Post-Tonal Analysis: Set Theory
Pitch classes and interval classes, transposition Tn and inversion TnI, normal form, prime form, interval-class vector and Forte numbers — up to Webern's row. Takes over from courses 27 and 37 outside tonality.
Building on the work of Milton Babbitt (who coined the notions of pitch class and interval class), Allen Forte (1926-2014) systematized in The Structure of Atonal Music (1973) a th…
À retenir : Set theory is a tool for analysis, not a method of composition; it does not replace the ear, it gives it an exact vocabulary. And it does not « dethrone » Schenker: each reigns over its own repertoire.
Why a new theory?
Around 1908-1909 — the finale of the Second Quartet, the Lieder of the Book of the Hanging Gardens, then Schoenberg's Three Pieces op. 11 — a music appears that no longer rests on a tonic. This is free atonality: no more hierarchy of degrees, no more dissonance relative to a consonance of reference (Schoenberg speaks of the « emancipation of the dissonance »), no more cadences articulating the form. Our tonal tools then fall silent: labelling a chord presupposes a root and stacked thirds it often no longer has; speaking of a « degree » presupposes a reference scale; and Schenkerian reduction (courses 27 and 37) presupposes precisely what this music abolishes — the prolongation of a tonic triad by consonant linear motions. Schenker himself explicitly reserved his theory for the tonal repertoire: for Webern or atonal Schoenberg, something else is needed.
This conceptual void was filled only belatedly. Building on the work of Milton Babbitt (who coined the notions of pitch class and interval class), Allen Forte systematized in The Structure of Atonal Music (1973) a theory of pitch-class sets (pitch-class set theory): instead of chords and degrees, sets of notes considered up to octave and enharmonic equivalence, comparable to one another under transposition and inversion, catalogued and named. The key idea: in free atonality, coherence no longer comes from tonal syntax but from the recurrence of small collections of intervals — cells the composer transposes, inverts, superimposes. Set theory is today the lingua franca of post-tonal analysis; the standard textbook is Joseph N. Straus's Introduction to Post-Tonal Theory, whose conventions this course follows.