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 }}

~~<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7187{{c}}~~

* [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7359{{c}} -->

[[Tuning ranges]]:

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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]]}.

[[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 }}

~~~~

[[Minimax tuning]]:

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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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* WE: ~2 = 1200.3089{{c}}, ~3/2 = 702.3377{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1562{{c}}

~~~~

Minimax tuning:

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* WE: ~2 = 1200.1703{{c}}, ~3/2 = 702.2122{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1135{{c}}

~~~~

Minimax tuning:

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* WE: ~2 = 1199.8140{{c}}, ~3/2 = 701.9833{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0909{{c}}

~~~~

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* WE: ~2 = 1199.9556{{c}}, ~3/2 = 702.0530{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0787{{c}}

~~~~

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* WE: ~2 = 1200.0046{{c}}, ~3/2 = 702.2167{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0962{{c}}

~~~~

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* WE: ~2 = 1200.2910{{c}}, ~3/2 = 702.2681{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0967{{c}}

~~~~

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* WE: ~2 = 1200.2970{{c}}, ~3/2 = 702.2697{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0943{{c}}

~~~~

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* WE: ~2 = 1200.1986{{c}}, ~3/2 = 702.2598{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1455{{c}}

~~~~

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* WE: ~2 = 1200.3057{{c}}, ~3/2 = 702.3138{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1373{{c}}

~~~~

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* WE: ~2 = 1200.1917{{c}}, ~3/2 = 702.4836{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3599{{c}}

~~~~

Minimax tuning:

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* WE: ~2 = 1200.3031{{c}}, ~3/2 = 702.7368{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.5420{{c}}

~~~~

Minimax tuning:

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* WE: ~2 = 1199.1984{{c}}, ~3/2 = 701.8424{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3384{{c}}

~~~~

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* WE: ~2 = 1199.5242{{c}}, ~3/2 = 702.0783{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3711{{c}}

~~~~

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* WE: ~2 = 1200.6122{{c}}, ~3/2 = 703.0830{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6968{{c}}

~~~~

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* WE: ~2 = 1200.7981{{c}}, ~3/2 = 703.2199{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.7221{{c}}

~~~~

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* WE: ~2 = 1201.3146{{c}}, ~3/2 = 703.4864{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6491{{c}}

~~~~

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* WE: ~2 = 1201.3140{{c}}, ~3/2 = 703.4860{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6578{{c}}

~~~~

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* WE: ~2 = 1199.7097{{c}}, ~3/2 = 701.5554{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7370{{c}}

~~~~

Minimax tuning:

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* WE: ~2 = 1199.7370{{c}}, ~3/2 = 701.5937{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7570{{c}}

~~~~

Minimax tuning:

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* WE: ~2 = 1199.2895{{c}}, ~3/2 = 701.2643{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6967{{c}}

~~~~

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* WE: ~2 = 1199.5280{{c}}, ~3/2 = 701.4290{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7149{{c}}

~~~~

Badness (Sintel): 1.18

~~=== Hemigari ===~~

~~Subgroup: 2.3.5.7.11~~

~~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~~

~~Optimal tunings:~~

* WE: ~2 = 1200.7303{{c}}, ~110/63 = 951.6605{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~110/63 = 951.0604{{c}}

~~~~

~~{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135ee }}~~

~~Badness (Sintel): 1.68~~

~~==== 13-limit ====~~

~~Subgroup: 2.3.5.7.11.13~~

~~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:~~

* WE: ~2 = 1200.8146{{c}}, ~26/15 = 951.7273{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.0574{{c}}

~~~~

~~{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135eef }}~~

~~Badness (Sintel): 1.13~~

=== Karadeniz ===

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* WE: ~2 = 1199.7351{{c}}, ~11/9 = 350.9167{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.9995{{c}}

~~~~

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* WE: ~2 = 1199.3042{{c}}, ~11/9 = 350.7533{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.9686{{c}}

~~~~

Badness (Sintel): 1.34

=== Hemigari ===

Subgroup: 2.3.5.7.11

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

Optimal tunings:

* WE: ~2 = 1200.7303{{c}}, ~110/63 = 951.6605{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~110/63 = 951.0604{{c}}

Badness (Sintel): 1.68

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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:

* WE: ~2 = 1200.8146{{c}}, ~26/15 = 951.7273{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.0574{{c}}

Badness (Sintel): 1.13

=== Sanjaab ===

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* WE: ~2 = 1200.1997{{c}}, ~11/10 = 166.0018{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9786{{c}}

~~~~

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* WE: ~2 = 1200.1224{{c}}, ~11/10 = 165.9800{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9659{{c}}

~~~~

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Badness (Sintel): 1.40

== ~~Schism~~ ==

== Pontiac ==

~~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.

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).

~~== Pontiac ==~~

~~{{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).

[[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 }}

~~~~

[[Minimax tuning]]:

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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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* WE: ~2 = 1200.3277{{c}}, ~3/2 = 701.9135{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7223{{c}}

~~~~

Minimax tuning:

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* WE: ~2 = 1200.1780{{c}}, ~3/2 = 701.8491{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7446{{c}}

~~~~

Minimax tuning:

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* WE: ~2 = 1200.1645{{c}}, ~3/2 = 701.8385{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7425{{c}}

~~~~

Minimax tuning:

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* WE: ~2 = 1200.5227{{c}}, ~3/2 = 702.0456{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7418{{c}}

~~~~

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* WE: ~2 = 1200.4988{{c}}, ~3/2 = 702.0218{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7332{{c}}

~~~~

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* WE: ~2 = 1200.5185{{c}}, ~3/2 = 702.0323{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7318{{c}}

~~~~

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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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* WE: ~2 = 1199.9814{{c}}, ~3/2 = 701.7725{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7834{{c}}

~~~~

Minimax tuning:

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* WE: ~2 = 1199.9601{{c}}, ~3/2 = 701.7610{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7845{{c}}

~~~~

Minimax tuning:

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* WE: ~2 = 1199.8850{{c}}, ~3/2 = 701.7101{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7775{{c}}

~~~~

Minimax tuning:

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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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* WE: ~2 = 1200.1259{{c}}, ~3/2 = 701.7980{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7256{{c}}

~~~~

Minimax tuning:

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* WE: ~2 = 1199.9254{{c}}, ~3/2 = 701.6945{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7378{{c}}

~~~~

Minimax tuning:

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* WE: ~2 = 1199.9454{{c}}, ~3/2 = 701.7085{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7401{{c}}

~~~~

Minimax tuning:

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* WE: ~2 = 1200.0897{{c}}, ~3/2 = 701.7874{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7356{{c}}

~~~~

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* WE: ~2 = 1200.1045{{c}}, ~3/2 = 701.7962{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7359{{c}}

~~~~

{{Optimal ET sequence|legend=0| 53ef, 118f, 171, 289, 460e, 749defg }}

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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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* WE: ~99/70 = 600.0500{{c}}, ~3/2 = 701.8153{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7584{{c}}

~~~~

{{Optimal ET sequence|legend=0| 106, 118, 224, 342, 1592c, 1934ce, 2276cde, 2618cde, 2960cde }}

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* WE: ~99/70 = 599.9939{{c}}, ~3/2 = 701.7657{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7728{{c}}

~~~~

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* WE: ~99/70 = 599.9839{{c}}, ~3/2 = 701.7463{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7649{{c}}

~~~~

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* WE: ~99/70 = 600.0405{{c}}, ~3/2 = 701.8160{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7697{{c}}

~~~~

{{Optimal ET sequence|legend=0| 106f, 118f, 224, 342f, 566, 1356cf }}

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* WE: ~99/70 = 600.0336{{c}}, ~3/2 = 701.8031{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7647{{c}}

~~~~

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* WE: ~99/70 = 600.0243{{c}}, ~3/2 = 701.7891{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7613{{c}}

~~~~

{{Optimal ET sequence|legend=0| 106fgh, 118f, 224, 342f, 566h, 908fgh }}

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* WE: ~208/175 = 300.0229{{c}}, ~3/2 = 701.8097{{c}}

* CWE: ~208/175 = 300.0000{{c}}, ~3/2 = 701.7578{{c}}

~~~~

{{Optimal ET sequence|legend=0| 224, 460, 684, 2276cde, 2960cde }}

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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 }}

~~~~

[[Minimax tuning]]:

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* WE: ~2 = 1199.7077{{c}}, ~3/2 = 701.0017{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.1804{{c}}

~~~~

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Line 977:

* WE: ~2 = 1199.7782{{c}}, ~3/2 = 701.0966{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2319{{c}}

~~~~

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* WE: ~2 = 1199.5839{{c}}, ~3/2 = 700.9632{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2137{{c}}

~~~~

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Line 1,007:

* WE: ~2 = 1199.7146{{c}}, ~3/2 = 701.0500{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2212{{c}}

~~~~

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Line 1,022:

* WE: ~2 = 1200.0060{{c}}, ~3/2 = 701.2202{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2167{{c}}

~~~~

Line 1,109:

Line 1,037:

* WE: ~2 = 1199.8388{{c}}, ~3/2 = 701.3071{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.4068{{c}}

~~~~

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Line 1,052:

* WE: ~2 = 1199.7329{{c}}, ~3/2 = 701.1918{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.3555{{c}}

~~~~

Line 1,141:

Line 1,067:

* WE: ~2 = 1199.8928{{c}}, ~3/2 = 701.4664{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.5327{{c}}

~~~~

Line 1,147:

Line 1,072:

Badness (Sintel): 2.01

== ~~Bischismic~~ ==

== 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]]: ~~3136~~/~~3125~~, ~~32805~~/~~32768~~

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

~~: mapping generators: ~567/400, ~3~~

[[Optimal tuning]]s:

* [[WE]]: ~~~567/400~~ = ~~600~~.~~0072~~{{c}}, ~3/2 = ~~701~~.~~6005~~{{c}}

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

: [[error map]]: {{val| +0.~~014 -0.340~~ +0.~~982~~ -0.~~629~~ }}

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

* [[CWE]]: ~~~567/400~~ = ~~600~~.0000{{c}}, ~3/2 = 701.~~5915~~{{c}}

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

: error map: {{val| 0.~~000 -0.364~~ +0.~~954~~ -0.~~656 }}~~

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

<!-~~- * [[CTE]]: ~567/400 = 600~~.~~0000{{c~~}}~~, ~3/2 = 701.5899{{c}} -->~~

~~[[Minimax tuning]]:~~

* [[7-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.7/3

* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7

{{Optimal ET sequence|legend=1| 12, …, ~~106d~~, ~~118~~, ~~130~~, ~~248, 378~~ }}

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

[[Badness]] (Sintel): 1.39

[[Badness]] (Sintel): 5.04

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal tunings:

* WE: ~~~99/70~~ = ~~600~~.~~0165~~{{c}}, ~3/2 = ~~701~~.~~6316~~{{c}}

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

* CWE: ~~~99/70 = 600.0000{{c}}, ~3/~~2 = ~~701.6110{{c}}~~

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

~~~~

{{Optimal ET sequence|legend=0| 12, …, ~~106de~~, ~~118, 130, 248~~ }}

Badness (Sintel): 0.~~931~~

Badness (Sintel): 3.83

==== 13-limit ====

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal tunings:

* WE: ~~~99/70~~ = ~~599~~.~~9610~~{{c}}, ~3/2 = ~~701~~.~~5445~~{{c}}

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

* CWE: ~~~99/70 = 600.0000{{c}}, ~3/~~2 = ~~701.5908{{c}}~~

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

~~~~

{{Optimal ET sequence|legend=0| 12, ~~118~~, ~~130~~, ~~248, 378~~ }}

Badness (Sintel): 1.19

Badness (Sintel): 2.13

==~~=== 17-limit ===~~==

== Schism ==

~~Subgroup: 2.3.5.7.11.13~~.17

See [[Archytas clan #Schism]].

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

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.

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

== 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.

~~Optimal tunings~~:

[[Subgroup]]: 2.3.5.7

* WE: ~99/70 = 600.0331{{c}}, ~3/2 ~~= 701.6387{{c}}~~

* CWE: ~99/70 = 600.~~0000{{c}}, ~~~3~~/2 = 701~~.~~5994{{c}}~~

~~~~

~~{{Optimal ET sequence|legend=0| 12~~, ~~118, 130, 248g }}~~

[[Comma list]]: 3136/3125, 32805/32768

~~Badness (Sintel)~~: ~~1.49~~

: mapping generators: ~567/400, ~3

==== ~~Bischis ====~~

[[Optimal tuning]]s:

~~Subgroup~~: ~~2.3~~.5.7.11.13

* [[WE]]: ~567/400 = 600.0072{{c}}, ~3/2 = 701.6005{{c}}

: [[error map]]: {{val| +0.014 -0.340 +0.982 -0.629 }}

* [[CWE]]: ~567/400 = 600.0000{{c}}, ~3/2 = 701.5915{{c}}

: error map: {{val| 0.000 -0.364 +0.954 -0.656 }}

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

[[Minimax tuning]]:

* [[7-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.7/3

* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7

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

{{Optimal ET sequence|legend=1| 12, …, 106d, 118, 130, 248, 378 }}

~~Optimal tunings~~:

[[Badness]] (Sintel): 1.39

* WE: ~55/39 = 599.9766{{c}}, ~3/2 = 701.5380{{c}}

* CWE: ~55/39 = 600.~~0000{{c}}, ~3/2 = 701.5670{{c}}~~

~~~~

~~{{Optimal ET sequence|legend~~=~~0| 12f, 106deff, 118f, 130 }}~~

=== 11-limit ===

Subgroup: 2.3.5.7.11

~~Badness (Sintel)~~: ~~1.21~~

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

~~===== 17-limit =====~~

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

~~Subgroup~~: 2~~.3.5.7.11.13.17~~

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

Optimal tunings:

* WE: ~99/70 = 600.0165{{c}}, ~3/2 = 701.6316{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.6110{{c}}

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

~~Optimal tunings~~:

Badness (Sintel): 0.931

* WE: ~55/39 = 600.0997{{c}}, ~3/2 = 701.7114{{c}}

* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.5899{{c}}

~~~~

~~{{Optimal ET sequence|legend~~=~~0| 12f, 106deff, 118f, 130, 248fg }}~~

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

~~Badness (Sintel)~~: ~~1.37~~

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

~~== Kleischismic ==~~

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

~~[[Subgroup]]~~: 2~~.3.5.7~~

~~[[Comma list]]~~: ~~32805~~/~~32768~~, ~~1500625~~/~~1492992~~

Optimal tunings:

* WE: ~99/70 = 599.9610{{c}}, ~3/2 = 701.5445{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5908{{c}}

~~: mapping generators: ~1225/864, ~35/24~~

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

Badness (Sintel): 1.19

* [[WE]]: ~1225/864 = 600.1246{{c}}, ~35/24 = 651.0550{{c}} (~36/35 = 50.9304{{c}})

~~: [[error map]]: {{val| +0.249 +0.280 -0.453 -0.650 }}~~

* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~35/24 = 650.9204{{c}} (~~~36/35 = 50.9204{{c}}~~)

: ~~error map: {{val| 0.000 -0.114 -1.041 -~~1.~~338 }}~~

~~~~

~~{{Optimal ET sequence|legend~~=~~1| 24, 94, 118, 212, 330, 542d, 872cdd, 1414ccddd }}~~

===== 17-limit =====

Subgroup: 2.3.5.7.11.13.17

~~[[Badness]] (Sintel)~~: ~~2.80~~

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

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

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

~~Subgroup: 2.3.5.7.11~~

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

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

Optimal tunings:

* WE: ~99/70 = 600.~~1645~~{{c}}, ~35/24 = ~~651~~.~~0963~~{{c}} ~~(~36/35 = 50.9319{{c}})~~

* WE: ~99/70 = 600.0331{{c}}, ~3/2 = 701.6387{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~~~35/24 = 650.9184{{c}} (~36/35 = 50.9184{{c}})~~

* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5994{{c}}

~~~~

{{Optimal ET sequence|legend=0| ~~24, 94~~, 118, ~~212, 330e, 542dee~~, ~~872cddeee~~ }}

Badness (Sintel): 1.21

Badness (Sintel): 1.49

==== ~~13-limit~~ ====

==== Bischis ====

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal tunings:

* WE: ~99/70 = ~~600~~.~~0696~~{{c}}, ~35/24 = ~~651~~.~~0136~~{{c}} ~~(~36/35 = 50.9440{{c}})~~

* WE: ~55/39 = 599.9766{{c}}, ~3/2 = 701.5380{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~~~35/24 = 650.9378{{c}} (~36/35 = 50.9378{{c}})~~

* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.5670{{c}}

~~~~

Badness (Sintel): 1.56

Badness (Sintel): 1.21

===== 17-limit =====

Subgroup: 2.3.5.7.11.13.17

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

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

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

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

Optimal tunings:

* WE: ~99/70 = 600.~~1134~~{{c}}, ~35/24 = ~~651~~.~~0646~~{{c}} ~~(~36/35 = 50.9512{{c}})~~

* WE: ~55/39 = 600.0997{{c}}, ~3/2 = 701.7114{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~~~35/24 = 650.9414{{c}} (~36/35 = 50.9414{{c}})~~

* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.5899{{c}}

~~~~

{{Optimal ET sequence|legend=0| 12f, 106deff, 118f, 130, 248fg }}

Badness (Sintel): 1.30

Badness (Sintel): 1.37

==~~== Kleischis ==~~==

== Kleischismic ==

~~Subgroup: 2~~.3.5.7.11.13

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.

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

[[Subgroup]]: 2.3.5.7

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

[[Comma list]]: 32805/32768, 1500625/1492992

~~Optimal tunings:~~

* WE: ~99/70 = 600.1909{{~~c}}, ~35/24~~ = ~~651.1578{{c~~}} ~~(~36/35 = 50.9670{{c}})~~

: mapping generators: ~1225/864, ~35/24

* CWE: ~~~99/70 = 600.0000{{c}}, ~35/24 = 650.9541{{c}} (~36/35 = 50.9541{{c}})~~

~~~~

{{~~Optimal ET sequence~~|~~legend~~=0~~| 24f, 94, 118f, 212~~ }}

[[Optimal tuning]]s:

* [[WE]]: ~1225/864 = 600.1246{{c}}, ~35/24 = 651.0550{{c}} (~36/35 = 50.9304{{c}})

: [[error map]]: {{val| +0.249 +0.280 -0.453 -0.650 }}

* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~35/24 = 650.9204{{c}} (~36/35 = 50.9204{{c}})

: error map: {{val| 0.000 -0.114 -1.041 -1.338 }}

Badness (Sintel): 1.55

{{Optimal ET sequence|legend=1| 24, 94, 118, 212, 330, 542d, 872cdd, 1414ccddd }}

[[Badness]] (Sintel): 2.80

===~~== 17~~-limit =====

=== 11-limit ===

Subgroup: 2.3.5.7.11~~.13.17~~

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal tunings:

* WE: ~99/70 = 600.~~2190~~{{c}}, ~35/24 = 651.~~1578~~{{c}} (~36/35 = 50.~~9670~~{{c}})

* WE: ~99/70 = 600.1645{{c}}, ~35/24 = 651.0963{{c}} (~36/35 = 50.9319{{c}})

* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.~~9518~~{{c}} (~36/35 = 50.~~9518~~{{c}})

* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9184{{c}} (~36/35 = 50.9184{{c}})

~~~~

{{Optimal ET sequence|legend=0| ~~24f~~, 94, ~~118f~~, ~~212g~~ }}

{{Optimal ET sequence|legend=0| 24, 94, 118, 212, 330e, 542dee, 872cddeee }}

Badness (Sintel): 1.26

Badness (Sintel): 1.21

== ~~Salsa~~ ==

==== 13-limit ====

[[Subgroup]]: 2.3.5.7

Subgroup: 2.3.5.7.11.13

[[Comma list]]: ~~245~~/~~243~~, ~~32805~~/~~32768~~

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 generators: ~2, ~128/105~~

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

Optimal tunings:

* [[WE]]: ~2 = ~~1200~~.~~7707~~{{c}}, ~~~128~~/~~105~~ = ~~351~~.~~2748~~{{c}}

* WE: ~99/70 = 600.0696{{c}}, ~35/24 = 651.0136{{c}} (~36/35 = 50.9440{{c}})

~~: [[error map]]:~~ {{~~val| +0.771 +1.365 -1.315 -3.024~~ }}

* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9378{{c}} (~36/35 = 50.9378{{c}})

* [[CWE]]: ~2 = ~~1200~~.0000{{c}}, ~~~128~~/~~105~~ = ~~351~~.~~0471~~{{c}}

~~: error map:~~ {{~~val| 0.000 +0.139 -3.068 -5.213~~ }}

{{Optimal ET sequence|legend=1| ~~17,~~ 24, ~~41, 106d~~, ~~147d~~, ~~188cd~~ }}

[[Badness]] (Sintel): 2.03

Badness (Sintel): 1.56

=== 11-limit ===

===== 17-limit =====

Subgroup: 2.3.5.7.11

Subgroup: 2.3.5.7.11.13.17

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

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

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

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

Optimal tunings:

* WE: ~2 = ~~1200~~.~~3891~~{{c}}, ~11/9 = ~~351~~.~~1275~~{{c}}

* WE: ~99/70 = 600.1134{{c}}, ~35/24 = 651.0646{{c}} (~36/35 = 50.9512{{c}})

* CWE: ~2 = ~~1200~~.0000{{c}}, ~11/9 = ~~351~~.~~0141~~{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9414{{c}} (~36/35 = 50.9414{{c}})

Badness (Sintel): 1.30

=== ~~13-limit~~ ===

==== Kleischis ====

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal tunings:

* WE: ~2 = ~~1199~~.~~9362~~{c}}, ~11/9 = ~~351~~.~~0061~~{{c}}

* WE: ~99/70 = 600.1909{{c}}, ~35/24 = 651.1578{{c}} (~36/35 = 50.9670{{c}})

* CWE: ~2 = ~~1200~~.0000{{c}}, ~11/9 = ~~351~~.~~0247~~{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9541{{c}} (~36/35 = 50.9541{{c}})

Badness (Sintel): 1.27

Badness (Sintel): 1.55

== ~~Hemischis~~ ==

===== 17-limit =====

[[Subgroup]]: 2.3.5.7

Subgroup: 2.3.5.7.11.13.17

[[Comma list]]: ~~6144~~/~~6125~~, ~~19683~~/~~19600~~

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 generators: ~2, ~140/81~~

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

Optimal tunings:

* [[WE]]: ~2 = ~~1199~~.~~8579~~{{c}}, ~~~140~~/81 = ~~951~~.~~6847~~{{c}}

* WE: ~99/70 = 600.2190{{c}}, ~35/24 = 651.1578{{c}} (~36/35 = 50.9670{{c}})

~~: [[error map]]:~~ {{~~val| -0.142 -0.586 +0.600 +0.708~~ }}

* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9518{{c}} (~36/35 = 50.9518{{c}})

* [[CWE]]: ~2 = ~~1200~~.0000{{c}}, ~~~140~~/~~81 = 951.7966{{c}}~~

~~: error map: {{val| 0.000 -0.362 +0.941 +1.088 }}~~

~~~~

[[Badness]] (Sintel): 1.16

Badness (Sintel): 1.26

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

== Salsa ==

~~Subgroup: 2~~.3.5.~~7.11~~

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]].

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

[[Subgroup]]: 2.3.5.7

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

[[Comma list]]: 245/243, 32805/32768

~~Optimal tunings:~~

* WE: ~2 = 1199.8482{{~~c}}, ~140/81~~ = ~~950.6809{{c~~}}

: mapping generators: ~2, ~128/105

* CWE: ~~~2 = 1200.0000{{c}}, ~140/81 = 950.8020{{c}}~~

~~~~

{{~~Optimal ET sequence~~|~~legend~~=0~~| 53, 130, 183, 313, 809cd~~ }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.7707{{c}}, ~128/105 = 351.2748{{c}}

: [[error map]]: {{val| +0.771 +1.365 -1.315 -3.024 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 351.0471{{c}}

: error map: {{val| 0.000 +0.139 -3.068 -5.213 }}

~~Badness (Sintel):~~ 1~~.20~~

~~=== 13-limit ===~~

[[Badness]] (Sintel): 2.03

~~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~~

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal tunings:

* WE: ~2 = ~~1199~~.~~9140~~{{c}}, ~~~140~~/81 = ~~950~~.~~7324~~{{c}}

* WE: ~2 = 1200.3891{{c}}, ~11/9 = 351.1275{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~~~140/81 = 950.8010{{c}}~~

* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0141{{c}}

~~~~

Badness (Sintel): 0.~~860~~

Badness (Sintel): 1.30

=== 17-limit ===

=== 13-limit ===

Subgroup: 2.3.5.7.11.13~~.17~~

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal tunings:

* WE: ~2 = 1199.~~9740~~{{c}}, ~26/15 = ~~950~~.~~7894~~{{c}}

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

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = ~~950~~.~~8100~~{{c}}

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

~~~~

Badness (Sintel): 1.07

Badness (Sintel): 1.27

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

== Hemischis ==

~~Subgroup: 2~~.~~3.5.7.11.13.17~~.19

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.

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

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.

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

[[Subgroup]]: 2.3.5.7

~~Optimal tunings~~:

[[Comma list]]: 6144/6125, 19683/19600

* WE: ~2 = 1200.0464{{c}}, ~26/15 = 950.8459{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/~~15 = 950.8091{{c}}~~

~~~~

: mapping generators: ~2, ~140/81

~~Badness (Sintel)~~: 1.11

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.8579{{c}}, ~140/81 = 951.6847{{c}}

: [[error map]]: {{val| -0.142 -0.586 +0.600 +0.708 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 951.7966{{c}}

: error map: {{val| 0.000 -0.362 +0.941 +1.088 }}

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

~~Subgroup: 2.3.5.7.11.13.17.19.23~~

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

[[Badness]] (Sintel): 1.16

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

=== 11-limit ===

Subgroup: 2.3.5.7.11

~~Optimal tunings~~:

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

* WE: ~2 = 1200.0215{{c}}, ~26/~~15 = 950.8239{{c}}~~

* CWE: ~2 = 1200.0000{{c}}, ~~~26~~/~~15 = 950.8069{{c}}~~

~~~~

{{~~Optimal ET sequence~~|~~legend=~~0| ~~53, 130, 183, 313h~~ }}

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

~~Badness (Sintel)~~: 1.06

Optimal tunings:

* WE: ~2 = 1199.8482{{c}}, ~140/81 = 950.6809{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~140/81 = 950.8020{{c}}

~~; Music~~

* ''HemischisMatic EP'' (2023) by [[User:Francium|~~Francium]] – [https://open.spotify.com/album/1Fx2shLclpNgFQJRw3ZHya Spotify]~~ | ~~[https://francium223.bandcamp.com/album/hemischismatic-ep Bandcamp] | [https://www.youtube.com/playlist?list=PLLZE7hMjEXRaiipPYK1InZBXTru_UtRsq YouTube] – 4-piece extended play~~

~~== Squirrel ==~~

Badness (Sintel): 1.20

~~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~~.

[[Subgroup]]: 2.3.5.7

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

[[Comma list]]: ~~686~~/675, ~~32805~~/~~32768~~

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 ~~tuning]]s~~:

Optimal tunings:

* [[WE]]: ~2 = ~~1200~~.~~7408~~{{c}}, ~~~160~~/~~147~~ = ~~166~~.~~2424~~{{c}}

* WE: ~2 = 1199.9140{{c}}, ~140/81 = 950.7324{{c}}

~~: [[error map]]: {{val| +0.741 +0.799 +2.763 -6.934~~ }}

* CWE: ~2 = 1200.0000{{c}}, ~140/81 = 950.8010{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~~~160/147 = 166.1597{{c}}~~

~~: error map: {{val| 0.000 -0.434 +1.518 -8.750 }}~~

~~~~

[[Badness]] (Sintel): 4.42

Badness (Sintel): 0.860

=== 11-limit ===

=== 17-limit ===

Subgroup: 2.3.5.7.11

Subgroup: 2.3.5.7.11.13.17

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

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

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

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

Optimal tunings:

* WE: ~2 = ~~1200~~.~~6379~~{{c}}, ~11/10 = ~~166~~.~~1853~~{{c}}

* WE: ~2 = 1199.9740{{c}}, ~26/15 = 950.7894{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~~~11/10 = 166.1157{{c}}~~

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8100{{c}}

~~~~

Badness (Sintel): 2.26

Badness (Sintel): 1.07

=== 13-limit ===

=== 19-limit ===

Subgroup: 2.3.5.7.11.13

Subgroup: 2.3.5.7.11.13.17.19

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

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

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

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

Optimal tunings:

* WE: ~2 = ~~1201~~.~~1361~~{{c}}, ~11/10 = ~~166~~.~~2110~~{{c}}

* WE: ~2 = 1200.0464{{c}}, ~26/15 = 950.8459{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~~~11/10 = 166.0833{{c}}~~

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8091{{c}}

~~~~

Badness (Sintel): 1.81

Badness (Sintel): 1.11

== ~~Tertiaschis~~ ==

=== 23-limit ===

~~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~~.

Subgroup: 2.3.5.7.11.13.17.19.23

~~[[Subgroup]]~~: ~~2.3.5.7~~

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

~~[[Comma list]]~~: ~~32805/32768, 1071875/1062882~~

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

Optimal tunings:

* WE: ~2 = 1200.0215{{c}}, ~26/15 = 950.8239{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8069{{c}}

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

* [[WE]]: ~2 = 1200.3627{{~~c}}, ~192/175~~ = ~~166.0691{{c}}~~

~~: [[error map]]: {{val~~| ~~+0.363 +0.563 -1.019 -0.790 }}~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~~~192/175 = 166.0172{{c}}~~

~~: error map: {{val| 0.000 -0.007 -1.901 -1.720 }}~~

~~~~

~~{{Optimal ET sequence|legend=~~1~~| 65, 94, 159, 253, 412cd }}~~

Badness (Sintel): 1.06

[[~~Badness~~]] ~~(Sintel)~~: 5.36

; Music

* ''HemischisMatic EP'' (2023) by [[User:Francium|Francium]] – [https://open.spotify.com/album/1Fx2shLclpNgFQJRw3ZHya Spotify] | [https://francium223.bandcamp.com/album/hemischismatic-ep Bandcamp] | [https://www.youtube.com/playlist?list=PLLZE7hMjEXRaiipPYK1InZBXTru_UtRsq YouTube] – 4-piece extended play

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

== Term ==

~~Subgroup: 2~~.3.5.7.11

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.

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

[[Subgroup]]: 2.3.5.7

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

[[Comma list]]: 32805/32768, 250047/250000

~~Optimal tunings:~~

* WE: ~2 = 1200.3379{{~~c}}, ~11/10~~ = ~~166.0638{{c~~}}

: mapping generators: ~63/50, ~3

* CWE: ~~~2 = 1200.0000{{c}}, ~11~~/~~10 = 166.0167{{c}}~~

~~~~

{{~~Optimal ET sequence~~|~~legend~~=~~0| 65~~, ~~94, 159, 253, 412cd, 665ccde~~ }}

[[Optimal tuning]]s:

* [[WE]]: ~63/50 = 400.0257{{c}}, ~3/2 = 701.7873{{c}}

~~Badness (Sintel)~~: 2.07

: [[error map]]: {{val| +0.077 -0.091 -0.072 +0.031 }}

* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 701.7383{{c}}

: error map: {{val| 0.000 -0.217 -0.220 -0.115 }}

~~=== 13~~-limit ~~===~~

[[Minimax tuning]]:

~~Subgroup~~: 2.3.5.7~~.11.13~~

* [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis)]]: 2.5/3

* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7

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

{{Optimal ET sequence|legend=1| 12, …, 159, 171, 867, 1038, 1209, 1380, 1551, 1722 }}

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

[[Badness]] (Sintel): 0.505

~~Optimal tunings:~~

=== Terminal ===

* WE: ~2 = ~~1200.3467{{c}}, ~11/10~~ = ~~166.0635{{c}}~~

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.

* CWE: ~2 = 1200.0000{{c}}, ~~~11/10 = 166.0142~~{{c}}

~~~~

~~{{Optimal ET sequence|legend=0| 65f, 94, 159, 253, 412cdf, 665ccdef }}~~

Subgroup: 2.3.5.7.11

~~Badness (Sintel)~~: ~~1.52~~

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

~~=== 17-limit ===~~

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

~~Subgroup~~: 2.3~~.5.7.11.13.17~~

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

Optimal tunings:

* WE: ~44/35 = 400.0464{{c}}, ~3/2 = 701.9053{{c}}

* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8178{{c}}

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

~~Optimal tunings~~:

Badness (Sintel): 1.97

* WE: ~2 = 1200.3019{{c}}, ~11/10 = 166.0535{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0114{{c}}

~~~~

~~{{Optimal ET sequence|legend~~=~~1| 65f, 94, 159, 253 }}~~

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

~~Badness (Sintel)~~: ~~1.35~~

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

~~== Countertertiaschis ==~~

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

~~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.~~

~~[[Subgroup]]~~: 2.3.~~5.7~~

Optimal tunings:

* WE: ~44/35 = 400.0449{{c}}, ~3/2 = 701.8995{{c}}

* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8156{{c}}

~~[[Comma list]]: 32805/32768~~, ~~244140625/243045684~~

~~{{Mapping|legend=~~1~~| 1 2 -1 -12 | 0 -3 24 107 }}~~

Badness (Sintel): 1.53

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

==== 17-limit ====

* [[WE]]: ~2 = ~~1200.1265{{c}}, ~625/567~~ = ~~166.0797{{c}}~~

Subgroup: 2.3.5.7.11.13.17

~~: [[error map]]: {{val| +0.127 +0.059~~ -~~0.529 +0.178 }}~~

* [[CWE]]: ~2 = ~~1200.0000{{c}}, ~625/567~~ = ~~166.0632{{c}}~~

: ~~error map: {{val| 0~~.~~000 -0~~.~~145 -0~~.~~797 -0~~.~~065 }}~~

~~~~

~~{{Optimal ET sequence|legend=1| 65d~~, ~~159~~, ~~224~~, ~~383~~, ~~607 }}~~

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

~~[[Badness]] (Sintel)~~: ~~4.76~~

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

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

Optimal tunings:

~~Subgroup~~: 2.3.~~5.7.11~~

* WE: ~34/27 = 400.0195{{c}}, ~3/2 = 701.8439{{c}}

* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 701.8081{{c}}

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

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

Badness (Sintel): 1.38

=== 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:

* WE: ~2 = ~~1200~~.~~0804~~{{c}}, ~11/10 = ~~166~~.~~0739~~{{c}}

* WE: ~63/50 = 399.9677{{c}}, ~3/2 = 701.6278{{c}}

* CWE: ~2 = ~~1200~~.0000{{c}}, ~11/~~10 = 166.0634{{c}}~~

* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6846{{c}}

~~~~

{{Optimal ET sequence|legend=0| ~~65d~~, ~~159~~, ~~224~~, ~~383~~, ~~607~~ }}

{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 537, 891de }}

Badness (Sintel): 1.62

Badness (Sintel): 2.21

=== 13-limit ===

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal tunings:

* WE: ~2 = ~~1200~~.~~0805~~{{c}}, ~11/10 = ~~166~~.~~0740~~{{c}}

* WE: ~63/50 = 399.9731{{c}}, ~3/2 = 701.6414{{c}}

* CWE: ~2 = ~~1200~~.0000{{c}}, ~11/~~10 = 166.0635{{c}}~~

* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}}

~~~~

{{Optimal ET sequence|legend=0| ~~65d~~, ~~159~~, ~~224~~, ~~383~~, ~~607~~ }}

Badness (Sintel): 1.01

Badness (Sintel): 1.47

== ~~Term~~ ==

==== 17-limit ====

~~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~~.

Subgroup: 2.3.5.7.11.13.17

~~[[Subgroup]]~~: ~~2.3.5.7~~

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

~~[[Comma list]]~~: ~~32805/32768, 250047/250000~~

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

Optimal tunings:

~~: mapping generators~~: ~63/50, ~3

* WE: ~63/50 = 399.9757{{c}}, ~3/2 = 701.6458{{c}}

* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}}

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

* [[WE]]: ~63/50 = 400.0257{{~~c}}, ~3/2~~ = ~~701.7873{{c}}~~

~~: [[error map]]: {{val~~| ~~+0.077 -0.091 -0.072 +0.031 }}~~

* [[CWE]]: ~63/50 = 400.0000{{c}}, ~~~3/2 = 701.7383{{c}}~~

~~: error map: {{val| 0.000 -0.217 -0.220 -0.115 }}~~

~~~~

~~[[Minimax tuning]]:~~

Badness (Sintel): 1.04

* [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis)]]: 2.5/3

* [[9-odd-limit]] unchanged-interval (~~eigenmonzo~~) ~~basis~~: 2.~~9/7~~

~~{{Optimal ET sequence|legend~~=~~1| 12, …, 159, 171, 867, 1038, 1209, 1380, 1551, 1722 }}~~

=== Semiterm ===

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.

~~[[Badness]] (Sintel): 0.505~~

==~~= 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.

Subgroup: 2.3.5.7.11

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

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

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

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

: mapping generators: ~55/49, ~3

Optimal tunings:

* WE: ~44/35 = ~~400~~.~~0464~~{{c}}, ~3/2 = 701.~~9053~~{{c}}

* WE: ~55/49 = 200.0134{{c}}, ~3/2 = 701.7931{{c}}

* CWE: ~44/35 = ~~400~~.0000{{c}}, ~3/2 = 701.~~8178~~{{c}}

* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7426{{c}}

~~~~

{{Optimal ET sequence|legend=0| 12, …, 330e, 342, 1380, 1722, 2064, 2406c, 5154bccdde }}

Badness (Sintel): 1.97

Badness (Sintel): 0.973

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal tunings:

* WE: ~44/35 = ~~400~~.~~0449~~{{c}}, ~3/2 = 701.~~8995~~{{c}}

* WE: ~55/49 = 200.0083{{c}}, ~3/2 = 701.7549{{c}}

* CWE: ~44/35 = ~~400~~.0000{{c}}, ~3/2 = 701.~~8156~~{{c}}

* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7238{{c}}

~~<!-- * POTE: ~44/35 = 400.000~~{{c}}~~, ~3~~/~~2 = 701.821{{c}} --~~>

{{Optimal ET sequence|legend=0| 12f, 330eff, 342f, 696f }} *

<nowiki>*</nowiki> optimal patent val: [[354edo|354]]

~~{{Optimal ET sequence|legend=0| 12f, …, 159, 330 }}~~

Badness (Sintel): 1.85

~~Badness~~ (~~Sintel~~)~~: 1~~.53

=== Hemiterm ===

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

~~==== 17-limit ====~~

Subgroup: 2.3.5.7.11

Subgroup: 2.3.5.7.11~~.13.17~~

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

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

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

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

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

Optimal tunings:

* WE: ~34/27 = 400.~~0195~~{{c}}, ~3/2 = ~~701~~.~~8439~~{{c}}

* WE: ~63/50 = 400.0309{{c}}, ~693/400 = 950.9458{{c}} (~12/11 = 150.8841{{c}})

* CWE: ~34/27 = ~~400~~.~~0000~~{{c}}, ~3/2 = ~~701~~.~~8081~~{{c}}

* CWE: ~63/50 = 400.0000{{c}}, ~693/400 = 950.8707{{c}} (~12/11 = 150.8707{{c}})

~~~~

{{Optimal ET sequence|legend=0| ~~12f~~, 159, ~~171~~, ~~330~~ }}

{{Optimal ET sequence|legend=0| 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce }}

Badness (Sintel): ~~1.38~~

Badness (Sintel): 0.684

~~=== Terminator ===~~

~~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:~~

* WE: ~63/50 = 399.9677{{c}}, ~3/2 = 701.6278{{c}}

* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6846{{c}}

~~~~

~~{{Optimal ET sequence|legend=~~0~~| 12e, 171, 183, 354, 537, 891de }}~~

~~Badness (Sintel): 2~~.21

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal tunings:

* WE: ~63/50 = ~~399~~.~~9731~~{{c}}, ~3/2 = ~~701~~.~~6414~~{{c}}

* WE: ~63/50 = 400.0541{{c}}, ~26/15 = 951.0013{{c}} (~12/11 = 150.8932{{c}})

* CWE: ~63/50 = 400.0000{{c}}, ~~~3/2 = 701.6881{{c}}~~

* CWE: ~63/50 = 400.0000{{c}}, ~26/15 = 950.8696{{c}} (~12/11 = 150.8696{{c}})

~~~~

{{Optimal ET sequence|legend=0| ~~12e~~, ~~171~~, 183, ~~354, 891de~~ }}

Badness (Sintel): 1.47

Badness (Sintel): 1.30

==== 17-limit ====

Subgroup: 2.3.5.7.11.13.17

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

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

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

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

Optimal tunings:

* WE: ~63/50 = ~~399~~.~~9757~~{{c}}, ~3/2 = ~~701~~.~~6458~~{{c}}

* WE: ~34/27 = 400.0373{{c}}, ~26/15 = 950.9556{{c}} (~12/11 = 150.8809{{c}})

* CWE: ~63/50 = ~~400~~.~~0000~~{{c}}, ~3/2 = ~~701~~.~~6881~~{{c}}

* CWE: ~34/27 = 400.0000{{c}}, ~26/15 = 950.8652{{c}} (~12/11 = 150.8652{{c}})

~~~~

{{Optimal ET sequence|legend=0| ~~12e~~, ~~171~~, 183, ~~354~~, ~~891de~~ }}

Badness (Sintel): 1.04

Badness (Sintel): 1.14

==~~= Semiterm =~~==

== Altinex ==

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

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~~.11~~

[[Subgroup]]: 2.3.5.7

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

[[Comma list]]: 32805/32768, 367653125/362797056

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

: mapping generators: ~55/49, ~3

: mapping generators: ~1536/1225, ~34300/19683

Optimal ~~tunings~~:

[[Optimal tuning]]s:

* WE: ~55/49 = ~~200~~.~~0134~~{{c}}, ~3/2 = ~~701~~.~~7931~~{{c}}

* [[WE]]: ~1536/1225 = 400.1360{{c}}, ~34300/19683 = 951.2867{{c}}

* CWE: ~55/49 = ~~200~~.0000{{c}}, ~3/2 = ~~701~~.~~7426~~{{c}}

: [[error map]]: {{val| +0.408 +0.618 -0.781 -1.304 }}

* [[CWE]]: ~1536/1225 = 400.0000{{c}}, ~34300/19683 = 950.9638{{c}}

: error map: {{val| 0.000 -0.027 -1.735 -2.441 }}

{{Optimal ET sequence|legend=0| 12, …, ~~330e~~, ~~342, 1380, 1722, 2064, 2406c, 5154bccdde~~ }}

Badness (Sintel): 0.~~973~~

[[Badness]] (Sintel): 10.7

===~~= 13~~-limit ====

=== 11-limit ===

Subgroup: 2.3.5.7.11~~.13~~

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal tunings:

* WE: ~55/49 = ~~200~~.~~0083~~{{c}}, ~3/2 = ~~701~~.~~7549~~{{c}}

* WE: ~44/35 = 400.1156{{c}}, ~121/70 = 951.2377{{c}}

* CWE: ~55/49 = ~~200~~.0000{{c}}, ~3/2 = ~~701~~.~~7238~~{{c}}

* CWE: ~44/35 = 400.0000{{c}}, ~121/70 = 950.9634{{c}}

~~~~

{{Optimal ET sequence|legend=0| ~~12f~~, ~~330eff~~, ~~342f, 696f~~ }} *

~~<nowiki>*</nowiki> optimal patent val~~: ~~[[354edo|354]]~~

Badness (Sintel): 3.35

~~Badness (Sintel)~~: 1.85

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

~~=== Hemiterm ===~~

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

~~Subgroup~~: ~~2.3.5.7.11~~

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

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

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

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

Optimal tunings:

* WE: ~63/50 = 400.~~0309~~{{c}}, ~~~693~~/~~400 = 950.9458{{c}} (~12/11~~ = ~~150~~.~~8841~~{{c}})

* WE: ~44/35 = 400.1396{{c}}, ~26/15 = 951.2799{{c}}

* CWE: ~63/50 = 400.0000{{c}}, ~~~693~~/~~400~~ = 950.~~8707{{c}} (~12/11 = 150.8707{{c}})~~

* CWE: ~44/35 = 400.0000{{c}}, ~26/15 = 950.9462{{c}}

~~~~

{{Optimal ET sequence|legend=0| ~~24d~~, 159~~, 183, 342, 1209, 1551, 1893e, 2235ce~~ }}

Badness (Sintel): 0.~~684~~

Badness (Sintel): 2.27

==~~== 13-limit ==~~==

== Squirrel ==

~~Subgroup: 2~~.3~~.5.~~7.11.13

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.

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

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 686/675, 32805/32768

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

Optimal ~~tunings~~:

[[Optimal tuning]]s:

* WE: ~~~63/50~~ = ~~400~~.~~0541~~{{c}}, ~26/15 = ~~951~~.~~0013~~{{c}} ~~(~12/11 = 150.8932~~{{c}})

* [[WE]]: ~2 = 1200.7408{{c}}, ~160/147 = 166.2424{{c}}

* CWE: ~~~63/50~~ = ~~400~~.0000{{c}}, ~26/15 = ~~950~~.~~8696~~{{c}} ~~(~12/11 = 150.8696~~{{~~c}})~~

: [[error map]]: {{val| +0.741 +0.799 +2.763 -6.934 }}

~~<!-~~- * POTE: ~63/50 = 400.~~000{{c}}, ~26/15 = 950~~.~~873{{c}} (~12/11 = 150~~.~~873{{c~~}}~~) -->~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~160/147 = 166.1597{{c}}

: error map: {{val| 0.000 -0.434 +1.518 -8.750 }}

{{Optimal ET sequence|legend=0| ~~24d~~, ~~159~~, ~~183, 342f~~ }}

Badness (Sintel): 1.30

[[Badness]] (Sintel): 4.42

===~~= 17~~-limit ====

=== 11-limit ===

Subgroup: 2.3.5.7.11~~.13.17~~

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal tunings:

* WE: ~~~34/27~~ = ~~400~~.~~0373~~{{c}}, ~~~26/15 = 950.9556{{c}} (~12~~/11 = ~~150~~.~~8809~~{{c}})

* WE: ~2 = 1200.6379{{c}}, ~11/10 = 166.1853{{c}}

* CWE: ~~~34/27~~ = ~~400~~.0000{{c}}, ~~~26/15 = 950.8652{{c}} (~12/~~11 ~~= 150.8652{{c}})~~

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.1157{{c}}

~~~~

{{Optimal ET sequence|legend=0| ~~24d~~, ~~159~~, ~~183, 342f, 525f~~ }}

Badness (Sintel): 1.14

Badness (Sintel): 2.26

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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:

* WE: ~2 = 1201.1361{{c}}, ~11/10 = 166.2110{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0833{{c}}

Badness (Sintel): 1.81

== Tertiaschis ==

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

~~== Altinex ==~~

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768, ~~367653125~~/~~362797056~~

[[Comma list]]: 32805/32768, 1071875/1062882

~~: mapping generators: ~1536/1225, ~34300/19683~~

[[Optimal tuning]]s:

* [[WE]]: ~~~1536/1225~~ = ~~400~~.~~1360~~{{c}}, ~~~34300~~/~~19683~~ = ~~951~~.~~2867~~{{c}}

* [[WE]]: ~2 = 1200.3627{{c}}, ~192/175 = 166.0691{{c}}

: [[error map]]: {{val| +0.~~408~~ +0.~~618~~ -0.~~781~~ -1.~~304~~ }}

: [[error map]]: {{val| +0.363 +0.563 -1.019 -0.790 }}

* [[CWE]]: ~~~1536/1225~~ = ~~400~~.0000{{c}}, ~~~34300~~/~~19683~~ = ~~950~~.~~9638~~{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~192/175 = 166.0172{{c}}

: error map: {{val| 0.000 -0.~~027~~ -1.~~735~~ -2.~~441~~ }}

: error map: {{val| 0.000 -0.007 -1.901 -1.720 }}

~~~~

[[Badness]] (Sintel): 10.7

[[Badness]] (Sintel): 5.36

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal tunings:

* WE: ~~~44/35~~ = ~~400~~.~~1156~~{{c}}, ~~~121~~/70 = ~~951~~.~~2377~~{{c}}

* WE: ~2 = 1200.3379{{c}}, ~11/10 = 166.0638{{c}}

* CWE: ~~~44/35~~ = ~~400.0000{{c}}, ~121/70 = 950.9634{{c}}~~

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0167{{c}}

~~~~

{{Optimal ET sequence|legend=0| 65, 94, 159, 253, 412cd, 665ccde }}

Badness (Sintel): 3.35

Badness (Sintel): 2.07

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal tunings:

* WE: ~~~44/35~~ = ~~400~~.~~1396~~{{c}}, ~26/15 = ~~951~~.~~2799~~{{c}}

* WE: ~2 = 1200.3467{{c}}, ~11/10 = 166.0635{{c}}

* CWE: ~~~44/35~~ = ~~400.0000{{c}}, ~26/15 = 950.9462{{c}}~~

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0142{{c}}

~~~~

{{Optimal ET sequence|legend=0| 65f, 94, 159, 253, 412cdf, 665ccdef }}

Badness (Sintel): 2.27

Badness (Sintel): 1.52

== ~~Sesquiquartififths~~ ==

=== 17-limit ===

[[Subgroup]]: 2.3.5.7

Subgroup: 2.3.5.7.11.13.17

[[Comma list]]: ~~2401~~/~~2400~~, ~~32805~~/~~32768~~

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 generators: ~2, ~448/405~~

[[Optimal ~~tuning]] ([[POTE]])~~: ~2 = 1200.~~000~~{{c}}, ~~~448~~/~~405~~ = ~~175~~.~~434~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.3019{{c}}, ~11/10 = 166.0535{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0114{{c}}

~~[[Minimax tuning]]:~~

* [[7-odd-limit]] [[Eigenmonzo basis|~~unchanged-interval (eigenmonzo) basis]]: 2.7/3~~

* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7

~~{{Optimal ET sequence|legend=~~1~~| 41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd }}~~

Badness (Sintel): 1.35

[[~~Badness~~]] ~~(Smith): 0~~.~~011244~~

== Countertertiaschis ==

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

~~=== Sesquart ===~~

[[Subgroup]]: 2.3.5.7

Subgroup: 2.3.5.7~~.11~~

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

[[Comma list]]: 32805/32768, 244140625/243045684

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

Optimal tuning ~~(POTE)~~: ~2 = 1200.~~000~~{{c}}, ~~~256~~/~~231~~ = ~~175~~.~~406~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.1265{{c}}, ~625/567 = 166.0797{{c}}

: [[error map]]: {{val| +0.127 +0.059 -0.529 +0.178 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/567 = 166.0632{{c}}

: error map: {{val| 0.000 -0.145 -0.797 -0.065 }}

{{Optimal ET sequence|legend=0| 41, 89, ~~130~~, ~~301e~~, ~~431e~~ }}

Badness (~~Smith~~): 0.~~029306~~

[[Badness]] (Sintel): 4.76

===~~= 13~~-limit ====

=== 11-limit ===

Subgroup: 2.3.5.7.11~~.13~~

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~72/65 = ~~175~~.~~409~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.0804{{c}}, ~11/10 = 166.0739{{c}}

{{~~Optimal ET sequence|legend~~=~~0| 41, 89, 130, 301e, 431e~~ }}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0634{{c}}

~~Badness (Smith):~~ 0~~.022396~~

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

Badness (Sintel): 1.62

~~Subgroup~~: 2.~~3.5.7.11.13.17~~

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

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

~~Optimal tuning (POTE)~~: ~~~2 = 1200.000~~{{~~c}}, ~72/65 = 175.424{{c~~}}

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

{{~~Optimal ET sequence|legend~~=~~0| 41~~, ~~89g, 130, 171, 301e~~ }}

Optimal tunings:

* WE: ~2 = 1200.0805{{c}}, ~11/10 = 166.0740{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0635{{c}}

~~Badness (Smith):~~ 0~~.023126~~

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

Badness (Sintel): 1.01

~~Subgroup~~: 2.~~3.5.7.11.13.17.19~~

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

== Quadrant ==

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

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

[[Subgroup]]: 2.3.5.7

~~Optimal tuning (POTE)~~: ~~~2 = 1200.000{{c}}~~, ~~~21~~/~~19 = 175.419{{c}}~~

[[Comma list]]: 32805/32768, 390625/388962

{{~~Optimal ET sequence~~|legend=0| ~~41, 89g, 130, 171, 301eh~~ }}

: mapping generators: ~25/21, ~3

~~Badness (Smith)~~: 0.~~020466~~

[[Optimal tuning]]s:

* [[WE]]: ~2 = 300.0255{{c}}, ~3/2 = 701.8831{{c}}

: [[error map]]: {{val| +0.102 +0.030 -0.664 +0.462 }}

* [[CWE]]: ~2 = 300.0000{{c}}, ~3/2 = 701.8180{{c}}

: error map: {{val| 0.000 -0.137 -0.858 +0.268 }}

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

{{Optimal ET sequence|legend=1| 12, …, 200, 212, 224, 436, 660 }}

~~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~~

[[Badness]] (Sintel): 2.79

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

=== 11-limit ===

Subgroup: 2.3.5.7.11

~~Optimal tuning (POTE)~~: ~~~2 = 1200.000{{c}}~~, ~~~21~~/~~19 = 175.412{{c}}~~

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

{{~~Optimal ET sequence~~|~~legend=~~0| ~~41i, 89gi, 130, 171, 301eh~~ }}

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

~~Badness (Smith)~~: 0.~~019043~~

Optimal tunings:

* WE: ~25/21 = 300.0244{{c}}, ~3/2 = 701.8759{{c}}

* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8145{{c}}

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

~~Subgroup: 2.3.5.7.11.13.17~~

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

Badness (Sintel): 1.51

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

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

~~Optimal tuning (POTE)~~: ~~~2 = 1200.000{{c}}~~, ~~~72~~/~~65 = 175.386{{c}}~~

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

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

~~Badness (Smith)~~: 0.~~028443~~

Optimal tunings:

* WE: ~25/21 = 300.0234{{c}}, ~3/2 = 701.8707{{c}}

* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8123{{c}}

=~~===== 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~~

Badness (Sintel): 1.13

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

== 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.

~~Optimal tuning (POTE)~~: ~2 ~~= 1200~~.~~000{{c}}, ~21/19 = 175~~.~~380{{c}}~~

[[Subgroup]]: 2.3.5.7

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

[[Comma list]]: 2401/2400, 32805/32768

~~Badness (Smith)~~: ~~0.023059~~

: mapping generators: ~2, ~448/405

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

[[Optimal tuning]]s:

~~Subgroup~~: 2.3.5.7.11.13.17

* [[WE]]: ~2 = 1200.0846{{c}}, ~448/405 = 175.4460{{c}}

: [[error map]]: {{val| +0.085 -0.086 +0.007 -0.093 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~448/405 = 175.4320{{c}}

: error map: {{val| 0.000 -0.227 -0.137 -0.306 }}

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

[[Minimax tuning]]:

* [[7-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3

* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7

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

{{Optimal ET sequence|legend=1| 41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd }}

~~Optimal tuning~~ (~~POTE~~): ~~~2 = 1200~~.~~000{{c}}, ~72/65 = 175.377{{c}}~~

[[Badness]] (Sintel): 0.285

~~{{Optimal ET sequence|legend~~=0| ~~41g~~, 89, ~~130~~, ~~609ceefgg }}~~

=== 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.

~~Badness (Smith)~~: 0.~~030680~~

Subgroup: 2.3.5.7.11

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

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

~~Subgroup~~: ~~2.3.5.7.11.13.17.19~~

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

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

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

Optimal tunings:

* WE: ~2 = 1199.8171{{c}}, ~256/231 = 175.3793{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~256/231 = 175.4081{{c}}

Optimal ~~tuning (POTE): ~2~~ = ~~1200.000{{c}}~~, ~~~21/19 = 175.377{{c~~}}

~~{{Optimal ET sequence|legend=~~0~~| 41g, 89, 130, 609ceefggh }}~~

Badness (Sintel): 0.969

~~Badness (Smith)~~: 0.~~022816~~

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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

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

~~Subgroup~~: ~~2.3.5.7.11.13.17.19.23~~

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

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

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

Optimal tunings:

* WE: ~2 = 1199.8352{{c}}, ~72/65 = 175.3852{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4095{{c}}

Optimal ~~tuning (POTE): ~2~~ = ~~1200.000{{c}}~~, ~~~21/19 = 175.376{{c~~}}

~~{{Optimal ET sequence|legend=~~0~~| 41g, 89, 130, 609ceefggh }}~~

Badness (Sintel): 0.925

~~Badness (Smith)~~: 0.~~019121~~

===== Heartia =====

Subgroup: 2.3.5.7.11.13.17

~~=== Bisesqui ===~~

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

~~Subgroup~~: ~~2.3.5.7.11~~

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

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

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

Optimal tunings:

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

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

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

Optimal ~~tuning (POTE): ~99/70~~ = ~~600.000{{c}}~~, ~~~448/405 = 175.435{{c~~}}

~~{{Optimal ET sequence|legend=~~1~~| 82e, 130, 212, 342, 1156, 1498, 1840d }}~~

Badness (Sintel): 1.45

~~Badness (Smith)~~: 0.~~016968~~

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

Subgroup: 2.3.5.7.11.13.17.19

~~== Quintilipyth ==~~

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

~~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.~~

~~[[Subgroup]]~~: 2~~.3.5.7~~

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

~~[[Comma list]]~~: ~~32805~~/~~32768~~, ~~9765625~~/~~9680832~~

Optimal tunings:

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

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

~~[[Optimal tuning]]~~ (~~[[POTE]]~~): ~~~2 = 1200~~.~~000{{c}}, ~625/588 = 99.625{{c}}~~

Badness (Sintel): 1.40

~~{{Optimal ET sequence|legend~~=~~1| 12, 253, 265 }}~~

===== Sesquartia =====

Subgroup: 2.3.5.7.11.13.17

~~[[Badness]] (Smith)~~: ~~0.253966~~

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

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

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

~~Subgroup:~~ 2~~.3.5.7.11~~

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

Optimal tunings:

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

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

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

~~Optimal tuning~~ (~~POTE~~): ~~~2 = 1200~~.~~000{{c}}, ~35/33 = 99.616{{c}}~~

Badness (Sintel): 1.18

~~{{Optimal ET sequence|legend~~=~~0| 12, 253, 265, 518c, 783cc }}~~

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

Subgroup: 2.3.5.7.11.13.17.19

~~Badness (Smith)~~: ~~0.113044~~

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

~~=== 13~~-~~limit ===~~

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

~~Subgroup:~~ 2~~.3.5.7.11.13~~

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

Optimal tunings:

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

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

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

~~Optimal tuning~~ (~~POTE~~): ~~~2 = 1200~~.~~000{{c}}, ~35/33 = 99.612{{c}}~~

Badness (Sintel): 1.24

~~{{Optimal ET sequence|legend~~=~~0| 12f, 253, 518c, 771cc }}~~

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

Subgroup: 2.3.5.7.11.13.17.19.23

~~Badness (Smith)~~: ~~0.069127~~

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

~~=== 17~~-~~limit ===~~

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

~~Subgroup:~~ 2~~.3.5.7.11.13.17~~

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

Optimal tunings:

* WE: ~2 = 1199.9606{{c}}, ~21/19 = 175.4067{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4123{{c}}

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

~~Optimal tuning~~ (~~POTE~~): ~~~2 = 1200~~.~~000{{c}}, ~18/17 = 99.612{{c}}~~

Badness (Sintel): 1.36

~~{{Optimal ET sequence|legend~~=~~0| 12f, 253, 518c, 771cc }}~~

===== Hearty =====

Subgroup: 2.3.5.7.11.13.17

~~Badness (Smith)~~: ~~0.045992~~

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

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

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

~~Subgroup:~~ 2~~.3.5.7.11.~~13~~.17.19~~

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

Optimal tunings:

* WE: ~2 = 1199.9458{{c}}, ~72/65 = 175.3689{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3770{{c}}

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

~~Optimal tuning~~ (~~POTE~~): ~~~2 = 1200~~.~~000{{c}}, ~18/17 = 99.615{{c}}~~

Badness (Sintel): 1.56

~~{{Optimal ET sequence|legend~~=~~0| 12f, 253, 265, 518ch }}~~

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

Subgroup: 2.3.5.7.11.13.17.19

~~Badness (Smith)~~: ~~0.038155~~

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

~~== Quintaschis ==~~

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

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

~~[[Subgroup]]~~: 2.3.5.7

Optimal tunings:

* WE: ~2 = 1200.0114{{c}}, ~72/65 = 175.3783{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3765{{c}}

~~[[Comma list]]: 32805/32768~~, ~~49009212/48828125~~

~~{{Mapping|legend=~~1~~| 1 2 -1 -5 | 0 -5 40 94 }}~~

Badness (Sintel): 1.39

~~[[Optimal tuning]] ([[POTE]])~~: ~2 ~~= 1200~~.~~000{{c}}, ~200/189 = 99~~.~~664{{c}}~~

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

Subgroup: 2.3.5.7.11.13.17.19.23

~~{{Optimal ET sequence|legend=1| 12~~, …, ~~289~~, ~~301~~, ~~590~~, ~~891~~, ~~1192 }}~~

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

~~[[Badness]] (Smith)~~: 0~~.132890~~

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

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

Optimal tunings:

~~Subgroup~~: 2.~~3.5.7~~.11

* WE: ~2 = 1200.0122{{c}}, ~72/65 = 175.3782{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3763{{c}}

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

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

Badness (Sintel): 1.37

~~Optimal tuning (POTE)~~: ~2 ~~= 1200~~.~~000{{c}}, ~35/33 = 99~~.~~653{{c}}~~

=== Bisesqui ===

Subgroup: 2.3.5.7.11

~~{{Optimal ET sequence|legend=0| 12~~, …, ~~277d, 289 }}~~

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

~~Badness (Smith)~~: 0~~.111477~~

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

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

==~~== 13-limit ====~~

Optimal tunings:

~~Subgroup~~: 2.~~3.5.7.11~~.13

* WE: ~99/70 = 600.0429{{c}}, ~448/405 = 175.4474{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~448/405 = 175.4334{{c}}

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

{{Optimal ET sequence|legend=1| 82e, 130, 212, 342, 1156, 1498, 1840d, 5862bbccdddee }}

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

Badness (Sintel): 0.561

~~Optimal tuning (POTE): ~2~~ = ~~1200.000{{c}}, ~35/33~~ = ~~99.658~~{{c}}

== Tsaharuk ==

{{~~Optimal ET sequence~~|~~legend=0~~| ~~12f~~, ~~…, 277dff, 289 }}~~

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.

~~Badness (Smith)~~: 0.~~074218~~

[[Subgroup]]: 2.3.5.7

~~==== 17-limit ====~~

[[Comma list]]: 32805/32768, 420175/419904

~~Subgroup~~: ~~2.3.5.7.11.13.17~~

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

: mapping generators: ~2, ~243/224

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

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.1039{{c}}, ~243/224 = 140.3620{{c}}

: [[error map]]: {{val| +0.104 -0.041 -0.067 -0.137 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/224 = 140.3496{{c}}

: error map: {{val| 0.000 -0.207 -0.296 -0.436 }}

Optimal ~~tuning (POTE): ~2~~ = ~~1200.000{{c}}~~, ~~~18/17 = 99.656{{c~~}}

~~{{Optimal ET sequence|legend=~~0~~| 12f, 277dff, 289 }}~~

[[Badness]] (Sintel): 0.777

~~Badness (Smith)~~: 0.~~050571~~

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

~~Subgroup~~: ~~2.3.5.7.11.13.17.19~~

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

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

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

Optimal tunings:

* WE: ~2 = 1200.3103{{c}}, ~88/81 = 140.4011{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.3649{{c}}

Optimal ~~tuning (POTE): ~2~~ = ~~1200.000{{c}}~~, ~~~18/17 = 99.659{{c~~}}

~~{{Optimal ET sequence|legend=0| 12f, 289 }}~~

Badness (Sintel): 2.10

~~Badness (Smith)~~: 0.~~042120~~

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

~~=== Quintahelenic ===~~

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

~~Subgroup~~: ~~2.3.5.7.11~~

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

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

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

Optimal tunings:

* WE: ~2 = 1200.1840{{c}}, ~13/12 = 140.3840{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.3627{{c}}

Optimal ~~tuning (POTE): ~2~~ = ~~1200.000{{c}}~~, ~~~200/189 = 99.671{{c~~}}

~~{{Optimal ET sequence|legend=0| 12, …, 289e, 301, 915 }}~~

Badness (Sintel): 1.57

~~Badness (Smith): 0~~.~~082225~~

== 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.

~~==== 13-limit ====~~

[[Subgroup]]: 2.3.5.7

Subgroup: 2.3.5.7~~.11.13~~

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

[[Comma list]]: 16875/16807, 32805/32768

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

: mapping generators: ~2, ~56/45

Optimal tuning ~~(POTE)~~: ~2 = 1200.~~000~~{{c}}, ~~~200~~/~~189~~ = 99.~~661~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0032{{c}}, ~56/45 = 380.3557{{c}}

: [[error map]]: {{val| +0.003 -0.177 -0.493 +0.898 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~56/45 = 380.3546{{c}}

: error map: {{val| 0.000 -0.182 -0.498 +0.890 }}

Badness (~~Smith~~): 0.~~055570~~

[[Badness]] (Sintel): 1.82

===~~== 17~~-limit =====

=== 11-limit ===

Subgroup: 2.3.5.7.11~~.13.17~~

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~18/17 = 99.~~665~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.9709{{c}}, ~56/45 = 380.3423{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3517{{c}}

Badness (~~Smith~~): 0.~~040412~~

Badness (Sintel): 1.04

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

=== 13-limit ===

Subgroup: 2.3.5.7.11.13~~.17.19~~

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~18/17 = 99.~~668~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.9663{{c}}, ~56/45 = 380.3403{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3509{{c}}

Badness (~~Smith~~): 0.~~036840~~

Badness (Sintel): 0.884

==~~== Quintahelenoid ==~~==

== Quintilipyth ==

~~Subgroup: 2.~~3~~.5.~~7.11.13

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.

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

[[Subgroup]]: 2.3.5.7

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

[[Comma list]]: 32805/32768, 9765625/9680832

~~Optimal tuning (POTE): ~2 = 1200.000~~{{c}}, ~~~200~~/~~189 = 99.672{{c}}~~

: mapping generators: ~2, ~625/588

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.1138{{c}}, ~625/588 = 99.6347{{c}}

: [[error map]]: {{val| +0.114 +0.099 -1.041 +0.761 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/588 = 99.6265{{c}}

: error map: {{val| 0.000 -0.087 -1.255 +0.544 }}

~~Badness (Smith): 0.066108~~

~~===== 17-limit =====~~

[[Badness]] (Sintel): 6.43

~~Subgroup~~: 2.~~3.5.7.11.13.17~~

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

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

~~Optimal tuning (POTE)~~: ~~~2 = 1200.000~~{{~~c}}, ~18/17 = 99.671{{c~~}}

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

Optimal tunings:

* WE: ~2 = 1200.1503{{c}}, ~35/33 = 99.6287{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6176{{c}}

~~Badness (Smith):~~ 0~~.047908~~

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

Badness (Sintel): 3.74

~~Subgroup~~: 2.3.~~5.7.11.13.17.19~~

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

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

~~Optimal tuning (POTE)~~: ~~~2 = 1200.000~~{{~~c}}, ~18/17 = 99.672{{c~~}}

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

Optimal tunings:

* WE: ~2 = 1200.1774{{c}}, ~35/33 = 99.6267{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6134{{c}}

~~Badness (Smith):~~ 0~~.039542~~

~~== Sextilifourths ==~~

Badness (Sintel): 2.86

~~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~~.

[[Subgroup]]: 2.3.5.7

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

[[Comma list]]: ~~32805~~/~~32768~~, ~~235298~~/~~234375~~

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 generators: ~2, ~21/2~~

[[Optimal ~~tuning]] ([[POTE]])~~: ~2 = 1200.~~000~~{{c}}, ~21/20 = 83.~~053~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.1745{{c}}, ~18/17 = 99.6265{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6131{{c}}

{{Optimal ET sequence|legend=1| 29, ~~72cd~~, ~~101, 130, 289, 419~~ }}

[[Badness]] (~~Smith~~): 0.~~108794~~

Badness (Sintel): 2.34

=== 11-limit ===

=== 19-limit ===

Subgroup: 2.3.5.7.11

Subgroup: 2.3.5.7.11.13.17.19

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

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~21/20 = 83.~~049~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.0713{{c}}, ~18/17 = 99.6208{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6152{{c}}

{{Optimal ET sequence|legend=0| 29, ~~72cde~~, ~~101e, 130, 289~~ }}

Badness (~~Smith~~): 0.~~045457~~

Badness (Sintel): 2.32

==~~= 13-limit =~~==

== Quintaschis ==

~~Subgroup: 2.~~3~~.5.~~7.11.13

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.

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

[[Subgroup]]: 2.3.5.7

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

[[Comma list]]: 32805/32768, 49009212/48828125

~~Optimal tuning (POTE): ~2 = 1200.000~~{{~~c}}, ~21/20~~ = ~~83.049{{c~~}}

{{~~Optimal ET sequence~~|~~legend~~=0~~| 29, 72cdef, 101e, 130, 289~~ }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0536{{c}}, ~200/189 = 99.6684{{c}}

: [[error map]]: {{val| +0.054 -0.190 +0.370 -0.262 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~200/189 = 99.6645{{c}}

: error map: {{val| 0.000 -0.277 +0.266 -0.363 }}

~~Badness (Smith): 0.025276~~

{{Optimal ET sequence|legend=1| 12, …, 289, 301, 590, 891, 1192 }}

~~== Septiquarschis ==~~

[[Badness]] (Sintel): 3.36

~~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~~).

[[Subgroup]]: 2.3.5.7

=== 11-limit ===

Subgroup: 2.3.5.7.11

[[Comma list]]: 32805/32768, ~~829440~~/~~823543~~

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

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

[[Optimal ~~tuning]] ([[POTE]])~~: ~2 = 1200.~~000~~{{c}}, ~~~147~~/~~128~~ = ~~242~~.~~614~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.0988{{c}}, ~35/33 = 99.6613{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6540{{c}}

{{Optimal ET sequence|legend=1| 89, 94, ~~183~~, ~~460d, 643d, 1103dd~~ }}

[[Badness]] (~~Smith~~): 0.~~187047~~

Badness (Sintel): 3.69

=== 11-limit ===

==== 13-limit ====

Subgroup: 2.3.5.7.11

Subgroup: 2.3.5.7.11.13

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

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

Mapping: {{mapping| 1 3 -~~9 2~~ -2 | 0 -~~7 -56 4 27~~ }}

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~~~147~~/~~128~~ = ~~242~~.~~616~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.0625{{c}}, ~35/33 = 99.6630{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6583{{c}}

{{Optimal ET sequence|legend=0| 89, 94, ~~183~~, ~~460d, 643d, 826dd~~ }}

Badness (~~Smith~~): 0.~~052002~~

Badness (Sintel): 3.07

=== 13-limit ===

==== 17-limit ====

Subgroup: 2.3.5.7.11.13

Subgroup: 2.3.5.7.11.13.17

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

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

Mapping: {{mapping| 1 3 -~~9 2~~ -~~2 13~~ | 0 -~~7 -56 4 27~~ -46 }}

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~~~147~~/~~128~~ = ~~242~~.~~610~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.1286{{c}}, ~18/17 = 99.6668{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6568{{c}}

Badness (~~Smith~~): 0.~~035315~~

Badness (Sintel): 2.58

== ~~Tsaharuk~~ ==

==== 19-limit ====

~~{{Main| Tsaharuk }}~~

Subgroup: 2.3.5.7.11.13.17.19

~~[[Subgroup]]~~: ~~2.3.5.7~~

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

~~[[Comma list]]~~: ~~32805/32768, 420175/419904~~

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

Optimal tunings:

~~: mapping generators~~: ~2, ~~~243~~/~~224~~

* WE: ~2 = 1200.0289{{c}}, ~18/17 = 99.6609{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6586{{c}}

[[Optimal ~~tuning]] ([[POTE]]): ~2~~ = ~~1200.000{{c}}~~, ~~~243/224 = 140.350{{c~~}}

~~{{Optimal ET sequence|legend=1| 17, 60c, 77, 94, 171 }}~~

Badness (Sintel): 2.56

~~[[Badness]] (Smith): 0.030697~~

=== Quintahelenic ===

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

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~88/81 = ~~140~~.~~365~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.0195{{c}}, ~200/189 = 99.6723{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6709{{c}}

{{Optimal ET sequence|legend=0| 17, ~~60ce~~, 77, 94, ~~171e, 265e, 436ee~~ }}

Badness (~~Smith~~): 0.~~063499~~

Badness (Sintel): 2.72

=== 13-limit ===

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~13/12 = ~~140~~.~~363~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.0442{{c}}, ~200/189 = 99.6709{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6675{{c}}

{{Optimal ET sequence|legend=0| 17, ~~60ce~~, 77, ~~94, 171e, 436ee~~ }}

Badness (~~Smith~~): 0.~~037886~~

Badness (Sintel): 2.30

== ~~Quanharuk~~ ==

===== 17-limit =====

[[Subgroup]]: 2.3.5.7

Subgroup: 2.3.5.7.11.13.17

[[Comma list]]: ~~16875~~/~~16807~~, ~~32805~~/~~32768~~

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 generators: ~2, ~56/45~~

[[Optimal ~~tuning]] ([[POTE]])~~: ~2 = 1200.~~000~~{{c}}, ~56/45 = ~~380~~.~~355~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.1227{{c}}, ~200/189 = 99.6753{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6658{{c}}

{{Optimal ET sequence|legend=1| 41, ~~142~~, ~~183, 224, 1303d, 1527cd, 1751cd, 1975cd~~ }}

[[Badness]] (~~Smith~~): 0.~~071950~~

Badness (Sintel): 2.06

=== 11-limit ===

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

Subgroup: 2.3.5.7.11

Subgroup: 2.3.5.7.11.13.17.19

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

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~56/45 = ~~380~~.~~352~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.0230{{c}}, ~200/189 = 99.6694{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6676{{c}}

{{Optimal ET sequence|legend=0| 41, ~~142, 183, 224, 631d, 855d, 1079d~~ }}

Badness (~~Smith~~): 0.~~031549~~

Badness (Sintel): 2.24

=== ~~13-limit~~ ===

==== Quintahelenoid ====

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~56/45 = ~~380~~.~~351~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.9919{{c}}, ~200/189 = 99.6712{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6718{{c}}

{{Optimal ET sequence|legend=0| 41, ~~142~~, ~~183~~, ~~224, 631d, 855d~~ }}

Badness (~~Smith~~): 0.~~021392~~

Badness (Sintel): 2.73

== ~~Quadrant~~ ==

===== 17-limit =====

~~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~~.

Subgroup: 2.3.5.7.11.13.17

~~[[Subgroup]]~~: ~~2.3.5.7~~

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

~~[[Comma list]]~~: ~~32805/32768, 390625/388962~~

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

Optimal tunings:

~~: mapping generators~~: ~~~25/21~~, ~3

* WE: ~2 = 1200.0469{{c}}, ~18/17 = 99.6749{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6710{{c}}

[[Optimal ~~tuning]] ([[POTE]]): ~25/21~~ = ~~300.0000{{c}}~~, ~~~3/2 = 701.8234{{c~~}}

~~{{Optimal ET sequence|legend=1| 212, 224, 436, 660, 1096c }}~~

Badness (Sintel): 2.44

~~[[Badness]] (Smith)~~: 0.~~110242~~

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

Subgroup: 2.3.5.7.11.13.17.19

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

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

~~Subgroup~~: ~~2.3.5.7.11~~

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

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

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

Optimal tunings:

* WE: ~2 = 1199.9925{{c}}, ~18/17 = 99.6710{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6716{{c}}

Optimal ~~tuning (POTE): ~25/21~~ = ~~300.0000{{c}}~~, ~~~3/2 = 701.8176{{c~~}}

~~{{Optimal ET sequence|legend=0| 212, 224, 436, 660 }}~~

Badness (Sintel): 2.41

~~Badness~~ (~~Smith~~)~~: 0~~.~~045738~~

== Sextilifourths ==

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.

~~=== 13-limit ===~~

[[Subgroup]]: 2.3.5.7

Subgroup: 2.3.5.7~~.11.13~~

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

[[Comma list]]: 32805/32768, 235298/234375

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

: mapping generators: ~2, ~21/20

Optimal tuning ~~(POTE)~~: ~25/21 = ~~300~~.0000{{c}}, ~3/2 = ~~701~~.~~8158~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0987{{c}}, ~21/20 = 83.0599{{c}}

: [[error map]]: {{val| +0.099 -0.117 +0.462 -0.630 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0543{{c}}

: error map: {{val| 0.000 -0.281 +0.295 -0.837 }}

{{Optimal ET sequence|legend=0| ~~212~~, ~~224~~, ~~436~~, ~~660~~ }}

Badness (~~Smith~~): 0.~~027243~~

[[Badness]] (Sintel): 2.75

== ~~Septant~~ ==

=== 11-limit ===

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

Subgroup: 2.3.5.7.11

~~[[Subgroup]]~~: ~~2.3.5.7~~

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

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

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

Optimal tunings:

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

* WE: ~2 = 1200.0424{{c}}, ~21/20 = 83.0520{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0497{{c}}

[[Optimal ~~tuning]] ([[POTE]]): ~8575/7776~~ = ~~171.429{{c}}~~, ~~~3/2 = 701.702{{c~~}}

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

Badness (Sintel): 1.50

~~[[Badness]] (Smith)~~: 0.~~111142~~

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

=== 11-limit ===

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:

* WE: ~2 = 1200.1056{{c}}, ~21/20 = 83.0566{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0508{{c}}

Badness (Sintel): 1.04

== Septant ==

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, 516560652/514714375

: mapping generators: ~8575/7776, ~3

[[Optimal tuning]]s:

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

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

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

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

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

[[Badness]] (Sintel): 2.81

=== 11-limit ===

Subgroup: 2.3.5.7.11

Line 2,550:

Line 2,568:

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

Optimal ~~tuning (POTE)~~: ~495/448 = 171.~~429~~{{c}}, ~3/2 = 701.~~719~~{{c}}

Optimal tunings:

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

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

Badness (~~Smith~~): 0.~~044122~~

Badness (Sintel): 1.46

=== 13-limit ===

Line 2,563:

Line 2,583:

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

Optimal ~~tuning (POTE)~~: ~495/448 = 171.~~429~~{{c}}, ~3/2 = 701.~~724~~{{c}}

Optimal tunings:

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

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

Badness (~~Smith~~): 0.~~024706~~

Badness (Sintel): 1.02

== Octant ==

~~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.

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

Line 2,579:

Line 2,601:

: mapping generators: ~42875/39366, ~3

[[Optimal tuning]] ([[~~POTE~~]]): ~42875/39366 = 150.~~000~~{{c}}, ~3/2 = 701.~~713~~{{c}}

[[Optimal tuning]]s:

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

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

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

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

[[Badness]] (~~Smith~~): 0.~~157186~~

[[Badness]] (Sintel): 3.98

=== 11-limit ===

Line 2,592:

Line 2,618:

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

Optimal ~~tuning (POTE)~~: ~12/11 = 150.~~000~~{{c}}, ~3/2 = 701.~~713~~{{c}}

Optimal tunings:

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

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

Badness (~~Smith~~): 0.~~044778~~

Badness (Sintel): 1.48

=== 13-limit ===

Line 2,605:

Line 2,633:

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

Optimal ~~tuning (POTE)~~: ~12/11 = 150.~~000~~{{c}}, ~3/2 = 701.~~725~~{{c}}

Optimal tunings:

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

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

Badness (~~Smith~~): 0.~~030425~~

Badness (Sintel): 1.26

== Nonant ==

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

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

Line 2,621:

Line 2,651:

: mapping generators: ~2592/2401, ~3

[[Optimal tuning]] ([[~~CTE~~]]): ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.~~7232~~{{c}}

[[Optimal tuning]]s:

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

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

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

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

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

[[Badness]] (~~Smith~~): 0.~~069896~~

[[Badness]] (Sintel): 1.77

=== 11-limit ===

Line 2,634:

Line 2,668:

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

Optimal ~~tuning (CTE)~~: ~242/225 = 133.3333{{c}}, ~3/2 = 701.~~8398~~{{c}}

Optimal tunings:

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

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

{{Optimal ET sequence|legend=0| 36~~, 99c~~, 135, 171~~, 477ce, 648cee~~ }}

Badness (~~Smith~~): 0.~~126910~~

Badness (Sintel): 4.20

=== 13-limit ===

Line 2,647:

Line 2,683:

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

Optimal ~~tuning (CTE)~~: ~242/225 = 133.3333{{c}}, ~3/2 = 701.~~7998~~{{c}}

Optimal tunings:

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

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

Badness (~~Smith~~): 0.~~076195~~

Badness (Sintel): 3.15

== ~~Tridecafifths~~ ==

== Septiquarschis ==

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

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

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

[[Comma list]]: 32805/32768, 829440/823543

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

: mapping generators: ~2, ~256/147

[[Optimal tuning]] ([[~~CTE~~]]): ~2 = 1200.0000{{c}}, ~~~1323~~/~~1280~~ = 53.~~9741~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.8855{{c}}, ~256/147 = 957.2944{{c}}

: [[error map]]: {{val| -0.114 -0.436 -0.182 +1.310 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~256/147 = 957.3867{{c}}

: error map: {{val| 0.000 -0.248 +0.032 +1.627 }}

[[Badness]] (~~Smith~~): 0.~~432580~~

[[Badness]] (Sintel): 4.73

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

Optimal tunings:

* WE: ~2 = 1199.9430{{c}}, ~256/147 = 957.3390{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3849{{c}}

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

Badness (Sintel): 1.72

~~Optimal tuning (CTE)~~: ~2 ~~= 1200~~.~~0000{{c}}, ~33/32 = 53~~.~~9744{{c}}~~

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

~~{{Optimal ET sequence|legend=0| 89~~, ~~200~~, ~~289 }}~~

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

Badness (~~Smith~~): 0.~~127820~~

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

Optimal tunings:

* WE: ~2 = 1200.0058{{c}}, ~256/147 = 957.3946{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3900{{c}}

Badness (Sintel): 1.46

== Subgroup extensions ==

=== Tridecaschismic (2.3.5.13) ===

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

Subgroup: 2.3.5.13

Comma list: 325/324, 32805/32768

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

Optimal tunings:

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

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

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

Badness (Sintel): 0.582

==== 2.3.5.13.19 subgroup ====

Subgroup: 2.3.5.13.19

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

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

Optimal tunings:

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

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

Badness (Sintel): 0.354

=== Photia (2.3.5.17) ===

Line 2,695:

Line 2,787:

: mapping generators: ~2, ~3

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~000~~{{c}}, ~3/2 = 701.~~491~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.5471{{c}}, ~3/2 = 701.2262{{c}}

: [[error map]]: {{val| -0.453 -1.182 +0.706 +3.628 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4976{{c}}

: error map: {{val| 0.000 -0.457 +1.705 +5.528 }}

[[~~Tp tuning #T2 tuning|RMS error~~]]: 0.~~4842 cents~~

[[Badness]] (Sintel): 0.479

==== 2.3.5.17.19 subgroup ====

Line 2,710:

Line 2,806:

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~3/2 = 701.~~470~~{{c}}

Optimal tunings:

* WE: ~2 = 1199.7225{{c}}, ~3/2 = 701.3077{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.4754{{c}}

~~RMS error~~: 0.~~5374 cents~~

Badness (Sintel): 0.332

=== Nestoria (2.3.5.19) ===

: ''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 2,730:

Line 2,828:

: mapping generators: ~2, ~3

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~000~~{{c}}, ~3/2 = 701.~~746~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.2250{{c}}, ~3/2 = 701.8776{{c}}

: [[error map]]: {{val| +0.225 +0.148 +0.240 -1.796 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7307{{c}}

: error map: {{val| 0.000 -0.224 -0.159 -2.705 }}

{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171, 460hh, 631hh }}

[[~~Tp tuning #T2 tuning|RMS error~~]]: 0.~~1763 cents~~

[[Badness]] (Sintel): 0.126

=== Taylor (2.3.5.13) ===

Line 2,748:

Line 2,850:

: mapping generators: ~2, ~26/15

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~000~~{{c}}, ~26/15 = 950.~~855~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.1497{{c}}, ~26/15 = 950.9740{{c}}

: [[error map]]: {{val| +0.150 -0.007 +0.348 -1.094 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 950.8493{{c}}

: error map: {{val| 0.000 -0.256 +0.098 -1.568 }}

{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236, 525f, 761ff }}

[[~~Tp tuning #T2 tuning|RMS error~~]]: 0.~~1485 cents~~

[[Badness]] (Sintel): 0.334

==== Dakota (2.3.5.13.19) ====

Line 2,761:

Line 2,867:

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

Optimal ~~tuning (CTE)~~: ~2 = 1200.0000{{c}}, ~26/15 = 950.~~8199~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.2611{{c}}, ~26/15 = 951.0703{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8532{{c}}

{{Optimal ET sequence|legend=0| 24, 29, 53, 130, 183, 236h, 289h }}

Badness (~~Smith~~): 0.~~00575~~

Badness (Sintel): 0.262

===== 2.3.5.13.19.37 subgroup =====

Line 2,774:

Line 2,882:

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

Optimal ~~tuning (CTE)~~: ~2 = 1200.0000{{c}}, ~26/15 = 950.~~8187~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.2987{{c}}, ~26/15 = 951.1060{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8595{{c}}

{{Optimal ET sequence|legend=0| 24, 29, 53, 183, 236h, 289hl, 631fhhll }}

Badness (~~Smith~~): 0.~~00357~~

Badness (Sintel): 0.223

=== Quintilischis (2.3.5.17) ===

Line 2,792:

Line 2,902:

: mapping generators: ~2, ~18/17

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~000~~{{c}}, ~18/17 = 99.~~649~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.1370{{c}}, ~18/17 = 99.6602{{c}}

: [[error map]]: {{val| +0.137 +0.018 -0.042 -0.533 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~18/17 = 99.6499{{c}}

: error map: {{val| 0.000 -0.205 -0.317 -1.104 }}

{{Optimal ET sequence|legend=1| 12, …, 253, 265, 277, 289, 566g, 855g }}

[[~~Tp tuning #T2 tuning|RMS error~~]]: 0.~~0719 cents~~

[[Badness]] (Sintel): 1.34

==== 2.3.5.17.19 subgroup ====

Line 2,807:

Line 2,921:

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~18/17 = 99.~~652~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.0350{{c}}, ~18/17 = 99.6550{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6520{{c}}

~~RMS error~~: 0.~~1636 cents~~

Badness (Sintel): 1.17

[[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 }}
-<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7187{{c}}
-* [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7359{{c}} -->
 [[Tuning ranges]]:
@@ Line 39: / 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 61: / 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 [[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]]}.
+[[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 72: / 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 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.085{{c}} -->
 [[Minimax tuning]]:
@@ Line 93: / 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.
 Subgroup: 2.3.5.7.11
@@ Line 104: / Line 92: @@
 * WE: ~2 = 1200.3089{{c}}, ~3/2 = 702.3377{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1562{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.157{{c}} -->
 Minimax tuning:
@@ Line 128: / Line 115: @@
 * WE: ~2 = 1200.1703{{c}}, ~3/2 = 702.2122{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1135{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.113{{c}} -->
 Minimax tuning:
@@ Line 153: / Line 139: @@
 * WE: ~2 = 1199.8140{{c}}, ~3/2 = 701.9833{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0909{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.092{{c}} -->
 {{Optimal ET sequence|legend=0| 12e, 41, 53, 94g }}
@@ Line 169: / Line 154: @@
 * WE: ~2 = 1199.9556{{c}}, ~3/2 = 702.0530{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0787{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.079{{c}} -->
 {{Optimal ET sequence|legend=0| 12e, 41, 53 }}
@@ Line 185: / Line 169: @@
 * WE: ~2 = 1200.0046{{c}}, ~3/2 = 702.2167{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0962{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.097{{c}} -->
 {{Optimal ET sequence|legend=0| 41g, 53, 94 }}
@@ Line 201: / Line 184: @@
 * WE: ~2 = 1200.2910{{c}}, ~3/2 = 702.2681{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0967{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.0978{{c}} -->
 {{Optimal ET sequence|legend=1| 41g, 53, 94 }}
@@ Line 217: / Line 199: @@
 * WE: ~2 = 1200.2970{{c}}, ~3/2 = 702.2697{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0943{{c}}
-<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0960{{c}} -->
 {{Optimal ET sequence|legend=0| 41g, 53, 94 }}
@@ Line 233: / Line 214: @@
 * WE: ~2 = 1200.1986{{c}}, ~3/2 = 702.2598{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1455{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.144{{c}} -->
 {{Optimal ET sequence|legend=0| 41, 53g, 94 }}
@@ Line 249: / Line 229: @@
 * WE: ~2 = 1200.3057{{c}}, ~3/2 = 702.3138{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1373{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.135{{c}} -->
 {{Optimal ET sequence|legend=0| 41, 53g, 94 }}
@@ Line 265: / Line 244: @@
 * WE: ~2 = 1200.1917{{c}}, ~3/2 = 702.4836{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3599{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.321{{c}} -->
 Minimax tuning:
@@ Line 289: / Line 267: @@
 * WE: ~2 = 1200.3031{{c}}, ~3/2 = 702.7368{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.5420{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.559{{c}} -->
 Minimax tuning:
@@ Line 314: / Line 291: @@
 * WE: ~2 = 1199.1984{{c}}, ~3/2 = 701.8424{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3384{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.312{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 29, 41 }}
@@ Line 330: / Line 306: @@
 * WE: ~2 = 1199.5242{{c}}, ~3/2 = 702.0783{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3711{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.357{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 29, 41 }}
@@ Line 346: / Line 321: @@
 * WE: ~2 = 1200.6122{{c}}, ~3/2 = 703.0830{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6968{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.725{{c}} -->
 {{Optimal ET sequence|legend=0| 12fg, 29g, 41, 70cd }}
@@ Line 362: / Line 336: @@
 * WE: ~2 = 1200.7981{{c}}, ~3/2 = 703.2199{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.7221{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.753{{c}} -->
 {{Optimal ET sequence|legend=0| 12fg, 29g, 41, 70cd }}
@@ Line 378: / Line 351: @@
 * WE: ~2 = 1201.3146{{c}}, ~3/2 = 703.4864{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6491{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.717{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 29g, 41g }}
@@ Line 394: / Line 366: @@
 * WE: ~2 = 1201.3140{{c}}, ~3/2 = 703.4860{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6578{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.716{{c}} -->
 {{Optimal ET sequence|legend=1| 12f, 29g, 41g }}
@@ Line 410: / Line 381: @@
 * WE: ~2 = 1199.7097{{c}}, ~3/2 = 701.5554{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7370{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.725{{c}} -->
 Minimax tuning:
@@ Line 430: / Line 400: @@
 * WE: ~2 = 1199.7370{{c}}, ~3/2 = 701.5937{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7570{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.747{{c}} -->
 Minimax tuning:
@@ Line 450: / Line 419: @@
 * WE: ~2 = 1199.2895{{c}}, ~3/2 = 701.2643{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6967{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.680{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 53, 65d, 118dg }}
@@ Line 466: / Line 434: @@
 * WE: ~2 = 1199.5280{{c}}, ~3/2 = 701.4290{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7149{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.705{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 53, 65d }}
 Badness (Sintel): 1.18
-=== Hemigari ===
-Subgroup: 2.3.5.7.11
-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
-Optimal tunings:
-* WE: ~2 = 1200.7303{{c}}, ~110/63 = 951.6605{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~110/63 = 951.0604{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~110/63 = 951.082{{c}} -->
-{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135ee }}
-Badness (Sintel): 1.68
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-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:
-* WE: ~2 = 1200.8146{{c}}, ~26/15 = 951.7273{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.0574{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~26/15 = 951.082{{c}} -->
-{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135eef }}
-Badness (Sintel): 1.13
 === Karadeniz ===
@@ Line 518: / Line 452: @@
 * WE: ~2 = 1199.7351{{c}}, ~11/9 = 350.9167{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.9995{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/9 = 350.994{{c}} -->
 {{Optimal ET sequence|legend=0| 24d, 41, 65d, 106, 147 }}
@@ Line 534: / Line 467: @@
 * WE: ~2 = 1199.3042{{c}}, ~11/9 = 350.7533{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.9686{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/9 = 351.014{{c}} -->
 {{Optimal ET sequence|legend=0| 24d, 41, 65d, 106f }}
 Badness (Sintel): 1.34
+=== Hemigari ===
+Subgroup: 2.3.5.7.11
+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
+Optimal tunings:
+* WE: ~2 = 1200.7303{{c}}, ~110/63 = 951.6605{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~110/63 = 951.0604{{c}}
+{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135ee }}
+Badness (Sintel): 1.68
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
+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:
+* WE: ~2 = 1200.8146{{c}}, ~26/15 = 951.7273{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.0574{{c}}
+{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135eef }}
+Badness (Sintel): 1.13
 === Sanjaab ===
@@ Line 551: / Line 514: @@
 * WE: ~2 = 1200.1997{{c}}, ~11/10 = 166.0018{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9786{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/10 = 165.974{{c}} -->
 {{Optimal ET sequence|legend=0| 29, 65d, 94 }}
@@ Line 567: / Line 529: @@
 * WE: ~2 = 1200.1224{{c}}, ~11/10 = 165.9800{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9659{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/10 = 165.963{{c}} -->
 {{Optimal ET sequence|legend=0| 29, 65d, 94 }}
@@ Line 573: / Line 534: @@
 Badness (Sintel): 1.40
-== Schism ==
+== Pontiac ==
-See [[Archytas clan #Schism]].
+{{Main| Pontiac }}
-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.
+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).
-== Pontiac ==
-{{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).
 [[Subgroup]]: 2.3.5.7
@@ Line 592: / Line 548: @@
 * [[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 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.757{{c}} -->
 [[Minimax tuning]]:
@@ Line 613: / Line 568: @@
 === 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 624: / Line 579: @@
 * WE: ~2 = 1200.3277{{c}}, ~3/2 = 701.9135{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7223{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.722{{c}} -->
 Minimax tuning:
@@ Line 644: / Line 598: @@
 * WE: ~2 = 1200.1780{{c}}, ~3/2 = 701.8491{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7446{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.745{{c}} -->
 Minimax tuning:
@@ Line 664: / Line 617: @@
 * WE: ~2 = 1200.1645{{c}}, ~3/2 = 701.8385{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7425{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.742{{c}} -->
 Minimax tuning:
@@ Line 684: / Line 636: @@
 * WE: ~2 = 1200.5227{{c}}, ~3/2 = 702.0456{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7418{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.740{{c}} -->
 {{Optimal ET sequence|legend=0| 53, 118f, 171ef }}
@@ Line 700: / Line 651: @@
 * WE: ~2 = 1200.4988{{c}}, ~3/2 = 702.0218{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7332{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.730{{c}} -->
 {{Optimal ET sequence|legend=0| 53, 118f, 171ef }}
@@ Line 716: / Line 666: @@
 * WE: ~2 = 1200.5185{{c}}, ~3/2 = 702.0323{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7318{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.729{{c}} -->
 {{Optimal ET sequence|legend=0| 53, 118f, 171ef }}
@@ Line 723: / Line 672: @@
 === 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 734: / Line 683: @@
 * WE: ~2 = 1199.9814{{c}}, ~3/2 = 701.7725{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7834{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.783{{c}} -->
 Minimax tuning:
@@ Line 754: / Line 702: @@
 * WE: ~2 = 1199.9601{{c}}, ~3/2 = 701.7610{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7845{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.784{{c}} -->
 Minimax tuning:
@@ Line 774: / Line 721: @@
 * WE: ~2 = 1199.8850{{c}}, ~3/2 = 701.7101{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7775{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.777{{c}} -->
 Minimax tuning:
@@ Line 785: / Line 731: @@
 === 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 796: / Line 742: @@
 * WE: ~2 = 1200.1259{{c}}, ~3/2 = 701.7980{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7256{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.724{{c}} -->
 Minimax tuning:
@@ Line 816: / Line 761: @@
 * WE: ~2 = 1199.9254{{c}}, ~3/2 = 701.6945{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7378{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.738{{c}} -->
 Minimax tuning:
@@ Line 836: / Line 780: @@
 * WE: ~2 = 1199.9454{{c}}, ~3/2 = 701.7085{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7401{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.740{{c}} -->
 Minimax tuning:
@@ Line 856: / Line 799: @@
 * WE: ~2 = 1200.0897{{c}}, ~3/2 = 701.7874{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7356{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.735{{c}} -->
 {{Optimal ET sequence|legend=0| 53ef, 118f, 171, 289 }}
@@ Line 872: / Line 814: @@
 * WE: ~2 = 1200.1045{{c}}, ~3/2 = 701.7962{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7359{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.735{{c}} -->
 {{Optimal ET sequence|legend=0| 53ef, 118f, 171, 289, 460e, 749defg }}
@@ Line 879: / Line 820: @@
 === 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 891: / Line 832: @@
 * WE: ~99/70 = 600.0500{{c}}, ~3/2 = 701.8153{{c}}
 * CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7584{{c}}
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~3/2 = 701.757{{c}} -->
 {{Optimal ET sequence|legend=0| 106, 118, 224, 342, 1592c, 1934ce, 2276cde, 2618cde, 2960cde }}
@@ Line 907: / Line 847: @@
 * WE: ~99/70 = 599.9939{{c}}, ~3/2 = 701.7657{{c}}
 * CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7728{{c}}
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~3/2 = 701.773{{c}} -->
 {{Optimal ET sequence|legend=0| 106, 118, 224, 566f, 790f }}
@@ Line 923: / Line 862: @@
 * WE: ~99/70 = 599.9839{{c}}, ~3/2 = 701.7463{{c}}
 * CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7649{{c}}
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~3/2 = 701.765{{c}} -->
 {{Optimal ET sequence|legend=0| 106g, 118, 224, 342, 566f }}
@@ Line 939: / Line 877: @@
 * WE: ~99/70 = 600.0405{{c}}, ~3/2 = 701.8160{{c}}
 * CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7697{{c}}
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~3/2 = 701.769{{c}} -->
 {{Optimal ET sequence|legend=0| 106f, 118f, 224, 342f, 566, 1356cf }}
@@ Line 955: / Line 892: @@
 * WE: ~99/70 = 600.0336{{c}}, ~3/2 = 701.8031{{c}}
 * CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7647{{c}}
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~3/2 = 701.764{{c}} -->
 {{Optimal ET sequence|legend=0| 106fg, 118f, 224, 342f, 566 }}
@@ Line 971: / Line 907: @@
 * WE: ~99/70 = 600.0243{{c}}, ~3/2 = 701.7891{{c}}
 * CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7613{{c}}
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~3/2 = 701.761{{c}} -->
 {{Optimal ET sequence|legend=0| 106fgh, 118f, 224, 342f, 566h, 908fgh }}
@@ Line 988: / Line 923: @@
 * WE: ~208/175 = 300.0229{{c}}, ~3/2 = 701.8097{{c}}
 * CWE: ~208/175 = 300.0000{{c}}, ~3/2 = 701.7578{{c}}
-<!-- * POTE: ~208/175 = 300.000{{c}}, ~3/2 = 701.756{{c}} -->
 {{Optimal ET sequence|legend=0| 224, 460, 684, 2276cde, 2960cde }}
@@ Line 995: / Line 929: @@
 == 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 1,007: / Line 941: @@
 * [[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 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.239{{c}} -->
 [[Minimax tuning]]:
@@ Line 1,029: / Line 962: @@
 * WE: ~2 = 1199.7077{{c}}, ~3/2 = 701.0017{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.1804{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.172{{c}} -->
 {{Optimal ET sequence|legend=0| 12, 65e, 77, 89, 166c }}
@@ Line 1,045: / Line 977: @@
 * WE: ~2 = 1199.7782{{c}}, ~3/2 = 701.0966{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2319{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.226{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 65ef, 77, 166cf }}
@@ Line 1,061: / Line 992: @@
 * WE: ~2 = 1199.5839{{c}}, ~3/2 = 700.9632{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2137{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.206{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 77, 89f, 166cf }}
@@ Line 1,077: / Line 1,007: @@
 * WE: ~2 = 1199.7146{{c}}, ~3/2 = 701.0500{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2212{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.217{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 77, 166cf }}
@@ Line 1,093: / Line 1,022: @@
 * WE: ~2 = 1200.0060{{c}}, ~3/2 = 701.2202{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2167{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.217{{c}} -->
 {{Optimal ET sequence|legend=0| 12, 77, 166c }}
@@ Line 1,109: / Line 1,037: @@
 * WE: ~2 = 1199.8388{{c}}, ~3/2 = 701.3071{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.4068{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.401{{c}} -->
 {{Optimal ET sequence|legend=0| 12, 53d, 65, 77e }}
@@ Line 1,125: / Line 1,052: @@
 * WE: ~2 = 1199.7329{{c}}, ~3/2 = 701.1918{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.3555{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.348{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 53dff, 65f, 77e }}
@@ Line 1,141: / Line 1,067: @@
 * WE: ~2 = 1199.8928{{c}}, ~3/2 = 701.4664{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.5327{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.529{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, …, 53d, 65 }}
@@ Line 1,147: / Line 1,072: @@
 Badness (Sintel): 2.01
-== Bischismic ==
+== 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]]: 3136/3125, 32805/32768
+[[Comma list]]: 32805/32768, 390625/387072
-{{Mapping|legend=1| 2 0 30 69 | 0 1 -8 -20 }}
+{{Mapping|legend=1| 1 0 15 109 | 0 1 -8 -67 }}
-: mapping generators: ~567/400, ~3
 [[Optimal tuning]]s:
-* [[WE]]: ~567/400 = 600.0072{{c}}, ~3/2 = 701.6005{{c}}
+* [[WE]]: ~2 = 1200.2569{{c}}, ~3/2 = 702.1149{{c}}
-: [[error map]]: {{val| +0.014 -0.340 +0.982 -0.629 }}
+: [[error map]]: {{val| +0.2569 +0.4168 -1.4342 +0.2685 }}
-* [[CWE]]: ~567/400 = 600.0000{{c}}, ~3/2 = 701.5915{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9615{{c}}
-: error map: {{val| 0.000 -0.364 +0.954 -0.656 }}
+: error map: {{val| 0.0000 +0.0065 -2.0054 -0.2437 }}
-<!-- * [[CTE]]: ~567/400 = 600.0000{{c}}, ~3/2 = 701.5899{{c}} -->
-[[Minimax tuning]]:
-* [[7-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.7/3
-* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
-{{Optimal ET sequence|legend=1| 12, …, 106d, 118, 130, 248, 378 }}
+{{Optimal ET sequence|legend=1| 53, 147d, 200, 253, 306c, 559c }}
-[[Badness]] (Sintel): 1.39
+[[Badness]] (Sintel): 5.04
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 441/440, 3136/3125, 8019/8000
+Comma list: 385/384, 19712/19683, 78125/77616
-Mapping: {{mapping| 2 0 30 69 102 | 0 1 -8 -20 -30 }}
+Mapping: {{mapping| 1 0 15 109 -117 | 0 1 -8 -67 76 }}
 Optimal tunings:
-* WE: ~99/70 = 600.0165{{c}}, ~3/2 = 701.6316{{c}}
+* WE: ~2 = 1200.3283{{c}}, ~3/2 = 702.1636{{c}}
-* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.6110{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9713{{c}}
-<!-- * CTE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.6077{{c}} -->
-{{Optimal ET sequence|legend=0| 12, …, 106de, 118, 130, 248 }}
+{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}
-Badness (Sintel): 0.931
+Badness (Sintel): 3.83
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 441/440, 729/728, 1001/1000, 3136/3125
+Comma list: 325/324, 385/384, 2200/2197, 19712/19683
-Mapping: {{mapping| 2 0 30 69 102 -75 | 0 1 -8 -20 -30 26 }}
+Mapping: {{mapping| 1 0 15 109 -117 -28 | 0 1 -8 -67 76 20 }}
 Optimal tunings:
-* WE: ~99/70 = 599.9610{{c}}, ~3/2 = 701.5445{{c}}
+* WE: ~2 = 1200.3229{{c}}, ~3/2 = 702.1603{{c}}
-* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5908{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9714{{c}}
-<!-- * CTE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5949{{c}} -->
-{{Optimal ET sequence|legend=0| 12, 118, 130, 248, 378 }}
+{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}
-Badness (Sintel): 1.19
+Badness (Sintel): 2.13
-===== 17-limit =====
+== Schism ==
-Subgroup: 2.3.5.7.11.13.17
+See [[Archytas clan #Schism]].
-Comma list: 289/288, 441/440, 561/560, 729/728, 3136/3125
+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.
-Mapping: {{mapping| 2 0 30 69 102 -75 5 | 0 1 -8 -20 -30 26 1 }}
+== 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.
-Optimal tunings:
+[[Subgroup]]: 2.3.5.7
-* WE: ~99/70 = 600.0331{{c}}, ~3/2 = 701.6387{{c}}
-* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5994{{c}}
-<!-- * CTE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5959{{c}} -->
-{{Optimal ET sequence|legend=0| 12, 118, 130, 248g }}
+[[Comma list]]: 3136/3125, 32805/32768
-Badness (Sintel): 1.49
+{{Mapping|legend=1| 2 0 30 69 | 0 1 -8 -20 }}
+: mapping generators: ~567/400, ~3
-==== Bischis ====
+[[Optimal tuning]]s:
-Subgroup: 2.3.5.7.11.13
+* [[WE]]: ~567/400 = 600.0072{{c}}, ~3/2 = 701.6005{{c}}
+: [[error map]]: {{val| +0.014 -0.340 +0.982 -0.629 }}
+* [[CWE]]: ~567/400 = 600.0000{{c}}, ~3/2 = 701.5915{{c}}
+: error map: {{val| 0.000 -0.364 +0.954 -0.656 }}
-Comma list: 351/350, 364/363, 441/440, 3136/3125
+[[Minimax tuning]]:
+* [[7-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.7/3
+* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
-Mapping: {{mapping| 2 0 30 69 102 131 | 0 1 -8 -20 -30 -39 }}
+{{Optimal ET sequence|legend=1| 12, …, 106d, 118, 130, 248, 378 }}
-Optimal tunings:
+[[Badness]] (Sintel): 1.39
-* WE: ~55/39 = 599.9766{{c}}, ~3/2 = 701.5380{{c}}
-* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.5670{{c}}
-<!-- * CTE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.5708{{c}} -->
-{{Optimal ET sequence|legend=0| 12f, 106deff, 118f, 130 }}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Badness (Sintel): 1.21
+Comma list: 441/440, 3136/3125, 8019/8000
-===== 17-limit =====
+Mapping: {{mapping| 2 0 30 69 102 | 0 1 -8 -20 -30 }}
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 221/220, 289/288, 351/350, 441/440, 3136/3125
+Optimal tunings:
+* WE: ~99/70 = 600.0165{{c}}, ~3/2 = 701.6316{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.6110{{c}}
-Mapping: {{mapping| 2 0 30 69 102 131 5 | 0 1 -8 -20 -30 -39 1 }}
+{{Optimal ET sequence|legend=0| 12, …, 106de, 118, 130, 248 }}
-Optimal tunings:
+Badness (Sintel): 0.931
-* WE: ~55/39 = 600.0997{{c}}, ~3/2 = 701.7114{{c}}
-* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.5899{{c}}
-<!-- * CTE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.5717{{c}} -->
-{{Optimal ET sequence|legend=0| 12f, 106deff, 118f, 130, 248fg }}
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Badness (Sintel): 1.37
+Comma list: 441/440, 729/728, 1001/1000, 3136/3125
-== Kleischismic ==
+Mapping: {{mapping| 2 0 30 69 102 -75 | 0 1 -8 -20 -30 26 }}
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 32805/32768, 1500625/1492992
+Optimal tunings:
+* WE: ~99/70 = 599.9610{{c}}, ~3/2 = 701.5445{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5908{{c}}
-{{Mapping|legend=1| 2 1 22 -15 | 0 2 -16 19 }}
+{{Optimal ET sequence|legend=0| 12, 118, 130, 248, 378 }}
-: mapping generators: ~1225/864, ~35/24
-[[Optimal tuning]]s:
+Badness (Sintel): 1.19
-* [[WE]]: ~1225/864 = 600.1246{{c}}, ~35/24 = 651.0550{{c}} (~36/35 = 50.9304{{c}})
-: [[error map]]: {{val| +0.249 +0.280 -0.453 -0.650 }}
-* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~35/24 = 650.9204{{c}} (~36/35 = 50.9204{{c}})
-: error map: {{val| 0.000 -0.114 -1.041 -1.338 }}
-<!-- * [[POTE]]: ~1225/864 = 600.000{{c}}, ~35/24 = 650.920{{c}} (~36/35 = 50.920{{c}}) -->
-{{Optimal ET sequence|legend=1| 24, 94, 118, 212, 330, 542d, 872cdd, 1414ccddd }}
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-[[Badness]] (Sintel): 2.80
+Comma list: 289/288, 441/440, 561/560, 729/728, 3136/3125
-=== 11-limit ===
+Mapping: {{mapping| 2 0 30 69 102 -75 5 | 0 1 -8 -20 -30 26 1 }}
-Subgroup: 2.3.5.7.11
-Comma list: 385/384, 9801/9800, 14641/14580
-Mapping: {{mapping| 2 1 22 -15 8 | 0 2 -16 19 -1 }}
 Optimal tunings:
-* WE: ~99/70 = 600.1645{{c}}, ~35/24 = 651.0963{{c}} (~36/35 = 50.9319{{c}})
+* WE: ~99/70 = 600.0331{{c}}, ~3/2 = 701.6387{{c}}
-* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9184{{c}} (~36/35 = 50.9184{{c}})
+* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5994{{c}}
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~35/24 = 650.918{{c}} (~36/35 = 50.918{{c}}) -->
-{{Optimal ET sequence|legend=0| 24, 94, 118, 212, 330e, 542dee, 872cddeee }}
+{{Optimal ET sequence|legend=0| 12, 118, 130, 248g }}
-Badness (Sintel): 1.21
+Badness (Sintel): 1.49
-==== 13-limit ====
+==== Bischis ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 352/351, 385/384, 729/728, 1575/1573
+Comma list: 351/350, 364/363, 441/440, 3136/3125
-Mapping: {{mapping| 2 1 22 -15 8 15 | 0 2 -16 19 -1 -7 }}
+Mapping: {{mapping| 2 0 30 69 102 131 | 0 1 -8 -20 -30 -39 }}
 Optimal tunings:
-* WE: ~99/70 = 600.0696{{c}}, ~35/24 = 651.0136{{c}} (~36/35 = 50.9440{{c}})
+* WE: ~55/39 = 599.9766{{c}}, ~3/2 = 701.5380{{c}}
-* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9378{{c}} (~36/35 = 50.9378{{c}})
+* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.5670{{c}}
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~35/24 = 650.938{{c}} (~36/35 = 50.938{{c}}) -->
-{{Optimal ET sequence|legend=0| 24, 94, 118, 212f }}
+{{Optimal ET sequence|legend=0| 12f, 106deff, 118f, 130 }}
-Badness (Sintel): 1.56
+Badness (Sintel): 1.21
 ===== 17-limit =====
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 170/169, 289/288, 352/351, 385/384, 561/560
+Comma list: 221/220, 289/288, 351/350, 441/440, 3136/3125
-Mapping: {{mapping| 2 1 22 -15 8 15 6 | 0 2 -16 19 -1 -7 2 }}
+Mapping: {{mapping| 2 0 30 69 102 131 5 | 0 1 -8 -20 -30 -39 1 }}
 Optimal tunings:
-* WE: ~99/70 = 600.1134{{c}}, ~35/24 = 651.0646{{c}} (~36/35 = 50.9512{{c}})
+* WE: ~55/39 = 600.0997{{c}}, ~3/2 = 701.7114{{c}}
-* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9414{{c}} (~36/35 = 50.9414{{c}})
+* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.5899{{c}}
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~35/24 = 650.942{{c}} (~36/35 = 50.942{{c}}) -->
-{{Optimal ET sequence|legend=0| 24, 94, 118 }}
+{{Optimal ET sequence|legend=0| 12f, 106deff, 118f, 130, 248fg }}
-Badness (Sintel): 1.30
+Badness (Sintel): 1.37
-==== Kleischis ====
+== Kleischismic ==
-Subgroup: 2.3.5.7.11.13
+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.
-Comma list: 325/324, 385/384, 1573/1568, 14641/14580
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 2 1 22 -15 8 -36 | 0 2 -16 19 -1 40 }}
+[[Comma list]]: 32805/32768, 1500625/1492992
-Optimal tunings:
+{{Mapping|legend=1| 2 1 22 -15 | 0 2 -16 19 }}
-* WE: ~99/70 = 600.1909{{c}}, ~35/24 = 651.1578{{c}} (~36/35 = 50.9670{{c}})
+: mapping generators: ~1225/864, ~35/24
-* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9541{{c}} (~36/35 = 50.9541{{c}})
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~35/24 = 650.951{{c}} (~36/35 = 50.951{{c}}) -->
-{{Optimal ET sequence|legend=0| 24f, 94, 118f, 212 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~1225/864 = 600.1246{{c}}, ~35/24 = 651.0550{{c}} (~36/35 = 50.9304{{c}})
+: [[error map]]: {{val| +0.249 +0.280 -0.453 -0.650 }}
+* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~35/24 = 650.9204{{c}} (~36/35 = 50.9204{{c}})
+: error map: {{val| 0.000 -0.114 -1.041 -1.338 }}
-Badness (Sintel): 1.55
+{{Optimal ET sequence|legend=1| 24, 94, 118, 212, 330, 542d, 872cdd, 1414ccddd }}
+[[Badness]] (Sintel): 2.80
-===== 17-limit =====
+=== 11-limit ===
-Subgroup: 2.3.5.7.11.13.17
+Subgroup: 2.3.5.7.11
-Comma list: 289/288, 325/324, 385/384, 442/441, 14641/14580
+Comma list: 385/384, 9801/9800, 14641/14580
-Mapping: {{mapping| 2 1 22 -15 8 -36 6 | 0 2 -16 19 -1 40 2 }}
+Mapping: {{mapping| 2 1 22 -15 8 | 0 2 -16 19 -1 }}
 Optimal tunings:
-* WE: ~99/70 = 600.2190{{c}}, ~35/24 = 651.1578{{c}} (~36/35 = 50.9670{{c}})
+* WE: ~99/70 = 600.1645{{c}}, ~35/24 = 651.0963{{c}} (~36/35 = 50.9319{{c}})
-* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9518{{c}} (~36/35 = 50.9518{{c}})
+* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9184{{c}} (~36/35 = 50.9184{{c}})
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~35/24 = 650.948{{c}} (~36/35 = 50.948{{c}}) -->
-{{Optimal ET sequence|legend=0| 24f, 94, 118f, 212g }}
+{{Optimal ET sequence|legend=0| 24, 94, 118, 212, 330e, 542dee, 872cddeee }}
-Badness (Sintel): 1.26
+Badness (Sintel): 1.21
-== Salsa ==
+==== 13-limit ====
-[[Subgroup]]: 2.3.5.7
+Subgroup: 2.3.5.7.11.13
-[[Comma list]]: 245/243, 32805/32768
+Comma list: 352/351, 385/384, 729/728, 1575/1573
-{{Mapping|legend=1| 1 1 7 -1 | 0 2 -16 13 }}
+Mapping: {{mapping| 2 1 22 -15 8 15 | 0 2 -16 19 -1 -7 }}
-: mapping generators: ~2, ~128/105
-[[Optimal tuning]]s:
+Optimal tunings:
-* [[WE]]: ~2 = 1200.7707{{c}}, ~128/105 = 351.2748{{c}}
+* WE: ~99/70 = 600.0696{{c}}, ~35/24 = 651.0136{{c}} (~36/35 = 50.9440{{c}})
-: [[error map]]: {{val| +0.771 +1.365 -1.315 -3.024 }}
+* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9378{{c}} (~36/35 = 50.9378{{c}})
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 351.0471{{c}}
-: error map: {{val| 0.000 +0.139 -3.068 -5.213 }}
-{{Optimal ET sequence|legend=1| 17, 24, 41, 106d, 147d, 188cd }}
+{{Optimal ET sequence|legend=0| 24, 94, 118, 212f }}
-[[Badness]] (Sintel): 2.03
+Badness (Sintel): 1.56
-=== 11-limit ===
+===== 17-limit =====
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13.17
-Comma list: 243/242, 245/242, 385/384
+Comma list: 170/169, 289/288, 352/351, 385/384, 561/560
-Mapping: {{mapping| 1 1 7 -1 2 | 0 2 -16 13 5 }}
+Mapping: {{mapping| 2 1 22 -15 8 15 6 | 0 2 -16 19 -1 -7 2 }}
 Optimal tunings:
-* WE: ~2 = 1200.3891{{c}}, ~11/9 = 351.1275{{c}}
+* WE: ~99/70 = 600.1134{{c}}, ~35/24 = 651.0646{{c}} (~36/35 = 50.9512{{c}})
-* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0141{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9414{{c}} (~36/35 = 50.9414{{c}})
-{{Optimal ET sequence|legend=0| 17, 24, 41, 106d }}
+{{Optimal ET sequence|legend=0| 24, 94, 118 }}
 Badness (Sintel): 1.30
-=== 13-limit ===
+==== Kleischis ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 105/104, 144/143, 243/242, 245/242
+Comma list: 325/324, 385/384, 1573/1568, 14641/14580
-Mapping: {{mapping| 1 1 7 -1 2 4 | 0 2 -16 13 5 -1 }}
+Mapping: {{mapping| 2 1 22 -15 8 -36 | 0 2 -16 19 -1 40 }}
 Optimal tunings:
-* WE: ~2 = 1199.9362{c}}, ~11/9 = 351.0061{{c}}
+* WE: ~99/70 = 600.1909{{c}}, ~35/24 = 651.1578{{c}} (~36/35 = 50.9670{{c}})
-* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0247{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9541{{c}} (~36/35 = 50.9541{{c}})
-{{Optimal ET sequence|legend=0| 17, 24, 41 }}
+{{Optimal ET sequence|legend=0| 24f, 94, 118f, 212 }}
-Badness (Sintel): 1.27
+Badness (Sintel): 1.55
-== Hemischis ==
+===== 17-limit =====
-[[Subgroup]]: 2.3.5.7
+Subgroup: 2.3.5.7.11.13.17
-[[Comma list]]: 6144/6125, 19683/19600
+Comma list: 289/288, 325/324, 385/384, 442/441, 14641/14580
-{{Mapping|legend=1| 1 0 15 -17 | 0 2 -16 25 }}
+Mapping: {{mapping| 2 1 22 -15 8 -36 6 | 0 2 -16 19 -1 40 2 }}
-: mapping generators: ~2, ~140/81
-[[Optimal tuning]]s:
+Optimal tunings:
-* [[WE]]: ~2 = 1199.8579{{c}}, ~140/81 = 951.6847{{c}}
+* WE: ~99/70 = 600.2190{{c}}, ~35/24 = 651.1578{{c}} (~36/35 = 50.9670{{c}})
-: [[error map]]: {{val| -0.142 -0.586 +0.600 +0.708 }}
+* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9518{{c}} (~36/35 = 50.9518{{c}})
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 951.7966{{c}}
-: error map: {{val| 0.000 -0.362 +0.941 +1.088 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~140/81 = 950.797{{c}} -->
-{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 313 }}
+{{Optimal ET sequence|legend=0| 24f, 94, 118f, 212g }}
-[[Badness]] (Sintel): 1.16
+Badness (Sintel): 1.26
-=== 11-limit ===
+== Salsa ==
-Subgroup: 2.3.5.7.11
+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]].
-Comma list: 540/539, 5632/5625, 8019/8000
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 1 0 15 -17 51 | 0 2 -16 25 -60 }}
+[[Comma list]]: 245/243, 32805/32768
-Optimal tunings:
+{{Mapping|legend=1| 1 1 7 -1 | 0 2 -16 13 }}
-* WE: ~2 = 1199.8482{{c}}, ~140/81 = 950.6809{{c}}
+: mapping generators: ~2, ~128/105
-* CWE: ~2 = 1200.0000{{c}}, ~140/81 = 950.8020{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~140/81 = 950.801{{c}} -->
-{{Optimal ET sequence|legend=0| 53, 130, 183, 313, 809cd }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.7707{{c}}, ~128/105 = 351.2748{{c}}
+: [[error map]]: {{val| +0.771 +1.365 -1.315 -3.024 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 351.0471{{c}}
+: error map: {{val| 0.000 +0.139 -3.068 -5.213 }}
-Badness (Sintel): 1.20
+{{Optimal ET sequence|legend=1| 17, 24, 41, 106d, 147d, 188cd }}
-=== 13-limit ===
+[[Badness]] (Sintel): 2.03
-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
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Comma list: 351/350, 540/539, 676/675, 4096/4095
+Comma list: 243/242, 245/242, 385/384
-Mapping: {{mapping| 1 0 15 -17 51 14 | 0 2 -16 25 -60 -13 }}
+Mapping: {{mapping| 1 1 7 -1 2 | 0 2 -16 13 5 }}
 Optimal tunings:
-* WE: ~2 = 1199.9140{{c}}, ~140/81 = 950.7324{{c}}
+* WE: ~2 = 1200.3891{{c}}, ~11/9 = 351.1275{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~140/81 = 950.8010{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0141{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~26/15 = 950.801{{c}} -->
-{{Optimal ET sequence|legend=0| 53, 130, 183, 313 }}
+{{Optimal ET sequence|legend=0| 17, 24, 41, 106d }}
-Badness (Sintel): 0.860
+Badness (Sintel): 1.30
-=== 17-limit ===
+=== 13-limit ===
-Subgroup: 2.3.5.7.11.13.17
+Subgroup: 2.3.5.7.11.13
-Comma list: 351/350, 442/441, 561/560, 676/675, 4096/4095
+Comma list: 105/104, 144/143, 243/242, 245/242
-Mapping: {{mapping| 1 0 15 -17 51 14 -49 | 0 2 -16 25 -60 -13 67 }}
+Mapping: {{mapping| 1 1 7 -1 2 4 | 0 2 -16 13 5 -1 }}
 Optimal tunings:
-* WE: ~2 = 1199.9740{{c}}, ~26/15 = 950.7894{{c}}
+* WE: ~2 = 1199.9362{{c}}, ~11/9 = 351.0061{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8100{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0247{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~26/15 = 950.810{{c}} -->
-{{Optimal ET sequence|legend=0| 53, 130, 183, 496d }}
+{{Optimal ET sequence|legend=0| 17, 24, 41 }}
-Badness (Sintel): 1.07
+Badness (Sintel): 1.27
-=== 19-limit ===
+== Hemischis ==
-Subgroup: 2.3.5.7.11.13.17.19
+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.
-Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 4096/4095
+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.
-Mapping: {{mapping| 1 0 15 -17 51 14 -49 9 | 0 2 -16 25 -60 -13 67 -6 }}
+[[Subgroup]]: 2.3.5.7
-Optimal tunings:
+[[Comma list]]: 6144/6125, 19683/19600
-* WE: ~2 = 1200.0464{{c}}, ~26/15 = 950.8459{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8091{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~26/15 = 950.809{{c}} -->
-{{Optimal ET sequence|legend=0| 53, 130, 183, 313h }}
+{{Mapping|legend=1| 1 0 15 -17 | 0 2 -16 25 }}
+: mapping generators: ~2, ~140/81
-Badness (Sintel): 1.11
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.8579{{c}}, ~140/81 = 951.6847{{c}}
+: [[error map]]: {{val| -0.142 -0.586 +0.600 +0.708 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 951.7966{{c}}
+: error map: {{val| 0.000 -0.362 +0.941 +1.088 }}
-=== 23-limit ===
+{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 313 }}
-Subgroup: 2.3.5.7.11.13.17.19.23
-Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 736/735, 4096/4095
+[[Badness]] (Sintel): 1.16
-Mapping: {{mapping| 1 0 15 -17 51 14 -49 9 -24 | 0 2 -16 25 -60 -13 67 -6 36 }}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Optimal tunings:
+Comma list: 540/539, 5632/5625, 8019/8000
-* WE: ~2 = 1200.0215{{c}}, ~26/15 = 950.8239{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8069{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~26/15 = 950.807{{c}} -->
-{{Optimal ET sequence|legend=0| 53, 130, 183, 313h }}
+Mapping: {{mapping| 1 0 15 -17 51 | 0 2 -16 25 -60 }}
-Badness (Sintel): 1.06
+Optimal tunings:
+* WE: ~2 = 1199.8482{{c}}, ~140/81 = 950.6809{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~140/81 = 950.8020{{c}}
-; Music
+{{Optimal ET sequence|legend=0| 53, 130, 183, 313, 809cd }}
-* ''HemischisMatic EP'' (2023) by [[User:Francium|Francium]] – [https://open.spotify.com/album/1Fx2shLclpNgFQJRw3ZHya Spotify] | [https://francium223.bandcamp.com/album/hemischismatic-ep Bandcamp] | [https://www.youtube.com/playlist?list=PLLZE7hMjEXRaiipPYK1InZBXTru_UtRsq YouTube] – 4-piece extended play
-== Squirrel ==
+Badness (Sintel): 1.20
-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.
-[[Subgroup]]: 2.3.5.7
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-[[Comma list]]: 686/675, 32805/32768
+Comma list: 351/350, 540/539, 676/675, 4096/4095
-{{Mapping|legend=1| 1 2 -1 1 | 0 -3 24 13 }}
+Mapping: {{mapping| 1 0 15 -17 51 14 | 0 2 -16 25 -60 -13 }}
-[[Optimal tuning]]s:
+Optimal tunings:
-* [[WE]]: ~2 = 1200.7408{{c}}, ~160/147 = 166.2424{{c}}
+* WE: ~2 = 1199.9140{{c}}, ~140/81 = 950.7324{{c}}
-: [[error map]]: {{val| +0.741 +0.799 +2.763 -6.934 }}
+* CWE: ~2 = 1200.0000{{c}}, ~140/81 = 950.8010{{c}}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~160/147 = 166.1597{{c}}
-: error map: {{val| 0.000 -0.434 +1.518 -8.750 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~160/147 = 166.140{{c}} -->
-{{Optimal ET sequence|legend=1| 29, 36, 65 }}
+{{Optimal ET sequence|legend=0| 53, 130, 183, 313 }}
-[[Badness]] (Sintel): 4.42
+Badness (Sintel): 0.860
-=== 11-limit ===
+=== 17-limit ===
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13.17
-Comma list: 245/242, 686/675, 896/891
+Comma list: 351/350, 442/441, 561/560, 676/675, 4096/4095
-Mapping: {{mapping| 1 2 -1 1 0 | 0 -3 24 13 25 }}
+Mapping: {{mapping| 1 0 15 -17 51 14 -49 | 0 2 -16 25 -60 -13 67 }}
 Optimal tunings:
-* WE: ~2 = 1200.6379{{c}}, ~11/10 = 166.1853{{c}}
+* WE: ~2 = 1199.9740{{c}}, ~26/15 = 950.7894{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.1157{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8100{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/10 = 166.097{{c}} -->
-{{Optimal ET sequence|legend=0| 29, 36, 65 }}
+{{Optimal ET sequence|legend=0| 53, 130, 183, 496d }}
-Badness (Sintel): 2.26
+Badness (Sintel): 1.07
-=== 13-limit ===
+=== 19-limit ===
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 91/90, 169/168, 245/242, 896/891
+Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 4096/4095
-Mapping: {{mapping| 1 2 -1 1 0 3 | 0 -3 24 13 25 5 }}
+Mapping: {{mapping| 1 0 15 -17 51 14 -49 9 | 0 2 -16 25 -60 -13 67 -6 }}
 Optimal tunings:
-* WE: ~2 = 1201.1361{{c}}, ~11/10 = 166.2110{{c}}
+* WE: ~2 = 1200.0464{{c}}, ~26/15 = 950.8459{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0833{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8091{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/10 = 166.054{{c}} -->
-{{Optimal ET sequence|legend=0| 29, 65f, 94df }}
+{{Optimal ET sequence|legend=0| 53, 130, 183, 313h }}
-Badness (Sintel): 1.81
+Badness (Sintel): 1.11
-== Tertiaschis ==
+=== 23-limit ===
-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.
+Subgroup: 2.3.5.7.11.13.17.19.23
-[[Subgroup]]: 2.3.5.7
+Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 736/735, 4096/4095
-[[Comma list]]: 32805/32768, 1071875/1062882
+Mapping: {{mapping| 1 0 15 -17 51 14 -49 9 -24 | 0 2 -16 25 -60 -13 67 -6 36 }}
-{{Mapping|legend=1| 1 2 -1 10 | 0 -3 24 -52 }}
+Optimal tunings:
+* WE: ~2 = 1200.0215{{c}}, ~26/15 = 950.8239{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8069{{c}}
-[[Optimal tuning]]s:
+{{Optimal ET sequence|legend=0| 53, 130, 183, 313h }}
-* [[WE]]: ~2 = 1200.3627{{c}}, ~192/175 = 166.0691{{c}}
-: [[error map]]: {{val| +0.363 +0.563 -1.019 -0.790 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~192/175 = 166.0172{{c}}
-: error map: {{val| 0.000 -0.007 -1.901 -1.720 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~192/175 = 166.019{{c}} -->
-{{Optimal ET sequence|legend=1| 65, 94, 159, 253, 412cd }}
+Badness (Sintel): 1.06
-[[Badness]] (Sintel): 5.36
+; Music
+* ''HemischisMatic EP'' (2023) by [[User:Francium|Francium]] – [https://open.spotify.com/album/1Fx2shLclpNgFQJRw3ZHya Spotify] | [https://francium223.bandcamp.com/album/hemischismatic-ep Bandcamp] | [https://www.youtube.com/playlist?list=PLLZE7hMjEXRaiipPYK1InZBXTru_UtRsq YouTube] – 4-piece extended play
-=== 11-limit ===
+== Term ==
-Subgroup: 2.3.5.7.11
+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.
-Comma list: 385/384, 4000/3993, 19712/19683
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 1 2 -1 10 0 | 0 -3 24 -52 25 }}
+[[Comma list]]: 32805/32768, 250047/250000
-Optimal tunings:
+{{Mapping|legend=1| 3 0 45 94 | 0 1 -8 -18 }}
-* WE: ~2 = 1200.3379{{c}}, ~11/10 = 166.0638{{c}}
+: mapping generators: ~63/50, ~3
-* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0167{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/10 = 166.017{{c}} -->
-{{Optimal ET sequence|legend=0| 65, 94, 159, 253, 412cd, 665ccde }}
+[[Optimal tuning]]s:
+* [[WE]]: ~63/50 = 400.0257{{c}}, ~3/2 = 701.7873{{c}}
-Badness (Sintel): 2.07
+: [[error map]]: {{val| +0.077 -0.091 -0.072 +0.031 }}
+* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 701.7383{{c}}
+: error map: {{val| 0.000 -0.217 -0.220 -0.115 }}
-=== 13-limit ===
+[[Minimax tuning]]:
-Subgroup: 2.3.5.7.11.13
+* [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis)]]: 2.5/3
+* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
-Comma list: 325/324, 385/384, 1575/1573, 10985/10976
+{{Optimal ET sequence|legend=1| 12, …, 159, 171, 867, 1038, 1209, 1380, 1551, 1722 }}
-Mapping: {{mapping| 1 2 -1 10 0 12 | 0 -3 24 -52 25 -60 }}
+[[Badness]] (Sintel): 0.505
-Optimal tunings:
+=== Terminal ===
-* WE: ~2 = 1200.3467{{c}}, ~11/10 = 166.0635{{c}}
+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.
-* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0142{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/10 = 166.016{{c}} -->
-{{Optimal ET sequence|legend=0| 65f, 94, 159, 253, 412cdf, 665ccdef }}
+Subgroup: 2.3.5.7.11
-Badness (Sintel): 1.52
+Comma list: 441/440, 4375/4356, 32805/32768
-=== 17-limit ===
+Mapping: {{mapping| 3 0 45 94 134 | 0 1 -8 -18 -26 }}
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
+Optimal tunings:
+* WE: ~44/35 = 400.0464{{c}}, ~3/2 = 701.9053{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8178{{c}}
-Mapping: {{mapping| 1 2 -1 10 0 12 -2 | 0 -3 24 -52 25 -60 44 }}
+{{Optimal ET sequence|legend=0| 12, …, 159, 330 }}
-Optimal tunings:
+Badness (Sintel): 1.97
-* WE: ~2 = 1200.3019{{c}}, ~11/10 = 166.0535{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0114{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/10 = 166.012{{c}} -->
-{{Optimal ET sequence|legend=1| 65f, 94, 159, 253 }}
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Badness (Sintel): 1.35
+Comma list: 364/363, 441/440, 625/624, 13720/13689
-== Countertertiaschis ==
+Mapping: {{mapping| 3 0 45 94 134 168 | 0 1 -8 -18 -26 -33 }}
-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.
-[[Subgroup]]: 2.3.5.7
+Optimal tunings:
+* WE: ~44/35 = 400.0449{{c}}, ~3/2 = 701.8995{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8156{{c}}
-[[Comma list]]: 32805/32768, 244140625/243045684
+{{Optimal ET sequence|legend=0| 12f, …, 159, 330 }}
-{{Mapping|legend=1| 1 2 -1 -12 | 0 -3 24 107 }}
+Badness (Sintel): 1.53
-[[Optimal tuning]]s:
+==== 17-limit ====
-* [[WE]]: ~2 = 1200.1265{{c}}, ~625/567 = 166.0797{{c}}
+Subgroup: 2.3.5.7.11.13.17
-: [[error map]]: {{val| +0.127 +0.059 -0.529 +0.178 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/567 = 166.0632{{c}}
-: error map: {{val| 0.000 -0.145 -0.797 -0.065 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~625/567 = 166.0621{{c}} -->
-{{Optimal ET sequence|legend=1| 65d, 159, 224, 383, 607 }}
+Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619
-[[Badness]] (Sintel): 4.76
+Mapping: {{mapping| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }}
-=== 11-limit ===
+Optimal tunings:
-Subgroup: 2.3.5.7.11
+* WE: ~34/27 = 400.0195{{c}}, ~3/2 = 701.8439{{c}}
+* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 701.8081{{c}}
-Comma list: 3025/3024, 4000/3993, 32805/32768
+{{Optimal ET sequence|legend=0| 12f, 159, 171, 330 }}
-Mapping: {{mapping| 1 2 -1 -12 0 | 0 -3 24 107 25 }}
+Badness (Sintel): 1.38
+=== 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:
-* WE: ~2 = 1200.0804{{c}}, ~11/10 = 166.0739{{c}}
+* WE: ~63/50 = 399.9677{{c}}, ~3/2 = 701.6278{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0634{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6846{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/10 = 166.0628{{c}} -->
-{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}
+{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 537, 891de }}
-Badness (Sintel): 1.62
+Badness (Sintel): 2.21
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976
+Comma list: 540/539, 729/728, 4096/4095, 31250/31213
-Mapping: {{mapping| 1 2 -1 -12 0 -10 | 0 -3 24 107 25 99 }}
+Mapping: {{mapping| 3 0 45 94 -137 -103 | 0 1 -8 -18 31 24 }}
 Optimal tunings:
-* WE: ~2 = 1200.0805{{c}}, ~11/10 = 166.0740{{c}}
+* WE: ~63/50 = 399.9731{{c}}, ~3/2 = 701.6414{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0635{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/10 = 166.0628{{c}} -->
-{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}
+{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }}
-Badness (Sintel): 1.01
+Badness (Sintel): 1.47
-== Term ==
+==== 17-limit ====
-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.
+Subgroup: 2.3.5.7.11.13.17
-[[Subgroup]]: 2.3.5.7
+Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095
-[[Comma list]]: 32805/32768, 250047/250000
+Mapping: {{mapping| 3 0 45 94 -137 -103 -2 | 0 1 -8 -18 31 24 3 }}
-{{Mapping|legend=1| 3 0 45 94 | 0 1 -8 -18 }}
+Optimal tunings:
-: mapping generators: ~63/50, ~3
+* WE: ~63/50 = 399.9757{{c}}, ~3/2 = 701.6458{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}}
-[[Optimal tuning]]s:
+{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }}
-* [[WE]]: ~63/50 = 400.0257{{c}}, ~3/2 = 701.7873{{c}}
-: [[error map]]: {{val| +0.077 -0.091 -0.072 +0.031 }}
-* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 701.7383{{c}}
-: error map: {{val| 0.000 -0.217 -0.220 -0.115 }}
-<!-- * [[POTE]]: ~63/50 = 400.000{{c}}, ~3/2 = 701.742{{c}} -->
-[[Minimax tuning]]:
+Badness (Sintel): 1.04
-* [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis)]]: 2.5/3
-* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
-{{Optimal ET sequence|legend=1| 12, …, 159, 171, 867, 1038, 1209, 1380, 1551, 1722 }}
+=== Semiterm ===
+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.
-[[Badness]] (Sintel): 0.505
-=== 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.
 Subgroup: 2.3.5.7.11
-Comma list: 441/440, 4375/4356, 32805/32768
+Comma list: 9801/9800, 32805/32768, 151263/151250
-Mapping: {{mapping| 3 0 45 94 134 | 0 1 -8 -18 -26 }}
+Mapping: {{mapping| 6 0 90 188 287 | 0 1 -8 -18 -28 }}
+: mapping generators: ~55/49, ~3
 Optimal tunings:
-* WE: ~44/35 = 400.0464{{c}}, ~3/2 = 701.9053{{c}}
+* WE: ~55/49 = 200.0134{{c}}, ~3/2 = 701.7931{{c}}
-* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8178{{c}}
+* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7426{{c}}
-<!-- * POTE: ~44/35 = 400.000{{c}}, ~3/2 = 701.824{{c}} -->
-{{Optimal ET sequence|legend=0| 12, …, 159, 330 }}
+{{Optimal ET sequence|legend=0| 12, …, 330e, 342, 1380, 1722, 2064, 2406c, 5154bccdde }}
-Badness (Sintel): 1.97
+Badness (Sintel): 0.973
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 364/363, 441/440, 625/624, 13720/13689
+Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375
-Mapping: {{mapping| 3 0 45 94 134 168 | 0 1 -8 -18 -26 -33 }}
+Mapping: {{mapping| 6 0 90 188 287 355 | 0 1 -8 -18 -28 -35 }}
 Optimal tunings:
-* WE: ~44/35 = 400.0449{{c}}, ~3/2 = 701.8995{{c}}
+* WE: ~55/49 = 200.0083{{c}}, ~3/2 = 701.7549{{c}}
-* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8156{{c}}
+* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7238{{c}}
-<!-- * POTE: ~44/35 = 400.000{{c}}, ~3/2 = 701.821{{c}} -->
+{{Optimal ET sequence|legend=0| 12f, 330eff, 342f, 696f }} *
+<nowiki>*</nowiki> optimal patent val: [[354edo|354]]
-{{Optimal ET sequence|legend=0| 12f, …, 159, 330 }}
+Badness (Sintel): 1.85
-Badness (Sintel): 1.53
+=== Hemiterm ===
+The hemiterm temperament tempers out [[3025/3024]] (lehmerisma), and may be described as {{nowrap| 159 & 183 }}. Its ploidacot is triploid alpha-dicot.
-==== 17-limit ====
+Subgroup: 2.3.5.7.11
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619
+Comma list: 3025/3024, 32805/32768, 102487/102400
-Mapping: {{mapping| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }}
+Mapping: {{mapping| 3 0 45 94 8 | 0 2 -16 -36 1 }}
+: mapping generators: ~63/50, ~693/400
 Optimal tunings:
-* WE: ~34/27 = 400.0195{{c}}, ~3/2 = 701.8439{{c}}
+* WE: ~63/50 = 400.0309{{c}}, ~693/400 = 950.9458{{c}} (~12/11 = 150.8841{{c}})
-* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 701.8081{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~693/400 = 950.8707{{c}} (~12/11 = 150.8707{{c}})
-<!-- * POTE: ~34/27 = 400.000{{c}}, ~3/2 = 701.810{{c}} -->
-{{Optimal ET sequence|legend=0| 12f, 159, 171, 330 }}
+{{Optimal ET sequence|legend=0| 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce }}
-Badness (Sintel): 1.38
+Badness (Sintel): 0.684
-=== Terminator ===
-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:
-* WE: ~63/50 = 399.9677{{c}}, ~3/2 = 701.6278{{c}}
-* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6846{{c}}
-<!-- * POTE: ~63/50 = 400.000{{c}}, ~3/2 = 701.685{{c}} -->
-{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 537, 891de }}
-Badness (Sintel): 2.21
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 729/728, 4096/4095, 31250/31213
+Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712
-Mapping: {{mapping| 3 0 45 94 -137 -103 | 0 1 -8 -18 31 24 }}
+Mapping: {{mapping| 3 0 45 94 8 42 | 0 2 -16 -36 1 -13 }}
 Optimal tunings:
-* WE: ~63/50 = 399.9731{{c}}, ~3/2 = 701.6414{{c}}
+* WE: ~63/50 = 400.0541{{c}}, ~26/15 = 951.0013{{c}} (~12/11 = 150.8932{{c}})
-* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~26/15 = 950.8696{{c}} (~12/11 = 150.8696{{c}})
-<!-- * POTE: ~63/50 = 400.000{{c}}, ~3/2 = 701.689{{c}} -->
-{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }}
+{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f }}
-Badness (Sintel): 1.47
+Badness (Sintel): 1.30
 ==== 17-limit ====
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095
+Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264
-Mapping: {{mapping| 3 0 45 94 -137 -103 -2 | 0 1 -8 -18 31 24 3 }}
+Mapping: {{mapping| 3 0 45 94 8 42 -2 | 0 2 -16 -36 1 -13 6 }}
 Optimal tunings:
-* WE: ~63/50 = 399.9757{{c}}, ~3/2 = 701.6458{{c}}
+* WE: ~34/27 = 400.0373{{c}}, ~26/15 = 950.9556{{c}} (~12/11 = 150.8809{{c}})
-* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}}
+* CWE: ~34/27 = 400.0000{{c}}, ~26/15 = 950.8652{{c}} (~12/11 = 150.8652{{c}})
-<!-- * POTE: ~63/50 = 400.000{{c}}, ~3/2 = 701.688{{c}} -->
-{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }}
+{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f, 525f }}
-Badness (Sintel): 1.04
+Badness (Sintel): 1.14
-=== Semiterm ===
+== Altinex ==
-The semiterm temperament ({{nowrap| 12 & 342 }}) has a period of 1/6 octave and tempers out [[9801/9800]] (kalisma) and 151263/151250 (odiheim comma).
+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: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7
-Comma list: 9801/9800, 32805/32768, 151263/151250
+[[Comma list]]: 32805/32768, 367653125/362797056
-Mapping: {{mapping| 6 0 90 188 287 | 0 1 -8 -18 -28 }}
+{{Mapping|legend=1| 3 0 45 -32 | 0 2 -16 17 }}
-: mapping generators: ~55/49, ~3
+: mapping generators: ~1536/1225, ~34300/19683
-Optimal tunings:
+[[Optimal tuning]]s:
-* WE: ~55/49 = 200.0134{{c}}, ~3/2 = 701.7931{{c}}
+* [[WE]]: ~1536/1225 = 400.1360{{c}}, ~34300/19683 = 951.2867{{c}}
-* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7426{{c}}
+: [[error map]]: {{val| +0.408 +0.618 -0.781 -1.304 }}
-<!-- * POTE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7460{{c}} -->
+* [[CWE]]: ~1536/1225 = 400.0000{{c}}, ~34300/19683 = 950.9638{{c}}
+: error map: {{val| 0.000 -0.027 -1.735 -2.441 }}
-{{Optimal ET sequence|legend=0| 12, …, 330e, 342, 1380, 1722, 2064, 2406c, 5154bccdde }}
+{{Optimal ET sequence|legend=1| 24, 135, 159, 612ccdd }}
-Badness (Sintel): 0.973
+[[Badness]] (Sintel): 10.7
-==== 13-limit ====
+=== 11-limit ===
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11
-Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375
+Comma list: 385/384, 14700/14641, 19712/19683
-Mapping: {{mapping| 6 0 90 188 287 355 | 0 1 -8 -18 -28 -35 }}
+Mapping: {{mapping| 3 0 45 -32 8 | 0 2 -16 17 1 }}
 Optimal tunings:
-* WE: ~55/49 = 200.0083{{c}}, ~3/2 = 701.7549{{c}}
+* WE: ~44/35 = 400.1156{{c}}, ~121/70 = 951.2377{{c}}
-* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7238{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~121/70 = 950.9634{{c}}
-<!-- * POTE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7256{{c}} -->
-{{Optimal ET sequence|legend=0| 12f, 330eff, 342f, 696f }} *
+{{Optimal ET sequence|legend=0| 24, 135, 159 }}
-<nowiki>*</nowiki> optimal patent val: [[354edo|354]]
+Badness (Sintel): 3.35
-Badness (Sintel): 1.85
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-=== Hemiterm ===
+Comma list: 364/363, 385/384, 676/675, 19712/19683
-Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 32805/32768, 102487/102400
+Mapping: {{mapping| 3 0 45 -32 8 42 | 0 2 -16 17 1 -13 }}
-Mapping: {{mapping| 3 0 45 94 8 | 0 2 -16 -36 1 }}
-: mapping generators: ~63/50, ~693/400
 Optimal tunings:
-* WE: ~63/50 = 400.0309{{c}}, ~693/400 = 950.9458{{c}} (~12/11 = 150.8841{{c}})
+* WE: ~44/35 = 400.1396{{c}}, ~26/15 = 951.2799{{c}}
-* CWE: ~63/50 = 400.0000{{c}}, ~693/400 = 950.8707{{c}} (~12/11 = 150.8707{{c}})
+* CWE: ~44/35 = 400.0000{{c}}, ~26/15 = 950.9462{{c}}
-<!-- * POTE: ~63/50 = 400.000{{c}}, ~693/400 = 950.872{{c}} (~12/11 = 150.872{{c}}) -->
-{{Optimal ET sequence|legend=0| 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce }}
+{{Optimal ET sequence|legend=0| 24, 135f, 159 }}
-Badness (Sintel): 0.684
+Badness (Sintel): 2.27
-==== 13-limit ====
+== Squirrel ==
-Subgroup: 2.3.5.7.11.13
+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.
-Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 686/675, 32805/32768
-Mapping: {{mapping| 3 0 45 94 8 42 | 0 2 -16 -36 1 -13 }}
+{{Mapping|legend=1| 1 2 -1 1 | 0 -3 24 13 }}
-Optimal tunings:
+[[Optimal tuning]]s:
-* WE: ~63/50 = 400.0541{{c}}, ~26/15 = 951.0013{{c}} (~12/11 = 150.8932{{c}})
+* [[WE]]: ~2 = 1200.7408{{c}}, ~160/147 = 166.2424{{c}}
-* CWE: ~63/50 = 400.0000{{c}}, ~26/15 = 950.8696{{c}} (~12/11 = 150.8696{{c}})
+: [[error map]]: {{val| +0.741 +0.799 +2.763 -6.934 }}
-<!-- * POTE: ~63/50 = 400.000{{c}}, ~26/15 = 950.873{{c}} (~12/11 = 150.873{{c}}) -->
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~160/147 = 166.1597{{c}}
+: error map: {{val| 0.000 -0.434 +1.518 -8.750 }}
-{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f }}
+{{Optimal ET sequence|legend=1| 29, 36, 65 }}
-Badness (Sintel): 1.30
+[[Badness]] (Sintel): 4.42
-==== 17-limit ====
+=== 11-limit ===
-Subgroup: 2.3.5.7.11.13.17
+Subgroup: 2.3.5.7.11
-Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264
+Comma list: 245/242, 686/675, 896/891
-Mapping: {{mapping| 3 0 45 94 8 42 -2 | 0 2 -16 -36 1 -13 6 }}
+Mapping: {{mapping| 1 2 -1 1 0 | 0 -3 24 13 25 }}
 Optimal tunings:
-* WE: ~34/27 = 400.0373{{c}}, ~26/15 = 950.9556{{c}} (~12/11 = 150.8809{{c}})
+* WE: ~2 = 1200.6379{{c}}, ~11/10 = 166.1853{{c}}
-* CWE: ~34/27 = 400.0000{{c}}, ~26/15 = 950.8652{{c}} (~12/11 = 150.8652{{c}})
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.1157{{c}}
-<!-- * POTE: ~34/27 = 400.000{{c}}, ~26/15 = 950.867{{c}} (~12/11 = 150.867{{c}}) -->
-{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f, 525f }}
+{{Optimal ET sequence|legend=0| 29, 36, 65 }}
-Badness (Sintel): 1.14
+Badness (Sintel): 2.26
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+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:
+* WE: ~2 = 1201.1361{{c}}, ~11/10 = 166.2110{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0833{{c}}
+{{Optimal ET sequence|legend=0| 29, 65f, 94df }}
+Badness (Sintel): 1.81
+== Tertiaschis ==
+Named by [[Xenllium]] in 2021, tertiaschis may be described as {{nowrap| 94 & 159 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 1071875/1062882 for prime 7. See also [[daemotertiaschis]] which is made using every other generator of tertiaschis.
-== Altinex ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 32805/32768, 367653125/362797056
+[[Comma list]]: 32805/32768, 1071875/1062882
-{{Mapping|legend=1| 3 0 45 -32 | 0 2 -16 17 }}
+{{Mapping|legend=1| 1 2 -1 10 | 0 -3 24 -52 }}
-: mapping generators: ~1536/1225, ~34300/19683
 [[Optimal tuning]]s:
-* [[WE]]: ~1536/1225 = 400.1360{{c}}, ~34300/19683 = 951.2867{{c}}
+* [[WE]]: ~2 = 1200.3627{{c}}, ~192/175 = 166.0691{{c}}
-: [[error map]]: {{val| +0.408 +0.618 -0.781 -1.304 }}
+: [[error map]]: {{val| +0.363 +0.563 -1.019 -0.790 }}
-* [[CWE]]: ~1536/1225 = 400.0000{{c}}, ~34300/19683 = 950.9638{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~192/175 = 166.0172{{c}}
-: error map: {{val| 0.000 -0.027 -1.735 -2.441 }}
+: error map: {{val| 0.000 -0.007 -1.901 -1.720 }}
-<!-- * [[CTE]]: ~1536/1225 = 400.000{{c}}, ~34300/19683 = 950.9654{{c}} -->
-{{Optimal ET sequence|legend=1| 24, 135, 159, 612ccdd }}
+{{Optimal ET sequence|legend=1| 65, 94, 159, 253, 412cd }}
-[[Badness]] (Sintel): 10.7
+[[Badness]] (Sintel): 5.36
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 385/384, 14700/14641, 19712/19683
+Comma list: 385/384, 4000/3993, 19712/19683
-Mapping: {{mapping| 3 0 45 -32 8 | 0 2 -16 17 1 }}
+Mapping: {{mapping| 1 2 -1 10 0 | 0 -3 24 -52 25 }}
 Optimal tunings:
-* WE: ~44/35 = 400.1156{{c}}, ~121/70 = 951.2377{{c}}
+* WE: ~2 = 1200.3379{{c}}, ~11/10 = 166.0638{{c}}
-* CWE: ~44/35 = 400.0000{{c}}, ~121/70 = 950.9634{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0167{{c}}
-<!-- * CTE: ~44/35 = 400.000{{c}}, ~121/70 = 950.9658{{c}} -->
-{{Optimal ET sequence|legend=0| 24, 135, 159 }}
+{{Optimal ET sequence|legend=0| 65, 94, 159, 253, 412cd, 665ccde }}
-Badness (Sintel): 3.35
+Badness (Sintel): 2.07
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 364/363, 385/384, 676/675, 19712/19683
+Comma list: 325/324, 385/384, 1575/1573, 10985/10976
-Mapping: {{mapping| 3 0 45 -32 8 42 | 0 2 -16 17 1 -13 }}
+Mapping: {{mapping| 1 2 -1 10 0 12 | 0 -3 24 -52 25 -60 }}
 Optimal tunings:
-* WE: ~44/35 = 400.1396{{c}}, ~26/15 = 951.2799{{c}}
+* WE: ~2 = 1200.3467{{c}}, ~11/10 = 166.0635{{c}}
-* CWE: ~44/35 = 400.0000{{c}}, ~26/15 = 950.9462{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0142{{c}}
-<!-- * CTE: ~44/35 = 400.000{{c}}, ~26/15 = 950.9360{{c}} -->
-{{Optimal ET sequence|legend=0| 24, 135f, 159 }}
+{{Optimal ET sequence|legend=0| 65f, 94, 159, 253, 412cdf, 665ccdef }}
-Badness (Sintel): 2.27
+Badness (Sintel): 1.52
-== Sesquiquartififths ==
+=== 17-limit ===
-[[Subgroup]]: 2.3.5.7
+Subgroup: 2.3.5.7.11.13.17
-[[Comma list]]: 2401/2400, 32805/32768
+Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
-{{Mapping|legend=1| 1 1 7 5 | 0 4 -32 -15 }}
+Mapping: {{mapping| 1 2 -1 10 0 12 -2 | 0 -3 24 -52 25 -60 44 }}
-: mapping generators: ~2, ~448/405
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~448/405 = 175.434{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.3019{{c}}, ~11/10 = 166.0535{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0114{{c}}
-[[Minimax tuning]]:
+{{Optimal ET sequence|legend=1| 65f, 94, 159, 253 }}
-* [[7-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
-* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
-{{Optimal ET sequence|legend=1| 41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd }}
+Badness (Sintel): 1.35
-[[Badness]] (Smith): 0.011244
+== Countertertiaschis ==
+Named by [[Flora Canou]] in 2021, Countertertiaschis may be described as {{nowrap| 159 & 224 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 244140625/243045684 for prime 7.
-=== Sesquart ===
+[[Subgroup]]: 2.3.5.7
-Subgroup: 2.3.5.7.11
-Comma list: 243/242, 441/440, 16384/16335
+[[Comma list]]: 32805/32768, 244140625/243045684
-Mapping: {{mapping| 1 1 7 5 2 | 0 4 -32 -15 10 }}
+{{Mapping|legend=1| 1 2 -1 -12 | 0 -3 24 107 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~256/231 = 175.406{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1265{{c}}, ~625/567 = 166.0797{{c}}
+: [[error map]]: {{val| +0.127 +0.059 -0.529 +0.178 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/567 = 166.0632{{c}}
+: error map: {{val| 0.000 -0.145 -0.797 -0.065 }}
-{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}
+{{Optimal ET sequence|legend=1| 65d, 159, 224, 383, 607 }}
-Badness (Smith): 0.029306
+[[Badness]] (Sintel): 4.76
-==== 13-limit ====
+=== 11-limit ===
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11
-Comma list: 243/242, 364/363, 441/440, 3584/3575
+Comma list: 3025/3024, 4000/3993, 32805/32768
-Mapping: {{mapping| 1 1 7 5 2 -2 | 0 4 -32 -15 10 39 }}
+Mapping: {{mapping| 1 2 -1 -12 0 | 0 -3 24 107 25 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~72/65 = 175.409{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0804{{c}}, ~11/10 = 166.0739{{c}}
-{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0634{{c}}
-Badness (Smith): 0.022396
+{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}
-===== Sesquartia =====
+Badness (Sintel): 1.62
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Mapping: {{mapping| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }}
+Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~72/65 = 175.424{{c}}
+Mapping: {{mapping| 1 2 -1 -12 0 -10 | 0 -3 24 107 25 99 }}
-{{Optimal ET sequence|legend=0| 41, 89g, 130, 171, 301e }}
+Optimal tunings:
+* WE: ~2 = 1200.0805{{c}}, ~11/10 = 166.0740{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0635{{c}}
-Badness (Smith): 0.023126
+{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}
-====== 19-limit ======
+Badness (Sintel): 1.01
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594
+== Quadrant ==
+Named by [[Xenllium]] in 2021, quadrant tempers out 390625/388962, the [[dimcomp comma]], and maps [[25/21]] to the 1/4-octave period. It may be described as the {{nowrap| 12 & 212 }} temperament; its ploidacot is tetraploid monocot. Just as [[#Term|term]] equates the syntonic~Pythagorean comma with three [[marvel comma]]s, quadrant equates the syntonic~Pythagorean comma with four. A [[septimal comma]] is then found as a stack of five marvel commas.
-Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }}
+[[Subgroup]]: 2.3.5.7
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/19 = 175.419{{c}}
+[[Comma list]]: 32805/32768, 390625/388962
-{{Optimal ET sequence|legend=0| 41, 89g, 130, 171, 301eh }}
+{{Mapping|legend=1| 4 0 60 119 | 0 1 -8 -17 }}
+: mapping generators: ~25/21, ~3
-Badness (Smith): 0.020466
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 300.0255{{c}}, ~3/2 = 701.8831{{c}}
+: [[error map]]: {{val| +0.102 +0.030 -0.664 +0.462 }}
+* [[CWE]]: ~2 = 300.0000{{c}}, ~3/2 = 701.8180{{c}}
+: error map: {{val| 0.000 -0.137 -0.858 +0.268 }}
-====== 23-limit ======
+{{Optimal ET sequence|legend=1| 12, …, 200, 212, 224, 436, 660 }}
-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
+[[Badness]] (Sintel): 2.79
-Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 -6 | 0 4 -32 -15 10 39 69 -12 72 }}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/19 = 175.412{{c}}
+Comma list: 1375/1372, 6250/6237, 32805/32768
-{{Optimal ET sequence|legend=0| 41i, 89gi, 130, 171, 301eh }}
+Mapping: {{mapping| 4 0 60 119 185 | 0 1 -8 -17 -27 }}
-Badness (Smith): 0.019043
+Optimal tunings:
+* WE: ~25/21 = 300.0244{{c}}, ~3/2 = 701.8759{{c}}
+* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8145{{c}}
-===== Heartia =====
+{{Optimal ET sequence|legend=0| 12, …, 212, 224, 436, 660 }}
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 243/242, 256/255, 273/272, 364/363, 441/440
+Badness (Sintel): 1.51
-Mapping: {{mapping| 1 1 7 5 2 -2 0 | 0 4 -32 -15 10 39 28 }}
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~72/65 = 175.386{{c}}
+Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647
-{{Optimal ET sequence|legend=0| 41, 89, 130g }}
+Mapping: {{mapping| 4 0 60 119 185 224 | 0 1 -8 -17 -27 -33 }}
-Badness (Smith): 0.028443
+Optimal tunings:
+* WE: ~25/21 = 300.0234{{c}}, ~3/2 = 701.8707{{c}}
+* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8123{{c}}
-====== 19-limit ======
+{{Optimal ET sequence|legend=0| 12f, …, 212, 224, 436, 660 }}
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 171/170, 243/242, 256/255, 273/272, 324/323, 441/440
+Badness (Sintel): 1.13
-Mapping: {{mapping| 1 1 7 5 2 -2 0 6 | 0 4 -32 -15 10 39 28 -12 }}
+== 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.
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/19 = 175.380{{c}}
+[[Subgroup]]: 2.3.5.7
-{{Optimal ET sequence|legend=0| 41, 89, 130g }}
+[[Comma list]]: 2401/2400, 32805/32768
-Badness (Smith): 0.023059
+{{Mapping|legend=1| 1 1 7 5 | 0 4 -32 -15 }}
+: mapping generators: ~2, ~448/405
-===== Hearty =====
+[[Optimal tuning]]s:
-Subgroup: 2.3.5.7.11.13.17
+* [[WE]]: ~2 = 1200.0846{{c}}, ~448/405 = 175.4460{{c}}
+: [[error map]]: {{val| +0.085 -0.086 +0.007 -0.093 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~448/405 = 175.4320{{c}}
+: error map: {{val| 0.000 -0.227 -0.137 -0.306 }}
-Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625
+[[Minimax tuning]]:
+* [[7-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
+* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
-Mapping: {{mapping| 1 1 7 5 2 -2 13 | 0 4 -32 -15 10 39 -61 }}
+{{Optimal ET sequence|legend=1| 41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~72/65 = 175.377{{c}}
+[[Badness]] (Sintel): 0.285
-{{Optimal ET sequence|legend=0| 41g, 89, 130, 609ceefgg }}
+=== 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.
-Badness (Smith): 0.030680
+Subgroup: 2.3.5.7.11
-====== 19-limit ======
+Comma list: 243/242, 441/440, 16384/16335
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 221/220, 243/242, 361/360, 364/363, 441/440, 456/455
+Mapping: {{mapping| 1 1 7 5 2 | 0 4 -32 -15 10 }}
-Mapping: {{mapping| 1 1 7 5 2 -2 13 6 | 0 4 -32 -15 10 39 -61 -12 }}
+Optimal tunings:
+* WE: ~2 = 1199.8171{{c}}, ~256/231 = 175.3793{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~256/231 = 175.4081{{c}}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/19 = 175.377{{c}}
+{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}
-{{Optimal ET sequence|legend=0| 41g, 89, 130, 609ceefggh }}
+Badness (Sintel): 0.969
-Badness (Smith): 0.022816
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-====== 23-limit ======
+Comma list: 243/242, 364/363, 441/440, 3584/3575
-Subgroup: 2.3.5.7.11.13.17.19.23
-Comma list: 221/220, 243/242, 276/275, 323/322, 361/360, 364/363, 441/440
+Mapping: {{mapping| 1 1 7 5 2 -2 | 0 4 -32 -15 10 39 }}
-Mapping: {{mapping| 1 1 7 5 2 -2 13 6 13 | 0 4 -32 -15 10 39 -61 -12 -58 }}
+Optimal tunings:
+* WE: ~2 = 1199.8352{{c}}, ~72/65 = 175.3852{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4095{{c}}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/19 = 175.376{{c}}
+{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}
-{{Optimal ET sequence|legend=0| 41g, 89, 130, 609ceefggh }}
+Badness (Sintel): 0.925
-Badness (Smith): 0.019121
+===== Heartia =====
+Subgroup: 2.3.5.7.11.13.17
-=== Bisesqui ===
+Comma list: 243/242, 256/255, 273/272, 364/363, 441/440
-Subgroup: 2.3.5.7.11
-Comma list: 2401/2400, 9801/9800, 32805/32768
+Mapping: {{mapping| 1 1 7 5 2 -2 0 | 0 4 -32 -15 10 39 28 }}
-Mapping: {{mapping| 2 2 14 10 23 | 0 4 -32 -15 -55 }}
+Optimal tunings:
-: mapping generators: ~99/70, ~448/405
+* WE: ~2 = 1199.6422{{c}}, ~72/65 = 175.3338{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3857{{c}}
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~448/405 = 175.435{{c}}
+{{Optimal ET sequence|legend=0| 41, 89, 130g }}
-{{Optimal ET sequence|legend=1| 82e, 130, 212, 342, 1156, 1498, 1840d }}
+Badness (Sintel): 1.45
-Badness (Smith): 0.016968
+====== 19-limit ======
+Subgroup: 2.3.5.7.11.13.17.19
-== Quintilipyth ==
+Comma list: 171/170, 243/242, 256/255, 273/272, 324/323, 441/440
-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.
-[[Subgroup]]: 2.3.5.7
+Mapping: {{mapping| 1 1 7 5 2 -2 0 6 | 0 4 -32 -15 10 39 28 -12 }}
-[[Comma list]]: 32805/32768, 9765625/9680832
+Optimal tunings:
+* WE: ~2 = 1199.7499{{c}}, ~21/19 = 175.3432{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.3797{{c}}
-{{Mapping|legend=1| 1 2 -1 -4 | 0 -5 40 82 }}
+{{Optimal ET sequence|legend=0| 41, 89, 130g }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~625/588 = 99.625{{c}}
+Badness (Sintel): 1.40
-{{Optimal ET sequence|legend=1| 12, 253, 265 }}
+===== Sesquartia =====
+Subgroup: 2.3.5.7.11.13.17
-[[Badness]] (Smith): 0.253966
+Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575
-=== 11-limit ===
+Mapping: {{mapping| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }}
-Subgroup: 2.3.5.7.11
-Comma list: 1375/1372, 4375/4356, 32805/32768
+Optimal tunings:
+* WE: ~2 = 1199.8902{{c}}, ~72/65 = 175.4077{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4234{{c}}
-Mapping: {{mapping| 1 2 -1 -4 -7 | 0 -5 40 82 126 }}
+{{Optimal ET sequence|legend=0| 41, 130, 171 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~35/33 = 99.616{{c}}
+Badness (Sintel): 1.18
-{{Optimal ET sequence|legend=0| 12, 253, 265, 518c, 783cc }}
+====== 19-limit ======
+Subgroup: 2.3.5.7.11.13.17.19
-Badness (Smith): 0.113044
+Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594
-=== 13-limit ===
+Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647
+Optimal tunings:
+* WE: ~2 = 1199.9864{{c}}, ~21/19 = 175.4169{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4189{{c}}
-Mapping: {{mapping| 1 2 -1 -4 -7 -9 | 0 -5 40 82 126 153 }}
+{{Optimal ET sequence|legend=0| 41, 130, 171 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~35/33 = 99.612{{c}}
+Badness (Sintel): 1.24
-{{Optimal ET sequence|legend=0| 12f, 253, 518c, 771cc }}
+====== 23-limit ======
+Subgroup: 2.3.5.7.11.13.17.19.23
-Badness (Smith): 0.069127
+Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594
-=== 17-limit ===
+Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 -6 | 0 4 -32 -15 10 39 69 -12 72 }}
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619
+Optimal tunings:
+* WE: ~2 = 1199.9606{{c}}, ~21/19 = 175.4067{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4123{{c}}
-Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 | 0 -5 40 82 126 153 -11 }}
+{{Optimal ET sequence|legend=0| 41i, 130, 171 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.612{{c}}
+Badness (Sintel): 1.36
-{{Optimal ET sequence|legend=0| 12f, 253, 518c, 771cc }}
+===== Hearty =====
+Subgroup: 2.3.5.7.11.13.17
-Badness (Smith): 0.045992
+Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625
-=== 19-limit ===
+Mapping: {{mapping| 1 1 7 5 2 -2 13 | 0 4 -32 -15 10 39 -61 }}
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971
+Optimal tunings:
+* WE: ~2 = 1199.9458{{c}}, ~72/65 = 175.3689{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3770{{c}}
-Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 4 | 0 -5 40 82 126 153 -11 3 }}
+{{Optimal ET sequence|legend=0| 41g, 89, 130 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.615{{c}}
+Badness (Sintel): 1.56
-{{Optimal ET sequence|legend=0| 12f, 253, 265, 518ch }}
+====== 19-limit ======
+Subgroup: 2.3.5.7.11.13.17.19
-Badness (Smith): 0.038155
+Comma list: 221/220, 243/242, 361/360, 364/363, 441/440, 456/455
-== Quintaschis ==
+Mapping: {{mapping| 1 1 7 5 2 -2 13 6 | 0 4 -32 -15 10 39 -61 -12 }}
-The quintaschis temperament ({{nowrap| 12 & 289 }}) slices the fourth (4/3) into five semitones and tempers out 49009212/48828125 in the 7-limit.
-[[Subgroup]]: 2.3.5.7
+Optimal tunings:
+* WE: ~2 = 1200.0114{{c}}, ~72/65 = 175.3783{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3765{{c}}
-[[Comma list]]: 32805/32768, 49009212/48828125
+{{Optimal ET sequence|legend=0| 41g, 89, 130 }}
-{{Mapping|legend=1| 1 2 -1 -5 | 0 -5 40 94 }}
+Badness (Sintel): 1.39
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~200/189 = 99.664{{c}}
+====== 23-limit ======
+Subgroup: 2.3.5.7.11.13.17.19.23
-{{Optimal ET sequence|legend=1| 12, …, 289, 301, 590, 891, 1192 }}
+Comma list: 221/220, 243/242, 276/275, 323/322, 361/360, 364/363, 441/440
-[[Badness]] (Smith): 0.132890
+Mapping: {{mapping| 1 1 7 5 2 -2 13 6 13 | 0 4 -32 -15 10 39 -61 -12 -58 }}
-=== 11-limit ===
+Optimal tunings:
-Subgroup: 2.3.5.7.11
+* WE: ~2 = 1200.0122{{c}}, ~72/65 = 175.3782{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3763{{c}}
-Comma list: 441/440, 32805/32768, 1953125/1951488
+{{Optimal ET sequence|legend=0| 41g, 89, 130 }}
-Mapping: {{mapping| 1 2 -1 -5 -8 | 0 -5 40 94 138 }}
+Badness (Sintel): 1.37
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~35/33 = 99.653{{c}}
+=== Bisesqui ===
+Subgroup: 2.3.5.7.11
-{{Optimal ET sequence|legend=0| 12, …, 277d, 289 }}
+Comma list: 2401/2400, 9801/9800, 32805/32768
-Badness (Smith): 0.111477
+Mapping: {{mapping| 2 2 14 10 23 | 0 4 -32 -15 -55 }}
+: mapping generators: ~99/70, ~448/405
-==== 13-limit ====
+Optimal tunings:
-Subgroup: 2.3.5.7.11.13
+* WE: ~99/70 = 600.0429{{c}}, ~448/405 = 175.4474{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~448/405 = 175.4334{{c}}
-Comma list: 364/363, 441/440, 32805/32768, 109512/109375
+{{Optimal ET sequence|legend=1| 82e, 130, 212, 342, 1156, 1498, 1840d, 5862bbccdddee }}
-Mapping: {{mapping| 1 2 -1 -5 -8 -11 | 0 -5 40 94 138 177 }}
+Badness (Sintel): 0.561
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~35/33 = 99.658{{c}}
+== Tsaharuk ==
+{{Main| Tsaharuk }}
-{{Optimal ET sequence|legend=0| 12f, …, 277dff, 289 }}
+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.
-Badness (Smith): 0.074218
+[[Subgroup]]: 2.3.5.7
-==== 17-limit ====
+[[Comma list]]: 32805/32768, 420175/419904
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768
+{{Mapping|legend=1| 1 1 7 0 | 0 5 -40 24 }}
+: mapping generators: ~2, ~243/224
-Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 | 0 -5 40 94 138 177 -11 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1039{{c}}, ~243/224 = 140.3620{{c}}
+: [[error map]]: {{val| +0.104 -0.041 -0.067 -0.137 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/224 = 140.3496{{c}}
+: error map: {{val| 0.000 -0.207 -0.296 -0.436 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.656{{c}}
+{{Optimal ET sequence|legend=1| 17, 77, 94, 171 }}
-{{Optimal ET sequence|legend=0| 12f, 277dff, 289 }}
+[[Badness]] (Sintel): 0.777
-Badness (Smith): 0.050571
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-==== 19-limit ====
+Comma list: 385/384, 1331/1323, 19712/19683
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859
+Mapping: {{mapping| 1 1 7 0 1 | 0 5 -40 24 21 }}
-Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 4 | 0 -5 40 94 138 177 -11 3 }}
+Optimal tunings:
+* WE: ~2 = 1200.3103{{c}}, ~88/81 = 140.4011{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.3649{{c}}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.659{{c}}
+{{Optimal ET sequence|legend=0| 17, 77, 94, 171e, 265e }}
-{{Optimal ET sequence|legend=0| 12f, 289 }}
+Badness (Sintel): 2.10
-Badness (Smith): 0.042120
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-=== Quintahelenic ===
+Comma list: 352/351, 385/384, 729/728, 1331/1323
-Subgroup: 2.3.5.7.11
-Comma list: 5632/5625, 8019/8000, 151263/151250
+Mapping: {{mapping| 1 1 7 0 1 3 | 0 5 -40 24 21 6 }}
-Mapping: {{mapping| 1 2 -1 -5 -9 | 0 -5 40 94 150 }}
+Optimal tunings:
+* WE: ~2 = 1200.1840{{c}}, ~13/12 = 140.3840{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.3627{{c}}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~200/189 = 99.671{{c}}
+{{Optimal ET sequence|legend=0| 17, 77, 94, 171e }}
-{{Optimal ET sequence|legend=0| 12, …, 289e, 301, 915 }}
+Badness (Sintel): 1.57
-Badness (Smith): 0.082225
+== 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.
-==== 13-limit ====
+[[Subgroup]]: 2.3.5.7
-Subgroup: 2.3.5.7.11.13
-Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000
+[[Comma list]]: 16875/16807, 32805/32768
-Mapping: {{mapping| 1 2 -1 -5 -9 -11 | 0 -5 40 94 150 177 }}
+{{Mapping|legend=1| 1 0 15 12 | 0 5 -40 -29 }}
+: mapping generators: ~2, ~56/45
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~200/189 = 99.661{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0032{{c}}, ~56/45 = 380.3557{{c}}
+: [[error map]]: {{val| +0.003 -0.177 -0.493 +0.898 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~56/45 = 380.3546{{c}}
+: error map: {{val| 0.000 -0.182 -0.498 +0.890 }}
-{{Optimal ET sequence|legend=0| 12f, …, 289e, 301 }}
+{{Optimal ET sequence|legend=1| 41, 142, 183, 224 }}
-Badness (Smith): 0.055570
+[[Badness]] (Sintel): 1.82
-===== 17-limit =====
+=== 11-limit ===
-Subgroup: 2.3.5.7.11.13.17
+Subgroup: 2.3.5.7.11
-Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750
+Comma list: 540/539, 1375/1372, 32805/32768
-Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 | 0 -5 40 94 150 177 -11 }}
+Mapping: {{mapping| 1 0 15 12 -7 | 0 5 -40 -29 33 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.665{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9709{{c}}, ~56/45 = 380.3423{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3517{{c}}
-{{Optimal ET sequence|legend=1| 12f, 289e, 301 }}
+{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}
-Badness (Smith): 0.040412
+Badness (Sintel): 1.04
-===== 19-limit =====
+=== 13-limit ===
-Subgroup: 2.3.5.7.11.13.17.19
+Subgroup: 2.3.5.7.11.13
-Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700
+Comma list: 540/539, 729/728, 1375/1372, 4096/4095
-Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 4 | 0 -5 40 94 150 177 -11 3 }}
+Mapping: {{mapping| 1 0 15 12 -7 -15 | 0 5 -40 -29 33 59 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.668{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9663{{c}}, ~56/45 = 380.3403{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3509{{c}}
-{{Optimal ET sequence|legend=0| 12f, 301 }}
+{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}
-Badness (Smith): 0.036840
+Badness (Sintel): 0.884
-==== Quintahelenoid ====
+== Quintilipyth ==
-Subgroup: 2.3.5.7.11.13
+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.
-Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 1 2 -1 -5 -9 14 | 0 -5 40 94 150 -124 }}
+[[Comma list]]: 32805/32768, 9765625/9680832
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~200/189 = 99.672{{c}}
+{{Mapping|legend=1| 1 2 -1 -4 | 0 -5 40 82 }}
+: mapping generators: ~2, ~625/588
-{{Optimal ET sequence|legend=0| 12, 301, 614, 915 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1138{{c}}, ~625/588 = 99.6347{{c}}
+: [[error map]]: {{val| +0.114 +0.099 -1.041 +0.761 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/588 = 99.6265{{c}}
+: error map: {{val| 0.000 -0.087 -1.255 +0.544 }}
-Badness (Smith): 0.066108
+{{Optimal ET sequence|legend=1| 12, …, 253, 265 }}
-===== 17-limit =====
+[[Badness]] (Sintel): 6.43
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Mapping: {{mapping| 1 2 -1 -5 -9 14 5 | 0 -5 40 94 150 -124 -11 }}
+Comma list: 1375/1372, 4375/4356, 32805/32768
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.671{{c}}
+Mapping: {{mapping| 1 2 -1 -4 -7 | 0 -5 40 82 126 }}
-{{Optimal ET sequence|legend=0| 12, 301 }}
+Optimal tunings:
+* WE: ~2 = 1200.1503{{c}}, ~35/33 = 99.6287{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6176{{c}}
-Badness (Smith): 0.047908
+{{Optimal ET sequence|legend=0| 12, …, 253, 265, 518c }}
-===== 19-limit =====
+Badness (Sintel): 3.74
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Mapping: {{mapping| 1 2 -1 -5 -9 14 5 4 | 0 -5 40 94 150 -124 -11 3 }}
+Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.672{{c}}
+Mapping: {{mapping| 1 2 -1 -4 -7 -9 | 0 -5 40 82 126 153 }}
-{{Optimal ET sequence|legend=0| 12, 301 }}
+Optimal tunings:
+* WE: ~2 = 1200.1774{{c}}, ~35/33 = 99.6267{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6134{{c}}
-Badness (Smith): 0.039542
+{{Optimal ET sequence|legend=0| 12f, …, 241cdef, 253 }}
-== Sextilifourths ==
+Badness (Sintel): 2.86
-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.
-[[Subgroup]]: 2.3.5.7
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
-[[Comma list]]: 32805/32768, 235298/234375
+Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619
-{{Mapping|legend=1| 1 2 -1 -1 | 0 -6 48 55 }}
+Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 | 0 -5 40 82 126 153 -11 }}
-: mapping generators: ~2, ~21/2
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~21/20 = 83.053{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.1745{{c}}, ~18/17 = 99.6265{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6131{{c}}
-{{Optimal ET sequence|legend=1| 29, 72cd, 101, 130, 289, 419 }}
+{{Optimal ET sequence|legend=0| 12f, 241cdef, 253 }}
-[[Badness]] (Smith): 0.108794
+Badness (Sintel): 2.34
-=== 11-limit ===
+=== 19-limit ===
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 441/440, 4000/3993, 235298/234375
+Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971
-Mapping: {{mapping| 1 2 -1 -1 0 | 0 -6 48 55 50 }}
+Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 4 | 0 -5 40 82 126 153 -11 3 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 83.049{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0713{{c}}, ~18/17 = 99.6208{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6152{{c}}
-{{Optimal ET sequence|legend=0| 29, 72cde, 101e, 130, 289 }}
+{{Optimal ET sequence|legend=0| 12f, 253, 265 }}
-Badness (Smith): 0.045457
+Badness (Sintel): 2.32
-=== 13-limit ===
+== Quintaschis ==
-Subgroup: 2.3.5.7.11.13
+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.
-Comma list: 364/363, 441/440, 676/675, 10985/10976
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 1 2 -1 -1 0 1 | 0 -6 48 55 50 39 }}
+[[Comma list]]: 32805/32768, 49009212/48828125
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 83.049{{c}}
+{{Mapping|legend=1| 1 2 -1 -5 | 0 -5 40 94 }}
-{{Optimal ET sequence|legend=0| 29, 72cdef, 101e, 130, 289 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0536{{c}}, ~200/189 = 99.6684{{c}}
+: [[error map]]: {{val| +0.054 -0.190 +0.370 -0.262 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~200/189 = 99.6645{{c}}
+: error map: {{val| 0.000 -0.277 +0.266 -0.363 }}
-Badness (Smith): 0.025276
+{{Optimal ET sequence|legend=1| 12, …, 289, 301, 590, 891, 1192 }}
-== Septiquarschis ==
+[[Badness]] (Sintel): 3.36
-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).
-[[Subgroup]]: 2.3.5.7
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-[[Comma list]]: 32805/32768, 829440/823543
+Comma list: 441/440, 32805/32768, 1953125/1951488
-{{Mapping|legend=1| 1 3 -9 2 | 0 -7 -56 4 }}
+Mapping: {{mapping| 1 2 -1 -5 -8 | 0 -5 40 94 138 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~147/128 = 242.614{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0988{{c}}, ~35/33 = 99.6613{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6540{{c}}
-{{Optimal ET sequence|legend=1| 89, 94, 183, 460d, 643d, 1103dd }}
+{{Optimal ET sequence|legend=0| 12, …, 277d, 289 }}
-[[Badness]] (Smith): 0.187047
+Badness (Sintel): 3.69
-=== 11-limit ===
+==== 13-limit ====
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 15488/15435, 32805/32768
+Comma list: 364/363, 441/440, 32805/32768, 109512/109375
-Mapping: {{mapping| 1 3 -9 2 -2 | 0 -7 -56 4 27 }}
+Mapping: {{mapping| 1 2 -1 -5 -8 -11 | 0 -5 40 94 138 177 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~147/128 = 242.616{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0625{{c}}, ~35/33 = 99.6630{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6583{{c}}
-{{Optimal ET sequence|legend=0| 89, 94, 183, 460d, 643d, 826dd }}
+{{Optimal ET sequence|legend=0| 12f, …, 277dff, 289 }}
-Badness (Smith): 0.052002
+Badness (Sintel): 3.07
-=== 13-limit ===
+==== 17-limit ====
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11.13.17
-Comma list: 540/539, 729/728, 1573/1568, 4096/4095
+Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768
-Mapping: {{mapping| 1 3 -9 2 -2 13 | 0 -7 -56 4 27 -46 }}
+Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 | 0 -5 40 94 138 177 -11 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~147/128 = 242.610{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.1286{{c}}, ~18/17 = 99.6668{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6568{{c}}
-{{Optimal ET sequence|legend=0| 89, 94, 183, 277, 460d }}
+{{Optimal ET sequence|legend=0| 12f, 277dff, 289 }}
-Badness (Smith): 0.035315
+Badness (Sintel): 2.58
-== Tsaharuk ==
+==== 19-limit ====
-{{Main| Tsaharuk }}
+Subgroup: 2.3.5.7.11.13.17.19
-[[Subgroup]]: 2.3.5.7
+Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859
-[[Comma list]]: 32805/32768, 420175/419904
+Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 4 | 0 -5 40 94 138 177 -11 3 }}
-{{Mapping|legend=1| 1 1 7 0 | 0 5 -40 24 }}
+Optimal tunings:
-: mapping generators: ~2, ~243/224
+* WE: ~2 = 1200.0289{{c}}, ~18/17 = 99.6609{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6586{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~243/224 = 140.350{{c}}
+{{Optimal ET sequence|legend=0| 12f, 289 }}
-{{Optimal ET sequence|legend=1| 17, 60c, 77, 94, 171 }}
+Badness (Sintel): 2.56
-[[Badness]] (Smith): 0.030697
+=== Quintahelenic ===
-=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 385/384, 1331/1323, 19712/19683
+Comma list: 5632/5625, 8019/8000, 151263/151250
-Mapping: {{mapping| 1 1 7 0 1 | 0 5 -40 24 21 }}
+Mapping: {{mapping| 1 2 -1 -5 -9 | 0 -5 40 94 150 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~88/81 = 140.365{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0195{{c}}, ~200/189 = 99.6723{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6709{{c}}
-{{Optimal ET sequence|legend=0| 17, 60ce, 77, 94, 171e, 265e, 436ee }}
+{{Optimal ET sequence|legend=0| 12, …, 289e, 301, 915 }}
-Badness (Smith): 0.063499
+Badness (Sintel): 2.72
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 352/351, 385/384, 729/728, 1331/1323
+Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000
-Mapping: {{mapping| 1 1 7 0 1 3 | 0 5 -40 24 21 6 }}
+Mapping: {{mapping| 1 2 -1 -5 -9 -11 | 0 -5 40 94 150 177 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/12 = 140.363{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0442{{c}}, ~200/189 = 99.6709{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6675{{c}}
-{{Optimal ET sequence|legend=0| 17, 60ce, 77, 94, 171e, 436ee }}
+{{Optimal ET sequence|legend=0| 12f, …, 289e, 301 }}
-Badness (Smith): 0.037886
+Badness (Sintel): 2.30
-== Quanharuk ==
+===== 17-limit =====
-[[Subgroup]]: 2.3.5.7
+Subgroup: 2.3.5.7.11.13.17
-[[Comma list]]: 16875/16807, 32805/32768
+Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750
-{{Mapping|legend=1| 1 0 15 12 | 0 5 -40 -29 }}
+Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 | 0 -5 40 94 150 177 -11 }}
-: mapping generators: ~2, ~56/45
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~56/45 = 380.355{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.1227{{c}}, ~200/189 = 99.6753{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6658{{c}}
-{{Optimal ET sequence|legend=1| 41, 142, 183, 224, 1303d, 1527cd, 1751cd, 1975cd }}
+{{Optimal ET sequence|legend=1| 12f, 289e, 301 }}
-[[Badness]] (Smith): 0.071950
+Badness (Sintel): 2.06
-=== 11-limit ===
+===== 19-limit =====
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 540/539, 1375/1372, 32805/32768
+Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700
-Mapping: {{mapping| 1 0 15 12 -7 | 0 5 -40 -29 33 }}
+Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 4 | 0 -5 40 94 150 177 -11 3 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~56/45 = 380.352{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0230{{c}}, ~200/189 = 99.6694{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6676{{c}}
-{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d, 1079d }}
+{{Optimal ET sequence|legend=0| 12f, 301 }}
-Badness (Smith): 0.031549
+Badness (Sintel): 2.24
-=== 13-limit ===
+==== Quintahelenoid ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 729/728, 1375/1372, 4096/4095
+Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436
-Mapping: {{mapping| 1 0 15 12 -7 -15 | 0 5 -40 -29 33 59 }}
+Mapping: {{mapping| 1 2 -1 -5 -9 14 | 0 -5 40 94 150 -124 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~56/45 = 380.351{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9919{{c}}, ~200/189 = 99.6712{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6718{{c}}
-{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}
+{{Optimal ET sequence|legend=0| 12, 301, 614, 915 }}
-Badness (Smith): 0.021392
+Badness (Sintel): 2.73
-== Quadrant ==
+===== 17-limit =====
-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.
+Subgroup: 2.3.5.7.11.13.17
-[[Subgroup]]: 2.3.5.7
+Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157
-[[Comma list]]: 32805/32768, 390625/388962
+Mapping: {{mapping| 1 2 -1 -5 -9 14 5 | 0 -5 40 94 150 -124 -11 }}
-{{Mapping|legend=1| 4 0 60 119 | 0 1 -8 -17 }}
+Optimal tunings:
-: mapping generators: ~25/21, ~3
+* WE: ~2 = 1200.0469{{c}}, ~18/17 = 99.6749{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6710{{c}}
-[[Optimal tuning]] ([[POTE]]): ~25/21 = 300.0000{{c}}, ~3/2 = 701.8234{{c}}
+{{Optimal ET sequence|legend=0| 12, 301 }}
-{{Optimal ET sequence|legend=1| 212, 224, 436, 660, 1096c }}
+Badness (Sintel): 2.44
-[[Badness]] (Smith): 0.110242
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-=== 11-limit ===
+Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137
-Subgroup: 2.3.5.7.11
-Comma list: 1375/1372, 6250/6237, 32805/32768
+Mapping: {{mapping| 1 2 -1 -5 -9 14 5 4 | 0 -5 40 94 150 -124 -11 3 }}
-Mapping: {{mapping| 4 0 60 119 185 | 0 1 -8 -17 -27 }}
+Optimal tunings:
+* WE: ~2 = 1199.9925{{c}}, ~18/17 = 99.6710{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6716{{c}}
-Optimal tuning (POTE): ~25/21 = 300.0000{{c}}, ~3/2 = 701.8176{{c}}
+{{Optimal ET sequence|legend=0| 12, 301 }}
-{{Optimal ET sequence|legend=0| 212, 224, 436, 660 }}
+Badness (Sintel): 2.41
-Badness (Smith): 0.045738
+== Sextilifourths ==
+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.
-=== 13-limit ===
+[[Subgroup]]: 2.3.5.7
-Subgroup: 2.3.5.7.11.13
-Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647
+[[Comma list]]: 32805/32768, 235298/234375
-Mapping: {{mapping| 4 0 60 119 185 224 | 0 1 -8 -17 -27 -33 }}
+{{Mapping|legend=1| 1 2 -1 -1 | 0 -6 48 55 }}
+: mapping generators: ~2, ~21/20
-Optimal tuning (POTE): ~25/21 = 300.0000{{c}}, ~3/2 = 701.8158{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0987{{c}}, ~21/20 = 83.0599{{c}}
+: [[error map]]: {{val| +0.099 -0.117 +0.462 -0.630 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0543{{c}}
+: error map: {{val| 0.000 -0.281 +0.295 -0.837 }}
-{{Optimal ET sequence|legend=0| 212, 224, 436, 660 }}
+{{Optimal ET sequence|legend=1| 29, 72cd, 101, 130, 289, 419 }}
-Badness (Smith): 0.027243
+[[Badness]] (Sintel): 2.75
-== Septant ==
+=== 11-limit ===
-The septant temperament ({{nowrap| 224 & 301 }}) has a period of 1/7 octave and tempers out the [[akjaysma]], {{monzo| 47 -7 -7 -7 }}.
+Subgroup: 2.3.5.7.11
-[[Subgroup]]: 2.3.5.7
+Comma list: 441/440, 4000/3993, 235298/234375
-[[Comma list]]: 32805/32768, 516560652/514714375
+Mapping: {{mapping| 1 2 -1 -1 0 | 0 -6 48 55 50 }}
-{{Mapping|legend=1| 7 0 105 -56 | 0 1 -8 7 }}
+Optimal tunings:
-: mapping generators: ~8575/7776, ~3
+* WE: ~2 = 1200.0424{{c}}, ~21/20 = 83.0520{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0497{{c}}
-[[Optimal tuning]] ([[POTE]]): ~8575/7776 = 171.429{{c}}, ~3/2 = 701.702{{c}}
+{{Optimal ET sequence|legend=0| 29, 72cde, 101e, 130, 289 }}
-{{Optimal ET sequence|legend=1| 77, 147, 224, 301, 525, 826, 1351 }}
+Badness (Sintel): 1.50
-[[Badness]] (Smith): 0.111142
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-=== 11-limit ===
+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:
+* WE: ~2 = 1200.1056{{c}}, ~21/20 = 83.0566{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0508{{c}}
+{{Optimal ET sequence|legend=0| 29, 72cdef, 101e, 130, 289 }}
+Badness (Sintel): 1.04
+== Septant ==
+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, 516560652/514714375
+{{Mapping|legend=1| 7 0 105 -56 | 0 1 -8 7 }}
+: mapping generators: ~8575/7776, ~3
+[[Optimal tuning]]s:
+* [[WE]]: ~8575/7776 = 171.4303{{c}}, ~3/2 = 701.7091{{c}}
+: [[error map]]: {{val| +0.012 -0.234 +0.096 +0.265 }}
+* [[CWE]]: ~8575/7776 = 171.4286{{c}}, ~3/2 = 701.7022{{c}}
+: error map: {{val| 0.000 -0.253 +0.069 +0.232 }}
+{{Optimal ET sequence|legend=1| 77, 147, 224, 301, 525, 826, 1351 }}
+[[Badness]] (Sintel): 2.81
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 2,550: / Line 2,568: @@
 Mapping: {{mapping| 7 0 105 -56 -120 | 0 1 -8 7 13 }}
-Optimal tuning (POTE): ~495/448 = 171.429{{c}}, ~3/2 = 701.719{{c}}
+Optimal tunings:
+* WE: ~495/448 = 171.4334{{c}}, ~3/2 = 701.7387{{c}}
+* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7198{{c}}
 {{Optimal ET sequence|legend=0| 77, 147, 224, 301, 525 }}
-Badness (Smith): 0.044122
+Badness (Sintel): 1.46
 === 13-limit ===
@@ Line 2,563: / Line 2,583: @@
 Mapping: {{mapping| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }}
-Optimal tuning (POTE): ~495/448 = 171.429{{c}}, ~3/2 = 701.724{{c}}
+Optimal tunings:
+* WE: ~495/448 = 171.4282{{c}}, ~3/2 = 701.7229{{c}}
+* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7242{{c}}
-{{Optimal ET sequence|legend=0| 77, 147, 224, 525 }}
+{{Optimal ET sequence|legend=0| 77, 147, 224, 525, 1274f }}
-Badness (Smith): 0.024706
+Badness (Sintel): 1.02
 == Octant ==
-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.
+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
@@ Line 2,579: / Line 2,601: @@
 : mapping generators: ~42875/39366, ~3
-[[Optimal tuning]] ([[POTE]]): ~42875/39366 = 150.000{{c}}, ~3/2 = 701.713{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~42875/39366 = 150.0048{{c}}, ~3/2 = 701.7356{{c}}
+: [[error map]]: {{val| +0.039 -0.181 +0.071 +0.127 }}
+* [[CWE]]: ~42875/39366 = 150.0000{{c}}, ~3/2 = 701.7134{{c}}
+: error map: {{val| 0.000 -0.242 -0.021 +0.022 }}
-{{Optimal ET sequence|legend=1| 24, 224, 472, 696, 1168 }}
+{{Optimal ET sequence|legend=1| 24, …, 224, 472, 696, 1168 }}
-[[Badness]] (Smith): 0.157186
+[[Badness]] (Sintel): 3.98
 === 11-limit ===
@@ Line 2,592: / Line 2,618: @@
 Mapping: {{mapping| 8 0 120 -117 15 | 0 1 -8 11 1 }}
-Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~3/2 = 701.713{{c}}
+Optimal tunings:
+* WE: ~12/11 = 150.0010{{c}}, ~3/2 = 701.7177{{c}}
+* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7131{{c}}
-{{Optimal ET sequence|legend=0| 24, 224, 472, 696, 1168 }}
+{{Optimal ET sequence|legend=0| 24, …, 224, 472, 696, 1168 }}
-Badness (Smith): 0.044778
+Badness (Sintel): 1.48
 === 13-limit ===
@@ Line 2,605: / Line 2,633: @@
 Mapping: {{mapping| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }}
-Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~3/2 = 701.725{{c}}
+Optimal tunings:
+* WE: ~12/11 = 149.9957{{c}}, ~3/2 = 701.7046{{c}}
+* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7247{{c}}
 {{Optimal ET sequence|legend=0| 24, 224, 472, 696 }}
-Badness (Smith): 0.030425
+Badness (Sintel): 1.26
 == Nonant ==
-The ''nonant'' temperament ({{nowrap| 36 & 135 }}) has a period of 1/9 octave and tempers out the [[septimal ennealimma]], {{monzo| -11 -9 0 9 }}.
+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
@@ Line 2,621: / Line 2,651: @@
 : mapping generators: ~2592/2401, ~3
-[[Optimal tuning]] ([[CTE]]): ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.7232{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2592/2401 = 133.3442{{c}}, ~3/2 = 701.8000{{c}}
+: [[error map]]: {{val| +0.098 -0.057 -0.027 -0.141 }}
+* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.7384{{c}}
+: error map: {{val| 0.000 -0.217 -0.221 -0.421 }}
-{{Optimal ET sequence|legend=1| 36, 99c, 135, 171 }}
+{{Optimal ET sequence|legend=1| 36, 99c, 135, 171, 2772bd, 2943bdd, …, 5166bccddd, 5337bccddd }}
-[[Badness]] (Smith): 0.069896
+[[Badness]] (Sintel): 1.77
 === 11-limit ===
@@ Line 2,634: / Line 2,668: @@
 Mapping: {{mapping| 9 0 135 11 131 | 0 1 -8 1 -7 }}
-Optimal tuning (CTE): ~242/225 = 133.3333{{c}}, ~3/2 = 701.8398{{c}}
+Optimal tunings:
+* WE: ~242/225 = 133.3308{{c}}, ~3/2 = 701.8205{{c}}
+* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.8351{{c}}
-{{Optimal ET sequence|legend=0| 36, 99c, 135, 171, 477ce, 648cee }}
+{{Optimal ET sequence|legend=0| 36, 135, 171 }}
-Badness (Smith): 0.126910
+Badness (Sintel): 4.20
 === 13-limit ===
@@ Line 2,647: / Line 2,683: @@
 Mapping: {{mapping| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }}
-Optimal tuning (CTE): ~242/225 = 133.3333{{c}}, ~3/2 = 701.7998{{c}}
+Optimal tunings:
+* WE: ~242/225 = 133.3180{{c}}, ~3/2 = 701.6956{{c}}
+* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.7800{{c}}
 {{Optimal ET sequence|legend=0| 36, 99cf, 135, 171 }}
-Badness (Smith): 0.076195
+Badness (Sintel): 3.15
-== Tridecafifths ==
+== Septiquarschis ==
-Tridecafifths divides the perfect 3/2 into 13 quartertones.
+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
-[[Comma list]]: 32805/32768, {{monzo| -14 -1 -9 13 }}
+[[Comma list]]: 32805/32768, 829440/823543
-{{Mapping|legend=1| 1 1 7 6 | 0 13 -104 -71 }}
+{{Mapping|legend=1| 1 -4 47 6 | 0 7 56 -4 }}
-: mapping generators: ~2, ~1323/1280
+: mapping generators: ~2, ~256/147
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~1323/1280 = 53.9741{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.8855{{c}}, ~256/147 = 957.2944{{c}}
+: [[error map]]: {{val| -0.114 -0.436 -0.182 +1.310 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~256/147 = 957.3867{{c}}
+: error map: {{val| 0.000 -0.248 +0.032 +1.627 }}
-{{Optimal ET sequence|legend=1| 89, 200, 289 }}
+{{Optimal ET sequence|legend=1| 89, 94, 183, 460d, 643d }}
-[[Badness]] (Smith): 0.432580
+[[Badness]] (Sintel): 4.73
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 441/440, 32805/32768, 55296000/55240493
+Comma list: 540/539, 15488/15435, 32805/32768
+Mapping: {{mapping| 1 -4 47 6 25 | 0 7 56 -4 -27 }}
+Optimal tunings:
+* WE: ~2 = 1199.9430{{c}}, ~256/147 = 957.3390{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3849{{c}}
+{{Optimal ET sequence|legend=0| 89, 94, 183, 460d }}
-Mapping: {{mapping| 1 1 7 6 4 | 0 13 -104 -71 -12 }}
+Badness (Sintel): 1.72
-Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~33/32 = 53.9744{{c}}
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-{{Optimal ET sequence|legend=0| 89, 200, 289 }}
+Comma list: 540/539, 729/728, 1573/1568, 4096/4095
-Badness (Smith): 0.127820
+Mapping: {{mapping| 1 -4 47 6 25 -33 | 0 7 56 -4 -27 46 }}
+Optimal tunings:
+* WE: ~2 = 1200.0058{{c}}, ~256/147 = 957.3946{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3900{{c}}
+{{Optimal ET sequence|legend=0| 89, 94, 183, 277, 460d }}
+Badness (Sintel): 1.46
 == Subgroup extensions ==
+=== Tridecaschismic (2.3.5.13) ===
+Proposed by [[Eufalesio]] in 2026, tridecaschismic adds the [[325/324|marveltwin comma]] to the comma list, or equivalently, the [[tridecapyth comma]]. It benefits from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is one of the lowest badness schismic extensions. It is also equivalent to the 2.3.5.13 [[restriction]] of 13-limit [[cassandra]].
+Subgroup: 2.3.5.13
+Comma list: 325/324, 32805/32768
+Subgroup-val mapping: {{mapping| 1 0 15 -28 | 0 1 -8 20 }}
+Optimal tunings:
+* WE: ~2 = 1200.3326{{c}} ~3/2 = 702.1092{{c}}
+* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9189{{c}}
+{{Optimal ET sequence|legend=0| 12, …, 41, 53, 412cf, 465cf, …, 783ccff, 836ccfff }}
+Badness (Sintel): 0.582
+==== 2.3.5.13.19 subgroup ====
+Subgroup: 2.3.5.13.19
+Comma list: 325/324, 361/360, 513/512
+Subgroup-val mapping: {{mapping| 1 0 15 -28 9 | 0 1 -8 20 -3 }}
+Optimal tunings:
+* WE: ~2 = 1200.4236{{c}}, ~3/2 = 702.1510{{c}}
+* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9064{{c}}
+{{Optimal ET sequence|legend=0| 12, …, 41, 53 }}
+Badness (Sintel): 0.354
 === Photia (2.3.5.17) ===
 {{See also| No-elevens subgroup temperaments #Garibaldia }}
@@ Line 2,695: / Line 2,787: @@
 : mapping generators: ~2, ~3
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.491{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.5471{{c}}, ~3/2 = 701.2262{{c}}
+: [[error map]]: {{val| -0.453 -1.182 +0.706 +3.628 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4976{{c}}
+: error map: {{val| 0.000 -0.457 +1.705 +5.528 }}
-{{Optimal ET sequence|legend=1| 12, 41, 53, 65 }}
+{{Optimal ET sequence|legend=1| 12, 41, 53, 65, 207g, 272gg }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.4842 cents
+[[Badness]] (Sintel): 0.479
 ==== 2.3.5.17.19 subgroup ====
@@ Line 2,710: / Line 2,806: @@
 Gencom mapping: {{mapping| 1 0 15 0 0 0 -7 9 | 0 1 -8 0 0 0 7 -3 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.470{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.7225{{c}}, ~3/2 = 701.3077{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.4754{{c}}
-{{Optimal ET sequence|legend=0| 12, 41, 53, 65 }}
+{{Optimal ET sequence|legend=0| 12, 41, 53, 65, 142g }}
-RMS error: 0.5374 cents
+Badness (Sintel): 0.332
 === Nestoria (2.3.5.19) ===
 : ''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 2,730: / Line 2,828: @@
 : mapping generators: ~2, ~3
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.746{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.2250{{c}}, ~3/2 = 701.8776{{c}}
+: [[error map]]: {{val| +0.225 +0.148 +0.240 -1.796 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7307{{c}}
+: error map: {{val| 0.000 -0.224 -0.159 -2.705 }}
-{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171 }}
+{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171, 460hh, 631hh }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.1763 cents
+[[Badness]] (Sintel): 0.126
 === Taylor (2.3.5.13) ===
@@ Line 2,748: / Line 2,850: @@
 : mapping generators: ~2, ~26/15
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~26/15 = 950.855{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1497{{c}}, ~26/15 = 950.9740{{c}}
+: [[error map]]: {{val| +0.150 -0.007 +0.348 -1.094 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 950.8493{{c}}
+: error map: {{val| 0.000 -0.256 +0.098 -1.568 }}
-{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236 }}
+{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236, 525f, 761ff }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.1485 cents
+[[Badness]] (Sintel): 0.334
 ==== Dakota (2.3.5.13.19) ====
@@ Line 2,761: / Line 2,867: @@
 Subgroup-val mapping: {{mapping| 1 0 15 14 9 | 0 2 -16 -13 -6 }}
-Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~26/15 = 950.8199{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.2611{{c}}, ~26/15 = 951.0703{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8532{{c}}
 {{Optimal ET sequence|legend=0| 24, 29, 53, 130, 183, 236h, 289h }}
-Badness (Smith): 0.00575
+Badness (Sintel): 0.262
 ===== 2.3.5.13.19.37 subgroup =====
@@ Line 2,774: / Line 2,882: @@
 Subgroup-val mapping: {{mapping| 1 0 15 14 9 6 | 0 2 -16 -13 -6 -1 }}
-Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~26/15 = 950.8187{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.2987{{c}}, ~26/15 = 951.1060{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8595{{c}}
 {{Optimal ET sequence|legend=0| 24, 29, 53, 183, 236h, 289hl, 631fhhll }}
-Badness (Smith): 0.00357
+Badness (Sintel): 0.223
 === Quintilischis (2.3.5.17) ===
@@ Line 2,792: / Line 2,902: @@
 : mapping generators: ~2, ~18/17
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~18/17 = 99.649{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1370{{c}}, ~18/17 = 99.6602{{c}}
+: [[error map]]: {{val| +0.137 +0.018 -0.042 -0.533 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~18/17 = 99.6499{{c}}
+: error map: {{val| 0.000 -0.205 -0.317 -1.104 }}
-{{Optimal ET sequence|legend=1| 12, 253, 265, 277, 289 }}
+{{Optimal ET sequence|legend=1| 12, …, 253, 265, 277, 289, 566g, 855g }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.0719 cents
+[[Badness]] (Sintel): 1.34
 ==== 2.3.5.17.19 subgroup ====
@@ Line 2,807: / Line 2,921: @@
 Gencom mapping: {{mapping| 1 2 -1 0 0 0 5 4 | 0 -5 40 0 0 0 -11 3 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.652{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0350{{c}}, ~18/17 = 99.6550{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6520{{c}}
-{{Optimal ET sequence|legend=0| 12, 253, 265, 277, 289 }}
+{{Optimal ET sequence|legend=0| 12, …, 253, 265, 277, 289 }}
-RMS error: 0.1636 cents
+Badness (Sintel): 1.17
+[[Category:Schismatic family| ]] <!-- main article -->
 [[Category:Temperament families]]
-[[Category:Schismatic family| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]