Schismatic family: Difference between revisions

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{{interwiki
{{interwiki
| en = Schismatic family
| de = Schismatische Temperaturen
| de = Schismatische Temperaturen
| en =
| es =  
| es =  
| ja =  
| ja =  
}}
}}
{{Technical data page}}
{{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]]. The generator is a fifth and 5/4 is represented by a diminished fourth.
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]].  
 
This defies the tradition of tertian harmony, as the just major triad on C is {{nowrap|{{dash|C, F♭, G|hair|med}}}}, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as {{nowrap|{{dash|C, vE, G|hair|med}}}}.  


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


The 5-limit version of the temperament is a [[microtemperament]], called ''schismic'', ''schismatic'', or ''helmholtz'', which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. [[53edo]] is a possible tuning for schismic, but you need [[118edo]] if you want to get the full effect. In exact analogy with [[1/4-comma meantone]] there is also 1/8 schismic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244{{cent}}, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better fifth, or 2/17 schisma, with both thirds flat by 1/17 of a schisma, although the differences would be very hard to distinguish unless using a large gamut. Simply leaving the fifths just would also make for a viable tuning, thus collapsing schismic to a simple relabeling of the 3-limit.  
The 5-limit version of the temperament is a [[microtemperament]], called ''schismic'', ''schismatic'', or ''helmholtz''. The generator is a fifth, flattened by a fraction of a schisma, and 5/4 is represented by a diminished fourth. This defies the tradition of {{w|tertian harmony}}, as the [[just major triad]] on C is C–F♭–G, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as C–vE–G.  
 
As a 5-limit system, schismic is far more accurate than [[meantone]] but still with manageable [[complexity]]. [[53edo]] is a possible tuning for schismic, but you need [[118edo]] if you want to get the full effect. In exact analogy with [[1/4-comma meantone]] there is also 1/8 schismic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244{{cent}}, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better fifth, or 2/17 schisma, with both thirds flat by 1/17 of a schisma, although the differences would be very hard to distinguish unless using a large gamut. Simply leaving the fifths just would also make for a viable tuning, thus collapsing schismic to a simple relabeling of the 3-limit.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 20: Line 20:


{{Mapping|legend=1| 1 0 15 | 0 1 -8 }}
{{Mapping|legend=1| 1 0 15 | 0 1 -8 }}
 
: mapping generators: ~2, ~3
: Mapping generators: ~2, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 701.7187
* [[WE]]: ~2 = 1200.0749{{c}}, ~3/2 = 701.7797{{c}}
* [[POTE]]: ~2 = 1\1, ~3/2 = 701.7359
: [[error map]]: {{val| +0.075 -0.100 -0.027 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7308{{c}}
: error map: {{val| 0.000 -0.224 -0.160 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* 5-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 705.882] (4\7 to 10\17)
* [[5-odd-limit]] [[diamond monotone]]: ~3/2 = [685.714, 705.882] (4\7 to 10\17)
* 5-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 701.955] (1/8-comma to untempered)
* 5-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 701.955] (1/8-comma to untempered)
* 5-odd-limit diamond monotone and tradeoff: ~3/2 = [701.711, 701.955]


{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bbc, 6452bbcc }}
{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bbc, 6452bbcc }}


[[Badness]]: 0.004259
[[Badness]] (Sintel): 0.0999


=== Overview to extensions ===
=== Overview to extensions ===
The second comma of the [[Normal lists #Normal interval list|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 }},  
[[#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.  
* [[#Schism|Schism]] adds [[64/63|{{monzo| 6 -2 0 -1 }}]],
 
* [[#Pontiac|Pontiac]] adds {{monzo| -59 39 0 -1 }}.  
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.  
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.
Temperaments discussed elsewhere include:
* [[#Hemischis|Hemischis]] adds {{monzo| -34 25 0 -2 }} and has a hemififth generator.
* ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
* [[Gamelismic clan #Guiron|Guiron]] adds [[1029/1024|{{monzo| -10 1 0 3 }}]], with an ~8/7 generator, three of which give the fifth.
* ''[[Pogo]]'' (+118098/117649) → [[Stearnsmic clan #Pogo|Stearnsmic clan]]
* [[#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
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.
* [[Sensamagic clan #Salsa|Salsa]]
* [[Gamelismic clan #Guiron|Guiron]]


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


== Garibaldi ==
== Garibaldi ==
{{Main| Garibaldi }}
{{Main| Garibaldi }}


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 67: Line 62:
{{Mapping|legend=1| 1 0 15 25 | 0 1 -8 -14 }}
{{Mapping|legend=1| 1 0 15 25 | 0 1 -8 -14 }}


: Mapping generators: ~2, ~3
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1200.1233{{c}}, ~3/2 = 702.1573{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.085
: [[error map]]: {{val| +0.123 +0.326 -2.709 +2.328 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0774{{c}}
: error map: {{val| 0.000 +0.122 -2.933 +2.090 }}


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[7-odd-limit]]: ~3/2 = {{monzo| 2/3 1/15 0 -1/15 }}
* [[7-odd-limit]]: ~3/2 = {{monzo| 2/3 1/15 0 -1/15 }}
: {{monzo list| 1 0 0 0 | 5/3 1/15 0 -1/15 | 5/3 -8/15 0 8/15 | 5/3 -14/15 0 14/15 }}
: {{monzo list| 1 0 0 0 | 5/3 1/15 0 -1/15 | 5/3 -8/15 0 8/15 | 5/3 -14/15 0 14/15 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/3
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
* [[9-odd-limit]]: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }}
* [[9-odd-limit]]: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }}
: {{monzo list| 1 0 0 0 | 25/16 1/8 0 -1/16 | 5/2 -1 0 1/2 | 25/8 -7/4 0 7/8 }}
: {{monzo list| 1 0 0 0 | 25/16 1/8 0 -1/16 | 5/2 -1 0 1/2 | 25/8 -7/4 0 7/8 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 83: Line 80:
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 702.915]
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 702.915]


{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 94, 241c, 335cd, 576ccd }}
{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 94 }}


[[Badness]]: 0.021644
[[Badness]] (Sintel): 0.548


=== Cassandra ===
=== Cassandra ===
Cassandra is one of the best extension of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup.  
Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup, even though it comes with a much higher complexity.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 96: Line 93:
Mapping: {{mapping| 1 0 15 25 -33 | 0 1 -8 -14 23 }}
Mapping: {{mapping| 1 0 15 25 -33 | 0 1 -8 -14 23 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.157
Optimal tunings:  
* WE: ~2 = 1200.3089{{c}}, ~3/2 = 702.3377{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1562{{c}}


Minimax tuning:
Minimax tuning:
* 11-odd-limit: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }}
* 11-odd-limit: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }}
: eigenmonzo (unchanged-interval) basis: 2.9/7
: unchanged-interval (eigenmonzo) basis: 2.9/7


Tuning ranges:  
Tuning ranges:  
Line 106: Line 105:
* 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 702.915]
* 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 702.915]


{{Optimal ET sequence|legend=1| 41, 53, 94, 229c, 323c, 417cce }}
{{Optimal ET sequence|legend=0| 12e, 41, 53, 94, 229c }}


Badness: 0.027396
Badness (Sintel): 0.906


==== 13-limit ====
==== 13-limit ====
Line 117: Line 116:
Mapping: {{mapping| 1 0 15 25 -33 -28 | 0 1 -8 -14 23 20 }}
Mapping: {{mapping| 1 0 15 25 -33 -28 | 0 1 -8 -14 23 20 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.113
Optimal tunings:  
* WE: ~2 = 1200.1703{{c}}, ~3/2 = 702.2122{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1135{{c}}


Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~3/2 = {{monzo| 19/34 0 0 -1/34 0 1/34 }}
* 13- and 15-odd-limit: ~3/2 = {{monzo| 19/34 0 0 -1/34 0 1/34 }}
: eigenmonzo (unchanged-interval) basis: 2.13/7
: unchanged-interval (eigenmonzo) basis: 2.13/7


Tuning ranges:  
Tuning ranges:  
Line 128: Line 129:
* 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 703.597]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 703.597]


{{Optimal ET sequence|legend=1| 41, 53, 94, 429ccdeef, 523ccdeef }}
{{Optimal ET sequence|legend=0| 41, 53, 94, 429ccdeef, 523ccdeef }}


Badness: 0.020676
Badness (Sintel): 0.854


===== Cassie =====
===== Cassie =====
Line 139: Line 140:
Mapping: {{mapping| 1 0 15 25 -33 -28 -7 | 0 1 -8 -14 23 20 7 }}
Mapping: {{mapping| 1 0 15 25 -33 -28 -7 | 0 1 -8 -14 23 20 7 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.092
Optimal tunings:
* WE: ~2 = 1199.8140{{c}}, ~3/2 = 701.9833{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0909{{c}}


{{Optimal ET sequence|legend=1| 41, 53, 94g }}
{{Optimal ET sequence|legend=0| 12e, 41, 53, 94g }}


Badness: 0.023270
Badness (Sintel): 1.19


====== 19-limit ======
====== 19-limit ======
Line 152: Line 155:
Mapping: {{mapping| 1 0 15 25 -33 -28 -7 9 | 0 1 -8 -14 23 20 7 -3 }}
Mapping: {{mapping| 1 0 15 25 -33 -28 -7 9 | 0 1 -8 -14 23 20 7 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.079
Optimal tunings:  
* WE: ~2 = 1199.9556{{c}}, ~3/2 = 702.0530{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0787{{c}}


{{Optimal ET sequence|legend=1| 41, 53, 94g }}
{{Optimal ET sequence|legend=0| 12e, 41, 53 }}


Badness: 0.018189
Badness (Sintel): 1.11


===== Cassandric =====
===== Cassandric =====
Line 165: Line 170:
Mapping: {{mapping| 1 0 15 25 -33 -28 77 | 0 1 -8 -14 23 20 -46 }}
Mapping: {{mapping| 1 0 15 25 -33 -28 77 | 0 1 -8 -14 23 20 -46 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.097
Optimal tunings:  
* WE: ~2 = 1200.0046{{c}}, ~3/2 = 702.2167{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0962{{c}}


{{Optimal ET sequence|legend=1| 41g, 53, 94, 241ce, 335cde }}
{{Optimal ET sequence|legend=0| 41g, 53, 94 }}


Badness: 0.023167
Badness (Sintel): 1.18


====== 19-limit ======
====== 19-limit ======
Line 179: Line 186:


Optimal tunings:
Optimal tunings:
* ~2 = 1200.2910, ~3/2 = -702.2681 ([[WE]])
* WE: ~2 = 1200.2910{{c}}, ~3/2 = 702.2681{{c}}
* ~2 = 1\1, ~3/2 = 702.0967 ([[CWE]])
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0967{{c}}
* ~2 = 1\1, ~3/2 = 702.0978 ([[POTE]])


{{Optimal ET sequence|legend=1| 41g, 53, 94, 241ceh, 335cdehh }}
{{Optimal ET sequence|legend=1| 41g, 53, 94 }}


Badness: 0.017635
Badness (Sintel): 1.07


====== 23-limit ======
====== 23-limit ======
Line 195: Line 201:


Optimal tunings:
Optimal tunings:
* ~2 = 1200.2970, ~3/2 = -702.2697 ([[WE]])
* WE: ~2 = 1200.2970{{c}}, ~3/2 = 702.2697{{c}}
* ~2 = 1\1, ~3/2 = 702.0943 ([[CWE]])
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0943{{c}}
* ~2 = 1\1, ~3/2 = 702.0960 ([[POTE]])


{{Optimal ET sequence|legend=1| 41g, 53, 94 }} <!-- fumica's calculator doesn't generate non-GPV ETs -->
{{Optimal ET sequence|legend=0| 41g, 53, 94 }}


Badness: 0.015072
Badness (Sintel): 1.08


===== Cassander =====
===== Cassander =====
Line 210: Line 215:
Mapping: {{mapping| 1 0 15 25 -33 -28 -72 | 0 1 -8 -14 23 20 48 }}
Mapping: {{mapping| 1 0 15 25 -33 -28 -72 | 0 1 -8 -14 23 20 48 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.144
Optimal tunings:
* WE: ~2 = 1200.1986{{c}}, ~3/2 = 702.2598{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1455{{c}}


{{Optimal ET sequence|legend=1| 41, 53g, 94 }}
{{Optimal ET sequence|legend=0| 41, 53g, 94 }}


Badness: 0.022454
Badness (Sintel): 1.14


====== 19-limit ======
====== 19-limit ======
Line 223: Line 230:
Mapping: {{mapping| 1 0 15 25 -33 -28 -72 9 | 0 1 -8 -14 23 20 48 -3 }}
Mapping: {{mapping| 1 0 15 25 -33 -28 -72 9 | 0 1 -8 -14 23 20 48 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.135
Optimal tunings:
* WE: ~2 = 1200.3057{{c}}, ~3/2 = 702.3138{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1373{{c}}


{{Optimal ET sequence|legend=1| 41, 53g, 94 }}
{{Optimal ET sequence|legend=0| 41, 53g, 94 }}


Badness: 0.017576
Badness (Sintel): 1.07


=== Andromeda ===
=== Andromeda ===
Line 236: Line 245:
Mapping: {{mapping| 1 0 15 25 32 | 0 1 -8 -14 -18 }}
Mapping: {{mapping| 1 0 15 25 32 | 0 1 -8 -14 -18 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.321
Optimal tunings:
* WE: ~2 = 1200.1917{{c}}, ~3/2 = 702.4836{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3599{{c}}


Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~3/2 = {{monzo| 3/5 1/10 0 0 -1/20 }}
* 11-odd-limit: ~3/2 = {{monzo| 3/5 1/10 0 0 -1/20 }}
: eigenmonzo (unchanged-interval) basis: 2.11/9
: unchanged-interval (eigenmonzo) basis: 2.11/9


Tuning ranges:  
Tuning ranges:  
Line 246: Line 257:
* 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]
* 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]


{{Optimal ET sequence|legend=1| 12, 29, 41, 217ce, 258ce }}
{{Optimal ET sequence|legend=0| 12, 29, 41 }}


Badness: 0.023556
Badness (Sintel): 0.779


==== 13-limit ====
==== 13-limit ====
Line 257: Line 268:
Mapping: {{mapping| 1 0 15 25 32 37 | 0 1 -8 -14 -18 -21 }}
Mapping: {{mapping| 1 0 15 25 32 37 | 0 1 -8 -14 -18 -21 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.559
Optimal tunings:
* WE: ~2 = 1200.3031{{c}}, ~3/2 = 702.7368{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.5420{{c}}


Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/23 2/23 0 0 0 -1/23 }}
* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/23 2/23 0 0 0 -1/23 }}
: eigenmonzo (unchanged-interval) basis: 2.13/9
: unchanged-interval (eigenmonzo) basis: 2.13/9


Tuning ranges:  
Tuning ranges:  
Line 268: Line 281:
* 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 704.377]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 704.377]


{{Optimal ET sequence|legend=1| 12f, 29, 41, 152cdf, 193cdf, 234cdf }}
{{Optimal ET sequence|legend=0| 12f, 29, 41 }}


Badness: 0.020749
Badness (Sintel): 0.857


===== 17-limit =====
===== 17-limit =====
Line 279: Line 292:
Mapping: {{mapping| 1 0 15 25 32 37 -7 | 0 1 -8 -14 -18 -21 7 }}
Mapping: {{mapping| 1 0 15 25 32 37 -7 | 0 1 -8 -14 -18 -21 7 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.312
Optimal tunings:
* WE: ~2 = 1199.1984{{c}}, ~3/2 = 701.8424{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3384{{c}}


{{Optimal ET sequence|legend=1| 12f, 29, 41 }}
{{Optimal ET sequence|legend=0| 12f, 29, 41 }}


Badness: 0.023406
Badness (Sintel): 1.19


====== 19-limit ======
====== 19-limit ======
Line 292: Line 307:
Mapping: {{mapping| 1 0 15 25 32 37 -7 9 | 0 1 -8 -14 -18 -21 7 -3 }}
Mapping: {{mapping| 1 0 15 25 32 37 -7 9 | 0 1 -8 -14 -18 -21 7 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.357
Optimal tunings:
* WE: ~2 = 1199.5242{{c}}, ~3/2 = 702.0783{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3711{{c}}


{{Optimal ET sequence|legend=1| 12f, 29, 41 }}
{{Optimal ET sequence|legend=0| 12f, 29, 41 }}


Badness: 0.019154
Badness (Sintel): 1.17


===== Schisicosiennic =====
===== Schisicosiennic =====
Line 305: Line 322:
Mapping: {{mapping| 1 0 15 25 32 37 58 | 0 1 -8 -14 -18 -21 -34 }}
Mapping: {{mapping| 1 0 15 25 32 37 58 | 0 1 -8 -14 -18 -21 -34 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.725
Optimal tunings:
* WE: ~2 = 1200.6122{{c}}, ~3/2 = 703.0830{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6968{{c}}


{{Optimal ET sequence|legend=1| 12fg, 29g, 41, 70cd, 111cd }}
{{Optimal ET sequence|legend=0| 12fg, 29g, 41, 70cd }}


Badness: 0.021758
Badness (Sintel): 1.11


====== 19-limit ======
====== 19-limit ======
Line 318: Line 337:
Mapping: {{mapping| 1 0 15 25 32 37 58 9 | 0 1 -8 -14 -18 -21 -34 -3 }}
Mapping: {{mapping| 1 0 15 25 32 37 58 9 | 0 1 -8 -14 -18 -21 -34 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.753
Optimal tunings:
* WE: ~2 = 1200.7981{{c}}, ~3/2 = 703.2199{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.7221{{c}}


{{Optimal ET sequence|legend=1| 12fg, 29g, 41, 70cd, 111cdh, 181ccddh }}
{{Optimal ET sequence|legend=0| 12fg, 29g, 41, 70cd }}


Badness: 0.017902
Badness (Sintel): 1.09


===== Schisicosiennoid =====
===== Schisicosiennoid =====
Line 331: Line 352:
Mapping: {{mapping| 1 0 15 25 32 37 12 | 0 1 -8 -14 -18 -21 -5 }}
Mapping: {{mapping| 1 0 15 25 32 37 12 | 0 1 -8 -14 -18 -21 -5 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.717
Optimal tunings:
* WE: ~2 = 1201.3146{{c}}, ~3/2 = 703.4864{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6491{{c}}


{{Optimal ET sequence|legend=1| 12f, 29g, 41g, 70cdgg }}
{{Optimal ET sequence|legend=0| 12f, 29g, 41g }}


Badness: 0.020895
Badness (Sintel): 1.06


====== 19-limit ======
====== 19-limit ======
Line 344: Line 367:
Mapping: {{mapping| 1 0 15 25 32 37 12 9 | 0 1 -8 -14 -18 -21 -5 -3 }}
Mapping: {{mapping| 1 0 15 25 32 37 12 9 | 0 1 -8 -14 -18 -21 -5 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.716
Optimal tunings:
* WE: ~2 = 1201.3140{{c}}, ~3/2 = 703.4860{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6578{{c}}


{{Optimal ET sequence|legend=1| 12f, 29g, 41g, 70cdgg }}
{{Optimal ET sequence|legend=1| 12f, 29g, 41g }}


Badness: 0.016773
Badness (Sintel): 1.02


=== Helenus ===
=== Helenus ===
Line 357: Line 382:
Mapping: {{mapping| 1 0 15 25 51 | 0 1 -8 -14 -30 }}
Mapping: {{mapping| 1 0 15 25 51 | 0 1 -8 -14 -30 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.725
Optimal tunings:
* WE: ~2 = 1199.7097{{c}}, ~3/2 = 701.5554{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7370{{c}}


Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }}
* 11-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }}
: eigenmonzo (unchanged-interval) basis: 2.11/9
: unchanged-interval (eigenmonzo) basis: 2.11/9


{{Optimal ET sequence|legend=1| 12, 41e, 53, 118d, 171de }}
{{Optimal ET sequence|legend=0| 12, 41e, 53, 118d }}


Badness: 0.035637
Badness (Sintel): 1.18


==== 13-limit ====
==== 13-limit ====
Line 374: Line 401:
Mapping: {{mapping| 1 0 15 25 51 56 | 0 1 -8 -14 -30 -33 }}
Mapping: {{mapping| 1 0 15 25 51 56 | 0 1 -8 -14 -30 -33 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.747
Optimal tunings:
* WE: ~2 = 1199.7370{{c}}, ~3/2 = 701.5937{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7570{{c}}


Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }}
* 13- and 15-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }}
: eigenmonzo (unchanged-interval) basis: 2.11/9
: unchanged-interval (eigenmonzo) basis: 2.11/9


{{Optimal ET sequence|legend=1| 12f, 41ef, 53, 118d, 171de }}
{{Optimal ET sequence|legend=0| 12f, …, 41ef, 53, 118d }}


Badness: 0.026284
Badness (Sintel): 1.09


==== 17-limit ====
==== 17-limit ====
Line 391: Line 420:
Mapping: {{mapping| 1 0 15 25 51 56 -7 | 0 1 -8 -14 -30 -33 7 }}
Mapping: {{mapping| 1 0 15 25 51 56 -7 | 0 1 -8 -14 -30 -33 7 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.680
Optimal tunings:
* WE: ~2 = 1199.2895{{c}}, ~3/2 = 701.2643{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6967{{c}}


{{Optimal ET sequence|legend=1| 12f, 41ef, 53, 65d, 118dg }}
{{Optimal ET sequence|legend=0| 12f, 53, 65d, 118dg }}


Badness: 0.023732
Badness (Sintel): 1.21


==== 19-limit ====
==== 19-limit ====
Line 404: Line 435:
Mapping: {{mapping| 1 0 15 25 51 56 -7 9 | 0 1 -8 -14 -30 -33 7 -3 }}
Mapping: {{mapping| 1 0 15 25 51 56 -7 9 | 0 1 -8 -14 -30 -33 7 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.705
Optimal tunings:
* WE: ~2 = 1199.5280{{c}}, ~3/2 = 701.4290{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7149{{c}}


{{Optimal ET sequence|legend=1| 12f, 41ef, 53, 65d, 118dg }}
{{Optimal ET sequence|legend=0| 12f, 53, 65d }}


Badness: 0.019411
Badness (Sintel): 1.18
 
=== Karadeniz ===
{{See also| Turkish maqam music temperaments #Karadeniz temperament }}


=== Hemigari ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 121/120, 225/224, 3125/3087
Comma list: 225/224, 243/242, 3125/3087


Mapping: {{mapping| 1 0 15 25 9 | 0 2 -16 -28 -7 }}
Mapping: {{mapping| 1 1 7 11 2 | 0 2 -16 -28 5 }}
: mapping generators: ~2, ~11/9


: Mapping generators: ~2, ~110/63
Optimal tunings:
 
* WE: ~2 = 1199.7351{{c}}, ~11/9 = 350.9167{{c}}
Optimal tuning (POTE): ~2 = 1\1, ~110/63 = 951.082 (~63/55 = 248.918)
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.9995{{c}}


{{Optimal ET sequence|legend=1| 29, 53, 82e, 135e, 188ce }}
{{Optimal ET sequence|legend=0| 24d, 41, 65d, 106, 147 }}


Badness: 0.050681
Badness (Sintel): 1.37


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


Comma list: 121/120, 169/168, 225/224, 275/273
Comma list: 225/224, 243/242, 325/324, 640/637


Mapping: {{mapping| 1 0 15 25 9 14 | 0 2 -16 -28 -7 -13 }}
Mapping: {{mapping| 1 1 7 11 2 -8 | 0 2 -16 -28 5 40 }}


Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 951.082 (~15/13 = 248.918)
Optimal tunings:
* WE: ~2 = 1199.3042{{c}}, ~11/9 = 350.7533{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.9686{{c}}


{{Optimal ET sequence|legend=1| 29, 53, 82e, 135ef, 188cef }}
{{Optimal ET sequence|legend=0| 24d, 41, 65d, 106f }}


Badness: 0.027464
Badness (Sintel): 1.34
 
=== Karadeniz ===
{{See also| Turkish maqam music temperaments #Karadeniz temperament }}


=== Hemigari ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 243/242, 3125/3087
Comma list: 121/120, 225/224, 3125/3087


Mapping: {{mapping| 1 1 7 11 2 | 0 2 -16 -28 5 }}
Mapping: {{mapping| 1 0 15 25 9 | 0 2 -16 -28 -7 }}
: mapping generators: ~2, ~110/63


: Mapping generators: ~2, ~11/9
Optimal tunings:
 
* WE: ~2 = 1200.7303{{c}}, ~110/63 = 951.6605{{c}}
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.994
* CWE: ~2 = 1200.0000{{c}}, ~110/63 = 951.0604{{c}}


{{Optimal ET sequence|legend=1| 41, 106, 147 }}
{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135ee }}


Badness: 0.041562
Badness (Sintel): 1.68


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


Comma list: 225/224, 243/242, 325/324, 640/637
Comma list: 121/120, 169/168, 225/224, 275/273


Mapping: {{mapping| 1 1 7 11 2 -8 | 0 2 -16 -28 5 40 }}
Mapping: {{mapping| 1 0 15 25 9 14 | 0 2 -16 -28 -7 -13 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.014
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=1| 41, 106, 147 }}
{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135eef }}


Badness: 0.042564
Badness (Sintel): 1.13


=== Sanjaab ===
=== Sanjaab ===
Line 474: Line 513:


Mapping: {{mapping| 1 2 -1 -3 0 | 0 -3 24 42 25 }}
Mapping: {{mapping| 1 2 -1 -3 0 | 0 -3 24 42 25 }}
: mapping generators: ~2, ~11/10
: mapping generators: ~2, ~11/10


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.974
Optimal tunings:
* WE: ~2 = 1200.1997{{c}}, ~11/10 = 166.0018{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9786{{c}}


{{Optimal ET sequence|legend=1| 29, 65d, 94, 441cde, 535cde, 629cde }}
{{Optimal ET sequence|legend=0| 29, 65d, 94 }}


Badness: 0.058040
Badness (Sintel): 1.92


==== 13-limit ====
==== 13-limit ====
Line 490: Line 530:
Mapping: {{mapping| 1 2 -1 -3 0 -1 | 0 -3 24 42 25 34 }}
Mapping: {{mapping| 1 2 -1 -3 0 -1 | 0 -3 24 42 25 34 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.963
Optimal tunings:
* WE: ~2 = 1200.1224{{c}}, ~11/10 = 165.9800{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9659{{c}}


{{Optimal ET sequence|legend=1| 29, 65d, 94 }}
{{Optimal ET sequence|legend=0| 29, 65d, 94 }}


Badness: 0.033849
Badness (Sintel): 1.40
 
== Schism ==
See [[Archytas clan #Schism]].
 
Schism is a relatively low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (C-Bb). 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.


== Pontiac ==
== Pontiac ==
{{Main| Pontiac }}
{{Main| Pontiac }}


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 512: Line 549:
{{Mapping|legend=1| 1 0 15 -59 | 0 1 -8 39 }}
{{Mapping|legend=1| 1 0 15 -59 | 0 1 -8 39 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.757
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0989{{c}}, ~3/2 = 701.8145{{c}}
: [[error map]]: {{val| +0.099 -0.042 -0.138 -0.038 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7579{{c}}
: error map: {{val| 0.000 -0.197 -0.377 -0.268 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]: ~3/2 = {{monzo| 27/47 0 -1/47 1/47 }}
* [[7-odd-limit]]: ~3/2 = {{monzo| 27/47 0 -1/47 1/47 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 74/47 0 -1/47 1/47 }}, {{monzo| 113/47 0 8/47 -8/47 }}, {{monzo| 113/47 0 -39/47 39/47 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 74/47 0 -1/47 1/47 }}, {{monzo| 113/47 0 8/47 -8/47 }}, {{monzo| 113/47 0 -39/47 39/47 }}]
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
* [[9-odd-limit]]: ~3/2 = {{monzo| 1/2 1/5 -1/10 }}
* [[9-odd-limit]]: ~3/2 = {{monzo| 1/2 1/5 -1/10 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 3/2 1/5 -1/10 0 }}, {{monzo| 3 -8/5 4/5 0 }}, {{monzo| -1/2 39/5 -39/10 0 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 3/2 1/5 -1/10 0 }}, {{monzo| 3 -8/5 4/5 0 }}, {{monzo| -1/2 39/5 -39/10 0 }}]
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [701.538, 701.886] (38\65 to 31\53)
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [701.538, 701.886] (38\65 to 31\53)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 701.955]
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 701.955]
* 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [701.711, 701.886]


{{Optimal ET sequence|legend=1| 53, 118, 171, 1592c, 1763c, 1934c, 2105c, 2276cd, 2447cd, 2618cd, 2789cd, 2960cd, 3131bcd }}
{{Optimal ET sequence|legend=1| 53, 118, 171, 1592c, 1763c, …, 2960cd, 3131bcd }}


[[Badness]]: 0.014133
[[Badness]] (Sintel): 0.358


=== Helenoid ===
=== Helenoid ===
The helenoid temperament ({{nowrap|53 &amp; 118}}) is closely related to the helenus temperament, but with the ragisma rather than the [[225/224|marvel comma]] tempered out.
Helenoid may be described as {{nowrap| 53 & 118 }}, and is closely related to the helenus temperament, differing only by the mapping of 7.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 540: Line 580:
Mapping: {{mapping| 1 0 15 -59 51 | 0 1 -8 39 -30 }}
Mapping: {{mapping| 1 0 15 -59 51 | 0 1 -8 39 -30 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.722
Optimal tunings:
* WE: ~2 = 1200.3277{{c}}, ~3/2 = 701.9135{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7223{{c}}


Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~3/2 = {{monzo| 41/69 0 0 1/69 -1/69 }}
* 11-odd-limit: ~3/2 = {{monzo| 41/69 0 0 1/69 -1/69 }}
: eigenmonzo (unchanged-interval) basis: 2.11/7
: unchanged-interval (eigenmonzo) basis: 2.11/7


{{Optimal ET sequence|legend=1| 53, 118, 289e, 407de }}
{{Optimal ET sequence|legend=0| 53, 118, 289e, 407de }}


Badness: 0.038863
Badness (Sintel): 1.28


==== 13-limit ====
==== 13-limit ====
Line 557: Line 599:
Mapping: {{mapping| 1 0 15 -59 51 56 | 0 1 -8 39 -30 -33 }}
Mapping: {{mapping| 1 0 15 -59 51 56 | 0 1 -8 39 -30 -33 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.745
Optimal tunings:
* WE: ~2 = 1200.1780{{c}}, ~3/2 = 701.8491{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7446{{c}}


Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~3/2 = {{monzo| 43/72 0 0 1/72 -1/72 }}
* 13- and 15-odd-limit: ~3/2 = {{monzo| 43/72 0 0 1/72 -1/72 }}
: eigenmonzo (unchanged-interval) basis: 2.13/7
: unchanged-interval (eigenmonzo) basis: 2.13/7


{{Optimal ET sequence|legend=1| 53, 118, 171e }}
{{Optimal ET sequence|legend=0| 53, 118, 171e }}


Badness: 0.033677
Badness (Sintel): 1.39


===== 17-limit =====
===== 17-limit =====
Line 574: Line 618:
Mapping: {{mapping| 1 0 15 -59 51 56 -91 | 0 1 -8 39 -30 -33 60 }}
Mapping: {{mapping| 1 0 15 -59 51 56 -91 | 0 1 -8 39 -30 -33 60 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.742
Optimal tunings:
* WE: ~2 = 1200.1645{{c}}, ~3/2 = 701.8385{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7425{{c}}


Minimax tuning:  
Minimax tuning:  
* 17-odd-limit: ~3/2 = {{monzo| 18/31 0 0 0 0 -1/93 1/93 }}
* 17-odd-limit: ~3/2 = {{monzo| 18/31 0 0 0 0 -1/93 1/93 }}
: eigenmonzo (unchanged-interval) basis: 2.17/13
: unchanged-interval (eigenmonzo) basis: 2.17/13


{{Optimal ET sequence|legend=1| 53, 118, 171e, 289ef, 460eef }}
{{Optimal ET sequence|legend=0| 53, 118, 171e }}


Badness: 0.028891
Badness (Sintel): 1.47


==== Helena ====
==== Helena ====
Line 591: Line 637:
Mapping: {{mapping| 1 0 15 -59 51 -28 | 0 1 -8 39 -30 20 }}
Mapping: {{mapping| 1 0 15 -59 51 -28 | 0 1 -8 39 -30 20 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.740
Optimal tunings:
* WE: ~2 = 1200.5227{{c}}, ~3/2 = 702.0456{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7418{{c}}


{{Optimal ET sequence|legend=1| 53, 118f, 171ef }}
{{Optimal ET sequence|legend=0| 53, 118f, 171ef }}


Badness: 0.036281
Badness (Sintel): 1.50


===== 17-limit =====
===== 17-limit =====
Line 604: Line 652:
Mapping: {{mapping| 1 0 15 -59 51 -28 -91 | 0 1 -8 39 -30 20 60 }}
Mapping: {{mapping| 1 0 15 -59 51 -28 -91 | 0 1 -8 39 -30 20 60 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.730
Optimal tunings:
* WE: ~2 = 1200.4988{{c}}, ~3/2 = 702.0218{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7332{{c}}


{{Optimal ET sequence|legend=1| 53, 118f, 171ef, 289eff }}
{{Optimal ET sequence|legend=0| 53, 118f, 171ef }}


Badness: 0.030688
Badness (Sintel): 1.56


===== 19-limit =====
===== 19-limit =====
Line 617: Line 667:
Mapping: {{mapping| 1 0 15 -59 51 -28 -91 9 | 0 1 -8 39 -30 20 60 -3 }}
Mapping: {{mapping| 1 0 15 -59 51 -28 -91 9 | 0 1 -8 39 -30 20 60 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.729
Optimal tunings:
* WE: ~2 = 1200.5185{{c}}, ~3/2 = 702.0323{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7318{{c}}


{{Optimal ET sequence|legend=1| 53, 118f, 171ef, 289effh }}
{{Optimal ET sequence|legend=0| 53, 118f, 171ef }}


Badness: 0.021892
Badness (Sintel): 1.33


=== Ponta ===
=== Ponta ===
The ponta temperament ({{nowrap|53 &amp; 171}}) tempers out the [[540/539|swetisma]] and the ragisma.
Ponta tempers out [[540/539]] and may be described as {{nowrap| 171 & 224 }}. [[224edo]] itself makes for an excellent tuning.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 632: Line 684:
Mapping: {{mapping| 1 0 15 -59 135 | 0 1 -8 39 -83 }}
Mapping: {{mapping| 1 0 15 -59 135 | 0 1 -8 39 -83 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.783
Optimal tunings:
* WE: ~2 = 1199.9814{{c}}, ~3/2 = 701.7725{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7834{{c}}


Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~3/2 = {{monzo| 36/61 0 0 1/122 -1/122 }}
* 11-odd-limit: ~3/2 = {{monzo| 36/61 0 0 1/122 -1/122 }}
: eigenmonzo (unchanged-interval) basis: 2.11/7
: unchanged-interval (eigenmonzo) basis: 2.11/7


{{Optimal ET sequence|legend=1| 53, 171, 224, 1291cde, 1515cde, 1739cddee, 1963cddee, 2187ccddee }}
{{Optimal ET sequence|legend=0| 53, 171, 224 }}


Badness: 0.048692
Badness (Sintel): 1.61


==== 13-limit ====
==== 13-limit ====
Line 649: Line 703:
Mapping: {{mapping| 1 0 15 -59 135 56 | 0 1 -8 39 -83 -33 }}
Mapping: {{mapping| 1 0 15 -59 135 56 | 0 1 -8 39 -83 -33 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.784
Optimal tunings:
* WE: ~2 = 1199.9601{{c}}, ~3/2 = 701.7610{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7845{{c}}


Minimax tuning:  
Minimax tuning:  
* 13 and 15-odd-limit: ~3/2 = {{monzo| 36/61 0 0 1/122 -1/122 }}
* 13 and 15-odd-limit: ~3/2 = {{monzo| 36/61 0 0 1/122 -1/122 }}
: Unchanged-interval (eigenmonzo) basis: 2.11/7
: unchanged-interval (eigenmonzo) basis: 2.11/7


{{Optimal ET sequence|legend=1| 53, 171, 224 }}
{{Optimal ET sequence|legend=0| 53, 171, 224 }}


Badness: 0.023616
Badness (Sintel): 0.976


==== 17-limit ====
==== 17-limit ====
Line 666: Line 722:
Mapping: {{mapping| 1 0 15 -59 135 56 -91 | 0 1 -8 39 -83 -33 60 }}
Mapping: {{mapping| 1 0 15 -59 135 56 -91 | 0 1 -8 39 -83 -33 60 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.777
Optimal tunings:
* WE: ~2 = 1199.8850{{c}}, ~3/2 = 701.7101{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7775{{c}}


Minimax tuning:  
Minimax tuning:  
* 17-odd-limit: ~3/2 = {{monzo| 83/143 0 0 0 -1/143 0 1/143 }}
* 17-odd-limit: ~3/2 = {{monzo| 83/143 0 0 0 -1/143 0 1/143 }}
: Unchanged-interval (eigenmonzo) basis: 2.17/11
: unchanged-interval (eigenmonzo) basis: 2.17/11


{{Optimal ET sequence|legend=1| 53, 171, 224, 395e, 619eg }}
{{Optimal ET sequence|legend=0| 53, 171, 224, 395e, 619eg }}


Badness: 0.022853
Badness (Sintel): 1.16


=== Pontic ===
=== Pontic ===
The pontic temperament ({{nowrap|118 &amp; 171}}) tempers out the [[441/440|werckisma]] and the ragisma.
Pontic temperament tempers out [[441/440]] and may be described as {{nowrap| 118 & 171 }}. [[289edo]] may be recommended as a tuning.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 685: Line 743:
Mapping: {{mapping| 1 0 15 -59 -136 | 0 1 -8 39 88 }}
Mapping: {{mapping| 1 0 15 -59 -136 | 0 1 -8 39 88 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.724
Optimal tunings:
* WE: ~2 = 1200.1259{{c}}, ~3/2 = 701.7980{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7256{{c}}


Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~3/2 = {{monzo| 6/11 0 0 0 1/88 }}
* 11-odd-limit: ~3/2 = {{monzo| 6/11 0 0 0 1/88 }}
: eigenmonzo (unchanged-interval) basis: 2.11
: unchanged-interval (eigenmonzo) basis: 2.11


{{Optimal ET sequence|legend=1| 53e, 118, 289, 407d, 696d }}
{{Optimal ET sequence|legend=0| 53e, 118, 289, 407d }}


Badness: 0.049573
Badness (Sintel): 1.64


==== 13-limit ====
==== 13-limit ====
Line 702: Line 762:
Mapping: {{mapping| 1 0 15 -59 -136 56 | 0 1 -8 39 88 -33 }}
Mapping: {{mapping| 1 0 15 -59 -136 56 | 0 1 -8 39 88 -33 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.738
Optimal tunings:
* WE: ~2 = 1199.9254{{c}}, ~3/2 = 701.6945{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7378{{c}}


Minimax tuning:  
Minimax tuning:  
* 13 and 15-odd-limit: ~3/2 = {{monzo| 71/121 0 0 0 1/121 -1/121 }}
* 13 and 15-odd-limit: ~3/2 = {{monzo| 71/121 0 0 0 1/121 -1/121 }}
: eigenmonzo (unchanged-interval) basis: 2.13/11
: unchanged-interval (eigenmonzo) basis: 2.13/11


{{Optimal ET sequence|legend=1| 53e, 118, 171, 289f, 460ef }}
{{Optimal ET sequence|legend=0| 53e, 118, 171, 289f }}


Badness: 0.045308
Badness (Sintel): 1.87


===== 17-limit =====
===== 17-limit =====
Line 719: Line 781:
Mapping: {{mapping| 1 0 15 -59 -136 56 -91 | 0 1 -8 39 88 -33 60 }}
Mapping: {{mapping| 1 0 15 -59 -136 56 -91 | 0 1 -8 39 88 -33 60 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.740
Optimal tunings:
* WE: ~2 = 1199.9454{{c}}, ~3/2 = 701.7085{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7401{{c}}


Minimax tuning:  
Minimax tuning:  
* 17-odd-limit: ~3/2 = {{monzo| 71/121 0 0 0 1/121 -1/121 }}
* 17-odd-limit: ~3/2 = {{monzo| 71/121 0 0 0 1/121 -1/121 }}
: Unchanged-interval (eigenmonzo) basis: 2.13/11
: unchanged-interval (eigenmonzo) basis: 2.13/11


{{Optimal ET sequence|legend=1| 53e, 118, 171, 289f, 460ef }}
{{Optimal ET sequence|legend=0| 53e, 118, 171, 289f }}


Badness: 0.029618
Badness (Sintel): 1.51


==== Pontoid ====
==== Pontoid ====
Line 736: Line 800:
Mapping: {{mapping| 1 0 15 -59 -136 -215 | 0 1 -8 39 88 138 }}
Mapping: {{mapping| 1 0 15 -59 -136 -215 | 0 1 -8 39 88 138 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.735
Optimal tunings:
* WE: ~2 = 1200.0897{{c}}, ~3/2 = 701.7874{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7356{{c}}


{{Optimal ET sequence|legend=1| 53ef, 118f, 171, 289, 460e, 749def }}
{{Optimal ET sequence|legend=0| 53ef, 118f, 171, 289 }}


Badness: 0.050188
Badness (Sintel): 2.07


===== 17-limit =====
===== 17-limit =====
Line 749: Line 815:
Mapping: {{mapping| 1 0 15 -59 -136 -215 -91 | 0 1 -8 39 88 138 60 }}
Mapping: {{mapping| 1 0 15 -59 -136 -215 -91 | 0 1 -8 39 88 138 60 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.735
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=1| 53ef, 118f, 171, 289, 460e, 749defg }}
{{Optimal ET sequence|legend=0| 53ef, 118f, 171, 289, 460e, 749defg }}


Badness: 0.029383
Badness (Sintel): 1.50


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


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 763: Line 831:


Mapping: {{mapping| 2 0 30 -118 -85 | 0 1 -8 39 29 }}
Mapping: {{mapping| 2 0 30 -118 -85 | 0 1 -8 39 29 }}
: mapping generators: ~99/70, ~3


: Mapping generators: ~99/70, ~3
Optimal tunings:
 
* WE: ~99/70 = 600.0500{{c}}, ~3/2 = 701.8153{{c}}
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.757
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7584{{c}}


{{Optimal ET sequence|legend=1| 106, 118, 224, 342, 1592c, 1934ce, 2276cde, 2618cde, 2960cde }}
{{Optimal ET sequence|legend=0| 106, 118, 224, 342, 1592c, 1934ce, 2276cde, 2618cde, 2960cde }}


Badness: 0.014629
Badness (Sintel): 0.484


==== 13-limit ====
==== 13-limit ====
Line 779: Line 848:
Mapping: {{mapping| 2 0 30 -118 -85 112 | 0 1 -8 39 29 -33 }}
Mapping: {{mapping| 2 0 30 -118 -85 112 | 0 1 -8 39 29 -33 }}


Mapping generators: ~99/70, ~3
Optimal tunings:
 
* WE: ~99/70 = 599.9939{{c}}, ~3/2 = 701.7657{{c}}
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.773
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7728{{c}}


{{Optimal ET sequence|legend=1| 106, 118, 224, 566f, 790f }}
{{Optimal ET sequence|legend=0| 106, 118, 224, 566f, 790f }}


Badness: 0.030172
Badness (Sintel): 1.25


===== 17-limit =====
===== 17-limit =====
Line 794: Line 863:
Mapping: {{mapping| 2 0 30 -118 -85 112 -182 | 0 1 -8 39 29 -33 60 }}
Mapping: {{mapping| 2 0 30 -118 -85 112 -182 | 0 1 -8 39 29 -33 60 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.765
Optimal tunings:
* WE: ~99/70 = 599.9839{{c}}, ~3/2 = 701.7463{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7649{{c}}


{{Optimal ET sequence|legend=1| 106g, 118, 224, 342, 566f }}
{{Optimal ET sequence|legend=0| 106g, 118, 224, 342, 566f }}


Badness: 0.027051
Badness (Sintel): 1.38


==== Counterbipont ====
==== Counterbipont ====
Line 807: Line 878:
Mapping: {{mapping| 2 0 30 -118 -85 -243 | 0 1 -8 39 29 79 }}
Mapping: {{mapping| 2 0 30 -118 -85 -243 | 0 1 -8 39 29 79 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.769
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=1| 106f, 118f, 224, 342f, 566, 1356cf, 1922cff }}
{{Optimal ET sequence|legend=0| 106f, 118f, 224, 342f, 566, 1356cf }}


Badness: 0.025547
Badness (Sintel): 1.06


===== 17-limit =====
===== 17-limit =====
Line 820: Line 893:
Mapping: {{mapping| 2 0 30 -118 -85 -243 -182 | 0 1 -8 39 29 79 60 }}
Mapping: {{mapping| 2 0 30 -118 -85 -243 -182 | 0 1 -8 39 29 79 60 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.764
Optimal tunings:
* WE: ~99/70 = 600.0336{{c}}, ~3/2 = 701.8031{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7647{{c}}


{{Optimal ET sequence|legend=1| 106fg, 118f, 224, 342f, 566, 908fg, 1474cffgg }}
{{Optimal ET sequence|legend=0| 106fg, 118f, 224, 342f, 566 }}


Badness: 0.025251
Badness (Sintel): 1.29


===== 19-limit =====
===== 19-limit =====
Line 833: Line 908:
Mapping: {{mapping| 2 0 30 -118 -85 -243 -182 -169 | 0 1 -8 39 29 79 60 56 }}
Mapping: {{mapping| 2 0 30 -118 -85 -243 -182 -169 | 0 1 -8 39 29 79 60 56 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.761
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=1| 106fgh, 118f, 224, 342f, 566h, 908fgh }}
{{Optimal ET sequence|legend=0| 106fgh, 118f, 224, 342f, 566h, 908fgh }}


Badness: 0.022267
Badness (Sintel): 1.35


==== Quadrapont ====
==== Quadrapont ====
Line 845: Line 922:


Mapping: {{mapping| 4 0 60 -236 -170 -131 | 0 1 -8 39 29 23 }}
Mapping: {{mapping| 4 0 60 -236 -170 -131 | 0 1 -8 39 29 23 }}
: mapping generators: ~208/175, ~3


: Mapping generators: ~208/175, ~3
Optimal tunings:
* WE: ~208/175 = 300.0229{{c}}, ~3/2 = 701.8097{{c}}
* CWE: ~208/175 = 300.0000{{c}}, ~3/2 = 701.7578{{c}}


Optimal tuning (POTE): ~208/175 = 1\4, ~3/2 = 701.756
{{Optimal ET sequence|legend=0| 224, 460, 684, 2276cde, 2960cde }}


{{Optimal ET sequence|legend=1| 224, 460, 684, 2276cde, 2960cde, 3644bccddee }}
Badness (Sintel): 0.869
 
Badness: 0.021025


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 862: Line 940:


{{Mapping|legend=1| 1 0 15 44 | 0 1 -8 -26 }}
{{Mapping|legend=1| 1 0 15 44 | 0 1 -8 -26 }}
: mapping generators: ~2, ~3


: Mapping generators: ~2, ~3
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1199.7974{{c}}, ~3/2 = 701.1210{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.239
: [[error map]]: {{val| -0.203 -1.037 +3.300 -1.618 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.2465{{c}}
: error map: {{val| 0.000 -0.709 +3.715 -1.234 }}


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


{{Optimal ET sequence|legend=1| 12, 53d, 65, 77, 166c, 243c }}
{{Optimal ET sequence|legend=1| 12, …, 65, 77, 166c }}


[[Badness]]: 0.070407
[[Badness]] (Sintel): 1.78


=== 11-limit ===
=== 11-limit ===
Line 882: Line 963:
Mapping: {{mapping| 1 0 15 44 70 | 0 1 -8 -26 -42 }}
Mapping: {{mapping| 1 0 15 44 70 | 0 1 -8 -26 -42 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.172
Optimal tunings:
* WE: ~2 = 1199.7077{{c}}, ~3/2 = 701.0017{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.1804{{c}}


{{Optimal ET sequence|legend=1| 12, 53dee, 65e, 77, 89, 166c, 255c }}
{{Optimal ET sequence|legend=0| 12, 65e, 77, 89, 166c }}


Badness: 0.048887
Badness (Sintel): 1.62


==== 13-limit ====
==== 13-limit ====
Line 895: Line 978:
Mapping: {{mapping| 1 0 15 44 70 75 | 0 1 -8 -26 -42 -45 }}
Mapping: {{mapping| 1 0 15 44 70 75 | 0 1 -8 -26 -42 -45 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.226
Optimal tunings:
* WE: ~2 = 1199.7782{{c}}, ~3/2 = 701.0966{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2319{{c}}


{{Optimal ET sequence|legend=1| 12f, 53deeff, 65ef, 77, 166cf, 243cf }}
{{Optimal ET sequence|legend=0| 12f, 65ef, 77, 166cf }}


Badness: 0.037859
Badness (Sintel): 1.56


===== 17-limit =====
===== 17-limit =====
Line 908: Line 993:
Mapping: {{mapping| 1 0 15 44 70 75 -7 | 0 1 -8 -26 -42 -45 7 }}
Mapping: {{mapping| 1 0 15 44 70 75 -7 | 0 1 -8 -26 -42 -45 7 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.206
Optimal tunings:
* WE: ~2 = 1199.5839{{c}}, ~3/2 = 700.9632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2137{{c}}


{{Optimal ET sequence|legend=1| 12f, 53deeff, 65ef, 77, 89f, 166cf }}
{{Optimal ET sequence|legend=0| 12f, 77, 89f, 166cf }}


Badness: 0.029864
Badness (Sintel): 1.52


===== 19-limit =====
===== 19-limit =====
Line 921: Line 1,008:
Mapping: {{mapping| 1 0 15 44 70 75 -7 9 | 0 1 -8 -26 -42 -45 7 -3 }}
Mapping: {{mapping| 1 0 15 44 70 75 -7 9 | 0 1 -8 -26 -42 -45 7 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.217
Optimal tunings:
* WE: ~2 = 1199.7146{{c}}, ~3/2 = 701.0500{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2212{{c}}


{{Optimal ET sequence|legend=1| 12f, 53deeff, 65ef, 77, 166cf }}
{{Optimal ET sequence|legend=0| 12f, 77, 166cf }}


Badness: 0.023096
Badness (Sintel): 1.40


==== Grackloid ====
==== Grackloid ====
Line 934: Line 1,023:
Mapping: {{mapping| 1 0 15 44 70 -47 | 0 1 -8 -26 -42 32 }}
Mapping: {{mapping| 1 0 15 44 70 -47 | 0 1 -8 -26 -42 32 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.217
Optimal tunings:
* WE: ~2 = 1200.0060{{c}}, ~3/2 = 701.2202{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2167{{c}}


{{Optimal ET sequence|legend=1| 12, 53deef, 65e, 77, 166c }}
{{Optimal ET sequence|legend=0| 12, 77, 166c }}


Badness: 0.048511
Badness (Sintel): 2.00


=== Grack ===
=== Grack ===
Line 947: Line 1,038:
Mapping: {{mapping| 1 0 15 44 51 | 0 1 -8 -26 -30 }}
Mapping: {{mapping| 1 0 15 44 51 | 0 1 -8 -26 -30 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.401
Optimal tunings:
* WE: ~2 = 1199.8388{{c}}, ~3/2 = 701.3071{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.4068{{c}}


{{Optimal ET sequence|legend=1| 12, 53d, 65, 77e, 142de }}
{{Optimal ET sequence|legend=0| 12, 53d, 65, 77e }}


Badness: 0.055908
Badness (Sintel): 1.85


==== 13-limit ====
==== 13-limit ====
Line 960: Line 1,053:
Mapping: {{mapping| 1 0 15 44 51 75 | 0 1 -8 -26 -30 -45 }}
Mapping: {{mapping| 1 0 15 44 51 75 | 0 1 -8 -26 -30 -45 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.348
Optimal tunings:
* WE: ~2 = 1199.7329{{c}}, ~3/2 = 701.1918{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.3555{{c}}


{{Optimal ET sequence|legend=1| 12f, 53dff, 65f, 77e }}
{{Optimal ET sequence|legend=0| 12f, 53dff, 65f, 77e }}


Badness: 0.044458
Badness (Sintel): 1.84


==== Catahelenic ====
==== Catahelenic ====
Line 973: Line 1,068:
Mapping: {{mapping| 1 0 15 44 51 56 | 0 1 -8 -26 -30 -33 }}
Mapping: {{mapping| 1 0 15 44 51 56 | 0 1 -8 -26 -30 -33 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.529
Optimal tunings:
* WE: ~2 = 1199.8928{{c}}, ~3/2 = 701.4664{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.5327{{c}}
 
{{Optimal ET sequence|legend=0| 12f, …, 53d, 65 }}


{{Optimal ET sequence|legend=1| 12f, 53df, 65 }}
Badness (Sintel): 2.01


Badness: 0.048524
== 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.  


== Bischismic ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3136/3125, 32805/32768
[[Comma list]]: 32805/32768, 390625/387072


{{Mapping|legend=1| 2 0 30 69 | 0 1 -8 -20 }}
{{Mapping|legend=1| 1 0 15 109 | 0 1 -8 -67 }}


: Mapping generators: ~567/400, ~3
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1200.2569{{c}}, ~3/2 = 702.1149{{c}}
[[Optimal tuning]] ([[CTE]]): ~567/400 = 1\2, ~3/2 = 701.5899
: [[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 }}


[[Minimax tuning]]:
{{Optimal ET sequence|legend=1| 53, 147d, 200, 253, 306c, 559c }}
* [[7-odd-limit]] eigenmonzo (unchanged-interval) basis: 2.7/3
* [[9-odd-limit]] eigenmonzo (unchanged-interval) basis: 2.9/7


{{Optimal ET sequence|legend=1| 12, 106d, 118, 130, 248, 378 }}
[[Badness]] (Sintel): 5.04
 
[[Badness]]: 0.054744


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


Comma list: 441/440, 3136/3125, 8019/8000
Comma list: 385/384, 19712/19683, 78125/77616


Mapping: {{mapping| 2 0 30 69 102 | 0 1 -8 -20 -30 }}
Mapping: {{mapping| 1 0 15 109 -117 | 0 1 -8 -67 76 }}


Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 701.6077
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=1| 12, 106de, 118, 130, 248 }}
{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}


Badness: 0.028160
Badness (Sintel): 3.83


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


Comma list: 441/440, 729/728, 1001/1000, 3136/3125
Comma list: 325/324, 385/384, 2200/2197, 19712/19683


Mapping: {{mapping| 2 0 30 69 102 -75 | 0 1 -8 -20 -30 26 }}
Mapping: {{mapping| 1 0 15 109 -117 -28 | 0 1 -8 -67 76 20 }}


Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 701.5949
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=1| 12, 106def, 118, 130, 248, 378 }}
{{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }}


Badness: 0.028722
Badness (Sintel): 2.13


===== 17-limit =====
== Schism ==
Subgroup: 2.3.5.7.11.13.17
See [[Archytas clan #Schism]].  


Comma list: 289/288, 441/440, 561/560, 729/728, 3136/3125
Schism is a relatively low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (C–B♭). 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.


Mapping: {{mapping| 2 0 30 69 102 -75 5 | 0 1 -8 -20 -30 26 1 }}
== Bischismic ==
Bischismic tempers out 3136/3125, the [[hemimean comma]], as well as 321489/320000, the [[varunisma]], and may be described as the {{nowrap| 118 & 130 }} temperament. The octave is split in halves, so the [[ploidacot]] of this temperament is diploid monocot. In schismic, -10 fifths make the interval class of 10/9. Bischismic then finds [[7/4]] by a stack of two [[10/9]]'s plus a semi-octave period, and in the [[11-limit]], it simply finds [[11/8]] by a stack of three [[10/9]]'s. [[248edo]] and [[378edo]] make for excellent tunings in both cases.


Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 701.5959
[[Subgroup]]: 2.3.5.7


{{Optimal ET sequence|legend=1| 12, 106def, 118, 130, 248g }}
[[Comma list]]: 3136/3125, 32805/32768


Badness: 0.029340
{{Mapping|legend=1| 2 0 30 69 | 0 1 -8 -20 }}
: mapping generators: ~567/400, ~3


==== Bischis ====
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13
* [[WE]]: ~567/400 = 600.0072{{c}}, ~3/2 = 701.6005{{c}}
: [[error map]]: {{val| +0.014 -0.340 +0.982 -0.629 }}
* [[CWE]]: ~567/400 = 600.0000{{c}}, ~3/2 = 701.5915{{c}}
: error map: {{val| 0.000 -0.364 +0.954 -0.656 }}


Comma list: 351/350, 364/363, 441/440, 3136/3125
[[Minimax tuning]]:  
* [[7-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.7/3
* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7


Mapping: {{mapping| 2 0 30 69 102 131 | 0 1 -8 -20 -30 -39 }}
{{Optimal ET sequence|legend=1| 12, …, 106d, 118, 130, 248, 378 }}


Optimal tuning (CTE): ~55/39 = 1\2, ~3/2 = 701.5708
[[Badness]] (Sintel): 1.39


{{Optimal ET sequence|legend=1| 12f, 106deff, 118f, 130 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.029321
Comma list: 441/440, 3136/3125, 8019/8000


===== 17-limit =====
Mapping: {{mapping| 2 0 30 69 102 | 0 1 -8 -20 -30 }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 221/220, 289/288, 351/350, 441/440, 3136/3125
Optimal tunings:  
* WE: ~99/70 = 600.0165{{c}}, ~3/2 = 701.6316{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.6110{{c}}


Mapping: {{mapping| 2 0 30 69 102 131 5 | 0 1 -8 -20 -30 -39 1 }}
{{Optimal ET sequence|legend=0| 12, …, 106de, 118, 130, 248 }}


Optimal tuning (CTE): ~55/39 = 1\2, ~3/2 = 701.5717
Badness (Sintel): 0.931


{{Optimal ET sequence|legend=1| 12f, 106deff, 118f, 130, 248fg }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.026894
Comma list: 441/440, 729/728, 1001/1000, 3136/3125


== Kleischismic ==
Mapping: {{mapping| 2 0 30 69 102 -75 | 0 1 -8 -20 -30 26 }}
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 32805/32768, 1500625/1492992
Optimal tunings:  
* WE: ~99/70 = 599.9610{{c}}, ~3/2 = 701.5445{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5908{{c}}


{{Mapping|legend=1| 2 1 22 -15 | 0 2 -16 19 }}
{{Optimal ET sequence|legend=0| 12, 118, 130, 248, 378 }}


: Mapping generators: ~1225/864, ~35/24
Badness (Sintel): 1.19


[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~35/24 = 650.920 (~36/35 = 50.920)
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


{{Optimal ET sequence|legend=1| 24, 70c, 94, 118, 212, 330, 542d, 872cd }}
Comma list: 289/288, 441/440, 561/560, 729/728, 3136/3125


[[Badness]]: 0.110583
Mapping: {{mapping| 2 0 30 69 102 -75 5 | 0 1 -8 -20 -30 26 1 }}


=== 11-limit ===
Optimal tunings:
Subgroup: 2.3.5.7.11
* WE: ~99/70 = 600.0331{{c}}, ~3/2 = 701.6387{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5994{{c}}


Comma list: 385/384, 9801/9800, 14641/14580
{{Optimal ET sequence|legend=0| 12, 118, 130, 248g }}


Mapping: {{mapping| 2 1 22 -15 8 | 0 2 -16 19 -1 }}
Badness (Sintel): 1.49


Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.918 (~36/35 = 50.918)
==== Bischis ====
 
{{Optimal ET sequence|legend=1| 24, 70c, 94, 118, 212, 330e, 542de }}
 
Badness: 0.036749
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 352/351, 385/384, 729/728, 1575/1573
Comma list: 351/350, 364/363, 441/440, 3136/3125


Mapping: {{mapping| 2 1 22 -15 8 15 | 0 2 -16 19 -1 -7 }}
Mapping: {{mapping| 2 0 30 69 102 131 | 0 1 -8 -20 -30 -39 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.938 (~36/35 = 50.938)
Optimal tunings:
* WE: ~55/39 = 599.9766{{c}}, ~3/2 = 701.5380{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.5670{{c}}


{{Optimal ET sequence|legend=1| 24, 70c, 94, 118, 212f }}
{{Optimal ET sequence|legend=0| 12f, 106deff, 118f, 130 }}


Badness: 0.037640
Badness (Sintel): 1.21


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


Comma list: 170/169, 289/288, 352/351, 385/384, 561/560
Comma list: 221/220, 289/288, 351/350, 441/440, 3136/3125
 
Mapping: {{mapping| 2 0 30 69 102 131 5 | 0 1 -8 -20 -30 -39 1 }}


Mapping: {{mapping| 2 1 22 -15 8 15 6 | 0 2 -16 19 -1 -7 2 }}
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 tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.942 (~36/35 = 50.942)
{{Optimal ET sequence|legend=0| 12f, 106deff, 118f, 130, 248fg }}


{{Optimal ET sequence|legend=1| 24, 70c, 94, 118, 212fg }}
Badness (Sintel): 1.37


Badness: 0.025615
== 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.  


==== Kleischis ====
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13


Comma list: 325/324, 385/384, 1573/1568, 14641/14580
[[Comma list]]: 32805/32768, 1500625/1492992


Mapping: {{mapping| 2 1 22 -15 8 -36 | 0 2 -16 19 -1 40 }}
{{Mapping|legend=1| 2 1 22 -15 | 0 2 -16 19 }}
: mapping generators: ~1225/864, ~35/24


Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.951 (~36/35 = 50.951)
[[Optimal tuning]]s:
* [[WE]]: ~1225/864 = 600.1246{{c}}, ~35/24 = 651.0550{{c}} (~36/35 = 50.9304{{c}})
: [[error map]]: {{val| +0.249 +0.280 -0.453 -0.650 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~35/24 = 650.9204{{c}} (~36/35 = 50.9204{{c}})
: error map: {{val| 0.000 -0.114 -1.041 -1.338 }}


{{Optimal ET sequence|legend=1| 24f, 70cf, 94, 118f, 212 }}
{{Optimal ET sequence|legend=1| 24, 94, 118, 212, 330, 542d, 872cdd, 1414ccddd }}


Badness: 0.037607
[[Badness]] (Sintel): 2.80


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


Comma list: 289/288, 325/324, 385/384, 442/441, 14641/14580
Comma list: 385/384, 9801/9800, 14641/14580


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


Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.948 (~36/35 = 50.948)
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=1| 24f, 70cf, 94, 118f, 212g }}
{{Optimal ET sequence|legend=0| 24, 94, 118, 212, 330e, 542dee, 872cddeee }}


Badness: 0.024734
Badness (Sintel): 1.21


== Hemischis ==
==== 13-limit ====
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 6144/6125, 19683/19600
Comma list: 352/351, 385/384, 729/728, 1575/1573


{{Mapping|legend=1| 1 0 15 -17 | 0 2 -16 25 }}
Mapping: {{mapping| 2 1 22 -15 8 15 | 0 2 -16 19 -1 -7 }}


: Mapping generators: ~2, ~140/81
Optimal tunings:  
* WE: ~99/70 = 600.0696{{c}}, ~35/24 = 651.0136{{c}} (~36/35 = 50.9440{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9378{{c}} (~36/35 = 50.9378{{c}})


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~140/81 = 950.797
{{Optimal ET sequence|legend=0| 24, 94, 118, 212f }}


{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 313 }}
Badness (Sintel): 1.56


[[Badness]]: 0.045817
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


=== 2.3.5.7.13.19.23 subgroup ===
Comma list: 170/169, 289/288, 352/351, 385/384, 561/560
Subgroup: 2.3.5.7.13.19.23


Comma list: 351/350, 456/455, 513/512, 576/575, 676/675
Mapping: {{mapping| 2 1 22 -15 8 15 6 | 0 2 -16 19 -1 -7 2 }}


Mapping: {{mapping| 1 0 15 -17 14 9 -24 | 0 2 -16 25 -13 -6 36 }}
Optimal tunings:  
* WE: ~99/70 = 600.1134{{c}}, ~35/24 = 651.0646{{c}} (~36/35 = 50.9512{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9414{{c}} (~36/35 = 50.9414{{c}})


Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.783
{{Optimal ET sequence|legend=0| 24, 94, 118 }}


{{Optimal ET sequence|legend=1| 24i, 53, 130 }}
Badness (Sintel): 1.30


Badness (Sintel): 0.583
==== Kleischis ====
Subgroup: 2.3.5.7.11.13


=== 11-limit ===
Comma list: 325/324, 385/384, 1573/1568, 14641/14580
Subgroup: 2.3.5.7.11


Comma list: 540/539, 5632/5625, 8019/8000
Mapping: {{mapping| 2 1 22 -15 8 -36 | 0 2 -16 19 -1 40 }}


Mapping: {{mapping| 1 0 15 -17 51 | 0 2 -16 25 -60 }}
Optimal tunings:  
* WE: ~99/70 = 600.1909{{c}}, ~35/24 = 651.1578{{c}} (~36/35 = 50.9670{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9541{{c}} (~36/35 = 50.9541{{c}})


Optimal tuning (POTE): ~2 = 1\1, ~140/81 = 950.801
{{Optimal ET sequence|legend=0| 24f, 94, 118f, 212 }}


{{Optimal ET sequence|legend=1| 24e, 53, 130, 183, 313 }}
Badness (Sintel): 1.55


Badness: 0.036289
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


=== 13-limit ===
Comma list: 289/288, 325/324, 385/384, 442/441, 14641/14580
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
Mapping: {{mapping| 2 1 22 -15 8 -36 6 | 0 2 -16 19 -1 40 2 }}


Comma list: 351/350, 540/539, 676/675, 4096/4095
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}})


Mapping: {{mapping| 1 0 15 -17 51 14 | 0 2 -16 25 -60 -13 }}
{{Optimal ET sequence|legend=0| 24f, 94, 118f, 212g }}


Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.801
Badness (Sintel): 1.26


{{Optimal ET sequence|legend=1| 24e, 53, 130, 183, 313 }}
== 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]].


Badness: 0.020816
[[Subgroup]]: 2.3.5.7


=== 17-limit ===
[[Comma list]]: 245/243, 32805/32768
Subgroup: 2.3.5.7.11.13.17


Comma list: 351/350, 442/441, 561/560, 676/675, 4096/4095
{{Mapping|legend=1| 1 1 7 -1 | 0 2 -16 13 }}
: mapping generators: ~2, ~128/105


Mapping: {{mapping| 1 0 15 -17 51 14 -49 | 0 2 -16 25 -60 -13 67 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.7707{{c}}, ~128/105 = 351.2748{{c}}
: [[error map]]: {{val| +0.771 +1.365 -1.315 -3.024 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 351.0471{{c}}
: error map: {{val| 0.000 +0.139 -3.068 -5.213 }}


Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.810
{{Optimal ET sequence|legend=1| 17, 24, 41, 106d, 147d, 188cd }}


{{Optimal ET sequence|legend=1| 53, 130, 183, 496d }}
[[Badness]] (Sintel): 2.03


Badness: 0.021073
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 19-limit ===
Comma list: 243/242, 245/242, 385/384
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 4096/4095
Mapping: {{mapping| 1 1 7 -1 2 | 0 2 -16 13 5 }}


Mapping: {{mapping| 1 0 15 -17 51 14 -49 9 | 0 2 -16 25 -60 -13 67 -6 }}
Optimal tunings:  
* WE: ~2 = 1200.3891{{c}}, ~11/9 = 351.1275{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0141{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.809
{{Optimal ET sequence|legend=0| 17, 24, 41, 106d }}


{{Optimal ET sequence|legend=1| 53, 130, 183, 313h }}
Badness (Sintel): 1.30


Badness : 0.018262
=== 13-limit ===
 
Subgroup: 2.3.5.7.11.13
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 736/735, 4096/4095
Comma list: 105/104, 144/143, 243/242, 245/242


Mapping: {{mapping| 1 0 15 -17 51 14 -49 9 -24 | 0 2 -16 25 -60 -13 67 -6 36 }}
Mapping: {{mapping| 1 1 7 -1 2 4 | 0 2 -16 13 5 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.807
Optimal tunings:
* WE: ~2 = 1199.9362{{c}}, ~11/9 = 351.0061{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0247{{c}}


{{Optimal ET sequence|legend=1| 53, 130, 183, 313h }}
{{Optimal ET sequence|legend=0| 17, 24, 41 }}


Badness (Sintel): 0.014819
Badness (Sintel): 1.27


; Music
== Hemischis ==
* ''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
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.


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


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


[[Comma list]]: 686/675, 32805/32768
[[Comma list]]: 6144/6125, 19683/19600


{{Mapping|legend=1| 1 2 -1 1 | 0 -3 24 13 }}
{{Mapping|legend=1| 1 0 15 -17 | 0 2 -16 25 }}
: mapping generators: ~2, ~140/81


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~160/147 = 166.140
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8579{{c}}, ~140/81 = 951.6847{{c}}
: [[error map]]: {{val| -0.142 -0.586 +0.600 +0.708 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 951.7966{{c}}
: error map: {{val| 0.000 -0.362 +0.941 +1.088 }}


{{Optimal ET sequence|legend=1| 29, 36, 65 }}
{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 313 }}


[[Badness]]: 0.174705
[[Badness]] (Sintel): 1.16


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


Comma list: 245/242, 686/675, 896/891
Comma list: 540/539, 5632/5625, 8019/8000


Mapping: {{mapping| 1 2 -1 1 0 | 0 -3 24 13 25 }}
Mapping: {{mapping| 1 0 15 -17 51 | 0 2 -16 25 -60 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.097
Optimal tunings:
* WE: ~2 = 1199.8482{{c}}, ~140/81 = 950.6809{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~140/81 = 950.8020{{c}}


{{Optimal ET sequence|legend=1| 29, 36, 65 }}
{{Optimal ET sequence|legend=0| 53, 130, 183, 313, 809cd }}


Badness: 0.068310
Badness (Sintel): 1.20


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


Comma list: 91/90, 169/168, 245/242, 896/891
Comma list: 351/350, 540/539, 676/675, 4096/4095


Mapping: {{mapping| 1 2 -1 1 0 3 | 0 -3 24 13 25 5 }}
Mapping: {{mapping| 1 0 15 -17 51 14 | 0 2 -16 25 -60 -13 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.054
Optimal tunings:
* WE: ~2 = 1199.9140{{c}}, ~140/81 = 950.7324{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~140/81 = 950.8010{{c}}


{{Optimal ET sequence|legend=1| 29, 36, 65f, 94df, 159df }}
{{Optimal ET sequence|legend=0| 53, 130, 183, 313 }}


Badness: 0.043750
Badness (Sintel): 0.860


== Tertiaschis ==
=== 17-limit ===
The tertiaschis temperament ({{nowrap|94 &amp; 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: 351/350, 442/441, 561/560, 676/675, 4096/4095


[[Comma list]]: 32805/32768, 1071875/1062882
Mapping: {{mapping| 1 0 15 -17 51 14 -49 | 0 2 -16 25 -60 -13 67 }}


{{Mapping|legend=1| 1 2 -1 10 | 0 -3 24 -52 }}
Optimal tunings:
* WE: ~2 = 1199.9740{{c}}, ~26/15 = 950.7894{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8100{{c}}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~192/175 = 166.019
{{Optimal ET sequence|legend=0| 53, 130, 183, 496d }}


{{Optimal ET sequence|legend=1| 65, 94, 159, 253, 412cd }}
Badness (Sintel): 1.07


[[Badness]]: 0.211859
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


=== 11-limit ===
Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 4096/4095
Subgroup: 2.3.5.7.11


Comma list: 385/384, 4000/3993, 19712/19683
Mapping: {{mapping| 1 0 15 -17 51 14 -49 9 | 0 2 -16 25 -60 -13 67 -6 }}


Mapping: {{mapping| 1 2 -1 10 0 | 0 -3 24 -52 25 }}
Optimal tunings:  
* WE: ~2 = 1200.0464{{c}}, ~26/15 = 950.8459{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8091{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.017
{{Optimal ET sequence|legend=0| 53, 130, 183, 313h }}


{{Optimal ET sequence|legend=1| 65, 94, 159, 253, 412cd }}
Badness (Sintel): 1.11


Badness: 0.061336
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


=== 13-limit ===
Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 736/735, 4096/4095
Subgroup: 2.3.5.7.11.13


Comma list: 325/324, 385/384, 1575/1573, 10985/10976
Mapping: {{mapping| 1 0 15 -17 51 14 -49 9 -24 | 0 2 -16 25 -60 -13 67 -6 36 }}


Mapping: {{mapping| 1 2 -1 10 0 12 | 0 -3 24 -52 25 -60 }}
Optimal tunings:  
* WE: ~2 = 1200.0215{{c}}, ~26/15 = 950.8239{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8069{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.016
{{Optimal ET sequence|legend=0| 53, 130, 183, 313h }}


{{Optimal ET sequence|legend=1| 65f, 94, 159, 253, 412cdf, 665ccdef }}
Badness (Sintel): 1.06


Badness: 0.036700
; Music
* ''HemischisMatic EP'' (2023) by [[User:Francium|Francium]] – [https://open.spotify.com/album/1Fx2shLclpNgFQJRw3ZHya Spotify] | [https://francium223.bandcamp.com/album/hemischismatic-ep Bandcamp] | [https://www.youtube.com/playlist?list=PLLZE7hMjEXRaiipPYK1InZBXTru_UtRsq YouTube] – 4-piece extended play


=== 17-limit ===
== Term ==
Subgroup: 2.3.5.7.11.13.17
Term tempers out the [[landscape comma]], mapping [[63/50]] to the 1/3-octave period. It can be described as {{nowrap| 12 & 171 }}, and is the unique temperament that equates a syntonic~Pythagorean comma with a stack of three [[marvel comma]]s. A [[septimal comma]] is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[171edo]] makes for an excellent tuning.  


Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 2 -1 10 0 12 -2 | 0 -3 24 -52 25 -60 44 }}
[[Comma list]]: 32805/32768, 250047/250000


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.012
{{Mapping|legend=1| 3 0 45 94 | 0 1 -8 -18 }}
: mapping generators: ~63/50, ~3


{{Optimal ET sequence|legend=1| 65f, 94, 159, 253 }}
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 400.0257{{c}}, ~3/2 = 701.7873{{c}}
: [[error map]]: {{val| +0.077 -0.091 -0.072 +0.031 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 701.7383{{c}}
: error map: {{val| 0.000 -0.217 -0.220 -0.115 }}


Badness: 0.026504
[[Minimax tuning]]:  
* [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis)]]: 2.5/3
* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7


== Countertertiaschis ==
{{Optimal ET sequence|legend=1| 12, …, 159, 171, 867, 1038, 1209, 1380, 1551, 1722 }}
The countertertiaschis temperament ({{nowrap|159 &amp; 224}}) has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel]], but tempers out 244140625/243045684 for prime 7.


[[Subgroup]]: 2.3.5.7
[[Badness]] (Sintel): 0.505


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


{{Mapping|legend=1| 1 2 -1 -12 | 0 -3 24 107 }}
Subgroup: 2.3.5.7.11


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~625/567 = 166.0621
Comma list: 441/440, 4375/4356, 32805/32768


{{Optimal ET sequence|legend=1| 65d, 159, 224, 383, 607 }}
Mapping: {{mapping| 3 0 45 94 134 | 0 1 -8 -18 -26 }}


[[Badness]]: 0.188043
Optimal tunings:  
* WE: ~44/35 = 400.0464{{c}}, ~3/2 = 701.9053{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8178{{c}}


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


Comma list: 3025/3024, 4000/3993, 32805/32768
Badness (Sintel): 1.97


Mapping: {{mapping| 1 2 -1 -12 0 | 0 -3 24 107 25 }}
==== 13-limit ====
 
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.0628
 
{{Optimal ET sequence|legend=1| 65d, 159, 224, 383, 607 }}
 
Badness: 0.048943
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976
Comma list: 364/363, 441/440, 625/624, 13720/13689


Mapping: {{mapping| 1 2 -1 -12 0 -10 | 0 -3 24 107 25 99 }}
Mapping: {{mapping| 3 0 45 94 134 168 | 0 1 -8 -18 -26 -33 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.0628
Optimal tunings:
* WE: ~44/35 = 400.0449{{c}}, ~3/2 = 701.8995{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8156{{c}}


{{Optimal ET sequence|legend=1| 65d, 159, 224, 383, 607 }}
{{Optimal ET sequence|legend=0| 12f, …, 159, 330 }}


Badness: 0.024506
Badness (Sintel): 1.53


== Pogo ==
==== 17-limit ====
{{See also| Stearnsmic clan #Pogo }}
Subgroup: 2.3.5.7.11.13.17


The pogo temperament ({{nowrap|94 &amp; 130}}) splits the period in two to address the difference between [[#Tertiaschis]] and [[#Countertertiaschis]]. The schismic tempering of the fifth is just about right for tempering out the stearnsma.
Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619


[[Subgroup]]: 2.3.5.7
Mapping: {{mapping| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }}


[[Comma list]]: 32805/32768, 118098/117649
Optimal tunings:  
* WE: ~34/27 = 400.0195{{c}}, ~3/2 = 701.8439{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 701.8081{{c}}


{{Mapping|legend=1| 2 1 22 2 | 0 3 -24 5 }}
{{Optimal ET sequence|legend=0| 12f, 159, 171, 330 }}


: Mapping generators: ~343/243, ~9/7
Badness (Sintel): 1.38


[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~9/7 = 433.901
=== Terminator ===
 
Terminator tempers out 540/539, and may be described as {{nowrap| 171 & 183 }}.  
{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354 }}
 
[[Badness]]: 0.079635


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


Comma list: 540/539, 4000/3993, 32805/32768
Comma list: 540/539, 32805/32768, 137781/137500


Mapping: {{mapping| 2 1 22 2 25 | 0 3 -24 5 -25 }}
Mapping: {{mapping| 3 0 45 94 -137 | 0 1 -8 -18 31 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911
Optimal tunings:
* WE: ~63/50 = 399.9677{{c}}, ~3/2 = 701.6278{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6846{{c}}


{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354, 578 }}
{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 537, 891de }}


Badness: 0.031857
Badness (Sintel): 2.21


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


Comma list: 540/539, 729/728, 1575/1573, 4096/4095
Comma list: 540/539, 729/728, 4096/4095, 31250/31213


Mapping: {{mapping| 2 1 22 2 25 -2 | 0 3 -24 5 -25 13 }}
Mapping: {{mapping| 3 0 45 94 -137 -103 | 0 1 -8 -18 31 24 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911
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=1| 36, 94, 130, 224, 354, 578 }}
{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }}


Badness: 0.017514
Badness (Sintel): 1.47


== Term ==
==== 17-limit ====
Term tempers out the [[landscape comma]], mapping ~63/50 to the 1/3-octave period. It can be described as {{nowrap|12 &amp; 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: 2.3.5.7.11.13.17


[[Subgroup]]: 2.3.5.7
Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095


[[Comma list]]: 32805/32768, 250047/250000
Mapping: {{mapping| 3 0 45 94 -137 -103 -2 | 0 1 -8 -18 31 24 3 }}


{{Mapping|legend=1| 3 0 45 94 | 0 1 -8 -18 }}
Optimal tunings:
* WE: ~63/50 = 399.9757{{c}}, ~3/2 = 701.6458{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}}


: Mapping generators: ~63/50, ~3
{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }}


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1＼3, ~3/2 = 701.742
Badness (Sintel): 1.04


[[Minimax tuning]]:
=== Semiterm ===
* [[7-odd-limit]] [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis)]]: 2.5/3
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.  
* [[9-odd-limit]] eigenmonzo (unchanged-interval) basis: 2.9/7
 
{{Optimal ET sequence|legend=1| 12, 147d, 159, 171, 867, 1038, 1209, 1380, 1551, 1722 }}
 
[[Badness]]: 0.019950
 
=== Terminal ===
The terminal temperament ({{nowrap|12 &amp; 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
Subgroup: 2.3.5.7.11


Comma list: 441/440, 4375/4356, 32805/32768
Comma list: 9801/9800, 32805/32768, 151263/151250


Mapping: {{mapping| 3 0 45 94 134 | 0 1 -8 -18 -26 }}
Mapping: {{mapping| 6 0 90 188 287 | 0 1 -8 -18 -28 }}
: mapping generators: ~55/49, ~3


Optimal tuning (POTE): ~44/35 = 1＼3, ~3/2 = 701.824
Optimal tunings:  
* WE: ~55/49 = 200.0134{{c}}, ~3/2 = 701.7931{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7426{{c}}


{{Optimal ET sequence|legend=0| 12, 147de, 159, 330 }}
{{Optimal ET sequence|legend=0| 12, …, 330e, 342, 1380, 1722, 2064, 2406c, 5154bccdde }}


Badness: 0.059502
Badness (Sintel): 0.973


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


Comma list: 364/363, 441/440, 625/624, 13720/13689
Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375


Mapping: {{mapping| 3 0 45 94 134 168 | 0 1 -8 -18 -26 -33 }}
Mapping: {{mapping| 6 0 90 188 287 355 | 0 1 -8 -18 -28 -35 }}


Optimal tuning (POTE): ~44/35 = 1＼3, ~3/2 = 701.821
Optimal tunings:  
* WE: ~55/49 = 200.0083{{c}}, ~3/2 = 701.7549{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7238{{c}}


{{Optimal ET sequence|legend=0| 12f, 147def, 159, 330 }}
{{Optimal ET sequence|legend=0| 12f, 330eff, 342f, 696f }} *


Badness: 0.037082
<nowiki>*</nowiki> optimal patent val: [[354edo|354]]


==== 17-limit ====
Badness (Sintel): 1.85
Subgroup: 2.3.5.7.11.13.17


Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619
=== Hemiterm ===
The hemiterm temperament tempers out [[3025/3024]] (lehmerisma), and may be described as {{nowrap| 159 & 183 }}. Its ploidacot is triploid alpha-dicot.


Mapping: {{mapping| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }}
Subgroup: 2.3.5.7.11


Optimal tuning (POTE): ~34/27 = 1＼3, ~3/2 = 701.810
Comma list: 3025/3024, 32805/32768, 102487/102400


{{Optimal ET sequence|legend=0| 12f, 147def, 159, 171, 330 }}
Mapping: {{mapping| 3 0 45 94 8 | 0 2 -16 -36 1 }}
: mapping generators: ~63/50, ~693/400


Badness: 0.027073
Optimal tunings:  
* WE: ~63/50 = 400.0309{{c}}, ~693/400 = 950.9458{{c}} (~12/11 = 150.8841{{c}})
* CWE: ~63/50 = 400.0000{{c}}, ~693/400 = 950.8707{{c}} (~12/11 = 150.8707{{c}})


=== Terminator ===
{{Optimal ET sequence|legend=0| 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce }}
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 32805/32768, 137781/137500
 
Mapping: {{mapping| 3 0 45 94 -137 | 0 1 -8 -18 31 }}


Optimal tuning (POTE): ~63/50 = 1＼3, ~3/2 = 701.685
Badness (Sintel): 0.684
 
{{Optimal ET sequence|legend=0| 12e, 159e, 171, 183, 354, 537, 891de }}
 
Badness: 0.066968


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


Comma list: 540/539, 729/728, 4096/4095, 31250/31213
Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712


Mapping: {{mapping| 3 0 45 94 -137 -103 | 0 1 -8 -18 31 24 }}
Mapping: {{mapping| 3 0 45 94 8 42 | 0 2 -16 -36 1 -13 }}


Optimal tuning (POTE): ~63/50 = 1＼3, ~3/2 = 701.689
Optimal tunings:
* WE: ~63/50 = 400.0541{{c}}, ~26/15 = 951.0013{{c}} (~12/11 = 150.8932{{c}})
* CWE: ~63/50 = 400.0000{{c}}, ~26/15 = 950.8696{{c}} (~12/11 = 150.8696{{c}})


{{Optimal ET sequence|legend=0| 171, 183, 354, 891de, 1245dee, 1599ddee }}
{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f }}


Badness: 0.035487
Badness (Sintel): 1.30


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


Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095
Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264


Mapping: {{mapping| 3 0 45 94 -137 -103 -2 | 0 1 -8 -18 31 24 3 }}
Mapping: {{mapping| 3 0 45 94 8 42 -2 | 0 2 -16 -36 1 -13 6 }}


Optimal tuning (POTE): ~63/50 = 1＼3, ~3/2 = 701.688
Optimal tunings:
* WE: ~34/27 = 400.0373{{c}}, ~26/15 = 950.9556{{c}} (~12/11 = 150.8809{{c}})
* CWE: ~34/27 = 400.0000{{c}}, ~26/15 = 950.8652{{c}} (~12/11 = 150.8652{{c}})


{{Optimal ET sequence|legend=0| 171, 183, 354, 891de, 1245dee, 1599ddee }}
{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f, 525f }}


Badness: 0.020434
Badness (Sintel): 1.14


=== Semiterm ===
== Altinex ==
The semiterm temperament ({{nowrap|12 &amp; 342}}) has a period of 1/6 octave and tempers out [[9801/9800]] (kalisma) and 151263/151250 (odiheim comma).
Named by [[Aura]] in 2021, altinex is an alternative to [[#Hemiterm|hemiterm]] and may be described as {{nowrap| 24 & 159 }}. [[159edo]] itself makes for a recommendable tuning.  


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7


Comma list: 9801/9800, 32805/32768, 151263/151250
[[Comma list]]: 32805/32768, 367653125/362797056


Mapping: {{mapping| 6 0 90 188 287 | 0 1 -8 -18 -28 }}
{{Mapping|legend=1| 3 0 45 -32 | 0 2 -16 17 }}
: mapping generators: ~1536/1225, ~34300/19683


: Mapping generators: ~55/49, ~3
[[Optimal tuning]]s:  
* [[WE]]: ~1536/1225 = 400.1360{{c}}, ~34300/19683 = 951.2867{{c}}
: [[error map]]: {{val| +0.408 +0.618 -0.781 -1.304 }}
* [[CWE]]: ~1536/1225 = 400.0000{{c}}, ~34300/19683 = 950.9638{{c}}
: error map: {{val| 0.000 -0.027 -1.735 -2.441 }}


Optimal tuning (POTE): ~55/49 = 1＼6, ~3/2 = 701.7460
{{Optimal ET sequence|legend=1| 24, 135, 159, 612ccdd }}


{{Optimal ET sequence|legend=0| 12, 330e, 342, 1380, 1722, 2064, 2406c }}
[[Badness]] (Sintel): 10.7


Badness: 0.029438
=== 11-limit ===
Subgroup: 2.3.5.7.11


==== 13-limit ====
Comma list: 385/384, 14700/14641, 19712/19683
Subgroup: 2.3.5.7.11.13


Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375
Mapping: {{mapping| 3 0 45 -32 8 | 0 2 -16 17 1 }}


Mapping: {{mapping| 6 0 90 188 287 355 | 0 1 -8 -18 -28 -35 }}
Optimal tunings:  
* WE: ~44/35 = 400.1156{{c}}, ~121/70 = 951.2377{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~121/70 = 950.9634{{c}}


Optimal tuning (POTE): ~55/49 = 1＼6, ~3/2 = 701.7256
{{Optimal ET sequence|legend=0| 24, 135, 159 }}


{{Optimal ET sequence|legend=0| 12f, 330eff, 342f, 696f }} *
Badness (Sintel): 3.35


<nowiki>*</nowiki> optimal patent val: [[354edo|354]]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.044657
Comma list: 364/363, 385/384, 676/675, 19712/19683


=== Hemiterm ===
Mapping: {{mapping| 3 0 45 -32 8 42 | 0 2 -16 17 1 -13 }}
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 32805/32768, 102487/102400
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 8 | 0 2 -16 -36 1 }}
{{Optimal ET sequence|legend=0| 24, 135f, 159 }}


: mapping generators: ~63/50, ~693/400
Badness (Sintel): 2.27


Optimal tuning (POTE): ~63/50 = 1＼3, ~693/400 = 950.872 (~12/11 = 150.872)
== 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.  


{{Optimal ET sequence|legend=0| 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce }}
[[Subgroup]]: 2.3.5.7


Badness: 0.020687
[[Comma list]]: 686/675, 32805/32768


==== 13-limit ====
{{Mapping|legend=1| 1 2 -1 1 | 0 -3 24 13 }}
Subgroup: 2.3.5.7.11.13


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


Mapping: {{mapping| 3 0 45 94 8 42 | 0 2 -16 -36 1 -13 }}
{{Optimal ET sequence|legend=1| 29, 36, 65 }}


Optimal tuning (POTE): ~63/50 = 1＼3, ~26/15 = 950.873 (~12/11 = 150.873)
[[Badness]] (Sintel): 4.42


{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f }}
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 245/242, 686/675, 896/891
 
Mapping: {{mapping| 1 2 -1 1 0 | 0 -3 24 13 25 }}
 
Optimal tunings:
* WE: ~2 = 1200.6379{{c}}, ~11/10 = 166.1853{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.1157{{c}}
 
{{Optimal ET sequence|legend=0| 29, 36, 65 }}


Badness: 0.031362
Badness (Sintel): 2.26


==== 17-limit ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 169/168, 245/242, 896/891


Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264
Mapping: {{mapping| 1 2 -1 1 0 3 | 0 -3 24 13 25 5 }}


Mapping: {{mapping| 3 0 45 94 8 42 -2 | 0 2 -16 -36 1 -13 6 }}
Optimal tunings:  
* WE: ~2 = 1201.1361{{c}}, ~11/10 = 166.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0833{{c}}


Optimal tuning (POTE): ~34/27 = 1＼3, ~26/15 = 950.867 (~12/11 = 150.867)
{{Optimal ET sequence|legend=0| 29, 65f, 94df }}


{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f, 525f, 867ff }}
Badness (Sintel): 1.81


Badness: 0.022316
== Tertiaschis ==
Named by [[Xenllium]] in 2021, tertiaschis may be described as {{nowrap| 94 & 159 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 1071875/1062882 for prime 7.  


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


[[Comma list]]: 32805/32768, 367653125/362797056
[[Comma list]]: 32805/32768, 1071875/1062882


{{Mapping|legend=1| 3 0 45 -32 | 0 2 -16 17 }}
{{Mapping|legend=1| 1 2 -1 10 | 0 -3 24 -52 }}


: mapping generators: ~1536/1225, ~34300/19683
[[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 tuning]] ([[CTE]]): ~1536/1225 = 1＼3, ~34300/19683 = 950.9654
{{Optimal ET sequence|legend=1| 65, 94, 159, 253, 412cd }}
 
{{Optimal ET sequence|legend=1| 24, …, 111c, 135, 159, 612ccdd, 771ccdd }}


[[Badness]]: 0.422026
[[Badness]] (Sintel): 5.36


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


Comma list: 385/384, 14700/14641, 19712/19683
Comma list: 385/384, 4000/3993, 19712/19683


Mapping: {{mapping| 3 0 45 -32 8 | 0 2 -16 17 1 }}
Mapping: {{mapping| 1 2 -1 10 0 | 0 -3 24 -52 25 }}


Optimal tuning (CTE): ~44/35 = 1＼3, ~121/70 = 950.9658
Optimal tunings:  
* WE: ~2 = 1200.3379{{c}}, ~11/10 = 166.0638{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0167{{c}}


Optimal ET sequence: {{Optimal ET sequence| 24, …, 111c, 135, 159, 612ccdd, 771ccdd }}
{{Optimal ET sequence|legend=0| 65, 94, 159, 253, 412cd, 665ccde }}


Badness: 0.101224
Badness (Sintel): 2.07


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


Comma list: 364/363, 385/384, 676/675, 19712/19683
Comma list: 325/324, 385/384, 1575/1573, 10985/10976


Mapping: {{mapping| 3 0 45 -32 8 42 | 0 2 -16 17 1 -13 }}
Mapping: {{mapping| 1 2 -1 10 0 12 | 0 -3 24 -52 25 -60 }}


Optimal tuning (CTE): ~44/35 = 1＼3, ~26/15 = 950.9360
Optimal tunings:  
* WE: ~2 = 1200.3467{{c}}, ~11/10 = 166.0635{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0142{{c}}


Optimal ET sequence: {{Optimal ET sequence| 24, …, 111cf, 135f, 159 }}
{{Optimal ET sequence|legend=0| 65f, 94, 159, 253, 412cdf, 665ccdef }}


Badness: 0.054894
Badness (Sintel): 1.52


== Sesquiquartififths ==
=== 17-limit ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17


[[Comma list]]: 2401/2400, 32805/32768
Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976


{{Mapping|legend=1| 1 1 7 5 | 0 4 -32 -15 }}
Mapping: {{mapping| 1 2 -1 10 0 12 -2 | 0 -3 24 -52 25 -60 44 }}


: Mapping generators: ~2, ~448/405
Optimal tunings:  
* WE: ~2 = 1200.3019{{c}}, ~11/10 = 166.0535{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0114{{c}}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~448/405 = 175.434
{{Optimal ET sequence|legend=1| 65f, 94, 159, 253 }}


[[Minimax tuning]]:
Badness (Sintel): 1.35
* [[7-odd-limit]] [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/3
* [[9-odd-limit]] eigenmonzo (unchanged-interval) basis: 2.9/7


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


[[Badness]]: 0.011244
[[Subgroup]]: 2.3.5.7


=== Sesquart ===
[[Comma list]]: 32805/32768, 244140625/243045684
Subgroup: 2.3.5.7.11


Comma list: 243/242, 441/440, 16384/16335
{{Mapping|legend=1| 1 2 -1 -12 | 0 -3 24 107 }}


Mapping: {{mapping| 1 1 7 5 2 | 0 4 -32 -15 10 }}
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1265{{c}}, ~625/567 = 166.0797{{c}}
: [[error map]]: {{val| +0.127 +0.059 -0.529 +0.178 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/567 = 166.0632{{c}}
: error map: {{val| 0.000 -0.145 -0.797 -0.065 }}


Optimal tuning (POTE): ~2 = 1\1, ~256/231 = 175.406
{{Optimal ET sequence|legend=1| 65d, 159, 224, 383, 607 }}


Optimal ET sequence: {{Optimal ET sequence| 41, 89, 130, 301e, 431e }}
[[Badness]] (Sintel): 4.76


Badness: 0.029306
=== 11-limit ===
Subgroup: 2.3.5.7.11


==== 13-limit ====
Comma list: 3025/3024, 4000/3993, 32805/32768
Subgroup: 2.3.5.7.11.13


Comma list: 243/242, 364/363, 441/440, 3584/3575
Mapping: {{mapping| 1 2 -1 -12 0 | 0 -3 24 107 25 }}


Mapping: {{mapping| 1 1 7 5 2 -2 | 0 4 -32 -15 10 39 }}
Optimal tunings:  
* WE: ~2 = 1200.0804{{c}}, ~11/10 = 166.0739{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0634{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.409
{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}


Optimal ET sequence: {{Optimal ET sequence| 41, 89, 130, 301e, 431e }}
Badness (Sintel): 1.62


Badness: 0.022396
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


===== Sesquartia =====
Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976
Subgroup: 2.3.5.7.11.13.17


Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575
Mapping: {{mapping| 1 2 -1 -12 0 -10 | 0 -3 24 107 25 99 }}


Mapping: {{mapping| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }}
Optimal tunings:  
* WE: ~2 = 1200.0805{{c}}, ~11/10 = 166.0740{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0635{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.424
{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}


Optimal ET sequence: {{Optimal ET sequence| 41, 89g, 130, 171, 301e }}
Badness (Sintel): 1.01


Badness: 0.023126
== Quadrant ==
Named by [[Xenllium]] in 2021, quadrant tempers out 390625/388962, the [[dimcomp comma]], and maps [[25/21]] to the 1/4-octave period. It may be described as the {{nowrap| 12 & 212 }} temperament; its ploidacot is tetraploid monocot. Just as [[#Term|term]] equates the syntonic~Pythagorean comma with three [[marvel comma]]s, quadrant equates the syntonic~Pythagorean comma with four. A [[septimal comma]] is then found as a stack of five marvel commas.  


====== 19-limit ======
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594
[[Comma list]]: 32805/32768, 390625/388962


Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }}
{{Mapping|legend=1| 4 0 60 119 | 0 1 -8 -17 }}
: mapping generators: ~25/21, ~3


Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.419
[[Optimal tuning]]s:
* [[WE]]: ~2 = 300.0255{{c}}, ~3/2 = 701.8831{{c}}
: [[error map]]: {{val| +0.102 +0.030 -0.664 +0.462 }}
* [[CWE]]: ~2 = 300.0000{{c}}, ~3/2 = 701.8180{{c}}
: error map: {{val| 0.000 -0.137 -0.858 +0.268 }}


Optimal ET sequence: {{Optimal ET sequence| 41, 89g, 130, 171, 301eh }}
{{Optimal ET sequence|legend=1| 12, …, 200, 212, 224, 436, 660 }}


Badness: 0.020466
[[Badness]] (Sintel): 2.79


====== 23-limit ======
=== 11-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11


Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594
Comma list: 1375/1372, 6250/6237, 32805/32768


Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 -6 | 0 4 -32 -15 10 39 69 -12 72 }}
Mapping: {{mapping| 4 0 60 119 185 | 0 1 -8 -17 -27 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.412
Optimal tunings:
* WE: ~25/21 = 300.0244{{c}}, ~3/2 = 701.8759{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8145{{c}}


Optimal ET sequence: {{Optimal ET sequence| 41i, 89gi, 130, 171, 301eh }}
{{Optimal ET sequence|legend=0| 12, …, 212, 224, 436, 660 }}


Badness: 0.019043
Badness (Sintel): 1.51


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


Comma list: 243/242, 256/255, 273/272, 364/363, 441/440
Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647


Mapping: {{mapping| 1 1 7 5 2 -2 0 | 0 4 -32 -15 10 39 28 }}
Mapping: {{mapping| 4 0 60 119 185 224 | 0 1 -8 -17 -27 -33 }}


Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.386
Optimal tunings:
* WE: ~25/21 = 300.0234{{c}}, ~3/2 = 701.8707{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8123{{c}}


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


Badness: 0.028443
Badness (Sintel): 1.13


====== 19-limit ======
== Sesquiquartififths ==
Subgroup: 2.3.5.7.11.13.17.19
Sesquiquartififths tempers out 2401/2400, the [[breedsma]], and may be described as the {{nowrap| 41 & 171 }} temperament. It splits the fifth into four; its [[ploidacot]] is thus tetracot.  


Comma list: 171/170, 243/242, 256/255, 273/272, 324/323, 441/440
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 1 7 5 2 -2 0 6 | 0 4 -32 -15 10 39 28 -12 }}
[[Comma list]]: 2401/2400, 32805/32768


Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.380
{{Mapping|legend=1| 1 1 7 5 | 0 4 -32 -15 }}
: mapping generators: ~2, ~448/405


Optimal ET sequence: {{Optimal ET sequence| 41, 89, 130g }}
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0846{{c}}, ~448/405 = 175.4460{{c}}
: [[error map]]: {{val| +0.085 -0.086 +0.007 -0.093 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~448/405 = 175.4320{{c}}
: error map: {{val| 0.000 -0.227 -0.137 -0.306 }}


Badness: 0.023059
[[Minimax tuning]]:  
* [[7-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7


===== Hearty =====
{{Optimal ET sequence|legend=1| 41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625
[[Badness]] (Sintel): 0.285


Mapping: {{mapping| 1 1 7 5 2 -2 13 | 0 4 -32 -15 10 39 -61 }}
=== Sesquart ===
Sesquart is the main [[11-limit|11-]] and [[13-limit]] extension of sesquiquartififths of practical interest, as it identifies the neutral third with [[11/9]], which is realized in [[41edo]], [[89edo]], [[130edo]], and [[171edo]] also makes for a possible tuning.


Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.377
Subgroup: 2.3.5.7.11


Optimal ET sequence: {{Optimal ET sequence| 41g, 89, 130, 609ceefgg }}
Comma list: 243/242, 441/440, 16384/16335


Badness: 0.030680
Mapping: {{mapping| 1 1 7 5 2 | 0 4 -32 -15 10 }}


====== 19-limit ======
Optimal tunings:
Subgroup: 2.3.5.7.11.13.17.19
* WE: ~2 = 1199.8171{{c}}, ~256/231 = 175.3793{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~256/231 = 175.4081{{c}}


Comma list: 221/220, 243/242, 361/360, 364/363, 441/440, 456/455
{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}


Mapping: {{mapping| 1 1 7 5 2 -2 13 6 | 0 4 -32 -15 10 39 -61 -12 }}
Badness (Sintel): 0.969


Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.377
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Optimal ET sequence: {{Optimal ET sequence| 41g, 89, 130, 609ceefggh }}
Comma list: 243/242, 364/363, 441/440, 3584/3575


Badness: 0.022816
Mapping: {{mapping| 1 1 7 5 2 -2 | 0 4 -32 -15 10 39 }}


====== 23-limit ======
Optimal tunings:
Subgroup: 2.3.5.7.11.13.17.19.23
* WE: ~2 = 1199.8352{{c}}, ~72/65 = 175.3852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4095{{c}}


Comma list: 221/220, 243/242, 276/275, 323/322, 361/360, 364/363, 441/440
{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}


Mapping: {{mapping| 1 1 7 5 2 -2 13 6 13 | 0 4 -32 -15 10 39 -61 -12 -58 }}
Badness (Sintel): 0.925


Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.376
===== Heartia =====
Subgroup: 2.3.5.7.11.13.17


Optimal ET sequence: {{Optimal ET sequence| 41g, 89, 130, 609ceefggh }}
Comma list: 243/242, 256/255, 273/272, 364/363, 441/440


Badness: 0.019121
Mapping: {{mapping| 1 1 7 5 2 -2 0 | 0 4 -32 -15 10 39 28 }}


=== Bisesqui ===
Optimal tunings:
Subgroup: 2.3.5.7.11
* WE: ~2 = 1199.6422{{c}}, ~72/65 = 175.3338{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3857{{c}}


Comma list: 2401/2400, 9801/9800, 32805/32768
{{Optimal ET sequence|legend=0| 41, 89, 130g }}


Mapping: {{mapping| 2 2 14 10 23 | 0 4 -32 -15 -55 }}
Badness (Sintel): 1.45


: Mapping generators: ~99/70, ~448/405
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19


Optimal tuning (POTE): ~99/70 = 1\2, ~448/405 = 175.435
Comma list: 171/170, 243/242, 256/255, 273/272, 324/323, 441/440


{{Optimal ET sequence|legend=1| 82e, 130, 212, 342, 1156, 1498, 1840d }}
Mapping: {{mapping| 1 1 7 5 2 -2 0 6 | 0 4 -32 -15 10 39 28 -12 }}


Badness: 0.016968
Optimal tunings:  
* WE: ~2 = 1199.7499{{c}}, ~21/19 = 175.3432{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.3797{{c}}


== Quintilipyth ==
{{Optimal ET sequence|legend=0| 41, 89, 130g }}
The quintilipyth temperament ({{nowrap|12 &amp; 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
Badness (Sintel): 1.40


[[Comma list]]: 32805/32768, 9765625/9680832
===== Sesquartia =====
Subgroup: 2.3.5.7.11.13.17


{{Mapping|legend=1| 1 2 -1 -4 | 0 -5 40 82 }}
Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~625/588 = 99.625
Mapping: {{mapping| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }}


{{Optimal ET sequence|legend=1| 12, 253, 265 }}
Optimal tunings:
* WE: ~2 = 1199.8902{{c}}, ~72/65 = 175.4077{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4234{{c}}


[[Badness]]: 0.253966
{{Optimal ET sequence|legend=0| 41, 130, 171 }}


=== 11-limit ===
Badness (Sintel): 1.18
Subgroup: 2.3.5.7.11


Comma list: 1375/1372, 4375/4356, 32805/32768
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594


Mapping: {{mapping| 1 2 -1 -4 -7 | 0 -5 40 82 126 }}
Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }}


Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.616
Optimal tunings:
* WE: ~2 = 1199.9864{{c}}, ~21/19 = 175.4169{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4189{{c}}


{{Optimal ET sequence|legend=0| 12, 253, 265, 518c, 783cc }}
{{Optimal ET sequence|legend=0| 41, 130, 171 }}


Badness: 0.113044
Badness (Sintel): 1.24


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


Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647
Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594


Mapping: {{mapping| 1 2 -1 -4 -7 -9 | 0 -5 40 82 126 153 }}
Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 -6 | 0 4 -32 -15 10 39 69 -12 72 }}


Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.612
Optimal tunings:
* WE: ~2 = 1199.9606{{c}}, ~21/19 = 175.4067{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4123{{c}}


{{Optimal ET sequence|legend=0| 12f, 253, 518c, 771cc }}
{{Optimal ET sequence|legend=0| 41i, 130, 171 }}


Badness: 0.069127
Badness (Sintel): 1.36


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


Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619
Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625


Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 | 0 -5 40 82 126 153 -11 }}
Mapping: {{mapping| 1 1 7 5 2 -2 13 | 0 4 -32 -15 10 39 -61 }}


Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.612
Optimal tunings:
* WE: ~2 = 1199.9458{{c}}, ~72/65 = 175.3689{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3770{{c}}


{{Optimal ET sequence|legend=0| 12f, 253, 518c, 771cc }}
{{Optimal ET sequence|legend=0| 41g, 89, 130 }}


Badness: 0.045992
Badness (Sintel): 1.56


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


Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971
Comma list: 221/220, 243/242, 361/360, 364/363, 441/440, 456/455


Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 4 | 0 -5 40 82 126 153 -11 3 }}
Mapping: {{mapping| 1 1 7 5 2 -2 13 6 | 0 4 -32 -15 10 39 -61 -12 }}


Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.615
Optimal tunings:
* WE: ~2 = 1200.0114{{c}}, ~72/65 = 175.3783{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3765{{c}}


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


Badness: 0.038155
Badness (Sintel): 1.39


== Quintaschis ==
====== 23-limit ======
The quintaschis temperament ({{nowrap|12 &amp; 289}}) slices the fourth (4/3) into five semitones and tempers out 49009212/48828125 in the 7-limit.
Subgroup: 2.3.5.7.11.13.17.19.23


[[Subgroup]]: 2.3.5.7
Comma list: 221/220, 243/242, 276/275, 323/322, 361/360, 364/363, 441/440


[[Comma list]]: 32805/32768, 49009212/48828125
Mapping: {{mapping| 1 1 7 5 2 -2 13 6 13 | 0 4 -32 -15 10 39 -61 -12 -58 }}
 
{{Mapping|legend=1| 1 2 -1 -5 | 0 -5 40 94 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~200/189 = 99.664
Optimal tunings:
* WE: ~2 = 1200.0122{{c}}, ~72/65 = 175.3782{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3763{{c}}


{{Optimal ET sequence|legend=1| 12, …, 289, 301, 590, 891, 1192 }}
{{Optimal ET sequence|legend=0| 41g, 89, 130 }}


[[Badness]]: 0.132890
Badness (Sintel): 1.37


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


Comma list: 441/440, 32805/32768, 1953125/1951488
Comma list: 2401/2400, 9801/9800, 32805/32768


Mapping: {{mapping| 1 2 -1 -5 -8 | 0 -5 40 94 138 }}
Mapping: {{mapping| 2 2 14 10 23 | 0 4 -32 -15 -55 }}
: mapping generators: ~99/70, ~448/405


Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.653
Optimal tunings:
* WE: ~99/70 = 600.0429{{c}}, ~448/405 = 175.4474{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~448/405 = 175.4334{{c}}


{{Optimal ET sequence|legend=1| 12, …, 277d, 289 }}
{{Optimal ET sequence|legend=1| 82e, 130, 212, 342, 1156, 1498, 1840d, 5862bbccdddee }}


Badness: 0.111477
Badness (Sintel): 0.561


==== 13-limit ====
== Tsaharuk ==
Subgroup: 2.3.5.7.11.13
{{Main| Tsaharuk }}


Comma list: 364/363, 441/440, 32805/32768, 109512/109375
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 -5 -8 -11 | 0 -5 40 94 138 177 }}
[[Subgroup]]: 2.3.5.7


Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.658
[[Comma list]]: 32805/32768, 420175/419904


{{Optimal ET sequence|legend=1| 12f, …, 277dff, 289 }}
{{Mapping|legend=1| 1 1 7 0 | 0 5 -40 24 }}
: mapping generators: ~2, ~243/224


Badness: 0.074218
[[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 }}


==== 17-limit ====
{{Optimal ET sequence|legend=1| 17, 77, 94, 171 }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768
[[Badness]] (Sintel): 0.777


Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 | 0 -5 40 94 138 177 -11 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.656
Comma list: 385/384, 1331/1323, 19712/19683


{{Optimal ET sequence|legend=1| 12f, 277dff, 289 }}
Mapping: {{mapping| 1 1 7 0 1 | 0 5 -40 24 21 }}


Badness: 0.050571
Optimal tunings:  
* WE: ~2 = 1200.3103{{c}}, ~88/81 = 140.4011{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.3649{{c}}


==== 19-limit ====
{{Optimal ET sequence|legend=0| 17, 77, 94, 171e, 265e }}
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859
Badness (Sintel): 2.10


Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 4 | 0 -5 40 94 138 177 -11 3 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.659
Comma list: 352/351, 385/384, 729/728, 1331/1323


{{Optimal ET sequence|legend=1| 12f, 289 }}
Mapping: {{mapping| 1 1 7 0 1 3 | 0 5 -40 24 21 6 }}


Badness: 0.042120
Optimal tunings:  
* WE: ~2 = 1200.1840{{c}}, ~13/12 = 140.3840{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.3627{{c}}


=== Quintahelenic ===
{{Optimal ET sequence|legend=0| 17, 77, 94, 171e }}
Subgroup: 2.3.5.7.11


Comma list: 5632/5625, 8019/8000, 151263/151250
Badness (Sintel): 1.57


Mapping: {{mapping| 1 2 -1 -5 -9 | 0 -5 40 94 150 }}
== Quanharuk ==
Quanharuk tempers out 16875/16807, the [[mirkwai]] comma, and may be described as the {{nowrap| 41 & 183 }} temperament. The generator is a slightly flat major third of [[~]][[56/45]], five of which make the [[3/1|3rd]] [[harmonic]], so the [[ploidacot]] of this temperament is alpha-pentacot. [[224edo]] makes for a recommendable tuning.


Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 99.671
[[Subgroup]]: 2.3.5.7


{{Optimal ET sequence|legend=1| 12, …, 289e, 301, 915 }}
[[Comma list]]: 16875/16807, 32805/32768


Badness: 0.082225
{{Mapping|legend=1| 1 0 15 12 | 0 5 -40 -29 }}
: mapping generators: ~2, ~56/45


==== 13-limit ====
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13
* [[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 }}


Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000
{{Optimal ET sequence|legend=1| 41, 142, 183, 224 }}


Mapping: {{mapping| 1 2 -1 -5 -9 -11 | 0 -5 40 94 150 177 }}
[[Badness]] (Sintel): 1.82


Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 99.661
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Optimal ET sequence|legend=1| 12f, …, 289e, 301 }}
Comma list: 540/539, 1375/1372, 32805/32768


Badness: 0.055570
Mapping: {{mapping| 1 0 15 12 -7 | 0 5 -40 -29 33 }}


===== 17-limit =====
Optimal tunings:
Subgroup: 2.3.5.7.11.13.17
* WE: ~2 = 1199.9709{{c}}, ~56/45 = 380.3423{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3517{{c}}


Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750
{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}


Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 | 0 -5 40 94 150 177 -11 }}
Badness (Sintel): 1.04


Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.665
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Optimal ET sequence|legend=1| 12f, 289e, 301 }}
Comma list: 540/539, 729/728, 1375/1372, 4096/4095


Badness: 0.040412
Mapping: {{mapping| 1 0 15 12 -7 -15 | 0 5 -40 -29 33 59 }}


===== 19-limit =====
Optimal tunings:
Subgroup: 2.3.5.7.11.13.17.19
* WE: ~2 = 1199.9663{{c}}, ~56/45 = 380.3403{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3509{{c}}


Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700
{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}


Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 4 | 0 -5 40 94 150 177 -11 3 }}
Badness (Sintel): 0.884


Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.668
== 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.  


{{Optimal ET sequence|legend=1| 12f, 301 }}
[[Subgroup]]: 2.3.5.7


Badness: 0.036840
[[Comma list]]: 32805/32768, 9765625/9680832


==== Quintahelenoid ====
{{Mapping|legend=1| 1 2 -1 -4 | 0 -5 40 82 }}
Subgroup: 2.3.5.7.11.13
: mapping generators: ~2, ~625/588


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


Mapping: {{mapping| 1 2 -1 -5 -9 14 | 0 -5 40 94 150 -124 }}
{{Optimal ET sequence|legend=1| 12, …, 253, 265 }}


Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 99.672
[[Badness]] (Sintel): 6.43


{{Optimal ET sequence|legend=1| 12, 301, 614, 915 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.066108
Comma list: 1375/1372, 4375/4356, 32805/32768


===== 17-limit =====
Mapping: {{mapping| 1 2 -1 -4 -7 | 0 -5 40 82 126 }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157
Optimal tunings:  
* WE: ~2 = 1200.1503{{c}}, ~35/33 = 99.6287{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6176{{c}}


Mapping: {{mapping| 1 2 -1 -5 -9 14 5 | 0 -5 40 94 150 -124 -11 }}
{{Optimal ET sequence|legend=0| 12, …, 253, 265, 518c }}


Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.671
Badness (Sintel): 3.74


{{Optimal ET sequence|legend=1| 12, 301 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.047908
Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647


===== 19-limit =====
Mapping: {{mapping| 1 2 -1 -4 -7 -9 | 0 -5 40 82 126 153 }}
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137
Optimal tunings:  
* WE: ~2 = 1200.1774{{c}}, ~35/33 = 99.6267{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6134{{c}}


Mapping: {{mapping| 1 2 -1 -5 -9 14 5 4 | 0 -5 40 94 150 -124 -11 3 }}
{{Optimal ET sequence|legend=0| 12f, …, 241cdef, 253 }}


Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.672
Badness (Sintel): 2.86


{{Optimal ET sequence|legend=1| 12, 301 }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.039542
Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619


== Sextilifourths ==
Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 | 0 -5 40 82 126 153 -11 }}
The sextilifourths ({{nowrap|130 &amp; 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
Optimal tunings:  
* WE: ~2 = 1200.1745{{c}}, ~18/17 = 99.6265{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6131{{c}}


[[Comma list]]: 32805/32768, 235298/234375
{{Optimal ET sequence|legend=0| 12f, 241cdef, 253 }}


{{Mapping|legend=1| 1 2 -1 -1 | 0 -6 48 55 }}
Badness (Sintel): 2.34


: Mapping generators: ~2, ~21/2
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 83.053
Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971


{{Optimal ET sequence|legend=1| 29, 72cd, 101, 130, 289, 419 }}
Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 4 | 0 -5 40 82 126 153 -11 3 }}


[[Badness]]: 0.108794
Optimal tunings:  
* WE: ~2 = 1200.0713{{c}}, ~18/17 = 99.6208{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6152{{c}}


=== 11-limit ===
{{Optimal ET sequence|legend=0| 12f, 253, 265 }}
Subgroup: 2.3.5.7.11


Comma list: 441/440, 4000/3993, 235298/234375
Badness (Sintel): 2.32


Mapping: {{mapping| 1 2 -1 -1 0 | 0 -6 48 55 50 }}
== Quintaschis ==
Named by [[Xenllium]] in 2021, quintaschis slices the [[4/3|perfect fourth]] into five semitones and tempers out 49009212/48828125 in the [[7-limit]]. It may be described as the {{nowrap| 12 & 289 }} temperament, and its [[ploidacot]] is omega-pentacot.


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.049
[[Subgroup]]: 2.3.5.7


{{Optimal ET sequence|legend=1| 29, 72cde, 101e, 130, 289 }}
[[Comma list]]: 32805/32768, 49009212/48828125


Badness: 0.045457
{{Mapping|legend=1| 1 2 -1 -5 | 0 -5 40 94 }}


=== 13-limit ===
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13
* [[WE]]: ~2 = 1200.0536{{c}}, ~200/189 = 99.6684{{c}}
: [[error map]]: {{val| +0.054 -0.190 +0.370 -0.262 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~200/189 = 99.6645{{c}}
: error map: {{val| 0.000 -0.277 +0.266 -0.363 }}


Comma list: 364/363, 441/440, 676/675, 10985/10976
{{Optimal ET sequence|legend=1| 12, …, 289, 301, 590, 891, 1192 }}


Mapping: {{mapping| 1 2 -1 -1 0 1 | 0 -6 48 55 50 39 }}
[[Badness]] (Sintel): 3.36


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.049
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Optimal ET sequence|legend=1| 29, 72cdef, 101e, 130, 289 }}
Comma list: 441/440, 32805/32768, 1953125/1951488


Badness: 0.025276
Mapping: {{mapping| 1 2 -1 -5 -8 | 0 -5 40 94 138 }}


== Septiquarschis ==
Optimal tunings:
The septiquarschis temperament ({{nowrap|89 &amp; 94}}) splits septimal minor seventh ([[7/4]]) into four generators and tempers out 829440/823543 (mynaslender comma) and 67108864/66706983 (septiness comma).
* WE: ~2 = 1200.0988{{c}}, ~35/33 = 99.6613{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6540{{c}}


[[Subgroup]]: 2.3.5.7
{{Optimal ET sequence|legend=0| 12, …, 277d, 289 }}


[[Comma list]]: 32805/32768, 829440/823543
Badness (Sintel): 3.69


{{Mapping|legend=1| 1 3 -9 2 | 0 -7 -56 4 }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~147/128 = 242.614
Comma list: 364/363, 441/440, 32805/32768, 109512/109375


{{Optimal ET sequence|legend=1| 89, 94, 183, 460d, 643d, 1103dd }}
Mapping: {{mapping| 1 2 -1 -5 -8 -11 | 0 -5 40 94 138 177 }}


[[Badness]]: 0.187047
Optimal tunings:  
* WE: ~2 = 1200.0625{{c}}, ~35/33 = 99.6630{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6583{{c}}


=== 11-limit ===
{{Optimal ET sequence|legend=0| 12f, …, 277dff, 289 }}
Subgroup: 2.3.5.7.11


Comma list: 540/539, 15488/15435, 32805/32768
Badness (Sintel): 3.07


Mapping: {{mapping| 1 3 -9 2 -2 | 0 -7 -56 4 27 }}
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Optimal tuning (POTE): ~2 = 1\1, ~147/128 = 242.616
Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768


{{Optimal ET sequence|legend=1| 89, 94, 183, 460d, 643d, 826dd }}
Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 | 0 -5 40 94 138 177 -11 }}


Badness: 0.052002
Optimal tunings:  
* WE: ~2 = 1200.1286{{c}}, ~18/17 = 99.6668{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6568{{c}}


=== 13-limit ===
{{Optimal ET sequence|legend=0| 12f, 277dff, 289 }}
Subgroup: 2.3.5.7.11.13


Comma list: 540/539, 729/728, 1573/1568, 4096/4095
Badness (Sintel): 2.58


Mapping: {{mapping| 1 3 -9 2 -2 13 | 0 -7 -56 4 27 -46 }}
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Optimal tuning (POTE): ~2 = 1\1, ~147/128 = 242.610
Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859


{{Optimal ET sequence|legend=1| 89, 94, 183, 277, 460d }}
Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 4 | 0 -5 40 94 138 177 -11 3 }}


Badness: 0.035315
Optimal tunings:  
* WE: ~2 = 1200.0289{{c}}, ~18/17 = 99.6609{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6586{{c}}


== Tsaharuk ==
{{Optimal ET sequence|legend=0| 12f, 289 }}
{{Main| Tsaharuk }}


[[Subgroup]]: 2.3.5.7
Badness (Sintel): 2.56


[[Comma list]]: 32805/32768, 420175/419904
=== Quintahelenic ===
Subgroup: 2.3.5.7.11


{{Mapping|legend=1| 1 1 7 0 | 0 5 -40 24 }}
Comma list: 5632/5625, 8019/8000, 151263/151250


: Mapping generators: ~2, ~243/224
Mapping: {{mapping| 1 2 -1 -5 -9 | 0 -5 40 94 150 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/224 = 140.350
Optimal tunings:
* WE: ~2 = 1200.0195{{c}}, ~200/189 = 99.6723{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6709{{c}}


{{Optimal ET sequence|legend=1| 17, 60c, 77, 94, 171 }}
{{Optimal ET sequence|legend=0| 12, …, 289e, 301, 915 }}


[[Badness]]: 0.030697
Badness (Sintel): 2.72


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


Comma list: 385/384, 1331/1323, 19712/19683
Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000


Mapping: {{mapping| 1 1 7 0 1 | 0 5 -40 24 21 }}
Mapping: {{mapping| 1 2 -1 -5 -9 -11 | 0 -5 40 94 150 177 }}


Optimal tuning (POTE): ~2 = 1\1, ~88/81 = 140.365
Optimal tunings:
* WE: ~2 = 1200.0442{{c}}, ~200/189 = 99.6709{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6675{{c}}


{{Optimal ET sequence|legend=1| 17, 60ce, 77, 94, 171e, 265e, 436ee }}
{{Optimal ET sequence|legend=0| 12f, …, 289e, 301 }}


Badness: 0.063499
Badness (Sintel): 2.30


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


Comma list: 352/351, 385/384, 729/728, 1331/1323
Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750


Mapping: {{mapping| 1 1 7 0 1 3 | 0 5 -40 24 21 6 }}
Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 | 0 -5 40 94 150 177 -11 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.363
Optimal tunings:
* WE: ~2 = 1200.1227{{c}}, ~200/189 = 99.6753{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6658{{c}}


{{Optimal ET sequence|legend=1| 17, 60ce, 77, 94, 171e, 436ee }}
{{Optimal ET sequence|legend=1| 12f, 289e, 301 }}


Badness: 0.037886
Badness (Sintel): 2.06


== Quanharuk ==
===== 19-limit =====
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17.19


[[Comma list]]: 16875/16807, 32805/32768
Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700


{{Mapping|legend=1| 1 0 15 12 | 0 5 -40 -29 }}
Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 4 | 0 -5 40 94 150 177 -11 3 }}


: Mapping generators: ~2, ~56/45
Optimal tunings:  
* WE: ~2 = 1200.0230{{c}}, ~200/189 = 99.6694{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6676{{c}}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~56/45 = 380.355
{{Optimal ET sequence|legend=0| 12f, 301 }}


{{Optimal ET sequence|legend=1| 41, 142, 183, 224, 1303d, 1527cd, 1751cd, 1975cd }}
Badness (Sintel): 2.24


[[Badness]]: 0.071950
==== Quintahelenoid ====
Subgroup: 2.3.5.7.11.13


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


Comma list: 540/539, 1375/1372, 32805/32768
Mapping: {{mapping| 1 2 -1 -5 -9 14 | 0 -5 40 94 150 -124 }}


Mapping: {{mapping| 1 0 15 12 -7 | 0 5 -40 -29 33 }}
Optimal tunings:  
* WE: ~2 = 1199.9919{{c}}, ~200/189 = 99.6712{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6718{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 380.352
{{Optimal ET sequence|legend=0| 12, 301, 614, 915 }}


{{Optimal ET sequence|legend=1| 41, 142, 183, 224, 631d, 855d, 1079d }}
Badness (Sintel): 2.73


Badness: 0.031549
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


=== 13-limit ===
Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157
Subgroup: 2.3.5.7.11.13


Comma list: 540/539, 729/728, 1375/1372, 4096/4095
Mapping: {{mapping| 1 2 -1 -5 -9 14 5 | 0 -5 40 94 150 -124 -11 }}


Mapping: {{mapping| 1 0 15 12 -7 -15 | 0 5 -40 -29 33 59 }}
Optimal tunings:  
* WE: ~2 = 1200.0469{{c}}, ~18/17 = 99.6749{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6710{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 380.351
{{Optimal ET sequence|legend=0| 12, 301 }}


{{Optimal ET sequence|legend=1| 41, 142, 183, 224, 631d, 855d }}
Badness (Sintel): 2.44


Badness: 0.021392
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


== Quadrant ==
Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137
The ''quadrant'' temperament ({{nowrap|12 &amp; 224}}) has a period of quarter octave and tempers out the [[dimcomp comma]], 390625/388962. In this temperament, 25/21 is mapped into quarter octave.


[[Subgroup]]: 2.3.5.7
Mapping: {{mapping| 1 2 -1 -5 -9 14 5 4 | 0 -5 40 94 150 -124 -11 3 }}


[[Comma list]]: 32805/32768, 390625/388962
Optimal tunings:  
* WE: ~2 = 1199.9925{{c}}, ~18/17 = 99.6710{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6716{{c}}


{{Mapping|legend=1| 4 0 60 119 | 0 1 -8 -17 }}
{{Optimal ET sequence|legend=0| 12, 301 }}


: Mapping generators: ~25/21, ~3
Badness (Sintel): 2.41


[[Optimal tuning]] ([[POTE]]): ~25/21 = 1\4, ~3/2 = 701.8234
== 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.  


{{Optimal ET sequence|legend=1| 212, 224, 436, 660, 1096c }}
[[Subgroup]]: 2.3.5.7


[[Badness]]: 0.110242
[[Comma list]]: 32805/32768, 235298/234375


=== 11-limit ===
{{Mapping|legend=1| 1 2 -1 -1 | 0 -6 48 55 }}
Subgroup: 2.3.5.7.11
: mapping generators: ~2, ~21/20


Comma list: 1375/1372, 6250/6237, 32805/32768
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0987{{c}}, ~21/20 = 83.0599{{c}}
: [[error map]]: {{val| +0.099 -0.117 +0.462 -0.630 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0543{{c}}
: error map: {{val| 0.000 -0.281 +0.295 -0.837 }}
 
{{Optimal ET sequence|legend=1| 29, 72cd, 101, 130, 289, 419 }}
 
[[Badness]] (Sintel): 2.75
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 4000/3993, 235298/234375


Mapping: {{mapping| 4 0 60 119 185 | 0 1 -8 -17 -27 }}
Mapping: {{mapping| 1 2 -1 -1 0 | 0 -6 48 55 50 }}


Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 701.8176
Optimal tunings:  
* WE: ~2 = 1200.0424{{c}}, ~21/20 = 83.0520{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0497{{c}}


{{Optimal ET sequence|legend=1| 212, 224, 436, 660 }}
{{Optimal ET sequence|legend=0| 29, 72cde, 101e, 130, 289 }}


Badness: 0.045738
Badness (Sintel): 1.50


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


Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647
Comma list: 364/363, 441/440, 676/675, 10985/10976


Mapping: {{mapping| 4 0 60 119 185 224 | 0 1 -8 -17 -27 -33 }}
Mapping: {{mapping| 1 2 -1 -1 0 1 | 0 -6 48 55 50 39 }}


Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 701.8158
Optimal tunings:  
* WE: ~2 = 1200.1056{{c}}, ~21/20 = 83.0566{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0508{{c}}


{{Optimal ET sequence|legend=1| 212, 224, 436, 660 }}
{{Optimal ET sequence|legend=0| 29, 72cdef, 101e, 130, 289 }}


Badness: 0.027243
Badness (Sintel): 1.04


== Septant ==
== Septant ==
The ''septant'' temperament ({{nowrap|224 &amp; 301}}) has a period of 1/7 octave and tempers out the [[akjaysma]], {{monzo| 47 -7 -7 -7 }}.
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
[[Subgroup]]: 2.3.5.7
Line 2,245: Line 2,553:


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


: Mapping generators: ~8575/7776, ~3
[[Optimal tuning]]s:  
 
* [[WE]]: ~8575/7776 = 171.4303{{c}}, ~3/2 = 701.7091{{c}}
[[Optimal tuning]] ([[POTE]]): ~8575/7776 = 1\7, ~3/2 = 701.702
: [[error map]]: {{val| +0.012 -0.234 +0.096 +0.265 }}
* [[CWE]]: ~8575/7776 = 171.4286{{c}}, ~3/2 = 701.7022{{c}}
: error map: {{val| 0.000 -0.253 +0.069 +0.232 }}


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


[[Badness]]: 0.111142
[[Badness]] (Sintel): 2.81


=== 11-limit ===
=== 11-limit ===
Line 2,261: Line 2,572:
Mapping: {{mapping| 7 0 105 -56 -120 | 0 1 -8 7 13 }}
Mapping: {{mapping| 7 0 105 -56 -120 | 0 1 -8 7 13 }}


Optimal tuning (POTE): ~495/448 = 1\7, ~3/2 = 701.719
Optimal tunings:
* WE: ~495/448 = 171.4334{{c}}, ~3/2 = 701.7387{{c}}
* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7198{{c}}


{{Optimal ET sequence|legend=1| 77, 147, 224, 301, 525 }}
{{Optimal ET sequence|legend=0| 77, 147, 224, 301, 525 }}


Badness: 0.044122
Badness (Sintel): 1.46


=== 13-limit ===
=== 13-limit ===
Line 2,274: Line 2,587:
Mapping: {{mapping| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }}
Mapping: {{mapping| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }}


Optimal tuning (POTE): ~495/448 = 1\7, ~3/2 = 701.724
Optimal tunings:
* WE: ~495/448 = 171.4282{{c}}, ~3/2 = 701.7229{{c}}
* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7242{{c}}


{{Optimal ET sequence|legend=1| 77, 147, 224, 525 }}
{{Optimal ET sequence|legend=0| 77, 147, 224, 525, 1274f }}


Badness: 0.024706
Badness (Sintel): 1.02


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 2,288: Line 2,603:


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


: Mapping generators: ~42875/39366, ~3
[[Optimal tuning]]s:  
 
* [[WE]]: ~42875/39366 = 150.0048{{c}}, ~3/2 = 701.7356{{c}}
[[Optimal tuning]] ([[POTE]]): ~42875/39366 = 1\8, ~3/2 = 701.713
: [[error map]]: {{val| +0.039 -0.181 +0.071 +0.127 }}
* [[CWE]]: ~42875/39366 = 150.0000{{c}}, ~3/2 = 701.7134{{c}}
: error map: {{val| 0.000 -0.242 -0.021 +0.022 }}


{{Optimal ET sequence|legend=1| 24, 224, 472, 696, 1168 }}
{{Optimal ET sequence|legend=1| 24, …, 224, 472, 696, 1168 }}


[[Badness]]: 0.157186
[[Badness]] (Sintel): 3.98


=== 11-limit ===
=== 11-limit ===
Line 2,304: Line 2,622:
Mapping: {{mapping| 8 0 120 -117 15 | 0 1 -8 11 1 }}
Mapping: {{mapping| 8 0 120 -117 15 | 0 1 -8 11 1 }}


Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 701.713
Optimal tunings:
* WE: ~12/11 = 150.0010{{c}}, ~3/2 = 701.7177{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7131{{c}}


{{Optimal ET sequence|legend=1| 24, 224, 472, 696, 1168 }}
{{Optimal ET sequence|legend=0| 24, …, 224, 472, 696, 1168 }}


Badness: 0.044778
Badness (Sintel): 1.48


=== 13-limit ===
=== 13-limit ===
Line 2,317: Line 2,637:
Mapping: {{mapping| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }}
Mapping: {{mapping| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }}


Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 701.725
Optimal tunings:
* WE: ~12/11 = 149.9957{{c}}, ~3/2 = 701.7046{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7247{{c}}


{{Optimal ET sequence|legend=1| 24, 224, 472, 696 }}
{{Optimal ET sequence|legend=0| 24, 224, 472, 696 }}


Badness: 0.030425
Badness (Sintel): 1.26


== Nonant ==
== Nonant ==
The ''nonant'' temperament ({{nowrap|36 &amp; 135}}) has a period of 1/9 octave and tempers out the [[septimal ennealimma]], {{monzo| -11 -9 0 9 }}.
Named by [[Xenllium]] in 2023, nonant tempers out the [[septimal ennealimma]] ({{monzo| -11 -9 0 9 }}) and may be described as the {{nowrap| 36 & 171 }} temperament. It has a period of 1/9 octave, and its [[ploidacot]] is enneaploid monocot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 2,331: Line 2,653:


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


: Mapping generators: ~2592/2401, ~3
[[Optimal tuning]]s:  
 
* [[WE]]: ~2592/2401 = 133.3442{{c}}, ~3/2 = 701.8000{{c}}
[[Optimal tuning]] ([[CTE]]): ~2592/2401 = 1\9, ~3/2 = 701.7232
: [[error map]]: {{val| +0.098 -0.057 -0.027 -0.141 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.7384{{c}}
: error map: {{val| 0.000 -0.217 -0.221 -0.421 }}


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


[[Badness]]: 0.069896
[[Badness]] (Sintel): 1.77


=== 11-limit ===
=== 11-limit ===
Line 2,347: Line 2,672:
Mapping: {{mapping| 9 0 135 11 131 | 0 1 -8 1 -7 }}
Mapping: {{mapping| 9 0 135 11 131 | 0 1 -8 1 -7 }}


Optimal tuning (CTE): ~242/225 = 1\9, ~3/2 = 701.8398
Optimal tunings:
* WE: ~242/225 = 133.3308{{c}}, ~3/2 = 701.8205{{c}}
* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.8351{{c}}


Optimal ET sequence: {{Optimal ET sequence| 36, 99c, 135, 171, 477ce, 648cee }}
{{Optimal ET sequence|legend=0| 36, 135, 171 }}


Badness: 0.126910
Badness (Sintel): 4.20


=== 13-limit ===
=== 13-limit ===
Line 2,360: Line 2,687:
Mapping: {{mapping| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }}
Mapping: {{mapping| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }}


Optimal tuning (CTE): ~242/225 = 1\9, ~3/2 = 701.7998
Optimal tunings:
* WE: ~242/225 = 133.3180{{c}}, ~3/2 = 701.6956{{c}}
* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.7800{{c}}


Optimal ET sequence: {{Optimal ET sequence| 36, 99cf, 135, 171 }}
{{Optimal ET sequence|legend=0| 36, 99cf, 135, 171 }}


Badness: 0.076195
Badness (Sintel): 3.15


== Tridecafifths ==
== Septiquarschis ==
Tridecafifths divides the perfect 3/2 into 13 quartertones.  
Named by [[Xenllium]] in 2021, septiquarschis tempers out [[829440/823543]] (mynaslender comma) and [[67108864/66706983]] (septiness comma), and may be described as the {{nowrap| 89 & 94 }} temperament. It splits septimal minor seventh ([[7/4]]) into four generators. Note that in the data below, the generator is the [[octave complement]] so that seven of them minus five octaves make a [[3/2|perfect fifth]]; its [[ploidacot]] is thus epsilon-heptacot.  


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


[[Comma list]]: 32805/32768, {{monzo| -14 -1 -9 13 }}
[[Comma list]]: 32805/32768, 829440/823543


{{Mapping|legend=1| 1 1 7 6 | 0 13 -104 -71 }}
{{Mapping|legend=1| 1 -4 47 6 | 0 7 56 -4 }}
: mapping generators: ~2, ~256/147


: Mapping generators: ~2, ~1323/1280
[[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 tuning]] ([[CTE]]): ~2 = 1\1, ~1323/1280 = 53.9741
{{Optimal ET sequence|legend=1| 89, 94, 183, 460d, 643d }}


{{Optimal ET sequence|legend=1| 89, 200, 289 }}
[[Badness]] (Sintel): 4.73
 
[[Badness]]: 0.432580


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


Comma list: 441/440, 32805/32768, 55296000/55240493
Comma list: 540/539, 15488/15435, 32805/32768


Mapping: {{mapping| 1 1 7 6 4 | 0 13 -104 -71 -12 }}
Mapping: {{mapping| 1 -4 47 6 25 | 0 7 56 -4 -27 }}


Optimal tuning (CTE): ~2 = 1\1, ~33/32 = 53.9744
Optimal tunings:
* WE: ~2 = 1199.9430{{c}}, ~256/147 = 957.3390{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3849{{c}}


Optimal ET sequence: {{Optimal ET sequence| 89, 200, 289 }}
{{Optimal ET sequence|legend=0| 89, 94, 183, 460d }}


Badness: 0.127820
Badness (Sintel): 1.72


== Subgroup extensions ==
=== 13-limit ===
=== Photia (2.3.5.17) ===
Subgroup: 2.3.5.7.11.13
{{See also| No-elevens subgroup temperaments #Garibaldia }}


[[Subgroup]]: 2.3.5.17
Comma list: 540/539, 729/728, 1573/1568, 4096/4095


[[Comma list]]: 256/255, 1458/1445
Mapping: {{mapping| 1 -4 47 6 25 -33 | 0 7 56 -4 -27 46 }}


{{Mapping|legend=2| 1 0 15 -7 | 0 1 -8 7 }}
Optimal tunings:
* WE: ~2 = 1200.0058{{c}}, ~256/147 = 957.3946{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3900{{c}}


{{Mapping|legend=3| 1 0 15 0 0 0 -7 | 0 1 -8 0 0 0 7 }}
{{Optimal ET sequence|legend=0| 89, 94, 183, 277, 460d }}


: [[gencom]]: [2 3; 256/255 1458/1445]
Badness (Sintel): 1.46


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.491
== Tridecafifths ==
Named by [[Eliora]] in 2023, tridecafifths may be described as the {{nowrap| 89 & 200 }} temperament. It divides the [[3/2|perfect fifth]] into thirteen quartertones, so its [[ploidacot]] is 13-cot. [[289edo]] gives a highly recommendable tuning.  


{{Optimal ET sequence|legend=1| 12, 41, 53, 65 }}
[[Subgroup]]: 2.3.5.7


[[Tp tuning #T2 tuning|RMS error]]: 0.4842 cents
[[Comma list]]: 32805/32768, {{monzo| -14 -1 -9 13 }}


==== 2.3.5.17.19 ====
{{Mapping|legend=1| 1 1 7 6 | 0 13 -104 -71 }}
[[Subgroup]]: 2.3.5.17.19
: mapping generators: ~2, ~1323/1280


[[Comma list]]: 171/170, 256/255, 324/323
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1431{{c}}, ~1323/1280 = 53.9838{{c}}
: [[error map]]: {{val| +0.143 -0.023 +0.375 -0.816 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1323/1280 = 53.9764{{c}}
: error map: {{val| 0.000 -0.261 -0.221 -0.421 }}


{{Mapping|legend=2| 1 0 15 -7 9 | 0 1 -8 7 -3 }}
{{Optimal ET sequence|legend=1| 89, 200, 289 }}


{{Mapping|legend=3| 1 0 15 0 0 0 -7 9 | 0 1 -8 0 0 0 7 -3 }}
[[Badness]] (Sintel): 10.9


: [[gencom]]: [2 3; 171/170 256/255 324/323]
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.470
Comma list: 441/440, 32805/32768, 55296000/55240493


{{Optimal ET sequence|legend=1| 12, 41, 53, 65 }}
Mapping: {{mapping| 1 1 7 6 4 | 0 13 -104 -71 -12 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.5374 cents
Optimal tunings:  
* WE: ~2 = 1200.0311{{c}}, ~33/32 = 53.9766{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 53.9750{{c}}


=== Nestoria (2.3.5.19) ===
{{Optimal ET sequence|legend=0| 89, 200, 289 }}
: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''


The [[S-expression]]-based comma list of this temperament is {[[1216/1215|S16/S18]], [[361/360|S19]] (, ''[[513/512|S15/S20]]'')}.  
Badness (Sintel): 4.23


[[Subgroup]]: 2.3.5.19
== Subgroup extensions ==
=== Maqamschismic (2.3.5.11) ===
Proposed by [[Eufalesio]] in 2026, maqamschismic is equivalent to the no-7 [[cassandra]]. The 2.3.5.11.13 subgroup adds [[352/351]] to the comma list and tempers 11/9~39/32 together (and 16/13~27/22), providing a very simple framework for tuning [[maqam]]at (especially the Turkish version), as outlined by [[Ozan Yarman]]. 41edo and 53edo are simplest, but 94edo is more optimized. It is only slightly worse than the no-7 [[helenus]].


[[Comma list]]: 361/360, 513/512
Subgroup: 2.3.5.11


{{Mapping|legend=2| 1 0 15 9 | 0 1 -8 -3 }}
Comma list: 2200/2187, 4125/4096


: mapping generators: ~2, ~3
Subgroup-val mapping: {{mapping| 1 0 15 -33 | 0 1 -8 23 }}


{{Mapping|legend=3| 1 0 15 0 0 0 0 9 | 0 1 -8 0 0 0 0 -3 }}
Optimal tunings:
* WE: ~2 = 1200.5458{{c}} ~3/2 = 702.4021{{c}}
* CWE: 2 = 1200.0000{{c}}, ~3/2 = 702.0906{{c}}
 
{{Optimal ET sequence|legend=0| 12e, …, 41, 53, 94, 147e, 241ce, 335ce }}
 
Badness (Sintel): 1.34
 
==== 2.3.5.11.13 subgroup ====
Subgroup: 2.3.5.11.13
 
Comma list: 325/324, 352/351, 4125/4096
 
Subgroup-val mapping: {{mapping| 1 0 15 -33 -28 | 0 1 -8 23 20 }}
 
Optimal tunings:
* WE: ~2 = 1200.4565{{c}} ~3/2 = 702.3057{{c}}
* CWE: 2 = 1200.0000{{c}}, ~3/2 = 702.0485{{c}}
 
{{Optimal ET sequence|legend=0| 12e, …, 41, 53, 94, 147e }}


: [[gencom]]: [2 3; 361/360 513/512]
Badness (Sintel): 0.862


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.746
=== Tridecaschismic (2.3.5.13) ===
Proposed by [[Eufalesio]] in 2026, tridecaschismic adds the [[325/324|marveltwin comma]] to the comma list, or equivalently, the [[tridecapyth comma]]. It benefits from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is one of the lowest badness schismic extensions. It is also equivalent to the 2.3.5.13 [[restriction]] of 13-limit [[cassandra]].


{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171 }}
Subgroup: 2.3.5.13


[[Tp tuning #T2 tuning|RMS error]]: 0.1763 cents
Comma list: 325/324, 32805/32768


=== Taylor (2.3.5.13) ===
Subgroup-val mapping: {{mapping| 1 0 15 -28 | 0 1 -8 20 }}
This is a 2.3.5.13 subgroup restriction of 13-limit hemischis.


[[Subgroup]]: 2.3.5.13
Optimal tunings:
* WE: ~2 = 1200.3326{{c}} ~3/2 = 702.1092{{c}}
* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9189{{c}}


[[Comma list]]: 676/675, 32805/32768
{{Optimal ET sequence|legend=0| 12, …, 41, 53, 412cf, 465cf, …, 783ccff, 836ccfff }}


{{Mapping|legend=2| 1 0 15 14 | 0 2 -16 -13 }}
Badness (Sintel): 0.582


: Mapping generators: ~2, ~26/15
==== 2.3.5.13.19 subgroup ====
Subgroup: 2.3.5.13.19


{{Mapping|legend=3| 1 0 15 0 0 14 | 0 2 -16 0 0 -13 }}
Comma list: 325/324, 361/360, 513/512


: [[gencom]]: [2 26/15; 676/675 32805/32768]
Subgroup-val mapping: {{mapping| 1 0 15 -28 9 | 0 1 -8 20 -3 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~26/15 = 950.855
Optimal tunings:
* WE: ~2 = 1200.4236{{c}}, ~3/2 = 702.1510{{c}}
* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9064{{c}}


{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236 }}
{{Optimal ET sequence|legend=0| 12, …, 41, 53 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.1485 cents
Badness (Sintel): 0.354


=== Quintilischis (2.3.5.17) ===
=== Photia (2.3.5.17) ===
: ''For full 17- and 19-limit extensions, see [[#Quintilipyth]] or [[#Quintaschis]].''
{{See also| No-elevens subgroup temperaments #Garibaldia }}


[[Subgroup]]: 2.3.5.17
[[Subgroup]]: 2.3.5.17


[[Comma list]]: 32805/32768, 1419857/1417176
[[Comma list]]: 256/255, 1458/1445
 
{{Mapping|legend=2| 1 0 15 -7 | 0 1 -8 7 }}


{{Mapping|legend=2| 1 2 -1 5 | 0 -5 40 -11 }}
{{Mapping|legend=3| 1 0 15 0 0 0 -7 | 0 1 -8 0 0 0 7 }}
: mapping generators: ~2, ~3


: Mapping generators: ~2, ~18/17
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.5471{{c}}, ~3/2 = 701.2262{{c}}
: [[error map]]: {{val| -0.453 -1.182 +0.706 +3.628 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4976{{c}}
: error map: {{val| 0.000 -0.457 +1.705 +5.528 }}


{{Mapping|legend=3| 1 2 -1 0 0 0 5 | 0 -5 40 0 0 0 -11 }}
{{Optimal ET sequence|legend=1| 12, 41, 53, 65, 207g, 272gg }}


: [[gencom]]: [2 18/17; 32805/32768 1419857/1417176]
[[Badness]] (Sintel): 0.479


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~18/17 = 99.649
==== 2.3.5.17.19 subgroup ====
Subgroup: 2.3.5.17.19


{{Optimal ET sequence|legend=1| 12, 253, 265, 277, 289 }}
Comma list: 171/170, 256/255, 324/323
 
Subgroup-val mapping: {{mapping| 1 0 15 -7 9 | 0 1 -8 7 -3 }}
 
Gencom mapping: {{mapping| 1 0 15 0 0 0 -7 9 | 0 1 -8 0 0 0 7 -3 }}
 
Optimal tunings:
* WE: ~2 = 1199.7225{{c}}, ~3/2 = 701.3077{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.4754{{c}}
 
{{Optimal ET sequence|legend=0| 12, 41, 53, 65, 142g }}
 
Badness (Sintel): 0.332
 
=== Nestoria (2.3.5.19) ===
: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''
 
Nestoria is notable for having one of the lowest-badness subgroup extensions of schismic. Note that despite prime [[19/1|19]] being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is generally not flatter than the fifth in optimal schismic due to its optimization considering intervals like [[19/10]] and [[19/15]].
 
[[Subgroup]]: 2.3.5.19
 
[[Comma list]]: 361/360, 513/512
 
{{Mapping|legend=2| 1 0 15 9 | 0 1 -8 -3 }}
 
{{Mapping|legend=3| 1 0 15 0 0 0 0 9 | 0 1 -8 0 0 0 0 -3 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2250{{c}}, ~3/2 = 701.8776{{c}}
: [[error map]]: {{val| +0.225 +0.148 +0.240 -1.796 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7307{{c}}
: error map: {{val| 0.000 -0.224 -0.159 -2.705 }}
 
{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171, 460hh, 631hh }}
 
[[Badness]] (Sintel): 0.126
 
=== Taylor (2.3.5.13) ===
This is a 2.3.5.13 subgroup restriction of 13-limit hemischis.
 
[[Subgroup]]: 2.3.5.13
 
[[Comma list]]: 676/675, 32805/32768
 
{{Mapping|legend=2| 1 0 15 14 | 0 2 -16 -13 }}
 
{{Mapping|legend=3| 1 0 15 0 0 14 | 0 2 -16 0 0 -13 }}
: mapping generators: ~2, ~26/15
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1497{{c}}, ~26/15 = 950.9740{{c}}
: [[error map]]: {{val| +0.150 -0.007 +0.348 -1.094 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 950.8493{{c}}
: error map: {{val| 0.000 -0.256 +0.098 -1.568 }}
 
{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236, 525f, 761ff }}
 
[[Badness]] (Sintel): 0.334
 
==== Dakota (2.3.5.13.19) ====
Subgroup: 2.3.5.13.19
 
Comma list: 361/360, 513/512, 676/675
 
Subgroup-val mapping: {{mapping| 1 0 15 14 9 | 0 2 -16 -13 -6 }}
 
Optimal tunings:
* WE: ~2 = 1200.2611{{c}}, ~26/15 = 951.0703{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8532{{c}}
 
{{Optimal ET sequence|legend=0| 24, 29, 53, 130, 183, 236h, 289h }}
 
Badness (Sintel): 0.262
 
===== 2.3.5.13.19.37 subgroup =====
Subgroup: 2.3.5.13.19.37
 
Comma list: 361/360, 481/480, 513/512, 676/675
 
Subgroup-val mapping: {{mapping| 1 0 15 14 9 6 | 0 2 -16 -13 -6 -1 }}
 
Optimal tunings:
* WE: ~2 = 1200.2987{{c}}, ~26/15 = 951.1060{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8595{{c}}
 
{{Optimal ET sequence|legend=0| 24, 29, 53, 183, 236h, 289hl, 631fhhll }}
 
Badness (Sintel): 0.223
 
=== Quintilischis (2.3.5.17) ===
: ''For full 17- and 19-limit extensions, see [[#Quintilipyth]] or [[#Quintaschis]].''
 
[[Subgroup]]: 2.3.5.17
 
[[Comma list]]: 32805/32768, 1419857/1417176
 
{{Mapping|legend=2| 1 2 -1 5 | 0 -5 40 -11 }}
 
{{Mapping|legend=3| 1 2 -1 0 0 0 5 | 0 -5 40 0 0 0 -11 }}
: mapping generators: ~2, ~18/17
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1370{{c}}, ~18/17 = 99.6602{{c}}
: [[error map]]: {{val| +0.137 +0.018 -0.042 -0.533 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~18/17 = 99.6499{{c}}
: error map: {{val| 0.000 -0.205 -0.317 -1.104 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0719 cents
{{Optimal ET sequence|legend=1| 12, …, 253, 265, 277, 289, 566g, 855g }}


==== 2.3.5.17.19 ====
[[Badness]] (Sintel): 1.34
[[Subgroup]]: 2.3.5.17.19


[[Comma list]]: 4624/4617, 6144/6137, 6885/6859
==== 2.3.5.17.19 subgroup ====
Subgroup: 2.3.5.17.19


{{Mapping|legend=2| 1 2 -1 5 4 | 0 -5 40 -11 3 }}
Comma list: 4624/4617, 6144/6137, 6885/6859


{{Mapping|legend=3| 1 2 -1 0 0 0 5 4 | 0 -5 40 0 0 0 -11 3 }}
Subgroup-val mapping: {{mapping| 1 2 -1 5 4 | 0 -5 40 -11 3 }}


: [[gencom]]: [2 18/17; 4624/4617 6144/6137 6885/6859]
Gencom mapping: {{mapping| 1 2 -1 0 0 0 5 4 | 0 -5 40 0 0 0 -11 3 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~18/17 = 99.652
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=1| 12, 253, 265, 277, 289 }}
{{Optimal ET sequence|legend=0| 12, …, 253, 265, 277, 289 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.1636 cents
Badness (Sintel): 1.17


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Schismatic family| ]] <!-- main article -->
[[Category:Schismatic family| ]] <!-- main article -->
[[Category:Schismatic| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 18:05, 6 May 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

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 just major third and a Pythagorean diminished fourth.

Schismic, schismatic, a.k.a. helmholtz

Main article: Schismic

The 5-limit version of the temperament is a microtemperament, called schismic, schismatic, or helmholtz. The generator is a fifth, flattened by a fraction of a schisma, and 5/4 is represented by a diminished fourth. This defies the tradition of tertian harmony, as the just major triad on C is C–F♭–G, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as C–vE–G.

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

Subgroup: 2.3.5

Comma list: 32805/32768

Mapping: [⟨1 0 15], ⟨0 1 -8]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1200.0749 ¢, ~3/2 = 701.7797 ¢
error map: ⟨+0.075 -0.100 -0.027]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7308 ¢
error map: ⟨0.000 -0.224 -0.160]

Tuning ranges:

Optimal ET sequence: 12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bbc, 6452bbcc

Badness (Sintel): 0.0999

Overview to extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at. Garibaldi adds [25 -14 0 -1⟩, grackle adds [-44 26 0 1⟩, pontiac adds [-59 39 0 -1⟩, and schism adds [6 -2 0 -1⟩. Those all have a fifth as generator.

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

Temperaments involving larger splits include tsaharuk, quanharuk, quintilipyth, quintaschis, altinex, pogo, sextilifourths, septant, octant, nonant, septiquarschis, and tridecafifths. Those split the schismic structure into five to thirteen parts.

Temperaments discussed elsewhere include:

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

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

Garibaldi

Main article: Garibaldi

Garibaldi tempers out the garischisma, equating the septimal comma with both the syntonic comma and the Pythagorean comma. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its S-expression-based comma list is {S8/S9, S15}.

Subgroup: 2.3.5.7

Comma list: 225/224, 3125/3087

Mapping: [⟨1 0 15 25], ⟨0 1 -8 -14]]

Optimal tunings:

  • WE: ~2 = 1200.1233 ¢, ~3/2 = 702.1573 ¢
error map: ⟨+0.123 +0.326 -2.709 +2.328]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0774 ¢
error map: ⟨0.000 +0.122 -2.933 +2.090]

Minimax tuning:

[[1 0 0 0⟩, [5/3 1/15 0 -1/15⟩, [5/3 -8/15 0 8/15⟩, [5/3 -14/15 0 14/15⟩]
unchanged-interval (eigenmonzo) basis: 2.7/3
[[1 0 0 0⟩, [25/16 1/8 0 -1/16⟩, [5/2 -1 0 1/2⟩, [25/8 -7/4 0 7/8⟩]
unchanged-interval (eigenmonzo) basis: 2.9/7

Tuning ranges:

Optimal ET sequence: 12, 29, 41, 53, 94

Badness (Sintel): 0.548

Cassandra

Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup, even though it comes with a much higher complexity.

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 2200/2187

Mapping: [⟨1 0 15 25 -33], ⟨0 1 -8 -14 23]]

Optimal tunings:

  • WE: ~2 = 1200.3089 ¢, ~3/2 = 702.3377 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1562 ¢

Minimax tuning:

  • 11-odd-limit: ~3/2 = [9/16 1/8 0 -1/16⟩
unchanged-interval (eigenmonzo) basis: 2.9/7

Tuning ranges:

  • 11-odd-limit diamond monotone: ~3/2 = [701.887, 702.439] (31\53 to 24\41)
  • 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 702.915]

Optimal ET sequence: 12e, 41, 53, 94, 229c

Badness (Sintel): 0.906

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 275/273, 325/324, 385/384

Mapping: [⟨1 0 15 25 -33 -28], ⟨0 1 -8 -14 23 20]]

Optimal tunings:

  • WE: ~2 = 1200.1703 ¢, ~3/2 = 702.2122 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1135 ¢

Minimax tuning:

  • 13- and 15-odd-limit: ~3/2 = [19/34 0 0 -1/34 0 1/34⟩
unchanged-interval (eigenmonzo) basis: 2.13/7

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~3/2 = [701.887, 702.439] (31\53 to 24\41)
  • 13-odd-limit diamond tradeoff: ~3/2 = [701.711, 703.597]
  • 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 703.597]

Optimal ET sequence: 41, 53, 94, 429ccdeef, 523ccdeef

Badness (Sintel): 0.854

Cassie

Subgroup: 2.3.5.7.11.13.17

Comma list: 120/119, 154/153, 225/224, 273/272, 325/324

Mapping: [⟨1 0 15 25 -33 -28 -7], ⟨0 1 -8 -14 23 20 7]]

Optimal tunings:

  • WE: ~2 = 1199.8140 ¢, ~3/2 = 701.9833 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0909 ¢

Optimal ET sequence: 12e, 41, 53, 94g

Badness (Sintel): 1.19

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 120/119, 154/153, 171/170, 190/189, 225/224, 273/272

Mapping: [⟨1 0 15 25 -33 -28 -7 9], ⟨0 1 -8 -14 23 20 7 -3]]

Optimal tunings:

  • WE: ~2 = 1199.9556 ¢, ~3/2 = 702.0530 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0787 ¢

Optimal ET sequence: 12e, 41, 53

Badness (Sintel): 1.11

Cassandric

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 275/273, 325/324, 375/374, 385/384

Mapping: [⟨1 0 15 25 -33 -28 77], ⟨0 1 -8 -14 23 20 -46]]

Optimal tunings:

  • WE: ~2 = 1200.0046 ¢, ~3/2 = 702.2167 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0962 ¢

Optimal ET sequence: 41g, 53, 94

Badness (Sintel): 1.18

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 190/189, 209/208, 225/224, 275/273, 325/324, 375/374

Mapping: [⟨1 0 15 25 -33 -28 77 9], ⟨0 1 -8 -14 23 20 -46 -3]]

Optimal tunings:

  • WE: ~2 = 1200.2910 ¢, ~3/2 = 702.2681 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0967 ¢

Optimal ET sequence: 41g, 53, 94

Badness (Sintel): 1.07

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 190/189, 209/208, 225/224, 253/252, 275/273, 325/324, 375/374

Mapping: [⟨1 0 15 25 -33 -28 77 9 60], ⟨0 1 -8 -14 23 20 -46 -3 -35]]

Optimal tunings:

  • WE: ~2 = 1200.2970 ¢, ~3/2 = 702.2697 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0943 ¢

Optimal ET sequence: 41g, 53, 94

Badness (Sintel): 1.08

Cassander

Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 225/224, 275/273, 325/324, 385/384

Mapping: [⟨1 0 15 25 -33 -28 -72], ⟨0 1 -8 -14 23 20 48]]

Optimal tunings:

  • WE: ~2 = 1200.1986 ¢, ~3/2 = 702.2598 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1455 ¢

Optimal ET sequence: 41, 53g, 94

Badness (Sintel): 1.14

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 170/169, 190/189, 209/208, 225/224, 275/273, 325/324

Mapping: [⟨1 0 15 25 -33 -28 -72 9], ⟨0 1 -8 -14 23 20 48 -3]]

Optimal tunings:

  • WE: ~2 = 1200.3057 ¢, ~3/2 = 702.3138 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1373 ¢

Optimal ET sequence: 41, 53g, 94

Badness (Sintel): 1.07

Andromeda

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 15 25 32], ⟨0 1 -8 -14 -18]]

Optimal tunings:

  • WE: ~2 = 1200.1917 ¢, ~3/2 = 702.4836 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3599 ¢

Minimax tuning:

  • 11-odd-limit: ~3/2 = [3/5 1/10 0 0 -1/20⟩
unchanged-interval (eigenmonzo) basis: 2.11/9

Tuning ranges:

  • 11-odd-limit diamond monotone: ~3/2 = [700.000, 703.448] (7\12 to 17\29)
  • 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]

Optimal ET sequence: 12, 29, 41

Badness (Sintel): 0.779

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 196/195, 245/242

Mapping: [⟨1 0 15 25 32 37], ⟨0 1 -8 -14 -18 -21]]

Optimal tunings:

  • WE: ~2 = 1200.3031 ¢, ~3/2 = 702.7368 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.5420 ¢

Minimax tuning:

  • 13- and 15-odd-limit: ~3/2 = [14/23 2/23 0 0 0 -1/23⟩
unchanged-interval (eigenmonzo) basis: 2.13/9

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~3/2 = [702.439, 703.448] (24\41 to 17\29)
  • 13-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]
  • 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 704.377]

Optimal ET sequence: 12f, 29, 41

Badness (Sintel): 0.857

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 120/119, 189/187, 196/195

Mapping: [⟨1 0 15 25 32 37 -7], ⟨0 1 -8 -14 -18 -21 7]]

Optimal tunings:

  • WE: ~2 = 1199.1984 ¢, ~3/2 = 701.8424 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3384 ¢

Optimal ET sequence: 12f, 29, 41

Badness (Sintel): 1.19

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 120/119, 133/132, 189/187, 196/195

Mapping: [⟨1 0 15 25 32 37 -7 9], ⟨0 1 -8 -14 -18 -21 7 -3]]

Optimal tunings:

  • WE: ~2 = 1199.5242 ¢, ~3/2 = 702.0783 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3711 ¢

Optimal ET sequence: 12f, 29, 41

Badness (Sintel): 1.17

Schisicosiennic

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 154/153, 170/169, 196/195

Mapping: [⟨1 0 15 25 32 37 58], ⟨0 1 -8 -14 -18 -21 -34]]

Optimal tunings:

  • WE: ~2 = 1200.6122 ¢, ~3/2 = 703.0830 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6968 ¢

Optimal ET sequence: 12fg, 29g, 41, 70cd

Badness (Sintel): 1.11

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 133/132, 154/153, 170/169, 190/189

Mapping: [⟨1 0 15 25 32 37 58 9], ⟨0 1 -8 -14 -18 -21 -34 -3]]

Optimal tunings:

  • WE: ~2 = 1200.7981 ¢, ~3/2 = 703.2199 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.7221 ¢

Optimal ET sequence: 12fg, 29g, 41, 70cd

Badness (Sintel): 1.09

Schisicosiennoid

Subgroup: 2.3.5.7.11.13.17

Comma list: 85/84, 100/99, 105/104, 119/117, 221/220

Mapping: [⟨1 0 15 25 32 37 12], ⟨0 1 -8 -14 -18 -21 -5]]

Optimal tunings:

  • WE: ~2 = 1201.3146 ¢, ~3/2 = 703.4864 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6491 ¢

Optimal ET sequence: 12f, 29g, 41g

Badness (Sintel): 1.06

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 85/84, 100/99, 105/104, 119/117, 133/132, 153/152

Mapping: [⟨1 0 15 25 32 37 12 9], ⟨0 1 -8 -14 -18 -21 -5 -3]]

Optimal tunings:

  • WE: ~2 = 1201.3140 ¢, ~3/2 = 703.4860 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6578 ¢

Optimal ET sequence: 12f, 29g, 41g

Badness (Sintel): 1.02

Helenus

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 3125/3087

Mapping: [⟨1 0 15 25 51], ⟨0 1 -8 -14 -30]]

Optimal tunings:

  • WE: ~2 = 1199.7097 ¢, ~3/2 = 701.5554 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7370 ¢

Minimax tuning:

  • 11-odd-limit: ~3/2 = [19/32 1/16 0 0 -1/32⟩
unchanged-interval (eigenmonzo) basis: 2.11/9

Optimal ET sequence: 12, 41e, 53, 118d

Badness (Sintel): 1.18

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 176/175, 275/273, 847/845

Mapping: [⟨1 0 15 25 51 56], ⟨0 1 -8 -14 -30 -33]]

Optimal tunings:

  • WE: ~2 = 1199.7370 ¢, ~3/2 = 701.5937 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7570 ¢

Minimax tuning:

  • 13- and 15-odd-limit: ~3/2 = [19/32 1/16 0 0 -1/32⟩
unchanged-interval (eigenmonzo) basis: 2.11/9

Optimal ET sequence: 12f, …, 41ef, 53, 118d

Badness (Sintel): 1.09

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 99/98, 120/119, 176/175, 275/273, 442/441

Mapping: [⟨1 0 15 25 51 56 -7], ⟨0 1 -8 -14 -30 -33 7]]

Optimal tunings:

  • WE: ~2 = 1199.2895 ¢, ~3/2 = 701.2643 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6967 ¢

Optimal ET sequence: 12f, 53, 65d, 118dg

Badness (Sintel): 1.21

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 99/98, 120/119, 176/175, 190/189, 209/208, 247/245

Mapping: [⟨1 0 15 25 51 56 -7 9], ⟨0 1 -8 -14 -30 -33 7 -3]]

Optimal tunings:

  • WE: ~2 = 1199.5280 ¢, ~3/2 = 701.4290 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7149 ¢

Optimal ET sequence: 12f, 53, 65d

Badness (Sintel): 1.18

Karadeniz

See also: Turkish maqam music temperaments § Karadeniz temperament

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 7 11 2], ⟨0 2 -16 -28 5]]

mapping generators: ~2, ~11/9

Optimal tunings:

  • WE: ~2 = 1199.7351 ¢, ~11/9 = 350.9167 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.9995 ¢

Optimal ET sequence: 24d, 41, 65d, 106, 147

Badness (Sintel): 1.37

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 7 11 2 -8], ⟨0 2 -16 -28 5 40]]

Optimal tunings:

  • WE: ~2 = 1199.3042 ¢, ~11/9 = 350.7533 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.9686 ¢

Optimal ET sequence: 24d, 41, 65d, 106f

Badness (Sintel): 1.34

Hemigari

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 15 25 9], ⟨0 2 -16 -28 -7]]

mapping generators: ~2, ~110/63

Optimal tunings:

  • WE: ~2 = 1200.7303 ¢, ~110/63 = 951.6605 ¢
  • CWE: ~2 = 1200.0000 ¢, ~110/63 = 951.0604 ¢

Optimal ET sequence: 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: [⟨1 0 15 25 9 14], ⟨0 2 -16 -28 -7 -13]]

Optimal tunings:

  • WE: ~2 = 1200.8146 ¢, ~26/15 = 951.7273 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.0574 ¢

Optimal ET sequence: 24d, 29, 53, 82e, 135eef

Badness (Sintel): 1.13

Sanjaab

Subgroup: 2.3.5.7.11

Comma list: 225/224, 1331/1323, 3125/3087

Mapping: [⟨1 2 -1 -3 0], ⟨0 -3 24 42 25]]

mapping generators: ~2, ~11/10

Optimal tunings:

  • WE: ~2 = 1200.1997 ¢, ~11/10 = 166.0018 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9786 ¢

Optimal ET sequence: 29, 65d, 94

Badness (Sintel): 1.92

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 275/273, 847/845, 1331/1323

Mapping: [⟨1 2 -1 -3 0 -1], ⟨0 -3 24 42 25 34]]

Optimal tunings:

  • WE: ~2 = 1200.1224 ¢, ~11/10 = 165.9800 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9659 ¢

Optimal ET sequence: 29, 65d, 94

Badness (Sintel): 1.40

Pontiac

Main article: Pontiac

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

Subgroup: 2.3.5.7

Comma list: 4375/4374, 32805/32768

Mapping: [⟨1 0 15 -59], ⟨0 1 -8 39]]

Optimal tunings:

  • WE: ~2 = 1200.0989 ¢, ~3/2 = 701.8145 ¢
error map: ⟨+0.099 -0.042 -0.138 -0.038]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7579 ¢
error map: ⟨0.000 -0.197 -0.377 -0.268]

Minimax tuning:

[[1 0 0 0⟩, [74/47 0 -1/47 1/47⟩, [113/47 0 8/47 -8/47⟩, [113/47 0 -39/47 39/47⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5
[[1 0 0 0⟩, [3/2 1/5 -1/10 0⟩, [3 -8/5 4/5 0⟩, [-1/2 39/5 -39/10 0⟩]
unchanged-interval (eigenmonzo) basis: 2.9/5

Tuning ranges:

Optimal ET sequence: 53, 118, 171, 1592c, 1763c, …, 2960cd, 3131bcd

Badness (Sintel): 0.358

Helenoid

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

Subgroup: 2.3.5.7.11

Comma list: 385/384, 3388/3375, 4375/4374

Mapping: [⟨1 0 15 -59 51], ⟨0 1 -8 39 -30]]

Optimal tunings:

  • WE: ~2 = 1200.3277 ¢, ~3/2 = 701.9135 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7223 ¢

Minimax tuning:

  • 11-odd-limit: ~3/2 = [41/69 0 0 1/69 -1/69⟩
unchanged-interval (eigenmonzo) basis: 2.11/7

Optimal ET sequence: 53, 118, 289e, 407de

Badness (Sintel): 1.28

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384, 625/624, 729/728

Mapping: [⟨1 0 15 -59 51 56], ⟨0 1 -8 39 -30 -33]]

Optimal tunings:

  • WE: ~2 = 1200.1780 ¢, ~3/2 = 701.8491 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7446 ¢

Minimax tuning:

  • 13- and 15-odd-limit: ~3/2 = [43/72 0 0 1/72 -1/72⟩
unchanged-interval (eigenmonzo) basis: 2.13/7

Optimal ET sequence: 53, 118, 171e

Badness (Sintel): 1.39

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 385/384, 561/560, 625/624, 729/728

Mapping: [⟨1 0 15 -59 51 56 -91], ⟨0 1 -8 39 -30 -33 60]]

Optimal tunings:

  • WE: ~2 = 1200.1645 ¢, ~3/2 = 701.8385 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7425 ¢

Minimax tuning:

  • 17-odd-limit: ~3/2 = [18/31 0 0 0 0 -1/93 1/93⟩
unchanged-interval (eigenmonzo) basis: 2.17/13

Optimal ET sequence: 53, 118, 171e

Badness (Sintel): 1.47

Helena

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 385/384, 3146/3125

Mapping: [⟨1 0 15 -59 51 -28], ⟨0 1 -8 39 -30 20]]

Optimal tunings:

  • WE: ~2 = 1200.5227 ¢, ~3/2 = 702.0456 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7418 ¢

Optimal ET sequence: 53, 118f, 171ef

Badness (Sintel): 1.50

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 273/272, 325/324, 385/384, 3146/3125

Mapping: [⟨1 0 15 -59 51 -28 -91], ⟨0 1 -8 39 -30 20 60]]

Optimal tunings:

  • WE: ~2 = 1200.4988 ¢, ~3/2 = 702.0218 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7332 ¢

Optimal ET sequence: 53, 118f, 171ef

Badness (Sintel): 1.56

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 273/272, 286/285, 325/324, 385/384, 627/625

Mapping: [⟨1 0 15 -59 51 -28 -91 9], ⟨0 1 -8 39 -30 20 60 -3]]

Optimal tunings:

  • WE: ~2 = 1200.5185 ¢, ~3/2 = 702.0323 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7318 ¢

Optimal ET sequence: 53, 118f, 171ef

Badness (Sintel): 1.33

Ponta

Ponta tempers out 540/539 and may be described as 171 & 224. 224edo itself makes for an excellent tuning.

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4374, 32805/32768

Mapping: [⟨1 0 15 -59 135], ⟨0 1 -8 39 -83]]

Optimal tunings:

  • WE: ~2 = 1199.9814 ¢, ~3/2 = 701.7725 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7834 ¢

Minimax tuning:

  • 11-odd-limit: ~3/2 = [36/61 0 0 1/122 -1/122⟩
unchanged-interval (eigenmonzo) basis: 2.11/7

Optimal ET sequence: 53, 171, 224

Badness (Sintel): 1.61

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 2200/2197

Mapping: [⟨1 0 15 -59 135 56], ⟨0 1 -8 39 -83 -33]]

Optimal tunings:

  • WE: ~2 = 1199.9601 ¢, ~3/2 = 701.7610 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7845 ¢

Minimax tuning:

  • 13 and 15-odd-limit: ~3/2 = [36/61 0 0 1/122 -1/122⟩
unchanged-interval (eigenmonzo) basis: 2.11/7

Optimal ET sequence: 53, 171, 224

Badness (Sintel): 0.976

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 375/374, 540/539, 625/624, 729/728, 2200/2197

Mapping: [⟨1 0 15 -59 135 56 -91], ⟨0 1 -8 39 -83 -33 60]]

Optimal tunings:

  • WE: ~2 = 1199.8850 ¢, ~3/2 = 701.7101 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7775 ¢

Minimax tuning:

  • 17-odd-limit: ~3/2 = [83/143 0 0 0 -1/143 0 1/143⟩
unchanged-interval (eigenmonzo) basis: 2.17/11

Optimal ET sequence: 53, 171, 224, 395e, 619eg

Badness (Sintel): 1.16

Pontic

Pontic temperament tempers out 441/440 and may be described as 118 & 171. 289edo may be recommended as a tuning.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4375/4374, 32805/32768

Mapping: [⟨1 0 15 -59 -136], ⟨0 1 -8 39 88]]

Optimal tunings:

  • WE: ~2 = 1200.1259 ¢, ~3/2 = 701.7980 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7256 ¢

Minimax tuning:

  • 11-odd-limit: ~3/2 = [6/11 0 0 0 1/88⟩
unchanged-interval (eigenmonzo) basis: 2.11

Optimal ET sequence: 53e, 118, 289, 407d

Badness (Sintel): 1.64

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 625/624, 729/728, 3584/3575

Mapping: [⟨1 0 15 -59 -136 56], ⟨0 1 -8 39 88 -33]]

Optimal tunings:

  • WE: ~2 = 1199.9254 ¢, ~3/2 = 701.6945 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7378 ¢

Minimax tuning:

  • 13 and 15-odd-limit: ~3/2 = [71/121 0 0 0 1/121 -1/121⟩
unchanged-interval (eigenmonzo) basis: 2.13/11

Optimal ET sequence: 53e, 118, 171, 289f

Badness (Sintel): 1.87

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 441/440, 595/594, 625/624, 729/728, 2880/2873

Mapping: [⟨1 0 15 -59 -136 56 -91], ⟨0 1 -8 39 88 -33 60]]

Optimal tunings:

  • WE: ~2 = 1199.9454 ¢, ~3/2 = 701.7085 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7401 ¢

Minimax tuning:

  • 17-odd-limit: ~3/2 = [71/121 0 0 0 1/121 -1/121⟩
unchanged-interval (eigenmonzo) basis: 2.13/11

Optimal ET sequence: 53e, 118, 171, 289f

Badness (Sintel): 1.51

Pontoid

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 4375/4374, 32805/32768

Mapping: [⟨1 0 15 -59 -136 -215], ⟨0 1 -8 39 88 138]]

Optimal tunings:

  • WE: ~2 = 1200.0897 ¢, ~3/2 = 701.7874 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7356 ¢

Optimal ET sequence: 53ef, 118f, 171, 289

Badness (Sintel): 2.07

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 1156/1155, 32805/32768

Mapping: [⟨1 0 15 -59 -136 -215 -91], ⟨0 1 -8 39 88 138 60]]

Optimal tunings:

  • WE: ~2 = 1200.1045 ¢, ~3/2 = 701.7962 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7359 ¢

Optimal ET sequence: 53ef, 118f, 171, 289, 460e, 749defg

Badness (Sintel): 1.50

Bipont

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

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 32805/32768

Mapping: [⟨2 0 30 -118 -85], ⟨0 1 -8 39 29]]

mapping generators: ~99/70, ~3

Optimal tunings:

  • WE: ~99/70 = 600.0500 ¢, ~3/2 = 701.8153 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.7584 ¢

Optimal ET sequence: 106, 118, 224, 342, 1592c, 1934ce, 2276cde, 2618cde, 2960cde

Badness (Sintel): 0.484

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 729/728, 1575/1573, 4096/4095

Mapping: [⟨2 0 30 -118 -85 112], ⟨0 1 -8 39 29 -33]]

Optimal tunings:

  • WE: ~99/70 = 599.9939 ¢, ~3/2 = 701.7657 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.7728 ¢

Optimal ET sequence: 106, 118, 224, 566f, 790f

Badness (Sintel): 1.25

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 625/624, 729/728, 1089/1088, 1225/1224, 2880/2873

Mapping: [⟨2 0 30 -118 -85 112 -182], ⟨0 1 -8 39 29 -33 60]]

Optimal tunings:

  • WE: ~99/70 = 599.9839 ¢, ~3/2 = 701.7463 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.7649 ¢

Optimal ET sequence: 106g, 118, 224, 342, 566f

Badness (Sintel): 1.38

Counterbipont

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 32805/32768

Mapping: [⟨2 0 30 -118 -85 -243], ⟨0 1 -8 39 29 79]]

Optimal tunings:

  • WE: ~99/70 = 600.0405 ¢, ~3/2 = 701.8160 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.7697 ¢

Optimal ET sequence: 106f, 118f, 224, 342f, 566, 1356cf

Badness (Sintel): 1.06

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 32805/32768

Mapping: [⟨2 0 30 -118 -85 -243 -182], ⟨0 1 -8 39 29 79 60]]

Optimal tunings:

  • WE: ~99/70 = 600.0336 ¢, ~3/2 = 701.8031 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.7647 ¢

Optimal ET sequence: 106fg, 118f, 224, 342f, 566

Badness (Sintel): 1.29

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 1540/1539, 4875/4864

Mapping: [⟨2 0 30 -118 -85 -243 -182 -169], ⟨0 1 -8 39 29 79 60 56]]

Optimal tunings:

  • WE: ~99/70 = 600.0243 ¢, ~3/2 = 701.7891 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.7613 ¢

Optimal ET sequence: 106fgh, 118f, 224, 342f, 566h, 908fgh

Badness (Sintel): 1.35

Quadrapont

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4225/4224, 4375/4374, 32805/32768

Mapping: [⟨4 0 60 -236 -170 -131], ⟨0 1 -8 39 29 23]]

mapping generators: ~208/175, ~3

Optimal tunings:

  • WE: ~208/175 = 300.0229 ¢, ~3/2 = 701.8097 ¢
  • CWE: ~208/175 = 300.0000 ¢, ~3/2 = 701.7578 ¢

Optimal ET sequence: 224, 460, 684, 2276cde, 2960cde

Badness (Sintel): 0.869

Grackle

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

Subgroup: 2.3.5.7

Comma list: 126/125, 32805/32768

Mapping: [⟨1 0 15 44], ⟨0 1 -8 -26]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.7974 ¢, ~3/2 = 701.1210 ¢
error map: ⟨-0.203 -1.037 +3.300 -1.618]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2465 ¢
error map: ⟨0.000 -0.709 +3.715 -1.234]

Minimax tuning:

  • 7-odd-limit unchanged-interval (eigenmonzo) basis: 2.7/3
  • 9-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7

Optimal ET sequence: 12, …, 65, 77, 166c

Badness (Sintel): 1.78

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 32805/32768

Mapping: [⟨1 0 15 44 70], ⟨0 1 -8 -26 -42]]

Optimal tunings:

  • WE: ~2 = 1199.7077 ¢, ~3/2 = 701.0017 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.1804 ¢

Optimal ET sequence: 12, 65e, 77, 89, 166c

Badness (Sintel): 1.62

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 5445/5408

Mapping: [⟨1 0 15 44 70 75], ⟨0 1 -8 -26 -42 -45]]

Optimal tunings:

  • WE: ~2 = 1199.7782 ¢, ~3/2 = 701.0966 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2319 ¢

Optimal ET sequence: 12f, 65ef, 77, 166cf

Badness (Sintel): 1.56

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 176/175, 196/195, 256/255, 2904/2873

Mapping: [⟨1 0 15 44 70 75 -7], ⟨0 1 -8 -26 -42 -45 7]]

Optimal tunings:

  • WE: ~2 = 1199.5839 ¢, ~3/2 = 700.9632 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2137 ¢

Optimal ET sequence: 12f, 77, 89f, 166cf

Badness (Sintel): 1.52

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 126/125, 171/170, 176/175, 196/195, 209/208, 324/323

Mapping: [⟨1 0 15 44 70 75 -7 9], ⟨0 1 -8 -26 -42 -45 7 -3]]

Optimal tunings:

  • WE: ~2 = 1199.7146 ¢, ~3/2 = 701.0500 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2212 ¢

Optimal ET sequence: 12f, 77, 166cf

Badness (Sintel): 1.40

Grackloid

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 729/728, 1287/1280

Mapping: [⟨1 0 15 44 70 -47], ⟨0 1 -8 -26 -42 32]]

Optimal tunings:

  • WE: ~2 = 1200.0060 ¢, ~3/2 = 701.2202 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2167 ¢

Optimal ET sequence: 12, 77, 166c

Badness (Sintel): 2.00

Grack

Subgroup: 2.3.5.7.11

Comma list: 126/125, 245/242, 896/891

Mapping: [⟨1 0 15 44 51], ⟨0 1 -8 -26 -30]]

Optimal tunings:

  • WE: ~2 = 1199.8388 ¢, ~3/2 = 701.3071 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.4068 ¢

Optimal ET sequence: 12, 53d, 65, 77e

Badness (Sintel): 1.85

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 245/242, 832/825

Mapping: [⟨1 0 15 44 51 75], ⟨0 1 -8 -26 -30 -45]]

Optimal tunings:

  • WE: ~2 = 1199.7329 ¢, ~3/2 = 701.1918 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.3555 ¢

Optimal ET sequence: 12f, 53dff, 65f, 77e

Badness (Sintel): 1.84

Catahelenic

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 126/125, 245/242, 352/351

Mapping: [⟨1 0 15 44 51 56], ⟨0 1 -8 -26 -30 -33]]

Optimal tunings:

  • WE: ~2 = 1199.8928 ¢, ~3/2 = 701.4664 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.5327 ¢

Optimal ET sequence: 12f, …, 53d, 65

Badness (Sintel): 2.01

Quasipyth

Named by Xenllium in 2026, quasipyth tempers out [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: [⟨1 0 15 109], ⟨0 1 -8 -67]]

Optimal tunings:

  • WE: ~2 = 1200.2569 ¢, ~3/2 = 702.1149 ¢
error map: ⟨+0.2569 +0.4168 -1.4342 +0.2685]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9615 ¢
error map: ⟨0.0000 +0.0065 -2.0054 -0.2437]

Optimal ET sequence: 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: [⟨1 0 15 109 -117], ⟨0 1 -8 -67 76]]

Optimal tunings:

  • WE: ~2 = 1200.3283 ¢, ~3/2 = 702.1636 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9713 ¢

Optimal ET sequence: 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: [⟨1 0 15 109 -117 -28], ⟨0 1 -8 -67 76 20]]

Optimal tunings:

  • WE: ~2 = 1200.3229 ¢, ~3/2 = 702.1603 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9714 ¢

Optimal ET sequence: 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 118 & 130 temperament. The octave is split in halves, so the ploidacot of this temperament is diploid monocot. In schismic, -10 fifths make the interval class of 10/9. Bischismic then finds 7/4 by a stack of two 10/9's plus a semi-octave period, and in the 11-limit, it simply finds 11/8 by a stack of three 10/9's. 248edo and 378edo make for excellent tunings in both cases.

Subgroup: 2.3.5.7

Comma list: 3136/3125, 32805/32768

Mapping: [⟨2 0 30 69], ⟨0 1 -8 -20]]

mapping generators: ~567/400, ~3

Optimal tunings:

  • WE: ~567/400 = 600.0072 ¢, ~3/2 = 701.6005 ¢
error map: ⟨+0.014 -0.340 +0.982 -0.629]
  • CWE: ~567/400 = 600.0000 ¢, ~3/2 = 701.5915 ¢
error map: ⟨0.000 -0.364 +0.954 -0.656]

Minimax tuning:

  • 7-odd-limit unchanged-interval (eigenmonzo) basis: 2.7/3
  • 9-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7

Optimal ET sequence: 12, …, 106d, 118, 130, 248, 378

Badness (Sintel): 1.39

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3136/3125, 8019/8000

Mapping: [⟨2 0 30 69 102], ⟨0 1 -8 -20 -30]]

Optimal tunings:

  • WE: ~99/70 = 600.0165 ¢, ~3/2 = 701.6316 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.6110 ¢

Optimal ET sequence: 12, …, 106de, 118, 130, 248

Badness (Sintel): 0.931

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 729/728, 1001/1000, 3136/3125

Mapping: [⟨2 0 30 69 102 -75], ⟨0 1 -8 -20 -30 26]]

Optimal tunings:

  • WE: ~99/70 = 599.9610 ¢, ~3/2 = 701.5445 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.5908 ¢

Optimal ET sequence: 12, 118, 130, 248, 378

Badness (Sintel): 1.19

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 441/440, 561/560, 729/728, 3136/3125

Mapping: [⟨2 0 30 69 102 -75 5], ⟨0 1 -8 -20 -30 26 1]]

Optimal tunings:

  • WE: ~99/70 = 600.0331 ¢, ~3/2 = 701.6387 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.5994 ¢

Optimal ET sequence: 12, 118, 130, 248g

Badness (Sintel): 1.49

Bischis

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 3136/3125

Mapping: [⟨2 0 30 69 102 131], ⟨0 1 -8 -20 -30 -39]]

Optimal tunings:

  • WE: ~55/39 = 599.9766 ¢, ~3/2 = 701.5380 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~3/2 = 701.5670 ¢

Optimal ET sequence: 12f, 106deff, 118f, 130

Badness (Sintel): 1.21

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 221/220, 289/288, 351/350, 441/440, 3136/3125

Mapping: [⟨2 0 30 69 102 131 5], ⟨0 1 -8 -20 -30 -39 1]]

Optimal tunings:

  • WE: ~55/39 = 600.0997 ¢, ~3/2 = 701.7114 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~3/2 = 701.5899 ¢

Optimal ET sequence: 12f, 106deff, 118f, 130, 248fg

Badness (Sintel): 1.37

Kleischismic

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

Subgroup: 2.3.5.7

Comma list: 32805/32768, 1500625/1492992

Mapping: [⟨2 1 22 -15], ⟨0 2 -16 19]]

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

Optimal tunings:

  • WE: ~1225/864 = 600.1246 ¢, ~35/24 = 651.0550 ¢ (~36/35 = 50.9304 ¢)
error map: ⟨+0.249 +0.280 -0.453 -0.650]
  • CWE: ~1225/864 = 600.0000 ¢, ~35/24 = 650.9204 ¢ (~36/35 = 50.9204 ¢)
error map: ⟨0.000 -0.114 -1.041 -1.338]

Optimal ET sequence: 24, 94, 118, 212, 330, 542d, 872cdd, 1414ccddd

Badness (Sintel): 2.80

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 1 22 -15 8], ⟨0 2 -16 19 -1]]

Optimal tunings:

  • WE: ~99/70 = 600.1645 ¢, ~35/24 = 651.0963 ¢ (~36/35 = 50.9319 ¢)
  • CWE: ~99/70 = 600.0000 ¢, ~35/24 = 650.9184 ¢ (~36/35 = 50.9184 ¢)

Optimal ET sequence: 24, 94, 118, 212, 330e, 542dee, 872cddeee

Badness (Sintel): 1.21

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 1 22 -15 8 15], ⟨0 2 -16 19 -1 -7]]

Optimal tunings:

  • WE: ~99/70 = 600.0696 ¢, ~35/24 = 651.0136 ¢ (~36/35 = 50.9440 ¢)
  • CWE: ~99/70 = 600.0000 ¢, ~35/24 = 650.9378 ¢ (~36/35 = 50.9378 ¢)

Optimal ET sequence: 24, 94, 118, 212f

Badness (Sintel): 1.56

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 289/288, 352/351, 385/384, 561/560

Mapping: [⟨2 1 22 -15 8 15 6], ⟨0 2 -16 19 -1 -7 2]]

Optimal tunings:

  • WE: ~99/70 = 600.1134 ¢, ~35/24 = 651.0646 ¢ (~36/35 = 50.9512 ¢)
  • CWE: ~99/70 = 600.0000 ¢, ~35/24 = 650.9414 ¢ (~36/35 = 50.9414 ¢)

Optimal ET sequence: 24, 94, 118

Badness (Sintel): 1.30

Kleischis

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 1573/1568, 14641/14580

Mapping: [⟨2 1 22 -15 8 -36], ⟨0 2 -16 19 -1 40]]

Optimal tunings:

  • WE: ~99/70 = 600.1909 ¢, ~35/24 = 651.1578 ¢ (~36/35 = 50.9670 ¢)
  • CWE: ~99/70 = 600.0000 ¢, ~35/24 = 650.9541 ¢ (~36/35 = 50.9541 ¢)

Optimal ET sequence: 24f, 94, 118f, 212

Badness (Sintel): 1.55

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨2 1 22 -15 8 -36 6], ⟨0 2 -16 19 -1 40 2]]

Optimal tunings:

  • WE: ~99/70 = 600.2190 ¢, ~35/24 = 651.1578 ¢ (~36/35 = 50.9670 ¢)
  • CWE: ~99/70 = 600.0000 ¢, ~35/24 = 650.9518 ¢ (~36/35 = 50.9518 ¢)

Optimal ET sequence: 24f, 94, 118f, 212g

Badness (Sintel): 1.26

Salsa

Salsa tempers out 245/243, the sensamagic comma, and may be described as the 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.

Subgroup: 2.3.5.7

Comma list: 245/243, 32805/32768

Mapping: [⟨1 1 7 -1], ⟨0 2 -16 13]]

mapping generators: ~2, ~128/105

Optimal tunings:

  • WE: ~2 = 1200.7707 ¢, ~128/105 = 351.2748 ¢
error map: ⟨+0.771 +1.365 -1.315 -3.024]
  • CWE: ~2 = 1200.0000 ¢, ~128/105 = 351.0471 ¢
error map: ⟨0.000 +0.139 -3.068 -5.213]

Optimal ET sequence: 17, 24, 41, 106d, 147d, 188cd

Badness (Sintel): 2.03

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 7 -1 2], ⟨0 2 -16 13 5]]

Optimal tunings:

  • WE: ~2 = 1200.3891 ¢, ~11/9 = 351.1275 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.0141 ¢

Optimal ET sequence: 17, 24, 41, 106d

Badness (Sintel): 1.30

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 243/242, 245/242

Mapping: [⟨1 1 7 -1 2 4], ⟨0 2 -16 13 5 -1]]

Optimal tunings:

  • WE: ~2 = 1199.9362 ¢, ~11/9 = 351.0061 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.0247 ¢

Optimal ET sequence: 17, 24, 41

Badness (Sintel): 1.27

Hemischis

Hemischis tempers out 6144/6125, the porwell comma, as well as 19683/19600, the cataharry comma, and may be described as the 53 & 130 temperament. Its ploidacot is alpha-dicot.

The S-expression-based comma list for 13-limit hemischis is {S12/S14, S13/S15 = S26, S27, S64, (S65)}. Tempering out 169/168 (S13), 225/224 (S15) or 625/624 (S25) leads to 53edo while tempering out 24192/24167 (S12/S13), 10985/10976 (S13/S14), 43904/43875 (S14/S15) or 2401/2400 (S49) leads to 130edo and implies S12, S13, S14, and S15 are tempered together.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 19683/19600

Mapping: [⟨1 0 15 -17], ⟨0 2 -16 25]]

mapping generators: ~2, ~140/81

Optimal tunings:

  • WE: ~2 = 1199.8579 ¢, ~140/81 = 951.6847 ¢
error map: ⟨-0.142 -0.586 +0.600 +0.708]
  • CWE: ~2 = 1200.0000 ¢, ~140/81 = 951.7966 ¢
error map: ⟨0.000 -0.362 +0.941 +1.088]

Optimal ET sequence: 24, 53, 130, 183, 313

Badness (Sintel): 1.16

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 15 -17 51], ⟨0 2 -16 25 -60]]

Optimal tunings:

  • WE: ~2 = 1199.8482 ¢, ~140/81 = 950.6809 ¢
  • CWE: ~2 = 1200.0000 ¢, ~140/81 = 950.8020 ¢

Optimal ET sequence: 53, 130, 183, 313, 809cd

Badness (Sintel): 1.20

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 676/675, 4096/4095

Mapping: [⟨1 0 15 -17 51 14], ⟨0 2 -16 25 -60 -13]]

Optimal tunings:

  • WE: ~2 = 1199.9140 ¢, ~140/81 = 950.7324 ¢
  • CWE: ~2 = 1200.0000 ¢, ~140/81 = 950.8010 ¢

Optimal ET sequence: 53, 130, 183, 313

Badness (Sintel): 0.860

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 442/441, 561/560, 676/675, 4096/4095

Mapping: [⟨1 0 15 -17 51 14 -49], ⟨0 2 -16 25 -60 -13 67]]

Optimal tunings:

  • WE: ~2 = 1199.9740 ¢, ~26/15 = 950.7894 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.8100 ¢

Optimal ET sequence: 53, 130, 183, 496d

Badness (Sintel): 1.07

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 4096/4095

Mapping: [⟨1 0 15 -17 51 14 -49 9], ⟨0 2 -16 25 -60 -13 67 -6]]

Optimal tunings:

  • WE: ~2 = 1200.0464 ¢, ~26/15 = 950.8459 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.8091 ¢

Optimal ET sequence: 53, 130, 183, 313h

Badness (Sintel): 1.11

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 736/735, 4096/4095

Mapping: [⟨1 0 15 -17 51 14 -49 9 -24], ⟨0 2 -16 25 -60 -13 67 -6 36]]

Optimal tunings:

  • WE: ~2 = 1200.0215 ¢, ~26/15 = 950.8239 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.8069 ¢

Optimal ET sequence: 53, 130, 183, 313h

Badness (Sintel): 1.06

Music

Term

Term tempers out the landscape comma, mapping 63/50 to the 1/3-octave period. It can be described as 12 & 171, and is the unique temperament that equates a syntonic~Pythagorean comma with a stack of three marvel commas. 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

Comma list: 32805/32768, 250047/250000

Mapping: [⟨3 0 45 94], ⟨0 1 -8 -18]]

mapping generators: ~63/50, ~3

Optimal tunings:

  • WE: ~63/50 = 400.0257 ¢, ~3/2 = 701.7873 ¢
error map: ⟨+0.077 -0.091 -0.072 +0.031]
  • CWE: ~63/50 = 400.0000 ¢, ~3/2 = 701.7383 ¢
error map: ⟨0.000 -0.217 -0.220 -0.115]

Minimax tuning:

Optimal ET sequence: 12, …, 159, 171, 867, 1038, 1209, 1380, 1551, 1722

Badness (Sintel): 0.505

Terminal

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

Subgroup: 2.3.5.7.11

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

Mapping: [⟨3 0 45 94 134], ⟨0 1 -8 -18 -26]]

Optimal tunings:

  • WE: ~44/35 = 400.0464 ¢, ~3/2 = 701.9053 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~3/2 = 701.8178 ¢

Optimal ET sequence: 12, …, 159, 330

Badness (Sintel): 1.97

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨3 0 45 94 134 168], ⟨0 1 -8 -18 -26 -33]]

Optimal tunings:

  • WE: ~44/35 = 400.0449 ¢, ~3/2 = 701.8995 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~3/2 = 701.8156 ¢

Optimal ET sequence: 12f, …, 159, 330

Badness (Sintel): 1.53

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨3 0 45 94 134 168 -2], ⟨0 1 -8 -18 -26 -33 3]]

Optimal tunings:

  • WE: ~34/27 = 400.0195 ¢, ~3/2 = 701.8439 ¢
  • CWE: ~34/27 = 400.0000 ¢, ~3/2 = 701.8081 ¢

Optimal ET sequence: 12f, 159, 171, 330

Badness (Sintel): 1.38

Terminator

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

Subgroup: 2.3.5.7.11

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

Mapping: [⟨3 0 45 94 -137], ⟨0 1 -8 -18 31]]

Optimal tunings:

  • WE: ~63/50 = 399.9677 ¢, ~3/2 = 701.6278 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~3/2 = 701.6846 ¢

Optimal ET sequence: 12e, 171, 183, 354, 537, 891de

Badness (Sintel): 2.21

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨3 0 45 94 -137 -103], ⟨0 1 -8 -18 31 24]]

Optimal tunings:

  • WE: ~63/50 = 399.9731 ¢, ~3/2 = 701.6414 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~3/2 = 701.6881 ¢

Optimal ET sequence: 12e, 171, 183, 354, 891de

Badness (Sintel): 1.47

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨3 0 45 94 -137 -103 -2], ⟨0 1 -8 -18 31 24 3]]

Optimal tunings:

  • WE: ~63/50 = 399.9757 ¢, ~3/2 = 701.6458 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~3/2 = 701.6881 ¢

Optimal ET sequence: 12e, 171, 183, 354, 891de

Badness (Sintel): 1.04

Semiterm

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

Subgroup: 2.3.5.7.11

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

Mapping: [⟨6 0 90 188 287], ⟨0 1 -8 -18 -28]]

mapping generators: ~55/49, ~3

Optimal tunings:

  • WE: ~55/49 = 200.0134 ¢, ~3/2 = 701.7931 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 701.7426 ¢

Optimal ET sequence: 12, …, 330e, 342, 1380, 1722, 2064, 2406c, 5154bccdde

Badness (Sintel): 0.973

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨6 0 90 188 287 355], ⟨0 1 -8 -18 -28 -35]]

Optimal tunings:

  • WE: ~55/49 = 200.0083 ¢, ~3/2 = 701.7549 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 701.7238 ¢

Optimal ET sequence: 12f, 330eff, 342f, 696f *

* optimal patent val: 354

Badness (Sintel): 1.85

Hemiterm

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

Subgroup: 2.3.5.7.11

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

Mapping: [⟨3 0 45 94 8], ⟨0 2 -16 -36 1]]

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

Optimal tunings:

  • WE: ~63/50 = 400.0309 ¢, ~693/400 = 950.9458 ¢ (~12/11 = 150.8841 ¢)
  • CWE: ~63/50 = 400.0000 ¢, ~693/400 = 950.8707 ¢ (~12/11 = 150.8707 ¢)

Optimal ET sequence: 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce

Badness (Sintel): 0.684

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨3 0 45 94 8 42], ⟨0 2 -16 -36 1 -13]]

Optimal tunings:

  • WE: ~63/50 = 400.0541 ¢, ~26/15 = 951.0013 ¢ (~12/11 = 150.8932 ¢)
  • CWE: ~63/50 = 400.0000 ¢, ~26/15 = 950.8696 ¢ (~12/11 = 150.8696 ¢)

Optimal ET sequence: 24d, 159, 183, 342f

Badness (Sintel): 1.30

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨3 0 45 94 8 42 -2], ⟨0 2 -16 -36 1 -13 6]]

Optimal tunings:

  • WE: ~34/27 = 400.0373 ¢, ~26/15 = 950.9556 ¢ (~12/11 = 150.8809 ¢)
  • CWE: ~34/27 = 400.0000 ¢, ~26/15 = 950.8652 ¢ (~12/11 = 150.8652 ¢)

Optimal ET sequence: 24d, 159, 183, 342f, 525f

Badness (Sintel): 1.14

Altinex

Named by Aura in 2021, altinex is an alternative to hemiterm and may be described as 24 & 159. 159edo itself makes for a recommendable tuning.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 367653125/362797056

Mapping: [⟨3 0 45 -32], ⟨0 2 -16 17]]

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

Optimal tunings:

  • WE: ~1536/1225 = 400.1360 ¢, ~34300/19683 = 951.2867 ¢
error map: ⟨+0.408 +0.618 -0.781 -1.304]
  • CWE: ~1536/1225 = 400.0000 ¢, ~34300/19683 = 950.9638 ¢
error map: ⟨0.000 -0.027 -1.735 -2.441]

Optimal ET sequence: 24, 135, 159, 612ccdd

Badness (Sintel): 10.7

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨3 0 45 -32 8], ⟨0 2 -16 17 1]]

Optimal tunings:

  • WE: ~44/35 = 400.1156 ¢, ~121/70 = 951.2377 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~121/70 = 950.9634 ¢

Optimal ET sequence: 24, 135, 159

Badness (Sintel): 3.35

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨3 0 45 -32 8 42], ⟨0 2 -16 17 1 -13]]

Optimal tunings:

  • WE: ~44/35 = 400.1396 ¢, ~26/15 = 951.2799 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~26/15 = 950.9462 ¢

Optimal ET sequence: 24, 135f, 159

Badness (Sintel): 2.27

Squirrel

Squirrel tempers out 686/675, the sengic comma, and may be described as 29 & 36. It has a ~11/10 generator, three of which give the fourth (4/3), and thirteen of which give 7/4 with octave reduction. Its ploidacot is omega-tricot.

Subgroup: 2.3.5.7

Comma list: 686/675, 32805/32768

Mapping: [⟨1 2 -1 1], ⟨0 -3 24 13]]

Optimal tunings:

  • WE: ~2 = 1200.7408 ¢, ~160/147 = 166.2424 ¢
error map: ⟨+0.741 +0.799 +2.763 -6.934]
  • CWE: ~2 = 1200.0000 ¢, ~160/147 = 166.1597 ¢
error map: ⟨0.000 -0.434 +1.518 -8.750]

Optimal ET sequence: 29, 36, 65

Badness (Sintel): 4.42

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 -1 1 0], ⟨0 -3 24 13 25]]

Optimal tunings:

  • WE: ~2 = 1200.6379 ¢, ~11/10 = 166.1853 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.1157 ¢

Optimal ET sequence: 29, 36, 65

Badness (Sintel): 2.26

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 -1 1 0 3], ⟨0 -3 24 13 25 5]]

Optimal tunings:

  • WE: ~2 = 1201.1361 ¢, ~11/10 = 166.2110 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.0833 ¢

Optimal ET sequence: 29, 65f, 94df

Badness (Sintel): 1.81

Tertiaschis

Named by Xenllium in 2021, tertiaschis may be described as 94 & 159. It 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

Comma list: 32805/32768, 1071875/1062882

Mapping: [⟨1 2 -1 10], ⟨0 -3 24 -52]]

Optimal tunings:

  • WE: ~2 = 1200.3627 ¢, ~192/175 = 166.0691 ¢
error map: ⟨+0.363 +0.563 -1.019 -0.790]
  • CWE: ~2 = 1200.0000 ¢, ~192/175 = 166.0172 ¢
error map: ⟨0.000 -0.007 -1.901 -1.720]

Optimal ET sequence: 65, 94, 159, 253, 412cd

Badness (Sintel): 5.36

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 -1 10 0], ⟨0 -3 24 -52 25]]

Optimal tunings:

  • WE: ~2 = 1200.3379 ¢, ~11/10 = 166.0638 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.0167 ¢

Optimal ET sequence: 65, 94, 159, 253, 412cd, 665ccde

Badness (Sintel): 2.07

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 -1 10 0 12], ⟨0 -3 24 -52 25 -60]]

Optimal tunings:

  • WE: ~2 = 1200.3467 ¢, ~11/10 = 166.0635 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.0142 ¢

Optimal ET sequence: 65f, 94, 159, 253, 412cdf, 665ccdef

Badness (Sintel): 1.52

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 2 -1 10 0 12 -2], ⟨0 -3 24 -52 25 -60 44]]

Optimal tunings:

  • WE: ~2 = 1200.3019 ¢, ~11/10 = 166.0535 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.0114 ¢

Optimal ET sequence: 65f, 94, 159, 253

Badness (Sintel): 1.35

Countertertiaschis

Named by Flora Canou in 2021, Countertertiaschis may be described as 159 & 224. It has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with squirrel, but tempers out 244140625/243045684 for prime 7.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 244140625/243045684

Mapping: [⟨1 2 -1 -12], ⟨0 -3 24 107]]

Optimal tunings:

  • WE: ~2 = 1200.1265 ¢, ~625/567 = 166.0797 ¢
error map: ⟨+0.127 +0.059 -0.529 +0.178]
  • CWE: ~2 = 1200.0000 ¢, ~625/567 = 166.0632 ¢
error map: ⟨0.000 -0.145 -0.797 -0.065]

Optimal ET sequence: 65d, 159, 224, 383, 607

Badness (Sintel): 4.76

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 -1 -12 0], ⟨0 -3 24 107 25]]

Optimal tunings:

  • WE: ~2 = 1200.0804 ¢, ~11/10 = 166.0739 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.0634 ¢

Optimal ET sequence: 65d, 159, 224, 383, 607

Badness (Sintel): 1.62

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 -1 -12 0 -10], ⟨0 -3 24 107 25 99]]

Optimal tunings:

  • WE: ~2 = 1200.0805 ¢, ~11/10 = 166.0740 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.0635 ¢

Optimal ET sequence: 65d, 159, 224, 383, 607

Badness (Sintel): 1.01

Quadrant

Named by Xenllium in 2021, quadrant tempers out 390625/388962, the dimcomp comma, and maps 25/21 to the 1/4-octave period. It may be described as the 12 & 212 temperament; its ploidacot is tetraploid monocot. Just as term equates the syntonic~Pythagorean comma with three marvel commas, quadrant equates the syntonic~Pythagorean comma with four. A septimal comma is then found as a stack of five marvel commas.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 390625/388962

Mapping: [⟨4 0 60 119], ⟨0 1 -8 -17]]

mapping generators: ~25/21, ~3

Optimal tunings:

  • WE: ~2 = 300.0255 ¢, ~3/2 = 701.8831 ¢
error map: ⟨+0.102 +0.030 -0.664 +0.462]
  • CWE: ~2 = 300.0000 ¢, ~3/2 = 701.8180 ¢
error map: ⟨0.000 -0.137 -0.858 +0.268]

Optimal ET sequence: 12, …, 200, 212, 224, 436, 660

Badness (Sintel): 2.79

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨4 0 60 119 185], ⟨0 1 -8 -17 -27]]

Optimal tunings:

  • WE: ~25/21 = 300.0244 ¢, ~3/2 = 701.8759 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~3/2 = 701.8145 ¢

Optimal ET sequence: 12, …, 212, 224, 436, 660

Badness (Sintel): 1.51

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨4 0 60 119 185 224], ⟨0 1 -8 -17 -27 -33]]

Optimal tunings:

  • WE: ~25/21 = 300.0234 ¢, ~3/2 = 701.8707 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~3/2 = 701.8123 ¢

Optimal ET sequence: 12f, …, 212, 224, 436, 660

Badness (Sintel): 1.13

Sesquiquartififths

Sesquiquartififths tempers out 2401/2400, the breedsma, and may be described as the 41 & 171 temperament. It splits the fifth into four; its ploidacot is thus tetracot.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 32805/32768

Mapping: [⟨1 1 7 5], ⟨0 4 -32 -15]]

mapping generators: ~2, ~448/405

Optimal tunings:

  • WE: ~2 = 1200.0846 ¢, ~448/405 = 175.4460 ¢
error map: ⟨+0.085 -0.086 +0.007 -0.093]
  • CWE: ~2 = 1200.0000 ¢, ~448/405 = 175.4320 ¢
error map: ⟨0.000 -0.227 -0.137 -0.306]

Minimax tuning:

Optimal ET sequence: 41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd

Badness (Sintel): 0.285

Sesquart

Sesquart is the main 11- and 13-limit extension of sesquiquartififths of practical interest, as it identifies the neutral third with 11/9, which is realized in 41edo, 89edo, 130edo, and 171edo also makes for a possible tuning.

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 7 5 2], ⟨0 4 -32 -15 10]]

Optimal tunings:

  • WE: ~2 = 1199.8171 ¢, ~256/231 = 175.3793 ¢
  • CWE: ~2 = 1200.0000 ¢, ~256/231 = 175.4081 ¢

Optimal ET sequence: 41, 89, 130, 301e, 431e

Badness (Sintel): 0.969

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 7 5 2 -2], ⟨0 4 -32 -15 10 39]]

Optimal tunings:

  • WE: ~2 = 1199.8352 ¢, ~72/65 = 175.3852 ¢
  • CWE: ~2 = 1200.0000 ¢, ~72/65 = 175.4095 ¢

Optimal ET sequence: 41, 89, 130, 301e, 431e

Badness (Sintel): 0.925

Heartia

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 256/255, 273/272, 364/363, 441/440

Mapping: [⟨1 1 7 5 2 -2 0], ⟨0 4 -32 -15 10 39 28]]

Optimal tunings:

  • WE: ~2 = 1199.6422 ¢, ~72/65 = 175.3338 ¢
  • CWE: ~2 = 1200.0000 ¢, ~72/65 = 175.3857 ¢

Optimal ET sequence: 41, 89, 130g

Badness (Sintel): 1.45

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 1 7 5 2 -2 0 6], ⟨0 4 -32 -15 10 39 28 -12]]

Optimal tunings:

  • WE: ~2 = 1199.7499 ¢, ~21/19 = 175.3432 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/19 = 175.3797 ¢

Optimal ET sequence: 41, 89, 130g

Badness (Sintel): 1.40

Sesquartia

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 1 7 5 2 -2 -6], ⟨0 4 -32 -15 10 39 69]]

Optimal tunings:

  • WE: ~2 = 1199.8902 ¢, ~72/65 = 175.4077 ¢
  • CWE: ~2 = 1200.0000 ¢, ~72/65 = 175.4234 ¢

Optimal ET sequence: 41, 130, 171

Badness (Sintel): 1.18

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 1 7 5 2 -2 -6 6], ⟨0 4 -32 -15 10 39 69 -12]]

Optimal tunings:

  • WE: ~2 = 1199.9864 ¢, ~21/19 = 175.4169 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/19 = 175.4189 ¢

Optimal ET sequence: 41, 130, 171

Badness (Sintel): 1.24

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

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

Mapping: [⟨1 1 7 5 2 -2 -6 6 -6], ⟨0 4 -32 -15 10 39 69 -12 72]]

Optimal tunings:

  • WE: ~2 = 1199.9606 ¢, ~21/19 = 175.4067 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/19 = 175.4123 ¢

Optimal ET sequence: 41i, 130, 171

Badness (Sintel): 1.36

Hearty

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 1 7 5 2 -2 13], ⟨0 4 -32 -15 10 39 -61]]

Optimal tunings:

  • WE: ~2 = 1199.9458 ¢, ~72/65 = 175.3689 ¢
  • CWE: ~2 = 1200.0000 ¢, ~72/65 = 175.3770 ¢

Optimal ET sequence: 41g, 89, 130

Badness (Sintel): 1.56

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 1 7 5 2 -2 13 6], ⟨0 4 -32 -15 10 39 -61 -12]]

Optimal tunings:

  • WE: ~2 = 1200.0114 ¢, ~72/65 = 175.3783 ¢
  • CWE: ~2 = 1200.0000 ¢, ~72/65 = 175.3765 ¢

Optimal ET sequence: 41g, 89, 130

Badness (Sintel): 1.39

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

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

Mapping: [⟨1 1 7 5 2 -2 13 6 13], ⟨0 4 -32 -15 10 39 -61 -12 -58]]

Optimal tunings:

  • WE: ~2 = 1200.0122 ¢, ~72/65 = 175.3782 ¢
  • CWE: ~2 = 1200.0000 ¢, ~72/65 = 175.3763 ¢

Optimal ET sequence: 41g, 89, 130

Badness (Sintel): 1.37

Bisesqui

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 2 14 10 23], ⟨0 4 -32 -15 -55]]

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

Optimal tunings:

  • WE: ~99/70 = 600.0429 ¢, ~448/405 = 175.4474 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~448/405 = 175.4334 ¢

Optimal ET sequence: 82e, 130, 212, 342, 1156, 1498, 1840d, 5862bbccdddee

Badness (Sintel): 0.561

Tsaharuk

Main article: Tsaharuk

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

Subgroup: 2.3.5.7

Comma list: 32805/32768, 420175/419904

Mapping: [⟨1 1 7 0], ⟨0 5 -40 24]]

mapping generators: ~2, ~243/224

Optimal tunings:

  • WE: ~2 = 1200.1039 ¢, ~243/224 = 140.3620 ¢
error map: ⟨+0.104 -0.041 -0.067 -0.137]
  • CWE: ~2 = 1200.0000 ¢, ~243/224 = 140.3496 ¢
error map: ⟨0.000 -0.207 -0.296 -0.436]

Optimal ET sequence: 17, 77, 94, 171

Badness (Sintel): 0.777

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 7 0 1], ⟨0 5 -40 24 21]]

Optimal tunings:

  • WE: ~2 = 1200.3103 ¢, ~88/81 = 140.4011 ¢
  • CWE: ~2 = 1200.0000 ¢, ~88/81 = 140.3649 ¢

Optimal ET sequence: 17, 77, 94, 171e, 265e

Badness (Sintel): 2.10

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 7 0 1 3], ⟨0 5 -40 24 21 6]]

Optimal tunings:

  • WE: ~2 = 1200.1840 ¢, ~13/12 = 140.3840 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.3627 ¢

Optimal ET sequence: 17, 77, 94, 171e

Badness (Sintel): 1.57

Quanharuk

Quanharuk tempers out 16875/16807, the mirkwai comma, and may be described as the 41 & 183 temperament. The generator is a slightly flat major third of ~56/45, five of which make the 3rd harmonic, so the ploidacot of this temperament is alpha-pentacot. 224edo makes for a recommendable tuning.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 32805/32768

Mapping: [⟨1 0 15 12], ⟨0 5 -40 -29]]

mapping generators: ~2, ~56/45

Optimal tunings:

  • WE: ~2 = 1200.0032 ¢, ~56/45 = 380.3557 ¢
error map: ⟨+0.003 -0.177 -0.493 +0.898]
  • CWE: ~2 = 1200.0000 ¢, ~56/45 = 380.3546 ¢
error map: ⟨0.000 -0.182 -0.498 +0.890]

Optimal ET sequence: 41, 142, 183, 224

Badness (Sintel): 1.82

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 15 12 -7], ⟨0 5 -40 -29 33]]

Optimal tunings:

  • WE: ~2 = 1199.9709 ¢, ~56/45 = 380.3423 ¢
  • CWE: ~2 = 1200.0000 ¢, ~56/45 = 380.3517 ¢

Optimal ET sequence: 41, 142, 183, 224, 631d, 855d

Badness (Sintel): 1.04

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 15 12 -7 -15], ⟨0 5 -40 -29 33 59]]

Optimal tunings:

  • WE: ~2 = 1199.9663 ¢, ~56/45 = 380.3403 ¢
  • CWE: ~2 = 1200.0000 ¢, ~56/45 = 380.3509 ¢

Optimal ET sequence: 41, 142, 183, 224, 631d, 855d

Badness (Sintel): 0.884

Quintilipyth

Named by Xenllium in 2021, quintilipyth (formerly quintilischis) slices the perfect fourth into five semitones and tempers out the compass comma (9765625/9680832) in the 7-limit. It may be described as the 12 & 253 temperament, and its ploidacot is omega-pentacot.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 9765625/9680832

Mapping: [⟨1 2 -1 -4], ⟨0 -5 40 82]]

mapping generators: ~2, ~625/588

Optimal tunings:

  • WE: ~2 = 1200.1138 ¢, ~625/588 = 99.6347 ¢
error map: ⟨+0.114 +0.099 -1.041 +0.761]
  • CWE: ~2 = 1200.0000 ¢, ~625/588 = 99.6265 ¢
error map: ⟨0.000 -0.087 -1.255 +0.544]

Optimal ET sequence: 12, …, 253, 265

Badness (Sintel): 6.43

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 -1 -4 -7], ⟨0 -5 40 82 126]]

Optimal tunings:

  • WE: ~2 = 1200.1503 ¢, ~35/33 = 99.6287 ¢
  • CWE: ~2 = 1200.0000 ¢, ~35/33 = 99.6176 ¢

Optimal ET sequence: 12, …, 253, 265, 518c

Badness (Sintel): 3.74

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 -1 -4 -7 -9], ⟨0 -5 40 82 126 153]]

Optimal tunings:

  • WE: ~2 = 1200.1774 ¢, ~35/33 = 99.6267 ¢
  • CWE: ~2 = 1200.0000 ¢, ~35/33 = 99.6134 ¢

Optimal ET sequence: 12f, …, 241cdef, 253

Badness (Sintel): 2.86

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 2 -1 -4 -7 -9 5], ⟨0 -5 40 82 126 153 -11]]

Optimal tunings:

  • WE: ~2 = 1200.1745 ¢, ~18/17 = 99.6265 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6131 ¢

Optimal ET sequence: 12f, 241cdef, 253

Badness (Sintel): 2.34

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 2 -1 -4 -7 -9 5 4], ⟨0 -5 40 82 126 153 -11 3]]

Optimal tunings:

  • WE: ~2 = 1200.0713 ¢, ~18/17 = 99.6208 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6152 ¢

Optimal ET sequence: 12f, 253, 265

Badness (Sintel): 2.32

Quintaschis

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

Subgroup: 2.3.5.7

Comma list: 32805/32768, 49009212/48828125

Mapping: [⟨1 2 -1 -5], ⟨0 -5 40 94]]

Optimal tunings:

  • WE: ~2 = 1200.0536 ¢, ~200/189 = 99.6684 ¢
error map: ⟨+0.054 -0.190 +0.370 -0.262]
  • CWE: ~2 = 1200.0000 ¢, ~200/189 = 99.6645 ¢
error map: ⟨0.000 -0.277 +0.266 -0.363]

Optimal ET sequence: 12, …, 289, 301, 590, 891, 1192

Badness (Sintel): 3.36

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 -1 -5 -8], ⟨0 -5 40 94 138]]

Optimal tunings:

  • WE: ~2 = 1200.0988 ¢, ~35/33 = 99.6613 ¢
  • CWE: ~2 = 1200.0000 ¢, ~35/33 = 99.6540 ¢

Optimal ET sequence: 12, …, 277d, 289

Badness (Sintel): 3.69

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 -1 -5 -8 -11], ⟨0 -5 40 94 138 177]]

Optimal tunings:

  • WE: ~2 = 1200.0625 ¢, ~35/33 = 99.6630 ¢
  • CWE: ~2 = 1200.0000 ¢, ~35/33 = 99.6583 ¢

Optimal ET sequence: 12f, …, 277dff, 289

Badness (Sintel): 3.07

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 2 -1 -5 -8 -11 5], ⟨0 -5 40 94 138 177 -11]]

Optimal tunings:

  • WE: ~2 = 1200.1286 ¢, ~18/17 = 99.6668 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6568 ¢

Optimal ET sequence: 12f, 277dff, 289

Badness (Sintel): 2.58

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 2 -1 -5 -8 -11 5 4], ⟨0 -5 40 94 138 177 -11 3]]

Optimal tunings:

  • WE: ~2 = 1200.0289 ¢, ~18/17 = 99.6609 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6586 ¢

Optimal ET sequence: 12f, 289

Badness (Sintel): 2.56

Quintahelenic

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 -1 -5 -9], ⟨0 -5 40 94 150]]

Optimal tunings:

  • WE: ~2 = 1200.0195 ¢, ~200/189 = 99.6723 ¢
  • CWE: ~2 = 1200.0000 ¢, ~200/189 = 99.6709 ¢

Optimal ET sequence: 12, …, 289e, 301, 915

Badness (Sintel): 2.72

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 -1 -5 -9 -11], ⟨0 -5 40 94 150 177]]

Optimal tunings:

  • WE: ~2 = 1200.0442 ¢, ~200/189 = 99.6709 ¢
  • CWE: ~2 = 1200.0000 ¢, ~200/189 = 99.6675 ¢

Optimal ET sequence: 12f, …, 289e, 301

Badness (Sintel): 2.30

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 2 -1 -5 -9 -11 5], ⟨0 -5 40 94 150 177 -11]]

Optimal tunings:

  • WE: ~2 = 1200.1227 ¢, ~200/189 = 99.6753 ¢
  • CWE: ~2 = 1200.0000 ¢, ~200/189 = 99.6658 ¢

Optimal ET sequence: 12f, 289e, 301

Badness (Sintel): 2.06

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 2 -1 -5 -9 -11 5 4], ⟨0 -5 40 94 150 177 -11 3]]

Optimal tunings:

  • WE: ~2 = 1200.0230 ¢, ~200/189 = 99.6694 ¢
  • CWE: ~2 = 1200.0000 ¢, ~200/189 = 99.6676 ¢

Optimal ET sequence: 12f, 301

Badness (Sintel): 2.24

Quintahelenoid

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 -1 -5 -9 14], ⟨0 -5 40 94 150 -124]]

Optimal tunings:

  • WE: ~2 = 1199.9919 ¢, ~200/189 = 99.6712 ¢
  • CWE: ~2 = 1200.0000 ¢, ~200/189 = 99.6718 ¢

Optimal ET sequence: 12, 301, 614, 915

Badness (Sintel): 2.73

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 2 -1 -5 -9 14 5], ⟨0 -5 40 94 150 -124 -11]]

Optimal tunings:

  • WE: ~2 = 1200.0469 ¢, ~18/17 = 99.6749 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6710 ¢

Optimal ET sequence: 12, 301

Badness (Sintel): 2.44

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 2 -1 -5 -9 14 5 4], ⟨0 -5 40 94 150 -124 -11 3]]

Optimal tunings:

  • WE: ~2 = 1199.9925 ¢, ~18/17 = 99.6710 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6716 ¢

Optimal ET sequence: 12, 301

Badness (Sintel): 2.41

Sextilifourths

Named by Xenllium in 2021, sextilifourths (also known as sextilischis, formerly sextilififths) slices the perfect fourth into six small semitones, which serves as both 21/20 and 22/21. It may be described as 130 & 159, and its ploidacot is omega-hexacot. 289edo gives a highly recommendable tuning.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 235298/234375

Mapping: [⟨1 2 -1 -1], ⟨0 -6 48 55]]

mapping generators: ~2, ~21/20

Optimal tunings:

  • WE: ~2 = 1200.0987 ¢, ~21/20 = 83.0599 ¢
error map: ⟨+0.099 -0.117 +0.462 -0.630]
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.0543 ¢
error map: ⟨0.000 -0.281 +0.295 -0.837]

Optimal ET sequence: 29, 72cd, 101, 130, 289, 419

Badness (Sintel): 2.75

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 -1 -1 0], ⟨0 -6 48 55 50]]

Optimal tunings:

  • WE: ~2 = 1200.0424 ¢, ~21/20 = 83.0520 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.0497 ¢

Optimal ET sequence: 29, 72cde, 101e, 130, 289

Badness (Sintel): 1.50

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 -1 -1 0 1], ⟨0 -6 48 55 50 39]]

Optimal tunings:

  • WE: ~2 = 1200.1056 ¢, ~21/20 = 83.0566 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.0508 ¢

Optimal ET sequence: 29, 72cdef, 101e, 130, 289

Badness (Sintel): 1.04

Septant

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

Subgroup: 2.3.5.7

Comma list: 32805/32768, 516560652/514714375

Mapping: [⟨7 0 105 -56], ⟨0 1 -8 7]]

mapping generators: ~8575/7776, ~3

Optimal tunings:

  • WE: ~8575/7776 = 171.4303 ¢, ~3/2 = 701.7091 ¢
error map: ⟨+0.012 -0.234 +0.096 +0.265]
  • CWE: ~8575/7776 = 171.4286 ¢, ~3/2 = 701.7022 ¢
error map: ⟨0.000 -0.253 +0.069 +0.232]

Optimal ET sequence: 77, 147, 224, 301, 525, 826, 1351

Badness (Sintel): 2.81

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨7 0 105 -56 -120], ⟨0 1 -8 7 13]]

Optimal tunings:

  • WE: ~495/448 = 171.4334 ¢, ~3/2 = 701.7387 ¢
  • CWE: ~495/448 = 171.4286 ¢, ~3/2 = 701.7198 ¢

Optimal ET sequence: 77, 147, 224, 301, 525

Badness (Sintel): 1.46

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨7 0 105 -56 -120 37], ⟨0 1 -8 7 13 -1]]

Optimal tunings:

  • WE: ~495/448 = 171.4282 ¢, ~3/2 = 701.7229 ¢
  • CWE: ~495/448 = 171.4286 ¢, ~3/2 = 701.7242 ¢

Optimal ET sequence: 77, 147, 224, 525, 1274f

Badness (Sintel): 1.02

Octant

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

Subgroup: 2.3.5.7

Comma list: 32805/32768, 2259436291848/2251875390625

Mapping: [⟨8 0 120 -117], ⟨0 1 -8 11]]

mapping generators: ~42875/39366, ~3

Optimal tunings:

  • WE: ~42875/39366 = 150.0048 ¢, ~3/2 = 701.7356 ¢
error map: ⟨+0.039 -0.181 +0.071 +0.127]
  • CWE: ~42875/39366 = 150.0000 ¢, ~3/2 = 701.7134 ¢
error map: ⟨0.000 -0.242 -0.021 +0.022]

Optimal ET sequence: 24, …, 224, 472, 696, 1168

Badness (Sintel): 3.98

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨8 0 120 -117 15], ⟨0 1 -8 11 1]]

Optimal tunings:

  • WE: ~12/11 = 150.0010 ¢, ~3/2 = 701.7177 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~3/2 = 701.7131 ¢

Optimal ET sequence: 24, …, 224, 472, 696, 1168

Badness (Sintel): 1.48

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨8 0 120 -117 15 93], ⟨0 1 -8 11 1 -5]]

Optimal tunings:

  • WE: ~12/11 = 149.9957 ¢, ~3/2 = 701.7046 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~3/2 = 701.7247 ¢

Optimal ET sequence: 24, 224, 472, 696

Badness (Sintel): 1.26

Nonant

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

Subgroup: 2.3.5.7

Comma list: 32805/32768, 40353607/40310784

Mapping: [⟨9 0 135 11], ⟨0 1 -8 1]]

mapping generators: ~2592/2401, ~3

Optimal tunings:

  • WE: ~2592/2401 = 133.3442 ¢, ~3/2 = 701.8000 ¢
error map: ⟨+0.098 -0.057 -0.027 -0.141]
  • CWE: ~2592/2401 = 133.3333 ¢, ~3/2 = 701.7384 ¢
error map: ⟨0.000 -0.217 -0.221 -0.421]

Optimal ET sequence: 36, 99c, 135, 171, 2772bd, 2943bdd, …, 5166bccddd, 5337bccddd

Badness (Sintel): 1.77

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨9 0 135 11 131], ⟨0 1 -8 1 -7]]

Optimal tunings:

  • WE: ~242/225 = 133.3308 ¢, ~3/2 = 701.8205 ¢
  • CWE: ~242/225 = 133.3333 ¢, ~3/2 = 701.8351 ¢

Optimal ET sequence: 36, 135, 171

Badness (Sintel): 4.20

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨9 0 135 11 131 -38], ⟨0 1 -8 1 -7 5]]

Optimal tunings:

  • WE: ~242/225 = 133.3180 ¢, ~3/2 = 701.6956 ¢
  • CWE: ~242/225 = 133.3333 ¢, ~3/2 = 701.7800 ¢

Optimal ET sequence: 36, 99cf, 135, 171

Badness (Sintel): 3.15

Septiquarschis

Named by Xenllium in 2021, septiquarschis tempers out 829440/823543 (mynaslender comma) and 67108864/66706983 (septiness comma), and may be described as the 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 perfect fifth; its ploidacot is thus epsilon-heptacot.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 829440/823543

Mapping: [⟨1 -4 47 6], ⟨0 7 56 -4]]

mapping generators: ~2, ~256/147

Optimal tunings:

  • WE: ~2 = 1199.8855 ¢, ~256/147 = 957.2944 ¢
error map: ⟨-0.114 -0.436 -0.182 +1.310]
  • CWE: ~2 = 1200.0000 ¢, ~256/147 = 957.3867 ¢
error map: ⟨0.000 -0.248 +0.032 +1.627]

Optimal ET sequence: 89, 94, 183, 460d, 643d

Badness (Sintel): 4.73

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -4 47 6 25], ⟨0 7 56 -4 -27]]

Optimal tunings:

  • WE: ~2 = 1199.9430 ¢, ~256/147 = 957.3390 ¢
  • CWE: ~2 = 1200.0000 ¢, ~256/147 = 957.3849 ¢

Optimal ET sequence: 89, 94, 183, 460d

Badness (Sintel): 1.72

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -4 47 6 25 -33], ⟨0 7 56 -4 -27 46]]

Optimal tunings:

  • WE: ~2 = 1200.0058 ¢, ~256/147 = 957.3946 ¢
  • CWE: ~2 = 1200.0000 ¢, ~256/147 = 957.3900 ¢

Optimal ET sequence: 89, 94, 183, 277, 460d

Badness (Sintel): 1.46

Tridecafifths

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

Subgroup: 2.3.5.7

Comma list: 32805/32768, [-14 -1 -9 13⟩

Mapping: [⟨1 1 7 6], ⟨0 13 -104 -71]]

mapping generators: ~2, ~1323/1280

Optimal tunings:

  • WE: ~2 = 1200.1431 ¢, ~1323/1280 = 53.9838 ¢
error map: ⟨+0.143 -0.023 +0.375 -0.816]
  • CWE: ~2 = 1200.0000 ¢, ~1323/1280 = 53.9764 ¢
error map: ⟨0.000 -0.261 -0.221 -0.421]

Optimal ET sequence: 89, 200, 289

Badness (Sintel): 10.9

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 7 6 4], ⟨0 13 -104 -71 -12]]

Optimal tunings:

  • WE: ~2 = 1200.0311 ¢, ~33/32 = 53.9766 ¢
  • CWE: ~2 = 1200.0000 ¢, ~33/32 = 53.9750 ¢

Optimal ET sequence: 89, 200, 289

Badness (Sintel): 4.23

Subgroup extensions

Maqamschismic (2.3.5.11)

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

Subgroup: 2.3.5.11

Comma list: 2200/2187, 4125/4096

Subgroup-val mapping: [⟨1 0 15 -33], ⟨0 1 -8 23]]

Optimal tunings:

  • WE: ~2 = 1200.5458 ¢ ~3/2 = 702.4021 ¢
  • CWE: 2 = 1200.0000 ¢, ~3/2 = 702.0906 ¢

Optimal ET sequence: 12e, …, 41, 53, 94, 147e, 241ce, 335ce

Badness (Sintel): 1.34

2.3.5.11.13 subgroup

Subgroup: 2.3.5.11.13

Comma list: 325/324, 352/351, 4125/4096

Subgroup-val mapping: [⟨1 0 15 -33 -28], ⟨0 1 -8 23 20]]

Optimal tunings:

  • WE: ~2 = 1200.4565 ¢ ~3/2 = 702.3057 ¢
  • CWE: 2 = 1200.0000 ¢, ~3/2 = 702.0485 ¢

Optimal ET sequence: 12e, …, 41, 53, 94, 147e

Badness (Sintel): 0.862

Tridecaschismic (2.3.5.13)

Proposed by Eufalesio in 2026, tridecaschismic adds the marveltwin comma to the comma list, or equivalently, the tridecapyth comma. It benefits from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is one of the lowest badness schismic extensions. It is also equivalent to the 2.3.5.13 restriction of 13-limit cassandra.

Subgroup: 2.3.5.13

Comma list: 325/324, 32805/32768

Subgroup-val mapping: [⟨1 0 15 -28], ⟨0 1 -8 20]]

Optimal tunings:

  • WE: ~2 = 1200.3326 ¢ ~3/2 = 702.1092 ¢
  • CWE: 2 = 1200.0000 ¢, ~3/2 = 701.9189 ¢

Optimal ET sequence: 12, …, 41, 53, 412cf, 465cf, …, 783ccff, 836ccfff

Badness (Sintel): 0.582

2.3.5.13.19 subgroup

Subgroup: 2.3.5.13.19

Comma list: 325/324, 361/360, 513/512

Subgroup-val mapping: [⟨1 0 15 -28 9], ⟨0 1 -8 20 -3]]

Optimal tunings:

  • WE: ~2 = 1200.4236 ¢, ~3/2 = 702.1510 ¢
  • CWE: 2 = 1200.0000 ¢, ~3/2 = 701.9064 ¢

Optimal ET sequence: 12, …, 41, 53

Badness (Sintel): 0.354

Photia (2.3.5.17)

See also: No-elevens subgroup temperaments § Garibaldia

Subgroup: 2.3.5.17

Comma list: 256/255, 1458/1445

Subgroup-val mapping: [⟨1 0 15 -7], ⟨0 1 -8 7]]

Gencom mapping: [⟨1 0 15 0 0 0 -7], ⟨0 1 -8 0 0 0 7]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.5471 ¢, ~3/2 = 701.2262 ¢
error map: ⟨-0.453 -1.182 +0.706 +3.628]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.4976 ¢
error map: ⟨0.000 -0.457 +1.705 +5.528]

Optimal ET sequence: 12, 41, 53, 65, 207g, 272gg

Badness (Sintel): 0.479

2.3.5.17.19 subgroup

Subgroup: 2.3.5.17.19

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

Subgroup-val mapping: [⟨1 0 15 -7 9], ⟨0 1 -8 7 -3]]

Gencom mapping: [⟨1 0 15 0 0 0 -7 9], ⟨0 1 -8 0 0 0 7 -3]]

Optimal tunings:

  • WE: ~2 = 1199.7225 ¢, ~3/2 = 701.3077 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.4754 ¢

Optimal ET sequence: 12, 41, 53, 65, 142g

Badness (Sintel): 0.332

Nestoria (2.3.5.19)

See also: No-elevens subgroup temperaments #Garibaldia and #Pontia

Nestoria is notable for having one of the lowest-badness subgroup extensions of schismic. Note that despite prime 19 being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is generally not flatter than the fifth in optimal schismic due to its optimization considering intervals like 19/10 and 19/15.

Subgroup: 2.3.5.19

Comma list: 361/360, 513/512

Subgroup-val mapping: [⟨1 0 15 9], ⟨0 1 -8 -3]]

Gencom mapping: [⟨1 0 15 0 0 0 0 9], ⟨0 1 -8 0 0 0 0 -3]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1200.2250 ¢, ~3/2 = 701.8776 ¢
error map: ⟨+0.225 +0.148 +0.240 -1.796]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7307 ¢
error map: ⟨0.000 -0.224 -0.159 -2.705]

Optimal ET sequence: 12, 29, 41, 53, 118, 171, 460hh, 631hh

Badness (Sintel): 0.126

Taylor (2.3.5.13)

This is a 2.3.5.13 subgroup restriction of 13-limit hemischis.

Subgroup: 2.3.5.13

Comma list: 676/675, 32805/32768

Subgroup-val mapping: [⟨1 0 15 14], ⟨0 2 -16 -13]]

Gencom mapping: [⟨1 0 15 0 0 14], ⟨0 2 -16 0 0 -13]]

mapping generators: ~2, ~26/15

Optimal tunings:

  • WE: ~2 = 1200.1497 ¢, ~26/15 = 950.9740 ¢
error map: ⟨+0.150 -0.007 +0.348 -1.094]
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.8493 ¢
error map: ⟨0.000 -0.256 +0.098 -1.568]

Optimal ET sequence: 24, 53, 130, 183, 236, 525f, 761ff

Badness (Sintel): 0.334

Dakota (2.3.5.13.19)

Subgroup: 2.3.5.13.19

Comma list: 361/360, 513/512, 676/675

Subgroup-val mapping: [⟨1 0 15 14 9], ⟨0 2 -16 -13 -6]]

Optimal tunings:

  • WE: ~2 = 1200.2611 ¢, ~26/15 = 951.0703 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.8532 ¢

Optimal ET sequence: 24, 29, 53, 130, 183, 236h, 289h

Badness (Sintel): 0.262

2.3.5.13.19.37 subgroup

Subgroup: 2.3.5.13.19.37

Comma list: 361/360, 481/480, 513/512, 676/675

Subgroup-val mapping: [⟨1 0 15 14 9 6], ⟨0 2 -16 -13 -6 -1]]

Optimal tunings:

  • WE: ~2 = 1200.2987 ¢, ~26/15 = 951.1060 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.8595 ¢

Optimal ET sequence: 24, 29, 53, 183, 236h, 289hl, 631fhhll

Badness (Sintel): 0.223

Quintilischis (2.3.5.17)

For full 17- and 19-limit extensions, see #Quintilipyth or #Quintaschis.

Subgroup: 2.3.5.17

Comma list: 32805/32768, 1419857/1417176

Subgroup-val mapping: [⟨1 2 -1 5], ⟨0 -5 40 -11]]

Gencom mapping: [⟨1 2 -1 0 0 0 5], ⟨0 -5 40 0 0 0 -11]]

mapping generators: ~2, ~18/17

Optimal tunings:

  • WE: ~2 = 1200.1370 ¢, ~18/17 = 99.6602 ¢
error map: ⟨+0.137 +0.018 -0.042 -0.533]
  • CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6499 ¢
error map: ⟨0.000 -0.205 -0.317 -1.104]

Optimal ET sequence: 12, …, 253, 265, 277, 289, 566g, 855g

Badness (Sintel): 1.34

2.3.5.17.19 subgroup

Subgroup: 2.3.5.17.19

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

Subgroup-val mapping: [⟨1 2 -1 5 4], ⟨0 -5 40 -11 3]]

Gencom mapping: [⟨1 2 -1 0 0 0 5 4], ⟨0 -5 40 0 0 0 -11 3]]

Optimal tunings:

  • WE: ~2 = 1200.0350 ¢, ~18/17 = 99.6550 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6520 ¢

Optimal ET sequence: 12, …, 253, 265, 277, 289

Badness (Sintel): 1.17

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