Schismatic family: Difference between revisions

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~~{{interwiki~~

~~| en = Schismatic family~~

~~| de = Schismatische Temperaturen~~

~~| es =~~

~~| ja =~~

}}

The [[5-limit]] parent comma for the '''schismatic''' (or '''schismic''') '''family''' is the [[schisma]] of 32805/32768, which is the amount by which the [[Pythagorean comma]] exceeds the [[syntonic comma]] (81/80), or alternatively put, the difference between a [[5/4|just major third]] and a [[8192/6561|Pythagorean diminished fourth]].

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

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

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

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

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

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

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

[[Tuning ranges]]:

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=== Overview to extensions ===

The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at.

The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at. [[#Garibaldi|Garibaldi]] adds [[garischisma|{{monzo| 25 -14 0 -1 }}]], [[#Grackle|grackle]] adds {{monzo| -44 26 0 1 }}, [[#Pontiac|pontiac]] adds {{monzo| -59 39 0 -1 }}, and [[#Schism|schism]] adds [[64/63|{{monzo| 6 -2 0 -1 }}]]. Those all have a fifth as generator.

* [[#Garibaldi|Garibaldi]] adds [[garischisma|{{monzo| 25 -14 0 -1 }}]],

* [[#Grackle|~~Grackle~~]] adds {{monzo| -44 26 0 1 }},

* [[#~~Schism~~|~~Schism~~]] adds ~~[[64/63|~~{{monzo| 6 -2 0 -1 }}]],

* [[#~~Pontiac~~|~~Pontiac~~]] adds {{monzo| -~~59 39~~ 0 -1 }}.

Those all have a fifth as generator.

* [[#Bischismic|Bischismic]] adds {{monzo| -69 40 0 2 }} and has a fifth generator with a half-octave period.

[[#Bischismic|Bischismic]] adds {{monzo| -69 40 0 2 }} and has a fifth generator with a half-octave period. [[#Salsa|Salsa]] adds [[parahemif comma|{{monzo| 15 -13 0 2 }}]] and has a hemififth generator. [[#Hemischis|Hemischis]] adds {{monzo| -34 25 0 -2 }} and has a hemitwelfth generator. [[Gamelismic clan #Guiron|Guiron]] adds [[1029/1024|{{monzo| -10 1 0 3 }}]], with an ~8/7 generator, three of which give the fifth. [[#Term|Term]] adds {{monzo| -94 54 0 3 }} with a 1/3-octave period. [[#Squirrel|Squirrel]], [[#Tertiaschis|tertiaschis]], and [[#Countertertiaschis|countertertiaschis]] each has a generator that is 1/3 of the fourth. [[#Quadrant|Quadrant]] adds {{monzo| -119 68 0 4 }} with a 1/4-octave period. [[#Kleischismic|Kleischismic]] adds {{monzo| 49 -38 0 4 }} with a half-octave period and also a bisect generator. [[#Sesquiquartififths|Sesquiquartififths]] adds {{monzo| -35 15 0 4 }} and slices the fifth in four.

* [[#Hemischis|Hemischis]] adds {{monzo| -34 25 0 -2 }} and has a ~~hemififth~~ generator.

* [[Gamelismic clan #Guiron|Guiron]] adds [[1029/1024|{{monzo| -10 1 0 3 }}]], with an ~8/7 generator, three of which give the fifth.

* [[#Term|Term]] adds {{monzo| -94 54 0 3 }} with a 1/3 octave period.

* [[#Sesquiquartififths|Sesquiquartififths]] adds {{monzo| -35 15 0 4 }} and slices the fifth in four.

Temperaments discussed elsewhere include:

* ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]

* ''[[Pogo]]'' (+118098/117649) → [[Stearnsmic clan #Pogo|Stearnsmic clan]]

Considered below are garibaldi, pontiac, grackle, schism, bischismic, kleischismic, salsa, hemischis, term, altinex, squirrel, tertiaschis, countertertiaschis, quadrant, sesquiquartififths, tsaharuk, quanharuk, quintilipyth, quintaschis, sextilifourths, septant, octant, nonant, septiquarschis, and tridecafifths.

The schismatic family boasts a variety of remarkable extensions to subgroups in high prime limits. These are listed at the bottom of this page, in [[#Subgroup extensions]].

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Garibaldi tempers out the [[~~garischisma~~]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double diminished octave (~~C-Cbb~~), or down-minor seventh (C-~~vBb~~) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}.

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

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

[[Subgroup]]: 2.3.5.7

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* [[WE]]: ~2 = 1200.1233{{c}}, ~3/2 = 702.1573{{c}}

: [[error map]]: {{val| +0.123 +0.326 -2.709 +2.328 }}

* [[CWE]]: ~2 = 1200.0000{{c]}, ~3/2 = 702.0774{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0774{{c}}

: error map: {{val| 0.000 +0.122 -2.933 +2.090 }}

~~~~

[[Minimax tuning]]:

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=== Cassandra ===

Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup.

Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup, even though it comes with a much higher complexity. It finds 11/8 as two commas above 4/3 and 13/8 a minor third above 11/8, or equivalently two commas above 128/81.

Subgroup: 2.3.5.7.11

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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:

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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:

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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}}

~~~~

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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 }}

{{Mapping|legend=0| 1 0 15 25 -33 -28 -7 9 | 0 1 -8 -14 23 20 7 -3 }}

Optimal tunings:

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

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

~~~~

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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 }}

{{Mapping|legend=0| 1 0 15 25 -33 -28 77 | 0 1 -8 -14 23 20 -46 }}

Optimal tunings:

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

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

~~~~

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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 }}

{{Mapping|legend=0| 1 0 15 25 -33 -28 77 9 | 0 1 -8 -14 23 20 -46 -3 }}

Optimal tunings:

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

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

~~~~

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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 }}

{{Mapping|legend=0| 1 0 15 25 -33 -28 77 9 60 | 0 1 -8 -14 23 20 -46 -3 -35 }}

Optimal tunings:

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

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

~~~~

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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 }}

{{Mapping|legend=0| 1 0 15 25 -33 -28 -72 | 0 1 -8 -14 23 20 48 }}

Optimal tunings:

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

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

~~~~

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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 }}

{{Mapping|legend=0| 1 0 15 25 -33 -28 -72 9 | 0 1 -8 -14 23 20 48 -3 }}

Optimal tunings:

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

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

~~~~

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=== Andromeda ===

Andromeda is the second-best extension to garibaldi, which is less complex but also more innacurate. 11/8 is two commas down from [[729/512]], as is 13/8 two commas down from [[27/16]].

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224, 245/242

~~Mapping:~~ {{~~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:

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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:

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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}}

~~~~

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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 }}

{{Mapping|legend=0| 1 0 15 25 32 37 -7 9 | 0 1 -8 -14 -18 -21 7 -3 }}

Optimal tunings:

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

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

~~~~

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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 }}

{{Mapping|legend=0| 1 0 15 25 32 37 58 | 0 1 -8 -14 -18 -21 -34 }}

Optimal tunings:

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

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

~~~~

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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 }}

{{Mapping|legend=0| 1 0 15 25 32 37 58 9 | 0 1 -8 -14 -18 -21 -34 -3 }}

Optimal tunings:

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

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

~~~~

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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 }}

{{Mapping|legend=0| 1 0 15 25 32 37 12 | 0 1 -8 -14 -18 -21 -5 }}

Optimal tunings:

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

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

~~~~

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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 }}

{{Mapping|legend=0| 1 0 15 25 32 37 12 9 | 0 1 -8 -14 -18 -21 -5 -3 }}

Optimal tunings:

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

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

~~~~

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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:

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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:

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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}}

~~~~

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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 }}

{{Mapping|legend=0| 1 0 15 25 51 56 -7 9 | 0 1 -8 -14 -30 -33 7 -3 }}

Optimal tunings:

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

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

~~~~

Badness (Sintel): 1.18

~~=== Hemigari ===~~

~~Subgroup: 2.3.5.7.11~~

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

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

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

~~Optimal tunings:~~

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

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

~~~~

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

~~Badness (Sintel): 1.68~~

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

~~Subgroup: 2.3.5.7.11.13~~

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

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

~~Optimal tunings:~~

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

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

~~~~

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

~~Badness (Sintel): 1.13~~

=== Karadeniz ===

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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

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

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

~~~~

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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}}

~~~~

Badness (Sintel): 1.34

=== Hemigari ===

Subgroup: 2.3.5.7.11

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

: mapping generators: ~2, ~110/63

Optimal tunings:

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

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

Badness (Sintel): 1.68

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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

Optimal tunings:

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

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

Badness (Sintel): 1.13

=== Sanjaab ===

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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

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

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

~~~~

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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}}

~~~~

Badness (Sintel): 1.40

~~== Schism ==~~

~~See [[Archytas clan #Schism]].~~

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

== Pontiac ==

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

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

[[Subgroup]]: 2.3.5.7

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

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

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

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

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

~~~~

[[Minimax tuning]]:

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=== Helenoid ===

~~The helenoid temperament (~~{{nowrap| 53 & 118 }}) is closely related to the helenus temperament, ~~but with the ragisma rather than~~ the ~~[[225/224|marvel comma]] tempered out~~.

Helenoid may be described as {{nowrap| 53 & 118 }}, and is closely related to the helenus temperament, differing only by the mapping of 7.

Subgroup: 2.3.5.7.11

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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:

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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:

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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 }}

{{Mapping|legend=0| 1 0 15 -59 51 56 -91 | 0 1 -8 39 -30 -33 60 }}

Optimal tunings:

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

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

~~~~

Minimax tuning:

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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}}

~~~~

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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 }}

{{Mapping|legend=0| 1 0 15 -59 51 -28 -91 | 0 1 -8 39 -30 20 60 }}

Optimal tunings:

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

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

~~~~

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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 }}

{{Mapping|legend=0| 1 0 15 -59 51 -28 -91 9 | 0 1 -8 39 -30 20 60 -3 }}

Optimal tunings:

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

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

~~~~

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=== Ponta ===

~~The ponta temperament (~~{{nowrap| 53 & ~~171~~ }}~~) tempers out the~~ [[~~540/539|swetisma~~]] ~~and the ragisma~~.

Ponta tempers out [[540/539]] and may be described as {{nowrap| 171 & 224 }}. [[224edo]] itself makes for an excellent tuning.

Subgroup: 2.3.5.7.11

Line 729:

Line 680:

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:

Line 749:

Line 699:

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:

Line 769:

Line 718:

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 }}

{{Mapping|legend=0| 1 0 15 -59 135 56 -91 | 0 1 -8 39 -83 -33 60 }}

Optimal tunings:

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

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

~~~~

Minimax tuning:

Line 785:

Line 733:

=== Pontic ===

~~The pontic~~ temperament ({{nowrap| 118 & 171 }}~~) tempers out the~~ [[~~441/440|werckisma~~]] ~~and the ragisma~~.

Pontic temperament tempers out [[441/440]] and may be described as {{nowrap| 118 & 171 }}. [[289edo]] may be recommended as a tuning.

Subgroup: 2.3.5.7.11

Line 791:

Line 739:

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:

Line 811:

Line 758:

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:

Line 831:

Line 777:

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 }}

{{Mapping|legend=0| 1 0 15 -59 -136 56 -91 | 0 1 -8 39 88 -33 60 }}

Optimal tunings:

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

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

~~~~

Minimax tuning:

Line 851:

Line 796:

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}}

~~~~

Line 867:

Line 811:

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 }}

{{Mapping|legend=0| 1 0 15 -59 -136 -215 -91 | 0 1 -8 39 88 138 60 }}

Optimal tunings:

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

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

~~~~

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

Line 879:

Line 822:

=== Bipont ===

~~The bipont temperament ({{nowrap| 118 & 224 }}) has a period of half octave and~~ tempers out the [[3025/3024|lehmerisma (3025/3024)]] and the [[9801/9800|kalisma (9801/9800)]].

Bipont tempers out the [[3025/3024|lehmerisma (3025/3024)]] and the [[9801/9800|kalisma (9801/9800)]]. It may be described as {{nowrap| 118 & 224 }}. It has a period of half octave and a ploidacot signature of diploid monocot. [[342edo]] may be recommended as a tuning.

Subgroup: 2.3.5.7.11

Line 885:

Line 828:

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

Line 891:

Line 834:

* 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 }}

Line 902:

Line 844:

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}}

~~~~

Line 918:

Line 859:

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 }}

{{Mapping|legend=0| 2 0 30 -118 -85 112 -182 | 0 1 -8 39 29 -33 60 }}

Optimal tunings:

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

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

~~~~

Line 934:

Line 874:

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 }}

Line 950:

Line 889:

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 }}

{{Mapping|legend=0| 2 0 30 -118 -85 -243 -182 | 0 1 -8 39 29 79 60 }}

Optimal tunings:

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

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

~~~~

Line 966:

Line 904:

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 }}

{{Mapping|legend=0| 2 0 30 -118 -85 -243 -182 -169 | 0 1 -8 39 29 79 60 56 }}

Optimal tunings:

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

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

~~~~

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

Line 982:

Line 919:

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

Line 988:

Line 925:

* 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 }}

Line 995:

Line 931:

== Grackle ==

Grackle tempers out {{monzo| -44 26 0 1 }}~~. The~~ 7/4 is found at -26 fifths, represented by the triple diminished ninth (~~C-Dbbbb~~), or double-down minor seventh (~~C-vvBb~~)~~, which is to say, two~~ comma steps are required to bend the Pythagorean minor seventh to the septimal one.

Grackle tempers out {{monzo| -44 26 0 1 }} so 7/4 is found at -26 fifths, represented by the triple-diminished ninth (C–D𝄫𝄫) or double-down minor seventh (C–vvB♭). Two comma steps are required to bend the Pythagorean minor seventh to the septimal one.

[[Subgroup]]: 2.3.5.7

Line 1,007:

Line 943:

* [[WE]]: ~2 = 1199.7974{{c}}, ~3/2 = 701.1210{{c}}

: [[error map]]: {{val| -0.203 -1.037 +3.300 -1.618 }}

* [[CWE]]: ~2 = 1200.0000{{c]}, ~3/2 = 701.2465{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.2465{{c}}

: error map: {{val| 0.000 -0.709 +3.715 -1.234 }}

~~~~

[[Minimax tuning]]:

Line 1,024:

Line 959:

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}}

~~~~

Line 1,040:

Line 974:

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}}

~~~~

Line 1,056:

Line 989:

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}}

~~~~

Line 1,072:

Line 1,004:

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 }}

{{Mapping|legend=0| 1 0 15 44 70 75 -7 9 | 0 1 -8 -26 -42 -45 7 -3 }}

Optimal tunings:

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

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

~~~~

Line 1,088:

Line 1,019:

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}}

~~~~

Line 1,104:

Line 1,034:

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}}

~~~~

Line 1,120:

Line 1,049:

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}}

~~~~

Line 1,136:

Line 1,064:

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}}

~~~~

Line 1,147:

Line 1,074:

Badness (Sintel): 2.01

== Bischismic ==

== Quasipyth ==

Named by [[Xenllium]] in 2026, quasipyth tempers out {{monzo| 109 -67 0 -1 }}, the [[nanisma]], as well as the [[catasyc comma]], 390625/387072. The 7/4 is found at −67 fifths, represented by the nonuple-diminished thirteenth.

[[Subgroup]]: 2.3.5.7

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

[[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

Optimal tunings:

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

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

Badness (Sintel): 3.83

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

Optimal tunings:

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

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

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

Line 1,160:

Line 1,143:

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

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

~~~~

[[Minimax tuning]]:

Line 1,175:

Line 1,157:

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}}

~~~~

Line 1,191:

Line 1,172:

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}}

~~~~

Line 1,207:

Line 1,187:

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}}

~~~~

Line 1,223:

Line 1,202:

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}}

~~~~

Line 1,239:

Line 1,217:

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 }}

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

Optimal tunings:

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

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

~~~~

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

Line 1,251:

Line 1,228:

== 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

Line 1,263:

Line 1,242:

* [[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 }}

Line 1,274:

Line 1,252:

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 }}

Line 1,290:

Line 1,267:

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}})

~~~~

Line 1,306:

Line 1,282:

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}})

~~~~

Line 1,322:

Line 1,297:

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}})

~~~~

Line 1,338:

Line 1,312:

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}})

~~~~

Line 1,350:

Line 1,323:

== 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

Line 1,372:

Line 1,347:

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

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

Optimal tunings:

Line 1,387:

Line 1,362:

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}}

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

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

Line 1,398:

Line 1,373:

== 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

Line 1,410:

Line 1,389:

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

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

~~~~

Line 1,421:

Line 1,399:

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}}

~~~~

Line 1,433:

Line 1,410:

=== 13-limit ===

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

Subgroup: 2.3.5.7.11.13

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}}

~~~~

Line 1,455:

Line 1,429:

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 }}

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

Optimal tunings:

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

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

~~~~

Line 1,471:

Line 1,444:

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 }}

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

Optimal tunings:

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

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

~~~~

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Line 1,459:

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 }}

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

Optimal tunings:

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

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

~~~~

Line 1,501:

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

== ~~Squirrel~~ ==

== Term ==

~~The squirrel temperament (~~{{nowrap| 29 & 36 }}~~) has~~ a ~~~11/10 generator,~~ three of ~~which give~~ the ~~fourth (~4/3)~~, ~~and thirteen of which give 7/4~~ with ~~octave reduction~~.

Term tempers out the [[landscape comma]], mapping [[63/50]] to the 1/3-octave period. It can be described as {{nowrap| 12 & 171 }}, and is the unique temperament that 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]]: ~~686~~/~~675~~, ~~32805~~/~~32768~~

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

: mapping generators: ~63/50, ~3

[[Optimal tuning]]s:

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

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

: [[error map]]: {{val| +0.~~741 +~~0.~~799~~ +2.~~763 -6.934~~ }}

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

* [[CWE]]: ~2 = ~~1200~~.0000{{c}}, ~~~160~~/~~147~~ = ~~166~~.~~1597~~{{c}}

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

: error map: {{val| 0.000 -0.~~434 +1~~.~~518~~ -8.~~750~~ }}

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

[[Minimax tuning]]:

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

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

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

~~{{Optimal ET sequence|legend=1| 29, 36, 65 }}~~

[[Badness]] (Sintel): 0.505

~~[[Badness]] (Sintel): 4~~.42

=== Terminal ===

Terminal tempers out 441/440 and 4375/4356, and may be described as {{nowrap| 159 & 171 }}. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave.

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

Subgroup: 2.3.5.7.11

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

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

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

Optimal tunings:

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

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

* CWE: ~2 = ~~1200~~.0000{{c}}, ~11/~~10 = 166.1157{{c}}~~

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

~~~~

Badness (Sintel): 2.26

Badness (Sintel): 1.97

=== 13-limit ===

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal tunings:

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

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

* CWE: ~2 = ~~1200~~.0000{{c}}, ~11/~~10 = 166.0833{{c}}~~

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

~~~~

Badness (Sintel): 1.81

Badness (Sintel): 1.53

== ~~Tertiaschis~~ ==

==== 17-limit ====

~~The tertiaschis temperament ({{nowrap| 94 & 159 }}) has a ~11/10 generator, sharing the same~~ 2.3.5.11 ~~subgroup with [[#Squirrel]], but tempers out 1071875/1062882 for prime 7~~.

Subgroup: 2.3.5.7.11.13.17

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

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

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

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

Optimal tunings:

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

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

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

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

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

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

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

~~~~

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

Badness (Sintel): 1.38

~~[[Badness]] (Sintel): 5~~.36

=== Terminator ===

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

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

Subgroup: 2.3.5.7.11

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

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

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

Optimal tunings:

* WE: ~2 = ~~1200~~.~~3379~~{{c}}, ~11/10 = ~~166~~.~~0638~~{{c}}

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

* CWE: ~2 = ~~1200~~.0000{{c}}, ~11/~~10 = 166.0167{{c}}~~

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

~~~~

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

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

Badness (Sintel): 2.07

Badness (Sintel): 2.21

=== 13-limit ===

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal tunings:

* WE: ~2 = ~~1200~~.~~3467~~{{c}}, ~11/10 = ~~166~~.~~0635~~{{c}}

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

* CWE: ~2 = ~~1200~~.0000{{c}}, ~11/~~10 = 166.0142{{c}}~~

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

~~~~

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

Badness (Sintel): 1.52

Badness (Sintel): 1.47

=== 17-limit ===

==== 17-limit ====

Subgroup: 2.3.5.7.11.13.17

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

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

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

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

Optimal tunings:

* WE: ~2 = ~~1200~~.~~3019~~{{c}}, ~11/10 = ~~166~~.~~0535~~{{c}}

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

* CWE: ~2 = ~~1200~~.0000{{c}}, ~11/~~10 = 166.0114{{c}}~~

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

~~~~

Badness (Sintel): 1.35

Badness (Sintel): 1.04

== ~~Countertertiaschis~~ ==

=== Semiterm ===

The ~~countertertiaschis~~ temperament ({{nowrap| ~~159~~ & ~~224~~ }}) has a ~~~11/10 generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel]], but tempers out 244140625~~/~~243045684 for prime 7~~.

The semiterm temperament tempers out [[9801/9800]] (kalisma) as well as [[151263/151250]] (odiheim comma), and may be described as {{nowrap| 12 & 342 }}. It has a period of 1/6 octave and its ploidacot is hexaploid monocot.

[[Subgroup]]: 2.3.5.7

Subgroup: 2.3.5.7.11

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

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

: mapping generators: ~55/49, ~3

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

Optimal tunings:

* [[WE]]: ~2 = ~~1200~~.~~1265~~{{c}}, ~~~625~~/~~567~~ = ~~166~~.~~0797~~{{c}}

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

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

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

* [[CWE]]: ~2 = ~~1200~~.0000{{c}}, ~~~625~~/~~567 = 166.0632{{c}}~~

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

~~~~

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

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

[[Badness]] (Sintel): 4.76

Badness (Sintel): 0.973

=== 11-limit ===

==== 13-limit ====

Subgroup: 2.3.5.7.11

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal tunings:

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

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

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

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

~~~~

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

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

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

Badness (Sintel): 1.85

~~Badness~~ (~~Sintel~~)~~: 1~~.62

=== Hemiterm ===

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

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

Subgroup: 2.3.5.7.11

Subgroup: 2.3.5.7.11~~.13~~

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

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

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

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

Optimal tunings:

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

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

* CWE: ~2 = ~~1200~~.~~0000~~{{c}}, ~11/10 = ~~166~~.~~0635~~{{c}}

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

~~~~

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

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

Badness (Sintel): 1.01

Badness (Sintel): 0.684

== ~~Term~~ ==

==== 13-limit ====

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

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal tunings:

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

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

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

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

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

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

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

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

~~~~

~~[[Minimax tuning]]:~~

Badness (Sintel): 1.30

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

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

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

==== 17-limit ====

Subgroup: 2.3.5.7.11.13.17

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

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

~~=== Terminal ===~~

~~The terminal temperament (~~{{~~nowrap~~| ~~12 & 159~~ }}~~) tempers out 441/440 and 4375/4356. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave.~~

~~Subgroup~~: 2.3.5.7.11

Optimal tunings:

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

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

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

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

Badness (Sintel): 1.14

~~Optimal tunings:~~

== Altinex ==

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

Named by [[Aura]] in 2021, altinex is an alternative to [[#Hemiterm|hemiterm]] and may be described as {{nowrap| 24 & 159 }}. [[159edo]] itself makes for a recommendable tuning.

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

~~~~

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

[[Subgroup]]: 2.3.5.7

~~Badness (Sintel)~~: ~~1.97~~

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

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

~~Subgroup~~: ~~2.3.5.7.11.13~~

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

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

[[Optimal tuning]]s:

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

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

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

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

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

~~Optimal tunings~~:

[[Badness]] (Sintel): 10.7

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

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

~~~~

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

=== 11-limit ===

Subgroup: 2.3.5.7.11

~~Badness (Sintel)~~: ~~1.53~~

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

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

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

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

Optimal tunings:

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

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

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

~~Optimal tunings~~:

Badness (Sintel): 3.35

* 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 }}~~

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

~~Badness (Sintel)~~: ~~1.38~~

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

=~~== Terminator ===~~

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

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

Optimal tunings:

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

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

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

~~Optimal tunings~~:

Badness (Sintel): 2.27

* 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 }}~~

== 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.

~~Badness (Sintel)~~: 2.21

[[Subgroup]]: 2.3.5.7

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

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

~~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 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 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): 4.42

~~Badness (Sintel)~~: 1.47

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

~~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}}

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

* CWE: ~~~63/50~~ = ~~400~~.0000{{c}}, ~3/~~2 = 701.6881{{c}}~~

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

~~~~

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

Badness (Sintel): 1.04

Badness (Sintel): 2.26

=== ~~Semiterm~~ ===

=== 13-limit ===

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

Subgroup: 2.3.5.7.11.13

~~Subgroup: 2.3.5.7.11~~

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

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

~~Mapping:~~ {{~~mapping~~| 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}}

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

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

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

~~~~

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

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

Badness (Sintel): 1.81

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

== Tertiaschis ==

~~Subgroup:~~ 2.3.5.7.11.13

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

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

[[Subgroup]]: 2.3.5.7

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

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

~~Optimal tunings:~~

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

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

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

[[Optimal tuning]]s:

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

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

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

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

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

Badness (Sintel): 1.85

[[Badness]] (Sintel): 5.36

=== ~~Hemiterm~~ ===

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal tunings:

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

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

* CWE: ~~~63/50~~ = ~~400~~.0000{{c}}, ~~~693/400 = 950.8707{{c}} (~12/~~11 ~~= 150.8707{{c}})~~

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

~~~~

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

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

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

Badness (Sintel): 2.07

==== 13-limit ====

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal tunings:

* WE: ~~~63/50~~ = ~~400~~.~~0541~~{{c}}, ~~~26/15 = 951.0013{{c}} (~12~~/11 = ~~150~~.~~8932~~{{c}})

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

* CWE: ~~~63/50~~ = ~~400~~.0000{{c}}, ~~~26/15 = 950.8696{{c}} (~12/~~11 ~~= 150.8696{{c}})~~

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

~~~~

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

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

Badness (Sintel): 1.30

Badness (Sintel): 1.52

==== 17-limit ====

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

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

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

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

Optimal tunings:

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

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

* CWE: ~~~34/27~~ = ~~400~~.0000{{c}}, ~26/15 = ~~950~~.~~8652~~{{c}} ~~(~12/11 = 150.8652~~{{~~c}})~~

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

~~~~

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

Badness (Sintel): 1.35

~~Badness (Sintel): 1~~.14

== Countertertiaschis ==

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

~~== Altinex ==~~

[[Subgroup]]: 2.3.5.7

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

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

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

[[Optimal tuning]]s:

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

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

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

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

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

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

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

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

~~~~

{{Optimal ET sequence|legend=1| 24, ~~135~~, ~~159~~, ~~612ccdd~~ }}

[[Badness]] (Sintel): 10.7

[[Badness]] (Sintel): 4.76

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

Optimal tunings:

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

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

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

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

~~~~

Badness (Sintel): 3.35

Badness (Sintel): 1.62

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal tunings:

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

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

* CWE: ~~~44/35~~ = ~~400~~.0000{{c}}, ~26/15 = ~~950~~.~~9462~~{{c}}

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

~~~~

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

Badness (Sintel): 1.01

~~Badness (Sintel): 2~~.27

== Quadrant ==

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

~~== Sesquiquartififths ==~~

[[Subgroup]]: 2.3.5.7

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

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

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

: mapping generators: ~25/21, ~3

[[Optimal tuning]]s:

* [[WE]]: ~2 = ~~1200~~.~~0846~~{{c}}, ~~~448~~/~~405~~ = ~~175~~.~~4460~~{{c}}

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

: [[error map]]: {{val| +0.~~085~~ -0.~~086~~ +0.~~007 -0.093~~ }}

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

* [[CWE]]: ~2 = ~~1200~~.0000{{c}}, ~~~448~~/~~405~~ = ~~175~~.~~4320~~{{c}}

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

: error map: {{val| 0.000 -0.~~227~~ -0.~~137 -~~0.~~306~~ }}

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

~~~~

~~[[Minimax tuning]]:~~

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

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

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

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

[[Badness]] (Sintel): 2.79

~~[[Badness]] (Sintel): 0.285~~

=== 11-limit ===

=== ~~Sesquart~~ ===

Subgroup: 2.3.5.7.11

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

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

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

Optimal tunings:

* WE: ~2 = ~~1199~~.~~8171~~{{c}}, ~~~256~~/~~231~~ = ~~175~~.~~3793~~{{c}}

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

* CWE: ~2 = ~~1200~~.0000{{c}}, ~~~256~~/~~231 = 175.4081{{c}}~~

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

~~~~

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

Badness (Sintel): 0.~~969~~

Badness (Sintel): 1.51

==== 13-limit ====

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal tunings:

* WE: ~2 = ~~1199~~.~~8352~~{{c}}, ~72/65 = ~~175~~.~~3852~~{{c}}

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

* CWE: ~2 = ~~1200~~.0000{{c}}, ~72/~~65 = 175.4095{{c}}~~

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

~~~~

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

Badness (Sintel): 0.~~925~~

Badness (Sintel): 1.13

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

== Sesquiquartififths ==

~~Subgroup: 2~~.3.~~5.7.11.13.17~~

Sesquiquartififths tempers out 2401/2400, the [[breedsma]], and may be described as the {{nowrap| 41 & 171 }} temperament. It splits the [[3/2|perfect fifth]] into four; its [[ploidacot]] is thus tetracot.

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

The name might have been a portmanteau of ''sesquiquart'' and ''quartififths'', each meaning dividing 3/2 into four.

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

[[Subgroup]]: 2.3.5.7

~~Optimal tunings~~:

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

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

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

~~~~

: mapping generators: ~2, ~448/405

~~Badness (Sintel)~~: 1.18

[[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 }}

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

[[Minimax tuning]]:

~~Subgroup~~: 2.3.5.7~~.11.13.17.19~~

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

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

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

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

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

[[Badness]] (Sintel): 0.285

~~Optimal tunings:~~

=== Sesquart ===

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

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.

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

Subgroup: 2.3.5.7.11

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

Optimal tunings:

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

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

Badness (Sintel): 1.24

Badness (Sintel): 0.969

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

==== 13-limit ====

Subgroup: 2.3.5.7.11.13~~.17.19.23~~

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal tunings:

* WE: ~2 = 1199.~~9606~~{{c}}, ~21/19 = 175.~~4067~~{{c}}

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

* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.~~4123~~{{c}}

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

~~~~

Badness (Sintel): 1.36

Badness (Sintel): 0.925

===== Heartia =====

Line 2,041:

Line 1,998:

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}}

~~~~

Line 2,057:

Line 2,013:

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 }}

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

Optimal tunings:

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

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

~~~~

Line 2,068:

Line 2,023:

Badness (Sintel): 1.40

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

===== Sesquartia =====

Subgroup: 2.3.5.7.11.13.17

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

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

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

Optimal tunings:

* WE: ~2 = 1199.~~9458~~{{c}}, ~72/65 = 175.~~3689~~{{c}}

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

* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.~~3770~~{{c}}

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

~~~~

Badness (Sintel): 1.56

Badness (Sintel): 1.18

====== 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

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

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

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

Optimal tunings:

* WE: ~2 = ~~1200~~.~~0114~~{{c}}, ~72/65 = 175.~~3783~~{{c}}

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

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

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

~~~~

Badness (Sintel): 1.39

Badness (Sintel): 1.24

====== 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

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

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

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

Optimal tunings:

* WE: ~2 = ~~1200~~.~~0122~~{{c}}, ~72/65 = 175.~~3782~~{{c}}

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

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

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

~~~~

Badness (Sintel): 1.37

Badness (Sintel): 1.36

=== ~~Bisesqui~~ ===

===== Hearty =====

Subgroup: 2.3.5.7.11

Subgroup: 2.3.5.7.11.13.17

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

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

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

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

Optimal tunings:

* WE: ~~~99/70~~ = ~~600~~.~~0429~~{{c}}, ~~~448~~/~~405~~ = 175.~~4474~~{{c}}

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

* CWE: ~~~99/70~~ = ~~600.0000{{c}}, ~448/405 = 175.4334{{c}}~~

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

~~~~

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

Badness (Sintel): 0.~~561~~

Badness (Sintel): 1.56

== ~~Quintilipyth~~ ==

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

~~The quintilipyth temperament ({{nowrap| 12 & 253 }}, formerly ''quintilischis'') slices the pythagorean fourth ([[4/~~3~~]]) into five semitones and tempers out the compass comma (9765625/9680832) in the~~ 7~~-limit~~.

Subgroup: 2.3.5.7.11.13.17.19

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

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

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

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

Optimal tunings:

~~: mapping generators~~: ~2, ~~~625~~/~~588~~

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

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

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

* [[WE]]: ~2 = 1200.1138{{~~c}}, ~625/588~~ = ~~99.6347{{c}}~~

~~: [[error map]]: {{val~~| ~~+0.114 +0.099 -1.041 +0.761 }}~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~~~625/588 = 99.6265{{c}}~~

~~: error map: {{val| 0.000 -0.087 -1.255 +0.544 }}~~

~~~~

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

Badness (Sintel): 1.39

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

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

Subgroup: 2.3.5.7.11.13.17.19.23

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

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

~~Subgroup: 2.3.5.7.11~~

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

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

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

Optimal tunings:

* WE: ~2 = 1200.~~1503~~{{c}}, ~35/33 = 99.~~6287~~{{c}}

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

* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.~~6176~~{{c}}

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

~~~~

Badness (Sintel): 3.74

Badness (Sintel): 1.37

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

=== Bisesqui ===

Subgroup: 2.3.5.7.11~~.13~~

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal tunings:

* WE: ~2 = ~~1200~~.~~1774~~{{c}}, ~35/33 = 99.~~6267~~{{c}}

* WE: ~99/70 = 600.0429{{c}}, ~448/405 = 175.4474{{c}}

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

* CWE: ~99/70 = 600.0000{{c}}, ~448/405 = 175.4334{{c}}

~~~~

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

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

Badness (Sintel): 2.86

Badness (Sintel): 0.561

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

== Tsaharuk ==

~~Subgroup~~: 2.3.5.~~7.11.13.17~~

[[File:Tsaharuk.jpg|alt=Tsaharuk.jpg|thumb|Parametric horagram of the tsaharuk temperament.]]

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

Tsaharuk tempers out 420175/419904, the [[wizma]], and may be described as the {{nowrap| 77 & 94 }} temperament. It is generated by a slightly flat neutral second of [[~]][[13/12]], five of which make the [[3/2|perfect fifth]], so its [[ploidacot]] is pentacot.

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

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

~~Optimal tunings:~~

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

* 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 }}~~

[[Subgroup]]: 2.3.5.7

~~Badness (Sintel)~~: ~~2.34~~

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

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

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

: mapping generators: ~2, ~243/224

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

[[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 }}

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

~~Optimal tunings~~:

[[Badness]] (Sintel): 0.777

* 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 }}~~

=== 11-limit ===

Subgroup: 2.3.5.7.11

~~Badness (Sintel)~~: ~~2.32~~

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

~~== Quintaschis ==~~

~~The quintaschis temperament (~~{{~~nowrap~~| ~~12 & 289~~ }}~~) slices the fourth (4/3) into five semitones and tempers out 49009212/48828125 in the 7-limit.~~

~~[[Subgroup]]~~: 2.3.5.7

Optimal tunings:

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

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

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

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

Badness (Sintel): 2.10

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

=== 13-limit ===

* [[WE]]: ~2 = ~~1200.0536{{c}}, ~200/189~~ = ~~99.6684{{c}}~~

Subgroup: 2.3.5.7.11.13

~~: [[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 }}~~

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

~~[[Badness]] (Sintel):~~ 3~~.36~~

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

Optimal tunings:

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

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

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

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

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

Badness (Sintel): 1.57

~~Optimal tunings:~~

== Quanharuk ==

* WE: ~2 = 1200.0988{{c}}, ~35/~~33 = 99.6613{{c}}~~

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.

* CWE: ~2 = 1200.0000{{c}}, ~~~35~~/~~33 = 99.6540{{c}}~~

~~<!-~~- * POTE: ~2 = 1200.~~000{{c}}, ~35/33 = 99~~.~~653{{c}} -->~~

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

[[Subgroup]]: 2.3.5.7

~~Badness (Sintel)~~: ~~3.69~~

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

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

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

: mapping generators: ~2, ~56/45

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

[[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 }}

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

~~Optimal tunings~~:

[[Badness]] (Sintel): 1.82

* 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 }}~~

=== 11-limit ===

Subgroup: 2.3.5.7.11

~~Badness (Sintel)~~: ~~3.07~~

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

~~==== 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}}

* WE: ~2 = 1199.9709{{c}}, ~56/45 = 380.3423{{c}}

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

* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3517{{c}}

~~~~

Badness (Sintel): 2.58

Badness (Sintel): 1.04

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

=== 13-limit ===

Subgroup: 2.3.5.7.11.13~~.17.19~~

Subgroup: 2.3.5.7.11.13

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

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

~~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}}

* WE: ~2 = 1199.9663{{c}}, ~56/45 = 380.3403{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~~~18/17 = 99.6586{{c}}~~

* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3509{{c}}

~~~~

Badness (Sintel): 2.56

Badness (Sintel): 0.884

=== ~~Quintahelenic~~ ===

== 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 generators: ~2, ~625/588

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.1138{{c}}, ~625/588 = 99.6347{{c}}

: [[error map]]: {{val| +0.114 +0.099 -1.041 +0.761 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/588 = 99.6265{{c}}

: error map: {{val| 0.000 -0.087 -1.255 +0.544 }}

[[Badness]] (Sintel): 6.43

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

Optimal tunings:

* WE: ~2 = 1200.~~0195~~{{c}}, ~~~200~~/~~189~~ = 99.~~6723~~{{c}}

* WE: ~2 = 1200.1503{{c}}, ~35/33 = 99.6287{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~~~200~~/~~189~~ = 99.~~6709~~{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6176{{c}}

~~~~

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

Badness (Sintel): 2.72

Badness (Sintel): 3.74

==== 13-limit ====

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

~~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}}

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

* CWE: ~2 = 1200.0000{{c}}, ~~~200~~/~~189~~ = 99.~~6675~~{{c}}

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

~~~~

Badness (Sintel): 2.30

Badness (Sintel): 2.86

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

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

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

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

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

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

Optimal tunings:

* WE: ~2 = 1200.~~1227~~{{c}}, ~~~200~~/~~189~~ = 99.~~6753~~{{c}}

* WE: ~2 = 1200.1745{{c}}, ~18/17 = 99.6265{{c}}

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

* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6131{{c}}

~~~~

Badness (Sintel): 2.06

Badness (Sintel): 2.34

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

=== 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~~

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

~~Mapping:~~ {{~~mapping~~| 1 2 -1 -5 -9 ~~-11~~ 5 4 | 0 -5 40 ~~94 150 177~~ -11 3 }}

{{Mapping|legend=0| 1 2 -1 -4 -7 -9 5 4 | 0 -5 40 82 126 153 -11 3 }}

Optimal tunings:

* WE: ~2 = 1200.~~0230~~{{c}}, ~~~200~~/~~189~~ = 99.~~6694~~{{c}}

* WE: ~2 = 1200.0713{{c}}, ~18/17 = 99.6208{{c}}

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

* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6152{{c}}

~~~~

Badness (Sintel): 2.24

Badness (Sintel): 2.32

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

== Quintaschis ==

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

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

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

[[Subgroup]]: 2.3.5.7

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

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

Optimal ~~tunings~~:

[[Optimal tuning]]s:

* WE: ~2 = ~~1199~~.~~9919~~{{c}}, ~200/189 = 99.~~6712~~{{c}}

* [[WE]]: ~2 = 1200.0536{{c}}, ~200/189 = 99.6684{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.~~6718~~{{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): 2.73

[[Badness]] (Sintel): 3.36

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

=== 11-limit ===

Subgroup: 2.3.5.7.11~~.13.17~~

Subgroup: 2.3.5.7.11

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

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

~~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}}

* WE: ~2 = 1200.0988{{c}}, ~35/33 = 99.6613{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.~~6710~~{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6540{{c}}

~~~~

Badness (Sintel): 2.44

Badness (Sintel): 3.69

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

==== 13-limit ====

Subgroup: 2.3.5.7.11.13~~.17.19~~

Subgroup: 2.3.5.7.11.13

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

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

~~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}}

* WE: ~2 = 1200.0625{{c}}, ~35/33 = 99.6630{{c}}

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

* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6583{{c}}

~~~~

Badness (Sintel): 2.41

Badness (Sintel): 3.07

== ~~Sextilifourths~~ ==

==== 17-limit ====

~~The sextilifourths ({{nowrap| 130 & 159 }}, also known as ''sextilischis'', formerly ''sextilififths'') temperament slices the fourth (4/~~3~~) into six small semitones, which serves as both 21/20 and 22/21~~.

Subgroup: 2.3.5.7.11.13.17

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

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

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

{{Mapping|legend=0| 1 2 -1 -5 -8 -11 5 | 0 -5 40 94 138 177 -11 }}

Optimal tunings:

~~: mapping generators~~: ~2, ~21/20

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

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

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

* [[WE]]: ~2 = 1200.0987{{~~c}}, ~21/20~~ = ~~83.0599{{c}}~~

~~: [[error map]]: {{val~~| ~~+0.099 -0.117 +0.462 -0.630 }}~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~~~21/20 = 83.0543{{c}}~~

~~: error map: {{val| 0.000 -0.281 +0.295 -0.837 }}~~

~~~~

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

Badness (Sintel): 2.58

~~[[Badness]] (Sintel)~~: 2.75

==== 19-limit ====

Subgroup: 2.3.5.7.11.13.17.19

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

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

~~Subgroup~~: ~~2.3.5.7.11~~

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

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

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

Optimal tunings:

* WE: ~2 = 1200.~~0424~~{{c}}, ~21/20 = 83.~~0520~~{{c}}

* WE: ~2 = 1200.0289{{c}}, ~18/17 = 99.6609{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.~~0497~~{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6586{{c}}

~~~~

Badness (Sintel): 1.50

Badness (Sintel): 2.56

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

=== Quintahelenic ===

Subgroup: 2.3.5.7.11~~.13~~

Subgroup: 2.3.5.7.11

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

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

~~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}}

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

* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.~~0508~~{{c}}

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

~~~~

{{Optimal ET sequence|legend=0| 29, ~~72cdef~~, ~~101e~~, ~~130~~, ~~289~~ }}

Badness (Sintel): 1.04

Badness (Sintel): 2.72

== ~~Septiquarschis~~ ==

==== 13-limit ====

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

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal tunings:

~~: mapping generators~~: ~2, ~~~256~~/~~147~~

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

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

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

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

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

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

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

~~~~

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

Badness (Sintel): 2.30

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

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

Subgroup: 2.3.5.7.11.13.17

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

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

~~Subgroup: 2.3.5.7.11~~

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

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

{{Mapping|legend=0| 1 2 -1 -5 -9 -11 5 | 0 -5 40 94 150 177 -11 }}

Optimal tunings:

* WE: ~2 = ~~1199~~.~~9430~~{{c}}, ~~~256~~/~~147~~ = ~~957~~.~~3390~~{{c}}

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

* CWE: ~2 = 1200.0000{{c}}, ~~~256/147 = 957.3849{{c}}~~

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

~~~~

Badness (Sintel): 1.72

Badness (Sintel): 2.06

=== 13-limit ===

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

Subgroup: 2.3.5.7.11.13

Subgroup: 2.3.5.7.11.13.17.19

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

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

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

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

Optimal tunings:

* WE: ~2 = 1200.~~0058~~{{c}}, ~~~256~~/~~147~~ = ~~957~~.~~3946~~{{c}}

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

* CWE: ~2 = 1200.0000{{c}}, ~~~256~~/~~147~~ = ~~957~~.~~3900~~{{c}}

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

~~~~

Badness (Sintel): 1.46

Badness (Sintel): 2.24

== ~~Tsaharuk~~ ==

==== Quintahelenoid ====

~~{{Main| Tsaharuk }}~~

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal tunings:

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

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

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

~~[[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): 2.73

~~[[Badness]] (Sintel)~~: 0.~~777~~

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

Subgroup: 2.3.5.7.11.13.17

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

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

~~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~~ }}

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

Optimal tunings:

* WE: ~2 = 1200.~~3103~~{{c}}, ~88/81 = ~~140~~.~~4011~~{{c}}

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

* CWE: ~2 = 1200.0000{{c}}, ~88/81 = ~~140~~.~~3649~~{{c}}

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

~~~~

Badness (Sintel): 2.10

Badness (Sintel): 2.44

=== 13-limit ===

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

Subgroup: 2.3.5.7.11.13

Subgroup: 2.3.5.7.11.13.17.19

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

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

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

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

Optimal tunings:

* WE: ~2 = ~~1200~~.~~1840~~{{c}}, ~13/12 = ~~140~~.~~3840~~{{c}}

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

* CWE: ~2 = 1200.0000{{c}}, ~13/12 = ~~140~~.~~3627~~{{c}}

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

~~~~

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

Badness (Sintel): 2.41

~~Badness~~ (~~Sintel~~)~~: 1~~.57

== 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.

~~== Quanharuk ==~~

[[Subgroup]]: 2.3.5.7

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

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

: mapping generators: ~2, ~56/45

: mapping generators: ~2, ~21/20

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.~~0032~~{{c}}, ~56/45 = ~~380~~.~~3557~~{{c}}

* [[WE]]: ~2 = 1200.0987{{c}}, ~21/20 = 83.0599{{c}}

: [[error map]]: {{val| +0.~~003~~ -0.~~177 -~~0.~~493 +~~0.~~898~~ }}

: [[error map]]: {{val| +0.099 -0.117 +0.462 -0.630 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~56/45 = ~~380~~.~~3546~~{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0543{{c}}

: error map: {{val| 0.000 -0.~~182 -0.498~~ +0.~~890 }}~~

: error map: {{val| 0.000 -0.281 +0.295 -0.837 }}

<!-~~- * [[POTE]]: ~2 = 1200~~.~~000{{c~~}}~~, ~56/45 = 380.355{{c}} -->~~

[[Badness]] (Sintel): 1.82

[[Badness]] (Sintel): 2.75

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

Optimal tunings:

* WE: ~2 = ~~1199~~.~~9709~~{{c}}, ~56/45 = ~~380~~.~~3423~~{{c}}

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

* CWE: ~2 = 1200.0000{{c}}, ~~~56/45 = 380.3517{{c}}~~

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

~~~~

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

Badness (Sintel): 1.04

Badness (Sintel): 1.50

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

~~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}}

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

* CWE: ~2 = 1200.0000{{c}}, ~~~56/45 = 380.3509{{c}}~~

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

~~~~

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

Badness (Sintel): 0.~~884~~

Badness (Sintel): 1.04

== ~~Quadrant~~ ==

== Septant ==

~~The ''quadrant'' temperament~~ ({{nowrap| 12 & ~~224~~ }}) has a period of ~~quarter~~ octave and ~~tempers out the~~ [[~~dimcomp comma~~]]~~, 390625/388962. In this temperament, 25/21~~ is ~~mapped into quarter octave~~.

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, ~~390625~~/~~388962~~

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

: mapping generators: ~25/21, ~3

: mapping generators: ~8575/7776, ~3

[[Optimal tuning]]s:

* [[WE]]: ~2 = ~~300~~.~~0255~~{{c}}, ~3/2 = 701.~~8831~~{{c}}

* [[WE]]: ~8575/7776 = 171.4303{{c}}, ~3/2 = 701.7091{{c}}

: [[error map]]: {{val| +0.~~102 +~~0.~~030 -~~0.~~664~~ +0.~~462~~ }}

: [[error map]]: {{val| +0.012 -0.234 +0.096 +0.265 }}

* [[CWE]]: ~2 = ~~300~~.~~0000~~{{c}}, ~3/2 = 701.~~8180~~{{c}}

* [[CWE]]: ~8575/7776 = 171.4286{{c}}, ~3/2 = 701.7022{{c}}

: error map: {{val| 0.000 -0.~~137 -~~0.~~858~~ +0.~~268~~ }}

: error map: {{val| 0.000 -0.253 +0.069 +0.232 }}

~~~~

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

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

[[Badness]] (Sintel): 2.79

[[Badness]] (Sintel): 2.81

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

Optimal tunings:

* WE: ~25/21 = ~~300~~.~~0244~~{{c}}, ~3/2 = 701.~~8759~~{{c}}

* WE: ~495/448 = 171.4334{{c}}, ~3/2 = 701.7387{{c}}

* CWE: ~25/21 = ~~300~~.~~0000~~{{c}}, ~3/2 = 701.~~8145~~{{c}}

* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7198{{c}}

~~~~

{{Optimal ET sequence|legend=0| 12, ~~…, 212~~, 224, ~~436~~, ~~660~~ }}

Badness (Sintel): 1.51

Badness (Sintel): 1.46

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal tunings:

* WE: ~25/21 = ~~300~~.~~0234~~{{c}}, ~3/2 = 701.~~8707~~{{c}}

* WE: ~495/448 = 171.4282{{c}}, ~3/2 = 701.7229{{c}}

* CWE: ~25/21 = ~~300~~.~~0000~~{{c}}, ~3/2 = 701.~~8123~~{{c}}

* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7242{{c}}

~~~~

{{Optimal ET sequence|legend=0| ~~12f~~, ~~…, 212~~, 224, ~~436~~, ~~660~~ }}

Badness (Sintel): 1.13

Badness (Sintel): 1.02

== ~~Septant~~ ==

== Octant ==

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

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

[[Subgroup]]: 2.3.5.7

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

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

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

: mapping generators: ~42875/39366, ~3

[[Optimal tuning]]s:

* [[WE]]: ~~~8575~~/~~7776~~ = ~~171~~.~~4303~~{{c}}, ~3/2 = 701.~~7091~~{{c}}

* [[WE]]: ~42875/39366 = 150.0048{{c}}, ~3/2 = 701.7356{{c}}

: [[error map]]: {{val| +0.~~012~~ -0.~~234~~ +0.~~096~~ +0.~~265~~ }}

: [[error map]]: {{val| +0.039 -0.181 +0.071 +0.127 }}

* [[CWE]]: ~~~8575~~/~~7776~~ = ~~171~~.~~4286~~{{c}}, ~3/2 = 701.~~7022~~{{c}}

* [[CWE]]: ~42875/39366 = 150.0000{{c}}, ~3/2 = 701.7134{{c}}

: error map: {{val| 0.000 -0.~~253 +~~0.~~069~~ +0.~~232~~ }}

: error map: {{val| 0.000 -0.242 -0.021 +0.022 }}

~~~~

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

[[Badness]] (Sintel): 2.81

[[Badness]] (Sintel): 3.98

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

Optimal tunings:

* WE: ~~~495~~/~~448~~ = ~~171~~.~~4334~~{{c}}, ~3/2 = 701.~~7387~~{{c}}

* WE: ~12/11 = 150.0010{{c}}, ~3/2 = 701.7177{{c}}

* CWE: ~~~495~~/~~448~~ = ~~171~~.~~4286~~{{c}}, ~3/2 = 701.~~7198~~{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7131{{c}}

~~~~

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

Badness (Sintel): 1.46

Badness (Sintel): 1.48

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal tunings:

* WE: ~~~495~~/~~448~~ = ~~171~~.~~4282~~{{c}}, ~3/2 = 701.~~7229~~{{c}}

* WE: ~12/11 = 149.9957{{c}}, ~3/2 = 701.7046{{c}}

* CWE: ~~~495~~/~~448~~ = ~~171~~.~~4286~~{{c}}, ~3/2 = 701.~~7242~~{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7247{{c}}

~~~~

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

Badness (Sintel): 1.02

Badness (Sintel): 1.26

== ~~Octant~~ ==

== Nonant ==

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

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

[[Subgroup]]: 2.3.5.7

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

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

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

: mapping generators: ~2592/2401, ~3

[[Optimal tuning]]s:

* [[WE]]: ~~~42875~~/~~39366~~ = ~~150~~.~~0048~~{{c}}, ~3/2 = 701.~~7356~~{{c}}

* [[WE]]: ~2592/2401 = 133.3442{{c}}, ~3/2 = 701.8000{{c}}

: [[error map]]: {{val| +0.~~039~~ -0.~~181 +~~0.~~071 +~~0.~~127~~ }}

: [[error map]]: {{val| +0.098 -0.057 -0.027 -0.141 }}

* [[CWE]]: ~~~42875~~/~~39366~~ = ~~150~~.~~0000~~{{c}}, ~3/2 = 701.~~7134~~{{c}}

* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.7384{{c}}

: error map: {{val| 0.000 -0.~~242~~ -0.~~021 +~~0.~~022 }}~~

: error map: {{val| 0.000 -0.217 -0.221 -0.421 }}

~~~~

{{Optimal ET sequence|legend=1| 24, …, ~~224~~, ~~472~~, ~~696~~, ~~1168~~ }}

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

[[Badness]] (Sintel): 3.98

[[Badness]] (Sintel): 1.77

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

Optimal tunings:

* WE: ~12/11 = ~~150~~.~~0010~~{{c}}, ~3/2 = 701.~~7177~~{{c}}

* WE: ~242/225 = 133.3308{{c}}, ~3/2 = 701.8205{{c}}

* CWE: ~12/11 = ~~150~~.~~0000~~{{c}}, ~3/2 = 701.~~7131~~{{c}}

* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.8351{{c}}

~~~~

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

Badness (Sintel): 1.48

Badness (Sintel): 4.20

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal tunings:

* WE: ~12/11 = ~~149~~.~~9957~~{{c}}, ~3/2 = 701.~~7046~~{{c}}

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

* CWE: ~12/11 = ~~150~~.~~0000~~{{c}}, ~3/2 = 701.~~7247~~{{c}}

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

~~~~

Badness (Sintel): 1.26

Badness (Sintel): 3.15

== ~~Nonant~~ ==

== Septiquarschis ==

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

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

[[Subgroup]]: 2.3.5.7

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

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

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

: mapping generators: ~2, ~256/147

[[Optimal tuning]]s:

* [[WE]]: ~~~2592/2401~~ = ~~133~~.~~3442~~{{c}}, ~3/2 = ~~701~~.~~8000~~{{c}}

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

: [[error map]]: {{val| +0.~~098~~ -0.~~057~~ -0.~~027 -0~~.~~141~~ }}

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

* [[CWE]]: ~~~2592/2401~~ = ~~133~~.~~3333~~{{c}}, ~3/2 = ~~701~~.~~7384~~{{c}}

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

: error map: {{val| 0.000 -0.~~217 -~~0.~~221 -0~~.~~421~~ }}

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

~~~~

{{Optimal ET sequence|legend=1| 36, ~~99c~~, ~~135~~, ~~171~~, ~~2772bd, 2943bdd, …, 5166bccddd, 5337bccddd~~ }}

[[Badness]] (Sintel): 1.77

[[Badness]] (Sintel): 4.73

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

Optimal tunings:

* WE: ~~~242/225~~ = ~~133~~.~~3308~~{{c}}, ~3/2 = ~~701~~.~~8205~~{{c}}

* WE: ~2 = 1199.9430{{c}}, ~256/147 = 957.3390{{c}}

* CWE: ~~~242/225 = 133.3333{{c}}, ~3/~~2 = ~~701~~.~~8351{{c}}~~

* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3849{{c}}

~~~~

Badness (Sintel): 4.20

Badness (Sintel): 1.72

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal tunings:

* WE: ~~~242/225~~ = ~~133~~.~~3180~~{{c}}, ~3/2 = ~~701~~.~~6956~~{{c}}

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

* CWE: ~~~242/225 = 133.3333{{c}}, ~3/~~2 = ~~701~~.~~7800{{c}}~~

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

~~~~

{{Optimal ET sequence|legend=0| 36, ~~99cf~~, ~~135~~, ~~171~~ }}

Badness (Sintel): 3.15

Badness (Sintel): 1.46

== ~~Tridecafifths~~ ==

== Subgroup extensions ==

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

=== Tridecaschismic (2.3.5.13) ===

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

[[Subgroup]]: 2.3.5.7

Subgroup: 2.3.5.13

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

Comma list: 325/324, 32805/32768

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

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

Optimal tunings:

* [[WE]]: ~2 = 1200.~~1431~~{{c}}, ~~~1323~~/~~1280~~ = 53.~~9838~~{{c}}

* WE: ~2 = 1200.3326{{c}} ~3/2 = 702.1092{{c}}

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

* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9189{{c}}

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

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

~~~~

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

[[Badness]] (Sintel): 10.9

Badness (Sintel): 0.582

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

==== 2.3.5.13.19 subgroup ====

Subgroup: 2.3.5.7.11

Subgroup: 2.3.5.13.19

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

Comma list: 325/324, 361/360, 513/512

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

Optimal tunings:

* WE: ~2 = 1200.~~0311~~{{c}}, ~33/32 = 53.~~9766~~{{c}}

* WE: ~2 = 1200.4236{{c}}, ~3/2 = 702.1510{{c}}

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

* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9064{{c}}

~~~~

Badness (Sintel): 4.23

Badness (Sintel): 0.354

~~== Subgroup extensions ==~~

=== Photia (2.3.5.17) ===

Line 2,881:

Line 2,790:

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

: mapping generators: ~2, ~3

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

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.5471{{c}}, ~3/2 = 701.2262{{c}}

: [[error map]]: {{val| -0.453 -1.182 +0.706 +3.628 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4976{{c}}

: error map: {{val| 0.000 -0.457 +1.705 +5.528 }}

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

[[Badness]] (Sintel): 0.479

==== 2.3.5.17.19 subgroup ====

Line 2,897:

Line 2,810:

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

Optimal tunings:

* WE: ~2 = 1199.7225{{c}}, ~3/2 = 701.3077{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.4754{{c}}

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

Badness (Sintel): 0.332

=== Nestoria (2.3.5.19) ===

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

{{See also| No-elevens subgroup temperaments #Garibaldia | No-elevens subgroup temperaments #Pontia }}

~~The [[S-expression]]~~-~~based comma list~~ of ~~this temperament is {~~[[~~1216~~/~~1215~~|~~S16/S18]], [[361/360|S19]] (, ''[[513/512|S15/S20~~]]~~'')}. Strangely, despite prime 19~~ being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is ~~actually sharper~~ than the fifth in optimal schismic~~. This is likely~~ due to its optimization considering intervals like 19/10 and 19/15.

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

[[Subgroup]]: 2.3.5.19

Line 2,916:

Line 2,831:

[[Comma list]]: 361/360, 513/512

: mapping generators: ~2, ~3

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~000~~{{c}}, ~3/2 = 701.~~746~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.2250{{c}}, ~3/2 = 701.8776{{c}}

: [[error map]]: {{val| +0.225 +0.148 +0.240 -1.796 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7307{{c}}

: error map: {{val| 0.000 -0.224 -0.159 -2.705 }}

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

[[~~Tp tuning #T2 tuning|RMS error~~]]: 0.~~1763 cents~~

[[Badness]] (Sintel): 0.126

=== Taylor (2.3.5.13) ===

Line 2,934:

Line 2,853:

[[Comma list]]: 676/675, 32805/32768

: mapping generators: ~2, ~26/15

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~000~~{{c}}, ~26/15 = 950.~~855~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.1497{{c}}, ~26/15 = 950.9740{{c}}

: [[error map]]: {{val| +0.150 -0.007 +0.348 -1.094 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 950.8493{{c}}

: error map: {{val| 0.000 -0.256 +0.098 -1.568 }}

{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236, 525f, 761ff }}

[[~~Tp tuning #T2 tuning|RMS error~~]]: 0.~~1485 cents~~

[[Badness]] (Sintel): 0.334

==== Dakota (2.3.5.13.19) ====

Line 2,950:

Line 2,873:

Comma list: 361/360, 513/512, 676/675

~~Subgroup-val mapping:~~ {{~~mapping~~| 1 0 15 14 9 | 0 2 -16 -13 -6 }}

Optimal ~~tuning (CTE)~~: ~2 = 1200.0000{{c}}, ~26/15 = 950.~~8199~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.2611{{c}}, ~26/15 = 951.0703{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8532{{c}}

{{Optimal ET sequence|legend=0| 24, 29, 53, 130, 183, 236h, 289h }}

Badness (~~Smith~~): 0.~~00575~~

Badness (Sintel): 0.262

===== 2.3.5.13.19.37 subgroup =====

Line 2,963:

Line 2,888:

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 ~~tuning (CTE)~~: ~2 = 1200.0000{{c}}, ~26/15 = 950.~~8187~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.2987{{c}}, ~26/15 = 951.1060{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8595{{c}}

{{Optimal ET sequence|legend=0| 24, 29, 53, 183, 236h, 289hl, 631fhhll }}

Badness (~~Smith~~): 0.~~00357~~

Badness (Sintel): 0.223

=== Quintilischis (2.3.5.17) ===

Line 2,978:

Line 2,905:

[[Comma list]]: 32805/32768, 1419857/1417176

: mapping generators: ~2, ~18/17

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~000~~{{c}}, ~18/17 = 99.~~649~~{{c}}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.1370{{c}}, ~18/17 = 99.6602{{c}}

: [[error map]]: {{val| +0.137 +0.018 -0.042 -0.533 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~18/17 = 99.6499{{c}}

: error map: {{val| 0.000 -0.205 -0.317 -1.104 }}

{{Optimal ET sequence|legend=1| 12, …, 253, 265, 277, 289, 566g, 855g }}

[[~~Tp tuning #T2 tuning|RMS error~~]]: 0.~~0719 cents~~

[[Badness]] (Sintel): 1.34

==== 2.3.5.17.19 subgroup ====

Line 2,994:

Line 2,925:

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

~~Subgroup-val mapping:~~ {{~~mapping~~| 1 2 -1 5 4 | 0 -5 40 -11 3 }}

~~Gencom mapping:~~ {{~~mapping~~| 1 2 -1 0 0 0 5 4 | 0 -5 40 0 0 0 -11 3 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~18/17 = 99.~~652~~{{c}}

Optimal tunings:

* WE: ~2 = 1200.0350{{c}}, ~18/17 = 99.6550{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6520{{c}}

~~RMS error~~: 0.~~1636 cents~~

Badness (Sintel): 1.17

== References ==

[[Category:Schismatic family| ]]

[[Category:Temperament families]]

[[Category:~~Schismatic family| ]] <!~~-~~- main article -->~~

[[Category:Catalogs of rank-2 temperaments]]

~~[[Category:Rank~~ 2]]

@@ Line 1: / Line 1: @@
-{{interwiki
-| en = Schismatic family
-| de = Schismatische Temperaturen
-| es =
-| ja =
-}}
 {{Technical data page}}
 The [[5-limit]] parent comma for the '''schismatic''' (or '''schismic''') '''family''' is the [[schisma]] of 32805/32768, which is the amount by which the [[Pythagorean comma]] exceeds the [[syntonic comma]] (81/80), or alternatively put, the difference between a [[5/4|just major third]] and a [[8192/6561|Pythagorean diminished fourth]].
@@ Line 25: / Line 19: @@
 * [[WE]]: ~2 = 1200.0749{{c}}, ~3/2 = 701.7797{{c}}
 : [[error map]]: {{val| +0.075 -0.100 -0.027 }}
-* [[CWE]]: ~2 = 1200.0000{{c]}, ~3/2 = 701.7308{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7308{{c}}
 : error map: {{val| 0.000 -0.224 -0.160 }}
-<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7187{{c}}
-* [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7359{{c}} -->
 [[Tuning ranges]]:
@@ Line 39: / Line 31: @@
 === Overview to extensions ===
-The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at.
+The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at. [[#Garibaldi|Garibaldi]] adds [[garischisma|{{monzo| 25 -14 0 -1 }}]], [[#Grackle|grackle]] adds {{monzo| -44 26 0 1 }}, [[#Pontiac|pontiac]] adds {{monzo| -59 39 0 -1 }}, and [[#Schism|schism]] adds [[64/63|{{monzo| 6 -2 0 -1 }}]]. Those all have a fifth as generator.
-* [[#Garibaldi|Garibaldi]] adds [[garischisma|{{monzo| 25 -14 0 -1 }}]],
-* [[#Grackle|Grackle]] adds {{monzo| -44 26 0 1 }},
-* [[#Schism|Schism]] adds [[64/63|{{monzo| 6 -2 0 -1 }}]],
-* [[#Pontiac|Pontiac]] adds {{monzo| -59 39 0 -1 }}.
-Those all have a fifth as generator.
-* [[#Bischismic|Bischismic]] adds {{monzo| -69 40 0 2 }} and has a fifth generator with a half-octave period.
+[[#Bischismic|Bischismic]] adds {{monzo| -69 40 0 2 }} and has a fifth generator with a half-octave period. [[#Salsa|Salsa]] adds [[parahemif comma|{{monzo| 15 -13 0 2 }}]] and has a hemififth generator. [[#Hemischis|Hemischis]] adds {{monzo| -34 25 0 -2 }} and has a hemitwelfth generator. [[Gamelismic clan #Guiron|Guiron]] adds [[1029/1024|{{monzo| -10 1 0 3 }}]], with an ~8/7 generator, three of which give the fifth. [[#Term|Term]] adds {{monzo| -94 54 0 3 }} with a 1/3-octave period. [[#Squirrel|Squirrel]], [[#Tertiaschis|tertiaschis]], and [[#Countertertiaschis|countertertiaschis]] each has a generator that is 1/3 of the fourth. [[#Quadrant|Quadrant]] adds {{monzo| -119 68 0 4 }} with a 1/4-octave period. [[#Kleischismic|Kleischismic]] adds {{monzo| 49 -38 0 4 }} with a half-octave period and also a bisect generator. [[#Sesquiquartififths|Sesquiquartififths]] adds {{monzo| -35 15 0 4 }} and slices the fifth in four.
-* [[#Hemischis|Hemischis]] adds {{monzo| -34 25 0 -2 }} and has a hemififth generator.
-* [[Gamelismic clan #Guiron|Guiron]] adds [[1029/1024|{{monzo| -10 1 0 3 }}]], with an ~8/7 generator, three of which give the fifth.
+Temperaments involving larger splits include [[#Tsaharuk|tsaharuk]], [[#Quanharuk|quanharuk]], [[#Quintilipyth|quintilipyth]], [[#Quintaschis|quintaschis]], [[#Altinex|altinex]], [[Stearnsmic clan #Pogo|pogo]], [[#Sextilifourths|sextilifourths]], [[#Septant|septant]], [[#Octant|octant]], [[#Nonant|nonant]], [[#Septiquarschis|septiquarschis]], and [[#Tridecafifths|tridecafifths]]. Those split the schismic structure into five to thirteen parts.
-* [[#Term|Term]] adds {{monzo| -94 54 0 3 }} with a 1/3 octave period.
-* [[#Sesquiquartififths|Sesquiquartififths]] adds {{monzo| -35 15 0 4 }} and slices the fifth in four.
 Temperaments discussed elsewhere include:
 * ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
 * ''[[Pogo]]'' (+118098/117649) → [[Stearnsmic clan #Pogo|Stearnsmic clan]]
+Considered below are garibaldi, pontiac, grackle, schism, bischismic, kleischismic, salsa, hemischis, term, altinex, squirrel, tertiaschis, countertertiaschis, quadrant, sesquiquartififths, tsaharuk, quanharuk, quintilipyth, quintaschis, sextilifourths, septant, octant, nonant, septiquarschis, and tridecafifths.
 The schismatic family boasts a variety of remarkable extensions to subgroups in high prime limits. These are listed at the bottom of this page, in [[#Subgroup extensions]].
@@ Line 61: / Line 48: @@
 {{Main| Garibaldi }}
-Garibaldi tempers out the [[garischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double diminished octave (C-Cbb), or down-minor seventh (C-vBb) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}.
+Garibaldi tempers out the [[septischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a very slightly sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}, which also makes it the unique intersection of [[marvel]] and [[aberschismic]], two very notable 7-limit temperament paradigms; however, their errors go in oposite directions as marvel tunes fifths flatter and aberschismic sharper.
+[[147edo]] (from 94+53) is a [[patent val]] tuning with 5 and 7 very close to equally out-of-tune in opposite directions s.t only 7/5 and 10/7 are inconsistent in the 9-odd-limit, so might be considered for (EG) a 41-note subset ([[12L 29s]]).
 [[Subgroup]]: 2.3.5.7
@@ Line 72: / Line 61: @@
 * [[WE]]: ~2 = 1200.1233{{c}}, ~3/2 = 702.1573{{c}}
 : [[error map]]: {{val| +0.123 +0.326 -2.709 +2.328 }}
-* [[CWE]]: ~2 = 1200.0000{{c]}, ~3/2 = 702.0774{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0774{{c}}
 : error map: {{val| 0.000 +0.122 -2.933 +2.090 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.085{{c}} -->
 [[Minimax tuning]]:
@@ Line 93: / Line 81: @@
 === Cassandra ===
-Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup.
+Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup, even though it comes with a much higher complexity. It finds 11/8 as two commas above 4/3 and 13/8 a minor third above 11/8, or equivalently two commas above 128/81.
 Subgroup: 2.3.5.7.11
@@ Line 99: / Line 87: @@
 Comma list: 225/224, 385/384, 2200/2187
-Mapping: {{mapping| 1 0 15 25 -33 | 0 1 -8 -14 23 }}
+{{Mapping|legend=0| 1 0 15 25 -33 | 0 1 -8 -14 23 }}
 Optimal tunings:
 * WE: ~2 = 1200.3089{{c}}, ~3/2 = 702.3377{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1562{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.157{{c}} -->
 Minimax tuning:
@@ Line 123: / Line 110: @@
 Comma list: 225/224, 275/273, 325/324, 385/384
-Mapping: {{mapping| 1 0 15 25 -33 -28 | 0 1 -8 -14 23 20 }}
+{{Mapping|legend=0| 1 0 15 25 -33 -28 | 0 1 -8 -14 23 20 }}
 Optimal tunings:
 * WE: ~2 = 1200.1703{{c}}, ~3/2 = 702.2122{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1135{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.113{{c}} -->
 Minimax tuning:
@@ Line 148: / Line 134: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 25 -33 -28 -7 | 0 1 -8 -14 23 20 7 }}
 Optimal tunings:
 * WE: ~2 = 1199.8140{{c}}, ~3/2 = 701.9833{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0909{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.092{{c}} -->
 {{Optimal ET sequence|legend=0| 12e, 41, 53, 94g }}
@@ Line 164: / Line 149: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 25 -33 -28 -7 9 | 0 1 -8 -14 23 20 7 -3 }}
 Optimal tunings:
 * WE: ~2 = 1199.9556{{c}}, ~3/2 = 702.0530{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0787{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.079{{c}} -->
 {{Optimal ET sequence|legend=0| 12e, 41, 53 }}
@@ Line 180: / Line 164: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 25 -33 -28 77 | 0 1 -8 -14 23 20 -46 }}
 Optimal tunings:
 * WE: ~2 = 1200.0046{{c}}, ~3/2 = 702.2167{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0962{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.097{{c}} -->
 {{Optimal ET sequence|legend=0| 41g, 53, 94 }}
@@ Line 196: / Line 179: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 25 -33 -28 77 9 | 0 1 -8 -14 23 20 -46 -3 }}
 Optimal tunings:
 * WE: ~2 = 1200.2910{{c}}, ~3/2 = 702.2681{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0967{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.0978{{c}} -->
 {{Optimal ET sequence|legend=1| 41g, 53, 94 }}
@@ Line 212: / Line 194: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 25 -33 -28 77 9 60 | 0 1 -8 -14 23 20 -46 -3 -35 }}
 Optimal tunings:
 * WE: ~2 = 1200.2970{{c}}, ~3/2 = 702.2697{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0943{{c}}
-<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0960{{c}} -->
 {{Optimal ET sequence|legend=0| 41g, 53, 94 }}
@@ Line 228: / Line 209: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 25 -33 -28 -72 | 0 1 -8 -14 23 20 48 }}
 Optimal tunings:
 * WE: ~2 = 1200.1986{{c}}, ~3/2 = 702.2598{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1455{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.144{{c}} -->
 {{Optimal ET sequence|legend=0| 41, 53g, 94 }}
@@ Line 244: / Line 224: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 25 -33 -28 -72 9 | 0 1 -8 -14 23 20 48 -3 }}
 Optimal tunings:
 * WE: ~2 = 1200.3057{{c}}, ~3/2 = 702.3138{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1373{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.135{{c}} -->
 {{Optimal ET sequence|legend=0| 41, 53g, 94 }}
@@ Line 256: / Line 235: @@
 === Andromeda ===
+Andromeda is the second-best extension to garibaldi, which is less complex but also more innacurate. 11/8 is two commas down from [[729/512]], as is 13/8 two commas down from [[27/16]].
 Subgroup: 2.3.5.7.11
 Comma list: 100/99, 225/224, 245/242
-Mapping: {{mapping| 1 0 15 25 32 | 0 1 -8 -14 -18 }}
+{{Mapping|legend=0| 1 0 15 25 32 | 0 1 -8 -14 -18 }}
 Optimal tunings:
 * WE: ~2 = 1200.1917{{c}}, ~3/2 = 702.4836{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3599{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.321{{c}} -->
 Minimax tuning:
@@ Line 284: / Line 264: @@
 Comma list: 100/99, 105/104, 196/195, 245/242
-Mapping: {{mapping| 1 0 15 25 32 37 | 0 1 -8 -14 -18 -21 }}
+{{Mapping|legend=0| 1 0 15 25 32 37 | 0 1 -8 -14 -18 -21 }}
 Optimal tunings:
 * WE: ~2 = 1200.3031{{c}}, ~3/2 = 702.7368{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.5420{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.559{{c}} -->
 Minimax tuning:
@@ Line 309: / Line 288: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 25 32 37 -7 | 0 1 -8 -14 -18 -21 7 }}
 Optimal tunings:
 * WE: ~2 = 1199.1984{{c}}, ~3/2 = 701.8424{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3384{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.312{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 29, 41 }}
@@ Line 325: / Line 303: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 25 32 37 -7 9 | 0 1 -8 -14 -18 -21 7 -3 }}
 Optimal tunings:
 * WE: ~2 = 1199.5242{{c}}, ~3/2 = 702.0783{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3711{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.357{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 29, 41 }}
@@ Line 341: / Line 318: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 25 32 37 58 | 0 1 -8 -14 -18 -21 -34 }}
 Optimal tunings:
 * WE: ~2 = 1200.6122{{c}}, ~3/2 = 703.0830{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6968{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.725{{c}} -->
 {{Optimal ET sequence|legend=0| 12fg, 29g, 41, 70cd }}
@@ Line 357: / Line 333: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 25 32 37 58 9 | 0 1 -8 -14 -18 -21 -34 -3 }}
 Optimal tunings:
 * WE: ~2 = 1200.7981{{c}}, ~3/2 = 703.2199{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.7221{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.753{{c}} -->
 {{Optimal ET sequence|legend=0| 12fg, 29g, 41, 70cd }}
@@ Line 373: / Line 348: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 25 32 37 12 | 0 1 -8 -14 -18 -21 -5 }}
 Optimal tunings:
 * WE: ~2 = 1201.3146{{c}}, ~3/2 = 703.4864{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6491{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.717{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 29g, 41g }}
@@ Line 389: / Line 363: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 25 32 37 12 9 | 0 1 -8 -14 -18 -21 -5 -3 }}
 Optimal tunings:
 * WE: ~2 = 1201.3140{{c}}, ~3/2 = 703.4860{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6578{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 702.716{{c}} -->
 {{Optimal ET sequence|legend=1| 12f, 29g, 41g }}
@@ Line 405: / Line 378: @@
 Comma list: 99/98, 176/175, 3125/3087
-Mapping: {{mapping| 1 0 15 25 51 | 0 1 -8 -14 -30 }}
+{{Mapping|legend=0| 1 0 15 25 51 | 0 1 -8 -14 -30 }}
 Optimal tunings:
 * WE: ~2 = 1199.7097{{c}}, ~3/2 = 701.5554{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7370{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.725{{c}} -->
 Minimax tuning:
@@ Line 425: / Line 397: @@
 Comma list: 99/98, 176/175, 275/273, 847/845
-Mapping: {{mapping| 1 0 15 25 51 56 | 0 1 -8 -14 -30 -33 }}
+{{Mapping|legend=0| 1 0 15 25 51 56 | 0 1 -8 -14 -30 -33 }}
 Optimal tunings:
 * WE: ~2 = 1199.7370{{c}}, ~3/2 = 701.5937{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7570{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.747{{c}} -->
 Minimax tuning:
@@ Line 445: / Line 416: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 25 51 56 -7 | 0 1 -8 -14 -30 -33 7 }}
 Optimal tunings:
 * WE: ~2 = 1199.2895{{c}}, ~3/2 = 701.2643{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6967{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.680{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 53, 65d, 118dg }}
@@ Line 461: / Line 431: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 25 51 56 -7 9 | 0 1 -8 -14 -30 -33 7 -3 }}
 Optimal tunings:
 * WE: ~2 = 1199.5280{{c}}, ~3/2 = 701.4290{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7149{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.705{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 53, 65d }}
 Badness (Sintel): 1.18
-=== Hemigari ===
-Subgroup: 2.3.5.7.11
-Comma list: 121/120, 225/224, 3125/3087
-Mapping: {{mapping| 1 0 15 25 9 | 0 2 -16 -28 -7 }}
-: mapping generators: ~2, ~110/63
-Optimal tunings:
-* WE: ~2 = 1200.7303{{c}}, ~110/63 = 951.6605{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~110/63 = 951.0604{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~110/63 = 951.082{{c}} -->
-{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135ee }}
-Badness (Sintel): 1.68
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 121/120, 169/168, 225/224, 275/273
-Mapping: {{mapping| 1 0 15 25 9 14 | 0 2 -16 -28 -7 -13 }}
-Optimal tunings:
-* WE: ~2 = 1200.8146{{c}}, ~26/15 = 951.7273{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.0574{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~26/15 = 951.082{{c}} -->
-{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135eef }}
-Badness (Sintel): 1.13
 === Karadeniz ===
@@ Line 512: / Line 448: @@
 Comma list: 225/224, 243/242, 3125/3087
-Mapping: {{mapping| 1 1 7 11 2 | 0 2 -16 -28 5 }}
+{{Mapping|legend=0| 1 1 7 11 2 | 0 2 -16 -28 5 }}
 : mapping generators: ~2, ~11/9
@@ Line 518: / Line 454: @@
 * WE: ~2 = 1199.7351{{c}}, ~11/9 = 350.9167{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.9995{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/9 = 350.994{{c}} -->
 {{Optimal ET sequence|legend=0| 24d, 41, 65d, 106, 147 }}
@@ Line 529: / Line 464: @@
 Comma list: 225/224, 243/242, 325/324, 640/637
-Mapping: {{mapping| 1 1 7 11 2 -8 | 0 2 -16 -28 5 40 }}
+{{Mapping|legend=0| 1 1 7 11 2 -8 | 0 2 -16 -28 5 40 }}
 Optimal tunings:
 * WE: ~2 = 1199.3042{{c}}, ~11/9 = 350.7533{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.9686{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/9 = 351.014{{c}} -->
 {{Optimal ET sequence|legend=0| 24d, 41, 65d, 106f }}
 Badness (Sintel): 1.34
+=== Hemigari ===
+Subgroup: 2.3.5.7.11
+Comma list: 121/120, 225/224, 3125/3087
+{{Mapping|legend=0| 1 0 15 25 9 | 0 2 -16 -28 -7 }}
+: mapping generators: ~2, ~110/63
+Optimal tunings:
+* WE: ~2 = 1200.7303{{c}}, ~110/63 = 951.6605{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~110/63 = 951.0604{{c}}
+{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135ee }}
+Badness (Sintel): 1.68
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 121/120, 169/168, 225/224, 275/273
+{{Mapping|legend=0| 1 0 15 25 9 14 | 0 2 -16 -28 -7 -13 }}
+Optimal tunings:
+* WE: ~2 = 1200.8146{{c}}, ~26/15 = 951.7273{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.0574{{c}}
+{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135eef }}
+Badness (Sintel): 1.13
 === Sanjaab ===
@@ Line 545: / Line 510: @@
 Comma list: 225/224, 1331/1323, 3125/3087
-Mapping: {{mapping| 1 2 -1 -3 0 | 0 -3 24 42 25 }}
+{{Mapping|legend=0| 1 2 -1 -3 0 | 0 -3 24 42 25 }}
 : mapping generators: ~2, ~11/10
@@ Line 551: / Line 516: @@
 * WE: ~2 = 1200.1997{{c}}, ~11/10 = 166.0018{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9786{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/10 = 165.974{{c}} -->
 {{Optimal ET sequence|legend=0| 29, 65d, 94 }}
@@ Line 562: / Line 526: @@
 Comma list: 225/224, 275/273, 847/845, 1331/1323
-Mapping: {{mapping| 1 2 -1 -3 0 -1 | 0 -3 24 42 25 34 }}
+{{Mapping|legend=0| 1 2 -1 -3 0 -1 | 0 -3 24 42 25 34 }}
 Optimal tunings:
 * WE: ~2 = 1200.1224{{c}}, ~11/10 = 165.9800{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9659{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/10 = 165.963{{c}} -->
 {{Optimal ET sequence|legend=0| 29, 65d, 94 }}
 Badness (Sintel): 1.40
-== Schism ==
-See [[Archytas clan #Schism]].
-Schism is a relatively low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (C-Bb). 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.
 == Pontiac ==
 {{Main| Pontiac }}
-Pontiac tempers out the [[ragisma]], rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple augmented third (C-Exx#), or triple-up major sixth (C-^<sup>3</sup>A).
+Pontiac tempers out the [[ragisma]], rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple-augmented third (C-E𝄪𝄪♯), or triple-up major sixth (C-^<sup>3</sup>A).
 [[Subgroup]]: 2.3.5.7
@@ Line 592: / Line 550: @@
 * [[WE]]: ~2 = 1200.0989{{c}}, ~3/2 = 701.8145{{c}}
 : [[error map]]: {{val| +0.099 -0.042 -0.138 -0.038 }}
-* [[CWE]]: ~2 = 1200.0000{{c]}, ~3/2 = 701.7579{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7579{{c}}
 : error map: {{val| 0.000 -0.197 -0.377 -0.268 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.757{{c}} -->
 [[Minimax tuning]]:
@@ Line 613: / Line 570: @@
 === Helenoid ===
-The helenoid temperament ({{nowrap| 53 & 118 }}) is closely related to the helenus temperament, but with the ragisma rather than the [[225/224|marvel comma]] tempered out.
+Helenoid may be described as {{nowrap| 53 & 118 }}, and is closely related to the helenus temperament, differing only by the mapping of 7.
 Subgroup: 2.3.5.7.11
@@ Line 619: / Line 576: @@
 Comma list: 385/384, 3388/3375, 4375/4374
-Mapping: {{mapping| 1 0 15 -59 51 | 0 1 -8 39 -30 }}
+{{Mapping|legend=0| 1 0 15 -59 51 | 0 1 -8 39 -30 }}
 Optimal tunings:
 * WE: ~2 = 1200.3277{{c}}, ~3/2 = 701.9135{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7223{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.722{{c}} -->
 Minimax tuning:
@@ Line 639: / Line 595: @@
 Comma list: 352/351, 385/384, 625/624, 729/728
-Mapping: {{mapping| 1 0 15 -59 51 56 | 0 1 -8 39 -30 -33 }}
+{{Mapping|legend=0| 1 0 15 -59 51 56 | 0 1 -8 39 -30 -33 }}
 Optimal tunings:
 * WE: ~2 = 1200.1780{{c}}, ~3/2 = 701.8491{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7446{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.745{{c}} -->
 Minimax tuning:
@@ Line 659: / Line 614: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 -59 51 56 -91 | 0 1 -8 39 -30 -33 60 }}
 Optimal tunings:
 * WE: ~2 = 1200.1645{{c}}, ~3/2 = 701.8385{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7425{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.742{{c}} -->
 Minimax tuning:
@@ Line 679: / Line 633: @@
 Comma list: 169/168, 325/324, 385/384, 3146/3125
-Mapping: {{mapping| 1 0 15 -59 51 -28 | 0 1 -8 39 -30 20 }}
+{{Mapping|legend=0| 1 0 15 -59 51 -28 | 0 1 -8 39 -30 20 }}
 Optimal tunings:
 * WE: ~2 = 1200.5227{{c}}, ~3/2 = 702.0456{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7418{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.740{{c}} -->
 {{Optimal ET sequence|legend=0| 53, 118f, 171ef }}
@@ Line 695: / Line 648: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 -59 51 -28 -91 | 0 1 -8 39 -30 20 60 }}
 Optimal tunings:
 * WE: ~2 = 1200.4988{{c}}, ~3/2 = 702.0218{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7332{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.730{{c}} -->
 {{Optimal ET sequence|legend=0| 53, 118f, 171ef }}
@@ Line 711: / Line 663: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 -59 51 -28 -91 9 | 0 1 -8 39 -30 20 60 -3 }}
 Optimal tunings:
 * WE: ~2 = 1200.5185{{c}}, ~3/2 = 702.0323{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7318{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.729{{c}} -->
 {{Optimal ET sequence|legend=0| 53, 118f, 171ef }}
@@ Line 723: / Line 674: @@
 === Ponta ===
-The ponta temperament ({{nowrap| 53 & 171 }}) tempers out the [[540/539|swetisma]] and the ragisma.
+Ponta tempers out [[540/539]] and may be described as {{nowrap| 171 & 224 }}. [[224edo]] itself makes for an excellent tuning.
 Subgroup: 2.3.5.7.11
@@ Line 729: / Line 680: @@
 Comma list: 540/539, 4375/4374, 32805/32768
-Mapping: {{mapping| 1 0 15 -59 135 | 0 1 -8 39 -83 }}
+{{Mapping|legend=0| 1 0 15 -59 135 | 0 1 -8 39 -83 }}
 Optimal tunings:
 * WE: ~2 = 1199.9814{{c}}, ~3/2 = 701.7725{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7834{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.783{{c}} -->
 Minimax tuning:
@@ Line 749: / Line 699: @@
 Comma list: 540/539, 625/624, 729/728, 2200/2197
-Mapping: {{mapping| 1 0 15 -59 135 56 | 0 1 -8 39 -83 -33 }}
+{{Mapping|legend=0| 1 0 15 -59 135 56 | 0 1 -8 39 -83 -33 }}
 Optimal tunings:
 * WE: ~2 = 1199.9601{{c}}, ~3/2 = 701.7610{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7845{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.784{{c}} -->
 Minimax tuning:
@@ Line 769: / Line 718: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 -59 135 56 -91 | 0 1 -8 39 -83 -33 60 }}
 Optimal tunings:
 * WE: ~2 = 1199.8850{{c}}, ~3/2 = 701.7101{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7775{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.777{{c}} -->
 Minimax tuning:
@@ Line 785: / Line 733: @@
 === Pontic ===
-The pontic temperament ({{nowrap| 118 & 171 }}) tempers out the [[441/440|werckisma]] and the ragisma.
+Pontic temperament tempers out [[441/440]] and may be described as {{nowrap| 118 & 171 }}. [[289edo]] may be recommended as a tuning.
 Subgroup: 2.3.5.7.11
@@ Line 791: / Line 739: @@
 Comma list: 441/440, 4375/4374, 32805/32768
-Mapping: {{mapping| 1 0 15 -59 -136 | 0 1 -8 39 88 }}
+{{Mapping|legend=0| 1 0 15 -59 -136 | 0 1 -8 39 88 }}
 Optimal tunings:
 * WE: ~2 = 1200.1259{{c}}, ~3/2 = 701.7980{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7256{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.724{{c}} -->
 Minimax tuning:
@@ Line 811: / Line 758: @@
 Comma list: 441/440, 625/624, 729/728, 3584/3575
-Mapping: {{mapping| 1 0 15 -59 -136 56 | 0 1 -8 39 88 -33 }}
+{{Mapping|legend=0| 1 0 15 -59 -136 56 | 0 1 -8 39 88 -33 }}
 Optimal tunings:
 * WE: ~2 = 1199.9254{{c}}, ~3/2 = 701.6945{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7378{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.738{{c}} -->
 Minimax tuning:
@@ Line 831: / Line 777: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 -59 -136 56 -91 | 0 1 -8 39 88 -33 60 }}
 Optimal tunings:
 * WE: ~2 = 1199.9454{{c}}, ~3/2 = 701.7085{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7401{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.740{{c}} -->
 Minimax tuning:
@@ Line 851: / Line 796: @@
 Comma list: 364/363, 441/440, 4375/4374, 32805/32768
-Mapping: {{mapping| 1 0 15 -59 -136 -215 | 0 1 -8 39 88 138 }}
+{{Mapping|legend=0| 1 0 15 -59 -136 -215 | 0 1 -8 39 88 138 }}
 Optimal tunings:
 * WE: ~2 = 1200.0897{{c}}, ~3/2 = 701.7874{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7356{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.735{{c}} -->
 {{Optimal ET sequence|legend=0| 53ef, 118f, 171, 289 }}
@@ Line 867: / Line 811: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 -59 -136 -215 -91 | 0 1 -8 39 88 138 60 }}
 Optimal tunings:
 * WE: ~2 = 1200.1045{{c}}, ~3/2 = 701.7962{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7359{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.735{{c}} -->
 {{Optimal ET sequence|legend=0| 53ef, 118f, 171, 289, 460e, 749defg }}
@@ Line 879: / Line 822: @@
 === Bipont ===
-The bipont temperament ({{nowrap| 118 & 224 }}) has a period of half octave and tempers out the [[3025/3024|lehmerisma (3025/3024)]] and the [[9801/9800|kalisma (9801/9800)]].
+Bipont tempers out the [[3025/3024|lehmerisma (3025/3024)]] and the [[9801/9800|kalisma (9801/9800)]]. It may be described as {{nowrap| 118 & 224 }}. It has a period of half octave and a ploidacot signature of diploid monocot. [[342edo]] may be recommended as a tuning.
 Subgroup: 2.3.5.7.11
@@ Line 885: / Line 828: @@
 Comma list: 3025/3024, 4375/4374, 32805/32768
-Mapping: {{mapping| 2 0 30 -118 -85 | 0 1 -8 39 29 }}
+{{Mapping|legend=0| 2 0 30 -118 -85 | 0 1 -8 39 29 }}
 : mapping generators: ~99/70, ~3
@@ Line 891: / Line 834: @@
 * WE: ~99/70 = 600.0500{{c}}, ~3/2 = 701.8153{{c}}
 * CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7584{{c}}
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~3/2 = 701.757{{c}} -->
 {{Optimal ET sequence|legend=0| 106, 118, 224, 342, 1592c, 1934ce, 2276cde, 2618cde, 2960cde }}
@@ Line 902: / Line 844: @@
 Comma list: 625/624, 729/728, 1575/1573, 4096/4095
-Mapping: {{mapping| 2 0 30 -118 -85 112 | 0 1 -8 39 29 -33 }}
+{{Mapping|legend=0| 2 0 30 -118 -85 112 | 0 1 -8 39 29 -33 }}
 Optimal tunings:
 * WE: ~99/70 = 599.9939{{c}}, ~3/2 = 701.7657{{c}}
 * CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7728{{c}}
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~3/2 = 701.773{{c}} -->
 {{Optimal ET sequence|legend=0| 106, 118, 224, 566f, 790f }}
@@ Line 918: / Line 859: @@
 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 }}
+{{Mapping|legend=0| 2 0 30 -118 -85 112 -182 | 0 1 -8 39 29 -33 60 }}
 Optimal tunings:
 * WE: ~99/70 = 599.9839{{c}}, ~3/2 = 701.7463{{c}}
 * CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7649{{c}}
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~3/2 = 701.765{{c}} -->
 {{Optimal ET sequence|legend=0| 106g, 118, 224, 342, 566f }}
@@ Line 934: / Line 874: @@
 Comma list: 1716/1715, 2080/2079, 3025/3024, 32805/32768
-Mapping: {{mapping| 2 0 30 -118 -85 -243 | 0 1 -8 39 29 79 }}
+{{Mapping|legend=0| 2 0 30 -118 -85 -243 | 0 1 -8 39 29 79 }}
 Optimal tunings:
 * WE: ~99/70 = 600.0405{{c}}, ~3/2 = 701.8160{{c}}
 * CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7697{{c}}
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~3/2 = 701.769{{c}} -->
 {{Optimal ET sequence|legend=0| 106f, 118f, 224, 342f, 566, 1356cf }}
@@ Line 950: / Line 889: @@
 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 }}
+{{Mapping|legend=0| 2 0 30 -118 -85 -243 -182 | 0 1 -8 39 29 79 60 }}
 Optimal tunings:
 * WE: ~99/70 = 600.0336{{c}}, ~3/2 = 701.8031{{c}}
 * CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7647{{c}}
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~3/2 = 701.764{{c}} -->
 {{Optimal ET sequence|legend=0| 106fg, 118f, 224, 342f, 566 }}
@@ Line 966: / Line 904: @@
 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 }}
+{{Mapping|legend=0| 2 0 30 -118 -85 -243 -182 -169 | 0 1 -8 39 29 79 60 56 }}
 Optimal tunings:
 * WE: ~99/70 = 600.0243{{c}}, ~3/2 = 701.7891{{c}}
 * CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7613{{c}}
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~3/2 = 701.761{{c}} -->
 {{Optimal ET sequence|legend=0| 106fgh, 118f, 224, 342f, 566h, 908fgh }}
@@ Line 982: / Line 919: @@
 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|legend=0| 4 0 60 -236 -170 -131 | 0 1 -8 39 29 23 }}
 : mapping generators: ~208/175, ~3
@@ Line 988: / Line 925: @@
 * WE: ~208/175 = 300.0229{{c}}, ~3/2 = 701.8097{{c}}
 * CWE: ~208/175 = 300.0000{{c}}, ~3/2 = 701.7578{{c}}
-<!-- * POTE: ~208/175 = 300.000{{c}}, ~3/2 = 701.756{{c}} -->
 {{Optimal ET sequence|legend=0| 224, 460, 684, 2276cde, 2960cde }}
@@ Line 995: / Line 931: @@
 == Grackle ==
-Grackle tempers out {{monzo| -44 26 0 1 }}. The 7/4 is found at -26 fifths, represented by the triple diminished ninth (C-Dbbbb), or double-down minor seventh (C-vvBb), which is to say, two comma steps are required to bend the Pythagorean minor seventh to the septimal one.
+Grackle tempers out {{monzo| -44 26 0 1 }} so 7/4 is found at -26 fifths, represented by the triple-diminished ninth (C–D𝄫𝄫) or double-down minor seventh (C–vvB♭). Two comma steps are required to bend the Pythagorean minor seventh to the septimal one.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,007: / Line 943: @@
 * [[WE]]: ~2 = 1199.7974{{c}}, ~3/2 = 701.1210{{c}}
 : [[error map]]: {{val| -0.203 -1.037 +3.300 -1.618 }}
-* [[CWE]]: ~2 = 1200.0000{{c]}, ~3/2 = 701.2465{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.2465{{c}}
 : error map: {{val| 0.000 -0.709 +3.715 -1.234 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.239{{c}} -->
 [[Minimax tuning]]:
@@ Line 1,024: / Line 959: @@
 Comma list: 126/125, 176/175, 32805/32768
-Mapping: {{mapping| 1 0 15 44 70 | 0 1 -8 -26 -42 }}
+{{Mapping|legend=0| 1 0 15 44 70 | 0 1 -8 -26 -42 }}
 Optimal tunings:
 * WE: ~2 = 1199.7077{{c}}, ~3/2 = 701.0017{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.1804{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.172{{c}} -->
 {{Optimal ET sequence|legend=0| 12, 65e, 77, 89, 166c }}
@@ Line 1,040: / Line 974: @@
 Comma list: 126/125, 176/175, 196/195, 5445/5408
-Mapping: {{mapping| 1 0 15 44 70 75 | 0 1 -8 -26 -42 -45 }}
+{{Mapping|legend=0| 1 0 15 44 70 75 | 0 1 -8 -26 -42 -45 }}
 Optimal tunings:
 * WE: ~2 = 1199.7782{{c}}, ~3/2 = 701.0966{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2319{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.226{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 65ef, 77, 166cf }}
@@ Line 1,056: / Line 989: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 44 70 75 -7 | 0 1 -8 -26 -42 -45 7 }}
 Optimal tunings:
 * WE: ~2 = 1199.5839{{c}}, ~3/2 = 700.9632{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2137{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.206{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 77, 89f, 166cf }}
@@ Line 1,072: / Line 1,004: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 44 70 75 -7 9 | 0 1 -8 -26 -42 -45 7 -3 }}
 Optimal tunings:
 * WE: ~2 = 1199.7146{{c}}, ~3/2 = 701.0500{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2212{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.217{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 77, 166cf }}
@@ Line 1,088: / Line 1,019: @@
 Comma list: 126/125, 176/175, 729/728, 1287/1280
-Mapping: {{mapping| 1 0 15 44 70 -47 | 0 1 -8 -26 -42 32 }}
+{{Mapping|legend=0| 1 0 15 44 70 -47 | 0 1 -8 -26 -42 32 }}
 Optimal tunings:
 * WE: ~2 = 1200.0060{{c}}, ~3/2 = 701.2202{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2167{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.217{{c}} -->
 {{Optimal ET sequence|legend=0| 12, 77, 166c }}
@@ Line 1,104: / Line 1,034: @@
 Comma list: 126/125, 245/242, 896/891
-Mapping: {{mapping| 1 0 15 44 51 | 0 1 -8 -26 -30 }}
+{{Mapping|legend=0| 1 0 15 44 51 | 0 1 -8 -26 -30 }}
 Optimal tunings:
 * WE: ~2 = 1199.8388{{c}}, ~3/2 = 701.3071{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.4068{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.401{{c}} -->
 {{Optimal ET sequence|legend=0| 12, 53d, 65, 77e }}
@@ Line 1,120: / Line 1,049: @@
 Comma list: 126/125, 196/195, 245/242, 832/825
-Mapping: {{mapping| 1 0 15 44 51 75 | 0 1 -8 -26 -30 -45 }}
+{{Mapping|legend=0| 1 0 15 44 51 75 | 0 1 -8 -26 -30 -45 }}
 Optimal tunings:
 * WE: ~2 = 1199.7329{{c}}, ~3/2 = 701.1918{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.3555{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.348{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 53dff, 65f, 77e }}
@@ Line 1,136: / Line 1,064: @@
 Comma list: 105/104, 126/125, 245/242, 352/351
-Mapping: {{mapping| 1 0 15 44 51 56 | 0 1 -8 -26 -30 -33 }}
+{{Mapping|legend=0| 1 0 15 44 51 56 | 0 1 -8 -26 -30 -33 }}
 Optimal tunings:
 * WE: ~2 = 1199.8928{{c}}, ~3/2 = 701.4664{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.5327{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~3/2 = 701.529{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, …, 53d, 65 }}
@@ Line 1,147: / Line 1,074: @@
 Badness (Sintel): 2.01
-== Bischismic ==
+== Quasipyth ==
+Named by [[Xenllium]] in 2026, quasipyth tempers out {{monzo| 109 -67 0 -1 }}, the [[nanisma]], as well as the [[catasyc comma]], 390625/387072. The 7/4 is found at −67 fifths, represented by the nonuple-diminished thirteenth.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 32805/32768, 390625/387072
+{{Mapping|legend=1| 1 0 15 109 | 0 1 -8 -67 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.2569{{c}}, ~3/2 = 702.1149{{c}}
+: [[error map]]: {{val| +0.2569 +0.4168 -1.4342 +0.2685 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9615{{c}}
+: error map: {{val| 0.0000 +0.0065 -2.0054 -0.2437 }}
+{{Optimal ET sequence|legend=1| 53, 147d, 200, 253, 306c, 559c }}
+[[Badness]] (Sintel): 5.04
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 385/384, 19712/19683, 78125/77616
+{{Mapping|legend=0| 1 0 15 109 -117 | 0 1 -8 -67 76 }}
+Optimal tunings:
+* WE: ~2 = 1200.3283{{c}}, ~3/2 = 702.1636{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9713{{c}}
+{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}
+Badness (Sintel): 3.83
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 325/324, 385/384, 2200/2197, 19712/19683
+{{Mapping|legend=0| 1 0 15 109 -117 -28 | 0 1 -8 -67 76 20 }}
+Optimal tunings:
+* WE: ~2 = 1200.3229{{c}}, ~3/2 = 702.1603{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9714{{c}}
+{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}
+Badness (Sintel): 2.13
+== Schism ==
+See [[Archytas clan #Schism]].
+Schism is a relatively low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (C–B♭). 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.
+== Bischismic ==
+Bischismic tempers out 3136/3125, the [[hemimean comma]], as well as 321489/320000, the [[varunisma]], and may be described as the {{nowrap| 118 & 130 }} temperament. The octave is split in halves, so the [[ploidacot]] of this temperament is diploid monocot. In schismic, -10 fifths make the interval class of 10/9. Bischismic then finds [[7/4]] by a stack of two [[10/9]]'s plus a semi-octave period, and in the [[11-limit]], it simply finds [[11/8]] by a stack of three [[10/9]]'s. [[248edo]] and [[378edo]] make for excellent tunings in both cases.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,160: / Line 1,143: @@
 * [[CWE]]: ~567/400 = 600.0000{{c}}, ~3/2 = 701.5915{{c}}
 : error map: {{val| 0.000 -0.364 +0.954 -0.656 }}
-<!-- * [[CTE]]: ~567/400 = 600.0000{{c}}, ~3/2 = 701.5899{{c}} -->
 [[Minimax tuning]]:
@@ Line 1,175: / Line 1,157: @@
 Comma list: 441/440, 3136/3125, 8019/8000
-Mapping: {{mapping| 2 0 30 69 102 | 0 1 -8 -20 -30 }}
+{{Mapping|legend=0| 2 0 30 69 102 | 0 1 -8 -20 -30 }}
 Optimal tunings:
 * WE: ~99/70 = 600.0165{{c}}, ~3/2 = 701.6316{{c}}
 * CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.6110{{c}}
-<!-- * CTE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.6077{{c}} -->
 {{Optimal ET sequence|legend=0| 12, …, 106de, 118, 130, 248 }}
@@ Line 1,191: / Line 1,172: @@
 Comma list: 441/440, 729/728, 1001/1000, 3136/3125
-Mapping: {{mapping| 2 0 30 69 102 -75 | 0 1 -8 -20 -30 26 }}
+{{Mapping|legend=0| 2 0 30 69 102 -75 | 0 1 -8 -20 -30 26 }}
 Optimal tunings:
 * WE: ~99/70 = 599.9610{{c}}, ~3/2 = 701.5445{{c}}
 * CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5908{{c}}
-<!-- * CTE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5949{{c}} -->
 {{Optimal ET sequence|legend=0| 12, 118, 130, 248, 378 }}
@@ Line 1,207: / Line 1,187: @@
 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 }}
+{{Mapping|legend=0| 2 0 30 69 102 -75 5 | 0 1 -8 -20 -30 26 1 }}
 Optimal tunings:
 * WE: ~99/70 = 600.0331{{c}}, ~3/2 = 701.6387{{c}}
 * CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5994{{c}}
-<!-- * CTE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5959{{c}} -->
 {{Optimal ET sequence|legend=0| 12, 118, 130, 248g }}
@@ Line 1,223: / Line 1,202: @@
 Comma list: 351/350, 364/363, 441/440, 3136/3125
-Mapping: {{mapping| 2 0 30 69 102 131 | 0 1 -8 -20 -30 -39 }}
+{{Mapping|legend=0| 2 0 30 69 102 131 | 0 1 -8 -20 -30 -39 }}
 Optimal tunings:
 * WE: ~55/39 = 599.9766{{c}}, ~3/2 = 701.5380{{c}}
 * CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.5670{{c}}
-<!-- * CTE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.5708{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 106deff, 118f, 130 }}
@@ Line 1,239: / Line 1,217: @@
 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 }}
+{{Mapping|legend=0| 2 0 30 69 102 131 5 | 0 1 -8 -20 -30 -39 1 }}
 Optimal tunings:
 * WE: ~55/39 = 600.0997{{c}}, ~3/2 = 701.7114{{c}}
 * CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.5899{{c}}
-<!-- * CTE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.5717{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 106deff, 118f, 130, 248fg }}
@@ Line 1,251: / Line 1,228: @@
 == 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
@@ Line 1,263: / Line 1,242: @@
 * [[CWE]]: ~1225/864 = 600.0000{{c}}, ~35/24 = 650.9204{{c}} (~36/35 = 50.9204{{c}})
 : error map: {{val| 0.000 -0.114 -1.041 -1.338 }}
-<!-- * [[POTE]]: ~1225/864 = 600.000{{c}}, ~35/24 = 650.920{{c}} (~36/35 = 50.920{{c}}) -->
 {{Optimal ET sequence|legend=1| 24, 94, 118, 212, 330, 542d, 872cdd, 1414ccddd }}
@@ Line 1,274: / Line 1,252: @@
 Comma list: 385/384, 9801/9800, 14641/14580
-Mapping: {{mapping| 2 1 22 -15 8 | 0 2 -16 19 -1 }}
+{{Mapping|legend=0| 2 1 22 -15 8 | 0 2 -16 19 -1 }}
 Optimal tunings:
 * WE: ~99/70 = 600.1645{{c}}, ~35/24 = 651.0963{{c}} (~36/35 = 50.9319{{c}})
 * CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9184{{c}} (~36/35 = 50.9184{{c}})
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~35/24 = 650.918{{c}} (~36/35 = 50.918{{c}}) -->
 {{Optimal ET sequence|legend=0| 24, 94, 118, 212, 330e, 542dee, 872cddeee }}
@@ Line 1,290: / Line 1,267: @@
 Comma list: 352/351, 385/384, 729/728, 1575/1573
-Mapping: {{mapping| 2 1 22 -15 8 15 | 0 2 -16 19 -1 -7 }}
+{{Mapping|legend=0| 2 1 22 -15 8 15 | 0 2 -16 19 -1 -7 }}
 Optimal tunings:
 * WE: ~99/70 = 600.0696{{c}}, ~35/24 = 651.0136{{c}} (~36/35 = 50.9440{{c}})
 * CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9378{{c}} (~36/35 = 50.9378{{c}})
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~35/24 = 650.938{{c}} (~36/35 = 50.938{{c}}) -->
 {{Optimal ET sequence|legend=0| 24, 94, 118, 212f }}
@@ Line 1,306: / Line 1,282: @@
 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 }}
+{{Mapping|legend=0| 2 1 22 -15 8 15 6 | 0 2 -16 19 -1 -7 2 }}
 Optimal tunings:
 * WE: ~99/70 = 600.1134{{c}}, ~35/24 = 651.0646{{c}} (~36/35 = 50.9512{{c}})
 * CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9414{{c}} (~36/35 = 50.9414{{c}})
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~35/24 = 650.942{{c}} (~36/35 = 50.942{{c}}) -->
 {{Optimal ET sequence|legend=0| 24, 94, 118 }}
@@ Line 1,322: / Line 1,297: @@
 Comma list: 325/324, 385/384, 1573/1568, 14641/14580
-Mapping: {{mapping| 2 1 22 -15 8 -36 | 0 2 -16 19 -1 40 }}
+{{Mapping|legend=0| 2 1 22 -15 8 -36 | 0 2 -16 19 -1 40 }}
 Optimal tunings:
 * WE: ~99/70 = 600.1909{{c}}, ~35/24 = 651.1578{{c}} (~36/35 = 50.9670{{c}})
 * CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9541{{c}} (~36/35 = 50.9541{{c}})
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~35/24 = 650.951{{c}} (~36/35 = 50.951{{c}}) -->
 {{Optimal ET sequence|legend=0| 24f, 94, 118f, 212 }}
@@ Line 1,338: / Line 1,312: @@
 Comma list: 289/288, 325/324, 385/384, 442/441, 14641/14580
-Mapping: {{mapping| 2 1 22 -15 8 -36 6 | 0 2 -16 19 -1 40 2 }}
+{{Mapping|legend=0| 2 1 22 -15 8 -36 6 | 0 2 -16 19 -1 40 2 }}
 Optimal tunings:
 * WE: ~99/70 = 600.2190{{c}}, ~35/24 = 651.1578{{c}} (~36/35 = 50.9670{{c}})
 * CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9518{{c}} (~36/35 = 50.9518{{c}})
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~35/24 = 650.948{{c}} (~36/35 = 50.948{{c}}) -->
 {{Optimal ET sequence|legend=0| 24f, 94, 118f, 212g }}
@@ Line 1,350: / Line 1,323: @@
 == 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
@@ Line 1,372: / Line 1,347: @@
 Comma list: 243/242, 245/242, 385/384
-Mapping: {{mapping| 1 1 7 -1 2 | 0 2 -16 13 5 }}
+{{Mapping|legend=0| 1 1 7 -1 2 | 0 2 -16 13 5 }}
 Optimal tunings:
@@ Line 1,387: / Line 1,362: @@
 Comma list: 105/104, 144/143, 243/242, 245/242
-Mapping: {{mapping| 1 1 7 -1 2 4 | 0 2 -16 13 5 -1 }}
+{{Mapping|legend=0| 1 1 7 -1 2 4 | 0 2 -16 13 5 -1 }}
 Optimal tunings:
-* WE: ~2 = 1199.9362{c}}, ~11/9 = 351.0061{{c}}
+* WE: ~2 = 1199.9362{{c}}, ~11/9 = 351.0061{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0247{{c}}
@@ Line 1,398: / Line 1,373: @@
 == 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
@@ Line 1,410: / Line 1,389: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 951.7966{{c}}
 : error map: {{val| 0.000 -0.362 +0.941 +1.088 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~140/81 = 950.797{{c}} -->
 {{Optimal ET sequence|legend=1| 24, 53, 130, 183, 313 }}
@@ Line 1,421: / Line 1,399: @@
 Comma list: 540/539, 5632/5625, 8019/8000
-Mapping: {{mapping| 1 0 15 -17 51 | 0 2 -16 25 -60 }}
+{{Mapping|legend=0| 1 0 15 -17 51 | 0 2 -16 25 -60 }}
 Optimal tunings:
 * WE: ~2 = 1199.8482{{c}}, ~140/81 = 950.6809{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~140/81 = 950.8020{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~140/81 = 950.801{{c}} -->
 {{Optimal ET sequence|legend=0| 53, 130, 183, 313, 809cd }}
@@ Line 1,433: / Line 1,410: @@
 === 13-limit ===
-Its [[S-expression]]-based comma list is {[[540/539|S12/S14]], [[676/675|S13/S15 = S26]], [[729/728|S27]], [[4096/4095|S64]](, [[4225/4224|S65]])}. Tempering out [[169/168|S13]], [[225/224|S15]] or [[625/624|S25]] leads to [[53edo]] (through [[Catakleismic]]) while tempering out [[24192/24167|S12/S13]], [[10985/10976|S13/S14]], [[43904/43875|S14/S15]] or [[2401/2400|S49]] (implying S12 = S13 = S14 = S15) leads to [[130edo]].
 Subgroup: 2.3.5.7.11.13
 Comma list: 351/350, 540/539, 676/675, 4096/4095
-Mapping: {{mapping| 1 0 15 -17 51 14 | 0 2 -16 25 -60 -13 }}
+{{Mapping|legend=0| 1 0 15 -17 51 14 | 0 2 -16 25 -60 -13 }}
 Optimal tunings:
 * WE: ~2 = 1199.9140{{c}}, ~140/81 = 950.7324{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~140/81 = 950.8010{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~26/15 = 950.801{{c}} -->
 {{Optimal ET sequence|legend=0| 53, 130, 183, 313 }}
@@ Line 1,455: / Line 1,429: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 -17 51 14 -49 | 0 2 -16 25 -60 -13 67 }}
 Optimal tunings:
 * WE: ~2 = 1199.9740{{c}}, ~26/15 = 950.7894{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8100{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~26/15 = 950.810{{c}} -->
 {{Optimal ET sequence|legend=0| 53, 130, 183, 496d }}
@@ Line 1,471: / Line 1,444: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 -17 51 14 -49 9 | 0 2 -16 25 -60 -13 67 -6 }}
 Optimal tunings:
 * WE: ~2 = 1200.0464{{c}}, ~26/15 = 950.8459{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8091{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~26/15 = 950.809{{c}} -->
 {{Optimal ET sequence|legend=0| 53, 130, 183, 313h }}
@@ Line 1,487: / Line 1,459: @@
 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 }}
+{{Mapping|legend=0| 1 0 15 -17 51 14 -49 9 -24 | 0 2 -16 25 -60 -13 67 -6 36 }}
 Optimal tunings:
 * WE: ~2 = 1200.0215{{c}}, ~26/15 = 950.8239{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8069{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~26/15 = 950.807{{c}} -->
 {{Optimal ET sequence|legend=0| 53, 130, 183, 313h }}
@@ Line 1,501: / Line 1,472: @@
 * ''HemischisMatic EP'' (2023) by [[User:Francium|Francium]] – [https://open.spotify.com/album/1Fx2shLclpNgFQJRw3ZHya Spotify] | [https://francium223.bandcamp.com/album/hemischismatic-ep Bandcamp] | [https://www.youtube.com/playlist?list=PLLZE7hMjEXRaiipPYK1InZBXTru_UtRsq YouTube] – 4-piece extended play
-== Squirrel ==
+== Term ==
-The squirrel temperament ({{nowrap| 29 & 36 }}) has a ~11/10 generator, three of which give the fourth (~4/3), and thirteen of which give 7/4 with octave reduction.
+Term tempers out the [[landscape comma]], mapping [[63/50]] to the 1/3-octave period. It can be described as {{nowrap| 12 & 171 }}, and is the unique temperament that 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]]: 686/675, 32805/32768
+[[Comma list]]: 32805/32768, 250047/250000
-{{Mapping|legend=1| 1 2 -1 1 | 0 -3 24 13 }}
+{{Mapping|legend=1| 3 0 45 94 | 0 1 -8 -18 }}
+: mapping generators: ~63/50, ~3
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1200.7408{{c}}, ~160/147 = 166.2424{{c}}
+* [[WE]]: ~63/50 = 400.0257{{c}}, ~3/2 = 701.7873{{c}}
-: [[error map]]: {{val| +0.741 +0.799 +2.763 -6.934 }}
+: [[error map]]: {{val| +0.077 -0.091 -0.072 +0.031 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~160/147 = 166.1597{{c}}
+* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 701.7383{{c}}
-: error map: {{val| 0.000 -0.434 +1.518 -8.750 }}
+: error map: {{val| 0.000 -0.217 -0.220 -0.115 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~160/147 = 166.140{{c}} -->
+[[Minimax tuning]]:
+* [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis)]]: 2.5/3
+* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
+{{Optimal ET sequence|legend=1| 12, …, 159, 171, 867, 1038, 1209, 1380, 1551, 1722 }}
-{{Optimal ET sequence|legend=1| 29, 36, 65 }}
+[[Badness]] (Sintel): 0.505
-[[Badness]] (Sintel): 4.42
+=== Terminal ===
+Terminal tempers out 441/440 and 4375/4356, and may be described as {{nowrap| 159 & 171 }}. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave.
-=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 245/242, 686/675, 896/891
+Comma list: 441/440, 4375/4356, 32805/32768
-Mapping: {{mapping| 1 2 -1 1 0 | 0 -3 24 13 25 }}
+{{Mapping|legend=0| 3 0 45 94 134 | 0 1 -8 -18 -26 }}
 Optimal tunings:
-* WE: ~2 = 1200.6379{{c}}, ~11/10 = 166.1853{{c}}
+* WE: ~44/35 = 400.0464{{c}}, ~3/2 = 701.9053{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.1157{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8178{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/10 = 166.097{{c}} -->
-{{Optimal ET sequence|legend=0| 29, 36, 65 }}
+{{Optimal ET sequence|legend=0| 12, …, 159, 330 }}
-Badness (Sintel): 2.26
+Badness (Sintel): 1.97
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 91/90, 169/168, 245/242, 896/891
+Comma list: 364/363, 441/440, 625/624, 13720/13689
-Mapping: {{mapping| 1 2 -1 1 0 3 | 0 -3 24 13 25 5 }}
+{{Mapping|legend=0| 3 0 45 94 134 168 | 0 1 -8 -18 -26 -33 }}
 Optimal tunings:
-* WE: ~2 = 1201.1361{{c}}, ~11/10 = 166.2110{{c}}
+* WE: ~44/35 = 400.0449{{c}}, ~3/2 = 701.8995{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0833{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8156{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/10 = 166.054{{c}} -->
-{{Optimal ET sequence|legend=0| 29, 65f, 94df }}
+{{Optimal ET sequence|legend=0| 12f, …, 159, 330 }}
-Badness (Sintel): 1.81
+Badness (Sintel): 1.53
-== Tertiaschis ==
+==== 17-limit ====
-The tertiaschis temperament ({{nowrap| 94 & 159 }}) has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel]], but tempers out 1071875/1062882 for prime 7.
+Subgroup: 2.3.5.7.11.13.17
-[[Subgroup]]: 2.3.5.7
+Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619
-[[Comma list]]: 32805/32768, 1071875/1062882
+{{Mapping|legend=0| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }}
-{{Mapping|legend=1| 1 2 -1 10 | 0 -3 24 -52 }}
+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 tuning]]s:
+{{Optimal ET sequence|legend=0| 12f, 159, 171, 330 }}
-* [[WE]]: ~2 = 1200.3627{{c}}, ~192/175 = 166.0691{{c}}
-: [[error map]]: {{val| +0.363 +0.563 -1.019 -0.790 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~192/175 = 166.0172{{c}}
-: error map: {{val| 0.000 -0.007 -1.901 -1.720 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~192/175 = 166.019{{c}} -->
-{{Optimal ET sequence|legend=1| 65, 94, 159, 253, 412cd }}
+Badness (Sintel): 1.38
-[[Badness]] (Sintel): 5.36
+=== Terminator ===
+Terminator tempers out 540/539, and may be described as {{nowrap| 171 & 183 }}.
-=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 385/384, 4000/3993, 19712/19683
+Comma list: 540/539, 32805/32768, 137781/137500
-Mapping: {{mapping| 1 2 -1 10 0 | 0 -3 24 -52 25 }}
+{{Mapping|legend=0| 3 0 45 94 -137 | 0 1 -8 -18 31 }}
 Optimal tunings:
-* WE: ~2 = 1200.3379{{c}}, ~11/10 = 166.0638{{c}}
+* WE: ~63/50 = 399.9677{{c}}, ~3/2 = 701.6278{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0167{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6846{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/10 = 166.017{{c}} -->
-{{Optimal ET sequence|legend=0| 65, 94, 159, 253, 412cd, 665ccde }}
+{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 537, 891de }}
-Badness (Sintel): 2.07
+Badness (Sintel): 2.21
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 325/324, 385/384, 1575/1573, 10985/10976
+Comma list: 540/539, 729/728, 4096/4095, 31250/31213
-Mapping: {{mapping| 1 2 -1 10 0 12 | 0 -3 24 -52 25 -60 }}
+{{Mapping|legend=0| 3 0 45 94 -137 -103 | 0 1 -8 -18 31 24 }}
 Optimal tunings:
-* WE: ~2 = 1200.3467{{c}}, ~11/10 = 166.0635{{c}}
+* WE: ~63/50 = 399.9731{{c}}, ~3/2 = 701.6414{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0142{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/10 = 166.016{{c}} -->
-{{Optimal ET sequence|legend=0| 65f, 94, 159, 253, 412cdf, 665ccdef }}
+{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }}
-Badness (Sintel): 1.52
+Badness (Sintel): 1.47
-=== 17-limit ===
+==== 17-limit ====
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
+Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095
-Mapping: {{mapping| 1 2 -1 10 0 12 -2 | 0 -3 24 -52 25 -60 44 }}
+{{Mapping|legend=0| 3 0 45 94 -137 -103 -2 | 0 1 -8 -18 31 24 3 }}
 Optimal tunings:
-* WE: ~2 = 1200.3019{{c}}, ~11/10 = 166.0535{{c}}
+* WE: ~63/50 = 399.9757{{c}}, ~3/2 = 701.6458{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0114{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/10 = 166.012{{c}} -->
-{{Optimal ET sequence|legend=1| 65f, 94, 159, 253 }}
+{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }}
-Badness (Sintel): 1.35
+Badness (Sintel): 1.04
-== Countertertiaschis ==
+=== Semiterm ===
-The countertertiaschis temperament ({{nowrap| 159 & 224 }}) has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel]], but tempers out 244140625/243045684 for prime 7.
+The semiterm temperament tempers out [[9801/9800]] (kalisma) as well as [[151263/151250]] (odiheim comma), and may be described as {{nowrap| 12 & 342 }}. It has a period of 1/6 octave and its ploidacot is hexaploid monocot.
-[[Subgroup]]: 2.3.5.7
+Subgroup: 2.3.5.7.11
-[[Comma list]]: 32805/32768, 244140625/243045684
+Comma list: 9801/9800, 32805/32768, 151263/151250
-{{Mapping|legend=1| 1 2 -1 -12 | 0 -3 24 107 }}
+{{Mapping|legend=0| 6 0 90 188 287 | 0 1 -8 -18 -28 }}
+: mapping generators: ~55/49, ~3
-[[Optimal tuning]]s:
+Optimal tunings:
-* [[WE]]: ~2 = 1200.1265{{c}}, ~625/567 = 166.0797{{c}}
+* WE: ~55/49 = 200.0134{{c}}, ~3/2 = 701.7931{{c}}
-: [[error map]]: {{val| +0.127 +0.059 -0.529 +0.178 }}
+* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7426{{c}}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/567 = 166.0632{{c}}
-: error map: {{val| 0.000 -0.145 -0.797 -0.065 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~625/567 = 166.0621{{c}} -->
-{{Optimal ET sequence|legend=1| 65d, 159, 224, 383, 607 }}
+{{Optimal ET sequence|legend=0| 12, …, 330e, 342, 1380, 1722, 2064, 2406c, 5154bccdde }}
-[[Badness]] (Sintel): 4.76
+Badness (Sintel): 0.973
-=== 11-limit ===
+==== 13-limit ====
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13
-Comma list: 3025/3024, 4000/3993, 32805/32768
+Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375
-Mapping: {{mapping| 1 2 -1 -12 0 | 0 -3 24 107 25 }}
+{{Mapping|legend=0| 6 0 90 188 287 355 | 0 1 -8 -18 -28 -35 }}
 Optimal tunings:
-* WE: ~2 = 1200.0804{{c}}, ~11/10 = 166.0739{{c}}
+* WE: ~55/49 = 200.0083{{c}}, ~3/2 = 701.7549{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0634{{c}}
+* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7238{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/10 = 166.0628{{c}} -->
-{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}
+{{Optimal ET sequence|legend=0| 12f, 330eff, 342f, 696f }} *
+<nowiki>*</nowiki> optimal patent val: [[354edo|354]]
+Badness (Sintel): 1.85
-Badness (Sintel): 1.62
+=== Hemiterm ===
+The hemiterm temperament tempers out [[3025/3024]] (lehmerisma), and may be described as {{nowrap| 159 & 183 }}. Its ploidacot is triploid alpha-dicot.
-=== 13-limit ===
+Subgroup: 2.3.5.7.11
-Subgroup: 2.3.5.7.11.13
-Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976
+Comma list: 3025/3024, 32805/32768, 102487/102400
-Mapping: {{mapping| 1 2 -1 -12 0 -10 | 0 -3 24 107 25 99 }}
+{{Mapping|legend=0| 3 0 45 94 8 | 0 2 -16 -36 1 }}
+: mapping generators: ~63/50, ~693/400
 Optimal tunings:
-* WE: ~2 = 1200.0805{{c}}, ~11/10 = 166.0740{{c}}
+* WE: ~63/50 = 400.0309{{c}}, ~693/400 = 950.9458{{c}} (~12/11 = 150.8841{{c}})
-* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0635{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~693/400 = 950.8707{{c}} (~12/11 = 150.8707{{c}})
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/10 = 166.0628{{c}} -->
-{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}
+{{Optimal ET sequence|legend=0| 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce }}
-Badness (Sintel): 1.01
+Badness (Sintel): 0.684
-== Term ==
+==== 13-limit ====
-Term tempers out the [[landscape comma]], mapping ~63/50 to the 1/3-octave period. It can be described as {{nowrap| 12 & 171 }}, and is the unique temperament that equates a syntonic~Pythagorean comma with a stack of three [[marvel comma]]s. A [[septimal comma]] is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[171edo]] makes for an excellent tuning.
+Subgroup: 2.3.5.7.11.13
-[[Subgroup]]: 2.3.5.7
+Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712
-[[Comma list]]: 32805/32768, 250047/250000
+{{Mapping|legend=0| 3 0 45 94 8 42 | 0 2 -16 -36 1 -13 }}
-{{Mapping|legend=1| 3 0 45 94 | 0 1 -8 -18 }}
+Optimal tunings:
-: mapping generators: ~63/50, ~3
+* WE: ~63/50 = 400.0541{{c}}, ~26/15 = 951.0013{{c}} (~12/11 = 150.8932{{c}})
+* CWE: ~63/50 = 400.0000{{c}}, ~26/15 = 950.8696{{c}} (~12/11 = 150.8696{{c}})
-[[Optimal tuning]]s:
+{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f }}
-* [[WE]]: ~63/50 = 400.0257{{c}}, ~3/2 = 701.7873{{c}}
-: [[error map]]: {{val| +0.077 -0.091 -0.072 +0.031 }}
-* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 701.7383{{c}}
-: error map: {{val| 0.000 -0.217 -0.220 -0.115 }}
-<!-- * [[POTE]]: ~63/50 = 400.000{{c}}, ~3/2 = 701.742{{c}} -->
-[[Minimax tuning]]:
+Badness (Sintel): 1.30
-* [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis)]]: 2.5/3
-* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
-{{Optimal ET sequence|legend=1| 12, …, 159, 171, 867, 1038, 1209, 1380, 1551, 1722 }}
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
-[[Badness]] (Sintel): 0.505
+Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264
-=== Terminal ===
+{{Mapping|legend=0| 3 0 45 94 8 42 -2 | 0 2 -16 -36 1 -13 6 }}
-The terminal temperament ({{nowrap| 12 & 159 }}) tempers out 441/440 and 4375/4356. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave.
-Subgroup: 2.3.5.7.11
+Optimal tunings:
+* WE: ~34/27 = 400.0373{{c}}, ~26/15 = 950.9556{{c}} (~12/11 = 150.8809{{c}})
+* CWE: ~34/27 = 400.0000{{c}}, ~26/15 = 950.8652{{c}} (~12/11 = 150.8652{{c}})
-Comma list: 441/440, 4375/4356, 32805/32768
+{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f, 525f }}
-Mapping: {{mapping| 3 0 45 94 134 | 0 1 -8 -18 -26 }}
+Badness (Sintel): 1.14
-Optimal tunings:
+== Altinex ==
-* WE: ~44/35 = 400.0464{{c}}, ~3/2 = 701.9053{{c}}
+Named by [[Aura]] in 2021, altinex is an alternative to [[#Hemiterm|hemiterm]] and may be described as {{nowrap| 24 & 159 }}. [[159edo]] itself makes for a recommendable tuning.
-* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8178{{c}}
-<!-- * POTE: ~44/35 = 400.000{{c}}, ~3/2 = 701.824{{c}} -->
-{{Optimal ET sequence|legend=0| 12, …, 159, 330 }}
+[[Subgroup]]: 2.3.5.7
-Badness (Sintel): 1.97
+[[Comma list]]: 32805/32768, 367653125/362797056
-==== 13-limit ====
+{{Mapping|legend=1| 3 0 45 -32 | 0 2 -16 17 }}
-Subgroup: 2.3.5.7.11.13
+: mapping generators: ~1536/1225, ~34300/19683
-Comma list: 364/363, 441/440, 625/624, 13720/13689
+[[Optimal tuning]]s:
+* [[WE]]: ~1536/1225 = 400.1360{{c}}, ~34300/19683 = 951.2867{{c}}
+: [[error map]]: {{val| +0.408 +0.618 -0.781 -1.304 }}
+* [[CWE]]: ~1536/1225 = 400.0000{{c}}, ~34300/19683 = 950.9638{{c}}
+: error map: {{val| 0.000 -0.027 -1.735 -2.441 }}
-Mapping: {{mapping| 3 0 45 94 134 168 | 0 1 -8 -18 -26 -33 }}
+{{Optimal ET sequence|legend=1| 24, 135, 159, 612ccdd }}
-Optimal tunings:
+[[Badness]] (Sintel): 10.7
-* WE: ~44/35 = 400.0449{{c}}, ~3/2 = 701.8995{{c}}
-* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8156{{c}}
-<!-- * POTE: ~44/35 = 400.000{{c}}, ~3/2 = 701.821{{c}} -->
-{{Optimal ET sequence|legend=0| 12f, …, 159, 330 }}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Badness (Sintel): 1.53
+Comma list: 385/384, 14700/14641, 19712/19683
-==== 17-limit ====
+{{Mapping|legend=0| 3 0 45 -32 8 | 0 2 -16 17 1 }}
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619
+Optimal tunings:
+* WE: ~44/35 = 400.1156{{c}}, ~121/70 = 951.2377{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~121/70 = 950.9634{{c}}
-Mapping: {{mapping| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }}
+{{Optimal ET sequence|legend=0| 24, 135, 159 }}
-Optimal tunings:
+Badness (Sintel): 3.35
-* WE: ~34/27 = 400.0195{{c}}, ~3/2 = 701.8439{{c}}
-* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 701.8081{{c}}
-<!-- * POTE: ~34/27 = 400.000{{c}}, ~3/2 = 701.810{{c}} -->
-{{Optimal ET sequence|legend=0| 12f, 159, 171, 330 }}
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Badness (Sintel): 1.38
+Comma list: 364/363, 385/384, 676/675, 19712/19683
-=== Terminator ===
+{{Mapping|legend=0| 3 0 45 -32 8 42 | 0 2 -16 17 1 -13 }}
-Subgroup: 2.3.5.7.11
-Comma list: 540/539, 32805/32768, 137781/137500
+Optimal tunings:
+* WE: ~44/35 = 400.1396{{c}}, ~26/15 = 951.2799{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~26/15 = 950.9462{{c}}
-Mapping: {{mapping| 3 0 45 94 -137 | 0 1 -8 -18 31 }}
+{{Optimal ET sequence|legend=0| 24, 135f, 159 }}
-Optimal tunings:
+Badness (Sintel): 2.27
-* WE: ~63/50 = 399.9677{{c}}, ~3/2 = 701.6278{{c}}
-* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6846{{c}}
-<!-- * POTE: ~63/50 = 400.000{{c}}, ~3/2 = 701.685{{c}} -->
-{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 537, 891de }}
+== 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.
-Badness (Sintel): 2.21
+[[Subgroup]]: 2.3.5.7
-==== 13-limit ====
+[[Comma list]]: 686/675, 32805/32768
-Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 729/728, 4096/4095, 31250/31213
+{{Mapping|legend=1| 1 2 -1 1 | 0 -3 24 13 }}
-Mapping: {{mapping| 3 0 45 94 -137 -103 | 0 1 -8 -18 31 24 }}
+[[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 tunings:
+{{Optimal ET sequence|legend=1| 29, 36, 65 }}
-* WE: ~63/50 = 399.9731{{c}}, ~3/2 = 701.6414{{c}}
-* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}}
-<!-- * POTE: ~63/50 = 400.000{{c}}, ~3/2 = 701.689{{c}} -->
-{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }}
+[[Badness]] (Sintel): 4.42
-Badness (Sintel): 1.47
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-==== 17-limit ====
+Comma list: 245/242, 686/675, 896/891
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095
+{{Mapping|legend=0| 1 2 -1 1 0 | 0 -3 24 13 25 }}
-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}}
+* WE: ~2 = 1200.6379{{c}}, ~11/10 = 166.1853{{c}}
-* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.1157{{c}}
-<!-- * POTE: ~63/50 = 400.000{{c}}, ~3/2 = 701.688{{c}} -->
-{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }}
+{{Optimal ET sequence|legend=0| 29, 36, 65 }}
-Badness (Sintel): 1.04
+Badness (Sintel): 2.26
-=== Semiterm ===
+=== 13-limit ===
-The semiterm temperament ({{nowrap| 12 & 342 }}) has a period of 1/6 octave and tempers out [[9801/9800]] (kalisma) and 151263/151250 (odiheim comma).
+Subgroup: 2.3.5.7.11.13
-Subgroup: 2.3.5.7.11
+Comma list: 91/90, 169/168, 245/242, 896/891
-Comma list: 9801/9800, 32805/32768, 151263/151250
-Mapping: {{mapping| 6 0 90 188 287 | 0 1 -8 -18 -28 }}
+{{Mapping|legend=0| 1 2 -1 1 0 3 | 0 -3 24 13 25 5 }}
-: mapping generators: ~55/49, ~3
 Optimal tunings:
-* WE: ~55/49 = 200.0134{{c}}, ~3/2 = 701.7931{{c}}
+* WE: ~2 = 1201.1361{{c}}, ~11/10 = 166.2110{{c}}
-* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7426{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0833{{c}}
-<!-- * POTE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7460{{c}} -->
-{{Optimal ET sequence|legend=0| 12, …, 330e, 342, 1380, 1722, 2064, 2406c, 5154bccdde }}
+{{Optimal ET sequence|legend=0| 29, 65f, 94df }}
-Badness (Sintel): 0.973
+Badness (Sintel): 1.81
-==== 13-limit ====
+== Tertiaschis ==
-Subgroup: 2.3.5.7.11.13
+Named by [[Xenllium]] in 2021, tertiaschis may be described as {{nowrap| 94 & 159 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 1071875/1062882 for prime 7. See also [[daemotertiaschis]] which is made using every other generator of tertiaschis.
-Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 6 0 90 188 287 355 | 0 1 -8 -18 -28 -35 }}
+[[Comma list]]: 32805/32768, 1071875/1062882
-Optimal tunings:
+{{Mapping|legend=1| 1 2 -1 10 | 0 -3 24 -52 }}
-* WE: ~55/49 = 200.0083{{c}}, ~3/2 = 701.7549{{c}}
-* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7238{{c}}
-<!-- * POTE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7256{{c}} -->
-{{Optimal ET sequence|legend=0| 12f, 330eff, 342f, 696f }} *
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.3627{{c}}, ~192/175 = 166.0691{{c}}
+: [[error map]]: {{val| +0.363 +0.563 -1.019 -0.790 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~192/175 = 166.0172{{c}}
+: error map: {{val| 0.000 -0.007 -1.901 -1.720 }}
-<nowiki>*</nowiki> optimal patent val: [[354edo|354]]
+{{Optimal ET sequence|legend=1| 65, 94, 159, 253, 412cd }}
-Badness (Sintel): 1.85
+[[Badness]] (Sintel): 5.36
-=== Hemiterm ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 32805/32768, 102487/102400
+Comma list: 385/384, 4000/3993, 19712/19683
-Mapping: {{mapping| 3 0 45 94 8 | 0 2 -16 -36 1 }}
+{{Mapping|legend=0| 1 2 -1 10 0 | 0 -3 24 -52 25 }}
-: mapping generators: ~63/50, ~693/400
 Optimal tunings:
-* WE: ~63/50 = 400.0309{{c}}, ~693/400 = 950.9458{{c}} (~12/11 = 150.8841{{c}})
+* WE: ~2 = 1200.3379{{c}}, ~11/10 = 166.0638{{c}}
-* CWE: ~63/50 = 400.0000{{c}}, ~693/400 = 950.8707{{c}} (~12/11 = 150.8707{{c}})
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0167{{c}}
-<!-- * POTE: ~63/50 = 400.000{{c}}, ~693/400 = 950.872{{c}} (~12/11 = 150.872{{c}}) -->
-{{Optimal ET sequence|legend=0| 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce }}
+{{Optimal ET sequence|legend=0| 65, 94, 159, 253, 412cd, 665ccde }}
-Badness (Sintel): 0.684
+Badness (Sintel): 2.07
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712
+Comma list: 325/324, 385/384, 1575/1573, 10985/10976
-Mapping: {{mapping| 3 0 45 94 8 42 | 0 2 -16 -36 1 -13 }}
+{{Mapping|legend=0| 1 2 -1 10 0 12 | 0 -3 24 -52 25 -60 }}
 Optimal tunings:
-* WE: ~63/50 = 400.0541{{c}}, ~26/15 = 951.0013{{c}} (~12/11 = 150.8932{{c}})
+* WE: ~2 = 1200.3467{{c}}, ~11/10 = 166.0635{{c}}
-* CWE: ~63/50 = 400.0000{{c}}, ~26/15 = 950.8696{{c}} (~12/11 = 150.8696{{c}})
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0142{{c}}
-<!-- * POTE: ~63/50 = 400.000{{c}}, ~26/15 = 950.873{{c}} (~12/11 = 150.873{{c}}) -->
-{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f }}
+{{Optimal ET sequence|legend=0| 65f, 94, 159, 253, 412cdf, 665ccdef }}
-Badness (Sintel): 1.30
+Badness (Sintel): 1.52
-==== 17-limit ====
+=== 17-limit ===
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264
+Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
-Mapping: {{mapping| 3 0 45 94 8 42 -2 | 0 2 -16 -36 1 -13 6 }}
+{{Mapping|legend=0| 1 2 -1 10 0 12 -2 | 0 -3 24 -52 25 -60 44 }}
 Optimal tunings:
-* WE: ~34/27 = 400.0373{{c}}, ~26/15 = 950.9556{{c}} (~12/11 = 150.8809{{c}})
+* WE: ~2 = 1200.3019{{c}}, ~11/10 = 166.0535{{c}}
-* CWE: ~34/27 = 400.0000{{c}}, ~26/15 = 950.8652{{c}} (~12/11 = 150.8652{{c}})
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0114{{c}}
-<!-- * POTE: ~34/27 = 400.000{{c}}, ~26/15 = 950.867{{c}} (~12/11 = 150.867{{c}}) -->
+{{Optimal ET sequence|legend=1| 65f, 94, 159, 253 }}
-{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f, 525f }}
+Badness (Sintel): 1.35
-Badness (Sintel): 1.14
+== Countertertiaschis ==
+Named by [[Flora Canou]] in 2021, countertertiaschis may be described as {{nowrap| 159 & 224 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 244140625/243045684 for prime 7.
-== Altinex ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 32805/32768, 367653125/362797056
+[[Comma list]]: 32805/32768, 244140625/243045684
-{{Mapping|legend=1| 3 0 45 -32 | 0 2 -16 17 }}
+{{Mapping|legend=1| 1 2 -1 -12 | 0 -3 24 107 }}
-: mapping generators: ~1536/1225, ~34300/19683
 [[Optimal tuning]]s:
-* [[WE]]: ~1536/1225 = 400.1360{{c}}, ~34300/19683 = 951.2867{{c}}
+* [[WE]]: ~2 = 1200.1265{{c}}, ~625/567 = 166.0797{{c}}
-: [[error map]]: {{val| +0.408 +0.618 -0.781 -1.304 }}
+: [[error map]]: {{val| +0.127 +0.059 -0.529 +0.178 }}
-* [[CWE]]: ~1536/1225 = 400.0000{{c}}, ~34300/19683 = 950.9638{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/567 = 166.0632{{c}}
-: error map: {{val| 0.000 -0.027 -1.735 -2.441 }}
+: error map: {{val| 0.000 -0.145 -0.797 -0.065 }}
-<!-- * [[CTE]]: ~1536/1225 = 400.000{{c}}, ~34300/19683 = 950.9654{{c}} -->
-{{Optimal ET sequence|legend=1| 24, 135, 159, 612ccdd }}
+{{Optimal ET sequence|legend=1| 65d, 159, 224, 383, 607 }}
-[[Badness]] (Sintel): 10.7
+[[Badness]] (Sintel): 4.76
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 385/384, 14700/14641, 19712/19683
+Comma list: 3025/3024, 4000/3993, 32805/32768
-Mapping: {{mapping| 3 0 45 -32 8 | 0 2 -16 17 1 }}
+{{Mapping|legend=0| 1 2 -1 -12 0 | 0 -3 24 107 25 }}
 Optimal tunings:
-* WE: ~44/35 = 400.1156{{c}}, ~121/70 = 951.2377{{c}}
+* WE: ~2 = 1200.0804{{c}}, ~11/10 = 166.0739{{c}}
-* CWE: ~44/35 = 400.0000{{c}}, ~121/70 = 950.9634{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0634{{c}}
-<!-- * CTE: ~44/35 = 400.000{{c}}, ~121/70 = 950.9658{{c}} -->
-{{Optimal ET sequence|legend=0| 24, 135, 159 }}
+{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}
-Badness (Sintel): 3.35
+Badness (Sintel): 1.62
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 364/363, 385/384, 676/675, 19712/19683
+Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976
-Mapping: {{mapping| 3 0 45 -32 8 42 | 0 2 -16 17 1 -13 }}
+{{Mapping|legend=0| 1 2 -1 -12 0 -10 | 0 -3 24 107 25 99 }}
 Optimal tunings:
-* WE: ~44/35 = 400.1396{{c}}, ~26/15 = 951.2799{{c}}
+* WE: ~2 = 1200.0805{{c}}, ~11/10 = 166.0740{{c}}
-* CWE: ~44/35 = 400.0000{{c}}, ~26/15 = 950.9462{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0635{{c}}
-<!-- * CTE: ~44/35 = 400.000{{c}}, ~26/15 = 950.9360{{c}} -->
+{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}
-{{Optimal ET sequence|legend=0| 24, 135f, 159 }}
+Badness (Sintel): 1.01
-Badness (Sintel): 2.27
+== Quadrant ==
+Named by [[Xenllium]] in 2021, quadrant tempers out 390625/388962, the [[dimcomp comma]], and maps [[25/21]] to the 1/4-octave period. It may be described as the {{nowrap| 12 & 212 }} temperament; its ploidacot is tetraploid monocot. Just as [[#Term|term]] equates the syntonic and Pythagorean commas for a generic comma step, and splits it into three [[marvel comma]]s, quadrant splits the same generic comma step into four. A [[septimal comma]] is then found as a stack of five marvel commas.
-== Sesquiquartififths ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 2401/2400, 32805/32768
+[[Comma list]]: 32805/32768, 390625/388962
-{{Mapping|legend=1| 1 1 7 5 | 0 4 -32 -15 }}
+{{Mapping|legend=1| 4 0 60 119 | 0 1 -8 -17 }}
-: mapping generators: ~2, ~448/405
+: mapping generators: ~25/21, ~3
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1200.0846{{c}}, ~448/405 = 175.4460{{c}}
+* [[WE]]: ~2 = 300.0255{{c}}, ~3/2 = 701.8831{{c}}
-: [[error map]]: {{val| +0.085 -0.086 +0.007 -0.093 }}
+: [[error map]]: {{val| +0.102 +0.030 -0.664 +0.462 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~448/405 = 175.4320{{c}}
+* [[CWE]]: ~2 = 300.0000{{c}}, ~3/2 = 701.8180{{c}}
-: error map: {{val| 0.000 -0.227 -0.137 -0.306 }}
+: error map: {{val| 0.000 -0.137 -0.858 +0.268 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~448/405 = 175.434{{c}} -->
-[[Minimax tuning]]:
+{{Optimal ET sequence|legend=1| 12, …, 200, 212, 224, 436, 660 }}
-* [[7-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
-* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
-{{Optimal ET sequence|legend=1| 41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd }}
+[[Badness]] (Sintel): 2.79
-[[Badness]] (Sintel): 0.285
+=== 11-limit ===
-=== Sesquart ===
 Subgroup: 2.3.5.7.11
-Comma list: 243/242, 441/440, 16384/16335
+Comma list: 1375/1372, 6250/6237, 32805/32768
-Mapping: {{mapping| 1 1 7 5 2 | 0 4 -32 -15 10 }}
+{{Mapping|legend=0| 4 0 60 119 185 | 0 1 -8 -17 -27 }}
 Optimal tunings:
-* WE: ~2 = 1199.8171{{c}}, ~256/231 = 175.3793{{c}}
+* WE: ~25/21 = 300.0244{{c}}, ~3/2 = 701.8759{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~256/231 = 175.4081{{c}}
+* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8145{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~256/231 = 175.406{{c}} -->
-{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}
+{{Optimal ET sequence|legend=0| 12, …, 212, 224, 436, 660 }}
-Badness (Sintel): 0.969
+Badness (Sintel): 1.51
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 243/242, 364/363, 441/440, 3584/3575
+Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647
-Mapping: {{mapping| 1 1 7 5 2 -2 | 0 4 -32 -15 10 39 }}
+{{Mapping|legend=0| 4 0 60 119 185 224 | 0 1 -8 -17 -27 -33 }}
 Optimal tunings:
-* WE: ~2 = 1199.8352{{c}}, ~72/65 = 175.3852{{c}}
+* WE: ~25/21 = 300.0234{{c}}, ~3/2 = 701.8707{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4095{{c}}
+* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8123{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~72/65 = 175.409{{c}} -->
-{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}
+{{Optimal ET sequence|legend=0| 12f, …, 212, 224, 436, 660 }}
-Badness (Sintel): 0.925
+Badness (Sintel): 1.13
-===== Sesquartia =====
+== Sesquiquartififths ==
-Subgroup: 2.3.5.7.11.13.17
+Sesquiquartififths tempers out 2401/2400, the [[breedsma]], and may be described as the {{nowrap| 41 & 171 }} temperament. It splits the [[3/2|perfect fifth]] into four; its [[ploidacot]] is thus tetracot.
-Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575
+The name might have been a portmanteau of ''sesquiquart'' and ''quartififths'', each meaning dividing 3/2 into four.
-Mapping: {{mapping| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }}
+[[Subgroup]]: 2.3.5.7
-Optimal tunings:
+[[Comma list]]: 2401/2400, 32805/32768
-* WE: ~2 = 1199.8902{{c}}, ~72/65 = 175.4077{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4234{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~72/65 = 175.424{{c}} -->
-{{Optimal ET sequence|legend=0| 41, 130, 171 }}
+{{Mapping|legend=1| 1 1 7 5 | 0 4 -32 -15 }}
+: mapping generators: ~2, ~448/405
-Badness (Sintel): 1.18
+[[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 }}
-====== 19-limit ======
+[[Minimax tuning]]:
-Subgroup: 2.3.5.7.11.13.17.19
+* [[7-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
+* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
-Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594
+{{Optimal ET sequence|legend=1| 41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd }}
-Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }}
+[[Badness]] (Sintel): 0.285
-Optimal tunings:
+=== Sesquart ===
-* WE: ~2 = 1199.9864{{c}}, ~21/19 = 175.4169{{c}}
+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.
-* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4189{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~21/19 = 175.419{{c}} -->
-{{Optimal ET sequence|legend=0| 41, 130, 171 }}
+Subgroup: 2.3.5.7.11
+Comma list: 243/242, 441/440, 16384/16335
+{{Mapping|legend=0| 1 1 7 5 2 | 0 4 -32 -15 10 }}
+Optimal tunings:
+* WE: ~2 = 1199.8171{{c}}, ~256/231 = 175.3793{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~256/231 = 175.4081{{c}}
+{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}
-Badness (Sintel): 1.24
+Badness (Sintel): 0.969
-====== 23-limit ======
+==== 13-limit ====
-Subgroup: 2.3.5.7.11.13.17.19.23
+Subgroup: 2.3.5.7.11.13
-Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594
+Comma list: 243/242, 364/363, 441/440, 3584/3575
-Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 -6 | 0 4 -32 -15 10 39 69 -12 72 }}
+{{Mapping|legend=0| 1 1 7 5 2 -2 | 0 4 -32 -15 10 39 }}
 Optimal tunings:
-* WE: ~2 = 1199.9606{{c}}, ~21/19 = 175.4067{{c}}
+* WE: ~2 = 1199.8352{{c}}, ~72/65 = 175.3852{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4123{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4095{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~21/19 = 175.412{{c}} -->
-{{Optimal ET sequence|legend=0| 41i, 130, 171 }}
+{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}
-Badness (Sintel): 1.36
+Badness (Sintel): 0.925
 ===== Heartia =====
@@ Line 2,041: / Line 1,998: @@
 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 }}
+{{Mapping|legend=0| 1 1 7 5 2 -2 0 | 0 4 -32 -15 10 39 28 }}
 Optimal tunings:
 * WE: ~2 = 1199.6422{{c}}, ~72/65 = 175.3338{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3857{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~72/65 = 175.386{{c}} -->
 {{Optimal ET sequence|legend=0| 41, 89, 130g }}
@@ Line 2,057: / Line 2,013: @@
 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 }}
+{{Mapping|legend=0| 1 1 7 5 2 -2 0 6 | 0 4 -32 -15 10 39 28 -12 }}
 Optimal tunings:
 * WE: ~2 = 1199.7499{{c}}, ~21/19 = 175.3432{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.3797{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~21/19 = 175.380{{c}} -->
 {{Optimal ET sequence|legend=0| 41, 89, 130g }}
@@ Line 2,068: / Line 2,023: @@
 Badness (Sintel): 1.40
-===== Hearty =====
+===== Sesquartia =====
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625
+Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575
-Mapping: {{mapping| 1 1 7 5 2 -2 13 | 0 4 -32 -15 10 39 -61 }}
+{{Mapping|legend=0| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }}
 Optimal tunings:
-* WE: ~2 = 1199.9458{{c}}, ~72/65 = 175.3689{{c}}
+* WE: ~2 = 1199.8902{{c}}, ~72/65 = 175.4077{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3770{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4234{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~72/65 = 175.377{{c}} -->
-{{Optimal ET sequence|legend=0| 41g, 89, 130 }}
+{{Optimal ET sequence|legend=0| 41, 130, 171 }}
-Badness (Sintel): 1.56
+Badness (Sintel): 1.18
 ====== 19-limit ======
 Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 221/220, 243/242, 361/360, 364/363, 441/440, 456/455
+Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594
-Mapping: {{mapping| 1 1 7 5 2 -2 13 6 | 0 4 -32 -15 10 39 -61 -12 }}
+{{Mapping|legend=0| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }}
 Optimal tunings:
-* WE: ~2 = 1200.0114{{c}}, ~72/65 = 175.3783{{c}}
+* WE: ~2 = 1199.9864{{c}}, ~21/19 = 175.4169{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3765{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4189{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~21/19 = 175.377{{c}} -->
-{{Optimal ET sequence|legend=0| 41g, 89, 130 }}
+{{Optimal ET sequence|legend=0| 41, 130, 171 }}
-Badness (Sintel): 1.39
+Badness (Sintel): 1.24
 ====== 23-limit ======
 Subgroup: 2.3.5.7.11.13.17.19.23
-Comma list: 221/220, 243/242, 276/275, 323/322, 361/360, 364/363, 441/440
+Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594
-Mapping: {{mapping| 1 1 7 5 2 -2 13 6 13 | 0 4 -32 -15 10 39 -61 -12 -58 }}
+{{Mapping|legend=0| 1 1 7 5 2 -2 -6 6 -6 | 0 4 -32 -15 10 39 69 -12 72 }}
 Optimal tunings:
-* WE: ~2 = 1200.0122{{c}}, ~72/65 = 175.3782{{c}}
+* WE: ~2 = 1199.9606{{c}}, ~21/19 = 175.4067{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3763{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4123{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~21/19 = 175.376{{c}} -->
-{{Optimal ET sequence|legend=0| 41g, 89, 130 }}
+{{Optimal ET sequence|legend=0| 41i, 130, 171 }}
-Badness (Sintel): 1.37
+Badness (Sintel): 1.36
-=== Bisesqui ===
+===== Hearty =====
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13.17
-Comma list: 2401/2400, 9801/9800, 32805/32768
+Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625
-Mapping: {{mapping| 2 2 14 10 23 | 0 4 -32 -15 -55 }}
+{{Mapping|legend=0| 1 1 7 5 2 -2 13 | 0 4 -32 -15 10 39 -61 }}
-: mapping generators: ~99/70, ~448/405
 Optimal tunings:
-* WE: ~99/70 = 600.0429{{c}}, ~448/405 = 175.4474{{c}}
+* WE: ~2 = 1199.9458{{c}}, ~72/65 = 175.3689{{c}}
-* CWE: ~99/70 = 600.0000{{c}}, ~448/405 = 175.4334{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3770{{c}}
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~448/405 = 175.435{{c}} -->
-{{Optimal ET sequence|legend=1| 82e, 130, 212, 342, 1156, 1498, 1840d, 5862bbccdddee }}
+{{Optimal ET sequence|legend=0| 41g, 89, 130 }}
-Badness (Sintel): 0.561
+Badness (Sintel): 1.56
-== Quintilipyth ==
+====== 19-limit ======
-The quintilipyth temperament ({{nowrap| 12 & 253 }}, formerly ''quintilischis'') slices the pythagorean fourth ([[4/3]]) into five semitones and tempers out the compass comma (9765625/9680832) in the 7-limit.
+Subgroup: 2.3.5.7.11.13.17.19
-[[Subgroup]]: 2.3.5.7
+Comma list: 221/220, 243/242, 361/360, 364/363, 441/440, 456/455
-[[Comma list]]: 32805/32768, 9765625/9680832
+{{Mapping|legend=0| 1 1 7 5 2 -2 13 6 | 0 4 -32 -15 10 39 -61 -12 }}
-{{Mapping|legend=1| 1 2 -1 -4 | 0 -5 40 82 }}
+Optimal tunings:
-: mapping generators: ~2, ~625/588
+* WE: ~2 = 1200.0114{{c}}, ~72/65 = 175.3783{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3765{{c}}
-[[Optimal tuning]]s:
+{{Optimal ET sequence|legend=0| 41g, 89, 130 }}
-* [[WE]]: ~2 = 1200.1138{{c}}, ~625/588 = 99.6347{{c}}
-: [[error map]]: {{val| +0.114 +0.099 -1.041 +0.761 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/588 = 99.6265{{c}}
-: error map: {{val| 0.000 -0.087 -1.255 +0.544 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~625/588 = 99.625{{c}} -->
-{{Optimal ET sequence|legend=1| 12, …, 253, 265 }}
+Badness (Sintel): 1.39
-[[Badness]] (Sintel): 6.43
+====== 23-limit ======
+Subgroup: 2.3.5.7.11.13.17.19.23
-=== 11-limit ===
+Comma list: 221/220, 243/242, 276/275, 323/322, 361/360, 364/363, 441/440
-Subgroup: 2.3.5.7.11
-Comma list: 1375/1372, 4375/4356, 32805/32768
-Mapping: {{mapping| 1 2 -1 -4 -7 | 0 -5 40 82 126 }}
+{{Mapping|legend=0| 1 1 7 5 2 -2 13 6 13 | 0 4 -32 -15 10 39 -61 -12 -58 }}
 Optimal tunings:
-* WE: ~2 = 1200.1503{{c}}, ~35/33 = 99.6287{{c}}
+* WE: ~2 = 1200.0122{{c}}, ~72/65 = 175.3782{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6176{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3763{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~35/33 = 99.616{{c}} -->
-{{Optimal ET sequence|legend=0| 12, …, 253, 265, 518c }}
+{{Optimal ET sequence|legend=0| 41g, 89, 130 }}
-Badness (Sintel): 3.74
+Badness (Sintel): 1.37
-=== 13-limit ===
+=== Bisesqui ===
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11
-Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647
+Comma list: 2401/2400, 9801/9800, 32805/32768
-Mapping: {{mapping| 1 2 -1 -4 -7 -9 | 0 -5 40 82 126 153 }}
+{{Mapping|legend=0| 2 2 14 10 23 | 0 4 -32 -15 -55 }}
+: mapping generators: ~99/70, ~448/405
 Optimal tunings:
-* WE: ~2 = 1200.1774{{c}}, ~35/33 = 99.6267{{c}}
+* WE: ~99/70 = 600.0429{{c}}, ~448/405 = 175.4474{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6134{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~448/405 = 175.4334{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~35/33 = 99.612{{c}} -->
-{{Optimal ET sequence|legend=0| 12f, …, 241cdef, 253 }}
+{{Optimal ET sequence|legend=1| 82e, 130, 212, 342, 1156, 1498, 1840d, 5862bbccdddee }}
-Badness (Sintel): 2.86
+Badness (Sintel): 0.561
-=== 17-limit ===
+== Tsaharuk ==
-Subgroup: 2.3.5.7.11.13.17
+[[File:Tsaharuk.jpg|alt=Tsaharuk.jpg|thumb|Parametric horagram of the tsaharuk temperament.]]
+{{See also| Tsaharuk24 }}
-Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619
+Tsaharuk tempers out 420175/419904, the [[wizma]], and may be described as the {{nowrap| 77 & 94 }} temperament. It is generated by a slightly flat neutral second of [[~]][[13/12]], five of which make the [[3/2|perfect fifth]], so its [[ploidacot]] is pentacot.
-Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 | 0 -5 40 82 126 153 -11 }}
+It was proposed by [[Jacques Dudon]] based on [[Julien Jalaleddine Weiss]] qanun tunings, which has [[mos]] of size 43, 60, 77 and 94 notes.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_100718.html Yahoo! Tuning Group | ''A temperament for Maqam'']</ref>
-Optimal tunings:
+It restricts to the [[2.3.7 subgroup]] as [[no-fives subgroup temperaments #Navy|navy]]. This relates tsaharuk to [[aberschismic temperaments #Quanic|quanic]], which favors a sharper fifth and works better in the higher limits, but which also has mos of size 43, 60, 77 and 94 notes. [[Bleu]] has, like tsaharuk, a generator of 1/5 fifth and makes for an instructive comparison. [[Edo]]s that support tsaharuk include [[77edo]], [[94edo]], [[171edo]] and [[248edo]], and edos supporting quanic are [[94edo]], [[111edo]] and [[205edo]]. Quanic and tsaharuk therefore become identical if 94edo is the tuning.
-* WE: ~2 = 1200.1745{{c}}, ~18/17 = 99.6265{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6131{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~18/17 = 99.612{{c}} -->
-{{Optimal ET sequence|legend=0| 12f, 241cdef, 253 }}
+[[Subgroup]]: 2.3.5.7
-Badness (Sintel): 2.34
+[[Comma list]]: 32805/32768, 420175/419904
-=== 19-limit ===
+{{Mapping|legend=1| 1 1 7 0 | 0 5 -40 24 }}
-Subgroup: 2.3.5.7.11.13.17.19
+: mapping generators: ~2, ~243/224
-Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971
+[[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 }}
-Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 4 | 0 -5 40 82 126 153 -11 3 }}
+{{Optimal ET sequence|legend=1| 17, 77, 94, 171 }}
-Optimal tunings:
+[[Badness]] (Sintel): 0.777
-* WE: ~2 = 1200.0713{{c}}, ~18/17 = 99.6208{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6152{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~18/17 = 99.615{{c}} -->
-{{Optimal ET sequence|legend=0| 12f, 253, 265 }}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Badness (Sintel): 2.32
+Comma list: 385/384, 1331/1323, 19712/19683
-== Quintaschis ==
+{{Mapping|legend=0| 1 1 7 0 1 | 0 5 -40 24 21 }}
-The quintaschis temperament ({{nowrap| 12 & 289 }}) slices the fourth (4/3) into five semitones and tempers out 49009212/48828125 in the 7-limit.
-[[Subgroup]]: 2.3.5.7
+Optimal tunings:
+* WE: ~2 = 1200.3103{{c}}, ~88/81 = 140.4011{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.3649{{c}}
-[[Comma list]]: 32805/32768, 49009212/48828125
+{{Optimal ET sequence|legend=0| 17, 77, 94, 171e, 265e }}
-{{Mapping|legend=1| 1 2 -1 -5 | 0 -5 40 94 }}
+Badness (Sintel): 2.10
-[[Optimal tuning]]s:
+=== 13-limit ===
-* [[WE]]: ~2 = 1200.0536{{c}}, ~200/189 = 99.6684{{c}}
+Subgroup: 2.3.5.7.11.13
-: [[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 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~200/189 = 99.664{{c}} -->
-{{Optimal ET sequence|legend=1| 12, …, 289, 301, 590, 891, 1192 }}
+Comma list: 352/351, 385/384, 729/728, 1331/1323
-[[Badness]] (Sintel): 3.36
+{{Mapping|legend=0| 1 1 7 0 1 3 | 0 5 -40 24 21 6 }}
-=== 11-limit ===
+Optimal tunings:
-Subgroup: 2.3.5.7.11
+* WE: ~2 = 1200.1840{{c}}, ~13/12 = 140.3840{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.3627{{c}}
-Comma list: 441/440, 32805/32768, 1953125/1951488
+{{Optimal ET sequence|legend=0| 17, 77, 94, 171e }}
-Mapping: {{mapping| 1 2 -1 -5 -8 | 0 -5 40 94 138 }}
+Badness (Sintel): 1.57
-Optimal tunings:
+== Quanharuk ==
-* WE: ~2 = 1200.0988{{c}}, ~35/33 = 99.6613{{c}}
+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.
-* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6540{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~35/33 = 99.653{{c}} -->
-{{Optimal ET sequence|legend=0| 12, …, 277d, 289 }}
+[[Subgroup]]: 2.3.5.7
-Badness (Sintel): 3.69
+[[Comma list]]: 16875/16807, 32805/32768
-==== 13-limit ====
+{{Mapping|legend=1| 1 0 15 12 | 0 5 -40 -29 }}
-Subgroup: 2.3.5.7.11.13
+: mapping generators: ~2, ~56/45
-Comma list: 364/363, 441/440, 32805/32768, 109512/109375
+[[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 }}
-Mapping: {{mapping| 1 2 -1 -5 -8 -11 | 0 -5 40 94 138 177 }}
+{{Optimal ET sequence|legend=1| 41, 142, 183, 224 }}
-Optimal tunings:
+[[Badness]] (Sintel): 1.82
-* WE: ~2 = 1200.0625{{c}}, ~35/33 = 99.6630{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6583{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~35/33 = 99.658{{c}} -->
-{{Optimal ET sequence|legend=0| 12f, …, 277dff, 289 }}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Badness (Sintel): 3.07
+Comma list: 540/539, 1375/1372, 32805/32768
-==== 17-limit ====
+{{Mapping|legend=0| 1 0 15 12 -7 | 0 5 -40 -29 33 }}
-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}}
+* WE: ~2 = 1199.9709{{c}}, ~56/45 = 380.3423{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6568{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3517{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~18/17 = 99.656{{c}} -->
-{{Optimal ET sequence|legend=0| 12f, 277dff, 289 }}
+{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}
-Badness (Sintel): 2.58
+Badness (Sintel): 1.04
-==== 19-limit ====
+=== 13-limit ===
-Subgroup: 2.3.5.7.11.13.17.19
+Subgroup: 2.3.5.7.11.13
-Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859
+Comma list: 540/539, 729/728, 1375/1372, 4096/4095
-Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 4 | 0 -5 40 94 138 177 -11 3 }}
+{{Mapping|legend=0| 1 0 15 12 -7 -15 | 0 5 -40 -29 33 59 }}
 Optimal tunings:
-* WE: ~2 = 1200.0289{{c}}, ~18/17 = 99.6609{{c}}
+* WE: ~2 = 1199.9663{{c}}, ~56/45 = 380.3403{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6586{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3509{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~18/17 = 99.659{{c}} -->
-{{Optimal ET sequence|legend=0| 12f, 289 }}
+{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}
-Badness (Sintel): 2.56
+Badness (Sintel): 0.884
-=== Quintahelenic ===
+== 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: 5632/5625, 8019/8000, 151263/151250
+Comma list: 1375/1372, 4375/4356, 32805/32768
-Mapping: {{mapping| 1 2 -1 -5 -9 | 0 -5 40 94 150 }}
+{{Mapping|legend=0| 1 2 -1 -4 -7 | 0 -5 40 82 126 }}
 Optimal tunings:
-* WE: ~2 = 1200.0195{{c}}, ~200/189 = 99.6723{{c}}
+* WE: ~2 = 1200.1503{{c}}, ~35/33 = 99.6287{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6709{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6176{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~200/189 = 99.671{{c}} -->
-{{Optimal ET sequence|legend=0| 12, …, 289e, 301, 915 }}
+{{Optimal ET sequence|legend=0| 12, …, 253, 265, 518c }}
-Badness (Sintel): 2.72
+Badness (Sintel): 3.74
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000
+Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647
-Mapping: {{mapping| 1 2 -1 -5 -9 -11 | 0 -5 40 94 150 177 }}
+{{Mapping|legend=0| 1 2 -1 -4 -7 -9 | 0 -5 40 82 126 153 }}
 Optimal tunings:
-* WE: ~2 = 1200.0442{{c}}, ~200/189 = 99.6709{{c}}
+* WE: ~2 = 1200.1774{{c}}, ~35/33 = 99.6267{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6675{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6134{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~200/189 = 99.661{{c}} -->
-{{Optimal ET sequence|legend=0| 12f, …, 289e, 301 }}
+{{Optimal ET sequence|legend=0| 12f, …, 241cdef, 253 }}
-Badness (Sintel): 2.30
+Badness (Sintel): 2.86
-===== 17-limit =====
+=== 17-limit ===
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750
+Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619
-Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 | 0 -5 40 94 150 177 -11 }}
+{{Mapping|legend=0| 1 2 -1 -4 -7 -9 5 | 0 -5 40 82 126 153 -11 }}
 Optimal tunings:
-* WE: ~2 = 1200.1227{{c}}, ~200/189 = 99.6753{{c}}
+* WE: ~2 = 1200.1745{{c}}, ~18/17 = 99.6265{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6658{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6131{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~18/17 = 99.665{{c}} -->
-{{Optimal ET sequence|legend=1| 12f, 289e, 301 }}
+{{Optimal ET sequence|legend=0| 12f, 241cdef, 253 }}
-Badness (Sintel): 2.06
+Badness (Sintel): 2.34
-===== 19-limit =====
+=== 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
+Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971
-Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 4 | 0 -5 40 94 150 177 -11 3 }}
+{{Mapping|legend=0| 1 2 -1 -4 -7 -9 5 4 | 0 -5 40 82 126 153 -11 3 }}
 Optimal tunings:
-* WE: ~2 = 1200.0230{{c}}, ~200/189 = 99.6694{{c}}
+* WE: ~2 = 1200.0713{{c}}, ~18/17 = 99.6208{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6676{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6152{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~18/17 = 99.668{{c}} -->
-{{Optimal ET sequence|legend=0| 12f, 301 }}
+{{Optimal ET sequence|legend=0| 12f, 253, 265 }}
-Badness (Sintel): 2.24
+Badness (Sintel): 2.32
-==== Quintahelenoid ====
+== Quintaschis ==
-Subgroup: 2.3.5.7.11.13
+Named by [[Xenllium]] in 2021, quintaschis slices the [[4/3|perfect fourth]] into five semitones and tempers out 49009212/48828125 in the [[7-limit]]. It may be described as the {{nowrap| 12 & 289 }} temperament, and its [[ploidacot]] is omega-pentacot.
-Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 32805/32768, 49009212/48828125
-Mapping: {{mapping| 1 2 -1 -5 -9 14 | 0 -5 40 94 150 -124 }}
+{{Mapping|legend=1| 1 2 -1 -5 | 0 -5 40 94 }}
-Optimal tunings:
+[[Optimal tuning]]s:
-* WE: ~2 = 1199.9919{{c}}, ~200/189 = 99.6712{{c}}
+* [[WE]]: ~2 = 1200.0536{{c}}, ~200/189 = 99.6684{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6718{{c}}
+: [[error map]]: {{val| +0.054 -0.190 +0.370 -0.262 }}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~200/189 = 99.672{{c}} -->
+* [[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=0| 12, 301, 614, 915 }}
+{{Optimal ET sequence|legend=1| 12, …, 289, 301, 590, 891, 1192 }}
-Badness (Sintel): 2.73
+[[Badness]] (Sintel): 3.36
-===== 17-limit =====
+=== 11-limit ===
-Subgroup: 2.3.5.7.11.13.17
+Subgroup: 2.3.5.7.11
-Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157
+Comma list: 441/440, 32805/32768, 1953125/1951488
-Mapping: {{mapping| 1 2 -1 -5 -9 14 5 | 0 -5 40 94 150 -124 -11 }}
+{{Mapping|legend=0| 1 2 -1 -5 -8 | 0 -5 40 94 138 }}
 Optimal tunings:
-* WE: ~2 = 1200.0469{{c}}, ~18/17 = 99.6749{{c}}
+* WE: ~2 = 1200.0988{{c}}, ~35/33 = 99.6613{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6710{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6540{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~18/17 = 99.671{{c}} -->
-{{Optimal ET sequence|legend=0| 12, 301 }}
+{{Optimal ET sequence|legend=0| 12, …, 277d, 289 }}
-Badness (Sintel): 2.44
+Badness (Sintel): 3.69
-===== 19-limit =====
+==== 13-limit ====
-Subgroup: 2.3.5.7.11.13.17.19
+Subgroup: 2.3.5.7.11.13
-Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137
+Comma list: 364/363, 441/440, 32805/32768, 109512/109375
-Mapping: {{mapping| 1 2 -1 -5 -9 14 5 4 | 0 -5 40 94 150 -124 -11 3 }}
+{{Mapping|legend=0| 1 2 -1 -5 -8 -11 | 0 -5 40 94 138 177 }}
 Optimal tunings:
-* WE: ~2 = 1199.9925{{c}}, ~18/17 = 99.6710{{c}}
+* WE: ~2 = 1200.0625{{c}}, ~35/33 = 99.6630{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6716{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6583{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~18/17 = 99.672{{c}} -->
-{{Optimal ET sequence|legend=0| 12, 301 }}
+{{Optimal ET sequence|legend=0| 12f, …, 277dff, 289 }}
-Badness (Sintel): 2.41
+Badness (Sintel): 3.07
-== Sextilifourths ==
+==== 17-limit ====
-The sextilifourths ({{nowrap| 130 & 159 }}, also known as ''sextilischis'', formerly ''sextilififths'') temperament slices the fourth (4/3) into six small semitones, which serves as both 21/20 and 22/21.
+Subgroup: 2.3.5.7.11.13.17
-[[Subgroup]]: 2.3.5.7
+Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768
-[[Comma list]]: 32805/32768, 235298/234375
+{{Mapping|legend=0| 1 2 -1 -5 -8 -11 5 | 0 -5 40 94 138 177 -11 }}
-{{Mapping|legend=1| 1 2 -1 -1 | 0 -6 48 55 }}
+Optimal tunings:
-: mapping generators: ~2, ~21/20
+* WE: ~2 = 1200.1286{{c}}, ~18/17 = 99.6668{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6568{{c}}
-[[Optimal tuning]]s:
+{{Optimal ET sequence|legend=0| 12f, 277dff, 289 }}
-* [[WE]]: ~2 = 1200.0987{{c}}, ~21/20 = 83.0599{{c}}
-: [[error map]]: {{val| +0.099 -0.117 +0.462 -0.630 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0543{{c}}
-: error map: {{val| 0.000 -0.281 +0.295 -0.837 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~21/20 = 83.053{{c}} -->
-{{Optimal ET sequence|legend=1| 29, 72cd, 101, 130, 289, 419 }}
+Badness (Sintel): 2.58
-[[Badness]] (Sintel): 2.75
+==== 19-limit ====
+Subgroup: 2.3.5.7.11.13.17.19
-=== 11-limit ===
+Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859
-Subgroup: 2.3.5.7.11
-Comma list: 441/440, 4000/3993, 235298/234375
+{{Mapping|legend=0| 1 2 -1 -5 -8 -11 5 4 | 0 -5 40 94 138 177 -11 3 }}
-Mapping: {{mapping| 1 2 -1 -1 0 | 0 -6 48 55 50 }}
 Optimal tunings:
-* WE: ~2 = 1200.0424{{c}}, ~21/20 = 83.0520{{c}}
+* WE: ~2 = 1200.0289{{c}}, ~18/17 = 99.6609{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0497{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6586{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~21/20 = 83.049{{c}} -->
-{{Optimal ET sequence|legend=0| 29, 72cde, 101e, 130, 289 }}
+{{Optimal ET sequence|legend=0| 12f, 289 }}
-Badness (Sintel): 1.50
+Badness (Sintel): 2.56
-=== 13-limit ===
+=== Quintahelenic ===
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11
-Comma list: 364/363, 441/440, 676/675, 10985/10976
+Comma list: 5632/5625, 8019/8000, 151263/151250
-Mapping: {{mapping| 1 2 -1 -1 0 1 | 0 -6 48 55 50 39 }}
+{{Mapping|legend=0| 1 2 -1 -5 -9 | 0 -5 40 94 150 }}
 Optimal tunings:
-* WE: ~2 = 1200.1056{{c}}, ~21/20 = 83.0566{{c}}
+* WE: ~2 = 1200.0195{{c}}, ~200/189 = 99.6723{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0508{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6709{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~21/20 = 83.049{{c}} -->
-{{Optimal ET sequence|legend=0| 29, 72cdef, 101e, 130, 289 }}
+{{Optimal ET sequence|legend=0| 12, …, 289e, 301, 915 }}
-Badness (Sintel): 1.04
+Badness (Sintel): 2.72
-== Septiquarschis ==
+==== 13-limit ====
-The septiquarschis temperament ({{nowrap| 89 & 94 }}) splits septimal minor seventh ([[7/4]]) into four generators and tempers out 829440/823543 (mynaslender comma) and 67108864/66706983 (septiness comma).
+Subgroup: 2.3.5.7.11.13
-[[Subgroup]]: 2.3.5.7
+Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000
-[[Comma list]]: 32805/32768, 829440/823543
+{{Mapping|legend=0| 1 2 -1 -5 -9 -11 | 0 -5 40 94 150 177 }}
-{{Mapping|legend=1| 1 -4 47 6 | 0 7 56 -4 }}
+Optimal tunings:
-: mapping generators: ~2, ~256/147
+* WE: ~2 = 1200.0442{{c}}, ~200/189 = 99.6709{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6675{{c}}
-[[Optimal tuning]]s:
+{{Optimal ET sequence|legend=0| 12f, …, 289e, 301 }}
-* [[WE]]: ~2 = 1199.8855{{c}}, ~256/147 = 957.2944{{c}}
-: [[error map]]: {{val| -0.114 -0.436 -0.182 +1.310 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~256/147 = 957.3867{{c}}
-: error map: {{val| 0.000 -0.248 +0.032 +1.627 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~256/147 = 957.386{{c}} -->
-{{Optimal ET sequence|legend=1| 89, 94, 183, 460d, 643d }}
+Badness (Sintel): 2.30
-[[Badness]] (Sintel): 4.73
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-=== 11-limit ===
+Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750
-Subgroup: 2.3.5.7.11
-Comma list: 540/539, 15488/15435, 32805/32768
-Mapping: {{mapping| 1 -4 47 6 25 | 0 7 56 -4 -27 }}
+{{Mapping|legend=0| 1 2 -1 -5 -9 -11 5 | 0 -5 40 94 150 177 -11 }}
 Optimal tunings:
-* WE: ~2 = 1199.9430{{c}}, ~256/147 = 957.3390{{c}}
+* WE: ~2 = 1200.1227{{c}}, ~200/189 = 99.6753{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3849{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6658{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~256/147 = 242.616{{c}} -->
-{{Optimal ET sequence|legend=0| 89, 94, 183, 460d }}
+{{Optimal ET sequence|legend=1| 12f, 289e, 301 }}
-Badness (Sintel): 1.72
+Badness (Sintel): 2.06
-=== 13-limit ===
+===== 19-limit =====
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 540/539, 729/728, 1573/1568, 4096/4095
+Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700
-Mapping: {{mapping| 1 -4 47 6 25 -33 | 0 7 56 -4 -27 46 }}
+{{Mapping|legend=0| 1 2 -1 -5 -9 -11 5 4 | 0 -5 40 94 150 177 -11 3 }}
 Optimal tunings:
-* WE: ~2 = 1200.0058{{c}}, ~256/147 = 957.3946{{c}}
+* WE: ~2 = 1200.0230{{c}}, ~200/189 = 99.6694{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3900{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6676{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~256/147 = 957.390{{c}} -->
-{{Optimal ET sequence|legend=0| 89, 94, 183, 277, 460d }}
+{{Optimal ET sequence|legend=0| 12f, 301 }}
-Badness (Sintel): 1.46
+Badness (Sintel): 2.24
-== Tsaharuk ==
+==== Quintahelenoid ====
-{{Main| Tsaharuk }}
+Subgroup: 2.3.5.7.11.13
-[[Subgroup]]: 2.3.5.7
+Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436
-[[Comma list]]: 32805/32768, 420175/419904
+{{Mapping|legend=0| 1 2 -1 -5 -9 14 | 0 -5 40 94 150 -124 }}
-{{Mapping|legend=1| 1 1 7 0 | 0 5 -40 24 }}
+Optimal tunings:
-: mapping generators: ~2, ~243/224
+* WE: ~2 = 1199.9919{{c}}, ~200/189 = 99.6712{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6718{{c}}
-[[Optimal tuning]]s:
+{{Optimal ET sequence|legend=0| 12, 301, 614, 915 }}
-* [[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 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~243/224 = 140.350{{c}} -->
-{{Optimal ET sequence|legend=1| 17, 77, 94, 171 }}
+Badness (Sintel): 2.73
-[[Badness]] (Sintel): 0.777
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-=== 11-limit ===
+Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157
-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 }}
+{{Mapping|legend=0| 1 2 -1 -5 -9 14 5 | 0 -5 40 94 150 -124 -11 }}
 Optimal tunings:
-* WE: ~2 = 1200.3103{{c}}, ~88/81 = 140.4011{{c}}
+* WE: ~2 = 1200.0469{{c}}, ~18/17 = 99.6749{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.3649{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6710{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~88/81 = 140.365{{c}} -->
-{{Optimal ET sequence|legend=0| 17, 77, 94, 171e, 265e }}
+{{Optimal ET sequence|legend=0| 12, 301 }}
-Badness (Sintel): 2.10
+Badness (Sintel): 2.44
-=== 13-limit ===
+===== 19-limit =====
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 352/351, 385/384, 729/728, 1331/1323
+Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137
-Mapping: {{mapping| 1 1 7 0 1 3 | 0 5 -40 24 21 6 }}
+{{Mapping|legend=0| 1 2 -1 -5 -9 14 5 4 | 0 -5 40 94 150 -124 -11 3 }}
 Optimal tunings:
-* WE: ~2 = 1200.1840{{c}}, ~13/12 = 140.3840{{c}}
+* WE: ~2 = 1199.9925{{c}}, ~18/17 = 99.6710{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.3627{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6716{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~13/12 = 140.363{{c}} -->
+{{Optimal ET sequence|legend=0| 12, 301 }}
-{{Optimal ET sequence|legend=0| 17, 77, 94, 171e }}
+Badness (Sintel): 2.41
-Badness (Sintel): 1.57
+== 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.
-== Quanharuk ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 16875/16807, 32805/32768
+[[Comma list]]: 32805/32768, 235298/234375
-{{Mapping|legend=1| 1 0 15 12 | 0 5 -40 -29 }}
+{{Mapping|legend=1| 1 2 -1 -1 | 0 -6 48 55 }}
-: mapping generators: ~2, ~56/45
+: mapping generators: ~2, ~21/20
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1200.0032{{c}}, ~56/45 = 380.3557{{c}}
+* [[WE]]: ~2 = 1200.0987{{c}}, ~21/20 = 83.0599{{c}}
-: [[error map]]: {{val| +0.003 -0.177 -0.493 +0.898 }}
+: [[error map]]: {{val| +0.099 -0.117 +0.462 -0.630 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~56/45 = 380.3546{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0543{{c}}
-: error map: {{val| 0.000 -0.182 -0.498 +0.890 }}
+: error map: {{val| 0.000 -0.281 +0.295 -0.837 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~56/45 = 380.355{{c}} -->
-{{Optimal ET sequence|legend=1| 41, 142, 183, 224 }}
+{{Optimal ET sequence|legend=1| 29, 72cd, 101, 130, 289, 419 }}
-[[Badness]] (Sintel): 1.82
+[[Badness]] (Sintel): 2.75
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 540/539, 1375/1372, 32805/32768
+Comma list: 441/440, 4000/3993, 235298/234375
-Mapping: {{mapping| 1 0 15 12 -7 | 0 5 -40 -29 33 }}
+{{Mapping|legend=0| 1 2 -1 -1 0 | 0 -6 48 55 50 }}
 Optimal tunings:
-* WE: ~2 = 1199.9709{{c}}, ~56/45 = 380.3423{{c}}
+* WE: ~2 = 1200.0424{{c}}, ~21/20 = 83.0520{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3517{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0497{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~56/45 = 380.352{{c}} -->
-{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}
+{{Optimal ET sequence|legend=0| 29, 72cde, 101e, 130, 289 }}
-Badness (Sintel): 1.04
+Badness (Sintel): 1.50
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 729/728, 1375/1372, 4096/4095
+Comma list: 364/363, 441/440, 676/675, 10985/10976
-Mapping: {{mapping| 1 0 15 12 -7 -15 | 0 5 -40 -29 33 59 }}
+{{Mapping|legend=0| 1 2 -1 -1 0 1 | 0 -6 48 55 50 39 }}
 Optimal tunings:
-* WE: ~2 = 1199.9663{{c}}, ~56/45 = 380.3403{{c}}
+* WE: ~2 = 1200.1056{{c}}, ~21/20 = 83.0566{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3509{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0508{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~56/45 = 380.351{{c}} -->
-{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}
+{{Optimal ET sequence|legend=0| 29, 72cdef, 101e, 130, 289 }}
-Badness (Sintel): 0.884
+Badness (Sintel): 1.04
-== Quadrant ==
+== Septant ==
-The ''quadrant'' temperament ({{nowrap| 12 & 224 }}) has a period of quarter octave and tempers out the [[dimcomp comma]], 390625/388962. In this temperament, 25/21 is mapped into quarter octave.
+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, 390625/388962
+[[Comma list]]: 32805/32768, 516560652/514714375
-{{Mapping|legend=1| 4 0 60 119 | 0 1 -8 -17 }}
+{{Mapping|legend=1| 7 0 105 -56 | 0 1 -8 7 }}
-: mapping generators: ~25/21, ~3
+: mapping generators: ~8575/7776, ~3
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 300.0255{{c}}, ~3/2 = 701.8831{{c}}
+* [[WE]]: ~8575/7776 = 171.4303{{c}}, ~3/2 = 701.7091{{c}}
-: [[error map]]: {{val| +0.102 +0.030 -0.664 +0.462 }}
+: [[error map]]: {{val| +0.012 -0.234 +0.096 +0.265 }}
-* [[CWE]]: ~2 = 300.0000{{c}}, ~3/2 = 701.8180{{c}}
+* [[CWE]]: ~8575/7776 = 171.4286{{c}}, ~3/2 = 701.7022{{c}}
-: error map: {{val| 0.000 -0.137 -0.858 +0.268 }}
+: error map: {{val| 0.000 -0.253 +0.069 +0.232 }}
-<!-- * [[POTE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8234{{c}} -->
-{{Optimal ET sequence|legend=1| 12, …, 200, 212, 224, 436, 660 }}
+{{Optimal ET sequence|legend=1| 77, 147, 224, 301, 525, 826, 1351 }}
-[[Badness]] (Sintel): 2.79
+[[Badness]] (Sintel): 2.81
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 1375/1372, 6250/6237, 32805/32768
+Comma list: 3025/3024, 24057/24010, 32805/32768
-Mapping: {{mapping| 4 0 60 119 185 | 0 1 -8 -17 -27 }}
+{{Mapping|legend=0| 7 0 105 -56 -120 | 0 1 -8 7 13 }}
 Optimal tunings:
-* WE: ~25/21 = 300.0244{{c}}, ~3/2 = 701.8759{{c}}
+* WE: ~495/448 = 171.4334{{c}}, ~3/2 = 701.7387{{c}}
-* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8145{{c}}
+* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7198{{c}}
-<!-- * POTE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8176{{c}} -->
-{{Optimal ET sequence|legend=0| 12, …, 212, 224, 436, 660 }}
+{{Optimal ET sequence|legend=0| 77, 147, 224, 301, 525 }}
-Badness (Sintel): 1.51
+Badness (Sintel): 1.46
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647
+Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024
-Mapping: {{mapping| 4 0 60 119 185 224 | 0 1 -8 -17 -27 -33 }}
+{{Mapping|legend=0| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }}
 Optimal tunings:
-* WE: ~25/21 = 300.0234{{c}}, ~3/2 = 701.8707{{c}}
+* WE: ~495/448 = 171.4282{{c}}, ~3/2 = 701.7229{{c}}
-* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8123{{c}}
+* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7242{{c}}
-<!-- * POTE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8158{{c}} -->
-{{Optimal ET sequence|legend=0| 12f, …, 212, 224, 436, 660 }}
+{{Optimal ET sequence|legend=0| 77, 147, 224, 525, 1274f }}
-Badness (Sintel): 1.13
+Badness (Sintel): 1.02
-== Septant ==
+== Octant ==
-The septant temperament ({{nowrap| 224 & 301 }}) has a period of 1/7 octave and tempers out the [[akjaysma]], {{monzo| 47 -7 -7 -7 }}.
+Octant may be described as the {{nowrap| 224 & 248 }} temperament. It has a period of 1/8 octave, and its [[ploidacot]] is octaploid monocot. In this temperament, [[12/11]], [[35/27]], and [[99/70]] are mapped to 1\8, 3\8, and 4\8 respectively.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 32805/32768, 516560652/514714375
+[[Comma list]]: 32805/32768, 2259436291848/2251875390625
-{{Mapping|legend=1| 7 0 105 -56 | 0 1 -8 7 }}
+{{Mapping|legend=1| 8 0 120 -117 | 0 1 -8 11 }}
-: mapping generators: ~8575/7776, ~3
+: mapping generators: ~42875/39366, ~3
 [[Optimal tuning]]s:
-* [[WE]]: ~8575/7776 = 171.4303{{c}}, ~3/2 = 701.7091{{c}}
+* [[WE]]: ~42875/39366 = 150.0048{{c}}, ~3/2 = 701.7356{{c}}
-: [[error map]]: {{val| +0.012 -0.234 +0.096 +0.265 }}
+: [[error map]]: {{val| +0.039 -0.181 +0.071 +0.127 }}
-* [[CWE]]: ~8575/7776 = 171.4286{{c}}, ~3/2 = 701.7022{{c}}
+* [[CWE]]: ~42875/39366 = 150.0000{{c}}, ~3/2 = 701.7134{{c}}
-: error map: {{val| 0.000 -0.253 +0.069 +0.232 }}
+: error map: {{val| 0.000 -0.242 -0.021 +0.022 }}
-<!-- * [[POTE]]: ~8575/7776 = 171.429{{c}}, ~3/2 = 701.702{{c}} -->
-{{Optimal ET sequence|legend=1| 77, 147, 224, 301, 525, 826, 1351 }}
+{{Optimal ET sequence|legend=1| 24, …, 224, 472, 696, 1168 }}
-[[Badness]] (Sintel): 2.81
+[[Badness]] (Sintel): 3.98
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 24057/24010, 32805/32768
+Comma list: 9801/9800, 32805/32768, 46656/46585
-Mapping: {{mapping| 7 0 105 -56 -120 | 0 1 -8 7 13 }}
+{{Mapping|legend=0| 8 0 120 -117 15 | 0 1 -8 11 1 }}
 Optimal tunings:
-* WE: ~495/448 = 171.4334{{c}}, ~3/2 = 701.7387{{c}}
+* WE: ~12/11 = 150.0010{{c}}, ~3/2 = 701.7177{{c}}
-* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7198{{c}}
+* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7131{{c}}
-<!-- * POTE: ~495/448 = 171.429{{c}}, ~3/2 = 701.719{{c}} -->
-{{Optimal ET sequence|legend=0| 77, 147, 224, 301, 525 }}
+{{Optimal ET sequence|legend=0| 24, …, 224, 472, 696, 1168 }}
-Badness (Sintel): 1.46
+Badness (Sintel): 1.48
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024
+Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655
-Mapping: {{mapping| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }}
+{{Mapping|legend=0| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }}
 Optimal tunings:
-* WE: ~495/448 = 171.4282{{c}}, ~3/2 = 701.7229{{c}}
+* WE: ~12/11 = 149.9957{{c}}, ~3/2 = 701.7046{{c}}
-* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7242{{c}}
+* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7247{{c}}
-<!-- * POTE: ~495/448 = 171.429{{c}}, ~3/2 = 701.724{{c}} -->
-{{Optimal ET sequence|legend=0| 77, 147, 224, 525, 1274f }}
+{{Optimal ET sequence|legend=0| 24, 224, 472, 696 }}
-Badness (Sintel): 1.02
+Badness (Sintel): 1.26
-== Octant ==
+== Nonant ==
-The octant temperament ({{nowrap| 224 & 472 }}) has a period of 1/8 octave. In this temperament, 12/11, 35/27, and 99/70 are mapped into 1\8, 3\8, and 4\8 respectively.
+Named by [[Xenllium]] in 2023, nonant tempers out the [[septimal ennealimma]] ({{monzo| -11 -9 0 9 }}) and may be described as the {{nowrap| 36 & 171 }} temperament. It has a period of 1/9 octave, and its [[ploidacot]] is enneaploid monocot.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 32805/32768, 2259436291848/2251875390625
+[[Comma list]]: 32805/32768, 40353607/40310784
-{{Mapping|legend=1| 8 0 120 -117 | 0 1 -8 11 }}
+{{Mapping|legend=1| 9 0 135 11 | 0 1 -8 1 }}
-: mapping generators: ~42875/39366, ~3
+: mapping generators: ~2592/2401, ~3
 [[Optimal tuning]]s:
-* [[WE]]: ~42875/39366 = 150.0048{{c}}, ~3/2 = 701.7356{{c}}
+* [[WE]]: ~2592/2401 = 133.3442{{c}}, ~3/2 = 701.8000{{c}}
-: [[error map]]: {{val| +0.039 -0.181 +0.071 +0.127 }}
+: [[error map]]: {{val| +0.098 -0.057 -0.027 -0.141 }}
-* [[CWE]]: ~42875/39366 = 150.0000{{c}}, ~3/2 = 701.7134{{c}}
+* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.7384{{c}}
-: error map: {{val| 0.000 -0.242 -0.021 +0.022 }}
+: error map: {{val| 0.000 -0.217 -0.221 -0.421 }}
-<!-- * [[POTE]]: ~42875/39366 = 150.000{{c}}, ~3/2 = 701.713{{c}} -->
-{{Optimal ET sequence|legend=1| 24, …, 224, 472, 696, 1168 }}
+{{Optimal ET sequence|legend=1| 36, 99c, 135, 171, 2772bd, 2943bdd, …, 5166bccddd, 5337bccddd }}
-[[Badness]] (Sintel): 3.98
+[[Badness]] (Sintel): 1.77
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 9801/9800, 32805/32768, 46656/46585
+Comma list: 540/539, 32805/32768, 42875/42592
-Mapping: {{mapping| 8 0 120 -117 15 | 0 1 -8 11 1 }}
+{{Mapping|legend=0| 9 0 135 11 131 | 0 1 -8 1 -7 }}
 Optimal tunings:
-* WE: ~12/11 = 150.0010{{c}}, ~3/2 = 701.7177{{c}}
+* WE: ~242/225 = 133.3308{{c}}, ~3/2 = 701.8205{{c}}
-* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7131{{c}}
+* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.8351{{c}}
-<!-- * POTE: ~12/11 = 150.000{{c}}, ~3/2 = 701.713{{c}} -->
-{{Optimal ET sequence|legend=0| 24, …, 224, 472, 696, 1168 }}
+{{Optimal ET sequence|legend=0| 36, 135, 171 }}
-Badness (Sintel): 1.48
+Badness (Sintel): 4.20
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655
+Comma list: 540/539, 729/728, 4096/4095, 16807/16731
-Mapping: {{mapping| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }}
+{{Mapping|legend=0| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }}
 Optimal tunings:
-* WE: ~12/11 = 149.9957{{c}}, ~3/2 = 701.7046{{c}}
+* WE: ~242/225 = 133.3180{{c}}, ~3/2 = 701.6956{{c}}
-* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7247{{c}}
+* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.7800{{c}}
-<!-- * POTE: ~12/11 = 150.000{{c}}, ~3/2 = 701.725{{c}} -->
-{{Optimal ET sequence|legend=0| 24, 224, 472, 696 }}
+{{Optimal ET sequence|legend=0| 36, 99cf, 135, 171 }}
-Badness (Sintel): 1.26
+Badness (Sintel): 3.15
-== Nonant ==
+== Septiquarschis ==
-The ''nonant'' temperament ({{nowrap| 36 & 135 }}) has a period of 1/9 octave and tempers out the [[septimal ennealimma]], {{monzo| -11 -9 0 9 }}.
+Named by [[Xenllium]] in 2021, septiquarschis tempers out [[829440/823543]] (mynaslender comma) and [[67108864/66706983]] (septiness comma), and may be described as the {{nowrap| 89 & 94 }} temperament. It splits septimal minor seventh ([[7/4]]) into four generators. Note that in the data below, the generator is the [[octave complement]] so that seven of them minus five octaves make a [[3/2|perfect fifth]]; its [[ploidacot]] is thus epsilon-heptacot.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 32805/32768, 40353607/40310784
+[[Comma list]]: 32805/32768, 829440/823543
-{{Mapping|legend=1| 9 0 135 11 | 0 1 -8 1 }}
+{{Mapping|legend=1| 1 -4 47 6 | 0 7 56 -4 }}
-: mapping generators: ~2592/2401, ~3
+: mapping generators: ~2, ~256/147
 [[Optimal tuning]]s:
-* [[WE]]: ~2592/2401 = 133.3442{{c}}, ~3/2 = 701.8000{{c}}
+* [[WE]]: ~2 = 1199.8855{{c}}, ~256/147 = 957.2944{{c}}
-: [[error map]]: {{val| +0.098 -0.057 -0.027 -0.141 }}
+: [[error map]]: {{val| -0.114 -0.436 -0.182 +1.310 }}
-* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.7384{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~256/147 = 957.3867{{c}}
-: error map: {{val| 0.000 -0.217 -0.221 -0.421 }}
+: error map: {{val| 0.000 -0.248 +0.032 +1.627 }}
-<!-- * [[CTE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.7232{{c}} -->
-{{Optimal ET sequence|legend=1| 36, 99c, 135, 171, 2772bd, 2943bdd, …, 5166bccddd, 5337bccddd }}
+{{Optimal ET sequence|legend=1| 89, 94, 183, 460d, 643d }}
-[[Badness]] (Sintel): 1.77
+[[Badness]] (Sintel): 4.73
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 540/539, 32805/32768, 42875/42592
+Comma list: 540/539, 15488/15435, 32805/32768
-Mapping: {{mapping| 9 0 135 11 131 | 0 1 -8 1 -7 }}
+{{Mapping|legend=0| 1 -4 47 6 25 | 0 7 56 -4 -27 }}
 Optimal tunings:
-* WE: ~242/225 = 133.3308{{c}}, ~3/2 = 701.8205{{c}}
+* WE: ~2 = 1199.9430{{c}}, ~256/147 = 957.3390{{c}}
-* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.8351{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3849{{c}}
-<!-- * CTE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.8398{{c}} -->
-{{Optimal ET sequence|legend=0| 36, 135, 171 }}
+{{Optimal ET sequence|legend=0| 89, 94, 183, 460d }}
-Badness (Sintel): 4.20
+Badness (Sintel): 1.72
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 729/728, 4096/4095, 16807/16731
+Comma list: 540/539, 729/728, 1573/1568, 4096/4095
-Mapping: {{mapping| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }}
+{{Mapping|legend=0| 1 -4 47 6 25 -33 | 0 7 56 -4 -27 46 }}
 Optimal tunings:
-* WE: ~242/225 = 133.3180{{c}}, ~3/2 = 701.6956{{c}}
+* WE: ~2 = 1200.0058{{c}}, ~256/147 = 957.3946{{c}}
-* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.7800{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3900{{c}}
-<!-- * CTE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.7998{{c}} -->
-{{Optimal ET sequence|legend=0| 36, 99cf, 135, 171 }}
+{{Optimal ET sequence|legend=0| 89, 94, 183, 277, 460d }}
-Badness (Sintel): 3.15
+Badness (Sintel): 1.46
-== Tridecafifths ==
+== Subgroup extensions ==
-Tridecafifths divides the perfect 3/2 into 13 quartertones.
+=== Tridecaschismic (2.3.5.13) ===
+Proposed by [[Eufalesio]] in 2026, tridecaschismic adds the [[325/324|marveltwin comma]] to the comma list, or equivalently, the [[tridecapyth comma]]. It benefits from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is one of the lowest badness schismic extensions. It is also equivalent to the [[2.3.5.13 subgroup|2.3.5.13-subgroup]] [[restriction]] of 13-limit [[cassandra]].
-[[Subgroup]]: 2.3.5.7
+Subgroup: 2.3.5.13
-[[Comma list]]: 32805/32768, {{monzo| -14 -1 -9 13 }}
+Comma list: 325/324, 32805/32768
-{{Mapping|legend=1| 1 1 7 6 | 0 13 -104 -71 }}
+{{Mapping|legend=2| 1 0 15 -28 | 0 1 -8 20 }}
-: mapping generators: ~2, ~1323/1280
-[[Optimal tuning]]s:
+Optimal tunings:
-* [[WE]]: ~2 = 1200.1431{{c}}, ~1323/1280 = 53.9838{{c}}
+* WE: ~2 = 1200.3326{{c}} ~3/2 = 702.1092{{c}}
-: [[error map]]: {{val| +0.143 -0.023 +0.375 -0.816 }}
+* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9189{{c}}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~1323/1280 = 53.9764{{c}}
-: error map: {{val| 0.000 -0.261 -0.221 -0.421 }}
-<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~1323/1280 = 53.9741{{c}} -->
-{{Optimal ET sequence|legend=1| 89, 200, 289 }}
+{{Optimal ET sequence|legend=0| 12, …, 41, 53, 412cf, 465cf, …, 783ccff, 836ccfff }}
-[[Badness]] (Sintel): 10.9
+Badness (Sintel): 0.582
-=== 11-limit ===
+==== 2.3.5.13.19 subgroup ====
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.13.19
-Comma list: 441/440, 32805/32768, 55296000/55240493
+Comma list: 325/324, 361/360, 513/512
-Mapping: {{mapping| 1 1 7 6 4 | 0 13 -104 -71 -12 }}
+{{Mapping|legend=2| 1 0 15 -28 9 | 0 1 -8 20 -3 }}
 Optimal tunings:
-* WE: ~2 = 1200.0311{{c}}, ~33/32 = 53.9766{{c}}
+* WE: ~2 = 1200.4236{{c}}, ~3/2 = 702.1510{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 53.9750{{c}}
+* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9064{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~33/32 = 53.9744{{c}} -->
-{{Optimal ET sequence|legend=0| 89, 200, 289 }}
+{{Optimal ET sequence|legend=0| 12, …, 41, 53 }}
-Badness (Sintel): 4.23
+Badness (Sintel): 0.354
-== Subgroup extensions ==
 === Photia (2.3.5.17) ===
 {{See also| No-elevens subgroup temperaments #Garibaldia }}
@@ Line 2,881: / Line 2,790: @@
 [[Comma list]]: 256/255, 1458/1445
-{{Mapping|legend=2| 1 0 15 -7 | 0 1 -8 7 }}
+{{Mapping|legend=3| 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|legend=5| 1 0 15 0 0 0 -7 | 0 1 -8 0 0 0 7 }}
 : mapping generators: ~2, ~3
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.491{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.5471{{c}}, ~3/2 = 701.2262{{c}}
+: [[error map]]: {{val| -0.453 -1.182 +0.706 +3.628 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4976{{c}}
+: error map: {{val| 0.000 -0.457 +1.705 +5.528 }}
-{{Optimal ET sequence|legend=1| 12, 41, 53, 65 }}
+{{Optimal ET sequence|legend=1| 12, 41, 53, 65, 207g, 272gg }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.4842 cents
+[[Badness]] (Sintel): 0.479
 ==== 2.3.5.17.19 subgroup ====
@@ Line 2,897: / Line 2,810: @@
 Comma list: 171/170, 256/255, 324/323
-Subgroup-val mapping: {{mapping| 1 0 15 -7 9 | 0 1 -8 7 -3 }}
+{{Mapping|legend=2| 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 }}
+{{Mapping|legend=4| 1 0 15 0 0 0 -7 9 | 0 1 -8 0 0 0 7 -3 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 701.470{{c}}
+Optimal tunings:
+* WE: ~2 = 1199.7225{{c}}, ~3/2 = 701.3077{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.4754{{c}}
-{{Optimal ET sequence|legend=0| 12, 41, 53, 65 }}
+{{Optimal ET sequence|legend=0| 12, 41, 53, 65, 142g }}
-RMS error: 0.5374 cents
+Badness (Sintel): 0.332
 === Nestoria (2.3.5.19) ===
-: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''
+{{See also| No-elevens subgroup temperaments #Garibaldia | No-elevens subgroup temperaments #Pontia }}
-The [[S-expression]]-based comma list of this temperament is {[[1216/1215|S16/S18]], [[361/360|S19]] (, ''[[513/512|S15/S20]]'')}. Strangely, despite prime 19 being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is actually sharper than the fifth in optimal schismic. This is likely due to its optimization considering intervals like 19/10 and 19/15.
+Nestoria is notable for having one of the lowest-badness subgroup extensions of schismic. Note that despite prime [[19/1|19]] being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is generally not flatter than the fifth in optimal schismic due to its optimization considering intervals like [[19/10]] and [[19/15]]. However, the dyadic tuning sensitivity of [[19/16]] suggests using tunings like [[65edo]] and [[77edo]] to optimize in favour of prime 19 (especially the minor triad ~16:19:24 which is equated with the Pythagorean minor triad), as [[171edo]] is already arguably undertempered for it despite being the optimal patent val.
 [[Subgroup]]: 2.3.5.19
@@ Line 2,916: / Line 2,831: @@
 [[Comma list]]: 361/360, 513/512
-{{Mapping|legend=2| 1 0 15 9 | 0 1 -8 -3 }}
+{{Mapping|legend=3| 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|legend=5| 1 0 15 0 0 0 0 9 | 0 1 -8 0 0 0 0 -3 }}
 : mapping generators: ~2, ~3
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.746{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.2250{{c}}, ~3/2 = 701.8776{{c}}
+: [[error map]]: {{val| +0.225 +0.148 +0.240 -1.796 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7307{{c}}
+: error map: {{val| 0.000 -0.224 -0.159 -2.705 }}
-{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171 }}
+{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171, 460hh, 631hh }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.1763 cents
+[[Badness]] (Sintel): 0.126
 === Taylor (2.3.5.13) ===
@@ Line 2,934: / Line 2,853: @@
 [[Comma list]]: 676/675, 32805/32768
-{{Mapping|legend=2| 1 0 15 14 | 0 2 -16 -13 }}
+{{Mapping|legend=3| 1 0 15 14 | 0 2 -16 -13 }}
-{{Mapping|legend=3| 1 0 15 0 0 14 | 0 2 -16 0 0 -13 }}
+{{Mapping|legend=5| 1 0 15 0 0 14 | 0 2 -16 0 0 -13 }}
 : mapping generators: ~2, ~26/15
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~26/15 = 950.855{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1497{{c}}, ~26/15 = 950.9740{{c}}
+: [[error map]]: {{val| +0.150 -0.007 +0.348 -1.094 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 950.8493{{c}}
+: error map: {{val| 0.000 -0.256 +0.098 -1.568 }}
-{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236 }}
+{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236, 525f, 761ff }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.1485 cents
+[[Badness]] (Sintel): 0.334
 ==== Dakota (2.3.5.13.19) ====
@@ Line 2,950: / Line 2,873: @@
 Comma list: 361/360, 513/512, 676/675
-Subgroup-val mapping: {{mapping| 1 0 15 14 9 | 0 2 -16 -13 -6 }}
+{{Mapping|legend=2| 1 0 15 14 9 | 0 2 -16 -13 -6 }}
-Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~26/15 = 950.8199{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.2611{{c}}, ~26/15 = 951.0703{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8532{{c}}
 {{Optimal ET sequence|legend=0| 24, 29, 53, 130, 183, 236h, 289h }}
-Badness (Smith): 0.00575
+Badness (Sintel): 0.262
 ===== 2.3.5.13.19.37 subgroup =====
@@ Line 2,963: / Line 2,888: @@
 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 }}
+{{Mapping|legend=2| 1 0 15 14 9 6 | 0 2 -16 -13 -6 -1 }}
-Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~26/15 = 950.8187{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.2987{{c}}, ~26/15 = 951.1060{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8595{{c}}
 {{Optimal ET sequence|legend=0| 24, 29, 53, 183, 236h, 289hl, 631fhhll }}
-Badness (Smith): 0.00357
+Badness (Sintel): 0.223
 === Quintilischis (2.3.5.17) ===
@@ Line 2,978: / Line 2,905: @@
 [[Comma list]]: 32805/32768, 1419857/1417176
-{{Mapping|legend=2| 1 2 -1 5 | 0 -5 40 -11 }}
+{{Mapping|legend=3| 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|legend=5| 1 2 -1 0 0 0 5 | 0 -5 40 0 0 0 -11 }}
 : mapping generators: ~2, ~18/17
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~18/17 = 99.649{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1370{{c}}, ~18/17 = 99.6602{{c}}
+: [[error map]]: {{val| +0.137 +0.018 -0.042 -0.533 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~18/17 = 99.6499{{c}}
+: error map: {{val| 0.000 -0.205 -0.317 -1.104 }}
-{{Optimal ET sequence|legend=1| 12, 253, 265, 277, 289 }}
+{{Optimal ET sequence|legend=1| 12, …, 253, 265, 277, 289, 566g, 855g }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.0719 cents
+[[Badness]] (Sintel): 1.34
 ==== 2.3.5.17.19 subgroup ====
@@ Line 2,994: / Line 2,925: @@
 Comma list: 4624/4617, 6144/6137, 6885/6859
-Subgroup-val mapping: {{mapping| 1 2 -1 5 4 | 0 -5 40 -11 3 }}
+{{Mapping|legend=2| 1 2 -1 5 4 | 0 -5 40 -11 3 }}
-Gencom mapping: {{mapping| 1 2 -1 0 0 0 5 4 | 0 -5 40 0 0 0 -11 3 }}
+{{Mapping|legend=4| 1 2 -1 0 0 0 5 4 | 0 -5 40 0 0 0 -11 3 }}
-Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~18/17 = 99.652{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.0350{{c}}, ~18/17 = 99.6550{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6520{{c}}
-{{Optimal ET sequence|legend=0| 12, 253, 265, 277, 289 }}
+{{Optimal ET sequence|legend=0| 12, …, 253, 265, 277, 289 }}
-RMS error: 0.1636 cents
+Badness (Sintel): 1.17
+== References ==
+<references/>
+[[Category:Schismatic family| ]] <!-- main article -->
 [[Category:Temperament families]]
-[[Category:Schismatic family| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]