Forte number

Inmusical set theory,aForte numberis the pair ofnumbersAllen Forteassigned to theprime formof eachpitch classsetof three or more members inThe Structure ofAtonalMusic(1973,ISBN0-300-02120-8). 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.

Set 3-1 has three possible rotations/inversions; the normal form (left) is the most compact, corresponding to the smallestsector

Major and minor chords on C.

In the12-TETtuning system (or in any other system of tuning that splits theoctaveinto twelvesemitones), 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 inlexicographic order) of either thenormal formof a set or of itsinversion.The normal form of a set is that which istransposedso as to be most compact. For example, asecond inversionmajor chordcontains the pitch classes 7, 0, and 4. The normal form would then be 0, 4, and 7. Its (transposed) inversion, which happens to be theminor chord,contains the pitch classes 0, 3, and 7; and is the prime form.

C major diatonic scale.

Locrian modeon C.

The major and minor chords are both given Forte number 3-11, indicating that it is the eleventh in Forte's ordering of pitch class sets with three pitches. In contrast, theViennese trichord,with pitch classes 0, 1, and 6, is given Forte number 3-5, indicating that it is the fifth in Forte's ordering of pitch class sets with three pitches. The normal form of thediatonic 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 number 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 identicalinterval 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).

Calculation

There are two prevailing methods of computing prime form. The first was described by Forte, and the second was introduced in John Rahn'sBasic Atonal Theoryand used in Joseph N. Straus'sIntroduction to Post-Tonal Theory,and is now generally more popular. For example, the Forte prime form for 6-31 is [0₁1₂3₂5₃8₁9₃] whereas the Rahn algorithm chooses [0₁1₃4₁5₂7₂9₃], where adjacency intervals are shown here by subscripts between pitch-class numerals. As seen, both versions of this set class have one of their largest adjacency intervals (3 semitones) at the right—i.e. they both have the smallest possible span—but, within that span, Forte chooses the version that is then most packed towards the left, whereas Rahn chooses the version that is most dispersed away from the right.

In the language ofcombinatorics,the Forte numbers correspond to the binarybraceletsof length 12: that is,equivalence classesofbinary sequencesof length 12 under the operations ofcyclic permutationand 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.

Elliott Carterhad earlier (1960–1967) produced a numbered listing of pitch class sets, or "chords", as Carter referred to them, for his own use.

See also

• List of set classes

External links

• "All About Set Theory: What is a Forte Number?",JayTomlin.com.
• "SetFinder: Prime Form Calculator",ComposerTools.com.
• "The Table of Pitch Class Sets",SolomonsMusic.net.
• "PC Set Calculator",MtA.Ca.