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 |
| 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 | 8–26 | 456562 |
| (0369) | 4–28 | 004002 | (0134679T) | 8–28 | 448444 |
| 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 |
| (01368) | 5–29 | 122131 | (0124679) | 7–29 | 344352 |
| (01369) | 5–31 | 114112 | (0134679) | 7–31 | 336333 |
| (01457) | 5–Z18 | 212221 | (0145679) 2 | 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 | 5–20 | 211231 | (0125679) 4 | 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 |
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 | (012467) | 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 | 6–Z29 | 224232 | (014679) | 6–Z50 | 224232 |
| (014568) | 6–16 | 322431 | |||
| (014579) 6 | 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 |
Forte prime form for 8–26: (0124579T)
Forte prime form for 7–Z18: (0123589)
Forte prime form for 5–20: (01378)
Forte prime form for 7–20: (0124789)
Forte prime form for 6–Z29: (013689)
Forte prime form for 6–31: (013589)
