List of set classes

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

Set 3-1 has three possible versions: [0₁1₁2_T], [0₁1_TE₁], and [0_TT₁E₁], where subscripts indicate adjacency intervals. The normal form is the smallest "slice of pie" (shaded) or most compact form; in this case, [0₁1₁2_T].

Sets are listed with links to their complements. For unsymmetrical sets, the prime form is marked with "A" and the inversion with "B"; sets without either are symmetrical. Sets marked with a "Z" refer to a pair of different set classes with identical interval class content unrelated by inversion, with one of each pair listed at the end of the respective list when they occur. ("Z" is derived from the prefix zygo-, from Ancient Greek ζυγόν, "yoke". Hence: zygosets.) "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.

There are two slightly different methods of obtaining the prime form—an earlier one by Allen Forte and a later but now generally more popular one 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 most dispersed from the right, whereas the precise description of the Forte spelling is to select the version most packed to the left within the smallest span.^[a] In the lists here, the Rahn spelling is used for the 17 out of 352 set classes where the two methods yield different results; the alternative Forte spellings are listed in the footnotes.^[3][4]

Before either (1960–67), Elliott Carter had 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 1961 article, "The Source Set and its Aggregate Formations".^[7]

The difference between the interval vectors of a set and its complement is <X, X, X, X, X, X/2>, where (in base-ten) X = 12 – 2C, where C is the smaller set's cardinality. In nearly all cases, complements of unsymmetrical sets are related by inversion—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

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ⓘ	Unison, octave	11-1
2-1	[0,1]	<1,0,0,0,0,0>	1	Playⓘ	Minor second, major seventh	10-1
2-2	[0,2]	<0,1,0,0,0,0>	2	Playⓘ	Major second, minor seventh	10-2
2-3	[0,3]	<0,0,1,0,0,0>	3	Playⓘ	Minor third, major sixth	10-3
2-4	[0,4]	<0,0,0,1,0,0>	4	Playⓘ	Major third, minor sixth	10-4
2-5	[0,5]	<0,0,0,0,1,0>	5	Playⓘ	Perfect fourth, perfect fifth	10-5
2-6	[0,6]	<0,0,0,0,0,1>	6	Playⓘ	Augmented fourth, diminished fifth, tritone	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ⓘ	Viennese trichord	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 (minor pentatonic subset)[8]	9-7A
3-8A	[0,2,6]	<0,1,0,1,0,1>	8	Playⓘ	Italian sixth chord	9-8B
3-8B	[0,4,6]			Playⓘ	...	9-8A
3-9	[0,2,7]	<0,1,0,0,2,0>	5	Playⓘ	Suspended chord	9-9
3-10	[0,3,6]	<0,0,2,0,0,1>	2	Playⓘ	Diminished chord	9-10
3-11A	[0,3,7]	<0,0,1,1,1,0>	6	Playⓘ	Minor chord, Neapolitan 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ⓘ	Augmented chord	9-12
4-1	[0,1,2,3]	<3,2,1,0,0,0>	1	Playⓘ	BACH motif	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ⓘ	Chromatic tetrachord	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ⓘ	Diatonic tetrachord	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ⓘ	Major seventh flat five chord	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ⓘ	Diminished major seventh 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ⓘ	Minor major seventh chord	8-19B
4-19B	[0,3,4,8]			Playⓘ	Augmented major seventh chord	8-19A
4-20	[0,1,5,8]	<1,0,1,2,2,0>	15	Playⓘ	Major seventh chord, minor flat sixth 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, seventh suspension four chord[9]	8-23
4-24	[0,2,4,8]	<0,2,0,3,0,1>	16	Playⓘ	Augmented seventh chord	8-24
4-25	[0,2,6,8]	<0,2,0,2,0,2>	12	Playⓘ	Dominant seventh flat five chord, French sixth chord	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, minor sixth chord, Tristan chord	8-27B
4-27B	[0,3,6,8]			Playⓘ	Dominant seventh chord, German sixth chord, harmonic seventh chord	8-27A
4-28	[0,3,6,9]	<0,0,4,0,0,2>	5	Playⓘ	Diminished seventh 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, minor major ninth 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ⓘ	Ryukyu scale	7-20B
5-20B	[0,2,3,7,8][c]			Playⓘ	Hirajōshi scale, in scale, iwato 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ⓘ	Half-diminished minor ninth chord	7-25B
5-25B	[0,3,5,6,8]			Playⓘ	Seven six chord	7-25A
5-26A	[0,2,4,5,8]	<1,2,2,3,1,1>		Playⓘ	Half-diminished ninth chord	7-26A
5-26B	[0,3,4,6,8]			Playⓘ	Augmented major ninth chord	7-26B
5-27A	[0,1,3,5,8]	<1,2,2,2,3,0>		Playⓘ	Major ninth chord	7-27B
5-27B	[0,3,5,7,8]			Playⓘ	Minor ninth chord	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ⓘ	Insen scale, minor sixth ninth chord	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ⓘ	Jazz sus chord, Lydian augmented chord	7-30A
5-31A	[0,1,3,6,9]	<1,1,4,1,1,2>		Playⓘ	Beta chord, diminished minor ninth chord	7-31B
5-31B	[0,2,3,6,9]			Playⓘ	Diminished ninth chord, dominant minor ninth chord	7-31A
5-32A	[0,1,4,6,9]	<1,1,3,2,2,1>		Playⓘ	Dominant seventh sharp ninth chord	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ⓘ	Augmented dominant ninth chord, ninth augmented fifth chord,[10] ninth flat fifth chord[10]	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ⓘ	Major 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ⓘ	Two-semitone tritone scale	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ⓘ	Istrian scale	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ⓘ	Augmented scale, "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ⓘ	...	6-22A
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ⓘ	Minor major eleventh chord	6-Z46B
6-Z24B	[0,2,4,5,7,8]			Playⓘ	Half-diminished eleventh chord	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ⓘ	Major ninth sharp eleventh chord	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ⓘ	Diminished eleventh chord	6-27A
6-Z28	[0,1,3,5,6,9]	<2,2,4,3,2,2>	21	Playⓘ	Augmented major eleventh chord	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, tritone scale	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, minor eleventh chord	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ⓘ	Augmented eleventh chord, dominant sharp eleventh chord, 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ⓘ	Scale of harmonics	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ⓘ	Natural minor scale	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, Hungarian minor 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ⓘ	Augmented thirteenth chord, 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ⓘ	Augmented major thirteenth chord, harmonic minor scale, Ukrainian Dorian scale	5-32B
7-32B	[0,1,3,5,6,8,9]			Playⓘ	Half-diminished thirteenth chord, harmonic major scale	5-32A
7-33	[0,1,2,4,6,8,T]	<2,6,2,6,2,3>	6	Playⓘ	Major Locrian scale, major Neapolitan scale	5-33
7-34	[0,1,3,4,6,8,T]	<2,5,4,4,4,2>	9	Playⓘ	Acoustic scale, altered scale, minor major thirteenth chord	5-34
7-35	[0,1,3,5,6,8,T]	<2,5,4,3,6,1>	7	Playⓘ	Diatonic scale, major thirteenth chord, minor thirteenth chord	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 dominant scale, bebop major 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ⓘ	Seventh flat five diminished scale	4-25
8-26	[0,1,3,4,5,7,8,T][n]	<4,5,6,5,6,2>		Playⓘ	Bebop harmonic minor scale	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ⓘ	Bebop melodic minor scale	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ⓘ	Northern lights chord	1-1
12-1	[0,1,2,3,4,5,6,7,8,9,T,E]	<C,C,C,C,C,6>		Playⓘ	Aggregate, all-interval twelve-tone row	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

• Interval vector
• List of chords
• Z-relation