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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* [[WE]]: ~2 = 1200.0749{{c}}, ~3/2 = 701.7797{{c}}

: [[error map]]: {{val| +0.075 -0.100 -0.027 }}

* [[CWE]]: ~2 = 1200.0000{{c]}, ~3/2 = 701.7308{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7308{{c}}

: error map: {{val| 0.000 -0.224 -0.160 }}

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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.

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.

* [[#Garibaldi|Garibaldi]] adds [[garischisma|{{monzo| 25 -14 0 -1 }}]],

* [[#Grackle|~~Grackle~~]] adds {{monzo| -44 26 0 1 }},

* [[#~~Schism~~|~~Schism~~]] adds ~~[[64/63|~~{{monzo| 6 -2 0 -1 }}]],

* [[#~~Pontiac~~|~~Pontiac~~]] adds {{monzo| -~~59 39~~ 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.

[[#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.

* [[#Hemischis|Hemischis]] adds {{monzo| -34 25 0 -2 }} and has a ~~hemififth~~ 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.

* [[#Sesquiquartififths|Sesquiquartififths]] adds {{monzo| -35 15 0 4 }} and slices the fifth in four.

Temperaments discussed elsewhere include:

* ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]

* ''[[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, 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-Cbb~~), or down-minor seventh (C-~~vBb~~) 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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* [[WE]]: ~2 = 1200.1233{{c}}, ~3/2 = 702.1573{{c}}

: [[error map]]: {{val| +0.123 +0.326 -2.709 +2.328 }}

* [[CWE]]: ~2 = 1200.0000{{c]}, ~3/2 = 702.0774{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0774{{c}}

: error map: {{val| 0.000 +0.122 -2.933 +2.090 }}

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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.

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

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

Optimal tunings:

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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:

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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:

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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:

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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:

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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:

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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:

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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:

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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:

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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:

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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:

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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:

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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:

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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 355:

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:

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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:

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

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

Optimal tunings:

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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:

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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:

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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:

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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

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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:

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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

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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:

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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

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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:

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Pontiac tempers out the [[ragisma]], rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple augmented third (C-~~Exx#~~), or triple-up major sixth (C-^<sup>3</sup>A).

Pontiac tempers out the [[ragisma]], rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple-augmented third (C-E𝄪𝄪♯), or triple-up major sixth (C-^<sup>3</sup>A).

[[Subgroup]]: 2.3.5.7

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* [[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 }}

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=== Helenoid ===

~~The helenoid temperament (~~{{nowrap| 53 & 118 }}) is closely related to the helenus temperament, ~~but with the ragisma rather than~~ the ~~[[225/224|marvel comma]] tempered out~~.

Helenoid may be described as {{nowrap| 53 & 118 }}, and is closely related to the helenus temperament, differing only by the mapping of 7.

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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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:

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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:

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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:

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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:

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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:

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=== Ponta ===

~~The ponta temperament (~~{{nowrap| 53 & ~~171~~ }}~~) tempers out the~~ [[~~540/539|swetisma~~]] ~~and the ragisma~~.

Ponta tempers out [[540/539]] and may be described as {{nowrap| 171 & 224 }}. [[224edo]] itself makes for an excellent tuning.

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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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:

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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:

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=== Pontic ===

~~The pontic~~ temperament ({{nowrap| 118 & 171 }}~~) tempers out the~~ [[~~441/440|werckisma~~]] ~~and the ragisma~~.

Pontic temperament tempers out [[441/440]] and may be described as {{nowrap| 118 & 171 }}. [[289edo]] may be recommended as a tuning.

Subgroup: 2.3.5.7.11

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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 765:

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:

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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:

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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:

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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:

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=== Bipont ===

~~The bipont temperament ({{nowrap| 118 & 224 }}) has a period of half octave and~~ tempers out the [[3025/3024|lehmerisma (3025/3024)]] and the [[9801/9800|kalisma (9801/9800)]].

Bipont tempers out the [[3025/3024|lehmerisma (3025/3024)]] and the [[9801/9800|kalisma (9801/9800)]]. It may be described as {{nowrap| 118 & 224 }}. It has a period of half octave and a ploidacot signature of diploid monocot. [[342edo]] may be recommended as a tuning.

Subgroup: 2.3.5.7.11

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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

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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:

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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:

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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:

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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:

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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:

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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

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== Grackle ==

Grackle tempers out {{monzo| -44 26 0 1 }}~~. The~~ 7/4 is found at -26 fifths, represented by the triple diminished ninth (~~C-Dbbbb~~), or double-down minor seventh (~~C-vvBb~~)~~, which is to say, two~~ comma steps are required to bend the Pythagorean minor seventh to the septimal one.

Grackle tempers out {{monzo| -44 26 0 1 }} so 7/4 is found at -26 fifths, represented by the triple-diminished ninth (C–D𝄫𝄫) or double-down minor seventh (C–vvB♭). Two comma steps are required to bend the Pythagorean minor seventh to the septimal one.

[[Subgroup]]: 2.3.5.7

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* [[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 966:

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 981:

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 996:

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 1,011:

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 1,026:

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 1,041:

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 1,056:

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 1,071:

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 1,080:

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 ==

See [[Archytas clan #Schism]].

Schism is a relatively low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (~~C-Bb~~). 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.

Schism is a relatively low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (C–B♭). 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.

== Bischismic ==

Bischismic tempers out 3136/3125, the [[hemimean comma]], as well as 321489/320000, the [[varunisma]], and may be described as the {{nowrap| 118 & 130 }} temperament. The octave is split in halves, so the [[ploidacot]] of this temperament is diploid monocot. In schismic, -10 fifths make the interval class of 10/9. Bischismic then finds [[7/4]] by a stack of two [[10/9]]'s plus a semi-octave period, and in the [[11-limit]], it simply finds [[11/8]] by a stack of three [[10/9]]'s. [[248edo]] and [[378edo]] make for excellent tunings in both cases.

[[Subgroup]]: 2.3.5.7

Line 1,113:

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 1,128:

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 1,143:

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 1,158:

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 1,173:

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 1,184:

Line 1,228:

== Kleischismic ==

Kleischismic tempers out 1500625/1492992, the [[uniwiz comma]], and may be described as the {{nowrap| 94 & 118 }} temperament. The generator is a infrafifth, two of which plus a semi-octave period make the [[3/1|3rd]] [[harmonic]]; its [[ploidacot]] is thus diploid alpha-dicot. In schismic, 10 fifths make the interval class of [[9/5]]. Kleischismic then finds [[7/4]] by that minus a [[36/35]] quartertone, which is the aforementioned generator minus a semi-octave period. The generator stands in for [[16/11]] and the quartertone stands in for [[33/32]] in the [[11-limit]]. [[212edo]] and [[330edo]] in the 330e val may be recommended as tunings.

[[Subgroup]]: 2.3.5.7

Line 1,206:

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 1,221:

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 1,236:

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 1,251:

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 1,266:

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 1,277:

Line 1,323:

== Salsa ==

Salsa tempers out 245/243, the [[sensamagic comma]], and may be described as the {{nowrap| 41 & 65 }} temperament. It has a neutral third as a generator; its [[ploidacot]] is dicot. In fact it is related to [[hemififths]], from which this less accurate temperament only differs by the mapping of [[5/1|5]].

[[Subgroup]]: 2.3.5.7

Line 1,299:

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 1,314:

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 1,325:

Line 1,373:

== Hemischis ==

Hemischis tempers out 6144/6125, the [[porwell comma]], as well as 19683/19600, the [[cataharry comma]], and may be described as the {{nowrap| 53 & 130 }} temperament. Its [[ploidacot]] is alpha-dicot.

The [[S-expression]]-based comma list for 13-limit hemischis is {[[540/539|S12/S14]], [[676/675|S13/S15 = S26]], [[729/728|S27]], [[4096/4095|S64]], ([[4225/4224|S65]])}. Tempering out [[169/168]] ({{S|13}}), [[225/224]] ({{S|15}}) or [[625/624]] ({{S|25}}) leads to [[53edo]] while tempering out [[24192/24167]] ([[S-expression|S12/S13]]), [[10985/10976]] ([[S-expression|S13/S14]]), [[43904/43875]] ([[S-expression|S14/S15]]) or [[2401/2400]] ([[S-expression|S49]]) leads to [[130edo]] and implies S12, S13, S14, and S15 are tempered together.

[[Subgroup]]: 2.3.5.7

Line 1,347:

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 1,358:

Line 1,410:

=== 13-limit ===

Its [[S-expression]]-based comma list is {[[540/539|S12/S14]], [[676/675|S13/S15 = S26]], [[729/728|S27]], [[4096/4095|S64]](, [[4225/4224|S65]])}. Tempering out [[169/168|S13]], [[225/224|S15]] or [[625/624|S25]] leads to [[53edo]] (through [[Catakleismic]]) while tempering out [[24192/24167|S12/S13]], [[10985/10976|S13/S14]], [[43904/43875|S14/S15]] or [[2401/2400|S49]] (implying S12 = S13 = S14 = S15) leads to [[130edo]].

Subgroup: 2.3.5.7.11.13

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 1,379:

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 1,394:

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 1,409:

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 1,423:

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 1,447:

Line 1,497:

=== Terminal ===

~~The terminal temperament ({{nowrap| 12 & 159 }})~~ tempers out 441/440 and 4375/4356. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave.

Terminal tempers out 441/440 and 4375/4356, and may be described as {{nowrap| 159 & 171 }}. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave.

Subgroup: 2.3.5.7.11

Line 1,453:

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 1,468:

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 1,483:

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 1,494:

Line 1,544:

=== Terminator ===

Terminator tempers out 540/539, and may be described as {{nowrap| 171 & 183 }}.

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

Line 1,513:

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 1,528:

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 1,539:

Line 1,591:

=== Semiterm ===

The semiterm temperament ({{nowrap| 12 & 342 }}) has a period of 1/6 octave and ~~tempers out [[9801/9800]] (kalisma) and 151263/151250 (odiheim comma)~~.

The semiterm temperament tempers out [[9801/9800]] (kalisma) as well as [[151263/151250]] (odiheim comma), and may be described as {{nowrap| 12 & 342 }}. It has a period of 1/6 octave and its ploidacot is hexaploid monocot.

Subgroup: 2.3.5.7.11

Line 1,545:

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 1,561:

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 1,574:

Line 1,626:

=== Hemiterm ===

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

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 1,594:

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 1,609:

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 1,620:

Line 1,674:

== Altinex ==

Named by [[Aura]] in 2021, altinex is an alternative to [[#Hemiterm|hemiterm]] and may be described as {{nowrap| 24 & 159 }}. [[159edo]] itself makes for a recommendable tuning.

[[Subgroup]]: 2.3.5.7

Line 1,642:

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 1,657:

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 1,668:

Line 1,724:

== Squirrel ==

~~The squirrel temperament (~~{{nowrap| 29 & 36 }}) has a ~11/10 generator, three of which give the fourth (~4/3), and thirteen of which give 7/4 with octave reduction.

Squirrel tempers out 686/675, the [[sengic comma]], and may be described as {{nowrap| 29 & 36 }}. It has a [[~]][[11/10]] generator, three of which give the fourth ([[4/3]]), and thirteen of which give [[7/4]] with octave reduction. Its [[ploidacot]] is omega-tricot.

[[Subgroup]]: 2.3.5.7

Line 1,691:

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 1,706:

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 1,717:

Line 1,773:

== Tertiaschis ==

~~The~~ tertiaschis ~~temperament (~~{{nowrap| 94 & 159 }}) has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with [[#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 1,740:

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 1,755:

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 1,770:

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 1,781:

Line 1,837:

== Countertertiaschis ==

~~The~~ countertertiaschis ~~temperament (~~{{nowrap| 159 & 224 }}) has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with [[#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 1,804:

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 1,819:

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 1,830:

Line 1,886:

== Quadrant ==

~~The ''~~quadrant~~'' temperament (~~{{nowrap| 12 & ~~224~~ }}~~) has~~ a ~~period of quarter octave~~ and ~~tempers out the~~ [[~~dimcomp~~ comma]], ~~390625/388962~~. ~~In this temperament, 25/21~~ is ~~mapped into quarter octave~~.

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 1,854:

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 1,869:

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 1,880:

Line 1,936:

== Sesquiquartififths ==

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 1,902:

Line 1,962:

=== Sesquart ===

Sesquart is the main [[11-limit|11-]] and [[13-limit]] extension of sesquiquartififths of practical interest, as it identifies the neutral third with [[11/9]], which is realized in [[41edo]], [[89edo]], [[130edo]], and [[171edo]] also makes for a possible tuning.

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

Line 1,921:

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 1,931:

Line 1,993:

Badness (Sintel): 0.925

===== ~~Sesquartia~~ =====

===== Heartia =====

Subgroup: 2.3.5.7.11.13.17

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

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

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

Optimal tunings:

* WE: ~2 = 1199.~~8902~~{{c}}, ~72/65 = 175.~~4077~~{{c}}

* WE: ~2 = 1199.6422{{c}}, ~72/65 = 175.3338{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.~~4234~~{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3857{{c}}

Badness (Sintel): 1.18

Badness (Sintel): 1.45

====== 19-limit ======

Subgroup: 2.3.5.7.11.13.17.19

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

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

~~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 0 6 | 0 4 -32 -15 10 39 28 -12 }}

Optimal tunings:

* WE: ~2 = 1199.~~9864~~{{c}}, ~21/19 = 175.~~4169~~{{c}}

* WE: ~2 = 1199.7499{{c}}, ~21/19 = 175.3432{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.~~4189~~{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.3797{{c}}

Badness (Sintel): 1.24

Badness (Sintel): 1.40

====== 23-limit ======

===== Sesquartia =====

Subgroup: 2.3.5.7.11.13.17

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

Optimal tunings:

* WE: ~2 = 1199.8902{{c}}, ~72/65 = 175.4077{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4234{{c}}

Badness (Sintel): 1.18

====== 19-limit ======

Subgroup: 2.3.5.7.11.13.17.19

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

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

Optimal tunings:

* WE: ~2 = 1199.9864{{c}}, ~21/19 = 175.4169{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4189{{c}}

Badness (Sintel): 1.24

====== 23-limit ======

Subgroup: 2.3.5.7.11.13.17.19.23

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 1,976:

Line 2,068:

Badness (Sintel): 1.36

===== ~~Heartia~~ =====

===== Hearty =====

Subgroup: 2.3.5.7.11.13.17

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

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

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

~~Optimal tunings:~~

* WE: ~2 = 1199.6422{{c}}, ~72/65 = 175.3338{{~~c}}~~

* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3857{{c}}

~~{{Optimal ET sequence|legend=0| 41, 89, 130g }}~~

~~Badness (Sintel): 1.45~~

~~====== 19-limit ======~~

~~Subgroup: 2.3.5.7.11.13.17.19~~

~~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 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1199.7499{{c}}, ~21/19 = 175.3432{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.3797{{c}}

~~{{Optimal ET sequence~~|legend=0~~| 41, 89, 130g }}~~

~~Badness (Sintel): 1.40~~

~~===== Hearty =====~~

~~Subgroup: 2.3.5.7.11.13.17~~

~~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 2,026:

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 2,041:

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 2,056:

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 2,068:

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 2,092:

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 2,107:

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 2,118:

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.

[[Subgroup]]: 2.3.5.7

Line 2,140:

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 2,155:

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 2,166:

Line 2,237:

== Quintilipyth ==

~~The~~ quintilipyth ~~temperament~~ (~~{{nowrap| 12 & 253 }},~~ formerly ''quintilischis'') slices the ~~pythagorean fourth (~~[[4/3]]) into five semitones and tempers out the compass comma (9765625/9680832) in the 7-limit.

Named by [[Xenllium]] in 2021, quintilipyth (formerly ''quintilischis'') slices the [[4/3|perfect fourth]] into five semitones and tempers out the [[compass comma]] (9765625/9680832) in the [[7-limit]]. It may be described as the {{nowrap| 12 & 253 }} temperament, and its [[ploidacot]] is omega-pentacot.

[[Subgroup]]: 2.3.5.7

Line 2,190:

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 2,205:

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 2,220:

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 2,235:

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 2,246:

Line 2,317:

== Quintaschis ==

~~The~~ quintaschis ~~temperament ({{nowrap| 12 & 289 }})~~ slices the ~~fourth (~~4/3) into five semitones and tempers out 49009212/48828125 in the 7-limit.

Named by [[Xenllium]] in 2021, quintaschis slices the [[4/3|perfect fourth]] into five semitones and tempers out 49009212/48828125 in the [[7-limit]]. It may be described as the {{nowrap| 12 & 289 }} temperament, and its [[ploidacot]] is omega-pentacot.

[[Subgroup]]: 2.3.5.7

Line 2,269:

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 2,284:

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 2,299:

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 2,314:

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 2,329:

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 2,344:

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 2,359:

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 2,374:

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 2,389:

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 2,404:

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 2,419:

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 2,430:

Line 2,501:

== Sextilifourths ==

~~The~~ sextilifourths (~~{{nowrap| 130 & 159 }},~~ also known as ''sextilischis'', formerly ''sextilififths'') ~~temperament~~ slices the ~~fourth (~~4/3) into six small semitones, which serves as both 21/20 and 22/21.

Named by [[Xenllium]] in 2021, sextilifourths (also known as ''sextilischis'', formerly ''sextilififths'') slices the [[4/3|perfect fourth]] into six small semitones, which serves as both [[21/20]] and [[22/21]]. It may be described as {{nowrap| 130 & 159 }}, and its [[ploidacot]] is omega-hexacot. [[289edo]] gives a highly recommendable tuning.

[[Subgroup]]: 2.3.5.7

Line 2,454:

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 2,469:

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 2,479:

Line 2,550:

Badness (Sintel): 1.04

== ~~Octant~~ ==

== Septant ==

~~The octant temperament~~ ({{nowrap| 224 & ~~472~~ }}) has a period of 1/8 octave~~. In this temperament, 12/11, 35/27, and 99/70 are mapped into 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 ==

~~The ''nonant'' temperament (~~{{nowrap| 36 & ~~135~~ }}) has a period of 1/9 octave and ~~tempers out the~~ [[~~septimal ennealimma~~]], ~~{{monzo| -~~11 ~~-9 0 9 }}~~.

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 ==

~~The septant temperament~~ ({{nowrap| ~~224~~ & ~~301~~ }}) has a period of 1/7 octave and ~~tempers out the~~ [[~~akjaysma~~]]~~, {{monzo| 47 -7 -7 -7 }}~~.

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 ==

~~The~~ septiquarschis ~~temperament~~ ({{nowrap| 89 & 94 }}) splits septimal minor seventh ([[7/4]]) into four generators ~~and tempers out 829440~~/~~823543 (mynaslender comma) and 67108864/66706983 (septiness comma)~~.

Named by [[Xenllium]] in 2021, septiquarschis tempers out [[829440/823543]] (mynaslender comma) and [[67108864/66706983]] (septiness comma), and may be described as the {{nowrap| 89 & 94 }} temperament. It splits septimal minor seventh ([[7/4]]) into four generators. Note that in the data below, the generator is the [[octave complement]] so that seven of them minus five octaves make a [[3/2|perfect fifth]]; its [[ploidacot]] is thus epsilon-heptacot.

[[Subgroup]]: 2.3.5.7

Line 2,654:

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 2,669:

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 2,679:

Line 2,750:

Badness (Sintel): 1.46

== ~~Tridecafifths~~ ==

== Subgroup extensions ==

~~Tridecafifths divides~~ the ~~perfect~~ 3/2 ~~into~~ 13 ~~quartertones~~.

=== 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 2,722:

Line 2,790:

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

: mapping generators: ~2, ~3

Line 2,742:

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 2,755:

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~~]]~~)}. Strangely, despite prime 19~~ being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is ~~actually sharper~~ than the fifth in optimal schismic~~. This is likely~~ 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 2,763:

Line 2,831:

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

: mapping generators: ~2, ~3

Line 2,785:

Line 2,853:

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

: mapping generators: ~2, ~26/15

Line 2,805:

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 2,820:

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 2,837:

Line 2,905:

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

: mapping generators: ~2, ~18/17

Line 2,857:

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 2,869:

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 25: / Line 19: @@
 * [[WE]]: ~2 = 1200.0749{{c}}, ~3/2 = 701.7797{{c}}
 : [[error map]]: {{val| +0.075 -0.100 -0.027 }}
-* [[CWE]]: ~2 = 1200.0000{{c]}, ~3/2 = 701.7308{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7308{{c}}
 : error map: {{val| 0.000 -0.224 -0.160 }}
@@ 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.
+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.
-* [[#Garibaldi|Garibaldi]] adds [[garischisma|{{monzo| 25 -14 0 -1 }}]],
-* [[#Grackle|Grackle]] adds {{monzo| -44 26 0 1 }},
-* [[#Schism|Schism]] adds [[64/63|{{monzo| 6 -2 0 -1 }}]],
-* [[#Pontiac|Pontiac]] adds {{monzo| -59 39 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.
+[[#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.
-* [[#Hemischis|Hemischis]] adds {{monzo| -34 25 0 -2 }} and has a hemififth generator.
-* [[Gamelismic clan #Guiron|Guiron]] adds [[1029/1024|{{monzo| -10 1 0 3 }}]], with an ~8/7 generator, three of which give the fifth.
+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.
-* [[#Term|Term]] adds {{monzo| -94 54 0 3 }} with a 1/3 octave period.
-* [[#Sesquiquartififths|Sesquiquartififths]] adds {{monzo| -35 15 0 4 }} and slices the fifth in four.
 Temperaments discussed elsewhere include:
 * ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
 * ''[[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, 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 59: / 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-Cbb), or down-minor seventh (C-vBb) 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 70: / Line 61: @@
 * [[WE]]: ~2 = 1200.1233{{c}}, ~3/2 = 702.1573{{c}}
 : [[error map]]: {{val| +0.123 +0.326 -2.709 +2.328 }}
-* [[CWE]]: ~2 = 1200.0000{{c]}, ~3/2 = 702.0774{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0774{{c}}
 : error map: {{val| 0.000 +0.122 -2.933 +2.090 }}
@@ Line 90: / 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.
+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 96: / 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 119: / 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 143: / 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 158: / 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 173: / 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 188: / 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 203: / 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 218: / 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 233: / 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 244: / 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 271: / 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 295: / 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 310: / 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 325: / 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 340: / 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 355: / 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 370: / 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 385: / 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 404: / 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 423: / 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 438: / 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 455: / 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 471: / 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 486: / 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 502: / 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 517: / 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 533: / 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 546: / Line 539: @@
 {{Main| Pontiac }}
-Pontiac tempers out the [[ragisma]], rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple augmented third (C-Exx#), or triple-up major sixth (C-^<sup>3</sup>A).
+Pontiac tempers out the [[ragisma]], rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple-augmented third (C-E𝄪𝄪♯), or triple-up major sixth (C-^<sup>3</sup>A).
 [[Subgroup]]: 2.3.5.7
@@ Line 557: / 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 577: / Line 570: @@
 === Helenoid ===
-The helenoid temperament ({{nowrap| 53 & 118 }}) is closely related to the helenus temperament, but with the ragisma rather than the [[225/224|marvel comma]] tempered out.
+Helenoid may be described as {{nowrap| 53 & 118 }}, and is closely related to the helenus temperament, differing only by the mapping of 7.
 Subgroup: 2.3.5.7.11
@@ Line 583: / 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 602: / 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 621: / 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 640: / 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 655: / 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 670: / 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 681: / Line 674: @@
 === Ponta ===
-The ponta temperament ({{nowrap| 53 & 171 }}) tempers out the [[540/539|swetisma]] and the ragisma.
+Ponta tempers out [[540/539]] and may be described as {{nowrap| 171 & 224 }}. [[224edo]] itself makes for an excellent tuning.
 Subgroup: 2.3.5.7.11
@@ Line 687: / 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 706: / 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 725: / 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 740: / Line 733: @@
 === Pontic ===
-The pontic temperament ({{nowrap| 118 & 171 }}) tempers out the [[441/440|werckisma]] and the ragisma.
+Pontic temperament tempers out [[441/440]] and may be described as {{nowrap| 118 & 171 }}. [[289edo]] may be recommended as a tuning.
 Subgroup: 2.3.5.7.11
@@ Line 746: / 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 765: / 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 784: / 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 803: / 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 818: / 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 829: / Line 822: @@
 === Bipont ===
-The bipont temperament ({{nowrap| 118 & 224 }}) has a period of half octave and tempers out the [[3025/3024|lehmerisma (3025/3024)]] and the [[9801/9800|kalisma (9801/9800)]].
+Bipont tempers out the [[3025/3024|lehmerisma (3025/3024)]] and the [[9801/9800|kalisma (9801/9800)]]. It may be described as {{nowrap| 118 & 224 }}. It has a period of half octave and a ploidacot signature of diploid monocot. [[342edo]] may be recommended as a tuning.
 Subgroup: 2.3.5.7.11
@@ Line 835: / 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 851: / 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 866: / 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 881: / 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 896: / 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 911: / 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 926: / 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 938: / Line 931: @@
 == Grackle ==
-Grackle tempers out {{monzo| -44 26 0 1 }}. The 7/4 is found at -26 fifths, represented by the triple diminished ninth (C-Dbbbb), or double-down minor seventh (C-vvBb), which is to say, two comma steps are required to bend the Pythagorean minor seventh to the septimal one.
+Grackle tempers out {{monzo| -44 26 0 1 }} so 7/4 is found at -26 fifths, represented by the triple-diminished ninth (C–D𝄫𝄫) or double-down minor seventh (C–vvB♭). Two comma steps are required to bend the Pythagorean minor seventh to the septimal one.
 [[Subgroup]]: 2.3.5.7
@@ Line 950: / 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 966: / 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 981: / 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 996: / 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 1,011: / 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 1,026: / 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 1,041: / 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 1,056: / 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 1,071: / 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 1,080: / 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 ==
 See [[Archytas clan #Schism]].
-Schism is a relatively low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (C-Bb). 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.
+Schism is a relatively low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (C–B♭). 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.
 == Bischismic ==
+Bischismic tempers out 3136/3125, the [[hemimean comma]], as well as 321489/320000, the [[varunisma]], and may be described as the {{nowrap| 118 & 130 }} temperament. The octave is split in halves, so the [[ploidacot]] of this temperament is diploid monocot. In schismic, -10 fifths make the interval class of 10/9. Bischismic then finds [[7/4]] by a stack of two [[10/9]]'s plus a semi-octave period, and in the [[11-limit]], it simply finds [[11/8]] by a stack of three [[10/9]]'s. [[248edo]] and [[378edo]] make for excellent tunings in both cases.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,113: / 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 1,128: / 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 1,143: / 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 1,158: / 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 1,173: / 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 1,184: / Line 1,228: @@
 == Kleischismic ==
+Kleischismic tempers out 1500625/1492992, the [[uniwiz comma]], and may be described as the {{nowrap| 94 & 118 }} temperament. The generator is a infrafifth, two of which plus a semi-octave period make the [[3/1|3rd]] [[harmonic]]; its [[ploidacot]] is thus diploid alpha-dicot. In schismic, 10 fifths make the interval class of [[9/5]]. Kleischismic then finds [[7/4]] by that minus a [[36/35]] quartertone, which is the aforementioned generator minus a semi-octave period. The generator stands in for [[16/11]] and the quartertone stands in for [[33/32]] in the [[11-limit]]. [[212edo]] and [[330edo]] in the 330e val may be recommended as tunings.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,206: / 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 1,221: / 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 1,236: / 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 1,251: / 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 1,266: / 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 1,277: / Line 1,323: @@
 == Salsa ==
+Salsa tempers out 245/243, the [[sensamagic comma]], and may be described as the {{nowrap| 41 & 65 }} temperament. It has a neutral third as a generator; its [[ploidacot]] is dicot. In fact it is related to [[hemififths]], from which this less accurate temperament only differs by the mapping of [[5/1|5]].
 [[Subgroup]]: 2.3.5.7
@@ Line 1,299: / 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 1,314: / 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 1,325: / Line 1,373: @@
 == Hemischis ==
+Hemischis tempers out 6144/6125, the [[porwell comma]], as well as 19683/19600, the [[cataharry comma]], and may be described as the {{nowrap| 53 & 130 }} temperament. Its [[ploidacot]] is alpha-dicot.
+The [[S-expression]]-based comma list for 13-limit hemischis is {[[540/539|S12/S14]], [[676/675|S13/S15 = S26]], [[729/728|S27]], [[4096/4095|S64]], ([[4225/4224|S65]])}. Tempering out [[169/168]] ({{S|13}}), [[225/224]] ({{S|15}}) or [[625/624]] ({{S|25}}) leads to [[53edo]] while tempering out [[24192/24167]] ([[S-expression|S12/S13]]), [[10985/10976]] ([[S-expression|S13/S14]]), [[43904/43875]] ([[S-expression|S14/S15]]) or [[2401/2400]] ([[S-expression|S49]]) leads to [[130edo]] and implies S12, S13, S14, and S15 are tempered together.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,347: / 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 1,358: / Line 1,410: @@
 === 13-limit ===
-Its [[S-expression]]-based comma list is {[[540/539|S12/S14]], [[676/675|S13/S15 = S26]], [[729/728|S27]], [[4096/4095|S64]](, [[4225/4224|S65]])}. Tempering out [[169/168|S13]], [[225/224|S15]] or [[625/624|S25]] leads to [[53edo]] (through [[Catakleismic]]) while tempering out [[24192/24167|S12/S13]], [[10985/10976|S13/S14]], [[43904/43875|S14/S15]] or [[2401/2400|S49]] (implying S12 = S13 = S14 = S15) leads to [[130edo]].
 Subgroup: 2.3.5.7.11.13
 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 1,379: / 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 1,394: / 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 1,409: / 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 1,423: / 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 1,447: / Line 1,497: @@
 === Terminal ===
-The terminal temperament ({{nowrap| 12 & 159 }}) tempers out 441/440 and 4375/4356. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave.
+Terminal tempers out 441/440 and 4375/4356, and may be described as {{nowrap| 159 & 171 }}. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave.
 Subgroup: 2.3.5.7.11
@@ Line 1,453: / 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 1,468: / 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 1,483: / 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 1,494: / Line 1,544: @@
 === Terminator ===
+Terminator tempers out 540/539, and may be described as {{nowrap| 171 & 183 }}.
 Subgroup: 2.3.5.7.11
 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 1,513: / 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 1,528: / 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 1,539: / Line 1,591: @@
 === Semiterm ===
-The semiterm temperament ({{nowrap| 12 & 342 }}) has a period of 1/6 octave and tempers out [[9801/9800]] (kalisma) and 151263/151250 (odiheim comma).
+The semiterm temperament tempers out [[9801/9800]] (kalisma) as well as [[151263/151250]] (odiheim comma), and may be described as {{nowrap| 12 & 342 }}. It has a period of 1/6 octave and its ploidacot is hexaploid monocot.
 Subgroup: 2.3.5.7.11
@@ Line 1,545: / 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 1,561: / 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 1,574: / Line 1,626: @@
 === Hemiterm ===
+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
 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 1,594: / 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 1,609: / 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 1,620: / Line 1,674: @@
 == Altinex ==
+Named by [[Aura]] in 2021, altinex is an alternative to [[#Hemiterm|hemiterm]] and may be described as {{nowrap| 24 & 159 }}. [[159edo]] itself makes for a recommendable tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,642: / 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 1,657: / 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 1,668: / Line 1,724: @@
 == Squirrel ==
-The squirrel temperament ({{nowrap| 29 & 36 }}) has a ~11/10 generator, three of which give the fourth (~4/3), and thirteen of which give 7/4 with octave reduction.
+Squirrel tempers out 686/675, the [[sengic comma]], and may be described as {{nowrap| 29 & 36 }}. It has a [[~]][[11/10]] generator, three of which give the fourth ([[4/3]]), and thirteen of which give [[7/4]] with octave reduction. Its [[ploidacot]] is omega-tricot.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,691: / 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 1,706: / 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 1,717: / Line 1,773: @@
 == Tertiaschis ==
-The tertiaschis temperament ({{nowrap| 94 & 159 }}) has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with [[#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 1,740: / 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 1,755: / 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 1,770: / 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 1,781: / Line 1,837: @@
 == Countertertiaschis ==
-The countertertiaschis temperament ({{nowrap| 159 & 224 }}) has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with [[#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 1,804: / 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 1,819: / 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 1,830: / Line 1,886: @@
 == Quadrant ==
-The ''quadrant'' temperament ({{nowrap| 12 & 224 }}) has a period of quarter octave and tempers out the [[dimcomp comma]], 390625/388962. In this temperament, 25/21 is mapped into quarter octave.
+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 1,854: / 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 1,869: / 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 1,880: / Line 1,936: @@
 == Sesquiquartififths ==
+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 1,902: / Line 1,962: @@
 === Sesquart ===
+Sesquart is the main [[11-limit|11-]] and [[13-limit]] extension of sesquiquartififths of practical interest, as it identifies the neutral third with [[11/9]], which is realized in [[41edo]], [[89edo]], [[130edo]], and [[171edo]] also makes for a possible tuning.
 Subgroup: 2.3.5.7.11
 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 1,921: / 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 1,931: / Line 1,993: @@
 Badness (Sintel): 0.925
-===== Sesquartia =====
+===== Heartia =====
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575
+Comma list: 243/242, 256/255, 273/272, 364/363, 441/440
-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 0 | 0 4 -32 -15 10 39 28 }}
 Optimal tunings:
-* WE: ~2 = 1199.8902{{c}}, ~72/65 = 175.4077{{c}}
+* WE: ~2 = 1199.6422{{c}}, ~72/65 = 175.3338{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4234{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3857{{c}}
-{{Optimal ET sequence|legend=0| 41, 130, 171 }}
+{{Optimal ET sequence|legend=0| 41, 89, 130g }}
-Badness (Sintel): 1.18
+Badness (Sintel): 1.45
 ====== 19-limit ======
 Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594
+Comma list: 171/170, 243/242, 256/255, 273/272, 324/323, 441/440
-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 0 6 | 0 4 -32 -15 10 39 28 -12 }}
 Optimal tunings:
-* WE: ~2 = 1199.9864{{c}}, ~21/19 = 175.4169{{c}}
+* WE: ~2 = 1199.7499{{c}}, ~21/19 = 175.3432{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4189{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.3797{{c}}
-{{Optimal ET sequence|legend=0| 41, 130, 171 }}
+{{Optimal ET sequence|legend=0| 41, 89, 130g }}
-Badness (Sintel): 1.24
+Badness (Sintel): 1.40
-====== 23-limit ======
+===== Sesquartia =====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575
+{{Mapping|legend=0| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }}
+Optimal tunings:
+* WE: ~2 = 1199.8902{{c}}, ~72/65 = 175.4077{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4234{{c}}
+{{Optimal ET sequence|legend=0| 41, 130, 171 }}
+Badness (Sintel): 1.18
+====== 19-limit ======
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594
+{{Mapping|legend=0| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }}
+Optimal tunings:
+* WE: ~2 = 1199.9864{{c}}, ~21/19 = 175.4169{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4189{{c}}
+{{Optimal ET sequence|legend=0| 41, 130, 171 }}
+Badness (Sintel): 1.24
+====== 23-limit ======
 Subgroup: 2.3.5.7.11.13.17.19.23
 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 1,976: / Line 2,068: @@
 Badness (Sintel): 1.36
-===== Heartia =====
+===== Hearty =====
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 243/242, 256/255, 273/272, 364/363, 441/440
+Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625
-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 13 | 0 4 -32 -15 10 39 -61 }}
-Optimal tunings:
-* WE: ~2 = 1199.6422{{c}}, ~72/65 = 175.3338{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3857{{c}}
-{{Optimal ET sequence|legend=0| 41, 89, 130g }}
-Badness (Sintel): 1.45
-====== 19-limit ======
-Subgroup: 2.3.5.7.11.13.17.19
-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 }}
-Optimal tunings:
-* WE: ~2 = 1199.7499{{c}}, ~21/19 = 175.3432{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.3797{{c}}
-{{Optimal ET sequence|legend=0| 41, 89, 130g }}
-Badness (Sintel): 1.40
-===== Hearty =====
-Subgroup: 2.3.5.7.11.13.17
-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 2,026: / 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 2,041: / 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 2,056: / 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 2,068: / 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 2,092: / 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 2,107: / 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 2,118: / 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.
 [[Subgroup]]: 2.3.5.7
@@ Line 2,140: / 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 2,155: / 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 2,166: / Line 2,237: @@
 == Quintilipyth ==
-The quintilipyth temperament ({{nowrap| 12 & 253 }}, formerly ''quintilischis'') slices the pythagorean fourth ([[4/3]]) into five semitones and tempers out the compass comma (9765625/9680832) in the 7-limit.
+Named by [[Xenllium]] in 2021, quintilipyth (formerly ''quintilischis'') slices the [[4/3|perfect fourth]] into five semitones and tempers out the [[compass comma]] (9765625/9680832) in the [[7-limit]]. It may be described as the {{nowrap| 12 & 253 }} temperament, and its [[ploidacot]] is omega-pentacot.
 [[Subgroup]]: 2.3.5.7
@@ Line 2,190: / 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 2,205: / 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 2,220: / 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 2,235: / 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 2,246: / Line 2,317: @@
 == Quintaschis ==
-The quintaschis temperament ({{nowrap| 12 & 289 }}) slices the fourth (4/3) into five semitones and tempers out 49009212/48828125 in the 7-limit.
+Named by [[Xenllium]] in 2021, quintaschis slices the [[4/3|perfect fourth]] into five semitones and tempers out 49009212/48828125 in the [[7-limit]]. It may be described as the {{nowrap| 12 & 289 }} temperament, and its [[ploidacot]] is omega-pentacot.
 [[Subgroup]]: 2.3.5.7
@@ Line 2,269: / 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 2,284: / 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 2,299: / 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 2,314: / 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 2,329: / 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 2,344: / 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 2,359: / 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 2,374: / 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 2,389: / 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 2,404: / 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 2,419: / 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 2,430: / Line 2,501: @@
 == Sextilifourths ==
-The sextilifourths ({{nowrap| 130 & 159 }}, also known as ''sextilischis'', formerly ''sextilififths'') temperament slices the fourth (4/3) into six small semitones, which serves as both 21/20 and 22/21.
+Named by [[Xenllium]] in 2021, sextilifourths (also known as ''sextilischis'', formerly ''sextilififths'') slices the [[4/3|perfect fourth]] into six small semitones, which serves as both [[21/20]] and [[22/21]]. It may be described as {{nowrap| 130 & 159 }}, and its [[ploidacot]] is omega-hexacot. [[289edo]] gives a highly recommendable tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 2,454: / 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 2,469: / 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 2,479: / Line 2,550: @@
 Badness (Sintel): 1.04
-== Octant ==
+== Septant ==
-The octant temperament ({{nowrap| 224 & 472 }}) has a period of 1/8 octave. In this temperament, 12/11, 35/27, and 99/70 are mapped into 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 ==
-The ''nonant'' temperament ({{nowrap| 36 & 135 }}) has a period of 1/9 octave and tempers out the [[septimal ennealimma]], {{monzo| -11 -9 0 9 }}.
+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 ==
-The septant temperament ({{nowrap| 224 & 301 }}) has a period of 1/7 octave and tempers out the [[akjaysma]], {{monzo| 47 -7 -7 -7 }}.
+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 ==
-The septiquarschis temperament ({{nowrap| 89 & 94 }}) splits septimal minor seventh ([[7/4]]) into four generators and tempers out 829440/823543 (mynaslender comma) and 67108864/66706983 (septiness comma).
+Named by [[Xenllium]] in 2021, septiquarschis tempers out [[829440/823543]] (mynaslender comma) and [[67108864/66706983]] (septiness comma), and may be described as the {{nowrap| 89 & 94 }} temperament. It splits septimal minor seventh ([[7/4]]) into four generators. Note that in the data below, the generator is the [[octave complement]] so that seven of them minus five octaves make a [[3/2|perfect fifth]]; its [[ploidacot]] is thus epsilon-heptacot.
 [[Subgroup]]: 2.3.5.7
@@ Line 2,654: / 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 2,669: / 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 2,679: / Line 2,750: @@
 Badness (Sintel): 1.46
-== Tridecafifths ==
+== Subgroup extensions ==
-Tridecafifths divides the perfect 3/2 into 13 quartertones.
+=== 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 2,722: / 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 2,742: / 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 2,755: / 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]])}. Strangely, despite prime 19 being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is actually sharper than the fifth in optimal schismic. This is likely 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 2,763: / 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 2,785: / 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 2,805: / 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 2,820: / 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 2,837: / 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 2,857: / 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 2,869: / 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]]