Definition
Musical set theory (pitch-class set theory), developed by Milton Babbitt and systematized by Allen Forte (The Structure of Atonal Music, 1973), describes pitches as numbers and chords as sets. Born to analyze atonal music (Schoenberg, Berg, Webern), it is now also useful for understanding symmetries, exotic scales and voicings in jazz and film music.
Pitch and pitch class
- A pitch is a specific sounding pitch (C4).
- A pitch class (pc) is the class of all notes with the same name regardless of octave and enharmonic spelling: C2, C4, B#3 and Dbb5 are all pc 0.
| pc | Notes |
|---|---|
| 0 | C, B#, Dbb |
| 1 | C#, Db |
| 2 | D |
| 3 | D#, Eb |
| 4 | E, Fb |
| 5 | F, E# |
| 6 | F#, Gb |
| 7 | G |
| 8 | G#, Ab |
| 9 | A |
| 10 (t or A) | A#, Bb |
| 11 (e or B) | B, Cb |
The arithmetic is modulo 12: 11 + 3 = 2 (B + minor third = D). For convenience, 10 and 11 are written t and e, or A and B.
Interval classes
An interval between pitch classes is measured in semitones (0–11). Since an interval and its inversion (e.g. minor 3rd = 3, major 6th = 9) are “the same” from a class perspective, we use interval classes (ic) from 0 to 6: ic = min(n, 12 − n).
| ic | Intervals |
|---|---|
| 1 | minor 2nd / major 7th |
| 2 | major 2nd / minor 7th |
| 3 | minor 3rd / major 6th |
| 4 | major 3rd / minor 6th |
| 5 | perfect 4th / perfect 5th |
| 6 | tritone |
Operations
- Transposition Tn: add n to every pc. T5 of {0, 4, 7} (C major) = {5, 9, 0} (F major).
- Inversion I (T0I): every pc becomes its opposite: x → −x mod 12. I of {0, 4, 7} = {0, 8, 5} = F minor (F Ab C). TnI: inversion followed by transposition: T7I of {0, 4, 7} = {7, 3, 0} = C minor.
An important result follows: the minor triad is the inversion of the major triad (the same interval structure flipped). This is the mathematical basis of negative harmony (Ernst Levy, Steve Coleman, popularized by Jacob Collier), in which every chord is reflected around an axis between the major and minor third of the tonic.
Symmetry and invariance
A set is symmetrical under transposition if Tn leaves it unchanged: the augmented triad {0, 4, 8} stays the same under T4 and T8; the diminished seventh {0, 3, 6, 9} under T3, T6, T9; the whole-tone scale under T2, T4…; the octatonic under T3, T6, T9. These are Messiaen's modes of limited transposition.
Musical examples
- Webern and the Second Viennese School often use the cell {0, 1, 6} (the “Viennese trichord”): semitone + tritone.
- Schoenberg, Drei Klavierstücke op. 11: the opening is one of the classic testing grounds of set analysis.
- In jazz: G7 and Db7 share the tritone {5, 11}; tritone substitution is T6 of a chord that keeps two of its notes (invariance).
- In modern film music the “colors” of Hans Zimmer or John Williams often alternate triads linked by neo-Riemannian transformations (P, L, R), which are inversions with two common tones.
Practical use for a producer
- Thinking in numbers makes it immediate to transpose and invert a voicing in FL Studio (each number = one semitone in the Piano roll).
- Symmetrical scales are built with numeric rules: whole-tone = steps of 2, diminished = 1-2 or 2-1, augmented = 3-1.
- Inversion around the negative-harmony axis in C (x → 7 − x mod 12, i.e. a reflection between Eb and E) turns C – Am – F – G into Cm – Eb – Gm – Fm: a “mirror” progression with the same voice movement.
Exercise
- Write in numbers the triads of C, Am, F#°, Eb+ and compute T3 and I of each.
- Take an 8-note melody, convert it to pcs, apply T0I and play it: it is the “upside-down” melody.
- Verify that {0, 3, 6, 9} transposed by 3 stays identical and explain why the diminished seventh has four resolutions.
Connections
Set theory (2): normal form, prime form and interval vector; Enharmonics; Messiaen's modes of limited transposition; Diminished scale; Whole-tone scale.