33.6: Lists of Set Classes
- Page ID
- 117600
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 |
| (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 | (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 | 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 |