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Meantone family: Difference between revisions

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The [[5-limit]] parent [[comma]] of the '''meantone family''' is the Didymus or [[Wikipedia: syntonic comma|syntonic comma]], [[81/80]]. This is the one they all temper out. The period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.
{{Technical data page}}
The '''meantone family''' is the family of [[rank-2 temperament]]s that [[tempering out|temper out]] the syntonic comma, [[81/80]], and thus can all be seen as [[extension]]s of [[meantone]].  


= Meantone (12&19, 2.3.5) =
== Meantone ==
{{main| Meantone }}
{{Main| Meantone }}


Period: 1\1
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]].


Optimal ([[POTE]]) generator: ~3/2 = 696.239
[[Subgroup]]: 2.3.5


EDO generators: [[12edo|7\12]], [[19edo|11\19]], [[31edo|18\31]], [[43edo|25\43]], [[50edo|29\50]]
[[Comma list]]: 81/80


Scales (Scala files): [[Meantone5]], [[Meantone7]], [[Meantone12]]
{{Mapping|legend=1| 1 0 -4 | 0 1 4 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
: mapping generators: ~2, ~3
<div style="line-height:1.6;">Interval table (7-note MOS, 2.3.5.7 POTE tuning)</div>
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
|+
|-
! #
! Cents<ref>octave-reduced</ref>
! class="unsortable"| Approximate ratios<ref>2.3.5, odd limit ≤ 27</ref>
|-
| 0
| 0.00
| 1/1
|-
| 1
| 696.2
| 3/2
|-
| 2
| 192.5
| 9/8, 10/9
|-
| 3
| 888.7
| 5/3
|-
| 4
| 385.0
| 5/4
|-
| 5
| 1081.2
| 15/8
|-
| 6
| 577.4
| 25/18
|}
<references/></div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 81/80
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1201.3906{{c}}, ~3/2 = 697.0455{{c}}
: [[error map]]: {{val| +1.391 -3.519 +1.868 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.6512{{c}}
: error map: {{val| 0.000 -5.304 +0.291 }}


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


Mapping generators: ~2, ~3
[[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.)


[[Wedgie]]: {{wedgie| 1 4 4 }}
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b }}


[[Tuning ranges]]:  
[[Badness]] (Sintel): 0.173


* valid range: [685.714, 720.000] (7 to 5)
=== Overview to extensions ===
* nice range: [694.786, 701.955] (1/3 comma to Pythagorean)
The second comma of the normal comma list defines which [[7-limit]] family member we are looking at.
* strict range: [694.786, 701.955]
* 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.


{{EDOs|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb }}
==== 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.


[[Badness]]: 0.00736
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.  
</div></div>


== Seven-limit extensions ==
Another way to extend meantone to higher limits involves decomposing the meantone comma into products of smaller commas, or expressing some other comma of interest in terms of the ratio between the meantone comma and another comma. However, this often results in [[weak extension]]s. Another opportunity given by the meantone fifth being flat is that the most obvious ways of dividing it into ''n'' parts leave the part closer to just than usual, because we can allow – and indeed want – more flatwards tempering on the fifth, so may be recommended for this reason.  
The [[7-limit]] [[extension]]s of meantone are:
* Septimal meantone, with normal comma list [{{Monzo| -4 4 -1 }}, [[Harrison's comma|{{Monzo| -13 10 0 -1 }}]]],  
* Flattone, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -17 9 0 1 }}],
* Dominant, with normal list [{{Monzo| -4 4 -1 }}, [[64/63|{{Monzo| 6 -2 0 -1 }}]]],  
* Sharptone, with normal list [{{Monzo| -4 4 -1 }}, [[28/27|{{Monzo| 2 -3 0 1 }}]]],
* Injera, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -7 8 0 -2 }}],
* Mohajira, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -23 11 0 2 }}],
* Godzilla, with normal list [{{Monzo| -4 4 -1 }}, [[49/48|{{Monzo| -4 -1 0 2 }}]]],
* Mothra, with normal list [{{Monzo| -4 4 -1 }}, [[1029/1024|{{Monzo| -10 1 0 3 }}]]],
* Squares, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -3 9 0 -4 }}], and
* Liese, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -9 11 0 -3 }}].


= Septimal meantone =
==== Splitting the meantone fifth into two (243/242) ====
<span style="display: block; text-align: right;">[[:de:septimal-mitteltönig|Deutsch]]</span>
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.


{{main| Meantone }}
==== Splitting the meantone fifth into three (1029/1024) ====
{{see also| Wikipedia: Septimal meantone temperament }}
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.


The [[7/4]] of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are [[7/6]], C-D#, the augmented second, [[7/5]], C-F#, the tritone, and [[21/16]], C-E#, the augmented third. Septimal meantone also tempers out the common 7-limit comma [[225/224]] and is in fact can be defined as the 7-limit temperament that tempers out 81/80 and 225/224.
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]]).


Period: 1\1
==== 31edo as splitting the fifth into two, three and nine ====
[[31edo]] is unique as combining all aforementioned tempering strategies into one elegant [[11-limit]] meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate [[5/4]] and [[7/4]] and an even more accurate [[35/32]]. A tempering strategy not mentioned is splitting a flattened [[3/2]] into nine sharpened [[25/24]]'s, resulting in the 5-limit version of [[valentine]] so that 31edo is the unique tuning that combines them. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle without tempering out [[225/224]], which interestingly, though a rank-2 temperament, only has 31edo as a [[patent val]] tuning (corresponding to also tempering out 225/224).


Optimal ([[POTE]]) generator: 696.495
Temperaments discussed elsewhere include
* ''[[Plutus]]'' (+15/14) → [[Very low accuracy temperaments #Plutus|Very low accuracy temperaments]]
* [[Godzilla]] (+49/48) → [[Semaphoresmic clan #Godzilla|Semaphoresmic clan]]
* [[Mothra]] (+1029/1024) → [[Gamelismic clan #Mothra|Gamelismic clan]]
* ''[[Mohaha]]'' (+121/120) → [[Rastmic clan #Mohaha|Rastmic clan]]


EDO generators: [[12edo|7\12]], [[19edo|11\19]], [[31edo|18\31]], [[43edo|25\43]], [[50edo|29\50]]
The rest are considered below.


Scales (Scala files): [[Meantone5]], [[Meantone7]], [[Meantone12]]
== Septimal meantone ==
<div style="float:right">[[:de:septimal-mitteltönig|Deutsch]]</div>
{{Main| Meantone #Septimal meantone}}
{{Wikipedia| Septimal meantone temperament }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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.  
<div style="line-height:1.6;">Interval table (12-note MOS, 2.3.5.7 POTE tuning)</div>
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
|+
|-
! #
! Cents<ref>octave-reduced</ref>
! class="unsortable"| Approximate ratios<ref>2.3.5.7, odd limit ≤ 27</ref>
|-
| 0
| 0.00
| 1/1
|-
| 1
| 696.5
| 3/2
|-
| 2
| 193.0
| 9/8, 10/9
|-
| 3
| 889.5
| 5/3
|-
| 4
| 386.0
| 5/4
|-
| 5
| 1082.5
| 15/8, 28/15
|-
| 6
| 579.0
| 7/5
|-
| 7
| 75.5
| 21/20, 25/24, 28/27
|-
| 8
| 772.0
| 14/9, 25/16
|-
| 9
| 268.5
| 7/6
|-
| 10
| 965.0
| 7/4
|-
| 11
| 461.4
| 21/16
|}
<references/></div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


[[Comma]] list: 81/80, 126/125
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val|1 0 -4 -13}}, {{val|0 1 4 10}}]
[[Comma list]]: 81/80, 126/125


Mapping generators: ~2, ~3
{{Mapping|legend=1| 1 0 -4 -13 | 0 1 4 10 }}


[[Wedgie]]: {{wedgie| 1 4 10 4 13 12 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2358{{c}}, ~3/2 = 697.2122{{c}}
: [[error map]]: {{val| +1.236 -3.507 +2.535 -0.412 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.6562{{c}}
: error map: {{val| 0.000 -5.299 +0.311 -2.264 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* 7- and [[9-odd-limit]]
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma)
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| -3 0 5/2 0 }}]
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | -3 0 5/2 0 }}
: [[Eigenmonzo]]s: 2, 5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* valid range: [694.737, 700.000] (19 to 12)
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
* nice range: [694.786, 701.955]
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 701.955] (1/3-comma to Pyth.)
* strict range: [694.786, 700.000]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
 
[[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.
 
{{Optimal ET sequence|legend=1| 12, 19, 31, 81, 112b, 143b }}
 
[[Badness]] (Sintel): 0.347
 
=== Undecimal meantone (huygens) ===
{{Redirect|Huygens|the Dutch mathematician, physicist and astronomer|Wikipedia: Christiaan Huygens}}
{{See also| Huygens vs meanpop }}
 
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.
 
Subgroup: 2.3.5.7.11
 
Comma list: 81/80, 99/98, 126/125


Algebraic generator: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, which comes to 503.4257 cents. The recurrence converges quickly.
Mapping: {{mapping| 1 0 -4 -13 -25 | 0 1 4 10 18 }}


{{EDOs|legend=1| 12, 19, 31, 81, 112b, 143b }}
Optimal tunings:
* WE: ~2 = 1200.7636{{c}}, ~3/2 = 697.4122{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.0315{{c}}


[[Badness]]: 0.0137
Minimax tuning:
</div></div>
* 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


== Bimeantone ==
Tuning ranges:
11/8 is mapped to half octave minus the [[meantone diesis]].
* 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.)


Commas: 81/80, 126/125, 245/242
Algebraic generator: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.


[[POTE generator]]: ~3/2 = 696.016
{{Optimal ET sequence|legend=0| 12, 19e, 31, 105, 136b }}


Map: [{{val|2 0 -8 -26 -31}}, {{val|0 1 4 10 12}}]
Badness (Sintel): 0.563


{{EDOs|legend=1| 12, 26de, 38d, 50 }}
; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3 ''Twinkle canon – 74 edo''] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]


Badness: 0.0381
==== 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]].  


=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Commas: 81/80, 105/104, 126/125, 245/242
 
Comma list: 81/80, 99/98, 126/125, 144/143
 
Mapping: {{mapping| 1 0 -4 -13 -25 29 | 0 1 4 10 18 -16 }}
 
Optimal tunings:
* WE: ~2 = 1199.9389{{c}}, ~3/2 = 697.2282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.2627{{c}}
 
Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 8/13 0 0 1/26 0 -1/26 }}
: eigenmonzo basis (unchanged-interval basis): 2.13/7
 
Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~3/2 = [696.774, 697.674] (18\31 to 25\43)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
 
{{Optimal ET sequence|legend=0| 12, 31, 43, 74 }}
 
Badness (Sintel): 1.07
 
===== 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.
 
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 81/80, 99/98, 120/119, 126/125, 144/143
 
Mapping: {{mapping| 1 0 -4 -13 -25 29 12 | 0 1 4 10 18 -16 -5 }}
 
Optimal tunings:
* WE: ~2 = 1199.5811{{c}}, ~3/2 = 697.0918{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3303{{c}}
 
{{Optimal ET sequence|legend=0| 12, 31, 43, 74g }}
 
Badness (Sintel): 1.06
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 81/80, 96/95, 99/98, 120/119, 126/125, 144/143
 
Mapping: {{mapping| 1 0 -4 -13 -25 29 12 9 | 0 1 4 10 18 -16 -5 -3 }}
 
Optimal tunings:
* WE: ~2 = 1199.2931{{c}}, ~3/2 = 696.9690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3736{{c}}
 
{{Optimal ET sequence|legend=0| 12, 31, 43, 74gh }}
 
Badness (Sintel): 1.07
 
==== 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.
 
This extension used to be known as ''tridecimal meantone'', but was decanonicalized in 2025.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 66/65, 81/80, 99/98, 105/104
 
Mapping: {{mapping| 1 0 -4 -13 -25 -20 | 0 1 4 10 18 15 }}
 
Optimal tunings:
* WE: ~2 = 1200.8149{{c}}, ~3/2 = 697.1155{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.7085{{c}}
 
Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}
: unchanged-interval (eigenmonzo) basis: 2.11/9
 
{{Optimal ET sequence|legend=0| 12f, 19e, 31 }}
 
Badness (Sintel): 0.746
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 66/65, 81/80, 99/98, 105/104, 120/119
 
Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 | 0 1 4 10 18 15 -5 }}
 
Optimal tunings:
* WE: ~2 = 1199.5548{{c}}, ~3/2 = 696.7449{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.9823{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 31 }}
 
Badness (Sintel): 1.02
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 66/65, 81/80, 96/95, 99/98, 105/104, 120/119
 
Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 9 | 0 1 4 10 18 15 -5 -3 }}
 
Optimal tunings:
* WE: ~2 = 1199.0408{{c}}, ~3/2 = 696.5824{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.1061{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 31 }}
 
Badness (Sintel): 1.10
 
==== Meridetone ====
Meridetone maps the 13/8 to the quadruple-augmented fourth (C–F𝄪𝄪). 43edo can be recommended as a tuning since it is the only 13-odd-limit diamond monotone tuning.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 78/77, 81/80, 99/98, 126/125
 
Mapping: {{mapping| 1 0 -4 -13 -25 -39 | 0 1 4 10 18 27 }}
 
Optimal tunings:
* WE: ~2 = 1199.9122{{c}}, ~3/2 = 697.4779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.5241{{c}}
 
Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/25 -2/25 0 0 0 1/25 }}
: unchanged-interval (eigenmonzo) basis: 2.13/9
 
{{Optimal ET sequence|legend=0| 12f, 31f, 43 }}
 
Badness (Sintel): 1.09
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 78/77, 81/80, 99/98, 120/119, 126/125
 
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 | 0 1 4 10 18 27 -5 }}
 
Optimal tunings:
* WE: ~2 = 1199.3793{{c}}, ~3/2 = 697.2833{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6222{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 43 }}
 
Badness (Sintel): 1.22
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


[[POTE generator]]: ~3/2 = 695.836
Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125


Map: [&lt;2 0 -8 -26 -31 -40|, &lt;0 1 4 10 12 15|]
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 9 | 0 1 4 10 18 27 -5 -3 }}


{{EDOs|legend=1| 12f, 26deff, 38df, 50 }}
Optimal tunings:
* WE: ~2 = 1199.0260{{c}}, ~3/2 = 697.1486{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6887{{c}}


Badness: 0.0288
{{Optimal ET sequence|legend=0| 12f, 43 }}


== Unidecimal meantone aka Huygens ==
Badness (Sintel): 1.25
{{see also| Meantone vs meanpop }}


[[Comma]]s: 81/80, 126/125, 99/98
==== Hemimeantone ====
Subgroup: 2.3.5.7.11.13


[[11-odd-limit|11-limit]] minimax
Comma list: 81/80, 99/98, 126/125, 169/168


[{{Monzo| 1 0 0 0 0 }}, {{Monzo| 25/16 -1/8 0 0 1/16 }}, {{Monzo| 9/4 -1/2 0 0 1/4 }},
Mapping: {{mapping| 1 0 -4 -13 -25 -5 | 0 2 8 20 36 11 }}
{{Monzo| 21/8 -5/4 0 0 5/8 }}, {{Monzo| 25/8 -9/4 0 0 9/8 }}<nowiki>]</nowiki>


[[Eigenmonzo]]s: 2, 11/9
: mapping generators: ~2, ~26/15


valid range: [696.774, 700.000] (31 to 12)
Optimal tunings:  
* WE: ~2 = 1201.0387{{c}}, ~26/15 = 949.2863{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5065{{c}}


nice range: [691.202, 701.955]
{{Optimal ET sequence|legend=0| 19e, 43, 62 }}


strict range: [696.774, 700.000]
Badness (Sintel): 1.30


[[POTE generator]]: ~3/2 = 696.967
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


[[Algebraic generator]]: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
Comma list: 81/80, 99/98, 126/125, 169/168, 221/220


[[Map]]: [&lt;1 0 -4 -13 -25|, &lt;0 1 4 10 18|]
Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 | 0 2 8 20 36 11 33 }}


Mapping generator: ~3
Optimal tunings:  
* WE: ~2 = 1201.0270{{c}}, ~26/15 = 949.2892{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5169{{c}}


{{EDOs|legend=1| 12, 19e, 31, 105, 136b, 167be, 198be }}
{{Optimal ET sequence|legend=0| 19eg, 43, 62 }}


[[Badness]]: 0.0170
Badness (Sintel): 1.19


* [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]
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


=== Tridecimal meantone ===
Comma list: 81/80, 99/98, 126/125, 153/152, 169/168, 221/220
Commas: 66/65, 81/80, 99/98, 105/104


[[POTE generator]]: ~3/2 = 696.642
Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 -25 | 0 2 8 20 36 11 33 37 }}


Mapping generator: ~3
Optimal tunings:  
* WE: ~2 = 1201.0339{{c}}, ~19/11 = 949.2902{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 948.5111{{c}}


Map: [&lt;1 0 -4 -13 -25 -20|, &lt;0 1 4 10 18 15|]
{{Optimal ET sequence|legend=0| 19egh, 43, 62 }}


{{EDOs|legend=1| 12f, 19e, 31 }}
Badness (Sintel): 1.15


Badness: 0.0180
==== Semimeantone ====
Subgroup: 2.3.5.7.11.13


=== Grosstone ===
Comma list: 81/80, 99/98, 126/125, 847/845
Commas: 81/80, 99/98, 126/125, 144/143


valid range: [696.774, 700.000] (31 to 12)
Mapping: {{mapping| 2 0 -8 -26 -50 -59 | 0 1 4 10 18 21 }}


nice range: [691.202, 701.955]
: mapping generators: ~55/39, ~3


strict range: [696.774, 700.000]
Optimal tunings:  
* WE: ~55/39 = 600.3606{{c}}, ~3/2 = 697.4241{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 697.0545{{c}}


[[POTE generator]]: ~3/2 = 697.264
{{Optimal ET sequence|legend=0| 12f, …, 50eff, 62, 136b }}


Mapping generator: ~3
Badness (Sintel): 1.68


Map: [&lt;1 0 -4 -13 -25 29|, &lt;0 1 4 10 18 -16|]
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


{{EDOs|legend=1| 12, 19ef, 31, 43, 74 }}
Comma list: 81/80, 99/98, 126/125, 221/220, 289/288


Badness: 0.0259
Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 | 0 1 4 10 18 21 1 }}


=== Meridetone ===
Optimal tunings:
Commas: 78/77, 81/80, 99/98, 126/125
* WE: ~17/12 = 600.5426{{c}}, ~3/2 = 697.5571{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9858{{c}}


[[POTE generator]]: ~3/2 = 697.529
{{Optimal ET sequence|legend=0| 12f, 50eff, 62, 136bg }}


Mapping generator: ~3
Badness (Sintel): 1.60


Map: [&lt;1 0 -4 -13 -25 -39|, &lt;0 1 4 10 18 27|]
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


{{EDOs|legend=1| 12f, 31f, 43 }}
Comma list: 81/80, 99/98, 126/125, 153/152, 209/208, 221/220


Badness: 0.0264
Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 -1 | 0 1 4 10 18 21 1 3 }}


=== Hemimeantone ===
Optimal tunings:
Commas: 81/80, 99/98, 126/125, 169/168
* WE: ~17/12 = 600.5959{{c}}, ~3/2 = 697.5985{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9638{{c}}


[[POTE generator]]: ~52/45 = 250.304
{{Optimal ET sequence|legend=0| 12f, 50eff, 62 }}


Mapping generator: ~26/15
Badness (Sintel): 1.47


Map: [&lt;1 0 -4 -13 -25 -5|, &lt;0 2 8 20 36 11|]
=== Meanpop ===
{{See also| Huygens vs meanpop }}


{{EDOs|legend=1| 19e, 43, 62, 167bef }}
Meanpop<ref name="meantone & meanpop 2003"/><ref name="meantone & meanpop 2004"/> maps the 11/8 to the double-diminished fifth (C–G𝄫), and tridecimal meanpop maps the 13/8 to the double-augmented fifth (C–G𝄪), tempering out 144/143 like in grosstone. Note also 11/10 is a double-diminished third; 12/11~13/12, double-augmented unison; and 14/13, minor second.


Badness: 0.0314
Subgroup: 2.3.5.7.11


== Meanpop ==
Comma list: 81/80, 126/125, 385/384
{{see also| Meantone vs meanpop }}


[[Comma]]s: 81/80, 126/125, 385/384
Mapping: {{mapping| 1 0 -4 -13 24 | 0 1 4 10 -13 }}


[[11-odd-limit|11-limit]] minimax 1/4 comma
: mapping generator: ~2, ~3


[{{Monzo| 1 0 0 0 0 }}, {{Monzo| 1 0 1/4 0 0 }}, {{Monzo| 0 0 1 0 0 }},
Optimal tunings:
{{Monzo| -3 0 5/2 0 0 }}, {{Monzo| 11 0 -13/4 0 0 }}]
* WE: ~2 = 1201.3464{{c}}, ~3/2 = 697.2159{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4509{{c}}


[[Eigenmonzo]]s: 2, 5
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


valid range: [694.737, 696.774] (19 to 31)
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.)


nice range: [691.202, 701.955]
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.


strict range: [694.737, 696.774]
{{Optimal ET sequence|legend=0| 12e, 19, 31, 81, 112b }}


[[POTE generator]]: 696.434
Badness (Sintel): 0.712


Mapping generator: ~3
; 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]


[[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.
==== Tridecimal meanpop ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 -4 -13 24|, &lt;0 1 4 10 -13|]
Comma list: 81/80, 105/104, 126/125, 144/143


{{EDOs|legend=1| 12e, 19, 31, 81 }}
Mapping: {{mapping| 1 0 -4 -13 24 -20 | 0 1 4 10 -13 15 }}


[[Badness]]: 0.0215
Optimal tunings:  
* WE: ~2 = 1201.0765{{c}}, ~3/2 = 696.8361{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2347{{c}}


* [http://soonlabel.com/xenharmonic/archives/607 Scott Joplin's "The Entertainer" tuned into meanpop]{{Dead link}}
Minimax tuning:
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3 Twinkle canon – 50 edo] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]
* 13- and 15-odd-limit: ~3/2 = {{monzo| 4/7 0 0 0 -1/28 1/28 }}
: unchanged-interval (eigenmonzo) basis: 2.13/11


=== 13-limit Meanpop ===
Tuning ranges:
Commas: 81/80, 105/104, 126/125, 144/143
* 13- and 15-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)


valid range: [694.737, 696.774] (19 to 31)
{{Optimal ET sequence|legend=0| 19, 31, 50, 81 }}


nice range: [691.202, 701.955]
Badness (Sintel): 0.863


strict range: [694.737, 696.774]
===== Meanpoppic =====
Subgroup: 2.3.5.7.11.13.17


[[POTE generator]]: ~3/2 = 696.211
Comma list: 81/80, 105/104, 126/125, 144/143, 273/272


Mapping generator: ~3
Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 | 0 1 4 10 -13 15 26 }}


Map: [&lt;1 0 -4 -13 24 -20|, &lt;0 1 4 10 -13 15|]
Optimal tunings:  
* WE: ~2 = 1201.0727{{c}}, ~3/2 = 696.8168{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2195{{c}}


{{EDOs|legend=1| 12ef, 19, 31, 50, 81, 131bd, 212bbddf }}
{{Optimal ET sequence|legend=0| 19g, 31, 50, 81, 131bd }}


Badness: 0.0209
Badness (Sintel): 1.02


=== Meanplop ===
====== 19-limit ======
Commas: 65/64, 78/77, 81/80, 91/90
Subgroup: 2.3.5.7.11.13.17.19


[[POTE generator]]: ~3/2 = 696.202
Comma list: 81/80, 105/104, 126/125, 144/143, 153/152, 273/272


Mapping generator: ~3
Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 -40 | 0 1 4 10 -13 15 26 28 }}


Map: [&lt;1 0 -4 -13 24 10|, &lt;0 1 4 10 -13 -4|]
Optimal tunings:  
* WE: ~2 = 1201.0719{{c}}, ~3/2 = 696.8101{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2137{{c}}


{{EDOs|legend=1| 12e, 19, 31f, 50ff, 81fff }}
{{Optimal ET sequence|legend=0| 19gh, 31, 50, 81 }}


Badness: 0.0277
Badness (Sintel): 1.08


== Meanenneadecal ==
===== Meanpoid =====
Commas: 45/44, 56/55, 81/80
Subgroup: 2.3.5.7.11.13.17


[[POTE generator]]: ~3/2 = 696.250
Comma list: 81/80, 105/104, 120/119, 126/125, 144/143


Mapping generator: ~3
Mapping: {{mapping| 1 0 -4 -13 24 -20 12 | 0 1 4 10 -13 15 -5 }}


Map: [&lt;1 0 -4 -13 -6|, &lt;0 1 4 10 6|]
Optimal tunings:  
* WE: ~2 = 1200.2768{{c}}, ~3/2 = 696.5683{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4114{{c}}


{{EDOs|legend=1| 7d, 12, 19, 31e, 50ee }}
{{Optimal ET sequence|legend=0| 19, 31 }}


Badness: 0.0214
Badness (Sintel): 1.17


=== 13-limit ===
====== 19-limit ======
Commas: 45/44, 56/55, 78/77, 81/80
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 77/76, 81/80, 96/95, 105/104, 120/119, 126/125


[[POTE generator]]: ~3/2 = 696.146
Mapping: {{mapping| 1 0 -4 -13 24 -20 12 9 | 0 1 4 10 -13 15 -5 -3 }}


Mapping generator: ~3
Optimal tunings:  
* WE: ~2 = 1199.7905{{c}}, ~3/2 = 696.3779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4973{{c}}


Map: [&lt;1 0 -4 -13 -6 -20|, &lt;0 1 4 10 6 15|]
{{Optimal ET sequence|legend=0| 19, 31 }}


{{EDOs|legend=1| 12f, 19, 31e, 50ee }}
Badness (Sintel): 1.25


Badness: 0.0212
==== Semimeanpop ====
Subgroup: 2.3.5.7.11.13


=== Vincenzo ===
Comma list: 81/80, 126/125, 385/384, 847/845
Commas: 45/44, 56/55, 65/64, 81/80


[[POTE generator]]: ~3/2 = 695.060
Mapping: {{mapping| 2 0 -8 -26 48 39 | 0 1 4 10 -13 -10 }}


Mapping generator: ~3
: mapping generators: ~55/39, ~3


Map: [&lt;1 0 -4 -13 -6 10|, &lt;0 1 4 10 6 -4|]
Optimal tunings:  
* WE: ~55/39 = 600.6704{{c}}, ~3/2 = 697.2151{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.4341{{c}}


{{EDOs|legend=1| 7d, 12, 19 }}
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112b }}


Badness: 0.0248
Badness (Sintel): 1.78


==== 17-limit ====
===== 17-limit =====
Commas: 45/44, 52/51, 56/55, 65/64, 81/80
Subgroup: 2.3.5.7.11.13.17


[[POTE generator]]: ~3/2 = 695.858
Comma list: 81/80, 126/125, 221/220, 273/272, 289/288


Map: [&lt;1 0 -4 -13 -6 10 12|, &lt;0 1 4 10 6 -4 -5|]
Mapping: {{mapping| 2 0 -8 -26 48 39 5 | 0 1 4 10 -13 -10 1 }}


{{EDOs|legend=1| 7d, 12, 19 }}
Optimal tunings:
* WE: ~17/12 = 600.7232{{c}}, ~3/2 = 697.2820{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4411{{c}}


Badness: 0.0255
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bg }}


==== 19-limit ====
Badness (Sintel): 1.45
Commas: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80


[[POTE generator]]: ~3/2 = 696.131
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;1 0 -4 -13 -6 10 12 9|, &lt;0 1 4 10 6 -4 -5 -3|]
Comma list: 81/80, 126/125, 153/152, 209/208, 221/220, 273/272


{{EDOs|legend=1| 7d, 12, 19 }}
Mapping: {{mapping| 2 0 -8 -26 48 39 5 -1 | 0 1 4 10 -13 -10 1 3 }}


Badness: 0.0223
Optimal tunings:  
* WE: ~17/12 = 600.7527{{c}}, ~3/2 = 697.3244{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4525{{c}}


==== 23-limit ====
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bgh }}
Commas: 39/38, 45/44, 52/51, 56/55, 65/64, 69/68, 81/80


[[POTE generator]]: ~3/2 = 696.044
Badness (Sintel): 1.28


Map: [&lt;1 0 -4 -13 -6 10 12 9 14|, &lt;0 1 4 10 6 -4 -5 -3 -6|]
=== Meanenneadecal ===
Meanenneadecal maps the 11/8 to the augmented fourth (C–F♯), and tridecimal meanenneadecal maps the 13/8 to the double-augmented fifth (C–G𝄪). Note also 11/10 is a major second; 12/11~14/13, minor second; and 13/12, double-augmented unison.


{{EDOs|legend=1| 7d, 12, 19 }}
Subgroup: 2.3.5.7.11


Badness: 0.0201
Comma list: 45/44, 56/55, 81/80


==== 29-limit ====
Mapping: {{mapping| 1 0 -4 -13 -6 | 0 1 4 10 6 }}
Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80


POTE generator: ~3/2 = 695.913
Optimal tunings:  
* WE: ~2 = 1199.6946{{c}}, ~3/2 = 696.0729{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2083{{c}}


Map: [&lt;1 0 -4 -13 -6 10 12 9 14 8|, &lt;0 1 4 10 6 -4 -5 -3 -6 -2|]
Tuning ranges:  
* 11-odd-limit diamond monotone: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 704.377]


{{EDOs|legend=1| 7d, 12, 19 }}
{{Optimal ET sequence|legend=0| 7d, 12, 19, 31e }}


Badness: 0.0182
Badness (Sintel): 0.708


==== 31-limit ====
==== 13-limit ====
Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80, 93/92
Subgroup: 2.3.5.7.11.13


POTE generator: ~3/2 = 695.750
Comma list: 45/44, 56/55, 78/77, 81/80


Map: [&lt;1 0 -4 -13 -6 10 12 9 14 8 16|, &lt;0 1 4 10 6 -4 -5 -3 -6 -2 -7|]
Mapping: {{mapping| 1 0 -4 -13 -6 -20 | 0 1 4 10 6 15 }}


{{EDOs|legend=1| 7d, 12, 19 }}
Optimal tunings:
* WE: ~2 = 1199.7931{{c}}, ~3/2 = 696.0258{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1241{{c}}


Badness: 0.0171
{{Optimal ET sequence|legend=0| 7df, 12f, 19, 31e }}


==== 37-limit ====
Badness (Sintel): 0.875
Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92


POTE generator: ~3/2 = 695.603
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 0 -4 -13 -6 10 12 9 14 8 16 -9|, &lt;0 1 4 10 6 -4 -5 -3 -6 -2 -7 9|]
Comma list: 45/44, 56/55, 78/77, 81/80, 120/119


{{EDOs|legend=1| 7d, 12, 19 }}
Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 | 0 1 4 10 6 15 -5 }}


Badness: 0.0161
Optimal tunings:  
* WE: ~2 = 1198.6665{{c}}, ~3/2 = 695.8010{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4998{{c}}


==== 41-limit ====
{{Optimal ET sequence|legend=0| 12f, 19, 31e }}
Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92, 124/123


POTE generator: ~3/2 = 695.696
Badness (Sintel): 1.17


Map: [&lt;1 0 -4 -13 -6 10 12 9 14 8 16 -9 18|, &lt;0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8|]
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


{{EDOs|legend=1| 7d, 12, 19 }}
Comma list: 45/44, 56/55, 78/77, 81/80, 96/95, 120/119


Badness: 0.0154
Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 9 | 0 1 4 10 6 15 -5 -3 }}


==== 43-limit ====
Optimal tunings:
Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 86/85, 93/92, 124/123
* WE: ~2 = 1198.2880{{c}}, ~3/2 = 695.7123{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.6370{{c}}


POTE generator: ~3/2 = 695.688
{{Optimal ET sequence|legend=0| 12f, 19, 31e }}


Map: [&lt;1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7|, &lt;0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1|]
Badness (Sintel): 1.23


{{EDOs|legend=1| 7d, 12, 19 }}
==== Vincenzo ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0139
Comma list: 45/44, 56/55, 65/64, 81/80


==== 47-limit ====
Mapping: {{mapping| 1 0 -4 -13 -6 10 | 0 1 4 10 6 -4 }}
Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 86/85, 93/92, 95/94, 124/123


POTE generator: ~3/2 = 695.676
Optimal tunings:  
* WE: ~2 = 1202.1684{{c}}, ~3/2 = 696.3160{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.2045{{c}}


Map: [&lt;1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7 4|, &lt;0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1 1|]
{{Optimal ET sequence|legend=0| 7d, 12, 19 }}


{{EDOs|legend=1| 7d, 12, 19 }}
Badness (Sintel): 1.02


Badness: 0.0138
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


=== Meanundec ===
Comma list: 45/44, 52/51, 56/55, 65/64, 81/80
Commas: 27/26, 40/39, 45/44, 56/55


POTE generator: ~3/2 = 697.254
Mapping: {{mapping| 1 0 -4 -13 -6 10 12 | 0 1 4 10 6 -4 -5 }}


Mapping generator: ~3
Optimal tunings:  
* WE: ~2 = 1200.5137{{c}}, ~3/2 = 696.1561{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.8771{{c}}


Map: [&lt;1 0 -4 -13 -6 -1|, &lt;0 1 4 10 6 3|]
{{Optimal ET sequence|legend=0| 12, 19 }}


{{EDOs|legend=1| 7d, 12f, 19f, 31eff }}
Badness (Sintel): 1.30


Badness: 0.0242
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


== Meanundeci ==
Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80
Commas: 33/32, 55/54, 77/75


POTE generator: ~3/2 = 694.689
Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 | 0 1 4 10 6 -4 -5 -3 }}


Mapping generator: ~3
Optimal tunings:  
* WE: ~2 = 1199.8261{{c}}, ~3/2 = 696.0298{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1262{{c}}


Map: [&lt;1 0 -4 -13 5|, &lt;0 1 4 10 -1|]
{{Optimal ET sequence|legend=0| 12, 19 }}


{{EDOs|legend=1| 7d, 12e, 19e }}
Badness (Sintel): 1.36


Badness: 0.0315
=== Bimeantone ===
11/8 is mapped to half octave minus the [[128/125|meantone diesis]].  


=== 13-limit ===
Subgroup: 2.3.5.7.11
Commas: 33/32, 55/54, 65/64, 77/75


POTE generator: ~3/2 = 694.764
Comma list: 81/80, 126/125, 245/242


Mapping generator: ~3
Mapping: {{mapping| 2 0 -8 -26 -31 | 0 1 4 10 12 }}


Map: [&lt;1 0 -4 -13 5 10|, &lt;0 1 4 10 -1 -4|]
: mapping generators: ~63/44, ~3


{{EDOs|legend=1| 7d, 12e, 19e }}
Optimal tunings:
* WE: ~63/44 = 600.7492{{c}}, ~3/2 = 696.8853{{c}}
* CWE: ~63/44 = 600.0000{{c}}, ~3/2 = 696.1908{{c}}


Badness: 0.0263
{{Optimal ET sequence|legend=0| 12, 26de, 38d, 50 }}


= Flattone =
Badness (Sintel): 1.26
In flattone, 9 generator steps of 4/3 get to the interval class for 7, meaning that [[7/4]] is a diminished seventh interval (C-Bbb). Other intervals are [[7/6]], a diminished third (C-Ebb), and [[7/5]], a doubly diminshed fifth (C-Gbb). Good tunings for flattone are [[26edo]], [[45edo]] and [[64edo]].


Period: 1\1
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Optimal ([[POTE]]) generator: ~3/2 = 693.779
Comma list: 81/80, 105/104, 126/125, 245/242


EDO generators: [[19edo|11\19]], [[26edo|15\26]], [[45edo|26\45]], [[64edo|37\64]]
Mapping: {{mapping| 2 0 -8 -26 -31 -40 | 0 1 4 10 12 15 }}


Scales (Scala files):
Optimal tunings:  
* WE: ~55/39 = 600.8309{{c}}, ~3/2 = 696.8000{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.0066{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}
<div style="line-height:1.6;">Interval table (12-note MOS, 2.3.5.7 POTE tuning)</div>
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
|+
|-
! #
! Cents<ref>octave-reduced</ref>
! class="unsortable"| Approximate ratios<ref>2.3.5.7, odd limit ≤ 27. JI readings in parentheses are outside the subgroup but are supported by the defining EDOs.</ref>
|-
| 0
| 0.00
| 1/1
|-
| 1
| 693.8
| 3/2
|-
| 2
| 187.6
| 9/8, 10/9
|-
| 3
| 881.3
| 5/3
|-
| 4
| 375.1
| 5/4, (16/13), (11/9)
|-
| 5
| 1068.9
| 15/8, (24/13), (11/6)
|-
| 6
| 562.7
| (18/13), (11/8)
|-
| 7
| 56.5
|
|-
| 8
| 750.2
| (20/13)
|-
| 9
| 244.0
| 8/7
|-
| 10
| 937.8
| 12/7
|-
| 11
| 431.6
| 9/7
|}
<references/></div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


[[Comma]] list: 81/80, 525/512
Badness (Sintel): 1.19


[[Mapping]]: [{{val|1 0 -4 17}}, {{val|0 1 4 -9}}]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Mapping generators: ~2, ~3
Comma list: 81/80, 105/104, 126/125, 189/187, 221/220


[[Wedgie]]: {{wedgie|1 4 -9 4 -17 -32}}
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 | 0 1 4 10 12 15 1 }}


[[Minimax tuning]]:  
Optimal tunings:  
* WE: ~17/12 = 600.9234{{c}}, ~3/2 = 696.8536{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.9317{{c}}


* [[7-odd-limit]]
{{Optimal ET sequence|legend=0| 12f, 38df, 50 }}
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 21/13 0 1/13 -1/13 }}, {{Monzo| 32/13 0 4/13 -4/13 }}, {{Monzo| 32/13 0 -9/13 9/13 }}]
: [[Eigenmonzo]]s: 2, 7/5


* [[9-odd-limit]]
Badness (Sintel): 1.15
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 17/11 2/11 0 -1/11 }}, {{Monzo| 24/11 8/11 0 -4/11 }}, {{Monzo| 34/11 -18/11 0 9/11 }}]
: Eigenmonzos: 2, 9/7


[[Tuning ranges]]:
==== 19-limit ====
* valid range: [692.308, 694.737] (26 to 19)
Subgroup: 2.3.5.7.11.13.17.19
* nice range: [692.353, 701.955]
* strict range: [692.353, 694.737]


Algebraic generator: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.
Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220


{{EDOs|legend=1| 7, 19, 26, 45 }}
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 -1 | 0 1 4 10 12 15 1 3 }}


[[Badness]]: 0.0386
Optimal tunings:  
</div></div>
* WE: ~17/12 = 600.9845{{c}}, ~3/2 = 696.8939{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.8947{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}
Commas: 45/44, 81/80, 385/384


valid range: [692.308, 694.737] (26 to 19)
Badness (Sintel): 1.08


nice range: [682.502, 701.955]
=== Trimean ===
{{See also| No-sevens subgroup temperaments #Superpine }}


strict range: [692.308, 694.737]
Subgroup: 2.3.5.7.11


POTE generator: ~3/2 = 693.126
Comma list: 81/80, 126/125, 1344/1331


Mapping generator: ~3
Mapping: {{mapping| 1 2 4 7 5 | 0 -3 -12 -30 -11 }}


Map: [&lt;1 0 -4 17 -6|, &lt;0 1 4 -9 6|]
: mapping generators: ~2, ~11/10


{{EDOs|legend=1| 7, 19, 26, 45, 71bc, 116bcde }}
Optimal tunings:
* WE: ~2 = 1200.7155{{c}}, ~11/10 = 167.9055{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7749{{c}}


Badness: 0.0338
{{Optimal ET sequence|legend=0| 7d, 36d, 43, 50, 93 }}


== 13-limit ==
Badness (Sintel): 1.68
Commas: 45/44, 65/64, 78/77, 81/80


valid range: [692.308, 694.737] (26 to 19)
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


nice range: [682.502, 701.955]
Comma list: 81/80, 126/125, 144/143, 364/363


strict range: [692.308, 694.737]
Mapping: {{mapping| 1 2 4 7 5 3 | 0 -3 -12 -30 -11 5 }}


POTE generator: ~3/2 = 693.058
Optimal tunings:
* WE: ~2 = 1200.6104{{c}}, ~11/10 = 167.8749{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7728{{c}}


Mapping generator: ~3
{{Optimal ET sequence|legend=0| 7d, 43, 50, 93 }}


Map: [&lt;1 0 -4 17 -6 10|, &lt;0 1 4 -9 6 -4|]
Badness (Sintel): 1.46


{{EDOs|legend=1| 7, 19, 26, 45f, 71bcf, 116bcdef }}
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0223
Comma list: 81/80, 126/125, 144/143, 189/187, 221/220


= Godzilla =
Mapping: {{mapping| 1 2 4 7 5 3 8 | 0 -3 -12 -30 -11 5 -28 }}
<span style="display: block; text-align: right;">[[:de:Semiphor,_Semaphor,_Godzilla|Deutsch]]</span>


{{main| Semaphore and Godzilla }}
Optimal tunings:
* WE: ~2 = 1200.6144{{c}}, ~11/10 = 167.8716{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7682{{c}}


Godzilla tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-quarter intervals these represent give a fourth, and so step-and-a-quarter generators generate godzilla. [[19edo]] is close to being the optimal generator tuning; hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes.
{{Optimal ET sequence|legend=0| 7dg, 43, 50, 93 }}


Period: 1\1
Badness (Sintel): 1.28


Optimal ([[POTE]]) generator: ~8/7 = 252.635
=== Migration ===
See [[Rastmic clan #Migration|Rastmic clan]].


EDO generators: [[19edo|4\19]], [[24edo|5\24]], [[43edo|9\43]], [[62edo|13\62]]
== Flattone ==
{{Main| Flattone }}


Scales (Scala files):
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]].


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Subgroup]]: 2.3.5.7
<div style="line-height:1.6;">Interval table (9-note MOS, 2.3.5.7 POTE tuning)</div>
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
|+
|-
! #
! Cents <ref>octave-reduced</ref>
! class="unsortable"| Approximate ratios<ref>2.3.5.7, odd limit ≤ 27. JI readings in parentheses are outside the subgroup but are supported by the defining EDOs.</ref>
|-
| 0
| 0.00
| 1/1
|-
| 1
| 252.6
| 7/6, 8/7, (15/13)
|-
| 2
| 505.3
| 4/3
|-
| 3
| 757.9
| 14/9, (20/13)
|-
| 4
| 1010.5
| 9/5, 16/9
|-
| 5
| 63.2
|
|-
| 6
| 315.8
| 6/5
|-
| 7
| 568.4
| 7/5, (18/13)
|-
| 8
| 821.1
| 8/5
|}
<references/></div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


[[Comma]] list: 49/48, 81/80
[[Comma list]]: 81/80, 525/512


[[Mapping]]: [{{val| 1 0 -4 2 }}, {{val| 0 2 8 1 }}]
{{Mapping|legend=1| 1 0 -4 17 | 0 1 4 -9 }}


Mapping generators: ~2, ~7/4
[[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 }}


[[Wedgie]]: {{wedgie| 2 8 1 8 -4 -20 }}
[[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]]:  
[[Tuning ranges]]:  
* valid range: [240.000, 257.143] (5 to 14c)
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* nice range: [231.174, 266.871]
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [692.353, 701.955]
* strict range: [240.000, 257.143]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]


{{EDOs|legend=1| 5, 14c, 19 }}
[[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.


[[Badness]]: 0.0267
{{Optimal ET sequence|legend=1| 7, 19, 26, 45 }}
</div></div>


== 11-limit ==
[[Badness]] (Sintel): 0.976
Commas: 45/44, 49/48, 81/80


valid range: [252.632, 257.143] (19 to 14c)
=== 11-limit ===
This can also be considered a no-sevens temperament: [[#Hypnotone|hypnotone]].


nice range: [231.174, 266.871]
Subgroup: 2.3.5.7.11


strict range: [252.632, 257.143]
Comma list: 45/44, 81/80, 385/384


POTE generator: ~8/7 = 254.027
Mapping: {{mapping| 1 0 -4 17 -6 | 0 1 4 -9 6 }}


Mapping generator: ~7/4
Optimal tuning:  
* WE: ~2 = 1202.3247{{c}}, ~3/2 = 694.4688{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.1467{{c}}


Map: [&lt;1 0 -4 2 -6|, &lt;0 2 8 1 12|]
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]


{{EDOs|legend=1| 14c, 19, 33cd, 52cd }}
{{Optimal ET sequence|legend=0| 7, 19, 26, 45, 71bc, 116bcde }}


Badness: 0.0290
Badness (Sintel): 1.12


=== 13-limit ===
==== 13-limit ====
Commas: 45/44, 49/48, 78/77, 81/80
Subgroup: 2.3.5.7.11.13
 
Comma list: 45/44, 65/64, 78/77, 81/80
 
Mapping: {{mapping| 1 0 -4 17 -6 10 | 0 1 4 -9 6 -4 }}
 
Optimal tunings:
* WE: ~2 = 1202.5156{{c}}, ~3/2 = 694.5107{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0538{{c}}
 
Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]
 
{{Optimal ET sequence|legend=0| 7, 19, 26, 45f, 71bcf, 116bcdef }}
 
Badness (Sintel): 0.920
 
=== Ptolemy ===
See [[Rastmic clan #Ptolemy|Rastmic clan]].
 
== Dominant ==
{{Main| Dominant (temperament) }}
{{See also| Archytas clan }}


valid range: 694.737 (19)
The interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh (C–Bb). The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is [[12edo]], but it also works well with the Pythagorean tuning of pure [[3/2]] fifths, and with [[29edo]], [[41edo]], or [[53edo]].


nice range: [621.581, 737.652]
Because dominant entails a near-pure perfect fifth, a small number of generators will not land on an interval close to prime 11. The canonical 11-limit extension identifies 11/8 with the diminished fifth. Domination tempers out 77/75 and identifies 11/8 with the augmented third. Domineering identifies 11/8 with the augmented fourth, which is a very inaccurate mapping; it is however, notable for having the lowest badness among the extensions. Arnold tempers out 33/32 and identifies 11/8 with the perfect fourth. None of them are nearly as good as the weak extension [[neutrominant]], splitting the fifth as well as the chromatic semitone in two like in all [[rastmic clan|rastmic]] temperaments.


strict range: 694.737
[[Subgroup]]: 2.3.5.7


POTE generator: ~8/7 = 253.603
[[Comma list]]: 36/35, 64/63


Mapping generator: ~7/4
{{Mapping|legend=1| 1 0 -4 6 | 0 1 4 -2 }}


Map: [&lt;1 0 -4 2 -6 -5|, &lt;0 2 8 1 12 11|]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1195.3384{{c}}, ~3/2 = 698.8478{{c}}
: [[error map]]: {{val| -4.662 -7.769 +9.077 +14.832 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.1125{{c}}
: error map: {{val| 0.000 -0.842 +18.136 +28.949 }}


{{EDOs|legend=1| 14cf, 19, 33cdff, 52cdf }}
[[Tuning ranges]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[diamond monotone]]: ~3/2 = [700.000, 720.000] (7\12 to 3\5)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 715.587]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]


Badness: 0.0225
{{Optimal ET sequence|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}


== Semafour ==
[[Badness]] (Sintel): 0.524
Commas: 33/32, 49/48, 55/54


POTE generator: ~8/7 = 254.042
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping generator: ~7/4
Comma list: 36/35, 56/55, 64/63


Map: [&lt;1 0 -4 2 5|, &lt;0 2 8 1 -2|]
Mapping: {{mapping| 1 0 -4 6 13 | 0 1 4 -2 -6 }}


{{EDOs|legend=1| 14c, 19e, 33cdee }}
Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 705.882] (7\12 to 10\17)
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]


Badness: 0.0285
Optimal tunings:  
* WE: ~2 = 1194.0169{{c}}, ~3/2 = 699.7473{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.2672{{c}}


== Varan ==
{{Optimal ET sequence|legend=0| 5, 12, 17c, 29cde }}
Commas: 49/48, 77/75, 81/80


POTE generator: ~8/7 = 251.079
Badness (Sintel): 0.799


Mapping generator: ~7/4
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 -4 2 -10|, &lt;0 2 8 1 17|]
Comma list: 36/35, 56/55, 64/63, 66/65


{{EDOs|legend=1| 19e, 24, 43de }}
Mapping: {{mapping| 1 0 -4 6 13 18 | 0 1 4 -2 -6 -9 }}


Badness: 0.0396
Optimal tunings:  
* WE: ~2 = 1193.8055{{c}}, ~3/2 = 700.0042{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.8254{{c}}


=== 13-limit ===
Tuning ranges:
Commas: 49/48, 66/65, 77/75, 81/80
* 13- and 15-odd-limit diamond monotone: ~3/2 = 705.882 (10\17)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
 
{{Optimal ET sequence|legend=0| 12f, 17c, 29cdef }}
 
Badness (Sintel): 0.996


POTE generator: ~8/7 = 251.165
==== Dominion ====
Subgroup: 2.3.5.7.11.13


Mapping generator: ~7/4
Comma list: 26/25, 36/35, 56/55, 64/63


Map: [&lt;1 0 -4 2 -10 -5|, &lt;0 2 8 1 17 11|]
Mapping: {{mapping| 1 0 -4 6 13 -9 | 0 1 4 -2 -6 8 }}


{{EDOs|legend=1| 19e, 24, 43de }}
Optimal tunings:
* WE: ~2 = 1195.0293{{c}}, ~3/2 = 701.9847{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7698{{c}}


Badness: 0.0257
{{Optimal ET sequence|legend=0| 5, 12, 17c }}


== Baragon ==
Badness (Sintel): 1.13
Commas: 49/48, 56/55, 81/80


POTE generator: ~8/7 = 251.173
=== Domination ===
Subgroup: 2.3.5.7.11


Mapping generator: ~7/4
Comma list: 36/35, 64/63, 77/75


Map: [&lt;1 0 -4 2 9|, &lt;0 2 8 1 -7|]
Mapping: {{mapping| 1 0 -4 6 -14 | 0 1 4 -2 11 }}


{{EDOs|legend=1| 19, 24, 43d }}
Optimal tunings:
* WE: ~2 = 1194.8645{{c}}, ~3/2 = 701.9872{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5945{{c}}


Badness: 0.0357
{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}


== Music ==
Badness (Sintel): 1.21
* [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 =
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{main|Mohajira}}
Comma list: 26/25, 36/35, 64/63, 66/65


[[Comma]]s: 81/80, 6144/6125
Mapping: {{mapping| 1 0 -4 6 -14 -9 | 0 1 4 -2 11 8 }}


Mohajira 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 = 1195.1324{{c}}, ~3/2 = 702.6343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.0791{{c}}


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.
{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}


[[7-odd-limit|7]] and [[9-odd-limit|9-limit]] minimax 1/4 comma
Badness (Sintel): 1.13


[{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 6 0 -11/8 0 }}]
=== Domineering ===
Subgroup: 2.3.5.7.11


[[Eigenmonzo]]s: 2, 5
Comma list: 36/35, 45/44, 64/63


[[POTE generator]]: ~128/105 = 348.415
Mapping: {{mapping| 1 0 -4 6 -6 | 0 1 4 -2 6 }}


Mapping generator: ~128/105
Optimal tunings:  
* WE: ~2 = 1194.7102{{c}}, ~3/2 = 695.6962{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1765{{c}}


Algebraic generator: Mohabis, real root of 3''x''<sup>3</sup> - 3''x''<sup>2</sup> - 1, 348.6067 cents. Corresponding recurrence converges quickly.
{{Optimal ET sequence|legend=0| 5e, 7, 12 }}


Map: [&lt;1 1 0 6|, &lt;0 2 8 -11|]
Badness (Sintel): 0.727


[[Generator]]s: 2, 128/105
=== Arnold ===
Subgroup: 2.3.5.7.11


[[Wedgie]]: &lt;&lt;2 8 -11 8 -23 -48||
Comma list: 22/21, 33/32, 36/35


{{EDOs|legend=1| 7, 24, 31, 38, 55, 69 }}
Mapping: {{mapping| 1 0 -4 6 5 | 0 1 4 -2 -1 }}


[[Badness]]: 0.0557
Optimal tunings:  
* WE: ~2 = 1199.8507{{c}}, ~3/2 = 698.4045{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.4822{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 5, 7, 12e }}
Period: 1\1


Optimal ([[POTE]]) generator: ~11/9 = 348.477
Badness (Sintel): 0.864


EDO generators: [[24edo|7\24]], [[31edo|9\31]]
=== Neutrominant ===
See [[Rastmic clan #Neutrominant|Rastmic clan]].


Scales (Scala files):
== Flattertone ==
In flattertone, 17 fifths get to the interval class for 7, so that [[7/4]] is a double-augmented sixth (C–Ax). The fifth in flattertone is typically at least as flat as [[26edo]]. Here, 26edo and [[33edo|33cd-edo]] are the two primary flattertone tunings. [[1/2-comma meantone]] is also encompassed within flattertone's range. Any flatter than this, the meantone mapping for 5/4 is too inaccurate (it becomes more of a [[16/13]] or [[27/22]]), and [[deeptone]] temperament's mapping is more logical.


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Flattertone was named by [[Flora Canou]] in 2024.  
<div style="line-height:1.6;">Interval table (10-note MOS, 2.3.5.7.11 POTE tuning)</div>
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
|+
|-
! #
! Cents<ref>octave-reduced</ref>
! class="unsortable"| Approximate ratios<ref>2.3.5.7.11, odd limit ≤ 27. JI readings in parentheses are outside the subgroup but are supported by the defining EDOs.</ref>
|-
| 0
| 0.00
| 1/1
|-
| 1
| 348.5
| 11/9
|-
| 2
| 697.0
| 3/2
|-
| 3
| 1045.4
| 11/6
|-
| 4
| 193.9
| 9/8
|-
| 5
| 542.4
| 11/8, 15/11
|-
| 6
| 890.9
| 5/3
|-
| 7
| 39.3
|
|-
| 8
| 387.8
| 5/4
|-
| 9
| 736.3
| 32/21
|}
<references/></div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 81/80, 121/120, 176/175
[[Subgroup]]: 2.3.5.7


Mapping: [{{val|1 1 0 6 2}}, {{val|0 2 8 -11 5}}]
[[Comma list]]: 81/80, 1875/1792


Mapping generators: ~2, ~11/9
{{Mapping|legend=1| 1 0 -4 -24 | 0 1 4 17 }}


Minimax tuning:  
: mapping generators: ~2, ~3
* [[11-odd-limit]]: 1/4 comma
: [{{Monzo| 1 0 0 0 0 }}, {{Monzo| 1 0 1/4 0 0 }}, {{Monzo| 0 0 1 0 0 }}, {{Monzo| 6 0 -11/8 0 0 }}, {{Monzo| 2 0 5/8 0 0 }}]
: Eigenmonzos: 2, 5


{{EDOs|legend=1| 7, 24, 31, 38, 55 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1204.4511{{c}}, ~3/2 = 694.3258{{c}}
: [[error map]]: {{val| +4.451 -3.178 -9.011 +3.554 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 692.0479{{c}}
: error map: {{val| 0.000 -9.907 -18.122 -4.012 }}


Badness: 0.0261
{{Optimal ET sequence|legend=1| 7d, 19d, 26, 59bcd, 85bccd }}
</div></div>


== 13-limit ==
[[Badness]] (Sintel): 2.43
Commas: 66/65, 81/80, 105/104, 121/120


POTE generator: ~11/9 = 348.558
==== 11-limit ====
Subgroup: 2.3.5.7.11


Mapping generator: ~11/9
Comma list: 45/44, 81/80, 1375/1344


Map: [&lt;1 1 0 6 2 4|, &lt;0 2 8 -11 5 -1|]
Mapping: {{mapping| 1 0 -4 -24 -6 | 0 1 4 17 6 }}


{{EDOs|legend=1| 7, 24, 31, 38, 55 }}
Optimal tunings:
* WE: ~2 = 1203.4653{{c}}, ~3/2 = 693.8144{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 692.0422{{c}}


Badness: 0.0234
{{Optimal ET sequence|legend=0| 7d, 19d, 26 }}


== 17-limit ==
Badness (Sintel): 1.53
Commas: 66/65, 81/80, 105/104, 121/120, 154/153


POTE generator: ~11/9 = 348.736
; Music
* [https://youtu.be/scCuGXnj5IY ''Music in 33EDO (33-Tone Equal Temperament) - Feb 2024''] by [[Budjarn Lambeth]] (2024)


Mapping generator: ~11/9
== Sharptone ==
Sharptone is a low-accuracy temperament tempering out [[21/20]] and [[28/27]]. In sharptone, 7/4 is a major sixth, 7/6 a whole tone, and 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo]] tuning does sharptone about as well as such a thing can be done, of course not in its patent val.


Map: [&lt;1 1 0 6 2 4 7|, &lt;0 2 8 -11 5 -1 -10|]
However, while 12edo ends up near-optimal, the only valid [[diamond monotone]] tuning for sharptone is [[5edo]]. Anything flat of it has ~12/7 and ~7/4 in the wrong order (and so should be dominant) and anything sharp of it has ~5/4 and ~4/3 in the wrong order (and so should not be meantone).


{{EDOs|legend=1| 7, 24, 31, 38g, 55 }}
The 11-limit extension was named by Gene Ward Smith in 2004<ref name="meantone & meanpop 2004"/>.


Badness: 0.0206
[[Subgroup]]: 2.3.5.7


== 19-limit ==
[[Comma list]]: 21/20, 28/27
Commas: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152


POTE generator: ~11/9 = 348.810
{{Mapping|legend=1| 1 0 -4 -2 | 0 1 4 3 }}


Mapping generator: ~11/9
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1204.2961{{c}}, ~3/2 = 702.6463{{c}}
: [[error map]]: {{val| +4.296 +4.987 +24.271 -56.591 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4928{{c}}
: error map: {{val| 0.000 -0.462 +19.657 -64.347 }}


Map: [&lt;1 1 0 6 2 4 7 6|, &lt;0 2 8 -11 5 -1 -10 -6|]
{{Optimal ET sequence|legend=1| 5, 7d, 12d }}


{{EDOs|legend=1| 7, 24, 31, 38gh, 55 }}
[[Badness]] (Sintel): 0.629


Badness: 0.0173
=== Meanertone ===
Subgroup: 2.3.5.7.11


= Dominant =
Comma list: 21/20, 28/27, 33/32
[[Comma]]s: 36/35, 64/63


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]].
Mapping: {{mapping| 1 0 -4 -2 5 | 0 1 4 3 -1 }}


valid range: [700.000, 720.000] (12 to 5)
Optimal tunings:  
* WE: ~2 = 1208.5304{{c}}, ~3/2 = 701.5669{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1117{{c}}


nice range: [694.786, 715.587]
{{Optimal ET sequence|legend=0| 5, 7d, 12de }}


strict range: [700.000, 715.587]
Badness (Sintel): 0.832


[[POTE generator]]: 701.573
== Mildtone ==
Mildtone tempers out [[16128/15625]] and finds the interval class of 7 at 22 generators up, as a triple-augmented fifth (C–G#x). [[55edo]] and [[67edo]] are among the possible tunings.  


Mapping generator: ~3
Mildtone was named by [[User: Lucius Chiaraviglio|Lucius Chiaraviglio]] in 2024.


Map: [&lt;1 0 -4 6|, &lt;0 1 4 -2|]
[[Subgroup]]: 2.3.5.7


[[Wedgie]]: &lt;&lt;1 4 -2 4 -6 -16||
[[Comma list]]: 81/80, 16128/15625


{{EDOs|legend=1| 5, 7, 12, 17c, 29cd }}
{{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 }}


[[Badness]]: 0.0207
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.7304{{c}}, ~3/2 = 698.3953{{c}}
: [[error map]]: {{val| -0.270 -3.829 +7.267 -1.434 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.5397{{c}}
: error map: {{val| 0.000 -3.415 +7.845 -0.952 }}


== 11-limit ==
{{Optimal ET sequence|legend=1| 12, 43d, 55, 67 }}
Commas: 36/35, 64/63, 56/55


valid range: [700.000, 705.882] (12 to 17)
[[Badness]] (Sintel): 2.67


nice range: [691.202, 715.587]
=== 11-limit ===


strict range: [700.000, 705.882]
[[Subgroup]]: 2.3.5.7.11


POTE generator: ~3/2 = 703.254
[[Comma list]]: 81/80, 176/175, 7056/6875


Mapping generator: ~3
{{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 30}}


Map: [&lt;1 0 -4 6 13|, &lt;0 1 4 -2 -6|]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.816{{c}}, ~3/2 = 698.355{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.455{{c}}


{{EDOs|legend=1| 5, 12, 17c, 29cde }}
{{Optimal ET sequence|legend=1| 12, 43de, 55, 67 }}


Badness: 0.0242
[[Badness]] (Sintel): 2.15


=== 13-limit ===
=== 13-limit ===
Commas: 36/35, 56/55, 64/63, 66/65


valid range: 705.882 (17)
[[Subgroup]]: 2.3.5.7.11.13
 
[[Comma list]]: 81/80, 176/175, 196/195, 832/825
 
{{Mapping|legend=1| 1 0 -4 -32 -44 | 0 1 4 22 30}}


nice range: [691.202, 715.587]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.788{{c}}, ~3/2 = 698.355{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.471{{c}}


strict range:705.882
{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}


POTE generator: ~3/2 = 703.636
[[Badness]] (Sintel): 2.04
 
=== 17-limit ===


Map: [&lt;1 0 -4 6 13 18|, &lt;0 1 4 -2 -6 -9|]
[[Subgroup]]: 2.3.5.7.11.13.17


{{EDOs|legend=1| 12f, 17c, 29cdef }}
[[Comma list]]: 81/80, 176/175, 189/187, 196/195, 832/825


Badness: 0.0241
{{Mapping|legend=1| 1 0 -4 -32 -44 12| 0 1 4 22 30 -5}}


=== Dominion ===
[[Optimal tuning]]s:
Commas: 26/25, 36/35, 56/55, 64/63
* [[WE]]: ~2 = 1199.655{{c}}, ~3/2 = 698.295{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.488{{c}}


POTE generator: ~3/2 = 704.905
{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}


Map: [&lt;1 0 -4 6 13 -9|, &lt;0 1 4 -2 -6 8|]
[[Badness]] (Sintel): 1.98


{{EDOs|legend=1| 5, 12, 17c, 46cde }}
=== 19-limit ===


Badness: 0.0273
[[Subgroup]]: 2.3.5.7.11.13.17.19


== Domineering ==
[[Comma list]]: 81/80, 96/95, 176/175, 189/187, 196/195, 832/825
Commas: 36/35, 45/44, 64/63


POTE generator: ~3/2 = 698.776
{{Mapping|legend=1| 1 0 -4 -32 -44 12 9| 0 1 4 22 30 -5 -3}}


Mapping generator: ~3
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.371{{c}}, ~3/2 = 698.164{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.519{{c}}


Map: [&lt;1 0 -4 6 -6|, &lt;0 1 4 -2 6|]
{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}


{{EDOs|legend=1| 5e, 7, 12, 19d, 43de }}
[[Badness]] (Sintel): 1.95


Badness: 0.0220
{{Todo|unify precision|review}}


=== 13-limit ===
== Supermean ==
Commas: 36/35, 45/44, 52/49, 64/63
Supermean tempers out 672/625 and finds the interval class of 7 at 15 generators up, as a double-augmented fifth (C–Gx). As such, it extends [[leapfrog]].


POTE generator: ~3/2 = 695.762
[[Subgroup]]: 2.3.5.7


Mapping generator: ~3
[[Comma list]]: 81/80, 672/625


Map: [&lt;1 0 -4 6 -6 10|, &lt;0 1 4 -2 6 -4|]
{{Mapping|legend=1| 1 0 -4 -21 | 0 1 4 15 }}


{{EDOs|legend=1| 5ef, 7, 12, 19d, 31def }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1195.4372{{c}}, ~3/2 = 702.2086{{c}}
: [[error map]]: {{val| -4.563 -4.309 +22.521 -8.319 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5375{{c}}
: error map: {{val| 0.000 +2.583 +31.836 -0.763 }}


Badness: 0.0270
{{Optimal ET sequence|legend=1| 5d, 12d, 17c }}


==== 17-limit ====
[[Badness]] (Sintel): 3.40
Commas: 36/35, 45/44, 51/49, 52/49, 64/63


POTE generator: ~3/2 = 696.115
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping generator: ~3
Comma list: 56/55, 81/80, 132/125


Map: [&lt;1 0 -4 6 -6 10 12|, &lt;0 1 4 -2 6 -4 -5|]
Mapping: {{mapping| 1 0 -4 -21 -14 | 0 1 4 15 11 }}


{{EDOs|legend=1| 5ef, 7, 12, 19d, 31def }}
Optimal tunings:
* WE: ~2 = 1195.7270{{c}}, ~3/2 = 702.5848{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7471{{c}}


Badness: 0.0245
{{Optimal ET sequence|legend=0| 5de, 12de, 17c }}


==== 19-limit ====
Badness (Sintel): 2.09
Commas: 36/35, 39/38, 45/44, 51/49, 52/49, 57/56


POTE generator: ~3/2 = 696.217
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Mapping generator: ~3
Comma list: 26/25, 56/55, 66/65, 81/80


Map: [&lt;1 0 -4 6 -6 10 12 9|, &lt;0 1 4 -2 6 -4 -5 -3|]
Mapping: {{mapping| 1 0 -4 -21 -14 -9 | 0 1 4 15 11 8 }}


{{EDOs|legend=1| 5ef, 7, 12, 19d, 31def }}
Optimal tunings:
* WE: ~2 = 1196.3958{{c}}, ~3/2 = 702.9766{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7940{{c}}


Badness: 0.0204
{{Optimal ET sequence|legend=0| 5de, 12de, 17c, 29c }}


=== Dominatrix ===
Badness (Sintel): 1.67
Commas: 27/26, 36/35, 45/44, 64/63


POTE generator: ~3/2 = 698.544
== Mohajira ==
{{Main| Mohajira }}


Mapping generator: ~3
Mohajira can be viewed as derived from [[mohaha]] which maps the interval half a [[chromatic semitone|chroma]] flat of the minor seventh to ~7/4 so that 7/4 is mapped to a semidiminished seventh (C–Bdb), although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the [[porwell comma]]. It can be described as {{nowrap| 24 & 31 }}; its ploidacot is dicot. [[31edo]] makes for an excellent mohajira tuning, with generator 9\31. Note that while 24 + 31 = [[55edo]] doesn't apear in the optimal ET sequence, it is a [[patent val]] tuning and recommendable if you prefer a light meantone tempering.


Map: [&lt;1 0 -4 6 -6 -1|, &lt;0 1 4 -2 6 3|]
[[Subgroup]]: 2.3.5.7


{{EDOs|legend=1| 5e, 7, 12f, 19df }}
[[Comma list]]: 81/80, 6144/6125


== Domination ==
{{Mapping|legend=1| 1 1 0 6 | 0 2 8 -11 }}
Commas: 36/35, 64/63, 77/75


POTE generator: ~3/2 = 705.004
: mapping generators: ~2, ~128/105


Mapping generator: ~3
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.8160{{c}}, ~128/105 = 348.6518{{c}}
: [[error map]]: {{val| +0.816 -3.835 +2.901 +0.900 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 348.4194{{c}}
: error map: {{val| 0.000 -5.116 +1.041 -1.439 }}


Map: [&lt;1 0 -4 6 -14|, &lt;0 1 4 -2 11|]
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~128/105 = {{monzo| 0 0 1/8 }}
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 6 0 -11/8 0 }}
: [[eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.5


{{EDOs|legend=1| 5e, 12e, 17c, 46cd }}
[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~128/105 = [347.368, 350.000] (11\38 to 7\24)
* 7-odd-limit [[diamond tradeoff]]: ~128/105 = [347.393, 350.978]
* 9-odd-limit diamond tradeoff: ~128/105 = [345.601, 350.978]


Badness: 0.0366
[[Algebraic generator]]: Mohabis, real root of 3''x''<sup>3</sup> - 3''x''<sup>2</sup> - 1, 348.6067 cents. Corresponding recurrence converges quickly.


=== 13-limit ===
{{Optimal ET sequence|legend=1| 7, 24, 31 }}
Commas: 26/25, 36/35, 64/63, 66/65


POTE generator: ~3/2 = 705.496
[[Badness]] (Sintel): 1.41


Mapping generator: ~3
Scales: [[mohaha7]], [[mohaha10]]


Map: [&lt;1 0 -4 6 -14 -9|, &lt;0 1 4 -2 11 8|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{EDOs|legend=1| 5e, 12e, 17c }}
Comma list: 81/80, 121/120, 176/175


Badness: 0.0274
Mapping: {{mapping| 1 1 0 6 2 | 0 2 8 -11 5 }}


== Arnold ==
Optimal tunings:
Commas: 22/21, 33/32, 36/35
* WE: ~2 = 1201.1562{{c}}, ~11/9 = 348.8124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.4910{{c}}


POTE generator: ~3/2 = 698.491
Minimax tuning:
* 11-odd-limit: ~11/9 = {{monzo| 0 0 1/8 }}
: projection map: [{{Monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 6 0 -11/8 0 0 }}, {{monzo| 2 0 5/8 0 0 }}]
: unchanged-interval (eigenmonzo) basis: 2.5


Mapping generator: ~3
Tuning ranges:
* 11-odd-limit diamond monotone: ~11/9 = [348.387, 350.000] (9\31 to 7\24)
* 11-odd-limit diamond tradeoff: ~11/9 = [344.999, 350.978]


Map: [&lt;1 0 -4 6 5|, &lt;0 1 4 -2 -1|]
{{Optimal ET sequence|legend=0| 7, 24, 31 }}


{{EDOs|legend=1| 5, 7, 12e }}
Badness (Sintel): 0.862


Badness: 0.0261
Scales: [[mohaha7]], [[mohaha10]]


=== 13-limit ===
=== 13-limit ===
Commas: 22/21, 27/26, 33/32, 36/35
Subgroup: 2.3.5.7.11.13
 
Comma list: 66/65, 81/80, 105/104, 121/120
 
Mapping: {{mapping| 1 1 0 6 2 4 | 0 2 8 -11 5 -1 }}


POTE generator: ~3/2 = 696.743
Optimal tunings:
* WE: ~2 = 1200.4256{{c}}, ~11/9 = 348.6819{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.5622{{c}}


Map: [&lt;1 0 -4 6 5 -1|, &lt;0 1 4 -2 -1 3|]
{{Optimal ET sequence|legend=0| 7, 24, 31 }}


{{EDOs|legend=1| 5, 7, 12ef, 19def }}
Badness (Sintel): 0.966


Badness: 0.0233
Scales: [[mohaha7]], [[mohaha10]]


=== 17-limit ===
=== 17-limit ===
Commas: 22/21, 27/26, 33/32, 36/35, 51/49
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 66/65, 81/80, 105/104, 121/120, 154/153


POTE generator: ~3/2 = 696.978
Mapping: {{mapping| 1 1 0 6 2 4 7 | 0 2 8 -11 5 -1 -10 }}


Map: [&lt;1 0 -4 6 5 -1 12|, &lt;0 1 4 -2 -1 3 -5|]
Optimal tunings:  
* WE: ~2 = 1200.0382{{c}}, ~11/9 = 348.7471{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.7360{{c}}


{{EDOs|legend=1| 5, 7, 12ef, 19def }}
{{Optimal ET sequence|legend=0| 7, 24, 31 }}


Badness: 0.0245
Badness (Sintel): 1.05
 
Scales: [[mohaha7]], [[mohaha10]]


=== 19-limit ===
=== 19-limit ===
Commas: 22/21, 27/26, 33/32, 36/35, 51/49, 57/56
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152
 
Mapping: {{mapping| 1 1 0 6 2 4 7 6 | 0 2 8 -11 5 -1 -10 -6 }}
 
Optimal tunings:
* WE: ~2 = 1199.7469{{c}}, ~11/9 = 348.7367{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.8117{{c}}
 
{{Optimal ET sequence|legend=0| 7, 24, 31, 55 }}
 
Badness (Sintel): 1.05
 
Scales: [[mohaha7]], [[mohaha10]]
 
== Mohamaq ==
Mohamaq is a lower-accuracy alternative to mohajira that favors tunings sharp of 24edo. It may be described as {{nowrap| 17c & 24 }}; its ploidacot is dicot, the same as mohajira.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 81/80, 392/375
 
{{Mapping|legend=1| 1 1 0 -1 | 0 2 8 13 }}
 
: mapping generators: ~2, ~25/21
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.0661{{c}}, ~25/21 = 350.3127{{c}}
: [[error map]]: {{val| -0.934 -2.264 +16.188 -13.827 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/21 = 350.4856{{c}}
: error map: {{val| 0.000 -0.984 +17.571 -12.513 }}
 
{{Optimal ET sequence|legend=1| 7d, 17c, 24 }}
 
[[Badness]] (Sintel): 1.97
 
Scales: [[mohaha7]], [[mohaha10]]
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 56/55, 77/75, 243/242
 
Mapping: {{mapping| 1 1 0 -1 2 | 0 2 8 13 5 }}
 
Optimal tunings:
* WE: ~2 = 1199.1924{{c}}, ~11/9 = 350.3286{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.4821{{c}}
 
{{Optimal ET sequence|legend=0| 7d, 17c, 24 }}


POTE generator: ~3/2 = 697.068
Badness (Sintel): 1.20


Map: [&lt;1 0 -4 6 5 -1 12 9|, &lt;0 1 4 -2 -1 3 -5 -3|]
Scales: [[mohaha7]], [[mohaha10]]


{{EDOs|legend=1| 5, 7, 12ef, 19def }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0211
Comma list: 56/55, 66/65, 77/75, 243/242


= Sharptone =
Mapping: {{mapping| 1 1 0 -1 2 4 | 0 2 8 13 5 -1 }}
[[Comma]]s: 21/20, 28/27


Sharptone is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo]] tuning does sharptone about as well as such a thing can be done, of course not in its patent val.
Optimal tunings:
* WE: ~2 = 1198.5986{{c}}, ~11/9 = 350.3353{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.6459{{c}}


[[POTE generator]]: ~3/2 = 700.140
{{Optimal ET sequence|legend=0| 7d, 17c, 24, 41c }}


Mapping generator: ~3
Badness (Sintel): 1.19


Map: [&lt;1 0 -4 -2|, &lt;0 1 4 3|]
Scales: [[mohaha7]], [[mohaha10]]


[[Wedgie]]: &lt;&lt;1 4 3 4 2 -4||
== Liese ==
<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>


{{EDOs|legend=1| 5, 7d, 12d }}
Liese splits the [[3/1|perfect twelfth]] into three generators of ~[[10/7]], using the comma [[1029/1000]]. It also tempers out [[686/675]], the senga. It may be described as {{nowrap| 17c & 19 }}; its ploidacot is alpha-tricot. It is a very natural 13-limit tuning, given the generator is so near 13/9. [[74edo]] makes for a good liese tuning, though [[19edo]] can be used. The tuning is well-supplied with mos scales: 7, 9, 11, 13, 15, 17, 19, 36, 55.


[[Badness]]: 0.0248
[[Subgroup]]: 2.3.5.7


== Meanertone ==
[[Comma list]]: 81/80, 686/675
Commas: 21/20, 28/27, 33/32


POTE generator: ~3/2 = 696.615
{{Mapping|legend=1| 1 0 -4 -3 | 0 3 12 11 }}


Map: [&lt;1 0 -4 -2 5|, &lt;0 1 4 3 -1|]
: mapping generators: ~2, ~10/7


{{EDOs|legend=1| 5, 7d, 12de }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.5548{{c}}, ~10/7 = 633.2251{{c}}
: [[error map]]: {{val| +1.555 -2.280 +6.168 -8.015 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 632.5640{{c}}
: error map: {{val| 0.000 -4.263 +4.454 -10.622 }}


Badness: 0.0252
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/7 = {{monzo| 1/3 0 1/12 }}
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 2/3 0 11/12 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


= Meansept =
[[Algebraic generator]]: Radix, the real root of ''x''<sup>5</sup> - 2''x''<sup>4</sup> + 2''x''<sup>3</sup> - 2''x''<sup>2</sup> + 2''x'' - 2, also a root of ''x''<sup>6</sup> - ''x''<sup>5</sup> - 2. The recurrence converges.
Commas: 15/14, 81/80


POTE generator: ~3/2 = 682.895
{{Optimal ET sequence|legend=1| 17c, 19, 55, 74d }}


Mapping generator: ~3
[[Badness]] (Sintel): 1.18


Map: [&lt;1 0 -4 -5|, &lt;0 1 4 5|]
=== Liesel ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;1 4 5 4 5 0||
Comma list: 56/55, 81/80, 540/539


{{EDOs|legend=1| 5d, 7, 12dd }}
Mapping: {{mapping| 1 0 -4 -3 4 | 0 3 12 11 -1 }}


Badness: 0.0453
Optimal tunings:  
* WE: ~2 = 1198.8507{{c}}, ~10/7 = 632.4668{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 632.9963{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 17c, 19, 36 }}
Commas: 15/14, 22/21, 81/80


POTE generator: ~3/2 = 685.234
Badness (Sintel): 1.35


Mapping generator: ~3
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 -4 -5 -6|, &lt;0 1 4 5 6|]
Comma list: 56/55, 78/77, 81/80, 91/90


{{EDOs|legend=1| 5de, 7, 12dd }}
Mapping: {{mapping| 1 0 -4 -3 4 0 | 0 3 12 11 -1 7 }}


Badness: 0.0325
Optimal tunings:  
* WE: ~2 = 1199.4968{{c}}, ~10/7 = 632.7766{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.0082{{c}}


= Supermean =
{{Optimal ET sequence|legend=0| 17c, 19, 36 }}
Commas: 81/80, 672/625


POTE generator: ~3/2 = 704.889
Badness (Sintel): 1.13


Map: [&lt;1 0 -4 -21|, &lt;0 1 4 15|]
=== Elisa ===
Subgroup: 2.3.5.7.11


{{EDOs|legend=1| 5d, 12d, 17c, 29c }}
Comma list: 77/75, 81/80, 99/98


Badness: 0.1342
Mapping: {{mapping| 1 0 -4 -3 -5 | 0 3 12 11 16 }}


== 11-limit ==
Optimal tunings:
Commas: 56/55, 81/80, 132/125
* WE: ~2 = 1201.0489{{c}}, ~10/7 = 633.6147{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1644{{c}}


POTE generator: ~3/2 = 705.096
{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}


Map: [&lt;1 0 -4 -21 -14|, &lt;0 1 4 15 11|]
Badness (Sintel): 1.37


{{EDOs|legend=1| 5de, 12de, 17c, 29c }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0633
Comma list: 66/65, 77/75, 81/80, 99/98


== 13-limit ==
Mapping: {{mapping| 1 0 -4 -3 -5 0 | 0 3 12 11 16 7 }}
Commas: 26/25, 56/55, 66/65, 81/80


POTE generator: ~3/2 = 705.094
Optimal tunings:
* WE: ~2 = 1201.4815{{c}}, ~10/7 = 633.7720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1281{{c}}


Map: [&lt;1 0 -4 -21 -14 -9|, &lt;0 1 4 15 11 8|]
{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}


{{EDOs|legend=1| 5de, 12de, 17c, 29c }}
Badness (Sintel): 1.11


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


Injera has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. [[38edo]], which is two parallel [[19edo]]s, is an excellent tuning for injera.
Comma list: 45/44, 81/80, 343/330


[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name]
Mapping: {{mapping| 1 0 -4 -3 -6 | 0 3 12 11 18 }}
 
Optimal tunings:
* WE: ~2 = 1202.6773{{c}}, ~10/7 = 632.7783{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.6175{{c}}
 
{{Optimal ET sequence|legend=0| 17cee, 19 }}
 
Badness (Sintel): 1.81
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


valid range: [685.714, 700.000] (14c to 12)
Comma list: 45/44, 81/80, 91/88, 147/143


nice range: [688.957, 701.955]
Mapping: {{mapping| 1 0 -4 -3 -6 0 | 0 3 12 11 18 7 }}


strict range: [688.957, 700.000]
Optimal tunings:  
* WE: ~2 = 1203.6086{{c}}, ~10/7 = 633.1193{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.5346{{c}}


[[POTE generator]]: 694.375
{{Optimal ET sequence|legend=0| 17cee, 19 }}


Mapping generator: ~3
Badness (Sintel): 1.49


Map: [&lt;2 0 -8 -7|, &lt;0 1 4 4|]
== Superpine ==
{{See also| No-sevens subgroup temperaments #Superpine }}


[[Wedgie]]: &lt;&lt;2 8 8 8 7 -4||
The superpine temperament is generated by 1/3 of a fourth, represented by [[~]][[35/32]], which resembles [[porcupine]], but it favors flat fifths instead of sharp ones. It may be described as {{nowrap| 36 & 43 }}; its ploidacot is omega-tricot. Unlike in porcupine, the minor third reached by 2 generators up is strongly neutral-flavored and does not represent [[6/5]] – harmonics other than 3 all require the 15-tone mos ([[7L 8s]]) to properly utilize. This temperament has an obvious 11-limit interpretation by treating the generator as [[11/10]] as in porcupine, which makes [[11/8]] high-[[complexity]] like the other harmonics, but in the 13-limit 5 generators up closely approximates [[13/8]]. [[43edo]] is a good tuning especially for the higher-limit extensions.


{{EDOs|legend=1| 12, 26, 38, 102bcd, 140bccd, 178bbccdd }}
[[Subgroup]]: 2.3.5.7


[[Badness]]: 0.0311
[[Comma list]]: 81/80, 1119744/1071875


== Music ==
{{Mapping|legend=1| 1 2 4 1 | 0 -3 -12 13 }}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 Two Pairs of Socks] (in [[26edo]]) by [[Igliashon Jones]]


== 11-limit ==
[[Optimal tuning]]s:
Commas: 45/44, 50/49, 81/80
* [[WE]]: ~2 = 1199.3652{{c}}, ~35/32 = 167.1615{{c}}
: [[error map]]: {{val| -0.635 -4.709 +5.209 +3.639 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/32 = 167.2561{{c}}
: error map: {{val| 0.000 -3.723 +6.613 +5.503 }}


valid range: [685.714, 700.000] (14c to 12)
{{Optimal ET sequence|legend=1| 7, 36, 43, 79c }}


nice range: [682.458, 701.955]
[[Badness]] (Sintel): 3.46


strict range: [685.714, 700.000]
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~3/2 = 692.840
Comma list: 81/80, 176/175, 864/847


Mapping generator: ~3
Mapping: {{mapping| 1 2 4 1 5 | 0 -3 -12 13 -11 }}


Map: [&lt;2 0 -8 -7 -12|, &lt;0 1 4 4 6|]
Optimal tunings:  
* WE: ~2 = 1199.0522{{c}}, ~11/10 = 167.1904{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.3382{{c}}


{{EDOs|legend=1| 12, 14c, 26, 90bce, 116bcce }}
{{Optimal ET sequence|legend=0| 7, 36, 43 }}


Badness: 0.0231
Badness (Sintel): 1.90


=== 13-limit ===
=== 13-limit ===
Commas: 45/44, 50/49, 78/77, 81/80
Subgroup: 2.3.5.7.11.13
 
Comma list: 78/77, 81/80, 144/143, 176/175


valid range: 692.308 (26)
Mapping: {{mapping| 1 2 4 1 5 3 | 0 -3 -12 13 -11 5 }}


nice range: [682.458, 701.955]
Optimal tunings:  
* WE: ~2 = 1199.4286{{c}}, ~11/10 = 167.3105{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.3958{{c}}


strict range: 692.308 (26)
{{Optimal ET sequence|legend=0| 7, 36, 43 }}


POTE generator: ~3/2 = 692.673
Badness (Sintel): 1.52


Mapping generator: ~3
== Lithium ==
Lithium is named after the 3rd element for having a 3rd-octave period (and also for lithium's molar mass of 6.9 g/mol since 69edo supports it). Its ploidacot is triploid monocot. It supports a [[3L 6s]] scale and thus intuitively can be thought of as "tcherepnin meantone" in that context.


Map: [&lt;2 0 -8 -7 -12 -21|, &lt;0 1 4 4 6 9|]
[[Subgroup]]: 2.3.5.7


{{EDOs|legend=1| 12f, 14cf, 26, 38e }}
[[Comma list]]: 81/80, 3125/3087


Badness: 0.0216
{{Mapping|legend=1| 3 0 -12 -20 | 0 1 4 6 }}


=== Enjera ===
: mapping generators: ~56/45, ~3
Commas: 27/26, 40/39, 45/44, 50/49


POTE generator: ~3/2 = 694.121
[[Optimal tuning]]s:  
* [[WE]]: ~56/45 = 400.6744{{c}}, ~3/2 = 695.8474{{c}} {~15/14 = 105.5015{{c}})
: [[error map]]: {{val| +2.023 -4.084 -2.924 +4.910 }}
* [[CWE]]: ~56/45 = 400.0000{{c}}, ~3/2 = 695.1413{{c}} {~15/14 = 104.8587{{c}})
: error map: {{val| 0.000 -6.814 -5.748 +2.022 }}


Mapping generator: ~3
{{Optimal ET sequence|legend=1| 12, 33cd, 45, 57 }}


Map: [&lt;2 0 -8 -7 -12 -2|, &lt;0 1 4 4 6 3|]
[[Badness]] (Sintel): 1.75


{{EDOs|legend=1| 12f, 14c, 26f, 38eff }}
== Squares ==
{{Main| Squares }}


Badness: 0.0265
Squares splits the [[6/1|6th harmonic]] into four subminor sixths of [[11/7]]~[[14/9]] (or splits a [[8/3|perfect eleventh]] into four supermajor thirds of [[9/7]]~[[14/11]]), and uses it for a generator. It may be described as {{nowrap| 14c & 17c }}; its ploidacot is beta-tetracot. [[31edo]], with a generator of 11/31, makes for a good squares tuning, with 8-, 11-, and 14-note mos scales available. Squares tempers out [[2401/2400]], the breedsma, as well as [[2430/2401]].


== Injerous ==
[[Subgroup]]: 2.3.5.7
Commas: 33/32, 50/49, 55/54


POTE generator: ~3/2 = 690.548
[[Comma list]]: 81/80, 2401/2400


Mapping generator: ~3
{{Mapping|legend=1| 1 -1 -8 -3 | 0 4 16 9 }}


Map: [&lt;2 0 -8 -7 10|, &lt;0 1 4 4 -1|]
: mapping generators: ~2, ~14/9


{{EDOs|legend=1| 12e, 14c, 26e, 40cee }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2488{{c}}, ~14/9 = 774.8640{{c}}
: [[error map]]: {{val| +1.249 -3.748 +1.520 +1.204 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 774.1560{{c}}
: error map: {{val| 0.000 -5.331 +0.183 -1.422 }}


Badness: 0.0386
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~9/7 = {{monzo| 1/2 0 -1/16 }}
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 3/2 0 9/16 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


== Lahoh ==
[[Algebraic generator]]: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.
Commas: 50/49, 56/55, 81/77


POTE generator: ~3/2 = 699.001
{{Optimal ET sequence|legend=1| 14c, 17c, 31, 169b, 200b }}


Mapping generator: ~3
[[Badness]] (Sintel): 1.16


Map: [&lt;2 0 -8 -7 7|, &lt;0 1 4 4 0|]
Scales: [[skwares8]], [[skwares11]], [[skwares14]]


{{EDOs|legend=1| 2cd, 12, 14ce }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0431
Comma list: 81/80, 99/98, 121/120


= Ptolemy =
Mapping: {{mapping| 1 -1 -8 -3 -3 | 0 4 16 9 10 }}
Commas: 81/80, 121/120, 525/512


POTE generator: ~11/9 = 346.922
Optimal tunings:  
* WE: ~2 = 1201.6657{{c}}, ~11/7 = 775.1171{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.1754{{c}}


Map: [&lt;1 1 0 8 2|, &lt;0 2 8 -18 5|]
{{Optimal ET sequence|legend=0| 14c, 17c, 31, 130bee, 169beee }}


{{EDOs|legend=1| 7, 31dd, 38d, 45e, 83bcddee }}
Badness (Sintel): 0.715


Badness: 0.0588
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


== 13-limit ==
Comma list: 66/65, 81/80, 99/98, 121/120
Commas: 65/64, 81/80, 105/104, 121/120


POTE generator: ~11/9 = 346.910
Mapping: {{mapping| 1 -1 -8 -3 -3 5 | 0 4 16 9 10 -2 }}


Map: [&lt;1 1 0 8 2 6|, &lt;0 2 8 -18 5 -8|]
Optimal tunings:  
* WE: ~2 = 1199.8419{{c}}, ~11/7 = 774.3484{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4422{{c}}


{{EDOs|legend=1| 7, 31ddf, 38df, 45ef, 83bcddeeff }}
{{Optimal ET sequence|legend=0| 14c, 17c, 31, 79cf }}


Badness: 0.0343
Badness (Sintel): 1.05


= Maqamic =
==== Squad ====
<span style="display: block; text-align: right;">[[:de:maqamisch|Deutsch]]</span>
Subgroup: 2.3.5.7.11.13


{{main| Maqamic }}
Comma list: 78/77, 81/80, 91/90, 99/98


[[Comma]]s: 81/80, 36/35, 121/120
Mapping: {{mapping| 1 -1 -8 -3 -3 -6 | 0 4 16 9 10 15 }}


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.
Optimal tunings:
* WE: ~2 = 1202.0312{{c}}, ~11/7 = 775.5589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4140{{c}}


[[POTE generator]]: ~11/9 = 350.934
{{Optimal ET sequence|legend=0| 14cf, 17c, 31f }}


Mapping generator: ~11/9
Badness (Sintel): 1.11


Map: [&lt;1 1 0 4 2|, &lt;0 2 8 -4 5|]
==== Agora ====
Subgroup: 2.3.5.7.11.13


[[generator|Generator]]s: 2, 11/9
Comma list: 81/80, 99/98, 105/104, 121/120


{{EDOs|legend=1| 7, 10c, 17c, 24d, 31d }}
Mapping: {{mapping| 1 -1 -8 -3 -3 -15 | 0 4 16 9 10 29 }}


== 13-limit ==
Optimal tunings:
[[Comma]]s: 81/80, 36/35, 121/120, 144/143
* WE: ~2 = 1202.3228{{c}}, ~11/7 = 775.2214{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8617{{c}}


[[POTE generator]]: ~11/9 = 350.816
{{Optimal ET sequence|legend=0| 14cf, 31, 45ef, 76e }}


Mapping generator: ~11/9
Badness (Sintel): 1.01


Map: [&lt;1 1 0 4 2 4|, &lt;0 2 8 -4 5 -1|]
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Generators: 2, 11/9
Comma list: 81/80, 99/98, 105/104, 120/119, 121/119


{{EDOs|legend=1| 7, 10c, 17c, 24d, 31d }}
Mapping: {{mapping| 1 -1 -8 -3 -3 -15 -3 | 0 4 16 9 10 29 11 }}


= Migration =
Optimal tunings:
Commas: 81/80, 121/120, 126/125
* WE: ~2 = 1201.4340{{c}}, ~11/7 = 774.7375{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8955{{c}}


POTE generator: ~11/9 = 348.182
{{Optimal ET sequence|legend=0| 14cf, 31 }}


Mapping generator: ~11/9
Badness (Sintel): 1.15


Map: [&lt;1 1 0 -3 2|, &lt;0 2 8 20 5|]
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


{{EDOs|legend=1| 7d, 31, 100de, 131bdee, 162bdee }}
Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119


Badness: 0.0255
Mapping: {{mapping| 1 -1 -8 -3 -3 -15 -3 -8 | 0 4 16 9 10 29 11 19 }}


== 13-limit ==
Optimal tunings:
Commas: 66/65, 81/80, 121/120, 126/125
* WE: ~2 = 1201.2461{{c}}, ~11/7 = 774.5783{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8479{{c}}


POTE generator: ~11/9 = 348.490
{{Optimal ET sequence|legend=0| 14cf, 31 }}


Map: [&lt;1 1 0 -3 2 4|, &lt;0 2 8 20 5 -1|]
Badness (Sintel): 1.15


{{EDOs|legend=1| 7d, 24d, 31, 55d }}
=== Cuboctahedra ===
Subgroup: 2.3.5.7.11


Badness: 0.0281
Comma list: 81/80, 385/384, 1375/1372


= Mohamaq =
Mapping: {{mapping| 1 -1 -8 -3 17 | 0 4 16 9 -21 }}
Commas: 81/80, 392/375


POTE generator: ~25/21 = 350.586
Optimal tunings:  
* WE: ~2 = 1201.4436{{c}}, ~14/9 = 774.9386{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 774.0243{{c}}


Mapping generator: ~25/21
{{Optimal ET sequence|legend=0| 31, 107b, 138b, 169be, 200be }}


Map: [&lt;1 1 0 -1|, &lt;0 2 8 13|]
Badness (Sintel): 1.88


{{EDOs|legend=1| 17c, 24, 65c, 89cd }}
== Jerome ==
Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5<sup>1/20</sup>, or 139.316 cents. It may be described as {{nowrap| 17c & 26 }}; its ploidacot is pentacot. While the generator represents both 13/12 and 12/11, the CTE/CWE and Hieronymus generators are close to 13/12 in size.


Badness: 0.0777
[[Subgroup]]: 2.3.5.7


== 11-limit ==
[[Comma list]]: 81/80, 17280/16807
Commas: 56/55, 77/75, 243/242


POTE generator: ~11/9 = 350.565
{{Mapping|legend=1| 1 1 0 2 | 0 5 20 7 }}


Mapping generator: ~11/9
: mapping generators: ~2, ~54/49


Map: [&lt;1 1 0 -1 2|, &lt;0 2 8 13 5|]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1640{{c}}, ~54/49 = 139.3624{{c}}
: [[error map]]: {{val| +0.164 -4.979 +0.934 +7.039 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~54/49 = 139.3528{{c}}
: error map: {{val| 0.000 -5.191 +0.741 +6.643 }}


{{EDOs|legend=1| 17c, 24, 65c, 89cd }}
{{Optimal ET sequence|legend=1| 17c, 26, 43 }}


Badness: 0.0362
[[Badness]] (Sintel): 2.75


== 13-limit ==
=== 11-limit ===
Commas: 56/55, 66/65, 77/75, 243/242
Subgroup: 2.3.5.7.11


POTE generator: ~11/9 = 350.745
Comma list: 81/80, 99/98, 864/847


Mapping generator: ~11/9
Mapping: {{mapping| 1 1 0 2 3 | 0 5 20 7 4 }}


Map: [&lt;1 1 0 -1 2 4|, &lt;0 2 8 13 5 -1|]
Optimal tunings:  
* WE: ~2 = 1201.4436{{c}}, ~12/11 = 139.3714{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 139.4038{{c}}


{{EDOs|legend=1| 17c, 24, 41c, 65c }}
{{Optimal ET sequence|legend=0| 17c, 26, 43 }}


Badness: 0.0287
Badness (Sintel): 1.58


= Orphic =
=== 13-limit ===
Commas: 81/80, 5898240/5764801
Subgroup: 2.3.5.7.11.13


POTE generator: ~7/6 = 275.794
Comma list: 78/77, 81/80, 99/98, 144/143


Mapping generator: ~343/288
Mapping: {{mapping| 1 1 0 2 3 3 | 0 5 20 7 4 6 }}


Map: [&lt;2 1 -4 4|, &lt;0 4 16 3|]
Optimal tunings:  
* WE: ~2 = 1199.8860{{c}}, ~13/12 = 139.3737{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3817{{c}}


Wedgie: &lt;&lt;8 32 6 32 -13 -76||
{{Optimal ET sequence|legend=0| 17c, 26, 43 }}


{{EDOs|legend=1| 26, 74, 174bd, 248bd }}
Badness (Sintel): 1.21


Badness: 0.2588
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


== 11-limit ==
Comma list: 78/77, 81/80, 99/98, 144/143, 189/187
Commas: 81/80, 99/98, 73728/73205


POTE generator: ~7/6 = 275.762
Mapping: {{mapping| 1 1 0 2 3 3 2 | 0 5 20 7 4 6 18 }}


Mapping generator: ~77/64
Optimal tunings:  
* WE: ~2 = 1199.8346{{c}}, ~13/12 = 139.3431{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3544{{c}}


Map: [&lt;2 1 -4 4 8|, &lt;0 4 16 3 -2|]
{{Optimal ET sequence|legend=0| 17cg, 26, 43 }}


{{EDOs|legend=1| 26, 48c, 74, 248bd, 322bd }}
Badness (Sintel): 1.06


Badness: 0.1015
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


== 13-limit ==
Comma list: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143
Commas: 81/80, 99/98, 144/143, 2200/2197


POTE generator: ~7/6 = 275.774
Mapping: {{mapping| 1 1 0 2 3 3 2 1 | 0 5 20 7 4 6 18 28 }}


Mapping generator: ~63/52
Optimal tunings:  
* WE: ~2 = 1199.8891{{c}}, ~13/12 = 139.3001{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3080{{c}}


Map: [&lt;2 1 -4 4 8 2|, &lt;0 4 16 3 -2 10|]
{{Optimal ET sequence|legend=0| 17cgh, 26, 43, 69 }}


{{EDOs|legend=1| 26, 48c, 74, 174bd, 248bd, 322bd }}
Badness (Sintel): 1.11


Badness: 0.0535
== Meantritone ==
The meantritone temperament tempers out the [[mirkwai comma]] (16875/16807) and [[trimyna comma]] (50421/50000) in the 7-limit. In this temperament, the 6th harmonic is split into five generators of ~10/7; the ploidacot of this temperament is beta-pentacot. The name ''meantritone'' is a portmanteau of ''meantone'' and ''tritone'', the latter is a generator of this temperament.


= Mothra =
[[Subgroup]]: 2.3.5.7
[[Comma]]s: 81/80, 1029/1024


Mothra splits the fifth into three 8/7 generators. It uses 1029/1024, the gamelisma, to accomplish this deed and also tempers out 1728/1715, the orwell comma. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra. In the 2.3.7-limit, mothra is identical to [[slendric]].
[[Comma list]]: 81/80, 16875/16807


Note that mothra can also be called cynder in the 7-limit, which can be a little confusing sometimes.
{{Mapping|legend=1| 1 -1 -8 -7 | 0 5 20 19 }}


[[7-odd-limit|7]] and [[9-odd-limit|9-limit]] minimax 1/4 comma
: mapping generators: ~2, ~10/7


[{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 3 0 -1/12 0 }}]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.3832{{c}}, ~10/7 = 619.9478{{c}}
: [[error map]]: {{val| +1.383 -3.599 +1.576 +0.499 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 619.3176{{c}}
: error map: {{val| 0.000 -5.367 +0.038 -1.791 }}


[[Eigenmonzo]]s: 2, 5
{{Optimal ET sequence|legend=1| 29cd, 31, 188bcd, 219bbcd }}


[[POTE generator]]: ~8/7 = 232.193
[[Badness]] (Sintel): 2.08


Mapping generator: ~8/7
=== 11-limit ===
Subgroup: 2.3.5.7.11


Algebraic generator: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.
Comma list: 81/80, 99/98, 2541/2500


Map: [&lt;1 1 0 3|, &lt;0 3 12 -1|]
Mapping: {{mapping| 1 -1 -8 -7 -11 | 0 5 20 19 28 }}


[[Generator]]s: 2, 8/7
Optimal tunings:  
* WE: ~2 = 1201.2054{{c}}, ~10/7 = 619.9752{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.4223{{c}}


[[Wedgie]]: &lt;&lt;3 12 -1 12 -10 -36||
{{Optimal ET sequence|legend=0| 29cde, 31 }}


{{EDOs|legend=1| 5, 26, 31, 57, 88 }}
Badness (Sintel): 1.42


[[Badness]]: 0.0371
== Injera ==
Injera has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a ~15/14 semitone difference between a half-octave and a perfect fifth. Injera may be described as {{nowrap| 12 & 26 }}; its ploidacot is diploid monocot. It tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. [[38edo]], which is two parallel [[19edo]]s, is an excellent tuning for injera.


== 11-limit ==
[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name]
Commas: 81/80, 99/98, 385/384


POTE generator: ~8/7 = 232.031
[[Subgroup]]: 2.3.5.7


Mapping generator: ~8/7
[[Comma list]]: 50/49, 81/80


Map: [&lt;1 1 0 3 5|, &lt;0 3 12 -1 -8|]
{{Mapping|legend=1| 2 0 -8 -7 | 0 1 4 4 }}


{{EDOs|legend=1| 5, 26, 31, 57, 88, 150be, 181bee }}
: mapping generators: ~7/5, ~3


Badness: 0.0256
[[Optimal tuning]]s:  
* [[WE]]: ~7/5 = 600.6662{{c}}, ~3/2 = 695.1463{{c}} (~21/20 = 94.4801{{c}})
: [[error map]]: {{val| +1.332 -5.476 -5.729 +12.425 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 694.7712{{c}} (~21/20 = 94.7712{{c}})
: error map: {{val| 0.000 -7.184 -7.229 +10.259 }}


=== 13-limit ===
[[Tuning ranges]]:
Commas: 81/80, 99/98, 105/104, 144/143
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [688.957, 701.955]
* 9-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]


POTE generator: ~8/7 = 231.811
{{Optimal ET sequence|legend=1| 12, 26, 38 }}


Mapping generator: ~8/7
[[Badness]] (Sintel): 0.788


Map: [&lt;1 1 0 3 5 1|, &lt;0 3 12 -1 -8 14|]
; Music
* [https://web.archive.org/web/20201127013520/http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 ''Two Pairs of Socks''] by [[Igliashon Jones]] – in [[26edo]] tuning


{{EDOs|legend=1| 5, 26, 31, 57, 88 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0240
Comma list: 45/44, 50/49, 81/80


== Cynder ==
Mapping: {{mapping| 2 0 -8 -7 -12 | 0 1 4 4 6 }}
Commas: 45/44, 81/80, 1029/1024


POTE generator: ~8/7 = 231.317
Optimal tunings:  
* WE: ~7/5 = 600.9350{{c}}, ~3/2 = 693.9198{{c}} (~21/20 = 92.9848{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 693.3539{{c}} (~21/20 = 93.3539{{c}})


Mapping generator: ~8/7
Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
* 11-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]


Map: [&lt;1 1 0 3 0|, &lt;0 3 12 -1 18|]
{{Optimal ET sequence|legend=0| 12, 26 }}


{{EDOs|legend=1| 5e, 26, 31e, 57e, 83bce }}
Badness (Sintel): 0.764


Badness: 0.0557
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== 13-limit ===
Comma list: 45/44, 50/49, 78/77, 81/80
Commas: 45/44, 78/77, 81/80, 640/637


POTE generator: ~8/7 = 231.293
Mapping: {{mapping| 2 0 -8 -7 -12 -21 | 0 1 4 4 6 9 }}


Mapping generator: ~8/7
Optimal tunings:  
* WE: ~7/5 = 600.9982{{c}}, ~3/2 = 693.8249{{c}} (~21/20 = 92.8267{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 693.0992{{c}} (~21/20 = 93.0992{{c}})


Map: [&lt;1 1 0 3 0 1|, &lt;0 3 12 -1 18 14|]
Tuning ranges:  
* 13-odd-limit diamond monotone: ~3/2 = 692.308 (15\26)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]


{{EDOs|legend=1| 5e, 26, 31e, 57e, 83bce }}
{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}


Badness: 0.0341
Badness (Sintel): 0.891


== Mosura ==
===== 17-limit =====
Commas: 81/80, 176/175, 540/539
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~8/7 = 232.419
Comma list: 45/44, 50/49, 78/77, 81/80, 85/84


Mapping generator: ~8/7
Mapping: {{mapping| 2 0 -8 -7 -12 -21 5 | 0 1 4 4 6 9 1 }}


Map: [&lt;1 1 0 3 -1|, &lt;0 3 12 -1 23|]
Optimal tunings:  
* WE: ~7/5 = 601.1757{{c}}, ~3/2 = 693.8441{{c}} (~21/20 = 92.6684{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 692.8879{{c}} (~21/20 = 92.8879{{c}})


{{EDOs|legend=1| 31, 36, 67, 98, 129, 160be, 191bce, 222bce, 253bcee }}
{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}


Badness: 0.0313
Badness (Sintel): 0.935


=== 13-limit ===
===== 19-limit =====
Commas: 81/80, 144/143, 176/175, 196/195
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~8/7 = 232.640
Comma list: 45/44, 50/49, 57/56, 78/77, 81/80, 85/84


Mapping generator: ~8/7
Mapping: {{mapping| 2 0 -8 -7 -12 -21 5 -1 | 0 1 4 4 6 9 1 3 }}


Map: [&lt;1 1 0 3 -1 7|, &lt;0 3 12 -1 23 -17|]
Optimal tunings:  
* WE: ~7/5 = 601.4245{{c}}, ~3/2 = 693.9426{{c}} (~21/20 = 92.5181{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 692.7606{{c}} (~21/20 = 92.7606{{c}})


{{EDOs|legend=1| 31, 36, 67, 98 }}
{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}


Badness: 0.0369
Badness (Sintel): 0.920


= Squares =
==== Enjera ====
[[Comma]]s: 81/80, 2401/2400
Subgroup: 2.3.5.7.11.13


Squares 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: 27/26, 40/39, 45/44, 50/49


7 and 9 limit minimax 1/4 comma
Mapping: {{mapping| 2 0 -8 -7 -12 -2 | 0 1 4 4 6 3 }}


[{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 3/2 0 9/16 0 }}]
Optimal tunings:
* WE: ~7/5 = 599.1863{{c}}, ~3/2 = 693.1791{{c}} (~21/20 = 93.9929{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 693.6809{{c}} (~21/20 = 93.6809{{c}})


[[Eigenmonzo]]s: 2, 5
{{Optimal ET sequence|legend=0| 10cdeef, 12f }}


[[POTE generator]]: ~9/7 = 425.942
Badness (Sintel): 1.10


Mapping generator: ~9/7
=== Injerous ===
Subgroup: 2.3.5.7.11


Algebraic generator: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.
Comma list: 33/32, 50/49, 55/54


Map: [&lt;1 3 8 6|, &lt;0 -4 -16 -9|]
Mapping: {{mapping| 2 0 -8 -7 10 | 0 1 4 4 -1 }}


[[Generator]]s: 2, 9/7
Optimal tunings:  
* WE: ~7/5 = 603.1682{{c}}, ~3/2 = 694.1945{{c}} (~21/20 = 91.0264{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 691.6107{{c}} (~21/20 = 91.6107{{c}})


{{EDOs|legend=1| 14c, 17c, 31, 45, 76 }}
{{Optimal ET sequence|legend=0| 12e, 14c, 26e, 40cee }}


[[Badness]]: 0.0460
Badness (Sintel): 1.28


Music:
=== Lahoh ===
Subgroup: 2.3.5.7.11


By [[Chris Vaisvil]]
Comma list: 50/49, 56/55, 81/77


* [http://clones.soonlabel.com/public/micro/tuning-survey/daily20100603-squares8piano.mp3 Square 8]
Mapping: {{mapping| 2 0 -8 -7 7 | 0 1 4 4 0 }}


== 11-limit ==
Optimal tunings:
Commas: 81/80, 99/98, 121/120
* WE: ~7/5 = 597.3179{{c}}, ~3/2 = 695.8759{{c}} (~21/20 = 98.5581{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 697.8757{{c}} (~21/20 = 97.8757{{c}})


POTE generator: ~9/7 = 425.957
{{Optimal ET sequence|legend=0| 10cd, 12 }}


Mapping generator: ~9/7
Badness (Sintel): 1.42


Map: [&lt;1 3 8 6 7|, &lt;0 -4 -16 -9 -10|]
=== Teff ===
{{Main| Teff }}


{{EDOs|legend=1| 14c, 17c, 31, 45e, 76e }}
Teff, found and named by [[Mason Green]], is to injera what mohajira is to meantone; it splits the generator in halves in order to accommodate higher-limit intervals, creating a half-octave quartertone temperament. Its ploidacot is diploid alpha-dicot.


Badness: 0.0216
Subgroup: 2.3.5.7.11


=== 13-limit ===
Comma list: 50/49, 81/80, 864/847
Commas: 66/65, 81/80, 99/98, 121/120


POTE generator: ~9/7 = 425.550
Mapping: {{mapping| 2 1 -4 -3 8 | 0 2 8 8 -1 }}


Mapping generator: ~9/7
: mapping generators: ~7/5, ~16/11


Map: [&lt;1 3 8 6 7 3|, &lt;0 -4 -16 -9 -10 2|]
Optimal tunings:  
* WE: ~7/5 = 600.2802{{c}}, ~16/11 = 647.7720{{c}} (~33/32 = 47.4918{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.5224{{c}} (~33/32 = 47.5224{{c}})


{{EDOs|legend=1| 14c, 17c, 31, 45e, 79cf }}
{{Optimal ET sequence|legend=0| 24d, 26, 50d }}


Badness: 0.0255
Badness (Sintel): 2.34


=== Agora ===
==== 13-limit ====
Commas: 81/80, 99/98, 105/104, 121/120
Subgroup: 2.3.5.7.11.13


POTE generator: ~9/7 = 426.276
Comma list: 50/49, 78/77, 81/80, 144/143


Mapping generator: ~9/7
Mapping: {{mapping| 2 1 -4 -3 8 2 | 0 2 8 8 -1 5 }}


Map: [&lt;1 3 8 6 7 14|, &lt;0 -4 -16 -9 -10 -29|]
Optimal tunings:  
* WE: ~7/5 = 600.3037{{c}}, ~16/11 = 647.7954{{c}} (~33/32 = 47.4917{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.5256{{c}} (~33/32 = 47.5256{{c}})


{{EDOs|legend=1| 14cf, 31, 45ef, 76e }}
{{Optimal ET sequence|legend=0| 24d, 26, 50d }}


Badness: 0.0245
Badness (Sintel): 1.65


==== 17-limit ====
==== 17-limit ====
Commas: 81/80, 99/98, 105/104, 120/119, 121/119
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 50/49, 78/77, 81/80, 85/84, 144/143


POTE generator: ~9/7 = 426.187
Mapping: {{mapping| 2 1 -4 -3 8 2 6 | 0 2 8 8 -1 5 2 }}


Mapping generator: ~9/7
Optimal tunings:  
* WE: ~7/5 = 600.5123{{c}}, ~16/11 = 647.8970{{c}} (~34/33 = 47.3846{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.4314{{c}} (~34/33 = 47.4314{{c}})


Map: [&lt;1 3 8 6 7 14 8|, &lt;0 -4 -16 -9 -10 -29 -11|]
{{Optimal ET sequence|legend=0| 24d, 26 }}


{{EDOs|legend=1| 14cf, 31, 45ef, 76e }}
Badness (Sintel): 1.50


==== 19-limit ====
==== 19-limit ====
Commas: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 50/49, 57/56, 78/77, 81/80, 85/84, 144/143
 
Mapping: {{mapping| 2 1 -4 -3 8 2 6 2 | 0 2 8 8 -1 5 2 6 }}


POTE generator: ~9/7 = 426.225
Optimal tunings:  
* WE: ~7/5 = 600.6308{{c}}, ~16/11 = 648.0424{{c}} (~34/33 = 47.4116{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.4715{{c}} (~34/33 = 47.4715{{c}})


Mapping generator: ~9/7
{{Optimal ET sequence|legend=0| 24d, 26 }}


Map: [&lt;1 3 8 6 7 14 8 11|, &lt;0 -4 -16 -9 -10 -29 -11 -19|]
Badness (Sintel): 1.41


{{EDOs|legend=1| 14cf, 31, 45ef, 76e }}
== Pombe ==
Pombe (named after the African millet beer) is a variant of [[#Teff]] by [[User:Kaiveran|Kaiveran Lugheidh]] that eschews the tempering of 50/49 to attain more accuracy in the 7-limit. Its ploidacot is diploid alpha-dicot, the same as teff. Oddly, the 7th harmonic has a lesser generator distance than in teff (-5 vs +8), but this combined with the fact that other harmonics are in the opposite direction means that the 7-limit diamond is more complex overall.


== Cuboctahedra ==
[[Subgroup]]: 2.3.5.7
[[Comma]]s: 81/80, 385/384, 1375/1372


[[POTE generator]]: ~9/7 = 425.993
[[Comma list]]: 81/80, 300125/294912


Mapping generator: ~9/7
{{Mapping|legend=1| 2 1 -4 11 | 0 2 8 -5 }}


Map: [&lt;1 3 8 6 -4|, &lt;0 -4 -16 -9 21|]
: mapping generators: ~735/512, ~35/24


{{EDOs|legend=1| 14ce, 17ce, 31, 45, 76, 107b }}
[[Optimal tuning]]s:
* [[WE]]: ~735/512 = 601.0652{{c}}, ~35/24 = 648.9295{{c}} (~36/35 = 47.8642{{c}})
: [[error map]]: {{val| +2.130 -3.031 +0.861 -1.756 }}
* [[CWE]]: ~735/512 = 600.0000{{c}}, ~35/24 = 647.8628{{c}} (~36/35 = 47.8628{{c}})
: error map: {{val| 0.000 -6.229 -3.411 -8.140 }}


[[Badness]]: 0.0568
{{Optimal ET sequence|legend=1| 24, 26, 50, 126bcd, 176bcdd, 226bbcdd }}


= Liese =
[[Badness]] (Sintel): 2.94
<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>


[[Comma]]s: 81/80, 686/675
=== 11-limit ===
Subgroup: 2.3.5.7.11


Liese splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. [[74edo]] 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.
Comma list: 81/80, 245/242, 385/384


7 and 9 limit minimax 1/4 comma
Mapping: {{mapping| 2 1 -4 11 8 | 0 2 8 -5 -1 }}


[{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 2/3 0 11/12 0 }}]
Optimal tunings:
* WE: ~99/70 = 600.7890{{c}}, ~16/11 = 648.7592{{c}} (~36/35 = 47.9701{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~16/11 = 647.9516{{c}} (~36/35 = 47.9516{{c}})


[[Eigenmonzo]]s: 2, 5
{{Optimal ET sequence|legend=0| 24, 26, 50 }}


[[POTE generator]]: ~10/7 = 632.406
Badness (Sintel): 1.72


Mapping generator: ~10/7
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Algebraic generator: Radix, the real root of ''x''<sup>5</sup> - 2''x''<sup>4</sup> + 2''x''<sup>3</sup> - 2''x''<sup>2</sup> + 2''x'' - 2, also a root of ''x''<sup>6</sup> - ''x''<sup>5</sup> - 2. The recurrence converges.
Comma list: 81/80, 105/104, 144/143, 245/242


Map: [&lt;1 0 -4 -3|, &lt;0 3 12 11|]
Mapping: {{mapping| 2 1 -4 11 8 2 | 0 2 8 -5 -1 5 }}


[[Generator]]s: 2, 10/7
Optimal tunings:  
* WE: ~99/70 = 600.6971{{c}}, ~16/11 = 648.6029{{c}} (~36/35 = 47.9058{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~16/11 = 647.8990{{c}} (~36/35 = 47.8990{{c}})


{{EDOs|legend=1| 17c, 19, 36, 55, 74d }}
{{Optimal ET sequence|legend=0| 24, 26, 50 }}


[[Badness]]: 0.0467
Badness (Sintel): 1.28


== Liesel ==
=== 17-limit ===
Commas: 56/55, 81/80, 540/539
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~10/7 = 633.073
Comma list: 81/80, 105/104, 144/143, 245/242, 273/272


Mapping generator: ~10/7
Mapping: {{mapping| 2 1 -4 11 8 2 6 | 0 2 8 -5 -1 5 2 }}


Map: [&lt;1 0 -4 -3 4|, &lt;0 3 12 11 -1|]
Optimal tunings:  
* WE: ~17/12 = 600.7610{{c}}, ~16/11 = 648.6638{{c}} (~36/35 = 47.9028{{c}})
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 647.8990{{c}} (~36/35 = 47.8990{{c}})


{{EDOs|legend=1| 17c, 19, 36, 55e, 91cee }}
{{Optimal ET sequence|legend=0| 24, 26, 50 }}


Badness: 0.0407
Badness (Sintel): 1.08


=== 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
 
Comma list: 81/80, 105/104, 133/132, 144/143, 171/170, 210/209


Commas: 56/55, 78/77, 81/80, 91/90
Mapping: {{mapping| 2 1 -4 11 8 2 6 2 | 0 2 8 -5 -1 5 2 6 }}


POTE generator: ~10/7 = ~13/9 = 633.042
Optimal tunings:  
* WE: ~17/12 = 600.8048{{c}}, ~16/11 = 648.7494{{c}} (~36/35 = 47.9446{{c}})
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 647.9425{{c}} (~36/35 = 47.9425{{c}})


Mapping generator: ~10/7
{{Optimal ET sequence|legend=0| 24, 26, 50 }}


Map: [&lt;1 0 -4 -3 4 0|, &lt;0 3 12 11 -1 7|]
Badness (Sintel): 1.01


{{EDOs|legend=1| 17c, 19, 36, 55ef, 91ceef }}
== Orphic ==
Orphic has a semi-octave period and four generators plus a period gives the 3rd harmonic; its ploidacot is diploid alpha-tetracot.


Badness: 0.0273
[[Subgroup]]: 2.3.5.7


== Elisa ==
[[Comma list]]: 81/80, 5898240/5764801
Commas: 77/75, 81/80, 99/98


POTE generator: ~10/7 = 633.061
{{Mapping|legend=1| 2 1 -4 4 | 0 4 16 3 }}


Mapping generator: ~10/7
: mapping generators: ~2401/1728, ~343/288


Map: [&lt;1 0 -4 -3 -5|, &lt;0 3 12 11 16|]
[[Optimal tuning]]s:  
* [[WE]]: ~2401/1728 = 600.1767{{c}}, ~343/288 = 324.3015{{c}} (~7/6 = 275.8751{{c}})
: [[error map]]: {{val| +0.353 -4.572 +1.804 +4.785 }}
* [[CWE]]: ~2401/1728 = 600.0000{{c}}, ~343/288 = 324.2285{{c}} (~7/6 = 275.7715{{c}})
: error map: {{val| 0.000 -5.041 +1.342 +3.860 }}


{{EDOs|legend=1| 17c, 19e, 36e }}
{{Optimal ET sequence|legend=1| 26, 48c, 74 }}


Badness: 0.0416
[[Badness]] (Sintel): 6.55


== Lisa ==
=== 11-limit ===
Commas: 45/44, 81/80, 343/330
Subgroup: 2.3.5.7.11


POTE generator: ~10/7 = 631.370
Comma list: 81/80, 99/98, 73728/73205


Mapping generator: ~10/7
Mapping: {{mapping| 2 1 -4 4 8 | 0 4 16 3 -2 }}


Map: [&lt;1 0 -4 -3 -6|, &lt;0 3 12 11 18|]
Optimal tunings:  
* WE: ~363/256 = 600.1011{{c}}, ~77/64 = 324.2923{{c}} (~7/6 = 275.8088{{c}})
* CWE: ~363/256 = 600.0000{{c}}, ~77/64 = 324.2463{{c}} (~7/6 = 275.7537{{c}})


{{EDOs|legend=1| 19 }}
{{Optimal ET sequence|legend=0| 26, 48c, 74 }}


Badness: 0.0548
Badness (Sintel): 3.36


=== 13-limit ===
=== 13-limit ===
Commas: 45/44, 81/80, 91/88, 147/143
Subgroup: 2.3.5.7.11.13
 
Comma list: 81/80, 99/98, 144/143, 2200/2197
 
Mapping: {{mapping| 2 1 -4 4 8 2 | 0 4 16 3 -2 10 }}
 
Optimal tunings:
* WE: ~55/39 = 600.0540{{c}}, ~77/64 = 324.2551{{c}} (~7/6 = 275.7989{{c}})
* CWE: ~55/39 = 600.0000{{c}}, ~77/64 = 324.2307{{c}} (~7/6 = 275.7693{{c}})
 
{{Optimal ET sequence|legend=0| 26, 48c, 74 }}
 
Badness (Sintel): 2.21
 
== Cloudtone ==
The cloudtone temperament tempers out the [[cloudy comma]], 16807/16384 and the [[syntonic comma]], 81/80 in the 7-limit. It may be described as {{nowrap| 5 & 50 }}; its ploidacot is pentaploid monocot. It can be extended to the 11- and 13-limit by adding 385/384 and 105/104 to the comma list in this order.


POTE generator: ~10/7 = 631.221
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 0 -4 -3 -6 0|, &lt;0 3 12 11 18 7|]
[[Comma list]]: 81/80, 16807/16384


{{EDOs|legend=1| 19 }}
{{Mapping|legend=1| 5 0 -20 14 | 0 1 4 0 }}


Badness: 0.0361
: mapping generators: ~8/7, ~3


= Jerome =
[[Optimal tuning]]s:
Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5<sup>1/20</sup>, or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size.
* [[WE]]: ~8/7 = 240.4267{{c}}, ~3/2 = 696.9566{{c}} (~49/48 = 24.3235{{c}})
: [[error map]]: {{val| +2.133 -2.865 +1.513 -2.852 }}
* [[CWE]]: ~8/7 = 240.0000{{c}}, ~3/2 = 696.1637{{c}} (~49/48 = 23.8373{{c}})
: error map: {{val| 0.000 -5.791 -1.659 -8.826 }}


Commas: 81/80, 17280/16807
{{Optimal ET sequence|legend=1| 5, 40c, 45, 50 }}


POTE generator: ~54/49 = 139.343
[[Badness]] (Sintel): 2.59


Mapping generator: ~54/49
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 1 0 2|, &lt;0 5 20 7|]
Comma list: 81/80, 385/384, 2401/2376


Wedgie: &lt;&lt;5 30 7 20 -3 -40||
Mapping: {{mapping| 5 0 -20 14 41 | 0 1 4 0 -3 }}


{{EDOs|legend=1| 9c, 17c, 26, 43, 69, 112bd }}
Optimal tunings:
* WE: ~8/7 = 240.2740{{c}}, ~3/2 = 697.3317{{c}} (~56/55 = 23.4904{{c}})
* CWE: ~8/7 = 240.0000{{c}}, ~3/2 = 696.6269{{c}} (~56/55 = 23.3731{{c}})


Badness: 0.1087
{{Optimal ET sequence|legend=0| 5, 45, 50 }}


== 11-limit ==
Badness (Sintel): 2.33
Commas: 81/80, 99/98, 864/847


POTE generator: ~12/11 = 139.428
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Mapping generator: ~12/11
Comma list: 81/80, 105/104, 144/143, 2401/2376


Map: [&lt;1 1 0 2 3|, &lt;0 5 20 7 4|]
Mapping: {{mapping| 5 0 -20 14 41 -21 | 0 1 4 0 -3 5 }}


{{EDOs|legend=1| 9c, 17c, 26, 43, 69 }}
Optimal tunings:
* WE: ~8/7 = 240.2435{{c}}, ~3/2 = 696.8686{{c}} (~91/90 = 23.8618{{c}})
* CWE: ~8/7 = 240.0000{{c}}, ~3/2 = 696.2653{{c}} (~91/90 = 23.7347{{c}})


Badness: 0.0479
{{Optimal ET sequence|legend=0| 5, 45f, 50 }}


== 13-limit ==
Badness (Sintel): 2.02
Commas: 78/77, 81/80, 99/98, 144/143


POTE generator: ~13/12 = 139.387
== Subgroup extensions ==
=== Stützel (2.3.5.19) ===
[[Subgroup]]: 2.3.5.19


Mapping generator: ~12/11
[[Comma list]]: 81/80, 96/95


Map: [&lt;1 1 0 2 3 3|, &lt;0 5 20 7 4 6|]
{{Mapping|legend=2| 1 0 -4 9 | 0 1 4 -3 }}


{{EDOs|legend=1| 9c, 17c, 26, 43, 69 }}
{{Mapping|legend=3| 1 0 -4 0 0 0 0 9 | 0 1 4 0 0 0 0 -3 }}


Badness: 0.0293
: mapping generators: ~2, ~3


== 17-limit ==
[[Optimal tuning]]s:
Commas: 78/77, 81/80, 99/98, 144/143, 189/187
* [[WE]]: ~2 = 1199.5513{{c}}, ~3/2 = 697.6058{{c}}
: [[error map]]: {{val| -0.448 -4.798 +4.110 +6.977 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.8222{{c}}
: error map: {{val| 0.000 -4.133 +4.975 +9.020 }}


POTE generator: ~13/12 = 139.362
{{Optimal ET sequence|legend=1| 5, 7, 12, 31, 43, 98h }}


Mapping generator: ~12/11
[[Badness]] (Sintel): 0.324


Map: [&lt;1 1 0 2 3 3 2|, &lt;0 5 20 7 4 6 18|]
=== Hypnotone ===
Hypnotone is no-sevens [[#Flattone|flattone]].


{{EDOs|legend=1| 26, 43, 69 }}
[[Subgroup]]: 2.3.5.11


Badness: 0.0209
[[Comma list]]: 45/44, 81/80


== 19-limit ==
{{Mapping|legend=2| 1 0 -4 -6 | 0 1 4 6 }}
Commas: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143


POTE generator: ~13/12 = 139.313
{{Mapping|legend=3| 1 0 -4 0 -6 | 0 1 4 0 6 }}


Mapping generator: ~12/11
: mapping generators: ~2, ~3


Map: [&lt;1 1 0 2 3 3 2 1|, &lt;0 5 20 7 4 6 18 28|]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1202.0621{{c}}, ~3/2 = 694.5448{{c}}
: [[error map]]: {{val| +2.062 -5.348 -8.135 +15.951 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.9085{{c}}
: error map: {{val| 0.000 -8.047 -10.680 +12.133 }}


{{EDOs|legend=1| 26, 43, 69 }}
{{Optimal ET sequence|legend=1| 7, 12, 19, 26, 45 }}


Badness: 0.0182
[[Badness]] (Sintel): 0.326


= Meanmag =
==== 2.3.5.11.13 subgroup ====
Commas: 81/80, 3125/3072
Subgroup: 2.3.5.11.13


POTE generator: ~8/7 = 238.396
Comma list: 45/44, 65/64, 81/80


Mapping generator: ~7
Subgroup-val mapping: {{mapping| 1 0 -4 -6 10 | 0 1 4 6 -4 }}


Map: [&lt;19 30 44 0|, &lt;0 0 0 1|]
Gencom mapping: {{mapping| 1 0 -4 0 -6 10 | 0 1 4 0 6 -4 }}


Wedgie: &lt;&lt;0 0 19 0 30 44||
Optimal tunings:  
* WE: ~2 = 1202.6916{{c}}, ~3/2 = 694.4181{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0870{{c}}


{{EDOs|legend=1| 19, 38, 57, 76, 95bc }}
{{Optimal ET sequence|legend=0| 7, 12, 19, 26, 45f }}


Badness: 0.0770
Badness (Sintel): 0.561


= Undevigintone =
=== Dequarter ===
Commas: 49/48, 81/80, 126/125
[[Subgroup]]: 2.3.5.11


POTE generator: ~11/8 = 538.047
[[Comma list]]: 33/32, 55/54


Mapping generator: ~11
{{Mapping|legend=2| 1 0 -4 5 | 0 1 4 -1 }}


Map: [&lt;19 30 44 53 0|, &lt;0 0 0 0 1|]
{{Mapping|legend=3| 1 0 -4 0 5 | 0 1 4 0 -1 }}


{{EDOs|legend=1| 19, 38d }}
: mapping generators: ~2, ~3


Badness: 0.0364
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1206.5832{{c}}, ~3/2 = 695.8763{{c}}
: [[error map]]: {{val| +6.583 +0.504 -2.809 -20.862 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 693.1206{{c}}
: error map: {{val| 0.000 -8.834 -13.831 -44.439 }}


== 13-limit ==
{{Optimal ET sequence|legend=1| 5, 7, 19e, 26e }}
`Commas: 49/48, 65/64, 81/80, 126/125


POTE generator: ~11/8 = 537.061
[[Badness]] (Sintel): 0.451


Map: [&lt;19 30 44 53 0 70|, &lt;0 0 0 0 1 0|]
==== Dreamtone ====
Subgroup: 2.3.5.11.13


{{EDOs|legend=1| 19, 38d }}
Comma list: 33/32, 55/54, 975/968


Badness: 0.0229
Subgroup-val mapping: {{mapping| 1 0 -4 5 21 | 0 1 4 -1 -11 }}


[[Category:Theory]]
Gencom mapping: {{mapping| 1 0 -4 0 5 21 | 0 1 4 0 -1 -11 }}
[[Category:Temperament family]]
 
[[Category:Meantone]]
Optimal tunings:
* WE: ~2 = 1207.8248{{c}}, ~3/2 = 694.7806{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 690.1826{{c}}
 
{{Optimal ET sequence|legend=0| 7, 19eff, 26eff, 33ceeff, 40ceeff }}
 
Badness (Sintel): 1.40
 
== References ==
<references/>
 
[[Category:Temperament families]]
[[Category:Meantone family| ]] <!-- main article -->
[[Category:Meantone| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Listen]]
[[Category:Listen]]
{{todo|review|improve readability}}
Retrieved from "https://en.xen.wiki/w/Meantone_family"