Meantone family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Inthar (talk | contribs)
autopatrolled, emailconfirmed
15,672 edits
m →Seven limit children: "parents" and "children" arent the most natural metaphor to describe temp families. temperaments don't form trees, they form posets.
m Text replacement - "octave" to "octave"
 
(287 intermediate revisions by 26 users not shown)
Line 5: Line 5:
| ja =  
| ja =  
}}
}}
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]].  


= 5-limit meantone =
== Meantone ==
Period: 1\1
{{Main| Meantone }}


Optimal ([[POTE]]) generator: ~3/2 = 696.239
Meantone is characterized by an [[octave]] [[period]], a [[3/2|fifth]] [[generator]], and the relationship that four fifths go to make up a [[5/1|5th harmonic]].


EDO generators: [[12edo|7\12]], [[19edo|11\19]], [[31edo|18\31]], [[43edo|25\43]], [[50edo|29\50]]
[[Subgroup]]: 2.3.5


Scales (Scala files): [[Meantone5]], [[Meantone7]], [[Meantone12]]
[[Comma list]]: 81/80


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Mapping|legend=1| 1 0 -4 | 0 1 4 }}
<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"
|+
|-
! #Gens up
! 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
| 577.434
| 15/18
|}
<references/></div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
[[Mappings|Period-generator mapping]]: [&lt;1 0 -4|, &lt;0 1 4|]


Comma: 81/80
: mapping generators: ~2, ~3


Mapping generator: ~3
[[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 }}


[[Tuning Ranges of Regular Temperaments|valid range]]: [685.714, 720.000] (7 to 5)
[[Minimax tuning]]:  
* [[5-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma)
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


[[Tuning Ranges of Regular Temperaments|nice range]]: [694.786, 701.955] (1/3 comma to Pythagorean)
[[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.)


[[Tuning Ranges of Regular Temperaments|strict range]]: [694.786, 701.955]
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b }}


{{EDOs|legend=1| 5, 7, 12, 19, 26, 31, 43, 45, 50, 55, 67, 69, 74, 81, 88, 98, 105, 117, 131b, 212bb, 293bb }}
[[Badness]] (Sintel): 0.173


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


[[Badness]]: 0.00736
==== Strong extensions ====
</div></div>
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.


== Seven-limit extensions ==
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.  
The [[7-limit]] extensions 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 =
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.
<span style="display: block; text-align: right;">[[:de:septimal-mitteltönig|Deutsch]]</span>


* [[Wikipedia: Septimal meantone temperament]]
==== Splitting the meantone fifth into two (243/242) ====
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.


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.
==== Splitting the meantone fifth into three (1029/1024) ====
By tempering out [[1029/1024]] we equate the distance from 7/6 to 8/7 (= [[49/48|S7]]) with the distance from 8/7 to 9/8 (= [[64/63|S8]]), so that ([[8/7]])<sup>3</sup> is equated with [[3/2]], because of being able to be rewritten as (9/8)(8/7)(7/6) – this observation can be generalized to define the family of [[ultraparticular]] commas. This is an unusually natural extension, with a surprising coincidence: ([[36/35]])/([[64/63]]) = [[81/80]], or using the shorthand notation, S6/S8 = S9. As S6/S8 is already tempered out, it is natural to want [[49/48]] (S7), which is bigger than S8 and smaller than S6 to be equated with both, to avoid inconsistent mappings. This has the surprising consequence of meaning that splitting the meantone fifth into three 8/7's is equivalent to splitting 8/5 into three 7/6's by tempering (8/5)/(7/6)<sup>3</sup> = [[1728/1715]] (S6/S7), the orwellisma.


Period: 1\1
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]]).


Optimal ([[POTE]]) generator: 696.495
==== 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).


EDO generators: [[12edo|7\12]], [[19edo|11\19]], [[31edo|18\31]], [[43edo|25\43]], [[50edo|29\50]]
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]]


Scales (Scala files): [[Meantone5]], [[Meantone7]], [[Meantone12]]
The rest are considered below.


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
== Septimal meantone ==
<div style="line-height:1.6;">Interval table (12-note MOS, 2.3.5.7 POTE tuning)</div>
<div style="float:right">[[:de:septimal-mitteltönig|Deutsch]]</div>
<div class="mw-collapsible-content">
{{Main| Meantone #Septimal meantone}}
{| class="wikitable right-1 right-2 sortable"
{{Wikipedia| Septimal meantone temperament }}
|+
|-
! #Gens up
! 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:400px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
[[Mappings|Period-generator mapping]]: [&lt;1 0 -4 -13|, &lt;0 1 4 10|]


[[Comma]]s: 81/80, 126/125
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.


7 and [[9-odd-limit|9-limit]] minimax
[[Subgroup]]: 2.3.5.7


[{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| -3 0 5/2 0 }}]
[[Comma list]]: 81/80, 126/125


[[Eigenmonzo]]s: 2, 5
{{Mapping|legend=1| 1 0 -4 -13 | 0 1 4 10 }}


[[Tuning Ranges of Regular Temperaments|valid range]]: [694.737, 700.000] (19 to 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 }}


nice range: [694.786, 701.955]
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma)
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | -3 0 5/2 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


strict range: [694.786, 700.000]
[[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.)


Mapping generator: ~3
[[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.


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


[[Wedgie]]: &lt;&lt;1 4 10 4 13 12||
[[Badness]] (Sintel): 0.347


[[Vals]]: 5, 7, 12, 19, 26, 31, 43, 45, 50, 55, 67, 69, 74, 81, 88, 98, 105, 117, 131b, 212bb, 293bb
=== Undecimal meantone (huygens) ===
{{Redirect|Huygens|the Dutch mathematician, physicist and astronomer|Wikipedia: Christiaan Huygens}}
{{See also| Huygens vs meanpop }}


[[Badness]]: 0.0137
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.
</div></div>
== Bimeantone ==
11/8 is mapped to half octave minus the [[meantone diesis]].


Commas: 81/80, 126/125, 245/242
Subgroup: 2.3.5.7.11


[[POTE generator]]: ~3/2 = 696.016
Comma list: 81/80, 99/98, 126/125


Map: [&lt;2 0 -8 -26 -31|, &lt;0 1 4 10 12|]
Mapping: {{mapping| 1 0 -4 -13 -25 | 0 1 4 10 18 }}


{{EDOs|legend=1| 12, 38d, 50 }}
Optimal tunings:
* WE: ~2 = 1200.7636{{c}}, ~3/2 = 697.4122{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.0315{{c}}


Badness: 0.0381
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


=== 13-limit ===
Tuning ranges:
Commas: 81/80, 105/104, 126/125, 245/242
* 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 ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
 
{{Optimal ET sequence|legend=0| 12, 19e, 31, 105, 136b }}
 
Badness (Sintel): 0.563
 
; 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]
 
==== 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: {{mapping| 1 0 -4 -13 -25 29 | 0 1 4 10 18 -16 }}
 
Optimal tunings:
* WE: ~2 = 1199.9389{{c}}, ~3/2 = 697.2282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.2627{{c}}
 
Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 8/13 0 0 1/26 0 -1/26 }}
: eigenmonzo basis (unchanged-interval basis): 2.13/7


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


Map: [&lt;2 0 -8 -26 -31 -40|, &lt;0 1 4 10 12 15|]
{{Optimal ET sequence|legend=0| 12, 31, 43, 74 }}


{{EDOs|legend=1| 12f, 50 }}
Badness (Sintel): 1.07


Badness: 0.0288
===== 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.  


== Unidecimal meantone aka Huygens ==
Subgroup: 2.3.5.7.11.13.17
{{see also| Meantone vs meanpop }}


[[Comma]]s: 81/80, 126/125, 99/98
Comma list: 81/80, 99/98, 120/119, 126/125, 144/143


[[11-odd-limit|11-limit]] minimax
Mapping: {{mapping| 1 0 -4 -13 -25 29 12 | 0 1 4 10 18 -16 -5 }}


[{{Monzo| 1 0 0 0 0 }}, {{Monzo| 25/16 -1/8 0 0 1/16 }}, {{Monzo| 9/4 -1/2 0 0 1/4 }},
Optimal tunings:
{{Monzo| 21/8 -5/4 0 0 5/8 }}, {{Monzo| 25/8 -9/4 0 0 9/8 }}<nowiki>]</nowiki>
* WE: ~2 = 1199.5811{{c}}, ~3/2 = 697.0918{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3303{{c}}


[[Eigenmonzo]]s: 2, 11/9
{{Optimal ET sequence|legend=0| 12, 31, 43, 74g }}


valid range: [696.774, 700.000] (31 to 12)
Badness (Sintel): 1.06


nice range: [691.202, 701.955]
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


strict range: [696.774, 700.000]
Comma list: 81/80, 96/95, 99/98, 120/119, 126/125, 144/143


[[POTE generator]]: ~3/2 = 696.967
Mapping: {{mapping| 1 0 -4 -13 -25 29 12 9 | 0 1 4 10 18 -16 -5 -3 }}


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


[[Algebraic generator]]: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
{{Optimal ET sequence|legend=0| 12, 31, 43, 74gh }}


[[Map]]: [&lt;1 0 -4 -13 -25|, &lt;0 1 4 10 18|]
Badness (Sintel): 1.07


[[Generator]]s: 2, 3
==== 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.


{{EDOs|legend=1| 12, 31, 43, 74, 105, 198be }}
This extension used to be known as ''tridecimal meantone'', but was decanonicalized in 2025.


[[Badness]]: 0.0170
Subgroup: 2.3.5.7.11.13


* [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]
Comma list: 66/65, 81/80, 99/98, 105/104


=== Tridecimal meantone ===
Mapping: {{mapping| 1 0 -4 -13 -25 -20 | 0 1 4 10 18 15 }}
Commas: 66/65, 81/80, 99/98, 105/104


[[POTE generator]]: ~3/2 = 696.642
Optimal tunings:  
* WE: ~2 = 1200.8149{{c}}, ~3/2 = 697.1155{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.7085{{c}}


Mapping generator: ~3
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


Map: [&lt;1 0 -4 -13 -25 -20|, &lt;0 1 4 10 18 15|]
{{Optimal ET sequence|legend=0| 12f, 19e, 31 }}


{{EDOs|legend=1| 12f, 31, 43f }}
Badness (Sintel): 0.746


Badness: 0.0180
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


=== Grosstone ===
Comma list: 66/65, 81/80, 99/98, 105/104, 120/119
Commas: 81/80, 99/98, 126/125, 144/143


valid range: [696.774, 700.000] (31 to 12)
Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 | 0 1 4 10 18 15 -5 }}


nice range: [691.202, 701.955]
Optimal tunings:  
* WE: ~2 = 1199.5548{{c}}, ~3/2 = 696.7449{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.9823{{c}}


strict range: [696.774, 700.000]
{{Optimal ET sequence|legend=0| 12f, 31 }}


[[POTE generator]]: ~3/2 = 697.264
Badness (Sintel): 1.02


Mapping generator: ~3
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;1 0 -4 -13 -25 29|, &lt;0 1 4 10 18 -16|]
Comma list: 66/65, 81/80, 96/95, 99/98, 105/104, 120/119


{{EDOs|legend=1| 12, 31, 43, 74, 105 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 9 | 0 1 4 10 18 15 -5 -3 }}


Badness: 0.0259
Optimal tunings:  
* 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 }}
Commas: 78/77, 81/80, 99/98, 126/125


[[POTE generator]]: ~3/2 = 697.529
Badness (Sintel): 1.10


Mapping generator: ~3
==== 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.


Map: [&lt;1 0 -4 -13 -25 -39|, &lt;0 1 4 10 18 27|]
Subgroup: 2.3.5.7.11.13


{{EDOs|legend=1| 12f, 31f, 43 }}
Comma list: 78/77, 81/80, 99/98, 126/125


Badness: 0.0264
Mapping: {{mapping| 1 0 -4 -13 -25 -39 | 0 1 4 10 18 27 }}


=== Hemimeantone ===
Optimal tunings:
Commas: 81/80, 99/98, 126/125, 169/168
* WE: ~2 = 1199.9122{{c}}, ~3/2 = 697.4779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.5241{{c}}


[[POTE generator]]: ~52/45 = 250.304
Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/25 -2/25 0 0 0 1/25 }}
: unchanged-interval (eigenmonzo) basis: 2.13/9


Mapping generator: ~26/15
{{Optimal ET sequence|legend=0| 12f, 31f, 43 }}


Map: [&lt;1 0 -4 -13 -25 -5|, &lt;0 2 8 20 36 11|]
Badness (Sintel): 1.09


{{EDOs|legend=1| 19e, 43, 62, 167bef }}
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0314
Comma list: 78/77, 81/80, 99/98, 120/119, 126/125


== Meanpop ==
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 | 0 1 4 10 18 27 -5 }}
{{see also| Meantone vs meanpop }}


[[Comma]]s: 81/80, 126/125, 385/384
Optimal tunings:  
* WE: ~2 = 1199.3793{{c}}, ~3/2 = 697.2833{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6222{{c}}


[[11-odd-limit|11-limit]] minimax 1/4 comma
{{Optimal ET sequence|legend=0| 12f, 43 }}


[{{Monzo| 1 0 0 0 0 }}, {{Monzo| 1 0 1/4 0 0 }}, {{Monzo| 0 0 1 0 0 }},
Badness (Sintel): 1.22
{{Monzo| -3 0 5/2 0 0 }}, {{Monzo| 11 0 -13/4 0 0 }}]


[[Eigenmonzo]]s: 2, 5
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


valid range: [694.737, 696.774] (19 to 31)
Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125


nice range: [691.202, 701.955]
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 9 | 0 1 4 10 18 27 -5 -3 }}


strict range: [694.737, 696.774]
Optimal tunings:  
* WE: ~2 = 1199.0260{{c}}, ~3/2 = 697.1486{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6887{{c}}


[[POTE generator]]: 696.434
{{Optimal ET sequence|legend=0| 12f, 43 }}


Mapping generator: ~3
Badness (Sintel): 1.25


[[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.
==== Hemimeantone ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 -4 -13 24|, &lt;0 1 4 10 -13|]
Comma list: 81/80, 99/98, 126/125, 169/168


[[Generator]]s: 2, 3
Mapping: {{mapping| 1 0 -4 -13 -25 -5 | 0 2 8 20 36 11 }}


{{EDOs|legend=1| 12, 19, 31, 50, 81 }}
: mapping generators: ~2, ~26/15


[[Badness]]: 0.0215
Optimal tunings:  
* WE: ~2 = 1201.0387{{c}}, ~26/15 = 949.2863{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5065{{c}}


* [http://soonlabel.com/xenharmonic/archives/607 Scott Joplin's "The Entertainer" tuned into meanpop]{{Dead link}}
{{Optimal ET sequence|legend=0| 19e, 43, 62 }}
* [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 ===
Badness (Sintel): 1.30
Commas: 81/80, 105/104, 126/125, 144/143


valid range: [694.737, 696.774] (19 to 31)
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


nice range: [691.202, 701.955]
Comma list: 81/80, 99/98, 126/125, 169/168, 221/220


strict range: [694.737, 696.774]
Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 | 0 2 8 20 36 11 33 }}


[[POTE generator]]: ~3/2 = 696.211
Optimal tunings:
* WE: ~2 = 1201.0270{{c}}, ~26/15 = 949.2892{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5169{{c}}


Mapping generator: ~3
{{Optimal ET sequence|legend=0| 19eg, 43, 62 }}


Map: [&lt;1 0 -4 -13 24 -20|, &lt;0 1 4 10 -13 15|]
Badness (Sintel): 1.19


{{EDOs|legend=1| 12ef, 19, 31, 50, 81 }}
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.0209
Comma list: 81/80, 99/98, 126/125, 153/152, 169/168, 221/220


=== Meanplop ===
Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 -25 | 0 2 8 20 36 11 33 37 }}
Commas: 65/64, 78/77, 81/80, 91/90


[[POTE generator]]: ~3/2 = 696.202
Optimal tunings:
* WE: ~2 = 1201.0339{{c}}, ~19/11 = 949.2902{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 948.5111{{c}}


Mapping generator: ~3
{{Optimal ET sequence|legend=0| 19egh, 43, 62 }}


Map: [&lt;1 0 -4 -13 24 10|, &lt;0 1 4 10 -13 -4|]
Badness (Sintel): 1.15


{{EDOs|legend=1| 12e, 19, 31f, 50ff }}
==== Semimeantone ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0277
Comma list: 81/80, 99/98, 126/125, 847/845


== Meanenneadecal ==
Mapping: {{mapping| 2 0 -8 -26 -50 -59 | 0 1 4 10 18 21 }}
Commas: 45/44, 56/55, 81/80


[[POTE generator]]: ~3/2 = 696.250
: mapping generators: ~55/39, ~3


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


Map: [&lt;1 0 -4 -13 -6|, &lt;0 1 4 10 6|]
{{Optimal ET sequence|legend=0| 12f, …, 50eff, 62, 136b }}


{{EDOs|legend=1| 7d, 12, 19, 31e, 50ee }}
Badness (Sintel): 1.68


Badness: 0.0214
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


=== 13-limit ===
Comma list: 81/80, 99/98, 126/125, 221/220, 289/288
Commas: 45/44, 56/55, 78/77, 81/80


[[POTE generator]]: ~3/2 = 696.146
Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 | 0 1 4 10 18 21 1 }}


Mapping generator: ~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}}


Map: [&lt;1 0 -4 -13 -6 -20|, &lt;0 1 4 10 6 15|]
{{Optimal ET sequence|legend=0| 12f, 50eff, 62, 136bg }}


{{EDOs|legend=1| 12f, 19, 31e, 50ee }}
Badness (Sintel): 1.60


Badness: 0.0212
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


=== Vincenzo ===
Comma list: 81/80, 99/98, 126/125, 153/152, 209/208, 221/220
Commas: 45/44, 56/55, 65/64, 81/80


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


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


Map: [&lt;1 0 -4 -13 -6 10|, &lt;0 1 4 10 6 -4|]
{{Optimal ET sequence|legend=0| 12f, 50eff, 62 }}


{{EDOs|legend=1| 7d, 12, 19, 26d }}
Badness (Sintel): 1.47


Badness: 0.0248
=== Meanpop ===
{{See also| Huygens vs meanpop }}


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


[[POTE generator]]: ~3/2 = 695.858
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 -4 -13 -6 10 12|, &lt;0 1 4 10 6 -4 -5|]
Comma list: 81/80, 126/125, 385/384


{{EDOs|legend=1| 7d, 12, 19 }}
Mapping: {{mapping| 1 0 -4 -13 24 | 0 1 4 10 -13 }}


Badness: 0.0255
: mapping generator: ~2, ~3


==== 19-limit ====
Optimal tunings:
Commas: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80
* WE: ~2 = 1201.3464{{c}}, ~3/2 = 697.2159{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4509{{c}}


[[POTE generator]]: ~3/2 = 696.131
Minimax tuning:
* 11-odd-limit: ~3/2 = {{monzo| 0 0 1/4 }}
: projection map: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| -3 0 5/2 0 0 }}, {{monzo| 11 0 -13/4 0 0 }}]
: unchanged-interval (eigenmonzo) basis: 2.5


Map: [&lt;1 0 -4 -13 -6 10 12 9|, &lt;0 1 4 10 6 -4 -5 -3|]
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.)


{{EDOs|legend=1| 7d, 12, 19 }}
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: 0.0223
{{Optimal ET sequence|legend=0| 12e, 19, 31, 81, 112b }}


==== 23-limit ====
Badness (Sintel): 0.712
Commas: 39/38, 45/44, 52/51, 56/55, 65/64, 69/68, 81/80


[[POTE generator]]: ~3/2 = 696.044
; 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]


Map: [&lt;1 0 -4 -13 -6 10 12 9 14|, &lt;0 1 4 10 6 -4 -5 -3 -6|]
==== Tridecimal meanpop ====
Subgroup: 2.3.5.7.11.13


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


Badness: 0.0201
Mapping: {{mapping| 1 0 -4 -13 24 -20 | 0 1 4 10 -13 15 }}


==== 29-limit ====
Optimal tunings:
Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80
* WE: ~2 = 1201.0765{{c}}, ~3/2 = 696.8361{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2347{{c}}


POTE generator: ~3/2 = 695.913
Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 4/7 0 0 0 -1/28 1/28 }}
: unchanged-interval (eigenmonzo) basis: 2.13/11


Map: [&lt;1 0 -4 -13 -6 10 12 9 14 8|, &lt;0 1 4 10 6 -4 -5 -3 -6 -2|]
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.)


{{EDOs|legend=1| 7d, 12, 19 }}
{{Optimal ET sequence|legend=0| 19, 31, 50, 81 }}


Badness: 0.0182
Badness (Sintel): 0.863


==== 31-limit ====
===== Meanpoppic =====
Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80, 93/92
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~3/2 = 695.750
Comma list: 81/80, 105/104, 126/125, 144/143, 273/272


Map: [&lt;1 0 -4 -13 -6 10 12 9 14 8 16|, &lt;0 1 4 10 6 -4 -5 -3 -6 -2 -7|]
Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 | 0 1 4 10 -13 15 26 }}


{{EDOs|legend=1| 7d, 12, 19 }}
Optimal tunings:
* WE: ~2 = 1201.0727{{c}}, ~3/2 = 696.8168{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2195{{c}}


Badness: 0.0171
{{Optimal ET sequence|legend=0| 19g, 31, 50, 81, 131bd }}


==== 37-limit ====
Badness (Sintel): 1.02
Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92


POTE generator: ~3/2 = 695.603
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;1 0 -4 -13 -6 10 12 9 14 8 16 -9|, &lt;0 1 4 10 6 -4 -5 -3 -6 -2 -7 9|]
Comma list: 81/80, 105/104, 126/125, 144/143, 153/152, 273/272


{{EDOs|legend=1| 7d, 12, 19 }}
Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 -40 | 0 1 4 10 -13 15 26 28 }}


Badness: 0.0161
Optimal tunings:  
* WE: ~2 = 1201.0719{{c}}, ~3/2 = 696.8101{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2137{{c}}


==== 41-limit ====
{{Optimal ET sequence|legend=0| 19gh, 31, 50, 81 }}
Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92, 124/123


POTE generator: ~3/2 = 695.696
Badness (Sintel): 1.08


Map: [&lt;1 0 -4 -13 -6 10 12 9 14 8 16 -9 18|, &lt;0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8|]
===== Meanpoid =====
Subgroup: 2.3.5.7.11.13.17


{{EDOs|legend=1| 7d, 12, 19 }}
Comma list: 81/80, 105/104, 120/119, 126/125, 144/143


Badness: 0.0154
Mapping: {{mapping| 1 0 -4 -13 24 -20 12 | 0 1 4 10 -13 15 -5 }}


==== 43-limit ====
Optimal tunings:
Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 86/85, 93/92, 124/123
* WE: ~2 = 1200.2768{{c}}, ~3/2 = 696.5683{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4114{{c}}


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


Map: [&lt;1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7|, &lt;0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1|]
Badness (Sintel): 1.17


{{EDOs|legend=1| 7d, 12, 19 }}
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.0139
Comma list: 77/76, 81/80, 96/95, 105/104, 120/119, 126/125


==== 47-limit ====
Mapping: {{mapping| 1 0 -4 -13 24 -20 12 9 | 0 1 4 10 -13 15 -5 -3 }}
Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 86/85, 93/92, 95/94, 124/123


POTE generator: ~3/2 = 695.676
Optimal tunings:  
* WE: ~2 = 1199.7905{{c}}, ~3/2 = 696.3779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4973{{c}}


Map: [&lt;1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7 4|, &lt;0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1 1|]
{{Optimal ET sequence|legend=0| 19, 31 }}


{{EDOs|legend=1| 7d, 12, 19 }}
Badness (Sintel): 1.25


Badness: 0.0138
==== Semimeanpop ====
Subgroup: 2.3.5.7.11.13


== Meanundeci ==
Comma list: 81/80, 126/125, 385/384, 847/845
Commas: 33/32, 55/54, 77/75


POTE generator: ~3/2 = 694.689
Mapping: {{mapping| 2 0 -8 -26 48 39 | 0 1 4 10 -13 -10 }}


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


Map: [&lt;1 0 -4 -13 5|, &lt;0 1 4 10 -1|]
Optimal tunings:  
* WE: ~55/39 = 600.6704{{c}}, ~3/2 = 697.2151{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.4341{{c}}


{{EDOs|legend=1| 5d, 7d, 12e, 19e }}
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112b }}


Badness: 0.0315
Badness (Sintel): 1.78


=== 13-limit ===
===== 17-limit =====
Commas: 33/32, 55/54, 65/64, 77/75
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~3/2 = 694.764
Comma list: 81/80, 126/125, 221/220, 273/272, 289/288


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


Map: [&lt;1 0 -4 -13 5 10|, &lt;0 1 4 10 -1 -4|]
Optimal tunings:  
* WE: ~17/12 = 600.7232{{c}}, ~3/2 = 697.2820{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4411{{c}}


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


Badness: 0.0263
Badness (Sintel): 1.45


=== Meanundec ===
===== 19-limit =====
Commas: 27/26, 40/39, 45/44, 56/55
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~3/2 = 697.254
Comma list: 81/80, 126/125, 153/152, 209/208, 221/220, 273/272


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


Map: [&lt;1 0 -4 -13 -6 -1|, &lt;0 1 4 10 6 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}}


{{EDOs|legend=1| 7d, 12f, 19f, 31eff }}
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bgh }}


Badness: 0.0242
Badness (Sintel): 1.28


= Flattone =
=== Meanenneadecal ===
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]].
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 = 693.779
Comma list: 45/44, 56/55, 81/80


EDO generators: [[19edo|11\19]], [[26edo|15\26]], [[45edo|26\45]], [[64edo|37\64]]
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;">Interval table (12-note MOS, 2.3.5.7 POTE tuning)</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]
{| class="wikitable right-1 right-2 sortable"
|+
|-
! #Gens up
! 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:400px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
[[Mappings|Period-generator mapping]]: [&lt;1 0 -4 17|, &lt;0 1 4 -9|]


[[7-odd-limit|7-limit]] minimax
{{Optimal ET sequence|legend=0| 7d, 12, 19, 31e }}


[{{Monzo| 1 0 0 0 }}, {{Monzo| 21/13 0 1/13 -1/13 }},
Badness (Sintel): 0.708
{{Monzo| 32/13 0 4/13 -4/13 }}, {{Monzo| 32/13 0 -9/13 9/13 }}<nowiki>]</nowiki>


[[Eigenmonzo]]s: 2, 7/5
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[9-odd-limit|9-limit]] minimax
Comma list: 45/44, 56/55, 78/77, 81/80


[{{Monzo| 1 0 0 0 }}, {{Monzo| 17/11 2/11 0 -1/11 }},
Mapping: {{mapping| 1 0 -4 -13 -6 -20 | 0 1 4 10 6 15 }}
{{Monzo| 24/11 8/11 0 -4/11 }}, {{Monzo| 34/11 -18/11 0 9/11 }}<nowiki>]</nowiki>


Eigenmonzos: 2, 9/7
Optimal tunings:  
* WE: ~2 = 1199.7931{{c}}, ~3/2 = 696.0258{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1241{{c}}


valid range: [692.308, 694.737] (26 to 19)
{{Optimal ET sequence|legend=0| 7df, 12f, 19, 31e }}


nice range: [692.353, 701.955]
Badness (Sintel): 0.875


strict range: [692.353, 694.737]
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Mapping generator: ~3
Comma list: 45/44, 56/55, 78/77, 81/80, 120/119


Algebraic generator: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.
Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 | 0 1 4 10 6 15 -5 }}


[[Wedgie]]: &lt;&lt;1 4 -9 4 -17 -32||
Optimal tunings:  
* WE: ~2 = 1198.6665{{c}}, ~3/2 = 695.8010{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4998{{c}}


[[Generator]]s: 2, 3
{{Optimal ET sequence|legend=0| 12f, 19, 31e }}


{{EDOs|legend=1| 7, 19, 26, 45 }}
Badness (Sintel): 1.17


[[Badness]]: 0.0386
===== 19-limit =====
</div></div>
Subgroup: 2.3.5.7.11.13.17.19


== 11-limit ==
Comma list: 45/44, 56/55, 78/77, 81/80, 96/95, 120/119
Commas: 45/44, 81/80, 385/384


valid range: [692.308, 694.737] (26 to 19)
Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 9 | 0 1 4 10 6 15 -5 -3 }}


nice range: [682.502, 701.955]
Optimal tunings:  
* WE: ~2 = 1198.2880{{c}}, ~3/2 = 695.7123{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.6370{{c}}


strict range: [692.308, 694.737]
{{Optimal ET sequence|legend=0| 12f, 19, 31e }}


POTE generator: ~3/2 = 693.126
Badness (Sintel): 1.23


Mapping generator: ~3
==== Vincenzo ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 -4 17 -6|, &lt;0 1 4 -9 6|]
Comma list: 45/44, 56/55, 65/64, 81/80


{{EDOs|legend=1| 7, 19, 26, 45, 71bc, 116bcde }}
Mapping: {{mapping| 1 0 -4 -13 -6 10 | 0 1 4 10 6 -4 }}


Badness: 0.0338
Optimal tunings:  
* WE: ~2 = 1202.1684{{c}}, ~3/2 = 696.3160{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.2045{{c}}


== 13-limit ==
{{Optimal ET sequence|legend=0| 7d, 12, 19 }}
45/44, 65/64, 78/77, 81/80


valid range: [692.308, 694.737] (26 to 19)
Badness (Sintel): 1.02


nice range: [682.502, 701.955]
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


strict range: [692.308, 694.737]
Comma list: 45/44, 52/51, 56/55, 65/64, 81/80


POTE generator: ~3/2 = 693.058
Mapping: {{mapping| 1 0 -4 -13 -6 10 12 | 0 1 4 10 6 -4 -5 }}


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


Map: [&lt;1 0 -4 17 -6 10|, &lt;0 1 4 -9 6 -4|]
{{Optimal ET sequence|legend=0| 12, 19 }}


{{EDOs|legend=1| 7, 19, 26, 45f, 71bcf, 116bcdef }}
Badness (Sintel): 1.30


Badness: 0.0223
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


= Dominant =
Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80
[[Comma]]s: 36/35, 64/63


The interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is [[12edo]], but it also works well with the Pythagorean tuning of pure [[3/2]] fifths, and with [[29edo]], [[41edo]], or [[53edo]].
Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 | 0 1 4 10 6 -4 -5 -3 }}


valid range: [700.000, 720.000] (12 to 5)
Optimal tunings:  
* WE: ~2 = 1199.8261{{c}}, ~3/2 = 696.0298{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1262{{c}}


nice range: [694.786, 715.587]
{{Optimal ET sequence|legend=0| 12, 19 }}


strict range: [700.000, 715.587]
Badness (Sintel): 1.36


[[POTE generator]]: 701.573
=== Bimeantone ===
11/8 is mapped to half octave minus the [[128/125|meantone diesis]].  


Mapping generator: ~3
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 -4 6|, &lt;0 1 4 -2|]
Comma list: 81/80, 126/125, 245/242


[[Wedgie]]: &lt;&lt;1 4 -2 4 -6 -16||
Mapping: {{mapping| 2 0 -8 -26 -31 | 0 1 4 10 12 }}


{{EDOs|legend=1| 5, 7, 12, 17c, 29cd }}
: mapping generators: ~63/44, ~3


[[Badness]]: 0.0207
Optimal tunings:  
* WE: ~63/44 = 600.7492{{c}}, ~3/2 = 696.8853{{c}}
* CWE: ~63/44 = 600.0000{{c}}, ~3/2 = 696.1908{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 12, 26de, 38d, 50 }}
Commas: 36/35, 64/63, 56/55


valid range: [700.000, 705.882] (12 to 17)
Badness (Sintel): 1.26


nice range: [691.202, 715.587]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


strict range: [700.000, 705.882]
Comma list: 81/80, 105/104, 126/125, 245/242


POTE generator: ~3/2 = 703.254
Mapping: {{mapping| 2 0 -8 -26 -31 -40 | 0 1 4 10 12 15 }}


Mapping generator: ~3
Optimal tunings:  
* WE: ~55/39 = 600.8309{{c}}, ~3/2 = 696.8000{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.0066{{c}}


Map: [&lt;1 0 -4 6 13|, &lt;0 1 4 -2 -6|]
{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}


{{EDOs|legend=1| 5, 12, 17c, 29cde }}
Badness (Sintel): 1.19


Badness: 0.0242
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


=== 13-limit ===
Comma list: 81/80, 105/104, 126/125, 189/187, 221/220
Commas: 36/35, 56/55, 64/63, 66/65


valid range: 705.882 (17)
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 | 0 1 4 10 12 15 1 }}


nice range: [691.202, 715.587]
Optimal tunings:  
* WE: ~17/12 = 600.9234{{c}}, ~3/2 = 696.8536{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.9317{{c}}


strict range:705.882
{{Optimal ET sequence|legend=0| 12f, 38df, 50 }}


POTE generator: ~3/2 = 703.636
Badness (Sintel): 1.15


Map: [&lt;1 0 -4 6 13 18|, &lt;0 1 4 -2 -6 -9|]
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


{{EDOs|legend=1| 12f, 17c, 29cdef }}
Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220


Badness: 0.0241
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 -1 | 0 1 4 10 12 15 1 3 }}


=== Dominion ===
Optimal tunings:
Commas: 26/25, 36/35, 56/55, 64/63
* WE: ~17/12 = 600.9845{{c}}, ~3/2 = 696.8939{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.8947{{c}}


POTE generator: ~3/2 = 704.905
{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}


Map: [&lt;1 0 -4 6 13 -9|, &lt;0 1 4 -2 -6 8|]
Badness (Sintel): 1.08


{{EDOs|legend=1| 5, 12, 17c, 46cde }}
=== Trimean ===
{{See also| No-sevens subgroup temperaments #Superpine }}


Badness: 0.0273
Subgroup: 2.3.5.7.11


== Domineering ==
Comma list: 81/80, 126/125, 1344/1331
Commas: 36/35, 45/44, 64/63


POTE generator: ~3/2 = 698.776
Mapping: {{mapping| 1 2 4 7 5 | 0 -3 -12 -30 -11 }}


Mapping generator: ~3
: mapping generators: ~2, ~11/10


Map: [&lt;1 0 -4 6 -6|, &lt;0 1 4 -2 6|]
Optimal tunings:  
* WE: ~2 = 1200.7155{{c}}, ~11/10 = 167.9055{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7749{{c}}


{{EDOs|legend=1| 5e, 7, 12, 19d, 43de }}
{{Optimal ET sequence|legend=0| 7d, 36d, 43, 50, 93 }}


Badness: 0.0220
Badness (Sintel): 1.68


=== 13-limit ===
==== 13-limit ====
Commas: 36/35, 45/44, 52/49, 64/63
Subgroup: 2.3.5.7.11.13


POTE generator: ~3/2 = 695.762
Comma list: 81/80, 126/125, 144/143, 364/363


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


Map: [&lt;1 0 -4 6 -6 10|, &lt;0 1 4 -2 6 -4|]
Optimal tunings:  
* WE: ~2 = 1200.6104{{c}}, ~11/10 = 167.8749{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7728{{c}}


{{EDOs|legend=1| 5ef, 7, 12, 19d, 31def }}
{{Optimal ET sequence|legend=0| 7d, 43, 50, 93 }}


Badness: 0.0270
Badness (Sintel): 1.46


==== 17-limit ====
==== 17-limit ====
Commas: 36/35, 45/44, 51/49, 52/49, 64/63
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 81/80, 126/125, 144/143, 189/187, 221/220
 
Mapping: {{mapping| 1 2 4 7 5 3 8 | 0 -3 -12 -30 -11 5 -28 }}
 
Optimal tunings:
* WE: ~2 = 1200.6144{{c}}, ~11/10 = 167.8716{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7682{{c}}
 
{{Optimal ET sequence|legend=0| 7dg, 43, 50, 93 }}
 
Badness (Sintel): 1.28
 
=== Migration ===
See [[Rastmic clan #Migration|Rastmic clan]].
 
== Flattone ==
{{Main| Flattone }}
 
In flattone, 9 fourths get to the interval class for 7, so that [[7/4]] is a diminished seventh (C–B𝄫), [[7/6]] is a diminished third (C–E𝄫), and [[7/5]] is a double-diminished fifth (C–G𝄫). In general, septimal subminor intervals are diminished and septimal supermajor intervals are augmented, which makes it quite easy to learn flattone notation. The fifth in flattone is typically flatter than that of [[19edo]]. Good tunings for flattone include [[45edo]], [[64edo]], and [[71edo]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 81/80, 525/512
 
{{Mapping|legend=1| 1 0 -4 17 | 0 1 4 -9 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.6308{{c}}, ~3/2 = 695.8782{{c}}
: [[error map]]: {{val| +3.631 -2.446 -2.801 -2.684 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.7334{{c}}
: error map: {{val| 0.000 -8.222 -11.380 -12.426 }}
 
[[Minimax tuning]]:
* [[7-odd-limit]]: ~3/2 = {{monzo| 8/13 0 1/13 -1/13 }}
: [[projection map]]: [{{monzo| 1 0 0 0 }}, {{monzo| 21/13 0 1/13 -1/13 }}, {{monzo| 32/13 0 4/13 -4/13 }}, {{monzo| 32/13 0 -9/13 9/13 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
* [[9-odd-limit]]: ~3/2 = {{monzo| 6/11 2/11 0 -1/11 }}
: [[projection map]]: [{{monzo| 1 0 0 0 }}, {{monzo| 17/11 2/11 0 -1/11 }}, {{monzo| 24/11 8/11 0 -4/11 }}, {{monzo| 34/11 -18/11 0 9/11 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7
 
[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [692.353, 701.955]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
 
[[Algebraic generator]]: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.
 
{{Optimal ET sequence|legend=1| 7, 19, 26, 45 }}
 
[[Badness]] (Sintel): 0.976
 
=== 11-limit ===
This can also be considered a no-sevens temperament: [[#Hypnotone|hypnotone]].
 
Subgroup: 2.3.5.7.11
 
Comma list: 45/44, 81/80, 385/384


POTE generator: ~3/2 = 696.115
Mapping: {{mapping| 1 0 -4 17 -6 | 0 1 4 -9 6 }}


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


Map: [&lt;1 0 -4 6 -6 10 12|, &lt;0 1 4 -2 6 -4 -5|]
Tuning ranges:  
* 11-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]


{{EDOs|legend=1| 5ef, 7, 12, 19d, 31def }}
{{Optimal ET sequence|legend=0| 7, 19, 26, 45, 71bc, 116bcde }}


Badness: 0.0245
Badness (Sintel): 1.12


==== 19-limit ====
==== 13-limit ====
Commas: 36/35, 39/38, 45/44, 51/49, 52/49, 57/56
Subgroup: 2.3.5.7.11.13
 
Comma list: 45/44, 65/64, 78/77, 81/80
 
Mapping: {{mapping| 1 0 -4 17 -6 10 | 0 1 4 -9 6 -4 }}
 
Optimal tunings:
* WE: ~2 = 1202.5156{{c}}, ~3/2 = 694.5107{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0538{{c}}


POTE generator: ~3/2 = 696.217
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]


Mapping generator: ~3
{{Optimal ET sequence|legend=0| 7, 19, 26, 45f, 71bcf, 116bcdef }}


Map: [&lt;1 0 -4 6 -6 10 12 9|, &lt;0 1 4 -2 6 -4 -5 -3|]
Badness (Sintel): 0.920


{{EDOs|legend=1| 5ef, 7, 12, 19d, 31def }}
=== Ptolemy ===
See [[Rastmic clan #Ptolemy|Rastmic clan]].


Badness: 0.0204
== Dominant ==
{{Main| Dominant (temperament) }}
{{See also| Archytas clan }}


=== Dominatrix ===
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]].
Commas: 27/26, 36/35, 45/44, 64/63


POTE generator: ~3/2 = 698.544
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.  


Mapping generator: ~3
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 0 -4 6 -6 -1|, &lt;0 1 4 -2 6 3|]
[[Comma list]]: 36/35, 64/63


{{EDOs|legend=1| 5e, 7, 12f, 19df }}
{{Mapping|legend=1| 1 0 -4 6 | 0 1 4 -2 }}


== Domination ==
[[Optimal tuning]]s:
Commas: 36/35, 64/63, 77/75
* [[WE]]: ~2 = 1195.3384{{c}}, ~3/2 = 698.8478{{c}}
: [[error map]]: {{val| -4.662 -7.769 +9.077 +14.832 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.1125{{c}}
: error map: {{val| 0.000 -0.842 +18.136 +28.949 }}


POTE generator: ~3/2 = 705.004
[[Tuning ranges]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[diamond monotone]]: ~3/2 = [700.000, 720.000] (7\12 to 3\5)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 715.587]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]


Mapping generator: ~3
{{Optimal ET sequence|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}


Map: [&lt;1 0 -4 6 -14|, &lt;0 1 4 -2 11|]
[[Badness]] (Sintel): 0.524


{{EDOs|legend=1| 5e, 12e, 17c, 46cd }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0366
Comma list: 36/35, 56/55, 64/63


=== 13-limit ===
Mapping: {{mapping| 1 0 -4 6 13 | 0 1 4 -2 -6 }}
Commas: 26/25, 36/35, 64/63, 66/65


POTE generator: ~3/2 = 705.496
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]


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


Map: [&lt;1 0 -4 6 -14 -9|, &lt;0 1 4 -2 11 8|]
{{Optimal ET sequence|legend=0| 5, 12, 17c, 29cde }}


{{EDOs|legend=1| 5e, 12e, 17c }}
Badness (Sintel): 0.799


Badness: 0.0274
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


== Arnold ==
Comma list: 36/35, 56/55, 64/63, 66/65
Commas: 22/21, 33/32, 36/35


POTE generator: ~3/2 = 698.491
Mapping: {{mapping| 1 0 -4 6 13 18 | 0 1 4 -2 -6 -9 }}


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


Map: [&lt;1 0 -4 6 5|, &lt;0 1 4 -2 -1|]
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]


{{EDOs|legend=1| 5, 7, 12e }}
{{Optimal ET sequence|legend=0| 12f, 17c, 29cdef }}


Badness: 0.0261
Badness (Sintel): 0.996


=== 13-limit ===
==== Dominion ====
Commas: 22/21, 27/26, 33/32, 36/35
Subgroup: 2.3.5.7.11.13


POTE generator: ~3/2 = 696.743
Comma list: 26/25, 36/35, 56/55, 64/63


Map: [&lt;1 0 -4 6 5 -1|, &lt;0 1 4 -2 -1 3|]
Mapping: {{mapping| 1 0 -4 6 13 -9 | 0 1 4 -2 -6 8 }}


{{EDOs|legend=1| 5, 7, 12ef, 19def }}
Optimal tunings:
* WE: ~2 = 1195.0293{{c}}, ~3/2 = 701.9847{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7698{{c}}


Badness: 0.0233
{{Optimal ET sequence|legend=0| 5, 12, 17c }}


=== 17-limit ===
Badness (Sintel): 1.13
Commas: 22/21, 27/26, 33/32, 36/35, 51/49


POTE generator: ~3/2 = 696.978
=== Domination ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 -4 6 5 -1 12|, &lt;0 1 4 -2 -1 3 -5|]
Comma list: 36/35, 64/63, 77/75


{{EDOs|legend=1| 5, 7, 12ef, 19def }}
Mapping: {{mapping| 1 0 -4 6 -14 | 0 1 4 -2 11 }}


Badness: 0.0245
Optimal tunings:  
* WE: ~2 = 1194.8645{{c}}, ~3/2 = 701.9872{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5945{{c}}


=== 19-limit ===
{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}
Commas: 22/21, 27/26, 33/32, 36/35, 51/49, 57/56


POTE generator: ~3/2 = 697.068
Badness (Sintel): 1.21


Map: [&lt;1 0 -4 6 5 -1 12 9|, &lt;0 1 4 -2 -1 3 -5 -3|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{EDOs|legend=1| 5, 7, 12ef, 19def }}
Comma list: 26/25, 36/35, 64/63, 66/65


Badness: 0.0211
Mapping: {{mapping| 1 0 -4 6 -14 -9 | 0 1 4 -2 11 8 }}


= Sharptone =
Optimal tunings:
[[Comma]]s: 21/20, 28/27
* WE: ~2 = 1195.1324{{c}}, ~3/2 = 702.6343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.0791{{c}}


Sharptone is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo]] tuning does sharptone about as well as such a thing can be done, of course not in its patent val.
{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}


[[POTE generator]]: ~3/2 = 700.140
Badness (Sintel): 1.13


Mapping generator: ~3
=== Domineering ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 -4 -2|, &lt;0 1 4 3|]
Comma list: 36/35, 45/44, 64/63


[[Wedgie]]: &lt;&lt;1 4 3 4 2 -4||
Mapping: {{mapping| 1 0 -4 6 -6 | 0 1 4 -2 6 }}


{{EDOs|legend=1| 5, 7d, 12d }}
Optimal tunings:
* WE: ~2 = 1194.7102{{c}}, ~3/2 = 695.6962{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1765{{c}}


[[Badness]]: 0.0248
{{Optimal ET sequence|legend=0| 5e, 7, 12 }}


== Meanertone ==
Badness (Sintel): 0.727
Commas: 21/20, 28/27, 33/32


POTE generator: ~3/2 = 696.615
=== Arnold ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 -4 -2 5|, &lt;0 1 4 3 -1|]
Comma list: 22/21, 33/32, 36/35


{{EDOs|legend=1| 5, 7d, 12de }}
Mapping: {{mapping| 1 0 -4 6 5 | 0 1 4 -2 -1 }}


Badness: 0.0252
Optimal tunings:  
* WE: ~2 = 1199.8507{{c}}, ~3/2 = 698.4045{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.4822{{c}}


= Meansept =
{{Optimal ET sequence|legend=0| 5, 7, 12e }}
Commas: 15/14, 81/80


POTE generator: ~3/2 = 682.895
Badness (Sintel): 0.864


Mapping generator: ~3
=== Neutrominant ===
See [[Rastmic clan #Neutrominant|Rastmic clan]].


Map: [&lt;1 0 -4 -5|, &lt;0 1 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.


Wedgie: &lt;&lt;1 4 5 4 5 0||
Flattertone was named by [[Flora Canou]] in 2024.


{{EDOs|legend=1| 5d, 7, 12dd }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0453
[[Comma list]]: 81/80, 1875/1792


== 11-limit ==
{{Mapping|legend=1| 1 0 -4 -24 | 0 1 4 17 }}
Commas: 15/14, 22/21, 81/80


POTE generator: ~3/2 = 685.234
: mapping generators: ~2, ~3


Mapping generator: ~3
[[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 }}


Map: [&lt;1 0 -4 -5 -6|, &lt;0 1 4 5 6|]
{{Optimal ET sequence|legend=1| 7d, 19d, 26, 59bcd, 85bccd }}


{{EDOs|legend=1| 5de, 7, 12dd }}
[[Badness]] (Sintel): 2.43


Badness: 0.0325
==== 11-limit ====
Subgroup: 2.3.5.7.11


= Supermean =
Comma list: 45/44, 81/80, 1375/1344
Commas: 81/80, 672/625


POTE generator: ~3/2 = 704.889
Mapping: {{mapping| 1 0 -4 -24 -6 | 0 1 4 17 6 }}


Map: [&lt;1 0 -4 -21|, &lt;0 1 4 15|]
Optimal tunings:  
* WE: ~2 = 1203.4653{{c}}, ~3/2 = 693.8144{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 692.0422{{c}}


{{EDOs|legend=1| 5d, 12d, 17c, 29c }}
{{Optimal ET sequence|legend=0| 7d, 19d, 26 }}


Badness: 0.1342
Badness (Sintel): 1.53


== 11-limit ==
; Music
Commas: 56/55, 81/80, 132/125
* [https://youtu.be/scCuGXnj5IY ''Music in 33EDO (33-Tone Equal Temperament) - Feb 2024''] by [[Budjarn Lambeth]] (2024)


POTE generator: ~3/2 = 705.096
== Sharptone ==
Sharptone is a low-accuracy temperament tempering out [[21/20]] and [[28/27]]. In sharptone, 7/4 is a major sixth, 7/6 a whole tone, and 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo]] tuning does sharptone about as well as such a thing can be done, of course not in its patent val.


Map: [&lt;1 0 -4 -21 -14|, &lt;0 1 4 15 11|]
However, while 12edo ends up near-optimal, the only valid [[diamond monotone]] tuning for sharptone is [[5edo]]. Anything flat of it has ~12/7 and ~7/4 in the wrong order (and so should be dominant) and anything sharp of it has ~5/4 and ~4/3 in the wrong order (and so should not be meantone).


{{EDOs|legend=1| 5de, 12de, 17c, 29c }}
The 11-limit extension was named by Gene Ward Smith in 2004<ref name="meantone & meanpop 2004"/>.


Badness: 0.0633
[[Subgroup]]: 2.3.5.7


== 13-limit ==
[[Comma list]]: 21/20, 28/27
Commas: 26/25, 56/55, 66/65, 81/80


POTE generator: ~3/2 = 705.094
{{Mapping|legend=1| 1 0 -4 -2 | 0 1 4 3 }}


Map: [&lt;1 0 -4 -21 -14 -9|, &lt;0 1 4 15 11 8|]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1204.2961{{c}}, ~3/2 = 702.6463{{c}}
: [[error map]]: {{val| +4.296 +4.987 +24.271 -56.591 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4928{{c}}
: error map: {{val| 0.000 -0.462 +19.657 -64.347 }}


{{EDOs|legend=1| 5de, 12de, 17c, 29c }}
{{Optimal ET sequence|legend=1| 5, 7d, 12d }}


= Injera =
[[Badness]] (Sintel): 0.629
[[Comma]]s: 50/49, 81/80


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.
=== Meanertone ===
Subgroup: 2.3.5.7.11


[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name]
Comma list: 21/20, 28/27, 33/32


valid range: [685.714, 700.000] (14c to 12)
Mapping: {{mapping| 1 0 -4 -2 5 | 0 1 4 3 -1 }}


nice range: [688.957, 701.955]
Optimal tunings:  
* WE: ~2 = 1208.5304{{c}}, ~3/2 = 701.5669{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1117{{c}}


strict range: [688.957, 700.000]
{{Optimal ET sequence|legend=0| 5, 7d, 12de }}


[[POTE generator]]: 694.375
Badness (Sintel): 0.832


Mapping generator: ~3
== 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.


Map: [&lt;2 0 -8 -7|, &lt;0 1 4 4|]
Mildtone was named by [[User: Lucius Chiaraviglio|Lucius Chiaraviglio]] in 2024.


[[Wedgie]]: &lt;&lt;2 8 8 8 7 -4||
[[Subgroup]]: 2.3.5.7


{{EDOs|legend=1| 12, 26, 38, 102bcd, 140bccd, 178bbccdd }}
[[Comma list]]: 81/80, 16128/15625


[[Badness]]: 0.0311
{{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 }}


== Music ==
[[Optimal tuning]]s:
* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 Two Pairs of Socks] (in [[26edo]]) by [[Igliashon Jones]]
* [[WE]]: ~2 = 1199.7304{{c}}, ~3/2 = 698.3953{{c}}
: [[error map]]: {{val| -0.270 -3.829 +7.267 -1.434 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.5397{{c}}
: error map: {{val| 0.000 -3.415 +7.845 -0.952 }}


== 11-limit ==
{{Optimal ET sequence|legend=1| 12, 43d, 55, 67 }}
Commas: 45/44, 50/49, 81/80


valid range: [685.714, 700.000] (14c to 12)
[[Badness]] (Sintel): 2.67


nice range: [682.458, 701.955]
=== 11-limit ===


strict range: [685.714, 700.000]
[[Subgroup]]: 2.3.5.7.11


POTE generator: ~3/2 = 692.840
[[Comma list]]: 81/80, 176/175, 7056/6875


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


Map: [&lt;2 0 -8 -7 -12|, &lt;0 1 4 4 6|]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.816{{c}}, ~3/2 = 698.355{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.455{{c}}


{{EDOs|legend=1| 12, 14c, 26, 90bce, 116bcce }}
{{Optimal ET sequence|legend=1| 12, 43de, 55, 67 }}


Badness: 0.0231
[[Badness]] (Sintel): 2.15


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


valid range: 692.308 (26)
[[Subgroup]]: 2.3.5.7.11.13
 
[[Comma list]]: 81/80, 176/175, 196/195, 832/825
 
{{Mapping|legend=1| 1 0 -4 -32 -44 | 0 1 4 22 30}}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.788{{c}}, ~3/2 = 698.355{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.471{{c}}
 
{{Optimal ET sequence|legend=1| 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|legend=1| 1 0 -4 -32 -44 12| 0 1 4 22 30 -5}}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.655{{c}}, ~3/2 = 698.295{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.488{{c}}
 
{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}
 
[[Badness]] (Sintel): 1.98
 
=== 19-limit ===


nice range: [682.458, 701.955]
[[Subgroup]]: 2.3.5.7.11.13.17.19


strict range: 692.308 (26)
[[Comma list]]: 81/80, 96/95, 176/175, 189/187, 196/195, 832/825


POTE generator: ~3/2 = 692.673
{{Mapping|legend=1| 1 0 -4 -32 -44 12 9| 0 1 4 22 30 -5 -3}}


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


Map: [&lt;2 0 -8 -7 -12 -21|, &lt;0 1 4 4 6 9|]
{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}


{{EDOs|legend=1| 12f, 14cf, 26, 38e }}
[[Badness]] (Sintel): 1.95


Badness: 0.0216
{{Todo|unify precision|review}}


=== Enjera ===
== Supermean ==
Commas: 27/26, 40/39, 45/44, 50/49
Supermean tempers out 672/625 and finds the interval class of 7 at 15 generators up, as a double-augmented fifth (C–Gx). As such, it extends [[leapfrog]].


POTE generator: ~3/2 = 694.121
[[Subgroup]]: 2.3.5.7


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


Map: [&lt;2 0 -8 -7 -12 -2|, &lt;0 1 4 4 6 3|]
{{Mapping|legend=1| 1 0 -4 -21 | 0 1 4 15 }}


{{EDOs|legend=1| 12f, 14c, 26f, 38eff }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1195.4372{{c}}, ~3/2 = 702.2086{{c}}
: [[error map]]: {{val| -4.563 -4.309 +22.521 -8.319 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5375{{c}}
: error map: {{val| 0.000 +2.583 +31.836 -0.763 }}


Badness: 0.0265
{{Optimal ET sequence|legend=1| 5d, 12d, 17c }}


== Injerous ==
[[Badness]] (Sintel): 3.40
Commas: 33/32, 50/49, 55/54


POTE generator: ~3/2 = 690.548
=== 11-limit ===
Subgroup: 2.3.5.7.11


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


Map: [&lt;2 0 -8 -7 10|, &lt;0 1 4 4 -1|]
Mapping: {{mapping| 1 0 -4 -21 -14 | 0 1 4 15 11 }}


{{EDOs|legend=1| 12e, 14c, 26e, 40cee }}
Optimal tunings:
* WE: ~2 = 1195.7270{{c}}, ~3/2 = 702.5848{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7471{{c}}


Badness: 0.0386
{{Optimal ET sequence|legend=0| 5de, 12de, 17c }}


== Lahoh ==
Badness (Sintel): 2.09
Commas: 50/49, 56/55, 81/77


POTE generator: ~3/2 = 699.001
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


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


Map: [&lt;2 0 -8 -7 7|, &lt;0 1 4 4 0|]
Mapping: {{mapping| 1 0 -4 -21 -14 -9 | 0 1 4 15 11 8 }}


{{EDOs|legend=1| 2cd, 12, 14ce }}
Optimal tunings:
* WE: ~2 = 1196.3958{{c}}, ~3/2 = 702.9766{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7940{{c}}


Badness: 0.0431
{{Optimal ET sequence|legend=0| 5de, 12de, 17c, 29c }}


= Godzilla =
Badness (Sintel): 1.67
<span style="display: block; text-align: right;">[[:de:Semiphor,_Semaphor,_Godzilla|Deutsch]]</span>


{{main|Semaphore and Godzilla}}
== Mohajira ==
{{Main| Mohajira }}


[[Comma]]s: 49/48, 81/80
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.


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 the perfect godzilla tuning, so much so that's there's not much point in looking elsewhere. Hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes.
[[Subgroup]]: 2.3.5.7


valid range: [240.000, 257.143] (5 to 14c)
[[Comma list]]: 81/80, 6144/6125


nice range: [231.174, 266.871]
{{Mapping|legend=1| 1 1 0 6 | 0 2 8 -11 }}


strict range: [240.000, 257.143]
: mapping generators: ~2, ~128/105


[[POTE generator]]: ~8/7 = 252.635
[[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 generator: ~7/4
[[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


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


[[Wedgie]]: &lt;&lt;2 8 1 8 -4 -20||
[[Algebraic generator]]: Mohabis, real root of 3''x''<sup>3</sup> - 3''x''<sup>2</sup> - 1, 348.6067 cents. Corresponding recurrence converges quickly.


{{EDOs|legend=1| 5, 9c, 14c, 19, 62d, 81d, 143bd }}
{{Optimal ET sequence|legend=1| 7, 24, 31 }}


[[Badness]]: 0.0267
[[Badness]] (Sintel): 1.41


== 11-limit ==
Scales: [[mohaha7]], [[mohaha10]]
Commas: 45/44, 49/48, 81/80


valid range: [252.632, 257.143] (19 to 14c)
=== 11-limit ===
Subgroup: 2.3.5.7.11


nice range: [231.174, 266.871]
Comma list: 81/80, 121/120, 176/175


strict range: [252.632, 257.143]
Mapping: {{mapping| 1 1 0 6 2 | 0 2 8 -11 5 }}


POTE generator: ~8/7 = 254.027
Optimal tunings:  
* WE: ~2 = 1201.1562{{c}}, ~11/9 = 348.8124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.4910{{c}}


Mapping generator: ~7/4
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


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


{{EDOs|legend=1| 14c, 19, 33cd, 52cd }}
{{Optimal ET sequence|legend=0| 7, 24, 31 }}


Badness: 0.0290
Badness (Sintel): 0.862
 
Scales: [[mohaha7]], [[mohaha10]]


=== 13-limit ===
=== 13-limit ===
Commas: 45/44, 49/48, 78/77, 81/80
Subgroup: 2.3.5.7.11.13
 
Comma list: 66/65, 81/80, 105/104, 121/120
 
Mapping: {{mapping| 1 1 0 6 2 4 | 0 2 8 -11 5 -1 }}
 
Optimal tunings:
* WE: ~2 = 1200.4256{{c}}, ~11/9 = 348.6819{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.5622{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 1 0 6 2 4 7 | 0 2 8 -11 5 -1 -10 }}
 
Optimal tunings:
* WE: ~2 = 1200.0382{{c}}, ~11/9 = 348.7471{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.7360{{c}}
 
{{Optimal ET sequence|legend=0| 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


valid range: 694.737 (19)
Mapping: {{mapping| 1 1 0 6 2 4 7 6 | 0 2 8 -11 5 -1 -10 -6 }}


nice range: [621.581, 737.652]
Optimal tunings:  
* WE: ~2 = 1199.7469{{c}}, ~11/9 = 348.7367{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.8117{{c}}


strict range: 694.737
{{Optimal ET sequence|legend=0| 7, 24, 31, 55 }}


POTE generator: ~8/7 = 253.603
Badness (Sintel): 1.05


Mapping generator: ~7/4
Scales: [[mohaha7]], [[mohaha10]]


Map: [&lt;1 0 -4 2 -6 -5|, &lt;0 2 8 1 12 11|]
== 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.


{{EDOs|legend=1| 14cf, 19, 33cdf, 52cdf }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0225
[[Comma list]]: 81/80, 392/375


== Semafour ==
{{Mapping|legend=1| 1 1 0 -1 | 0 2 8 13 }}
Commas: 33/32, 49/48, 55/54


POTE generator: ~8/7 = 254.042
: mapping generators: ~2, ~25/21


Mapping generator: ~7/4
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.0661{{c}}, ~25/21 = 350.3127{{c}}
: [[error map]]: {{val| -0.934 -2.264 +16.188 -13.827 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/21 = 350.4856{{c}}
: error map: {{val| 0.000 -0.984 +17.571 -12.513 }}


Map: [&lt;1 0 -4 2 5|, &lt;0 2 8 1 -2|]
{{Optimal ET sequence|legend=1| 7d, 17c, 24 }}


{{EDOs|legend=1| 5, 14c, 19e, 33cde }}
[[Badness]] (Sintel): 1.97


Badness: 0.0285
Scales: [[mohaha7]], [[mohaha10]]


== Varan ==
=== 11-limit ===
Commas: 49/48, 77/75, 81/80
Subgroup: 2.3.5.7.11


POTE generator: ~8/7 = 251.079
Comma list: 56/55, 77/75, 243/242


Mapping generator: ~7/4
Mapping: {{mapping| 1 1 0 -1 2 | 0 2 8 13 5 }}


Map: [&lt;1 0 -4 2 -10|, &lt;0 2 8 1 17|]
Optimal tunings:  
* WE: ~2 = 1199.1924{{c}}, ~11/9 = 350.3286{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.4821{{c}}


{{EDOs|legend=1| 19e, 24, 43de }}
{{Optimal ET sequence|legend=0| 7d, 17c, 24 }}


Badness: 0.0396
Badness (Sintel): 1.20
 
Scales: [[mohaha7]], [[mohaha10]]


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


POTE generator: ~8/7 = 251.165
Comma list: 56/55, 66/65, 77/75, 243/242


Mapping generator: ~7/4
Mapping: {{mapping| 1 1 0 -1 2 4 | 0 2 8 13 5 -1 }}


Map: [&lt;1 0 -4 2 -10 -5|, &lt;0 2 8 1 17 11|]
Optimal tunings:  
* WE: ~2 = 1198.5986{{c}}, ~11/9 = 350.3353{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.6459{{c}}


{{EDOs|legend=1| 19e, 24, 43de }}
{{Optimal ET sequence|legend=0| 7d, 17c, 24, 41c }}


Badness: 0.0257
Badness (Sintel): 1.19


== Baragon ==
Scales: [[mohaha7]], [[mohaha10]]
Commas: 49/48, 56/55, 81/80


POTE generator: ~8/7 = 251.173
== Liese ==
<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>


Mapping generator: ~7/4
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.


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


{{EDOs|legend=1| 19, 24, 43d }}
[[Comma list]]: 81/80, 686/675


Badness: 0.0357
{{Mapping|legend=1| 1 0 -4 -3 | 0 3 12 11 }}


== Music ==
: mapping generators: ~2, ~10/7
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Bobro/GodzillaExample.mp3 Godzilla Example] by [[Cameron Bobro]]
* [http://tinyurl.com/4uyumk9 "Change is on the Wind"] in Godzilla[9] by [[Igliashon Jones]]


= Mohajira =
[[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 }}


{{main|Mohajira}}
[[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


[[Comma]]s: 81/80, 6144/6125
[[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.


Mohajira really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. [[31edo]] makes for an excellent (7-limit) mohajira tuning, with generator 9/31. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs.
{{Optimal ET sequence|legend=1| 17c, 19, 55, 74d }}


Mohajira can also be thought of, intuitively, as "meantone with quarter tones"; as is the 3/2 generator subdivided in half, so is the 25/24 chromatic semitone divided into two equal ~33/32 quarter tones (in the 11-limit). Within this paradigm, mohajira is the temperament that splits the 3/2 into two equal 11/9's, that splits the 6/5 into two equal 11/10's, that maps four 3/2's to 5/1, and that maps the interval one quarter tone flat of 16/9 to 7/4.
[[Badness]] (Sintel): 1.18


[[7-odd-limit|7]] and [[9-odd-limit|9-limit]] minimax 1/4 comma
=== Liesel ===
Subgroup: 2.3.5.7.11


[{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 6 0 -11/8 0 }}]
Comma list: 56/55, 81/80, 540/539


[[Eigenmonzo]]s: 2, 5
Mapping: {{mapping| 1 0 -4 -3 4 | 0 3 12 11 -1 }}


[[POTE generator]]: ~128/105 = 348.415
Optimal tunings:  
* WE: ~2 = 1198.8507{{c}}, ~10/7 = 632.4668{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 632.9963{{c}}


Mapping generator: ~128/105
{{Optimal ET sequence|legend=0| 17c, 19, 36 }}


Algebraic generator: Mohabis, real root of 3''x''<sup>3</sup> - 3''x''<sup>2</sup> - 1, 348.6067 cents. Corresponding recurrence converges quickly.
Badness (Sintel): 1.35


Map: [&lt;1 1 0 6|, &lt;0 2 8 -11|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[Generator]]s: 2, 128/105
Comma list: 56/55, 78/77, 81/80, 91/90


[[Wedgie]]: &lt;&lt;2 8 -11 8 -23 -48||
Mapping: {{mapping| 1 0 -4 -3 4 0 | 0 3 12 11 -1 7 }}


{{EDOs|legend=1| 7, 24, 31, 38, 55, 69 }}
Optimal tunings:
* WE: ~2 = 1199.4968{{c}}, ~10/7 = 632.7766{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.0082{{c}}


[[Badness]]: 0.0557
{{Optimal ET sequence|legend=0| 17c, 19, 36 }}


== 11-limit ==
Badness (Sintel): 1.13
Commas: 81/80, 121/120, 176/175


[[11-odd-limit|11-limit]] minimax 1/4 comma
=== Elisa ===
Subgroup: 2.3.5.7.11


[{{Monzo| 1 0 0 0 0 }}, {{Monzo| 1 0 1/4 0 0 }}, {{Monzo| 0 0 1 0 0 }},
Comma list: 77/75, 81/80, 99/98
{{Monzo| 6 0 -11/8 0 0 }}, {{Monzo| 2 0 5/8 0 0 }}<nowiki>]</nowiki>


Eigenmonzos: 2, 5
Mapping: {{mapping| 1 0 -4 -3 -5 | 0 3 12 11 16 }}


POTE generator: ~11/9 = 348.477
Optimal tunings:  
* WE: ~2 = 1201.0489{{c}}, ~10/7 = 633.6147{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1644{{c}}


Mapping generator: ~11/9
{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}


Map: [&lt;1 1 0 6 2|, &lt;0 2 8 -11 5|]
Badness (Sintel): 1.37


Generators: 2, 11/9
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{EDOs|legend=1| 7, 24, 31, 38, 55 }}
Comma list: 66/65, 77/75, 81/80, 99/98


Badness: 0.0261
Mapping: {{mapping| 1 0 -4 -3 -5 0 | 0 3 12 11 16 7 }}


== 13-limit ==
Optimal tunings:
Commas: 66/65, 81/80, 105/104, 121/120
* WE: ~2 = 1201.4815{{c}}, ~10/7 = 633.7720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1281{{c}}


POTE generator: ~11/9 = 348.558
{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}


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


Map: [&lt;1 1 0 6 2 4|, &lt;0 2 8 -11 5 -1|]
=== Lisa ===
Subgroup: 2.3.5.7.11


{{EDOs|legend=1| 7, 24, 31, 38, 55 }}
Comma list: 45/44, 81/80, 343/330


Badness: 0.0234
Mapping: {{mapping| 1 0 -4 -3 -6 | 0 3 12 11 18 }}


== 17-limit ==
Optimal tunings:
Commas: 66/65, 81/80, 105/104, 121/120, 154/153
* WE: ~2 = 1202.6773{{c}}, ~10/7 = 632.7783{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.6175{{c}}


POTE generator: ~11/9 = 348.736
{{Optimal ET sequence|legend=0| 17cee, 19 }}


Mapping generator: ~11/9
Badness (Sintel): 1.81


Map: [&lt;1 1 0 6 2 4 7|, &lt;0 2 8 -11 5 -1 -10|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{EDOs|legend=1| 7, 24, 31, 38g, 55 }}
Comma list: 45/44, 81/80, 91/88, 147/143


Badness: 0.0206
Mapping: {{mapping| 1 0 -4 -3 -6 0 | 0 3 12 11 18 7 }}


== 19-limit ==
Optimal tunings:
Commas: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152
* WE: ~2 = 1203.6086{{c}}, ~10/7 = 633.1193{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.5346{{c}}


POTE generator: ~11/9 = 348.810
{{Optimal ET sequence|legend=0| 17cee, 19 }}


Mapping generator: ~11/9
Badness (Sintel): 1.49


Map: [&lt;1 1 0 6 2 4 7 6|, &lt;0 2 8 -11 5 -1 -10 -6|]
== Superpine ==
{{See also| No-sevens subgroup temperaments #Superpine }}


{{EDOs|legend=1| 7, 24, 31, 38gh, 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.


Badness: 0.0173
[[Subgroup]]: 2.3.5.7


= Ptolemy =
[[Comma list]]: 81/80, 1119744/1071875
Commas: 81/80, 121/120, 525/512


POTE generator: ~11/9 = 346.922
{{Mapping|legend=1| 1 2 4 1 | 0 -3 -12 13 }}


Map: [&lt;1 1 0 8 2|, &lt;0 2 8 -18 5|]
[[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 }}


{{EDOs|legend=1| 7, 31dd, 38d, 45e, 83bcddee }}
{{Optimal ET sequence|legend=1| 7, 36, 43, 79c }}


Badness: 0.0588
[[Badness]] (Sintel): 3.46


== 13-limit ==
=== 11-limit ===
Commas: 65/64, 81/80, 105/104, 121/120
Subgroup: 2.3.5.7.11


POTE generator: ~11/9 = 346.910
Comma list: 81/80, 176/175, 864/847


Map: [&lt;1 1 0 8 2 6|, &lt;0 2 8 -18 5 -8|]
Mapping: {{mapping| 1 2 4 1 5 | 0 -3 -12 13 -11 }}


{{EDOs|legend=1| 7, 31ddf, 38df, 45ef, 83bcddeeff }}
Optimal tunings:
* WE: ~2 = 1199.0522{{c}}, ~11/10 = 167.1904{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.3382{{c}}


Badness: 0.0343
{{Optimal ET sequence|legend=0| 7, 36, 43 }}


= Maqamic =
Badness (Sintel): 1.90
<span style="display: block; text-align: right;">[[:de:maqamisch|Deutsch]]</span>


{{main| Maqamic }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Comma]]s: 81/80, 36/35, 121/120
Comma list: 78/77, 81/80, 144/143, 176/175


Maqamic temperament is much like Mohajira, except in that it 36/35 vanishes instead of 176/175. It makes the most sense if viewed as an adaptive temperament, whereby 7/4 and 9/5 simply share an equivalence class in the resulting scales, but don't need to share a particular tempered "middle-of-the-road" intonation.
Mapping: {{mapping| 1 2 4 1 5 3 | 0 -3 -12 13 -11 5 }}


[[POTE generator]]: ~11/9 = 350.934
Optimal tunings:  
* WE: ~2 = 1199.4286{{c}}, ~11/10 = 167.3105{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.3958{{c}}


Mapping generator: ~11/9
{{Optimal ET sequence|legend=0| 7, 36, 43 }}


Map: [&lt;1 1 0 4 2|, &lt;0 2 8 -4 5|]
Badness (Sintel): 1.52


[[generator|Generator]]s: 2, 11/9
== 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.


{{EDOs|legend=1| 7, 10c, 17c, 24d, 31d }}
[[Subgroup]]: 2.3.5.7


== 13-limit ==
[[Comma list]]: 81/80, 3125/3087
[[Comma]]s: 81/80, 36/35, 121/120, 144/143


[[POTE generator]]: ~11/9 = 350.816
{{Mapping|legend=1| 3 0 -12 -20 | 0 1 4 6 }}


Mapping generator: ~11/9
: mapping generators: ~56/45, ~3


Map: [&lt;1 1 0 4 2 4|, &lt;0 2 8 -4 5 -1|]
[[Optimal tuning]]s:  
* [[WE]]: ~56/45 = 400.6744{{c}}, ~3/2 = 695.8474{{c}} {~15/14 = 105.5015{{c}})
: [[error map]]: {{val| +2.023 -4.084 -2.924 +4.910 }}
* [[CWE]]: ~56/45 = 400.0000{{c}}, ~3/2 = 695.1413{{c}} {~15/14 = 104.8587{{c}})
: error map: {{val| 0.000 -6.814 -5.748 +2.022 }}


Generators: 2, 11/9
{{Optimal ET sequence|legend=1| 12, 33cd, 45, 57 }}


{{EDOs|legend=1| 7, 10c, 17c, 24d, 31d }}
[[Badness]] (Sintel): 1.75


= Migration =
== Squares ==
Commas: 81/80, 121/120, 126/125
{{Main| Squares }}


POTE generator: ~11/9 = 348.182
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]].


Mapping generator: ~11/9
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 1 0 -3 2|, &lt;0 2 8 20 5|]
[[Comma list]]: 81/80, 2401/2400


{{EDOs|legend=1| 7d, 31, 100de, 131bdee, 162bdee }}
{{Mapping|legend=1| 1 -1 -8 -3 | 0 4 16 9 }}


Badness: 0.0255
: mapping generators: ~2, ~14/9


== 13-limit ==
[[Optimal tuning]]s:
Commas: 66/65, 81/80, 121/120, 126/125
* [[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 }}


POTE generator: ~11/9 = 348.490
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~9/7 = {{monzo| 1/2 0 -1/16 }}
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 3/2 0 9/16 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


Map: [&lt;1 1 0 -3 2 4|, &lt;0 2 8 20 5 -1|]
[[Algebraic generator]]: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.


{{EDOs|legend=1| 7d, 24d, 31, 55d }}
{{Optimal ET sequence|legend=1| 14c, 17c, 31, 169b, 200b }}


Badness: 0.0281
[[Badness]] (Sintel): 1.16


= Mohamaq =
Scales: [[skwares8]], [[skwares11]], [[skwares14]]
Commas: 81/80, 392/375


POTE generator: ~25/21 = 350.586
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping generator: ~25/21
Comma list: 81/80, 99/98, 121/120


Map: [&lt;1 1 0 -1|, &lt;0 2 8 13|]
Mapping: {{mapping| 1 -1 -8 -3 -3 | 0 4 16 9 10 }}


{{EDOs|legend=1| 17c, 24, 65c, 89cd }}
Optimal tunings:
* WE: ~2 = 1201.6657{{c}}, ~11/7 = 775.1171{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.1754{{c}}


Badness: 0.0777
{{Optimal ET sequence|legend=0| 14c, 17c, 31, 130bee, 169beee }}


== 11-limit ==
Badness (Sintel): 0.715
Commas: 56/55, 77/75, 243/242


POTE generator: ~11/9 = 350.565
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Mapping generator: ~11/9
Comma list: 66/65, 81/80, 99/98, 121/120


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


{{EDOs|legend=1| 17c, 24, 65c, 89cd }}
Optimal tunings:
* WE: ~2 = 1199.8419{{c}}, ~11/7 = 774.3484{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4422{{c}}


Badness: 0.0362
{{Optimal ET sequence|legend=0| 14c, 17c, 31, 79cf }}


== 13-limit ==
Badness (Sintel): 1.05
Commas: 56/55, 66/65, 77/75, 243/242


POTE generator: ~11/9 = 350.745
==== Squad ====
Subgroup: 2.3.5.7.11.13


Mapping generator: ~11/9
Comma list: 78/77, 81/80, 91/90, 99/98


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


{{EDOs|legend=1| 17c, 24, 41c, 65c }}
Optimal tunings:
* WE: ~2 = 1202.0312{{c}}, ~11/7 = 775.5589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4140{{c}}


Badness: 0.0287
{{Optimal ET sequence|legend=0| 14cf, 17c, 31f }}


= Orphic =
Badness (Sintel): 1.11
Commas: 81/80, 5898240/5764801


POTE generator: ~7/6 = 275.794
==== Agora ====
Subgroup: 2.3.5.7.11.13


Mapping generator: ~343/288
Comma list: 81/80, 99/98, 105/104, 121/120


Map: [&lt;2 1 -4 4|, &lt;0 4 16 3|]
Mapping: {{mapping| 1 -1 -8 -3 -3 -15 | 0 4 16 9 10 29 }}


Wedgie: &lt;&lt;8 32 6 32 -13 -76||
Optimal tunings:  
* WE: ~2 = 1202.3228{{c}}, ~11/7 = 775.2214{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8617{{c}}


{{EDOs|legend=1| 26, 74, 174bd, 248bd }}
{{Optimal ET sequence|legend=0| 14cf, 31, 45ef, 76e }}


Badness: 0.2588
Badness (Sintel): 1.01


== 11-limit ==
===== 17-limit =====
Commas: 81/80, 99/98, 73728/73205
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~7/6 = 275.762
Comma list: 81/80, 99/98, 105/104, 120/119, 121/119


Mapping generator: ~77/64
Mapping: {{mapping| 1 -1 -8 -3 -3 -15 -3 | 0 4 16 9 10 29 11 }}


Map: [&lt;2 1 -4 4 8|, &lt;0 4 16 3 -2|]
Optimal tunings:  
* WE: ~2 = 1201.4340{{c}}, ~11/7 = 774.7375{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8955{{c}}


{{EDOs|legend=1| 26, 48c, 74, 248bd, 322bd }}
{{Optimal ET sequence|legend=0| 14cf, 31 }}


Badness: 0.1015
Badness (Sintel): 1.15


== 13-limit ==
===== 19-limit =====
Commas: 81/80, 99/98, 144/143, 2200/2197
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~7/6 = 275.774
Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119


Mapping generator: ~63/52
Mapping: {{mapping| 1 -1 -8 -3 -3 -15 -3 -8 | 0 4 16 9 10 29 11 19 }}


Map: [&lt;2 1 -4 4 8 2|, &lt;0 4 16 3 -2 10|]
Optimal tunings:  
* WE: ~2 = 1201.2461{{c}}, ~11/7 = 774.5783{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8479{{c}}


{{EDOs|legend=1| 26, 48c, 74, 174bd, 248bd, 322bd }}
{{Optimal ET sequence|legend=0| 14cf, 31 }}


Badness: 0.0535
Badness (Sintel): 1.15


= Mothra =
=== Cuboctahedra ===
[[Comma]]s: 81/80, 1029/1024
Subgroup: 2.3.5.7.11


Mothra splits the fifth into three 8/7 generators. It uses 1029/1024, the gamelisma, to accomplish this deed and also tempers out 1728/1715, the orwell comma. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra. In the 2.3.7-limit, mothra is identical to [[slendric]].
Comma list: 81/80, 385/384, 1375/1372


Note that mothra can also be called cynder in the 7-limit, which can be a little confusing sometimes.
Mapping: {{mapping| 1 -1 -8 -3 17 | 0 4 16 9 -21 }}


[[7-odd-limit|7]] and [[9-odd-limit|9-limit]] minimax 1/4 comma
Optimal tunings:
* WE: ~2 = 1201.4436{{c}}, ~14/9 = 774.9386{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 774.0243{{c}}


[{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 3 0 -1/12 0 }}]
{{Optimal ET sequence|legend=0| 31, 107b, 138b, 169be, 200be }}


[[Eigenmonzo]]s: 2, 5
Badness (Sintel): 1.88


[[POTE generator]]: ~8/7 = 232.193
== 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.


Mapping generator: ~8/7
[[Subgroup]]: 2.3.5.7


Algebraic generator: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.
[[Comma list]]: 81/80, 17280/16807


Map: [&lt;1 1 0 3|, &lt;0 3 12 -1|]
{{Mapping|legend=1| 1 1 0 2 | 0 5 20 7 }}


[[Generator]]s: 2, 8/7
: mapping generators: ~2, ~54/49


[[Wedgie]]: &lt;&lt;3 12 -1 12 -10 -36||
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1640{{c}}, ~54/49 = 139.3624{{c}}
: [[error map]]: {{val| +0.164 -4.979 +0.934 +7.039 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~54/49 = 139.3528{{c}}
: error map: {{val| 0.000 -5.191 +0.741 +6.643 }}


{{EDOs|legend=1| 5, 26, 31, 57, 88 }}
{{Optimal ET sequence|legend=1| 17c, 26, 43 }}


[[Badness]]: 0.0371
[[Badness]] (Sintel): 2.75


== 11-limit ==
=== 11-limit ===
Commas: 81/80, 99/98, 385/384
Subgroup: 2.3.5.7.11


POTE generator: ~8/7 = 232.031
Comma list: 81/80, 99/98, 864/847


Mapping generator: ~8/7
Mapping: {{mapping| 1 1 0 2 3 | 0 5 20 7 4 }}


Map: [&lt;1 1 0 3 5|, &lt;0 3 12 -1 -8|]
Optimal tunings:  
* WE: ~2 = 1201.4436{{c}}, ~12/11 = 139.3714{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 139.4038{{c}}


{{EDOs|legend=1| 5, 26, 31, 57, 88, 150be, 181bee }}
{{Optimal ET sequence|legend=0| 17c, 26, 43 }}


Badness: 0.0256
Badness (Sintel): 1.58


=== 13-limit ===
=== 13-limit ===
Commas: 81/80, 99/98, 105/104, 144/143
Subgroup: 2.3.5.7.11.13
 
Comma list: 78/77, 81/80, 99/98, 144/143
 
Mapping: {{mapping| 1 1 0 2 3 3 | 0 5 20 7 4 6 }}
 
Optimal tunings:
* WE: ~2 = 1199.8860{{c}}, ~13/12 = 139.3737{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3817{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 1 0 2 3 3 2 | 0 5 20 7 4 6 18 }}
 
Optimal tunings:
* WE: ~2 = 1199.8346{{c}}, ~13/12 = 139.3431{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3544{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 1 0 2 3 3 2 1 | 0 5 20 7 4 6 18 28 }}
 
Optimal tunings:
* WE: ~2 = 1199.8891{{c}}, ~13/12 = 139.3001{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3080{{c}}
 
{{Optimal ET sequence|legend=0| 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|legend=1| 1 -1 -8 -7 | 0 5 20 19 }}
 
: mapping generators: ~2, ~10/7
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 -1 -8 -7 -11 | 0 5 20 19 28 }}
 
Optimal tunings:
* WE: ~2 = 1201.2054{{c}}, ~10/7 = 619.9752{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.4223{{c}}
 
{{Optimal ET sequence|legend=0| 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 {{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.
 
[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name]


POTE generator: ~8/7 = 231.811
[[Subgroup]]: 2.3.5.7


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


Map: [&lt;1 1 0 3 5 1|, &lt;0 3 12 -1 -8 14|]
{{Mapping|legend=1| 2 0 -8 -7 | 0 1 4 4 }}


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


Badness: 0.0240
[[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 }}


== Cynder ==
[[Tuning ranges]]:
Commas: 45/44, 81/80, 1029/1024
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [688.957, 701.955]
* 9-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]


POTE generator: ~8/7 = 231.317
{{Optimal ET sequence|legend=1| 12, 26, 38 }}


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


Map: [&lt;1 1 0 3 0|, &lt;0 3 12 -1 18|]
; Music
* [https://web.archive.org/web/20201127013520/http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 ''Two Pairs of Socks''] by [[Igliashon Jones]] – in [[26edo]] tuning


{{EDOs|legend=1| 5e, 26, 31e, 57e, 83bce }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0557
Comma list: 45/44, 50/49, 81/80


=== 13-limit ===
Mapping: {{mapping| 2 0 -8 -7 -12 | 0 1 4 4 6 }}
Commas: 45/44, 78/77, 81/80, 640/637
 
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}})
 
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|legend=0| 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


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


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


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


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


Badness: 0.0341
Badness (Sintel): 0.891


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


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


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


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


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


Badness: 0.0313
Badness (Sintel): 0.935


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


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


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


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


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


Badness: 0.0369
Badness (Sintel): 0.920


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


Squares splits the interval of an eleventh, or 8/3, into four supermajor third ([[9/7]]) intervals, and uses it for a generator. [[31edo]], with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out 2401/2400, the breedsma, as well as 2430/2401.
Comma list: 27/26, 40/39, 45/44, 50/49


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


Badness: 0.0216
Subgroup: 2.3.5.7.11


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


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


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


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


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


Badness: 0.0255
Badness (Sintel): 2.34


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


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


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


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


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


Badness: 0.0245
Badness (Sintel): 1.65


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


Liese splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. [[74edo]] makes for a good liese tuning, though [[19edo]] can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55.
Comma list: 81/80, 245/242, 385/384


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


Badness: 0.0407
Badness (Sintel): 1.08
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


=== 13-limit ===
Comma list: 81/80, 105/104, 133/132, 144/143, 171/170, 210/209
Liesel is a very natural 13-limit tuning, given the generator is so near 13/9.


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


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


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


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


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


Badness: 0.0273
[[Subgroup]]: 2.3.5.7


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


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


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


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


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


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


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


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


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


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


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


Badness: 0.0548
Badness (Sintel): 3.36


=== 13-limit ===
=== 13-limit ===
Commas: 45/44, 81/80, 91/88, 147/143
Subgroup: 2.3.5.7.11.13


POTE generator: ~10/7 = 631.221
Comma list: 81/80, 99/98, 144/143, 2200/2197


Map: [&lt;1 0 -4 -3 -6 0|, &lt;0 3 12 11 18 7|]
Mapping: {{mapping| 2 1 -4 4 8 2 | 0 4 16 3 -2 10 }}


{{EDOs|legend=1| 19 }}
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: 0.0361
{{Optimal ET sequence|legend=0| 26, 48c, 74 }}


= Jerome =
Badness (Sintel): 2.21
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.


Commas: 81/80, 17280/16807
== Cloudtone ==
The cloudtone temperament tempers out the [[cloudy comma]], 16807/16384 and the [[syntonic comma]], 81/80 in the 7-limit. It may be described as {{nowrap| 5 & 50 }}; its ploidacot is pentaploid monocot. It can be extended to the 11- and 13-limit by adding 385/384 and 105/104 to the comma list in this order.


POTE generator: ~54/49 = 139.343
[[Subgroup]]: 2.3.5.7


Mapping generator: ~54/49
[[Comma list]]: 81/80, 16807/16384


Map: [&lt;1 1 0 2|, &lt;0 5 20 7|]
{{Mapping|legend=1| 5 0 -20 14 | 0 1 4 0 }}


Wedgie: &lt;&lt;5 30 7 20 -3 -40||
: mapping generators: ~8/7, ~3


{{EDOs|legend=1| 9c, 17c, 26, 43, 69, 112bd }}
[[Optimal tuning]]s:
* [[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 }}


Badness: 0.1087
{{Optimal ET sequence|legend=1| 5, 40c, 45, 50 }}


== 11-limit ==
[[Badness]] (Sintel): 2.59
Commas: 81/80, 99/98, 864/847


POTE generator: ~12/11 = 139.428
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping generator: ~12/11
Comma list: 81/80, 385/384, 2401/2376


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


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


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


== 13-limit ==
Badness (Sintel): 2.33
Commas: 78/77, 81/80, 99/98, 144/143


POTE generator: ~13/12 = 139.387
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


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


Map: [&lt;1 1 0 2 3 3|, &lt;0 5 20 7 4 6|]
Mapping: {{mapping| 5 0 -20 14 41 -21 | 0 1 4 0 -3 5 }}


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


Badness: 0.0293
{{Optimal ET sequence|legend=0| 5, 45f, 50 }}


== 17-limit ==
Badness (Sintel): 2.02
Commas: 78/77, 81/80, 99/98, 144/143, 189/187


POTE generator: ~13/12 = 139.362
== Subgroup extensions ==
=== Stützel (2.3.5.19) ===
[[Subgroup]]: 2.3.5.19


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


Map: [&lt;1 1 0 2 3 3 2|, &lt;0 5 20 7 4 6 18|]
{{Mapping|legend=2| 1 0 -4 9 | 0 1 4 -3 }}


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


Badness: 0.0209
: mapping generators: ~2, ~3


== 19-limit ==
[[Optimal tuning]]s:
Commas: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143
* [[WE]]: ~2 = 1199.5513{{c}}, ~3/2 = 697.6058{{c}}
: [[error map]]: {{val| -0.448 -4.798 +4.110 +6.977 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.8222{{c}}
: error map: {{val| 0.000 -4.133 +4.975 +9.020 }}


POTE generator: ~13/12 = 139.313
{{Optimal ET sequence|legend=1| 5, 7, 12, 31, 43, 98h }}


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


Map: [&lt;1 1 0 2 3 3 2 1|, &lt;0 5 20 7 4 6 18 28|]
=== Hypnotone ===
Hypnotone is no-sevens [[#Flattone|flattone]].


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


Badness: 0.0182
[[Comma list]]: 45/44, 81/80


= Meanmag =
{{Mapping|legend=2| 1 0 -4 -6 | 0 1 4 6 }}
Commas: 81/80, 3125/3072


POTE generator: ~8/7 = 238.396
{{Mapping|legend=3| 1 0 -4 0 -6 | 0 1 4 0 6 }}


Mapping generator: ~7
: mapping generators: ~2, ~3


Map: [&lt;19 30 44 0|, &lt;0 0 0 1|]
[[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 }}


Wedgie: &lt;&lt;0 0 19 0 30 44||
{{Optimal ET sequence|legend=1| 7, 12, 19, 26, 45 }}


{{EDOs|legend=1| 19, 38, 57, 76, 95bc }}
[[Badness]] (Sintel): 0.326


Badness: 0.0770
==== 2.3.5.11.13 subgroup ====
Subgroup: 2.3.5.11.13


= Undevigintone =
Comma list: 45/44, 65/64, 81/80
Commas: 49/48, 81/80, 126/125


POTE generator: ~11/8 = 538.047
Subgroup-val mapping: {{mapping| 1 0 -4 -6 10 | 0 1 4 6 -4 }}


Mapping generator: ~11
Gencom mapping: {{mapping| 1 0 -4 0 -6 10 | 0 1 4 0 6 -4 }}


Map: [&lt;19 30 44 53 0|, &lt;0 0 0 0 1|]
Optimal tunings:  
* WE: ~2 = 1202.6916{{c}}, ~3/2 = 694.4181{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0870{{c}}


{{EDOs|legend=1| 19, 38d }}
{{Optimal ET sequence|legend=0| 7, 12, 19, 26, 45f }}


Badness: 0.0364
Badness (Sintel): 0.561


== 13-limit ==
=== Dequarter ===
Commas: 49/48, 65/64, 81/80, 126/125
[[Subgroup]]: 2.3.5.11


POTE generator: ~11/8 = 537.061
[[Comma list]]: 33/32, 55/54


Map: [&lt;19 30 44 53 0 70|, &lt;0 0 0 0 1 0|]
{{Mapping|legend=2| 1 0 -4 5 | 0 1 4 -1 }}


{{EDOs|legend=1| 19, 38d }}
{{Mapping|legend=3| 1 0 -4 0 5 | 0 1 4 0 -1 }}


Badness: 0.0229
: mapping generators: ~2, ~3


[[Category:Theory]]
[[Optimal tuning]]s:
[[Category:Temperament family]]
* [[WE]]: ~2 = 1206.5832{{c}}, ~3/2 = 695.8763{{c}}
[[Category:Meantone]]
: [[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 }}
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 0 -4 5 21 | 0 1 4 -1 -11 }}
 
Gencom mapping: {{mapping| 1 0 -4 0 5 21 | 0 1 4 0 -1 -11 }}
 
Optimal tunings:
* WE: ~2 = 1207.8248{{c}}, ~3/2 = 694.7806{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 690.1826{{c}}
 
{{Optimal ET sequence|legend=0| 7, 19eff, 26eff, 33ceeff, 40ceeff }}
 
Badness (Sintel): 1.40
 
== References ==
<references/>
 
[[Category:Temperament families]]
[[Category:Meantone family| ]] <!-- main article -->
[[Category:Meantone| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Listen]]
[[Category:Listen]]
{{todo|review|improve readability}}

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

Deutsch
Main article: Meantone § Septimal meantone
English Wikipedia has an article on:

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
Retrieved from "https://en.xen.wiki/index.php?title=Meantone_family&oldid=233821"