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__FORCETOC__
{{Technical data page}}
<span style="display: block; text-align: right;">Other languages: [[:de:Schismatische_Temperaturen|Deutsch]]</span>
The [[5-limit]] parent comma for the '''schismatic''' (or '''schismic''') '''family''' is the [[schisma]] of 32805/32768, which is the amount by which the [[Pythagorean comma]] exceeds the [[syntonic comma]] (81/80), or alternatively put, the difference between a [[5/4|just major third]] and a [[8192/6561|Pythagorean diminished fourth]].


=Five limit=
== Schismic, schismatic, a.k.a. helmholtz ==
The 5-limit parent comma for the schismatic family is the [[schisma]] of 32805/32768, which is the amount by which the Pythagorean comma exceeds the [[Didymus comma]] ([[81/80]]), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its [[monzo]] is |-15 8 1&gt;, and flipping that yields &lt;&lt;1 -8 -15|| for the [[Wedgies_and_Multivals|wedgie]]. This tells us the generator is a fifth and that we will need eight fourths in succession to reach the pitch class of a major third. In fact, 10 = (4/3)^8 * 32805/32768.
{{Main| Schismic }}


The 5-limit version of the temperament is a [[Microtempering|microtemperament]], sometimes called '''Helmholtz''' or schismatic, which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. [[53edo]] is a possible tuning for schismatic, but you need [[118edo]] if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering.
The 5-limit version of the temperament is a [[microtemperament]], called ''schismic'', ''schismatic'', or ''helmholtz''. The generator is a fifth, flattened by a fraction of a schisma, and 5/4 is represented by a diminished fourth. This defies the tradition of {{w|tertian harmony}}, as the [[just major triad]] on C is C–F♭–G, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as C–vE–G.  


[[POTE_tuning|POTE generator]]: ~3/2 = 701.736
As a 5-limit system, schismic is far more accurate than [[meantone]] but still with manageable [[complexity]]. [[53edo]] is a possible tuning for schismic, but you need [[118edo]] if you want to get the full effect. In exact analogy with [[1/4-comma meantone]] there is also 1/8 schismic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244{{cent}}, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better fifth, or 2/17 schisma, with both thirds flat by 1/17 of a schisma, although the differences would be very hard to distinguish unless using a large gamut. Simply leaving the fifths just would also make for a viable tuning, thus collapsing schismic to a simple relabeling of the 3-limit.  


Mapping generator: ~3
[[Subgroup]]: 2.3.5


Map: [&lt;1 0 15|, &lt;0 1 -8|]
[[Comma list]]: 32805/32768


EDOs: [[12edo|12]], [[29edo|29]], [[41edo|41]], [[53edo|53]], [[65edo|65]], [[77edo|77]], [[94edo|94]], [[118edo|118]], [[171edo|171]], [[183edo|183]], [[224edo|224]], [[289edo|289]], 407, 460, 749, 3456bc, 4205bc, 4954bc, 5703bc
{{Mapping|legend=1| 1 0 15 | 0 1 -8 }}
: mapping generators: ~2, ~3


Badness: 0.00426
[[Optimal tuning]]s:  
* [[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}}
: error map: {{val| 0.000 -0.224 -0.160 }}


=Seven limit children=
[[Tuning ranges]]:
The second comma of the [[Normal_lists|normal comma list]] defines which 7-limit family member we are looking at. Adding |25 -14 0 -1&gt; gives garibaldi, |-44 26 0 1&gt; grackle, |6 -2 0 -1&gt; schism and |-59 39 0 -1&gt; pontiac; these all have a fifth as generator. Bischismic adds |-69 40 0 2&gt; and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = |-10 1 0 3&gt;, with an 8/7 generator, three of which give the fifth, and term adds |-94 54 0 3&gt; with a 1/3 octave period. Sesquiquartififths adds |-35 15 0 4&gt; and slices the fifth in four.
* [[5-odd-limit]] [[diamond monotone]]: ~3/2 = [685.714, 705.882] (4\7 to 10\17)
* 5-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 701.955] (1/8-comma to untempered)


=Garibaldi=
{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bbc, 6452bbcc }}
Commas: 225/224, 3125/3087


7-limit minimax tuning:
[[Badness]] (Sintel): 0.0999


7-limit: [|1 0 0 0&gt;, |5/3 1/15 0 -1/15&gt;,
=== Overview to extensions ===
|5/3 -8/15 0 8/15&gt;, |5/3 -14/15 0 14/15&gt;]
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.


[[Eigenmonzo|Eigenmonzo]]s: 2, 7/6
[[#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.


9-limit: [|1 0 0 0&gt;, |25/16 1/8 0 -1/16&gt;,
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.
|5/2 -1 0 1/2&gt;, |25/8 -7/4 0 7/8&gt;]


Eigenmonzos: 2, 9/7
Temperaments discussed elsewhere include:
* ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
* ''[[Pogo]]'' (+118098/117649) → [[Stearnsmic clan #Pogo|Stearnsmic clan]]


[[POTE_tuning|POTE generator]]: ~3/2 = 702.085
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.  


Mapping generator: ~3
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]].


Map: [&lt;1 0 15 25|, &lt;0 1 -8 -14|]
== Garibaldi ==
{{Main| Garibaldi }}


Wedgie: &lt;&lt;1 -8 -14 -15 -25 -10||
Garibaldi tempers out the [[septischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a very slightly sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}, which also makes it the unique intersection of [[marvel]] and [[aberschismic]], two very notable 7-limit temperament paradigms; however, their errors go in oposite directions as marvel tunes fifths flatter and aberschismic sharper.


EDOs: 12, 29, 41, 53, [[94edo|94]], 241c, 335cd, 576cd
[[147edo]] (from 94+53) is a [[patent val]] tuning with 5 and 7 very close to equally out-of-tune in opposite directions s.t only 7/5 and 10/7 are inconsistent in the 9-odd-limit, so might be considered for (EG) a 41-note subset ([[12L 29s]]).


Badness: 0.0216
[[Subgroup]]: 2.3.5.7


==11-limit==
[[Comma list]]: 225/224, 3125/3087
Commas: 225/224, 385/384, 2200/2187
 
{{Mapping|legend=1| 1 0 15 25 | 0 1 -8 -14 }}
 
[[Optimal tuning]]s:
* [[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}}
: error map: {{val| 0.000 +0.122 -2.933 +2.090 }}
 
[[Minimax tuning]]:
* [[7-odd-limit]]: ~3/2 = {{monzo| 2/3 1/15 0 -1/15 }}
: {{monzo list| 1 0 0 0 | 5/3 1/15 0 -1/15 | 5/3 -8/15 0 8/15 | 5/3 -14/15 0 14/15 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
* [[9-odd-limit]]: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }}
: {{monzo list| 1 0 0 0 | 25/16 1/8 0 -1/16 | 5/2 -1 0 1/2 | 25/8 -7/4 0 7/8 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7
 
[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 703.448] (7\12 to 17\29)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 702.915]
 
{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 94 }}
 
[[Badness]] (Sintel): 0.548
 
=== 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, even though it comes with a much higher complexity. It finds 11/8 as two commas above 4/3 and 13/8 a minor third above 11/8, or equivalently two commas above 128/81.
 
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 2200/2187
 
{{Mapping|legend=0| 1 0 15 25 -33 | 0 1 -8 -14 23 }}
 
Optimal tunings:
* WE: ~2 = 1200.3089{{c}}, ~3/2 = 702.3377{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1562{{c}}


Minimax tuning:
Minimax tuning:
* 11-odd-limit: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }}
: unchanged-interval (eigenmonzo) basis: 2.9/7
Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [701.887, 702.439] (31\53 to 24\41)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 702.915]
{{Optimal ET sequence|legend=0| 12e, 41, 53, 94, 229c }}
Badness (Sintel): 0.906
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 275/273, 325/324, 385/384
{{Mapping|legend=0| 1 0 15 25 -33 -28 | 0 1 -8 -14 23 20 }}
Optimal tunings:
* WE: ~2 = 1200.1703{{c}}, ~3/2 = 702.2122{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1135{{c}}
Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 19/34 0 0 -1/34 0 1/34 }}
: unchanged-interval (eigenmonzo) basis: 2.13/7
Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~3/2 = [701.887, 702.439] (31\53 to 24\41)
* 13-odd-limit diamond tradeoff: ~3/2 = [701.711, 703.597]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 703.597]
{{Optimal ET sequence|legend=0| 41, 53, 94, 429ccdeef, 523ccdeef }}
Badness (Sintel): 0.854
===== Cassie =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 120/119, 154/153, 225/224, 273/272, 325/324
{{Mapping|legend=0| 1 0 15 25 -33 -28 -7 | 0 1 -8 -14 23 20 7 }}
Optimal tunings:
* WE: ~2 = 1199.8140{{c}}, ~3/2 = 701.9833{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0909{{c}}
{{Optimal ET sequence|legend=0| 12e, 41, 53, 94g }}
Badness (Sintel): 1.19
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 120/119, 154/153, 171/170, 190/189, 225/224, 273/272
{{Mapping|legend=0| 1 0 15 25 -33 -28 -7 9 | 0 1 -8 -14 23 20 7 -3 }}
Optimal tunings:
* WE: ~2 = 1199.9556{{c}}, ~3/2 = 702.0530{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0787{{c}}
{{Optimal ET sequence|legend=0| 12e, 41, 53 }}
Badness (Sintel): 1.11
===== Cassandric =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 225/224, 275/273, 325/324, 375/374, 385/384
{{Mapping|legend=0| 1 0 15 25 -33 -28 77 | 0 1 -8 -14 23 20 -46 }}
Optimal tunings:
* WE: ~2 = 1200.0046{{c}}, ~3/2 = 702.2167{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0962{{c}}
{{Optimal ET sequence|legend=0| 41g, 53, 94 }}
Badness (Sintel): 1.18
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 190/189, 209/208, 225/224, 275/273, 325/324, 375/374
{{Mapping|legend=0| 1 0 15 25 -33 -28 77 9 | 0 1 -8 -14 23 20 -46 -3 }}
Optimal tunings:
* WE: ~2 = 1200.2910{{c}}, ~3/2 = 702.2681{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0967{{c}}
{{Optimal ET sequence|legend=1| 41g, 53, 94 }}
Badness (Sintel): 1.07
====== 23-limit ======
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 190/189, 209/208, 225/224, 253/252, 275/273, 325/324, 375/374
{{Mapping|legend=0| 1 0 15 25 -33 -28 77 9 60 | 0 1 -8 -14 23 20 -46 -3 -35 }}
Optimal tunings:
* WE: ~2 = 1200.2970{{c}}, ~3/2 = 702.2697{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0943{{c}}
{{Optimal ET sequence|legend=0| 41g, 53, 94 }}
Badness (Sintel): 1.08
===== Cassander =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 170/169, 225/224, 275/273, 325/324, 385/384
{{Mapping|legend=0| 1 0 15 25 -33 -28 -72 | 0 1 -8 -14 23 20 48 }}
Optimal tunings:
* WE: ~2 = 1200.1986{{c}}, ~3/2 = 702.2598{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1455{{c}}
{{Optimal ET sequence|legend=0| 41, 53g, 94 }}
Badness (Sintel): 1.14
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 170/169, 190/189, 209/208, 225/224, 275/273, 325/324
{{Mapping|legend=0| 1 0 15 25 -33 -28 -72 9 | 0 1 -8 -14 23 20 48 -3 }}
Optimal tunings:
* WE: ~2 = 1200.3057{{c}}, ~3/2 = 702.3138{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1373{{c}}
{{Optimal ET sequence|legend=0| 41, 53g, 94 }}
Badness (Sintel): 1.07
=== Andromeda ===
Andromeda is the second-best extension to garibaldi, which is less complex but also more innacurate. 11/8 is two commas down from [[729/512]], as is 13/8 two commas down from [[27/16]].
Subgroup: 2.3.5.7.11
Comma list: 100/99, 225/224, 245/242
{{Mapping|legend=0| 1 0 15 25 32 | 0 1 -8 -14 -18 }}
Optimal tunings:
* WE: ~2 = 1200.1917{{c}}, ~3/2 = 702.4836{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3599{{c}}
Minimax tuning:
* 11-odd-limit: ~3/2 = {{monzo| 3/5 1/10 0 0 -1/20 }}
: unchanged-interval (eigenmonzo) basis: 2.11/9
Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 703.448] (7\12 to 17\29)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]
{{Optimal ET sequence|legend=0| 12, 29, 41 }}
Badness (Sintel): 0.779
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 105/104, 196/195, 245/242
{{Mapping|legend=0| 1 0 15 25 32 37 | 0 1 -8 -14 -18 -21 }}
Optimal tunings:
* WE: ~2 = 1200.3031{{c}}, ~3/2 = 702.7368{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.5420{{c}}
Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/23 2/23 0 0 0 -1/23 }}
: unchanged-interval (eigenmonzo) basis: 2.13/9
Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~3/2 = [702.439, 703.448] (24\41 to 17\29)
* 13-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 704.377]
{{Optimal ET sequence|legend=0| 12f, 29, 41 }}
Badness (Sintel): 0.857
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 100/99, 105/104, 120/119, 189/187, 196/195
{{Mapping|legend=0| 1 0 15 25 32 37 -7 | 0 1 -8 -14 -18 -21 7 }}
Optimal tunings:
* WE: ~2 = 1199.1984{{c}}, ~3/2 = 701.8424{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3384{{c}}
{{Optimal ET sequence|legend=0| 12f, 29, 41 }}
Badness (Sintel): 1.19
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 100/99, 105/104, 120/119, 133/132, 189/187, 196/195
{{Mapping|legend=0| 1 0 15 25 32 37 -7 9 | 0 1 -8 -14 -18 -21 7 -3 }}
Optimal tunings:
* WE: ~2 = 1199.5242{{c}}, ~3/2 = 702.0783{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3711{{c}}
{{Optimal ET sequence|legend=0| 12f, 29, 41 }}
Badness (Sintel): 1.17
===== Schisicosiennic =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 100/99, 105/104, 154/153, 170/169, 196/195
{{Mapping|legend=0| 1 0 15 25 32 37 58 | 0 1 -8 -14 -18 -21 -34 }}
Optimal tunings:
* WE: ~2 = 1200.6122{{c}}, ~3/2 = 703.0830{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6968{{c}}
{{Optimal ET sequence|legend=0| 12fg, 29g, 41, 70cd }}
Badness (Sintel): 1.11
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 100/99, 105/104, 133/132, 154/153, 170/169, 190/189
{{Mapping|legend=0| 1 0 15 25 32 37 58 9 | 0 1 -8 -14 -18 -21 -34 -3 }}
Optimal tunings:
* WE: ~2 = 1200.7981{{c}}, ~3/2 = 703.2199{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.7221{{c}}
{{Optimal ET sequence|legend=0| 12fg, 29g, 41, 70cd }}
Badness (Sintel): 1.09
===== Schisicosiennoid =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 85/84, 100/99, 105/104, 119/117, 221/220
{{Mapping|legend=0| 1 0 15 25 32 37 12 | 0 1 -8 -14 -18 -21 -5 }}
Optimal tunings:
* WE: ~2 = 1201.3146{{c}}, ~3/2 = 703.4864{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6491{{c}}
{{Optimal ET sequence|legend=0| 12f, 29g, 41g }}
Badness (Sintel): 1.06
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 85/84, 100/99, 105/104, 119/117, 133/132, 153/152
{{Mapping|legend=0| 1 0 15 25 32 37 12 9 | 0 1 -8 -14 -18 -21 -5 -3 }}
Optimal tunings:
* WE: ~2 = 1201.3140{{c}}, ~3/2 = 703.4860{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6578{{c}}
{{Optimal ET sequence|legend=1| 12f, 29g, 41g }}
Badness (Sintel): 1.02
=== Helenus ===
Subgroup: 2.3.5.7.11
Comma list: 99/98, 176/175, 3125/3087
{{Mapping|legend=0| 1 0 15 25 51 | 0 1 -8 -14 -30 }}
Optimal tunings:
* WE: ~2 = 1199.7097{{c}}, ~3/2 = 701.5554{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7370{{c}}
Minimax tuning:
* 11-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }}
: unchanged-interval (eigenmonzo) basis: 2.11/9
{{Optimal ET sequence|legend=0| 12, 41e, 53, 118d }}
Badness (Sintel): 1.18
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 99/98, 176/175, 275/273, 847/845
{{Mapping|legend=0| 1 0 15 25 51 56 | 0 1 -8 -14 -30 -33 }}
Optimal tunings:
* WE: ~2 = 1199.7370{{c}}, ~3/2 = 701.5937{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7570{{c}}
Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }}
: unchanged-interval (eigenmonzo) basis: 2.11/9
{{Optimal ET sequence|legend=0| 12f, …, 41ef, 53, 118d }}
Badness (Sintel): 1.09
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 99/98, 120/119, 176/175, 275/273, 442/441
{{Mapping|legend=0| 1 0 15 25 51 56 -7 | 0 1 -8 -14 -30 -33 7 }}
Optimal tunings:
* WE: ~2 = 1199.2895{{c}}, ~3/2 = 701.2643{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6967{{c}}
{{Optimal ET sequence|legend=0| 12f, 53, 65d, 118dg }}
Badness (Sintel): 1.21
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 99/98, 120/119, 176/175, 190/189, 209/208, 247/245
{{Mapping|legend=0| 1 0 15 25 51 56 -7 9 | 0 1 -8 -14 -30 -33 7 -3 }}
Optimal tunings:
* WE: ~2 = 1199.5280{{c}}, ~3/2 = 701.4290{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7149{{c}}
{{Optimal ET sequence|legend=0| 12f, 53, 65d }}
Badness (Sintel): 1.18
=== Karadeniz ===
{{See also| Turkish maqam music temperaments #Karadeniz temperament }}
Subgroup: 2.3.5.7.11
Comma list: 225/224, 243/242, 3125/3087
{{Mapping|legend=0| 1 1 7 11 2 | 0 2 -16 -28 5 }}
: mapping generators: ~2, ~11/9
Optimal tunings:
* WE: ~2 = 1199.7351{{c}}, ~11/9 = 350.9167{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.9995{{c}}
{{Optimal ET sequence|legend=0| 24d, 41, 65d, 106, 147 }}
Badness (Sintel): 1.37
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 243/242, 325/324, 640/637
{{Mapping|legend=0| 1 1 7 11 2 -8 | 0 2 -16 -28 5 40 }}
Optimal tunings:
* WE: ~2 = 1199.3042{{c}}, ~11/9 = 350.7533{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.9686{{c}}
{{Optimal ET sequence|legend=0| 24d, 41, 65d, 106f }}
Badness (Sintel): 1.34
=== Hemigari ===
Subgroup: 2.3.5.7.11
Comma list: 121/120, 225/224, 3125/3087
{{Mapping|legend=0| 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|legend=0| 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
Comma list: 225/224, 1331/1323, 3125/3087
{{Mapping|legend=0| 1 2 -1 -3 0 | 0 -3 24 42 25 }}
: mapping generators: ~2, ~11/10
Optimal tunings:
* WE: ~2 = 1200.1997{{c}}, ~11/10 = 166.0018{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9786{{c}}
{{Optimal ET sequence|legend=0| 29, 65d, 94 }}
Badness (Sintel): 1.92
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 275/273, 847/845, 1331/1323
{{Mapping|legend=0| 1 2 -1 -3 0 -1 | 0 -3 24 42 25 34 }}
Optimal tunings:
* WE: ~2 = 1200.1224{{c}}, ~11/10 = 165.9800{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9659{{c}}
{{Optimal ET sequence|legend=0| 29, 65d, 94 }}
Badness (Sintel): 1.40
== 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-E𝄪𝄪♯), or triple-up major sixth (C-^<sup>3</sup>A).
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 4375/4374, 32805/32768
{{Mapping|legend=1| 1 0 15 -59 | 0 1 -8 39 }}
[[Optimal tuning]]s:
* [[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}}
: error map: {{val| 0.000 -0.197 -0.377 -0.268 }}
[[Minimax tuning]]:
* [[7-odd-limit]]: ~3/2 = {{monzo| 27/47 0 -1/47 1/47 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 74/47 0 -1/47 1/47 }}, {{monzo| 113/47 0 8/47 -8/47 }}, {{monzo| 113/47 0 -39/47 39/47 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
* [[9-odd-limit]]: ~3/2 = {{monzo| 1/2 1/5 -1/10 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 3/2 1/5 -1/10 0 }}, {{monzo| 3 -8/5 4/5 0 }}, {{monzo| -1/2 39/5 -39/10 0 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5
[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [701.538, 701.886] (38\65 to 31\53)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 701.955]
{{Optimal ET sequence|legend=1| 53, 118, 171, 1592c, 1763c, …, 2960cd, 3131bcd }}
[[Badness]] (Sintel): 0.358
=== Helenoid ===
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
Comma list: 385/384, 3388/3375, 4375/4374
{{Mapping|legend=0| 1 0 15 -59 51 | 0 1 -8 39 -30 }}
Optimal tunings:
* WE: ~2 = 1200.3277{{c}}, ~3/2 = 701.9135{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7223{{c}}
Minimax tuning:
* 11-odd-limit: ~3/2 = {{monzo| 41/69 0 0 1/69 -1/69 }}
: unchanged-interval (eigenmonzo) basis: 2.11/7
{{Optimal ET sequence|legend=0| 53, 118, 289e, 407de }}
Badness (Sintel): 1.28
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 385/384, 625/624, 729/728
{{Mapping|legend=0| 1 0 15 -59 51 56 | 0 1 -8 39 -30 -33 }}
Optimal tunings:
* WE: ~2 = 1200.1780{{c}}, ~3/2 = 701.8491{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7446{{c}}
Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 43/72 0 0 1/72 -1/72 }}
: unchanged-interval (eigenmonzo) basis: 2.13/7
{{Optimal ET sequence|legend=0| 53, 118, 171e }}
Badness (Sintel): 1.39
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 352/351, 385/384, 561/560, 625/624, 729/728
{{Mapping|legend=0| 1 0 15 -59 51 56 -91 | 0 1 -8 39 -30 -33 60 }}
Optimal tunings:
* WE: ~2 = 1200.1645{{c}}, ~3/2 = 701.8385{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7425{{c}}
Minimax tuning:
* 17-odd-limit: ~3/2 = {{monzo| 18/31 0 0 0 0 -1/93 1/93 }}
: unchanged-interval (eigenmonzo) basis: 2.17/13
{{Optimal ET sequence|legend=0| 53, 118, 171e }}
Badness (Sintel): 1.47
==== Helena ====
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 325/324, 385/384, 3146/3125
{{Mapping|legend=0| 1 0 15 -59 51 -28 | 0 1 -8 39 -30 20 }}
Optimal tunings:
* WE: ~2 = 1200.5227{{c}}, ~3/2 = 702.0456{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7418{{c}}
{{Optimal ET sequence|legend=0| 53, 118f, 171ef }}
Badness (Sintel): 1.50
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 169/168, 273/272, 325/324, 385/384, 3146/3125
{{Mapping|legend=0| 1 0 15 -59 51 -28 -91 | 0 1 -8 39 -30 20 60 }}
Optimal tunings:
* WE: ~2 = 1200.4988{{c}}, ~3/2 = 702.0218{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7332{{c}}
{{Optimal ET sequence|legend=0| 53, 118f, 171ef }}
Badness (Sintel): 1.56
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 169/168, 273/272, 286/285, 325/324, 385/384, 627/625
{{Mapping|legend=0| 1 0 15 -59 51 -28 -91 9 | 0 1 -8 39 -30 20 60 -3 }}
Optimal tunings:
* WE: ~2 = 1200.5185{{c}}, ~3/2 = 702.0323{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7318{{c}}
{{Optimal ET sequence|legend=0| 53, 118f, 171ef }}
Badness (Sintel): 1.33
=== Ponta ===
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
Comma list: 540/539, 4375/4374, 32805/32768
{{Mapping|legend=0| 1 0 15 -59 135 | 0 1 -8 39 -83 }}
Optimal tunings:
* WE: ~2 = 1199.9814{{c}}, ~3/2 = 701.7725{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7834{{c}}
Minimax tuning:
* 11-odd-limit: ~3/2 = {{monzo| 36/61 0 0 1/122 -1/122 }}
: unchanged-interval (eigenmonzo) basis: 2.11/7
{{Optimal ET sequence|legend=0| 53, 171, 224 }}
Badness (Sintel): 1.61
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 625/624, 729/728, 2200/2197
{{Mapping|legend=0| 1 0 15 -59 135 56 | 0 1 -8 39 -83 -33 }}
Optimal tunings:
* WE: ~2 = 1199.9601{{c}}, ~3/2 = 701.7610{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7845{{c}}
Minimax tuning:
* 13 and 15-odd-limit: ~3/2 = {{monzo| 36/61 0 0 1/122 -1/122 }}
: unchanged-interval (eigenmonzo) basis: 2.11/7
{{Optimal ET sequence|legend=0| 53, 171, 224 }}
Badness (Sintel): 0.976
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 375/374, 540/539, 625/624, 729/728, 2200/2197
{{Mapping|legend=0| 1 0 15 -59 135 56 -91 | 0 1 -8 39 -83 -33 60 }}
Optimal tunings:
* WE: ~2 = 1199.8850{{c}}, ~3/2 = 701.7101{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7775{{c}}
Minimax tuning:
* 17-odd-limit: ~3/2 = {{monzo| 83/143 0 0 0 -1/143 0 1/143 }}
: unchanged-interval (eigenmonzo) basis: 2.17/11
{{Optimal ET sequence|legend=0| 53, 171, 224, 395e, 619eg }}
Badness (Sintel): 1.16
=== Pontic ===
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
Comma list: 441/440, 4375/4374, 32805/32768
{{Mapping|legend=0| 1 0 15 -59 -136 | 0 1 -8 39 88 }}
Optimal tunings:
* WE: ~2 = 1200.1259{{c}}, ~3/2 = 701.7980{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7256{{c}}
Minimax tuning:
* 11-odd-limit: ~3/2 = {{monzo| 6/11 0 0 0 1/88 }}
: unchanged-interval (eigenmonzo) basis: 2.11
{{Optimal ET sequence|legend=0| 53e, 118, 289, 407d }}
Badness (Sintel): 1.64
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 441/440, 625/624, 729/728, 3584/3575
{{Mapping|legend=0| 1 0 15 -59 -136 56 | 0 1 -8 39 88 -33 }}
Optimal tunings:
* WE: ~2 = 1199.9254{{c}}, ~3/2 = 701.6945{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7378{{c}}
Minimax tuning:
* 13 and 15-odd-limit: ~3/2 = {{monzo| 71/121 0 0 0 1/121 -1/121 }}
: unchanged-interval (eigenmonzo) basis: 2.13/11
{{Optimal ET sequence|legend=0| 53e, 118, 171, 289f }}
Badness (Sintel): 1.87
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 441/440, 595/594, 625/624, 729/728, 2880/2873
{{Mapping|legend=0| 1 0 15 -59 -136 56 -91 | 0 1 -8 39 88 -33 60 }}
Optimal tunings:
* WE: ~2 = 1199.9454{{c}}, ~3/2 = 701.7085{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7401{{c}}
Minimax tuning:
* 17-odd-limit: ~3/2 = {{monzo| 71/121 0 0 0 1/121 -1/121 }}
: unchanged-interval (eigenmonzo) basis: 2.13/11
{{Optimal ET sequence|legend=0| 53e, 118, 171, 289f }}
Badness (Sintel): 1.51
==== Pontoid ====
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 4375/4374, 32805/32768
{{Mapping|legend=0| 1 0 15 -59 -136 -215 | 0 1 -8 39 88 138 }}
Optimal tunings:
* WE: ~2 = 1200.0897{{c}}, ~3/2 = 701.7874{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7356{{c}}
{{Optimal ET sequence|legend=0| 53ef, 118f, 171, 289 }}
Badness (Sintel): 2.07
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 364/363, 441/440, 595/594, 1156/1155, 32805/32768
{{Mapping|legend=0| 1 0 15 -59 -136 -215 -91 | 0 1 -8 39 88 138 60 }}
Optimal tunings:
* 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 }}
Badness (Sintel): 1.50
=== Bipont ===
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
Comma list: 3025/3024, 4375/4374, 32805/32768
{{Mapping|legend=0| 2 0 30 -118 -85 | 0 1 -8 39 29 }}
: mapping generators: ~99/70, ~3
Optimal tunings:
* 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 }}
Badness (Sintel): 0.484
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 729/728, 1575/1573, 4096/4095
{{Mapping|legend=0| 2 0 30 -118 -85 112 | 0 1 -8 39 29 -33 }}
Optimal tunings:
* WE: ~99/70 = 599.9939{{c}}, ~3/2 = 701.7657{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7728{{c}}
{{Optimal ET sequence|legend=0| 106, 118, 224, 566f, 790f }}
Badness (Sintel): 1.25
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 625/624, 729/728, 1089/1088, 1225/1224, 2880/2873
{{Mapping|legend=0| 2 0 30 -118 -85 112 -182 | 0 1 -8 39 29 -33 60 }}
Optimal tunings:
* WE: ~99/70 = 599.9839{{c}}, ~3/2 = 701.7463{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7649{{c}}
{{Optimal ET sequence|legend=0| 106g, 118, 224, 342, 566f }}
Badness (Sintel): 1.38
==== Counterbipont ====
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 3025/3024, 32805/32768
{{Mapping|legend=0| 2 0 30 -118 -85 -243 | 0 1 -8 39 29 79 }}
Optimal tunings:
* 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 }}
Badness (Sintel): 1.06
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 32805/32768
{{Mapping|legend=0| 2 0 30 -118 -85 -243 -182 | 0 1 -8 39 29 79 60 }}
Optimal tunings:
* WE: ~99/70 = 600.0336{{c}}, ~3/2 = 701.8031{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7647{{c}}
{{Optimal ET sequence|legend=0| 106fg, 118f, 224, 342f, 566 }}
Badness (Sintel): 1.29
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 1540/1539, 4875/4864
{{Mapping|legend=0| 2 0 30 -118 -85 -243 -182 -169 | 0 1 -8 39 29 79 60 56 }}
Optimal tunings:
* 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 }}
Badness (Sintel): 1.35
==== Quadrapont ====
Subgroup: 2.3.5.7.11.13
Comma list: 3025/3024, 4225/4224, 4375/4374, 32805/32768
{{Mapping|legend=0| 4 0 60 -236 -170 -131 | 0 1 -8 39 29 23 }}
: mapping generators: ~208/175, ~3
Optimal tunings:
* 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 }}
Badness (Sintel): 0.869
== Grackle ==
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
[[Comma list]]: 126/125, 32805/32768
{{Mapping|legend=1| 1 0 15 44 | 0 1 -8 -26 }}
: mapping generators: ~2, ~3
[[Optimal tuning]]s:
* [[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}}
: error map: {{val| 0.000 -0.709 +3.715 -1.234 }}
[[Minimax tuning]]:
* [[7-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.7/3
* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
{{Optimal ET sequence|legend=1| 12, …, 65, 77, 166c }}
[[Badness]] (Sintel): 1.78
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 126/125, 176/175, 32805/32768
{{Mapping|legend=0| 1 0 15 44 70 | 0 1 -8 -26 -42 }}
Optimal tunings:
* WE: ~2 = 1199.7077{{c}}, ~3/2 = 701.0017{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.1804{{c}}
{{Optimal ET sequence|legend=0| 12, 65e, 77, 89, 166c }}
Badness (Sintel): 1.62
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 176/175, 196/195, 5445/5408
{{Mapping|legend=0| 1 0 15 44 70 75 | 0 1 -8 -26 -42 -45 }}
Optimal tunings:
* WE: ~2 = 1199.7782{{c}}, ~3/2 = 701.0966{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2319{{c}}
{{Optimal ET sequence|legend=0| 12f, 65ef, 77, 166cf }}
Badness (Sintel): 1.56
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 126/125, 176/175, 196/195, 256/255, 2904/2873
{{Mapping|legend=0| 1 0 15 44 70 75 -7 | 0 1 -8 -26 -42 -45 7 }}
Optimal tunings:
* WE: ~2 = 1199.5839{{c}}, ~3/2 = 700.9632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2137{{c}}
{{Optimal ET sequence|legend=0| 12f, 77, 89f, 166cf }}
Badness (Sintel): 1.52
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 126/125, 171/170, 176/175, 196/195, 209/208, 324/323
{{Mapping|legend=0| 1 0 15 44 70 75 -7 9 | 0 1 -8 -26 -42 -45 7 -3 }}
Optimal tunings:
* WE: ~2 = 1199.7146{{c}}, ~3/2 = 701.0500{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2212{{c}}
{{Optimal ET sequence|legend=0| 12f, 77, 166cf }}
Badness (Sintel): 1.40
==== Grackloid ====
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 176/175, 729/728, 1287/1280
{{Mapping|legend=0| 1 0 15 44 70 -47 | 0 1 -8 -26 -42 32 }}
Optimal tunings:
* WE: ~2 = 1200.0060{{c}}, ~3/2 = 701.2202{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2167{{c}}
{{Optimal ET sequence|legend=0| 12, 77, 166c }}
Badness (Sintel): 2.00
=== Grack ===
Subgroup: 2.3.5.7.11
Comma list: 126/125, 245/242, 896/891
{{Mapping|legend=0| 1 0 15 44 51 | 0 1 -8 -26 -30 }}
Optimal tunings:
* WE: ~2 = 1199.8388{{c}}, ~3/2 = 701.3071{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.4068{{c}}
{{Optimal ET sequence|legend=0| 12, 53d, 65, 77e }}
Badness (Sintel): 1.85
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 196/195, 245/242, 832/825
{{Mapping|legend=0| 1 0 15 44 51 75 | 0 1 -8 -26 -30 -45 }}
Optimal tunings:
* WE: ~2 = 1199.7329{{c}}, ~3/2 = 701.1918{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.3555{{c}}
{{Optimal ET sequence|legend=0| 12f, 53dff, 65f, 77e }}
Badness (Sintel): 1.84
==== Catahelenic ====
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 126/125, 245/242, 352/351
{{Mapping|legend=0| 1 0 15 44 51 56 | 0 1 -8 -26 -30 -33 }}
Optimal tunings:
* WE: ~2 = 1199.8928{{c}}, ~3/2 = 701.4664{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.5327{{c}}
{{Optimal ET sequence|legend=0| 12f, …, 53d, 65 }}
Badness (Sintel): 2.01
== Quasipyth ==
Named by [[Xenllium]] in 2026, quasipyth tempers out {{monzo| 109 -67 0 -1 }}, the [[nanisma]], as well as the [[catasyc comma]], 390625/387072. The 7/4 is found at −67 fifths, represented by the nonuple-diminished thirteenth.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 32805/32768, 390625/387072
{{Mapping|legend=1| 1 0 15 109 | 0 1 -8 -67 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2569{{c}}, ~3/2 = 702.1149{{c}}
: [[error map]]: {{val| +0.2569 +0.4168 -1.4342 +0.2685 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9615{{c}}
: error map: {{val| 0.0000 +0.0065 -2.0054 -0.2437 }}
{{Optimal ET sequence|legend=1| 53, 147d, 200, 253, 306c, 559c }}
[[Badness]] (Sintel): 5.04
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 385/384, 19712/19683, 78125/77616
{{Mapping|legend=0| 1 0 15 109 -117 | 0 1 -8 -67 76 }}
Optimal tunings:
* WE: ~2 = 1200.3283{{c}}, ~3/2 = 702.1636{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9713{{c}}
{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}
Badness (Sintel): 3.83
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 385/384, 2200/2197, 19712/19683
{{Mapping|legend=0| 1 0 15 109 -117 -28 | 0 1 -8 -67 76 20 }}
Optimal tunings:
* WE: ~2 = 1200.3229{{c}}, ~3/2 = 702.1603{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9714{{c}}
{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}
Badness (Sintel): 2.13
== Schism ==
See [[Archytas clan #Schism]].
Schism is a relatively low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (C–B♭). 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.
== Bischismic ==
Bischismic tempers out 3136/3125, the [[hemimean comma]], as well as 321489/320000, the [[varunisma]], and may be described as the {{nowrap| 118 & 130 }} temperament. The octave is split in halves, so the [[ploidacot]] of this temperament is diploid monocot. In schismic, -10 fifths make the interval class of 10/9. Bischismic then finds [[7/4]] by a stack of two [[10/9]]'s plus a semi-octave period, and in the [[11-limit]], it simply finds [[11/8]] by a stack of three [[10/9]]'s. [[248edo]] and [[378edo]] make for excellent tunings in both cases.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 3136/3125, 32805/32768
{{Mapping|legend=1| 2 0 30 69 | 0 1 -8 -20 }}
: mapping generators: ~567/400, ~3
[[Optimal tuning]]s:
* [[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 }}
[[Minimax tuning]]:
* [[7-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.7/3
* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
{{Optimal ET sequence|legend=1| 12, …, 106d, 118, 130, 248, 378 }}
[[Badness]] (Sintel): 1.39
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 441/440, 3136/3125, 8019/8000
{{Mapping|legend=0| 2 0 30 69 102 | 0 1 -8 -20 -30 }}
Optimal tunings:
* WE: ~99/70 = 600.0165{{c}}, ~3/2 = 701.6316{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.6110{{c}}
{{Optimal ET sequence|legend=0| 12, …, 106de, 118, 130, 248 }}
Badness (Sintel): 0.931
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 441/440, 729/728, 1001/1000, 3136/3125
{{Mapping|legend=0| 2 0 30 69 102 -75 | 0 1 -8 -20 -30 26 }}
Optimal tunings:
* WE: ~99/70 = 599.9610{{c}}, ~3/2 = 701.5445{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5908{{c}}
{{Optimal ET sequence|legend=0| 12, 118, 130, 248, 378 }}
Badness (Sintel): 1.19
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 289/288, 441/440, 561/560, 729/728, 3136/3125
{{Mapping|legend=0| 2 0 30 69 102 -75 5 | 0 1 -8 -20 -30 26 1 }}
Optimal tunings:
* WE: ~99/70 = 600.0331{{c}}, ~3/2 = 701.6387{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5994{{c}}
{{Optimal ET sequence|legend=0| 12, 118, 130, 248g }}
Badness (Sintel): 1.49
==== Bischis ====
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 364/363, 441/440, 3136/3125
{{Mapping|legend=0| 2 0 30 69 102 131 | 0 1 -8 -20 -30 -39 }}
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| 12f, 106deff, 118f, 130 }}
Badness (Sintel): 1.21
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 221/220, 289/288, 351/350, 441/440, 3136/3125
{{Mapping|legend=0| 2 0 30 69 102 131 5 | 0 1 -8 -20 -30 -39 1 }}
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| 12f, 106deff, 118f, 130, 248fg }}
Badness (Sintel): 1.37
== 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
[[Comma list]]: 32805/32768, 1500625/1492992
{{Mapping|legend=1| 2 1 22 -15 | 0 2 -16 19 }}
: mapping generators: ~1225/864, ~35/24
[[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 }}
{{Optimal ET sequence|legend=1| 24, 94, 118, 212, 330, 542d, 872cdd, 1414ccddd }}
[[Badness]] (Sintel): 2.80
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 385/384, 9801/9800, 14641/14580
{{Mapping|legend=0| 2 1 22 -15 8 | 0 2 -16 19 -1 }}
Optimal tunings:
* WE: ~99/70 = 600.1645{{c}}, ~35/24 = 651.0963{{c}} (~36/35 = 50.9319{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9184{{c}} (~36/35 = 50.9184{{c}})
{{Optimal ET sequence|legend=0| 24, 94, 118, 212, 330e, 542dee, 872cddeee }}
Badness (Sintel): 1.21
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 385/384, 729/728, 1575/1573
{{Mapping|legend=0| 2 1 22 -15 8 15 | 0 2 -16 19 -1 -7 }}
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}})
{{Optimal ET sequence|legend=0| 24, 94, 118, 212f }}
Badness (Sintel): 1.56
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 170/169, 289/288, 352/351, 385/384, 561/560
{{Mapping|legend=0| 2 1 22 -15 8 15 6 | 0 2 -16 19 -1 -7 2 }}
Optimal tunings:
* WE: ~99/70 = 600.1134{{c}}, ~35/24 = 651.0646{{c}} (~36/35 = 50.9512{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9414{{c}} (~36/35 = 50.9414{{c}})
{{Optimal ET sequence|legend=0| 24, 94, 118 }}
Badness (Sintel): 1.30
==== Kleischis ====
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 385/384, 1573/1568, 14641/14580
{{Mapping|legend=0| 2 1 22 -15 8 -36 | 0 2 -16 19 -1 40 }}
Optimal tunings:
* WE: ~99/70 = 600.1909{{c}}, ~35/24 = 651.1578{{c}} (~36/35 = 50.9670{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9541{{c}} (~36/35 = 50.9541{{c}})
{{Optimal ET sequence|legend=0| 24f, 94, 118f, 212 }}
Badness (Sintel): 1.55
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 289/288, 325/324, 385/384, 442/441, 14641/14580
{{Mapping|legend=0| 2 1 22 -15 8 -36 6 | 0 2 -16 19 -1 40 2 }}
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=0| 24f, 94, 118f, 212g }}
Badness (Sintel): 1.26
== 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
[[Comma list]]: 245/243, 32805/32768
{{Mapping|legend=1| 1 1 7 -1 | 0 2 -16 13 }}
: mapping generators: ~2, ~128/105
[[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 }}
{{Optimal ET sequence|legend=1| 17, 24, 41, 106d, 147d, 188cd }}
[[Badness]] (Sintel): 2.03
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 243/242, 245/242, 385/384
{{Mapping|legend=0| 1 1 7 -1 2 | 0 2 -16 13 5 }}
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| 17, 24, 41, 106d }}
Badness (Sintel): 1.30
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 144/143, 243/242, 245/242
{{Mapping|legend=0| 1 1 7 -1 2 4 | 0 2 -16 13 5 -1 }}
Optimal tunings:
* WE: ~2 = 1199.9362{{c}}, ~11/9 = 351.0061{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0247{{c}}
{{Optimal ET sequence|legend=0| 17, 24, 41 }}
Badness (Sintel): 1.27
== Hemischis ==
Hemischis tempers out 6144/6125, the [[porwell comma]], as well as 19683/19600, the [[cataharry comma]], and may be described as the {{nowrap| 53 & 130 }} temperament. Its [[ploidacot]] is alpha-dicot.
The [[S-expression]]-based comma list for 13-limit hemischis is {[[540/539|S12/S14]], [[676/675|S13/S15 = S26]], [[729/728|S27]], [[4096/4095|S64]], ([[4225/4224|S65]])}. Tempering out [[169/168]] ({{S|13}}), [[225/224]] ({{S|15}}) or [[625/624]] ({{S|25}}) leads to [[53edo]] while tempering out [[24192/24167]] ([[S-expression|S12/S13]]), [[10985/10976]] ([[S-expression|S13/S14]]), [[43904/43875]] ([[S-expression|S14/S15]]) or [[2401/2400]] ([[S-expression|S49]]) leads to [[130edo]] and implies S12, S13, S14, and S15 are tempered together.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 6144/6125, 19683/19600
{{Mapping|legend=1| 1 0 15 -17 | 0 2 -16 25 }}
: mapping generators: ~2, ~140/81
[[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 }}
{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 313 }}
[[Badness]] (Sintel): 1.16
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 540/539, 5632/5625, 8019/8000
{{Mapping|legend=0| 1 0 15 -17 51 | 0 2 -16 25 -60 }}
Optimal tunings:
* WE: ~2 = 1199.8482{{c}}, ~140/81 = 950.6809{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~140/81 = 950.8020{{c}}
{{Optimal ET sequence|legend=0| 53, 130, 183, 313, 809cd }}
Badness (Sintel): 1.20
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 540/539, 676/675, 4096/4095
{{Mapping|legend=0| 1 0 15 -17 51 14 | 0 2 -16 25 -60 -13 }}
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=0| 53, 130, 183, 313 }}
Badness (Sintel): 0.860
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 351/350, 442/441, 561/560, 676/675, 4096/4095
{{Mapping|legend=0| 1 0 15 -17 51 14 -49 | 0 2 -16 25 -60 -13 67 }}
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| 53, 130, 183, 496d }}
Badness (Sintel): 1.07
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 4096/4095
{{Mapping|legend=0| 1 0 15 -17 51 14 -49 9 | 0 2 -16 25 -60 -13 67 -6 }}
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| 53, 130, 183, 313h }}
Badness (Sintel): 1.11
=== 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
{{Mapping|legend=0| 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 ET sequence|legend=0| 53, 130, 183, 313h }}
Badness (Sintel): 1.06
; 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
== Term ==
Term tempers out the [[landscape comma]], mapping [[63/50]] to the 1/3-octave period. It can be described as {{nowrap| 12 & 171 }}, and is the unique temperament that tempers together the syntonic and Pythagorean commas and equates it with a stack of three [[marvel comma]]s. A [[septimal comma]] is then found as a stack of four marvel commas. In certain 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma #As an interval region|kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[171edo]] makes for an excellent tuning.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 32805/32768, 250047/250000
{{Mapping|legend=1| 3 0 45 94 | 0 1 -8 -18 }}
: mapping generators: ~63/50, ~3
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 400.0257{{c}}, ~3/2 = 701.7873{{c}}
: [[error map]]: {{val| +0.077 -0.091 -0.072 +0.031 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 701.7383{{c}}
: error map: {{val| 0.000 -0.217 -0.220 -0.115 }}
[[Minimax tuning]]:
* [[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 }}
[[Badness]] (Sintel): 0.505
=== 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.
Subgroup: 2.3.5.7.11
Comma list: 441/440, 4375/4356, 32805/32768
{{Mapping|legend=0| 3 0 45 94 134 | 0 1 -8 -18 -26 }}
Optimal tunings:
* WE: ~44/35 = 400.0464{{c}}, ~3/2 = 701.9053{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8178{{c}}
{{Optimal ET sequence|legend=0| 12, …, 159, 330 }}
Badness (Sintel): 1.97
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 625/624, 13720/13689
{{Mapping|legend=0| 3 0 45 94 134 168 | 0 1 -8 -18 -26 -33 }}
Optimal tunings:
* 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 }}
Badness (Sintel): 1.53
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619
{{Mapping|legend=0| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }}
Optimal tunings:
* WE: ~34/27 = 400.0195{{c}}, ~3/2 = 701.8439{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 701.8081{{c}}
{{Optimal ET sequence|legend=0| 12f, 159, 171, 330 }}
Badness (Sintel): 1.38
=== Terminator ===
Terminator tempers out 540/539, and may be described as {{nowrap| 171 & 183 }}.
Subgroup: 2.3.5.7.11
Comma list: 540/539, 32805/32768, 137781/137500
{{Mapping|legend=0| 3 0 45 94 -137 | 0 1 -8 -18 31 }}
Optimal tunings:
* WE: ~63/50 = 399.9677{{c}}, ~3/2 = 701.6278{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6846{{c}}
{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 537, 891de }}
Badness (Sintel): 2.21
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 4096/4095, 31250/31213
{{Mapping|legend=0| 3 0 45 94 -137 -103 | 0 1 -8 -18 31 24 }}
Optimal tunings:
* WE: ~63/50 = 399.9731{{c}}, ~3/2 = 701.6414{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}}
{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }}
Badness (Sintel): 1.47
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095
{{Mapping|legend=0| 3 0 45 94 -137 -103 -2 | 0 1 -8 -18 31 24 3 }}
Optimal tunings:
* WE: ~63/50 = 399.9757{{c}}, ~3/2 = 701.6458{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}}
{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }}
Badness (Sintel): 1.04
=== 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: 9801/9800, 32805/32768, 151263/151250
{{Mapping|legend=0| 6 0 90 188 287 | 0 1 -8 -18 -28 }}
: mapping generators: ~55/49, ~3
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, …, 330e, 342, 1380, 1722, 2064, 2406c, 5154bccdde }}
Badness (Sintel): 0.973
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375
{{Mapping|legend=0| 6 0 90 188 287 355 | 0 1 -8 -18 -28 -35 }}
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, 330eff, 342f, 696f }} *
<nowiki>*</nowiki> optimal patent val: [[354edo|354]]
Badness (Sintel): 1.85
=== Hemiterm ===
The hemiterm temperament tempers out [[3025/3024]] (lehmerisma), and may be described as {{nowrap| 159 & 183 }}. Its ploidacot is triploid alpha-dicot.
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 32805/32768, 102487/102400
{{Mapping|legend=0| 3 0 45 94 8 | 0 2 -16 -36 1 }}
: mapping generators: ~63/50, ~693/400
Optimal tunings:
* WE: ~63/50 = 400.0309{{c}}, ~693/400 = 950.9458{{c}} (~12/11 = 150.8841{{c}})
* CWE: ~63/50 = 400.0000{{c}}, ~693/400 = 950.8707{{c}} (~12/11 = 150.8707{{c}})
{{Optimal ET sequence|legend=0| 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce }}
Badness (Sintel): 0.684
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712
{{Mapping|legend=0| 3 0 45 94 8 42 | 0 2 -16 -36 1 -13 }}
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| 24d, 159, 183, 342f }}
Badness (Sintel): 1.30
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264
{{Mapping|legend=0| 3 0 45 94 8 42 -2 | 0 2 -16 -36 1 -13 6 }}
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| 24d, 159, 183, 342f, 525f }}
Badness (Sintel): 1.14
== Altinex ==
Named by [[Aura]] in 2021, altinex is an alternative to [[#Hemiterm|hemiterm]] and may be described as {{nowrap| 24 & 159 }}. [[159edo]] itself makes for a recommendable tuning.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 32805/32768, 367653125/362797056
{{Mapping|legend=1| 3 0 45 -32 | 0 2 -16 17 }}
: mapping generators: ~1536/1225, ~34300/19683
[[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=1| 24, 135, 159, 612ccdd }}
[[Badness]] (Sintel): 10.7
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 385/384, 14700/14641, 19712/19683
{{Mapping|legend=0| 3 0 45 -32 8 | 0 2 -16 17 1 }}
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| 24, 135, 159 }}
Badness (Sintel): 3.35
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 385/384, 676/675, 19712/19683
{{Mapping|legend=0| 3 0 45 -32 8 42 | 0 2 -16 17 1 -13 }}
Optimal tunings:
* WE: ~44/35 = 400.1396{{c}}, ~26/15 = 951.2799{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~26/15 = 950.9462{{c}}
{{Optimal ET sequence|legend=0| 24, 135f, 159 }}
Badness (Sintel): 2.27
== 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.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 686/675, 32805/32768
{{Mapping|legend=1| 1 2 -1 1 | 0 -3 24 13 }}
[[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=1| 29, 36, 65 }}
[[Badness]] (Sintel): 4.42
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 245/242, 686/675, 896/891
{{Mapping|legend=0| 1 2 -1 1 0 | 0 -3 24 13 25 }}
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| 29, 36, 65 }}
Badness (Sintel): 2.26
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 169/168, 245/242, 896/891
{{Mapping|legend=0| 1 2 -1 1 0 3 | 0 -3 24 13 25 5 }}
Optimal tunings:
* WE: ~2 = 1201.1361{{c}}, ~11/10 = 166.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0833{{c}}
{{Optimal ET sequence|legend=0| 29, 65f, 94df }}
Badness (Sintel): 1.81
== Tertiaschis ==
Named by [[Xenllium]] in 2021, tertiaschis may be described as {{nowrap| 94 & 159 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 1071875/1062882 for prime 7. See also [[daemotertiaschis]] which is made using every other generator of tertiaschis.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 32805/32768, 1071875/1062882
{{Mapping|legend=1| 1 2 -1 10 | 0 -3 24 -52 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3627{{c}}, ~192/175 = 166.0691{{c}}
: [[error map]]: {{val| +0.363 +0.563 -1.019 -0.790 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~192/175 = 166.0172{{c}}
: error map: {{val| 0.000 -0.007 -1.901 -1.720 }}
{{Optimal ET sequence|legend=1| 65, 94, 159, 253, 412cd }}
[[Badness]] (Sintel): 5.36
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 385/384, 4000/3993, 19712/19683
{{Mapping|legend=0| 1 2 -1 10 0 | 0 -3 24 -52 25 }}
Optimal tunings:
* WE: ~2 = 1200.3379{{c}}, ~11/10 = 166.0638{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0167{{c}}
{{Optimal ET sequence|legend=0| 65, 94, 159, 253, 412cd, 665ccde }}
Badness (Sintel): 2.07
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 385/384, 1575/1573, 10985/10976
{{Mapping|legend=0| 1 2 -1 10 0 12 | 0 -3 24 -52 25 -60 }}
Optimal tunings:
* WE: ~2 = 1200.3467{{c}}, ~11/10 = 166.0635{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0142{{c}}
{{Optimal ET sequence|legend=0| 65f, 94, 159, 253, 412cdf, 665ccdef }}
Badness (Sintel): 1.52
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
{{Mapping|legend=0| 1 2 -1 10 0 12 -2 | 0 -3 24 -52 25 -60 44 }}
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=1| 65f, 94, 159, 253 }}
Badness (Sintel): 1.35
== 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.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 32805/32768, 244140625/243045684
{{Mapping|legend=1| 1 2 -1 -12 | 0 -3 24 107 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1265{{c}}, ~625/567 = 166.0797{{c}}
: [[error map]]: {{val| +0.127 +0.059 -0.529 +0.178 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/567 = 166.0632{{c}}
: error map: {{val| 0.000 -0.145 -0.797 -0.065 }}
{{Optimal ET sequence|legend=1| 65d, 159, 224, 383, 607 }}
[[Badness]] (Sintel): 4.76
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4000/3993, 32805/32768
{{Mapping|legend=0| 1 2 -1 -12 0 | 0 -3 24 107 25 }}
Optimal tunings:
* WE: ~2 = 1200.0804{{c}}, ~11/10 = 166.0739{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0634{{c}}
{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}
Badness (Sintel): 1.62
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976
{{Mapping|legend=0| 1 2 -1 -12 0 -10 | 0 -3 24 107 25 99 }}
Optimal tunings:
* WE: ~2 = 1200.0805{{c}}, ~11/10 = 166.0740{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0635{{c}}
{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}
Badness (Sintel): 1.01
== 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 and Pythagorean commas for a generic comma step, and splits it into three [[marvel comma]]s, quadrant splits the same generic comma step into four. A [[septimal comma]] is then found as a stack of five marvel commas.
[[Subgroup]]: 2.3.5.7
[[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:
* [[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 }}
{{Optimal ET sequence|legend=1| 12, …, 200, 212, 224, 436, 660 }}
[[Badness]] (Sintel): 2.79
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 1375/1372, 6250/6237, 32805/32768
{{Mapping|legend=0| 4 0 60 119 185 | 0 1 -8 -17 -27 }}
Optimal tunings:
* WE: ~25/21 = 300.0244{{c}}, ~3/2 = 701.8759{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8145{{c}}
{{Optimal ET sequence|legend=0| 12, …, 212, 224, 436, 660 }}
Badness (Sintel): 1.51
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647
{{Mapping|legend=0| 4 0 60 119 185 224 | 0 1 -8 -17 -27 -33 }}
Optimal tunings:
* WE: ~25/21 = 300.0234{{c}}, ~3/2 = 701.8707{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8123{{c}}
{{Optimal ET sequence|legend=0| 12f, …, 212, 224, 436, 660 }}
Badness (Sintel): 1.13
== Sesquiquartififths ==
Sesquiquartififths tempers out 2401/2400, the [[breedsma]], and may be described as the {{nowrap| 41 & 171 }} temperament. It splits the [[3/2|perfect fifth]] into four; its [[ploidacot]] is thus tetracot.
The name might have been a portmanteau of ''sesquiquart'' and ''quartififths'', each meaning dividing 3/2 into four.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 2401/2400, 32805/32768
{{Mapping|legend=1| 1 1 7 5 | 0 4 -32 -15 }}
: mapping generators: ~2, ~448/405
[[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 }}
[[Minimax tuning]]:
* [[7-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
{{Optimal ET sequence|legend=1| 41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd }}
[[Badness]] (Sintel): 0.285
=== Sesquart ===
Sesquart is the main [[11-limit|11-]] and [[13-limit]] extension of sesquiquartififths of practical interest, as it identifies the neutral third with [[11/9]], which is realized in [[41edo]], [[89edo]], [[130edo]], and [[171edo]] also makes for a possible tuning.
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 16384/16335
{{Mapping|legend=0| 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
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 364/363, 441/440, 3584/3575
{{Mapping|legend=0| 1 1 7 5 2 -2 | 0 4 -32 -15 10 39 }}
Optimal tunings:
* WE: ~2 = 1199.8352{{c}}, ~72/65 = 175.3852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4095{{c}}
{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}
Badness (Sintel): 0.925
===== Heartia =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 243/242, 256/255, 273/272, 364/363, 441/440
{{Mapping|legend=0| 1 1 7 5 2 -2 0 | 0 4 -32 -15 10 39 28 }}
Optimal tunings:
* WE: ~2 = 1199.6422{{c}}, ~72/65 = 175.3338{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3857{{c}}
{{Optimal ET sequence|legend=0| 41, 89, 130g }}
Badness (Sintel): 1.45
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 171/170, 243/242, 256/255, 273/272, 324/323, 441/440
{{Mapping|legend=0| 1 1 7 5 2 -2 0 6 | 0 4 -32 -15 10 39 28 -12 }}
Optimal tunings:
* WE: ~2 = 1199.7499{{c}}, ~21/19 = 175.3432{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.3797{{c}}
{{Optimal ET sequence|legend=0| 41, 89, 130g }}
Badness (Sintel): 1.40
===== Sesquartia =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575
{{Mapping|legend=0| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }}
Optimal tunings:
* WE: ~2 = 1199.8902{{c}}, ~72/65 = 175.4077{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4234{{c}}
{{Optimal ET sequence|legend=0| 41, 130, 171 }}
Badness (Sintel): 1.18
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594
{{Mapping|legend=0| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }}
Optimal tunings:
* WE: ~2 = 1199.9864{{c}}, ~21/19 = 175.4169{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4189{{c}}
{{Optimal ET sequence|legend=0| 41, 130, 171 }}
Badness (Sintel): 1.24
====== 23-limit ======
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594
{{Mapping|legend=0| 1 1 7 5 2 -2 -6 6 -6 | 0 4 -32 -15 10 39 69 -12 72 }}
Optimal tunings:
* WE: ~2 = 1199.9606{{c}}, ~21/19 = 175.4067{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4123{{c}}
{{Optimal ET sequence|legend=0| 41i, 130, 171 }}
Badness (Sintel): 1.36
===== Hearty =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625
{{Mapping|legend=0| 1 1 7 5 2 -2 13 | 0 4 -32 -15 10 39 -61 }}
Optimal tunings:
* WE: ~2 = 1199.9458{{c}}, ~72/65 = 175.3689{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3770{{c}}
{{Optimal ET sequence|legend=0| 41g, 89, 130 }}
Badness (Sintel): 1.56
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 221/220, 243/242, 361/360, 364/363, 441/440, 456/455
{{Mapping|legend=0| 1 1 7 5 2 -2 13 6 | 0 4 -32 -15 10 39 -61 -12 }}


[|1 0 0 0 0&gt;, |25/16 1/8 0 -1/16 0&gt;, |5/2 -1 0 1/2 0&gt;,
Optimal tunings:
|25/8 -7/4 0 7/8 0&gt;, |47/16 23/8 0 -23/16 0&gt;]
* WE: ~2 = 1200.0114{{c}}, ~72/65 = 175.3783{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3765{{c}}


Eigenmonzos: 2, 9/7
{{Optimal ET sequence|legend=0| 41g, 89, 130 }}


[[POTE_tuning|POTE generator]]: ~3/2 = 702.157
Badness (Sintel): 1.39


Mapping generator: ~3
====== 23-limit ======
Subgroup: 2.3.5.7.11.13.17.19.23


Map: [&lt;1 0 15 25 -33|, &lt;0 1 -8 -14 23|]
Comma list: 221/220, 243/242, 276/275, 323/322, 361/360, 364/363, 441/440


[[EDO|Edos]]: [[41edo|41]], [[53edo|53]], [[94edo|94]], [[135edo|135]], 229c, 323c, 417ce
{{Mapping|legend=0| 1 1 7 5 2 -2 13 6 13 | 0 4 -32 -15 10 39 -61 -12 -58 }}


Badness: 0.0274
Optimal tunings:  
* WE: ~2 = 1200.0122{{c}}, ~72/65 = 175.3782{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3763{{c}}


===13-limit===
{{Optimal ET sequence|legend=0| 41g, 89, 130 }}
Commas: 225/224, 275/273, 325/324, 385/384


[[POTE_tuning|POTE generator]]: ~3/2 = 702.113
Badness (Sintel): 1.37


Mapping generator: ~3
=== Bisesqui ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 15 25 -33 -28|, &lt;0 1 -8 -14 23 20|]
Comma list: 2401/2400, 9801/9800, 32805/32768


EDOs: 12e, 41, 53, 94, 135, 429cdef, 523cdef
{{Mapping|legend=0| 2 2 14 10 23 | 0 4 -32 -15 -55 }}
: mapping generators: ~99/70, ~448/405


Badness: 0.0207
Optimal tunings:  
* WE: ~99/70 = 600.0429{{c}}, ~448/405 = 175.4474{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~448/405 = 175.4334{{c}}


==Cassandra==
{{Optimal ET sequence|legend=1| 82e, 130, 212, 342, 1156, 1498, 1840d, 5862bbccdddee }}
Commas: 100/99, 225/224, 245/242


[[POTE_tuning|POTE generator]]: ~3/2 = 702.321
Badness (Sintel): 0.561


Mapping generator: ~3
== Tsaharuk ==
[[File:Tsaharuk.jpg|alt=Tsaharuk.jpg|thumb|Parametric horagram of the tsaharuk temperament.]]
{{See also| Tsaharuk24 }}


Map: [&lt;1 0 15 25 32|, &lt;0 1 -8 -14 -18|]
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.


EDOs: 12, 29, 41, 217ce, 258ce
It was proposed by [[Jacques Dudon]] based on [[Julien Jalaleddine Weiss]] qanun tunings, which has [[mos]] of size 43, 60, 77 and 94 notes.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_100718.html Yahoo! Tuning Group | ''A temperament for Maqam'']</ref>


Badness: 0.0236
It restricts to the [[2.3.7 subgroup]] as [[no-fives subgroup temperaments #Navy|navy]]. This relates tsaharuk to [[aberschismic temperaments #Quanic|quanic]], which favors a sharper fifth and works better in the higher limits, but which also has mos of size 43, 60, 77 and 94 notes. [[Bleu]] has, like tsaharuk, a generator of 1/5 fifth and makes for an instructive comparison. [[Edo]]s that support tsaharuk include [[77edo]], [[94edo]], [[171edo]] and [[248edo]], and edos supporting quanic are [[94edo]], [[111edo]] and [[205edo]]. Quanic and tsaharuk therefore become identical if 94edo is the tuning.


===13-limit===
[[Subgroup]]: 2.3.5.7
Commas: 100/99, 105/104, 196/195, 245/242


[[POTE_tuning|POTE generator]]: ~3/2 = 702.559
[[Comma list]]: 32805/32768, 420175/419904


Mapping generator: ~3
{{Mapping|legend=1| 1 1 7 0 | 0 5 -40 24 }}
: mapping generators: ~2, ~243/224


Map: [&lt;1 0 15 25 32 37|, &lt;0 1 -8 -14 -18 -21|]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1039{{c}}, ~243/224 = 140.3620{{c}}
: [[error map]]: {{val| +0.104 -0.041 -0.067 -0.137 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/224 = 140.3496{{c}}
: error map: {{val| 0.000 -0.207 -0.296 -0.436 }}


EDOs: 12f, 29, 41, 152cdf, 193cdf, 234cdf
{{Optimal ET sequence|legend=1| 17, 77, 94, 171 }}


Badness: 0.0207
[[Badness]] (Sintel): 0.777


==Helenus==
=== 11-limit ===
Commas: 99/98 176/175 3125/3087
Subgroup: 2.3.5.7.11


POTE generator: ~3/2 = 701.725
Comma list: 385/384, 1331/1323, 19712/19683


Mapping generator: ~3
{{Mapping|legend=0| 1 1 7 0 1 | 0 5 -40 24 21 }}


Map: [&lt;1 0 15 25 51|, &lt;0 1 -8 -14 -30|]
Optimal tunings:  
* WE: ~2 = 1200.3103{{c}}, ~88/81 = 140.4011{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.3649{{c}}


EDOs: 12, 53, 118d, 171de
{{Optimal ET sequence|legend=0| 17, 77, 94, 171e, 265e }}


Badness: 0.0356
Badness (Sintel): 2.10


===13-limit Helenus===
=== 13-limit ===
Commas: 99/98, 176/175, 275/273, 847/845
Subgroup: 2.3.5.7.11.13


POTE generator: ~3/2 = 701.747
Comma list: 352/351, 385/384, 729/728, 1331/1323


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


Map: [&lt;1 0 15 25 51 56|, &lt;0 1 -8 -14 -30 -33|]
Optimal tunings:  
* WE: ~2 = 1200.1840{{c}}, ~13/12 = 140.3840{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.3627{{c}}


EDOs: 53, 118d, 171de
{{Optimal ET sequence|legend=0| 17, 77, 94, 171e }}


Badness: 0.0263
Badness (Sintel): 1.57


=Hemigari=
== Quanharuk ==
Commas: 121/120, 225/224, 3125/3087
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.


POTE generator: ~63/55 = 248.918
[[Subgroup]]: 2.3.5.7


Mapping generator: ~110/63
[[Comma list]]: 16875/16807, 32805/32768


Map: [&lt;1 0 15 25 9|, &lt;0 2 -16 -28 -7|]
{{Mapping|legend=1| 1 0 15 12 | 0 5 -40 -29 }}
: mapping generators: ~2, ~56/45


EDOs: 29, 53, 82e, 135e, 188ce
[[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: 0.0507
{{Optimal ET sequence|legend=1| 41, 142, 183, 224 }}


==13-limit==
[[Badness]] (Sintel): 1.82
Commas: 121/120, 169/168, 225/224, 275/273


POTE generator: ~15/13 = 248.918
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping generator: ~26/15
Comma list: 540/539, 1375/1372, 32805/32768


Map: [&lt;1 0 15 25 9 14|, &lt;0 2 -16 -28 -7 -13|]
{{Mapping|legend=0| 1 0 15 12 -7 | 0 5 -40 -29 33 }}


EDOs: 29, 53, 82e, 135ef, 188cef
Optimal tunings:  
* WE: ~2 = 1199.9709{{c}}, ~56/45 = 380.3423{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3517{{c}}


Badness: 0.027464
{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}


=Guiron=
Badness (Sintel): 1.04
[[Comma|Commas]]: 1029/1024, 10976/10935


[[Minimax tuning]]:
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 1375/1372, 4096/4095
 
{{Mapping|legend=0| 1 0 15 12 -7 -15 | 0 5 -40 -29 33 59 }}
 
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| 41, 142, 183, 224, 631d, 855d }}
 
Badness (Sintel): 0.884
 
== Quintilipyth ==
Named by [[Xenllium]] in 2021, quintilipyth (formerly ''quintilischis'') slices the [[4/3|perfect fourth]] into five semitones and tempers out the [[compass comma]] (9765625/9680832) in the [[7-limit]]. It may be described as the {{nowrap| 12 & 253 }} temperament, and its [[ploidacot]] is omega-pentacot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 32805/32768, 9765625/9680832
 
{{Mapping|legend=1| 1 2 -1 -4 | 0 -5 40 82 }}
: 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 }}
 
{{Optimal ET sequence|legend=1| 12, …, 253, 265 }}
 
[[Badness]] (Sintel): 6.43
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 1375/1372, 4375/4356, 32805/32768
 
{{Mapping|legend=0| 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}}
 
{{Optimal ET sequence|legend=0| 12, …, 253, 265, 518c }}
 
Badness (Sintel): 3.74
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647
 
{{Mapping|legend=0| 1 2 -1 -4 -7 -9 | 0 -5 40 82 126 153 }}
 
Optimal tunings:
* WE: ~2 = 1200.1774{{c}}, ~35/33 = 99.6267{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6134{{c}}
 
{{Optimal ET sequence|legend=0| 12f, …, 241cdef, 253 }}
 
Badness (Sintel): 2.86
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619
 
{{Mapping|legend=0| 1 2 -1 -4 -7 -9 5 | 0 -5 40 82 126 153 -11 }}
 
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| 12f, 241cdef, 253 }}
 
Badness (Sintel): 2.34
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971
 
{{Mapping|legend=0| 1 2 -1 -4 -7 -9 5 4 | 0 -5 40 82 126 153 -11 3 }}


7+9 limit: [|1 0 0 0&gt;, |15/8 0 -1/8 0&gt;,
Optimal tunings:  
|0 0 1 0&gt;, |65/24 0 1/24 0&gt;]
* WE: ~2 = 1200.0713{{c}}, ~18/17 = 99.6208{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6152{{c}}


[[Eigenmonzo|Eigenmonzos]]: 2, 5/4
{{Optimal ET sequence|legend=0| 12f, 253, 265 }}


[[POTE_tuning|POTE generator]]: ~8/7 = 233.930
Badness (Sintel): 2.32


Mapping generator: ~8/7
== Quintaschis ==
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.


Map: [&lt;1 1 7 3|, &lt;0 3 -24 -1|]
[[Subgroup]]: 2.3.5.7


Wedgie: &lt;&lt;3 -24 -1 -45 -10 65||
[[Comma list]]: 32805/32768, 49009212/48828125


EDOs: 36, 41, [[77edo|77]], 118, 277d
{{Mapping|legend=1| 1 2 -1 -5 | 0 -5 40 94 }}


Badness: 0.0475
[[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 }}


==11-limit==
{{Optimal ET sequence|legend=1| 12, …, 289, 301, 590, 891, 1192 }}
[[Comma|Commas]]: 385/384, 441/440, 10976/10935


[[Minimax tuning]]:
[[Badness]] (Sintel): 3.36


[|1 0 0 0 0&gt;, |15/8 0 -1/8 0 0&gt;,
=== 11-limit ===
|0 0 1 0 0&gt;, |65/24 0 1/24 0 0&gt;,
Subgroup: 2.3.5.7.11
|37/6 0 -7/6 0 0&gt;]


[[Eigenmonzo|Eigenmonzos]]: 2, 5/4
Comma list: 441/440, 32805/32768, 1953125/1951488


[[POTE_tuning|POTE generator]]: ~8/7 = 233.931
{{Mapping|legend=0| 1 2 -1 -5 -8 | 0 -5 40 94 138 }}


Mapping generator: ~8/7
Optimal tunings:  
* WE: ~2 = 1200.0988{{c}}, ~35/33 = 99.6613{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6540{{c}}


Map: [&lt;1 1 7 3 -2|, &lt;0 3 -24 -1 28|]
{{Optimal ET sequence|legend=0| 12, …, 277d, 289 }}


[[EDO|Edos]]: [[41edo|41]], [[77edo|77]], [[118edo|118]], [[159edo|159]], [[200edo|200]], [[277edo|277d]]
Badness (Sintel): 3.69


Badness: 0.0266
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


==13-limit==
Comma list: 364/363, 441/440, 32805/32768, 109512/109375
Commas: 196/195, 352/351, 385/384, 729/728


POTE generator: ~8/7 = 233.890
{{Mapping|legend=0| 1 2 -1 -5 -8 -11 | 0 -5 40 94 138 177 }}


Mapping generator: ~8/7
Optimal tunings:  
* WE: ~2 = 1200.0625{{c}}, ~35/33 = 99.6630{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6583{{c}}


Map: [&lt;1 1 7 3 -2 0|, &lt;0 3 -24 -1 28 19|]
{{Optimal ET sequence|legend=0| 12f, …, 277dff, 289 }}


EDOs: 41, 77, 118
Badness (Sintel): 3.07


Badness: 0.0284
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


=Pogo=
Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768
Commas: 32805/32768, 118098/117649


POTE generator: ~9/7 = 433.901
{{Mapping|legend=0| 1 2 -1 -5 -8 -11 5 | 0 -5 40 94 138 177 -11 }}


Mapping generator: ~9/7
Optimal tunings:  
* WE: ~2 = 1200.1286{{c}}, ~18/17 = 99.6668{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6568{{c}}


Map: [&lt;2 1 22 2|, &lt;0 3 -24 5|]
{{Optimal ET sequence|legend=0| 12f, 277dff, 289 }}


EDOs: 36, 94, 130, 171, 183, 224, 354
Badness (Sintel): 2.58


Badness: 0.0796
==== 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
Commas: 540/539, 4000/3993, 32805/32768


POTE generator: ~9/7 = 433.911
{{Mapping|legend=0| 1 2 -1 -5 -8 -11 5 4 | 0 -5 40 94 138 177 -11 3 }}


Mapping generator: ~9/7
Optimal tunings:  
* WE: ~2 = 1200.0289{{c}}, ~18/17 = 99.6609{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6586{{c}}


Map: [&lt;2 1 22 2 25|, &lt;0 3 -24 5 -25|]
{{Optimal ET sequence|legend=0| 12f, 289 }}


EDOs: 36, 94, 130, 224, 354, 578
Badness (Sintel): 2.56


Badness: 0.0319
=== Quintahelenic ===
Subgroup: 2.3.5.7.11


==13-limit==
Comma list: 5632/5625, 8019/8000, 151263/151250
Commas: 540/539, 729/728, 4000/3993, 4225/4224


POTE generator: ~9/7 = 433.911
{{Mapping|legend=0| 1 2 -1 -5 -9 | 0 -5 40 94 150 }}


Map: [&lt;2 1 22 2 25 -2|, &lt;0 3 -24 5 -25 13|]
Optimal tunings:  
* WE: ~2 = 1200.0195{{c}}, ~200/189 = 99.6723{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6709{{c}}


EDOs: 36, 94, 130, 224, 354, 578
{{Optimal ET sequence|legend=0| 12, …, 289e, 301, 915 }}


Badness: 0.0175
Badness (Sintel): 2.72


=Sanjaab=
==== 13-limit ====
Commas: 225/224, 3125/3087, 1331/1323
Subgroup: 2.3.5.7.11.13


POTE generator: ~11/10 = 165.974
Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000


Map: [&lt;1 2 -1 -3 0|, &lt;0 -3 24 42 25|]
{{Mapping|legend=0| 1 2 -1 -5 -9 -11 | 0 -5 40 94 150 177 }}


EDOs: 29, 65d, 94, 441cde, 535cde, 629cde
Optimal tunings:  
* WE: ~2 = 1200.0442{{c}}, ~200/189 = 99.6709{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6675{{c}}


Badness: 0.0580
{{Optimal ET sequence|legend=0| 12f, …, 289e, 301 }}


==13-limit==
Badness (Sintel): 2.30
Commas: 225/224, 275/273, 847/845, 1331/1323


POTE generator: ~11/10 = 165.963
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 2 -1 -3 0 -1|, &lt;0 -3 24 42 25 34|]
Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750


EDOs: 29, 65d, 94
{{Mapping|legend=0| 1 2 -1 -5 -9 -11 5 | 0 -5 40 94 150 177 -11 }}


Badness: 0.0338
Optimal tunings:  
* WE: ~2 = 1200.1227{{c}}, ~200/189 = 99.6753{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6658{{c}}


=Squirrel=
{{Optimal ET sequence|legend=1| 12f, 289e, 301 }}
Commmas: 686/675, 32805/32768


POTE generator: ~160/147 = 166.140
Badness (Sintel): 2.06


Map: [&lt;1 2 -1 1|, &lt;0 -3 24 13|]
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


EDOs: 29, 36, 65
Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700


Badness: 0.1747
{{Mapping|legend=0| 1 2 -1 -5 -9 -11 5 4 | 0 -5 40 94 150 177 -11 3 }}


==11-limit==
Optimal tunings:
Commas: 245/242, 686/675, 896/891
* WE: ~2 = 1200.0230{{c}}, ~200/189 = 99.6694{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6676{{c}}


POTE generator: ~11/10 = 166.097
{{Optimal ET sequence|legend=0| 12f, 301 }}


Map: [&lt;1 2 -1 1 0|, &lt;0 -3 24 13 25|]
Badness (Sintel): 2.24


EDOs: 29, 36, 65
==== Quintahelenoid ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0683
Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436


==13-limit==
{{Mapping|legend=0| 1 2 -1 -5 -9 14 | 0 -5 40 94 150 -124 }}
Commas: 91/90, 169/168, 245/242, 896/891


POTE generator: ~11/10 = 166.054
Optimal tunings:  
* WE: ~2 = 1199.9919{{c}}, ~200/189 = 99.6712{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6718{{c}}


Map: [&lt;1 2 -1 1 0 3|, &lt;0 -3 24 13 25 5|]
{{Optimal ET sequence|legend=0| 12, 301, 614, 915 }}


EDOs: 29, 36, 65f, 94df, 159df
Badness (Sintel): 2.73


Badness: 0.0437
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


=Schism=
Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157
Commas: 64/63, 360/343


POTE generator: ~3/2 = 701.556
{{Mapping|legend=0| 1 2 -1 -5 -9 14 5 | 0 -5 40 94 150 -124 -11 }}


Mapping generator: ~3
Optimal tunings:  
* WE: ~2 = 1200.0469{{c}}, ~18/17 = 99.6749{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6710{{c}}


Map: [&lt;1 0 15 6|, &lt;0 1 -8 -2|]
{{Optimal ET sequence|legend=0| 12, 301 }}


Wedgie: &lt;&lt;1 -8 -2 -15 -6 18||
Badness (Sintel): 2.44


EDOs: 12, 29d, 41d, 53d
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.0566
Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137


==11-limit==
{{Mapping|legend=0| 1 2 -1 -5 -9 14 5 4 | 0 -5 40 94 150 -124 -11 3 }}
Commas: 45/44, 64/63, 99/98


POTE generator ~3/2 = 702.136
Optimal tunings:
* WE: ~2 = 1199.9925{{c}}, ~18/17 = 99.6710{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6716{{c}}


Mapping generator: ~3
{{Optimal ET sequence|legend=0| 12, 301 }}


Map: [&lt;1 0 15 6 13|, &lt;0 1 -8 -2 -6|]
Badness (Sintel): 2.41


EDOs: 12, 29de, 41de
== 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: 0.0375
[[Subgroup]]: 2.3.5.7


=Pontiac=
[[Comma list]]: 32805/32768, 235298/234375
Commas: 32805/32768, 4375/4374


7-limit minimax:
{{Mapping|legend=1| 1 2 -1 -1 | 0 -6 48 55 }}
: mapping generators: ~2, ~21/20


[|1 0 0 0&gt;, |74/47 0 -1/47 1/47&gt;, |113/47 0 8/47 -8/47&gt;,
[[Optimal tuning]]s:
|113/47 0 -39/47 39/47&gt;]
* [[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 }}


Eigenmonzos: 2, 7/5
{{Optimal ET sequence|legend=1| 29, 72cd, 101, 130, 289, 419 }}


9-limit minimax:
[[Badness]] (Sintel): 2.75


[|1 0 0 0&gt;, |3/2 1/5 -1/10 0&gt;,
=== 11-limit ===
|3 -8/5 4/5 0&gt;, |-1/2 39/5 -39/10 0&gt;]
Subgroup: 2.3.5.7.11


Eigenmonzos: 2, 10/9
Comma list: 441/440, 4000/3993, 235298/234375


[[POTE_tuning|POTE generator]]: 701.757
{{Mapping|legend=0| 1 2 -1 -1 0 | 0 -6 48 55 50 }}


Mapping generator: ~3
Optimal tunings:  
* WE: ~2 = 1200.0424{{c}}, ~21/20 = 83.0520{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0497{{c}}


Map: [&lt;1 0 15 -59|, &lt;0 1 -8 39|]
{{Optimal ET sequence|legend=0| 29, 72cde, 101e, 130, 289 }}


Wedgie: &lt;&lt;1 -8 39 -15 59 113||
Badness (Sintel): 1.50


[[EDO|Edos]]: [[53edo|53]], [[118edo|118]], [[171edo|171]], [[1079edo|1079]], [[1250edo|1250]], [[1421edo|1421]]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0141
Comma list: 364/363, 441/440, 676/675, 10985/10976


=Grackle=
{{Mapping|legend=0| 1 2 -1 -1 0 1 | 0 -6 48 55 50 39 }}
Commas: 126/125, 32805/32768


7-limit minimax
Optimal tunings:
* WE: ~2 = 1200.1056{{c}}, ~21/20 = 83.0566{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0508{{c}}


Eigenmonzos: 2, 7/6
{{Optimal ET sequence|legend=0| 29, 72cdef, 101e, 130, 289 }}


9-limit minimax
Badness (Sintel): 1.04


Eigenmonzos: 2, 9/7
== Septant ==
Named by [[Xenllium]] in 2021, septant notably tempers out the [[akjaysma]] ({{monzo| 47 -7 -7 -7 }}) and may be described as the {{nowrap| 224 & 301 }} temperament. It has a period of 1/7 octave, and its [[ploidacot]] is heptaploid monocot.


[[POTE_tuning|POTE generator]]: 701.239
[[Subgroup]]: 2.3.5.7


Mapping generator: ~3
[[Comma list]]: 32805/32768, 516560652/514714375


Map: [&lt;1 0 15 -44|, &lt;0 1 -8 -26|]
{{Mapping|legend=1| 7 0 105 -56 | 0 1 -8 7 }}
: mapping generators: ~8575/7776, ~3


Wedgie: &lt;&lt;1 -8 -26 -15 -44 -38||
[[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 }}


[[EDO|Edos]]: [[77edo|77]], [[89edo|89]], [[101edo|101]], [[166edo|166c]], [[243edo|243c]]
{{Optimal ET sequence|legend=1| 77, 147, 224, 301, 525, 826, 1351 }}


Badness: 0.0704
[[Badness]] (Sintel): 2.81


=Bischismic=
=== 11-limit ===
Commas: 3136/3125, 32805/32768
Subgroup: 2.3.5.7.11


7-limit minimax
Comma list: 3025/3024, 24057/24010, 32805/32768


Eigenmonzos: 2, 7/6
{{Mapping|legend=0| 7 0 105 -56 -120 | 0 1 -8 7 13 }}


9-limit minimax
Optimal tunings:
* WE: ~495/448 = 171.4334{{c}}, ~3/2 = 701.7387{{c}}
* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7198{{c}}


Eigenmonzos: 2, 9/7
{{Optimal ET sequence|legend=0| 77, 147, 224, 301, 525 }}


[[POTE_tuning|POTE generator]]: 701.592
Badness (Sintel): 1.46


Mapping generator: ~3
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 0 30 69|, &lt;0 1 -8 -20|]
Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024


Wedgie: &lt;&lt;2 -16 -40 -30 -69 -48||
{{Mapping|legend=0| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }}


[[EDO|Edos]]: [[118edo|118]], [[130edo|130]], [[248edo|248]], [[378edo|378]], [[508edo|508]]
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: 0.0547
{{Optimal ET sequence|legend=0| 77, 147, 224, 525, 1274f }}


==11-limit==
Badness (Sintel): 1.02
Commas: 441/440, 3136/3125, 8019/8000


POTE generator: ~3/2 = 701.612
== Octant ==
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.


Mapping generator: ~3
[[Subgroup]]: 2.3.5.7


Map: [&lt;2 0 30 69 102|, &lt;0 1 -8 -20 -30|]
[[Comma list]]: 32805/32768, 2259436291848/2251875390625


EDOs: 12, 118, 130, 248
{{Mapping|legend=1| 8 0 120 -117 | 0 1 -8 11 }}
: mapping generators: ~42875/39366, ~3


Badness: 0.0282
[[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 }}


==13-limit==
{{Optimal ET sequence|legend=1| 24, …, 224, 472, 696, 1168 }}
Commas: 441/440, 729/728, 1001/1000, 3136/3125


POTE generator: ~3/2 = 701.590
[[Badness]] (Sintel): 3.98


Mapping generator: ~3
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;2 0 30 69 102 -75|, &lt;0 1 -8 -20 -30 26|]
Comma list: 9801/9800, 32805/32768, 46656/46585


EDOs: 12, 118, 130, 248, 378
{{Mapping|legend=0| 8 0 120 -117 15 | 0 1 -8 11 1 }}


Badness: 0.0287
Optimal tunings:  
* WE: ~12/11 = 150.0010{{c}}, ~3/2 = 701.7177{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7131{{c}}


=Kleischismic=
{{Optimal ET sequence|legend=0| 24, …, 224, 472, 696, 1168 }}
Commas: 32805/32768, 1500625/1492992


POTE generator: ~36/35 = 50.920
Badness (Sintel): 1.48


Mapping generator: ~35/24
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 1 22 -15|, &lt;0 2 -16 19|]
Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655


Wedgie: &lt;&lt;4 -32 38 -60 49 178||
{{Mapping|legend=0| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }}


EDOs: 24, 94, 118, 212, 330, 542d, 872cd
Optimal tunings:  
* WE: ~12/11 = 149.9957{{c}}, ~3/2 = 701.7046{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7247{{c}}


Badness: 0.1106
{{Optimal ET sequence|legend=0| 24, 224, 472, 696 }}


==11-limit==
Badness (Sintel): 1.26
Commas: 385/384, 9801/9800, 14641/14580


POTE generator: ~36/35 = 50.918
== Nonant ==
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.  


Mapping generator: ~16/11
[[Subgroup]]: 2.3.5.7


Map: [&lt;2 1 22 -15 8|, &lt;0 2 -16 19 -1|]
[[Comma list]]: 32805/32768, 40353607/40310784


EDOs: 24, 94, 118, 212, 330e, 542de
{{Mapping|legend=1| 9 0 135 11 | 0 1 -8 1 }}
: mapping generators: ~2592/2401, ~3


Badness: 0.0367
[[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 }}


==13-limit==
{{Optimal ET sequence|legend=1| 36, 99c, 135, 171, 2772bd, 2943bdd, …, 5166bccddd, 5337bccddd }}
Commas: 352/351, 385/384, 729/728, 1575/1573


POTE generator: ~36/35 = 50.938
[[Badness]] (Sintel): 1.77


Mapping generator: ~16/11
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;2 1 22 -15 8 15|, &lt;0 2 -16 19 -1 -7|]
Comma list: 540/539, 32805/32768, 42875/42592


EDOs: 24, 94, 118, 212f
{{Mapping|legend=0| 9 0 135 11 131 | 0 1 -8 1 -7 }}


Badness: 0.0376
Optimal tunings:  
* WE: ~242/225 = 133.3308{{c}}, ~3/2 = 701.8205{{c}}
* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.8351{{c}}


=Term=
{{Optimal ET sequence|legend=0| 36, 135, 171 }}
Commas: 32805/32768, 250047/250000


7-limit minimax
Badness (Sintel): 4.20


Eigenmonzos: 2, 6/5
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


9-limit minimax
Comma list: 540/539, 729/728, 4096/4095, 16807/16731


Eigenmonzos: 2, 9/7
{{Mapping|legend=0| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }}


[[POTE_tuning|POTE generator]]: ~3/2 = 701.742
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 generator: ~3
{{Optimal ET sequence|legend=0| 36, 99cf, 135, 171 }}


Map: [&lt;3 0 45 94|, &lt;0 1 -8 -18|]
Badness (Sintel): 3.15


Wedgie: &lt;&lt;3 -24 -54 -45 -94 -58||
== 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.


[[EDO|Edos]]: [[171edo|171]], [[1038edo|1038]], [[1209edo|1209]]
[[Subgroup]]: 2.3.5.7


Badness: 0.0200
[[Comma list]]: 32805/32768, 829440/823543


=Sesquiquartififths=
{{Mapping|legend=1| 1 -4 47 6 | 0 7 56 -4 }}
Commas: 2401/2400, 32805/32768
: mapping generators: ~2, ~256/147


7-limit minimax
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8855{{c}}, ~256/147 = 957.2944{{c}}
: [[error map]]: {{val| -0.114 -0.436 -0.182 +1.310 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~256/147 = 957.3867{{c}}
: error map: {{val| 0.000 -0.248 +0.032 +1.627 }}


Eigenmonzos: 2, 7/6
{{Optimal ET sequence|legend=1| 89, 94, 183, 460d, 643d }}


9-limit minimax
[[Badness]] (Sintel): 4.73


Eigenmonzos: 2, 9/7
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[POTE_tuning|POTE generator]]: ~448/405 = 175.434
Comma list: 540/539, 15488/15435, 32805/32768


Mapping generator: ~448/405
{{Mapping|legend=0| 1 -4 47 6 25 | 0 7 56 -4 -27 }}


Map: [&lt;1 1 7 5|, &lt;0 4 -32 -15|]
Optimal tunings:  
* WE: ~2 = 1199.9430{{c}}, ~256/147 = 957.3390{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3849{{c}}


Wedgie: &lt;&lt;4 -32 -15 -60 -35 55||
{{Optimal ET sequence|legend=0| 89, 94, 183, 460d }}


[[EDO|Edos]]: [[171edo|171]], [[643edo|643]], [[2100edo|2100edo]]
Badness (Sintel): 1.72


Badness: 0.0112
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==Sesquart==
Comma list: 540/539, 729/728, 1573/1568, 4096/4095
Commas: 243/242, 441/440, 16384/16335


POTE generator: ~256/231 = 175.406
{{Mapping|legend=0| 1 -4 47 6 25 -33 | 0 7 56 -4 -27 46 }}


Mapping generator: ~256/231
Optimal tunings:  
* WE: ~2 = 1200.0058{{c}}, ~256/147 = 957.3946{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3900{{c}}


Map: [&lt;1 1 7 5 2|, &lt;0 4 -32 -15 10|]
{{Optimal ET sequence|legend=0| 89, 94, 183, 277, 460d }}


EDOs: 41, 89, 130
Badness (Sintel): 1.46


Badness: 0.0293
== Subgroup extensions ==
=== Tridecaschismic (2.3.5.13) ===
Proposed by [[Eufalesio]] in 2026, tridecaschismic adds the [[325/324|marveltwin comma]] to the comma list, or equivalently, the [[tridecapyth comma]]. It benefits from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is one of the lowest badness schismic extensions. It is also equivalent to the [[2.3.5.13 subgroup|2.3.5.13-subgroup]] [[restriction]] of 13-limit [[cassandra]].


===13-limit===
Subgroup: 2.3.5.13
Commas: 243/242, 364/363, 441/440, 105644/105625


POTE generator: ~256/231 = 175.409
Comma list: 325/324, 32805/32768


Map: [&lt;1 1 7 5 2 -2|, &lt;0 4 -32 -15 10 39|]
{{Mapping|legend=2| 1 0 15 -28 | 0 1 -8 20 }}


EDOs: 41, 89, 130, 301e, 431e
Optimal tunings:
* WE: ~2 = 1200.3326{{c}} ~3/2 = 702.1092{{c}}
* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9189{{c}}


Badness: 0.0224
{{Optimal ET sequence|legend=0| 12, …, 41, 53, 412cf, 465cf, …, 783ccff, 836ccfff }}


==Bisesqui==
Badness (Sintel): 0.582
Commas: 2401/2400, 9801/9800, 32805/32768


[[POTE_tuning|POTE generator]]: ~448/405 = 175.435
==== 2.3.5.13.19 subgroup ====
Subgroup: 2.3.5.13.19


Map: [&lt;2 2 14 10 23|, &lt;0 4 -32 -15 -55|]
Comma list: 325/324, 361/360, 513/512


EDOs: 130, 212, 342, 1156, 1498
{{Mapping|legend=2| 1 0 15 -28 9 | 0 1 -8 20 -3 }}


Badness: 0.0170
Optimal tunings:
* WE: ~2 = 1200.4236{{c}}, ~3/2 = 702.1510{{c}}
* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9064{{c}}


=Sextilififths=
{{Optimal ET sequence|legend=0| 12, …, 41, 53 }}
Commas: 32768/32805, 235298/234375


POTE generator: ~21/20 = 83.053
Badness (Sintel): 0.354


Mapping generator: ~21/20
=== Photia (2.3.5.17) ===
{{See also| No-elevens subgroup temperaments #Garibaldia }}


Map: [&lt;1 2 -1 -1|, &lt;0 -6 48 55|]
[[Subgroup]]: 2.3.5.17


EDOs: 29, 101, 130, 289, 419
[[Comma list]]: 256/255, 1458/1445


Badness: 0.1088
{{Mapping|legend=3| 1 0 15 -7 | 0 1 -8 7 }}


==11-limit==
{{Mapping|legend=5| 1 0 15 0 0 0 -7 | 0 1 -8 0 0 0 7 }}
Commas: 441/440, 4000/3993, 235298/234375
: mapping generators: ~2, ~3


POTE generator: ~21/20 = 83.049
[[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 }}


Mapping generator: ~21/20
{{Optimal ET sequence|legend=1| 12, 41, 53, 65, 207g, 272gg }}


Map: [&lt;1 2 -1 -1 0|, &lt;0 -6 48 55 50|]
[[Badness]] (Sintel): 0.479


EDOs: 29, 130, 289
==== 2.3.5.17.19 subgroup ====
Subgroup: 2.3.5.17.19


Badness: 0.0455
Comma list: 171/170, 256/255, 324/323


==13-limit==
{{Mapping|legend=2| 1 0 15 -7 9 | 0 1 -8 7 -3 }}
Commas: 363/363, 441/440, 676/675, 10985/10976


POTE generator: ~21/20 = 83.049
{{Mapping|legend=4| 1 0 15 0 0 0 -7 9 | 0 1 -8 0 0 0 7 -3 }}


Mapping generator: ~21/20
Optimal tunings:  
* WE: ~2 = 1199.7225{{c}}, ~3/2 = 701.3077{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.4754{{c}}


Map: [&lt;1 2 -1 -1 0 1|, &lt;0 -6 48 55 50 39|]
{{Optimal ET sequence|legend=0| 12, 41, 53, 65, 142g }}


EDOs: 29, 130, 289
Badness (Sintel): 0.332


Badness: 0.0253
=== Nestoria (2.3.5.19) ===
{{See also| No-elevens subgroup temperaments #Garibaldia | No-elevens subgroup temperaments #Pontia }}


=Tsaharuk=
Nestoria is notable for having one of the lowest-badness subgroup extensions of schismic. Note that despite prime [[19/1|19]] being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is generally not flatter than the fifth in optimal schismic due to its optimization considering intervals like [[19/10]] and [[19/15]]. However, the dyadic tuning sensitivity of [[19/16]] suggests using tunings like [[65edo]] and [[77edo]] to optimize in favour of prime 19 (especially the minor triad ~16:19:24 which is equated with the Pythagorean minor triad), as [[171edo]] is already arguably undertempered for it despite being the optimal patent val.
Commas: 32805/32768, 420175/419904


POTE generator: ~243/224 = 140.350
[[Subgroup]]: 2.3.5.19


Mapping generator: ~243/224
[[Comma list]]: 361/360, 513/512


Map: [&lt;1 1 7 0|, &lt;0 5 -40 24|]
{{Mapping|legend=3| 1 0 15 9 | 0 1 -8 -3 }}


Wedgie: &lt;&lt;5 -40 24 -75 24 168||
{{Mapping|legend=5| 1 0 15 0 0 0 0 9 | 0 1 -8 0 0 0 0 -3 }}
: mapping generators: ~2, ~3


EDOs: 17, 60c, 77, 94, 171
[[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 }}


Badness: 0.0307
{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171, 460hh, 631hh }}


=Quanharuk=
[[Badness]] (Sintel): 0.126
Commas: 16875/16807, 32805/32768


POTE generator: ~56/45 = 380.355
=== Taylor (2.3.5.13) ===
This is a 2.3.5.13 subgroup restriction of 13-limit hemischis.


Mapping generator: ~56/45
[[Subgroup]]: 2.3.5.13


Map: [&lt;1 0 15 12|, &lt;0 5 -40 -29|]
[[Comma list]]: 676/675, 32805/32768


Wedgie: &lt;&lt;5 -40 -29 -75 -60 45||
{{Mapping|legend=3| 1 0 15 14 | 0 2 -16 -13 }}


EDOs: 41, 142, 183, 224, 1303d, 1527cd, 1751cd, 1975cd
{{Mapping|legend=5| 1 0 15 0 0 14 | 0 2 -16 0 0 -13 }}
: mapping generators: ~2, ~26/15


Badness: 0.0720
[[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 }}


==11-limit==
{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236, 525f, 761ff }}
Commas: 540/539, 1375/1372, 32805/32768


POTE generator: ~56/45 = 380.352
[[Badness]] (Sintel): 0.334


Mapping generator: ~56/45
==== Dakota (2.3.5.13.19) ====
Subgroup: 2.3.5.13.19


Map: [&lt;1 0 15 12 -7|, &lt;0 5 -40 -29 33|]
Comma list: 361/360, 513/512, 676/675


EDOs: 41, 142, 183, 224, 631d, 855d, 1079d
{{Mapping|legend=2| 1 0 15 14 9 | 0 2 -16 -13 -6 }}


Badness: 0.0315
Optimal tunings:  
* WE: ~2 = 1200.2611{{c}}, ~26/15 = 951.0703{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8532{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 24, 29, 53, 130, 183, 236h, 289h }}
Commas: 540/539, 1375/1372, 4096/4095, 6656/6655


POTE generator: ~56/45 = 380.351
Badness (Sintel): 0.262


Map: [&lt;1 0 15 12 -7 -15|, &lt;0 5 -40 -29 33 59|]
===== 2.3.5.13.19.37 subgroup =====
Subgroup: 2.3.5.13.19.37


EDOs: 41, 142, 183, 224, 631d, 855d
Comma list: 361/360, 481/480, 513/512, 676/675


Badness: 0.0214
{{Mapping|legend=2| 1 0 15 14 9 6 | 0 2 -16 -13 -6 -1 }}


=Octant=
Optimal tunings:
Commas: 32805/32768, 2259436291848/2251875390625
* WE: ~2 = 1200.2987{{c}}, ~26/15 = 951.1060{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8595{{c}}


POTE generator: ~3/2 = 701.713
{{Optimal ET sequence|legend=0| 24, 29, 53, 183, 236h, 289hl, 631fhhll }}


Mapping generator: ~3
Badness (Sintel): 0.223


Map: [&lt;8 0 120 -117|, &lt;0 1 -8 11|]
=== Quintilischis (2.3.5.17) ===
: ''For full 17- and 19-limit extensions, see [[#Quintilipyth]] or [[#Quintaschis]].''


Wedgie: &lt;&lt;8 -64 88 -120 117 384||
[[Subgroup]]: 2.3.5.17


EDOs: 24, 224, 472, 696, 1168
[[Comma list]]: 32805/32768, 1419857/1417176


Badness: 0.1572
{{Mapping|legend=3| 1 2 -1 5 | 0 -5 40 -11 }}


==11-limit==
{{Mapping|legend=5| 1 2 -1 0 0 0 5 | 0 -5 40 0 0 0 -11 }}
Commas: 9801/9800, 32805/32768, 46656/46585
: mapping generators: ~2, ~18/17


POTE generator: ~3/2 = 701.713
[[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 }}


Mapping generator: ~3
{{Optimal ET sequence|legend=1| 12, …, 253, 265, 277, 289, 566g, 855g }}


Map: [&lt;8 0 120 -117 15|, &lt;0 1 -8 11 1|]
[[Badness]] (Sintel): 1.34


EDOs: 24, 224, 472, 696, 1168
==== 2.3.5.17.19 subgroup ====
Subgroup: 2.3.5.17.19


Badness: 0.0448
Comma list: 4624/4617, 6144/6137, 6885/6859


==13-limit==
{{Mapping|legend=2| 1 2 -1 5 4 | 0 -5 40 -11 3 }}
Commas: 729/728, 1575/1573, 2200/2197, 6656/6655


POTE generator: ~3/2 = 701.725
{{Mapping|legend=4| 1 2 -1 0 0 0 5 4 | 0 -5 40 0 0 0 -11 3 }}


Mapping generator: ~3
Optimal tunings:  
* WE: ~2 = 1200.0350{{c}}, ~18/17 = 99.6550{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6520{{c}}


Map: [&lt;8 0 120 -117 15 93|, &lt;0 1 -8 11 1 -5|]
{{Optimal ET sequence|legend=0| 12, …, 253, 265, 277, 289 }}


EDOs: 24, 224, 472, 696
Badness (Sintel): 1.17


Badness: 0.0304
== References ==
<references/>


[[Category:definition]]
[[Category:Schismatic family| ]] <!-- main article -->
[[Category:family]]
[[Category:Temperament families]]
[[Category:index]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:list]]
[[Category:overview]]
[[Category:schismatic]]
[[Category:theory]]
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