Meantone family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Xenwolf (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, Interface administrators, Sysops
8,705 edits
→Injera: added sub-heading for music
Hkm (talk | contribs)
autopatrolled
977 edits
 
(338 intermediate revisions by 28 users not shown)
Line 1: Line 1:
<span style="display: block; text-align: right;">
{{interwiki
__FORCETOC__
| de = Mitteltönig
[[de:mitteltönig]]</span>
| en = Meantone family
| es =
| ja =  
}}
{{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]].


The [[5-limit|5-limit]] parent [[Comma|comma]] of the [[Meantone|meantone]] family is the Didymus or [http://en.wikipedia.org/wiki/Syntonic_comma syntonic comma], 81/80. This is the one they all temper out. The [[Monzos_and_Interval_Space|monzo]] for 81/80 goes |-4 4 -1&gt;, and that can be flipped around to the corresponding [[Wedgies_and_Multivals|wedgie]], &lt;&lt;1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.
== Meantone ==
{{Main| Meantone }}


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


Mapping generator: ~3
[[Subgroup]]: 2.3.5


[[Tuning_Ranges_of_Regular_Temperaments|valid range]]: [685.714, 720.000] (7 to 5)
[[Comma list]]: 81/80


nice range: [694.786, 701.955] (1/3 comma to Pythagorean)
{{Mapping|legend=1| 1 0 -4 | 0 1 4 }}


strict range: [694.786, 701.955]
: mapping generators: ~2, ~3


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


EDOs: [[5edo|5]], [[7edo|7]], [[12edo|12]], [[19edo|19]], [[24edo|24]], [[26edo|26]], [[31edo|31]], [[36edo|36]], [[38edo|38]], [[43edo|43]], [[45edo|45]], [[50edo|50]], [[55edo|55]], [[57edo|57]], [[62edo|62]], [[67edo|67]], [[69edo|69]], [[74edo|74]], [[76edo|76]], [[81edo|81]], [[86edo|86]], [[88edo|88]], [[93edo|93]], [[98edo|98]], [[100edo|100]], [[105edo|105]], [[117edo|117]], [[129edo|129]], [[212edo|212b]]
[[Minimax tuning]]:
* [[5-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma)
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


[[Badness|Badness]]: 0.00736
[[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.)


==Seven limit children==
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b }}
The [[7-limit|7-limit]] children of 81/80 are septimal meantone, with normal comma list [|-4 4 -1&gt;, |-13 10 0 -1&gt;], flattone, with normal list [|-4 4 -1&gt;, |-17 9 0 1&gt;], dominant, with normal list [|-4 4 -1&gt;, |6 -2 0 -1&gt;], sharptone, with normal list [|-4 4 -1&gt;, |2 -3 0 1&gt;], injera, with normal list [|-4 4 -1&gt;, |-7 8 0 -2&gt;], mohajira, with normal list [|-4 4 -1&gt;, |-23 11 0 2&gt;], godzilla, with normal list [|-4 4 -1&gt;, |-4 -1 0 2&gt;], mothra, with normal list [|-4 4 -1&gt;, |-10 1 0 3&gt;], squares, with normal list [|-4 4 -1&gt;, |-3 9 0 -4&gt;], and liese, with normal list [|-4 4 -1&gt;, |-9 11 0 -3&gt;].


=Septimal meantone=
[[Badness]] (Sintel): 0.173
<span style="display: block; text-align: right;">[[:de:septimal-mitteltönig|Deutsch]]</span>


[https://en.wikipedia.org/wiki/Septimal_meantone_temperament Wikipedia article]
=== 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.


The comma |-13 10 0 -1&gt; for septimal meantone tells us that the interval class for 7 is 10 generator steps up. Hence, the [[7/4|7/4]] of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, and [[7/5|7/5]], C-F#, the tritone. The [[Wedgies_and_Multivals|wedgie]] for septimal meantone is &lt;&lt;1 4 10 4 13 12||, again telling us how to get to 5 and 7 in terms of generator steps. The temperament, aside from what is on the normal list, tempers out 126/125 and 225/224, and [[31edo|31edo]] is a good tuning for it.
==== Strong extensions ====
For any meantone generator tuning between 7\12 and 11\19, the augmented sixth is sharper than the diminished seventh and flatter than the minor seventh, befitting an approximation to interval class of 7. This coincides with interpreting the tritone (~9/8)<sup>3</sup> as [[7/5]], leading to septimal meantone, a very elegant extension to the 7-limit.  


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


7 and [[9-limit|9-limit]] minimax
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.


[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |-3 0 5/2 0&gt;]
==== 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.


[[Eigenmonzo|Eigenmonzo]]s: 2, 5
==== 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.


[[Tuning_Ranges_of_Regular_Temperaments|valid range]]: [694.737, 700.000] (19 to 12)
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]]).


nice range: [694.786, 701.955]
==== 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).


strict range: [694.786, 700.000]
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]]


[[POTE_tuning|POTE generator]]: 696.495
The rest are considered below.


Mapping generator: ~3
== Septimal meantone ==
<div style="float:right">[[:de:septimal-mitteltönig|Deutsch]]</div>
{{Main| Meantone #Septimal meantone}}
{{Wikipedia| Septimal meantone temperament }}


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


[[Map|Map]]: [&lt;1 0 -4 -13|, &lt;0 1 4 10|]
[[Subgroup]]: 2.3.5.7


[[generator|Generator]]s: 2, 3
[[Comma list]]: 81/80, 126/125


[[wedgie|Wedgie]]: &lt;&lt;1 4 10 4 13 12||
{{Mapping|legend=1| 1 0 -4 -13 | 0 1 4 10 }}


EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[43edo|43]], [[50edo|50]], [[62edo|62]], [[74edo|74]], [[81edo|81]], [[93edo|93]], [[105edo|105]], [[143edo|143b]]
[[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 }}


[[Badness|Badness]]: 0.0137
[[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


==Bimeantone==
[[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.)


11/8 is mapped to half octave minus the meantone diesis.
[[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.


Commas: 81/80, 126/125, 245/242
{{Optimal ET sequence|legend=1| 12, 19, 31, 81, 112b, 143b }}


[[POTE_tuning|POTE generator]]: ~3/2 = 696.016
[[Badness]] (Sintel): 0.347


Map: [&lt;2 0 -8 -26 -31|, &lt;0 1 4 10 12|]
=== Undecimal meantone (huygens) ===
{{Redirect|Huygens|the Dutch mathematician, physicist and astronomer|Wikipedia: Christiaan Huygens}}
{{See also| Huygens vs meanpop }}


EDOs: 12, 38d, 50
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.


Badness: 0.0381
Subgroup: 2.3.5.7.11


===13-limit===
Comma list: 81/80, 99/98, 126/125
Commas: 81/80, 105/104, 126/125, 245/242


[[POTE_tuning|POTE generator]]: ~3/2 = 695.836
Mapping: {{mapping| 1 0 -4 -13 -25 | 0 1 4 10 18 }}


Map: [&lt;2 0 -8 -26 -31 -40|, &lt;0 1 4 10 12 15|]
Optimal tunings:  
* WE: ~2 = 1200.7636{{c}}, ~3/2 = 697.4122{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.0315{{c}}


EDOs: 12f, 50
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


Badness: 0.0288
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.)


==Unidecimal meantone aka Huygens==
Algebraic generator: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
See also [[Meantone_vs_meanpop|Meantone vs meanpop]]


[[Comma|Comma]]s: 81/80, 126/125, 99/98
{{Optimal ET sequence|legend=0| 12, 19e, 31, 105, 136b }}


[[11-limit|11-limit]] minimax
Badness (Sintel): 0.563


[|1 0 0 0 0&gt;, |25/16 -1/8 0 0 1/16&gt;, |9/4 -1/2 0 0 1/4&gt;,
; Music
|21/8 -5/4 0 0 5/8&gt;, |25/8 -9/4 0 0 9/8&gt;]
* [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]


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


valid range: [696.774, 700.000] (31 to 12)
Subgroup: 2.3.5.7.11.13


nice range: [691.202, 701.955]
Comma list: 81/80, 99/98, 126/125, 144/143


strict range: [696.774, 700.000]
Mapping: {{mapping| 1 0 -4 -13 -25 29 | 0 1 4 10 18 -16 }}


[[POTE_tuning|POTE generator]]: 696.967
Optimal tunings:  
* WE: ~2 = 1199.9389{{c}}, ~3/2 = 697.2282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.2627{{c}}


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


[[Algebraic_generator|Algebraic generator]]: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents.
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|Map]]: [&lt;1 0 -4 -13 -25|, &lt;0 1 4 10 18|]
{{Optimal ET sequence|legend=0| 12, 31, 43, 74 }}


[[generator|Generator]]s: 2, 3
Badness (Sintel): 1.07


EDOs: [[12edo|12]], [[31edo|31]], [[43edo|43]], [[62edo|62]], [[74edo|74]], [[105edo|105]], [[198edo|198be]]
===== 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.


[[Badness|Badness]]: 0.0170
Subgroup: 2.3.5.7.11.13.17


[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: 81/80, 99/98, 120/119, 126/125, 144/143


===Tridecimal meantone===
Mapping: {{mapping| 1 0 -4 -13 -25 29 12 | 0 1 4 10 18 -16 -5 }}
[[Comma|Comma]]s: 66/65, 81/80, 99/98, 105/104


valid range: 697.674 (43)
Optimal tunings:  
* WE: ~2 = 1199.5811{{c}}, ~3/2 = 697.0918{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3303{{c}}


nice range: [691.202, 701.955]
{{Optimal ET sequence|legend=0| 12, 31, 43, 74g }}


strict range: 697.674
Badness (Sintel): 1.06


[[POTE_tuning|POTE generator]]: ~3/2 = 696.642
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Mapping generator: ~3
Comma list: 81/80, 96/95, 99/98, 120/119, 126/125, 144/143


Map: [&lt;1 0 -4 -13 -25 -20|, &lt;0 1 4 10 18 15|]
Mapping: {{mapping| 1 0 -4 -13 -25 29 12 9 | 0 1 4 10 18 -16 -5 -3 }}


EDOs: [[31edo|31]]
Optimal tunings:  
* WE: ~2 = 1199.2931{{c}}, ~3/2 = 696.9690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3736{{c}}


[[Badness|Badness]]: 0.0180
{{Optimal ET sequence|legend=0| 12, 31, 43, 74gh }}


===Grosstone===
Badness (Sintel): 1.07
Commas: 81/80, 99/98, 126/125, 144/143


POTE generator: ~3/2 = 697.264
==== 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.  


Mapping generator: ~3
This extension used to be known as ''tridecimal meantone'', but was decanonicalized in 2025.


Map: [&lt;1 0 -4 -13 -25 29|, &lt;0 1 4 10 18 -16|]
Subgroup: 2.3.5.7.11.13


EDOs: [[12edo|12]], [[31edo|31]], [[43edo|43]], [[74edo|74]], [[105edo|105]]
Comma list: 66/65, 81/80, 99/98, 105/104


Badness: 0.0259
Mapping: {{mapping| 1 0 -4 -13 -25 -20 | 0 1 4 10 18 15 }}


===Meridetone===
Optimal tunings:
Commas: 78/77, 81/80, 99/98, 126/125
* WE: ~2 = 1200.8149{{c}}, ~3/2 = 697.1155{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.7085{{c}}


POTE generator: ~3/2 = 697.529
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


Mapping generator: ~3
{{Optimal ET sequence|legend=0| 12f, 19e, 31 }}


Map: [&lt;1 0 -4 -13 -25 -39|, &lt;0 1 4 10 18 27|]
Badness (Sintel): 0.746


EDOs: 43, 117df, 160bdf, 203bcdef
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0264
Comma list: 66/65, 81/80, 99/98, 105/104, 120/119


===Hemimeantone===
Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 | 0 1 4 10 18 15 -5 }}
Commas: 81/80, 99/98, 126/125, 169/168


POTE generator: ~52/45 = 250.304
Optimal tunings:  
* WE: ~2 = 1199.5548{{c}}, ~3/2 = 696.7449{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.9823{{c}}


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


Map: [&lt;1 0 -4 -13 -25 -5|, &lt;0 2 8 20 36 11|]
Badness (Sintel): 1.02


EDOs: 43, 62, 167bef, 229bef
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.0314
Comma list: 66/65, 81/80, 96/95, 99/98, 105/104, 120/119


==Meanpop==
Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 9 | 0 1 4 10 18 15 -5 -3 }}
See also [[Meantone_vs_meanpop|Meantone vs meanpop]]


[[Comma|Comma]]s: 81/80, 126/125, 385/384
Optimal tunings:  
* WE: ~2 = 1199.0408{{c}}, ~3/2 = 696.5824{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.1061{{c}}


[[11-limit|11-limit]] [[minimax|minimax]] 1/4 comma
{{Optimal ET sequence|legend=0| 12f, 31 }}


[|1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;,
Badness (Sintel): 1.10
|-3 0 5/2 0 0&gt;, |11 0 -13/4 0 0&gt;]


[[Eigenmonzo|Eigenmonzo]]s: 2, 5
==== 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.


valid range: [694.737, 696.774] (19 to 31)
Subgroup: 2.3.5.7.11.13


nice range: [691.202, 701.955]
Comma list: 78/77, 81/80, 99/98, 126/125


strict range: [694.737, 696.774]
Mapping: {{mapping| 1 0 -4 -13 -25 -39 | 0 1 4 10 18 27 }}


[[POTE_tuning|POTE generator]]: 696.434
Optimal tunings:  
* WE: ~2 = 1199.9122{{c}}, ~3/2 = 697.4779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.5241{{c}}


Mapping generator: ~3
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


[[Algebraic_generator|Algebraic generator]]: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
{{Optimal ET sequence|legend=0| 12f, 31f, 43 }}


[http://soonlabel.com/xenharmonic/archives/607 Scott Joplin's "The Entertainer" tuned into meanpop]
Badness (Sintel): 1.09


Map: [&lt;1 0 -4 -13 24|, &lt;0 1 4 10 -13|]
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


[[generator|Generator]]s: 2, 3
Comma list: 78/77, 81/80, 99/98, 120/119, 126/125


EDOs: [[19edo|19]], [[31edo|31]], [[50edo|50]], [[62edo|62]], [[81edo|81]]
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 | 0 1 4 10 18 27 -5 }}


[[Badness|Badness]]: 0.0215
Optimal tunings:  
* WE: ~2 = 1199.3793{{c}}, ~3/2 = 697.2833{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6222{{c}}


[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]
{{Optimal ET sequence|legend=0| 12f, 43 }}


===13-limit Meanpop===
Badness (Sintel): 1.22
[[Comma|Comma]]s: 81/80, 105/104, 144/143, 196/195


valid range: [694.737, 696.774] (19 to 31)
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


nice range: [691.202, 701.955]
Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125


strict range: [694.737, 696.774]
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 9 | 0 1 4 10 18 27 -5 -3 }}


POTE generator: ~3/2 = 696.211
Optimal tunings:  
* WE: ~2 = 1199.0260{{c}}, ~3/2 = 697.1486{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6887{{c}}


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


Map: [&lt;1 0 -4 -13 24 -20|, &lt;0 1 4 10 -13 15|]
Badness (Sintel): 1.25


EDOS: [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]], [[131edo|131bd]], [[212edo|212bdf]]
==== Hemimeantone ====
Subgroup: 2.3.5.7.11.13


[[Badness|Badness]]: 0.0209
Comma list: 81/80, 99/98, 126/125, 169/168


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


POTE generator: ~3/2 = 696.202
: mapping generators: ~2, ~26/15


Mapping generator: ~3
Optimal tunings:  
* WE: ~2 = 1201.0387{{c}}, ~26/15 = 949.2863{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5065{{c}}


Map: [&lt;1 0 -4 -13 24 10|, &lt;0 1 4 10 -13 -4|]
{{Optimal ET sequence|legend=0| 19e, 43, 62 }}


EDOs: 12e, 19, 31f, 50f
Badness (Sintel): 1.30


Badness: 0.0277
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


==Meanenneadecal==
Comma list: 81/80, 99/98, 126/125, 169/168, 221/220
[[Comma|Comma]]s: 45/44, 56/55, 81/80


[[POTE_tuning|POTE generator]]: ~3/2 = 696.250
Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 | 0 2 8 20 36 11 33 }}


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


Map: [&lt;1 0 -4 -13 -6|, &lt;0 1 4 10 6|]
{{Optimal ET sequence|legend=0| 19eg, 43, 62 }}


EDOs: [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31e]], [[50edo|50e]]
Badness (Sintel): 1.19


[[Badness|Badness]]: 0.0214
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


===13-limit===
Comma list: 81/80, 99/98, 126/125, 153/152, 169/168, 221/220
[[Comma|Comma]]s: 45/44, 56/55, 78/77, 81/80


[[POTE_tuning|POTE generator]]: ~3/2 = 696.146
Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 -25 | 0 2 8 20 36 11 33 37 }}


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


Map: [&lt;1 0 -4 -13 -6 -20|, &lt;0 1 4 10 6 15|]
{{Optimal ET sequence|legend=0| 19egh, 43, 62 }}


EDOs: [[19edo|19]], [[31edo|31e]], [[50edo|50e]]]
Badness (Sintel): 1.15


[[Badness|Badness]]: 0.0212
==== Semimeantone ====
Subgroup: 2.3.5.7.11.13


===Vincenzo===
Comma list: 81/80, 99/98, 126/125, 847/845
Commas: 81/80 126/125 45/44 65/64 256/255 153/152 23/22


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


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


Map: [&lt;1 0 -4 -13 ... |, &lt;0 1 4 10 6 -4 -5 -3 -6|]
Optimal tunings:  
* WE: ~55/39 = 600.3606{{c}}, ~3/2 = 697.4241{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 697.0545{{c}}


EDOs: 12
{{Optimal ET sequence|legend=0| 12f, …, 50eff, 62, 136b }}


Badness:
Badness (Sintel): 1.68


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


POTE generator: ~3/2 = 694.689
Comma list: 81/80, 99/98, 126/125, 221/220, 289/288


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


Map: [&lt;1 0 -4 -13 5|, &lt;0 1 4 10 -1|]
Optimal tunings:  
* WE: ~17/12 = 600.5426{{c}}, ~3/2 = 697.5571{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9858{{c}}


EDOs: 12e, 19e
{{Optimal ET sequence|legend=0| 12f, 50eff, 62, 136bg }}


Badness: 0.0315
Badness (Sintel): 1.60


===13-limit===
===== 19-limit =====
Commas: 33/32, 55/54, 77/75, 729/728
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~3/2 = 694.764
Comma list: 81/80, 99/98, 126/125, 153/152, 209/208, 221/220


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


Map: [&lt;1 0 -4 -13 5 10|, &lt;0 1 4 10 -1 -4|]
Optimal tunings:  
* WE: ~17/12 = 600.5959{{c}}, ~3/2 = 697.5985{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9638{{c}}


EDOs: 12e, 19e
{{Optimal ET sequence|legend=0| 12f, 50eff, 62 }}


Badness: 0.0263
Badness (Sintel): 1.47


==Meanundec==
=== Meanpop ===
Commas: 27/26, 40/39, 45/44, 56/55
{{See also| Huygens vs meanpop }}


POTE generator: ~3/2 = 697.254
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.  


Mapping generator: ~3
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 -4 -13 -6 -1|, &lt;0 1 4 10 6 3|]
Comma list: 81/80, 126/125, 385/384


EDOS: 12f, 19f, 31ef
Mapping: {{mapping| 1 0 -4 -13 24 | 0 1 4 10 -13 }}


Badness: 0.0242
: mapping generator: ~2, ~3


=Flattone=
Optimal tunings:
[[Comma|Comma]]s: 81/80, 525/512
* WE: ~2 = 1201.3464{{c}}, ~3/2 = 697.2159{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4509{{c}}


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


[[7-limit|7-limit]] minimax
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.)


[|1 0 0 0&gt;, |21/13 0 1/13 -1/13&gt;,
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.
|32/13 0 4/13 -4/13&gt;, |32/13 0 -9/13 9/13&gt;]


[[Eigenmonzo|Eigenmonzo]]s: 2, 7/5
{{Optimal ET sequence|legend=0| 12e, 19, 31, 81, 112b }}


[[9-limit|9-limit]] minimax
Badness (Sintel): 0.712


[|1 0 0 0&gt;, |17/11 2/11 0 -1/11&gt;,
; Music
|24/11 8/11 0 -4/11&gt;, |34/11 -18/11 0 9/11&gt;]
* [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]


[[Eigenmonzo|Eigenmonzo]]s: 2, 9/7
==== Tridecimal meanpop ====
Subgroup: 2.3.5.7.11.13


valid range: [692.308, 694.737] (26 to 19)
Comma list: 81/80, 105/104, 126/125, 144/143


nice range: [692.353, 701.955]
Mapping: {{mapping| 1 0 -4 -13 24 -20 | 0 1 4 10 -13 15 }}


strict range: [692.353, 694.737]
Optimal tunings:  
* WE: ~2 = 1201.0765{{c}}, ~3/2 = 696.8361{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2347{{c}}


[[POTE_tuning|POTE generator]]: 693.779
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


Mapping generator: ~3
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.)


Algebraic generator: Squarto, the positive root of 8x^2-4x-9, at 506.3239 cents, equal to (1+sqrt(19))/4.
{{Optimal ET sequence|legend=0| 19, 31, 50, 81 }}


Map: [&lt;1 0 -4 17|, &lt;0 1 4 -9|]
Badness (Sintel): 0.863


[[wedgie|Wedgie]]: &lt;&lt;1 4 -9 4 -17 -32||
===== Meanpoppic =====
Subgroup: 2.3.5.7.11.13.17


[[generator|Generator]]s: 2, 3
Comma list: 81/80, 105/104, 126/125, 144/143, 273/272


EDOs: [[7edo|7]], [[19edo|19]], [[45edo|45]], [[64edo|64]]
Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 | 0 1 4 10 -13 15 26 }}


[[Badness|Badness]]: 0.0386
Optimal tunings:  
* WE: ~2 = 1201.0727{{c}}, ~3/2 = 696.8168{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2195{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 19g, 31, 50, 81, 131bd }}
Commas: 45/44, 81/80, 385/384


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


nice range: [682.502, 701.955]
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19


strict range: [692.308, 694.737]
Comma list: 81/80, 105/104, 126/125, 144/143, 153/152, 273/272


POTE generator: ~3/2 = 693.126
Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 -40 | 0 1 4 10 -13 15 26 28 }}


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


Map: [&lt;1 0 -4 17 -6|, &lt;0 1 4 -9 6|]
{{Optimal ET sequence|legend=0| 19gh, 31, 50, 81 }}


EDOs: 7, 19, 26, 45, 71bc, 116bcde
Badness (Sintel): 1.08


Badness: 0.0338
===== Meanpoid =====
Subgroup: 2.3.5.7.11.13.17


==13-limit==
Comma list: 81/80, 105/104, 120/119, 126/125, 144/143
45/44, 65/64, 78/77, 81/80


valid range: [692.308, 694.737] (26 to 19)
Mapping: {{mapping| 1 0 -4 -13 24 -20 12 | 0 1 4 10 -13 15 -5 }}


nice range: [682.502, 701.955]
Optimal tunings:  
* WE: ~2 = 1200.2768{{c}}, ~3/2 = 696.5683{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4114{{c}}


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


POTE generator: ~3/2 = 693.058
Badness (Sintel): 1.17


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


Map: [&lt;1 0 -4 17 -6 10|, &lt;0 1 4 -9 6 -4|]
Comma list: 77/76, 81/80, 96/95, 105/104, 120/119, 126/125


EDOs: 7, 19, 26, 45f, 71bcf, 116bcdef
Mapping: {{mapping| 1 0 -4 -13 24 -20 12 9 | 0 1 4 10 -13 15 -5 -3 }}


Badness: 0.0223
Optimal tunings:  
* WE: ~2 = 1199.7905{{c}}, ~3/2 = 696.3779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4973{{c}}


=Dominant=
{{Optimal ET sequence|legend=0| 19, 31 }}
[[Comma|Comma]]s: 36/35, 64/63


The wedgie for dominant is &lt;&lt;1 4 -2 4 -6 -16||. Now the interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is [[12edo|12edo]], but it also works well with the Pythagorean tuning of pure [[3/2|3/2]] fifths, and with [[29edo|29edo]], [[41edo|41edo]], or [[53edo|53edo]].
Badness (Sintel): 1.25


valid range: [700.000, 720.000] (12 to 5)
==== Semimeanpop ====
Subgroup: 2.3.5.7.11.13


nice range: [694.786, 715.587]
Comma list: 81/80, 126/125, 385/384, 847/845


strict range: [700.000, 715.587]
Mapping: {{mapping| 2 0 -8 -26 48 39 | 0 1 4 10 -13 -10 }}


[[POTE_tuning|POTE generator]]: 701.573
: mapping generators: ~55/39, ~3


Mapping generator: ~3
Optimal tunings:  
* WE: ~55/39 = 600.6704{{c}}, ~3/2 = 697.2151{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.4341{{c}}


Map: [&lt;1 0 -4 6|, &lt;0 1 4 -2|]
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112b }}


[[wedgie|Wedgie]]: &lt;&lt;1 4 -2 4 -6 -16||
Badness (Sintel): 1.78


EDOs: [[5edo|5]], [[7edo|7]], [[12edo|12]], [[53edo|53]], [[65edo|65]]
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


[[Badness|Badness]]: 0.0207
Comma list: 81/80, 126/125, 221/220, 273/272, 289/288


==11-limit==
Mapping: {{mapping| 2 0 -8 -26 48 39 5 | 0 1 4 10 -13 -10 1 }}
Commas: 36/35, 64/63, 56/55


valid range: [700.000, 705.882] (12 to 17)
Optimal tunings:  
* WE: ~17/12 = 600.7232{{c}}, ~3/2 = 697.2820{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4411{{c}}


nice range: [691.202, 715.587]
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bg }}


strict range: [700.000, 705.882]
Badness (Sintel): 1.45


POTE generator: ~3/2 = 703.254
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


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


Map: [&lt;1 0 -4 6 13|, &lt;0 1 4 -2 -6|]
Mapping: {{mapping| 2 0 -8 -26 48 39 5 -1 | 0 1 4 10 -13 -10 1 3 }}


EDOs: 5, 12, 17c, 29cde
Optimal tunings:  
* WE: ~17/12 = 600.7527{{c}}, ~3/2 = 697.3244{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4525{{c}}


Badness: 0.0242
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bgh }}


==13-limit==
Badness (Sintel): 1.28
Commas: 36/35, 56/55, 64/63, 66/65


valid range: 705.882 (17)
=== 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.


nice range: [691.202, 715.587]
Subgroup: 2.3.5.7.11


strict range:705.882
Comma list: 45/44, 56/55, 81/80


POTE generator: ~3/2 = 703.636
Mapping: {{mapping| 1 0 -4 -13 -6 | 0 1 4 10 6 }}


Map: [&lt;1 0 -4 6 13 18|, &lt;0 1 4 -2 -6 -9|]
Optimal tunings:  
* WE: ~2 = 1199.6946{{c}}, ~3/2 = 696.0729{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2083{{c}}


EDOs: 12f, 17c, 29cdef
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]


Badness: 0.0241
{{Optimal ET sequence|legend=0| 7d, 12, 19, 31e }}


==Dominion==
Badness (Sintel): 0.708
Commas: 26/25, 36/35, 56/55, 64/63


POTE generator: ~3/2 = 704.905
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 -4 6 13 -9|, &lt;0 1 4 -2 -6 8|]
Comma list: 45/44, 56/55, 78/77, 81/80


EDOs: 5, 12, 17c, 46cde
Mapping: {{mapping| 1 0 -4 -13 -6 -20 | 0 1 4 10 6 15 }}


Badness: 0.0273
Optimal tunings:  
* WE: ~2 = 1199.7931{{c}}, ~3/2 = 696.0258{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1241{{c}}


==Domineering==
{{Optimal ET sequence|legend=0| 7df, 12f, 19, 31e }}
Commas: 36/35, 45/44, 64/63


POTE generator: ~3/2 = 698.776
Badness (Sintel): 0.875


Mapping generator: ~3
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 0 -4 6 -6|, &lt;0 1 4 -2 6|]
Comma list: 45/44, 56/55, 78/77, 81/80, 120/119


EDOs: 7, 12, 43de
Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 | 0 1 4 10 6 15 -5 }}


Badness: 0.0220
Optimal tunings:  
* WE: ~2 = 1198.6665{{c}}, ~3/2 = 695.8010{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4998{{c}}


==Domination==
{{Optimal ET sequence|legend=0| 12f, 19, 31e }}
Commas: 36/35, 64/63, 77/75


POTE generator: ~3/2 = 705.004
Badness (Sintel): 1.17


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


Map: [&lt;1 0 -4 6 -14|, &lt;0 1 4 -2 11|]
Comma list: 45/44, 56/55, 78/77, 81/80, 96/95, 120/119


EDOs: 17c, 46cd
Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 9 | 0 1 4 10 6 15 -5 -3 }}


Badness: 0.0366
Optimal tunings:  
* WE: ~2 = 1198.2880{{c}}, ~3/2 = 695.7123{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.6370{{c}}


===13-limit===
{{Optimal ET sequence|legend=0| 12f, 19, 31e }}
Commas: 26/25, 36/35, 64/63, 66/65


POTE generator: ~3/2 = 705.496
Badness (Sintel): 1.23


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


Map: [&lt;1 0 -4 6 -14 -9|, &lt;0 1 4 -2 11 8|]
Comma list: 45/44, 56/55, 65/64, 81/80


EDOs: 17c
Mapping: {{mapping| 1 0 -4 -13 -6 10 | 0 1 4 10 6 -4 }}


Badness: 0.0274
Optimal tunings:  
* WE: ~2 = 1202.1684{{c}}, ~3/2 = 696.3160{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.2045{{c}}


==Twelve==
{{Optimal ET sequence|legend=0| 7d, 12, 19 }}
Commas: 81/80 64/63 45/44 65/64 256/255 153/152


POTE generator: ~3/2 = 696.217
Badness (Sintel): 1.02


Mapping generator: ~3
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 0 -4 6 -6 10 12 9|, &lt;0 1 4 -2 6 -4 -5 -3|]
Comma list: 45/44, 52/51, 56/55, 65/64, 81/80


EDOs: 7, 12, 19d, 31def
Mapping: {{mapping| 1 0 -4 -13 -6 10 12 | 0 1 4 10 6 -4 -5 }}


Badness: 0.0204
Optimal tunings:  
* WE: ~2 = 1200.5137{{c}}, ~3/2 = 696.1561{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.8771{{c}}


==Arnold==
{{Optimal ET sequence|legend=0| 12, 19 }}
Commas: 22/21, 33/32, 36/35


POTE generator: ~3/2 = 698.491
Badness (Sintel): 1.30


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


Map: [&lt;1 0 -4 6 5|, &lt;0 1 4 -2 -1|]
Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80


EDOs: 5, 7, 12e
Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 | 0 1 4 10 6 -4 -5 -3 }}


Badness: 0.0261
Optimal tunings:  
* WE: ~2 = 1199.8261{{c}}, ~3/2 = 696.0298{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1262{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 12, 19 }}
Commas: 22/21, 27/26, 33/32, 40/39


POTE generator: ~3/2 = 696.743
Badness (Sintel): 1.36


Mapping generator: ~3
=== Bimeantone ===
11/8 is mapped to half octave minus the [[128/125|meantone diesis]].


Map: [&lt;1 0 -4 6 5 -1|, &lt;0 1 4 -2 -1 3|]
Subgroup: 2.3.5.7.11


EDOs: 5, 7, 12ef, 19def, 31def
Comma list: 81/80, 126/125, 245/242


Badness: 0.0233
Mapping: {{mapping| 2 0 -8 -26 -31 | 0 1 4 10 12 }}


==Dominatrix==
: mapping generators: ~63/44, ~3
Commas: 27/26 36/35 45/44 64/63


POTE generator: ~3/2 = 698.544
Optimal tunings:  
* WE: ~63/44 = 600.7492{{c}}, ~3/2 = 696.8853{{c}}
* CWE: ~63/44 = 600.0000{{c}}, ~3/2 = 696.1908{{c}}


Mapping generator: ~3
{{Optimal ET sequence|legend=0| 12, 26de, 38d, 50 }}


Map: [&lt;1 0 -4 6 -6 -1|, &lt;0 1 4 -2 6 3|]
Badness (Sintel): 1.26


EDOs: 7, 12f
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0183
Comma list: 81/80, 105/104, 126/125, 245/242


=Sharptone=
Mapping: {{mapping| 2 0 -8 -26 -31 -40 | 0 1 4 10 12 15 }}
[[Comma|Comma]]s: 21/20, 28/27


Sharptone, with a wedgie &lt;&lt;1 4 3 4 2 -4||, is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo|12edo]] tuning does sharptone about as well as such a thing can be done, of course not in its patent val.
Optimal tunings:
* WE: ~55/39 = 600.8309{{c}}, ~3/2 = 696.8000{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.0066{{c}}


[[POTE_tuning|POTE generator]]: 700.140
{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}


Mapping generator: ~3
Badness (Sintel): 1.19


Map: [&lt;1 0 -4 -2|, &lt;0 1 4 3|]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


[[wedgie|Wedgie]]: &lt;&lt;1 4 3 4 2 -4||
Comma list: 81/80, 105/104, 126/125, 189/187, 221/220


EDOs: [[5edo|5]], [[12edo|12]]
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 | 0 1 4 10 12 15 1 }}


[[Badness|Badness]]: 0.0248
Optimal tunings:  
* WE: ~17/12 = 600.9234{{c}}, ~3/2 = 696.8536{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.9317{{c}}


=Meansept=
{{Optimal ET sequence|legend=0| 12f, 38df, 50 }}
Commas: 15/14, 81/80


POTE generator: ~3/2 = 682.895
Badness (Sintel): 1.15


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


Map: [&lt;1 0 -4 -5|, &lt;0 1 4 5|]
Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220


Wedgie: &lt;&lt;1 4 5 4 5 0||
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 -1 | 0 1 4 10 12 15 1 3 }}


EDOs: 7
Optimal tunings:  
* WE: ~17/12 = 600.9845{{c}}, ~3/2 = 696.8939{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.8947{{c}}


Badness: 0.0453
{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}


==11-limit==
Badness (Sintel): 1.08
Commas: 15/14, 22/21, 125/121


POTE generator: ~3/2 = 685.234
=== Trimean ===
{{See also| No-sevens subgroup temperaments #Superpine }}


Mapping generator: ~3
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 -4 -5 -6|, &lt;0 1 4 5 6|]
Comma list: 81/80, 126/125, 1344/1331


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


Badness: 0.0325
: mapping generators: ~2, ~11/10


=Supermean=
Optimal tunings:
Commas: 81/80, 672/625
* WE: ~2 = 1200.7155{{c}}, ~11/10 = 167.9055{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7749{{c}}


POTE generator: ~3/2 = 704.889
{{Optimal ET sequence|legend=0| 7d, 36d, 43, 50, 93 }}


Map: [&lt;1 0 -4 -21|, &lt;0 1 4 15|]
Badness (Sintel): 1.68


EDOs: 17c, 46c
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.1342
Comma list: 81/80, 126/125, 144/143, 364/363


==11-limit==
Mapping: {{mapping| 1 2 4 7 5 3 | 0 -3 -12 -30 -11 5 }}
Commas: 56/55, 81/80, 132/125


POTE generator: ~3/2 = 705.096
Optimal tunings:
* WE: ~2 = 1200.6104{{c}}, ~11/10 = 167.8749{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7728{{c}}


Map: [&lt;1 0 -4 -21 -14|, &lt;0 1 4 15 11|]
{{Optimal ET sequence|legend=0| 7d, 43, 50, 93 }}


EDOs: 17c, 46c
Badness (Sintel): 1.46


Badness: 0.0633
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


==13-limit==
Comma list: 81/80, 126/125, 144/143, 189/187, 221/220
Commas: 26/25, 56/55, 66/65, 81/80


POTE generator: ~3/2 = 705.094
Mapping: {{mapping| 1 2 4 7 5 3 8 | 0 -3 -12 -30 -11 5 -28 }}


Map: [&lt;1 0 -4 -21 -14 -9|, &lt;0 1 4 15 11 8|]
Optimal tunings:  
* WE: ~2 = 1200.6144{{c}}, ~11/10 = 167.8716{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7682{{c}}


EDOs: 17c, 46c
{{Optimal ET sequence|legend=0| 7dg, 43, 50, 93 }}


=Injera=
Badness (Sintel): 1.28
[[Comma|Comma]]s: 50/49, 81/80


The wedgie for injera is &lt;&lt;2 8 8 8 7 -4||, which tells us it has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. [[38edo|38edo]], which is two parallel [[19edo|19edo]]s, is an excellent tuning for injera.
=== Migration ===
See [[Rastmic clan #Migration|Rastmic clan]].


[http://tech.groups.yahoo.com/group/tuning-math/message/3091 Origin of the name]
== Flattone ==
{{Main| Flattone }}


valid range: [685.714, 700.000] (14c to 12)
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]].


nice range: [688.957, 701.955]
[[Subgroup]]: 2.3.5.7


strict range: [688.957, 700.000]
[[Comma list]]: 81/80, 525/512


[[POTE_tuning|POTE generator]]: 694.375
{{Mapping|legend=1| 1 0 -4 17 | 0 1 4 -9 }}


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


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


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


EDOs: [[12edo|12]], [[26edo|26]], [[38edo|38]], [[102edo|102bcd]], [[140edo|140bcd]], [[178edo|178bcd]]
[[Algebraic generator]]: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.


[[Badness|Badness]]: 0.0311
{{Optimal ET sequence|legend=1| 7, 19, 26, 45 }}


== Music ==
[[Badness]] (Sintel): 0.976
* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 Two Pairs of Socks] (in [[26edo]]) by [[Igliashon Jones]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20Injera%20Jam.mp3 Injera Jam] (in [[26edo]]) by [[Zach Curley]]


==11-limit==
=== 11-limit ===
Commas: 45/44, 50/49, 81/80
This can also be considered a no-sevens temperament: [[#Hypnotone|hypnotone]].


valid range: [685.714, 700.000] (14c to 12)
Subgroup: 2.3.5.7.11


nice range: [682.458, 701.955]
Comma list: 45/44, 81/80, 385/384


strict range: [685.714, 700.000]
Mapping: {{mapping| 1 0 -4 17 -6 | 0 1 4 -9 6 }}


POTE generator: ~3/2 = 692.840
Optimal tuning:  
* WE: ~2 = 1202.3247{{c}}, ~3/2 = 694.4688{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.1467{{c}}


Mapping generator: ~3
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]


Map: [&lt;2 0 -8 -7 -12|, &lt;0 1 4 4 6|]
{{Optimal ET sequence|legend=0| 7, 19, 26, 45, 71bc, 116bcde }}


EDOs: 12, 14c, 26. 90bce, 116bce
Badness (Sintel): 1.12


Badness: 0.0231
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


==13-limit==
Comma list: 45/44, 65/64, 78/77, 81/80
Commas: 45/44, 50/49, 81/80, 78/77


valid range: 692.308 (26)
Mapping: {{mapping| 1 0 -4 17 -6 10 | 0 1 4 -9 6 -4 }}


nice range: [682.458, 701.955]
Optimal tunings:  
* WE: ~2 = 1202.5156{{c}}, ~3/2 = 694.5107{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0538{{c}}


strict range: 692.308 (26)
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]


POTE generator: ~3/2 = 692.673
{{Optimal ET sequence|legend=0| 7, 19, 26, 45f, 71bcf, 116bcdef }}


Mapping generator: ~3
Badness (Sintel): 0.920


Map: [&lt;2 0 -8 -7 -12 -21|, &lt;0 1 4 4 6 9|]
=== Ptolemy ===
See [[Rastmic clan #Ptolemy|Rastmic clan]].


EDOs: 26, 104bcf
== Dominant ==
{{Main| Dominant (temperament) }}
{{See also| Archytas clan }}


Badness: 0.0216
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]].


==Enjera==
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.
Commas: 27/26, 40/39, 45/44, 99/98


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


Mapping generator: ~3
[[Comma list]]: 36/35, 64/63


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


EDOs: 12f, 26f, 38ef
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1195.3384{{c}}, ~3/2 = 698.8478{{c}}
: [[error map]]: {{val| -4.662 -7.769 +9.077 +14.832 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.1125{{c}}
: error map: {{val| 0.000 -0.842 +18.136 +28.949 }}


Badness: 0.0265
[[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]


==Injerous==
{{Optimal ET sequence|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}
Commas: 33/32, 50/49, 55/54


POTE generator: ~3/2 = 690.548
[[Badness]] (Sintel): 0.524


Mapping generator: ~3
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;2 0 -8 -7 10|, &lt;0 1 4 4 -1|]
Comma list: 36/35, 56/55, 64/63


EDOs: 12e, 14c, 26e, 40ce
Mapping: {{mapping| 1 0 -4 6 13 | 0 1 4 -2 -6 }}


Badness: 0.0386
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]


==Lahoh==
Optimal tunings:
Commas: 50/49, 56/55, 81/77
* WE: ~2 = 1194.0169{{c}}, ~3/2 = 699.7473{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.2672{{c}}


POTE generator: ~3/2 = 699.001
{{Optimal ET sequence|legend=0| 5, 12, 17c, 29cde }}


Mapping generator: ~3
Badness (Sintel): 0.799


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


EDOs: 12
Comma list: 36/35, 56/55, 64/63, 66/65


Badness: 0.0431
Mapping: {{mapping| 1 0 -4 6 13 18 | 0 1 4 -2 -6 -9 }}


=Godzilla=
Optimal tunings:
<span style="display: block; text-align: right;">[[:de:Semiphor,_Semaphor,_Godzilla|Deutsch]]</span>
* WE: ~2 = 1193.8055{{c}}, ~3/2 = 700.0042{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.8254{{c}}


Main article: [[Semaphore_and_Godzilla|Semaphore and Godzilla]]
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]


[[Comma|Comma]]s: 49/48, 81/80
{{Optimal ET sequence|legend=0| 12f, 17c, 29cdef }}


Godzilla has wedgie &lt;&lt;2 8 1 8 -4 -20||, and tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-quarter intervals these represent give a fourth, and so step-and-a-quarter generators generate godzilla. [[19edo|19edo]] is the perfect godzilla tuning, so much so that's there's not much point in looking elsewhere. Hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes.
Badness (Sintel): 0.996


valid range: [240.000, 257.143] (5 to 14c)
==== Dominion ====
Subgroup: 2.3.5.7.11.13


nice range: [231.174, 266.871]
Comma list: 26/25, 36/35, 56/55, 64/63


strict range: [240.000, 257.143]
Mapping: {{mapping| 1 0 -4 6 13 -9 | 0 1 4 -2 -6 8 }}


[[POTE_tuning|POTE generator]]: ~8/7 = 252.635
Optimal tunings:  
* WE: ~2 = 1195.0293{{c}}, ~3/2 = 701.9847{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7698{{c}}


Mapping generator: ~7/4
{{Optimal ET sequence|legend=0| 5, 12, 17c }}


Map: [&lt;1 0 -4 2|, &lt;0 2 8 1|]
Badness (Sintel): 1.13


[[wedgie|Wedgie]]: &lt;&lt;2 8 1 8 -4 -20||
=== Domination ===
Subgroup: 2.3.5.7.11


EDOs: [[5edo|5]], [[9edo|9c]], [[14edo|14c]], [[19edo|19]], [[62edo|62d]], [[81edo|81d]], 143bd
Comma list: 36/35, 64/63, 77/75


[[Badness|Badness]]: 0.0267
Mapping: {{mapping| 1 0 -4 6 -14 | 0 1 4 -2 11 }}


==11-limit==
Optimal tunings:
Commas: 45/44, 49/48, 81/80
* WE: ~2 = 1194.8645{{c}}, ~3/2 = 701.9872{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5945{{c}}


valid range: [252.632, 257.143] (19 to 14c)
{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}


nice range: [231.174, 266.871]
Badness (Sintel): 1.21


strict range: [252.632, 257.143]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


POTE generator: ~8/7 = 254.027
Comma list: 26/25, 36/35, 64/63, 66/65


Mapping generator: ~7/4
Mapping: {{mapping| 1 0 -4 6 -14 -9 | 0 1 4 -2 11 8 }}


Map: [&lt;1 0 -4 2 -6|, &lt;0 2 8 1 12|]
Optimal tunings:  
* WE: ~2 = 1195.1324{{c}}, ~3/2 = 702.6343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.0791{{c}}


EDOs: 14c, 19, 33cd, 52cd
{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}


Badness: 0.0290
Badness (Sintel): 1.13


==13-limit==
=== Domineering ===
Commas: 45/44, 49/48, 78/77, 81/80
Subgroup: 2.3.5.7.11


valid range: 694.737 (19)
Comma list: 36/35, 45/44, 64/63


nice range: [621.581, 737.652]
Mapping: {{mapping| 1 0 -4 6 -6 | 0 1 4 -2 6 }}


strict range: 694.737
Optimal tunings:  
* WE: ~2 = 1194.7102{{c}}, ~3/2 = 695.6962{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1765{{c}}


POTE generator: ~8/7 = 253.603
{{Optimal ET sequence|legend=0| 5e, 7, 12 }}


Mapping generator: ~7/4
Badness (Sintel): 0.727


Map: [&lt;1 0 -4 2 -6 -5|, &lt;0 2 8 1 12 11|]
=== Arnold ===
Subgroup: 2.3.5.7.11


EDOs: 14cf, 19, 33cdf, 52cdf
Comma list: 22/21, 33/32, 36/35


Badness: 0.0225
Mapping: {{mapping| 1 0 -4 6 5 | 0 1 4 -2 -1 }}


==Semafour==
Optimal tunings:
Commas: 33/32, 49/48, 55/54
* WE: ~2 = 1199.8507{{c}}, ~3/2 = 698.4045{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.4822{{c}}


POTE generator: ~8/7 = 254.042
{{Optimal ET sequence|legend=0| 5, 7, 12e }}


Mapping generator: ~7/4
Badness (Sintel): 0.864


Map: [&lt;1 0 -4 2 5|, &lt;0 2 8 1 -2|]
=== Neutrominant ===
See [[Rastmic clan #Neutrominant|Rastmic clan]].


EDOs: 5, 14c, 19e, 33cde
== 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.


Badness: 0.0285
Flattertone was named by [[Flora Canou]] in 2024.  


==Varan==
[[Subgroup]]: 2.3.5.7
Commas: 49/48, 77/75, 81/80


POTE generator: ~8/7 = 251.079
[[Comma list]]: 81/80, 1875/1792


Mapping generator: ~7/4
{{Mapping|legend=1| 1 0 -4 -24 | 0 1 4 17 }}


Map: [&lt;1 0 -4 2 -10|, &lt;0 2 8 1 17|]
: mapping generators: ~2, ~3


EDOs: 19e, 24, 43de
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1204.4511{{c}}, ~3/2 = 694.3258{{c}}
: [[error map]]: {{val| +4.451 -3.178 -9.011 +3.554 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 692.0479{{c}}
: error map: {{val| 0.000 -9.907 -18.122 -4.012 }}


Badness: 0.0396
{{Optimal ET sequence|legend=1| 7d, 19d, 26, 59bcd, 85bccd }}


===13-limit===
[[Badness]] (Sintel): 2.43
Commas: 49/48, 66/65, 77/75, 81/80


POTE generator: ~8/7 = 251.165
==== 11-limit ====
Subgroup: 2.3.5.7.11


Mapping generator: ~7/4
Comma list: 45/44, 81/80, 1375/1344


Map: [&lt;1 0 -4 2 -10 -5|, &lt;0 2 8 1 17 11|]
Mapping: {{mapping| 1 0 -4 -24 -6 | 0 1 4 17 6 }}


EDOs: 19e, 24, 43de
Optimal tunings:  
* WE: ~2 = 1203.4653{{c}}, ~3/2 = 693.8144{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 692.0422{{c}}


Badness: 0.0257
{{Optimal ET sequence|legend=0| 7d, 19d, 26 }}


==Baragon==
Badness (Sintel): 1.53
Commas: 49/48, 56/55, 81/80


POTE generator: ~8/7 = 251.173
; Music
* [https://youtu.be/scCuGXnj5IY ''Music in 33EDO (33-Tone Equal Temperament) - Feb 2024''] by [[Budjarn Lambeth]] (2024)


Mapping generator: ~7/4
== 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 2 9|, &lt;0 2 8 1 -7|]
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: 19, 24, 43d
The 11-limit extension was named by Gene Ward Smith in 2004<ref name="meantone & meanpop 2004"/>.


Badness: 0.0357
[[Subgroup]]: 2.3.5.7


==Music==
[[Comma list]]: 21/20, 28/27
[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Bobro/GodzillaExample.mp3 Godzilla Example] by [[Cameron_Bobro|Cameron Bobro]]


[http://tinyurl.com/4uyumk9 "Change is on the Wind"] in Godzilla[9] by [[Igliashon_Jones|Igliashon Jones]]
{{Mapping|legend=1| 1 0 -4 -2 | 0 1 4 3 }}


=Mohajira=
[[Optimal tuning]]s:
<span style="display: block; text-align: right;">[[:de:Mohajira|Deutsch]]</span>
* [[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 }}


[[Comma|Comma]]s: 81/80, 6144/6125
{{Optimal ET sequence|legend=1| 5, 7d, 12d }}


Mohajira, with wedgie &lt;&lt;2 8 -11 8 -23 -48||, really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. [[31edo|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.
[[Badness]] (Sintel): 0.629


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.
=== Meanertone ===
Subgroup: 2.3.5.7.11


[[7-limit|7]] and [[9-limit|9-limit]] minimax 1/4 comma
Comma list: 21/20, 28/27, 33/32


[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |6 0 -11/8 0&gt;]
Mapping: {{mapping| 1 0 -4 -2 5 | 0 1 4 3 -1 }}


[[Eigenmonzo|Eigenmonzo]]s: 2, 5
Optimal tunings:  
* WE: ~2 = 1208.5304{{c}}, ~3/2 = 701.5669{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1117{{c}}


[[POTE_tuning|POTE generator]]: ~128/105 = 348.415
{{Optimal ET sequence|legend=0| 5, 7d, 12de }}


Mapping generator: ~128/105
Badness (Sintel): 0.832


Algebraic generator: Mohabis, real root of 3x^3-3x^2-1, 348.6067 cents. Corresponding recurrence converges quickly.
== 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;1 1 0 6|, &lt;0 2 8 -11|]
Mildtone was named by [[User: Lucius Chiaraviglio|Lucius Chiaraviglio]] in 2024.


[[generator|Generator]]s: 2, 128/105
[[Subgroup]]: 2.3.5.7


[[wedgie|Wedgie]]: &lt;&lt;2 8 -11 8 -23 -48||
[[Comma list]]: 81/80, 16128/15625


EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]]
{{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 }}


[[Badness|Badness]]: 0.0557
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7304{{c}}, ~3/2 = 698.3953{{c}}
: [[error map]]: {{val| -0.270 -3.829 +7.267 -1.434 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.5397{{c}}
: error map: {{val| 0.000 -3.415 +7.845 -0.952 }}


==11-limit==
{{Optimal ET sequence|legend=1| 12, 43d, 55, 67 }}
[[Comma|Comma]]s: 81/80, 121/120, 176/175


[[11-limit|11-limit]] minimax 1/4 comma
[[Badness]] (Sintel): 2.67


[|1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;,
=== 11-limit ===
|6 0 -11/8 0 0&gt;, |2 0 5/8 0 0&gt;]


[[Eigenmonzo|Eigenmonzo]]s: 2, 5
[[Subgroup]]: 2.3.5.7.11


[[POTE_tuning|POTE generator]]: ~11/9 = 348.477
[[Comma list]]: 81/80, 176/175, 7056/6875


Mapping generator: ~11/9
{{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 30}}


Map: [&lt;1 1 0 6 2|, &lt;0 2 8 -11 5|]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.816{{c}}, ~3/2 = 698.355{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.455{{c}}


[[generator|Generator]]s: 2, 11/9
{{Optimal ET sequence|legend=1| 12, 43de, 55, 67 }}


EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]]
[[Badness]] (Sintel): 2.15


[[Badness|Badness]]: 0.0261
=== 13-limit ===


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


POTE generator: ~11/9 = 348.558
[[Comma list]]: 81/80, 176/175, 196/195, 832/825


Mapping generator: ~11/9
{{Mapping|legend=1| 1 0 -4 -32 -44 | 0 1 4 22 30}}


Map: [&lt;1 1 0 6 2 4|, &lt;0 2 8 -11 5 -1|]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.788{{c}}, ~3/2 = 698.355{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.471{{c}}


EDOs: 7, 24, 31, 117ef, 148bef
{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}


Badness: 0.0234
[[Badness]] (Sintel): 2.04


=Ptolemy=
=== 17-limit ===
Commas: 81/80, 121/120, 525/512


POTE generator: ~11/9 = 346.922
[[Subgroup]]: 2.3.5.7.11.13.17


Map: [&lt;1 1 0 8 2|, &lt;0 2 8 -18 5|]
[[Comma list]]: 81/80, 176/175, 189/187, 196/195, 832/825


EDOs: 7, 38d, 45e, 83bcde
{{Mapping|legend=1| 1 0 -4 -32 -44 12| 0 1 4 22 30 -5}}


Badness: 0.0588
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.655{{c}}, ~3/2 = 698.295{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.488{{c}}


==13-limit==
{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}
Commas: 65/64, 81/80, 105/104, 121/120


POTE generator: ~11/9 = 346.910
[[Badness]] (Sintel): 1.98


Map: [&lt;1 1 0 8 2 6|, &lt;0 2 8 -18 5 -8|]
=== 19-limit ===


EDOs: 7, 38df, 45ef, 83bcdef
[[Subgroup]]: 2.3.5.7.11.13.17.19


Badness: 0.0343
[[Comma list]]: 81/80, 96/95, 176/175, 189/187, 196/195, 832/825


=Maqamic=
{{Mapping|legend=1| 1 0 -4 -32 -44 12 9| 0 1 4 22 30 -5 -3}}
<span style="display: block; text-align: right;">[[:de:maqamisch|Deutsch]]</span>


Main article: [[maqamic|Maqamic]]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.371{{c}}, ~3/2 = 698.164{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.519{{c}}


[[Comma|Comma]]s: 81/80, 36/35, 121/120
{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}


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.
[[Badness]] (Sintel): 1.95


[[POTE_tuning|POTE generator]]: ~11/9 = 350.934
{{Todo|unify precision|review}}


Mapping generator: ~11/9
== 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]].


Map: [&lt;1 1 0 4 2|, &lt;0 2 8 -4 5|]
[[Subgroup]]: 2.3.5.7


[[generator|Generator]]s: 2, 11/9
[[Comma list]]: 81/80, 672/625


EDOs: [[7edo|7]], [[10edo|10c]], [[17edo|17c]], [[24edo|24d]], [[31edo|31d]]
{{Mapping|legend=1| 1 0 -4 -21 | 0 1 4 15 }}


==13-limit==
[[Optimal tuning]]s:
[[Comma|Comma]]s: 81/80, 36/35, 121/120, 144/143
* [[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 }}


[[POTE_tuning|POTE generator]]: ~11/9 = 350.816
{{Optimal ET sequence|legend=1| 5d, 12d, 17c }}


Mapping generator: ~11/9
[[Badness]] (Sintel): 3.40


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


Generators: 2, 11/9
Comma list: 56/55, 81/80, 132/125


EDOs: [[7edo|7]], [[10edo|10c]], [[17edo|17c]], [[24edo|24d]],[[31edo| 31d]]
Mapping: {{mapping| 1 0 -4 -21 -14 | 0 1 4 15 11 }}


=Migration=
Optimal tunings:
Commas: 81/80, 121/120, 126/125
* WE: ~2 = 1195.7270{{c}}, ~3/2 = 702.5848{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7471{{c}}


POTE generator: ~11/9 = 348.182
{{Optimal ET sequence|legend=0| 5de, 12de, 17c }}


Mapping generator: ~11/9
Badness (Sintel): 2.09


Map: [&lt;1 1 0 -3 2|, &lt;0 2 8 20 5|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 31, 100de, 131bde, 162bde
Comma list: 26/25, 56/55, 66/65, 81/80


Badness: 0.0255
Mapping: {{mapping| 1 0 -4 -21 -14 -9 | 0 1 4 15 11 8 }}


=Mohamaq=
Optimal tunings:
Commas: 81/80, 392/375
* WE: ~2 = 1196.3958{{c}}, ~3/2 = 702.9766{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7940{{c}}


POTE generator: ~25/21 = 350.586
{{Optimal ET sequence|legend=0| 5de, 12de, 17c, 29c }}


Mapping generator: ~25/21
Badness (Sintel): 1.67


Map: [&lt;1 1 0 -1|, &lt;0 2 8 13|]
== Mohajira ==
{{Main| Mohajira }}


EDOs: 17c, 24, 65c, 89cd
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.


Badness: 0.0777
[[Subgroup]]: 2.3.5.7


==11-limit==
[[Comma list]]: 81/80, 6144/6125
Commas: 56/55, 77/75, 243/242


POTE generator: ~11/9 = 350.565
{{Mapping|legend=1| 1 1 0 6 | 0 2 8 -11 }}


Mapping generator: ~11/9
: mapping generators: ~2, ~128/105


Map: [&lt;1 1 0 -1 2|, &lt;0 2 8 13 5|]
[[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 }}


EDOs: 17c, 24, 65c, 89cd
[[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


Badness: 0.0362
[[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]


==13-limit==
[[Algebraic generator]]: Mohabis, real root of 3''x''<sup>3</sup> - 3''x''<sup>2</sup> - 1, 348.6067 cents. Corresponding recurrence converges quickly.
Commas: 56/55, 66/65, 77/75, 243/242


POTE generator: ~11/9 = 350.745
{{Optimal ET sequence|legend=1| 7, 24, 31 }}


Mapping generator: ~11/9
[[Badness]] (Sintel): 1.41


Map: [&lt;1 1 0 -1 2 4|, &lt;0 2 8 13 5 -1|]
Scales: [[mohaha7]], [[mohaha10]]


EDOs: 17c, 24, 41c, 65c
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0287
Comma list: 81/80, 121/120, 176/175


=Orphic=
Mapping: {{mapping| 1 1 0 6 2 | 0 2 8 -11 5 }}
Commas: 81/80, 5898240/5764801


POTE generator: ~7/6 = 275.794
Optimal tunings:  
* WE: ~2 = 1201.1562{{c}}, ~11/9 = 348.8124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.4910{{c}}


Mapping generator: ~343/288
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;2 1 -4 4|, &lt;0 4 16 3|]
Tuning ranges:
* 11-odd-limit diamond monotone: ~11/9 = [348.387, 350.000] (9\31 to 7\24)
* 11-odd-limit diamond tradeoff: ~11/9 = [344.999, 350.978]


Wedgie: &lt;&lt;8 32 6 32 -13 -76||
{{Optimal ET sequence|legend=0| 7, 24, 31 }}


EDOs: 26, 74, 174bd, 248bd
Badness (Sintel): 0.862


Badness: 0.2588
Scales: [[mohaha7]], [[mohaha10]]


==11-limit==
=== 13-limit ===
Commas: 81/80, 99/98, 73728/73205
Subgroup: 2.3.5.7.11.13


POTE generator: ~7/6 = 275.762
Comma list: 66/65, 81/80, 105/104, 121/120


Mapping generator: ~77/64
Mapping: {{mapping| 1 1 0 6 2 4 | 0 2 8 -11 5 -1 }}


Map: [&lt;2 1 -4 4 8|, &lt;0 4 16 3 -2|]
Optimal tunings:  
* WE: ~2 = 1200.4256{{c}}, ~11/9 = 348.6819{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.5622{{c}}


EDOs: 26, 48c, 74, 248bd, 322bd
{{Optimal ET sequence|legend=0| 7, 24, 31 }}


Badness: 0.1015
Badness (Sintel): 0.966


==13-limit==
Scales: [[mohaha7]], [[mohaha10]]
Commas: 81/80, 99/98, 144/143, 2200/2197


POTE generator: ~7/6 = 275.774
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Mapping generator: ~63/52
Comma list: 66/65, 81/80, 105/104, 121/120, 154/153


Map: [&lt;2 1 -4 4 8 2|, &lt;0 4 16 3 -2 10|]
Mapping: {{mapping| 1 1 0 6 2 4 7 | 0 2 8 -11 5 -1 -10 }}


EDOs: 26, 48c, 74, 174bd, 248bd, 322bd
Optimal tunings:  
* WE: ~2 = 1200.0382{{c}}, ~11/9 = 348.7471{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.7360{{c}}


Badness: 0.0535
{{Optimal ET sequence|legend=0| 7, 24, 31 }}


=Mothra=
Badness (Sintel): 1.05
[[Comma|Comma]]s: 81/80, 1029/1024


Mothra, with wedgie &lt;&lt;3 12 -1 12 -10 -36||, splits the fifth into three 8/7 generators. It uses 1029/1024, the gamelisma, to accomplish this deed and also tempers out 1728/1715, the orwell comma. Using [[31edo|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|slendric]].
Scales: [[mohaha7]], [[mohaha10]]


Note that mothra can also be called cynder in the 7-limit, which can be a little confusing sometimes.
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


[[7-limit|7]] and [[9-limit|9-limit]] minimax 1/4 comma
Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152


[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |3 0 -1/12 0&gt;]
Mapping: {{mapping| 1 1 0 6 2 4 7 6 | 0 2 8 -11 5 -1 -10 -6 }}


[[Eigenmonzo|Eigenmonzo]]s: 2, 5
Optimal tunings:  
* WE: ~2 = 1199.7469{{c}}, ~11/9 = 348.7367{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.8117{{c}}


[[POTE_tuning|POTE generator]]: ~8/7 = 232.193
{{Optimal ET sequence|legend=0| 7, 24, 31, 55 }}


Mapping generator: ~8/7
Badness (Sintel): 1.05


Algebraic generator: Rabrindanath, largest real root of x^8-3x^2+1, or 232.0774 cents.
Scales: [[mohaha7]], [[mohaha10]]


Map: [&lt;1 1 0 3|, &lt;0 3 12 -1|]
== 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.


[[generator|Generator]]s: 2, 8/7
[[Subgroup]]: 2.3.5.7


[[wedgie|Wedgie]]: &lt;&lt;3 12 -1 12 -10 -36||
[[Comma list]]: 81/80, 392/375


EDOs: [[5edo|5]], [[26edo|26]], [[31edo|31]]
{{Mapping|legend=1| 1 1 0 -1 | 0 2 8 13 }}


[[Badness|Badness]]: 0.0371
: mapping generators: ~2, ~25/21


==11-limit==
[[Optimal tuning]]s:
[[Comma|Comma]]s: 81/80, 99/98, 385/384
* [[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 }}


POTE generator: ~8/7 = 232.031
{{Optimal ET sequence|legend=1| 7d, 17c, 24 }}


Mapping generator: ~8/7
[[Badness]] (Sintel): 1.97


Map: [&lt;1 1 0 3 5|, &lt;0 3 12 -1 -8|]
Scales: [[mohaha7]], [[mohaha10]]


EDOs: [[5edo|5]], [[26edo|26]], [[31edo|31]], [[88edo|88]], [[150edo|150]], [[181edo|181]]
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Badness|Badness]]: 0.0256
Comma list: 56/55, 77/75, 243/242


==13-limit==
Mapping: {{mapping| 1 1 0 -1 2 | 0 2 8 13 5 }}
Commas: 81/80, 99/98, 105/104, 144/143


POTE generator: ~8/7 = 231.811
Optimal tunings:  
* WE: ~2 = 1199.1924{{c}}, ~11/9 = 350.3286{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.4821{{c}}


Mapping generator: ~8/7
{{Optimal ET sequence|legend=0| 7d, 17c, 24 }}


Map: [&lt;1 1 0 3 5 1|, &lt;0 3 12 -1 -8 14|]
Badness (Sintel): 1.20


EDOs: 5, 26, 31, 57, 88
Scales: [[mohaha7]], [[mohaha10]]


Badness: 0.0240
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==Cynder==
Comma list: 56/55, 66/65, 77/75, 243/242
Commas: 45/44, 81/80, 1029/1024


POTE generator: ~8/7 = 231.317
Mapping: {{mapping| 1 1 0 -1 2 4 | 0 2 8 13 5 -1 }}


Mapping generator: ~8/7
Optimal tunings:  
* WE: ~2 = 1198.5986{{c}}, ~11/9 = 350.3353{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.6459{{c}}


Map: [&lt;1 1 0 3 0|, &lt;0 3 12 -1 18|]
{{Optimal ET sequence|legend=0| 7d, 17c, 24, 41c }}


EDOs: 26, 57e, 83bce
Badness (Sintel): 1.19


Badness: 0.0557
Scales: [[mohaha7]], [[mohaha10]]


===13-limit===
== Liese ==
Commas: 45/44, 78/77, 81/80, 640/637
<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>
 
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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 81/80, 686/675
 
{{Mapping|legend=1| 1 0 -4 -3 | 0 3 12 11 }}
 
: mapping generators: ~2, ~10/7
 
[[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 }}
 
[[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
 
[[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.
 
{{Optimal ET sequence|legend=1| 17c, 19, 55, 74d }}
 
[[Badness]] (Sintel): 1.18
 
=== Liesel ===
Subgroup: 2.3.5.7.11
 
Comma list: 56/55, 81/80, 540/539
 
Mapping: {{mapping| 1 0 -4 -3 4 | 0 3 12 11 -1 }}
 
Optimal tunings:
* WE: ~2 = 1198.8507{{c}}, ~10/7 = 632.4668{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 632.9963{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 0 -4 -3 4 0 | 0 3 12 11 -1 7 }}
 
Optimal tunings:
* WE: ~2 = 1199.4968{{c}}, ~10/7 = 632.7766{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.0082{{c}}
 
{{Optimal ET sequence|legend=0| 17c, 19, 36 }}
 
Badness (Sintel): 1.13
 
=== Elisa ===
Subgroup: 2.3.5.7.11
 
Comma list: 77/75, 81/80, 99/98
 
Mapping: {{mapping| 1 0 -4 -3 -5 | 0 3 12 11 16 }}
 
Optimal tunings:
* WE: ~2 = 1201.0489{{c}}, ~10/7 = 633.6147{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1644{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 0 -4 -3 -5 0 | 0 3 12 11 16 7 }}
 
Optimal tunings:
* WE: ~2 = 1201.4815{{c}}, ~10/7 = 633.7720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1281{{c}}
 
{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}
 
Badness (Sintel): 1.11
 
=== Lisa ===
Subgroup: 2.3.5.7.11
 
Comma list: 45/44, 81/80, 343/330
 
Mapping: {{mapping| 1 0 -4 -3 -6 | 0 3 12 11 18 }}
 
Optimal tunings:
* WE: ~2 = 1202.6773{{c}}, ~10/7 = 632.7783{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.6175{{c}}
 
{{Optimal ET sequence|legend=0| 17cee, 19 }}
 
Badness (Sintel): 1.81
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 45/44, 81/80, 91/88, 147/143
 
Mapping: {{mapping| 1 0 -4 -3 -6 0 | 0 3 12 11 18 7 }}
 
Optimal tunings:
* WE: ~2 = 1203.6086{{c}}, ~10/7 = 633.1193{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.5346{{c}}
 
{{Optimal ET sequence|legend=0| 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 {{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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 81/80, 1119744/1071875
 
{{Mapping|legend=1| 1 2 4 1 | 0 -3 -12 13 }}
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 2 4 1 5 | 0 -3 -12 13 -11 }}
 
Optimal tunings:
* WE: ~2 = 1199.0522{{c}}, ~11/10 = 167.1904{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.3382{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 2 4 1 5 3 | 0 -3 -12 13 -11 5 }}
 
Optimal tunings:
* WE: ~2 = 1199.4286{{c}}, ~11/10 = 167.3105{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.3958{{c}}
 
{{Optimal ET sequence|legend=0| 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|legend=1| 3 0 -12 -20 | 0 1 4 6 }}
 
: mapping generators: ~56/45, ~3
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 12, 33cd, 45, 57 }}
 
[[Badness]] (Sintel): 1.75
 
== Squares ==
{{Main| Squares }}
 
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]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 81/80, 2401/2400
 
{{Mapping|legend=1| 1 -1 -8 -3 | 0 4 16 9 }}
 
: mapping generators: ~2, ~14/9
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2488{{c}}, ~14/9 = 774.8640{{c}}
: [[error map]]: {{val| +1.249 -3.748 +1.520 +1.204 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 774.1560{{c}}
: error map: {{val| 0.000 -5.331 +0.183 -1.422 }}
 
[[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
 
[[Algebraic generator]]: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 -1 -8 -3 -3 | 0 4 16 9 10 }}
 
Optimal tunings:
* WE: ~2 = 1201.6657{{c}}, ~11/7 = 775.1171{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.1754{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -1 -8 -3 -3 5 | 0 4 16 9 10 -2 }}
 
Optimal tunings:
* WE: ~2 = 1199.8419{{c}}, ~11/7 = 774.3484{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4422{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -1 -8 -3 -3 -6 | 0 4 16 9 10 15 }}
 
Optimal tunings:
* WE: ~2 = 1202.0312{{c}}, ~11/7 = 775.5589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4140{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -1 -8 -3 -3 -15 | 0 4 16 9 10 29 }}
 
Optimal tunings:
* WE: ~2 = 1202.3228{{c}}, ~11/7 = 775.2214{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8617{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -1 -8 -3 -3 -15 -3 | 0 4 16 9 10 29 11 }}
 
Optimal tunings:
* WE: ~2 = 1201.4340{{c}}, ~11/7 = 774.7375{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8955{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -1 -8 -3 -3 -15 -3 -8 | 0 4 16 9 10 29 11 19 }}


POTE generator: ~8/7 = 231.293
Optimal tunings:  
* WE: ~2 = 1201.2461{{c}}, ~11/7 = 774.5783{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8479{{c}}


Mapping generator: ~8/7
{{Optimal ET sequence|legend=0| 14cf, 31 }}


Map: [&lt;1 1 0 3 0 1|, &lt;0 3 12 -1 18 14|]
Badness (Sintel): 1.15


EDOs: 26, 57e, 83bce
=== Cuboctahedra ===
Subgroup: 2.3.5.7.11


Badness: 0.0341
Comma list: 81/80, 385/384, 1375/1372


==Mosura==
Mapping: {{mapping| 1 -1 -8 -3 17 | 0 4 16 9 -21 }}
Commas: 81/80, 176/175, 1029/1024


POTE generator: ~8/7 = 232.419
Optimal tunings:  
* WE: ~2 = 1201.4436{{c}}, ~14/9 = 774.9386{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 774.0243{{c}}


Mapping generator: ~8/7
{{Optimal ET sequence|legend=0| 31, 107b, 138b, 169be, 200be }}


Map: [&lt;1 1 0 3 -1|, &lt;0 3 12 -1 23|]
Badness (Sintel): 1.88


EDOs: 31, 129, 136b, 148be, 160be, 191bce, 222bce, 253bce
== Jerome ==
Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5<sup>1/20</sup>, or 139.316 cents. It may be described as {{nowrap| 17c & 26 }}; its ploidacot is pentacot. While the generator represents both 13/12 and 12/11, the CTE/CWE and Hieronymus generators are close to 13/12 in size.


Badness: 0.0313
[[Subgroup]]: 2.3.5.7


===13-limit===
[[Comma list]]: 81/80, 17280/16807
Commas: 81/80, 144/143, 176/175, 1029/1024


POTE generator: ~8/7 = 232.640
{{Mapping|legend=1| 1 1 0 2 | 0 5 20 7 }}


Mapping generator: ~8/7
: mapping generators: ~2, ~54/49


Map: [&lt;1 1 0 3 -1 7|, &lt;0 3 12 -1 23 -17|]
[[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: 31, 67, 98
{{Optimal ET sequence|legend=1| 17c, 26, 43 }}


Badness: 0.0369
[[Badness]] (Sintel): 2.75


=Squares=
=== 11-limit ===
[[Comma|Comma]]s: 81/80, 2401/2400
Subgroup: 2.3.5.7.11


Squares, with wedgie &lt;&lt;4 16 9 16 3 -24||, splits the interval of an eleventh, or 8/3, into four supermajor third ([[9/7|9/7]]) intervals, and uses it for a generator. [[31edo|31edo]], with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out 2401/2400, the breedsma, as well as 2430/2401.
Comma list: 81/80, 99/98, 864/847


7 and 9 limit minimax 1/4 comma
Mapping: {{mapping| 1 1 0 2 3 | 0 5 20 7 4 }}


[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |3/2 0 9/16 0&gt;]
Optimal tunings:
* WE: ~2 = 1201.4436{{c}}, ~12/11 = 139.3714{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 139.4038{{c}}


[[Eigenmonzo|Eigenmonzo]]s: 2, 5
{{Optimal ET sequence|legend=0| 17c, 26, 43 }}


[[POTE_tuning|POTE generator]]: ~9/7 = 425.942
Badness (Sintel): 1.58


Mapping generator: ~9/7
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Algebraic generator: Sceptre2, the positive root of 9x^2+x-16, or (sqrt(577)-1)/18, which is 425.9311 cents.
Comma list: 78/77, 81/80, 99/98, 144/143


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


[[generator|Generator]]s: 2, 9/7
Optimal tunings:  
* WE: ~2 = 1199.8860{{c}}, ~13/12 = 139.3737{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3817{{c}}


EDOs: [[14edo|14]], [[31edo|31]], [[262edo|262]], [[293edo|293]]
{{Optimal ET sequence|legend=0| 17c, 26, 43 }}


[[Badness|Badness]]: 0.0460
Badness (Sintel): 1.21


Music:
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


By [[Chris_Vaisvil|Chris Vaisvil]]
Comma list: 78/77, 81/80, 99/98, 144/143, 189/187


[http://clones.soonlabel.com/public/micro/tuning-survey/daily20100603-squares8piano.mp3 Square 8]
Mapping: {{mapping| 1 1 0 2 3 3 2 | 0 5 20 7 4 6 18 }}


==11-limit==
Optimal tunings:
Commas: 81/80, 99/98, 121/120
* WE: ~2 = 1199.8346{{c}}, ~13/12 = 139.3431{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3544{{c}}


POTE generator: ~9/7 = 425.957
{{Optimal ET sequence|legend=0| 17cg, 26, 43 }}


Mapping generator: ~9/7
Badness (Sintel): 1.06


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


EDOs: [[5edo|5]], [[8edo|8]], [[11edo|11]], [[14edo|14]], [[17edo|17]], [[31edo|31]]
Comma list: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143


[[Badness|Badness]]: 0.0216
Mapping: {{mapping| 1 1 0 2 3 3 2 1 | 0 5 20 7 4 6 18 28 }}


==13-limit==
Optimal tunings:
Commas: 81/80, 99/98, 121/120, 66/65
* WE: ~2 = 1199.8891{{c}}, ~13/12 = 139.3001{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3080{{c}}


POTE generator: ~9/7 = 425.550
{{Optimal ET sequence|legend=0| 17cgh, 26, 43, 69 }}


Mapping generator: ~9/7
Badness (Sintel): 1.11


Map: [&lt;1 3 8 6 7 3|, &lt;0 -4 -16 -9 -10 2|]
== Meantritone ==
The meantritone temperament tempers out the [[mirkwai comma]] (16875/16807) and [[trimyna comma]] (50421/50000) in the 7-limit. In this temperament, the 6th harmonic is split into five generators of ~10/7; the ploidacot of this temperament is beta-pentacot. The name ''meantritone'' is a portmanteau of ''meantone'' and ''tritone'', the latter is a generator of this temperament.


EDOs: 17c, 31, 79cf, 110cef, 141cef
[[Subgroup]]: 2.3.5.7


[[Badness|Badness]]: 0.0255
[[Comma list]]: 81/80, 16875/16807


==Agora==
{{Mapping|legend=1| 1 -1 -8 -7 | 0 5 20 19 }}
Commas: 81/80, 99/98, 105/104, 121/120


POTE generator: ~9/7 = 426.276
: mapping generators: ~2, ~10/7


Mapping generator: ~9/7
[[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 }}


Map: [&lt;1 3 8 6 7 14|, &lt;0 -4 -16 -9 -10 -29|]
{{Optimal ET sequence|legend=1| 29cd, 31, 188bcd, 219bbcd }}


EDOs: 31, 45ef, 76e
[[Badness]] (Sintel): 2.08


Badness: 0.0245
=== 11-limit ===
Subgroup: 2.3.5.7.11


=Cuboctahedra=
Comma list: 81/80, 99/98, 2541/2500


==11-limit==
Mapping: {{mapping| 1 -1 -8 -7 -11 | 0 5 20 19 28 }}
[[Comma|Comma]]s: 81/80, 385/384, 1375/1372


[[POTE_tuning|POTE generator]]: ~9/7 = 425.993
Optimal tunings:  
* WE: ~2 = 1201.2054{{c}}, ~10/7 = 619.9752{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.4223{{c}}


Mapping generator: ~9/7
{{Optimal ET sequence|legend=0| 29cde, 31 }}


Map: [&lt;1 3 8 6 -4|, &lt;0 -4 -16 -9 21|]
Badness (Sintel): 1.42


EDOs: [[14edo|14]], [[31edo|31]], [[45edo|45]], [[200edo|200]]
== 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.


[[Badness|Badness]]: 0.0568
[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name]


=Liese=
[[Subgroup]]: 2.3.5.7
<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>
 
[[Comma list]]: 50/49, 81/80
 
{{Mapping|legend=1| 2 0 -8 -7 | 0 1 4 4 }}
 
: mapping generators: ~7/5, ~3
 
[[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 }}
 
[[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|legend=1| 12, 26, 38 }}
 
[[Badness]] (Sintel): 0.788
 
; 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
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 45/44, 50/49, 81/80
 
Mapping: {{mapping| 2 0 -8 -7 -12 | 0 1 4 4 6 }}
 
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
 
Mapping: {{mapping| 2 0 -8 -7 -12 -21 | 0 1 4 4 6 9 }}
 
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}})
 
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|legend=0| 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: {{mapping| 2 0 -8 -7 -12 -21 5 | 0 1 4 4 6 9 1 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 0 -8 -7 -12 -21 5 -1 | 0 1 4 4 6 9 1 3 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 0 -8 -7 -12 -2 | 0 1 4 4 6 3 }}
 
Optimal tunings:
* WE: ~7/5 = 599.1863{{c}}, ~3/2 = 693.1791{{c}} (~21/20 = 93.9929{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 693.6809{{c}} (~21/20 = 93.6809{{c}})
 
{{Optimal ET sequence|legend=0| 10cdeef, 12f }}
 
Badness (Sintel): 1.10
 
=== Injerous ===
Subgroup: 2.3.5.7.11
 
Comma list: 33/32, 50/49, 55/54
 
Mapping: {{mapping| 2 0 -8 -7 10 | 0 1 4 4 -1 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 12e, 14c, 26e, 40cee }}
 
Badness (Sintel): 1.28
 
=== Lahoh ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 56/55, 81/77
 
Mapping: {{mapping| 2 0 -8 -7 7 | 0 1 4 4 0 }}
 
Optimal tunings:
* WE: ~7/5 = 597.3179{{c}}, ~3/2 = 695.8759{{c}} (~21/20 = 98.5581{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 697.8757{{c}} (~21/20 = 97.8757{{c}})
 
{{Optimal ET sequence|legend=0| 10cd, 12 }}
 
Badness (Sintel): 1.42
 
=== Teff ===
{{Main| 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: {{mapping| 2 1 -4 -3 8 | 0 2 8 8 -1 }}
 
: mapping generators: ~7/5, ~16/11
 
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}})
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 1 -4 -3 8 2 | 0 2 8 8 -1 5 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 1 -4 -3 8 2 6 | 0 2 8 8 -1 5 2 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 1 -4 -3 8 2 6 2 | 0 2 8 8 -1 5 2 6 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 24d, 26 }}
 
Badness (Sintel): 1.41
 
== 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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 81/80, 300125/294912
 
{{Mapping|legend=1| 2 1 -4 11 | 0 2 8 -5 }}
 
: mapping generators: ~735/512, ~35/24
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 2 1 -4 11 8 | 0 2 8 -5 -1 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 1 -4 11 8 2 | 0 2 8 -5 -1 5 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 1 -4 11 8 2 6 | 0 2 8 -5 -1 5 2 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 24, 26, 50 }}
 
Badness (Sintel): 1.08


[[Comma|Comma]]s: 81/80, 686/675
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Liese, with wedgie &lt;&lt;3 12 11 12 9 -8||, splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. [[74edo|74edo]] makes for a good liese tuning, though [[19edo|19edo]] can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55.
Comma list: 81/80, 105/104, 133/132, 144/143, 171/170, 210/209


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


[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |2/3 0 11/12 0&gt;]
Optimal tunings:
* WE: ~17/12 = 600.8048{{c}}, ~16/11 = 648.7494{{c}} (~36/35 = 47.9446{{c}})
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 647.9425{{c}} (~36/35 = 47.9425{{c}})


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


[[POTE_tuning|POTE generator]]: ~10/7 = 632.406
Badness (Sintel): 1.01


Mapping generator: ~10/7
== Orphic ==
Orphic has a semi-octave period and four generators plus a period gives the 3rd harmonic; its ploidacot is diploid alpha-tetracot.


Algebraic generator: Radix, the real root of x^5-2x^4+2x^3-2x^2+2x-2, also a root of x^6-x^5-2. The recurrence converges.
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 0 -4 -3|, &lt;0 3 12 11|]
[[Comma list]]: 81/80, 5898240/5764801


[[generator|Generator]]s: 2, 10/7
{{Mapping|legend=1| 2 1 -4 4 | 0 4 16 3 }}


EDOs: [[17edo|17]], [[19edo|19]], [[55edo|55]], [[74edo|74]]
: mapping generators: ~2401/1728, ~343/288


[[Badness|Badness]]: 0.0467
[[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 }}


==Liesel==
{{Optimal ET sequence|legend=1| 26, 48c, 74 }}
Commas: 56/55, 81/80, 540/539


POTE generator: ~10/7 = 633.073
[[Badness]] (Sintel): 6.55


Mapping generator: ~10/7
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 -4 -3 4|, &lt;0 3 12 11 -1|]
Comma list: 81/80, 99/98, 73728/73205


EDOs: 17c, 19, 36, 91ce
Mapping: {{mapping| 2 1 -4 4 8 | 0 4 16 3 -2 }}


Badness: 0.0407
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}})


==13-limit==
{{Optimal ET sequence|legend=0| 26, 48c, 74 }}
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
Badness (Sintel): 3.36


POTE generator: ~10/7 = ~13/9 = 633.042
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


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


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


EDOs: 17c, 19, 36, 91cef
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: 0.0273
{{Optimal ET sequence|legend=0| 26, 48c, 74 }}


==Elisa==
Badness (Sintel): 2.21
Commas: 77/75, 81/80, 99/98


POTE generator: ~10/7 = 633.061
== Cloudtone ==
The cloudtone temperament tempers out the [[cloudy comma]], 16807/16384 and the [[syntonic comma]], 81/80 in the 7-limit. It may be described as {{nowrap| 5 & 50 }}; its ploidacot is pentaploid monocot. It can be extended to the 11- and 13-limit by adding 385/384 and 105/104 to the comma list in this order.


Mapping generator: ~10/7
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 0 -4 -3 -5|, &lt;0 3 12 11 16|]
[[Comma list]]: 81/80, 16807/16384


EDOs: 19e, 36e
{{Mapping|legend=1| 5 0 -20 14 | 0 1 4 0 }}


Badness: 0.0416
: mapping generators: ~8/7, ~3


==Lisa==
[[Optimal tuning]]s:
Commas: 45/44, 81/80, 343/330
* [[WE]]: ~8/7 = 240.4267{{c}}, ~3/2 = 696.9566{{c}} (~49/48 = 24.3235{{c}})
: [[error map]]: {{val| +2.133 -2.865 +1.513 -2.852 }}
* [[CWE]]: ~8/7 = 240.0000{{c}}, ~3/2 = 696.1637{{c}} (~49/48 = 23.8373{{c}})
: error map: {{val| 0.000 -5.791 -1.659 -8.826 }}


POTE generator: ~10/7 = 631.370
{{Optimal ET sequence|legend=1| 5, 40c, 45, 50 }}


Mapping generator: ~10/7
[[Badness]] (Sintel): 2.59


Map: [&lt;1 0 -4 -3 -6|, &lt;0 3 12 11 18|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 19
Comma list: 81/80, 385/384, 2401/2376


Badness: 0.0548
Mapping: {{mapping| 5 0 -20 14 41 | 0 1 4 0 -3 }}


==13-limit==
Optimal tunings:
Commas: 45/44, 81/80, 91/88, 147/143
* 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}})


POTE generator: ~10/7 = 631.221
{{Optimal ET sequence|legend=0| 5, 45, 50 }}


Mapping generator: ~10/7
Badness (Sintel): 2.33


Map: [&lt;1 0 -4 -3 -6 0|, &lt;0 3 12 11 18 7|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 19
Comma list: 81/80, 105/104, 144/143, 2401/2376


Badness: 0.0361
Mapping: {{mapping| 5 0 -20 14 41 -21 | 0 1 4 0 -3 5 }}


=Jerome=
Optimal tunings:
Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5^(1/20), or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size.
* 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}})


Commas: 81/80, 17280/16807
{{Optimal ET sequence|legend=0| 5, 45f, 50 }}


POTE generator: ~54/49 = 139.343
Badness (Sintel): 2.02


Mapping generator: ~54/49
== Subgroup extensions ==
=== Stützel (2.3.5.19) ===
[[Subgroup]]: 2.3.5.19


Map: [&lt;1 1 0 2|, &lt;0 5 20 7|]
[[Comma list]]: 81/80, 96/95


Wedgie: &lt;&lt;5 30 7 20 -3 -40||
{{Mapping|legend=2| 1 0 -4 9 | 0 1 4 -3 }}


EDOs: 8, 9, 17, 26, 43, 112
{{Mapping|legend=3| 1 0 -4 0 0 0 0 9 | 0 1 4 0 0 0 0 -3 }}


Badness: 0.1087
: mapping generators: ~2, ~3


==11-limit==
[[Optimal tuning]]s:
Commas: 81/80, 99/98, 864/847
* [[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: ~12/11 = 139.428
{{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|, &lt;0 5 20 7 4|]
=== Hypnotone ===
Hypnotone is no-sevens [[#Flattone|flattone]].


EDOs: 8, 9, 17, 26, 43, 241
[[Subgroup]]: 2.3.5.11


Badness: 0.0479
[[Comma list]]: 45/44, 81/80


==13-limit==
{{Mapping|legend=2| 1 0 -4 -6 | 0 1 4 6 }}
Commas: 77/78, 81/80, 99/98, 144/143


POTE generator: ~13/12 = 139.387
{{Mapping|legend=3| 1 0 -4 0 -6 | 0 1 4 0 6 }}


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


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


EDOs: 8, 9, 17, 26, 43, 155, 198
{{Optimal ET sequence|legend=1| 7, 12, 19, 26, 45 }}


Badness: 0.0293
[[Badness]] (Sintel): 0.326


==17-limit==
==== 2.3.5.11.13 subgroup ====
Commas: 78/77, 81/80, 99/98, 144/143, 189/187
Subgroup: 2.3.5.11.13


POTE generator: ~13/12 = 139.362
Comma list: 45/44, 65/64, 81/80


Mapping generator: ~12/11
Subgroup-val mapping: {{mapping| 1 0 -4 -6 10 | 0 1 4 6 -4 }}


Map: [&lt;1 1 0 2 3 3 2|, &lt;0 5 20 7 4 6 18|]
Gencom mapping: {{mapping| 1 0 -4 0 -6 10 | 0 1 4 0 6 -4 }}


EDOs: 8, 9, 17, 26, 43, 155
Optimal tunings:  
* WE: ~2 = 1202.6916{{c}}, ~3/2 = 694.4181{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0870{{c}}


Badness: 0.0209
{{Optimal ET sequence|legend=0| 7, 12, 19, 26, 45f }}


=Meanmag=
Badness (Sintel): 0.561
Commas: 81/80, 3125/3072


POTE generator: ~8/7 = 238.396
=== Dequarter ===
[[Subgroup]]: 2.3.5.11


Mapping generator: ~7
[[Comma list]]: 33/32, 55/54


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


Wedgie: &lt;&lt;0 0 19 0 30 44||
{{Mapping|legend=3| 1 0 -4 0 5 | 0 1 4 0 -1 }}


EDOs: 19, 57, 76, 171bcd
: mapping generators: ~2, ~3


Badness: 0.0770
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1206.5832{{c}}, ~3/2 = 695.8763{{c}}
: [[error map]]: {{val| +6.583 +0.504 -2.809 -20.862 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 693.1206{{c}}
: error map: {{val| 0.000 -8.834 -13.831 -44.439 }}


=Undevigintone=
{{Optimal ET sequence|legend=1| 5, 7, 19e, 26e }}
Commas: 49/48, 81/80, 126/125


POTE generator: ~11/8 = 538.047
[[Badness]] (Sintel): 0.451


Mapping generator: ~11
==== Dreamtone ====
Subgroup: 2.3.5.11.13


Map: [&lt;19 30 44 53 0|, &lt;0 0 0 0 1|]
Comma list: 33/32, 55/54, 975/968


EDOs: 19, 38d
Subgroup-val mapping: {{mapping| 1 0 -4 5 21 | 0 1 4 -1 -11 }}


Badness: 0.0364
Gencom mapping: {{mapping| 1 0 -4 0 5 21 | 0 1 4 0 -1 -11 }}


==13-limit==
Optimal tunings:
Commas: 49/48, 65/64, 81/80, 126/125
* WE: ~2 = 1207.8248{{c}}, ~3/2 = 694.7806{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 690.1826{{c}}


POTE generator: ~11/8 = 537.061
{{Optimal ET sequence|legend=0| 7, 19eff, 26eff, 33ceeff, 40ceeff }}


Map: [&lt;19 30 44 53 0 70|, &lt;0 0 0 0 1 0|]
Badness (Sintel): 1.40


EDOs: 19, 38d
== References ==
<references/>


Badness: 0.0229
[[Category:Temperament families]]
[[Category:catalog]]
[[Category:Meantone family| ]] <!-- main article -->
[[Category:family]]
[[Category:Meantone| ]] <!-- key article -->
[[Category:listen]]
[[Category:Rank 2]]
[[Category:meantone]]
[[Category:Listen]]
[[Category:rank_2]]
[[Category:theory]]

Latest revision as of 17:44, 8 June 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=232032"