
12 (335 out of 351, or about 95%), and there are only 16 chords with periods other than 12. Of these, only 10 are reasonably playable (having 6 or fewer notes). These 10 chords are shown in Table 3.12. Devising finger positions for each of these chords on the guitar makes an interesting exercise. Figure 3.12 shows how to play the unique p = 2 chord (which is a bit difficult, as it requires two "bar" fingerings). Incidentally, the subject of guitar fingerings contains many other intriguing questions. For example, since it is possible to sound at most 6 notes, at most 5 adjacencies are possible in a guitar chord. Is this value of a = 5 actually achievable? The other chord depicted in Figure 3.12 shows that it is: by transposing the base notes {0,5, 10,3,7,0} by {4,0,4,0,0,6}, we produce {4,5,2,3, 7,6} (mod 12), or {2,3,4,5,6, 7}, which consists of 6 adjacent notes. This must certainly be one of the more bizarre chords that can be played on a guitar.
Span (and Maximum Interval) Analysis Just as the period of a chord, discussed in the previous section, is important when playing chords on a stringed instrument, there is another property of chords which relates to chord-playing on a keyboard instrument. This property is the span of a chord, defined as follows:
Definition. The span of a chord consisting of notes {mt. m2, ... , mn} (where mt < m2 < ... < mn) is equal to mn- mt; in other words, it is the number of semitones separating the first (lowest) and last (highest) note in the chord. However, since inversions of the same chord are considered isomorphic, we are really interested in the minimum span of any inversion of the chord. For example, consider a 7th chord (see Table 3.5), which has intervals (4,3,3,2). The four possible inversions, with their spans,
are: Inversion 04 710 0368 0359 0269
Span 10 8 9 9
Thus, the span of a 7th chord is 8, the minimum among the spans of its 4 inversions. Note that this corresponds to the inversion {0 3 6 8}, or {E G Bb C} in the key of C, which is not the most common inversion of the
Chapter 3
64
7th chord ( {C E G Bb} is the usual way of playing a C7 chord). This illustrates the fact that the minimum-span inversion may be different from the most commonly-used inversion. The connection with keyboard instruments is this: The span of a chord is roughly the physical width whicl: must be covered when playing a chord with one hand on a keyboard. We say "roughly" because not all semitone intervals are the same width on a piano keyboard. For example, if the distance from C to C# is 1 unit, the distance from E to F, which is also 1 semitone, is 2 units. In fact, 10 of the 12 intervals are 1 unit wide, with the other two (E to F and B to C) being 2 units wide. For simplicity, however, we ignore this distinction, and simply consider the numerical span in number of semitones. In general, chords with larger spans are "harder" to play on a keyboard. Notice that, by the defmition of a chord (with octaves isomorphic), the maximum span value possible is 11.
Span and Maximum Interval. The minimum interval problem, discussed earlier in this chapter, asks for the number of chords with minimum interval equal to m. The dual property, which asks for the number of chords with maximum interval equal to some value, is also solved in this section, because we claim that:
The number of n-note chords with span s is equal to the number of n-note chords with maximum interval L-s. The proof of this is simple. By the definition of span as the minimum span of any rotation (inversion) of the chord, the chord cannot contain an interval larger than L-s. If it did the chord could be rotated to place this interval at the end, which would mean that the chord has a span of smaller than s, which is a contradiction. · This observation becomes useful in the next chapter, because, unlike chords, where minimum interval is musically significant, one of the important properties of a scale is its maximum interval.
Counting chords by span. We define S(L,n,s) as the number of
s Figure 3.13. Generic necklace with L beads having n black beads and spans.
65
Counting Chords
chords with n notes in a scale of length L having span exactly equal to s, and seek a fonnula for S(L,n,s). To do this we use the full generality of the m-color necklace-counting theorem from Chapter 2. Consider a generic necklace having L beads (with n black beads) and span s, as shown in Figure 3.13. We have rotated the necklace so that all black beads are located in the first s+ 1 positions (which must be possible, by the definition of span). Now, we consider the meta-necklace composed of n meta-beads, where each metabead consists of a single black bead followed by ki-1 white beads. Note that k1 + k2 + ... + kn-1 =s, by the definition of the span, which means that kn =L- s. For a given set of {ki} values, we can determine the number of distinct meta-necklaces as follows. Noting that each ki is a number between 1 and n, we consider these possible values for the {k;} as n different colors, and count the number of meta-beads of each color that occur. Suppose there are 11(r) meta-beads of color r present (1 s; r s; m); then the total number of meta-necklaces is simply N(n, 110), 11(2), ... , 11(n))
where N(n, ... ) is the general neckhice-counting formula from Chapter 2. At this point we invoke the now-familiar argument: since no type of metabead is isomorphic (when considered as a string of beads) to another, the number of meta-necklaces equals the number of necklaces. In the above discussion we considered merely a generic set of {ki} values. In order to count all the possible necklaces, we must consider all possible sets of {ki} values that can occur. But this is simple: by the definition of the {ki}, we see that k1 + k2 + ... + kn = L; that is, the {ki} are simply a partition of L into n parts. Furthennore, ki s; L-s for all i, since if any ki were greater than L-s this would imply a span smaller than s. Finally, we must have at least one ki = L-s. To summarize: the possible {ki} sets are all possible partitions of L into n parts with maximum part L-s. For each set we count the metanecklaces with that distribution of colors; the total number of necklaces with span s is simply the sum over all {ki} sets. Therefore we arrive at the following fonnula for S(L,n,s), the number of n-note chords having a span of exactly s:
S(L,n,s)
=
L
N(n, 11(1), 11(2), ... , 11(n))
k]+ ...+kn=L 1 SkiS 12-s at least one kj =12-s
where 11(r) is the number of r's in the set {kJ ,... ,kn}.
Example. What is the value of S(12,4,8)? First, we must compute all the possible {ki} sets, which are all possible partitions of 12 into 4 parts with maximum part equal to 12-8 = 4. Such a partition is of the form {4,x,y,z}, where x+y+z = 8; i.e. we need to know all partitions of 8 into exactly 3 parts. There are three such partitions: {4,3, 1}, {3,3,2}, and {4,2,2}, which means that the four {kil sets are {4,4,3,1}, {4,3,3,2}, {4,4,2,2}. Applying the span formula, we get: S(12,4,8)
=N(4, 1,0,1,2) + N(4, 0,1,2,1) + N(4, = 2N(3, 1,1,2) + N(2, 2,2) = 2·3 + 2 = 8.
0,2,0,2)
The values of S(12,n,s), computed using this formula, are shown in Table 3.13. There are several notable features of this table. First, notice the geometric pattern formed by the non-zero values in the table (bounded s = Span
n = Number of notes
1 0 1 2 3 4 5 6
2
3
4
5
6
7
8
9
10
11
12
Total
1
1 1 2 4 8 16 32 59 96 101 30 1
1
351
1 1 1 1 1 1 1
7
8 9 10
1 2 3 4 5 3 1
1 3 6 10 14 8 1
1 4 10 20 25 6
1 5 15 33 25 1
1 6 21 35 3
1 7
25 10
1 8 10
1 5
1
11
Total 1
6
19
43
66
80
66
43
19
6
1
Table 3.13. Values of S{12,n,s): number of n-note chords with spans.
67
Counting Chords
by a diagonal line on top but by something else on the bottom). This is due to the fact that the {kil are partitions of Linton parts with maximum partL-s. There will be no such partitions if the minimum possible value of 'I:.ki is less than L, or if the maximum possible value of 'I:.ki is greater than L. The maximum possible value of 'I:.ki occurs when all ki = L-s, and is n·(L-s); the minimum value occurs when one ki = L-s and all the rest= 1, and is L-s+(n-1). Thus, S(L,n,s) = 0 if
n·(L-s) < L
or L-s+(n-1) > L
which can be sirnplifiedto L
n < L-s
or n > s+ 1
which are, as illustrated by Table 3.13, a hyperbola and a diagonal line, respectively, in (n,s)-space. Another somewhat startling feature of Table 3.13 is that many of the entries (such as the first 7 rows, and parts of other rows) look like parts of Pascal's triangle; i.e., binomial coefficients. Is this just coincidence, or is there a deeper reason for this? To see the explanation for this, consider the value of S(L,n,s) where s is "small" (say, somewhat less than L/2). Recalling Figure 3.13, we see that such a chord has all its notes bunched up near the beginning of the necklace. Two notes (the base notes) of the chord are determined by the value of s (the first position in the necklace, and another ones beads away), and the remaining n-2 notes are all located between these two. Thus, the total number of such chords is (~~~). except for the possibility that two such chords may be isomorphic under rotation. But, this is impossible, because this requires another long string of consecutive unused notes (besides the one present at the right end of the unrotated necklace), and the span is too small to permit such a string to occur between the two base notes. Therefore, the number of chords is simply equal to
(~~~).
More generally, it is impossible to have a second string of L-s-1 unused notes if s-1 (the number of necklace positions between the two base notes) minus n-2 (the number of notes of the chord between the base notes) is less then L-s-1, because that would mean that there are less than L-s-1 unused notes between the base notes, which obviously means that
n = Number of notes
s = Span
1 0 1 2 3 4 5 6 7 8 9 10
2
3
4
5
6
7
8
9
10
11
12
1
1 1 2 4 8 16 32 59 96 101 30 1
1
351
1* 1* 1* 1* 1* 1* 1.
1* 2* 1* 3* 3* 1* 4* 6* 4* 1* 5* 10* 10* 5* 1* 3 14 20* 15* 6* 1* 8 25 33 21* 7* 1* 1 8* 1 6 25 35 25 3 10 10 1
1* 5
1*
11
Total 1
6
19
43
66
80
66
43
19
6
1
Total
Table 3.14. Values of S(l2,n,s). Asterisked values are those due to the binomial span theorem.
there cannot be a string of L-s-1 consecutive unused notes. So we see that S(L,n,s) is equal to a binomial coefficient if
(s-1)- (n-2) < L- s- 1 which gives us the Binomial Span Theorem:
S(L,n,s)
= ( s-1) n-2
if n > 2(s+ 1) - L.
In particular, if s < L/2 - 1, the right hand side of the inequality is ~ 0, which means that the theorem holds for all n. This explains the first five rows of Table 3.13; for succeeding rows, the binomial span theorem only applies for sufficiently large n. Table 3.14 shows the distribution of these special val.ues of S(L,n,s); in fact, the majority of the values are binomial coefficients. There is also an interesting connection between span and minimum interval, which can be seen by comparing the s=lO row of Table 3.14 with the m=2 row of Table 3.3: S(12,6-11,10) = 1 3 10 10 5 1 /(12, 1-6,2)
=1
5 10 10 3 1
69
Counting Chords
Is this just a coincidence, or is there some reason why S(12,n,10)
= /(12,12-n,2),
that is, the number of n-note chords with span = 10 is the same as the number of n-note chords with minimum interval~ 2? It turns out that again this is not just coincidence. For a chord to have a span of exactly 10, it cannot have two consecutive unused notes, since if it did it could be rotated to place those at the end, and then would have a span of less than 10. The dual of this set of chords (obtained by changing all notes to unused notes, and vice versa) is the set of chords with 12-n notes with no two consecutive notes, which is precisely /(12,12-n,2). In general, we have S(L,n,L-2) = l(L,L-n,2) = A(L,L-n,O). The second equality results from the definition of the A() function: the number of chords with 0 adjacencies is the same as the number with minimum interval~ 2.
Interval Sets In previous sections of this chapter, we considered the classification of n-note chords by several properties related to intervals: the minimum interval, the number of 1 intervals (adjacencies), and the maximum interval. These questions are all special cases of the more general problem of classifyin-& all chords by their interval set, which is defined as the set of all intervals m the chord (including, of course, the interval from the last note back to the first; i.e., considering the chord as a necklace). More precisely, the interval set for a chord consisting of the notes {m 1, m2, ... , m 0 } (where m 1 < m2 < ... < m 0 ) is the (unordered) set {m2- mt. m3- m2, ... , m 0 - mn-t. L- m 0 }. From this definition, it is clear that the sum of the intervals is L; therefore, the number of possible interval sets for a given L is simply the number of partitions of L into n parts, or p(L,n), using the notation from Chapter 2. From Table 2.4, we see that for L = 12, the number of interval sets for each value of n forms the sequence p(12,*) = 1, 6, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1. which sums to 77. In other words, in the 12-note equal-tempered scale:
There are 77 distinct interval sets. Using the methods described in Chapter 2, we can generate a list of the partitions of 12, or the interval sets in the 12-note scale; this list is
Chapter 3
70
given in Table 3.15. This table also shows how many distinct chords are generated by each partition, which can be calculated using the m-color necklace formula, using the same technique as in the previous section (i.e., by considering the different interval values as different colors).
K-atonic interval sets. An interesting type of interval set is one in which all intervals between 1 and k (inclusive) are represented, for some k. Such a set is referred to as k-atonic, after the musical term diatonic which is used for the case k = 2. We refer to k = 3, 4, ... as triatonic, quadratonic, etc. At least two interesting questions arise: (1) For a given value of L, which k-atonic interval sets are possible? (2) For each possible k value, how many k-atonic interval sets exist? To answer the first question, let a(k) denote the sum of the integers from 1 through k. Then clearly we must have a(k) ~ L, which, since a(k)
k(k + 1)
2
means that k(k + 1)/2 ~ L. Solving fork, we see that k-atonic interval sets are possible only for
k~["8L~1-1]. For L=12, we find that k
~
4, as verified by Table 3.15.
To answer the second question, notice that a k-atonic interval set must contain the elements {1,2, ... k}, which sum to a(k). This means that the remaining elements in the partition must sum to exactly L- a(k) (and must all, of course, be ~ k). Thus,. the number of k-atonic interval sets is equal to q(L- a(k), k). For L=12, this means there are q(9,2) = 5 diatonic sets q(6,3) = 7 triatonic sets and q(2,4) = 2 quadratonic sets. These are marked (as "Di", "Tri", or "Quad") in Table 3.15. Note that there is a triatonic and quadratonic set on as few as 5 notes (3 3 3 2 1 and 4 3 2 2 1, respectively). The intervals 4 3 2 2 1 (in that order) form one of the most playable quadratonic chords: a major 7th with
71
Counting Chords
n=1 (1) 12 n=2 66 75 n=3 444 (1) 552 (1) 543 g~ 651 n=4 3333 (1) 4431 (3) 4422 (2) 4332 (3) 5511 - (2) n=5 33321 (4) Tri 33222 (2) 44211 (6) 4 3 3 1 1 (6) (12) Quad 43221 n=6 2 2 2 2 2 2 (1) 3 3 3 1 1 1 (4) 3 3 2 2 11 (16) Tri 3 2 2 2 2 1 (5) Tri n=7 2222211 332 1111 3222111 n=8 22221111 33 111111 n=9 222 111111 n=10 221 1 1 1 1 1 11 n=ll n=12
m
84 93
m
10 2 111
64 2 633 74 1 732
(2) (1) (2) (2)
831 822 921 10 11
5421 5331 5332 64 11 6321
(6) (3) (3) (3) (6)
6222 7 3 11 7221 8211 9 111
(1) (3) (3) (3)
42222 54 111 53211 52221
(1) (4) (12) (4)
63111 62211 72 111 81111
(4) (6) (4) (1)
441111 (3) 522111 4 3 2 1 1 1 (20)Quad 6 2 1 1 1 1 4 2 2 2 1 1 (10) 711111 5 3 1 1 1 1 (5) 4311111 422 1111 (3) Di 5211111 (15) Tri (20) Tri 6111111 322 11 111 42111111 (10) Di (4) 5 1111111 32 1111111 (10) Di 4 11111111 (5) Di
3 111111111 2 1111111111 111111111111
(1) (1)
(2) (1)
m (1)
(10) (5)
(1)
(6) (15)
~~
(21) Tri
m (8) Tri (1) (1) (1)
Di
(1)
Table 3.15. The 77 interval sets for L = 12. The number in parentheses next to each interval set is the number of distinct chords it generates.
Chapter 3
72
added 6th, or C E G A B in the key of C. This interval set generates 11 other chords, which are also musically useful. Table 3.15 provides a rich area to explore for chords containing certain interval sets. We have already discussed several notable interval sets, such as those consisting of a single repeated number (of which there are exactly 6, corresponding to the 6 divisors of 12). As another somewhat frivolous example, suppose we ask about prime chords, in which all intervals are prime numbers (and we are strict in the definition of primes, so that 1 is excluded). We see that the only intervals sets consisting entirely of primes are (7 5), (5 5 2), (7 3 2), (3 3 3 3), (5 3 2 2) so that there are exactly 1 + 1 + 2 + 1 + 3 = 8 prime chords. Of these, the sustained chord (52 5) and the diminished (3 3 3 3) are perhaps the most common.
Graceful Chords An even more musically useful way of analyzing intervals in a chord is to realize that when an n-note chord is sounded there are actually (~). intervals present (not just n, as we assumed in the previous section). Since all n notes are sounded simultaneously, a musical interval is formed by every pair of notes in the chord - not just pairs of adjacent notes. The "sound" of a chord is influenced by all these intervals. This motivates the definition of the full interval set of a chord. The full interval set of a chord consisting of the notes {m t. m2, ... , mn} (where m1 < m2 < ... < mn) is the set of values mi- mj, fori> j. The property we are interested in is the variety of a chord, whtch is defined as the number of distinct intervals in the full interval set. We denote the number of n-note chords in an L-note scale having exactly V'different intervals in the full interval set by V(L,n,v), and seek the values of V(L,n,v) for all nand v. The first thing to notice is that the value of v is not the same for all inversions of a chord; for example, the full interval set for the chord
(·-·--·--·---) is {2,3,8,3,6,8} (or, eliminating the extra 3, {2,3,5,6,8 }), whence a value of v = 5, whereas an inversion of this chord is
(•---·-·--·--)
73
Counting Chords
which yields the set {4,6,9,2,5,3} or {2,3,4,5,6,9} and thus a value of v = 6. In computing the variety measure of a chord, therefore, we shall ask for the maximum v value of any inversion of the chord. The variety value is, loosely speaking, a measure of the "richness" of a chord. The larger the v value, the more different intervals that are present; and hence the more complex the physical waveform, due to the interaction of periodic waveforms with frequencies in different ratios.
The graceful connection. The variety value of a chord is related to a famous concept in mathematical graph theory - graceful graph numbering. In this context, a graph is a collection of points (nodes) connected in some way by a set of lines (edges). The graceful graph numbering problem asks for a numbering of the nodes of a graph with different integers ~ 0 so that (a) All the edge numbers (where an edge number is defined as the difference between the edge's two node numbers) are distinct, and (b) The largest node number is as small as possible. Figure 3.14 shows a graceful numbering of the Ks, the complete graph on 5 points. (The complete graph on n points, K n. consists of n points with an edge connecting every pair of points.) The question of interest in graceful numbering is: what is the minimum possiblevalue for the largest node number (which we will denote by M)? Clearly, M must be at least equal toe, the number of edges in the graph. Why? If M is smaller than e, then we have less than e numbers to assign to the e edges of the graph, which the pigeonhole
9
11
4
Figure 3.14. A graceful numbering of Ks, the complete graph on 5 points.
Chapter 3
74
principle tells us is impossible. The "most graceful" numbering of a graph occurs when M = e, which means that the edge numbers are not only distinct, but they are exactly the set {1,2, ... ,e}. Note, of course, that the minimwn node number in a graceful graph is always 0 (if it were k > 0, we could reduce all node numbers by k and produce a graph with smaller M, which is a contradiction since the graph is graceful to start with.) So, the largest vertex number Min a graceful graph is always at least equal toe, the number of edges in the graph. However, it turns out that M = e is not usually achievable; in particular, the graceful numbering of Ks shown in Figure 3.14 does not have M = e, since e = 10 whereas M = 11. In fact, M = 11 is known to be the minimum value achievable for this graph. The connection with the variety value of a chord is now clear. A chord is equivalent to a graph with node numbers equal to note numbers and edge numbers equal to intervals (since intervals are differences between notes). Asking for an n-note chord with as great a v value as possible (i.e., as many different intervals as possible) is simply asking for a graceful numbering of the complete graph Kn. For small n, the largest v value possible is(~). for the case where all the intervals in the full interval set are distinct. However, in a scale of length L, the largest note number possible is L-1; this maximum v value will be possible only if the corresponding graceful graph numbering has M
3
4
M: 1
3
6 11 17 25 34
5
6
7
8
We see that M < 12 for n :s; 5; thus graceful chords (chords with all intervals in the full interval set distinct) are possible only for n :s; 5. For n > 5, the maximum v value is simply L-1 = 11. Table 3.16 shows the v values for all 351 chords. Graceful graph numbering is a notoriously difficult mathematical problem; there are no simple formulas known for calculating the numbers in Table 3.16; instead, a simple computer program was used. The asterisked numbers where v = (~) identify the graceful chords - those with all distinct intervals. In all, there are 6 + 18 + 33 + 4 = 61 graceful chords.
Counting Chords
v Variety 1
75
n = Number of notes
~
0 1 2 3
2
3
4
5
6
7
8
9
10
11
12
1
7 1 19 1 9 34 4* 42 26
59
1 42
19
6
1
1
1 6 1 19 2 9 34 8 20 43 54 154
80
66
43
19
6
1
1
351
6* 1 18*
5 6
1 2 7
33*
7
8 9 10 11
Total 1
6
19
43
Total
1 1
66
1
·. 7
Table 3.16. Values of V(12,n,v): number of n-note chords with v different intervals in the full interval set. Asterisks'indicate the graceful chords.
Perhaps the most interesting of these are the 4 large graceful chords One of these graceful chords is given by the notes (n
=5), corresponding to the 4 graceful numberings of K5.
··--·----·-·
which yields the interval set { 1,2,3,4,5,7 ,8,9, 10,11}. This chord corresponds to the numbering of K5 given in Figure 3.14. A second chord is given by reversing the underlined notes, which changes the 10 interval to a 6. The final two are obtained by simply taking the first two and reversing the entire string (which, of course, does not change the interval set). The four resulting chords are shown in Table 3.17. Why do these four graceful chords divide into two pairs, where the chords in one pair are simply the reversals of the chords in the other? Because a graceful graph numbering can be turned into another graceful numbering by simply replacing each node number k by M-k. But notice that there is an odd number of n =4 graceful chords; in other words, they Chord inC Partition 1 3 52 CC#EAB 3152 CEbEAB CDGBbB 2531 2513 CDGG#B Table 3.17. The 4 graceful chords containing 5 notes.
Chapter 3
76
cannot be divided into two isomorphic sets in this way. The reason for this is that some chords (such as • - - - - • • - - - - •) are palindromes (i.e., equal to their own reversal).
Harmonic Simplicity Even with all the chord properties at our disposal from the discussion so far in this chapter, there is still one annoying problem: none of the properties discussed so far explain why the major chord (4 3 5, or C E G in the key of C) is so ubiquitous in music. Why do entire genres of music (such as rock and pop) use this chord almost to the exclusion of all 350 others? One reason is the fact that the major chord is one of the 30 chords with no adjacencies, as shown in Table 3.4. But this does not explain why it is to be favored over the other chords with a =0.
Interval simplicity. The reason has to do with the physical meanings of the various intervals (1 through 11 semitones) that can appear in a chord. Recall, as discussed in Chapter 1, that in the twelve-tone equaltempered scale these intervals correspond to frequency ratios of the form
't2 ,.
1
1.0595 is where i is the number of semitones in the interval. Since a number just slight larger than 1, a single semitone interval produces an acoustical waveform consisting of the superimposition of two waveforms of slightly different frequencies. As we discussed in relation to Figure 3.2, this produces a complicated waveform containing a "beat frequency", which sounds somewhat harsh to the listener. As the interval is increased, one might expect the resulting waveform to become simpler and simpler, until the octave interval (12 semitones) is reached, at which point the two waveforms are in the very simple ratio of 2:1. However, a little thought shows that this is not the case. For example, a 7-semitone interval (which corresponds to a musical fifth) has frequency ratio equal to (1.0595)7 = 1.4983, which is very close to 1.5; thus, the 7-semitone interval also has (very nearly) a simple frequency ratio; namely, 3:2. In general, we can define the simplicity of an interval as the smallest integer d such that a fraction of the form n/d exists which is closer to the interval in question than any other interval. In other words, the value
Counting Chords
77
2i/12 can be approximated by a fraction with denominator d. This means
that two notes separated by that interval have a frequency ratio of approximately n!d to 1. Or, multiplying by d, this means they are in a ratio of n to d (which we write as n:d). The simplicity of an interval is crucial in determining the sound that occurs when two· notes with that interval are played together. This is illustrated in Figure 3.15, which shows the result of playing a musical fifth (a 7-semitone interval) which has frequency ratio 3:2. Because the denominator is so small (2), the period of the composite waveform is small and the waveform itself is simple in structure. Thus the resulting waveform is very simple and sounds pleasing to the ear. The smallest-denominator fractions for each interval (from 1 to 12; i.e., including the octave interval) are shown in Table 3.18. These numbers allow us to order the various intervals by their simplicity, as determined by the size of the denominator in each rational approximation to the interval. Where two intervals use the same denominator, ties are broken by using the error value shown in the last column, which is the difference between the rational approximation and the exact value. Table 3.19 lists the intervals in order of simplicity. The simplest interval is the octave (with ratio 2:1), followed by the musical fifth (with ratio 3:2), followed by the musical fourth (with ratio 4:3), and so on. Not surprisingly, the "worst" interval is the single semitone, which further justifies the relevance of the adjacency analysis presented earlier in this chapter.
rii~ ~(\
n ~n
Figure 3.15. Harmonic simplicity. The sum of two waveforms whose frequencies are in a simple ratio (such as 3:2) is a simple periodic waveform.
Chapter 3
78
Interval Exact Value 1 1.0595 2 1.1225 1.1892 3 4 1.2599 5 1.3348 6 1.4142 7 1.4983 8 1.5874 9 1.6818 10 1.7818 11 1.8877 12 2.0000
Fraction 12/11 8n 6/5 5/4 4/3 7/5 3/2 8/5 5/3 7/4 13n 2/1
E110r 0.0314 0.0204 0.0108 0.0099 0.0015 0.0142 0.0017 0.0126 0.0151 0.0318 0.0306 0.0000
Value 1.0909 1.1429 1.2000 1.2500 1.3333 1.4000 1.5000 1.6000 1.6667 1.7500 1.8571 2.0000
Table 3.18. Simplest-fraction approximations to the 12 different semitone intervals
Chord simplicity. We can now return to our original question: what is special about the major chord? To answer this, we can generalize the notion of interval simplicity to chord simplicity, which is defined as the smallest integer d such that the frequency ratio between the first note of the chord and each of the other notes can be expressed by a fraction with denominator d. As in the previous section, chord simplicity is invers~on dependent, so we define the simplicity of a chord as the simplicity of the simplest inversion. We now claim the following: The major chord is the harmonically simplest 3-note chord.
To see that this is true, note that the smallest denominator possible for a Interval 12 7
5 9 4 10 3 8 6 2 11 1
Denominator 1 2 3 3 4 4
5 5 5 7 7 11
Ratio 2:1 3:2 4:3 5:3 5:4 7:4 6:5 8:5 7:5 8:7 13:7 12:11
Name Octave 5th 4th 6th 3rd min? min3
dim5 aug5
200 Maj7 semitone
Table 3.19. The musical intervals in order of harmonic simplicity.
Counting Chords
79
n
d =
Denominator 1 2 1 2 3 4 5
3
4
=
5
Number of notes 6
7
8
9
10
11
12
Total
1
1 1 2 5 9 14 42 52 91 89 30 14 0 1
1
351
1 1 1
2 1
6
7 8 9 10
1
1 2 4 3 6 3
1 3 6
14 10 9
1 4 14 17 21 9
11
1 6
1
15 31 23 4
21 29 9
6
12 13 14 Total 1
1 8 20 11
3
6
19
43
66
so
66
43
1 7 5 6
19
1 1 4
6
1
1
Table 3.20. The harmonic simplicity of the 351 chords.
three-note chord is 3, since there are not two intervals in Table 3.19 with denominators less than 3. But there are two intervals with denominators equal to 3 -the 4th (ratio 4:3) and the 6th (ratio 5:3). Now, a note combined with a 4th and a 6th (5 and 9 semitones, respectively) is simply the chord {0 5 9) or G C E, which is an inversion of the major chord. The notes in this chord are in a ratio of 3:4:5. In fact, we see from Table 3.19 that all three inversion of the major chord have very simple ratios: (5,4,3) = {0 59} is 3:4:5 (4,3,5) = {0 4 7} is 4:5:6 (3,5,4) = {0 3 8} is 5:6:8 which further confmns the uniquely simple character of the major chord.
A full analysis. There are several subtleties involved in the computation of simplicity values (that is, the denominator values d). By definition, d is the smallest denominator which can represent the given interval or note, which means that no smaller d value will do. However, this does not imply that all larger d values will also work. As a simple example, the musical 5th, which can be represented with d = 2 by the fraction 3/2, cannot be represented with d = 3, since the two nearest fractions of the form n/3 are 4/3 and 5/3, which are both closer to other
Chapter3
80
intervals. Second, there is no simple relationship between the d value of a chord and the d values of the individual intervals. We can, however, use a simple computer program to calculate the d values for all 351 chords. The result is shown in Table 3.20. Of particular interest is the fact that there is a unique simplest chord for every value of n. Here are these uniquely simple chords for the first few values of n, notated beginning with the note C: Chord .. CG CFA CEGBb CEbF# AbBb
Name 5th interval (F) major chord (C) 7th chord (Ab) 9th chord
Ratios 2:3
3:4:5 4:5:6:7 5:6:7:8:9
Not surprisingly, 7th and 9th chords are also quite common in music.
Poly chords Finally, we consider another class of musical chords which forms the basis for the title problem of this book. A polychord is a separation of the notes of a chord into one or more (simpler) subchords; i.e., a subdivision of the set of notes in the chord into a collection of disjoint subsets. A classic example of a polychord, in the traditional musical sense, is the chord (221232), which is listed in Table 3.7as one of the 55 chords with one adjacency. A slight rotation of this chord produces (222123), which in the key of Cis the chord {C, D,E, F#, G, A}. This chord is the union of the two chords {C, E, G} and {D, F#, A}; i.e., two major chords (4 3 5) in two different keys (C and D). Conceptually, this complex 6-note chord is "really" just two ordinary major chords combined, although the auditory effect is far from simple, requiring a trained ear to detect the fact that two major chords are being played at once. More generally, a k-fold polychord is one in which an entire n-note chord is subdivided into k non-empty subchords, with 1 ~ k~ n. The case k = 2 is of particular musical interest, since it is by far the most commonly encountered case. (The musical effect is more confusing than interesting when k > 2.) The case k = 2 is also musically significant because it directly relates to keyboard (e.g., piano) playil}g. There are usually two hands involved in playing the piano, and 2-fold polychords c~m be created trivially by simply playing two (possibly different) chords (possibly in different keys), one with each hand.
Counting Chords
81
The mathematical problem which we ask is the following: In an nnote scale, how many distinct chords are required so than any of the Ch(n,*) chords can be formed by combining k chords from the given set into a k-fold polychord? When playing the k chords, each chord can be independently transposed. A minimal-sized set of such chords is called a set of elementary k-fold polychords, and we. denote the number of elementary k-fold polychords (in an n-note scale) by 7t(n,k).
A necklace formulation. Like most chord problems we have encountered, we can formulate this as a problem about necklaces. As usual, represent chords as 2-color necklaces where each black bead indicates a note included in the chord. Then, combining the separate chords in a polychord into the composite chord can be represented by laying the separate necklaces on top of each other, and creating a composite necklace which has a black bead in a given position if and only if at least one of the separate necklaces has a black bead in that position. Allowing the black beads to "overlap" in this way models the fact that the same note is allowed to appear in more than one of the separate chords constituting a polychord. Thus, for example, the following three necklaces combine into the composite necklace shown on the right.
This combining rule is also equivalent to representing each necklace as a binary string (with black bead = the digit 1, white bead= 0) and then performing a bitwise logical OR of the separate necklaces to produce the composite. The polychord problem can now be expressed as follows: Find a minimal set P of n-bead necklaces such that any of the N(n, *)-1 n-bead necklaces (not counting the one consisting of all white beads) can be produced by combining k (not necessarily different) necklaces from the set P. When combining, each subnecklace can be rotated to any position. Since we seek to "cover" all possible n-bead necklaces, we refer to this as the polynecklace covering problem. The number of elements in P is denoted 7t(n,k), with k ~ n. Consider a simple example: what are the values of 7t(5,k), for k = 1, ... , 5? Of course, 7t(5,1) is simply Ch(5,*) = N(5,*)-1 = 7; these seven 5-bead necklaces are shown on the left of Figure 3.16. Also,
Chapter 3
82
0000 00 0 0 0 000 1t(5,2) = 3
1t(5,1)= 7
1t(5,3) = 2 1t(5,4)= 2
Figure 3.16. Sets of elementary k-fold polychords for n=5 and k=l,2,3,4.
clearly, 1t(5,5) = 1, since every 5-bead necklace can be made by combining 5 copies of the unique one-bead necklace. So, the only interesting values are 1t(5,2), 1t(5,3), and 1t(5,4). Let's look at 1t(5,2) first, in which we seek to cover all 7 5-bead necklaces with pairs of necklaces from some set P, Clearly, the one-bead necklace must be in the set P, since otherwise there will be no way to make the one-bead necklace in the full set. With this single-bead necklace we can also make the necklaces (• • - - -) and (• - • - -) but no others. To make the necklace (• • • - -) we must add another necklace to the set P: either (• • - - -), (• - • - -) or (• • • - -). But whichever one we choose, we will still not be able to make the necklace (• • • • ·• ). This shows that a' least 3 necklaces are required; in fact, 3 are sufficient: the necklaces (• - - - -), (• • - - -), and (• • • - -) suffice, as shown in Figure 3.16. For 1t(5,3) and 1t(5,4), we.see that at least two necklaces are required, since (• - - - -) alone is not sufficient, and · in fact two suffice: (•--- -)and (• •-- -). In summary, we have shown that 1t(5,1) = 7 1t(5,2) = 3 1t(5,3) = 2 1t(5,4) = 2 1t(5,5) = 1.
The general case. The general polynecklace covering problem is to find the values of 1t(n,k) for all n and k. This seems to be a new and quite challenging combinatorial problem; in particular, we do not know of a general formula for 1t(n,k). However, it is possible to derive a simple upper bound for the value of 1t(n,k).
83
Counting Chords
Of course, 1t(n, 1) = C h(n, *). Now consider the case k > 2. As shown in Figure 3.17(a), a necklace of length n can be divided into k parts, each with length ~ rntkl, where rntkl denotes the smallest integer~ nlk. To cover all possible necklaces, we will construct the set P of all necklaces having the following properties: (a) At most the first rntkl beads are black beads (b) Every possible rn/kl-bead chain can be made to appear in the first rntkl beads of some necklace, by rotating the necklace appropriately. Every necklace can then be covered by considering each segment of the composite necklace as a chain, and choosing the correct necklace from P which can cover that chain. How many necklaces are in the set P? That is, how many distinct necklaces are required to cover all r-bead chains? The answer is 2'-1; simply take all necklaces starting with a black bead and with all 2r-1 possible combinations for the next r-1 beads. Since in our case r = rntkl, this means that: 1t(n,k) ~ 2rntk1-1.
Figure 3.17(b) shows that we can cover all necklaces in another way, by k parts
(a)
rnfklbeads kparts (b)
[n!k]beads
n (mod k) beads
Figure 3.17. Two ways of obtaining an upper bound for 1t(n,k).
Chapter 3
84
dividing into k parts of length [n/k] plus a leftover part of length n mod k. All necklaces with the leftover part containing all white beads can be covered by a set which covers alllength-[n/k] chains, which has 2[n/k1- 1 members, plus an explicit list of all the necklaces not yet covered (i.e., those which black beads in some of the last n mod k positions), which has 2n mod k - 1 members. Therefore, we also have 1t(n,k) ~ 2[n/k)-1 + 2n mod k _ 1
which means that 7t(n,k) ~ min(2rn/k1-1, 2[n/k)-1 + 2n mod k _ 1)
(k ~ 2).
If we apply this formula, for example, to the values of 7t(5,k) which we derived above by logical argument, we get
7t(5,2) 7t(5,3) 7t(5,4) 7t(5,5)
~ ~
22-1 + 2121-1 + 2221-1 + 21 21-1 + 20-
min(23-1, min(22-1, ~ min(22-1, ~ min(21-1,
1) = min(4,3) = 3 1) = min(2,4) = 2 1) =min(2,2) =2 1) = min(1,1) = 1
which agrees exactly with the values obtained earlier; in other words, the upper bounds are equal to the actual values. Indeed, this upper bound is relatively tight, and can be easily seen to be exact in certain special cases. For example, we already know that 1t(n,k) = 1 if an only if n =k. Now, suppose that k < n < 2k. Then the above formula says that 7t(n,k) ~ .min(2rn/k1-1, 2[n/k]-1 + 2n mod k _ 1).
If k < n < 2k, then 2rntk1-1 = 2[n/k1- 1 = 2, and so the first term is always minimal, which means 7t(n,k) ~ 2. However, 7t(n,k) = 1 is impossible since n -:# k, so 7t(n,k) must 2. This applies only to n strictly less than 2k; however, if n =2k, then the formula also yields 7t(n,k) ~ 2. Thus, we have shown that
=
7t(n,k)
=
1 for n =k 2 for k < n
~
2k.
The upper bound is not always exact, however. The smallest example for which the upper bound can be improved upon is n =8, k =2. The upper bound formula says 7t(8,2) ~ 8, whereas in fact 7t(8,2) =6. All 35 8-bead necklaces can be covered by pairs of necklaces from the following set: (•-------) (••------) (•-•-----) (•••-----) (••-•----) (••••----)
85
Counting Chords
k
1
2
3
4
5
6
7
8
9
10
11
12
1 2 2 2 2 3 3* 4 4
1 2 2 2 2 2 3 3*
1 2 2 2 2 2 2
1 2 2 2 2 2
1 2 2 2 2
1 2 2 2
1 2 2
1 2
1
n 1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 5 7 13 19 35 59 107 187 351
1 2 2 3 4 5 6* 9 13* 17 23*
1 2 2 2 3 3* 5 6* 8
Table 3.21. The best known values of 1t(n,k), the number of distinct k-fold polychords in an equal~tempered scale of n notes. An asterisk indicates a value which improves on the simple upper bound.
The Values of 7t(n,k). Table 3.21 shows the best values of 7t(n,k) that we have been able to construct~ for n,k ~ 12. The values marked with an asterisk are the only ones which are smaller than the simple upper bound. The most musically important value in this table is, of course, 7t(12,2) = 23. This says that only 23 chords are required to create any of the 351 L = 12 chords using 2-fold polychords. That is, only 23 different hand positions are required to play every possible chord on a piano keyboard. These 23 chords are listed in Table 3.22. This set is not unique; there are many other 23-chord sets (some of them containing more interesting chords than this set, whose main feature is that alLits chords have small span). It is not known for sure that 23 is the minimal number, nor whether other values in Table 3.21 can be reduced.
.·--·---------------- -- .. --- .. ------ .-.-------- - ... -·- ----.... ··-·-------- ·-----·----- ·-··-------····-------·---·------- ··--·------- ·-·-·------···-·------- ·--··------·---·•------ ··---·-----··-··------·-···------·····------····-·------ ···-··------ ······------ ··-·-·-----Table 3.22. A set of 23 (12,2)-polychords. Is this the smallest such set?
86
Chapter 3
Musical Applications The various methods of classifying chords described in this chapter have many applications in human or computer composition of music. Several experiments readily suggest themselves, such as: • Use the a value of chords to measure or control the degree of strangeness in the harmonic structure of a composition. • More generally, use the harmonic simplicity measure to describe or control the degree of harmonic strangeness. The d value can be varied over time using some function, d(t) to precisely control the strangeness (or, conversely, the simplicity) of a piece. One interesting scheme for doing this is to generalize the notion of musical forms such as the A B A sonata form. But instead of having discrete steps (such as 0 1 0, if we interpret A and B as 0 and 1), the d value can be made to vary continuously between some minimum value up to a maximum value and back again. Many other interesting functions suggest themselves, such as fractal functions containing self-similarity. • Compose pieces using only chords from certain interval sets shown in Table 3.15. For example, use only triatonic chords. Or use chords with certain maximum and minimum intervals. Or chords of a certain type with a certain value of n. • Vary the v value (Table 3.16) of chords used in a piece to control the number of intervals present at a given time. Compose pieces using just low-v chords or high-v chords (such as the graceful chords). • In any of the above schemes, use the various measures as analysis tools as well as synthesis tools. In other words, analyze existing compositions to determine their characteristics, and use the resulting data to produce compositions of similar character to the analyzed piece or pieces. As an illustration of the possibilities, the following page shows a simple piece composed using the 12 quadratonic chords formed from the interval set (4 3 2 2 1). That there are 12 such chords suggests a 12-bar jazz form, in which each of the 12 chord types is used exactly once (with different tonics) in the course of the piece. These 12 chords are balanced precariously between the ordinary and the strange; they contain several large intervals, but also contain one adjacency (i.e., a = 1). This means that the piece has a minimum value of a = 1 over time, which is quite high for a minimum value, but also has a maximum value of a= 1, which is only modestly high. Because of this, it has an intriguing quality: not too normal, but not too strange.
Counting Chords
87
Complete Chord List See Appendix A for a complete table of the 351 chords, with an explicit listing of the various properties discussed in this chapter: number of adjacencies, minimum interval, period, span, chromaticity type, variety, and simplicity.
Chapter 3
88
Quadratonic Quodlibet M. Keith
Am7+9 (21432)
~
.
{·
~tf>i.9fl~)
C7sus (41232)
~~ ~·
j,~ ~
.fl.
-
E7+b5 (42132)
l>t:
1....-l .... .Q.
:! :s: r~: -: t~·-:·M·l p
'-
--
~
Bmf>l.b9/F# (23124)
{!0
G7+aug5 (43122)
D7+6 (43212)
0
Fmaj7+9 (22341)
f
I
Dm4r.p-.(32241)
p
t
I: :
1
~: : i Ul I
~Ma
11
4 Scale-Counting Problems And, when alllwpe seem'd desp'rate, [he] wildly hurl'd himself into the scale, and sav'd a world. -Thomas Moore, The Veiled Prophet.
... Thus, unencumber'd, indefinite, Each his own melody hails... Ah, you might live to a grand age Ere you could play all the Scales. -"Theme With Variations" Punch, January 5, 1889.
In this chapter we consider various combinatorial questions involving musical scales, concentrating primarily on the 462 seven-note scales, which are of great importance in music. We consider analogous properties to those discussed in Chapter 3 for chords, including minimum and maximum interval, period, and number of adjacencies, as well as new properties specific to scales, such as idealness, chord containment, spelling, and so on.
Basic Results Fundamental properties. Recall that a scale is the same as a chord but without the property of rotational isomorphism. That is, a scale is simply a subset of the L notes in a musical scale of length L, or, equivalently, a subset of the set of integers {0, 1, ... , L-1 }. We also require that the element {0} always be present; the element {0} represents the tonic or key of the scale, which by definition must be the first note. The fundamental scale-counting problem is: how many n-note scales are there in an equal-tempered musical scale of L notes (where n::; L)? The answer to this question, unlike the corresponding question for chords, is completely trivial. The first note, {0} is fixed, and the other n-1 notes can be chosen arbitrarily from the remaining L-1 notes; thus: The number of distinct n-note scales is
(~~f).
The total number of scales in an equal-tempered musical scale of L notes is therefore
Chapter4
90
Number of scales Number of notes (n) 1 or 12 1 11 2 or 11 3 or 10 55 4 or9 165 5 or 8 330 6or7 462 .. Table 4.1. Values of Sc(12,n): the number of d•stmct n-note scales within the 12-tone equal-tempered scale.
Notice that this is exactly half of 2L, the number of binary strings of length L, which is as expected since the scales are just those strings that begin with a 1 because we require that the first note always be included. These comprise half of the available binary strings (the other half are those that begin with a "0''). 11
11
,
This formula tells us, for the case we are most interested in (L = 12), that: There are precisely 2048 essentially different scales.
We denote the number of scales by Sc(L,n); the values of Sc(12,n) are shown in Table 4.1. As with chords, these numbers are symmetrical - the number of scales with 4 notes, for example, is the same as the number with 9 notes. But note that the symmetry is different for scales than for chords (compare with Table 3.1)- with chords, Ch(L,n)
= Ch(L,L-n),
whereas Table 4.1 suggests that Sc(L,n)
= Sc(L,L+ 1-n).
Why is this? It is because the number of chords is based on the necklace formula which involves binomial coefficients of the form
(f). whereas
1 the number of scales is based on coefficients of the form ( L; ); in particular, Sc(L,L+1-n)
L-1 ) ( L-1 ) (L-1) = ( L+1-n-1 = L-1-(n-1) = n-.1 = Sc(L,n).
Scale-Counting Problems
91
Chords and scales. Is there a relation between the number of n-note chords and the number of n-note scales? For instance, why is the number of 6- and 7-note chords 80 and 66, respectively, and yet the number of 6and 7-note scales are both 462? The answer has to do with how the set of n-note chords (in which rotationally-isomorphic objects are not counted separately) generates the corresponding set of n-note scales (in which rotationally-isomorphic objects are counted separately). To generate the set of n-note scales, we can start with the set of n-note chords and rotate each chord n times to place a note at the beginning. However, some of these may produce duplicates - for instance, the chord (3,3,3,3) = (• --• --• --• --) only produces one distinct scale, not four, since all four rotations are the same. The key point is that we do not have to worry about a rotation of one chord producing the same scale as the rotation of a different chord, by the rotational-isomorphism property of chords. However, we do have to worry about two rotations of the same chord producing the same scale. We already have a means of doing this - the period property of chords. The period tells us the number of positions of rotation after which the chord repeats exactly, or after which no more distinct scales will be generated. Referring back to Figure 3.11, we see that such a chord has L/p pieces, which means that each piece has n/(L/p) black beads. Each of these beads can be placed at the beginning, to generate a different scale. Therefore, we arrive at the following formula which relates the number of n-note scales with period p, which we denote by Sp(L,n,p), and the number of n-note chords with period p, P(L,n,p):
Sp(L,n,p)
L
=
L?p · P(L,n,p)
(1)
lSpSL
If p = L, the first factor inside the summation reduces to simply n; that is, each n-note chord generates a full n different scales. Only if p < L do duplications occur, so that fewer than n scales are generated. As we know from Chapter 3, if n is relatively prime to L, only p =L is possible, which means that the above formula reduces to simply
=n · P(L,n,p)
Sp(L,n,p) or, since p takes on only one value,
Sc(L,n)
= n · Ch(L,n)
In other words, if n is relatively prime to L, the number of n-note scales is equal to the number of n-note chords times n. This special case is quite
Chapter4
92
n = Number of notes
p=
Period 1
2
3
4
5
6
7
8
9
10
11
1
1 1
2
1
1
3 4 6
1 1
5
18
40
66
75
66
40
18
Total 1 Chords
6
19
43
66
80
66
43
19
6
1
2
2
1
1
1
2 1
6
12
1
351
1
1 1
3
2
4
1
3 6
2
1
4
9
8
1 1 2
335
1
3
Total
3 9
1
1 3
1 5
12
12
27 2010
5
1
10
54 160 330 450 462 320 162
50
11
Total 1 Scales
11
55 165 330 462 462 330 165
55
11
1
2048
Table 4.2. Values of P{l2,n,p) and Sp{l2,n,p): number of n-note chords and scales with period p.
important, since, as we shall see shortly, the case n relatively prime to L = 12, is of particular musical interest.
= 7,
which is
To illustrate these relations between chords and scales, Table 4.2 shows all the values of both P(l2,n,p), which we computed earlier in Chapter 3, and Sp(l2,n,p), computed with formula (1). Note, for example, that 6 x 80 = 480, whereas there are actually only 462 6-note scales. This is because not all 80 6-note chords generate distinct scales, as predicted by the formula.
The 7-note Scales In Chapter 3, we considered various properties of n-note chords formed from a scale of L notes and presented specific results for the case L = 12, which corresponds to the commonly-used 12-note equal-tempered scale. In this chapter, we will further specialize in presenting numerical results and tables, in· that we will also concentrate on the case n = 7 as well as
Scale~Counting
93
Problems
L = 12. That is, we shall primarily study the properties of 7 ~note scales formed from the notes of the 12~note equal~tempered musical scale.
There are several reasons why the case n interest:
= 7 is of special musical
(1) A very high percentage of all music uses ?~note scales. In particular, the two scales that probably account for 90% of all (at least Western) music- the major and minor scale- are ?~note scales.
(2) The major scale is so ubiquitous that there are well-known syllables (used in singing) for the 7 notes of the scale: do, re,
mi, fa, sol, Ia, ti, do. (3) There are 7 white keys per octave on a piano, corresponding to the major scale in the key of C. (4) There are 7 letter names for the musical notes: A, B, C, D, E, F, G. This means that 7-note scales can be conveniently "spelled" using each letter once (possibly including accidentals, such as "C#"). The most prominent exception to the popularity of 7-note scales is the 5note scale {0,2,4,7,9} which corresponds to the black keys on a piano and is known as the pentatonic scale. This scale is used, for example, in the well-known theme from Dvorak's New World Symphony. It is also the basis of much Eastern music. In the remainder of this chapter, therefore, we shall concentrate on the case n = 7, with occasional references to other values of n (such as n = 5) or to the general case.
Fundamental properties. Since n = 7 is relatively prime to L
= 12, we know from the previous section that there are 462 7-note scales generated by the 7 rotations of the 66 chords. Since these 66 chords generate all 462 scales, we also refer to them as the 66 scale types. In addition, we know from Chapter 3 that there are precisely 7 interval sets for n = 7, which generate the 66 chords (or scale types); we also refer to these 7 interval sets as the 7 scale classes. In summary: Within the 7-note scales, there are: 7 scale classes 66 scale types 462 scales
94
Chapter4
The other case of interest is n = 5, which is also relatively prime to = 12; therefore, its structure is similarly simple. For n = 5, there are 13 scale classes, 66 scale types, and 330 scales.
L
Since the scale classes are isomorphic to the 7 chord interval sets, and the scale types are isomorphic to the 66 chords, we immediately know many properties of the 66 scale types from the results in Chapter 3. These properties are summarized in Table 4.3. k-atonic scales. The k-atonic scales - those that contain all intervals in the set {l, ... k}- are of particular importance in music. As noted above, the major and minor scales are among the most commonly used ones; more generally, the diatonic (of which the major scale is an example) and triatonic (of which the minor is an example) scales are the most common. From Table 4.3, we see that there are 3 x7 35 x 7
=21 diatonic and =245 triatonic
scales in all. Table 4.4 gives a complete list of the 21 diatonic scales, which are of great importance in music. The diatonic scales in Table 4.4 are grouped into the 3 scale types, labelled group A, B, and C. In the right column we show the traditional musical name for each scale, or, where there is none, a symbolic name which we will use to refer to these scales in future discussions. Group A is the most commonly used, and includes the major scale. The scales in group B are somewhat less frequently heard, and group C is the least common of all. Note that the number of scales having traditional musical names is the greatest for group A, smaller for group B, and none for group C. This is an indication of the fact that the groups are ordered correctly by "popularity". A mathematical explanation for this ordering
All have minimum interval = 1 All have period= 12 Number of scale classes (interval sets) = 7 a
Number with a adiacencies
m
2 3 2 3
3 20 3 35 3 35
4 30 4 21
5 12 5 6
6 1 6 1
Number with max interval = m k 2 Number that are k-atonic 3 Table 4.3. The basic properties of the 66 scale types.
Scale-Counting Problems
Intervals 2212221 2122212 1222122 2221221 2212212 2122122 1221222 1212222 2122221 1222212 2222121 2221212 2212122 2121222 1122222 1222221 2222211 2222112 2221122 2211222 2112222
95
Scale in tbe key of C Name Group A c D E F G A B Major c D Eb F G A Bb Dorian c Db Eb F G Ab Bb Phrygian c D E F# G A B Lydian c D E F G A Bb Mixolydian c D Eb F G Ab Bb Natural Minor c Db Eb F Gb Ab Bb Locrian GroupB c Db Eb Fb Gb Ab Bb Super-Locrian c D Eb F G A B Melodic Minor c Db Eb F G A Bb B3 c D E F# G# A B B4 c D E F# G A Bb B5 c D E F G Ab Bb B6 c D Eb F Gb Ab Bb B7 Groupe c Db Ebb Fb Gb Ab Bb C1 c Db Eb F G A B C2 c D E F# G# A# B C3 c D E F# G# A Bb C4 c D E F# G Ab Bb C5 c D E F Gb Ab Bb C6 c D Eb Fb Gb Ab Bb C7 Table 4.4. The 21 diatonic scales
will be given in the next section, using the concept of idea/ness of a scale. The next most common scales encountered in music are the triatonic scales. A few of the triatonic scales which are common enough to have names are shown in Table 4.5. The triatonic scales include the common minor scale (also known as the Harmonic Minor, to distinguish it from the other minor scales in Table 4.4 and 4.5).
Evenness of an interval set., Of the 7 interval sets for L Intervals Scale in the key of C 2122131 c D Eb F G Ab B 2131131 c D Eb F# G Ab B 1312131 c Db E F G Ab B 3112311 c D# E F G A# B 1322211 c Db E F# G# A# B Table 4.5. Some triatonic scales with their
= 12
Name Minor [Harmonic] Hungarian Minor Lydian Chromatic Phrygian Chromatic Enigmatic musical names.
and
96
Chapter4
n = 7, why is one set (the diatonic set, (2222211)) the overwhelming favorite in music, followed by the two triatonic sets ((3321111) and (3222111))? What property do these sets have that others, such as (5211111) do not?
One answer is to observe that the intervals in the diatonic set, (2222211), are the most nearly equal of the intervals in any of the interval sets. Apparently, the ear likes to hear scales that are as balanced, or even, as possible, in which the intervals between successive notes of the scale are as similar as possible. This is related to the use of scales in forming melodies, which often involve short sequences of consecutive notes of the scale (either ascending or descending). In such a series of consecutive notes, it is somewhat unsettling to have a series of similar, small, intervals, followed by a large jump due to the unevenness of a scale, followed by some more small intervals. Usually it is not possible to have a perfectly balanced scale, because n is not necessarily a divisor of L, whereas the intervals must by definition be integers. For L = 12 and n = 7, for example, the ideal scale would be one with all intervals equal to L/n =12n. But since 12n is not an integer, the diatonic interval set, (2222211), is as close as we can come to being perfectly balanced. In general, we can define the evenness of an interval set as the "distance" between it and the ideal interval set of (L/n, L/n, ... ,Lin). The measure of distance we will use is the mean-square error, defined as the average of the squares of the term-wise differences. Thus, the evenness of an interval set (/1, /2, .. ./0 ) is the value of '
where a smaller value indicates a more even scale, and a value of 0 indicates perfect evenness. For convenience, we make this value an integer by multiplying by n2; this yields the following definition for the evenness value, e:
From the second form of the formula, it is clear that the value of e does not depend on the order of the /i, which is as it should be, since interval sets are unordered.
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97
The e value of an interval set provides a function similar to that provided by the a value of a chord; namely: it provides an ordering of the interval sets by their degree of strangeness. The more uneven an interval set, the more musically strange are the scales which can be constructed from it. Table 4.6 shows the e values for the seven n =7, L = 12 interval sets, ordered. by their degree of strangeness. This order is identical to the frequency with which these intervals sets are used in music, with the diatonic set being first, followed by the two triatonic sets, followed by. the others. Note that the minor scale, which is the most commonly-used tria tonic scale, is, as expected, constructed from the more even of the two triatonic sets (3222111). The mean error per interval is the square root of the mean-square error. Since we multiplied by n2 in the definition of e, this means that the mean error is equal to ...Jetn2.
For the diatonic interval set, this equals ..J 10/49 = 0.45. In other words, each interval of a diatonic scale is, on average, slightly less than half a semitone away from the ideal interval of 12fl.
Idealness of a Scale The evenness measure, e, successfully quantifies the degree of strangeness of the 7 interval sets, or scale classes. However, the value of e is the same for all scales within a scale class, whereas we see from Table 4.4 that even within the diatonic scale class the three scale types (A, B, and C) are not equally popular. Even within one of the 66 scale types, some scales are more popular than others. For example, in the most Interval Set
e
2222211 3222111 332 1111 4221111 4 311111 52 11111 6 111111
10
24 38 52 66
94 150
Table 4.6. The seven n =7 interval sets, ordered by their degree of evenness.
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98
popular class, class A, the Dorian and Mixolydian scales are more frequently used than the Lydian and Locrian. So we ask: is there a stronger measure than the e value that can be used to order all 462 scales by their degree of strangeness? The answer is yes. The problem with the e value is that it measures the evenness of each interval, whereas a better measure of the smoothness of a scale is the distance between each note and the ideal. Each note in an ideal scale with all intervals equal to Lin is at an interval equal to a multiple of Lin from the first note of the scale, so let us define the cumulative mean-square error of a scale with intervals (/1, /2, ... , / 0 ) as
1L (
-
IJ· i·L) -
n
OSj
n
OSi
or, again multiplying by n2,
E
2 (n ~j - i·L ) n ~ OSJ
= !_ ~
OSi
We refer toE as a measure of the idea/ness of a scale. The geometric interpretation of E is illuminating. Consider a two. dimensional coordinate system where each unit on the x axis represents one note of the n-note scale, and each unit on the y axis represents one note of the L-tone equal-tempered scale. Then the ideal scale with all intervals = Lin is simply a straight line with slope Lin in this coordinate system, since then notes of the scale(= n units on the x axis) must span one octave(= L units on they axis). An actual scale with integer intervals thus corresponds to a discrete approximation to a line with slope Lin. The value of E is simply a measure of the error between the points of the integer-approximation line and the ideal line with slope Lin. It is simple to compute the value of E for a given scale, but the more interesting question is: for a given L and n, which scale or scales attain the smallest value of E; that is: which are the most ideal scales? The answer to this question is simple. To minimize the mean-square error, we can locally minimize the error of each term in the outer summation; i.e., the error between each note and the·line Lin. To do this we simply choose the closest integer-lattice point to each ·ideal note iLin (for 0 ~ i < n). These are just the values ofiLin rounded to the nearest
Scale-Counting Problems
99
integer. The most ideal line may not be unique, since one or more values of iL!n may have a fractional part exactly equal to 1/2, which may be either rounded up or down with the same resulting error. The only reason this procedure might not work is if two adjacent notes in the scale turned out to be the same, which is not allowed since all notes of the scale must by definition be distinct. However, two adjacent notes cannot turn out the same, because L ~ n, and hence the slope of the line is~ 1, which means that each pair of adjacent numbers differs by at least one. This algorithm is essentially the same as a well-known algorithm in computer graphics (Bresenham's line-drawing algorithm) which is used to draw lines on raster-scan displays (which are also required to have integer coordinates). It can also be viewed as somewhat of a dual to the classical statistics problem of least-squares linear regression (or line-fitting). Instead of picking the best straight line to approximate a set of points, we are picking the best set of points to approximate a straight line.
The Dorian scale is best. If we apply this procedure to our favorite case of L = 12 and n = 7, we find that the multiples of Lin= 1217 (the ideal note locations) are 5
3
1
6
4
2
{0, 1:7, 37, 57, ~· 87, 10¥} which, rounded to the nearest integers, are {0,2,3,5,7,9,10} = (2,1,2,2,2,1,2) or, the Dorian scale. The value of E is~0+22+32+12+12+32+22) = 2817 = 4. The mean error per note is ....J Efn2 = ;/ 4/49 = 217 "" .29. Thus, the smallest E value among the 462 scales is .29, as compared to the smallest achievable e value of .45. Now consider other values of n ~ 7. Since n = 1,2,3,4 and 6 are all divisors of L = 12, a perfect scale with E=O is possible in these cases, since Lin is an integer. For n =.5 we get {0, 2.4, 4.8, 7.2, 9.6} which rounds to {0,2,5,7,10} = (2,3,2,3,2). This 5-note scale, in the key of Ab, is {Ab,Bb,Db,Eb,Gb }, which can be played using the 5 black keys on a piano. Note that the Dorian scale is also monochromatic: it can be played using the 7 white keys, in the key of D: {D,E,F,G,A,B,C}! Thus, the unique ideal scales on 5 and 7 notes are playable using only black and white keys, respectively.
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100
Finally, we observe that theE value for a scale is unchanged if the order of the intervals is reversed. If the notes of a scale are Ni and the notes of the reversed scale are Mi, then Mi = L - Nn-i. which means that E(M)
=!1:(nMi- iL) 2 n i =! 1:(n(L-Nn-i) - iL) 2
n.I
= ! 1:[-(nNn-i- (n-i)L)P n . I
=E(N).
Thus, scales with a given value of E come in reversal pairs, except for those scales whose interval sets are palindromic. The two scales with minimum E for n =5 and n =7, derived above, are both palindromic, and therefore unique.
The "Top 10" scales. The Dorian scale is the most nearly perfect L
= 12, n = 7 scale.
What are the next few scales with sman·values of
E? The answer to this question is illustrated graphically in Figure 4.1, which shows the top 10 best scales as judged by their E values. Satisfyingly, both the major and minor scales, which comprise so much of Western music, are among the top 10 of the 462 scales. This figure illustrates the graphical interpretation of a scale as a discrete approximation to a line of slope Lin (in this case, 12n). The solid circles represent the best' approximation, corresponding to the Dorian scale. The open circles represent the next best points for several of the notes (namely, the 2nd, 3rd, 6th, and 7th). By choosing either one or two of these alternate notes, the 9 next best scales after the Dorian are obtained, with E values ranging from 5 to 8, as compared to E =4 for the Dorian scale. Notice the perfect symmetry of this diagram, which is due to the fact, demonstrated above, that the E value is unchanged under scale reversal. The Dorian scale and the scale B6 are both palindromic; the other 8 scales come in 4 mirror-reversal pairs. Given the 4 alternate notes, and the fact that we choose either 0, 1 or 2 of them to form other scales, we should expect
(ri) + (i) + (i) = 11
Scale-Counting Problems
101
scales in Figure 4.1, when in fact there are only 10. The missing scale is the one obtained by taking the alternate note in the 2nd and 7th position, which yields the scale (1222221), or C2. The reason this scale is missing is that it is not the 11th best! The increase in error from choosing both the 2nd and 7th alternate notes is too large; there are 4 other scales, obtained by choosing three alternate notes, which have a smaller E value than the scale C2, which is in 15th place among all scales. We see from Figure 4.1 that, whereas the Dorian is the most ideal diatonic scale, the most ideal triatonic scale is the ordinary minor scale (and, of course, its reversal). Also of interest, but not shown in Figure 4.1, are the worst diatonic scales: (2222211) and (1122222), which have E =35. Not surprisingly, these scales are, musically, pretty strange (even though they are diatonic).
Distance Measures The set of all scales, Sc(L,n), and the idealness measure E, can be viewed in a. geometrical fashion that leads to many useful musical applications. Since the first note of a scale is always 0, each scale is of the form {O,at,az, ... ,an-tl. which can be interpreted as a point in (n-1)dimensional space- namely, the point {at,az, ... ,an-tl· In fact, it is a
e
Dorian (4)
- -
r-
Major(8)
\
1,/
I
\.
(1312212) (8)
<
B3(7)
' ,.....\ ..t :-
.....
ll ..... 'rf
1.--
..,. ~
~
'"
~
-
I"'""
M elodic Minor (7)
/' "~ Minor(8) ~
I
Mixolydian (5)
_j_
:-
Natural Minor (5)
t:>
~
B6 (6)
Phrygian (8)
Figure 4.1. The 10 most ideal scales. The most nearly perfect scale is the Dorian scale, which is equivalent to the best discrete approximation to a line of slope 12n. The value of E is shown in parentheses after each scale name. The italicized scales differ from the Dorian by two notes, the others by just one.
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102
lattice point in (n-1)-dimensional space, since it has all integer coordinates. The set of all possible scales, Sc(L,n), consists of all sets of the form {O,at,a2, ... ,an-tl with 0 < a1 < a2 < ...
This is illustrated in Figure 4.2a for the case n =3, which corresponds to a 2-D picture. The triangular region corresponds to the intersection of the regions {0
Ideal scale 000 0
Best scale
t (a)
5 (b)
Figure 4.2. (a) The region of the 2-D plane corresponding to then= 3 scales, for arbitrary L. (b) The specific case n 3, L 5.
=
=
Scale-Counting Problems
103
The best scale is simply the nearest lattice point toP. (Actually, E is not quite the Euclidean distance, but rather the Euclidean distance squared, times a constant (lin). However, all the properties we are interested in are. unaffected by these distinctions.) The situation is illustrated, for the specific case L = 5 and n = 3, in Figure 4.2b. The point {2,3} is the nearest lattice point to the ideal point, and therefore {0,2,3} is the most ideal L =5, n =3 scale. The Taxicab Metric. When we compare actual scales (lattice points) to the ideal scale (in general, a non-lattice point), the Euclidean distance is an obvious choice. However, another musically interesting question is to ask for the distance between two (L,n) scales or to find all the scales "near" a certain scale (such as the major scale).
When comparing two scales (both of which are lattice points), a more musically useful metric is the so-called taxicab metric. Instead of the Euclidean metric,
the taxicab metric is
where the summation, in both cases, is over all coordinates of the two points X= {Xt.X2 •... .xnl andY= {yt.Y2·····Ynl being compared. The way the taxicab metric measures distance can be described as follows. Connect all lattice points with the edges of an n-dimensional rectangular lattice (i.e., two points are connected if and only if they differ by 1 in exactly one coordinate). Then the taxicab metric counts the minimum number of lattice edges that must be traversed to get from point X to point Y (by stepping along edges of the rectangular lattice). The metric gets its name from the case n = 2: it is the distance travelled by a taxicab in going from point X to point Y in a city with a rectangular grid of streets. Why is the taxicab metric musically important? Because it is a good measure of the perceptual "closeness" of musical scales, particularly for small distances. If a particular scale (say, a major scale) is played, followed by a scale that is nearby (as measured by the taxicab metric), it is quite easy to detect the different scale, and even possible to tell the difference. between a scale that is distance-1 away and a scale that is distance-2 away. As the taxicab distance increases, the second scale sounds less similar to the original, and, if the original was a common
Chapter4
104
T 0 1 2 3 4
Number Number T T 62 12 8 6 49 6 13 7 14 32 3 8 2 25 15 9 1 16 16 10 72 12 1 17 11 5 75 Table 4.7. The number of scales at taxicab distance Tfrom the major scale. Number 1 9 31 57
scale, more strange. This provides another useful measure of the strangeness of a scale: the taxicab distance of the scale from some common scale, such as the major scale. An obvious question arises: Given a particular (L,n) scale, how many scales are taxicab.;distance d away? In particular, what is the answer for L = 12, n = 7, and the major scale? Consider, for example, the case d represented as
= 1.
The major scale can be
01234567891011
* * ** * *
*
from which it is clear that there are exactly 9 scales distance-! away, since the note "2" can be moved by one position in 2 ways (to "1" or "3), "4" can be moved in 1 way, "5" in 1 way, "7" and "9" in 2 ways, and "11" in 1 way. Exactly 3 of these 9 nearby scales are also diatonic; the other6 are tria tonic. A complete table of values- showing how many scales are distance d away from the major scale, for all d - is given in Table 4. 7. It is not known if there is a simple formula for these yalues, or for the more general case of arbitrary L;n, and base scale. Of the 40 scales with taxicab distance 1 or 2, all but one are either diatonic or triatonic. The only exception is the distance-2 scale (2211411). Note that the ordinary minor scale, (2122131), is among those with T = 2. Also interesting is the farthest diatonic scale: (1122222), which is taxicab distance 7 away from the major scale. Figure 4.3 shows the portion of the lattice of scales nearest the major scale. All 9 scales with T = 1 are shown, as well as a few selected scales with T= 2. Figure 4.3 also answers an interesting musical question relating to the major and minor scales. It is well-known, almost to the point of being a musical cliche, that the major and minor scales have particular emotional
Scale-Counting Problems
105
qualities; in particular, the major scale sounds "happy" and the minor scale sounds "sad". Since the minor scale is distance-2 away from the major, we can use the taxicab metric to find all scales that are halfway between the major and minor, which we might call schizophrenic scales (since they are half happy and half sad). We see from Figure 4.3 that there are exactly two scales with "mixed emotions":
am
2122221 (the Melodic Minor) =CD Eb F GAB (in the key of C) 2212131 =C DE FG Ab B.
These two scales are, in one musical respect, duals of each other. In the first scale, the chord based on the letters C, E, G (which is known as the Tonic chord) is a minor chord (C Eb G) whereas the chord based on the letters F A C (which is known as the Subdominant chord) is a major chord. In the second scale, the Tonic chord is major (C E G) and the Subdominant chord is minor (F Ab C). Both of these scales can be used to create intriguing compositions.
The Oulipo +N algorithm. The ability to quantify the distance between scales permits us to apply a famous algorithm, originally invented for transforming text in interesting ways, to music. This algorithm, known as the +N algorithm, was invented in the 1960's by the French group Oulipo (from "Ouvroir de Litterature Potentielle"). Applied to text (in a given language), the algorithm is to take a passage of text and replace each word (of a certain type) with the (or a) word that is distanceN away from the original word (according to some metric). The classic +N algorithm is to use the Nth succeeding word in some given dictionary;
Figure 4.3. A portion of the lattice ofall462 scales, sho.wing the 9 scales one step away (rounded rectangles) and 6 of the 31 scaies 2 steps away (rectangles) from the major scale.
Chapter4
106
a more interesting version, however, is to use the Nth succeeding word in a thesaurus, so that words are replaced with words that are "nearby" in meaning, where N controls the degree of nearness. For example, here is a famous passage from literature with all words except prepositions replaced by the word +5 away in Roget's Thesaurus: To happen or not to happen- that is the point: Whether 'tis saintlier in the senses to meet with The bows and spears of raging doom Or to steal armor counter to waters of riots, And, by differing, expire them.
The obvious analogue of the +N algorithm in music would simply be to replace each note with the note N steps higher (or lower), which is not at all interesting since it merely transposes the piece into a different key. A much more intriguing algorithm is replace the scale used at a given point in a piece by a scale that is taxicab-distance N away from the original. This is a generalization of the musical cliche of taking a piece written using the major scale and replacing the major scale with a minor one, thus changing the mood of the piece from "happy" to "sad". There are 461 scales that can be used to replace the major scale, which opens a vast area for experimentation. Obvious candidates include all the scales a small distance away (such as those shown in Figure 4.3) especially the two schizophrenic scales - as well as other more exotic scales such as those shown in Table 4.5 (especially the Enigmatic, which almost always produces interesting results). See the Musical Applications section at the end of this chapter for other variations on this idea, and an example of a musical piece transformed using the +N algorithm.
Chord Containment In this section we consider a property relating scales to chords, which we call chord containment. The chord containment problem asks: for a given (L,n) scale, how many chords are contained within it; i.e., are playable using only notes contained in the scale itself. In other words, we ask for the number of chords that can be formed from a subset of the n notes of the scale, rather from a subset of the full L notes available. This number is somewhat analogous to the variety value of a chord, in that the variety of a scale is related to the number of different types of chords that are "compatible" with it (i.e., contained within the notes of the scale).
The subnecklace connection. For the two main cases we are interested in- (L,n)
= (12,7) or (12,5)- n is relatively prime to L, and
107
Scale-Counting Problems
so, as we have seen, the (L,n) scales are generated by then rotations of the Ch(L,n) chords. Now note that the number of chords contained in a scale is unchanged under rotation of the scale (since the chords contained within it are isomorphic under rotation). Therefore, if n is relatively prime to L, we are simply asking for the number of subnecklaces of a given necklace.
In this context, a subnecklace of a 2-color necklace is any necklace that results from changing some subset of the black beads into white beads. The original necklace corresponds to the n-note scale, and the subnecklace with r black beads (r:::;; n) corresponds to the r-note chord contained within it. The general subnecklace problems asks for the number of subnecklaces, containing r black beads of a given necklace of length n with k black beads. (We have changed terminology here, from chords having length L and n notes, to necklaces having length n and k black beads, which correspond to notes). For the same reason as in Chapter 3, we do not count the trivial necklace with all white beads, which is clearly a subnecklace of any necklace. The obvious question arises: for given values of n, k, which (n,k) necklace contains the maximum total number of subnecklaces (including all r 2: 1)? This is equivalent to asking: how interesting (from a chord containment standpoint) can a scale be? We denote this maximum value by M(n,k), and we also ask for the minimum value, which we denote by m(n,k).
The subnecklace problems seems rather difficult. In particular, we do not know a formula for either m(n,k) or M(n,k). However, it is easy to establish a simple upper bound, by noting that the number of subnecklaces with r black beads is limited by two things: the number of ways of choosing r objects from k (which corresponds to choosing which of the k black beads in the original necklace are to remain black, or equivalently, choosing which notes of the scale to use in the chord), and the number of (n,r) necklaces (since all the subnecklaces are (n,r) necklaces). An upper bound on the total number of subnecklaces is the sum of this bound over all r; that is: M(n,k):::;;
L
min [
(~). N(n,r)] = U(n,k).
r;;::l
For small n and k, this upper bound is fairly tight. For example, of the 78 cases with n,k:::;; 12, only 16 fail to achieve the upper bound.
108
Chapter4
(~)
r 1 2 3
7 21 35 35 21 7 1
4 5
6 7 Total
N(12,r) 1 6 19
min
1 6 19
43
35
66
21 7 1 90
80 66
Table 4.8. Calculation of U(12,7) = 90.
k
=
Number of black beads 1 2
1 2 3 4 5 6 7 8 9 10 11
12
1 1
1 1 2 2
n = number of beads 3
4
5
6
1 1 2 2 3 3
1 1 2 2 4 4 5 5
1 1 2 2 4 4 6 6 7 7
1 1 2 2 3 5 7 9 12 12 13 13
7
8
1 1 2 2 4 5 8 9 14 14/1 18 18 19 19
1 1 2 2 4
9
10
1 1 1 1 2 2 2 2 3 4 5 5 5 6 5 8 10 10 11 16 16 7 18 20/1 21/1 24 20 26 27/2 34/2 39/1 34 49 59 34 51 65/4 35 58 81 35 58 93/4 59 106 59 106 107 107
11 1 1 2 2 4 5 8 11 16. 22 32 43 62 75/5 112 123/7 168 168/7 186 186 187 187
12 1 1 2 2 3 5 5 12 12 23 13 4 6/2 59 86/4 64 153/9 145 240/21 288 327/8 350 350 351 351
Table 4.9. M(n.}c) and M(n.}c): Minimum and maximum number of necklaces contained in an (n,k) necklace. The symbol/m indicates that the maximum is m less than the simple upper bound.
Scale-Counting Problems
109
The computation of U(12,7) = 90 is illustrated in Table 4.8. In turns out that M(12,7) = 86, which illustrates the fact that the upper bound is not always achieved. Table 4.9 shows the values of M(n,k}, m(n,k}, and (indirectly) U(n,k) for all n,k ~ 12. Since M(n,k) = U(n,k) for most of these cases, instead of U(n,k) we show the value of U(n,k)-M(n,k) as a suffix on the value of M(n,k). For example, 86/4, the entry for (n,k) = (12,7), means that M(12,7) = 86, and it falls 4 short of the upper bound U(12,7) = 90. The minimum number of chords contained in a 7-note scale is m(12,7) =59. Unlike the (12,7) scales, some (12,5) scales do achieve the upper bound. There are in fact many 5-note scales that contain the maximum possible 23 chords; an example is (12125), or
··--··-·---Finally, Table 4.10 shows a complete breakdown of all 66 (12,7) necklaces (scale types) by the number of subnecklaces (chords) contained. From this table we see that there are exactly 4 scale types that achieve the maximum of 86 chords contained, and exactly 2 that have the minimum of 59. The four maximum-variety scale types are (4112121) and (2111313) plus their reversals. The minimum variety scale types are (3121212) and its reversal. Note that these scales come in reversal pairs, for the usual reason: the chord containment of a scale is invariant under reversal. The total number of scales is, of course, the number of scale types times 7. Therefore, there are 28 maximum-variety scales and 14 minimum-variety scales. Which of the entries in Table 4.10 correspond to the major and minor scales? It turns out that: Number of chords in major scale = 64 Number of chords in minor scale = 82 The major scale has the second smallest value possible - second only to the minimal-value scales with value 59! The minor scale, by contrast, has the third largest value possible of82. Thus, there is.one sense in which the minor scale is much more interesting than the major.
Spelling Analysis Another musical combinatorial problem arises from a connection between musical scales artd written musical notation. This problem, which we call
110
Chapter4
Number Number n Number n 10 2 81 2 77 22 82 64 5 2 78 85 5 73 79 2 3 4 86 75 2 80 7 Table 4.10. The number of (12,7) necklaces (scale types) containing n subnecklaces (chords). n
59
the spelling problem, asks for the degree of accidentals required to spell the scale using the 7 musical note names (A,B,C,D,E,F,G) once each. In order to understand this problem, recall the definition of accidentals from Chapter 1. Accidentals are the suffixes applied to note names to indicate a note that is ±n steps away from the note indicated by the letter · name. In standard music notation, there are exactly 4 possibilities for ±n:
+ 1, denoted by # (sharp) -1, denoted by b (flat) +2, denoted by x (double sharp, also sometimes##) ~2. denoted by bb (double flat) The seven letter names are distributed among the 12 notes as follows: C•D•EF•G•A•B which means that the twelve notes can be "spelled" in the following different ways using either just the note names or accidentals: C C# Dbb Db B# Bx
D
D#
E
Ebb
Eb Fbb
Fb
Cx
Dx
F F# Gbb Gb E# Ex
G Abb Fx
G# Ab
A Bbb Gx
A# Bb Cbb
B Cb Ax·
Recall that the scale is circular; this is why, for example,'C = B#. Also observe that the note G#/Ab is unique in that it can only be spelled in two ways; all the others can be spelled in three ways. Although musical notation only has symbols for accidentals of magnitude up to 2, we can consider spelling notes using even more accidentals. For example, E could also be written as Gbbb, which uses a "non-standard" accidental of magnitude 3 (the triple flat, bbb). Now that we know how to spell individual notes, we can define the
spelling of a scale as a representation of the notes of the scale in the above notation, with the spelling of the individual notes following two rules: (1) The 7 letter names are each used exactly once, and in cyclic order. The name for the first note must be the same as the key of the scale.
Scale-Counting Problems
111
(2) Each note is spelled using the smallest magnitude accidental possible. Denote by a the largest magnitude accidental used in spelling a scale. We are interested in classifying the 462 scales by their value of a; in particular, scales with a ~ 2 can be spelled using ordinary musical notation. The first thing to notice is that the statement of the problem is incomplete, since the value of a is dependent not just on which of the 462 scales is being spelled, but also on which of the 13 keys (see Chapter 1) it is being spelled in. For example, the major scale, (2212221), in the key of Cis spelled as CD E F GAB, which has a= 0 (no accidentals), whereas in the key of G it is G A B C D E F#, which has a = 1. In fact, the major scale has a = 1 in all12 non-C keys In other words:
There are 462 x 13 = 6006 notationally-distinct scales In general, let us define Ac(L,n,k,a) as the number of n-note scales in a musical scale with L notes that require accidentals of magnitude a to spell in key k. Each key k is defined by specifying a subset of the L notes that corresponds to then letter-named notes. Note that we require the number of letter names to be the same as the number of notes in the scale. We seek the values of Ac(L,n,k,a), especially for the easeL= 12, n = 7, a ~ 2, and the 13 musical keys.
A formula for Ac(L,n,k,a). The determination of Ac(L,n,k,a) is one of those interesting problems which can be easily solved by making it harder. Instead of asking for the number of n-note scales, let us ask for the number of m-note subscales of an n-note scale, where the first note of the subscale iss, with 0 ~ s ~ L. In addition, we change the condition "using accidentals with magnitude a" to "using accidentals with magnitude ~ a". This yields a new function, a(L,n,m,B,a,s): the number of m-note subscales starting on notes (0 ~ s < L), of ann-note scale in an L-note musical scale with n letter names, whose spelling requires accidentals of magnitude ~ a. The B term is a replacement for the key k, and denotes a set B = {bi} which specifies the note numbers corresponding to then letter names, which we call the base notes of the key. For example, a(12,7 ,7, {0,2,3,5,7 ,9,10} ,2,0)
Chapter4
112
denotes the number of scales that require accidentals of magnitude ~ 2 in the key of D, since the letter names starting with D are arranged thus: D•EF•G•A•BC• or, {0,2,3,5,7,9,10}. If we can solve this harder problem, then the answer to our original problem is given by Ac(L,n;B,a),= a(L,n,n,B,a,O)- a(L,n,n,B,a-1,0) a(L,n,n,B ,a,O)
(a> 0) (a 0)
=
(1)
The (a> 0) formula is derived from the obvious fact that. the number of scales requiring magnitude-a accidentals is equal to the number requiring ' magnitude~ a minus the number requiring magnitude~ a-1; the other formula is the degenerate case when a =0. Although no explicit formula for a(L,n,m,B,a,s) is known, we can easily derive a recurrence relation that allows us to compute values for various cases of interest. The recurrence relation is obtained by realizing that all possible m-note subscales starting on note s can be generated by taking all possible choices for the second note (not the frrst, since the frrst note is, by definition, s) and counting the number of (m-1)-note subscales · generated by each of these choices. What are the possible choices for the next note? Because of the requirement that the scales have accidentals with magnitude ~ a, the next note must be the next available base note ± a. The choice is also limited by two other things, however: (1) The next note must be> s, since by definition the subscale starts on notes.
(2) The, next note must be <: L, the total length of the equal-tempered scale from which the scales are formed.
ILthe base notes are numbered bo, ... ,bn-I; the index of the next available base note is 1 (the seconp one) plus n-m (the number of notes of the scale chosen so far). Therefore, the above conditions can be written as:
bn-m+ I-a ~ t ~ bn-m+ 1+a . s < t
11;3
Scale-Counting Problems
a(L,n,1)J,a,s) is simply equal to 1, since the ftrst note of the (1-note) subscale is required to be s. Finally, note that if a= 0, the value of a will simply be either 1 or 0, depending on whether or not s is a base note (because if it is not, then no a =0 scales are possible, and if it is, exactly 1 is possible: the scale with all notes equal to base notes). In summary, we have the following formula for a(L,n,m,B,a,s):
a (L ,n,m,B ,0 ,s)
B ={01 otifhse . erwtse
a(L,n,m,B,a,s)
=
L a(L,n,m-1,B,a,t) }
bn-m+ }-a ~ t ~ bn-m+ 1+a s< t
a(L,n,1,B ,a,s)
a>O
=1
Table 4.11 shows the values of Ac(12,7,B,a) for all 13 keys and all possible a, computed using this formula and equation (1) above. As expected, there are exactly 7 scales with a= 0, corresponding to the 7 cyclic permutations of the 7 base notes. The values of a(l) and a(2) - the number of scales that can be spelled using no more than. single or double accidentals - are also shown. These· numbers vary quite substantially between the different keys. The best keys, for spelling, are A and G, for which 396 out of 462 (86%) scales can be spelled using standard musical notation; the worst key is Gb, of which only 277 (60%) can be spelled in this way. Also, observe that there are four pairs of identical columns in Table 4.11. The pairs of keys (C,E), (F,B), (Bb,F#), and (G,A) have exactly the same statistics. Even more surprising is that two of these pairs (F,B) and (Bb,F#)- are almost identical to each other, differing only in the a =0 and a = 1 positions. What is the reason for these phenomena? The explanation has to do with the positions of the base notes in these pairs of keys, which are arranged as follows: C
-. 1-•-••-•-•-•
F
-. 1-•-•-••-•-•
E
-.1•-•-•-••-•-
B
-.1•-•-••-•-•-
Bb F#
--1.•-•-••-•-• •- I•- •- • • - • -• -
G -.1-•-••-•-••A - .. 1-••-•-••-•-
In these diagrams,"." represents the location of the first base note, and • represents the remaining base notes. The vertical bar is positioned after the first note of the scale (which may or may not coincide with the ftrst base note).
114
Chapter4
(a) Number of accident als req'd 0 l f ls c F G 0 1 2 3 5
l<ey 2f Bb
2s D
3f Eb
3s A
4f Ab
4s E
Sf Db
Ss B
Sf Gb
Ss Fit
1 1 0 1 0 0 0 1 1 0 1 0 1 154 152 176 153 188 149 176 105 154 87 152 81 153 227 215 219 215 191 192 219 241 227 232 215 196 215 68 82 59 82 70 95 59 91 68 102 82 144 82 11 11 7 11 12 25 I 24 11 35 11 35 11 6 1 6 1 0 1 0 1 0 1 1 1 1
a(lJ 155 153 177 153 189 149 177 105 155
87 153
81 153
a(2J 382 368 396 368 380 341 396 346 382 319 368 277 368
Table 4.11. The number of scales requiring accidentals of magnitude a, in each of the 13 keys, plus a.(1) and a.{2)- the number requiring magnitude $1 or $2.
From the formula, we see that for a > 0, the value of a. is dependent only on the • base notes. From the above diagram, we see that for these pairs of keys, the sequence of • base notes are reversals of one another. By symmetry, it is clear that the subscales with a accidentals on a given set of base notes B can be mapped one-to-one to the subscales with a accidentals on the reversed set B'. This explains the isomorphism between these pairs of keys. In addition, we see that the sequence of • base notes is identical for the (F,B) and (Bb,F#) pairs. This explains why their statistics are almost identical - only the case a =0 is different, since the first note of the scale coincides with the first base note for the keys (F,B), but does not for the keys (Bb,F#).
Special cases. We can specialize the formula for a.(L,n,m,B,a,s) to count only the most commonly used scales, the diatonic ones, by requiring the next note chosen, t, to be within two steps of the first note of the subscale, s; that is, by imposing the additional condition t~s+2
in the summation. In addition, note that a.(L,n, 1,B ,a,s) is not always equal to 1, but is equal to
a.(L,n, 1,B,a,s)
={
1 if s ~ L-2 0 otherwise
Scale-Counting Problems
(a) Number of accidentals req'd 0 l f ls c F G 0 1 2 3
1 19 1 0
1 17
3 0
1 19 1 0
115
Key 2f Bb
2s D
3f Eb
3s A
Ab
4s E
Sf Db
Ss B
Sf Gb
F#
0 18 3 0
1 20 0 0
0 1S
1 19 1 0
0 12 9 0
1 19 1 0
0
1 17 3 0
0 7 13 1
0 18 3 0
6
0
4f
9 12 0
Ss
Table 4.12. The number of diatonic scales requiring magnitude-a accidentals.
where the ftrst clause ("1 if s ~ L-2") is required to ensure that the last interval in the scale (from the last note circularly back to the ftrst) is =:;; 2, as required by the diatonic condition. Table 4.12 shows the values of this revised formula for. all 273 (= 21 x 13) notationally-distinct diatonic scales. Of particular interest are
the numbers for a =2 and a= 3. For a =2, we observe that there is only one scale with no diatonic scales requiring a= 2 (or a= 3): the key of D. All 21 diatonic scales in the key of D require only single accidentals. The next best keys are the keys of C, G, A, and E, which only have one a = 2 scale. These four unique diatonic scales with a =2 are shown below. C G A E
Db Ebb Fb Gb Ab Ab Bbb Cb Db Eb
Bb
F B C# D# E# F## G# F# G# A# B# C## D#
The a = 3 row in Table 4.12 provides quite a shock: exactly one of the 273 diatonic scales cannot be spelled using standard musical notation! This unique scale is the scale (1122222) in the key ofGb, or: Gb Abb Bbbb Cbb Dbb Ebb Fb
which requires a triple accidental. Recall that the scale (1122222) is one of the two "strangest" diatonic scales, having E = 35, the maximum value possible for a diatonic scale. Another interesting special case is the 7 scales formed from the strangest interval set: (6111111 ). Table 4.13 shows the a distribution for these scales. The most amazing feature of this table is that there is exactly one (6111111) scale in each key that can be spelled using standard notation, since it requires only double accidentals. For example, in the key of C the scale is (1116111), or
116
Chapter4
C Db Ebb Fbb Gx A# B This phenomenon is quite remarkable, because such a scale has a gap of 6 between two of its notes, and yet all of its notes are within 2 notes of the 7 base notes (which have a gap of at most 2 between them). Finally, we can ask about the number of accidentals required to spell a scale in any key. For a given one of the 462 scales, define A as the maximum value of a over all keys. The number of scales with each possible value of Ais shown below.
INurn~ I ~ 119~ Iw! I4: I ~ I::~ I
Of course, there are no scales with A = 0. There are only four scales (all diatonic) with A = 1: the Major, Lydian, Mixolydian, and BS (or (2221212)). The scale B5 is remarkable in being the only one of the 462 scales that has the same a value (namely, a = 1) in all13 keys.
Musical Applications Musical applications of the analyses presented in this.chapter include: • Composing pieces based on a certain one of the 462 scales. Only a very few of these scales are commonly used in music, so there is a huge area of exploration here. • Using various measures, such as idealness, to select scales for use in composition. Different scales could be used within the same piece, with E values chosen to select either relatively ideal or relatively strange scales. • Exploring variations on the Oulipo +N algorithm. The simplest scheme is to take a piece using only one kind of scale (such as major) and replace (a)
Number of. accidentals req'd 0 1f 1s c F G 2 3 4
s
1 3 2 1
1 3 2 1
1 3 3 0
Key 2f Bb
2s D
3f Eb
3s
4f
A
Ab
1 3 2 1
1 2 4
1 2
1 3 3
0
3 1
0
1 3 2 1
4s E
Sf Db
Ss B
Sf Gb
Ss F#
1 3 2 1
1 3 1 2
1 3 2 1
1 3 1 2
1 3 2 1
Table 4.13. The number of (6111111) scales requiring a accidentals.
Scale-Counting Problems
117
the major scale throughout with another scale at taxicab distance N from the original. Notice that there is some ambiguity here since the scale is only 7 notes whereas all 12 different chromatic tones might be used in the course of the piece. One simple method is to leave all notes that are not part of the 7-note scale unchanged, but other algorithms are possible. The next step would be, for example, to replace with a different scale in each measure of the piece, but with all scales still being distance N from the original. Scales could be chosen randomly or with some more sophisticated algorithm. An even further extension is to vary the value of N over the course of the piece - for instance, start with N = 0, build to some maximum value, and then decrease back to zero. This is another analogue of the ABA sonata form. Notice that if the +N algorithm is carried to an extreme, all that is left is the rhythm of a piece and a somewhat vague indication of the original melody (for instance, ordering is preserved: if note n is below note m before the transformation, it will still be below it after the +N algorithm is applied). It is interesting to see how far a piece can be altered and still be recognizable. • Using the connection between scales and chords provided by the chord containment measure. For example, compose a piece using a certain scale and using all the chords contained in that scale (or all chords with certain properties). Or compose a piece using one of the 4 maximal-containment scales. Or one of the 2 minimal ones. Two illustrations of these ideas follow. The first piece, Shifted Sonata, is the familiar beginning of Mozart's Sonata in C Major for Piano, subjected to the Oulipo +N algorithm. The major scale in measure number n of the piece (0 ~ n < 12) is replaced by a randomly-chosen scale at taxicab distance T
= 6 sin (mt/11)
from the major scale. This function is zero for the first and last measure, and attains a maximum ofT= 5 in the middle of the piece. The scale used in each measure and its taxicab distance are shown in the score. The second piece, Schizophrenic Scherzo, is based on the scale (2212131) which is halfway (in taxicab distance) between the major and minor scales. Being situated midway between the merry major and the mournful minor, this scale lends itself to ambiguous, emotionallyconfused compositions, such as this one.
118
Chapter4
Complete Scales List See Appendix B for a complete table of the 462 scales with L = 12, n = 7, including a list of the various properties discussed in this chapter such as Euclidean distance from the ideal, taxicab distance from the major scale, chord containment, and spelling characteristics.
Scale-Counting Problems
119
Schizophrenic Scherzo M. Keith
Chapter4
120
Shifted Sonata W. A. Mozart (sort of)
Major(f=ll)
2121132 (f=S)
1312221 (f=1)
2213121 (f=1)
2311311 (f=3)
1221231 (f=4)
3221121 (f=S)
Major(f=ll)
5
Combinatorics of Rhythms Keeping time, time, time, In a sort of Runic rhyme. - Edgar Allen Poe, The Bells
Rhythm is nothing more than melody deprived of its pitch. - Schopenhauer
In this chapter we .consider some combinatorial problems involving musical rhythms. We first discover that there is an intimate connection between the number of essentially distinct metrical patterns and the "delayed Fibonacci sequence". The second half of the chapter considers the combinatorics of rhythmic patterns, and in particular the property of syncopation, which provides a measure of the complexity of a rhythm.
Metrical Patterns Introduction. Most musical compositions are characterized by a simple rhythmic pattern, consisting of some number of "beats", which repeats over and over. In music, the basic pattern of n beats is referred to as a measure, and there are said to ben beats per measure (or per bar, since the end of each measure is denoted in the musical score by a vertical bar). Common values of n are n = 4 in jazz and rock music and n = 3 in certain types of music such as the waltz.
The basicrhythmic !;tructure of a composition, referred to as its meter, is described in musical notation by a time signature, as shown in Figure 5.1. The time signature is a fraction, nld, in which the numerator n denotes the number of beats per measure and the denominator d specifies which type of musical note (1 =whole note, 2 =half note, 4 =quarter note, 8 =eighth note, etc.) corresponds to one beat. Observe that musical note types all have durations of the form 112n, and so the denominator d is
Jl y Figure 5.1. Some musical time signatures. The numerator denotes the number of beats per measure; the denominator specifies the basic unit of time.
122
Chapter 5
always a power of 2. The first thing to note - a fact not realized by a surprising number of musicians, even - is that the denominator of the time signature is, mathematically, not meaningful. The reason is that the duration of musical notes (whole, half, quarter, etc.) is not in absolute time units but rather in relative units. So, for example, a piece that is in 3/4 time can equivalently written in 3/8 time by halving all the durations of the written notes (half notes become quarter notes, etc.) and then doubling the absolute measure of time. Thus, we concentrate solely on n, the numerator of the time signature (=the number of beats per measure). We ask: how many essentially different metrical patterns are there, for a given value of n? At first glance, the answer may seem trivial - there is exactly one metrical pattern, consisting of n beats in a row. However, this answer misses a psychoacoustically important fact: as the value of n becomes larger, the ear tends to subdivide the meter into smaller units. For some reason, the primitive perceptual units of rhythm seem to be "2" and "3". Any meter with n > 3 tends to get interpreted by musicians and listeners as a combination of 2's and 3's.l For example, a musical measure with 6 beats (n = 6) is usually "heard" as 3+3 or as 2+2+2, depending on the way the musical phrase is written, orchestrated, or played. This can also be viewed as specifying where the accents fall within each measure. The first beat within each group is usually accented in some way in order to make the subdivision clear. The example of n = 6 as either 3+3 or 2+2+2 is nicely illustrated by the song "America" from West Side Story by Leonard Bernstein. In this song, written in 6/8 time, the meter alternates between these two possibilities: even measures are 3+3 and odd measures are 2+2+2. This produces a quite intriguing musical effect. We define r(n) as the number of metrical patterns on n beats, which, by the above discussion, is just the number of partitions of n into 2's and 3's. What are the values of r(n), and is there a simple formula?
A formula for r(n). A formula for r(n) is easily derived by asking: how many 3's can a partition ofn into 2's and 3's have? Clearly, the minimum number of 3's is zero if n is even (corresponding to the 1There is an analogy here to the metrical "feet" which form the basic units of rhythm in poetry. The most common feet are those with 2 or 3 syllables (= "beats").
Combinatorics of Rhythms
123
partition (2 2 ... 2)) or 1 if n is odd (corresponding to the partition (3 2 2 ... 2)). We can successively increase the number of 3's (and decrease the number of 2's) by replacing three 2's by two 3's. Thus the minimum number of 3's is n mod 2, and increases by steps of 2 up to a maximum of [n/3]. For a given number of 3's, the number of metrical patterns is simply the number of strings of 3's and 2's with a given number of 3's and 2's. If we denote by k the number of 3's, then there are n- 3k.
2 2's, which means the total number of digits in the string is n - 3k n- k 2 +k =-2-.
How many strings of (n-k)/2 2's and 3's with exactly k 3's are there? It is simply the number of ways of choosing k objects from a set of (n-k)/2 objects, which corresponds to choosing the positions of the k 3's 12
in the string. This number is, of course, (
=
L ((nt)/2) k = n mod2+2i i ~ 0
Notice the similarity between this formula and the formula for A(£,*,1) from Chapter 3, which was:
We saw that these values A(£,*,1) are sums of d~agonals of Pascal's triangle, which we showed were also equal to the values of the Fibonacci sequence defined by F(n)
=F(n-1) + F(n:.2).
The values of r(n ), instead of sums of diagonals (lines with slope = 1) of Pascal's triangle, are sums of lines with slope = 1/2 in Pascal's triangle. Figure 5.2 shows that these values satisfy the equation r(n)
=r(n-2) + r(n-3).
Chapter 5
124
This is the same equation as the Fibonacci sequence except that the addition of the two previous terms is delayed by one step- we take n-2 and n-3 instead of n-1 and n-2. Thus, we also refer to this sequence as the delayed Fibonacci sequence Ft(n), defined as follows: (n > 2)
Ft(n) =Ft(n-2) +Ft(n-3) Ft(O) = 1 Ft(l) = 1 Ft(2) = 1.
(The "1" subscript in Ft(n) indicates that the sum is delayed by one step. More generally, Fs(n) is the s-step delayed Fibonacci sequence, in which the sum is delayed by s steps, as follows: Fs(n) = Fs(n-s-1) + Fs(n-s-2) Fs(n) = 1
(n > s+ 1) (n ~ s+1).
Exercise: Show that the values of Fs(n) are equal to the sums of lines with slope l!(s+ 1) in Pascal's triangle.) Since the sequence r(n) has initial values r(2) = r(3) = r(4) = 1, we see that r(n) = Ft(n-2).
-k-
0
n
0
0
0
0
0
0
0
0
0
0
0
<~~~)
0
~
0
ei/)
G)
0 0 "-... Pascal's Triangle 0 r(n-1) r(n)
0
0 0
0 0
\ 0
Figure 5.2. Sums of lines with slope 1{2 in Pascal's triangle satisfy the equation r(n) = r(n-2) + r(n-3).
Combinatorics of Rhythms
125
That is, the number of n-beat metrical patterns is equal to the (n-2)th delayed Fibonacci number. The simple formula for F1(n) allows us to easily compute the values of r(n), shown in Table 5.1. N:ote that n = 5, the first really unusual musical meter, is also the first one with r(n) > 1. Table 5.1 also contains the remarkable pair of values r(24) = 351
r(25) =465 in which the first is exactly equal to the number of distinct chords from Chapter 3 (351), and the second is near\y equal to the number of distinct scales from Chapter 4 (462)! These coincidences suggest several curious possibilities such as a composition in a. time signature of 24/8 in which the metrical pattern in each measure is determined by the chord type used in that measure. A shnilar (but not quite one-to-one) mapping could be made between scales and metrical patterns in 25/8 time.
Some examples.. The .musical implications of the values of r(n) are similar to the implications of our earlier analyses of chords and scales; namely, that when we begin to systematically list all the rhythmic possibilities in this way, we realize that the vast majority of all (at least Western) music only uses a very few of the available rhythms. In particular, almost all music uses either n =2, 3, or 4! Note that these are the only ones with r(n) = 1. In Table 5.2 we explicitly list all the possibilities for 2 ~ n ~ 11, with examples of musical pieces that use some of the more unusual rhythms. Naturally, as n becomes larger it becomes harder to find examples, although jazz trumpeter Don Ellis (who is also represented in Table 5.2) is n
r(n)
n
r(n)
n
r(n)
n
r(n)
2 1 18 616 10 7 65 26 3 1 11 9 19 816 86 27 4 1 12 12 114 20 1081 28 5 2 16 21 151 29 1432 13 6 14 21 2 22 200 1897 30 7 3 15 28 265 31 2513 23 4 8 16 37 24 351 32 3329 9 17 49 25 5 465 4410 33 Table 5.1. Value of r(n), the number of metrical patterns of n beats, or the number of partitions of n into 2's and 3's, or values of the delayed Fibonacci sequence.
Chapter 5
126
responsible for very listenable compositions in such exotic meters as:
19 = 3 3 2 2 3 2 2 2 33=222232222232322 Metrical complexity. We have referred several times to "unusual" or n 2 3 4
5
6 7
8
9
10
11
Partition 2 3 22 32 23 33 222 223 322 232 332 323 233 2222 333 3222 2223 2322 2232 3322 2233 3232 3223 2332 2323 22222 3332 2333 3323 3233 22223 32222 23222 22322 22232
Example common common common
Take Five, Dave Brubeck Mars, from The Planets, Gustav Holst Symphony No. 6 Tchaikovsky (2nd Movement) common common Money, Pink Floyd
The Remembering, Yes Traditional bluegrass banjo rhythm common common
Strawberry Soup, Don Ellis Blue Rondo a Ia Turk, Dave Brubeck Indian Lady, Don Ellis
Outside Now, Frank Zappa
Pictures at an Exhibition, Moussorgsky
Table 5.2. The metrical patterns up to n=ll, with some example musical pieces.
Combinatorics of Rhythms
127
"exotic" meters without d~fining this term precisely. Clearly, it is not just a matter of small n's being more common than large n's, since n = 8 is much more common than n = 7, even though it is larger. Even within a given value of n, certain meters (i.e., partitions of n into 2's and 3's) are more common than others. Why is this? One clue is provided by noting that the most commonly-used meters in Table 5.2 are those of the form (3+3+ ... +3)+(2+2+ ... +2) or (2+2+ ...+2)+(3+3+ ... +3); that is, a string of 3's followed by a string of 2's, or vice versa. The reason for this seems to be that the listener can temporarily be fooled into thinking of the meter as a simple "3", and then, temporarily, as a simple "2". In other words, the brain has to shift gears between "3 mode" and "2 mode" as little as possible- namely, at most once. The simplest meters of all are those consisting of just 2's or 3's. This suggests that the most complex meters are those that switch between "2" and "3" as often as possible; i.e.: 2+3+2+3+ ... +3. However, when we do this an interesting thing happens: the ear now begins to hear, primarily, the repeated 2+3 units! That is, such a meter is perceived as being in "5 mode". A similar thing happens with even larger groups; for example, 14 = 3+2+2+3+2+2 is perceived as "7's": (3+2+2)+(3+2+2). This motivates a definition of the complexity of a meter based on its "sub-meters" (subdivision of its partition into groups, as in the examples above), where sub-meters can occur at all possible grouping resolutions. Specificaliy, letS be a string of 2's and 3's. PartitionS into a disjoint set of subunits, where each subunit is a substring of length~ 1, and the union of all the subunits is the stringS. Then define a unit as one or more identical contiguous subunits. Denote the complexity value of of a string S by c(S). The complexity value of a unit G consisting of a number of identical subunits H is defined as c(G) =max( number of subunits, c(H)) and the complexity value of S for a given division into units (denoted by Su). is defined as c(Su)
L c(G).
= G
E
s
Finally, the complexity value of Sis defined as the minimum complexity value for any division of S into units: c(S)
=min(Su).
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128
The key part of this definition is the definition for the complexity value of a unit: c(G) = max(number of subunits, c(H)). This definition encapsulates the fact noted earlier that the complexity of a repeated pattern (unit) is roughly equal to the complexity of one of the repeated patterns (subunits). However, it also recognizes the fact that as the numberofrepetitions becomes larger, the repetitions themselves begin to impose an additional rhythmic structure. So, for example, a unit consisting of 5 repetitions of the subunit '2' has a value of max(5,2) = 5, since the 5 repetitions becomes a stronger rhythmic factor that the basic rhythmic subunit of '2'. · Since the computation of c(S) depends on the absolute positions of the events within a musical measure, there is an implicit assumption that the beginning of each measure is well-defined. Ifwe are looking at written music, this is a good assumption since measure boundaries are indicated in the musical notation. It might seem that there is considerable ambiguity, as far as measure boundaries is concerned, if we are trying to analyze a piece of music by simply listening to it. However, in practice this is usually not a problem, since the beginnings of measures are usually apparent even when listening. They can be indicated in several ways: by an accent (slightly louder sound) on the first beat of the measure, by the way the melody of the musical piece is phrased, and so on. There are several other subtleties in the definition of complexity, c(S), and no doubt other definitions are possible. It does, however, correspond reasonably well with our expectations about which meters are simple and which are complex. For example, any meter consisting of 2k 2's yields a value of c =2, since it can be divided into a single unit with two subunits containing 2k-l 2's, which have value 2, and so max(2,2) = 2. This is the smallest value of c possible, and so agrees with our intuition that these are the simplest possible meters. As a more complicated example, the 5 meters with n = 9 break down as follows: Meter
333 3222 2223 2322 2232
c 3 6 6 7 7
129
Combinatorics of Rhythms
which corresponds exactly with our expectations, since the meter (3 3 3) is by far the most common, and (3 2 2 2), (2 2 2 3) are more common than (2 3 2 2), (2 2 3 2), as illustrated in Table 5.2.
Numerical Results. Using the above definition, we can calculate (somewhat laboriously) the c value for a given meter. Define 1e(n,c) as the number of n-beat meters with complexity value c. These numbers provide a rough classification of all meters by their complexity. The values of 1e(n,c) for n s; 20 are given in Table 5.3. At best, the values of 1e(n,c) are quite mysterious, and many interesting questions arise from study of Table 5.3. The larger dots represent the boundary of the c > n region, which clearly contains ~1 zeros since c must be s; n. (Why? The largest possible c value occurs when the stringS is composed of only one unit and subunit with value n). In fact, it seems clear from Table 5.3 that the maximum attainable c value for a given n, which we denote A(n), is, for large n, strictly less than n - note how each row tends to come less near the c > n diagonal. Number of notes (n) 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Complexity 2
3
1
. .
4
5
6
7
8
(c)
9 10 11 12 13 14 15
1
1 2
.
2 2 2
1
1
..
1 2
1 1
4 4
2 2 7 3 1 2 2 1
2 8
2 1 1 2 5 5 6 4 4 9 6 6 16 11 6
1 4 1 7 6 14 17 3 16 27
5 1 2 11 1 11
2 2 2 12 5 2 12 6 4 20 3 28 13 12 27 13 10
1 4 3 4
3
4
Total 1 1 1 2 2 3 4 5 7 9 12 16 21 28 37 49 65 86 114
Table 5.3. Values of K(n,c): Number of n-beat metrical patterns with complexity = c (n :;; 20).
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130
The values of A(n) for n :S 20 are: A(n)
= 2, 3, 2, 5, 3, 7, 8, 7, 8, 8, 10, 10, 11, 10, 13, 12, 13, 13,
15,...
which differ from n by n- A(n)
= 0, 0, 2, 0, 3, 0, 0, 2, 2, 3, 2, 3, 3, 5, 3, 5, 5, 6, 5, ... n = 2,3,5,7,8 contain the only perfectly complex meters,
The values which attain c =n. These special meters are:
2 = (2) 3 = (3) 5 = (2 3) or (3 2) 7 = (2 3 2) 8 = (3 2 3)
It is easy to prove inductively that these are the only values of n with A(n) = n, or, equivalently, that A(n) < n for sufficiently large n. Consider a string S for n sufficiently large; the largest c(S) can possibly be is the first element of S (say, k) plus c(T), where Tis the substring obtained by deleting the first element of S. Now, c(T) :S A(n-k). Since k = 2 or 3, A(n-k) < n-k by the induction hypothesis. Thus, c(T) < n-k and so c(S) < (n-k) + k, hence c(S) < n. Since this holds for any S, we have A(n) < n.
Table 5.4 lists the simplest and most complex meters for all n :S 20. The notation [xyz] in this table denotes both the partition 'xyz' and its reversal. Since the c value of a meter is unchanged under reversal, each meter in Table 5.4 appears with its reversal, unless, of course, it is palindromic. Notice that the majority of the simplest and most complex meters are either unique (exactly 1 possibility listed in Table 5.4) or essentially unique (exactly 2 possibilities listed in Table 5.4- a reversal pair). The only ones which are not are the simplest meters for n = 11, 14, 15 and the most complex meters for n = 11, 15, and 17-20. The least unique entries in Table 5.4 are the si~plest meter for n = 14, for which there are 7 possibilities, and the most complex meter for n = 15, of which there are 12. These patterns are: Simplest n = 14 (c
2222222 [3 2 2 3 2 2] [3 3 2 2 2 2] [3 3 3 3 2]
=5)
Most complex n = 15 (c = 10)
{3 3 2 [3 2] 2) {3223 [23]) {3 2 3 [3 2 2])
131
Combinatorics of Rhythms
n 2 3 4
Simplest meter(s)
c
Most complex meter(s)
c
*2 2 *3 3 2 22 *5 5 5 [3 2] 6 3 3 2 2 2, 3 3 *7 7 5 232 *8 8 2 323 7 9 3 [2 3 2 2] 8 10 4 3223 11 8 5 [2 3 2 2 2], [3 2 3 3] 12 10 3 [3 2 2 3 2] .13 10 5 [3 2 3 3 2] 14 11 5 [3 2 2 2 3 2] 15 10 5 see text (12) 13 16 2 2322232 12 17 5 [2 3 3 2 [2 3] 2] 18 13 3 [3 2 2 3 [2 3] 3] 13 19 5 [2 3 3 2 2 2 3 2], 3 2 3 3 3 2 3 . 20 4 r2 3 a 2 3 b 2 31 (a b)= (2 3) or (3 2) 15 Table 5.4. The simplest and most complex metrical patterns for n ~ 20. Those marked with * are the only ones which achieve A(n) n. 2 3 22 [3 2] 2 2 2, 3 3 [3 2 2] 2222 333 22222 [3 2 2 2 2], [3 3 3 2] 2 2 2 2 2 2, 3 3 3 [3 3 3 2 2] see text (7) [3 2 3 2 3 2], 3 3 3 3 3 22222222 [3 3 3 2 2 2 2] 2 2 2 2 2 2 2 2 2, 3 3 3 3 3 3 [3 2 2 2 2 2 2 2 2] 2222222222
2 3 2
=
(The notation {xyz} means to take the string 'xyz' as well as its complement obtained by interchanging 2's and 3's, and not to take its reversal).
Rhythmic Patterns Introduction. Given one of the r(n) metrical patterns containing n "beats", there is another layer of rhythmic construction which remains to be considered, which we refer to as rhythmic patterns. A rhythmic pattern is a series of musical events superimposed on the basic n-beat metrical pattern. These events can be interpreted in at least two ways: (1) As rhythmic events- for example, sounds made by percussion instruments. Particular combinations of rhythmic events define different kinds of rhythms, such as rock, jazz, Latin, etc. (2) As melodic events - for example, a series of notes and rests defining the melody of a composition. In this case, the "rhythm" may extend over many measures, unlike case (1) in which the rhythm usually repeats on a one- or two-measure basis.
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132
As an example of the difference between meter and rhythm, Figure 5.3 shows the meter and rhythm for Dave Brubeck's Take Five. The meter is 5/4, with the 5 beats in each measure subdivided as 3+2. The basic rhythm, however, is:
(~ + 1 + ~ + 1) + (1 + 1) which is the rhythm of chords played by the piano in the usual arrangements of this piece. In musical notation, an eighth note is half the duration of a quarter note. Since the meter is 5/4, one "beat" equals a quarter note, and therefore the eighth notes shown in Figure 5.3 are "half a beat" long. This is the reason for the l/2's in the description of this rhythm. Similarly, 1/16 notes would be equal to 1/4 of a beat, l/32 notes would equal 1/8 of a beat, and so on. For simplicity, we will always describe durations of rhythmic components in terms of beats rather than musical notes, with a beat also being equal to the basic unit of meter.
We can represent rhythms symbolically by dividing the meter into 2's and 3's as described in the previous section and then dividing each 2 and 3 into 2(2d) or 3(2d) parts, respectively, where d is determined by the granularity needed to express the rhythm in question. In particular, d is the integer such that the smallest fraction of a beat occurring in the durations of the rhythmic elements is 1f2d . For the above example, d=1 since we need to use l/2's in the description of the rhythm. The symbolic representation of the Take Five rhythm is: X X
Meter:
•
X X
5=
,
I
•
3
X
•
X
.
I
+ 2 '~
I~ J fltJJ)J J J Rhythm: 5 = 1/2 + 1 + 1/2 + 1 + 1 + 1 Figure 5.3. Meter vs. rhythm. The meter of Dave Brubeck's Take Five is 5 = 3+2, but the rhythm is 5 = 1/2 + 1 + 1/2 + 1 + 1 + 1. The "tie" between the second and third notes indicates that they are to be treated as a single quarter note.
..
133
Combinatorics of Rhythms
In this symbolism, each 'x' or'.' represents a unit of time, which the x's denoting the locations of musical events. The '/' marks indicate the metrical divisions into 2's and 3's. The meter in this example is 3+2, and we have divided time into half-beat units (as required by the fact that some events occur at half-beat resolution), so there are 6 time units in the first part and 4 in the second part. An 'x' immediately followed by an 'x' represents an eighth note, whereas an 'x' followed by a'.' represents a quarter note. In other words, each 'x' represents the beginning of a musical event. Each event implicitly ends when the next event begins. In general, we can defme a rhythm as a placement of musical events into a series of m measures. Each measure contains n(2d) time slots, where n is the number of beats in each measure and dis the amount by which the beats are subdivided by the use of 1/8 notes, 1/16 notes, etc. A musical event can be thought of as either a musical note or a rest. (If one or more rests occur in a row, they are considered a single event.) Note, therefore, that the first time slot always contains an event, since even if there are no musical notes for a while, there is still an initial event consisting of a rest.
For given values of m, n, and d, how many different rhythms are there? The total number of time slots (or positions) available is p =m · n(2d)
since there are m measures available and each contains n(2d) time positions. Each time position can either contain a musical event ("x") or the absence of an event(".''), which means that the number of distinct rhythms equals the number of (p-1)-bead chains. (The "-1" comes from the fact that we insist on the first bead of the chain being an event.) This number is, of course, 2P-1.
This trivial question becomes much more interesting when we consider the additional musical concept of syncopation. In addition to uncovering difficult mathematical questions, the idea of syncopation provides a method for ordering rhythms by their degree of strangeness - just like the other ordering methods we have discussed so far for chords, .scales, and metrical patterns. Thus it is of musical as well as mathematical interest.
Syncopation. Although syncopation in music is relatively easy to perceive, it is more than a little difficult to define precisely. Roughly speaking, syncopation occurs when events start or end· "off the beat". However, we need to be careful to define. "beat" in a manner which encapsulates all possible manifestations of the syncopation concept.
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134
As a first illustration, Figure 5.4 (a) and (b) illustrate two d = 1, n = 4 measures that do not contain any syncopation. In these measures, no musical event "straddles" a beat boundary. For example, each quarter note begins and ends on a quarter note boundary. By contrast, one event in Figure 5.4(c) contains the simplest form of syncopation, which might be termed a hesitation. The next to last event starts on a quarter note boundary but ends on an eighth-note boundary. As compared to the end of Figure 5.4(a), there is a hesitation in this last note prior to sounding the final eighth note. Figure 5.4(d) illustrates the opposite case, where the beginning of the second event is "off the beat" (not on a quarter-note boundary) but the end of the event is on the beat. We call this an anticipation. Even more dramatic is the second event of Figure 5.4(e), which is the Take Five rhythm again. This event, which is a quarter-note in duration, begins and ends on a non-quarter-note boundary; i.e., it contains both an anticipation and a hesitation, which combined produce a syncopation. The musical effect of a syncopation is best illustrated when there are several notes of this kind in a row. In such a case, the "beat" of the underlying metrical pattern does not "line up" with the implied beat of the musical events, resulting in an interesting musical ambiguity. Taking syncopation to extremes can almost entirely subvert the underlying metrical beat. Deciding on the relative rhythmical "strength" of a hesitation, anticipation, and syncopation is an interesting philosophical problem. Most listeners would probably agree that an anticipation is stronger than a
if i Jh J J if i JJ1."11 I (b)lfi iJ j j hI if~ iti JJ1<~ • • ••• if ~ l V l J J J I (a)
(c)
(e)
Figure 5.4. (a), (b): No syncopation; (c) hesitation; (d) anticipation; (e) syncopation. The dots represent the underlying metrical "beats".
135
Combinatorics of Rhythms
hesitation, and a syncopation is strongest of all. For simplicity, therefore, we define the syncopation value (s) of these three types of events as follows: hesitation =1 anticipation = 2 syncopation = 3 (=anticipation+ hesitation) We still need to define "off the beat" precisely. Let us consider only the following restricted subset of all meters: those with n = a power of 2, and a partition consisting of all 2's. In this case, we define "off the beat" as follows: Let obe the duration of the event expressed as a multiple of lf2d beats, and let S be the start time of the event, expressed in the same units. Then the start (end) of the event is "off the beat" if S (S + S) is not a multiple of D, where D = o rounded down to the nearest power of 2. Then the syncopation value (s) for the event is defined as:
s = (2 if start is off the beat) + ( 1 if end is off the beat). It is easily verified that this rule works for the examples in Figure 5.4. It also correctly handles more complicated cases such as X
X
o1
••
I ..
2 3 I 4
s
X
•
6 7
In this example, the second note "feels" truly syncopated, and should therefore have a value of s=3, even though its event ends "on the beat". Its duration (5), when rounded down to a power of 2, is 4, which means that S + o= 6 is not a multiple of 4, and so the above rule correctly gives s=3. Note that the time positions are numbered starting with 0. Now we can define the syncopation values of an entire rhythmic pattern as the sum of the s values for the individual notes. The s value of a rhythm is a good measure of its musical sophistication. Simple, relatively boring rhythms have s =0 (no syncopation). As the s value increases, the rhythms become more complex and "modern" sounding.
Rhythmic classification.
In general, let us define Sy(p,n,s) as the number of p-position, n-event rhythms with a syncopation value of s, and seek a formula for the values of Sy(p,n,s). Of particular interest are the values of Sy(p,n,O): the number of rhythms with no syncopation.
Chapter 5
136
To begin, define an auxiliary function CJ(p,n,m,s): the number of nevent sub-rhythms starting on position m with syncopation value~ s (as opposed to = s). Then, Sy(p,n,s)
=
CJ(p,n,O,s)- CJ(p,n,O,s-1) CJ(p,n,O,s)
(s >0) (s =0)
(1)
A recurrence formula for CJ(p,n,m,s) can be derived by trying all possibilities for the duration of the next event (which starts at position m) and summing up the (n-1)-event sub-rhythms generated by each choice. The duration of the next event, d, must be ~ 1 and ~ p-m, since it must not extend past the end of the p positions available. For a given d, the number of sub-rhythms is CJ(p,n-1,m,s-S(d)), where S(d) is the syncopation value of an event of duration d starting at position m, which can be calculated using the definition of the s value of an event: S(d) = 2 (if D(d) f. m) + 1 (if D(d) f. (m +d)),
where D(d) = d rounded down to the nearest power of 2. Of course, if n = 0, the value of CJ(p,n,m,s) is 1 if and only if m = p (i.e., the last event ends exactly at the end of the p positions) and s ~ 0 (i.e., the available amount of s has not been "used up"). Thus, we have the following recurrence formula for CJ(p,n,m,s): CJ(p,n,m,s)
L CJ(p,n-1,m+d,s-S(d))
=
lSdSp-m
O CJ(p, ,m,s)
=
{ 1 if m =p and s ;::: 0 0 otherwise
with S(d) as defined above. For example, let us compute the value of Sy(8,4,0) = CJ(8,4,0,0). Note that we need only choose d values that have S(d): 0, since any other d will yield CJ(p,n-1,m+d,s-S(d)) = 0 (since s-S(d) will be< 0). Therefore: C1(8,4,0,0) = C1(8,3,1,0) + C1(8,3,2,0) + C1(8,3,4,0) + C1(8,3,8,0) =1+2+2+0 = 5. Table 5.5(a) shows the values of Sy(4,n,s), calculated using the above formula and equation (1). This table says that of the 8 rhythmic patterns on 4 positions, 5 have no syncopation (s = 0), and there is exactly one each with s = 1, 2, and 3. These are pictured in Table 5.5(b).
Combinatorics of Rhythms
n
=
s
1
0 1 2 3
1
137
Number of events 2 3 4 Total
Pattern
n
s
X
1 2 3 3
0 0 0 0 0 1 2 3
X
1 1 1
2
1
5 1 1 1
1
X
X X X X
X X
X X X X
4
X
2 2 3
X
X X
Total
1
3
3
1
8
X X
X
(a)
(b)
Table 5.5. (a) Values of Sy(4,n,s): number of 4-position rhythms with n events and syncopation value= s. (b) Explicit list of the 8 rhythmic patterns.
A more complex example is shown in Table 5.6: the values of Sy(S,n,s). One striking feature of this table is the appearance, along a
diagonal, of a row of Pascal's triangle! Reading from the upper right corner to the middle of the bottom row we see the numbers 1,3,3,1,
(i).
which are the values of or row 3 of Pascal's triangle. Is this just a coincidence? Does a similar phenomenon occur for all tables of Sy(p,n,s) values with p equal to a power of 2? Observe that these numbers are the right-most numbers in rows with row number equal to a multiple of 3; i.e., with s =3t and n =p - t. Such rhythms must consist of t events of length 2, each starting on an odd position and thus having values of s = 3. The remaining p - 2t events n = Number of notes s
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6
1
1 2 2 2
2 4 4 6 2 2 1
5 4 5 12 3 2 2 1 1
6 6 6 6 4 4 2
6 3 3 6
4
1
7
8
Total
1
7
21
35
35
20 35 9 8 8 1 1 1
3
3
21
26 19
1
9
Total
7
1
128
Table 5.6. Values of Sy(8,n,s): number of 8-position rhythms with n events and syncopation value = s.
Chapter 5
138
each have length 1. Now, since pis even, there are exactly p/2- 1 odd positions available, and the t length-2 events can be placed independently at any of these positions. Therefore, Sy(p, p-t, 3t)
= ( p/2t
1)
(p = 2k),
so that, indeed each such table of Sy(p,n,s) will contain a diagonal of binomial coefficients. The preceding analysis also tells us that the maximum s value for a rhythm with p positions is 3(p/2 - 1), for the unique rhythm consisting of the maximum possible number of s = 3 events. For example, for p=8 this value is 3(8/2 - 1) = 9, as illustrated by Table 5.6. Another interesting subset of the Sy(p,n,s) array is the values of Sy(p,n,O): the number of non-syncopated rhythms for each value of p and n. These are shown in Table 5.7, for values of p = 2k. (Note that this table consists of the first rows of tables such as Table 5.5(a) and 5.6.) The numbers in the "totals" column of Table 5.7 suggest that
= Sy(2k-l,*,0)2 + 1.
Sy(2k,*,O)
(2)
To see that this equation is in fact true, observe that all the possible Sy(2k, *,0) rhythms can be constructed by taking all possible pairs of Sy(2k-l, *,0) rhythms plus the one additional rhythm consisting of a single event spanning all 2k positions. No other additional rhythms are possible because any other event which spans the middle position will have a value of s > 0. This recurrence allows us to calculate a musically-significant number: the number of non-syncopated 8-measure rhythms in 4/4 time, which is one of the common lengths for jazz, blues, and popular tunes. If the n = number of events 1
2
1 1 1 1 1
1 1 1 1
3
4
5
6
7
8
9
10
11
12
13
14
15
16 Total
p
1 2 4 8 16
2 2 2
1 5 5
6 14
6 26
4 44
1 69
94 114 116
94
60
28
8
1
1 2 5 26 677
Table 5.7. Values of Sy(p,n,O): number of non-syncopated n-event rhythmic patterns on
p =2k positions.
Combinatorics of Rhythms
139
rhythm uses events no shorter than quarter notes, then we have p and
= 32,
= Sy(16,*,0)2 + 1 = 6772 + 1 = 458330. whereas at l/8-note resolution (p = 64) we obtain Sy(64,*,0) = 210066388901. Sy(32,*,0)
This equation also says that the number of non-syncopated rhythmic patterns is much smaller than the total number of patterns. From equation (2), .
Sy(2k' * ,0) "" 22k-1 whereas the total number of metrical patterns is simply the number of nonempty binary strings of length 2k, since each position in the string can be either "x" or ". ", and we disallow the string consisting of all ". "'s. Hence:
Sy('J..k,*,*)
= 22k-t
so that, asymptotically,
Sy(2k,*,O) _ 1 Sy(2k,*,*) - 2(2k-l.t)" For example, for p = 32, there are 458330 non-syncopated patterns out of 2147883648 total patterns. Non-syncopated rhythms and melodies are in a very small minority, even though they dominated music for centuries. Before leaving Table 5.7 we cannot resist a little puzzle for the reader: Why are the first few numbers in each row of Table 5.7 Catalan numbers? Catalan numbers are given by
1 (2k) CJc=m k with Ck (k;;:: 0) = 1, 1, 2, 5, 14, 42, 132, 429, 1430, ... These numbers have many combinatorial interpretations, including the number of paths on an integer-lattice grid from (0,0) to (k,k) which do not cross the line y =x, or the number of rooted planar binary trees with k + 1 endpoints. (Here, "planar" means that reflections are counted as distinct.) Incidentally, the pattern does continue beyond the rows shown in Table 5.7: the first few values of Sy(32,n,O) are 1, 1, 2, 5, 14, 42, 100, ...
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140
Musical Applications The primary application for the meters and rhythms discussed in this chapter is their use in creating musical compositions with certain rhythmic characteristics. The metrical complexity measure, c, can be used to choose time signatures from the simple to the complex, according to taste. Although composers have used unusual time signatures for centuries, there are still many unexplored meters, even with low values of n, as illustrated by Table 5.2. The listing of all possible meters also suggests a rhythmic analogue of the Oulipo +N algorithm: take a composition written in a given meter and replace the meter with a different one, according to some criterion. For example, take a composition written in common 4/4 time and transform it to 7/4 time by taking each pair of measures (4+4) and dropping one beat from, say, the second measure, to yield 4+3 =7. Or, change it to 13/4 by adding one beat to 4+4+4 to make 4+4+5 = 13. The effect of this simple change can be dramatic. As a simple example of the use of mathematically interesting rhythms, we offer the following piece, Nonadecimal Nocturne. This piece uses the time signature 19/4, subdivided as 19
= 2+3+3+2+2+2+3+2,
which is one of the maximally-complex n = 19 meters, and is ori.e of the eight most complex meters for all n ~ 19 (see Tables 5.3 and 5.4). Superimposed against this rather bizarre time signature is a harmonicallysimple piece in the classical style. The combination is somewhat unnerving, but not without musical interest.
Combinatorics of Rhythms
141
Nonadecimal Nocturne M. Keith
(19
=2+3+3+2+2+2+3+2)
{!; : y. f : t ; (!;, : r : t
J :
E ::
t : :
~if f f : I
tL : Jlr t
~t
E
r
0
E
t)
I
(~:: : f fft : r r : r : f f : : : ; : fZ I (!/, : f ::tt: f f : j : f j f : ~ t t r I ~~;,
f' t'Q ;
r r ; :: ' l
J
iQ: I ....__,
Appendix A The 351 Chords
This appendix gives an explicit list of the 351 essentially distinct chords in the 12-tone scale. Each chord is shown in two forms: a symbolic form showing which of the twelve notes of the scale are used by the chord (with * indicating notes used), and also as a list of intervals (the distances between successive notes of the chord). For each chord, the following properties are tabulated: a: Number of adjacencies (equivalently, "1" intervals) in the chord.
m: Minimum interval in the chord. p: Period of the chord (smallest amount of transposition that brings
the chord into coincidence with itself).
s: Span of the chord (smallest span of any inversion). ' v: Variety value (number of distinct intervals in full interval set). d: Harmonic simplicity (minimum value of d such that all intervals in the chord can be represented by fractions with denominator d). We also list the "chromaticity" type of the chord when played on a piano. There are three possible types: B Can be played (in some key) using only black keys and (in some other key) using only white keys. W Can be played using only white keys, but not using only black keys. P
Is "polychromatic"; i.e., requires both colors in all 12 keys.
The list is sorted by the value of n, the number of notes in the chord (from 1 to 12). Within each value of n, chords are sorted by the value of a, which is perhaps the most important property listed here. See Chapter 3 for a discussion of interesting subsets of this list, such as the 30 chords with a= 0, the 55 chords with a= 1, the 15 non-trivial chords with p -:¢: 12, and so on.
143
The 351 Chords
Notes
Intervals
a
m
p
s
Type v
d
Comments
(12)
0
12
12
0
B
0
1
Single note
2(10) 39 48 57 66 1{11)
0 0 0 0 0 1
2 3 4
2 3 4
B B B B
6 1
w w
1 1 1 1 1 1
4 3 4 2
6 1
12 12 12 12 6 12
228 237 327 246 336 426 525 345 435 444 129 219 138 318 147 417 156 516 11{10)
0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2
2 2 2 2 3 2 2 3 3 4 1 1 1 1 1 1 1 1 1
12 12 12 12 12 12 12 12 12 4 12 12 12 12 12 12 12 12 12
4
B B B
3 3 3 3 3 3 3 3 3 2 3 3 3 3 3 3 3 3 3
5 5
2226 2235 2325 3225 2244 2334 4323 2424 4332 3333 1227 2127 2217 1236 2136 1326
0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2 3 1 1 1 1 1 1
12 12 12 12 12 12 12 6 12 3 12 12 12 12 12 12
n=l
• n=2 ••.......... •.......... • •.......... • •.......... • •.......... • •• .......... n=3 ......... ••• ......... ••• •...• •...... ••.... •..... •..•.. •..... •......... •• •......... •• • • •......... •...•..•.... •......... • • ••• ••• ........ •• • •.. •• • •...•• ...... •• • •....•• ..... ••• n=4 ........ •••• •........ •• • ........ •••• •........ ••• •........ •• • •........ ••• •........ ••• •........ • •• •...•..•.. •. •........ ••• •••• •.••.• ...... •••• ...... •••• ••• • •• • • ••<
5
5
5 5 6 6 6 7 7 7 8 3 3 4 4
5 5 6 6 2 6 7 7 7 8 8 8 8 8 9
5 5 5 6 6 6
w w w
B B B p
w w p p
w w w w p
5
4 4
5 5 6 6 3 7 6 7 7 7 7 8 8 7 8
w
5
B
6
B B
6
p
5
6 6 7
6 6 4 6 3 6 6 6 6 6 6
6 7 4 9 7 6 9 6 7 7
w B p
w p
w w w w p p
5
Only n=2,p<12
7
Sustained Minor Major Augmented; p = 4 (smallest P chord)
5 5 Minor 11th
5 6th p < 12 7th Diminished
Appendix A
144
Notes
•••• ... ····· ••••..... •••• ••••····· ••• • •• • • •..•• ..•.... •• •• •... •• .•.... •• •• .... • • •• .... • ••• .... •••• ••• • • •• • •••• •••• ....... •••• ....... ••• • •••• • ••• ••• • •• ...•• ..... • ••• ..... ••• • •• •• •••• n=5 ....... ••••• •........ •• • • •........ •• •• •• ....... ••• •....... •• • • •..• ••.•.... •... • • •• .... •• ....... •• • • ••...... •• •..• •• ..•... •• ....... • •• • ••...... •• •• ....... ••• •.. •• .•.•... •... • ••.•... •...• •• .•... •.... • • •• ... •.... • • •• ...
Intervals 3126 2316 3216 1245 2145 1335 3135 1425 4125 2415 3315 4215 1344 3144 4341 1128 1218 2118 1137 1317 3117 1146 1416 4116 1155 1515 1119 22224 22332 22323 12225 21225 22125 22215 12234 21234 22134 12324 21324 13224 31224 23124 32124 22314 23214
a
1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 2 2 2 2 2 2 2 2 2 2 2 3
0 0 0 1 1 1 1 1 1 I
1 1 1 1 1 1 1 1
m 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
12 12 12 12 12 12 12 12 12 12 12 12 I2 12 12 12 12 12 12 12 12 12 12 12 12 6 12
2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12
p
s 6 6 6 7 7 7 7 7 7 7 7 7 8 8 8 4 4 4
5 5 5 6 6 6 7 7 3 8 9 9 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8
5
d 7 7 7 7 7 9 9 7 6 8 9 8 7 7 8 8 7 8 9 8 8 8 8 8 9 9 9
5
7
8 6 9 8 8 9 9 9 9 9 8 9 9 8 9 9 9
6 7 7 6 8 8 6 7 6 7 7 7 7 7 7 7
Type v p 6 p 6
w w w
w
6 6 6 6 6 6 6 6 6 6
p
5 5
p p
w w w p p
w p p p p p p
p
w p p p p
p
w B
w w w w w w p
w p p
p p
w p
w
6 6 6 6 6 6 6 6 6 6
5 4
5
Comments
Minor 9th (m=1)
Major 7th (most common m=1)
p < 12
145
The 351 Chords
Notes
*.. *.*.** ... **.* .. *..*.. *....... ** * * ** ....... ** * *....... * ** * *** * * ** ....... ** * *.*** .*..... ** * ** ..... * ** ** ..... * * *** ..... *** * * ** ....... ** * *.*** ...... * *** * * ** ........ ** * *.. *** .*.... ** * ** .... * ** ** ..... ** * ** .... * ** ** .... * * *** * * *** *** * * ** ** * * *** * *** * * ** ....... ** * * *** * *** ** ** ... ** * * ** ...** ... ** * ** ... * ** ** ... * * *** **** * *** ** ...... **.*** * **** **** * *** ** ..... ** *** * **** **** * *** ... ** .... ** ... ***
Intervals 322I4 I2333 21333 13233 23133 11226 12126 21I26 I2216 21216 22116 11235 I2135 21135 11325 13I25 31125 12315 21315 13215 31215 23115 32115 11244 12144 21144 11334 13134 31134 11424 14124 21414 13314 31314 33114 11127 11217 12117 21117 11136 11316 13116 31116 11145 11415 14115
a
I 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3
m I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
p
s
I2 12 12 12 12 12 12 12 12 12 12 I2 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12
8 9 9 9 9 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 5 5 5 5 6 6 6 6 7 7 7
Type v w 9 p 7 p 7 p 9 p 9 p 9 p 8 p 8 w 9 p 8 p 9 p 9 p 9 p 10 p 9 p 10 p 10 p 8 p IO p 8 p 9 p 9 p p p p p p p p
w w p p p p p p p p p p p p p p
9 9 9 9 9 7 8 8 8 8 8 7 9 8 8 8 8 9 9 9 9 9 7 7
d 10
9 9 9 9 8 7 8 9 7 8 9 9 9 9 8 8 9 10 9 9 10
9 8 7 8 9 IO 10
8 8 8 10
8 8 9 9 8 10
9 8 8 9 10 10
9
Comments
Appendix A
146
Notes
Intervals
p
* **** *****
a 3 4
m
41115 11118
1 1
12 12
222222 122223 212223 221223 222123 222213 112224 121224 211224 122124 212124 221124 122214 212214 221214 222114 112233 121233 211233 122133 212133 221133 112323 121323 211323 113223 131223 123123 213123 132123 221313 213213 111225 112125 121125 211125 112215 121215 211215 122115 212115 221115 111234
0 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3
2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 6 12 12 12 6 12 12 12 12 12 12 12 12 12 12 12
s 7 4
n=6
****** ** ...... **** * ** ..... *** *...... * ** * * *...... * * ** * *.... * * * ** .. *** * * * ** ...... ** * * * *** * * ** ...... * ** * *...... ** ** * * * *** * ** ... * * ** ... * ** .. * ** ... * * ** ** ... * * * *** *** * * * ** ** ..... ** * *** * * **.*.** ..*.. *.**.** ..*.. *.*.*** .... * *** * * * ** ...... ** * * * *** * * *** * * * ** ...... ** * * ** ...... * ** * * ** .. ** .*.. ** * ** * * * ** ** .. * ** * ** .. **** * * *** ** * ** *** * * **** * *** * ** .... ** ** ** .... * *** ** ** * *** * ** *** .... * * **** **** * * '
•
t
•••
'
'
t
•
••
10
9 9 9 9 9 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 7 7 7 7 7 7 7 7 7 7 8
Type v
d
p p
9 7
9 9
p p
5 10
8 7 7 10 6 7 8 7 8 9 7 8 8 10 7 8 8 9 9 9 9 9 9 9 9 10 8 9 9 9 10 9 10 9 8 10 9 9 9 9 10 10 9
w w w p p p p
w p p
w w p p p p p p p p p p p p p p p p p p p p p p p p p p p p p
10
9 10 10 10 10
10 10
io 10 10 10
10 10 11 10 10
10 10
11 11 9 11 10
10 9 10 9 10 9 11 10 11 10
11 10 11 11 10 11 11
Comments
Whole-tone scale (largest a=O)
p < 12
p < 12
147
The 351 Chords
a 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 5
m 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
p
s
*** ** * ** .*** ..... * * **** * **** * * *** ** * ** .. *** .*... *..**** .*... *** * ** ... ** ** ** ... * *** ** ... *** ..*.** ... ** ** ** ... * *** ** ... ** * *** * ** *** ... ** * *** ... * ** *** ... *... * **** ... *..*.**** ... **** * * *** ** * ** .. *** ..*.. ** ** ** .. *****.* ..... **** ** ..... *** *** ..... ** **** * ***** ..... ***** * **** ** .... *** *** .... ** .. **** .... * ***** .... ***** * **** ... ** ... *** ... *** ... ****** ......
Intervals 112134 121134 211134 111324 113124 131124 311124 112314 121314 211314 113214 131214 311214 123114 213114 132114 312114 231114 321114 111333 113133 131133 131313 111126 111216 112116 121116 211116 111135 111315 113115 131115 311115 111144 111414 114114 111117
12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 4 12 12 12 12 12 12 12 12 12 12 12 12 6 12
8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9
*** ..... **** ** **.... *** ** ..... * ** * * **** * * * ***..... ** * * ** .*** .*... * *.**** .*... * *** ..... * ** *
1122222 1212222 1221222 1112223 1121223 1211223 2111223 1122123
2 2 2 3 3 3 3 3
1 1 1 1 1 1 1 1
12 12 12 12 12 12 12 12
10 10 10
Notes
6 6 6 6 6 7 7 7 7 7
8 8 8 5
Type v p 11 p 10 p 11 p 11 p 10 p 11 p 11 p 10 p 10 p 11 p 10 p 10 p 10 p 10 p 10 p 10 p 11 p 11 p 11 p 11 p 9 p 9 p 7 p 10 p 10 p 10 p 10 p 10 p 11 p 11 p 10 p 11 p 11 p 11 p 9 p 8 p 9
d
Comments
9 10 10
9 8 8 9 9 10
8 9 10 10 10
8 10
8 10 10
9 10 10 11
9 9 8 10
9 9 10 10
9 9 11 11 11 10
n=7
9 9 9 9 9
p p
w p p p p p
11 10 10 11 11
11 11 11
8 7 10 10 10
8 10
9
LastWchord
Appendix. A
148
Notes
**.**.**.* .. *.*** .** ... * ** * *** * * ** *** * * * **** * *** * * ** .. ** ** * ** .. *.***.*.** .. **.*.**.** .. * ** ** ** .. * * *** ** .. ** * * *** *.**.*.*** .. * * ** *** * * * **** ***** * * **** ** * *** *** * **.****.* ... * ***** * **** * ** ... *** ** ** ... **.***.** ... *.****.** ... ***.*.*** **.**.*** * *** *** ** * **** * ** **** *.*.***** ... ***** * * ****.** ..*.. ***.*** ..*.. **.**** * *.***** * ***** * * **** ** * *** ..***.* .. ** ..***** ***.** ..** .. ** *** ** * **** ** .. **** ..*.** .. *** ** ** .. * *** *** ****** *
Intervals 1212123 2112123 1221123 2121123 2211123 1122213 1212213 2112213 1221213 2121213 2211213 1222113 2122113 2212113 2221113 1111224 1112124 1121124 1211124 2111124 1112214 1121214 1211214 2111214 1122114 1212114 2112114 1221114 2121114 2211114 1111233 1112133 1121133 1211133 2111133 1111323 1113123 1131123 1311123 1121313 1211313 2111313 1113213 1131213 2113113 1111125
a
m
p
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12
5
s 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 7
Type v p 10 p 11 p 11 p 11 p 11 p 11 p 10 p 11 p 10 p 10 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 10 p 11 p 11
d 9 9 9 9 10 8 10 9 9 9 9 8 9 10 10 10 9 8 10 9
iO
9 10 10 11 10 8 11 10 11 9 9 10 10 9 9 10 10 9 10 11 10 9 10 10 10
Comments
149
The 351 Chords
Notes
Intervals
a
***** ** .... **** *** .... *** **** .... ** ***** .... * ****** ****** * ***** ** ... **** *** ... *** **** ** ***** * ****** ... ******* .....
1111215 1112115 1121115 1211115 2111115 1111134 1111314 1113114 1131114 1311114 3111114 1111116
5 5 5 5 5 5 5 5 5 5 5
***** * * * **** ** * * *** .*** .*.. * ** .**** .*.. * **** * ** * ***.... ** ** * ** .*** .**.. * *** .*.*** .*. **.**.***.*. ** ** ** ** ******.*.* .. ***** ** * **** *** * *** .**** .*.. ** .***** .*.. *.****** .*.. ***** * ** **** ** ** *** *** ** ** **** ** .. * ***** ** **** * *** .. *** ** *** .. ** *** *** .. * **** *** *** * **** ** ** **** .. *.*** .**** ** * ***** .. * ** ***** * * ****** .. ******* * ****** ** ...
11112222 11121222 11211222 12111222 11122122 11212122 12112122 11221122 12121122 12121212 11111223 11112123 11121123 11211123 12111123 21111123 11112213 11121213 11211213 12111213 21111213 11122113 11212113 12112113 21112113 11221113 12121113 21121113 12211113 21211113 22111113 11111124 11111214
6
m 1 1 1 1 1 1 1 1 1 1 1 1
12 12 12 12 12 12 12 12 12 12 12 12
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
12 12 12 12 12 12 12 6 12 3 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12
p
s 7 7 7 7 7 8 8 8 8 8 8 6
n=B
4 4 4 4 4 4 4 4 4 4
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6
10 10 10 10 10 10 10 10 10 10
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 8 8
Type v p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11
d 9 10 10 9 10 10 11 11 11 11 11 10
p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p
12 10
11 11 11 11 11 11 11 11 11 10 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11
8 10 10
9 10 12
9 9 10
9 10 10
9 10 10
9 10 10
9 11 10 11 10 11 10 10 11
9 10 10 10
Comments
Appendix. A
150
Type v p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11 p 11
d 11 11 11
a
m
p
8
***** *** **** **** *** ***** ** ****** * ******* ... ******* * ****** ** .. ***** *** **** **** ********
Intervals 11112114 11121114 11211114 12111114 21111114 11111133 11111313 11ll3113 11131113 11111115
6 6 6 6 6 6 6 6 6 7
1 1 1 1 1 1 1 1 1 1
12 12 12 12 12 12 12 12 6 12
8 8 8 8 8 9 9 9 9 7
******* * * ****** ** * ***** *** * **** **** * *** ***** * ** .****** .*. ***** ** ** **** *** ** *** **** ** *** *** *** ******** * ******* ** ****** *** ***** **** .. **** ***** *** ****** ** ******* * ******** *********
111111222 111112122 111121122 111211122 112111122 121111122 111121212 111211212 112111212 112112112 111111123 111111213 111112113 111121113 111211113 112111113 121111113 211111113 111111114
6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
12 12 12 12 12 12 12 12 12 4 12 12 12 12 12 12 12 12 12
10 10 10 10 10 10 10 10 10 10 9 9 9 9 9 9 9 9 8
p p p p p p p p p p p p p p p p p p p
11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11
12 10 12 11 12 10 9 10 10 12 10 10 11 11
********* * ******** ** ******* *** ****** **** ***** ***** **********
1111111122 1111111212 1111112112 1111121112 1111211112 1111111113
8 8 8 8 8 9
1 1 1 1 1 1
12 12 12 12 6 12
10 10 10 10 10 9
p p p p p p
11 11 11 11 11 11
12 10 12 11 12 12
***********
11111111112 10
12
10
p
11
12
************
11111111111112
1
11
p
11
14
Notes
n=9
n =10
n =11
n =12
1
11
12 10 11 11 11 10
11
11 12 10 12
Comments
Appendix B The 462 Scales
In this appendix we list the 462 seven-note scales. The format is similar to the chord list in Appendix A: Each scale is listed by notes as well as by partition (intervals), followed by a list of various properties: E: Mean-square error between the scale and the perfect scale having all intervals = 12n. T
Taxicab distance between the scale and the major scale.
c Number of chords contained within the scale (i.e., formable using only notes in the scale). The fourth property, listed as a 13-digit number, gives the spelling characteristics of each scale. The 13 digits denote the degree of accidental required to spell the scale in each of the 13 keys, listed in the order C, F (1 flat), G (1 sharp), ... , up to Gb (6 flats), F# (6 sharps). The organization of the list is as follows. The 462 scales are grouped into 66 groups of 7, where each group corresponds to a scale type consisting of a specific set of intervals in a particular cyclical order. The 7 scales in each group are generated by the 7 cyclic permutations of the intervals of the first scale listed. As described in Chapter 4, the 66 scale types can be further combined into just 7 scale classes, where the scales in each class use the same (unordered) interval set. The 7 scale classes are marked with bold-face headings. The 7 scale classes are ordered by their average evenness value e, then within each class the scale types are ordered by their average E value, then within each scale type the 7 scales are ordered by their exact E value. This yields a rough ordering of the 462 scales by their "simplicity", with the scales most commonly used in music appearing near the top of the list. Note that scales with names tend to appear near the beginning of each scale class, indicating the success of this ordering measure. See Chapter 4 for a discussion of interesting subsets of this list, such as the diatonic and triatonic scales, scales with small E value, scales near the major scale (i.e., with small T value), scales with unique spelling properties, and so on.
Appendix B
152
E T Intervals Scale class (2222211) - Diatonic
Notes
c
Spelling
64 64 64 64 64 64 64
1111011111121 1101111111111 1111110112121 0111111111111 1111111202121 1011111111111 1111121212021
6
80 80 80 80 80 80 80
1111111112121 1111111111121 1111111212121 1111111111111 1111121212121 1111111111212 1212121212121
2 3 3 3 5 3 7
73 73 73 73 73 73 73
1111111212121 1111111112121 1111121212121 1111111111212 1212121212121 1111112121212 2222121212131
82 82 82 82 82 82 82
1111111212121 1111111112121 1111121212121 1111111111121 1212121212121 2212121212121 1111111121212
33
1 2 7
82 82 82 82 82 82 82
1111111112121 1111111111121 1111111212121 1111121212121 1111111111212 1111111121212 2212121212121
10 11
2 1
82 82
1111121212121 1111111212121
* ** * * ** *.. * ** .*.** *.**.*.**.*. *.*.**.*.*.* ** ..... * * ** * *..... * * ** * * ** ..... * ** * *
2122212 2212212 2122122 2212221 1222122 2221221 1221222
*..... * ** ** * *.**.*.*.*.* **.*.*.*.** * * * ** ** *.**.**.*.*. *.*.*.*.**.* ** ..... ** * * * **.*.*.*.*.* *.*.*.*** .*. *.*.*** .*.. * *.*.*.*.*** * *** * * * *.*.*.*.*.** ***..... ****
2212122 2122221 1222212 2221212 2121222 2222121 1212222
7 7 10 10 22 22
1222221 2221122 2211222 2222112 2112222 2222211 1122222
10 11 11 19 19 35 35
4
5 5 8 8 13 13 6
2 1 3 0 4
1 5 2 1 3 2 4
2
Dorian Mixolydian Natural Minor Major Phrygian Lydian
Locrian B6 Melodic Minor; schizo.
B3 B5; note spelling! B7 B4 Super-Locrian
Only a=3 diatonic (Gb)
Scale class (3222111) - Triatonic
** ** * ** *.*.**.** ..* * ** ** ** *.** ..**.*.* ** ..... ** ** * **.*.**.** .. *.. ** .*. ** .*
1312212 2212131 2121312 2131221 1213122 1221213 3122121
*.**.*.** .. * * ** ** ** ** ..... ** ** * ** * ** ** *..... * ** ** * *.. **.**.*.* ** **.. * ** ..
2122131 2131212 1312122 1221312 2213121 3121221 1212213
* * *** ** ** .. **.*.*.*
2211312 1312221
8 9 9 12 17
2 1 3 2 5
24
6
33
3
8 9 9 12 17
2 3 3
24
4
a schizophrenic scale
Minor
The 462 Scales
Notes
153
T 2 4
* * * *** * * *** ** * ** * * *** *** ..... ** * * *..... ** * * **
Intervals 2221131 2113122 1222113 1131222 3122211
* ** *** * ** ..... * * ** * * * ** *** ** ..*** .*.. * *..***.*.*.* *.*.*.** ..** *** * * ** ..
2131122 1222131 2213112 1311222 3112221 2221311 1122213
10 11 14 14 19 26 46
4 3 2 4 1 2 8
82 82 82 82 82 82 82
1111111112121 1111111212121 1111111111212 1111121212121 1111111121212 1111112121212 2222121212131
*.*.*** ... ** * *** * ** *.**.*.*** .. ** ..... * ** * * ***..... * ** * ** * *** * *..... * ** * **
2211321 2113212 2122113 1321221 1132122 1221132 3212211
13 13 16 16 21 28 61
1 3 4 2
81 81 81 81 81 81 81
1111121212121 1212121212121 2212121212121 1111111212121 2222121212131 2222221213131 1111212121212
* ** ** ... ** ** * ** ** **.** ..*.** * ** ** ** .. *..... ** * ** * ** ..... ** ** * *..... * ** ** *
2121321 1321212 1213212 2121213 2132121 1212132 3212121
12 13 16 21 21 37 48
2 3 4
59 59 59 59 59
1111121212121 1111111212121 1212121212121 2212121212121 1111111111222 2222221213131 1111212121212
Minimum c value
** * ** ** *..... ** ** * * *..... ** * ** * * ** ** ** ** ** * ** *..... * ** ** * ** ..... ** ** *
1231212 2121231 2123121 3121212 1212312 2312121 1212123
12 13 16 21 21 37 48
4 3 2 3
59
Minimum c value
5
59
3 8
59 59
1111111212121 1111121212121 1111111111222 1111111121212 1212121212121 1111212121212 2222221213131
* ** * *** ** * *** * * *** * ** ** ..... * ** * * *..... * ** * ** * * *** * * *** .. * ** .*..
2123112 1231122 3112212 1221231 2212311 2311221 1122123
13 13 16 16 21 28 61
3
81 81 81 81 81 81 81
1111111111222 1111111212121 1111111121212 1111121212121 1111112121212 1111212121212 2222221213131
E
14 14 19 26 46
6 4
c 82 82 82 82 82
Spelling 1111111112121 1212121212121 2212121212121 2222121212131 1111112121212
5
5 6
5
5 3 7 4
5 2 4 1 2 9
59
59 59 59 59
AppendixB
154
c
Notes
Intervals
** .. *.*** .*. ** ..... * ** * * * ** * *** * *** * ** .. *..... * ** * ** *.. *.*** .*.* *** .. * ** ..*.
1321122 1221321 2132112 2112213 2213211 3211221 1122132
14 15 18 30 30 39 50
4 3 4 6 2 3 8
79 79 79 79 79 79 79
Spelling 1111111212121 1111121212121 1111111111222 2212121212121 1111112121212 1111212121212 2222221213131
*.*.*** .*..* ** ..... * ** * * *.***.* ..** * ** * *** ***..... * ** * ** .*.*** .*.. *..... * ** * **
2211231 1231221 2112312 3122112 1123122 1221123 2312211
14 15 18 30 30 39 50
2 3 4 4 6 7 4
79 79 79 79 79 79 79
1111121212121 1111111212121 1212121212121 1111111121212 2222121212131 2222221213131 1111212121212
*.*** .... *** * * ** *** *** ..*.*.** *.**.*** .. *. ** ..*.*.**.* ** .*** .... ** *.. *.*.**.**
2113221 2212113 1132212 2121132 1322121 1211322 3221211
16 17 20 25 25 41 80
2 3 4 5 3 7 6
82 82 82 82 82 82 82
1212121212121 2212121212121 2222121212131 2222221213131 1111111212222 2222232323131 2121212121212
** * * *** *.. ***.**.*. *.**.*.* .. ** **.**.*.* .. * * * *** ** *.*.* .. ***.* *** ..... ** * *
1223112 3112122 2122311 1212231 2311212 2231121 1121223
16 17 20 25 25 41 80
4 3 2 5 3 3 10
82 82 82 82 82 82 82
1111111212222 1111111122222 1111112121222 1212121212121 1111212121212 2121212121212 2222232323131
** ..... ** * * * *.*.***.** .. ** .. *.*.*** *.***.** ..*. *.** ..*.*.** *** ..... ** * * *.. *.*.*** .*
1213221 2211213 1322112 2112132 2132211 1121322 3221121
19 22 22 34 34 54 67
3 4 4 6 4 8 5
82 82 82 82 82 82 82
1212121212121 2212121212121 1111111212222 2222221213131 1111112121222 2222232323131 2121212121212
** ..... * * ** * *.. **.*** * *.***.*.* ..* *** * * ** *.* .. **.*** *..... * * ** **
1223121 3121122 2112231 1122312 2312112 2231211
19 22 22 34 34 54
3 4 4 6 4 4
82 82 82 82 82 82
1111111212222 111111f122222 1212121212121 2222121212131 1111212121212 2121212121212
E
T
The 462 Scales
Notes
155
Intervals
E
T
c 82
2222232323131 1212121212121 1111111212222 2222121212131 2222221213131 1111212121212 2222232323131 2121212121212
Spelling
** .*** .*... *
1211223
67
9
*.*** .*... ** **.* ..*.**.* *** * * ** * ** *** * * * ** *** ** .*** .*... * *..... * * ** **
2112321 1232121 1123212 2121123 3212112 1211232 2321211
21 24 29 36 45 56 69
3 4 5 6 5 5
82 82 82 82 82 82 82
**.* ..*.*** **.**.* ..*.* *..... ** * * ** * * *** ** *.*** .** ... * *.* .. *.***.* ***.**.* ..*.
1232112 1212321 2123211 3211212 2112123 2321121 1121232
21 24 29 36 45 56 69
5 4 3 4 7 4 9
82 82 82 82 82 82 82
1111111212222 1212121212121 1111112121222 1111212121212 2222221213131 2121212121212 2222232323131
*.*.*.**** .. *** ..*.*.*.* * * **** * * **** * * ** ..*.*.*.** **** * * * *.. *.*.*.***
2221113 1132221 2211132 2111322 1322211 1113222 3222111
22 23 26 38 38 58 103
4 3 4 6
82 82 82 82 82 82 82
2212121212121 2222121212131 2222221213131 2222232323131 1111112222222 2323232323232 2222212121313
*.. **** .*.. * **.*.*.* .. ** *.* .. ****.*. *** * * * * *.*.* .. **** * * * * *** ****.*.*.* ..
3111222 1222311 2311122 1122231 2231112 2223111 1112223
22 23 26 38 38 58 103
82 82 82 82 82 82 82
1111121222222 1111112222222 1111212122222 2222121212131 2121212121212 2222212121313 2323232323232
*** * * * * *.*.****.* .. ** * * * ** * **** * * * * * **** **** * * * * * * * ***
1123221 2211123 1232211 2111232 3221112 1112322 2322111
32 37 37 53
78 78 78 78 78 78 78
2222121212131 2222221213131 1111112222222 2222232323131 2121212121212 2323232323232 2222212121313
**.*.* ..*.** * * **** * *** * * * * * * * ****
1223211 3211122 1122321 2321112
78 78 78 78
1111112222222 1111212122222 2222121212131 2121212121212
8
4 8
7 4
3 4
6 4 4 11 4
64 77
5 5 7 6 9·
92
6
32 37 37 53
4
5 5 5
Enigmatic
AppendixB
156
Notes
*.**** .*... * * * * * *** **** * * *
Intervals 2111223 2232111 1112232
E
64 77 92
c
T
8
5 10
Spelling
78 78 78
2222232323131 2222212121313 2323232323232
Scale class (3321111) - Triatonic
** ..... ** ** * *.** .. *** .. * ** *** ** *..*** ... ** * ** ** *** *** ** ** .. *.. **.** .. **
1312131 2131131 1311312 3113121 1213113 1131213 3121311
12 13 13 28 28 37 37
2 3 3 2 6 7 3
80 80 80 80 80 80 80
1111111212121 1111111112121 1111121212121 1111111121212 2212121212121 2222121212131 1111112121212
** .*..*** ..* ** ..***.* ..* *..*** .. *** *.. ***.* .. ** *.* ..*** ..** *** * *** *** ..*** .*..
1231131 1311231 3113112 3112311 2311311 1123113 1131123
16 17 25 32 41 41 52
4 3 3 2 3 7 8
81 81 81 81 81 81 81
1111111212121 1111121212121 1111111121212 1111112121212 1111212121212 2222121212131 2222221213131
** ** *** *..***.** ..* **.** ..** ..* *.** ..** ..** ** ..***.** .. *.. ** .. ***.* *** ** ..** .. *.*** ..** ..* ** ..** .. **.* ** ** *** .. *** ** ** *.. **.*** .. * ** *** ** *.. ** .. **.**
1313112 3112131 1213131 2131311 1311213 3131121 1121313
17 20 20 25 25 52
3 2 4 3
75 75 75 75 75 75 75
1111111212222 1111111122222 1212121212121 1111112121222 2212121212121 2121212121212 2222232323131
2113131 1313121 1312113 1131312 3121131 1211313 3131211
17 20 20 25 25 52
3 2 4
65
5
75 75 75 75 75 75 75
1212121212121 1111111212222 2212121212121 2222121212131 1111111122222 2222232323131 2121212121212
** ..*** ..*.* ** .. *.*** .. * *.*** ..*** *** * *** .. *** ..*** ..*. *.. *** .. *.** *.. *.*** ..**
1311321 1321131 2113113 1132113 1131132 3113211 3211311
16 17 25 32 41 41 52
2 3
81 81 81 81 81 81 81
1111121212121 1111111212121 2212121212121 2222121212131 2222221213131 1111112121212 1111212121212
65
5 4 9
5 3 8
5 6 7 3 4
Lydian Chromatic Hungarian Minor
Phrygian Chromati¢'
The 462 Scales
Notes
157
Intervals
E
* **** ** ** ** * ** ** * **** .. *.*.. **** .. * *** ** * * * ** * *** **** ** *
3111312 1312311 1231113 2311131 1131231 3123111 1113123
21 24 24 29 29 69
* ** **** .. * **** * * *** * ** * ** **** * ** * ** ** **** * ** .. * * ** ***
T
c
Spelling
84
3 2 6 3 5 5 10
77 77 77 77 77 77 77
1111121222222 1111112222222 2212121212121 1111212122222 2222121212131 2222212121313 2323232323232
2131113 3111321 1132131 1311132 1321311 1113213 3213111
21 24 24 29 29 69 84
5 2 4 5 3 9 6
77 77 77 77 77 77 77
2212121212121 1111121222222 2222121212131 2222221213131 1111112222222 2323232323232 2222212121313
* **** * * ** * ** ** *** * ** * ** .. **** .*.. * ** **** * * ** *** **** * **
3111231 1231311 1123131 1311123 3131112 2313111 1112313
25 28 33 40 49 73 88
3 4 5 6 5 5 10
86 86 86 86 86 86 86
1111121222222 1111112222222 2222121212131 2222221213131 2121212121212 2222212121313 2323232323232
Maximum c value
** * **** *** ** * * ** ** * ** * .. *.**** .. * * **** ** .. **** ** * * ** * ***
1321113 1131321 1313211 3211131 2111313 1113132 3132111
25 28 33 40 49 73 88
5 4 3 4 7 9 6
86 86 86 86 86 86 86
2212121212121 2222121212131 1111112222222 1111212122222 2222232323131 2323232323232 2222212121313
Maximum c value
* *** *** * *** * ** *** * *** ** ..*.. *** .* ** .*** .... * * * * *** ** *** .*** .... *
3112113 2113311 1133112 1331121 .1211331. 3311211 1121133
28 29 29 109 109
4 3 5 4 6 7 11
81 81 81 81 81 81 81
2212121222222 1212122222222 2222121212232 2121212222222 2222232323131 2222323232313 3333332324242
** ** * ** * **** ** ** * **** * ** * *** *** ** * * *.. *.. **** .*
1213311 3111213 1331112 2133111 1121331 331ll2l
32 33 41 57 57 96
4 5 5 5 7 6
82 82 82 82 82 82
1212122222222 2212121222222 2121212222222 2222212121323 2222232323131 2222323232313
44 44
AppendixB
158
Notes
Intervals
E
T
**** ** *
1112133
132
12
82
3333332324242
*** ..*.. **.* *.. **.**** .. *.**** ..*..* **** .. *.. ** ** ..*..**.** **.**** ..*.. *.. *..**.***
1133121 3121113 2111331 1113312 1331211 1211133 3312111
32 33 41 57 57 96 132
4 5 5 7 5 8
82 82 82 82 82 82 82
2222121212232 2212121222222 2222232323131 2323232323232 2121212222222 3333332324242 2222323232313
*** .. *.* .. ** *.* ..***** .. *.. ***** ..*. **** ..*.* ..* ** ..*.* ..*** ***** .. *.* .. *.. *.* ..****
1132311 2311113 3111132 1113231 1323111 1111323 3231111
36 37 37 61 61 136 136
4 5 5 7 5 12 8
85 85 85 85 85 85 85
2222122222232 2212222222222 2222221223232 2323232323232 2222212222323 3333343434242 3232323232323
***.* ..*..** *.. *****.* .. **.* ..*..*** ****.* ..*..* *.. *.. ***** *.* .. *..**** ***** * * *** .. *.. *.** *.. *.***** .. **** .. *..*.* ** ..*.. *.*** *.***** .. *.. ***** ..*..*. * * * ****
1123311 3111123 1233111 1112331 3311112 2331111 1111233
45 48
5 6 6 8 7 13
82 82 82 82 82 82 82
2222122222232 2222221223232 2222212222323 2323232323232 2222323232313 3232323232323 3333343434242
5 6 6 6 9 11 9
82 82 82 82 82 82 82
2222122222232 2212222222222 2323232323232 2222212222323 3333332324242 3333343434242 3232323232323
4 2 6 5 3 6
1212121212222 1111122222222 2222121212131 2222221213131 1111212222222 2222222131313 3323232323232 1212121212222 2222121212131
1133211 3211113 1113321 1332111 2111133 1111332 3321111
60
80 93 125 165 45 48 60
80 93 125 165
10
7
c
Spelling
Scale class (4221111)
*.*** ... *** *.*.*** ... ** *** ...***.*. **.*.***...* ** ... ***.*.* *...***.*.** ***.*.*** ...
2114112 2211411 1141122 1221141 1411221 4112211 1122114
22 26 26 31 31 86 86
10
78 78 78 78 78 78 78
*.***... ** * *** ...** **
2114121 1141212
25 25
3 5
86 86
Maximum c value
159
The 462 Scales
Notes
Intervals
E
T
* ** *** * ** *** ** ** ... ** ** * **.**.*** ... *...** ** **
2121141 12.11412 1412121 1212114 4121211
28 40 40 73 105
4 6 4 9 7
* ** ** ... ** ** ** ... *** ** ... *** ** ** ** ** ...* * ** ... *** * *...*** ** * *** ** ** ...
2121411 1214112 1411212 1212141 2141121 4112121 1121214
25 25 28 40 40 73 105
3
*** ...**.*.* *.*.**** ...* * **** ... ** **** ...** * ** ... ** * ** ** * **** *...** * ***
1141221 2211141 2111412 1114122 1412211 1221114 4122111
28 29 37 53 53
4 3
64
128
8 8
** ** ... ** * * *** ** ... * ** ... ** *** *** ** ...** * ** ... ** ** *...** *** * ** *** ** ** * ** ... ** ** **** * * ** ... **** *** * ** ... * * * ** ... *** *...**** * * **** * ** ...
1214121 2112141 1412112 1121412 2141211 4121121 1211214
28 37 37 53 53 92 92
1221411 1411122 2141112 1122141 2214111 4111221 1112214
28 29 37 53 53
*** * *** * *** * ** ** *** * * ** ...*.***.* * *** *** *** *** * * * *** **
1142112 2114211 1211421 1421121 2112114 1121142 4211211
64
128 34 38 43
5
4 6 4 5 11
5
7 5
4 5 5
7 5
6 10 4 5 5
7 3 4 12 6 4
59
5 5
70 98 134
8 10 8
c 86
Spelling
86 86 86 86
2222221213131 2222232323131 1111212222222 3323232323232 2222222131313
86 86 86 86 86 86 86
1111122222222 1212121212222 1111212222222 2222221213131 2121212121222 2222222131313 3323232323232
80 80 80 80 80 80 80
2222121212131 2222221213131 2222232323131 2323232323232 1111212222222 3323232323232 2222222131313
81 81 81 81 81 81 81
1212121212222 2222221213131 1111212222222 2222232323131 2121212121222 2222222131313 3323232323232
80 80 80 80 80 80 80
1111122222222 1111212222222 2121212121222 2222221213131 2222212121313 2222222131313 3323232323232
82 82 82 82 82 82 82
2222121212232 1212122222222 2222232323131 2121212222222 3323232323232 3333332324242 2222323232313
Maximum c value
AppendixB
160
4 6 5 7 6 6 12
1212122222222 2222121212232 2121212222222 2222232323131 2222222131313 2222323232313 3333332324242
5 4 6 7 6 9 9
85 85 85 85 85 85 85
2222121212232 2222232323131 2323232323232 3323232323232 2121212222222 3333332324242 2222323232313
85 85 85 85 85 85 85
1212122222222 2121212222222 2222212121323 2222222131313 2222232323131 2222323232313 3333332324242
Intervals
E
T
. . ... •• ••••• •••• ••• •• • •••• . . ...• •••••• •...••• ••• •• ••••• ••••••• ••• • •• • ••••• •• •••• ••• ••••••• ... •• • •• •• •••••• • • • •• ••• ••••• •• •• • •••• • •• • ••• • •••• •• •••••• • •• ••••• ••••••• •••• • •• ••••• • • •• • •••• ••• • ••• ••••••• ... •...•.•••••. •••••• • •••• •• • . . ...• •••••• ••• •• •• ••••• •• • •• •••• ••••••• •• •• ***
2112411 1124112 1241121 1211241 4112112 2411211 1121124
34 38 43 59 70 98 134
1142121 2111421 1114212 2121114 1421211 1211142 4212111
37 40 52 61
•••
•• **
•••• ••• ••••••• ... • ••••• • ** * * *** ***** ••
72
85 157
Spelling
c 82 82 82 82 82 82 82
Notes
1212411 1241112 2124111 4111212 1121241 2411121 1112124
85 157
5 6 4 5 8 5 13
1214211 1121421 1421112 2142111 2111214 4211121 1112142
41 56 56 76 89 121 121
5 6 6 6 9 7 11
85 85 85 85 85 85 85
1212122222222 2222232323131 2121212222222 2222212121323 3323232323232 2222323232313 3333332324242
1124121 2111241 1241211 1112412 4121112 1211124 2412111
41 56 56 76 89 121 121
5 6 6 8 7 11 7
85 85 85 85 85 85 85
2222121212232 2222232323131 2121212222222 2323232323232 2222222131313 3333332324242 2222323232313
1142211 1114221 2211114 2111142 1422111 1111422
50 55 62 82 95 110
6 5 6 8 7 10
82 82 82 82 82 82
2222122222232 2323232323232 3323232323232 3333332324242 2222212222323 3333343434242
37 40 52 61 72
161
The 462 Scales
Notes
* * * ****
4221111
190
10
c 82
*** * * ** ** * * *** *...***** * * * ***** **** * * * *** **** ***** * *
1122411 1224111 4111122 2411112 1112241 2241111 1111224
50 55 62 82 95 110 190
6 5 6 6 9 6 14
82 82 82 82 82 82 82
2222122222232 2222212222323 2222222132323 2222323232313 2323232323232 3232323232323 3333343434242
*** * * ** **** * * * ** * * *** *...*.***** *.***** .*... ***** * * * * * ****
1124211 1112421 1242111 4211112 2111124 1111242 2421111
54 79 79 118 118 154 154
6 7 7 8 10 12 8
80 80 80 80 80 80 80
2222122222232 2323232323232 2222212222323 2222323232313 3333332324242 3333343434242 3232323232323
29 32 37
5 4 3 4 8 9 7
73 73 73 73 73 73 73
2222121212131 2222221213131 1111122222222 1111212222222 2323232323232 3323232323232 2222222131313 1111122222222 1111212222222 2222121212131 2222221213131 2222212121313 2222222131313 3323232323232
Intervals
E
T
Spelling
3232323232323
Scale class (4311111)
*** ...*** * ** **** * * **** ** ** ... *** ** **** *** *** **** *... *** ***
1141131 1311141 3111411 1411311 1114113 1131114 4113111
** *** ... ** ** ... **** * *** ...**** *** *** * * *** ... *** *... **** ** **** ***
1311411 1411131 1141113 1131141 3114111 4111311 1113114
77
3 4 7 6 4 5
109
11
73 73 73 73 73 73 73
* ***** * *** ** ** **** ...** * ** ***** ** ** *** ***** ...** .. *... ** ****
3111141 1141311 1114131 1311114 1413111 1111413 4131111
40 41 56 65 76 121 161
4 5 6 7 6 11 9
81 81 81 81 81 81 81
2222221223232 2222122222232 2323232323232 3323232323232 2222212222323 3333343434242 3232323232323
** ... ***** *** ** ... **
1411113 1131411
40 41
6 5
81 81
2212222222222 2222122222232
44 64
77
109 29 32 37 44 64
AppendixB
162
Notes
Intervals
E
T
c
Spelling
** ** *** *... ***** * **** ** ... * *.. ** ...**** ***** ** ...
1314111 4111131 1113141 3141111 1111314
56 65 76 121 161
4 5 8 7 13
81 81 81 81 81
2222212222323 2222222132323 2323232323232 3232323232323 3333343434242
**** * ** * ****** *** * *** ***** * * ** * **** ****** * * * *****
1114311 3111114 1143111 1111431 1431111 1111143 4311111
68 73 73 113 128 208 233
6 7 7 9 8 14 11
82 82 82 82 82 82 82
2323232323232 3323232323232 2222222222333 3333343434242 3232323232323 4444443435353 3333434343424
*** * *** *... ****** **** * ** ** * **** ***** * * * * ***** ****** *
1134111 4111113 1113411 1341111 1111341 3411111 1111134
68 73 73 113 128 208 233
6 7 7 7 10 10 15
82 82 82 82 82 82 82
2222222222333 2222222232323 2323232323232 3232323232323 3333343434242 3333434343424 4444443435353
Scale class (5211111)
* **** ** *** *** * **** *** ** **** * ** *** ** *** **** .... *.... *** ***
2111511 1151121 1115112 1211151 1511211 1121115 5112111
53 56 61 88 101 173 196
5 6 7 8 7 13 10
80 80 80 80 80 80 80
2222232323232 2222222222232 2323232323232 3333332324242 2222323232323 4434343434343 3333333242424
*** .... **** ** *** ** * *** .... *** ** **** * *** *** * *.... **** ** **** ***
1151112 1211511 2115111 1511121 1121151 5111211 1112115
53 56 61 88 101 173 196
7 6 5 6 9 9 14
80 80 80 80 80 80 80
2222222222232 2222232323232 2222222222323 2222323232323 3333332324242 3333333242424 4434343434343
**** ** * *** ** ** *.***** ....* ***** ** ** ** *** ** ***** ....
1115121 1151211 2111151 1111512 1512111 1211115
64
6 7 7 9 8 12
81 81 81 81 81 81
2323232323232 2222222222232 3333332324242 3333343434242 2222323232323 4434343434343
69 85 109 124 160
The 462 Scales
163
Notes *....** ****
Intervals 5121111
** ** .... *** *** ** .... ** ** .... ***** * ** ....**** **** ** .... * *.... ***** * ***** ** ....
1215111 1121511 1511112 2151111 1112151 5111121 1111215
69 85 109 124 160 229
**** * ** *** * *** ***** * * * ****** ** * **** ****** * * * *****
1115211 1152111 1111521 2111115 1521111 1111152 5211111
92 112 157 157 197 272
*** * *** **** * ** ** * **** *....****** ***** * * ** ***** ****** *
1125111 1112511 1251111 5111112 1111251 2511111 1111125
c 81
Spelling 3333333242424
81 81 81 81 81 81 81
2222222222323 2222232323232 2222323232323 3232323232323 3333332324242 3333333242424 4434343434343
12
78 78 78 78 78 78 78
2323232323232 2222222222333 3333343434242 4434343434343 3232323232323 4444443435353 3333434343424
77 92 112 157 157 197 272
7 8 8 9 11 9 16
78 78 78 78 78 78 78
2222222222333 2323232323232 3232323232323 3333333242424 3333343434242 3333434343424 4444443435353
100 125 125 200 200 325 325
8 9 9 12
64 64 64 64 64 64 64
2323232323333 3333343434242 3232323232333 4444443435353 3333434343424 4444444353535 5545454545454
E
T
229
11
64
6 7 7 7
77
10
8 15 7 8 8 11 9 13
Scale class (6111111)
**** ..... *** ***** ..... ** *** ..... **** ****** .....* ** ..... ***** *..... ****** ******* .....
1116111 1111611 1161111 1111161 1611111 6111111 1111116
10 13
17
Some have a=2
Appendix C Some Musical Numbers
The table below summarizes some of the important musical integers encountered in our study of the combinatorics of musical objects. The chord and scale numbers all apply to the 12-tone equal-tempered scale.
3
Number of diatonic scale classes Number of note names (A-G); number of notes in a common scale; number of white keys per octave on a piano; number of scale classes(= 7-note interval sets). 9 Number of scales one note different from the Major scale. 12 Number of notes in the most common equalctempered scale. 13 Number of common musical keys (C + 1-6 flats/sharps). 21 Number of7-note diatonic scales(= 7 x 3). 23 Number of elementary 2-fold polychords. 55 Number of chords with a single "1" interval(= F9). 66 Number of7-note chords(= 7-note scale types). 77 Number of distinct interval sets(= partitions of 12). 245 Number of 7-note triatonic scales(= 7 x 35). 273 Number of notationally-distinct diatonic scales(= 13 x 21). 351 Number of distinct chord types(= N(12)-1). 462 Number of (7-note) musical scales(= 7 x 66). 2048 Number of n-note scales for all n (= 211). 4095 Number of chords without rotational isomorphism(= 212-1). 6006 Number of notationally-distinct scales (= 13 x 462). 458330 Number of non-syncopated 8-bar 1/4-note rhythmic patterns. 210066388901 Number of non-syncopated 8-bar 1/8-note rhythmic patterns.
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Index Accidental4 Adjacency analysis 47 Adjacency table 54 Beat frequency 42 Binomial coefficients 22 Binomial Span Theorem 68
Ideal scales 101 Idealness of a scale 97 Interval2 Interval set table 71 Interval sets 69 Interval simplicity 77 Inversion 7 Isomorphic 3
Catalan numbers 139 Chain 15 Chains, 2-color 17 · Chains, counting 17 Chains, m-color 18 Chord 6 Chord containment 106 Chord isomorphism 8 Chord simplicity 78 Chords and necklaces 40 Chords with no adjacencies 45 Chords with one adjacency 53 Chords, counting 39 Chords, the 351 distinct 142 Combination number 17 Configurations 23 Cycle-index polynomial26 Cyclic group 29
Maximum interval 63 Meter, metrical patterns 121 Metrical complexity 126 Metrical complexity table 129 Metrical pattern formula 125 Minimum interval formula 44 Minimum interval of a chord 41 Mono/polychromatic table 57 Monochromatic chords 55 Multinomial coefficients 22 Musical numbers 164
Delayed Fibonacci sequence 124 Diatonic spelling 115 Dorian scale 99
Necklace 15 Nonadecimal Nocturne 141 Note 1
Equal-tempered scale 3 Euclidean distance 103 Euler's phi function 30 Evenness of an interval set 96
Octave2 One-adjacency formula 51 Oulipo 106
Fibonacci sequence 51 Frequency 1 Graceful chords 72 Graceful graph 73 Guitar chords 63
K-atonic interval sets 70 Key 11 Keys, notationally-distinct 12 Keys, the 13 common 12
Partition 15 Partitions, counting 34 Pascal's triangle 19 Period formula 60 Period of a chord 58 Period table 61 Permutation group 25
166
Permutations 16 Pitch 1 Polya's theorem 26 Polychord counting 80 Polychord counting table 85 Polychords, set of 2-fold 86 Polynecklace covering problem 81 Quadratonic Qucxnibet 88 Rhythmic patterns 131 Scale 10 Scale classes 94 Scale isomorphism 11 Scale types 93 Scales, 7-note 92 Scales, counting 89 Scales, k-atonic 94 Scales, notationally-distinct 111 Scales, the 462 seven-note 151 Schizophrenic scales 105 Schizophrenic Scherzo 120 Shifted Sonata 119 Simplicity 76 Simplicity table 79 Span 63 Span formula 66 Span table 66 Spelling scales 109 Spelling table 114 Subnecklace table 108 Subnecklaces 107 Syncopation 133 Taxicab distance 103 Time signature 121 Variety of a chord 72 Variety table 75
Index
