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-The [[5-limit]] parent [[comma]] of the '''meantone family''' is the Didymus or [[Wikipedia: syntonic comma|syntonic comma]], [[81/80]]. This is the one they all temper out. The period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.
+{{Technical data page}}
+The '''meantone family''' is the family of [[rank-2 temperament]]s that [[tempering out|temper out]] the syntonic comma, [[81/80]], and thus can all be seen as [[extension]]s of [[meantone]].
-== Meantone (12&amp;19, 2.3.5) ==
+== Meantone ==
-{{main| Meantone }}
+{{Main| Meantone }}
-Subgroup: 2.3.5
+Meantone is characterized by an [[octave]] [[period]], a [[3/2|fifth]] [[generator]], and the relationship that four fifths go to make up a [[5/1|5th harmonic]].
+[[Subgroup]]: 2.3.5
 [[Comma list]]: 81/80
-[[Mapping]]: [{{val| 1 0 -4 }}, {{val| 0 1 4 }}]
+{{Mapping|legend=1| 1 0 -4 | 0 1 4 }}
-Mapping generators: ~2, ~3
+: mapping generators: ~2, ~3
-{{multival|legend=1| 1 4 4 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.3906{{c}}, ~3/2 = 697.0455{{c}}
-[[POTE generator]]: ~3/2 = 696.239
+: [[error map]]: {{val| +1.391 -3.519 +1.868 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.6512{{c}}
+: error map: {{val| 0.000 -5.304 +0.291 }}
 [[Minimax tuning]]:
-* [[5-odd-limit]]: ~3/2 = {{Monzo| 0 0 1/4 }}
+* [[5-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma)
-: [[Eigenmonzo]]s (unchanged intervals): 2, 5
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
 [[Tuning ranges]]:
 * 5-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 720.000] (4\7 to 3\5)
-* 5-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 701.955]
+* 5-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 701.955] (1/3-comma to Pyth.)
-* 5-odd-limit diamond monotone and tradeoff: ~3/2 = [694.786, 701.955]
+{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b }}
+[[Badness]] (Sintel): 0.173
-{{Val list|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb }}
+=== Overview to extensions ===
+The second comma of the normal comma list defines which [[7-limit]] family member we are looking at.
+* Flattertone adds {{monzo| -24 17 0 -1 }}, finding the [[~]][[7/4]] at the double-augmented sixth, for a tuning between 33edo and 26edo.
+* Flattone adds {{monzo| -17 9 0 1 }}, finding the ~7/4 at the diminished seventh, for a tuning between 26edo and 19edo.
+* Septimal meantone adds [[Harrison's comma|{{monzo| -13 10 0 -1 }}]], finding the ~7/4 at the augmented sixth, for a tuning between 19edo and 12edo.
+* Dominant adds [[64/63|{{monzo| 6 -2 0 -1 }}]], finding the ~7/4 at the minor seventh, for a tuning between 12edo and 5edo.
+* Sharptone adds [[28/27|{{monzo| 2 -3 0 1 }}]], finding the ~7/4 at the major sixth, for an [[exotemperament]] never exactly well-tuned, and where 5edo is the only [[diamond monotone]] tuning, with a terrible 5-limit part.
+Those all have a fifth as generator.
+* Injera adds {{monzo| -7 8 0 -2 }} with a half-octave period.
+* Mohajira adds {{monzo| -23 11 0 2 }} and splits the fifth in two.
+* Godzilla adds [[49/48|{{monzo| -4 -1 0 2 }}]] with an ~[[8/7]] generator, two of which give the [[4/3|fourth]].
+* Mothra adds [[1029/1024|{{monzo| -10 1 0 3 }}]] with an ~8/7 generator, three of which give the fifth.
+* Liese adds {{monzo| -9 11 0 -3 }} with a ~[[10/7]] generator, three of which give the [[3/1|twelfth]].
+* Squares adds {{monzo| -3 9 0 -4 }} with a ~[[9/7]] generator, four of which give the [[8/3|eleventh]].
+* Jerome adds {{monzo| 3 7 0 -5 }} and slices the fifth in five.
-[[Badness]]: 0.007381
+==== Strong extensions ====
+For any meantone generator tuning between 7\12 and 11\19, the augmented sixth is sharper than the diminished seventh and flatter than the minor seventh, befitting an approximation to interval class of 7. This coincides with interpreting the tritone (~9/8)<sup>3</sup> as [[7/5]], leading to septimal meantone, a very elegant extension to the 7-limit.
-Scales: [[meantone5]], [[meantone7]], [[meantone12]]
+For any tuning flatter than 11\19, the augmented sixth and diminished seventh swap their orders, so the diminished seventh becomes a better approximation to the interval class of 7, resulting in flattone. Likewise, for any tuning sharper than 7\12, the minor seventh is the proper approximation instead, resulting in dominant.
-=== Seven-limit extensions ===
+Another way to extend meantone to higher limits involves decomposing the meantone comma into products of smaller commas, or expressing some other comma of interest in terms of the ratio between the meantone comma and another comma. However, this often results in [[weak extension]]s. Another opportunity given by the meantone fifth being flat is that the most obvious ways of dividing it into ''n'' parts leave the part closer to just than usual, because we can allow – and indeed want – more flatwards tempering on the fifth, so may be recommended for this reason.
-The [[7-limit]] [[extension]]s of meantone are:
-* Septimal meantone, with normal comma list [{{Monzo| -4 4 -1 }}, [[Harrison's comma|{{Monzo| -13 10 0 -1 }}]]],
+==== Splitting the meantone fifth into two (243/242) ====
-* Flattone, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -17 9 0 1 }}],
+By tempering out [[243/242]] we equate the distance from 9/8 to 10/9 (= [[81/80|S9]]) with the distance between 11/10 to 12/11 (= [[121/120|S11]]), leading to [[mohaha]] which is in some sense thus a trivial tuning of [[rastmic]] (as 81/80 and 121/120 vanish), but an important one, as it leads to the 11/9 being a more in-tune "hemififth" than in non-meantone [[rastmic]] temperaments (which require sharper fifths in good tunings), and it has a natural extension to the full [[11-limit]] by finding [[7/4]] as the semi-diminished seventh, leading to [[mohajira]], which inflates [[64/63]] to equate it with a small quarter-tone, which is characteristic. Mohajira can also be thought of as equating a slightly sharpened [[25/16|(5/4)<sup>2</sup>]] with [[11/7]], which is also natural as meantone tempering usually has [[5/4]] slightly sharp. There is also the consideration that tempering out [[121/120]] leads to similarly high damage in the 11-limit as tempering [[81/80]] in the 5-limit, because both erase key distinctions of their respective JI subgroups.
-* Dominant, with normal list [{{Monzo| -4 4 -1 }}, [[64/63|{{Monzo| 6 -2 0 -1 }}]]],
-* Sharptone, with normal list [{{Monzo| -4 4 -1 }}, [[28/27|{{Monzo| 2 -3 0 1 }}]]],
+==== Splitting the meantone fifth into three (1029/1024) ====
-* Injera, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -7 8 0 -2 }}],
+By tempering out [[1029/1024]] we equate the distance from 7/6 to 8/7 (= [[49/48|S7]]) with the distance from 8/7 to 9/8 (= [[64/63|S8]]), so that ([[8/7]])<sup>3</sup> is equated with [[3/2]], because of being able to be rewritten as (9/8)(8/7)(7/6) – this observation can be generalized to define the family of [[ultraparticular]] commas. This is an unusually natural extension, with a surprising coincidence: ([[36/35]])/([[64/63]]) = [[81/80]], or using the shorthand notation, S6/S8 = S9. As S6/S8 is already tempered out, it is natural to want [[49/48]] (S7), which is bigger than S8 and smaller than S6 to be equated with both, to avoid inconsistent mappings. This has the surprising consequence of meaning that splitting the meantone fifth into three 8/7's is equivalent to splitting 8/5 into three 7/6's by tempering (8/5)/(7/6)<sup>3</sup> = [[1728/1715]] (S6/S7), the orwellisma.
-* Mohajira, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -23 11 0 2 }}],
-* Godzilla, with normal list [{{Monzo| -4 4 -1 }}, [[49/48|{{Monzo| -4 -1 0 2 }}]]],
+This strategy leads to the 7-limit version of [[mothra]], which is also sometimes called '''cynder'''. Though undecimal mothra is the simplest extension by a small margin, when measured in terms of generators required to reach 11, there is another extension that is perhaps more obvious, by noticing that because we have S6~S7~S8 with S9 tempered out, we can try S8~S10 by tempering out [[176/175]] (S8/S10), which is (11/7)/(5/4)<sup>2</sup>, taking advantage of 10/9 being tempered sharp in meantone so that we can distinguish 11/10 from it, thus finding 16/11 at 100/99 above the meantone diminished fifth, ([[6/5]])<sup>2</sup> = [[36/25]] = ([[3/2]])/([[25/24]]).
-* Mothra, with normal list [{{Monzo| -4 4 -1 }}, [[1029/1024|{{Monzo| -10 1 0 3 }}]]],
-* Squares, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -3 9 0 -4 }}], and
+==== 31edo as splitting the fifth into two, three and nine ====
-* Liese, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -9 11 0 -3 }}].
+[[31edo]] is unique as combining all aforementioned tempering strategies into one elegant [[11-limit]] meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate [[5/4]] and [[7/4]] and an even more accurate [[35/32]]. A tempering strategy not mentioned is splitting a flattened [[3/2]] into nine sharpened [[25/24]]'s, resulting in the 5-limit version of [[valentine]] so that 31edo is the unique tuning that combines them. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle without tempering out [[225/224]], which interestingly, though a rank-2 temperament, only has 31edo as a [[patent val]] tuning (corresponding to also tempering out 225/224).
+Temperaments discussed elsewhere include
+* ''[[Plutus]]'' (+15/14) → [[Very low accuracy temperaments #Plutus|Very low accuracy temperaments]]
+* [[Godzilla]] (+49/48) → [[Semaphoresmic clan #Godzilla|Semaphoresmic clan]]
+* [[Mothra]] (+1029/1024) → [[Gamelismic clan #Mothra|Gamelismic clan]]
+* ''[[Mohaha]]'' (+121/120) → [[Rastmic clan #Mohaha|Rastmic clan]]
+The rest are considered below.
 == Septimal meantone ==
 <div style="float:right">[[:de:septimal-mitteltönig|Deutsch]]</div>
-{{see also| Meantone }}
+{{Main| Meantone #Septimal meantone}}
-{{see also| Wikipedia: Septimal meantone temperament }}
+{{Wikipedia| Septimal meantone temperament }}
-The [[7/4]] of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are [[7/6]], C-D#, the augmented second, [[7/5]], C-F#, the tritone, and [[21/16]], C-E#, the augmented third. Septimal meantone tempers out the common 7-limit commas [[126/125]] and [[225/224]] and in fact can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125 and 225/224.
+In septimal meantone, ten fifths get to the interval class for 7, so that [[7/4]] is an augmented sixth (C–A♯), [[7/6]] is an augmented second (C–D♯), [[7/5]] is an augmented fourth (C–F♯), and [[21/16]] is an augmented third (C–E♯). This mapping is rationalized by the fact that 81/80 factors as ([[126/125]])⋅([[225/224]]), and septimal meantone tempers out both of these commas as well as their difference, [[3136/3125]]. In fact it can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125, 225/224, and 3136/3125.
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 81/80, 126/125
-[[Mapping]]: [{{val| 1 0 -4 -13 }}, {{val| 0 1 4 10 }}]
+{{Mapping|legend=1| 1 0 -4 -13 | 0 1 4 10 }}
-{{Multival|legend=1| 1 4 10 4 13 12 }}
-[[POTE generator]]: ~3/2 = 696.495
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.2358{{c}}, ~3/2 = 697.2122{{c}}
+: [[error map]]: {{val| +1.236 -3.507 +2.535 -0.412 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.6562{{c}}
+: error map: {{val| 0.000 -5.299 +0.311 -2.264 }}
 [[Minimax tuning]]:
-* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~3/2 = {{Monzo| 0 0 1/4 }}
+* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma)
-: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| -3 0 5/2 0 }}]
+: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | -3 0 5/2 0 }}
-: [[Eigenmonzo]]s (unchanged intervals): 2, 5
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
 [[Tuning ranges]]:
 * 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
-* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 701.955]
+* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 701.955] (1/3-comma to Pyth.)
-* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
+* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
-* 7-odd-limit diamond monotone and tradeoff: ~3/2 = [694.786, 700.000]
-* 9-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 700.000]
 [[Algebraic generator]]: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, 503.4257 cents. The recurrence converges quickly.
-{{Val list|legend=1| 12, 19, 31, 81, 112b, 143b }}
+{{Optimal ET sequence|legend=1| 12, 19, 31, 81, 112b, 143b }}
-[[Badness]]: 0.013707
+[[Badness]] (Sintel): 0.347
-Scales: [[meantone5]], [[meantone7]], [[meantone12]]
+=== Undecimal meantone (huygens) ===
+{{Redirect|Huygens|the Dutch mathematician, physicist and astronomer|Wikipedia: Christiaan Huygens}}
+{{See also| Huygens vs meanpop }}
-=== Unidecimal meantone aka Huygens ===
+Undecimal meantone<ref name="meantone & meanpop 2003">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_6048.html#6052 Yahoo! Tuning Group | ''good 11-limit meantones'']</ref> a.k.a. huygens<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10437.html Yahoo! Tuning Group | ''The meantone family'']</ref><ref name="meantone & meanpop 2004">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10864.html#10870 Yahoo! Tuning Group | ''names and definitions: meantone'']</ref> maps the [[11/8]] to the double-augmented third (C–E𝄪). See [[chords of huygens]] for a list of dyadic chords in this temperament.
-{{see also| Meantone vs meanpop }}
 Subgroup: 2.3.5.7.11
@@ Line 94: / Line 126: @@
 Comma list: 81/80, 99/98, 126/125
-Mapping: [{{val| 1 0 -4 -13 -25 }}, {{val| 0 1 4 10 18 }}]
+Mapping: {{mapping| 1 0 -4 -13 -25 | 0 1 4 10 18 }}
-POTE generator: ~3/2 = 696.967
+Optimal tunings:
+* WE: ~2 = 1200.7636{{c}}, ~3/2 = 697.4122{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.0315{{c}}
 Minimax tuning:
-* [[11-odd-limit]]: ~3/2 = {{Monzo| 9/16 -1/8 0 0 1/16 }}
+* 11-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}
-: [{{Monzo| 1 0 0 0 0 }}, {{Monzo| 25/16 -1/8 0 0 1/16 }}, {{Monzo| 9/4 -1/2 0 0 1/4 }}, {{Monzo| 21/8 -5/4 0 0 5/8 }}, {{Monzo| 25/8 -9/4 0 0 9/8 }}]
+: projection map: [{{monzo| 1 0 0 0 0 }}, {{monzo| 25/16 -1/8 0 0 1/16 }}, {{monzo| 9/4 -1/2 0 0 1/4 }}, {{monzo| 21/8 -5/4 0 0 5/8 }}, {{monzo| 25/8 -9/4 0 0 9/8 }}]
-: Eigenmonzos (unchanged intervals): 2, 11/9
+: unchanged-interval (eigenmonzo) basis: 2.11/9
 Tuning ranges:
 * 11-odd-limit diamond monotone: ~3/2 = [696.774, 700.000] (18\31 to 7\12)
-* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
+* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
-* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [696.774, 700.000]
 Algebraic generator: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
-Vals: {{Val list| 12, 19e, 31, 105, 136b, 167be, 198be }}
+{{Optimal ET sequence|legend=0| 12, 19e, 31, 105, 136b }}
-Badness: 0.017027
+Badness (Sintel): 0.563
 ; Music
-* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3 Twinkle canon – 74 edo] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]
+* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3 ''Twinkle canon – 74 edo''] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]
-==== Tridecimal meantone ====
+==== Grosstone ====
-Subgroup: 2.3.5.7.11.13
+Grosstone, named for tempering out the [[grossma]], is the main extension of interest that extends undecimal meantone to the 13-limit. It maps 13/8 to the double-diminished seventh (C–B♭♭♭). Note also that 11/10 is a double-augmented unison; 12/11~13/12 is a double-diminished third; and 14/13 is a triple-augmented seventh octave reduced. Grosstone is flexible with its tunings; among the good tunings are [[31edo]], [[43edo]], and [[74edo]].
-Comma list: 66/65, 81/80, 99/98, 105/104
-Mapping: [{{val| 1 0 -4 -13 -25 -20 }}, {{val| 0 1 4 10 18 15 }}]
-POTE generator: ~3/2 = 696.642
-Minimax tuning:
-* [[13-odd-limit|13-]] and [[15-odd-limit]]: ~3/2 = {{Monzo| 9/16 -1/8 0 0 1/16 }}
-: Eigenmonzos (unchanged intervals): 2, 11/9
-Vals: {{Val list| 12f, 19e, 31 }}
-Badness: 0.018048
-==== Grosstone ====
 Subgroup: 2.3.5.7.11.13
 Comma list: 81/80, 99/98, 126/125, 144/143
-Mapping: [{{val| 1 0 -4 -13 -25 29 }}, {{val| 0 1 4 10 18 -16 }}]
+Mapping: {{mapping| 1 0 -4 -13 -25 29 | 0 1 4 10 18 -16 }}
-POTE generator: ~3/2 = 697.264
+Optimal tunings:
+* WE: ~2 = 1199.9389{{c}}, ~3/2 = 697.2282{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.2627{{c}}
 Minimax tuning:
-* 13- and 15-odd-limit: ~3/2 = {{Monzo| 8/13 0 0 1/26 0 -1/26 }}
+* 13- and 15-odd-limit: ~3/2 = {{monzo| 8/13 0 0 1/26 0 -1/26 }}
-: Eigenmonzos (unchanged intervals): 2, 14/13
+: eigenmonzo basis (unchanged-interval basis): 2.13/7
 Tuning ranges:
 * 13- and 15-odd-limit diamond monotone: ~3/2 = [696.774, 697.674] (18\31 to 25\43)
-* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
+* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
-* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [696.774, 697.674]
-Vals: {{Val list| 12, 19ef, 31, 43, 74 }}
+{{Optimal ET sequence|legend=0| 12, 31, 43, 74 }}
-Badness: 0.025899
+Badness (Sintel): 1.07
-==== Meridetone ====
+===== 17-limit =====
-Subgroup: 2.3.5.7.11.13
+This extension maps 17/16 to the minor second (C–D♭), and 19/16 to the minor third (C–E♭), suitable for a system generated by a mildly tempered fifth.
-Comma list: 78/77, 81/80, 99/98, 126/125
+Subgroup: 2.3.5.7.11.13.17
-Mapping: [{{val| 1 0 -4 -13 -25 -39 }}, {{val| 0 1 4 10 18 27 }}]
+Comma list: 81/80, 99/98, 120/119, 126/125, 144/143
-POTE generator: ~3/2 = 697.529
+Mapping: {{mapping| 1 0 -4 -13 -25 29 12 | 0 1 4 10 18 -16 -5 }}
-Minimax tuning:
+Optimal tunings:
-* 13- and 15-odd-limit: ~3/2 = {{Monzo| 14/25 -2/25 0 0 0 1/25 }}
+* WE: ~2 = 1199.5811{{c}}, ~3/2 = 697.0918{{c}}
-: Eigenmonzos (unchanged intervals): 2, 18/13
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3303{{c}}
-Vals: {{Val list| 12f, 31f, 43 }}
+{{Optimal ET sequence|legend=0| 12, 31, 43, 74g }}
-Badness: 0.026421
+Badness (Sintel): 1.06
-==== Hemimeantone ====
+===== 19-limit =====
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 81/80, 99/98, 126/125, 169/168
+Comma list: 81/80, 96/95, 99/98, 120/119, 126/125, 144/143
-Mapping: [{{val| 1 0 -4 -13 -25 -5 }}, {{val| 0 2 8 20 36 11 }}]
+Mapping: {{mapping| 1 0 -4 -13 -25 29 12 9 | 0 1 4 10 18 -16 -5 -3 }}
-Mapping generators: ~2, ~26/15
+Optimal tunings:
+* WE: ~2 = 1199.2931{{c}}, ~3/2 = 696.9690{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3736{{c}}
-POTE generator: ~15/13 = 251.535
+{{Optimal ET sequence|legend=0| 12, 31, 43, 74gh }}
-Vals: {{Val list| 19e, 43, 62, 167bef }}
+Badness (Sintel): 1.07
-Badness: 0.031433
+==== Fokkertone ====
+Fokkertone maps the [[13/8]] to the double-augmented fifth (C–G𝄪). Note that the minor third conflates 13/11 with 6/5, and that 11/10~13/12 is a double-augmented unison; 12/11 is a double-diminished third; and 14/13 is a minor second. 31edo can be recommended as a tuning since it is the only 13-odd-limit diamond monotone tuning.
-=== Meanpop ===
+This extension used to be known as ''tridecimal meantone'', but was decanonicalized in 2025.
-{{see also| Meantone vs meanpop }}
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13
-Comma list: 81/80, 126/125, 385/384
+Comma list: 66/65, 81/80, 99/98, 105/104
-Mapping: [{{val| 1 0 -4 -13 24 }}, {{val| 0 1 4 10 -13 }}]
+Mapping: {{mapping| 1 0 -4 -13 -25 -20 | 0 1 4 10 18 15 }}
-Mapping generator: ~2, ~3
+Optimal tunings:
+* WE: ~2 = 1200.8149{{c}}, ~3/2 = 697.1155{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.7085{{c}}
-POTE generator: ~3/2 = 696.434
+Minimax tuning:
+* 13- and 15-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}
+: unchanged-interval (eigenmonzo) basis: 2.11/9
-Minimax tuning:
+{{Optimal ET sequence|legend=0| 12f, 19e, 31 }}
-* 11-odd-limit: ~3/2 = {{Monzo| 0 0 1/4 }}
-: [{{Monzo| 1 0 0 0 0 }}, {{Monzo| 1 0 1/4 0 0 }}, {{Monzo| 0 0 1 0 0 }}, {{Monzo| -3 0 5/2 0 0 }}, {{Monzo| 11 0 -13/4 0 0 }}]
-: Eigenmonzos (unchanged intervals): 2, 5
-Tuning ranges:
+Badness (Sintel): 0.746
-* 11-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
-* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
-* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 696.774]
-Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-Vals: {{Val list| 12e, 19, 31, 81 }}
+Comma list: 66/65, 81/80, 99/98, 105/104, 120/119
-Badness: 0.021543
+Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 | 0 1 4 10 18 15 -5 }}
-; Music
+Optimal tunings:
-* [http://soonlabel.com/xenharmonic/archives/607 Scott Joplin's "The Entertainer" tuned into meanpop]{{Dead link}}
+* WE: ~2 = 1199.5548{{c}}, ~3/2 = 696.7449{{c}}
-* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3 Twinkle canon – 50 edo] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.9823{{c}}
-==== Tridecimal meanpop ====
+{{Optimal ET sequence|legend=0| 12f, 31 }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 81/80, 105/104, 126/125, 144/143
+Badness (Sintel): 1.02
-Mapping: [{{val| 1 0 -4 -13 24 -20 }}, {{val| 0 1 4 10 -13 15 }}]
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-Mapping generator: ~2, ~3
+Comma list: 66/65, 81/80, 96/95, 99/98, 105/104, 120/119
-POTE generator: ~3/2 = 696.211
+Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 9 | 0 1 4 10 18 15 -5 -3 }}
-Minimax tuning:
+Optimal tunings:
-* 13- and 15-odd-limit: ~3/2 = {{Monzo| 4/7 0 0 0 -1/28 1/28 }}
+* WE: ~2 = 1199.0408{{c}}, ~3/2 = 696.5824{{c}}
-: Eigenmonzos (unchanged intervals): 2, 13/11
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.1061{{c}}
-Tuning ranges:
+{{Optimal ET sequence|legend=0| 12f, 31 }}
-* 13- and 15-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
-* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
-* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 696.774]
-Vals: {{Val list| 12ef, 19, 31, 50, 81, 131bd, 212bbddf }}
+Badness (Sintel): 1.10
-Badness: 0.020883
+==== Meridetone ====
+Meridetone maps the 13/8 to the quadruple-augmented fourth (C–F𝄪𝄪). 43edo can be recommended as a tuning since it is the only 13-odd-limit diamond monotone tuning.
-==== Meanplop ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 65/64, 78/77, 81/80, 91/90
+Comma list: 78/77, 81/80, 99/98, 126/125
-Mapping: [{{val| 1 0 -4 -13 24 10 }}, {{val| 0 1 4 10 -13 -4 }}]
+Mapping: {{mapping| 1 0 -4 -13 -25 -39 | 0 1 4 10 18 27 }}
-POTE generator: ~3/2 = 696.202
+Optimal tunings:
+* WE: ~2 = 1199.9122{{c}}, ~3/2 = 697.4779{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.5241{{c}}
 Minimax tuning:
-* 13- and 15-odd-limit: ~3/2 = {{Monzo| 11/13 0 0 0 -1/13 }}
+* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/25 -2/25 0 0 0 1/25 }}
-: Eigenmonzos (unchanged intervals): 2, 11
+: unchanged-interval (eigenmonzo) basis: 2.13/9
-Vals: {{Val list| 12e, 19, 31f, 50ff, 81fff }}
+{{Optimal ET sequence|legend=0| 12f, 31f, 43 }}
-Badness: 0.027666
+Badness (Sintel): 1.09
-=== Meanenneadecal ===
+===== 17-limit =====
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13.17
-Comma list: 45/44, 56/55, 81/80
+Comma list: 78/77, 81/80, 99/98, 120/119, 126/125
-Mapping: [{{val| 1 0 -4 -13 -6 }}, {{val| 0 1 4 10 6 }}]
+Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 | 0 1 4 10 18 27 -5 }}
-POTE generator: ~3/2 = 696.250
+Optimal tunings:
+* WE: ~2 = 1199.3793{{c}}, ~3/2 = 697.2833{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6222{{c}}
-Tuning ranges:
+{{Optimal ET sequence|legend=0| 12f, 43 }}
-* 11-odd-limit diamond monotone: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
-* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 704.377]
-* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 700.000]
-Vals: {{Val list| 7d, 12, 19, 31e, 50ee }}
+Badness (Sintel): 1.22
-Badness: 0.021423
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-==== 13-limit ====
+Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125
-Subgroup: 2.3.5.7.11.13
-Comma list: 45/44, 56/55, 78/77, 81/80
+Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 9 | 0 1 4 10 18 27 -5 -3 }}
-Mapping: [{{val| 1 0 -4 -13 -6 -20 }}, {{val| 0 1 4 10 6 15 }}]
+Optimal tunings:
+* WE: ~2 = 1199.0260{{c}}, ~3/2 = 697.1486{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6887{{c}}
-POTE generator: ~3/2 = 696.146
+{{Optimal ET sequence|legend=0| 12f, 43 }}
-Vals: {{Val list| 12f, 19, 31e, 50ee }}
+Badness (Sintel): 1.25
-Badness: 0.021182
+==== Hemimeantone ====
+Subgroup: 2.3.5.7.11.13
-==== Vincenzo ====
+Comma list: 81/80, 99/98, 126/125, 169/168
-Subgroup: 2.3.5.7.11.13
-Comma list: 45/44, 56/55, 65/64, 81/80
+Mapping: {{mapping| 1 0 -4 -13 -25 -5 | 0 2 8 20 36 11 }}
-Mapping: [{{val| 1 0 -4 -13 -6 10 }}, {{val| 0 1 4 10 6 -4 }}]
+: mapping generators: ~2, ~26/15
-POTE generator: ~3/2 = 695.060
+Optimal tunings:
+* WE: ~2 = 1201.0387{{c}}, ~26/15 = 949.2863{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5065{{c}}
-Vals: {{Val list| 7d, 12, 19 }}
+{{Optimal ET sequence|legend=0| 19e, 43, 62 }}
-Badness: 0.024763
+Badness (Sintel): 1.30
 ===== 17-limit =====
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 45/44, 52/51, 56/55, 65/64, 81/80
+Comma list: 81/80, 99/98, 126/125, 169/168, 221/220
-Mapping: [{{val| 1 0 -4 -13 -6 10 12 }}, {{val| 0 1 4 10 6 -4 -5 }}]
+Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 | 0 2 8 20 36 11 33 }}
-POTE generator: ~3/2 = 695.858
+Optimal tunings:
+* WE: ~2 = 1201.0270{{c}}, ~26/15 = 949.2892{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5169{{c}}
-Vals: {{Val list| 7d, 12, 19 }}
+{{Optimal ET sequence|legend=0| 19eg, 43, 62 }}
-Badness: 0.025535
+Badness (Sintel): 1.19
 ===== 19-limit =====
 Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80
+Comma list: 81/80, 99/98, 126/125, 153/152, 169/168, 221/220
-Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 }}, {{val| 0 1 4 10 6 -4 -5 -3 }}]
+Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 -25 | 0 2 8 20 36 11 33 37 }}
-POTE generator: ~3/2 = 696.131
+Optimal tunings:
+* WE: ~2 = 1201.0339{{c}}, ~19/11 = 949.2902{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 948.5111{{c}}
-Vals: {{Val list| 7d, 12, 19 }}
+{{Optimal ET sequence|legend=0| 19egh, 43, 62 }}
-Badness: 0.022302
+Badness (Sintel): 1.15
-===== 23-limit =====
+==== Semimeantone ====
-Subgroup: 2.3.5.7.11.13.17.19.23
+Subgroup: 2.3.5.7.11.13
-Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 69/68, 81/80
+Comma list: 81/80, 99/98, 126/125, 847/845
-Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 14 }}, {{val| 0 1 4 10 6 -4 -5 -3 -6 }}]
+Mapping: {{mapping| 2 0 -8 -26 -50 -59 | 0 1 4 10 18 21 }}
-POTE generator: ~3/2 = 696.044
+: mapping generators: ~55/39, ~3
-Vals: {{Val list| 7d, 12, 19 }}
+Optimal tunings:
+* WE: ~55/39 = 600.3606{{c}}, ~3/2 = 697.4241{{c}}
+* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 697.0545{{c}}
-Badness: 0.020139
+{{Optimal ET sequence|legend=0| 12f, …, 50eff, 62, 136b }}
-===== 29-limit =====
+Badness (Sintel): 1.68
-Subgroup: 2.3.5.7.11.13.17.19.23.29
-Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 14 8 }}, {{val| 0 1 4 10 6 -4 -5 -3 -6 -2 }}]
+Comma list: 81/80, 99/98, 126/125, 221/220, 289/288
-POTE generator: ~3/2 = 695.913
+Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 | 0 1 4 10 18 21 1 }}
-Vals: {{Val list| 7d, 12, 19 }}
+Optimal tunings:
+* WE: ~17/12 = 600.5426{{c}}, ~3/2 = 697.5571{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9858{{c}}
-Badness: 0.018168
+{{Optimal ET sequence|legend=0| 12f, 50eff, 62, 136bg }}
-===== 31-limit =====
+Badness (Sintel): 1.60
-Subgroup: 2.3.5.7.11.13.17.19.23.29.31
-Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80, 93/92
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 14 8 16 }}, {{val| 0 1 4 10 6 -4 -5 -3 -6 -2 -7 }}]
+Comma list: 81/80, 99/98, 126/125, 153/152, 209/208, 221/220
-POTE generator: ~3/2 = 695.750
+Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 -1 | 0 1 4 10 18 21 1 3 }}
-Vals: {{Val list| 7d, 12, 19 }}
+Optimal tunings:
+* WE: ~17/12 = 600.5959{{c}}, ~3/2 = 697.5985{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9638{{c}}
-Badness: 0.017069
+{{Optimal ET sequence|legend=0| 12f, 50eff, 62 }}
-===== 37-limit =====
+Badness (Sintel): 1.47
-Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37
-Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92
+=== Meanpop ===
+{{See also| Huygens vs meanpop }}
-Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 14 8 16 -9 }}, {{val| 0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 }}]
+Meanpop<ref name="meantone & meanpop 2003"/><ref name="meantone & meanpop 2004"/> maps the 11/8 to the double-diminished fifth (C–G𝄫), and tridecimal meanpop maps the 13/8 to the double-augmented fifth (C–G𝄪), tempering out 144/143 like in grosstone. Note also 11/10 is a double-diminished third; 12/11~13/12, double-augmented unison; and 14/13, minor second.
-POTE generator: ~3/2 = 695.603
+Subgroup: 2.3.5.7.11
-Vals: {{Val list| 7d, 12, 19 }}
+Comma list: 81/80, 126/125, 385/384
-Badness: 0.016129
+Mapping: {{mapping| 1 0 -4 -13 24 | 0 1 4 10 -13 }}
-===== 41-limit =====
+: mapping generator: ~2, ~3
-Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41
-Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92, 124/123
+Optimal tunings:
+* WE: ~2 = 1201.3464{{c}}, ~3/2 = 697.2159{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4509{{c}}
-Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 }}, {{val| 0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 }}]
+Minimax tuning:
+* 11-odd-limit: ~3/2 = {{monzo| 0 0 1/4 }}
+: projection map: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| -3 0 5/2 0 0 }}, {{monzo| 11 0 -13/4 0 0 }}]
+: unchanged-interval (eigenmonzo) basis: 2.5
-POTE generator: ~3/2 = 695.696
+Tuning ranges:
+* 11-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
+* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
-Vals: {{Val list| 7d, 12, 19 }}
+Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
-Badness: 0.015356
+{{Optimal ET sequence|legend=0| 12e, 19, 31, 81, 112b }}
-===== 43-limit =====
+Badness (Sintel): 0.712
-Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41.43
-Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 86/85, 93/92, 124/123
+; Music
+* [http://soonlabel.com/xenharmonic/archives/607 Scott Joplin's "The Entertainer" tuned into meanpop]{{dead link}}
+* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3 ''Twinkle canon – 50 edo''] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]
-Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7 }}, {{val| 0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1 }}]
+==== Tridecimal meanpop ====
+Subgroup: 2.3.5.7.11.13
-POTE generator: ~3/2 = 695.688
+Comma list: 81/80, 105/104, 126/125, 144/143
-Vals: {{Val list| 7d, 12, 19 }}
+Mapping: {{mapping| 1 0 -4 -13 24 -20 | 0 1 4 10 -13 15 }}
-Badness: 0.013906
+Optimal tunings:
+* WE: ~2 = 1201.0765{{c}}, ~3/2 = 696.8361{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2347{{c}}
-===== 47-limit =====
+Minimax tuning:
-Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41.43.47
+* 13- and 15-odd-limit: ~3/2 = {{monzo| 4/7 0 0 0 -1/28 1/28 }}
+: unchanged-interval (eigenmonzo) basis: 2.13/11
-Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 86/85, 93/92, 95/94, 124/123
+Tuning ranges:
+* 13- and 15-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
+* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
-Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7 4 }}, {{val| 0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1 1 }}]
+{{Optimal ET sequence|legend=0| 19, 31, 50, 81 }}
-POTE generator: ~3/2 = 695.676
+Badness (Sintel): 0.863
-Vals: {{Val list| 7d, 12, 19 }}
+===== Meanpoppic =====
+Subgroup: 2.3.5.7.11.13.17
-Badness: 0.013818
+Comma list: 81/80, 105/104, 126/125, 144/143, 273/272
-==== Meanundec ====
+Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 | 0 1 4 10 -13 15 26 }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 27/26, 40/39, 45/44, 56/55
+Optimal tunings:
+* WE: ~2 = 1201.0727{{c}}, ~3/2 = 696.8168{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2195{{c}}
-Mapping: [{{val| 1 0 -4 -13 -6 -1 }}, {{val| 0 1 4 10 6 3 }}]
+{{Optimal ET sequence|legend=0| 19g, 31, 50, 81, 131bd }}
-POTE generator: ~3/2 = 697.254
+Badness (Sintel): 1.02
-Vals: {{Val list| 7d, 12f, 19f, 31eff }}
+====== 19-limit ======
+Subgroup: 2.3.5.7.11.13.17.19
-Badness: 0.024243
+Comma list: 81/80, 105/104, 126/125, 144/143, 153/152, 273/272
-=== Meanundeci ===
+Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 -40 | 0 1 4 10 -13 15 26 28 }}
-Subgroup: 2.3.5.7.11
-Comma list: 33/32, 55/54, 77/75
+Optimal tunings:
+* WE: ~2 = 1201.0719{{c}}, ~3/2 = 696.8101{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2137{{c}}
-Mapping: [{{val|1 0 -4 -13 5 }}, {{val| 0 1 4 10 -1 }}]
+{{Optimal ET sequence|legend=0| 19gh, 31, 50, 81 }}
-POTE generator: ~3/2 = 694.689
+Badness (Sintel): 1.08
-Vals: {{Val list| 7d, 12e, 19e }}
+===== Meanpoid =====
+Subgroup: 2.3.5.7.11.13.17
-Badness: 0.031539
+Comma list: 81/80, 105/104, 120/119, 126/125, 144/143
-==== 13-limit ====
+Mapping: {{mapping| 1 0 -4 -13 24 -20 12 | 0 1 4 10 -13 15 -5 }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 33/32, 55/54, 65/64, 77/75
+Optimal tunings:
+* WE: ~2 = 1200.2768{{c}}, ~3/2 = 696.5683{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4114{{c}}
-Mapping: [{{val|1 0 -4 -13 5 10 }}, {{val| 0 1 4 10 -1 -4 }}]
+{{Optimal ET sequence|legend=0| 19, 31 }}
-POTE generator: ~3/2 = 694.764
+Badness (Sintel): 1.17
-Vals: {{Val list| 7d, 12e, 19e }}
+====== 19-limit ======
+Subgroup: 2.3.5.7.11.13.17.19
-Badness: 0.026288
-=== Migration ===
-Subgroup: 2.3.5.7.11
-Comma list: 81/80, 121/120, 126/125
-Mapping: [{{val| 1 1 0 -3 2 }}, {{val| 0 2 8 20 5 }}]
+Comma list: 77/76, 81/80, 96/95, 105/104, 120/119, 126/125
-Mapping generators: ~2, ~11/9
+Mapping: {{mapping| 1 0 -4 -13 24 -20 12 9 | 0 1 4 10 -13 15 -5 -3 }}
-POTE generator: ~11/9 = 348.182
+Optimal tunings:
+* WE: ~2 = 1199.7905{{c}}, ~3/2 = 696.3779{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4973{{c}}
-Vals: {{Val list| 7d, 24d, 31, 100de, 131bdee, 162bdee }}
+{{Optimal ET sequence|legend=0| 19, 31 }}
-Badness: 0.025516
+Badness (Sintel): 1.25
-==== 13-limit ====
+==== Semimeanpop ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 66/65, 81/80, 121/120, 126/125
+Comma list: 81/80, 126/125, 385/384, 847/845
-Mapping: [{{val| 1 1 0 -3 2 4 }}, {{val| 0 2 8 20 5 -1 }}]
+Mapping: {{mapping| 2 0 -8 -26 48 39 | 0 1 4 10 -13 -10 }}
-Mapping generators: ~2, ~11/9
+: mapping generators: ~55/39, ~3
-POTE generator: ~11/9 = 348.490
+Optimal tunings:
+* WE: ~55/39 = 600.6704{{c}}, ~3/2 = 697.2151{{c}}
+* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.4341{{c}}
-Vals: {{Val list| 7d, 24d, 31, 55d }}
+{{Optimal ET sequence|legend=0| 12e, 50, 62, 112b }}
-Badness: 0.028071
+Badness (Sintel): 1.78
-=== Bimeantone ===
+===== 17-limit =====
-/8 is mapped to half octave minus the [[128/125|meantone diesis]].
+Subgroup: 2.3.5.7.11.13.17
-Subgroup: 2.3.5.7.11
+Comma list: 81/80, 126/125, 221/220, 273/272, 289/288
-Comma list: 81/80, 126/125, 245/242
+Mapping: {{mapping| 2 0 -8 -26 48 39 5 | 0 1 4 10 -13 -10 1 }}
-Mapping: [{{val| 2 0 -8 -26 -31 }}, {{val| 0 1 4 10 12 }}]
+Optimal tunings:
+* WE: ~17/12 = 600.7232{{c}}, ~3/2 = 697.2820{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4411{{c}}
-Mapping generators: ~63/44, ~3
+{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bg }}
-POTE generator: ~3/2 = 696.016
+Badness (Sintel): 1.45
-Vals: {{Val list| 12, 26de, 38d, 50 }}
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-Badness: 0.038122
+Comma list: 81/80, 126/125, 153/152, 209/208, 221/220, 273/272
-==== 13-limit ====
+Mapping: {{mapping| 2 0 -8 -26 48 39 5 -1 | 0 1 4 10 -13 -10 1 3 }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 81/80, 105/104, 126/125, 245/242
+Optimal tunings:
+* WE: ~17/12 = 600.7527{{c}}, ~3/2 = 697.3244{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4525{{c}}
-Mapping: [{{val| 2 0 -8 -26 -31 -40 }}, {{val| 0 1 4 10 12 15 }}]
+{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bgh }}
-Mapping generators: ~55/39, ~3
+Badness (Sintel): 1.28
-POTE generator: ~3/2 = 695.836
+=== Meanenneadecal ===
+Meanenneadecal maps the 11/8 to the augmented fourth (C–F♯), and tridecimal meanenneadecal maps the 13/8 to the double-augmented fifth (C–G𝄪). Note also 11/10 is a major second; 12/11~14/13, minor second; and 13/12, double-augmented unison.
-Vals: {{Val list| 12f, 26deff, 38df, 50 }}
+Subgroup: 2.3.5.7.11
-Badness: 0.028817
+Comma list: 45/44, 56/55, 81/80
-==== 17-limit ====
+Mapping: {{mapping| 1 0 -4 -13 -6 | 0 1 4 10 6 }}
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 81/80, 105/104, 126/125, 189/187, 221/220
+Optimal tunings:
+* WE: ~2 = 1199.6946{{c}}, ~3/2 = 696.0729{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2083{{c}}
-Mapping: [{{val| 2 0 -8 -26 -31 -40 5 }}, {{val| 0 1 4 10 12 15 1 }}]
+Tuning ranges:
+* 11-odd-limit diamond monotone: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
+* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 704.377]
-Mapping generators: ~17/12, ~3
+{{Optimal ET sequence|legend=0| 7d, 12, 19, 31e }}
-POTE generator: ~3/2 = 695.783
+Badness (Sintel): 0.708
-Vals: {{Val list| 12f, 26deff, 38df, 50 }}
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Badness: 0.022666
+Comma list: 45/44, 56/55, 78/77, 81/80
-== Flattone ==
+Mapping: {{mapping| 1 0 -4 -13 -6 -20 | 0 1 4 10 6 15 }}
-In flattone, 9 generator steps of 4/3 get to the interval class for 7, meaning that [[7/4]] is a diminished seventh interval (C-Bbb). Other intervals are [[7/6]], a diminished third (C-Ebb), and [[7/5]], a doubly diminshed fifth (C-Gbb). Good tunings for flattone are [[26edo|26EDO]], [[45edo|45EDO]] and [[64edo|64EDO]].
-Subgroup: 2.3.5.7
+Optimal tunings:
+* WE: ~2 = 1199.7931{{c}}, ~3/2 = 696.0258{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1241{{c}}
-[[Comma list]]: 81/80, 525/512
+{{Optimal ET sequence|legend=0| 7df, 12f, 19, 31e }}
-[[Mapping]]: [{{val| 1 0 -4 17 }}, {{val| 0 1 4 -9 }}]
+Badness (Sintel): 0.875
-{{Multival|legend=1| 1 4 -9 4 -17 -32 }}
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-[[POTE generator]]: ~3/2 = 693.779
+Comma list: 45/44, 56/55, 78/77, 81/80, 120/119
-[[Minimax tuning]]:
+Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 | 0 1 4 10 6 15 -5 }}
-* [[7-odd-limit]]: ~3/2 = {{Monzo| 8/13 0 1/13 -1/13 }}
-: [{{Monzo| 1 0 0 0 }}, {{Monzo| 21/13 0 1/13 -1/13 }}, {{Monzo| 32/13 0 4/13 -4/13 }}, {{Monzo| 32/13 0 -9/13 9/13 }}]
-: [[Eigenmonzo]]s (unchanged intervals): 2, 7/5
-* [[9-odd-limit]]: ~3/2 = {{Monzo| 6/11 2/11 0 -1/11 }}
-: [{{Monzo| 1 0 0 0 }}, {{Monzo| 17/11 2/11 0 -1/11 }}, {{Monzo| 24/11 8/11 0 -4/11 }}, {{Monzo| 34/11 -18/11 0 9/11 }}]
-: Eigenmonzos (unchanged intervals): 2, 9/7
-[[Tuning ranges]]:
-* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
-* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [692.353, 701.955]
-* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
-* 7-odd-limit diamond monotone and tradeoff: ~3/2 = [692.353, 694.737]
-* 9-odd-limit diamond monotone and tradeoff: ~3/2 = [692.308, 694.737]
-Algebraic generator: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.
+Optimal tunings:
+* WE: ~2 = 1198.6665{{c}}, ~3/2 = 695.8010{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4998{{c}}
-{{Val list|legend=1| 7, 19, 26, 45 }}
+{{Optimal ET sequence|legend=0| 12f, 19, 31e }}
-[[Badness]]: 0.038553
+Badness (Sintel): 1.17
-Scales: [[flattone12]]
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-=== 11-limit ===
+Comma list: 45/44, 56/55, 78/77, 81/80, 96/95, 120/119
-Subgroup: 2.3.5.7.11
-Comma list: 45/44, 81/80, 385/384
+Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 9 | 0 1 4 10 6 15 -5 -3 }}
-Mapping: [{{val| 1 0 -4 17 -6 }}, {{val| 0 1 4 -9 6 }}]
+Optimal tunings:
+* WE: ~2 = 1198.2880{{c}}, ~3/2 = 695.7123{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.6370{{c}}
-POTE generator: ~3/2 = 693.126
+{{Optimal ET sequence|legend=0| 12f, 19, 31e }}
-Tuning ranges:
+Badness (Sintel): 1.23
-* 11-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
-* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]
-* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [692.308, 694.737]
-Vals: {{Val list| 7, 19, 26, 45, 71bc, 116bcde }}
+==== Vincenzo ====
-Badness: 0.033839
-Scales: [[flattone12]]
-==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 45/44, 65/64, 78/77, 81/80
+Comma list: 45/44, 56/55, 65/64, 81/80
-Mapping: [{{val| 1 0 -4 17 -6 10 }}, {{val| 0 1 4 -9 6 -4 }}]
+Mapping: {{mapping| 1 0 -4 -13 -6 10 | 0 1 4 10 6 -4 }}
-POTE generator: ~3/2 = 693.058
+Optimal tunings:
+* WE: ~2 = 1202.1684{{c}}, ~3/2 = 696.3160{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.2045{{c}}
-Tuning ranges:
+{{Optimal ET sequence|legend=0| 7d, 12, 19 }}
-* 13- and 15-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
-* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]
-* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [692.308, 694.737]
-Vals: {{Val list| 7, 19, 26, 45f, 71bcf, 116bcdef }}
+Badness (Sintel): 1.02
-Badness: 0.022260
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-Scales: [[flattone12]]
+Comma list: 45/44, 52/51, 56/55, 65/64, 81/80
-=== Ptolemy ===
+Mapping: {{mapping| 1 0 -4 -13 -6 10 12 | 0 1 4 10 6 -4 -5 }}
-Subgroup: 2.3.5.7.11
-Comma list: 81/80, 121/120, 525/512
+Optimal tunings:
+* WE: ~2 = 1200.5137{{c}}, ~3/2 = 696.1561{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.8771{{c}}
-Mapping: [{{val| 1 1 0 8 2 }}, {{val| 0 2 8 -18 5 }}]
+{{Optimal ET sequence|legend=0| 12, 19 }}
-POTE generator: ~11/9 = 346.922
+Badness (Sintel): 1.30
-Vals: {{Val list| 7, 31dd, 38d, 45e, 83bcddee }}
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-Badness: 0.058785
+Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80
-==== 13-limit ====
+Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 | 0 1 4 10 6 -4 -5 -3 }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 65/64, 81/80, 105/104, 121/120
+Optimal tunings:
+* WE: ~2 = 1199.8261{{c}}, ~3/2 = 696.0298{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1262{{c}}
-Mapping: [{{val| 1 1 0 8 2 6 }}, {{val| 0 2 8 -18 5 -8 }}]
+{{Optimal ET sequence|legend=0| 12, 19 }}
-POTE generator: ~11/9 = 346.910
+Badness (Sintel): 1.36
-Vals: {{Val list| 7, 31ddf, 38df, 45ef, 83bcddeeff }}
+=== Bimeantone ===
+/8 is mapped to half octave minus the [[128/125|meantone diesis]].
-Badness: 0.034316
+Subgroup: 2.3.5.7.11
-== Dominant ==
+Comma list: 81/80, 126/125, 245/242
-The interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is [[12edo|12EDO]], but it also works well with the Pythagorean tuning of pure [[3/2]] fifths, and with [[29edo|29EDO]], [[41edo|41EDO]], or [[53edo|53EDO]].
-Subgroup: 2.3.5.7
+Mapping: {{mapping| 2 0 -8 -26 -31 | 0 1 4 10 12 }}
-[[Comma list]]: 36/35, 64/63
+: mapping generators: ~63/44, ~3
-[[Mapping]]: [{{val| 1 0 -4 6 }}, {{val| 0 1 4 -2 }}]
+Optimal tunings:
+* WE: ~63/44 = 600.7492{{c}}, ~3/2 = 696.8853{{c}}
+* CWE: ~63/44 = 600.0000{{c}}, ~3/2 = 696.1908{{c}}
-{{Multival|legend=1| 1 4 -2 4 -6 -16 }}
+{{Optimal ET sequence|legend=0| 12, 26de, 38d, 50 }}
-[[POTE generator]]: ~3/2 = 701.573
+Badness (Sintel): 1.26
-[[Tuning ranges]]:
+==== 13-limit ====
-* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 720.000] (7\12 to 3\5)
+Subgroup: 2.3.5.7.11.13
-* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 715.587]
-* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
-* 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [700.000, 715.587]
-{{Val list|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}
+Comma list: 81/80, 105/104, 126/125, 245/242
-[[Badness]]: 0.020690
-=== 11-limit ===
+Mapping: {{mapping| 2 0 -8 -26 -31 -40 | 0 1 4 10 12 15 }}
-Subgroup: 2.3.5.7.11
-Comma list: 36/35, 56/55, 64/63
+Optimal tunings:
+* WE: ~55/39 = 600.8309{{c}}, ~3/2 = 696.8000{{c}}
+* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.0066{{c}}
-Mapping: [{{val| 1 0 -4 6 13 }}, {{val| 0 1 4 -2 -6 }}]
+{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}
-Tuning ranges:
+Badness (Sintel): 1.19
-* 11-odd-limit diamond monotone: ~3/2 = [700.000, 705.882] (7\12 to 10\17)
-* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
-* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [700.000, 705.882]
-POTE generator: ~3/2 = 703.254
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
-Vals: {{Val list| 5, 12, 17c, 29cde }}
+Comma list: 81/80, 105/104, 126/125, 189/187, 221/220
-Badness: 0.024180
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 36/35, 56/55, 64/63, 66/65
+Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 | 0 1 4 10 12 15 1 }}
-Mapping: [{{val| 1 0 -4 6 13 18 }}, {{val| 0 1 4 -2 -6 -9 }}]
+Optimal tunings:
+* WE: ~17/12 = 600.9234{{c}}, ~3/2 = 696.8536{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.9317{{c}}
-POTE generator: ~3/2 = 703.636
+{{Optimal ET sequence|legend=0| 12f, 38df, 50 }}
-Tuning ranges:
+Badness (Sintel): 1.15
-* 13- and 15-odd-limit diamond monotone: ~3/2 = 705.882 (10\17)
-* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
-* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = 705.882
-Vals: {{Val list| 12f, 17c, 29cdef }}
+==== 19-limit ====
+Subgroup: 2.3.5.7.11.13.17.19
-Badness: 0.024108
+Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220
-==== Dominion ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 26/25, 36/35, 56/55, 64/63
+Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 -1 | 0 1 4 10 12 15 1 3 }}
-Mapping: [{{val| 1 0 -4 6 13 -9 }}, {{val| 0 1 4 -2 -6 8 }}]
+Optimal tunings:
+* WE: ~17/12 = 600.9845{{c}}, ~3/2 = 696.8939{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.8947{{c}}
-Vals: {{Val list| 5, 12, 17c, 46cde }}
+{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}
-POTE generator: ~3/2 = 704.905
+Badness (Sintel): 1.08
-Badness: 0.027295
+=== Trimean ===
+{{See also| No-sevens subgroup temperaments #Superpine }}
-=== Domineering ===
 Subgroup: 2.3.5.7.11
-Comma list: 36/35, 45/44, 64/63
+Comma list: 81/80, 126/125, 1344/1331
+Mapping: {{mapping| 1 2 4 7 5 | 0 -3 -12 -30 -11 }}
-Mapping: [{{val| 1 0 -4 6 -6 }}, {{val| 0 1 4 -2 6 }}]
+: mapping generators: ~2, ~11/10
-POTE generator: ~3/2 = 698.776
+Optimal tunings:
+* WE: ~2 = 1200.7155{{c}}, ~11/10 = 167.9055{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7749{{c}}
-Vals: {{Val list| 5e, 7, 12, 19d, 43de }}
+{{Optimal ET sequence|legend=0| 7d, 36d, 43, 50, 93 }}
-Badness: 0.021978
+Badness (Sintel): 1.68
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 36/35, 45/44, 52/49, 64/63
+Comma list: 81/80, 126/125, 144/143, 364/363
-Mapping: [{{val| 1 0 -4 6 -6 10 }}, {{val| 0 1 4 -2 6 -4 }}]
+Mapping: {{mapping| 1 2 4 7 5 3 | 0 -3 -12 -30 -11 5 }}
-POTE generator: ~3/2 = 695.762
+Optimal tunings:
+* WE: ~2 = 1200.6104{{c}}, ~11/10 = 167.8749{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7728{{c}}
-Vals: {{Val list| 5ef, 7, 12, 19d, 31def }}
+{{Optimal ET sequence|legend=0| 7d, 43, 50, 93 }}
-Badness: 0.027039
+Badness (Sintel): 1.46
-===== 17-limit =====
+==== 17-limit ====
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 36/35, 45/44, 51/49, 52/49, 64/63
+Comma list: 81/80, 126/125, 144/143, 189/187, 221/220
-Mapping: [{{val| 1 0 -4 6 -6 10 12 }}, {{val| 0 1 4 -2 6 -4 -5 }}]
+Mapping: {{mapping| 1 2 4 7 5 3 8 | 0 -3 -12 -30 -11 5 -28 }}
-POTE generator: ~3/2 = 696.115
+Optimal tunings:
+* WE: ~2 = 1200.6144{{c}}, ~11/10 = 167.8716{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7682{{c}}
-Vals: {{Val list| 5ef, 7, 12, 19d, 31def }}
+{{Optimal ET sequence|legend=0| 7dg, 43, 50, 93 }}
-Badness: 0.024539
+Badness (Sintel): 1.28
-===== 19-limit =====
+=== Migration ===
-Subgroup: 2.3.5.7.11.13.17.19
+See [[Rastmic clan #Migration|Rastmic clan]].
-Comma list: 36/35, 39/38, 45/44, 51/49, 52/49, 57/56
+== Flattone ==
+{{Main| Flattone }}
-Mapping: [{{val| 1 0 -4 6 -6 10 12 9 }}, {{val| 0 1 4 -2 6 -4 -5 -3 }}]
+In flattone, 9 fourths get to the interval class for 7, so that [[7/4]] is a diminished seventh (C–B𝄫), [[7/6]] is a diminished third (C–E𝄫), and [[7/5]] is a double-diminished fifth (C–G𝄫). In general, septimal subminor intervals are diminished and septimal supermajor intervals are augmented, which makes it quite easy to learn flattone notation. The fifth in flattone is typically flatter than that of [[19edo]]. Good tunings for flattone include [[45edo]], [[64edo]], and [[71edo]].
-POTE generator: ~3/2 = 696.217
+[[Subgroup]]: 2.3.5.7
-Vals: {{Val list| 5ef, 7, 12, 19d, 31def }}
+[[Comma list]]: 81/80, 525/512
-Badness: 0.020398
+{{Mapping|legend=1| 1 0 -4 17 | 0 1 4 -9 }}
-==== Dominatrix ====
+[[Optimal tuning]]s:
-Subgroup: 2.3.5.7.11.13
+* [[WE]]: ~2 = 1203.6308{{c}}, ~3/2 = 695.8782{{c}}
+: [[error map]]: {{val| +3.631 -2.446 -2.801 -2.684 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.7334{{c}}
+: error map: {{val| 0.000 -8.222 -11.380 -12.426 }}
-Comma list: 27/26, 36/35, 45/44, 64/63
+[[Minimax tuning]]:
+* [[7-odd-limit]]: ~3/2 = {{monzo| 8/13 0 1/13 -1/13 }}
+: [[projection map]]: [{{monzo| 1 0 0 0 }}, {{monzo| 21/13 0 1/13 -1/13 }}, {{monzo| 32/13 0 4/13 -4/13 }}, {{monzo| 32/13 0 -9/13 9/13 }}]
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
+* [[9-odd-limit]]: ~3/2 = {{monzo| 6/11 2/11 0 -1/11 }}
+: [[projection map]]: [{{monzo| 1 0 0 0 }}, {{monzo| 17/11 2/11 0 -1/11 }}, {{monzo| 24/11 8/11 0 -4/11 }}, {{monzo| 34/11 -18/11 0 9/11 }}]
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7
-Mapping: [{{val| 1 0 -4 6 -6 -1 }}, {{val| 0 1 4 -2 6 3 }}]
+[[Tuning ranges]]:
+* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
+* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [692.353, 701.955]
+* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
-POTE generator: ~3/2 = 698.544
+[[Algebraic generator]]: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.
-Vals: {{Val list| 5e, 7, 12f, 19df }}
+{{Optimal ET sequence|legend=1| 7, 19, 26, 45 }}
-Badness: 0.018289
+[[Badness]] (Sintel): 0.976
+=== 11-limit ===
+This can also be considered a no-sevens temperament: [[#Hypnotone|hypnotone]].
-=== Domination ===
 Subgroup: 2.3.5.7.11
-Comma list: 36/35, 64/63, 77/75
+Comma list: 45/44, 81/80, 385/384
-Mapping: [{{val| 1 0 -4 6 -14 }}, {{val| 0 1 4 -2 11 }}]
+Mapping: {{mapping| 1 0 -4 17 -6 | 0 1 4 -9 6 }}
-POTE generator: ~3/2 = 705.004
+Optimal tuning:
+* WE: ~2 = 1202.3247{{c}}, ~3/2 = 694.4688{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.1467{{c}}
-Vals: {{Val list| 5e, 12e, 17c, 46cd }}
+Tuning ranges:
+* 11-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
+* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]
-Badness: 0.036562
+{{Optimal ET sequence|legend=0| 7, 19, 26, 45, 71bc, 116bcde }}
-==== 13-limit ====
+Badness (Sintel): 1.12
-Subgroup: 2.3.5.7.11.13
-Comma list: 26/25, 36/35, 64/63, 66/65
-Mapping: [{{val| 1 0 -4 6 -14 -9 }}, {{val| 0 1 4 -2 11 8 }}]
-POTE generator: ~3/2 = 705.496
-Vals: {{Val list| 5e, 12e, 17c }}
-Badness: 0.027435
-=== Arnold ===
-Subgroup: 2.3.5.7.11
-Comma list: 22/21, 33/32, 36/35
-Mapping: [{{val| 1 0 -4 6 5 }}, {{val| 0 1 4 -2 -1 }}]
-POTE generator: ~3/2 = 698.491
-Vals: {{Val list| 5, 7, 12e }}
-Badness: 0.026141
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 22/21, 27/26, 33/32, 36/35
+Comma list: 45/44, 65/64, 78/77, 81/80
-Mapping: [{{val| 1 0 -4 6 5 -1 }}, {{val| 0 1 4 -2 3 }}]
+Mapping: {{mapping| 1 0 -4 17 -6 10 | 0 1 4 -9 6 -4 }}
-POTE generator: ~3/2 = 696.743
+Optimal tunings:
+* WE: ~2 = 1202.5156{{c}}, ~3/2 = 694.5107{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0538{{c}}
-Vals: {{Val list| 5, 7, 12ef, 19def }}
+Tuning ranges:
+* 13- and 15-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
+* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]
-Badness: 0.023300
+{{Optimal ET sequence|legend=0| 7, 19, 26, 45f, 71bcf, 116bcdef }}
-==== 17-limit ====
+Badness (Sintel): 0.920
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 22/21, 27/26, 33/32, 36/35, 51/49
+=== Ptolemy ===
+See [[Rastmic clan #Ptolemy|Rastmic clan]].
-Mapping: [{{val| 1 0 -4 6 5 -1 12 }}, {{val| 0 1 4 -2 3 -5 }}]
+== Dominant ==
+{{Main| Dominant (temperament) }}
+{{See also| Archytas clan }}
-POTE generator: ~3/2 = 696.978
+The interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh (C–Bb). The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is [[12edo]], but it also works well with the Pythagorean tuning of pure [[3/2]] fifths, and with [[29edo]], [[41edo]], or [[53edo]].
-Vals: {{Val list| 5, 7, 12ef, 19def }}
+Because dominant entails a near-pure perfect fifth, a small number of generators will not land on an interval close to prime 11. The canonical 11-limit extension identifies 11/8 with the diminished fifth. Domination tempers out 77/75 and identifies 11/8 with the augmented third. Domineering identifies 11/8 with the augmented fourth, which is a very inaccurate mapping; it is however, notable for having the lowest badness among the extensions. Arnold tempers out 33/32 and identifies 11/8 with the perfect fourth. None of them are nearly as good as the weak extension [[neutrominant]], splitting the fifth as well as the chromatic semitone in two like in all [[rastmic clan|rastmic]] temperaments.
-Badness: 0.024535
+[[Subgroup]]: 2.3.5.7
-==== 19-limit ====
+[[Comma list]]: 36/35, 64/63
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 22/21, 27/26, 33/32, 36/35, 51/49, 57/56
+{{Mapping|legend=1| 1 0 -4 6 | 0 1 4 -2 }}
-Mapping: [{{val| 1 0 -4 6 5 -1 12 9 }}, {{val| 0 1 4 -2 3 -5 -3 }}]
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1195.3384{{c}}, ~3/2 = 698.8478{{c}}
+: [[error map]]: {{val| -4.662 -7.769 +9.077 +14.832 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.1125{{c}}
+: error map: {{val| 0.000 -0.842 +18.136 +28.949 }}
-POTE generator: ~3/2 = 697.068
+[[Tuning ranges]]:
+* [[7-odd-limit|7-]] and [[9-odd-limit]] [[diamond monotone]]: ~3/2 = [700.000, 720.000] (7\12 to 3\5)
+* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 715.587]
+* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
-Vals: {{Val list| 5, 7, 12ef, 19def }}
+{{Optimal ET sequence|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}
-Badness: 0.021098
+[[Badness]] (Sintel): 0.524
-=== Maqamic ===
-<span style="display: block; text-align: right;">[[:de:maqamisch|Deutsch]]</span>
-{{main| Maqamic }}
-Maqamic has a hemififth generator (~11/9) and tempers out 36/35 and 121/120. It makes the most sense if viewed as an adaptive temperament, whereby 7/4 and 9/5 simply share an equivalence class in the resulting scales, but don't need to share a particular tempered "middle-of-the-road" intonation.
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 36/35, 64/63, 121/120
+Comma list: 36/35, 56/55, 64/63
-Mapping: [{{val| 1 1 0 4 2 }}, {{val| 0 2 8 -4 5 }}]
+Mapping: {{mapping| 1 0 -4 6 13 | 0 1 4 -2 -6 }}
-Mapping generators: ~2, ~11/9
+Tuning ranges:
+* 11-odd-limit diamond monotone: ~3/2 = [700.000, 705.882] (7\12 to 10\17)
+* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
-POTE generator: ~11/9 = 350.934
+Optimal tunings:
+* WE: ~2 = 1194.0169{{c}}, ~3/2 = 699.7473{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.2672{{c}}
-Vals: {{Val list| 7, 17c, 24d, 41cd }}
+{{Optimal ET sequence|legend=0| 5, 12, 17c, 29cde }}
-Badness: 0.040240
+Badness (Sintel): 0.799
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 36/35, 64/63, 66/65, 121/120
+Comma list: 36/35, 56/55, 64/63, 66/65
-Mapping: [{{val| 1 1 0 4 2 4 }}, {{val| 0 2 8 -4 5 -1 }}]
+Mapping: {{mapping| 1 0 -4 6 13 18 | 0 1 4 -2 -6 -9 }}
-Mapping generators: ~2, ~11/9
+Optimal tunings:
+* WE: ~2 = 1193.8055{{c}}, ~3/2 = 700.0042{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.8254{{c}}
-POTE generator: ~11/9 = 350.816
+Tuning ranges:
+* 13- and 15-odd-limit diamond monotone: ~3/2 = 705.882 (10\17)
+* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
-Vals: {{Val list| 7, 17c, 24d, 41cd }}
+{{Optimal ET sequence|legend=0| 12f, 17c, 29cdef }}
-Badness: 0.027214
+Badness (Sintel): 0.996
-== Sharptone ==
+==== Dominion ====
-Sharptone is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo|12EDO]] tuning does sharptone about as well as such a thing can be done, of course not in its patent val.
+Subgroup: 2.3.5.7.11.13
-Subgroup: 2.3.5.7
+Comma list: 26/25, 36/35, 56/55, 64/63
-[[Comma list]]: 21/20, 28/27
+Mapping: {{mapping| 1 0 -4 6 13 -9 | 0 1 4 -2 -6 8 }}
-[[Mapping]]: [{{val| 1 0 -4 -2 }}, {{val| 0 1 4 3 }}]
+Optimal tunings:
+* WE: ~2 = 1195.0293{{c}}, ~3/2 = 701.9847{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7698{{c}}
-{{Multival|legend=1| 1 4 3 4 2 -4 }}
+{{Optimal ET sequence|legend=0| 5, 12, 17c }}
-[[POTE generator]]: ~3/2 = 700.140
+Badness (Sintel): 1.13
-{{Val list|legend=1| 5, 7d, 12d }}
+=== Domination ===
-[[Badness]]: 0.024848
-=== Meanertone ===
 Subgroup: 2.3.5.7.11
-Comma list: 21/20, 28/27, 33/32
+Comma list: 36/35, 64/63, 77/75
-Mapping: [{{val| 1 0 -4 -2 5 }}, {{val| 0 1 4 3 -1 }}]
+Mapping: {{mapping| 1 0 -4 6 -14 | 0 1 4 -2 11 }}
-POTE generator: ~3/2 = 696.615
+Optimal tunings:
+* WE: ~2 = 1194.8645{{c}}, ~3/2 = 701.9872{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5945{{c}}
-Vals: {{Val list| 5, 7d, 12de }}
+{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}
-Badness: 0.025167
+Badness (Sintel): 1.21
-== Plutus ==
+==== 13-limit ====
-Subgroup: 2.3.5.7
+Subgroup: 2.3.5.7.11.13
-[[Comma list]]: 15/14, 81/80
+Comma list: 26/25, 36/35, 64/63, 66/65
-[[Mapping]]: [{{val| 1 0 -4 -5 }}, {{val| 0 1 4 5 }}]
+Mapping: {{mapping| 1 0 -4 6 -14 -9 | 0 1 4 -2 11 8 }}
-{{Multival|legend=1| 1 4 5 4 5 0 }}
+Optimal tunings:
+* WE: ~2 = 1195.1324{{c}}, ~3/2 = 702.6343{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.0791{{c}}
-[[POTE generator]]: ~3/2 = 682.895
+{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}
-{{Val list|legend=1| 2cd, 5d, 7 }}
+Badness (Sintel): 1.13
-[[Badness]]: 0.045275
+=== Domineering ===
-=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 15/14, 22/21, 81/80
+Comma list: 36/35, 45/44, 64/63
-Mapping: [{{val| 1 0 -4 -5 -6 }}, {{val| 0 1 4 5 6 }}]
+Mapping: {{mapping| 1 0 -4 6 -6 | 0 1 4 -2 6 }}
-POTE generator: ~3/2 = 685.234
+Optimal tunings:
+* WE: ~2 = 1194.7102{{c}}, ~3/2 = 695.6962{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1765{{c}}
-Vals: {{Val list| 2cde, 5de, 7 }}
+{{Optimal ET sequence|legend=0| 5e, 7, 12 }}
-Badness: 0.032521
+Badness (Sintel): 0.727
-== Supermean ==
+=== Arnold ===
-Subgroup: 2.3.5.7
+Subgroup: 2.3.5.7.11
-[[Comma list]]: 81/80, 672/625
+Comma list: 22/21, 33/32, 36/35
-[[Mapping]]: [{{val| 1 0 -4 -21 }}, {{val| 0 1 4 15 }}]
+Mapping: {{mapping| 1 0 -4 6 5 | 0 1 4 -2 -1 }}
-[[POTE generator]]: ~3/2 = 704.889
+Optimal tunings:
+* WE: ~2 = 1199.8507{{c}}, ~3/2 = 698.4045{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.4822{{c}}
-{{Val list|legend=1| 5d, 12d, 17c, 29c }}
+{{Optimal ET sequence|legend=0| 5, 7, 12e }}
-[[Badness]]: 0.134204
+Badness (Sintel): 0.864
-=== 11-limit ===
+=== Neutrominant ===
-Subgroup: 2.3.5.7.11
+See [[Rastmic clan #Neutrominant|Rastmic clan]].
-Comma list: 56/55, 81/80, 132/125
+== Flattertone ==
+In flattertone, 17 fifths get to the interval class for 7, so that [[7/4]] is a double-augmented sixth (C–Ax). The fifth in flattertone is typically at least as flat as [[26edo]]. Here, 26edo and [[33edo|33cd-edo]] are the two primary flattertone tunings. [[1/2-comma meantone]] is also encompassed within flattertone's range. Any flatter than this, the meantone mapping for 5/4 is too inaccurate (it becomes more of a [[16/13]] or [[27/22]]), and [[deeptone]] temperament's mapping is more logical.
-Mapping: [{{val| 1 0 -4 -21 -14 }}, {{val| 0 1 4 15 11 }}]
+Flattertone was named by [[Flora Canou]] in 2024.
-POTE generator: ~3/2 = 705.096
+[[Subgroup]]: 2.3.5.7
-Vals: {{Val list| 5de, 12de, 17c, 29c }}
+[[Comma list]]: 81/80, 1875/1792
-Badness: 0.063262
+{{Mapping|legend=1| 1 0 -4 -24 | 0 1 4 17 }}
-=== 13-limit ===
+: mapping generators: ~2, ~3
-Subgroup: 2.3.5.7.11.13
-Comma list: 26/25, 56/55, 66/65, 81/80
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1204.4511{{c}}, ~3/2 = 694.3258{{c}}
+: [[error map]]: {{val| +4.451 -3.178 -9.011 +3.554 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 692.0479{{c}}
+: error map: {{val| 0.000 -9.907 -18.122 -4.012 }}
-Mapping: [{{val| 1 0 -4 -21 -14 -9 }}, {{val| 0 1 4 15 11 8 }}]
+{{Optimal ET sequence|legend=1| 7d, 19d, 26, 59bcd, 85bccd }}
-POTE generator: ~3/2 = 705.094
+[[Badness]] (Sintel): 2.43
-Vals: {{Val list| 5de, 12de, 17c, 29c }}
+==== 11-limit ====
+Subgroup: 2.3.5.7.11
-Badness: 0.040324
+Comma list: 45/44, 81/80, 1375/1344
-== Godzilla ==
+Mapping: {{mapping| 1 0 -4 -24 -6 | 0 1 4 17 6 }}
-<span style="display: block; text-align: right;">[[:de:Semiphor,_Semaphor,_Godzilla|Deutsch]]</span>
-{{main| Semaphore and Godzilla }}
-Godzilla tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-quarter intervals these represent give a fourth, and so step-and-a-quarter generators generate godzilla. [[19edo|19EDO]] is close to being the optimal generator tuning; hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes.
+Optimal tunings:
+* WE: ~2 = 1203.4653{{c}}, ~3/2 = 693.8144{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 692.0422{{c}}
-Subgroup: 2.3.5.7
+{{Optimal ET sequence|legend=0| 7d, 19d, 26 }}
-[[Comma list]]: 49/48, 81/80
+Badness (Sintel): 1.53
-[[Mapping]]: [{{val| 1 0 -4 2 }}, {{val| 0 2 8 1 }}]
+; Music
+* [https://youtu.be/scCuGXnj5IY ''Music in 33EDO (33-Tone Equal Temperament) - Feb 2024''] by [[Budjarn Lambeth]] (2024)
-Mapping generators: ~2, ~7/4
+== Sharptone ==
+Sharptone is a low-accuracy temperament tempering out [[21/20]] and [[28/27]]. In sharptone, 7/4 is a major sixth, 7/6 a whole tone, and 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo]] tuning does sharptone about as well as such a thing can be done, of course not in its patent val.
-{{Multival|legend=1| 2 8 1 8 -4 -20 }}
+However, while 12edo ends up near-optimal, the only valid [[diamond monotone]] tuning for sharptone is [[5edo]]. Anything flat of it has ~12/7 and ~7/4 in the wrong order (and so should be dominant) and anything sharp of it has ~5/4 and ~4/3 in the wrong order (and so should not be meantone).
-[[POTE generator]]: ~8/7 = 252.635
+The 11-limit extension was named by Gene Ward Smith in 2004<ref name="meantone & meanpop 2004"/>.
-[[Tuning ranges]]:
+[[Subgroup]]: 2.3.5.7
-* 7- and 9-odd-limit [[diamond monotone]]: ~7/6 = [240.000, 257.143] (1\5 to 3\14)
-* 7- and 9-odd-limit [[diamond tradeoff]]: ~7/6 = [231.174, 266.871]
-* 7- and 9-odd-limit diamond monotone and tradeoff: ~7/6 = [240.000, 257.143]
-{{Val list|legend=1| 5, 14c, 19 }}
+[[Comma list]]: 21/20, 28/27
-[[Badness]]: 0.026747
+{{Mapping|legend=1| 1 0 -4 -2 | 0 1 4 3 }}
-=== 11-limit ===
+[[Optimal tuning]]s:
-Subgroup: 2.3.5.7.11
+* [[WE]]: ~2 = 1204.2961{{c}}, ~3/2 = 702.6463{{c}}
+: [[error map]]: {{val| +4.296 +4.987 +24.271 -56.591 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4928{{c}}
+: error map: {{val| 0.000 -0.462 +19.657 -64.347 }}
-Comma list: 45/44, 49/48, 81/80
+{{Optimal ET sequence|legend=1| 5, 7d, 12d }}
-Mapping: [{{val| 1 0 -4 2 -6 }}, {{val| 0 2 8 1 12 }}]
+[[Badness]] (Sintel): 0.629
-Mapping generators: ~2, ~7/4
+=== Meanertone ===
+Subgroup: 2.3.5.7.11
-POTE generator: ~8/7 = 254.027
+Comma list: 21/20, 28/27, 33/32
-Tuning ranges:
+Mapping: {{mapping| 1 0 -4 -2 5 | 0 1 4 3 -1 }}
-* 11-odd-limit diamond monotone: ~7/6 = [252.632, 257.143] (4\19 to 3\14)
-* 11-odd-limit diamond tradeoff: ~7/6 = [231.174, 266.871]
-* 11-odd-limit diamond monotone and tradeoff: ~7/6 = [252.632, 257.143]
-Vals: {{Val list| 14c, 19, 33cd, 52cd }}
+Optimal tunings:
+* WE: ~2 = 1208.5304{{c}}, ~3/2 = 701.5669{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1117{{c}}
-Badness: 0.028947
+{{Optimal ET sequence|legend=0| 5, 7d, 12de }}
-==== 13-limit ====
+Badness (Sintel): 0.832
-Subgroup: 2.3.5.7.11.13
-Comma list: 45/44, 49/48, 78/77, 81/80
+== Mildtone ==
+Mildtone tempers out [[16128/15625]] and finds the interval class of 7 at 22 generators up, as a triple-augmented fifth (C–G#x). [[55edo]] and [[67edo]] are among the possible tunings.
-Mapping: [{{val| 1 0 -4 2 -6 -5 }}, {{val| 0 2 8 1 12 11 }}]
+Mildtone was named by [[User: Lucius Chiaraviglio|Lucius Chiaraviglio]] in 2024.
-Mapping generators: ~2, ~7/4
+[[Subgroup]]: 2.3.5.7
-POTE generator: ~8/7 = 253.603
+[[Comma list]]: 81/80, 16128/15625
-Tuning ranges:
+{{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 }}
-* 13- and 15-odd-limit diamond monotone: ~7/6 = 252.632 (4\19)
-* 13- and 15-odd-limit diamond tradeoff: ~7/6 = [231.174, 289.210]
-* 13- and 15-odd-limit diamond monotone and tradeoff: ~7/6 = 252.632
-Vals: {{Val list| 14cf, 19, 33cdff, 52cdff }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.7304{{c}}, ~3/2 = 698.3953{{c}}
+: [[error map]]: {{val| -0.270 -3.829 +7.267 -1.434 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.5397{{c}}
+: error map: {{val| 0.000 -3.415 +7.845 -0.952 }}
-Badness: 0.022503
+{{Optimal ET sequence|legend=1| 12, 43d, 55, 67 }}
-=== Semafour ===
+[[Badness]] (Sintel): 2.67
-Subgroup: 2.3.5.7.11
-Comma list: 33/32, 49/48, 55/54
+=== 11-limit ===
-Mapping: [{{val| 1 0 -4 2 5 }}, {{val| 0 2 8 1 -2 }}]
+[[Subgroup]]: 2.3.5.7.11
-Mapping generators: ~2, ~7/4
+[[Comma list]]: 81/80, 176/175, 7056/6875
-POTE generator: ~8/7 = 254.042
+{{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 30}}
-Vals: {{Val list| 14c, 19e, 33cdee }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.816{{c}}, ~3/2 = 698.355{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.455{{c}}
-Badness: 0.028510
+{{Optimal ET sequence|legend=1| 12, 43de, 55, 67 }}
-=== Varan ===
+[[Badness]] (Sintel): 2.15
-Subgroup: 2.3.5.7.11
-Comma list: 49/48, 77/75, 81/80
+=== 13-limit ===
-Mapping: [{{val| 1 0 -4 2 -10 }}, {{val| 0 2 8 1 17 }}]
+[[Subgroup]]: 2.3.5.7.11.13
-Mapping generators: ~2, ~7/4
+[[Comma list]]: 81/80, 176/175, 196/195, 832/825
-POTE generator: ~8/7 = 251.079
+{{Mapping|legend=1| 1 0 -4 -32 -44 | 0 1 4 22 30}}
-Vals: {{Val list| 19e, 24, 43de }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.788{{c}}, ~3/2 = 698.355{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.471{{c}}
-Badness: 0.039647
+{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}
-==== 13-limit ====
+[[Badness]] (Sintel): 2.04
-Subgroup: 2.3.5.7.11.13
-Comma list: 49/48, 66/65, 77/75, 81/80
+=== 17-limit ===
-Mapping: [{{val| 1 0 -4 2 -10 -5 }}, {{val| 0 2 8 1 17 11 }}]
+[[Subgroup]]: 2.3.5.7.11.13.17
-Mapping generators: ~2, ~7/4
+[[Comma list]]: 81/80, 176/175, 189/187, 196/195, 832/825
-POTE generator: ~8/7 = 251.165
+{{Mapping|legend=1| 1 0 -4 -32 -44 12| 0 1 4 22 30 -5}}
-Vals: {{Val list| 19e, 24, 43de }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.655{{c}}, ~3/2 = 698.295{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.488{{c}}
-Badness: 0.025676
+{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}
-=== Baragon ===
+[[Badness]] (Sintel): 1.98
-Subgroup: 2.3.5.7.11
-Comma list: 49/48, 56/55, 81/80
+=== 19-limit ===
-Mapping: [{{val| 1 0 -4 2 9 }}, {{val| 0 2 8 1 -7 }}]
+[[Subgroup]]: 2.3.5.7.11.13.17.19
-Mapping generators: ~2, ~7/4
+[[Comma list]]: 81/80, 96/95, 176/175, 189/187, 196/195, 832/825
-POTE generator: ~8/7 = 251.173
+{{Mapping|legend=1| 1 0 -4 -32 -44 12 9| 0 1 4 22 30 -5 -3}}
-Vals: {{Val list| 19, 24, 43d }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.371{{c}}, ~3/2 = 698.164{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.519{{c}}
-Badness: 0.035673
+{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}
-== Injera ==
+[[Badness]] (Sintel): 1.95
-Injera has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. [[38edo|38EDO]], which is two parallel [[19edo|19EDOs]], is an excellent tuning for injera.
-[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name]
+{{Todo|unify precision|review}}
-Subgroup: 2.3.5.7
+== Supermean ==
+Supermean tempers out 672/625 and finds the interval class of 7 at 15 generators up, as a double-augmented fifth (C–Gx). As such, it extends [[leapfrog]].
-[[Comma list]]: 50/49, 81/80
+[[Subgroup]]: 2.3.5.7
-[[Mapping]]: [{{val| 2 0 -8 -7 }}, {{val| 0 1 4 4 }}]
+[[Comma list]]: 81/80, 672/625
-Mapping generators: ~7/5, ~3
+{{Mapping|legend=1| 1 0 -4 -21 | 0 1 4 15 }}
-{{Multival|legend=1| 2 8 8 8 7 -4 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1195.4372{{c}}, ~3/2 = 702.2086{{c}}
+: [[error map]]: {{val| -4.563 -4.309 +22.521 -8.319 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5375{{c}}
+: error map: {{val| 0.000 +2.583 +31.836 -0.763 }}
-[[POTE generator]]: ~3/2 = 694.375
+{{Optimal ET sequence|legend=1| 5d, 12d, 17c }}
-[[Tuning ranges]]:
+[[Badness]] (Sintel): 3.40
-* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
-* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [688.957, 701.955]
-* 9-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
-* 7-odd-limit diamond monotone and tradeoff: ~3/2 = [688.957, 700.000]
-* 9-odd-limit diamond monotone and tradeoff: ~3/2 = [685.714, 700.000]
-{{Val list|legend=1| 12, 26, 38, 102bcd, 140bccd, 178bbccdd }}
-[[Badness]]: 0.031130
-; Music
-* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 Two Pairs of Socks] (in [[26edo|26EDO]]) by [[Igliashon Jones]]
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 45/44, 50/49, 81/80
+Comma list: 56/55, 81/80, 132/125
-Mapping: [{{val| 2 0 -8 -7 -12 }}, {{val| 0 1 4 4 6 }}]
-Mapping generators: ~7/5, ~3
-POTE generator: ~3/2 = 692.840
+Mapping: {{mapping| 1 0 -4 -21 -14 | 0 1 4 15 11 }}
-Tuning ranges:
+Optimal tunings:
-* 11-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
+* WE: ~2 = 1195.7270{{c}}, ~3/2 = 702.5848{{c}}
-* 11-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7471{{c}}
-* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [685.714, 700.000]
-Vals: {{Val list| 12, 14c, 26, 90bce, 116bcce }}
+{{Optimal ET sequence|legend=0| 5de, 12de, 17c }}
-Badness: 0.023124
+Badness (Sintel): 2.09
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 45/44, 50/49, 78/77, 81/80
+Comma list: 26/25, 56/55, 66/65, 81/80
-Mapping: [{{val| 2 0 -8 -7 -12 -21 }}, {{val| 0 1 4 4 6 9 }}]
+Mapping: {{mapping| 1 0 -4 -21 -14 -9 | 0 1 4 15 11 8 }}
-Mapping generators: ~7/5, ~3
+Optimal tunings:
+* WE: ~2 = 1196.3958{{c}}, ~3/2 = 702.9766{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7940{{c}}
-POTE generator: ~3/2 = 692.673
+{{Optimal ET sequence|legend=0| 5de, 12de, 17c, 29c }}
-Tuning ranges:
+Badness (Sintel): 1.67
-* 13- and 15-odd-limit diamond monotone: ~3/2 = 692.308 (15\26)
-* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
-* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = 692.308
-Vals: {{Val list| 12f, 14cf, 26, 38e }}
+== Mohajira ==
+{{Main| Mohajira }}
-Badness: 0.021565
+Mohajira can be viewed as derived from [[mohaha]] which maps the interval half a [[chromatic semitone|chroma]] flat of the minor seventh to ~7/4 so that 7/4 is mapped to a semidiminished seventh (C–Bdb), although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the [[porwell comma]]. It can be described as {{nowrap| 24 & 31 }}; its ploidacot is dicot. [[31edo]] makes for an excellent mohajira tuning, with generator 9\31. Note that while 24 + 31 = [[55edo]] doesn't apear in the optimal ET sequence, it is a [[patent val]] tuning and recommendable if you prefer a light meantone tempering.
-==== Enjera ====
+[[Subgroup]]: 2.3.5.7
-Subgroup: 2.3.5.7.11.13
-Comma list: 27/26, 40/39, 45/44, 50/49
+[[Comma list]]: 81/80, 6144/6125
-Mapping: [{{val| 2 0 -8 -7 -12 -2 }}, {{val| 0 1 4 4 6 3 }}]
+{{Mapping|legend=1| 1 1 0 6 | 0 2 8 -11 }}
-Mapping generators: ~7/5, ~3
+: mapping generators: ~2, ~128/105
-POTE generator: ~3/2 = 694.121
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.8160{{c}}, ~128/105 = 348.6518{{c}}
+: [[error map]]: {{val| +0.816 -3.835 +2.901 +0.900 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 348.4194{{c}}
+: error map: {{val| 0.000 -5.116 +1.041 -1.439 }}
-Vals: {{Val list| 12f, 14c, 26f, 38eff }}
+[[Minimax tuning]]:
+* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~128/105 = {{monzo| 0 0 1/8 }}
+: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 6 0 -11/8 0 }}
+: [[eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.5
-Badness: 0.026542
+[[Tuning ranges]]:
+* 7- and 9-odd-limit [[diamond monotone]]: ~128/105 = [347.368, 350.000] (11\38 to 7\24)
+* 7-odd-limit [[diamond tradeoff]]: ~128/105 = [347.393, 350.978]
+* 9-odd-limit diamond tradeoff: ~128/105 = [345.601, 350.978]
-=== Injerous ===
+[[Algebraic generator]]: Mohabis, real root of 3''x''<sup>3</sup> - 3''x''<sup>2</sup> - 1, 348.6067 cents. Corresponding recurrence converges quickly.
-Subgroup: 2.3.5.7.11
-Comma list: 33/32, 50/49, 55/54
+{{Optimal ET sequence|legend=1| 7, 24, 31 }}
-Mapping: [{{val| 2 0 -8 -7 10 }}, {{val| 0 1 4 4 -1 }}]
+[[Badness]] (Sintel): 1.41
-Mapping generators: ~7/5, ~3
+Scales: [[mohaha7]], [[mohaha10]]
-POTE generator: ~3/2 = 690.548
+=== 11-limit ===
-Vals: {{Val list| 12e, 14c, 26e, 40cee }}
-Badness: 0.038577
-=== Lahoh ===
 Subgroup: 2.3.5.7.11
-Comma list: 50/49, 56/55, 81/77
+Comma list: 81/80, 121/120, 176/175
-Mapping: [{{val| 2 0 -8 -7 7 }}, {{val| 0 1 4 4 0 }}]
+Mapping: {{mapping| 1 1 0 6 2 | 0 2 8 -11 5 }}
-Mapping generators: ~7/5, ~3
+Optimal tunings:
+* WE: ~2 = 1201.1562{{c}}, ~11/9 = 348.8124{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.4910{{c}}
-POTE generator: ~3/2 = 699.001
+Minimax tuning:
+* 11-odd-limit: ~11/9 = {{monzo| 0 0 1/8 }}
+: projection map: [{{Monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 6 0 -11/8 0 0 }}, {{monzo| 2 0 5/8 0 0 }}]
+: unchanged-interval (eigenmonzo) basis: 2.5
-Vals: {{Val list| 2cd, 10cd, 12 }}
+Tuning ranges:
+* 11-odd-limit diamond monotone: ~11/9 = [348.387, 350.000] (9\31 to 7\24)
+* 11-odd-limit diamond tradeoff: ~11/9 = [344.999, 350.978]
-Badness: 0.043062
+{{Optimal ET sequence|legend=0| 7, 24, 31 }}
-=== Teff ===
+Badness (Sintel): 0.862
-{{Main| Teff }}
-Teff (found by Mason Green) is to injera what mohajira is to meantone; it splits the generator in half in order to accommodate higher limit intervals, creating a half-octave quarter-tone temperament.
+Scales: [[mohaha7]], [[mohaha10]]
-Subgroup: 2.3.5.7.11
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Comma list: 50/49, 81/80, 864/847
+Comma list: 66/65, 81/80, 105/104, 121/120
-Mapping: [{{val| 2 1 -4 -3 8 }}, {{val| 0 2 8 8 -1 }}]
+Mapping: {{mapping| 1 1 0 6 2 4 | 0 2 8 -11 5 -1 }}
-Mapping generators: ~7/5, ~16/11
+Optimal tunings:
+* WE: ~2 = 1200.4256{{c}}, ~11/9 = 348.6819{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.5622{{c}}
-POTE generator: ~11/8 = 552.5303
+{{Optimal ET sequence|legend=0| 7, 24, 31 }}
-Vals: {{Val list| 24d, 26, 50d }}
+Badness (Sintel): 0.966
-Badness: 0.0707
+Scales: [[mohaha7]], [[mohaha10]]
-==== 13-limit ====
+=== 17-limit ===
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11.13.17
-Comma list: 50/49, 78/77, 81/80, 144/143
+Comma list: 66/65, 81/80, 105/104, 121/120, 154/153
-Mapping: [{{val| 2 1 -4 -3 8 2 }}, {{val| 0 2 8 8 -1 5 }}]
+Mapping: {{mapping| 1 1 0 6 2 4 7 | 0 2 8 -11 5 -1 -10 }}
-POTE generator: ~11/8 = 552.5324
+Optimal tunings:
+* WE: ~2 = 1200.0382{{c}}, ~11/9 = 348.7471{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.7360{{c}}
-Vals: {{Val list| 24d, 26, 50d }}
+{{Optimal ET sequence|legend=0| 7, 24, 31 }}
-Badness: 0.0400
+Badness (Sintel): 1.05
-==== 17-limit ====
+Scales: [[mohaha7]], [[mohaha10]]
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 50/49, 78/77, 81/80, 85/84, 144/143
+=== 19-limit ===
+Subgroup: 2.3.5.7.11.13.17.19
-Mapping: [{{val| 2 1 -4 -3 8 2 6 }}, {{val| 0 2 8 8 -1 5 2 }}]
+Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152
-POTE generator: ~11/8 = 552.6558
+Mapping: {{mapping| 1 1 0 6 2 4 7 6 | 0 2 8 -11 5 -1 -10 -6 }}
-Vals: {{Val list| 24d, 26 }}
+Optimal tunings:
+* WE: ~2 = 1199.7469{{c}}, ~11/9 = 348.7367{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.8117{{c}}
-Badness: 0.0295
+{{Optimal ET sequence|legend=0| 7, 24, 31, 55 }}
-==== 19-limit ====
+Badness (Sintel): 1.05
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 50/49, 57/56, 78/77, 81/80, 85/84, 144/143
+Scales: [[mohaha7]], [[mohaha10]]
-Mapping: [{{val| 2 1 -4 -3 8 2 6 2 }}, {{val| 0 2 8 8 -1 5 2 6 }}]
+== Mohamaq ==
+Mohamaq is a lower-accuracy alternative to mohajira that favors tunings sharp of 24edo. It may be described as {{nowrap| 17c & 24 }}; its ploidacot is dicot, the same as mohajira.
-POTE generator: ~11/8 = 552.6382
+[[Subgroup]]: 2.3.5.7
-Vals: {{Val list| 24d, 26 }}
+[[Comma list]]: 81/80, 392/375
-Badness: 0.0231
+{{Mapping|legend=1| 1 1 0 -1 | 0 2 8 13 }}
-== Pombe ==
-Pombe (named after the African millet beer) is a variant of [[#Teff]] by Kaiveran Lugheidh that eschews the tempering of 50/49 to attain more accuracy in the 7-limit. Oddly, the 7th harmonic has a lesser generator distance than in teff (-5 vs +8), but this combined with the fact that other harmonics are in the opposite direction means that the 7-limit diamond is more complex overall.
-Subgroup: 2.3.5.7.11.13
+: mapping generators: ~2, ~25/21
-[[Comma list]]: 81/80, 300125/294912
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.0661{{c}}, ~25/21 = 350.3127{{c}}
+: [[error map]]: {{val| -0.934 -2.264 +16.188 -13.827 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/21 = 350.4856{{c}}
+: error map: {{val| 0.000 -0.984 +17.571 -12.513 }}
-[[Mapping]]: [{{val| 2 1 -4 11 }}, {{val| 0 2 8 -5 }}]
+{{Optimal ET sequence|legend=1| 7d, 17c, 24 }}
-Mapping generators: ~735/512, ~35/24
+[[Badness]] (Sintel): 1.97
-{{Multival|legend=1| 4 16 -10 16 -27 -68 }}
+Scales: [[mohaha7]], [[mohaha10]]
-[[POTE generator]]: ~48/35 = 552.2206
-{{Val list|legend=1| 24, 26, 50, 126bcd, 176bcdd, 226bbcdd }}
-[[Badness]]: 0.1161
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 81/80, 245/242, 385/384
+Comma list: 56/55, 77/75, 243/242
+Mapping: {{mapping| 1 1 0 -1 2 | 0 2 8 13 5 }}
-Mapping: [{{val| 2 1 -4 11 8 }}, {{val| 0 2 8 -5 -1 }}]
+Optimal tunings:
+* WE: ~2 = 1199.1924{{c}}, ~11/9 = 350.3286{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.4821{{c}}
-POTE generator: ~11/8 = 552.0929
+{{Optimal ET sequence|legend=0| 7d, 17c, 24 }}
-Vals: {{Val list| 24, 26, 50 }}
+Badness (Sintel): 1.20
-Badness: 0.0521
+Scales: [[mohaha7]], [[mohaha10]]
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 81/80, 105/104, 144/143, 245/242
+Comma list: 56/55, 66/65, 77/75, 243/242
-Mapping: [{{val| 2 1 -4 11 8 2 }}, {{val| 0 2 8 -5 -1 5 }}]
+Mapping: {{mapping| 1 1 0 -1 2 4 | 0 2 8 13 5 -1 }}
-POTE generator: ~11/8 = 552.1498
+Optimal tunings:
+* WE: ~2 = 1198.5986{{c}}, ~11/9 = 350.3353{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.6459{{c}}
-Vals: {{Val list| 24, 26, 50 }}
+{{Optimal ET sequence|legend=0| 7d, 17c, 24, 41c }}
-Badness: 0.0310
+Badness (Sintel): 1.19
-=== 17-limit ===
+Scales: [[mohaha7]], [[mohaha10]]
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 81/80, 105/104, 144/143, 245/242, 273/272
+== Liese ==
+<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>
-Mapping: [{{val| 2 1 -4 11 8 2 6 }}, {{val| 0 2 8 -5 -1 5 2 }}]
+Liese splits the [[3/1|perfect twelfth]] into three generators of ~[[10/7]], using the comma [[1029/1000]]. It also tempers out [[686/675]], the senga. It may be described as {{nowrap| 17c & 19 }}; its ploidacot is alpha-tricot. It is a very natural 13-limit tuning, given the generator is so near 13/9. [[74edo]] makes for a good liese tuning, though [[19edo]] can be used. The tuning is well-supplied with mos scales: 7, 9, 11, 13, 15, 17, 19, 36, 55.
-POTE generator: ~11/8 = 552.1579
+[[Subgroup]]: 2.3.5.7
-Vals: {{Val list| 24, 26, 50 }}
+[[Comma list]]: 81/80, 686/675
-Badness: 0.0213
+{{Mapping|legend=1| 1 0 -4 -3 | 0 3 12 11 }}
-=== 19-limit ===
+: mapping generators: ~2, ~10/7
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 81/80, 105/104, 133/132, 144/143, 171/170, 210/209
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.5548{{c}}, ~10/7 = 633.2251{{c}}
+: [[error map]]: {{val| +1.555 -2.280 +6.168 -8.015 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 632.5640{{c}}
+: error map: {{val| 0.000 -4.263 +4.454 -10.622 }}
-Mapping: [{{val| 2 1 -4 11 8 2 6 2 }}, {{val| 0 2 8 -5 -1 5 2 6 }}]
+[[Minimax tuning]]:
+* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/7 = {{monzo| 1/3 0 1/12 }}
+: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 2/3 0 11/12 0 }}
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
-POTE generator: ~11/8 = 552.1196
+[[Algebraic generator]]: Radix, the real root of ''x''<sup>5</sup> - 2''x''<sup>4</sup> + 2''x''<sup>3</sup> - 2''x''<sup>2</sup> + 2''x'' - 2, also a root of ''x''<sup>6</sup> - ''x''<sup>5</sup> - 2. The recurrence converges.
-Vals: {{Val list| 24, 26, 50 }}
+{{Optimal ET sequence|legend=1| 17c, 19, 55, 74d }}
-Badness: 0.0165
+[[Badness]] (Sintel): 1.18
-== Mohaha ==
+=== Liesel ===
-{{see also| Subgroup temperaments #Mohaha }}
+Subgroup: 2.3.5.7.11
-Mohaha is the 2.3.5.11 subgroup temperament with a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 11/9. Mohaha can be thought of, intuitively, as "meantone with quarter tones"; as is the 3/2 generator subdivided in half, so is the 25/24 chromatic semitone divided into two equal ~33/32 quarter tones (in the 2.3.5.11 subgroup). Within this paradigm, mohaha is the temperament that splits the 3/2 into two equal 11/9's, that splits the 6/5 into two equal 11/10's, and that maps four 3/2's to 5/1. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs. Taking septimal meantone mapping of 7 leads migration, flattone mapping of 7 leads ptolemy, and dominant mapping of 7 leads maqamic.
+Comma list: 56/55, 81/80, 540/539
-Subgroup: 2.3.5.11
+Mapping: {{mapping| 1 0 -4 -3 4 | 0 3 12 11 -1 }}
-[[Comma list]]: 81/80, 121/120
+Optimal tunings:
+* WE: ~2 = 1198.8507{{c}}, ~10/7 = 632.4668{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 632.9963{{c}}
-[[Sval]] [[mapping]]: [{{val|1 1 0 2}}, {{val|0 2 8 5}}]
+{{Optimal ET sequence|legend=0| 17c, 19, 36 }}
-Sval mapping generators: ~2, ~11/9
+Badness (Sintel): 1.35
-Gencom mapping: [{{val|1 1 0 0 2}}, {{val|0 2 8 0 5}}]
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-[[Gencom]]: [2 11/9; 81/80 121/120]
+Comma list: 56/55, 78/77, 81/80, 91/90
-[[POTE generator]]: ~11/9 = 348.0938
+Mapping: {{mapping| 1 0 -4 -3 4 0 | 0 3 12 11 -1 7 }}
-{{Val list|legend=1| 7, 17c, 24, 31, 100e, 131bee }}
+Optimal tunings:
+* WE: ~2 = 1199.4968{{c}}, ~10/7 = 632.7766{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.0082{{c}}
-Scales: [[mohaha7]], [[mohaha10]]
+{{Optimal ET sequence|legend=0| 17c, 19, 36 }}
-=== Mohoho ===
+Badness (Sintel): 1.13
-Subgroup: 2.3.5.11.13
-Comma list: 66/65, 81/80, 121/120
+=== Elisa ===
+Subgroup: 2.3.5.7.11
-Sval mapping: [{{val|1 1 0 2 4}}, {{val|0 2 8 5 -1}}]
+Comma list: 77/75, 81/80, 99/98
-Sval mapping generators: ~2, ~11/9
+Mapping: {{mapping| 1 0 -4 -3 -5 | 0 3 12 11 16 }}
-Gencom mapping: [{{val|1 1 0 0 2 4}}, {{val|0 2 8 0 5 -1}}]
+Optimal tunings:
+* WE: ~2 = 1201.0489{{c}}, ~10/7 = 633.6147{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1644{{c}}
-Gencom: [2 11/9; 66/65 81/80 121/120]
+{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}
-POTE generator: ~11/9 = 348.9155
+Badness (Sintel): 1.37
-Vals: {{Val list| 7, 17c, 24, 31, 55, 86ef, 141ceff }}
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Scales: [[mohaha7]], [[mohaha10]]
+Comma list: 66/65, 77/75, 81/80, 99/98
-== Mohajira ==
+Mapping: {{mapping| 1 0 -4 -3 -5 0 | 0 3 12 11 16 7 }}
-{{main|Mohajira}}
-Mohajira can be viewed as derived from mohaha which maps the interval one quarter tone flat of 16/9 to 7/4, although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the porwell comma. [[31edo|31EDO]] makes for an excellent (7-limit) mohajira tuning, with generator 9/31.
+Optimal tunings:
+* WE: ~2 = 1201.4815{{c}}, ~10/7 = 633.7720{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1281{{c}}
-Subgroup: 2.3.5.7
+{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}
-[[Comma list]]: 81/80, 6144/6125
+Badness (Sintel): 1.11
-[[Mapping]]: [{{val| 1 1 0 6 }}, {{val| 0 2 8 -11 }}]
+=== Lisa ===
+Subgroup: 2.3.5.7.11
-Mapping generators: ~2, ~128/105
+Comma list: 45/44, 81/80, 343/330
-{{Multival|legend=1| 2 8 -11 8 -23 -48 }}
+Mapping: {{mapping| 1 0 -4 -3 -6 | 0 3 12 11 18 }}
-[[POTE generator]]: ~128/105 = 348.415
+Optimal tunings:
+* WE: ~2 = 1202.6773{{c}}, ~10/7 = 632.7783{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.6175{{c}}
-[[Minimax tuning]]:
+{{Optimal ET sequence|legend=0| 17cee, 19 }}
-* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~128/105 = {{Monzo| 0 0 1/8 }}
-: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 6 0 -11/8 0 }}]
-: [[Eigenmonzo]]s (unchanged intervals): 2, 5
-[[Tuning ranges]]:
+Badness (Sintel): 1.81
-* 7- and 9-odd-limit [[diamond monotone]]: ~128/105 = [347.368, 350.000] (11\38 to 7\24)
-* 7-odd-limit [[diamond tradeoff]]: ~128/105 = [347.393, 350.978]
-* 9-odd-limit [[diamond tradeoff]]: ~128/105 = [345.601, 350.978]
-* 7-odd-limit diamond monotone and tradeoff: ~128/105 = [347.393, 350.000]
-* 9-odd-limit diamond monotone and tradeoff: ~128/105 = [347.368, 350.000]
-[[Algebraic generator]]: Mohabis, real root of 3''x''<sup>3</sup> - 3''x''<sup>2</sup> - 1, 348.6067 cents. Corresponding recurrence converges quickly.
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-{{Val list|legend=1| 7, 24, 31 }}
+Comma list: 45/44, 81/80, 91/88, 147/143
-[[Badness]]: 0.055714
+Mapping: {{mapping| 1 0 -4 -3 -6 0 | 0 3 12 11 18 7 }}
-Scales: [[mohaha7]], [[mohaha10]]
+Optimal tunings:
+* WE: ~2 = 1203.6086{{c}}, ~10/7 = 633.1193{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.5346{{c}}
-=== 11-limit ===
+{{Optimal ET sequence|legend=0| 17cee, 19 }}
-Subgroup: 2.3.5.7.11
-Comma list: 81/80, 121/120, 176/175
+Badness (Sintel): 1.49
-Mapping: [{{val| 1 1 0 6 2 }}, {{val| 0 2 8 -11 5 }}]
+== Superpine ==
+{{See also| No-sevens subgroup temperaments #Superpine }}
-Mapping generators: ~2, ~11/9
+The superpine temperament is generated by 1/3 of a fourth, represented by [[~]][[35/32]], which resembles [[porcupine]], but it favors flat fifths instead of sharp ones. It may be described as {{nowrap| 36 & 43 }}; its ploidacot is omega-tricot. Unlike in porcupine, the minor third reached by 2 generators up is strongly neutral-flavored and does not represent [[6/5]] – harmonics other than 3 all require the 15-tone mos ([[7L 8s]]) to properly utilize. This temperament has an obvious 11-limit interpretation by treating the generator as [[11/10]] as in porcupine, which makes [[11/8]] high-[[complexity]] like the other harmonics, but in the 13-limit 5 generators up closely approximates [[13/8]]. [[43edo]] is a good tuning especially for the higher-limit extensions.
-POTE generator: ~11/9 = 348.477
+[[Subgroup]]: 2.3.5.7
-Minimax tuning:
+[[Comma list]]: 81/80, 1119744/1071875
-* [[11-odd-limit]]: ~11/9 = {{Monzo| 0 0 1/8 }}
-: [{{Monzo| 1 0 0 0 0 }}, {{Monzo| 1 0 1/4 0 0 }}, {{Monzo| 0 0 1 0 0 }}, {{Monzo| 6 0 -11/8 0 0 }}, {{Monzo| 2 0 5/8 0 0 }}]
-: Eigenmonzos (unchanged intervals): 2, 5
-[[Tuning ranges]]:
+{{Mapping|legend=1| 1 2 4 1 | 0 -3 -12 13 }}
-* 11-odd-limit diamond monotone: ~11/9 = [348.387, 350.000] (9\31 to 7\24)
-* 11-odd-limit diamond tradeoff: ~11/9 = [344.999, 350.978]
-* 11-odd-limit diamond monotone and tradeoff: ~11/9 = [348.387, 350.000]
-Vals: {{Val list| 7, 24, 31 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.3652{{c}}, ~35/32 = 167.1615{{c}}
+: [[error map]]: {{val| -0.635 -4.709 +5.209 +3.639 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/32 = 167.2561{{c}}
+: error map: {{val| 0.000 -3.723 +6.613 +5.503 }}
-Badness: 0.026064
+{{Optimal ET sequence|legend=1| 7, 36, 43, 79c }}
-Scales: [[mohaha7]], [[mohaha10]]
+[[Badness]] (Sintel): 3.46
-=== 13-limit ===
+=== 11-limit ===
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11
-Comma list: 66/65, 81/80, 105/104, 121/120
+Comma list: 81/80, 176/175, 864/847
-Mapping: [{{val| 1 1 0 6 2 4 }}, {{val| 0 2 8 -11 5 -1 }}]
+Mapping: {{mapping| 1 2 4 1 5 | 0 -3 -12 13 -11 }}
-Mapping generators: ~2, ~11/9
+Optimal tunings:
+* WE: ~2 = 1199.0522{{c}}, ~11/10 = 167.1904{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.3382{{c}}
-POTE generator: ~11/9 = 348.558
+{{Optimal ET sequence|legend=0| 7, 36, 43 }}
-Vals: {{Val list| 7, 24, 31 }}
+Badness (Sintel): 1.90
-Badness: 0.023388
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Scales: [[mohaha7]], [[mohaha10]]
+Comma list: 78/77, 81/80, 144/143, 176/175
-=== 17-limit ===
+Mapping: {{mapping| 1 2 4 1 5 3 | 0 -3 -12 13 -11 5 }}
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 66/65, 81/80, 105/104, 121/120, 154/153
+Optimal tunings:
+* WE: ~2 = 1199.4286{{c}}, ~11/10 = 167.3105{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.3958{{c}}
-Mapping: [{{val| 1 1 0 6 2 4 7 }}, {{val| 0 2 8 -11 5 -1 -10 }}]
+{{Optimal ET sequence|legend=0| 7, 36, 43 }}
-Mapping generators: ~2, ~11/9
+Badness (Sintel): 1.52
-POTE generator: ~11/9 = 348.736
+== Lithium ==
+Lithium is named after the 3rd element for having a 3rd-octave period (and also for lithium's molar mass of 6.9 g/mol since 69edo supports it). Its ploidacot is triploid monocot. It supports a [[3L 6s]] scale and thus intuitively can be thought of as "tcherepnin meantone" in that context.
-Vals: {{Val list| 7, 24, 31, 86ef }}
+[[Subgroup]]: 2.3.5.7
-Badness: 0.020576
+[[Comma list]]: 81/80, 3125/3087
-Scales: [[mohaha7]], [[mohaha10]]
+{{Mapping|legend=1| 3 0 -12 -20 | 0 1 4 6 }}
-=== 19-limit ===
+: mapping generators: ~56/45, ~3
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152
+[[Optimal tuning]]s:
+* [[WE]]: ~56/45 = 400.6744{{c}}, ~3/2 = 695.8474{{c}} {~15/14 = 105.5015{{c}})
+: [[error map]]: {{val| +2.023 -4.084 -2.924 +4.910 }}
+* [[CWE]]: ~56/45 = 400.0000{{c}}, ~3/2 = 695.1413{{c}} {~15/14 = 104.8587{{c}})
+: error map: {{val| 0.000 -6.814 -5.748 +2.022 }}
-Mapping: [{{val| 1 1 0 6 2 4 7 6 }}, {{val| 0 2 8 -11 5 -1 -10 -6 }}]
+{{Optimal ET sequence|legend=1| 12, 33cd, 45, 57 }}
-Mapping generators: ~2, ~11/9
+[[Badness]] (Sintel): 1.75
-POTE generator: ~11/9 = 348.810
+== Squares ==
+{{Main| Squares }}
-Vals: {{Val list| 7, 24, 31, 55, 86efh }}
+Squares splits the [[6/1|6th harmonic]] into four subminor sixths of [[11/7]]~[[14/9]] (or splits a [[8/3|perfect eleventh]] into four supermajor thirds of [[9/7]]~[[14/11]]), and uses it for a generator. It may be described as {{nowrap| 14c & 17c }}; its ploidacot is beta-tetracot. [[31edo]], with a generator of 11/31, makes for a good squares tuning, with 8-, 11-, and 14-note mos scales available. Squares tempers out [[2401/2400]], the breedsma, as well as [[2430/2401]].
-Badness: 0.017302
+[[Subgroup]]: 2.3.5.7
-Scales: [[mohaha7]], [[mohaha10]]
+[[Comma list]]: 81/80, 2401/2400
-== Mohamaq ==
+{{Mapping|legend=1| 1 -1 -8 -3 | 0 4 16 9 }}
-Subgroup: 2.3.5.7
-[[Comma list]]: 81/80, 392/375
+: mapping generators: ~2, ~14/9
-[[Mapping]]: [{{val| 1 1 0 -1 }}, {{val| 0 2 8 13 }}]
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.2488{{c}}, ~14/9 = 774.8640{{c}}
+: [[error map]]: {{val| +1.249 -3.748 +1.520 +1.204 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 774.1560{{c}}
+: error map: {{val| 0.000 -5.331 +0.183 -1.422 }}
-Mapping generators: ~2, ~25/21
+[[Minimax tuning]]:
+* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~9/7 = {{monzo| 1/2 0 -1/16 }}
+: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 3/2 0 9/16 0 }}
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
-[[POTE generator]]: ~25/21 = 350.586
+[[Algebraic generator]]: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.
-{{Val list|legend=1| 7d, 17c, 24, 65cc, 89ccd }}
+{{Optimal ET sequence|legend=1| 14c, 17c, 31, 169b, 200b }}
-[[Badness]]: 0.077734
+[[Badness]] (Sintel): 1.16
-Scales: [[mohaha7]], [[mohaha10]]
+Scales: [[skwares8]], [[skwares11]], [[skwares14]]
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 56/55, 77/75, 243/242
+Comma list: 81/80, 99/98, 121/120
-Mapping: [{{val| 1 1 0 -1 2 }}, {{val| 0 2 8 13 5 }}]
+Mapping: {{mapping| 1 -1 -8 -3 -3 | 0 4 16 9 10 }}
-Mapping generators: ~2, ~11/9
+Optimal tunings:
+* WE: ~2 = 1201.6657{{c}}, ~11/7 = 775.1171{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.1754{{c}}
-POTE generator: ~11/9 = 350.565
+{{Optimal ET sequence|legend=0| 14c, 17c, 31, 130bee, 169beee }}
-Vals: {{Val list| 7d, 17c, 24, 65cc, 89ccd }}
+Badness (Sintel): 0.715
-Badness: 0.036207
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Scales: [[mohaha7]], [[mohaha10]]
+Comma list: 66/65, 81/80, 99/98, 121/120
-=== 13-limit ===
+Mapping: {{mapping| 1 -1 -8 -3 -3 5 | 0 4 16 9 10 -2 }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 56/55, 66/65, 77/75, 243/242
+Optimal tunings:
+* WE: ~2 = 1199.8419{{c}}, ~11/7 = 774.3484{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4422{{c}}
-Mapping: [{{val| 1 1 0 -1 2 4 }}, {{val| 0 2 8 13 5 -1 }}]
+{{Optimal ET sequence|legend=0| 14c, 17c, 31, 79cf }}
-Mapping generators: ~2, ~11/9
+Badness (Sintel): 1.05
-POTE generator: ~11/9 = 350.745
+==== Squad ====
+Subgroup: 2.3.5.7.11.13
-Vals: {{Val list| 7d, 17c, 24, 41c, 65cc }}
+Comma list: 78/77, 81/80, 91/90, 99/98
-Badness: 0.028738
+Mapping: {{mapping| 1 -1 -8 -3 -3 -6 | 0 4 16 9 10 15 }}
-Scales: [[mohaha7]], [[mohaha10]]
+Optimal tunings:
+* WE: ~2 = 1202.0312{{c}}, ~11/7 = 775.5589{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4140{{c}}
-== Orphic ==
+{{Optimal ET sequence|legend=0| 14cf, 17c, 31f }}
-Subgroup: 2.3.5.7
-[[Comma list]]: 81/80, 5898240/5764801
+Badness (Sintel): 1.11
-[[Mapping]]: [{{val| 2 5 12 7 }}, {{val| 0 -4 -16 -3 }}]
+==== Agora ====
+Subgroup: 2.3.5.7.11.13
-Mapping generators: ~2401/1728, ~7/6
+Comma list: 81/80, 99/98, 105/104, 121/120
-{{Multival|legend=1| 8 32 6 32 -13 -76 }}
+Mapping: {{mapping| 1 -1 -8 -3 -3 -15 | 0 4 16 9 10 29 }}
-[[POTE generator]]: ~7/6 = 275.794
+Optimal tunings:
+* WE: ~2 = 1202.3228{{c}}, ~11/7 = 775.2214{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8617{{c}}
-{{Val list|legend=1| 26, 48c, 74, 174bd, 248bbd }}
+{{Optimal ET sequence|legend=0| 14cf, 31, 45ef, 76e }}
-[[Badness]]: 0.258825
+Badness (Sintel): 1.01
-=== 11-limit ===
+===== 17-limit =====
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13.17
-Comma list: 81/80, 99/98, 73728/73205
+Comma list: 81/80, 99/98, 105/104, 120/119, 121/119
-Mapping: [{{val| 2 5 12 7 6 }}, {{val| 0 -4 -16 -3 2 }}]
+Mapping: {{mapping| 1 -1 -8 -3 -3 -15 -3 | 0 4 16 9 10 29 11 }}
-Mapping generators: ~363/256, ~7/6
+Optimal tunings:
+* WE: ~2 = 1201.4340{{c}}, ~11/7 = 774.7375{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8955{{c}}
-POTE generator: ~7/6 = 275.762
+{{Optimal ET sequence|legend=0| 14cf, 31 }}
-Vals: {{Val list| 26, 48c, 74, 248bbd, 322bbdd }}
+Badness (Sintel): 1.15
-Badness: 0.101499
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-=== 13-limit ===
+Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119
-Subgroup: 2.3.5.7.11.13
-Comma list: 81/80, 99/98, 144/143, 2200/2197
+Mapping: {{mapping| 1 -1 -8 -3 -3 -15 -3 -8 | 0 4 16 9 10 29 11 19 }}
-Mapping: [{{val| 2 5 12 7 6 12 }}, {{val| 0 -4 -16 -3 2 -10 }}]
+Optimal tunings:
+* WE: ~2 = 1201.2461{{c}}, ~11/7 = 774.5783{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8479{{c}}
-Mapping generators: ~55/39, ~7/6
+{{Optimal ET sequence|legend=0| 14cf, 31 }}
-POTE generator: ~7/6 = 275.774
+Badness (Sintel): 1.15
-Vals: {{Val list| 26, 48c, 74, 174bd, 248bbd, 322bbdd }}
+=== Cuboctahedra ===
+Subgroup: 2.3.5.7.11
-Badness: 0.053482
+Comma list: 81/80, 385/384, 1375/1372
-== Mothra ==
+Mapping: {{mapping| 1 -1 -8 -3 17 | 0 4 16 9 -21 }}
-Mothra splits the fifth into three 8/7 generators. It uses [[1029/1024]], the gamelisma, to accomplish this deed and also tempers out [[1728/1715]], the orwell comma. Using [[31edo|31EDO]] with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra. In the 2.3.7 subgroup, mothra is identical to [[slendric]].
-Note that mothra can also be called '''cynder''' in the 7-limit, which can be a little confusing sometimes.
+Optimal tunings:
+* WE: ~2 = 1201.4436{{c}}, ~14/9 = 774.9386{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 774.0243{{c}}
-Subgroup: 2.3.5.7
+{{Optimal ET sequence|legend=0| 31, 107b, 138b, 169be, 200be }}
-[[Comma list]]: 81/80, 1029/1024
+Badness (Sintel): 1.88
-[[Mapping]]: [{{val| 1 1 0 3 }}, {{val| 0 3 12 -1 }}]
+== Jerome ==
+Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5<sup>1/20</sup>, or 139.316 cents. It may be described as {{nowrap| 17c & 26 }}; its ploidacot is pentacot. While the generator represents both 13/12 and 12/11, the CTE/CWE and Hieronymus generators are close to 13/12 in size.
-Mapping generators: ~2, ~8/7
+[[Subgroup]]: 2.3.5.7
-{{Multival|legend=1| 3 12 -1 12 -10 -36 }}
+[[Comma list]]: 81/80, 17280/16807
-[[POTE generator]]: ~8/7 = 232.193
+{{Mapping|legend=1| 1 1 0 2 | 0 5 20 7 }}
-[[Algebraic generator]]: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.
+: mapping generators: ~2, ~54/49
-[[Minimax tuning]]:
+[[Optimal tuning]]s:
-* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{Monzo| 0 0 1/12 }}
+* [[WE]]: ~2 = 1200.1640{{c}}, ~54/49 = 139.3624{{c}}
-: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 3 0 -1/12 0 }}]
+: [[error map]]: {{val| +0.164 -4.979 +0.934 +7.039 }}
-: [[Eigenmonzo]]s (unchanged intervals): 2, 5
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~54/49 = 139.3528{{c}}
+: error map: {{val| 0.000 -5.191 +0.741 +6.643 }}
-{{Val list|legend=1| 5, 26, 31 }}
+{{Optimal ET sequence|legend=1| 17c, 26, 43 }}
-[[Badness]]: 0.037146
+[[Badness]] (Sintel): 2.75
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 81/80, 99/98, 385/384
+Comma list: 81/80, 99/98, 864/847
-Mapping: [{{val| 1 1 0 3 5 }}, {{val| 0 3 12 -1 -8 }}]
+Mapping: {{mapping| 1 1 0 2 3 | 0 5 20 7 4 }}
-Mapping generators: ~2, ~8/7
+Optimal tunings:
+* WE: ~2 = 1201.4436{{c}}, ~12/11 = 139.3714{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 139.4038{{c}}
-POTE generator: ~8/7 = 232.031
+{{Optimal ET sequence|legend=0| 17c, 26, 43 }}
-Vals: {{Val list| 5, 26, 31, 88, 150be, 181bee }}
+Badness (Sintel): 1.58
-Badness: 0.025642
+=== 13-limit ===
-==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 81/80, 99/98, 105/104, 144/143
+Comma list: 78/77, 81/80, 99/98, 144/143
-Mapping: [{{val| 1 1 0 3 5 1 }}, {{val| 0 3 12 -1 -8 14 }}]
+Mapping: {{mapping| 1 1 0 2 3 3 | 0 5 20 7 4 6 }}
-Mapping generators: ~2, ~8/7
+Optimal tunings:
+* WE: ~2 = 1199.8860{{c}}, ~13/12 = 139.3737{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3817{{c}}
-POTE generator: ~8/7 = 231.811
+{{Optimal ET sequence|legend=0| 17c, 26, 43 }}
-Vals: {{Val list| 5, 26, 31, 57, 88 }}
+Badness (Sintel): 1.21
-Badness: 0.023954
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
-; Music:
+Comma list: 78/77, 81/80, 99/98, 144/143, 189/187
-* [http://micro.soonlabel.com/16-ET/mothra/20141028_mothra16br4.mp3 Prelude for solo piano in mothra16, brat 4 tuning] by [http://chrisvaisvil.com/prelude-for-solo-piano-in-mothra16-brat-4-tuning/ Chris Vaisvil]
-=== Cynder ===
+Mapping: {{mapping| 1 1 0 2 3 3 2 | 0 5 20 7 4 6 18 }}
-Subgroup: 2.3.5.7.11
-Comma list: 45/44, 81/80, 1029/1024
+Optimal tunings:
+* WE: ~2 = 1199.8346{{c}}, ~13/12 = 139.3431{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3544{{c}}
-Mapping: [{{val| 1 1 0 3 0 }}, {{val| 0 3 12 -1 18 }}]
+{{Optimal ET sequence|legend=0| 17cg, 26, 43 }}
-Mapping generators: ~2, ~8/7
+Badness (Sintel): 1.06
-POTE generator: ~8/7 = 231.317
+=== 19-limit ===
+Subgroup: 2.3.5.7.11.13.17.19
-Vals: {{Val list| 5e, 26, 57e, 83bce }}
+Comma list: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143
-Badness: 0.055706
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 45/44, 78/77, 81/80, 640/637
+Mapping: {{mapping| 1 1 0 2 3 3 2 1 | 0 5 20 7 4 6 18 28 }}
-Mapping: [{{val| 1 1 0 3 0 1 }}, {{val| 0 3 12 -1 18 14 }}]
+Optimal tunings:
+* WE: ~2 = 1199.8891{{c}}, ~13/12 = 139.3001{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3080{{c}}
-Mapping generators: ~2, ~8/7
+{{Optimal ET sequence|legend=0| 17cgh, 26, 43, 69 }}
-POTE generator: ~8/7 = 231.293
+Badness (Sintel): 1.11
-Vals: {{Val list| 5e, 26, 57e, 83bce }}
+== Meantritone ==
+The meantritone temperament tempers out the [[mirkwai comma]] (16875/16807) and [[trimyna comma]] (50421/50000) in the 7-limit. In this temperament, the 6th harmonic is split into five generators of ~10/7; the ploidacot of this temperament is beta-pentacot. The name ''meantritone'' is a portmanteau of ''meantone'' and ''tritone'', the latter is a generator of this temperament.
-Badness: 0.034124
+[[Subgroup]]: 2.3.5.7
-=== Mosura ===
+[[Comma list]]: 81/80, 16875/16807
-Subgroup: 2.3.5.7.11
-Comma list: 81/80, 176/175, 540/539
+{{Mapping|legend=1| 1 -1 -8 -7 | 0 5 20 19 }}
-Mapping: [{{val| 1 1 0 3 -1 }}, {{val| 0 3 12 -1 23 }}]
+: mapping generators: ~2, ~10/7
-Mapping generators: ~2, ~8/7
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.3832{{c}}, ~10/7 = 619.9478{{c}}
+: [[error map]]: {{val| +1.383 -3.599 +1.576 +0.499 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 619.3176{{c}}
+: error map: {{val| 0.000 -5.367 +0.038 -1.791 }}
-POTE generator: ~8/7 = 232.419
+{{Optimal ET sequence|legend=1| 29cd, 31, 188bcd, 219bbcd }}
-Vals: {{Val list| 31, 129, 160be, 191bce, 222bce, 253bcee }}
+[[Badness]] (Sintel): 2.08
-Badness: 0.0313
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-==== 13-limit ====
+Comma list: 81/80, 99/98, 2541/2500
-Subgroup: 2.3.5.7.11.13
-Comma list: 81/80, 144/143, 176/175, 196/195
+Mapping: {{mapping| 1 -1 -8 -7 -11 | 0 5 20 19 28 }}
-Mapping: [{{val| 1 1 0 3 -1 7 }}, {{val| 0 3 12 -1 23 -17 }}]
+Optimal tunings:
+* WE: ~2 = 1201.2054{{c}}, ~10/7 = 619.9752{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.4223{{c}}
-Mapping generators: ~2, ~8/7
+{{Optimal ET sequence|legend=0| 29cde, 31 }}
-POTE generator: ~8/7 = 232.640
+Badness (Sintel): 1.42
-Vals: {{Val list| 31, 36, 67, 98 }}
+== Injera ==
+Injera has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a ~15/14 semitone difference between a half-octave and a perfect fifth. Injera may be described as {{nowrap| 12 & 26 }}; its ploidacot is diploid monocot. It tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. [[38edo]], which is two parallel [[19edo]]s, is an excellent tuning for injera.
-Badness: 0.036857
+[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name]
-== Squares ==
+[[Subgroup]]: 2.3.5.7
-{{Main| Squares }}
-Squares splits the interval of an eleventh, or 8/3, into four supermajor third ([[9/7]]) intervals, and uses it for a generator. [[31edo|31EDO]], with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out [[2401/2400]], the breedsma, as well as [[2430/2401]].
+[[Comma list]]: 50/49, 81/80
-Subgroup: 2.3.5.7
+{{Mapping|legend=1| 2 0 -8 -7 | 0 1 4 4 }}
-[[Comma list]]: 81/80, 2401/2400
+: mapping generators: ~7/5, ~3
-[[Mapping]]: [{{val| 1 3 8 6 }}, {{val| 0 -4 -16 -9 }}]
+[[Optimal tuning]]s:
+* [[WE]]: ~7/5 = 600.6662{{c}}, ~3/2 = 695.1463{{c}} (~21/20 = 94.4801{{c}})
+: [[error map]]: {{val| +1.332 -5.476 -5.729 +12.425 }}
+* [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 694.7712{{c}} (~21/20 = 94.7712{{c}})
+: error map: {{val| 0.000 -7.184 -7.229 +10.259 }}
-Mapping generators: ~2, ~9/7
+[[Tuning ranges]]:
+* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
+* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [688.957, 701.955]
+* 9-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
-{{Multival|legend=1| 4 16 9 16 3 -24 }}
+{{Optimal ET sequence|legend=1| 12, 26, 38 }}
-[[POTE generator]]: ~9/7 = 425.942
+[[Badness]] (Sintel): 0.788
-[[Minimax tuning]]:
+; Music
-* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~9/7 = {{monzo| 1/2 0 -1/16 }}
+* [https://web.archive.org/web/20201127013520/http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 ''Two Pairs of Socks''] by [[Igliashon Jones]] – in [[26edo]] tuning
-: [{{monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 3/2 0 9/16 0 }}]
-: [[Eigenmonzo]]s (unchanged intervals): 2, 5
-[[Algebraic generator]]: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.
-{{Val list|legend=1| 14c, 17c, 31 }}
-[[Badness]]: 0.045993
-Scales: [[skwares8]], [[skwares11]], [[skwares14]]
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 81/80, 99/98, 121/120
+Comma list: 45/44, 50/49, 81/80
-Mapping: [{{val| 1 3 8 6 7 }}, {{val| 0 -4 -16 -9 -10 }}]
+Mapping: {{mapping| 2 0 -8 -7 -12 | 0 1 4 4 6 }}
-POTE generator: ~9/7 = 425.957
+Optimal tunings:
+* WE: ~7/5 = 600.9350{{c}}, ~3/2 = 693.9198{{c}} (~21/20 = 92.9848{{c}})
+* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 693.3539{{c}} (~21/20 = 93.3539{{c}})
-Vals: {{Val list| 14c, 17c, 31 }}
+Tuning ranges:
+* 11-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
+* 11-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
-Badness: 0.021636
+{{Optimal ET sequence|legend=0| 12, 26 }}
+Badness (Sintel): 0.764
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 66/65, 81/80, 99/98, 121/120
+Comma list: 45/44, 50/49, 78/77, 81/80
-Mapping: [{{val| 1 3 8 6 7 3 }}, {{val| 0 -4 -16 -9 -10 2 }}]
+Mapping: {{mapping| 2 0 -8 -7 -12 -21 | 0 1 4 4 6 9 }}
-POTE generator: ~9/7 = 425.550
+Optimal tunings:
+* WE: ~7/5 = 600.9982{{c}}, ~3/2 = 693.8249{{c}} (~21/20 = 92.8267{{c}})
+* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 693.0992{{c}} (~21/20 = 93.0992{{c}})
-Vals: {{Val list| 14c, 17c, 31, 79cf }}
+Tuning ranges:
+* 13-odd-limit diamond monotone: ~3/2 = 692.308 (15\26)
-Badness: 0.025514
+* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
-==== Squad ====
+{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 78/77, 81/80, 91/90, 99/98
+Badness (Sintel): 0.891
-Mapping: [{{val| 1 3 8 6 7 9 }}, {{val| 0 -4 -16 -9 -10 -15 }}]
-POTE generator: ~9/7 = 425.7516
-Vals: {{Val list| 14cf, 17c, 31f }}
-Badness: 0.026877
-==== Agora ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 81/80, 99/98, 105/104, 121/120
-Mapping: [{{val| 1 3 8 6 7 14 }}, {{val| 0 -4 -16 -9 -10 -29 }}]
-POTE generator: ~9/7 = 426.276
-Vals: {{Val list| 14cf, 31, 45ef, 76e }}
-Badness: 0.024522
 ===== 17-limit =====
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 81/80, 99/98, 105/104, 120/119, 121/119
+Comma list: 45/44, 50/49, 78/77, 81/80, 85/84
-Mapping: [{{val| 1 3 8 6 7 14 8 }}, {{val| 0 -4 -16 -9 -10 -29 -11 }}]
+Mapping: {{mapping| 2 0 -8 -7 -12 -21 5 | 0 1 4 4 6 9 1 }}
-POTE generator: ~9/7 = 426.187
+Optimal tunings:
+* WE: ~7/5 = 601.1757{{c}}, ~3/2 = 693.8441{{c}} (~21/20 = 92.6684{{c}})
+* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 692.8879{{c}} (~21/20 = 92.8879{{c}})
-Vals: {{Val list| 14cf, 31, 76e }}
+{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}
-Badness: 0.022573
+Badness (Sintel): 0.935
 ===== 19-limit =====
 Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119
+Comma list: 45/44, 50/49, 57/56, 78/77, 81/80, 85/84
+Mapping: {{mapping| 2 0 -8 -7 -12 -21 5 -1 | 0 1 4 4 6 9 1 3 }}
+Optimal tunings:
+* WE: ~7/5 = 601.4245{{c}}, ~3/2 = 693.9426{{c}} (~21/20 = 92.5181{{c}})
+* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 692.7606{{c}} (~21/20 = 92.7606{{c}})
-Mapping: [{{val| 1 3 8 6 7 14 8 11 }}, {{val| 0 -4 -16 -9 -10 -29 -11 -19 }}]
+{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}
-POTE generator: ~9/7 = 426.225
+Badness (Sintel): 0.920
-Vals: {{Val list| 14cf, 31, 76e }}
+==== Enjera ====
+Subgroup: 2.3.5.7.11.13
-Badness: 0.018839
+Comma list: 27/26, 40/39, 45/44, 50/49
-=== Cuboctahedra ===
+Mapping: {{mapping| 2 0 -8 -7 -12 -2 | 0 1 4 4 6 3 }}
-Subgroup: 2.3.5.7.11
-Comma list: 81/80, 385/384, 1375/1372
+Optimal tunings:
+* WE: ~7/5 = 599.1863{{c}}, ~3/2 = 693.1791{{c}} (~21/20 = 93.9929{{c}})
+* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 693.6809{{c}} (~21/20 = 93.6809{{c}})
-Mapping: [{{val| 1 3 8 6 -4 }}, {{val| 0 -4 -16 -9 21 }}]
+{{Optimal ET sequence|legend=0| 10cdeef, 12f }}
-POTE generator: ~9/7 = 425.993
+Badness (Sintel): 1.10
-Vals: {{Val list| 14ce, 17ce, 31, 107b, 138b, 169be, 200be }}
+=== Injerous ===
+Subgroup: 2.3.5.7.11
-Badness: 0.056826
+Comma list: 33/32, 50/49, 55/54
-== Liese ==
+Mapping: {{mapping| 2 0 -8 -7 10 | 0 1 4 4 -1 }}
-<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>
-Liese splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. [[74edo|74EDO]] makes for a good liese tuning, though [[19edo|19EDO]] can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55.
+Optimal tunings:
+* WE: ~7/5 = 603.1682{{c}}, ~3/2 = 694.1945{{c}} (~21/20 = 91.0264{{c}})
+* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 691.6107{{c}} (~21/20 = 91.6107{{c}})
-Subgroup: 2.3.5.7
+{{Optimal ET sequence|legend=0| 12e, 14c, 26e, 40cee }}
-[[Comma list]]: 81/80, 686/675
+Badness (Sintel): 1.28
-[[Mapping]]: [{{val| 1 0 -4 -3 }}, {{val| 0 3 12 11 }}]
+=== Lahoh ===
+Subgroup: 2.3.5.7.11
-Mapping generators: ~2, ~10/7
+Comma list: 50/49, 56/55, 81/77
-{{Multival|legend=1| 3 12 11 12 9 -8 }}
+Mapping: {{mapping| 2 0 -8 -7 7 | 0 1 4 4 0 }}
-[[POTE generator]]: ~10/7 = 632.406
+Optimal tunings:
+* WE: ~7/5 = 597.3179{{c}}, ~3/2 = 695.8759{{c}} (~21/20 = 98.5581{{c}})
+* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 697.8757{{c}} (~21/20 = 97.8757{{c}})
-Minimax tuning:
+{{Optimal ET sequence|legend=0| 10cd, 12 }}
-* 7- and 9-odd-limit: ~10/7 = {{Monzo| 1/3 0 1/12 }}
-: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 2/3 0 11/12 0 }}]
-: [[Eigenmonzo]]s (unchanged intervals): 2, 5
-[[Algebraic generator]]: Radix, the real root of ''x''<sup>5</sup> - 2''x''<sup>4</sup> + 2''x''<sup>3</sup> - 2''x''<sup>2</sup> + 2''x'' - 2, also a root of ''x''<sup>6</sup> - ''x''<sup>5</sup> - 2. The recurrence converges.
+Badness (Sintel): 1.42
-{{Val list|legend=1| 17c, 19, 55, 74d }}
+=== Teff ===
+{{Main| Teff }}
-[[Badness]]: 0.046706
+Teff, found and named by [[Mason Green]], is to injera what mohajira is to meantone; it splits the generator in halves in order to accommodate higher-limit intervals, creating a half-octave quartertone temperament. Its ploidacot is diploid alpha-dicot.
-=== Liesel ===
 Subgroup: 2.3.5.7.11
-Comma list: 56/55, 81/80, 540/539
+Comma list: 50/49, 81/80, 864/847
-Mapping: [{{val| 1 0 -4 -3 4 }}, {{val| 0 3 12 11 -1 }}]
+Mapping: {{mapping| 2 1 -4 -3 8 | 0 2 8 8 -1 }}
-Mapping generators: ~2, ~10/7
+: mapping generators: ~7/5, ~16/11
-POTE generator: ~10/7 = 633.073
+Optimal tunings:
+* WE: ~7/5 = 600.2802{{c}}, ~16/11 = 647.7720{{c}} (~33/32 = 47.4918{{c}})
+* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.5224{{c}} (~33/32 = 47.5224{{c}})
-Vals: {{Val list| 17c, 19, 36, 91cee }}
+{{Optimal ET sequence|legend=0| 24d, 26, 50d }}
-Badness: 0.040721
+Badness (Sintel): 2.34
 ==== 13-limit ====
-Liesel is a very natural 13-limit tuning, given the generator is so near 13/9.
 Subgroup: 2.3.5.7.11.13
-Comma list: 56/55, 78/77, 81/80, 91/90
+Comma list: 50/49, 78/77, 81/80, 144/143
-Mapping: [{{val| 1 0 -4 -3 4 0 }}, {{val| 0 3 12 11 -1 7 }}]
+Mapping: {{mapping| 2 1 -4 -3 8 2 | 0 2 8 8 -1 5 }}
-Mapping generators: ~2, ~10/7
+Optimal tunings:
+* WE: ~7/5 = 600.3037{{c}}, ~16/11 = 647.7954{{c}} (~33/32 = 47.4917{{c}})
+* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.5256{{c}} (~33/32 = 47.5256{{c}})
-POTE generator: ~10/7 = 633.042
+{{Optimal ET sequence|legend=0| 24d, 26, 50d }}
-Vals: {{Val list| 17c, 19, 36, 91ceef }}
+Badness (Sintel): 1.65
-Badness: 0.027304
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
-=== Elisa ===
+Comma list: 50/49, 78/77, 81/80, 85/84, 144/143
-Subgroup: 2.3.5.7.11
-Comma list: 77/75, 81/80, 99/98
+Mapping: {{mapping| 2 1 -4 -3 8 2 6 | 0 2 8 8 -1 5 2 }}
-Mapping: [{{val| 1 0 -4 -3 -5 }}, {{val| 0 3 12 11 16 }}]
+Optimal tunings:
+* WE: ~7/5 = 600.5123{{c}}, ~16/11 = 647.8970{{c}} (~34/33 = 47.3846{{c}})
+* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.4314{{c}} (~34/33 = 47.4314{{c}})
-Mapping generators: ~2, ~10/7
+{{Optimal ET sequence|legend=0| 24d, 26 }}
-POTE generator: ~10/7 = 633.061
+Badness (Sintel): 1.50
-Vals: {{Val list| 17c, 19e, 36e }}
+==== 19-limit ====
+Subgroup: 2.3.5.7.11.13.17.19
-Badness: 0.041592
+Comma list: 50/49, 57/56, 78/77, 81/80, 85/84, 144/143
-==== 13-limit ====
+Mapping: {{mapping| 2 1 -4 -3 8 2 6 2 | 0 2 8 8 -1 5 2 6 }}
-Subgroup: 2.3.5.7.11
-Comma list: 66/65, 77/75, 81/80, 99/98
+Optimal tunings:
+* WE: ~7/5 = 600.6308{{c}}, ~16/11 = 648.0424{{c}} (~34/33 = 47.4116{{c}})
+* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.4715{{c}} (~34/33 = 47.4715{{c}})
-Mapping: [{{val| 1 0 -4 -3 -5 0 }}, {{val| 0 3 12 11 16 7 }}]
+{{Optimal ET sequence|legend=0| 24d, 26 }}
-Mapping generators: ~2, ~10/7
+Badness (Sintel): 1.41
-POTE generator: ~10/7 = 632.991
+== Pombe ==
+Pombe (named after the African millet beer) is a variant of [[#Teff]] by [[User:Kaiveran|Kaiveran Lugheidh]] that eschews the tempering of 50/49 to attain more accuracy in the 7-limit. Its ploidacot is diploid alpha-dicot, the same as teff. Oddly, the 7th harmonic has a lesser generator distance than in teff (-5 vs +8), but this combined with the fact that other harmonics are in the opposite direction means that the 7-limit diamond is more complex overall.
-Vals: {{Val list| 17c, 19e, 36e }}
+[[Subgroup]]: 2.3.5.7
-Badness: 0.026922
+[[Comma list]]: 81/80, 300125/294912
-=== Lisa ===
+{{Mapping|legend=1| 2 1 -4 11 | 0 2 8 -5 }}
-Subgroup: 2.3.5.7.11
-Comma list: 45/44, 81/80, 343/330
+: mapping generators: ~735/512, ~35/24
-Mapping: [{{val| 1 0 -4 -3 -6 }}, {{val| 0 3 12 11 18 }}]
+[[Optimal tuning]]s:
+* [[WE]]: ~735/512 = 601.0652{{c}}, ~35/24 = 648.9295{{c}} (~36/35 = 47.8642{{c}})
+: [[error map]]: {{val| +2.130 -3.031 +0.861 -1.756 }}
+* [[CWE]]: ~735/512 = 600.0000{{c}}, ~35/24 = 647.8628{{c}} (~36/35 = 47.8628{{c}})
+: error map: {{val| 0.000 -6.229 -3.411 -8.140 }}
-Mapping generators: ~2, ~10/7
+{{Optimal ET sequence|legend=1| 24, 26, 50, 126bcd, 176bcdd, 226bbcdd }}
-POTE generator: ~10/7 = 631.370
+[[Badness]] (Sintel): 2.94
-Vals: {{Val list| 17cee, 19 }}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Badness: 0.054829
+Comma list: 81/80, 245/242, 385/384
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 45/44, 81/80, 91/88, 147/143
+Mapping: {{mapping| 2 1 -4 11 8 | 0 2 8 -5 -1 }}
-Mapping: [{{val| 1 0 -4 -3 -6 0 }}, {{val| 0 3 12 11 18 7 }}]
+Optimal tunings:
+* WE: ~99/70 = 600.7890{{c}}, ~16/11 = 648.7592{{c}} (~36/35 = 47.9701{{c}})
+* CWE: ~99/70 = 600.0000{{c}}, ~16/11 = 647.9516{{c}} (~36/35 = 47.9516{{c}})
-Mapping generators: ~2, ~10/7
+{{Optimal ET sequence|legend=0| 24, 26, 50 }}
-POTE generator: ~10/7 = 631.221
+Badness (Sintel): 1.72
-Vals: {{Val list| 17cee, 19 }}
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Badness: 0.036144
+Comma list: 81/80, 105/104, 144/143, 245/242
-== Jerome ==
+Mapping: {{mapping| 2 1 -4 11 8 2 | 0 2 8 -5 -1 5 }}
-Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5<sup>1/20</sup>, or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size.
-Subgroup: 2.3.5.7
+Optimal tunings:
+* WE: ~99/70 = 600.6971{{c}}, ~16/11 = 648.6029{{c}} (~36/35 = 47.9058{{c}})
+* CWE: ~99/70 = 600.0000{{c}}, ~16/11 = 647.8990{{c}} (~36/35 = 47.8990{{c}})
-[[Comma list]]: 81/80, 17280/16807
+{{Optimal ET sequence|legend=0| 24, 26, 50 }}
-[[Mapping]]: [{{val| 1 1 0 2 }}, {{val| 0 5 20 7 }}]
+Badness (Sintel): 1.28
-Mapping generators: ~2, ~54/49
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
-{{Multival|legend=1| 5 20 7 20 -3 -40 }}
+Comma list: 81/80, 105/104, 144/143, 245/242, 273/272
-[[POTE generator]]: ~54/49 = 139.343
+Mapping: {{mapping| 2 1 -4 11 8 2 6 | 0 2 8 -5 -1 5 2 }}
-{{Val list|legend=1| 17c, 26, 43, 69, 112bd }}
+Optimal tunings:
+* WE: ~17/12 = 600.7610{{c}}, ~16/11 = 648.6638{{c}} (~36/35 = 47.9028{{c}})
+* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 647.8990{{c}} (~36/35 = 47.8990{{c}})
-[[Badness]]: 0.108656
+{{Optimal ET sequence|legend=0| 24, 26, 50 }}
-=== 11-limit ===
+Badness (Sintel): 1.08
-Subgroup: 2.3.5.7.11
-Comma list: 81/80, 99/98, 864/847
+=== 19-limit ===
+Subgroup: 2.3.5.7.11.13.17.19
-Mapping: [{{val| 1 1 0 2 3 }}, {{val| 0 5 20 7 4 }}]
+Comma list: 81/80, 105/104, 133/132, 144/143, 171/170, 210/209
-Mapping generators: ~2, ~12/11
+Mapping: {{mapping| 2 1 -4 11 8 2 6 2 | 0 2 8 -5 -1 5 2 6 }}
-POTE generator: ~12/11 = 139.428
+Optimal tunings:
+* WE: ~17/12 = 600.8048{{c}}, ~16/11 = 648.7494{{c}} (~36/35 = 47.9446{{c}})
+* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 647.9425{{c}} (~36/35 = 47.9425{{c}})
-Vals: {{Val list| 17c, 26, 43, 69 }}
+{{Optimal ET sequence|legend=0| 24, 26, 50 }}
-Badness: 0.047914
+Badness (Sintel): 1.01
-=== 13-limit ===
+== Orphic ==
-Subgroup: 2.3.5.7.11.13
+Orphic has a semi-octave period and four generators plus a period gives the 3rd harmonic; its ploidacot is diploid alpha-tetracot.
-Comma list: 78/77, 81/80, 99/98, 144/143
+[[Subgroup]]: 2.3.5.7
-Mapping: [{{val| 1 1 0 2 3 3 }}, {{val| 0 5 20 7 4 6 }}]
+[[Comma list]]: 81/80, 5898240/5764801
-Mapping generators: ~2, ~12/11
+{{Mapping|legend=1| 2 1 -4 4 | 0 4 16 3 }}
-POTE generator: ~12/11 = 139.387
+: mapping generators: ~2401/1728, ~343/288
-Vals: {{Val list| 17c, 26, 43, 69 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2401/1728 = 600.1767{{c}}, ~343/288 = 324.3015{{c}} (~7/6 = 275.8751{{c}})
+: [[error map]]: {{val| +0.353 -4.572 +1.804 +4.785 }}
+* [[CWE]]: ~2401/1728 = 600.0000{{c}}, ~343/288 = 324.2285{{c}} (~7/6 = 275.7715{{c}})
+: error map: {{val| 0.000 -5.041 +1.342 +3.860 }}
-Badness: 0.029285
+{{Optimal ET sequence|legend=1| 26, 48c, 74 }}
-=== 17-limit ===
+[[Badness]] (Sintel): 6.55
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 78/77, 81/80, 99/98, 144/143, 189/187
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Mapping: [{{val| 1 1 0 2 3 3 2 }}, {{val| 0 5 20 7 4 6 18 }}]
+Comma list: 81/80, 99/98, 73728/73205
-Mapping generators: ~2, ~12/11
+Mapping: {{mapping| 2 1 -4 4 8 | 0 4 16 3 -2 }}
-POTE generator: ~12/11 = 139.362
+Optimal tunings:
+* WE: ~363/256 = 600.1011{{c}}, ~77/64 = 324.2923{{c}} (~7/6 = 275.8088{{c}})
+* CWE: ~363/256 = 600.0000{{c}}, ~77/64 = 324.2463{{c}} (~7/6 = 275.7537{{c}})
-Vals: {{Val list| 17cg, 26, 43, 69 }}
+{{Optimal ET sequence|legend=0| 26, 48c, 74 }}
-Badness: 0.020878
+Badness (Sintel): 3.36
-=== 19-limit ===
+=== 13-limit ===
-Subgroup: 2.3.5.7.11.13.17.19
+Subgroup: 2.3.5.7.11.13
-Comma list: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143
+Comma list: 81/80, 99/98, 144/143, 2200/2197
-Mapping: [{{val| 1 1 0 2 3 3 2 1 }}, {{val| 0 5 20 7 4 6 18 28 }}]
+Mapping: {{mapping| 2 1 -4 4 8 2 | 0 4 16 3 -2 10 }}
-Mapping generators: ~2, ~12/11
+Optimal tunings:
+* WE: ~55/39 = 600.0540{{c}}, ~77/64 = 324.2551{{c}} (~7/6 = 275.7989{{c}})
+* CWE: ~55/39 = 600.0000{{c}}, ~77/64 = 324.2307{{c}} (~7/6 = 275.7693{{c}})
-POTE generator: ~12/11 = 139.313
+{{Optimal ET sequence|legend=0| 26, 48c, 74 }}
-Vals: {{Val list| 17cgh, 26, 43, 69 }}
+Badness (Sintel): 2.21
-Badness: 0.018229
 == Cloudtone ==
-The ''cloudtone'' temperament (5&amp;50) tempers out the [[cloudy comma]], 16807/16384 and the [[81/80|syntonic comma]], 81/80 in the 7-limit. It can be extended to the 11- and 13-limit by adding 385/384 and 105/104 to the comma list in this order.
+The cloudtone temperament tempers out the [[cloudy comma]], 16807/16384 and the [[syntonic comma]], 81/80 in the 7-limit. It may be described as {{nowrap| 5 & 50 }}; its ploidacot is pentaploid monocot. It can be extended to the 11- and 13-limit by adding 385/384 and 105/104 to the comma list in this order.
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 81/80, 16807/16384
-[[Mapping]]: [{{val| 5 0 -20 14 }}, {{val| 0 1 4 0 }}]
+{{Mapping|legend=1| 5 0 -20 14 | 0 1 4 0 }}
-Mapping generators: ~8/7, ~3
+: mapping generators: ~8/7, ~3
-{{Multival|legend=1| 5 20 0 20 -14 -56 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~8/7 = 240.4267{{c}}, ~3/2 = 696.9566{{c}} (~49/48 = 24.3235{{c}})
+: [[error map]]: {{val| +2.133 -2.865 +1.513 -2.852 }}
+* [[CWE]]: ~8/7 = 240.0000{{c}}, ~3/2 = 696.1637{{c}} (~49/48 = 23.8373{{c}})
+: error map: {{val| 0.000 -5.791 -1.659 -8.826 }}
-[[POTE generator]]: ~3/2 = 695.720
+{{Optimal ET sequence|legend=1| 5, 40c, 45, 50 }}
-{{Val list|legend=1| 5, 45, 50 }}
+[[Badness]] (Sintel): 2.59
-[[Badness]]: 0.102256
 === 11-limit ===
@@ Line 2,080: / Line 2,222: @@
 Comma list: 81/80, 385/384, 2401/2376
-Mapping: [{{val| 5 0 -20 14 41 }}, {{val| 0 1 4 0 -3 }}]
+Mapping: {{mapping| 5 0 -20 14 41 | 0 1 4 0 -3 }}
-Mapping generators: ~8/7, ~3
+Optimal tunings:
+* WE: ~8/7 = 240.2740{{c}}, ~3/2 = 697.3317{{c}} (~56/55 = 23.4904{{c}})
+* CWE: ~8/7 = 240.0000{{c}}, ~3/2 = 696.6269{{c}} (~56/55 = 23.3731{{c}})
-POTE generator: ~3/2 = 696.536
+{{Optimal ET sequence|legend=0| 5, 45, 50 }}
-Vals: {{Val list| 5, 45, 50, 155bdd, 205bddd }}
+Badness (Sintel): 2.33
-Badness: 0.070378
 === 13-limit ===
@@ Line 2,095: / Line 2,237: @@
 Comma list: 81/80, 105/104, 144/143, 2401/2376
-Mapping: [{{val| 5 0 -20 14 41 -21 }}, {{val| 0 1 4 0 -3 5 }}]
+Mapping: {{mapping| 5 0 -20 14 41 -21 | 0 1 4 0 -3 5 }}
-Mapping generators: ~8/7, ~3
+Optimal tunings:
+* WE: ~8/7 = 240.2435{{c}}, ~3/2 = 696.8686{{c}} (~91/90 = 23.8618{{c}})
+* CWE: ~8/7 = 240.0000{{c}}, ~3/2 = 696.2653{{c}} (~91/90 = 23.7347{{c}})
-POTE generator: ~3/2 = 696.162
+{{Optimal ET sequence|legend=0| 5, 45f, 50 }}
-Vals: {{Val list| 5, 45f, 50 }}
+Badness (Sintel): 2.02
-Badness: 0.048829
+== Subgroup extensions ==
+=== Stützel (2.3.5.19) ===
+[[Subgroup]]: 2.3.5.19
-== Meanmag ==
+[[Comma list]]: 81/80, 96/95
-Subgroup: 2.3.5.7
-[[Comma list]]: 81/80, 3125/3072
+{{Mapping|legend=2| 1 0 -4 9 | 0 1 4 -3 }}
-[[Mapping]]: [{{val| 19 30 44 0 }}, {{val| 0 0 0 1 }}]
+{{Mapping|legend=3| 1 0 -4 0 0 0 0 9 | 0 1 4 0 0 0 0 -3 }}
-Mapping generators: ~25/24, ~7
+: mapping generators: ~2, ~3
-{{Multival|legend=1| 0 0 19 0 30 44 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.5513{{c}}, ~3/2 = 697.6058{{c}}
+: [[error map]]: {{val| -0.448 -4.798 +4.110 +6.977 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.8222{{c}}
+: error map: {{val| 0.000 -4.133 +4.975 +9.020 }}
-[[POTE generator]]: ~8/7 = 238.396
+{{Optimal ET sequence|legend=1| 5, 7, 12, 31, 43, 98h }}
-{{Val list|legend=1| 19, 38, 57, 76, 95bc }}
+[[Badness]] (Sintel): 0.324
-[[Badness]]: 0.077023
+=== Hypnotone ===
+Hypnotone is no-sevens [[#Flattone|flattone]].
-=== 11-limit ===
+[[Subgroup]]: 2.3.5.11
-Subgroup: 2.3.5.7.11
+[[Comma list]]: 45/44, 81/80
+{{Mapping|legend=2| 1 0 -4 -6 | 0 1 4 6 }}
-Comma list: 81/80, 385/384, 625/616
+{{Mapping|legend=3| 1 0 -4 0 -6 | 0 1 4 0 6 }}
-Mapping: [{{val| 19 30 44 0 119 }}, {{val| 0 0 0 1 -1 }}]
+: mapping generators: ~2, ~3
-Mapping generators: ~25/24, ~7
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1202.0621{{c}}, ~3/2 = 694.5448{{c}}
+: [[error map]]: {{val| +2.062 -5.348 -8.135 +15.951 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.9085{{c}}
+: error map: {{val| 0.000 -8.047 -10.680 +12.133 }}
-POTE generator: ~8/7 = 233.486
+{{Optimal ET sequence|legend=1| 7, 12, 19, 26, 45 }}
-Vals: {{Val list| 19, 38, 57, 76 }}
+[[Badness]] (Sintel): 0.326
-Badness: 0.066829
+==== 2.3.5.11.13 subgroup ====
+Subgroup: 2.3.5.11.13
-=== 13-limit ===
+Comma list: 45/44, 65/64, 81/80
-Subgroup: 2.3.5.7.11.13
+Subgroup-val mapping: {{mapping| 1 0 -4 -6 10 | 0 1 4 6 -4 }}
-Comma list: 81/80, 105/104, 144/143, 625/616
+Gencom mapping: {{mapping| 1 0 -4 0 -6 10 | 0 1 4 0 6 -4 }}
-Mapping: [{{val| 19 30 44 0 119 17 }}, {{val| 0 0 0 1 -1 1 }}]
+Optimal tunings:
+* WE: ~2 = 1202.6916{{c}}, ~3/2 = 694.4181{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0870{{c}}
-Mapping generators: ~25/24, ~7
+{{Optimal ET sequence|legend=0| 7, 12, 19, 26, 45f }}
-POTE generator: ~8/7 = 234.890
+Badness (Sintel): 0.561
-Vals: {{Val list| 19, 38, 57, 76 }}
+=== Dequarter ===
+[[Subgroup]]: 2.3.5.11
-Badness: 0.045844
+[[Comma list]]: 33/32, 55/54
-== Undevigintone ==
+{{Mapping|legend=2| 1 0 -4 5 | 0 1 4 -1 }}
-Subgroup: 2.3.5.7.11
-[[Comma list]]: 49/48, 81/80, 126/125
+{{Mapping|legend=3| 1 0 -4 0 5 | 0 1 4 0 -1 }}
-[[Mapping]]: [{{val| 19 30 44 53 0 }}, {{val| 0 0 0 0 1 }}]
+: mapping generators: ~2, ~3
-Mapping generators: ~21/20, ~11
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1206.5832{{c}}, ~3/2 = 695.8763{{c}}
+: [[error map]]: {{val| +6.583 +0.504 -2.809 -20.862 }}
+* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 693.1206{{c}}
+: error map: {{val| 0.000 -8.834 -13.831 -44.439 }}
-[[POTE generator]]: ~11/8 = 538.047
+{{Optimal ET sequence|legend=1| 5, 7, 19e, 26e }}
-{{Val list|legend=1| 19, 38d }}
+[[Badness]] (Sintel): 0.451
-[[Badness]]: 0.036387
+==== Dreamtone ====
+Subgroup: 2.3.5.11.13
-=== 13-limit ===
+Comma list: 33/32, 55/54, 975/968
-Subgroup: 2.3.5.7.11.13
-Comma list: 49/48, 65/64, 81/80, 126/125
+Subgroup-val mapping: {{mapping| 1 0 -4 5 21 | 0 1 4 -1 -11 }}
-Mapping: [{{val| 19 30 44 53 0 70 }}, {{val| 0 0 0 0 1 0 }}]
+Gencom mapping: {{mapping| 1 0 -4 0 5 21 | 0 1 4 0 -1 -11 }}
-Mapping generators: ~21/20, ~11
+Optimal tunings:
+* WE: ~2 = 1207.8248{{c}}, ~3/2 = 694.7806{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 690.1826{{c}}
-POTE generator: ~11/8 = 537.061
+{{Optimal ET sequence|legend=0| 7, 19eff, 26eff, 33ceeff, 40ceeff }}
-Vals: {{Val list| 19, 38df }}
+Badness (Sintel): 1.40
-Badness: 0.022933
+== References ==
+<references/>
-[[Category:Regular temperament theory]]
+[[Category:Temperament families]]
-[[Category:Temperament family]]
 [[Category:Meantone family| ]] <!-- main article -->
-[[Category:Meantone|#]] <!-- key article -->
+[[Category:Meantone| ]] <!-- key article -->
 [[Category:Rank 2]]
 [[Category:Listen]]

Meantone family: Difference between revisions