Schismatic family: Difference between revisions

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

[[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.

Subgroup: 2.3.5.7.11

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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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=== 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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=== 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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=== 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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== 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 }}

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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

Mapping: {{mapping| 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}}

Badness (Sintel): 3.83

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

* 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

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== 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

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== 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

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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}}

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== 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

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=== 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

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== 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

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=== 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

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=== Terminator ===

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

Subgroup: 2.3.5.7.11

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=== 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

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=== 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

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== 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

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== 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

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== 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.

[[Subgroup]]: 2.3.5.7

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== 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

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== 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~Pythagorean comma with three [[marvel comma]]s, quadrant equates the syntonic~Pythagorean comma with four. A [[septimal comma]] is then found as a stack of five marvel commas.

[[Subgroup]]: 2.3.5.7

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== Sesquiquartififths ==

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

[[Subgroup]]: 2.3.5.7

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=== 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

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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: {{mapping| 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}}

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: {{mapping| 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

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}}

* 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: {{mapping| 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

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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 }}

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== Tsaharuk ==

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.

[[Subgroup]]: 2.3.5.7

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== 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

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== 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

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== 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

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== 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

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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~~ }}

Mapping: {{mapping| 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~~ }}

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: {{mapping| 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}}

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 }}

Mapping: {{mapping| 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}}

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: {{mapping| 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~~ }}

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 }}

Mapping: {{mapping| 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~~ }}

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: {{mapping| 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~~ }}

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

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Badness (Sintel): 1.46

== ~~Tridecafifths~~ ==

== Subgroup extensions ==

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

[[~~Subgroup~~]]: 2.3.5.7

=== Tridecaschismic (2.3.5.13) ===

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

Subgroup: 2.3.5.13

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

Comma list: 325/324, 32805/32768

Subgroup-val mapping: {{mapping| 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=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 }}

Subgroup-val mapping: {{mapping| 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}}

Badness (Sintel): 4.23

Badness (Sintel): 0.354

~~== Subgroup extensions ==~~

=== Photia (2.3.5.17) ===

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: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#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 25: / Line 25: @@
 * [[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 37: @@
 === 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 54: @@
 {{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 [[garischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}.
 [[Subgroup]]: 2.3.5.7
@@ Line 70: / Line 65: @@
 * [[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 85: @@
 === 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.
 Subgroup: 2.3.5.7.11
@@ Line 546: / Line 541: @@
 {{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 552: @@
 * [[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 572: @@
 === 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 681: / Line 676: @@
 === 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 740: / Line 735: @@
 === 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 829: / Line 824: @@
 === 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 938: / Line 933: @@
 == 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 945: @@
 * [[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 1,080: / Line 1,075: @@
 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: {{mapping| 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: {{mapping| 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 1,184: / Line 1,230: @@
 == 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 1,277: / Line 1,325: @@
 == 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 1,317: / Line 1,367: @@
 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 1,375: @@
 == 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 1,358: / Line 1,412: @@
 === 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
@@ Line 1,423: / Line 1,475: @@
 == 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 1,499: @@
 === 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 1,494: / Line 1,546: @@
 === Terminator ===
+Terminator tempers out 540/539, and may be described as {{nowrap| 171 & 183 }}.
 Subgroup: 2.3.5.7.11
@@ Line 1,539: / Line 1,593: @@
 === 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 1,574: / Line 1,628: @@
 === 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
@@ Line 1,620: / Line 1,676: @@
 == 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 1,668: / Line 1,726: @@
 == 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 1,717: / Line 1,775: @@
 == 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.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,781: / Line 1,839: @@
 == 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 1,830: / Line 1,888: @@
 == 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~Pythagorean comma with three [[marvel comma]]s, quadrant equates the syntonic~Pythagorean comma with four. A [[septimal comma]] is then found as a stack of five marvel commas.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,880: / Line 1,938: @@
 == Sesquiquartififths ==
+Sesquiquartififths tempers out 2401/2400, the [[breedsma]], and may be described as the {{nowrap| 41 & 171 }} temperament. It splits the fifth into four; its [[ploidacot]] is thus tetracot.
 [[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
@@ 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: {{mapping| 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: {{mapping| 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: {{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}}
+* 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: {{mapping| 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
@@ 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 }}
@@ Line 2,069: / Line 2,131: @@
 == Tsaharuk ==
 {{Main| Tsaharuk }}
+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.
 [[Subgroup]]: 2.3.5.7
@@ Line 2,118: / Line 2,182: @@
 == 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 2,166: / Line 2,232: @@
 == 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 2,246: / Line 2,312: @@
 == 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 2,430: / Line 2,496: @@
 == 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 2,479: / Line 2,545: @@
 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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 2,679: / Line 2,745: @@
 Badness (Sintel): 1.46
-== Tridecafifths ==
+== Subgroup extensions ==
-Tridecafifths divides the perfect 3/2 into 13 quartertones.
-[[Subgroup]]: 2.3.5.7
+=== Tridecaschismic (2.3.5.13) ===
+Proposed by [[Eufalesio]] in 2026, tridecaschismic adds the [[325/324|marveltwin comma]] to the comma list, or equivalently, the [[tridecapyth comma]]. It benefits from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is one of the lowest badness schismic extensions. It is also equivalent to the 2.3.5.13 [[restriction]] of 13-limit [[cassandra]].
+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 }}
+Subgroup-val mapping: {{mapping| 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 }}
+Subgroup-val mapping: {{mapping| 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 2,757: / Line 2,821: @@
 : ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#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