Download Chapter 1: Introduction to Pairwise Well-Formed Scales

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Chapter 1: Introduction to Pairwise Well-Formed Scales
This dissertation arises from my efforts during the past decade to understand how
it is that certain collections in the pitch domain have supported vast musical repertories,
across cultures and historical eras. In these attempts, I have regarded the underlying pitch
collections as the embodiments of certain relationships which composers, or perhaps even
entire cultures, may come to exploit.
The first stage of this project has been to
characterize some of the structural features of these collections. I emphasize that only
some of these features have been singled out for investigation: the most important
musical entities, such as the diatonic scale and the harmonic triad, seem to be overdetermined, supported by multiple rationales. Some of these rationales are mutually
implicated, some are incommensurable with one another, and others are in conflict.
The immediate inspiration and technical basis for the research presented here is
the collaborative work I have done with my colleague Norman Carey, first and foremost,
and some of the ideas that arose during the Working Group in Music Theory held at
SUNY Buffalo, July 28-29, 1993, a symposium organized by John Clough to discuss a
relation suggested by Richard Cohn, the P-relation.1 In this introductory section, I wish
1The
symposium was attended by Norman Carey, David Clampitt, John Clough, Richard
Cohn, Jack Douthett, Dan Harrison, Martha Hyde, Carol Krumhansl, David Lewin, and
Charles Smith. Cohn’s initial notions were presented in “Generalized Cycles of Fifths,
Some Late-Nineteenth Century Applications, and Some Extensions to Microtonal and
Beat-Class Spaces,” the keynote address for the Fifth Annual Meeting of Music Theory
Midwest/Eighth Biennial Symposium of the Indiana University Graduate Theory
Association, Bloomington, Indiana, May 14, 1994. Some of Cohn’s ideas are contained
in “Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic
Triadic Progressions,” Music Analysis vol. 15, no. 1 (1996): 9-40. David Lewin’s “Cohn
Functions,” submitted to Journal of Music Theory, elaborates work Lewin presented at
the symposium.
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to present a thumbnail sketch of the proximate history of the development of my ideas,
and to acknowledge my immediate intellectual debts to others.
The more general
background, in the form of a review of the relevant literature, is a part of Chapter 2.
Norman Carey and I took as our point of departure commonalities between
diatonic and pentatonic scales. We investigated the mathematical conditions underlying
these commonalities, and derived the set of equivalences and implications that constitute
the theory of well-formed scales, as we call them. The category of well-formed scales
was described by Carey and me in our 1989 paper.2 We generally speak of scales rather
than sets because our formulation has the strongest claim to musical validity when
musical intuition supports the notion of a generic scale-step interval. A scale is wellformed if it is generated by a single interval that spans the same number of scale-steps
whenever the interval occurs. For example, the usual pentatonic is generated by the
perfect fifth, and all perfect “fifths” within the pentatonic span three pentatonic scalesteps. The pentatonic scale is considered a non-degenerate well-formed scale, because it
does not simply divide the octave into equal parts; that is, there is a remainder interval,
also spanning three pentatonic scale-steps, but not of the same size as the perfect fifth.
The theory of well-formed scales is a prerequisite for the theory of pairwise well-formed
scales, and the necessary foundation will be provided in Chapter 2, along with a review of
the pertinent diatonic set theory literature. Where proofs of statements are in published
material, the references will be provided. Otherwise, proofs will be offered in this
dissertation. The material on well-formed scales, however, unless otherwise stated, is
attributable to both Carey and myself.
2Norman
Carey and David Clampitt, “Aspects of Well-Formed Scales,” Music Theory
Spectrum vol. 11, no. 2 (1989): 187-206. Further development of the theory is to be
found in sections V and VI of our paper, “Regions: A Theory of Tonal Spaces in Early
Medieval Treatises,” Journal of Music Theory vol. 40, no. 1 (1996): 113-47; and in our
paper “Self-Similar Pitch Structures, Their Duals, and Rhythmic Analogues,” to appear in
Perspectives of New Music.
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The notion of a pairwise well-formed scale is my own extension of the theory to a
class that includes (but is not limited to) some scales that incorporate the step interval of
the augmented second. These scales are prominent in world musics, and have been
appropriated by composers of Western art music, especially for exotic effects. A pairwise
well-formed scale is one with three distinct step-interval sizes which is well-formed when
any two step-interval sizes are taken to be equivalent. Examples of such scales include
the Japanese hemitonic pentatonics such as hira-joshi, e.g., E F A B C (E); the octatonicminus-one-note or Romanian major, e.g., C D# E F# G A Bb (C); and the so-called
Hungarian or Gypsy minor scale, modes of which are also prominent in Indian, Arabic,
and Jewish music, e.g., C D Eb F# G Ab B (C).
Cohn’s point of departure was to observe a feature shared by diatonic and
pentatonic scales and by harmonic triads, considered as pitch-class sets. In each case, it is
possible to form a non-trivial cycle of sets all of the same set class such that adjacent sets
in the cycle have all but one pitch class in common and the remaining pitch classes differ
by interval class 1. The adjacent sets are said to be P-related, and the cycle is called a Pcycle (assuming it is non-trivial, i.e., embraces at least 3 distinct sets).3 The three
conditions for the P-relation are thus: (1) preservation of set class; (2) maximal retention
of common tones (3) minimal motion by the moving pitch class. In the case of the
diatonic and pentatonic sets, the P-cycle simply follows the familiar circle of fifths, while
cycles of six harmonic triads move alternately according to the Parallel transform and the
Leittonwechsel transform, the two mode-altering quasi-Riemannian transformations that
move a single note by a semitone. Both can be thought of as contextually defined
inversions, the former inverting a triad by altering its third, the latter inverting a triad by
altering its root or fifth, respectively, depending on whether the move is from a major
triad to a minor triad or the reverse.4
3The
P-cycle is called a “maximally smooth cycle” in Cohn’s 1996 article.
addition to Cohn, “Maximally Smooth Cycles,” modern applications of these and
other Riemannian transformations are presented in David Lewin, Generalized Musical
4In
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There are number of differences between the pentatonic/diatonic situations and
the case of the harmonic triad, some of them noted by Cohn himself, although he wished
to emphasize the unification he had achieved under the aegis of the P-cycle. Cohn called
the pentatonic/diatonic type unidirectional, because the sets in the cycle are transposed
(or inverted) by means of pitch classes moving always in the same direction as one moves
through the cycle. The triad-type he referred to as toggling, because the sets are inverted
(not transposed) by moving pitch classes in alternating directions as one proceeds through
the cycle. For example, a C-major triad moves to a C-minor triad under the Parallel
transform (P), followed by a move to an Ab-major triad under the Leittonwechsel
transform (L); thus E descends to Eb and G ascends to Ab. The directions the pitch
classes move turns out not to be essential, but the distinction between cycles of
transpositions and cycles where adjacent sets are inverted does prove essential.
I noted that, although in the modulo 12 system all the variable notes moved by the
same interval, in other tuning systems, including open tuning systems such as the
Pythagorean or
1
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-comma meantone temperament, what one may call the voice-leading
interval differs according to the case. In the diatonic case, the voice-leading motion is by
a chromatic semitone; in the pentatonic, the variable note moves by a diatonic semitone;
while the harmonic triad involves two distinct voice-leading motions, differing not only
in direction but in size: the diatonic semitone for L and the chromatic semitone for P. If
one considers the line of fifths, the diatonic semitone descent is 5 fifths and the chromatic
semitone ascent is 7 fifths in the same direction on the line:
. . . -3 -2 -1 0 1 2 3 4 5 6 7 8 9 . . .
Ab Eb Bb F C G D A E B F# C# G#
Intervals and Transformations (New Haven and London: Yale University Press, 1987);
Lewin, “Amfortas’s Prayer to Titurel and the Role of D in Parsifal: The Tonal Spaces of
the Drama and the Enharmonic Cb/B,” 19th-Century Music vol. 7, no. 3 (1984): 336-49;
Brian Hyer, “Reimag(in)ing Riemann,” Journal of Music Theory vol. 39, no. 1 (1995):
101-38; and Henry Klumpenhouwer, “Some Remarks on the Use of Riemann
Transformations,” Music Theory Online vol. 0, no. 9 (1994).
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F C
C
A
A E
A E
C#
Of course, these motions are always in different directions, and only of the same
magnitude when -5  7, that is, in a modulo 12 system. It does not matter whether the unit
of measurement is fifths or semitones, in any case this equivalence means equal
temperament. P-relations are defined only in chromatic universes (or systems) modulo N.
What this suggested to me was that the condition that restricted the voice-leading
interval to interval class 1 was unnecessary, that the P-relation might be better thought of
as a special case of a more general Q-relation in which the variable pitch class was free to
move by any interval class, keeping the other conditions for a P-relation: preservation of
set class and maximal retention of common tones. Pending further clarifications below, a
Q-cycle is defined analogously to a P-cycle, that is, a cycle embracing at least three
distinct sets where adjacent sets in the cycle are Q-related.5 Since Q is the weaker
relation, it would be logical to shelter P-cycles under the Q-cycle umbrella, but for a
while I will preserve the distinction and reserve the term “Q-cycle” for a cycle where the
interval of motion must at some point be greater than 1 semitone. In later chapters the
term Q-cycle will be used to include P-cycles.
There are aspects of the P-relation that give it a privileged status: on practical
musical grounds because of voice-leading norms (Schoenberg’s Law of the Shortest Path)
and formally, among other reasons, because the restriction to interval class 1 makes
unnecessary a further condition that otherwise must be stipulated. If the moving note can
only move by interval class 1, it necessarily must slide into an open position, rather than
leap over stationary pitch classes. This turns out to be a necessary formal constraint, so to
5Lewin
treats a similar generalization. In “Cohn Functions,” Lewin credits me with
identifying the set classes that support what I am calling Q-cycles. Other generalizations
have been studied extensively by Jack Douthett, in work that has been disseminated
informally.
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preserve it I insist that the moving pitch class not be permitted to leap over one or more
stationary pitch classes. If there is such a leap, I will call it a Q*-relation. Thus, in a
proper Q-relation, the move effects an exchange of adjacent intervals in the descriptions
of the sets as a sequence of step intervals. For example, in Forte’s 5-7:{01267} set class,
{01267} or <11415> and {01567} or <14115> are Q-related, while {01267} or <11415>
and {0167e} or <15141> are Q*-related.6 Similarly, the cycle of six 7-33 (whole-toneplus-one) sets below is a Q*-cycle:
{02468te}  {012468t}  {023468t}  {024568t}  {024678t}  {024689t} 
({02468te})
In this case the whole-tone motion consistently involves a leap over a stationary
pitch class. Both of these examples of Q*-relations have their own interest, and the
former will be put in play in an analytical example in Chapter 6.
Another formal point that must be taken up in establishing the proper definition of
a Q-cycle involves trichords. We will find that some trichordal sets can support P- or Qcycles in several different ways, but that all larger sets that support P- or Q-cycles can do
so in only one way. For trichords, I will insist that for each Q-cycle, at most two types of
voice-leading moves be permitted. For example, in the case of the harmonic triad, one
may form a cycle by alternating the Parallel transform with the Relative transform, but
not by mixing Parallel, Relative, and Leittonwechsel in a single cycle.7
The connection to well-formed and pairwise well-formed scales is suggested by
the fact that diatonic and pentatonic scales are the examples par excellence of wellformedness, while the harmonic triad is, albeit somewhat trivially, an example of
6Pitch-class
sets are labeled according to Forte’s system. See Allen Forte, The Structure
of Atonal Music (New Haven: Yale University Press, 1973).
7Here and elsewhere, I intend by the terms Parallel and Relative the meanings they have
in English parlance, not the reversed meanings they have in German theory.
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pairwise well-formedness.
One is unlikely to consider the notion of pairwise well-
formedness on the basis of the triad alone, but when one generalizes to Q-cycles,
involving sets like the 5-7 set mentioned above, one finds pairwise well-formed scales
among the sets that support Q-cycles. The situation turns out to be more complicated
than a simple equivalence, but as this dissertation will show, the (mutually exclusive)
properties well-formedness and pairwise well-formedness interact in definite and
interesting ways with the ability to support P- and Q-cycles (or infinite P- or Q-chains, in
the case of open tuning systems such as just intonation, Pythagorean tuning, etc.). Indeed,
all and only non-degenerate well-formed sets are able to support one type of P- or Q-cycle
(or infinite chain), while pairwise well-formedness is all but equivalent, in a well-defined
way, to support for another type of P- or Q-cycle.
Furthermore, as I will show, the pairwise well-formed sets that comprise a P- or
Q-cycle can be construed as a Lewinian Generalized Interval System (GIS) in multiple
ways.8 The cycle of pairwise well-formed sets can be understood as a commutative GIS
with an associated cyclic group, and an interval function that measures intervallic
distance in terms of distance along the P- or Q-cycle; or as a (generally) non-commutative
GIS, where the associated group that operates on the sets in a simply transitive way may
be either a subgroup of the Tn/In group, or a related group that is generated by the
contextual inversions that form the cycle. These GIS constructions will be introduced
later in Chapter 1, discussed further in Chapter 5, and made use of in analytical situations
in Chapter 6.
This concludes the evolutionary sketch I wished to provide. The remainder of
Chapter 1 will further motivate the notion of pairwise well-formedness through a series of
examples, state some of the definitions, theorems, and corollaries, but without proofs or
undue formalism, and provide a guide to the rest of the dissertation. Chapter 2 will, as
stated above, provide a review of the pertinent literature, especially the literature of
8Lewin,
Generalized Musical Intervals and Transformations, 26.
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diatonic theory, and in particular will develop the required material from the theory of
well-formed scales and the necessary formal apparatus towards the treatment of pairwise
well-formed scales. Chapter 3 will give that formal treatment, providing proofs of the
assertions made in this chapter concerning the structure of pairwise well-formed scales.
Chapter 4 develops the transformational theory of well-formed and pairwise well-formed
scales, while Chapter 5 is devoted to Generalized Interval Systems arising from pairwise
well-formed scales, and in particular elaborates the multiple descriptions available
through commutative and non-commutative GISs. Chapter 6 is a series of analytical
applications of some of the material developed.
***
Since pairwise well-formedness is a generalization of well-formedness, part of the
motivation for considering the more general category depends upon an understanding of
the musical significance of the notion of a well-formed scale. This significance is
suggested by the list of scales or sets generated by the perfect fifth that are well-formed:
two notes a fifth apart, three notes that are in the same relation as are tonic, subdominant,
and dominant in a major or minor diatonic scale, the usual pentatonic scale, the usual
diatonic scale, and the chromatic scale. All equal divisions of the octave trivially satisfy
the definition of well-formedness, and are called degenerate well-formed scales, but if the
perfect fifth is the interval given by the harmonic series, with frequency ratio 3/2, then the
chromatic scale is a non-degenerate well-formed scale, with two different types of step
intervals, diatonic and chromatic semitones. The next scale in the hierarchy also has a
place in the history of theory, the 17-note division of the octave discussed in Arabic
theory by Al-F a r a b i and in Renaissance theories of chromaticism such as that of
Prosdocimo de’ Beldomandi.9 Although the perfect fourth/fifth is the generator for these
9Curt
Sachs, The Rise of Music in the Ancient World (New York: W. W. Norton & Co.,
1943), 177; Jan Herlinger, “Fractional Divisions of the Whole Tone,” Music Theory
Spectrum vol. 3 (1981): 74-83; idem, ed. Prosdocimo de’ Beldomandi Brevis summula
proportionum . . . (Lincoln, Nebraska, and London: University of Nebraska Press, 1987),
139-47.
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significant examples, well-formed sets generated by other intervals are possible: in the
usual 12-note chromatic, there are 21 non-degenerate well-formed sets in all, 17 of which
are generated by interval classes 1 to 4. Microtonal well-formed scales are also a possible
compositional resource: examples include the underlying sets in some of the microtonal
études by Blackwood, and theoretical constructions by Mandelbaum, Balzano, Gamer,
and others. Finally, well-formed scales may be interpreted as cyclic or quasi-cyclic
rhythmic patterns, as discussed in Carey’s and my paper “Self-Similar Pitch Structures,
Their Duals, and Rhythmic Analogues,” as well as in Marc Wooldridge’s dissertation.10
Most work in diatonic set theory takes seriously the distinction between generic
and specific measure: generic measure counts the number of scale-steps that the interval
spans, while specific measure is the actual size (quality) of the interval. Recall that in the
definition of a well-formed scale, a specific interval, the generating interval, always has a
constant generic description, spanning the same number of scale-steps whenever it occurs
in the set. In the context of well-formed scales, specific measure is expressed as an
integer multiple of the generating interval—the fifth, or its replacement, the formal
fifth—reduced modulo the octave—or some interval replacing it in its role of determining
octave equivalence. For example, in the usual diatonic set, 2 fifths modulo the octave is
the major second, and -5 fifths is the minor second. The musical significance of wellformedness lies principally in its systematic distinction between steps and leaps, and
generally between intervals of different generic measure: in a well-formed set, not only
does the fifth (or formal fifth, the generating interval) always retain its generic identity, by
definition, but also, the interval determined by n fifths always retains its generic identity,
no matter where in the set it may be located. In particular, this means that if n fifths is a
step interval somewhere in the set, it remains unambiguously a step everywhere in the
system. In the hexachordal scale that is generated by perfect fifths, C D E F G A (C), the
10Marc
Wooldridge, Rhythmic Implications of Diatonic Theory: A Study of Scott Joplin’s
Ragtime Piano Works, Ph.D. diss. State University of New York at Buffalo, 1992.
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minor third (-3 fifths) is sometimes a step (at A to C) and sometimes a skip (at D to F and
E to G). Of course, in this set some of the perfect fifths span 4 step intervals, such as C to
G, and others, such as G to D, span only three step intervals, so the set is not well-formed.
As Carey and I pointed out in “Regions,” this distinction between the well-formed
diatonic and pentatonic and the hexachord was drawn by informally by Dahlhaus: “The
pentatonic and heptatonic scales are systems. In comparison, the hexachord is a mere
auxiliary construction. As a system it would be self-contradictory. . . . would . . . lead to
the absurd consequence that the listener would have to alternate between the idea of the
minor third as a ‘step’ and as a ‘leap.’”11
This coordination between generic and specific intervals in well-formed scales
entails other consequences that signify an efficient information system. A property
defined by Clough and Myerson, called cardinality equals variety for lines, holds for all
non-degenerate well-formed scales, limiting the number of possible types of melodic
fragments of a given generic description to the number of distinct pitch classes that
participate in the line. Thus, in particular, all non-zero generic intervals (two-note lines)
come in two specific varieties. Clough and Myerson call this special case Myhill’s
Property.12 This will be discussed more precisely in Chapter 2, but for now it suffices to
observe, impressionistically at least, that melodic constructions in well-formed scales are
not too boring, not too exciting, which may be a good thing in systems of communication
as well as in modes of political organization.
Along related lines is the property of self-similarity, which is one of several ways
of characterizing well-formed sets. The diatonic scale is self-similar in the following
respect: the distribution of semitones within any diatonic interval is approximately equal
to the overall distribution of semitones within the octave, namely 2 in 7. Consider two
11Carl
Dalhaus, Studies on the Origin of Harmonic Tonality, Robert Gjerdingen, trans.
(Princeton: Princeton University Press, 1990), 72.
12John Clough and Gerald Myerson, “Variety and Multiplicity in Diatonic Systems,”
Journal of Music Theory vol. 29, no. 2 (1985): 249-70.
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diatonic segments, for example C D E F G A and E F G A B C, spanning major and
minor sixths, respectively. Of the five steps in the major sixth, one is a half step, while
the ratio is two in five in the case of the minor sixth. Now 1/5 < 2/7 < 2/5, and 1/5 and 2/5
are, in fact, the closest approximations to 2/7 with denominator 5. The same holds true
for seconds, thirds, fourths, fifths, and sevenths as well. The self-similarity property
balances diversity and uniformity, amplifying the discussion of the dialectic of pattern
matching and position finding by Richmond Browne. The property is also suggestive of
the maximal evenness property defined by Clough and Douthett, to be discussed in
Chapter 2.13
It is useful to have a procedure for determining whether a given scale is wellformed or not, given its sequence of step intervals. By virtue of octave equivalence, (or
whatever plays the role of octave equivalence in a given context), the structure of a scale
can be determined by Richard Crisman’s “successive-interval array” or Robert Morris’s
CINT1, what I will informally call its sequence of step intervals.14 Typically, I will
consider all modes and inversions of a scale to be equivalent, thus a sequence of step
intervals is determined only up to rotation and retrogression. For example, a diatonic
scale is characterized by a sequence of step intervals in which semitones are separated
alternately by two and three whole tones; thus, any rotation of <baabaaa> represents a
diatonic scale, where b represents a semitone interval, a a whole tone interval. This
example points up another level of abstraction that will frequently be invoked: it is often
13Richmond
Browne, “Tonal Implications of the Diatonic Set,” In Theory Only vol. 5,
nos. 6-7 (1981): 3-21. John Clough and Jack Douthett, “Maximally Even Sets,” Journal
of Music Theory vol. 35, nos. 1-2 (1991): 93-173. These articles will be summarized in
the review of the literature in Chapter 2.
14Richard Chrisman, “Identification and Correlation of Pitch-Sets,” Journal of Music
Theory vol. 15 (1971): 58-83; idem, “Describing Structural Aspects of Pitch-Sets Using
Successive-Interval Arrays,” Journal of Music Theory vol. 21, no. 1 (1977): 1-28. Robert
Morris, Composition with Pitch-Classes (New Haven and London: Yale University Press,
1987), 107. See also, Eric Regener, “On Allen Forte’s Theory of Chords,” Perspectives
of New Music vol. 13, no. 1 (1974): 191-214.
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of little structural moment to specify what the tokens, here a and b, represent. Many of
the abstract or combinatorial features of the set depend only upon the pattern.
Among the prerequisite results developed in Chapter 2 is one that allows the
determination of whether a scale or set is well-formed or not by examination of its
sequence of step intervals: in a well-formed scale of cardinality N, the number of step
intervals spanned by the generating interval is the multiplicative inverse of the
multiplicity one of the two step intervals. (That there are necessarily exactly two distinct
step interval types is a special case of Myhill’s Property.) For example, consider the
diatonic scale pattern, <baabaaa>.
Count one of the step intervals; this is the
multiplicity g of that step interval. In this case, for example, there are five as. If N is the
cardinality of the scale, find the multiplicative inverse mod N of the step-interval
multiplicity, that is, solve the congruence gx  1mod N. In this instance, that means solve
the congruence 5x  1mod 7; thus, x=3. If it is impossible to find such an inverse, because
the multiplicity is not relatively prime to N, then the scale is not well-formed. If it is
possible to solve the congruence, then check the intervals that span x steps. If N-1 of
them are all of the same size (i.e., enclose the same number of as and bs), then the scale
is generated by an interval of constant span, hence is well-formed; if not, the scale is not
well-formed. In the case of the diatonic pattern, the first six fourths all enclose two as
and one b, that is, they are “perfect fourths.” Therefore, the scale is generated by a single
interval that spans the same number of scale-steps whenever the interval occurs, thus, by
definition, the scale is well-formed. Of course, we could have started by setting g=2, the
number of half steps. Then we would have determined that the diatonic set is also
generated by the perfect fifth.
In Chapter 2 I will present an algorithm for determining all possible well-formed
scales for a given generating interval, from a theorem attributable to Carey and Clampitt
and stated in our 1989 article. As I will prove in Chapter 3, however, pairwise wellformed scales are (with a well-defined exception) not generated sets. The procedure
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outlined above will nevertheless permit us to determine whether or not any given
sequence of step intervals is a pairwise well-formed scale. The formal definition of
pairwise well-formedness is as follows:
Definition 1.1: A scale is pairwise well-formed if, when any pair of step
intervals is equivalenced, the resulting pattern is that of a non-degenerate
well-formed scale.
From the definition it is immediate that there are at least three step-interval sizes: any
scale with two step interval sizes is reduced to a degenerate well-formed scale when its
step intervals are equivalenced. It turns out that in a pairwise well-formed scale, there are
also at most three step-interval sizes; again, this is a formal result in Chapter 3. Thus, if
a , b , c are the step sizes, considering a  b with c , a  c with b , and b  c with a
yields a well-formed set in each case.
For example, given a scale with a pattern of three step intervals, say, <abcabac>,
one can determine whether it is pairwise well-formed by equivalencing each pair of step
intervals and checking whether the resulting scale is well-formed, using the procedure
described above, if necessary. In this case, equivalencing a and b, the resulting pattern is
<aacaaac>, a diatonic pattern, which is well-formed.
Equivalencing a and c, the
resulting pattern is <abaabaa>; again the pattern is a diatonic pattern and well-formed.
Finally, equivalencing b and c, the resulting pattern is <abbabab>, which is again wellformed: there are 4 b steps, 2 is the multiplicative inverse of 4 mod 7, and the interval
that spans 2 steps and encloses 1 a step and 1 b step can be seen to be the generating
interval for the scale, so the scale is pairwise well-formed.15
15The
generator by definition has multiplicity N-1. In this case the six ab (or ba)
intervals from <abbabab> can be found comprising order positions 1 and 2, 3 and 4, 4
and 5, 5 and 6, 6 and 7, and 7 and 1.
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Pairwise well-formedness is a second-order construction, undeniably rather
abstract. I am not saying that one experiences, perceptually, any of these equivalences,
only that if one performs these mental operations, one obtains the formal result. As in the
case of simple well-formedness, what is of interest is the logical structure of the set of
properties possessed by this class of scales. In particular, the transformational properties
of almost all of these scales are of more immediate musical salience.
One of the structural properties of pairwise well-formed scales is trivalence: not
only do steps come in three sizes, but all generic intervals—thirds, fourths, and so on—
are of three types. This is reminiscent of Myhill’s Property defined by Clough and
Myerson, wherein all generic intervals come in two sizes. Since, as we will see in
Chapter 2, Myhill’s Property is actually equivalent to non-degenerate well-formedness,
one might conjecture that trivalence is equivalent to pairwise well-formedness. In the
present context, however, trivalence is by no means so strong a property as Myhill’s
Property: there are many pitch-class sets with trivalence that are not pairwise wellformed, so the implication goes just one way. An example of a trivalent set that is not
pairwise well-formed is the pentachord 5-22:{01478}, Anatol Vieru’s Bacovia mode.16
If, however, N is the cardinality of a pairwise well-formed scale, and d and N are
relatively prime, we will see that the interval cycles of generic intervals spanning d steps
(cycles of thirds, fourths, etc.) themselves form pairwise well-formed patterns. In this
respect, the property of well-formed scales carries over to all pairwise well-formed scales,
mutatis mutandi.
The transformational properties of pairwise well-formed scales involve the P- and
Q-relations discussed in the introduction, that have been studied by Richard Cohn and
David Lewin in various musical contexts. My treatment relates the categories well-
16Anatol
Vieru, The Book of Modes, Yvonne Petrescu, trans. (Bucharest: Editura
Muzicala, 1993).
15
formed and pairwise well-formed to P- and Q-cycles and -chains.
The relevant
definitions follow.
Def. 1.2. P-relation (Cohn): Two pitch-class sets are in the P-relation if
there exists a Tn or In mapping one set onto the other that leaves all but
one pitch class of the set invariant and moves the remaining pitch class by
interval class 1. Only defined in chromatic universes mod N.
Def. 1.3. P-cycle (Cohn): A cycle of length greater than 2 in which
adjacent sets are P-related.
Def. 1.4. Q-relation: Two pitch-class sets are in a Q-relation if there exists
a Tn or In mapping one set onto the other that leaves all but one pitch class
of the set invariant and moves the remaining pitch class by any interval
class, where the moving pitch class slides between frozen pitch classes,
rather than leaping over them. For example, {01267} and {01567} are Qrelated (2 slides to 5), whereas {01267} and {01267e} are not (2 leaps
over stationary pitch classes to 11).
Def. 1.5. Q-cycle: A cycle of length greater than 2 in which adjacent sets
are Q-related. For trichords, in addition, it is required that only one or
two of the three possible interval exchanges take place in the course
of the cycle. That is, if the trichord has sequence of step intervals <abc>,
and the cycle proceeds by moving one pitch class to effect the exchange
of a and b, and then by moving another to effect the exchange of b and
c, it may not also proceed by exchanging a and c.
16
Among the interesting aspects of well-formedness and pairwise well-formedness
are the ways these properties interact with support for P- and Q-cycles (or infinite chains
of P- or Q-related sets). Table 1.1 lists the set classes in the usual 12-note universe that
are well-formed (WF) or pairwise well-formed (PWWF) and/or support P- or Q-cycles.
17
Table 1.1
The one- and two-note set classes are trivial examples. All are trivially wellformed (WF) and support Q-cycles; the singleton class trivially supports P- or Q-cycles.
3-1, 3-6, 3-9, 3-10 are WF and support Q-cycles, and 3-2, 3-3, 3-4, 3-5, 3-7, and
3-8 are trivially pairwise well-formed (PWWF) and support Q-cycles. 3-11 is PWWF
and supports a P- cycle and Q-cycles. (3-12 is degenerate well-formed.)
Label
Prime Form
4-1
4-21
0,1,2,3
0,2,4,6
1119
2226
WF, Q-cyclic
WF, Q-cyclic
5-1
5-3
5-6
5-7
5-10
5-20
5-21
5-23
5-32
5-33
5-35
0,1,2,3,4
0,1,2,4,5
0,1,2,5,6
0,1,2,6,7
0,1,3,4,6
0,1,3,7,8
0,1,4,5,8
0,2,3,5,7
0,1,4,6,9
0,2,4,6,8
0,2,4,7,9
11118
11217
11316
11415
12126
12414
13134
21225
13233
22224
22323
WF, Q-cyclic
PWWF, Q-cyclic
PWWF, Q-cycic
PWWF, Q-cyclic Webern op. 5/4
PWWF, Q-cyclic
PWWF, Q-cyclic Japanese In-scale
PWWF, Q-cyclic
PWWF, Q-cyclic
PWWF, Q-cyclic
WF, Q-cyclic whole-tone minus 1
WF, P-cyclic usual pentatonic
6-1
6-Z44
0,1,2,3,4,5
0,1,2,5,6,9
111117
113133
WF, Q-cyclic
Q-cyclic Schoenberg signature
7-1
7-5
7-7
7-22
7-31
7-35
7-Z37
0,1,2,3,4,5,6
0,1,2,3,5,6,7
0,1,2,3,6,7,8
0,1,2,5,6,8,9
0,1,3,4,6,7,9
0,1,3,5,6,8,t
0,1,3,4,5,7,8
1111116
1112115
1113114
1131213
1212123
1221222
1211214
WF, Q-cyclic
PWWF, Q-cyclic
PWWF, Q-cyclic Webern op.5/4
PWWF Hungarian (gypsy) minor
PWWF, Q-cyclic folk scale
WF, P-cyclic usual diatonic
PWWF
8-1
0,1,2,3,4,5,6,7
11111115
WF, Q-cyclic
9-1
9-5
9-11
0,1,2,3,4,5,6,7,8
0,1,2,3,4,6,7,8,9
0,1,2,3,5,6,7,9,t
111111114
111121113
111211212
WF, Q-cyclic
PWWF, Q-cyclic
P-cyclic harmonic triad complement
1111111113
WF, Q-cyclic
10-1 0,1,2,3,4,5,6,7,8,9
Interval Form
Properties
11-1 0,1,2,3,4,5,6,7,8,9,t 11111111112 WF, P-cyclic
18
Non-degenerate WF, PWWF, and P- and Q-cyclic set classes
Of course, all these objects are defined for chromatic universes of any cardinality.
Moreover, it is not necessary to conceive of scales as embedded in chromatic universes.
Thus, the diatonic scale in Pythagorean tuning is well-formed under the definition, but
instead of a P-cycle it supports an infinite Q-chain, where the variable pitch class moves
consistently by a Pythagorean chromatic semitone (C major to G major is pitch class F
moving to F#). The diatonic scale in just intonation is pairwise well-formed and supports
an infinite Q-chain. Here the intervals of motion are alternately syntonic commas and
larger limmas. The just major scale has step intervals (in frequency ratios) as follows:
do
re
9
8
mi
10
9
fa
16
15
sol
9
8
la
10
9
ti
9
8
(do)
16
15
If re is lowered by a syntonic comma (81/80), this produces an inverted form of
9 80 10
the scale: <10/9, 9/8, 16/15, 9/8, 10/9, 9/8, 16/15>. (Because 
. Since the

8 81 9
intervals are expressed here in frequency ratios, “subtracting a syntonic comma” is
expressed as division by 81/80, or multiplication by its inverse.) If we follow this
operation by lowering ti by a larger limma (multiplication by 128/135), the result is an
inversion again. The composition of the two operations is a transposition by a just
(Pythagorean) perfect fourth. Alternating these two operations forms an infinite Q-chain.
The just scale is pairwise well-formed, because equivalencing the 9/8 and 10/9
whole steps, the resulting scale is of the same formal type as the usual diatonic, with 5
whole steps and 2 half steps; similarly if 9/8 and 16/15 are equivalenced, and if 10/9 and
16/15 are equivalenced, again a well-formed scale results, but in this case the generating
interval is a third (or a sixth). Note that we need not have checked for pairwise well-
19
formedness in this particular case, because the just scale is a realization of the abstract
step-interval sequence studied above, <abcabac>. Because the just scale is pairwise
well-formed, and because Easley Blackwood in his microtonal études chooses subsets of
the equal-tempered divisions of the octave of 13 to 24 notes that best approximate the
(just) diatonic scale, some of his études are based upon sets that are pairwise well-formed.
One of the Indian forms of the diatonic, the sa-grama, is also pairwise wellformed. Presenting it as a partition of 22 srutis (which may be an artificial imposition of
the modulo 22 system)17 the sa-grama is:
3
sa
2
ri
4
ga
3
ma
4
pa
2
dha ni
4
(sa)
As in the case of the just scale, one can invert sa-grama either by lowering sa by
one sruti, or by lowering ga by two srutis. By alternating these two moves, a Q-cycle is
formed. Again, the scale belongs to the class represented abstractly by the step-interval
sequence <abcabac>.
Other pairwise well-formed scales are prominent in world musics. One is a
Japanese hemitonic pentatonic, the In-scale, which may be notated as E F A B C (E).
Like the usual pentatonic, it can be considered a connected segment of the diatonic cycle
of fifths, but unlike the usual pentatonic, this subset embraces the diminished fifth as
well: A E B F C. Its cyclic step-interval sequence modulo 12 is <1 4 2 1 4>. It is
possible to modulate to an inverted form of the scale either by moving one note down a
(chromatic) semitone, or by moving one note up a tone, and if one assumes 12-note equal
temperament, through all 24 members of the set class 5-20: E F A B C  E F A Bb C
 E F A Bb D  . . .  B C E F# G  B C E F G  (B C E F A). Here the
17See
the discussion of this representation of the srutis in John Clough, Jack Douthett, N.
Ramanathan, Lewis Rowell, “Early Indian Heptatonic Scales and Recent Diatonic
Theory,” Music Theory Spectrum vol. 15, no. 1 (1993): 36-58.
20
product of two succesive inversions is a T5-transpose of the original set. As an exercise,
the reader may readily verify that this Japanese pentatonic scale satisfies the definition as
a pairwise well-formed scale.
The set classes 5-7:{01267} and 7-7:{0123678} are both pairwise well-formed.
The step-interval sequences for these sets are <11415> and <1113114>. Both support Qcycles, where the product of two successive inversions is T1/T11. The Q-relation cycles
for 5-5 and 7-7, as well as for 5-20, exhaust their respective set classes (since 1, 11, 5 and
7 are units mod 12, i.e. relatively prime to 12). The cycle through 7-7 is shown below:
Q
Q
Q
Q
{0123678} 

{0125678} 

{012567e} 

{014567e} . . . 

. . .
Q
{1236789} 

({0123678})
As table 1.1 shows, 5-7 and 7-7 are the only pentachord/heptachord complements
in the usual 12-note chromatic universe that are pairwise well-formed. Q-relation chains
of these sets figure in an analysis of Webern op. 5, no. 4, in Chapter 6.
The octatonic-minus-one set 7-31: {0, 1, 3, 4, 6, 7, 9} is also pairwise wellformed, and supports a Q-cycle of length 8. In this cycle, the voice-leading moves are
alternately by whole steps and semitones, that is, the sets are alternately Q- and P-related.
The three distinct Q-cycles partition the 7-31 set class: each of the cycles embraces the 8
seven-note subsets of each of the three octatonic pitch-class sets, as shown below.
{0134679} 

{t134679} 

{t034679} 

{t014679} 


Q
P
Q
P
{t013679} 

{t013479} 

{t013469} 

{t013467} 


Q
P
{0134679}
Q
P
21
Although the set figures in much of Stravinsky’s octatonic music, it is especially
prominent in Les Noces. The set also appears in some Eastern European folk music;
Vieru calls one mode of it Romanian major.
As table 1.1 suggests, the properties well-formedness and pairwise wellformedness and P- or Q-cyclicity are closely related, but in complicated ways. In the type
of P- or Q-cycle that Cohn calls unidirectional, closely related to what Lewin calls a twoplace generated Cohn function, adjacent sets in the cycle are transpositionally related
(e.g., the diatonic set). One of the formal results proven in this dissertation is that a set is
non-degenerate well-formed if and only if it supports a unidirectional P- or Q-cycle or
infinite chain. This is another characterization of well-formedness.
There is a type of P- or Q-cycle where the participating set is neither well-formed
nor pairwise well-formed. The only examples in the 12-note universe are the Schoenberg
signature hexachord, 6Z-44, and the complement of the harmonic triad, 9-11. This
supports a P-cycle, but is neither well-formed nor pairwise well-formed. Lewin calls this
type a two-place antithesis-pair Cohn function. This class of Q-cyclic sets will not be
analyzed in this dissertation.
There is a another type of P- or Q-cycle, affiliated with what Lewin calls a
generated 3-place Cohn function, that is all but equivalent to pairwise well-formedness.18
That is, all of the sets that support this type of P- or Q-cycle are pairwise well-formed, but
there is one class of sets—it is not too much to say that there is essentially only one set—
that is pairwise well-formed but does not support a P- or Q-cycle. This set has cardinality
7, and has the interval pattern <abacaba>. As unlikely as it might seem, this is the only
18Since
Lewin’s study has not appeared in print as of this writing, I will mention that
well-formedness and pairwise well-formedness are not categories that he addresses.
Thus, we often consider similar objects of thought, but approach them from different
points of view. Cohn cycles and Q-cycles are closely linked, but are distinct in that Cohn
cycles take place at the level of the step-interval sequence, involving rotations or
retrogressions of the sequence, whereas Q-cycles involve the actual transpositions or
inversions of the set. There will be more about this in Chapter 4.
22
set type, of whatever cardinality, that is pairwise well-formed but is frozen in terms of
Cohn-type voice-leading moves. Alone among pairwise well-formed scales, scales of this
type do not have distinct inversions (that is, are symmetric under inversion).
To check the pairwise well-formedness of such scales, note that equivalencing a
and b results in the pattern <aaaaaac>, which is obviously well-formed, following the
pattern of the diatonic cycle of fifths; equivalencing a and c results in the pattern
<abaaaba>, the pattern of the diatonic scale itself; and equivalencing b and c results in
the pattern <abababa>, the pattern we have shown before to be well-formed, and which
follows the pattern of the diatonic cycle of thirds. These are all of the essentially
different well-formed patterns for scales of cardinality 7.
As table 1.1 shows, there are two set-class representatives of this type in the
universe of twelve pitch classes: 7-22 and 7Z-37. Of these, 7-22 is the more interesting,
since the step intervals differ by one or two semitones, thus better satisfying our intuitive
notion of what a “scale” should be. It may be construed as a harmonic minor scale with
raised fourth degree, the so-called Hungarian or Gypsy minor.19 The first eight measures
at the solo entrance in the first movement of the Brahms Violin Concerto, for example,
use all and only notes of this scale. (Interestingly, it is usually the last movement of the
Concerto that is cited in discussions of the Hungarian flavor of the piece and the reference
to Joachim’s origins.) Similarly, the first two measures of Mozart’s celebrated C-minor
Fantasy, K. 475, employ all and only notes of this scale. The same collection and mode,
C D Eb F# G Ab B (C), is the South Indian me la Si m hendramadhyama , the scale for


Ra ga Si m hendramadhyama . Another mode, a North Indian that, is called bhairav: C

Db E F G Ab B (C). It is also the Afghan mode Beiru, a word related to the Hindustani
bhairav. Bartók’s “Arabian Song,” no. 42 of his 44 Duos for Two Violins, is in this
mode, withholding the modal center until almost nine measures have been played.
Undoubtedly the symmetric disposition of this mode about its modal center appealed to
19According
to Bartók, this scale is unknown in traditional Hungarian music.
23
Bartók. It is one of the Arabic maqams, though an unusual one, but perhaps was once
more common, because it is also a Spanish type, most likely due to Moorish influence.
Examples in Spanish-flavored European art music are found in Ravel’s Rapsodie
Espagnole and Debussy’s La Soirée dans Grenade. Hungarian minor is also one variant
of a Jewish mode called Avannah Roboah, and arises in much twentieth-century and
some nineteenth-century repertory by composers affecting an Hebraic style. The scale is
prominent in the music of Ernest Bloch, and has been used evocatively by Shulamit Ran.
A curious harmonic occurence of the pitch-class set 7-22 is in Le Sacre du
printemps: the string harmonics chord at the end of the 4-measure Kiss of the Earth
section, 1 before R72. The chord fascinated Ernest Ansermet, for whom it represented
“sans doute le maximum de tension harmonique que puisse se signifier la conscience
musicale.” It bothered Allen Forte, because it was a prominent harmonic entity that did
not fit any of his set-complexes for the work. Richard Taruskin, in his review of Forte’s
book (p. 124), pointed out that the chord could be realized as a stack of major and minor
thirds: Ab C Eb G B D F# (Ab). That even F#-Ab looks like a third is an artifact of our
notational system, but in fact these are all generic thirds in this scale.20
Sets of this type, what I will call the class of singular pairwise well-formed scales,
have the property that all of their generic interval cycles are formally identical. For
example, 7-22 has the step-interval sequence <1131213>, while its cycle of thirds is
<2434434> and its cycle of fourths is <5456556>, all equivalent up to rotation to the
abstract cyclic form <aabacab>. Note that in each case, the multiplicities of the three
specific varieties of intervals of a given generic length are 1, 2, and 4.
In few of the examples of the use of this scale does the knowledge that it is a
pairwise well-formed scale—and one with special properties—lead to analytical insight,
20Ernest
Ansermet, Les Fondements de la Musique dans la Conscience Humaine
(Neuchâtel: Editions de la Baconnière, 1961), 2:187; Allen Forte, The Harmonic
Organization of the Rite of Spring (New Haven: Yale University Press, 1978); Richard
Taruskin, Review of The Harmonic Organization of the Rite of Spring by Allen Forte,
Current Musicology vol. 28 (1979): 114-29.
24
apparently. Similarly—even more strongly—the knowledge that a diatonic scale provides
the structural tones for a particular piece of music, and that the diatonic scale is wellformed, is almost never of analytical moment. The property is so general, while the
particularity of a piece is what an analysis usually seeks to address. In the case of the
well-formedness of the diatonic scale, however, one can point to the salience of the
property for the potency of the set: tonal music, and diatonic music generally, privileges
step-wise motion, and the well-formedness of the diatonic set permits steps and leaps to
be unambiguously identified, and allows intervals of every generic description to be
systematically and coherently organized in relation to the powerful interval of the perfect
fifth. A similar justification for the relative ubiquity of the Hungarian minor is more
elusive; certainly part of its appeal lies in the way it can be used to support traditional
functional harmony.
Nonetheless, its fascinating formal properties also suggest an
attractive musical potential.
All other pairwise well-formed scales are set apart from scales of the singular type
by the potential analytical tool and compositional resource that it uniquely lacks: the
ability to support P- or Q-cycles. Any sets that participate in a P- or Q-cycle can be
organized into a tightly organized system, what Lewin has defined as a Generalized
Interval System (GIS).21 Moreover, the sets embraced by a given P- or Q-cycle may be
viewed as the elements of a GIS in at least two, if not three, different but natural and
reasonable ways. This yields a double or multiple description, in the sense of Gregory
Bateson, who indicates its value in binocular vision and other domains.22 The possibility
of invoking a particular GIS in the environment of a piece is first of all a warrant that one
can say or hear something coherent in those intervallic or transformational terms. This
presupposes that the GIS is well grounded in musical reality. In the analyses Chapter 6,
21Lewin,
Generalized Musical Intervals and Transformations, passim. In the interest of
clarifying what I am borrowing from Lewin: the notion of a GIS is due to Lewin, but in
his work on Cohn functions Lewin does not invoke the GIS concept.
22Gregory Bateson, Mind and Nature (New York: Dutton, 1979).
25
the transformational networks acquire a firm basis in the underlying multiple descriptions
that these GIS structures afford. The formal theory of Generalized Interval Systems in the
context of pairwise well-formed scales will be the subject addressed in Chapter 5, but in
the interest of motivating the study of pairwise well-formed scales I will foreshadow that
discussion by introducing the basic material and some examples at this point.23
Generalized Interval Systems and Q-cycles
A Lewinian Generalized Interval System is defined as a set or space of musical
objects, S, together with a group, IVLS,  —(the group of intervals, with the binary
operation  )—and a function int, that maps the set of all ordered pairs (s,t), where s and t
are elements of S, into the group IVLS in such a way that (1) for all r, s, and t in S,
int(r,s)  int(s,t) = int(r,t); (2) for every s in S and every i in IVLS, there is a unique t in S
which is the interval i from s, that is, a unique t which satisfies the equation int(s,t)=i.24
The conditions on the int function ensure that its outputs, the elements of IVLS that span
between elements of S, act in ways our intuition leads us to expect musical intervals to
act. Because of the uniqueness in condition 2, in any GIS we can define a transposition
by the interval i as a mapping Ti: S 
 S: s 
 t where t is the unique element that
lies the interval i from s. Ti is thus a transformation of S, and Lewin shows it to be a
permutation of S, (or as he calls it, an operation on S), that is, a one-to-one function of S
onto itself: if s and s' are distinct elements of s, then Ti(s)  Ti(s'), and if t is any element
of S, there is a (unique) element s such that Ti(s) = t.
Later in his book, Lewin takes a transformational point of view, reinterpreting
intervals as transposition operations on a set. It turns out that Generalized Interval
23The
mathematical machinery employed below is discussed in more detail in the first
part of Chapter 5.
24With very slight (non-substantive) modifications, this is the definition Lewin gives in
Generalized Musical Intervals and Transformations, 26.
26
Systems are equivalent to simply transitive group actions on a set. A group G of
permutations of S (operations on S) acts on a set S in a simply transitive way when, for
any pair of elements s and t in S, there is a unique element g of G such g(s)=t. The
elements of the group G are exactly the transpositions; the intervals from s to t and s' to t'
are the same if and only if g(s)=t and g(s')=t'.
27
Example 1. 1
Schubert, Sonatina op. 137, no. 3
D. 408 (1816), Andante, mm. 32-39
To introduce the notion of multiple description GIS structures in pairwise wellformed scales, I will consider the hexatonic systems studied by Cohn. In “Maximally
Smooth Cycles,” Cohn introduces the P-cycles that are supported in the usual 12-note
chromatic, one for each odd cardinality less than 12. In particular, he considers the Pcycles of harmonic triads. (Keep in mind that harmonic triads are pairwise well-formed,
though trivially so: any three-element set with three distinct step intervals is pairwise
well-formed. Pairwise well-formed trichords are special, as we will see, because they can
support three distinct P- or Q-cycles.) The 24 major and minor triads are partitioned into
four groups of six by the P-cycles, which Cohn calls Hexatonic Systems.
Consider the Schubert Sonatina excerpt in example 1.1. In the second half of this
slow movement, starting from m. 32, the music moves harmonically through a chain of
five P-relations: B major to B minor, to G major for three bars; G minor in m. 37, to Eb
major. Since Eb major is the eventual harmonic goal, one can reasonably conclude the
chain here, although in the next eight measures, leading into the recapitulation, Schubert
adverts to Eb minor, forming a complete P-cycle through a hexatonic system of triads.
We will see that the six triads can be considered to be the elements of two mutually
reinforcing Generalized Interval Systems. One can consider the cycle to be primary, and
measure intervals between triads in terms of the number of pitch classes displaced or
turned over. Thus, the interval between B major and G major is the same as the interval
28
between B minor and G minor. This does not seem at all strange to us, because in both
cases the triads are related by the same transposition operator, T8. But in this GIS, the
interval from B major to G minor is the same as the interval from B minor to Eb major.
In both cases, the triads have no common tones; they are thus as far apart as they can be
in this hexatonic system. In this interpretation, which Cohn uses in his analyses, the cycle
is the foreground figure, against a background that takes for granted that the sets are
members of the same class, here the harmonic triad. If we reverse this perspective, and
take the existence of the P-cycle as background, another GIS is brought to the foreground.
Here the six triads are again the elements of the GIS, and the intervals are the unique
transposition and inversion operations that take one triad to another. Here, for example,
the interval between the B major and B minor triads is given by I5, which is also the
interval between G minor and Eb major, and between G major and Eb minor. Here, what
the ear attends to are the triads between which D and Eb (or D#) are exchanged.
Similarly, I9 is a measure of transformation for triads in which pitch classes B and Bb are
exchanged. Finally, one can consider how the triadic configuration is altered by the
quasi-Riemannian transformations, Parallel and Leittonwechsel, or P and L, and
combinations of these transformations. Taking the dualist description of triads, wherein
the Riemannian root of a minor triad is what otherwise is its fifth, P transforms triads by
sending the third of the triad a semitone away from its fifth, towards its (Riemannian)
root. In other words, fifth and root exchange roles under P. Thus, P sends B major to B
minor by sending pitch class D# to pitch class D, and B minor to B major by sending
pitch class D away from its fifth B, to pitch class D#, towards its root F#, which becomes
the fifth of B major. L transforms triads by sending the root of the triad a semitone away
from the other two constituents, third and fifth. When L and P are combined to form LP,
performing P first, B major is sent to G major, while B minor is sent to Eb minor.25
25Henry
Klumpenhouwer, in “Some Remarks on the Use of Riemann Transformations,”
discusses in depth the relationship between transformations such as P and L and a dualist
29
Under this interpretation, attention is turned to which constituents of the triadic
configuration are altered and in what way. The way these GIS structures are related will
now be considered in more detail and from a somewhat more abstract point of view.
The P-cycle through one of the hexatonic systems, Hex1, is shown below; each
element of the cycle is a major or minor triad, designated by a traditional root with plus or
minus sign indicating quality:
A- 
 A+ 
 C#- 
 C#+ 
 F- 
 F+ 
 (A-)
The rationale for the name “hexatonic” is that the six triads of each system
embrace the six pitch classes of a set of set class 6-20: {0, 1, 4, 5, 8, 9}; in the case of the
cycle above, the six pitch classes are those of the prime form of the set. The justification
of the word “system” is that the six triads actually form a Generalized Interval System.
The GIS that Cohn chooses as the setting for his analyses of some nineteenth-century
music takes the triads of a hexatonic system as the elements of the set S, and takes the
cyclic group induced by the P-cycle as the group that acts in a simply transitive way on
the set of triads. The measure of intervallic distance is simply the number of pitch classes
displaced (or common tones retained), that is, distance along the P-cycle. Thus the six
transpositions in this group are: (1) the identity operation, R0, which maps every triad to
itself; (2) one move forward along the P-cycle, the transposition R1, which maps A- to
A+, A+ to C#-, and so on; (3) two moves along the P-cycle, R2, which takes A- to C#-,
A+ to C#+, and so on; R2 has the same effect on the triads as the ordinary transposition
T4; (4) three moves along the P-cycle, R3; (5) four moves along the P-cycle, R4, which
acts on the triads in the same way that the usual T4 does, and five moves along P-cycle,
R5. Because the transpositions combine as elements of the cyclic group mod 6, (to wit:
RmRn = RnRm = R(m+n)mod6), it is possible to write R4 as R(-2) and R5 as R(-1), so that
the if we class together an index and its complement, the smaller of the values (if there
conception of triads.
30
are two), in other words, the mod 6 interval class, reflects the number of tones displaced.
For example, the triads transposed by R3 = R(-3) have three tones displaced (zero
common tones). This GIS arises in a natural and reasonable way, and has a simple
structure; the GIS is commutative, because the binary operation in the associated group is
commutative. The only counterintuitive aspect of this group is that the transpositions
with odd indices reflect a non-standard notion of interval: the “interval” from the A minor
triad to the C# major triad is the same as the “interval” from the A major triad to the F
minor triad, for example. The reader can check that the basic notion of an interval is
satisfied, however, because the group is simply transitive on the set of triads. The
transformational graphs in Cohn’s analyses are based on this GIS.26
There is, however, a pair of complementary Generalized Interval Systems,
separate from the foregoing GIS, that this and every other pairwise well-formed P- or Qcyclic set gives rise to in a very natural and reasonable way. In this instance, the two
GISs again have as their space S the set of six triads of Hex1. On the one hand, the same
subgroup of atonal operations that maps the underlying hexatonic set of pitch classes onto
itself (in a simply transitive way) and thus makes the hexatonic set into the space of a
GIS, also acts as the group for the six associated triads in a GIS. This is a slight abus de
langage, because it presumes to identify the group that acts on the pitch classes, a
subgroup of the Tn/In group, with the group that acts on the triads, a subgroup of what I
will designate as the Tn/In group (see footnote 26). Of course, in this case the group
acting on the triads can be viewed as transforming the triads by operating on their pitchclass constituents. One can see that this identification is a very natural one if one
associates pitch classes and triads in the following way (identifying a triad with its third):
26Cohn
symbolizes the transpositions of this GIS by Ti; I prefer to reserve this notation
for the usual transpositions. I designate groups that are presumed to act on triads with a
different font, thus: Ti. The usual font designates groups that act on pitch classes. In later
chapters, this somewhat fussy distinction is abandoned.
31
0
1
4
5
8
9
A-
A+
C# -
C# +
F-
F+
I will use the permutation notation that decomposes a permutation into disjoint
cycles.27 For example, T4 acts on the hexatonic pitch classes by sending 0 to 4, 4 to 8,
and 8 to 0 (one cycle, represented by (048)), and by sending 1 to 5, 5 to 9, and 9 to 1
(another cycle, represented by (159)). Thus the operation may be represented as the
product of these two cycles: (048)(159). The second GIS on the triads (and the associated
one on the hexatonic set) arises from the operations shown below:
T0: ( )
T0: ( )
T4: (048)(159)
T4: (A- C#- F-)(A+ C#+ F+)
T8: (084)(195)
T8: (A- F- C#-)(A+ F+ C#+)
I1: (01)(49)(58)
I1: (A- A+)(C#- F+)(C#+ F-)
I5: (05)(14)(89)
I5: (A- C#+)(A+ C#-)(F- F+)
I9: (09)(18)(45)
I9: (A- F+)(A+ F-)(C#- C#+)
The group table (for “both” groups) is as follows:
° T0 T4 T8 I1 I5 I9
T0 T0 T4 T8 I1 I5 I9
T4 T4 T8 T0 I9 I1 I5
T8 T8 T0 T4 I5 I9 I1
I1 I1 I5 I9 T0 T4 T8
I5 I5 I9 I1 T8 T0 T4
I9 I9 I1 I5 T4 T8 T0
27By
a theorem of elementary group theory, every permutation of N objects can be
uniquely expressed as the composition of disjoint cyclic permutations. The empty cycle,
denoted by ( ), leaves all objects fixed. See I. N. Herstein, Topics in Algebra (Waltham
(Massachusetts), Toronto, London: Ginn & Co., 1964), 66-67.
32
Again, the notion of interval that this GIS presumes may seem counterintuitive,
e.g., the interval between C# minor and F major is the same as the interval between A
minor and A major, since the transposition between them (in either direction) is I1, in
both cases. What the ear attends to in this GIS with triads that lie the interval I1 apart is
the exchange between pitch classes C and C#: this is what distinguishes A minor from A
major and C# major and F minor, and is one of the pitch classes that move between C#
minor and F major. Another aspect of this GIS that perhaps makes it less intuitive than
the GIS associated with the cyclic group is that this group is nonabelian, i.e., the group
operation is non-commutative, as the group table shows. As we will see, the existence of
yet another hexatonic GIS is indissolubly linked to the nonabelian character of the Tn/In
subgroup associated with this GIS.
Recall that the hexatonic cycle proceeds by the application, in alternation, of the
Parallel transform and the Leittonwechsel transform, designated here by P and L.28 These
Riemannian operations are contextually defined inversions, in that they provide rules for
inverting triads that depend upon the triadic configuration itself for their definitions.29
Thus, P inverts by altering the third of the triad, while L inverts by altering the
Riemannian root of a triad, that is, the root of a major triad and the fifth of a minor triad.
It is clear that P and L define permutations of the six triads of Hex1, and together they
generate a group that acts in a simply transitive way on these six triads. The group
{I,
P, L, PL, LP, PLP} acts on the set Hex1 as follows:
28There
is an interesting hemiolic relationship between the hexatonic cycle as described
by the Riemannian transformations and as described by the Tn/In operations, since P
and L alternate three times in the course of a cycle, while the Tn/In operations proceed
through two cycles of length three.
29For an understanding of contextually defined inversions as a general concept I am
indebted to discussions of them that arose in a seminar taught by John Clough, Fall 1996.
In particular, Jon Kochavi and Nora Engebretsen, along with Professor Clough,
contributed to an emerging notion of contextual operations.
33
I: ( )
P: (A+ A-)(C#+ C#-)(F+ F-)
L: (A- F+)(C#- A+)(C#+ F-)
PL: (F+ A+ C#+)(F- C#- A-)
LP: (F+ C#+ A+)(F- A- C#-)
PLP: (A+ F-)(F+ C#-)(C#+ A-)
One can check that these permutations form a group: (1) the identity
transformation is included; (2) each of the elements has an inverse (left and right inverse),
since P and L are their own inverses, as is PLP, and PL and LP are inverses of each other;
(3) the operation, composition of permutations, is closed and associative.
As Lewin demonstrates, every non-commutative GIS (S, IVLS1, int1) is paired
with another GIS, (S, IVLS2, int2), where all the elements of the group IVLS2 commute
with all the elements of the group IVLS1. The P/L Hex1 GIS is linked to the Tn/In Hex1
GIS in that the elements of the P/L group commute with the elements of the Tn/In group
that acts on Hex1. For example, I9PL(A+) = I9(C#+) = C#- = PLI9(A+) = PL(F-) = C#-.
The groups that commute with each other are isomorphic to each other. As we will see,
however, there is a natural mapping between the groups that is an anti-isomorphism, i.e.,
a one-to-one correspondence i between the groups such that i(AB)=i(B)i(A), for all
elements A and B.30 An explicit anti-isomorphism between the groups is displayed
below:
i: I  T0, P  I1, L  I5, PLP  I9, PL  T4, LP  T8.
30As
we will see in Chapter 5, if there exists an anti-isomorphism between two groups,
the groups are isomorphic, and conversely, given an isomorphism, one can construct an
anti-isomorphism.
34
Thus, for example, i(I1I5) = i(T8) = i(I5)i(I1) = LP, and i(I1I9) = i(T4) = i(I9)i(I1) =
(PLP)P = (PL)PP= PL. Also, i(I9I5) = i(T4) = i(I5)i(I9) = L(PLP) = (LP)(LP) = PL, and
i(I5I9) = i(T8) = i(I9)i(I5) = (PLP)L = (PL)(PL) = LP.
In the GIS with Hex1={A+, A-, F+, F-, C#+, C#-} as the set S and the group
IVLS1 = {T0, T4, T8, I1, I5, I9}, the group generated by P and L is the group of intervalpreserving operations for the GIS, and vice versa, taking the P/L group as IVLS2. That
is, if h1 and h2 are elements of Hex1, and X is an operation in the P/L group acting on
Hex1, then int1(h1,h2) = int1(X(h1), X(h2)), and if Y is an operation in the Tn/In group,
int2(h1, h2) = int2(Y(h1), Y(h2)). For example, the interval between A+ and F- is PLP;
I1(A+) = A-, and I1(F-) = C#+, and the interval between A- and C#+ is also PLP.
The P/L group can be defined on any of the hexatonic systems, because its
operations are not sensitive to the labeling of the pitch-class constituents of the triads, but
act identically on all major triads, on the one hand, and on all minor triads, on the other.
The P/L group is also the commuting group and the group of interval-preserving
operations for {T0, T4, T8, I1, I5, I9}, when it acts on Hex3 = {B-, B+, Eb-, Eb+, G-, G+},
amd for the other Tn/In subgroup for the hexatonic systems, {T0, T4, T8, I3, I7, I11},
which acts simply transitively on the sets of triads Hex2 = {Bb-, Bb+, D-, D+, F#-, F#+}
and Hex4 = {C-, C+, E-, E+, Ab-, Ab+}.31
One could argue, with good arguments on both sides, as to which of the GISs
associated with the triads is the most “natural,” the cyclic group or either one of the
nonabelian permutation groups.
The more important point, I believe, is one of
foreground and background. The cyclic group {R0, R1, . . ., R5} has providing it with a
very firm background the GIS of six permutations that are identified with the subgroup of
Tn/TnI operations that make the hexatonic pitch-class set itself a GIS. At least in part,
that is implicit in the set-class consistency condition of the P-cycle, but it is worth
31Cohn,
in “Maximally Smooth Cycles,” labels the hexatonic systems according to the
cardinal points of the compass.
35
spelling out.
What the cyclic group favored by Cohn foregrounds is precisely the
existence of the P-cycle (maximally smooth cycle), with the measure of intervallic
distance the number of pitch classes displaced between triads. The fact that in a P-cycle
(as opposed to a Q-cycle) the moving pitch class moves by a minimal distance plays no
formal role in terms of the GIS (it is a very strong formal constraint, though, as we will
see) but it is a further musical justification of the whole construction’s analytical validity
because of voice-leading norms.
On the other hand, if one chooses the Tn/In group as the basis for the hexatonic
Generalized Interval System, one foregrounds the power of the Tn/In operations not only
to ensure the set-class consistency condition but to form a group; moreover, a group that
acts in a simply transitive way on both the hexatonic pitch classes and on the six triads.
In this Generalized Interval System, it is the motion of particular pitch classes that is
placed in the foreground. For example, I1 is characterized by the exchange of pitch
classes C and C#, in the transformations between A-major and A-minor, F-minor and C#major, and F-major and C#-minor triads. What provides this structure with further
musical justification is what is pushed to the background, namely, the P-cycle and the
cyclic group GIS that it suggests. For that reason, if one were to relabel Cohn’s analyses
with the Tn/In operations, one must not lose sight of the fact that these particular Tn/In
operations coexist in a tightly organized group of order 6, not just in the big group of
order 24.
For this reason, I favor keeping in mind all of these perspectives—the cyclic GIS
that measures distance along the P- or Q-cycle, the Tn/In GIS that tracks the motion of
particular pcs, and the associated commuting-group GIS that is expressed in terms of the
contextually defined inversions that alternate to form the P- or Q-cycle—in any analytical
application of P- or Q-cyclic sets associated with pairwise well-formed sets. The multiple
description does not arise in well-formed sets, as we will see, because there the group
induced by the P- or Q-cycle is the same as a cyclic set of ordinary transpositions.
36
The relative impoverishment of a GIS of six triads that is more one-dimensional,
associated with the Tn/In group introduced above, is shown by the following example,
suggested by Stephen Soderberg in a personal communication. The set of six triads
{Bb+, Ab-, D+, C-, F#+, E-} is the space for a GIS where again the group acting on the
triads in a simply transitive way is {T0, T4, T8, I1, I5, I9}, ˚. I would not dismiss the
musical significance of this GIS—in fact, I will invoke it in the discussion of a passage by
Wagner in Chapter 6—but it provides a contrast to the richer hexatonic system. Here, all
12 pitch classes are engaged by the set of triads, so the group of six Tn/In operations does
not form a GIS with the set of pitch classes. More significantly, the justification for
invoking the cyclic group is weaker here, (although John Clough has pointed out to me
that the whole-tone scale ordering is tonally suggestive), therefore a well-behaved
commutative GIS does not arise in such a natural way. Finally, the commuting group for
this GIS is somewhat more complicated than the group generated by P and L. Here, it is
the group obtained by conjugating the P/L group by the relative transform R, that is, the
group R(P/L)R-1 = R(P/L)R = {I, LP, PL, RPR, RLR, RPLPR}.32
Having seen that the Tn/In group that acts simply transitively on the hexatonic
pitch classes is congruent with the group that does so on the triads, that is, it preserves
triads and transforms them according to the P-cycle while operating on their constituent
hexatonic pitch classes, one might ask if there is a cyclic group that acts on the pitch
classes and preserves triads. It turns out that there is. The cyclic group acting simply
transitively on the pitch classes of Hex1 is defined as follows:
R0 : ( )
R1: (094185)
R2: (048)(159) (= usual T4)
R3: (01)(45)(89)
32This
example and the concept of conjugation will be discussed further in Chapter 6.
37
R4: (084)(195) (= usual T8)
R5: (058149)
This group preserves harmonic triads, and when applied to the triads yields the Ri
group previously defined, but most set classes as usually understood are not preserved
(are expanded), since although semitones and major thirds are preserved, minor thirds and
perfect fourths may exchange, that is, they are in the same “interval class” here. For
example, in this GIS, the “(014) set class” determined by the transpositions of the GIS is:
{014},{015},{458},{459},{089},{189}.
The P-cycle of triads provided a first exemplification of the availability of
multiple descriptions through commutative and non-commutative GISs in pairwise wellformed scales that support P- or Q-cycles, that is, in all cases except those of the singular
Hungarian minor type. In general, there is only one P- or Q-cycle associated with a given
pairwise well-formed set class. Pairwise well-formed trichords, however, including the
harmonic triad class, support three distinct P- or Q-relations, and these relations pair up
in three different ways to form P- or Q-cycles. In the case of the harmonic triads, the
three relations are the Riemannian P-relations P and L and the Q-relation R. Applying P
and R in alternation produces a Q-cycle of triads of length 8:
P
R
P
R
P
R
P
R
C- 
C+ 
A+ 
F#+ 
Eb+ 


A- 


F#- 


Eb- 


(C-)
As before, the eight triads form a GIS with the cyclic group, as well as with the
Tn/In group acting on the octatonic triads, here {T0, T3, T6, T9, I1, I4, I7, I10}, and with
the group generated by R and P. Again, these two nonabelian groups are commuting
38
groups for each other, and each provides the interval-preserving operations for the GIS of
the other group.
Applying R and L in alternation, on the other hand, exhausts the entire set class,
and again, a cyclic GIS can be formed, or non-commutative GISs with either the whole
Tn/In group or the R/L group.
For pairwise well-formed trichords in general, three distinct P- or Q-cycles arise
from the three pairs of contextually defined operations. If the trichord has prime form
<abc>, with a<b<c, (a, b, c are necessarily distinct), the three operations are Qa/b,
Qb/c, and Qa/c, where Qx/y exchanges intervals x and y. These operations can be
readily seen to produce P- or Q-relations. In the usual 12-note universe, there are seven
pairwise well-formed trichords, each supporting three P- or Q-cycles. As Table 1.1
confirms, these trichords are the set classes 3-2:{013}, 3-3:{014}, 3-4:{015}, 3-5:{016},
3-7:{025}, 3-8:{026}, and 3-11:{037}. The harmonic triad, 3-11, is the only trichord that
supports a strict P-cycle, the cycle alternating P and L discussed above, so in the
remainder of this discussion I will refer to Q-cycles.
There are two other hexatonic cycles, for example, one arising from 3-3, with
Qa/c followed by Qb/c (both Q-relations involving motion by interval class 5), the other
from 3-4, with Qa/b followed by Qb/c (both Q-relations, with motion by interval class 3).
These cycles are shown below, embracing the set Hex1.
014
145
458
589
890
901
015
045
459
489
891
801
The first statement of the row of Webern’s Concerto, op. 24, is shown below:33
33Here
and elsewhere, I use the pitch notation suggested by the Acoustical Society of
America. The number following a note indexes the register of the note. Middle C is C4,
39
Oboe: B5 Bb4 D5
Flute:
Eb6 G6 F#5
Trumpet, muted:
G#4 E4 F5
Clarinet:
C5 C#6 A5
The row, as is well known, comprises four serial transformations of the 3-3:{014}
trichord, played by four instruments, with the last notes of the first three trichords
overlapping the first notes of the second, third, and fourth trichords. In Chapter 6, the
organization of the row and of the row forms in terms of the 3-3 trichord will be
discussed. Here only a few observations on the 3-4:{015} trichords embedded in the first
statement of the row will be entertained.
The first two trichords embrace a hexatonic set, {te2367}. The overlapping 3rd
and 4th row elements suggest a chain of four Q-related 3-4:{015} trichords:
I5
3
e t
3
t 2
3 7
2
7 6
2
I1
I5
I9
Qa/b
Qb/c
Q a/b
e+3=2
t-3=7
3+3=6
Figure 1.1. Q-related 3-4 trichords in the first hexachord of op. 24
the C an octave above is C5, and so forth. Inflected notes are labelled according to their
position on the staff; thus, B# above middle C is B#4, enharmonically C5.
40
The chain ends when the turnover of pitch classes is complete, and the hexachord
is completed. Thus, the hexachord is partitioned into two disjoint 3-4 trichords (bold and
underlined in the equations above): the first trichord, representing those pitch classes cast
off in the course of the chain, and the last, repesenting those picked up in the course of
the chain. Were the chain to continue, with a Qb/c relation, here matched with I1 (a
manifestation of the hemiolic property), the first note of the row would reappear,
violating the row structure. Note that I5 is the “interval” between the disjoint trichords
and between the adjacent trichords. There are other aspects of the compositional design
that help set in relief the Q-chain of 3-4 trichords. Because the pattern of order numbers
for these trichords is symmetrical, and because the arrangement of the row in pitch-space
creates symmetries for each of the hexachords, the trichords are organized symmetrically
in pitch space. The row is shown below, together with the directed pitch intervals
connecting adjacent row elements:
e
t
-13
2
+4
3
+13
7
+4
6 ||
-13
(-10)
8
4
-4
5
+13
0
-5
1
+13
9
-4
The four 3-4 trichords in the first hexachord thus involve the following symmetrical
pattern of pitch intervals:
-13 +17
+4 +13
+13 +4
+17 -13
If the pitch space of the first hexachord, Bb4 to G6, is divided into two registral bands of
equal width, with each move along the Q-chain the moving pitch class changes register:
41
HLH
LHL
HLH
LHL
Because the serial operations on the 3-3:{014} trichords are fundamental, in the
second hexachord the Q-chain of 3-4 trichords does not follow chronological order in the
outer trichord-pairs (the chronological order is indicated by the unidirectional arrows in
figure 1.2 below). The pitch-space symmetries discussed above carry over to the second
hexachord.
T4
0 1
8
I1
Qb/c
I5
0 1
5
T4
0
4 5
I5
Qa/b
9
4 5
I9
Qb/c
Figure 1.2. Q-related 3-4 trichords in the second hexachord of op. 24
Table 1.2 displays information about the Q-cycles available for the seven pairwise
well-formed trichords in the usual 12-note universe. For each trichord there are three
cycles. Each of the trichordal set classes has 24 members, and the lengths of the cycles
divide 24 (for reasons to be discussed in Chapter 5). Because the cycles are non-trivial,
their lengths are necessarily greater than two. The Forte labels beside the entries for each
trichord-cycle indicate the set of pitch classes embraced in the course of the cycle. Thus,
12-1 is the aggregate, and is obviously the set of pitch classes embraced by any cycle that
exhausts the trichordal set class; 8-28 is the octatonic; 6-35 is the whole-tone hexachord;
42
6-20 is the hexatonic; 4-25 is the French sixth; and 4-9 is the tetrachord formerly known
as Z, {0167}. The length of the cycle is also the number of elements in whichever GIS
may be under consideration, as well as the order of the group for that GIS.
Table 1.2
Cycle length:
24
12
8
3-2: {013}
1 (12-1)
1 (12-1)
1 (8-28)
3-3: {014}
1 (12-1)
3-4: {015}
2 (12-1)
3-5: {016}
2 (12-1)
3-7: {025}
1 (12-1)
6
4
Trichord:
3-8: {026}
3-11:{037}
1 (8-28)
1 (6-20)
1 (4-9)
1 (12-1)
1 (8-28)
1 (6-35)
1 (12-1)
1 (6-20)
1 (6-35)
1 (8-28)
Trichordal P- and Q-cycles
1 (6-20)
1 (4-25)
43
In the mod 7 universe of the diatonic pitch classes, there is only one pairwise
well-formed trichord, the all-interval <124> trichord.34 This trichord has some special
properties, to be discussed in Chapter 3.
Of course, all this discussion of trichords should not cause the reader to forget that
all non-singular pairwise well-formed sets also participate in Q-cycles, and the remarks
concerning multiple GIS descriptions hold equally for these sets. An example is provided
by the “Romanian major” or octatonic-minus-one set, 7-31, which participates in
Octatonic Systems, analogous to Cohn’s Hexatonic Systems. Here we have the Q-cycle
through the 8 seven-note subsets of an octatonic set, previously introduced. In this case
an octatonic set of pitch classes is engaged, and along with a cyclic group of order 8
acting on the 7-31 sets that is induced by the Q-cycle, there is also a nonabelian subgroup
of order 8 of the Tn/In group that acts simply transitively on the octatonic set to form a
GIS, and an associated Tn/In group that is understood as acting on the 8 forms of 7-31.
To conclude this introduction to pairwise well-formed scales, I will sketch how
the unification of well-formedness and pairwise well-formedness will be achieved in the
formal development of the theory, via lumpiness vs. smoothness.
Lumpiness vs.
smoothness provides a generative scheme for well-formed and pairwise well-formed sets.
Figure 1.3 is a 7X7 matrix of as and bs. If one focuses first on the columns, one
observes that each is a maximally lumpy distribution of as and bs, or a partition of those
elements into separate connected segments. The clump of as moves cyclically, two
rotations downward each time.
One can consider this a kind of abacus, effecting
multiplication by 2 modulo 7: the leading a takes on successively positions 2, 4, 6, 1, 3,
5, 7 or 0 mod 7.
34The
context here is the generic interval measure discussed in John Clough, “Aspects of
Diatonic Sets,” Journal of Music Theory vol. 23, no. 1 (1979): 45-61, and applied in
Clough and Myerson, “Variety and Multiplicity.”
44
2  4  1mod 7
N=7
m=2
v=4
abba bbb
abbba bb
babba bb
babbba b
bbabba b
bbabbba
bbbabba
Figure 1.3. “Lumpiness is orthogonal to smoothness.”
Now looking at the rows, and interpreting the as and bs as semitones and whole
steps, respectively, one can see that these rows represent the modes of the diatonic scale,
in circle of fifths order, from Lydian on the bottom to Locrian at the top. Note that in the
rows, of course, the semitones are maximally evenly distributed. Because both the rows
and columns are understood cyclicly, this matrix might be superimposed on a torus,
identifying top and bottom edges and left and right edges.
Now m=2 is the multiplicity of the half step, and 4 is its multiplicative inverse
mod 7. Note the interval bracketed under v=4: two half steps separated by two whole
steps, the diminished fifth. With this construction, a unique interval of diatonic span or
generic length v will always occur on the top row of the array: it will enclose one more a
step interval than every other interval of that diatonic length; every other interval of that
length will enclose one more b step interval, and one less a interval. We have thus
constructed a generating interval of constant diatonic length, and therefore we have a
mechanism for constructing a well-formed set. We have illustrated in doing this what
45
will be proven algebraically in Chapter 2, that in a well-formed set the “multiplicity of a
step interval is the multiplicative inverse of the diatonic length of a generator,” the fact
that provided us with an algorithm for determining the status of a given sequence of step
intervals as a well-formed or pairwise well-formed scale.
Figure 1.4 shows the general case. Each row represents a mode of the scale, from
the bottom, each one beginning v notes above the next. The interval enclosing steps in
positions 1 to v contains one more a interval, and one less b interval, than every other
interval of length v.
mv  1mod N
v
m
1
2 3
1-v .
.
.
.
.
.
.
.
2v 2v+1 .
v v+1 .
0 1 .
.
.
.
.
.
.
2v+1 .
.
v+1 v+2 .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. v .
. 0 .
. . .
. . .
. . .
. . .
. . .
. . .
. 1 .
. . . .
. . . .
. . 0
. . .
. . .
. . .
. . .
. . .
. . .
. . 1
. . .
. . .
. . v
Figure 1.4. “The multiplicity of a step interval is the multiplicative inverse of the
diatonic length of a generator.”
46
Turning now to figure 1.5, we show how a similar generative scheme produces
modes of a pairwise well-formed set of cardinality 7, of the type represented by the
diatonic in just intonation, or by the sa-grama, <a b c a c b c>. For the moment we will
disregard the array. Consider one row, say the top row. Equivalencing a and c, we have a
b a a a b a, in other words something formally of the type of the Dorian mode.
Equivalencing b and c, a b b a b b b, thus formally like the Locrian mode. Now
equivalencing a and b, a a c a c a c is not a diatonic pattern, but it is well-formed. Thus,
the top row represents a mode of the pairwise well-formed set, and each of the rows in the
array is a mode of the pairwise well-formed set. Taking into consideration the whole
array, equivalencing a and b will produce modes of <a a c a c a c>. In a sense, this is the
least problematic case, since equivalencing a and b results in the lumpy vs. smooth array
similar to the diatonic case in figure 1.3, but now with m=4 and v=2. The generative
scheme works to produce well-formed sets in the cases of the other two equivalences
because the matrix can be automorphically transformed into the lumpy vs. smooth
diatonic matrix that we saw above, as figure 1.6 suggests.
m=4
N=7
v=2
abc acbc
bca bcac
ac bcabc
bca cbca
ca bcacb
cbc abca
ca cbcab
Figure 1.5. Generative scheme for a pairwise well-formed set
nij=[j+5(i-1)]mod 7
nij=[j+3(i-1)]mod 7
47
row i
1
2
3
4
5
6
7
1234560
6012345
4560123
2345601
0123456
5601234
3456012
X
1234560
4560123
0123456
3456012
6012345
2345601
5601234


Y
1234560
4560123
0123456
3456012
6012345
2345601
5601234


Z
Figure 1.6 Transformation of generative scheme for the pairwise well-formed set
Matrix X in figure 1.6 reproduces the pairwise well-formed set in figure 1.5, with
numbers representing order positions of the step intervals of the scale in the top row,
where bold font represents step intervals of type a, italic font represents step intervals of
type b, and the unmarked font represents type c step intervals.
Matrix Y is an
automorphic transformation of matrix X: the entry nij in the ith row and the jth column of
matrix X is nij=[j+5(i-1)]mod
nij=[j+3(i-1)]mod7.
7,
while the comparable entry in matrix Y is
In matrix Z, step intervals of types b and c are equivalenced,
eliminating italic font, and matrix Z is now in the lumpy vs. smooth formation of the
usual diatonic, with N=7, m=2, and v=4. Generalizing and making precise this heuristic
example will be the burden of Chapter 4.
It is hoped that this introductory chapter has suggested enough of the structural
interest and transformational power of pairwise well-formed sets to motivate the
formation of a theory for them. To lay the groundwork for such a theory, we turn next to
a survey of scale theory.