Post-Tonal and Set-Theory Analysis

How twentieth-century music without a tonal center is described using pitch-class sets and related tools.

Definition

The body of analytical methods — chiefly pitch-class set theory and twelve-tone theory — developed to describe music that organizes pitch without a tonal center.

Scope

Covers the analytical apparatus for music that abandons functional tonality: pitch-class sets, set classes and normal/prime form, interval-class vectors, transposition and inversion operations, and the twelve-tone (serial) method. Treats the analytical theory; the historical repertoire of modernism belongs to music history.

Core questions

• What is a pitch-class set, and how are normal and prime form found?
• How do transposition and inversion relate sets to one another?
• What does an interval-class vector summarize?
• How does the twelve-tone method organize pitch through a row?
• What can set theory reveal about atonal music that tonal analysis cannot?

Key theories

Pitch-class set theory

Forte systematized the analysis of atonal music by classifying collections of pitch classes into set classes related by transposition and inversion, characterizing each by its interval-class content, providing a rigorous vocabulary for music without a tonal center.

Twelve-tone (serial) method

Schoenberg's method organizes the twelve chromatic pitch classes into an ordered row deployed in its prime, inversion, retrograde, and retrograde-inversion forms, providing a non-tonal basis for pitch organization.

History

As tonality dissolved in the early twentieth century, Schoenberg devised the twelve-tone method to organize atonal pitch material; American theorists, notably Allen Forte and Milton Babbitt, later built a rigorous set-theoretic and transformational apparatus for analyzing this repertoire.

Debates

Whether set theory captures how atonal music is heard

Set theory offers a precise classification of pitch collections, but critics question whether its categories correspond to perceptible features of the music or impose abstract relationships that a listener cannot hear.

Key figures

• Allen Forte
• Arnold Schoenberg
• Milton Babbitt
• Joseph N. Straus

Seminal works

• forte1973
• straus2016
• schoenberg1975

Frequently asked questions

What is a pitch-class set?

An unordered collection of pitch classes (notes regardless of octave) treated as a unit; sets related by transposition or inversion belong to the same set class.

Is twelve-tone music the same as atonal music?

Not exactly. Atonal music lacks a tonal center generally; twelve-tone music is a specific systematic method, devised by Schoenberg, for organizing such music around an ordered row of all twelve pitch classes.