Pitch Class Set Theory
- ppittman2010
- Feb 20, 2024
- 3 min read
Pitch class set theory offers a unique analytical approach for examining a specific set of notes. A pitch class set, also known as a pc set, comprises a group of distinct integers that represent pitch classes without any duplicates. Allen Forte in his book The Structure of Atonal Music identified 220 distinct pc sets. Allen Forte was a prominent music theorist who pioneered the study of pitch class sets. Forte's contribution to the field was the creation of pitch class sets, which are identified by a number with a dash. The first number in the set represents the cardinality of the set, with 3-11, for instance, denoting the 11th set on Forte's list of three-note sets.
To differentiate between pitches, we utilize octave and enharmonic equivalence. For instance, middle C and C one octave above are considered octave equivalent, while in post-tonal music, D# and Eb are regarded as enharmonically equivalent, though they are distinct in tonal music.
With this knowledge, we can refer to a set of pitches sharing the same or enharmonic name as a pitch class. This is a unique concept as pitch classes cannot be notated in music, but rather serve as a grouping of pitches. For example, pitch class C could refer to both middle C (C4) and C5. There are twelve pitch classes, each distinguished by integers ranging from 0 to 11. If we designate middle C as zero, the pitch notation for each pitch class would be as follows:
The pitch class interval is the measurement between two pitch classes and cannot exceed eleven semitones. Moving up or down an octave results in a pitch class with the same designation. To ensure that any number greater than 11 or less than 0 is reduced to an integer between 0-11, we apply the mod 12 formula. As a result, -13, 1, 13, and 25 are all equivalent.
In music theory, a group of pitch classes is known as a pitch class set, which can be either in an unordered or ordered arrangement, such as [4,1,8]. When analyzing these sets, it's crucial to simplify them into a normal order - the most condensed form - and then into a prime form. To achieve a normal order, circular permutation is applied, where the first element in the set is placed at the end. For instance, the permutation of [1,4,8] is [4,8,13], and the following permutation is [8,13,16]. To maintain ascending order, we add 12 to the first element before appending it to the end. To obtain the prime form and establish normal order, follow these steps:
1. Place the set in ascending order: [1,4,8]
2. Conduct the circular permutation of the set:
P0: [1,4,8]
P1: [4,8,13]
P2: [8,13,16]
3. Next subtract the first integer from the last integer among the permutations. The permutation with the least difference is the normal order.
P0: [1,4,8] 8-1 = 7
P1: [4,8,13] 13-4 = 9
P2: [8,13,16] 16-8 = 8
4. If there is a tie then subtract the first integer from the second integer. If you have greater than 3 and another tie exists, subtract the first integer from the third integer to determine normal order. If each time the differences are the same, select arbitrarily one order as the normal order.
5. To establish the prime form, begin by setting the first integer to zero, and then determining the interval classes of the subsequent integers in relation to the first integer. Using the example provided, the normal order of P0 [1,4,8] becomes the prime form [037] 1 is set to 0, 4 is 3 away from 1 and 8 is 7 away from 1. This is pc set 3-11 in Forte numbering. It's worth noting that arriving at the prime form may require operations such as transposition or inversion.
Presented below is the Pitch Class Set List which includes T and E as notations for 10 and 11 in the Prime Form, respectively. The interval vectors depict all potential occurrences of intervals in the pitch class set. The first vector represents minor 2nds, followed by Major 2nds, Minor 3rds, Major 3rds, Perfect 4/5ths, and finally tritones. Some sets are denoted with a Z for example 4-Z15 and 4-Z29. The Z denotes that the pitch class shares the same interval vectors but are not equivalent with regard to transposition or inversion. Both of these pc sets have interval vectors of 111111 but are distinct as a pc set.
PrimeForm | Forte Num. | Interval Vect. | Prime Form | Forte Num. | Interval Vect. |
(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 |
PrimeForm | Forte Num. | Interval Vect. | Prime Form | Forte Num. | Interval Vect. |
(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) | 8–26 | 456562 |
(0369) | 4–28 | 004002 | (0134679T) | 8–28 | 448444 |
PrimeForm | Forte Num. | Interval Vect. | Prime Form | Forte Num. | Interval Vect. |
(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 | (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 |
(01378) | 5–20 | 211231 | (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 |
PrimeForm | Forte Num. | Interval Vect. | Prime Form | Forte Num. | Interval Vect. |
(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 | |||
(013689) | 6–Z29 | 224232 | (014679) | 6–Z50 | 224232 |
(014568) | 6–16 | 322431 | |||
(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 |
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