4115 Hexadecatonic PC Sets

This page presents, with a few statistical notes, the 4115 PC Sets for a 16 note division of the octave. Enjoy.

interval vector 16TET index permutation: (71)(53) consonancy 7 2 5 4 3 6 1 8 Interval Vector component reordering

For 16TET, as for 12TET, the interval vector is relatively easily transformed into the more useful consonancy. All even slots (2, 4, 6 and 8) remain as they are. Slots 1 and 7 are exchanged, as are slots 3 and 5. Thus a quick search for high values of the seventh slot of the interval vector (the second slot already counting pitch classes in secondary dominant relationships) is often sufficient to pick out sets rich in dominants and secondary dominants (if that's your taste).

Sets are organised mostly in order of increasing Forte Interval vector (which in this case has 8 components), excepting that the symmetric sets (blue = those equal to their own inverses) have been floated to the top. The (rare - and only of the hemi-cyclic 'yin-yang' type turning up only amongst cardinalities 6 and 10) green colouring applies to sets with purely cyclic (i.e. no inversional) symmetries. Thus, if you wish to hunt by interval vector (which seems unlikely, but we don't have any idea which properties of a set may excite you), bear in mind that each cardinality division is blocked up first by such symmetry considerations.

Each set is shown in prime interval path form, with - à la Forte - the largest gaps between pitch classes at the end. The interval path, or interval string, representation is preferred as it confers many advantages:

  1. It provides a structurally precise, and unique (within rotations), set identifier - sans any cultural and historical baggage
  2. The length of the path (in characters) is the (k) size of the set
  3. Its k rotations (tail-head character shifts) exactly describe all k 'modes' of the k-set
  4. The sum of the characters in the path is the (n) size of the PC 'universe'
  5. Inversions of sets are simply the (renormalised) reverse of the path
  6. It's easy to tell at a glance what the prime form would be (highest skips at end)
  7. It's easy to tell if a set is symmetric by mentally reversing the path and spotting (in)equality
  8. It's easy to see - for asymmetric sets - which of two inversions for normal form is 'best'

If you absolutely must see the path as the traditional, visually uninformative, unapprehendable brace-enclosed comma-separated list of pitch classes, there's a javascript tool to unwrap the path for you (and also show any related inverse and complementary/dual sets).

1 16-setsymmetries
 ivecpathrotaflip
 <GGGGGGG8>1111111111111111a168a28+8a12a27
1 1-set & 15-setsymmetries
ivecpathivecpathrotaflip
<00000000>G<EEEEEEE7>111111111111112–a12a27
8 2-sets & 14-setssymmetries
ivecpathivecpathrotaflip
<00000001>88<CCCCCCC7>11111121111112a822a12a27
<00000010>79<CCCCCCD6>11111112111112–a28
<00000100>6A<CCCCCDC6>11111111211112–a12a27
<00001000>5B<CCCCDCC6>11111111121112–a28
<00010000>4C<CCCDCCC6>11111111112112–a12a27
<00100000>3D<CCDCCCC6>11111111111212–a28
<01000000>2E<CDCCCCC6>11111111111122–a12a27
<10000000>1F<DCCCCCC6>11111111111113–a28
35 3-sets & 13-sets (21 distinct ivecs each)symmetries
ivecpathivecpathrotaflip
<00002100>556<AAAACBA5>1111211121112–a12a27
<00010200>466<AAABACA5>1111211112112–a12a27
<00020001>448<AAACAAA6>1111112112112–a12a27
<00200100>33A<AACAABA5>1111111121212–a12a27
<01000020>277<ABAAAAC5>1111121111122–a12a27
<02010000>22C<ACABAAA5>1111111111222–a12a27
<21000000>11E<CBAAAAA5>1111111111114–a12a27
<00011010>457<AAABBAB5>1112111112112–
5471111121112112–
<00100110>367<AABAABB5>1111211111212–
6371111121111212–
<00101001>358<AABABAA6>1112111111212–
5381111112111212–
<00110010>349<AABBAAB5>1121111111212–
4391111111211212–
<01000101>268<ABAAABA6>1111211111122–
6281111112111122–
<01001010>259<ABAABAB5>1112111111122–
5291111111211122–
<01010100>24A<ABABABA5>1121111111122–
42A1111111121122–
<01101000>23B<ABBABAA5>1211111111122–
32B1111111112122–
<10000011>178<BAAAAAB6>1111121111113–
7181111112111113–
<10000110>169<BAAAABB5>1111211111113–
6191111111211113–
<10001100>15A<BAAABBA5>1112111111113–
51A1111111121113–
<10011000>14B<BAABBAA5>1121111111113–
41B1111111112113–
<10110000>13C<BABBAAA5>1211111111113–
31C1111111111213–
<11100000>12D<BBBAAAA5>2111111111113–
21D1111111111123–
116 4-sets & 12-sets (70 distinct ivecs each)symmetries
ivecpathivecpathrotaflip
<00040002>4444<888C8886>112112112112a444a12a27
<00121020>4345<889A98A4>112111211212–a28
<00202002>3535<88A8A886>111212111212a822a28
<00202101>3355<88A8A985>111211121212–a12a27
<00210120>3436<88A989A4>111121211212–a12a27
<00300210>3337<88B88A94>111112121212–a28
<01012020>2545<8989A8A4>111211211122–a12a27
<01020300>4246<898A8B84>112111121122–a12a27
<01202001>3238<89A8A885>121111112122–a12a27
<02000202>2626<8A888A86>111122111122a822a12a27
<02001030>2527<8A8898B4>111112211122–a28
<02010201>2266<8A898A85>111121111222–a12a27
2428111111221122–a12a27
<02102010>2329<8A98A894>111111122122–a28
<03020100>222A<8B8A8984>111111112222–a12a27
<10003200>1555<9888BA84>111211121113–a28
<10022010>4147<988AA894>112111112113–a28
<10100220>1636<98988AA4>111121211113–a28
<10220010>3139<98AA8894>121111111213–a28
<12201000>212B<9AA89884>211111111123–a28
<20000022>1717<A88888A6>111113111113a822a28
<20000121>1618<A88889A5>111111311113–a12a27
<20001210>1519<A8889A94>111111131113–a28
<20012100>141A<A889A984>111111113113–a12a27
<20121000>131B<A89A9884>111111111313–a28
<21000021>1177<A98888A5>111112111114–a12a27
<21210000>121C<A9A98884>111111111133–a12a27
<32100000>111D<BA988884>111111111115–a28
<00121011>3445<889A9895>112112111212–
4435111211211212–
<00210210>3346<88A98A94>112111121212–
4336111121121212–
<01012110>2455<8989A994>112111211122–
4255111211121122–
<01020201>2446<898A8A85>112112111122–
4426111121121122–
<01101111>2536<89989995>111212111122–
3526111121211122–
<01102101>2356<8998A985>121112111122–
5326111121112122–
<01102110>3256<8998A994>111211112122–
5236121111211122–
<01110120>2437<899989A4>112121111122–
3427111112121122–
<01111020>2347<899998A4>121121111122–
4327111112112122–
<01111110>3247<89999994>112111112122–
4237121111121122–
<01201101>2338<89A89985>121211111122–
3328111111212122–
<02011020>2257<8A8998A4>111211111222–
5227111112111222–
<02020101>2248<8A8A8985>112111111222–
4228111111211222–
<02111010>2239<8A999894>121111111222–
3229111111121222–
<10011210>1546<98899A94>111211211113–
4516111121121113–
<10012110>1456<9889A994>112111211113–
5416111121112113–
<10012200>4156<9889AA84>111211112113–
5146112111121113–
<10021011>1447<988A9895>112112111113–
4417111112112113–
<10101111>1537<98989995>111212111113–
3517111112121113–
<10110210>1366<98998A94>121111211113–
3166111121111213–
<10111110>3157<98999994>111211111213–
5137121111121113–
<10120011>1348<989A8895>121121111113–
4318111111211213–
<10121001>3148<989A9885>112111111213–
4138121111112113–
<10210110>1339<98A98994>121211111113–
3319111111121213–
<11000121>1627<998889A5>111122111113–
2617111112211113–
<11001111>1528<99889995>111221111113–
2518111111221113–
<11011110>1429<99899994>112211111113–
2419111111122113–
<11100111>1267<99988995>211112111113–
6217111112111123–
<11100120>2167<999889A4>111121111123–
6127211111211113–
<11101011>1258<99989895>211121111113–
5218111111211123–
<11101101>2158<99989985>111211111123–
5128211111121113–
<11110110>1249<99998994>211211111113–
4219111111121123–
<11111010>2149<99999894>112111111123–
4129211111112113–
<11111100>132A<99999984>122111111113–
231A111111112213–
<11201100>123A<99A89984>212111111113–
321A111111112123–
<11210100>213A<99A98984>121111111123–
312A211111111213–
<12111000>122B<9A999884>221111111113–
221B111111111223–
<21000111>1168<A9888995>111121111114–
6118111111211114–
<21001110>1159<A9889994>111211111114–
5119111111121114–
<21011100>114A<A9899984>112111111114–
411A111111112114–
<21111000>113B<A9999884>121111111114–
311B111111111214–
<22110000>112C<AA998884>211111111114–
211C111111111124–
<10111011>1357<98999895>121112111113–
1438112121111113–
3418111111212113–
5317111112111213–
273 5-sets & 11-sets (137 distinct ivecs each)symmetries
ivecpathivecpathrotaflip
<00410320>33334<66A76983>11212121212–a12a27
<01220320>24334<67886983>11212121122–a12a27
<01222021>32344<67888684>12112112122–a12a27
<02030401>24244<68696A64>11211221122–a12a27
<02040202>22444<686A6865>11211211222–a12a27
<02202202>23326<68868865>11112212122–a12a27
<02212120>32236<68878783>12111121222–a12a27
<03012040>22525<696786A3>11122111222–a12a27
<03022120>22255<69688783>11121112222–a12a27
<04030201>22228<6A696864>11111122222–a12a27
<20014300>14155<8667A963>11121113113–a12a27
<20022121>14416<86688784>11113112113–a12a27
<20220121>13318<86886784>11111131213–a12a27
<21012220>11545<87678883>11121121114–a12a27
<21022300>41146<87688963>11211112114–a12a27
<21210220>12166<87876883>11112111133–a12a27
<21222001>31138<87888664>12111111214–a12a27
<22000222>11626<88666885>11112211114–a12a27
<22212100>1221A<88878763>11111111323–a12a27
<23220100>2112A<89886763>21111111124–a12a27
<43210000>1111C<A9876663>11111111116–a12a27
<01221130>23434<67887793>12112121122–
3243411212112122–
<01221211>23344<67887874>12121121122–
3324411211212122–
<01302211>23335<67968874>12121211122–
3332511121212122–
<01303102>32335<67969765>12121112122–
3323512111212122–
<02112130>23425<68778793>11122121122–
2432511122112122–
<02112211>24235<68778874>12111221122–
3242511221112122–
<02113120>23245<68779783>11211122122–
4232511121122122–
<02121121>22435<68787784>11212111222–
3422511121211222–
<02122030>22345<68788693>12112111222–
4322511121121222–
<02122111>32245<68788774>11211121222–
4223512111211222–
<02203111>23236<68869774>12111122122–
3232611112122122–
<02211211>22336<68877874>12121111222–
3322611112121222–
<03020302>22426<69686965>11221111222–
2422611112211222–
<03030301>22246<69696964>11211112222–
4222611112112222–
<03112030>22327<69778693>12211111222–
2322711111221222–
<03121120>22237<69787783>12111112222–
3222711111212222–
<10132021>14344<76798684>11212112113–
3414411211212113–
<10141012>13444<767A7675>12112112113–
3144411211211213–
<10212211>14335<76878874>11212121113–
3341511121212113–
<10221121>13435<76887784>12112121113–
3431511121211213–
<10221220>13345<76887883>12121121113–
4331511121121213–
<10222012>31435<76888675>11212111213–
3413512111212113–
<10222111>33145<76888774>11211121213–
4133512121112113–
<10231021>31345<76897684>12112111213–
4313512111211213–
<10310320>13336<76976983>12121211113–
3331611112121213–
<10320220>31336<76986883>12121111213–
3313612111121213–
<11022211>14425<77688874>11211221113–
2441511122112113–
<11022310>14245<77688973>11221121113–
4241511121122113–
<11023120>24145<77689783>11211122113–
4142511221112113–
<11103211>15235<77769874>11122121113–
1532511121221113–
<11112220>14236<77778883>11221211113–
3241611112122113–
<11113210>13255<77779873>12211121113–
2315511121112213–
<11120221>13426<77786884>12112211113–
2431611112211213–
<11121310>13246<77787973>12211211113–
4231611112112213–
<11122021>21445<77788684>11211211123–
4412521112112113–
<11122120>41245<77788783>21121112113–
4214511211121123–
<11122201>23146<77788864>11211112213–
4132612211112113–
<11202112>12535<77868775>21112121113–
2153511121211123–
<11203201>12355<77869864>21211121113–
3215511121112123–
<11210230>12436<77876893>21121211113–
3421611112121123–
<11211220>12346<77877883>21211211113–
4321611112112123–
<11212210>32146<77878873>11211112123–
4123621211112113–
<11220211>21346<77886874>12112111123–
4312621111211213–
<11220310>31246<77886973>21121111213–
4213612111121123–
<11221120>23137<77887783>12111112213–
3132712211111213–
<11301211>12337<77967874>21212111113–
3321711111212123–
<11310220>21337<77976883>12121111123–
3312721111121213–
<11311210>31237<77977873>21211111213–
3213712111112123–
<12011221>15226<78677884>11122211113–
2251611112221113–
<12021121>14227<78687784>11222111113–
2241711111222113–
<12101131>12526<78767794>21112211113–
2521611112211123–
<12101212>21526<78767875>11122111123–
2512621111221113–
<12110221>12427<78776884>21122111113–
2421711111221123–
<12111130>21427<78777793>11221111123–
2412721111122113–
<12112120>12256<78778783>22111211113–
5221611112111223–
<12112201>22156<78778864>11121111223–
5122622111121113–
<12122020>22147<78788683>11211111223–
4122722111112113–
<12211120>21247<78877783>21121111123–
4212721111121123–
<12211201>21328<78877864>12211111123–
2312821111112213–
<12212011>12238<78878674>22121111113–
3221811111121223–
<12221101>22138<78887764>12111111223–
3122822111111213–
<12302101>21238<78968764>21211111123–
3212821111112123–
<13121110>12229<79787773>22211111113–
2221911111112223–
<13212010>21229<79878673>22111111123–
2212921111111223–
<20013310>15145<86679973>11211131113–
4151511131112113–
<20023210>14146<86689873>11211113113–
4141611112113113–
<20111221>13516<86777884>11113121113–
1531611113111213–
<20111320>15136<86777983>12111131113–
3151611131111213–
<20121022>13417<86787685>11111312113–
1431711111311213–
<20122111>14137<86788774>12111113113–
3141711311111213–
<20122210>13156<86788873>11121111313–
5131611112111313–
<20132011>13147<86798674>11211111313–
4131711111211313–
<20231011>13138<86897674>12111111313–
3131811111121313–
<21013210>11455<87679873>11211121114–
4115511121112114–
<21021211>11446<87687874>11211211114–
4411611112112114–
<21100231>16126<87766894>21111311113–
2161611113111123–
<21101122>12517<87767785>11111321113–
1521711111311123–
<21112210>31156<87778873>11121111214–
5113612111121114–
<21121021>11347<87787684>12112111114–
4311711111211214–
<21122110>31147<87788773>11211111214–
4113712111112114–
<21211210>12319<87877873>11111113213–
1321911111113123–
<21221011>12148<87887674>11211111133–
4121811111121133–
<21221110>13129<87887773>21111111313–
2131913111111123–
<21320110>12139<87986773>12111111133–
3121911111112133–
<22001131>11527<88667794>11122111114–
2511711111221114–
<22011211>11428<88677874>11221111114–
2411811111122114–
<22110211>11266<88776874>21111211114–
2116611112111124–
<22111021>11257<88777684>21112111114–
5211711111211124–
<22111120>21157<88777783>11121111124–
5112721111121114–
<22112110>11329<88778773>12211111114–
2311911111112214–
<22120111>11248<88786774>21121111114–
4211811111121124–
<22121101>21148<88787764>11211111124–
4112821111112114–
<22211110>11239<88877773>21211111114–
3211911111112124–
<22221010>21139<88887673>12111111124–
3112921111111214–
<22311100>1212A<88977763>21111111133–
2121A11111111233–
<23121100>1122A<89787763>22111111114–
2211A11111111224–
<31000132>11617<97666795>11113111114–
1611711111311114–
<31001221>11518<97667884>11131111114–
1511811111131114–
<31012210>11419<97678873>11311111114–
1411911111113114–
<31122100>1131A<97788763>13111111114–
1311A11111111314–
<32100121>11167<98766784>11112111115–
6111711111211115–
<32101111>11158<98767774>11121111115–
5111811111121115–
<32111110>11149<98777773>11211111115–
4111911111112115–
<32211100>1113A<98877763>12111111115–
3111A11111111215–
<32221000>1121B<98887663>31111111114–
1211B11111111134–
<33211000>1112B<99877663>21111111115–
2111B11111111125–
<11112121>14326<77778784>11212211113–
2341611112212113–
2513512111221113–
3152511122111213–
<11121211>12445<77787874>21121121113–
2413612111122113–
3142611221111213–
4421511121121123–
<11211121>13327<77877784>12122111113–
2143611212111123–
2331711111221213–
3412621111212113–
<11212111>13237<77878774>12212111113–
2135512111211123–
3125521112111213–
3231711111212213–
<12121111>12247<78787774>22112111113–
1322812221111113–
2231811111122213–
4221711111211223–
<12202111>12328<78868774>21221111113–
2125621112111123–
2321811111122123–
5212621111211123–
<21101221>11536<87767884>11121211114–
1512721111131113–
2151711131111123–
3511611112121114–
<21111121>11437<87777784>11212111114–
1241811111132113–
1421811111131123–
3411711111212114–
<21112111>11356<87778774>12111211114–
1412821111113113–
2141811311111123–
5311611112111214–
<21211111>11338<87877774>12121111114–
1215711121111133–
3311811111121214–
5121711111211133–
504 6-sets & 10-sets (246 distinct ivecs each)symmetries
ivecpathivecpathrotaflip
<02404203>233233<46848645>1212212122a822a12a27
<02404212>232333<46848654>1212122122–a28
<02412321>223333<46856763>1212121222–a12a27
<03214221>232324<47658663>1122122122–a12a27
<03232140>322234<47676582>1211212222–a12a27
<04040403>224224<48484845>1122211222a822a12a27
<04040502>222424<48484944>1122112222–a12a27
<04050402>222244<48494844>1121122222–a12a27
<04123050>223225<48567492>1112221222–a28
<05040402>222226<49484844>1111222222–a12a27
<10520430>133333<54964872>1212121213–a28
<12124230>142324<56568672>1122122113–a28
<12223230>231325<56667672>1221112213–a28
<12321240>212434<56765682>2112121123–a28
<12404202>321235<56848644>2121112123–a28
<14223030>221227<58667472>2211111223–a28
<20242032>143134<64686474>1211311213–a28
<20243022>131434<64687464>1121211313–a28
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<32123220>411145<76567662>1121112115–a28
<32202222>111535<76646664>1112121115–a28
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<34322010>211129<78766452>2111111125–a28
<42000243>116116<86444685>1111411114a822a12a27
<42001242>115117<86445684>1111141114–a28
<42012321>114118<86456763>1111114114–a12a27
<42100242>111616<86544684>1111311115–a28
<42123210>113119<86567652>1111111414–a28
<43210221>111166<87654663>1111211116–a12a27
<43232100>11211A<87676542>1111111144–a12a27
<54321000>11111B<98765442>1111111117–a28
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3141341131211213–a12a27
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3113441211211214–a12a27
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1323161111312213–a12a27
1421351211131123–
3124152113111213–
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3143141121311213a82–
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2152151112311123a82–
<22212222>113326<66656664>1212211114–
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1215251112211133–a12a27
2141261132111123–a12a27
<23222121>112237<67666563>2212111114–
3221171111121224–
1222181111113223–a12a27
2112552111211124–a12a27
<03222312>223324<47666754>1212211222–
2332241122121222–
<03223131>223234<47667573>1221211222–
3232241121221222–
<03223140>232234<47667582>1211221222–
3223241221121222–
<03231321>222334<47675763>1212112222–
3322241121212222–
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2322251112212222–
<04132221>222235<48576663>1211122222–
3222251112122222–
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2333141122121213–
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3233141121221213–
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3323141121212213–
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3133241212211213–
<11331330>231334<55775772>1212112213–
3313241221121213–
<11332131>313234<55776573>1221211213–
3231341211212213–
<11412321>123334<55856763>2121212113–
3332141121212123–
<11420430>213334<55864872>1212121123–
3331242112121213–
<11421321>312334<55865763>2121211213–
3321341211212123–
<11421330>321334<55865772>1212112123–
3312342121121213–
<12132411>132424<56576853>1221122113–
2314241122112213–
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1432241121222113–
<12141222>224134<56585664>1211222113–
3142241122211213–
<12142212>132244<56586654>1222112113–
2231441121122213–
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2323151112212213–
<12222240>124324<56666682>2112122113–
2142341122121123–
<12222330>241234<56666772>2121122113–
3214241122112123–
<12230421>242134<56674863>1211221123–
3124242112211213–
<12231411>213244<56675853>1221121123–
2312442112112213–
<12233031>122344<56677473>2212112113–
3221441121121223–
<12241212>221344<56685654>1211211223–
3122442211211213–
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2332151112212123–
<12304212>123235<56748654>2122121113–
3232151112122123–
<12312231>213325<56756673>1212211123–
2331252111221213–
<12313221>232135<56757663>1211122123–
3123252122111213–
<12313311>231235<56757753>2121112213–
3213251221112123–
<12323112>312235<56767554>2212111213–
3221351211121223–
<12403212>212335<56847654>2121211123–
3321252111212123–
<13032321>142225<57476763>1122221113–
2224151112222113–
<13121322>124225<57565764>2112221113–
2242151112221123–
<13122141>214225<57566583>1122211123–
2241252111222113–
<13122312>122425<57566754>2211221113–
2422151112211223–
<13123140>221425<57567582>1122111223–
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2223161111222213–
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4222151112112223–
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2322161111221223–
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2231262111122213–
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3222161111212223–
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2312262211112213–
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4221252111211223–
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4212252211121123–
<13231311>222136<57675753>1211112223–
3122262221111213–
<13303212>212326<57747654>2122111123–
2321262111122123–
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3221262111121223–
<13313211>221236<57757653>2121111223–
3212262211112123–
<14131221>122227<58575663>2222111113–
2222171111122223–
<14222130>212227<58666572>2221111123–
2221272111112223–
<20233221>141334<64677663>1212113113–
3314141131121213–
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3131441121121313–
<20331231>133135<64775673>1211131213–
3133151112131213–
<20332221>131335<64776663>1212111313–
3313151112121313–
<21123420>132415<65567862>1113122113–
1423151113112213–
<21124311>141325<65568753>1221113113–
2314151131112213–
<21133131>124144<65577573>1121132113–
1421441121131123–
<21213321>123415<65657763>1113212113–
1432151113112123–
<21214311>141235<65658753>2121113113–
3214151131112123–
<21221331>124315<65665773>1113211213–
1342151113121123–
<21222132>134125<65666574>2111312113–
2143151121311123–
<21223122>214135<65667564>1131211123–
3141251132111213–
<21223311>123145<65667753>1121113213–
4132151112113123–
<21223320>132145<65667762>1121113123–
4123151112113213–
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3114341121211214–
<21232320>131245<65676762>2112111313–
4213151311121123–
<21233121>213145<65677563>1311211123–
4131252111211313–
<21311331>123316<65755773>1111321213–
1332161111312123–
<21312321>113335<65756763>1212121114–
3331151112121214–
<21320331>133126<65764773>2111131213–
2133161213111123–
<21321330>123136<65765772>1211113213–
3132161111213123–
<21321420>132136<65765862>1211113123–
3123161111213213–
<21322122>121435<65766564>1121211133–
3412151112121133–
<21322320>131236<65766762>2121111313–
3213161311112123–
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3311351211121214–
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4121351211121133–
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2411441121122114–
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2341151112212114–
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4231151112112214–
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4113251221112114–
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2114441121121124–
<22211241>124126<66655683>2111132113–
2142161111231123–
<22212231>142126<66656673>2111131123–
2124162113111123–
<22213221>113236<66657663>1221211114–
3231161111212214–
<22222230>112345<66666672>2121121114–
4321151112112124–
<22223220>122146<66667662>1121111323–
4122161111211323–
<22232031>211345<66676473>1211211124–
4311252111211214–
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2123172131111123–
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2121551112111233–
<22321320>121246<66765762>2112111133–
4212161111211233–
<22322220>311236<66766662>2121111214–
3211361211112124–
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3212171111121233–
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3121272111112133–
<23021322>114226<67465764>1122211114–
2241161111222114–
<23112141>112525<67556583>2111221114–
2115251112211124–
<23120322>112426<67564764>2112211114–
2421161111221124–
<23121312>211426<67565754>1122111124–
2411262111122114–
<23122131>113227<67566573>1222111114–
2231171111122214–
<23123220>112255<67567662>2211121114–
2211551112111224–
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<23132301>221146<67576743>1121111224–
4112262211112114–
<23212131>112327<67656573>2122111114–
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<23222130>211327<67666572>1221111124–
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<23230311>211246<67674753>2112111124–
4211262111121124–
<23232120>221137<67676562>1211111224–
3112272211111214–
<23313111>122128<67757553>2111111323–
2122181111112323–
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3211272111112124–
<23322111>121228<67766553>2211111133–
2212181111112233–
<24131211>112228<68575653>2221111114–
2221181111112224–
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2211282111111224–
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1413161111311313–
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1441151113112114–
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4141151112113114–
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1153151112131114–
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3115151113111214–
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3141161111213114–
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1311551112111314–
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4131161111211314–
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2151161111231114–
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2115161113111124–
<32112231>114127<76556673>1132111114–
2141171111123114–
<32121132>112417<76565574>2113111114–
1421171111131124–
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3411161111212115–
<32212221>113218<76656663>1231111114–
1231181111113214–
<32213211>111355<76657653>1211121115–
3111551112111215–
<32222220>121156<76666662>1112111134–
5112163111121114–
<32222310>311146<76666752>1121111215–
4111361211112115–
<32232021>112147<76676463>3112111114–
4121171111121134–
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3311171111121215–
<32331111>112138<76775553>3121111114–
3121181111112134–
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3112183111111214–
<33101232>111526<77545674>1112211115–
2511161111221115–
<33111231>111427<77555673>1122111115–
2411171111122115–
<33212121>111256<77656563>2111211115–
5211161111211125–
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4211171111121125–
<33222120>211147<77666562>1121111125–
4111272111112115–
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1221191111111324–
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3211181111112125–
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3111282111111215–
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1311191111111315–
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5111171111121116–
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4111181111112116–
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3111191111111216–
<43322100>11121A<87766542>3111111115–
12111A1111111135–
<44321100>11112A<88765542>2111111116–
21111A1111111126–
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1332251212221113–
2143241121221123–
2233151112221213–
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1322351222121113–
2321441121122123–
3223151112122213–
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2214341121211223–
2231351211122213–
3132251222111213–
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2132351221211123–
3231252111212213–
3322151112121223–
<12322221>212344<56766663>2121121123–
2213351212111223–
3212442112112123–
3312252211121213–
<13212231>123226<57656673>2122211113–
2124252112211123–
2232161111222123–
2421252111221123–
<21133221>131425<65577663>1122111313–
1412442112113113–
2141441131121123–
2413151112211313–
<21222222>124135<65666664>1211132113–
1431252111311213–
2134151211311123–
3142151112131123–
<21232221>113344<65676663>1212112114–
1313261221111313–
2313161111221313–
3311441121121214–
<21331221>121345<65775663>1211211133–
2131361312111123–
3131262111121313–
4312151112112133–
<22113321>114235<66557763>1122121114–
1512252211131113–
2215151113111223–
3241151112122114–
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1224161111322113–
1422161111311223–
2431151112211214–
<22123221>141226<66567663>2211113113–
2214161131111223–
2411351211122114–
3114251122111214–
<22221231>112435<66665673>2112121114–
1214261122111133–
2412161111221133–
3421151112121124–
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1322171111131223–
2114351121211124–
3411252111212114–
<22223211>231136<66667653>1211112214–
3113261221111214–
3211451121112124–
4112352121112114–
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2213171311111223–
3112452112111214–
4211351211121124–
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1231272111113213–
2132171111123123–
3321161111212124–
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2113361212111124–
2312171111122133–
3311262111121214–
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2121461121111233–
3122171111121323–
4121262111121133–
<32111232>112516<76555674>2111311114–
1142171123111114–
1241171111132114–
1521161111311124–
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1411272111113114–
2114171131111124–
4411151112112115–
<32221221>111346<76665663>1211211115–
1123182131111114–
1321181111113124–
4311161111211215–
<32222211>112156<76666653>3111211114–
1131281321111114–
2131181111112314–
5121161111211134–
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1311282111111314–
2113181311111124–
4112173111112114–
<33212211>111328<77656653>1221111115–
2111561112111125–
2311181111112215–
5111262111121115–
715 7-sets & 9-sets (327 distinct ivecs each)symmetries
ivecpathivecpathrotaflip
<05234160>2223223<27456381>122212222–a12a27
<05234241>2222323<27456462>122122222–a12a27
<05242422>2222233<27464643>121222222–a12a27
<06060603>2222224<28282824>112222222–a12a27
<21442341>1323133<43664562>121312213–a12a27
<21630540>1213333<43852761>121212133–a12a27
<22244322>1322314<44466543>113122213–a12a27
<22414422>1233214<44636643>113212123–a12a27
<22424223>1133233<44646444>121221214–a12a27
<22430541>2133124<44652762>121321123–a12a27
<22442322>3122134<44664543>121121323–a12a27
<23234340>1221424<45456561>112211323–a12a27
<23234421>2311324<45456642>122112214–a12a27
<23242242>1214224<45464463>112221133–a12a27
<23244141>1222144<45466362>112113223–a12a27
<23432340>3211234<45654561>212112124–a12a27
<23432421>2121244<45654642>211211233–a12a27
<24042522>1142224<46264743>112222114–a12a27
<24232341>1222216<46454562>111132223–a12a27
<24240522>2112424<46462743>211221124–a12a27
<24414222>2122126<46636443>211112323–a12a27
<25242402>2211226<47464623>221111224–a12a27
<41036520>1141414<63258741>113113114–a12a27
<41244321>1311316<63466542>131111314–a12a27
<42034521>1141144<64256742>112114114–a12a27
<42212442>1151215<64434663>111331114–a12a27
<42222243>1133116<64444464>111141214–a12a27
<43234320>1121155<65456541>111211144–a12a27
<43242222>1111444<65464443>112112116–a12a27
<43432320>3111136<65654541>121111216–a12a27
<43442121>1211218<65664342>311111134–a12a27
<44212242>1111525<66434463>111221116–a12a27
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1214314112131133–
1331134121131214–
3113314121311214–
<31343232>1313125<53565453>211131313–
1412134121131133–
2131315131311123–
3121414113112133–
<31353132>1213144<53575353>131121133–
1312144112113133–
3113134131211214–
3131134121121314–
<32224431>1132315<54446652>122131114–
1141234113212114–
1143214112123114–
1323115111312214–
<32322342>1133125<54544563>121321114–
1214215112311133–
1241215111321133–
2133115111231214–
<32323431>1132135<54545652>123121114–
1212415211311133–
1421215111311233–
3123115111213214–
<32324322>1231135<54546543>121113214–
1412125211131133–
2121415113111233–
3113215123111214–
<32333331>1221316<54555552>131111323–
1312216111131323–
1321135121113124–
3112315213111214–
<32343132>1121434<54565353>311212114–
1211434112121134–
2113135131211124–
3131125211121314–
<32422431>1113334<54644652>121212115–
1212316213111133–
1321216111131233–
3331114112121215–
<33223242>1132216<55445463>122311114–
1223116111132214–
2114125113211124–
2141125211123114–
<33232332>1122316<55454553>221311114–
1322116111131224–
1421125211131124–
2112415211311124–
<33233322>1131226<55455543>132211114–
1211425112211134–
2213116111122314–
2411215311122114–
<33313332>1113325<55535553>121221115–
1132126123211114–
2123116111123214–
2331115111221215–
<33332322>1121326<55554543>312211114–
2113126132111124–
2131126211112314–
2312116111122134–
<33333231>1112344<55555452>212112115–
1211326122111134–
2311216311112214–
3211144112112125–
<33333330>1121245<55555551>321121114–
1221136121111324–
3112216231111214–
4212115111211234–
<33423222>1212217<55645443>231111133–
1221217111113233–
2111335121211125–
3311125211121215–
<34222332>1112425<56444553>211221115–
1113226122211115–
2231116111122215–
2421115111221125–
<34323231>1112236<56545452>221211115–
1122127232111114–
2122117111112324–
3221116111121225–
<34333131>1121227<56555352>322111114–
2111245211211125–
2212117111112234–
4211125211121125–
<42222342>1114315<64444563>112131115–
1141216113311114–
1214116111133114–
1341115111312115–
<42223332>1113415<64445553>121131115–
1114135113121115–
1431115111311215–
3141115111213115–
<42234321>1113145<64456542>131121115–
1131136121111414–
3113116111121414–
4131115111211315–
<43222242>1112416<65444463>211311115–
1123117111114214–
1132117111114124–
1421116111131125–
<43322232>1111435<65544453>112121116–
1112317213111115–
1321117111113125–
3411115111212116–
<43432221>1112137<65654442>312111115–
1121218331111114–
1212118111111334–
3121117111112135–
<44323221>1111255<66545442>211121116–
1112218231111115–
1221118111111325–
2111155111211126–
<23333241>2113234<45555462>122121124–
2131225132211123–
2213125132111223–
2321134121122124–
3112324212211214–
3231124211212214–
<32234331>1131325<54456552>131221114–
1132144123112114–
1231144112113214–
1411234212113114–
2313115111221314–
3211414113112124–
<32432331>1121335<54654552>312121114–
1312126211113133–
2121316131111233–
3111334121211215–
3311134121121215–
3312115111212134–
<43333221>1112146<65555442>311211115–
1311127211111315–
2111317131111125–
3111145112111216–
4111135121112116–
4121116111121135–

The 8-sets, being comprised of those which divide the 16 tone space exactly into two complementary parts, admit the possibility that the dual of the set is the same set, or that it is the inverse to the set (or both, in the case of symmetric sets). The first 12 sets listed are such self-inverse-duals.

The next 29 sets listed are those symmetric 8-sets (i.e. self-inverses) which are not the same shape as their duals. These are thus 'Z-related' in Fortean terms.

Of the remaining - all non-symmetric - sets, 60 are single pairs where the inverse is also the dual. There follow 148 quartets of 8-sets (each quartet with a common interval vector) where the set's inverse is not its dual but its dual's inverse's dual. There are a final 6 quartets of similarly related 8-sets but where 3 pairs of quartets (i.e. 3 octets of 8-sets - confused yet?) share each of three interval vectors.

Because each of these 8-sets have duals which are also 8-sets, and consequently may be used to complete a 16TET tone row (after appropriate transposition from prime-form), they may be of some interest to those intrigued by Hauer's Tropes (but with the 254 tropes - rather more numerous - available to Hexadecaphonics).

810 8-sets (254 distinct ivecs)symmetries
ivecpathrotaflip
<76543210>11111119–a28
<64202464>11151115a822a28
<56543230>21111127–a28
<52345630>13111315–a28
<52147630>11311414–a28
<40484044>13131313a444a28
<36345250>22111225–a28
<32741650>12121333–a28
<32543650>13213123–a28
<24606424>21232123a822a28
<16345270>22212223–a28
<08080804>22222222a288a12a27
ivecprim
path		dual
path		rotaflip
<24444523>1221322312221323–a12a27
<24445252>1322122322131223–a28
<24452542>1213222312222133–a12a27
<24616423>2121232321221233–a12a27
<25254442>1132222312222214–a12a27
<25434361>2112322321222124–a12a27
<25444423>2211223322121224–a12a27
<32544442>1313212321313123–a28
<34426432>1113232312212215–a28
<34624432>2111233321212125–a28
<41464342>1311313313121314–a12a27
<42444244>1133113312141214a822a12a27
<42445432>1131133312121414–a28
<42542452>1113313312131215–a28
<43236541>1132311411412214–a12a27
<43464142>1211214431121134–a12a27
<43632541>1111333312121216–a12a27
<44242543>1142112421141124–a12a27
<44254522>1121142411221144–a12a27
<44323462>1114212411232115–a28
<44344252>1421112421131125–a28
<45242443>1111422411222116–a12a27
<45452422>2111124421121126–a12a27
<52245442>1114131411313115–a28
<54443242>1111143411212117–a28
<54544222>1211121731111135–a28
<64212463>1111511511141116–a12a27
<64323442>1111151511131117–a28
<65434321>1111115511121118–a12a27
<16262623>1222222322222213–
<16344442>2122222322222123–
<16345261>2212222322221223–
<24363442>1312222322221313–
<24443632>1231222322213213–
<24525442>2132122322123123–
<24526423>1221232321221323–
<24543451>1212223322212133–
<24625423>2121223322121233–
<26162623>1122222422222114–
<26261623>2112222422221124–
<26262613>2211222422211224–
<32464243>1313122322131313–
<32543641>1312312321321313–
<32641651>1213213312312133–
<32643442>1312123321213133–
<34245451>1132221412223114–
<34264432>1311222422211314–
<34344262>1223112421132214–
<34344451>1231122422113214–
<34344622>1122132423122114–
<34345450>1221132412211324–
<34424443>1123212421232114–
<34441642>2112312421321124–
<34445242>1121223432212114–
<34462432>2211213431211224–
<34543441>2112123432121124–
<34543450>2121123421211234–
<36243442>1112222522221115–
<36344251>2111222522211125–
<42345631>1131231413213114–
<42346432>1231131413113214–
<42364243>1313112421131314–
<42443452>1133121412133114–
<42463243>1121313431312114–
<42464233>1311213431211314–
<44144632>1141122422114114–
<44324443>1113221512231115–
<44345431>1311122522111315–
<44363422>1121124421121144–
<44423443>1112312521321115–
<44424433>1231112521113215–
<44525422>1112123532121115–
<44544232>2111213531211125–
<46342432>1111222622211116–
<46443412>2111122622111126–
<52146631>1141131413114114–
<52344442>1113131513131115–
<54222463>1114211511241115–
<54224452>1141112521114115–
<54345421>1112114541121115–
<54345430>1121114511211145–
<54422452>1111321612311116–
<54443431>1311112621111316–
<54542431>1111213631211116–
<54543430>1211113612111136–
<56443231>1111122722111117–
<64222453>1111411611411116–
<64443232>1111131713111117–
<66543211>1111112821111118–
<24444451>1322212321312223–
1321222322213123–
<24525433>2123122321322123–
1232122322132123–
<24534442>1222123321213223–
1212322321222133–
<24535342>1221223322121323–
1212232322122133–
<25344361>1222212421132223–
1123222321222214–
<25345261>1222122422113223–
1122322322122214–
<25345351>1221222422211323–
1122232322212214–
<25353442>1212222422221133–
1122223322221214–
<25435342>2122122422112323–
2112232322122124–
<25443532>2121222422211233–
2112223322212124–
<32553442>1221313313121323–
1213132313122133–
<32642551>1213123313212133–
1212313321312133–
<33355342>1222131413113223–
1131322313122214–
<33435532>1221321412311323–
1132132312312214–
<33436432>1221231421311323–
1131232313212214–
<33444343>1213221412231133–
1132213312231214–
<33444451>1221312413211323–
1123132321312214–
<33445333>1212231422131133–
1131223313221214–
<33445342>1312212421131323–
1311232321221314–
<33453352>1213122413221133–
1122313322131214–
<33454342>1312122422113133–
1311223322121314–
<33525433>1212321421231133–
1132123312321214–
<33533452>1213212412321133–
1123213321231214–
<33533542>1212312421321133–
1123123321321214–
<33533551>1231212421132133–
1231123321213214–
<33534532>2113123313212124–
1321212421131233–
<33542551>2112313321312124–
1321123321213124–
<33543541>1212132412211333–
1122133323121214–
<33544333>1212213423121133–
1121323331221214–
<33544342>1221213412113233–
1211323312212134–
<33634432>1212123421211333–
1121233332121214–
<33642541>2112133331212124–
1211233321212134–
<34254532>1131222413222114–
1122231422213114–
<34335352>1132212412232114–
1123221421223114–
<34335442>1132122412322114–
1122321422123114–
<34343533>1123122421322114–
1122312422132114–
<34353532>2131122422112314–
1321122422113124–
<34354252>1222113412113224–
1211322412221134–
<34433452>2112321421231124–
1232112421132124–
<34434442>1212221522311133–
1113223312221215–
<34435432>1122123423212114–
1121232432122114–
<34444351>2122113412112324–
1211232421221134–
<34444441>2121132412211234–
1221123421211324–
<34452532>2112213423121124–
2112132431221124–
<34454242>2212113412112234–
1211223422121134–
<34525333>1212212523211133–
1112323321221215–
<34525423>2111323312212125–
1221212521113233–
<34534432>1212122522111333–
1112233322121215–
<35243542>1122221522231114–
1113222412222115–
<35334352>1122212522321114–
1112322421222115–
<35335351>1122122523221114–
1112232422122115–
<35343433>2111322412221125–
1222112521113224–
<35344351>1121222532221114–
1112223422212115–
<35344522>2211132412211225–
1221122522111324–
<35353432>2221113412112225–
1211222522211134–
<35434342>2112212523211124–
2111232421221125–
<35444251>2112122532211124–
2111223422121125–
<35444341>2211123421211225–
2121122522111234–
<41464243>1213131413131133–
1131313313131214–
<42345541>1132131412313114–
1131321413123114–
<42354442>1131312413132114–
1123131421313114–
<42355432>1311312413211314–
1311231421311314–
<42453442>1131213413312114–
1121331431213114–
<42454243>1213113412113314–
1211331412131134–
<42464143>1312113412113134–
1211313413121134–
<42543541>1212131513111333–
1113133313121215–
<43246531>1131132412211414–
1122141423113114–
<43334542>1133211411412124–
1123311411421214–
<43335532>1131221513231114–
1113231412213115–
<43343353>1214112421133114–
1133112421141214–
<43344343>1123113412114214–
1121412431132114–
<43344352>1132113412114124–
1121421431123114–
<43344541>1122131523131114–
1113132413122115–
<43345441>1411212432113114–
1131123421211414–
<43345531>1311132412211315–
1221131513111324–
<43354342>2112141431131124–
1412112421131134–
<43424542>1132121512331114–
1113321412123115–
<43432552>1123121521331114–
1113312412132115–
<43433542>1121321531231114–
1113213412312115–
<43434541>1121231532131114–
1113123413212115–
<43443343>1331112421131215–
1213112521113314–
<43443352>1121312531321114–
1112313421312115–
<43443541>1321121531113124–
1211231521311134–
<43444333>1321113412113125–
1211312513211134–
<43445332>1311212532111314–
1311123421211315–
<43453342>2111313413121125–
1312112521113134–
<43454332>1121214433112114–
1121133412121144–
<43543441>1121213533121114–
1112133431212115–
<43544332>1212113512111334–
1211133412121135–
<43553332>3111213431211215–
1211213531211134–
<44234452>1114221411223115–
1114122411322115–
<44245441>1411122422113115–
1131122522111414–
<44333362>1123112521114214–
1112412421132115–
<44333452>2111412411321125–
1132112521114124–
<44334442>1113122513221115–
1112231522131115–
<44344441>1122113512111424–
1112142431122115–
<44345341>1121132512211144–
1112214423112115–
<44414443>1113212512321115–
1112321521231115–
<44433442>1122121623311114–
1111332412122116–
<44434432>1112213523121115–
1112132531221115–
<44435422>1221113512111325–
1211132512211135–
<44443441>1221121631111324–
1211221623111134–
<44444431>1121123521211144–
1112124432112115–
<44454232>2111214431121125–
1211124421121135–
<44533432>1121212633211114–
1111233421212116–
<44535322>2121113512111235–
1211123521211135–
<44542441>2111133412121126–
1212112621111334–
<44543341>2121121631111234–
1211212632111134–
<45333352>1112221622311115–
1111322512221116–
<45342532>1122112621111424–
1111242421122116–
<45353422>1121122622111144–
1111224422112116–
<45424342>1112212623211115–
1111232521221116–
<45434332>1112122632211115–
1111223522121116–
<45434341>2111132512211126–
1221112621111325–
<45444331>2211113512111226–
1211122622111135–
<45534322>2111212632111125–
2111123521211126–
<52246531>1131131513111414–
1113141413113115–
<53234452>1114311411214115–
1114113411412115–
<53234542>1141121531114114–
1141113412114115–
<53245531>1121141511311144–
1113114414112115–
<53323453>1114121511331115–
1113311512141115–
<53335432>1113113514121115–
1112141531131115–
<53335531>1211141511311135–
1131113512111415–
<53343343>1111431411213116–
1111413411312116–
<53344342>1211311614111134–
1131121631111414–
<53354332>1121131613111144–
1111314413112116–
<53432452>1113121613311115–
1111331512131116–
<53443342>1112131631311115–
1111313513121116–
<53444431>1311113512111316–
1211131613111135–
<54223453>1114112511421115–
1112411521141115–
<54322453>1113211612411115–
1111421511231116–
<54323353>1112311621411115–
1111412511321116–
<54333442>1113112614211115–
1111241521131116–
<54334432>1411112521113116–
1131112621111415–
<54432442>1111312613211116–
1111231621311116–
<54444331>1112113641211115–
1111214531121116–
<54444421>1211114511211136–
1121113612111145–
<54453232>1121121731111144–
1111134412112117–
<54533332>1112121733111115–
1111133512121117–
<55333342>1112211724111115–
1111142511221117–
<55433332>1111221723111116–
1111132612211117–
<55443331>1112112742111115–
1111124521121117–
<55444231>2111114511211127–
1121112721111145–
<55533331>1111212732111116–
1111123621211117–
<55543321>2111113612111127–
1211112721111136–
<64333342>1111311714111116–
1111141611311117–
<65443321>1111211841111116–
1111114611211118–
<65543221>1111121831111117–
1111113712111118–
<34353433>1121322431222114–
1122213422312114–
1322112421131224–
2112231422131124–
<43434442>1112331421213115–
1131212513321114–
1211321512311134–
1231113412113215–
<44434342>1111323412212116–
1121221632311114–
1221211611113234–
1321112521113125–

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