Pitch-Class Set Theory
A comprehensive catalog of all 224 set classes in 12-tone equal temperament, based on Allen Forte's classification system.
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Cardinality Distribution
Number of distinct set classes for each cardinality (0-12)
Featured Set Classes
Key Concepts
Prime Form
The prime form is the most compact representation of a pitch-class set, transposed to start on 0 and inverted if necessary. Two sets with the same prime form belong to the same set class.
Interval Vector
The interval vector counts how many of each interval class (1-6) appear in a set. Sets with the same interval vector but different prime forms are called Z-related.
Quick Reference
| Forte # ↑ | Prime Form | IC Vector | Card ↕ |
|---|---|---|---|
| 3-1 | [0,1,2] | <2,1,0,0,0,0> | 3 |
| 3-2 | [0,1,3] | <1,1,1,0,0,0> | 3 |
| 3-3 | [0,1,4] | <1,0,1,1,0,0> | 3 |
| 3-4 | [0,1,5] | <1,0,0,1,1,0> | 3 |
| 3-5 | [0,1,6] | <1,0,0,0,1,1> | 3 |
| 3-6 | [0,2,4] | <0,2,0,1,0,0> | 3 |
| 3-7 | [0,2,5] | <0,1,1,0,1,0> | 3 |
| 3-8 | [0,2,6] | <0,1,0,1,0,1> | 3 |
| 3-9 | [0,2,7] | <0,1,0,0,2,0> | 3 |
| 3-10 | [0,3,6] | <0,0,2,0,0,1> | 3 |
| 3-11 | [0,3,7] | <0,0,1,1,1,0> | 3 |
| 3-12 | [0,4,8] | <0,0,0,3,0,0> | 3 |
T-Sym: Transpositional symmetry (how many transpositions map the set to itself)
I-Sym: Inversional symmetry (how many inversions map the set to itself)
Z-Mate: Z-related set class (same interval vector, different prime form)