In musical set theory, a Forte number is the pair of numbers Allen Forte assigned to the prime form of each pitch class set of three or more members in The Structure of Atonal Music (1977, ISBN 0300021208). The first number indicates the number of pitch classes in the pitch class set and the second number indicates the set's sequence in Forte's ordering of all pitch class sets containing that number of pitches.
In the 12-TET tuning system (or in any other system of tuning that splits the octave into twelve semitones), each pitch class may be denoted by an integer in the range from 0 to 11 (inclusive), and a pitch class set may be denoted by a set of these integers. The prime form of a pitch class set is the most compact (i.e., leftwards packed or smallest in lexicographic order) of either the normal form of a set or of its inversion. The normal form of a set is that which is transposed so as to be most compact. For example, a (second inversion) major chord contains the pitch classes 7, 0, and 4. The normal form would then be 0, 4 and 7. Its (transposed) inversion, which happens to be the minor chord, contains the pitch classes 0, 3, and 7; and is the prime form.
The major and minor chords are both given Forte # 3-11, indicating that it is the eleventh in Forte's ordering of pitch class sets with three pitches. In contrast, the Viennese trichord, with pitch classes 0,1, and 6, is given Forte # 3-5, indicating that it is the fifth in Forte's ordering of pitch class sets with three pitches. The normal form of the diatonic scale, such as C major; 0, 2, 4, 5, 7, 9, and 11; is 11, 0, 2, 4, 5, 7, and 9; while its prime form is 0, 1, 3, 5, 6, 8, and 10; and its Forte # is 7-35, indicating that it is the thirty-fifth of the seven-member pitch class sets.
In the language of combinatorics, the Forte numbers correspond to the binary bracelets of length 12: that is, equivalence classes of binary sequences of length 12 under the operations of cyclic permutation and reversal. In this correspondence, a one in a binary sequence corresponds to a pitch that is present in a pitch class set, and a zero in a binary sequence corresponds to a pitch that is absent. The rotation of binary sequences corresponds to transposition of chords, and the reversal of binary sequences corresponds to inversion of chords. The most compact form of a pitch class set is the lexicographically maximal sequence within the corresponding equivalence class of sequences.
| Forte Number | Pitch-class Set | Interval Vector | Forte Number | Pitch-class Set | Interval Vector |
|---|---|---|---|---|---|
| 3-1 | [0,1,2] | <2,1,0,0,0,0> | 9-1 | [0,1,2,3,4,5,6,7,8] | <8,7,6,6,6,3> |
| 3-2 | [0,1,3] | <1,1,1,0,0,0> | 9-2 | [0,1,2,3,4,5,6,7,9] | <7,7,7,6,6,3> |
| 3-3 | [0,1,4] | <1,0,1,1,0,0> | 9-3 | [0,1,2,3,4,5,6,8,9] | <7,6,7,7,6,3> |
| 3-4 | [0,1,5] | <1,0,0,1,1,0> | 9-4 | [0,1,2,3,4,5,7,8,9] | <7,6,6,7,7,3> |
| 3-5 | [0,1,6] | <1,0,0,0,1,1> | 9-5 | [0,1,2,3,4,6,7,8,9] | <7,6,6,6,7,4> |
| 3-6 | [0,2,4] | <0,2,0,1,0,0> | 9-6 | [0,1,2,3,4,5,6,8,10] | <6,8,6,7,6,3> |
| 3-7 | [0,2,5] | <0,1,1,0,1,0> | 9-7 | [0,1,2,3,4,5,7,8,10] | <6,7,7,6,7,3> |
| 3-8 | [0,2,6] | <0,1,0,1,0,1> | 9-8 | [0,1,2,3,4,6,7,8,10] | <6,7,6,7,6,4> |
| 3-9 | [0,2,7] | <0,1,0,0,2,0> | 9-9 | [0,1,2,3,5,6,7,8,10] | <6,7,6,6,8,3> |
| 3-10 | [0,3,6] | <0,0,2,0,0,1> | 9-10 | [0,1,2,3,4,6,7,9,10] | <6,6,8,6,6,4> |
| 3-11 | [0,3,7] | <0,0,1,1,1,0> | 9-11 | [0,1,2,3,5,6,7,9,10] | <6,6,7,7,7,3> |
| 3-12 | [0,4,8] | <0,0,0,3,0,0> | 9-12 | [0,1,2,4,5,6,8,9,10] | <6,6,6,9,6,3> |
| 4-1 | [0,1,2,3] | <3,2,1,0,0,0> | 8-1 | [0,1,2,3,4,5,6,7] | <7,6,5,4,4,2> |
| 4-2 | [0,1,2,4] | <2,2,1,1,0,0> | 8-2 | [0,1,2,3,4,5,6,8] | <6,6,5,5,4,2> |
| 4-3 | [0,1,3,4] | <2,1,2,1,0,0> | 8-3 | [0,1,2,3,4,5,6,9] | <6,5,6,5,4,2> |
| 4-4 | [0,1,2,5] | <2,1,1,1,1,0> | 8-4 | [0,1,2,3,4,5,7,8] | <6,5,5,5,5,2> |
| 4-5 | [0,1,2,6] | <2,1,0,1,1,1> | 8-5 | [0,1,2,3,4,6,7,8] | <6,5,4,5,5,3> |
| 4-6 | [0,1,2,7] | <2,1,0,0,2,1> | 8-6 | [0,1,2,3,5,6,7,8] | <6,5,4,4,6,3> |
| 4-7 | [0,1,4,5] | <2,0,1,2,1,0> | 8-7 | [0,1,2,3,4,5,8,9] | <6,4,5,6,5,2> |
| 4-8 | [0,1,5,6] | <2,0,0,1,2,1> | 8-8 | [0,1,2,3,4,7,8,9] | <6,4,4,5,6,3> |
| 4-9 | [0,1,6,7] | <2,0,0,0,2,2> | 8-9 | [0,1,2,3,6,7,8,9] | <6,4,4,4,6,4> |
| 4-10 | [0,2,3,5] | <1,2,2,0,1,0> | 8-10 | [0,2,3,4,5,6,7,9] | <5,6,6,4,5,2> |
| 4-11 | [0,1,3,5] | <1,2,1,1,1,0> | 8-11 | [0,1,2,3,4,5,7,9] | <5,6,5,5,5,2> |
| 4-12 | [0,2,3,6] | <1,1,2,1,0,1> | 8-12 | [0,1,3,4,5,6,7,9] | <5,5,6,5,4,3> |
| 4-13 | [0,1,3,6] | <1,1,2,0,1,1> | 8-13 | [0,1,2,3,4,6,7,9] | <5,5,6,4,5,3> |
| 4-14 | [0,2,3,7] | <1,1,1,1,2,0> | 8-14 | [0,1,2,4,5,6,7,9] | <5,5,5,5,6,2> |
| 4-z15 | [0,1,4,6] | <1,1,1,1,1,1> | 8-z15 | [0,1,2,3,4,6,8,9] | <5,5,5,5,5,3> |
| 4-16 | [0,1,5,7] | <1,1,0,1,2,1> | 8-16 | [0,1,2,3,5,7,8,9] | <5,5,4,5,6,3> |
| 4-17 | [0,3,4,7] | <1,0,2,2,1,0> | 8-17 | [0,1,3,4,5,6,8,9] | <5,4,6,6,5,2> |
| 4-18 | [0,1,4,7] | <1,0,2,1,1,1> | 8-18 | [0,1,2,3,5,6,8,9] | <5,4,6,5,5,3> |
| 4-19 | [0,1,4,8] | <1,0,1,3,1,0> | 8-19 | [0,1,2,4,5,6,8,9] | <5,4,5,7,5,2> |
| 4-20 | [0,1,5,8] | <1,0,1,2,2,0> | 8-20 | [0,1,2,4,5,7,8,9] | <5,4,5,6,6,2> |
| 4-21 | [0,2,4,6] | <0,3,0,2,0,1> | 8-21 | [0,1,2,3,4,6,8,10] | <4,7,4,6,4,3> |
| 4-22 | [0,2,4,7] | <0,2,1,1,2,0> | 8-22 | [0,1,2,3,5,6,8,10] | <4,6,5,5,6,2> |
| 4-23 | [0,2,5,7] | <0,2,1,0,3,0> | 8-23 | [0,1,2,3,5,7,8,10] | <4,6,5,4,7,2> |
| 4-24 | [0,2,4,8] | <0,2,0,3,0,1> | 8-24 | [0,1,2,4,5,6,8,10] | <4,6,4,7,4,3> |
| 4-25 | [0,2,6,8] | <0,2,0,2,0,2> | 8-25 | [0,1,2,4,6,7,8,10] | <4,6,4,6,4,4> |
| 4-26 | [0,3,5,8] | <0,1,2,1,2,0> | 8-26 | [0,1,3,4,5,7,8,10] | <4,5,6,5,6,2> |
| 4-27 | [0,2,5,8] | <0,1,2,1,1,1> | 8-27 | [0,1,2,4,5,7,8,10] | <4,5,6,5,5,3> |
| 4-28 | [0,3,6,9] | <0,0,4,0,0,2> | 8-28 | [0,1,3,4,6,7,9,10] | <4,4,8,4,4,4> |
| 4-z29 | [0,1,3,7] | <1,1,1,1,1,1> | 8-z29 | [0,1,2,3,5,6,7,9] | <5,5,5,5,5,3> |
| 5-1 | [0,1,2,3,4] | <4,3,2,1,0,0> | 7-1 | [0,1,2,3,4,5,6] | <6,5,4,3,2,1> |
| 5-2 | [0,1,2,3,5] | <3,3,2,1,1,0> | 7-2 | [0,1,2,3,4,5,7] | <5,5,4,3,3,1> |
| 5-3 | [0,1,2,4,5] | <3,2,2,2,1,0> | 7-3 | [0,1,2,3,4,5,8] | <5,4,4,4,3,1> |
| 5-4 | [0,1,2,3,6] | <3,2,2,1,1,1> | 7-4 | [0,1,2,3,4,6,7] | <5,4,4,3,3,2> |
| 5-5 | [0,1,2,3,7] | <3,2,1,1,2,1> | 7-5 | [0,1,2,3,5,6,7] | <5,4,3,3,4,2> |
| 5-6 | [0,1,2,5,6] | <3,1,1,2,2,1> | 7-6 | [0,1,2,3,4,7,8] | <5,3,3,4,4,2> |
| 5-7 | [0,1,2,6,7] | <3,1,0,1,3,2> | 7-7 | [0,1,2,3,6,7,8] | <5,3,2,3,5,3> |
| 5-8 | [0,2,3,4,6] | <2,3,2,2,0,1> | 7-8 | [0,2,3,4,5,6,8] | <4,5,4,4,2,2> |
| 5-9 | [0,1,2,4,6] | <2,3,1,2,1,1> | 7-9 | [0,1,2,3,4,6,8] | <4,5,3,4,3,2> |
| 5-10 | [0,1,3,4,6] | <2,2,3,1,1,1> | 7-10 | [0,1,2,3,4,6,9] | <4,4,5,3,3,2> |
| 5-11 | [0,2,3,4,7] | <2,2,2,2,2,0> | 7-11 | [0,1,3,4,5,6,8] | <4,4,4,4,4,1> |
| 5-z12 | [0,1,3,5,6] | <2,2,2,1,2,1> | 7-z12 | [0,1,2,3,4,7,9] | <4,4,4,3,4,2> |
| 5-13 | [0,1,2,4,8] | <2,2,1,3,1,1> | 7-13 | [0,1,2,4,5,6,8] | <4,4,3,5,3,2> |
| 5-14 | [0,1,2,5,7] | <2,2,1,1,3,1> | 7-14 | [0,1,2,3,5,7,8] | <4,4,3,3,5,2> |
| 5-15 | [0,1,2,6,8] | <2,2,0,2,2,2> | 7-15 | [0,1,2,4,6,7,8] | <4,4,2,4,4,3> |
| 5-16 | [0,1,3,4,7] | <2,1,3,2,1,1> | 7-16 | [0,1,2,3,5,6,9] | <4,3,5,4,3,2> |
| 5-z17 | [0,1,3,4,8] | <2,1,2,3,2,0> | 7-z17 | [0,1,2,4,5,6,9] | <4,3,4,5,4,1> |
| 5-z18 | [0,1,4,5,7] | <2,1,2,2,2,1> | 7-z18 | [0,1,4,5,6,7,9] | <4,3,4,4,4,2> |
| 5-19 | [0,1,3,6,7] | <2,1,2,1,2,2> | 7-19 | [0,1,2,3,6,7,9] | <4,3,4,3,4,3> |
| 5-20 | [0,1,5,6,8] | <2,1,1,2,3,1> | 7-20 | [0,1,2,5,6,7,9] | <4,3,3,4,5,2> |
| 5-21 | [0,1,4,5,8] | <2,0,2,4,2,0> | 7-21 | [0,1,2,4,5,8,9] | <4,2,4,6,4,1> |
| 5-22 | [0,1,4,7,8] | <2,0,2,3,2,1> | 7-22 | [0,1,2,5,6,8,9] | <4,2,4,5,4,2> |
| 5-23 | [0,2,3,5,7] | <1,3,2,1,3,0> | 7-23 | [0,2,3,4,5,7,9] | <3,5,4,3,5,1> |
| 5-24 | [0,1,3,5,7] | <1,3,1,2,2,1> | 7-24 | [0,1,2,3,5,7,9] | <3,5,3,4,4,2> |
| 5-25 | [0,2,3,5,8] | <1,2,3,1,2,1> | 7-25 | [0,2,3,4,6,7,9] | <3,4,5,3,4,2> |
| 5-26 | [0,2,4,5,8] | <1,2,2,3,1,1> | 7-26 | [0,1,3,4,5,7,9] | <3,4,4,5,3,2> |
| 5-27 | [0,1,3,5,8] | <1,2,2,2,3,0> | 7-27 | [0,1,2,4,5,7,9] | <3,4,4,4,5,1> |
| 5-28 | [0,2,3,6,8] | <1,2,2,2,1,2> | 7-28 | [0,1,3,5,6,7,9] | <3,4,4,4,3,3> |
| 5-29 | [0,1,3,6,8] | <1,2,2,1,3,1> | 7-29 | [0,1,2,4,6,7,9] | <3,4,4,3,5,2> |
| 5-30 | [0,1,4,6,8] | <1,2,1,3,2,1> | 7-30 | [0,1,2,4,6,8,9] | <3,4,3,5,4,2> |
| 5-31 | [0,1,3,6,9] | <1,1,4,1,1,2> | 7-31 | [0,1,3,4,6,7,9] | <3,3,6,3,3,3> |
| 5-32 | [0,1,4,6,9] | <1,1,3,2,2,1> | 7-32 | [0,1,3,4,6,8,9] | <3,3,5,4,4,2> |
| 5-33 | [0,2,4,6,8] | <0,4,0,4,0,2> | 7-33 | [0,1,2,4,6,8,10] | <2,6,2,6,2,3> |
| 5-34 | [0,2,4,6,9] | <0,3,2,2,2,1> | 7-34 | [0,1,3,4,6,8,10] | <2,5,4,4,4,2> |
| 5-35 | [0,2,4,7,9] | <0,3,2,1,4,0> | 7-35 | [0,1,3,5,6,8,10] | <2,5,4,3,6,1> |
| 5-z36 | [0,1,2,4,7] | <2,2,2,1,2,1> | 7-z36 | [0,1,2,3,5,6,8] | <4,4,4,3,4,2> |
| 5-z37 | [0,3,4,5,8] | <2,1,2,3,2,0> | 7-z37 | [0,1,3,4,5,7,8] | <4,3,4,5,4,1> |
| 5-z38 | [0,1,2,5,8] | <2,1,2,2,2,1> | 7-z38 | [0,1,2,4,5,7,8] | <4,3,4,4,4,2> |
| 6-1 | [0,1,2,3,4,5] | <5,4,3,2,1,0> | |||
| 6-2 | [0,1,2,3,4,6] | <4,4,3,2,1,1> | |||
| 6-z3 | [0,1,2,3,5,6] | <4,3,3,2,2,1> | |||
| 6-z4 | [0,1,2,4,5,6] | <4,3,2,3,2,1> | |||
| 6-5 | [0,1,2,3,6,7] | <4,2,2,2,3,2> | |||
| 6-z6 | [0,1,2,5,6,7] | <4,2,1,2,4,2> | |||
| 6-7 | [0,1,2,6,7,8] | <4,2,0,2,4,3> | |||
| 6-8 | [0,2,3,4,5,7] | <3,4,3,2,3,0> | |||
| 6-9 | [0,1,2,3,5,7] | <3,4,2,2,3,1> | |||
| 6-z10 | [0,1,3,4,5,7] | <3,3,3,3,2,1> | |||
| 6-z11 | [0,1,2,4,5,7] | <3,3,3,2,3,1> | |||
| 6-z12 | [0,1,2,4,6,7] | <3,3,2,2,3,2> | |||
| 6-z13 | [0,1,3,4,6,7] | <3,2,4,2,2,2> | |||
| 6-14 | [0,1,3,4,5,8] | <3,2,3,4,3,0> | |||
| 6-15 | [0,1,2,4,5,8] | <3,2,3,4,2,1> | |||
| 6-16 | [0,1,4,5,6,8] | <3,2,2,4,3,1> | |||
| 6-z17 | [0,1,2,4,7,8] | <3,2,2,3,3,2> | |||
| 6-18 | [0,1,2,5,7,8] | <3,2,2,2,4,2> | |||
| 6-z19 | [0,1,3,4,7,8] | <3,1,3,4,3,1> | |||
| 6-20 | [0,1,4,5,8,9] | <3,0,3,6,3,0> | |||
| 6-21 | [0,2,3,4,6,8] | <2,4,2,4,1,2> | |||
| 6-22 | [0,1,2,4,6,8] | <2,4,1,4,2,2> | |||
| 6-z23 | [0,2,3,5,6,8] | <2,3,4,2,2,2> | |||
| 6-z24 | [0,1,3,4,6,8] | <2,3,3,3,3,1> | |||
| 6-z25 | [0,1,3,5,6,8] | <2,3,3,2,4,1> | |||
| 6-z26 | [0,1,3,5,7,8] | <2,3,2,3,4,1> | |||
| 6-27 | [0,1,3,4,6,9] | <2,2,5,2,2,2> | |||
| 6-z28 | [0,1,3,5,6,9] | <2,2,4,3,2,2> | |||
| 6-z29 | [0,2,3,6,7,9] | <2,2,4,2,3,2> | |||
| 6-30 | [0,1,3,6,7,9] | <2,2,4,2,2,3> | |||
| 6-31 | [0,1,4,5,7,9] | <2,2,3,4,3,1> | |||
| 6-32 | [0,2,4,5,7,9] | <1,4,3,2,5,0> | |||
| 6-33 | [0,2,3,5,7,9] | <1,4,3,2,4,1> | |||
| 6-34 | [0,1,3,5,7,9] | <1,4,2,4,2,2> | |||
| 6-35 | [0,2,4,6,8,10] | <0,6,0,6,0,3> | |||
| 6-z36 | [0,1,2,3,4,7] | <4,3,3,2,2,1> | |||
| 6-z37 | [0,1,2,3,4,8] | <4,3,2,3,2,1> | |||
| 6-z38 | [0,1,2,3,7,8] | <4,2,1,2,4,2> | |||
| 6-z39 | [0,2,3,4,5,8] | <3,3,3,3,2,1> | |||
| 6-z40 | [0,1,2,3,5,8] | <3,3,3,2,3,1> | |||
| 6-z41 | [0,1,2,3,6,8] | <3,3,2,2,3,2> | |||
| 6-z42 | [0,1,2,3,6,9] | <3,2,4,2,2,2> | |||
| 6-z43 | [0,1,2,5,6,8] | <3,2,2,3,3,2> | |||
| 6-z44 | [0,1,2,5,6,9] | <3,1,3,4,3,1> | |||
| 6-z45 | [0,2,3,4,6,9] | <2,3,4,2,2,2> | |||
| 6-z46 | [0,1,2,4,6,9] | <2,3,3,3,3,1> | |||
| 6-z47 | [0,1,2,4,7,9] | <2,3,3,2,4,1> | |||
| 6-z48 | [0,1,2,5,7,9] | <2,3,2,3,4,1> | |||
| 6-z49 | [0,1,3,4,7,9] | <2,2,4,3,2,2> | |||
| 6-z50 | [0,1,4,6,7,9] | <2,2,4,2,3,2> |