@@ Line 1: / Line 1: @@
 {{interwiki
+| en = Schismatic family
 | de = Schismatische Temperaturen
-| en =
 | es =
 | ja =
 }}
-The 5-limit parent comma for the '''schismatic''' (or '''schismic''') '''family''' is the [[schisma]] of 32805/32768, which is the amount by which the [[Pythagorean comma]] exceeds the [[Didymus 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.
+{{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]].
-This defies the tradition of tertian harmony, as the just major triad on C is C-Fb-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.
+== Schismic, schismatic, a.k.a. helmholtz ==
+{{Main| Schismic }}
-== Schismatic aka helmholtz ==
+The 5-limit version of the temperament is a [[microtemperament]], called ''schismic'', ''schismatic'', or ''helmholtz''. The generator is a fifth, flattened by a fraction of a schisma, and 5/4 is represented by a diminished fourth. This defies the tradition of {{w|tertian harmony}}, as the [[just major triad]] on C is C–F♭–G, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as C–vE–G.
-The 5-limit version of the temperament is a [[microtemperament]], sometimes called '''helmholtz''', '''schismic''' or '''schismatic''', which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. [[53edo]] is a possible tuning for schismatic, but you need [[118edo]] if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better 5th, 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.
+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
@@ Line 17: / Line 20: @@
 {{Mapping|legend=1| 1 0 15 | 0 1 -8 }}
 : mapping generators: ~2, ~3
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.736
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0749{{c}}, ~3/2 = 701.7797{{c}}
+: [[error map]]: {{val| +0.075 -0.100 -0.027 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7308{{c}}
+: error map: {{val| 0.000 -0.224 -0.160 }}
 [[Tuning ranges]]:
-* 5-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 705.882] (4\7 to 10\17)
+* [[5-odd-limit]] [[diamond 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 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 }}
-[[Badness]]: 0.004259
+[[Badness]] (Sintel): 0.0999
 === 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 ==
-{{Main| Garibaldi temperament }}
+{{Main| 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-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
@@ Line 62: / Line 62: @@
 {{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}}
-{{Multival|legend=1| 1 -8 -14 -15 -25 -10 }}
+: [[error map]]: {{val| +0.123 +0.326 -2.709 +2.328 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0774{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.085
+: error map: {{val| 0.000 +0.122 -2.933 +2.090 }}
 [[Minimax tuning]]:
 * [[7-odd-limit]]: ~3/2 = {{monzo| 2/3 1/15 0 -1/15 }}
 : {{monzo list| 1 0 0 0 | 5/3 1/15 0 -1/15 | 5/3 -8/15 0 8/15 | 5/3 -14/15 0 14/15 }}
-: [[Eigenmonzo basis|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 }}
 : {{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]]:
@@ Line 80: / Line 80: @@
 * 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 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
@@ Line 93: / Line 93: @@
 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:
 * 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:
@@ Line 103: / Line 105: @@
 * 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 ====
@@ Line 114: / Line 116: @@
 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:
 * 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:
@@ Line 125: / Line 129: @@
 * 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 =====
@@ Line 136: / Line 140: @@
 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 ======
@@ Line 149: / Line 155: @@
 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 =====
@@ Line 162: / Line 170: @@
 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 ======
@@ Line 176: / Line 186: @@
 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 ======
@@ Line 192: / Line 201: @@
 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 =====
@@ Line 207: / Line 215: @@
 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 ======
@@ Line 220: / Line 230: @@
 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 ===
@@ Line 233: / Line 245: @@
 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:
 * 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:
@@ Line 243: / Line 257: @@
 * 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 ====
@@ Line 254: / Line 268: @@
 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:
 * 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:
@@ Line 265: / Line 281: @@
 * 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 =====
@@ Line 276: / Line 292: @@
 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 ======
@@ Line 289: / Line 307: @@
 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 =====
@@ Line 302: / Line 322: @@
 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 ======
@@ Line 315: / Line 337: @@
 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 =====
@@ Line 328: / Line 352: @@
 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 ======
@@ Line 341: / Line 367: @@
 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 ===
@@ Line 354: / Line 382: @@
 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:
 * 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 ====
@@ Line 371: / Line 401: @@
 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:
 * 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 ====
@@ Line 388: / Line 420: @@
 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 ====
@@ Line 401: / Line 435: @@
 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
-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 ====
 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
-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}}
+* CWE: ~2 = 1200.0000{{c}}, ~110/63 = 951.0604{{c}}
-Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.994
+{{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135ee }}
-{{Optimal ET sequence|legend=1| 41, 106, 147 }}
+Badness (Sintel): 1.68
-Badness: 0.041562
 ==== 13-limit ====
 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 ===
@@ Line 471: / Line 513: @@
 Mapping: {{mapping| 1 2 -1 -3 0 | 0 -3 24 42 25 }}
 : 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 ====
@@ Line 487: / Line 530: @@
 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}}
-{{Optimal ET sequence|legend=1| 29, 65d, 94 }}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9659{{c}}
-Badness: 0.033849
-== Schism ==
-{{See also| 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.
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 64/63, 360/343
-{{Mapping|legend=1| 1 0 15 6 | 0 1 -8 -2 }}
-[[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1\1, ~3/2 = 702.2696
-* [[POTE]]: ~2 = 1\1, ~3/2 = 701.556
-{{Multival|legend=1| 1 -8 -2 -15 -6 18 }}
-{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}
-[[Badness]]: 0.056648
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 45/44, 64/63, 99/98
+{{Optimal ET sequence|legend=0| 29, 65d, 94 }}
-Mapping: {{mapping| 1 0 15 6 13 | 0 1 -8 -2 -6 }}
+Badness (Sintel): 1.40
-Optimal tunings:
-* CTE: ~2 = 1\1, ~3/2 = 703.3833
-* POTE: ~2 = 1\1, ~3/2 = 702.136
-{{Optimal ET sequence|legend=1| 5c, 7ce, 12, 29de }}
-Badness: 0.037482
 == Pontiac ==
 {{Main| Pontiac }}
-Pontiac tempers out the [[ragisma]], rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple augmented third (C-Exx#), or triple-up major sixth (C-^<sup>3</sup>A).
+Pontiac tempers out the [[ragisma]], rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple-augmented third (C-E𝄪𝄪♯), or triple-up major sixth (C-^<sup>3</sup>A).
 [[Subgroup]]: 2.3.5.7
@@ Line 540: / Line 549: @@
 {{Mapping|legend=1| 1 0 15 -59 | 0 1 -8 39 }}
-{{Multival|legend=1| 1 -8 39 -15 59 113 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0989{{c}}, ~3/2 = 701.8145{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.757
+: [[error map]]: {{val| +0.099 -0.042 -0.138 -0.038 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7579{{c}}
+: error map: {{val| 0.000 -0.197 -0.377 -0.268 }}
 [[Minimax tuning]]:
 * [[7-odd-limit]]: ~3/2 = {{monzo| 27/47 0 -1/47 1/47 }}
 : [{{monzo| 1 0 0 0 }}, {{monzo| 74/47 0 -1/47 1/47 }}, {{monzo| 113/47 0 8/47 -8/47 }}, {{monzo| 113/47 0 -39/47 39/47 }}]
-: [[Eigenmonzo basis|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 }}
 : [{{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]]:
 * 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 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 ===
-The helenoid temperament (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
@@ Line 570: / Line 580: @@
 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:
 * 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 ====
@@ Line 587: / Line 599: @@
 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:
 * 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 =====
@@ Line 604: / Line 618: @@
 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:
 * 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 ====
@@ Line 621: / Line 637: @@
 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 =====
@@ Line 634: / Line 652: @@
 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 =====
@@ Line 647: / Line 667: @@
 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 ===
-The ponta temperament (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
@@ Line 662: / Line 684: @@
 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:
 * 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 ====
@@ Line 679: / Line 703: @@
 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:
 * 13 and 15-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 }}
+{{Optimal ET sequence|legend=0| 53, 171, 224 }}
-Badness: 0.023616
+Badness (Sintel): 0.976
 ==== 17-limit ====
@@ Line 696: / Line 722: @@
 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:
 * 17-odd-limit: ~3/2 = {{monzo| 83/143 0 0 0 -1/143 0 1/143 }}
-: Eigenmonzo (unchanged-interval) 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 ===
-The pontic temperament (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
@@ Line 715: / Line 743: @@
 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:
 * 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 ====
@@ Line 732: / Line 762: @@
 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:
 * 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 =====
@@ Line 749: / Line 781: @@
 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:
 * 17-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: 0.029618
+Badness (Sintel): 1.51
 ==== Pontoid ====
@@ Line 766: / Line 800: @@
 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 =====
@@ Line 779: / Line 815: @@
 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 ===
-The bipont temperament (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
@@ Line 793: / Line 831: @@
 Mapping: {{mapping| 2 0 30 -118 -85 | 0 1 -8 39 29 }}
 : mapping generators: ~99/70, ~3
-Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.757
+Optimal tunings:
+* WE: ~99/70 = 600.0500{{c}}, ~3/2 = 701.8153{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7584{{c}}
-{{Optimal ET sequence|legend=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 ====
@@ Line 809: / Line 848: @@
 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 =====
@@ Line 824: / Line 863: @@
 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 ====
@@ Line 837: / Line 878: @@
 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 =====
@@ Line 850: / Line 893: @@
 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 =====
@@ Line 863: / Line 908: @@
 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 ====
@@ Line 875: / Line 922: @@
 Mapping: {{mapping| 4 0 60 -236 -170 -131 | 0 1 -8 39 29 23 }}
 : mapping generators: ~208/175, ~3
-Optimal tuning (POTE): ~208/175 = 1\4, ~3/2 = 701.756
+Optimal tunings:
+* WE: ~208/175 = 300.0229{{c}}, ~3/2 = 701.8097{{c}}
+* CWE: ~208/175 = 300.0000{{c}}, ~3/2 = 701.7578{{c}}
-{{Optimal ET sequence|legend=1| 224, 460, 684, 2276cde, 2960cde, 3644bccddee }}
+{{Optimal ET sequence|legend=0| 224, 460, 684, 2276cde, 2960cde }}
-Badness: 0.021025
+Badness (Sintel): 0.869
 == Grackle ==
-Grackle tempers out {{monzo| -44 26 0 1 }}. The 7/4 is found at -26 fifths, represented by the triple diminished ninth (C-Dbbbb), or double-down minor seventh (C-vvBb), which is to say, two comma steps are required to bend the Pythagorean minor seventh to the septimal one.
+Grackle tempers out {{monzo| -44 26 0 1 }} so 7/4 is found at -26 fifths, represented by the triple-diminished ninth (C–D𝄫𝄫) or double-down minor seventh (C–vvB♭). Two comma steps are required to bend the Pythagorean minor seventh to the septimal one.
 [[Subgroup]]: 2.3.5.7
@@ Line 892: / Line 940: @@
 {{Mapping|legend=1| 1 0 15 44 | 0 1 -8 -26 }}
 : mapping generators: ~2, ~3
-{{Multival|legend=1| 1 -8 -26 -15 -44 -38 }}
+[[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]]:
-* [[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 ===
@@ Line 914: / Line 963: @@
 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 ====
@@ Line 927: / Line 978: @@
 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 =====
@@ Line 940: / Line 993: @@
 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 =====
@@ Line 953: / Line 1,008: @@
 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 ====
@@ Line 966: / Line 1,023: @@
 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 ===
@@ Line 979: / Line 1,038: @@
 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 ====
@@ Line 992: / Line 1,053: @@
 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 ====
@@ Line 1,005: / Line 1,068: @@
 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
-[[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}}
-{{Multival|legend=1| 2 -16 -40 -30 -69 -48 }}
+: [[error map]]: {{val| +0.2569 +0.4168 -1.4342 +0.2685 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9615{{c}}
-[[Optimal tuning]] ([[CTE]]): ~567/400 = 1\2, ~3/2 = 701.5899
+: 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 ===
 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
-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
-{{Multival|legend=1| 4 -32 38 -60 49 178 }}
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~35/24 = 650.920 (~36/35 = 50.920)
+Comma list: 289/288, 441/440, 561/560, 729/728, 3136/3125
-{{Optimal ET sequence|legend=1| 24, 70c, 94, 118, 212, 330, 542d, 872cd }}
+Mapping: {{mapping| 2 0 30 69 102 -75 5 | 0 1 -8 -20 -30 26 1 }}
-[[Badness]]: 0.110583
+Optimal tunings:
+* WE: ~99/70 = 600.0331{{c}}, ~3/2 = 701.6387{{c}}
-=== 11-limit ===
+* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5994{{c}}
-Subgroup: 2.3.5.7.11
-Comma list: 385/384, 9801/9800, 14641/14580
-Mapping: {{mapping| 2 1 22 -15 8 | 0 2 -16 19 -1 }}
+{{Optimal ET sequence|legend=0| 12, 118, 130, 248g }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.918 (~36/35 = 50.918)
+Badness (Sintel): 1.49
-{{Optimal ET sequence|legend=1| 24, 70c, 94, 118, 212, 330e, 542de }}
+==== Bischis ====
-Badness: 0.036749
-==== 13-limit ====
 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 =====
 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 1 22 -15 8 15 6 | 0 2 -16 19 -1 -7 2 }}
+Mapping: {{mapping| 2 0 30 69 102 131 5 | 0 1 -8 -20 -30 -39 1 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.942 (~36/35 = 50.942)
+Optimal tunings:
+* WE: ~55/39 = 600.0997{{c}}, ~3/2 = 701.7114{{c}}
+* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.5899{{c}}
-{{Optimal ET sequence|legend=1| 24, 70c, 94, 118, 212fg }}
+{{Optimal ET sequence|legend=0| 12f, 106deff, 118f, 130, 248fg }}
-Badness: 0.025615
+Badness (Sintel): 1.37
-==== Kleischis ====
+== Kleischismic ==
-Subgroup: 2.3.5.7.11.13
+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.
-Comma list: 325/324, 385/384, 1573/1568, 14641/14580
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 2 1 22 -15 8 -36 | 0 2 -16 19 -1 40 }}
+[[Comma list]]: 32805/32768, 1500625/1492992
-Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.951 (~36/35 = 50.951)
+{{Mapping|legend=1| 2 1 22 -15 | 0 2 -16 19 }}
+: mapping generators: ~1225/864, ~35/24
-{{Optimal ET sequence|legend=1| 24f, 70cf, 94, 118f, 212 }}
+[[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 }}
-Badness: 0.037607
+{{Optimal ET sequence|legend=1| 24, 94, 118, 212, 330, 542d, 872cdd, 1414ccddd }}
-===== 17-limit =====
+[[Badness]] (Sintel): 2.80
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 289/288, 325/324, 385/384, 442/441, 14641/14580
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Mapping: {{mapping| 2 1 22 -15 8 -36 6 | 0 2 -16 19 -1 40 2 }}
+Comma list: 385/384, 9801/9800, 14641/14580
-Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.948 (~36/35 = 50.948)
+Mapping: {{mapping| 2 1 22 -15 8 | 0 2 -16 19 -1 }}
-{{Optimal ET sequence|legend=1| 24f, 70cf, 94, 118f, 212g }}
+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}})
-Badness: 0.024734
+{{Optimal ET sequence|legend=0| 24, 94, 118, 212, 330e, 542dee, 872cddeee }}
-== Hemischis ==
+Badness (Sintel): 1.21
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 6144/6125, 19683/19600
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-{{Mapping|legend=1| 1 0 15 -17 | 0 2 -16 25 }}
+Comma list: 352/351, 385/384, 729/728, 1575/1573
-: mapping generators: ~2, ~140/81
+Mapping: {{mapping| 2 1 22 -15 8 15 | 0 2 -16 19 -1 -7 }}
-{{Multival|legend=1| 2 -16 25 -30 34 103 }}
+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 }} <!-- 443h is not a GPV -->
+Badness (Sintel): 1.30
-Badness (Dirichlet): 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]].
+Mapping: {{mapping| 2 1 22 -15 8 -36 6 | 0 2 -16 19 -1 40 2 }}
-Subgroup: 2.3.5.7.11.13
+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}})
-Comma list: 351/350, 540/539, 676/675, 4096/4095
+{{Optimal ET sequence|legend=0| 24f, 94, 118f, 212g }}
-Mapping: {{mapping| 1 0 15 -17 51 14 | 0 2 -16 25 -60 -13 }}
+Badness (Sintel): 1.26
-Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.801
+== 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]].
-{{Optimal ET sequence|legend=1| 24e, 53, 130, 183, 313 }}
+[[Subgroup]]: 2.3.5.7
-Badness: 0.020816
+[[Comma list]]: 245/243, 32805/32768
-=== 17-limit ===
+{{Mapping|legend=1| 1 1 7 -1 | 0 2 -16 13 }}
-Subgroup: 2.3.5.7.11.13.17
+: mapping generators: ~2, ~128/105
-Comma list: 351/350, 442/441, 561/560, 676/675, 4096/4095
+[[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 }}
-Mapping: {{mapping| 1 0 15 -17 51 14 -49 | 0 2 -16 25 -60 -13 67 }}
+{{Optimal ET sequence|legend=1| 17, 24, 41, 106d, 147d, 188cd }}
-Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.810
+[[Badness]] (Sintel): 2.03
-{{Optimal ET sequence|legend=1| 53, 130, 183, 496d }} <!-- 679df is not a GPV -->
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Badness: 0.021073
+Comma list: 243/242, 245/242, 385/384
-=== 19-limit ===
+Mapping: {{mapping| 1 1 7 -1 2 | 0 2 -16 13 5 }}
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 4096/4095
+Optimal tunings:
+* WE: ~2 = 1200.3891{{c}}, ~11/9 = 351.1275{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0141{{c}}
-Mapping: {{mapping| 1 0 15 -17 51 14 -49 9 | 0 2 -16 25 -60 -13 67 -6 }}
+{{Optimal ET sequence|legend=0| 17, 24, 41, 106d }}
-Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.809
+Badness (Sintel): 1.30
-{{Optimal ET sequence|legend=1| 53, 130, 183, 313h }} <!-- 496dh is not a GPV -->
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Badness (Dirichlet): 1.111
+Comma list: 105/104, 144/143, 243/242, 245/242
-=== 23-limit ===
+Mapping: {{mapping| 1 1 7 -1 2 4 | 0 2 -16 13 5 -1 }}
-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
+Optimal tunings:
+* WE: ~2 = 1199.9362{{c}}, ~11/9 = 351.0061{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0247{{c}}
-Mapping: {{mapping| 1 0 15 -17 51 14 -49 9 -24 | 0 2 -16 25 -60 -13 67 -6 36 }}
+{{Optimal ET sequence|legend=0| 17, 24, 41 }}
-Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.807
+Badness (Sintel): 1.27
-{{Optimal ET sequence|legend=1| 53, 130, 183, 313h }} <!-- 496dh is not a GPV -->
+== Hemischis ==
+Hemischis tempers out 6144/6125, the [[porwell comma]], as well as 19683/19600, the [[cataharry comma]], and may be described as the {{nowrap| 53 & 130 }} temperament. Its [[ploidacot]] is alpha-dicot.
-Badness (Dirichlet): 1.061
+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.
-; 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
-== Squirrel ==
-The squirrel temperament (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
-[[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
-{{Multival|legend=1| 3 -24 -13 -45 -29 37 }}
+[[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 tuning]] ([[POTE]]): ~2 = 1\1, ~160/147 = 166.140
+{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 313 }}
-{{Optimal ET sequence|legend=1| 29, 36, 65 }}
-[[Badness]]: 0.174705
+[[Badness]] (Sintel): 1.16
 === 11-limit ===
 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 ===
 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 (94 &amp; 159) has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel]], but tempers out 1071785/1062882 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}}
-{{Multival|legend=1| 3 -24 52 -45 74 188 }}
+{{Optimal ET sequence|legend=0| 53, 130, 183, 496d }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~192/175 = 166.019
+Badness (Sintel): 1.07
-{{Optimal ET sequence|legend=1| 65, 94, 159, 253, 412cd }}
+=== 19-limit ===
+Subgroup: 2.3.5.7.11.13.17.19
-[[Badness]]: 0.211859
+Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 4096/4095
-=== 11-limit ===
+Mapping: {{mapping| 1 0 15 -17 51 14 -49 9 | 0 2 -16 25 -60 -13 67 -6 }}
-Subgroup: 2.3.5.7.11
-Comma list: 385/384, 4000/3993, 19712/19683
+Optimal tunings:
+* WE: ~2 = 1200.0464{{c}}, ~26/15 = 950.8459{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8091{{c}}
-Mapping: {{mapping| 1 2 -1 10 0 | 0 -3 24 -52 25 }}
+{{Optimal ET sequence|legend=0| 53, 130, 183, 313h }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.017
+Badness (Sintel): 1.11
-{{Optimal ET sequence|legend=1| 65, 94, 159, 253, 412cd }}
+=== 23-limit ===
+Subgroup: 2.3.5.7.11.13.17.19.23
-Badness: 0.061336
+Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 736/735, 4096/4095
-=== 13-limit ===
+Mapping: {{mapping| 1 0 15 -17 51 14 -49 9 -24 | 0 2 -16 25 -60 -13 67 -6 36 }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 325/324, 385/384, 1575/1573, 10985/10976
+Optimal tunings:
+* WE: ~2 = 1200.0215{{c}}, ~26/15 = 950.8239{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8069{{c}}
-Mapping: {{mapping| 1 2 -1 10 0 12 | 0 -3 24 -52 25 -60 }}
+{{Optimal ET sequence|legend=0| 53, 130, 183, 313h }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.016
+Badness (Sintel): 1.06
-{{Optimal ET sequence|legend=1| 65f, 94, 159, 253, 412cdf, 665ccdef }}
+; 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
-Badness: 0.036700
+== Term ==
+Term tempers out the [[landscape comma]], mapping [[63/50]] to the 1/3-octave period. It can be described as {{nowrap| 12 & 171 }}, and is the unique temperament that tempers together the syntonic and Pythagorean commas and equates it with a stack of three [[marvel comma]]s. A [[septimal comma]] is then found as a stack of four marvel commas. In certain 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma #As an interval region|kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[171edo]] makes for an excellent tuning.
-=== 17-limit ===
+[[Subgroup]]: 2.3.5.7
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
+[[Comma list]]: 32805/32768, 250047/250000
-Mapping: {{mapping| 1 2 -1 10 0 12 -2 | 0 -3 24 -52 25 -60 44 }}
+{{Mapping|legend=1| 3 0 45 94 | 0 1 -8 -18 }}
+: mapping generators: ~63/50, ~3
-Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.012
+[[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 }}
-{{Optimal ET sequence|legend=1| 65f, 94, 159, 253 }}
+[[Minimax tuning]]:
+* [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis)]]: 2.5/3
+* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
-Badness: 0.026504
+{{Optimal ET sequence|legend=1| 12, …, 159, 171, 867, 1038, 1209, 1380, 1551, 1722 }}
-== Countertertiaschis ==
+[[Badness]] (Sintel): 0.505
-The countertertiaschis temperament (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
+=== 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.
-[[Comma list]]: 32805/32768, 244140625/243045684
+Subgroup: 2.3.5.7.11
-{{Mapping|legend=1| 1 2 -1 -12 | 0 -3 24 107 }}
+Comma list: 441/440, 4375/4356, 32805/32768
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~625/567 = 166.0621
+Mapping: {{mapping| 3 0 45 94 134 | 0 1 -8 -18 -26 }}
-{{Optimal ET sequence|legend=1| 65d, 159, 224, 383, 607 }}
+Optimal tunings:
+* WE: ~44/35 = 400.0464{{c}}, ~3/2 = 701.9053{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8178{{c}}
-[[Badness]]: 0.188043
+{{Optimal ET sequence|legend=0| 12, …, 159, 330 }}
-=== 11-limit ===
+Badness (Sintel): 1.97
-Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 4000/3993, 32805/32768
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Mapping: {{mapping| 1 2 -1 -12 0 | 0 -3 24 107 25 }}
+Comma list: 364/363, 441/440, 625/624, 13720/13689
-Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.0628
+Mapping: {{mapping| 3 0 45 94 134 168 | 0 1 -8 -18 -26 -33 }}
-{{Optimal ET sequence|legend=1| 65d, 159, 224, 383, 607 }}
+Optimal tunings:
+* WE: ~44/35 = 400.0449{{c}}, ~3/2 = 701.8995{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8156{{c}}
-Badness: 0.048943
+{{Optimal ET sequence|legend=0| 12f, …, 159, 330 }}
-=== 13-limit ===
+Badness (Sintel): 1.53
-Subgroup: 2.3.5.7.11.13
-Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
-Mapping: {{mapping| 1 2 -1 -12 0 -10 | 0 -3 24 107 25 99 }}
+Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619
-Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.0628
+Mapping: {{mapping| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }}
-{{Optimal ET sequence|legend=1| 65d, 159, 224, 383, 607 }}
+Optimal tunings:
+* WE: ~34/27 = 400.0195{{c}}, ~3/2 = 701.8439{{c}}
+* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 701.8081{{c}}
-Badness: 0.024506
+{{Optimal ET sequence|legend=0| 12f, 159, 171, 330 }}
-== Pogo ==
+Badness (Sintel): 1.38
-{{See also| Stearnsmic clan #Pogo }}
-The pogo temperament (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.
+=== Terminator ===
+Terminator tempers out 540/539, and may be described as {{nowrap| 171 & 183 }}.
-[[Subgroup]]: 2.3.5.7
+Subgroup: 2.3.5.7.11
-[[Comma list]]: 32805/32768, 118098/117649
+Comma list: 540/539, 32805/32768, 137781/137500
-{{Mapping|legend=1| 2 1 22 2 | 0 3 -24 5 }}
+Mapping: {{mapping| 3 0 45 94 -137 | 0 1 -8 -18 31 }}
-: mapping generators: ~343/243, ~9/7
+Optimal tunings:
+* WE: ~63/50 = 399.9677{{c}}, ~3/2 = 701.6278{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6846{{c}}
-{{Multival|legend=1| 6 -48 10 -90 -1 158 }}
+{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 537, 891de }}
-[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~9/7 = 433.901
+Badness (Sintel): 2.21
-{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354 }}
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-[[Badness]]: 0.079635
+Comma list: 540/539, 729/728, 4096/4095, 31250/31213
-=== 11-limit ===
+Mapping: {{mapping| 3 0 45 94 -137 -103 | 0 1 -8 -18 31 24 }}
-Subgroup: 2.3.5.7.11
-Comma list: 540/539, 4000/3993, 32805/32768
+Optimal tunings:
+* WE: ~63/50 = 399.9731{{c}}, ~3/2 = 701.6414{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}}
-Mapping: {{mapping| 2 1 22 2 25 | 0 3 -24 5 -25 }}
+{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911
+Badness (Sintel): 1.47
-{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354, 578 }}
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
-Badness: 0.031857
+Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095
-=== 13-limit ===
+Mapping: {{mapping| 3 0 45 94 -137 -103 -2 | 0 1 -8 -18 31 24 3 }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 729/728, 1575/1573, 4096/4095
+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: {{mapping| 2 1 22 2 25 -2 | 0 3 -24 5 -25 13 }}
+{{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911
+Badness (Sintel): 1.04
-{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354, 578 }}
+=== Semiterm ===
+The semiterm temperament tempers out [[9801/9800]] (kalisma) as well as [[151263/151250]] (odiheim comma), and may be described as {{nowrap| 12 & 342 }}. It has a period of 1/6 octave and its ploidacot is hexaploid monocot.
-Badness: 0.017514
+Subgroup: 2.3.5.7.11
-== Term ==
+Comma list: 9801/9800, 32805/32768, 151263/151250
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 32805/32768, 250047/250000
+Mapping: {{mapping| 6 0 90 188 287 | 0 1 -8 -18 -28 }}
+: mapping generators: ~55/49, ~3
-{{Mapping|legend=1| 3 0 45 94 | 0 1 -8 -18 }}
+Optimal tunings:
+* WE: ~55/49 = 200.0134{{c}}, ~3/2 = 701.7931{{c}}
+* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7426{{c}}
-: mapping generators: ~63/50, ~3
+{{Optimal ET sequence|legend=0| 12, …, 330e, 342, 1380, 1722, 2064, 2406c, 5154bccdde }}
-{{Multival|legend=1| 3 -24 -54 -45 -94 -58 }}
+Badness (Sintel): 0.973
-[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~3/2 = 701.742
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-[[Minimax tuning]]:
+Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375
-* [[7-odd-limit]] [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis)]]: 2.5/3
-* [[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 }}
+Mapping: {{mapping| 6 0 90 188 287 355 | 0 1 -8 -18 -28 -35 }}
-[[Badness]]: 0.019950
+Optimal tunings:
+* WE: ~55/49 = 200.0083{{c}}, ~3/2 = 701.7549{{c}}
+* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7238{{c}}
-=== Terminal ===
+{{Optimal ET sequence|legend=0| 12f, 330eff, 342f, 696f }} *
-The terminal temperament (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
+<nowiki>*</nowiki> optimal patent val: [[354edo|354]]
-Comma list: 441/440, 4375/4356, 32805/32768
+Badness (Sintel): 1.85
-Mapping: {{mapping| 3 0 45 94 134 | 0 1 -8 -18 -26 }}
+=== Hemiterm ===
+The hemiterm temperament tempers out [[3025/3024]] (lehmerisma), and may be described as {{nowrap| 159 & 183 }}. Its ploidacot is triploid alpha-dicot.
-Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 701.824
+Subgroup: 2.3.5.7.11
-{{Optimal ET sequence|legend=1| 12, 147de, 159, 330 }}
+Comma list: 3025/3024, 32805/32768, 102487/102400
-Badness: 0.059502
+Mapping: {{mapping| 3 0 45 94 8 | 0 2 -16 -36 1 }}
+: mapping generators: ~63/50, ~693/400
+Optimal tunings:
+* WE: ~63/50 = 400.0309{{c}}, ~693/400 = 950.9458{{c}} (~12/11 = 150.8841{{c}})
+* CWE: ~63/50 = 400.0000{{c}}, ~693/400 = 950.8707{{c}} (~12/11 = 150.8707{{c}})
+{{Optimal ET sequence|legend=0| 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce }}
+Badness (Sintel): 0.684
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 364/363, 441/440, 625/624, 13720/13689
+Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712
-Mapping: {{mapping| 3 0 45 94 134 168 | 0 1 -8 -18 -26 -33 }}
+Mapping: {{mapping| 3 0 45 94 8 42 | 0 2 -16 -36 1 -13 }}
-Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 701.821
+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=1| 12f, 147def, 159, 330 }}
+{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f }}
-Badness: 0.037082
+Badness (Sintel): 1.30
 ==== 17-limit ====
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619
+Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264
-Mapping: {{mapping| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }}
+Mapping: {{mapping| 3 0 45 94 8 42 -2 | 0 2 -16 -36 1 -13 6 }}
-Optimal tuning (POTE): ~34/27 = 1\3, ~3/2 = 701.810
+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=1| 12f, 147def, 159, 171, 330 }}
+{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f, 525f }}
-Badness: 0.027073
+Badness (Sintel): 1.14
-=== Terminator ===
+== Altinex ==
-Subgroup: 2.3.5.7.11
+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.
-Comma list: 540/539, 32805/32768, 137781/137500
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 3 0 45 94 -137 | 0 1 -8 -18 31 }}
+[[Comma list]]: 32805/32768, 367653125/362797056
-Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.685
+{{Mapping|legend=1| 3 0 45 -32 | 0 2 -16 17 }}
+: mapping generators: ~1536/1225, ~34300/19683
-{{Optimal ET sequence|legend=1| 12e, 159e, 171, 183, 354, 537, 891de }}
+[[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 }}
-Badness: 0.066968
+{{Optimal ET sequence|legend=1| 24, 135, 159, 612ccdd }}
-==== 13-limit ====
+[[Badness]] (Sintel): 10.7
-Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 729/728, 4096/4095, 31250/31213
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Mapping: {{mapping| 3 0 45 94 -137 -103 | 0 1 -8 -18 31 24 }}
+Comma list: 385/384, 14700/14641, 19712/19683
-Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.689
+Mapping: {{mapping| 3 0 45 -32 8 | 0 2 -16 17 1 }}
-{{Optimal ET sequence|legend=1| 171, 183, 354, 891de, 1245dee, 1599ddee }}
+Optimal tunings:
+* WE: ~44/35 = 400.1156{{c}}, ~121/70 = 951.2377{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~121/70 = 950.9634{{c}}
-Badness: 0.035487
+{{Optimal ET sequence|legend=0| 24, 135, 159 }}
-==== 17-limit ====
+Badness (Sintel): 3.35
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Mapping: {{mapping| 3 0 45 94 -137 -103 -2 | 0 1 -8 -18 31 24 3 }}
+Comma list: 364/363, 385/384, 676/675, 19712/19683
-Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.688
+Mapping: {{mapping| 3 0 45 -32 8 42 | 0 2 -16 17 1 -13 }}
-{{Optimal ET sequence|legend=1| 171, 183, 354, 891de, 1245dee, 1599ddee }}
+Optimal tunings:
+* WE: ~44/35 = 400.1396{{c}}, ~26/15 = 951.2799{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~26/15 = 950.9462{{c}}
-Badness: 0.020434
+{{Optimal ET sequence|legend=0| 24, 135f, 159 }}
-=== Semiterm ===
+Badness (Sintel): 2.27
-The semiterm temperament (12 &amp; 342) has a period of 1/6 octave and tempers out [[9801/9800]] (kalisma) and 151263/151250 (odiheim comma).
-Subgroup: 2.3.5.7.11
+== 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.
-Comma list: 9801/9800, 32805/32768, 151263/151250
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 6 0 90 188 287 | 0 1 -8 -18 -28 }}
+[[Comma list]]: 686/675, 32805/32768
-: mapping generators: ~55/49, ~3
+{{Mapping|legend=1| 1 2 -1 1 | 0 -3 24 13 }}
-Optimal tuning (POTE): ~55/49 = 1\6, ~3/2 = 701.7460
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.7408{{c}}, ~160/147 = 166.2424{{c}}
+: [[error map]]: {{val| +0.741 +0.799 +2.763 -6.934 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~160/147 = 166.1597{{c}}
+: error map: {{val| 0.000 -0.434 +1.518 -8.750 }}
-{{Optimal ET sequence|legend=1| 12, 330e, 342, 1380, 1722, 2064, 2406c }}
+{{Optimal ET sequence|legend=1| 29, 36, 65 }}
-Badness: 0.029438
+[[Badness]] (Sintel): 4.42
-==== 13-limit ====
+=== 11-limit ===
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11
-Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375
+Comma list: 245/242, 686/675, 896/891
-Mapping: {{mapping| 6 0 90 188 287 355 | 0 1 -8 -18 -28 -35 }}
+Mapping: {{mapping| 1 2 -1 1 0 | 0 -3 24 13 25 }}
-Optimal tuning (POTE): ~55/49 = 1\6, ~3/2 = 701.7256
+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=1| 12f, 330eff, 342f, 696f }} *
+{{Optimal ET sequence|legend=0| 29, 36, 65 }}
-<nowiki>*</nowiki> optimal patent val: [[354edo|354]]
+Badness (Sintel): 2.26
-Badness: 0.044657
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-=== Hemiterm ===
+Comma list: 91/90, 169/168, 245/242, 896/891
-Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 32805/32768, 102487/102400
+Mapping: {{mapping| 1 2 -1 1 0 3 | 0 -3 24 13 25 5 }}
-Mapping: {{mapping| 3 0 45 94 8 | 0 2 -16 -36 1 }}
+Optimal tunings:
+* WE: ~2 = 1201.1361{{c}}, ~11/10 = 166.2110{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0833{{c}}
-: mapping generators: ~63/50, ~693/400
+{{Optimal ET sequence|legend=0| 29, 65f, 94df }}
-Optimal tuning (POTE): ~63/50 = 1\3, ~693/400 = 950.872 (~12/11 = 150.872)
+Badness (Sintel): 1.81
-{{Optimal ET sequence|legend=1| 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce }}
+== 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.
-Badness: 0.020687
+[[Subgroup]]: 2.3.5.7
-==== 13-limit ====
+[[Comma list]]: 32805/32768, 1071875/1062882
-Subgroup: 2.3.5.7.11.13
-Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712
+{{Mapping|legend=1| 1 2 -1 10 | 0 -3 24 -52 }}
-Mapping: {{mapping| 3 0 45 94 8 42 | 0 2 -16 -36 1 -13 }}
+[[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 (POTE): ~63/50 = 1\3, ~26/15 = 950.873 (~12/11 = 150.873)
+{{Optimal ET sequence|legend=1| 65, 94, 159, 253, 412cd }}
-{{Optimal ET sequence|legend=1| 24d, 159, 183, 342f }}
+[[Badness]] (Sintel): 5.36
-Badness: 0.031362
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-==== 17-limit ====
+Comma list: 385/384, 4000/3993, 19712/19683
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264
+Mapping: {{mapping| 1 2 -1 10 0 | 0 -3 24 -52 25 }}
-Mapping: {{mapping| 3 0 45 94 8 42 -2 | 0 2 -16 -36 1 -13 6 }}
+Optimal tunings:
+* WE: ~2 = 1200.3379{{c}}, ~11/10 = 166.0638{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0167{{c}}
-Optimal tuning (POTE): ~34/27 = 1\3, ~26/15 = 950.867 (~12/11 = 150.867)
+{{Optimal ET sequence|legend=0| 65, 94, 159, 253, 412cd, 665ccde }}
-{{Optimal ET sequence|legend=1| 24d, 159, 183, 342f, 525f, 867ff }}
+Badness (Sintel): 2.07
-Badness: 0.022316
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-== Altinex ==
+Comma list: 325/324, 385/384, 1575/1573, 10985/10976
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 32805/32768, 367653125/362797056
+Mapping: {{mapping| 1 2 -1 10 0 12 | 0 -3 24 -52 25 -60 }}
-{{Mapping|legend=1| 3 0 45 -32 | 0 2 -16 17 }}
+Optimal tunings:
+* WE: ~2 = 1200.3467{{c}}, ~11/10 = 166.0635{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0142{{c}}
-: mapping generators: ~1536/1225, ~34300/19683
+{{Optimal ET sequence|legend=0| 65f, 94, 159, 253, 412cdf, 665ccdef }}
-[[Optimal tuning]] ([[CTE]]): ~1536/1225 = 1\3, ~34300/19683 = 950.9654
+Badness (Sintel): 1.52
-{{Optimal ET sequence|legend=1| 24, …, 111c, 135, 159, 612ccdd, 771ccdd }}
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
-[[Badness]]: 0.422026
+Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
-=== 11-limit ===
+Mapping: {{mapping| 1 2 -1 10 0 12 -2 | 0 -3 24 -52 25 -60 44 }}
-Subgroup: 2.3.5.7.11
-Comma list: 385/384, 14700/14641, 19712/19683
+Optimal tunings:
+* WE: ~2 = 1200.3019{{c}}, ~11/10 = 166.0535{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0114{{c}}
-Mapping: {{mapping| 3 0 45 -32 8 | 0 2 -16 17 1 }}
+{{Optimal ET sequence|legend=1| 65f, 94, 159, 253 }}
-Optimal tuning (CTE): ~44/35 = 1\3, ~121/70 = 950.9658
+Badness (Sintel): 1.35
-Optimal ET sequence: {{Optimal ET sequence| 24, …, 111c, 135, 159, 612ccdd, 771ccdd }}
+== 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: 0.101224
+[[Subgroup]]: 2.3.5.7
-=== 13-limit ===
+[[Comma list]]: 32805/32768, 244140625/243045684
-Subgroup: 2.3.5.7.11.13
-Comma list: 364/363, 385/384, 676/675, 19712/19683
+{{Mapping|legend=1| 1 2 -1 -12 | 0 -3 24 107 }}
-Mapping: {{mapping| 3 0 45 -32 8 42 | 0 2 -16 17 1 -13 }}
+[[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 (CTE): ~44/35 = 1\3, ~26/15 = 950.9360
+{{Optimal ET sequence|legend=1| 65d, 159, 224, 383, 607 }}
-Optimal ET sequence: {{Optimal ET sequence| 24, …, 111cf, 135f, 159 }}
+[[Badness]] (Sintel): 4.76
-Badness: 0.054894
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-== Sesquiquartififths ==
+Comma list: 3025/3024, 4000/3993, 32805/32768
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 2401/2400, 32805/32768
+Mapping: {{mapping| 1 2 -1 -12 0 | 0 -3 24 107 25 }}
-{{Mapping|legend=1| 1 1 7 5 | 0 4 -32 -15 }}
+Optimal tunings:
+* WE: ~2 = 1200.0804{{c}}, ~11/10 = 166.0739{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0634{{c}}
-: mapping generators: ~2, ~448/405
+{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}
-{{Multival|legend=1| 4 -32 -15 -60 -35 55 }}
+Badness (Sintel): 1.62
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~448/405 = 175.434
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-[[Minimax tuning]]:
+Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976
-* [[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 }}
+Mapping: {{mapping| 1 2 -1 -12 0 -10 | 0 -3 24 107 25 99 }}
-[[Badness]]: 0.011244
+Optimal tunings:
+* WE: ~2 = 1200.0805{{c}}, ~11/10 = 166.0740{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0635{{c}}
-=== Sesquart ===
+{{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }}
-Subgroup: 2.3.5.7.11
-Comma list: 243/242, 441/440, 16384/16335
+Badness (Sintel): 1.01
-Mapping: {{mapping| 1 1 7 5 2 | 0 4 -32 -15 10 }}
+== 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.
-Optimal tuning (POTE): ~2 = 1\1, ~256/231 = 175.406
+[[Subgroup]]: 2.3.5.7
-Optimal ET sequence: {{Optimal ET sequence| 41, 89, 130, 301e, 431e }}
+[[Comma list]]: 32805/32768, 390625/388962
-Badness: 0.029306
+{{Mapping|legend=1| 4 0 60 119 | 0 1 -8 -17 }}
+: mapping generators: ~25/21, ~3
-==== 13-limit ====
+[[Optimal tuning]]s:
-Subgroup: 2.3.5.7.11.13
+* [[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 }}
-Comma list: 243/242, 364/363, 441/440, 3584/3575
+{{Optimal ET sequence|legend=1| 12, …, 200, 212, 224, 436, 660 }}
-Mapping: {{mapping| 1 1 7 5 2 -2 | 0 4 -32 -15 10 39 }}
+[[Badness]] (Sintel): 2.79
-Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.409
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Optimal ET sequence: {{Optimal ET sequence| 41, 89, 130, 301e, 431e }}
+Comma list: 1375/1372, 6250/6237, 32805/32768
-Badness: 0.022396
+Mapping: {{mapping| 4 0 60 119 185 | 0 1 -8 -17 -27 }}
-===== Sesquartia =====
+Optimal tunings:
-Subgroup: 2.3.5.7.11.13.17
+* WE: ~25/21 = 300.0244{{c}}, ~3/2 = 701.8759{{c}}
+* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8145{{c}}
-Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575
+{{Optimal ET sequence|legend=0| 12, …, 212, 224, 436, 660 }}
-Mapping: {{mapping| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }}
+Badness (Sintel): 1.51
-Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.424
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Optimal ET sequence: {{Optimal ET sequence| 41, 89g, 130, 171, 301e }}
+Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647
-Badness: 0.023126
+Mapping: {{mapping| 4 0 60 119 185 224 | 0 1 -8 -17 -27 -33 }}
-====== 19-limit ======
+Optimal tunings:
-Subgroup: 2.3.5.7.11.13.17.19
+* WE: ~25/21 = 300.0234{{c}}, ~3/2 = 701.8707{{c}}
+* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8123{{c}}
-Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594
+{{Optimal ET sequence|legend=0| 12f, …, 212, 224, 436, 660 }}
-Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }}
+Badness (Sintel): 1.13
-Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.419
+== Sesquiquartififths ==
+Sesquiquartififths tempers out 2401/2400, the [[breedsma]], and may be described as the {{nowrap| 41 & 171 }} temperament. It splits the fifth into four; its [[ploidacot]] is thus tetracot.
-Optimal ET sequence: {{Optimal ET sequence| 41, 89g, 130, 171, 301eh }}
+[[Subgroup]]: 2.3.5.7
-Badness: 0.020466
+[[Comma list]]: 2401/2400, 32805/32768
-====== 23-limit ======
+{{Mapping|legend=1| 1 1 7 5 | 0 4 -32 -15 }}
-Subgroup: 2.3.5.7.11.13.17.19.23
+: mapping generators: ~2, ~448/405
-Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594
+[[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 }}
-Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 -6 | 0 4 -32 -15 10 39 69 -12 72 }}
+[[Minimax tuning]]:
+* [[7-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
+* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7
-Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.412
+{{Optimal ET sequence|legend=1| 41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd }}
-Optimal ET sequence: {{Optimal ET sequence| 41i, 89gi, 130, 171, 301eh }}
+[[Badness]] (Sintel): 0.285
-Badness: 0.019043
+=== 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.
-===== Heartia =====
+Subgroup: 2.3.5.7.11
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 243/242, 256/255, 273/272, 364/363, 441/440
+Comma list: 243/242, 441/440, 16384/16335
-Mapping: {{mapping| 1 1 7 5 2 -2 0 | 0 4 -32 -15 10 39 28 }}
+Mapping: {{mapping| 1 1 7 5 2 | 0 4 -32 -15 10 }}
-Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.386
+Optimal tunings:
+* WE: ~2 = 1199.8171{{c}}, ~256/231 = 175.3793{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~256/231 = 175.4081{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 41, 89, 130g }}
+{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}
-Badness: 0.028443
+Badness (Sintel): 0.969
-====== 19-limit ======
+==== 13-limit ====
-Subgroup: 2.3.5.7.11.13.17.19
+Subgroup: 2.3.5.7.11.13
-Comma list: 171/170, 243/242, 256/255, 273/272, 324/323, 441/440
+Comma list: 243/242, 364/363, 441/440, 3584/3575
-Mapping: {{mapping| 1 1 7 5 2 -2 0 6 | 0 4 -32 -15 10 39 28 -12 }}
+Mapping: {{mapping| 1 1 7 5 2 -2 | 0 4 -32 -15 10 39 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.380
+Optimal tunings:
+* WE: ~2 = 1199.8352{{c}}, ~72/65 = 175.3852{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4095{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 41, 89, 130g }}
+{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }}
-Badness: 0.023059
+Badness (Sintel): 0.925
-===== Hearty =====
+===== Heartia =====
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625
+Comma list: 243/242, 256/255, 273/272, 364/363, 441/440
-Mapping: {{mapping| 1 1 7 5 2 -2 13 | 0 4 -32 -15 10 39 -61 }}
+Mapping: {{mapping| 1 1 7 5 2 -2 0 | 0 4 -32 -15 10 39 28 }}
-Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.377
+Optimal tunings:
+* WE: ~2 = 1199.6422{{c}}, ~72/65 = 175.3338{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3857{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 41g, 89, 130, 609ceefgg }}
+{{Optimal ET sequence|legend=0| 41, 89, 130g }}
-Badness: 0.030680
+Badness (Sintel): 1.45
 ====== 19-limit ======
 Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 221/220, 243/242, 361/360, 364/363, 441/440, 456/455
+Comma list: 171/170, 243/242, 256/255, 273/272, 324/323, 441/440
-Mapping: {{mapping| 1 1 7 5 2 -2 13 6 | 0 4 -32 -15 10 39 -61 -12 }}
+Mapping: {{mapping| 1 1 7 5 2 -2 0 6 | 0 4 -32 -15 10 39 28 -12 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.377
+Optimal tunings:
+* WE: ~2 = 1199.7499{{c}}, ~21/19 = 175.3432{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.3797{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 41g, 89, 130, 609ceefggh }}
+{{Optimal ET sequence|legend=0| 41, 89, 130g }}
-Badness: 0.022816
+Badness (Sintel): 1.40
-====== 23-limit ======
+===== Sesquartia =====
-Subgroup: 2.3.5.7.11.13.17.19.23
+Subgroup: 2.3.5.7.11.13.17
-Comma list: 221/220, 243/242, 276/275, 323/322, 361/360, 364/363, 441/440
+Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575
-Mapping: {{mapping| 1 1 7 5 2 -2 13 6 13 | 0 4 -32 -15 10 39 -61 -12 -58 }}
+Mapping: {{mapping| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.376
+Optimal tunings:
+* WE: ~2 = 1199.8902{{c}}, ~72/65 = 175.4077{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4234{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 41g, 89, 130, 609ceefggh }}
+{{Optimal ET sequence|legend=0| 41, 130, 171 }}
-Badness: 0.019121
+Badness (Sintel): 1.18
-=== Bisesqui ===
+====== 19-limit ======
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 2401/2400, 9801/9800, 32805/32768
+Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594
-Mapping: {{mapping| 2 2 14 10 23 | 0 4 -32 -15 -55 }}
+Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }}
-: mapping generators: ~99/70, ~448/405
+Optimal tunings:
+* WE: ~2 = 1199.9864{{c}}, ~21/19 = 175.4169{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4189{{c}}
-Optimal tuning (POTE): ~99/70 = 1\2, ~448/405 = 175.435
+{{Optimal ET sequence|legend=0| 41, 130, 171 }}
-{{Optimal ET sequence|legend=1| 82e, 130, 212, 342, 1156, 1498, 1840d }}
+Badness (Sintel): 1.24
-Badness: 0.016968
+====== 23-limit ======
+Subgroup: 2.3.5.7.11.13.17.19.23
-== Quintilipyth ==
+Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594
-The quintilipyth temperament (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
+Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 -6 | 0 4 -32 -15 10 39 69 -12 72 }}
-[[Comma list]]: 32805/32768, 9765625/9680832
+Optimal tunings:
+* WE: ~2 = 1199.9606{{c}}, ~21/19 = 175.4067{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4123{{c}}
-{{Mapping|legend=1| 1 2 -1 -4 | 0 -5 40 82 }}
+{{Optimal ET sequence|legend=0| 41i, 130, 171 }}
-{{Multival|legend=1| 5 -40 -82 -75 -144 -78 }}
+Badness (Sintel): 1.36
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~625/588 = 99.625
+===== Hearty =====
+Subgroup: 2.3.5.7.11.13.17
-{{Optimal ET sequence|legend=1| 12, 253, 265 }}
+Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625
-[[Badness]]: 0.253966
+Mapping: {{mapping| 1 1 7 5 2 -2 13 | 0 4 -32 -15 10 39 -61 }}
-=== 11-limit ===
+Optimal tunings:
-Subgroup: 2.3.5.7.11
+* WE: ~2 = 1199.9458{{c}}, ~72/65 = 175.3689{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3770{{c}}
-Comma list: 1375/1372, 4375/4356, 32805/32768
+{{Optimal ET sequence|legend=0| 41g, 89, 130 }}
-Mapping: {{mapping| 1 2 -1 -4 -7 | 0 -5 40 82 126 }}
+Badness (Sintel): 1.56
-Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.616
+====== 19-limit ======
+Subgroup: 2.3.5.7.11.13.17.19
-{{Optimal ET sequence|legend=1| 12, 253, 265, 518c, 783cc }}
+Comma list: 221/220, 243/242, 361/360, 364/363, 441/440, 456/455
-Badness: 0.113044
+Mapping: {{mapping| 1 1 7 5 2 -2 13 6 | 0 4 -32 -15 10 39 -61 -12 }}
-=== 13-limit ===
+Optimal tunings:
-Subgroup: 2.3.5.7.11.13
+* WE: ~2 = 1200.0114{{c}}, ~72/65 = 175.3783{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3765{{c}}
-Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647
+{{Optimal ET sequence|legend=0| 41g, 89, 130 }}
-Mapping: {{mapping| 1 2 -1 -4 -7 -9 | 0 -5 40 82 126 153 }}
+Badness (Sintel): 1.39
-Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.612
+====== 23-limit ======
+Subgroup: 2.3.5.7.11.13.17.19.23
-{{Optimal ET sequence|legend=1| 12f, 253, 518c, 771cc }}
+Comma list: 221/220, 243/242, 276/275, 323/322, 361/360, 364/363, 441/440
-Badness: 0.069127
+Mapping: {{mapping| 1 1 7 5 2 -2 13 6 13 | 0 4 -32 -15 10 39 -61 -12 -58 }}
-=== 17-limit ===
+Optimal tunings:
-Subgroup: 2.3.5.7.11.13.17
+* WE: ~2 = 1200.0122{{c}}, ~72/65 = 175.3782{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3763{{c}}
-Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619
+{{Optimal ET sequence|legend=0| 41g, 89, 130 }}
-Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 | 0 -5 40 82 126 153 -11 }}
+Badness (Sintel): 1.37
-Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.612
+=== Bisesqui ===
+Subgroup: 2.3.5.7.11
-{{Optimal ET sequence|legend=1| 12f, 253, 518c, 771cc }}
+Comma list: 2401/2400, 9801/9800, 32805/32768
-Badness: 0.045992
+Mapping: {{mapping| 2 2 14 10 23 | 0 4 -32 -15 -55 }}
+: mapping generators: ~99/70, ~448/405
-=== 19-limit ===
+Optimal tunings:
-Subgroup: 2.3.5.7.11.13.17.19
+* WE: ~99/70 = 600.0429{{c}}, ~448/405 = 175.4474{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~448/405 = 175.4334{{c}}
-Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971
+{{Optimal ET sequence|legend=1| 82e, 130, 212, 342, 1156, 1498, 1840d, 5862bbccdddee }}
-Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 4 | 0 -5 40 82 126 153 -11 3 }}
+Badness (Sintel): 0.561
-Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.615
+== Tsaharuk ==
+{{Main| Tsaharuk }}
-{{Optimal ET sequence|legend=1| 12f, 253, 265, 518ch }}
+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.
-Badness: 0.038155
-== Quintaschis ==
-The quintaschis temperament (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
-[[Comma list]]: 32805/32768, 49009212/48828125
+[[Comma list]]: 32805/32768, 420175/419904
-{{Mapping|legend=1| 1 2 -1 -5 | 0 -5 40 94 }}
+{{Mapping|legend=1| 1 1 7 0 | 0 5 -40 24 }}
+: mapping generators: ~2, ~243/224
-{{Multival|legend=1| 5 -40 -94 -75 -163 -106 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1039{{c}}, ~243/224 = 140.3620{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~200/189 = 99.664
+: [[error map]]: {{val| +0.104 -0.041 -0.067 -0.137 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/224 = 140.3496{{c}}
+: error map: {{val| 0.000 -0.207 -0.296 -0.436 }}
-{{Optimal ET sequence|legend=1| 12, …, 289, 301, 590, 891, 1192 }}
+{{Optimal ET sequence|legend=1| 17, 77, 94, 171 }}
-[[Badness]]: 0.132890
+[[Badness]] (Sintel): 0.777
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 441/440, 32805/32768, 1953125/1951488
+Comma list: 385/384, 1331/1323, 19712/19683
-Mapping: {{mapping| 1 2 -1 -5 -8 | 0 -5 40 94 138 }}
+Mapping: {{mapping| 1 1 7 0 1 | 0 5 -40 24 21 }}
-Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.653
+Optimal tunings:
+* WE: ~2 = 1200.3103{{c}}, ~88/81 = 140.4011{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.3649{{c}}
-{{Optimal ET sequence|legend=1| 12, …, 277d, 289 }}
+{{Optimal ET sequence|legend=0| 17, 77, 94, 171e, 265e }}
-Badness: 0.111477
+Badness (Sintel): 2.10
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 364/363, 441/440, 32805/32768, 109512/109375
+Comma list: 352/351, 385/384, 729/728, 1331/1323
-Mapping: {{mapping| 1 2 -1 -5 -8 -11 | 0 -5 40 94 138 177 }}
+Mapping: {{mapping| 1 1 7 0 1 3 | 0 5 -40 24 21 6 }}
-Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.658
+Optimal tunings:
+* WE: ~2 = 1200.1840{{c}}, ~13/12 = 140.3840{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.3627{{c}}
-{{Optimal ET sequence|legend=1| 12f, …, 277dff, 289 }}
+{{Optimal ET sequence|legend=0| 17, 77, 94, 171e }}
-Badness: 0.074218
+Badness (Sintel): 1.57
-==== 17-limit ====
+== Quanharuk ==
-Subgroup: 2.3.5.7.11.13.17
+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.
-Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 | 0 -5 40 94 138 177 -11 }}
+[[Comma list]]: 16875/16807, 32805/32768
-Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.656
+{{Mapping|legend=1| 1 0 15 12 | 0 5 -40 -29 }}
+: mapping generators: ~2, ~56/45
-{{Optimal ET sequence|legend=1| 12f, 277dff, 289 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0032{{c}}, ~56/45 = 380.3557{{c}}
+: [[error map]]: {{val| +0.003 -0.177 -0.493 +0.898 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~56/45 = 380.3546{{c}}
+: error map: {{val| 0.000 -0.182 -0.498 +0.890 }}
-Badness: 0.050571
+{{Optimal ET sequence|legend=1| 41, 142, 183, 224 }}
-==== 19-limit ====
+[[Badness]] (Sintel): 1.82
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 4 | 0 -5 40 94 138 177 -11 3 }}
+Comma list: 540/539, 1375/1372, 32805/32768
-Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.659
+Mapping: {{mapping| 1 0 15 12 -7 | 0 5 -40 -29 33 }}
-{{Optimal ET sequence|legend=1| 12f, 289 }}
+Optimal tunings:
+* WE: ~2 = 1199.9709{{c}}, ~56/45 = 380.3423{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3517{{c}}
-Badness: 0.042120
+{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}
-=== Quintahelenic ===
+Badness (Sintel): 1.04
-Subgroup: 2.3.5.7.11
-Comma list: 5632/5625, 8019/8000, 151263/151250
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Mapping: {{mapping| 1 2 -1 -5 -9 | 0 -5 40 94 150 }}
+Comma list: 540/539, 729/728, 1375/1372, 4096/4095
-Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 99.671
+Mapping: {{mapping| 1 0 15 12 -7 -15 | 0 5 -40 -29 33 59 }}
-{{Optimal ET sequence|legend=1| 12, …, 289e, 301, 915 }}
+Optimal tunings:
+* WE: ~2 = 1199.9663{{c}}, ~56/45 = 380.3403{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3509{{c}}
-Badness: 0.082225
+{{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }}
-==== 13-limit ====
+Badness (Sintel): 0.884
-Subgroup: 2.3.5.7.11.13
-Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000
+== 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.
-Mapping: {{mapping| 1 2 -1 -5 -9 -11 | 0 -5 40 94 150 177 }}
+[[Subgroup]]: 2.3.5.7
-Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 99.661
+[[Comma list]]: 32805/32768, 9765625/9680832
-{{Optimal ET sequence|legend=1| 12f, …, 289e, 301 }}
+{{Mapping|legend=1| 1 2 -1 -4 | 0 -5 40 82 }}
+: mapping generators: ~2, ~625/588
-Badness: 0.055570
+[[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 }}
-===== 17-limit =====
+{{Optimal ET sequence|legend=1| 12, …, 253, 265 }}
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750
+[[Badness]] (Sintel): 6.43
-Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 | 0 -5 40 94 150 177 -11 }}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.665
+Comma list: 1375/1372, 4375/4356, 32805/32768
-{{Optimal ET sequence|legend=1| 12f, 289e, 301 }}
+Mapping: {{mapping| 1 2 -1 -4 -7 | 0 -5 40 82 126 }}
-Badness: 0.040412
+Optimal tunings:
+* WE: ~2 = 1200.1503{{c}}, ~35/33 = 99.6287{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6176{{c}}
-===== 19-limit =====
+{{Optimal ET sequence|legend=0| 12, …, 253, 265, 518c }}
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700
+Badness (Sintel): 3.74
-Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 4 | 0 -5 40 94 150 177 -11 3 }}
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.668
+Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647
-{{Optimal ET sequence|legend=1| 12f, 301 }}
+Mapping: {{mapping| 1 2 -1 -4 -7 -9 | 0 -5 40 82 126 153 }}
-Badness: 0.036840
+Optimal tunings:
+* WE: ~2 = 1200.1774{{c}}, ~35/33 = 99.6267{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6134{{c}}
-==== Quintahelenoid ====
+{{Optimal ET sequence|legend=0| 12f, …, 241cdef, 253 }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436
+Badness (Sintel): 2.86
-Mapping: {{mapping| 1 2 -1 -5 -9 14 | 0 -5 40 94 150 -124 }}
+=== 17-limit ===
-Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 99.672
-{{Optimal ET sequence|legend=1| 12, 301, 614, 915 }}
-Badness: 0.066108
-===== 17-limit =====
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157
+Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619
-Mapping: {{mapping| 1 2 -1 -5 -9 14 5 | 0 -5 40 94 150 -124 -11 }}
+Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 | 0 -5 40 82 126 153 -11 }}
-Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.671
+Optimal tunings:
+* WE: ~2 = 1200.1745{{c}}, ~18/17 = 99.6265{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6131{{c}}
-{{Optimal ET sequence|legend=1| 12, 301 }}
+{{Optimal ET sequence|legend=0| 12f, 241cdef, 253 }}
-Badness: 0.047908
+Badness (Sintel): 2.34
-===== 19-limit =====
+=== 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
+Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971
-Mapping: {{mapping| 1 2 -1 -5 -9 14 5 4 | 0 -5 40 94 150 -124 -11 3 }}
+Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 4 | 0 -5 40 82 126 153 -11 3 }}
-Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.672
+Optimal tunings:
+* WE: ~2 = 1200.0713{{c}}, ~18/17 = 99.6208{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6152{{c}}
-{{Optimal ET sequence|legend=1| 12, 301 }}
+{{Optimal ET sequence|legend=0| 12f, 253, 265 }}
-Badness: 0.039542
+Badness (Sintel): 2.32
-== Sextilififths ==
+== Quintaschis ==
-The sextilififths (130 &amp; 159, also known as ''sextilischis'') slices the fourth (4/3) into six small semitones, which serves as both 21/20 and 22/21.
+Named by [[Xenllium]] in 2021, quintaschis slices the [[4/3|perfect fourth]] into five semitones and tempers out 49009212/48828125 in the [[7-limit]]. It may be described as the {{nowrap| 12 & 289 }} temperament, and its [[ploidacot]] is omega-pentacot.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 32805/32768, 235298/234375
+[[Comma list]]: 32805/32768, 49009212/48828125
-{{Mapping|legend=1| 1 2 -1 -1 | 0 -6 48 55 }}
+{{Mapping|legend=1| 1 2 -1 -5 | 0 -5 40 94 }}
-: mapping generators: ~2, ~21/20
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0536{{c}}, ~200/189 = 99.6684{{c}}
+: [[error map]]: {{val| +0.054 -0.190 +0.370 -0.262 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~200/189 = 99.6645{{c}}
+: error map: {{val| 0.000 -0.277 +0.266 -0.363 }}
-{{Multival|legend=1| 6 -48 -55 -90 -104 7 }}
+{{Optimal ET sequence|legend=1| 12, …, 289, 301, 590, 891, 1192 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 83.053
+[[Badness]] (Sintel): 3.36
-{{Optimal ET sequence|legend=1| 29, 72cd, 101, 130, 289, 419 }}
-[[Badness]]: 0.108794
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 441/440, 4000/3993, 235298/234375
+Comma list: 441/440, 32805/32768, 1953125/1951488
-Mapping: {{mapping| 1 2 -1 -1 0 | 0 -6 48 55 50 }}
+Mapping: {{mapping| 1 2 -1 -5 -8 | 0 -5 40 94 138 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.049
+Optimal tunings:
+* WE: ~2 = 1200.0988{{c}}, ~35/33 = 99.6613{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6540{{c}}
-{{Optimal ET sequence|legend=1| 29, 72cde, 101e, 130, 289 }}
+{{Optimal ET sequence|legend=0| 12, …, 277d, 289 }}
-Badness: 0.045457
+Badness (Sintel): 3.69
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 364/363, 441/440, 676/675, 10985/10976
+Comma list: 364/363, 441/440, 32805/32768, 109512/109375
-Mapping: {{mapping| 1 2 -1 -1 0 1 | 0 -6 48 55 50 39 }}
+Mapping: {{mapping| 1 2 -1 -5 -8 -11 | 0 -5 40 94 138 177 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.049
+Optimal tunings:
+* WE: ~2 = 1200.0625{{c}}, ~35/33 = 99.6630{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6583{{c}}
-{{Optimal ET sequence|legend=1| 29, 72cdef, 101e, 130, 289 }}
+{{Optimal ET sequence|legend=0| 12f, …, 277dff, 289 }}
-Badness: 0.025276
+Badness (Sintel): 3.07
-== Septiquarschis ==
+==== 17-limit ====
-The septiquarschis temperament (89 &amp; 94) splits septimal minor seventh ([[7/4]]) into four generators and tempers out 829440/823543 (mynaslender comma) and 67108864/66706983 (septiness comma).
+Subgroup: 2.3.5.7.11.13.17
-[[Subgroup]]: 2.3.5.7
+Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768
-[[Comma list]]: 32805/32768, 829440/823543
+Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 | 0 -5 40 94 138 177 -11 }}
-{{Mapping|legend=1| 1 3 -9 2 | 0 -7 -56 4 }}
+Optimal tunings:
+* WE: ~2 = 1200.1286{{c}}, ~18/17 = 99.6668{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6568{{c}}
-{{Multival|legend=1| 7 56 -4 231 -26 -76 }}
+{{Optimal ET sequence|legend=0| 12f, 277dff, 289 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~147/128 = 242.614
+Badness (Sintel): 2.58
-{{Optimal ET sequence|legend=1| 89, 94, 183, 460d, 643d, 1103dd }}
+==== 19-limit ====
+Subgroup: 2.3.5.7.11.13.17.19
-[[Badness]]: 0.187047
+Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859
-=== 11-limit ===
+Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 4 | 0 -5 40 94 138 177 -11 3 }}
-Subgroup: 2.3.5.7.11
-Comma list: 540/539, 15488/15435, 32805/32768
+Optimal tunings:
+* WE: ~2 = 1200.0289{{c}}, ~18/17 = 99.6609{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6586{{c}}
-Mapping: {{mapping| 1 3 -9 2 -2 | 0 -7 -56 4 27 }}
+{{Optimal ET sequence|legend=0| 12f, 289 }}
-Optimal tuning (POTE): ~2 = 1\1, ~147/128 = 242.616
+Badness (Sintel): 2.56
-{{Optimal ET sequence|legend=1| 89, 94, 183, 460d, 643d, 826dd }}
+=== Quintahelenic ===
+Subgroup: 2.3.5.7.11
-Badness: 0.052002
+Comma list: 5632/5625, 8019/8000, 151263/151250
-=== 13-limit ===
+Mapping: {{mapping| 1 2 -1 -5 -9 | 0 -5 40 94 150 }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 729/728, 1573/1568, 4096/4095
+Optimal tunings:
+* WE: ~2 = 1200.0195{{c}}, ~200/189 = 99.6723{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6709{{c}}
-Mapping: {{mapping| 1 3 -9 2 -2 13 | 0 -7 -56 4 27 -46 }}
+{{Optimal ET sequence|legend=0| 12, …, 289e, 301, 915 }}
-Optimal tuning (POTE): ~2 = 1\1, ~147/128 = 242.610
+Badness (Sintel): 2.72
-{{Optimal ET sequence|legend=1| 89, 94, 183, 277, 460d }}
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Badness: 0.035315
+Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000
-== Tsaharuk ==
+Mapping: {{mapping| 1 2 -1 -5 -9 -11 | 0 -5 40 94 150 177 }}
-{{Main| Tsaharuk }}
-[[Subgroup]]: 2.3.5.7
+Optimal tunings:
+* WE: ~2 = 1200.0442{{c}}, ~200/189 = 99.6709{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6675{{c}}
-[[Comma list]]: 32805/32768, 420175/419904
+{{Optimal ET sequence|legend=0| 12f, …, 289e, 301 }}
-{{Mapping|legend=1| 1 1 7 0 | 0 5 -40 24 }}
+Badness (Sintel): 2.30
-: mapping generators: ~2, ~243/224
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-{{Multival|legend=1| 5 -40 24 -75 24 168 }}
+Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/224 = 140.350
+Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 | 0 -5 40 94 150 177 -11 }}
-{{Optimal ET sequence|legend=1| 17, 60c, 77, 94, 171 }}
+Optimal tunings:
+* WE: ~2 = 1200.1227{{c}}, ~200/189 = 99.6753{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6658{{c}}
-[[Badness]]: 0.030697
+{{Optimal ET sequence|legend=1| 12f, 289e, 301 }}
-=== 11-limit ===
+Badness (Sintel): 2.06
-Subgroup: 2.3.5.7.11
-Comma list: 385/384, 1331/1323, 19712/19683
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700
-Mapping: {{mapping| 1 1 7 0 1 | 0 5 -40 24 21 }}
+Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 4 | 0 -5 40 94 150 177 -11 3 }}
-Optimal tuning (POTE): ~2 = 1\1, ~88/81 = 140.365
+Optimal tunings:
+* WE: ~2 = 1200.0230{{c}}, ~200/189 = 99.6694{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6676{{c}}
-{{Optimal ET sequence|legend=1| 17, 60ce, 77, 94, 171e, 265e, 436ee }}
+{{Optimal ET sequence|legend=0| 12f, 301 }}
-Badness: 0.063499
+Badness (Sintel): 2.24
-=== 13-limit ===
+==== Quintahelenoid ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 352/351, 385/384, 729/728, 1331/1323
+Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436
-Mapping: {{mapping| 1 1 7 0 1 3 | 0 5 -40 24 21 6 }}
+Mapping: {{mapping| 1 2 -1 -5 -9 14 | 0 -5 40 94 150 -124 }}
-Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.363
+Optimal tunings:
+* WE: ~2 = 1199.9919{{c}}, ~200/189 = 99.6712{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6718{{c}}
-{{Optimal ET sequence|legend=1| 17, 60ce, 77, 94, 171e, 436ee }}
+{{Optimal ET sequence|legend=0| 12, 301, 614, 915 }}
-Badness: 0.037886
+Badness (Sintel): 2.73
-== Quanharuk ==
+===== 17-limit =====
-[[Subgroup]]: 2.3.5.7
+Subgroup: 2.3.5.7.11.13.17
-[[Comma list]]: 16875/16807, 32805/32768
+Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157
-{{Mapping|legend=1| 1 0 15 12 | 0 5 -40 -29 }}
+Mapping: {{mapping| 1 2 -1 -5 -9 14 5 | 0 -5 40 94 150 -124 -11 }}
-: mapping generators: ~2, ~56/45
+Optimal tunings:
+* WE: ~2 = 1200.0469{{c}}, ~18/17 = 99.6749{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6710{{c}}
-{{Multival|legend=1| 5 -40 -29 -75 -60 45 }}
+{{Optimal ET sequence|legend=0| 12, 301 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~56/45 = 380.355
+Badness (Sintel): 2.44
-{{Optimal ET sequence|legend=1| 41, 142, 183, 224, 1303d, 1527cd, 1751cd, 1975cd }}
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-[[Badness]]: 0.071950
+Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137
-=== 11-limit ===
+Mapping: {{mapping| 1 2 -1 -5 -9 14 5 4 | 0 -5 40 94 150 -124 -11 3 }}
-Subgroup: 2.3.5.7.11
-Comma list: 540/539, 1375/1372, 32805/32768
+Optimal tunings:
+* WE: ~2 = 1199.9925{{c}}, ~18/17 = 99.6710{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6716{{c}}
-Mapping: {{mapping| 1 0 15 12 -7 | 0 5 -40 -29 33 }}
+{{Optimal ET sequence|legend=0| 12, 301 }}
-Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 380.352
+Badness (Sintel): 2.41
-{{Optimal ET sequence|legend=1| 41, 142, 183, 224, 631d, 855d, 1079d }}
+== Sextilifourths ==
+Named by [[Xenllium]] in 2021, sextilifourths (also known as ''sextilischis'', formerly ''sextilififths'') slices the [[4/3|perfect fourth]] into six small semitones, which serves as both [[21/20]] and [[22/21]]. It may be described as {{nowrap| 130 & 159 }}, and its [[ploidacot]] is omega-hexacot. [[289edo]] gives a highly recommendable tuning.
-Badness: 0.031549
+[[Subgroup]]: 2.3.5.7
-=== 13-limit ===
+[[Comma list]]: 32805/32768, 235298/234375
-Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 729/728, 1375/1372, 4096/4095
+{{Mapping|legend=1| 1 2 -1 -1 | 0 -6 48 55 }}
+: mapping generators: ~2, ~21/20
-Mapping: {{mapping| 1 0 15 12 -7 -15 | 0 5 -40 -29 33 59 }}
+[[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 tuning (POTE): ~2 = 1\1, ~56/45 = 380.351
+{{Optimal ET sequence|legend=1| 29, 72cd, 101, 130, 289, 419 }}
-{{Optimal ET sequence|legend=1| 41, 142, 183, 224, 631d, 855d }}
+[[Badness]] (Sintel): 2.75
-Badness: 0.021392
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-== Quadrant ==
+Comma list: 441/440, 4000/3993, 235298/234375
-The ''quadrant'' temperament (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 -1 0 | 0 -6 48 55 50 }}
-[[Comma list]]: 32805/32768, 390625/388962
+Optimal tunings:
+* WE: ~2 = 1200.0424{{c}}, ~21/20 = 83.0520{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0497{{c}}
-{{Mapping|legend=1| 4 0 60 119 | 0 1 -8 -17 }}
+{{Optimal ET sequence|legend=0| 29, 72cde, 101e, 130, 289 }}
-: mapping generators: ~25/21, ~3
+Badness (Sintel): 1.50
-{{Multival|legend=1| 4 -32 -68 -60 -119 -68 }}
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-[[Optimal tuning]] ([[POTE]]): ~25/21 = 1\4, ~3/2 = 701.8234
+Comma list: 364/363, 441/440, 676/675, 10985/10976
-{{Optimal ET sequence|legend=1| 212, 224, 436, 660, 1096c }}
+Mapping: {{mapping| 1 2 -1 -1 0 1 | 0 -6 48 55 50 39 }}
-[[Badness]]: 0.110242
+Optimal tunings:
+* WE: ~2 = 1200.1056{{c}}, ~21/20 = 83.0566{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0508{{c}}
-=== 11-limit ===
+{{Optimal ET sequence|legend=0| 29, 72cdef, 101e, 130, 289 }}
-Subgroup: 2.3.5.7.11
-Comma list: 1375/1372, 6250/6237, 32805/32768
+Badness (Sintel): 1.04
-Mapping: {{mapping| 4 0 60 119 185 | 0 1 -8 -17 -27 }}
-Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 701.8176
-{{Optimal ET sequence|legend=1| 212, 224, 436, 660 }}
-Badness: 0.045738
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647
-Mapping: {{mapping| 4 0 60 119 185 224 | 0 1 -8 -17 -27 -33 }}
-Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 701.8158
-{{Optimal ET sequence|legend=1| 212, 224, 436, 660 }}
-Badness: 0.027243
 == Septant ==
-The ''septant'' temperament (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
@@ Line 2,305: / Line 2,553: @@
 {{Mapping|legend=1| 7 0 105 -56 | 0 1 -8 7 }}
 : mapping generators: ~8575/7776, ~3
-{{Multival|legend=1| 7 -56 49 -105 58 271 }}
+[[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 }}
-[[Badness]]: 0.111142
+[[Badness]] (Sintel): 2.81
 === 11-limit ===
@@ Line 2,323: / Line 2,572: @@
 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 ===
@@ Line 2,336: / Line 2,587: @@
 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 ==
-The octant temperament (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
@@ Line 2,350: / Line 2,603: @@
 {{Mapping|legend=1| 8 0 120 -117 | 0 1 -8 11 }}
 : mapping generators: ~42875/39366, ~3
-{{Multival|legend=1| 8 -64 88 -120 117 384 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~42875/39366 = 150.0048{{c}}, ~3/2 = 701.7356{{c}}
+: [[error map]]: {{val| +0.039 -0.181 +0.071 +0.127 }}
+* [[CWE]]: ~42875/39366 = 150.0000{{c}}, ~3/2 = 701.7134{{c}}
+: error map: {{val| 0.000 -0.242 -0.021 +0.022 }}
-[[Optimal tuning]] ([[POTE]]): ~42875/39366 = 1\8, ~3/2 = 701.713
+{{Optimal ET sequence|legend=1| 24, …, 224, 472, 696, 1168 }}
-{{Optimal ET sequence|legend=1| 24, 224, 472, 696, 1168 }}
+[[Badness]] (Sintel): 3.98
-[[Badness]]: 0.157186
 === 11-limit ===
@@ Line 2,368: / Line 2,622: @@
 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 ===
@@ Line 2,381: / Line 2,637: @@
 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 ==
-The ''nonant'' temperament (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
@@ Line 2,395: / Line 2,653: @@
 {{Mapping|legend=1| 9 0 135 11 | 0 1 -8 1 }}
 : mapping generators: ~2592/2401, ~3
-[[Optimal tuning]] ([[CTE]]): ~2592/2401 = 1\9, ~3/2 = 701.7232
+[[Optimal tuning]]s:
+* [[WE]]: ~2592/2401 = 133.3442{{c}}, ~3/2 = 701.8000{{c}}
+: [[error map]]: {{val| +0.098 -0.057 -0.027 -0.141 }}
+* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.7384{{c}}
+: error map: {{val| 0.000 -0.217 -0.221 -0.421 }}
-{{Optimal ET sequence|legend=1| 36, 99c, 135, 171 }}
+{{Optimal ET sequence|legend=1| 36, 99c, 135, 171, 2772bd, 2943bdd, …, 5166bccddd, 5337bccddd }}
-[[Badness]]: 0.069896
+[[Badness]] (Sintel): 1.77
 === 11-limit ===
@@ Line 2,411: / Line 2,672: @@
 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 ===
@@ Line 2,424: / Line 2,687: @@
 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
-[[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}}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~1323/1280 = 53.9741
+: [[error map]]: {{val| -0.114 -0.436 -0.182 +1.310 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~256/147 = 957.3867{{c}}
+: error map: {{val| 0.000 -0.248 +0.032 +1.627 }}
-{{Optimal ET sequence|legend=1| 89, 200, 289 }}
+{{Optimal ET sequence|legend=1| 89, 94, 183, 460d, 643d }}
-[[Badness]]: 0.432580
+[[Badness]] (Sintel): 4.73
 === 11-limit ===
 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
+== Subgroup extensions ==
-{{Optimal ET sequence|legend=1| 12, 41, 53, 65 }}
+=== 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]].
-[[Tp tuning #T2 tuning|RMS error]]: 0.4842 cents
+Subgroup: 2.3.5.13
-==== 2.3.5.17.19 ====
+Comma list: 325/324, 32805/32768
-[[Subgroup]]: 2.3.5.17.19
-[[Comma list]]: 171/170, 256/255, 324/323
+Subgroup-val mapping: {{mapping| 1 0 15 -28 | 0 1 -8 20 }}
-{{Mapping|legend=2| 1 0 15 -7 9 | 0 1 -8 7 -3 }}
+Optimal tunings:
+* WE: ~2 = 1200.3326{{c}} ~3/2 = 702.1092{{c}}
+* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9189{{c}}
-{{Mapping|legend=3| 1 0 15 0 0 0 -7 9 | 0 1 -8 0 0 0 7 -3 }}
+{{Optimal ET sequence|legend=0| 12, …, 41, 53, 412cf, 465cf, …, 783ccff, 836ccfff }}
-: [[gencom]]: [2 3; 171/170 256/255 324/323]
+Badness (Sintel): 0.582
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.470
+==== 2.3.5.13.19 subgroup ====
+Subgroup: 2.3.5.13.19
-{{Optimal ET sequence|legend=1| 12, 41, 53, 65 }}
+Comma list: 325/324, 361/360, 513/512
-[[Tp tuning #T2 tuning|RMS error]]: 0.5374 cents
+Subgroup-val mapping: {{mapping| 1 0 15 -28 9 | 0 1 -8 20 -3 }}
-=== Nestoria (2.3.5.19) ===
+Optimal tunings:
-: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''
+* WE: ~2 = 1200.4236{{c}}, ~3/2 = 702.1510{{c}}
+* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9064{{c}}
-The [[S-expression]]-based comma list of this temperament is {[[1216/1215|S16/S18]], [[361/360|S19]] (, ''[[513/512|S15/S20]]'')}.
+{{Optimal ET sequence|legend=0| 12, …, 41, 53 }}
-[[Subgroup]]: 2.3.5.19
+Badness (Sintel): 0.354
-[[Comma list]]: 361/360, 513/512
+=== Photia (2.3.5.17) ===
+{{See also| No-elevens subgroup temperaments #Garibaldia }}
-{{Mapping|legend=2| 1 0 15 9 | 0 1 -8 -3 }}
+[[Subgroup]]: 2.3.5.17
-: mapping generators: ~2, ~3
+[[Comma list]]: 256/255, 1458/1445
+{{Mapping|legend=2| 1 0 15 -7 | 0 1 -8 7 }}
+{{Mapping|legend=3| 1 0 15 0 0 0 -7 | 0 1 -8 0 0 0 7 }}
+: mapping generators: ~2, ~3
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.5471{{c}}, ~3/2 = 701.2262{{c}}
+: [[error map]]: {{val| -0.453 -1.182 +0.706 +3.628 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4976{{c}}
+: error map: {{val| 0.000 -0.457 +1.705 +5.528 }}
+{{Optimal ET sequence|legend=1| 12, 41, 53, 65, 207g, 272gg }}
+[[Badness]] (Sintel): 0.479
+==== 2.3.5.17.19 subgroup ====
+Subgroup: 2.3.5.17.19
+Comma list: 171/170, 256/255, 324/323
+Subgroup-val mapping: {{mapping| 1 0 15 -7 9 | 0 1 -8 7 -3 }}
+Gencom mapping: {{mapping| 1 0 15 0 0 0 -7 9 | 0 1 -8 0 0 0 7 -3 }}
+Optimal tunings:
+* WE: ~2 = 1199.7225{{c}}, ~3/2 = 701.3077{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.4754{{c}}
+{{Optimal ET sequence|legend=0| 12, 41, 53, 65, 142g }}
+Badness (Sintel): 0.332
+=== Nestoria (2.3.5.19) ===
+: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''
+Nestoria is notable for having one of the lowest-badness subgroup extensions of schismic. Note that despite prime [[19/1|19]] being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is generally not flatter than the fifth in optimal schismic due to its optimization considering intervals like [[19/10]] and [[19/15]]. However, the dyadic tuning sensitivity of [[19/16]] suggests using tunings like [[65edo]] and [[77edo]] to optimize in favour of prime 19 (especially the minor triad ~16:19:24 which is equated with the Pythagorean minor triad), as [[171edo]] is already arguably undertempered for it despite being the optimal patent val.
+[[Subgroup]]: 2.3.5.19
+[[Comma list]]: 361/360, 513/512
+{{Mapping|legend=2| 1 0 15 9 | 0 1 -8 -3 }}
 {{Mapping|legend=3| 1 0 15 0 0 0 0 9 | 0 1 -8 0 0 0 0 -3 }}
+: mapping generators: ~2, ~3
-: [[gencom]]: [2 3; 361/360 513/512]
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.2250{{c}}, ~3/2 = 701.8776{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.746
+: [[error map]]: {{val| +0.225 +0.148 +0.240 -1.796 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7307{{c}}
+: error map: {{val| 0.000 -0.224 -0.159 -2.705 }}
-{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171 }}
+{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171, 460hh, 631hh }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.1763 cents
+[[Badness]] (Sintel): 0.126
 === Taylor (2.3.5.13) ===
@@ Line 2,529: / Line 2,851: @@
 {{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
-{{Mapping|legend=3| 1 0 15 0 0 14 | 0 2 -16 0 0 -13 }}
+[[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
-: [[gencom]]: [2 26/15; 676/675 32805/32768]
+Subgroup-val mapping: {{mapping| 1 0 15 14 9 6 | 0 2 -16 -13 -6 -1 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~26/15 = 950.855
+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=1| 24, 53, 130, 183, 236 }}
+{{Optimal ET sequence|legend=0| 24, 29, 53, 183, 236h, 289hl, 631fhhll }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.1485 cents
+Badness (Sintel): 0.223
 === Quintilischis (2.3.5.17) ===
@@ Line 2,550: / Line 2,903: @@
 {{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
-{{Mapping|legend=3| 1 2 -1 0 0 0 5 | 0 -5 40 0 0 0 -11 }}
+[[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 }}
-: [[gencom]]: [2 18/17; 32805/32768 1419857/1417176]
+{{Optimal ET sequence|legend=1| 12, …, 253, 265, 277, 289, 566g, 855g }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~18/17 = 99.649
+[[Badness]] (Sintel): 1.34
-{{Optimal ET sequence|legend=1| 12, 253, 265, 277, 289 }}
+==== 2.3.5.17.19 subgroup ====
+Subgroup: 2.3.5.17.19
-[[Tp tuning #T2 tuning|RMS error]]: 0.0719 cents
+Comma list: 4624/4617, 6144/6137, 6885/6859
-==== 2.3.5.17.19 ====
+Subgroup-val mapping: {{mapping| 1 2 -1 5 4 | 0 -5 40 -11 3 }}
-[[Subgroup]]: 2.3.5.17.19
-[[Comma list]]: 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 }}
-{{Mapping|legend=2| 1 2 -1 5 4 | 0 -5 40 -11 3 }}
+Optimal tunings:
+* WE: ~2 = 1200.0350{{c}}, ~18/17 = 99.6550{{c}}
-{{Mapping|legend=3| 1 2 -1 0 0 0 5 4 | 0 -5 40 0 0 0 -11 3 }}
+* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6520{{c}}
-: [[gencom]]: [2 18/17; 4624/4617 6144/6137 6885/6859]
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~18/17 = 99.652
-{{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:Schismatic family| ]] <!-- main article -->
-[[Category:Schismatic| ]] <!-- key article -->
 [[Category:Rank 2]]

Schismatic family: Difference between revisions