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Schismatic family: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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| de = Schismatische Temperaturen
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<h4>Original Wikitext content:</h4>
{{Technical data page}}
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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 just 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.  
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]].  


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 schisma 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.
== Schismic, schismatic, a.k.a. helmholtz ==
{{Main| Schismic }}


==Seven limit children==
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.  
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.


===Garibaldi===
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.
Commas: {225/224, 3125/3087}


7-limit minimax tuning:
[[Subgroup]]: 2.3.5
7-limit: [|1 0 0 0&gt;, |5/3 1/15 0 -1/15&gt;,
|5/3 -8/15 0 8/15&gt;, |5/3 -14/15 0 14/15&gt;]
Eigenmonzos: 2, 7/6


9-limit: [|1 0 0 0&gt;, |25/16 1/8 0 -1/16&gt;,
[[Comma list]]: 32805/32768
|5/2 -1 0 1/2&gt;, |25/8 -7/4 0 7/8&gt;]
Eigenmonzos: 2, 9/7


11-limit
{{Mapping|legend=1| 1 0 15 | 0 1 -8 }}
Commas: {225/224, 385/384, 2200/2187}
: mapping generators: ~2, ~3
 
[[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 }}
 
[[Tuning ranges]]:
* [[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)
 
{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bbc, 6452bbcc }}
 
[[Badness]] (Sintel): 0.0999
 
=== Overview to extensions ===
The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at. [[#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.
 
[[#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.
 
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.
 
Temperaments discussed elsewhere include:
* ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
* ''[[Pogo]]'' (+118098/117649) → [[Stearnsmic clan #Pogo|Stearnsmic clan]]
 
Considered below are garibaldi, pontiac, grackle, schism, bischismic, kleischismic, salsa, hemischis, term, altinex, squirrel, tertiaschis, countertertiaschis, quadrant, sesquiquartififths, tsaharuk, quanharuk, quintilipyth, quintaschis, sextilifourths, septant, octant, nonant, septiquarschis, and tridecafifths.
 
The schismatic family boasts a variety of remarkable extensions to subgroups in high prime limits. These are listed at the bottom of this page, in [[#Subgroup extensions]].
 
== Garibaldi ==
{{Main| Garibaldi }}
 
Garibaldi tempers out the [[garischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 3125/3087
 
{{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.
 
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 2200/2187
 
Mapping: {{mapping| 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:
[|1 0 0 0 0&gt;, |25/16 1/8 0 -1/16 0&gt;, |5/2 -1 0 1/2 0&gt;,
* 11-odd-limit: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }}
|25/8 -7/4 0 7/8 0&gt;, |47/16 23/8 0 -23/16 0&gt;]
: unchanged-interval (eigenmonzo) basis: 2.9/7
Eigenmonzos: 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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 ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 225/224, 245/242
 
Mapping: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 1 0 15 25 9 | 0 2 -16 -28 -7 }}
: mapping generators: ~2, ~110/63
 
Optimal tunings:
* WE: ~2 = 1200.7303{{c}}, ~110/63 = 951.6605{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~110/63 = 951.0604{{c}}
 
{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135ee }}
 
Badness (Sintel): 1.68
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 169/168, 225/224, 275/273
 
Mapping: {{mapping| 1 0 15 25 9 14 | 0 2 -16 -28 -7 -13 }}
 
Optimal tunings:
* WE: ~2 = 1200.8146{{c}}, ~26/15 = 951.7273{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.0574{{c}}
 
{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135eef }}
 
Badness (Sintel): 1.13
 
=== Sanjaab ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 1331/1323, 3125/3087
 
Mapping: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 1 0 15 109 -117 | 0 1 -8 -67 76 }}
 
Optimal tunings:
* WE: ~2 = 1200.3283{{c}}, ~3/2 = 702.1636{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9713{{c}}
 
{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}
 
Badness (Sintel): 3.83
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 385/384, 2200/2197, 19712/19683
 
Mapping: {{mapping| 1 0 15 109 -117 -28 | 0 1 -8 -67 76 20 }}
 
Optimal tunings:
* WE: ~2 = 1200.3229{{c}}, ~3/2 = 702.1603{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9714{{c}}
 
{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}
 
Badness (Sintel): 2.13
 
== Schism ==
See [[Archytas clan #Schism]].
 
Schism is a relatively low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (C–B♭). 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.
 
== Bischismic ==
Bischismic 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 2 1 22 -15 8 | 0 2 -16 19 -1 }}
 
Optimal tunings:
* WE: ~99/70 = 600.1645{{c}}, ~35/24 = 651.0963{{c}} (~36/35 = 50.9319{{c}})
* 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 1 0 15 -17 51 14 -49 9 -24 | 0 2 -16 25 -60 -13 67 -6 36 }}
 
Optimal tunings:
* WE: ~2 = 1200.0215{{c}}, ~26/15 = 950.8239{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8069{{c}}
 
{{Optimal 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 3 0 45 94 -137 | 0 1 -8 -18 31 }}
 
Optimal tunings:
* WE: ~63/50 = 399.9677{{c}}, ~3/2 = 701.6278{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6846{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 537, 891de }}
 
Badness (Sintel): 2.21
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 4096/4095, 31250/31213
 
Mapping: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 3 0 45 94 8 | 0 2 -16 -36 1 }}
: mapping generators: ~63/50, ~693/400
 
Optimal tunings:
* WE: ~63/50 = 400.0309{{c}}, ~693/400 = 950.9458{{c}} (~12/11 = 150.8841{{c}})
* 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 1 2 -1 1 0 3 | 0 -3 24 13 25 5 }}
 
Optimal tunings:
* WE: ~2 = 1201.1361{{c}}, ~11/10 = 166.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0833{{c}}
 
{{Optimal ET sequence|legend=0| 29, 65f, 94df }}
 
Badness (Sintel): 1.81
 
== Tertiaschis ==
Named by [[Xenllium]] in 2021, tertiaschis may be described as {{nowrap| 94 & 159 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 1071875/1062882 for prime 7.
 
[[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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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~Pythagorean comma with three [[marvel comma]]s, quadrant equates the syntonic~Pythagorean comma with four. A [[septimal comma]] is then found as a stack of five marvel commas.
 
[[Subgroup]]: 2.3.5.7
 
[[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: {{mapping| 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: {{mapping| 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 fifth into four; its [[ploidacot]] is thus tetracot.
 
[[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: {{mapping| 1 1 7 5 2 | 0 4 -32 -15 10 }}
 
Optimal tunings:
* WE: ~2 = 1199.8171{{c}}, ~256/231 = 175.3793{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~256/231 = 175.4081{{c}}
 
{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}
 
Badness (Sintel): 0.969
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 364/363, 441/440, 3584/3575
 
Mapping: {{mapping| 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: {{mapping| 1 1 7 5 2 -2 0 | 0 4 -32 -15 10 39 28 }}
 
Optimal tunings:
* WE: ~2 = 1199.6422{{c}}, ~72/65 = 175.3338{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3857{{c}}
 
{{Optimal ET sequence|legend=0| 41, 89, 130g }}
 
Badness (Sintel): 1.45
 
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 171/170, 243/242, 256/255, 273/272, 324/323, 441/440
 
Mapping: {{mapping| 1 1 7 5 2 -2 0 6 | 0 4 -32 -15 10 39 28 -12 }}
 
Optimal tunings:
* WE: ~2 = 1199.7499{{c}}, ~21/19 = 175.3432{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.3797{{c}}
 
{{Optimal ET sequence|legend=0| 41, 89, 130g }}
 
Badness (Sintel): 1.40
 
===== Sesquartia =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575
 
Mapping: {{mapping| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }}
 
Optimal tunings:
* WE: ~2 = 1199.8902{{c}}, ~72/65 = 175.4077{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4234{{c}}
 
{{Optimal ET sequence|legend=0| 41, 130, 171 }}
 
Badness (Sintel): 1.18
 
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594
 
Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }}
 
Optimal tunings:
* WE: ~2 = 1199.9864{{c}}, ~21/19 = 175.4169{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4189{{c}}
 
{{Optimal ET sequence|legend=0| 41, 130, 171 }}
 
Badness (Sintel): 1.24
 
====== 23-limit ======
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594
 
Mapping: {{mapping| 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: {{mapping| 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: {{mapping| 1 1 7 5 2 -2 13 6 | 0 4 -32 -15 10 39 -61 -12 }}
 
Optimal tunings:
* WE: ~2 = 1200.0114{{c}}, ~72/65 = 175.3783{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3765{{c}}
 
{{Optimal ET sequence|legend=0| 41g, 89, 130 }}
 
Badness (Sintel): 1.39
 
====== 23-limit ======
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 221/220, 243/242, 276/275, 323/322, 361/360, 364/363, 441/440
 
Mapping: {{mapping| 1 1 7 5 2 -2 13 6 13 | 0 4 -32 -15 10 39 -61 -12 -58 }}
 
Optimal tunings:
* WE: ~2 = 1200.0122{{c}}, ~72/65 = 175.3782{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3763{{c}}
 
{{Optimal ET sequence|legend=0| 41g, 89, 130 }}
 
Badness (Sintel): 1.37
 
=== Bisesqui ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 9801/9800, 32805/32768
 
Mapping: {{mapping| 2 2 14 10 23 | 0 4 -32 -15 -55 }}
: mapping generators: ~99/70, ~448/405
 
Optimal tunings:
* WE: ~99/70 = 600.0429{{c}}, ~448/405 = 175.4474{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~448/405 = 175.4334{{c}}
 
{{Optimal ET sequence|legend=1| 82e, 130, 212, 342, 1156, 1498, 1840d, 5862bbccdddee }}
 
Badness (Sintel): 0.561
 
== Tsaharuk ==
{{Main| Tsaharuk }}
 
Tsaharuk tempers out 420175/419904, the [[wizma]], and may be described as the {{nowrap| 77 & 94 }} temperament. It is generated by a slightly flat neutral second of [[~]][[13/12]], five of which make the [[3/2|perfect fifth]], so its [[ploidacot]] is pentacot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 32805/32768, 420175/419904
 
{{Mapping|legend=1| 1 1 7 0 | 0 5 -40 24 }}
: mapping generators: ~2, ~243/224
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1039{{c}}, ~243/224 = 140.3620{{c}}
: [[error map]]: {{val| +0.104 -0.041 -0.067 -0.137 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/224 = 140.3496{{c}}
: error map: {{val| 0.000 -0.207 -0.296 -0.436 }}
 
{{Optimal ET sequence|legend=1| 17, 77, 94, 171 }}
 
[[Badness]] (Sintel): 0.777
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 1331/1323, 19712/19683
 
Mapping: {{mapping| 1 1 7 0 1 | 0 5 -40 24 21 }}
 
Optimal tunings:
* WE: ~2 = 1200.3103{{c}}, ~88/81 = 140.4011{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.3649{{c}}
 
{{Optimal ET sequence|legend=0| 17, 77, 94, 171e, 265e }}
 
Badness (Sintel): 2.10
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 385/384, 729/728, 1331/1323
 
Mapping: {{mapping| 1 1 7 0 1 3 | 0 5 -40 24 21 6 }}
 
Optimal tunings:
* WE: ~2 = 1200.1840{{c}}, ~13/12 = 140.3840{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.3627{{c}}
 
{{Optimal ET sequence|legend=0| 17, 77, 94, 171e }}
 
Badness (Sintel): 1.57
 
== Quanharuk ==
Quanharuk tempers out 16875/16807, the [[mirkwai]] comma, and may be described as the {{nowrap| 41 & 183 }} temperament. The generator is a slightly flat major third of [[~]][[56/45]], five of which make the [[3/1|3rd]] [[harmonic]], so the [[ploidacot]] of this temperament is alpha-pentacot. [[224edo]] makes for a recommendable tuning.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 16875/16807, 32805/32768
 
{{Mapping|legend=1| 1 0 15 12 | 0 5 -40 -29 }}
: mapping generators: ~2, ~56/45
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0032{{c}}, ~56/45 = 380.3557{{c}}
: [[error map]]: {{val| +0.003 -0.177 -0.493 +0.898 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~56/45 = 380.3546{{c}}
: error map: {{val| 0.000 -0.182 -0.498 +0.890 }}
 
{{Optimal ET sequence|legend=1| 41, 142, 183, 224 }}
 
[[Badness]] (Sintel): 1.82
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 32805/32768
 
Mapping: {{mapping| 1 0 15 12 -7 | 0 5 -40 -29 33 }}
 
Optimal tunings:
* WE: ~2 = 1199.9709{{c}}, ~56/45 = 380.3423{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3517{{c}}
 
{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}
 
Badness (Sintel): 1.04
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 1375/1372, 4096/4095
 
Mapping: {{mapping| 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: {{mapping| 1 2 -1 -4 -7 | 0 -5 40 82 126 }}
 
Optimal tunings:
* WE: ~2 = 1200.1503{{c}}, ~35/33 = 99.6287{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6176{{c}}
 
{{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: {{mapping| 1 2 -1 -4 -7 -9 | 0 -5 40 82 126 153 }}
 
Optimal tunings:
* WE: ~2 = 1200.1774{{c}}, ~35/33 = 99.6267{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6134{{c}}
 
{{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: {{mapping| 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: {{mapping| 1 2 -1 -4 -7 -9 5 4 | 0 -5 40 82 126 153 -11 3 }}
 
Optimal tunings:
* WE: ~2 = 1200.0713{{c}}, ~18/17 = 99.6208{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6152{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 253, 265 }}
 
Badness (Sintel): 2.32
 
== 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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 32805/32768, 49009212/48828125
 
{{Mapping|legend=1| 1 2 -1 -5 | 0 -5 40 94 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0536{{c}}, ~200/189 = 99.6684{{c}}
: [[error map]]: {{val| +0.054 -0.190 +0.370 -0.262 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~200/189 = 99.6645{{c}}
: error map: {{val| 0.000 -0.277 +0.266 -0.363 }}
 
{{Optimal ET sequence|legend=1| 12, …, 289, 301, 590, 891, 1192 }}
 
[[Badness]] (Sintel): 3.36
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 32805/32768, 1953125/1951488
 
Mapping: {{mapping| 1 2 -1 -5 -8 | 0 -5 40 94 138 }}
 
Optimal tunings:
* WE: ~2 = 1200.0988{{c}}, ~35/33 = 99.6613{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6540{{c}}
 
{{Optimal ET sequence|legend=0| 12, …, 277d, 289 }}
 
Badness (Sintel): 3.69
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 441/440, 32805/32768, 109512/109375
 
Mapping: {{mapping| 1 2 -1 -5 -8 -11 | 0 -5 40 94 138 177 }}
 
Optimal tunings:
* WE: ~2 = 1200.0625{{c}}, ~35/33 = 99.6630{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6583{{c}}
 
{{Optimal ET sequence|legend=0| 12f, …, 277dff, 289 }}
 
Badness (Sintel): 3.07
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768
 
Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 | 0 -5 40 94 138 177 -11 }}
 
Optimal tunings:
* WE: ~2 = 1200.1286{{c}}, ~18/17 = 99.6668{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6568{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 277dff, 289 }}
 
Badness (Sintel): 2.58
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859
 
Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 4 | 0 -5 40 94 138 177 -11 3 }}
 
Optimal tunings:
* WE: ~2 = 1200.0289{{c}}, ~18/17 = 99.6609{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6586{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 289 }}
 
Badness (Sintel): 2.56
 
=== Quintahelenic ===
Subgroup: 2.3.5.7.11
 
Comma list: 5632/5625, 8019/8000, 151263/151250
 
Mapping: {{mapping| 1 2 -1 -5 -9 | 0 -5 40 94 150 }}
 
Optimal tunings:
* WE: ~2 = 1200.0195{{c}}, ~200/189 = 99.6723{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6709{{c}}
 
{{Optimal ET sequence|legend=0| 12, …, 289e, 301, 915 }}
 
Badness (Sintel): 2.72
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000
 
Mapping: {{mapping| 1 2 -1 -5 -9 -11 | 0 -5 40 94 150 177 }}
 
Optimal tunings:
* WE: ~2 = 1200.0442{{c}}, ~200/189 = 99.6709{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6675{{c}}
 
{{Optimal ET sequence|legend=0| 12f, …, 289e, 301 }}
 
Badness (Sintel): 2.30
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750
 
Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 | 0 -5 40 94 150 177 -11 }}
 
Optimal tunings:
* WE: ~2 = 1200.1227{{c}}, ~200/189 = 99.6753{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6658{{c}}
 
{{Optimal ET sequence|legend=1| 12f, 289e, 301 }}
 
Badness (Sintel): 2.06
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700
 
Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 4 | 0 -5 40 94 150 177 -11 3 }}
 
Optimal tunings:
* WE: ~2 = 1200.0230{{c}}, ~200/189 = 99.6694{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6676{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 301 }}
 
Badness (Sintel): 2.24
 
==== Quintahelenoid ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436
 
Mapping: {{mapping| 1 2 -1 -5 -9 14 | 0 -5 40 94 150 -124 }}
 
Optimal tunings:
* WE: ~2 = 1199.9919{{c}}, ~200/189 = 99.6712{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6718{{c}}
 
{{Optimal ET sequence|legend=0| 12, 301, 614, 915 }}
 
Badness (Sintel): 2.73
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157
 
Mapping: {{mapping| 1 2 -1 -5 -9 14 5 | 0 -5 40 94 150 -124 -11 }}
 
Optimal tunings:
* WE: ~2 = 1200.0469{{c}}, ~18/17 = 99.6749{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6710{{c}}
 
{{Optimal ET sequence|legend=0| 12, 301 }}
 
Badness (Sintel): 2.44
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137
 
Mapping: {{mapping| 1 2 -1 -5 -9 14 5 4 | 0 -5 40 94 150 -124 -11 3 }}
 
Optimal tunings:
* WE: ~2 = 1199.9925{{c}}, ~18/17 = 99.6710{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6716{{c}}
 
{{Optimal ET sequence|legend=0| 12, 301 }}
 
Badness (Sintel): 2.41
 
== 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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 32805/32768, 235298/234375
 
{{Mapping|legend=1| 1 2 -1 -1 | 0 -6 48 55 }}
: mapping generators: ~2, ~21/20
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0987{{c}}, ~21/20 = 83.0599{{c}}
: [[error map]]: {{val| +0.099 -0.117 +0.462 -0.630 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0543{{c}}
: error map: {{val| 0.000 -0.281 +0.295 -0.837 }}
 
{{Optimal ET sequence|legend=1| 29, 72cd, 101, 130, 289, 419 }}
 
[[Badness]] (Sintel): 2.75
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 4000/3993, 235298/234375
 
Mapping: {{mapping| 1 2 -1 -1 0 | 0 -6 48 55 50 }}
 
Optimal tunings:
* WE: ~2 = 1200.0424{{c}}, ~21/20 = 83.0520{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0497{{c}}
 
{{Optimal ET sequence|legend=0| 29, 72cde, 101e, 130, 289 }}
 
Badness (Sintel): 1.50
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 441/440, 676/675, 10985/10976
 
Mapping: {{mapping| 1 2 -1 -1 0 1 | 0 -6 48 55 50 39 }}
 
Optimal tunings:
* WE: ~2 = 1200.1056{{c}}, ~21/20 = 83.0566{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0508{{c}}
 
{{Optimal ET sequence|legend=0| 29, 72cdef, 101e, 130, 289 }}
 
Badness (Sintel): 1.04
 
== Septant ==
Named by [[Xenllium]] in 2021, septant notably tempers out the [[akjaysma]] ({{monzo| 47 -7 -7 -7 }}) and may be described as the {{nowrap| 224 & 301 }} temperament. It has a period of 1/7 octave, and its [[ploidacot]] is heptaploid monocot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 32805/32768, 516560652/514714375
 
{{Mapping|legend=1| 7 0 105 -56 | 0 1 -8 7 }}
: mapping generators: ~8575/7776, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~8575/7776 = 171.4303{{c}}, ~3/2 = 701.7091{{c}}
: [[error map]]: {{val| +0.012 -0.234 +0.096 +0.265 }}
* [[CWE]]: ~8575/7776 = 171.4286{{c}}, ~3/2 = 701.7022{{c}}
: error map: {{val| 0.000 -0.253 +0.069 +0.232 }}
 
{{Optimal ET sequence|legend=1| 77, 147, 224, 301, 525, 826, 1351 }}
 
[[Badness]] (Sintel): 2.81
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 24057/24010, 32805/32768
 
Mapping: {{mapping| 7 0 105 -56 -120 | 0 1 -8 7 13 }}
 
Optimal tunings:
* WE: ~495/448 = 171.4334{{c}}, ~3/2 = 701.7387{{c}}
* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7198{{c}}
 
{{Optimal ET sequence|legend=0| 77, 147, 224, 301, 525 }}
 
Badness (Sintel): 1.46
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024
 
Mapping: {{mapping| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }}
 
Optimal tunings:
* WE: ~495/448 = 171.4282{{c}}, ~3/2 = 701.7229{{c}}
* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7242{{c}}
 
{{Optimal ET sequence|legend=0| 77, 147, 224, 525, 1274f }}
 
Badness (Sintel): 1.02
 
== 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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 32805/32768, 2259436291848/2251875390625
 
{{Mapping|legend=1| 8 0 120 -117 | 0 1 -8 11 }}
: mapping generators: ~42875/39366, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~42875/39366 = 150.0048{{c}}, ~3/2 = 701.7356{{c}}
: [[error map]]: {{val| +0.039 -0.181 +0.071 +0.127 }}
* [[CWE]]: ~42875/39366 = 150.0000{{c}}, ~3/2 = 701.7134{{c}}
: error map: {{val| 0.000 -0.242 -0.021 +0.022 }}
 
{{Optimal ET sequence|legend=1| 24, …, 224, 472, 696, 1168 }}
 
[[Badness]] (Sintel): 3.98
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 32805/32768, 46656/46585
 
Mapping: {{mapping| 8 0 120 -117 15 | 0 1 -8 11 1 }}
 
Optimal tunings:
* WE: ~12/11 = 150.0010{{c}}, ~3/2 = 701.7177{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7131{{c}}
 
{{Optimal ET sequence|legend=0| 24, …, 224, 472, 696, 1168 }}
 
Badness (Sintel): 1.48
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655
 
Mapping: {{mapping| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }}
 
Optimal tunings:
* WE: ~12/11 = 149.9957{{c}}, ~3/2 = 701.7046{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7247{{c}}
 
{{Optimal ET sequence|legend=0| 24, 224, 472, 696 }}
 
Badness (Sintel): 1.26
 
== 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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 32805/32768, 40353607/40310784
 
{{Mapping|legend=1| 9 0 135 11 | 0 1 -8 1 }}
: mapping generators: ~2592/2401, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~2592/2401 = 133.3442{{c}}, ~3/2 = 701.8000{{c}}
: [[error map]]: {{val| +0.098 -0.057 -0.027 -0.141 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.7384{{c}}
: error map: {{val| 0.000 -0.217 -0.221 -0.421 }}
 
{{Optimal ET sequence|legend=1| 36, 99c, 135, 171, 2772bd, 2943bdd, …, 5166bccddd, 5337bccddd }}
 
[[Badness]] (Sintel): 1.77
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 32805/32768, 42875/42592
 
Mapping: {{mapping| 9 0 135 11 131 | 0 1 -8 1 -7 }}
 
Optimal tunings:
* WE: ~242/225 = 133.3308{{c}}, ~3/2 = 701.8205{{c}}
* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.8351{{c}}
 
{{Optimal ET sequence|legend=0| 36, 135, 171 }}
 
Badness (Sintel): 4.20
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 4096/4095, 16807/16731
 
Mapping: {{mapping| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }}
 
Optimal tunings:
* WE: ~242/225 = 133.3180{{c}}, ~3/2 = 701.6956{{c}}
* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.7800{{c}}
 
{{Optimal ET sequence|legend=0| 36, 99cf, 135, 171 }}
 
Badness (Sintel): 3.15
 
== Septiquarschis ==
Named by [[Xenllium]] in 2021, septiquarschis tempers out [[829440/823543]] (mynaslender comma) and [[67108864/66706983]] (septiness comma), and may be described as the {{nowrap| 89 & 94 }} temperament. It splits septimal minor seventh ([[7/4]]) into four generators. Note that in the data below, the generator is the [[octave complement]] so that seven of them minus five octaves make a [[3/2|perfect fifth]]; its [[ploidacot]] is thus epsilon-heptacot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 32805/32768, 829440/823543
 
{{Mapping|legend=1| 1 -4 47 6 | 0 7 56 -4 }}
: mapping generators: ~2, ~256/147
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8855{{c}}, ~256/147 = 957.2944{{c}}
: [[error map]]: {{val| -0.114 -0.436 -0.182 +1.310 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~256/147 = 957.3867{{c}}
: error map: {{val| 0.000 -0.248 +0.032 +1.627 }}
 
{{Optimal ET sequence|legend=1| 89, 94, 183, 460d, 643d }}
 
[[Badness]] (Sintel): 4.73
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 15488/15435, 32805/32768
 
Mapping: {{mapping| 1 -4 47 6 25 | 0 7 56 -4 -27 }}
 
Optimal tunings:
* WE: ~2 = 1199.9430{{c}}, ~256/147 = 957.3390{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3849{{c}}
 
{{Optimal ET sequence|legend=0| 89, 94, 183, 460d }}
 
Badness (Sintel): 1.72
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 1573/1568, 4096/4095
 
Mapping: {{mapping| 1 -4 47 6 25 -33 | 0 7 56 -4 -27 46 }}
 
Optimal tunings:
* WE: ~2 = 1200.0058{{c}}, ~256/147 = 957.3946{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3900{{c}}
 
{{Optimal ET sequence|legend=0| 89, 94, 183, 277, 460d }}
 
Badness (Sintel): 1.46
 
== Subgroup extensions ==
 
=== Tridecaschismic (2.3.5.13) ===
Proposed by [[Eufalesio]] in 2026, tridecaschismic adds the [[325/324|marveltwin comma]] to the comma list, or equivalently, the [[tridecapyth comma]]. It benefits from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is one of the lowest badness schismic extensions. It is also equivalent to the 2.3.5.13 [[restriction]] of 13-limit [[cassandra]].
 
Subgroup: 2.3.5.13
 
Comma list: 325/324, 32805/32768
 
Subgroup-val mapping: {{mapping| 1 0 15 -28 | 0 1 -8 20 }}
 
Optimal tunings:
* WE: ~2 = 1200.3326{{c}} ~3/2 = 702.1092{{c}}
* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9189{{c}}
 
{{Optimal ET sequence|legend=0| 12, …, 41, 53, 412cf, 465cf, …, 783ccff, 836ccfff }}
 
Badness (Sintel): 0.582
 
==== 2.3.5.13.19 subgroup ====
Subgroup: 2.3.5.13.19
 
Comma list: 325/324, 361/360, 513/512
 
Subgroup-val mapping: {{mapping| 1 0 15 -28 9 | 0 1 -8 20 -3 }}
 
Optimal tunings:
* WE: ~2 = 1200.4236{{c}}, ~3/2 = 702.1510{{c}}
* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9064{{c}}
 
{{Optimal ET sequence|legend=0| 12, …, 41, 53 }}
 
Badness (Sintel): 0.354
 
=== Photia (2.3.5.17) ===
{{See also| No-elevens subgroup temperaments #Garibaldia }}
 
[[Subgroup]]: 2.3.5.17
 
[[Comma list]]: 256/255, 1458/1445
 
{{Mapping|legend=2| 1 0 15 -7 | 0 1 -8 7 }}
 
{{Mapping|legend=3| 1 0 15 0 0 0 -7 | 0 1 -8 0 0 0 7 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5471{{c}}, ~3/2 = 701.2262{{c}}
: [[error map]]: {{val| -0.453 -1.182 +0.706 +3.628 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4976{{c}}
: error map: {{val| 0.000 -0.457 +1.705 +5.528 }}
 
{{Optimal ET sequence|legend=1| 12, 41, 53, 65, 207g, 272gg }}
 
[[Badness]] (Sintel): 0.479
 
==== 2.3.5.17.19 subgroup ====
Subgroup: 2.3.5.17.19
 
Comma list: 171/170, 256/255, 324/323
 
Subgroup-val mapping: {{mapping| 1 0 15 -7 9 | 0 1 -8 7 -3 }}
 
Gencom mapping: {{mapping| 1 0 15 0 0 0 -7 9 | 0 1 -8 0 0 0 7 -3 }}
 
Optimal tunings:
* WE: ~2 = 1199.7225{{c}}, ~3/2 = 701.3077{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.4754{{c}}
 
{{Optimal ET sequence|legend=0| 12, 41, 53, 65, 142g }}
 
Badness (Sintel): 0.332
 
=== Nestoria (2.3.5.19) ===
: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''
 
Nestoria is notable for having one of the lowest-badness subgroup extensions of schismic. Note that despite prime [[19/1|19]] being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is generally not flatter than the fifth in optimal schismic due to its optimization considering intervals like [[19/10]] and [[19/15]]. However, the dyadic tuning sensitivity of [[19/16]] suggests using tunings like [[65edo]] and [[77edo]] to optimize in favour of prime 19 (especially the minor triad ~16:19:24 which is equated with the Pythagorean minor triad), as [[171edo]] is already arguably undertempered for it despite being the optimal patent val.
 
[[Subgroup]]: 2.3.5.19
 
[[Comma list]]: 361/360, 513/512
 
{{Mapping|legend=2| 1 0 15 9 | 0 1 -8 -3 }}
 
{{Mapping|legend=3| 1 0 15 0 0 0 0 9 | 0 1 -8 0 0 0 0 -3 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2250{{c}}, ~3/2 = 701.8776{{c}}
: [[error map]]: {{val| +0.225 +0.148 +0.240 -1.796 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7307{{c}}
: error map: {{val| 0.000 -0.224 -0.159 -2.705 }}
 
{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171, 460hh, 631hh }}
 
[[Badness]] (Sintel): 0.126
 
=== Taylor (2.3.5.13) ===
This is a 2.3.5.13 subgroup restriction of 13-limit hemischis.
 
[[Subgroup]]: 2.3.5.13
 
[[Comma list]]: 676/675, 32805/32768
 
{{Mapping|legend=2| 1 0 15 14 | 0 2 -16 -13 }}
 
{{Mapping|legend=3| 1 0 15 0 0 14 | 0 2 -16 0 0 -13 }}
: mapping generators: ~2, ~26/15
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1497{{c}}, ~26/15 = 950.9740{{c}}
: [[error map]]: {{val| +0.150 -0.007 +0.348 -1.094 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 950.8493{{c}}
: error map: {{val| 0.000 -0.256 +0.098 -1.568 }}
 
{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236, 525f, 761ff }}
 
[[Badness]] (Sintel): 0.334
 
==== Dakota (2.3.5.13.19) ====
Subgroup: 2.3.5.13.19
 
Comma list: 361/360, 513/512, 676/675
 
Subgroup-val mapping: {{mapping| 1 0 15 14 9 | 0 2 -16 -13 -6 }}
 
Optimal tunings:
* WE: ~2 = 1200.2611{{c}}, ~26/15 = 951.0703{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8532{{c}}
 
{{Optimal ET sequence|legend=0| 24, 29, 53, 130, 183, 236h, 289h }}
 
Badness (Sintel): 0.262
 
===== 2.3.5.13.19.37 subgroup =====
Subgroup: 2.3.5.13.19.37
 
Comma list: 361/360, 481/480, 513/512, 676/675
 
Subgroup-val mapping: {{mapping| 1 0 15 14 9 6 | 0 2 -16 -13 -6 -1 }}
 
Optimal tunings:
* WE: ~2 = 1200.2987{{c}}, ~26/15 = 951.1060{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8595{{c}}
 
{{Optimal ET sequence|legend=0| 24, 29, 53, 183, 236h, 289hl, 631fhhll }}
 
Badness (Sintel): 0.223
 
=== Quintilischis (2.3.5.17) ===
: ''For full 17- and 19-limit extensions, see [[#Quintilipyth]] or [[#Quintaschis]].''
 
[[Subgroup]]: 2.3.5.17
 
[[Comma list]]: 32805/32768, 1419857/1417176
 
{{Mapping|legend=2| 1 2 -1 5 | 0 -5 40 -11 }}
 
{{Mapping|legend=3| 1 2 -1 0 0 0 5 | 0 -5 40 0 0 0 -11 }}
: mapping generators: ~2, ~18/17
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1370{{c}}, ~18/17 = 99.6602{{c}}
: [[error map]]: {{val| +0.137 +0.018 -0.042 -0.533 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~18/17 = 99.6499{{c}}
: error map: {{val| 0.000 -0.205 -0.317 -1.104 }}
 
{{Optimal ET sequence|legend=1| 12, …, 253, 265, 277, 289, 566g, 855g }}
 
[[Badness]] (Sintel): 1.34
 
==== 2.3.5.17.19 subgroup ====
Subgroup: 2.3.5.17.19
 
Comma list: 4624/4617, 6144/6137, 6885/6859
 
Subgroup-val mapping: {{mapping| 1 2 -1 5 4 | 0 -5 40 -11 3 }}


Map: [&lt;1 0 15 25 -33|, &lt;0 1 -8 -14 23|]
Gencom mapping: {{mapping| 1 2 -1 0 0 0 5 4 | 0 -5 40 0 0 0 -11 3 }}
Edos: 94, 135


===Pontiac===
Optimal tunings:
Commas: {32805/32768, 4375/4374}
* WE: ~2 = 1200.0350{{c}}, ~18/17 = 99.6550{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6520{{c}}


7-limit minimax:
{{Optimal ET sequence|legend=0| 12, …, 253, 265, 277, 289 }}
[|1 0 0 0&gt;, |74/47 0 -1/47 1/47&gt;, |113/47 0 8/47 -8/47&gt;,  
|113/47 0 -39/47 39/47&gt;]
Eigenmonzos: 2, 7/5


9-limit minimax:
Badness (Sintel): 1.17
[|1 0 0 0&gt;, |3/2 1/5 -1/10 0&gt;,
|3 -8/5 4/5 0&gt;, |-1/2 39/5 -39/10 0&gt;]
Eigenmonzos: 2, 10/9


Map: [&lt;1 0 15 -59|, &lt;0 1 -8 39|]
[[Category:Temperament families]]
Edos: 171, 1079, 1250, 1421</pre></div>
[[Category:Schismatic family| ]] <!-- main article -->
<h4>Original HTML content:</h4>
[[Category:Rank 2]]
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Schismatic family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;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 just diminished fourth. Its &lt;a class="wiki_link" href="/monzo"&gt;monzo&lt;/a&gt; is |-15 8 1&amp;gt;, and flipping that yields &amp;lt;&amp;lt;1 -8 -15|| for the &lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;wedgie&lt;/a&gt;. 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. &lt;br /&gt;
&lt;br /&gt;
The 5-limit version of the temperament is a &lt;a class="wiki_link" href="/Microtempering"&gt;microtemperament&lt;/a&gt;, 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. &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt; is a possible tuning for schismatic, but you need &lt;a class="wiki_link" href="/118edo"&gt;118edo&lt;/a&gt; if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schisma 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.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Seven limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Seven limit children&lt;/h2&gt;
The second comma of the &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; defines which 7-limit family member we are looking at. Adding |25 -14 0 -1&amp;gt; gives garibaldi, |-44 26 0 1&amp;gt; grackle, |6 -2 0 -1&amp;gt; schism and |-59 39 0 -1&amp;gt; pontiac; these all have a fifth as generator. Bischismic adds |-69 40 0 2&amp;gt; and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = |-10 1 0 3&amp;gt;, with an 8/7 generator, three of which give the fifth, and term adds |-94 54 0 3&amp;gt; with a 1/3 octave period. Sesquiquartififths adds |-35 15 0 4&amp;gt; and slices the fifth in four.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-Seven limit children-Garibaldi"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Garibaldi&lt;/h3&gt;
Commas: {225/224, 3125/3087}&lt;br /&gt;
&lt;br /&gt;
7-limit minimax tuning:&lt;br /&gt;
7-limit: [|1 0 0 0&amp;gt;, |5/3 1/15 0 -1/15&amp;gt;,&lt;br /&gt;
|5/3 -8/15 0 8/15&amp;gt;, |5/3 -14/15 0 14/15&amp;gt;]&lt;br /&gt;
Eigenmonzos: 2, 7/6&lt;br /&gt;
&lt;br /&gt;
9-limit: [|1 0 0 0&amp;gt;, |25/16 1/8 0 -1/16&amp;gt;, &lt;br /&gt;
|5/2 -1 0 1/2&amp;gt;, |25/8 -7/4 0 7/8&amp;gt;]&lt;br /&gt;
Eigenmonzos: 2, 9/7&lt;br /&gt;
&lt;br /&gt;
11-limit&lt;br /&gt;
Commas: {225/224, 385/384, 2200/2187}&lt;br /&gt;
&lt;br /&gt;
Minimax tuning:&lt;br /&gt;
[|1 0 0 0 0&amp;gt;, |25/16 1/8 0 -1/16 0&amp;gt;, |5/2 -1 0 1/2 0&amp;gt;,&lt;br /&gt;
|25/8 -7/4 0 7/8 0&amp;gt;, |47/16 23/8 0 -23/16 0&amp;gt;]&lt;br /&gt;
Eigenmonzos: 2, 9/7&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 15 25 -33|, &amp;lt;0 1 -8 -14 23|]&lt;br /&gt;
Edos: 94, 135&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-Seven limit children-Pontiac"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Pontiac&lt;/h3&gt;
Commas: {32805/32768, 4375/4374}&lt;br /&gt;
&lt;br /&gt;
7-limit minimax:&lt;br /&gt;
[|1 0 0 0&amp;gt;, |74/47 0 -1/47 1/47&amp;gt;, |113/47 0 8/47 -8/47&amp;gt;, &lt;br /&gt;
|113/47 0 -39/47 39/47&amp;gt;]&lt;br /&gt;
Eigenmonzos: 2, 7/5&lt;br /&gt;
&lt;br /&gt;
9-limit minimax:&lt;br /&gt;
[|1 0 0 0&amp;gt;, |3/2 1/5 -1/10 0&amp;gt;, &lt;br /&gt;
|3 -8/5 4/5 0&amp;gt;, |-1/2 39/5 -39/10 0&amp;gt;]&lt;br /&gt;
Eigenmonzos: 2, 10/9&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 15 -59|, &amp;lt;0 1 -8 39|]&lt;br /&gt;
Edos: 171, 1079, 1250, 1421&lt;/body&gt;&lt;/html&gt;</pre></div>
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