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Icositetratonic PC Sets 1-5

This is the first part of a huge catalogue of 24-tone music's PC set shapes, an analogue of Allen Forte's similar classification for 12-tone music. It is quite impossible to present such a vast amount of data on a single web page. It will take some experimentation to chunk it up both usefully and without upsetting google. It is probably not terribly useful, in any case, to present this material on a web page. It's not something that could be 'integrated by eye' by a human being. A normal one, anyway.

Labelling uses a 'base 36' system of numbering where 0 to 9 stand for the usual number values and A to Z take on values from 10 to 36 (although the largest such 'digit' in this collection will never exceed 'O' for 24). This obviates the need for separating numeric quantities by commas, thus compressing the space needed for representing tuples and also allowing one to sort such tuples alphabetically in a meaningful way. The table columns are:

1. the intervals found between all distinct PC pairs in the set - akin to Forte's 'interval vector'
2. the shape of the set in terms of its internal interval (in quartertones) jumps
3. the symmetries of the set - both cyclic and dihedral - which can be found amongst these sets

Unlike 12 tone music where over a quarter of the sets exhibit at least bilateral symmetry, by the time we get to 24 tone symmetric PC sets are rare indeed. Only 6143 out of the 699251 sets, less than 1%, are properly invertible.

Counts of interval classes (1 to 12) between pairs of distinct pitch classes in each set range from 0 to a maximum of M (i.e. 22).

Having parts doth not a vector make

×

We have great difficulty with the terminology. It's not a vector, it's a frequency distribution in the form of a tuple. It makes no sense at all to 'add' such creatures. Or 'multiply' them. They don't have useful 'lengths'. There's nothing corresponding to any kind of vector space. Where's the covariance or contravariance?

There are subdivisions according to the number of pitch classes in the set:

PC set cardinality (Number of pitch classes in PC set)
123456789101112
2322212019181716151413
Number of distinct PC set shapes
112854461771562014421306675448481752104006112720

Each sub-table is grouped by PC set cardinality and is ordered, in yet finer-grained groups, by their common interval vector, tabulated in ascending alphabetical order. Such ordering serves no purpose beyond providing such kinship groupings.

Thus the single PC set with 1 pitch class (paired with the corresponding dual PC set with 23 pitch classes) appears first. This is followed by the 12 members of the 2/22 PC sets group, each row therein paired with its dual. When we reach 3/21 sets, the interval vector starts to cover more than one shape, as non-symmetric PC sets begin to turn up alongside duals which, although differently shaped, share the interval vector value (Forte's Z forms).

The shape 'paths' are - as far as we know - how Forte would have presented them, with the largest intervals towards the end. Such paths may of course be 'rotated' without altering the essential shape of the set, although it will then not be regarded as being in 'prime form'. For example the prime form 5 PC set attached to the shape '11A1B' stands for the PC set {0,1,2,12,13} (the final B taking you 'up' 11 to 24).

Thus the entire collection, the five 'modes' of such a scale would be

• 11A1B ≡ {0,1,2,12,13}
• 1A1B1 ≡ {0,1,11,12,23}
• A1B11 ≡ {0,10,11,22,23}
• 1B11A ≡ {0,1,12,13,14}
• B11A1 ≡ {0,11,12,13,23}

Since this is not an symmetric set, its inversion - though still congruent mirror-imagewise - is prime-form indexed as 1A11B (the set {0,1,11,12,13}). This 1A11B is 'reverse-flipped-alphabetically earlier' than 11A1B (because B11A1 < B1A11 in that exceedingly computationally awkward way that the prime form works) and so gets the secondary 'B' (not to be confused with the As and Bs in the path description!) place in the Fortean formalism.

It's interesting to speculate on what a nomenclature for the symmetric 7/24 PC set 2442444 might have been, by suitable historical accident, just to drive home the analogy between 24 and 12:

• 2442444 ≡ {0,2,6,10,12,16,20} - Icositetralocrian?
• 4424442 ≡ {0,4,8,10,14,18,22} - Icositetrionian?
• 4244424 ≡ {0,4,6,10,14,18,20} - Icositetradorian?
• 2444244 ≡ {0,2,6,10,14,16,20} - Icositetraphrygian?
• 4442442 ≡ {0,4,8,12,14,18,22} - Icositetralydian?
• 4424424 ≡ {0,4,8,10,14,18,22} - Icositetramixolydian?
• 4244244 ≡ {0,4,6,10,14,15,20} - Icositetraeolian?

These are, of course, indistinguishable from the classical modes (only surely so in equal temperament, strictly speaking) except that your fingers must cover twice the distance in quartertones than you would need to cover in semitones.

The whole caboodle is available in a downloadable csv file organised (in 699251 entries) more suitably for import into a relational database. A typical set of related lines is:

``` … 07,122114113122,3216246,d208a0,c14412,fbaf96,C_0,F_0 … 07,122114113122,4261236,c14412,d208a0,fb4fae,C_0,F_0 … 17,BCCBBEBBDBC7,11112211211112123,fbaf96,fb4fae,d208a0,C_0,F_0 … ```

where in each case the first datum is the number of pitch classes in the set, the second is the PC set's interval distribution (interval vector, length 12), the third is the interval path along the chord/scale/set.

The next three data are the keys (six hex characters each) which relate the three entiries. In order these are, PRIMARY (unique row identifier), INVERSION (link to PRIMARY of the PC set's inversion), DUAL (link to PRIMARY of PC set's dual or complementary set, the PCs left out, as it were). The above example would thus take you to its inverse entry (the 2nd one above) with, perforce, the exact same value (122114113122) for the interval distribution (each entry's second, INVERSION, index cross-referencing the other's PRIMARY). The third, dual, entry has a corresponding 17 character interval path. Note that it's third index (d208a0) is the PRIMARY index of the first, i.e. each is dual to each other, in the same way as inverses are relatively anticlockwise to each other. And of course the dual has its own 17-chord inverse, referenced by fb4fae, which we've not bothered to present.

Bits and Pieces

As a technical aside, it's perhaps worth noting that the 6 hexadecimal digit character key is a 24 bit representation of the set (or, more accurately, of its inverse). E.g. the key of the first set, of cardinality 7, is d208a0 which, in binary, is 110100100000100010100000 (the 1 bits indicating set membership with the 0 class on the left, the 23 on the right). This one picks out members 0, 1, 3, 6, 12, 16, 18 which would produce an intervalic path of 1236426. This is a reversed binary 23 least significant binary digit most significant bit preserving rotation (one feels certain that Deutsch hat ein Wort dafür) of the prime form path 3216246.

In case this seems obnoxiously perverse a mapping, do note that the number was used to generate the prime form path - a relatively easy operation - and not the other way round. It's easy to find the largest binary number with a particular distribution of k on and 24 - k off bits by bitwise rotation. Once one has this, one replaces each of the k runs of 0s ending with a 1 (including final wraparound) by the length of that sequence and thus generates a k-character label which can itself be repeatedly rotated, characterwise shifting characters off one end and on to the other, to find the version of it which sorts highest. This will then turn out to be the reverse of the prime form path of the set's inverse (with all the highest alphabetical characters, representing the largest interval gaps - the way Forte and consequently all PC set theorists, want it - packed on the right).

Don't blame me. This is all Forte. As Collins remarked (somewhat understatedly) in 2004, "In general, Forte’s set class representative list would differ in some cases from that produced by the natural coding, and Forte’s prime form algorithm is thus unnecessarily complicated as a computational entity compared to the coding described here."

Also available is a csv file containing the distribution of the interval vectors.

Notes on symmetries

The vast majority of sets have no symmetries at all. Such symmetries as may be present are of two kinds - cyclic, or rotational, symmetries denoted by the usual C2, C3, C4, C6, C8 and C12, pretty much what you'd expect to see turn up within a 24 sided polygon. Dihedral symmetries (the various 'flips' Fi) are shown below in terms of their cycle index (monomials or polynomials indicating opposite vertex flips (na12a211) or opposite edge-centred flips (na212), or both, where n indicates how many). The table shows how many shapes (of the 699251 distinct shapes in general) have these symmetries.

SymmetriesPC Set Cardinality
cyclicflipindexindex1/232/223/214/205/196/187/178/169/1510/1411/1312total
(absent any symmetries)743801716538614256301445415080904103544111732692840
F1a2126301082403964502010
F2a12a211151030551041652403273864624504020
C2a212'yin yang'14285658254
C2F3a2122a21236927
C2F4a2122a12a211124610854
C3a38'manx'4412
C3F5a383a212226
C3F6a383a12a211113212
C4a46'fylfot'22
C4F7a464a21212
C4F8a464a12a2111115
C6F10a646a21211
C6F11a646a12a21112
C8F9a834a212+4a12a21112
C12F12a12212a12a21111
totals:2241708923542112402884261334108968163504208012112720699250

Note that, when totalling, PC set cardinality frequencies 1 to 11 count twice; (e.g.) once for the #5 set and again for the #19 set. The #12 sized set frequencies contribute only once to the totals.

Notes on Harmonics

The 'fifth' of the 24TET series is found at interval class 10 (in the same way as that of 12TET is found at interval class 5) as the frequency of the fifth is - ideally - 3/2 times the frequency of the root and 2^(10/24) ≅ 4/3. (These are actually 'fourths' but symmetry of inversion class results in 5 ≡ 7 for 12TET and 10 ≡ 14 for 24TET).

With the table of 5-pitchclass chords below, one may use this to find all which maximise the number of fifths found in the set. The maximum possible number of fourths/fifths between pairs of notes in a 5 note chord is 4, so we can hunt for those entries with interval vectors matching xxxxxxxx4xx (x is don't care). There is only the single entry 000302010400,44646 (which also matches its dual, a 19-chord) and it is also F2 flip symmetric (not too surprisingly). As its interval (quartertone) path sequence (44646) comprises only even numbers, one may as well 'rewrite' as semitones 22323 - it's essentially, therefore, a 12TET chord - and could comprise the real-ised notes C, D, E, G, A were one to equate (as one usually does) pitch class 0 with C. It's exactly what you'd expect - a harmonious pentatonic chord.

And now, the tables themselves.

1 1-chord1 23-chordsym
000000000000OMMMMMMMMMMMB11111111111111111111112-F2
12 2-chords12 22-chordssym
000000000001CCKKKKKKKKKKKB1111111111211111111112C2F4
000000000010BDKKKKKKKKKKLA1111111111121111111112-F1
000000000100AEKKKKKKKKKLKA1111111111112111111112-F2
0000000010009FKKKKKKKKLKKA1111111111111211111112-F1
0000000100008GKKKKKKKLKKKA1111111111111121111112-F2
0000001000007HKKKKKKLKKKKA1111111111111112111112-F1
0000010000006IKKKKKLKKKKKA1111111111111111211112-F2
0000100000005JKKKKLKKKKKKA1111111111111111121112-F1
0001000000004KKKKLKKKKKKKA1111111111111111112112-F2
0010000000003LKKLKKKKKKKKA1111111111111111111212-F1
0100000000002MKLKKKKKKKKKA1111111111111111111122-F2
1000000000001NLKKKKKKKKKKA1111111111111111111113-F1
85 3-chords85 21-chordssym
000000030000888IIIIIIILIII9111111211111121111112C3F6
000000111000789IIIIIIJJJII9111111211111112111112--
879111111121111112111112--
00000020010077AIIIIIIKIIJI9111111112111112111112-F2
000001002000699IIIIIJIIKII9111111121111111211112-F2
00000101010068AIIIIIJIJIJI9111111211111111211112--
86A111111112111111211112--
00000110001067BIIIIIJJIIIJ9111112111111111211112--
76B111111111211111211112--
00000200000166CIIIIIKIIIIIA111111111121111211112-F2
00001000110059AIIIIJIIIJJI9111111121111111121112--
95A111111112111111121112--
00001001001058BIIIIJIIJIIJ9111111211111111121112--
85B111111111211111121112--
00001010000157CIIIIJIJIIIIA111112111111111121112--
75C111111111121111121112--
00001100001056DIIIIJJIIIIJ9111121111111111121112--
65D111111111112111121112--
00002000010055EIIIIKIIIIJI9111111111111211121112-F2
0001000002004AAIIIJIIIIIKI9111111112111111112112-F2
00010000101049BIIIJIIIIJIJ9111111121111111112112--
94B111111111211111112112--
00010001000148CIIIJIIIJIIIA111111211111111112112--
84C111111111121111112112--
00010010001047DIIIJIIJIIIJ9111112111111111112112--
74D111111111112111112112--
00010100010046EIIIJIJIIIJI9111121111111111112112--
64E111111111111211112112--
00011000100045FIIIJJIIIJII9111211111111111112112--
54F111111111111121112112--
00020001000044GIIIKIIIJIII9111111111111112112112-F2
0010000001103ABIIJIIIIIIJJ9111111112111111111212--
A3B111111111211111111212--
00100000100139CIIJIIIIIJIIA111111121111111111212--
93C111111111121111111212--
00100001001038DIIJIIIIJIIJ9111111211111111111212--
83D111111111112111111212--
00100010010037EIIJIIIJIIJI9111112111111111111212--
73E111111111111211111212--
00100100100036FIIJIIJIIJII9111121111111111111212--
63F111111111111121111212--
00101001000035GIIJIJIIJIII9111211111111111111212--
53G111111111111112111212--
00110010000034HIIJJIIJIIII9112111111111111111212--
43H111111111111111211212--
00200100000033IIIKIIJIIIII9111111111111111121212-F2
0100000000202BBIJIIIIIIIIK9111111111211111111122-F2
0100000001012ACIJIIIIIIIJIA111111112111111111122--
A2C111111111121111111122--
01000000101029DIJIIIIIIJIJ9111111121111111111122--
92D111111111112111111122--
01000001010028EIJIIIIIJIJI9111111211111111111122--
82E111111111111211111122--
01000010100027FIJIIIIJIJII9111112111111111111122--
72F111111111111121111122--
01000101000026GIJIIIJIJIII9111121111111111111122--
62G111111111111112111122--
01001010000025HIJIIJIJIIII9111211111111111111122--
52H111111111111111211122--
01010100000024IIJIJIJIIIII9112111111111111111122--
42I111111111111111121122--
01101000000023JIJJIJIIIIII9121111111111111111122--
32J111111111111111112122--
02010000000022KIKIJIIIIIII9111111111111111111222-F2
1000000000111BCJIIIIIIIIIJA111111111211111111113--
B1C111111111121111111113--
A1D111111111112111111113--
10000000110019EJIIIIIIIJJI9111111121111111111113--
91E111111111111211111113--
10000001100018FJIIIIIIJJII9111111211111111111113--
81F111111111111121111113--
10000011000017GJIIIIIJJIII9111112111111111111113--
71G111111111111112111113--
10000110000016HJIIIIJJIIII9111121111111111111113--
61H111111111111111211113--
10001100000015IJIIIJJIIIII9111211111111111111113--
51I111111111111111121113--
10011000000014JJIIJJIIIIII9112111111111111111113--
41J111111111111111112113--
10110000000013KJIJJIIIIIII9121111111111111111113--
31K111111111111111111213--
11100000000012LJJJIIIIIIII9211111111111111111113--
21L111111111111111111123--
21000000000011MKJIIIIIIIII9111111111111111111114-F2
446 4-chords446 20-chordssym
0000040000026666GGGGGKGGGGGA11112111121111211112C4F8
0000121000115667GGGGHIHGGGH911112111121111121112--
665711111211112111121112--
0000121000206567GGGGHIHGGGI811112111112111121112-F1
0000202000025757GGGGIGIGGGGA11111211121111121112C2F3
0000202001015577GGGGIGIGGHG911111211111211121112-F2
0000210100205658GGGGIHGHGGI811111121112111121112-F2
0000210101105568GGGGIHGHGHH811112111111211121112--
655811111121111211121112--
0000300012005559GGGGJGGGHIG811111112111211121112-F1
0001012000204767GGGHGHIGGGI811111211112111112112-F2
0001012001104677GGGHGHIGGHH811112111112111112112--
647711111211111211112112--
0001020101014668GGGHGIGHGHG911112111121111112112--
664811111121111211112112--
0001020102006468GGGHGIGHGIG811112111111211112112-F2
0001101100114758GGGHHGHHGGH911111211121111112112--
574811111121112111112112--
0001101110014578GGGHHGHHHGG911121111121111112112--
754811111121111121112112--
0001101110105478GGGHHGHHHGH811111211111121112112--
745811121111112111112112--
0001110011104659GGGHHHGGHHH811112111211111112112--
564911111112111211112112--
0001110020104569GGGHHHGGIGH811121111211111112112--
654911111112111121112112--
0001110021005469GGGHHHGGIHG811112111111121112112--
645911121111111211112112--
000120001200455AGGGHIGGGHIG811121112111111112112--
554A11111111211121112112--
000120002100545AGGGHIGGGIHG811121111111121112112-F2
0002000200024848GGGIGGGIGGGA11111121121111112112C2F4
0002000300014488GGGIGGGJGGG911111121111112112112-F2
0002001010204749GGGIGGHGHGI811111112112111112112-F1
0002001110104479GGGIGGHHHGH811111211111112112112--
744911111112111112112112--
000201000300464AGGGIGHGGGJG811111111211211112112-F2
000201010200446AGGGIGHGHGIG811112111111112112112--
644A11111111211112112112--
000210002010454BGGGIHGGGIGH811111111121121112112-F1
000210011010445BGGGIHGGHHGH811121111111112112112--
544B11111111121112112112--
000300020001444CGGGJGGGIGGG911111111112112112112-F2
0010003002003777GGHGGGJGGIG811111211111211111212-F1
0010011101103768GGHGGHHHGHH811111211112111111212--
673811111121111211111212--
0010011110103678GGHGGHHHHGH811112111112111111212--
763811111121111121111212--
0010011111006378GGHGGHHHHHG811111211111121111212--
736811112111111211111212--
0010020020013669GGHGGIGGIGG911112111121111111212--
663911111112111121111212--
0010020030006369GGHGGIGGJGG811112111111121111212-F1
0010100200203858GGHGHGGIGGI811111121112111111212-F1
0010100300103588GGHGHGGJGGH811121111112111111212--
538811111121111112111212--
0010101011013759GGHGHGHGHHG911111211121111111212--
573911111112111211111212--
0010101110013579GGHGHGHHHGG911121111121111111212--
753911111112111112111212--
0010101111005379GGHGHGHHHHG811111211111112111212--
735911121111111211111212--
001011001110365AGGHGHHGGHHH811112111211111111212--
563A11111111211121111212--
001011010110356AGGHGHHGHGHH811121111211111111212--
653A11111111211112111212--
001011011100536AGGHGHHGHHHG811112111111112111212--
635A11121111111121111212--
001020010110355BGGHGIGGHGHH811121112111111111212--
553B11111111121112111212--
001020020010535BGGHGIGGIGGH811121111111112111212-F1
0011000110113849GGHHGGGHHGH911111121121111111212--
483911111112112111111212--
001100100210374AGGHHGGHGGIH811111211211111111212--
473A11111111211211111212--
0011001110013489GGHHGGHHHGG911211111121111111212--
843911111112111111211212--
0011001110104389GGHHGGHHHGH811111121111111211212--
834911211111112111111212--
001100200110347AGGHHGGIGGHH811211111211111111212--
743A11111111211111211212--
001100200200437AGGHHGGIGGIG811111211111111211212--
734A11211111111211111212--
001101001110364BGGHHGHGGHHH811112112111111111212--
463B11111111121121111212--
001101100110346BGGHHGHHGGHH811211112111111111212--
643B11111111121111211212--
001101101010436BGGHHGHHGHGH811112111111111211212--
634B11211111111121111212--
001110011001354CGGHHHGGHHGG911121121111111111212--
453C11111111112112111212--
001110101001345CGGHHHGHGHGG911211121111111111212--
543C11111111112111211212--
001110110001435CGGHHHGHHGGG911121111111111211212--
534C11211111111112111212--
001200110010344DGGHIGGHHGGH811211211111111111212--
443D11111111111211211212--
001200200010434DGGHIGGIGGGH811211111111111211212-F1
0020000020023939GGIGGGGGIGGA11111112121111111212C2F3
002000010120383AGGIGGGGHGHI811111111212111111212-F2
002000100210373BGGIGGGHGGIH811111111121211111212-F1
0020010020013399GGIGGHGGIGG911111112111111121212-F2
363C11111111112121111212-F2
002001010110338AGGIGGHGHGHH811111121111111121212--
833A11111111211111121212--
002001100110337BGGIGGHHGGHH811111211111111121212--
733B11111111121111121212--
002002001001336CGGIGGIGGHGG911112111111111121212--
633C11111111112111121212--
002010020010353DGGIGHGGIGGH811111111111212111212-F1
002011010010335DGGIGHHGHGGH811121111111111121212--
533D11111111111211121212--
002100200100343EGGIHGGIGGHG811111111111121211212-F2
002101100100334EGGIHGHHGGHG811211111111111121212--
433E11111111111121121212--
003002001000333FGGJGGIGGHGG811111111111112121212-F1
0100002111002778GHGGGGIHHHG811111211111211111122--
772811111121111121111122--
0100002120007278GHGGGGIHIGG811111211111121111122-F2
0100010202002868GHGGGHGIGIG811111121111211111122-F2
0100010301002688GHGGGHGJGHG811112111111211111122--
628811111121111112111122--
0100011020102769GHGGGHHGIGH811111211112111111122--
672911111112111121111122--
0100011110102679GHGGGHHHHGH811112111112111111122--
762911111112111112111122--
0100011120006279GHGGGHHHIGG811111211111112111122--
726911112111111121111122--
010002010101266AGHGGGIGHGHG911112111121111111122--
662A11111111211112111122--
010002020100626AGHGGGIGIGHG811112111111112111122-F2
0100100111102859GHGGHGGHHHH811111121112111111122--
582911111112111211111122--
010010101101275AGHGGHGHGHHG911111211121111111122--
572A11111111211121111122--
0100101110102589GHGGHGHHHGH811121111112111111122--
852911111112111111211122--
0100101111005289GHGGHGHHHHG811111121111111211122--
825911121111111211111122--
010010200101257AGHGGHGIGGHG911121111121111111122--
752A11111111211111211122--
010010201100527AGHGGHGIGHHG811111211111111211122--
725A11121111111121111122--
010011010020265BGHGGHHGHGGI811112111211111111122--
562B11111111121112111122--
010011100020256BGHGGHHHGGGI811121111211111111122--
652B11111111121111211122--
010011110010526BGHGGHHHHGGH811112111111111211122--
625B11121111111112111122--
010020100101255CGHGGIGHGGHG911121112111111111122--
552C11111111112111211122--
010020200001525CGHGGIGIGGGG911121111111111211122-F2
0101000020202949GHGHGGGGIGI811111112112111111122-F2
010100010201284AGHGHGGGHGIG911111121121111111122--
482A11111111211211111122--
010100101020274BGHGHGGHGHGI811111211211111111122--
472B11111111121121111122--
0101010020102499GHGHGHGGIGH811211111112111111122--
429911111112111111121122--
010101010101248AGHGHGHGHGHG911211111121111111122--
264C11112112111111111122--
462C11111111112112111122--
842A11111111211111121122--
010101010200428AGHGHGHGHGIG811111121111111121122--
824A11211111111211111122--
010101100020247BGHGHGHHGGGI811211111211111111122--
742B11111111121111121122--
010101101010427BGHGHGHHGHGH811111211111111121122--
724B11211111111121111122--
010102000101246CGHGHGIGGGHG911211112111111111122--
642C11111111112111121122--
010102010001426CGHGHGIGHGGG911112111111111121122--
624C11211111111112111122--
010110101010254DGHGHHGHGHGH811121121111111111122--
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1771 5-chords1771 19-chordssym
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