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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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{{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">[[toc|flat]]
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]] 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 }}


[[POTE tuning|POTE generator]]: 696.239
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]].


Map: [&lt;1 0 -4|, &lt;0 1 4|]
[[Subgroup]]: 2.3.5
EDOs: [[5edo|5]], [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]]
[[Badness]]: 0.00736


==Seven limit children==
[[Comma list]]: 81/80
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;].


=Septimal meantone=
{{Mapping|legend=1| 1 0 -4 | 0 1 4 }}
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.


[[Comma]]s: 81/80, 126/125
: mapping generators: ~2, ~3


7 and [[9-limit]] minimax
[[Optimal tuning]]s:
[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |-3 0 5/2 0&gt;]
* [[WE]]: ~2 = 1201.3906{{c}}, ~3/2 = 697.0455{{c}}
[[Eigenmonzo]]s: 2, 5
: [[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 }}


[[POTE tuning|POTE generator]]: 696.495
[[Minimax tuning]]:
* [[5-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma)
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


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


Map: [&lt;1 0 -4 -13|, &lt;0 1 4 10|]
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b }}
[[Generator]]s: 2, 3
Wedgie: &lt;&lt;1 4 10 4 13 12||
EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[81edo|81]]
[[Badness]]: 0.0137


==Unidecimal meantone aka Huygens==
[[Badness]] (Sintel): 0.173
[[Comma]]s: 81/80, 126/125, 99/98


[[11-limit]] minimax
=== Overview to extensions ===
[|1 0 0 0 0&gt;, |25/16 -1/8 0 0 1/16&gt;, |9/4 -1/2 0 0 1/4&gt;,  
The second comma of the normal comma list defines which [[7-limit]] family member we are looking at.
|21/8 -5/4 0 0 5/8&gt;, |25/8 -9/4 0 0 9/8&gt;]
* Flattertone adds {{monzo| -24 17 0 -1 }}, finding the [[~]][[7/4]] at the double-augmented sixth, for a tuning between 33edo and 26edo.
[[Eigenmonzo]]s: 2, 11/9
* 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.


[[POTE tuning|POTE generator]]: 696.967
==== 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.  


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


Map: [&lt;1 0 -4 -13 -25|, &lt;0 1 4 10 18|]
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.  
[[Generator]]s: 2, 3
EDOs: [[7edo|7]], [[12edo|12]], [[31edo|31]], [[105edo|105]], [[198edo|198]]
[[Badness]]: 0.0170


===Tridecimal meantone===
==== Splitting the meantone fifth into two (243/242) ====
[[Comma]]s: 66/65, 81/80, 99/98, 105/104
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.


POTE generator: ~3/2 = 696.642
==== 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.


Map: Map: [&lt;1 0 -4 -13 -25 -20|, &lt;0 1 4 10 18 15|]
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]]).
EDOs: [[12edo|12]], [[19edo|19], [[31edo|31]], [[267edo|267]], [[298edo|298]]
[[Badness]]: 0.0180


==Meanpop==
==== 31edo as splitting the fifth into two, three and nine ====
[[Comma]]s: 81/80, 126/125, 385/384
[[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).


11-limit minimax 1/4 comma
Temperaments discussed elsewhere include
[|1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;,
* ''[[Plutus]]'' (+15/14) → [[Very low accuracy temperaments #Plutus|Very low accuracy temperaments]]
|-3 0 5/2 0 0&gt;, |11 0 -13/4 0 0&gt;]
* [[Godzilla]] (+49/48) → [[Semaphoresmic clan #Godzilla|Semaphoresmic clan]]
[[Eigenmonzo]]s: 2, 5
* [[Mothra]] (+1029/1024) → [[Gamelismic clan #Mothra|Gamelismic clan]]
* ''[[Mohaha]]'' (+121/120) → [[Rastmic clan #Mohaha|Rastmic clan]]


[[POTE tuning|POTE generator]]: 696.434
The rest are considered below.


Algebraic generator: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
== Septimal meantone ==
<div style="float:right">[[:de:septimal-mitteltönig|Deutsch]]</div>
{{Main| Meantone #Septimal meantone}}
{{Wikipedia| Septimal meantone temperament }}


Map: [&lt;1 0 -4 -13 24|, &lt;0 1 4 10 -13|]
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.  
[[Generator]]s: 2, 3
EDOs: 12, 19, 31, 81, [[112edo|112]]
[[Badness]]: 0.0215


===13-limit Meanpop===
[[Subgroup]]: 2.3.5.7
[[Comma]]s: 81/80, 105/104, 144/143, 196/195


POTE generator: ~3/2 = 696.211
[[Comma list]]: 81/80, 126/125


Map: [&lt;1 0 -4 -13 24|, &lt;0 1 4 10 -13|]
{{Mapping|legend=1| 1 0 -4 -13 | 0 1 4 10 }}
EDOS: 7, 12, 19, 31, 50, 81, [[131edo|131]]
[[Badness]]: 0.0209


==Meanenneadecal==
[[Optimal tuning]]s:
[[Comma]]s: 45/44, 56/55, 81/80
* [[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 }}


[[POTE tuning|POTE generator]]: ~3/2 = 696.250
[[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


Map: [&lt;1 0 -4 -13 -6|, &lt;0 1 4 10 6|]
[[Tuning ranges]]:  
EDOs: 7, 12, 19, 31, 50, 81
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
[[Badness]]: 0.0214
* 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.)


===13-limit===
[[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: 45/44, 56/55, 78/77, 81/80


[[POTE tuning|POTE generator]]: ~3/2 = 696.146
{{Optimal ET sequence|legend=1| 12, 19, 31, 81, 112b, 143b }}


Map: [&lt;1 0 -4 -13 -6 -20|, &lt;0 1 4 10 6 15|]
[[Badness]] (Sintel): 0.347
EDOs: 7, 12, 19, 31, 50, [[131edo|131]], [[181edo|181]]
[[Badness]]: 0.0212


=Flattone=
=== Undecimal meantone (huygens) ===
[[Comma]]s: 81/80, 525/512
{{Redirect|Huygens|the Dutch mathematician, physicist and astronomer|Wikipedia: Christiaan Huygens}}
{{See also| Huygens vs meanpop }}


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 minor seventh interval. Other intervals are 7/6, a diminished minor third, and 7/5, a doubly diminshed fifth. Good tunings for flattone are [[26edo]], [[45edo]] and [[64edo]].
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.


[[7-limit]] minimax
Subgroup: 2.3.5.7.11
[|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
Comma list: 81/80, 99/98, 126/125
[|1 0 0 0&gt;, |17/11 2/11 0 -1/11&gt;,  
|24/11 8/11 0 -4/11&gt;, |34/11 -18/11 0 9/11&gt;]
[[Eigenmonzo]]s: 2, 9/7


[[POTE tuning|POTE generator]]: 693.779
Mapping: {{mapping| 1 0 -4 -13 -25 | 0 1 4 10 18 }}


Algebraic generator: Squarto, the positive root of 8x^2-4x-9, at 506.3239 cents, equal to (1+sqrt(19))/4.
Optimal tunings:  
* 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 17|, &lt;0 1 4 -9|]
Minimax tuning:  
[[Wedgie]]: &lt;&lt;1 4 -9 4 -17 -32||
* 11-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}
[[Generator]]s: 2, 3
: 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 }}]
EDOs: 7, 19, [[45edo|45]], [[64edo|64]]
: unchanged-interval (eigenmonzo) basis: 2.11/9
[[Badness]]: 0.0386


=Dominant=
Tuning ranges:
[[Comma]]s: 36/35, 64/63
* 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.)


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]].
Algebraic generator: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.


[[POTE tuning|POTE generator]]: 701.573
{{Optimal ET sequence|legend=0| 12, 19e, 31, 105, 136b }}


Map: [&lt;1 0 -4 6|, &lt;0 1 4 -2|]
Badness (Sintel): 0.563
[[Wedgie]]: &lt;&lt;1 4 -2 4 -6 -16||
EDOs: 5, 7, 12, [[53edo|53]], [[65edo|65]]
[[Badness]]: 0.0207


=Sharptone=
; Music
[[Comma]]s: 21/20, 28/27
* [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]


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.
==== 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]].  


[[POTE tuning|POTE generator]]: 700.140
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 -4 -2|, &lt;0 1 4 3|]
Comma list: 81/80, 99/98, 126/125, 144/143
[[Wedgie]]: &lt;&lt;1 4 3 4 2 -4||
EDOs: 5, 12
[[Badness]]: 0.0248


=Injera=
Mapping: {{mapping| 1 0 -4 -13 -25 29 | 0 1 4 10 18 -16 }}
[[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 19edos, is an excellent tuning for injera.
Optimal tunings:
* WE: ~2 = 1199.9389{{c}}, ~3/2 = 697.2282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.2627{{c}}


[[POTE tuning|POTE generator]]: 694.375
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


Map: [&lt;2 0 -8 -7|, &lt;0 1 4 4|]
Tuning ranges:  
[[Wedgie]]: &lt;&lt;2 8 8 8 7 -4||
* 13- and 15-odd-limit diamond monotone: ~3/2 = [696.774, 697.674] (18\31 to 25\43)
EDOs: [[12edo|12]], [[26edo|26]], [[38edo|38]], [[140edo|140]], [[178edo|178]]
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
[[Badness]]: 0.0311


=Godzilla=
{{Optimal ET sequence|legend=0| 12, 31, 43, 74 }}
[[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-half intervals these represent give a fourth, and so step-and-a-half 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.
Badness (Sintel): 1.07


[[POTE tuning|POTE generator]]: 252.635
===== 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.  


Map: [&lt;1 0 -4 2|, &lt;0 2 8 1|]
Subgroup: 2.3.5.7.11.13.17
[[Wedgie]]: &lt;&lt;2 8 1 8 -4 -20||
EDOs: [[5edo|5]], [[9edo|9]], [[14edo|14]], 19
[[Badness]]: 0.0267


==Music==
Comma list: 81/80, 99/98, 120/119, 126/125, 144/143
[[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=
Mapping: {{mapping| 1 0 -4 -13 -25 29 12 | 0 1 4 10 18 -16 -5 }}
[[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.
Optimal tunings:
* WE: ~2 = 1199.5811{{c}}, ~3/2 = 697.0918{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3303{{c}}


7 and 9-limit minimax 1/4 comma
{{Optimal ET sequence|legend=0| 12, 31, 43, 74g }}
[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |6 0 -11/8 0&gt;]
[[Eigenmonzo]]s: 2, 5


[[POTE tuning|POTE generator]]: 348.415
Badness (Sintel): 1.06


Algebraic generator: Mohabis, real root of 3x^3-3x^2-1, 348.6067 cents. Corresponding recurrence converges quickly.
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;1 1 0 6|, &lt;0 2 8 -11|]
Comma list: 81/80, 96/95, 99/98, 120/119, 126/125, 144/143
[[Generator]]s: 2, 128/105
[[Wedgie]]: &lt;&lt;2 8 -11 8 -23 -48||
EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]]
[[Badness]]: 0.0557


==11-limit==
Mapping: {{mapping| 1 0 -4 -13 -25 29 12 9 | 0 1 4 10 18 -16 -5 -3 }}
[[Comma]]s: 81/80, 121/120, 176/175


[[11-limit]] minimax 1/4 comma
Optimal tunings:
[|1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;,  
* WE: ~2 = 1199.2931{{c}}, ~3/2 = 696.9690{{c}}
|6 0 -11/8 0 0&gt;, |2 0 5/8 0 0&gt;]
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3736{{c}}
[[Eigenmonzo]]s: 2, 5


[[POTE tuning|POTE generator]]: 348.477
{{Optimal ET sequence|legend=0| 12, 31, 43, 74gh }}


Map: [&lt;1 1 0 6 2|, &lt;0 2 8 -11 5|]
Badness (Sintel): 1.07
[[Generator]]s: 2, 11/9
EDOs: 7, 24, 31
[[Badness]]: 0.0261


=Mothra=
==== Fokkertone ====
Commas: 81/80, 1029/1024
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.


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.
This extension used to be known as ''tridecimal meantone'', but was decanonicalized in 2025.  


7 and 9 limit minimax 1/4 comma
Subgroup: 2.3.5.7.11.13
[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |3 0 -1/12 0&gt;]
Eigenmonzos: 2, 5


[[POTE tuning|POTE generator]]: 232.193
Comma list: 66/65, 81/80, 99/98, 105/104


Algebraic generator: Rabrindanath, largest real root of x^8-3x^2+1, or 232.0774 cents.
Mapping: {{mapping| 1 0 -4 -13 -25 -20 | 0 1 4 10 18 15 }}


Map: [&lt;1 1 0 3|, &lt;0 3 12 -1|]
Optimal tunings:  
[[Generator]]s: 2, 8/7
* WE: ~2 = 1200.8149{{c}}, ~3/2 = 697.1155{{c}}
[[Wedgie]]: &lt;&lt;3 12 -1 12 -10 -36||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.7085{{c}}
EDOs: 5, [[26edo|26]], 31
[[Badness]]: 0.0371


==11-limit==
Minimax tuning:
[[Comma]]s: 81/80, 99/98, 385/384
* 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: ~63/55 = 232.031
{{Optimal ET sequence|legend=0| 12f, 19e, 31 }}


Map: [&lt;1 1 0 3 5|, &lt;0 3 12 -1 -8|]
Badness (Sintel): 0.746
EDOs: 5, [[26edo|26]], 31, [[88edo|88]], [[150edo|150]], [[181edo|181]]
[[Badness]]: 0.0256


=Squares=
===== 17-limit =====
[[Comma]]s: 81/80, 2401/2400
Subgroup: 2.3.5.7.11.13.17


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.
Comma list: 66/65, 81/80, 99/98, 105/104, 120/119


7 and 9 limit minimax 1/4 comma
Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 | 0 1 4 10 18 15 -5 }}
[|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]]: 425.942
Optimal tunings:  
* WE: ~2 = 1199.5548{{c}}, ~3/2 = 696.7449{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.9823{{c}}


Algebraic generator: Sceptre2, the positive root of 9x^2+x-16, or (sqrt(577)-1)/18, which is 425.9311 cents.
{{Optimal ET sequence|legend=0| 12f, 31 }}


Map: [&lt;1 3 8 6|, &lt;0 -4 -16 -9|]
Badness (Sintel): 1.02
[[Generator]]s: 2, 9/7
EDOs: [[14edo|14]], 31, [[262edo|262]], [[293edo|293]]
[[Badness]]: 0.0460


Music:
===== 19-limit =====
By [[Chris Vaisvil]]
Subgroup: 2.3.5.7.11.13.17.19
http://tinyurl.com/25kv7cq
http://tinyurl.com/24cbxse


==11-limit==
Comma list: 66/65, 81/80, 96/95, 99/98, 105/104, 120/119
[[Comma]]s: 81/80, 385/384, 1375/1372


[[POTE tuning|POTE generator]]: 425.993
Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 9 | 0 1 4 10 18 15 -5 -3 }}


Map: [&lt;1 3 8 6 -4|, &lt;0 -4 -16 -9 21|]
Optimal tunings:  
EDOs: [[14edo|14]], 31, [[200edo|200]]
* WE: ~2 = 1199.0408{{c}}, ~3/2 = 696.5824{{c}}
[[Badness]]: 0.0568
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.1061{{c}}


=Liese=
{{Optimal ET sequence|legend=0| 12f, 31 }}
[[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.
Badness (Sintel): 1.10


7 and 9 limit minimax 1/4 comma
==== Meridetone ====
[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |2/3 0 11/12 0&gt;]
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.
[[Eigenmonzo]]s: 2, 5


[[POTE tuning|POTE generator]]: 632.406
Subgroup: 2.3.5.7.11.13


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: 78/77, 81/80, 99/98, 126/125


Map: [&lt;1 0 -4 -3|, &lt;0 3 12 11|]
Mapping: {{mapping| 1 0 -4 -13 -25 -39 | 0 1 4 10 18 27 }}
[[Generator]]s: 2, 10/7
EDOs: [[17edo|17]], [[19edo|19]], [[55edo|55]], [[74edo|74]]
[[Badness]]: 0.0467


=Squares=
Optimal tunings:
Commas: 81/80, 2401/2400
* WE: ~2 = 1199.9122{{c}}, ~3/2 = 697.4779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.5241{{c}}


POTE generator: ~9/7 = 425.942
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


Map: [&lt;1 3 8 6|, &lt;0 -4 -16 -9|]
{{Optimal ET sequence|legend=0| 12f, 31f, 43 }}
Wedgie: &lt;&lt;4 16 9 16 3 -24||
EDOs: 5, 8, 11, 14, 17, 31
Badness: 0.0460


==11-limit==
Badness (Sintel): 1.09
Commas: 81/80, 99/98, 121/120


POTE generator: ~9/7 = 425.957
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 3 8 6 7|, &lt;0 -4 -16 -9 -10|]
Comma list: 78/77, 81/80, 99/98, 120/119, 126/125
EDOs: 5, 8, 11, 14, 17, 31
Badness: 0.0216


==13-limit==
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 | 0 1 4 10 18 27 -5 }}
Commas: 81/80, 99/98, 121/120, 66/65


POTE generator: ~9/7 = 425.550
Optimal tunings:  
* WE: ~2 = 1199.3793{{c}}, ~3/2 = 697.2833{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6222{{c}}


Map: [&lt;1 3 8 6 7 3|, &lt;0 -4 -16 -9 -10 2|]
{{Optimal ET sequence|legend=0| 12f, 43 }}
EDOs: 5, 8, 11, 14, 17, 31
 
Badness: 0.0255</pre></div>
Badness (Sintel): 1.22
<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;!-- ws:start:WikiTextTocRule:48:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!--
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125
 
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 9 | 0 1 4 10 18 27 -5 -3 }}
 
Optimal tunings:
* WE: ~2 = 1199.0260{{c}}, ~3/2 = 697.1486{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6887{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 43 }}
 
Badness (Sintel): 1.25
 
==== Hemimeantone ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 81/80, 99/98, 126/125, 169/168
 
Mapping: {{mapping| 1 0 -4 -13 -25 -5 | 0 2 8 20 36 11 }}
 
: mapping generators: ~2, ~26/15
 
Optimal tunings:
* WE: ~2 = 1201.0387{{c}}, ~26/15 = 949.2863{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5065{{c}}
 
{{Optimal ET sequence|legend=0| 19e, 43, 62 }}
 
Badness (Sintel): 1.30
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 81/80, 99/98, 126/125, 169/168, 221/220
 
Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 | 0 2 8 20 36 11 33 }}
 
Optimal tunings:
* WE: ~2 = 1201.0270{{c}}, ~26/15 = 949.2892{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5169{{c}}
 
{{Optimal ET sequence|legend=0| 19eg, 43, 62 }}
 
Badness (Sintel): 1.19
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 81/80, 99/98, 126/125, 153/152, 169/168, 221/220
 
Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 -25 | 0 2 8 20 36 11 33 37 }}
 
Optimal tunings:
* WE: ~2 = 1201.0339{{c}}, ~19/11 = 949.2902{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 948.5111{{c}}
 
{{Optimal ET sequence|legend=0| 19egh, 43, 62 }}
 
Badness (Sintel): 1.15
 
==== Semimeantone ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 81/80, 99/98, 126/125, 847/845
 
Mapping: {{mapping| 2 0 -8 -26 -50 -59 | 0 1 4 10 18 21 }}
 
: mapping generators: ~55/39, ~3
 
Optimal tunings:
* WE: ~55/39 = 600.3606{{c}}, ~3/2 = 697.4241{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 697.0545{{c}}
 
{{Optimal ET sequence|legend=0| 12f, …, 50eff, 62, 136b }}
 
Badness (Sintel): 1.68
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 81/80, 99/98, 126/125, 221/220, 289/288
 
Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 | 0 1 4 10 18 21 1 }}
 
Optimal tunings:
* WE: ~17/12 = 600.5426{{c}}, ~3/2 = 697.5571{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9858{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 50eff, 62, 136bg }}
 
Badness (Sintel): 1.60
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 81/80, 99/98, 126/125, 153/152, 209/208, 221/220
 
Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 -1 | 0 1 4 10 18 21 1 3 }}
 
Optimal tunings:
* WE: ~17/12 = 600.5959{{c}}, ~3/2 = 697.5985{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9638{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 50eff, 62 }}
 
Badness (Sintel): 1.47
 
=== Meanpop ===
{{See also| Huygens vs meanpop }}
 
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.
 
Subgroup: 2.3.5.7.11
 
Comma list: 81/80, 126/125, 385/384
 
Mapping: {{mapping| 1 0 -4 -13 24 | 0 1 4 10 -13 }}
 
: mapping generator: ~2, ~3
 
Optimal tunings:
* WE: ~2 = 1201.3464{{c}}, ~3/2 = 697.2159{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4509{{c}}
 
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
 
Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
 
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.
 
{{Optimal ET sequence|legend=0| 12e, 19, 31, 81, 112b }}
 
Badness (Sintel): 0.712
 
; 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]
 
==== Tridecimal meanpop ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 81/80, 105/104, 126/125, 144/143
 
Mapping: {{mapping| 1 0 -4 -13 24 -20 | 0 1 4 10 -13 15 }}
 
Optimal tunings:
* WE: ~2 = 1201.0765{{c}}, ~3/2 = 696.8361{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2347{{c}}
 
Minimax tuning
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