Why they are needed
A chord can appear in many transpositions, inversions and spacings. The normal form and the prime form let you reduce every set to a single representative and compare chords and scales objectively. The Forte number (e.g. 3-11) is the catalog label.
Normal form (algorithm)
- Remove repetitions and octaves, write the pcs in ascending order.
- Write all the rotations (cycles) of the set.
- Choose the rotation with the smallest interval between the first and the last note (the most compact).
- In case of a tie, choose the one with the smallest interval between the first and the second-to-last note, and so on (Rahn's criterion; Forte uses a slightly different criterion in the rare cases of a further tie).
Example: C major in the position E–G–C = {4, 7, 0}. Rotations: [0 4 7] span 7; [4 7 0] span 8; [7 0 4] span 9. Normal form: [0, 4, 7].
Prime form
- Find the normal form of the set and transpose it so that it starts from 0.
- Do the same with the inversion of the set.
- Choose the one that is most “packed to the left” (smallest intervals at the beginning). The result is written in parentheses without commas.
Example: C major → (047). Inversion {0, 8, 5} → normal form [5, 8, 0] → transposed [0, 3, 7]. Between (047) and (037), (037) wins. So the major triad and the minor triad have the same prime form (037): they are the same set up to inversion. Forte number: 3-11.
The interval vector
The interval vector counts how many times each interval class (ic1…ic6) appears among all the pairs of notes in the set. It is written in angle brackets.
- Major/minor triad (037): pairs 0–3 (ic3), 0–7 (ic5), 3–7 (ic4) → <001110>.
- Major / diatonic scale, 7-35 (013568t): <254361>. It has the maximum number of fifths (6) and only one tritone: that is why it is so “tonal”.
- Pentatonic, 5-35 (02479): <032140>: no semitone and no tritone, hence no strong tension (see the card on pentatonic scales).
- Whole-tone, 6-35 (02468t): <060603>: only major seconds, major thirds and tritones; no semitone, no perfect fifth.
- Octatonic, 8-28 (0134679t): <448444>.
- Diminished seventh, 4-28 (0369): <004002>. Augmented triad, 3-12 (048): <000300>.
Notable sets and Forte numbers
| Forte | Prime form | Example |
|---|---|---|
| 3-11 | (037) | major and minor triads |
| 3-10 | (036) | diminished triad |
| 3-12 | (048) | augmented triad |
| 3-5 | (016) | “Viennese trichord” (Webern) |
| 4-27 | (0258) | dominant seventh and half-diminished seventh (they are inversions) |
| 4-26 | (0358) | minor seventh (and major sixth: Am7 = C6) |
| 4-20 | (0158) | major seventh |
| 4-28 | (0369) | diminished seventh |
| 5-35 | (02479) | pentatonic |
| 6-35 | (02468t) | whole-tone |
| 7-35 | (013568t) | diatonic (all the modes of the major scale) |
| 7-34 | (013468t) | melodic minor (all its modes) |
| 7-32 | (0134689) | harmonic minor and harmonic major (inversions) |
| 8-28 | (0134679t) | octatonic |
Note: G7 and Bø7 have the same prime form (0258): the half-diminished is the flip side of the dominant. And harmonic minor and harmonic major are inversions of each other (7-32).
Z-relation
Two different sets (not related by Tn or TnI) with the same vector are called Z-related. The famous case is the two all-interval tetrachords 4-Z15 (0146) and 4-Z29 (0137), both with vector <111111>: they contain each interval class exactly once. Elliott Carter used them as basic material in several works.
Practical use
- Classify an “exotic scale” found online: compute its prime form and you will often discover that it is a mode of a known scale (for example, the Hungarian minor is a mode of the double harmonic scale, 7-22).
- Choose voicings with a certain “color”: a set with many ic5 sounds open (quartal), one with many ic1 sounds tense (cluster).
- Read 20th-century analysis texts and modern theory (Tymoczko, A Geometry of Music).
Exercise
- Compute normal form, prime form and vector of: D F A C (Dm7), B D F A (Bø7), C E G# (C+), C Db F# (Viennese trichord).
- Verify that Dm7 and F6 have the same prime form.
- Take the enigmatic scale (C Db E F# G# A# B) and compute its vector: how many perfect fifths does it contain?
Connections
Set theory (1): pitch classes and operations; Messiaen's modes of limited transposition; Whole-tone scale; Diminished scale; Seventh chords: catalog.