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List of set classes

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Set 3-1 has three possible versions: [01112T], [011TE1], and [0TT1E1], where the subscripts indicate adjacency intervals. The normal form is the smallest "slice of pie" (shaded) or most compact form, in this case: [01112T].

This is a list of set classes, by Forte number.[1] A set class (an abbreviation of pitch-class-set class) in music theory is an ascending collection of pitch classes, transposed to begin at zero. For a list of ordered collections, see: list of tone rows and series.

Sets are listed with links to their complements. The prime form of unsymmetrical sets is marked "A". Inversions are marked "B" (sets not marked "A" or "B" are symmetrical). "T" and "E" are conventionally used in sets to notate ten and eleven, respectively, as single characters. The ordering of sets in the lists is based on the string of numerals in the interval vector treated as an integer, decreasing in value, following the strategy used by Forte in constructing his numbering system. Numbers marked with a "Z" refer to a pair of different set classes with identical interval class content that are not related by inversion, with one of each pair listed at the end of the respective list when they occur. [The "Z" derives from the prefix "zygo"—from the ancient Greek, meaning yoked or paired. Hence: zygosets.]

There are two slightly different methods of obtaining the prime form—an earlier one due to Allen Forte and a later (and generally now more popular) one devised by John Rahn—both often confusingly described as "most packed to the left". However, a more precise description of the Rahn spelling is to select the version that is most dispersed from the right. The precise description of the Forte spelling is to select the version that is most packed to the left within the smallest span. [a] This results in two different prime form sets for the same Forte number in a number of cases. The lists here use the Rahn spelling. The alternative notations for those set classes where the Forte spelling differs are listed in the footnotes.[3][4]

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.[5][6] Donald Martino had produced tables of hexachords, tetrachords, trichords, and pentachords for combinatoriality in his article, "The Source Set and its Aggregate Formations" (1961).[7]

The difference between the interval vector of a set and that of its complement is <X, X, X, X, X, X/2>, where (in base-ten) X = 12 – 2C, and C is the cardinality of the smaller set. In nearly all cases, complements of unsymmetrical sets are inversionally related—i.e. the complement of an "A" version of a set of cardinality C is (usually) the "B" version of the respective set of cardinality 12 – C. The most significant exceptions are the sets 4-14/8-14, 5-11/7-11, and 6-14, which are all closely related in terms of subset/superset structure.

List

[edit]
Forte no. Prime form Interval vector Carter no. Audio Possible spacings Complement
0-1 [] <0,0,0,0,0,0> empty set 12-1
1-1 [0] <0,0,0,0,0,0> Playⓘ PU, P8 11-1
2-1 [0,1] <1,0,0,0,0,0> 1 Playⓘ m2, M7 10-1
2-2 [0,2] <0,1,0,0,0,0> 2 Playⓘ M2, m7 10-2
2-3 [0,3] <0,0,1,0,0,0> 3 Playⓘ m3, M6 10-3
2-4 [0,4] <0,0,0,1,0,0> 4 Playⓘ M3, m6 10-4
2-5 [0,5] <0,0,0,0,1,0> 5 Playⓘ P4, P5 10-5
2-6 [0,6] <0,0,0,0,0,1> 6 Playⓘ A4, d5 10-6
3-1 [0,1,2] <2,1,0,0,0,0> 4 Playⓘ ... 9-1
3-2A [0,1,3] <1,1,1,0,0,0> 12 Playⓘ ... 9-2B
3-2B [0,2,3] Playⓘ ... 9-2A
3-3A [0,1,4] <1,0,1,1,0,0> 11 Playⓘ ... 9-3B
3-3B [0,3,4] Playⓘ ... 9-3A
3-4A [0,1,5] <1,0,0,1,1,0> 9 Playⓘ ... 9-4B
3-4B [0,4,5] Playⓘ ... 9-4A
3-5A [0,1,6] <1,0,0,0,1,1> 7 Playⓘ Viennese trichord 9-5B
3-5B [0,5,6] Playⓘ ... 9-5A
3-6 [0,2,4] <0,2,0,1,0,0> 3 Playⓘ ... 9-6
3-7A [0,2,5] <0,1,1,0,1,0> 10 Playⓘ ... 9-7B
3-7B [0,3,5] Playⓘ Blues trichord (min. pentatonic subset)[8] 9-7A
3-8A [0,2,6] <0,1,0,1,0,1> 8 Playⓘ It6 9-8B
3-8B [0,4,6] Playⓘ ... 9-8A
3-9 [0,2,7] <0,1,0,0,2,0> 5 Playⓘ sus. chord 9-9
3-10 [0,3,6] <0,0,2,0,0,1> 2 Playⓘ dim. chord 9-10
3-11A [0,3,7] <0,0,1,1,1,0> 6 Playⓘ minor chord 9-11B
3-11B [0,4,7] Playⓘ major chord 9-11A
3-12 [0,4,8] <0,0,0,3,0,0> 1 Playⓘ Aug. chord 9-12
4-1 [0,1,2,3] <3,2,1,0,0,0> 1 Playⓘ ... 8-1
4-2A [0,1,2,4] <2,2,1,1,0,0> 17 Playⓘ ... 8-2B
4-2B [0,2,3,4] Playⓘ ... 8-2A
4-3 [0,1,3,4] <2,1,2,1,0,0> 9 Playⓘ DSCH motif 8-3
4-4A [0,1,2,5] <2,1,1,1,1,0> 20 Playⓘ ... 8-4B
4-4B [0,3,4,5] Playⓘ ... 8-4A
4-5A [0,1,2,6] <2,1,0,1,1,1> 22 Playⓘ ... 8-5B
4-5B [0,4,5,6] Playⓘ ... 8-5A
4-6 [0,1,2,7] <2,1,0,0,2,1> 6 Playⓘ dream chord 8-6
4-7 [0,1,4,5] <2,0,1,2,1,0> 8 Playⓘ ... 8-7
4-8 [0,1,5,6] <2,0,0,1,2,1> 10 Playⓘ ... 8-8
4-9 [0,1,6,7] <2,0,0,0,2,2> 2 Playⓘ distance model 8-9
4-10 [0,2,3,5] <1,2,2,0,1,0> 3 Playⓘ ... 8-10
4-11A [0,1,3,5] <1,2,1,1,1,0> 26 Playⓘ ... 8-11B
4-11B [0,2,4,5] Playⓘ ... 8-11A
4-12A [0,2,3,6] <1,1,2,1,0,1> 28 Playⓘ ... 8-12A
4-12B [0,3,4,6] Playⓘ ... 8-12B
4-13A [0,1,3,6] <1,1,2,0,1,1> 7 Playⓘ ... 8-13B
4-13B [0,3,5,6] Playⓘ ... 8-13A
4-14A [0,2,3,7] <1,1,1,1,2,0> 25 Playⓘ ... 8-14A
4-14B [0,4,5,7] Playⓘ ... 8-14B
4-Z15A [0,1,4,6] <1,1,1,1,1,1> 18 Playⓘ all-interval tetrachord 8-Z15B
4-Z15B [0,2,5,6] Playⓘ all-interval tetrachord 8-Z15A
4-16A [0,1,5,7] <1,1,0,1,2,1> 19 Playⓘ ... 8-16B
4-16B [0,2,6,7] Playⓘ ... 8-16A
4-17 [0,3,4,7] <1,0,2,2,1,0> 13 Playⓘ alpha chord 8-17
4-18A [0,1,4,7] <1,0,2,1,1,1> 21 Playⓘ dim. M7 chord 8-18B
4-18B [0,3,6,7] Playⓘ ... 8-18A
4-19A [0,1,4,8] <1,0,1,3,1,0> 24 Playⓘ mM7 chord 8-19B
4-19B [0,3,4,8] Playⓘ ... 8-19A
4-20 [0,1,5,8] <1,0,1,2,2,0> 15 Playⓘ M7 chord 8-20
4-21 [0,2,4,6] <0,3,0,2,0,1> 11 Playⓘ ... 8-21
4-22A [0,2,4,7] <0,2,1,1,2,0> 27 Playⓘ mu chord 8-22B
4-22B [0,3,5,7] Playⓘ ... 8-22A
4-23 [0,2,5,7] <0,2,1,0,3,0> 4 Playⓘ quartal chord 8-23
4-24 [0,2,4,8] <0,2,0,3,0,1> 16 Playⓘ A7 chord 8-24
4-25 [0,2,6,8] <0,2,0,2,0,2> 12 Playⓘ Fr6 8-25
4-26 [0,3,5,8] <0,1,2,1,2,0> 14 Playⓘ minor seventh chord 8-26
4-27A [0,2,5,8] <0,1,2,1,1,1> 29 Playⓘ Half-diminished seventh chord 8-27B
4-27B [0,3,6,8] Playⓘ dominant 7th chord 8-27A
4-28 [0,3,6,9] <0,0,4,0,0,2> 5 Playⓘ dim. 7th chord 8-28
4-Z29A [0,1,3,7] <1,1,1,1,1,1> 23 Playⓘ all-interval tetrachord 8-Z29B
4-Z29B [0,4,6,7] Playⓘ all-interval tetrachord 8-Z29A
5-1 [0,1,2,3,4] <4,3,2,1,0,0> Playⓘ ... 7-1
5-2A [0,1,2,3,5] <3,3,2,1,1,0> Playⓘ ... 7-2B
5-2B [0,2,3,4,5] Playⓘ ... 7-2A
5-3A [0,1,2,4,5] <3,2,2,2,1,0> Playⓘ ... 7-3B
5-3B [0,1,3,4,5] Playⓘ ... 7-3A
5-4A [0,1,2,3,6] <3,2,2,1,1,1> Playⓘ ... 7-4B
5-4B [0,3,4,5,6] Playⓘ ... 7-4A
5-5A [0,1,2,3,7] <3,2,1,1,2,1> Playⓘ ... 7-5B
5-5B [0,4,5,6,7] Playⓘ ... 7-5A
5-6A [0,1,2,5,6] <3,1,1,2,2,1> Playⓘ ... 7-6B
5-6B [0,1,4,5,6] Playⓘ ... 7-6A
5-7A [0,1,2,6,7] <3,1,0,1,3,2> Playⓘ ... 7-7B
5-7B [0,1,5,6,7] Playⓘ ... 7-7A
5-8 [0,2,3,4,6] <2,3,2,2,0,1> Playⓘ ... 7-8
5-9A [0,1,2,4,6] <2,3,1,2,1,1> Playⓘ ... 7-9B
5-9B [0,2,4,5,6] Playⓘ ... 7-9A
5-10A [0,1,3,4,6] <2,2,3,1,1,1> Playⓘ ... 7-10B
5-10B [0,2,3,5,6] Playⓘ ... 7-10A
5-11A [0,2,3,4,7] <2,2,2,2,2,0> Playⓘ ... 7-11A
5-11B [0,3,4,5,7] Playⓘ ... 7-11B
5-Z12 [0,1,3,5,6] <2,2,2,1,2,1> Playⓘ ... 7-Z12
5-13A [0,1,2,4,8] <2,2,1,3,1,1> Playⓘ ... 7-13B
5-13B [0,2,3,4,8] Playⓘ ... 7-13A
5-14A [0,1,2,5,7] <2,2,1,1,3,1> Playⓘ ... 7-14B
5-14B [0,2,5,6,7] Playⓘ ... 7-14A
5-15 [0,1,2,6,8] <2,2,0,2,2,2> Playⓘ ... 7-15
5-16A [0,1,3,4,7] <2,1,3,2,1,1> Playⓘ ... 7-16B
5-16B [0,3,4,6,7] Playⓘ ... 7-16A
5-Z17 [0,1,3,4,8] <2,1,2,3,2,0> Playⓘ Farben chord 7-Z17
5-Z18A [0,1,4,5,7] <2,1,2,2,2,1> Playⓘ ... 7-Z18B
5-Z18B [0,2,3,6,7] Playⓘ ... 7-Z18A
5-19A [0,1,3,6,7] <2,1,2,1,2,2> Playⓘ ... 7-19B
5-19B [0,1,4,6,7] Playⓘ ... 7-19A
5-20A [0,1,5,6,8][b] <2,1,1,2,3,1> Playⓘ ... 7-20B
5-20B [0,2,3,7,8][c] Playⓘ In scale 7-20A
5-21A [0,1,4,5,8] <2,0,2,4,2,0> Playⓘ ... 7-21B
5-21B [0,3,4,7,8] Playⓘ ... 7-21A
5-22 [0,1,4,7,8] <2,0,2,3,2,1> Playⓘ ... 7-22
5-23A [0,2,3,5,7] <1,3,2,1,3,0> Playⓘ ... 7-23B
5-23B [0,2,4,5,7] Playⓘ ... 7-23A
5-24A [0,1,3,5,7] <1,3,1,2,2,1> Playⓘ ... 7-24B
5-24B [0,2,4,6,7] Playⓘ ... 7-24A
5-25A [0,2,3,5,8] <1,2,3,1,2,1> Playⓘ Seven six chord 7-25B
5-25B [0,3,5,6,8] Playⓘ ... 7-25A
5-26A [0,2,4,5,8] <1,2,2,3,1,1> Playⓘ ... 7-26A
5-26B [0,3,4,6,8] Playⓘ ... 7-26B
5-27A [0,1,3,5,8] <1,2,2,2,3,0> Playⓘ ... 7-27B
5-27B [0,3,5,7,8] Playⓘ ... 7-27A
5-28A [0,2,3,6,8] <1,2,2,2,1,2> Playⓘ ... 7-28A
5-28B [0,2,5,6,8] Playⓘ ... 7-28B
5-29A [0,1,3,6,8] <1,2,2,1,3,1> Playⓘ ... 7-29B
5-29B [0,2,5,7,8] Playⓘ ... 7-29A
5-30A [0,1,4,6,8] <1,2,1,3,2,1> Playⓘ ... 7-30B
5-30B [0,2,4,7,8] Playⓘ ... 7-30A
5-31A [0,1,3,6,9] <1,1,4,1,1,2> Playⓘ beta chord 7-31B
5-31B [0,2,3,6,9] Playⓘ Dominant minor ninth chord 7-31A
5-32A [0,1,4,6,9] <1,1,3,2,2,1> Playⓘ ... 7-32B
5-32B [0,2,5,6,9][d] Playⓘ Elektra chord, gamma chord 7-32A
5-33 [0,2,4,6,8] <0,4,0,4,0,2> Playⓘ ... 7-33
5-34 [0,2,4,6,9] <0,3,2,2,2,1> Playⓘ Dominant ninth chord 7-34
5-35 [0,2,4,7,9] <0,3,2,1,4,0> Playⓘ M pentatonic scale 7-35
5-Z36A [0,1,2,4,7] <2,2,2,1,2,1> Playⓘ ... 7-Z36B
5-Z36B [0,3,5,6,7] Playⓘ ... 7-Z36A
5-Z37 [0,3,4,5,8] <2,1,2,3,2,0> Playⓘ ... 7-Z37
5-Z38A [0,1,2,5,8] <2,1,2,2,2,1> Playⓘ ... 7-Z38B
5-Z38B [0,3,6,7,8] Playⓘ ... 7-Z38A
6-1 [0,1,2,3,4,5] <5,4,3,2,1,0> 4 Playⓘ chromatic hexachord 6-1
6-2A [0,1,2,3,4,6] <4,4,3,2,1,1> 19 Playⓘ ... 6-2B
6-2B [0,2,3,4,5,6] Playⓘ ... 6-2A
6-Z3A [0,1,2,3,5,6] <4,3,3,2,2,1> 49 Playⓘ ... 6-Z36B
6-Z3B [0,1,3,4,5,6] Playⓘ ... 6-Z36A
6-Z4 [0,1,2,4,5,6] <4,3,2,3,2,1> 24 Playⓘ ... 6-Z37
6-5A [0,1,2,3,6,7] <4,2,2,2,3,2> 16 Playⓘ ... 6-5B
6-5B [0,1,4,5,6,7] Playⓘ ... 6-5A
6-Z6 [0,1,2,5,6,7] <4,2,1,2,4,2> 33 Playⓘ ... 6-Z38
6-7 [0,1,2,6,7,8] <4,2,0,2,4,3> 7 Playⓘ ... 6-7
6-8 [0,2,3,4,5,7] <3,4,3,2,3,0> 5 Playⓘ ... 6-8
6-9A [0,1,2,3,5,7] <3,4,2,2,3,1> 20 Playⓘ ... 6-9B
6-9B [0,2,4,5,6,7] Playⓘ ... 6-9A
6-Z10A [0,1,3,4,5,7] <3,3,3,3,2,1> 42 Playⓘ ... 6-Z39A
6-Z10B [0,2,3,4,6,7] Playⓘ ... 6-Z39B
6-Z11A [0,1,2,4,5,7] <3,3,3,2,3,1> 47 Playⓘ ... 6-Z40B
6-Z11B [0,2,3,5,6,7] Playⓘ Sacher hexachord 6-Z40A
6-Z12A [0,1,2,4,6,7] <3,3,2,2,3,2> 46 Playⓘ ... 6-Z41B
6-Z12B [0,1,3,5,6,7] Playⓘ ... 6-Z41A
6-Z13 [0,1,3,4,6,7] <3,2,4,2,2,2> 29 Playⓘ ... 6-Z42
6-14A [0,1,3,4,5,8] <3,2,3,4,3,0> 3 Playⓘ ... 6-14A
6-14B [0,3,4,5,7,8] Playⓘ ... 6-14B
6-15A [0,1,2,4,5,8] <3,2,3,4,2,1> 13 Playⓘ ... 6-15B
6-15B [0,3,4,6,7,8] Playⓘ ... 6-15A
6-16A [0,1,4,5,6,8] <3,2,2,4,3,1> 11 Playⓘ ... 6-16B
6-16B [0,2,3,4,7,8] Playⓘ ... 6-16A
6-Z17A [0,1,2,4,7,8] <3,2,2,3,3,2> 35 Playⓘ all-trichord hexachord 6-Z43B
6-Z17B [0,1,4,6,7,8] Playⓘ ... 6-Z43A
6-18A [0,1,2,5,7,8] <3,2,2,2,4,2> 17 Playⓘ ... 6-18B
6-18B [0,1,3,6,7,8] Playⓘ ... 6-18A
6-Z19A [0,1,3,4,7,8] <3,1,3,4,3,1> 37 Playⓘ ... 6-Z44B
6-Z19B [0,1,4,5,7,8] Playⓘ ... 6-Z44A
6-20 [0,1,4,5,8,9] <3,0,3,6,3,0> 2 Playⓘ "Ode-to-Napoleon" hexachord 6-20
6-21A [0,2,3,4,6,8] <2,4,2,4,1,2> 12 Playⓘ ... 6-21B
6-21B [0,2,4,5,6,8] Playⓘ ... 6-21A
6-22A [0,1,2,4,6,8] <2,4,1,4,2,2> 10 Playⓘ ... 6-22B
6-22B [0,2,4,6,7,8] Playⓘ ... 6-21A
6-Z23 [0,2,3,5,6,8] <2,3,4,2,2,2> 27 Playⓘ ... 6-Z45
6-Z24A [0,1,3,4,6,8] <2,3,3,3,3,1> 39 Playⓘ ... 6-Z46B
6-Z24B [0,2,4,5,7,8] Playⓘ ... 6-Z46A
6-Z25A [0,1,3,5,6,8] <2,3,3,2,4,1> 43 Playⓘ Major eleventh chord 6-Z47B
6-Z25B [0,2,3,5,7,8] Playⓘ ... 6-Z47A
6-Z26 [0,1,3,5,7,8] <2,3,2,3,4,1> 26 Playⓘ ... 6-Z48
6-27A [0,1,3,4,6,9] <2,2,5,2,2,2> 14 Playⓘ ... 6-27B
6-27B [0,2,3,5,6,9] Playⓘ ... 6-27A
6-Z28 [0,1,3,5,6,9] <2,2,4,3,2,2> 21 Playⓘ ... 6-Z49
6-Z29 [0,2,3,6,7,9][e] <2,2,4,2,3,2> 32 Playⓘ Bridge chord 6-Z50
6-30A [0,1,3,6,7,9] <2,2,4,2,2,3> 15 Playⓘ ... 6-30B
6-30B [0,2,3,6,8,9] Playⓘ Petrushka chord 6-30A
6-31A [0,1,4,5,7,9][f] <2,2,3,4,3,1> 8 Playⓘ ... 6-31B
6-31B [0,2,4,5,8,9][g] Playⓘ ... 6-31A
6-32 [0,2,4,5,7,9] <1,4,3,2,5,0> 6 Playⓘ diatonic hexachord 6-32
6-33A [0,2,3,5,7,9] <1,4,3,2,4,1> 18 Playⓘ ... 6-33B
6-33B [0,2,4,6,7,9] Playⓘ Dominant eleventh chord 6-33A
6-34A [0,1,3,5,7,9] <1,4,2,4,2,2> 9 Playⓘ mystic chord 6-34B
6-34B [0,2,4,6,8,9] Playⓘ Prélude chord 6-34A
6-35 [0,2,4,6,8,T] <0,6,0,6,0,3> 1 Playⓘ whole tone scale 6-35
6-Z36A [0,1,2,3,4,7] <4,3,3,2,2,1> 50 Playⓘ ... 6-Z3B
6-Z36B [0,3,4,5,6,7] Playⓘ ... 6-Z3A
6-Z37 [0,1,2,3,4,8] <4,3,2,3,2,1> 23 Playⓘ ... 6-Z4
6-Z38 [0,1,2,3,7,8] <4,2,1,2,4,2> 34 Playⓘ ... 6-Z6
6-Z39A [0,2,3,4,5,8] <3,3,3,3,2,1> 41 Playⓘ ... 6-Z10A
6-Z39B [0,3,4,5,6,8] Playⓘ ... 6-Z10B
6-Z40A [0,1,2,3,5,8] <3,3,3,2,3,1> 48 Playⓘ ... 6-Z11B
6-Z40B [0,3,5,6,7,8] Playⓘ ... 6-Z11A
6-Z41A [0,1,2,3,6,8] <3,3,2,2,3,2> 45 Playⓘ ... 6-Z12B
6-Z41B [0,2,5,6,7,8] Playⓘ ... 6-Z12A
6-Z42 [0,1,2,3,6,9] <3,2,4,2,2,2> 30 Playⓘ ... 6-Z13
6-Z43A [0,1,2,5,6,8] <3,2,2,3,3,2> 36 Playⓘ ... 6-Z17B
6-Z43B [0,2,3,6,7,8] Playⓘ ... 6-Z17A
6-Z44A [0,1,2,5,6,9] <3,1,3,4,3,1> 38 Playⓘ Schoenberg hexachord 6-Z19B
6-Z44B [0,1,4,5,6,9][h] Playⓘ ... 6-Z19A
6-Z45 [0,2,3,4,6,9] <2,3,4,2,2,2> 28 Playⓘ ... 6-Z23
6-Z46A [0,1,2,4,6,9] <2,3,3,3,3,1> 40 Playⓘ ... 6-Z24B
6-Z46B [0,2,4,5,6,9] Playⓘ ... 6-Z24A
6-Z47A [0,1,2,4,7,9] <2,3,3,2,4,1> 44 Playⓘ ... 6-Z25B
6-Z47B [0,2,3,4,7,9] Playⓘ blues scale 6-Z25A
6-Z48 [0,1,2,5,7,9] <2,3,2,3,4,1> 25 Playⓘ ... 6-Z26
6-Z49 [0,1,3,4,7,9] <2,2,4,3,2,2> 22 Playⓘ ... 6-Z28
6-Z50 [0,1,4,6,7,9] <2,2,4,2,3,2> 31 Playⓘ ... 6-Z29
7-1 [0,1,2,3,4,5,6] <6,5,4,3,2,1> 1 Playⓘ ... 5-1
7-2A [0,1,2,3,4,5,7] <5,5,4,3,3,1> 11 Playⓘ ... 5-2B
7-2B [0,2,3,4,5,6,7] Playⓘ ... 5-2A
7-3A [0,1,2,3,4,5,8] <5,4,4,4,3,1> 14 Playⓘ ... 5-3B
7-3B [0,3,4,5,6,7,8] Playⓘ ... 5-3A
7-4A [0,1,2,3,4,6,7] <5,4,4,3,3,2> 12 Playⓘ ... 5-4B
7-4B [0,1,3,4,5,6,7] Playⓘ ... 5-4A
7-5A [0,1,2,3,5,6,7] <5,4,3,3,4,2> 13 Playⓘ ... 5-5B
7-5B [0,1,2,4,5,6,7] Playⓘ ... 5-5A
7-6A [0,1,2,3,4,7,8] <5,3,3,4,4,2> 27 Playⓘ ... 5-6B
7-6B [0,1,4,5,6,7,8] Playⓘ ... 5-6A
7-7A [0,1,2,3,6,7,8] <5,3,2,3,5,3> 30 Playⓘ ... 5-7B
7-7B [0,1,2,5,6,7,8] Playⓘ ... 5-7A
7-8 [0,2,3,4,5,6,8] <4,5,4,4,2,2> 2 Playⓘ ... 5-8
7-9A [0,1,2,3,4,6,8] <4,5,3,4,3,2> 15 Playⓘ ... 5-9B
7-9B [0,2,4,5,6,7,8] Playⓘ ... 5-9A
7-10A [0,1,2,3,4,6,9] <4,4,5,3,3,2> 19 Playⓘ ... 5-10B
7-10B [0,2,3,4,5,6,9] Playⓘ ... 5-10A
7-11A [0,1,3,4,5,6,8] <4,4,4,4,4,1> 18 Playⓘ ... 5-11A
7-11B [0,2,3,4,5,7,8] Playⓘ ... 5-11B
7-Z12 [0,1,2,3,4,7,9] <4,4,4,3,4,2> 5 Playⓘ ... 5-Z12
7-13A [0,1,2,4,5,6,8] <4,4,3,5,3,2> 17 Playⓘ ... 5-13B
7-13B [0,2,3,4,6,7,8] Playⓘ ... 5-13A
7-14A [0,1,2,3,5,7,8] <4,4,3,3,5,2> 28 Playⓘ ... 5-14B
7-14B [0,1,3,5,6,7,8] Playⓘ ... 5-14A
7-15 [0,1,2,4,6,7,8] <4,4,2,4,4,3> 4 Playⓘ ... 5-15
7-16A [0,1,2,3,5,6,9] <4,3,5,4,3,2> 20 Playⓘ ... 5-16B
7-16B [0,1,3,4,5,6,9] Playⓘ ... 5-16A
7-Z17 [0,1,2,4,5,6,9] <4,3,4,5,4,1> 10 Playⓘ ... 5-Z17
7-Z18A [0,1,4,5,6,7,9][i] <4,3,4,4,4,2> 35 Playⓘ ... 5-Z18B
7-Z18B [0,2,3,4,5,8,9] [j] Playⓘ ... 5-Z18A
7-19A [0,1,2,3,6,7,9] <4,3,4,3,4,3> 31 Playⓘ ... 5-19B
7-19B [0,1,2,3,6,8,9] Playⓘ ... 5-19A
7-20A [0,1,2,5,6,7,9][k] <4,3,3,4,5,2> 34 Playⓘ Persian scale 5-20B
7-20B [0,2,3,4,7,8,9][l] Playⓘ ... 5-20A
7-21A [0,1,2,4,5,8,9] <4,2,4,6,4,1> 21 Playⓘ ... 5-21B
7-21B [0,1,3,4,5,8,9] Playⓘ ... 5-21A
7-22 [0,1,2,5,6,8,9] <4,2,4,5,4,2> 8 Playⓘ double harmonic scale 5-22
7-23A [0,2,3,4,5,7,9] <3,5,4,3,5,1> 25 Playⓘ ... 5-23B
7-23B [0,2,4,5,6,7,9] Playⓘ ... 5-23A
7-24A [0,1,2,3,5,7,9] <3,5,3,4,4,2> 22 Playⓘ ... 5-24B
7-24B [0,2,4,6,7,8,9] Playⓘ enigmatic scale 5-24A
7-25A [0,2,3,4,6,7,9] <3,4,5,3,4,2> 24 Playⓘ ... 5-25B
7-25B [0,2,3,5,6,7,9] Playⓘ ... 5-25A
7-26A [0,1,3,4,5,7,9] <3,4,4,5,3,2> 26 Playⓘ ... 5-26A
7-26B [0,2,4,5,6,8,9] Playⓘ ... 5-26B
7-27A [0,1,2,4,5,7,9] <3,4,4,4,5,1> 23 Playⓘ ... 5-27B
7-27B [0,2,4,5,7,8,9] Playⓘ ... 5-27A
7-28A [0,1,3,5,6,7,9] <3,4,4,4,3,3> 36 Playⓘ ... 5-28A
7-28B [0,2,3,4,6,8,9] Playⓘ ... 5-28B
7-29A [0,1,2,4,6,7,9] <3,4,4,3,5,2> 32 Playⓘ ... 5-29B
7-29B [0,2,3,5,7,8,9] Playⓘ ... 5-29A
7-30A [0,1,2,4,6,8,9] <3,4,3,5,4,2> 37 Playⓘ minor Neapolitan scale 5-30B
7-30B [0,1,3,5,7,8,9] Playⓘ ... 5-30A
7-31A [0,1,3,4,6,7,9] <3,3,6,3,3,3> 38 Playⓘ Hungarian major scale 5-31B
7-31B [0,2,3,5,6,8,9] Playⓘ Romanian major scale 5-31A
7-32A [0,1,3,4,6,8,9] <3,3,5,4,4,2> 33 Playⓘ harmonic minor scale 5-32B
7-32B [0,1,3,5,6,8,9] Playⓘ harmonic major scale 5-32A
7-33 [0,1,2,4,6,8,T] <2,6,2,6,2,3> 6 Playⓘ M Locrian scale 5-33
7-34 [0,1,3,4,6,8,T] <2,5,4,4,4,2> 9 Playⓘ altered scale 5-34
7-35 [0,1,3,5,6,8,T] <2,5,4,3,6,1> 7 Playⓘ diatonic scale 5-35
7-Z36A [0,1,2,3,5,6,8] <4,4,4,3,4,2> 16 Playⓘ ... 5-Z36B
7-Z36B [0,2,3,5,6,7,8] Playⓘ ... 5-Z36A
7-Z37 [0,1,3,4,5,7,8] <4,3,4,5,4,1> 3 Playⓘ ... 5-Z37
7-Z38A [0,1,2,4,5,7,8] <4,3,4,4,4,2> 29 Playⓘ ... 5-Z38B
7-Z38B [0,1,3,4,6,7,8] Playⓘ ... 5-Z38A
8-1 [0,1,2,3,4,5,6,7] <7,6,5,4,4,2> Playⓘ ... 4-1
8-2A [0,1,2,3,4,5,6,8] <6,6,5,5,4,2> Playⓘ ... 4-2B
8-2B [0,2,3,4,5,6,7,8] Playⓘ ... 4-2A
8-3 [0,1,2,3,4,5,6,9] <6,5,6,5,4,2> Playⓘ ... 4-3
8-4A [0,1,2,3,4,5,7,8] <6,5,5,5,5,2> Playⓘ ... 4-4B
8-4B [0,1,3,4,5,6,7,8] Playⓘ ... 4-4A
8-5A [0,1,2,3,4,6,7,8] <6,5,4,5,5,3> Playⓘ ... 4-5B
8-5B [0,1,2,4,5,6,7,8] Playⓘ ... 4-5A
8-6 [0,1,2,3,5,6,7,8] <6,5,4,4,6,3> Playⓘ ... 4-6
8-7 [0,1,2,3,4,5,8,9] <6,4,5,6,5,2> Playⓘ ... 4-7
8-8 [0,1,2,3,4,7,8,9] <6,4,4,5,6,3> Playⓘ ... 4-8
8-9 [0,1,2,3,6,7,8,9] <6,4,4,4,6,4> Playⓘ ... 4-9
8-10 [0,2,3,4,5,6,7,9] <5,6,6,4,5,2> Playⓘ ... 4-10
8-11A [0,1,2,3,4,5,7,9] <5,6,5,5,5,2> Playⓘ ... 4-11B
8-11B [0,2,4,5,6,7,8,9] Playⓘ ... 4-11A
8-12A [0,1,3,4,5,6,7,9] <5,5,6,5,4,3> Playⓘ ... 4-12A
8-12B [0,2,3,4,5,6,8,9] Playⓘ ... 4-12B
8-13A [0,1,2,3,4,6,7,9] <5,5,6,4,5,3> Playⓘ ... 4-13B
8-13B [0,2,3,5,6,7,8,9] Playⓘ ... 4-13A
8-14A [0,1,2,4,5,6,7,9] <5,5,5,5,6,2> Playⓘ ... 4-14A
8-14B [0,2,3,4,5,7,8,9] Playⓘ ... 4-14B
8-Z15A [0,1,2,3,4,6,8,9] <5,5,5,5,5,3> Playⓘ ... 4-Z15B
8-Z15B [0,1,3,5,6,7,8,9] Playⓘ ... 4-Z15A
8-16A [0,1,2,3,5,7,8,9] <5,5,4,5,6,3> Playⓘ ... 4-16B
8-16B [0,1,2,4,6,7,8,9] Playⓘ ... 4-16A
8-17 [0,1,3,4,5,6,8,9] <5,4,6,6,5,2> Playⓘ ... 4-17
8-18A [0,1,2,3,5,6,8,9] <5,4,6,5,5,3> Playⓘ ... 4-18B
8-18B [0,1,3,4,6,7,8,9] Playⓘ ... 4-18A
8-19A [0,1,2,4,5,6,8,9] <5,4,5,7,5,2> Playⓘ ... 4-19B
8-19B [0,1,3,4,5,7,8,9] Playⓘ ... 4-19A
8-20 [0,1,2,4,5,7,8,9] <5,4,5,6,6,2> Playⓘ ... 4-20
8-21 [0,1,2,3,4,6,8,T] <4,7,4,6,4,3> Playⓘ ... 4-21
8-22A [0,1,2,3,5,6,8,T] <4,6,5,5,6,2> Playⓘ ... 4-22B
8-22B [0,1,3,4,5,6,8,T] [m] Playⓘ ... 4-22A
8-23 [0,1,2,3,5,7,8,T] <4,6,5,4,7,2> Playⓘ bebop scale 4-23
8-24 [0,1,2,4,5,6,8,T] <4,6,4,7,4,3> Playⓘ ... 4-24
8-25 [0,1,2,4,6,7,8,T] <4,6,4,6,4,4> Playⓘ ... 4-25
8-26 [0,1,3,4,5,7,8,T][n] <4,5,6,5,6,2> Playⓘ ... 4-26
8-27A [0,1,2,4,5,7,8,T] <4,5,6,5,5,3> Playⓘ ... 4-27B
8-27B [0,1,3,4,6,7,8,T] [o] Playⓘ ... 4-27A
8-28 [0,1,3,4,6,7,9,T] <4,4,8,4,4,4> Playⓘ octatonic scale 4-28
8-Z29A [0,1,2,3,5,6,7,9] <5,5,5,5,5,3> Playⓘ ... 4-Z29B
8-Z29B [0,2,3,4,6,7,8,9] Playⓘ ... 4-Z29A
9-1 [0,1,2,3,4,5,6,7,8] <8,7,6,6,6,3> Playⓘ ... 3-1
9-2A [0,1,2,3,4,5,6,7,9] <7,7,7,6,6,3> Playⓘ ... 3-2B
9-2B [0,2,3,4,5,6,7,8,9] Playⓘ ... 3-2A
9-3A [0,1,2,3,4,5,6,8,9] <7,6,7,7,6,3> Playⓘ ... 3-3B
9-3B [0,1,3,4,5,6,7,8,9] Playⓘ ... 3-3A
9-4A [0,1,2,3,4,5,7,8,9] <7,6,6,7,7,3> Playⓘ ... 3-4B
9-4B [0,1,2,4,5,6,7,8,9] Playⓘ ... 3-4A
9-5A [0,1,2,3,4,6,7,8,9] <7,6,6,6,7,4> Playⓘ ... 3-5B
9-5B [0,1,2,3,5,6,7,8,9] Playⓘ ... 3-5A
9-6 [0,1,2,3,4,5,6,8,T] <6,8,6,7,6,3> Playⓘ ... 3-6
9-7A [0,1,2,3,4,5,7,8,T] <6,7,7,6,7,3> Playⓘ ... 3-7B
9-7B [0,1,3,4,5,6,7,8,T] [p] Playⓘ ... 3-7A
9-8A [0,1,2,3,4,6,7,8,T] <6,7,6,7,6,4> Playⓘ ... 3-8B
9-8B [0,1,2,4,5,6,7,8,T] [q] Playⓘ ... 3-8A
9-9 [0,1,2,3,5,6,7,8,T] <6,7,6,6,8,3> Playⓘ blues scale 3-9
9-10 [0,1,2,3,4,6,7,9,T] <6,6,8,6,6,4> Playⓘ ... 3-10
9-11A [0,1,2,3,5,6,7,9,T] <6,6,7,7,7,3> Playⓘ ... 3-11B
9-11B [0,1,2,4,5,6,7,9,T] [r] Playⓘ ... 3-11A
9-12 [0,1,2,4,5,6,8,9,T] <6,6,6,9,6,3> Playⓘ ... 3-12
10-1 [0,1,2,3,4,5,6,7,8,9] <9,8,8,8,8,4> Playⓘ ... 2-1
10-2 [0,1,2,3,4,5,6,7,8,T] <8,9,8,8,8,4> Playⓘ ... 2-2
10-3 [0,1,2,3,4,5,6,7,9,T] <8,8,9,8,8,4> Playⓘ ... 2-3
10-4 [0,1,2,3,4,5,6,8,9,T] <8,8,8,9,8,4> Playⓘ ... 2-4
10-5 [0,1,2,3,4,5,7,8,9,T] <8,8,8,8,9,4> Playⓘ ... 2-5
10-6 [0,1,2,3,4,6,7,8,9,T] <8,8,8,8,8,5> Playⓘ ... 2-6
11-1 [0,1,2,3,4,5,6,7,8,9,T] <T,T,T,T,T,5> Playⓘ ... 1-1
12-1 [0,1,2,3,4,5,6,7,8,9,T,E] <C,C,C,C,C,6> Playⓘ aggregate 0-1

There is an anomaly in Allen Forte's book concerning the numbering of the pair of hexachords 6-Z28, [011232516393], and 6-Z49, [011231437293], where adjacency intervals are shown here by subscripts. They both have the same span, with a minor-third at the right. But, within that span, the hexachord [0,1,3,4,7,9] is "more packed to the left" than [0,1,3,5,6,9], as seen by inspecting the left-hand adjacency-interval sequences, and therefore, according to Forte's own rule, the set [0,1,3,4,7,9] should have been assigned the lower number 6-Z28, with [0,1,3,5,6,9] given the higher number 6-Z49.

See also

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References

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Notes

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  1. ^ Forte and Rahn both list prime forms as the most left-packed possible version of a set. However, Forte packs from the left and Rahn packs from the right ("making the small numbers smaller," versus making, "the larger numbers ... smaller"[2]).
  2. ^ Forte 5-20A: [0,1,3,7,8]
  3. ^ Forte 5-20B: [0,1,5,7,8]
  4. ^ Forte 5-32B: [0,1,4,7,9]
  5. ^ Forte 6-Z29: [0,1,3,6,8,9]
  6. ^ Forte 6-31A: [0,1,3,5,8,9]
  7. ^ Forte 6-31B: [0,1,4,6,8,9]
  8. ^ Forte 6-Z44B: [0,1,2,5,8,9]
  9. ^ Forte 7-Z18A: [0,1,2,3,5,8,9]
  10. ^ Forte 7-Z18B: [0,1,4,6,7,8,9]
  11. ^ Forte 7-20A: [0,1,2,4,7,8,9]
  12. ^ Forte 7-20B: [0,1,2,5,7,8,9]
  13. ^ Forte 8-22B: [0,1,2,3,5,7,9,T]
  14. ^ Forte 8-26: [0,1,2,4,5,7,9,T]
  15. ^ Forte 8-27B: [0,1,2,4,6,7,9,T]
  16. ^ Forte 9-7B: [0,1,2,3,4,5,7,9,T]
  17. ^ Forte 9-8B: [0,1,2,3,4,6,8,9,T]
  18. ^ Forte 9-11B: [0,1,2,3,5,6,8,9,T]

Sources

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  1. ^ Forte, Allen (1973). The Structure of Atonal Music. Yale University Press. ISBN 0-300-02120-8.
  2. ^ Nelson, Paul (2004). "Two Algorithms for Computing the Prime Form", ComposerTools.com.
  3. ^ Rahn, John (1980). Basic Atonal Theory. New York: Longman. ISBN 978-0028731605.
  4. ^ Straus, Joseph N. (1990). Introduction to Post-Tonal Theory. Prentice-Hall. ISBN 9780131898905.
  5. ^ Schiff, David (1983/1998). The Music of Elliott Carter.
  6. ^ Carter, Elliott (2002). The Harmony Book, "Appendix 1". ISBN 9780825845949.
  7. ^ Schuijer, Michael (2008). Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts, p.97. University of Rochester. ISBN 978-1-58046-270-9.
  8. ^ Everett, Walter (2008). The Foundations of Rock, p.169. Oxford. ISBN 9780199718702.
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Online lists

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