Wikipedia

Forte number

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 (1973, ISBN 0-300-02120-8). 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.

Sets of pitches which share the same Forte number have identical interval vectors. Those that have different Forte numbers have different interval vectors with the exception of z-related sets (for example 6-Z44 and 6-Z19).

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.[citation needed]

There are three prevailing methods of computing prime form. The first was described by Forte, and the second was introduced in John Rahn's Basic Atonal Theory and used in Joseph N. Straus's Introduction to Post-Tonal Theory. The article, "List of pitch-class sets", appears to use the Rahn algorithm. For example, the Forte prime for 6-31 is {0,1,3,5,8,9} whereas the Rahn algorithm chooses {0,1,4,5,7,9}. The third method of computing prime was invented by Ian Ring during the early development of PHPMusicTools, an open-source music theory and analysis library. 

In 2017, Ian Ring discovered (through exhaustive brute force calculation) that there are exactly six primes which differ between the Forte and Rahn algorithms (not five, as had been previously published). They are 5-20, 6-Z29, 6-31, 7-20, 7-Z18, and 8-26. The prime that had previously gone unnoticed was 7-Z18. Ian invented a simpler way to compute the prime of a set, and showed through exhaustive computation that the output from his algorithm was identical to Rahn's more complex method, and so those primes are alternately cited as "Rahn primes" or "Ring primes". 

Forte number Prime according to Forte Prime according to Rahn/Ring
5-20 {0,1,3,7,8} {0,1,5,6,8}
6-Z29 {0,1,3,6,8,9} {0,2,3,6,7,9}
6-31 {0,1,3,5,8,9} {0,1,4,5,7,9}
7-Z18 {0,1,2,3,5,8,9} {0,1,4,5,6,7,9}
7-20 {0,1,2,4,7,8,9} {0,1,2,5,6,7,9}
8-26 {0,1,2,4,5,7,9,10} {0,1,3,4,5,7,8,10}

Ian Ring's formula for calculating the prime of a set is this: To get a prime, convert the tones in the set into a binary bitmask, where an "on" bit represents a pitch class that is present in the set, and an "off" bit represents one that is not. The prime of a set is the rotational transposition with the lowest numeric value. The result of this simple algorithm produces the same results as Rahn, and because it employs a binary representation it has distinct advantages when manipulating primes using computation.

Elliott Carter had earlier (1960-67) produced a numbered listing of pitch class sets, or "chords", as Carter referred to them, for his own use.