Music Is Not a "Notational System" William E. Webster The Journal of Aesthetics and Art Criticism, Vol. 29, No. 4. (Summer, 1971), pp. 489-497. Stable URL: http://links.jstor.org/sici?sici=0021-8529%28197122%2929%3A4%3C489%3AMINA%22S%3E2.0.CO%3B2-%23 The Journal of Aesthetics and Art Criticism is currently published by The American Society for Aesthetics.
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W I L L I A M E. W E B S T E R
Music Is Not a ggNotationalSystem"
I. AUTOGRAPHIC/ALLOGRAPHIC DISTINCTION
A. Introduction
ture are examples of "autographic works of art"; music and literature are "allographic works of art."
B. Forgery book Languages of Art Goodman explicitly warns us that this is rich with ideas and provocative suggestions distinction of the autographic from the allofor the construction of a theory of musical graphic as a criterion of the forgeable and the notation. He categorizes the arts as being either "autographic" or "allographic." He unforgeable, is only preliminary to the acthen characterizes allographic arts in terms count of allographic arts in terms of notational of notational systems. In this paper I will es- schemes. (fn. p. 113) Yet, questions arise as tablish that Goodman did not sufficiently ex- to whether the forthcoming definition of plicate the notion of "forgery" to support the allographic works in terms of the concept of distinction between what he calls the "auto- "notational system" would coincide with the graphic" and "allographic" arts. I will pre- idea of allographic works that we receive sent counterexamples to show that Goodman's from his discussions of forgery. While forgery is not explicitly defined, from semantic requirements do not hold for musical Goodman's uses of the term3 we can construct notation in the way that he asserts they do; an approximate definition with which Goodfrom this it will follow that while music is a notational system, it is not the kind of nota- man might concur: P is a forgery of a work of tional system that Goodman asserts that it is. art W, if and only if P is a copy, reproduction, For Goodman no work of art is both or imitation of W and P is intended to deceive "autographic" and "allographic." He defines by falsely purporting to be genuine. A work "autographic work of art" in two ways: a of art W is a genuine work G, by Y, if and work of art is "autographic if and only if the only if W is a work of art and the historical distinction between original and forgery of it fact can be established that G was actually is significant; or better, if and only if even the created by Y, and that W = G. (p. 116) However, not all of his examples satisfy most exact duplication of it [the original work]' does not thereby count as genuine." the conditions of the autographic-allographic (p. 113)2X is an "allographic work of art," distinction. Goodman mentions at least the if and only if X is a work of art and X is not following examples of forgery: Lucretia and "a superlative imitation an "autographic work of art," or "in music . . there is no such thing as a forgery of a of it." (p. 100) This is the strongest sense of known work." (p. 112) Painting and sculp- the term forgery, i.e., the forgery is a copy or a reproduction of a specific work of art and it WILLIAM E. WEBSTER teaches philosophy at tht Phil- purports "to have the history of production requisite for the (or an) original of the work." adelphia College of Art,
NELSON GOODMAN'S
.
Music Is Not a "Notational System" i.e., stylistic imitation. The supposed lost works of various Masters which are discovered in someone's attic are often forgeries in this sense. In such a manner any Master's style could be forged. M2) The publication of the Flute Sonatas by Johann Joachim Quantz under the name of Frederick the Great could be regarded as forgery. Normally a person forges by claiming possession, or knowledge, of a work purported to have been done by an artist when it actually was not; it is a reversal of this situation when an artist gives the credit for the production of a work to another. I t was possibly out of respect and admiration for his patron and flute student that Quantz made this gracious tribute. M3) Is A's completing the unfinished work of B and purveying the work as having been completed by B a stylistic forgery? Surely it is at least that. This sense of "forgery" might lie between the sense used for (PI) and sense (P2) because the work forged is a space-time p a r t i c ~ l a rIf. ~this is true, it weakens the claim that the stronger sense of forgery is limited to (Pl), i.e., the plastic arts (or space-time particulars). If someone unadmittedly finished a CCzanne painting, it would be an instance of forgery in the same manner as it would have been had Deryck Cooke failed to admit the authorship of Mahler's Symphony no. 10. Since Goodman regards music as allographic and painting as autographic, and yet there can be in painting forgeries of exactly the sort described immediately above and there can also be in music that same kind of forgery, it follows either that the possibility of forgery does not distinguish the allographic from the autographic, or (as we have maintained at several earlier points) both terms are equivocal, or the only relevant sense of forgery is (PI), in which a space-time particular can be forged. M4) When Btla Bart6k went into the hills of Hungary, tape recorded folk songs, then published them unchanged under his own name, could he be said to have forged them? This also could be answered affirmatively or negatively: they are forgeries if he is publishing notational inscriptions of the same works the peasants are singing; they are not forgeries if the notational inscription of the published
score and the vocal sound events produced by the natives are different works. Either answer can be justified on Goodman's theory. This kind of forgery is different from the kinds considered heretofore. In this case, one is taking the credit himself for artistic creation of that which rightfully belongs to another. This may be more a case of plagiarism than forgery. M5) A composer's reproducing his own work is much more prevalent in music than in the plastic arts, i.e. (P6). This happens, on the one hand, with the admission of duplication and, on the other hand, deceptively. Bach often used identical works in different contexts or with slight changes in instrumentation, e.g., the Sinfonia for oboe and strings in Cantata no. 156 is precisely the same as the Air on a G String. This is not a case of "quoting." Composers may also publish a "new" edition of a composition, with little or no change, simply for additional royalties. These cases, if called "forgeries," could be forging in an extended sense. Thus, it is false that there are no forgeries of musical compositions. If Goodman's assertion about forgery is true, it is contrary to ordinary usage and based upon stipulation. Granted, he can hold that the only meaning of forgery that is of interest to him is the strongest sense ( P l ) : that there can be no forging of a work of music like a forgery of Lucretia. But, as was said above, this would be evidently true, because the forgery of Lucretia would be a physical object which is purported to be the physical object identical with the genuine Lucretia, whereas there is no physical object identical with any work of music. Perhaps there are deeper considerations: why can there be forgeries of Lucretia and none of the London Symphony? Goodman answers that it is because the latter is expressed in a notational system and, thus, can be said to have constitutive and contingent properties. For Lucretia, no pictorial property which it possesses can justifiably be said to be contingent in it because there is no comparable notational system with respect to which the distinction between constitutive and contingent properties can be made : (cf. p. 1 16) but for the London Symphony, its expression in a notational system is both sufficient and necessary for its identity: nothing is more
WILLIAM E.
the original than a correct copy. (p. 116) The point here is that a score which is not exactly equivalent, notationally, to the score of the London Symphony, cannot be a forgery of it since it is incorrect, and if it is correct it counts as the original. This argument of Goodman's depends upon the assumption that music really is expressed in a notational system as he defines that expression.
WEBSTER
objects which one deserves the pedigree in that all would be genuine.
D. Musical Performance
Goodman's position is that for the autographic arts there is no criterion for distinguishing constitutive properties from contingent properties; for the allographic arts there is. Both music and literature are allographic; and both are notational; it is the C. Duplication notation which provides the distinction beConsidering Goodman's "preferred" defi- tween autographic and allographic arts. His nition of autographic (cf. p. 113) in terms of point and its difficulty become clearer if we "duplication," I would argue against him as iurn to musical performance. follows: if every atom were duplicated in a Goodman claims that some arts are "onework of art, then the duplication would "count stage" (e.g., painting and writing) and others as genuine." Why is this the case? Because the are "two-stage" (e.g., music). He contrasts possibility of perceptible difference (which music to literature, "What the writer prowould be aesthetically relevant) is ruled out duces is ultimate; the text is not merely a by the exclusion of any perceptible difference means to oral readings as a score is a means at all. If this is true, it follows that painting to performances in music." (p. 114) The sug(Goodman's paradigm of the autographic gestion here is that "a score is merely a means arts) either is or could be allographic and that to performance in music." This is false for a this is a contingent matter dependent only score may be ultimate, i.e., read in a similar upon the excellence of one's replicative tech- manner to a work of literature. nology. Goodman treats performances as "instances Goodman attempts to escape the force of of a work," i.e., as comparable to a copy of a this argument by employing a technical work of literature. He says, "There may indefinition of genuine, i.e., "The only way of deed be forgeries of performances. Such forascertaining that the Lucretia before us is geries are performances that purport to be by genuine is thus to establish the historical fact a certain musician, etc.; but these, if in acthat it is the actual object made by Rem- cordance with the score, are nevertheless brandt." (p. 116) This, however, is at the cost genuine instances of the work." (fn. 113) of eliminating the distinction of meaning be- What is of primary interest to Goodman is tween the terms genuine and original. For "whether there can be forgeries of work, Goodman, "to be genuine" seems to be "to not. . . whether there can be forgeries of inhave a correct and authoritative pedigree." stances of works." (Ibid.) If he regards a However, given the conditions of reproduc- musical performance as an "instance of a tion proposed above, were the objects ran- work," analogously to a copy of a novel being domly interchanged, there would be no an "instance of a work," then this notion is physically possible way of determining to incorrect. Copies of novels and poems are which object the pedigree belongs. Thus, it tokens or instances of a work of art. Likewise, would seem to be better to regard any two copies of drama are instances of a work and works of art as genuine that satisfy the condi- co@es of scores are instances of compositional tion that it is physically impossible to deter- works of art in music. These senses seem to be mine to which of them the pedigree belongs. analogous to one another. However, it is false From which it would follow that being "au- that a performance of Hamlet by the Old Vic, or tographic" in sense (Pl) is a contingent the Eroica performed by the Philadelphia orproperty of painting, because it could happen chestra is an instance of these works of art in that there will be cases where it is physically any sense that is analogous to a copy of impossible to determine among competing Portnoy's Complaint, which is an instance of that
Music Is Not a "Notational System" work. The expression "instance of the work" cannot be univocally applied to copies of literature and to performances of drama and music because they are of different logical types. Goodman's unusual use of the term herformance may account for the following strange locution: there are "not performances that are forgeries of the London Symphony." (p. 118) This assertion seems to involve a category mistake: a performance cannot be said to be a forgery of (or denied to be a forgery of) anything except another performance; it could never be a "forgery of a compositional work." Inscriptions may be forgeries of inscriptions; performances, forgeries of performances. We can no more mix the categories of performance and composition, as Goodman does, then we can assert that a painting can be a forgery of a piece of scuplture. 11. MUSIC A S A N O T A T I O N A L SYSTEM
A. Syntactic Requirements for a Notational System In this section I will state those syntactic elements of Goodman's theory of music as a notational system. 1. CHARACTERS IN MUSICAL NOTATION. Goodman asserts, "A necessary condition for a notation, then, is character indzference among the instances of each character. Two marks are character indifferent if each is an inscription (i.e., belongs to some character) and neither one belongs to any character the other does not." (p. 132) And further, "marks correctly judged to be joint members of a character will always be true copies of one another." (p. 134) "The second requirement upon a notational scheme, then, is that the characters b e j n i t e l y dzferentiated, or articulate." ( P 135) 2. VIOLATION OF "DISJOINTNESS". There is an example which occurs in musical literature which violates the condition of disjointness (and character indifferen~e).~ Consider:
A problematic case can be found between
the fourth beat of the first measure and the first beat of the second measure. There is no assurance that these seemingly identical characters are indeed the same. The problem is created by the rule which states that the bar line dissolves all accidentals which have held through the previous measure. Assume for the present that the fourth beat of measure one is D natural. The identity of the following character will be determined bv whether the line above the note, which connects one to the other, is a "slur" or a "tie." There is no way of determining this from the score; the two score markings (tie and slur) are perceptually indistinguishable. If the line is a slur, then the second character will be different from the fourth beat D natural; if the line is a tie, the two inscriptions will be of the same character. Hence, if there are not clear harmonic indications of which alternative to choose, the exact identity of the character may be undecidable. Goodman's condition of character indifference is violated because there is no conclusive way of determining whether the inscription belongs to the character "tie" or the character "slur." I t seems to be the case that Goodman's syntactic requirements may be formulated so that they will hold for musical notation. However, given his syntactic requirements, I will show that his semantic requirements for musical notation do not hold.
B. Semantic Requirements for a Notational System 1. SEMANTIC UNAMBIGUITY. Goodman asserts: "the first semantic requirement upon notational systems is that they be unambiguous"; (p. 148) "a character is ambiguous if any inscription of it is; . . . the character is ambiguous unless all its inscriptions have the same compliance-class." (p. 147) This condition is often violated in musical scores by the << tritone." In a passage of harmonic modulation, one might have the following harmonic progression-from a V7 chord in B major (F#, A#, C#, E) to a Vi chord in F major (G, Bb, C , E). The A# (or Bb) and E would be pivotal notes, and on a keyboard instrument they would remain the same in both chords (see ex. 1 below). For the sake of simplicity the
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WILLIAM
editor notates the second chord: G, A#, C, E; (see ex. 2) however, the second note would not be read as A# by a person applying the rules of traditional harmony to determine which chord was scored because within the harmonic possibilities of classical music, a fact established by context, there is no V7 chord, C, E, G, A#. This kind of ambiguity can be highlighted more subtly if it were stated with the incription on the fourth line of the staff. The progression would be: Bb, D, F, Ab to B, D, E, G#; but written: Cb, D, Fb, Ab to keep the spelling of the D and Ab the same for ease of performance (see exs. 3 and 4, respectively). Notice, the D and Ab are tied across, thus they are supposedly the same character; however, the two seemingly identical characters have different compliance-classes.
Ex. 1
..
Ex. 2
Ex. 3-
This leads to consideration of the curious musical problem of the tritone which came to light with the rise of diatonicism. Tritone indicates the interval of an augmented fourth. (Tritone denotes three whole steps.) The interval was avoided prior to diatonic writing, but it became one of the most powerful harmonic phenomena of that music. I t is interesting to note that this interval is one half an octave, i.e., when the interval is inverted, it remains an augmented fourth (e.g., C to F# is the same size interval as F# to C). The tritone is the interval between the third and the seventh of any dominant seventh chord (e.g., in the dominant seventh V7-C, E, G, Bb, the tritone is E-Bb; in the third inversion, it would be Bb-E.) The behavior of this interval in harmonic progression violates Goodman's semantic requirements given his syntax. I n the tradition of diatonic composition this interval is considered to be active or moving toward resolution, generally to its tonic (see exs. 5 and 6). The manner in which the tritone resolves is instrumental in determining the aurally apparent width of the interval
E .
WEBSTER
itself; for instance, in ex. 3, chord "i" resolves inward to the chord Eb-G, as illustrated in ex. 5, while chord "ii" in ex. 3 resolves outward to C#-A as in ex. 6. The rules governing these resolutions is: intervals consisting of a note that is "sharp" (e.g., G#) resolve outward and those containing a note that is "flat" resolve inward. The rule seems to apply to all diatonic keys expressed by the simplest correct enharmonic pel ling.^ In the progression (not resolution) in ex. 3, resolution to the tonic of chord "i" (as in ex. 5) is delayed with chord "ii," and the necessity of resolving chord "i" is thus removed by chord "ii." The fact that the progression notated in ex. 3 can be notated as in ex. 4 without violation of any rule for notating music (and it may even be preferable for performance) violates Goodman's condition of un-
Ex. 4
Ex. 5
Ex. 6
ambiguity (i.e., it is false that every complete Ab inscription has the samecompliance-class). While the two inscriptions of Ab in ex. 4 appear to constitute the same character, and while the two D inscriptions in that example appear to consititue the same character, in both cases the compliance-classes are different. O n a non-tempered instrument this is evident and it is also clarified by the way the rules of resolution are applied;I0 other considerations urged below support the same conclusion. Exs. 5 and 6 are intended to show that the tritones (D-Ab and D-G#) are not the same aurally perceived interval, even on a tempered instrument, i.e., the interval containing the sharp note is perceptually wider (anticipating outward resolution) than the interval with the flat (anticipating inward resolution). Thus, given chord "i," the perceptual width of the tritone D-Ab is determined by the expected resolution (as in ex. 5); however, when that resolution is preempted by chord "ii," the width of the tritone perceptually expands in expectation of its resolution, as in ex. 6. This can be heard, even on the piano; and
Music Is Not a "Notational System" there would be no question of "sameness" of the interval, even the physical interval, on instruments playing the natural scale. I t follows that two characters apparently identical because of the identity of the character inscription may have different compliance-classes. 2. SEMANTIC DISJOINTNESS. Goodman asserts: "We may consider the disjointness requirement to stipulate that no two characters have any compliant in common." (p. 151) Let us postulate a compliant which is a physical sound of 440 cps. (cycles per second). What characters of musical notation will denote this sound? Within each of the twelve transpositions .for each of the seven clefs there is a character which denotes the physical sound 440 cps.; for instance, in the G clef a score for French horn in F, in order to require the sound A = 440 cps., must contain the character on the fourth &ace of the staff. Thus, there are eighty-four different characters in musical notation that may have this compliant in common, not counting enharmonic equivalences on non-tempered instruments which would at least double the number. Since a peculiarly elaborate score might use a substantial number of these characters to denote A = 440 cps., a musical notation does not have the disjointness that Goodman reauires. Also. since no one of these eighty-four particular characters is uniquely recoverable from a performance of the sound called A (440 cps.), it seems implausible that a score can be recovered as precisely as Goodman supposes, i.e., the score correctly recovered would very probably not be considered a true copyn of the original score, but merelv an acce~tablealternative. ~ o o d m a n aimits that redundancy destroys his condition of semantic disjointness: "In a redundant system, some inscription will have a compliant that also complies with a second inscription that is not a true copy of the first.'' (p. 151) He attempts to avoid the force of this counterexample to his theory with the assertion, "And of course redundancy in a system is easily removed by discarding all but one of any set of coextensive terms." (Ibid.) That the latter can be accomplished is very doubtful. 3. INTERSECTION AND TRANSPOSING INSTRUMENTS. Another argument against "semantic
-
disjointness" is based upon the intersection of different compliance-classes. Goodman says, "Any intersection of different complianceclasses defeats the primary purpose of a notational system" (p. 151), i.e., the possibility of two-way mapping-from inscription to compliance-class and from compliance-class to inscription. Consider the following: Gbb, F, E##, Gb, F#, A b, G, F$#. These are the names of eight characters listed in ascending order of pitch, having eight compliance-classes spanning a musical interval of a major second, or one whole step. These are legitimate notational characters denoting eight different pitches perceptually distinguishable on a non-tempered instrument. However, there is not one adjacent pair of these complianceclasses that does not intersect, and in all probability any adjacent four will intersect with one another, thus violating this semantic requirement. When any transposing instrument correctly reads a notational character, the compliance-class of that symbol will be different from what it would be for non-transposing instruments reading the same character; for example, when a trumpet or clarinet reads the character C, third space of the treble clef, its compliance-class is Bb, one step lower. Thus, the compliance-class of all transposing instruments is different from the complianceclass of non-transposing instruments relative to the same notational inscription. Returning to the troublesome French horn, this phenomenon is increased twelve-fold: any notational character may have any one of twelve compliance-classes according. to what the transposition is. Thus, the sound may be named and/or thought of as the pitch sounded, the inscribed character, or the charactor as read and transposed. Disjointness is violated every time a transposing instrument and a non-transposing instrument correctly read the same notational score in unison. 4. SEMANTIC DENSITY. Consideration of the nature of the classes of sounds denoted by the characters of musical notation will take us to the heart of the subject of semantic requirements. The set of sound events which make up all compliance-classes is a set of dense series: between any two pitches, there is a third; between any two durations, there is a
W I L L I A M E.
WEBSTER
third; between any two intensities, there is a MM ] = 100 years? third. Given everything we need to know to AND EQUAL 5. PITCH (AS SCALE FUNCTION) identify a character in notation, i.e., treble TEMPERAMENT. Goodman is aware of the probclef, C major, common time, the inscription A as a quarter note, etc., there is still an in- lem of enharmonic equivalences, (e.g., Db finite number of sound events which will com- and C#) relative to their compliants: "In a ply with it. This is because each dimension violin score the characters for C# and Db (pitch, duration, intensity, etc.) associated have no compliant in common." (pp. 181with the character is as indefinite and as dense 82) His account of this hen omen on is clever as is the theoretical variation of pitch. Thus, but not entirely clear, i.e., the characters are because of this density an infinite number of to be considered "vacant atomic characters" alternative sound events would comply with which combine with the instrumental instruction "to form different prime characters." any score. Granting that the "syntactic requirements" Goodman says that the condition of "sedo hold, even if the above explanation were mantic differentiation" is violated by any sufficient to account for simple pairs of system which is semantically dense. (p. 153) enharmonics, it opens a Pandora's box. In tra. Compare the following two inscriptions in a ditional diatonicism, for non-tempered inscore: "allegro" and struments there are fifteen different compliance-classes for every single character, e.g., consider the character A
Goodman armes that since there is no de" terminate relation between "allegro" and its compliance-class, the inscription is not notational. I will grant Goodman's point, then as a tonic12 (in C major) it has one complihasten to assert that the same argument ap- ance-class; as a supertonic (in Bb major) it plies to the note inscription "A." There is an has another compliance-class; similarly as a infinite number of pitches complying with mediant (Ab major), sub-dominant (G mathis character. There is no more guarantee jor), dominant (F major), as a sub-mediant (Eb major), and finally, as a leading tone that the (Db major). In addition to its functioning in the primary positions of the diatonic scale, this C also functions chromaticallv for all kevs which do not have the character C natural in their natural scale. Thus, this one character is recoverable from a performance than there has seven primary scale tone functions and is that the "allegro" is recoverable, and vice eight chromatic functions,13all of which may versa. These considerations suggest that ap- have different pitch compliance. Further pealing to the fact that "allegro" and similar compliance variety would be introduced by inscriptions in the score fail to have discrete characters arranged in modes other than the compliance-classesin order to determine that major mode. such markings are not constitutive of the The reason why we speak of only fifteen score is mistaken, because by the same reason- compliance-classes for a single character is ing it can be shown that the note inscriptions that-we tacitly assume that there is some abthemselves are not constitutive either. solute pitch which constitutes the point of deMoreover, it seems hyperbolic to say that parture for the cycle stated above, when C is "No departure from the indicated tempo dis- taken as a tonic. If we take C as tonic at qualifies a performance as an instance523.3 cps. and guarantee that all C's funchowever wretched-of the work defined by tioning as tonic will have the same cps. (or the score." (p. 185) What about a perform- cps. in the ratio of 1:2:4, etc.), then the numance at the tempo ber of compliance-classes can be limited to
.,
Muric Is Not a "Notational System" the fifteen indicated. As a matter of fact, this cannot be done: temperature, humidity, and altitude all affect pitch and these are constantly changing. Attempting to isolate compliance-classes based on pitch (as cps.) is futile. Goodman might mislead one with the statement, "After all, one hardly expects chemical purity outside the laboratory." (p. 186) The laboratory of aesthetics is the empirical world of the artist, and a theory of musical notation which is inconsistent with the empirical state of that art fails its test for chemical purity.
Phrases in [ ] are explanatory interpolations of the text. aAll page references are to Nelson Goodman, Languages of Art, (New York, 1968). a a) "The question is most strikingly illustrated by the case of a given work and a forgery or copy or reproduction of it" (p. 99). b) Immediately following, he compares the original painting Lucretia to "a superlative imitation of it" (p. 100). c) "A forgery of a work of art is an object falsely purporting to have the history of production requisite for the (or an) original of the work" (p. 122). (One might think this states the necessary and sufficient conditions, but in fact it is not sufficient, i.e., a work could have the same "history of production requisite for. ." and not be the same work.) d) "There can be no deceptive imitation, no forgery, of that work" (p. 116). e) "Where there is a theoretically decisive test for determining that an object has all the constitutive properties of the work in question without determining how or by whom the object was produced, there is no requisite history of production and hence no forgery of any given work" (p. 122).
.
'Numbers prefaced with a (P) refer to works that Goodman calls "autographic" (or space-time particulars); thwe numbers prefaced with an (M) refer to works that Goodman calls "allographic" (namely, music). "Something" should not be taken to refer only to space-time particulars here. Thus taken, the assertion is rendered false. Ideas for a work of art may be forged; an organizational scheme or plan may be forged. This is a case of what was alluded to in (5) above. This requirement is as follows: "Two marks are character indifferent if each is an inscription (i.e., belongs to some character) and neither one belongs to any character the other does not." (p. 132) Stated more accurately: in harmonic progressions from dominant to tonic function, the subdominant resolves downward to the mediant and the leading tone resolves upward to the tonic. Since the notes in the tritone are virtually enharmonically interchangeable, the harmonic function of the dominant chord will determine which note is the subdominant and which is the leading tone, thus what the correct enharmonic spelling of each is, and how the tritone should resolve (inward or outward). I mean here notation for intervals and progressions which is consistent with the clef and key for the section and with the rules for harmonic progression in traditional diatonicism. l o In example 3, chord ii is not a resolution of chord i. II "conformed copy." la Tonic is the name of the first degree of the natural scale. The names of the remaining six degrees follow in the text. l3 This number is determined by the standard fifteen major scales used in diatonic writing (the key of no sharps or flats, flat keys through seven flats, and sharp keys through seven sharps). The fact that there are enharmonic equivalences is acceptable since my argument is based on the possibility of there being as many complianceclasses for each character as there are possible scale tone and chromatic functions. Q