@@ Line 1: / Line 1: @@
-<span style="display: block; text-align: right;">
+{{interwiki
-__FORCETOC__
+| de = Mitteltönig
-[[de:mitteltönig]]</span>
+| en = Meantone family
+| es =
+| ja =
+}}
+{{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]].
-The [[5-limit]] parent [[Comma|comma]] of the [[Meantone|meantone]] family is the Didymus or [http://en.wikipedia.org/wiki/Syntonic_comma syntonic comma], 81/80. This is the one they all temper out. The [[Monzos_and_Interval_Space|monzo]] for 81/80 goes |-4 4 -1&gt;, and that can be flipped around to the corresponding [[Wedgies_and_Multivals|wedgie]], &lt;&lt;1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.
+== Meantone ==
+{{Main| Meantone }}
-=5-limit meantone=
+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]].
-Comma: 81/80
-[[POTE_tuning|POTE generator]]: ~3/2 = 696.239
+[[Subgroup]]: 2.3.5
-Mapping generator: ~3
+[[Comma list]]: 81/80
-[[Tuning_Ranges_of_Regular_Temperaments|valid range]]: [685.714, 720.000] (7 to 5)
+{{Mapping|legend=1| 1 0 -4 | 0 1 4 }}
-nice range: [694.786, 701.955] (1/3 comma to Pythagorean)
+: mapping generators: ~2, ~3
-strict range: [694.786, 701.955]
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.3906{{c}}, ~3/2 = 697.0455{{c}}
+: [[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 }}
-[[Map|Map]]: [&lt;1 0 -4|, &lt;0 1 4|]
+[[Minimax tuning]]:
+* [[5-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma)
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
-EDOs: [[5edo|5]], [[7edo|7]], [[12edo|12]], [[19edo|19]], [[26edo|26]], [[31edo|31]], [[43edo|43]], [[45edo|45]], [[50edo|50]], [[55edo|55]], [[67edo|67]], [[69edo|69]], [[74edo|74]], [[81edo|81]], [[88edo|88]], [[98edo|98]], [[105edo|105]], [[117edo|117]], [[205edo|205b]]
+[[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] (1/3-comma to Pyth.)
-[[Badness|Badness]]: 0.00736
+{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b }}
-==Seven limit children==
+[[Badness]] (Sintel): 0.173
-The [[7-limit|7-limit]] children of 81/80 are septimal meantone, with normal comma list [|-4 4 -1&gt;, |-13 10 0 -1&gt;], flattone, with normal list [|-4 4 -1&gt;, |-17 9 0 1&gt;], dominant, with normal list [|-4 4 -1&gt;, |6 -2 0 -1&gt;], sharptone, with normal list [|-4 4 -1&gt;, |2 -3 0 1&gt;], injera, with normal list [|-4 4 -1&gt;, |-7 8 0 -2&gt;], mohajira, with normal list [|-4 4 -1&gt;, |-23 11 0 2&gt;], godzilla, with normal list [|-4 4 -1&gt;, |-4 -1 0 2&gt;], mothra, with normal list [|-4 4 -1&gt;, |-10 1 0 3&gt;], squares, with normal list [|-4 4 -1&gt;, |-3 9 0 -4&gt;], and liese, with normal list [|-4 4 -1&gt;, |-9 11 0 -3&gt;].
-=Septimal meantone=
+=== Overview to extensions ===
-<span style="display: block; text-align: right;">[[:de:septimal-mitteltönig|Deutsch]]</span>
+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.
-[https://en.wikipedia.org/wiki/Septimal_meantone_temperament Wikipedia article]
+==== 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.
-The comma |-13 10 0 -1&gt; for septimal meantone tells us that the interval class for 7 is 10 generator steps up. Hence, the [[7/4]] of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, and [[7/5]], C-F#, the tritone. The [[Wedgies_and_Multivals|wedgie]] for septimal meantone is &lt;&lt;1 4 10 4 13 12||, again telling us how to get to 5 and 7 in terms of generator steps. The temperament, aside from what is on the normal list, tempers out 126/125 and 225/224, and [[31edo]] is a good tuning for it.
+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.
-[[Comma]]s: 81/80, 126/125
+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.
-and [[9-odd-limit|9-limit]] minimax
+==== Splitting the meantone fifth into two (243/242) ====
+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.
-[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |-3 0 5/2 0&gt;]
+==== Splitting the meantone fifth into three (1029/1024) ====
+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.
-[[Eigenmonzo]]s: 2, 5
+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]]).
-[[Tuning_Ranges_of_Regular_Temperaments|valid range]]: [694.737, 700.000] (19 to 12)
+==== 31edo as splitting the fifth into two, three and nine ====
+[[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).
-nice range: [694.786, 701.955]
+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]]
-strict range: [694.786, 700.000]
+The rest are considered below.
-[[POTE_tuning|POTE generator]]: 696.495
+== Septimal meantone ==
+<div style="float:right">[[:de:septimal-mitteltönig|Deutsch]]</div>
+{{Main| Meantone #Septimal meantone}}
+{{Wikipedia| Septimal meantone temperament }}
-Mapping generator: ~3
+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.
-Algebraic generator: Cybozem, the real root of 15x^3-10x^2-18, which comes to 503.4257 cents. The recurrence converges quickly.
+[[Subgroup]]: 2.3.5.7
-[[Map]]: [&lt;1 0 -4 -13|, &lt;0 1 4 10|]
+[[Comma list]]: 81/80, 126/125
-[[Generator]]s: 2, 3
+{{Mapping|legend=1| 1 0 -4 -13 | 0 1 4 10 }}
-[[Wedgie]]: &lt;&lt;1 4 10 4 13 12||
+[[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 }}
-EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[43edo|43]], [[50edo|50]], [[74edo|74]], [[81edo|81]], [[105edo|105]], [[143edo|143b]]
+[[Minimax tuning]]:
+* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma)
+: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | -3 0 5/2 0 }}
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
-[[Badness]]: 0.0137
+[[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] (1/3-comma to Pyth.)
+* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
-==Bimeantone==
+[[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.
-/8 is mapped to half octave minus the meantone diesis.
-Commas: 81/80, 126/125, 245/242
+{{Optimal ET sequence|legend=1| 12, 19, 31, 81, 112b, 143b }}
-[[POTE_tuning|POTE generator]]: ~3/2 = 696.016
+[[Badness]] (Sintel): 0.347
-Map: [&lt;2 0 -8 -26 -31|, &lt;0 1 4 10 12|]
+=== Undecimal meantone (huygens) ===
+{{Redirect|Huygens|the Dutch mathematician, physicist and astronomer|Wikipedia: Christiaan Huygens}}
+{{See also| Huygens vs meanpop }}
-EDOs: 12, 38d, 50
+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.
-Badness: 0.0381
+Subgroup: 2.3.5.7.11
-===13-limit===
+Comma list: 81/80, 99/98, 126/125
-Commas: 81/80, 105/104, 126/125, 245/242
-[[POTE_tuning|POTE generator]]: ~3/2 = 695.836
+Mapping: {{mapping| 1 0 -4 -13 -25 | 0 1 4 10 18 }}
-Map: [&lt;2 0 -8 -26 -31 -40|, &lt;0 1 4 10 12 15|]
+Optimal tunings:
+* WE: ~2 = 1200.7636{{c}}, ~3/2 = 697.4122{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.0315{{c}}
-EDOs: 12f, 50
+Minimax tuning:
+* 11-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}
+: 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 }}]
+: unchanged-interval (eigenmonzo) basis: 2.11/9
-Badness: 0.0288
+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] (1/2-comma to Pyth.)
-==Unidecimal meantone aka Huygens==
+Algebraic generator: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
-See also [[Meantone_vs_meanpop|Meantone vs meanpop]]
-[[Comma]]s: 81/80, 126/125, 99/98
+{{Optimal ET sequence|legend=0| 12, 19e, 31, 105, 136b }}
-[[11-odd-limit|11-limit]] minimax
+Badness (Sintel): 0.563
-[|1 0 0 0 0&gt;, |25/16 -1/8 0 0 1/16&gt;, |9/4 -1/2 0 0 1/4&gt;,
+; Music
-|21/8 -5/4 0 0 5/8&gt;, |25/8 -9/4 0 0 9/8&gt;]
+* [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]
-[[Eigenmonzo]]s: 2, 11/9
+==== Grosstone ====
+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]].
-valid range: [696.774, 700.000] (31 to 12)
+Subgroup: 2.3.5.7.11.13
-nice range: [691.202, 701.955]
+Comma list: 81/80, 99/98, 126/125, 144/143
-strict range: [696.774, 700.000]
+Mapping: {{mapping| 1 0 -4 -13 -25 29 | 0 1 4 10 18 -16 }}
-[[POTE_tuning|POTE generator]]: 696.967
+Optimal tunings:
+* WE: ~2 = 1199.9389{{c}}, ~3/2 = 697.2282{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.2627{{c}}
-Mapping generator: ~3
+Minimax tuning:
+* 13- and 15-odd-limit: ~3/2 = {{monzo| 8/13 0 0 1/26 0 -1/26 }}
+: eigenmonzo basis (unchanged-interval basis): 2.13/7
-[[Algebraic_generator|Algebraic generator]]: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents.
+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] (1/2-comma to Pyth.)
-[[Map]]: [&lt;1 0 -4 -13 -25|, &lt;0 1 4 10 18|]
+{{Optimal ET sequence|legend=0| 12, 31, 43, 74 }}
-[[Generator]]s: 2, 3
+Badness (Sintel): 1.07
-EDOs: [[12edo|12]], [[31edo|31]], [[43edo|43]], [[74edo|74]], [[105edo|105]], [[198edo|198be]]
+===== 17-limit =====
+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.
-[[Badness]]: 0.0170
+Subgroup: 2.3.5.7.11.13.17
-[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]
+Comma list: 81/80, 99/98, 120/119, 126/125, 144/143
-===Tridecimal meantone===
+Mapping: {{mapping| 1 0 -4 -13 -25 29 12 | 0 1 4 10 18 -16 -5 }}
-Commas: 66/65, 81/80, 99/98, 105/104
-valid range: 697.674 (43)
+Optimal tunings:
+* WE: ~2 = 1199.5811{{c}}, ~3/2 = 697.0918{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3303{{c}}
-nice range: [691.202, 701.955]
+{{Optimal ET sequence|legend=0| 12, 31, 43, 74g }}
-strict range: 697.674
+Badness (Sintel): 1.06
-[[POTE_tuning|POTE generator]]: ~3/2 = 696.642
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-Mapping generator: ~3
+Comma list: 81/80, 96/95, 99/98, 120/119, 126/125, 144/143
-Map: [&lt;1 0 -4 -13 -25 -20|, &lt;0 1 4 10 18 15|]
+Mapping: {{mapping| 1 0 -4 -13 -25 29 12 9 | 0 1 4 10 18 -16 -5 -3 }}
-EDOs: 12f, 31, 43f
+Optimal tunings:
+* WE: ~2 = 1199.2931{{c}}, ~3/2 = 696.9690{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3736{{c}}
-Badness: 0.0180
+{{Optimal ET sequence|legend=0| 12, 31, 43, 74gh }}
-===Grosstone===
+Badness (Sintel): 1.07
-Commas: 81/80, 99/98, 126/125, 144/143
-POTE generator: ~3/2 = 697.264
+==== 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.
-Mapping generator: ~3
+This extension used to be known as ''tridecimal meantone'', but was decanonicalized in 2025.
-Map: [&lt;1 0 -4 -13 -25 29|, &lt;0 1 4 10 18 -16|]
+Subgroup: 2.3.5.7.11.13
-EDOs: [[12edo|12]], [[31edo|31]], [[43edo|43]], [[74edo|74]], [[105edo|105]]
+Comma list: 66/65, 81/80, 99/98, 105/104
-Badness: 0.0259
+Mapping: {{mapping| 1 0 -4 -13 -25 -20 | 0 1 4 10 18 15 }}
-===Meridetone===
+Optimal tunings:
-Commas: 78/77, 81/80, 99/98, 126/125
+* WE: ~2 = 1200.8149{{c}}, ~3/2 = 697.1155{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.7085{{c}}
-POTE generator: ~3/2 = 697.529
+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
-Mapping generator: ~3
+{{Optimal ET sequence|legend=0| 12f, 19e, 31 }}
-Map: [&lt;1 0 -4 -13 -25 -39|, &lt;0 1 4 10 18 27|]
+Badness (Sintel): 0.746
-EDOs: 12f, 31f, 43, 117df, 160bdf, 203bcdef
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-Badness: 0.0264
+Comma list: 66/65, 81/80, 99/98, 105/104, 120/119
-===Hemimeantone===
+Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 | 0 1 4 10 18 15 -5 }}
-Commas: 81/80, 99/98, 126/125, 169/168
-POTE generator: ~52/45 = 250.304
+Optimal tunings:
+* WE: ~2 = 1199.5548{{c}}, ~3/2 = 696.7449{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.9823{{c}}
-Mapping generator: ~26/15
+{{Optimal ET sequence|legend=0| 12f, 31 }}
-Map: [&lt;1 0 -4 -13 -25 -5|, &lt;0 2 8 20 36 11|]
+Badness (Sintel): 1.02
-EDOs: 43, 62, 167bef, 229bef
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-Badness: 0.0314
+Comma list: 66/65, 81/80, 96/95, 99/98, 105/104, 120/119
-==Meanpop==
+Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 9 | 0 1 4 10 18 15 -5 -3 }}
-See also [[Meantone_vs_meanpop|Meantone vs meanpop]]
-[[Comma]]s: 81/80, 126/125, 385/384
+Optimal tunings:
+* WE: ~2 = 1199.0408{{c}}, ~3/2 = 696.5824{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.1061{{c}}
-[[11-odd-limit|11-limit]] minimax 1/4 comma
+{{Optimal ET sequence|legend=0| 12f, 31 }}
-[|1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;,
+Badness (Sintel): 1.10
-|-3 0 5/2 0 0&gt;, |11 0 -13/4 0 0&gt;]
-[[Eigenmonzo]]s: 2, 5
+==== 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.
-valid range: [694.737, 696.774] (19 to 31)
+Subgroup: 2.3.5.7.11.13
-nice range: [691.202, 701.955]
+Comma list: 78/77, 81/80, 99/98, 126/125
-strict range: [694.737, 696.774]
+Mapping: {{mapping| 1 0 -4 -13 -25 -39 | 0 1 4 10 18 27 }}
-[[POTE_tuning|POTE generator]]: 696.434
+Optimal tunings:
+* WE: ~2 = 1199.9122{{c}}, ~3/2 = 697.4779{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.5241{{c}}
-Mapping generator: ~3
+Minimax tuning:
+* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/25 -2/25 0 0 0 1/25 }}
+: unchanged-interval (eigenmonzo) basis: 2.13/9
-[[Algebraic_generator|Algebraic generator]]: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
+{{Optimal ET sequence|legend=0| 12f, 31f, 43 }}
-[http://soonlabel.com/xenharmonic/archives/607 Scott Joplin's "The Entertainer" tuned into meanpop]
+Badness (Sintel): 1.09
-Map: [&lt;1 0 -4 -13 24|, &lt;0 1 4 10 -13|]
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-[[Generator]]s: 2, 3
+Comma list: 78/77, 81/80, 99/98, 120/119, 126/125
-EDOs: [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]]
+Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 | 0 1 4 10 18 27 -5 }}
-[[Badness]]: 0.0215
+Optimal tunings:
+* WE: ~2 = 1199.3793{{c}}, ~3/2 = 697.2833{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6222{{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]
+{{Optimal ET sequence|legend=0| 12f, 43 }}
-===13-limit Meanpop===
+Badness (Sintel): 1.22
-Commas: 81/80, 105/104, 126/125, 144/143
-valid range: [694.737, 696.774] (19 to 31)
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-nice range: [691.202, 701.955]
+Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125
-strict range: [694.737, 696.774]
+Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 9 | 0 1 4 10 18 27 -5 -3 }}
-POTE generator: ~3/2 = 696.211
+Optimal tunings:
+* WE: ~2 = 1199.0260{{c}}, ~3/2 = 697.1486{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6887{{c}}
-Mapping generator: ~3
+{{Optimal ET sequence|legend=0| 12f, 43 }}
-Map: [&lt;1 0 -4 -13 24 -20|, &lt;0 1 4 10 -13 15|]
+Badness (Sintel): 1.25
-EDOS: [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]]
+==== Hemimeantone ====
+Subgroup: 2.3.5.7.11.13
-Badness: 0.0209
+Comma list: 81/80, 99/98, 126/125, 169/168
-===Meanplop===
+Mapping: {{mapping| 1 0 -4 -13 -25 -5 | 0 2 8 20 36 11 }}
-Commas: 65/64, 78/77, 81/80, 91/90
-POTE generator: ~3/2 = 696.202
+: mapping generators: ~2, ~26/15
-Mapping generator: ~3
+Optimal tunings:
+* WE: ~2 = 1201.0387{{c}}, ~26/15 = 949.2863{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5065{{c}}
-Map: [&lt;1 0 -4 -13 24 10|, &lt;0 1 4 10 -13 -4|]
+{{Optimal ET sequence|legend=0| 19e, 43, 62 }}
-EDOs: 12e, 19, 31f, 50f
+Badness (Sintel): 1.30
-Badness: 0.0277
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-==Meanenneadecal==
+Comma list: 81/80, 99/98, 126/125, 169/168, 221/220
-Commas: 45/44, 56/55, 81/80
-POTE generator: ~3/2 = 696.250
+Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 | 0 2 8 20 36 11 33 }}
-Mapping generator: ~3
+Optimal tunings:
+* WE: ~2 = 1201.0270{{c}}, ~26/15 = 949.2892{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5169{{c}}
-Map: [&lt;1 0 -4 -13 -6|, &lt;0 1 4 10 6|]
+{{Optimal ET sequence|legend=0| 19eg, 43, 62 }}
-EDOs: [[7edo|7d]], [[12edo|12]], [[19edo|19]], [[31edo|31e]], [[50edo|50ee]]
+Badness (Sintel): 1.19
-Badness: 0.0214
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-===13-limit===
+Comma list: 81/80, 99/98, 126/125, 153/152, 169/168, 221/220
-Commas: 45/44, 56/55, 78/77, 81/80
-POTE generator: ~3/2 = 696.146
+Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 -25 | 0 2 8 20 36 11 33 37 }}
-Mapping generator: ~3
+Optimal tunings:
+* WE: ~2 = 1201.0339{{c}}, ~19/11 = 949.2902{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 948.5111{{c}}
-Map: [&lt;1 0 -4 -13 -6 -20|, &lt;0 1 4 10 6 15|]
+{{Optimal ET sequence|legend=0| 19egh, 43, 62 }}
-EDOs: 12f, 19, 31e, 50ee
+Badness (Sintel): 1.15
-Badness: 0.0212
+==== Semimeantone ====
+Subgroup: 2.3.5.7.11.13
-==Vincenzo==
+Comma list: 81/80, 99/98, 126/125, 847/845
-Commas: 45/44, 56/55, 65/64, 81/80
-POTE generator: ~3/2 = 695.060
+Mapping: {{mapping| 2 0 -8 -26 -50 -59 | 0 1 4 10 18 21 }}
-Mapping generator: ~3
+: mapping generators: ~55/39, ~3
-Map: [&lt;1 0 -4 -13 -6 10|, &lt;0 1 4 10 6 -4|]
+Optimal tunings:
+* WE: ~55/39 = 600.3606{{c}}, ~3/2 = 697.4241{{c}}
+* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 697.0545{{c}}
-EDOs: 7d, 12, 19, 26d
+{{Optimal ET sequence|legend=0| 12f, …, 50eff, 62, 136b }}
-Badness: 0.0248
+Badness (Sintel): 1.68
-===17-limit===
+===== 17-limit =====
-Commas: 45/44, 52/51, 56/55, 65/64, 81/80
+Subgroup: 2.3.5.7.11.13.17
-POTE generator: ~3/2 = 695.858
+Comma list: 81/80, 99/98, 126/125, 221/220, 289/288
-Map: [&lt;1 0 -4 -13 -6 10 12|, &lt;0 1 4 10 6 -4 -5|]
+Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 | 0 1 4 10 18 21 1 }}
-EDOs: 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: 0.0255
+{{Optimal ET sequence|legend=0| 12f, 50eff, 62, 136bg }}
-===19-limit===
+Badness (Sintel): 1.60
-Commas: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80
-POTE generator: ~3/2 = 696.131
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-Map: [&lt;1 0 -4 -13 -6 10 12 9|, &lt;0 1 4 10 6 -4 -5 -3|]
+Comma list: 81/80, 99/98, 126/125, 153/152, 209/208, 221/220
-EDOs: 7d, 12, 19
+Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 -1 | 0 1 4 10 18 21 1 3 }}
-Badness: 0.0223
+Optimal tunings:
+* WE: ~17/12 = 600.5959{{c}}, ~3/2 = 697.5985{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9638{{c}}
-===23-limit===
+{{Optimal ET sequence|legend=0| 12f, 50eff, 62 }}
-Commas: 39/38, 45/44, 52/51, 56/55, 65/64, 69/68, 81/80
-POTE generator: ~3/2 = 696.044
+Badness (Sintel): 1.47
-Map: [&lt;1 0 -4 -13 -6 10 12 9 14|, &lt;0 1 4 10 6 -4 -5 -3 -6|]
+=== Meanpop ===
+{{See also| Huygens vs meanpop }}
-EDOs: 7d, 12, 19
+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.
-Badness: 0.0201
+Subgroup: 2.3.5.7.11
-===29-limit===
+Comma list: 81/80, 126/125, 385/384
-Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80
-POTE generator: ~3/2 = 695.913
+Mapping: {{mapping| 1 0 -4 -13 24 | 0 1 4 10 -13 }}
-Map: [&lt;1 0 -4 -13 -6 10 12 9 14 8|, &lt;0 1 4 10 6 -4 -5 -3 -6 -2|]
+: mapping generator: ~2, ~3
-EDOs: 7d, 12, 19
+Optimal tunings:
+* WE: ~2 = 1201.3464{{c}}, ~3/2 = 697.2159{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4509{{c}}
-Badness: 0.0182
+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
-===31-limit===
+Tuning ranges:
-Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80, 93/92
+* 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.)
-POTE generator: ~3/2 = 695.750
+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.
-Map: [&lt;1 0 -4 -13 -6 10 12 9 14 8 16|, &lt;0 1 4 10 6 -4 -5 -3 -6 -2 -7|]
+{{Optimal ET sequence|legend=0| 12e, 19, 31, 81, 112b }}
-EDOs: 7d, 12, 19
+Badness (Sintel): 0.712
-Badness: 0.0171
+; 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]
-===37-limit===
+==== Tridecimal meanpop ====
-Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92
+Subgroup: 2.3.5.7.11.13
-POTE generator: ~3/2 = 695.603
+Comma list: 81/80, 105/104, 126/125, 144/143
-Map: [&lt;1 0 -4 -13 -6 10 12 9 14 8 16 -9|, &lt;0 1 4 10 6 -4 -5 -3 -6 -2 -7 9|]
+Mapping: {{mapping| 1 0 -4 -13 24 -20 | 0 1 4 10 -13 15 }}
-EDOs: 7d, 12, 19
+Optimal tunings:
+* WE: ~2 = 1201.0765{{c}}, ~3/2 = 696.8361{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2347{{c}}
-Badness: 0.0161
+Minimax tuning:
+* 13- and 15-odd-limit: ~3/2 = {{monzo| 4/7 0 0 0 -1/28 1/28 }}
+: unchanged-interval (eigenmonzo) basis: 2.13/11
-===41-limit===
+Tuning ranges:
-Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92, 124/123
+* 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.)
-POTE generator: ~3/2 = 695.696
+{{Optimal ET sequence|legend=0| 19, 31, 50, 81 }}
-Map: [&lt;1 0 -4 -13 -6 10 12 9 14 8 16 -9 18|, &lt;0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8|]
+Badness (Sintel): 0.863
-EDOs: 7d, 12, 19
+===== Meanpoppic =====
+Subgroup: 2.3.5.7.11.13.17
-Badness: 0.0154
+Comma list: 81/80, 105/104, 126/125, 144/143, 273/272
-==Meanundeci==
+Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 | 0 1 4 10 -13 15 26 }}
-Commas: 33/32, 55/54, 77/75
-POTE generator: ~3/2 = 694.689
+Optimal tunings:
+* WE: ~2 = 1201.0727{{c}}, ~3/2 = 696.8168{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2195{{c}}
-Mapping generator: ~3
+{{Optimal ET sequence|legend=0| 19g, 31, 50, 81, 131bd }}
-Map: [&lt;1 0 -4 -13 5|, &lt;0 1 4 10 -1|]
+Badness (Sintel): 1.02
-EDOs: 12e, 19e
+====== 19-limit ======
+Subgroup: 2.3.5.7.11.13.17.19
-Badness: 0.0315
+Comma list: 81/80, 105/104, 126/125, 144/143, 153/152, 273/272
-===13-limit===
+Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 -40 | 0 1 4 10 -13 15 26 28 }}
-Commas: 33/32, 55/54, 77/75, 729/728
-POTE generator: ~3/2 = 694.764
+Optimal tunings:
+* WE: ~2 = 1201.0719{{c}}, ~3/2 = 696.8101{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2137{{c}}
-Mapping generator: ~3
+{{Optimal ET sequence|legend=0| 19gh, 31, 50, 81 }}
-Map: [&lt;1 0 -4 -13 5 10|, &lt;0 1 4 10 -1 -4|]
+Badness (Sintel): 1.08
-EDOs: 12e, 19e
+===== Meanpoid =====
+Subgroup: 2.3.5.7.11.13.17
-Badness: 0.0263
+Comma list: 81/80, 105/104, 120/119, 126/125, 144/143
-==Meanundec==
+Mapping: {{mapping| 1 0 -4 -13 24 -20 12 | 0 1 4 10 -13 15 -5 }}
-Commas: 27/26, 40/39, 45/44, 56/55
-POTE generator: ~3/2 = 697.254
+Optimal tunings:
+* WE: ~2 = 1200.2768{{c}}, ~3/2 = 696.5683{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4114{{c}}
-Mapping generator: ~3
+{{Optimal ET sequence|legend=0| 19, 31 }}
-Map: [&lt;1 0 -4 -13 -6 -1|, &lt;0 1 4 10 6 3|]
+Badness (Sintel): 1.17
-EDOS: 12f, 19f, 31ef
+====== 19-limit ======
+Subgroup: 2.3.5.7.11.13.17.19
-Badness: 0.0242
+Comma list: 77/76, 81/80, 96/95, 105/104, 120/119, 126/125
-=Flattone=
+Mapping: {{mapping| 1 0 -4 -13 24 -20 12 9 | 0 1 4 10 -13 15 -5 -3 }}
-[[Comma]]s: 81/80, 525/512
-The [[wedgie]] for flattone is &lt;&lt;1 4 -9 4 -17 -32||, which tells us among other things that 9 generator steps of 4/3 get to the interval class for 7, meaning that [[7/4]] is a diminished seventh interval. Other intervals are [[7/6]], a diminished third, and [[7/5]], a doubly diminshed fifth. Good tunings for flattone are [[26edo]], [[45edo]] and [[64edo]].
+Optimal tunings:
+* WE: ~2 = 1199.7905{{c}}, ~3/2 = 696.3779{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4973{{c}}
-[[7-odd-limit|7-limit]] minimax
+{{Optimal ET sequence|legend=0| 19, 31 }}
-[|1 0 0 0&gt;, |21/13 0 1/13 -1/13&gt;,
+Badness (Sintel): 1.25
-|32/13 0 4/13 -4/13&gt;, |32/13 0 -9/13 9/13&gt;]
-[[Eigenmonzo]]s: 2, 7/5
+==== Semimeanpop ====
+Subgroup: 2.3.5.7.11.13
-[[9-odd-limit|9-limit]] minimax
+Comma list: 81/80, 126/125, 385/384, 847/845
-[|1 0 0 0&gt;, |17/11 2/11 0 -1/11&gt;,
+Mapping: {{mapping| 2 0 -8 -26 48 39 | 0 1 4 10 -13 -10 }}
-|24/11 8/11 0 -4/11&gt;, |34/11 -18/11 0 9/11&gt;]
-Eigenmonzos: 2, 9/7
+: mapping generators: ~55/39, ~3
-valid range: [692.308, 694.737] (26 to 19)
+Optimal tunings:
+* WE: ~55/39 = 600.6704{{c}}, ~3/2 = 697.2151{{c}}
+* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.4341{{c}}
-nice range: [692.353, 701.955]
+{{Optimal ET sequence|legend=0| 12e, 50, 62, 112b }}
-strict range: [692.353, 694.737]
+Badness (Sintel): 1.78
-[[POTE_tuning|POTE generator]]: 693.779
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-Mapping generator: ~3
+Comma list: 81/80, 126/125, 221/220, 273/272, 289/288
-Algebraic generator: Squarto, the positive root of 8x^2-4x-9, at 506.3239 cents, equal to (1+sqrt(19))/4.
+Mapping: {{mapping| 2 0 -8 -26 48 39 5 | 0 1 4 10 -13 -10 1 }}
-Map: [&lt;1 0 -4 17|, &lt;0 1 4 -9|]
+Optimal tunings:
+* WE: ~17/12 = 600.7232{{c}}, ~3/2 = 697.2820{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4411{{c}}
-[[Wedgie]]: &lt;&lt;1 4 -9 4 -17 -32||
+{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bg }}
-[[Generator]]s: 2, 3
+Badness (Sintel): 1.45
-EDOs: [[7edo|7]], [[19edo|19]], [[26edo|26]], [[45edo|45]]
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-[[Badness]]: 0.0386
+Comma list: 81/80, 126/125, 153/152, 209/208, 221/220, 273/272
-==11-limit==
+Mapping: {{mapping| 2 0 -8 -26 48 39 5 -1 | 0 1 4 10 -13 -10 1 3 }}
-Commas: 45/44, 81/80, 385/384
-valid range: [692.308, 694.737] (26 to 19)
+Optimal tunings:
+* WE: ~17/12 = 600.7527{{c}}, ~3/2 = 697.3244{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4525{{c}}
-nice range: [682.502, 701.955]
+{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bgh }}
-strict range: [692.308, 694.737]
+Badness (Sintel): 1.28
-POTE generator: ~3/2 = 693.126
+=== 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.
-Mapping generator: ~3
+Subgroup: 2.3.5.7.11
-Map: [&lt;1 0 -4 17 -6|, &lt;0 1 4 -9 6|]
+Comma list: 45/44, 56/55, 81/80
-EDOs: 7, 19, 26, 45, 71bc, 116bcde
+Mapping: {{mapping| 1 0 -4 -13 -6 | 0 1 4 10 6 }}
-Badness: 0.0338
+Optimal tunings:
+* WE: ~2 = 1199.6946{{c}}, ~3/2 = 696.0729{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2083{{c}}
-==13-limit==
+Tuning ranges:
-/44, 65/64, 78/77, 81/80
+* 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]
-valid range: [692.308, 694.737] (26 to 19)
+{{Optimal ET sequence|legend=0| 7d, 12, 19, 31e }}
-nice range: [682.502, 701.955]
+Badness (Sintel): 0.708
-strict range: [692.308, 694.737]
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-POTE generator: ~3/2 = 693.058
+Comma list: 45/44, 56/55, 78/77, 81/80
-Mapping generator: ~3
+Mapping: {{mapping| 1 0 -4 -13 -6 -20 | 0 1 4 10 6 15 }}
-Map: [&lt;1 0 -4 17 -6 10|, &lt;0 1 4 -9 6 -4|]
+Optimal tunings:
+* WE: ~2 = 1199.7931{{c}}, ~3/2 = 696.0258{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1241{{c}}
-EDOs: 7, 19, 26, 45f, 71bcf, 116bcdef
+{{Optimal ET sequence|legend=0| 7df, 12f, 19, 31e }}
-Badness: 0.0223
+Badness (Sintel): 0.875
-=Dominant=
+===== 17-limit =====
-[[Comma|Comma]]s: 36/35, 64/63
+Subgroup: 2.3.5.7.11.13.17
-The wedgie for dominant is &lt;&lt;1 4 -2 4 -6 -16||. Now 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]], but it also works well with the Pythagorean tuning of pure [[3/2]] fifths, and with [[29edo]], [[41edo]], or [[53edo]].
+Comma list: 45/44, 56/55, 78/77, 81/80, 120/119
-valid range: [700.000, 720.000] (12 to 5)
+Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 | 0 1 4 10 6 15 -5 }}
-nice range: [694.786, 715.587]
+Optimal tunings:
+* WE: ~2 = 1198.6665{{c}}, ~3/2 = 695.8010{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4998{{c}}
-strict range: [700.000, 715.587]
+{{Optimal ET sequence|legend=0| 12f, 19, 31e }}
-[[POTE_tuning|POTE generator]]: 701.573
+Badness (Sintel): 1.17
-Mapping generator: ~3
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-Map: [&lt;1 0 -4 6|, &lt;0 1 4 -2|]
+Comma list: 45/44, 56/55, 78/77, 81/80, 96/95, 120/119
-[[Wedgie]]: &lt;&lt;1 4 -2 4 -6 -16||
+Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 9 | 0 1 4 10 6 15 -5 -3 }}
-EDOs: [[5edo|5]], [[7edo|7]], [[12edo|12]], [[17edo|17c]], [[29edo|29cd]]
+Optimal tunings:
+* WE: ~2 = 1198.2880{{c}}, ~3/2 = 695.7123{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.6370{{c}}
-[[Badness]]: 0.0207
+{{Optimal ET sequence|legend=0| 12f, 19, 31e }}
-==11-limit==
+Badness (Sintel): 1.23
-Commas: 36/35, 64/63, 56/55
-valid range: [700.000, 705.882] (12 to 17)
+==== Vincenzo ====
+Subgroup: 2.3.5.7.11.13
-nice range: [691.202, 715.587]
+Comma list: 45/44, 56/55, 65/64, 81/80
-strict range: [700.000, 705.882]
+Mapping: {{mapping| 1 0 -4 -13 -6 10 | 0 1 4 10 6 -4 }}
-POTE generator: ~3/2 = 703.254
+Optimal tunings:
+* WE: ~2 = 1202.1684{{c}}, ~3/2 = 696.3160{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.2045{{c}}
-Mapping generator: ~3
+{{Optimal ET sequence|legend=0| 7d, 12, 19 }}
-Map: [&lt;1 0 -4 6 13|, &lt;0 1 4 -2 -6|]
+Badness (Sintel): 1.02
-EDOs: 5, 12, 17c, 29cde
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-Badness: 0.0242
+Comma list: 45/44, 52/51, 56/55, 65/64, 81/80
-==13-limit==
+Mapping: {{mapping| 1 0 -4 -13 -6 10 12 | 0 1 4 10 6 -4 -5 }}
-Commas: 36/35, 56/55, 64/63, 66/65
-valid range: 705.882 (17)
+Optimal tunings:
+* WE: ~2 = 1200.5137{{c}}, ~3/2 = 696.1561{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.8771{{c}}
-nice range: [691.202, 715.587]
+{{Optimal ET sequence|legend=0| 12, 19 }}
-strict range:705.882
+Badness (Sintel): 1.30
-POTE generator: ~3/2 = 703.636
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-Map: [&lt;1 0 -4 6 13 18|, &lt;0 1 4 -2 -6 -9|]
+Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80
-EDOs: 12f, 17c, 29cdef
+Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 | 0 1 4 10 6 -4 -5 -3 }}
-Badness: 0.0241
+Optimal tunings:
+* WE: ~2 = 1199.8261{{c}}, ~3/2 = 696.0298{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1262{{c}}
-==Dominion==
+{{Optimal ET sequence|legend=0| 12, 19 }}
-Commas: 26/25, 36/35, 56/55, 64/63
-POTE generator: ~3/2 = 704.905
+Badness (Sintel): 1.36
-Map: [&lt;1 0 -4 6 13 -9|, &lt;0 1 4 -2 -6 8|]
+=== Bimeantone ===
+/8 is mapped to half octave minus the [[128/125|meantone diesis]].
-EDOs: 5, 12, 17c, 46cde
+Subgroup: 2.3.5.7.11
-Badness: 0.0273
+Comma list: 81/80, 126/125, 245/242
-==Domineering==
+Mapping: {{mapping| 2 0 -8 -26 -31 | 0 1 4 10 12 }}
-Commas: 36/35, 45/44, 64/63
-POTE generator: ~3/2 = 698.776
+: mapping generators: ~63/44, ~3
-Mapping generator: ~3
+Optimal tunings:
+* WE: ~63/44 = 600.7492{{c}}, ~3/2 = 696.8853{{c}}
+* CWE: ~63/44 = 600.0000{{c}}, ~3/2 = 696.1908{{c}}
-Map: [&lt;1 0 -4 6 -6|, &lt;0 1 4 -2 6|]
+{{Optimal ET sequence|legend=0| 12, 26de, 38d, 50 }}
-EDOs: 5e, 7, 12, 19d, 43de
+Badness (Sintel): 1.26
-Badness: 0.0220
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-===13-limit===
+Comma list: 81/80, 105/104, 126/125, 245/242
-Commas: 36/35, 45/44, 52/49, 64/63
-POTE generator: ~3/2 = 695.762
+Mapping: {{mapping| 2 0 -8 -26 -31 -40 | 0 1 4 10 12 15 }}
-Mapping generator: ~3
+Optimal tunings:
+* WE: ~55/39 = 600.8309{{c}}, ~3/2 = 696.8000{{c}}
+* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.0066{{c}}
-Map: [&lt;1 0 -4 6 -6 10|, &lt;0 1 4 -2 6 -4|]
+{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}
-EDOs: 5ef, 7, 12, 19d, 31def
+Badness (Sintel): 1.19
-Badness: 0.0270
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
-===17-limit===
+Comma list: 81/80, 105/104, 126/125, 189/187, 221/220
-Commas: 36/35, 45/44, 51/49, 52/49, 64/63
-POTE generator: ~3/2 = 696.115
+Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 | 0 1 4 10 12 15 1 }}
-Mapping generator: ~3
+Optimal tunings:
+* WE: ~17/12 = 600.9234{{c}}, ~3/2 = 696.8536{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.9317{{c}}
-Map: [&lt;1 0 -4 6 -6 10 12|, &lt;0 1 4 -2 6 -4 -5|]
+{{Optimal ET sequence|legend=0| 12f, 38df, 50 }}
-EDOs: 5ef, 7, 12, 19d, 31def
+Badness (Sintel): 1.15
-Badness: 0.0245
+==== 19-limit ====
+Subgroup: 2.3.5.7.11.13.17.19
-===19-limit===
+Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220
-Commas: 36/35, 39/38, 45/44, 51/49, 52/49, 57/56
-POTE generator: ~3/2 = 696.217
+Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 -1 | 0 1 4 10 12 15 1 3 }}
-Mapping generator: ~3
+Optimal tunings:
+* WE: ~17/12 = 600.9845{{c}}, ~3/2 = 696.8939{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.8947{{c}}
-Map: [&lt;1 0 -4 6 -6 10 12 9|, &lt;0 1 4 -2 6 -4 -5 -3|]
+{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}
-EDOs: 5ef, 7, 12, 19d, 31def
+Badness (Sintel): 1.08
-Badness: 0.0204
+=== Trimean ===
+{{See also| No-sevens subgroup temperaments #Superpine }}
-===Dominatrix===
+Subgroup: 2.3.5.7.11
-Commas: 27/26, 36/35, 45/44, 64/63
-POTE generator: ~3/2 = 698.544
+Comma list: 81/80, 126/125, 1344/1331
-Mapping generator: ~3
+Mapping: {{mapping| 1 2 4 7 5 | 0 -3 -12 -30 -11 }}
-Map: [&lt;1 0 -4 6 -6 -1|, &lt;0 1 4 -2 6 3|]
+: mapping generators: ~2, ~11/10
-EDOs: 5e, 7, 12f, 19df
+Optimal tunings:
+* WE: ~2 = 1200.7155{{c}}, ~11/10 = 167.9055{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7749{{c}}
-==Domination==
+{{Optimal ET sequence|legend=0| 7d, 36d, 43, 50, 93 }}
-Commas: 36/35, 64/63, 77/75
-POTE generator: ~3/2 = 705.004
+Badness (Sintel): 1.68
-Mapping generator: ~3
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Map: [&lt;1 0 -4 6 -14|, &lt;0 1 4 -2 11|]
+Comma list: 81/80, 126/125, 144/143, 364/363
-EDOs: 5e, 12e, 17c, 46cd
+Mapping: {{mapping| 1 2 4 7 5 3 | 0 -3 -12 -30 -11 5 }}
-Badness: 0.0366
+Optimal tunings:
+* WE: ~2 = 1200.6104{{c}}, ~11/10 = 167.8749{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7728{{c}}
-===13-limit===
+{{Optimal ET sequence|legend=0| 7d, 43, 50, 93 }}
-Commas: 26/25, 36/35, 64/63, 66/65
-POTE generator: ~3/2 = 705.496
+Badness (Sintel): 1.46
-Mapping generator: ~3
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
-Map: [&lt;1 0 -4 6 -14 -9|, &lt;0 1 4 -2 11 8|]
+Comma list: 81/80, 126/125, 144/143, 189/187, 221/220
-EDOs: 5e, 12e, 17c
+Mapping: {{mapping| 1 2 4 7 5 3 8 | 0 -3 -12 -30 -11 5 -28 }}
-Badness: 0.0274
+Optimal tunings:
+* WE: ~2 = 1200.6144{{c}}, ~11/10 = 167.8716{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7682{{c}}
-==Arnold==
+{{Optimal ET sequence|legend=0| 7dg, 43, 50, 93 }}
-Commas: 22/21, 33/32, 36/35
-POTE generator: ~3/2 = 698.491
+Badness (Sintel): 1.28
-Mapping generator: ~3
+=== Migration ===
+See [[Rastmic clan #Migration|Rastmic clan]].
-Map: [&lt;1 0 -4 6 5|, &lt;0 1 4 -2 -1|]
+== Flattone ==
+{{Main| Flattone }}
-EDOs: 5, 7, 12e
+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]].
-Badness: 0.0261
+[[Subgroup]]: 2.3.5.7
-==13-limit==
+[[Comma list]]: 81/80, 525/512
-Commas: 22/21, 27/26, 33/32, 40/39
-POTE generator: ~3/2 = 696.743
+{{Mapping|legend=1| 1 0 -4 17 | 0 1 4 -9 }}
-Mapping generator: ~3
+[[Optimal tuning]]s:
+* [[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 }}
-Map: [&lt;1 0 -4 6 5 -1|, &lt;0 1 4 -2 -1 3|]
+[[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
-EDOs: 5, 7, 12ef, 19def, 31def
+[[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]
-Badness: 0.0233
+[[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.
-=Sharptone=
+{{Optimal ET sequence|legend=1| 7, 19, 26, 45 }}
-[[Comma]]s: 21/20, 28/27
-Sharptone, with a wedgie &lt;&lt;1 4 3 4 2 -4||, 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]] tuning does sharptone about as well as such a thing can be done, of course not in its patent val.
+[[Badness]] (Sintel): 0.976
-[[POTE_tuning|POTE generator]]: ~3/2 = 700.140
+=== 11-limit ===
+This can also be considered a no-sevens temperament: [[#Hypnotone|hypnotone]].
-Mapping generator: ~3
+Subgroup: 2.3.5.7.11
-Map: [&lt;1 0 -4 -2|, &lt;0 1 4 3|]
+Comma list: 45/44, 81/80, 385/384
-[[Wedgie]]: &lt;&lt;1 4 3 4 2 -4||
+Mapping: {{mapping| 1 0 -4 17 -6 | 0 1 4 -9 6 }}
-EDOs: [[5edo|5]], [[7edo|7d]], [[12edo|12d]]
+Optimal tuning:
+* WE: ~2 = 1202.3247{{c}}, ~3/2 = 694.4688{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.1467{{c}}
-[[Badness]]: 0.0248
+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]
-==Meanertone==
+{{Optimal ET sequence|legend=0| 7, 19, 26, 45, 71bc, 116bcde }}
-Commas: 21/20, 28/27, 33/32
-POTE generator: ~3/2 = 696.615
+Badness (Sintel): 1.12
-Map: [&lt;1 0 -4 -2 5|, &lt;0 1 4 3 -1|]
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-EDOs: 5, 7d, 12de
+Comma list: 45/44, 65/64, 78/77, 81/80
-=Meansept=
+Mapping: {{mapping| 1 0 -4 17 -6 10 | 0 1 4 -9 6 -4 }}
-Commas: 15/14, 81/80
-POTE generator: ~3/2 = 682.895
+Optimal tunings:
+* WE: ~2 = 1202.5156{{c}}, ~3/2 = 694.5107{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0538{{c}}
-Mapping generator: ~3
+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]
-Map: [&lt;1 0 -4 -5|, &lt;0 1 4 5|]
+{{Optimal ET sequence|legend=0| 7, 19, 26, 45f, 71bcf, 116bcdef }}
-Wedgie: &lt;&lt;1 4 5 4 5 0||
+Badness (Sintel): 0.920
-EDOs: 5d, 7, 12dd
+=== Ptolemy ===
+See [[Rastmic clan #Ptolemy|Rastmic clan]].
-Badness: 0.0453
+== Dominant ==
+{{Main| Dominant (temperament) }}
+{{See also| Archytas clan }}
-==11-limit==
+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]].
-Commas: 15/14, 22/21, 81/80
-POTE generator: ~3/2 = 685.234
+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.
-Mapping generator: ~3
+[[Subgroup]]: 2.3.5.7
-Map: [&lt;1 0 -4 -5 -6|, &lt;0 1 4 5 6|]
+[[Comma list]]: 36/35, 64/63
-EDOs: 5de, 7, 12dd
+{{Mapping|legend=1| 1 0 -4 6 | 0 1 4 -2 }}
-Badness: 0.0325
+[[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 }}
-=Supermean=
+[[Tuning ranges]]:
-Commas: 81/80, 672/625
+* [[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]
-POTE generator: ~3/2 = 704.889
+{{Optimal ET sequence|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}
-Map: [&lt;1 0 -4 -21|, &lt;0 1 4 15|]
+[[Badness]] (Sintel): 0.524
-EDOs: 17c, 46c
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Badness: 0.1342
+Comma list: 36/35, 56/55, 64/63
-==11-limit==
+Mapping: {{mapping| 1 0 -4 6 13 | 0 1 4 -2 -6 }}
-Commas: 56/55, 81/80, 132/125
-POTE generator: ~3/2 = 705.096
+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]
-Map: [&lt;1 0 -4 -21 -14|, &lt;0 1 4 15 11|]
+Optimal tunings:
+* WE: ~2 = 1194.0169{{c}}, ~3/2 = 699.7473{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.2672{{c}}
-EDOs: 17c, 46c
+{{Optimal ET sequence|legend=0| 5, 12, 17c, 29cde }}
-Badness: 0.0633
+Badness (Sintel): 0.799
-==13-limit==
+==== 13-limit ====
-Commas: 26/25, 56/55, 66/65, 81/80
+Subgroup: 2.3.5.7.11.13
-POTE generator: ~3/2 = 705.094
+Comma list: 36/35, 56/55, 64/63, 66/65
-Map: [&lt;1 0 -4 -21 -14 -9|, &lt;0 1 4 15 11 8|]
+Mapping: {{mapping| 1 0 -4 6 13 18 | 0 1 4 -2 -6 -9 }}
-EDOs: 17c, 46c
+Optimal tunings:
+* WE: ~2 = 1193.8055{{c}}, ~3/2 = 700.0042{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.8254{{c}}
-=Injera=
+Tuning ranges:
-[[Comma|Comma]]s: 50/49, 81/80
+* 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]
-The wedgie for injera is &lt;&lt;2 8 8 8 7 -4||, which tells us it 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|19edo]]s, is an excellent tuning for injera.
+{{Optimal ET sequence|legend=0| 12f, 17c, 29cdef }}
-[http://tech.groups.yahoo.com/group/tuning-math/message/3091 Origin of the name]
+Badness (Sintel): 0.996
-valid range: [685.714, 700.000] (14c to 12)
+==== Dominion ====
+Subgroup: 2.3.5.7.11.13
-nice range: [688.957, 701.955]
+Comma list: 26/25, 36/35, 56/55, 64/63
-strict range: [688.957, 700.000]
+Mapping: {{mapping| 1 0 -4 6 13 -9 | 0 1 4 -2 -6 8 }}
-[[POTE_tuning|POTE generator]]: 694.375
+Optimal tunings:
+* WE: ~2 = 1195.0293{{c}}, ~3/2 = 701.9847{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7698{{c}}
-Mapping generator: ~3
+{{Optimal ET sequence|legend=0| 5, 12, 17c }}
-Map: [&lt;2 0 -8 -7|, &lt;0 1 4 4|]
+Badness (Sintel): 1.13
-[[wedgie|Wedgie]]: &lt;&lt;2 8 8 8 7 -4||
+=== Domination ===
+Subgroup: 2.3.5.7.11
-EDOs: [[12edo|12]], [[26edo|26]], [[38edo|38]], [[102edo|102bcd]], [[140edo|140bcd]], [[178edo|178bcd]]
+Comma list: 36/35, 64/63, 77/75
-[[Badness|Badness]]: 0.0311
+Mapping: {{mapping| 1 0 -4 6 -14 | 0 1 4 -2 11 }}
-== Music ==
+Optimal tunings:
-* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 Two Pairs of Socks] (in [[26edo]]) by [[Igliashon Jones]]
+* WE: ~2 = 1194.8645{{c}}, ~3/2 = 701.9872{{c}}
-* [http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20Injera%20Jam.mp3 Injera Jam] (in [[26edo]]) by [[Zach Curley]]
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5945{{c}}
-==11-limit==
+{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}
-Commas: 45/44, 50/49, 81/80
-valid range: [685.714, 700.000] (14c to 12)
+Badness (Sintel): 1.21
-nice range: [682.458, 701.955]
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-strict range: [685.714, 700.000]
+Comma list: 26/25, 36/35, 64/63, 66/65
-POTE generator: ~3/2 = 692.840
+Mapping: {{mapping| 1 0 -4 6 -14 -9 | 0 1 4 -2 11 8 }}
-Mapping generator: ~3
+Optimal tunings:
+* WE: ~2 = 1195.1324{{c}}, ~3/2 = 702.6343{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.0791{{c}}
-Map: [&lt;2 0 -8 -7 -12|, &lt;0 1 4 4 6|]
+{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}
-EDOs: 12, 14c, 26. 90bce, 116bce
+Badness (Sintel): 1.13
-Badness: 0.0231
+=== Domineering ===
+Subgroup: 2.3.5.7.11
-==13-limit==
+Comma list: 36/35, 45/44, 64/63
-Commas: 45/44, 50/49, 81/80, 78/77
-valid range: 692.308 (26)
+Mapping: {{mapping| 1 0 -4 6 -6 | 0 1 4 -2 6 }}
-nice range: [682.458, 701.955]
+Optimal tunings:
+* WE: ~2 = 1194.7102{{c}}, ~3/2 = 695.6962{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1765{{c}}
-strict range: 692.308 (26)
+{{Optimal ET sequence|legend=0| 5e, 7, 12 }}
-POTE generator: ~3/2 = 692.673
+Badness (Sintel): 0.727
-Mapping generator: ~3
+=== Arnold ===
+Subgroup: 2.3.5.7.11
-Map: [&lt;2 0 -8 -7 -12 -21|, &lt;0 1 4 4 6 9|]
+Comma list: 22/21, 33/32, 36/35
-EDOs: 26, 104bcf
+Mapping: {{mapping| 1 0 -4 6 5 | 0 1 4 -2 -1 }}
-Badness: 0.0216
+Optimal tunings:
+* WE: ~2 = 1199.8507{{c}}, ~3/2 = 698.4045{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.4822{{c}}
-==Enjera==
+{{Optimal ET sequence|legend=0| 5, 7, 12e }}
-Commas: 27/26, 40/39, 45/44, 99/98
-POTE generator: ~3/2 = 694.121
+Badness (Sintel): 0.864
-Mapping generator: ~3
+=== Neutrominant ===
+See [[Rastmic clan #Neutrominant|Rastmic clan]].
-Map: [&lt;2 0 -8 -7 -12 -2|, &lt;0 1 4 4 6 3|]
+== 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.
-EDOs: 12f, 26f, 38ef
+Flattertone was named by [[Flora Canou]] in 2024.
-Badness: 0.0265
+[[Subgroup]]: 2.3.5.7
-==Injerous==
+[[Comma list]]: 81/80, 1875/1792
-Commas: 33/32, 50/49, 55/54
-POTE generator: ~3/2 = 690.548
+{{Mapping|legend=1| 1 0 -4 -24 | 0 1 4 17 }}
-Mapping generator: ~3
+: mapping generators: ~2, ~3
-Map: [&lt;2 0 -8 -7 10|, &lt;0 1 4 4 -1|]
+[[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 }}
-EDOs: 12e, 14c, 26e, 40ce
+{{Optimal ET sequence|legend=1| 7d, 19d, 26, 59bcd, 85bccd }}
-Badness: 0.0386
+[[Badness]] (Sintel): 2.43
-==Lahoh==
+==== 11-limit ====
-Commas: 50/49, 56/55, 81/77
+Subgroup: 2.3.5.7.11
-POTE generator: ~3/2 = 699.001
+Comma list: 45/44, 81/80, 1375/1344
-Mapping generator: ~3
+Mapping: {{mapping| 1 0 -4 -24 -6 | 0 1 4 17 6 }}
-Map: [&lt;2 0 -8 -7 7|, &lt;0 1 4 4 0|]
+Optimal tunings:
+* WE: ~2 = 1203.4653{{c}}, ~3/2 = 693.8144{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 692.0422{{c}}
-EDOs: 12
+{{Optimal ET sequence|legend=0| 7d, 19d, 26 }}
-Badness: 0.0431
+Badness (Sintel): 1.53
-=Godzilla=
+; Music
-<span style="display: block; text-align: right;">[[:de:Semiphor,_Semaphor,_Godzilla|Deutsch]]</span>
+* [https://youtu.be/scCuGXnj5IY ''Music in 33EDO (33-Tone Equal Temperament) - Feb 2024''] by [[Budjarn Lambeth]] (2024)
-Main article: [[Semaphore_and_Godzilla|Semaphore and Godzilla]]
+== 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.
-[[Comma|Comma]]s: 49/48, 81/80
+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).
-Godzilla has wedgie &lt;&lt;2 8 1 8 -4 -20||, and 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 the perfect godzilla tuning, so much so that's there's not much point in looking elsewhere. Hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes.
+The 11-limit extension was named by Gene Ward Smith in 2004<ref name="meantone & meanpop 2004"/>.
-valid range: [240.000, 257.143] (5 to 14c)
+[[Subgroup]]: 2.3.5.7
-nice range: [231.174, 266.871]
+[[Comma list]]: 21/20, 28/27
-strict range: [240.000, 257.143]
+{{Mapping|legend=1| 1 0 -4 -2 | 0 1 4 3 }}
-[[POTE_tuning|POTE generator]]: ~8/7 = 252.635
+[[Optimal tuning]]s:
+* [[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 }}
-Mapping generator: ~7/4
+{{Optimal ET sequence|legend=1| 5, 7d, 12d }}
-Map: [&lt;1 0 -4 2|, &lt;0 2 8 1|]
+[[Badness]] (Sintel): 0.629
-[[wedgie|Wedgie]]: &lt;&lt;2 8 1 8 -4 -20||
+=== Meanertone ===
+Subgroup: 2.3.5.7.11
-EDOs: [[5edo|5]], [[9edo|9c]], [[14edo|14c]], [[19edo|19]], [[62edo|62d]], [[81edo|81d]], 143bd
+Comma list: 21/20, 28/27, 33/32
-[[Badness|Badness]]: 0.0267
+Mapping: {{mapping| 1 0 -4 -2 5 | 0 1 4 3 -1 }}
-==11-limit==
+Optimal tunings:
-Commas: 45/44, 49/48, 81/80
+* WE: ~2 = 1208.5304{{c}}, ~3/2 = 701.5669{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1117{{c}}
-valid range: [252.632, 257.143] (19 to 14c)
+{{Optimal ET sequence|legend=0| 5, 7d, 12de }}
-nice range: [231.174, 266.871]
+Badness (Sintel): 0.832
-strict range: [252.632, 257.143]
+== 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.
-POTE generator: ~8/7 = 254.027
+Mildtone was named by [[User: Lucius Chiaraviglio|Lucius Chiaraviglio]] in 2024.
-Mapping generator: ~7/4
+[[Subgroup]]: 2.3.5.7
-Map: [&lt;1 0 -4 2 -6|, &lt;0 2 8 1 12|]
+[[Comma list]]: 81/80, 16128/15625
-EDOs: 14c, 19, 33cd, 52cd
+{{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 }}
-Badness: 0.0290
+[[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 }}
-==13-limit==
+{{Optimal ET sequence|legend=1| 12, 43d, 55, 67 }}
-Commas: 45/44, 49/48, 78/77, 81/80
-valid range: 694.737 (19)
+[[Badness]] (Sintel): 2.67
-nice range: [621.581, 737.652]
+=== 11-limit ===
-strict range: 694.737
+[[Subgroup]]: 2.3.5.7.11
-POTE generator: ~8/7 = 253.603
+[[Comma list]]: 81/80, 176/175, 7056/6875
-Mapping generator: ~7/4
+{{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 30}}
-Map: [&lt;1 0 -4 2 -6 -5|, &lt;0 2 8 1 12 11|]
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.816{{c}}, ~3/2 = 698.355{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.455{{c}}
-EDOs: 14cf, 19, 33cdf, 52cdf
+{{Optimal ET sequence|legend=1| 12, 43de, 55, 67 }}
-Badness: 0.0225
+[[Badness]] (Sintel): 2.15
-==Semafour==
+=== 13-limit ===
-Commas: 33/32, 49/48, 55/54
-POTE generator: ~8/7 = 254.042
+[[Subgroup]]: 2.3.5.7.11.13
-Mapping generator: ~7/4
+[[Comma list]]: 81/80, 176/175, 196/195, 832/825
-Map: [&lt;1 0 -4 2 5|, &lt;0 2 8 1 -2|]
+{{Mapping|legend=1| 1 0 -4 -32 -44 | 0 1 4 22 30}}
-EDOs: 5, 14c, 19e, 33cde
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.788{{c}}, ~3/2 = 698.355{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.471{{c}}
-Badness: 0.0285
+{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}
-==Varan==
+[[Badness]] (Sintel): 2.04
-Commas: 49/48, 77/75, 81/80
-POTE generator: ~8/7 = 251.079
+=== 17-limit ===
-Mapping generator: ~7/4
+[[Subgroup]]: 2.3.5.7.11.13.17
-Map: [&lt;1 0 -4 2 -10|, &lt;0 2 8 1 17|]
+[[Comma list]]: 81/80, 176/175, 189/187, 196/195, 832/825
-EDOs: 19e, 24, 43de
+{{Mapping|legend=1| 1 0 -4 -32 -44 12| 0 1 4 22 30 -5}}
-Badness: 0.0396
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.655{{c}}, ~3/2 = 698.295{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.488{{c}}
-===13-limit===
+{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}
-Commas: 49/48, 66/65, 77/75, 81/80
-POTE generator: ~8/7 = 251.165
+[[Badness]] (Sintel): 1.98
-Mapping generator: ~7/4
+=== 19-limit ===
-Map: [&lt;1 0 -4 2 -10 -5|, &lt;0 2 8 1 17 11|]
+[[Subgroup]]: 2.3.5.7.11.13.17.19
-EDOs: 19e, 24, 43de
+[[Comma list]]: 81/80, 96/95, 176/175, 189/187, 196/195, 832/825
-Badness: 0.0257
+{{Mapping|legend=1| 1 0 -4 -32 -44 12 9| 0 1 4 22 30 -5 -3}}
-==Baragon==
+[[Optimal tuning]]s:
-Commas: 49/48, 56/55, 81/80
+* [[WE]]: ~2 = 1199.371{{c}}, ~3/2 = 698.164{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.519{{c}}
-POTE generator: ~8/7 = 251.173
+{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}
-Mapping generator: ~7/4
+[[Badness]] (Sintel): 1.95
-Map: [&lt;1 0 -4 2 9|, &lt;0 2 8 1 -7|]
+{{Todo|unify precision|review}}
-EDOs: 19, 24, 43d
+== 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]].
-Badness: 0.0357
+[[Subgroup]]: 2.3.5.7
-==Music==
+[[Comma list]]: 81/80, 672/625
-[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Bobro/GodzillaExample.mp3 Godzilla Example] by [[Cameron_Bobro|Cameron Bobro]]
-[http://tinyurl.com/4uyumk9 "Change is on the Wind"] in Godzilla[9] by [[Igliashon_Jones|Igliashon Jones]]
+{{Mapping|legend=1| 1 0 -4 -21 | 0 1 4 15 }}
-=Mohajira=
+[[Optimal tuning]]s:
-<span style="display: block; text-align: right;">[[:de:Mohajira|Deutsch]]</span>
+* [[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 }}
-[[Comma]]s: 81/80, 6144/6125
+{{Optimal ET sequence|legend=1| 5d, 12d, 17c }}
-Mohajira, with wedgie &lt;&lt;2 8 -11 8 -23 -48||, really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. [[31edo]] makes for an excellent (7-limit) mohajira tuning, with generator 9/31. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs.
+[[Badness]] (Sintel): 3.40
-Mohajira can also 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 11-limit). Within this paradigm, mohajira 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, that maps four 3/2's to 5/1, and that maps the interval one quarter tone flat of 16/9 to 7/4.
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-[[7-odd-limit|7]] and [[9-odd-limit|9-limit]] minimax 1/4 comma
+Comma list: 56/55, 81/80, 132/125
-[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |6 0 -11/8 0&gt;]
+Mapping: {{mapping| 1 0 -4 -21 -14 | 0 1 4 15 11 }}
-[[Eigenmonzo]]s: 2, 5
+Optimal tunings:
+* WE: ~2 = 1195.7270{{c}}, ~3/2 = 702.5848{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7471{{c}}
-[[POTE_tuning|POTE generator]]: ~128/105 = 348.415
+{{Optimal ET sequence|legend=0| 5de, 12de, 17c }}
-Mapping generator: ~128/105
+Badness (Sintel): 2.09
-Algebraic generator: Mohabis, real root of 3x^3-3x^2-1, 348.6067 cents. Corresponding recurrence converges quickly.
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Map: [&lt;1 1 0 6|, &lt;0 2 8 -11|]
+Comma list: 26/25, 56/55, 66/65, 81/80
-[[Generator]]s: 2, 128/105
+Mapping: {{mapping| 1 0 -4 -21 -14 -9 | 0 1 4 15 11 8 }}
-[[Wedgie]]: &lt;&lt;2 8 -11 8 -23 -48||
+Optimal tunings:
+* WE: ~2 = 1196.3958{{c}}, ~3/2 = 702.9766{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7940{{c}}
-EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]], [[38edo|38]], [[55edo|55]], [[69edo|69]]
+{{Optimal ET sequence|legend=0| 5de, 12de, 17c, 29c }}
-[[Badness]]: 0.0557
+Badness (Sintel): 1.67
-==11-limit==
+== Mohajira ==
-Commas: 81/80, 121/120, 176/175
+{{Main| Mohajira }}
-[[11-odd-limit|11-limit]] minimax 1/4 comma
+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.
-[|1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;,
+[[Subgroup]]: 2.3.5.7
-|6 0 -11/8 0 0&gt;, |2 0 5/8 0 0&gt;]
-Eigenmonzos: 2, 5
+[[Comma list]]: 81/80, 6144/6125
-POTE generator: ~11/9 = 348.477
+{{Mapping|legend=1| 1 1 0 6 | 0 2 8 -11 }}
-Mapping generator: ~11/9
+: mapping generators: ~2, ~128/105
-Map: [&lt;1 1 0 6 2|, &lt;0 2 8 -11 5|]
+[[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 }}
-Generators: 2, 11/9
+[[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
-EDOs: 7, 24, 31, 38, 55
+[[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]
-Badness: 0.0261
+[[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==
+{{Optimal ET sequence|legend=1| 7, 24, 31 }}
-Commas: 66/65, 81/80, 105/104, 121/120
-POTE generator: ~11/9 = 348.558
+[[Badness]] (Sintel): 1.41
-Mapping generator: ~11/9
+Scales: [[mohaha7]], [[mohaha10]]
-Map: [&lt;1 1 0 6 2 4|, &lt;0 2 8 -11 5 -1|]
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-EDOs: 7, 24, 31, 38, 55
+Comma list: 81/80, 121/120, 176/175
-Badness: 0.0234
+Mapping: {{mapping| 1 1 0 6 2 | 0 2 8 -11 5 }}
-==17-limit==
+Optimal tunings:
-Commas: 66/65, 81/80, 105/104, 121/120, 154/153
+* WE: ~2 = 1201.1562{{c}}, ~11/9 = 348.8124{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.4910{{c}}
-POTE generator: ~11/9 = 348.736
+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
-Mapping generator: ~11/9
+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]
-Map: [&lt;1 1 0 6 2 4 7|, &lt;0 2 8 -11 5 -1 -10|]
+{{Optimal ET sequence|legend=0| 7, 24, 31 }}
-EDOs: 7, 24, 31, 38g, 55
+Badness (Sintel): 0.862
-Badness: 0.0206
+Scales: [[mohaha7]], [[mohaha10]]
-==19-limit==
+=== 13-limit ===
-Commas: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152
+Subgroup: 2.3.5.7.11.13
-POTE generator: ~11/9 = 348.810
+Comma list: 66/65, 81/80, 105/104, 121/120
-Mapping generator: ~11/9
+Mapping: {{mapping| 1 1 0 6 2 4 | 0 2 8 -11 5 -1 }}
-Map: [&lt;1 1 0 6 2 4 7 6|, &lt;0 2 8 -11 5 -1 -10 -6|]
+Optimal tunings:
+* WE: ~2 = 1200.4256{{c}}, ~11/9 = 348.6819{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.5622{{c}}
-EDOs: 7, 24, 31, 38gh, 55
+{{Optimal ET sequence|legend=0| 7, 24, 31 }}
-Badness: 0.0173
+Badness (Sintel): 0.966
-=Ptolemy=
+Scales: [[mohaha7]], [[mohaha10]]
-Commas: 81/80, 121/120, 525/512
-POTE generator: ~11/9 = 346.922
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
-Map: [&lt;1 1 0 8 2|, &lt;0 2 8 -18 5|]
+Comma list: 66/65, 81/80, 105/104, 121/120, 154/153
-EDOs: 7, 31dd, 38d, 45e, 83bcddee
+Mapping: {{mapping| 1 1 0 6 2 4 7 | 0 2 8 -11 5 -1 -10 }}
-Badness: 0.0588
+Optimal tunings:
+* WE: ~2 = 1200.0382{{c}}, ~11/9 = 348.7471{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.7360{{c}}
-==13-limit==
+{{Optimal ET sequence|legend=0| 7, 24, 31 }}
-Commas: 65/64, 81/80, 105/104, 121/120
-POTE generator: ~11/9 = 346.910
+Badness (Sintel): 1.05
-Map: [&lt;1 1 0 8 2 6|, &lt;0 2 8 -18 5 -8|]
+Scales: [[mohaha7]], [[mohaha10]]
-EDOs: 7, 31ddf, 38df, 45ef, 83bcddeeff
+=== 19-limit ===
+Subgroup: 2.3.5.7.11.13.17.19
-Badness: 0.0343
+Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152
-=Maqamic=
+Mapping: {{mapping| 1 1 0 6 2 4 7 6 | 0 2 8 -11 5 -1 -10 -6 }}
-<span style="display: block; text-align: right;">[[:de:maqamisch|Deutsch]]</span>
-Main article: [[maqamic|Maqamic]]
+Optimal tunings:
+* WE: ~2 = 1199.7469{{c}}, ~11/9 = 348.7367{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.8117{{c}}
-[[Comma|Comma]]s: 81/80, 36/35, 121/120
+{{Optimal ET sequence|legend=0| 7, 24, 31, 55 }}
-Maqamic temperament is much like Mohajira, except in that it 36/35 vanishes instead of 176/175. 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.
+Badness (Sintel): 1.05
-[[POTE_tuning|POTE generator]]: ~11/9 = 350.934
+Scales: [[mohaha7]], [[mohaha10]]
-Mapping generator: ~11/9
+== 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.
-Map: [&lt;1 1 0 4 2|, &lt;0 2 8 -4 5|]
+[[Subgroup]]: 2.3.5.7
-[[generator|Generator]]s: 2, 11/9
+[[Comma list]]: 81/80, 392/375
-EDOs: [[7edo|7]], [[10edo|10c]], [[17edo|17c]], [[24edo|24d]], [[31edo|31d]]
+{{Mapping|legend=1| 1 1 0 -1 | 0 2 8 13 }}
-==13-limit==
+: mapping generators: ~2, ~25/21
-[[Comma|Comma]]s: 81/80, 36/35, 121/120, 144/143
-[[POTE_tuning|POTE generator]]: ~11/9 = 350.816
+[[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 generator: ~11/9
+{{Optimal ET sequence|legend=1| 7d, 17c, 24 }}
-Map: [&lt;1 1 0 4 2 4|, &lt;0 2 8 -4 5 -1|]
+[[Badness]] (Sintel): 1.97
-Generators: 2, 11/9
+Scales: [[mohaha7]], [[mohaha10]]
-EDOs: [[7edo|7]], [[10edo|10c]], [[17edo|17c]], [[24edo|24d]],[[31edo| 31d]]
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-=Migration=
+Comma list: 56/55, 77/75, 243/242
-Commas: 81/80, 121/120, 126/125
-POTE generator: ~11/9 = 348.182
+Mapping: {{mapping| 1 1 0 -1 2 | 0 2 8 13 5 }}
-Mapping generator: ~11/9
+Optimal tunings:
+* WE: ~2 = 1199.1924{{c}}, ~11/9 = 350.3286{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.4821{{c}}
-Map: [&lt;1 1 0 -3 2|, &lt;0 2 8 20 5|]
+{{Optimal ET sequence|legend=0| 7d, 17c, 24 }}
-EDOs: 31, 100de, 131bde, 162bde
+Badness (Sintel): 1.20
-Badness: 0.0255
+Scales: [[mohaha7]], [[mohaha10]]
-=Mohamaq=
+=== 13-limit ===
-Commas: 81/80, 392/375
+Subgroup: 2.3.5.7.11.13
-POTE generator: ~25/21 = 350.586
+Comma list: 56/55, 66/65, 77/75, 243/242
-Mapping generator: ~25/21
+Mapping: {{mapping| 1 1 0 -1 2 4 | 0 2 8 13 5 -1 }}
-Map: [&lt;1 1 0 -1|, &lt;0 2 8 13|]
+Optimal tunings:
+* WE: ~2 = 1198.5986{{c}}, ~11/9 = 350.3353{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.6459{{c}}
-EDOs: 17c, 24, 65c, 89cd
+{{Optimal ET sequence|legend=0| 7d, 17c, 24, 41c }}
-Badness: 0.0777
+Badness (Sintel): 1.19
-==11-limit==
+Scales: [[mohaha7]], [[mohaha10]]
-Commas: 56/55, 77/75, 243/242
-POTE generator: ~11/9 = 350.565
+== Liese ==
+<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>
+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.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 81/80, 686/675
+{{Mapping|legend=1| 1 0 -4 -3 | 0 3 12 11 }}
+: mapping generators: ~2, ~10/7
+[[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 }}
+[[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
+[[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.
+{{Optimal ET sequence|legend=1| 17c, 19, 55, 74d }}
+[[Badness]] (Sintel): 1.18
+=== Liesel ===
+Subgroup: 2.3.5.7.11
+Comma list: 56/55, 81/80, 540/539
+Mapping: {{mapping| 1 0 -4 -3 4 | 0 3 12 11 -1 }}
+Optimal tunings:
+* WE: ~2 = 1198.8507{{c}}, ~10/7 = 632.4668{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 632.9963{{c}}
+{{Optimal ET sequence|legend=0| 17c, 19, 36 }}
+Badness (Sintel): 1.35
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 56/55, 78/77, 81/80, 91/90
+Mapping: {{mapping| 1 0 -4 -3 4 0 | 0 3 12 11 -1 7 }}
+Optimal tunings:
+* WE: ~2 = 1199.4968{{c}}, ~10/7 = 632.7766{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.0082{{c}}
+{{Optimal ET sequence|legend=0| 17c, 19, 36 }}
+Badness (Sintel): 1.13
+=== Elisa ===
+Subgroup: 2.3.5.7.11
+Comma list: 77/75, 81/80, 99/98
-Mapping generator: ~11/9
+Mapping: {{mapping| 1 0 -4 -3 -5 | 0 3 12 11 16 }}
-Map: [&lt;1 1 0 -1 2|, &lt;0 2 8 13 5|]
+Optimal tunings:
+* WE: ~2 = 1201.0489{{c}}, ~10/7 = 633.6147{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1644{{c}}
-EDOs: 17c, 24, 65c, 89cd
+{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}
-Badness: 0.0362
+Badness (Sintel): 1.37
-==13-limit==
+==== 13-limit ====
-Commas: 56/55, 66/65, 77/75, 243/242
+Subgroup: 2.3.5.7.11.13
-POTE generator: ~11/9 = 350.745
+Comma list: 66/65, 77/75, 81/80, 99/98
-Mapping generator: ~11/9
+Mapping: {{mapping| 1 0 -4 -3 -5 0 | 0 3 12 11 16 7 }}
-Map: [&lt;1 1 0 -1 2 4|, &lt;0 2 8 13 5 -1|]
+Optimal tunings:
+* WE: ~2 = 1201.4815{{c}}, ~10/7 = 633.7720{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1281{{c}}
-EDOs: 17c, 24, 41c, 65c
+{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}
-Badness: 0.0287
+Badness (Sintel): 1.11
-=Orphic=
+=== Lisa ===
-Commas: 81/80, 5898240/5764801
+Subgroup: 2.3.5.7.11
-POTE generator: ~7/6 = 275.794
+Comma list: 45/44, 81/80, 343/330
-Mapping generator: ~343/288
+Mapping: {{mapping| 1 0 -4 -3 -6 | 0 3 12 11 18 }}
-Map: [&lt;2 1 -4 4|, &lt;0 4 16 3|]
+Optimal tunings:
+* WE: ~2 = 1202.6773{{c}}, ~10/7 = 632.7783{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.6175{{c}}
-Wedgie: &lt;&lt;8 32 6 32 -13 -76||
+{{Optimal ET sequence|legend=0| 17cee, 19 }}
-EDOs: 26, 74, 174bd, 248bd
+Badness (Sintel): 1.81
-Badness: 0.2588
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-==11-limit==
+Comma list: 45/44, 81/80, 91/88, 147/143
-Commas: 81/80, 99/98, 73728/73205
-POTE generator: ~7/6 = 275.762
+Mapping: {{mapping| 1 0 -4 -3 -6 0 | 0 3 12 11 18 7 }}
-Mapping generator: ~77/64
+Optimal tunings:
+* WE: ~2 = 1203.6086{{c}}, ~10/7 = 633.1193{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.5346{{c}}
-Map: [&lt;2 1 -4 4 8|, &lt;0 4 16 3 -2|]
+{{Optimal ET sequence|legend=0| 17cee, 19 }}
-EDOs: 26, 48c, 74, 248bd, 322bd
+Badness (Sintel): 1.49
-Badness: 0.1015
+== Superpine ==
+{{See also| No-sevens subgroup temperaments #Superpine }}
-==13-limit==
+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.
-Commas: 81/80, 99/98, 144/143, 2200/2197
-POTE generator: ~7/6 = 275.774
+[[Subgroup]]: 2.3.5.7
-Mapping generator: ~63/52
+[[Comma list]]: 81/80, 1119744/1071875
-Map: [&lt;2 1 -4 4 8 2|, &lt;0 4 16 3 -2 10|]
+{{Mapping|legend=1| 1 2 4 1 | 0 -3 -12 13 }}
-EDOs: 26, 48c, 74, 174bd, 248bd, 322bd
+[[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: 0.0535
+{{Optimal ET sequence|legend=1| 7, 36, 43, 79c }}
-=Mothra=
+[[Badness]] (Sintel): 3.46
-[[Comma|Comma]]s: 81/80, 1029/1024
-Mothra, with wedgie &lt;&lt;3 12 -1 12 -10 -36||, 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-limit, mothra is identical to [[Slendric|slendric]].
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Note that mothra can also be called cynder in the 7-limit, which can be a little confusing sometimes.
+Comma list: 81/80, 176/175, 864/847
-[[7-limit|7]] and [[9-limit|9-limit]] minimax 1/4 comma
+Mapping: {{mapping| 1 2 4 1 5 | 0 -3 -12 13 -11 }}
-[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |3 0 -1/12 0&gt;]
+Optimal tunings:
+* WE: ~2 = 1199.0522{{c}}, ~11/10 = 167.1904{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.3382{{c}}
-[[Eigenmonzo|Eigenmonzo]]s: 2, 5
+{{Optimal ET sequence|legend=0| 7, 36, 43 }}
-[[POTE_tuning|POTE generator]]: ~8/7 = 232.193
+Badness (Sintel): 1.90
-Mapping generator: ~8/7
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Algebraic generator: Rabrindanath, largest real root of x^8-3x^2+1, or 232.0774 cents.
+Comma list: 78/77, 81/80, 144/143, 176/175
-Map: [&lt;1 1 0 3|, &lt;0 3 12 -1|]
+Mapping: {{mapping| 1 2 4 1 5 3 | 0 -3 -12 13 -11 5 }}
-[[generator|Generator]]s: 2, 8/7
+Optimal tunings:
+* WE: ~2 = 1199.4286{{c}}, ~11/10 = 167.3105{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.3958{{c}}
-[[wedgie|Wedgie]]: &lt;&lt;3 12 -1 12 -10 -36||
+{{Optimal ET sequence|legend=0| 7, 36, 43 }}
-EDOs: [[5edo|5]], [[26edo|26]], [[31edo|31]]
+Badness (Sintel): 1.52
-[[Badness|Badness]]: 0.0371
+== 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.
-==11-limit==
+[[Subgroup]]: 2.3.5.7
-[[Comma|Comma]]s: 81/80, 99/98, 385/384
-POTE generator: ~8/7 = 232.031
+[[Comma list]]: 81/80, 3125/3087
-Mapping generator: ~8/7
+{{Mapping|legend=1| 3 0 -12 -20 | 0 1 4 6 }}
-Map: [&lt;1 1 0 3 5|, &lt;0 3 12 -1 -8|]
+: mapping generators: ~56/45, ~3
-EDOs: [[5edo|5]], [[26edo|26]], [[31edo|31]], [[88edo|88]], [[150edo|150]], [[181edo|181]]
+[[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 }}
-[[Badness|Badness]]: 0.0256
+{{Optimal ET sequence|legend=1| 12, 33cd, 45, 57 }}
-==13-limit==
+[[Badness]] (Sintel): 1.75
-Commas: 81/80, 99/98, 105/104, 144/143
-POTE generator: ~8/7 = 231.811
+== Squares ==
+{{Main| Squares }}
-Mapping generator: ~8/7
+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]].
-Map: [&lt;1 1 0 3 5 1|, &lt;0 3 12 -1 -8 14|]
+[[Subgroup]]: 2.3.5.7
-EDOs: 5, 26, 31, 57, 88
+[[Comma list]]: 81/80, 2401/2400
-Badness: 0.0240
+{{Mapping|legend=1| 1 -1 -8 -3 | 0 4 16 9 }}
-==Cynder==
+: mapping generators: ~2, ~14/9
-Commas: 45/44, 81/80, 1029/1024
-POTE generator: ~8/7 = 231.317
+[[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 generator: ~8/7
+[[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
-Map: [&lt;1 1 0 3 0|, &lt;0 3 12 -1 18|]
+[[Algebraic generator]]: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.
-EDOs: 26, 57e, 83bce
+{{Optimal ET sequence|legend=1| 14c, 17c, 31, 169b, 200b }}
-Badness: 0.0557
+[[Badness]] (Sintel): 1.16
-===13-limit===
+Scales: [[skwares8]], [[skwares11]], [[skwares14]]
-Commas: 45/44, 78/77, 81/80, 640/637
-POTE generator: ~8/7 = 231.293
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Mapping generator: ~8/7
+Comma list: 81/80, 99/98, 121/120
-Map: [&lt;1 1 0 3 0 1|, &lt;0 3 12 -1 18 14|]
+Mapping: {{mapping| 1 -1 -8 -3 -3 | 0 4 16 9 10 }}
-EDOs: 26, 57e, 83bce
+Optimal tunings:
+* WE: ~2 = 1201.6657{{c}}, ~11/7 = 775.1171{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.1754{{c}}
-Badness: 0.0341
+{{Optimal ET sequence|legend=0| 14c, 17c, 31, 130bee, 169beee }}
-==Mosura==
+Badness (Sintel): 0.715
-Commas: 81/80, 176/175, 1029/1024
-POTE generator: ~8/7 = 232.419
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Mapping generator: ~8/7
+Comma list: 66/65, 81/80, 99/98, 121/120
-Map: [&lt;1 1 0 3 -1|, &lt;0 3 12 -1 23|]
+Mapping: {{mapping| 1 -1 -8 -3 -3 5 | 0 4 16 9 10 -2 }}
-EDOs: 31, 129, 136b, 148be, 160be, 191bce, 222bce, 253bce
+Optimal tunings:
+* WE: ~2 = 1199.8419{{c}}, ~11/7 = 774.3484{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4422{{c}}
-Badness: 0.0313
+{{Optimal ET sequence|legend=0| 14c, 17c, 31, 79cf }}
-===13-limit===
+Badness (Sintel): 1.05
-Commas: 81/80, 144/143, 176/175, 1029/1024
-POTE generator: ~8/7 = 232.640
+==== Squad ====
+Subgroup: 2.3.5.7.11.13
-Mapping generator: ~8/7
+Comma list: 78/77, 81/80, 91/90, 99/98
-Map: [&lt;1 1 0 3 -1 7|, &lt;0 3 12 -1 23 -17|]
+Mapping: {{mapping| 1 -1 -8 -3 -3 -6 | 0 4 16 9 10 15 }}
-EDOs: 31, 67, 98
+Optimal tunings:
+* WE: ~2 = 1202.0312{{c}}, ~11/7 = 775.5589{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4140{{c}}
-Badness: 0.0369
+{{Optimal ET sequence|legend=0| 14cf, 17c, 31f }}
-=Squares=
+Badness (Sintel): 1.11
-[[Comma]]s: 81/80, 2401/2400
-Squares, with wedgie &lt;&lt;4 16 9 16 3 -24||, splits the interval of an eleventh, or 8/3, into four supermajor third ([[9/7]]) intervals, and uses it for a generator. [[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.
+==== Agora ====
+Subgroup: 2.3.5.7.11.13
-and 9 limit minimax 1/4 comma
+Comma list: 81/80, 99/98, 105/104, 121/120
-[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |3/2 0 9/16 0&gt;]
+Mapping: {{mapping| 1 -1 -8 -3 -3 -15 | 0 4 16 9 10 29 }}
-[[Eigenmonzo]]s: 2, 5
+Optimal tunings:
+* WE: ~2 = 1202.3228{{c}}, ~11/7 = 775.2214{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8617{{c}}
-[[POTE_tuning|POTE generator]]: ~9/7 = 425.942
+{{Optimal ET sequence|legend=0| 14cf, 31, 45ef, 76e }}
-Mapping generator: ~9/7
+Badness (Sintel): 1.01
-Algebraic generator: Sceptre2, the positive root of 9x^2+x-16, or (sqrt(577)-1)/18, which is 425.9311 cents.
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
-Map: [&lt;1 3 8 6|, &lt;0 -4 -16 -9|]
+Comma list: 81/80, 99/98, 105/104, 120/119, 121/119
-[[Generator]]s: 2, 9/7
+Mapping: {{mapping| 1 -1 -8 -3 -3 -15 -3 | 0 4 16 9 10 29 11 }}
-EDOs: [[14edo|14c]], [[17edo|17c]], [[31edo|31]], [[45edo|45]], [[76edo|76]]
+Optimal tunings:
+* WE: ~2 = 1201.4340{{c}}, ~11/7 = 774.7375{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8955{{c}}
-[[Badness]]: 0.0460
+{{Optimal ET sequence|legend=0| 14cf, 31 }}
-Music:
+Badness (Sintel): 1.15
-By [[Chris_Vaisvil|Chris Vaisvil]]
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-[http://clones.soonlabel.com/public/micro/tuning-survey/daily20100603-squares8piano.mp3 Square 8]
+Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119
-==11-limit==
+Mapping: {{mapping| 1 -1 -8 -3 -3 -15 -3 -8 | 0 4 16 9 10 29 11 19 }}
-Commas: 81/80, 99/98, 121/120
-POTE generator: ~9/7 = 425.957
+Optimal tunings:
+* WE: ~2 = 1201.2461{{c}}, ~11/7 = 774.5783{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8479{{c}}
-Mapping generator: ~9/7
+{{Optimal ET sequence|legend=0| 14cf, 31 }}
-Map: [&lt;1 3 8 6 7|, &lt;0 -4 -16 -9 -10|]
+Badness (Sintel): 1.15
-EDOs: [[14edo|14c]], [[17edo|17c]], [[31edo|31]], [[45edo|45e]], [[76edo|76e]]
+=== Cuboctahedra ===
+Subgroup: 2.3.5.7.11
-Badness: 0.0216
+Comma list: 81/80, 385/384, 1375/1372
-==13-limit==
+Mapping: {{mapping| 1 -1 -8 -3 17 | 0 4 16 9 -21 }}
-Commas: 66/65, 81/80, 99/98, 121/120
-POTE generator: ~9/7 = 425.550
+Optimal tunings:
+* WE: ~2 = 1201.4436{{c}}, ~14/9 = 774.9386{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 774.0243{{c}}
-Mapping generator: ~9/7
+{{Optimal ET sequence|legend=0| 31, 107b, 138b, 169be, 200be }}
-Map: [&lt;1 3 8 6 7 3|, &lt;0 -4 -16 -9 -10 2|]
+Badness (Sintel): 1.88
-EDOs: 14c, 17c, 31, 45e, 79cf
+== 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.
-Badness: 0.0255
+[[Subgroup]]: 2.3.5.7
-==Agora==
+[[Comma list]]: 81/80, 17280/16807
-Commas: 81/80, 99/98, 105/104, 121/120
-POTE generator: ~9/7 = 426.276
+{{Mapping|legend=1| 1 1 0 2 | 0 5 20 7 }}
-Mapping generator: ~9/7
+: mapping generators: ~2, ~54/49
-Map: [&lt;1 3 8 6 7 14|, &lt;0 -4 -16 -9 -10 -29|]
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1640{{c}}, ~54/49 = 139.3624{{c}}
+: [[error map]]: {{val| +0.164 -4.979 +0.934 +7.039 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~54/49 = 139.3528{{c}}
+: error map: {{val| 0.000 -5.191 +0.741 +6.643 }}
-EDOs: 14cf, 31, 45ef, 76e
+{{Optimal ET sequence|legend=1| 17c, 26, 43 }}
-Badness: 0.0245
+[[Badness]] (Sintel): 2.75
-===17-limit===
+=== 11-limit ===
-Commas: 81/80, 99/98, 105/104, 120/119, 121/119
+Subgroup: 2.3.5.7.11
-POTE generator: ~9/7 = 426.187
+Comma list: 81/80, 99/98, 864/847
-Mapping generator: ~9/7
+Mapping: {{mapping| 1 1 0 2 3 | 0 5 20 7 4 }}
-Map: [&lt;1 3 8 6 7 14 8|, &lt;0 -4 -16 -9 -10 -29 -11|]
+Optimal tunings:
+* WE: ~2 = 1201.4436{{c}}, ~12/11 = 139.3714{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 139.4038{{c}}
-EDOs: 14cf, 31, 45ef, 76e
+{{Optimal ET sequence|legend=0| 17c, 26, 43 }}
-===19-limit===
+Badness (Sintel): 1.58
-Commas: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119
-POTE generator: ~9/7 = 426.225
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Mapping generator: ~9/7
+Comma list: 78/77, 81/80, 99/98, 144/143
-Map: [&lt;1 3 8 6 7 14 8 11|, &lt;0 -4 -16 -9 -10 -29 -11 -19|]
+Mapping: {{mapping| 1 1 0 2 3 3 | 0 5 20 7 4 6 }}
-EDOs: 14cf, 31, 45ef, 76e
+Optimal tunings:
+* WE: ~2 = 1199.8860{{c}}, ~13/12 = 139.3737{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3817{{c}}
-=Cuboctahedra=
+{{Optimal ET sequence|legend=0| 17c, 26, 43 }}
-[[Comma]]s: 81/80, 385/384, 1375/1372
-[[POTE_tuning|POTE generator]]: ~9/7 = 425.993
+Badness (Sintel): 1.21
-Mapping generator: ~9/7
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
-Map: [&lt;1 3 8 6 -4|, &lt;0 -4 -16 -9 21|]
+Comma list: 78/77, 81/80, 99/98, 144/143, 189/187
-EDOs: [[14edo|14ce]], [[17edo|17ce]], [[31edo|31]], [[45edo|45]], [[76edo|76]], [[107edo|107b]]
+Mapping: {{mapping| 1 1 0 2 3 3 2 | 0 5 20 7 4 6 18 }}
-[[Badness]]: 0.0568
+Optimal tunings:
+* WE: ~2 = 1199.8346{{c}}, ~13/12 = 139.3431{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3544{{c}}
-=Liese=
+{{Optimal ET sequence|legend=0| 17cg, 26, 43 }}
-<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>
+Badness (Sintel): 1.06
+=== 19-limit ===
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143
+Mapping: {{mapping| 1 1 0 2 3 3 2 1 | 0 5 20 7 4 6 18 28 }}
+Optimal tunings:
+* WE: ~2 = 1199.8891{{c}}, ~13/12 = 139.3001{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3080{{c}}
+{{Optimal ET sequence|legend=0| 17cgh, 26, 43, 69 }}
+Badness (Sintel): 1.11
+== 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.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 81/80, 16875/16807
+{{Mapping|legend=1| 1 -1 -8 -7 | 0 5 20 19 }}
+: mapping generators: ~2, ~10/7
+[[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 }}
+{{Optimal ET sequence|legend=1| 29cd, 31, 188bcd, 219bbcd }}
+[[Badness]] (Sintel): 2.08
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 81/80, 99/98, 2541/2500
+Mapping: {{mapping| 1 -1 -8 -7 -11 | 0 5 20 19 28 }}
+Optimal tunings:
+* WE: ~2 = 1201.2054{{c}}, ~10/7 = 619.9752{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.4223{{c}}
+{{Optimal ET sequence|legend=0| 29cde, 31 }}
+Badness (Sintel): 1.42
+== 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.
+[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name]
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 50/49, 81/80
+{{Mapping|legend=1| 2 0 -8 -7 | 0 1 4 4 }}
+: mapping generators: ~7/5, ~3
+[[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 }}
+[[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]
+{{Optimal ET sequence|legend=1| 12, 26, 38 }}
+[[Badness]] (Sintel): 0.788
+; Music
+* [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
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 45/44, 50/49, 81/80
+Mapping: {{mapping| 2 0 -8 -7 -12 | 0 1 4 4 6 }}
+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}})
+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]
+{{Optimal ET sequence|legend=0| 12, 26 }}
+Badness (Sintel): 0.764
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 45/44, 50/49, 78/77, 81/80
+Mapping: {{mapping| 2 0 -8 -7 -12 -21 | 0 1 4 4 6 9 }}
+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}})
+Tuning ranges:
+* 13-odd-limit diamond monotone: ~3/2 = 692.308 (15\26)
+* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
+{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}
+Badness (Sintel): 0.891
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 45/44, 50/49, 78/77, 81/80, 85/84
+Mapping: {{mapping| 2 0 -8 -7 -12 -21 5 | 0 1 4 4 6 9 1 }}
+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}})
+{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}
+Badness (Sintel): 0.935
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
+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}})
+{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}
+Badness (Sintel): 0.920
+==== Enjera ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 27/26, 40/39, 45/44, 50/49
+Mapping: {{mapping| 2 0 -8 -7 -12 -2 | 0 1 4 4 6 3 }}
+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}})
+{{Optimal ET sequence|legend=0| 10cdeef, 12f }}
+Badness (Sintel): 1.10
+=== Injerous ===
+Subgroup: 2.3.5.7.11
+Comma list: 33/32, 50/49, 55/54
+Mapping: {{mapping| 2 0 -8 -7 10 | 0 1 4 4 -1 }}
+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}})
+{{Optimal ET sequence|legend=0| 12e, 14c, 26e, 40cee }}
+Badness (Sintel): 1.28
+=== Lahoh ===
+Subgroup: 2.3.5.7.11
+Comma list: 50/49, 56/55, 81/77
+Mapping: {{mapping| 2 0 -8 -7 7 | 0 1 4 4 0 }}
+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}})
+{{Optimal ET sequence|legend=0| 10cd, 12 }}
+Badness (Sintel): 1.42
+=== Teff ===
+{{Main| Teff }}
+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.
+Subgroup: 2.3.5.7.11
+Comma list: 50/49, 81/80, 864/847
+Mapping: {{mapping| 2 1 -4 -3 8 | 0 2 8 8 -1 }}
+: mapping generators: ~7/5, ~16/11
+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}})
+{{Optimal ET sequence|legend=0| 24d, 26, 50d }}
+Badness (Sintel): 2.34
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 50/49, 78/77, 81/80, 144/143
+Mapping: {{mapping| 2 1 -4 -3 8 2 | 0 2 8 8 -1 5 }}
+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}})
+{{Optimal ET sequence|legend=0| 24d, 26, 50d }}
+Badness (Sintel): 1.65
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 50/49, 78/77, 81/80, 85/84, 144/143
+Mapping: {{mapping| 2 1 -4 -3 8 2 6 | 0 2 8 8 -1 5 2 }}
+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}})
+{{Optimal ET sequence|legend=0| 24d, 26 }}
+Badness (Sintel): 1.50
+==== 19-limit ====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 50/49, 57/56, 78/77, 81/80, 85/84, 144/143
+Mapping: {{mapping| 2 1 -4 -3 8 2 6 2 | 0 2 8 8 -1 5 2 6 }}
+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}})
+{{Optimal ET sequence|legend=0| 24d, 26 }}
+Badness (Sintel): 1.41
+== 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.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 81/80, 300125/294912
+{{Mapping|legend=1| 2 1 -4 11 | 0 2 8 -5 }}
+: mapping generators: ~735/512, ~35/24
+[[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 }}
+{{Optimal ET sequence|legend=1| 24, 26, 50, 126bcd, 176bcdd, 226bbcdd }}
+[[Badness]] (Sintel): 2.94
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 81/80, 245/242, 385/384
+Mapping: {{mapping| 2 1 -4 11 8 | 0 2 8 -5 -1 }}
+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}})
+{{Optimal ET sequence|legend=0| 24, 26, 50 }}
+Badness (Sintel): 1.72
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 81/80, 105/104, 144/143, 245/242
+Mapping: {{mapping| 2 1 -4 11 8 2 | 0 2 8 -5 -1 5 }}
+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}})
+{{Optimal ET sequence|legend=0| 24, 26, 50 }}
+Badness (Sintel): 1.28
-[[Comma|Comma]]s: 81/80, 686/675
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
-Liese, with wedgie &lt;&lt;3 12 11 12 9 -8||, 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.
+Comma list: 81/80, 105/104, 144/143, 245/242, 273/272
-and 9 limit minimax 1/4 comma
+Mapping: {{mapping| 2 1 -4 11 8 2 6 | 0 2 8 -5 -1 5 2 }}
-[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |2/3 0 11/12 0&gt;]
+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}})
-[[Eigenmonzo|Eigenmonzo]]s: 2, 5
+{{Optimal ET sequence|legend=0| 24, 26, 50 }}
-[[POTE_tuning|POTE generator]]: ~10/7 = 632.406
+Badness (Sintel): 1.08
-Mapping generator: ~10/7
+=== 19-limit ===
+Subgroup: 2.3.5.7.11.13.17.19
-Algebraic generator: Radix, the real root of x^5-2x^4+2x^3-2x^2+2x-2, also a root of x^6-x^5-2. The recurrence converges.
+Comma list: 81/80, 105/104, 133/132, 144/143, 171/170, 210/209
-Map: [&lt;1 0 -4 -3|, &lt;0 3 12 11|]
+Mapping: {{mapping| 2 1 -4 11 8 2 6 2 | 0 2 8 -5 -1 5 2 6 }}
-[[generator|Generator]]s: 2, 10/7
+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}})
-EDOs: [[17edo|17]], [[19edo|19]], [[55edo|55]], [[74edo|74]]
+{{Optimal ET sequence|legend=0| 24, 26, 50 }}
-[[Badness|Badness]]: 0.0467
+Badness (Sintel): 1.01
-==Liesel==
+== Orphic ==
-Commas: 56/55, 81/80, 540/539
+Orphic has a semi-octave period and four generators plus a period gives the 3rd harmonic; its ploidacot is diploid alpha-tetracot.
-POTE generator: ~10/7 = 633.073
+[[Subgroup]]: 2.3.5.7
-Mapping generator: ~10/7
+[[Comma list]]: 81/80, 5898240/5764801
-Map: [&lt;1 0 -4 -3 4|, &lt;0 3 12 11 -1|]
+{{Mapping|legend=1| 2 1 -4 4 | 0 4 16 3 }}
-EDOs: 17c, 19, 36, 91ce
+: mapping generators: ~2401/1728, ~343/288
-Badness: 0.0407
+[[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 }}
-==13-limit==
+{{Optimal ET sequence|legend=1| 26, 48c, 74 }}
-Liesel is a very natural 13-limit tuning, given the generator is so near 13/9.
-Commas: 56/55, 78/77, 81/80, 91/90
+[[Badness]] (Sintel): 6.55
-POTE generator: ~10/7 = ~13/9 = 633.042
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Mapping generator: ~10/7
+Comma list: 81/80, 99/98, 73728/73205
-Map: [&lt;1 0 -4 -3 4 0|, &lt;0 3 12 11 -1 7|]
+Mapping: {{mapping| 2 1 -4 4 8 | 0 4 16 3 -2 }}
-EDOs: 17c, 19, 36, 91cef
+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}})
-Badness: 0.0273
+{{Optimal ET sequence|legend=0| 26, 48c, 74 }}
-==Elisa==
+Badness (Sintel): 3.36
-Commas: 77/75, 81/80, 99/98
-POTE generator: ~10/7 = 633.061
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Mapping generator: ~10/7
+Comma list: 81/80, 99/98, 144/143, 2200/2197
-Map: [&lt;1 0 -4 -3 -5|, &lt;0 3 12 11 16|]
+Mapping: {{mapping| 2 1 -4 4 8 2 | 0 4 16 3 -2 10 }}
-EDOs: 19e, 36e
+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}})
-Badness: 0.0416
+{{Optimal ET sequence|legend=0| 26, 48c, 74 }}
-==Lisa==
+Badness (Sintel): 2.21
-Commas: 45/44, 81/80, 343/330
-POTE generator: ~10/7 = 631.370
+== Cloudtone ==
+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.
-Mapping generator: ~10/7
+[[Subgroup]]: 2.3.5.7
-Map: [&lt;1 0 -4 -3 -6|, &lt;0 3 12 11 18|]
+[[Comma list]]: 81/80, 16807/16384
-EDOs: 19
+{{Mapping|legend=1| 5 0 -20 14 | 0 1 4 0 }}
-Badness: 0.0548
+: mapping generators: ~8/7, ~3
-==13-limit==
+[[Optimal tuning]]s:
-Commas: 45/44, 81/80, 91/88, 147/143
+* [[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: ~10/7 = 631.221
+{{Optimal ET sequence|legend=1| 5, 40c, 45, 50 }}
-Mapping generator: ~10/7
+[[Badness]] (Sintel): 2.59
-Map: [&lt;1 0 -4 -3 -6 0|, &lt;0 3 12 11 18 7|]
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-EDOs: 19
+Comma list: 81/80, 385/384, 2401/2376
-Badness: 0.0361
+Mapping: {{mapping| 5 0 -20 14 41 | 0 1 4 0 -3 }}
-=Jerome=
+Optimal tunings:
-Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5^(1/20), 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.
+* 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}})
-Commas: 81/80, 17280/16807
+{{Optimal ET sequence|legend=0| 5, 45, 50 }}
-POTE generator: ~54/49 = 139.343
+Badness (Sintel): 2.33
-Mapping generator: ~54/49
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Map: [&lt;1 1 0 2|, &lt;0 5 20 7|]
+Comma list: 81/80, 105/104, 144/143, 2401/2376
-Wedgie: &lt;&lt;5 30 7 20 -3 -40||
+Mapping: {{mapping| 5 0 -20 14 41 -21 | 0 1 4 0 -3 5 }}
-EDOs: 9c, 17c, 26, 43, 69, 112bd
+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}})
-Badness: 0.1087
+{{Optimal ET sequence|legend=0| 5, 45f, 50 }}
-==11-limit==
+Badness (Sintel): 2.02
-Commas: 81/80, 99/98, 864/847
-POTE generator: ~12/11 = 139.428
+== Subgroup extensions ==
+=== Stützel (2.3.5.19) ===
+[[Subgroup]]: 2.3.5.19
-Mapping generator: ~12/11
+[[Comma list]]: 81/80, 96/95
-Map: [&lt;1 1 0 2 3|, &lt;0 5 20 7 4|]
+{{Mapping|legend=2| 1 0 -4 9 | 0 1 4 -3 }}
-EDOs: 9c, 17c, 26, 43, 69
+{{Mapping|legend=3| 1 0 -4 0 0 0 0 9 | 0 1 4 0 0 0 0 -3 }}
-Badness: 0.0479
+: mapping generators: ~2, ~3
-==13-limit==
+[[Optimal tuning]]s:
-Commas: 78/77, 81/80, 99/98, 144/143
+* [[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: ~13/12 = 139.387
+{{Optimal ET sequence|legend=1| 5, 7, 12, 31, 43, 98h }}
-Mapping generator: ~12/11
+[[Badness]] (Sintel): 0.324
-Map: [&lt;1 1 0 2 3 3|, &lt;0 5 20 7 4 6|]
+=== Hypnotone ===
+Hypnotone is no-sevens [[#Flattone|flattone]].
-EDOs: 9c, 17c, 26, 43, 69
+[[Subgroup]]: 2.3.5.11
-Badness: 0.0293
+[[Comma list]]: 45/44, 81/80
-==17-limit==
+{{Mapping|legend=2| 1 0 -4 -6 | 0 1 4 6 }}
-Commas: 78/77, 81/80, 99/98, 144/143, 189/187
-POTE generator: ~13/12 = 139.362
+{{Mapping|legend=3| 1 0 -4 0 -6 | 0 1 4 0 6 }}
-Mapping generator: ~12/11
+: mapping generators: ~2, ~3
-Map: [&lt;1 1 0 2 3 3 2|, &lt;0 5 20 7 4 6 18|]
+[[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 }}
-EDOs: 26, 43, 69
+{{Optimal ET sequence|legend=1| 7, 12, 19, 26, 45 }}
-Badness: 0.0209
+[[Badness]] (Sintel): 0.326
-==19-limit==
+==== 2.3.5.11.13 subgroup ====
-Commas: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143
+Subgroup: 2.3.5.11.13
-POTE generator: ~13/12 = 139.313
+Comma list: 45/44, 65/64, 81/80
-Mapping generator: ~12/11
+Subgroup-val mapping: {{mapping| 1 0 -4 -6 10 | 0 1 4 6 -4 }}
-Map: [&lt;1 1 0 2 3 3 2 1|, &lt;0 5 20 7 4 6 18 28|]
+Gencom mapping: {{mapping| 1 0 -4 0 -6 10 | 0 1 4 0 6 -4 }}
-EDOs: 26, 43, 69
+Optimal tunings:
+* WE: ~2 = 1202.6916{{c}}, ~3/2 = 694.4181{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0870{{c}}
-Badness: 0.0182
+{{Optimal ET sequence|legend=0| 7, 12, 19, 26, 45f }}
-=Meanmag=
+Badness (Sintel): 0.561
-Commas: 81/80, 3125/3072
-POTE generator: ~8/7 = 238.396
+=== Dequarter ===
+[[Subgroup]]: 2.3.5.11
-Mapping generator: ~7
+[[Comma list]]: 33/32, 55/54
-Map: [&lt;19 30 44 0|, &lt;0 0 0 1|]
+{{Mapping|legend=2| 1 0 -4 5 | 0 1 4 -1 }}
-Wedgie: &lt;&lt;0 0 19 0 30 44||
+{{Mapping|legend=3| 1 0 -4 0 5 | 0 1 4 0 -1 }}
-EDOs: 19, 57, 76, 171bcd
+: mapping generators: ~2, ~3
-Badness: 0.0770
+[[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 }}
-=Undevigintone=
+{{Optimal ET sequence|legend=1| 5, 7, 19e, 26e }}
-Commas: 49/48, 81/80, 126/125
-POTE generator: ~11/8 = 538.047
+[[Badness]] (Sintel): 0.451
-Mapping generator: ~11
+==== Dreamtone ====
+Subgroup: 2.3.5.11.13
-Map: [&lt;19 30 44 53 0|, &lt;0 0 0 0 1|]
+Comma list: 33/32, 55/54, 975/968
-EDOs: 19, 38d
+Subgroup-val mapping: {{mapping| 1 0 -4 5 21 | 0 1 4 -1 -11 }}
-Badness: 0.0364
+Gencom mapping: {{mapping| 1 0 -4 0 5 21 | 0 1 4 0 -1 -11 }}
-==13-limit==
+Optimal tunings:
-Commas: 49/48, 65/64, 81/80, 126/125
+* 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 }}
-Map: [&lt;19 30 44 53 0 70|, &lt;0 0 0 0 1 0|]
+Badness (Sintel): 1.40
-EDOs: 19, 38d
+== References ==
+<references/>
-Badness: 0.0229
+[[Category:Temperament families]]
-[[Category:catalog]]
+[[Category:Meantone family| ]] <!-- main article -->
-[[Category:family]]
+[[Category:Meantone| ]] <!-- key article -->
-[[Category:listen]]
+[[Category:Rank 2]]
-[[Category:meantone]]
+[[Category:Listen]]
-[[Category:rank_2]]
-[[Category:theory]]

Meantone family: Difference between revisions