Xenharmonic Wiki

Meantone family: Difference between revisions

← Older edit
VisualWikitext
Wikispaces>hstraub
**Imported revision 448780312 - Original comment: **
m Text replacement - "octave" to "octave"
 
(373 intermediate revisions by 39 users not shown)
Line 1: Line 1:
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| de = Mitteltönig
: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2013-09-05 10:33:27 UTC</tt>.<br>
| en = Meantone family
: The original revision id was <tt>448780312</tt>.<br>
| es =
: The revision comment was: <tt></tt><br>
| ja =  
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
}}
<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]].  
Other languages: [[xenharmonie/mitteltönig|Deutsch]]
&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
== Meantone ==
Mapping generator: ~3
{{Main| Meantone }}


[[Map]]: [&lt;1 0 -4|, &lt;0 1 4|]
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]].
EDOs: [[5edo|5]], [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]], [[212edo|212b]]
[[Badness]]: 0.00736


==Seven limit children==
[[Subgroup]]: 2.3.5
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=
[[Comma list]]: 81/80
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
{{Mapping|legend=1| 1 0 -4 | 0 1 4 }}


7 and [[9-limit]] minimax
: mapping generators: ~2, ~3
[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |-3 0 5/2 0&gt;]
[[Eigenmonzo]]s: 2, 5


[[POTE tuning|POTE generator]]: 696.495
[[Optimal tuning]]s:
Mapping generator: ~3
* [[WE]]: ~2 = 1201.3906{{c}}, ~3/2 = 697.0455{{c}}
: [[error map]]: {{val| +1.391 -3.519 +1.868 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.6512{{c}}
: error map: {{val| 0.000 -5.304 +0.291 }}


Algebraic generator: Cybozem, the real root of 15x^3-10x^2-18, which comes to 503.4257 cents. The recurrence converges quickly.
[[Minimax tuning]]:  
* [[5-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma)
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


[[Map]]: [&lt;1 0 -4 -13|, &lt;0 1 4 10|]
[[Tuning ranges]]:  
[[Generator]]s: 2, 3
* 5-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 720.000] (4\7 to 3\5)
[[Wedgie]]: &lt;&lt;1 4 10 4 13 12||
* 5-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 701.955] (1/3-comma to Pyth.)
EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[81edo|81]], [[143edo|143b]]
[[Badness]]: 0.0137


==Unidecimal meantone aka Huygens==
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b }}
See also [[Meantone vs meanpop]]
[[Comma]]s: 81/80, 126/125, 99/98


[[11-limit]] minimax
[[Badness]] (Sintel): 0.173
[|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


[[POTE tuning|POTE generator]]: 696.967
=== Overview to extensions ===
Mapping generator: ~3
The second comma of the normal comma list defines which [[7-limit]] family member we are looking at.
* Flattertone adds {{monzo| -24 17 0 -1 }}, finding the [[~]][[7/4]] at the double-augmented sixth, for a tuning between 33edo and 26edo.
* Flattone adds {{monzo| -17 9 0 1 }}, finding the ~7/4 at the diminished seventh, for a tuning between 26edo and 19edo.
* Septimal meantone adds [[Harrison's comma|{{monzo| -13 10 0 -1 }}]], finding the ~7/4 at the augmented sixth, for a tuning between 19edo and 12edo.
* Dominant adds [[64/63|{{monzo| 6 -2 0 -1 }}]], finding the ~7/4 at the minor seventh, for a tuning between 12edo and 5edo.
* Sharptone adds [[28/27|{{monzo| 2 -3 0 1 }}]], finding the ~7/4 at the major sixth, for an [[exotemperament]] never exactly well-tuned, and where 5edo is the only [[diamond monotone]] tuning, with a terrible 5-limit part.
Those all have a fifth as generator.
* Injera adds {{monzo| -7 8 0 -2 }} with a half-octave period.
* Mohajira adds {{monzo| -23 11 0 2 }} and splits the fifth in two.
* Godzilla adds [[49/48|{{monzo| -4 -1 0 2 }}]] with an ~[[8/7]] generator, two of which give the [[4/3|fourth]].
* Mothra adds [[1029/1024|{{monzo| -10 1 0 3 }}]] with an ~8/7 generator, three of which give the fifth.
* Liese adds {{monzo| -9 11 0 -3 }} with a ~[[10/7]] generator, three of which give the [[3/1|twelfth]].
* Squares adds {{monzo| -3 9 0 -4 }} with a ~[[9/7]] generator, four of which give the [[8/3|eleventh]].
* Jerome adds {{monzo| 3 7 0 -5 }} and slices the fifth in five.


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


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


[[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]]
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.  


===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 tuning|POTE generator]]: ~3/2 = 696.642
==== Splitting the meantone fifth into three (1029/1024) ====
Mapping generator: ~3
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: [&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


===Grosstone===  
==== 31edo as splitting the fifth into two, three and nine ====
Commas: 81/80, 99/98, 126/125, 144/143
[[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).


POTE generator: ~3/2 = 697.264
Temperaments discussed elsewhere include
Mapping generator: ~3
* ''[[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]]


Map: [&lt;1 0 -4 -13 -25 29|, &lt;0 1 4 10 18 -16|]
The rest are considered below.
EDOs: 12, 31, 43, 74
Badness: 0.0259


===Meridetone===
== Septimal meantone ==
Commas: 78/77, 81/80, 99/98, 126/125
<div style="float:right">[[:de:septimal-mitteltönig|Deutsch]]</div>
{{Main| Meantone #Septimal meantone}}
{{Wikipedia| Septimal meantone temperament }}


POTE generator: ~3/2 = 697.529
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.  
Mapping generator: ~3


Map: [&lt;1 0 -4 -13 -25 -39|, &lt;0 1 4 10 18 27|]
[[Subgroup]]: 2.3.5.7
EDOs: 43, 117df, 160bdf, 203bcdef
Badness: 0.0264


===Hemimeantone===
[[Comma list]]: 81/80, 126/125
Commas: 81/80, 99/98, 126/125, 169/168


POTE generator: ~52/45 = 250.304
{{Mapping|legend=1| 1 0 -4 -13 | 0 1 4 10 }}
Mapping generator: ~26/15


Map: [&lt;1 0 -4 -13 -25 -5|, &lt;0 2 8 20 36 11|]
[[Optimal tuning]]s:
EDOs: 43, 62, 167bef, 229bef
* [[WE]]: ~2 = 1201.2358{{c}}, ~3/2 = 697.2122{{c}}
Badness: 0.0314
: [[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 }}


==Meanpop==
[[Minimax tuning]]:
See also [[Meantone vs meanpop]]
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma)
[[Comma]]s: 81/80, 126/125, 385/384
: [[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


[[11-limit]] [[minimax]] 1/4 comma
[[Tuning ranges]]:
[|1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;,
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
|-3 0 5/2 0 0&gt;, |11 0 -13/4 0 0&gt;]
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 701.955] (1/3-comma to Pyth.)
[[Eigenmonzo]]s: 2, 5
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)


[[POTE tuning|POTE generator]]: 696.434
[[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.
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.
{{Optimal ET sequence|legend=1| 12, 19, 31, 81, 112b, 143b }}


[[@http://soonlabel.com/xenharmonic/archives/607|Scott Joplin's "The Entertainer" tuned into meanpop]]
[[Badness]] (Sintel): 0.347


Map: [&lt;1 0 -4 -13 24|, &lt;0 1 4 10 -13|]
=== Undecimal meantone (huygens) ===
[[Generator]]s: 2, 3
{{Redirect|Huygens|the Dutch mathematician, physicist and astronomer|Wikipedia: Christiaan Huygens}}
EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[81edo|81]], [[112edo|112]]
{{See also| Huygens vs meanpop }}
[[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]]
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.


===13-limit Meanpop===
Subgroup: 2.3.5.7.11
[[Comma]]s: 81/80, 105/104, 144/143, 196/195


POTE generator: ~3/2 = 696.211
Comma list: 81/80, 99/98, 126/125
Mapping generator: ~3


Map: [&lt;1 0 -4 -13 24 -20|, &lt;0 1 4 10 -13 15|]
Mapping: {{mapping| 1 0 -4 -13 -25 | 0 1 4 10 18 }}
EDOS: [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]], [[131edo|131bd]], [[212edo|212bdf]]
[[Badness]]: 0.0209


===Meanplop===
Optimal tunings:
Commas: 65/64, 78/77, 81/80, 91/90
* WE: ~2 = 1200.7636{{c}}, ~3/2 = 697.4122{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.0315{{c}}


POTE generator: ~3/2 = 696.202
Minimax tuning:
Mapping generator: ~3
* 11-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}
: projection map: [{{monzo| 1 0 0 0 0 }}, {{monzo| 25/16 -1/8 0 0 1/16 }}, {{monzo| 9/4 -1/2 0 0 1/4 }}, {{monzo| 21/8 -5/4 0 0 5/8 }}, {{monzo| 25/8 -9/4 0 0 9/8 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/9


Map: [&lt;1 0 -4 -13 24 10|, &lt;0 1 4 10 -13 -4|]
Tuning ranges:  
EDOs: 12e, 19, 31f, 50f
* 11-odd-limit diamond monotone: ~3/2 = [696.774, 700.000] (18\31 to 7\12)
Badness: 0.0277
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)


==Meanenneadecal==
Algebraic generator: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
[[Comma]]s: 45/44, 56/55, 81/80


[[POTE tuning|POTE generator]]: ~3/2 = 696.250
{{Optimal ET sequence|legend=0| 12, 19e, 31, 105, 136b }}
Mapping generator: ~3


Map: [&lt;1 0 -4 -13 -6|, &lt;0 1 4 10 6|]
Badness (Sintel): 0.563
EDOs: [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31e]], [[50edo|50e]]
[[Badness]]: 0.0214


===13-limit===
; Music
[[Comma]]s: 45/44, 56/55, 78/77, 81/80
* [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]


[[POTE tuning|POTE generator]]: ~3/2 = 696.146
==== Grosstone ====
Mapping generator: ~3
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]].  


Map: [&lt;1 0 -4 -13 -6 -20|, &lt;0 1 4 10 6 15|]
Subgroup: 2.3.5.7.11.13
EDOs: [[19edo|19]], [[31edo|31e]], [[50edo|50e]]]
[[Badness]]: 0.0212


===Vincenzo===
Comma list: 81/80, 99/98, 126/125, 144/143
Commas: 81/80 126/125 45/44 65/64 256/255 153/152 23/22


POTE generator: ~3/2
Mapping: {{mapping| 1 0 -4 -13 -25 29 | 0 1 4 10 18 -16 }}
Mapping generator: ~3


Map: [&lt;1 0 -4 -13 ... |, &lt;0 1 4 10 6 -4 -5 -3 -6|]
Optimal tunings:  
EDOs: 12
* WE: ~2 = 1199.9389{{c}}, ~3/2 = 697.2282{{c}}
Badness:
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.2627{{c}}


==Meanundeci==
Minimax tuning:
Commas: 33/32, 55/54, 77/75
* 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


POTE generator: ~3/2 = 694.689
Tuning ranges:
Mapping generator: ~3
* 13- and 15-odd-limit diamond monotone: ~3/2 = [696.774, 697.674] (18\31 to 25\43)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)


Map: [&lt;1 0 -4 -13 -6 5|, &lt;0 1 4 10 6 -|]
{{Optimal ET sequence|legend=0| 12, 31, 43, 74 }}
EDOs: 12e, 19e
Badness: 0.0315


=Flattone=
Badness (Sintel): 1.07
[[Comma]]s: 81/80, 525/512


The [[wedgie]] for flattone is &lt;&lt;1 4 -9 4 -17 -32||, which tells us among other things that 9 generator steps of 4/3 get to the interval class for 7, meaning that [[7_4|7/4]] is a diminished minor seventh interval. Other intervals are [[7_6|7/6]], a diminished minor third, and [[7_5|7/5]], a doubly diminshed fifth. Good tunings for flattone are [[26edo]], [[45edo]] and [[64edo]].
===== 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.  


[[7-limit]] minimax
Subgroup: 2.3.5.7.11.13.17
[|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, 120/119, 126/125, 144/143
[|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 29 12 | 0 1 4 10 18 -16 -5 }}
Mapping generator: ~3


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 = 1199.5811{{c}}, ~3/2 = 697.0918{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3303{{c}}


Map: [&lt;1 0 -4 17|, &lt;0 1 4 -9|]
{{Optimal ET sequence|legend=0| 12, 31, 43, 74g }}
[[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==
Badness (Sintel): 1.06
Commas: 45/44, 81/80, 385/384


POTE generator: ~3/2 = 693.126
===== 19-limit =====
Mapping generator: ~3
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;1 0 -4 17 -6|, &lt;0 1 4 -9 6|]
Comma list: 81/80, 96/95, 99/98, 120/119, 126/125, 144/143
EDOs: 7, 19, 26, 45, 71bc, 116bcde
Badness: 0.0338


==13-limit==
Mapping: {{mapping| 1 0 -4 -13 -25 29 12 9 | 0 1 4 10 18 -16 -5 -3 }}
45/44, 65/64, 78/77, 81/80


POTE generator: ~3/2 = 693.058
Optimal tunings:
Mapping generator: ~3
* WE: ~2 = 1199.2931{{c}}, ~3/2 = 696.9690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3736{{c}}


Map: [&lt;1 0 -4 17 -6 10|, &lt;0 1 4 -9 6 -4|]
{{Optimal ET sequence|legend=0| 12, 31, 43, 74gh }}
EDOs: 7, 19, 26, 45f, 71bcf, 116bcdef
Badness: 0.0223


=Dominant=
Badness (Sintel): 1.07
[[Comma]]s: 36/35, 64/63


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]].
==== Fokkertone ====
Fokkertone maps the [[13/8]] to the double-augmented fifth (C–G𝄪). Note that the minor third conflates 13/11 with 6/5, and that 11/10~13/12 is a double-augmented unison; 12/11 is a double-diminished third; and 14/13 is a minor second. 31edo can be recommended as a tuning since it is the only 13-odd-limit diamond monotone tuning.  


[[POTE tuning|POTE generator]]: 701.573
This extension used to be known as ''tridecimal meantone'', but was decanonicalized in 2025.  
Mapping generator: ~3


Map: [&lt;1 0 -4 6|, &lt;0 1 4 -2|]
Subgroup: 2.3.5.7.11.13
[[Wedgie]]: &lt;&lt;1 4 -2 4 -6 -16||
EDOs: [[5edo|5]], [[7edo|7]], [[12edo|12]], [[53edo|53]], [[65edo|65]]
[[Badness]]: 0.0207


==11-limit==
Comma list: 66/65, 81/80, 99/98, 105/104
Commas: 36/35, 64/63, 56/55


POTE generator: ~3/2 = 703.254
Mapping: {{mapping| 1 0 -4 -13 -25 -20 | 0 1 4 10 18 15 }}
Mapping generator: ~3


Map: [&lt;1 0 -4 6 13|, &lt;0 1 4 -2 -6|]
Optimal tunings:  
EDOs: 5, 12, 17c, 29cde
* WE: ~2 = 1200.8149{{c}}, ~3/2 = 697.1155{{c}}
Badness: 0.0242
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.7085{{c}}


==Domineering==
Minimax tuning:
Commas: 36/35, 45/44, 64/63
* 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 = 698.776
{{Optimal ET sequence|legend=0| 12f, 19e, 31 }}
Mapping generator: ~3


Map: [&lt;1 0 -4 6 -6|, &lt;0 1 4 -2 6|]
Badness (Sintel): 0.746
EDOs: 7, 12, 43de
Badness: 0.0220


==Domination==  
===== 17-limit =====
Commas: 36/35, 64/63, 77/75
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~3/2 = 705.004
Comma list: 66/65, 81/80, 99/98, 105/104, 120/119
Mapping generator: ~3


Map: [&lt;1 0 -4 6 -14|, &lt;0 1 4 -2 11|]
Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 | 0 1 4 10 18 15 -5 }}
EDOs: 17c, 46cd
Badness: 0.0366


===13-limit===
Optimal tunings:
Commas: 26/25, 36/35, 64/63, 66/65
* WE: ~2 = 1199.5548{{c}}, ~3/2 = 696.7449{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.9823{{c}}


POTE generator: ~3/2 = 705.496
{{Optimal ET sequence|legend=0| 12f, 31 }}
Mapping generator: ~3


Map: [&lt;1 0 -4 6 -14 -9|, &lt;0 1 4 -2 11 8|]
Badness (Sintel): 1.02
EDOs: 17c
Badness: 0.0274


==Twelve==  
===== 19-limit =====
Commas: 81/80 64/63 45/44 65/64 256/255 153/152
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~3/2 = 696.217
Comma list: 66/65, 81/80, 96/95, 99/98, 105/104, 120/119
Mapping generator: ~3


Map: [&lt;1 0 -4 6 -6 10 12 9|, &lt;0 1 4 -2 6 -4 -5 -3|]
Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 9 | 0 1 4 10 18 15 -5 -3 }}
EDOs: 7, 12, 19d, 31def
Badness: 0.0204


==Arnold==
Optimal tunings:
Commas: 22/21, 33/32, 36/35
* WE: ~2 = 1199.0408{{c}}, ~3/2 = 696.5824{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.1061{{c}}


POTE generator: ~3/2 = 698.491
{{Optimal ET sequence|legend=0| 12f, 31 }}
Mapping generator: ~3


Map: [&lt;1 0 -4 6 5|, &lt;0 1 4 -2 -1|]
Badness (Sintel): 1.10
EDOs: 5, 7, 12e
Badness: 0.0261


==13-limit==  
==== Meridetone ====
Commas: 22/21, 27/26, 33/32, 40/39
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.


POTE generator: ~3/2 = 696.743
Subgroup: 2.3.5.7.11.13
Mapping generator: ~3


Map: [&lt;1 0 -4 6 5 -1|, &lt;0 1 4 -2 -1 3|]
Comma list: 78/77, 81/80, 99/98, 126/125
EDOs: 5, 7, 12ef, 19def, 31def
Badness: 0.0233


==Dominatrix==
Mapping: {{mapping| 1 0 -4 -13 -25 -39 | 0 1 4 10 18 27 }}
Commas: 27/26 36/35 45/44 64/63


POTE generator: ~3/2 = 698.544
Optimal tunings:
Mapping generator: ~3
* WE: ~2 = 1199.9122{{c}}, ~3/2 = 697.4779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.5241{{c}}


Map: [&lt;1 0 -4 6 -6 -1|, &lt;0 1 4 -2 6 3|]
Minimax tuning:  
EDOs: 7, 12f
* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/25 -2/25 0 0 0 1/25 }}
Badness: 0.0183
: unchanged-interval (eigenmonzo) basis: 2.13/9


=Sharptone=
{{Optimal ET sequence|legend=0| 12f, 31f, 43 }}
[[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.
Badness (Sintel): 1.09


[[POTE tuning|POTE generator]]: 700.140
===== 17-limit =====
Mapping generator: ~3
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 0 -4 -2|, &lt;0 1 4 3|]
Comma list: 78/77, 81/80, 99/98, 120/119, 126/125
[[Wedgie]]: &lt;&lt;1 4 3 4 2 -4||
EDOs: [[5edo|5]], [[12edo|12]]
[[Badness]]: 0.0248


=Meansept=
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 | 0 1 4 10 18 27 -5 }}
Commas: 15/14, 81/80


POTE generator: ~3/2 = 682.895
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 -5|, &lt;0 1 4 5|]
{{Optimal ET sequence|legend=0| 12f, 43 }}
Wedgie: &lt;&lt;1 4 5 4 5 0||
EDOs: 7
Badness: 0.0453


==11-limit==
Badness (Sintel): 1.22
Commas: 15/14, 22/21, 125/121


POTE generator: ~3/2 = 685.234
===== 19-limit =====
Mapping generator: ~3
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;1 0 -4 -5 -6|, &lt;0 1 4 5 6|]
Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125
EDOs: 7
Badness: 0.0325


=Injera=
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 9 | 0 1 4 10 18 27 -5 -3 }}
[[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.
Optimal tunings:
* WE: ~2 = 1199.0260{{c}}, ~3/2 = 697.1486{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6887{{c}}


[[http://tech.groups.yahoo.com/group/tuning-math/message/3091|Origin of the name]]
{{Optimal ET sequence|legend=0| 12f, 43 }}


[[POTE tuning|POTE generator]]: 694.375
Badness (Sintel): 1.25
Mapping generator: ~3


Map: [&lt;2 0 -8 -7|, &lt;0 1 4 4|]
==== Hemimeantone ====
[[Wedgie]]: &lt;&lt;2 8 8 8 7 -4||
Subgroup: 2.3.5.7.11.13
EDOs: [[12edo|12]], [[26edo|26]], [[38edo|38]], [[102edo|102bcd]], [[140edo|140bcd]], [[178edo|178bcd]]
[[Badness]]: 0.0311


[[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]]
Comma list: 81/80, 99/98, 126/125, 169/168
[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20Injera%20Jam.mp3|Injera Jam]] (in [[26edo]]) by [[Zach Curley]]


==11-limit==
Mapping: {{mapping| 1 0 -4 -13 -25 -5 | 0 2 8 20 36 11 }}
Commas: 45/44, 50/49, 81/80


POTE generator: ~3/2 = 692.840
: mapping generators: ~2, ~26/15
Mapping generator: ~3


Map: [&lt;2 0 -8 -7 -12|, &lt;0 1 4 4 6|]
Optimal tunings:  
EDOs: 12, 14c, 26. 90bce, 116bce
* WE: ~2 = 1201.0387{{c}}, ~26/15 = 949.2863{{c}}
Badness: 0.0231
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5065{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 19e, 43, 62 }}
Commas: 45/44, 50/49, 81/80, 78/77


POTE generator: ~3/2 = 692.673
Badness (Sintel): 1.30
Mapping generator: ~3


Map: [&lt;2 0 -8 -7 -12 -21|, &lt;0 1 4 4 6 9|]
===== 17-limit =====
EDOs: 26, 104bcf
Subgroup: 2.3.5.7.11.13.17
Badness: 0.0216


==Enjera==
Comma list: 81/80, 99/98, 126/125, 169/168, 221/220
Commas: 27/26, 40/39, 45/44, 99/98


POTE generator: ~3/2 = 694.121
Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 | 0 2 8 20 36 11 33 }}
Mapping generator: ~3


Map: [&lt;2 0 -8 -7 -12 -2|, &lt;0 1 4 4 6 3|]
Optimal tunings:  
EDOs: 12f, 26f, 38ef
* WE: ~2 = 1201.0270{{c}}, ~26/15 = 949.2892{{c}}
Badness: 0.0265
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5169{{c}}


==Injerous==
{{Optimal ET sequence|legend=0| 19eg, 43, 62 }}
Commas: 33/32, 50/49, 55/54


POTE generator: ~3/2 = 690.548
Badness (Sintel): 1.19
Mapping generator: ~3


Map: [&lt;2 0 -8 -7 10|, &lt;0 1 4 4 -1|]
===== 19-limit =====
EDOs: 12e, 14c, 26e, 40ce
Subgroup: 2.3.5.7.11.13.17.19
Badness: 0.0386


==Lahoh==
Comma list: 81/80, 99/98, 126/125, 153/152, 169/168, 221/220
Commas: 50/49, 56/55, 81/77


POTE generator: ~3/2 = 699.001
Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 -25 | 0 2 8 20 36 11 33 37 }}
Mapping generator: ~3


Map: [&lt;2 0 -8 -7 7|, &lt;0 1 4 4 0|]
Optimal tunings:  
EDOs: 12
* WE: ~2 = 1201.0339{{c}}, ~19/11 = 949.2902{{c}}
Badness: 0.0431
* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 948.5111{{c}}


=Godzilla=
{{Optimal ET sequence|legend=0| 19egh, 43, 62 }}
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.
Badness (Sintel): 1.15


[[POTE tuning|POTE generator]]: ~8/7 = 252.635
==== Semimeantone ====
Mapping generator: ~7/4
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 -4 2|, &lt;0 2 8 1|]
Comma list: 81/80, 99/98, 126/125, 847/845
[[Wedgie]]: &lt;&lt;2 8 1 8 -4 -20||
EDOs: [[5edo|5]], [[9edo|9]], [[14edo|14]], [[19edo|19]], [[31edo|31]], [[81edo|81]], 143b
[[Badness]]: 0.0267


==11-limit==
Mapping: {{mapping| 2 0 -8 -26 -50 -59 | 0 1 4 10 18 21 }}
Commas: 45/44, 49/48, 81/80


POTE generator: ~8/7 = 254.027
: mapping generators: ~55/39, ~3
Mapping generator: ~7/4


Map: [&lt;1 0 -4 2 -6|, &lt;0 2 8 1 12|]
Optimal tunings:  
EDOs: 14c, 19, 33cd, 52cd
* WE: ~55/39 = 600.3606{{c}}, ~3/2 = 697.4241{{c}}
Badness: 0.0290
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 697.0545{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 12f, …, 50eff, 62, 136b }}
Commas: 45/44, 49/48, 78/77, 81/80


POTE generator: ~8/7 = 253.603
Badness (Sintel): 1.68
Mapping generator: ~7/4


Map: [&lt;1 0 -4 2 -6 -5|, &lt;0 2 8 1 12 11|]
===== 17-limit =====
EDOs: 14cf, 19, 33cdf, 52cdf
Subgroup: 2.3.5.7.11.13.17
Badness: 0.0225


==Semafour==
Comma list: 81/80, 99/98, 126/125, 221/220, 289/288
Commas: 33/32, 49/48, 55/54


POTE generator: ~8/7 = 254.042
Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 | 0 1 4 10 18 21 1 }}
Mapping generator: ~7/4


Map: [&lt;1 0 -4 2 5|, &lt;0 2 8 1 -2|]
Optimal tunings:  
EDOs: 5, 14c, 19e, 33cde
* WE: ~17/12 = 600.5426{{c}}, ~3/2 = 697.5571{{c}}
Badness: 0.0285
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9858{{c}}


==Varan==
{{Optimal ET sequence|legend=0| 12f, 50eff, 62, 136bg }}
Commas: 49/48, 77/75, 81/80


POTE generator: ~8/7 = 251.079
Badness (Sintel): 1.60
Mapping generator: ~7/4


Map: [&lt;1 0 -4 2 -10|, &lt;0 2 8 1 17|]
===== 19-limit =====
EDOs: 19e, 24, 43de
Subgroup: 2.3.5.7.11.13.17.19
Badness: 0.0396


===13-limit===
Comma list: 81/80, 99/98, 126/125, 153/152, 209/208, 221/220
Commas: 49/48, 66/65, 77/75, 81/80


POTE generator: ~8/7 = 251.165
Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 -1 | 0 1 4 10 18 21 1 3 }}
Mapping generator: ~7/4


Map: [&lt;1 0 -4 2 -10 -5|, &lt;0 2 8 1 17 11|]
Optimal tunings:  
EDOs: 19e, 24, 43de
* WE: ~17/12 = 600.5959{{c}}, ~3/2 = 697.5985{{c}}
Badness: 0.0257
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9638{{c}}


==Baragon==
{{Optimal ET sequence|legend=0| 12f, 50eff, 62 }}
Commas: 49/48, 56/55, 81/80


POTE generator: ~8/7 = 251.173
Badness (Sintel): 1.47
Mapping generator: ~7/4


Map: [&lt;1 0 -4 2 9|, &lt;0 2 8 1 -7|]
=== Meanpop ===
EDOs: 19, 24, 43d
{{See also| Huygens vs meanpop }}
Badness: 0.0357


==Music==
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.  
[[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=
Subgroup: 2.3.5.7.11
[[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, 385/384


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| 1 0 -4 -13 24 | 0 1 4 10 -13 }}


[[7-limit|7]] and [[9-limit]] minimax 1/4 comma
: mapping generator: ~2, ~3
[|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]]: ~128/105 = 348.415
Optimal tunings:  
Mapping generator: ~128/105
* WE: ~2 = 1201.3464{{c}}, ~3/2 = 697.2159{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4509{{c}}


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


Map: [&lt;1 1 0 6|, &lt;0 2 8 -11|]
Tuning ranges:  
[[Generator]]s: 2, 128/105
* 11-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
[[Wedgie]]: &lt;&lt;2 8 -11 8 -23 -48||
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]]
[[Badness]]: 0.0557


==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.
[[Comma]]s: 81/80, 121/120, 176/175


[[11-limit]] minimax 1/4 comma
{{Optimal ET sequence|legend=0| 12e, 19, 31, 81, 112b }}
[|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
Badness (Sintel): 0.712
Mapping generator: ~11/9


Map: [&lt;1 1 0 6 2|, &lt;0 2 8 -11 5|]
; Music
[[Generator]]s: 2, 11/9
* [http://soonlabel.com/xenharmonic/archives/607 Scott Joplin's "The Entertainer" tuned into meanpop]{{dead link}}
EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]]
* [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]
[[Badness]]: 0.0261


==13-limit==  
==== Tridecimal meanpop ====
Commas: 81/80, 121/120, 105/104, 66/65
Subgroup: 2.3.5.7.11.13


POTE generator: ~11/9 = 348.558
Comma list: 81/80, 105/104, 126/125, 144/143
Mapping generator: ~11/9


Map: [&lt;1 1 0 6 2 4|, &lt;0 2 8 -11 5 -1|]
Mapping: {{mapping| 1 0 -4 -13 24 -20 | 0 1 4 10 -13 15 }}
EDOs: 7, 24, 31, 117ef, 148bef
Badness: 0.0234


=Ptolemy=  
Optimal tunings:
Commas: 81/80, 121/120, 525/512
* WE: ~2 = 1201.0765{{c}}, ~3/2 = 696.8361{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2347{{c}}


POTE generator: ~11/9 = 346.922
Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 4/7 0 0 0 -1/28 1/28 }}
: unchanged-interval (eigenmonzo) basis: 2.13/11


Map: [&lt;1 1 0 8 2|, &lt;0 2 8 -18 5|]
Tuning ranges:  
EDOs: 7, 38d, 45e, 83bcde
* 13- and 15-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
Badness: 0.0588
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)


==13-limit==
{{Optimal ET sequence|legend=0| 19, 31, 50, 81 }}
Commas: 65/64, 81/80, 105/104, 121/120


POTE generator: ~11/9 = 346.910
Badness (Sintel): 0.863


Map: [&lt;1 1 0 8 2 6|, &lt;0 2 8 -18 5 -8|]
===== Meanpoppic =====
EDOs: 7, 38df, 45ef, 83bcdef
Subgroup: 2.3.5.7.11.13.17
Badness: 0.0343


=Maqamic=
Comma list: 81/80, 105/104, 126/125, 144/143, 273/272
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.
Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 | 0 1 4 10 -13 15 26 }}


[[POTE tuning|POTE generator]]: ~11/9 = 350.934
Optimal tunings:  
Mapping generator: ~11/9
* WE: ~2 = 1201.0727{{c}}, ~3/2 = 696.8168{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2195{{c}}


Map: [&lt;1 1 0 4 2|, &lt;0 2 8 -4 5|]
{{Optimal ET sequence|legend=0| 19g, 31, 50, 81, 131bd }}
[[Generator]]s: 2, 11/9
EDOs: [[7edo|7]], [[10edo|10c]], [[17edo|17c]], [[24edo|24d]], [[31edo|31d]]


==13-limit==
Badness (Sintel): 1.02
[[Comma]]s: 81/80, 36/35, 121/120, 144/143


[[POTE tuning|POTE generator]]: ~11/9 = 350.816
====== 19-limit ======
Mapping generator: ~11/9
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;1 1 0 4 2 4|, &lt;0 2 8 -4 5 -1|]
Comma list: 81/80, 105/104, 126/125, 144/143, 153/152, 273/272
Generators: 2, 11/9
EDOs: [[7edo|7]], [[10edo|10c]], [[17edo|17c]], [[24edo|24d]],[[31edo| 31d]]


=Migration=
Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 -40 | 0 1 4 10 -13 15 26 28 }}
Commas: 81/80, 121/120, 126/125


POTE generator: ~11/9 = 348.182
Optimal tunings:  
Mapping generator: ~11/9
* WE: ~2 = 1201.0719{{c}}, ~3/2 = 696.8101{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2137{{c}}


Map: [&lt;1 1 0 -3 2|, &lt;0 2 8 20 5|]
{{Optimal ET sequence|legend=0| 19gh, 31, 50, 81 }}
EDOs: 31, 100de, 131bde, 162bde
Badness: 0.0255


=Mohamaq=
Badness (Sintel): 1.08
Commas: 81/80, 392/375


POTE generator: ~25/21 = 350.586
===== Meanpoid =====
Mapping generator: ~25/21
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 1 0 -1|, &lt;0 2 8 13|]
Comma list: 81/80, 105/104, 120/119, 126/125, 144/143
EDOs: 17c, 24, 65c, 89cd
Badness: 0.0777


==11-limit==
Mapping: {{mapping| 1 0 -4 -13 24 -20 12 | 0 1 4 10 -13 15 -5 }}
Commas: 56/55, 77/75, 243/242


POTE generator: ~11/9 = 350.565
Optimal tunings:  
Mapping generator: ~11/9
* WE: ~2 = 1200.2768{{c}}, ~3/2 = 696.5683{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4114{{c}}


Map: [&lt;1 1 0 -1 2|, &lt;0 2 8 13 5|]
{{Optimal ET sequence|legend=0| 19, 31 }}
EDOs: 17c, 24, 65c, 89cd
Badness: 0.0362


==13-limit==
Badness (Sintel): 1.17
Commas: 56/55, 66/65, 77/75, 243/242


POTE generator: ~11/9 = 350.745
====== 19-limit ======
Mapping generator: ~11/9
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;1 1 0 -1 2 4|, &lt;0 2 8 13 5 -1|]
Comma list: 77/76, 81/80, 96/95, 105/104, 120/119, 126/125
EDOs: 17c, 24, 41c, 65c
Badness: 0.0287


=Orphic=
Mapping: {{mapping| 1 0 -4 -13 24 -20 12 9 | 0 1 4 10 -13 15 -5 -3 }}
Commas: 81/80, 5898240/5764801


POTE generator: ~7/6 = 275.794
Optimal tunings:  
Mapping generator: ~343/288
* WE: ~2 = 1199.7905{{c}}, ~3/2 = 696.3779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4973{{c}}


Map: [&lt;2 1 -4 4|, &lt;0 4 16 3|]
{{Optimal ET sequence|legend=0| 19, 31 }}
Wedgie: &lt;&lt;8 32 6 32 -13 -76||
EDOs: 26, 74, 174bd, 248bd
Badness: 0.2588


==11-limit==
Badness (Sintel): 1.25
Commas: 81/80, 99/98, 73728/73205


POTE generator: ~7/6 = 275.762
==== Semimeanpop ====
Mapping generator: ~77/64
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 1 -4 4 8|, &lt;0 4 16 3 -2|]
Comma list: 81/80, 126/125, 385/384, 847/845
EDOs: 26, 48c, 74, 248bd, 322bd
Badness: 0.1015


==13-limit==
Mapping: {{mapping| 2 0 -8 -26 48 39 | 0 1 4 10 -13 -10 }}
Commas: 81/80, 99/98, 144/143, 2200/2197


POTE generator: ~7/6 = 275.774
: mapping generators: ~55/39, ~3
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: ~55/39 = 600.6704{{c}}, ~3/2 = 697.2151{{c}}
Badness: 0.0535
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.4341{{c}}


=Mothra=
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112b }}
[[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.
Badness (Sintel): 1.78


[[7-limit|7]] and [[9-limit]] minimax 1/4 comma
===== 17-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
[[Eigenmonzo]]s: 2, 5


[[POTE tuning|POTE generator]]: ~8/7 = 232.193
Comma list: 81/80, 126/125, 221/220, 273/272, 289/288
Mapping generator: ~8/7


Algebraic generator: Rabrindanath, largest real root of x^8-3x^2+1, or 232.0774 cents.
Mapping: {{mapping| 2 0 -8 -26 48 39 5 | 0 1 4 10 -13 -10 1 }}


Map: [&lt;1 1 0 3|, &lt;0 3 12 -1|]
Optimal tunings:  
[[Generator]]s: 2, 8/7
* WE: ~17/12 = 600.7232{{c}}, ~3/2 = 697.2820{{c}}
[[Wedgie]]: &lt;&lt;3 12 -1 12 -10 -36||
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4411{{c}}
EDOs: [[5edo|5]], [[26edo|26]], [[31edo|31]]
[[Badness]]: 0.0371


==11-limit==
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bg }}
[[Comma]]s: 81/80, 99/98, 385/384


POTE generator: ~8/7 = 232.031
Badness (Sintel): 1.45
Mapping generator: ~8/7


Map: [&lt;1 1 0 3 5|, &lt;0 3 12 -1 -8|]
===== 19-limit =====
EDOs: [[5edo|5]], [[26edo|26]], [[31edo|31]], [[88edo|88]], [[150edo|150]], [[181edo|181]]
Subgroup: 2.3.5.7.11.13.17.19
[[Badness]]: 0.0256


==13-limit==
Comma list: 81/80, 126/125, 153/152, 209/208, 221/220, 273/272
Commas: 81/80, 99/98, 105/104, 144/143


POTE generator: ~8/7 = 231.811
Mapping: {{mapping| 2 0 -8 -26 48 39 5 -1 | 0 1 4 10 -13 -10 1 3 }}
Mapping generator: ~8/7


Map: [&lt;1 1 0 3 5 1|, &lt;0 3 12 -1 -8 14|]
Optimal tunings:  
EDOs: 5, 26, 31, 57, 88
* WE: ~17/12 = 600.7527{{c}}, ~3/2 = 697.3244{{c}}
Badness: 0.0240
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4525{{c}}


==Cynder==
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bgh }}
Commas: 45/44, 81/80, 1029/1024


POTE generator: ~8/7 = 231.317
Badness (Sintel): 1.28
Mapping generator: ~8/7


Map: [&lt;1 1 0 3 0|, &lt;0 3 12 -1 18|]
=== Meanenneadecal ===
EDOs: 26, 57e, 83bce
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.  
Badness: 0.0557


===13-limit===
Subgroup: 2.3.5.7.11
Commas: 45/44, 78/77, 81/80, 640/637


POTE generator: ~8/7 = 231.293
Comma list: 45/44, 56/55, 81/80
Mapping generator: ~8/7


Map: [&lt;1 1 0 3 0 1|, &lt;0 3 12 -1 18 14|]
Mapping: {{mapping| 1 0 -4 -13 -6 | 0 1 4 10 6 }}
EDOs: 26, 57e, 83bce
Badness: 0.0341


==Mosura==
Optimal tunings:
Commas: 81/80, 176/175, 1029/1024
* WE: ~2 = 1199.6946{{c}}, ~3/2 = 696.0729{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2083{{c}}


POTE generator: ~8/7 = 232.419
Tuning ranges:
Mapping generator: ~8/7
* 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]


Map: [&lt;1 1 0 3 -1|, &lt;0 3 12 -1 23|]
{{Optimal ET sequence|legend=0| 7d, 12, 19, 31e }}
EDOs: 31, 129, 136b, 148be, 160be, 191bce, 222bce, 253bce
Badness: 0.0313


===13-limit===
Badness (Sintel): 0.708
Commas: 81/80, 144/143, 176/175, 1029/1024


POTE generator: ~8/7 = 232.640
==== 13-limit ====
Mapping generator: ~8/7
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 1 0 3 -1 7|, &lt;0 3 12 -1 23 -17|]
Comma list: 45/44, 56/55, 78/77, 81/80
EDOs: 31, 55, 67, 98
Badness: 0.0369


=Squares=
Mapping: {{mapping| 1 0 -4 -13 -6 -20 | 0 1 4 10 6 15 }}
[[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: ~2 = 1199.7931{{c}}, ~3/2 = 696.0258{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1241{{c}}


7 and 9 limit minimax 1/4 comma
{{Optimal ET sequence|legend=0| 7df, 12f, 19, 31e }}
[|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): 0.875
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.
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 3 8 6|, &lt;0 -4 -16 -9|]
Comma list: 45/44, 56/55, 78/77, 81/80, 120/119
[[Generator]]s: 2, 9/7
EDOs: [[14edo|14]], [[31edo|31]], [[262edo|262]], [[293edo|293]]
[[Badness]]: 0.0460


Music:
Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 | 0 1 4 10 6 15 -5 }}
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: ~2 = 1198.6665{{c}}, ~3/2 = 695.8010{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4998{{c}}


POTE generator: ~9/7 = 425.957
{{Optimal ET sequence|legend=0| 12f, 19, 31e }}
Mapping generator: ~9/7


Map: [&lt;1 3 8 6 7|, &lt;0 -4 -16 -9 -10|]
Badness (Sintel): 1.17
EDOs: [[5edo|5]], [[8edo|8]], [[11edo|11]], [[14edo|14]], [[17edo|17]], [[31edo|31]]
[[Badness]]: 0.0216


==13-limit==  
===== 19-limit =====
Commas: 81/80, 99/98, 121/120, 66/65
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~9/7 = 425.550
Comma list: 45/44, 56/55, 78/77, 81/80, 96/95, 120/119
Mapping generator: ~9/7


Map: [&lt;1 3 8 6 7 3|, &lt;0 -4 -16 -9 -10 2|]
Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 9 | 0 1 4 10 6 15 -5 -3 }}
EDOs: 17c, 31, 79cf, 110cef, 141cef
[[Badness]]: 0.0255


==Agora==
Optimal tunings:
Commas: 81/80, 99/98, 105/104, 121/120
* WE: ~2 = 1198.2880{{c}}, ~3/2 = 695.7123{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.6370{{c}}


POTE generator: ~9/7 = 426.276
{{Optimal ET sequence|legend=0| 12f, 19, 31e }}
Mapping generator: ~9/7


Map: [&lt;1 3 8 6 7 14|, &lt;0 -4 -16 -9 -10 -29|]
Badness (Sintel): 1.23
EDOs: 31, 45ef, 76e
Badness: 0.0245


=Cuboctahedra=  
==== Vincenzo ====
==11-limit==  
Subgroup: 2.3.5.7.11.13
[[Comma]]s: 81/80, 385/384, 1375/1372


[[POTE tuning|POTE generator]]: ~9/7 = 425.993
Comma list: 45/44, 56/55, 65/64, 81/80
Mapping generator: ~9/7


Map: [&lt;1 3 8 6 -4|, &lt;0 -4 -16 -9 21|]
Mapping: {{mapping| 1 0 -4 -13 -6 10 | 0 1 4 10 6 -4 }}
EDOs: [[14edo|14]], [[31edo|31]], [[45edo|45]], [[200edo|200]]
[[Badness]]: 0.0568


=Liese=  
Optimal tunings:
[[Comma]]s: 81/80, 686/675
* WE: ~2 = 1202.1684{{c}}, ~3/2 = 696.3160{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.2045{{c}}


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.
{{Optimal ET sequence|legend=0| 7d, 12, 19 }}


7 and 9 limit minimax 1/4 comma
Badness (Sintel): 1.02
[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |2/3 0 11/12 0&gt;]
[[Eigenmonzo]]s: 2, 5


[[POTE tuning|POTE generator]]: ~10/7 = 632.406
===== 17-limit =====
Mapping generator: ~10/7
Subgroup: 2.3.5.7.11.13.17


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: 45/44, 52/51, 56/55, 65/64, 81/80


Map: [&lt;1 0 -4 -3|, &lt;0 3 12 11|]
Mapping: {{mapping| 1 0 -4 -13 -6 10 12 | 0 1 4 10 6 -4 -5 }}
[[Generator]]s: 2, 10/7
EDOs: [[17edo|17]], [[19edo|19]], [[55edo|55]], [[74edo|74]]
[[Badness]]: 0.0467


==Liesel==
Optimal tunings:
Commas: 56/55, 81/80, 540/539
* WE: ~2 = 1200.5137{{c}}, ~3/2 = 696.1561{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.8771{{c}}


POTE generator: ~10/7 = 633.073
{{Optimal ET sequence|legend=0| 12, 19 }}
Mapping generator: ~10/7


Map: [&lt;1 0 -4 -3 4|, &lt;0 3 12 11 -1|]
Badness (Sintel): 1.30
EDOs: 17c, 19, 36, 91ce
Badness: 0.0407


==13-limit==  
===== 19-limit =====
Liesel is a very natural 13-limit tuning, given the generator is so near 13/9.
Subgroup: 2.3.5.7.11.13.17.19


Commas: 56/55, 78/77, 81/80, 91/90
Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80


POTE generator: ~10/7 = ~13/9 = 633.042
Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 | 0 1 4 10 6 -4 -5 -3 }}
Mapping generator: ~10/7


Map: [&lt;1 0 -4 -3 4 0|, &lt;0 3 12 11 -1 7|]
Optimal tunings:  
EDOs: 17c, 19, 36, 91cef
* WE: ~2 = 1199.8261{{c}}, ~3/2 = 696.0298{{c}}
Badness: 0.0273
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1262{{c}}


==Elisa==
{{Optimal ET sequence|legend=0| 12, 19 }}
Commas: 77/75, 81/80, 99/98


POTE generator: ~10/7 = 633.061
Badness (Sintel): 1.36
Mapping generator: ~10/7


Map: [&lt;1 0 -4 -3 -5|, &lt;0 3 12 11 16|]
=== Bimeantone ===
EDOs: 19e, 36e
11/8 is mapped to half octave minus the [[128/125|meantone diesis]].  
Badness: 0.0416


==Lisa==
Subgroup: 2.3.5.7.11
Commas: 45/44, 81/80, 343/330


POTE generator: ~10/7 = 631.370
Comma list: 81/80, 126/125, 245/242
Mapping generator: ~10/7


Map: [&lt;1 0 -4 -3 -6|, &lt;0 3 12 11 18|]
Mapping: {{mapping| 2 0 -8 -26 -31 | 0 1 4 10 12 }}
EDOs: 19
Badness: 0.0548


==13-limit==
: mapping generators: ~63/44, ~3
Commas: 45/44, 81/80, 91/88, 147/143


POTE generator: ~10/7 = 631.221
Optimal tunings:  
Mapping generator: ~10/7
* 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 0 -4 -3 -6 0|, &lt;0 3 12 11 18 7|]
{{Optimal ET sequence|legend=0| 12, 26de, 38d, 50 }}
EDOs: 19
Badness: 0.0361


=Jerome=
Badness (Sintel): 1.26
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
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


POTE generator: ~54/49 = 139.343
Comma list: 81/80, 105/104, 126/125, 245/242
Mapping generator: ~54/49


Map: [&lt;1 1 0 2|, &lt;0 5 20 7|]
Mapping: {{mapping| 2 0 -8 -26 -31 -40 | 0 1 4 10 12 15 }}
Wedgie: &lt;&lt;5 30 7 20 -3 -40||
EDOs: 8, 9, 17, 26, 43, 112
Badness: 0.1087


==11-limit==
Optimal tunings:
Commas: 81/80, 99/98, 864/847
* WE: ~55/39 = 600.8309{{c}}, ~3/2 = 696.8000{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.0066{{c}}


POTE generator: ~12/11 = 139.428
{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}
Mapping generator: ~12/11


Map: [&lt;1 1 0 2 3|, &lt;0 5 20 7 4|]
Badness (Sintel): 1.19
EDOs: 8, 9, 17, 26, 43, 241
Badness: 0.0479


==13-limit==  
==== 17-limit ====
Commas: 77/78, 81/80, 99/98, 144/143
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~13/12 = 139.387
Comma list: 81/80, 105/104, 126/125, 189/187, 221/220
Mapping generator: ~12/11


Map: [&lt;1 1 0 2 3 3|, &lt;0 5 20 7 4 6|]
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 | 0 1 4 10 12 15 1 }}
EDOs: 8, 9, 17, 26, 43, 155, 198
Badness: 0.0293


==17-limit==  
Optimal tunings:
Commas: 78/77, 81/80, 99/98, 144/143, 189/187
* WE: ~17/12 = 600.9234{{c}}, ~3/2 = 696.8536{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.9317{{c}}


POTE generator: ~13/12 = 139.362
{{Optimal ET sequence|legend=0| 12f, 38df, 50 }}
Mapping generator: ~12/11


Map: [&lt;1 1 0 2 3 3 2|, &lt;0 5 20 7 4 6 18|]
Badness (Sintel): 1.15
EDOs: 8, 9, 17, 26, 43, 155
Badness: 0.0209


=Meanmag=  
==== 19-limit ====
Commas: 81/80, 3125/3072
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~8/7 = 238.396
Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220
Mapping generator: ~7


Map: [&lt;19 30 44 0|, &lt;0 0 0 1|]
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 -1 | 0 1 4 10 12 15 1 3 }}
Wedgie: &lt;&lt;0 0 19 0 30 44||
EDOs: 19, 57, 76, 171bcd
Badness: 0.0770


=Undevigintone=  
Optimal tunings:
Commas: 49/48, 81/80, 126/125
* WE: ~17/12 = 600.9845{{c}}, ~3/2 = 696.8939{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.8947{{c}}


POTE generator: ~11/8 = 538.047
{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}
Mapping generator: ~11


Map: [&lt;19 30 44 53 0|, &lt;0 0 0 0 1|]
Badness (Sintel): 1.08
EDOs: 19, 38d
Badness: 0.0364


==13-limit==  
=== Trimean ===
Commas: 49/48, 65/64, 81/80, 126/125
{{See also| No-sevens subgroup temperaments #Superpine }}


POTE generator: ~11/8 = 537.061
Subgroup: 2.3.5.7.11


Map: [&lt;19 30 44 53 0 70|, &lt;0 0 0 0 1 0|]
Comma list: 81/80, 126/125, 1344/1331
EDOs: 19, 38d
 
Badness: 0.0229</pre></div>
Mapping: {{mapping| 1 2 4 7 5 | 0 -3 -12 -30 -11 }}
<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:166:&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:166 --&gt;&lt;!-- ws:start:WikiTextTocRule:167: --&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;
: mapping generators: ~2, ~11/10
&lt;!-- ws:end:WikiTextTocRule:167 --&gt;&lt;!-- ws:start:WikiTextTocRule:168: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Septimal meantone"&gt;Septimal meantone&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:168 --&gt;&lt;!-- ws:start:WikiTextTocRule:169: --&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;
Optimal tunings:
&lt;!-- ws:end:WikiTextTocRule:169 --&gt;&lt;!-- ws:start:WikiTextTocRule:170: --&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;
* WE: ~2 = 1200.7155{{c}}, ~11/10 = 167.9055{{c}}
&lt;!-- ws:end:WikiTextTocRule:170 --&gt;&lt;!-- ws:start:WikiTextTocRule:171: --&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;
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7749{{c}}
&lt;!-- ws:end:WikiTextTocRule:171 --&gt;&lt;!-- ws:start:WikiTextTocRule:172: --&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;
 
&lt;!-- ws:end:WikiTextTocRule:172 --&gt;&lt;!-- ws:start:WikiTextTocRule:173: --&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;
{{Optimal ET sequence|legend=0| 7d, 36d, 43, 50, 93 }}
&lt;!-- ws:end:WikiTextTocRule:173 --&gt;&lt;!-- ws:start
 
Badness (Sintel): 1.68
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 81/80, 126/125, 144/143, 364/363
 
Mapping: {{mapping| 1 2 4 7 5 3 | 0 -3 -12 -30 -11 5 }}
 
Optimal tunings:
* WE: ~2 = 1200.6104{{c}}, ~11/10 = 167.8749{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7728{{c}}
 
{{Optimal ET sequence|legend=0| 7d, 43, 50, 93 }}
 
Badness (Sintel): 1.46
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 81/80, 126/125, 144/143, 189/187, 221/220
 
Mapping: {{mapping| 1 2 4 7 5 3 8 | 0 -3 -12 -30 -11 5 -28 }}
 
Optimal tunings:
* WE: ~2 = 1200.6144{{c}}, ~11/10 = 167.8716{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7682{{c}}
 
{{Optimal ET sequence|legend=0| 7dg, 43, 50, 93 }}
 
Badness (Sintel): 1.28
 
=== Migration ===
See [[Rastmic clan #Migration|Rastmic clan]].
 
== Flattone ==
{{Main| Flattone }}
 
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]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 81/80, 525/512
 
{{Mapping|legend=1| 1 0 -4 17 | 0 1 4 -9 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.6308{{c}}, ~3/2 = 695.8782{{c}}
: [[error map]]: {{val| +3.631 -2.446 -2.801 -2.684 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.7334{{c}}
: error map: {{val| 0.000 -8.222 -11.380 -12.426 }}
 
[[Minimax tuning]]:
* [[7-odd-limit]]: ~3/2 = {{monzo| 8/13 0 1/13 -1/13 }}
: [[projection map]]: [{{monzo| 1 0 0 0 }}, {{monzo| 21/13 0 1/13 -1/13 }}, {{monzo| 32/13 0 4/13 -4/13 }}, {{monzo| 32/13 0 -9/13 9/13 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
* [[9-odd-limit]]: ~3/2 = {{monzo| 6/11 2/11 0 -1/11 }}
: [[projection map]]: [{{monzo| 1 0 0 0 }}, {{monzo| 17/11 2/11 0 -1/11 }}, {{monzo| 24/11 8/11 0 -4/11 }}, {{monzo| 34/11 -18/11 0 9/11 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7
 
[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [692.353, 701.955]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
 
[[Algebraic generator]]: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.
 
{{Optimal ET sequence|legend=1| 7, 19, 26, 45 }}
 
[[Badness]] (Sintel): 0.976
 
=== 11-limit ===
This can also be considered a no-sevens temperament: [[#Hypnotone|hypnotone]].
 
Subgroup: 2.3.5.7.11
 
Comma list: 45/44, 81/80, 385/384
 
Mapping: {{mapping| 1 0 -4 17 -6 | 0 1 4 -9 6 }}
 
Optimal tuning:
* WE: ~2 = 1202.3247{{c}}, ~3/2 = 694.4688{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.1467{{c}}
 
Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]
 
{{Optimal ET sequence|legend=0| 7, 19, 26, 45, 71bc, 116bcde }}
 
Badness (Sintel): 1.12
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 45/44, 65/64, 78/77, 81/
Retrieved from "https://en.xen.wiki/w/Meantone_family"