Set Theory and the 20th-Century Mathematization — Analyzing Music with Mathematics

How Do You Analyze Atonality?

After Schoenberg (Article 8) discarded tonality, theorists faced a new problem — how do you analyze music that has no tonality? Functional harmony (T/S/D) and Schenkerian analysis (Ursatz) both presuppose tonality, so they cannot be used. Atonal music has neither a “tonic” nor a “chord function.”

The answer was the most mathematical theory in music history — pitch-class set theory (musical set theory). 2500 years after Pythagoras (Article 1) grasped music with “number,” theory is at last written in the language of mathematics.

Representing Notes as Numbers — Pitch Classes

The starting point of set theory is to represent notes as the numbers 0-11:

• C=0, C♯=1, D=2, …, B=11 (0-11 per semitone)
• notes an octave apart are identified (a C in any octave is “0”) → this is called a pitch class
• enharmonics (C♯ and D♭) are also identified (equivalent under equal temperament, Article 5)

Then a chord or a cluster of notes becomes a set of numbers. For example, C-E-G (the C major triad) is {0, 4, 7}, and a cluster of three semitones is {0, 1, 2}. The core of set theory is treating chords as “pitch-class sets.”

Group-Theoretic Operations — Transposition and Inversion

Set theory captures relations among sets with group-theoretic operations:

• transposition (Tn) — add the same number to every note (= parallel shift / transposition). {0,4,7} + 2 = {2,6,9}
• inversion (TnI) — flip the notes up-down (subtract from 12)
• sets that map to one another under these are regarded as the “same set class”

The Forte number — Allen Forte classified all pitch-class sets and gave them numbers (e.g., 3-11 is the class of major and minor triads). Analyses like “these two chords look different but are the same set class = structurally equivalent” become possible. The “shape” of music can be treated, detached from note names and keys, as pure structure (a type of set).

Milton Babbitt — Thorough Mathematization

Milton Babbitt (1916-2011) — a composer-theorist with a background in mathematics. He pushed the twelve-tone technique toward total serialism (systematizing not only pitch but rhythm, dynamics, and timbre) and brought rigorous mathematics (group theory, combinatorics) into music theory. He symbolizes the stance that “music theory should have the same rigor as science.”

What Was Gained — and the Limits

The significance and limits of set theory:

What was gained:

• a universal analytical language not presupposing tonality — atonal, twelve-tone, and experimental music can be objectively described
• a perspective that frees musical structure from “note name/key” and treats it as an abstract type
• reproducibility and objectivity of analysis (anyone arrives at the same set class)

Limits/criticisms:

• divergence from listening — even with “the same set class,” things can sound entirely different to the ear. Number vs. ear (the thread of Article 1) reignites here
• excessive for tonal music — there is little need to use set theory on pieces explainable by functional harmony
• prone to “analysis for analysis’s sake” (the criticism that it doesn’t explain a piece’s appeal)

Set theory showed one extreme of “how far music theory can become mathematics.” It is the sharpest culmination of the “number” lineage since Pythagoras, and at the same time the theory that most acutely poses the old question of “the divergence between what the ear hears and theory.”

Ethnomusicology — “Western Theory Is Not Universal”

Another important 20th-century movement is the development of ethnomusicology — the study of the world’s music through fieldwork and recording. The recognition it brought was decisive for theory’s history:

Western music theory (functional harmony, equal temperament, the five-line staff) is not a universal truth but merely one local cultural system.

India’s raga, the Arab maqam, and Indonesian gamelan each have their own refined theoretical systems (detailed in Article 10). The unconscious premise that “music theory = Western harmony” was relativized. While set theory “abstracted Western theory to the limit,” ethnomusicology “demoted Western theory to one example” — two contrasting vectors of 20th-century theory.

From This Article to the Next

Set theory pushed the mathematization of Western theory to its peak, and ethnomusicology relativized Western theory. The next article looks beyond that relativization — non-Western music theory. India’s raga, the Arab maqam, China’s twelve lü, Japan’s ritsu-ryo, Indonesian gamelan. We approach ways of grasping “the order of sound” utterly different from the West.