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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.


{{Vals|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: ~3/2 = 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, which comes to 503.4257 cents. The recurrence converges quickly.
[[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.


{{Vals|legend=1| 12, 19, 31, 81, 112b, 143b }}
{{Optimal ET sequence|legend=1| 12, 19, 31, 81, 112b, 143b }}


[[Badness]]: 0.0137
[[Badness]] (Sintel): 0.347
</div></div>


== Bimeantone ==
=== Undecimal meantone (huygens) ===
11/8 is mapped to half octave minus the [[meantone diesis]].
{{Redirect|Huygens|the Dutch mathematician, physicist and astronomer|Wikipedia: Christiaan Huygens}}
{{See also| Huygens vs meanpop }}


Period: 1\2
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.


Optimal ([[POTE]]) generator: ~3/2 = 696.016
Subgroup: 2.3.5.7.11


EDO generators: [[38edo|22\38]], [[50edo|29\50]]
Comma list: 81/80, 99/98, 126/125


Scales (Scala files):  
Mapping: {{mapping| 1 0 -4 -13 -25 | 0 1 4 10 18 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:  
<div style="line-height:1.6;">Technical data</div>
* WE: ~2 = 1200.7636{{c}}, ~3/2 = 697.4122{{c}}
<div class="mw-collapsible-content">
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.0315{{c}}
Comma list: 81/80, 126/125, 245/242


Mapping: [{{val| 2 0 -8 -26 -31 }}, {{val| 0 1 4 10 12 }}]
Minimax tuning:
* 11-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}
: projection map: [{{monzo| 1 0 0 0 0 }}, {{monzo| 25/16 -1/8 0 0 1/16 }}, {{monzo| 9/4 -1/2 0 0 1/4 }}, {{monzo| 21/8 -5/4 0 0 5/8 }}, {{monzo| 25/8 -9/4 0 0 9/8 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/9


Mapping generators: ~63/44, ~3
Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [696.774, 700.000] (18\31 to 7\12)
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)


{{Vals|legend=1| 12, 26de, 38d, 50 }}
Algebraic generator: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.


Badness: 0.0381
{{Optimal ET sequence|legend=0| 12, 19e, 31, 105, 136b }}
</div></div>


=== 13-limit ===
Badness (Sintel): 0.563


Period: 1\2
; 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]


Optimal ([[POTE]]) generator: ~3/2 = 695.836
==== 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]].  


EDO generators: [[38edo|22\38]], [[50edo|29\50]]
Subgroup: 2.3.5.7.11.13


Scales (Scala files):  
Comma list: 81/80, 99/98, 126/125, 144/143


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 1 0 -4 -13 -25 29 | 0 1 4 10 18 -16 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 81/80, 105/104, 126/125, 245/242


Mapping: [{{val| 2 0 -8 -26 -31 -40 }}, {{val| 0 1 4 10 12 15}}]
Optimal tunings:  
* WE: ~2 = 1199.9389{{c}}, ~3/2 = 697.2282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.2627{{c}}


Mapping generators: ~55/39, ~3
Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 8/13 0 0 1/26 0 -1/26 }}
: eigenmonzo basis (unchanged-interval basis): 2.13/7


{{Vals|legend=1| 12f, 26deff, 38df, 50 }}
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.)


Badness: 0.0288
{{Optimal ET sequence|legend=0| 12, 31, 43, 74 }}
</div></div>


== Unidecimal meantone aka Huygens ==
Badness (Sintel): 1.07
{{see also| Meantone vs meanpop }}


Period: 1\1
===== 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.


Optimal ([[POTE]]) generator: ~3/2 = 696.967
Subgroup: 2.3.5.7.11.13.17


EDO generators: [[31edo|18\31]], [[43edo|25\43]]
Comma list: 81/80, 99/98, 120/119, 126/125, 144/143


Scales (Scala files):  
Mapping: {{mapping| 1 0 -4 -13 -25 29 12 | 0 1 4 10 18 -16 -5 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:  
<div style="line-height:1.6;">Technical data</div>
* WE: ~2 = 1199.5811{{c}}, ~3/2 = 697.0918{{c}}
<div class="mw-collapsible-content">
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3303{{c}}
Comma list: 81/80, 126/125, 99/98


Mapping: [{{val| 1 0 -4 -13 -25 }}, {{val| 0 1 4 10 18 }}]
{{Optimal ET sequence|legend=0| 12, 31, 43, 74g }}


Mapping generators: ~2, ~3
Badness (Sintel): 1.06


Minimax tuning:
===== 19-limit =====
* [[11-odd-limit]]
Subgroup: 2.3.5.7.11.13.17.19
: [{{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 }}]
: [[Eigenmonzo]]s: 2, 11/9


Tuning ranges:  
Comma list: 81/80, 96/95, 99/98, 120/119, 126/125, 144/143
* valid range: [696.774, 700.000] (31 to 12)
* nice range: [691.202, 701.955]
* strict range: [696.774, 700.000]
 
[[Algebraic generator]]: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
 
{{Vals|legend=1| 12, 19e, 31, 105, 136b, 167be, 198be }}


Badness: 0.0170
Mapping: {{mapping| 1 0 -4 -13 -25 29 12 9 | 0 1 4 10 18 -16 -5 -3 }}
</div></div>


* [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]
Optimal tunings:
* WE: ~2 = 1199.2931{{c}}, ~3/2 = 696.9690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3736{{c}}


=== Tridecimal meantone ===
{{Optimal ET sequence|legend=0| 12, 31, 43, 74gh }}


Period: 1\1
Badness (Sintel): 1.07


Optimal ([[POTE]]) generator: ~3/2 = 696.642
==== 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.  


EDO generators: [[31edo|18\31]]
This extension used to be known as ''tridecimal meantone'', but was decanonicalized in 2025.


Scales (Scala files):  
Subgroup: 2.3.5.7.11.13


<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: 66/65, 81/80, 99/98, 105/104
Comma list: 66/65, 81/80, 99/98, 105/104


Mapping: [{{val| 1 0 -4 -13 -25 -20 }}, {{val| 0 1 4 10 18 15 }}]
Mapping: {{mapping| 1 0 -4 -13 -25 -20 | 0 1 4 10 18 15 }}


Mapping generators: ~2, ~3
Optimal tunings:
* WE: ~2 = 1200.8149{{c}}, ~3/2 = 697.1155{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.7085{{c}}


{{Vals|legend=1| 12f, 19e, 31 }}
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


Badness: 0.0180
{{Optimal ET sequence|legend=0| 12f, 19e, 31 }}
</div></div>


=== Grosstone ===
Badness (Sintel): 0.746


Period: 1\1
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Optimal ([[POTE]]) generator: ~3/2 = 697.264
Comma list: 66/65, 81/80, 99/98, 105/104, 120/119


EDO generators: [[31edo|18\31]], [[43edo|25\43]]
Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 | 0 1 4 10 18 15 -5 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~2 = 1199.5548{{c}}, ~3/2 = 696.7449{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.9823{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 12f, 31 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 81/80, 99/98, 126/125, 144/143


Mapping: [{{val| 1 0 -4 -13 -25 29 }}, {{val| 0 1 4 10 18 -16 }}]
Badness (Sintel): 1.02


Mapping generators: ~2, ~3
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Tuning ranges:  
Comma list: 66/65, 81/80, 96/95, 99/98, 105/104, 120/119
* valid range: [696.774, 700.000] (31 to 12)
* nice range: [691.202, 701.955]
* strict range: [696.774, 700.000]


{{Vals|legend=1| 12, 19ef, 31, 43, 74 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 9 | 0 1 4 10 18 15 -5 -3 }}


Badness: 0.0259
Optimal tunings:  
</div></div>
* WE: ~2 = 1199.0408{{c}}, ~3/2 = 696.5824{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.1061{{c}}


=== Meridetone ===
{{Optimal ET sequence|legend=0| 12f, 31 }}


Period: 1\1
Badness (Sintel): 1.10


Optimal ([[POTE]]) generator: ~3/2 = 697.529
==== 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.


EDO generators: [[43edo|25\43]]
Subgroup: 2.3.5.7.11.13


Scales (Scala files):
<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: 78/77, 81/80, 99/98, 126/125
Comma list: 78/77, 81/80, 99/98, 126/125


Mapping: [{{val| 1 0 -4 -13 -25 -39 }}, {{val| 0 1 4 10 18 27 }}]
Mapping: {{mapping| 1 0 -4 -13 -25 -39 | 0 1 4 10 18 27 }}
 
Mapping generators: ~2, ~3
 
{{Vals|legend=1| 12f, 31f, 43 }}
 
Badness: 0.0264
</div></div>
 
=== Hemimeantone ===
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~15/13 = 250.304
 
EDO generators: [[43edo|9\43]], [[62edo|13\62]]
 
Scales (Scala files):
 
<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, 99/98, 126/125, 169/168
 
Mapping: [{{val| 1 0 -4 -13 -25 -5 }}, {{val| 0 2 8 20 36 11 }}]
 
Mapping generators: ~2, ~26/15
 
{{Vals|legend=1| 19e, 43, 62, 167bef }}
 
Badness: 0.0314
</div></div>


== Meanpop ==
Optimal tunings:
{{see also| Meantone vs meanpop }}
* WE: ~2 = 1199.9122{{c}}, ~3/2 = 697.4779{{c}}
 
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.5241{{c}}
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 696.434
 
EDO generators: [[19edo|11\19]], [[31edo|18\31]], [[50edo|29\50]]
 
Scales (Scala files):
 
<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, 385/384
 
Mapping: [{{val| 1 0 -4 -13 24 }}, {{val| 0 1 4 10 -13 }}]
 
Mapping generator: ~2, ~3


Minimax tuning:  
Minimax tuning:  
* [[11-odd-limit]]: 1/4 comma
* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/25 -2/25 0 0 0 1/25 }}
: [{{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: 2.13/9
: [[Eigenmonzo]]s: 2, 5


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


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.
Badness (Sintel): 1.09


{{Vals|legend=1| 12e, 19, 31, 81 }}
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


[[Badness]]: 0.0215
Comma list: 78/77, 81/80, 99/98, 120/119, 126/125
</div></div>


* [http://soonlabel.com/xenharmonic/archives/607 Scott Joplin's "The Entertainer" tuned into meanpop]{{Dead link}}
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 | 0 1 4 10 18 27 -5 }}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3 Twinkle canon – 50 edo] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]


=== 13-limit Meanpop ===
Optimal tunings:
* WE: ~2 = 1199.3793{{c}}, ~3/2 = 697.2833{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6222{{c}}


Period: 1\1
{{Optimal ET sequence|legend=0| 12f, 43 }}


Optimal ([[POTE]]) generator: ~3/2 = 696.211
Badness (Sintel): 1.22


EDO generators: [[19edo|11\19]], [[31edo|18\31]], [[50edo|29\50]]
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Scales (Scala files):  
Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 9 | 0 1 4 10 18 27 -5 -3 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 81/80, 105/104, 126/125, 144/143


Mapping: [{{val| 1 0 -4 -13 24 -20 }}, {{val| 0 1 4 10 -13 15 }}]
Optimal tunings:  
* WE: ~2 = 1199.0260{{c}}, ~3/2 = 697.1486{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6887{{c}}


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


Tuning ranges:
Badness (Sintel): 1.25
* valid range: [694.737, 696.774] (19 to 31)
* nice range: [691.202, 701.955]
* strict range: [694.737, 696.774]


{{Vals|legend=1| 12ef, 19, 31, 50, 81, 131bd, 212bbddf }}
==== Hemimeantone ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0209
Comma list: 81/80, 99/98, 126/125, 169/168
</div></div>


=== Meanplop ===
Mapping: {{mapping| 1 0 -4 -13 -25 -5 | 0 2 8 20 36 11 }}


Period: 1\1
: mapping generators: ~2, ~26/15


Optimal ([[POTE]]) generator: ~3/2 = 696.202
Optimal tunings:
* WE: ~2 = 1201.0387{{c}}, ~26/15 = 949.2863{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5065{{c}}


EDO generators: [[19edo|11\19]], [[31edo|18\31]]
{{Optimal ET sequence|legend=0| 19e, 43, 62 }}


Scales (Scala files):  
Badness (Sintel): 1.30


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
===== 17-limit =====
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11.13.17
<div class="mw-collapsible-content">
Comma list: 65/64, 78/77, 81/80, 91/90


Mapping: [{{val| 1 0 -4 -13 24 10 }}, {{val| 0 1 4 10 -13 -4 }}]
Comma list: 81/80, 99/98, 126/125, 169/168, 221/220


Mapping generators: ~2, ~3
Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 | 0 2 8 20 36 11 33 }}


{{Vals|legend=1| 12e, 19, 31f, 50ff, 81fff }}
Optimal tunings:
* WE: ~2 = 1201.0270{{c}}, ~26/15 = 949.2892{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5169{{c}}


Badness: 0.0277
{{Optimal ET sequence|legend=0| 19eg, 43, 62 }}
</div></div>


== Meanenneadecal ==
Badness (Sintel): 1.19


Period: 1\1
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Optimal ([[POTE]]) generator: ~3/2 = 696.250
Comma list: 81/80, 99/98, 126/125, 153/152, 169/168, 221/220


EDO generators: [[12edo|7\12]], [[19edo|11\19]]
Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 -25 | 0 2 8 20 36 11 33 37 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~2 = 1201.0339{{c}}, ~19/11 = 949.2902{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 948.5111{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 19egh, 43, 62 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 45/44, 56/55, 81/80


Mapping: [{{val| 1 0 -4 -13 -6 }}, {{val| 0 1 4 10 6 }}]
Badness (Sintel): 1.15


Mapping generators: ~2, ~3
==== Semimeantone ====
Subgroup: 2.3.5.7.11.13


{{Vals|legend=1| 7d, 12, 19, 31e, 50ee }}
Comma list: 81/80, 99/98, 126/125, 847/845


Badness: 0.0214
Mapping: {{mapping| 2 0 -8 -26 -50 -59 | 0 1 4 10 18 21 }}
</div></div>


=== 13-limit ===
: mapping generators: ~55/39, ~3


Period: 1\1
Optimal tunings:  
* WE: ~55/39 = 600.3606{{c}}, ~3/2 = 697.4241{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 697.0545{{c}}


Optimal ([[POTE]]) generator: ~3/2 = 696.146
{{Optimal ET sequence|legend=0| 12f, …, 50eff, 62, 136b }}


EDO generators: [[12edo|7\12]], [[19edo|11\19]]
Badness (Sintel): 1.68


Scales (Scala files):  
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 81/80, 99/98, 126/125, 221/220, 289/288
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 45/44, 56/55, 78/77, 81/80


Mapping: [{{val| 1 0 -4 -13 -6 -20 }}, {{val| 0 1 4 10 6 15 }}]
Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 | 0 1 4 10 18 21 1 }}


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


{{Vals|legend=1| 12f, 19, 31e, 50ee }}
{{Optimal ET sequence|legend=0| 12f, 50eff, 62, 136bg }}


Badness: 0.0212
Badness (Sintel): 1.60
</div></div>


=== Vincenzo ===
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Period: 1\1
Comma list: 81/80, 99/98, 126/125, 153/152, 209/208, 221/220


Optimal ([[POTE]]) generator: ~3/2 = 695.060
Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 -1 | 0 1 4 10 18 21 1 3 }}


EDO generators: [[12edo|7\12]], [[19edo|11\19]]
Optimal tunings:  
* WE: ~17/12 = 600.5959{{c}}, ~3/2 = 697.5985{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9638{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 12f, 50eff, 62 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.47
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 45/44, 56/55, 65/64, 81/80


Mapping: [{{val| 1 0 -4 -13 -6 10 }}, {{val| 0 1 4 10 6 -4 }}]
=== Meanpop ===
{{See also| Huygens vs meanpop }}


Mapping generators: ~2, ~3
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.


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


Badness: 0.0248
Comma list: 81/80, 126/125, 385/384
</div></div>


==== 17-limit ====
Mapping: {{mapping| 1 0 -4 -13 24 | 0 1 4 10 -13 }}


Period: 1\1
: mapping generator: ~2, ~3


Optimal ([[POTE]]) generator: ~3/2 = 695.858
Optimal tunings:  
* WE: ~2 = 1201.3464{{c}}, ~3/2 = 697.2159{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4509{{c}}


EDO generators: [[12edo|7\12]], [[19edo|11\19]]
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


Scales (Scala files):  
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.)


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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.
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 45/44, 52/51, 56/55, 65/64, 81/80


Mapping: [{{val| 1 0 -4 -13 -6 10 12 }}, {{val| 0 1 4 10 6 -4 -5 }}]
{{Optimal ET sequence|legend=0| 12e, 19, 31, 81, 112b }}


Mapping generators: ~2, ~3
Badness (Sintel): 0.712


{{Vals|legend=1| 7d, 12, 19 }}
; 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]


Badness: 0.0255
==== Tridecimal meanpop ====
</div></div>
Subgroup: 2.3.5.7.11.13


==== 19-limit ====
Comma list: 81/80, 105/104, 126/125, 144/143


Period: 1\1
Mapping: {{mapping| 1 0 -4 -13 24 -20 | 0 1 4 10 -13 15 }}


Optimal ([[POTE]]) generator: ~3/2 = 696.131
Optimal tunings:  
* WE: ~2 = 1201.0765{{c}}, ~3/2 = 696.8361{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2347{{c}}


EDO generators: [[12edo|7\12]], [[19edo|11\19]]
Minimax tuning:  
* 13- and 15-odd-limit: ~3/2 = {{monzo| 4/7 0 0 0 -1/28 1/28 }}
: unchanged-interval (eigenmonzo) basis: 2.13/11


Scales (Scala files):  
Tuning ranges:
* 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.)


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 19, 31, 50, 81 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80


Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 }}, {{val| 0 1 4 10 6 -4 -5 -3 }}]
Badness (Sintel): 0.863


Mapping generators: ~2, ~3
===== Meanpoppic =====
Subgroup: 2.3.5.7.11.13.17


{{Vals|legend=1| 7d, 12, 19 }}
Comma list: 81/80, 105/104, 126/125, 144/143, 273/272


Badness: 0.0223
Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 | 0 1 4 10 -13 15 26 }}
</div></div>


==== 23-limit ====
Optimal tunings:
* WE: ~2 = 1201.0727{{c}}, ~3/2 = 696.8168{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2195{{c}}


Period: 1\1
{{Optimal ET sequence|legend=0| 19g, 31, 50, 81, 131bd }}


Optimal ([[POTE]]) generator: ~3/2 = 696.044
Badness (Sintel): 1.02


EDO generators: [[12edo|7\12]], [[19edo|11\19]]
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19


Scales (Scala files):  
Comma list: 81/80, 105/104, 126/125, 144/143, 153/152, 273/272


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 -40 | 0 1 4 10 -13 15 26 28 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 69/68, 81/80


Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 14 }}, {{val| 0 1 4 10 6 -4 -5 -3 -6 }}]
Optimal tunings:  
* WE: ~2 = 1201.0719{{c}}, ~3/2 = 696.8101{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2137{{c}}


Mapping generators: ~2, ~3
{{Optimal ET sequence|legend=0| 19gh, 31, 50, 81 }}


{{Vals|legend=1| 7d, 12, 19 }}
Badness (Sintel): 1.08


Badness: 0.0201
===== Meanpoid =====
</div></div>
Subgroup: 2.3.5.7.11.13.17


==== 29-limit ====
Comma list: 81/80, 105/104, 120/119, 126/125, 144/143


Period: 1\1
Mapping: {{mapping| 1 0 -4 -13 24 -20 12 | 0 1 4 10 -13 15 -5 }}


Optimal ([[POTE]]) generator: ~3/2 = 695.913
Optimal tunings:  
* WE: ~2 = 1200.2768{{c}}, ~3/2 = 696.5683{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4114{{c}}


EDO generators: [[12edo|7\12]], [[19edo|11\19]]
{{Optimal ET sequence|legend=0| 19, 31 }}


Scales (Scala files):  
Badness (Sintel): 1.17


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
====== 19-limit ======
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11.13.17.19
<div class="mw-collapsible-content">
Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80


Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 14 8 }}, {{val| 0 1 4 10 6 -4 -5 -3 -6 -2 }}]
Comma list: 77/76, 81/80, 96/95, 105/104, 120/119, 126/125


Mapping generators: ~2, ~3
Mapping: {{mapping| 1 0 -4 -13 24 -20 12 9 | 0 1 4 10 -13 15 -5 -3 }}


{{Vals|legend=1| 7d, 12, 19 }}
Optimal tunings:
* WE: ~2 = 1199.7905{{c}}, ~3/2 = 696.3779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4973{{c}}


Badness: 0.0182
{{Optimal ET sequence|legend=0| 19, 31 }}
</div></div>


==== 31-limit ====
Badness (Sintel): 1.25


Period: 1\1
==== Semimeanpop ====
Subgroup: 2.3.5.7.11.13


Optimal ([[POTE]]) generator: ~3/2 = 695.750
Comma list: 81/80, 126/125, 385/384, 847/845


EDO generators: [[12edo|7\12]], [[19edo|11\19]]
Mapping: {{mapping| 2 0 -8 -26 48 39 | 0 1 4 10 -13 -10 }}


Scales (Scala files):  
: mapping generators: ~55/39, ~3


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:  
<div style="line-height:1.6;">Technical data</div>
* WE: ~55/39 = 600.6704{{c}}, ~3/2 = 697.2151{{c}}
<div class="mw-collapsible-content">
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.4341{{c}}
Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80, 93/92


Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 14 8 16 }}, {{val| 0 1 4 10 6 -4 -5 -3 -6 -2 -7 }}]
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112b }}


Mapping generators: ~2, ~3
Badness (Sintel): 1.78


{{Vals|legend=1| 7d, 12, 19 }}
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0171
Comma list: 81/80, 126/125, 221/220, 273/272, 289/288
</div></div>


==== 37-limit ====
Mapping: {{mapping| 2 0 -8 -26 48 39 5 | 0 1 4 10 -13 -10 1 }}


Period: 1\1
Optimal tunings:  
* WE: ~17/12 = 600.7232{{c}}, ~3/2 = 697.2820{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4411{{c}}


Optimal ([[POTE]]) generator: ~3/2 = 695.603
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bg }}


EDO generators: [[12edo|7\12]], [[19edo|11\19]]
Badness (Sintel): 1.45


Scales (Scala files):  
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 81/80, 126/125, 153/152, 209/208, 221/220, 273/272
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92


Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 14 8 16 -9 }}, {{val| 0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 }}]
Mapping: {{mapping| 2 0 -8 -26 48 39 5 -1 | 0 1 4 10 -13 -10 1 3 }}


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


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


Badness: 0.0161
Badness (Sintel): 1.28
</div></div>


==== 41-limit ====
=== 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.


Period: 1\1
Subgroup: 2.3.5.7.11


Optimal ([[POTE]]) generator: ~3/2 = 695.696
Comma list: 45/44, 56/55, 81/80


EDO generators: [[12edo|7\12]], [[19edo|11\19]]
Mapping: {{mapping| 1 0 -4 -13 -6 | 0 1 4 10 6 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~2 = 1199.6946{{c}}, ~3/2 = 696.0729{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2083{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Tuning ranges:
<div style="line-height:1.6;">Technical data</div>
* 11-odd-limit diamond monotone: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
<div class="mw-collapsible-content">
* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 704.377]
Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92, 124/123


Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 }}, {{val| 0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 }}]
{{Optimal ET sequence|legend=0| 7d, 12, 19, 31e }}


Mapping generators: ~2, ~3
Badness (Sintel): 0.708


{{Vals|legend=1| 7d, 12, 19 }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0154
Comma list: 45/44, 56/55, 78/77, 81/80
</div></div>


==== 43-limit ====
Mapping: {{mapping| 1 0 -4 -13 -6 -20 | 0 1 4 10 6 15 }}


Period: 1\1
Optimal tunings:  
* WE: ~2 = 1199.7931{{c}}, ~3/2 = 696.0258{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1241{{c}}


Optimal ([[POTE]]) generator: ~3/2 = 695.688
{{Optimal ET sequence|legend=0| 7df, 12f, 19, 31e }}


EDO generators: [[12edo|7\12]], [[19edo|11\19]]
Badness (Sintel): 0.875


Scales (Scala files):  
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 45/44, 56/55, 78/77, 81/80, 120/119
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 86/85, 93/92, 124/123


Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7 }}, {{val| 0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1 }}]
Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 | 0 1 4 10 6 15 -5 }}


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


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


Badness: 0.0139
Badness (Sintel): 1.17
</div></div>


==== 47-limit ====
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Period: 1\1
Comma list: 45/44, 56/55, 78/77, 81/80, 96/95, 120/119


Optimal ([[POTE]]) generator: ~3/2 = 695.676
Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 9 | 0 1 4 10 6 15 -5 -3 }}


EDO generators: [[12edo|7\12]], [[19edo|11\19]]
Optimal tunings:  
* WE: ~2 = 1198.2880{{c}}, ~3/2 = 695.7123{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.6370{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 12f, 19, 31e }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.23
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 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


Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7 4 }}, {{val| 0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1 1 }}]
==== Vincenzo ====
Subgroup: 2.3.5.7.11.13


Mapping generators: ~2, ~3
Comma list: 45/44, 56/55, 65/64, 81/80


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


Badness: 0.0138
Optimal tunings:  
</div></div>
* WE: ~2 = 1202.1684{{c}}, ~3/2 = 696.3160{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.2045{{c}}


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


Period: 1\1
Badness (Sintel): 1.02


Optimal ([[POTE]]) generator: ~3/2 = 697.254
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


EDO generators: [[12edo|7\12]]
Comma list: 45/44, 52/51, 56/55, 65/64, 81/80


Scales (Scala files):  
Mapping: {{mapping| 1 0 -4 -13 -6 10 12 | 0 1 4 10 6 -4 -5 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:  
<div style="line-height:1.6;">Technical data</div>
* WE: ~2 = 1200.5137{{c}}, ~3/2 = 696.1561{{c}}
<div class="mw-collapsible-content">
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.8771{{c}}
Comma list: 27/26, 40/39, 45/44, 56/55


Mapping: [{{val| 1 0 -4 -13 -6 -1 }}, {{val| 0 1 4 10 6 3 }}]
{{Optimal ET sequence|legend=0| 12, 19 }}


Mapping generators: ~2, ~3
Badness (Sintel): 1.30


{{Vals|legend=1| 7d, 12f, 19f, 31eff }}
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.0242
Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80
</div></div>


== Meanundeci ==
Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 | 0 1 4 10 6 -4 -5 -3 }}


Period: 1\1
Optimal tunings:  
* WE: ~2 = 1199.8261{{c}}, ~3/2 = 696.0298{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1262{{c}}


Optimal ([[POTE]]) generator: ~3/2 = 694.689
{{Optimal ET sequence|legend=0| 12, 19 }}


EDO generators: [[12edo|7\12]], [[19edo|11\19]]
Badness (Sintel): 1.36


Scales (Scala files):
=== Bimeantone ===
11/8 is mapped to half octave minus the [[128/125|meantone diesis]].


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Subgroup: 2.3.5.7.11
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 33/32, 55/54, 77/75


POTE generator: ~3/2 = 694.689
Comma list: 81/80, 126/125, 245/242


Mapping: [{{val|1 0 -4 -13 5 }}, {{val| 0 1 4 10 -1 }}]
Mapping: {{mapping| 2 0 -8 -26 -31 | 0 1 4 10 12 }}


Mapping generator: ~3
: mapping generators: ~63/44, ~3


{{Vals|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: 0.0315
{{Optimal ET sequence|legend=0| 12, 26de, 38d, 50 }}
</div></div>


=== 13-limit ===
Badness (Sintel): 1.26


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


Optimal ([[POTE]]) generator: ~3/2 = 694.764
Comma list: 81/80, 105/104, 126/125, 245/242


EDO generators: [[12edo|7\12]], [[19edo|11\19]]
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;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 33/32, 55/54, 65/64, 77/75


Mapping: [{{val|1 0 -4 -13 5 10 }}, {{val| 0 1 4 10 -1 -4 }}]
Badness (Sintel): 1.19


Mapping generators: ~2, ~3
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


{{Vals|legend=1| 7d, 12e, 19e }}
Comma list: 81/80, 105/104, 126/125, 189/187, 221/220


Badness: 0.0263
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 | 0 1 4 10 12 15 1 }}
</div></div>


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


Period: 1\1
{{Optimal ET sequence|legend=0| 12f, 38df, 50 }}


Optimal ([[POTE]]) generator: ~3/2 = 693.779
Badness (Sintel): 1.15


EDO generators: [[19edo|11\19]], [[26edo|15\26]], [[45edo|26\45]], [[64edo|37\64]]
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Scales (Scala files): [[Flattone12]]
Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 -1 | 0 1 4 10 12 15 1 3 }}
<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
Optimal tunings:  
* WE: ~17/12 = 600.9845{{c}}, ~3/2 = 696.8939{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.8947{{c}}


[[Mapping]]: [{{val|1 0 -4 17}}, {{val|0 1 4 -9}}]
{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}


Mapping generators: ~2, ~3
Badness (Sintel): 1.08


[[Wedgie]]: {{wedgie|1 4 -9 4 -17 -32}}
=== Trimean ===
{{See also| No-sevens subgroup temperaments #Superpine }}


[[Minimax tuning]]:  
Subgroup: 2.3.5.7.11


* [[7-odd-limit]]
Comma list: 81/80, 126/125, 1344/1331
: [{{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]]
Mapping: {{mapping| 1 2 4 7 5 | 0 -3 -12 -30 -11 }}
: [{{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]]:  
: mapping generators: ~2, ~11/10
* valid range: [692.308, 694.737] (26 to 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.
Optimal tunings:  
* WE: ~2 = 1200.7155{{c}}, ~11/10 = 167.9055{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7749{{c}}


{{Vals|legend=1| 7, 19, 26, 45 }}
{{Optimal ET sequence|legend=0| 7d, 36d, 43, 50, 93 }}


[[Badness]]: 0.0386
Badness (Sintel): 1.68
</div></div>


== 11-limit ==
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Period: 1\1
Comma list: 81/80, 126/125, 144/143, 364/363


Optimal ([[POTE]]) generator: ~3/2 = 693.126
Mapping: {{mapping| 1 2 4 7 5 3 | 0 -3 -12 -30 -11 5 }}


EDO generators: [[19edo|11\19]], [[26edo|15\26]], [[45edo|26\45]], [[64edo|37\64]]
Optimal tunings:  
* WE: ~2 = 1200.6104{{c}}, ~11/10 = 167.8749{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7728{{c}}


Scales (Scala files): [[Flattone12]]
{{Optimal ET sequence|legend=0| 7d, 43, 50, 93 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.46
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 45/44, 81/80, 385/384


Mapping: [{{val| 1 0 -4 17 -6 }}, {{val| 0 1 4 -9 6 }}]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Mapping generators: ~2, ~3
Comma list: 81/80, 126/125, 144/143, 189/187, 221/220


Tuning ranges:  
Mapping: {{mapping| 1 2 4 7 5 3 8 | 0 -3 -12 -30 -11 5 -28 }}
* valid range: [692.308, 694.737] (26 to 19)
* nice range: [682.502, 701.955]
* strict range: [692.308, 694.737]


{{Vals|legend=1| 7, 19, 26, 45, 71bc, 116bcde }}
Optimal tunings:
* WE: ~2 = 1200.6144{{c}}, ~11/10 = 167.8716{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7682{{c}}


Badness: 0.0338
{{Optimal ET sequence|legend=0| 7dg, 43, 50, 93 }}
</div></div>


== 13-limit ==
Badness (Sintel): 1.28


Period: 1\1
=== Migration ===
See [[Rastmic clan #Migration|Rastmic clan]].


Optimal ([[POTE]]) generator: ~3/2 = 693.058
== Flattone ==
{{Main| Flattone }}


EDO generators: [[19edo|11\19]], [[26edo|15\26]], [[45edo|26\45]], [[64edo|37\64]]
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]].


Scales (Scala files): [[Flattone12]]
[[Subgroup]]: 2.3.5.7


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Comma list]]: 81/80, 525/512
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 45/44, 65/64, 78/77, 81/80


Mapping: [{{val| 1 0 -4 17 -6 10 }}, {{val| 0 1 4 -9 6 -4 }}]
{{Mapping|legend=1| 1 0 -4 17 | 0 1 4 -9 }}


Mapping generators: ~2, ~3
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.6308{{c}}, ~3/2 = 695.8782{{c}}
: [[error map]]: {{val| +3.631 -2.446 -2.801 -2.684 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.7334{{c}}
: error map: {{val| 0.000 -8.222 -11.380 -12.426 }}


Tuning ranges:
[[Minimax tuning]]:
* valid range: [692.308, 694.737] (26 to 19)
* [[7-odd-limit]]: ~3/2 = {{monzo| 8/13 0 1/13 -1/13 }}
* nice range: [682.502, 701.955]
: [[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 }}]
* strict range: [692.308, 694.737]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
 
* [[9-odd-limit]]: ~3/2 = {{monzo| 6/11 2/11 0 -1/11 }}
{{Vals|legend=1| 7, 19, 26, 45f, 71bcf, 116bcdef }}
: [[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
Badness: 0.0223
</div></div>
 
= Godzilla =
<span style="display: block; text-align: right;">[[:de:Semiphor,_Semaphor,_Godzilla|Deutsch]]</span>
 
{{main| Semaphore and Godzilla }}
 
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.
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~8/7 = 252.635
 
EDO generators: [[14edo|3\14]], [[19edo|4\19]], [[24edo|5\24]], [[33edo|7\33]], [[43edo|9\43]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<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
| 21/20, 28/27
|-
| 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
 
[[Mapping]]: [{{val| 1 0 -4 2 }}, {{val| 0 2 8 1 }}]
 
Mapping generators: ~2, ~7/4
 
[[Wedgie]]: {{wedgie| 2 8 1 8 -4 -20 }}


[[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]


{{Vals|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


Period: 1\1
=== 11-limit ===
This can also be considered a no-sevens temperament: [[#Hypnotone|hypnotone]].


Optimal ([[POTE]]) generator: ~8/7 = 254.027
Subgroup: 2.3.5.7.11


EDO generators: [[14edo|3\14]], [[19edo|4\19]], [[33edo|7\33]]
Comma list: 45/44, 81/80, 385/384
 
Scales (Scala files):
 
<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: 45/44, 49/48, 81/80


Mapping: [{{val| 1 0 -4 2 -6 }}, {{val| 0 2 8 1 12 }}]
Mapping: {{mapping| 1 0 -4 17 -6 | 0 1 4 -9 6 }}


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


Tuning ranges:  
Tuning ranges:  
* valid range: [252.632, 257.143] (19 to 14c)
* 11-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* nice range: [231.174, 266.871]
* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]
* strict range: [252.632, 257.143]


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


Badness: 0.0290
Badness (Sintel): 1.12
</div></div>


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Period: 1\1
Comma list: 45/44, 65/64, 78/77, 81/80
 
Optimal ([[POTE]]) generator: ~8/7 = 253.603
 
EDO generators: [[19edo|4\19]]


Scales (Scala files):
Mapping: {{mapping| 1 0 -4 17 -6 10 | 0 1 4 -9 6 -4 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:  
<div style="line-height:1.6;">Technical data</div>
* WE: ~2 = 1202.5156{{c}}, ~3/2 = 694.5107{{c}}
<div class="mw-collapsible-content">
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0538{{c}}
Comma list: 45/44, 49/48, 78/77, 81/80
 
Mapping: [{{val| 1 0 -4 2 -6 -5 }}, {{val| 0 2 8 1 12 11 }}]
 
Mapping generators: ~2, ~7/4


Tuning ranges:  
Tuning ranges:  
* valid range: 694.737 (19)
* 13- and 15-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* nice range: [621.581, 737.652]
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]
* strict range: 694.737
 
{{Vals|legend=1| 14cf, 19, 33cdff, 52cdf }}
 
Badness: 0.0225
</div></div>
 
== Semafour ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~8/7 = 254.042
 
EDO generators: [[14edo|3\14]], [[19edo|4\19]]
 
Scales (Scala files):
 
<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: 33/32, 49/48, 55/54
 
Mapping: [{{val| 1 0 -4 2 5 }}, {{val| 0 2 8 1 -2 }}]
 
Mapping generators: ~2, ~7/4
 
{{Vals|legend=1| 14c, 19e, 33cdee }}
 
Badness: 0.0285
</div></div>
 
== Varan ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~8/7 = 251.079
 
EDO generators: [[19edo|4\19]], [[24edo|5\24]], [[43edo|9\43]]
 
Scales (Scala files):
 
<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, 77/75, 81/80
 
Mapping: [{{val| 1 0 -4 2 -10 }}, {{val| 0 2 8 1 17 }}]
 
Mapping generators: ~2, ~7/4
 
{{Vals|legend=1| 19e, 24, 43de }}


Badness: 0.0396
{{Optimal ET sequence|legend=0| 7, 19, 26, 45f, 71bcf, 116bcdef }}
</div></div>


=== 13-limit ===
Badness (Sintel): 0.920


Period: 1\1
=== Ptolemy ===
See [[Rastmic clan #Ptolemy|Rastmic clan]].


Optimal ([[POTE]]) generator: ~8/7 = 251.165
== Dominant ==
{{Main| Dominant (temperament) }}
{{See also| Archytas clan }}


EDO generators: [[19edo|4\19]], [[24edo|5\24]], [[43edo|9\43]]
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]].


Scales (Scala files):
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.


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Subgroup]]: 2.3.5.7
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 49/48, 66/65, 77/75, 81/80


Mapping: [{{val| 1 0 -4 2 -10 -5 }}, {{val| 0 2 8 1 17 11 }}]
[[Comma list]]: 36/35, 64/63


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


{{Vals|legend=1| 19e, 24, 43de }}
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1195.3384{{c}}, ~3/2 = 698.8478{{c}}
Badness: 0.0257
: [[error map]]: {{val| -4.662 -7.769 +9.077 +14.832 }}
</div></div>
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.1125{{c}}
 
: error map: {{val| 0.000 -0.842 +18.136 +28.949 }}
== Baragon ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~8/7 = 251.173
 
EDO generators: [[19edo|4\19]], [[24edo|5\24]], [[43edo|9\43]]
 
Scales (Scala files):
 
<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, 56/55, 81/80
 
Mapping: [{{val| 1 0 -4 2 9 }}, {{val| 0 2 8 1 -7 }}]
 
Mapping generators: ~2, ~7/4
 
{{Vals|legend=1| 19, 24, 43d }}
 
Badness: 0.0357
</div></div>
 
== Music ==
* [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]]
 
= Dominant =
 
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]].
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 701.573
 
EDO generators: [[5edo|3\5]], [[7edo|4\7]], [[12edo|7\12]], [[17edo|10\17]]
 
Scales (Scala files):
 
<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: 36/35, 64/63
 
Mapping: [{{val| 1 0 -4 6 }}, {{val| 0 1 4 -2 }}]
 
Mapping generators: ~2, ~3
 
[[Wedgie]]: {{wedgie| 1 4 -2 4 -6 -16 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* valid range: [700.000, 720.000] (12 to 5)
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[diamond monotone]]: ~3/2 = [700.000, 720.000] (7\12 to 3\5)
* nice range: [694.786, 715.587]
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 715.587]
* strict range: [700.000, 715.587]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]


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


[[Badness]]: 0.0207
[[Badness]] (Sintel): 0.524
</div></div>


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11


Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 703.254
EDO generators: [[12edo|7\12]], [[17edo|10\17]]
Scales (Scala files):
<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: 36/35, 56/55, 64/63
Comma list: 36/35, 56/55, 64/63


Mapping: [{{val| 1 0 -4 6 13 }}, {{val| 0 1 4 -2 -6 }}]
Mapping: {{mapping| 1 0 -4 6 13 | 0 1 4 -2 -6 }}
 
Mapping generators: ~2, ~3


Tuning ranges:  
Tuning ranges:  
* valid range: [700.000, 705.882] (12 to 17)
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 705.882] (7\12 to 10\17)
* nice range: [691.202, 715.587]
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
* strict range: [700.000, 705.882]
 
{{Vals|legend=1| 5, 12, 17c, 29cde }}


Badness: 0.0242
Optimal tunings:  
</div></div>
* WE: ~2 = 1194.0169{{c}}, ~3/2 = 699.7473{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.2672{{c}}


=== 13-limit ===
{{Optimal ET sequence|legend=0| 5, 12, 17c, 29cde }}


Period: 1\1
Badness (Sintel): 0.799


Optimal ([[POTE]]) generator: ~3/2 = 703.636
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDO generators: [[12edo|7\12]], [[17edo|10\17]]
Scales (Scala files):
<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: 36/35, 56/55, 64/63, 66/65
Comma list: 36/35, 56/55, 64/63, 66/65


Mapping: [{{val| 1 0 -4 6 13 18 }}, {{val| 0 1 4 -2 -6 -9 }}]
Mapping: {{mapping| 1 0 -4 6 13 18 | 0 1 4 -2 -6 -9 }}


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


Tuning ranges:  
Tuning ranges:  
* valid range: 705.882 (17)
* 13- and 15-odd-limit diamond monotone: ~3/2 = 705.882 (10\17)
* nice range: [691.202, 715.587]
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
* strict range: 705.882


{{Vals|legend=1| 12f, 17c, 29cdef }}
{{Optimal ET sequence|legend=0| 12f, 17c, 29cdef }}


Badness: 0.0241
Badness (Sintel): 0.996
</div></div>


=== Dominion ===
==== Dominion ====
Subgroup: 2.3.5.7.11.13


Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 704.905
EDO generators: [[17edo|10\17]]
Scales (Scala files):
<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: 26/25, 36/35, 56/55, 64/63
Comma list: 26/25, 36/35, 56/55, 64/63


Mapping: [{{val| 1 0 -4 6 13 -9 }}, {{val| 0 1 4 -2 -6 8 }}]
Mapping: {{mapping| 1 0 -4 6 13 -9 | 0 1 4 -2 -6 8 }}


{{Vals|legend=1| 5, 12, 17c, 46cde }}
Optimal tunings:
* WE: ~2 = 1195.0293{{c}}, ~3/2 = 701.9847{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7698{{c}}


Badness: 0.0273
{{Optimal ET sequence|legend=0| 5, 12, 17c }}
</div></div>


== Domineering ==
Badness (Sintel): 1.13


Period: 1\1
=== Domination ===
Subgroup: 2.3.5.7.11


Optimal ([[POTE]]) generator: ~3/2 = 698.776
Comma list: 36/35, 64/63, 77/75


EDO generators: [[7edo|4\7]], [[12edo|7\12]]
Mapping: {{mapping| 1 0 -4 6 -14 | 0 1 4 -2 11 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~2 = 1194.8645{{c}}, ~3/2 = 701.9872{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5945{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 36/35, 45/44, 64/63


Mapping: [{{val| 1 0 -4 6 -6 }}, {{val| 0 1 4 -2 6 }}]
Badness (Sintel): 1.21


Mapping generators: ~2, ~3
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Vals|legend=1| 5e, 7, 12, 19d, 43de }}
Comma list: 26/25, 36/35, 64/63, 66/65


Badness: 0.0220
Mapping: {{mapping| 1 0 -4 6 -14 -9 | 0 1 4 -2 11 8 }}
</div></div>


=== 13-limit ===
Optimal tunings:
* WE: ~2 = 1195.1324{{c}}, ~3/2 = 702.6343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.0791{{c}}


Period: 1\1
{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}


Optimal ([[POTE]]) generator: ~3/2 = 695.762
Badness (Sintel): 1.13


EDO generators: [[7edo|4\7]], [[12edo|7\12]]
=== Domineering ===
Subgroup: 2.3.5.7.11


Scales (Scala files):  
Comma list: 36/35, 45/44, 64/63


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 1 0 -4 6 -6 | 0 1 4 -2 6 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 36/35, 45/44, 52/49, 64/63


Mapping: [{{val| 1 0 -4 6 -6 10 }}, {{val| 0 1 4 -2 6 -4 }}]
Optimal tunings:  
* WE: ~2 = 1194.7102{{c}}, ~3/2 = 695.6962{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1765{{c}}


Mapping generators: ~2, ~3
{{Optimal ET sequence|legend=0| 5e, 7, 12 }}


{{Vals|legend=1| 5ef, 7, 12, 19d, 31def }}
Badness (Sintel): 0.727


Badness: 0.0270
=== Arnold ===
</div></div>
Subgroup: 2.3.5.7.11


==== 17-limit ====
Comma list: 22/21, 33/32, 36/35


Period: 1\1
Mapping: {{mapping| 1 0 -4 6 5 | 0 1 4 -2 -1 }}


Optimal ([[POTE]]) generator: ~3/2 = 696.115
Optimal tunings:  
* WE: ~2 = 1199.8507{{c}}, ~3/2 = 698.4045{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.4822{{c}}


EDO generators: [[7edo|4\7]], [[12edo|7\12]]
{{Optimal ET sequence|legend=0| 5, 7, 12e }}


Scales (Scala files):  
Badness (Sintel): 0.864


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
=== Neutrominant ===
<div style="line-height:1.6;">Technical data</div>
See [[Rastmic clan #Neutrominant|Rastmic clan]].
<div class="mw-collapsible-content">
Comma list: 36/35, 45/44, 51/49, 52/49, 64/63


Mapping: [{{val| 1 0 -4 6 -6 10 12 }}, {{val| 0 1 4 -2 6 -4 -5 }}]
== 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.


Mapping generators: ~2, ~3
Flattertone was named by [[Flora Canou]] in 2024.


{{Vals|legend=1| 5ef, 7, 12, 19d, 31def }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0245
[[Comma list]]: 81/80, 1875/1792
</div></div>


==== 19-limit ====
{{Mapping|legend=1| 1 0 -4 -24 | 0 1 4 17 }}


Period: 1\1
: mapping generators: ~2, ~3


Optimal ([[POTE]]) generator: ~3/2 = 696.217
[[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 }}


EDO generators: [[7edo|4\7]], [[12edo|7\12]]
{{Optimal ET sequence|legend=1| 7d, 19d, 26, 59bcd, 85bccd }}


Scales (Scala files):  
[[Badness]] (Sintel): 2.43


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
==== 11-limit ====
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11
<div class="mw-collapsible-content">
Comma list: 36/35, 39/38, 45/44, 51/49, 52/49, 57/56


Mapping: [{{val| 1 0 -4 6 -6 10 12 9 }}, {{val| 0 1 4 -2 6 -4 -5 -3 }}]
Comma list: 45/44, 81/80, 1375/1344


Mapping generators: ~2, ~3
Mapping: {{mapping| 1 0 -4 -24 -6 | 0 1 4 17 6 }}


{{Vals|legend=1| 5ef, 7, 12, 19d, 31def }}
Optimal tunings:
* WE: ~2 = 1203.4653{{c}}, ~3/2 = 693.8144{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 692.0422{{c}}


Badness: 0.0204
{{Optimal ET sequence|legend=0| 7d, 19d, 26 }}
</div></div>


=== Dominatrix ===
Badness (Sintel): 1.53


Period: 1\1
; Music
* [https://youtu.be/scCuGXnj5IY ''Music in 33EDO (33-Tone Equal Temperament) - Feb 2024''] by [[Budjarn Lambeth]] (2024)


Optimal ([[POTE]]) generator: ~3/2 = 698.544
== 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.


EDO generators: [[7edo|4\7]], [[12edo|7\12]]
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).


Scales (Scala files):
The 11-limit extension was named by Gene Ward Smith in 2004<ref name="meantone & meanpop 2004"/>.


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Subgroup]]: 2.3.5.7
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 27/26, 36/35, 45/44, 64/63


Mapping: [{{val| 1 0 -4 6 -6 -1 }}, {{val| 0 1 4 -2 6 3 }}]
[[Comma list]]: 21/20, 28/27


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


{{Vals|legend=1| 5e, 7, 12f, 19df }}
[[Optimal tuning]]s:
</div></div>
* [[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 }}


== Domination ==
{{Optimal ET sequence|legend=1| 5, 7d, 12d }}


Period: 1\1
[[Badness]] (Sintel): 0.629


Optimal ([[POTE]]) generator: ~3/2 = 705.004
=== Meanertone ===
Subgroup: 2.3.5.7.11


EDO generators: [[12edo|7\12]], [[17edo|10\17]]
Comma list: 21/20, 28/27, 33/32
 
Scales (Scala files):
 
<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: 36/35, 64/63, 77/75
 
Mapping: [{{val| 1 0 -4 6 -14 }}, {{val| 0 1 4 -2 11 }}]
 
Mapping generators: ~2, ~3


{{Vals|legend=1| 5e, 12e, 17c, 46cd }}
Mapping: {{mapping| 1 0 -4 -2 5 | 0 1 4 3 -1 }}


Badness: 0.0366
Optimal tunings:  
</div></div>
* WE: ~2 = 1208.5304{{c}}, ~3/2 = 701.5669{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1117{{c}}


=== 13-limit ===
{{Optimal ET sequence|legend=0| 5, 7d, 12de }}


Period: 1\1
Badness (Sintel): 0.832


Optimal ([[POTE]]) generator: ~3/2 = 705.496
== 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.  


EDO generators: [[12edo|7\12]], [[17edo|10\17]]
Mildtone was named by [[User: Lucius Chiaraviglio|Lucius Chiaraviglio]] in 2024.


Scales (Scala files):  
[[Subgroup]]: 2.3.5.7


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Comma list]]: 81/80, 16128/15625
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 26/25, 36/35, 64/63, 66/65


Mapping: [{{val| 1 0 -4 6 -14 -9 }}, {{val| 0 1 4 -2 11 8 }}]
{{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 }}


Mapping generators: ~2, ~3
[[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 }}


{{Vals|legend=1| 5e, 12e, 17c }}
{{Optimal ET sequence|legend=1| 12, 43d, 55, 67 }}


Badness: 0.0274
[[Badness]] (Sintel): 2.67
</div></div>


== Arnold ==
=== 11-limit ===


Period: 1\1
[[Subgroup]]: 2.3.5.7.11


Optimal ([[POTE]]) generator: ~3/2 = 698.491
[[Comma list]]: 81/80, 176/175, 7056/6875


EDO generators: [[5edo|3\5]], [[7edo|4\7]], [[12edo|7\12]]
{{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 30}}


Scales (Scala files):  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.816{{c}}, ~3/2 = 698.355{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.455{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=1| 12, 43de, 55, 67 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 22/21, 33/32, 36/35


Mapping: [{{val| 1 0 -4 6 5 }}, {{val| 0 1 4 -2 -1 }}]
[[Badness]] (Sintel): 2.15
 
Mapping generators: ~2, ~3
 
{{Vals|legend=1| 5, 7, 12e }}
 
Badness: 0.0261
</div></div>


=== 13-limit ===
=== 13-limit ===


Period: 1\1
[[Subgroup]]: 2.3.5.7.11.13
 
Optimal ([[POTE]]) generator: ~3/2 = 696.743
 
EDO generators: [[5edo|3\5]], [[7edo|4\7]], [[12edo|7\12]]
 
Scales (Scala files):


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Comma list]]: 81/80, 176/175, 196/195, 832/825
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Commas: 22/21, 27/26, 33/32, 36/35
{{Mapping|legend=1| 1 0 -4 -32 -44 | 0 1 4 22 30}}


Mapping: [{{val| 1 0 -4 6 5 -1 }}, {{val| 0 1 4 -2 3 }}]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.788{{c}}, ~3/2 = 698.355{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.471{{c}}


Mapping generators: ~2, ~3
{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}


{{Vals|legend=1| 5, 7, 12ef, 19def }}
[[Badness]] (Sintel): 2.04
 
Badness: 0.0233
</div></div>


=== 17-limit ===
=== 17-limit ===


Period: 1\1
[[Subgroup]]: 2.3.5.7.11.13.17
 
Optimal ([[POTE]]) generator: ~3/2 = 696.978
 
EDO generators: [[5edo|3\5]], [[7edo|4\7]], [[12edo|7\12]]


Scales (Scala files):  
[[Comma list]]: 81/80, 176/175, 189/187, 196/195, 832/825


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Mapping|legend=1| 1 0 -4 -32 -44 12| 0 1 4 22 30 -5}}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Commas: 22/21, 27/26, 33/32, 36/35, 51/49


Mapping: [{{val| 1 0 -4 6 5 -1 12 }}, {{val| 0 1 4 -2 3 -5 }}]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.655{{c}}, ~3/2 = 698.295{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.488{{c}}


Mapping generators: ~2, ~3
{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}


{{Vals|legend=1| 5, 7, 12ef, 19def }}
[[Badness]] (Sintel): 1.98
 
Badness: 0.0245
</div></div>


=== 19-limit ===
=== 19-limit ===


Period: 1\1
[[Subgroup]]: 2.3.5.7.11.13.17.19
 
Optimal ([[POTE]]) generator: ~3/2 = 697.068
 
EDO generators: [[5edo|3\5]], [[7edo|4\7]], [[12edo|7\12]]
 
Scales (Scala files):
 
<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">
Commas: 22/21, 27/26, 33/32, 36/35, 51/49, 57/56
 
Mapping: [{{val| 1 0 -4 6 5 -1 12 9 }}, {{val| 0 1 4 -2 3 -5 -3 }}]
 
Mapping generators: ~2, ~3
 
{{Vals|legend=1| 5, 7, 12ef, 19def }}
 
Badness: 0.0211
</div></div>
 
= Sharptone =
 
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.
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 700.140
 
EDO generators: [[5edo|3\5]], [[7edo|4\7]], [[12edo|7\12]]
 
Scales (Scala files):
 
<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: 21/20, 28/27
 
Mapping: [{{val| 1 0 -4 -2 }}, {{val| 0 1 4 3 }}]
 
Mapping generators: ~2, ~3
 
[[Wedgie]]: {{wedgie| 1 4 3 4 2 -4 }}
 
{{Vals|legend=1| 5, 7d, 12d }}
 
[[Badness]]: 0.0248
</div></div>
 
== Meanertone ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 696.615
 
EDO generators: [[5edo|3\5]], [[7edo|4\7]], [[12edo|7\12]]
 
Scales (Scala files):
 
<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: 21/20, 28/27, 33/32
 
Mapping: [{{val| 1 0 -4 -2 5 }}, {{val| 0 1 4 3 -1 }}]


{{Vals|legend=1| 5, 7d, 12de }}
[[Comma list]]: 81/80, 96/95, 176/175, 189/187, 196/195, 832/825


Badness: 0.0252
{{Mapping|legend=1| 1 0 -4 -32 -44 12 9| 0 1 4 22 30 -5 -3}}
</div></div>


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


Period: 1\1
{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}


Optimal ([[POTE]]) generator: ~3/2 = 682.895
[[Badness]] (Sintel): 1.95


EDO generators: [[7edo|4\7]]
{{Todo|unify precision|review}}


Scales (Scala files):
== Supermean ==
Supermean tempers out 672/625 and finds the interval class of 7 at 15 generators up, as a double-augmented fifth (C–Gx). As such, it extends [[leapfrog]].


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Subgroup]]: 2.3.5.7
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 15/14, 81/80


Mapping: [{{val| 1 0 -4 -5 }}, {{val| 0 1 4 5 }}]
[[Comma list]]: 81/80, 672/625


Mapping generators: ~2, ~3
{{Mapping|legend=1| 1 0 -4 -21 | 0 1 4 15 }}


Wedgie: {{wedgie| 1 4 5 4 5 0 }}
[[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 }}


{{Vals|legend=1| 5d, 7, 12dd }}
{{Optimal ET sequence|legend=1| 5d, 12d, 17c }}


Badness: 0.0453
[[Badness]] (Sintel): 3.40
</div></div>


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11


Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 685.234
EDO generators: [[7edo|4\7]]
Scales (Scala files):
<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: 15/14, 22/21, 81/80
Mapping: [{{val| 1 0 -4 -5 -6 }}, {{val| 0 1 4 5 6 }}]
Mapping generators: ~2, ~3
{{Vals|legend=1| 5de, 7, 12dd }}
Badness: 0.0325
</div></div>
= Supermean =
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 704.889
EDO generators: [[12edo|7\12]], [[17edo|10\17]], [[29edo|17\29]]
Scales (Scala files):
<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, 672/625
Mapping: [{{val| 1 0 -4 -21 }}, {{val| 0 1 4 15 }}]
{{Vals|legend=1| 5d, 12d, 17c, 29c }}
Badness: 0.1342
</div></div>
== 11-limit ==
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 705.096
EDO generators: [[12edo|7\12]], [[17edo|10\17]], [[29edo|17\29]]
Scales (Scala files):
<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: 56/55, 81/80, 132/125
Comma list: 56/55, 81/80, 132/125


Mapping: [{{val| 1 0 -4 -21 -14 }}, {{val| 0 1 4 15 11 }}]
Mapping: {{mapping| 1 0 -4 -21 -14 | 0 1 4 15 11 }}


{{Vals|legend=1| 5de, 12de, 17c, 29c }}
Optimal tunings:
* WE: ~2 = 1195.7270{{c}}, ~3/2 = 702.5848{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7471{{c}}


Badness: 0.0633
{{Optimal ET sequence|legend=0| 5de, 12de, 17c }}
</div></div>


== 13-limit ==
Badness (Sintel): 2.09


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


Optimal ([[POTE]]) generator: ~3/2 = 705.094
EDO generators: [[12edo|7\12]], [[17edo|10\17]], [[29edo|17\29]]
Scales (Scala files):
<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: 26/25, 56/55, 66/65, 81/80
Comma list: 26/25, 56/55, 66/65, 81/80


Mapping: [{{val| 1 0 -4 -21 -14 -9 }}, {{val| 0 1 4 15 11 8 }}]
Mapping: {{mapping| 1 0 -4 -21 -14 -9 | 0 1 4 15 11 8 }}


{{Vals|legend=1| 5de, 12de, 17c, 29c }}
Optimal tunings:
</div></div>
* WE: ~2 = 1196.3958{{c}}, ~3/2 = 702.9766{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7940{{c}}


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


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.
Badness (Sintel): 1.67


[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name]
== Mohajira ==
{{Main| Mohajira }}


Period: 1\2
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.


Optimal ([[POTE]]) generator: ~3/2 = 694.375
[[Subgroup]]: 2.3.5.7


EDO generators: [[12edo|7\12]], [[14edo|8\14]], [[26edo|15\26]], [[38edo|22\38]]
[[Comma list]]: 81/80, 6144/6125


Scales (Scala files):
{{Mapping|legend=1| 1 1 0 6 | 0 2 8 -11 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
: mapping generators: ~2, ~128/105
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
[[Comma]] list: 50/49, 81/80


Mapping: [{{val| 2 0 -8 -7 }}, {{val| 0 1 4 4 }}]
[[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 }}


Mapping generators: ~7/5, ~3
[[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


[[Tuning ranges]]:  
[[Tuning ranges]]:
* valid range: [685.714, 700.000] (14c to 12)
* 7- and 9-odd-limit [[diamond monotone]]: ~128/105 = [347.368, 350.000] (11\38 to 7\24)
* nice range: [688.957, 701.955]
* 7-odd-limit [[diamond tradeoff]]: ~128/105 = [347.393, 350.978]
* strict range: [688.957, 700.000]
* 9-odd-limit diamond tradeoff: ~128/105 = [345.601, 350.978]


[[Wedgie]]: {{wedgie| 2 8 8 8 7 -4 }}
[[Algebraic generator]]: Mohabis, real root of 3''x''<sup>3</sup> - 3''x''<sup>2</sup> - 1, 348.6067 cents. Corresponding recurrence converges quickly.


{{Vals|legend=1| 12, 26, 38, 102bcd, 140bccd, 178bbccdd }}
{{Optimal ET sequence|legend=1| 7, 24, 31 }}


[[Badness]]: 0.0311
[[Badness]] (Sintel): 1.41
</div></div>


== Music ==
Scales: [[mohaha7]], [[mohaha10]]
* [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 ==
=== 11-limit ===
Subgroup: 2.3.5.7.11


Period: 1\2
Comma list: 81/80, 121/120, 176/175


Optimal ([[POTE]]) generator: ~3/2 = 692.840
Mapping: {{mapping| 1 1 0 6 2 | 0 2 8 -11 5 }}


EDO generators: [[12edo|7\12]], [[14edo|8\14]], [[26edo|15\26]], [[38edo|22\38]]
Optimal tunings:  
* WE: ~2 = 1201.1562{{c}}, ~11/9 = 348.8124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.4910{{c}}


Scales (Scala files):  
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


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Tuning ranges:
<div style="line-height:1.6;">Technical data</div>
* 11-odd-limit diamond monotone: ~11/9 = [348.387, 350.000] (9\31 to 7\24)
<div class="mw-collapsible-content">
* 11-odd-limit diamond tradeoff: ~11/9 = [344.999, 350.978]
Comma list: 45/44, 50/49, 81/80


Mapping: [{{val| 2 0 -8 -7 -12 }}, {{val| 0 1 4 4 6 }}]
{{Optimal ET sequence|legend=0| 7, 24, 31 }}


Mapping generators: ~7/5, ~3
Badness (Sintel): 0.862


Tuning ranges:
Scales: [[mohaha7]], [[mohaha10]]
* valid range: [685.714, 700.000] (14c to 12)
* nice range: [682.458, 701.955]
* strict range: [685.714, 700.000]
 
{{Vals|legend=1| 12, 14c, 26, 90bce, 116bcce }}
 
Badness: 0.0231
</div></div>


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Period: 1\2
Comma list: 66/65, 81/80, 105/104, 121/120


Optimal ([[POTE]]) generator: ~3/2 = 692.673
Mapping: {{mapping| 1 1 0 6 2 4 | 0 2 8 -11 5 -1 }}


EDO generators: [[12edo|7\12]], [[14edo|8\14]], [[26edo|15\26]], [[38edo|22\38]]
Optimal tunings:  
* WE: ~2 = 1200.4256{{c}}, ~11/9 = 348.6819{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.5622{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 7, 24, 31 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 0.966
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 45/44, 50/49, 78/77, 81/80


Mapping: [{{val| 2 0 -8 -7 -12 -21 }}, {{val| 0 1 4 4 6 9 }}]
Scales: [[mohaha7]], [[mohaha10]]


Mapping generators: ~7/5, ~3
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Tuning ranges:  
Comma list: 66/65, 81/80, 105/104, 121/120, 154/153
* valid range: 692.308 (26)
* nice range: [682.458, 701.955]
* strict range: 692.308 (26)


{{Vals|legend=1| 12f, 14cf, 26, 38e }}
Mapping: {{mapping| 1 1 0 6 2 4 7 | 0 2 8 -11 5 -1 -10 }}


Badness: 0.0216
Optimal tunings:  
</div></div>
* WE: ~2 = 1200.0382{{c}}, ~11/9 = 348.7471{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.7360{{c}}


=== Enjera ===
{{Optimal ET sequence|legend=0| 7, 24, 31 }}


Period: 1\2
Badness (Sintel): 1.05


Optimal ([[POTE]]) generator: ~3/2 = 694.121
Scales: [[mohaha7]], [[mohaha10]]


EDO generators: [[12edo|7\12]], [[14edo|8\14]], [[26edo|15\26]]
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Scales (Scala files):  
Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 1 1 0 6 2 4 7 6 | 0 2 8 -11 5 -1 -10 -6 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 27/26, 40/39, 45/44, 50/49


Mapping: [{{val| 2 0 -8 -7 -12 -2 }}, {{val| 0 1 4 4 6 3 }}]
Optimal tunings:  
* WE: ~2 = 1199.7469{{c}}, ~11/9 = 348.7367{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.8117{{c}}


Mapping generators: ~7/5, ~3
{{Optimal ET sequence|legend=0| 7, 24, 31, 55 }}


{{Vals|legend=1| 12f, 14c, 26f, 38eff }}
Badness (Sintel): 1.05


Badness: 0.0265
Scales: [[mohaha7]], [[mohaha10]]
</div></div>


== Injerous ==
== 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.


Period: 1\2
[[Subgroup]]: 2.3.5.7


Optimal ([[POTE]]) generator: ~3/2 = 690.548
[[Comma list]]: 81/80, 392/375


EDO generators: [[12edo|7\12]], [[14edo|8\14]]
{{Mapping|legend=1| 1 1 0 -1 | 0 2 8 13 }}


Scales (Scala files):  
: mapping generators: ~2, ~25/21


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Optimal tuning]]s:
<div style="line-height:1.6;">Technical data</div>
* [[WE]]: ~2 = 1199.0661{{c}}, ~25/21 = 350.3127{{c}}
<div class="mw-collapsible-content">
: [[error map]]: {{val| -0.934 -2.264 +16.188 -13.827 }}
Comma list: 33/32, 50/49, 55/54
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/21 = 350.4856{{c}}
: error map: {{val| 0.000 -0.984 +17.571 -12.513 }}


Mapping: [{{val| 2 0 -8 -7 10 }}, {{val| 0 1 4 4 -1 }}]
{{Optimal ET sequence|legend=1| 7d, 17c, 24 }}


Mapping generators: ~7/5, ~3
[[Badness]] (Sintel): 1.97


{{Vals|legend=1| 12e, 14c, 26e, 40cee }}
Scales: [[mohaha7]], [[mohaha10]]


Badness: 0.0386
=== 11-limit ===
</div></div>
Subgroup: 2.3.5.7.11


== Lahoh ==
Comma list: 56/55, 77/75, 243/242
 
Period: 1\2


Optimal ([[POTE]]) generator: ~3/2 = 699.001
Mapping: {{mapping| 1 1 0 -1 2 | 0 2 8 13 5 }}


EDO generators: [[12edo|7\12]], [[14edo|8\14]]
Optimal tunings:  
* WE: ~2 = 1199.1924{{c}}, ~11/9 = 350.3286{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.4821{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 7d, 17c, 24 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.20
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 50/49, 56/55, 81/77


Mapping: [{{val| 2 0 -8 -7 7 }}, {{val| 0 1 4 4 0 }}]
Scales: [[mohaha7]], [[mohaha10]]


Mapping generators: ~7/5, ~3
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Vals|legend=1| 12, 14ce }}
Comma list: 56/55, 66/65, 77/75, 243/242


Badness: 0.0431
Mapping: {{mapping| 1 1 0 -1 2 4 | 0 2 8 13 5 -1 }}
</div></div>


= Mohaha =
Optimal tunings:
{{see also| Subgroup temperaments #Mohaha }}
* WE: ~2 = 1198.5986{{c}}, ~11/9 = 350.3353{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.6459{{c}}


Mohaha is the 2.3.5.11 subgroup temperament with a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 11/9. Mohaha can 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 2.3.5.11 subgroup). Within this paradigm, mohaha 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, and that maps four 3/2's to 5/1. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs.
{{Optimal ET sequence|legend=0| 7d, 17c, 24, 41c }}


Period: 1\1
Badness (Sintel): 1.19


Optimal ([[POTE]]) generator: ~11/9 = 348.0938
Scales: [[mohaha7]], [[mohaha10]]


EDO generators: [[17edo|5\17]], [[24edo|7\24]], [[31edo|9\31]], [[38edo|11\38]], [[55edo|16\55]]
== Liese ==
<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>


Scales (Scala files): [[Mohaha7]], [[Mohaha10]]
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.


<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 (10-note MOS, 2.3.5.11 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>2.3.5.11, 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
| 348.1
| 11/9
|-
| 2
| 696.2
| 3/2
|-
| 3
| 1044.3
| 11/6
|-
| 4
| 192.4
| 9/8
|-
| 5
| 540.5
| 11/8, 15/11
|-
| 6
| 888.6
| 5/3
|-
| 7
| 36.7
|
|-
| 8
| 384.8
| 5/4
|-
| 9
| 732.8
| (32/21)
|}
<references/></div></div>


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Comma list]]: 81/80, 686/675
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 81/80, 121/120
{{Mapping|legend=1| 1 0 -4 -3 | 0 3 12 11 }}


Mapping: [{{val|1 1 0 0 2}}, {{val|0 2 8 0 5}}]
: mapping generators: ~2, ~10/7


Mapping generators: ~2, ~11/9
[[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 }}


{{Vals|legend=1| 7, 17c, 24, 31, 100e, 131bee }}
[[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


Badness: 0.0261
[[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.
</div></div>


== Mohoho ==
{{Optimal ET sequence|legend=1| 17c, 19, 55, 74d }}


Period: 1\1
[[Badness]] (Sintel): 1.18


Optimal ([[POTE]]) generator: ~11/9 = 348.9155
=== Liesel ===
Subgroup: 2.3.5.7.11


EDO generators: [[17edo|5\17]], [[24edo|7\24]], [[31edo|9\31]], [[38edo|11\38]], [[55edo|16\55]]
Comma list: 56/55, 81/80, 540/539


Scales (Scala files): [[Mohaha7]], [[Mohaha10]]
Mapping: {{mapping| 1 0 -4 -3 4 | 0 3 12 11 -1 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:
<div style="line-height:1.6;">Technical data</div>
* WE: ~2 = 1198.8507{{c}}, ~10/7 = 632.4668{{c}}
<div class="mw-collapsible-content">
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 632.9963{{c}}


Comma list: 66/65, 81/80, 121/120
{{Optimal ET sequence|legend=0| 17c, 19, 36 }}


Mapping: [{{val|1 1 0 0 2 4}}, {{val|0 2 8 0 5 -1}}]
Badness (Sintel): 1.35


Mapping generators: ~2, ~11/9
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Vals|legend=1| 7, 17c, 24, 31, 55, 86ef, 141ceff }}
Comma list: 56/55, 78/77, 81/80, 91/90


Badness: 0.0261
Mapping: {{mapping| 1 0 -4 -3 4 0 | 0 3 12 11 -1 7 }}
</div></div>


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


{{main|Mohajira}}
{{Optimal ET sequence|legend=0| 17c, 19, 36 }}


Mohajira can be viewed as derived from mohaha which maps the interval one quarter tone flat of 16/9 to 7/4, although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the porwell comma. [[31edo]] makes for an excellent (7-limit) mohajira tuning, with generator 9/31.  
Badness (Sintel): 1.13


== 7-limit ==
=== Elisa ===
Period: 1\1
Subgroup: 2.3.5.7.11


Optimal ([[POTE]]) generator: ~128/105 = 348.415
Comma list: 77/75, 81/80, 99/98


EDO generators: [[24edo|7\24]], [[31edo|9\31]], [[38edo|11\38]], [[55edo|16\55]]
Mapping: {{mapping| 1 0 -4 -3 -5 | 0 3 12 11 16 }}


Scales (Scala files): [[Mohaha7]], [[Mohaha10]]
Optimal tunings:  
* WE: ~2 = 1201.0489{{c}}, ~10/7 = 633.6147{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1644{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
[[Comma]] list: 81/80, 6144/6125


Mapping: [{{val| 1 1 0 6 }}, {{val| 0 2 8 -11 }}]
Badness (Sintel): 1.37


Mapping generators: ~2, ~128/105
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[Wedgie]]: {{wedgie| 2 8 -11 8 -23 -48 }}
Comma list: 66/65, 77/75, 81/80, 99/98


[[Minimax tuning]]:  
Mapping: {{mapping| 1 0 -4 -3 -5 0 | 0 3 12 11 16 7 }}
* [[7-odd-limit|7]]- and [[9-odd-limit]]: 1/4 comma
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 6 0 -11/8 0 }}]
: [[Eigenmonzo]]s: 2, 5


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 tunings:  
* WE: ~2 = 1201.4815{{c}}, ~10/7 = 633.7720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1281{{c}}


{{Vals|legend=1| 7, 24, 31 }}
{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}


[[Badness]]: 0.0557
Badness (Sintel): 1.11
</div></div>


== 11-limit ==
=== Lisa ===
Period: 1\1
Subgroup: 2.3.5.7.11


Optimal ([[POTE]]) generator: ~11/9 = 348.477
Comma list: 45/44, 81/80, 343/330


EDO generators: [[24edo|7\24]], [[31edo|9\31]], [[38edo|11\38]], [[55edo|16\55]]
Mapping: {{mapping| 1 0 -4 -3 -6 | 0 3 12 11 18 }}


Scales (Scala files): [[Mohaha7]], [[Mohaha10]]
Optimal tunings:  
* WE: ~2 = 1202.6773{{c}}, ~10/7 = 632.7783{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.6175{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 17cee, 19 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 81/80, 121/120, 176/175
Badness (Sintel): 1.81


Mapping: [{{val|1 1 0 6 2}}, {{val|0 2 8 -11 5}}]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Mapping generators: ~2, ~11/9
Comma list: 45/44, 81/80, 91/88, 147/143


Minimax tuning:  
Mapping: {{mapping| 1 0 -4 -3 -6 0 | 0 3 12 11 18 7 }}
* [[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


{{Vals|legend=1| 7, 24, 31 }}
Optimal tunings:
* WE: ~2 = 1203.6086{{c}}, ~10/7 = 633.1193{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.5346{{c}}


Badness: 0.0261
{{Optimal ET sequence|legend=0| 17cee, 19 }}
</div></div>


== 13-limit ==
Badness (Sintel): 1.49
Period: 1\1


Optimal ([[POTE]]) generator: ~11/9 = 348.558
== Superpine ==
{{See also| No-sevens subgroup temperaments #Superpine }}


EDO generators: [[24edo|7\24]], [[31edo|9\31]], [[38edo|11\38]], [[55edo|16\55]]
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.


Scales (Scala files): [[Mohaha7]], [[Mohaha10]]
[[Subgroup]]: 2.3.5.7


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Comma list]]: 81/80, 1119744/1071875
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 66/65, 81/80, 105/104, 121/120


Mapping: [{{val| 1 1 0 6 2 4 }}, {{val| 0 2 8 -11 5 -1 }}]
{{Mapping|legend=1| 1 2 4 1 | 0 -3 -12 13 }}


Mapping generators: ~2, ~11/9
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3652{{c}}, ~35/32 = 167.1615{{c}}
: [[error map]]: {{val| -0.635 -4.709 +5.209 +3.639 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/32 = 167.2561{{c}}
: error map: {{val| 0.000 -3.723 +6.613 +5.503 }}


{{Vals|legend=1| 7, 24, 31 }}
{{Optimal ET sequence|legend=1| 7, 36, 43, 79c }}


Badness: 0.0234
[[Badness]] (Sintel): 3.46
</div></div>


== 17-limit ==
=== 11-limit ===
Period: 1\1
Subgroup: 2.3.5.7.11


Optimal ([[POTE]]) generator: ~11/9 = 348.736
Comma list: 81/80, 176/175, 864/847


EDO generators: [[24edo|7\24]], [[31edo|9\31]], [[55edo|16\55]]
Mapping: {{mapping| 1 2 4 1 5 | 0 -3 -12 13 -11 }}


Scales (Scala files): [[Mohaha7]], [[Mohaha10]]
Optimal tunings:  
* WE: ~2 = 1199.0522{{c}}, ~11/10 = 167.1904{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.3382{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 7, 36, 43 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 66/65, 81/80, 105/104, 121/120, 154/153


Mapping: [{{val| 1 1 0 6 2 4 7 }}, {{val| 0 2 8 -11 5 -1 -10 }}]
Badness (Sintel): 1.90


Mapping generators: ~2, ~11/9
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Vals|legend=1| 7, 24, 31, 86ef }}
Comma list: 78/77, 81/80, 144/143, 176/175


Badness: 0.0206
Mapping: {{mapping| 1 2 4 1 5 3 | 0 -3 -12 13 -11 5 }}
</div></div>


== 19-limit ==
Optimal tunings:
Period: 1\1
* WE: ~2 = 1199.4286{{c}}, ~11/10 = 167.3105{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.3958{{c}}


Optimal ([[POTE]]) generator: ~11/9 = 348.810
{{Optimal ET sequence|legend=0| 7, 36, 43 }}


EDO generators: [[24edo|7\24]], [[31edo|9\31]], [[55edo|16\55]]
Badness (Sintel): 1.52


Scales (Scala files): [[Mohaha7]], [[Mohaha10]]
== 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.


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Subgroup]]: 2.3.5.7
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152


Mapping: [{{val| 1 1 0 6 2 4 7 6 }}, {{val| 0 2 8 -11 5 -1 -10 -6 }}]
[[Comma list]]: 81/80, 3125/3087


Mapping generators: ~2, ~11/9
{{Mapping|legend=1| 3 0 -12 -20 | 0 1 4 6 }}


{{Vals|legend=1| 7, 24, 31, 55, 86efh }}
: mapping generators: ~56/45, ~3


Badness: 0.0173
[[Optimal tuning]]s:  
</div></div>
* [[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 }}


= Migration =
{{Optimal ET sequence|legend=1| 12, 33cd, 45, 57 }}
Migration takes [[#Septimal meantone]] mapping of 7 and [[#Mohaha]] mapping of 11.


Period: 1\1
[[Badness]] (Sintel): 1.75


Optimal ([[POTE]]) generator: ~11/9 = 348.182
== Squares ==
{{Main| Squares }}


EDO generators: [[24edo|7\24]], [[31edo|9\31]]
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]].


Scales (Scala files):  
[[Subgroup]]: 2.3.5.7


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Comma list]]: 81/80, 2401/2400
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 81/80, 121/120, 126/125


Mapping: [{{val| 1 1 0 -3 2 }}, {{val| 0 2 8 20 5 }}]
{{Mapping|legend=1| 1 -1 -8 -3 | 0 4 16 9 }}


Mapping generators: ~2, ~11/9
: mapping generators: ~2, ~14/9


{{Vals|legend=1| 7d, 31, 100de, 131bdee, 162bdee }}
[[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: 0.0255
[[Minimax tuning]]:  
</div></div>
* [[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


== 13-limit ==
[[Algebraic generator]]: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.


Period: 1\1
{{Optimal ET sequence|legend=1| 14c, 17c, 31, 169b, 200b }}


Optimal ([[POTE]]) generator: ~11/9 = 348.490
[[Badness]] (Sintel): 1.16


EDO generators: [[24edo|7\24]], [[31edo|9\31]]
Scales: [[skwares8]], [[skwares11]], [[skwares14]]


Scales (Scala files):  
=== 11-limit ===
Subgroup: 2.3.5.7.11


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 81/80, 99/98, 121/120
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 66/65, 81/80, 121/120, 126/125


Mapping: [{{val| 1 1 0 -3 2 4 }}, {{val| 0 2 8 20 5 -1 }}]
Mapping: {{mapping| 1 -1 -8 -3 -3 | 0 4 16 9 10 }}


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


{{Vals|legend=1| 7d, 24d, 31, 55d }}
{{Optimal ET sequence|legend=0| 14c, 17c, 31, 130bee, 169beee }}


Badness: 0.0281
Badness (Sintel): 0.715
</div></div>


= Ptolemy =
==== 13-limit ====
Ptolemy takes [[#Flattone]] mapping of 7 and [[#Mohaha]] mapping of 11.  
Subgroup: 2.3.5.7.11.13


Period: 1\1
Comma list: 66/65, 81/80, 99/98, 121/120


Optimal ([[POTE]]) generator: ~11/9 = 346.922
Mapping: {{mapping| 1 -1 -8 -3 -3 5 | 0 4 16 9 10 -2 }}


EDO generators: [[38edo|11\38]], [[45edo|13\45]]
Optimal tunings:  
* WE: ~2 = 1199.8419{{c}}, ~11/7 = 774.3484{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4422{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 14c, 17c, 31, 79cf }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.05
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 81/80, 121/120, 525/512


Mapping: [{{val| 1 1 0 8 2 }}, {{val| 0 2 8 -18 5 }}]
==== Squad ====
Subgroup: 2.3.5.7.11.13


{{Vals|legend=1| 7, 31dd, 38d, 45e, 83bcddee }}
Comma list: 78/77, 81/80, 91/90, 99/98


Badness: 0.0588
Mapping: {{mapping| 1 -1 -8 -3 -3 -6 | 0 4 16 9 10 15 }}
</div></div>


== 13-limit ==
Optimal tunings:
* WE: ~2 = 1202.0312{{c}}, ~11/7 = 775.5589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4140{{c}}


Period: 1\1
{{Optimal ET sequence|legend=0| 14cf, 17c, 31f }}


Optimal ([[POTE]]) generator: ~11/9 = 346.910
Badness (Sintel): 1.11


EDO generators: [[38edo|11\38]], [[45edo|13\45]]
==== Agora ====
Subgroup: 2.3.5.7.11.13


Scales (Scala files):  
Comma list: 81/80, 99/98, 105/104, 121/120


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 1 -1 -8 -3 -3 -15 | 0 4 16 9 10 29 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 65/64, 81/80, 105/104, 121/120


Mapping: [{{val| 1 1 0 8 2 6 }}, {{val| 0 2 8 -18 5 -8 }}]
Optimal tunings:  
* WE: ~2 = 1202.3228{{c}}, ~11/7 = 775.2214{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8617{{c}}


{{Vals|legend=1| 7, 31ddf, 38df, 45ef, 83bcddeeff }}
{{Optimal ET sequence|legend=0| 14cf, 31, 45ef, 76e }}


Badness: 0.0343
Badness (Sintel): 1.01
</div></div>


= Maqamic =
===== 17-limit =====
<span style="display: block; text-align: right;">[[:de:maqamisch|Deutsch]]</span>
Subgroup: 2.3.5.7.11.13.17


{{main| Maqamic }}
Comma list: 81/80, 99/98, 105/104, 120/119, 121/119


Maqamic takes [[#Dominant]] mapping of 7 and [[#Mohaha]] mapping of 11, so it is [[36/35]] that vanishes instead of [[176/175]] as in mohajira. It makes the most sense if viewed as an adaptive temperament, whereby 7/4 and 9/5 simply share an equivalence class in the resulting scales, but don't need to share a particular tempered "middle-of-the-road" intonation.
Mapping: {{mapping| 1 -1 -8 -3 -3 -15 -3 | 0 4 16 9 10 29 11 }}


Period: 1\1
Optimal tunings:  
* WE: ~2 = 1201.4340{{c}}, ~11/7 = 774.7375{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8955{{c}}


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


EDO generators: [[7edo|2\7]], [[10edo|3\10]], [[17edo|5\17]], [[24edo|7\24]]
Badness (Sintel): 1.15


Scales (Scala files):  
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
[[Comma]] list: 81/80, 36/35, 121/120


Mapping: [{{val| 1 1 0 4 2 }}, {{val| 0 2 8 -4 5 }}]
Mapping: {{mapping| 1 -1 -8 -3 -3 -15 -3 -8 | 0 4 16 9 10 29 11 19 }}


Mapping generators: ~2, ~11/9
Optimal tunings:
* WE: ~2 = 1201.2461{{c}}, ~11/7 = 774.5783{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8479{{c}}


{{Vals|legend=1| 7, 10c, 17c, 24d, 31d }}
{{Optimal ET sequence|legend=0| 14cf, 31 }}
</div></div>


== 13-limit ==
Badness (Sintel): 1.15


Period: 1\1
=== Cuboctahedra ===
Subgroup: 2.3.5.7.11


Optimal ([[POTE]]) generator: ~11/9 = 350.816
Comma list: 81/80, 385/384, 1375/1372


EDO generators: [[7edo|2\7]], [[10edo|3\10]], [[17edo|5\17]], [[24edo|7\24]]
Mapping: {{mapping| 1 -1 -8 -3 17 | 0 4 16 9 -21 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~2 = 1201.4436{{c}}, ~14/9 = 774.9386{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 774.0243{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 31, 107b, 138b, 169be, 200be }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
[[Comma]] list: 81/80, 36/35, 121/120, 144/143


Mapping: [{{val| 1 1 0 4 2 4 }}, {{val| 0 2 8 -4 5 -1 }}]
Badness (Sintel): 1.88


Mapping generators: ~2, ~11/9
== 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.


{{Vals|legend=1| 7, 10c, 17c, 24d, 31d }}
[[Subgroup]]: 2.3.5.7
</div></div>


= Mohamaq =
[[Comma list]]: 81/80, 17280/16807
== 7-limit ==
Period: 1\1


Optimal ([[POTE]]) generator: ~25/21 = 350.586
{{Mapping|legend=1| 1 1 0 2 | 0 5 20 7 }}


EDO generators: [[17edo|5\17]], [[24edo|7\24]]
: mapping generators: ~2, ~54/49


Scales (Scala files):  
[[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 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=1| 17c, 26, 43 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 81/80, 392/375


Mapping: [{{val| 1 1 0 -1 }}, {{val| 0 2 8 13 }}]
[[Badness]] (Sintel): 2.75


Mapping generators: ~2, ~25/21
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Vals|legend=1| 17c, 24, 65c, 89cd }}
Comma list: 81/80, 99/98, 864/847


Badness: 0.0777
Mapping: {{mapping| 1 1 0 2 3 | 0 5 20 7 4 }}
</div></div>


== 11-limit ==
Optimal tunings:
* WE: ~2 = 1201.4436{{c}}, ~12/11 = 139.3714{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 139.4038{{c}}


Period: 1\1
{{Optimal ET sequence|legend=0| 17c, 26, 43 }}


Optimal ([[POTE]]) generator: ~11/9 = 350.565
Badness (Sintel): 1.58


EDO generators: [[17edo|5\17]], [[24edo|7\24]]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Scales (Scala files):  
Comma list: 78/77, 81/80, 99/98, 144/143


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 1 1 0 2 3 3 | 0 5 20 7 4 6 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 56/55, 77/75, 243/242


Mapping: [{{val| 1 1 0 -1 2 }}, {{val| 0 2 8 13 5 }}]
Optimal tunings:  
* WE: ~2 = 1199.8860{{c}}, ~13/12 = 139.3737{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3817{{c}}


Mapping generators: ~2, ~11/9
{{Optimal ET sequence|legend=0| 17c, 26, 43 }}


{{Vals|legend=1| 17c, 24, 65c, 89cd }}
Badness (Sintel): 1.21


Badness: 0.0362
=== 17-limit ===
</div></div>
Subgroup: 2.3.5.7.11.13.17


== 13-limit ==
Comma list: 78/77, 81/80, 99/98, 144/143, 189/187


Period: 1\1
Mapping: {{mapping| 1 1 0 2 3 3 2 | 0 5 20 7 4 6 18 }}


Optimal ([[POTE]]) generator: ~11/9 = 350.745
Optimal tunings:  
* WE: ~2 = 1199.8346{{c}}, ~13/12 = 139.3431{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3544{{c}}


EDO generators: [[17edo|5\17]], [[24edo|7\24]]
{{Optimal ET sequence|legend=0| 17cg, 26, 43 }}


Scales (Scala files):  
Badness (Sintel): 1.06


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
=== 19-limit ===
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11.13.17.19
<div class="mw-collapsible-content">
Comma list: 56/55, 66/65, 77/75, 243/242


Mapping: [{{val| 1 1 0 -1 2 4 }}, {{val| 0 2 8 13 5 -1 }}]
Comma list: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143


Mapping generators: ~2, ~11/9
Mapping: {{mapping| 1 1 0 2 3 3 2 1 | 0 5 20 7 4 6 18 28 }}


{{Vals|legend=1| 17c, 24, 41c, 65c }}
Optimal tunings:
* WE: ~2 = 1199.8891{{c}}, ~13/12 = 139.3001{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3080{{c}}


Badness: 0.0287
{{Optimal ET sequence|legend=0| 17cgh, 26, 43, 69 }}
</div></div>


= Orphic =
Badness (Sintel): 1.11


Period: 1\2
== 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.


Optimal (POTE) generator: ~7/6 = 275.794
[[Subgroup]]: 2.3.5.7


EDO generators: [[26edo|6\26]], [[48edo|11\48]]
[[Comma list]]: 81/80, 16875/16807


Scales (Scala files):
{{Mapping|legend=1| 1 -1 -8 -7 | 0 5 20 19 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
: mapping generators: ~2, ~10/7
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 81/80, 5898240/5764801
[[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 }}


[[Mapping]]: [{{val| 2 1 -4 4 }}, {{val|0 4 16 3 }}]
{{Optimal ET sequence|legend=1| 29cd, 31, 188bcd, 219bbcd }}


Mapping generators: ~2401/1728, ~343/288
[[Badness]] (Sintel): 2.08


[[Wedgie]]: {{wedgie| 8 32 6 32 -13 -76 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Vals|legend=1| 26, 48c, 74, 174bd, 248bd }}
Comma list: 81/80, 99/98, 2541/2500


[[Badness]]: 0.2588
Mapping: {{mapping| 1 -1 -8 -7 -11 | 0 5 20 19 28 }}


</div></div>
Optimal tunings:
* WE: ~2 = 1201.2054{{c}}, ~10/7 = 619.9752{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.4223{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 29cde, 31 }}


Period: 1\2
Badness (Sintel): 1.42


Optimal (POTE) generator: ~7/6 = 275.762
== 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.


EDO generators: [[26edo|6\26]], [[48edo|11\48]]
[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name]


Scales (Scala files):
[[Subgroup]]: 2.3.5.7


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Comma list]]: 50/49, 81/80
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 81/80, 99/98, 73728/73205
{{Mapping|legend=1| 2 0 -8 -7 | 0 1 4 4 }}


Mapping: [{{val| 2 1 -4 4 8 }}, {{val|0 4 16 3 -2 }}]
: mapping generators: ~7/5, ~3


Mapping generators: ~363/256, ~77/64
[[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 }}


{{Vals|legend=1| 26, 48c, 74, 248bd, 322bd }}
[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [688.957, 701.955]
* 9-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]


Badness: 0.1015
{{Optimal ET sequence|legend=1| 12, 26, 38 }}


</div></div>
[[Badness]] (Sintel): 0.788


== 13-limit ==
; 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


Period: 1\2
=== 11-limit ===
Subgroup: 2.3.5.7.11


Optimal (POTE) generator: ~7/6 = 275.774
Comma list: 45/44, 50/49, 81/80


EDO generators: [[26edo|6\26]], [[48edo|11\48]]
Mapping: {{mapping| 2 0 -8 -7 -12 | 0 1 4 4 6 }}


Scales (Scala files):
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}})


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Tuning ranges:
<div style="line-height:1.6;">Technical data</div>
* 11-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
<div class="mw-collapsible-content">
* 11-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]


Comma list: 81/80, 99/98, 144/143, 2200/2197
{{Optimal ET sequence|legend=0| 12, 26 }}


Mapping: [{{val| 2 1 -4 4 8 2 }}, {{val|0 4 16 3 -2 10 }}]
Badness (Sintel): 0.764


Mapping generators: ~55/39, ~63/52
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Vals|legend=1| 26, 48c, 74, 174bd, 248bd, 322bd }}
Comma list: 45/44, 50/49, 78/77, 81/80


Badness: 0.0535
Mapping: {{mapping| 2 0 -8 -7 -12 -21 | 0 1 4 4 6 9 }}


</div></div>
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}})


= Mothra =
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]


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]].
{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}


Note that mothra can also be called '''cynder''' in the 7-limit, which can be a little confusing sometimes.  
Badness (Sintel): 0.891


Period: 1\1
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Optimal (POTE) generator: ~8/7 = 232.193
Comma list: 45/44, 50/49, 78/77, 81/80, 85/84


EDO generators: [[26edo|5\26]], [[31edo|6\31]], [[36edo|7\36]], [[57edo|11\57]], [[67edo|13\67]]
Mapping: {{mapping| 2 0 -8 -7 -12 -21 5 | 0 1 4 4 6 9 1 }}


Scales (Scala files):
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}})


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


[[Comma]] list: 81/80, 1029/1024
Badness (Sintel): 0.935


[[Mapping]]: [{{val| 1 1 0 3 }}, {{val| 0 3 12 -1 }}]
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Mapping generators: ~2, ~8/7
Comma list: 45/44, 50/49, 57/56, 78/77, 81/80, 85/84


Algebraic generator: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.
Mapping: {{mapping| 2 0 -8 -7 -12 -21 5 -1 | 0 1 4 4 6 9 1 3 }}


[[Wedgie]]: {{wedgie| 3 12 -1 12 -10 -36 }}
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}})


[[Minimax tuning]]:
{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}
* [[7-odd-limit|7-]] and [[9-odd-limit]]: 1/4 comma
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 3 0 -1/12 0 }}]
: [[Eigenmonzo]]s: 2, 5


{{Vals|legend=1| 5, 26, 31 }}
Badness (Sintel): 0.920


[[Badness]]: 0.0371
==== Enjera ====
Subgroup: 2.3.5.7.11.13


</div></div>
Comma list: 27/26, 40/39, 45/44, 50/49


== 11-limit ==
Mapping: {{mapping| 2 0 -8 -7 -12 -2 | 0 1 4 4 6 3 }}


Period: 1\1
Optimal tunings:  
* WE: ~7/5 = 599.1863{{c}}, ~3/2 = 693.1791{{c}} (~21/20 = 93.9929{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 693.6809{{c}} (~21/20 = 93.6809{{c}})


Optimal (POTE) generator: ~8/7 = 232.031
{{Optimal ET sequence|legend=0| 10cdeef, 12f }}


EDO generators: [[26edo|5\26]], [[31edo|6\31]], [[57edo|11\57]]
Badness (Sintel): 1.10


Scales (Scala files):
=== Injerous ===
Subgroup: 2.3.5.7.11


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 33/32, 50/49, 55/54
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 81/80, 99/98, 385/384
Mapping: {{mapping| 2 0 -8 -7 10 | 0 1 4 4 -1 }}


Mapping: [{{val| 1 1 0 3 5 }}, {{val| 0 3 12 -1 -8 }}]
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}})


Mapping generators: ~2, ~8/7
{{Optimal ET sequence|legend=0| 12e, 14c, 26e, 40cee }}


{{Vals|legend=1| 5, 26, 31, 88, 150be, 181bee }}
Badness (Sintel): 1.28


Badness: 0.0256
=== Lahoh ===
Subgroup: 2.3.5.7.11


</div></div>
Comma list: 50/49, 56/55, 81/77


=== 13-limit ===
Mapping: {{mapping| 2 0 -8 -7 7 | 0 1 4 4 0 }}


Period: 1\1
Optimal tunings:  
* WE: ~7/5 = 597.3179{{c}}, ~3/2 = 695.8759{{c}} (~21/20 = 98.5581{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 697.8757{{c}} (~21/20 = 97.8757{{c}})


Optimal (POTE) generator: ~8/7 = 231.811
{{Optimal ET sequence|legend=0| 10cd, 12 }}


EDO generators: [[26edo|5\26]], [[31edo|6\31]], [[57edo|11\57]]
Badness (Sintel): 1.42


Scales (Scala files):
=== Teff ===
{{Main| Teff }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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.  
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 81/80, 99/98, 105/104, 144/143
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 1 0 3 5 1 }}, {{val| 0 3 12 -1 -8 14 }}]
Comma list: 50/49, 81/80, 864/847


Mapping generators: ~2, ~8/7
Mapping: {{mapping| 2 1 -4 -3 8 | 0 2 8 8 -1 }}


{{Vals|legend=1| 5, 26, 31, 57, 88 }}
: mapping generators: ~7/5, ~16/11


Badness: 0.0240
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}})


</div></div>
{{Optimal ET sequence|legend=0| 24d, 26, 50d }}


== Cynder ==
Badness (Sintel): 2.34


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


Optimal (POTE) generator: ~8/7 = 231.317
Comma list: 50/49, 78/77, 81/80, 144/143


EDO generators: [[26edo|5\26]], [[31edo|6\31]], [[57edo|11\57]]
Mapping: {{mapping| 2 1 -4 -3 8 2 | 0 2 8 8 -1 5 }}


Scales (Scala files):
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}})


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 24d, 26, 50d }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 45/44, 81/80, 1029/1024
Badness (Sintel): 1.65


Mapping: [{{val| 1 1 0 3 0 }}, {{val| 0 3 12 -1 18 }}]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Mapping generators: ~2, ~8/7
Comma list: 50/49, 78/77, 81/80, 85/84, 144/143


{{Vals|legend=1| 5e, 26, 57e, 83bce }}
Mapping: {{mapping| 2 1 -4 -3 8 2 6 | 0 2 8 8 -1 5 2 }}


Badness: 0.0557
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}})


</div></div>
{{Optimal ET sequence|legend=0| 24d, 26 }}


=== 13-limit ===
Badness (Sintel): 1.50


Period: 1\1
==== 19-limit ====
 
Subgroup: 2.3.5.7.11.13.17.19
Optimal (POTE) generator: ~8/7 = 232.293
 
EDO generators: [[26edo|5\26]], [[31edo|6\31]], [[57edo|11\57]]
 
Scales (Scala files):


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 50/49, 57/56, 78/77, 81/80, 85/84, 144/143
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Commas: 45/44, 78/77, 81/80, 640/637
Mapping: {{mapping| 2 1 -4 -3 8 2 6 2 | 0 2 8 8 -1 5 2 6 }}


Mapping: [{{val| 1 1 0 3 0 1 }}, {{val| 0 3 12 -1 18 14 }}]
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 generators: ~2, ~8/7
{{Optimal ET sequence|legend=0| 24d, 26 }}


{{Vals|legend=1| 5e, 26, 57e, 83bce }}
Badness (Sintel): 1.41


Badness: 0.0341
== 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.


</div></div>
[[Subgroup]]: 2.3.5.7


== Mosura ==
[[Comma list]]: 81/80, 300125/294912


Period: 1\1
{{Mapping|legend=1| 2 1 -4 11 | 0 2 8 -5 }}


Optimal (POTE) generator: ~8/7 = 232.419
: mapping generators: ~735/512, ~35/24


EDO generators: [[31edo|6\31]], [[36edo|7\36]], [[67edo|13\67]]
[[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 }}


Scales (Scala files):
{{Optimal ET sequence|legend=1| 24, 26, 50, 126bcd, 176bcdd, 226bbcdd }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 2.94
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 81/80, 176/175, 540/539
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 1 0 3 -1 }}, {{val| 0 3 12 -1 23 }}]
Comma list: 81/80, 245/242, 385/384


Mapping generators: ~2, ~8/7
Mapping: {{mapping| 2 1 -4 11 8 | 0 2 8 -5 -1 }}


{{Vals|legend=1| 31, 129, 160be, 191bce, 222bce, 253bcee }}
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}})


Badness: 0.0313
{{Optimal ET sequence|legend=0| 24, 26, 50 }}


</div></div>
Badness (Sintel): 1.72


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Period: 1\1
Comma list: 81/80, 105/104, 144/143, 245/242


Optimal (POTE) generator: ~8/7 = 232.640
Mapping: {{mapping| 2 1 -4 11 8 2 | 0 2 8 -5 -1 5 }}


EDO generators: [[31edo|6\31]], [[36edo|7\36]], [[67edo|13\67]]
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}})


Scales (Scala files):
{{Optimal ET sequence|legend=0| 24, 26, 50 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.28
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 81/80, 144/143, 176/175, 196/195
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Mapping: [{{val| 1 1 0 3 -1 7 }}, {{val| 0 3 12 -1 23 -17 }}]
Comma list: 81/80, 105/104, 144/143, 245/242, 273/272


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


{{Vals|legend=1| 31, 36, 67, 98 }}
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}})


Badness: 0.0369
{{Optimal ET sequence|legend=0| 24, 26, 50 }}


</div></div>
Badness (Sintel): 1.08


= Squares =
=== 19-limit ===
[[Comma]]s: 81/80, 2401/2400
Subgroup: 2.3.5.7.11.13.17.19


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


7 and 9 limit minimax 1/4 comma
Mapping: {{mapping| 2 1 -4 11 8 2 6 2 | 0 2 8 -5 -1 5 2 6 }}


[{{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: ~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}})


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


[[POTE generator]]: ~9/7 = 425.942
Badness (Sintel): 1.01


Mapping generator: ~9/7
== Orphic ==
Orphic has a semi-octave period and four generators plus a period gives the 3rd harmonic; its ploidacot is diploid alpha-tetracot.


Algebraic generator: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 3 8 6|, &lt;0 -4 -16 -9|]
[[Comma list]]: 81/80, 5898240/5764801


[[Generator]]s: 2, 9/7
{{Mapping|legend=1| 2 1 -4 4 | 0 4 16 3 }}


{{Vals|legend=1| 14c, 17c, 31, 45, 76 }}
: mapping generators: ~2401/1728, ~343/288


[[Badness]]: 0.0460
[[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 }}


Music:
{{Optimal ET sequence|legend=1| 26, 48c, 74 }}


By [[Chris Vaisvil]]
[[Badness]] (Sintel): 6.55


* [http://clones.soonlabel.com/public/micro/tuning-survey/daily20100603-squares8piano.mp3 Square 8]
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 81/80, 99/98, 73728/73205
Commas: 81/80, 99/98, 121/120
 
POTE generator: ~9/7 = 425.957


Mapping generator: ~9/7
Mapping: {{mapping| 2 1 -4 4 8 | 0 4 16 3 -2 }}


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


{{Vals|legend=1| 14c, 17c, 31, 45e, 76e }}
{{Optimal ET sequence|legend=0| 26, 48c, 74 }}


Badness: 0.0216
Badness (Sintel): 3.36


=== 13-limit ===
=== 13-limit ===
Commas: 66/65, 81/80, 99/98, 121/120
Subgroup: 2.3.5.7.11.13


POTE generator: ~9/7 = 425.550
Comma list: 81/80, 99/98, 144/143, 2200/2197
 
Mapping generator: ~9/7


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


{{Vals|legend=1| 14c, 17c, 31, 45e, 79cf }}
Optimal tunings:
* WE: ~55/39 = 600.0540{{c}}, ~77/64 = 324.2551{{c}} (~7/6 = 275.7989{{c}})
* CWE: ~55/39 = 600.0000{{c}}, ~77/64 = 324.2307{{c}} (~7/6 = 275.7693{{c}})


Badness: 0.0255
{{Optimal ET sequence|legend=0| 26, 48c, 74 }}


=== Agora ===
Badness (Sintel): 2.21
Commas: 81/80, 99/98, 105/104, 121/120


POTE generator: ~9/7 = 426.276
== Cloudtone ==
The cloudtone temperament tempers out the [[cloudy comma]], 16807/16384 and the [[syntonic comma]], 81/80 in the 7-limit. It may be described as {{nowrap| 5 & 50 }}; its ploidacot is pentaploid monocot. It can be extended to the 11- and 13-limit by adding 385/384 and 105/104 to the comma list in this order.


Mapping generator: ~9/7
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 3 8 6 7 14|, &lt;0 -4 -16 -9 -10 -29|]
[[Comma list]]: 81/80, 16807/16384


{{Vals|legend=1| 14cf, 31, 45ef, 76e }}
{{Mapping|legend=1| 5 0 -20 14 | 0 1 4 0 }}


Badness: 0.0245
: mapping generators: ~8/7, ~3


==== 17-limit ====
[[Optimal tuning]]s:
Commas: 81/80, 99/98, 105/104, 120/119, 121/119
* [[WE]]: ~8/7 = 240.4267{{c}}, ~3/2 = 696.9566{{c}} (~49/48 = 24.3235{{c}})
: [[error map]]: {{val| +2.133 -2.865 +1.513 -2.852 }}
* [[CWE]]: ~8/7 = 240.0000{{c}}, ~3/2 = 696.1637{{c}} (~49/48 = 23.8373{{c}})
: error map: {{val| 0.000 -5.791 -1.659 -8.826 }}


POTE generator: ~9/7 = 426.187
{{Optimal ET sequence|legend=1| 5, 40c, 45, 50 }}


Mapping generator: ~9/7
[[Badness]] (Sintel): 2.59


Map: [&lt;1 3 8 6 7 14 8|, &lt;0 -4 -16 -9 -10 -29 -11|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Vals|legend=1| 14cf, 31, 45ef, 76e }}
Comma list: 81/80, 385/384, 2401/2376


==== 19-limit ====
Mapping: {{mapping| 5 0 -20 14 41 | 0 1 4 0 -3 }}
Commas: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119


POTE generator: ~9/7 = 426.225
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}})


Mapping generator: ~9/7
{{Optimal ET sequence|legend=0| 5, 45, 50 }}


Map: [&lt;1 3 8 6 7 14 8 11|, &lt;0 -4 -16 -9 -10 -29 -11 -19|]
Badness (Sintel): 2.33
 
{{Vals|legend=1| 14cf, 31, 45ef, 76e }}
 
== Cuboctahedra ==
[[Comma]]s: 81/80, 385/384, 1375/1372
 
[[POTE generator]]: ~9/7 = 425.993
 
Mapping generator: ~9/7
 
Map: [&lt;1 3 8 6 -4|, &lt;0 -4 -16 -9 21|]
 
{{Vals|legend=1| 14ce, 17ce, 31, 45, 76, 107b }}
 
[[Badness]]: 0.0568
 
= Liese =
<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>
 
[[Comma]]s: 81/80, 686/675
 
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.
 
7 and 9 limit minimax 1/4 comma
 
[{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 2/3 0 11/12 0 }}]
 
[[Eigenmonzo]]s: 2, 5
 
[[POTE generator]]: ~10/7 = 632.406
 
Mapping generator: ~10/7
 
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.
 
Map: [&lt;1 0 -4 -3|, &lt;0 3 12 11|]
 
[[Generator]]s: 2, 10/7
 
{{Vals|legend=1| 17c, 19, 36, 55, 74d }}
 
[[Badness]]: 0.0467
 
== Liesel ==
Commas: 56/55, 81/80, 540/539
 
POTE generator: ~10/7 = 633.073
 
Mapping generator: ~10/7
 
Map: [&lt;1 0 -4 -3 4|, &lt;0 3 12 11 -1|]
 
{{Vals|legend=1| 17c, 19, 36, 55e, 91cee }}
 
Badness: 0.0407


=== 13-limit ===
=== 13-limit ===
Liesel is a very natural 13-limit tuning, given the generator is so near 13/9.
Subgroup: 2.3.5.7.11.13
 
Commas: 56/55, 78/77, 81/80, 91/90


POTE generator: ~10/7 = ~13/9 = 633.042
Comma list: 81/80, 105/104, 144/143, 2401/2376


Mapping generator: ~10/7
Mapping: {{mapping| 5 0 -20 14 41 -21 | 0 1 4 0 -3 5 }}


Map: [&lt;1 0 -4 -3 4 0|, &lt;0 3 12 11 -1 7|]
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}})


{{Vals|legend=1| 17c, 19, 36, 55ef, 91ceef }}
{{Optimal ET sequence|legend=0| 5, 45f, 50 }}


Badness: 0.0273
Badness (Sintel): 2.02


== Elisa ==
== Subgroup extensions ==
Commas: 77/75, 81/80, 99/98
=== Stützel (2.3.5.19) ===
[[Subgroup]]: 2.3.5.19


POTE generator: ~10/7 = 633.061
[[Comma list]]: 81/80, 96/95


Mapping generator: ~10/7
{{Mapping|legend=2| 1 0 -4 9 | 0 1 4 -3 }}


Map: [&lt;1 0 -4 -3 -5|, &lt;0 3 12 11 16|]
{{Mapping|legend=3| 1 0 -4 0 0 0 0 9 | 0 1 4 0 0 0 0 -3 }}


{{Vals|legend=1| 17c, 19e, 36e }}
: mapping generators: ~2, ~3


Badness: 0.0416
[[Optimal tuning]]s:  
* [[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 }}


== Lisa ==
{{Optimal ET sequence|legend=1| 5, 7, 12, 31, 43, 98h }}
Commas: 45/44, 81/80, 343/330


POTE generator: ~10/7 = 631.370
[[Badness]] (Sintel): 0.324


Mapping generator: ~10/7
=== Hypnotone ===
Hypnotone is no-sevens [[#Flattone|flattone]].


Map: [&lt;1 0 -4 -3 -6|, &lt;0 3 12 11 18|]
[[Subgroup]]: 2.3.5.11


{{Vals|legend=1| 19 }}
[[Comma list]]: 45/44, 81/80


Badness: 0.0548
{{Mapping|legend=2| 1 0 -4 -6 | 0 1 4 6 }}


=== 13-limit ===
{{Mapping|legend=3| 1 0 -4 0 -6 | 0 1 4 0 6 }}
Commas: 45/44, 81/80, 91/88, 147/143


POTE generator: ~10/7 = 631.221
: mapping generators: ~2, ~3


Map: [&lt;1 0 -4 -3 -6 0|, &lt;0 3 12 11 18 7|]
[[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 }}


{{Vals|legend=1| 19 }}
{{Optimal ET sequence|legend=1| 7, 12, 19, 26, 45 }}


Badness: 0.0361
[[Badness]] (Sintel): 0.326


= Jerome =
==== 2.3.5.11.13 subgroup ====
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.
Subgroup: 2.3.5.11.13


Commas: 81/80, 17280/16807
Comma list: 45/44, 65/64, 81/80


POTE generator: ~54/49 = 139.343
Subgroup-val mapping: {{mapping| 1 0 -4 -6 10 | 0 1 4 6 -4 }}


Mapping generator: ~54/49
Gencom mapping: {{mapping| 1 0 -4 0 -6 10 | 0 1 4 0 6 -4 }}


Map: [&lt;1 1 0 2|, &lt;0 5 20 7|]
Optimal tunings:  
* WE: ~2 = 1202.6916{{c}}, ~3/2 = 694.4181{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0870{{c}}


Wedgie: &lt;&lt;5 30 7 20 -3 -40||
{{Optimal ET sequence|legend=0| 7, 12, 19, 26, 45f }}


{{Vals|legend=1| 9c, 17c, 26, 43, 69, 112bd }}
Badness (Sintel): 0.561


Badness: 0.1087
=== Dequarter ===
[[Subgroup]]: 2.3.5.11


== 11-limit ==
[[Comma list]]: 33/32, 55/54
Commas: 81/80, 99/98, 864/847


POTE generator: ~12/11 = 139.428
{{Mapping|legend=2| 1 0 -4 5 | 0 1 4 -1 }}


Mapping generator: ~12/11
{{Mapping|legend=3| 1 0 -4 0 5 | 0 1 4 0 -1 }}


Map: [&lt;1 1 0 2 3|, &lt;0 5 20 7 4|]
: mapping generators: ~2, ~3


{{Vals|legend=1| 9c, 17c, 26, 43, 69 }}
[[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 }}


Badness: 0.0479
{{Optimal ET sequence|legend=1| 5, 7, 19e, 26e }}


== 13-limit ==
[[Badness]] (Sintel): 0.451
Commas: 78/77, 81/80, 99/98, 144/143


POTE generator: ~13/12 = 139.387
==== Dreamtone ====
Subgroup: 2.3.5.11.13


Mapping generator: ~12/11
Comma list: 33/32, 55/54, 975/968


Map: [&lt;1 1 0 2 3 3|, &lt;0 5 20 7 4 6|]
Subgroup-val mapping: {{mapping| 1 0 -4 5 21 | 0 1 4 -1 -11 }}


{{Vals|legend=1| 9c, 17c, 26, 43, 69 }}
Gencom mapping: {{mapping| 1 0 -4 0 5 21 | 0 1 4 0 -1 -11 }}


Badness: 0.0293
Optimal tunings:  
* WE: ~2 = 1207.8248{{c}}, ~3/2 = 694.7806{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 690.1826{{c}}


== 17-limit ==
{{Optimal ET sequence|legend=0| 7, 19eff, 26eff, 33ceeff, 40ceeff }}
Commas: 78/77, 81/80, 99/98, 144/143, 189/187


POTE generator: ~13/12 = 139.362
Badness (Sintel): 1.40


Mapping generator: ~12/11
== References ==
<references/>


Map: [&lt;1 1 0 2 3 3 2|, &lt;0 5 20 7 4 6 18|]
[[Category:Temperament families]]
 
[[Category:Meantone family| ]] <!-- main article -->
{{Vals|legend=1| 26, 43, 69 }}
[[Category:Meantone| ]] <!-- key article -->
 
Badness: 0.0209
 
== 19-limit ==
Commas: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143
 
POTE generator: ~13/12 = 139.313
 
Mapping generator: ~12/11
 
Map: [&lt;1 1 0 2 3 3 2 1|, &lt;0 5 20 7 4 6 18 28|]
 
{{Vals|legend=1| 26, 43, 69 }}
 
Badness: 0.0182
 
= Meanmag =
Commas: 81/80, 3125/3072
 
POTE generator: ~8/7 = 238.396
 
Mapping generator: ~7
 
Map: [&lt;19 30 44 0|, &lt;0 0 0 1|]
 
Wedgie: &lt;&lt;0 0 19 0 30 44||
 
{{Vals|legend=1| 19, 38, 57, 76, 95bc }}
 
Badness: 0.0770
 
= Undevigintone =
Commas: 49/48, 81/80, 126/125
 
POTE generator: ~11/8 = 538.047
 
Mapping generator: ~11
 
Map: [&lt;19 30 44 53 0|, &lt;0 0 0 0 1|]
 
{{Vals|legend=1| 19, 38d }}
 
Badness: 0.0364
 
== 13-limit ==
`Commas: 49/48, 65/64, 81/80, 126/125
 
POTE generator: ~11/8 = 537.061
 
Map: [&lt;19 30 44 53 0 70|, &lt;0 0 0 0 1 0|]
 
{{Vals|legend=1| 19, 38d }}
 
Badness: 0.0229
 
[[Category:Theory]]
[[Category:Temperament family]]
[[Category:Meantone]]
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Listen]]
[[Category:Listen]]
{{todo|review|improve readability}}

Latest revision as of 14:15, 14 July 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The meantone family is the family of rank-2 temperaments that temper out the syntonic comma, 81/80, and thus can all be seen as extensions of meantone.

Meantone

Main article: Meantone

Meantone is characterized by an octave period, a fifth generator, and the relationship that four fifths go to make up a 5th harmonic.

Subgroup: 2.3.5

Comma list: 81/80

Mapping: [⟨1 0 -4], ⟨0 1 4]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1201.3906 ¢, ~3/2 = 697.0455 ¢
error map: ⟨+1.391 -3.519 +1.868]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.6512 ¢
error map: ⟨0.000 -5.304 +0.291]

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.5

Tuning ranges:

Optimal ET sequence: 5, 7, 12, 19, 31, 50, 81, 131b

Badness (Sintel): 0.173

Overview to extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at.

  • Flattertone adds [-24 17 0 -1⟩, finding the ~7/4 at the double-augmented sixth, for a tuning between 33edo and 26edo.
  • Flattone adds [-17 9 0 1⟩, finding the ~7/4 at the diminished seventh, for a tuning between 26edo and 19edo.
  • Septimal meantone adds [-13 10 0 -1⟩, finding the ~7/4 at the augmented sixth, for a tuning between 19edo and 12edo.
  • Dominant adds [6 -2 0 -1⟩, finding the ~7/4 at the minor seventh, for a tuning between 12edo and 5edo.
  • Sharptone adds [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 [-7 8 0 -2⟩ with a half-octave period.
  • Mohajira adds [-23 11 0 2⟩ and splits the fifth in two.
  • Godzilla adds [-4 -1 0 2⟩ with an ~8/7 generator, two of which give the fourth.
  • Mothra adds [-10 1 0 3⟩ with an ~8/7 generator, three of which give the fifth.
  • Liese adds [-9 11 0 -3⟩ with a ~10/7 generator, three of which give the twelfth.
  • Squares adds [-3 9 0 -4⟩ with a ~9/7 generator, four of which give the eleventh.
  • Jerome adds [3 7 0 -5⟩ and slices the fifth in five.

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)3 as 7/5, leading to septimal meantone, a very elegant extension to the 7-limit.

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.

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 extensions. 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.

Splitting the meantone fifth into two (243/242)

By tempering out 243/242 we equate the distance from 9/8 to 10/9 (= S9) with the distance between 11/10 to 12/11 (= 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 (5/4)2 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.

Splitting the meantone fifth into three (1029/1024)

By tempering out 1029/1024 we equate the distance from 7/6 to 8/7 (= S7) with the distance from 8/7 to 9/8 (= S8), so that (8/7)3 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)3 = 1728/1715 (S6/S7), the orwellisma.

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)2, 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)2 = 36/25 = (3/2)/(25/24).

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).

Temperaments discussed elsewhere include

The rest are considered below.

Septimal meantone

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Main article: Meantone § Septimal meantone
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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.

Subgroup: 2.3.5.7

Comma list: 81/80, 126/125

Mapping: [⟨1 0 -4 -13], ⟨0 1 4 10]]

Optimal tunings:

  • WE: ~2 = 1201.2358 ¢, ~3/2 = 697.2122 ¢
error map: ⟨+1.236 -3.507 +2.535 -0.412]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.6562 ¢
error map: ⟨0.000 -5.299 +0.311 -2.264]

Minimax tuning:

projection map: [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [-3 0 5/2 0⟩]
unchanged-interval (eigenmonzo) basis: 2.5

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
  • 7-odd-limit diamond tradeoff: ~3/2 = [694.786, 701.955] (1/3-comma to Pyth.)
  • 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)

Algebraic generator: Cybozem, the real root of 15x3 - 10x2 - 18, 503.4257 cents. The recurrence converges quickly.

Optimal ET sequence: 12, 19, 31, 81, 112b, 143b

Badness (Sintel): 0.347

Undecimal meantone (huygens)

"Huygens" redirects here. For the Dutch mathematician, physicist and astronomer, see Wikipedia: Christiaan Huygens.
See also: Huygens vs meanpop

Undecimal meantone[1] a.k.a. huygens[2][3] 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

Mapping: [⟨1 0 -4 -13 -25], ⟨0 1 4 10 18]]

Optimal tunings:

  • WE: ~2 = 1200.7636 ¢, ~3/2 = 697.4122 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.0315 ¢

Minimax tuning:

  • 11-odd-limit: ~3/2 = [9/16 -1/8 0 0 1/16⟩
projection map: [[1 0 0 0 0⟩, [25/16 -1/8 0 0 1/16⟩, [9/4 -1/2 0 0 1/4⟩, [21/8 -5/4 0 0 5/8⟩, [25/8 -9/4 0 0 9/8⟩]
unchanged-interval (eigenmonzo) basis: 2.11/9

Tuning ranges:

  • 11-odd-limit diamond monotone: ~3/2 = [696.774, 700.000] (18\31 to 7\12)
  • 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)

Algebraic generator: Traverse, the positive real root of x4 + 2x - 13, or 696.9529 cents.

Optimal ET sequence: 12, 19e, 31, 105, 136b

Badness (Sintel): 0.563

Music

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.

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 126/125, 144/143

Mapping: [⟨1 0 -4 -13 -25 29], ⟨0 1 4 10 18 -16]]

Optimal tunings:

  • WE: ~2 = 1199.9389 ¢, ~3/2 = 697.2282 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.2627 ¢

Minimax tuning:

  • 13- and 15-odd-limit: ~3/2 = [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: 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: [⟨1 0 -4 -13 -25 29 12], ⟨0 1 4 10 18 -16 -5]]

Optimal tunings:

  • WE: ~2 = 1199.5811 ¢, ~3/2 = 697.0918 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.3303 ¢

Optimal ET sequence: 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: [⟨1 0 -4 -13 -25 29 12 9], ⟨0 1 4 10 18 -16 -5 -3]]

Optimal tunings:

  • WE: ~2 = 1199.2931 ¢, ~3/2 = 696.9690 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.3736 ¢

Optimal ET sequence: 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: [⟨1 0 -4 -13 -25 -20], ⟨0 1 4 10 18 15]]

Optimal tunings:

  • WE: ~2 = 1200.8149 ¢, ~3/2 = 697.1155 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.7085 ¢

Minimax tuning:

  • 13- and 15-odd-limit: ~3/2 = [9/16 -1/8 0 0 1/16⟩
unchanged-interval (eigenmonzo) basis: 2.11/9

Optimal ET sequence: 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: [⟨1 0 -4 -13 -25 -20 12], ⟨0 1 4 10 18 15 -5]]

Optimal tunings:

  • WE: ~2 = 1199.5548 ¢, ~3/2 = 696.7449 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.9823 ¢

Optimal ET sequence: 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: [⟨1 0 -4 -13 -25 -20 12 9], ⟨0 1 4 10 18 15 -5 -3]]

Optimal tunings:

  • WE: ~2 = 1199.0408 ¢, ~3/2 = 696.5824 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.1061 ¢

Optimal ET sequence: 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: [⟨1 0 -4 -13 -25 -39], ⟨0 1 4 10 18 27]]

Optimal tunings:

  • WE: ~2 = 1199.9122 ¢, ~3/2 = 697.4779 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.5241 ¢

Minimax tuning:

  • 13- and 15-odd-limit: ~3/2 = [14/25 -2/25 0 0 0 1/25⟩
unchanged-interval (eigenmonzo) basis: 2.13/9

Optimal ET sequence: 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: [⟨1 0 -4 -13 -25 -39 12], ⟨0 1 4 10 18 27 -5]]

Optimal tunings:

  • WE: ~2 = 1199.3793 ¢, ~3/2 = 697.2833 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.6222 ¢

Optimal ET sequence: 12f, 43

Badness (Sintel): 1.22

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125

Mapping: [⟨1 0 -4 -13 -25 -39 12 9], ⟨0 1 4 10 18 27 -5 -3]]

Optimal tunings:

  • WE: ~2 = 1199.0260 ¢, ~3/2 = 697.1486 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.6887 ¢

Optimal ET sequence: 12f, 43

Badness (Sintel): 1.25

Hemimeantone

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 126/125, 169/168

Mapping: [⟨1 0 -4 -13 -25 -5], ⟨0 2 8 20 36 11]]

mapping generators: ~2, ~26/15

Optimal tunings:

  • WE: ~2 = 1201.0387 ¢, ~26/15 = 949.2863 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 948.5065 ¢

Optimal ET sequence: 19e, 43, 62

Badness (Sintel): 1.30

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 99/98, 126/125, 169/168, 221/220

Mapping: [⟨1 0 -4 -13 -25 -5 -22], ⟨0 2 8 20 36 11 33]]

Optimal tunings:

  • WE: ~2 = 1201.0270 ¢, ~26/15 = 949.2892 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 948.5169 ¢

Optimal ET sequence: 19eg, 43, 62

Badness (Sintel): 1.19

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 99/98, 126/125, 153/152, 169/168, 221/220

Mapping: [⟨1 0 -4 -13 -25 -5 -22 -25], ⟨0 2 8 20 36 11 33 37]]

Optimal tunings:

  • WE: ~2 = 1201.0339 ¢, ~19/11 = 949.2902 ¢
  • CWE: ~2 = 1200.0000 ¢, ~19/11 = 948.5111 ¢

Optimal ET sequence: 19egh, 43, 62

Badness (Sintel): 1.15

Semimeantone

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 126/125, 847/845

Mapping: [⟨2 0 -8 -26 -50 -59], ⟨0 1 4 10 18 21]]

mapping generators: ~55/39, ~3

Optimal tunings:

  • WE: ~55/39 = 600.3606 ¢, ~3/2 = 697.4241 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~3/2 = 697.0545 ¢

Optimal ET sequence: 12f, …, 50eff, 62, 136b

Badness (Sintel): 1.68

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 99/98, 126/125, 221/220, 289/288

Mapping: [⟨2 0 -8 -26 -50 -59 5], ⟨0 1 4 10 18 21 1]]

Optimal tunings:

  • WE: ~17/12 = 600.5426 ¢, ~3/2 = 697.5571 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 696.9858 ¢

Optimal ET sequence: 12f, 50eff, 62, 136bg

Badness (Sintel): 1.60

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 99/98, 126/125, 153/152, 209/208, 221/220

Mapping: [⟨2 0 -8 -26 -50 -59 5 -1], ⟨0 1 4 10 18 21 1 3]]

Optimal tunings:

  • WE: ~17/12 = 600.5959 ¢, ~3/2 = 697.5985 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 696.9638 ¢

Optimal ET sequence: 12f, 50eff, 62

Badness (Sintel): 1.47

Meanpop

See also: Huygens vs meanpop

Meanpop[1][3] maps the 11/8 to the double-diminished fifth (C–G𝄫), and tridecimal meanpop maps the 13/8 to the double-augmented fifth (C–G𝄪), tempering out 144/143 like in grosstone. Note also 11/10 is a double-diminished third; 12/11~13/12, double-augmented unison; and 14/13, minor second.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 126/125, 385/384

Mapping: [⟨1 0 -4 -13 24], ⟨0 1 4 10 -13]]

mapping generator: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1201.3464 ¢, ~3/2 = 697.2159 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.4509 ¢

Minimax tuning:

  • 11-odd-limit: ~3/2 = [0 0 1/4⟩
projection map: [[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [-3 0 5/2 0 0⟩, [11 0 -13/4 0 0⟩]
unchanged-interval (eigenmonzo) basis: 2.5

Tuning ranges:

  • 11-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
  • 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)

Algebraic generator: Cybozem; or else Radieubiz, the real root of 3x3 + 6x - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.

Optimal ET sequence: 12e, 19, 31, 81, 112b

Badness (Sintel): 0.712

Music

Tridecimal meanpop

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 126/125, 144/143

Mapping: [⟨1 0 -4 -13 24 -20], ⟨0 1 4 10 -13 15]]

Optimal tunings:

  • WE: ~2 = 1201.0765 ¢, ~3/2 = 696.8361 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.2347 ¢

Minimax tuning:

  • 13- and 15-odd-limit: ~3/2 = [4/7 0 0 0 -1/28 1/28⟩
unchanged-interval (eigenmonzo) basis: 2.13/11

Tuning ranges:

  • 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.)

Optimal ET sequence: 19, 31, 50, 81

Badness (Sintel): 0.863

Meanpoppic

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 105/104, 126/125, 144/143, 273/272

Mapping: [⟨1 0 -4 -13 24 -20 -37], ⟨0 1 4 10 -13 15 26]]

Optimal tunings:

  • WE: ~2 = 1201.0727 ¢, ~3/2 = 696.8168 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.2195 ¢

Optimal ET sequence: 19g, 31, 50, 81, 131bd

Badness (Sintel): 1.02

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 105/104, 126/125, 144/143, 153/152, 273/272

Mapping: [⟨1 0 -4 -13 24 -20 -37 -40], ⟨0 1 4 10 -13 15 26 28]]

Optimal tunings:

  • WE: ~2 = 1201.0719 ¢, ~3/2 = 696.8101 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.2137 ¢

Optimal ET sequence: 19gh, 31, 50, 81

Badness (Sintel): 1.08

Meanpoid

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 105/104, 120/119, 126/125, 144/143

Mapping: [⟨1 0 -4 -13 24 -20 12], ⟨0 1 4 10 -13 15 -5]]

Optimal tunings:

  • WE: ~2 = 1200.2768 ¢, ~3/2 = 696.5683 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.4114 ¢

Optimal ET sequence: 19, 31

Badness (Sintel): 1.17

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 77/76, 81/80, 96/95, 105/104, 120/119, 126/125

Mapping: [⟨1 0 -4 -13 24 -20 12 9], ⟨0 1 4 10 -13 15 -5 -3]]

Optimal tunings:

  • WE: ~2 = 1199.7905 ¢, ~3/2 = 696.3779 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.4973 ¢

Optimal ET sequence: 19, 31

Badness (Sintel): 1.25

Semimeanpop

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 126/125, 385/384, 847/845

Mapping: [⟨2 0 -8 -26 48 39], ⟨0 1 4 10 -13 -10]]

mapping generators: ~55/39, ~3

Optimal tunings:

  • WE: ~55/39 = 600.6704 ¢, ~3/2 = 697.2151 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~3/2 = 696.4341 ¢

Optimal ET sequence: 12e, 50, 62, 112b

Badness (Sintel): 1.78

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨2 0 -8 -26 48 39 5], ⟨0 1 4 10 -13 -10 1]]

Optimal tunings:

  • WE: ~17/12 = 600.7232 ¢, ~3/2 = 697.2820 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 696.4411 ¢

Optimal ET sequence: 12e, 50, 62, 112bg

Badness (Sintel): 1.45

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 126/125, 153/152, 209/208, 221/220, 273/272

Mapping: [⟨2 0 -8 -26 48 39 5 -1], ⟨0 1 4 10 -13 -10 1 3]]

Optimal tunings:

  • WE: ~17/12 = 600.7527 ¢, ~3/2 = 697.3244 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 696.4525 ¢

Optimal ET sequence: 12e, 50, 62, 112bgh

Badness (Sintel): 1.28

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.

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 -4 -13 -6], ⟨0 1 4 10 6]]

Optimal tunings:

  • WE: ~2 = 1199.6946 ¢, ~3/2 = 696.0729 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.2083 ¢

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]

Optimal ET sequence: 7d, 12, 19, 31e

Badness (Sintel): 0.708

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 -4 -13 -6 -20], ⟨0 1 4 10 6 15]]

Optimal tunings:

  • WE: ~2 = 1199.7931 ¢, ~3/2 = 696.0258 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.1241 ¢

Optimal ET sequence: 7df, 12f, 19, 31e

Badness (Sintel): 0.875

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 45/44, 56/55, 78/77, 81/80, 120/119

Mapping: [⟨1 0 -4 -13 -6 -20 12], ⟨0 1 4 10 6 15 -5]]

Optimal tunings:

  • WE: ~2 = 1198.6665 ¢, ~3/2 = 695.8010 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.4998 ¢

Optimal ET sequence: 12f, 19, 31e

Badness (Sintel): 1.17

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 45/44, 56/55, 78/77, 81/80, 96/95, 120/119

Mapping: [⟨1 0 -4 -13 -6 -20 12 9], ⟨0 1 4 10 6 15 -5 -3]]

Optimal tunings:

  • WE: ~2 = 1198.2880 ¢, ~3/2 = 695.7123 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.6370 ¢

Optimal ET sequence: 12f, 19, 31e

Badness (Sintel): 1.23

Vincenzo

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 -4 -13 -6 10], ⟨0 1 4 10 6 -4]]

Optimal tunings:

  • WE: ~2 = 1202.1684 ¢, ~3/2 = 696.3160 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 695.2045 ¢

Optimal ET sequence: 7d, 12, 19

Badness (Sintel): 1.02

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 0 -4 -13 -6 10 12], ⟨0 1 4 10 6 -4 -5]]

Optimal tunings:

  • WE: ~2 = 1200.5137 ¢, ~3/2 = 696.1561 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 695.8771 ¢

Optimal ET sequence: 12, 19

Badness (Sintel): 1.30

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80

Mapping: [⟨1 0 -4 -13 -6 10 12 9], ⟨0 1 4 10 6 -4 -5 -3]]

Optimal tunings:

  • WE: ~2 = 1199.8261 ¢, ~3/2 = 696.0298 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.1262 ¢

Optimal ET sequence: 12, 19

Badness (Sintel): 1.36

Bimeantone

11/8 is mapped to half octave minus the meantone diesis.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 126/125, 245/242

Mapping: [⟨2 0 -8 -26 -31], ⟨0 1 4 10 12]]

mapping generators: ~63/44, ~3

Optimal tunings:

  • WE: ~63/44 = 600.7492 ¢, ~3/2 = 696.8853 ¢
  • CWE: ~63/44 = 600.0000 ¢, ~3/2 = 696.1908 ¢

Optimal ET sequence: 12, 26de, 38d, 50

Badness (Sintel): 1.26

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 126/125, 245/242

Mapping: [⟨2 0 -8 -26 -31 -40], ⟨0 1 4 10 12 15]]

Optimal tunings:

  • WE: ~55/39 = 600.8309 ¢, ~3/2 = 696.8000 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~3/2 = 696.0066 ¢

Optimal ET sequence: 12f, 26deff, 38df, 50

Badness (Sintel): 1.19

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 105/104, 126/125, 189/187, 221/220

Mapping: [⟨2 0 -8 -26 -31 -40 5], ⟨0 1 4 10 12 15 1]]

Optimal tunings:

  • WE: ~17/12 = 600.9234 ¢, ~3/2 = 696.8536 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 695.9317 ¢

Optimal ET sequence: 12f, 38df, 50

Badness (Sintel): 1.15

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220

Mapping: [⟨2 0 -8 -26 -31 -40 5 -1], ⟨0 1 4 10 12 15 1 3]]

Optimal tunings:

  • WE: ~17/12 = 600.9845 ¢, ~3/2 = 696.8939 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 695.8947 ¢

Optimal ET sequence: 12f, 26deff, 38df, 50

Badness (Sintel): 1.08

Trimean

See also: No-sevens subgroup temperaments § Superpine

Subgroup: 2.3.5.7.11

Comma list: 81/80, 126/125, 1344/1331

Mapping: [⟨1 2 4 7 5], ⟨0 -3 -12 -30 -11]]

mapping generators: ~2, ~11/10

Optimal tunings:

  • WE: ~2 = 1200.7155 ¢, ~11/10 = 167.9055 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 167.7749 ¢

Optimal ET sequence: 7d, 36d, 43, 50, 93

Badness (Sintel): 1.68

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 126/125, 144/143, 364/363

Mapping: [⟨1 2 4 7 5 3], ⟨0 -3 -12 -30 -11 5]]

Optimal tunings:

  • WE: ~2 = 1200.6104 ¢, ~11/10 = 167.8749 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 167.7728 ¢

Optimal ET sequence: 7d, 43, 50, 93

Badness (Sintel): 1.46

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 2 4 7 5 3 8], ⟨0 -3 -12 -30 -11 5 -28]]

Optimal tunings:

  • WE: ~2 = 1200.6144 ¢, ~11/10 = 167.8716 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 167.7682 ¢

Optimal ET sequence: 7dg, 43, 50, 93

Badness (Sintel): 1.28

Migration

See Rastmic clan.

Flattone

Main article: Flattone

In flattone, 9 fourths get to the interval class for 7, so that 7/4 is a diminished seventh (C–B𝄫), 7/6 is a diminished third (C–E𝄫), and 7/5 is a double-diminished fifth (C–G𝄫). In general, septimal subminor intervals are diminished and septimal supermajor intervals are augmented, which makes it quite easy to learn flattone notation. The fifth in flattone is typically flatter than that of 19edo. Good tunings for flattone include 45edo, 64edo, and 71edo.

Subgroup: 2.3.5.7

Comma list: 81/80, 525/512

Mapping: [⟨1 0 -4 17], ⟨0 1 4 -9]]

Optimal tunings:

  • WE: ~2 = 1203.6308 ¢, ~3/2 = 695.8782 ¢
error map: ⟨+3.631 -2.446 -2.801 -2.684]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 693.7334 ¢
error map: ⟨0.000 -8.222 -11.380 -12.426]

Minimax tuning:

projection map: [[1 0 0 0⟩, [21/13 0 1/13 -1/13⟩, [32/13 0 4/13 -4/13⟩, [32/13 0 -9/13 9/13⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5
projection map: [[1 0 0 0⟩, [17/11 2/11 0 -1/11⟩, [24/11 8/11 0 -4/11⟩, [34/11 -18/11 0 9/11⟩]
unchanged-interval (eigenmonzo) basis: 2.9/7

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
  • 7-odd-limit diamond tradeoff: ~3/2 = [692.353, 701.955]
  • 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]

Algebraic generator: Squarto, the positive root of 8x2 - 4x - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.

Optimal ET sequence: 7, 19, 26, 45

Badness (Sintel): 0.976

11-limit

This can also be considered a no-sevens temperament: hypnotone.

Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80, 385/384

Mapping: [⟨1 0 -4 17 -6], ⟨0 1 4 -9 6]]

Optimal tuning:

  • WE: ~2 = 1202.3247 ¢, ~3/2 = 694.4688 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 693.1467 ¢

Tuning ranges:

  • 11-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
  • 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]

Optimal ET sequence: 7, 19, 26, 45, 71bc, 116bcde

Badness (Sintel): 1.12

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 65/64, 78/77, 81/80

Mapping: [⟨1 0 -4 17 -6 10], ⟨0 1 4 -9 6 -4]]

Optimal tunings:

  • WE: ~2 = 1202.5156 ¢, ~3/2 = 694.5107 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 693.0538 ¢

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: 7, 19, 26, 45f, 71bcf, 116bcdef

Badness (Sintel): 0.920

Ptolemy

See Rastmic clan.

Dominant

Main article: Dominant (temperament)
See also: Archytas clan

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.

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 temperaments.

Subgroup: 2.3.5.7

Comma list: 36/35, 64/63

Mapping: [⟨1 0 -4 6], ⟨0 1 4 -2]]

Optimal tunings:

  • WE: ~2 = 1195.3384 ¢, ~3/2 = 698.8478 ¢
error map: ⟨-4.662 -7.769 +9.077 +14.832]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.1125 ¢
error map: ⟨0.000 -0.842 +18.136 +28.949]

Tuning ranges:

Optimal ET sequence: 5, 7, 12, 41cd, 53cdd, 65ccddd

Badness (Sintel): 0.524

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 64/63

Mapping: [⟨1 0 -4 6 13], ⟨0 1 4 -2 -6]]

Tuning ranges:

  • 11-odd-limit diamond monotone: ~3/2 = [700.000, 705.882] (7\12 to 10\17)
  • 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]

Optimal tunings:

  • WE: ~2 = 1194.0169 ¢, ~3/2 = 699.7473 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.2672 ¢

Optimal ET sequence: 5, 12, 17c, 29cde

Badness (Sintel): 0.799

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 56/55, 64/63, 66/65

Mapping: [⟨1 0 -4 6 13 18], ⟨0 1 4 -2 -6 -9]]

Optimal tunings:

  • WE: ~2 = 1193.8055 ¢, ~3/2 = 700.0042 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.8254 ¢

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~3/2 = 705.882 (10\17)
  • 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]

Optimal ET sequence: 12f, 17c, 29cdef

Badness (Sintel): 0.996

Dominion

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 36/35, 56/55, 64/63

Mapping: [⟨1 0 -4 6 13 -9], ⟨0 1 4 -2 -6 8]]

Optimal tunings:

  • WE: ~2 = 1195.0293 ¢, ~3/2 = 701.9847 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.7698 ¢

Optimal ET sequence: 5, 12, 17c

Badness (Sintel): 1.13

Domination

Subgroup: 2.3.5.7.11

Comma list: 36/35, 64/63, 77/75

Mapping: [⟨1 0 -4 6 -14], ⟨0 1 4 -2 11]]

Optimal tunings:

  • WE: ~2 = 1194.8645 ¢, ~3/2 = 701.9872 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5945 ¢

Optimal ET sequence: 5e, 12e, 17c

Badness (Sintel): 1.21

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 36/35, 64/63, 66/65

Mapping: [⟨1 0 -4 6 -14 -9], ⟨0 1 4 -2 11 8]]

Optimal tunings:

  • WE: ~2 = 1195.1324 ¢, ~3/2 = 702.6343 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 705.0791 ¢

Optimal ET sequence: 5e, 12e, 17c

Badness (Sintel): 1.13

Domineering

Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44, 64/63

Mapping: [⟨1 0 -4 6 -6], ⟨0 1 4 -2 6]]

Optimal tunings:

  • WE: ~2 = 1194.7102 ¢, ~3/2 = 695.6962 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 698.1765 ¢

Optimal ET sequence: 5e, 7, 12

Badness (Sintel): 0.727

Arnold

Subgroup: 2.3.5.7.11

Comma list: 22/21, 33/32, 36/35

Mapping: [⟨1 0 -4 6 5], ⟨0 1 4 -2 -1]]

Optimal tunings:

  • WE: ~2 = 1199.8507 ¢, ~3/2 = 698.4045 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 698.4822 ¢

Optimal ET sequence: 5, 7, 12e

Badness (Sintel): 0.864

Neutrominant

See Rastmic clan.

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 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.

Flattertone was named by Flora Canou in 2024.

Subgroup: 2.3.5.7

Comma list: 81/80, 1875/1792

Mapping: [⟨1 0 -4 -24], ⟨0 1 4 17]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1204.4511 ¢, ~3/2 = 694.3258 ¢
error map: ⟨+4.451 -3.178 -9.011 +3.554]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 692.0479 ¢
error map: ⟨0.000 -9.907 -18.122 -4.012]

Optimal ET sequence: 7d, 19d, 26, 59bcd, 85bccd

Badness (Sintel): 2.43

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80, 1375/1344

Mapping: [⟨1 0 -4 -24 -6], ⟨0 1 4 17 6]]

Optimal tunings:

  • WE: ~2 = 1203.4653 ¢, ~3/2 = 693.8144 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 692.0422 ¢

Optimal ET sequence: 7d, 19d, 26

Badness (Sintel): 1.53

Music

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.

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).

The 11-limit extension was named by Gene Ward Smith in 2004[3].

Subgroup: 2.3.5.7

Comma list: 21/20, 28/27

Mapping: [⟨1 0 -4 -2], ⟨0 1 4 3]]

Optimal tunings:

  • WE: ~2 = 1204.2961 ¢, ~3/2 = 702.6463 ¢
error map: ⟨+4.296 +4.987 +24.271 -56.591]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.4928 ¢
error map: ⟨0.000 -0.462 +19.657 -64.347]

Optimal ET sequence: 5, 7d, 12d

Badness (Sintel): 0.629

Meanertone

Subgroup: 2.3.5.7.11

Comma list: 21/20, 28/27, 33/32

Mapping: [⟨1 0 -4 -2 5], ⟨0 1 4 3 -1]]

Optimal tunings:

  • WE: ~2 = 1208.5304 ¢, ~3/2 = 701.5669 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 698.1117 ¢

Optimal ET sequence: 5, 7d, 12de

Badness (Sintel): 0.832

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.

Mildtone was named by Lucius Chiaraviglio in 2024.

Subgroup: 2.3.5.7

Comma list: 81/80, 16128/15625

Mapping: [⟨1 0 -4 -32], ⟨0 1 4 22]]

Optimal tunings:

  • WE: ~2 = 1199.7304 ¢, ~3/2 = 698.3953 ¢
error map: ⟨-0.270 -3.829 +7.267 -1.434]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 698.5397 ¢
error map: ⟨0.000 -3.415 +7.845 -0.952]

Optimal ET sequence: 12, 43d, 55, 67

Badness (Sintel): 2.67

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 176/175, 7056/6875

Mapping: [⟨1 0 -4 -32], ⟨0 1 4 22 30]]

Optimal tunings:

  • WE: ~2 = 1199.816 ¢, ~3/2 = 698.355 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 698.455 ¢

Optimal ET sequence: 12, 43de, 55, 67

Badness (Sintel): 2.15

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 176/175, 196/195, 832/825

Mapping: [⟨1 0 -4 -32 -44], ⟨0 1 4 22 30]]

Optimal tunings:

  • WE: ~2 = 1199.788 ¢, ~3/2 = 698.355 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 698.471 ¢

Optimal ET sequence: 12f, 55f, 67

Badness (Sintel): 2.04

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 176/175, 189/187, 196/195, 832/825

Mapping: [⟨1 0 -4 -32 -44 12], ⟨0 1 4 22 30 -5]]

Optimal tunings:

  • WE: ~2 = 1199.655 ¢, ~3/2 = 698.295 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 698.488 ¢

Optimal ET sequence: 12f, 55f, 67

Badness (Sintel): 1.98

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 96/95, 176/175, 189/187, 196/195, 832/825

Mapping: [⟨1 0 -4 -32 -44 12 9], ⟨0 1 4 22 30 -5 -3]]

Optimal tunings:

  • WE: ~2 = 1199.371 ¢, ~3/2 = 698.164 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 698.519 ¢

Optimal ET sequence: 12f, 55f, 67

Badness (Sintel): 1.95

Supermean

Supermean tempers out 672/625 and finds the interval class of 7 at 15 generators up, as a double-augmented fifth (C–Gx). As such, it extends leapfrog.

Subgroup: 2.3.5.7

Comma list: 81/80, 672/625

Mapping: [⟨1 0 -4 -21], ⟨0 1 4 15]]

Optimal tunings:

  • WE: ~2 = 1195.4372 ¢, ~3/2 = 702.2086 ¢
error map: ⟨-4.563 -4.309 +22.521 -8.319]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5375 ¢
error map: ⟨0.000 +2.583 +31.836 -0.763]

Optimal ET sequence: 5d, 12d, 17c

Badness (Sintel): 3.40

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 81/80, 132/125

Mapping: [⟨1 0 -4 -21 -14], ⟨0 1 4 15 11]]

Optimal tunings:

  • WE: ~2 = 1195.7270 ¢, ~3/2 = 702.5848 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.7471 ¢

Optimal ET sequence: 5de, 12de, 17c

Badness (Sintel): 2.09

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 56/55, 66/65, 81/80

Mapping: [⟨1 0 -4 -21 -14 -9], ⟨0 1 4 15 11 8]]

Optimal tunings:

  • WE: ~2 = 1196.3958 ¢, ~3/2 = 702.9766 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.7940 ¢

Optimal ET sequence: 5de, 12de, 17c, 29c

Badness (Sintel): 1.67

Mohajira

Main article: Mohajira

Mohajira can be viewed as derived from mohaha which maps the interval half a 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 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.

Subgroup: 2.3.5.7

Comma list: 81/80, 6144/6125

Mapping: [⟨1 1 0 6], ⟨0 2 8 -11]]

mapping generators: ~2, ~128/105

Optimal tunings:

  • WE: ~2 = 1200.8160 ¢, ~128/105 = 348.6518 ¢
error map: ⟨+0.816 -3.835 +2.901 +0.900]
  • CWE: ~2 = 1200.0000 ¢, ~128/105 = 348.4194 ¢
error map: ⟨0.000 -5.116 +1.041 -1.439]

Minimax tuning:

projection map: [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [6 0 -11/8 0⟩]
Unchanged-interval (eigenmonzo) basis: 2.5

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]

Algebraic generator: Mohabis, real root of 3x3 - 3x2 - 1, 348.6067 cents. Corresponding recurrence converges quickly.

Optimal ET sequence: 7, 24, 31

Badness (Sintel): 1.41

Scales: mohaha7, mohaha10

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 121/120, 176/175

Mapping: [⟨1 1 0 6 2], ⟨0 2 8 -11 5]]

Optimal tunings:

  • WE: ~2 = 1201.1562 ¢, ~11/9 = 348.8124 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.4910 ¢

Minimax tuning:

  • 11-odd-limit: ~11/9 = [0 0 1/8⟩
projection map: [[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [6 0 -11/8 0 0⟩, [2 0 5/8 0 0⟩]
unchanged-interval (eigenmonzo) basis: 2.5

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]

Optimal ET sequence: 7, 24, 31

Badness (Sintel): 0.862

Scales: mohaha7, mohaha10

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 81/80, 105/104, 121/120

Mapping: [⟨1 1 0 6 2 4], ⟨0 2 8 -11 5 -1]]

Optimal tunings:

  • WE: ~2 = 1200.4256 ¢, ~11/9 = 348.6819 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.5622 ¢

Optimal ET sequence: 7, 24, 31

Badness (Sintel): 0.966

Scales: mohaha7, mohaha10

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 66/65, 81/80, 105/104, 121/120, 154/153

Mapping: [⟨1 1 0 6 2 4 7], ⟨0 2 8 -11 5 -1 -10]]

Optimal tunings:

  • WE: ~2 = 1200.0382 ¢, ~11/9 = 348.7471 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.7360 ¢

Optimal ET sequence: 7, 24, 31

Badness (Sintel): 1.05

Scales: mohaha7, mohaha10

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152

Mapping: [⟨1 1 0 6 2 4 7 6], ⟨0 2 8 -11 5 -1 -10 -6]]

Optimal tunings:

  • WE: ~2 = 1199.7469 ¢, ~11/9 = 348.7367 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.8117 ¢

Optimal ET sequence: 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 17c & 24; its ploidacot is dicot, the same as mohajira.

Subgroup: 2.3.5.7

Comma list: 81/80, 392/375

Mapping: [⟨1 1 0 -1], ⟨0 2 8 13]]

mapping generators: ~2, ~25/21

Optimal tunings:

  • WE: ~2 = 1199.0661 ¢, ~25/21 = 350.3127 ¢
error map: ⟨-0.934 -2.264 +16.188 -13.827]
  • CWE: ~2 = 1200.0000 ¢, ~25/21 = 350.4856 ¢
error map: ⟨0.000 -0.984 +17.571 -12.513]

Optimal ET sequence: 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: [⟨1 1 0 -1 2], ⟨0 2 8 13 5]]

Optimal tunings:

  • WE: ~2 = 1199.1924 ¢, ~11/9 = 350.3286 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.4821 ¢

Optimal ET sequence: 7d, 17c, 24

Badness (Sintel): 1.20

Scales: mohaha7, mohaha10

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 0 -1 2 4], ⟨0 2 8 13 5 -1]]

Optimal tunings:

  • WE: ~2 = 1198.5986 ¢, ~11/9 = 350.3353 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.6459 ¢

Optimal ET sequence: 7d, 17c, 24, 41c

Badness (Sintel): 1.19

Scales: mohaha7, mohaha10

Liese

Deutsch

Liese splits the 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 17c & 19; its ploidacot is alpha-tricot. It is a very natural 13-limit tuning, given the generator is so near 13/9. 74edo makes for a good liese tuning, though 19edo can be used. The tuning is well-supplied with mos scales: 7, 9, 11, 13, 15, 17, 19, 36, 55.

Subgroup: 2.3.5.7

Comma list: 81/80, 686/675

Mapping: [⟨1 0 -4 -3], ⟨0 3 12 11]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 1201.5548 ¢, ~10/7 = 633.2251 ¢
error map: ⟨+1.555 -2.280 +6.168 -8.015]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 632.5640 ¢
error map: ⟨0.000 -4.263 +4.454 -10.622]

Minimax tuning:

projection map: [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [2/3 0 11/12 0⟩]
unchanged-interval (eigenmonzo) basis: 2.5

Algebraic generator: Radix, the real root of x5 - 2x4 + 2x3 - 2x2 + 2x - 2, also a root of x6 - x5 - 2. The recurrence converges.

Optimal ET sequence: 17c, 19, 55, 74d

Badness (Sintel): 1.18

Liesel

Subgroup: 2.3.5.7.11

Comma list: 56/55, 81/80, 540/539

Mapping: [⟨1 0 -4 -3 4], ⟨0 3 12 11 -1]]

Optimal tunings:

  • WE: ~2 = 1198.8507 ¢, ~10/7 = 632.4668 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 632.9963 ¢

Optimal ET sequence: 17c, 19, 36

Badness (Sintel): 1.35

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 78/77, 81/80, 91/90

Mapping: [⟨1 0 -4 -3 4 0], ⟨0 3 12 11 -1 7]]

Optimal tunings:

  • WE: ~2 = 1199.4968 ¢, ~10/7 = 632.7766 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 633.0082 ¢

Optimal ET sequence: 17c, 19, 36

Badness (Sintel): 1.13

Elisa

Subgroup: 2.3.5.7.11

Comma list: 77/75, 81/80, 99/98

Mapping: [⟨1 0 -4 -3 -5], ⟨0 3 12 11 16]]

Optimal tunings:

  • WE: ~2 = 1201.0489 ¢, ~10/7 = 633.6147 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 633.1644 ¢

Optimal ET sequence: 17c, 19e, 36e

Badness (Sintel): 1.37

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 -4 -3 -5 0], ⟨0 3 12 11 16 7]]

Optimal tunings:

  • WE: ~2 = 1201.4815 ¢, ~10/7 = 633.7720 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 633.1281 ¢

Optimal ET sequence: 17c, 19e, 36e

Badness (Sintel): 1.11

Lisa

Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80, 343/330

Mapping: [⟨1 0 -4 -3 -6], ⟨0 3 12 11 18]]

Optimal tunings:

  • WE: ~2 = 1202.6773 ¢, ~10/7 = 632.7783 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 631.6175 ¢

Optimal ET sequence: 17cee, 19

Badness (Sintel): 1.81

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 81/80, 91/88, 147/143

Mapping: [⟨1 0 -4 -3 -6 0], ⟨0 3 12 11 18 7]]

Optimal tunings:

  • WE: ~2 = 1203.6086 ¢, ~10/7 = 633.1193 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 631.5346 ¢

Optimal ET sequence: 17cee, 19

Badness (Sintel): 1.49

Superpine

See also: No-sevens subgroup temperaments § Superpine

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 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.

Subgroup: 2.3.5.7

Comma list: 81/80, 1119744/1071875

Mapping: [⟨1 2 4 1], ⟨0 -3 -12 13]]

Optimal tunings:

  • WE: ~2 = 1199.3652 ¢, ~35/32 = 167.1615 ¢
error map: ⟨-0.635 -4.709 +5.209 +3.639]
  • CWE: ~2 = 1200.0000 ¢, ~35/32 = 167.2561 ¢
error map: ⟨0.000 -3.723 +6.613 +5.503]

Optimal ET sequence: 7, 36, 43, 79c

Badness (Sintel): 3.46

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 176/175, 864/847

Mapping: [⟨1 2 4 1 5], ⟨0 -3 -12 13 -11]]

Optimal tunings:

  • WE: ~2 = 1199.0522 ¢, ~11/10 = 167.1904 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 167.3382 ¢

Optimal ET sequence: 7, 36, 43

Badness (Sintel): 1.90

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 81/80, 144/143, 176/175

Mapping: [⟨1 2 4 1 5 3], ⟨0 -3 -12 13 -11 5]]

Optimal tunings:

  • WE: ~2 = 1199.4286 ¢, ~11/10 = 167.3105 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 167.3958 ¢

Optimal ET sequence: 7, 36, 43

Badness (Sintel): 1.52

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.

Subgroup: 2.3.5.7

Comma list: 81/80, 3125/3087

Mapping: [⟨3 0 -12 -20], ⟨0 1 4 6]]

mapping generators: ~56/45, ~3

Optimal tunings:

  • WE: ~56/45 = 400.6744 ¢, ~3/2 = 695.8474 ¢ {~15/14 = 105.5015 ¢)
error map: ⟨+2.023 -4.084 -2.924 +4.910]
  • CWE: ~56/45 = 400.0000 ¢, ~3/2 = 695.1413 ¢ {~15/14 = 104.8587 ¢)
error map: ⟨0.000 -6.814 -5.748 +2.022]

Optimal ET sequence: 12, 33cd, 45, 57

Badness (Sintel): 1.75

Squares

Main article: Squares

Squares splits the 6th harmonic into four subminor sixths of 11/7~14/9 (or splits a perfect eleventh into four supermajor thirds of 9/7~14/11), and uses it for a generator. It may be described as 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.

Subgroup: 2.3.5.7

Comma list: 81/80, 2401/2400

Mapping: [⟨1 -1 -8 -3], ⟨0 4 16 9]]

mapping generators: ~2, ~14/9

Optimal tunings:

  • WE: ~2 = 1201.2488 ¢, ~14/9 = 774.8640 ¢
error map: ⟨+1.249 -3.748 +1.520 +1.204]
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 774.1560 ¢
error map: ⟨0.000 -5.331 +0.183 -1.422]

Minimax tuning:

projection map: [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [3/2 0 9/16 0⟩]
unchanged-interval (eigenmonzo) basis: 2.5

Algebraic generator: Sceptre2, the positive root of 9x2 + x - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.

Optimal ET sequence: 14c, 17c, 31, 169b, 200b

Badness (Sintel): 1.16

Scales: skwares8, skwares11, skwares14

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -1 -8 -3 -3], ⟨0 4 16 9 10]]

Optimal tunings:

  • WE: ~2 = 1201.6657 ¢, ~11/7 = 775.1171 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 774.1754 ¢

Optimal ET sequence: 14c, 17c, 31, 130bee, 169beee

Badness (Sintel): 0.715

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 81/80, 99/98, 121/120

Mapping: [⟨1 -1 -8 -3 -3 5], ⟨0 4 16 9 10 -2]]

Optimal tunings:

  • WE: ~2 = 1199.8419 ¢, ~11/7 = 774.3484 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 774.4422 ¢

Optimal ET sequence: 14c, 17c, 31, 79cf

Badness (Sintel): 1.05

Squad

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 81/80, 91/90, 99/98

Mapping: [⟨1 -1 -8 -3 -3 -6], ⟨0 4 16 9 10 15]]

Optimal tunings:

  • WE: ~2 = 1202.0312 ¢, ~11/7 = 775.5589 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 774.4140 ¢

Optimal ET sequence: 14cf, 17c, 31f

Badness (Sintel): 1.11

Agora

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 105/104, 121/120

Mapping: [⟨1 -1 -8 -3 -3 -15], ⟨0 4 16 9 10 29]]

Optimal tunings:

  • WE: ~2 = 1202.3228 ¢, ~11/7 = 775.2214 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 773.8617 ¢

Optimal ET sequence: 14cf, 31, 45ef, 76e

Badness (Sintel): 1.01

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 99/98, 105/104, 120/119, 121/119

Mapping: [⟨1 -1 -8 -3 -3 -15 -3], ⟨0 4 16 9 10 29 11]]

Optimal tunings:

  • WE: ~2 = 1201.4340 ¢, ~11/7 = 774.7375 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 773.8955 ¢

Optimal ET sequence: 14cf, 31

Badness (Sintel): 1.15

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119

Mapping: [⟨1 -1 -8 -3 -3 -15 -3 -8], ⟨0 4 16 9 10 29 11 19]]

Optimal tunings:

  • WE: ~2 = 1201.2461 ¢, ~11/7 = 774.5783 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 773.8479 ¢

Optimal ET sequence: 14cf, 31

Badness (Sintel): 1.15

Cuboctahedra

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -1 -8 -3 17], ⟨0 4 16 9 -21]]

Optimal tunings:

  • WE: ~2 = 1201.4436 ¢, ~14/9 = 774.9386 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 774.0243 ¢

Optimal ET sequence: 31, 107b, 138b, 169be, 200be

Badness (Sintel): 1.88

Jerome

Jerome is related to Hieronymus' tuning; the Hieronymus generator is 51/20, or 139.316 cents. It may be described as 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.

Subgroup: 2.3.5.7

Comma list: 81/80, 17280/16807

Mapping: [⟨1 1 0 2], ⟨0 5 20 7]]

mapping generators: ~2, ~54/49

Optimal tunings:

  • WE: ~2 = 1200.1640 ¢, ~54/49 = 139.3624 ¢
error map: ⟨+0.164 -4.979 +0.934 +7.039]
  • CWE: ~2 = 1200.0000 ¢, ~54/49 = 139.3528 ¢
error map: ⟨0.000 -5.191 +0.741 +6.643]

Optimal ET sequence: 17c, 26, 43

Badness (Sintel): 2.75

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 864/847

Mapping: [⟨1 1 0 2 3], ⟨0 5 20 7 4]]

Optimal tunings:

  • WE: ~2 = 1201.4436 ¢, ~12/11 = 139.3714 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/11 = 139.4038 ¢

Optimal ET sequence: 17c, 26, 43

Badness (Sintel): 1.58

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 81/80, 99/98, 144/143

Mapping: [⟨1 1 0 2 3 3], ⟨0 5 20 7 4 6]]

Optimal tunings:

  • WE: ~2 = 1199.8860 ¢, ~13/12 = 139.3737 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.3817 ¢

Optimal ET sequence: 17c, 26, 43

Badness (Sintel): 1.21

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 78/77, 81/80, 99/98, 144/143, 189/187

Mapping: [⟨1 1 0 2 3 3 2], ⟨0 5 20 7 4 6 18]]

Optimal tunings:

  • WE: ~2 = 1199.8346 ¢, ~13/12 = 139.3431 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.3544 ¢

Optimal ET sequence: 17cg, 26, 43

Badness (Sintel): 1.06

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143

Mapping: [⟨1 1 0 2 3 3 2 1], ⟨0 5 20 7 4 6 18 28]]

Optimal tunings:

  • WE: ~2 = 1199.8891 ¢, ~13/12 = 139.3001 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.3080 ¢

Optimal ET sequence: 17cgh, 26, 43, 69

Badness (Sintel): 1.11

Meantritone

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

Subgroup: 2.3.5.7

Comma list: 81/80, 16875/16807

Mapping: [⟨1 -1 -8 -7], ⟨0 5 20 19]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 1201.3832 ¢, ~10/7 = 619.9478 ¢
error map: ⟨+1.383 -3.599 +1.576 +0.499]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.3176 ¢
error map: ⟨0.000 -5.367 +0.038 -1.791]

Optimal ET sequence: 29cd, 31, 188bcd, 219bbcd

Badness (Sintel): 2.08

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 2541/2500

Mapping: [⟨1 -1 -8 -7 -11], ⟨0 5 20 19 28]]

Optimal tunings:

  • WE: ~2 = 1201.2054 ¢, ~10/7 = 619.9752 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.4223 ¢

Optimal ET sequence: 29cde, 31

Badness (Sintel): 1.42

Injera

Injera has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a ~15/14 semitone difference between a half-octave and a perfect fifth. Injera may be described as 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 19edos, is an excellent tuning for injera.

Origin of the name

Subgroup: 2.3.5.7

Comma list: 50/49, 81/80

Mapping: [⟨2 0 -8 -7], ⟨0 1 4 4]]

mapping generators: ~7/5, ~3

Optimal tunings:

  • WE: ~7/5 = 600.6662 ¢, ~3/2 = 695.1463 ¢ (~21/20 = 94.4801 ¢)
error map: ⟨+1.332 -5.476 -5.729 +12.425]
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 694.7712 ¢ (~21/20 = 94.7712 ¢)
error map: ⟨0.000 -7.184 -7.229 +10.259]

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
  • 7-odd-limit diamond tradeoff: ~3/2 = [688.957, 701.955]
  • 9-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]

Optimal ET sequence: 12, 26, 38

Badness (Sintel): 0.788

Music

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 0 -8 -7 -12], ⟨0 1 4 4 6]]

Optimal tunings:

  • WE: ~7/5 = 600.9350 ¢, ~3/2 = 693.9198 ¢ (~21/20 = 92.9848 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 693.3539 ¢ (~21/20 = 93.3539 ¢)

Tuning ranges:

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

Optimal ET sequence: 12, 26

Badness (Sintel): 0.764

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 50/49, 78/77, 81/80

Mapping: [⟨2 0 -8 -7 -12 -21], ⟨0 1 4 4 6 9]]

Optimal tunings:

  • WE: ~7/5 = 600.9982 ¢, ~3/2 = 693.8249 ¢ (~21/20 = 92.8267 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 693.0992 ¢ (~21/20 = 93.0992 ¢)

Tuning ranges:

  • 13-odd-limit diamond monotone: ~3/2 = 692.308 (15\26)
  • 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]

Optimal ET sequence: 12f, 14cf, 26

Badness (Sintel): 0.891

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 45/44, 50/49, 78/77, 81/80, 85/84

Mapping: [⟨2 0 -8 -7 -12 -21 5], ⟨0 1 4 4 6 9 1]]

Optimal tunings:

  • WE: ~7/5 = 601.1757 ¢, ~3/2 = 693.8441 ¢ (~21/20 = 92.6684 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 692.8879 ¢ (~21/20 = 92.8879 ¢)

Optimal ET sequence: 12f, 14cf, 26

Badness (Sintel): 0.935

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 45/44, 50/49, 57/56, 78/77, 81/80, 85/84

Mapping: [⟨2 0 -8 -7 -12 -21 5 -1], ⟨0 1 4 4 6 9 1 3]]

Optimal tunings:

  • WE: ~7/5 = 601.4245 ¢, ~3/2 = 693.9426 ¢ (~21/20 = 92.5181 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 692.7606 ¢ (~21/20 = 92.7606 ¢)

Optimal ET sequence: 12f, 14cf, 26

Badness (Sintel): 0.920

Enjera

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 40/39, 45/44, 50/49

Mapping: [⟨2 0 -8 -7 -12 -2], ⟨0 1 4 4 6 3]]

Optimal tunings:

  • WE: ~7/5 = 599.1863 ¢, ~3/2 = 693.1791 ¢ (~21/20 = 93.9929 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 693.6809 ¢ (~21/20 = 93.6809 ¢)

Optimal ET sequence: 10cdeef, 12f

Badness (Sintel): 1.10

Injerous

Subgroup: 2.3.5.7.11

Comma list: 33/32, 50/49, 55/54

Mapping: [⟨2 0 -8 -7 10], ⟨0 1 4 4 -1]]

Optimal tunings:

  • WE: ~7/5 = 603.1682 ¢, ~3/2 = 694.1945 ¢ (~21/20 = 91.0264 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 691.6107 ¢ (~21/20 = 91.6107 ¢)

Optimal ET sequence: 12e, 14c, 26e, 40cee

Badness (Sintel): 1.28

Lahoh

Subgroup: 2.3.5.7.11

Comma list: 50/49, 56/55, 81/77

Mapping: [⟨2 0 -8 -7 7], ⟨0 1 4 4 0]]

Optimal tunings:

  • WE: ~7/5 = 597.3179 ¢, ~3/2 = 695.8759 ¢ (~21/20 = 98.5581 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 697.8757 ¢ (~21/20 = 97.8757 ¢)

Optimal ET sequence: 10cd, 12

Badness (Sintel): 1.42

Teff

Main article: Teff

Teff, found and named by Mason Green, is to injera what mohajira is to meantone; it splits the generator in halves in order to accommodate higher-limit intervals, creating a half-octave quartertone temperament. Its ploidacot is diploid alpha-dicot.

Subgroup: 2.3.5.7.11

Comma list: 50/49, 81/80, 864/847

Mapping: [⟨2 1 -4 -3 8], ⟨0 2 8 8 -1]]

mapping generators: ~7/5, ~16/11

Optimal tunings:

  • WE: ~7/5 = 600.2802 ¢, ~16/11 = 647.7720 ¢ (~33/32 = 47.4918 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~16/11 = 647.5224 ¢ (~33/32 = 47.5224 ¢)

Optimal ET sequence: 24d, 26, 50d

Badness (Sintel): 2.34

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 78/77, 81/80, 144/143

Mapping: [⟨2 1 -4 -3 8 2], ⟨0 2 8 8 -1 5]]

Optimal tunings:

  • WE: ~7/5 = 600.3037 ¢, ~16/11 = 647.7954 ¢ (~33/32 = 47.4917 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~16/11 = 647.5256 ¢ (~33/32 = 47.5256 ¢)

Optimal ET sequence: 24d, 26, 50d

Badness (Sintel): 1.65

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 78/77, 81/80, 85/84, 144/143

Mapping: [⟨2 1 -4 -3 8 2 6], ⟨0 2 8 8 -1 5 2]]

Optimal tunings:

  • WE: ~7/5 = 600.5123 ¢, ~16/11 = 647.8970 ¢ (~34/33 = 47.3846 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~16/11 = 647.4314 ¢ (~34/33 = 47.4314 ¢)

Optimal ET sequence: 24d, 26

Badness (Sintel): 1.50

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 50/49, 57/56, 78/77, 81/80, 85/84, 144/143

Mapping: [⟨2 1 -4 -3 8 2 6 2], ⟨0 2 8 8 -1 5 2 6]]

Optimal tunings:

  • WE: ~7/5 = 600.6308 ¢, ~16/11 = 648.0424 ¢ (~34/33 = 47.4116 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~16/11 = 647.4715 ¢ (~34/33 = 47.4715 ¢)

Optimal ET sequence: 24d, 26

Badness (Sintel): 1.41

Pombe

Pombe (named after the African millet beer) is a variant of #Teff by Kaiveran Lugheidh that eschews the tempering of 50/49 to attain more accuracy in the 7-limit. Its ploidacot is diploid alpha-dicot, the same as teff. Oddly, the 7th harmonic has a lesser generator distance than in teff (-5 vs +8), but this combined with the fact that other harmonics are in the opposite direction means that the 7-limit diamond is more complex overall.

Subgroup: 2.3.5.7

Comma list: 81/80, 300125/294912

Mapping: [⟨2 1 -4 11], ⟨0 2 8 -5]]

mapping generators: ~735/512, ~35/24

Optimal tunings:

  • WE: ~735/512 = 601.0652 ¢, ~35/24 = 648.9295 ¢ (~36/35 = 47.8642 ¢)
error map: ⟨+2.130 -3.031 +0.861 -1.756]
  • CWE: ~735/512 = 600.0000 ¢, ~35/24 = 647.8628 ¢ (~36/35 = 47.8628 ¢)
error map: ⟨0.000 -6.229 -3.411 -8.140]

Optimal ET sequence: 24, 26, 50, 126bcd, 176bcdd, 226bbcdd

Badness (Sintel): 2.94

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 245/242, 385/384

Mapping: [⟨2 1 -4 11 8], ⟨0 2 8 -5 -1]]

Optimal tunings:

  • WE: ~99/70 = 600.7890 ¢, ~16/11 = 648.7592 ¢ (~36/35 = 47.9701 ¢)
  • CWE: ~99/70 = 600.0000 ¢, ~16/11 = 647.9516 ¢ (~36/35 = 47.9516 ¢)

Optimal ET sequence: 24, 26, 50

Badness (Sintel): 1.72

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 144/143, 245/242

Mapping: [⟨2 1 -4 11 8 2], ⟨0 2 8 -5 -1 5]]

Optimal tunings:

  • WE: ~99/70 = 600.6971 ¢, ~16/11 = 648.6029 ¢ (~36/35 = 47.9058 ¢)
  • CWE: ~99/70 = 600.0000 ¢, ~16/11 = 647.8990 ¢ (~36/35 = 47.8990 ¢)

Optimal ET sequence: 24, 26, 50

Badness (Sintel): 1.28

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 105/104, 144/143, 245/242, 273/272

Mapping: [⟨2 1 -4 11 8 2 6], ⟨0 2 8 -5 -1 5 2]]

Optimal tunings:

  • WE: ~17/12 = 600.7610 ¢, ~16/11 = 648.6638 ¢ (~36/35 = 47.9028 ¢)
  • CWE: ~17/12 = 600.0000 ¢, ~16/11 = 647.8990 ¢ (~36/35 = 47.8990 ¢)

Optimal ET sequence: 24, 26, 50

Badness (Sintel): 1.08

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨2 1 -4 11 8 2 6 2], ⟨0 2 8 -5 -1 5 2 6]]

Optimal tunings:

  • WE: ~17/12 = 600.8048 ¢, ~16/11 = 648.7494 ¢ (~36/35 = 47.9446 ¢)
  • CWE: ~17/12 = 600.0000 ¢, ~16/11 = 647.9425 ¢ (~36/35 = 47.9425 ¢)

Optimal ET sequence: 24, 26, 50

Badness (Sintel): 1.01

Orphic

Orphic has a semi-octave period and four generators plus a period gives the 3rd harmonic; its ploidacot is diploid alpha-tetracot.

Subgroup: 2.3.5.7

Comma list: 81/80, 5898240/5764801

Mapping: [⟨2 1 -4 4], ⟨0 4 16 3]]

mapping generators: ~2401/1728, ~343/288

Optimal tunings:

  • WE: ~2401/1728 = 600.1767 ¢, ~343/288 = 324.3015 ¢ (~7/6 = 275.8751 ¢)
error map: ⟨+0.353 -4.572 +1.804 +4.785]
  • CWE: ~2401/1728 = 600.0000 ¢, ~343/288 = 324.2285 ¢ (~7/6 = 275.7715 ¢)
error map: ⟨0.000 -5.041 +1.342 +3.860]

Optimal ET sequence: 26, 48c, 74

Badness (Sintel): 6.55

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 73728/73205

Mapping: [⟨2 1 -4 4 8], ⟨0 4 16 3 -2]]

Optimal tunings:

  • WE: ~363/256 = 600.1011 ¢, ~77/64 = 324.2923 ¢ (~7/6 = 275.8088 ¢)
  • CWE: ~363/256 = 600.0000 ¢, ~77/64 = 324.2463 ¢ (~7/6 = 275.7537 ¢)

Optimal ET sequence: 26, 48c, 74

Badness (Sintel): 3.36

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 144/143, 2200/2197

Mapping: [⟨2 1 -4 4 8 2], ⟨0 4 16 3 -2 10]]

Optimal tunings:

  • WE: ~55/39 = 600.0540 ¢, ~77/64 = 324.2551 ¢ (~7/6 = 275.7989 ¢)
  • CWE: ~55/39 = 600.0000 ¢, ~77/64 = 324.2307 ¢ (~7/6 = 275.7693 ¢)

Optimal ET sequence: 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 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.

Subgroup: 2.3.5.7

Comma list: 81/80, 16807/16384

Mapping: [⟨5 0 -20 14], ⟨0 1 4 0]]

mapping generators: ~8/7, ~3

Optimal tunings:

  • WE: ~8/7 = 240.4267 ¢, ~3/2 = 696.9566 ¢ (~49/48 = 24.3235 ¢)
error map: ⟨+2.133 -2.865 +1.513 -2.852]
  • CWE: ~8/7 = 240.0000 ¢, ~3/2 = 696.1637 ¢ (~49/48 = 23.8373 ¢)
error map: ⟨0.000 -5.791 -1.659 -8.826]

Optimal ET sequence: 5, 40c, 45, 50

Badness (Sintel): 2.59

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 385/384, 2401/2376

Mapping: [⟨5 0 -20 14 41], ⟨0 1 4 0 -3]]

Optimal tunings:

  • WE: ~8/7 = 240.2740 ¢, ~3/2 = 697.3317 ¢ (~56/55 = 23.4904 ¢)
  • CWE: ~8/7 = 240.0000 ¢, ~3/2 = 696.6269 ¢ (~56/55 = 23.3731 ¢)

Optimal ET sequence: 5, 45, 50

Badness (Sintel): 2.33

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 144/143, 2401/2376

Mapping: [⟨5 0 -20 14 41 -21], ⟨0 1 4 0 -3 5]]

Optimal tunings:

  • WE: ~8/7 = 240.2435 ¢, ~3/2 = 696.8686 ¢ (~91/90 = 23.8618 ¢)
  • CWE: ~8/7 = 240.0000 ¢, ~3/2 = 696.2653 ¢ (~91/90 = 23.7347 ¢)

Optimal ET sequence: 5, 45f, 50

Badness (Sintel): 2.02

Subgroup extensions

Stützel (2.3.5.19)

Subgroup: 2.3.5.19

Comma list: 81/80, 96/95

Subgroup-val mapping: [⟨1 0 -4 9], ⟨0 1 4 -3]]

Gencom mapping: [⟨1 0 -4 0 0 0 0 9], ⟨0 1 4 0 0 0 0 -3]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.5513 ¢, ~3/2 = 697.6058 ¢
error map: ⟨-0.448 -4.798 +4.110 +6.977]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.8222 ¢
error map: ⟨0.000 -4.133 +4.975 +9.020]

Optimal ET sequence: 5, 7, 12, 31, 43, 98h

Badness (Sintel): 0.324

Hypnotone

Hypnotone is no-sevens flattone.

Subgroup: 2.3.5.11

Comma list: 45/44, 81/80

Subgroup-val mapping: [⟨1 0 -4 -6], ⟨0 1 4 6]]

Gencom mapping: [⟨1 0 -4 0 -6], ⟨0 1 4 0 6]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1202.0621 ¢, ~3/2 = 694.5448 ¢
error map: ⟨+2.062 -5.348 -8.135 +15.951]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 693.9085 ¢
error map: ⟨0.000 -8.047 -10.680 +12.133]

Optimal ET sequence: 7, 12, 19, 26, 45

Badness (Sintel): 0.326

2.3.5.11.13 subgroup

Subgroup: 2.3.5.11.13

Comma list: 45/44, 65/64, 81/80

Subgroup-val mapping: [⟨1 0 -4 -6 10], ⟨0 1 4 6 -4]]

Gencom mapping: [⟨1 0 -4 0 -6 10], ⟨0 1 4 0 6 -4]]

Optimal tunings:

  • WE: ~2 = 1202.6916 ¢, ~3/2 = 694.4181 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 693.0870 ¢

Optimal ET sequence: 7, 12, 19, 26, 45f

Badness (Sintel): 0.561

Dequarter

Subgroup: 2.3.5.11

Comma list: 33/32, 55/54

Subgroup-val mapping: [⟨1 0 -4 5], ⟨0 1 4 -1]]

Gencom mapping: [⟨1 0 -4 0 5], ⟨0 1 4 0 -1]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1206.5832 ¢, ~3/2 = 695.8763 ¢
error map: ⟨+6.583 +0.504 -2.809 -20.862]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 693.1206 ¢
error map: ⟨0.000 -8.834 -13.831 -44.439]

Optimal ET sequence: 5, 7, 19e, 26e

Badness (Sintel): 0.451

Dreamtone

Subgroup: 2.3.5.11.13

Comma list: 33/32, 55/54, 975/968

Subgroup-val mapping: [⟨1 0 -4 5 21], ⟨0 1 4 -1 -11]]

Gencom mapping: [⟨1 0 -4 0 5 21], ⟨0 1 4 0 -1 -11]]

Optimal tunings:

  • WE: ~2 = 1207.8248 ¢, ~3/2 = 694.7806 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 690.1826 ¢

Optimal ET sequence: 7, 19eff, 26eff, 33ceeff, 40ceeff

Badness (Sintel): 1.40

References

  1. ↑ 1.0 1.1 Yahoo! Tuning Group | good 11-limit meantones
  2. ↑ Yahoo! Tuning Group | The meantone family
  3. ↑ 3.0 3.1 3.2 Yahoo! Tuning Group | names and definitions: meantone
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