Schismatic family: Difference between revisions

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~~{{interwiki~~

~~| en = Schismatic family~~

~~| de = Schismatische Temperaturen~~

~~| es =~~

~~| ja =~~

}}

The [[5-limit]] parent comma for the '''schismatic''' (or '''schismic''') '''family''' is the [[schisma]] of 32805/32768, which is the amount by which the [[Pythagorean comma]] exceeds the [[syntonic comma]] (81/80), or alternatively put, the difference between a [[5/4|just major third]] and a [[8192/6561|Pythagorean diminished fourth]].

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=== Overview to extensions ===

The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at. [[#Garibaldi|Garibaldi]] adds [[garischisma|{{monzo| 25 -14 0 -1 }}]]. [[#Grackle|~~Grackle~~]] adds {{monzo| -44 26 0 1 }}. [[#Pontiac|~~Pontiac~~]] adds {{monzo| -59 39 0 -1 }}. [[#Schism|~~Schism~~]] adds [[64/63|{{monzo| 6 -2 0 -1 }}]]. Those all have a fifth as generator.

The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at. [[#Garibaldi|Garibaldi]] adds [[garischisma|{{monzo| 25 -14 0 -1 }}]], [[#Grackle|grackle]] adds {{monzo| -44 26 0 1 }}, [[#Pontiac|pontiac]] adds {{monzo| -59 39 0 -1 }}, and [[#Schism|schism]] adds [[64/63|{{monzo| 6 -2 0 -1 }}]]. Those all have a fifth as generator.

[[#Bischismic|Bischismic]] adds {{monzo| -69 40 0 2 }} and has a fifth generator with a half-octave period. [[#Salsa|Salsa]] adds [[parahemif comma|{{monzo| 15 -13 0 2 }}]] and has a hemififth generator. [[#Hemischis|Hemischis]] adds {{monzo| -34 25 0 -2 }} and has a hemitwelfth generator. [[Gamelismic clan #Guiron|Guiron]] adds [[1029/1024|{{monzo| -10 1 0 3 }}]], with an ~8/7 generator, three of which give the fifth. [[#Term|Term]] adds {{monzo| -94 54 0 3 }} with a 1/3-octave period. [[#Squirrel|Squirrel]], [[#Tertiaschis|tertiaschis]], and [[#Countertertiaschis|countertertiaschis]] each has a generator that is 1/3 of the fourth. [[#Quadrant|Quadrant]] adds {{monzo| -119 68 0 4 }} with a 1/4-octave period. [[#Kleischismic|Kleischismic]] adds {{monzo| 49 -38 0 4 }} with a half-octave period and also a bisect generator. [[#Sesquiquartififths|Sesquiquartififths]] adds {{monzo| -35 15 0 4 }} and slices the fifth in four.

Temperaments discussed elsewhere include:

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* ''[[Pogo]]'' (+118098/117649) → [[Stearnsmic clan #Pogo|Stearnsmic clan]]

Considered below are garibaldi, pontiac, grackle, schism, bischismic, kleischismic, salsa, hemischis, term, altinex, squirrel, tertiaschis, countertertiaschis, quadrant, sesquiquartififths, tsaharuk, quanharuk, quintilipyth, quintaschis, sextilifourths, octant, nonant~~, septant~~, septiquarschis, and tridecafifths.

Considered below are garibaldi, pontiac, grackle, schism, bischismic, kleischismic, salsa, hemischis, term, altinex, squirrel, tertiaschis, countertertiaschis, quadrant, sesquiquartififths, tsaharuk, quanharuk, quintilipyth, quintaschis, sextilifourths, septant, octant, nonant, septiquarschis, and tridecafifths.

The schismatic family boasts a variety of remarkable extensions to subgroups in high prime limits. These are listed at the bottom of this page, in [[#Subgroup extensions]].

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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

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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 114:

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 138:

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 153:

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 168:

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 183:

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 198:

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 213:

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 228:

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:

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=== 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 266:

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 290:

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 305:

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 320:

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 335:

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 350:

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 365:

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 380:

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 399:

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 418:

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 433:

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 450:

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 466:

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 481:

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 497:

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 512:

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 528:

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 552:

Line 550:

* [[WE]]: ~2 = 1200.0989{{c}}, ~3/2 = 701.8145{{c}}

: [[error map]]: {{val| +0.099 -0.042 -0.138 -0.038 }}

* [[CWE]]: ~2 = 1200.0000{{c]}, ~3/2 = 701.7579{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7579{{c}}

: error map: {{val| 0.000 -0.197 -0.377 -0.268 }}

Line 578:

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 597:

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 616:

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 635:

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 650:

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 665:

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 682:

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 701:

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 720:

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 741:

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 760:

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 779:

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 798:

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 813:

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 830:

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 846:

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 861:

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 876:

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 891:

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 906:

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 921:

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 945:

Line 943:

* [[WE]]: ~2 = 1199.7974{{c}}, ~3/2 = 701.1210{{c}}

: [[error map]]: {{val| -0.203 -1.037 +3.300 -1.618 }}

* [[CWE]]: ~2 = 1200.0000{{c]}, ~3/2 = 701.2465{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.2465{{c}}

: error map: {{val| 0.000 -0.709 +3.715 -1.234 }}

Line 961:

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 976:

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 991:

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 1,006:

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 1,021:

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 1,036:

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 1,051:

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 1,066:

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 1,075:

Line 1,073:

Badness (Sintel): 2.01

== Quasipyth ==

Named by [[Xenllium]] in 2026, quasipyth tempers out {{monzo| 109 -67 0 -1 }}, the [[nanisma]], as well as the [[catasyc comma]], 390625/387072. The 7/4 is found at −67 fifths, represented by the nonuple-diminished thirteenth.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768, 390625/387072

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.2569{{c}}, ~3/2 = 702.1149{{c}}

: [[error map]]: {{val| +0.2569 +0.4168 -1.4342 +0.2685 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9615{{c}}

: error map: {{val| 0.0000 +0.0065 -2.0054 -0.2437 }}

{{Optimal ET sequence|legend=1| 53, 147d, 200, 253, 306c, 559c }}

[[Badness]] (Sintel): 5.04

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

Optimal tunings:

* WE: ~2 = 1200.3283{{c}}, ~3/2 = 702.1636{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9713{{c}}

Badness (Sintel): 3.83

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

Optimal tunings:

* WE: ~2 = 1200.3229{{c}}, ~3/2 = 702.1603{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9714{{c}}

Badness (Sintel): 2.13

== Schism ==

Line 1,110:

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 1,125:

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 1,140:

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 1,155:

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 1,170:

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 1,205:

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 1,220:

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 1,235:

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 1,250:

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 1,265:

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 1,300:

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 1,315:

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:

* WE: ~2 = 1199.9362{c}}, ~11/9 = 351.0061{{c}}

* WE: ~2 = 1199.9362{{c}}, ~11/9 = 351.0061{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0247{{c}}

Line 1,352:

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 1,367:

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 1,382:

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 1,397:

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 1,412:

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 1,426:

Line 1,473:

== Term ==

Term tempers out the [[landscape comma]], mapping [[63/50]] to the 1/3-octave period. It can be described as {{nowrap| 12 & 171 }}, and is the unique temperament that ~~equates a~~ syntonic~Pythagorean ~~comma~~ with a stack of three [[marvel comma]]s. A [[septimal comma]] is then found as a stack of four marvel commas. In ~~some~~ 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[171edo]] makes for an excellent tuning.

Term tempers out the [[landscape comma]], mapping [[63/50]] to the 1/3-octave period. It can be described as {{nowrap| 12 & 171 }}, and is the unique temperament that tempers together the syntonic and Pythagorean commas and equates it with a stack of three [[marvel comma]]s. A [[septimal comma]] is then found as a stack of four marvel commas. In certain 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma #As an interval region|kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[171edo]] makes for an excellent tuning.

[[Subgroup]]: 2.3.5.7

Line 1,456:

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 1,471:

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 1,486:

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 1,503:

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 1,518:

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 1,533:

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 1,550:

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 1,566:

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 1,579:

Line 1,626:

=== Hemiterm ===

The hemiterm temperament tempers out [[3025/3024]] (lehmerisma), and may be described as {{nowrap| 159 & 183 }}. Its ploidacot is triploid ~~beta~~-dicot.

The hemiterm temperament tempers out [[3025/3024]] (lehmerisma), and may be described as {{nowrap| 159 & 183 }}. Its ploidacot is triploid alpha-dicot.

Subgroup: 2.3.5.7.11

Line 1,585:

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 1,601:

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 1,616:

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 1,651:

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 1,666:

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 1,700:

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 1,715:

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 1,726:

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.

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 1,749:

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 1,764:

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 1,779:

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 1,790:

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 1,813:

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 1,828:

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 1,839:

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 1,863:

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 1,878:

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 1,889:

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 1,919:

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 1,934:

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 1,949:

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 1,964:

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 1,979:

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 1,994:

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 2,009:

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 2,024:

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 2,039:

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 2,054:

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 2,069:

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 2,081:

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 2,107:

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 2,122:

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 2,133:

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 2,157:

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 2,172:

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 2,207:

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 2,222:

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 2,237:

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 2,252:

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 2,286:

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 2,301:

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 2,316:

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 2,331:

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 2,346:

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 2,361:

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 2,376:

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 2,391:

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 2,406:

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 2,421:

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 2,436:

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 2,471:

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 2,486:

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 2,496:

Line 2,550:

Badness (Sintel): 1.04

== ~~Octant~~ ==

== Septant ==

~~Octant~~ may be described as the {{nowrap| 224 & ~~248~~ }} temperament. It has a period of 1/8 octave, and its [[ploidacot]] is ~~octaploid~~ monocot~~. In this temperament, [[12/11]], [[35/27]], and [[99/70]] are mapped to 1\8, 3\8, and 4\8 respectively~~.

Named by [[Xenllium]] in 2021, septant notably tempers out the [[akjaysma]] ({{monzo| 47 -7 -7 -7 }}) and may be described as the {{nowrap| 224 & 301 }} temperament. It has a period of 1/7 octave, and its [[ploidacot]] is heptaploid monocot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768, ~~2259436291848~~/~~2251875390625~~

[[Comma list]]: 32805/32768, 516560652/514714375

: mapping generators: ~~~42875~~/~~39366~~, ~3

: mapping generators: ~8575/7776, ~3

[[Optimal tuning]]s:

* [[WE]]: ~~~42875~~/~~39366~~ = ~~150~~.~~0048~~{{c}}, ~3/2 = 701.~~7356~~{{c}}

* [[WE]]: ~8575/7776 = 171.4303{{c}}, ~3/2 = 701.7091{{c}}

: [[error map]]: {{val| +0.~~039~~ -0.~~181~~ +0.~~071~~ +0.~~127~~ }}

: [[error map]]: {{val| +0.012 -0.234 +0.096 +0.265 }}

* [[CWE]]: ~~~42875~~/~~39366~~ = ~~150~~.~~0000~~{{c}}, ~3/2 = 701.~~7134~~{{c}}

* [[CWE]]: ~8575/7776 = 171.4286{{c}}, ~3/2 = 701.7022{{c}}

: error map: {{val| 0.000 -0.~~242 -~~0.~~021~~ +0.~~022~~ }}

: error map: {{val| 0.000 -0.253 +0.069 +0.232 }}

{{Optimal ET sequence|legend=1| 24, …, 224, ~~472~~, ~~696~~, ~~1168~~ }}

{{Optimal ET sequence|legend=1| 77, 147, 224, 301, 525, 826, 1351 }}

[[Badness]] (Sintel): 3.98

[[Badness]] (Sintel): 2.81

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

Optimal tunings:

* WE: ~12/11 = ~~150~~.~~0010~~{{c}}, ~3/2 = 701.~~7177~~{{c}}

* WE: ~495/448 = 171.4334{{c}}, ~3/2 = 701.7387{{c}}

* CWE: ~12/11 = ~~150~~.~~0000~~{{c}}, ~3/2 = 701.~~7131~~{{c}}

* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7198{{c}}

{{Optimal ET sequence|legend=0| 24, …, 224, ~~472~~, ~~696, 1168~~ }}

Badness (Sintel): 1.48

Badness (Sintel): 1.46

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal tunings:

* WE: ~12/11 = ~~149~~.~~9957~~{{c}}, ~3/2 = 701.~~7046~~{{c}}

* WE: ~495/448 = 171.4282{{c}}, ~3/2 = 701.7229{{c}}

* CWE: ~12/11 = ~~150~~.~~0000~~{{c}}, ~3/2 = 701.~~7247~~{{c}}

* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7242{{c}}

Badness (Sintel): 1.26

Badness (Sintel): 1.02

== ~~Nonant~~ ==

== Octant ==

~~Named by [[Xenllium]] in 2023, nonant tempers out the [[septimal ennealimma]] ({{monzo| -11 -9 0 9 }}) and~~ may be described as the {{nowrap| 36 & ~~171~~ }} temperament. It has a period of 1/9 octave, and its [[ploidacot]] is ~~enneaploid~~ monocot.

Octant may be described as the {{nowrap| 224 & 248 }} temperament. It has a period of 1/8 octave, and its [[ploidacot]] is octaploid monocot. In this temperament, [[12/11]], [[35/27]], and [[99/70]] are mapped to 1\8, 3\8, and 4\8 respectively.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768, ~~40353607~~/~~40310784~~

[[Comma list]]: 32805/32768, 2259436291848/2251875390625

: mapping generators: ~~~2592~~/~~2401~~, ~3

: mapping generators: ~42875/39366, ~3

[[Optimal tuning]]s:

* [[WE]]: ~~~2592~~/~~2401~~ = ~~133~~.~~3442~~{{c}}, ~3/2 = 701.~~8000~~{{c}}

* [[WE]]: ~42875/39366 = 150.0048{{c}}, ~3/2 = 701.7356{{c}}

: [[error map]]: {{val| +0.~~098~~ -0.~~057 -~~0.~~027 -~~0.~~141~~ }}

: [[error map]]: {{val| +0.039 -0.181 +0.071 +0.127 }}

* [[CWE]]: ~~~2592~~/~~2401~~ = ~~133~~.~~3333~~{{c}}, ~3/2 = 701.~~7384~~{{c}}

* [[CWE]]: ~42875/39366 = 150.0000{{c}}, ~3/2 = 701.7134{{c}}

: error map: {{val| 0.000 -0.~~217~~ -0.~~221 -~~0.~~421~~ }}

: error map: {{val| 0.000 -0.242 -0.021 +0.022 }}

{{Optimal ET sequence|legend=1| 36, ~~99c~~, ~~135~~, ~~171, 2772bd, 2943bdd, …~~, ~~5166bccddd~~, ~~5337bccddd~~ }}

[[Badness]] (Sintel): 1.77

[[Badness]] (Sintel): 3.98

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

Optimal tunings:

* WE: ~~~242~~/~~225~~ = ~~133~~.~~3308~~{{c}}, ~3/2 = 701.~~8205~~{{c}}

* WE: ~12/11 = 150.0010{{c}}, ~3/2 = 701.7177{{c}}

* CWE: ~~~242~~/~~225~~ = ~~133~~.~~3333~~{{c}}, ~3/2 = 701.~~8351~~{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7131{{c}}

Badness (Sintel): 4.20

Badness (Sintel): 1.48

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal tunings:

* WE: ~~~242~~/~~225~~ = ~~133~~.~~3180~~{{c}}, ~3/2 = 701.~~6956~~{{c}}

* WE: ~12/11 = 149.9957{{c}}, ~3/2 = 701.7046{{c}}

* CWE: ~~~242~~/~~225~~ = ~~133~~.~~3333~~{{c}}, ~3/2 = 701.~~7800~~{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7247{{c}}

{{Optimal ET sequence|legend=0| 36, ~~99cf~~, ~~135~~, ~~171~~ }}

Badness (Sintel): 3.15

Badness (Sintel): 1.26

== ~~Septant~~ ==

== Nonant ==

Named by [[Xenllium]] in ~~2021~~, ~~septant notably~~ tempers out the [[~~akjaysma~~]] ({{monzo| 47 -7 -~~7 -7~~ }}) and may be described as the {{nowrap| ~~224~~ & ~~301~~ }} temperament. It has a period of 1/7 octave, and its [[ploidacot]] is ~~heptaploid~~ monocot.

Named by [[Xenllium]] in 2023, nonant tempers out the [[septimal ennealimma]] ({{monzo| -11 -9 0 9 }}) and may be described as the {{nowrap| 36 & 171 }} temperament. It has a period of 1/9 octave, and its [[ploidacot]] is enneaploid monocot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768, ~~516560652~~/~~514714375~~

[[Comma list]]: 32805/32768, 40353607/40310784

: mapping generators: ~~~8575~~/~~7776~~, ~3

: mapping generators: ~2592/2401, ~3

[[Optimal tuning]]s:

* [[WE]]: ~~~8575~~/~~7776~~ = ~~171~~.~~4303~~{{c}}, ~3/2 = 701.~~7091~~{{c}}

* [[WE]]: ~2592/2401 = 133.3442{{c}}, ~3/2 = 701.8000{{c}}

: [[error map]]: {{val| +0.~~012~~ -0.~~234 +~~0.~~096 +~~0.~~265~~ }}

: [[error map]]: {{val| +0.098 -0.057 -0.027 -0.141 }}

* [[CWE]]: ~~~8575~~/~~7776~~ = ~~171~~.~~4286~~{{c}}, ~3/2 = 701.~~7022~~{{c}}

* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.7384{{c}}

: error map: {{val| 0.000 -0.~~253 +~~0.~~069 +~~0.~~232~~ }}

: error map: {{val| 0.000 -0.217 -0.221 -0.421 }}

{{Optimal ET sequence|legend=1| 77, ~~147~~, ~~224~~, ~~301~~, ~~525~~, ~~826~~, ~~1351~~ }}

{{Optimal ET sequence|legend=1| 36, 99c, 135, 171, 2772bd, 2943bdd, …, 5166bccddd, 5337bccddd }}

[[Badness]] (Sintel): 2.81

[[Badness]] (Sintel): 1.77

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

Optimal tunings:

* WE: ~~~495~~/~~448~~ = ~~171~~.~~4334~~{{c}}, ~3/2 = 701.~~7387~~{{c}}

* WE: ~242/225 = 133.3308{{c}}, ~3/2 = 701.8205{{c}}

* CWE: ~~~495~~/~~448~~ = ~~171~~.~~4286~~{{c}}, ~3/2 = 701.~~7198~~{{c}}

* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.8351{{c}}

{{Optimal ET sequence|legend=0| 77, ~~147~~, ~~224, 301, 525~~ }}

Badness (Sintel): 1.46

Badness (Sintel): 4.20

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal tunings:

* WE: ~~~495~~/~~448~~ = ~~171~~.~~4282~~{{c}}, ~3/2 = 701.~~7229~~{{c}}

* WE: ~242/225 = 133.3180{{c}}, ~3/2 = 701.6956{{c}}

* CWE: ~~~495~~/~~448~~ = ~~171~~.~~4286~~{{c}}, ~3/2 = 701.~~7242~~{{c}}

* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.7800{{c}}

{{Optimal ET sequence|legend=0| 77, ~~147~~, ~~224~~, ~~525, 1274f~~ }}

Badness (Sintel): 1.02

Badness (Sintel): 3.15

== Septiquarschis ==

Line 2,671:

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 2,686:

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 2,696:

Line 2,750:

Badness (Sintel): 1.46

== ~~Tridecafifths~~ ==

== Subgroup extensions ==

~~Named~~ by [[~~Eliora~~]] in ~~2023~~, ~~tridecafifths may be described as~~ the ~~{{nowrap~~| ~~89 & 200 }} temperament~~. It ~~divides~~ the [[3/2~~|perfect fifth~~]] ~~into thirteen quartertones, so its~~ [[~~ploidacot~~]] is 13-~~cot.~~ [[~~289edo~~]] ~~gives a highly recommendable tuning~~.

=== 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 subgroup|2.3.5.13-subgroup]] [[restriction]] of 13-limit [[cassandra]].

[[Subgroup]]: 2.3.5.7

Subgroup: 2.3.5.13

[[Comma list]]: 32805/32768~~, {{monzo| -14 -1 -9 13 }}~~

Comma list: 325/324, 32805/32768

~~: mapping generators: ~2, ~1323/1280~~

[[Optimal ~~tuning]]s~~:

Optimal tunings:

* [[WE]]: ~2 = 1200.~~1431~~{{c}}, ~~~1323~~/~~1280~~ = 53.~~9838~~{{c}}

* WE: ~2 = 1200.3326{{c}} ~3/2 = 702.1092{{c}}

~~: [[error map]]: {{val| +0.143 -0.023 +0.375 -0.816~~ }}

* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9189{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~~~1323~~/~~1280~~ = 53.~~9764~~{{c}}

~~: error map: {{val| 0.000 -0.261 -0.221 -0.421~~ }}

{{Optimal ET sequence|legend=0| 12, …, 41, 53, 412cf, 465cf, …, 783ccff, 836ccfff }}

[[Badness]] (Sintel): 10.9

Badness (Sintel): 0.582

=== ~~11-limit~~ ===

==== 2.3.5.13.19 subgroup ====

Subgroup: 2.3.5.7.11

Subgroup: 2.3.5.13.19

Comma list: ~~441~~/~~440~~, ~~32805~~/~~32768~~, ~~55296000~~/~~55240493~~

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

~~Mapping:~~ {{~~mapping~~| 1 ~~1 7 6 4~~ | 0 ~~13 -104~~ -71 -12 }}

Optimal tunings:

* WE: ~2 = 1200.~~0311~~{{c}}, ~33/32 = 53.~~9766~~{{c}}

* WE: ~2 = 1200.4236{{c}}, ~3/2 = 702.1510{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 53.~~9750~~{{c}}

* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9064{{c}}

Badness (Sintel): 4.23

Badness (Sintel): 0.354

~~== Subgroup extensions ==~~

=== Photia (2.3.5.17) ===

Line 2,739:

Line 2,790:

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

: mapping generators: ~2, ~3

Line 2,759:

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 2,772:

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 }}

~~The [[S~~-~~expression]]-based comma list~~ of ~~this temperament is {[[1216/1215|S16/S18]], [[361/360|S19]], ([[513/512|S15/S20]])}~~. 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]].

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.

[[Subgroup]]: 2.3.5.19

Line 2,780:

Line 2,831:

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

: mapping generators: ~2, ~3

Line 2,802:

Line 2,853:

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

: mapping generators: ~2, ~26/15

Line 2,822:

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 2,837:

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 2,854:

Line 2,905:

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

: mapping generators: ~2, ~18/17

Line 2,874:

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 2,886:

Line 2,937:

Badness (Sintel): 1.17

== References ==

[[Category:Schismatic family| ]]

[[Category:Temperament families]]

[[Category:~~Schismatic family| ]] <!~~-~~- main article -->~~

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

~~[[Category:Rank~~ 2]]

@@ Line 1: / Line 1: @@
-{{interwiki
-| en = Schismatic family
-| de = Schismatische Temperaturen
-| es =
-| ja =
-}}
 {{Technical data page}}
 The [[5-limit]] parent comma for the '''schismatic''' (or '''schismic''') '''family''' is the [[schisma]] of 32805/32768, which is the amount by which the [[Pythagorean comma]] exceeds the [[syntonic comma]] (81/80), or alternatively put, the difference between a [[5/4|just major third]] and a [[8192/6561|Pythagorean diminished fourth]].
@@ Line 37: / Line 31: @@
 === Overview to extensions ===
-The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at. [[#Garibaldi|Garibaldi]] adds [[garischisma|{{monzo| 25 -14 0 -1 }}]]. [[#Grackle|Grackle]] adds {{monzo| -44 26 0 1 }}. [[#Pontiac|Pontiac]] adds {{monzo| -59 39 0 -1 }}. [[#Schism|Schism]] adds [[64/63|{{monzo| 6 -2 0 -1 }}]]. Those all have a fifth as generator.
+The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at. [[#Garibaldi|Garibaldi]] adds [[garischisma|{{monzo| 25 -14 0 -1 }}]], [[#Grackle|grackle]] adds {{monzo| -44 26 0 1 }}, [[#Pontiac|pontiac]] adds {{monzo| -59 39 0 -1 }}, and [[#Schism|schism]] adds [[64/63|{{monzo| 6 -2 0 -1 }}]]. Those all have a fifth as generator.
 [[#Bischismic|Bischismic]] adds {{monzo| -69 40 0 2 }} and has a fifth generator with a half-octave period. [[#Salsa|Salsa]] adds [[parahemif comma|{{monzo| 15 -13 0 2 }}]] and has a hemififth generator. [[#Hemischis|Hemischis]] adds {{monzo| -34 25 0 -2 }} and has a hemitwelfth generator. [[Gamelismic clan #Guiron|Guiron]] adds [[1029/1024|{{monzo| -10 1 0 3 }}]], with an ~8/7 generator, three of which give the fifth. [[#Term|Term]] adds {{monzo| -94 54 0 3 }} with a 1/3-octave period. [[#Squirrel|Squirrel]], [[#Tertiaschis|tertiaschis]], and [[#Countertertiaschis|countertertiaschis]] each has a generator that is 1/3 of the fourth. [[#Quadrant|Quadrant]] adds {{monzo| -119 68 0 4 }} with a 1/4-octave period. [[#Kleischismic|Kleischismic]] adds {{monzo| 49 -38 0 4 }} with a half-octave period and also a bisect generator. [[#Sesquiquartififths|Sesquiquartififths]] adds {{monzo| -35 15 0 4 }} and slices the fifth in four.
-Temperaments involving larger splits include [[#Tsaharuk|tsaharuk]], [[#Quanharuk|quanharuk]], [[#Quintilipyth|quintilipyth]], [[#Quintaschis|quintaschis]], [[#Altinex|altinex]], [[Stearnsmic clan #Pogo|pogo]], [[#Sextilifourths|sextilifourths]], [[#Octant|octant]], [[#Nonant|nonant]], [[#Septant|septant]], [[#Septiquarschis|septiquarschis]], and [[#Tridecafifths|tridecafifths]]. Those split the schismic structure into five to thirteen parts.
+Temperaments involving larger splits include [[#Tsaharuk|tsaharuk]], [[#Quanharuk|quanharuk]], [[#Quintilipyth|quintilipyth]], [[#Quintaschis|quintaschis]], [[#Altinex|altinex]], [[Stearnsmic clan #Pogo|pogo]], [[#Sextilifourths|sextilifourths]], [[#Septant|septant]], [[#Octant|octant]], [[#Nonant|nonant]], [[#Septiquarschis|septiquarschis]], and [[#Tridecafifths|tridecafifths]]. Those split the schismic structure into five to thirteen parts.
 Temperaments discussed elsewhere include:
@@ Line 47: / Line 41: @@
 * ''[[Pogo]]'' (+118098/117649) → [[Stearnsmic clan #Pogo|Stearnsmic clan]]
-Considered below are garibaldi, pontiac, grackle, schism, bischismic, kleischismic, salsa, hemischis, term, altinex, squirrel, tertiaschis, countertertiaschis, quadrant, sesquiquartififths, tsaharuk, quanharuk, quintilipyth, quintaschis, sextilifourths, octant, nonant, septant, septiquarschis, and tridecafifths.
+Considered below are garibaldi, pontiac, grackle, schism, bischismic, kleischismic, salsa, hemischis, term, altinex, squirrel, tertiaschis, countertertiaschis, quadrant, sesquiquartififths, tsaharuk, quanharuk, quintilipyth, quintaschis, sextilifourths, septant, octant, nonant, septiquarschis, and tridecafifths.
 The schismatic family boasts a variety of remarkable extensions to subgroups in high prime limits. These are listed at the bottom of this page, in [[#Subgroup extensions]].
@@ Line 54: / 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 85: / 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 91: / 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 114: / 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 138: / 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 153: / 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 168: / 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 183: / 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 198: / 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 213: / 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 228: / 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 239: / 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 266: / 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 290: / 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 305: / 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 320: / 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 335: / 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 350: / 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 365: / 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 380: / 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 399: / 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 418: / 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 433: / 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 450: / 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 466: / 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 481: / 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 497: / 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 512: / 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 528: / 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 552: / Line 550: @@
 * [[WE]]: ~2 = 1200.0989{{c}}, ~3/2 = 701.8145{{c}}
 : [[error map]]: {{val| +0.099 -0.042 -0.138 -0.038 }}
-* [[CWE]]: ~2 = 1200.0000{{c]}, ~3/2 = 701.7579{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7579{{c}}
 : error map: {{val| 0.000 -0.197 -0.377 -0.268 }}
@@ Line 578: / 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 597: / 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 616: / 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 635: / 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 650: / 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 665: / 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 682: / 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 701: / 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 720: / 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 741: / 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 760: / 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 779: / 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 798: / 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 813: / 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 830: / 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 846: / 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 861: / 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 876: / 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 891: / 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 906: / 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 921: / 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 945: / Line 943: @@
 * [[WE]]: ~2 = 1199.7974{{c}}, ~3/2 = 701.1210{{c}}
 : [[error map]]: {{val| -0.203 -1.037 +3.300 -1.618 }}
-* [[CWE]]: ~2 = 1200.0000{{c]}, ~3/2 = 701.2465{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.2465{{c}}
 : error map: {{val| 0.000 -0.709 +3.715 -1.234 }}
@@ Line 961: / 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 976: / 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 991: / 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 1,006: / 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 1,021: / 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 1,036: / 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 1,051: / 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 1,066: / 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 1,075: / Line 1,073: @@
 Badness (Sintel): 2.01
+== Quasipyth ==
+Named by [[Xenllium]] in 2026, quasipyth tempers out {{monzo| 109 -67 0 -1 }}, the [[nanisma]], as well as the [[catasyc comma]], 390625/387072. The 7/4 is found at −67 fifths, represented by the nonuple-diminished thirteenth.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 32805/32768, 390625/387072
+{{Mapping|legend=1| 1 0 15 109 | 0 1 -8 -67 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.2569{{c}}, ~3/2 = 702.1149{{c}}
+: [[error map]]: {{val| +0.2569 +0.4168 -1.4342 +0.2685 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9615{{c}}
+: error map: {{val| 0.0000 +0.0065 -2.0054 -0.2437 }}
+{{Optimal ET sequence|legend=1| 53, 147d, 200, 253, 306c, 559c }}
+[[Badness]] (Sintel): 5.04
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 385/384, 19712/19683, 78125/77616
+{{Mapping|legend=0| 1 0 15 109 -117 | 0 1 -8 -67 76 }}
+Optimal tunings:
+* WE: ~2 = 1200.3283{{c}}, ~3/2 = 702.1636{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9713{{c}}
+{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}
+Badness (Sintel): 3.83
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 325/324, 385/384, 2200/2197, 19712/19683
+{{Mapping|legend=0| 1 0 15 109 -117 -28 | 0 1 -8 -67 76 20 }}
+Optimal tunings:
+* WE: ~2 = 1200.3229{{c}}, ~3/2 = 702.1603{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9714{{c}}
+{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}
+Badness (Sintel): 2.13
 == Schism ==
@@ Line 1,110: / 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 1,125: / 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 1,140: / 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 1,155: / 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 1,170: / 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 1,205: / 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 1,220: / 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 1,235: / 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 1,250: / 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 1,265: / 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 1,300: / 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 1,315: / 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:
-* WE: ~2 = 1199.9362{c}}, ~11/9 = 351.0061{{c}}
+* WE: ~2 = 1199.9362{{c}}, ~11/9 = 351.0061{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0247{{c}}
@@ Line 1,352: / 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 1,367: / 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 1,382: / 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 1,397: / 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 1,412: / 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 1,426: / Line 1,473: @@
 == Term ==
-Term tempers out the [[landscape comma]], mapping [[63/50]] to the 1/3-octave period. It can be described as {{nowrap| 12 & 171 }}, and is the unique temperament that equates a syntonic~Pythagorean comma with a stack of three [[marvel comma]]s. A [[septimal comma]] is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[171edo]] makes for an excellent tuning.
+Term tempers out the [[landscape comma]], mapping [[63/50]] to the 1/3-octave period. It can be described as {{nowrap| 12 & 171 }}, and is the unique temperament that tempers together the syntonic and Pythagorean commas and equates it with a stack of three [[marvel comma]]s. A [[septimal comma]] is then found as a stack of four marvel commas. In certain 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma #As an interval region|kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[171edo]] makes for an excellent tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,456: / 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 1,471: / 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 1,486: / 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 1,503: / 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 1,518: / 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 1,533: / 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 1,550: / 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 1,566: / 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 1,579: / Line 1,626: @@
 === Hemiterm ===
-The hemiterm temperament tempers out [[3025/3024]] (lehmerisma), and may be described as {{nowrap| 159 & 183 }}. Its ploidacot is triploid beta-dicot.
+The hemiterm temperament tempers out [[3025/3024]] (lehmerisma), and may be described as {{nowrap| 159 & 183 }}. Its ploidacot is triploid alpha-dicot.
 Subgroup: 2.3.5.7.11
@@ Line 1,585: / 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 1,601: / 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 1,616: / 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 1,651: / 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 1,666: / 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 1,700: / 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 1,715: / 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 1,726: / 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.
+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 1,749: / 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 1,764: / 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 1,779: / 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 1,790: / 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 1,813: / 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 1,828: / 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 1,839: / 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 1,863: / 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 1,878: / 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 1,889: / 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 1,919: / 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 1,934: / 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 1,949: / 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 1,964: / 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 1,979: / 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 1,994: / 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 2,009: / 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 2,024: / 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 2,039: / 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 2,054: / 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 2,069: / 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 2,081: / 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 2,107: / 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 2,122: / 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 2,133: / 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 2,157: / 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 2,172: / 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 2,207: / 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 2,222: / 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 2,237: / 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 2,252: / 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 2,286: / 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 2,301: / 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 2,316: / 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 2,331: / 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 2,346: / 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 2,361: / 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 2,376: / 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 2,391: / 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 2,406: / 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 2,421: / 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 2,436: / 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 2,471: / 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 2,486: / 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 2,496: / Line 2,550: @@
 Badness (Sintel): 1.04
-== Octant ==
+== Septant ==
-Octant may be described as the {{nowrap| 224 & 248 }} temperament. It has a period of 1/8 octave, and its [[ploidacot]] is octaploid monocot. In this temperament, [[12/11]], [[35/27]], and [[99/70]] are mapped to 1\8, 3\8, and 4\8 respectively.
+Named by [[Xenllium]] in 2021, septant notably tempers out the [[akjaysma]] ({{monzo| 47 -7 -7 -7 }}) and may be described as the {{nowrap| 224 & 301 }} temperament. It has a period of 1/7 octave, and its [[ploidacot]] is heptaploid monocot.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 32805/32768, 2259436291848/2251875390625
+[[Comma list]]: 32805/32768, 516560652/514714375
-{{Mapping|legend=1| 8 0 120 -117 | 0 1 -8 11 }}
+{{Mapping|legend=1| 7 0 105 -56 | 0 1 -8 7 }}
-: mapping generators: ~42875/39366, ~3
+: mapping generators: ~8575/7776, ~3
 [[Optimal tuning]]s:
-* [[WE]]: ~42875/39366 = 150.0048{{c}}, ~3/2 = 701.7356{{c}}
+* [[WE]]: ~8575/7776 = 171.4303{{c}}, ~3/2 = 701.7091{{c}}
-: [[error map]]: {{val| +0.039 -0.181 +0.071 +0.127 }}
+: [[error map]]: {{val| +0.012 -0.234 +0.096 +0.265 }}
-* [[CWE]]: ~42875/39366 = 150.0000{{c}}, ~3/2 = 701.7134{{c}}
+* [[CWE]]: ~8575/7776 = 171.4286{{c}}, ~3/2 = 701.7022{{c}}
-: error map: {{val| 0.000 -0.242 -0.021 +0.022 }}
+: error map: {{val| 0.000 -0.253 +0.069 +0.232 }}
-{{Optimal ET sequence|legend=1| 24, …, 224, 472, 696, 1168 }}
+{{Optimal ET sequence|legend=1| 77, 147, 224, 301, 525, 826, 1351 }}
-[[Badness]] (Sintel): 3.98
+[[Badness]] (Sintel): 2.81
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 9801/9800, 32805/32768, 46656/46585
+Comma list: 3025/3024, 24057/24010, 32805/32768
-Mapping: {{mapping| 8 0 120 -117 15 | 0 1 -8 11 1 }}
+{{Mapping|legend=0| 7 0 105 -56 -120 | 0 1 -8 7 13 }}
 Optimal tunings:
-* WE: ~12/11 = 150.0010{{c}}, ~3/2 = 701.7177{{c}}
+* WE: ~495/448 = 171.4334{{c}}, ~3/2 = 701.7387{{c}}
-* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7131{{c}}
+* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7198{{c}}
-{{Optimal ET sequence|legend=0| 24, …, 224, 472, 696, 1168 }}
+{{Optimal ET sequence|legend=0| 77, 147, 224, 301, 525 }}
-Badness (Sintel): 1.48
+Badness (Sintel): 1.46
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655
+Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024
-Mapping: {{mapping| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }}
+{{Mapping|legend=0| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }}
 Optimal tunings:
-* WE: ~12/11 = 149.9957{{c}}, ~3/2 = 701.7046{{c}}
+* WE: ~495/448 = 171.4282{{c}}, ~3/2 = 701.7229{{c}}
-* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7247{{c}}
+* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7242{{c}}
-{{Optimal ET sequence|legend=0| 24, 224, 472, 696 }}
+{{Optimal ET sequence|legend=0| 77, 147, 224, 525, 1274f }}
-Badness (Sintel): 1.26
+Badness (Sintel): 1.02
-== Nonant ==
+== Octant ==
-Named by [[Xenllium]] in 2023, nonant tempers out the [[septimal ennealimma]] ({{monzo| -11 -9 0 9 }}) and may be described as the {{nowrap| 36 & 171 }} temperament. It has a period of 1/9 octave, and its [[ploidacot]] is enneaploid monocot.
+Octant may be described as the {{nowrap| 224 & 248 }} temperament. It has a period of 1/8 octave, and its [[ploidacot]] is octaploid monocot. In this temperament, [[12/11]], [[35/27]], and [[99/70]] are mapped to 1\8, 3\8, and 4\8 respectively.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 32805/32768, 40353607/40310784
+[[Comma list]]: 32805/32768, 2259436291848/2251875390625
-{{Mapping|legend=1| 9 0 135 11 | 0 1 -8 1 }}
+{{Mapping|legend=1| 8 0 120 -117 | 0 1 -8 11 }}
-: mapping generators: ~2592/2401, ~3
+: mapping generators: ~42875/39366, ~3
 [[Optimal tuning]]s:
-* [[WE]]: ~2592/2401 = 133.3442{{c}}, ~3/2 = 701.8000{{c}}
+* [[WE]]: ~42875/39366 = 150.0048{{c}}, ~3/2 = 701.7356{{c}}
-: [[error map]]: {{val| +0.098 -0.057 -0.027 -0.141 }}
+: [[error map]]: {{val| +0.039 -0.181 +0.071 +0.127 }}
-* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.7384{{c}}
+* [[CWE]]: ~42875/39366 = 150.0000{{c}}, ~3/2 = 701.7134{{c}}
-: error map: {{val| 0.000 -0.217 -0.221 -0.421 }}
+: error map: {{val| 0.000 -0.242 -0.021 +0.022 }}
-{{Optimal ET sequence|legend=1| 36, 99c, 135, 171, 2772bd, 2943bdd, …, 5166bccddd, 5337bccddd }}
+{{Optimal ET sequence|legend=1| 24, …, 224, 472, 696, 1168 }}
-[[Badness]] (Sintel): 1.77
+[[Badness]] (Sintel): 3.98
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 540/539, 32805/32768, 42875/42592
+Comma list: 9801/9800, 32805/32768, 46656/46585
-Mapping: {{mapping| 9 0 135 11 131 | 0 1 -8 1 -7 }}
+{{Mapping|legend=0| 8 0 120 -117 15 | 0 1 -8 11 1 }}
 Optimal tunings:
-* WE: ~242/225 = 133.3308{{c}}, ~3/2 = 701.8205{{c}}
+* WE: ~12/11 = 150.0010{{c}}, ~3/2 = 701.7177{{c}}
-* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.8351{{c}}
+* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7131{{c}}
-{{Optimal ET sequence|legend=0| 36, 135, 171 }}
+{{Optimal ET sequence|legend=0| 24, …, 224, 472, 696, 1168 }}
-Badness (Sintel): 4.20
+Badness (Sintel): 1.48
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 729/728, 4096/4095, 16807/16731
+Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655
-Mapping: {{mapping| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }}
+{{Mapping|legend=0| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }}
 Optimal tunings:
-* WE: ~242/225 = 133.3180{{c}}, ~3/2 = 701.6956{{c}}
+* WE: ~12/11 = 149.9957{{c}}, ~3/2 = 701.7046{{c}}
-* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.7800{{c}}
+* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7247{{c}}
-{{Optimal ET sequence|legend=0| 36, 99cf, 135, 171 }}
+{{Optimal ET sequence|legend=0| 24, 224, 472, 696 }}
-Badness (Sintel): 3.15
+Badness (Sintel): 1.26
-== Septant ==
+== Nonant ==
-Named by [[Xenllium]] in 2021, septant notably tempers out the [[akjaysma]] ({{monzo| 47 -7 -7 -7 }}) and may be described as the {{nowrap| 224 & 301 }} temperament. It has a period of 1/7 octave, and its [[ploidacot]] is heptaploid monocot.
+Named by [[Xenllium]] in 2023, nonant tempers out the [[septimal ennealimma]] ({{monzo| -11 -9 0 9 }}) and may be described as the {{nowrap| 36 & 171 }} temperament. It has a period of 1/9 octave, and its [[ploidacot]] is enneaploid monocot.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 32805/32768, 516560652/514714375
+[[Comma list]]: 32805/32768, 40353607/40310784
-{{Mapping|legend=1| 7 0 105 -56 | 0 1 -8 7 }}
+{{Mapping|legend=1| 9 0 135 11 | 0 1 -8 1 }}
-: mapping generators: ~8575/7776, ~3
+: mapping generators: ~2592/2401, ~3
 [[Optimal tuning]]s:
-* [[WE]]: ~8575/7776 = 171.4303{{c}}, ~3/2 = 701.7091{{c}}
+* [[WE]]: ~2592/2401 = 133.3442{{c}}, ~3/2 = 701.8000{{c}}
-: [[error map]]: {{val| +0.012 -0.234 +0.096 +0.265 }}
+: [[error map]]: {{val| +0.098 -0.057 -0.027 -0.141 }}
-* [[CWE]]: ~8575/7776 = 171.4286{{c}}, ~3/2 = 701.7022{{c}}
+* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.7384{{c}}
-: error map: {{val| 0.000 -0.253 +0.069 +0.232 }}
+: error map: {{val| 0.000 -0.217 -0.221 -0.421 }}
-{{Optimal ET sequence|legend=1| 77, 147, 224, 301, 525, 826, 1351 }}
+{{Optimal ET sequence|legend=1| 36, 99c, 135, 171, 2772bd, 2943bdd, …, 5166bccddd, 5337bccddd }}
-[[Badness]] (Sintel): 2.81
+[[Badness]] (Sintel): 1.77
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 24057/24010, 32805/32768
+Comma list: 540/539, 32805/32768, 42875/42592
-Mapping: {{mapping| 7 0 105 -56 -120 | 0 1 -8 7 13 }}
+{{Mapping|legend=0| 9 0 135 11 131 | 0 1 -8 1 -7 }}
 Optimal tunings:
-* WE: ~495/448 = 171.4334{{c}}, ~3/2 = 701.7387{{c}}
+* WE: ~242/225 = 133.3308{{c}}, ~3/2 = 701.8205{{c}}
-* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7198{{c}}
+* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.8351{{c}}
-{{Optimal ET sequence|legend=0| 77, 147, 224, 301, 525 }}
+{{Optimal ET sequence|legend=0| 36, 135, 171 }}
-Badness (Sintel): 1.46
+Badness (Sintel): 4.20
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024
+Comma list: 540/539, 729/728, 4096/4095, 16807/16731
-Mapping: {{mapping| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }}
+{{Mapping|legend=0| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }}
 Optimal tunings:
-* WE: ~495/448 = 171.4282{{c}}, ~3/2 = 701.7229{{c}}
+* WE: ~242/225 = 133.3180{{c}}, ~3/2 = 701.6956{{c}}
-* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7242{{c}}
+* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.7800{{c}}
-{{Optimal ET sequence|legend=0| 77, 147, 224, 525, 1274f }}
+{{Optimal ET sequence|legend=0| 36, 99cf, 135, 171 }}
-Badness (Sintel): 1.02
+Badness (Sintel): 3.15
 == Septiquarschis ==
@@ Line 2,671: / 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 2,686: / 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 2,696: / Line 2,750: @@
 Badness (Sintel): 1.46
-== Tridecafifths ==
+== Subgroup extensions ==
-Named by [[Eliora]] in 2023, tridecafifths may be described as the {{nowrap| 89 & 200 }} temperament. It divides the [[3/2|perfect fifth]] into thirteen quartertones, so its [[ploidacot]] is 13-cot. [[289edo]] gives a highly recommendable tuning.
+=== 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 subgroup|2.3.5.13-subgroup]] [[restriction]] of 13-limit [[cassandra]].
-[[Subgroup]]: 2.3.5.7
+Subgroup: 2.3.5.13
-[[Comma list]]: 32805/32768, {{monzo| -14 -1 -9 13 }}
+Comma list: 325/324, 32805/32768
-{{Mapping|legend=1| 1 1 7 6 | 0 13 -104 -71 }}
+{{Mapping|legend=2| 1 0 15 -28 | 0 1 -8 20 }}
-: mapping generators: ~2, ~1323/1280
-[[Optimal tuning]]s:
+Optimal tunings:
-* [[WE]]: ~2 = 1200.1431{{c}}, ~1323/1280 = 53.9838{{c}}
+* WE: ~2 = 1200.3326{{c}} ~3/2 = 702.1092{{c}}
-: [[error map]]: {{val| +0.143 -0.023 +0.375 -0.816 }}
+* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9189{{c}}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~1323/1280 = 53.9764{{c}}
-: error map: {{val| 0.000 -0.261 -0.221 -0.421 }}
-{{Optimal ET sequence|legend=1| 89, 200, 289 }}
+{{Optimal ET sequence|legend=0| 12, …, 41, 53, 412cf, 465cf, …, 783ccff, 836ccfff }}
-[[Badness]] (Sintel): 10.9
+Badness (Sintel): 0.582
-=== 11-limit ===
+==== 2.3.5.13.19 subgroup ====
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.13.19
-Comma list: 441/440, 32805/32768, 55296000/55240493
+Comma list: 325/324, 361/360, 513/512
-Mapping: {{mapping| 1 1 7 6 4 | 0 13 -104 -71 -12 }}
+{{Mapping|legend=2| 1 0 15 -28 9 | 0 1 -8 20 -3 }}
 Optimal tunings:
-* WE: ~2 = 1200.0311{{c}}, ~33/32 = 53.9766{{c}}
+* WE: ~2 = 1200.4236{{c}}, ~3/2 = 702.1510{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 53.9750{{c}}
+* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9064{{c}}
-{{Optimal ET sequence|legend=0| 89, 200, 289 }}
+{{Optimal ET sequence|legend=0| 12, …, 41, 53 }}
-Badness (Sintel): 4.23
+Badness (Sintel): 0.354
-== Subgroup extensions ==
 === Photia (2.3.5.17) ===
 {{See also| No-elevens subgroup temperaments #Garibaldia }}
@@ Line 2,739: / 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 2,759: / 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 2,772: / 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 }}
-The [[S-expression]]-based comma list of this temperament is {[[1216/1215|S16/S18]], [[361/360|S19]], ([[513/512|S15/S20]])}. 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]].
+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.
 [[Subgroup]]: 2.3.5.19
@@ Line 2,780: / 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 2,802: / 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 2,822: / 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 2,837: / 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 2,854: / 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 2,874: / 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 2,886: / Line 2,937: @@
 Badness (Sintel): 1.17
+== References ==
+<references/>
+[[Category:Schismatic family| ]] <!-- main article -->
 [[Category:Temperament families]]
-[[Category:Schismatic family| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]