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* [[WE]]: ~2 = 1200.0749{{c}}, ~3/2 = 701.7797{{c}}
* [[WE]]: ~2 = 1200.0749{{c}}, ~3/2 = 701.7797{{c}}
: [[error map]]: {{val| +0.075 -0.100 -0.027 }}
: [[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 }}
: error map: {{val| 0.000 -0.224 -0.160 }}


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=== Overview to extensions ===
=== 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:
Temperaments discussed elsewhere include:
* ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
* ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
* ''[[Pogo]]'' (+118098/117649) → [[Stearnsmic clan #Pogo|Stearnsmic 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]].
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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{{Main| Garibaldi }}
{{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
[[Subgroup]]: 2.3.5.7
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* [[WE]]: ~2 = 1200.1233{{c}}, ~3/2 = 702.1573{{c}}
* [[WE]]: ~2 = 1200.1233{{c}}, ~3/2 = 702.1573{{c}}
: [[error map]]: {{val| +0.123 +0.326 -2.709 +2.328 }}
: [[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 }}
: error map: {{val| 0.000 +0.122 -2.933 +2.090 }}


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=== Cassandra ===
=== 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
Subgroup: 2.3.5.7.11
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Badness (Sintel): 1.18
Badness (Sintel): 1.18
=== Hemigari ===
Subgroup: 2.3.5.7.11
Comma list: 121/120, 225/224, 3125/3087
Mapping: {{mapping| 1 0 15 25 9 | 0 2 -16 -28 -7 }}
: mapping generators: ~2, ~110/63
Optimal tunings:
* WE: ~2 = 1200.7303{{c}}, ~110/63 = 951.6605{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~110/63 = 951.0604{{c}}
{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135ee }}
Badness (Sintel): 1.68
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 169/168, 225/224, 275/273
Mapping: {{mapping| 1 0 15 25 9 14 | 0 2 -16 -28 -7 -13 }}
Optimal tunings:
* WE: ~2 = 1200.8146{{c}}, ~26/15 = 951.7273{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.0574{{c}}
{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135eef }}
Badness (Sintel): 1.13


=== Karadeniz ===
=== Karadeniz ===
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Badness (Sintel): 1.34
Badness (Sintel): 1.34


=== Sanjaab ===
=== Hemigari ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 225/224, 3125/3087
 
Mapping: {{mapping| 1 0 15 25 9 | 0 2 -16 -28 -7 }}
: mapping generators: ~2, ~110/63
 
Optimal tunings:
* WE: ~2 = 1200.7303{{c}}, ~110/63 = 951.6605{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~110/63 = 951.0604{{c}}
 
{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135ee }}
 
Badness (Sintel): 1.68
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 169/168, 225/224, 275/273
 
Mapping: {{mapping| 1 0 15 25 9 14 | 0 2 -16 -28 -7 -13 }}
 
Optimal tunings:
* WE: ~2 = 1200.8146{{c}}, ~26/15 = 951.7273{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.0574{{c}}
 
{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135eef }}
 
Badness (Sintel): 1.13
 
=== Sanjaab ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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Badness (Sintel): 1.40
Badness (Sintel): 1.40
== Schism ==
See [[Archytas clan #Schism]].
Schism is a relatively low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (C-Bb). 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.


== Pontiac ==
== Pontiac ==
{{Main| 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
[[Subgroup]]: 2.3.5.7
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* [[WE]]: ~2 = 1200.0989{{c}}, ~3/2 = 701.8145{{c}}
* [[WE]]: ~2 = 1200.0989{{c}}, ~3/2 = 701.8145{{c}}
: [[error map]]: {{val| +0.099 -0.042 -0.138 -0.038 }}
: [[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 }}
: error map: {{val| 0.000 -0.197 -0.377 -0.268 }}


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=== Helenoid ===
=== 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
Subgroup: 2.3.5.7.11
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=== Ponta ===
=== 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
Subgroup: 2.3.5.7.11
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=== Pontic ===
=== 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
Subgroup: 2.3.5.7.11
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=== Bipont ===
=== 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
Subgroup: 2.3.5.7.11
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== Grackle ==
== 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
[[Subgroup]]: 2.3.5.7
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* [[WE]]: ~2 = 1199.7974{{c}}, ~3/2 = 701.1210{{c}}
* [[WE]]: ~2 = 1199.7974{{c}}, ~3/2 = 701.1210{{c}}
: [[error map]]: {{val| -0.203 -1.037 +3.300 -1.618 }}
: [[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 }}
: error map: {{val| 0.000 -0.709 +3.715 -1.234 }}


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Badness (Sintel): 2.01
Badness (Sintel): 2.01
== Quasipyth ==
Named by [[Xenllium]] in 2026, quasipyth tempers out {{monzo| 109 -67 0 -1 }}, the [[nanisma]], as well as the [[catasyc comma]], 390625/387072. The 7/4 is found at −67 fifths, represented by the nonuple-diminished thirteenth.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 32805/32768, 390625/387072
{{Mapping|legend=1| 1 0 15 109 | 0 1 -8 -67 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2569{{c}}, ~3/2 = 702.1149{{c}}
: [[error map]]: {{val| +0.2569 +0.4168 -1.4342 +0.2685 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9615{{c}}
: error map: {{val| 0.0000 +0.0065 -2.0054 -0.2437 }}
{{Optimal ET sequence|legend=1| 53, 147d, 200, 253, 306c, 559c }}
[[Badness]] (Sintel): 5.04
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 385/384, 19712/19683, 78125/77616
Mapping: {{mapping| 1 0 15 109 -117 | 0 1 -8 -67 76 }}
Optimal tunings:
* WE: ~2 = 1200.3283{{c}}, ~3/2 = 702.1636{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9713{{c}}
{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}
Badness (Sintel): 3.83
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 385/384, 2200/2197, 19712/19683
Mapping: {{mapping| 1 0 15 109 -117 -28 | 0 1 -8 -67 76 20 }}
Optimal tunings:
* WE: ~2 = 1200.3229{{c}}, ~3/2 = 702.1603{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9714{{c}}
{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}
Badness (Sintel): 2.13
== Schism ==
See [[Archytas clan #Schism]].
Schism is a relatively low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (C–B♭). 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.


== Bischismic ==
== Bischismic ==
Bischismic tempers out 3136/3125, the [[hemimean comma]], as well as 321489/320000, the [[varunisma]], and may be described as the {{nowrap| 118 & 130 }} temperament. The octave is split in halves, so the [[ploidacot]] of this temperament is diploid monocot. In schismic, -10 fifths make the interval class of 10/9. Bischismic then finds [[7/4]] by a stack of two [[10/9]]'s plus a semi-octave period, and in the [[11-limit]], it simply finds [[11/8]] by a stack of three [[10/9]]'s. [[248edo]] and [[378edo]] make for excellent tunings in both cases.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 1,184: Line 1,230:


== Kleischismic ==
== Kleischismic ==
Kleischismic tempers out 1500625/1492992, the [[uniwiz comma]], and may be described as the {{nowrap| 94 & 118 }} temperament. The generator is a infrafifth, two of which plus a semi-octave period make the [[3/1|3rd]] [[harmonic]]; its [[ploidacot]] is thus diploid alpha-dicot. In schismic, 10 fifths make the interval class of [[9/5]]. Kleischismic then finds [[7/4]] by that minus a [[36/35]] quartertone, which is the aforementioned generator minus a semi-octave period. The generator stands in for [[16/11]] and the quartertone stands in for [[33/32]] in the [[11-limit]]. [[212edo]] and [[330edo]] in the 330e val may be recommended as tunings.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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== Salsa ==
== Salsa ==
Salsa tempers out 245/243, the [[sensamagic comma]], and may be described as the {{nowrap| 41 & 65 }} temperament. It has a neutral third as a generator; its [[ploidacot]] is dicot. In fact it is related to [[hemififths]], from which this less accurate temperament only differs by the mapping of [[5/1|5]].
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 1,317: Line 1,367:


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9362{c}}, ~11/9 = 351.0061{{c}}
* WE: ~2 = 1199.9362{{c}}, ~11/9 = 351.0061{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0247{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0247{{c}}


Line 1,325: Line 1,375:


== Hemischis ==
== Hemischis ==
[[Subgroup]]: 2.3.5.7
Hemischis tempers out 6144/6125, the [[porwell comma]], as well as 19683/19600, the [[cataharry comma]], and may be described as the {{nowrap| 53 & 130 }} temperament. Its [[ploidacot]] is alpha-dicot.
 
The [[S-expression]]-based comma list for 13-limit hemischis is {[[540/539|S12/S14]], [[676/675|S13/S15 = S26]], [[729/728|S27]], [[4096/4095|S64]], ([[4225/4224|S65]])}. Tempering out [[169/168]] ({{S|13}}), [[225/224]] ({{S|15}}) or [[625/624]] ({{S|25}}) leads to [[53edo]] while tempering out [[24192/24167]] ([[S-expression|S12/S13]]), [[10985/10976]] ([[S-expression|S13/S14]]), [[43904/43875]] ([[S-expression|S14/S15]]) or [[2401/2400]] ([[S-expression|S49]]) leads to [[130edo]] and implies S12, S13, S14, and S15 are tempered together.
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 6144/6125, 19683/19600
[[Comma list]]: 6144/6125, 19683/19600
Line 1,358: Line 1,412:


=== 13-limit ===
=== 13-limit ===
Its [[S-expression]]-based comma list is {[[540/539|S12/S14]], [[676/675|S13/S15 = S26]], [[729/728|S27]], [[4096/4095|S64]](, [[4225/4224|S65]])}. Tempering out [[169/168|S13]], [[225/224|S15]] or [[625/624|S25]] leads to [[53edo]] (through [[Catakleismic]]) while tempering out [[24192/24167|S12/S13]], [[10985/10976|S13/S14]], [[43904/43875|S14/S15]] or [[2401/2400|S49]] (implying S12 = S13 = S14 = S15) leads to [[130edo]].
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


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


[[Comma list]]: 686/675, 32805/32768
[[Comma list]]: 32805/32768, 250047/250000


{{Mapping|legend=1| 1 2 -1 1 | 0 -3 24 13 }}
{{Mapping|legend=1| 3 0 45 94 | 0 1 -8 -18 }}
: mapping generators: ~63/50, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.7408{{c}}, ~160/147 = 166.2424{{c}}
* [[WE]]: ~63/50 = 400.0257{{c}}, ~3/2 = 701.7873{{c}}
: [[error map]]: {{val| +0.741 +0.799 +2.763 -6.934 }}
: [[error map]]: {{val| +0.077 -0.091 -0.072 +0.031 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~160/147 = 166.1597{{c}}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 701.7383{{c}}
: error map: {{val| 0.000 -0.434 +1.518 -8.750 }}
: error map: {{val| 0.000 -0.217 -0.220 -0.115 }}
 
[[Minimax tuning]]:
* [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis)]]: 2.5/3
* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
 
{{Optimal ET sequence|legend=1| 12, …, 159, 171, 867, 1038, 1209, 1380, 1551, 1722 }}


{{Optimal ET sequence|legend=1| 29, 36, 65 }}
[[Badness]] (Sintel): 0.505


[[Badness]] (Sintel): 4.42
=== 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.  


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 245/242, 686/675, 896/891
Comma list: 441/440, 4375/4356, 32805/32768


Mapping: {{mapping| 1 2 -1 1 0 | 0 -3 24 13 25 }}
Mapping: {{mapping| 3 0 45 94 134 | 0 1 -8 -18 -26 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.6379{{c}}, ~11/10 = 166.1853{{c}}
* WE: ~44/35 = 400.0464{{c}}, ~3/2 = 701.9053{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.1157{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8178{{c}}


{{Optimal ET sequence|legend=0| 29, 36, 65 }}
{{Optimal ET sequence|legend=0| 12, …, 159, 330 }}


Badness (Sintel): 2.26
Badness (Sintel): 1.97


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 91/90, 169/168, 245/242, 896/891
Comma list: 364/363, 441/440, 625/624, 13720/13689


Mapping: {{mapping| 1 2 -1 1 0 3 | 0 -3 24 13 25 5 }}
Mapping: {{mapping| 3 0 45 94 134 168 | 0 1 -8 -18 -26 -33 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.1361{{c}}, ~11/10 = 166.2110{{c}}
* WE: ~44/35 = 400.0449{{c}}, ~3/2 = 701.8995{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0833{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8156{{c}}
 
{{Optimal ET sequence|legend=0| 12f, …, 159, 330 }}


{{Optimal ET sequence|legend=0| 29, 65f, 94df }}
Badness (Sintel): 1.53


Badness (Sintel): 1.81
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


== Tertiaschis ==
Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619
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
Mapping: {{mapping| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }}


[[Comma list]]: 32805/32768, 1071875/1062882
Optimal tunings:  
* WE: ~34/27 = 400.0195{{c}}, ~3/2 = 701.8439{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 701.8081{{c}}


{{Mapping|legend=1| 1 2 -1 10 | 0 -3 24 -52 }}
{{Optimal ET sequence|legend=0| 12f, 159, 171, 330 }}


[[Optimal tuning]]s:  
Badness (Sintel): 1.38
* [[WE]]: ~2 = 1200.3627{{c}}, ~192/175 = 166.0691{{c}}
: [[error map]]: {{val| +0.363 +0.563 -1.019 -0.790 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~192/175 = 166.0172{{c}}
: error map: {{val| 0.000 -0.007 -1.901 -1.720 }}


{{Optimal ET sequence|legend=1| 65, 94, 159, 253, 412cd }}
=== Terminator ===
 
Terminator tempers out 540/539, and may be described as {{nowrap| 171 & 183 }}.  
[[Badness]] (Sintel): 5.36


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 385/384, 4000/3993, 19712/19683
Comma list: 540/539, 32805/32768, 137781/137500


Mapping: {{mapping| 1 2 -1 10 0 | 0 -3 24 -52 25 }}
Mapping: {{mapping| 3 0 45 94 -137 | 0 1 -8 -18 31 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.3379{{c}}, ~11/10 = 166.0638{{c}}
* WE: ~63/50 = 399.9677{{c}}, ~3/2 = 701.6278{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0167{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6846{{c}}


{{Optimal ET sequence|legend=0| 65, 94, 159, 253, 412cd, 665ccde }}
{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 537, 891de }}


Badness (Sintel): 2.07
Badness (Sintel): 2.21


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 325/324, 385/384, 1575/1573, 10985/10976
Comma list: 540/539, 729/728, 4096/4095, 31250/31213


Mapping: {{mapping| 1 2 -1 10 0 12 | 0 -3 24 -52 25 -60 }}
Mapping: {{mapping| 3 0 45 94 -137 -103 | 0 1 -8 -18 31 24 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.3467{{c}}, ~11/10 = 166.0635{{c}}
* WE: ~63/50 = 399.9731{{c}}, ~3/2 = 701.6414{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0142{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}}


{{Optimal ET sequence|legend=0| 65f, 94, 159, 253, 412cdf, 665ccdef }}
{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }}


Badness (Sintel): 1.52
Badness (Sintel): 1.47


=== 17-limit ===
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095


Mapping: {{mapping| 1 2 -1 10 0 12 -2 | 0 -3 24 -52 25 -60 44 }}
Mapping: {{mapping| 3 0 45 94 -137 -103 -2 | 0 1 -8 -18 31 24 3 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.3019{{c}}, ~11/10 = 166.0535{{c}}
* WE: ~63/50 = 399.9757{{c}}, ~3/2 = 701.6458{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0114{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}}


{{Optimal ET sequence|legend=1| 65f, 94, 159, 253 }}
{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }}


Badness (Sintel): 1.35
Badness (Sintel): 1.04


== Countertertiaschis ==
=== Semiterm ===
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.  
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
Subgroup: 2.3.5.7.11


[[Comma list]]: 32805/32768, 244140625/243045684
Comma list: 9801/9800, 32805/32768, 151263/151250


{{Mapping|legend=1| 1 2 -1 -12 | 0 -3 24 107 }}
Mapping: {{mapping| 6 0 90 188 287 | 0 1 -8 -18 -28 }}
: mapping generators: ~55/49, ~3


[[Optimal tuning]]s:  
Optimal tunings:  
* [[WE]]: ~2 = 1200.1265{{c}}, ~625/567 = 166.0797{{c}}
* WE: ~55/49 = 200.0134{{c}}, ~3/2 = 701.7931{{c}}
: [[error map]]: {{val| +0.127 +0.059 -0.529 +0.178 }}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7426{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/567 = 166.0632{{c}}
: error map: {{val| 0.000 -0.145 -0.797 -0.065 }}


{{Optimal ET sequence|legend=1| 65d, 159, 224, 383, 607 }}
{{Optimal ET sequence|legend=0| 12, …, 330e, 342, 1380, 1722, 2064, 2406c, 5154bccdde }}


[[Badness]] (Sintel): 4.76
Badness (Sintel): 0.973


=== 11-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 3025/3024, 4000/3993, 32805/32768
Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375


Mapping: {{mapping| 1 2 -1 -12 0 | 0 -3 24 107 25 }}
Mapping: {{mapping| 6 0 90 188 287 355 | 0 1 -8 -18 -28 -35 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0804{{c}}, ~11/10 = 166.0739{{c}}
* WE: ~55/49 = 200.0083{{c}}, ~3/2 = 701.7549{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0634{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7238{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 330eff, 342f, 696f }} *
 
<nowiki>*</nowiki> optimal patent val: [[354edo|354]]


{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}
Badness (Sintel): 1.85


Badness (Sintel): 1.62
=== Hemiterm ===
The hemiterm temperament tempers out [[3025/3024]] (lehmerisma), and may be described as {{nowrap| 159 & 183 }}. Its ploidacot is triploid alpha-dicot.  


=== 13-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976
Comma list: 3025/3024, 32805/32768, 102487/102400


Mapping: {{mapping| 1 2 -1 -12 0 -10 | 0 -3 24 107 25 99 }}
Mapping: {{mapping| 3 0 45 94 8 | 0 2 -16 -36 1 }}
: mapping generators: ~63/50, ~693/400


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0805{{c}}, ~11/10 = 166.0740{{c}}
* WE: ~63/50 = 400.0309{{c}}, ~693/400 = 950.9458{{c}} (~12/11 = 150.8841{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0635{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~693/400 = 950.8707{{c}} (~12/11 = 150.8707{{c}})


{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}
{{Optimal ET sequence|legend=0| 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce }}


Badness (Sintel): 1.01
Badness (Sintel): 0.684


== Term ==
==== 13-limit ====
Term tempers out the [[landscape comma]], mapping ~63/50 to the 1/3-octave period. It can be described as {{nowrap| 12 & 171 }}, and is the unique temperament that equates a syntonic~Pythagorean comma with a stack of three [[marvel comma]]s. A [[septimal comma]] is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[171edo]] makes for an excellent tuning.  
Subgroup: 2.3.5.7.11.13


[[Subgroup]]: 2.3.5.7
Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712


[[Comma list]]: 32805/32768, 250047/250000
Mapping: {{mapping| 3 0 45 94 8 42 | 0 2 -16 -36 1 -13 }}


{{Mapping|legend=1| 3 0 45 94 | 0 1 -8 -18 }}
Optimal tunings:
: mapping generators: ~63/50, ~3
* 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 tuning]]s:
{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f }}
* [[WE]]: ~63/50 = 400.0257{{c}}, ~3/2 = 701.7873{{c}}
: [[error map]]: {{val| +0.077 -0.091 -0.072 +0.031 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 701.7383{{c}}
: error map: {{val| 0.000 -0.217 -0.220 -0.115 }}


[[Minimax tuning]]:
Badness (Sintel): 1.30
* [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis)]]: 2.5/3
* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7


{{Optimal ET sequence|legend=1| 12, …, 159, 171, 867, 1038, 1209, 1380, 1551, 1722 }}
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


[[Badness]] (Sintel): 0.505
Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264


=== Terminal ===
Mapping: {{mapping| 3 0 45 94 8 42 -2 | 0 2 -16 -36 1 -13 6 }}
The terminal temperament ({{nowrap| 12 & 159 }}) tempers out 441/440 and 4375/4356. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave.


Subgroup: 2.3.5.7.11
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}})


Comma list: 441/440, 4375/4356, 32805/32768
{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f, 525f }}


Mapping: {{mapping| 3 0 45 94 134 | 0 1 -8 -18 -26 }}
Badness (Sintel): 1.14


Optimal tunings:
== Altinex ==
* WE: ~44/35 = 400.0464{{c}}, ~3/2 = 701.9053{{c}}
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.  
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8178{{c}}


{{Optimal ET sequence|legend=0| 12, …, 159, 330 }}
[[Subgroup]]: 2.3.5.7


Badness (Sintel): 1.97
[[Comma list]]: 32805/32768, 367653125/362797056


==== 13-limit ====
{{Mapping|legend=1| 3 0 45 -32 | 0 2 -16 17 }}
Subgroup: 2.3.5.7.11.13
: mapping generators: ~1536/1225, ~34300/19683


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


Mapping: {{mapping| 3 0 45 94 134 168 | 0 1 -8 -18 -26 -33 }}
{{Optimal ET sequence|legend=1| 24, 135, 159, 612ccdd }}


Optimal tunings:  
[[Badness]] (Sintel): 10.7
* WE: ~44/35 = 400.0449{{c}}, ~3/2 = 701.8995{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8156{{c}}


{{Optimal ET sequence|legend=0| 12f, …, 159, 330 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness (Sintel): 1.53
Comma list: 385/384, 14700/14641, 19712/19683


==== 17-limit ====
Mapping: {{mapping| 3 0 45 -32 8 | 0 2 -16 17 1 }}
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619
 
Mapping: {{mapping| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~34/27 = 400.0195{{c}}, ~3/2 = 701.8439{{c}}
* WE: ~44/35 = 400.1156{{c}}, ~121/70 = 951.2377{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 701.8081{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~121/70 = 950.9634{{c}}


{{Optimal ET sequence|legend=0| 12f, 159, 171, 330 }}
{{Optimal ET sequence|legend=0| 24, 135, 159 }}


Badness (Sintel): 1.38
Badness (Sintel): 3.35


=== Terminator ===
=== 13-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 540/539, 32805/32768, 137781/137500
Comma list: 364/363, 385/384, 676/675, 19712/19683


Mapping: {{mapping| 3 0 45 94 -137 | 0 1 -8 -18 31 }}
Mapping: {{mapping| 3 0 45 -32 8 42 | 0 2 -16 17 1 -13 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~63/50 = 399.9677{{c}}, ~3/2 = 701.6278{{c}}
* WE: ~44/35 = 400.1396{{c}}, ~26/15 = 951.2799{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6846{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~26/15 = 950.9462{{c}}


{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 537, 891de }}
{{Optimal ET sequence|legend=0| 24, 135f, 159 }}


Badness (Sintel): 2.21
Badness (Sintel): 2.27


==== 13-limit ====
== Squirrel ==
Subgroup: 2.3.5.7.11.13
Squirrel tempers out 686/675, the [[sengic comma]], and may be described as {{nowrap| 29 & 36 }}. It has a [[~]][[11/10]] generator, three of which give the fourth ([[4/3]]), and thirteen of which give [[7/4]] with octave reduction. Its [[ploidacot]] is omega-tricot.  


Comma list: 540/539, 729/728, 4096/4095, 31250/31213
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 3 0 45 94 -137 -103 | 0 1 -8 -18 31 24 }}
[[Comma list]]: 686/675, 32805/32768
 
{{Mapping|legend=1| 1 2 -1 1 | 0 -3 24 13 }}


Optimal tunings:  
[[Optimal tuning]]s:  
* WE: ~63/50 = 399.9731{{c}}, ~3/2 = 701.6414{{c}}
* [[WE]]: ~2 = 1200.7408{{c}}, ~160/147 = 166.2424{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{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| 12e, 171, 183, 354, 891de }}
{{Optimal ET sequence|legend=1| 29, 36, 65 }}


Badness (Sintel): 1.47
[[Badness]] (Sintel): 4.42


==== 17-limit ====
=== 11-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11


Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095
Comma list: 245/242, 686/675, 896/891


Mapping: {{mapping| 3 0 45 94 -137 -103 -2 | 0 1 -8 -18 31 24 3 }}
Mapping: {{mapping| 1 2 -1 1 0 | 0 -3 24 13 25 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~63/50 = 399.9757{{c}}, ~3/2 = 701.6458{{c}}
* WE: ~2 = 1200.6379{{c}}, ~11/10 = 166.1853{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.1157{{c}}


{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }}
{{Optimal ET sequence|legend=0| 29, 36, 65 }}


Badness (Sintel): 1.04
Badness (Sintel): 2.26


=== Semiterm ===
=== 13-limit ===
The semiterm temperament ({{nowrap| 12 & 342 }}) has a period of 1/6 octave and tempers out [[9801/9800]] (kalisma) and 151263/151250 (odiheim comma).
Subgroup: 2.3.5.7.11.13


Subgroup: 2.3.5.7.11
Comma list: 91/90, 169/168, 245/242, 896/891


Comma list: 9801/9800, 32805/32768, 151263/151250
Mapping: {{mapping| 1 2 -1 1 0 3 | 0 -3 24 13 25 5 }}
 
Mapping: {{mapping| 6 0 90 188 287 | 0 1 -8 -18 -28 }}
: mapping generators: ~55/49, ~3


Optimal tunings:  
Optimal tunings:  
* WE: ~55/49 = 200.0134{{c}}, ~3/2 = 701.7931{{c}}
* WE: ~2 = 1201.1361{{c}}, ~11/10 = 166.2110{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7426{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0833{{c}}


{{Optimal ET sequence|legend=0| 12, …, 330e, 342, 1380, 1722, 2064, 2406c, 5154bccdde }}
{{Optimal ET sequence|legend=0| 29, 65f, 94df }}


Badness (Sintel): 0.973
Badness (Sintel): 1.81


==== 13-limit ====
== Tertiaschis ==
Subgroup: 2.3.5.7.11.13
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: 1716/1715, 2080/2079, 32805/32768, 34398/34375
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 6 0 90 188 287 355 | 0 1 -8 -18 -28 -35 }}
[[Comma list]]: 32805/32768, 1071875/1062882


Optimal tunings:
{{Mapping|legend=1| 1 2 -1 10 | 0 -3 24 -52 }}
* 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, 330eff, 342f, 696f }} *
[[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 }}


<nowiki>*</nowiki> optimal patent val: [[354edo|354]]
{{Optimal ET sequence|legend=1| 65, 94, 159, 253, 412cd }}


Badness (Sintel): 1.85
[[Badness]] (Sintel): 5.36


=== Hemiterm ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 32805/32768, 102487/102400
Comma list: 385/384, 4000/3993, 19712/19683


Mapping: {{mapping| 3 0 45 94 8 | 0 2 -16 -36 1 }}
Mapping: {{mapping| 1 2 -1 10 0 | 0 -3 24 -52 25 }}
: mapping generators: ~63/50, ~693/400


Optimal tunings:  
Optimal tunings:  
* WE: ~63/50 = 400.0309{{c}}, ~693/400 = 950.9458{{c}} (~12/11 = 150.8841{{c}})
* WE: ~2 = 1200.3379{{c}}, ~11/10 = 166.0638{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~693/400 = 950.8707{{c}} (~12/11 = 150.8707{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0167{{c}}


{{Optimal ET sequence|legend=0| 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce }}
{{Optimal ET sequence|legend=0| 65, 94, 159, 253, 412cd, 665ccde }}


Badness (Sintel): 0.684
Badness (Sintel): 2.07


==== 13-limit ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712
Comma list: 325/324, 385/384, 1575/1573, 10985/10976


Mapping: {{mapping| 3 0 45 94 8 42 | 0 2 -16 -36 1 -13 }}
Mapping: {{mapping| 1 2 -1 10 0 12 | 0 -3 24 -52 25 -60 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~63/50 = 400.0541{{c}}, ~26/15 = 951.0013{{c}} (~12/11 = 150.8932{{c}})
* WE: ~2 = 1200.3467{{c}}, ~11/10 = 166.0635{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~26/15 = 950.8696{{c}} (~12/11 = 150.8696{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0142{{c}}


{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f }}
{{Optimal ET sequence|legend=0| 65f, 94, 159, 253, 412cdf, 665ccdef }}


Badness (Sintel): 1.30
Badness (Sintel): 1.52


==== 17-limit ====
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264
Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976


Mapping: {{mapping| 3 0 45 94 8 42 -2 | 0 2 -16 -36 1 -13 6 }}
Mapping: {{mapping| 1 2 -1 10 0 12 -2 | 0 -3 24 -52 25 -60 44 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~34/27 = 400.0373{{c}}, ~26/15 = 950.9556{{c}} (~12/11 = 150.8809{{c}})
* WE: ~2 = 1200.3019{{c}}, ~11/10 = 166.0535{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~26/15 = 950.8652{{c}} (~12/11 = 150.8652{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0114{{c}}
 
{{Optimal ET sequence|legend=1| 65f, 94, 159, 253 }}


{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f, 525f }}
Badness (Sintel): 1.35


Badness (Sintel): 1.14
== Countertertiaschis ==
Named by [[Flora Canou]] in 2021, Countertertiaschis may be described as {{nowrap| 159 & 224 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 244140625/243045684 for prime 7.  


== Altinex ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 32805/32768, 367653125/362797056
[[Comma list]]: 32805/32768, 244140625/243045684


{{Mapping|legend=1| 3 0 45 -32 | 0 2 -16 17 }}
{{Mapping|legend=1| 1 2 -1 -12 | 0 -3 24 107 }}
: mapping generators: ~1536/1225, ~34300/19683


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~1536/1225 = 400.1360{{c}}, ~34300/19683 = 951.2867{{c}}
* [[WE]]: ~2 = 1200.1265{{c}}, ~625/567 = 166.0797{{c}}
: [[error map]]: {{val| +0.408 +0.618 -0.781 -1.304 }}
: [[error map]]: {{val| +0.127 +0.059 -0.529 +0.178 }}
* [[CWE]]: ~1536/1225 = 400.0000{{c}}, ~34300/19683 = 950.9638{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/567 = 166.0632{{c}}
: error map: {{val| 0.000 -0.027 -1.735 -2.441 }}
: error map: {{val| 0.000 -0.145 -0.797 -0.065 }}


{{Optimal ET sequence|legend=1| 24, 135, 159, 612ccdd }}
{{Optimal ET sequence|legend=1| 65d, 159, 224, 383, 607 }}


[[Badness]] (Sintel): 10.7
[[Badness]] (Sintel): 4.76


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 385/384, 14700/14641, 19712/19683
Comma list: 3025/3024, 4000/3993, 32805/32768


Mapping: {{mapping| 3 0 45 -32 8 | 0 2 -16 17 1 }}
Mapping: {{mapping| 1 2 -1 -12 0 | 0 -3 24 107 25 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~44/35 = 400.1156{{c}}, ~121/70 = 951.2377{{c}}
* WE: ~2 = 1200.0804{{c}}, ~11/10 = 166.0739{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~121/70 = 950.9634{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0634{{c}}


{{Optimal ET sequence|legend=0| 24, 135, 159 }}
{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}


Badness (Sintel): 3.35
Badness (Sintel): 1.62


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 364/363, 385/384, 676/675, 19712/19683
Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976


Mapping: {{mapping| 3 0 45 -32 8 42 | 0 2 -16 17 1 -13 }}
Mapping: {{mapping| 1 2 -1 -12 0 -10 | 0 -3 24 107 25 99 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~44/35 = 400.1396{{c}}, ~26/15 = 951.2799{{c}}
* WE: ~2 = 1200.0805{{c}}, ~11/10 = 166.0740{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~26/15 = 950.9462{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0635{{c}}
 
{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}


{{Optimal ET sequence|legend=0| 24, 135f, 159 }}
Badness (Sintel): 1.01


Badness (Sintel): 2.27
== 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.  


== Sesquiquartififths ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 32805/32768
[[Comma list]]: 32805/32768, 390625/388962


{{Mapping|legend=1| 1 1 7 5 | 0 4 -32 -15 }}
{{Mapping|legend=1| 4 0 60 119 | 0 1 -8 -17 }}
: mapping generators: ~2, ~448/405
: mapping generators: ~25/21, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0846{{c}}, ~448/405 = 175.4460{{c}}
* [[WE]]: ~2 = 300.0255{{c}}, ~3/2 = 701.8831{{c}}
: [[error map]]: {{val| +0.085 -0.086 +0.007 -0.093 }}
: [[error map]]: {{val| +0.102 +0.030 -0.664 +0.462 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~448/405 = 175.4320{{c}}
* [[CWE]]: ~2 = 300.0000{{c}}, ~3/2 = 701.8180{{c}}
: error map: {{val| 0.000 -0.227 -0.137 -0.306 }}
: error map: {{val| 0.000 -0.137 -0.858 +0.268 }}


[[Minimax tuning]]:
{{Optimal ET sequence|legend=1| 12, …, 200, 212, 224, 436, 660 }}
* [[7-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7


{{Optimal ET sequence|legend=1| 41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd }}
[[Badness]] (Sintel): 2.79


[[Badness]] (Sintel): 0.285
=== 11-limit ===
 
=== Sesquart ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 243/242, 441/440, 16384/16335
Comma list: 1375/1372, 6250/6237, 32805/32768


Mapping: {{mapping| 1 1 7 5 2 | 0 4 -32 -15 10 }}
Mapping: {{mapping| 4 0 60 119 185 | 0 1 -8 -17 -27 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.8171{{c}}, ~256/231 = 175.3793{{c}}
* WE: ~25/21 = 300.0244{{c}}, ~3/2 = 701.8759{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~256/231 = 175.4081{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8145{{c}}


{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}
{{Optimal ET sequence|legend=0| 12, …, 212, 224, 436, 660 }}


Badness (Sintel): 0.969
Badness (Sintel): 1.51


==== 13-limit ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 243/242, 364/363, 441/440, 3584/3575
Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647


Mapping: {{mapping| 1 1 7 5 2 -2 | 0 4 -32 -15 10 39 }}
Mapping: {{mapping| 4 0 60 119 185 224 | 0 1 -8 -17 -27 -33 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.8352{{c}}, ~72/65 = 175.3852{{c}}
* WE: ~25/21 = 300.0234{{c}}, ~3/2 = 701.8707{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4095{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8123{{c}}


{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}
{{Optimal ET sequence|legend=0| 12f, …, 212, 224, 436, 660 }}


Badness (Sintel): 0.925
Badness (Sintel): 1.13


===== Sesquartia =====
== Sesquiquartififths ==
Subgroup: 2.3.5.7.11.13.17
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.  


Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }}
[[Comma list]]: 2401/2400, 32805/32768


Optimal tunings:
{{Mapping|legend=1| 1 1 7 5 | 0 4 -32 -15 }}
* WE: ~2 = 1199.8902{{c}}, ~72/65 = 175.4077{{c}}
: mapping generators: ~2, ~448/405
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4234{{c}}


{{Optimal ET sequence|legend=0| 41, 130, 171 }}
[[Optimal tuning]]s:
* [[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 }}


Badness (Sintel): 1.18
[[Minimax tuning]]:
* [[7-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7


====== 19-limit ======
{{Optimal ET sequence|legend=1| 41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd }}
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594
[[Badness]] (Sintel): 0.285


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


Optimal tunings:  
Subgroup: 2.3.5.7.11
* WE: ~2 = 1199.9864{{c}}, ~21/19 = 175.4169{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4189{{c}}


{{Optimal ET sequence|legend=0| 41, 130, 171 }}
Comma list: 243/242, 441/440, 16384/16335


Badness (Sintel): 1.24
Mapping: {{mapping| 1 1 7 5 2 | 0 4 -32 -15 10 }}
 
Optimal tunings:
* WE: ~2 = 1199.8171{{c}}, ~256/231 = 175.3793{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~256/231 = 175.4081{{c}}
 
{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}
 
Badness (Sintel): 0.969


====== 23-limit ======
==== 13-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13


Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594
Comma list: 243/242, 364/363, 441/440, 3584/3575


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9606{{c}}, ~21/19 = 175.4067{{c}}
* WE: ~2 = 1199.8352{{c}}, ~72/65 = 175.3852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4123{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4095{{c}}


{{Optimal ET sequence|legend=0| 41i, 130, 171 }}
{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}


Badness (Sintel): 1.36
Badness (Sintel): 0.925


===== Heartia =====
===== Heartia =====
Line 1,956: Line 2,023:
Badness (Sintel): 1.40
Badness (Sintel): 1.40


===== Hearty =====
===== Sesquartia =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625
Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575


Mapping: {{mapping| 1 1 7 5 2 -2 13 | 0 4 -32 -15 10 39 -61 }}
Mapping: {{mapping| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9458{{c}}, ~72/65 = 175.3689{{c}}
* WE: ~2 = 1199.8902{{c}}, ~72/65 = 175.4077{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3770{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4234{{c}}


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


Badness (Sintel): 1.56
Badness (Sintel): 1.18


====== 19-limit ======
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 221/220, 243/242, 361/360, 364/363, 441/440, 456/455
Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594


Mapping: {{mapping| 1 1 7 5 2 -2 13 6 | 0 4 -32 -15 10 39 -61 -12 }}
Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0114{{c}}, ~72/65 = 175.3783{{c}}
* WE: ~2 = 1199.9864{{c}}, ~21/19 = 175.4169{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3765{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4189{{c}}


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


Badness (Sintel): 1.39
Badness (Sintel): 1.24


====== 23-limit ======
====== 23-limit ======
Subgroup: 2.3.5.7.11.13.17.19.23
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
Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594


Mapping: {{mapping| 1 1 7 5 2 -2 13 6 13 | 0 4 -32 -15 10 39 -61 -12 -58 }}
Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 -6 | 0 4 -32 -15 10 39 69 -12 72 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0122{{c}}, ~72/65 = 175.3782{{c}}
* WE: ~2 = 1199.9606{{c}}, ~21/19 = 175.4067{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3763{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4123{{c}}


{{Optimal ET sequence|legend=0| 41g, 89, 130 }}
{{Optimal ET sequence|legend=0| 41i, 130, 171 }}


Badness (Sintel): 1.37
Badness (Sintel): 1.36


=== Bisesqui ===
===== Hearty =====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17


Comma list: 2401/2400, 9801/9800, 32805/32768
Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625


Mapping: {{mapping| 2 2 14 10 23 | 0 4 -32 -15 -55 }}
Mapping: {{mapping| 1 1 7 5 2 -2 13 | 0 4 -32 -15 10 39 -61 }}
: mapping generators: ~99/70, ~448/405


Optimal tunings:  
Optimal tunings:  
* WE: ~99/70 = 600.0429{{c}}, ~448/405 = 175.4474{{c}}
* WE: ~2 = 1199.9458{{c}}, ~72/65 = 175.3689{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~448/405 = 175.4334{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3770{{c}}


{{Optimal ET sequence|legend=1| 82e, 130, 212, 342, 1156, 1498, 1840d, 5862bbccdddee }}
{{Optimal ET sequence|legend=0| 41g, 89, 130 }}


Badness (Sintel): 0.561
Badness (Sintel): 1.56


== Quintilipyth ==
====== 19-limit ======
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: 2.3.5.7.11.13.17.19


[[Subgroup]]: 2.3.5.7
Comma list: 221/220, 243/242, 361/360, 364/363, 441/440, 456/455


[[Comma list]]: 32805/32768, 9765625/9680832
Mapping: {{mapping| 1 1 7 5 2 -2 13 6 | 0 4 -32 -15 10 39 -61 -12 }}


{{Mapping|legend=1| 1 2 -1 -4 | 0 -5 40 82 }}
Optimal tunings:
: mapping generators: ~2, ~625/588
* WE: ~2 = 1200.0114{{c}}, ~72/65 = 175.3783{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3765{{c}}


[[Optimal tuning]]s:
{{Optimal ET sequence|legend=0| 41g, 89, 130 }}
* [[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 }}


{{Optimal ET sequence|legend=1| 12, …, 253, 265 }}
Badness (Sintel): 1.39


[[Badness]] (Sintel): 6.43
====== 23-limit ======
Subgroup: 2.3.5.7.11.13.17.19.23


=== 11-limit ===
Comma list: 221/220, 243/242, 276/275, 323/322, 361/360, 364/363, 441/440
Subgroup: 2.3.5.7.11


Comma list: 1375/1372, 4375/4356, 32805/32768
Mapping: {{mapping| 1 1 7 5 2 -2 13 6 13 | 0 4 -32 -15 10 39 -61 -12 -58 }}
 
Mapping: {{mapping| 1 2 -1 -4 -7 | 0 -5 40 82 126 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.1503{{c}}, ~35/33 = 99.6287{{c}}
* WE: ~2 = 1200.0122{{c}}, ~72/65 = 175.3782{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6176{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3763{{c}}


{{Optimal ET sequence|legend=0| 12, …, 253, 265, 518c }}
{{Optimal ET sequence|legend=0| 41g, 89, 130 }}


Badness (Sintel): 3.74
Badness (Sintel): 1.37


=== 13-limit ===
=== Bisesqui ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11


Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647
Comma list: 2401/2400, 9801/9800, 32805/32768


Mapping: {{mapping| 1 2 -1 -4 -7 -9 | 0 -5 40 82 126 153 }}
Mapping: {{mapping| 2 2 14 10 23 | 0 4 -32 -15 -55 }}
: mapping generators: ~99/70, ~448/405


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.1774{{c}}, ~35/33 = 99.6267{{c}}
* WE: ~99/70 = 600.0429{{c}}, ~448/405 = 175.4474{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6134{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~448/405 = 175.4334{{c}}


{{Optimal ET sequence|legend=0| 12f, …, 241cdef, 253 }}
{{Optimal ET sequence|legend=1| 82e, 130, 212, 342, 1156, 1498, 1840d, 5862bbccdddee }}


Badness (Sintel): 2.86
Badness (Sintel): 0.561


=== 17-limit ===
== Tsaharuk ==
Subgroup: 2.3.5.7.11.13.17
{{Main| Tsaharuk }}


Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619
Tsaharuk tempers out 420175/419904, the [[wizma]], and may be described as the {{nowrap| 77 & 94 }} temperament. It is generated by a slightly flat neutral second of [[~]][[13/12]], five of which make the [[3/2|perfect fifth]], so its [[ploidacot]] is pentacot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 32805/32768, 420175/419904


Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 | 0 -5 40 82 126 153 -11 }}
{{Mapping|legend=1| 1 1 7 0 | 0 5 -40 24 }}
: mapping generators: ~2, ~243/224


Optimal tunings:  
[[Optimal tuning]]s:  
* WE: ~2 = 1200.1745{{c}}, ~18/17 = 99.6265{{c}}
* [[WE]]: ~2 = 1200.1039{{c}}, ~243/224 = 140.3620{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6131{{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, 241cdef, 253 }}
{{Optimal ET sequence|legend=1| 17, 77, 94, 171 }}


Badness (Sintel): 2.34
[[Badness]] (Sintel): 0.777


=== 19-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11


Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971
Comma list: 385/384, 1331/1323, 19712/19683


Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 4 | 0 -5 40 82 126 153 -11 3 }}
Mapping: {{mapping| 1 1 7 0 1 | 0 5 -40 24 21 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0713{{c}}, ~18/17 = 99.6208{{c}}
* WE: ~2 = 1200.3103{{c}}, ~88/81 = 140.4011{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.3649{{c}}


{{Optimal ET sequence|legend=0| 12f, 253, 265 }}
{{Optimal ET sequence|legend=0| 17, 77, 94, 171e, 265e }}


Badness (Sintel): 2.32
Badness (Sintel): 2.10
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 385/384, 729/728, 1331/1323
 
Mapping: {{mapping| 1 1 7 0 1 3 | 0 5 -40 24 21 6 }}
 
Optimal tunings:
* WE: ~2 = 1200.1840{{c}}, ~13/12 = 140.3840{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.3627{{c}}


== Quintaschis ==
{{Optimal ET sequence|legend=0| 17, 77, 94, 171e }}
The quintaschis temperament ({{nowrap| 12 & 289 }}) slices the fourth (4/3) into five semitones and tempers out 49009212/48828125 in the 7-limit.
 
Badness (Sintel): 1.57
 
== Quanharuk ==
Quanharuk tempers out 16875/16807, the [[mirkwai]] comma, and may be described as the {{nowrap| 41 & 183 }} temperament. The generator is a slightly flat major third of [[~]][[56/45]], five of which make the [[3/1|3rd]] [[harmonic]], so the [[ploidacot]] of this temperament is alpha-pentacot. [[224edo]] makes for a recommendable tuning.  


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


[[Comma list]]: 32805/32768, 49009212/48828125
[[Comma list]]: 16875/16807, 32805/32768


{{Mapping|legend=1| 1 2 -1 -5 | 0 -5 40 94 }}
{{Mapping|legend=1| 1 0 15 12 | 0 5 -40 -29 }}
: mapping generators: ~2, ~56/45


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0536{{c}}, ~200/189 = 99.6684{{c}}
* [[WE]]: ~2 = 1200.0032{{c}}, ~56/45 = 380.3557{{c}}
: [[error map]]: {{val| +0.054 -0.190 +0.370 -0.262 }}
: [[error map]]: {{val| +0.003 -0.177 -0.493 +0.898 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~200/189 = 99.6645{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~56/45 = 380.3546{{c}}
: error map: {{val| 0.000 -0.277 +0.266 -0.363 }}
: error map: {{val| 0.000 -0.182 -0.498 +0.890 }}


{{Optimal ET sequence|legend=1| 12, …, 289, 301, 590, 891, 1192 }}
{{Optimal ET sequence|legend=1| 41, 142, 183, 224 }}


[[Badness]] (Sintel): 3.36
[[Badness]] (Sintel): 1.82


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 441/440, 32805/32768, 1953125/1951488
Comma list: 540/539, 1375/1372, 32805/32768


Mapping: {{mapping| 1 2 -1 -5 -8 | 0 -5 40 94 138 }}
Mapping: {{mapping| 1 0 15 12 -7 | 0 5 -40 -29 33 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0988{{c}}, ~35/33 = 99.6613{{c}}
* WE: ~2 = 1199.9709{{c}}, ~56/45 = 380.3423{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6540{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3517{{c}}


{{Optimal ET sequence|legend=0| 12, …, 277d, 289 }}
{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}


Badness (Sintel): 3.69
Badness (Sintel): 1.04


==== 13-limit ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 364/363, 441/440, 32805/32768, 109512/109375
Comma list: 540/539, 729/728, 1375/1372, 4096/4095


Mapping: {{mapping| 1 2 -1 -5 -8 -11 | 0 -5 40 94 138 177 }}
Mapping: {{mapping| 1 0 15 12 -7 -15 | 0 5 -40 -29 33 59 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0625{{c}}, ~35/33 = 99.6630{{c}}
* WE: ~2 = 1199.9663{{c}}, ~56/45 = 380.3403{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6583{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3509{{c}}


{{Optimal ET sequence|legend=0| 12f, …, 277dff, 289 }}
{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}


Badness (Sintel): 3.07
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: 364/363, 441/440, 595/594, 3757/3750, 32805/32768
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 | 0 -5 40 94 138 177 -11 }}
[[Comma list]]: 32805/32768, 9765625/9680832


Optimal tunings:  
{{Mapping|legend=1| 1 2 -1 -4 | 0 -5 40 82 }}
* WE: ~2 = 1200.1286{{c}}, ~18/17 = 99.6668{{c}}
: mapping generators: ~2, ~625/588
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6568{{c}}
 
[[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 }}


{{Optimal ET sequence|legend=0| 12f, 277dff, 289 }}
{{Optimal ET sequence|legend=1| 12, …, 253, 265 }}


Badness (Sintel): 2.58
[[Badness]] (Sintel): 6.43


==== 19-limit ====
=== 11-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11


Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859
Comma list: 1375/1372, 4375/4356, 32805/32768


Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 4 | 0 -5 40 94 138 177 -11 3 }}
Mapping: {{mapping| 1 2 -1 -4 -7 | 0 -5 40 82 126 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0289{{c}}, ~18/17 = 99.6609{{c}}
* WE: ~2 = 1200.1503{{c}}, ~35/33 = 99.6287{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6586{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6176{{c}}


{{Optimal ET sequence|legend=0| 12f, 289 }}
{{Optimal ET sequence|legend=0| 12, …, 253, 265, 518c }}


Badness (Sintel): 2.56
Badness (Sintel): 3.74


=== Quintahelenic ===
=== 13-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 5632/5625, 8019/8000, 151263/151250
Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647


Mapping: {{mapping| 1 2 -1 -5 -9 | 0 -5 40 94 150 }}
Mapping: {{mapping| 1 2 -1 -4 -7 -9 | 0 -5 40 82 126 153 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0195{{c}}, ~200/189 = 99.6723{{c}}
* WE: ~2 = 1200.1774{{c}}, ~35/33 = 99.6267{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6709{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6134{{c}}


{{Optimal ET sequence|legend=0| 12, …, 289e, 301, 915 }}
{{Optimal ET sequence|legend=0| 12f, …, 241cdef, 253 }}


Badness (Sintel): 2.72
Badness (Sintel): 2.86


==== 13-limit ====
=== 17-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17


Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000
Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619


Mapping: {{mapping| 1 2 -1 -5 -9 -11 | 0 -5 40 94 150 177 }}
Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 | 0 -5 40 82 126 153 -11 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0442{{c}}, ~200/189 = 99.6709{{c}}
* WE: ~2 = 1200.1745{{c}}, ~18/17 = 99.6265{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6675{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6131{{c}}


{{Optimal ET sequence|legend=0| 12f, …, 289e, 301 }}
{{Optimal ET sequence|legend=0| 12f, 241cdef, 253 }}


Badness (Sintel): 2.30
Badness (Sintel): 2.34


===== 17-limit =====
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750
Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971


Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 | 0 -5 40 94 150 177 -11 }}
Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 4 | 0 -5 40 82 126 153 -11 3 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.1227{{c}}, ~200/189 = 99.6753{{c}}
* WE: ~2 = 1200.0713{{c}}, ~18/17 = 99.6208{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6658{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6152{{c}}


{{Optimal ET sequence|legend=1| 12f, 289e, 301 }}
{{Optimal ET sequence|legend=0| 12f, 253, 265 }}


Badness (Sintel): 2.06
Badness (Sintel): 2.32


===== 19-limit =====
== Quintaschis ==
Subgroup: 2.3.5.7.11.13.17.19
Named by [[Xenllium]] in 2021, quintaschis slices the [[4/3|perfect fourth]] into five semitones and tempers out 49009212/48828125 in the [[7-limit]]. It may be described as the {{nowrap| 12 & 289 }} temperament, and its [[ploidacot]] is omega-pentacot.  


Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 4 | 0 -5 40 94 150 177 -11 3 }}
[[Comma list]]: 32805/32768, 49009212/48828125


Optimal tunings:
{{Mapping|legend=1| 1 2 -1 -5 | 0 -5 40 94 }}
* WE: ~2 = 1200.0230{{c}}, ~200/189 = 99.6694{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6676{{c}}


{{Optimal ET sequence|legend=0| 12f, 301 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0536{{c}}, ~200/189 = 99.6684{{c}}
: [[error map]]: {{val| +0.054 -0.190 +0.370 -0.262 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~200/189 = 99.6645{{c}}
: error map: {{val| 0.000 -0.277 +0.266 -0.363 }}


Badness (Sintel): 2.24
{{Optimal ET sequence|legend=1| 12, …, 289, 301, 590, 891, 1192 }}


==== Quintahelenoid ====
[[Badness]] (Sintel): 3.36
Subgroup: 2.3.5.7.11.13


Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: {{mapping| 1 2 -1 -5 -9 14 | 0 -5 40 94 150 -124 }}
Comma list: 441/440, 32805/32768, 1953125/1951488
 
Mapping: {{mapping| 1 2 -1 -5 -8 | 0 -5 40 94 138 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9919{{c}}, ~200/189 = 99.6712{{c}}
* WE: ~2 = 1200.0988{{c}}, ~35/33 = 99.6613{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6718{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6540{{c}}


{{Optimal ET sequence|legend=0| 12, 301, 614, 915 }}
{{Optimal ET sequence|legend=0| 12, …, 277d, 289 }}


Badness (Sintel): 2.73
Badness (Sintel): 3.69


===== 17-limit =====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13


Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157
Comma list: 364/363, 441/440, 32805/32768, 109512/109375


Mapping: {{mapping| 1 2 -1 -5 -9 14 5 | 0 -5 40 94 150 -124 -11 }}
Mapping: {{mapping| 1 2 -1 -5 -8 -11 | 0 -5 40 94 138 177 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0469{{c}}, ~18/17 = 99.6749{{c}}
* WE: ~2 = 1200.0625{{c}}, ~35/33 = 99.6630{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6710{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6583{{c}}


{{Optimal ET sequence|legend=0| 12, 301 }}
{{Optimal ET sequence|legend=0| 12f, …, 277dff, 289 }}


Badness (Sintel): 2.44
Badness (Sintel): 3.07


===== 19-limit =====
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17


Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137
Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768


Mapping: {{mapping| 1 2 -1 -5 -9 14 5 4 | 0 -5 40 94 150 -124 -11 3 }}
Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 | 0 -5 40 94 138 177 -11 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9925{{c}}, ~18/17 = 99.6710{{c}}
* WE: ~2 = 1200.1286{{c}}, ~18/17 = 99.6668{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6716{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6568{{c}}


{{Optimal ET sequence|legend=0| 12, 301 }}
{{Optimal ET sequence|legend=0| 12f, 277dff, 289 }}


Badness (Sintel): 2.41
Badness (Sintel): 2.58


== Sextilifourths ==
==== 19-limit ====
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: 2.3.5.7.11.13.17.19


[[Subgroup]]: 2.3.5.7
Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859


[[Comma list]]: 32805/32768, 235298/234375
Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 4 | 0 -5 40 94 138 177 -11 3 }}


{{Mapping|legend=1| 1 2 -1 -1 | 0 -6 48 55 }}
Optimal tunings:
: mapping generators: ~2, ~21/20
* WE: ~2 = 1200.0289{{c}}, ~18/17 = 99.6609{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6586{{c}}


[[Optimal tuning]]s:
{{Optimal ET sequence|legend=0| 12f, 289 }}
* [[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| 29, 72cd, 101, 130, 289, 419 }}
Badness (Sintel): 2.56


[[Badness]] (Sintel): 2.75
=== Quintahelenic ===
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 441/440, 4000/3993, 235298/234375
Comma list: 5632/5625, 8019/8000, 151263/151250


Mapping: {{mapping| 1 2 -1 -1 0 | 0 -6 48 55 50 }}
Mapping: {{mapping| 1 2 -1 -5 -9 | 0 -5 40 94 150 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0424{{c}}, ~21/20 = 83.0520{{c}}
* WE: ~2 = 1200.0195{{c}}, ~200/189 = 99.6723{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0497{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6709{{c}}


{{Optimal ET sequence|legend=0| 29, 72cde, 101e, 130, 289 }}
{{Optimal ET sequence|legend=0| 12, …, 289e, 301, 915 }}


Badness (Sintel): 1.50
Badness (Sintel): 2.72


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 364/363, 441/440, 676/675, 10985/10976
Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000


Mapping: {{mapping| 1 2 -1 -1 0 1 | 0 -6 48 55 50 39 }}
Mapping: {{mapping| 1 2 -1 -5 -9 -11 | 0 -5 40 94 150 177 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.1056{{c}}, ~21/20 = 83.0566{{c}}
* WE: ~2 = 1200.0442{{c}}, ~200/189 = 99.6709{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0508{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6675{{c}}


{{Optimal ET sequence|legend=0| 29, 72cdef, 101e, 130, 289 }}
{{Optimal ET sequence|legend=0| 12f, …, 289e, 301 }}


Badness (Sintel): 1.04
Badness (Sintel): 2.30


== Septiquarschis ==
===== 17-limit =====
The septiquarschis temperament ({{nowrap| 89 & 94 }}) splits septimal minor seventh ([[7/4]]) into four generators and tempers out 829440/823543 (mynaslender comma) and 67108864/66706983 (septiness comma).
Subgroup: 2.3.5.7.11.13.17


[[Subgroup]]: 2.3.5.7
Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750


[[Comma list]]: 32805/32768, 829440/823543
Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 | 0 -5 40 94 150 177 -11 }}


{{Mapping|legend=1| 1 -4 47 6 | 0 7 56 -4 }}
Optimal tunings:
: mapping generators: ~2, ~256/147
* WE: ~2 = 1200.1227{{c}}, ~200/189 = 99.6753{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6658{{c}}


[[Optimal tuning]]s:
{{Optimal ET sequence|legend=1| 12f, 289e, 301 }}
* [[WE]]: ~2 = 1199.8855{{c}}, ~256/147 = 957.2944{{c}}
: [[error map]]: {{val| -0.114 -0.436 -0.182 +1.310 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~256/147 = 957.3867{{c}}
: error map: {{val| 0.000 -0.248 +0.032 +1.627 }}


{{Optimal ET sequence|legend=1| 89, 94, 183, 460d, 643d }}
Badness (Sintel): 2.06


[[Badness]] (Sintel): 4.73
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


=== 11-limit ===
Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700
Subgroup: 2.3.5.7.11


Comma list: 540/539, 15488/15435, 32805/32768
Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 4 | 0 -5 40 94 150 177 -11 3 }}
 
Mapping: {{mapping| 1 -4 47 6 25 | 0 7 56 -4 -27 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9430{{c}}, ~256/147 = 957.3390{{c}}
* WE: ~2 = 1200.0230{{c}}, ~200/189 = 99.6694{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3849{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6676{{c}}


{{Optimal ET sequence|legend=0| 89, 94, 183, 460d }}
{{Optimal ET sequence|legend=0| 12f, 301 }}


Badness (Sintel): 1.72
Badness (Sintel): 2.24


=== 13-limit ===
==== Quintahelenoid ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 540/539, 729/728, 1573/1568, 4096/4095
Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436


Mapping: {{mapping| 1 -4 47 6 25 -33 | 0 7 56 -4 -27 46 }}
Mapping: {{mapping| 1 2 -1 -5 -9 14 | 0 -5 40 94 150 -124 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0058{{c}}, ~256/147 = 957.3946{{c}}
* WE: ~2 = 1199.9919{{c}}, ~200/189 = 99.6712{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3900{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6718{{c}}


{{Optimal ET sequence|legend=0| 89, 94, 183, 277, 460d }}
{{Optimal ET sequence|legend=0| 12, 301, 614, 915 }}


Badness (Sintel): 1.46
Badness (Sintel): 2.73


== Tsaharuk ==
===== 17-limit =====
{{Main| Tsaharuk }}
Subgroup: 2.3.5.7.11.13.17


[[Subgroup]]: 2.3.5.7
Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157


[[Comma list]]: 32805/32768, 420175/419904
Mapping: {{mapping| 1 2 -1 -5 -9 14 5 | 0 -5 40 94 150 -124 -11 }}


{{Mapping|legend=1| 1 1 7 0 | 0 5 -40 24 }}
Optimal tunings:
: mapping generators: ~2, ~243/224
* WE: ~2 = 1200.0469{{c}}, ~18/17 = 99.6749{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6710{{c}}


[[Optimal tuning]]s:
{{Optimal ET sequence|legend=0| 12, 301 }}
* [[WE]]: ~2 = 1200.1039{{c}}, ~243/224 = 140.3620{{c}}
: [[error map]]: {{val| +0.104 -0.041 -0.067 -0.137 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/224 = 140.3496{{c}}
: error map: {{val| 0.000 -0.207 -0.296 -0.436 }}


{{Optimal ET sequence|legend=1| 17, 77, 94, 171 }}
Badness (Sintel): 2.44


[[Badness]] (Sintel): 0.777
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


=== 11-limit ===
Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137
Subgroup: 2.3.5.7.11


Comma list: 385/384, 1331/1323, 19712/19683
Mapping: {{mapping| 1 2 -1 -5 -9 14 5 4 | 0 -5 40 94 150 -124 -11 3 }}
 
Mapping: {{mapping| 1 1 7 0 1 | 0 5 -40 24 21 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.3103{{c}}, ~88/81 = 140.4011{{c}}
* WE: ~2 = 1199.9925{{c}}, ~18/17 = 99.6710{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.3649{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6716{{c}}


{{Optimal ET sequence|legend=0| 17, 77, 94, 171e, 265e }}
{{Optimal ET sequence|legend=0| 12, 301 }}


Badness (Sintel): 2.10
Badness (Sintel): 2.41


=== 13-limit ===
== Sextilifourths ==
Subgroup: 2.3.5.7.11.13
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.  


Comma list: 352/351, 385/384, 729/728, 1331/1323
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 1 7 0 1 3 | 0 5 -40 24 21 6 }}
[[Comma list]]: 32805/32768, 235298/234375


Optimal tunings:
{{Mapping|legend=1| 1 2 -1 -1 | 0 -6 48 55 }}
* WE: ~2 = 1200.1840{{c}}, ~13/12 = 140.3840{{c}}
: mapping generators: ~2, ~21/20
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.3627{{c}}


{{Optimal ET sequence|legend=0| 17, 77, 94, 171e }}
[[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 }}


Badness (Sintel): 1.57
{{Optimal ET sequence|legend=1| 29, 72cd, 101, 130, 289, 419 }}


== Quanharuk ==
[[Badness]] (Sintel): 2.75
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 16875/16807, 32805/32768
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Mapping|legend=1| 1 0 15 12 | 0 5 -40 -29 }}
Comma list: 441/440, 4000/3993, 235298/234375
: mapping generators: ~2, ~56/45
 
Mapping: {{mapping| 1 2 -1 -1 0 | 0 -6 48 55 50 }}


[[Optimal tuning]]s:  
Optimal tunings:  
* [[WE]]: ~2 = 1200.0032{{c}}, ~56/45 = 380.3557{{c}}
* WE: ~2 = 1200.0424{{c}}, ~21/20 = 83.0520{{c}}
: [[error map]]: {{val| +0.003 -0.177 -0.493 +0.898 }}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0497{{c}}
* [[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| 41, 142, 183, 224 }}
{{Optimal ET sequence|legend=0| 29, 72cde, 101e, 130, 289 }}


[[Badness]] (Sintel): 1.82
Badness (Sintel): 1.50


=== 11-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 540/539, 1375/1372, 32805/32768
Comma list: 364/363, 441/440, 676/675, 10985/10976


Mapping: {{mapping| 1 0 15 12 -7 | 0 5 -40 -29 33 }}
Mapping: {{mapping| 1 2 -1 -1 0 1 | 0 -6 48 55 50 39 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9709{{c}}, ~56/45 = 380.3423{{c}}
* WE: ~2 = 1200.1056{{c}}, ~21/20 = 83.0566{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3517{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0508{{c}}


{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}
{{Optimal ET sequence|legend=0| 29, 72cdef, 101e, 130, 289 }}


Badness (Sintel): 1.04
Badness (Sintel): 1.04


=== 13-limit ===
== Septant ==
Subgroup: 2.3.5.7.11.13
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.  


Comma list: 540/539, 729/728, 1375/1372, 4096/4095
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 0 15 12 -7 -15 | 0 5 -40 -29 33 59 }}
[[Comma list]]: 32805/32768, 516560652/514714375


Optimal tunings:
{{Mapping|legend=1| 7 0 105 -56 | 0 1 -8 7 }}
* WE: ~2 = 1199.9663{{c}}, ~56/45 = 380.3403{{c}}
: mapping generators: ~8575/7776, ~3
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3509{{c}}
 
{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}
 
Badness (Sintel): 0.884
 
== Quadrant ==
The ''quadrant'' temperament ({{nowrap| 12 & 224 }}) has a period of quarter octave and tempers out the [[dimcomp comma]], 390625/388962. In this temperament, 25/21 is mapped into quarter octave.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 32805/32768, 390625/388962
 
{{Mapping|legend=1| 4 0 60 119 | 0 1 -8 -17 }}
: mapping generators: ~25/21, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 300.0255{{c}}, ~3/2 = 701.8831{{c}}
* [[WE]]: ~8575/7776 = 171.4303{{c}}, ~3/2 = 701.7091{{c}}
: [[error map]]: {{val| +0.102 +0.030 -0.664 +0.462 }}
: [[error map]]: {{val| +0.012 -0.234 +0.096 +0.265 }}
* [[CWE]]: ~2 = 300.0000{{c}}, ~3/2 = 701.8180{{c}}
* [[CWE]]: ~8575/7776 = 171.4286{{c}}, ~3/2 = 701.7022{{c}}
: error map: {{val| 0.000 -0.137 -0.858 +0.268 }}
: error map: {{val| 0.000 -0.253 +0.069 +0.232 }}


{{Optimal ET sequence|legend=1| 12, …, 200, 212, 224, 436, 660 }}
{{Optimal ET sequence|legend=1| 77, 147, 224, 301, 525, 826, 1351 }}


[[Badness]] (Sintel): 2.79
[[Badness]] (Sintel): 2.81


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 1375/1372, 6250/6237, 32805/32768
Comma list: 3025/3024, 24057/24010, 32805/32768


Mapping: {{mapping| 4 0 60 119 185 | 0 1 -8 -17 -27 }}
Mapping: {{mapping| 7 0 105 -56 -120 | 0 1 -8 7 13 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~25/21 = 300.0244{{c}}, ~3/2 = 701.8759{{c}}
* WE: ~495/448 = 171.4334{{c}}, ~3/2 = 701.7387{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8145{{c}}
* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7198{{c}}


{{Optimal ET sequence|legend=0| 12, …, 212, 224, 436, 660 }}
{{Optimal ET sequence|legend=0| 77, 147, 224, 301, 525 }}


Badness (Sintel): 1.51
Badness (Sintel): 1.46


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647
Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024


Mapping: {{mapping| 4 0 60 119 185 224 | 0 1 -8 -17 -27 -33 }}
Mapping: {{mapping| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~25/21 = 300.0234{{c}}, ~3/2 = 701.8707{{c}}
* WE: ~495/448 = 171.4282{{c}}, ~3/2 = 701.7229{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8123{{c}}
* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7242{{c}}


{{Optimal ET sequence|legend=0| 12f, …, 212, 224, 436, 660 }}
{{Optimal ET sequence|legend=0| 77, 147, 224, 525, 1274f }}


Badness (Sintel): 1.13
Badness (Sintel): 1.02


== Septant ==
== Octant ==
The septant temperament ({{nowrap| 224 & 301 }}) has a period of 1/7 octave and tempers out the [[akjaysma]], {{monzo| 47 -7 -7 -7 }}.
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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 32805/32768, 516560652/514714375
[[Comma list]]: 32805/32768, 2259436291848/2251875390625


{{Mapping|legend=1| 7 0 105 -56 | 0 1 -8 7 }}
{{Mapping|legend=1| 8 0 120 -117 | 0 1 -8 11 }}
: mapping generators: ~8575/7776, ~3
: mapping generators: ~42875/39366, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~8575/7776 = 171.4303{{c}}, ~3/2 = 701.7091{{c}}
* [[WE]]: ~42875/39366 = 150.0048{{c}}, ~3/2 = 701.7356{{c}}
: [[error map]]: {{val| +0.012 -0.234 +0.096 +0.265 }}
: [[error map]]: {{val| +0.039 -0.181 +0.071 +0.127 }}
* [[CWE]]: ~8575/7776 = 171.4286{{c}}, ~3/2 = 701.7022{{c}}
* [[CWE]]: ~42875/39366 = 150.0000{{c}}, ~3/2 = 701.7134{{c}}
: error map: {{val| 0.000 -0.253 +0.069 +0.232 }}
: error map: {{val| 0.000 -0.242 -0.021 +0.022 }}


{{Optimal ET sequence|legend=1| 77, 147, 224, 301, 525, 826, 1351 }}
{{Optimal ET sequence|legend=1| 24, …, 224, 472, 696, 1168 }}


[[Badness]] (Sintel): 2.81
[[Badness]] (Sintel): 3.98


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 24057/24010, 32805/32768
Comma list: 9801/9800, 32805/32768, 46656/46585


Mapping: {{mapping| 7 0 105 -56 -120 | 0 1 -8 7 13 }}
Mapping: {{mapping| 8 0 120 -117 15 | 0 1 -8 11 1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~495/448 = 171.4334{{c}}, ~3/2 = 701.7387{{c}}
* WE: ~12/11 = 150.0010{{c}}, ~3/2 = 701.7177{{c}}
* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7198{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7131{{c}}


{{Optimal ET sequence|legend=0| 77, 147, 224, 301, 525 }}
{{Optimal ET sequence|legend=0| 24, …, 224, 472, 696, 1168 }}


Badness (Sintel): 1.46
Badness (Sintel): 1.48


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024
Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655


Mapping: {{mapping| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }}
Mapping: {{mapping| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~495/448 = 171.4282{{c}}, ~3/2 = 701.7229{{c}}
* WE: ~12/11 = 149.9957{{c}}, ~3/2 = 701.7046{{c}}
* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7242{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7247{{c}}


{{Optimal ET sequence|legend=0| 77, 147, 224, 525, 1274f }}
{{Optimal ET sequence|legend=0| 24, 224, 472, 696 }}


Badness (Sintel): 1.02
Badness (Sintel): 1.26


== Octant ==
== Nonant ==
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 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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 32805/32768, 2259436291848/2251875390625
[[Comma list]]: 32805/32768, 40353607/40310784


{{Mapping|legend=1| 8 0 120 -117 | 0 1 -8 11 }}
{{Mapping|legend=1| 9 0 135 11 | 0 1 -8 1 }}
: mapping generators: ~42875/39366, ~3
: mapping generators: ~2592/2401, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~42875/39366 = 150.0048{{c}}, ~3/2 = 701.7356{{c}}
* [[WE]]: ~2592/2401 = 133.3442{{c}}, ~3/2 = 701.8000{{c}}
: [[error map]]: {{val| +0.039 -0.181 +0.071 +0.127 }}
: [[error map]]: {{val| +0.098 -0.057 -0.027 -0.141 }}
* [[CWE]]: ~42875/39366 = 150.0000{{c}}, ~3/2 = 701.7134{{c}}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.7384{{c}}
: error map: {{val| 0.000 -0.242 -0.021 +0.022 }}
: error map: {{val| 0.000 -0.217 -0.221 -0.421 }}


{{Optimal ET sequence|legend=1| 24, …, 224, 472, 696, 1168 }}
{{Optimal ET sequence|legend=1| 36, 99c, 135, 171, 2772bd, 2943bdd, …, 5166bccddd, 5337bccddd }}


[[Badness]] (Sintel): 3.98
[[Badness]] (Sintel): 1.77


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 9801/9800, 32805/32768, 46656/46585
Comma list: 540/539, 32805/32768, 42875/42592


Mapping: {{mapping| 8 0 120 -117 15 | 0 1 -8 11 1 }}
Mapping: {{mapping| 9 0 135 11 131 | 0 1 -8 1 -7 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~12/11 = 150.0010{{c}}, ~3/2 = 701.7177{{c}}
* WE: ~242/225 = 133.3308{{c}}, ~3/2 = 701.8205{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7131{{c}}
* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.8351{{c}}


{{Optimal ET sequence|legend=0| 24, …, 224, 472, 696, 1168 }}
{{Optimal ET sequence|legend=0| 36, 135, 171 }}


Badness (Sintel): 1.48
Badness (Sintel): 4.20


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655
Comma list: 540/539, 729/728, 4096/4095, 16807/16731


Mapping: {{mapping| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }}
Mapping: {{mapping| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~12/11 = 149.9957{{c}}, ~3/2 = 701.7046{{c}}
* WE: ~242/225 = 133.3180{{c}}, ~3/2 = 701.6956{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7247{{c}}
* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.7800{{c}}


{{Optimal ET sequence|legend=0| 24, 224, 472, 696 }}
{{Optimal ET sequence|legend=0| 36, 99cf, 135, 171 }}


Badness (Sintel): 1.26
Badness (Sintel): 3.15


== Nonant ==
== Septiquarschis ==
The ''nonant'' temperament ({{nowrap| 36 & 135 }}) has a period of 1/9 octave and tempers out the [[septimal ennealimma]], {{monzo| -11 -9 0 9 }}.
Named by [[Xenllium]] in 2021, septiquarschis tempers out [[829440/823543]] (mynaslender comma) and [[67108864/66706983]] (septiness comma), and may be described as the {{nowrap| 89 & 94 }} temperament. It splits septimal minor seventh ([[7/4]]) into four generators. Note that in the data below, the generator is the [[octave complement]] so that seven of them minus five octaves make a [[3/2|perfect fifth]]; its [[ploidacot]] is thus epsilon-heptacot.  


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


[[Comma list]]: 32805/32768, 40353607/40310784
[[Comma list]]: 32805/32768, 829440/823543


{{Mapping|legend=1| 9 0 135 11 | 0 1 -8 1 }}
{{Mapping|legend=1| 1 -4 47 6 | 0 7 56 -4 }}
: mapping generators: ~2592/2401, ~3
: mapping generators: ~2, ~256/147


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2592/2401 = 133.3442{{c}}, ~3/2 = 701.8000{{c}}
* [[WE]]: ~2 = 1199.8855{{c}}, ~256/147 = 957.2944{{c}}
: [[error map]]: {{val| +0.098 -0.057 -0.027 -0.141 }}
: [[error map]]: {{val| -0.114 -0.436 -0.182 +1.310 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.7384{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~256/147 = 957.3867{{c}}
: error map: {{val| 0.000 -0.217 -0.221 -0.421 }}
: error map: {{val| 0.000 -0.248 +0.032 +1.627 }}


{{Optimal ET sequence|legend=1| 36, 99c, 135, 171, 2772bd, 2943bdd, …, 5166bccddd, 5337bccddd }}
{{Optimal ET sequence|legend=1| 89, 94, 183, 460d, 643d }}


[[Badness]] (Sintel): 1.77
[[Badness]] (Sintel): 4.73


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 540/539, 32805/32768, 42875/42592
Comma list: 540/539, 15488/15435, 32805/32768


Mapping: {{mapping| 9 0 135 11 131 | 0 1 -8 1 -7 }}
Mapping: {{mapping| 1 -4 47 6 25 | 0 7 56 -4 -27 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~242/225 = 133.3308{{c}}, ~3/2 = 701.8205{{c}}
* WE: ~2 = 1199.9430{{c}}, ~256/147 = 957.3390{{c}}
* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.8351{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3849{{c}}


{{Optimal ET sequence|legend=0| 36, 135, 171 }}
{{Optimal ET sequence|legend=0| 89, 94, 183, 460d }}


Badness (Sintel): 4.20
Badness (Sintel): 1.72


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 540/539, 729/728, 4096/4095, 16807/16731
Comma list: 540/539, 729/728, 1573/1568, 4096/4095


Mapping: {{mapping| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }}
Mapping: {{mapping| 1 -4 47 6 25 -33 | 0 7 56 -4 -27 46 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~242/225 = 133.3180{{c}}, ~3/2 = 701.6956{{c}}
* WE: ~2 = 1200.0058{{c}}, ~256/147 = 957.3946{{c}}
* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.7800{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3900{{c}}


{{Optimal ET sequence|legend=0| 36, 99cf, 135, 171 }}
{{Optimal ET sequence|legend=0| 89, 94, 183, 277, 460d }}


Badness (Sintel): 3.15
Badness (Sintel): 1.46


== Tridecafifths ==
== Tridecafifths ==
Tridecafifths divides the perfect 3/2 into 13 quartertones.  
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
[[Subgroup]]: 2.3.5.7
Line 2,715: Line 2,781:


== Subgroup extensions ==
== 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) ===
=== Photia (2.3.5.17) ===
{{See also| No-elevens subgroup temperaments #Garibaldia }}
{{See also| No-elevens subgroup temperaments #Garibaldia }}
Line 2,757: Line 2,887:
: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''
: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''


The [[S-expression]]-based comma list of this temperament is {[[1216/1215|S16/S18]], [[361/360|S19]] , ([[513/512|S15/S20]])}. Strangely, despite prime 19 being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is actually sharper than the fifth in optimal schismic. This is likely due to its optimization considering intervals like 19/10 and 19/15.  
Nestoria is notable for having one of the lowest-badness subgroup extensions of schismic. Note that despite prime [[19/1|19]] being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is generally not flatter than the fifth in optimal schismic due to its optimization considering intervals like [[19/10]] and [[19/15]].  


[[Subgroup]]: 2.3.5.19
[[Subgroup]]: 2.3.5.19
Retrieved from "https://en.xen.wiki/w/Schismatic_family"