1 A GENERALIZED INTERVALLIC APPROACH TO TRIADS ANDREW AZIZ AND TREVOR HAUGHTON Chordal distance is a topic of great theoretical debate, as it suggests space in both the physical and perceptual realms. The current paper introduces new tools for understanding chordal distance, applying the concept of a Lewinian “interval.” On page 1 of Generalized Musical Intervals and Transformations, Lewin defines interval as “a directed measurement, distance, or motion from endpoint s to endpoint t.” Figure 1 reproduces an interval from s to t: Figure 1: Lewinnian Interval Intervals are the centerpiece of a Lewinian “generalized interval system” (GIS). 1 In our paper, the endpoints of Figure 1 are represented by triads. Rather than focusing on physical space, we aim to build a theory that depicts the perceptual distance between two triads. This is particularly relevant to chromatic music of the nineteenth century, in which composers regularly juxtapose triads incorporating several neo-Riemannian operations, such as P, L, and R. 2 If these transformations represent “paths” between triads, an open question is: how do we hear these paths? Is it possible that for any path, there are multiple “distances”? First, we summarize some of the major theoretical works on chordal distance; we then introduce several mathematical definitions and define our GIS; finally, we provide musical examples of Beethoven (in his “Hammerklavier” Sonata, Op. 106) and Schubert (in his Piano Sonata in Bb major, D. 960) that apply our new theoretical framework, underscoring the value of the approach within a nineteenth-century context.
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A GENERALIZED INTERVALLIC APPROACH TO TRIADS
ANDREW AZIZ AND TREVOR HAUGHTON
Chordal distance is a topic of great theoretical debate, as it suggests space in both the
physical and perceptual realms. The current paper introduces new tools for understanding
chordal distance, applying the concept of a Lewinian “interval.” On page 1 of Generalized
Musical Intervals and Transformations, Lewin defines interval as “a directed measurement,
distance, or motion from endpoint s to endpoint t.” Figure 1 reproduces an interval from s to t:
Figure 1: Lewinnian Interval
Intervals are the centerpiece of a Lewinian “generalized interval system” (GIS).1
In our paper, the endpoints of Figure 1 are represented by triads. Rather than focusing on
physical space, we aim to build a theory that depicts the perceptual distance between two triads.
This is particularly relevant to chromatic music of the nineteenth century, in which composers
regularly juxtapose triads incorporating several neo-Riemannian operations, such as P, L, and
R.2 If these transformations represent “paths” between triads, an open question is: how do we
hear these paths? Is it possible that for any path, there are multiple “distances”?
First, we summarize some of the major theoretical works on chordal distance; we then
introduce several mathematical definitions and define our GIS; finally, we provide musical
examples of Beethoven (in his “Hammerklavier” Sonata, Op. 106) and Schubert (in his Piano
Sonata in Bb major, D. 960) that apply our new theoretical framework, underscoring the value of
the approach within a nineteenth-century context.
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Prior Theoretical Approaches
In his landmark 2001 book Tonal Pitch Space, Fred Lerdahl defines algorithms for
measuring distance between tonal chords. As an example, we illustrate how the pair of triads Bb
major (Bb+) and F# minor (F#-) are calculated within the diatonic “basic space,” which for Bb+
is the following:
Figure 2a: Basic Space of Bb+
a: Bb F
b: Bb F
c: Bb D F
d: Bb C D Eb F G A
e: Bb B C Db D Eb E F Gb G Ab A
Employing five levels, ^1 and ^5 are given a “weight” of 5; ^3 is given a weight of 3; other
diatonic scale degrees a weight of 2; and other chromatic notes a weight of 1.3 The distance to
F#- from Bb+ is measured the following way:
Figure 2b: Basic Space of F#- in terms of Bb+ (I bvi)
a: C# F#
b: C# F#
c: C# F# A
d: B C# D E F# G# A
e: A# B C C# D D# E E# F# G G# A
For Lerdahl, there are twelve distinctive pcs (as underlined) in the basic space of F#- compared
to those in the basic space of Bb+. In effect, this is mapping the relationship between any I and
bvi; Lerdahl shows that this is equivalent to one parallel move (I i) + one diatonic move (i
bVI) + one parallel move (bVI bvi). He concludes that the distance between these two
chords—in the context of the diatonic basic space—is a particularly high number (23).4
The distance between these two chords changes substantially when, instead of using the
diatonic basic space, Lerdahl uses the triadic / hexatonic space, tapping into the hexatonic
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properties of triads.5 The basic space weights the tonic triad most heavily (^1, ^3, and ^5), but
instead of prioritizing diatonic scale degrees, it highlights the remaining members of the
hexatonic collection. As such, there are only six additional distinctive pcs between Bb major /
hexatonic and F# minor / hexatonic, as shown in Figure 2c:
Figure 2c: Bb Major to F# Minor Tonic / Hexatonic
a: F#
b: C# F#
c: C# F# A
d: A# C# D E# F# A
e: A# B C C# D D# E E# F# G G# A
In fact, the total distance between these two chords in hexatonic space is 7, a significantly
smaller number than 23. From this, we already observe the possibility of viewing multiple
distances, each through a different scalar lens.
In his 2011 book A Geometry of Music, Dmitri Tymoczko claims that voice-leading size
should depend “only on how far the individual voices move, with the larger motion leading to
larger voice leadings.”6 This seems fairly intuitive, as one could assume that when one travels
more distance, it requires more “work” to move from one triad to the next. So, Tymoczko’s
conclusion is based purely on counterpoint and de-emphasizes scalar and functional context.
This is illuminated in an important footnote:
“In principle, voice leading provides just one of many possible notions of musical distance. We might sometimes want to conceive of musical distance harmonically (based on common tones or shared interval content) or in a way that privileges membership in the same diatonic scale—so that the F major triad is closer to C major than E major is.”7
This, however, generates an interesting case study. Take the following two pairs of chords, both
carrying voice-leading distances of three semitones under the taxicab metric:
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Figure 3a: C+ triad with its neighbor 6/4 chord (F+)
Figure 3b: Bb+ triad with its hexatonic pole, F#-, an “uncanny” juxtaposition8
Although the total semitonal displacement between these two pairs of triads is the same, their
perceptual distances are qualitatively different, with the former possessing a diatonic
juxtaposition and the latter possessing a hexatonic one. Considering the Euclidean distance for
these triads, the results are even more “uncanny,” in that the functionally unambiguous neighbor
6/4 (Neo-Riemannian operation of “D”) possesses a larger distance (√02 + 12 + 22 = √5) than
the paradoxical hexatonic poles (Neo-Riemannian operations “PLP”), representing a distance of
√12 + 12 + 12 = √3.
Naturally, these geometric distances display an incomplete picture. Cognitive studies by
Krumhansl (1998) investigate the psychological paths of neo-Riemannian operations required to
map any triad (major or minor) into any other triad—in essence measuring how “far apart” two
triads are perceptually using different combinations of D, P, R, and L. This generates two
important conclusions: 1) including L in a model with D, P, and R improved the “fit” between
pairs of triads; 2) D is not perceptually equivalent to a combination of R and L operations (and
therefore is not redundant).9 Of course, certain operations, such as “Slide,” (S) that rely on
compositions of these “basic” transformations (that is, the generators of the Schritt/Wechsel-
group), require larger amounts of perceptual work, despite the fact that only two semitones are
being displaced.10 Studies by Rogers and Callender (2006) delve more deeply into the issues of
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perceptual distance between trichords. While they conclude that the total sum of voice-leading
motion correlates with perceptual distance, this is shaded by the number of common tones. In the
case of triadic relationships, Neo-Riemannian operations P, L, and R are heard as being
“especially close.”11 They believe that an accurate model for perceived musical distance “cannot
rely on a straightforward metric that only combines features in a linear manner.”12
Thus, the primary goal of this paper is to provide a rigorous alternative method for
measuring voice-leading distances that accounts not only for geometric distance, but also
perceptual distance. The framework for our discussion was developed by neo-Riemannian
theorists over the last several decades, but in particular Richard Cohn, whose exploration of the
triadic universe and of Weitzmann and Hexatonic spaces generated the impetus for our current
paper.13
A Generalized Intervallic Approach to Triads
As a brief review, let us examine the famous “Cube Dance” graph, as established by
Douthett and Steinbach, shown in Figure 4.14 The twenty-four consonant triads can be
partitioned into eight pitch-class (pc) sums (1, 2, 4, 5, 7, 8, 10, 11); the remaining four slots
belong to augmented triads (sums 0, 3, 6, and 9).
Figure 4: Cube Dance Graph15
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Each triad has two pitch classes in common with one augmented triad, as well as with two other
consonant triads. For example, F-, with pc sum = 1, has two pc’s in common with Ab aug (pc
sum = 0); additionally, F- has two pcs in common with F+ and Db+ (pc sum = 2). In addition,
each region is a group of six triads generated by the neo-Riemannian operations P and L, which
partitions the triads into four groups of six; each is called a “hexatonic region.” Ergo, our
original pair of triads—Bb+ and F#--are in the same hexatonic region.16
Figure 5: Hexatonic Regions
Alternatively—and not to the exclusion of the hexatonic perspective—each consonant
triad can be viewed as either an upshift or a downshift of an augmented triad, generating four
different Weitzmann regions; for example, a Bb+ triad, with pc sum = 5, can be viewed as a
downshift from a Bb-D-F# augmented triad, located at pc sum = 6. As such, these two triads—
previously contained in the same hexatonic region—are now located in disparate Weitzmann
regions, as shown in Figure 6.
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Figure 6: Weitzmann Regions
A natural question arises: how might we coordinate triadic transformations in a tonal
context, in which dual functional identities exist for any consonant triad—as up- or down-shifts
of one of the four augmented triads, or, on the other hand, as “floating” in one of the four
hexatonic subspaces?17 Our solution is to impart contextual function to triads as members of
both W- and H-spaces, thus granting any consonant triad a dual identity (an extension of the
historical concept of Mehrdeutigkeit, or multiple meaning).18
Our methodology for developing analytical applications using both W- and H-spaces is
inspired by Cohn’s 2012 book Audacious Euphony, in which he analyzes the first movement of
Schubert’s D. 960 in a similar fashion,19 following his earlier work (1999)20 as well as Michael
Siciliano’s 2002 dissertation.21 Our main argument is expressed in the following axiom: if every
triad can possess multiple meaning, then there exist multiple perceptual voice-leading distances
between each triad, any one of which can be chosen to model specific analytical intuitions.
To this end, we return to our GIS framework. Every GIS contains a space 𝑆 of musical
elements; a mathematical group 𝐼𝑉𝐿𝑆 of intervals; and an interval function 𝑖𝑛𝑡 which is a map
from the Cartesian product 𝑆 × 𝑆 into the group of intervals. In our framework, the space S
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contains each of the twenty-four major and minor triads, with each triad adopting coordinates in
both Weitzmann and Hexatonic spaces, for a total of forty-eight elements. Additionally, every
pair of elements in 𝑆 (a major or minor triad with either a “W” or “H” identity) is assigned a
unique distance—an interval from IVLS. Represented as a Weitzmann element, a triad is
contextualized as a single-semitone displacement of an augmented triad; represented as a
Hexatonic element, a triad is contextualized as derived from one of the four hexatonic
collections. Each coordinate is represented as an ordered triple; the first entry is the pitch class
sum, where 3n represents the pitch-class sum of the nearest augmented triad (0, 3, 6, 9); the
second entry represents the semitone displacement (k = +/-1) from the nearest augmented triad;
and the third entry represents the pitch-class root r, allowing for the coordination of root-motion
among triadic voice-leadings. This can be expressed in the following way:
W = (3n, k, r); H = (3n + k, 0, r)
(n = {0, 1, 2, 3}; k = {1, -1})
Of course, when the set of elements is specifically major and minor triads (set class (037)), W
coordinates will always have a non-zero increment “k,” whereas H coordinates will contain a
middle entry of 0.
Figure 7 presents a complete chart that contains all major and minor (037) triads,
including their pitch-class sums and Weitzmann / Hexatonic coordinates. Using this chart as a
reference, let us return to the original pair of triads, Bb+ and F#-. Their pc sums are 5 and 4,
respectively, resulting in Hexatonic coordinates of (5, 0, 10) and (4, 0, 6). Their Weitzmann
coordinates, however, factor in the upshift and downshift from augmented triads; 5 is a
downshifted (k = -1) augmented triad of value 6, and 4 is an upshifted (k = +1) augmented triad
of value 3. This results in Weitzmann coordinates of (6, -1, 10) and (3, 1, 6), respectively.
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Figure 7. Table of W and H coordinates for all major and minor triads (set class (037))
Triad VL Sum Weitzmann Hexatonic
C E G# 0
C+ 11 (0, -1, 0) (11, 0, 0)
E+ 11 (0, -1, 4) (11, 0, 4)
Ab+ 11 (0, -1, 8) (11, 0, 8)
e- 10 (9, 1, 4) (10, 0, 4)
c- 10 (9, 1, 0) (10, 0, 0)
g#- 10 (9, 1, 8) (10, 0, 8)
Eb G B 9
G+ 8 (9, -1, 7) (8, 0, 7)
B+ 8 (9, -1, 11) (8, 0, 11)
Eb+ 8 (9, -1, 3) (8, 0, 3)
b- 7 (6, 1, 11) (7, 0, 11)
g- 7 (6, 1, 7) (7, 0, 7)
eb- 7 (6, 1, 3) (7, 0, 3)
D F# Bb 6
D+ 5 (6, -1, 2) (5, 0, 2)
F#+ 5 (6, -1, 6) (5, 0, 6)
Bb+ 5 (6, -1, 10) (5, 0, 10)
f#- 4 (3, 1, 6) (4, 0, 6)
d- 4 (3, 1, 2) (4, 0, 2)
bb- 4 (3, 1, 10) (4, 0, 10)
C# F A 3
A+ 2 (3, -1, 9) (2, 0, 9)
C#+ 2 (3, -1, 1) (2, 0, 1)
F+ 2 (3, -1, 5) (2, 0, 5)
c#- 1 (0, 1, 1) (1, 0, 1)
a- 1 (0, 1, 9) (1, 0, 9)
f- 1 (0, 1, 5) (1, 0, 5)
We should also note that the first two values in our triples are independent, despite their
appearance in the chart; e.g., when the first coordinate is 3n, the second coordinate is always
nonzero. This is the consequence of considering only (037) set classes. If we were to expand our
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harmonic vocabulary beyond (037) set classes, however, then it would open up the possibility of
triples that are not being articulated in the current paper: for example, one can think of a C
augmented triad as an upshifted major triad, carrying the ordered triple (11, 1, 0) —just one
possibility that would open up the domain for our three values.
Because each pair of triads contains four possible paths: W1 to W2 ; H1 to H2 ; W1 to H2 ;
and H1 to W2, we can consider the four pairs as a compound interval, which can be written as a
In an expanded version of this paper, we would consider the long-range relationship of
Bb major with its harmonic foil of the first movement, B minor, as well as considering long
range juxtaposition in later movements, including but not limited to the end of the second
movement and the beginning of the third movement, bearing the hexatonic relationship of Bb
major and F# minor, and, of course, endless possibilities in the fourth movement.
Further Considerations
We have demonstrated that any consonant triad can be conceptualized as possessing both
W- or H-identities, and that it is possible to measure four different types of perceptual voice-
leading intervals among these multiple meanings, thus reintroducing a specific notion of function
into neo-Riemannian transformations. A natural question arises: might equivalent GISes be
constructed to model voice-leading intervals in spaces of different dimension?
Consider seventh chords. Seventh-chord 4-space is an overlapping union of minimal
displacements of diminished seventh chords (termed “Boretz Spiders by Cohn22) and the
intervening spaces (the three octatonic pools). While set-class (0258)—that is, half-diminished
sevenths and major-minor sevenths—cannot participate directly in the perfectly parsimonious
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voice-leading enjoyed by the privileged members of set-class (037), we can bridge the 2-
semitone gap by including either or both minor sevenths or French sixths.
Figure 8. Boretz spiders (from Cohn 2012, p. 157)
Thus, in the same manner as earlier with regard to triads, any seventh chord can be characterized
as being derived from a diminished seventh chord (that is, Boretz-flavored), or as floating in an
octatonic pool (that is, derived from the referential collection of one of the three octatonic
scales). Consequently, one can construct an equivalent GIS to model perceptual distances among
functionally contextualized seventh chords; the Boretz regions are akin to the Weitzmann
regions, and the octatonic pools are akin to the hexatonic spaces.23
In summary, we assert that sonorities in voice-leading space can be contextualized as
having dual functional identities: one related to its derivation from a scalar collection (Hexatonic
or Octatonic), and one related to minimal displacements of an equal division of the octave (an
augmented triad or a diminished seventh chord). This produces a multidimensional spectrum of
perceptual intervallic distances, and enriches our hearing of the spaces between chords.
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1 See David Lewin, Generalized Musical Intervals and Transformations (New Haven, CT: Yale University Press, 1987; reprint: Oxford University Press, 2007), see p. xxix. “A Generalized Interval System (GIS) is an ordered triple (S, IVLS, int), where S, the space of the GIS, is a family of elements, IVLS, the group of intervals for the GIS, is a mathematical group, and int is a function mapping S x S into IVLS, all subject to the two conditions (A) and (B) following. (A): For all r, s, and t in S, int (r, s) int (s, t) = int (r, t). (B): For every s in S and every i in IVLS, there is a unique t in S which lies the interval i from s, that is a unique t which satisfies the equation int (s, t) = I” (p. 26). 2 To name a few: Richard Cohn, “Neo-Riemannian Operations, Parsimonious Trichords, and Their ‘Tonnetz’ Representations,” Journal of Music Theory 41/1 (1997): 1-66; Julian Hook, “Uniform Triadic Transformations,” Journal of Music Theory 46/1-2 (2002), 57-126; Brian Hyer, “Reimag(in)ing Riemann,” Journal of Music Theory 39/1 (1995): 101-138; David Kopp, Chromatic Transformations in Nineteenth-Century Music (Cambridge University Press, 2002). 3 See Fred Lerdahl, Tonal Pitch Space (Oxford University Press, 2001). Specifically: “Level a is octave (or root) space, level b is fifth space, level c is triadic space, level d is diatonic space, and level e is chromatic space. In contrast to Deutsch and Feroe, a fifth space is included here, because the fifth is more stable than the third of a triad and because the fifth becomes the basis for shifting the space. The seventh-chord level is excluded because in Classical music seventh chords have little independent status, the interval of a seventh usually behaving as a local dissonance governed by voice-leading principles. If however, a seventh is judged to be harmonic, it can be added at the triadic level” (p. 47). 4 Ibid., p. 69. 5 Ibid., pp. 258-63. 6 See Dmitri Tymoczko, A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice (Oxford University Press), p. 50. 7 Ibid., p. 51, n. 30. 8 This juxtaposition is the central topic in Richard Cohn’s 2004 article “Uncanny Resemblances: Tonal Signification in the Freudian Age,” Journal of the American Musicological Society, 57/2 (2004), pp. 285-323. 9 See Carol Krumhansl, “Perceived Triad Distance; Evidence Supporting the Psychological Reading of Neo-Riemannian Transformations,” Journal of Music Theory 42/2 (1998): pp. 265-282. 10 Of course, including other Schritt and Wechsel operations of Riemann (as noted in Krumhansl 1998, p. 275) would generate different psychological “paths.” As in her article, the “neo”-Riemannian operations are most central to this paper. 11 See Nancy Rogers and Clifton Callender, “Judgments of Distance Between Trichords,” Proceeding of the 9th Annual International Conference of Perception and Cognition (2006), University of Bologna, p. 1687. 12 Ibid., p. 1691. 13 See Richard Cohn, “Square Dances with Cubes” Journal of Music Theory 42/2 (1998): 283-296; “Weitzmann’s Regions, My Cycles, and Douthett’s Dancing Cubes,” Music Theory Spectrum 22/1 (2000): 89-103; Audacious Euphony: Chromatic Harmony and the Triad’s Second Nature (Oxford University Press, 2012). 14 See Jack Douthett and Peter Steinbach. “Parsimonious Graphs: A Study in Parsimony, Contextual Transformations, and Modes of Limited Transposition,” Journal of Music Theory 42/2 (1998): pp. 241-263. 15 Ibid., p. 254. 16 Figures 5 and 6 are alternative illustrations of Cohn’s original figures (2000, p. 97).
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17 Cohn 2012, p. 105: “We define three such transformation classes: an H class, consisting of the three members of the hexatonic group (L, P, and H); a W class, consisting of the three members of the Weitzmann group (R, N, and S); and an E class, consisting of the transformations that map triads within their own zone (LP, PL, and the identity operation E).” 18 See Janna K. Saslaw and James P. Walsh, “Musical Invariance as a Cognitive Structure: ‘Multiple Meaning’ in the Early Nineteenth Century,” in Music Theory in the Age of Romanticism, ed. Ian Bent, pp. 211-232 (Cambridge University Press, 1996). Saslaw and Walsh state that for Vogler, “Multiple Meaning is an important compositional tool to be used to lead the listener into unexpected territory,” (p. 217); the augmented triad (as well as the diminished seventh) is among those that possess this quality (p. 218). Weber also treats these two chords, with the “modern-day augmented-sixth chords as transformations of the half-diminished-seventh chord on scale degree 2 in minor” (p. 220). In summary: “For Weber, Multiple Meaning is used by composers to create variety and richness, and to smooth the transition between distantly related keys. For Vogler, Multiple Meaning creates a sense of surprise or deception—allowing the composer to “remodel” the listener’s heart at will (p. 222). 19 See Cohn 2012, p. 128. 20 See Richard Cohn, “As Wonderful as Star Clusters: Instruments for Gazing at Tonality in Schubert,” Nineteenth-Century Music 22/3 (1999): pp. 213-232. 21 See Michael Siciliano, “Neo-Riemannian Transformation and the Harmony of Franz Schubert,” Ph.D. diss., University of Chicago, 2002. Siciliano comments on Cohn’s early (1999) conception of traversing W- and H- spaces at once, highlight the paradoxical nature of voice-leading “function”: “It is not self-contradictory because the aspects occur in two distinct ‘spaces’: a harmonic/triadic space and a motivic/melodic space. However, we are asked to interpret the same phenomenon, shared pitch classes, in both spaces. In the triadic space F#- shares no pitch classes with Bb+, and thus although both still in the tonic region, is as far removed from Bb+ as possible. At the same time in the melodic space, F#- shares two pitch classes with the augmented triad that includes F, and thus has a (melodic?) affinity for the dominant. These spaces are not well defined or differentiated. The augmented triad appealed to by the melodic space is defined by the roots of the triads in the dominant LP region. Conversely, the harmonic space of the LP regions is defined by melodic relations (the minimal displacements). Further, the melodic sharing of pitch classes with an augmented triad associated with a region creates the paradoxical result that ALL the minor triads in the tonic region have a greater affinity to the dominant region than ANY of the minor triads in the dominant region” (p. 96). 22 See Cohn 2012, p. 157. 23 A similar approach could be adopted for every possible space generated by equal division of the octave. For instance, we could generate a whole-tone division in which the minimal displacements are Mystic chords (but, of course, again encountering the reality of (037)’s privileged status, in that there are no efficient voice-leadings among Mystic chords derived from different whole-tone scales).