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Section 33.6 Lists of Set Classes

Below are lists of all set classes with prime form, Forte number, and interval vectors shown. Allen Forte published the original list of set classes in The Structure of Atonal Music in 1973. These lists use prime forms as calculated using the Rahn method. Prime forms of sets are ordered from most packed to the left to least packed to the left, as is found in the list of set classes in both John Rahn's Basic Atonal Theory and Joseph Straus' Introduction to Post-Tonal Theory. Sets are listed across from their complements. When taken together, complements can complete the 12-note chromatic scale when correctly transposed (and sometimes inverted).

Prime
Form
Forte
Number
Interval
Vector
Prime
Form
Forte
Number
Interval
Vector
(012) 3–1 210000 (012345678) 9–1 876663
(013) 3–2 111000 (012345679) 9–2 777663
(014) 3–3 101100 (012345689) 9–3 767763
(015) 3–4 100110 (012345789) 9–4 766773
(016) 3–5 100011 (012346789) 9–5 766674
(024) 3–6 020100 (01234568T) 9–6 686763
(025) 3–7 011010 (01234578T) 9–7 677673
(026) 3–8 010101 (01234678T) 9–8 676764
(027) 3–9 010020 (01235678T) 9–9 676683
(036) 3–10 002001 (01234679T) 9–10 668664
(037) 3–11 001110 (01235679T) 9–11 667773
(048) 3–12 000300 (01245689T) 9–12 666963
Table 33.6.1. List of Set Classes for 3- and 9-note sets (Trichords and Nonachords)
Prime
Form
Forte
Number
Interval
Vector
Prime
Form
Forte
Number
Interval
Vector
(0123) 4–1 321000 (01234567) 8–1 765442
(0124) 4–2 221100 (01234568) 8–2 665542
(0125) 4–4 211110 (01234578) 8–4 655552
(0126) 4–5 210111 (01234678) 8–5 654553
(0127) 4–6 210021 (01235678) 8–6 654463
(0134) 4–3 212100 (01234569) 8–3 656542
(0135) 4–11 121110 (01234579) 8–11 565552
(0136) 4–13 112011 (01234679) 8–13 556453
(0137) 4–Z29 111111 (01235679) 8–Z29 555553
(0145) 4–7 201210 (01234589) 8–7 645652
(0146) 4–Z15 111111 (01234689) 8–Z15 555553
(0147) 4–18 102111 (01235689) 8–18 546553
(0148) 4–19 101310 (01245689) 8–19 545752
(0156) 4–8 200121 (01234789) 8–8 644563
(0157) 4–16 110121 (01235789) 8–16 554563
(0158) 4–20 101220 (01245789) 8–20 545662
(0167) 4–9 200022 (01236789) 8–9 644464
(0235) 4–10 122010 (02345679) 8–10 566452
(0236) 4–12 112101 (01345679) 8–12 556543
(0237) 4–14 111120 (01245679) 8–14 555562
(0246) 4–21 030201 (0123468T) 8–21 474643
(0247) 4–22 021120 (0123568T) 8–22 465562
(0248) 4–24 020301 (0124568T) 8–24 464743
(0257) 4–23 021030 (0123578T) 8–23 465472
(0258) 4–27 012111 (0124578T) 8–27 456553
(0268) 4–25 020202 (0124678T) 8–25 464644
(0347) 4–17 102210 (01345689) 8–17 546652
(0358) 4–26 012120 (0134578T)  1 Forte prime form for 8–26: (0124579T) 8–26 456562
(0369) 4–28 004002 (0134679T) 8–28 448444
Table 33.6.2. List of Set Classes for 4– and 8–note sets (Tetrachords and Octachords)
Prime
Form
Forte
Number
Interval
Vector
Prime
Form
Forte
Number
Interval
Vector
(01234) 5–1 432100 (0123456) 7–1 654321
(01235) 5–2 332110 (0123457) 7–2 554331
(01236) 5–4 322111 (0123467) 7–4 544332
(01237) 5–5 321121 (0123567) 7–5 543342
(01245) 5–3 322210 (0123458) 7–3 544431
(01246) 5–9 231211 (0123468) 7–9 453432
(01247) 5–Z36 222121 (0123568) 7–Z36 444342
(01248) 5–13 2221311 (0124568) 7–13 443532
(01256) 5–6 311221 (0123478) 7–6 533442
(01257) 5–14 221131 (0123578) 7–14 443352
(01258) 5–Z38 212221 (0124578) 7–Z38 434442
(01267) 5–7 310132 (0123678) 7–7 532353
(01268) 5–15 220222 (0124678) 7–15 442443
(01346) 5–10 223111 (0123469) 7–10 445332
(01347) 5–16 213211 (0123569) 7–16 435432
(01348) 5–Z17 212320 (0124569) 7–Z17 434541
(01356) 5–Z12 222121 (0123479) 7–Z12 444342
(01357) 5–24 131221 (0123579) 7–24 353442
(01358) 5–27 122230 (0124579) 7–27 344451
(01367) 5–19 212122 (0123679) 7–19 434343
(01369) 5–31 114112 (0134679) 7–31 336333
(01457) 5–Z18 212221 (0145679)  2 Forte prime form for 7–Z18: (0123589) 7–Z18 434442
(01458) 5–21 202420 (0124589) 7–21 424641
(01468) 5–30 121321 (0124689) 7–30 343542
(01469) 5–32 113221 (0134689) 7–32 335442
(01478) 5–22 202321 (0125689) 7–22 424542
(01568)  3 Forte prime form for 5–20: (01378) 5–20 211231 (0125679)  4 Forte prime form for 7–20: (0124789) 7–20 433452
(02346) 5–8 232201 (0234568) 7–8 454422
(02347) 5–11 222220 (0134568) 7–11 444441
(02357) 5–23 132130 (0234579) 7–23 354351
(02358) 5–25 123121 (0234679) 7–25 345342
(02368) 5–28 122212 (0135679) 7–28 344433
(02458) 5–26 122311 (0134579) 7–26 344532
(02468) 5–33 040402 (012468T) 7–33 262623
(02469) 5–34 032221 (013468T) 7–34 254442
(02479) 5–35 032140 (013568T) 7–35 254361
(03458) 5–Z37 212320 (0134578) 7–Z37 434541
Table 33.6.3. List of Set Classes for 5– and 7–note sets (Pentachords and Septachords)

In the table below, when no set is listed across from a six–note set, it is self–complementary (that is, it can combine with a transposed and possibly inverted set of itself to complete a 12-note chromatic scale.

Prime
Form
Forte
Number
Interval
Vector
Prime
Form
Forte
Number
Interval
Vector
(012345) 6–1 543210
(012346) 6–2 4443211
(012347) 6–Z36 433221 (012356) 6–Z3 433221
(012348) 6–Z37 432321 (012456) 6–Z4 432321
(012357) 6–9 342231
(012358) 6–Z40 333231 (012457) 6–Z11 333231
(012367) 6–5 422232
(012368) 6–Z41 332232 (012457) 6–Z12 332232
(012369) 6–Z42 324222 (013467) 6–Z13 324222
(012378) 6–Z38 421242 (012567) 6–Z6 421242
(012458) 6–15 323421
(012468) 6–22 241422
(012469) 6–Z46 233331 (013468) 6–Z24 233331
(012478) 6–Z17 322332 (012568) 6–Z43 233331
(012479) 6–Z47 233241 (013568) 6–Z25 233241
(012569) 6–Z44 313431 (013478) 6–Z19 313431
(012578) 6–18 322242
(012579) 6–Z48 232341 (013578) 6–Z26 232341
(012678) 6–7 420243
(013457) 6–Z10 333321 (023458) 6–Z39 333321
(013458) 6–14 323430
(013469) 6–27 225222
(013479) 6–Z49 224322 (013569) 6–Z28 224322
(013579) 6–34 142422
(013679) 6–30 224223
(023679)  5 Forte prime form for 6–Z29: (013689) 6–Z29 224232 (014679) 6–Z50 224232
(014568) 6–16 322431
(014579)  6 Forte prime form for 6–31: (013589) 6–31 223431
(014589) 6–20 303630
(023457) 6–8 343230
(023468) 6–21 242412
(023469) 6–Z45 234222 (023568) 6–Z23 234222
(023579) 6–33 143241
(024579) 6–32 143250
(02468T) 6–35 060603
Table 33.6.4. List of Set Classes for 6-note sets (Hexachords)