Schismatic family: Difference between revisions

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Garibaldi tempers out the [[~~garischisma~~]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}.

Garibaldi tempers out the [[septischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a very slightly sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}, which also makes it the unique intersection of [[marvel]] and [[aberschismic]], two very notable 7-limit temperament paradigms; however, their errors go in oposite directions as marvel tunes fifths flatter and aberschismic sharper.

[[147edo]] (from 94+53) is a [[patent val]] tuning with 5 and 7 very close to equally out-of-tune in opposite directions s.t only 7/5 and 10/7 are inconsistent in the 9-odd-limit, so might be considered for (EG) a 41-note subset ([[12L 29s]]).

[[Subgroup]]: 2.3.5.7

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=== Cassandra ===

Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup, even though it comes with a much higher complexity.

Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup, even though it comes with a much higher complexity. It finds 11/8 as two commas above 4/3 and 13/8 a minor third above 11/8, or equivalently two commas above 128/81.

Subgroup: 2.3.5.7.11

Line 85:

Line 87:

Comma list: 225/224, 385/384, 2200/2187

~~Mapping:~~ {{~~mapping~~| 1 0 15 25 -33 | 0 1 -8 -14 23 }}

Optimal tunings:

Line 108:

Line 110:

Comma list: 225/224, 275/273, 325/324, 385/384

~~Mapping:~~ {{~~mapping~~| 1 0 15 25 -33 -28 | 0 1 -8 -14 23 20 }}

Optimal tunings:

Line 132:

Line 134:

Comma list: 120/119, 154/153, 225/224, 273/272, 325/324

~~Mapping:~~ {{~~mapping~~| 1 0 15 25 -33 -28 -7 | 0 1 -8 -14 23 20 7 }}

Optimal tunings:

Line 147:

Line 149:

Comma list: 120/119, 154/153, 171/170, 190/189, 225/224, 273/272

~~Mapping:~~ {{~~mapping~~| 1 0 15 25 -33 -28 -7 9 | 0 1 -8 -14 23 20 7 -3 }}

{{Mapping|legend=0| 1 0 15 25 -33 -28 -7 9 | 0 1 -8 -14 23 20 7 -3 }}

Optimal tunings:

Line 162:

Line 164:

Comma list: 225/224, 275/273, 325/324, 375/374, 385/384

~~Mapping:~~ {{~~mapping~~| 1 0 15 25 -33 -28 77 | 0 1 -8 -14 23 20 -46 }}

{{Mapping|legend=0| 1 0 15 25 -33 -28 77 | 0 1 -8 -14 23 20 -46 }}

Optimal tunings:

Line 177:

Line 179:

Comma list: 190/189, 209/208, 225/224, 275/273, 325/324, 375/374

~~Mapping:~~ {{~~mapping~~| 1 0 15 25 -33 -28 77 9 | 0 1 -8 -14 23 20 -46 -3 }}

{{Mapping|legend=0| 1 0 15 25 -33 -28 77 9 | 0 1 -8 -14 23 20 -46 -3 }}

Optimal tunings:

Line 192:

Line 194:

Comma list: 190/189, 209/208, 225/224, 253/252, 275/273, 325/324, 375/374

~~Mapping:~~ {{~~mapping~~| 1 0 15 25 -33 -28 77 9 60 | 0 1 -8 -14 23 20 -46 -3 -35 }}

{{Mapping|legend=0| 1 0 15 25 -33 -28 77 9 60 | 0 1 -8 -14 23 20 -46 -3 -35 }}

Optimal tunings:

Line 207:

Line 209:

Comma list: 170/169, 225/224, 275/273, 325/324, 385/384

~~Mapping:~~ {{~~mapping~~| 1 0 15 25 -33 -28 -72 | 0 1 -8 -14 23 20 48 }}

{{Mapping|legend=0| 1 0 15 25 -33 -28 -72 | 0 1 -8 -14 23 20 48 }}

Optimal tunings:

Line 222:

Line 224:

Comma list: 170/169, 190/189, 209/208, 225/224, 275/273, 325/324

~~Mapping:~~ {{~~mapping~~| 1 0 15 25 -33 -28 -72 9 | 0 1 -8 -14 23 20 48 -3 }}

{{Mapping|legend=0| 1 0 15 25 -33 -28 -72 9 | 0 1 -8 -14 23 20 48 -3 }}

Optimal tunings:

Line 233:

Line 235:

=== Andromeda ===

Andromeda is the second-best extension to garibaldi, which is less complex but also more innacurate. 11/8 is two commas down from [[729/512]], as is 13/8 two commas down from [[27/16]].

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224, 245/242

~~Mapping:~~ {{~~mapping~~| 1 0 15 25 32 | 0 1 -8 -14 -18 }}

Optimal tunings:

Line 260:

Line 264:

Comma list: 100/99, 105/104, 196/195, 245/242

~~Mapping:~~ {{~~mapping~~| 1 0 15 25 32 37 | 0 1 -8 -14 -18 -21 }}

Optimal tunings:

Line 284:

Line 288:

Comma list: 100/99, 105/104, 120/119, 189/187, 196/195

~~Mapping:~~ {{~~mapping~~| 1 0 15 25 32 37 -7 | 0 1 -8 -14 -18 -21 7 }}

Optimal tunings:

Line 299:

Line 303:

Comma list: 100/99, 105/104, 120/119, 133/132, 189/187, 196/195

~~Mapping:~~ {{~~mapping~~| 1 0 15 25 32 37 -7 9 | 0 1 -8 -14 -18 -21 7 -3 }}

{{Mapping|legend=0| 1 0 15 25 32 37 -7 9 | 0 1 -8 -14 -18 -21 7 -3 }}

Optimal tunings:

Line 314:

Line 318:

Comma list: 100/99, 105/104, 154/153, 170/169, 196/195

~~Mapping:~~ {{~~mapping~~| 1 0 15 25 32 37 58 | 0 1 -8 -14 -18 -21 -34 }}

{{Mapping|legend=0| 1 0 15 25 32 37 58 | 0 1 -8 -14 -18 -21 -34 }}

Optimal tunings:

Line 329:

Line 333:

Comma list: 100/99, 105/104, 133/132, 154/153, 170/169, 190/189

~~Mapping:~~ {{~~mapping~~| 1 0 15 25 32 37 58 9 | 0 1 -8 -14 -18 -21 -34 -3 }}

{{Mapping|legend=0| 1 0 15 25 32 37 58 9 | 0 1 -8 -14 -18 -21 -34 -3 }}

Optimal tunings:

Line 344:

Line 348:

Comma list: 85/84, 100/99, 105/104, 119/117, 221/220

~~Mapping:~~ {{~~mapping~~| 1 0 15 25 32 37 12 | 0 1 -8 -14 -18 -21 -5 }}

{{Mapping|legend=0| 1 0 15 25 32 37 12 | 0 1 -8 -14 -18 -21 -5 }}

Optimal tunings:

Line 359:

Line 363:

Comma list: 85/84, 100/99, 105/104, 119/117, 133/132, 153/152

~~Mapping:~~ {{~~mapping~~| 1 0 15 25 32 37 12 9 | 0 1 -8 -14 -18 -21 -5 -3 }}

{{Mapping|legend=0| 1 0 15 25 32 37 12 9 | 0 1 -8 -14 -18 -21 -5 -3 }}

Optimal tunings:

Line 374:

Line 378:

Comma list: 99/98, 176/175, 3125/3087

~~Mapping:~~ {{~~mapping~~| 1 0 15 25 51 | 0 1 -8 -14 -30 }}

Optimal tunings:

Line 393:

Line 397:

Comma list: 99/98, 176/175, 275/273, 847/845

~~Mapping:~~ {{~~mapping~~| 1 0 15 25 51 56 | 0 1 -8 -14 -30 -33 }}

Optimal tunings:

Line 412:

Line 416:

Comma list: 99/98, 120/119, 176/175, 275/273, 442/441

~~Mapping:~~ {{~~mapping~~| 1 0 15 25 51 56 -7 | 0 1 -8 -14 -30 -33 7 }}

Optimal tunings:

Line 427:

Line 431:

Comma list: 99/98, 120/119, 176/175, 190/189, 209/208, 247/245

~~Mapping:~~ {{~~mapping~~| 1 0 15 25 51 56 -7 9 | 0 1 -8 -14 -30 -33 7 -3 }}

{{Mapping|legend=0| 1 0 15 25 51 56 -7 9 | 0 1 -8 -14 -30 -33 7 -3 }}

Optimal tunings:

Line 444:

Line 448:

Comma list: 225/224, 243/242, 3125/3087

~~Mapping:~~ {{~~mapping~~| 1 1 7 11 2 | 0 2 -16 -28 5 }}

: mapping generators: ~2, ~11/9

Line 460:

Line 464:

Comma list: 225/224, 243/242, 325/324, 640/637

~~Mapping:~~ {{~~mapping~~| 1 1 7 11 2 -8 | 0 2 -16 -28 5 40 }}

Optimal tunings:

Line 475:

Line 479:

Comma list: 121/120, 225/224, 3125/3087

~~Mapping:~~ {{~~mapping~~| 1 0 15 25 9 | 0 2 -16 -28 -7 }}

: mapping generators: ~2, ~110/63

Line 491:

Line 495:

Comma list: 121/120, 169/168, 225/224, 275/273

~~Mapping:~~ {{~~mapping~~| 1 0 15 25 9 14 | 0 2 -16 -28 -7 -13 }}

Optimal tunings:

Line 506:

Line 510:

Comma list: 225/224, 1331/1323, 3125/3087

~~Mapping:~~ {{~~mapping~~| 1 2 -1 -3 0 | 0 -3 24 42 25 }}

: mapping generators: ~2, ~11/10

Line 522:

Line 526:

Comma list: 225/224, 275/273, 847/845, 1331/1323

~~Mapping:~~ {{~~mapping~~| 1 2 -1 -3 0 -1 | 0 -3 24 42 25 34 }}

Optimal tunings:

Line 572:

Line 576:

Comma list: 385/384, 3388/3375, 4375/4374

~~Mapping:~~ {{~~mapping~~| 1 0 15 -59 51 | 0 1 -8 39 -30 }}

Optimal tunings:

Line 591:

Line 595:

Comma list: 352/351, 385/384, 625/624, 729/728

~~Mapping:~~ {{~~mapping~~| 1 0 15 -59 51 56 | 0 1 -8 39 -30 -33 }}

Optimal tunings:

Line 610:

Line 614:

Comma list: 352/351, 385/384, 561/560, 625/624, 729/728

~~Mapping:~~ {{~~mapping~~| 1 0 15 -59 51 56 -91 | 0 1 -8 39 -30 -33 60 }}

{{Mapping|legend=0| 1 0 15 -59 51 56 -91 | 0 1 -8 39 -30 -33 60 }}

Optimal tunings:

Line 629:

Line 633:

Comma list: 169/168, 325/324, 385/384, 3146/3125

~~Mapping:~~ {{~~mapping~~| 1 0 15 -59 51 -28 | 0 1 -8 39 -30 20 }}

Optimal tunings:

Line 644:

Line 648:

Comma list: 169/168, 273/272, 325/324, 385/384, 3146/3125

~~Mapping:~~ {{~~mapping~~| 1 0 15 -59 51 -28 -91 | 0 1 -8 39 -30 20 60 }}

{{Mapping|legend=0| 1 0 15 -59 51 -28 -91 | 0 1 -8 39 -30 20 60 }}

Optimal tunings:

Line 659:

Line 663:

Comma list: 169/168, 273/272, 286/285, 325/324, 385/384, 627/625

~~Mapping:~~ {{~~mapping~~| 1 0 15 -59 51 -28 -91 9 | 0 1 -8 39 -30 20 60 -3 }}

{{Mapping|legend=0| 1 0 15 -59 51 -28 -91 9 | 0 1 -8 39 -30 20 60 -3 }}

Optimal tunings:

Line 676:

Line 680:

Comma list: 540/539, 4375/4374, 32805/32768

~~Mapping:~~ {{~~mapping~~| 1 0 15 -59 135 | 0 1 -8 39 -83 }}

Optimal tunings:

Line 695:

Line 699:

Comma list: 540/539, 625/624, 729/728, 2200/2197

~~Mapping:~~ {{~~mapping~~| 1 0 15 -59 135 56 | 0 1 -8 39 -83 -33 }}

Optimal tunings:

Line 714:

Line 718:

Comma list: 375/374, 540/539, 625/624, 729/728, 2200/2197

~~Mapping:~~ {{~~mapping~~| 1 0 15 -59 135 56 -91 | 0 1 -8 39 -83 -33 60 }}

{{Mapping|legend=0| 1 0 15 -59 135 56 -91 | 0 1 -8 39 -83 -33 60 }}

Optimal tunings:

Line 735:

Line 739:

Comma list: 441/440, 4375/4374, 32805/32768

~~Mapping:~~ {{~~mapping~~| 1 0 15 -59 -136 | 0 1 -8 39 88 }}

Optimal tunings:

Line 754:

Line 758:

Comma list: 441/440, 625/624, 729/728, 3584/3575

~~Mapping:~~ {{~~mapping~~| 1 0 15 -59 -136 56 | 0 1 -8 39 88 -33 }}

Optimal tunings:

Line 773:

Line 777:

Comma list: 441/440, 595/594, 625/624, 729/728, 2880/2873

~~Mapping:~~ {{~~mapping~~| 1 0 15 -59 -136 56 -91 | 0 1 -8 39 88 -33 60 }}

{{Mapping|legend=0| 1 0 15 -59 -136 56 -91 | 0 1 -8 39 88 -33 60 }}

Optimal tunings:

Line 792:

Line 796:

Comma list: 364/363, 441/440, 4375/4374, 32805/32768

~~Mapping:~~ {{~~mapping~~| 1 0 15 -59 -136 -215 | 0 1 -8 39 88 138 }}

Optimal tunings:

Line 807:

Line 811:

Comma list: 364/363, 441/440, 595/594, 1156/1155, 32805/32768

~~Mapping:~~ {{~~mapping~~| 1 0 15 -59 -136 -215 -91 | 0 1 -8 39 88 138 60 }}

{{Mapping|legend=0| 1 0 15 -59 -136 -215 -91 | 0 1 -8 39 88 138 60 }}

Optimal tunings:

Line 824:

Line 828:

Comma list: 3025/3024, 4375/4374, 32805/32768

~~Mapping:~~ {{~~mapping~~| 2 0 30 -118 -85 | 0 1 -8 39 29 }}

: mapping generators: ~99/70, ~3

Line 840:

Line 844:

Comma list: 625/624, 729/728, 1575/1573, 4096/4095

~~Mapping:~~ {{~~mapping~~| 2 0 30 -118 -85 112 | 0 1 -8 39 29 -33 }}

Optimal tunings:

Line 855:

Line 859:

Comma list: 625/624, 729/728, 1089/1088, 1225/1224, 2880/2873

~~Mapping:~~ {{~~mapping~~| 2 0 30 -118 -85 112 -182 | 0 1 -8 39 29 -33 60 }}

{{Mapping|legend=0| 2 0 30 -118 -85 112 -182 | 0 1 -8 39 29 -33 60 }}

Optimal tunings:

Line 870:

Line 874:

Comma list: 1716/1715, 2080/2079, 3025/3024, 32805/32768

~~Mapping:~~ {{~~mapping~~| 2 0 30 -118 -85 -243 | 0 1 -8 39 29 79 }}

Optimal tunings:

Line 885:

Line 889:

Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 32805/32768

~~Mapping:~~ {{~~mapping~~| 2 0 30 -118 -85 -243 -182 | 0 1 -8 39 29 79 60 }}

{{Mapping|legend=0| 2 0 30 -118 -85 -243 -182 | 0 1 -8 39 29 79 60 }}

Optimal tunings:

Line 900:

Line 904:

Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 1540/1539, 4875/4864

~~Mapping:~~ {{~~mapping~~| 2 0 30 -118 -85 -243 -182 -169 | 0 1 -8 39 29 79 60 56 }}

{{Mapping|legend=0| 2 0 30 -118 -85 -243 -182 -169 | 0 1 -8 39 29 79 60 56 }}

Optimal tunings:

Line 915:

Line 919:

Comma list: 3025/3024, 4225/4224, 4375/4374, 32805/32768

~~Mapping:~~ {{~~mapping~~| 4 0 60 -236 -170 -131 | 0 1 -8 39 29 23 }}

: mapping generators: ~208/175, ~3

Line 955:

Line 959:

Comma list: 126/125, 176/175, 32805/32768

~~Mapping:~~ {{~~mapping~~| 1 0 15 44 70 | 0 1 -8 -26 -42 }}

Optimal tunings:

Line 970:

Line 974:

Comma list: 126/125, 176/175, 196/195, 5445/5408

~~Mapping:~~ {{~~mapping~~| 1 0 15 44 70 75 | 0 1 -8 -26 -42 -45 }}

Optimal tunings:

Line 985:

Line 989:

Comma list: 126/125, 176/175, 196/195, 256/255, 2904/2873

~~Mapping:~~ {{~~mapping~~| 1 0 15 44 70 75 -7 | 0 1 -8 -26 -42 -45 7 }}

Optimal tunings:

Line 1,000:

Line 1,004:

Comma list: 126/125, 171/170, 176/175, 196/195, 209/208, 324/323

~~Mapping:~~ {{~~mapping~~| 1 0 15 44 70 75 -7 9 | 0 1 -8 -26 -42 -45 7 -3 }}

{{Mapping|legend=0| 1 0 15 44 70 75 -7 9 | 0 1 -8 -26 -42 -45 7 -3 }}

Optimal tunings:

Line 1,015:

Line 1,019:

Comma list: 126/125, 176/175, 729/728, 1287/1280

~~Mapping:~~ {{~~mapping~~| 1 0 15 44 70 -47 | 0 1 -8 -26 -42 32 }}

Optimal tunings:

Line 1,030:

Line 1,034:

Comma list: 126/125, 245/242, 896/891

~~Mapping:~~ {{~~mapping~~| 1 0 15 44 51 | 0 1 -8 -26 -30 }}

Optimal tunings:

Line 1,045:

Line 1,049:

Comma list: 126/125, 196/195, 245/242, 832/825

~~Mapping:~~ {{~~mapping~~| 1 0 15 44 51 75 | 0 1 -8 -26 -30 -45 }}

Optimal tunings:

Line 1,060:

Line 1,064:

Comma list: 105/104, 126/125, 245/242, 352/351

~~Mapping:~~ {{~~mapping~~| 1 0 15 44 51 56 | 0 1 -8 -26 -30 -33 }}

Optimal tunings:

Line 1,094:

Line 1,098:

Comma list: 385/384, 19712/19683, 78125/77616

~~Mapping:~~ {{~~mapping~~| 1 0 15 109 -117 | 0 1 -8 -67 76 }}

Optimal tunings:

Line 1,109:

Line 1,113:

Comma list: 325/324, 385/384, 2200/2197, 19712/19683

~~Mapping:~~ {{~~mapping~~| 1 0 15 109 -117 -28 | 0 1 -8 -67 76 20 }}

Optimal tunings:

Line 1,153:

Line 1,157:

Comma list: 441/440, 3136/3125, 8019/8000

~~Mapping:~~ {{~~mapping~~| 2 0 30 69 102 | 0 1 -8 -20 -30 }}

Optimal tunings:

Line 1,168:

Line 1,172:

Comma list: 441/440, 729/728, 1001/1000, 3136/3125

~~Mapping:~~ {{~~mapping~~| 2 0 30 69 102 -75 | 0 1 -8 -20 -30 26 }}

Optimal tunings:

Line 1,183:

Line 1,187:

Comma list: 289/288, 441/440, 561/560, 729/728, 3136/3125

~~Mapping:~~ {{~~mapping~~| 2 0 30 69 102 -75 5 | 0 1 -8 -20 -30 26 1 }}

Optimal tunings:

Line 1,198:

Line 1,202:

Comma list: 351/350, 364/363, 441/440, 3136/3125

~~Mapping:~~ {{~~mapping~~| 2 0 30 69 102 131 | 0 1 -8 -20 -30 -39 }}

Optimal tunings:

Line 1,213:

Line 1,217:

Comma list: 221/220, 289/288, 351/350, 441/440, 3136/3125

~~Mapping:~~ {{~~mapping~~| 2 0 30 69 102 131 5 | 0 1 -8 -20 -30 -39 1 }}

{{Mapping|legend=0| 2 0 30 69 102 131 5 | 0 1 -8 -20 -30 -39 1 }}

Optimal tunings:

Line 1,248:

Line 1,252:

Comma list: 385/384, 9801/9800, 14641/14580

~~Mapping:~~ {{~~mapping~~| 2 1 22 -15 8 | 0 2 -16 19 -1 }}

Optimal tunings:

Line 1,263:

Line 1,267:

Comma list: 352/351, 385/384, 729/728, 1575/1573

~~Mapping:~~ {{~~mapping~~| 2 1 22 -15 8 15 | 0 2 -16 19 -1 -7 }}

Optimal tunings:

Line 1,278:

Line 1,282:

Comma list: 170/169, 289/288, 352/351, 385/384, 561/560

~~Mapping:~~ {{~~mapping~~| 2 1 22 -15 8 15 6 | 0 2 -16 19 -1 -7 2 }}

Optimal tunings:

Line 1,293:

Line 1,297:

Comma list: 325/324, 385/384, 1573/1568, 14641/14580

~~Mapping:~~ {{~~mapping~~| 2 1 22 -15 8 -36 | 0 2 -16 19 -1 40 }}

Optimal tunings:

Line 1,308:

Line 1,312:

Comma list: 289/288, 325/324, 385/384, 442/441, 14641/14580

~~Mapping:~~ {{~~mapping~~| 2 1 22 -15 8 -36 6 | 0 2 -16 19 -1 40 2 }}

Optimal tunings:

Line 1,343:

Line 1,347:

Comma list: 243/242, 245/242, 385/384

~~Mapping:~~ {{~~mapping~~| 1 1 7 -1 2 | 0 2 -16 13 5 }}

Optimal tunings:

Line 1,358:

Line 1,362:

Comma list: 105/104, 144/143, 243/242, 245/242

~~Mapping:~~ {{~~mapping~~| 1 1 7 -1 2 4 | 0 2 -16 13 5 -1 }}

Optimal tunings:

Line 1,395:

Line 1,399:

Comma list: 540/539, 5632/5625, 8019/8000

~~Mapping:~~ {{~~mapping~~| 1 0 15 -17 51 | 0 2 -16 25 -60 }}

Optimal tunings:

Line 1,410:

Line 1,414:

Comma list: 351/350, 540/539, 676/675, 4096/4095

~~Mapping:~~ {{~~mapping~~| 1 0 15 -17 51 14 | 0 2 -16 25 -60 -13 }}

Optimal tunings:

Line 1,425:

Line 1,429:

Comma list: 351/350, 442/441, 561/560, 676/675, 4096/4095

~~Mapping:~~ {{~~mapping~~| 1 0 15 -17 51 14 -49 | 0 2 -16 25 -60 -13 67 }}

{{Mapping|legend=0| 1 0 15 -17 51 14 -49 | 0 2 -16 25 -60 -13 67 }}

Optimal tunings:

Line 1,440:

Line 1,444:

Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 4096/4095

~~Mapping:~~ {{~~mapping~~| 1 0 15 -17 51 14 -49 9 | 0 2 -16 25 -60 -13 67 -6 }}

{{Mapping|legend=0| 1 0 15 -17 51 14 -49 9 | 0 2 -16 25 -60 -13 67 -6 }}

Optimal tunings:

Line 1,455:

Line 1,459:

Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 736/735, 4096/4095

~~Mapping:~~ {{~~mapping~~| 1 0 15 -17 51 14 -49 9 -24 | 0 2 -16 25 -60 -13 67 -6 36 }}

{{Mapping|legend=0| 1 0 15 -17 51 14 -49 9 -24 | 0 2 -16 25 -60 -13 67 -6 36 }}

Optimal tunings:

Line 1,499:

Line 1,503:

Comma list: 441/440, 4375/4356, 32805/32768

~~Mapping:~~ {{~~mapping~~| 3 0 45 94 134 | 0 1 -8 -18 -26 }}

Optimal tunings:

Line 1,514:

Line 1,518:

Comma list: 364/363, 441/440, 625/624, 13720/13689

~~Mapping:~~ {{~~mapping~~| 3 0 45 94 134 168 | 0 1 -8 -18 -26 -33 }}

Optimal tunings:

Line 1,529:

Line 1,533:

Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619

~~Mapping:~~ {{~~mapping~~| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }}

{{Mapping|legend=0| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }}

Optimal tunings:

Line 1,546:

Line 1,550:

Comma list: 540/539, 32805/32768, 137781/137500

~~Mapping:~~ {{~~mapping~~| 3 0 45 94 -137 | 0 1 -8 -18 31 }}

Optimal tunings:

Line 1,561:

Line 1,565:

Comma list: 540/539, 729/728, 4096/4095, 31250/31213

~~Mapping:~~ {{~~mapping~~| 3 0 45 94 -137 -103 | 0 1 -8 -18 31 24 }}

Optimal tunings:

Line 1,576:

Line 1,580:

Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095

~~Mapping:~~ {{~~mapping~~| 3 0 45 94 -137 -103 -2 | 0 1 -8 -18 31 24 3 }}

{{Mapping|legend=0| 3 0 45 94 -137 -103 -2 | 0 1 -8 -18 31 24 3 }}

Optimal tunings:

Line 1,593:

Line 1,597:

Comma list: 9801/9800, 32805/32768, 151263/151250

~~Mapping:~~ {{~~mapping~~| 6 0 90 188 287 | 0 1 -8 -18 -28 }}

: mapping generators: ~55/49, ~3

Line 1,609:

Line 1,613:

Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375

~~Mapping:~~ {{~~mapping~~| 6 0 90 188 287 355 | 0 1 -8 -18 -28 -35 }}

Optimal tunings:

Line 1,628:

Line 1,632:

Comma list: 3025/3024, 32805/32768, 102487/102400

~~Mapping:~~ {{~~mapping~~| 3 0 45 94 8 | 0 2 -16 -36 1 }}

: mapping generators: ~63/50, ~693/400

Line 1,644:

Line 1,648:

Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712

~~Mapping:~~ {{~~mapping~~| 3 0 45 94 8 42 | 0 2 -16 -36 1 -13 }}

Optimal tunings:

Line 1,659:

Line 1,663:

Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264

~~Mapping:~~ {{~~mapping~~| 3 0 45 94 8 42 -2 | 0 2 -16 -36 1 -13 6 }}

Optimal tunings:

Line 1,694:

Line 1,698:

Comma list: 385/384, 14700/14641, 19712/19683

~~Mapping:~~ {{~~mapping~~| 3 0 45 -32 8 | 0 2 -16 17 1 }}

Optimal tunings:

Line 1,709:

Line 1,713:

Comma list: 364/363, 385/384, 676/675, 19712/19683

~~Mapping:~~ {{~~mapping~~| 3 0 45 -32 8 42 | 0 2 -16 17 1 -13 }}

Optimal tunings:

Line 1,743:

Line 1,747:

Comma list: 245/242, 686/675, 896/891

~~Mapping:~~ {{~~mapping~~| 1 2 -1 1 0 | 0 -3 24 13 25 }}

Optimal tunings:

Line 1,758:

Line 1,762:

Comma list: 91/90, 169/168, 245/242, 896/891

~~Mapping:~~ {{~~mapping~~| 1 2 -1 1 0 3 | 0 -3 24 13 25 5 }}

Optimal tunings:

Line 1,769:

Line 1,773:

== Tertiaschis ==

Named by [[Xenllium]] in 2021, tertiaschis may be described as {{nowrap| 94 & 159 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 1071875/1062882 for prime 7. See also [[~~daemotertiachis~~]] which is made using every other generator of tertiaschis.

Named by [[Xenllium]] in 2021, tertiaschis may be described as {{nowrap| 94 & 159 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 1071875/1062882 for prime 7. See also [[daemotertiaschis]] which is made using every other generator of tertiaschis.

[[Subgroup]]: 2.3.5.7

Line 1,792:

Line 1,796:

Comma list: 385/384, 4000/3993, 19712/19683

~~Mapping:~~ {{~~mapping~~| 1 2 -1 10 0 | 0 -3 24 -52 25 }}

Optimal tunings:

Line 1,807:

Line 1,811:

Comma list: 325/324, 385/384, 1575/1573, 10985/10976

~~Mapping:~~ {{~~mapping~~| 1 2 -1 10 0 12 | 0 -3 24 -52 25 -60 }}

Optimal tunings:

Line 1,822:

Line 1,826:

Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976

~~Mapping:~~ {{~~mapping~~| 1 2 -1 10 0 12 -2 | 0 -3 24 -52 25 -60 44 }}

Optimal tunings:

Line 1,833:

Line 1,837:

== Countertertiaschis ==

Named by [[Flora Canou]] in 2021, ~~Countertertiaschis~~ may be described as {{nowrap| 159 & 224 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 244140625/243045684 for prime 7.

Named by [[Flora Canou]] in 2021, countertertiaschis may be described as {{nowrap| 159 & 224 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 244140625/243045684 for prime 7.

[[Subgroup]]: 2.3.5.7

Line 1,856:

Line 1,860:

Comma list: 3025/3024, 4000/3993, 32805/32768

~~Mapping:~~ {{~~mapping~~| 1 2 -1 -12 0 | 0 -3 24 107 25 }}

Optimal tunings:

Line 1,871:

Line 1,875:

Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976

~~Mapping:~~ {{~~mapping~~| 1 2 -1 -12 0 -10 | 0 -3 24 107 25 99 }}

Optimal tunings:

Line 1,882:

Line 1,886:

== Quadrant ==

Named by [[Xenllium]] in 2021, quadrant tempers out 390625/388962, the [[dimcomp comma]], and maps [[25/21]] to the 1/4-octave period. It may be described as the {{nowrap| 12 & 212 }} temperament; its ploidacot is tetraploid monocot. Just as [[#Term|term]] equates the syntonic~Pythagorean comma ~~with~~ three [[marvel comma]]s, quadrant ~~equates~~ the ~~syntonic~Pythagorean~~ comma ~~with~~ four. A [[septimal comma]] is then found as a stack of five marvel commas.

Named by [[Xenllium]] in 2021, quadrant tempers out 390625/388962, the [[dimcomp comma]], and maps [[25/21]] to the 1/4-octave period. It may be described as the {{nowrap| 12 & 212 }} temperament; its ploidacot is tetraploid monocot. Just as [[#Term|term]] equates the syntonic and Pythagorean commas for a generic comma step, and splits it into three [[marvel comma]]s, quadrant splits the same generic comma step into four. A [[septimal comma]] is then found as a stack of five marvel commas.

[[Subgroup]]: 2.3.5.7

Line 1,906:

Line 1,910:

Comma list: 1375/1372, 6250/6237, 32805/32768

~~Mapping:~~ {{~~mapping~~| 4 0 60 119 185 | 0 1 -8 -17 -27 }}

Optimal tunings:

Line 1,921:

Line 1,925:

Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647

~~Mapping:~~ {{~~mapping~~| 4 0 60 119 185 224 | 0 1 -8 -17 -27 -33 }}

Optimal tunings:

Line 1,932:

Line 1,936:

== Sesquiquartififths ==

Sesquiquartififths tempers out 2401/2400, the [[breedsma]], and may be described as the {{nowrap| 41 & 171 }} temperament. It splits the fifth into four; its [[ploidacot]] is thus tetracot.

Sesquiquartififths tempers out 2401/2400, the [[breedsma]], and may be described as the {{nowrap| 41 & 171 }} temperament. It splits the [[3/2|perfect fifth]] into four; its [[ploidacot]] is thus tetracot.

The name might have been a portmanteau of ''sesquiquart'' and ''quartififths'', each meaning dividing 3/2 into four.

[[Subgroup]]: 2.3.5.7

Line 1,962:

Line 1,968:

Comma list: 243/242, 441/440, 16384/16335

~~Mapping:~~ {{~~mapping~~| 1 1 7 5 2 | 0 4 -32 -15 10 }}

Optimal tunings:

Line 1,977:

Line 1,983:

Comma list: 243/242, 364/363, 441/440, 3584/3575

~~Mapping:~~ {{~~mapping~~| 1 1 7 5 2 -2 | 0 4 -32 -15 10 39 }}

Optimal tunings:

Line 1,992:

Line 1,998:

Comma list: 243/242, 256/255, 273/272, 364/363, 441/440

~~Mapping:~~ {{~~mapping~~| 1 1 7 5 2 -2 0 | 0 4 -32 -15 10 39 28 }}

Optimal tunings:

Line 2,007:

Line 2,013:

Comma list: 171/170, 243/242, 256/255, 273/272, 324/323, 441/440

~~Mapping:~~ {{~~mapping~~| 1 1 7 5 2 -2 0 6 | 0 4 -32 -15 10 39 28 -12 }}

{{Mapping|legend=0| 1 1 7 5 2 -2 0 6 | 0 4 -32 -15 10 39 28 -12 }}

Optimal tunings:

Line 2,022:

Line 2,028:

Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575

~~Mapping:~~ {{~~mapping~~| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }}

Optimal tunings:

Line 2,037:

Line 2,043:

Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594

~~Mapping:~~ {{~~mapping~~| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }}

{{Mapping|legend=0| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }}

Optimal tunings:

Line 2,052:

Line 2,058:

Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594

~~Mapping:~~ {{~~mapping~~| 1 1 7 5 2 -2 -6 6 -6 | 0 4 -32 -15 10 39 69 -12 72 }}

{{Mapping|legend=0| 1 1 7 5 2 -2 -6 6 -6 | 0 4 -32 -15 10 39 69 -12 72 }}

Optimal tunings:

Line 2,067:

Line 2,073:

Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625

~~Mapping:~~ {{~~mapping~~| 1 1 7 5 2 -2 13 | 0 4 -32 -15 10 39 -61 }}

Optimal tunings:

Line 2,082:

Line 2,088:

Comma list: 221/220, 243/242, 361/360, 364/363, 441/440, 456/455

~~Mapping:~~ {{~~mapping~~| 1 1 7 5 2 -2 13 6 | 0 4 -32 -15 10 39 -61 -12 }}

{{Mapping|legend=0| 1 1 7 5 2 -2 13 6 | 0 4 -32 -15 10 39 -61 -12 }}

Optimal tunings:

Line 2,097:

Line 2,103:

Comma list: 221/220, 243/242, 276/275, 323/322, 361/360, 364/363, 441/440

~~Mapping:~~ {{~~mapping~~| 1 1 7 5 2 -2 13 6 13 | 0 4 -32 -15 10 39 -61 -12 -58 }}

{{Mapping|legend=0| 1 1 7 5 2 -2 13 6 13 | 0 4 -32 -15 10 39 -61 -12 -58 }}

Optimal tunings:

Line 2,112:

Line 2,118:

Comma list: 2401/2400, 9801/9800, 32805/32768

~~Mapping:~~ {{~~mapping~~| 2 2 14 10 23 | 0 4 -32 -15 -55 }}

: mapping generators: ~99/70, ~448/405

Line 2,124:

Line 2,130:

== Tsaharuk ==

[[File:Tsaharuk.jpg|alt=Tsaharuk.jpg|thumb|Parametric horagram of the tsaharuk temperament.]]

Tsaharuk tempers out 420175/419904, the [[wizma]], and may be described as the {{nowrap| 77 & 94 }} temperament. It is generated by a slightly flat neutral second of [[~]][[13/12]], five of which make the [[3/2|perfect fifth]], so its [[ploidacot]] is pentacot.

It was proposed by [[Jacques Dudon]] based on [[Julien Jalaleddine Weiss]] qanun tunings, which has [[mos]] of size 43, 60, 77 and 94 notes.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_100718.html Yahoo! Tuning Group | ''A temperament for Maqam'']</ref>

It restricts to the [[2.3.7 subgroup]] as [[no-fives subgroup temperaments #Navy|navy]]. This relates tsaharuk to [[aberschismic temperaments #Quanic|quanic]], which favors a sharper fifth and works better in the higher limits, but which also has mos of size 43, 60, 77 and 94 notes. [[Bleu]] has, like tsaharuk, a generator of 1/5 fifth and makes for an instructive comparison. [[Edo]]s that support tsaharuk include [[77edo]], [[94edo]], [[171edo]] and [[248edo]], and edos supporting quanic are [[94edo]], [[111edo]] and [[205edo]]. Quanic and tsaharuk therefore become identical if 94edo is the tuning.

[[Subgroup]]: 2.3.5.7

Line 2,150:

Line 2,161:

Comma list: 385/384, 1331/1323, 19712/19683

~~Mapping:~~ {{~~mapping~~| 1 1 7 0 1 | 0 5 -40 24 21 }}

Optimal tunings:

Line 2,165:

Line 2,176:

Comma list: 352/351, 385/384, 729/728, 1331/1323

~~Mapping:~~ {{~~mapping~~| 1 1 7 0 1 3 | 0 5 -40 24 21 6 }}

Optimal tunings:

Line 2,176:

Line 2,187:

== Quanharuk ==

Quanharuk tempers out 16875/16807, the [[mirkwai]] comma, and may be described as the {{nowrap| 41 & 183 }} temperament. The generator is a slightly flat major third of [[~]][[56/45]], five of which make the [[3/1|3rd~~]] [[~~harmonic]], so the [[ploidacot]] of this temperament is alpha-pentacot. [[224edo]] makes for a recommendable tuning.

Quanharuk tempers out 16875/16807, the [[mirkwai]] comma, and may be described as the {{nowrap| 41 & 183 }} temperament. The generator is a slightly flat major third of [[~]][[56/45]], five of which make the [[3/1|3rd harmonic]], so the [[ploidacot]] of this temperament is alpha-pentacot. [[224edo]] makes for a recommendable tuning.

[[Subgroup]]: 2.3.5.7

Line 2,200:

Line 2,211:

Comma list: 540/539, 1375/1372, 32805/32768

~~Mapping:~~ {{~~mapping~~| 1 0 15 12 -7 | 0 5 -40 -29 33 }}

Optimal tunings:

Line 2,215:

Line 2,226:

Comma list: 540/539, 729/728, 1375/1372, 4096/4095

~~Mapping:~~ {{~~mapping~~| 1 0 15 12 -7 -15 | 0 5 -40 -29 33 59 }}

Optimal tunings:

Line 2,250:

Line 2,261:

Comma list: 1375/1372, 4375/4356, 32805/32768

~~Mapping:~~ {{~~mapping~~| 1 2 -1 -4 -7 | 0 -5 40 82 126 }}

Optimal tunings:

Line 2,265:

Line 2,276:

Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647

~~Mapping:~~ {{~~mapping~~| 1 2 -1 -4 -7 -9 | 0 -5 40 82 126 153 }}

Optimal tunings:

Line 2,280:

Line 2,291:

Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619

~~Mapping:~~ {{~~mapping~~| 1 2 -1 -4 -7 -9 5 | 0 -5 40 82 126 153 -11 }}

{{Mapping|legend=0| 1 2 -1 -4 -7 -9 5 | 0 -5 40 82 126 153 -11 }}

Optimal tunings:

Line 2,295:

Line 2,306:

Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971

~~Mapping:~~ {{~~mapping~~| 1 2 -1 -4 -7 -9 5 4 | 0 -5 40 82 126 153 -11 3 }}

{{Mapping|legend=0| 1 2 -1 -4 -7 -9 5 4 | 0 -5 40 82 126 153 -11 3 }}

Optimal tunings:

Line 2,329:

Line 2,340:

Comma list: 441/440, 32805/32768, 1953125/1951488

~~Mapping:~~ {{~~mapping~~| 1 2 -1 -5 -8 | 0 -5 40 94 138 }}

Optimal tunings:

Line 2,344:

Line 2,355:

Comma list: 364/363, 441/440, 32805/32768, 109512/109375

~~Mapping:~~ {{~~mapping~~| 1 2 -1 -5 -8 -11 | 0 -5 40 94 138 177 }}

Optimal tunings:

Line 2,359:

Line 2,370:

Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768

~~Mapping:~~ {{~~mapping~~| 1 2 -1 -5 -8 -11 5 | 0 -5 40 94 138 177 -11 }}

{{Mapping|legend=0| 1 2 -1 -5 -8 -11 5 | 0 -5 40 94 138 177 -11 }}

Optimal tunings:

Line 2,374:

Line 2,385:

Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859

~~Mapping:~~ {{~~mapping~~| 1 2 -1 -5 -8 -11 5 4 | 0 -5 40 94 138 177 -11 3 }}

{{Mapping|legend=0| 1 2 -1 -5 -8 -11 5 4 | 0 -5 40 94 138 177 -11 3 }}

Optimal tunings:

Line 2,389:

Line 2,400:

Comma list: 5632/5625, 8019/8000, 151263/151250

~~Mapping:~~ {{~~mapping~~| 1 2 -1 -5 -9 | 0 -5 40 94 150 }}

Optimal tunings:

Line 2,404:

Line 2,415:

Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000

~~Mapping:~~ {{~~mapping~~| 1 2 -1 -5 -9 -11 | 0 -5 40 94 150 177 }}

Optimal tunings:

Line 2,419:

Line 2,430:

Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750

~~Mapping:~~ {{~~mapping~~| 1 2 -1 -5 -9 -11 5 | 0 -5 40 94 150 177 -11 }}

{{Mapping|legend=0| 1 2 -1 -5 -9 -11 5 | 0 -5 40 94 150 177 -11 }}

Optimal tunings:

Line 2,434:

Line 2,445:

Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700

~~Mapping:~~ {{~~mapping~~| 1 2 -1 -5 -9 -11 5 4 | 0 -5 40 94 150 177 -11 3 }}

{{Mapping|legend=0| 1 2 -1 -5 -9 -11 5 4 | 0 -5 40 94 150 177 -11 3 }}

Optimal tunings:

Line 2,449:

Line 2,460:

Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436

~~Mapping:~~ {{~~mapping~~| 1 2 -1 -5 -9 14 | 0 -5 40 94 150 -124 }}

Optimal tunings:

Line 2,464:

Line 2,475:

Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157

~~Mapping:~~ {{~~mapping~~| 1 2 -1 -5 -9 14 5 | 0 -5 40 94 150 -124 -11 }}

{{Mapping|legend=0| 1 2 -1 -5 -9 14 5 | 0 -5 40 94 150 -124 -11 }}

Optimal tunings:

Line 2,479:

Line 2,490:

Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137

~~Mapping:~~ {{~~mapping~~| 1 2 -1 -5 -9 14 5 4 | 0 -5 40 94 150 -124 -11 3 }}

{{Mapping|legend=0| 1 2 -1 -5 -9 14 5 4 | 0 -5 40 94 150 -124 -11 3 }}

Optimal tunings:

Line 2,514:

Line 2,525:

Comma list: 441/440, 4000/3993, 235298/234375

~~Mapping:~~ {{~~mapping~~| 1 2 -1 -1 0 | 0 -6 48 55 50 }}

Optimal tunings:

Line 2,529:

Line 2,540:

Comma list: 364/363, 441/440, 676/675, 10985/10976

~~Mapping:~~ {{~~mapping~~| 1 2 -1 -1 0 1 | 0 -6 48 55 50 39 }}

Optimal tunings:

Line 2,564:

Line 2,575:

Comma list: 3025/3024, 24057/24010, 32805/32768

~~Mapping:~~ {{~~mapping~~| 7 0 105 -56 -120 | 0 1 -8 7 13 }}

Optimal tunings:

Line 2,579:

Line 2,590:

Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024

~~Mapping:~~ {{~~mapping~~| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }}

Optimal tunings:

Line 2,614:

Line 2,625:

Comma list: 9801/9800, 32805/32768, 46656/46585

~~Mapping:~~ {{~~mapping~~| 8 0 120 -117 15 | 0 1 -8 11 1 }}

Optimal tunings:

Line 2,629:

Line 2,640:

Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655

~~Mapping:~~ {{~~mapping~~| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }}

Optimal tunings:

Line 2,664:

Line 2,675:

Comma list: 540/539, 32805/32768, 42875/42592

~~Mapping:~~ {{~~mapping~~| 9 0 135 11 131 | 0 1 -8 1 -7 }}

Optimal tunings:

Line 2,679:

Line 2,690:

Comma list: 540/539, 729/728, 4096/4095, 16807/16731

~~Mapping:~~ {{~~mapping~~| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }}

Optimal tunings:

Line 2,714:

Line 2,725:

Comma list: 540/539, 15488/15435, 32805/32768

~~Mapping:~~ {{~~mapping~~| 1 -4 47 6 25 | 0 7 56 -4 -27 }}

Optimal tunings:

Line 2,729:

Line 2,740:

Comma list: 540/539, 729/728, 1573/1568, 4096/4095

~~Mapping:~~ {{~~mapping~~| 1 -4 47 6 25 -33 | 0 7 56 -4 -27 46 }}

Optimal tunings:

Line 2,740:

Line 2,751:

== Subgroup extensions ==

=== Tridecaschismic (2.3.5.13) ===

Proposed by [[Eufalesio]] in 2026, tridecaschismic adds the [[325/324|marveltwin comma]] to the comma list, or equivalently, the [[tridecapyth comma]]. It benefits from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is one of the lowest badness schismic extensions. It is also equivalent to the 2.3.5.13 [[restriction]] of 13-limit [[cassandra]].

Proposed by [[Eufalesio]] in 2026, tridecaschismic adds the [[325/324|marveltwin comma]] to the comma list, or equivalently, the [[tridecapyth comma]]. It benefits from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is one of the lowest badness schismic extensions. It is also equivalent to the [[2.3.5.13 subgroup|2.3.5.13-subgroup]] [[restriction]] of 13-limit [[cassandra]].

Subgroup: 2.3.5.13

Line 2,748:

Line 2,758:

Comma list: 325/324, 32805/32768

~~Subgroup-val mapping:~~ {{~~mapping~~| 1 0 15 -28 | 0 1 -8 20 }}

Optimal tunings:

Line 2,763:

Line 2,773:

Comma list: 325/324, 361/360, 513/512

~~Subgroup-val mapping:~~ {{~~mapping~~| 1 0 15 -28 9 | 0 1 -8 20 -3 }}

Optimal tunings:

Line 2,780:

Line 2,790:

[[Comma list]]: 256/255, 1458/1445

: mapping generators: ~2, ~3

Line 2,800:

Line 2,810:

Comma list: 171/170, 256/255, 324/323

~~Subgroup-val mapping:~~ {{~~mapping~~| 1 0 15 -7 9 | 0 1 -8 7 -3 }}

~~Gencom mapping:~~ {{~~mapping~~| 1 0 15 0 0 0 -7 9 | 0 1 -8 0 0 0 7 -3 }}

Optimal tunings:

Line 2,813:

Line 2,823:

=== Nestoria (2.3.5.19) ===

~~: ''~~See also~~: [[~~No-elevens subgroup temperaments #Garibaldia~~]] and [[~~No-elevens subgroup temperaments #Pontia~~|#Pontia]]''~~

{{See also| No-elevens subgroup temperaments #Garibaldia | No-elevens subgroup temperaments #Pontia }}

Nestoria is notable for having one of the lowest-badness subgroup extensions of schismic. Note that despite prime [[19/1|19]] being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is generally not flatter than the fifth in optimal schismic due to its optimization considering intervals like [[19/10]] and [[19/15]]. However, the dyadic tuning sensitivity of [[19/16]] suggests using tunings like [[65edo]] and [[77edo]] to optimize in favour of prime 19 (especially the minor triad ~16:19:24 which is equated with the Pythagorean minor triad), as [[171edo]] is already arguably undertempered for it despite being the optimal patent val.

Line 2,821:

Line 2,831:

[[Comma list]]: 361/360, 513/512

: mapping generators: ~2, ~3

Line 2,843:

Line 2,853:

[[Comma list]]: 676/675, 32805/32768

: mapping generators: ~2, ~26/15

Line 2,863:

Line 2,873:

Comma list: 361/360, 513/512, 676/675

~~Subgroup-val mapping:~~ {{~~mapping~~| 1 0 15 14 9 | 0 2 -16 -13 -6 }}

Optimal tunings:

Line 2,878:

Line 2,888:

Comma list: 361/360, 481/480, 513/512, 676/675

~~Subgroup-val mapping:~~ {{~~mapping~~| 1 0 15 14 9 6 | 0 2 -16 -13 -6 -1 }}

Optimal tunings:

Line 2,895:

Line 2,905:

[[Comma list]]: 32805/32768, 1419857/1417176

: mapping generators: ~2, ~18/17

Line 2,915:

Line 2,925:

Comma list: 4624/4617, 6144/6137, 6885/6859

~~Subgroup-val mapping:~~ {{~~mapping~~| 1 2 -1 5 4 | 0 -5 40 -11 3 }}

~~Gencom mapping:~~ {{~~mapping~~| 1 2 -1 0 0 0 5 4 | 0 -5 40 0 0 0 -11 3 }}

Optimal tunings:

Line 2,926:

Line 2,936:

Badness (Sintel): 1.17

== References ==

[[Category:Schismatic family| ]]

[[Category:Temperament families]]

[[Category:Catalogs of rank-2 temperaments]]

@@ Line 48: / Line 48: @@
 {{Main| Garibaldi }}
-Garibaldi tempers out the [[garischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}.
+Garibaldi tempers out the [[septischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a very slightly sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}, which also makes it the unique intersection of [[marvel]] and [[aberschismic]], two very notable 7-limit temperament paradigms; however, their errors go in oposite directions as marvel tunes fifths flatter and aberschismic sharper.
+[[147edo]] (from 94+53) is a [[patent val]] tuning with 5 and 7 very close to equally out-of-tune in opposite directions s.t only 7/5 and 10/7 are inconsistent in the 9-odd-limit, so might be considered for (EG) a 41-note subset ([[12L 29s]]).
 [[Subgroup]]: 2.3.5.7
@@ Line 79: / Line 81: @@
 === Cassandra ===
-Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup, even though it comes with a much higher complexity.
+Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup, even though it comes with a much higher complexity. It finds 11/8 as two commas above 4/3 and 13/8 a minor third above 11/8, or equivalently two commas above 128/81.
 Subgroup: 2.3.5.7.11
@@ Line 85: / Line 87: @@
 Comma list: 225/224, 385/384, 2200/2187
-Mapping: {{mapping| 1 0 15 25 -33 | 0 1 -8 -14 23 }}
+{{Mapping|legend=0| 1 0 15 25 -33 | 0 1 -8 -14 23 }}
 Optimal tunings:
@@ Line 108: / Line 110: @@
 Comma list: 225/224, 275/273, 325/324, 385/384
-Mapping: {{mapping| 1 0 15 25 -33 -28 | 0 1 -8 -14 23 20 }}
+{{Mapping|legend=0| 1 0 15 25 -33 -28 | 0 1 -8 -14 23 20 }}
 Optimal tunings:
@@ Line 132: / Line 134: @@
 Comma list: 120/119, 154/153, 225/224, 273/272, 325/324
-Mapping: {{mapping| 1 0 15 25 -33 -28 -7 | 0 1 -8 -14 23 20 7 }}
+{{Mapping|legend=0| 1 0 15 25 -33 -28 -7 | 0 1 -8 -14 23 20 7 }}
 Optimal tunings:
@@ Line 147: / Line 149: @@
 Comma list: 120/119, 154/153, 171/170, 190/189, 225/224, 273/272
-Mapping: {{mapping| 1 0 15 25 -33 -28 -7 9 | 0 1 -8 -14 23 20 7 -3 }}
+{{Mapping|legend=0| 1 0 15 25 -33 -28 -7 9 | 0 1 -8 -14 23 20 7 -3 }}
 Optimal tunings:
@@ Line 162: / Line 164: @@
 Comma list: 225/224, 275/273, 325/324, 375/374, 385/384
-Mapping: {{mapping| 1 0 15 25 -33 -28 77 | 0 1 -8 -14 23 20 -46 }}
+{{Mapping|legend=0| 1 0 15 25 -33 -28 77 | 0 1 -8 -14 23 20 -46 }}
 Optimal tunings:
@@ Line 177: / Line 179: @@
 Comma list: 190/189, 209/208, 225/224, 275/273, 325/324, 375/374
-Mapping: {{mapping| 1 0 15 25 -33 -28 77 9 | 0 1 -8 -14 23 20 -46 -3 }}
+{{Mapping|legend=0| 1 0 15 25 -33 -28 77 9 | 0 1 -8 -14 23 20 -46 -3 }}
 Optimal tunings:
@@ Line 192: / Line 194: @@
 Comma list: 190/189, 209/208, 225/224, 253/252, 275/273, 325/324, 375/374
-Mapping: {{mapping| 1 0 15 25 -33 -28 77 9 60 | 0 1 -8 -14 23 20 -46 -3 -35 }}
+{{Mapping|legend=0| 1 0 15 25 -33 -28 77 9 60 | 0 1 -8 -14 23 20 -46 -3 -35 }}
 Optimal tunings:
@@ Line 207: / Line 209: @@
 Comma list: 170/169, 225/224, 275/273, 325/324, 385/384
-Mapping: {{mapping| 1 0 15 25 -33 -28 -72 | 0 1 -8 -14 23 20 48 }}
+{{Mapping|legend=0| 1 0 15 25 -33 -28 -72 | 0 1 -8 -14 23 20 48 }}
 Optimal tunings:
@@ Line 222: / Line 224: @@
 Comma list: 170/169, 190/189, 209/208, 225/224, 275/273, 325/324
-Mapping: {{mapping| 1 0 15 25 -33 -28 -72 9 | 0 1 -8 -14 23 20 48 -3 }}
+{{Mapping|legend=0| 1 0 15 25 -33 -28 -72 9 | 0 1 -8 -14 23 20 48 -3 }}
 Optimal tunings:
@@ Line 233: / Line 235: @@
 === Andromeda ===
+Andromeda is the second-best extension to garibaldi, which is less complex but also more innacurate. 11/8 is two commas down from [[729/512]], as is 13/8 two commas down from [[27/16]].
 Subgroup: 2.3.5.7.11
 Comma list: 100/99, 225/224, 245/242
-Mapping: {{mapping| 1 0 15 25 32 | 0 1 -8 -14 -18 }}
+{{Mapping|legend=0| 1 0 15 25 32 | 0 1 -8 -14 -18 }}
 Optimal tunings:
@@ Line 260: / Line 264: @@
 Comma list: 100/99, 105/104, 196/195, 245/242
-Mapping: {{mapping| 1 0 15 25 32 37 | 0 1 -8 -14 -18 -21 }}
+{{Mapping|legend=0| 1 0 15 25 32 37 | 0 1 -8 -14 -18 -21 }}
 Optimal tunings:
@@ Line 284: / Line 288: @@
 Comma list: 100/99, 105/104, 120/119, 189/187, 196/195
-Mapping: {{mapping| 1 0 15 25 32 37 -7 | 0 1 -8 -14 -18 -21 7 }}
+{{Mapping|legend=0| 1 0 15 25 32 37 -7 | 0 1 -8 -14 -18 -21 7 }}
 Optimal tunings:
@@ Line 299: / Line 303: @@
 Comma list: 100/99, 105/104, 120/119, 133/132, 189/187, 196/195
-Mapping: {{mapping| 1 0 15 25 32 37 -7 9 | 0 1 -8 -14 -18 -21 7 -3 }}
+{{Mapping|legend=0| 1 0 15 25 32 37 -7 9 | 0 1 -8 -14 -18 -21 7 -3 }}
 Optimal tunings:
@@ Line 314: / Line 318: @@
 Comma list: 100/99, 105/104, 154/153, 170/169, 196/195
-Mapping: {{mapping| 1 0 15 25 32 37 58 | 0 1 -8 -14 -18 -21 -34 }}
+{{Mapping|legend=0| 1 0 15 25 32 37 58 | 0 1 -8 -14 -18 -21 -34 }}
 Optimal tunings:
@@ Line 329: / Line 333: @@
 Comma list: 100/99, 105/104, 133/132, 154/153, 170/169, 190/189
-Mapping: {{mapping| 1 0 15 25 32 37 58 9 | 0 1 -8 -14 -18 -21 -34 -3 }}
+{{Mapping|legend=0| 1 0 15 25 32 37 58 9 | 0 1 -8 -14 -18 -21 -34 -3 }}
 Optimal tunings:
@@ Line 344: / Line 348: @@
 Comma list: 85/84, 100/99, 105/104, 119/117, 221/220
-Mapping: {{mapping| 1 0 15 25 32 37 12 | 0 1 -8 -14 -18 -21 -5 }}
+{{Mapping|legend=0| 1 0 15 25 32 37 12 | 0 1 -8 -14 -18 -21 -5 }}
 Optimal tunings:
@@ Line 359: / Line 363: @@
 Comma list: 85/84, 100/99, 105/104, 119/117, 133/132, 153/152
-Mapping: {{mapping| 1 0 15 25 32 37 12 9 | 0 1 -8 -14 -18 -21 -5 -3 }}
+{{Mapping|legend=0| 1 0 15 25 32 37 12 9 | 0 1 -8 -14 -18 -21 -5 -3 }}
 Optimal tunings:
@@ Line 374: / Line 378: @@
 Comma list: 99/98, 176/175, 3125/3087
-Mapping: {{mapping| 1 0 15 25 51 | 0 1 -8 -14 -30 }}
+{{Mapping|legend=0| 1 0 15 25 51 | 0 1 -8 -14 -30 }}
 Optimal tunings:
@@ Line 393: / Line 397: @@
 Comma list: 99/98, 176/175, 275/273, 847/845
-Mapping: {{mapping| 1 0 15 25 51 56 | 0 1 -8 -14 -30 -33 }}
+{{Mapping|legend=0| 1 0 15 25 51 56 | 0 1 -8 -14 -30 -33 }}
 Optimal tunings:
@@ Line 412: / Line 416: @@
 Comma list: 99/98, 120/119, 176/175, 275/273, 442/441
-Mapping: {{mapping| 1 0 15 25 51 56 -7 | 0 1 -8 -14 -30 -33 7 }}
+{{Mapping|legend=0| 1 0 15 25 51 56 -7 | 0 1 -8 -14 -30 -33 7 }}
 Optimal tunings:
@@ Line 427: / Line 431: @@
 Comma list: 99/98, 120/119, 176/175, 190/189, 209/208, 247/245
-Mapping: {{mapping| 1 0 15 25 51 56 -7 9 | 0 1 -8 -14 -30 -33 7 -3 }}
+{{Mapping|legend=0| 1 0 15 25 51 56 -7 9 | 0 1 -8 -14 -30 -33 7 -3 }}
 Optimal tunings:
@@ Line 444: / Line 448: @@
 Comma list: 225/224, 243/242, 3125/3087
-Mapping: {{mapping| 1 1 7 11 2 | 0 2 -16 -28 5 }}
+{{Mapping|legend=0| 1 1 7 11 2 | 0 2 -16 -28 5 }}
 : mapping generators: ~2, ~11/9
@@ Line 460: / Line 464: @@
 Comma list: 225/224, 243/242, 325/324, 640/637
-Mapping: {{mapping| 1 1 7 11 2 -8 | 0 2 -16 -28 5 40 }}
+{{Mapping|legend=0| 1 1 7 11 2 -8 | 0 2 -16 -28 5 40 }}
 Optimal tunings:
@@ Line 475: / Line 479: @@
 Comma list: 121/120, 225/224, 3125/3087
-Mapping: {{mapping| 1 0 15 25 9 | 0 2 -16 -28 -7 }}
+{{Mapping|legend=0| 1 0 15 25 9 | 0 2 -16 -28 -7 }}
 : mapping generators: ~2, ~110/63
@@ Line 491: / Line 495: @@
 Comma list: 121/120, 169/168, 225/224, 275/273
-Mapping: {{mapping| 1 0 15 25 9 14 | 0 2 -16 -28 -7 -13 }}
+{{Mapping|legend=0| 1 0 15 25 9 14 | 0 2 -16 -28 -7 -13 }}
 Optimal tunings:
@@ Line 506: / Line 510: @@
 Comma list: 225/224, 1331/1323, 3125/3087
-Mapping: {{mapping| 1 2 -1 -3 0 | 0 -3 24 42 25 }}
+{{Mapping|legend=0| 1 2 -1 -3 0 | 0 -3 24 42 25 }}
 : mapping generators: ~2, ~11/10
@@ Line 522: / Line 526: @@
 Comma list: 225/224, 275/273, 847/845, 1331/1323
-Mapping: {{mapping| 1 2 -1 -3 0 -1 | 0 -3 24 42 25 34 }}
+{{Mapping|legend=0| 1 2 -1 -3 0 -1 | 0 -3 24 42 25 34 }}
 Optimal tunings:
@@ Line 572: / Line 576: @@
 Comma list: 385/384, 3388/3375, 4375/4374
-Mapping: {{mapping| 1 0 15 -59 51 | 0 1 -8 39 -30 }}
+{{Mapping|legend=0| 1 0 15 -59 51 | 0 1 -8 39 -30 }}
 Optimal tunings:
@@ Line 591: / Line 595: @@
 Comma list: 352/351, 385/384, 625/624, 729/728
-Mapping: {{mapping| 1 0 15 -59 51 56 | 0 1 -8 39 -30 -33 }}
+{{Mapping|legend=0| 1 0 15 -59 51 56 | 0 1 -8 39 -30 -33 }}
 Optimal tunings:
@@ Line 610: / Line 614: @@
 Comma list: 352/351, 385/384, 561/560, 625/624, 729/728
-Mapping: {{mapping| 1 0 15 -59 51 56 -91 | 0 1 -8 39 -30 -33 60 }}
+{{Mapping|legend=0| 1 0 15 -59 51 56 -91 | 0 1 -8 39 -30 -33 60 }}
 Optimal tunings:
@@ Line 629: / Line 633: @@
 Comma list: 169/168, 325/324, 385/384, 3146/3125
-Mapping: {{mapping| 1 0 15 -59 51 -28 | 0 1 -8 39 -30 20 }}
+{{Mapping|legend=0| 1 0 15 -59 51 -28 | 0 1 -8 39 -30 20 }}
 Optimal tunings:
@@ Line 644: / Line 648: @@
 Comma list: 169/168, 273/272, 325/324, 385/384, 3146/3125
-Mapping: {{mapping| 1 0 15 -59 51 -28 -91 | 0 1 -8 39 -30 20 60 }}
+{{Mapping|legend=0| 1 0 15 -59 51 -28 -91 | 0 1 -8 39 -30 20 60 }}
 Optimal tunings:
@@ Line 659: / Line 663: @@
 Comma list: 169/168, 273/272, 286/285, 325/324, 385/384, 627/625
-Mapping: {{mapping| 1 0 15 -59 51 -28 -91 9 | 0 1 -8 39 -30 20 60 -3 }}
+{{Mapping|legend=0| 1 0 15 -59 51 -28 -91 9 | 0 1 -8 39 -30 20 60 -3 }}
 Optimal tunings:
@@ Line 676: / Line 680: @@
 Comma list: 540/539, 4375/4374, 32805/32768
-Mapping: {{mapping| 1 0 15 -59 135 | 0 1 -8 39 -83 }}
+{{Mapping|legend=0| 1 0 15 -59 135 | 0 1 -8 39 -83 }}
 Optimal tunings:
@@ Line 695: / Line 699: @@
 Comma list: 540/539, 625/624, 729/728, 2200/2197
-Mapping: {{mapping| 1 0 15 -59 135 56 | 0 1 -8 39 -83 -33 }}
+{{Mapping|legend=0| 1 0 15 -59 135 56 | 0 1 -8 39 -83 -33 }}
 Optimal tunings:
@@ Line 714: / Line 718: @@
 Comma list: 375/374, 540/539, 625/624, 729/728, 2200/2197
-Mapping: {{mapping| 1 0 15 -59 135 56 -91 | 0 1 -8 39 -83 -33 60 }}
+{{Mapping|legend=0| 1 0 15 -59 135 56 -91 | 0 1 -8 39 -83 -33 60 }}
 Optimal tunings:
@@ Line 735: / Line 739: @@
 Comma list: 441/440, 4375/4374, 32805/32768
-Mapping: {{mapping| 1 0 15 -59 -136 | 0 1 -8 39 88 }}
+{{Mapping|legend=0| 1 0 15 -59 -136 | 0 1 -8 39 88 }}
 Optimal tunings:
@@ Line 754: / Line 758: @@
 Comma list: 441/440, 625/624, 729/728, 3584/3575
-Mapping: {{mapping| 1 0 15 -59 -136 56 | 0 1 -8 39 88 -33 }}
+{{Mapping|legend=0| 1 0 15 -59 -136 56 | 0 1 -8 39 88 -33 }}
 Optimal tunings:
@@ Line 773: / Line 777: @@
 Comma list: 441/440, 595/594, 625/624, 729/728, 2880/2873
-Mapping: {{mapping| 1 0 15 -59 -136 56 -91 | 0 1 -8 39 88 -33 60 }}
+{{Mapping|legend=0| 1 0 15 -59 -136 56 -91 | 0 1 -8 39 88 -33 60 }}
 Optimal tunings:
@@ Line 792: / Line 796: @@
 Comma list: 364/363, 441/440, 4375/4374, 32805/32768
-Mapping: {{mapping| 1 0 15 -59 -136 -215 | 0 1 -8 39 88 138 }}
+{{Mapping|legend=0| 1 0 15 -59 -136 -215 | 0 1 -8 39 88 138 }}
 Optimal tunings:
@@ Line 807: / Line 811: @@
 Comma list: 364/363, 441/440, 595/594, 1156/1155, 32805/32768
-Mapping: {{mapping| 1 0 15 -59 -136 -215 -91 | 0 1 -8 39 88 138 60 }}
+{{Mapping|legend=0| 1 0 15 -59 -136 -215 -91 | 0 1 -8 39 88 138 60 }}
 Optimal tunings:
@@ Line 824: / Line 828: @@
 Comma list: 3025/3024, 4375/4374, 32805/32768
-Mapping: {{mapping| 2 0 30 -118 -85 | 0 1 -8 39 29 }}
+{{Mapping|legend=0| 2 0 30 -118 -85 | 0 1 -8 39 29 }}
 : mapping generators: ~99/70, ~3
@@ Line 840: / Line 844: @@
 Comma list: 625/624, 729/728, 1575/1573, 4096/4095
-Mapping: {{mapping| 2 0 30 -118 -85 112 | 0 1 -8 39 29 -33 }}
+{{Mapping|legend=0| 2 0 30 -118 -85 112 | 0 1 -8 39 29 -33 }}
 Optimal tunings:
@@ Line 855: / Line 859: @@
 Comma list: 625/624, 729/728, 1089/1088, 1225/1224, 2880/2873
-Mapping: {{mapping| 2 0 30 -118 -85 112 -182 | 0 1 -8 39 29 -33 60 }}
+{{Mapping|legend=0| 2 0 30 -118 -85 112 -182 | 0 1 -8 39 29 -33 60 }}
 Optimal tunings:
@@ Line 870: / Line 874: @@
 Comma list: 1716/1715, 2080/2079, 3025/3024, 32805/32768
-Mapping: {{mapping| 2 0 30 -118 -85 -243 | 0 1 -8 39 29 79 }}
+{{Mapping|legend=0| 2 0 30 -118 -85 -243 | 0 1 -8 39 29 79 }}
 Optimal tunings:
@@ Line 885: / Line 889: @@
 Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 32805/32768
-Mapping: {{mapping| 2 0 30 -118 -85 -243 -182 | 0 1 -8 39 29 79 60 }}
+{{Mapping|legend=0| 2 0 30 -118 -85 -243 -182 | 0 1 -8 39 29 79 60 }}
 Optimal tunings:
@@ Line 900: / Line 904: @@
 Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 1540/1539, 4875/4864
-Mapping: {{mapping| 2 0 30 -118 -85 -243 -182 -169 | 0 1 -8 39 29 79 60 56 }}
+{{Mapping|legend=0| 2 0 30 -118 -85 -243 -182 -169 | 0 1 -8 39 29 79 60 56 }}
 Optimal tunings:
@@ Line 915: / Line 919: @@
 Comma list: 3025/3024, 4225/4224, 4375/4374, 32805/32768
-Mapping: {{mapping| 4 0 60 -236 -170 -131 | 0 1 -8 39 29 23 }}
+{{Mapping|legend=0| 4 0 60 -236 -170 -131 | 0 1 -8 39 29 23 }}
 : mapping generators: ~208/175, ~3
@@ Line 955: / Line 959: @@
 Comma list: 126/125, 176/175, 32805/32768
-Mapping: {{mapping| 1 0 15 44 70 | 0 1 -8 -26 -42 }}
+{{Mapping|legend=0| 1 0 15 44 70 | 0 1 -8 -26 -42 }}
 Optimal tunings:
@@ Line 970: / Line 974: @@
 Comma list: 126/125, 176/175, 196/195, 5445/5408
-Mapping: {{mapping| 1 0 15 44 70 75 | 0 1 -8 -26 -42 -45 }}
+{{Mapping|legend=0| 1 0 15 44 70 75 | 0 1 -8 -26 -42 -45 }}
 Optimal tunings:
@@ Line 985: / Line 989: @@
 Comma list: 126/125, 176/175, 196/195, 256/255, 2904/2873
-Mapping: {{mapping| 1 0 15 44 70 75 -7 | 0 1 -8 -26 -42 -45 7 }}
+{{Mapping|legend=0| 1 0 15 44 70 75 -7 | 0 1 -8 -26 -42 -45 7 }}
 Optimal tunings:
@@ Line 1,000: / Line 1,004: @@
 Comma list: 126/125, 171/170, 176/175, 196/195, 209/208, 324/323
-Mapping: {{mapping| 1 0 15 44 70 75 -7 9 | 0 1 -8 -26 -42 -45 7 -3 }}
+{{Mapping|legend=0| 1 0 15 44 70 75 -7 9 | 0 1 -8 -26 -42 -45 7 -3 }}
 Optimal tunings:
@@ Line 1,015: / Line 1,019: @@
 Comma list: 126/125, 176/175, 729/728, 1287/1280
-Mapping: {{mapping| 1 0 15 44 70 -47 | 0 1 -8 -26 -42 32 }}
+{{Mapping|legend=0| 1 0 15 44 70 -47 | 0 1 -8 -26 -42 32 }}
 Optimal tunings:
@@ Line 1,030: / Line 1,034: @@
 Comma list: 126/125, 245/242, 896/891
-Mapping: {{mapping| 1 0 15 44 51 | 0 1 -8 -26 -30 }}
+{{Mapping|legend=0| 1 0 15 44 51 | 0 1 -8 -26 -30 }}
 Optimal tunings:
@@ Line 1,045: / Line 1,049: @@
 Comma list: 126/125, 196/195, 245/242, 832/825
-Mapping: {{mapping| 1 0 15 44 51 75 | 0 1 -8 -26 -30 -45 }}
+{{Mapping|legend=0| 1 0 15 44 51 75 | 0 1 -8 -26 -30 -45 }}
 Optimal tunings:
@@ Line 1,060: / Line 1,064: @@
 Comma list: 105/104, 126/125, 245/242, 352/351
-Mapping: {{mapping| 1 0 15 44 51 56 | 0 1 -8 -26 -30 -33 }}
+{{Mapping|legend=0| 1 0 15 44 51 56 | 0 1 -8 -26 -30 -33 }}
 Optimal tunings:
@@ Line 1,094: / Line 1,098: @@
 Comma list: 385/384, 19712/19683, 78125/77616
-Mapping: {{mapping| 1 0 15 109 -117 | 0 1 -8 -67 76 }}
+{{Mapping|legend=0| 1 0 15 109 -117 | 0 1 -8 -67 76 }}
 Optimal tunings:
@@ Line 1,109: / Line 1,113: @@
 Comma list: 325/324, 385/384, 2200/2197, 19712/19683
-Mapping: {{mapping| 1 0 15 109 -117 -28 | 0 1 -8 -67 76 20 }}
+{{Mapping|legend=0| 1 0 15 109 -117 -28 | 0 1 -8 -67 76 20 }}
 Optimal tunings:
@@ Line 1,153: / Line 1,157: @@
 Comma list: 441/440, 3136/3125, 8019/8000
-Mapping: {{mapping| 2 0 30 69 102 | 0 1 -8 -20 -30 }}
+{{Mapping|legend=0| 2 0 30 69 102 | 0 1 -8 -20 -30 }}
 Optimal tunings:
@@ Line 1,168: / Line 1,172: @@
 Comma list: 441/440, 729/728, 1001/1000, 3136/3125
-Mapping: {{mapping| 2 0 30 69 102 -75 | 0 1 -8 -20 -30 26 }}
+{{Mapping|legend=0| 2 0 30 69 102 -75 | 0 1 -8 -20 -30 26 }}
 Optimal tunings:
@@ Line 1,183: / Line 1,187: @@
 Comma list: 289/288, 441/440, 561/560, 729/728, 3136/3125
-Mapping: {{mapping| 2 0 30 69 102 -75 5 | 0 1 -8 -20 -30 26 1 }}
+{{Mapping|legend=0| 2 0 30 69 102 -75 5 | 0 1 -8 -20 -30 26 1 }}
 Optimal tunings:
@@ Line 1,198: / Line 1,202: @@
 Comma list: 351/350, 364/363, 441/440, 3136/3125
-Mapping: {{mapping| 2 0 30 69 102 131 | 0 1 -8 -20 -30 -39 }}
+{{Mapping|legend=0| 2 0 30 69 102 131 | 0 1 -8 -20 -30 -39 }}
 Optimal tunings:
@@ Line 1,213: / Line 1,217: @@
 Comma list: 221/220, 289/288, 351/350, 441/440, 3136/3125
-Mapping: {{mapping| 2 0 30 69 102 131 5 | 0 1 -8 -20 -30 -39 1 }}
+{{Mapping|legend=0| 2 0 30 69 102 131 5 | 0 1 -8 -20 -30 -39 1 }}
 Optimal tunings:
@@ Line 1,248: / Line 1,252: @@
 Comma list: 385/384, 9801/9800, 14641/14580
-Mapping: {{mapping| 2 1 22 -15 8 | 0 2 -16 19 -1 }}
+{{Mapping|legend=0| 2 1 22 -15 8 | 0 2 -16 19 -1 }}
 Optimal tunings:
@@ Line 1,263: / Line 1,267: @@
 Comma list: 352/351, 385/384, 729/728, 1575/1573
-Mapping: {{mapping| 2 1 22 -15 8 15 | 0 2 -16 19 -1 -7 }}
+{{Mapping|legend=0| 2 1 22 -15 8 15 | 0 2 -16 19 -1 -7 }}
 Optimal tunings:
@@ Line 1,278: / Line 1,282: @@
 Comma list: 170/169, 289/288, 352/351, 385/384, 561/560
-Mapping: {{mapping| 2 1 22 -15 8 15 6 | 0 2 -16 19 -1 -7 2 }}
+{{Mapping|legend=0| 2 1 22 -15 8 15 6 | 0 2 -16 19 -1 -7 2 }}
 Optimal tunings:
@@ Line 1,293: / Line 1,297: @@
 Comma list: 325/324, 385/384, 1573/1568, 14641/14580
-Mapping: {{mapping| 2 1 22 -15 8 -36 | 0 2 -16 19 -1 40 }}
+{{Mapping|legend=0| 2 1 22 -15 8 -36 | 0 2 -16 19 -1 40 }}
 Optimal tunings:
@@ Line 1,308: / Line 1,312: @@
 Comma list: 289/288, 325/324, 385/384, 442/441, 14641/14580
-Mapping: {{mapping| 2 1 22 -15 8 -36 6 | 0 2 -16 19 -1 40 2 }}
+{{Mapping|legend=0| 2 1 22 -15 8 -36 6 | 0 2 -16 19 -1 40 2 }}
 Optimal tunings:
@@ Line 1,343: / Line 1,347: @@
 Comma list: 243/242, 245/242, 385/384
-Mapping: {{mapping| 1 1 7 -1 2 | 0 2 -16 13 5 }}
+{{Mapping|legend=0| 1 1 7 -1 2 | 0 2 -16 13 5 }}
 Optimal tunings:
@@ Line 1,358: / Line 1,362: @@
 Comma list: 105/104, 144/143, 243/242, 245/242
-Mapping: {{mapping| 1 1 7 -1 2 4 | 0 2 -16 13 5 -1 }}
+{{Mapping|legend=0| 1 1 7 -1 2 4 | 0 2 -16 13 5 -1 }}
 Optimal tunings:
@@ Line 1,395: / Line 1,399: @@
 Comma list: 540/539, 5632/5625, 8019/8000
-Mapping: {{mapping| 1 0 15 -17 51 | 0 2 -16 25 -60 }}
+{{Mapping|legend=0| 1 0 15 -17 51 | 0 2 -16 25 -60 }}
 Optimal tunings:
@@ Line 1,410: / Line 1,414: @@
 Comma list: 351/350, 540/539, 676/675, 4096/4095
-Mapping: {{mapping| 1 0 15 -17 51 14 | 0 2 -16 25 -60 -13 }}
+{{Mapping|legend=0| 1 0 15 -17 51 14 | 0 2 -16 25 -60 -13 }}
 Optimal tunings:
@@ Line 1,425: / Line 1,429: @@
 Comma list: 351/350, 442/441, 561/560, 676/675, 4096/4095
-Mapping: {{mapping| 1 0 15 -17 51 14 -49 | 0 2 -16 25 -60 -13 67 }}
+{{Mapping|legend=0| 1 0 15 -17 51 14 -49 | 0 2 -16 25 -60 -13 67 }}
 Optimal tunings:
@@ Line 1,440: / Line 1,444: @@
 Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 4096/4095
-Mapping: {{mapping| 1 0 15 -17 51 14 -49 9 | 0 2 -16 25 -60 -13 67 -6 }}
+{{Mapping|legend=0| 1 0 15 -17 51 14 -49 9 | 0 2 -16 25 -60 -13 67 -6 }}
 Optimal tunings:
@@ Line 1,455: / Line 1,459: @@
 Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 736/735, 4096/4095
-Mapping: {{mapping| 1 0 15 -17 51 14 -49 9 -24 | 0 2 -16 25 -60 -13 67 -6 36 }}
+{{Mapping|legend=0| 1 0 15 -17 51 14 -49 9 -24 | 0 2 -16 25 -60 -13 67 -6 36 }}
 Optimal tunings:
@@ Line 1,499: / Line 1,503: @@
 Comma list: 441/440, 4375/4356, 32805/32768
-Mapping: {{mapping| 3 0 45 94 134 | 0 1 -8 -18 -26 }}
+{{Mapping|legend=0| 3 0 45 94 134 | 0 1 -8 -18 -26 }}
 Optimal tunings:
@@ Line 1,514: / Line 1,518: @@
 Comma list: 364/363, 441/440, 625/624, 13720/13689
-Mapping: {{mapping| 3 0 45 94 134 168 | 0 1 -8 -18 -26 -33 }}
+{{Mapping|legend=0| 3 0 45 94 134 168 | 0 1 -8 -18 -26 -33 }}
 Optimal tunings:
@@ Line 1,529: / Line 1,533: @@
 Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619
-Mapping: {{mapping| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }}
+{{Mapping|legend=0| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }}
 Optimal tunings:
@@ Line 1,546: / Line 1,550: @@
 Comma list: 540/539, 32805/32768, 137781/137500
-Mapping: {{mapping| 3 0 45 94 -137 | 0 1 -8 -18 31 }}
+{{Mapping|legend=0| 3 0 45 94 -137 | 0 1 -8 -18 31 }}
 Optimal tunings:
@@ Line 1,561: / Line 1,565: @@
 Comma list: 540/539, 729/728, 4096/4095, 31250/31213
-Mapping: {{mapping| 3 0 45 94 -137 -103 | 0 1 -8 -18 31 24 }}
+{{Mapping|legend=0| 3 0 45 94 -137 -103 | 0 1 -8 -18 31 24 }}
 Optimal tunings:
@@ Line 1,576: / Line 1,580: @@
 Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095
-Mapping: {{mapping| 3 0 45 94 -137 -103 -2 | 0 1 -8 -18 31 24 3 }}
+{{Mapping|legend=0| 3 0 45 94 -137 -103 -2 | 0 1 -8 -18 31 24 3 }}
 Optimal tunings:
@@ Line 1,593: / Line 1,597: @@
 Comma list: 9801/9800, 32805/32768, 151263/151250
-Mapping: {{mapping| 6 0 90 188 287 | 0 1 -8 -18 -28 }}
+{{Mapping|legend=0| 6 0 90 188 287 | 0 1 -8 -18 -28 }}
 : mapping generators: ~55/49, ~3
@@ Line 1,609: / Line 1,613: @@
 Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375
-Mapping: {{mapping| 6 0 90 188 287 355 | 0 1 -8 -18 -28 -35 }}
+{{Mapping|legend=0| 6 0 90 188 287 355 | 0 1 -8 -18 -28 -35 }}
 Optimal tunings:
@@ Line 1,628: / Line 1,632: @@
 Comma list: 3025/3024, 32805/32768, 102487/102400
-Mapping: {{mapping| 3 0 45 94 8 | 0 2 -16 -36 1 }}
+{{Mapping|legend=0| 3 0 45 94 8 | 0 2 -16 -36 1 }}
 : mapping generators: ~63/50, ~693/400
@@ Line 1,644: / Line 1,648: @@
 Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712
-Mapping: {{mapping| 3 0 45 94 8 42 | 0 2 -16 -36 1 -13 }}
+{{Mapping|legend=0| 3 0 45 94 8 42 | 0 2 -16 -36 1 -13 }}
 Optimal tunings:
@@ Line 1,659: / Line 1,663: @@
 Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264
-Mapping: {{mapping| 3 0 45 94 8 42 -2 | 0 2 -16 -36 1 -13 6 }}
+{{Mapping|legend=0| 3 0 45 94 8 42 -2 | 0 2 -16 -36 1 -13 6 }}
 Optimal tunings:
@@ Line 1,694: / Line 1,698: @@
 Comma list: 385/384, 14700/14641, 19712/19683
-Mapping: {{mapping| 3 0 45 -32 8 | 0 2 -16 17 1 }}
+{{Mapping|legend=0| 3 0 45 -32 8 | 0 2 -16 17 1 }}
 Optimal tunings:
@@ Line 1,709: / Line 1,713: @@
 Comma list: 364/363, 385/384, 676/675, 19712/19683
-Mapping: {{mapping| 3 0 45 -32 8 42 | 0 2 -16 17 1 -13 }}
+{{Mapping|legend=0| 3 0 45 -32 8 42 | 0 2 -16 17 1 -13 }}
 Optimal tunings:
@@ Line 1,743: / Line 1,747: @@
 Comma list: 245/242, 686/675, 896/891
-Mapping: {{mapping| 1 2 -1 1 0 | 0 -3 24 13 25 }}
+{{Mapping|legend=0| 1 2 -1 1 0 | 0 -3 24 13 25 }}
 Optimal tunings:
@@ Line 1,758: / Line 1,762: @@
 Comma list: 91/90, 169/168, 245/242, 896/891
-Mapping: {{mapping| 1 2 -1 1 0 3 | 0 -3 24 13 25 5 }}
+{{Mapping|legend=0| 1 2 -1 1 0 3 | 0 -3 24 13 25 5 }}
 Optimal tunings:
@@ Line 1,769: / Line 1,773: @@
 == Tertiaschis ==
-Named by [[Xenllium]] in 2021, tertiaschis may be described as {{nowrap| 94 & 159 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 1071875/1062882 for prime 7. See also [[daemotertiachis]] which is made using every other generator of tertiaschis.
+Named by [[Xenllium]] in 2021, tertiaschis may be described as {{nowrap| 94 & 159 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 1071875/1062882 for prime 7. See also [[daemotertiaschis]] which is made using every other generator of tertiaschis.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,792: / Line 1,796: @@
 Comma list: 385/384, 4000/3993, 19712/19683
-Mapping: {{mapping| 1 2 -1 10 0 | 0 -3 24 -52 25 }}
+{{Mapping|legend=0| 1 2 -1 10 0 | 0 -3 24 -52 25 }}
 Optimal tunings:
@@ Line 1,807: / Line 1,811: @@
 Comma list: 325/324, 385/384, 1575/1573, 10985/10976
-Mapping: {{mapping| 1 2 -1 10 0 12 | 0 -3 24 -52 25 -60 }}
+{{Mapping|legend=0| 1 2 -1 10 0 12 | 0 -3 24 -52 25 -60 }}
 Optimal tunings:
@@ Line 1,822: / Line 1,826: @@
 Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
-Mapping: {{mapping| 1 2 -1 10 0 12 -2 | 0 -3 24 -52 25 -60 44 }}
+{{Mapping|legend=0| 1 2 -1 10 0 12 -2 | 0 -3 24 -52 25 -60 44 }}
 Optimal tunings:
@@ Line 1,833: / Line 1,837: @@
 == Countertertiaschis ==
-Named by [[Flora Canou]] in 2021, Countertertiaschis may be described as {{nowrap| 159 & 224 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 244140625/243045684 for prime 7.
+Named by [[Flora Canou]] in 2021, countertertiaschis may be described as {{nowrap| 159 & 224 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 244140625/243045684 for prime 7.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,856: / Line 1,860: @@
 Comma list: 3025/3024, 4000/3993, 32805/32768
-Mapping: {{mapping| 1 2 -1 -12 0 | 0 -3 24 107 25 }}
+{{Mapping|legend=0| 1 2 -1 -12 0 | 0 -3 24 107 25 }}
 Optimal tunings:
@@ Line 1,871: / Line 1,875: @@
 Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976
-Mapping: {{mapping| 1 2 -1 -12 0 -10 | 0 -3 24 107 25 99 }}
+{{Mapping|legend=0| 1 2 -1 -12 0 -10 | 0 -3 24 107 25 99 }}
 Optimal tunings:
@@ Line 1,882: / Line 1,886: @@
 == Quadrant ==
-Named by [[Xenllium]] in 2021, quadrant tempers out 390625/388962, the [[dimcomp comma]], and maps [[25/21]] to the 1/4-octave period. It may be described as the {{nowrap| 12 & 212 }} temperament; its ploidacot is tetraploid monocot. Just as [[#Term|term]] equates the syntonic~Pythagorean comma with three [[marvel comma]]s, quadrant equates the syntonic~Pythagorean comma with four. A [[septimal comma]] is then found as a stack of five marvel commas.
+Named by [[Xenllium]] in 2021, quadrant tempers out 390625/388962, the [[dimcomp comma]], and maps [[25/21]] to the 1/4-octave period. It may be described as the {{nowrap| 12 & 212 }} temperament; its ploidacot is tetraploid monocot. Just as [[#Term|term]] equates the syntonic and Pythagorean commas for a generic comma step, and splits it into three [[marvel comma]]s, quadrant splits the same generic comma step into four. A [[septimal comma]] is then found as a stack of five marvel commas.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,906: / Line 1,910: @@
 Comma list: 1375/1372, 6250/6237, 32805/32768
-Mapping: {{mapping| 4 0 60 119 185 | 0 1 -8 -17 -27 }}
+{{Mapping|legend=0| 4 0 60 119 185 | 0 1 -8 -17 -27 }}
 Optimal tunings:
@@ Line 1,921: / Line 1,925: @@
 Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647
-Mapping: {{mapping| 4 0 60 119 185 224 | 0 1 -8 -17 -27 -33 }}
+{{Mapping|legend=0| 4 0 60 119 185 224 | 0 1 -8 -17 -27 -33 }}
 Optimal tunings:
@@ Line 1,932: / Line 1,936: @@
 == Sesquiquartififths ==
-Sesquiquartififths tempers out 2401/2400, the [[breedsma]], and may be described as the {{nowrap| 41 & 171 }} temperament. It splits the fifth into four; its [[ploidacot]] is thus tetracot.
+Sesquiquartififths tempers out 2401/2400, the [[breedsma]], and may be described as the {{nowrap| 41 & 171 }} temperament. It splits the [[3/2|perfect fifth]] into four; its [[ploidacot]] is thus tetracot.
+The name might have been a portmanteau of ''sesquiquart'' and ''quartififths'', each meaning dividing 3/2 into four.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,962: / Line 1,968: @@
 Comma list: 243/242, 441/440, 16384/16335
-Mapping: {{mapping| 1 1 7 5 2 | 0 4 -32 -15 10 }}
+{{Mapping|legend=0| 1 1 7 5 2 | 0 4 -32 -15 10 }}
 Optimal tunings:
@@ Line 1,977: / Line 1,983: @@
 Comma list: 243/242, 364/363, 441/440, 3584/3575
-Mapping: {{mapping| 1 1 7 5 2 -2 | 0 4 -32 -15 10 39 }}
+{{Mapping|legend=0| 1 1 7 5 2 -2 | 0 4 -32 -15 10 39 }}
 Optimal tunings:
@@ Line 1,992: / Line 1,998: @@
 Comma list: 243/242, 256/255, 273/272, 364/363, 441/440
-Mapping: {{mapping| 1 1 7 5 2 -2 0 | 0 4 -32 -15 10 39 28 }}
+{{Mapping|legend=0| 1 1 7 5 2 -2 0 | 0 4 -32 -15 10 39 28 }}
 Optimal tunings:
@@ Line 2,007: / Line 2,013: @@
 Comma list: 171/170, 243/242, 256/255, 273/272, 324/323, 441/440
-Mapping: {{mapping| 1 1 7 5 2 -2 0 6 | 0 4 -32 -15 10 39 28 -12 }}
+{{Mapping|legend=0| 1 1 7 5 2 -2 0 6 | 0 4 -32 -15 10 39 28 -12 }}
 Optimal tunings:
@@ Line 2,022: / Line 2,028: @@
 Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575
-Mapping: {{mapping| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }}
+{{Mapping|legend=0| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }}
 Optimal tunings:
@@ Line 2,037: / Line 2,043: @@
 Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594
-Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }}
+{{Mapping|legend=0| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }}
 Optimal tunings:
@@ Line 2,052: / Line 2,058: @@
 Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594
-Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 -6 | 0 4 -32 -15 10 39 69 -12 72 }}
+{{Mapping|legend=0| 1 1 7 5 2 -2 -6 6 -6 | 0 4 -32 -15 10 39 69 -12 72 }}
 Optimal tunings:
@@ Line 2,067: / Line 2,073: @@
 Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625
-Mapping: {{mapping| 1 1 7 5 2 -2 13 | 0 4 -32 -15 10 39 -61 }}
+{{Mapping|legend=0| 1 1 7 5 2 -2 13 | 0 4 -32 -15 10 39 -61 }}
 Optimal tunings:
@@ Line 2,082: / Line 2,088: @@
 Comma list: 221/220, 243/242, 361/360, 364/363, 441/440, 456/455
-Mapping: {{mapping| 1 1 7 5 2 -2 13 6 | 0 4 -32 -15 10 39 -61 -12 }}
+{{Mapping|legend=0| 1 1 7 5 2 -2 13 6 | 0 4 -32 -15 10 39 -61 -12 }}
 Optimal tunings:
@@ Line 2,097: / Line 2,103: @@
 Comma list: 221/220, 243/242, 276/275, 323/322, 361/360, 364/363, 441/440
-Mapping: {{mapping| 1 1 7 5 2 -2 13 6 13 | 0 4 -32 -15 10 39 -61 -12 -58 }}
+{{Mapping|legend=0| 1 1 7 5 2 -2 13 6 13 | 0 4 -32 -15 10 39 -61 -12 -58 }}
 Optimal tunings:
@@ Line 2,112: / Line 2,118: @@
 Comma list: 2401/2400, 9801/9800, 32805/32768
-Mapping: {{mapping| 2 2 14 10 23 | 0 4 -32 -15 -55 }}
+{{Mapping|legend=0| 2 2 14 10 23 | 0 4 -32 -15 -55 }}
 : mapping generators: ~99/70, ~448/405
@@ Line 2,124: / Line 2,130: @@
 == Tsaharuk ==
-{{Main| Tsaharuk }}
+[[File:Tsaharuk.jpg|alt=Tsaharuk.jpg|thumb|Parametric horagram of the tsaharuk temperament.]]
+{{See also| Tsaharuk24 }}
 Tsaharuk tempers out 420175/419904, the [[wizma]], and may be described as the {{nowrap| 77 & 94 }} temperament. It is generated by a slightly flat neutral second of [[~]][[13/12]], five of which make the [[3/2|perfect fifth]], so its [[ploidacot]] is pentacot.
+It was proposed by [[Jacques Dudon]] based on [[Julien Jalaleddine Weiss]] qanun tunings, which has [[mos]] of size 43, 60, 77 and 94 notes.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_100718.html Yahoo! Tuning Group | ''A temperament for Maqam'']</ref>
+It restricts to the [[2.3.7 subgroup]] as [[no-fives subgroup temperaments #Navy|navy]]. This relates tsaharuk to [[aberschismic temperaments #Quanic|quanic]], which favors a sharper fifth and works better in the higher limits, but which also has mos of size 43, 60, 77 and 94 notes. [[Bleu]] has, like tsaharuk, a generator of 1/5 fifth and makes for an instructive comparison. [[Edo]]s that support tsaharuk include [[77edo]], [[94edo]], [[171edo]] and [[248edo]], and edos supporting quanic are [[94edo]], [[111edo]] and [[205edo]]. Quanic and tsaharuk therefore become identical if 94edo is the tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 2,150: / Line 2,161: @@
 Comma list: 385/384, 1331/1323, 19712/19683
-Mapping: {{mapping| 1 1 7 0 1 | 0 5 -40 24 21 }}
+{{Mapping|legend=0| 1 1 7 0 1 | 0 5 -40 24 21 }}
 Optimal tunings:
@@ Line 2,165: / Line 2,176: @@
 Comma list: 352/351, 385/384, 729/728, 1331/1323
-Mapping: {{mapping| 1 1 7 0 1 3 | 0 5 -40 24 21 6 }}
+{{Mapping|legend=0| 1 1 7 0 1 3 | 0 5 -40 24 21 6 }}
 Optimal tunings:
@@ Line 2,176: / Line 2,187: @@
 == Quanharuk ==
-Quanharuk tempers out 16875/16807, the [[mirkwai]] comma, and may be described as the {{nowrap| 41 & 183 }} temperament. The generator is a slightly flat major third of [[~]][[56/45]], five of which make the [[3/1|3rd]] [[harmonic]], so the [[ploidacot]] of this temperament is alpha-pentacot. [[224edo]] makes for a recommendable tuning.
+Quanharuk tempers out 16875/16807, the [[mirkwai]] comma, and may be described as the {{nowrap| 41 & 183 }} temperament. The generator is a slightly flat major third of [[~]][[56/45]], five of which make the [[3/1|3rd harmonic]], so the [[ploidacot]] of this temperament is alpha-pentacot. [[224edo]] makes for a recommendable tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 2,200: / Line 2,211: @@
 Comma list: 540/539, 1375/1372, 32805/32768
-Mapping: {{mapping| 1 0 15 12 -7 | 0 5 -40 -29 33 }}
+{{Mapping|legend=0| 1 0 15 12 -7 | 0 5 -40 -29 33 }}
 Optimal tunings:
@@ Line 2,215: / Line 2,226: @@
 Comma list: 540/539, 729/728, 1375/1372, 4096/4095
-Mapping: {{mapping| 1 0 15 12 -7 -15 | 0 5 -40 -29 33 59 }}
+{{Mapping|legend=0| 1 0 15 12 -7 -15 | 0 5 -40 -29 33 59 }}
 Optimal tunings:
@@ Line 2,250: / Line 2,261: @@
 Comma list: 1375/1372, 4375/4356, 32805/32768
-Mapping: {{mapping| 1 2 -1 -4 -7 | 0 -5 40 82 126 }}
+{{Mapping|legend=0| 1 2 -1 -4 -7 | 0 -5 40 82 126 }}
 Optimal tunings:
@@ Line 2,265: / Line 2,276: @@
 Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647
-Mapping: {{mapping| 1 2 -1 -4 -7 -9 | 0 -5 40 82 126 153 }}
+{{Mapping|legend=0| 1 2 -1 -4 -7 -9 | 0 -5 40 82 126 153 }}
 Optimal tunings:
@@ Line 2,280: / Line 2,291: @@
 Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619
-Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 | 0 -5 40 82 126 153 -11 }}
+{{Mapping|legend=0| 1 2 -1 -4 -7 -9 5 | 0 -5 40 82 126 153 -11 }}
 Optimal tunings:
@@ Line 2,295: / Line 2,306: @@
 Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971
-Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 4 | 0 -5 40 82 126 153 -11 3 }}
+{{Mapping|legend=0| 1 2 -1 -4 -7 -9 5 4 | 0 -5 40 82 126 153 -11 3 }}
 Optimal tunings:
@@ Line 2,329: / Line 2,340: @@
 Comma list: 441/440, 32805/32768, 1953125/1951488
-Mapping: {{mapping| 1 2 -1 -5 -8 | 0 -5 40 94 138 }}
+{{Mapping|legend=0| 1 2 -1 -5 -8 | 0 -5 40 94 138 }}
 Optimal tunings:
@@ Line 2,344: / Line 2,355: @@
 Comma list: 364/363, 441/440, 32805/32768, 109512/109375
-Mapping: {{mapping| 1 2 -1 -5 -8 -11 | 0 -5 40 94 138 177 }}
+{{Mapping|legend=0| 1 2 -1 -5 -8 -11 | 0 -5 40 94 138 177 }}
 Optimal tunings:
@@ Line 2,359: / Line 2,370: @@
 Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768
-Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 | 0 -5 40 94 138 177 -11 }}
+{{Mapping|legend=0| 1 2 -1 -5 -8 -11 5 | 0 -5 40 94 138 177 -11 }}
 Optimal tunings:
@@ Line 2,374: / Line 2,385: @@
 Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859
-Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 4 | 0 -5 40 94 138 177 -11 3 }}
+{{Mapping|legend=0| 1 2 -1 -5 -8 -11 5 4 | 0 -5 40 94 138 177 -11 3 }}
 Optimal tunings:
@@ Line 2,389: / Line 2,400: @@
 Comma list: 5632/5625, 8019/8000, 151263/151250
-Mapping: {{mapping| 1 2 -1 -5 -9 | 0 -5 40 94 150 }}
+{{Mapping|legend=0| 1 2 -1 -5 -9 | 0 -5 40 94 150 }}
 Optimal tunings:
@@ Line 2,404: / Line 2,415: @@
 Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000
-Mapping: {{mapping| 1 2 -1 -5 -9 -11 | 0 -5 40 94 150 177 }}
+{{Mapping|legend=0| 1 2 -1 -5 -9 -11 | 0 -5 40 94 150 177 }}
 Optimal tunings:
@@ Line 2,419: / Line 2,430: @@
 Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750
-Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 | 0 -5 40 94 150 177 -11 }}
+{{Mapping|legend=0| 1 2 -1 -5 -9 -11 5 | 0 -5 40 94 150 177 -11 }}
 Optimal tunings:
@@ Line 2,434: / Line 2,445: @@
 Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700
-Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 4 | 0 -5 40 94 150 177 -11 3 }}
+{{Mapping|legend=0| 1 2 -1 -5 -9 -11 5 4 | 0 -5 40 94 150 177 -11 3 }}
 Optimal tunings:
@@ Line 2,449: / Line 2,460: @@
 Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436
-Mapping: {{mapping| 1 2 -1 -5 -9 14 | 0 -5 40 94 150 -124 }}
+{{Mapping|legend=0| 1 2 -1 -5 -9 14 | 0 -5 40 94 150 -124 }}
 Optimal tunings:
@@ Line 2,464: / Line 2,475: @@
 Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157
-Mapping: {{mapping| 1 2 -1 -5 -9 14 5 | 0 -5 40 94 150 -124 -11 }}
+{{Mapping|legend=0| 1 2 -1 -5 -9 14 5 | 0 -5 40 94 150 -124 -11 }}
 Optimal tunings:
@@ Line 2,479: / Line 2,490: @@
 Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137
-Mapping: {{mapping| 1 2 -1 -5 -9 14 5 4 | 0 -5 40 94 150 -124 -11 3 }}
+{{Mapping|legend=0| 1 2 -1 -5 -9 14 5 4 | 0 -5 40 94 150 -124 -11 3 }}
 Optimal tunings:
@@ Line 2,514: / Line 2,525: @@
 Comma list: 441/440, 4000/3993, 235298/234375
-Mapping: {{mapping| 1 2 -1 -1 0 | 0 -6 48 55 50 }}
+{{Mapping|legend=0| 1 2 -1 -1 0 | 0 -6 48 55 50 }}
 Optimal tunings:
@@ Line 2,529: / Line 2,540: @@
 Comma list: 364/363, 441/440, 676/675, 10985/10976
-Mapping: {{mapping| 1 2 -1 -1 0 1 | 0 -6 48 55 50 39 }}
+{{Mapping|legend=0| 1 2 -1 -1 0 1 | 0 -6 48 55 50 39 }}
 Optimal tunings:
@@ Line 2,564: / Line 2,575: @@
 Comma list: 3025/3024, 24057/24010, 32805/32768
-Mapping: {{mapping| 7 0 105 -56 -120 | 0 1 -8 7 13 }}
+{{Mapping|legend=0| 7 0 105 -56 -120 | 0 1 -8 7 13 }}
 Optimal tunings:
@@ Line 2,579: / Line 2,590: @@
 Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024
-Mapping: {{mapping| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }}
+{{Mapping|legend=0| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }}
 Optimal tunings:
@@ Line 2,614: / Line 2,625: @@
 Comma list: 9801/9800, 32805/32768, 46656/46585
-Mapping: {{mapping| 8 0 120 -117 15 | 0 1 -8 11 1 }}
+{{Mapping|legend=0| 8 0 120 -117 15 | 0 1 -8 11 1 }}
 Optimal tunings:
@@ Line 2,629: / Line 2,640: @@
 Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655
-Mapping: {{mapping| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }}
+{{Mapping|legend=0| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }}
 Optimal tunings:
@@ Line 2,664: / Line 2,675: @@
 Comma list: 540/539, 32805/32768, 42875/42592
-Mapping: {{mapping| 9 0 135 11 131 | 0 1 -8 1 -7 }}
+{{Mapping|legend=0| 9 0 135 11 131 | 0 1 -8 1 -7 }}
 Optimal tunings:
@@ Line 2,679: / Line 2,690: @@
 Comma list: 540/539, 729/728, 4096/4095, 16807/16731
-Mapping: {{mapping| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }}
+{{Mapping|legend=0| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }}
 Optimal tunings:
@@ Line 2,714: / Line 2,725: @@
 Comma list: 540/539, 15488/15435, 32805/32768
-Mapping: {{mapping| 1 -4 47 6 25 | 0 7 56 -4 -27 }}
+{{Mapping|legend=0| 1 -4 47 6 25 | 0 7 56 -4 -27 }}
 Optimal tunings:
@@ Line 2,729: / Line 2,740: @@
 Comma list: 540/539, 729/728, 1573/1568, 4096/4095
-Mapping: {{mapping| 1 -4 47 6 25 -33 | 0 7 56 -4 -27 46 }}
+{{Mapping|legend=0| 1 -4 47 6 25 -33 | 0 7 56 -4 -27 46 }}
 Optimal tunings:
@@ Line 2,740: / Line 2,751: @@
 == Subgroup extensions ==
 === Tridecaschismic (2.3.5.13) ===
-Proposed by [[Eufalesio]] in 2026, tridecaschismic adds the [[325/324|marveltwin comma]] to the comma list, or equivalently, the [[tridecapyth comma]]. It benefits from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is one of the lowest badness schismic extensions. It is also equivalent to the 2.3.5.13 [[restriction]] of 13-limit [[cassandra]].
+Proposed by [[Eufalesio]] in 2026, tridecaschismic adds the [[325/324|marveltwin comma]] to the comma list, or equivalently, the [[tridecapyth comma]]. It benefits from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is one of the lowest badness schismic extensions. It is also equivalent to the [[2.3.5.13 subgroup|2.3.5.13-subgroup]] [[restriction]] of 13-limit [[cassandra]].
 Subgroup: 2.3.5.13
@@ Line 2,748: / Line 2,758: @@
 Comma list: 325/324, 32805/32768
-Subgroup-val mapping: {{mapping| 1 0 15 -28 | 0 1 -8 20 }}
+{{Mapping|legend=2| 1 0 15 -28 | 0 1 -8 20 }}
 Optimal tunings:
@@ Line 2,763: / Line 2,773: @@
 Comma list: 325/324, 361/360, 513/512
-Subgroup-val mapping: {{mapping| 1 0 15 -28 9 | 0 1 -8 20 -3 }}
+{{Mapping|legend=2| 1 0 15 -28 9 | 0 1 -8 20 -3 }}
 Optimal tunings:
@@ Line 2,780: / Line 2,790: @@
 [[Comma list]]: 256/255, 1458/1445
-{{Mapping|legend=2| 1 0 15 -7 | 0 1 -8 7 }}
+{{Mapping|legend=3| 1 0 15 -7 | 0 1 -8 7 }}
-{{Mapping|legend=3| 1 0 15 0 0 0 -7 | 0 1 -8 0 0 0 7 }}
+{{Mapping|legend=5| 1 0 15 0 0 0 -7 | 0 1 -8 0 0 0 7 }}
 : mapping generators: ~2, ~3
@@ Line 2,800: / Line 2,810: @@
 Comma list: 171/170, 256/255, 324/323
-Subgroup-val mapping: {{mapping| 1 0 15 -7 9 | 0 1 -8 7 -3 }}
+{{Mapping|legend=2| 1 0 15 -7 9 | 0 1 -8 7 -3 }}
-Gencom mapping: {{mapping| 1 0 15 0 0 0 -7 9 | 0 1 -8 0 0 0 7 -3 }}
+{{Mapping|legend=4| 1 0 15 0 0 0 -7 9 | 0 1 -8 0 0 0 7 -3 }}
 Optimal tunings:
@@ Line 2,813: / Line 2,823: @@
 === Nestoria (2.3.5.19) ===
-: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''
+{{See also| No-elevens subgroup temperaments #Garibaldia | No-elevens subgroup temperaments #Pontia }}
 Nestoria is notable for having one of the lowest-badness subgroup extensions of schismic. Note that despite prime [[19/1|19]] being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is generally not flatter than the fifth in optimal schismic due to its optimization considering intervals like [[19/10]] and [[19/15]]. However, the dyadic tuning sensitivity of [[19/16]] suggests using tunings like [[65edo]] and [[77edo]] to optimize in favour of prime 19 (especially the minor triad ~16:19:24 which is equated with the Pythagorean minor triad), as [[171edo]] is already arguably undertempered for it despite being the optimal patent val.
@@ Line 2,821: / Line 2,831: @@
 [[Comma list]]: 361/360, 513/512
-{{Mapping|legend=2| 1 0 15 9 | 0 1 -8 -3 }}
+{{Mapping|legend=3| 1 0 15 9 | 0 1 -8 -3 }}
-{{Mapping|legend=3| 1 0 15 0 0 0 0 9 | 0 1 -8 0 0 0 0 -3 }}
+{{Mapping|legend=5| 1 0 15 0 0 0 0 9 | 0 1 -8 0 0 0 0 -3 }}
 : mapping generators: ~2, ~3
@@ Line 2,843: / Line 2,853: @@
 [[Comma list]]: 676/675, 32805/32768
-{{Mapping|legend=2| 1 0 15 14 | 0 2 -16 -13 }}
+{{Mapping|legend=3| 1 0 15 14 | 0 2 -16 -13 }}
-{{Mapping|legend=3| 1 0 15 0 0 14 | 0 2 -16 0 0 -13 }}
+{{Mapping|legend=5| 1 0 15 0 0 14 | 0 2 -16 0 0 -13 }}
 : mapping generators: ~2, ~26/15
@@ Line 2,863: / Line 2,873: @@
 Comma list: 361/360, 513/512, 676/675
-Subgroup-val mapping: {{mapping| 1 0 15 14 9 | 0 2 -16 -13 -6 }}
+{{Mapping|legend=2| 1 0 15 14 9 | 0 2 -16 -13 -6 }}
 Optimal tunings:
@@ Line 2,878: / Line 2,888: @@
 Comma list: 361/360, 481/480, 513/512, 676/675
-Subgroup-val mapping: {{mapping| 1 0 15 14 9 6 | 0 2 -16 -13 -6 -1 }}
+{{Mapping|legend=2| 1 0 15 14 9 6 | 0 2 -16 -13 -6 -1 }}
 Optimal tunings:
@@ Line 2,895: / Line 2,905: @@
 [[Comma list]]: 32805/32768, 1419857/1417176
-{{Mapping|legend=2| 1 2 -1 5 | 0 -5 40 -11 }}
+{{Mapping|legend=3| 1 2 -1 5 | 0 -5 40 -11 }}
-{{Mapping|legend=3| 1 2 -1 0 0 0 5 | 0 -5 40 0 0 0 -11 }}
+{{Mapping|legend=5| 1 2 -1 0 0 0 5 | 0 -5 40 0 0 0 -11 }}
 : mapping generators: ~2, ~18/17
@@ Line 2,915: / Line 2,925: @@
 Comma list: 4624/4617, 6144/6137, 6885/6859
-Subgroup-val mapping: {{mapping| 1 2 -1 5 4 | 0 -5 40 -11 3 }}
+{{Mapping|legend=2| 1 2 -1 5 4 | 0 -5 40 -11 3 }}
-Gencom mapping: {{mapping| 1 2 -1 0 0 0 5 4 | 0 -5 40 0 0 0 -11 3 }}
+{{Mapping|legend=4| 1 2 -1 0 0 0 5 4 | 0 -5 40 0 0 0 -11 3 }}
 Optimal tunings:
@@ Line 2,926: / Line 2,936: @@
 Badness (Sintel): 1.17
+== References ==
+<references/>
 [[Category:Schismatic family| ]] <!-- main article -->
 [[Category:Temperament families]]
 [[Category:Catalogs of rank-2 temperaments]]