Schismatic family: Difference between revisions

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

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

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

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

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

~~<!-- * [[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]]}.

[[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}}

~~~~

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Badness (Sintel): 1.18

=== ~~Hemigari~~ ===

=== Karadeniz ===

{{See also| Turkish maqam music temperaments #Karadeniz temperament }}

Subgroup: 2.3.5.7.11

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

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

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

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

: mapping generators: ~2, ~~~110~~/63

: mapping generators: ~2, ~11/9

Optimal tunings:

* WE: ~2 = ~~1200~~.~~7303~~{{c}}, ~~~110~~/63 = ~~951~~.~~6605~~{{c}}

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

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

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

~~~~

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

Badness (Sintel): 1.68

Badness (Sintel): 1.37

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal tunings:

* WE: ~2 = ~~1200~~.~~8146~~{{c}}, ~26/15 = ~~951~~.~~7273~~{{c}}

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

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

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

~~~~

Badness (Sintel): 1.13

Badness (Sintel): 1.34

~~=== Karadeniz ===~~

~~{{See also| Turkish maqam music temperaments #Karadeniz temperament }}~~

=== Hemigari ===

Subgroup: 2.3.5.7.11

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

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

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

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

: mapping generators: ~2, ~11/9

: mapping generators: ~2, ~110/63

Optimal tunings:

* WE: ~2 = ~~1199~~.~~7351~~{{c}}, ~11/9 = ~~350~~.~~9167~~{{c}}

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

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

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

~~~~

{{Optimal ET sequence|legend=0| 24d, 41, ~~65d~~, ~~106~~, ~~147~~ }}

Badness (Sintel): 1.37

Badness (Sintel): 1.68

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal tunings:

* WE: ~2 = ~~1199~~.~~3042~~{{c}}, ~11/9 = ~~350~~.~~7533~~{{c}}

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

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

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

~~~~

Badness (Sintel): 1.34

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 ==

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

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

[[Subgroup]]: 2.3.5.7

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

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

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

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

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

~~~~

[[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}}

~~~~

Line 700:

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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]]:

Line 1,029:

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

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

~~~~

Line 1,045:

Line 981:

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

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

~~~~

Line 1,061:

Line 996:

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

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

~~~~

Line 1,077:

Line 1,011:

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

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

~~~~

Line 1,093:

Line 1,026:

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

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

~~~~

Line 1,109:

Line 1,041:

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

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

~~~~

Line 1,125:

Line 1,056:

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

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

~~~~

Line 1,141:

Line 1,071:

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

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

~~~~

Line 1,147:

Line 1,076:

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

~~[[Optimal tuning]] ([[CTE]]): ~567/400 = 600.0000~~{{~~c}}, ~3/2~~ = ~~701.5899{{c~~}}

[[~~Minimax~~ tuning]]:

[[Optimal tuning]]s:

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

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

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

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

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

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

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

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

[[Badness]] (~~Smith~~): 0.~~054744~~

[[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 ~~tuning (CTE)~~: ~99/70 = ~~600~~.0000{{c}}, ~3/2 = 701.~~6077~~{{c}}

Optimal tunings:

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

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

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

Badness (~~Smith~~): 0.~~028160~~

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 ~~tuning (CTE)~~: ~99/70 = ~~600~~.0000{{c}}, ~3/2 = 701.~~5949~~{{c}}

Optimal tunings:

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

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

{{Optimal ET sequence|legend=0| 12, ~~106def~~, ~~118~~, ~~130, 248, 378~~ }}

Badness (~~Smith~~): 0.~~028722~~

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 tuning (CTE)~~: ~~~99/70 = 600~~.~~0000{{c}}, ~~~3~~/2 = 701~~.~~5959{{c}}~~

[[Subgroup]]: 2.3.5.7

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

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

~~Badness (Smith)~~: ~~0.029340~~

: 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 tuning~~ (~~CTE~~): ~~~55/~~39 ~~= 600.0000{{c}}, ~3/2 = 701.5708{{c}}~~

[[Badness]] (Sintel): 1.39

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

=== 11-limit ===

Subgroup: 2.3.5.7.11

~~Badness (Smith)~~: ~~0.029321~~

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 tuning~~ (~~CTE~~): ~~~55/39 = 600~~.~~0000{{c}}, ~3/2 = 701.5717{{c}}~~

Badness (Sintel): 0.931

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

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

~~Badness (Smith)~~: ~~0.026894~~

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]]~~ (~~[[POTE]]~~): ~~~1225/864 = 600~~.~~000{{c}}, ~35/24 = 650.920{{c}} (~36/35 = 50.920{{c}})~~

Badness (Sintel): 1.19

~~{{Optimal ET sequence|legend~~=~~1| 24, 70c, 94, 118, 212, 330, 542d, 872cd }}~~

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

Subgroup: 2.3.5.7.11.13.17

~~[[Badness]] (Smith)~~: ~~0.110583~~

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~~

Optimal tunings:

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

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

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

Optimal ~~tuning (POTE): ~99/70~~ = ~~600.000{{c}}~~, ~~~35/24 = 650.918{{c}} (~36/35 = 50.918{{c~~}})

~~{{Optimal ET sequence|legend=0| 24, 70c, 94, 118, 212, 330e, 542de }}~~

Badness (Sintel): 1.49

~~Badness (Smith): 0.036749~~

==== Bischis ====

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

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 ~~tuning (POTE)~~: ~99/70 = ~~600~~.~~000~~{{c}}, ~35/24 = ~~650~~.~~938~~{{c}} (~36/35 = 50.~~938~~{{c}})

Optimal tunings:

* 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| 24, ~~70c~~, 94, ~~118, 212f~~ }}

Badness (~~Smith~~): 0.~~037640~~

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 ~~tuning (POTE)~~: ~99/70 = 600.~~000~~{{c}}, ~35/24 = ~~650~~.~~942~~{{c}} (~36/35 = 50.~~942~~{{c}})

Optimal tunings:

* 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| 24, ~~70c~~, 94, ~~118~~, ~~212fg~~ }}

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

Badness (~~Smith~~): 0.~~025615~~

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 tuning (POTE): ~99/70 = 600.000~~{{c}}, ~35/24 ~~= 650.951{{c}} (~36/35 = 50.951{{c}})~~

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

{{~~Optimal ET sequence~~|~~legend~~=0~~| 24f, 70cf, 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 (Smith): 0.037607~~

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

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

[[Badness]] (Sintel): 2.80

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

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

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

~~Optimal tuning (POTE)~~: ~~~99/70 = 600.000~~{{c}}~~, ~35/24 = 650.948{{c}} (~36/35 = 50.948{{c}})~~

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

{{~~Optimal ET sequence|legend~~=~~0| 24f~~, ~~70cf, 94, 118f, 212g~~ }}

Optimal tunings:

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

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

~~Badness (Smith):~~ 0~~.024734~~

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

~~== Salsa ==~~

Badness (Sintel): 1.21

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

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

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

~~{{Mapping|legend=1| 1 1 7 -1 | 0 2 -16 13 }}~~

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

: ~~mapping generators: ~2~~, ~~~128~~/~~105~~

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

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

* [[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~~ }}

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

Optimal tunings:

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

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

~~[[Badness]] (Sintel): 2.03~~

=== 11-limit ===

Badness (Sintel): 1.56

Subgroup: 2.3.5.7.11

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

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]]~~ (~~[[POTE]]~~): ~2 = ~~1200~~.~~000~~{{c}}, ~~~140~~/81 = ~~950~~.~~797~~{{c}}

Optimal tunings:

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

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

[[Badness]] (~~Smith~~): 0.~~045817~~

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 tuning (POTE): ~2 = 1200.000~~{{c}}, ~~~140~~/~~81 = 950.801{{c}}~~

: mapping generators: ~2, ~128/105

{{~~Optimal ET sequence~~|~~legend~~=0~~| 24e, 53, 130, 183, 313~~ }}

[[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 (Smith): 0.036289~~

~~=== 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 ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~26/15 = ~~950~~.~~801~~{{c}}

Optimal tunings:

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

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

{{Optimal ET sequence|legend=0| ~~24e~~, 53, ~~130~~, ~~183, 313~~ }}

Badness (~~Smith~~): 0.~~020816~~

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 ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~26/15 = ~~950~~.~~810~~{{c}}

Optimal tunings:

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

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

Badness (~~Smith~~): 0.~~021073~~

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 tuning (POTE)~~: ~~~2 = 1200.000{{c}}~~, ~~~26~~/~~15 = 950.809{{c}}~~

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

: mapping generators: ~2, ~140/81

~~Badness (Smith)~~: 0.~~018262~~

[[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 tuning (POTE)~~: ~~~2 = 1200.000{{c}}~~, ~~~26~~/~~15 = 950.807{{c}}~~

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

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

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

~~Badness (Smith)~~: 0.~~014819~~

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]] ([[POTE]])~~: ~2 = 1200.~~000~~{{c}}, ~~~160~~/~~147~~ = ~~166~~.~~140~~{{c}}

Optimal tunings:

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

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

[[Badness]] (~~Smith~~): 0.~~174705~~

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 ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~11/10 = ~~166~~.~~097~~{{c}}

Optimal tunings:

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

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

Badness (~~Smith~~): 0.~~068310~~

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 ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~11/10 = ~~166~~.~~054~~{{c}}

Optimal tunings:

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

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

{{Optimal ET sequence|legend=0| 29, 36, ~~65f~~, ~~94df, 159df~~ }}

Badness (~~Smith~~): 0.~~043750~~

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]] ([[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~~]] ~~(Smith)~~: 0.~~211859~~

; 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 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.

~~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 tuning (POTE): ~2 = 1200.000~~{{c}}, ~~~11/10 = 166.017{{c}}~~

: mapping generators: ~63/50, ~3

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

[[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 }}

~~Badness~~ (~~Smith~~): 0.~~061336~~

[[Minimax tuning]]:

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

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

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

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

~~Subgroup: 2.3.5.7.11.13~~

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

[[Badness]] (Sintel): 0.505

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

=== Terminal ===

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.

~~Optimal tuning (POTE)~~: ~2 ~~= 1200~~.~~000{{c}}, ~~~11~~/10 = 166.016{{c}}~~

Subgroup: 2.3.5.7.11

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

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

~~Badness (Smith)~~: 0~~.036700~~

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

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

Optimal tunings:

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

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

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

~~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 }}~~

Badness (Sintel): 1.97

~~Optimal tuning (POTE)~~: ~2 ~~= 1200~~.~~000{{c}}, ~~~11~~/10 = 166~~.~~012{{c}}~~

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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

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

~~Badness (Smith)~~: 0~~.026504~~

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

=~~= Countertertiaschis ==~~

Optimal tunings:

~~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~~.

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

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

~~[[Subgroup]]: 2.3.5.7~~

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

Badness (Sintel): 1.53

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

==== 17-limit ====

Subgroup: 2.3.5.7.11.13.17

~~[[Optimal tuning]] ([[POTE]])~~: ~~~2 = 1200.000{{c}}~~, ~~~625~~/~~567 = 166.0621{{c}}~~

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

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

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

~~[[Badness]] (Smith)~~: 0.~~188043~~

Optimal tunings:

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

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

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

~~Subgroup: 2.3.5.7.11~~

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

Badness (Sintel): 1.38

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

=== Terminator ===

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

~~Optimal tuning (POTE)~~: ~2 ~~= 1200~~.~~000{{c}}, ~~~11~~/10 = 166.0628{{c}}~~

Subgroup: 2.3.5.7.11

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

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

~~Badness (Smith)~~: 0~~.048943~~

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

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

Optimal tunings:

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

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

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

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

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

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

Badness (Sintel): 2.21

~~Optimal tuning (POTE)~~: ~2 ~~= 1200~~.~~000{{c}}, ~~~11~~/10 = 166~~.~~0628{{c}}~~

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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

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

~~Badness (Smith)~~: 0~~.024506~~

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

~~== Term ==~~

Optimal tunings:

~~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~~.

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

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

~~[[Subgroup]]: 2.3.5.7~~

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

Badness (Sintel): 1.47

~~{{Mapping|legend~~=~~1| 3 0 45 94 | 0 1~~ -~~8 -18 }}~~

==== 17-limit ====

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

Subgroup: 2.3.5.7.11.13.17

~~[[Optimal tuning]] ([[POTE]])~~: ~~~63~~/~~50 = 400.000{{c}}~~, ~3/~~2 = 701.742{{c}}~~

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

~~[[Minimax tuning]]~~:

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

* [[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~~, ~~147d, 159, 171, 867, 1038, 1209, 1380, 1551, 1722~~ }}

Optimal tunings:

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

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

~~[[Badness]] (Smith):~~ 0~~.019950~~

=== ~~Terminal~~ ===

Badness (Sintel): 1.04

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.

=== 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.

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 ~~tuning (POTE)~~: ~44/35 = ~~400~~.~~000~~{{c}}, ~3/2 = 701.~~824~~{{c}}

Optimal tunings:

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

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

{{Optimal ET sequence|legend=0| 12, ~~147de~~, ~~159~~, ~~330~~ }}

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

Badness (~~Smith~~): 0.~~059502~~

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 ~~tuning (POTE)~~: ~44/35 = ~~400~~.~~000~~{{c}}, ~3/2 = 701.~~821~~{{c}}

Optimal tunings:

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

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

{{Optimal ET sequence|legend=0| 12f, ~~147def~~, ~~159~~, ~~330~~ }}

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

~~Badness (Smith)~~: ~~0.037082~~

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

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

Badness (Sintel): 1.85

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

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

=== Hemiterm ===

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

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

Subgroup: 2.3.5.7.11

~~Optimal tuning (POTE)~~: ~~~34~~/~~27 = 400.000{{c}}~~, ~3/~~2 = 701.810{{c}}~~

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

{{~~Optimal ET sequence~~|~~legend=~~0| ~~12f, 147def, 159, 171, 330~~ }}

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

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

~~Badness~~ (~~Smith~~): 0.~~027073~~

Optimal tunings:

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

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

=~~== Terminator ===~~

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

~~Subgroup: 2.3.5.7.11~~

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

Badness (Sintel): 0.684

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

==== 13-limit ====

~~Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~3/2 = 701.685{{c}}~~

~~{{Optimal ET sequence|legend=0| 12e, 159e, 171, 183, 354, 537, 891de }}~~

~~Badness (Smith): 0.066968~~

==== 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 ~~tuning~~ (~~POTE~~): ~63/50 = 400.~~000~~{{c}}, ~3/2 = ~~701~~.~~689~~{{c}}

Optimal tunings:

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

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

{{Optimal ET sequence|legend=0| ~~171~~, 183, ~~354, 891de, 1245dee, 1599ddee~~ }}

Badness (~~Smith~~): 0.~~035487~~

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 ~~tuning~~ (~~POTE~~): ~63/50 = 400.~~000~~{{c}}, ~3/2 = ~~701~~.~~688~~{{c}}

Optimal tunings:

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

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

{{Optimal ET sequence|legend=0| ~~171~~, 183, ~~354, 891de, 1245dee~~, ~~1599ddee~~ }}

Badness (~~Smith~~): 0.~~020434~~

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 tuning ~~(POTE)~~: ~55/49 = ~~200~~.0000{{c}}, ~3/2 = ~~701~~.~~7460~~{{c}}

[[Optimal tuning]]s:

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

Badness (~~Smith~~): 0.~~029438~~

[[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 ~~tuning (POTE)~~: ~55/49 = ~~200~~.0000{{c}}, ~3/2 = ~~701~~.~~7256~~{{c}}

Optimal tunings:

* WE: ~44/35 = 400.1156{{c}}, ~121/70 = 951.2377{{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 (Smith)~~: 0.~~044657~~

=== 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~~ }}

Optimal tunings:

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

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

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

Optimal ~~tuning (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 }}~~

Badness (Sintel): 2.27

~~Badness~~ (~~Smith~~)~~: 0~~.~~020687~~

== Squirrel ==

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.

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

[[Subgroup]]: 2.3.5.7

Subgroup: 2.3.5.7~~.11.13~~

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

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

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

Optimal tuning ~~(POTE)~~: ~~~63/50~~ = ~~400~~.~~000~~{{c}}, ~26/15 = ~~950~~.~~873~~{{c}} (~12/11 = ~~150~~.~~873~~{{c}})

[[Optimal tuning]]s:

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

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

* [[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 (~~Smith~~): 0.~~031362~~

[[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 ~~tuning (POTE)~~: ~~~34/27~~ = ~~400~~.~~000~~{{c}}, ~26/15 = ~~950~~.~~867~~{{c}} (~12/11 = ~~150~~.~~867~~{{c}})

Optimal tunings:

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

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

{{Optimal ET sequence|legend=0| ~~24d~~, ~~159~~, ~~183, 342f, 525f, 867ff~~ }}

Badness (~~Smith~~): 0.~~022316~~

Badness (Sintel): 2.26

== ~~Altinex~~ ==

=== 13-limit ===

[[Subgroup]]: 2.3.5.7

Subgroup: 2.3.5.7.11.13

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

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

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

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

[[Optimal ~~tuning]] ([[CTE]])~~: ~~~1536~~/~~1225~~ = ~~400~~.~~000~~{{c}}, ~~~34300~~/~~19683~~ = ~~950~~.~~9654~~{{c}}

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=1| 24, …, ~~111c, 135, 159, 612ccdd, 771ccdd~~ }}

[[Badness]] (~~Smith~~): 0.~~422026~~

Badness (Sintel): 1.81

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

== Tertiaschis ==

~~Subgroup:~~ 2.3.5.7.11

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.

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

[[Subgroup]]: 2.3.5.7

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

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

~~Optimal tuning (CTE): ~44/35 = 400.000~~{{~~c}}, ~121/70~~ = ~~950.9658{{c~~}}

{{~~Optimal ET sequence~~|~~legend~~=0~~| 24, …, 111c, 135, 159, 612ccdd, 771ccdd~~ }}

[[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 }}

~~Badness (Smith): 0.101224~~

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

[[Badness]] (Sintel): 5.36

~~Subgroup~~: ~~2.3.~~5.~~7.11.13~~

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

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

~~Optimal tuning (CTE)~~: ~~~44/35 = 400.000~~{{~~c}}, ~26/15 = 950.9360{{c~~}}

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

{{~~Optimal ET sequence|legend~~=~~0| 24~~, ~~…, 111cf, 135f, 159~~ }}

Optimal tunings:

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

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

~~Badness (Smith):~~ 0~~.054894~~

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

~~== Sesquiquartififths ==~~

Badness (Sintel): 2.07

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

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

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

~~{{Mapping|legend=1| 1 1 7 5 | 0 4 -32 -15 }}~~

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

: ~~napping generators: ~2~~, ~~~448~~/~~405~~

~~[[Optimal tuning]] ([[POTE]])~~: ~~~2 = 1200.000~~{{~~c}}, ~448/405 = 175.434{{c~~}}

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

~~[[Minimax tuning]]~~:

Optimal tunings:

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

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

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

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

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

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

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

Badness (Sintel): 1.52

=== ~~Sesquart~~ ===

=== 17-limit ===

Subgroup: 2.3.5.7.11

Subgroup: 2.3.5.7.11.13.17

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

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

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

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

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

Optimal tunings:

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

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

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

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

Badness (Sintel): 1.35

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

== Countertertiaschis ==

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

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.

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

[[Subgroup]]: 2.3.5.7

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

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

~~Optimal tuning (POTE): ~2 = 1200.000~~{{~~c}}, ~72/65~~ = ~~175.409{{c~~}}

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

[[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 }}

~~Badness (Smith): 0.022396~~

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

[[Badness]] (Sintel): 4.76

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

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

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

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

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

Optimal tunings:

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

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

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

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

Badness (Sintel): 1.62

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

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

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

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

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

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~~.020466~~

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

Badness (Sintel): 1.01

~~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~~

== 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 -6 | 0 4 -32 -15 10 39 69 -12 72 }}~~

[[Subgroup]]: 2.3.5.7

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

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

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

: mapping generators: ~25/21, ~3

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

[[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 }}

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

{{Optimal ET sequence|legend=1| 12, …, 200, 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): 2.79

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

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal tunings:

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

* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8145{{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): 1.51

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

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

~~Optimal tuning (POTE)~~: ~~~2 = 1200.000{{c}}~~, ~~~21~~/~~19 = 175.380{{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.~~023059~~

Optimal tunings:

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

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

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

~~Subgroup: 2.3.5.7.11.13.17~~

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

Badness (Sintel): 1.13

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

== 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}}, ~72/65 = 175~~.~~377{{c}}~~

[[Subgroup]]: 2.3.5.7

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

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

~~Badness (Smith)~~: ~~0.030680~~

: mapping generators: ~2, ~448/405

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

[[Optimal tuning]]s:

~~Subgroup~~: 2.3.5.7.11.~~13.17~~.19

* [[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, 361/360, 364/363, 441/440, 456~~/~~455~~

[[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 6~~ | ~~0 4 -32 -15 10 39 -61 -12~~ }}

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

~~Optimal tuning~~ (~~POTE~~): ~~~2 = 1200~~.~~000{{c}}, ~21/19 = 175.377{{c}}~~

[[Badness]] (Sintel): 0.285

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

=== 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.~~022816~~

Subgroup: 2.3.5.7.11

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

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

~~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 | 0 4 -32 -15 10 }}

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

Optimal tunings:

* WE: ~2 = 1199.8171{{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 = 175.376{{c~~}}

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

Badness (Sintel): 0.969

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

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

~~=== Bisesqui ===~~

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

~~Subgroup~~: ~~2.3.5.7.11~~

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

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

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

Optimal tunings:

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

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

* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4095{{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): 0.925

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

===== Heartia =====

Subgroup: 2.3.5.7.11.13.17

~~== Quintilipyth ==~~

Comma list: 243/242, 256/255, 273/272, 364/363, 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 | 0 4 -32 -15 10 39 28 }}

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

Optimal tunings:

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

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

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

Badness (Sintel): 1.45

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

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

Subgroup: 2.3.5.7.11.13.17.19

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

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

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

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

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

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

Optimal tunings:

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

* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.3797{{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): 1.40

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

===== Sesquartia =====

Subgroup: 2.3.5.7.11.13.17

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

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

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

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

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

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

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 -9~~ | ~~0 -5 40 82 126 153~~ }}

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

Badness (Sintel): 1.18

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

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

Subgroup: 2.3.5.7.11.13.17.19

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

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

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

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

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

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

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 5~~ | ~~0 -5 40 82 126 153 -11~~ }}

~~Optimal tuning~~ (~~POTE~~): ~~~2 = 1200~~.~~000{{c}}, ~18/17 = 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)~~: ~~0.045992~~

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

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

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

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

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

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 4~~ | ~~0 -5 40 82 126 153 -11 3~~ }}

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

Badness (Sintel): 1.36

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

===== Hearty =====

Subgroup: 2.3.5.7.11.13.17

~~Badness (Smith)~~: ~~0.038155~~

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

~~== Quintaschis ==~~

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

~~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 = 1199.9458{{c}}, ~72/65 = 175.3689{{c}}

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

~~[[Comma list]]: 32805/32768~~, ~~49009212/48828125~~

~~{{Mapping|legend=~~1~~| 1 2 -1 -5 | 0 -5 40 94 }}~~

Badness (Sintel): 1.56

~~[[Optimal tuning]] ([[POTE]])~~: ~2 ~~= 1200~~.~~000{{c}}, ~200/189 = 99~~.~~664{{c}}~~

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

Subgroup: 2.3.5.7.11.13.17.19

~~{{Optimal ET sequence|legend=1| 12~~, …, ~~289~~, ~~301~~, ~~590~~, ~~891, 1192 }}~~

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

~~[[Badness]] (Smith)~~: 0~~.132890~~

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

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

Optimal tunings:

~~Subgroup~~: 2.~~3.5.7~~.11

* WE: ~2 = 1200.0114{{c}}, ~72/65 = 175.3783{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3765{{c}}

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

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

Badness (Sintel): 1.39

~~Optimal tuning (POTE)~~: ~2 ~~= 1200~~.~~000{{c}}, ~35/33 = 99~~.~~653{{c}}~~

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

Subgroup: 2.3.5.7.11.13.17.19.23

~~{{Optimal ET sequence|legend=0| 12~~, …, ~~277d~~, ~~289 }}~~

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

~~Badness (Smith)~~: 0~~.111477~~

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

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

Optimal tunings:

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

* WE: ~2 = 1200.0122{{c}}, ~72/65 = 175.3782{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3763{{c}}

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

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

Badness (Sintel): 1.37

~~Optimal tuning (POTE)~~: ~2 ~~= 1200~~.~~000{{c}}, ~35/33 = 99~~.~~658{{c}}~~

=== Bisesqui ===

Subgroup: 2.3.5.7.11

~~{{Optimal ET sequence|legend=0| 12f~~, …, ~~277dff, 289 }}~~

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

~~Badness (Smith)~~: 0~~.074218~~

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

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

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

Optimal tunings:

~~Subgroup~~: 2.~~3.5.7.11.13~~.17

* 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~~, ~~595/594~~, ~~3757/3750~~, ~~32805/32768~~

{{Optimal ET sequence|legend=1| 82e, 130, 212, 342, 1156, 1498, 1840d, 5862bbccdddee }}

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

Badness (Sintel): 0.561

~~Optimal tuning (POTE): ~2~~ = ~~1200.000{{c}}, ~18/17~~ = ~~99.656~~{{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.~~050571~~

[[Subgroup]]: 2.3.5.7

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

[[Comma list]]: 32805/32768, 420175/419904

~~Subgroup~~: ~~2.3.5.7.11.13.17.19~~

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

: mapping generators: ~2, ~243/224

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

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.1039{{c}}, ~243/224 = 140.3620{{c}}

~~Optimal tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~18/17 = 99.~~659~~{{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 }}

Badness (~~Smith~~): 0.~~042120~~

[[Badness]] (Sintel): 0.777

=== ~~Quintahelenic~~ ===

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~~~200~~/~~189~~ = 99.~~671~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.3103{{c}}, ~88/81 = 140.4011{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.3649{{c}}

{{Optimal ET sequence|legend=0| 12, …, ~~289e~~, ~~301~~, ~~915~~ }}

Badness (~~Smith~~): 0.~~082225~~

Badness (Sintel): 2.10

==== 13-limit ====

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~~~200~~/~~189~~ = 99.~~661~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.1840{{c}}, ~13/12 = 140.3840{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.3627{{c}}

Badness (Sintel): 1.57

~~Badness (Smith): 0~~.~~055570~~

== 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.

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

[[Subgroup]]: 2.3.5.7

Subgroup: 2.3.5.7~~.11.13.17~~

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

[[Comma list]]: 16875/16807, 32805/32768

~~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}}, ~18/17 = 99.~~665~~{{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.~~040412~~

[[Badness]] (Sintel): 1.82

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

=== 11-limit ===

Subgroup: 2.3.5.7.11~~.13.17.19~~

Subgroup: 2.3.5.7.11

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

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

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 | 0 5 -40 -29 33 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~18/17 = 99.~~668~~{{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.~~036840~~

Badness (Sintel): 1.04

===~~= Quintahelenoid =~~===

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~~~200~~/~~189~~ = 99.~~672~~{{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.~~066108~~

Badness (Sintel): 0.884

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

== Quintilipyth ==

~~Subgroup: 2.~~3~~.5.~~7.~~11.13~~.17

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

[[Subgroup]]: 2.3.5.7

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

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

~~Optimal tuning (POTE): ~2 = 1200.000~~{{c}}, ~18/~~17 = 99.671{{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): 0.047908~~

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

[[Badness]] (Sintel): 6.43

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

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

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

~~Optimal tuning (POTE)~~: ~~~2 = 1200.000~~{{~~c}}, ~18/17 = 99.672{{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~~.039542~~

~~== Sextilifourths ==~~

Badness (Sintel): 3.74

~~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

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

[[Comma list]]: ~~32805~~/~~32768~~, ~~235298~~/~~234375~~

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

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

~~: mapping generators: ~2, ~21/2~~

[[Optimal ~~tuning]] ([[POTE]])~~: ~2 = 1200.~~000~~{{c}}, ~21/20 = 83.~~053~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.1774{{c}}, ~35/33 = 99.6267{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6134{{c}}

{{Optimal ET sequence|legend=1| 29, ~~72cd~~, ~~101~~, ~~130, 289, 419~~ }}

[[Badness]] (~~Smith~~): 0.~~108794~~

Badness (Sintel): 2.86

=== 11-limit ===

=== 17-limit ===

Subgroup: 2.3.5.7.11

Subgroup: 2.3.5.7.11.13.17

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

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~21/20 = 83.~~049~~{{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=0| 29, ~~72cde~~, ~~101e, 130, 289~~ }}

Badness (~~Smith~~): 0.~~045457~~

Badness (Sintel): 2.34

=== 13-limit ===

=== 19-limit ===

Subgroup: 2.3.5.7.11.13

Subgroup: 2.3.5.7.11.13.17.19

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

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

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

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, ~~72cdef~~, ~~101e, 130, 289~~ }}

Badness (~~Smith~~): 0.~~025276~~

Badness (Sintel): 2.32

== ~~Septiquarschis~~ ==

== Quintaschis ==

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

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

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768, ~~829440~~/~~823543~~

[[Comma list]]: 32805/32768, 49009212/48828125

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~000~~{{c}}, ~~~147~~/~~128~~ = ~~242~~.~~614~~{{c}}

[[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 }}

{{Optimal ET sequence|legend=1| 89, 94, ~~183~~, ~~460d~~, ~~643d~~, ~~1103dd~~ }}

{{Optimal ET sequence|legend=1| 12, …, 289, 301, 590, 891, 1192 }}

[[Badness]] (~~Smith~~): 0.~~187047~~

[[Badness]] (Sintel): 3.36

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

Mapping: {{mapping| 1 3 -~~9 2~~ -2 | 0 -~~7 -56 4 27~~ }}

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~~~147~~/~~128~~ = ~~242~~.~~616~~{{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=0| 89, 94, ~~183~~, ~~460d, 643d, 826dd~~ }}

Badness (~~Smith~~): 0.~~052002~~

Badness (Sintel): 3.69

=== 13-limit ===

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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

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

Mapping: {{mapping| 1 3 -~~9 2~~ -~~2 13~~ | 0 -~~7 -56 4 27 -46~~ }}

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

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~~~147~~/~~128~~ = ~~242~~.~~610~~{{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~~, ~~277, 460d~~ }}

Badness (~~Smith~~): 0.~~035315~~

Badness (Sintel): 3.07

== ~~Tsaharuk~~ ==

==== 17-limit ====

~~{{Main| Tsaharuk }}~~

Subgroup: 2.3.5.7.11.13.17

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

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

~~[[Comma list]]~~: ~~32805/32768, 420175/419904~~

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

Optimal tunings:

~~: mapping generators~~: ~2, ~~~243~~/~~224~~

* WE: ~2 = 1200.1286{{c}}, ~18/17 = 99.6668{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6568{{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): 2.58

~~[[Badness]] (Smith)~~: 0.~~030697~~

==== 19-limit ====

Subgroup: 2.3.5.7.11.13.17.19

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

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

~~Subgroup~~: ~~2.3.5.7.11~~

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

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

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

Optimal tunings:

* 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}}~~, ~~~88/81 = 140.365{{c~~}}

~~{{Optimal ET sequence|legend=0| 17, 60ce, 77, 94, 171e, 265e, 436ee }}~~

Badness (Sintel): 2.56

~~Badness (Smith)~~: 0.~~063499~~

=== Quintahelenic ===

Subgroup: 2.3.5.7.11

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

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

~~Subgroup~~: ~~2.3.5.7.11.13~~

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

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

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

Optimal tunings:

* WE: ~2 = 1200.0195{{c}}, ~200/189 = 99.6723{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6709{{c}}

Optimal ~~tuning (POTE): ~2~~ = ~~1200.000{{c}}~~, ~~~13/12 = 140.363{{c~~}}

~~{{Optimal ET sequence|legend=0| 17, 60ce, 77, 94, 171e, 436ee }}~~

Badness (Sintel): 2.72

~~Badness (Smith)~~: 0.~~037886~~

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

~~== Quanharuk ==~~

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

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

~~[[Comma list]]~~: ~~16875/16807, 32805/32768~~

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

Optimal tunings:

~~: mapping generators~~: ~2, ~56/45

* WE: ~2 = 1200.0442{{c}}, ~200/189 = 99.6709{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6675{{c}}

[[Optimal ~~tuning]] ([[POTE]]): ~2~~ = ~~1200.000{{c}}~~, ~~~56/45 = 380.355{{c~~}}

~~{{Optimal ET sequence|legend=1| 41, 142, 183, 224, 1303d, 1527cd, 1751cd, 1975cd }}~~

Badness (Sintel): 2.30

~~[[Badness]] (Smith)~~: 0.~~071950~~

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

Subgroup: 2.3.5.7.11.13.17

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

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

~~Subgroup~~: ~~2.3.5.7.11~~

~~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 tunings:

* WE: ~2 = 1200.1227{{c}}, ~200/189 = 99.6753{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6658{{c}}

Optimal ~~tuning (POTE): ~2~~ = ~~1200.000{{c}}~~, ~~~56/45 = 380.352{{c~~}}

~~{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d, 1079d }}~~

Badness (Sintel): 2.06

~~Badness (Smith)~~: 0.~~031549~~

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

Subgroup: 2.3.5.7.11.13.17.19

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

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

~~Subgroup~~: ~~2.3.5.7.11.13~~

~~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 tunings:

* WE: ~2 = 1200.0230{{c}}, ~200/189 = 99.6694{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6676{{c}}

Optimal ~~tuning (POTE): ~2~~ = ~~1200.000{{c}}~~, ~~~56/45 = 380.351{{c~~}}

~~{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}~~

Badness (Sintel): 2.24

~~Badness (Smith)~~: 0.~~021392~~

==== Quintahelenoid ====

Subgroup: 2.3.5.7.11.13

~~== Quadrant ==~~

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

~~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

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

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

Optimal tunings:

* WE: ~2 = 1199.9919{{c}}, ~200/189 = 99.6712{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6718{{c}}

~~: mapping generators: ~25/21, ~3~~

~~[[Optimal tuning]]~~ (~~[[POTE]]~~): ~~~25/21 = 300.0000{{c}}, ~3/~~2 ~~= 701~~.~~8234{{c}}~~

Badness (Sintel): 2.73

~~{{Optimal ET sequence|legend~~=~~1| 212, 224, 436, 660, 1096c }}~~

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

Subgroup: 2.3.5.7.11.13.17

~~[[Badness]] (Smith)~~: ~~0.110242~~

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

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

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

~~Subgroup~~: 2~~.3.~~5~~.7.~~11

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

Optimal tunings:

* WE: ~2 = 1200.0469{{c}}, ~18/17 = 99.6749{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6710{{c}}

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

~~Optimal tuning~~ (~~POTE~~): ~~~25/21 = 300.0000{{c}}, ~3/~~2 ~~= 701~~.~~8176{{c}}~~

Badness (Sintel): 2.44

~~{{Optimal ET sequence|legend~~=~~0| 212, 224, 436, 660 }}~~

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

Subgroup: 2.3.5.7.11.13.17.19

~~Badness (Smith)~~: ~~0.045738~~

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

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

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

~~Subgroup~~: 2~~.3.~~5~~.7.~~11~~.13~~

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

Optimal tunings:

* WE: ~2 = 1199.9925{{c}}, ~18/17 = 99.6710{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6716{{c}}

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

~~Optimal tuning~~ (~~POTE~~): ~~~25/21 = 300.0000{{c}}, ~3/~~2 ~~= 701~~.~~8158{{c}}~~

Badness (Sintel): 2.41

~~{{Optimal ET sequence|legend=0| 212, 224, 436, 660 }}~~

== 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.

~~Badness (Smith): 0.027243~~

== ~~Septant~~ ==

~~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

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

[[Comma list]]: 32805/32768, 235298/234375

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

: mapping generators: ~2, ~21/20

[[Optimal tuning]] ([[~~POTE~~]]): ~~~8575~~/~~7776~~ = ~~171~~.~~429~~{{c}}, ~3/2 = ~~701~~.~~702~~{{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=1| 77, ~~147~~, ~~224~~, ~~301~~, ~~525~~, ~~826, 1351~~ }}

[[Badness]] (~~Smith~~): 0.~~111142~~

[[Badness]] (Sintel): 2.75

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal ~~tuning (POTE)~~: ~~~495~~/~~448~~ = ~~171~~.~~429~~{{c}}, ~3/2 = ~~701~~.~~719~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.0424{{c}}, ~21/20 = 83.0520{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0497{{c}}

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

Badness (~~Smith~~): 0.~~044122~~

Badness (Sintel): 1.50

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal ~~tuning (POTE)~~: ~~~495~~/~~448~~ = ~~171~~.~~429~~{{c}}, ~3/2 = ~~701~~.~~724~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.1056{{c}}, ~21/20 = 83.0566{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0508{{c}}

Badness (~~Smith~~): 0.~~024706~~

Badness (Sintel): 1.04

== ~~Octant~~ ==

== Septant ==

~~The octant temperament~~ ({{nowrap| 224 & ~~472~~ }}) has a period of 1/8 octave~~. In this temperament, 12/11, 35/27, and 99/70 are mapped into 1\8, 3\8~~, and ~~4\8 respectively~~.

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

[[Subgroup]]: 2.3.5.7

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

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

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

: mapping generators: ~8575/7776, ~3

[[Optimal tuning]] ([[~~POTE~~]]): ~~~42875~~/~~39366~~ = ~~150~~.~~000~~{{c}}, ~3/2 = 701.~~713~~{{c}}

[[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| 24, 224, ~~472~~, ~~696~~, ~~1168~~ }}

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

[[Badness]] (~~Smith~~): 0.~~157186~~

[[Badness]] (Sintel): 2.81

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal ~~tuning (POTE)~~: ~12/11 = ~~150~~.~~000~~{{c}}, ~3/2 = 701.~~713~~{{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| 24, 224, ~~472, 696~~, ~~1168~~ }}

Badness (~~Smith~~): 0.~~044778~~

Badness (Sintel): 1.46

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal ~~tuning (POTE)~~: ~12/11 = ~~150~~.~~000~~{{c}}, ~3/2 = 701.~~725~~{{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.~~030425~~

Badness (Sintel): 1.02

== ~~Nonant~~ ==

== Octant ==

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

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

[[Subgroup]]: 2.3.5.7

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

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

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

: mapping generators: ~42875/39366, ~3

[[Optimal tuning]] ([[~~CTE~~]]): ~~~2592~~/~~2401~~ = ~~133~~.~~3333~~{{c}}, ~3/2 = 701.~~7232~~{{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.~~069896~~

[[Badness]] (Sintel): 3.98

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal ~~tuning (CTE)~~: ~~~242~~/~~225~~ = ~~133~~.~~3333~~{{c}}, ~3/2 = 701.~~8398~~{{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| 36, ~~99c~~, ~~135~~, ~~171~~, ~~477ce~~, ~~648cee~~ }}

Badness (~~Smith~~): 0.~~126910~~

Badness (Sintel): 1.48

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal ~~tuning (CTE)~~: ~~~242~~/~~225~~ = ~~133~~.~~3333~~{{c}}, ~3/2 = 701.~~7998~~{{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| 36, ~~99cf~~, ~~135~~, ~~171~~ }}

Badness (~~Smith~~): 0.~~076195~~

Badness (Sintel): 1.26

== ~~Tridecafifths~~ ==

== Nonant ==

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

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

[[Subgroup]]: 2.3.5.7

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

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

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

: mapping generators: ~2592/2401, ~3

[[Optimal tuning]] ([[~~CTE~~]]): ~2 = ~~1200~~.~~0000~~{{c}}, ~~~1323~~/~~1280~~ = 53.~~9741~~{{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.~~432580~~

[[Badness]] (Sintel): 1.77

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal ~~tuning (CTE)~~: ~2 = ~~1200~~.~~0000~~{{c}}, ~33/32 = 53.~~9744~~{{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}}

Badness (~~Smith~~): 0.~~127820~~

Badness (Sintel): 4.20

== Subgroup extensions ==

=== 13-limit ===

=== Photia (2.3.5.17) ===

Subgroup: 2.3.5.7.11.13

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

[[Subgroup]]: 2.3.5.17

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

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

Optimal tunings:

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

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

: mapping generators: ~2, ~3

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~000~~{{c}}, ~3/2 = 701.~~491~~{{c}}

Badness (Sintel): 3.15

== Septiquarschis ==

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.

[[~~Tp tuning #T2 tuning|RMS error~~]]: 0.~~4842 cents~~

[[Subgroup]]: 2.3.5.7

==== 2.3.5.17.19 subgroup ====

Subgroup: 2.3.5.17.19

[[Comma list]]: 32805/32768, 829440/823543

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

: mapping generators: ~2, ~256/147

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

[[Optimal tuning]]s:

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

* [[WE]]: ~2 = 1199.8855{{c}}, ~256/147 = 957.2944{{c}}

: [[error map]]: {{val| -0.114 -0.436 -0.182 +1.310 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~3/2 = 701.~~470~~{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~256/147 = 957.3867{{c}}

: error map: {{val| 0.000 -0.248 +0.032 +1.627 }}

~~RMS error~~: 0.~~5374 cents~~

[[Badness]] (Sintel): 4.73

=== Nestoria (2.3.5.19) ===

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

=== 11-limit ===

Subgroup: 2.3.5.7.11

~~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.

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

[[Subgroup]]: 2.3.5.19

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}}

Badness (Sintel): 1.72

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

* WE: ~2 = 1200.0058{{c}}, ~256/147 = 957.3946{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3900{{c}}

Badness (Sintel): 1.46

== Tridecafifths ==

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

[[Subgroup]]: 2.3.5.7

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

: mapping generators: ~2, ~1323/1280

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.1431{{c}}, ~1323/1280 = 53.9838{{c}}

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

* [[CWE]]: ~2 = 1200.0000{{c}}, ~1323/1280 = 53.9764{{c}}

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

[[Badness]] (Sintel): 10.9

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

* WE: ~2 = 1200.0311{{c}}, ~33/32 = 53.9766{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 53.9750{{c}}

Badness (Sintel): 4.23

== Subgroup extensions ==

=== Maqamschismic (2.3.5.11) ===

Proposed by [[Eufalesio]] in 2026, maqamschismic is equivalent to the no-7 [[cassandra]]. The 2.3.5.11.13 subgroup adds [[352/351]] to the comma list and tempers 11/9~39/32 together (and 16/13~27/22), providing a very simple framework for tuning [[maqam]]at (especially the Turkish version), as outlined by [[Ozan Yarman]]. 41edo and 53edo are simplest, but 94edo is more optimized. It is only slightly worse than the no-7 [[helenus]].

Subgroup: 2.3.5.11

Comma list: 2200/2187, 4125/4096

Subgroup-val mapping: {{mapping| 1 0 15 -33 | 0 1 -8 23 }}

Optimal tunings:

* WE: ~2 = 1200.5458{{c}} ~3/2 = 702.4021{{c}}

* CWE: 2 = 1200.0000{{c}}, ~3/2 = 702.0906{{c}}

{{Optimal ET sequence|legend=0| 12e, …, 41, 53, 94, 147e, 241ce, 335ce }}

Badness (Sintel): 1.34

==== 2.3.5.11.13 subgroup ====

Subgroup: 2.3.5.11.13

Comma list: 325/324, 352/351, 4125/4096

Subgroup-val mapping: {{mapping| 1 0 15 -33 -28 | 0 1 -8 23 20 }}

Optimal tunings:

* WE: ~2 = 1200.4565{{c}} ~3/2 = 702.3057{{c}}

* CWE: 2 = 1200.0000{{c}}, ~3/2 = 702.0485{{c}}

Badness (Sintel): 0.862

=== 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) ===

[[Subgroup]]: 2.3.5.17

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

: mapping generators: ~2, ~3

[[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 }}

[[Badness]] (Sintel): 0.479

==== 2.3.5.17.19 subgroup ====

Subgroup: 2.3.5.17.19

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

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

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

Optimal tunings:

* WE: ~2 = 1199.7225{{c}}, ~3/2 = 701.3077{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.4754{{c}}

Badness (Sintel): 0.332

=== Nestoria (2.3.5.19) ===

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

Nestoria is notable for having one of the lowest-badness subgroup extensions of schismic. Note that despite prime [[19/1|19]] being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is generally not flatter than the fifth in optimal schismic due to its optimization considering intervals like [[19/10]] and [[19/15]]. However, the dyadic tuning sensitivity of [[19/16]] suggests using tunings like [[65edo]] and [[77edo]] to optimize in favour of prime 19, as [[53edo]] is already arguably undertempered for it despite being the flattest tuning appearing in the optimal ET sequence.

[[Subgroup]]: 2.3.5.19

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

: mapping generators: ~2, ~3

[[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 }}

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

{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171, 460hh, 631hh }}

~~{{Mapping|legend=2| 1 0 15 9 | 0 1 -8 -3 }}~~

[[Badness]] (Sintel): 0.126

~~{{Mapping|legend=3| 1 0 15 0 0 0 0 9 | 0 1 -8 0 0 0 0 -3 }}~~

~~: mapping generators: ~2, ~3~~

[[~~Optimal tuning~~]] (~~[[POTE]]~~)~~: ~2 = 1200.000{{c}}, ~3/2 = 701.746{{c}}~~

~~{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171 }}~~

~~[[Tp tuning #T2 tuning|RMS error]]~~: 0.~~1763 cents~~

=== Taylor (2.3.5.13) ===

Line 2,603:

Line 2,920:

: 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 2,616:

Line 2,937:

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 2,629:

Line 2,952:

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 2,647:

Line 2,972:

: 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 2,662:

Line 2,991:

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:Temperament families]]

[[Category:Schismatic family| ]]

[[Category:Rank 2]]

@@ Line 25: / Line 25: @@
 * [[WE]]: ~2 = 1200.0749{{c}}, ~3/2 = 701.7797{{c}}
 : [[error map]]: {{val| +0.075 -0.100 -0.027 }}
-* [[CWE]]: ~2 = 1200.0000{{c]}, ~3/2 = 701.7308{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7308{{c}}
 : error map: {{val| 0.000 -0.224 -0.160 }}
-<!-- * [[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 37: @@
 === Overview to extensions ===
-The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at.
+The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at. [[#Garibaldi|Garibaldi]] adds [[garischisma|{{monzo| 25 -14 0 -1 }}]], [[#Grackle|grackle]] adds {{monzo| -44 26 0 1 }}, [[#Pontiac|pontiac]] adds {{monzo| -59 39 0 -1 }}, and [[#Schism|schism]] adds [[64/63|{{monzo| 6 -2 0 -1 }}]]. Those all have a fifth as generator.
-* [[#Garibaldi|Garibaldi]] adds [[garischisma|{{monzo| 25 -14 0 -1 }}]],
-* [[#Grackle|Grackle]] adds {{monzo| -44 26 0 1 }},
-* [[#Schism|Schism]] adds [[64/63|{{monzo| 6 -2 0 -1 }}]],
-* [[#Pontiac|Pontiac]] adds {{monzo| -59 39 0 -1 }}.
-Those all have a fifth as generator.
-* [[#Bischismic|Bischismic]] adds {{monzo| -69 40 0 2 }} and has a fifth generator with a half-octave period.
+[[#Bischismic|Bischismic]] adds {{monzo| -69 40 0 2 }} and has a fifth generator with a half-octave period. [[#Salsa|Salsa]] adds [[parahemif comma|{{monzo| 15 -13 0 2 }}]] and has a hemififth generator. [[#Hemischis|Hemischis]] adds {{monzo| -34 25 0 -2 }} and has a hemitwelfth generator. [[Gamelismic clan #Guiron|Guiron]] adds [[1029/1024|{{monzo| -10 1 0 3 }}]], with an ~8/7 generator, three of which give the fifth. [[#Term|Term]] adds {{monzo| -94 54 0 3 }} with a 1/3-octave period. [[#Squirrel|Squirrel]], [[#Tertiaschis|tertiaschis]], and [[#Countertertiaschis|countertertiaschis]] each has a generator that is 1/3 of the fourth. [[#Quadrant|Quadrant]] adds {{monzo| -119 68 0 4 }} with a 1/4-octave period. [[#Kleischismic|Kleischismic]] adds {{monzo| 49 -38 0 4 }} with a half-octave period and also a bisect generator. [[#Sesquiquartififths|Sesquiquartififths]] adds {{monzo| -35 15 0 4 }} and slices the fifth in four.
-* [[#Hemischis|Hemischis]] adds {{monzo| -34 25 0 -2 }} and has a hemififth generator.
-* [[Gamelismic clan #Guiron|Guiron]] adds [[1029/1024|{{monzo| -10 1 0 3 }}]], with an ~8/7 generator, three of which give the fifth.
+Temperaments involving larger splits include [[#Tsaharuk|tsaharuk]], [[#Quanharuk|quanharuk]], [[#Quintilipyth|quintilipyth]], [[#Quintaschis|quintaschis]], [[#Altinex|altinex]], [[Stearnsmic clan #Pogo|pogo]], [[#Sextilifourths|sextilifourths]], [[#Septant|septant]], [[#Octant|octant]], [[#Nonant|nonant]], [[#Septiquarschis|septiquarschis]], and [[#Tridecafifths|tridecafifths]]. Those split the schismic structure into five to thirteen parts.
-* [[#Term|Term]] adds {{monzo| -94 54 0 3 }} with a 1/3 octave period.
-* [[#Sesquiquartififths|Sesquiquartififths]] adds {{monzo| -35 15 0 4 }} and slices the fifth in four.
 Temperaments discussed elsewhere include:
 * ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
 * ''[[Pogo]]'' (+118098/117649) → [[Stearnsmic clan #Pogo|Stearnsmic clan]]
+Considered below are garibaldi, pontiac, grackle, schism, bischismic, kleischismic, salsa, hemischis, term, altinex, squirrel, tertiaschis, countertertiaschis, quadrant, sesquiquartififths, tsaharuk, quanharuk, quintilipyth, quintaschis, sextilifourths, septant, octant, nonant, septiquarschis, and tridecafifths.
 The schismatic family boasts a variety of remarkable extensions to subgroups in high prime limits. These are listed at the bottom of this page, in [[#Subgroup extensions]].
@@ Line 61: / Line 54: @@
 {{Main| Garibaldi }}
-Garibaldi tempers out the [[garischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double diminished octave (C-Cbb), or down-minor seventh (C-vBb) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}.
+Garibaldi tempers out the [[garischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}.
 [[Subgroup]]: 2.3.5.7
@@ Line 72: / Line 65: @@
 * [[WE]]: ~2 = 1200.1233{{c}}, ~3/2 = 702.1573{{c}}
 : [[error map]]: {{val| +0.123 +0.326 -2.709 +2.328 }}
-* [[CWE]]: ~2 = 1200.0000{{c]}, ~3/2 = 702.0774{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0774{{c}}
 : error map: {{val| 0.000 +0.122 -2.933 +2.090 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.085{{c}} -->
 [[Minimax tuning]]:
@@ Line 93: / Line 85: @@
 === Cassandra ===
-Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup.
+Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup, even though it comes with a much higher complexity.
 Subgroup: 2.3.5.7.11
@@ Line 104: / Line 96: @@
 * 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 119: @@
 * 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 143: @@
 * 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 158: @@
 * 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 173: @@
 * 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 188: @@
 * 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 203: @@
 * 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 218: @@
 * 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 233: @@
 * 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 248: @@
 * 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 271: @@
 * 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 295: @@
 * 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 310: @@
 * 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 325: @@
 * 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 340: @@
 * 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 355: @@
 * 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 370: @@
 * 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 385: @@
 * 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 404: @@
 * 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 423: @@
 * 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 438: @@
 * 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 }}
@@ Line 472: / Line 443: @@
 Badness (Sintel): 1.18
-=== Hemigari ===
+=== Karadeniz ===
+{{See also| Turkish maqam music temperaments #Karadeniz temperament }}
 Subgroup: 2.3.5.7.11
-Comma list: 121/120, 225/224, 3125/3087
+Comma list: 225/224, 243/242, 3125/3087
-Mapping: {{mapping| 1 0 15 25 9 | 0 2 -16 -28 -7 }}
+Mapping: {{mapping| 1 1 7 11 2 | 0 2 -16 -28 5 }}
-: mapping generators: ~2, ~110/63
+: mapping generators: ~2, ~11/9
 Optimal tunings:
-* WE: ~2 = 1200.7303{{c}}, ~110/63 = 951.6605{{c}}
+* WE: ~2 = 1199.7351{{c}}, ~11/9 = 350.9167{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~110/63 = 951.0604{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.9995{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~110/63 = 951.082{{c}} -->
-{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135ee }}
+{{Optimal ET sequence|legend=0| 24d, 41, 65d, 106, 147 }}
-Badness (Sintel): 1.68
+Badness (Sintel): 1.37
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 121/120, 169/168, 225/224, 275/273
+Comma list: 225/224, 243/242, 325/324, 640/637
-Mapping: {{mapping| 1 0 15 25 9 14 | 0 2 -16 -28 -7 -13 }}
+Mapping: {{mapping| 1 1 7 11 2 -8 | 0 2 -16 -28 5 40 }}
 Optimal tunings:
-* WE: ~2 = 1200.8146{{c}}, ~26/15 = 951.7273{{c}}
+* WE: ~2 = 1199.3042{{c}}, ~11/9 = 350.7533{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.0574{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.9686{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~26/15 = 951.082{{c}} -->
-{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135eef }}
+{{Optimal ET sequence|legend=0| 24d, 41, 65d, 106f }}
-Badness (Sintel): 1.13
+Badness (Sintel): 1.34
-=== Karadeniz ===
-{{See also| Turkish maqam music temperaments #Karadeniz temperament }}
+=== Hemigari ===
 Subgroup: 2.3.5.7.11
-Comma list: 225/224, 243/242, 3125/3087
+Comma list: 121/120, 225/224, 3125/3087
-Mapping: {{mapping| 1 1 7 11 2 | 0 2 -16 -28 5 }}
+Mapping: {{mapping| 1 0 15 25 9 | 0 2 -16 -28 -7 }}
-: mapping generators: ~2, ~11/9
+: mapping generators: ~2, ~110/63
 Optimal tunings:
-* WE: ~2 = 1199.7351{{c}}, ~11/9 = 350.9167{{c}}
+* WE: ~2 = 1200.7303{{c}}, ~110/63 = 951.6605{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.9995{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~110/63 = 951.0604{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/9 = 350.994{{c}} -->
-{{Optimal ET sequence|legend=0| 24d, 41, 65d, 106, 147 }}
+{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135ee }}
-Badness (Sintel): 1.37
+Badness (Sintel): 1.68
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 225/224, 243/242, 325/324, 640/637
+Comma list: 121/120, 169/168, 225/224, 275/273
-Mapping: {{mapping| 1 1 7 11 2 -8 | 0 2 -16 -28 5 40 }}
+Mapping: {{mapping| 1 0 15 25 9 14 | 0 2 -16 -28 -7 -13 }}
 Optimal tunings:
-* WE: ~2 = 1199.3042{{c}}, ~11/9 = 350.7533{{c}}
+* WE: ~2 = 1200.8146{{c}}, ~26/15 = 951.7273{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.9686{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.0574{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/9 = 351.014{{c}} -->
-{{Optimal ET sequence|legend=0| 24d, 41, 65d, 106f }}
+{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135eef }}
-Badness (Sintel): 1.34
+Badness (Sintel): 1.13
 === Sanjaab ===
@@ Line 551: / Line 518: @@
 * 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 533: @@
 * 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 538: @@
 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 ==
 {{Main| Pontiac }}
-Pontiac tempers out the [[ragisma]], rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple augmented third (C-Exx#), or triple-up major sixth (C-^<sup>3</sup>A).
+Pontiac tempers out the [[ragisma]], rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple-augmented third (C-E𝄪𝄪♯), or triple-up major sixth (C-^<sup>3</sup>A).
 [[Subgroup]]: 2.3.5.7
@@ Line 592: / Line 552: @@
 * [[WE]]: ~2 = 1200.0989{{c}}, ~3/2 = 701.8145{{c}}
 : [[error map]]: {{val| +0.099 -0.042 -0.138 -0.038 }}
-* [[CWE]]: ~2 = 1200.0000{{c]}, ~3/2 = 701.7579{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7579{{c}}
 : error map: {{val| 0.000 -0.197 -0.377 -0.268 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.757{{c}} -->
 [[Minimax tuning]]:
@@ Line 613: / Line 572: @@
 === Helenoid ===
-The helenoid temperament ({{nowrap| 53 & 118 }}) is closely related to the helenus temperament, but with the ragisma rather than the [[225/224|marvel comma]] tempered out.
+Helenoid may be described as {{nowrap| 53 & 118 }}, and is closely related to the helenus temperament, differing only by the mapping of 7.
 Subgroup: 2.3.5.7.11
@@ Line 624: / Line 583: @@
 * 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 602: @@
 * 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 621: @@
 * 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 640: @@
 * 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 655: @@
 * 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 670: @@
 * 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 676: @@
 === Ponta ===
-The ponta temperament ({{nowrap| 53 & 171 }}) tempers out the [[540/539|swetisma]] and the ragisma.
+Ponta tempers out [[540/539]] and may be described as {{nowrap| 171 & 224 }}. [[224edo]] itself makes for an excellent tuning.
 Subgroup: 2.3.5.7.11
@@ Line 734: / Line 687: @@
 * 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 706: @@
 * 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 725: @@
 * 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 735: @@
 === Pontic ===
-The pontic temperament ({{nowrap| 118 & 171 }}) tempers out the [[441/440|werckisma]] and the ragisma.
+Pontic temperament tempers out [[441/440]] and may be described as {{nowrap| 118 & 171 }}. [[289edo]] may be recommended as a tuning.
 Subgroup: 2.3.5.7.11
@@ Line 796: / Line 746: @@
 * 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 765: @@
 * 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 784: @@
 * 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 803: @@
 * 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 818: @@
 * 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 824: @@
 === Bipont ===
-The bipont temperament ({{nowrap| 118 & 224 }}) has a period of half octave and tempers out the [[3025/3024|lehmerisma (3025/3024)]] and the [[9801/9800|kalisma (9801/9800)]].
+Bipont tempers out the [[3025/3024|lehmerisma (3025/3024)]] and the [[9801/9800|kalisma (9801/9800)]]. It may be described as {{nowrap| 118 & 224 }}. It has a period of half octave and a ploidacot signature of diploid monocot. [[342edo]] may be recommended as a tuning.
 Subgroup: 2.3.5.7.11
@@ Line 891: / Line 836: @@
 * 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 851: @@
 * 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 866: @@
 * 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 881: @@
 * 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 896: @@
 * 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 911: @@
 * 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 927: @@
 * 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 933: @@
 == Grackle ==
-Grackle tempers out {{monzo| -44 26 0 1 }}. The 7/4 is found at -26 fifths, represented by the triple diminished ninth (C-Dbbbb), or double-down minor seventh (C-vvBb), which is to say, two comma steps are required to bend the Pythagorean minor seventh to the septimal one.
+Grackle tempers out {{monzo| -44 26 0 1 }} so 7/4 is found at -26 fifths, represented by the triple-diminished ninth (C–D𝄫𝄫) or double-down minor seventh (C–vvB♭). Two comma steps are required to bend the Pythagorean minor seventh to the septimal one.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,007: / Line 945: @@
 * [[WE]]: ~2 = 1199.7974{{c}}, ~3/2 = 701.1210{{c}}
 : [[error map]]: {{val| -0.203 -1.037 +3.300 -1.618 }}
-* [[CWE]]: ~2 = 1200.0000{{c]}, ~3/2 = 701.2465{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.2465{{c}}
 : error map: {{val| 0.000 -0.709 +3.715 -1.234 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.239{{c}} -->
 [[Minimax tuning]]:
@@ Line 1,029: / Line 966: @@
 * 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 981: @@
 * 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 996: @@
 * 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 1,011: @@
 * 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 1,026: @@
 * 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 1,041: @@
 * 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 1,056: @@
 * 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 1,071: @@
 * 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 1,076: @@
 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 generators: ~567/400, ~3
-[[Optimal tuning]] ([[CTE]]): ~567/400 = 600.0000{{c}}, ~3/2 = 701.5899{{c}}
+{{Mapping|legend=1| 1 0 15 109 | 0 1 -8 -67 }}
-[[Minimax tuning]]:
+[[Optimal tuning]]s:
-* [[7-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.7/3
+* [[WE]]: ~2 = 1200.2569{{c}}, ~3/2 = 702.1149{{c}}
-* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
+: [[error map]]: {{val| +0.2569 +0.4168 -1.4342 +0.2685 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9615{{c}}
+: error map: {{val| 0.0000 +0.0065 -2.0054 -0.2437 }}
-{{Optimal ET sequence|legend=1| 12, 106d, 118, 130, 248, 378 }}
+{{Optimal ET sequence|legend=1| 53, 147d, 200, 253, 306c, 559c }}
-[[Badness]] (Smith): 0.054744
+[[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 tuning (CTE): ~99/70 = 600.0000{{c}}, ~3/2 = 701.6077{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.3283{{c}}, ~3/2 = 702.1636{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9713{{c}}
-{{Optimal ET sequence|legend=0| 12, 106de, 118, 130, 248 }}
+{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}
-Badness (Smith): 0.028160
+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 tuning (CTE): ~99/70 = 600.0000{{c}}, ~3/2 = 701.5949{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.3229{{c}}, ~3/2 = 702.1603{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9714{{c}}
-{{Optimal ET sequence|legend=0| 12, 106def, 118, 130, 248, 378 }}
+{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}
-Badness (Smith): 0.028722
+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 tuning (CTE): ~99/70 = 600.0000{{c}}, ~3/2 = 701.5959{{c}}
+[[Subgroup]]: 2.3.5.7
-{{Optimal ET sequence|legend=0| 12, 106def, 118, 130, 248g }}
+[[Comma list]]: 3136/3125, 32805/32768
-Badness (Smith): 0.029340
+{{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 tuning (CTE): ~55/39 = 600.0000{{c}}, ~3/2 = 701.5708{{c}}
+[[Badness]] (Sintel): 1.39
-{{Optimal ET sequence|legend=0| 12f, 106deff, 118f, 130 }}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Badness (Smith): 0.029321
+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 tuning (CTE): ~55/39 = 600.0000{{c}}, ~3/2 = 701.5717{{c}}
+Badness (Sintel): 0.931
-{{Optimal ET sequence|legend=0| 12f, 106deff, 118f, 130, 248fg }}
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Badness (Smith): 0.026894
+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]] ([[POTE]]): ~1225/864 = 600.000{{c}}, ~35/24 = 650.920{{c}} (~36/35 = 50.920{{c}})
+Badness (Sintel): 1.19
-{{Optimal ET sequence|legend=1| 24, 70c, 94, 118, 212, 330, 542d, 872cd }}
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-[[Badness]] (Smith): 0.110583
+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
+Optimal tunings:
+* WE: ~99/70 = 600.0331{{c}}, ~3/2 = 701.6387{{c}}
-Mapping: {{mapping| 2 1 22 -15 8 | 0 2 -16 19 -1 }}
+* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5994{{c}}
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~35/24 = 650.918{{c}} (~36/35 = 50.918{{c}})
+{{Optimal ET sequence|legend=0| 12, 118, 130, 248g }}
-{{Optimal ET sequence|legend=0| 24, 70c, 94, 118, 212, 330e, 542de }}
+Badness (Sintel): 1.49
-Badness (Smith): 0.036749
+==== Bischis ====
-==== 13-limit ====
 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 tuning (POTE): ~99/70 = 600.000{{c}}, ~35/24 = 650.938{{c}} (~36/35 = 50.938{{c}})
+Optimal tunings:
+* 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| 24, 70c, 94, 118, 212f }}
+{{Optimal ET sequence|legend=0| 12f, 106deff, 118f, 130 }}
-Badness (Smith): 0.037640
+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 tuning (POTE): ~99/70 = 600.000{{c}}, ~35/24 = 650.942{{c}} (~36/35 = 50.942{{c}})
+Optimal tunings:
+* 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| 24, 70c, 94, 118, 212fg }}
+{{Optimal ET sequence|legend=0| 12f, 106deff, 118f, 130, 248fg }}
-Badness (Smith): 0.025615
+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 tuning (POTE): ~99/70 = 600.000{{c}}, ~35/24 = 650.951{{c}} (~36/35 = 50.951{{c}})
+{{Mapping|legend=1| 2 1 22 -15 | 0 2 -16 19 }}
+: mapping generators: ~1225/864, ~35/24
-{{Optimal ET sequence|legend=0| 24f, 70cf, 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 (Smith): 0.037607
+{{Optimal ET sequence|legend=1| 24, 94, 118, 212, 330, 542d, 872cdd, 1414ccddd }}
-===== 17-limit =====
+[[Badness]] (Sintel): 2.80
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 289/288, 325/324, 385/384, 442/441, 14641/14580
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Mapping: {{mapping| 2 1 22 -15 8 -36 6 | 0 2 -16 19 -1 40 2 }}
+Comma list: 385/384, 9801/9800, 14641/14580
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~35/24 = 650.948{{c}} (~36/35 = 50.948{{c}})
+Mapping: {{mapping| 2 1 22 -15 8 | 0 2 -16 19 -1 }}
-{{Optimal ET sequence|legend=0| 24f, 70cf, 94, 118f, 212g }}
+Optimal tunings:
+* WE: ~99/70 = 600.1645{{c}}, ~35/24 = 651.0963{{c}} (~36/35 = 50.9319{{c}})
+* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9184{{c}} (~36/35 = 50.9184{{c}})
-Badness (Smith): 0.024734
+{{Optimal ET sequence|legend=0| 24, 94, 118, 212, 330e, 542dee, 872cddeee }}
-== Salsa ==
+Badness (Sintel): 1.21
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 245/243, 32805/32768
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-{{Mapping|legend=1| 1 1 7 -1 | 0 2 -16 13 }}
+Comma list: 352/351, 385/384, 729/728, 1575/1573
-: mapping generators: ~2, ~128/105
-[[Optimal tuning]]s:
+Mapping: {{mapping| 2 1 22 -15 8 15 | 0 2 -16 19 -1 -7 }}
-* [[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 }}
-{{Optimal ET sequence|legend=1| 17, 24, 41, 106d, 147d, 188cd }}
+Optimal tunings:
+* WE: ~99/70 = 600.0696{{c}}, ~35/24 = 651.0136{{c}} (~36/35 = 50.9440{{c}})
+* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9378{{c}} (~36/35 = 50.9378{{c}})
-[[Badness]] (Sintel): 2.03
+{{Optimal ET sequence|legend=0| 24, 94, 118, 212f }}
-=== 11-limit ===
+Badness (Sintel): 1.56
-Subgroup: 2.3.5.7.11
+===== 17-limit =====
+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]] ([[POTE]]): ~2 = 1200.000{{c}}, ~140/81 = 950.797{{c}}
+Optimal tunings:
+* WE: ~99/70 = 600.2190{{c}}, ~35/24 = 651.1578{{c}} (~36/35 = 50.9670{{c}})
+* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9518{{c}} (~36/35 = 50.9518{{c}})
-{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 313 }}
+{{Optimal ET sequence|legend=0| 24f, 94, 118f, 212g }}
-[[Badness]] (Smith): 0.045817
+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 tuning (POTE): ~2 = 1200.000{{c}}, ~140/81 = 950.801{{c}}
+{{Mapping|legend=1| 1 1 7 -1 | 0 2 -16 13 }}
+: mapping generators: ~2, ~128/105
-{{Optimal ET sequence|legend=0| 24e, 53, 130, 183, 313 }}
+[[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 (Smith): 0.036289
+{{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 tuning (POTE): ~2 = 1200.000{{c}}, ~26/15 = 950.801{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.3891{{c}}, ~11/9 = 351.1275{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0141{{c}}
-{{Optimal ET sequence|legend=0| 24e, 53, 130, 183, 313 }}
+{{Optimal ET sequence|legend=0| 17, 24, 41, 106d }}
-Badness (Smith): 0.020816
+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 tuning (POTE): ~2 = 1200.000{{c}}, ~26/15 = 950.810{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9362{{c}}, ~11/9 = 351.0061{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0247{{c}}
-{{Optimal ET sequence|legend=0| 53, 130, 183, 496d }}
+{{Optimal ET sequence|legend=0| 17, 24, 41 }}
-Badness (Smith): 0.021073
+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 tuning (POTE): ~2 = 1200.000{{c}}, ~26/15 = 950.809{{c}}
+[[Comma list]]: 6144/6125, 19683/19600
-{{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 (Smith): 0.018262
+[[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 tuning (POTE): ~2 = 1200.000{{c}}, ~26/15 = 950.807{{c}}
+Comma list: 540/539, 5632/5625, 8019/8000
-{{Optimal ET sequence|legend=0| 53, 130, 183, 313h }}
+Mapping: {{mapping| 1 0 15 -17 51 | 0 2 -16 25 -60 }}
-Badness (Smith): 0.014819
+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]] ([[POTE]]): ~2 = 1200.000{{c}}, ~160/147 = 166.140{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9140{{c}}, ~140/81 = 950.7324{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~140/81 = 950.8010{{c}}
-{{Optimal ET sequence|legend=1| 29, 36, 65 }}
+{{Optimal ET sequence|legend=0| 53, 130, 183, 313 }}
-[[Badness]] (Smith): 0.174705
+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 tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 166.097{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.9740{{c}}, ~26/15 = 950.7894{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8100{{c}}
-{{Optimal ET sequence|legend=0| 29, 36, 65 }}
+{{Optimal ET sequence|legend=0| 53, 130, 183, 496d }}
-Badness (Smith): 0.068310
+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 tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 166.054{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0464{{c}}, ~26/15 = 950.8459{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8091{{c}}
-{{Optimal ET sequence|legend=0| 29, 36, 65f, 94df, 159df }}
+{{Optimal ET sequence|legend=0| 53, 130, 183, 313h }}
-Badness (Smith): 0.043750
+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]] ([[POTE]]): ~2 = 1200.000{{c}}, ~192/175 = 166.019{{c}}
+{{Optimal ET sequence|legend=0| 53, 130, 183, 313h }}
-{{Optimal ET sequence|legend=1| 65, 94, 159, 253, 412cd }}
+Badness (Sintel): 1.06
-[[Badness]] (Smith): 0.211859
+; 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 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.
-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 tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 166.017{{c}}
+{{Mapping|legend=1| 3 0 45 94 | 0 1 -8 -18 }}
+: mapping generators: ~63/50, ~3
-{{Optimal ET sequence|legend=0| 65, 94, 159, 253, 412cd }}
+[[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 }}
-Badness (Smith): 0.061336
+[[Minimax tuning]]:
+* [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis)]]: 2.5/3
+* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
-=== 13-limit ===
+{{Optimal ET sequence|legend=1| 12, …, 159, 171, 867, 1038, 1209, 1380, 1551, 1722 }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 325/324, 385/384, 1575/1573, 10985/10976
+[[Badness]] (Sintel): 0.505
-Mapping: {{mapping| 1 2 -1 10 0 12 | 0 -3 24 -52 25 -60 }}
+=== Terminal ===
+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.
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 166.016{{c}}
+Subgroup: 2.3.5.7.11
-{{Optimal ET sequence|legend=0| 65f, 94, 159, 253, 412cdf, 665ccdef }}
+Comma list: 441/440, 4375/4356, 32805/32768
-Badness (Smith): 0.036700
+Mapping: {{mapping| 3 0 45 94 134 | 0 1 -8 -18 -26 }}
-=== 17-limit ===
+Optimal tunings:
-Subgroup: 2.3.5.7.11.13.17
+* WE: ~44/35 = 400.0464{{c}}, ~3/2 = 701.9053{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8178{{c}}
-Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
+{{Optimal ET sequence|legend=0| 12, …, 159, 330 }}
-Mapping: {{mapping| 1 2 -1 10 0 12 -2 | 0 -3 24 -52 25 -60 44 }}
+Badness (Sintel): 1.97
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 166.012{{c}}
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-{{Optimal ET sequence|legend=1| 65f, 94, 159, 253 }}
+Comma list: 364/363, 441/440, 625/624, 13720/13689
-Badness (Smith): 0.026504
+Mapping: {{mapping| 3 0 45 94 134 168 | 0 1 -8 -18 -26 -33 }}
-== Countertertiaschis ==
+Optimal tunings:
-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.
+* WE: ~44/35 = 400.0449{{c}}, ~3/2 = 701.8995{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8156{{c}}
-[[Subgroup]]: 2.3.5.7
+{{Optimal ET sequence|legend=0| 12f, …, 159, 330 }}
-[[Comma list]]: 32805/32768, 244140625/243045684
+Badness (Sintel): 1.53
-{{Mapping|legend=1| 1 2 -1 -12 | 0 -3 24 107 }}
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~625/567 = 166.0621{{c}}
+Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619
-{{Optimal ET sequence|legend=1| 65d, 159, 224, 383, 607 }}
+Mapping: {{mapping| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }}
-[[Badness]] (Smith): 0.188043
+Optimal tunings:
+* WE: ~34/27 = 400.0195{{c}}, ~3/2 = 701.8439{{c}}
+* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 701.8081{{c}}
-=== 11-limit ===
+{{Optimal ET sequence|legend=0| 12f, 159, 171, 330 }}
-Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 4000/3993, 32805/32768
+Badness (Sintel): 1.38
-Mapping: {{mapping| 1 2 -1 -12 0 | 0 -3 24 107 25 }}
+=== Terminator ===
+Terminator tempers out 540/539, and may be described as {{nowrap| 171 & 183 }}.
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 166.0628{{c}}
+Subgroup: 2.3.5.7.11
-{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}
+Comma list: 540/539, 32805/32768, 137781/137500
-Badness (Smith): 0.048943
+Mapping: {{mapping| 3 0 45 94 -137 | 0 1 -8 -18 31 }}
-=== 13-limit ===
+Optimal tunings:
-Subgroup: 2.3.5.7.11.13
+* WE: ~63/50 = 399.9677{{c}}, ~3/2 = 701.6278{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6846{{c}}
-Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976
+{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 537, 891de }}
-Mapping: {{mapping| 1 2 -1 -12 0 -10 | 0 -3 24 107 25 99 }}
+Badness (Sintel): 2.21
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 166.0628{{c}}
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}
+Comma list: 540/539, 729/728, 4096/4095, 31250/31213
-Badness (Smith): 0.024506
+Mapping: {{mapping| 3 0 45 94 -137 -103 | 0 1 -8 -18 31 24 }}
-== Term ==
+Optimal tunings:
-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.
+* WE: ~63/50 = 399.9731{{c}}, ~3/2 = 701.6414{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}}
-[[Subgroup]]: 2.3.5.7
+{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }}
-[[Comma list]]: 32805/32768, 250047/250000
+Badness (Sintel): 1.47
-{{Mapping|legend=1| 3 0 45 94 | 0 1 -8 -18 }}
+==== 17-limit ====
-: mapping generators: ~63/50, ~3
+Subgroup: 2.3.5.7.11.13.17
-[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~3/2 = 701.742{{c}}
+Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095
-[[Minimax tuning]]:
+Mapping: {{mapping| 3 0 45 94 -137 -103 -2 | 0 1 -8 -18 31 24 3 }}
-* [[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, 147d, 159, 171, 867, 1038, 1209, 1380, 1551, 1722 }}
+Optimal tunings:
+* WE: ~63/50 = 399.9757{{c}}, ~3/2 = 701.6458{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}}
-[[Badness]] (Smith): 0.019950
+{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }}
-=== Terminal ===
+Badness (Sintel): 1.04
-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.
+=== 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.
 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 tuning (POTE): ~44/35 = 400.000{{c}}, ~3/2 = 701.824{{c}}
+Optimal tunings:
+* WE: ~55/49 = 200.0134{{c}}, ~3/2 = 701.7931{{c}}
+* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7426{{c}}
-{{Optimal ET sequence|legend=0| 12, 147de, 159, 330 }}
+{{Optimal ET sequence|legend=0| 12, …, 330e, 342, 1380, 1722, 2064, 2406c, 5154bccdde }}
-Badness (Smith): 0.059502
+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 tuning (POTE): ~44/35 = 400.000{{c}}, ~3/2 = 701.821{{c}}
+Optimal tunings:
+* WE: ~55/49 = 200.0083{{c}}, ~3/2 = 701.7549{{c}}
+* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7238{{c}}
-{{Optimal ET sequence|legend=0| 12f, 147def, 159, 330 }}
+{{Optimal ET sequence|legend=0| 12f, 330eff, 342f, 696f }} *
-Badness (Smith): 0.037082
+<nowiki>*</nowiki> optimal patent val: [[354edo|354]]
-==== 17-limit ====
+Badness (Sintel): 1.85
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619
+=== Hemiterm ===
+The hemiterm temperament tempers out [[3025/3024]] (lehmerisma), and may be described as {{nowrap| 159 & 183 }}. Its ploidacot is triploid alpha-dicot.
-Mapping: {{mapping| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }}
+Subgroup: 2.3.5.7.11
-Optimal tuning (POTE): ~34/27 = 400.000{{c}}, ~3/2 = 701.810{{c}}
+Comma list: 3025/3024, 32805/32768, 102487/102400
-{{Optimal ET sequence|legend=0| 12f, 147def, 159, 171, 330 }}
+Mapping: {{mapping| 3 0 45 94 8 | 0 2 -16 -36 1 }}
+: mapping generators: ~63/50, ~693/400
-Badness (Smith): 0.027073
+Optimal tunings:
+* WE: ~63/50 = 400.0309{{c}}, ~693/400 = 950.9458{{c}} (~12/11 = 150.8841{{c}})
+* CWE: ~63/50 = 400.0000{{c}}, ~693/400 = 950.8707{{c}} (~12/11 = 150.8707{{c}})
-=== Terminator ===
+{{Optimal ET sequence|legend=0| 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce }}
-Subgroup: 2.3.5.7.11
-Comma list: 540/539, 32805/32768, 137781/137500
+Badness (Sintel): 0.684
-Mapping: {{mapping| 3 0 45 94 -137 | 0 1 -8 -18 31 }}
+==== 13-limit ====
-Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~3/2 = 701.685{{c}}
-{{Optimal ET sequence|legend=0| 12e, 159e, 171, 183, 354, 537, 891de }}
-Badness (Smith): 0.066968
-==== 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 tuning (POTE): ~63/50 = 400.000{{c}}, ~3/2 = 701.689{{c}}
+Optimal tunings:
+* WE: ~63/50 = 400.0541{{c}}, ~26/15 = 951.0013{{c}} (~12/11 = 150.8932{{c}})
+* CWE: ~63/50 = 400.0000{{c}}, ~26/15 = 950.8696{{c}} (~12/11 = 150.8696{{c}})
-{{Optimal ET sequence|legend=0| 171, 183, 354, 891de, 1245dee, 1599ddee }}
+{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f }}
-Badness (Smith): 0.035487
+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 tuning (POTE): ~63/50 = 400.000{{c}}, ~3/2 = 701.688{{c}}
+Optimal tunings:
+* WE: ~34/27 = 400.0373{{c}}, ~26/15 = 950.9556{{c}} (~12/11 = 150.8809{{c}})
+* CWE: ~34/27 = 400.0000{{c}}, ~26/15 = 950.8652{{c}} (~12/11 = 150.8652{{c}})
-{{Optimal ET sequence|legend=0| 171, 183, 354, 891de, 1245dee, 1599ddee }}
+{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f, 525f }}
-Badness (Smith): 0.020434
+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 tuning (POTE): ~55/49 = 200.0000{{c}}, ~3/2 = 701.7460{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~1536/1225 = 400.1360{{c}}, ~34300/19683 = 951.2867{{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 }}
+{{Optimal ET sequence|legend=1| 24, 135, 159, 612ccdd }}
-Badness (Smith): 0.029438
+[[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 tuning (POTE): ~55/49 = 200.0000{{c}}, ~3/2 = 701.7256{{c}}
+Optimal tunings:
+* WE: ~44/35 = 400.1156{{c}}, ~121/70 = 951.2377{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~121/70 = 950.9634{{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 (Smith): 0.044657
+=== 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 }}
+Optimal tunings:
-: mapping generators: ~63/50, ~693/400
+* WE: ~44/35 = 400.1396{{c}}, ~26/15 = 951.2799{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~26/15 = 950.9462{{c}}
-Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~693/400 = 950.872{{c}} (~12/11 = 150.872{{c}})
+{{Optimal ET sequence|legend=0| 24, 135f, 159 }}
-{{Optimal ET sequence|legend=0| 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce }}
+Badness (Sintel): 2.27
-Badness (Smith): 0.020687
+== Squirrel ==
+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.
-==== 13-limit ====
+[[Subgroup]]: 2.3.5.7
-Subgroup: 2.3.5.7.11.13
-Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712
+[[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 tuning (POTE): ~63/50 = 400.000{{c}}, ~26/15 = 950.873{{c}} (~12/11 = 150.873{{c}})
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.7408{{c}}, ~160/147 = 166.2424{{c}}
+: [[error map]]: {{val| +0.741 +0.799 +2.763 -6.934 }}
+* [[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 (Smith): 0.031362
+[[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 tuning (POTE): ~34/27 = 400.000{{c}}, ~26/15 = 950.867{{c}} (~12/11 = 150.867{{c}})
+Optimal tunings:
+* WE: ~2 = 1200.6379{{c}}, ~11/10 = 166.1853{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.1157{{c}}
-{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f, 525f, 867ff }}
+{{Optimal ET sequence|legend=0| 29, 36, 65 }}
-Badness (Smith): 0.022316
+Badness (Sintel): 2.26
-== Altinex ==
+=== 13-limit ===
-[[Subgroup]]: 2.3.5.7
+Subgroup: 2.3.5.7.11.13
-[[Comma list]]: 32805/32768, 367653125/362797056
+Comma list: 91/90, 169/168, 245/242, 896/891
-{{Mapping|legend=1| 3 0 45 -32 | 0 2 -16 17 }}
+Mapping: {{mapping| 1 2 -1 1 0 3 | 0 -3 24 13 25 5 }}
-: mapping generators: ~1536/1225, ~34300/19683
-[[Optimal tuning]] ([[CTE]]): ~1536/1225 = 400.000{{c}}, ~34300/19683 = 950.9654{{c}}
+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=1| 24, …, 111c, 135, 159, 612ccdd, 771ccdd }}
+{{Optimal ET sequence|legend=0| 29, 65f, 94df }}
-[[Badness]] (Smith): 0.422026
+Badness (Sintel): 1.81
-=== 11-limit ===
+== Tertiaschis ==
-Subgroup: 2.3.5.7.11
+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.
-Comma list: 385/384, 14700/14641, 19712/19683
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 3 0 45 -32 8 | 0 2 -16 17 1 }}
+[[Comma list]]: 32805/32768, 1071875/1062882
-Optimal tuning (CTE): ~44/35 = 400.000{{c}}, ~121/70 = 950.9658{{c}}
+{{Mapping|legend=1| 1 2 -1 10 | 0 -3 24 -52 }}
-{{Optimal ET sequence|legend=0| 24, …, 111c, 135, 159, 612ccdd, 771ccdd }}
+[[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 }}
-Badness (Smith): 0.101224
+{{Optimal ET sequence|legend=1| 65, 94, 159, 253, 412cd }}
-=== 13-limit ===
+[[Badness]] (Sintel): 5.36
-Subgroup: 2.3.5.7.11.13
-Comma list: 364/363, 385/384, 676/675, 19712/19683
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Mapping: {{mapping| 3 0 45 -32 8 42 | 0 2 -16 17 1 -13 }}
+Comma list: 385/384, 4000/3993, 19712/19683
-Optimal tuning (CTE): ~44/35 = 400.000{{c}}, ~26/15 = 950.9360{{c}}
+Mapping: {{mapping| 1 2 -1 10 0 | 0 -3 24 -52 25 }}
-{{Optimal ET sequence|legend=0| 24, …, 111cf, 135f, 159 }}
+Optimal tunings:
+* WE: ~2 = 1200.3379{{c}}, ~11/10 = 166.0638{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0167{{c}}
-Badness (Smith): 0.054894
+{{Optimal ET sequence|legend=0| 65, 94, 159, 253, 412cd, 665ccde }}
-== Sesquiquartififths ==
+Badness (Sintel): 2.07
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 2401/2400, 32805/32768
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-{{Mapping|legend=1| 1 1 7 5 | 0 4 -32 -15 }}
+Comma list: 325/324, 385/384, 1575/1573, 10985/10976
-: napping generators: ~2, ~448/405
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~448/405 = 175.434{{c}}
+Mapping: {{mapping| 1 2 -1 10 0 12 | 0 -3 24 -52 25 -60 }}
-[[Minimax tuning]]:
+Optimal tunings:
-* [[7-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
+* WE: ~2 = 1200.3467{{c}}, ~11/10 = 166.0635{{c}}
-* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0142{{c}}
-{{Optimal ET sequence|legend=1| 41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd }}
+{{Optimal ET sequence|legend=0| 65f, 94, 159, 253, 412cdf, 665ccdef }}
-[[Badness]] (Smith): 0.011244
+Badness (Sintel): 1.52
-=== Sesquart ===
+=== 17-limit ===
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13.17
-Comma list: 243/242, 441/440, 16384/16335
+Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
-Mapping: {{mapping| 1 1 7 5 2 | 0 4 -32 -15 10 }}
+Mapping: {{mapping| 1 2 -1 10 0 12 -2 | 0 -3 24 -52 25 -60 44 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~256/231 = 175.406{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.3019{{c}}, ~11/10 = 166.0535{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0114{{c}}
-{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}
+{{Optimal ET sequence|legend=1| 65f, 94, 159, 253 }}
-Badness (Smith): 0.029306
+Badness (Sintel): 1.35
-==== 13-limit ====
+== Countertertiaschis ==
-Subgroup: 2.3.5.7.11.13
+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.
-Comma list: 243/242, 364/363, 441/440, 3584/3575
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 1 1 7 5 2 -2 | 0 4 -32 -15 10 39 }}
+[[Comma list]]: 32805/32768, 244140625/243045684
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~72/65 = 175.409{{c}}
+{{Mapping|legend=1| 1 2 -1 -12 | 0 -3 24 107 }}
-{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}
+[[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 }}
-Badness (Smith): 0.022396
+{{Optimal ET sequence|legend=1| 65d, 159, 224, 383, 607 }}
-===== Sesquartia =====
+[[Badness]] (Sintel): 4.76
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Mapping: {{mapping| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }}
+Comma list: 3025/3024, 4000/3993, 32805/32768
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~72/65 = 175.424{{c}}
+Mapping: {{mapping| 1 2 -1 -12 0 | 0 -3 24 107 25 }}
-{{Optimal ET sequence|legend=0| 41, 89g, 130, 171, 301e }}
+Optimal tunings:
+* WE: ~2 = 1200.0804{{c}}, ~11/10 = 166.0739{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0634{{c}}
-Badness (Smith): 0.023126
+{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}
-====== 19-limit ======
+Badness (Sintel): 1.62
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }}
+Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/19 = 175.419{{c}}
+Mapping: {{mapping| 1 2 -1 -12 0 -10 | 0 -3 24 107 25 99 }}
-{{Optimal ET sequence|legend=0| 41, 89g, 130, 171, 301eh }}
+Optimal tunings:
+* WE: ~2 = 1200.0805{{c}}, ~11/10 = 166.0740{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0635{{c}}
-Badness (Smith): 0.020466
+{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}
-====== 23-limit ======
+Badness (Sintel): 1.01
-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
+== 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 -6 | 0 4 -32 -15 10 39 69 -12 72 }}
+[[Subgroup]]: 2.3.5.7
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/19 = 175.412{{c}}
+[[Comma list]]: 32805/32768, 390625/388962
-{{Optimal ET sequence|legend=0| 41i, 89gi, 130, 171, 301eh }}
+{{Mapping|legend=1| 4 0 60 119 | 0 1 -8 -17 }}
+: mapping generators: ~25/21, ~3
-Badness (Smith): 0.019043
+[[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 }}
-===== Heartia =====
+{{Optimal ET sequence|legend=1| 12, …, 200, 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): 2.79
-Mapping: {{mapping| 1 1 7 5 2 -2 0 | 0 4 -32 -15 10 39 28 }}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~72/65 = 175.386{{c}}
+Comma list: 1375/1372, 6250/6237, 32805/32768
-{{Optimal ET sequence|legend=0| 41, 89, 130g }}
+Mapping: {{mapping| 4 0 60 119 185 | 0 1 -8 -17 -27 }}
-Badness (Smith): 0.028443
+Optimal tunings:
+* WE: ~25/21 = 300.0244{{c}}, ~3/2 = 701.8759{{c}}
+* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8145{{c}}
-====== 19-limit ======
+{{Optimal ET sequence|legend=0| 12, …, 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): 1.51
-Mapping: {{mapping| 1 1 7 5 2 -2 0 6 | 0 4 -32 -15 10 39 28 -12 }}
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/19 = 175.380{{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): 0.023059
+Optimal tunings:
+* WE: ~25/21 = 300.0234{{c}}, ~3/2 = 701.8707{{c}}
+* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8123{{c}}
-===== Hearty =====
+{{Optimal ET sequence|legend=0| 12f, …, 212, 224, 436, 660 }}
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625
+Badness (Sintel): 1.13
-Mapping: {{mapping| 1 1 7 5 2 -2 13 | 0 4 -32 -15 10 39 -61 }}
+== 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}}, ~72/65 = 175.377{{c}}
+[[Subgroup]]: 2.3.5.7
-{{Optimal ET sequence|legend=0| 41g, 89, 130, 609ceefgg }}
+[[Comma list]]: 2401/2400, 32805/32768
-Badness (Smith): 0.030680
+{{Mapping|legend=1| 1 1 7 5 | 0 4 -32 -15 }}
+: mapping generators: ~2, ~448/405
-====== 19-limit ======
+[[Optimal tuning]]s:
-Subgroup: 2.3.5.7.11.13.17.19
+* [[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, 361/360, 364/363, 441/440, 456/455
+[[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 6 | 0 4 -32 -15 10 39 -61 -12 }}
+{{Optimal ET sequence|legend=1| 41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/19 = 175.377{{c}}
+[[Badness]] (Sintel): 0.285
-{{Optimal ET sequence|legend=0| 41g, 89, 130, 609ceefggh }}
+=== 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): 0.022816
+Subgroup: 2.3.5.7.11
-====== 23-limit ======
+Comma list: 243/242, 441/440, 16384/16335
-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 | 0 4 -32 -15 10 }}
-Mapping: {{mapping| 1 1 7 5 2 -2 13 6 13 | 0 4 -32 -15 10 39 -61 -12 -58 }}
+Optimal tunings:
+* WE: ~2 = 1199.8171{{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 = 175.376{{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): 0.019121
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-=== Bisesqui ===
+Comma list: 243/242, 364/363, 441/440, 3584/3575
-Subgroup: 2.3.5.7.11
-Comma list: 2401/2400, 9801/9800, 32805/32768
+Mapping: {{mapping| 1 1 7 5 2 -2 | 0 4 -32 -15 10 39 }}
-Mapping: {{mapping| 2 2 14 10 23 | 0 4 -32 -15 -55 }}
+Optimal tunings:
-: mapping generators: ~99/70, ~448/405
+* WE: ~2 = 1199.8352{{c}}, ~72/65 = 175.3852{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4095{{c}}
-Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~448/405 = 175.435{{c}}
+{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}
-{{Optimal ET sequence|legend=1| 82e, 130, 212, 342, 1156, 1498, 1840d }}
+Badness (Sintel): 0.925
-Badness (Smith): 0.016968
+===== Heartia =====
+Subgroup: 2.3.5.7.11.13.17
-== Quintilipyth ==
+Comma list: 243/242, 256/255, 273/272, 364/363, 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 | 0 4 -32 -15 10 39 28 }}
-[[Comma list]]: 32805/32768, 9765625/9680832
+Optimal tunings:
+* WE: ~2 = 1199.6422{{c}}, ~72/65 = 175.3338{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3857{{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): 1.45
-{{Optimal ET sequence|legend=1| 12, 253, 265 }}
+====== 19-limit ======
+Subgroup: 2.3.5.7.11.13.17.19
-[[Badness]] (Smith): 0.253966
+Comma list: 171/170, 243/242, 256/255, 273/272, 324/323, 441/440
-=== 11-limit ===
+Mapping: {{mapping| 1 1 7 5 2 -2 0 6 | 0 4 -32 -15 10 39 28 -12 }}
-Subgroup: 2.3.5.7.11
-Comma list: 1375/1372, 4375/4356, 32805/32768
+Optimal tunings:
+* WE: ~2 = 1199.7499{{c}}, ~21/19 = 175.3432{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.3797{{c}}
-Mapping: {{mapping| 1 2 -1 -4 -7 | 0 -5 40 82 126 }}
+{{Optimal ET sequence|legend=0| 41, 89, 130g }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~35/33 = 99.616{{c}}
+Badness (Sintel): 1.40
-{{Optimal ET sequence|legend=0| 12, 253, 265, 518c, 783cc }}
+===== Sesquartia =====
+Subgroup: 2.3.5.7.11.13.17
-Badness (Smith): 0.113044
+Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575
-=== 13-limit ===
+Mapping: {{mapping| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647
+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 -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): 1.18
-{{Optimal ET sequence|legend=0| 12f, 253, 518c, 771cc }}
+====== 19-limit ======
+Subgroup: 2.3.5.7.11.13.17.19
-Badness (Smith): 0.069127
+Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594
-=== 17-limit ===
+Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }}
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619
+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 5 | 0 -5 40 82 126 153 -11 }}
+{{Optimal ET sequence|legend=0| 41, 130, 171 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 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): 0.045992
+Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594
-=== 19-limit ===
+Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 -6 | 0 4 -32 -15 10 39 69 -12 72 }}
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971
+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 4 | 0 -5 40 82 126 153 -11 3 }}
+{{Optimal ET sequence|legend=0| 41i, 130, 171 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.615{{c}}
+Badness (Sintel): 1.36
-{{Optimal ET sequence|legend=0| 12f, 253, 265, 518ch }}
+===== Hearty =====
+Subgroup: 2.3.5.7.11.13.17
-Badness (Smith): 0.038155
+Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625
-== Quintaschis ==
+Mapping: {{mapping| 1 1 7 5 2 -2 13 | 0 4 -32 -15 10 39 -61 }}
-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 = 1199.9458{{c}}, ~72/65 = 175.3689{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3770{{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): 1.56
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~200/189 = 99.664{{c}}
+====== 19-limit ======
+Subgroup: 2.3.5.7.11.13.17.19
-{{Optimal ET sequence|legend=1| 12, …, 289, 301, 590, 891, 1192 }}
+Comma list: 221/220, 243/242, 361/360, 364/363, 441/440, 456/455
-[[Badness]] (Smith): 0.132890
+Mapping: {{mapping| 1 1 7 5 2 -2 13 6 | 0 4 -32 -15 10 39 -61 -12 }}
-=== 11-limit ===
+Optimal tunings:
-Subgroup: 2.3.5.7.11
+* WE: ~2 = 1200.0114{{c}}, ~72/65 = 175.3783{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3765{{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): 1.39
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~35/33 = 99.653{{c}}
+====== 23-limit ======
+Subgroup: 2.3.5.7.11.13.17.19.23
-{{Optimal ET sequence|legend=0| 12, …, 277d, 289 }}
+Comma list: 221/220, 243/242, 276/275, 323/322, 361/360, 364/363, 441/440
-Badness (Smith): 0.111477
+Mapping: {{mapping| 1 1 7 5 2 -2 13 6 13 | 0 4 -32 -15 10 39 -61 -12 -58 }}
-==== 13-limit ====
+Optimal tunings:
-Subgroup: 2.3.5.7.11.13
+* WE: ~2 = 1200.0122{{c}}, ~72/65 = 175.3782{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3763{{c}}
-Comma list: 364/363, 441/440, 32805/32768, 109512/109375
+{{Optimal ET sequence|legend=0| 41g, 89, 130 }}
-Mapping: {{mapping| 1 2 -1 -5 -8 -11 | 0 -5 40 94 138 177 }}
+Badness (Sintel): 1.37
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~35/33 = 99.658{{c}}
+=== Bisesqui ===
+Subgroup: 2.3.5.7.11
-{{Optimal ET sequence|legend=0| 12f, …, 277dff, 289 }}
+Comma list: 2401/2400, 9801/9800, 32805/32768
-Badness (Smith): 0.074218
+Mapping: {{mapping| 2 2 14 10 23 | 0 4 -32 -15 -55 }}
+: mapping generators: ~99/70, ~448/405
-==== 17-limit ====
+Optimal tunings:
-Subgroup: 2.3.5.7.11.13.17
+* 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, 595/594, 3757/3750, 32805/32768
+{{Optimal ET sequence|legend=1| 82e, 130, 212, 342, 1156, 1498, 1840d, 5862bbccdddee }}
-Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 | 0 -5 40 94 138 177 -11 }}
+Badness (Sintel): 0.561
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.656{{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): 0.050571
+[[Subgroup]]: 2.3.5.7
-==== 19-limit ====
+[[Comma list]]: 32805/32768, 420175/419904
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859
+{{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 4 | 0 -5 40 94 138 177 -11 3 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1039{{c}}, ~243/224 = 140.3620{{c}}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.659{{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 ET sequence|legend=0| 12f, 289 }}
+{{Optimal ET sequence|legend=1| 17, 77, 94, 171 }}
-Badness (Smith): 0.042120
+[[Badness]] (Sintel): 0.777
-=== Quintahelenic ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 5632/5625, 8019/8000, 151263/151250
+Comma list: 385/384, 1331/1323, 19712/19683
-Mapping: {{mapping| 1 2 -1 -5 -9 | 0 -5 40 94 150 }}
+Mapping: {{mapping| 1 1 7 0 1 | 0 5 -40 24 21 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~200/189 = 99.671{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.3103{{c}}, ~88/81 = 140.4011{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.3649{{c}}
-{{Optimal ET sequence|legend=0| 12, …, 289e, 301, 915 }}
+{{Optimal ET sequence|legend=0| 17, 77, 94, 171e, 265e }}
-Badness (Smith): 0.082225
+Badness (Sintel): 2.10
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000
+Comma list: 352/351, 385/384, 729/728, 1331/1323
-Mapping: {{mapping| 1 2 -1 -5 -9 -11 | 0 -5 40 94 150 177 }}
+Mapping: {{mapping| 1 1 7 0 1 3 | 0 5 -40 24 21 6 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~200/189 = 99.661{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.1840{{c}}, ~13/12 = 140.3840{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.3627{{c}}
-{{Optimal ET sequence|legend=0| 12f, …, 289e, 301 }}
+{{Optimal ET sequence|legend=0| 17, 77, 94, 171e }}
+Badness (Sintel): 1.57
-Badness (Smith): 0.055570
+== 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.
-===== 17-limit =====
+[[Subgroup]]: 2.3.5.7
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750
+[[Comma list]]: 16875/16807, 32805/32768
-Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 | 0 -5 40 94 150 177 -11 }}
+{{Mapping|legend=1| 1 0 15 12 | 0 5 -40 -29 }}
+: mapping generators: ~2, ~56/45
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.665{{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=1| 12f, 289e, 301 }}
+{{Optimal ET sequence|legend=1| 41, 142, 183, 224 }}
-Badness (Smith): 0.040412
+[[Badness]] (Sintel): 1.82
-===== 19-limit =====
+=== 11-limit ===
-Subgroup: 2.3.5.7.11.13.17.19
+Subgroup: 2.3.5.7.11
-Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700
+Comma list: 540/539, 1375/1372, 32805/32768
-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 | 0 5 -40 -29 33 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.668{{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=0| 12f, 301 }}
+{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}
-Badness (Smith): 0.036840
+Badness (Sintel): 1.04
-==== Quintahelenoid ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436
+Comma list: 540/539, 729/728, 1375/1372, 4096/4095
-Mapping: {{mapping| 1 2 -1 -5 -9 14 | 0 -5 40 94 150 -124 }}
+Mapping: {{mapping| 1 0 15 12 -7 -15 | 0 5 -40 -29 33 59 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~200/189 = 99.672{{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| 12, 301, 614, 915 }}
+{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}
-Badness (Smith): 0.066108
+Badness (Sintel): 0.884
-===== 17-limit =====
+== Quintilipyth ==
-Subgroup: 2.3.5.7.11.13.17
+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: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 1 2 -1 -5 -9 14 5 | 0 -5 40 94 150 -124 -11 }}
+[[Comma list]]: 32805/32768, 9765625/9680832
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.671{{c}}
+{{Mapping|legend=1| 1 2 -1 -4 | 0 -5 40 82 }}
+: mapping generators: ~2, ~625/588
-{{Optimal ET sequence|legend=0| 12, 301 }}
+[[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): 0.047908
+{{Optimal ET sequence|legend=1| 12, …, 253, 265 }}
-===== 19-limit =====
+[[Badness]] (Sintel): 6.43
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Mapping: {{mapping| 1 2 -1 -5 -9 14 5 4 | 0 -5 40 94 150 -124 -11 3 }}
+Comma list: 1375/1372, 4375/4356, 32805/32768
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.672{{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): 0.039542
+{{Optimal ET sequence|legend=0| 12, …, 253, 265, 518c }}
-== Sextilifourths ==
+Badness (Sintel): 3.74
-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
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-[[Comma list]]: 32805/32768, 235298/234375
+Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647
-{{Mapping|legend=1| 1 2 -1 -1 | 0 -6 48 55 }}
+Mapping: {{mapping| 1 2 -1 -4 -7 -9 | 0 -5 40 82 126 153 }}
-: mapping generators: ~2, ~21/2
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~21/20 = 83.053{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.1774{{c}}, ~35/33 = 99.6267{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6134{{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): 2.86
-=== 11-limit ===
+=== 17-limit ===
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13.17
-Comma list: 441/440, 4000/3993, 235298/234375
+Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619
-Mapping: {{mapping| 1 2 -1 -1 0 | 0 -6 48 55 50 }}
+Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 | 0 -5 40 82 126 153 -11 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 83.049{{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=0| 29, 72cde, 101e, 130, 289 }}
+{{Optimal ET sequence|legend=0| 12f, 241cdef, 253 }}
-Badness (Smith): 0.045457
+Badness (Sintel): 2.34
-=== 13-limit ===
+=== 19-limit ===
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 364/363, 441/440, 676/675, 10985/10976
+Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971
-Mapping: {{mapping| 1 2 -1 -1 0 1 | 0 -6 48 55 50 39 }}
+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, 72cdef, 101e, 130, 289 }}
+{{Optimal ET sequence|legend=0| 12f, 253, 265 }}
-Badness (Smith): 0.025276
+Badness (Sintel): 2.32
-== Septiquarschis ==
+== Quintaschis ==
-The septiquarschis temperament ({{nowrap| 89 & 94 }}) splits septimal minor seventh ([[7/4]]) into four generators and tempers out 829440/823543 (mynaslender comma) and 67108864/66706983 (septiness comma).
+Named by [[Xenllium]] in 2021, quintaschis slices the [[4/3|perfect fourth]] into five semitones and tempers out 49009212/48828125 in the [[7-limit]]. It may be described as the {{nowrap| 12 & 289 }} temperament, and its [[ploidacot]] is omega-pentacot.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 32805/32768, 829440/823543
+[[Comma list]]: 32805/32768, 49009212/48828125
-{{Mapping|legend=1| 1 3 -9 2 | 0 -7 -56 4 }}
+{{Mapping|legend=1| 1 2 -1 -5 | 0 -5 40 94 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~147/128 = 242.614{{c}}
+[[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 }}
-{{Optimal ET sequence|legend=1| 89, 94, 183, 460d, 643d, 1103dd }}
+{{Optimal ET sequence|legend=1| 12, …, 289, 301, 590, 891, 1192 }}
-[[Badness]] (Smith): 0.187047
+[[Badness]] (Sintel): 3.36
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 540/539, 15488/15435, 32805/32768
+Comma list: 441/440, 32805/32768, 1953125/1951488
-Mapping: {{mapping| 1 3 -9 2 -2 | 0 -7 -56 4 27 }}
+Mapping: {{mapping| 1 2 -1 -5 -8 | 0 -5 40 94 138 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~147/128 = 242.616{{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=0| 89, 94, 183, 460d, 643d, 826dd }}
+{{Optimal ET sequence|legend=0| 12, …, 277d, 289 }}
-Badness (Smith): 0.052002
+Badness (Sintel): 3.69
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 729/728, 1573/1568, 4096/4095
+Comma list: 364/363, 441/440, 32805/32768, 109512/109375
-Mapping: {{mapping| 1 3 -9 2 -2 13 | 0 -7 -56 4 27 -46 }}
+Mapping: {{mapping| 1 2 -1 -5 -8 -11 | 0 -5 40 94 138 177 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~147/128 = 242.610{{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, 277, 460d }}
+{{Optimal ET sequence|legend=0| 12f, …, 277dff, 289 }}
-Badness (Smith): 0.035315
+Badness (Sintel): 3.07
-== Tsaharuk ==
+==== 17-limit ====
-{{Main| Tsaharuk }}
+Subgroup: 2.3.5.7.11.13.17
-[[Subgroup]]: 2.3.5.7
+Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768
-[[Comma list]]: 32805/32768, 420175/419904
+Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 | 0 -5 40 94 138 177 -11 }}
-{{Mapping|legend=1| 1 1 7 0 | 0 5 -40 24 }}
+Optimal tunings:
-: mapping generators: ~2, ~243/224
+* WE: ~2 = 1200.1286{{c}}, ~18/17 = 99.6668{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6568{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~243/224 = 140.350{{c}}
+{{Optimal ET sequence|legend=0| 12f, 277dff, 289 }}
-{{Optimal ET sequence|legend=1| 17, 60c, 77, 94, 171 }}
+Badness (Sintel): 2.58
-[[Badness]] (Smith): 0.030697
+==== 19-limit ====
+Subgroup: 2.3.5.7.11.13.17.19
-=== 11-limit ===
+Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859
-Subgroup: 2.3.5.7.11
-Comma list: 385/384, 1331/1323, 19712/19683
+Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 4 | 0 -5 40 94 138 177 -11 3 }}
-Mapping: {{mapping| 1 1 7 0 1 | 0 5 -40 24 21 }}
+Optimal tunings:
+* 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}}, ~88/81 = 140.365{{c}}
+{{Optimal ET sequence|legend=0| 12f, 289 }}
-{{Optimal ET sequence|legend=0| 17, 60ce, 77, 94, 171e, 265e, 436ee }}
+Badness (Sintel): 2.56
-Badness (Smith): 0.063499
+=== Quintahelenic ===
+Subgroup: 2.3.5.7.11
-=== 13-limit ===
+Comma list: 5632/5625, 8019/8000, 151263/151250
-Subgroup: 2.3.5.7.11.13
-Comma list: 352/351, 385/384, 729/728, 1331/1323
+Mapping: {{mapping| 1 2 -1 -5 -9 | 0 -5 40 94 150 }}
-Mapping: {{mapping| 1 1 7 0 1 3 | 0 5 -40 24 21 6 }}
+Optimal tunings:
+* WE: ~2 = 1200.0195{{c}}, ~200/189 = 99.6723{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6709{{c}}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/12 = 140.363{{c}}
+{{Optimal ET sequence|legend=0| 12, …, 289e, 301, 915 }}
-{{Optimal ET sequence|legend=0| 17, 60ce, 77, 94, 171e, 436ee }}
+Badness (Sintel): 2.72
-Badness (Smith): 0.037886
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-== Quanharuk ==
+Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000
-[[Subgroup]]: 2.3.5.7
-[[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 }}
+Optimal tunings:
-: mapping generators: ~2, ~56/45
+* WE: ~2 = 1200.0442{{c}}, ~200/189 = 99.6709{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6675{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~56/45 = 380.355{{c}}
+{{Optimal ET sequence|legend=0| 12f, …, 289e, 301 }}
-{{Optimal ET sequence|legend=1| 41, 142, 183, 224, 1303d, 1527cd, 1751cd, 1975cd }}
+Badness (Sintel): 2.30
-[[Badness]] (Smith): 0.071950
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-=== 11-limit ===
+Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750
-Subgroup: 2.3.5.7.11
-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 tunings:
+* WE: ~2 = 1200.1227{{c}}, ~200/189 = 99.6753{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6658{{c}}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~56/45 = 380.352{{c}}
+{{Optimal ET sequence|legend=1| 12f, 289e, 301 }}
-{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d, 1079d }}
+Badness (Sintel): 2.06
-Badness (Smith): 0.031549
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-=== 13-limit ===
+Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700
-Subgroup: 2.3.5.7.11.13
-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 tunings:
+* WE: ~2 = 1200.0230{{c}}, ~200/189 = 99.6694{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6676{{c}}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~56/45 = 380.351{{c}}
+{{Optimal ET sequence|legend=0| 12f, 301 }}
-{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}
+Badness (Sintel): 2.24
-Badness (Smith): 0.021392
+==== Quintahelenoid ====
+Subgroup: 2.3.5.7.11.13
-== Quadrant ==
+Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436
-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
+Mapping: {{mapping| 1 2 -1 -5 -9 14 | 0 -5 40 94 150 -124 }}
-[[Comma list]]: 32805/32768, 390625/388962
+Optimal tunings:
+* WE: ~2 = 1199.9919{{c}}, ~200/189 = 99.6712{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6718{{c}}
-{{Mapping|legend=1| 4 0 60 119 | 0 1 -8 -17 }}
+{{Optimal ET sequence|legend=0| 12, 301, 614, 915 }}
-: mapping generators: ~25/21, ~3
-[[Optimal tuning]] ([[POTE]]): ~25/21 = 300.0000{{c}}, ~3/2 = 701.8234{{c}}
+Badness (Sintel): 2.73
-{{Optimal ET sequence|legend=1| 212, 224, 436, 660, 1096c }}
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-[[Badness]] (Smith): 0.110242
+Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157
-=== 11-limit ===
+Mapping: {{mapping| 1 2 -1 -5 -9 14 5 | 0 -5 40 94 150 -124 -11 }}
-Subgroup: 2.3.5.7.11
-Comma list: 1375/1372, 6250/6237, 32805/32768
+Optimal tunings:
+* WE: ~2 = 1200.0469{{c}}, ~18/17 = 99.6749{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6710{{c}}
-Mapping: {{mapping| 4 0 60 119 185 | 0 1 -8 -17 -27 }}
+{{Optimal ET sequence|legend=0| 12, 301 }}
-Optimal tuning (POTE): ~25/21 = 300.0000{{c}}, ~3/2 = 701.8176{{c}}
+Badness (Sintel): 2.44
-{{Optimal ET sequence|legend=0| 212, 224, 436, 660 }}
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-Badness (Smith): 0.045738
+Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137
-=== 13-limit ===
+Mapping: {{mapping| 1 2 -1 -5 -9 14 5 4 | 0 -5 40 94 150 -124 -11 3 }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647
+Optimal tunings:
+* WE: ~2 = 1199.9925{{c}}, ~18/17 = 99.6710{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6716{{c}}
-Mapping: {{mapping| 4 0 60 119 185 224 | 0 1 -8 -17 -27 -33 }}
+{{Optimal ET sequence|legend=0| 12, 301 }}
-Optimal tuning (POTE): ~25/21 = 300.0000{{c}}, ~3/2 = 701.8158{{c}}
+Badness (Sintel): 2.41
-{{Optimal ET sequence|legend=0| 212, 224, 436, 660 }}
+== 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.
-Badness (Smith): 0.027243
-== Septant ==
-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
-[[Comma list]]: 32805/32768, 516560652/514714375
+[[Comma list]]: 32805/32768, 235298/234375
-{{Mapping|legend=1| 7 0 105 -56 | 0 1 -8 7 }}
+{{Mapping|legend=1| 1 2 -1 -1 | 0 -6 48 55 }}
-: mapping generators: ~8575/7776, ~3
+: mapping generators: ~2, ~21/20
-[[Optimal tuning]] ([[POTE]]): ~8575/7776 = 171.429{{c}}, ~3/2 = 701.702{{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=1| 77, 147, 224, 301, 525, 826, 1351 }}
+{{Optimal ET sequence|legend=1| 29, 72cd, 101, 130, 289, 419 }}
-[[Badness]] (Smith): 0.111142
+[[Badness]] (Sintel): 2.75
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 24057/24010, 32805/32768
+Comma list: 441/440, 4000/3993, 235298/234375
-Mapping: {{mapping| 7 0 105 -56 -120 | 0 1 -8 7 13 }}
+Mapping: {{mapping| 1 2 -1 -1 0 | 0 -6 48 55 50 }}
-Optimal tuning (POTE): ~495/448 = 171.429{{c}}, ~3/2 = 701.719{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0424{{c}}, ~21/20 = 83.0520{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0497{{c}}
-{{Optimal ET sequence|legend=0| 77, 147, 224, 301, 525 }}
+{{Optimal ET sequence|legend=0| 29, 72cde, 101e, 130, 289 }}
-Badness (Smith): 0.044122
+Badness (Sintel): 1.50
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024
+Comma list: 364/363, 441/440, 676/675, 10985/10976
-Mapping: {{mapping| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }}
+Mapping: {{mapping| 1 2 -1 -1 0 1 | 0 -6 48 55 50 39 }}
-Optimal tuning (POTE): ~495/448 = 171.429{{c}}, ~3/2 = 701.724{{c}}
+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| 77, 147, 224, 525 }}
+{{Optimal ET sequence|legend=0| 29, 72cdef, 101e, 130, 289 }}
-Badness (Smith): 0.024706
+Badness (Sintel): 1.04
-== Octant ==
+== Septant ==
-The octant temperament ({{nowrap| 224 & 472 }}) has a period of 1/8 octave. In this temperament, 12/11, 35/27, and 99/70 are mapped into 1\8, 3\8, and 4\8 respectively.
+Named by [[Xenllium]] in 2021, septant notably tempers out the [[akjaysma]] ({{monzo| 47 -7 -7 -7 }}) and may be described as the {{nowrap| 224 & 301 }} temperament. It has a period of 1/7 octave, and its [[ploidacot]] is heptaploid monocot.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 32805/32768, 2259436291848/2251875390625
+[[Comma list]]: 32805/32768, 516560652/514714375
-{{Mapping|legend=1| 8 0 120 -117 | 0 1 -8 11 }}
+{{Mapping|legend=1| 7 0 105 -56 | 0 1 -8 7 }}
-: mapping generators: ~42875/39366, ~3
+: mapping generators: ~8575/7776, ~3
-[[Optimal tuning]] ([[POTE]]): ~42875/39366 = 150.000{{c}}, ~3/2 = 701.713{{c}}
+[[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| 24, 224, 472, 696, 1168 }}
+{{Optimal ET sequence|legend=1| 77, 147, 224, 301, 525, 826, 1351 }}
-[[Badness]] (Smith): 0.157186
+[[Badness]] (Sintel): 2.81
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 9801/9800, 32805/32768, 46656/46585
+Comma list: 3025/3024, 24057/24010, 32805/32768
-Mapping: {{mapping| 8 0 120 -117 15 | 0 1 -8 11 1 }}
+Mapping: {{mapping| 7 0 105 -56 -120 | 0 1 -8 7 13 }}
-Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~3/2 = 701.713{{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| 24, 224, 472, 696, 1168 }}
+{{Optimal ET sequence|legend=0| 77, 147, 224, 301, 525 }}
-Badness (Smith): 0.044778
+Badness (Sintel): 1.46
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655
+Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024
-Mapping: {{mapping| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }}
+Mapping: {{mapping| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }}
-Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~3/2 = 701.725{{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| 24, 224, 472, 696 }}
+{{Optimal ET sequence|legend=0| 77, 147, 224, 525, 1274f }}
-Badness (Smith): 0.030425
+Badness (Sintel): 1.02
-== Nonant ==
+== Octant ==
-The ''nonant'' temperament ({{nowrap| 36 & 135 }}) has a period of 1/9 octave and tempers out the [[septimal ennealimma]], {{monzo| -11 -9 0 9 }}.
+Octant may be described as the {{nowrap| 224 & 248 }} temperament. It has a period of 1/8 octave, and its [[ploidacot]] is octaploid monocot. In this temperament, [[12/11]], [[35/27]], and [[99/70]] are mapped to 1\8, 3\8, and 4\8 respectively.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 32805/32768, 40353607/40310784
+[[Comma list]]: 32805/32768, 2259436291848/2251875390625
-{{Mapping|legend=1| 9 0 135 11 | 0 1 -8 1 }}
+{{Mapping|legend=1| 8 0 120 -117 | 0 1 -8 11 }}
-: mapping generators: ~2592/2401, ~3
+: mapping generators: ~42875/39366, ~3
-[[Optimal tuning]] ([[CTE]]): ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.7232{{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| 36, 99c, 135, 171 }}
+{{Optimal ET sequence|legend=1| 24, …, 224, 472, 696, 1168 }}
-[[Badness]] (Smith): 0.069896
+[[Badness]] (Sintel): 3.98
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 540/539, 32805/32768, 42875/42592
+Comma list: 9801/9800, 32805/32768, 46656/46585
-Mapping: {{mapping| 9 0 135 11 131 | 0 1 -8 1 -7 }}
+Mapping: {{mapping| 8 0 120 -117 15 | 0 1 -8 11 1 }}
-Optimal tuning (CTE): ~242/225 = 133.3333{{c}}, ~3/2 = 701.8398{{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| 36, 99c, 135, 171, 477ce, 648cee }}
+{{Optimal ET sequence|legend=0| 24, …, 224, 472, 696, 1168 }}
-Badness (Smith): 0.126910
+Badness (Sintel): 1.48
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 729/728, 4096/4095, 16807/16731
+Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655
-Mapping: {{mapping| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }}
+Mapping: {{mapping| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }}
-Optimal tuning (CTE): ~242/225 = 133.3333{{c}}, ~3/2 = 701.7998{{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| 36, 99cf, 135, 171 }}
+{{Optimal ET sequence|legend=0| 24, 224, 472, 696 }}
-Badness (Smith): 0.076195
+Badness (Sintel): 1.26
-== Tridecafifths ==
+== Nonant ==
-Tridecafifths divides the perfect 3/2 into 13 quartertones.
+Named by [[Xenllium]] in 2023, nonant tempers out the [[septimal ennealimma]] ({{monzo| -11 -9 0 9 }}) and may be described as the {{nowrap| 36 & 171 }} temperament. It has a period of 1/9 octave, and its [[ploidacot]] is enneaploid monocot.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 32805/32768, {{monzo| -14 -1 -9 13 }}
+[[Comma list]]: 32805/32768, 40353607/40310784
-{{Mapping|legend=1| 1 1 7 6 | 0 13 -104 -71 }}
+{{Mapping|legend=1| 9 0 135 11 | 0 1 -8 1 }}
-: mapping generators: ~2, ~1323/1280
+: mapping generators: ~2592/2401, ~3
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~1323/1280 = 53.9741{{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| 89, 200, 289 }}
+{{Optimal ET sequence|legend=1| 36, 99c, 135, 171, 2772bd, 2943bdd, …, 5166bccddd, 5337bccddd }}
-[[Badness]] (Smith): 0.432580
+[[Badness]] (Sintel): 1.77
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 441/440, 32805/32768, 55296000/55240493
+Comma list: 540/539, 32805/32768, 42875/42592
-Mapping: {{mapping| 1 1 7 6 4 | 0 13 -104 -71 -12 }}
+Mapping: {{mapping| 9 0 135 11 131 | 0 1 -8 1 -7 }}
-Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~33/32 = 53.9744{{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| 89, 200, 289 }}
+{{Optimal ET sequence|legend=0| 36, 135, 171 }}
-Badness (Smith): 0.127820
+Badness (Sintel): 4.20
-== Subgroup extensions ==
+=== 13-limit ===
-=== Photia (2.3.5.17) ===
+Subgroup: 2.3.5.7.11.13
-{{See also| No-elevens subgroup temperaments #Garibaldia }}
+Comma list: 540/539, 729/728, 4096/4095, 16807/16731
-[[Subgroup]]: 2.3.5.17
+Mapping: {{mapping| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }}
-[[Comma list]]: 256/255, 1458/1445
+Optimal tunings:
-{{Mapping|legend=2| 1 0 15 -7 | 0 1 -8 7 }}
+* WE: ~242/225 = 133.3180{{c}}, ~3/2 = 701.6956{{c}}
+* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.7800{{c}}
-{{Mapping|legend=3| 1 0 15 0 0 0 -7 | 0 1 -8 0 0 0 7 }}
-: mapping generators: ~2, ~3
+{{Optimal ET sequence|legend=0| 36, 99cf, 135, 171 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.491{{c}}
+Badness (Sintel): 3.15
-{{Optimal ET sequence|legend=1| 12, 41, 53, 65 }}
+== Septiquarschis ==
+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.
-[[Tp tuning #T2 tuning|RMS error]]: 0.4842 cents
+[[Subgroup]]: 2.3.5.7
-==== 2.3.5.17.19 subgroup ====
-Subgroup: 2.3.5.17.19
+[[Comma list]]: 32805/32768, 829440/823543
-Comma list: 171/170, 256/255, 324/323
+{{Mapping|legend=1| 1 -4 47 6 | 0 7 56 -4 }}
+: mapping generators: ~2, ~256/147
-Subgroup-val mapping: {{mapping| 1 0 15 -7 9 | 0 1 -8 7 -3 }}
+[[Optimal tuning]]s:
-Gencom mapping: {{mapping| 1 0 15 0 0 0 -7 9 | 0 1 -8 0 0 0 7 -3 }}
+* [[WE]]: ~2 = 1199.8855{{c}}, ~256/147 = 957.2944{{c}}
+: [[error map]]: {{val| -0.114 -0.436 -0.182 +1.310 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.470{{c}}
+* [[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=0| 12, 41, 53, 65 }}
+{{Optimal ET sequence|legend=1| 89, 94, 183, 460d, 643d }}
-RMS error: 0.5374 cents
+[[Badness]] (Sintel): 4.73
-=== Nestoria (2.3.5.19) ===
-: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-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.
+Comma list: 540/539, 15488/15435, 32805/32768
-[[Subgroup]]: 2.3.5.19
+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 }}
+Badness (Sintel): 1.72
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 540/539, 729/728, 1573/1568, 4096/4095
+Mapping: {{mapping| 1 -4 47 6 25 -33 | 0 7 56 -4 -27 46 }}
+Optimal tunings:
+* 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
+== Tridecafifths ==
+Named by [[Eliora]] in 2023, tridecafifths may be described as the {{nowrap| 89 & 200 }} temperament. It divides the [[3/2|perfect fifth]] into thirteen quartertones, so its [[ploidacot]] is 13-cot. [[289edo]] gives a highly recommendable tuning.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 32805/32768, {{monzo| -14 -1 -9 13 }}
+{{Mapping|legend=1| 1 1 7 6 | 0 13 -104 -71 }}
+: mapping generators: ~2, ~1323/1280
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1431{{c}}, ~1323/1280 = 53.9838{{c}}
+: [[error map]]: {{val| +0.143 -0.023 +0.375 -0.816 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~1323/1280 = 53.9764{{c}}
+: error map: {{val| 0.000 -0.261 -0.221 -0.421 }}
+{{Optimal ET sequence|legend=1| 89, 200, 289 }}
+[[Badness]] (Sintel): 10.9
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 441/440, 32805/32768, 55296000/55240493
+Mapping: {{mapping| 1 1 7 6 4 | 0 13 -104 -71 -12 }}
+Optimal tunings:
+* WE: ~2 = 1200.0311{{c}}, ~33/32 = 53.9766{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 53.9750{{c}}
+{{Optimal ET sequence|legend=0| 89, 200, 289 }}
+Badness (Sintel): 4.23
+== Subgroup extensions ==
+=== Maqamschismic (2.3.5.11) ===
+Proposed by [[Eufalesio]] in 2026, maqamschismic is equivalent to the no-7 [[cassandra]]. The 2.3.5.11.13 subgroup adds [[352/351]] to the comma list and tempers 11/9~39/32 together (and 16/13~27/22), providing a very simple framework for tuning [[maqam]]at (especially the Turkish version), as outlined by [[Ozan Yarman]]. 41edo and 53edo are simplest, but 94edo is more optimized. It is only slightly worse than the no-7 [[helenus]].
+Subgroup: 2.3.5.11
+Comma list: 2200/2187, 4125/4096
+Subgroup-val mapping: {{mapping| 1 0 15 -33 | 0 1 -8 23 }}
+Optimal tunings:
+* WE: ~2 = 1200.5458{{c}} ~3/2 = 702.4021{{c}}
+* CWE: 2 = 1200.0000{{c}}, ~3/2 = 702.0906{{c}}
+{{Optimal ET sequence|legend=0| 12e, …, 41, 53, 94, 147e, 241ce, 335ce }}
+Badness (Sintel): 1.34
+==== 2.3.5.11.13 subgroup ====
+Subgroup: 2.3.5.11.13
+Comma list: 325/324, 352/351, 4125/4096
+Subgroup-val mapping: {{mapping| 1 0 15 -33 -28 | 0 1 -8 23 20 }}
+Optimal tunings:
+* WE: ~2 = 1200.4565{{c}} ~3/2 = 702.3057{{c}}
+* CWE: 2 = 1200.0000{{c}}, ~3/2 = 702.0485{{c}}
+{{Optimal ET sequence|legend=0| 12e, …, 41, 53, 94, 147e }}
+Badness (Sintel): 0.862
+=== 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 }}
+[[Subgroup]]: 2.3.5.17
+[[Comma list]]: 256/255, 1458/1445
+{{Mapping|legend=2| 1 0 15 -7 | 0 1 -8 7 }}
+{{Mapping|legend=3| 1 0 15 0 0 0 -7 | 0 1 -8 0 0 0 7 }}
+: mapping generators: ~2, ~3
+[[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, 207g, 272gg }}
+[[Badness]] (Sintel): 0.479
+==== 2.3.5.17.19 subgroup ====
+Subgroup: 2.3.5.17.19
+Comma list: 171/170, 256/255, 324/323
+Subgroup-val mapping: {{mapping| 1 0 15 -7 9 | 0 1 -8 7 -3 }}
+Gencom mapping: {{mapping| 1 0 15 0 0 0 -7 9 | 0 1 -8 0 0 0 7 -3 }}
+Optimal tunings:
+* 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, 142g }}
+Badness (Sintel): 0.332
+=== Nestoria (2.3.5.19) ===
+: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''
+Nestoria is notable for having one of the lowest-badness subgroup extensions of schismic. Note that despite prime [[19/1|19]] being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is generally not flatter than the fifth in optimal schismic due to its optimization considering intervals like [[19/10]] and [[19/15]]. However, the dyadic tuning sensitivity of [[19/16]] suggests using tunings like [[65edo]] and [[77edo]] to optimize in favour of prime 19, as [[53edo]] is already arguably undertempered for it despite being the flattest tuning appearing in the optimal ET sequence.
+[[Subgroup]]: 2.3.5.19
+[[Comma list]]: 361/360, 513/512
+{{Mapping|legend=2| 1 0 15 9 | 0 1 -8 -3 }}
+{{Mapping|legend=3| 1 0 15 0 0 0 0 9 | 0 1 -8 0 0 0 0 -3 }}
+: mapping generators: ~2, ~3
+[[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 }}
-[[Comma list]]: 361/360, 513/512
+{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171, 460hh, 631hh }}
-{{Mapping|legend=2| 1 0 15 9 | 0 1 -8 -3 }}
+[[Badness]] (Sintel): 0.126
-{{Mapping|legend=3| 1 0 15 0 0 0 0 9 | 0 1 -8 0 0 0 0 -3 }}
-: mapping generators: ~2, ~3
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.746{{c}}
-{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171 }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.1763 cents
 === Taylor (2.3.5.13) ===
@@ Line 2,603: / Line 2,920: @@
 : 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 2,616: / Line 2,937: @@
 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 2,629: / Line 2,952: @@
 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 2,647: / Line 2,972: @@
 : 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 2,662: / Line 2,991: @@
 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:Temperament families]]
 [[Category:Schismatic family| ]] <!-- main article -->
 [[Category:Rank 2]]