MUSIC 377: Post-Tonal Music Theory

Estimated study time: 2 hr 14 min

Table of contents

These notes draw on Joseph N. Straus’s Introduction to Post-Tonal Theory (4th ed., 2016), Allen Forte’s The Structure of Atonal Music (1973), David Lewin’s Generalized Musical Intervals and Transformations (1987), Richard Cohn’s Audacious Euphony: Chromatic Harmony and the Triad’s Second Nature (2012), and supplementary material from Indiana University MUS T556 (Analysis of Music Since 1900) and University of Chicago MUSI 31300 (Analysis of 20th-Century Music).

Chapter 1: Pitch Classes and Intervals in Post-Tonal Theory

1.1 From Pitch to Pitch Class

The tonal system that governed Western art music for roughly three centuries rested on a pair of assumptions so deeply embedded in compositional practice that they were rarely articulated explicitly: first, that pitches separated by one or more octaves are functionally equivalent, and second, that the twelve chromatic pitch classes suffice to describe the harmonic and melodic content of a composition. The first assumption is older than modern tonality itself, traceable to the ancient Greek doctrine of antiphon and to the medieval habit of writing chants with ambitus spanning more than an octave by treating the upper and lower registers as presenting the same modal material. The second assumption crystallized in the early seventeenth century as equal temperament gradually displaced meantone tuning and as chromatic harmony came to occupy an increasingly prominent role in the expressive vocabulary of composers from Gesualdo to Monteverdi.

Post-tonal music, the repertoire that emerges roughly after 1900 and whose analysis is the subject of this course, does not abandon either assumption. It radicalizes them. If tonal music uses the twelve pitch classes within a hierarchical framework — one pitch class as tonic, others as more or less stable scale degrees, the whole system organized by the gravitational pull of functional harmony — then post-tonal music removes that hierarchy while retaining the equivalence. The result is a space of twelve pitch classes in which no pitch class is privileged over any other, and in which the relationships between pitches must be described not by their positions in a scale but by the intervals between them and by the abstract properties of the collections they form. The analytical apparatus that serves this purpose — pitch-class arithmetic, set theory, twelve-tone theory, transformational theory — constitutes the subject matter of MUSIC 377.

Definition 1.1 (Pitch Class). Two pitches are pitch-class equivalent if they differ by zero or more octaves. The pitch class of a pitch is the equivalence class of all pitches related to it by octave equivalence. There are exactly twelve distinct pitch classes, corresponding to the twelve tones of the chromatic scale. We represent pitch classes by integers \(0, 1, 2, \ldots, 11\), assigning \(0\) to C, \(1\) to C\(\sharp\)/D\(\flat\), \(2\) to D, \(3\) to D\(\sharp\)/E\(\flat\), \(4\) to E, \(5\) to F, \(6\) to F\(\sharp\)/G\(\flat\), \(7\) to G, \(8\) to G\(\sharp\)/A\(\flat\), \(9\) to A, \(10\) to A\(\sharp\)/B\(\flat\), and \(11\) to B.

This integer notation, standard in post-tonal theory since Forte’s landmark treatise of 1973, encodes octave equivalence directly into the arithmetic: pitch class \(p\) and pitch class \(p + 12\) are the same element of our system. The underlying algebraic structure is the integers modulo 12.

Definition 1.2 (Pitch-Class Space \(\mathbb{Z}_{12}\)). Pitch-class space is the set \(\mathbb{Z}_{12} = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}\) equipped with the operation of addition modulo 12. That is, for \(x, y \in \mathbb{Z}_{12}\), we define \(x + y = (x + y) \bmod 12\), where \(\bmod\) denotes the remainder upon division by 12. The set \(\mathbb{Z}_{12}\) forms a cyclic group of order 12 under this operation, generated by any element \(n\) coprime to 12 (i.e., \(n \in \{1, 5, 7, 11\}\)).

The group structure of \(\mathbb{Z}_{12}\) is not merely a mathematical convenience. It captures the topology of pitch-class space: the fact that the “chromatic circle” wraps around, so that moving twelve semitones in any direction returns one to the starting pitch class. This circularity is absent from pitch space (where the piano keyboard extends finitely in both directions) but is fundamental to the pitch-class space in which post-tonal analysis operates. One should picture pitch-class space as a clock face: C at twelve o’clock, C\(\sharp\) at one o’clock, D at two, and so on around to B at eleven, after which twelve o’clock (C) is reached again. The clock metaphor is imperfect — clocks have twelve positions, not a continuous range — but it captures the essential topology.

Remark 1.1 (Enharmonic Equivalence). The integer notation treats C\(\sharp\) and D\(\flat\) as the same pitch class (integer 1), and similarly for all other enharmonic pairs. This is the assumption of enharmonic equivalence, which is exact in equal temperament. In just intonation or meantone tuning, C\(\sharp\) and D\(\flat\) are slightly different pitches (separated by the syntonic comma, approximately 21.5 cents). Post-tonal theory, following Forte and Straus, assumes twelve-tone equal temperament and enharmonic equivalence throughout. Spectral music (Chapter 6) is the primary post-tonal idiom that challenges this assumption by requiring pitches tuned to irrational fractions of the octave.

1.2 Pitch-Class Intervals

Definition 1.3 (Directed Pitch-Class Interval). The directed pitch-class interval from pitch class \(x\) to pitch class \(y\) is \[ i(x, y) = y - x \pmod{12}. \] This is an element of \(\mathbb{Z}_{12}\) and represents the number of semitones one must ascend, mod 12, to move from \(x\) to \(y\).

Note that \(i(x, y) \neq i(y, x)\) in general: the interval from C (0) to G (7) is \(i(0,7) = 7\) (a perfect fifth ascending), while the interval from G (7) to C (0) is \(i(7,0) = 0 - 7 \equiv 5 \pmod{12}\) (a perfect fourth ascending, or equivalently a perfect fifth descending). The directed interval is sensitive to order and to the “direction” of motion around the chromatic circle.

Example 1.1. Some illustrative directed pitch-class intervals: \[ i(0, 4) = 4 \quad (\text{C to E: major third}), \qquad i(4, 0) = 8 \quad (\text{E to C: minor sixth}), \] \[ i(9, 2) = 2 - 9 \equiv -7 \equiv 5 \pmod{12} \quad (\text{A to D: perfect fourth}), \] \[ i(11, 3) = 3 - 11 \equiv -8 \equiv 4 \pmod{12} \quad (\text{B to E\flat: major third}). \]

For many analytical purposes, however, we wish to abstract away from direction and ask only about the “size” of the interval, regardless of which pitch class is on top. This motivates the concept of interval class.

Definition 1.4 (Interval Class). The interval class of two pitch classes \(x\) and \(y\) is \[ \text{ic}(x, y) = \min\bigl(i(x,y),\; 12 - i(x,y)\bigr) = \min\bigl(i(x,y),\; i(y,x)\bigr). \] Interval classes take values in \(\{1, 2, 3, 4, 5, 6\}\). The interval class 6 (the tritone) satisfies \(\min(6, 12-6) = \min(6,6) = 6\), confirming it is self-inverse.
Remark 1.2 (The Six Interval Classes). Interval class 0 (the unison) is excluded from the standard listing because it arises only between a pitch class and itself — between identical elements. The six interval classes correspond to the following acoustical intervals and their inversional complements: ic 1 = minor second / major seventh; ic 2 = major second / minor seventh; ic 3 = minor third / major sixth; ic 4 = major third / minor sixth; ic 5 = perfect fourth / perfect fifth; ic 6 = tritone (augmented fourth = diminished fifth, self-inverting). The reduction from twelve directed intervals to six interval classes reflects two layers of equivalence: octave equivalence (already built into \(\mathbb{Z}_{12}\)) and inversional equivalence — the principle that a perfect fifth and a perfect fourth are, in atonal contexts, analytically equivalent because one is the complement of the other in the octave.

The analytical motivation for interval classes becomes clear when one reflects on the nature of atonal music. In tonal music, a perfect fourth and a perfect fifth have very different harmonic functions: the fifth is the dominant relation, the fourth the subdominant. In atonal music, where no interval has a privileged functional meaning, the distinction between “a fifth up” and “a fourth up” — which differ only in the direction one chooses to travel around the chromatic circle — is far less analytically significant than the underlying “size” of the interval. The interval class captures exactly this abstract size.

1.3 The Circle of Fifths and the Structure of \(\mathbb{Z}_{12}\)

Before turning to the operations of transposition and inversion, it is worth pausing to examine the internal arithmetic structure of \(\mathbb{Z}_{12}\) in some detail. The twelve elements of \(\mathbb{Z}_{12}\) can be arranged not only in the ascending chromatic ordering \(0, 1, 2, \ldots, 11\) but also in the ordering generated by repeated addition of 7 (a perfect fifth in semitones): \(0, 7, 2, 9, 4, 11, 6, 1, 8, 3, 10, 5, 0\). Since \(\gcd(7, 12) = 1\), the element 7 generates all of \(\mathbb{Z}_{12}\), and this sequence visits all twelve pitch classes before returning to the start. The resulting ordering is the familiar circle of fifths: C, G, D, A, E, B, F\(\sharp\), D\(\flat\), A\(\flat\), E\(\flat\), B\(\flat\), F, C.

Remark 1.3 (Generators of \(\mathbb{Z}_{12}\)). The generators of \(\mathbb{Z}_{12}\) — the elements \(n\) for which \(\gcd(n, 12) = 1\) — are \(\{1, 5, 7, 11\}\). In musical terms: ic 1 (semitone), ic 5 (fourth), ic 7 = ic 5 by inversional equivalence (fifth), and ic 11 = ic 1 by inversional equivalence (major seventh). These four interval classes are precisely the ones that, when applied repeatedly, cycle through all twelve pitch classes. The non-generators — \(\{2, 3, 4, 6, 8, 9, 10\}\) — generate proper subgroups of \(\mathbb{Z}_{12}\): the whole-tone scale (\(\langle 2 \rangle = \{0,2,4,6,8,10\}\)), the diminished seventh chord (\(\langle 3 \rangle = \{0,3,6,9\}\)), the augmented triad (\(\langle 4 \rangle = \{0,4,8\}\)), and the tritone pair (\(\langle 6 \rangle = \{0,6\}\)). Each of these subgroups corresponds to a pitch-class set of limited transposition — exactly the collections Messiaen called "modes of limited transposition."

The subgroup structure of \(\mathbb{Z}_{12}\) thus provides the algebraic explanation for the existence of Messiaen’s modes: they arise because the divisors of 12 greater than 1 (namely 2, 3, 4, 6, 12) correspond to proper subgroups of \(\mathbb{Z}_{12}\), and the orbits of repeated addition by these non-generating elements visit only fractions of the full chromatic circle. The musical consequence — that whole-tone, diminished, and augmented collections “come back to where they started” after fewer than twelve transpositions — is a theorem about the arithmetic of \(\mathbb{Z}_{12}\), not an aesthetic convention.

1.4 Transposition and Inversion

The two most fundamental operations on pitch-class space are transposition and inversion. Together they generate the group of symmetries that underlies all of post-tonal set theory.

Definition 1.5 (Transposition). For \(n \in \mathbb{Z}_{12}\), the transposition operation \(T_n : \mathbb{Z}_{12} \to \mathbb{Z}_{12}\) is defined by \[ T_n(p) = p + n \pmod{12}. \] \(T_0\) is the identity; \(T_n \circ T_m = T_{n+m}\). The set \(\{T_0, T_1, \ldots, T_{11}\}\) forms a group isomorphic to \(\mathbb{Z}_{12}\).
Definition 1.6 (Inversion). The inversion operation \(I : \mathbb{Z}_{12} \to \mathbb{Z}_{12}\) is defined by \[ I(p) = -p \pmod{12} = 12 - p \pmod{12}. \] More generally, the combined operation \(T_nI\) is defined by \[ T_nI(p) = n - p \pmod{12}. \] Note: \(T_nI\) first inverts (\(p \mapsto -p\)) then transposes (\(-p \mapsto -p + n\)). Each \(T_nI\) is an involution: \((T_nI)^2 = T_0 = \text{id}\). The unique fixed point of \(T_nI\) is \(p = n/2\) when \(n\) is even (two fixed points: \(n/2\) and \(n/2 + 6\)); when \(n\) is odd, \(T_nI\) has no fixed points in \(\mathbb{Z}_{12}\).
Theorem 1.1 (The \(T/I\) Group). The set of operations \(\{T_n : n \in \mathbb{Z}_{12}\} \cup \{T_nI : n \in \mathbb{Z}_{12}\}\) forms a group of order 24 under composition, called the \(T/I\) group, isomorphic to the dihedral group \(D_{12}\). Explicitly: \[ T_n \circ T_m = T_{n+m}, \quad T_nI \circ T_m = T_{n+m}I, \quad T_m \circ T_nI = T_{n-m}I, \quad T_mI \circ T_nI = T_{m-n}. \] The transpositions \(\{T_n\}\) form a normal subgroup isomorphic to \(\mathbb{Z}_{12}\), and \(D_{12} / \mathbb{Z}_{12} \cong \mathbb{Z}_2\).
Example 1.2 (Computing Transformations). Apply \(T_3I\) to the pitch class A (9): \[ T_3I(9) = 3 - 9 \equiv -6 \equiv 6 \pmod{12} = \text{F\sharp}. \] Apply \(T_7\) to the major triad \(\{0, 4, 7\}\) (C major): \[ T_7\{0,4,7\} = \{7, 11, 2\} = \text{\{G, B, D\}} = \text{G major.} \] Apply \(T_0I\) to C major: \(I\{0,4,7\} = \{0, 8, 5\} = \{0,5,8\} = \text{\{C, F, A\flat\}} = \text{F minor.}\)

The twelve transpositions and twelve inversions exhaust the symmetries under which pc sets are considered equivalent in set-class theory. Chapter 2 will build the entire theory of set classes on this foundation.

Chapter 2: Pitch-Class Sets

2.1 Sets, Normal Form, and Prime Form

A pitch-class set (pc set) is a subset of \(\mathbb{Z}_{12}\) — an unordered collection of distinct pitch classes. The cardinality of a pc set is the number of distinct pitch classes it contains; sets of cardinality 2, 3, 4, 5, 6 are called dyads, trichords, tetrachords, pentachords, hexachords. Because we deal with unordered sets, the notations \(\{0, 4, 7\}\), \(\{4, 7, 0\}\), and \(\{7, 0, 4\}\) all refer to the same pc set. Post-tonal analysts require a canonical representation to enable systematic comparison of sets across a score, and two such canonical forms have become standard: normal form and prime form.

Definition 2.1 (Normal Form). The normal form of a pitch-class set \(S = \{s_1, s_2, \ldots, s_n\}\) is the ordering \([s_{(1)}, s_{(2)}, \ldots, s_{(n)}]\) such that:
  1. All elements are arranged in ascending order mod 12.
  2. Among all \(n\) rotations of this ascending ordering, the one minimizing the interval \(s_{(n)} - s_{(1)} \pmod{12}\) (the span) is chosen.
  3. Ties in span are broken by choosing the rotation that minimizes \(s_{(n-1)} - s_{(1)}\), then \(s_{(n-2)} - s_{(1)}\), and so on (the "most packed to the left" criterion).
Definition 2.2 (Prime Form). The prime form of a pc set is obtained by:
  1. Computing the normal form \([a_0, a_1, \ldots, a_{n-1}]\) of \(S\).
  2. Computing the normal form of the inversion \(I(S) = \{12 - s \bmod 12 : s \in S\}\).
  3. Transposing each to begin on 0: \([0, a_1 - a_0, a_2 - a_0, \ldots, a_{n-1} - a_0]\).
  4. Choosing the more compact form by lexicographic comparison from the left.
Prime form is denoted with parentheses: \((0\,1\,4)\), \((0\,3\,7)\), etc.

The prime form of a pc set is invariant under both transposition and inversion: all transpositions and inversions of a set have the same prime form. Allen Forte’s catalogue in The Structure of Atonal Music (1973) lists all prime forms for sets of cardinality 3 through 9 and assigns each a label [cardinality-ordinal]. The catalogue contains 12 trichord classes, 29 tetrachord classes, 38 pentachord classes, 50 hexachord classes, 38 heptachord classes, 29 octachord classes, and 12 nonachord classes — 208 set classes in total (excluding the trivial cases of cardinalities 0, 1, 2, 10, 11, 12).

Example 2.1 (Normal Form Computation). Find the normal form of \(\{3, 7, 10, 2\}\) (E\(\flat\), G, B\(\flat\), D — a minor seventh chord). Arrange in ascending order: \([2, 3, 7, 10]\). Four rotations and their spans: \[ [2,3,7,10]: \text{span } 10-2=8; \quad [3,7,10,2]: \text{span } 14-3 \equiv 11; \quad [7,10,2,3]: \text{span } 15-7\equiv 8; \quad [10,2,3,7]: \text{span } 17-10\equiv 9. \] Two rotations tie at span 8: \([2,3,7,10]\) and \([7,10,2,3]\). Tiebreak: compare \(s_{(2)} - s_{(0)}\): \(7-2 = 5\) vs. \(2-7 \equiv 7\). Since \(5 < 7\), normal form is \([2,3,7,10]\). Prime form: subtract 2 from each: \([0,1,5,8]\). Check inversion: \(I\{2,3,7,10\} = \{10,9,5,2\}\), normal form \([2,5,9,10]\), transposed: \([0,3,7,8]\). Compare \([0,1,5,8]\) vs. \([0,3,7,8]\) lexicographically: 1 < 3, so prime form is \((0\,1\,5\,8)\), Forte label [4-19].

2.2 The Interval Vector

Definition 2.3 (Interval Vector). Let \(S\) be a pitch-class set of cardinality \(n\). The interval vector of \(S\) is the ordered 6-tuple \(\langle f_1, f_2, f_3, f_4, f_5, f_6 \rangle\), where \[ f_k = \#\bigl\{\{x, y\} \subseteq S : \text{ic}(x, y) = k\bigr\} \] counts the unordered pairs in \(S\) with interval class \(k\). The total number of pairs is \(\binom{n}{2} = \sum_{k=1}^{6} f_k\) (with ic-6 pairs counted once since the tritone is its own complement).
Example 2.2 (Major Triad). The major triad \(\{0, 4, 7\}\) has three pairs: \(\{0,4\}\) ic 4, \(\{4,7\}\) ic 3, \(\{0,7\}\) ic 5. Interval vector: \(\langle 0\,0\,1\,1\,1\,0 \rangle\). The minor triad \(\{0, 3, 7\}\) has pairs \(\{0,3\}\) ic 3, \(\{3,7\}\) ic 4, \(\{0,7\}\) ic 5 — the same vector, confirming [3-11] as a single set class.
Example 2.3 (Forte's Catalogue Entries). A selection of important set classes with their interval vectors: \[ \begin{array}{lll} \text{[3-1]} & (0\,1\,2) & \langle 2\,1\,0\,0\,0\,0 \rangle \quad \text{(chromatic trichord)} \\ \text{[3-5]} & (0\,1\,6) & \langle 1\,0\,0\,0\,1\,1 \rangle \quad \text{(``split'')} \\ \text{[3-11]} & (0\,3\,7) & \langle 0\,0\,1\,1\,1\,0 \rangle \quad \text{(major/minor triad)} \\ \text{[4-28]} & (0\,3\,6\,9) & \langle 0\,0\,4\,0\,0\,2 \rangle \quad \text{(fully dim.\ seventh)} \\ \text{[6-35]} & (0\,2\,4\,6\,8\,10) & \langle 0\,6\,0\,6\,0\,3 \rangle \quad \text{(whole-tone scale)} \\ \end{array} \]

The interval vector is a compact fingerprint of a set’s harmonic character. A set with large \(f_5\) contains many perfect fourths and fifths, giving it a resonant, “open” quality; a set with large \(f_1\) is saturated with semitones, producing the dense chromatic clusters characteristic of Webern and Ligeti. The interval vector allows the analyst to characterize a set’s harmonic color without reference to transposition or inversion.

Definition 2.4 (Z-Relation). Two pc sets \(S\) and \(T\) are Z-related if they have identical interval vectors but are not related by any \(T_n\) or \(T_nI\) — identical interval-class content, distinct set classes. Z-related sets bear a "Z" in Forte's label, e.g., [4-Z15] = \((0\,1\,4\,6)\) and [4-Z29] = \((0\,1\,3\,7)\), both with interval vector \(\langle 1\,1\,1\,1\,1\,1 \rangle\).

Z-related pairs are relatively rare — Forte’s catalogue contains 23 Z-pairs — but analytically significant: they demonstrate that the interval vector does not uniquely determine a set class. The set \((0\,1\,4\,6)\) and the set \((0\,1\,3\,7)\) sound equally “rich” in interval-class terms but are genuinely distinct abstract objects. The existence of Z-pairs is a consequence of the combinatorial structure of \(\mathbb{Z}_{12}\) and has no simple geometric explanation; it is, in some sense, a numerical coincidence of modular arithmetic.

Definition 2.5 (Complement). The complement of a pc set \(S \subseteq \mathbb{Z}_{12}\) is \(\bar{S} = \mathbb{Z}_{12} \setminus S\). If \(\#S = n\), then \(\#\bar{S} = 12 - n\).
Theorem 2.1 (Complement Theorem). If \(T\) is a transposition or inversion of \(S\), then \(\bar{T}\) is the corresponding transposition or inversion of \(\bar{S}\). Hence complementation sends set classes to set classes. Moreover, if \(S\) has interval vector \(\langle f_1, \ldots, f_6 \rangle\), then \(\bar{S}\) has interval vector \[ \langle f_1 + A, f_2 + A, f_3 + A, f_4 + A, f_5 + A, f_6 + B \rangle, \] where \(A = \binom{12-n}{2} - \binom{n}{2}\cdot 0\) ... (the exact formula depends on the set sizes via the combinatorial identity). Concretely: the complement of a trichord is a nonachord, and their interval vectors are related by a fixed additive constant in each ic position.

The complement relation provides an important bridge between locally occurring collections and the aggregate. In twelve-tone music (Chapter 4), the complement relation becomes especially salient: the row partitions the twelve pitch classes into two hexachords, and the relationship between those hexachords is governed by exactly the set-class theory of complements.

2.3b Similarity Relations Between Set Classes

Beyond the binary relationship of “same set class” vs. “different set class,” several metrics have been proposed to measure the degree of similarity between set classes that are not identical.

Definition 2.6b (Interval-Vector Similarity). Let \(S\) and \(T\) be pc sets with interval vectors \(\mathbf{v}(S) = \langle a_1, \ldots, a_6 \rangle\) and \(\mathbf{v}(T) = \langle b_1, \ldots, b_6 \rangle\). The interval-vector similarity of \(S\) and \(T\) is \[ \text{IVsim}(S, T) = \sum_{k=1}^{6} |a_k - b_k|. \] Two set classes are Rp-related (Forte's "relative prime" relation) if they share a common subset of cardinality one less than the smaller set's cardinality. They are Rn-related if they are Z-related (same interval vector, different set class).

Forte’s original similarity relations (R0, R1, R2, Rp) were the first systematic attempt to measure inter-set-class proximity. Subsequent theorists (Eric Isaacson, Robert Morris, Larry Solomon) proposed alternatives: Isaacson’s “IVSIMn” measures the cosine similarity between interval vectors; Morris’s “similarity index” measures the proportion of subset relations shared. All of these measures agree in their extremes — two set classes with the same interval vector are maximally similar (within the framework), while a trichord and a hexachord are maximally dissimilar — but they disagree in the middle range, reflecting genuine uncertainty about what “harmonic similarity” means for abstract set classes.

2.4 Symmetry and Invariance Under Transposition and Inversion

Definition 2.6 (Transpositional Symmetry). A pc set \(S\) has transpositional symmetry at level \(n\) if \(T_n(S) = S\). The degree of transpositional symmetry is the number of values \(n \in \mathbb{Z}_{12}\) for which \(T_n(S) = S\). Every set has \(T_0(S) = S\); a set with additional transpositional symmetries has special musical properties.
Example 2.4 (Symmetrical Sets). The whole-tone scale \(\{0,2,4,6,8,10\}\) satisfies \(T_2(S) = S\), \(T_4(S) = S\), \(T_6(S) = S\), \(T_8(S) = S\), \(T_{10}(S) = S\) — six transpositional symmetries. The fully diminished seventh chord \(\{0,3,6,9\}\) satisfies \(T_3(S) = T_6(S) = T_9(S) = S\) — four transpositional symmetries. The augmented triad \(\{0,4,8\}\) has \(T_4(S) = T_8(S) = S\). These high-symmetry sets are the ones that generate the "limited transposition" modes described by Messiaen in his *Traité de rythme, de couleur, et d'ornithologie*.
Definition 2.7 (Inversional Symmetry). A pc set \(S\) has inversional symmetry at axis \(n\) if \(T_nI(S) = S\). The axis of symmetry in this case consists of the pitch classes \(n/2\) and \(n/2 + 6\) (when \(n\) is even).

Inversionally symmetric sets play a special role in twelve-tone and atonal composition because they can be “paired” with themselves under inversion, creating the possibility of invariant pitches under the transformational operations the composer deploys.

Chapter 3: Analyzing Atonal Music with Set Theory

3.1 Segmentation and the Analytical Problem

The application of set-class theory to an actual musical score requires the analyst to decide which groups of pitches constitute meaningful segments. The score presents a continuous stream of pitches articulated by dynamics, register, rhythm, timbre, and articulation; the analyst must identify which groupings of pitches form the “meaningful” sets that constitute the building blocks of the composition’s structure. This decision is called segmentation, and it is the most interpretive — and therefore the most contested — step in set-class analysis.

Definition 3.1 (Segmentation). Segmentation is the analytical procedure of partitioning a musical passage into segments — contiguous or simultaneous groups of pitches — each of which is analysed as a pc set. A segmentation is motivated if the boundaries of each segment are reinforced by at least one of the following: (1) registral isolation (the segment occupies a distinct register), (2) rhythmic articulation (a segment is bounded by rests or by a distinctive rhythmic pattern), (3) melodic contour (a segment is defined by a local apex or nadir in the melodic line), (4) dynamic or timbral differentiation, (5) voice or instrument assignments in a polyphonic texture.

The absence of a unique correct segmentation has been one of the primary criticisms of Fortean set theory. Without a principled criterion, an analyst can find almost any set class in almost any passage. The appropriate response is to discipline the analysis: a good analysis finds segmentations that are musically motivated, reveals recurring set classes across multiple passages, and shows how those recurring sets are related by \(T_n\) or \(T_nI\). When the same set class recurs many times in a passage under different transformations, the analysis gains credibility precisely because the recurrences are mutually confirming.

3.2 Analytical Method: Identifying T_n and T_nI Relations

Given two pc sets \(A\) and \(B\) of the same cardinality, the analyst wishes to determine whether \(B = T_n(A)\) or \(B = T_nI(A)\) for some \(n\).

Theorem 3.1 (Finding \(T_n\) and \(T_nI\)). Let \(A = [a_0, a_1, \ldots, a_{n-1}]\) and \(B = [b_0, b_1, \ldots, b_{n-1}]\) be pc sets in normal form.
  • If \(B = T_n(A)\) for some \(n\), then \(n = b_0 - a_0 \pmod{12}\), and we verify by checking \(b_k = a_k + n \pmod{12}\) for all \(k\).
  • If \(B = T_nI(A)\), then \(n = b_0 + a_0 \pmod{12}\), and we verify by checking \(b_k = n - a_k \pmod{12}\) for all \(k\) (in possibly reordered fashion).
Example 3.1. Determine the relationship between \(A = \{0, 1, 4\}\) and \(B = \{5, 6, 9\}\). Since \(B\) and \(A\) have the same prime form \((0\,1\,4)\), they are in the same set class. Try \(T_n\): \(n = 5 - 0 = 5\); check \(0+5=5\) ✓, \(1+5=6\) ✓, \(4+5=9\) ✓. So \(B = T_5(A)\). Now try \(A' = \{0, 1, 4\}\) and \(C = \{3, 6, 7\}\). Try \(T_n\): \(n = 3\); check \(0+3=3\) ✓, \(1+3=4\) but \(4 \notin C\). Try \(T_nI\): \(n = 0+3 = 3\); \(T_3I(A) = \{3-0, 3-1, 3-4\} = \{3, 2, 11\}\) — not \(C\). Try \(n = 1+6 = 7\): \(T_7I\{0,1,4\} = \{7,6,3\} = \{3,6,7\}\) = \(C\). So \(C = T_7I(A)\).

3.3 Case Study: Schoenberg, Op. 11, No. 1

Schoenberg’s Three Piano Pieces, Op. 11 (1909) are among the earliest works of “free atonality” — pieces that abandon tonal syntax without yet adopting the systematic twelve-tone method. The opening of No. 1 is one of the most-analyzed passages in the post-tonal repertoire, and for good reason: it demonstrates with exceptional clarity how a single pc set can serve as the generative cell from which an entire composition is built.

The opening right-hand melody presents the pitches B (11), G\(\sharp\) (8), and G (7) in succession, forming the set \(\{7, 8, 11\}\). Normal form: \([7, 8, 11]\) with span 4. The inversion: \(\{12-7, 12-8, 12-11\} = \{5, 4, 1\} = \{1, 4, 5\}\), span 4. Transposing to 0: original gives \([0, 1, 4]\), inversion gives \([0, 1, 4]\) — the same! So both the set and its inversion have prime form \((0\,1\,4)\), Forte label [3-3], interval vector \(\langle 1\,0\,1\,1\,0\,0 \rangle\).

Example 3.2 (Trichordal Saturation in Op. 11, No. 1). The opening melody \(\{7, 8, 11\}\) is [3-3] at transposition level \(T_7\) (since \(T_7(0,1,4) = (7,8,11)\)). The left-hand chord in m. 2 contains \(\{4, 5, 8\}\): this is \(T_4(0,1,4) = (4,5,8)\). The ascending figure in mm. 3–4 yields \(\{9, 10, 1\}\) = \(T_9(0,1,4)\). The middle section introduces the inversion: \(T_3I(0,1,4) = (3-0,3-1,3-4) = (3,2,11) \equiv \{11,2,3\}\) — also [3-3] since this set class is inversionally symmetric.

In each case the three-note “motto” is recognizable as the same abstract structure \((0\,1\,4)\) appearing under different transpositions and inversions. The piece achieves its sense of motivic unity not through shared melody (as in tonal music) but through shared set-class identity across a variety of pitch-class levels.

3.4 Case Study: Webern, Op. 5, No. 4

Webern’s Five Movements for String Quartet, Op. 5 (1909) exhibit a sparser, more concentrated texture than Schoenberg’s Op. 11, yet its harmonic language is equally saturated by a small number of set classes. The fourth movement, marked Sehr langsam (Very slow), is only 13 measures long but contains some of the most refined set-class writing in the early atonal repertoire.

Example 3.3 (Webern Op. 5, No. 4 — Set Class [3-5]). The set \(\{0, 1, 6\}\), Forte label [3-5], has prime form \((0\,1\,6)\) and interval vector \(\langle 1\,0\,0\,0\,1\,1 \rangle\): one semitone (ic 1), one perfect fourth/fifth (ic 5), and one tritone (ic 6). This highly distinctive set — characterized by the extreme opposition of a semitone and a tritone — dominates Op. 5, No. 4.

Its inversional symmetry: \(T_7I(0,1,6) = (7,6,1)\) = \(\{1,6,7\}\) = \(T_1(0,5,6)\) — wait, check: \(\{1,6,7\}\) has intervals 5 (from 1 to 6), 1 (from 6 to 7), 6 (from 1 to 7). Same interval vector \(\langle 1\,0\,0\,0\,1\,1 \rangle\). Normal form of \(\{1,6,7\}\): span of \([1,6,7] = 6\), span of \([6,7,1]\equiv [6,7,13]\) is 7, span of \([7,1,6]\) is \(1-7 \equiv 6\). Tie between \([1,6,7]\) and \([7,1,6]\equiv[7,13,18]\). Tiebreak: second-interval \(6-1=5\) vs. \(13-7=6\); choose \([1,6,7]\), transposed to 0: \([0,5,6]\). Prime form: compare \([0,5,6]\) and \([0,1,6]\) (inversion): \(1 < 5\) so prime form is \((0\,1\,6)\). Confirmed: [3-5] appears as both \(\{0,1,6\}\) and its transpositions \(\{1,6,7\}\), \(\{2,7,8\}\), etc.

The inversional symmetry of [3-5] — the fact that it maps to itself under certain \(T_nI\) operations — means that as the set is transposed and inverted across the movement, invariant pitch classes appear between successive statements. Webern exploits this carefully: the isolated pizzicato chord in m. 7 shares two pitch classes with both the preceding and the following set statements, creating a fragile continuity from within the silence.

3.5 Case Study: Bartók, Music for Strings, Percussion and Celesta

Bartók’s approach to pitch organisation in his mature works differs from both Schoenberg and Webern. Rather than saturating a texture with a single set class, Bartók typically employs intervallic patterns — interval cycles, axis symmetry, and the “acoustic scale” — to organise pitch at multiple structural levels simultaneously.

The fugue subject of the first movement of the Music for Strings, Percussion and Celesta (1936) begins on A (9) and unfolds a chromatic-fifths wedge: 9, 10, 8, 11, 7, 0, 6, 1, 5, 2, 4, 3 — alternating upward semitone and downward perfect fifth (or, equivalently, upward minor second and upward tritone), gradually expanding outward to reach E\(\flat\) (3), the tritone of A.

Example 3.4 (Bartók Fugue Subject). The six successive tetrachords extracted from the fugue subject: \(\{9,10,8,11\}\), \(\{8,11,7,0\}\), \(\{7,0,6,1\}\), \(\{6,1,5,2\}\), \(\{5,2,4,3\}\), and the final dyad \(\{4,3\}\). Each tetrachord: normal form of \(\{9,10,8,11\} = [8,9,10,11]\), span 3, prime form \((0\,1\,2\,3)\) [4-1]. Next: \(\{7,8,11,0\}\) in order \([7,8,11,0]\), span \(0-7+12=5\); vs. \([8,11,0,7]\) span \(7-8+12=11\); vs. \([11,0,7,8]\) span \(8-11+12=9\); vs. \([0,7,8,11]\) span \(11-0=11\). Minimum span is 5 for \([7,8,11,0]\), transposed: \([0,1,4,5]\) — prime form \((0\,1\,4\,5)\) [4-8], interval vector \(\langle 2\,0\,1\,2\,0\,1 \rangle\). The consistent use of ic 1 and ic 5 (semitone and fourth/fifth) reflects Bartók's characteristic "fifths plus semitones" intervallic language, a post-tonal adaptation of folk-modal scales.

Chapter 4: Twelve-Tone Technique

4.1 The Historical Context of the Row

By the early 1920s Schoenberg recognized that the “freely atonal” music of his Expressionist period (1908–1923) lacked an underlying structural principle comparable to the hierarchies of tonal harmony. The works of this period achieved tremendous expressive power — the concentrated violence of Erwartung (1909), the austere economy of the Piano Pieces Op. 19 (1911), the fevered expressionism of Pierrot Lunaire (1912) — but at the cost of the long-range formal coherence that traditional forms had previously supplied. The twelve-tone method, which Schoenberg articulated in 1921–1923 and first deployed systematically in the Piano Suite Op. 25, was his solution: a compositional procedure that imposes order on the chromatic aggregate without reinstating tonal hierarchy.

Definition 4.1 (Tone Row). A tone row (also: series, twelve-tone set) is an ordered sequence \((p_0, p_1, p_2, \ldots, p_{11})\) of all twelve pitch classes, each appearing exactly once — a permutation of \(\mathbb{Z}_{12}\). A tone row is not a pc set (which is unordered) but an ordered tuple. The fundamental principle of twelve-tone composition is that all pitch material in a work is derived from the row and its transformations; no pitch class may be "repeated" (re-sounded as a new structural event) until all twelve have appeared.
Remark 4.1 (Twelve-Tone "Rules" in Practice). The prohibition on pitch-class repetition is a guideline, not an absolute rule enforced identically by all twelve-tone composers. Schoenberg frequently allows pitch classes to be sustained (as pedal tones or inner voices) while other row members sound, effectively exempting sustained tones from the "no repetition" principle. Berg treats the row with even greater freedom, sometimes using rows that outline diatonic scales or tonal harmonies and moving between rows mid-passage. Webern adheres most strictly to the row principle but exploits symmetrical rows to produce highly concentrated textures in which the row's abstract structure is directly audible.

4.2 Row Forms and the 12×12 Matrix

Definition 4.2 (Row Forms). Given a prime row \(P_0 = (p_0, p_1, \ldots, p_{11})\), the four canonical row forms are:
  • Prime \(P_n\): \((p_0 + n,\, p_1 + n,\, \ldots,\, p_{11} + n) \pmod{12}\).
  • Inversion \(I_n\): \((n - p_0,\, n - p_1,\, \ldots,\, n - p_{11}) \pmod{12}\).
  • Retrograde \(R_n\): \((p_{11} + n,\, p_{10} + n,\, \ldots,\, p_0 + n) \pmod{12}\).
  • Retrograde-Inversion \(RI_n\): \((n - p_{11},\, n - p_{10},\, \ldots,\, n - p_0) \pmod{12}\).
For \(n \in \{0,1,\ldots,11\}\), this yields \(4 \times 12 = 48\) row forms. If the row has symmetries, some forms coincide; the maximum for a maximally symmetric row is 24 distinct forms (when \(P_n = R_m\) for some \(n,m\)).
Definition 4.3 (Row Matrix). The row matrix (or "magic square") of \(P_0\) is the \(12 \times 12\) array \(M\) where row \(i\) is \(P_{I_0(i)}\) (the prime whose first element equals the \(i\)th element of \(I_0\)). Equivalently, \(M_{i,j} = p_j + (I_{p_0}(i) - p_0) \pmod{12}\). Reading left-to-right across row \(i\) gives \(P_{n_i}\); right-to-left gives \(R_{n_i}\); top-to-bottom in column \(j\) gives \(I_{m_j}\); bottom-to-top gives \(RI_{m_j}\).
Example 4.1 (Webern's Row, Op. 24). The row of Webern's Concerto Op. 24 is celebrated for its extreme motivic concentration. In integer notation: \((11, 10, 2, 3, 7, 6, 8, 4, 5, 0, 1, 9)\). The row's first three-note segment \((11,10,2)\) has interval content \([-1, +4] = [+11, +4]\) (a descending semitone and an ascending major third). The next trichord \((3,7,6)\) has intervals \([+4, -1]\) — the retrograde inversion of the first trichord's interval pattern. The four trichords in the row \((11,10,2)\), \((3,7,6)\), \((8,4,5)\), \((0,1,9)\) are all in the same set class and are related by the four \(T/I\) operations in a consistent pattern. This "derived row" — built from a single generating trichord — is characteristic of Webern's late style and explains the extraordinary motivic density of the Op. 24 Concerto.

4.3 Invariance and Combinatoriality

Definition 4.4 (Invariance). A pitch class \(p\) is invariant under \(T_n\) if \(p + n \equiv p \pmod{12}\), i.e., \(n \equiv 0\). Hence no pitch class is invariant under a non-trivial transposition. However, a pitch class \(p\) is invariant under \(T_nI\) if \(n - p \equiv p \pmod{12}\), i.e., \(2p \equiv n \pmod{12}\). When \(n\) is even, there are exactly two invariant pitch classes: \(p = n/2\) and \(p = n/2 + 6\). An invariant subset of a row under a transformation is a collection of pitch classes preserved (as a set) by that transformation.
Definition 4.5 (Combinatoriality). A row \(P_0\) is semi-combinatorial if there exists a row form (some \(I_n\), \(R_n\), or \(RI_n\)) whose first hexachord is the complement of \(P_0\)'s first hexachord. It is all-combinatorial if it is semi-combinatorial with respect to transposition at multiple levels and all three operations. Milton Babbitt identified six hexachord types (out of the 35 hexachord set classes in Forte's catalogue) that generate all-combinatorial rows: [6-1], [6-7], [6-8], [6-20], [6-32], and [6-33].
Remark 4.2 (Aggregate Formation). When a row is combinatorially paired with a transposition of its inversion — say, \(P_0\) with \(I_5\) — the two forms together state all twelve pitch classes in their first hexachords simultaneously. This creates a "secondary aggregate": the aggregate is heard not only horizontally (across the full row) but vertically (across the two voices at any hexachordal moment). Babbitt's term for this property is aggregate completion or aggregate formation. All-combinatorial rows allow aggregate formation in all four row-form types and at multiple transposition levels, creating the maximum saturation of the aggregate across multiple voices.

4.4 Schoenberg’s Piano Suite Op. 25

The Piano Suite Op. 25 (1921–1923) is the first composition Schoenberg completed using the twelve-tone method and the first complete twelve-tone work in the repertoire. Its row in integer notation: \((4, 5, 7, 1, 6, 3, 8, 2, 11, 0, 9, 10)\) (E, F, G, D\(\flat\), F\(\sharp\), E\(\flat\), G\(\sharp\), D, B, C, A, B\(\flat\)).

Example 4.2 (Op. 25 Row Analysis). The first hexachord of \(P_0\) is \(\{4,5,7,1,6,3\} = \{1,3,4,5,6,7\}\), prime form \((0\,1\,2\,3\,4\,6)\), Forte label [6-1] — the chromatic hexachord. The second hexachord \(\{8,2,11,0,9,10\} = \{0,2,8,9,10,11\}\), prime form — its complement in \(\mathbb{Z}_{12}\) is \(\{1,3,4,5,6,7\}\) = the first hexachord. So the two hexachords are complements: \(P_0\) is semi-combinatorial with \(I\) forms. Specifically, \(I_5\) has first hexachord \(\{1,0,10,4,11,2\} = \{0,1,2,4,10,11\}\)... (verifying combinatoriality requires the full calculation). The six movements of Op. 25 (Prelude, Gavotte, Musette, Intermezzo, Menuett, Gigue) each deploy the row in characteristic ways: the Gavotte uses \(P_4\) and \(RI_4\) in counterpoint, exploiting the symmetry of \(P_4 = RI_4\) (up to ordering) to produce a near-palindromic texture.

4.4b Schoenberg’s Wind Quintet Op. 26 and Hexachordal Combinatoriality

The Wind Quintet Op. 26 (1924) is Schoenberg’s first large-scale twelve-tone work, and it demonstrates a more sophisticated deployment of the row than the earlier Suite Op. 25. The row \((9, 10, 0, 11, 4, 3, 8, 7, 6, 1, 5, 2)\) — A, B\(\flat\), C, B, E, E\(\flat\), G\(\sharp\), G, F\(\sharp\), C\(\sharp\), F, D — has the property that its first hexachord \(\{9,10,0,11,4,3\} = \{0,3,4,9,10,11\}\) belongs to set class [6-20] (prime form \((0\,1\,4\,5\,8\,9)\)), one of Babbitt’s all-combinatorial hexachords.

Example 4.2b (Hexachordal Regions in Op. 26). Because [6-20] is all-combinatorial, the row \(P_9\) can be paired with \(I_3\) to form secondary aggregates: the first hexachord of \(P_9\) = \(\{9,10,0,11,4,3\}\) and the first hexachord of \(I_3\) = \(\{3-9, 3-10, 3-0, 3-11, 3-4, 3-3\} \pmod{12} = \{6,5,3,4,11,0\} = \{0,3,4,5,6,11\}\). Are these complements? \(\{9,10,0,11,4,3\} \cup \{0,3,4,5,6,11\}\)... they share elements 0, 3, 4, 11 — not complements. The correct inversion level for semi-combinatoriality requires that the first hexachord of \(I_n\) be the complement of the first hexachord of \(P_0\) (not \(P_9\)). Since [6-20] is all-combinatorial, there exist \(I\), \(R\), and \(RI\) forms at specific transposition levels that complement the first hexachord at those levels. Schoenberg exploits these pairings throughout the Quintet to create a texture in which each six-measure phrase states the aggregate in both horizontal (sequential row statements) and vertical (simultaneous hexachord complements) dimensions.

4.5 Berg’s Free Use of the Row

Alban Berg’s approach to twelve-tone composition stands in deliberate contrast to Schoenberg’s. In Lulu (1929–1935) and the Violin Concerto (1935), Berg employs rows that contain tonal references — the Violin Concerto’s row \((7, 11, 2, 6, 9, 0, 4, 8, 11, 0, 2, 5)\) consists of four overlapping minor and major thirds (outlining the open strings of the violin) followed by a segment of the whole-tone scale. The last four pitch classes \((11, 0, 2, 5)\) = (B, C, D, F) quote the opening four notes of the Bach chorale “Es ist genug,” which Berg sets explicitly in the Concerto’s final movement. This calculated tonal allusion within a twelve-tone framework is characteristic of Berg’s “dialectical” serialism — the row is both a post-tonal serial structure and a vehicle for tonal reminiscence and autobiographical reference.

Chapter 5: Total Serialism and Its Extensions

5.1 The Postwar Serial Revolution

The decade following World War II witnessed a radical extension of Schoenberg’s twelve-tone idea to all musical parameters. The twin impulses were Messiaen’s “Mode de valeurs et d’intensités” (Étude de rythme No. 2, 1949) and Babbitt’s “Some Aspects of Twelve-Tone Composition” (1955). Both responded, in different ways, to the question: if pitch can be serialized, why not everything else?

Messiaen’s “Mode de valeurs” employs three “modes” — ordered series of 36, 24, and 12 elements respectively — each assigning to each pitch a specific duration value, dynamic level (\(\textit{ppp}\) through \(\textit{fff}\)), and articulation (no articulation, staccato, accent, etc.). The three modes proceed simultaneously in three independent strands, creating a texture of extraordinary complexity: no two adjacent events share all four parameters (pitch, duration, dynamic, articulation) because each parameter follows its own series independently.

Definition 5.1 (Total Serialism). Total serialism (or integral serialism) is a compositional method in which multiple musical parameters are each organized by an independent serial ordering. Formally: let parameters \(\pi_1, \pi_2, \ldots, \pi_k\) have finite ordered sets of values \(V_1, V_2, \ldots, V_k\) with \(|V_i| = n_i\). Each parameter is governed by a permutation \(\sigma_i\) of \(V_i\) (the "series" for that parameter), and the composition unfolds by cycling through the permutations, possibly with transformations (retrograde, inversion, transposition) applied to each independently.

5.2 Boulez’s Structures Ia and the Critique of Total Serialism

Pierre Boulez’s Structures Ia for two pianos (1952) is the most systematic realization of total serialism’s theoretical program. Boulez derived his pitch series directly from Messiaen’s “Mode de valeurs”: E\(\flat\), D, A, A\(\flat\), G, F\(\sharp\), E, C\(\sharp\), C, B\(\flat\), F, B (integers \(3,2,9,8,7,6,4,1,0,10,5,11\)). The duration series assigns values proportional to ordinal position (1 unit through 12 units). Dynamic and articulation “series” are independently derived. Piano I and Piano II unfold different forms of the row matrix simultaneously, so that at any moment both pitch and duration in both pianos are determined by serial operations on the row.

Remark 5.1 (The Perceptual Paradox). The paradox of total serialism, widely acknowledged by the mid-1950s, is that a system designed to maximize structural rigor produces, in performance, what many listeners experience as indistinguishable from randomness. When every parameter is independently serialized, no single parameter can carry the burden of perceptual organisation; the listener, unable to track 12-element series simultaneously across pitch, duration, dynamic, and articulation, hears a complex, apparently arbitrary surface. Ligeti's celebrated 1960 analysis of *Structures Ia* demonstrates precisely this: he shows that despite the elaborate serial organisation, the piece is perceptually "white noise" — all events equally probable, all contrasts equally intense, all "gestures" equally undifferentiated. This critique does not invalidate serial procedure as a compositional tool but reveals the gap between structural organisation and perceptual coherence.

5.3 Babbitt’s Time-Point Sets

Milton Babbitt’s response to the challenge of serializing rhythm was more sophisticated than simple duration series. The concept of the time-point set, developed in “Twelve-Tone Rhythmic Structure and the Electronic Medium” (1962), treats rhythmic position within a measure as a pitch-class-like object.

Definition 5.2 (Time-Point Set). Divide a measure into \(n\) equal time-points (positions \(0, 1, \ldots, n-1\)). A time-point set is an ordered sequence of time-points in \(\mathbb{Z}_n\) analogous to a pitch-class set. For twelve-tone rhythm, \(n = 12\), and each position represents one twelfth of the tactus. Transposition \(T_k\) of a time-point set shifts all time-points by \(k\) positions mod 12 — a temporal shift equivalent to a change of metric position. Inversion maps time-point \(t\) to \(-t \pmod{12}\) — a rhythmic retrograde. The full \(T/I\) group acts on time-point sets exactly as on pitch-class sets.

The formal isomorphism between pitch-class space and time-point space is Babbitt’s central insight. By treating rhythm as an instance of the same mathematical structure that governs pitch, one achieves genuine integration of pitch and rhythm — not the parallel but independent serialization of Boulez, but a single structural logic that governs both dimensions. Babbitt exploited this isomorphism extensively in his electronic works (Composition for Synthesizer, 1961; Philomel, 1964) and in his instrumental chamber music (Semi-Simple Variations, 1956; the String Quartets), creating textures of extraordinary rhythmic complexity that nonetheless derive from the same twelve-tone logic as the pitch organization.

5.4 Xenakis and Stochastic Music

The composer Iannis Xenakis, trained as an architect under Le Corbusier and as a mathematician, proposed a radically different response to the crisis of total serialism: rather than imposing deterministic serial order, he governed musical events by probability distributions.

Definition 5.3 (Stochastic Music). In Xenakis's formulation, a stochastic music is one governed by probability distributions over musical parameters. If events arrive according to a Poisson process with rate \(\lambda\) (events per unit time), the probability of exactly \(k\) events in an interval of length \(t\) is \[ P(N(t) = k) = \frac{(\lambda t)^k e^{-\lambda t}}{k!}. \] The mean number of events in time \(t\) is \(\lambda t\); the variance is also \(\lambda t\). Pitch and duration can be drawn from Gaussian, exponential, or uniform distributions. The composer specifies distributions and their parameters; the actual pitch-duration sequence is determined by (pseudo-)random sampling.
Example 5.1 (*Achorripsis*, 1957). Xenakis divided *Achorripsis* into a \(7 \times 28 = 196\)-cell matrix (7 instrument groups, 28 time sections). The mean event density in each cell was determined by a Poisson process with \(\lambda = 0.37\) events per cell (the value he calculated by minimising a complexity measure). The actual pitch classes within each event were drawn from a uniform distribution over the chromatic scale; glissandi followed exponential distributions for slope. The result: a score of quasi-random events whose global statistical properties (density, register distribution) are under precise compositional control, even though no individual event is predetermined.

5.4b Carter’s Metric Modulation

Elliott Carter’s solution to the challenge of rhythmic complexity was neither serial (like Babbitt’s time-point sets) nor stochastic (like Xenakis’s Poisson processes) but structural: the technique of metric modulation, in which the tempo changes smoothly by a rational ratio, with a note value of the old tempo becoming a note value of the new tempo.

Definition 5.4b (Metric Modulation). A metric modulation occurs when the performer reinterprets a note value \(v_1\) in the old tempo \(T_1\) beats-per-minute as a different note value \(v_2\) in the new tempo \(T_2\) beats-per-minute, such that the duration of \(v_1\) in the old tempo equals the duration of \(v_2\) in the new tempo: \[ \frac{60}{T_1} \cdot d_1 = \frac{60}{T_2} \cdot d_2 \quad \Longrightarrow \quad \frac{T_2}{T_1} = \frac{d_2}{d_1}, \] where \(d_1\) and \(d_2\) are the durations of \(v_1\) and \(v_2\) in their respective beat units. The new tempo is a rational multiple of the old, determined by the note values chosen for the modulation.
Example 5.2b (Metric Modulation). Suppose the current tempo is \(T_1 = 120\) bpm (quarter note = 120). The composer introduces a triplet quarter-note (\(d_1 = 2/3\) of a beat). In the new tempo, this triplet quarter becomes the new quarter note (\(d_2 = 1\) beat). Then \(T_2 = T_1 \cdot (d_2 / d_1) = 120 \cdot (3/2) = 180\) bpm. The new tempo is 180 bpm — a tempo increase of \(3:2\). Carter uses such modulations to create a sense of continuous tempo acceleration or deceleration that remains precisely notated and reproducible, unlike the gradual rubato of Romantic performance practice.

Carter’s String Quartet No. 1 (1951) and Concerto for Orchestra (1969) are among the most extensively metric-modulated works in the repertoire. The analytical task of tracing the tempo structure through a Carter work — mapping the sequence of metric modulations, computing the exact tempo at each moment, and understanding how the multiple simultaneous tempo layers interact — requires exactly the rational arithmetic of Definition 5.4b, applied iteratively across hundreds of measure-boundaries.

5.5 Ligeti’s Micropolyphony

György Ligeti’s Atmosphères (1961) and Lontano (1967) represent a different response: borrowing the density and complexity of total serialism while abandoning its deterministic underpinning. The concept Ligeti called micropolyphony involves writing dense canons in many voices — sometimes 56 or more simultaneous lines — each moving so slowly and at such similar pitch levels that individual melodic contours become imperceptible.

Remark 5.2 (Statistical Analysis of Micropolyphony). Micropolyphonic textures are most accurately described by statistical properties rather than by set-class labels or row analyses. The relevant analytical quantities are: (1) the registral centroid — the mean pitch class weighted by simultaneity density; (2) the spectral bandwidth — the standard deviation of pitch classes around the centroid; (3) the rate of centroid change over time. *Atmosphères* opens with a 56-voice saturated chromatic cluster spanning four octaves (mean density: all 12 pitch classes present throughout), then gradually thins and polarizes toward isolated pitch classes at the close. The analysis of this temporal process is properly a matter of time-varying spectral density, not of set-class succession.

Chapter 6: Spectral Music and Acoustic Analysis

6.1 The Spectral School: Origins and Premises

The “spectral” school of composition emerged from the work of Gérard Grisey and Tristan Murail in Paris in the late 1970s, associated with the ensemble L’Itinéraire (founded 1973). The central premise of spectralism is that the natural overtone series — the acoustic phenomenon underlying every sustained musical tone — should serve as the primary generative material of musical composition. Where serialism derived its harmonic language from abstract permutational mathematics applied to the chromatic scale, spectralism derives it from the physics of vibrating strings and columns of air.

Definition 6.1 (Harmonic Series). A tone at fundamental frequency \(f_0\) Hz produces a harmonic series consisting of sinusoidal components at frequencies \(f_k = k \cdot f_0\) for \(k = 1, 2, 3, \ldots\). The \(k\)th partial has frequency \(k f_0\). The interval from the \(m\)th to the \(n\)th partial (\(n > m\)) has frequency ratio \(n : m\) and measures \[ 1200 \cdot \log_2\!\left(\frac{n}{m}\right) \text{ cents} \] in equal temperament. The fundamental is the first partial (\(k = 1\)); the higher partials are the overtones.

The spectral composer observes that the harmonic series does not align with equal temperament above the first several partials. The seventh partial at \(7f_0\) lies approximately 31 cents flat of the equal-tempered minor seventh above the fundamental. The eleventh partial lies approximately 49 cents sharp of the equal-tempered tritone. The thirteenth partial lies roughly between the equal-tempered major sixth and minor sixth. These “deviations” from equal temperament are not imprecisions but the actual acoustic content of the tone, and the spectral composer embraces them by writing in quarter-tones or eighth-tones.

6.2 Grisey’s Partiels: Orchestral Synthesis

Gérard Grisey’s Partiels (1975), the third work in the six-work cycle Les Espaces Acoustiques, is the paradigmatic text of spectral music. The work opens with a solo trombone on E2 (approximately 82.4 Hz), then gradually introduces other instruments, each assigned to a specific partial of that E2 tone, constructing an orchestral “synthesis” of the harmonic spectrum.

Example 6.1 (Spectrum of E2 \(\approx 82.4\) Hz, Partials 1–14). \[ \begin{array}{crlr} k & f_k \text{ (Hz)} & \text{Nearest ET pitch} & \text{Deviation (cents)} \\ \hline 1 & 82.4 & \text{E2} & 0 \\ 2 & 164.8 & \text{E3} & 0 \\ 3 & 247.2 & \text{B3} & +2 \\ 4 & 329.6 & \text{E4} & 0 \\ 5 & 412.0 & \text{G\sharp4} & -14 \\ 6 & 494.4 & \text{B4} & +2 \\ 7 & 576.8 & \text{D5} & -31 \\ 8 & 659.2 & \text{E5} & 0 \\ 9 & 741.6 & \text{F\sharp5} & +4 \\ 10 & 824.0 & \text{G\sharp5} & -14 \\ 11 & 906.4 & \text{A\sharp5} & -49 \\ 12 & 988.8 & \text{B5} & +2 \\ 13 & 1071.2 & \text{C\sharp6} & +41 \\ 14 & 1153.6 & \text{D6} & -31 \\ \end{array} \] Grisey scored each partial for a specific orchestral instrument, with dynamics proportional to the natural amplitude envelope of that partial (lower partials louder, higher partials softer), reconstructing the acoustic spectrum as an orchestral texture. The result is that the orchestra "becomes" a single, enormous resonating body — the E2 trombone tone heard at a temporal magnification of several orders of magnitude.

6.3 Spectral Time and Temporal Envelopes

The concept of “spectral time” in Grisey’s work refers to the temporal processes through which the harmonic spectrum transforms — processes that mirror the acoustics of real sounds at vastly expanded time scales.

Definition 6.2 (Spectral Time). Spectral time is the organisation of musical duration by analogy with acoustic temporal processes: the attack, sustain, and decay envelope of a physical sound. A "spectral" work is structured so that its large-scale temporal arch — its rate of harmonic change, its density of events, its dynamic trajectory — mirrors the temporal envelope of the acoustic material from which its pitch content is derived. In *Partiels*, the opening measures (orchestra assembling the harmonic series) correspond to the "attack" of the original trombone tone; the central section (full spectrum sustained) to the "sustain"; and the gradual dissolution of partials in the coda to the "decay."

6.4 Murail’s Gondwana and Spectral Transformation

Tristan Murail’s Gondwana (1980) for orchestra demonstrates the technique of spectral transformation: the systematic morphing of one acoustic spectrum into another.

Definition 6.3 (Stretched Spectrum). Given a harmonic series with partials at \(f_0, 2f_0, 3f_0, \ldots\), a stretched spectrum with stretch factor \(\alpha > 1\) places partials at \[ f_k = f_0 \cdot \alpha^{k-1}, \quad k = 1, 2, 3, \ldots. \] When \(\alpha = 2\), adjacent partials are an octave apart — a "maximally stretched" harmonic series (all pitches octave-equivalent). For \(\alpha\) slightly greater than 2, the spectrum resembles the inharmonic spectrum of a bell, whose vibration modes produce non-integer-multiple frequencies due to the physical stiffness of the bell's shell. As \(\alpha \to 1\), the spectrum compresses toward a single pitch-class cluster.

Gondwana begins with bell-like inharmonic sonorities (\(\alpha > 2\)), derived from the acoustic analysis of a bell struck at a specific pitch. Over the course of the work, the stretch factor decreases toward 2, and the spectrum gradually “harmonicizes” — acquiring the character of a brass-like harmonic series. This systematic transformation of spectral type is the structural skeleton of the piece, replacing the thematic development of Classical form with a physically motivated acoustic metamorphosis.

6.4b Haas and Extended Spectral Technique

Georg Friedrich Haas (b. 1953) represents a second generation of spectral composers, working in Vienna and deeply influenced by Grisey and Murail. Haas’s most distinctive contribution is the systematic use of combination tones — psychoacoustic phenomena that arise when two tones sound simultaneously.

Definition 6.4b (Combination Tones). When two tones at frequencies \(f_1\) and \(f_2\) (with \(f_2 > f_1\)) sound simultaneously at sufficient intensity, the auditory system produces additional perceived tones at the difference frequency \(f_2 - f_1\) and the summation frequency \(f_1 + f_2\), as well as higher-order combination tones at \(2f_1 - f_2\), \(3f_1 - 2f_2\), etc. These combination tones are not present in the acoustic signal but arise from the non-linear response of the basilar membrane.

Haas exploits combination tones in works like in vain (2000) for 24 instruments: he writes pairs of simultaneous pitches whose difference tones fall on specific harmonically important pitch classes. The combination tones are not notated — they cannot be, since they are heard inside the listener’s auditory system rather than sounded by any instrument — but they are calculated and composed with the same precision as the written notes. The result is a harmonic texture that changes as the listener’s distance from the ensemble changes (since combination tones depend on the relative intensity of the two source tones) and that literally sounds different from different positions in the concert hall.

6.5 Microtonality as Spectral Consequence

The microtonality required by spectral music — the quarter-tones, eighth-tones, and arbitrary pitch deviations seen in Example 6.1 — is not an expressive device chosen for novelty but a direct consequence of accurately representing the physics of the harmonic series. The spectral composer is, in a sense, doing empirical science: measuring the frequencies of the partials of a real physical sound, converting them to musical notation, and demanding that performers reproduce those frequencies as precisely as possible.

Remark 6.1 (Timbre and Harmony in Spectral Music). Spectral music dissolves the traditional distinction between timbre and harmony. In conventional orchestral writing, timbre is a property of individual instruments (strings sound differently from winds), while harmony is the relationship between simultaneously sounding pitches. In spectral music, harmony is constructed by distributing the partials of a single tone across multiple instruments: the "chord" and the "tone" are the same thing at different scales of observation. The harmonic series of an E2 trombone, when played by the full orchestra with each instrument assigned one partial, is both a particular timbral color (the color of E2) and a specific harmony (the spectral chord built on E2). This identification of timbre and harmony is the most radical proposition of the spectral school.

Chapter 7: Transformational Theory

7.1 Lewin’s Reconception of Musical Space

David Lewin’s Generalized Musical Intervals and Transformations (GMIT, 1987) is one of the most philosophically ambitious works in the history of music theory. Its central argument is simple but radical: the traditional way of thinking about musical intervals — as distances measured between two static points in a musical space — should be replaced by a way of thinking about musical transformations — the operations that move us from one object to another.

In Lewin’s words, the traditional theoretical question is “what is the interval \(i(s, t)\) from \(s\) to \(t\)?”, a question that treats musical space as a geometry and musical objects as static points. Lewin proposes replacing this with the question “what is the transformation \(g\) such that \(g(s) = t\)?”, emphasizing agency, process, and directed motion rather than distance and location. The difference is not merely philosophical: it has direct analytical consequences, enabling a unified treatment of pitch, rhythm, timbre, and any other musical parameter within a single algebraic framework.

Definition 7.1 (Generalized Interval System). A Generalized Interval System (GIS) is a triple \((S, IVLS, \text{int})\) where:
  • \(S\) is a non-empty set (the space of musical objects).
  • \((IVLS, \cdot)\) is a group (the interval group).
  • \(\text{int} : S \times S \to IVLS\) is a function (the interval function) satisfying:
    1. Composition law: For all \(r, s, t \in S\), \(\;\text{int}(r, t) = \text{int}(r, s) \cdot \text{int}(s, t)\).
    2. Unique transposition: For each \(s \in S\) and \(i \in IVLS\), there exists a unique \(t \in S\) satisfying \(\text{int}(s, t) = i\).
Example 7.1 (Pitch-Class GIS). The standard pitch-class GIS: \(S = \mathbb{Z}_{12}\), \(IVLS = (\mathbb{Z}_{12}, +)\), \(\text{int}(x, y) = y - x \pmod{12}\). Verify axioms: (1) \(\text{int}(r,t) = t - r = (s - r) + (t - s) = \text{int}(r,s) + \text{int}(s,t)\) ✓; (2) given \(s\) and \(i\), set \(t = s + i\) — unique ✓. The interval from C (0) to G (7) is 7; from G (7) to D (2) is 7; both are "perfect fifth" in the GIS sense, confirming that \(\text{int}\) captures the notion of directed interval.
Example 7.2 (Rhythmic GIS). Set \(S = \mathbb{Z}_{12}\) (time-points within a measure, as in Definition 5.2), \(IVLS = (\mathbb{Z}_{12}, +)\), \(\text{int}(t_1, t_2) = t_2 - t_1 \pmod{12}\). This is formally identical to the pitch-class GIS, confirming the isomorphism between pitch-class space and time-point space that Babbitt exploited.

7.2 Transformation Networks

Definition 7.2 (Transformation Network). A transformation network is a directed graph \((V, E, \ell_V, \ell_E)\) where \(V\) is a finite set of nodes, \(E \subseteq V \times V\) is a set of directed edges, \(\ell_V : V \to S\) assigns musical objects to nodes, and \(\ell_E : E \to \text{End}(S)\) assigns transformations to edges, such that for each \((u, v) \in E\): \[ \ell_E(u,v)\bigl(\ell_V(u)\bigr) = \ell_V(v). \] A network is internally consistent if for any two directed paths from node \(a\) to node \(b\), the composed transformations along both paths are equal.

Transformation networks make explicit the web of relationships that structure a musical passage. Where a Roman-numeral analysis shows the function of each chord within a tonal hierarchy, and a set-class analysis shows the interval-class content of each collection, a transformation network shows the operations that connect successive or simultaneous objects — the “grammar” of the passage at the level of process rather than content.

Example 7.3 (Network for a Twelve-Tone Passage). Consider a passage where a tetrachord \(A = \{0,1,4,6\}\) is followed by \(B = \{3,4,7,9\} = T_3(A)\), then \(C = \{6,7,10,0\} = T_3(B) = T_6(A)\), then \(D = \{9,10,1,3\} = T_3(C) = T_9(A)\). The transformation network has four nodes labelled \(A, B, C, D\) with three \(T_3\) arrows \(A \to B \to C \to D\). Internal consistency: the composed arrow \(A \to D\) (via two paths: directly and via \(B\) and \(C\)) both give \(T_9\). ✓ The network reveals the cyclic structure of the passage: applying \(T_3\) four times returns to the original set (\(T_{12} = T_0 = \text{id}\)).

7.3 Klumpenhouwer Networks

A particularly influential extension of Lewin’s transformational theory is the theory of Klumpenhouwer networks (K-nets), developed by Henry Klumpenhouwer and elaborated by Lewin.

Definition 7.3 (Klumpenhouwer Network). A Klumpenhouwer network (K-net) is a transformation network \((V, E, \ell_V, \ell_E)\) where:
  • \(S = \mathbb{Z}_{12}\) (pitch-class space).
  • Each node is labelled with a single pitch class.
  • Each edge is labelled with either a transposition \(T_n\) or an inversion \(T_nI\).
  • The network is internally consistent.
Two K-nets \(\mathbf{K}_1\) and \(\mathbf{K}_2\) (with the same graph structure) are isographic if there exists an automorphism of the \(T/I\) group mapping each edge label of \(\mathbf{K}_1\) to the corresponding edge label of \(\mathbf{K}_2\). Positive isography preserves all \(T_n\) labels and scales all \(T_nI\) axes by the same constant; negative isography inverts the \(T_n\) direction.

K-nets are especially useful for analysing passages where a single chord or collection is related to others by a mixture of transpositions and inversions, and where the internal structure (the relationships between notes within a chord) mirrors the external structure (the relationships between chords). This “recursive” or “self-similar” quality — in which the same transformational logic operates at multiple structural levels — is characteristic of much late-twentieth-century post-tonal music.

7.3b RICH Chains in Webern and Berg

The RICH transformation (Definition 7.3) generates chains of overlapping row forms that share boundary pitch classes, creating a specific kind of melodic continuity. In Webern’s serial works, RICH chains are a structural principle: successive row statements are connected by pitch-class invariance at their boundaries.

Example 7.4 (RICH Chain in Webern Op. 27). In the Variations for Piano Op. 27 (1936), the row \(P_0 = (0, 11, 3, 4, 8, 7, 9, 6, 2, 1, 5, 10)\) is frequently chained by RICH: \(\text{RICH}(P_0) = RI_{0+10} = RI_{10}\). The first element of \(P_0\) is 0 and the last is 10, so RICH maps these boundary elements to each other, and the next row statement begins on 10 (the last element of \(P_0\)), connecting the two statements by shared boundary pitch class. Applying RICH again: \(\text{RICH}(RI_{10})\) shares the boundary \(\{10, \ldots, 0\}\) (since \(RI_{10}\) is the retrograde inversion whose first element maps from 10 and last maps to 0). The chain \(P_0 \to RI_{10} \to P_0 \to \cdots\) (alternating) is a two-element RICH cycle — a palindromic oscillation that is entirely characteristic of Webern's late style.
Remark 7.2 (Contextual Transformations and Musical Meaning). Transformations like RICH and FLIP are called contextual because they are defined not by a fixed formula (\(T_7\) always adds 7) but by the context — specifically, by the content of the musical object being transformed (the boundary pitch classes of the row in the case of RICH). Contextual transformations are more analytically powerful than fixed transformations in one respect: they can reveal structural relationships that depend on the specific pitches present, not merely on the abstract interval. But they require the analyst to specify the "context" — what counts as the relevant boundary, what constitutes the "structure" of the object being transformed — which reintroduces the interpretive ambiguity that fixed transformations avoid.

7.4 Applying Transformational Theory: Lewin on Brahms

Lewin’s own extended analyses apply transformational theory not only to the atonal and twelve-tone repertoire but also to tonal music. His analysis of Brahms’s song “Der Wunsch” from Liebeslieder-Walzer Op. 47 demonstrates that transformational thinking illuminates aspects of tonal music that Roman-numeral analysis overlooks.

Remark 7.1 (Transformational vs. Functional Analysis). In a functional-harmonic analysis of tonal music, each chord is labelled by its scale-degree function (I, V, IV, ii, etc.) and its relationship to adjacent chords is described by root motion. In Lewin's transformational analysis, each chord is labelled by a pitch-class set or individual pitch, and the relationships between chords are described by the specific \(T_n\) or \(T_nI\) that connects them. The two approaches are not contradictory but complementary: functional analysis emphasises the harmonic hierarchy (the tonal center and the gravitational field around it), while transformational analysis emphasises the symmetry of the voice-leading operations (which transformations recur, which are avoided, which create the large-scale shape). For post-tonal music, where functional hierarchy is absent, transformational analysis is often the primary analytical tool; for tonal music, it provides a supplementary lens that reveals structural features invisible to functional analysis.

Chapter 8: Neo-Riemannian Theory and Chromatic Harmony

8.1 Riemann Revisited: Dualism and Parsimonious Transformations

The closing chapter of this course brings us, in a certain sense, back to the beginning: we return to the triad, the most familiar object of tonal music, and find that it conceals a mathematical structure quite different from functional-harmonic logic. Neo-Riemannian theory, developed by Brian Hyer, Richard Cohn, and others in the early 1990s, mines Hugo Riemann’s nineteenth-century dualist theory for a set of triadic transformations that operate entirely within pitch-class space, making no reference to tonal function.

Riemann’s original dualism held that major and minor triads are acoustic “mirror images”: the major triad is generated upward from a root by a major third and a perfect fifth, while the minor triad is generated downward from a “root” (placed at the top) by the same intervals. This metaphysically motivated theory found few adherents in the twentieth century, but it contains an insight of lasting value: the major and minor triads are related by inversion in pitch-class space, and this inversional relationship generates a family of transformations with remarkable mathematical properties.

Definition 8.1 (Triadic Transformations P, L, R). A consonant triad is a pitch-class set of the form \(\{r, r+4, r+7\}\) (major) or \(\{r, r+3, r+7\}\) (minor) for root \(r \in \mathbb{Z}_{12}\). The three basic neo-Riemannian transformations are:
  • P (Parallel): Maps each major triad to its parallel minor, and each minor triad to its parallel major, by moving the third by one semitone. In pitch-class terms: \(P\{r, r+4, r+7\} = \{r, r+3, r+7\}\).
  • L (Leading-tone exchange): Maps each major triad to a minor triad by moving the root down by semitone; maps each minor triad to a major triad by moving the fifth up by semitone. \(L\{r, r+4, r+7\} = \{r+4, r+7, r+11\} = \{r-1, r+4, r+7\} \pmod{12}\).
  • R (Relative): Maps each major triad to its relative minor, and vice versa, moving one pitch class by a whole tone. \(R\{r, r+4, r+7\} = \{r+4, r+7, r+9\} \pmod{12}\).
Each of P, L, R is an involution: \(P^2 = L^2 = R^2 = \text{id}\). Each preserves two pitch classes and moves one by a semitone (P, L) or a whole tone (R).
Remark 8.1 (Parsimony). The defining property of P, L, and R is voice-leading parsimony: each transformation moves only one voice, and that voice moves by the smallest possible interval (one or two semitones). This parsimony makes neo-Riemannian progressions acoustically smooth — the listener perceives a stepwise connection between triads that are harmonically distant. It is precisely this combination of harmonic distance (the two triads may share no root, third, or fifth relationship in tonal terms) and voice-leading proximity (only one voice moves, and minimally) that gives chromatic triadic music of the nineteenth century its characteristic quality: distant yet smooth, adventurous yet connected.

8.2 The Group Generated by P, L, R

Theorem 8.1 (The Neo-Riemannian Group). The group \(\mathcal{G}\) generated by \(\{P, L, R\}\), acting on the set of 24 consonant triads, is isomorphic to the dihedral group \(D_{12}\) of order 24. This group acts simply transitively on the 24 triads: for any two triads \(\tau_1, \tau_2\), there is a unique \(g \in \mathcal{G}\) with \(g(\tau_1) = \tau_2\).

Proof sketch: The 24 triads biject naturally with the 24 elements of \(D_{12}\) via the identification of a triad with the coset of its root in \(\mathbb{Z}_{12} / \langle 4 \rangle\) (mod the major-triad structure). The generators satisfy the dihedral group relations: \(P^2 = L^2 = R^2 = (LP)^{12} = \ldots = \text{id}\). Simple transitivity follows from the fact that any triad can be reached from any other by a unique product of \(P, L, R\) operations. \(\square\)

Example 8.1 (LR Cycle and Hexatonic Systems). Starting from C major: \[ \begin{array}{lll} \text{C major} & \xrightarrow{L} & \text{E minor} \xrightarrow{R} \text{G major} \xrightarrow{L} \text{B minor} \xrightarrow{R} \text{D major} \xrightarrow{L} \text{F\sharp minor} \xrightarrow{R} \text{A major} \xrightarrow{L} \text{C\sharp minor} \xrightarrow{R} \cdots \end{array} \] Applying \(LR\) six times returns to C major. The six triads \{\text{C major, E minor, G major, B minor, D major, F\sharp minor}\} form an \(LR\)-chain cycling through two "hexatonic systems." Alternatively: the PL cycle. \[ C \text{ maj} \xrightarrow{P} C\text{ min} \xrightarrow{L} A\flat\text{ maj} \xrightarrow{P} A\flat\text{ min} \xrightarrow{L} E\text{ maj} \xrightarrow{P} E\text{ min} \xrightarrow{L} C\text{ maj}. \]

Applying PL three times returns to the start. The six triads {C maj, C min, A\(\flat\) maj, A\(\flat\) min, E maj, E min} form a hexatonic system — Cohn’s term for the four-triad orbit of the PL group, together with their parallel minor partners.

Definition 8.2 (Hexatonic Systems). Cohn identifies four hexatonic systems, each a set of six triads (three major, three minor) closed under the PL subgroup of the neo-Riemannian group: \[ \begin{array}{ll} \text{Northern: } & \{C\text{ maj, C min, A\flat maj, A\flat min, E maj, E min}\} \\ \text{Eastern: } & \{E\text{ maj, E min, C maj, C min, A\flat maj, A\flat min}\} \end{array} \] (and two more). Each hexatonic system contains triads that collectively exhaust only six of the twelve pitch classes — hence "hexatonic." The four systems partition the 24 triads into four groups of six. The "hexatonic pole" of a triad is the triad in the opposite hexatonic system related by the PLR transformation — a maximally voice-leading-distant triad connected by a smooth three-step path.

8.3 The Tonnetz

Definition 8.3 (Tonnetz). The Tonnetz is a planar graph (tiling a torus due to pitch-class equivalence) with:
  • Vertices: the 12 pitch classes \(\mathbb{Z}_{12}\).
  • Horizontal edges: connecting pitch classes related by perfect fifth (ic 5), so the row \(\ldots F\text{-}C\text{-}G\text{-}D\text{-}A\text{-}E\text{-}B\text{-}F\sharp\text{-}\ldots\)
  • Northeast-diagonal edges: connecting pitch classes related by major third (ic 4): \(C\text{-}E\text{-}G\sharp\text{-}C\), \(D\text{-}F\sharp\text{-}B\flat\text{-}D\), etc.
  • Southeast-diagonal edges: connecting pitch classes related by minor third (ic 3): \(C\text{-}E\flat\text{-}F\sharp\text{-}A\text{-}C\), etc.
Every triangular face of the Tonnetz represents a consonant triad. P is a reflection across a horizontal edge (preserving the perfect fifth, moving the third); L is a reflection across a northeast-diagonal edge (preserving the major third, moving the root/fifth); R is a reflection across a southeast-diagonal edge (preserving the minor third, moving the fifth).
Theorem 8.2 (Tonnetz as a Torus). Because \(3 \times 4 = 12 \equiv 0 \pmod{12}\) (three major thirds = octave) and \(4 \times 3 = 12 \equiv 0\) (four minor thirds = octave), and \(12 \times 7 \equiv 0\) (twelve fifths = seven octaves by the circle of fifths), the Tonnetz lattice tiles the plane with periods \((12, 0)\) in the fifth-direction and \((0, 3)\) in the third-direction. Quotienting by these periods yields a torus of finite area. The 12 pitch classes become the 12 vertices of this toroidal graph, the 24 triangular faces correspond to the 24 consonant triads, and P, L, R act as reflections of the triangular tiling.

The Tonnetz representation makes the geometry of P, L, R completely transparent: each transformation reflects a triangle across one of its three sides, flipping it to the adjacent triangle. A sequence of transformations traces a path across the Tonnetz surface, and the overall geometric shape of that path encodes the large-scale harmonic structure of a passage.

8.4 Applications to the Repertoire

Example 8.2 (Beethoven, String Quartet Op. 131 in C\(\sharp\) Minor). The first movement of Op. 131 (1827) opens with a fugue subject on C\(\sharp\) minor. In the recapitulation, the countersubject traverses the following triadic sequence: \[ C\sharp\text{ min} \xrightarrow{R} E\text{ maj} \xrightarrow{L} G\sharp\text{ min} \xrightarrow{R} B\text{ maj} \xrightarrow{L} D\sharp\text{ min} \xrightarrow{R} F\sharp\text{ maj}. \]

This alternating \(RL\) chain visits six triads forming a complete LR-chain (an orbit of the \(\langle LR \rangle\) subgroup). On the Tonnetz, this traces a zig-zag path moving consistently northeast — a straight-line trajectory on the toral surface. The functional-harmonic analyst, forced to interpret each chord as a Roman numeral, finds successive modulations to E major, G\(\sharp\) minor, B major, D\(\sharp\) minor, F\(\sharp\) major — an increasingly remote series of keys. The neo-Riemannian analyst sees a single coherent transformation sequence — \(RL\) applied five times — describing a straight path on the torus.

Example 8.3 (Schubert, String Quintet D. 956, Adagio). The slow movement of Schubert's String Quintet (1828) contains one of the most celebrated passages in the neo-Riemannian repertoire: the middle section's oscillation between E major and F minor. These two triads are separated by a semitone root motion but are not diatonically related in any key; the progression defies conventional Roman-numeral analysis.

In the Tonnetz: E major occupies the triangle \(\{E, G\sharp, B\} = \{4, 8, 11\}\). F minor occupies the triangle \(\{F, A\flat, C\} = \{5, 8, 0\}\). These two triangles share the edge \(\{G\sharp, C\} = \{8, 0\}\)… wait: G\(\sharp\) = 8 and C = 0, and \(\text{ic}(8, 0) = 4\) — the major-third edge. Reflecting E major across its major-third edge yields: the edge \(\{4, 8\}\) (E-G\(\sharp\)) is the northeast diagonal; reflecting across it maps \(B (11)\) to \(F (5)\), yielding triangle \(\{4, 8, 5\} = \{4, 5, 8\} = \{E, F, G\sharp\}\) — that’s an augmented triad, not F minor. Let me reconsider: E major and F minor share pitch class A\(\flat\) = G\(\sharp\) (8). This single shared tone is the “common tone” across the hexatonic pole. Cohn calls these “hexatonic poles” — the most distant relationship within the neo-Riemannian system (no two of the three voice-leading transformations connect them directly; it takes PLR or equivalently LPR to reach from E major to F minor). Despite this harmonic distance, the voice leading is maximally parsimonious: each voice moves by one semitone. This paradox — harmonic distance combined with voice-leading proximity — is the defining character of Schubert’s most adventurous chromatic progressions.

Example 8.4 (Wagner, Tristan und Isolde Prelude — Opening). The opening of the Tristan Prelude (1857) has generated more analytical literature than perhaps any other passage in Western music. The famous "Tristan chord" — spelled F, B, D\(\sharp\), G\(\sharp\) in the standard analysis of mm. 2–3 — has been identified as a half-diminished seventh chord (root F, minor third A\(\flat\), diminished fifth B, major seventh E\(\sharp\) enharmonically) or equivalently as a French augmented sixth chord. In integer notation: \(\{5, 11, 3, 8\} = \{3, 5, 8, 11\} = \{D\sharp, F, A\flat, B\}\). Prime form: \((0\,2\,5\,8)\), Forte label [4-27], interval vector \(\langle 0\,1\,2\,1\,1\,1 \rangle\).

The chord resolves (incompletely, deceptively) to an E dominant seventh \(\{4, 8, 11, 2\}\) = [4-27] again — the “resolution” chord is in the same set class as the “dissonance.” On the Tonnetz: the motion from the Tristan chord to the E\(^7\) can be read as motion from the triangle \(\{B, D\sharp, G\sharp\}\) (B major triad) toward \(\{G\sharp, B, E\}\) (E major triad) via \(R\). The Prelude never stabilizes in a single hexatonic system; it navigates through three of the four systems across its 111 measures, touching the fourth system only at the climactic statement of the main theme at the structural apex. Cohn’s analysis in Audacious Euphony (2012) demonstrates that this large-scale hexatonic navigation is not accidental but encodes the dramatic arc of desire, displacement, and perpetual non-resolution that is the opera’s metaphysical subject.

8.5 Mathematical Unification and the Future of Post-Tonal Theory

Theorem 8.3 (Unification via Group Actions). The three major analytical frameworks of this course — Fortean set-class theory, Lewinian transformational theory, and Cohn's neo-Riemannian theory — are all instances of a group \(G\) acting on a set \(S\) of musical objects:
  • Set-class theory: \(G = D_{12}\), \(S = \mathcal{P}(\mathbb{Z}_{12})\) (subsets of pitch-class space). Set classes are the orbits of this action.
  • Transformational theory: \(G = IVLS\) (the interval group of a GIS), \(S\) is the space of musical objects. The GIS is the data of the group action plus the interval function.
  • Neo-Riemannian theory: \(G = D_{12}\), \(S\) = the 24 consonant triads. The Tonnetz encodes the geometry of this action.
The deep unity of post-tonal theory is the unity of group actions: wherever music exhibits symmetry — octave equivalence, transposability, invertibility, voice-leading parsimony — group theory provides the language to describe it precisely.

The student who completes this course should be equipped with a versatile analytical toolkit: the ability to compute normal forms, prime forms, and interval vectors of pc sets; to construct and read row matrices; to identify recurring set classes across a passage and trace their \(T_n\) and \(T_nI\) relationships; to understand the aesthetic and perceptual implications of serial, stochastic, and spectral compositional methods; to construct transformation networks for short passages; and to apply the PLR transformations and Tonnetz to chromatic triadic music. More fundamentally, the student should have acquired a way of hearing — a sensitivity to the interval-class content of post-tonal harmonies, the transformational logic connecting successive musical objects, and the voice-leading geometry that connects apparently remote triadic sonorities. Post-tonal music demands an engaged, analytically informed listener, and the mathematics of this course is in service of that listening.

Remark 8.2 (Expanding Scope of Post-Tonal Methods). Post-tonal theory was developed primarily to analyse music of the early and mid-twentieth century — Schoenberg, Webern, Berg, Bartók, Stravinsky, Messiaen, Boulez, Babbitt, Xenakis. But its methods are not historically confined. The same set-class apparatus that illuminates Webern's Op. 5 applies to the harmonic language of jazz: set class [4-27] (the half-diminished seventh) and [4-26] (the dominant seventh) are the primary harmonic objects of post-bop jazz, and the voice-leading relationships between them follow exactly the neo-Riemannian transformations P and L. Neo-Riemannian theory has been applied to film music (the work of Deborah Mawer and Guy Capuzzo on John Williams, Hans Zimmer), to progressive rock (Dmitri Tymoczko's analysis of Pink Floyd and Tool), and to contemporary pop (Christopher White's work on common-practice and post-tonal elements in popular music). The GIS framework of Lewin is so general that it encompasses all of these applications and more — any musical parameter that can be modelled as a set with a group action admits a GIS description. Post-tonal theory is not a historical museum but a living mathematical language, one whose deepest structures continue to illuminate music written yesterday and music not yet imagined.
Remark 8.3 (Computational Music Theory). The computational tractability of the mathematical structures in this course has made them central to the emerging field of computational music analysis. The 208 set classes in Forte's catalogue, the 48 row forms of any twelve-tone row, the 24 triads and 24 neo-Riemannian transformations — these are finite, enumerable objects that can be exhaustively computed and searched. Algorithms for computing normal form and prime form run in \(O(n \log n)\) time (where \(n\) is the cardinality of the set). Algorithms for finding all row forms related by combinatoriality, for computing the Tonnetz path between two triads, for analysing a full score for set-class recurrences — all of these are feasible computations. The Music21 library (Python) and the music analysis toolkit in Mathematica both implement the core algorithms of this course. The computational perspective reinforces the mathematical one: post-tonal theory is not merely a descriptive vocabulary but a precise formal system, amenable to algorithmic implementation and capable of generating analytical insights at scales beyond manual computation.

Supplement A: Set-Class Tables and Reference Material

A.1 Complete Trichord Set-Class Table

The following table lists all twelve trichord set classes (cardinality-3 pc sets) with their Forte labels, prime forms, interval vectors, and representative pitch-class collections. Familiarity with these twelve trichords is essential for analytical work in the early atonal repertoire, where trichordal structure is almost universally the primary level of motivic organisation.

Definition A.1 (Trichord Catalogue). The twelve trichord set classes are: \[ \begin{array}{lllp{5cm}} \text{Label} & \text{Prime Form} & \text{IV} & \text{Common Names / Character} \\ \hline \text{[3-1]} & (0\,1\,2) & \langle 2\,1\,0\,0\,0\,0 \rangle & \text{Chromatic cluster; ic-1 saturated} \\ \text{[3-2]} & (0\,1\,3) & \langle 1\,1\,1\,0\,0\,0 \rangle & \text{Common in atonal music; asymmetric} \\ \text{[3-3]} & (0\,1\,4) & \langle 1\,0\,1\,1\,0\,0 \rangle & \text{Schoenberg Op. 11, No. 1 motto trichord} \\ \text{[3-4]} & (0\,1\,5) & \langle 1\,0\,0\,1\,1\,0 \rangle & \text{Asymmetric; ic-1, ic-4, ic-5} \\ \text{[3-5]} & (0\,1\,6) & \langle 1\,0\,0\,0\,1\,1 \rangle & \text{``Split'' trichord; Webern Op. 5, No. 4} \\ \text{[3-6]} & (0\,2\,4) & \langle 0\,2\,0\,1\,0\,0 \rangle & \text{Whole-tone trichord; all ic-2 and ic-4} \\ \text{[3-7]} & (0\,2\,5) & \langle 0\,1\,1\,0\,1\,0 \rangle & \text{Sus2/sus4 chord fragment} \\ \text{[3-8]} & (0\,2\,6) & \langle 0\,1\,0\,1\,0\,1 \rangle & \text{Dominant seventh fragment; asymmetric} \\ \text{[3-9]} & (0\,2\,7) & \langle 0\,1\,0\,0\,2\,0 \rangle & \text{Quartal trichord; open, resonant} \\ \text{[3-10]} & (0\,3\,6) & \langle 0\,0\,2\,0\,0\,1 \rangle & \text{Diminished triad} \\ \text{[3-11]} & (0\,3\,7) & \langle 0\,0\,1\,1\,1\,0 \rangle & \text{Major/minor triad; the consonant triad} \\ \text{[3-12]} & (0\,4\,8) & \langle 0\,0\,0\,3\,0\,0 \rangle & \text{Augmented triad; transpositionally symmetric} \\ \end{array} \]

Notice the extreme cases: [3-1] \((0\,1\,2)\) is the most compact possible trichord (all three notes adjacent on the chromatic scale), while [3-12] \((0\,4\,8)\) is the most evenly distributed (augmented triad), dividing the octave into three equal major thirds. The augmented triad [3-12] is the unique trichord with three identical interval classes between adjacent members and three transpositional symmetries (\(T_4\) and \(T_8\) in addition to \(T_0\)). It is, in a precise sense, the trichordal analogue of the whole-tone scale (which divides the octave into six equal major seconds).

A.2 Hexachord Set-Class Table: All-Combinatorial Types

The following six hexachords are the all-combinatorial source hexachords identified by Babbitt. A row whose first hexachord belongs to one of these set classes is all-combinatorial — it can be paired with its own inversion, retrograde, and retrograde-inversion (at appropriate transposition levels) to form secondary aggregates in all three combinatorial operations.

Definition A.2 (All-Combinatorial Hexachords). Babbitt's six source sets: \[ \begin{array}{lll} \text{Forte Label} & \text{Prime Form} & \text{Interval Vector} \\ \hline \text{[6-1]} & (0\,1\,2\,3\,4\,5) & \langle 5\,4\,3\,2\,1\,0 \rangle \\ \text{[6-7]} & (0\,1\,2\,6\,7\,8) & \langle 4\,2\,0\,2\,4\,3 \rangle \\ \text{[6-8]} & (0\,2\,3\,4\,5\,7) & \langle 3\,4\,3\,2\,3\,0 \rangle \\ \text{[6-20]} & (0\,1\,4\,5\,8\,9) & \langle 3\,0\,3\,6\,3\,0 \rangle \\ \text{[6-32]} & (0\,2\,4\,5\,7\,9) & \langle 1\,4\,3\,2\,5\,0 \rangle \\ \text{[6-35]} & (0\,2\,4\,6\,8\,10) & \langle 0\,6\,0\,6\,0\,3 \rangle \\ \end{array} \]

Set [6-35] is the whole-tone hexachord — the entire whole-tone scale — and it is maximally symmetric: it has six transpositional symmetries (\(T_0, T_2, T_4, T_6, T_8, T_{10}\)) and six inversional symmetries, for a total symmetry group of order 12 (half the order of the full \(T/I\) group). Set [6-20] is Babbitt’s favourite source hexachord, appearing in works from his Three Compositions for Piano (1947) onward: its distinctive interval vector \(\langle 3\,0\,3\,6\,3\,0 \rangle\) — all ic-3 and ic-4 content, no ic-1, ic-2, or ic-6 — gives it a distinctive harmonic color, simultaneously resonant (from the ic-4 major thirds) and ambiguous (the ic-3 minor thirds hover between consonance and dissonance).

A.3 Messiaen’s Modes of Limited Transposition

Olivier Messiaen’s “modes of limited transposition,” described in his treatise La Technique de mon langage musical (1944), are pitch-class sets whose transpositional symmetry groups are non-trivial — sets that return to themselves under transpositions other than \(T_0\). The name “limited transposition” reflects the fact that such a set can be transposed to only a limited number of distinct transposition levels (rather than the usual 12).

Definition A.3 (Mode of Limited Transposition). A pc set \(S\) is a mode of limited transposition if \(T_n(S) = S\) for some \(n \neq 0\) — equivalently, if the transpositional symmetry group of \(S\) is non-trivial. The number of distinct transpositions of \(S\) equals \(12 / k\), where \(k = \#\{n : T_n(S) = S\}\) is the order of the transpositional symmetry group.
Example A.1 (Messiaen's Seven Modes). Messiaen identified seven modes: \[ \begin{array}{llll} \text{Mode} & \text{pc set} & \text{Sym. order} & \text{Distinct transpositions} \\ \hline 1 & \{0,2,4,6,8,10\} & 6 & 2 \quad\text{(whole-tone scale)} \\ 2 & \{0,1,3,4,6,7,9,10\} & 4 & 3 \quad\text{(octatonic scale)} \\ 3 & \{0,2,3,4,6,7,8,10,11\} & 3 & 4 \\ 4 & \{0,1,2,5,6,7,8,11\} & 2 & 6 \\ 5 & \{0,1,5,6,7,11\} & 2 & 6 \\ 6 & \{0,2,4,5,6,8,10,11\} & 2 & 6 \\ 7 & \{0,1,2,3,5,6,7,8,9,11\} & 2 & 6 \\ \end{array} \]

Mode 2 (the octatonic scale) is the most important for analytical purposes. It consists of alternating semitones and whole tones: \([0,1,3,4,6,7,9,10]\). Its symmetry group has order 4 (\(T_0, T_3, T_6, T_9\)), yielding only three distinct transpositions of the scale. This means that any two octatonic collections are identical under one of \(T_3, T_6, T_9\), and the “distance” between any two octatonic-derived harmonies can be measured solely in terms of which of the three octatonic collections they belong to. Stravinsky exploited this property extensively in his neoclassical works (Petrushka, The Rite of Spring) and Messiaen used it throughout his compositional career.

A.4 Voice-Leading Geometry and Chord Spaces

A significant development in post-tonal theory since the 1990s has been the geometrization of voice-leading. Dmitri Tymoczko’s A Geometry of Music (2011) argues that the voice-leading relationships between chords can be represented as distances in a geometric space whose topology is determined by the cardinality of the chords.

Definition A.4 (Chord Space). The chord space for \(n\)-voice chords is the orbifold \[ \mathcal{O}_n = \mathbb{T}^n / S_n, \] where \(\mathbb{T}^n = (\mathbb{R}/12\mathbb{Z})^n\) is the \(n\)-dimensional torus of pitch-class \(n\)-tuples (ordered), and \(S_n\) is the symmetric group acting by permutation of coordinates (encoding the equivalence of ordering within an unordered chord). The voice-leading distance between two chords \(A\) and \(B\) (as points in \(\mathcal{O}_n\)) is the minimum \(L^1\) distance over all bijections \(\phi : A \to B\): \[ d_{VL}(A, B) = \min_{\phi : A \to B} \sum_{a \in A} |a - \phi(a)| \pmod{12\text{-optimal}}. \]

This geometric framework provides an alternative to the neo-Riemannian lattice for visualizing voice-leading structure. In 2-voice chord space (\(\mathcal{O}_2 \cong\) a Möbius strip) and 3-voice chord space (\(\mathcal{O}_3 \cong\) an orbifold with triangular cross-sections), the voice-leading distances between chords correspond to Euclidean distances in the orbifold. Smooth voice leading — the kind that characterizes both Classical part-writing and Romantic chromatic harmony — corresponds to short paths in chord space. The “efficient voice leading” between two common-tone-related triads (like C major and A minor) is a very short path; the “hexatonic pole” connection (C major to E minor via PLPL) is longer but still traverses a compact region of the orbifold.

Supplement B: Extended Analytical Examples

B.1 Schoenberg’s Pierrot Lunaire, Op. 21 — “Mondestrunken”

The opening song of Pierrot Lunaire (1912) presents a texture of extraordinary motivic density. The piano introduction (mm. 1–4) is built entirely from two set classes: [3-3] \((0\,1\,4)\) and [3-9] \((0\,2\,7)\). The alternation of these two set classes — the first saturated with small intervals (ic 1, ic 3, ic 4), the second open and resonant (two ic-5 intervals) — establishes a motivic dialectic that permeates the entire song cycle.

Example B.1 (Mondestrunken, mm. 1–4 — Set-Class Analysis). The right-hand piano line in mm. 1–2 presents: B (11), G (7), E (4) — set \(\{4,7,11\}\). Normal form: \([4,7,11]\) span 7; \([7,11,4]\) span 9; \([11,4,7]\) span 8. Minimum span 7 for \([4,7,11]\), transposed: \([0,3,7]\). Prime form \((0\,3\,7)\) = [3-11] — a minor triad! But immediately following: B\(\flat\) (10), E (4), F\(\sharp\) (6) — set \(\{4,6,10\}\). Normal form: \([4,6,10]\) span 6; \([6,10,4]\) span 10; \([10,4,6]\) span 8. Minimum 6 for \([4,6,10]\), transposed: \([0,2,6]\). Prime form: compare \([0,2,6]\) with its inversion \([0,4,6]\) — lexicographically \(2 < 4\) so prime form is \((0\,2\,6)\) = [3-8]. The systematic alternation of [3-11] and [3-8] throughout the opening creates the characteristic "misty" quality of the song — tonal harmony (the minor triad) perpetually shadowed by its atonal near-double (the half-diminished fragment).

B.2 Stravinsky’s Rite of Spring — “Augurs of Spring”

The famous “Augurs of Spring” chord from Stravinsky’s Le Sacre du Printemps (1913) is one of the most celebrated sonorities in the post-tonal repertoire. The chord consists of an E\(\flat\) major triad in root position superimposed against a dominant seventh chord on E natural: \(\{3, 7, 10\} \cup \{4, 8, 11, 2\} = \{2, 3, 4, 7, 8, 10, 11\}\) — seven distinct pitch classes. The set class of this heptachord is [7-32] with prime form \((0\,1\,3\,4\,6\,8\,9)\) and interval vector \(\langle 3\,3\,5\,5\,3\,2 \rangle\).

Example B.2 (The Octatonic Context of the "Augurs" Chord). The "Augurs" chord \(\{2,3,4,7,8,10,11\}\) is not an arbitrary pitch collection: it belongs to the octatonic scale on E\(\flat\)/D\(\sharp\). The octatonic collection containing E\(\flat\) (3) is \(\{1,3,4,6,7,9,10,0\}\) = \(\{0,1,3,4,6,7,9,10\}\) (Messiaen's mode 2, transposition 1). The "Augurs" chord uses seven of these eight octatonic pitch classes (omitting 1 = C\(\sharp\)/D\(\flat\)): \(\{0,3,4,7,8,10,11\}\) — but wait, the chord contains pc 2 (D) which is not in this octatonic collection. Let us check: octatonic on E = \(\{0,1,3,4,6,7,9,10\}\). The "Augurs" chord \(\{2,3,4,7,8,10,11\}\): pcs 2, 11 are not in the E-octatonic collection. The chord in fact straddles two octatonic collections, which is characteristic of Stravinsky's polytonal technique: the E\(\flat\) major triad \(\{3,7,10\}\) is from one octatonic collection, and the E dominant seventh \(\{4,8,11,2\}\) is from a different one (the D-octatonic: \(\{2,3,5,6,8,9,11,0\}\) — check: 4 and... not all present). In fact the polytonal superimposition of two diatonically coherent triads/seventh chords from different tonal regions, rather than octatonic derivation, is Stravinsky's primary pitch-organizing technique in this passage.

B.3 Webern’s Symphony Op. 21 — Row Structure and Palindrome

Webern’s Symphony Op. 21 (1928) is one of the most mathematically sophisticated works in the twelve-tone literature. Its row \((0, 11, 3, 4, 8, 7, 9, 6, 2, 1, 5, 10)\) (using C = 0) has an extraordinary property: it is its own retrograde at the tritone transposition — \(P_0 = R_6\). This means the row is a palindrome at the pitch-class level (when the tritone transformation is applied), and it implies that the row matrix has only 24 distinct forms rather than the usual 48.

Example B.4 (Op. 21 Row Symmetry). The interval sequence of the row: \((11-0, 3-11, 4-3, 8-4, 7-8, 9-7, 6-9, 2-6, 1-2, 5-1, 10-5) \pmod{12} = (11, 4, 1, 4, 11, 2, 9, 8, 11, 4, 5)\). The retrograde of the row has interval sequence \((5, 4, 11, 8, 9, 2, 11, 4, 1, 4, 11)\) — the retrograde of the original interval sequence plus 12-complement. The row's palindrome property means that the first hexachord \(\{0,11,3,4,8,7\} = \{0,3,4,7,8,11\}\) and the second hexachord \(\{9,6,2,1,5,10\} = \{1,2,5,6,9,10\}\) are complements (partition \(\mathbb{Z}_{12}\)) and are related by the tritone transposition \(T_6\): \(T_6\{0,3,4,7,8,11\} = \{6,9,10,1,2,5\} = \{1,2,5,6,9,10\}\) ✓. The first hexachord has prime form \((0\,1\,4\,5\,8\,9)\) — it is set class [6-20], one of Babbitt's all-combinatorial hexachords! The Symphony's extraordinary architectural clarity — each movement structured as a series of canons whose entries and cadences are determined by the row's symmetries — flows directly from this mathematical property.

B.4 Messiaen’s Quartet for the End of Time — “Danse de la fureur”

Olivier Messiaen’s Quartet for the End of Time (1940–41) represents a synthesis of his two primary compositional techniques: his “modes of limited transposition” (modal pitch organisation) and his complex rhythmic procedures, including “non-retrogradable rhythms” (palindromic rhythms) and the use of Greek and Hindu rhythmic models.

Example B.5 (Modes and Non-Retrogradable Rhythms). The fifth movement, "Danse de la fureur pour les sept trompettes" (Dance of fury for the seven trumpets), scored for all four instruments in unison, uses only Mode 2 (the octatonic scale) throughout. The pitch collection is restricted to a single octatonic transposition: \(\{0,1,3,4,6,7,9,10\}\). Within this restriction, every melodic figure is drawn from this mode's eight pitch classes, creating a harmonic consistency that coexists with metric complexity. The rhythmic organisation employs a "non-retrogradable" (palindromic) rhythmic cell: a duration sequence \((d_1, d_2, d_3, d_2, d_1)\) reads the same forwards and backwards. Messiaen believed these palindromic rhythms had a quality of "impossible retracing" — they create a sense of temporal stasis despite constant motion. The combination of octatonic pitch organisation (with its three-fold symmetry under \(T_3\)) and palindromic rhythmic organisation (with its bilateral temporal symmetry) produces the movement's characteristic quality of feverish intensity without forward momentum.

B.5 Ligeti’s Piano Études — Micropolyphony Reduced to Two Voices

Ligeti’s Piano Études (1985–2001) apply many of the same density-and-texture principles as Atmosphères but reduce them to the two-hand, ten-finger medium of the solo piano. The first étude, “Désordre,” is particularly revealing: the two hands play essentially the same pattern in parallel, but the right hand uses only white keys (the C major / A minor pentatonic set \(\{0,2,4,7,9\}\) plus passing tones) while the left hand uses only black keys (\(\{1,3,6,8,10\}\) = the pentatonic complement). Each hand plays a repeated rhythmic pattern of 8 + 5 = 13 pulses, but the two hands begin each cycle at different points, creating a shifting polyrhythm whose phase relationship changes at every cycle.

Example B.6 (Désordre: Set-Class Analysis of the Bimodal Structure). The right-hand pitch collection \(\{0,2,4,7,9\}\): prime form \((0\,2\,4\,7\,9)\), Forte label [5-35] (the pentatonic scale), interval vector \(\langle 0\,3\,2\,1\,4\,0 \rangle\). The left-hand collection \(\{1,3,6,8,10\}\) = \(T_1\{0,2,5,7,9\}\): also prime form \((0\,2\,4\,7\,9)\) = [5-35]. The two pentatonic collections are Z-related? No — they are in fact transpositions of each other (\(T_6\{0,2,4,7,9\} = \{6,8,10,1,3\} = \{1,3,6,8,10\}\) ✓), hence in the same set class. More precisely, the two-hand combination yields all twelve pitch classes: \(\{0,2,4,7,9\} \cup \{1,3,6,8,10\} = \{0,1,2,3,4,6,7,8,9,10\} = \mathbb{Z}_{12} \setminus \{5,11\}\) — wait, that's only 10 pitch classes, missing 5 and 11. Actually \(\{0,2,4,7,9\} \cup \{1,3,6,8,10\} = \{0,1,2,3,4,6,7,8,9,10\}\), which omits 5 (F) and 11 (B). The two pentatonic scales together do not cover the aggregate — the tritone pair \(\{5, 11\}\) is absent. This is not an error in Ligeti's design but a consequence of the specific pentatonic scales he chose: the complement of [5-35] in \(\mathbb{Z}_{12}\) is a 7-note collection [7-35] = the diatonic scale, not another pentatonic scale. Ligeti's two pentatonic scales (white-key and black-key) are the two whole-tone "complements" of the pentatonic world, and their union covers 10 of 12 pitch classes, leaving F and B as the "absent" tritone pair that never sounds.

B.6 Theoretical Connections: Pitch-Class Sets and Fourier Analysis

A recent and mathematically sophisticated development in post-tonal theory applies the Discrete Fourier Transform (DFT) to pitch-class sets, providing a new way to measure the “tonal”, “whole-tone”, and “octatonic” content of any arbitrary collection.

Definition B.1 (DFT of a Pitch-Class Set). Let \(S \subseteq \mathbb{Z}_{12}\) be a pitch-class set. Define its characteristic function \(f_S : \mathbb{Z}_{12} \to \{0,1\}\) by \(f_S(k) = 1\) if \(k \in S\) and \(f_S(k) = 0\) otherwise. The Discrete Fourier Transform of \(f_S\) is \[ \hat{f}_S(n) = \sum_{k=0}^{11} f_S(k) \cdot e^{2\pi i k n / 12} = \sum_{k \in S} e^{2\pi i k n / 12}, \quad n = 0, 1, \ldots, 11. \] The coefficient \(\hat{f}_S(n)\) is a complex number; its magnitude \(|\hat{f}_S(n)|\) measures how strongly \(S\) "resonates" with the \(n\)-fold symmetry of the chromatic circle. Specifically:
  • \(|\hat{f}_S(0)| = \#S\) (the cardinality, always real and positive).
  • \(|\hat{f}_S(1)|\) measures "diatonicity" (resonance with the circle of fifths — large for diatonic scales).
  • \(|\hat{f}_S(2)|\) measures "whole-tone content" (large for whole-tone-adjacent collections).
  • \(|\hat{f}_S(3)|\) measures "diminished" or "octatonic" content (large for octatonic collections).
  • \(|\hat{f}_S(4)|\) measures "augmented" or "hexatonic" content (large for hexatonic collections).
  • \(|\hat{f}_S(6)|\) measures "tritone" content (large for tritone-saturated collections).
Example B.7 (DFT of the Major Scale). The major scale \(S = \{0,2,4,5,7,9,11\}\): \[ \hat{f}_S(1) = e^0 + e^{2\pi i \cdot 2/12} + e^{2\pi i \cdot 4/12} + e^{2\pi i \cdot 5/12} + e^{2\pi i \cdot 7/12} + e^{2\pi i \cdot 9/12} + e^{2\pi i \cdot 11/12}. \] Computing: this sum has magnitude approximately 6.05, the largest possible value for a 7-note set at frequency 1 (approached by diatonic and near-diatonic collections). By contrast, the octatonic scale \(\{0,1,3,4,6,7,9,10\}\) has \(|\hat{f}_S(3)| = 8\) (its maximum), \(|\hat{f}_S(1)| = 0\) (no diatonic resonance), confirming that octatonic and diatonic collections are at "opposite ends" of the Fourier spectrum.

The DFT of pitch-class sets, developed theoretically by Ian Quinn (“General Equal-Tempered Harmony,” 2006–2007) and Dmitri Tymoczko, provides a coordinate system for describing the “harmonic character” of any collection in terms of its resonance with the fundamental symmetries of the chromatic circle. This approach offers a continuous, quantitative alternative to the discrete catalogue of Forte’s set-class theory: rather than assigning a set to one of 208 set classes, the DFT assigns it a point in a 6-dimensional space (one complex coordinate per Fourier coefficient), and proximity in this space corresponds to similarity of harmonic character. A C major scale and a G major scale are close in this space (differing only in one pitch class, and therefore in one small DFT perturbation); a C major scale and an octatonic scale are far apart (their DFT profiles are nearly orthogonal, differing in every Fourier coefficient).

B.7 Summary: The Mathematical Structure of Post-Tonal Theory

This course has covered four major analytical frameworks — set-class theory, twelve-tone theory, transformational theory, and neo-Riemannian theory — and several extended topics including total serialism, stochastic composition, spectral music, chord-space geometry, and Fourier analysis of pitch-class sets. We close by summarizing the mathematical structures underlying each framework, emphasizing the common thread.

Theorem B.1 (Summary of Mathematical Structures). Post-tonal music theory employs the following mathematical frameworks:
  1. Modular arithmetic (\(\mathbb{Z}_{12}\)): the foundation of pitch-class equivalence and interval arithmetic.
  2. Group theory (\(D_{12}\), the symmetric group \(S_n\), cyclic groups): the language of transposition, inversion, and the symmetries of pitch-class sets and twelve-tone rows.
  3. Combinatorics: the enumeration of set classes, the catalogue of 208 prime forms, the counting of Z-pairs.
  4. Graph theory: transformation networks (directed graphs encoding musical relationships), the Tonnetz (a triangulated toroidal graph).
  5. Geometry and topology: chord spaces as orbifolds, the Tonnetz as a torus, voice-leading distances as geodesic distances in these spaces.
  6. Probability theory: Poisson processes in Xenakis, Gaussian pitch distributions in stochastic music.
  7. Fourier analysis (DFT on \(\mathbb{Z}_{12}\)): spectral characterization of pitch-class sets, measurement of diatonic/octatonic/whole-tone content.
  8. Physics (acoustic theory): the harmonic series, spectral analysis, stretched spectra, and the derivation of microtonal pitch content from physical acoustics.
The unity of these frameworks is not superficial: they all arise from the attempt to describe, with mathematical precision, the relationships among the finite set of twelve pitch classes and the infinite variety of musical structures built from them. Post-tonal music theory is, in this sense, a branch of applied mathematics — one whose domain is the aesthetic and perceptual world of twentieth-century music.

Every analysis performed in this course — from the simple computation of a normal form to the construction of a Tonnetz path for a Wagnerian chromatic progression — is an instance of this mathematical structure applied to a musical phenomenon. The student who understands the mathematical structure will find the analytical techniques not merely useful but illuminating: each analysis reveals something about the music that resists description in any other language, and the mathematics is the reason why.

Supplement C: Analytical Exercises and Problem Sets

C.1 Pitch-Class Arithmetic

The following exercises develop fluency with the modular arithmetic of \(\mathbb{Z}_{12}\).

Exercise C.1. Compute the following directed pitch-class intervals: (a) \(i(3, 9)\); (b) \(i(9, 3)\); (c) \(i(11, 2)\); (d) \(i(7, 0)\); (e) \(i(6, 6)\).

Solution. (a) \(i(3,9) = 9 - 3 = 6\) (tritone ascending). (b) \(i(9,3) = 3 - 9 = -6 \equiv 6 \pmod{12}\) (tritone ascending). Note: the tritone is its own directed inverse since \(\min(6, 12-6) = 6\). (c) \(i(11,2) = 2 - 11 = -9 \equiv 3 \pmod{12}\) (minor third ascending). (d) \(i(7,0) = 0 - 7 = -7 \equiv 5 \pmod{12}\) (perfect fourth ascending). (e) \(i(6,6) = 0\) (unison).

Exercise C.2. Compute \(T_5(3)\), \(T_9(10)\), \(T_3I(4)\), \(T_7I(7)\).

Solution. \(T_5(3) = 8\). \(T_9(10) = 19 \equiv 7\). \(T_3I(4) = 3 - 4 = -1 \equiv 11\). \(T_7I(7) = 7 - 7 = 0\).

Exercise C.3. Find the interval class between each pair: (a) \(\text{ic}(0, 7)\); (b) \(\text{ic}(3, 10)\); (c) \(\text{ic}(4, 10)\); (d) \(\text{ic}(1, 8)\).

Solution. (a) \(i(0,7)=7\), \(12-7=5\), \(\text{ic}=5\). (b) \(i(3,10)=7\), \(\text{ic}=5\). (c) \(i(4,10)=6\), \(\text{ic}=6\) (tritone). (d) \(i(1,8)=7\), \(\text{ic}=5\).

C.2 Normal Form and Prime Form

Exercise C.4. Find the normal form and prime form of: (a) \(\{2, 5, 9\}\); (b) \(\{0, 3, 6, 9\}\); (c) \(\{1, 2, 6, 7\}\); (d) \(\{0, 2, 4, 6, 8\}\).

Solution. (a) \(\{2,5,9\}\). Rotations and spans: \([2,5,9]\) span 7; \([5,9,2]\) span 9; \([9,2,5]\) span 8. Normal form \([2,5,9]\), transposed: \([0,3,7]\). Inversion: \(\{10,7,3\}\) normal form \([3,7,10]\) transposed \([0,4,7]\). Compare \([0,3,7]\) vs. \([0,4,7]\): \(3 < 4\), prime form \((0\,3\,7)\) = [3-11] (minor triad).

(b) \(\{0,3,6,9\}\). All spans equal 9 (symmetric). Normal form by tiebreaking: \([0,3,6,9]\), span 9, internal \(3-0=3 \le 6-0=6 \le 9-0=9\). Similarly for all rotations. Inversion = \(\{0,3,6,9\}\) itself (self-inverse). Transposed: \([0,3,6,9]\). Prime form \((0\,3\,6\,9)\) = [4-28].

(c) \(\{1,2,6,7\}\). Rotations: \([1,2,6,7]\) span 6; \([2,6,7,1]\) span 11; \([6,7,1,2]\) span 8; \([7,1,2,6]\) span 11. Normal form \([1,2,6,7]\), transposed \([0,1,5,6]\). Inversion \(\{11,10,6,5\}\) = \(\{5,6,10,11\}\), normal form \([5,6,10,11]\) transposed \([0,1,5,6]\). Same! Prime form \((0\,1\,5\,6)\) = [4-9].

(d) \(\{0,2,4,6,8\}\). The pentatonic whole-tone set: all rotations span 8. Tiebreak: \([0,2,4,6,8]\) gives second interval 2. All rotations identical after transposition. Prime form \((0\,2\,4\,6\,8)\) = [5-33] (or [5-B]). Interval vector \(\langle 0\,4\,0\,4\,0\,2 \rangle\).

Exercise C.5. Compute the interval vector of \(\{0,1,3,5,6,8\}\) and identify its Forte label.

Solution. The set has \(\binom{6}{2} = 15\) pairs. List all pairs and their interval classes: \(\{0,1\}\): ic 1; \(\{0,3\}\): ic 3; \(\{0,5\}\): ic 5; \(\{0,6\}\): ic 6; \(\{0,8\}\): ic 4; \(\{1,3\}\): ic 2; \(\{1,5\}\): ic 4; \(\{1,6\}\): ic 5; \(\{1,8\}\): ic 5 (since \(i(1,8)=7\), \(\text{ic}=5\)); \(\{3,5\}\): ic 2; \(\{3,6\}\): ic 3; \(\{3,8\}\): ic 5; \(\{5,6\}\): ic 1; \(\{5,8\}\): ic 3; \(\{6,8\}\): ic 2.

Count: ic1 = 2, ic2 = 3, ic3 = 3, ic4 = 2, ic5 = 4, ic6 = 1. Interval vector: \(\langle 2\,3\,3\,2\,4\,1 \rangle\).

Prime form: span of \([0,1,3,5,6,8]\) is 8; check all rotations for minimum span. \([0,1,3,5,6,8]\) span 8; \([1,3,5,6,8,0]\) span 11; \([3,5,6,8,0,1]\) span 10; \([5,6,8,0,1,3]\) span 10; \([6,8,0,1,3,5]\) span 11; \([8,0,1,3,5,6]\) span 10. Minimum 8 for \([0,1,3,5,6,8]\), transposed: \([0,1,3,5,6,8]\). Check inversion: \(\{0,11,9,7,6,4\} = \{0,4,6,7,9,11\}\), normal form \([0,4,6,7,9,11]\) transposed \([0,4,6,7,9,11]\). Compare \([0,1,3,5,6,8]\) vs. \([0,4,6,7,9,11]\): \(1 < 4\), prime form \((0\,1\,3\,5\,6\,8)\) = [6-Z28] (or check Forte’s catalogue — the interval vector \(\langle 2\,3\,3\,2\,4\,1 \rangle\) matches [6-Z28] and its Z-partner [6-Z49]).

C.3 Row Analysis

Exercise C.6. Given the row \(P_0 = (2, 9, 10, 0, 11, 4, 3, 8, 7, 1, 6, 5)\), compute \(I_4\), \(R_0\), and \(RI_7\).

Solution. \(I_4 = (4-2, 4-9, 4-10, 4-0, 4-11, 4-4, 4-3, 4-8, 4-7, 4-1, 4-6, 4-5) \pmod{12}\) \(= (2, 7, 6, 4, 5, 0, 1, 8, 9, 3, 10, 11)\).

\(R_0 = (5, 6, 1, 7, 8, 3, 4, 11, 0, 10, 9, 2)\) (reverse of \(P_0\)).

\(RI_7 = (7-5, 7-6, 7-1, 7-7, 7-8, 7-3, 7-4, 7-11, 7-0, 7-10, 7-9, 7-2) \pmod{12}\) \(= (2, 1, 6, 0, 11, 4, 3, 8, 7, 9, 10, 5)\).

Exercise C.7. Determine whether the row in Exercise C.6 is semi-combinatorial. (Hint: check whether the inversion at some level has a first hexachord complementary to \(P_0\)’s first hexachord.)

Solution. First hexachord of \(P_0\): \(\{2,9,10,0,11,4\} = \{0,2,4,9,10,11\}\). Complement: \(\{1,3,5,6,7,8\}\). We need \(I_n\) such that the first hexachord of \(I_n\) is \(\{1,3,5,6,7,8\}\). The first element of \(I_n\) is \(n - 2 \pmod{12}\). The first hexachord of \(I_n\) is \(\{n-2, n-9, n-10, n-0, n-11, n-4\} \pmod{12}\). For this to equal \(\{1,3,5,6,7,8\}\), we need \(n\) such that these six values produce \(\{1,3,5,6,7,8\}\). Trying \(n = 9\): \(\{7, 0, 11, 9, 10, 5\} = \{0,5,7,9,10,11\}\) — not \(\{1,3,5,6,7,8\}\). Trying \(n = 3\): \(\{1, 6, 5, 3, 4, 11\} = \{1,3,4,5,6,11\}\) — not matching. The row is not trivially semi-combinatorial from inspection; a full systematic search would be required. (Answer: this particular row does not have a complement hexachord among \(I_n\) forms; it is not \(I\)-combinatorial, though it may be \(RI\)-combinatorial for some \(n\).)

C.4 Neo-Riemannian Exercises

Exercise C.8. Starting from A\(\flat\) major \(\{8, 0, 3\}\), apply the sequence of transformations \(L, R, P, L\) and identify the resulting triad after each step.

Solution. A\(\flat\) major = \(\{8, 0, 3\}\) (A\(\flat\), C, E\(\flat\)). \(L\{8,0,3\}\): move the root (8 = A\(\flat\)) down a semitone to 7 (G): \(\{7, 0, 3\}\) = G minor (G, B\(\flat\), D — but 3 = E\(\flat\) not D; let me recheck). A\(\flat\) major = {A\(\flat\), C, E\(\flat\)} = {8, 0, 3}. \(L\) maps major to minor by moving the root down by semitone: root A\(\flat\) (8) → G (7), yielding {7, 0, 3} = G minor? Check: G minor = {G, B\(\flat\), D} = {7, 10, 2} ≠ {7,0,3}. Correction: \(L\{r, r+4, r+7\} = \{r+4, r+7, r+11\}\). So \(L\{8,0,3\} = \{0, 3, 7\}\) = C minor (C, E\(\flat\), G).

\(R\{0,3,7\}\) (C minor): \(R\{r,r+3,r+7\} = \{r-2, r, r+3\} \pmod{12}\). Wait — for minor: \(R\{r,r+3,r+7\} = \{r+3, r+7, r+10\}\)? Let me use the consistent definition: \(R\) maps minor to major by moving the fifth by whole tone. C minor \(\{0,3,7\}\): move 7 (G) up to 9 (A), yielding \(\{0,3,9\}\) = A minor (A, C, E\(\flat\))? Check: A minor = {A, C, E} = {9, 0, 4} ≠ {0,3,9}. This exercise illustrates that consistent application of P, L, R requires careful attention to the definition’s details. Using Cohn’s standard formulation: \(R\) maps C major to A minor by moving the fifth of C major (G = 7) up by whole tone to A (9): \(\{0,4,7\} \to \{9,0,4\}\). For C minor \(\{0,3,7\}\): \(R\) moves the root (0 = C) down by whole tone to B\(\flat\) (10): \(\{0,3,7\} \to \{10, 0, 3\}\) = B\(\flat\) major.

\(P\{10,0,3\}\) (B\(\flat\) major = {10,2,5}… recheck: B\(\flat\) major \(\{10,2,5\} = \{\text{B}\flat, \text{D}, \text{F}\}\). But \(\{10,0,3\}\) = {\text{B}\flat, \text{C}, \text{E}\flat}) is not a consonant triad. The inconsistency indicates the exercise needs to be reworked with the correct pitch classes for A\(\flat\) major. A\(\flat\) major correctly: \(\{8, 0, 3\} = \{G\sharp, C, E\flat\}\) — but A\(\flat\) major is {A\(\flat\), C, E\(\flat\)} = {8, 0, 3} where 8 = A\(\flat\). That is correct in pc notation. The definitional details of P, L, R for the general form \(\{r, r+4, r+7\}\) require \(r\) to be the root: A\(\flat\) major has root \(r = 8\). So the set is \(\{8, 12, 15\} = \{8, 0, 3\}\) mod 12. \(L\{8, 0, 3\} = \{0, 3, 7\}\) = C minor. \(R\{0,3,7\}\): C minor has root 0, so \(R\) of C minor = E\(\flat\) major = {3, 7, 10}. \(P\{3,7,10\}\): E\(\flat\) major → E\(\flat\) minor = {3, 6, 10}. \(L\{3,6,10\}\): E\(\flat\) minor → B\(\flat\) major = {10, 2, 5} (= {B\flat, D, F}). Summary: A\(\flat\) maj → C min → E\(\flat\) maj → E\(\flat\) min → B\(\flat\) maj.

Exercise C.9. On the Tonnetz, identify the Tonnetz triangle representing E major \(\{4, 8, 11\}\) and its three neighbors obtained by P, L, R.

Solution. E major \(\{4, 8, 11\} = \{\text{E}, \text{G}\sharp, \text{B}\}\): \(P\{4,8,11\} = \{4, 7, 11\}\) = E minor (E, G, B). Tonnetz: shares edge \(\{4, 11\}\) (E–B, a perfect fifth — horizontal edge). \(L\{4,8,11\} = \{8, 11, 3\}\) = C\(\sharp\) minor (C\(\sharp\), E, G\(\sharp\)). Shares edge \(\{4, 8\}\) (E–G\(\sharp\), major third — NE diagonal). \(R\{4,8,11\} = \{4, 8, 1\} = \{1, 4, 8\}\) = C\(\sharp\) major (C\(\sharp\), F, A\(\flat\) — but check: \(1+4=5\) ≠ 8). Correction: R of E major moves the root E (4) by adding 9: \(\{8, 11, 4\} \to \{4+9, 8, 11\}\) … Using the formula: \(R\{r,r+4,r+7\} = \{r+4, r+7, r+9\} = \{4+4, 4+7, 4+9\} = \{8, 11, 1\}\) = C\(\sharp\) minor. The three Tonnetz neighbors of the E major triangle are E minor, C\(\sharp\) minor, and G\(\sharp\) minor, obtained by reflecting across the triangle’s three edges.

These exercises demonstrate both the power and the care required in applying the neo-Riemannian operations. The Tonnetz makes the geometry transparent: each of P, L, R reflects the triangle across one of its three sides, and the three neighbors of E major are exactly the three triangles adjacent to it on the Tonnetz surface.

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