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University Microfilms International 300 N. Zeeb Road Ann Arbor, Ml 48106
8518911
Roush, Dean K.
ON CHORDS GENERATING SCALES AND "THREE COMPOSITIONS FOR ORCHESTRA." (ORIGINAL WORKS)
The Ohio Siate University D.M.A. 1985
University Microfilms International300 N. Zeeb Road, Ann Arbor, Ml 48106
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University Microfilms International
On Chords Generating Scales
D.M.A. Document
Presented in Partial Fulfillment of the Requirements
For the Degree Doctor of Musical Arts
in the Graduate School of The Ohio State University
By
Dean K. Roush, B.F.A., M.M.
The Ohio State University
1985
Reading Committee: Approved by
Professor Thomas Wells
Professor Gregory Proctor Advisor Professor David Butler School of Music Copyright by Dean K. Roush © 1985 V ita
January 26, 1952 ' Born: Richmond, Virginia
1973 B.F.A., Music Composition: Ohio University, Athens, Ohio
1973-1975 ...... Graduate Teaching Assistant: Bowling Green State University, Bowling Green, Ohio
1975 M.M., Music Composition: Bowling Green State University
1975-1981 ...... Instructor: College of Musical Arts, Bowling Green State University
1981-1984 ...... Graduate Teaching Associate: School of Music, The Ohio State University, Columbus, Ohio
1984-1985 ...... Graduate Research Associate: Department of Computer Science, The Ohio State University
Publications
Sonatina and Suite for harp. Published by Salvi International. Recorded on Orion Master Recordings, ORS 83458.
Fields of Study
Composition Tonal and Atonal Theory Digital Sound Synthesis Music History Performance: Harp, Organ, Piano
ii Table of Contents
Page
V i t a ...... ii
List of Tables ...... iv
List of F igures ...... v
Introduction ...... 1
Chapter
I. Definition of Basic Concepts, Operations, and Equivalence Systems . . 3
II. Pitch-CIaes Set Relations
Inclusion R elations ...... 13
Similarity Relations ...... 19
Common-Tone Relations ...... 21
III. The Generation of Scales by Configurations of Chords
Scale Types in Pair-Generation ...... 25
Multiple-Generation ...... 30
Exceptional Chord Configurations ...... 37
Voice-Leading Considerations ...... 42
IV. Examples of Set-Complexes for Composition ...... 45
List of References ...... 59
iii List of Tables
Table Page
1. SG(2) Set-Classes...... 11 2. Self-partition vectors for trichords ...... 24
3. Numbera of distinct self-partition vectors for classes of size 3-9 ...... 24
4. Self-partition vectors for trichords, modified ...... 26
5. Numbers of distinct scale classes generated by SG(2) pair-configurations .... 27
6. Ratios of distinct pair-generating classes to distinct subset classes ...... 29
7. Scales successively generated by T„-configurations of 3 - 1 1 ...... 34
8. Exceptional Tn-configurations where Ml and M2 are of the same class ...... 37
9. Exceptional Tn-configurations where CT, Ml, and M2 are of the same class . . 39
10. Configurations of 3-5 for which CT and MU are of the same class ...... 40
11. Other trichords with the same pro p erty ...... 40 12. Hexachords with this p r o p e r t y ...... 41
13. Self-configurations for which MU=2-1 ...... 43 14. Self-configurations with possible semitonal connections in contrary motion . . . 44
15. Sizes of scale-complexes for 3-11 45
16. Self-configurations where MU is an all-interval tetrach o rd ...... 51
17. Tetrachordal groups of Example E ...... 55
18. Prime forms of hexachords of Example F ...... 56
19. Vector complex of Example F ...... 57
iv List of Figures
Figure Page 1. CTF matrices for 7-35 and two forms of 3-11 . 21 2. Tn-configurations of pairs of 3-I I P ...... 33 3. Symmetrical trichords formed by complementary transpositions ...... 33 4. Generalized model of a closed cycle of configurations ...... 35 5. A closed cycle of configurations of 3 - 4 ...... 35 6. A closed cycle of 4-Z15 ...... 35 7. A corresponding closed cycle of 4 -Z 2 9 ...... 36 8. A closed cycle of triads with intermediate scales of different s iz e s ...... 36 9. A closed cycle of four configurations of 4 - 1 4 ...... 36 10. An exceptional configuration of 6 -Z 4 3 P ...... 40 11. Additional exceptional configurations of 6-Z43P ...... 41 12. Exceptional configurations of 6-34 . 44 13. Pair-configurations of 2-1 and their M 7-transform s ...... 46 14. Configurations of 2-3 with 2 - 5 ...... 47 15. Partial set of configurations of 2-1 with 2-4 and their M7-transforms ...... 47 16. Other common dyads in tetrachords pair-generated by 2-1 . . . 1 ...... 48 17. SG(1} prime forms of all-interval te tr a c h o r d s ...... 48 18. Transpositions of 01. .4.6 sharing PC 6 ...... 49 19. All-interval tetrachords sharing forms of 2-1, 2-2, 2-4, and 2-5 ...... 49 20. All-interval tetrachords sharing forms of 2-3 and 2 -6 ...... 49 21. M-configurations of all-interval tetrachords sharing trichords ...... 50 22. Pair-configurations of all-interval tetrachords generating 8-28 ...... 50 23. A complex of 3-11 and 5-32 with successions of all-interval tetrachords ...... 51 24. Hexachordal aggregate partitions of Example D ...... 52 25. Closed cycle of configurations of 3 -4 ...... 52 26. Transitions of Example D ...... 53 27. Set-complex of Example D ...... 53 28. Alternate trichordal pair-generators of hexachords D, B, and E ...... 54 29. An alternate combinatorial cycle of Example D ...... 54 30. An alternate trichordal generation of hexachord B ...... 54 31. Set-complex of Example E ...... 56 32. Examples of pairings of Exampte F sets with maximum common tones ...... 57 33. Generation of hexachord O as a succession ...... 58 34. Generation of hexachord P as a succession ...... 58
v Introduction
Harmonic coherence in composition is often achieved by the recurrence of pitch col lections, or chords, whose content with respect to some measure or whose one or more particular patterns of simultaneity or succession function as harmonic references. Scales are the sets of pitches, of local or global significance, resulting from two or more distinct instances of chords regarded as equivalent, and may be viewed as fields of available pitches within which harmonic activity takes place, perhaps, though not necessarily, serving as har monic references themselves at other hierarchical levels of an extended pitch-compositional structure.
Several equivalence relations are imposed on sets of pitches; most far-reaching is the grouping of all pitches into twelve pitch-classes, each representing any or all of a number of octave-related pitches, along with the further generalization that any set of pitch-classes represents any or all of its ordinal permutations. Though this notion of unordered pitch-class set has wide enough currency in cwentieth-century music theory not to need justification for its use, it bears mention that since pitches cannot occur in music without a specific register or ordering, the concern here is with whatever properties of pitch collections are independent of their registral and permutational attributes, whose perceptual immediacy, incidentally, suggests that the construct may be more appropriate for compositional than analytical application: the creation of registral and permutational context for pitch-class sets is constructive, while the abstraction from pitches in a composition to unordered pitch-class sets can be problematic.1 Equivalence among unordered pitch-class sets is defined as relation under one or more of the following operations: transposition, preserving directed pitch-class interval content, inversion, preserving at least undirected pitch-class interval content, and certain multiplicative operations, preserving structural characteristics but not necessarily interval content. Nonequivalent pitch-class sets may be considered more or less similar to each other on the basis of mutually shared subsets or similar combinational properties.
Central to this study, as in other pitch-class set theory, is the concept of inclusion, either literal, or abstract with respect to some equivalence relation, but here the concern goes beyond simply inclusion to the specific manner in which pitch-class sets contain their subsets or are contained by their supersets, with particular emphasis on the multiple inclusion, in scales, of chords deemed equivalent or similar in some context, and on the various ways that chords can be combined to generate such scales. Chords and scales are not further restricted as to their other properties, but some informal treatment is given to contrapuntal possibilities between chords as they generate scales. Several means of scale generation are considered, combining two or more chords with or without pitch-class intersection. In some cases the resulting scales are used to generate yet larger scales, or a particular scale is generated in several different ways. Throughout, more attention is given to chords and scales which are exceptional in some way, with less specific treatment of those lying between extremes with respect to a particular property.
1 See particularly Browne [1974]. 2
In large part this investigation is a generalization of the generation of the diatonic scale by particular configurations of triads in a contrapuntal relation, and as such it is a no more complete description of any harmonic system than is the triadic ally-generated diatonic scale of classical tonality or of any other system employing it. Furthermore, no analytical validity is claimed for these procedures, though there may be some; they are intended to be suggestive of compositionally useful set-complexes, some examples of which are offered in the concluding chapter. C h a p te r I
Definition of Basic Concepts, Operations, and Equivalence Systems
Pitch —Available pitches are those of the tempered chromatic scale, with twelve distinct pitches per octave, uniformly spaced with respect to interval, or ratios of their fundamental frequencies. The unit of distance is the semitone, and enharmonic equivalence is assumed in the naming and notation of pitches.
Pitch-class (PC) —A pitch-class is a collection of pitches related by octave equivalence, such that members of each class differ by integral multiples of twelve semitones. Whether or how octave equivalence is asserted compositionally 13 not an issue. There are twelve PCs and any pitch belongs to just one PC.
Integer model of pitch-clas3 —Since pitches are equally spaced, integers provide a suit able model for dealing with pitch structures, and thus the integers 0... 11, modulo 12, for pitch-classes. Results of all arithmetic operations on these integers are to be converted to residue classes mod. 12 such that the norma/ form or representative of a PC is always its smallest nonnegative member. The convention adhered to here is that the PC containing any Cl] (or Bit, Dlt, etc.) is arbitrarily denoted by the integer 0. Single-character abbrevi ations for PCs 10 and 11 are T and E, respectively. The next “higher” pitch-class from PC n is PC (n + l) since each of its members is a semitone higher than some member of PC n, but since PCs are registfally'indeterminate there is no sense in which the compositional instance of the latter is presumed to be a higher pitch than that of the former.
Directed PC interval or interval —When the ordering of two PCs is regarded as signifi cant, the interval from PC a to PC 6 may be taken as (6 — a), thus always in an “upward” direction, yielding an interval from 0 ... 11 inclusive.
Complementary intervals —Any interval n and (12 —n) are complementary with respect to the pitch-class octave, thus the interval (6—a) above is complementary to (a — b). Intervals 0 and 0 are their own complements, and all other intervals have distinct complements.
Undirected PC interval or interval-class (IC) —The systems of primary concern here are combinational, meaning that a PC collection’s content alone is regarded as significant.' In such systems there are necessarily two (complementary) intervals between two PCs, so the smaller of the two is taken as the normal form or representative of an interval-class containing both. Thus ICs range from 0 ... 6 inclusive, seven in all.
PCset or set—An unordered collection of distinct PCs is called a PCset. There are 2t2 or 4096 PCsets.
- Babbitt [i960).
3 4
Size or cardinality —The size or cardinality of a PCset called A is its number of mem bers, abbreviated # A. Generic names for sets of size 0 ... 12, respectively, are null set, monad, dyad, trichord, tetrachord, pentachord, hexachord, heptachord, octachord, nona- chord, decachord, undecachord, and dodecachord or aggregate.
Notations for PCsets —In integer notation, integers representing PCs of a set are strung together, as 30ET47 (“Es-C-H-B-E-G”) By convention, the integers are placed in ascending numerical order, but not necessarily beginning with the smallest integer, thus, for example,
30ET47 -+ 7TE034.
An interval series (IS) is the sequence of directed intervals between successive PCs of an integer notation, including between the last and first, separated by hyphens:3
7TE034 -* 3-1-1-3-1-3.
The number of successive intervals is equal to the size of the PCset, and their sum is always 12. A quasi-binary array can be constructed from an interval series by replacing each interval n with a 1 followed by (n — 1) zeros:4
3-1-1-3-1-3 -* 100111001100.
In a truly binary notation proposed by Starr5, each of the 4096 PCsets is represented by a unique twelve-digit binary integer in which for each PC n of a set the digit or bit corresponding to 2n is “flipped on:”
7TE034 — 110010011001.
The system of notation for PCsets introduced here is a special, form of integer notation in corporating features of the other notations, thus simplifying conversion from one to another: each integer n of an integer notation is placed in the (n + l)th position of a twelve-element array, the unoccupied positions filled with dots, e.g., for the same set,
0..34..7..TE
With respect to the position of the elements, this amounts to a reversal of the binary integer notation, every combination of integers forming a unique visual pattern. But unlike the binary integer, the array is understood to be circular and may begin with any position, e-g-. 7 ..TE0..34.. Occasionally, “trailing” dots may be omitted:
7..TE or 0. .34
Normal form —The normal form of a PCset in integer or interval notation is the one circular permutation or rotation determined by the following conditions, steps after the first to be invoked if the current step fails to produce a unique rotation:
3 Chrism an [1971]. 4 Ibid. 5 Starr [1978]. 5
1) within the smallest span, i.e., with the largest successive (directed) interval from the last to the first member 2) most closely packed to the left, i.e., a) with the smallest successive interval between the first and second member b) if a) yields more than one rotation, with the smallest successive interval between the second and third member, and so forth 3) with the smallest initial PC number. For example, for the set 0347TE, after step 1 remain the rotations
7TE034 3-1-1-3-1-3 TE0347 1-1-3-1-3-3 347TE0 1-3-3-1-1-3.
Step 2a eliminates the first, and 2b the third, and Step 3 does not apply in this case. Since the procedure is expressed mostly in terms of successive intervals, it is obviously easiest in interval notation. These rules do not apply at all to Starr’s position-dependent binary integer, but in the modification to be used here the normal form of thin example can be expressed: TEO..34..7..
Inclusion —If every PC of set A is also a member of set B, then B is said to literally include or contain A, and A is a literal subset of B , and B is a literal superset of A . If there is some relation X which maps each member of A into a member of B, then B is said to abstractly include or contain A in a system which admits X as an equivalence relation.
Complementary PCsets —The literal complement of PCset A, taken with respect to the aggregate, is A', containing all and only the PCs which are not members of A. If A is of size n, then A' is of size (12 — n). The abstract complement of A is B if there is some equivalence relation X in the system which maps the PC content of B into A'. Unless made explicit to the contrary, both inclusion and complementation are assumed to be abstract. Though complementary sets have many common properties, complementation itself is not considered an equivalence relation here.
Union —The union S of two sets A and B is the set of all distinct members of A and B, and A and B are said to generate S. In such cases A and B are arbitrarily called chords and S a scale.
Intersection —The intersection of sets A and B is the set C of PCs contained in both A and B. A member of C will be called a common tone with respect to A and B, and the size of C will be called the common-tone index fCTIJ of A and B. If A and B are both of size n, the common-tone index can range from 0 ... n for n < 6, and from (2n — 12)... n for n > 6. This means, for example, that two heptachords cannot have a CTI less than 2, since the maximum number of non-common tones is 5, the size of the complement of either.
Numbers of subsets and sizes —A set of size n has 2n distinct subsets, including the null set and the set itself as (non-proper) subsets. A set of size n contains x distinct subsets of size m where
_ nl or n(n - l)(n - 2) ... (n - m +1) m!(n — m)! ml 6
Values of x are highest at m = ~ (or ^ — 1 and ^ +1 where n is odd), and decrease sym metrically for higher or lower values of m, with values of x shared by m = (^ - 1) and (j + 1), — 2) and + 2 ) , . . and (n — 1) and 1.
Operations on pitch-classes (PC-operationsJ —There are 48 arithmetic operations which produce one-to-one mappings of PCs, i.e., mapping the aggregate into itself, thus preserving distinctions among PCs.
PC Transposition (Tn)—The transposition by an interval of n semitones (thus n = 0...11) of PCset A (Tn(A) or An) is the substitution, for each PC m of A, of the PC (m + n). Tn preserves interval-class and directed intervals, so the interval series remains unchanged, and PCs related by T„ have a constant difference of n. To is an identity operation. The inverse of Tn, i.e., the operation that “undoes” the effect of Tn, is T_n, transposition by the complementary interval. To and T q are their own inverse operations.
PC Inversion (I) —The inversion of A (1(A)) is the substitution, for each PC m of A, of the PC (—m). I preserves IC but reverses directed intervals, so the interval series is reversed, to within rotation. When combined with T„, I is commutative only with To, but no confusion arises if the two operations are applied in a consistent order, which by convention is to transpose last. Still, there are advantages to defining inversion in terms of the compound operation transposed inversion (TnI), the substitution, for each PC m of A, of the PC (n — m). T„I for any n is its own inverse operation, and pairs of PCs related by TnI have a constant sum of n, called the l’ntiersuma/ index. A pair of PCs so related has a dual axis of symmetry of PCs j and ^ + 6, where if the index n is odd, each of the two axes lies between adjacent PCs, and may be taken to comprise both.
PC Multiplicative operations or M-operations —M s is the substitution, for each PC m of A, of the PC (m X n). These operations preserve ratios of pitch-intervals but not necessarily even distance of PC-intervals.° Four values of n preserve the distinctness of PCs; each is its own inverse operation, and any pair is commutative:
Ml is the identity operation, equivalent to To.
M il is equivalent to I, since m x 11= — m .
M5, known as the “cycle-of-fourths transform,” maps the cycle of semitones and the cycle of fourths into each other:
m 0123456789TE 5m 05T3816E49 2 7
M7, known as the “cycle-of-fifths transform,” is equivalent to IM5:
m 0123456789TE 7m 07294E6183T 5
No other values of n for Mn produce one-tc-one mappings: M2 or M10, M3 or M9, M4 or M8, M6, and MO map the aggregate into, respectively, a whole-tone scale, a diminished seventh chord, an augmented triad, a tritone, and one PC. Although M5 and M7 produce
G These operations are discussed by Howe [1965], Winham [1970], Starr [1978], Rahn [ 1980], and Morris [1982a] and [l982b[. 7 distinct results with respect to interval, they operate identically on interval-class, exchanging 10s 1 and 5 and preserving all other ICs. Since IMS is identical to M7, the operations to be considered, in addition to T„ and TnI, can be limited to TnM5 and T„IM5. Since neither M5 nor IM5 commutes with T„, it is necessary to state that the inverse operation of T„M5 is T7„M5, and that of T„IM5, Ts„IM5. EVom this it follows that, of these 24 M-operations, only ten are their own inverse operations: T„M5 for n = 0,2,4,6,8, or 10, and T„.IM5 for n = 0,3,6, or 9. These are the only ones which, like To, To, or T„I for any n, either map a PC into itself or produce cycles comprising pairs of PCs such that two successive applications of the operation return the original PC. Other values of n for T„M5 or T„IM5 produce longer cycles.7
PCset equivalence class or set-class (SC) or simply class or type —The domain of PCsets is usually partitioned into nonintersecting classes whose members are systematically defined as equivalent. For example, sets of the same size are grouped on this basis into cardinality types . Additionally, an equivalence class of PCsets containing A may contain all and only PCsets derivable from A by one or more canom'cal transformations, i.e., operations admitted to a given system as defining equivalence among PCsets.8 The representative or prime form of a set-class is its member which exhibits the “best” normal form according to the above rules, considering all applicable canonical transformations.
Set-group (SG) equivalence systems —Set-group systems are defined by the operations they admit as canonical transformations. The three principal SG systems in use are denoted by SG(1), SG{2), and SG(3),° each subsuming the previous. SG(l) is the finest partition, i.e., has the greatest number of SCs, while SG(3) has the smallest number of SCs, each in general relating a greater number of PCsets and up to four SG(l) classes.
SG(1) includes T„, i.e., the 12 operations preserving interval. Strictly speaking, SG(l) does not include the concept of interval-class.
SG(2) includes T„ and TnI, i.e., the 24 operations preserving IC.
SG(3) includes Tn, T„I, TnM5, and TnIM5, i.e., all 48 PC-operations. In SG(3), SG(2) ICs 1 and 5 can be considered as one, reducing the number of ICs by one.
Alt three SGs will be invoked from time to time, according to whether the properties under consideration are shared by sets of a class in the particular SG. SG(2) is our primary system, but distinctions made in SG(1) not preserved in SG(2) are crucial when dealing, even in unordered sets of registrally indeterminate PCs, with the structure of sets of size 3 or greater, since the inversion of sets which cannot be mapped into themselves under TnI reverses their entire structure with respect to “direction” in the modular pitch-class space. On the other hand, since there is a structural equivalence between M-related sets, when certain combinational properties are shared by two distinct but M-related SG(2) classes, such as the number of distinct SG(2) subset types where the specific content of those types is not at issue, there is no reason not to consider those properties in terms of a single SG(3) class.
Sym m etry—If some canonical transformation X maps PCset A into itself, i.e., preserv ing its PC content, then A is symmetrical with respect to X, or X-symmetrical.
7 See Starr [1978] for a discussion of these cycles. 8 Lew in [1977|. 0 These labels are borrowed from Morris [I982a|. 8
Transpositional symmetry (Tn-symmetry or simply T'Symmetry)—Since every set is To-symmetrical, normally this term will be used in connection with sets symmetrical under Tn for one or more nonzero n. In this case a set’s interval series contains one or more repeating segments; multiple subscripts to a set's notation indicate this kind of symmetry. For example,
(0 1 ----- 6 7 ____ )2/8 -* ..2 3 ------89.. or (0.. .4.. .8.. .)1 /5 /!). 1.. .5. . .9..
Inversional symmetry (I-symmetry) and transposed-inversional symmetry (TnI-sym- metry)—A set is inversionally symmetrical to within Tn if its interval series is equivalent to its own retrograde at some rotation. Transposed-inversional symmetry is symmetry under TnI for one or more values of n. If the integer notation of an inversionally symmetrical set is rotated so that its interval series is in palindromic form, then the specific operation that maps the set into itself is TnI where n is the sum of the first and last PC. In some cases there are several such rotations. When the size of the set is an even number, to make the IS palindromic one of its members must be dropped (when the set contains zero or one of its axes of symmetry) or duplicated (when the set contains both its axes).
M -symmetry—M-symmetry to within a single SG(2) class obtains if a set's PC content can be mapped into itself under T„M5 or TnIM5 for one or more n. Specifically, with only several exceptions, an M-symmetrical set can be mapped into itself only under one or more of the M-operations that is its own inverse, i.e., T„M5 where n is even, or T„IM5 where n is an integral multiple of 3, both cases including n = O.10 For M-symmetrical sets with inversionally distinct forms (i.e., SG(l)-nonequivalent), some are T„M5-symmetrical and are mapped into a T„I-related form by T„IM5, while others are TnIM5-symmetrical and are mapped into a TnI-related form by TnM5; it is important to note that, because of the heirarchical relation of the three SG systems such that inversionally related SG(l) classes are already merged into a single SG(2) class, SG(3) as defined here does not preserve this distinction.
Degree o f symmetry f* S )—The degree of symmetry of a set is the total number of canonical transformations of the SG system that map the set’s PC content into itself.11 In SG(2) this number may be broken down as degree of transpositional symmetry (°TnS) and degree of inversional symmetry (°IS, always either 0 or 1) or transposed-inversional symmetry (°TrtIS), where, since if inversional symmetry obtains it pairs one T„-symmetry with one T„I-symmetry,
°S = °T„S x (°IS+ 1) and °S = °T„S + °T„IS.
The number of members of a particular SG equivalence class can be found by dividing the total number of canonical transformations of the SG system (e.g., 24 in SG(2)) by the degree of symmetry of the class. °S can also be used to calculate the number of members of each SC in an undecachord: in SG(2), where n is the size of the sets in the class,
24 - (2 X n)
10 Besides the aggregate itself, certain other T-symmetrical sets are the exceptions, for some values of n: for TnM5, 4-9 and 4-28 and their complements, and for TnIM5, any union of one or more augmented triads, i.e., 3-12, 6-20, 6-35, and 9-12. (These names are explained in connection with Table 1 below.) 11 Rahn [1980]. 9
In either case, if the °S of the subset class is 1, there are equal numbers of members of I-related SG(l) classes.
Interval vector (V) —The construct widely known as interval vector, a term retained here, is more accurately an interval-class vector, displaying a set’s multiplicity of occurrence of each nonzero IC from 1...6, in order.13 This is equivalent, in SG(2), to a display of multiplicity of dyadic chord types.
Z-relation —The Z-relation13 in general holds between two sets which share an interval vector but which are not members of the same class in a particular SG system. For present purposes, only the SG(2) Z-relation is relevant. Specifically, two sets of identical IC content are Z-related if they are not T„- or T„I-related, and classes so related will be called Z- pairings. Although the relation in general is independent of PC-operations, about one-third of SG(2) Z-pairings are M-related, i.e., are members of the same SG(3) class. In fact, all but one nonhexachordal Z-relation is resolved by SG(3): 5-Z12 and 5-Z36 and their complements. This is also the only Z-pairing whose members differ in degree of (inversional) symmetry, and 5-Z12 is further distinguished as the only SG(2) class of a size smaller than 6 not abstractly contained by its complement.
Table of SG(S) Set-Classes —Table 1, listing SG(2) equivalence classes, is a modifica tion of that of Forte [1973a], whose naming conventions are retained. In addition to its prime form representative, a set-class carries an arbitrary name comprising a cardinal and ordinal component separated by a hyphen. Complementary SCs are displayed in horizontal alignment and, except hexachords, have the same ordinal number. Ordinality is determined by treating the interval vector as a number and listing in (descending) order, as this is the only method giving complements the same ordinal number. This rule has been followed in this listing for the ordering of Z-related sets as well; the inconsistency in ordinal numbers for cardinality types 4, 5, 7, and 8 is due to Forte's placement of some of these at the ends of the lists after he stopped considering IC-content identity an equivalence relation. A key to the entries of the table follows:
P SG(2) prime form.
V Interval vector. Vectors shared by Z-related sets, including complementary hexachords, are shown only once.
I Interval series of the prime form, given for classes of size 2-6 only.
S Degree of symmetry. The single entry applies to complementary classes, including hexa chords. Where not given, °S = 1. With the sole exception of 6-30, transpositionally but not inversionally symmetrical, a value greater than 1 indicates inversional symmetry, and 4 or greater both inversional and transpositional symmetry.
M Again, the single entry applies to complementary classes, except hexachords. Given is the ordinal number of the M-related SG(2) class of M-nonsymmetrical classes.
Index Since the order of SCs in the main table is determined by IC content, and since inver sions of prime forms are not listed, it is sometimes difficult to locate a prime form on
12 Even this description is slightly confusing, since each interval class stands for the two complementary directed intervals between a distinct pair of PCs, and, since one is self- complementary, the six ICs of the interval vector stand for eleven directed intervals. 13 Forte [1964],[1973a]. 10
the basis of PC content alone, and most particularly when one is dealing with SG(l) equivalence. The index to the left of the main table gives ordinal numbers for all rep resentations of SCs of sizes 3-6 which begin on PC 0 and occur within the smallest span, listed in increasing numerical order as though each string of PCs were a number, ignoring the dots. Thus one need only rotate a given PCset so that its largest succes sive interval is rightmost and transpose it to begin on PC 0, dispensing with the other rules for normal form. Listed in general is the one best normal form of an inversionally symmetrical class, and one each for prime (P) and inverted (I) forms of an inversionally nonsymmetrical class, but there are several cases for which there are other members of a class within the smallest span beginning on PC 0—the condition for this is that the largest number of the interval series occurs twice or more, for transposition ally nonsymmetrical classes. For these cases the index indicates rotation (R) following the ordinal number.
Naming conventions for 3G(1) and SG(S) SCs —SG(1) SCs without inversional symme try are denoted by their SG(2) names followed by P or I: for example, 4-Z15P and 4-Z15I are distinct SG(1) classes. Inversionally symmetrical SCs have the same name in both systems: for example, 4-1 or 4-23. SG(3) SCs combining two SG(2) SCs are given a compound SG(2) ordinal number: for example, 4-1/23 cr 4-Z15/Z29. hO, _ i # t H h i ,B O h H I . J, . .
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O00OOQ0OOO009OO OOPoaOQDOOQOOOQOOOqOOOQOOOQOOOOoSSeQOOOOOOQOOOOOOOO Chapter II Pitch-Class Set Relations Inclusion Relations A fundamental premise of this study is that a set’s structure is characterized completely and uniquely by nothing less than its total subset content and that its combinational possibilities as a generating set, though somewhat dependent upon its structure, are similarly character ized by the set’s total superset “content.” Inclusion, either literal or abstract with respect to some canonical transformation, is the principal criterion for relations among nonequivalent sets—this is based on the notion that we hear a set of pitches in at least some contexts as the combination of all its structured subsets, and not, for example, solely on the basis of its interval or interval-class content. The distinction is particularly germane to Z-related pairs of sets, which have identical dyadic content but whose constituent dyads are structured differently with respect to their transpositions relative to one another, and it turns out that no two Z-related (or any other) SG(2) types share even their trichordal content.14 There are useful correlations between various kinds of symmetry and a set’s subset content. The transposition T„ of a set transposes its subsets by n, so a transpositionally symmetrical set must contain each of its SG(1) subsets a number of times equal to or integral multiples of its degree of transpositional symmetry divided by the degree of transpositional symmetry of the particular subset, where, if the latter equals 1, a number of groups of subsets of that type is present in which the members of each group are related at the same Tn or T„’s that map the set into itself, or where, if the degree of transpositional symmetry of the subset is greater than 1, this subset has the same degree and levels of transpositional symmetry as the set itself. Inverting a set inverts all its subsets, so subsets of inversionally symmetrical sets are either inversionally symmetrical themselves (to within T„) or are contained as pairs of subsets which are mapped into each other at the same index n of TnI that maps the set into itself. It follows that the union of any set and any member of its inversionally related SG(1) class is inversionally symmetrical. Any M-transform of a set is also the same M-transform of its subsets, so M-symmetrical subsets are mapped into T„- or T„I-related forms of themselves, i.e., members of the same SG(2) class, while M-nonsymmetrical subsets are mapped into members of their M-related SG(2) class, the multiplicities of these two classes exchanged. Within this, if two subsets of the same SG(2) class in the original set are related by, for example, Tn, then in the ToM5-transform of the set the corresponding subsets are related by Tmsh< It follows that subsets of M-symmetrical sets are either M-symmetrical themselves or are contained as pairs of subsets related under the same M-operation(s) mapping the set into itself; this does not mean, however, that the 14 Only one SG(2) Z-pair of size < 6 shares trichordal type content, but not the multiplicity of each type. Many other non-Z-pairs of sets share trichordal type content in this way, but of course not dyadic content. 13 14 union of a set with any member of its M-related SG(2) class is M-symmetrical, since in general only ten of the M-operations apply to M-symmetry, as mentioned above, but it is true that the union of any two sets related by one of these ten16 is M-symmetrical. Tn general, the higher a set’s degree of symmetry, the fewer its type-distinct subsets and supersets. In addition, regardless of a set’s overall symmetrical properties, its total subset and superset content with respect to distinct types is partly dependent upon the symmetrical properties of its individual subsets, with the total number of subset and superset types reduced by the presence of symmetrical subsets. The relations developed in the remainder of thi3 section deal more specifically with the multiplicity of subset and superset types. As a generalization of Forte’s interval vector, Lewin [1977] develops the concept of embedding number , a measure of subset content, in which, for PCsets A and B, EMB(A,B) is the number of distinct forms of A, i.e., dis tinct members of A ’s SC, which are subsets of B, or, in other words, the multiplicity of occurrence of type A in set B. A and B may both represent SCs, but still in the sense that EMB counts the number of members of A ’s SC in any one member of B ’s SC, In a later article,10 Lewin proposes a related covering number, a measure of superset content, in which COV(A,B) is the number of members of SC B containing PCset A (or a particular member of A where A is understood as a SC). For example, in classical tonality, it may be important to know how many different diatonic scales could contain a given triad. Also compositionally useful, though the idea will not be pursued until later, is Lewin’s sandwich ing number. SNDW(A,B,C) is the number of forms of class B literally containing A and literally contained in G, thus dependent upon the specific structure of C with respect to its subsets of types A and B. The following examples are borrowed from Lewin [1979|: 1) EMB(2-4,3-12) = 3, since a given augmented triad contains three forms of major third. 2) COV(2-4,3-12) = 1, since a given major third is contained by only one form of aug mented triad. 3) SNDW(C major triad, major seventh chord, C major scale) = 1, but SNDW{G major triad, major seventh chord, C major scale) = 0. If B includes A, its literal complement B ' is included by A', and since this can of course be extended to canonical transformations, and thus to abstract complementation, then the following relation holds:17 COV(A.B) = EMB(B', A'). For example, 1) COV(9-12,10-4) = EMB(2-4,3-12) = 3, and 2) EMB(9-12,10-4) = COV(2-4,3-12) = 1. The extension of this is that the total subset content of A is completely equivalent to the total superset content of A’, and vice versa, with respect to the complementary set-classes contained. This leads to observations such as the following: 16 TnM5 for even n or TnIM5 for n = 0,3 ,6, or 9. 10 Lewin [1979]; the line of inquiry in both articles is probabilistic. 17 Ibid. 15 1) 10-1 includes only the five nonachordal types whose trichordal complements include 2-1, and trichordal types (all twelve, in this case) whose nonachordal complements include 2- 1, and so forth. 2) Since all pentachordal types contain at least one instance of the dyadic type 2-4,18 then every heptachordal type is contained by 10-4. 3) Since every heptachordal type contains at least one instance of every dyadic type, then every pentachordal type is contained in every'decachordal type. 4) Since 4-Z15 and 4-Z29 are the only tetrachordal types containing every dyadic type, their complements are the only octachordal types which have every decachordal type as supersets. 5) The subset types of any particular hexachord are complementary to the superset types of its complementary hexachord, and vice versa; if the hexachord is self-complementary, then all its own supersets are complementary to its own subsets, but if the hexachord is one of a Z-pair, then they are so only for the subset types shared by the pair. In general, this relation between complements and inclusion is intuitively straightforward in that if a set A has a particular superset type B, then some member of B ’s type is usually a subset of A', and this is partly the basis of Forte’s theory of set-complexes.10 But it is crucial to note that, following from observation 5} above, this cannot be the case when B is any hexachord of a Z-pair.30 Thus Forte’s emphasis on complementation is directly at odds with the very inclusion relation which is ostensibly the basis of his system, creating a hexachordal bias of debatable musical significance.21 There is a direct relation between EMB(A,B) and COV(A,B), not given explicitly by Lewin: COV(A,B) = EMB|A:S% , °5(A1 Replacing COV(A,B) above with the equivalent EMB(B',A'), the same formula applies to the relation of embedding numbers of complementary pairs of sets. This means, for example, to extend observation 3) above, that a list of the multiplicity, for all decachordal types, of a particular pentachordal subset type, reads the same as the interval vector (dyadic subsets) of the complementary heptachordal type, after adjustments are made for degrees of symmetry: for example, 10-1 through 10-6 contain 5-35 respectively 2, 5, 4, 3, 6, and 2 times, and the interval vector of 7-35 is 254361. In applying the above formula, if any of the sets involved has a degree of symmetry of 1, the partition between SG(2) P and I forms is preserved, but P and I may be exchanged due to SG(2) normal order conventions: for example, EMB(4-12,7-31P) = 4 (at P1/I3 In Ig), therefore EMB(5-3I,8-12P) = 4 (but at Ic/Po Ps Po). 18 Morris [1979] otfers an intuitive proof of this: since the four augmented triad trichords of 3-12 partition (without intersection) the aggregate, it is impossible to select five distinct PCs without choosing at least two PCs from one of them. The only dyadic type contained in 3-12 is 2-4; therefore every pentachord contains at least one instance of 2-4. 10 Forte [1964|,[1973a]. 20 The unique extreme case is 5-Z12, with no hexachordal superset class which is a subset class of its complement. 21 See, in this connection, Regener [1975], 208-210. 16 The relation between EMB(A,B) and EMB(A,B') can be demonstrated only for the case of #A = 2. This generalization of Babbitt's "hexachord theorem” is well known, and can be expressed in our terms as e m b (a , b ') . EMB(A'Bi : ,y B,-*tB| a(A J for # A = 2 and where °S(A) = 4 for the tritone and 2 for all other type3. This works because for each of the 12 (or 6) dyads of each type, each of its two constituent PCs can participate in a total of only two (or one) members of the type. With respect to the partitioning of the aggregate by B and B' (literal complementation is meant here), the two PCs of each dyad of the type must lie either both within B, both within B', or one each in B and B ', and if the dyad 0 . .3, for example, is in B, then 3. .6, the other dyad of the same class involving PC 3, cannot be in B ', so must lie either also in B or “between” members of B and B'. Further, if 3. .6 is in B, then 6 . .9 cannot be in B', and so forth, and there is therefore always an equal number of PCs of each set paired “across” the partition for each dyad type. Because of this one-to-one correspondence, the difference between EMB(A,B) and EMB(A,B') for #A = 2 is directly related to the difference in the sizes of B and B'.22 Unfortunately, or perhaps fortunately, since much of the diversity of inclusion patterns in the system is due to just this, the relation between EMB(A,B) and EMB(A,B') for #A > 2 is far more complex, such that if B embeds A, there is no guarantee, even if # B ' > #B and particularly if #B ' = #B, that B ' embeds any member of A ’s type at all, let alone with related multiplicity, and it is primarily for this reason that complementation is not considered here as an equivalence relation, despite the many other properties shared by complementary types. As an illustration of the independence of EMB(A,B) and EMB(A,B') for #A > 2, consider the simplest trichordal case of EMB(3-12,B) = 1 for, say, # B = 5. This would at first appear to be a simpler case than for dyads, since each PC can participate in only one of the four members of 3-12. But we can determine EMB(3-12,B') only by considering whether or not IC 4 occurs between the two PCs of B which are not members of the instance of 3-12, and this information is not given by EMB(3-12,B). The situation with other trichords is even more complex. Assume now that EMB(3-2,B) = 2 for # B = 5. Each PC of each of the two forms of 3-2 in B can be involved in a total of six of the 24 members of the type, none of the other five of which could therefore be in B ', but any of which could be “split between” B and B' in one of two ways, or be the other form of 3-2 in B, since every trichord except members of 3-12 can share zero, one, or two PCs with another member of its type. It turns out that for most cases of # A > 2 we can determine EMB(A,B') by considering nothing less than the total subset content of B. Of course, when #B ' > #B and B 5-Z12, then EMB(B,B') > 0, so for any A such that EMB(A,B) > 0, then EMB(A,B'j > 0, though still not with predictable multiplicity; but because of its larger size, B' can additionally contain many types of A ’s size not in B. As a measure of the combinational flexibility of PCsets, the embedding and covering constructs will be extended to encompass a set’s total subset or superset content. For sets of a given size, the total number of distinct subsets, as well as the number of subsets of a given size, can be derived from the formula given in Chapter I, and by application of the rules for complements given in this section, the number of supersets can be found. Since all these numbers are the same for sets of a given size, the measures are to be based on the number of distinct classes included as subsets or supersets, with the idea that in general the fewer types involved, the more multiply a set’s fixed number of subsets or supersets accumulate 22 The conceptual model is borrowed from Starr (1978|. 17 into the same classes, and the more “unified” the set with respect to its subset or superset content. In the spirit of some useful relations developed by Rahn [1979], to be taken up in turn, the concept of “class embedding” of subsets will be abbreviated as CEMB, and that of “class coverage” of supersets, CCOV. CEMB(A,B) indicates whether at least one member of SC A is embedded in B, otherwise ignoring its multiplicity: CEMB(A.B) = 1 if EMB(A,B) > 0; otherwise CEMB(A,B) = 0. Similarly, CCOV(A.B) = 1 if COV(A;B) > 0; otherwise CCOV(A.B) = 0. Since EMB(A,B) = COV(B',A'j, CEMB(A,B) = CCOV(B',A'). CEMB„(B), the number of distinct classes of size n embedded in B, is simply a count of the number of classes A of size n for which CEMB(A,B) = 1, and similarly for CCOV„(B): CEM Bn (B) = sum of all nonzero CEMB(A,B) for every SC A of the SG where #A = n CCOVn(B) = sum of all nonzero CCOV(A,B) for every SC A of the SG where # A= n Because of the complement relations noted above, CEMBn(B) = CCOVis-nfB') and CCOVn(B) = CEMBi2-n(B'), but there is no direct correlation between CEMBn(B) and CCOV„ (B)- The maximum value of CEMBn(B) b the smaller of the following: 1) the number of classes of size n in the SG 2) the number of subsets of size n contained in a set of B ’s size and the maximum value of CCOV„ (B) is the maximum value of CEMBi2- n(B'), not a trivially imposed limit. In calculating the total number of subset classes for all sizes of subsets of B, TCEMB(B), we wish to exclude, as not making distinctions among set3, subsets of sizes 0, 1, and # B, thus # B — 1 TCEMB(B) = £ CEMB„(B). re=2 Likewise, since every class b included by 12-1, 11-1, and itself, we exclude these as superset classes counted by TCCOV: 10 TCCOV(B) = ^ 2 CC O V „(B). n = # B + l Normalizing these values to between 0 and 1 by dividing by the maximum possible values of TCEMB and TCCOV is useful in comparing sets of different sizes with respect to these properties. The same values of CEMBn(B) used to derive TCEMB(B) can be displayed as a vector, called CEMBnV(B), and similarly for CCOV„V(B). For example, for B = 6-20, containing 2-1, 2-3, 2-4, 2-5; 3-3, 3-4, 3-11, 3-12; 4-7, 4-17, 4-19, 4-20; and 5-21, and contained by the complements of these, CEMB„V(B) = 4441 CCOV„V(B) = 1444 Both TCEMB(B) and TCCOV(B) = 13 in thb case. 18 In general, the higher a set’s degree of symmetry, the lower the value of CEMB„ and CCOVn for each n; further generalization from °S alone is not possible since the exact values of these functions are dependent upon sometimes quite complex combinations of symmetries with respect to the total subset content of the set. Nevertheless some relations can be generalized, to within only a few exceptions, for subsets of size (n — 1), of which there are n: 1) If °S(A)= 1, then usually CEMB„_i(A)= n, meaning that there b one instance each of n different classes; for these cases CCOVn+1(A) =CEM B„_i (A’) = (12 — n). There are four exceptions for which CEMB„_i(A)= (n — 1), meaning that there are two (n — l)-sized subsets of the same class, always transpositionally related members of an I-nonsymmetrical SG(1) class, though CCOV„+i(A) still equals (12 — n) in these cases: the exceptions are 5-21, 6-27, 7-31, and 8-19, three of which are themselves (n — 1)- sized subsets of sets with multiple levels of T-symmetry. It b also interesting that for #A 6 and °S(A) = 1, at least one subset of size (n — 1) b I-symmetrical, with the sole exceptions of the “all-interval tetrachords” (4-Z15/Z29) and their complements; for hexachords, 9 of the 29 SG(2) types fof which °S = 1 are excepted,23 and so b the special case of 6-30, whose °S = 2 but which b I-nonsymmetrical. 2) For °S(A) > 1, i.e., all I-symmetrical classes and abo 6-30, usually CEMB„_i(A)= n/°S(A), where if the result b fractional it b rounded up to the next integer.24 For °S = 2, thb reduction of classes b due to I-nonsymmetrical subsets accumulating by I-related pairs into the same SG(2) class, except 6-30, whose pairs are Tc-related; for °S > 2, a similar situation occurs with respect to T-symmetry. An exception occurs if a set contains both its axes of symmetry and the sum of these axes is even: in thb case there b one fewer such P/I pairing and at least two separate I-symmetrical classes contained once each, so one class must be added to the above result. If thb condition holds for a set, it cannot hold for its complement. Following from this, since the value of CCOVn+1(A) is in a sense dependent, for inversionally symmetrical sets, on the structure of A’s complement with respect to its axes of symmetry, apparent discrepancies for CCOVn+i(A) for sets of the same size and °S are resolved. For example, there are five distinct trichordal classes that can be formed by adding one PC to 2-1/5 or 2-3 (odd sum), but six for 2-2 or 2-4 (even sum)—2-6 has only three because of its T-symmetry. The extremes of these measures are of interest. An example of a class displaying, for its size, minimum values of CEMBrt and CCOVn for all n b the “octatonic scale” (8-28), which excludes five of the twelve trichordal type3, and of course any supersets of these, thus containing only six hexachordal types and one heptachordal type, and furthermore b the only type of size 8 to exclude more than one trichordal type. The result b TCEMB = 40, where the next higher value for an octachordal class b 60, and the highest b 118. The feature of 8-28 such that C E M B #a-i = 1 b shared by 3-12, 4-9, 4-25, 4-28, 6-20, and 6-35, all with °S > 4. It b particularly interesting that the four “maximal subcollections” (6-1/32 and 7-1/35, containing unique multiplicity of each IC) have the lowest value of TMEMB for their size and degree of symmetry (°S = 2), i.e., the lowest possible values short of T-symmetry; thb is due to their inclusion of more than the usual number of I-symmetrical subsets for sets of their size, a property shared by all other sets of ordinal 1 (chromatic scale segments) and their M-transforms (cycle-of-fifths segments). The opposite extreme, 23 Their ordinal numbers are 5, 9, Z11/Z40, 15, 16, 18, 27, and 31. 24 In any of these cases, CCOVn+l(A)=CEMB„_i(A'). An I-symmetrical set may or may not contain any I-symmetrical subsets of size n - 1. 19 maximum diversity of subset or superset class content, is where CEMB„(A) reaches a maximum value for some n and where that maximum value is not shared by other classes of A ’s size: that CEMBs(4-Z15/Z29)= 6 is well known; this is the SG(3) class comprising the two SG(2) “all-interval (-class) tetrachords.” In addition, since CCOVs(4-Zl5/Z29) equals a unique maximum of 29, these tetrachords are the only ones to be contained in all octachordal classes, and that 8-Z15/Z29 are “all-tetrachord octachords.” By extension, the two all-interval tetrachords are the only ones that can be combined with some member of any tetrachord class, including their own, without PC intersection. Also, since CEMB Similarity Relations Many similarity relations have been developed for sets of different classes. Most of these are based on comparisons of interval vectors, sometimes with results scaled to compensate for sets of different sizes.25 The embedding constructs provide an adequate framework to model these relations when the size of the embedded sets is 2, and have the advantage of making possible the extension to embedded sets of larger sizes. Rahn’s extension of Lewin’s embedding construct is the similarity relation most strongly based on inclusion, taking into account the total subset content of both sets being compared.20 For the sake of consistency, a slight modification is proposed, without damage to the concepts embodied by Rahn’s MEMBn(X,A,B) and related constructs. MEMB(X,A,B) counts the forms of X mutually embedded in A and B (or in some form of each class A and B). MEMB(X,A,B) = EMB(X,A) + EMB(X,B) such that EMB(X,A) > 0 and EMB(X,B) > 0. A form of X must be embedded in both A and B to be counted, then all forms of X in both are counted, including the case, for example, of X = A, for which MEMB(A,A,B) = EMB(A,B) + 1. Generalizing to all shared subsets of a given size, MEMBn(A,B) is the sum of all nonzero MEMB(X,A,B) for #X = n. TMEMB(A,B), the total number of mutually embedded sets in A and B excluding the null set, monadic sets, and sets larger than the smaller of A and B ,27 is given by m TMEMB(A,B) = MEMB„(A,B) where m = the lesser of #A and #B. n=2 25 See, for example, Forte [1973a] and Morris [1979]. 20 R ahn [1979]. 27 Rahn then normalizes these values to between 0 and 1. 20 The similar extension of COV(A,X), counting mutual supersets of two sets, follows: MCOV{X,A,B) = COV(X,A) + COV(X,B) such that COV(X,A) > 0 and COV(X.B) > 0. . MCOVn(A,B) is the sum of all nonzero MCOV(X,A,B) for # X = n. 10 TMCOV(A,B) = ^ MCOV„(A,B) where m = the greater of #A and #B . n=m The complement relation is TMEMB(A.B) = TMCOV(B', A') and all these measures reach a maximum when A = B. The group of mutual embedding functions can be generalized to a count of mutually embedded classes, ignoring the multiplicity of the mutual containment of each class. These relations are useful for comparing different pairings of classes. CMEMB(X,A,B) indicates whether at least one member of SC X is mutually embedded in A and B, i.e., for which MEMB(X,A,B)> 0. CEMB„(A,B) is the number of distinct classes X of size n for which CMEMB(X,A,B) is positive. The maximum value of CMEMBn(A,B) is the smaller of CEMBn(A) and CEMBn(B). TCMEMB(A,B) is the total number of classes mutually em bedded in A and B of sizes from 2 through the smaller of A and B. The same values of CMEMBn(A,B) can be displayed as a vector, called CMEMBnV(A,B). For example, for A=4-Z15, containing one each of 3-3, 3-5, 3-7, and 3-8, and B=4-Z29, with 3-2, 3-5, 3-8, and 3-11, both containing all dyad types once each, CMEMB„V(A,B)= 62, indicating six shared dyadic types and two shared trichordal types. The corresponding ex tension of the MCOV relations to CMCOV, etc., should be evident. Examples in Chapter IV illustrate the use of many of these constructs. 21 Comxnon-Tone Relations Though pitch'd asses are of course not “tones” in the usual sense, the term “common-tone” seems less unwieldy than, for example, Forte’s “invariant pitch-dasses” when referring to the set of intersection of two PCsets, and no confusion arises in a context that otherwise assumes pitch-class. A common-tone in this sense, then, is only potentially a common pitch. Eric Regener, in his criticism of Forte [1973a],28 develops several useful common- tone relations and vectors which will be presented here, with slight changes in terminology and notation, and extended to encompass equivalence systems other than Regener’s SG(1). These relations may be used to compare a set to transformations of itself, or to any other set whatsoever. The common-tone function of PCsets A and B, CTF(A,B), is the number of common tones, i.e., the common-tone index or CTI, between A and each transposition of B: CTF(A,B) = CTI(A,T„(B)) for every n. Each (A,Tn(B)) pair is a configuration of A and B, that is, a relative transposition of A and B, each configuration standing for the twelve possibilities (Tm(A),Tm+,i(B)), for m = 0 ... 11 for a given n. Configurations of A and T„(B) are notated as A:T„(B) or, where A and B are contextually dear, simply as Tn, or, where Tn is clear, as A:B. The common- tone vector of A and B, CTV(A,B), displays the values of the common-tone function of A and B for n = 0 ... 11. The CTI for a given n of the CTF is the same as the multiplicity of occurrence of the directed interval n from any PC of To(B) to any PC of A. Why this is so becomes clear when we consider that since the interval n from PC b of set B to PC a of set A is (a — 6), then Tn(6)= a, and thus there is one common tone between Tn(B) and A for every such interval n between A and B. A matrix or subtraction table is practical for calculating these # A x # B intervals.20 For example, for A = <0.2.45.7.9.E (7-35) and B = 0 . . .4. .7. . . . (3-111), Figure 1 shows these matrices for CTF(A,B) and CTF(A,I(B)).30 - 0 2 4 5 7 9 E 0 (° 2 4 5 7 9 E T 0 1 3 5 7 4 1 8 7 U 7 9 T 0 2 4 — 0 2 4 5 7 9 E 0 (° 2 4 5 7 9 E 8 6 8 9 E 1 3 4 5 V7 9 E 0 2 4 6 F ig u re 1. CTF matrices for 7-35 and two forms of 3-11. 28 Regener [1975]; see also Lewin [1977] and Rahn [1979]. 20 Gamer At Lansky [1976], 30 Since ToI(6)=T_t, where b is a PC in B, and since if a — b = n, then a + (—6) = n, the same results can be obtained by summing each PC of A and each of the inversion of B (ToI(B)) for CTF(A,B), and summing each PC of A and each of the original form of B (Tn(B)) for CTF(A,ToI(B)). This is useful in hand calculations, where mod. 12 addition is somewhat less likely to result in errors than subtraction. 22 The common-tone vectors are 312123031221 and 212131221303, respectively. Beginning with a different transposition of A or B produces a rotation of the common-tone vector, so for comparing sets of different types, the “normal form” of the vector is defined to be that for which A and B are the prime forms of their respective SG(l) classes.31 This would affect the ordering of both the vectors above. As can be seen by comparing these two vectors, inverting A or B (but not both) produces a retrograde of the vector, to within rotation, and therefore if both A and B are I-symmetrical the vector Is circularly palindromic.33 The partition function of A and B counts the number of transpositions which produce each distinct CTI of the CTF, and the partition vector PV(A,B) lists these for each CTI = 0 ,1,.. .n where n is the size of the smaller of A and B. For either example of Figure 1, the partition vector is 1443, indicating one transposition of B producing zero common tones with A, tour each producing one and two common tones, and three producing three common tones. Clearly, the sum of the entries in the PV is always 12. An entry in the final position is the number of times the smaller is embedded in the larger, if no transpositional symmetry other than To is involved. Since the partition vector is independent of the permutation of the entries in the common-tone vector, it is useful as a similarity relation for comparing pairings of set types . The procedures outlined just above can be applied to any pair- of PCsets, but we are particularly interested in the common-tone functions of pairings of two members-of the same class of some SG. After Regener, for these the constructs will be called generally self-common-tone /unctions, self-common-tone vectors, and self-partition vectors ,33 and to indicate specifically which group of twelve transpositionatly related transformations is in volved in self-configurations, a prefix will be adjoined, e.g., TnCTV(A), T„IPV(A), etc. The “normal form" of these vectors is defined as CTV(A,ToX‘(A))1 where X is a null operation, I, M5, or IM5. The common-tone function between some set A and itself yields a self-common-tone vector (TnCTVfA)) which is a “true” interval vector, i.e., a display of directed interval content. This “long” form of interval vector includes a number of zero-sized intervals equal to the size of the set, and, since complementary intervals are present in equal numbers, is (circularly) symmetrical around this entry and the one for tritones, and can be viewed as an “unfolding” of the “doubling" of the “short" interval vector, with an extra initial entry.34 For example, for A = 01. .4 .6 ...... (4-Z15P), an all-interval tetrachord whose vector is 111111, T„CTV(A) = CTV(A, A) = 411111211111 The T„-partition vector in this case is 0T101. Because of the entry for To in the T„CTV, every TnPV’s last entry is the degree of transpositional symmetry of the set. In this case, ten of the eleven other n’s of Tn(A) produce one common tone; the exception is for To, which always produces an even number of common tones, since if an IC 6 is present To maps both its PCs into each other. 31 This differs significantly from Regener’s normalization; also, we will later make an exception for when B is a transformation of A. 32 Regener notes some exceptional cases in which the vector is symmetrical in this way for pairs of sets from different SG(1) classes, where neither is I-symmetrical. 33 Regener uses only the latter term, and since his canonical group is SG(l), it applies only when A=A. 34 The TnCTV of sets whose °TS > 1 is periodic; this is generally obscured in the short vector. For example, for 4-9, V=200022, the T„CTV= 420002 420002. 23 . The TnI-common-tone vector of A (Rahn’s “TICS" vector)35 is the common-tone vector of A and its inversion, and yields the T„I-partition vector: T„ICTF(A) = CTF(A,T0I(A)). For the example above, T T„IM5CTV(A) = CTV(A,T0IM5(A)) Continuing the example of A = 4-Z15, T„M5CTV(A)= 220021222021 and TnM5PV(A)= 32700 (i.e., the same as T„IPV(A)); T„IM5CTV(A)= 311011311211 and T„IM5PV(A)= 18120. Assuming #A ' > #A , where both are members of complementary types, then because of the relation of interval content of complementary types, every entry of TnCTV(A') equals #A'—# A added to the corresponding entry of T„CTV(A), with no exception for the tritone entry, since the entries correspond to directed intervals and not ICs. Not as obviously, the same holds for T„ICTV(A'), including for Z-related hexachords, except that corresponding entries are rotated with respect to the vector of A, and often in retrograde as well. In either the Tn or TnI case the last (#A+1) entries of the corresponding self-partition vectors of A ' are identical to' the entire partition vector of A; in other words, since no A' can have fewer than (#A '—#A ) common tones with another set of its size, its self-partition vector naturally has this number of insignificant leading zeros, which, if dropped, yield its complement’s corresponding self-partition vector. Self-common-tone vectors of M-related types are permutations of each other, the entries rearranged as though each interval of transposition were subjected to the same M-operation that relates the two types. The two vectors may or may not be related by some circular permutation, but in either case both the Tn- and TnI-self-partition vectors are identical for M-related types. All Z-related sets must have the same TnCTV, and hence the same TnPV, since by definition they have the same interval content, but this does not mean that they share rotationally equivalent T„I-common-tone vectors. AH Z-related hexachords do, but because they are complementary, not because they share interval content. Of the nonhexachordal Z-relations, 4-Z15/Z29 (and their complements) have rotationally equiva lent TnI-common-tone vectors, and are incidentally M-related, and likewise for 5-Z17/Z37. However, the same does not hold for 5-Z18/Z38, despite the M-relation, and, finally, 5-Z12 and 5-Z36 cannot share a TnICTV because one is I-symmetrical and the other is not. Thus the property of sharing a rotationally equivalent T„ICTV is independent of both the Z- relation and any M-relation, but in all these cases except the last, the T„I-self-parii/ion vector is the same, because of the M-relation. Disregarding now the actual ordering of the entries in the self-common-tone vectors, i.e., the exact intervals of transposition involved in each common-tone index, a given T„- or Tn I-partition vector may be shared by a number cf members of a cardinality type, and can as such be a criterion for similarity, at least with respect to the particular combinational properties involved. Types which share a T„PV often share a TnIPV, but the situation cannot be generalized any further: there are many cases where a T„PV is shared by types 35 R ahn [1980[. 24 of different degrees of inversional symmetry, so must have different T„IPVs. To some extent the self-partition vectors reflect the symmetries of the individual subsets of set types. For example, a nonzero penultimate entry in the T„IPV for a set of size n indicates the presence of at least one I-symmetrical subset of size (n —. 1). Table 2 displays the five distinct cases (A-E) of trichordal self-partition vectors. The distinctness of each case is produced by combinations of two factors relating to subset content: 1) multiple inclusion of a single subset type—for trichords this produces I-symmetry, and in case A T-symmetry as well, so T„PV = T„IPV for cases A, B, and C 2) inclusion of the T-symmetrical tritone. The class labeled B has both properties, those labeled C and D have one or the other, and E neither. Table 2. Self-partition vectors for trichorda . Ordinal T„PV T „IPV Case 12 9003 <- A 10 8031 +- B 1 /9 ,6 7221 *- C 5, 8 . 6411 7140 D 2/7, 3/11, 4 6600 6330 E The complexity of the situation increases geometrically with the size of the sets, since for tetrachords, for example, we must consider, in addition to dyadic combinations, how each of our five trichordal cases may be contained. Still, the total number of distinct self-partition vectors, shown in Table 3, is relatively limited compared to the number of possible partitions of 12 objects into 4, 5, €, or 7 groups. T a b le 3. Num bers o f distinct self-partition vectors for classes of size 3-9. Distinct pairings Distinct T„IPVs of T„PVs and T„IPVs Cardinal Distinct T„PVs where different from TnPV incl. TnPV with itself i.e., for I-nonsym. classes for I-sym. classes 3/9 5 2 5 4 /8 10 5 11 5/7 13 9 16 6 14 12 20 Partition vectors shared by relatively few types indicate combinational properties unique to those types, as compared to larger groups of more “ordinary” types. AH multiply-T- symmetrical types are in this category, as are the all-interval tetrachords, and so are all types of ordinal 1 of sizes 4-8, i.e., some of the chromatic sets and their M-transforms, including the diatonic scale. Most of the remaining types with unique or near-unique partition vectors contain some T-symmetrical subset with one additional PC, as in 4-19 (01. .4. . .8.. .) or 5-31 (01.3. .6 . .9. .). Chapter III The Generation of Scales by Configurations of Chords By chord, is meant a set, whose content is presumed to function as a harmonic reference in a composition, which is combined with one or more other such sets to generate larger sets. The term may also be used to mean the one or more c/asses of the generating sets, but it is normally understood that specific configurations of class members are involved. In much of what follows, generating chords are of the same SG(2) class, forming Tn-configurations or TnI-configurations. A scale is the union of two or more chords, that is, the generated set. This perhaps differs from traditional use of the term in that there are no further constraints limiting what might be called a scale, such as some restriction on size or the imposition of "adjacency” criteria, nor is it assumed that a scale is in itself a harmonic reference (though it may be), but rather a (perhaps momentary) quasi-aggrcgate within which harmonic activity takes place. A configuration of two chords is said to pair-generate a scale, abbreviated S-generate, In successive scale generation, a pair-generated scale may itself be treated as a chord pair-generating a larger scale, in which case it is an intermediate scale, and the original chords may be said to S-generate or ^-generate the larger scale. In other cases chords may multiply generate a scale without first generating an intermediate scale. In any of these cases, the chord configurations involved may or may not produce common tones; if not, the generating chords partition the scale, and if so, the common tones have a pivotal function within the scale with respect to their position in the generating chords, and it may be argued that for systems which, rather than assuming the aggregate at the outset, have as an initial premise a chord of specific interval content (perhaps not using equally-tempered semitones), common tones are a necessary reference point in scale generation, particularly when those scales are smaller (or larger) than the aggregate. All voice-leading possibilities within a pair-configuration are defined by the common-tone function between one chord and the other, taken as PCsets, not classes, and if common tones are treated separately, the voice-leading possibilities for the changing or “melodic” tones are similarly defined by the common-tone function between the non-common tones of one chord and those of the other. Scale Types in Pair-Generation For generating chords of the same class, the self-common tone vectors show which opera tions produce which CTI, and the self-partition vectors show how many of these operations produce each particular CTI, and in both of these the CTI can be interpreted as giving the size of the resulting scales, where #(Scale) = (2 x #(Chord)) - CTI. 25 26 On the other hand, they do not show the ctasa of the scale, and there are most often two or more configurations of a chord producing scales of the same class and thus fewer total classes generatable than the number of configurations. For the TnCTF, a reduction of classes always occurs, since transpositions by complementary intervals produce common tones and scales of the same SG(1) class, or, in other words, the intersection of A and Tn(A) is a transposition, by Tn, of the intersection of A and T -n(A), and the same for the respective unions. This means that there are only six "classes” of Tn-configurations (where n is a nonzero interval class), and that interpreting the "short” interval vector as a common-tone vector gives a better indication of the number of distinct classes of scales. But such a reduction does not apply to the TnICTV, and any further reductions in either are due to internal symmetries or potential symmetries specific to the chord class. Table 4 gives a modification of the T„- and TnI-self-partition vectors for trichords, showing the number of distinct SG(2) scale types of each size greater than itself generatable by pairs related by Tn or TnI (cf. Table 2). The three entries of the modified vector show the number of classes of scales of size 6, 5, and 4, respectively. T a b le 4. Self-partition vectors for trichords, modified to show number of distinct scale types. Ordinal Modified Number of T„ Modified Number of T„I Total distinct numbers TnPV scale types T„IPV scale types scale types 12 200 2 - - 2 10 401 5 - 5 1/9,6 411 6 - - 6 8 14 311 S 713 11 5 15 2/7 17 330 633 12 3/11, 4 18 For inversionally nonsymmetrical chords, there is often a further reduction in the total number of distinct scale types caused by shared types between one or more T„- and T„I- configurations. Note, in Table 4, that there are two such shared classes for 3-8, one each for 3-5, 3-2, and 3-7, and none for 3-3, 3-11, or 3-4. In general, the greater the size of the chord, the fewer the total number of scale types it generates, if only because there are fewer type3 of each of the largest sizes of scales. For example, whereas the three T„-configurations of 3-11 with zero common tones produce three distinct hexachordal classes, the corresponding configurations of 9-11, with six common tones, all generate the aggregate. Table 5 summarizes the total numbers of distinct SG(2) classes for both T„- and combined T „/T reI configurations. Maximums, of course, are 6 and 18, respectively, but only seven chord.classes have this property. For each size of chord, the first figure in the table is the number of distinct scale types resulting from the six Tn-configurations; the second figure gives the total distinct scale classes available from 27 T a b le 5. Numbers of distinct scale classes generated by Tn- and Tn I-pair-conGgurations. Total Number Total Number Distinct distinct of SG(2) Distinct distinct of SG(2) Size of T„ Tn and TnI chord Size of T„ T n and T „I chord chords scale scale classes with chords scale scale classes with types types these totals types types these totals 3 — 1 (6) 4 12 2 6 - 5 5 12 5 6 13 1 2 - 1 5 14 3 5 - 1 6 14 6 6 - 3 6 15 2 5 14 1 5 16 3 5 15 1 6 16 2 6 17 2 6 . 17 1 6 18 3 7 4 - 1 1 - 1 5 - 3 3 - 2 6 - 6 5 - 2 2 7 1 6 - 10 3 8 1 4 • 13 1 4 8 2 6 16 2 4 9 1 6 17 7 5 9 2 6 18 4 4 10 6 5 10 11 4 - 1 6 10 3 5 - 1 6 11 1 6 - 8 3 10 1 8 1 - 1 4 12 1 2 - 2 5 12 1 4 - 10 4 13 1 5 _ 2 5 14 1 3 6 1 6 14 5 3 7 3 6 15 12 4 7 4 6 16 6 2 8 2 4 8 4 1 - 1 2 - 1 9 1 - 1 3 - 1 3 - 4 5 - 10 2 5 5 2 5 1 3 5 2 6 - 7 3 9 1 10 1 - 1 6 10 3 2 - 5 28 both these and the twelve Tn [-configurations, where this is different from the first figure, i.e., for I-nonsymmetrical chords. In general, the number of scale types resulting from the T„I-configurations alone is not deducible from the difference between the two figures. The figure in the last column gives the number of SG(2) chord classes whose configurations yield these totals. Table 5 shows that within each size of chord, the numbers of scale types are distributed fairly evenly between the extremes. In general, the higher the degree of symmetry of the chord, the more symmetrical the subsets contained, and the fewer the distinct scale types generated. But it is also generally true that the more symmetries contained, the fewer the possible supersets, and if this is taken into account the range of variation within each chord size is narrower. Table 6 summarizes the corresponding situation for scale types, that is, what propor tion of the distinct subset classes of a size equal to or greater than half the size of each can pair-generate it. This is best expressed as a number between 0. and 1., obtained by dividing the total number of generating classes by the totals for CEMB„ (scale) for n = #(scale)/2. . .#(scale)—1, where the first number is rounded up to the next integer if frac tional, i.e., for odd sizes of scales. Note that this figure does not take into account the fact there is often more than one pair-configuration of a chord class generating a particular scale type. The second column gives the number of SG(2) classes, with a given proportion or range of proportions. From Table 6 it can be seen that 61 (28%) of these 216 classes have no pair-generating chord classes at all, and another 77 (36%) are at .32 or below, so most of the pair-copfigurations of the 220 chords accounted for in Table 5 accumulate into the remaining 78 (36%) scale types, nearly all of which have at least inversional symmetry.30 Thus, given two chord classes of similar size, each generating up to 18 different scale types, the probability that they can mutually generate at least one scale is significantly great, in fact virtually certain for some chord sizes. Also compositionally suggestive is that once one of these relatively few scale types is generated, advantage can be taken of its sometimes many other generating chord classes. 30 Incidentally, the scale with the most symmetry, the aggregate, does not have the highest proportion of pair-generators under these restrictions, because its 30 Z-hexachord types and 7-Z12 cannot pair-generate it; some smaller scales, such as the octatonic scale, with .93, or the two hexachordal types with 1.00, are more integrated with respect to this property. 29 T a b le 6. Ratios of distinct pair-generating classes to distinct subset classes of half their sise or greater. Size of Number of Size of Number of scales Proportion SG(2) classes scales Proportion SG{2) classes 3 1.00 1 (7) .05-.09 14 .50 4 .02 7 .00 7 .00 6 4 1.00 3 8 .93 1 .80 3 .82 1 .67 7 .60 2 .14-. 17 2 .5 5 -5 7 7 .00 14 .2S-.26 2 .1 9 -2 0 3 5 .67 1 .0 2 -0 7 9 .57 2 .00 4 .38-.50 8 .0 8 -1 0 12 9 .58 1 o 1ft .00 15 f 4 .13-. 14 4 6 1.00 2 .0 8 -1 1 2 .85 1 .00 1 .75 2 .62-.68 12 10 .72 1 .30 1 .6 5 -6 7 3 .17 4 .32 1 .0 7 -0 8 7 .27 1 .03-.04 7 .00 14 11 .67 1 7 .48-.50 3 12 .77 I .3 7 -4 2 7 .27 1 30 M ultiple-Generation The preceding takes into account the generation of scales by configurations consisting of pairs of subsets, and there is no guarantee that EMB(Chord,Scale) is greater than 2. But it may be that in a given context we wish to use scales with a higher saturation of a given generating chord type, perhaps with other constraints operative as well. Four broad categories of multiple inclusion are of interest: 1) Scales which incidentally contain multiple instances of a given pair-generator’s type, and particularly when there is more than one generating configuration of that type. 2) Scales which, though not pair-generatable by a given chord type, can be generated by configurations of three or four members of that type. 3) A special case of the second situation in which a particular pair-configuration is repre sented at more than one transpositional level. This amounts to successive scale gener ation, a large scale resulting from a pair-configuration of an intermediate scale which is itself generated by a pair-configuration of chords. 4) Combination of several different configurations of a chord type to generate scales, but in such a way that a closed cycle of configurations results, within which no new config urations arise. In the first category, there are a few scales which contain a pair-generating chord type more than twice, even in SG(1), i.e., T„-related. For example, a configuration (A:B) of 3-5P: A = 3-5P 01 ____ 6 ...... +B = Ti(A) .12 -----7 ------generate 5-7P 012...67 ____ also containing C = To (A) 0 ...... 67.... B:C is another generating configuration, but A:C is not. More typical is that additional members of a transpositionally-generating class occur in their inverted forms, for only SG(2) equivalence; the scale in this example contains two of these as well: T ilfA ) 0 1 ...... 7 ------T 7I(A ) . 1....67.... If SG(2) is generally operative, admitting T„I-related pairs as generators, it is quite common that such a configuration produces a scale containing other TnI-related but nongenerating pairs of the class: A = 3-7P 0.2..5...... + T 9I(A) ___ 4..7.9.. = 6-32 0.2.45.7.9.. also containing T2(A) ..2.4..7___ T7I(A) ..2..5.7___ and T7(A) 0 ...... 7.9. . T2I(A) 0 . 2 ...... 9 . . 31 Some scales have the special property of generability by both a T„- and a T„I-configuration of the same class (A:B and A:C here): A =3-7P 0 . 2 . . 5 ...... A 0 . 2 . .5 ...... B = T 6(A) ...... 5.7. .T, C=T0I(A) 0 ...... 7..T. (5-35) ia 0.2..5.7..T. 0.2..5.7..T. Scales with this property for at least one generator (and sometimes many) are 4-9 and 4- 25; 5-1/35 and 5-33; 6-1/32, 6-7, and 6-20; 7-1/35, 7-8/34, 7-Z17/Z37, and 7-22; and all I-symmetrical scales of size 8 or greater. In the particular case of 5-35 there is an additional instance of one of the generating configurations; D = T 7I(A) ..2..5.7___ T 6(D )= C 0 ...... 7..T. for a total of three distinct generating pair-configurations among only four members of 3-7. Some of the other scales listed above have the property with respect to several of their generators, and in some cases there are more generating configurations than there are members of the generating class, i.e., where any pairing of subsets of the class generates the scale. The second possibility for multiple containment of generating subsets involves config urations of three or more members of a type, for which the simplest case is that of triples of dyad types: any scale generatable from two T„-related trichords can be viewed as gener- atable from three transpositions, at intervals of that trichord class, of the dyad type of the same class as the interval n of T„. For example, since 6-32 can be generated by a pair of 3-6 at Tfi, or a pair of 3-9 at T3, it can also be generated by triples of 2-5 or 2-3 at intervals of 3-6 and 3-9, respectively: 6-32 0.2.45.7.9.. 6-32 0.2.45.7.9.. (3-6)o 0.2.4...... (3-9)2 .2.4___ 9.. (3-6)5 ...... 5.7.9.. (3-9)5 0 ____ 5.7____ i I (2-5)o 0 ____ 5 ...... (2-3)2 . . 2 . . 5 ...... (2-5)2 . 2___ 7 ____ (2-3)4______4..7____ (2-5)4 4 9.. (2-3)9 0...... 9.. Extending this to quadruples of dyads, since the same hexachord is generated by a pair of 4-23 at T2, it can be split into a quadruple of 2-2 at intervals of 4-23: (4-23),, 0.2. .5.7____ (4-23)2 ..2 .4..7.9.. i (2-2)o 0 .2 ...... (2-2)2 -.2.4...... (2-2)6 5.7 ------(2-2)7 ...... 7.9.. and also two ways as quadruples of 2-5, at intervals of 4-22 (0.2.4. .7... .) and 4-26 (0.2. .5. . .9..). This can be conceptualized as a “multiplication” of one chord by another in the sense of transpositionally replicating one of the chords at each of the PCs of the other. 32 If one of the two lacks I-symmetry, the resulting scale will still always be SG(2)-equivalent to that generated by substituting an inversionally related form, but when neither is I* symmetrical, scales generated in this way by different SG(l) pairs are not usually even SG(2)- equivalent, and the situation is even more complicated if one or more of the replications is to be inversionally related to the others, because in this case the order of operations makes a difference in the class of scale. Thus it seems best to limit the discussion to transposition only, and to 3-generation by trichords and tetrachords only, where up to 19 SG(2) scales result, one corresponding to each SG(l) trichord type. For the case of trichords 3-generating hexachords not already transpositionally pair-generatable by those trichords, each of the five SG(2) trichord types without I-symmetry and without the tritone generates two new hexachords in this way, and 3-8 generates one; nine hexachordal scales are involved, one of which, for example, is 6-32 as generated by 3-7 (0.2. .5...... ) at intervals of 3-9 (0 . 2 . . . .7 . . ..). Every trichord but two can generate from 3 to 8 heptachordal scales and from 3 to 9 octachordal scales, and from one to five nonachordal scales, most just one.37 Only one tetrachord 3-generates a hexachordal scale not already 2-generatable by it: 4-19 at intervals of 3-12 yields 6-20 because 4-19 contains the transpositionally symmetrical 3-12 plus one PC.38 Transpositional triples of eight tetrachordal types yield but four new heptachordal scales, yet all but seven generate octachords in this way. The greatest number of possibilities is for triples of tetrachords generating nonachords and decachords; only seven types can 3- generate the aggregate, and by definition these are the six all-combinatorial tetrachords of ordinal 1/23, 6, 9, 10, and 28, and the single T-only combinatorial tetrachord 4-13. Put another way, these seven are the only tetrachords which can transpositionally pair-generate their complements. The third and perhaps most compositionally fruitful possibility for multiple inclusion is a variant of the second, with the additional limitation that a particular pair-configuration is to occur at more than one transpositional level within a scale. A procedure that guarantees this is to treat a scale resulting from a particular pair-configuration of generating chords as an intermediate scale pair-generating a larger scale containing two instances of the original configuration. In the following it is assumed that this pair of pair-configurations will assume its most parsimonious form, that is, the smallest scale satisfying the conditions, unless further constraints are imposed. In Figure 2 the generating chord is 3-IIP, i.e., SG(1) “minor triad” or SG(2) “triad,” with interval vector 001110. Six transpositionally related pair-configurations are possible, the first three with one common tone and the last three with zero. In this case none of the resultant scales is inversionally symmetrical, though F is trans positionally symmetrical. Incidentally or not, depending on the SG in use, A, B, C, and E each embed an inverted form of the generator, and D embeds two. It is clear that in treating these pentachords and hexachords as intermediate scales that any subsequent Tn- configuration of them would replicate the original triad configuration, and that the most parsimonious scale would be the result of the transposition-class with the highest common- tone index, i.e., Ts(A), T4 (B), TS(C), T4(D), T$ or Ts(E), and T0(F), except that the last of these yields “too many” common tones (i.e., all six) to produce another configuration. 37 The one that generates 5 is 3-10, the only one which cannot 4-generate the aggregate. 38 It is also notable that 4-19, for the same reason, is the only tetrachord for which three combined forms, one inversionally related to the others, produces any new hexachords: 6-14, 6-Z19, and 6-Z44. The tetrachord is pervasive in early atonal music, and both 6-20 and 6-Z44 are common hexachords in Schoenberg, the latter his “motto.” 6-Z19 is both the complement and M-transform of 6-Z44, and 6-14 is the unique T-only combinatorial hexachord, i.e., the only Tn- and 1-nonsymmetrical hexachord to exclude an IC. 33 A To 0 ..3 ...7 ___ t 3 ...3 ..6 . . .T. 5-32P 0..3..67..T. V — 113221 B To 0 ..3 ...7___ t 4 ___ 4..7...E 5-21P 0..34..7...E V = 202420 C To 0..3 ...7___ To 0___ 5. .8 .. . 5-271 0..3.5.78... V = 122230 D T„ 0..3 ...7 ___ Ti .1 ..4 ...8 .. . 6-Z19P 01.34..78... V = 313431 E To 0 ..3 ...7___ t 2 ..2..5...9.. 6-33P 0.23.5.7.9.. V = 143241 F T„ 0 ..3 ...7 ___ Tc .1-----6 . .9.. 6-30P 01.3..67.9.. V = 224223 Figure 2. Tn-configurations of pairs of 3-llP. But assume now that we wish to replicate each configuration in a symmetrical way, such that one of the chords, say, the original 0. .3.. .7.. .., is involved in both configurations and thus has the same relationship to a third chord that the second chord has to it. This is the same as transposing the configuration keeping the 0..3...7.... in common, or trans posing the intermediate scale by the interval complementary to the one which produced the original configuration, and at least in A, B, and C this is also the one with the highest common-tone index. An equivalent description of these operations is that of “multiplica tion” of 0 .. 3... 7... . by each of the five symmetrical trichords and the tritone, shown in Figure 3. A ...9 . . 0 . .3... 3-10 B . . 8 ...0...4.. 3-12 C .7____ 0____S. 3-9 D E01...... 3-1 E T.0.2----- 3-6 F 6 ...... 0 ...... 6 2-6 Figure 3. Symmetrical trichords formed by complementary transpositions. The resulting scales will of course not only contain these trichords but also be generatable by them at the triad’s intervals of transposition. Now for each scale there are two trans positionally equivalent configurations involving 0..3...7...., but there is also a third configuration between the other two transpositions of 0. .3. . .7 always related by an even n of T„. In F this is To, so still no new configuration arises, but in B there is necessarily 34 a third instance of the original T4 configuration. For A, C, D, and E this third configuration differs from the original one, and since these are at To, T3, T3, and T4, respectively, then A’s scale will contain 6-30P, C’s and D’s 6-33P, and E’s 5-21P. Table 7 lists these resulting scales, their interval vectors, the number of minor triads (P) and incidental major triads (I) embedded, and the number of times the original configuration is embedded. (To replicate F’s configuration we settle for the smallest scale containing two such configurations, at T3 generating 8-28, which incidentally contains A’s original configuration four times.) T a b le 7. Scales successively generated by Tn-conSguration3 of 3-11. Scale (S) V EMB(S,3-11) Configs, A 7-3 IP 0..34.67.9T. 336333 3P, 21 2 B 6-20 0..34..78..E 303630 3P, 31 3 C 7-35 0.23.S.78.T. 254361 3P, 31 2 D 9-4P 01234.678..E 766773 5P, 51 2 E “■•221 0123.B.7.9T. 465562 4P, 31 2 F 8-28 01.34.67.9T. 448444 4P, 41 2 In each case, if the scale is I-symmetrical (B, C, F), then the major triad has all the same relations within the same scale, and if not (A, D, E), the major triad could similarly gen erate the scales's inversion. If a scale generated in this way is not already symmetrical, then in many cases the number of configurations can be doubled by the addition of only one PC, as in A, here, to make 8-28. Continuing the process always yields scales of high degrees of symmetry; this point was reached immediately for B in 6-20, but in some cases it would not be reached until the aggregate—here the loss of parsimony could be overcome in compositional deployment through judicious use of register. C’s scale is of course the diatonic scale of classical triadic tonality, and has other special properties, including unique multiplicity of each IC (and thus CTI) and only two intervals of adjacency, both distinct from and smaller than any harmonic (i.e., triadic) interval; such properties may or may not be significant in a given context. Suppose now that a scale is to be successively generated from a configuration of two inversionally related but transpositionally nonequivalent forms of a chord. In general we cannot create a pair of transpositionally equivalent T„I-configurations such that a single chord is involved in both, since there are ordinarily up to twelve distinct classes resulting from T„I-configurations. The exceptions, of course, are interesting, and particularly when a Tn- and T„I-configuration can produce the same scale, but normally we settle for one of up to six different scales that can contain a nonoverlapping pair of TnI-configurations, much as with F above, and often there are several “most parsimonious” possibilities, generally producing scales slightly larger than with Tn-configurations. This procedure can as well be extended to configurations of M-related chords. As the fourth and last possibility for multiple generation, consider now a combination of aspects of several of the cases above, in which a particular generating chord is involved in three or more different configurations generating clas3-distinct intermediate scales, but in such a way that the same large scale always results, and, as in the case of Ts:Tq:T4 above, no new configurations arise between any pair of the generating chords, and thus combinations of any two always generate one of the intermediate scales, yielding a closed cycle of configurations. In other words (see Figure 4), configurations C0:C1, C0:C2, and C0:C3 generate scales SI, S2, and S3, respectively, in such a way that configuration Cl:C2 is transpositionally equivalent to C0:C3, generating an intermediate scale of S3’s class, and likewise C2:C3 to C0:C1, generating S i’s class, and C3:C1 to C0:C2, generating S2. The 35 C l T si /■ i \ SS CO S3 / / ' \ \ S3 S2 ,/ \ C 3 -— SI — ► C 2 F ig u re 4 . Generalized model of a closed cycle of configurations. necessary relation among the configurations (if there are three configurations involving four chords of a type) is that one be at To, and the other two be at any T„I and Tn+oI.39 For example, if C0=3-4P, with configurations at T C 0=3-4P 0 1 ...5 C0:Cl=6-7 01...567...E C0:C2=6-8 0123.5 T. C l= T 0(CO)...... 67...E C0:C3=6-20 01..45..89.. C2=T3I(C0) ..23...... T. C2:C3=6-7 ..234...89T. C l:C 3= 6-8 ----- 4 .6 7 8 9 .E C3=ToI(C0) ___ 4...89.. Cl:C2=6-20 ..23..67..TE F ig u re 5. A closed cycle of configurations o f 3-4. Any two configurations of Figure 5 involving three different chords combine to generate 9*4, and since the. four chords together generate the aggregate, and each of the intermediate scales is an all-combinatorial hexachord, this is obviously related to the trichordal row derivation of serial theory, well documented in the literature.40 But the point here is that either member of any configuration can be held as common tones, the other moved to form another configuration, and in so moving from one intermediate scale to another within the large scale 9-4, the third intermediate scale results from the two sets of non-common tones, i.e., is a residue of the voice-leading process. Not even this last feature is lost when common tones are present in the configurations, as Figure 6 shows. C0=4-Z15P 01..4.6...... C0:C1=6-30I 01..4.67..T. C0:C2=6-Z5O 01..4.67.9.. C 1=T g(C0) 0 ...... 67..T. C0:C3=6-Z13 0 1 .3 4 .6 7 ___ C2=TiI(C0) 0 1 7.9. . C2:C3=6-30P 01.3..67.9.. C1:C3=6-Z50 01.3..67..T. C3= T tI(C0) .1.3..67___ C1:C2=6-Z13 0 1 ___ 6 7 .9T. Figure 6. A cycle of 4-Z15. 39 Even for I-symmetrical sets, the appropriate TnI must be considered. Also, it is clear that SG{2) equivalence is necessary here, for I-nonsymmetrical chords. 40 See particularly Babbitt (1955], [1961], [1973], and Martino [1961]. 36 7-31 is the result of the combination of any two configurations, and the “aggregate” here is the octatonic scale 8-28. Since one scale (6-30) is M-symmetrical and the other two are M-related, the same complex must be generatable by the M-related class (Figure 7). C0=4-Z29P 01.3... 7 ____ C0:C1=6-30P 01.3..67.9.. C0:C2=6-Z50 01.3. . 67. . T. C l= T 0(CO) .1 67.9.. C0:C3=6-Z13 01.34.67 ____ C 2 = T 1I(C0) 0 1 ____ 6 ...T. C2:C3=6-30I 01..4.67..T. C1:C3=6-Z50 01. . 4.67.9. . C3=T7I(C0) 0. ._.4.67 _ C1:C2=6-Z13 01 ____ 6 7 .9T. F ig u re 7. A corresponding closed cycle of 4-Z29. Nor are the sizes of the intermediate scales necessarily the same (Figure 8). C0=3-11P 0 . .3...7___ C0:C1=6-30P 01.3..67.9.. C0:C2=5-34 0..3.S.7.9.. C1=T0(C0) .1___ 6 . .9. . C0:C3=5-22 0..3..67...E C 2= T oI(C0) 0 ___ 5. . .9.. C2:C3=6-30I 0..3.56.,9.E C l:C 3=5-34 .1.3..6 . .9.E C 3= T oI(C0) . . .3 . . 6 ___ E C l:C 2=5-22 01...56..9.. F ig u re 8. A closed cycle of triads with intermediate scales of different sixes. For any of the cases shown in Figures 5-8 there are five other cycles producing similar results, though with different resultant scale types. This principle can be extended to encompass similar groups of four configurations (in volving five members of a SG(2) chord type) by combining a chord with its T4 and each of one of the following groups of T„I-configurations: T„I, Tn+4I, and Tn+gl, where n = 0, 1, 2, or 3. Such cycles are somewhat les3 “dosed,” since one of the intermediate scales is always left out of the immediate process, and since there is most often more than one duplication of a single configuration within the group. Figure 9 gives an example. C0:C1=7-Z37 0.234.67...E C0=4-14P 0.23... 7 ____ C0:C2=5-Z17 0.23...7...E C0:C3=7-Z37 0.234.67...E C1=T4(C0) 4.67.. .E C0:C4=6-Z26 0.23. . .78.T. C2=T3I(C0) 0.2 ____ 7...E C2:C3=7-Z37 0.234.67...E C1:C3=5-Z17 ...34.67...E C3=ToI(C0) ...34.6 E Cl:C2=6-Z26 0.2.4.67...E C1:C4=7-Z37 . . . 34.678. TE C 4 = T 10I(CO) ...3...78.T. . C2:C4=7-Z37 0.23... 78. TE C3:C4=7-Z37 .. . 34.678. TE F ig u re 9. A closed cycle of four configurations o f 4-14. In Figure 9, T0:T4 and ToiTqI are distinct configurations, both generating 7-Z37, but the former occurs a total of four times within pairs of members of the group—the “aggregate” here is 9-12. For six configurations to form this sort of cycle, To is combined with both T3 and To and each of one of the following groups: T„I, T n^I, T„+oI, Tn+oI, where n = 0, 1, or 2. For nine, To is combined with each of T3, T4, and To, and each of the group of either the even-numbered transposed inversions, or the odd ones. 37 Exceptional Chord Configurations For any configuration of two chords generating a scale, there are seven subsets of the scale whose relations to one another with respect to symmetry or class may be considered: the generating chords themselves (Cl and C2, of class C), the scale itself (S), the set of common tones (CT), the set of non-common or "melodic" tones in each chord (Ml and M2), and their union (MU). The discussion of these relations will be limited to cases in which the generators are related by T„ or T„I, and since #C 1 = #C 2, then #M 1 = #M 2, and #MU is always an even number. The size of the set of common tones, i.e., the CTI, is (#C x 2) — #S. Since CT and MU (or CT, Ml, and M2) partition the scale, #M U = #S — CTI, and if either CT or MU is null, the other is the same set as S. The symmetry of the generating chords affects the symmetry of the other sets. If the class of the chords is I-nonsymmetrical, then T-configurations may produce I-nonsymmetri- cal CT and MU, regardless of the degree of symmetry of the scale, though more often one or both of CT and MU are I-symmetrical, and necessarily so for sizes of 0, 1, or 2 for CT and MU. In this case Ml and M2 may differ in degree of symmetry, and generally do differ in class as well. Excepted is any configuration at Tg» in which both CT and MU are sets generatable by some number of tritones (including none) and are thus always T g- symmetrical (if nonnull); in this case Ml and M2 are always SG(l)-related. If the two chords of the configuration are TnI~related, including for T-configurations of I-symmetrical chords, CT and MU are always I-symmetrical, rnd Ml and M2 are of the same SG(2) class, either both I-symmetrical themselves or TnI-related I-nonsymmetrical sets. There are several exceptional cases for which M l and M2 are of the same SG(l) class in configurations which do not ordinarily produce this result, i.e., Tn-configurations of I-nonsymmetrical chords where n ^ 6. These are listed in Table 8.41 T a b le 8. Exceptional Tn-conBgurations where M l and M2 are of the same class. Chord class Configuration Chord class Configuration 5-13/30 t 4 7-19 t 3 5-19 t 3 7-21 t 2 5-26 t 4 7-26 t 4 5-28 t 3 7-28 t 3 6-Z10/Z46 t 4 8-2/22 Ts/Ti 6-Z39/Z24 t 4 8-5/16 t 3 6-Z12 t 4 8-12/27 T c /T i 6-Z41 t 4 8-13 t 4 6-Z17 t 4 8-18 t 2 G-Z43 t 4 8-19 t 2 6-Z19/Z44 T i,T 3,T5 ni w w m 6-14 9-2/7 T 4,T3/ T i 6-30 T3 9-3/11 Ts.Ta/Ti 9-4 t 2, t 3 7-7 t 3 9-5 t 2, t 3, t 4 7-13/30 t 4 9-8 T i,T 3,T 5 41 Excluded from Table 8 is the case of CTI = 0, since here both Ml and M2 are the generating chords themselves, and also the case of CTI = #C —1, i.e., where Ml and M2 are monadic. 38 For configurations, at the same transpositions, of complementary or M-related chord types, there are several relations which can be useful, particularly for self-complementary hexachords. It is simpler to state these entirely in SG(2) terms, since in this system a set’s abstract complement’s literal complement is always a member of the class of the original set. For a configuration A of CI:C2 at some T„ and a configuration B of Cl':C2' at the same T„, A’s scale and B’s common tones are (abstract) complements, and vice versa; Ml, M2, and MU are of the same size and class in both A and B. The same holds for T„I- configurations, except that the exact values of n producing the correspondences depend on the structure of the chords; in any case the relation holds between both groups of twelve T„I- configurations to within rotational equivalence. For M-symmetrical chords, configurations at T2, T3, T4, and Tu produce M-symmetrical S, CT, Ml, M2, and MU, while these five sets for Ti-configurations and those for Ts-configurations are M-related to each other. Likewise, for T„I-configurations of M-symmetrical chords, resulting sets are either M-symmetrical themselves or paired with M-related sets of another T„I-configuration of the same chords. Two M-related configurations of M-symmetrical chords at Tj, T3, Tj, or To yield M- related classes for their sets of S, CT, Ml, M2, and MU; the M-relations also hold between one’s Ti- and the other’s Tj-configurations. The meaning of all this for the twenty self- complementary hexachordal chord types is that for Tn-configurations, the common tones always comprise the abstract complement of the scale generated. For TuI-configurations of the I-nonsymmetrical ones, this is true only for the uniquely T-self-compleinentary 6- 14, but in other cases, it is true for two tritone-related T„I-configurations, one of which is the literal-complement-mapping operation, and for each of the other T„I-configurations producing a particular scale, some other Tn[-configuration holds that scale’s complement as common tones, and has the same MU class. If the (self-complementary) hexachord is M- symmetrical, the MU of the configuration at Ti is M-related to the MU of the configuration at T&, but this is not the case if the hexachord is M-nonsymmetrical. Hexachordal types involved in a Z-pairing are no different in any of these respects than are complementary sets in general, except that there are more than the usual number of correspondences in configurations of the six Z-hexachordal types which are M-related to their complements. For example, for any T„-configuration except Ti and Tc, which are exchanged, CT is always SC(3)-related to the complement of the scale, and often this is the SG(2) complement as well. In generating scales from combinations of chords, particularly chords of the same class, it may be of compositional importance to achieve maximum unity of “class content’’ within a configuration. Clearly, CT and Ml are literal subsets of Cl, as are CT and M2 of C2. and both Ml and M2 are abstractly subsets of both Cl and C2 when the latter are considered as a class C, but there are not necessarily any subsets in common between CT and Ml or M2, or CT and MU. Also, MU is generally not an abstract subset of C—one reason is that for this to be possible more than half a chord’s members must be common tones, but there is also no general correspondence between the content of the chords and the common-tone function of M l and M2, representing the voice-leading possibilities within the configuration’s scale, and this of course suggests that a functional differentiation be made between simultaneous and successive events. Still, there are many exceptions, particularly with respect to partially shared subset content among these sets (not to be pursued here), and there are a few exceptional configurations for which the sets of one of the following groups are of the same class: I) CT, Ml, and M2, a few of which also have MU and C of the same class, or 2) CT and MU. The first applies to generating chords of size 2, 4, 6, or 8, and the second to chords of size 3, 6, or 9, as these are the situations that produce sets of the same size within each group. In the first case, while C1:C2 generates S, Cl is being generated by CT:M1 and C2 by CT:M2, each with no common tones and at complementary transpositions, such that a configuration of Cl and C2 can hold CT in common with no intersection of Ml 39 and M2. This is usually a special case of the successive scale generation described earlier, except that here the intermediate scales are the ones actually being treated as chords, and in most cases all the components are symmetrical. Thus, every dyad except 2*6 has this property (almost trivially, since all monads are of the same type) at its configuration with one common tone, producing one of the five I-symmetrical trichords. Every T-nonsymmetrical tetrachord pair-generatable by a dyad type, i.e., every tetrachord with °S = 2 except 4-6 and 4-24, can be in at least one self-configuration generating, with CT, Ml, and M2 of the same class, an I-symmetrical hexachordal scale. Likewise, ten of the hexachords pair- generatable by symmetrical trichords can, with this property, further generate one of the five symmetrical nonachords, at the same T„-configuration that relates the trichordal generators if that configuration has three common tones, and the six octachordal complements of all- combinatorial tetrachords have this property at some configuration generating the aggregate. In each of these the results are due to symmetries originating in the class to which CT, Ml, and M2 belong and to the use of complementary transpositions, but there are a few additional cases for which none of the sets involved (except the scale) is symmetrical, and the property here is truly exceptional, a result of the specific structure of the combination of several sets. Table 9. Exceptional T„-con6gurations where CT, M l, and M 2 are of the same class, for I-nonsymmetrical chords. Chord Con fig. Scale C T /M 1/M 2 MU 6-Z10P t 4 9-12 3-2P 6-Z10P 6-Z46P t 4 9-12 3-71 6-Z46P 6-14P t 3 9-41 3-4P 6-7 6-Z19P Tx 9-4P 3-11P 6-33P 6-Z41P T2 9-6 3-5P 6-Z43I 6-Z43P t 4 9-12 3-51 6-Z43P 8-13P t 4 12-1 4-131 8-13P Table 9 shows these; note that the equivalence is limited to SG(l). The first two form an M-related pair of classes; none of the complements of these sets has the property (except the self-complementary 6-14), even though 6-Z19 is M-related to its complement/ 3 8-13 is the complement of the T-only combinatorial tetrachord. Of the seven configurations of Table 9, and the symmetrical cases described just above, the ones which generate, but do not already contain, ‘‘multiples” of augmented triads (3-12, 6-20, 6-35, 9-12, and 12-1),43 all at T4, also result in MU of the same class as the generating chord class itself. The configuration of Figure 10 has both CT, Ml, M2, and C, MU of the same SG(l) class. 42 Concerning this configuration of 6-Z19, Forte [1973a], p. 35, cites the first measure of the Introduction to Part II of Stravinsky’s Lt sac re du printemps, stating that interval content (i.e., 6-Z19) remains constant as the common tones are held and the “remaining elements change,” but does not make the point that both sets of changing elements are transpositionally related to each other and to the set of common tones, all triadic in this case. 43 These generators are, namely, 2-4, 4-7/20, 4-21, 4-25, 6-Z4, 6-Z10/Z46, 6-Z26, 6-Z43, 6-Z49, 8-1/23, 8-6, 8-9, 8-10, 8-13, and 8-28, which are all of the classes generatable by some number of nonintersecting instances of 2-4. 40 C l 012..56.8... To C2 0...456..91. t 4 S 012.4S6.89T. 9-12 CT 0... .56..... 3-51 I M l .1 2 ...... 8 . . . 3-51 I M2 ___ 4 ____ 9T. 3-51 MU .12.4...89T. Ts(Cl) Figure 10. An exceptional configuration o f 6-Z43P. In the second case, for chords of size 3, 6, or 9 to produce configurations with CT and MU of the same class, the resulting scale is always T-symmetrical, specifically 4-9, 4-25, 4- 28, 8-9, 8-28, or the aggregate. The configurations involved produce triangular closed cycles of configurations, i.e, with only one exception for 6-30, a To-configuration and one or more pairs of TnI/T n+ 3-5P To 4-9 2-6 1-1 2-6 T XI 4-9 2-1 1-1 2-1 t 7i 4-9 2-5 1-1 2-5 In the first configuration of Table 10, 1-1 at T q :T g generates 2-6 in both CT and MU, and 2-6 at T 3-8P To, T2I, TaI 4-25 2-6, 2-2, 2-4 1-1 3-10 T0, T3 (=T0I), T0 (=T3I) 4-28 2-6, 2-3, 2-3 1-1 Since 3-10 is I-symmetrical, its third configuration is equivalent to its second. Of course, all three of these tetrachordal scales have these generating chords as their only trichordal subset, so perhaps the property is not unexpected here. There are two groups of hexachords 41 with the property: 1) All nonsymmetrical hexachordal generators of 8-9 and 2) all six hexachordal subsets of the octatonic scale 8-28, regardless of symmetry.44 In Table 12, the first pair of each group is M-related. T a b le 12. Hexnchords with this property. Chord Configurations Scale Respective CT (=MU) M 6-5P T g, T 3I, To I 8-9 4-9, 4-1, 4-7 2-1 6-18P To, Til, TrI 8-9 4-9, 4-20, 4-23 2-5 6-Z41P Tg, T 3I, TqI 8-9 4-25, 4-1, 4-23 2-2 &-Z43P To, T J , T 7I 8-9 4-25, 4-20, 4-7 2-4 6-Z13 Tg, T 3 8-28 4-9, 4-3 2-1 6-Z50 T c,T 3 8-28 4-9, 4-26 2-5 6-Z23 To, T3 8-28 4-25, 4-10 2-2 6-Z49 To, T 3 8-28 4-25, 4-17 2-4 6-27P To, Til, T7I, T4I, Tml 8-28 4-28, 4-17, 4-10, 4-3, 4-20 2-3 6-30 P T3, T il, T 7I, T 4I, T 10I 8-28 4-28, 4-9, 4-9, 4-25, 4-25 2-6 ire 11 shows explicitly these three configurations of 6-Z43, the only chord type in the system to have all of the properties under consideration, i.e., CT, M l, M2 of the same class with the exceptional MU of the same class as itself (see Figure 10), and here, CT and MU of the same class, though of course at different configurations.45 C l .012..56.8.. To .0 1 2 . .5 6 .8 .. To .012..56.8.. To C2 E0.2...6.8.. To E01. . .5.78.. T il E.12..567... T 7I S E012..5678.. 8-9 E012. .5678.. 8-9 E012..5678.. 8-9 CT .0.2...6.8.. 4-25 .01. ..5..8.. 4-20 .. 1 2 . . 5 6 ___ 4-7 II I MU E.l...5.7... 4-25 E. .2. ..67... 4-20 E 0...... 7 8 .. 4-7 M l . . 1 . . . 5 ...... 2-4 .. . 2 . . .6 ___ 2-4 . 0 ...... 8 . . 2-4 M2 E...... 7. . . 2-4 E ___ . . . 7 . . . 2-4 E ...... 7. . . 2-4 Figure 11. Additional exceptional configurations o f 0-Z43. - i —.t .. - ■ ■ — 44 The 8-28 group represents all hexachordal generating configurations, and the property here results from the octatonic scale’s 4-generability by any dyadic type. 8-25, the one other T-symmetrical octachord besides 8-9 and 8-28, contains no hexachord generating it at more than two configurations. 45 Recall that this class, the complement of the "all-trichord hexachord” 6-Z17, also has the unique (for hexachords) maximum value of 12 for CCOV9. 6-Z43 is also one of three hexachordal classes with the maximum number of “basic interval patterns” by Forte’s defi nition [1973b], 241; it occurs frequently in the preserial works of Webern, perhaps because its only pair-generator is 3-5, prominent in this music. 42 Nine of the twelve nonachordal types, when treated as chords, have this property at each of some group of configurations generating the aggregate. In each case the hexachordal type of CT and MU is either all-combinatorial (ordinals 1/32, 7, 8, 20, or 35) or the “almost” all- combinatorial 6-30, and M, the class of the small trichordal generator, is the complement of the chord. Of the three remaining nonachords, two (9-5 and 9-8) have only one configuration with CT and MU of the same class, and the other, 9-10, cannot have the property since its trichordal complement cannot 4-generate the aggregate.40 This situation, incidentally, is comparable to that of the hexachords generating the octatonic scale: if the scale is treated as an eight-member “aggregate,” then in each case above, M is the dy&dic complement of the hexachord in each set of configurations, and the seven tetrachordal types involved can be considered “all-combinatorial” since each is I-symmetrical and can be mapped into its tetrachordal complement under T„. and T„I.47 Voice-Leading Considerations Assuming a distinction between “harmonic” and “melodic” pitch events, the former may be either simultaneous or successive, but the latter is taken to be a succession of pitches, and it may be that a particular pattern of succession is to be made to provide consistency within a composition, perhaps in addition to one or more chord types as harmonic reference structures. There might even be a distinction made between the function of intervals with respect to these functions; for example, in classical tonality, the triad is the harmonic referent, and “intervals of adjacency” are those smaller than the smallest triadic interval, and the smaller of these two, the semitone, has other special functions within the system. While there need be no limitations on intervals of succession, or a clear distinction between these and intervals of simultaneity, at least it is true that successive motion by semitone is “maximum adjacency” in the chromatic system, and we may even wish to distinguish between, say, pitch-class semitones stated in the same register when successive, and as some other member of the interval-class when simultaneous, whether or not the pitches involved belong to the same harmony, and it is to this extent that the interval-class construct is relaxed at this point. The intent in this section is to provide, by example, a sense of the scope of possibilities for a particular voice-leading pattern, and in the following examples it is assumed that the semitone is the referential interval of succession. There are many possibilities for progressing from one chord to another of the same size, holding common tones and moving one or two voices by semitone, but let us also assume, as in most of the foregoing, that harmonic consistency is to be maintained by progressing to another chord with at least the same interval-class content. 40 The more general situation for nonachords is that at any configuration generating the aggregate, M is the chord’s trichordal complement and CT and MU are complementary hex achords, usually either Z-related, or T„I-related. The only hexachordal types not involved are those with no trichordal pair-generator which are not Z-related to one that does. 47 The idea of complement mapping can be extended to the four “l-combinatorial” tetra- chords with respect to this scale; there are also two Z-related half-aggregates in this system— not surprisingly, these are the all-interval tetrachords, Z-related in the chromatic system as well. These are the only tetrachordal subsets of the scale which cannot pair-generate it, but they are complementary to one another. Even if the scale is considered as a quasi-equally tempered mod. 8 system, the tetrachords corresp.onding to these (01.34.. . and 012. .5. .) are still Z-related, and the only classes in the entire system to be so related. (See Clough [1983[ for an exposition of a mod. 8 pitch-class system.) 43 Table 13 lists self-configurations of chords with all but one of their tones in common and MU=2-1.48 Self-configurations of chords of even sues have even interval-classes for MU when #M U=2, so do not have this property. In each case the complement of the chord has a configuration with the same MU, though of course the scales generated are not complementary.40 The triad class 3-11 is the only one with two such configurations, and the pentatonic scale class 5-35 the only one to have the property at a T„-configuration, and the only symmetrical class. T a b le 13. Self-conBgurations for which MU—2-1, indicating possible sewitonal voice-leading connection. Chord Config. Scale CT 1-1 Tr 2-1 — 3-2 t 3i 4-1 2-3 3-5 T il 4-9 2-1 3-7 t 6i 4-10 2-5 3-11 t 3i 4-20 2-3 T r I 4-17 2-5 5-3 T*I 6-1 4-7 5-5 t 3i 6-Z38 4-1 5-19 T rI 6-Z13 4-9 5-20 T 3I 6-Z38 4-20 5-21 Tfil 6-20 4-7 5-23 . T r I 6-8 4-23 5-25 T sI 6-Z50 4-10 5-31 t 3i 6-Z42 4-28 5-32 T il 6-Z50 4-17 5-35 t 5 6-32 4-23 Suppose now that we wish for two melodic voices to produce semitonal successions in self configurations, with the further constraint that they move in contrary motion (in the pitch domain). The first constraint limits MU to one of the five (I-symmetrical) tetrachords that can be partitioned into a pair of dyad type 2-1, and where this partition does not represent Ml and M2, since the 2-1 pair is to be available as successions. The contrary motion constraint further limits Ml and M2 to the dyadic partition of the tetrachord in which the dyads differ in class, at least for four of the five (4-1, 4-3, 4-7, and 4-8), thus making the property available only at T„-configurations (n ^ 6) of I-nonsymmetrical chords. Because of the additional symmetries of the fifth tetrachord (4-9), its partition into a pair of 2-5 yields semitonal contrary motion, and since Ml and M2 are of the same class, the property is available at both To and T„f-configurations, regardless of the symmetry of the chord. Because of the second constraint, we have limited the possibility of MU=4-8 out of existence, and nearly so for the others except 4-9 (see Table 14). 48 M-transforms of these configurations do not share the property, since for these the corresponding MU is 2-5. 40 But recall that the scale generated by the complement of a configuration A is the complement of A’s CT, and vice versa. 44 T a b le 14. Self-conGgurations with possible-semitonal successions in contrary motion. Chord Config. Scale CT M l M2 MU 6-34P t 2 8-21 4-21 2-1 2-3 4-1 4-27 P t 3 6-27P 2-3 2-2 2-4 4-3 3-11P t 4 5-271 1-1 2-3 2-5 4-7 6-34P t 4 8-24 4-24 2-3 2-5 4-7 2-5 ■ To 4-9 ... 2-5 2-5 4-9 4-14P t 3i 6-Z38 2-3 i i i 4-16P To 6-7 2-6 4-26 t 3 6-Z50 2-5 4-27P To 6-30P 2-6 6-14P T sI 8-7 4-7 6-18P To 8-9 4-9 6-Z29 To 8-9 4-28 6-Z50 To 8-28 4-9 6-32 t 3 8-23 4-23 6-34P To 8-25 4-25 Again the triad class has the property, and most of the other chords in Table 14 contain 3-11. 4-27, the dominant-seventh or half-diminished seventh chord class, has two configurations with the property, and 6-34 is the only chord in the system with three—this is Skryabin’s “mystic chord,” shown in Figure 12 at a transposition often encountered in his works.50 To 0.2.4.6. .9T. t 2 0.2.4.6.8..E To 0.2.4.6..9T. t 4 .12.4.6.8.T. To 0.2.4.6..9T. To 0..34.6.8.T. Figure 12. Exceptional configurations o f 6-34. There are many other configurations which produce, for two moving voices, parallel dyads (in the pitch domain) or some pairing of dyadic types. Even taking into account configurations involving chords of different classes, there are for a given set of constraints imposed on a voice-leading pattern normally a small number of configurations which pro duce it. 50 These configurations are discussed by Proctor [1983]. Chapter IV Examples of Set-Complexes for Composition This chapter illustrates the application of many of the foregoing principles to some specific sets, sometimes several at a time. It is intended to be not exhaustive but indicative of the variety and flexibility of techniques available for scale generation. The term sef- complex is used in a general sense to mean a collection of PCsets or types resulting from a group of given procedures or constraints, the concurrent use of which thus relates those sets or types. E x a m p le A A set-complex in this sense might be defined, for example, by either mutual inclusion of or mutual generability by a particular chord type, in this example a major or minor triad (3-11), and further constraints might be imposed on the sort of inclusion or generation involved. Table 15 shows the effect of certain such constraints on the size of resulting complexes, where the given figure is the number of SG(2) types of sizes 4 through 10 that share the listed properties with respect to the triad (or its M-transform 3-3). T a b le 15. Sizes of scale-complexes for 3-1J. Size 4 5 6 7 8 9 10 Totals Total classes of each size 29 38 50 38 29 12 6 202 As subsets: one or more triads 9 25 43 37 29 12 6 161 two or more SG(1) triads 0 3 18 29 28 12 6 96 two or more SG(2) triads 3 13 36 36 29 12 6 134 three or more SG(l) triads 0 0 2 13 22 12 6 55 three or more 5G{2) triads 0 3 17 29 28 12 6 95 As generators: 2-configurations: T„/Tn 0 3 3 6 T „ /T „ I 3 3 6 12 3-configurations: T„/T„/T„ 2 8 6 I 17 t „/ t „/ t „ i 3 16 13 3 35 The first row of figures gives CCOV„(3-ll) for each of these sizes, and in the last column TCCOV(3-ll), indicating the number of types generatable by the triad with some other chord or chords, including itself. In the next two rows, the limitation on the scales is that the triad be embedded twice or more, the instances Tn-related (in the second row) or either Tn- or TnI-related (in the third), and the same in the following two rows for three or more triads. It is evident that for all of these complexes larger scales are less well 45 46 distinguished from other scales of their size by the property of triad containment. The six scales sharing pair-generability by Tn-configurations of triads were treated in detail above in connection with Figure 2, and are all three pentachordal types and three of the eighteen hexachordal types from the second row of Table 15. Likewise, there are twelve scales generated by TnI-configurations, and in this case there is no overlap with the set of T„-generated scales. Many of these scales contain more than two instances of triads, but where other pairs do not generate the scale. Among new scales 3-generatable by Tn-related triads are the diatonic scale and seven other heptachordal types, and but one nonachordal type, namely 9-4, mentioned earlier in connection with a configuration of 6-Z19 in v/hich CT=Ml=M2=triad. Last is the number of scales generated by 3-configurations involving a triad, its transposition, and its transposed inversion; one of these scales is th$ triad's complementary type. Exam ple B As a simple example of a complex resulting from all pair-coufigurations of a dyadic type and an illustration of the effect of M-operations, in Figure 13 a semitonal dyad is held while another slides past it in the chromatic space to form alt configuration-classes. The non-common tones between one scale and the next form an interval of two semitones,51 and of course the process continues in reverse with T7 until the dyads re-merge at To- The M7-transform of the whole construct slides perfect fifths through a cycle-of-fifths space, and although all but the last two resultant sets are SG(2) nonequivalent to those of the original, the “ascending” melodic motion by two semitones is preserved, because of the M7-symmetry of this interval. The two complexes are structurally identical with respect to their generating subsets. To 0 1 ...... 2-1 To 0 ...... 7 ____ 2-5 Ti 0 1 2 ...... 3-1 t 7 0 . 2 ___ 7 ____ 3-9 t 2 0123 ...... 4-1 t 2 0 . 2 ___ 7 .9 . . 4-23 t 3 0 1 .3 4 ...... 4-3 To 0...4..7.9.. 4-26 t 4 0 1 ..4 5 ...... 4-7 t 4 0...4..7...E 4-20 t 5 0 1 . . . 5 6 ...... 4-8 Tu 0 ...... 6 7 .. .E 4-8 To 0 1 ___ 6 7 ____ 4-9 Tc 0 1 ----- 6 7 ____ 4-9 Figure 13. Pair-configurations of 2-1 and their M7-transforms. Either of the dyads of Figure 13 can 3-generate four pentachordal and seven hexachordal types, but here we will pursue, as a similarity relation, an extension of the idea that the tetrachordal types above are all and only those that can be partitioned into discrete Tn- related dyads of 2-1 or 2-5. For such partitions into like dyads, five tetrachords share the property with respect to a given generating dyad (except 2-6, generating only three), and all are as a result I-symmetrical. The extension is to consider all three dyadic partitions of all tetrachords and to assert that any two tetrachords generatable by some configuration of the same two dyadic types are thereby equivalent. Of course the condition-that the dyads be discrete, forming a partition of the tetrachord, is more restrictive than that of mutual 51 Webern, in Op. 5 # 4, first two measures, makes light of this melodic relation (here EJ-Ftl) between 4-8 and 4-9. 47 - embedding of the two dyadic types. Each of the fifteen other combinations (two at a time) of the six dyadic classes produces five tetrachord types if the the dyadic types are both even or both odd (only three if the tritone i3 involved), and four if one each is even and odd (two if the tritone is involved52). For example, 4-7 of Figure 13 above has a partition into a pair of 2-1 and a pair of 2-4, but also one into a 2-3 and a 2-5, whose configurations produce four additional tetrachordal types. Figure 14 shows the entire cycle of configurations of 2-3 and 2-5, including the trichords resulting from PC intersections of dyads. In this case, 2-5 is held, and the moving 2-3 is shown in boldface. 0 1 ..4 5 ...... •4-7 0 . 2 . . 5 ...... 3-7P 0 . . 3 . 5 6 ...... 4-131 0 . . . 4 5 . 7 ___ 4-141 0 ___ 5..8... 3-1 IP 0 ___ 5 6 . .9 . . 4-18P 0 ___ 5.7. .T. •4-23 0 ___ S. .8 . .E 4-181 0 ___ 5.. .9. . 3-111 0 1 . . . 5 ___ T . 4-14P 0 .2 . . 5 ...... E 4-13P 0 . . 3 . 5 ...... 3-71 Figure 14. Configurations of 2-3 with 2-5 . Notice that the sequence is retrograde-equivalent to itself with respect to the SG(2) classes produced, pivoting, in this case, around the I-symmetrical “axis” types 4-7 and 4-23, each represented once, and this occurs when both dyads are even or odd. When one even and one odd dyad are the generators, these “axes” fall “between” configurations, and the total number of classes of the complex is reduced by one. For example, in Figure 15 a segment of the cycle of configurations of 2-1 and 2-4 is shown, and, to the right, its M7-trhnsform. In both, each of the middle pair of adjacent configurations produces 4-19. 0 1 2 . . . 6 ...... 4-5P 0.2...67 ___ 4-161 0 1 . 3 . . . 7 ___ 4-Z29P 0 1 ...... 7 .9 . . 4-Z15I o 00 •4-19P 0...4..78... •4-191 01...5...9.. •4-191 0..3...7...E •4-19P 0 1 ___ 6 . . .T . 4-Z29I 0 ...... 6 7 . .T . 4-Z15P 0 1 ...... 7 . . .E 4-51 0 1 . . . 5 . 7 ___ 4-16P Figure 15. Partial set of conBgurations of 2-1 with 2-4 and their M 7- transforms. Notice, in Figure 15, that the MU set of pairs adjacent in the list undergoes M7-transforma- tion as well, from 4-7 to 4-20, in such a way that there is still one semitonal succession possible. This sort of equivalence can be even further restricted by relating tetrachordal classes which could have the same three ICs between registrally or temporally adjacent PCs, disre garding the other three ICs in each. This means, for any particular dyadic partition, that 62 The pairs of dyadic classes whose nonintersecting configurations always produce one of only two tetrachordal classes are 2-3 and 2-6, generating one of the all-interval tetrachords 4- Z15 or 4-Z29, M-related to each other, 2-1 and 2-6, generating 4-5 or 4-12, and the M-related 2-5 and 2-6, generating 4-16 or 4-27. 48 one of the four remaining ICs among the members of the dyads is considered.53 Returning to the original example of Figure 13, in the partition of tetrachords into a pair of 2-1 the IC relating a PC of one dyad to a PC of the other ranges from IC 1 to IC 6, and for each there are as many as three of these tetrachords54 which could be ordered in such a way as to highlight the relation (see Figure 16). Interval pattern: 111 121 131 141 151 161 Common dyad: 2-1 2-2 2-3 2-4 2-5 2-6 Chords: 4-1 0123 0123 0123 4-3 0 1 .3 4 0 1 .3 4 0 1 .3 4 4-7 0 1 ..4 5 01..45 01..45 4-8 0 1 ...5 6 01...56 01...56 4-9 0 1 ___ 67 0 1 ___ 67 Figure 16. Other common dyads in tetrachords pair-generated by 2-1. Any of these relations could be easily enough extended to dyadic 3-generation, or to config urations of two trichords or even larger chords of different classes. E x a m p le C The next example, from the second of my Three Compositions for Orchestra , illustrates some interesting properties of all-interval tetrachords, and shows that it need not be contradictory to make simultaneous U3e of more than one kind of SG-equivalence. These two Z- and M- related SG(2) classes form a single SG(3) class with 48 .forms, twelve each of four SG(l) classes whose prime forms are given in Figure 17. a l 0 1 . . 4 . 6 ...... A 1 0 . 2 . . 5 6 ...... a2 0 1 . 3 . . . 7 ____ A2 0 . . . 4 . 6 7 ------ Figure 17. SG(1) prime forms of all-interval tetrachords. In other words, each SG(2) type has a “minor” and “major” SG(1) type, whose SG(3) equivalence allows for certain structural regularities when taken together.55 Beginning with any form, consider its four partitions into a PC and a trichord, i.e., isolating each chord member; because of the all-interval-class property, transposition such 53 This differs from Forte’s “bip” or basic interval pattern in that the central IC here remains so; Forte normalizes, for example, 151 (second-to-last in Figure 16) to 115, which also takes in 4-4 (012. .5) and 4-6 (012. . . .7), but ignores the original partition. This is useful for relating sets only if the ordering of the ICs in the presentation of the sets is not of significance. See Forte [l973b| and Forte (1973a|, 63-73. 54 For other dyadic partitions there are as many as four tetrachords related in this way. 55 Common-tone vectors for all pairings of these four SG(1) types are given in Chapter II in the section on common-tone relations, page 23. 49 that the monadic member of each partition is held as a common tone generates the aggregate with only one other PC duplication. Figure 18 shows this for a form of al. T 0 0 1 . . 4 . 6 ...... T2 ..23..6.8... T s ...... 5 6 . . 9 . E To 0 ...... 6 7 . . T. Figure 18. Transpositions of 01. .4.6 sharing PC 6. Holding as common a particular dyad and fitting it into the one member of each SG(1) class containing that dyad produces an octatonic scale in the case of 2-1, 2-2, 2-4, and 2-5, providing many possibilities for connections between pairs (Figure 19). a l 01 . 4 . 6 ...... a l 0 1 . . 4 . 6 ...... A l 01 ___ 7 . 9 . . A l 0. .3 4 ...... T. a2 01 3 . . . 7 ___ a2 ' 0 . . . 4 ___ 9T. A2 01...6...T. A2 0 . . . 4 . 6 7 ___ a l 01 . 4 . 6 ...... a l 0 1 ..4 .6 . A l ' , , .4 .6 ..9 1 . A l . 1 . 3 . . 6 7 ___ a2 , , 34.6...T. a2 . 1 ___ 6 7 .9 . . A2 0 . .4 .6 7 ___ A2 0 1 ___ 6 . . .T. Figure 19. All-interval tetrachords sharing forms of 2-1, 2-2, 2-4, and 2-5. In each case of Figure 19 the changing PCs produce the hexachordal octatonic scale comple ments of these dyads, which happen to be all and only the I-symmetrical ones. In the case of 2-3 and 2-6, since these can partition both SG(2) types, and there are no other dyadic partitions involving either, the changing PCs produce octatonic scales and the total scale i3 the aggregate complement of the common dyad, as shown in Figure 20. (Because of its T-symmetry, any 2-6 is contained by two members of each SG(1) type.) a l 0 1 . 4 . 6 ...... A l 0 .2 .5 6 ...... a l 0 1 . . 4 . 6 ...... a2 0 . . . 5 6 . 8 . . . A l .1 . .4 5 ...... E A2 0 . . 4 .6 7 ___ a2 .12.4...8... a l 0 . . . . 6 7 . .T. A2 .1 .3 4 ___ 9 . . A l 0 . . . . 6 . 8 . .E a2 0 .2 . . 6 . . . .E A2 01. ..6...T. Figure 20. All-interval tetrachords sharing forms of 2-3 and 2-6. The situation is different for the case of maximum (three) common tones, because since all four trichordal subsets are I-nonsymmetrical a given SG(l) trichord can be shared with only one other member of the SG(3) group, and this only if it is one of the two trichordal classes shared between SG(2) types, i.e., 3-5 or 3-8. Sharing the member of 3-8 produces the octatonic complement of 3-5, and vice vers$ (Figure 21). 50 al 01..4.6 al ..23..6.8... A2 0.. .4.67 ____ A2 0 .2 3 ------8 . . . 5-191 0 1 ..4 .6 7 ____ 5-28P 0.23..6.8... or or Al .1.3..67 ____ Al 0.2..56 ...... a2 01.3...7 a2 0 ____ 5 6 . 8 . . . 5-i9P 01.3..67 ____ 5-281 0 . 2 . . 5 6 . 8 . . . Figure 21. M-configurations of all-interval tetrachords sharing trichords. Notice that the members of the shared trichord class not held as common tones combine to form the highly symmetrical 4-9 (01. . . .67) or 4-25 (0.2. . .6.8). The pairs of types 3-5 and 3-8, 4-9 and 4-25, and 5-19 and 5-28 share (within pairings) many combinational properties, e.g., identical partition vectors unique to their cardinality type, but are not M-related. Among the other possible pair-configurations among these 48 SG(3) forms are seven with no common tones, in which the generating trichords partition the resultant scale. Of course there are no such Trt-con figurations because of the all-interval property, but three T„I-configurations of 5G(2) forms produce octachordal scales (the same three for both): 8-9, 8-10, and 8-17. Three more are possible with pairings of al:a2 or Al:A2 (T„M5- related): 8-3, 8-25, 8-26. Finally, one configuration each of al:A2 and Al:a2 (i.e., the same T„M7-related pairs that yield maximum common tones) is a partition of the octatonic scale (Figure 22). a l 0 1 . . 4 . 6 ...... A l ____ 4 .6 ..9 T . A2 ...3...7.9T. a2 01.3...7...: Figure 22. Pair-configurations of all-interval tetrachords generating 8-28. The seven octachords just mentioned are the aggregate complements of the seven tetrachords pair-generatable by either 2-3 or 2-6. Suppose now that we wish for a form of all-interval tetrachord, rather than to be a generating chord, to result from the melodic motion of two voices, i.e., to be the MU of pair-configurations of whatever chord types are required to produce this. The conditions are that Ml and M2 be one of the three dyadic partitions of one of the tetrachords: for 4-Z15, one of two configurations of 2-1 and 2-2 or 2-4 and 2-5, or one of four configurations of 2-3 and 2-6, and for 4-Z29, one of two configurations of 2-1 and 2-4 or 2-2 and 2-5, or one of the four remaining nonintersecting configurations of 2-3 and 2-6. Self-configurations with this property are shown in Table 16. Adjacent pairs of configurations are M-related. Note the relation of the chords and scales of the first two pairs to the common tones and chords, respectively, of the last two pairs; all are octatonic subsets, and, as it turns out, in these eight configurations all the sets involved belong to the SG(1) subset/superset complex of SG(3) 4-Z15/Z29. Figure 23 shows a realization of the specific case of 3-1 IP from Table 16 as generator, and, through successive scale generation, 5-32P as generator, such that all-interval tetrachords are em bedded in multiple ways as successions, at more than one rhythmic level. In Figure 23, 51 T ab le 16. Self-configurations where MU is a m ember of SG(3) 4-Z15/Z29. Chord Config. Scale CT M l, M2 MU M c* e* t 3-2P t 3 5-10P 1-1 1 a l 3-7P t 3 5-25P 1-1 2-2, 2-5 A2 3-3P t 3 5-16P 1-1 2-1, 2-4 a2 3 -llP t 3 5-32P 1-1 . 2-5, 2-4 a l 3-4P T i 5-6P 1-1 2-5, 2-4 A l 3-4 P T s 5-201 1-1 2-1, 2-4 A2 5-4P T! 7-4P 3-1 2-6, 2-3 A2 5-29P Ts 7-29P 3-9 2-3, 2-6 a l 5-10P t 3 7-31P 3-2P 2-1, 2-2 A l 5-25 P t 3 7-3 IP 3-7P 2-2, 2-5 a2 5-16P t 3 7-311 3-3P 2-1, 2-4 A2 5-32P Ta 7-3 IP 3 -U P 2-5. 2-4 A l Chords Voice 1 Voice 2 A l J, a l | a) To 0..3...7.... al . .3 ...... 7 ____ To 0 . . . 4 . . . . 9 ...... 9 .. ___ 4 ...... ' b) To I 0 ___ 5. . , 9 . . Al ___ 5 ...... 9 .. T il 0..3....8...... 8 . . . .. , 3 ...... c) To 0..34..7.9.. t 3 0..3..67..T. Chords C T = a)-b) Voice 3 Voice 4 a2 1 a2 i d) Tn 0..34..7.9.. 0..3...7.... Al . . .4 ...... 9 . . Ta 0..3..67..T...... T...... 6 ...... To 0. .34..7.9.. 0...4....9.. Al . . 3 ...... 7 ____ To 01..4.6..9...... 6 . ____ .1 ...... A2 I A2 I To I 0..3.5..89.. 0 ___ S . . .9 .. a l ...... 8...... 3 ...... To I 0.2..56..9.. To I 0 . . 3 . S . . 8 9 . . 0 . .3 . a l T 3I 0 . . 3 . . 6 . 8 . . E .9 . Figure 23. A complex of 3-11 and 5-32 producing multiple and leveled successions of all-interval tetrachords. 52 moving vertically down the page represents succession in time. At a), a Ta-configuration of 3-11P produces al (34. .7.9) as MU. Since the simultaneities Ml and M2 are 2-4 and 2-5, respectively, then either of the other two partitions of this al is available as successions: 2-1 and 2-2 (3—*4 and 7—*9) or 2-3 and 2-6 (7—»4 and 3—*9). At b) the inversion of the entire configuration of a), with MU of Al (3.5. .89), is shown. If a) and b) taken together are a succession of four triads, and the two moving voices are limited to successions of 2-3 or 2-6, then because of the shared tritone (3 ...... 9) each of these voices (labeled 1 and 2 in Figure 23) can outline one of the same forms of al or Al that form the respective MU of each pair-configuration of triads. At c) the entire scale of a) is itself treated as a chord whose T3-configuration, producing MU of A1 (4.6. .9T), happens to hold the original 0 . . 3. ,. 7 as CT; the resulting scale, 7-31P, embeds a total of eight all-interval tetrachords, two of each SG(1) type. At d) four To-configurations of SG(2) 5-32, each with MU of Al or al, are strung together in such a way that the entire four-triad succession of a)-b), with voices 1 and 2, is embedded as the common tones within each pair-configuration and such that voices 3 and 4 can produce successions of a2 or A2 as they move through each pair of pair-configurations.6C The first half of d) forms an octatonic scale, and the second half forms the octatonic scale sharing 0. .3. .6 . .9. . with the first, the group of eight chords exhausting the aggregate. Similar results are produced by 3-2 and 5-10 or the M-related 3-7 and 5-25, and by 3-3 and 5-16, M-related to Figure 23, and any of these complexes is SG(l)-nonequivalent to its inversion. When this entire complex of complexes is considered, each SG(1) type of all-interval tetrachord occurs in each position of the structure. E x a m p le D In my Nocturne for piano, the principle of the closed triangular cycle of configurations of a common generator is extended to produce trichordally derived, hexachordally combinatorial, but nonserial aggregate generation, also utilizing several distinct classes of generator within each intermediate hexachordal scale. As in the earlier example introducing the principle (Figure 5), the trichordal generator is 3-4 and the three intermediate scales are 6-7, 6-8, and 6-20, all-combinatorial hexachords called “D,” “B", and “E," respectively,67 and, though the aggregate Is in constant circulation, its distinct hexachordal partitions are delineated by registral proximity of forms of the generating trichord; here it will suffice to distinguish, in Figure 24, between “upper” and “lower” registers. Upper D„/0 012...678... B2 ..2.4567.9.. E„/4/a 01..45..89.. Lower D3/0 ...345...9TE Ba 01.3 8.TE Eo/o/in ..23..67..TE Figure 24. Hexachordal aggregate partitions of Example D. The four forms of 3-4 generating the entire group are shown in Figure 25. b:d = D„/C TJ = b o 00 p:q = D3/»9 t 7i = d . .2. . .67. . . . where d:q = B2 Tm = p . . . 3 ...... TE b:p = Ba t 4 = q ___ 4 5 . . . 9 . . b:q = Eh/4/8 d:p = Ei/c/m Figure 25. Closed cycle of configurations of 3-4. 60 In the composition itself distinctions among these four voices are articulated through instrumentation and register, and a pitch represented here by PC 0 is a common tone throughout. 67 After Martino [1961|. 33 Each generation of the aggregate is represented in Figure 26 by a Roman numeral, and the transition from one registrally adjacent hexachordal type to another is accomplished by holding one trichord in each register as common tones while exchanging the registers of the other two. I II III 1 Upper b \ q — q \ d d — d \ b —* b Lower PP p p q / b / d /' q Figure 26. Transitions o f Example D. Any three of the four trichords generate the closed cycle, so, for example, moving from I to II in the upper register from Do/o (h:d) to (q:d) with CT=d, yields Ed/4/b (b:q) as MU. This property is not unique to these sets since any group of three sets with a common generator at particular configurations can be made to do this, but here, because of the combinatoriality, moving from D3/0 to Bg in the lower register also yields M U =Eo/4/ 8, while the union of the common tones (CTU) of both registers is the complementary En/a/m (d:p), and thus every PC is involved in all three hexachordal types. For the entire group of transitions, Figure 27 shows the resultant hexachords. MU MU MU I CTU II CTU III CTU I Upper hexachord Do/o B>> Eo/^/g D»/c E.,/4/8 Do/g B 2 ^2/0/10 D3/9 Bg Lower hexachord D3/9 Bg E2/C/l(( D3/9 Figure 27. Set-complex of Example D. Because of transpositional symmetry, each form of D and E has. respectively, one or two additional generating 3-4 configurations, so several structures, with different forms of 3-4 as b, d, p, and q, are possible when beginning with a given form of either D or E. Each configuration of hexachords is composed out as a harmonic area to form a section of the piece, and distinguished from the others by sub-registral and ordinal emphasis on the hexachord’s pair-generators other than. 3-4, none of which is shared by either of the others (excepting dyadic generators, which are multiple). One example of each additional trichordal generating class is shown in Figure 28 (for D, and particularly for E, there are in some cases other generating configurations of a clas3 present.) 54 J 0/G B2 E q / 4 / 8 0 1 2 ...6 7 8 . ..2.4567.9.. 01..45..89.. 0-1 u i z . . . .41.40...... 0-0 U l. .6 7 8 ...... 6 7 .9 .. . .5. .89. . 3-5 0 1 . . . . 6 ...... 3-6 . . 2 . 4 . 6 ...... 3-11 0 . . 5 . . 9 . . . . 2 . . . . 7 8 ...... 5 .7 . 9 . . .1 .4 . . . 8 . . . 3-9 0 . 2 . . . . 7 ___ 3-7 . . 2 . . 5 . 7 ___ 3-12 0 . . 4 . 8 . . . .1... .6.8... ___ 4.6..9.. .1. . .5. . .9. . Figure 28. A/tern ate trichordal pair-generators of hexachords D, B, and E. Since for any of these a parallel but complementary structure can exist in the other register, new groups similar to the original can arise between the registers. For example, Figure 29 shows a cycle of 3*11 generating a form of E and its complement, resulting in two new hexachords 6-32 and 6-30.68 00 to b:p — Eo/4/8 o it* cn T i = b CO d:q = Ea/o/io . .23..67..TE T 0I = P 0....5...9.. where b:q = (6-32)ii .1.34.6.8..E t 7 = d . .2 ___ 7 . .T. p:d = (6-32)c 0.2..5.7.9T. TCI = q . . .3 . .6 ___ E b:d = (6-30P)i ,12.4..78.T. p:q = (6-30I)o 0. .3.56..9.E Figure 29. An alternate combinatorial cycle cf Exam ple D. “Modulations” to new transpositional levels of the entire D /B /E system are accomplished by exchanging the registers of a 3-6 pair generating complementary B hexachords, shown in Figure 30. 5 . 7 . 9 . . — 5 . 7 . 9 . . B2 . . 2 . 4 . 6 ...... \ 0 ...... 8 .T. BS Bs 0 8.T. /" . . 2 . 4 . 6 ...... B u . 1 . 3 ...... E — . 1 . 3 ...... E Figure 30. An alternate trichordal generation of hexachord B. This permits an even distribution, throughout the piece, of the twelve forms of B, the six of D, and the four of E. 58 6-32 is Martino’s “C" hexachord; 6-30 is the “Petrushka” hexachord. 55 Exam ple E This example, from the third of my Three Compositions for Orchestra, illustrates the gen eration. of symmetrical scales by nonsymmetrical chords, several at a time, where the intermediate-sized scales of the complex do not necessarily generate the larger ones; the total PC content is limited, and symmetry is a structural determinant. Two contrasting groups of three tetrachord types each generate the smallest possible (pentachordal) scales containing a form of each member of the group, which in turn T„I-generate two hexachordal scales also T„ I-generatable by each tetrachord of the group and in general excluding the tetrachords of the other group. A synthesis of the two groups is accomplished by a hep- tachordal scale containing all six tetrachords, but where, in order to maintain a context of minimal PC change, the latter scale does not contain all of the intermediate hexachordal or pentachordal scales, and is not generatable by any of them. Nine PCs are used, no more than seven at a time; the total PC material of the movement can be viewed as a diatonic scale on C with two flats whose second and seventh steps D/Bb may be inflected as Db/Bb or Dlj/Btj (but not Db/Btj), and where, for the smaller scales, C, F, and G may or may not be present. For each class of tetrachords, then, a limited number of its members is available. T a b le 17. Tetrachordal groups of Example E. Group I: Chord IS V Group II: Chord IS V 4-Z29 1-2-4-5 111111 4-Z15 1-3-2-6 111111 4-14 2-1-4-5 111120 4-12 2-1-3-6 112101 4-16 1-4-2-5 110121 4-13 1-2-3-6 112011 The six tetrachord types are those which have four distinct intervals in their interval series (IS). All are I-nonsymmetrical, and the IS is a permutation of 1-2-4-5 in Group I and of 1-3-2-6 in Group II (Table 17). Within each group, each of the four trichordal classes of each tetrachord is shared with one or the other of the group, but not both, allowing with either a connection with but one moving voice. There are nine trichordal classes in the whole system, only three of which occur in both groups; these three are all in 4-Z29, so this tetrachord assumes a position somewhat more central than that of the others. All Group I chords are diatonic subsets, with five IC 5 among them, and all Group II chords are octatonic subsets, with five IC 3 but no triads among them. There i3 for each group of tetrachordal types one pentachordal type containing all three, shown at a) in Figure 31. Each of these, with its inversion, and thus a T„I-related form of each tetrachord, gen erates a symmetrical hexachord, shown at b). This form of 6-Z23 contains contains and there is another form of 6-Z23, shown at c), containing Db/Bb and having the same number of common tones with this form of 6-Z26, and these are the determinants of the PC inflections mentioned above. Each hexachordal type excludes tetrachords of the other group except 4-Z29, which is contained in both. 6-Z26 contains neither of its axes of symmetry, nor does 6-Z23, but each of these forms of 6-Z23 contains both the other’s axes. Adding to either its missing axis which is also a member of the one form of 6-Z26 produces, at d), one of two forms of the “melodic minor” scale. Each of these contains one or the other form of 6-Z23, but excludes 6-Z26 and 5-20. However, 7-34 does contain a T„I-related pair of each Group I tetrachordal type, and thus exactly one such pair of every tetrachordal type of the system. The quasi-subdominant/dominant duality of the conflicting axes is resolved by adding to the form of 6-Z26 its missing axis (PC 0) which is also a member of both forms of 6-Z23 and 7-34, to form a diatonic scale 7-35, at e), containing one of the axes of every symmetrical scale of the system. 7-35 embeds only one Group II type, so the resolution is essentially to Group I, containing the “central” tetrachord. 56 G roup I G roup II 4-Z29I ...3 ...7.9T. 4-Z15P 0 . . 3 . 5 . ___ E +4-14P . . 2 ___ 7.9T . +4-121 ..2 3 .5 . ___ E +4-16P ..23...7.9.. +4-13P 0 . 2 . . 5 . ___ E a) =5-20P ..23...7.9T. =5-10P 0.23.5. ___ E +5-201 ..23.5...9T. +5-101 0.23... . .9.E b) =6-Z26 . .23.5.7.9T. (Axis 0/6) =6-Z23 0.23.5. . .9.E (Axis 1/7) c) 6-Z23 0 1 .3 . ... 7.9T. (Axis 11/5) d) 7-34 01.3.5.7.9T. (Axis 11/5) 7-34 .23.5.7.9.E0 (Axis 1/7) e) 7-35 0.23.S.7.9T. (Axis 0/6) 7-35 .23.5.7.9T.0 (Axis 0/6) Figure 31. Set-complex of Example E. E x a m p le F The final example, from my Pastorale for flute, viola, and harp, illustrates the use of several of the relations developed in Chapter II, as applied to several nonequivalent sets. The principal harmonic reference, used as both chord and scale, is 6-Z24I, here arbitrarily labeled “P”. The SG(2) prime form is not used, nor are its complement or M-transform, though all of these have related combinational properties. Because of its subset content and common- tone properties, this set has a mediating function in progressions involving three symmetrical formations: 6-20 (“E”), 6-Z13 (“O"), and 6-32 (“C”).69 Table 18 shows the prime forms of these four hexachords. T ab le 18. Prime forms of hexachords of Example F. Hexachord V °S P 0.2.45.78... 233331 1 E 01..45..89.. 303630 6 O 0 1 .3 4 .6 7 ___ 324222 2 C 0.2.45.7.9.. 143250 2 Table 19 shows common-tone vectors, partition vectors, class-embedding or class-mutual- embedding vectors, and the total number of classes embedded or mutually embedded, for pairings of each with forms of itself and with forms each of the others.01* so E and C are as in Example D; O is an octatonic hexachord. 00 E is symmetrical at Tu/i/a, so any common-tone vector involving it is periodic in three segments. 57 T a b le 19. Vector complex of Example F. Self T„CTV T„PV CEMB„VTCEMB P:P 623333233332 0038001 6 11 13 6 36 E:E 630363036303 3006003 4 4 4 1 13 0:0 632422422423 0062301 6 7 9 3 25 C:C 614325052341 1222221 5 6 7 3 21 Pairs CTV PV CMEMB„VTCMEMB P:E 422442244224 0060600 4 4 10 9 P :0 353243233233 0037110 6 7 5 1 19 P:C 523424143242 0142410 5 6 5 1 17 E:0 343234323432 0036300 4 2 1 0 7 0 :0 343333323333 001T 100 5 3 1 0 9 C:E 432343234323 0036300 4 2 1 0 7 P’s role as mediator is supported by its containment of all the trichordal types of E, O, and C, even sharing a pentachordal type with each of O and C, and overall P contains a relatively large percentage of each of the others’ TCEMB classes, from 69% to 81%, while the proportions of any other pair's TCMEMB to the respective TCEMB of each member of the pair range from only 28% to 54%. When there is only one mutually embedded class of a given size, then the set of common tones (CT) must be of this class in any configuration with a CTI of that size. This is clearly true of the shared pentachords in the configurations Po:Oi or Po'.Co, but also, for example, in Po:E„ for any of the six n where CTI= 4, the common tones form the one mutually embedded tetrachordal class 4-19, though not all have the same Ml, M2, and MU. In general many of the mutually embedded classes are available as CT when there are CTIs of the appropriate value, but at least they can all occur as subsets of some configuration’s CT. For example, there is no configuration of P:E such that CTI= 3, but all four mutually embedded trichordal classes are embedded by 4-19. Here the "sandwich function” can be useful as well. To continue the same example, SNDW(C major triad,4-19,Po)= 1, and SNDW(C major triad,4-19,£3/7/^)= 1, meaning that though EMB(4-19,E)= 6, only one form of E contains the form of 4-19 con taining the particular major triad. On the other hand, SNDW(C augmented triad,4-19,Po)= 2, and SNDW(C augmented triad,4-19,E3/7/n) = 3, so there is a somewhat wider choice of configurations of P:E holding the particular aug mented triad as common tones. Figure 32 shows a configuration of P with each of the other hexachords, each with maximum common tones. P0 0.2.45.78... Pn 0.2.45.78... P0 0.2.45.78... Eo/4/8 01..45..89.. Oi .12.45.78... C0 0.2.45.7.9.. Figure 32. Examples of pairings of Example F sets with maximum common tones, A melodic statement of one of these sets as resulting from a harmonic procedure applied to another is an additional unifying device. In Figure 33, successive applications of To to triadically-generated C’s, each with three tones in common with the next, produces Oo as a succession (the “thirds” of the triads). 58 C2 4..T...E ..2...6..9.. C u .1..4...8...... 3. .6 E C8 .1...5 T. 0 . .3 ___ 8 . . . O0 T (0 1 .3 4 .6 7 ) Figure 33. Generation of hexachord O as a succession. A final example from this complex shows three configurations of E, O, and C with each other; in each the melodic connections are by IC 1 or, in the last case, IC 2, in “contrary motion,” and a melodic statement of P0 results: Os 0 .2 3 ___ 89. E c7 0.2.4..7.9.E El/6/0 .12..56..9T. Co .12.4.6..9.E E l/5/9 .12..66..9T. Oc 0 1 ___ 6 7 .9T. Pu T (0.2.45.78) Figure 34. Generation of hexachord P as a succession. Conclusion These examples of upitch-clas3 compositions,” while not music, are harmonic relations that can be realized as pitches, given an appropriate ordering in musical time and registral space; nevertheless, it is assumed that such relations retain, to varying extents, some in dependence of order and register, and are thus articulatable in multiple ways in the time and pitch domains. That every pitch-class of the foregoing examples is a member of one or more referential sets by no means implies that this is necessary for all pitches in order that a composition maintain harmonic coherence, and indeed it is quite possible that a relatively small proportion of pitches be so involved, as in much classically tonal music, where set-class analysis of pitches adjacent in time or register may not reveal much about a composition’s harmonic structure. Among the remaining pitches, each as integral a part of a composition as any other, there may be functional differentiation to the extent that, for example, the role of a pitch in relation to one or more others may be so strongly dependent upon its register that there is no sense in which we would wish to consider it a member of a pitch-class. List of References Babbitt, Milton [1955]. “Some Aspects of Twelve-Tone Composition.” The Score and I.M.A. Magazine 12, 53-61. Babbitt [i960]. “Twelve-Tone Invariants as Compositional Determinants.” Musical Quar terly 46, 246-59. (Reprinted in P. H. Lang, ed. Problems of Modern Music . New York: W. W. Norton [1962], 108-21.) Babbitt [l96l|. “Set Structure as a Compositional Determinant.” Journal of Music Theory 5/2, 72-94. (Reprinted in B. Boretz and E. Cone, eds. Perspectives on Contem porary Music Theory. New York: W. W. Norton [1972], 129-47.) B abbitt [1973]. “Since Schoenberg.” Perspectives of New Music 12/1—2 (double issue), 3-28. Browne, Richmond [1974]. Review of Forte [1973a]. Journal of Music Theory 18/2, 390-415. Chrisman, Richard [197lj. “Identification and Correlation of Pitch-Sets.” Journal of Music Theory 15/1—2 (double issue), 58-83. Chrisman [1977]. “Describing Structural Aspects of Pitch-Sets Using Successive-Interval Arrays.” Journal of Music Theory 21/1, 1-28. Clough, John [1983]. “Use of the Exclusion Relation to Profile Pitch-Class Sets.” Journal of Music Theory 2,1/2, 181-201. Forte, Allen [ 1964]. “A Theory of Set-Complexes for Music.” Journal o f Afustc Theory 8 /2 , 136-83. Forte [1973a]. 27ie Structure of Atonal Music. New Haven: Yale University Press. Forte [1973b]. “The Basic Interval Patterns.” Journal of Music Theory 17/2, 237-73. Gamer, Carlton, and Paul Lansky [1976]. “Fanfares for the Common Tone." Perspectives of New Music 14/2-15/1 (double issue), 229-35. Howe, Hubert [1965]. “Some Combinational Properties of Pitch Structures.” Perspectives of New Music 4/1, 45-61. Lewin, David [1977[. “Forte’s Interval Vector, My Interval Function, and Regener’s Common- Note Function.” Journal of Music Theory 21/2, 194-237. Lewin [1979]. “Some New Constructs Involving Abstract PCSets, and Probabilistic Appli cations.” Perspectives of New Music 18/1-2 (double issue), 433-44. Addendum in same issue: “A Response to a Response: On PCSet Relatedness.” 498-502. 59 Martino, Donald [1961], “The Source Set and its Aggregate Formations.” Journal of Music Theory 5/2, 224-73. Addendum in JMT 6 /2 [1962], 322-23. Morris, Robert [1979]. “A Similarity Index for Pitch-Class Sets." Perspectives of New Music 18/1-2 (double issue), 445-60. Morris [1982a]. “Set Groups, Complementation, and Mappings among Pitch-Class Sets." Journal of Music Theory 26/1, 101-44. Morris [1982b]. Review of Rahn [1980]. Music Theory Spectrum 4, 138-54. Proctor, Gregory [1983]. “Some Notes on Skryabin’s Harmony." Paper read at Eighteenth National Festival-Conference of the American Society of University Composers, Baton Rouge, Louisiana. Rahn, John [1979]. “Relating Sets.” Perspectives of New Music 18/1—2 (double issue), 483-98. Rahn [1980]. Basic Atonal Theory. New York: Longman. Regener, Eric [1975]. “On Allen Forte’s Theory of Chords." Perspectives of New Music 1 3 /1 , 191-212. Starr, Daniel [1978]. “Sets, Invariance, and Partitions.” Journo/ of Music Theory 2 2 /1 , 1-42. Winham, Godfrey [1970]. “Composition with Arrays." Perspectives of New Music 9 /1 , 43- 67. (Reprinted in B. Boretz and E. Cone, eds. Perspectives on Contemporary Music Theory. New York: W. W. Norton [1972], 261-85.) Three Compositions for Orchestra D.M.A. Composition Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Musical Arts in the Graduate School of The Ohio State University By Dean K. RouBh, B.F.A., M.M. ■k k k k it -k k The Ohio State University 1985 Reading Committee: Approved by Professor Thomas Wells Professor Gregory Proctor ----- Advisor Professor David Butler School of Music Three Compositions for Orchestra (1985) Chaconne Pattndrone Reverie Dean K. Roush Concert Pitch Score Duration: ca. 14 minutes Instrumentation: 3 Flutes (3nl doubling piccolo) 2 Oboes Engtisli Horn 2 Clarinets in Bb 2 Bassoons 4 Horns in F 3 Trumpets in C 2 Trombones Bass Trombone Tuba 3 Percussion: Timpani tn E, Bb, Db Tuned metal windciiimes (actual pitches): or handbells Tubular Chimes Tam-Tam Medium Suspended Cymbal Triangle Bass Drum Crotales (sounding two octaves higher): Glockenspiel (sounding two octaves higher) Vibraphone (motor off) Harp (harmonics sounding an octave higher) Violins I and II Violas Violoncellos Contrabasses (sounding an octave lower) T h r e e C ompositions f o r O r c h e st r a lor Carolina I. CHACONHE Andante sostenulo J ■ e 72 Dean K. Roush Piccolo F lu t it ppp EngUift Homi t, c»n im i l ; l -■* PPP Tub* PPP PPP Andante sostenulo J - c 72 V iolin* PPP V lolai Contrabaaam PPP 1SS5 Daan K. 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