Pitch-Class Set Theory

Introduction: A Language for Atonal Music

Much music written after about 1908 (Schoenberg, Berg, and Webern's "free atonal" works, much of Bartók and Stravinsky, and countless later pieces) doesn't use keys, triads, or Roman numerals. Yet it isn't random: composers build it from small, recurring groups of pitches whose interval content gives the music its sound and coherence. Pitch-class set theory, developed by Milton Babbitt, Allen Forte, John Rahn, and others, gives us tools to name those groups, compare them, and hear how they're transformed.

Pitch Classes and Integer Notation

Set theory treats all Cs as the same pitch class (octave equivalence), and treats enharmonic spellings as identical (C♯ = D♭). The twelve pitch classes are numbered from C = 0, using T for 10 and E for 11 so that every pitch class is one character:

Intervals: Pitch, Pitch Class, and Interval Class

• A pitch interval counts semitones between two actual pitches (C4 up to E5 = +16).
• An ordered pitch-class interval measures upward from one pitch class to another, mod 12 (C to E = 4; E to C = 8).
• An interval class (ic) is the smaller of an interval and its inversion: 1 through 6. So ic 4 includes major thirds, minor sixths, and all their compounds.

Normal Form

A pitch-class set is an unordered collection of pitch classes. To compare sets, we write each in a standard order called normal form, the most compact ascending arrangement:

1. List the distinct pitch classes in ascending numerical order.
2. Write every rotation (start on each pc in turn, wrapping around).
3. Choose the rotation with the smallest span from first to last note.
4. If two rotations tie, choose the one with the smallest interval from the first to the second-to-last note, then the third-to-last, and so on.
5. If they still tie (a symmetrical set), choose the one that starts on the smallest pitch class.

Example: the pitches A♭, C, E, F give pitch classes {8, 0, 4, 5}. Sorted: [0458]. Its rotations:

[0458] and [4580] both span 8 semitones. The tie-breaker (first to second-to-last) is 5 for [0458] and 4 for [4580], so the normal form is [4580].

Transposition and Inversion

• Transposition (T_n) adds n to every pitch class (mod 12). T₄[0,1,4] = [4,5,8].
• Inversion (T_nI) subtracts every pitch class from n (inverts around 0, then transposes): T_nI(x) = n − x. T₀I[0,1,4] = {0, 11, 8} = [8,E,0] in normal form.
• Sets related by T_n or T_nI are members of the same set class. They have the same interval content and a closely related sound, the way all major and minor triads belong to one family.

A quick test: two sets are related by T_n if adding the same n to every pc of one gives the other. They're related by T_nI if every pair of corresponding pcs (first with last, and so on, in normal form) adds up to the same n.

Prime Form and Forte Names

Each set class is named by its prime form, its most compact representative, beginning on 0:

1. Find the normal form of the set and transpose it to begin on 0.
2. Invert the set, find the normal form of the inversion, and transpose it to begin on 0.
3. Compare the two: the prime form is the one with smaller numbers at the left (more packed to the left). Write it in parentheses: (014).

Allen Forte's catalog also gives each set class a name: cardinality–ordinal number. The twelve trichord classes are:

The Interval-Class Vector

The interval-class vector (ICV) counts how many times each interval class (1 through 6) occurs among all pairs of notes in a set. The major/minor triad (037) has the vector <001110>: one minor third, one major third, one perfect fourth/fifth. The vector summarizes the set's sound: (012) <210000> is crunchy; (027) <010020> is open; (048) <000300> has only major thirds.

Occasionally two set classes that are not related by T_n or T_nI share the same vector. These are Z-related sets, marked with a Z in Forte's names. The smallest pair is 4-Z15 (0146) and 4-Z29 (0137): both have the "all-interval" vector <111111>.

Segmentation: Analyzing a Passage

To analyze a passage, divide it into meaningful segments (motives, chords, phrases, registral or timbral groups) and find each segment's set class. Repeated set classes, and the T_n/T_nI operations that connect them, reveal the passage's structure.

Check it: segment 1 is [145], segment 2 is [TE2], segment 3 is [478]. For a T_n relation, the same number is added to every pc; for T_nI, corresponding pcs (first with last, second with second-to-last…) all sum to n. Try the operations in the app below.

Try It: Set Calculator

Summary

• Pitch classes are numbered 0–11 (C = 0, T = 10, E = 11); octave and enharmonic equivalence apply.
• Interval classes 1–6 combine an interval with its inversion.
• Normal form is the most compact ascending ordering (smallest span, then packed from the right).
• T_n adds n; T_nI subtracts from n. Sets related by either belong to the same set class, named by its prime form and Forte name.
• The interval-class vector counts interval classes; Z-related sets share a vector without being T/I related.
• Segmentation finds recurring set classes and the operations that connect them.

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