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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-02-08 18:17:33 UTC</tt>.<br>
| en = Meantone family
: The original revision id was <tt>605816603</tt>.<br>
| es =
: The revision comment was: <tt>Intervals are either diminished or minor, never both at once.</tt><br>
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<h4>Original Wikitext content:</h4>
{{Technical data page}}
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">&lt;span style="display: block; text-align: right;"&gt;[[toc]]
The '''meantone family''' is the family of [[rank-2 temperament]]s that [[tempering out|temper out]] the syntonic comma, [[81/80]], and thus can all be seen as [[extension]]s of [[meantone]].


== Meantone ==
{{Main| 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]].


[[Subgroup]]: 2.3.5


[[Comma list]]: 81/80


[[xenharmonie/mitteltönig|Deutsch]]
{{Mapping|legend=1| 1 0 -4 | 0 1 4 }}
&lt;/span&gt;
The [[5-limit]] parent [[Comma|comma]] of the [[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.


[[POTE tuning|POTE generator]]: ~3/2 = 696.239
: mapping generators: ~2, ~3
Mapping generator: ~3


[[Tuning Ranges of Regular Temperaments|valid range]]: [685.714, 720.000] (7 to 5)
[[Optimal tuning]]s:  
nice range: [694.786, 701.955] (1/3 comma to Pythagorean)
* [[WE]]: ~2 = 1201.3906{{c}}, ~3/2 = 697.0455{{c}}
strict range: [694.786, 701.955]
: [[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]]: [&lt;1 0 -4|, &lt;0 1 4|]
[[Minimax tuning]]:  
EDOs (patent val edo list is complete): [[5edo|5]], [[7edo|7]], [[12edo|12]], [[19edo|19]], [[24edo|24]], [[26edo|26]], [[31edo|31]], [[36edo|36]], [[38edo|38]], [[43edo|43]], [[45edo|45]], [[50edo|50]], [[55edo|55]], [[57edo|57]], [[62edo|62]], [[67edo|67]], [[69edo|69]], [[74edo|74]], [[76edo|76]], [[81edo|81]], [[86edo|86]], [[88edo|88]], [[93edo|93]], [[98edo|98]], [[100edo|100]], [[105edo|105]], [[117edo|117]], [[129edo|129]], [[212edo|212b]]
* [[5-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma)
[[Badness]]: 0.00736
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


==Seven limit children==
[[Tuning ranges]]:
The [[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;].
* 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.)


=Septimal meantone=
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b }}
&lt;span style="display: block; text-align: right;"&gt;[[xenharmonie/septimal-mitteltönig|Deutsch]]
&lt;/span&gt;
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|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|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.


[[Comma]]s: 81/80, 126/125
[[Badness]] (Sintel): 0.173


7 and [[9-limit]] minimax
=== Overview to extensions ===
[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |-3 0 5/2 0&gt;]
The second comma of the normal comma list defines which [[7-limit]] family member we are looking at.
[[Eigenmonzo]]s: 2, 5
* 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.


[[Tuning Ranges of Regular Temperaments|valid range]]: [694.737, 700.000] (19 to 12)
==== Strong extensions ====
nice range: [694.786, 701.955]
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.  
strict range: [694.786, 700.000]


[[POTE tuning|POTE generator]]: 696.495
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.  
Mapping generator: ~3


Algebraic generator: Cybozem, the real root of 15x^3-10x^2-18, which comes to 503.4257 cents. The recurrence converges quickly.
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.  


[[Map]]: [&lt;1 0 -4 -13|, &lt;0 1 4 10|]
==== Splitting the meantone fifth into two (243/242) ====
[[Generator]]s: 2, 3
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.
[[Wedgie]]: &lt;&lt;1 4 10 4 13 12||
EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[81edo|81]], [[143edo|143b]]
[[Badness]]: 0.0137


==Bimeantone==  
==== Splitting the meantone fifth into three (1029/1024) ====
Commas: 81/80, 126/125, 245/242
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.


[[POTE tuning|POTE generator]]: ~3/2 = 696.016
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]]).


Map: [&lt;2 0 -8 -26 -31|, &lt;0 1 4 10 12|]
==== 31edo as splitting the fifth into two, three and nine ====
EDOs: 12, 38d, 50
[[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).
Badness: 0.0381


===13-limit===
Temperaments discussed elsewhere include
Commas: 81/80, 105/104, 126/125, 245/242
* ''[[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]]


[[POTE tuning|POTE generator]]: ~3/2 = 695.836
The rest are considered below.


Map: [&lt;2 0 -8 -26 -31 -40|, &lt;0 1 4 10 12 15|]
== Septimal meantone ==
EDOs: 12f, 50
<div style="float:right">[[:de:septimal-mitteltönig|Deutsch]]</div>
Badness: 0.0288
{{Main| Meantone #Septimal meantone}}
{{Wikipedia| Septimal meantone temperament }}


==Unidecimal meantone aka Huygens==
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.
See also [[Meantone vs meanpop]]
[[Comma]]s: 81/80, 126/125, 99/98


[[11-limit]] minimax
[[Subgroup]]: 2.3.5.7
[|1 0 0 0 0&gt;, |25/16 -1/8 0 0 1/16&gt;, |9/4 -1/2 0 0 1/4&gt;,
|21/8 -5/4 0 0 5/8&gt;, |25/8 -9/4 0 0 9/8&gt;]
[[Eigenmonzo]]s: 2, 11/9


valid range: [696.774, 700.000] (31 to 12)
[[Comma list]]: 81/80, 126/125
nice range: [691.202, 701.955]
strict range: [696.774, 700.000]


[[POTE tuning|POTE generator]]: 696.967
{{Mapping|legend=1| 1 0 -4 -13 | 0 1 4 10 }}
Mapping generator: ~3


[[Algebraic generator]]: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents.
[[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 }}


[[Map]]: [&lt;1 0 -4 -13 -25|, &lt;0 1 4 10 18|]
[[Minimax tuning]]:  
[[Generator]]s: 2, 3
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma)
EDOs: [[7edo|7]], [[12edo|12]], [[31edo|31]], [[105edo|105]], [[198edo|198be]]
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | -3 0 5/2 0 }}
[[Badness]]: 0.0170
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


[[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]]
[[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.)


===Tridecimal meantone===
[[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.
[[Comma]]s: 66/65, 81/80, 99/98, 105/104


valid range: 697.674 (43)
{{Optimal ET sequence|legend=1| 12, 19, 31, 81, 112b, 143b }}
nice range: [691.202, 701.955]
strict range: 697.674


[[POTE tuning|POTE generator]]: ~3/2 = 696.642
[[Badness]] (Sintel): 0.347
Mapping generator: ~3


Map: [&lt;1 0 -4 -13 -25 -20|, &lt;0 1 4 10 18 15|]
=== Undecimal meantone (huygens) ===
EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[267edo|267]], [[298edo|298]]
{{Redirect|Huygens|the Dutch mathematician, physicist and astronomer|Wikipedia: Christiaan Huygens}}
[[Badness]]: 0.0180
{{See also| Huygens vs meanpop }}


===Grosstone===
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.
Commas: 81/80, 99/98, 126/125, 144/143


POTE generator: ~3/2 = 697.264
Subgroup: 2.3.5.7.11
Mapping generator: ~3


Map: [&lt;1 0 -4 -13 -25 29|, &lt;0 1 4 10 18 -16|]
Comma list: 81/80, 99/98, 126/125
EDOs: 12, 31, 43, 74
Badness: 0.0259


===Meridetone===
Mapping: {{mapping| 1 0 -4 -13 -25 | 0 1 4 10 18 }}
Commas: 78/77, 81/80, 99/98, 126/125


POTE generator: ~3/2 = 697.529
Optimal tunings:  
Mapping generator: ~3
* WE: ~2 = 1200.7636{{c}}, ~3/2 = 697.4122{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.0315{{c}}


Map: [&lt;1 0 -4 -13 -25 -39|, &lt;0 1 4 10 18 27|]
Minimax tuning:
EDOs: 43, 117df, 160bdf, 203bcdef
* 11-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}
Badness: 0.0264
: 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


===Hemimeantone===
Tuning ranges:
Commas: 81/80, 99/98, 126/125, 169/168
* 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.)


POTE generator: ~52/45 = 250.304
Algebraic generator: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
Mapping generator: ~26/15


Map: [&lt;1 0 -4 -13 -25 -5|, &lt;0 2 8 20 36 11|]
{{Optimal ET sequence|legend=0| 12, 19e, 31, 105, 136b }}
EDOs: 43, 62, 167bef, 229bef
Badness: 0.0314


==Meanpop==
Badness (Sintel): 0.563
See also [[Meantone vs meanpop]]
[[Comma]]s: 81/80, 126/125, 385/384


[[11-limit]] [[minimax]] 1/4 comma
; Music
[|1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&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]
|-3 0 5/2 0 0&gt;, |11 0 -13/4 0 0&gt;]
[[Eigenmonzo]]s: 2, 5


valid range: [694.737, 696.774] (19 to 31)
==== Grosstone ====
nice range: [691.202, 701.955]
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]].  
strict range: [694.737, 696.774]


[[POTE tuning|POTE generator]]: 696.434
Subgroup: 2.3.5.7.11.13
Mapping generator: ~3


[[Algebraic generator]]: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
Comma list: 81/80, 99/98, 126/125, 144/143


[[@http://soonlabel.com/xenharmonic/archives/607|Scott Joplin's "The Entertainer" tuned into meanpop]]
Mapping: {{mapping| 1 0 -4 -13 -25 29 | 0 1 4 10 18 -16 }}


Map: [&lt;1 0 -4 -13 24|, &lt;0 1 4 10 -13|]
Optimal tunings:  
[[Generator]]s: 2, 3
* WE: ~2 = 1199.9389{{c}}, ~3/2 = 697.2282{{c}}
EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[81edo|81]], [[112edo|112]]
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.2627{{c}}
[[Badness]]: 0.0215


[[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]]
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


===13-limit Meanpop===  
Tuning ranges:
[[Comma]]s: 81/80, 105/104, 144/143, 196/195
* 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.)


valid range: [694.737, 696.774] (19 to 31)
{{Optimal ET sequence|legend=0| 12, 31, 43, 74 }}
nice range: [691.202, 701.955]
strict range: [694.737, 696.774]


POTE generator: ~3/2 = 696.211
Badness (Sintel): 1.07
Mapping generator: ~3


Map: [&lt;1 0 -4 -13 24 -20|, &lt;0 1 4 10 -13 15|]
===== 17-limit =====
EDOS: [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]], [[131edo|131bd]], [[212edo|212bdf]]
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]]: 0.0209


===Meanplop===
Subgroup: 2.3.5.7.11.13.17
Commas: 65/64, 78/77, 81/80, 91/90


POTE generator: ~3/2 = 696.202
Comma list: 81/80, 99/98, 120/119, 126/125, 144/143
Mapping generator: ~3


Map: [&lt;1 0 -4 -13 24 10|, &lt;0 1 4 10 -13 -4|]
Mapping: {{mapping| 1 0 -4 -13 -25 29 12 | 0 1 4 10 18 -16 -5 }}
EDOs: 12e, 19, 31f, 50f
Badness: 0.0277


==Meanenneadecal==
Optimal tunings:
[[Comma]]s: 45/44, 56/55, 81/80
* WE: ~2 = 1199.5811{{c}}, ~3/2 = 697.0918{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3303{{c}}


[[POTE tuning|POTE generator]]: ~3/2 = 696.250
{{Optimal ET sequence|legend=0| 12, 31, 43, 74g }}
Mapping generator: ~3


Map: [&lt;1 0 -4 -13 -6|, &lt;0 1 4 10 6|]
Badness (Sintel): 1.06
EDOs: [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31e]], [[50edo|50e]]
[[Badness]]: 0.0214


===13-limit===  
===== 19-limit =====
[[Comma]]s: 45/44, 56/55, 78/77, 81/80
Subgroup: 2.3.5.7.11.13.17.19


[[POTE tuning|POTE generator]]: ~3/2 = 696.146
Comma list: 81/80, 96/95, 99/98, 120/119, 126/125, 144/143
Mapping generator: ~3


Map: [&lt;1 0 -4 -13 -6 -20|, &lt;0 1 4 10 6 15|]
Mapping: {{mapping| 1 0 -4 -13 -25 29 12 9 | 0 1 4 10 18 -16 -5 -3 }}
EDOs: [[19edo|19]], [[31edo|31e]], [[50edo|50e]]]
[[Badness]]: 0.0212


===Vincenzo===
Optimal tunings:
Commas: 81/80 126/125 45/44 65/64 256/255 153/152 23/22
* WE: ~2 = 1199.2931{{c}}, ~3/2 = 696.9690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3736{{c}}


POTE generator: ~3/2
{{Optimal ET sequence|legend=0| 12, 31, 43, 74gh }}
Mapping generator: ~3


Map: [&lt;1 0 -4 -13 ... |, &lt;0 1 4 10 6 -4 -5 -3 -6|]
Badness (Sintel): 1.07
EDOs: 12
Badness:


==Meanundeci==  
==== Fokkertone ====
Commas: 33/32, 55/54, 77/75
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.


POTE generator: ~3/2 = 694.689
This extension used to be known as ''tridecimal meantone'', but was decanonicalized in 2025.  
Mapping generator: ~3


Map: [&lt;1 0 -4 -13 5|, &lt;0 1 4 10 -1|]
Subgroup: 2.3.5.7.11.13
EDOs: 12e, 19e
Badness: 0.0315


===13-limit===
Comma list: 66/65, 81/80, 99/98, 105/104
Commas: 33/32, 55/54, 77/75, 729/728


POTE generator: ~3/2 = 694.764
Mapping: {{mapping| 1 0 -4 -13 -25 -20 | 0 1 4 10 18 15 }}
Mapping generator: ~3


Map: [&lt;1 0 -4 -13 5 10|, &lt;0 1 4 10 -1 -4|]
Optimal tunings:  
EDOs: 12e, 19e
* WE: ~2 = 1200.8149{{c}}, ~3/2 = 697.1155{{c}}
Badness: 0.0263
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.7085{{c}}


==Meanundec==
Minimax tuning:
Commas: 27/26, 40/39, 45/44, 56/55
* 13- and 15-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}
: unchanged-interval (eigenmonzo) basis: 2.11/9


POTE generator: ~3/2 = 697.254
{{Optimal ET sequence|legend=0| 12f, 19e, 31 }}
Mapping generator: ~3


Map: [&lt;1 0 -4 -13 -6 -1|, &lt;0 1 4 10 6 3|]
Badness (Sintel): 0.746
EDOS: 12f, 19f, 31ef
Badness: 0.0242


=Flattone=  
===== 17-limit =====
[[Comma]]s: 81/80, 525/512
Subgroup: 2.3.5.7.11.13.17


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|7/4]] is a diminished seventh interval. Other intervals are [[7_6|7/6]], a diminished third, and [[7_5|7/5]], a doubly diminshed fifth. Good tunings for flattone are [[26edo]], [[45edo]] and [[64edo]].
Comma list: 66/65, 81/80, 99/98, 105/104, 120/119


[[7-limit]] minimax
Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 | 0 1 4 10 18 15 -5 }}
[|1 0 0 0&gt;, |21/13 0 1/13 -1/13&gt;,
|32/13 0 4/13 -4/13&gt;, |32/13 0 -9/13 9/13&gt;]
[[Eigenmonzo]]s: 2, 7/5


[[9-limit]] minimax
Optimal tunings:
[|1 0 0 0&gt;, |17/11 2/11 0 -1/11&gt;,
* WE: ~2 = 1199.5548{{c}}, ~3/2 = 696.7449{{c}}
|24/11 8/11 0 -4/11&gt;, |34/11 -18/11 0 9/11&gt;]
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.9823{{c}}
[[Eigenmonzo]]s: 2, 9/7


valid range: [692.308, 694.737] (26 to 19)
{{Optimal ET sequence|legend=0| 12f, 31 }}
nice range: [692.353, 701.955]
strict range: [692.353, 694.737]


[[POTE tuning|POTE generator]]: 693.779
Badness (Sintel): 1.02
Mapping generator: ~3


Algebraic generator: Squarto, the positive root of 8x^2-4x-9, at 506.3239 cents, equal to (1+sqrt(19))/4.
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;1 0 -4 17|, &lt;0 1 4 -9|]
Comma list: 66/65, 81/80, 96/95, 99/98, 105/104, 120/119
[[Wedgie]]: &lt;&lt;1 4 -9 4 -17 -32||
[[Generator]]s: 2, 3
EDOs: [[7edo|7]], [[19edo|19]], [[45edo|45]], [[64edo|64]]
[[Badness]]: 0.0386


==11-limit==
Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 9 | 0 1 4 10 18 15 -5 -3 }}
Commas: 45/44, 81/80, 385/384


valid range: [692.308, 694.737] (26 to 19)
Optimal tunings:  
nice range: [682.502, 701.955]
* WE: ~2 = 1199.0408{{c}}, ~3/2 = 696.5824{{c}}
strict range: [692.308, 694.737]
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.1061{{c}}


POTE generator: ~3/2 = 693.126
{{Optimal ET sequence|legend=0| 12f, 31 }}
Mapping generator: ~3


Map: [&lt;1 0 -4 17 -6|, &lt;0 1 4 -9 6|]
Badness (Sintel): 1.10
EDOs: 7, 19, 26, 45, 71bc, 116bcde
Badness: 0.0338


==13-limit==  
==== Meridetone ====
45/44, 65/64, 78/77, 81/80
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: [692.308, 694.737] (26 to 19)
Subgroup: 2.3.5.7.11.13
nice range: [682.502, 701.955]
strict range: [692.308, 694.737]


POTE generator: ~3/2 = 693.058
Comma list: 78/77, 81/80, 99/98, 126/125
Mapping generator: ~3


Map: [&lt;1 0 -4 17 -6 10|, &lt;0 1 4 -9 6 -4|]
Mapping: {{mapping| 1 0 -4 -13 -25 -39 | 0 1 4 10 18 27 }}
EDOs: 7, 19, 26, 45f, 71bcf, 116bcdef
Badness: 0.0223


=Dominant=  
Optimal tunings:
[[Comma]]s: 36/35, 64/63
* WE: ~2 = 1199.9122{{c}}, ~3/2 = 697.4779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.5241{{c}}


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|3/2]] fifths, and with [[29edo]], [[41edo]], or [[53edo]].
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


valid range: [700.000, 720.000] (12 to 5)
{{Optimal ET sequence|legend=0| 12f, 31f, 43 }}
nice range: [694.786, 715.587]
strict range: [700.000, 715.587]


[[POTE tuning|POTE generator]]: 701.573
Badness (Sintel): 1.09
Mapping generator: ~3


Map: [&lt;1 0 -4 6|, &lt;0 1 4 -2|]
===== 17-limit =====
[[Wedgie]]: &lt;&lt;1 4 -2 4 -6 -16||
Subgroup: 2.3.5.7.11.13.17
EDOs: [[5edo|5]], [[7edo|7]], [[12edo|12]], [[53edo|53]], [[65edo|65]]
[[Badness]]: 0.0207


==11-limit==
Comma list: 78/77, 81/80, 99/98, 120/119, 126/125
Commas: 36/35, 64/63, 56/55


valid range: [700.000, 705.882] (12 to 17)
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 | 0 1 4 10 18 27 -5 }}
nice range: [691.202, 715.587]
strict range: [700.000, 705.882]


POTE generator: ~3/2 = 703.254
Optimal tunings:
Mapping generator: ~3
* WE: ~2 = 1199.3793{{c}}, ~3/2 = 697.2833{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6222{{c}}


Map: [&lt;1 0 -4 6 13|, &lt;0 1 4 -2 -6|]
{{Optimal ET sequence|legend=0| 12f, 43 }}
EDOs: 5, 12, 17c, 29cde
Badness: 0.0242


==13-limit==
Badness (Sintel): 1.22
Commas: 36/35, 56/55, 64/63, 66/65


valid range: 705.882 (17)
===== 19-limit =====
nice range: [691.202, 715.587]
Subgroup: 2.3.5.7.11.13.17.19
strict range:705.882


POTE generator: ~3/2 = 703.636
Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125


Map: [&lt;1 0 -4 6 13 18|, &lt;0 1 4 -2 -6 -9|]
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 9 | 0 1 4 10 18 27 -5 -3 }}
EDOs: 12f, 17c, 29cdef
Badness: 0.0241


==Dominion==
Optimal tunings:
Commas: 26/25, 36/35, 56/55, 64/63
* WE: ~2 = 1199.0260{{c}}, ~3/2 = 697.1486{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6887{{c}}


POTE generator: ~3/2 = 704.905
{{Optimal ET sequence|legend=0| 12f, 43 }}


Map: [&lt;1 0 -4 6 13 -9|, &lt;0 1 4 -2 -6 8|]
Badness (Sintel): 1.25
EDOs: 5, 12, 17c, 46cde
Badness: 0.0273


==Domineering==  
==== Hemimeantone ====
Commas: 36/35, 45/44, 64/63
Subgroup: 2.3.5.7.11.13


POTE generator: ~3/2 = 698.776
Comma list: 81/80, 99/98, 126/125, 169/168
Mapping generator: ~3


Map: [&lt;1 0 -4 6 -6|, &lt;0 1 4 -2 6|]
Mapping: {{mapping| 1 0 -4 -13 -25 -5 | 0 2 8 20 36 11 }}
EDOs: 7, 12, 43de
Badness: 0.0220


==Domination==
: mapping generators: ~2, ~26/15
Commas: 36/35, 64/63, 77/75


POTE generator: ~3/2 = 705.004
Optimal tunings:
Mapping generator: ~3
* 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 6 -14|, &lt;0 1 4 -2 11|]
{{Optimal ET sequence|legend=0| 19e, 43, 62 }}
EDOs: 17c, 46cd
Badness: 0.0366


===13-limit===
Badness (Sintel): 1.30
Commas: 26/25, 36/35, 64/63, 66/65


POTE generator: ~3/2 = 705.496
===== 17-limit =====
Mapping generator: ~3
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, 99/98, 126/125, 169/168, 221/220
EDOs: 17c
Badness: 0.0274


==Twelve==
Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 | 0 2 8 20 36 11 33 }}
Commas: 81/80 64/63 45/44 65/64 256/255 153/152


POTE generator: ~3/2 = 696.217
Optimal tunings:
Mapping generator: ~3
* 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 6 -6 10 12 9|, &lt;0 1 4 -2 6 -4 -5 -3|]
{{Optimal ET sequence|legend=0| 19eg, 43, 62 }}
EDOs: 7, 12, 19d, 31def
Badness: 0.0204


==Arnold==
Badness (Sintel): 1.19
Commas: 22/21, 33/32, 36/35


POTE generator: ~3/2 = 698.491
===== 19-limit =====
Mapping generator: ~3
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;1 0 -4 6 5|, &lt;0 1 4 -2 -1|]
Comma list: 81/80, 99/98, 126/125, 153/152, 169/168, 221/220
EDOs: 5, 7, 12e
Badness: 0.0261


==13-limit==
Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 -25 | 0 2 8 20 36 11 33 37 }}
Commas: 22/21, 27/26, 33/32, 40/39


POTE generator: ~3/2 = 696.743
Optimal tunings:
Mapping generator: ~3
* 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 6 5 -1|, &lt;0 1 4 -2 -1 3|]
{{Optimal ET sequence|legend=0| 19egh, 43, 62 }}
EDOs: 5, 7, 12ef, 19def, 31def
Badness: 0.0233


==Dominatrix==
Badness (Sintel): 1.15
Commas: 27/26 36/35 45/44 64/63


POTE generator: ~3/2 = 698.544
==== Semimeantone ====
Mapping generator: ~3
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 -4 6 -6 -1|, &lt;0 1 4 -2 6 3|]
Comma list: 81/80, 99/98, 126/125, 847/845
EDOs: 7, 12f
Badness: 0.0183


=Sharptone=
Mapping: {{mapping| 2 0 -8 -26 -50 -59 | 0 1 4 10 18 21 }}
[[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.
: mapping generators: ~55/39, ~3


[[POTE tuning|POTE generator]]: 700.140
Optimal tunings:  
Mapping generator: ~3
* WE: ~55/39 = 600.3606{{c}}, ~3/2 = 697.4241{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 697.0545{{c}}


Map: [&lt;1 0 -4 -2|, &lt;0 1 4 3|]
{{Optimal ET sequence|legend=0| 12f, …, 50eff, 62, 136b }}
[[Wedgie]]: &lt;&lt;1 4 3 4 2 -4||
EDOs: [[5edo|5]], [[12edo|12]]
[[Badness]]: 0.0248


=Meansept=
Badness (Sintel): 1.68
Commas: 15/14, 81/80


POTE generator: ~3/2 = 682.895
===== 17-limit =====
Mapping generator: ~3
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 0 -4 -5|, &lt;0 1 4 5|]
Comma list: 81/80, 99/98, 126/125, 221/220, 289/288
Wedgie: &lt;&lt;1 4 5 4 5 0||
EDOs: 7
Badness: 0.0453


==11-limit==
Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 | 0 1 4 10 18 21 1 }}
Commas: 15/14, 22/21, 125/121


POTE generator: ~3/2 = 685.234
Optimal tunings:
Mapping generator: ~3
* WE: ~17/12 = 600.5426{{c}}, ~3/2 = 697.5571{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9858{{c}}


Map: [&lt;1 0 -4 -5 -6|, &lt;0 1 4 5 6|]
{{Optimal ET sequence|legend=0| 12f, 50eff, 62, 136bg }}
EDOs: 7
Badness: 0.0325


=Supermean=
Badness (Sintel): 1.60
Commas: 81/80, 672/625


POTE generator: ~3/2 = 704.889
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;1 0 -4 -21|, &lt;0 1 4 15|]
Comma list: 81/80, 99/98, 126/125, 153/152, 209/208, 221/220
EDOs: 17c, 46c
Badness: 0.1342


==11-limit==
Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 -1 | 0 1 4 10 18 21 1 3 }}
Commas: 56/55, 81/80, 132/125


POTE generator: ~3/2 = 705.096
Optimal tunings:  
* WE: ~17/12 = 600.5959{{c}}, ~3/2 = 697.5985{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9638{{c}}


Map: [&lt;1 0 -4 -21 -14|, &lt;0 1 4 15 11|]
{{Optimal ET sequence|legend=0| 12f, 50eff, 62 }}
EDOs: 17c, 46c
Badness: 0.0633


==13-limit==
Badness (Sintel): 1.47
Commas: 26/25, 56/55, 66/65, 81/80


POTE generator: ~3/2 = 705.094
=== Meanpop ===
{{See also| Huygens vs meanpop }}


Map: [&lt;1 0 -4 -21 -14 -9|, &lt;0 1 4 15 11 8|]
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.
EDOs: 17c, 46c


=Injera=
Subgroup: 2.3.5.7.11
[[Comma]]s: 50/49, 81/80


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]], which is two parallel [[19edo]]s, is an excellent tuning for injera.
Comma list: 81/80, 126/125, 385/384


[[http://tech.groups.yahoo.com/group/tuning-math/message/3091|Origin of the name]]
Mapping: {{mapping| 1 0 -4 -13 24 | 0 1 4 10 -13 }}


valid range: [685.714, 700.000] (14c to 12)
: mapping generator: ~2, ~3
nice range: [688.957, 701.955]
strict range: [688.957, 700.000]


[[POTE tuning|POTE generator]]: 694.375
Optimal tunings:  
Mapping generator: ~3
* WE: ~2 = 1201.3464{{c}}, ~3/2 = 697.2159{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4509{{c}}


Map: [&lt;2 0 -8 -7|, &lt;0 1 4 4|]
Minimax tuning:  
[[Wedgie]]: &lt;&lt;2 8 8 8 7 -4||
* 11-odd-limit: ~3/2 = {{monzo| 0 0 1/4 }}
EDOs: [[12edo|12]], [[26edo|26]], [[38edo|38]], [[102edo|102bcd]], [[140edo|140bcd]], [[178edo|178bcd]]
: 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 }}]
[[Badness]]: 0.0311
: unchanged-interval (eigenmonzo) basis: 2.5


[[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3|Two Pairs of Socks]] (in [[26edo]]) by [[Igliashon Jones|Igliashon Calvin Jones-Coolidge]]
Tuning ranges:
[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20Injera%20Jam.mp3|Injera Jam]] (in [[26edo]]) by [[Zach Curley]]
* 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.)


==11-limit==
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.
Commas: 45/44, 50/49, 81/80


valid range: [685.714, 700.000] (14c to 12)
{{Optimal ET sequence|legend=0| 12e, 19, 31, 81, 112b }}
nice range: [682.458, 701.955]
strict range: [685.714, 700.000]


POTE generator: ~3/2 = 692.840
Badness (Sintel): 0.712
Mapping generator: ~3


Map: [&lt;2 0 -8 -7 -12|, &lt;0 1 4 4 6|]
; Music
EDOs: 12, 14c, 26. 90bce, 116bce
* [http://soonlabel.com/xenharmonic/archives/607 Scott Joplin's "The Entertainer" tuned into meanpop]{{dead link}}
Badness: 0.0231
* [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]


==13-limit==  
==== Tridecimal meanpop ====
Commas: 45/44, 50/49, 81/80, 78/77
Subgroup: 2.3.5.7.11.13


valid range: 692.308 (26)
Comma list: 81/80, 105/104, 126/125, 144/143
nice range: [682.458, 701.955]
strict range: 692.308 (26)


POTE generator: ~3/2 = 692.673
Mapping: {{mapping| 1 0 -4 -13 24 -20 | 0 1 4 10 -13 15 }}
Mapping generator: ~3


Map: [&lt;2 0 -8 -7 -12 -21|, &lt;0 1 4 4 6 9|]
Optimal tunings:  
EDOs: 26, 104bcf
* WE: ~2 = 1201.0765{{c}}, ~3/2 = 696.8361{{c}}
Badness: 0.0216
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2347{{c}}


==Enjera==
Minimax tuning:
Commas: 27/26, 40/39, 45/44, 99/98
* 13- and 15-odd-limit: ~3/2 = {{monzo| 4/7 0 0 0 -1/28 1/28 }}
: unchanged-interval (eigenmonzo) basis: 2.13/11


POTE generator: ~3/2 = 694.121
Tuning ranges:
Mapping generator: ~3
* 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.)


Map: [&lt;2 0 -8 -7 -12 -2|, &lt;0 1 4 4 6 3|]
{{Optimal ET sequence|legend=0| 19, 31, 50, 81 }}
EDOs: 12f, 26f, 38ef
Badness: 0.0265


==Injerous==
Badness (Sintel): 0.863
Commas: 33/32, 50/49, 55/54


POTE generator: ~3/2 = 690.548
===== Meanpoppic =====
Mapping generator: ~3
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;2 0 -8 -7 10|, &lt;0 1 4 4 -1|]
Comma list: 81/80, 105/104, 126/125, 144/143, 273/272
EDOs: 12e, 14c, 26e, 40ce
Badness: 0.0386


==Lahoh==
Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 | 0 1 4 10 -13 15 26 }}
Commas: 50/49, 56/55, 81/77


POTE generator: ~3/2 = 699.001
Optimal tunings:
Mapping generator: ~3
* WE: ~2 = 1201.0727{{c}}, ~3/2 = 696.8168{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2195{{c}}


Map: [&lt;2 0 -8 -7 7|, &lt;0 1 4 4 0|]
{{Optimal ET sequence|legend=0| 19g, 31, 50, 81, 131bd }}
EDOs: 12
Badness: 0.0431


=Godzilla=
Badness (Sintel): 1.02
Main article: [[Semaphore and Godzilla]]
[[Comma]]s: 49/48, 81/80


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]] 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.
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19


valid range: [240.000, 257.143] (5 to 14c)
Comma list: 81/80, 105/104, 126/125, 144/143, 153/152, 273/272
nice range: [231.174, 266.871]
strict range: [240.000, 257.143]


[[POTE tuning|POTE generator]]: ~8/7 = 252.635
Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 -40 | 0 1 4 10 -13 15 26 28 }}
Mapping generator: ~7/4


Map: [&lt;1 0 -4 2|, &lt;0 2 8 1|]
Optimal tunings:  
[[Wedgie]]: &lt;&lt;2 8 1 8 -4 -20||
* WE: ~2 = 1201.0719{{c}}, ~3/2 = 696.8101{{c}}
EDOs: [[5edo|5]], [[9edo|9c]], [[14edo|14c]], [[19edo|19]], [[62edo|62d]], [[81edo|81d]], 143bd
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2137{{c}}
[[Badness]]: 0.0267


==11-limit==
{{Optimal ET sequence|legend=0| 19gh, 31, 50, 81 }}
Commas: 45/44, 49/48, 81/80


valid range: [252.632, 257.143] (19 to 14c)
Badness (Sintel): 1.08
nice range: [231.174, 266.871]
strict range: [252.632, 257.143]


POTE generator: ~8/7 = 254.027
===== Meanpoid =====
Mapping generator: ~7/4
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 0 -4 2 -6|, &lt;0 2 8 1 12|]
Comma list: 81/80, 105/104, 120/119, 126/125, 144/143
EDOs: 14c, 19, 33cd, 52cd
Badness: 0.0290


==13-limit==
Mapping: {{mapping| 1 0 -4 -13 24 -20 12 | 0 1 4 10 -13 15 -5 }}
Commas: 45/44, 49/48, 78/77, 81/80


valid range: 694.737 (19)
Optimal tunings:  
nice range: [621.581, 737.652]
* WE: ~2 = 1200.2768{{c}}, ~3/2 = 696.5683{{c}}
strict range: 694.737
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4114{{c}}


POTE generator: ~8/7 = 253.603
{{Optimal ET sequence|legend=0| 19, 31 }}
Mapping generator: ~7/4


Map: [&lt;1 0 -4 2 -6 -5|, &lt;0 2 8 1 12 11|]
Badness (Sintel): 1.17
EDOs: 14cf, 19, 33cdf, 52cdf
Badness: 0.0225


==Semafour==  
====== 19-limit ======
Commas: 33/32, 49/48, 55/54
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~8/7 = 254.042
Comma list: 77/76, 81/80, 96/95, 105/104, 120/119, 126/125
Mapping generator: ~7/4


Map: [&lt;1 0 -4 2 5|, &lt;0 2 8 1 -2|]
Mapping: {{mapping| 1 0 -4 -13 24 -20 12 9 | 0 1 4 10 -13 15 -5 -3 }}
EDOs: 5, 14c, 19e, 33cde
Badness: 0.0285


==Varan==
Optimal tunings:
Commas: 49/48, 77/75, 81/80
* WE: ~2 = 1199.7905{{c}}, ~3/2 = 696.3779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4973{{c}}


POTE generator: ~8/7 = 251.079
{{Optimal ET sequence|legend=0| 19, 31 }}
Mapping generator: ~7/4


Map: [&lt;1 0 -4 2 -10|, &lt;0 2 8 1 17|]
Badness (Sintel): 1.25
EDOs: 19e, 24, 43de
Badness: 0.0396


===13-limit===  
==== Semimeanpop ====
Commas: 49/48, 66/65, 77/75, 81/80
Subgroup: 2.3.5.7.11.13


POTE generator: ~8/7 = 251.165
Comma list: 81/80, 126/125, 385/384, 847/845
Mapping generator: ~7/4


Map: [&lt;1 0 -4 2 -10 -5|, &lt;0 2 8 1 17 11|]
Mapping: {{mapping| 2 0 -8 -26 48 39 | 0 1 4 10 -13 -10 }}
EDOs: 19e, 24, 43de
Badness: 0.0257


==Baragon==
: mapping generators: ~55/39, ~3
Commas: 49/48, 56/55, 81/80


POTE generator: ~8/7 = 251.173
Optimal tunings:  
Mapping generator: ~7/4
* WE: ~55/39 = 600.6704{{c}}, ~3/2 = 697.2151{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.4341{{c}}


Map: [&lt;1 0 -4 2 9|, &lt;0 2 8 1 -7|]
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112b }}
EDOs: 19, 24, 43d
Badness: 0.0357


==Music==
Badness (Sintel): 1.78
[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Bobro/GodzillaExample.mp3|Godzilla Example]] by [[Cameron Bobro]]
[[http://tinyurl.com/4uyumk9|"Change is on the Wind"]] in Godzilla[9] by [[Igliashon Jones]]


=Mohajira=  
===== 17-limit =====
&lt;span style="display: block; text-align: right;"&gt;[[xenharmonie/Mohajira|Deutsch]]
Subgroup: 2.3.5.7.11.13.17
&lt;/span&gt;
[[Comma]]s: 81/80, 6144/6125


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.
Comma list: 81/80, 126/125, 221/220, 273/272, 289/288


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.
Mapping: {{mapping| 2 0 -8 -26 48 39 5 | 0 1 4 10 -13 -10 1 }}


[[7-limit|7]] and [[9-limit]] minimax 1/4 comma
Optimal tunings:
[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |6 0 -11/8 0&gt;]
* WE: ~17/12 = 600.7232{{c}}, ~3/2 = 697.2820{{c}}
[[Eigenmonzo]]s: 2, 5
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4411{{c}}


[[POTE tuning|POTE generator]]: ~128/105 = 348.415
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bg }}
Mapping generator: ~128/105


Algebraic generator: Mohabis, real root of 3x^3-3x^2-1, 348.6067 cents. Corresponding recurrence converges quickly.
Badness (Sintel): 1.45


Map: [&lt;1 1 0 6|, &lt;0 2 8 -11|]
===== 19-limit =====
[[Generator]]s: 2, 128/105
Subgroup: 2.3.5.7.11.13.17.19
[[Wedgie]]: &lt;&lt;2 8 -11 8 -23 -48||
EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]]
[[Badness]]: 0.0557


==11-limit==
Comma list: 81/80, 126/125, 153/152, 209/208, 221/220, 273/272
[[Comma]]s: 81/80, 121/120, 176/175


[[11-limit]] minimax 1/4 comma
Mapping: {{mapping| 2 0 -8 -26 48 39 5 -1 | 0 1 4 10 -13 -10 1 3 }}
[|1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;,
|6 0 -11/8 0 0&gt;, |2 0 5/8 0 0&gt;]
[[Eigenmonzo]]s: 2, 5


[[POTE tuning|POTE generator]]: ~11/9 = 348.477
Optimal tunings:  
Mapping generator: ~11/9
* WE: ~17/12 = 600.7527{{c}}, ~3/2 = 697.3244{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4525{{c}}


Map: [&lt;1 1 0 6 2|, &lt;0 2 8 -11 5|]
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bgh }}
[[Generator]]s: 2, 11/9
EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]]
[[Badness]]: 0.0261


==13-limit==
Badness (Sintel): 1.28
Commas: 81/80, 121/120, 105/104, 66/65


POTE generator: ~11/9 = 348.558
=== Meanenneadecal ===
Mapping generator: ~11/9
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.


Map: [&lt;1 1 0 6 2 4|, &lt;0 2 8 -11 5 -1|]
Subgroup: 2.3.5.7.11
EDOs: 7, 24, 31, 117ef, 148bef
Badness: 0.0234


=Ptolemy=
Comma list: 45/44, 56/55, 81/80
Commas: 81/80, 121/120, 525/512


POTE generator: ~11/9 = 346.922
Mapping: {{mapping| 1 0 -4 -13 -6 | 0 1 4 10 6 }}


Map: [&lt;1 1 0 8 2|, &lt;0 2 8 -18 5|]
Optimal tunings:  
EDOs: 7, 38d, 45e, 83bcde
* WE: ~2 = 1199.6946{{c}}, ~3/2 = 696.0729{{c}}
Badness: 0.0588
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2083{{c}}


==13-limit==
Tuning ranges:
Commas: 65/64, 81/80, 105/104, 121/120
* 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]


POTE generator: ~11/9 = 346.910
{{Optimal ET sequence|legend=0| 7d, 12, 19, 31e }}


Map: [&lt;1 1 0 8 2 6|, &lt;0 2 8 -18 5 -8|]
Badness (Sintel): 0.708
EDOs: 7, 38df, 45ef, 83bcdef
Badness: 0.0343


=Maqamic=  
==== 13-limit ====
&lt;span style="display: block; text-align: right;"&gt;[[xenharmonie/maqamisch|Deutsch]]
Subgroup: 2.3.5.7.11.13
&lt;/span&gt;
Main article: [[Maqamic]]
[[Comma]]s: 81/80, 36/35, 121/120


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.
Comma list: 45/44, 56/55, 78/77, 81/80


[[POTE tuning|POTE generator]]: ~11/9 = 350.934
Mapping: {{mapping| 1 0 -4 -13 -6 -20 | 0 1 4 10 6 15 }}
Mapping generator: ~11/9


Map: [&lt;1 1 0 4 2|, &lt;0 2 8 -4 5|]
Optimal tunings:  
[[Generator]]s: 2, 11/9
* WE: ~2 = 1199.7931{{c}}, ~3/2 = 696.0258{{c}}
EDOs: [[7edo|7]], [[10edo|10c]], [[17edo|17c]], [[24edo|24d]], [[31edo|31d]]
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1241{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 7df, 12f, 19, 31e }}
[[Comma]]s: 81/80, 36/35, 121/120, 144/143


[[POTE tuning|POTE generator]]: ~11/9 = 350.816
Badness (Sintel): 0.875
Mapping generator: ~11/9


Map: [&lt;1 1 0 4 2 4|, &lt;0 2 8 -4 5 -1|]
===== 17-limit =====
Generators: 2, 11/9
Subgroup: 2.3.5.7.11.13.17
EDOs: [[7edo|7]], [[10edo|10c]], [[17edo|17c]], [[24edo|24d]],[[31edo| 31d]]


=Migration=
Comma list: 45/44, 56/55, 78/77, 81/80, 120/119
Commas: 81/80, 121/120, 126/125


POTE generator: ~11/9 = 348.182
Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 | 0 1 4 10 6 15 -5 }}
Mapping generator: ~11/9


Map: [&lt;1 1 0 -3 2|, &lt;0 2 8 20 5|]
Optimal tunings:  
EDOs: 31, 100de, 131bde, 162bde
* WE: ~2 = 1198.6665{{c}}, ~3/2 = 695.8010{{c}}
Badness: 0.0255
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4998{{c}}


=Mohamaq=
{{Optimal ET sequence|legend=0| 12f, 19, 31e }}
Commas: 81/80, 392/375


POTE generator: ~25/21 = 350.586
Badness (Sintel): 1.17
Mapping generator: ~25/21


Map: [&lt;1 1 0 -1|, &lt;0 2 8 13|]
===== 19-limit =====
EDOs: 17c, 24, 65c, 89cd
Subgroup: 2.3.5.7.11.13.17.19
Badness: 0.0777


==11-limit==
Comma list: 45/44, 56/55, 78/77, 81/80, 96/95, 120/119
Commas: 56/55, 77/75, 243/242


POTE generator: ~11/9 = 350.565
Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 9 | 0 1 4 10 6 15 -5 -3 }}
Mapping generator: ~11/9


Map: [&lt;1 1 0 -1 2|, &lt;0 2 8 13 5|]
Optimal tunings:  
EDOs: 17c, 24, 65c, 89cd
* WE: ~2 = 1198.2880{{c}}, ~3/2 = 695.7123{{c}}
Badness: 0.0362
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.6370{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 12f, 19, 31e }}
Commas: 56/55, 66/65, 77/75, 243/242


POTE generator: ~11/9 = 350.745
Badness (Sintel): 1.23
Mapping generator: ~11/9


Map: [&lt;1 1 0 -1 2 4|, &lt;0 2 8 13 5 -1|]
==== Vincenzo ====
EDOs: 17c, 24, 41c, 65c
Subgroup: 2.3.5.7.11.13
Badness: 0.0287


=Orphic=
Comma list: 45/44, 56/55, 65/64, 81/80
Commas: 81/80, 5898240/5764801


POTE generator: ~7/6 = 275.794
Mapping: {{mapping| 1 0 -4 -13 -6 10 | 0 1 4 10 6 -4 }}
Mapping generator: ~343/288


Map: [&lt;2 1 -4 4|, &lt;0 4 16 3|]
Optimal tunings:  
Wedgie: &lt;&lt;8 32 6 32 -13 -76||
* WE: ~2 = 1202.1684{{c}}, ~3/2 = 696.3160{{c}}
EDOs: 26, 74, 174bd, 248bd
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.2045{{c}}
Badness: 0.2588


==11-limit==
{{Optimal ET sequence|legend=0| 7d, 12, 19 }}
Commas: 81/80, 99/98, 73728/73205


POTE generator: ~7/6 = 275.762
Badness (Sintel): 1.02
Mapping generator: ~77/64


Map: [&lt;2 1 -4 4 8|, &lt;0 4 16 3 -2|]
===== 17-limit =====
EDOs: 26, 48c, 74, 248bd, 322bd
Subgroup: 2.3.5.7.11.13.17
Badness: 0.1015


==13-limit==
Comma list: 45/44, 52/51, 56/55, 65/64, 81/80
Commas: 81/80, 99/98, 144/143, 2200/2197


POTE generator: ~7/6 = 275.774
Mapping: {{mapping| 1 0 -4 -13 -6 10 12 | 0 1 4 10 6 -4 -5 }}
Mapping generator: ~63/52


Map: [&lt;2 1 -4 4 8 2|, &lt;0 4 16 3 -2 10|]
Optimal tunings:  
EDOs: 26, 48c, 74, 174bd, 248bd, 322bd
* WE: ~2 = 1200.5137{{c}}, ~3/2 = 696.1561{{c}}
Badness: 0.0535
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.8771{{c}}


=Mothra=
{{Optimal ET sequence|legend=0| 12, 19 }}
[[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]] 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]].
Badness (Sintel): 1.30
Note that mothra can also be called cynder in the 7-limit, which can be a little confusing sometimes.


[[7-limit|7]] and [[9-limit]] minimax 1/4 comma
===== 19-limit =====
[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |3 0 -1/12 0&gt;]
Subgroup: 2.3.5.7.11.13.17.19
[[Eigenmonzo]]s: 2, 5


[[POTE tuning|POTE generator]]: ~8/7 = 232.193
Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80
Mapping generator: ~8/7


Algebraic generator: Rabrindanath, largest real root of x^8-3x^2+1, or 232.0774 cents.
Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 | 0 1 4 10 6 -4 -5 -3 }}


Map: [&lt;1 1 0 3|, &lt;0 3 12 -1|]
Optimal tunings:  
[[Generator]]s: 2, 8/7
* WE: ~2 = 1199.8261{{c}}, ~3/2 = 696.0298{{c}}
[[Wedgie]]: &lt;&lt;3 12 -1 12 -10 -36||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1262{{c}}
EDOs: [[5edo|5]], [[26edo|26]], [[31edo|31]]
[[Badness]]: 0.0371


==11-limit==
{{Optimal ET sequence|legend=0| 12, 19 }}
[[Comma]]s: 81/80, 99/98, 385/384


POTE generator: ~8/7 = 232.031
Badness (Sintel): 1.36
Mapping generator: ~8/7


Map: [&lt;1 1 0 3 5|, &lt;0 3 12 -1 -8|]
=== Bimeantone ===
EDOs: [[5edo|5]], [[26edo|26]], [[31edo|31]], [[88edo|88]], [[150edo|150]], [[181edo|181]]
11/8 is mapped to half octave minus the [[128/125|meantone diesis]].  
[[Badness]]: 0.0256


==13-limit==
Subgroup: 2.3.5.7.11
Commas: 81/80, 99/98, 105/104, 144/143


POTE generator: ~8/7 = 231.811
Comma list: 81/80, 126/125, 245/242
Mapping generator: ~8/7


Map: [&lt;1 1 0 3 5 1|, &lt;0 3 12 -1 -8 14|]
Mapping: {{mapping| 2 0 -8 -26 -31 | 0 1 4 10 12 }}
EDOs: 5, 26, 31, 57, 88
Badness: 0.0240


==Cynder==
: mapping generators: ~63/44, ~3
Commas: 45/44, 81/80, 1029/1024


POTE generator: ~8/7 = 231.317
Optimal tunings:  
Mapping generator: ~8/7
* 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 1 0 3 0|, &lt;0 3 12 -1 18|]
{{Optimal ET sequence|legend=0| 12, 26de, 38d, 50 }}
EDOs: 26, 57e, 83bce
Badness: 0.0557


===13-limit===
Badness (Sintel): 1.26
Commas: 45/44, 78/77, 81/80, 640/637


POTE generator: ~8/7 = 231.293
==== 13-limit ====
Mapping generator: ~8/7
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 1 0 3 0 1|, &lt;0 3 12 -1 18 14|]
Comma list: 81/80, 105/104, 126/125, 245/242
EDOs: 26, 57e, 83bce
Badness: 0.0341


==Mosura==
Mapping: {{mapping| 2 0 -8 -26 -31 -40 | 0 1 4 10 12 15 }}
Commas: 81/80, 176/175, 1029/1024


POTE generator: ~8/7 = 232.419
Optimal tunings:  
Mapping generator: ~8/7
* 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 1 0 3 -1|, &lt;0 3 12 -1 23|]
{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}
EDOs: 31, 129, 136b, 148be, 160be, 191bce, 222bce, 253bce
Badness: 0.0313


===13-limit===
Badness (Sintel): 1.19
Commas: 81/80, 144/143, 176/175, 1029/1024


POTE generator: ~8/7 = 232.640
==== 17-limit ====
Mapping generator: ~8/7
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 1 0 3 -1 7|, &lt;0 3 12 -1 23 -17|]
Comma list: 81/80, 105/104, 126/125, 189/187, 221/220
EDOs: 31, 67, 98
Badness: 0.0369


=Squares=
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 | 0 1 4 10 12 15 1 }}
[[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|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.
Optimal tunings:
* WE: ~17/12 = 600.9234{{c}}, ~3/2 = 696.8536{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.9317{{c}}


7 and 9 limit minimax 1/4 comma
{{Optimal ET sequence|legend=0| 12f, 38df, 50 }}
[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |3/2 0 9/16 0&gt;]
[[Eigenmonzo]]s: 2, 5


[[POTE tuning|POTE generator]]: ~9/7 = 425.942
Badness (Sintel): 1.15
Mapping generator: ~9/7


Algebraic generator: Sceptre2, the positive root of 9x^2+x-16, or (sqrt(577)-1)/18, which is 425.9311 cents.
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;1 3 8 6|, &lt;0 -4 -16 -9|]
Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220
[[Generator]]s: 2, 9/7
EDOs: [[14edo|14]], [[31edo|31]], [[262edo|262]], [[293edo|293]]
[[Badness]]: 0.0460


Music:
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 -1 | 0 1 4 10 12 15 1 3 }}
By [[Chris Vaisvil]]
[[http://clones.soonlabel.com/public/micro/tuning-survey/daily20100603-squares8piano.mp3|Square 8]]


==11-limit==
Optimal tunings:
Commas: 81/80, 99/98, 121/120
* WE: ~17/12 = 600.9845{{c}}, ~3/2 = 696.8939{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.8947{{c}}


POTE generator: ~9/7 = 425.957
{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}
Mapping generator: ~9/7


Map: [&lt;1 3 8 6 7|, &lt;0 -4 -16 -9 -10|]
Badness (Sintel): 1.08
EDOs: [[5edo|5]], [[8edo|8]], [[11edo|11]], [[14edo|14]], [[17edo|17]], [[31edo|31]]
[[Badness]]: 0.0216


==13-limit==  
=== Trimean ===
Commas: 81/80, 99/98, 121/120, 66/65
{{See also| No-sevens subgroup temperaments #Superpine }}


POTE generator: ~9/7 = 425.550
Subgroup: 2.3.5.7.11
Mapping generator: ~9/7


Map: [&lt;1 3 8 6 7 3|, &lt;0 -4 -16 -9 -10 2|]
Comma list: 81/80, 126/125, 1344/1331
EDOs: 17c, 31, 79cf, 110cef, 141cef
[[Badness]]: 0.0255


==Agora==
Mapping: {{mapping| 1 2 4 7 5 | 0 -3 -12 -30 -11 }}
Commas: 81/80, 99/98, 105/104, 121/120


POTE generator: ~9/7 = 426.276
: mapping generators: ~2, ~11/10
Mapping generator: ~9/7


Map: [&lt;1 3 8 6 7 14|, &lt;0 -4 -16 -9 -10 -29|]
Optimal tunings:  
EDOs: 31, 45ef, 76e
* WE: ~2 = 1200.7155{{c}}, ~11/10 = 167.9055{{c}}
Badness: 0.0245
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7749{{c}}


=Cuboctahedra=
{{Optimal ET sequence|legend=0| 7d, 36d, 43, 50, 93 }}
==11-limit==
[[Comma]]s: 81/80, 385/384, 1375/1372


[[POTE tuning|POTE generator]]: ~9/7 = 425.993
Badness (Sintel): 1.68
Mapping generator: ~9/7


Map: [&lt;1 3 8 6 -4|, &lt;0 -4 -16 -9 21|]
==== 13-limit ====
EDOs: [[14edo|14]], [[31edo|31]], [[45edo|45]], [[200edo|200]]
Subgroup: 2.3.5.7.11.13
[[Badness]]: 0.0568


=Liese=
Comma list: 81/80, 126/125, 144/143, 364/363
[[Comma]]s: 81/80, 686/675


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]] makes for a good liese tuning, though [[19edo]] can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55.
Mapping: {{mapping| 1 2 4 7 5 3 | 0 -3 -12 -30 -11 5 }}


7 and 9 limit minimax 1/4 comma
Optimal tunings:
[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |2/3 0 11/12 0&gt;]
* WE: ~2 = 1200.6104{{c}}, ~11/10 = 167.8749{{c}}
[[Eigenmonzo]]s: 2, 5
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7728{{c}}


[[POTE tuning|POTE generator]]: ~10/7 = 632.406
{{Optimal ET sequence|legend=0| 7d, 43, 50, 93 }}
Mapping generator: ~10/7


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.
Badness (Sintel): 1.46


Map: [&lt;1 0 -4 -3|, &lt;0 3 12 11|]
==== 17-limit ====
[[Generator]]s: 2, 10/7
Subgroup: 2.3.5.7.11.13.17
EDOs: [[17edo|17]], [[19edo|19]], [[55edo|55]], [[74edo|74]]
[[Badness]]: 0.0467


==Liesel==
Comma list: 81/80, 126/125, 144/143, 189/187, 221/220
Commas: 56/55, 81/80, 540/539


POTE generator: ~10/7 = 633.073
Mapping: {{mapping| 1 2 4 7 5 3 8 | 0 -3 -12 -30 -11 5 -28 }}
Mapping generator: ~10/7


Map: [&lt;1 0 -4 -3 4|, &lt;0 3 12 11 -1|]
Optimal tunings:  
EDOs: 17c, 19, 36, 91ce
* WE: ~2 = 1200.6144{{c}}, ~11/10 = 167.8716{{c}}
Badness: 0.0407
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7682{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 7dg, 43, 50, 93 }}
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): 1.28


POTE generator: ~10/7 = ~13/9 = 633.042
=== Migration ===
Mapping generator: ~10/7
See [[Rastmic clan #Migration|Rastmic clan]].


Map: [&lt;1 0 -4 -3 4 0|, &lt;0 3 12 11 -1 7|]
== Flattone ==
EDOs: 17c, 19, 36, 91cef
{{Main| Flattone }}
Badness: 0.0273


==Elisa==
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]].
Commas: 77/75, 81/80, 99/98


POTE generator: ~10/7 = 633.061
[[Subgroup]]: 2.3.5.7
Mapping generator: ~10/7


Map: [&lt;1 0 -4 -3 -5|, &lt;0 3 12 11 16|]
[[Comma list]]: 81/80, 525/512
EDOs: 19e, 36e
Badness: 0.0416


==Lisa==
{{Mapping|legend=1| 1 0 -4 17 | 0 1 4 -9 }}
Commas: 45/44, 81/80, 343/330


POTE generator: ~10/7 = 631.370
[[Optimal tuning]]s:  
Mapping generator: ~10/7
* [[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 -3 -6|, &lt;0 3 12 11 18|]
[[Minimax tuning]]:
EDOs: 19
* [[7-odd-limit]]: ~3/2 = {{monzo| 8/13 0 1/13 -1/13 }}
Badness: 0.0548
: [[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


==13-limit==
[[Tuning ranges]]:
Commas: 45/44, 81/80, 91/88, 147/143
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [692.353, 701.955]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]


POTE generator: ~10/7 = 631.221
[[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.
Mapping generator: ~10/7


Map: [&lt;1 0 -4 -3 -6 0|, &lt;0 3 12 11 18 7|]
{{Optimal ET sequence|legend=1| 7, 19, 26, 45 }}
EDOs: 19
Badness: 0.0361


=Jerome=
[[Badness]] (Sintel): 0.976
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.


Commas: 81/80, 17280/16807
=== 11-limit ===
This can also be considered a no-sevens temperament: [[#Hypnotone|hypnotone]].


POTE generator: ~54/49 = 139.343
Subgroup: 2.3.5.7.11
Mapping generator: ~54/49


Map: [&lt;1 1 0 2|, &lt;0 5 20 7|]
Comma list: 45/44, 81/80, 385/384
Wedgie: &lt;&lt;5 30 7 20 -3 -40||
EDOs: 8, 9, 17, 26, 43, 112
Badness: 0.1087


==11-limit==
Mapping: {{mapping| 1 0 -4 17 -6 | 0 1 4 -9 6 }}
Commas: 81/80, 99/98, 864/847


POTE generator: ~12/11 = 139.428
Optimal tuning:  
Mapping generator: ~12/11
* WE: ~2 = 1202.3247{{c}}, ~3/2 = 694.4688{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.1467{{c}}


Map: [&lt;1 1 0 2 3|, &lt;0 5 20 7 4|]
Tuning ranges:  
EDOs: 8, 9, 17, 26, 43, 241
* 11-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
Badness: 0.0479
* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]


==13-limit==
{{Optimal ET sequence|legend=0| 7, 19, 26, 45, 71bc, 116bcde }}
Commas: 77/78, 81/80, 99/98, 144/143


POTE generator: ~13/12 = 139.387
Badness (Sintel): 1.12
Mapping generator: ~12/11


Map: [&lt;1 1 0 2 3 3|, &lt;0 5 20 7 4 6|]
==== 13-limit ====
EDOs: 8, 9, 17, 26, 43, 155, 198
Subgroup: 2.3.5.7.11.13
Badness: 0.0293


==17-limit==
Comma list: 45/44, 65/64, 78/77, 81/80
Commas: 78/77, 81/80, 99/98, 144/143, 189/187


POTE generator: ~13/12 = 139.362
Mapping: {{mapping| 1 0 -4 17 -6 10 | 0 1 4 -9 6 -4 }}
Mapping generator: ~12/11


Map: [&lt;1 1 0 2 3 3 2|, &lt;0 5 20 7 4 6 18|]
Optimal tunings:  
EDOs: 8, 9, 17, 26, 43, 155
* WE: ~2 = 1202.5156{{c}}, ~3/2 = 694.5107{{c}}
Badness: 0.0209
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0538{{c}}


=Meanmag=
Tuning ranges:
Commas: 81/80, 3125/3072
* 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]


POTE generator: ~8/7 = 238.396
{{Optimal ET sequence|legend=0| 7, 19, 26, 45f, 71bcf, 116bcdef }}
Mapping generator: ~7


Map: [&lt;19 30 44 0|, &lt;0 0 0 1|]
Badness (Sintel): 0.920
Wedgie: &lt;&lt;0 0 19 0 30 44||
EDOs: 19, 57, 76, 171bcd
Badness: 0.0770


=Undevigintone=  
=== Ptolemy ===
Commas: 49/48, 81/80, 126/125
See [[Rastmic clan #Ptolemy|Rastmic clan]].


POTE generator: ~11/8 = 538.047
== Dominant ==
Mapping generator: ~11
{{Main| Dominant (temperament) }}
{{See also| Archytas clan }}


Map: [&lt;19 30 44 53 0|, &lt;0 0 0 0 1|]
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]].
EDOs: 19, 38d
Badness: 0.0364


==13-limit==
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.
Commas: 49/48, 65/64, 81/80, 126/125


POTE generator: ~11/8 = 537.061
[[Subgroup]]: 2.3.5.7


Map: [&lt;19 30 44 53 0 70|, &lt;0 0 0 0 1 0|]
[[Comma list]]: 36/35, 64/63
EDOs: 19, 38d
 
Badness: 0.0229</pre></div>
{{Mapping|legend=1| 1 0 -4 6 | 0 1 4 -2 }}
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Meantone family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="display: block; text-align: right;"&gt;&lt;!-- ws:start:WikiTextTocRule:184:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:184 --&gt;&lt;!-- ws:start:WikiTextTocRule:185: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#x-Seven limit children"&gt;Seven limit children&lt;/a&gt;&lt;/div&gt;
[[Optimal tuning]]s:
&lt;!-- ws:end:WikiTextTocRule:185 --&gt;&lt;!-- ws:start:WikiTextTocRule:186: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Septimal meantone"&gt;Septimal meantone&lt;/a&gt;&lt;/div&gt;
* [[WE]]: ~2 = 1195.3384{{c}}, ~3/2 = 698.8478{{c}}
&lt;!-- ws:end:WikiTextTocRule:186 --&gt;&lt;!-- ws:start:WikiTextTocRule:187: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Septimal meantone-Bimeantone"&gt;Bimeantone&lt;/a&gt;&lt;/div&gt;
: [[error map]]: {{val| -4.662 -7.769 +9.077 +14.832 }}
&lt;!-- ws:end:WikiTextTocRule:187 --&gt;&lt;!-- ws:start:WikiTextTocRule:188: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Septimal meantone-Bimeantone-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.1125{{c}}
&lt;!-- ws:end:WikiTextTocRule:188 --&gt;&lt;!-- ws:start:WikiTextTocRule:189: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Septimal meantone-Unidecimal meantone aka Huygens"&gt;Unidecimal meantone aka Huygens&lt;/a&gt;&lt;/div&gt;
: error map: {{val| 0.000 -0.842 +18.136 +28.949 }}
&lt;!-- ws:end:WikiTextTocRule:189 --&gt;&lt;!-- ws:start:WikiTextTocRule:190: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Septimal meantone-Unidecimal meantone aka Huygens-Tridecimal meantone"&gt;Tridecimal meantone&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:190 --&gt;&lt;!-- ws:start:WikiTextTocRule:191: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Septimal meantone-Unidecimal meantone aka Huygens-Grosstone"&gt;Grosstone&lt;/a&gt;&lt;/div&gt;
[[Tuning ranges]]:
&lt;!-- ws:end:WikiTextTocRule:191 --&gt;&lt;!-- ws:start:WikiTextTocRule:192: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Septimal meantone-Unidecimal meantone aka Huygens-Meridetone"&gt;Meridetone&lt;/a&gt;&lt;/div&gt;
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[diamond monotone]]: ~3/2 = [700.000, 720.000] (7\12 to 3\5)
&lt;!-- ws:end:WikiTextTocRule:192 --&gt;&lt;!-- ws:start:WikiTextTocRule:193: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Septimal meantone-Unidecimal meantone aka Huygens-Hemimeantone"&gt;Hemimeantone&lt;/a&gt;&lt;/div&gt;
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 715.587]
&lt;!-- ws:end:WikiTextTocRule:193 --&gt;&lt
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
 
{{Optimal ET sequence|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}
 
[[Badness]] (Sintel): 0.524
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 36/35, 56/55, 64/63
 
Mapping: {{mapping| 1 0 -4 6 13 | 0 1 4 -2 -6 }}
 
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]
 
Optimal tunings:
* WE: ~2 = 1194.0169{{c}}, ~3/2 = 699.7473{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.2672{{c}}
 
{{Optimal ET sequence|legend=0| 5, 12, 17c, 29cde }}
 
Badness (Sintel): 0.799
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 36/35, 56/55, 64/63, 66/65
 
Mapping: {{mapping| 1 0 -4 6 13 18 | 0 1 4 -2 -6 -9 }}
 
Optimal tunings:
* WE: ~2 = 1193.8055{{c}}, ~3/2 = 700.0042{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.8254{{c}}
 
Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~3/2 = 705.882 (10\17)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
 
{{Optimal ET sequence|legend=0| 12f, 17c, 29cdef }}
 
Badness (Sintel): 0.996
 
==== Dominion ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 26/25, 36/35, 56/55, 64/63
 
Mapping: {{mapping| 1 0 -4 6 13 -9 | 0 1 4 -2 -6 8 }}
 
Optimal tunings:
* WE: ~2 = 1195.0293{{c}}, ~3/2 = 701.9847{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7698{{c}}
 
{{Optimal ET sequence|legend=0| 5, 12, 17c }}
 
Badness (Sintel): 1.13
 
=== Domination ===
Subgroup: 2.3.5.7.11
 
Comma list: 36/35, 64/63, 77/75
 
Mapping: {{mapping| 1 0 -4 6 -14 | 0 1 4 -2 11 }}
 
Optimal tunings:
* WE: ~2 = 1194.8645{{c}}, ~3/2 = 701.9872{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5945{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}
 
Badness (Sintel): 1.21
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 26/25, 36/35, 64/63, 66/65
 
Mapping: {{mapping| 1 0 -4 6 -14 -9 | 0 1 4 -2 11 8 }}
 
Optimal tunings:
* WE: ~2 = 1195.1324{{c}}, ~3/2 = 702.6343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.0791{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}
 
Badness (Sintel): 1.13
 
=== Domineering ===
Subgroup: 2.3.5.7.11
 
Comma list: 36/35, 45/44, 64/63
 
Mapping: {{mapping| 1 0 -4 6 -6 | 0 1 4 -2 6 }}
 
Optimal tunings:
* WE: ~2 = 1194.7102{{c}}, ~3/2 = 695.6962{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1765{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 7, 12 }}
 
Badness (Sintel): 0.727
 
=== Arnold ===
Subgroup: 2.3.5.7.11
 
Comma list: 22/21, 33/32, 36/35
 
Mapping: {{mapping| 1 0 -4 6 5 | 0 1 4 -2 -1 }}
 
Optimal tunings
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