Meantone family: Difference between revisions

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The [[5-limit]] parent [[comma]] of the '''meantone family''' is the syntonic comma, [[81/80]]. This is the one they all temper out. The [[period]] is an [[octave]], the [[generator]] is a [[3/2|fifth]], and four fifths go to make up a [[5/1]] interval.
{{Technical data page}}
The '''meantone family''' is the family of [[rank-2 temperament]]s that [[tempering out|temper out]] the syntonic comma, [[81/80]], and thus can all be seen as [[extension]]s of [[meantone]].  


== Meantone ==
== Meantone ==
{{Main| Meantone }}
{{Main| Meantone }}
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]].


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 17: Line 20:


: mapping generators: ~2, ~3
: mapping generators: ~2, ~3
{{Multival|legend=1| 1 4 4 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 697.2143
* [[WE]]: ~2 = 1201.3906{{c}}, ~3/2 = 697.0455{{c}}
* [[POTE]]: ~2 = 1\1, ~3/2 = 696.239
: [[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 }}


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


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 34: Line 37:
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b }}
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b }}


[[Badness]]: 0.007381
[[Badness]] (Sintel): 0.173


=== Extensions ===
=== Overview to extensions ===
The second comma of the normal comma list defines which [[7-limit]] family member we are looking at.
The second comma of the normal comma list defines which [[7-limit]] family member we are looking at.
* Septimal meantone adds [[Harrison's comma|{{monzo| -13 10 0 -1 }}]], finding the ~7/4 at the augmented sixth,  
* 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,  
* Flattone adds {{monzo| -17 9 0 1 }}, finding the ~7/4 at the diminished seventh, for a tuning between 26edo and 19edo.
* Dominant adds [[64/63|{{monzo| 6 -2 0 -1 }}]], finding the ~7/4 at the minor seventh,  
* 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.
* Flattertone adds {{monzo| -24 17 0 -1 }}, finding the ~7/4 at the double-augmented sixth,  
* 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,  
* 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.
Those all have a fifth as generator.
* Injera adds {{monzo| -7 8 0 -2 }} with a half-octave period.
* 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.
* 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]].
* 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.
* 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]].
* 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]].
* 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.
* Jerome adds {{monzo| 3 7 0 -5 }} and slices the fifth in five.


Line 63: Line 66:


==== Splitting the meantone fifth into three (1029/1024) ====
==== Splitting the meantone fifth into three (1029/1024) ====
By tempering [[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.
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.


This strategy leads to the 7-limit version of [[mothra]], which is also sometimes called [[cynder]], though confusingly cynder has a different mapping for 11 in the 11-limit. Though 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]]).
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]]).


==== 31edo as splitting the fifth into two, three and nine ====
==== 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 225/224, which interestingly, though a rank 2 temperament, only has 31edo as a [[patent val]] tuning (corresponding to also tempering 225/224).
[[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
Temperaments discussed elsewhere include
* ''[[Plutus]]'' → [[Very low accuracy temperaments #Plutus|Very low accuracy temperaments]]
* ''[[Plutus]]'' (+15/14) → [[Very low accuracy temperaments #Plutus|Very low accuracy temperaments]]
* [[Godzilla]] → [[Slendro clan #Godzilla|Slendro clan]]
* [[Godzilla]] (+49/48) → [[Semaphoresmic clan #Godzilla|Semaphoresmic clan]]
* [[Mothra]] → [[Gamelismic clan #Mothra|Gamelismic clan]]
* [[Mothra]] (+1029/1024) → [[Gamelismic clan #Mothra|Gamelismic clan]]
* [[Mohaha]] → [[Rastmic clan #Mohaha|Rastmic clan]]
* ''[[Mohaha]]'' (+121/120) → [[Rastmic clan #Mohaha|Rastmic clan]]
* [[No-sevens subgroup temperaments#Dequarter|Dequarter]] → [[No-sevens subgroup temperaments#Dequarter|No-sevens subgroup temperaments]]


The rest are considered below.
The rest are considered below.
Line 84: Line 86:
{{Wikipedia| Septimal meantone temperament }}
{{Wikipedia| Septimal meantone temperament }}


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♯). Septimal meantone tempers out the common 7-limit commas [[126/125]], [[225/224]], and [[3136/3125]] and in fact can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125 and 225/224.  
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
[[Subgroup]]: 2.3.5.7
Line 91: Line 93:


{{Mapping|legend=1| 1 0 -4 -13 | 0 1 4 10 }}
{{Mapping|legend=1| 1 0 -4 -13 | 0 1 4 10 }}
{{Multival|legend=1| 1 4 10 4 13 12 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 696.9521
* [[WE]]: ~2 = 1201.2358{{c}}, ~3/2 = 697.2122{{c}}
* [[POTE]]: ~2 = 1\1, ~3/2 = 696.495
: [[error map]]: {{val| +1.236 -3.507 +2.535 -0.412 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.6562{{c}}
: error map: {{val| 0.000 -5.299 +0.311 -2.264 }}


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


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 112: Line 114:
{{Optimal ET sequence|legend=1| 12, 19, 31, 81, 112b, 143b }}
{{Optimal ET sequence|legend=1| 12, 19, 31, 81, 112b, 143b }}


[[Badness]]: 0.013707
[[Badness]] (Sintel): 0.347


=== Undecimal meantone (huygens) ===
=== Undecimal meantone (huygens) ===
{{Redirect|Huygens|the Dutch mathematician, physicist and astronomer|Wikipedia: Christiaan Huygens}}
{{Redirect|Huygens|the Dutch mathematician, physicist and astronomer|Wikipedia: Christiaan Huygens}}
{{See also| Meantone vs meanpop }}
{{See also| Huygens vs meanpop }}


Undecimal meantone maps the [[11/8]] to the double augmented third (C-E𝄪), and tridecimal meantone 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 the double augmented unison; 12/11 is a double diminished third; and 14/13 is a minor second.  
Undecimal meantone<ref name="meantone & meanpop 2003">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_6048.html#6052 Yahoo! Tuning Group | ''good 11-limit meantones'']</ref> a.k.a. huygens<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10437.html Yahoo! Tuning Group | ''The meantone family'']</ref><ref name="meantone & meanpop 2004">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10864.html#10870 Yahoo! Tuning Group | ''names and definitions: meantone'']</ref> maps the [[11/8]] to the double-augmented third (C–E𝄪). See [[chords of huygens]] for a list of dyadic chords in this temperament.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 127: Line 129:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 697.1676
* WE: ~2 = 1200.7636{{c}}, ~3/2 = 697.4122{{c}}
* POTE: ~2 = 1\1, ~3/2 = 696.967
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.0315{{c}}


Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}
* 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 }}]
: 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 }}]
: eigenmonzo (unchanged-interval) basis: 2.11/9
: unchanged-interval (eigenmonzo) basis: 2.11/9


Tuning ranges:  
Tuning ranges:  
Line 141: Line 143:
Algebraic generator: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
Algebraic generator: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.


{{Optimal ET sequence|legend=1| 12, 19e, 31, 105, 136b }}
{{Optimal ET sequence|legend=0| 12, 19e, 31, 105, 136b }}


Badness: 0.017027
Badness (Sintel): 0.563


; Music
; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3 ''Twinkle canon – 74 edo''] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]
* [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]


==== Tridecimal meantone ====
==== 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
Subgroup: 2.3.5.7.11.13


Comma list: 66/65, 81/80, 99/98, 105/104
Comma list: 81/80, 99/98, 126/125, 144/143


Mapping: {{mapping| 1 0 -4 -13 -25 -20 | 0 1 4 10 18 15 }}
Mapping: {{mapping| 1 0 -4 -13 -25 29 | 0 1 4 10 18 -16 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 696.8552
* WE: ~2 = 1199.9389{{c}}, ~3/2 = 697.2282{{c}}
* POTE: ~2 = 1\1, ~3/2 = 696.642
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.2627{{c}}


Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}
* 13- and 15-odd-limit: ~3/2 = {{monzo| 8/13 0 0 1/26 0 -1/26 }}
: eigenmonzo (unchanged-interval) basis: 2.11/9
: eigenmonzo basis (unchanged-interval basis): 2.13/7
 
Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~3/2 = [696.774, 697.674] (18\31 to 25\43)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)


{{Optimal ET sequence|legend=1| 12f, 19e, 31 }}
{{Optimal ET sequence|legend=0| 12, 31, 43, 74 }}


Badness: 0.018048
Badness (Sintel): 1.07


===== Meantonic =====
===== 17-limit =====
Dubbed ''meantonic'' here, this extension maps the 17/16 to the octave-reduced triple augmented seventh (C-B𝄪♯), and 19/16 to the quadruple augmented unison (C-C𝄪𝄪). The major second is now 19/17, and 17/16 is conflated with 19/18, as do all the other extensions discussed below. 31edo also conflates 17/16~19/18 with 16/15 whereas 50edo conflates all of 17/16, 18/17, 19/18, and 20/19, so a good tuning would be somewhere in this range.  
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
Subgroup: 2.3.5.7.11.13.17


Comma list: 66/65, 81/80, 99/98, 105/104, 121/119
Comma list: 81/80, 99/98, 120/119, 126/125, 144/143


Mapping: {{mapping| 1 0 -4 -13 -25 -20 -37 | 0 1 4 10 18 15 26 }}
Mapping: {{mapping| 1 0 -4 -13 -25 29 12 | 0 1 4 10 18 -16 -5 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 696.6486
* WE: ~2 = 1199.5811{{c}}, ~3/2 = 697.0918{{c}}
* POTE: ~2 = 1\1, ~3/2 = 696.377
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3303{{c}}


{{Optimal ET sequence|legend=1| 12fg, 19eg, 31, 50e }}
{{Optimal ET sequence|legend=0| 12, 31, 43, 74g }}


Badness: 0.019037
Badness (Sintel): 1.06


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


Comma list: 66/65, 77/76, 81/80, 99/98, 105/104, 121/119
Comma list: 81/80, 96/95, 99/98, 120/119, 126/125, 144/143


Mapping: {{mapping| 1 0 -4 -13 -25 -20 -37 -40 | 0 1 4 10 18 15 26 28 }}
Mapping: {{mapping| 1 0 -4 -13 -25 29 12 9 | 0 1 4 10 18 -16 -5 -3 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 696.5551
* WE: ~2 = 1199.2931{{c}}, ~3/2 = 696.9690{{c}}
* POTE: ~2 = 1\1, ~3/2 = 696.273
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3736{{c}}
 
{{Optimal ET sequence|legend=0| 12, 31, 43, 74gh }}


{{Optimal ET sequence|legend=1| 12fghh, 19egh, 31, 50e }}
Badness (Sintel): 1.07


Badness: 0.017846
==== 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.  


===== Meantoid =====
This extension used to be known as ''tridecimal meantone'', but was decanonicalized in 2025.  
Dubbed ''meantoid'' here, this extension maps 17/16~19/18 to the augmented unison (C-C♯) and 19/16 to the augmented second (C-D♯). For any tuning flatter than 12edo, the sizes of 17/16 (augmented unison) and 18/17 (minor second) are inverted, so genuine septendecimal and undevicesimal harmony cannot be expected.  


Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13


Comma list: 51/50, 66/65, 81/80, 85/84, 99/98
Comma list: 66/65, 81/80, 99/98, 105/104


Mapping: {{mapping| 1 0 -4 -13 -25 -20 -7 | 0 1 4 10 18 15 7 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -20 | 0 1 4 10 18 15 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 697.0360
* WE: ~2 = 1200.8149{{c}}, ~3/2 = 697.1155{{c}}
* POTE: ~2 = 1\1, ~3/2 = 696.448
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.7085{{c}}


{{Optimal ET sequence|legend=1| 12f, 19eg, 31g }}
Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}
: unchanged-interval (eigenmonzo) basis: 2.11/9


Badness: 0.019433
{{Optimal ET sequence|legend=0| 12f, 19e, 31 }}


====== 19-limit ======
Badness (Sintel): 0.746
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 51/50, 57/56, 66/65, 81/80, 85/84, 99/98
 
Mapping: {{mapping| 1 0 -4 -13 -25 -20 -7 -10 | 0 1 4 10 18 15 7 9 }}
 
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 697.2161
* POTE: ~2 = 1\1, ~3/2 = 696.394
 
{{Optimal ET sequence|legend=1| 12f, 19egh, 31gh }}
 
Badness: 0.017437
 
===== Huygens =====
Dubbed ''huygens'' here, this extension is perhaps the most practical, as it 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.


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


Line 241: Line 238:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 696.9080
* WE: ~2 = 1199.5548{{c}}, ~3/2 = 696.7449{{c}}
* POTE: ~2 = 1\1, ~3/2 = 697.003
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.9823{{c}}


{{Optimal ET sequence|legend=1| 12f, 31 }}
{{Optimal ET sequence|legend=0| 12f, 31 }}


Badness: 0.019982
Badness (Sintel): 1.02


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


Line 256: Line 253:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 696.9308
* WE: ~2 = 1199.0408{{c}}, ~3/2 = 696.5824{{c}}
* POTE: ~2 = 1\1, ~3/2 = 697.140
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.1061{{c}}


{{Optimal ET sequence|legend=1| 12f, 31 }}
{{Optimal ET sequence|legend=0| 12f, 31 }}


Badness: 0.018047
Badness (Sintel): 1.10


==== Grosstone ====
==== Meridetone ====
Grosstone maps 13/8 to the double diminished seventh (C-B♭♭♭).  
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
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 99/98, 126/125, 144/143
Comma list: 78/77, 81/80, 99/98, 126/125


Mapping: {{mapping| 1 0 -4 -13 -25 29 | 0 1 4 10 18 -16 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -39 | 0 1 4 10 18 27 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 697.2582
* WE: ~2 = 1199.9122{{c}}, ~3/2 = 697.4779{{c}}
* POTE: ~2 = 1\1, ~3/2 = 697.264
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.5241{{c}}


Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~3/2 = {{monzo| 8/13 0 0 1/26 0 -1/26 }}
* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/25 -2/25 0 0 0 1/25 }}
: eigenmonzo basis (unchanged-interval basis): 2.13/7
: unchanged-interval (eigenmonzo) basis: 2.13/9


Tuning ranges:
{{Optimal ET sequence|legend=0| 12f, 31f, 43 }}
* 13- and 15-odd-limit diamond monotone: ~3/2 = [696.774, 697.674] (18\31 to 25\43)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)


{{Optimal ET sequence|legend=1| 12, 31, 43, 74 }}
Badness (Sintel): 1.09
 
Badness: 0.025899


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


Comma list: 81/80, 99/98, 120/119, 126/125, 144/143
Comma list: 78/77, 81/80, 99/98, 120/119, 126/125


Mapping: {{mapping| 1 0 -4 -13 -25 29 12 | 0 1 4 10 18 -16 -5 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 | 0 1 4 10 18 27 -5 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 697.2996
* WE: ~2 = 1199.3793{{c}}, ~3/2 = 697.2833{{c}}
* POTE: ~2 = 1\1, ~3/2 = 697.335
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6222{{c}}


{{Optimal ET sequence|legend=1| 12, 31, 43, 74g }}
{{Optimal ET sequence|legend=0| 12f, 43 }}


Badness: 0.020889
Badness (Sintel): 1.22


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


Comma list: 81/80, 96/95, 99/98, 120/119, 126/125, 144/143
Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125


Mapping: {{mapping| 1 0 -4 -13 -25 29 12 9 | 0 1 4 10 18 -16 -5 -3 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 9 | 0 1 4 10 18 27 -5 -3 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 697.3271
* WE: ~2 = 1199.0260{{c}}, ~3/2 = 697.1486{{c}}
* POTE: ~2 = 1\1, ~3/2 = 697.380
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6887{{c}}


{{Optimal ET sequence|legend=1| 12, 31, 43, 74gh }}
{{Optimal ET sequence|legend=0| 12f, 43 }}


Badness: 0.017611
Badness (Sintel): 1.25


==== Meridetone ====
==== Hemimeantone ====
Meridetone maps the 13/8 to the quadruple augmented fourth (C-F𝄪𝄪).
Subgroup: 2.3.5.7.11.13


Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 99/98, 126/125, 169/168


Comma list: 78/77, 81/80, 99/98, 126/125
Mapping: {{mapping| 1 0 -4 -13 -25 -5 | 0 2 8 20 36 11 }}


Mapping: {{mapping| 1 0 -4 -13 -25 -39 | 0 1 4 10 18 27 }}
: mapping generators: ~2, ~26/15


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 697.5155
* WE: ~2 = 1201.0387{{c}}, ~26/15 = 949.2863{{c}}
* POTE: ~2 = 1\1, ~3/2 = 697.529
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5065{{c}}


Minimax tuning:
{{Optimal ET sequence|legend=0| 19e, 43, 62 }}
* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/25 -2/25 0 0 0 1/25 }}
: eigenmonzo (unchanged-interval) basis: 2.13/9


{{Optimal ET sequence|legend=1| 12f, 31f, 43 }}
Badness (Sintel): 1.30


Badness: 0.026421
===== 17-limit =====
 
===== Meridetonic =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 78/77, 81/80, 99/98, 126/125, 273/272
Comma list: 81/80, 99/98, 126/125, 169/168, 221/220


Mapping: {{mapping| 1 0 -4 -13 -25 -39 -56 | 0 1 4 10 18 27 38 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 | 0 2 8 20 36 11 33 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 697.5076
* WE: ~2 = 1201.0270{{c}}, ~26/15 = 949.2892{{c}}
* POTE: ~2 = 1\1, ~3/2 = 697.514
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5169{{c}}


{{Optimal ET sequence|legend=1| 12fg, 31fg, 43 }}
{{Optimal ET sequence|legend=0| 19eg, 43, 62 }}


Badness: 0.027706
Badness (Sintel): 1.19


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


Comma list: 78/77, 81/80, 99/98, 126/125, 153/152, 273/272
Comma list: 81/80, 99/98, 126/125, 153/152, 169/168, 221/220


Mapping: {{mapping| 1 0 -4 -13 -25 -39 -56 -59 | 0 1 4 10 18 27 38 40 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 -25 | 0 2 8 20 36 11 33 37 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 697.4848
* WE: ~2 = 1201.0339{{c}}, ~19/11 = 949.2902{{c}}
* POTE: ~2 = 1\1, ~3/2 = 697.481
* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 948.5111{{c}}


{{Optimal ET sequence|legend=1| 12fghh, 31fgh, 43 }}
{{Optimal ET sequence|legend=0| 19egh, 43, 62 }}


Badness: 0.025315
Badness (Sintel): 1.15


===== Meridetoid =====
==== Semimeantone ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13


Comma list: 51/50, 78/77, 81/80, 85/84, 99/98
Comma list: 81/80, 99/98, 126/125, 847/845


Mapping: {{mapping| 1 0 -4 -13 -25 -39 -7 | 0 1 4 10 18 27 7 }}
Mapping: {{mapping| 2 0 -8 -26 -50 -59 | 0 1 4 10 18 21 }}
 
: mapping generators: ~55/39, ~3


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 697.6098
* WE: ~55/39 = 600.3606{{c}}, ~3/2 = 697.4241{{c}}
* POTE: ~2 = 1\1, ~3/2 = 697.376
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 697.0545{{c}}


{{Optimal ET sequence|legend=1| 12f, 31fg, 43g }}
{{Optimal ET sequence|legend=0| 12f, …, 50eff, 62, 136b }}


Badness: 0.027518
Badness (Sintel): 1.68


====== 19-limit ======
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17


Comma list: 51/50, 57/56, 78/77, 81/80, 85/84, 99/98
Comma list: 81/80, 99/98, 126/125, 221/220, 289/288


Mapping: {{mapping| 1 0 -4 -13 -25 -39 -7 -10 | 0 1 4 10 18 27 7 9 }}
Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 | 0 1 4 10 18 21 1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 697.7012
* WE: ~17/12 = 600.5426{{c}}, ~3/2 = 697.5571{{c}}
* POTE: ~2 = 1\1, ~3/2 = 697.316
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9858{{c}}


{{Optimal ET sequence|legend=1| 12f, 19effgh, 31fgh, 43gh }}
{{Optimal ET sequence|legend=0| 12f, 50eff, 62, 136bg }}


Badness: 0.023613
Badness (Sintel): 1.60


===== Sauveuric =====
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 78/77, 81/80, 99/98, 120/119, 126/125
Comma list: 81/80, 99/98, 126/125, 153/152, 209/208, 221/220


Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 | 0 1 4 10 18 27 -5 }}
Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 -1 | 0 1 4 10 18 21 1 3 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 697.5384
* WE: ~17/12 = 600.5959{{c}}, ~3/2 = 697.5985{{c}}
* POTE: ~2 = 1\1, ~3/2 = 697.644
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9638{{c}}


{{Optimal ET sequence|legend=1| 12f, 43 }}
{{Optimal ET sequence|legend=0| 12f, 50eff, 62 }}


Badness: 0.023881
Badness (Sintel): 1.47


====== 19-limit ======
=== Meanpop ===
Subgroup: 2.3.5.7.11.13.17.19
{{See also| Huygens vs meanpop }}


Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125
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: {{mapping| 1 0 -4 -13 -25 -39 12 9 | 0 1 4 10 18 27 -5 -3 }}
Subgroup: 2.3.5.7.11


Optimal tunings:  
Comma list: 81/80, 126/125, 385/384
* CTE: ~2 = 1\1, ~3/2 = 697.5550
* POTE: ~2 = 1\1, ~3/2 = 697.715


{{Optimal ET sequence|legend=1| 12f, 43 }}
Mapping: {{mapping| 1 0 -4 -13 24 | 0 1 4 10 -13 }}


Badness: 0.020540
: mapping generator: ~2, ~3


==== Hemimeantone ====
Optimal tunings:
Subgroup: 2.3.5.7.11.13
* WE: ~2 = 1201.3464{{c}}, ~3/2 = 697.2159{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4509{{c}}


Comma list: 81/80, 99/98, 126/125, 169/168
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


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


: mapping generators: ~2, ~26/15
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.


Optimal tunings:
{{Optimal ET sequence|legend=0| 12e, 19, 31, 81, 112b }}
* CTE: ~2 = 1\1, ~26/15 = 948.6109
* POTE: ~2 = 1\1, ~26/15 = 948.465


{{Optimal ET sequence|legend=1| 19e, 43, 62 }}
Badness (Sintel): 0.712


Badness: 0.031433
; Music
* [http://soonlabel.com/xenharmonic/archives/607 Scott Joplin's "The Entertainer" tuned into meanpop]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3 ''Twinkle canon – 50 edo''] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]


===== 17-limit =====
==== Tridecimal meanpop ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 99/98, 126/125, 169/168, 221/220
Comma list: 81/80, 105/104, 126/125, 144/143


Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 | 0 2 8 20 36 11 33 }}
Mapping: {{mapping| 1 0 -4 -13 24 -20 | 0 1 4 10 -13 15 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~26/15 = 948.6173
* WE: ~2 = 1201.0765{{c}}, ~3/2 = 696.8361{{c}}
* POTE: ~2 = 1\1, ~26/15 = 948.477
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2347{{c}}


{{Optimal ET sequence|legend=1| 19eg, 43, 62 }}
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


Badness: 0.023380
Tuning ranges:
 
* 13- and 15-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
===== 19-limit =====
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
Subgroup: 2.3.5.7.11.13.17.19
 
{{Optimal ET sequence|legend=0| 19, 31, 50, 81 }}
 
Badness (Sintel): 0.863
 
===== Meanpoppic =====
Subgroup: 2.3.5.7.11.13.17


Comma list: 81/80, 99/98, 126/125, 153/152, 169/168, 221/220
Comma list: 81/80, 105/104, 126/125, 144/143, 273/272


Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 -25 | 0 2 8 20 36 11 33 37 }}
Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 | 0 1 4 10 -13 15 26 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~19/11 = 948.6088
* WE: ~2 = 1201.0727{{c}}, ~3/2 = 696.8168{{c}}
* POTE: ~2 = 1\1, ~19/11 = 948.473
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2195{{c}}


{{Optimal ET sequence|legend=1| 19egh, 43, 62 }}
{{Optimal ET sequence|legend=0| 19g, 31, 50, 81, 131bd }}


Badness: 0.018952
Badness (Sintel): 1.02


==== Semimeantone ====
====== 19-limit ======
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 81/80, 99/98, 126/125, 847/845
Comma list: 81/80, 105/104, 126/125, 144/143, 153/152, 273/272


Mapping: {{mapping| 2 0 -8 -26 -50 -59 | 0 1 4 10 18 21 }}
Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 -40 | 0 1 4 10 -13 15 26 28 }}
 
: mapping generators: ~55/39, ~3


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/39 = 1\2, ~3/2 = 697.1678
* WE: ~2 = 1201.0719{{c}}, ~3/2 = 696.8101{{c}}
* POTE: ~55/39 = 1\2, ~3/2 = 697.005
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2137{{c}}


{{Optimal ET sequence|legend=1| 12f, 38deefff, 50eff, 62, 136b }}
{{Optimal ET sequence|legend=0| 19gh, 31, 50, 81 }}


Badness: 0.040668
Badness (Sintel): 1.08


===== 17-limit =====
===== Meanpoid =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 81/80, 99/98, 126/125, 221/220, 289/288
Comma list: 81/80, 105/104, 120/119, 126/125, 144/143


Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 | 0 1 4 10 18 21 1 }}
Mapping: {{mapping| 1 0 -4 -13 24 -20 12 | 0 1 4 10 -13 15 -5 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 1\2, ~3/2 = 697.1740
* WE: ~2 = 1200.2768{{c}}, ~3/2 = 696.5683{{c}}
* POTE: ~17/12 = 1\2, ~3/2 = 696.927
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4114{{c}}


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


Badness: 0.031491
Badness (Sintel): 1.17


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


Comma list: 81/80, 99/98, 126/125, 153/152, 209/208, 221/220
Comma list: 77/76, 81/80, 96/95, 105/104, 120/119, 126/125


Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 -1 | 0 1 4 10 18 21 1 3 }}
Mapping: {{mapping| 1 0 -4 -13 24 -20 12 9 | 0 1 4 10 -13 15 -5 -3 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 1\2, ~3/2 = 697.1871
* WE: ~2 = 1199.7905{{c}}, ~3/2 = 696.3779{{c}}
* POTE: ~17/12 = 1\2, ~3/2 = 696.906
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4973{{c}}


{{Optimal ET sequence|legend=1| 12f, 50eff, 62 }}
{{Optimal ET sequence|legend=0| 19, 31 }}


Badness: 0.024206
Badness (Sintel): 1.25


=== Meanpop ===
==== Semimeanpop ====
{{See also| Meantone vs meanpop }}
Subgroup: 2.3.5.7.11.13


Meanpop maps the 11/8 to the double diminished fifth (C-G𝄫), and tridecimal meanpop still maps the 13/8 to the double augmented fifth (C-G𝄪). Note also 11/10 is the double diminished third; 12/11~13/12, double augmented unison; and 14/13, minor second.
Comma list: 81/80, 126/125, 385/384, 847/845


Subgroup: 2.3.5.7.11
Mapping: {{mapping| 2 0 -8 -26 48 39 | 0 1 4 10 -13 -10 }}


Comma list: 81/80, 126/125, 385/384
: mapping generators: ~55/39, ~3


Mapping: {{mapping| 1 0 -4 -13 24 | 0 1 4 10 -13 }}
Optimal tunings:  
* WE: ~55/39 = 600.6704{{c}}, ~3/2 = 697.2151{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.4341{{c}}


: mapping generator: ~2, ~3
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112b }}


Optimal tunings:  
Badness (Sintel): 1.78
* CTE: ~2 = 1\1, ~3/2 = 696.5311
* POTE: ~2 = 1\1, ~3/2 = 696.434


Minimax tuning:
===== 17-limit =====
* 11-odd-limit: ~3/2 = {{monzo| 0 0 1/4 }}
Subgroup: 2.3.5.7.11.13.17
: [{{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 }}]
: eigenmonzo (unchanged-interval) basis: 2.5


Tuning ranges:  
Comma list: 81/80, 126/125, 221/220, 273/272, 289/288
* 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 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
Mapping: {{mapping| 2 0 -8 -26 48 39 5 | 0 1 4 10 -13 -10 1 }}


{{Optimal ET sequence|legend=1| 12e, 19, 31, 81, 112b }}
Optimal tunings:
* WE: ~17/12 = 600.7232{{c}}, ~3/2 = 697.2820{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4411{{c}}


Badness: 0.021543
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bg }}


; Music
Badness (Sintel): 1.45
* [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]


==== Tridecimal meanpop ====
===== 19-limit =====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 81/80, 105/104, 126/125, 144/143
Comma list: 81/80, 126/125, 153/152, 209/208, 221/220, 273/272


Mapping: {{mapping| 1 0 -4 -13 24 -20 | 0 1 4 10 -13 15 }}
Mapping: {{mapping| 2 0 -8 -26 48 39 5 -1 | 0 1 4 10 -13 -10 1 3 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 696.3563
* WE: ~17/12 = 600.7527{{c}}, ~3/2 = 697.3244{{c}}
* POTE: ~2 = 1\1, ~3/2 = 696.211
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4525{{c}}


Minimax tuning:
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bgh }}
* 13- and 15-odd-limit: ~3/2 = {{monzo| 4/7 0 0 0 -1/28 1/28 }}
: eigenmonzo (unchanged-interval) basis: 2.13/11


Tuning ranges:
Badness (Sintel): 1.28
* 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|legend=1| 19, 31, 50, 81 }}
=== 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.


Badness: 0.020883
Subgroup: 2.3.5.7.11


===== Meanpoppic =====
Comma list: 45/44, 56/55, 81/80
Subgroup: 2.3.5.7.11.13.17


Comma list: 81/80, 105/104, 126/125, 144/143, 273/272
Mapping: {{mapping| 1 0 -4 -13 -6 | 0 1 4 10 6 }}
 
Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 | 0 1 4 10 -13 15 26 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 696.3508
* WE: ~2 = 1199.6946{{c}}, ~3/2 = 696.0729{{c}}
* POTE: ~2 = 1\1, ~3/2 = 696.194
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2083{{c}}


{{Optimal ET sequence|legend=1| 19g, 31, 50, 81, 131bd }}
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: 0.019953
{{Optimal ET sequence|legend=0| 7d, 12, 19, 31e }}


====== 19-limit ======
Badness (Sintel): 0.708
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 81/80, 105/104, 126/125, 144/143, 153/152, 273/272
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 -40 | 0 1 4 10 -13 15 26 28 }}
Comma list: 45/44, 56/55, 78/77, 81/80
 
Mapping: {{mapping| 1 0 -4 -13 -6 -20 | 0 1 4 10 6 15 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 696.3471
* WE: ~2 = 1199.7931{{c}}, ~3/2 = 696.0258{{c}}
* POTE: ~2 = 1\1, ~3/2 = 696.188
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1241{{c}}


{{Optimal ET sequence|legend=1| 19gh, 31, 50, 81 }}
{{Optimal ET sequence|legend=0| 7df, 12f, 19, 31e }}


Badness: 0.017791
Badness (Sintel): 0.875


===== Meanpoid =====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 81/80, 105/104, 120/119, 126/125, 144/143
Comma list: 45/44, 56/55, 78/77, 81/80, 120/119


Mapping: {{mapping| 1 0 -4 -13 24 -20 12 | 0 1 4 10 -13 15 -5 }}
Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 | 0 1 4 10 6 15 -5 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 696.4388
* WE: ~2 = 1198.6665{{c}}, ~3/2 = 695.8010{{c}}
* POTE: ~2 = 1\1, ~3/2 = 696.408
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4998{{c}}


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


Badness: 0.022870
Badness (Sintel): 1.17


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


Comma list: 77/76, 81/80, 96/95, 105/104, 120/119, 126/125
Comma list: 45/44, 56/55, 78/77, 81/80, 96/95, 120/119


Mapping: {{mapping| 1 0 -4 -13 24 -20 12 9 | 0 1 4 10 -13 15 -5 -3 }}
Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 9 | 0 1 4 10 6 15 -5 -3 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 696.4838
* WE: ~2 = 1198.2880{{c}}, ~3/2 = 695.7123{{c}}
* POTE: ~2 = 1\1, ~3/2 = 696.499
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.6370{{c}}


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


Badness: 0.020488
Badness (Sintel): 1.23


==== Meanplop ====
==== Vincenzo ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 65/64, 78/77, 81/80, 91/90
Comma list: 45/44, 56/55, 65/64, 81/80


Mapping: {{mapping| 1 0 -4 -13 24 10 | 0 1 4 10 -13 -4 }}
Mapping: {{mapping| 1 0 -4 -13 -6 10 | 0 1 4 10 6 -4 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 696.2827
* WE: ~2 = 1202.1684{{c}}, ~3/2 = 696.3160{{c}}
* POTE: ~2 = 1\1, ~3/2 = 696.202
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.2045{{c}}
 
Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 11/13 0 0 0 -1/13 }}
: Eigenmonzo (unchanged-interval) basis: 2.11


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


Badness: 0.027666
Badness (Sintel): 1.02


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


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


Mapping: {{mapping| 1 0 -4 -13 24 10 12 | 0 1 4 10 -13 -4 -5 }}
Mapping: {{mapping| 1 0 -4 -13 -6 10 12 | 0 1 4 10 6 -4 -5 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 696.4069
* WE: ~2 = 1200.5137{{c}}, ~3/2 = 696.1561{{c}}
* POTE: ~2 = 1\1, ~3/2 = 696.414
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.8771{{c}}


{{Optimal ET sequence|legend=1| 12e, 19 }}
{{Optimal ET sequence|legend=0| 12, 19 }}


Badness: 0.026836
Badness (Sintel): 1.30


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


Comma list: 39/38, 52/51, 65/64, 77/76, 81/80, 91/90
Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80


Mapping: {{mapping| 1 0 -4 -13 24 10 12 9 | 0 1 4 10 -13 -4 -5 -3 }}
Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 | 0 1 4 10 6 -4 -5 -3 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 696.4731
* WE: ~2 = 1199.8261{{c}}, ~3/2 = 696.0298{{c}}
* POTE: ~2 = 1\1, ~3/2 = 696.497
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1262{{c}}


{{Optimal ET sequence|legend=1| 12e, 19 }}
{{Optimal ET sequence|legend=0| 12, 19 }}


Badness: 0.023540
Badness (Sintel): 1.36


===== Meanploid =====
=== Bimeantone ===
Subgroup: 2.3.5.7.11.13.17
11/8 is mapped to half octave minus the [[128/125|meantone diesis]].
 
Subgroup: 2.3.5.7.11
 
Comma list: 81/80, 126/125, 245/242


Comma list: 51/50, 65/64, 78/77, 81/80, 85/84
Mapping: {{mapping| 2 0 -8 -26 -31 | 0 1 4 10 12 }}


Mapping: {{mapping| 1 0 -4 -13 24 10 -7 | 0 1 4 10 -13 -4 7 }}
: mapping generators: ~63/44, ~3


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 696.6614
* WE: ~63/44 = 600.7492{{c}}, ~3/2 = 696.8853{{c}}
* POTE: ~2 = 1\1, ~3/2 = 696.415
* CWE: ~63/44 = 600.0000{{c}}, ~3/2 = 696.1908{{c}}


{{Optimal ET sequence|legend=1| 12e, 19g, 31fg }}
{{Optimal ET sequence|legend=0| 12, 26de, 38d, 50 }}


Badness: 0.026094
Badness (Sintel): 1.26


====== 19-limit ======
==== 13-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13


Comma list: 51/50, 57/56, 65/64, 76/75, 78/77, 81/80
Comma list: 81/80, 105/104, 126/125, 245/242


Mapping: {{mapping| 1 0 -4 -13 24 10 -7 -10 | 0 1 4 10 -13 -4 7 9 }}
Mapping: {{mapping| 2 0 -8 -26 -31 -40 | 0 1 4 10 12 15 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 697.0160
* WE: ~55/39 = 600.8309{{c}}, ~3/2 = 696.8000{{c}}
* POTE: ~2 = 1\1, ~3/2 = 696.583
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.0066{{c}}


{{Optimal ET sequence|legend=1| 12e, 19gh, 31fgh }}
{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}


Badness: 0.023104
Badness (Sintel): 1.19


=== Meanenneadecal ===
==== 17-limit ====
Meanenneadecal maps the 11/8 to the augmented fourth (C-F♯), and tridecimal meanenneadecal still maps the 13/8 to the double augmented fifth (C-G𝄪). Note also 11/10 is the major second; 12/11~14/13, minor second; and 13/12, double augmented unison.  
Subgroup: 2.3.5.7.11.13.17


Subgroup: 2.3.5.7.11
Comma list: 81/80, 105/104, 126/125, 189/187, 221/220


Comma list: 45/44, 56/55, 81/80
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 | 0 1 4 10 12 15 1 }}
 
Mapping: {{mapping| 1 0 -4 -13 -6 | 0 1 4 10 6 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 696.1527
* WE: ~17/12 = 600.9234{{c}}, ~3/2 = 696.8536{{c}}
* POTE: ~2 = 1\1, ~3/2 = 696.250
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.9317{{c}}


Tuning ranges:
{{Optimal ET sequence|legend=0| 12f, 38df, 50 }}
* 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|legend=1| 7d, 12, 19, 31e }}
Badness (Sintel): 1.15


Badness: 0.021423
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


==== 13-limit ====
Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220
Subgroup: 2.3.5.7.11.13
 
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 -1 | 0 1 4 10 12 15 1 3 }}


Comma list: 45/44, 56/55, 78/77, 81/80
Optimal tunings:  
* WE: ~17/12 = 600.9845{{c}}, ~3/2 = 696.8939{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.8947{{c}}


Mapping: {{mapping| 1 0 -4 -13 -6 -20 | 0 1 4 10 6 15 }}
{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}


Optimal tunings:  
Badness (Sintel): 1.08
* CTE: ~2 = 1\1, ~3/2 = 696.0983
* POTE: ~2 = 1\1, ~3/2 = 696.146


{{Optimal ET sequence|legend=1| 7df, 12f, 19, 31e }}
=== Trimean ===
{{See also| No-sevens subgroup temperaments #Superpine }}


Badness: 0.021182
Subgroup: 2.3.5.7.11


===== 17-limit =====
Comma list: 81/80, 126/125, 1344/1331
Subgroup: 2.3.5.7.11.13.17


Comma list: 45/44, 56/55, 78/77, 81/80, 120/119
Mapping: {{mapping| 1 2 4 7 5 | 0 -3 -12 -30 -11 }}


Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 | 0 1 4 10 6 15 -5 }}
: mapping generators: ~2, ~11/10


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 696.2161
* WE: ~2 = 1200.7155{{c}}, ~11/10 = 167.9055{{c}}
* POTE: ~2 = 1\1, ~3/2 = 696.575
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7749{{c}}


{{Optimal ET sequence|legend=1| 12f, 19, 31e }}
{{Optimal ET sequence|legend=0| 7d, 36d, 43, 50, 93 }}


Badness: 0.022980
Badness (Sintel): 1.68


====== 19-limit ======
==== 13-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13


Comma list: 45/44, 56/55, 78/77, 81/80, 96/95, 120/119
Comma list: 81/80, 126/125, 144/143, 364/363


Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 9 | 0 1 4 10 6 15 -5 -3 }}
Mapping: {{mapping| 1 2 4 7 5 3 | 0 -3 -12 -30 -11 5 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 696.2774
* WE: ~2 = 1200.6104{{c}}, ~11/10 = 167.8749{{c}}
* POTE: ~2 = 1\1, ~3/2 = 696.706
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7728{{c}}


{{Optimal ET sequence|legend=1| 12f, 19, 31e }}
{{Optimal ET sequence|legend=0| 7d, 43, 50, 93 }}


Badness: 0.020293
Badness (Sintel): 1.46


===== Meanenneadecoid =====
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 34/33, 45/44, 51/50, 56/55, 78/77
Comma list: 81/80, 126/125, 144/143, 189/187, 221/220


Mapping: {{mapping| 1 0 -4 -13 -6 -20 -7 | 0 1 4 10 6 15 7 }}
Mapping: {{mapping| 1 2 4 7 5 3 8 | 0 -3 -12 -30 -11 5 -28 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 696.4501
* WE: ~2 = 1200.6144{{c}}, ~11/10 = 167.8716{{c}}
* POTE: ~2 = 1\1, ~3/2 = 696.025
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7682{{c}}


{{Optimal ET sequence|legend=1| 7dfg, 12f, 19g }}
{{Optimal ET sequence|legend=0| 7dg, 43, 50, 93 }}


Badness: 0.020171
Badness (Sintel): 1.28


====== 19-limit ======
=== Migration ===
Subgroup: 2.3.5.7.11.13.17.19
See [[Rastmic clan #Migration|Rastmic clan]].


Comma list: 34/33, 45/44, 51/50, 56/55, 57/55, 78/77
== Flattone ==
{{Main| Flattone }}


Mapping: {{mapping| 1 0 -4 -13 -6 -20 -7 -10 | 0 1 4 10 6 15 7 9 }}
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]].


Optimal tunings:  
[[Subgroup]]: 2.3.5.7
* CTE: ~2 = 1\1, ~3/2 = 696.7925
* POTE: ~2 = 1\1, ~3/2 = 696.121


{{Optimal ET sequence|legend=1| 7dfgh, 12f, 19gh }}
[[Comma list]]: 81/80, 525/512


Badness: 0.018045
{{Mapping|legend=1| 1 0 -4 17 | 0 1 4 -9 }}


==== Vincenzo ====
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13
* [[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 }}


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


Mapping: {{mapping| 1 0 -4 -13 -6 10 | 0 1 4 10 6 -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]


Optimal tunings:  
[[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.
* CTE: ~2 = 1\1, ~3/2 = 695.7897
* POTE: ~2 = 1\1, ~3/2 = 695.060


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


Badness: 0.024763
[[Badness]] (Sintel): 0.976


===== 17-limit =====
=== 11-limit ===
Subgroup: 2.3.5.7.11.13.17
This can also be considered a no-sevens temperament: [[#Hypnotone|hypnotone]].


Comma list: 45/44, 52/51, 56/55, 65/64, 81/80
Subgroup: 2.3.5.7.11


Mapping: {{mapping| 1 0 -4 -13 -6 10 12 | 0 1 4 10 6 -4 -5 }}
Comma list: 45/44, 81/80, 385/384


Optimal tunings:  
Mapping: {{mapping| 1 0 -4 17 -6 | 0 1 4 -9 6 }}
* CTE: ~2 = 1\1, ~3/2 = 696.0114
* POTE: ~2 = 1\1, ~3/2 = 695.858


{{Optimal ET sequence|legend=1| 12, 19 }}
Optimal tuning:
* WE: ~2 = 1202.3247{{c}}, ~3/2 = 694.4688{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.1467{{c}}


Badness: 0.025535
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]


====== 19-limit ======
{{Optimal ET sequence|legend=0| 7, 19, 26, 45, 71bc, 116bcde }}
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80
Badness (Sintel): 1.12


Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 | 0 1 4 10 6 -4 -5 -3 }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Optimal tunings:  
Comma list: 45/44, 65/64, 78/77, 81/80
* CTE: ~2 = 1\1, ~3/2 = 696.1196
* POTE: ~2 = 1\1, ~3/2 = 696.131


{{Optimal ET sequence|legend=1| 12, 19 }}
Mapping: {{mapping| 1 0 -4 17 -6 10 | 0 1 4 -9 6 -4 }}


Badness: 0.022302
Optimal tunings:  
* WE: ~2 = 1202.5156{{c}}, ~3/2 = 694.5107{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0538{{c}}


====== 23-limit ======
Tuning ranges:
Subgroup: 2.3.5.7.11.13.17.19.23
* 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]


Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 69/68, 81/80
{{Optimal ET sequence|legend=0| 7, 19, 26, 45f, 71bcf, 116bcdef }}


Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 14 | 0 1 4 10 6 -4 -5 -3 -6 }}
Badness (Sintel): 0.920


Optimal tunings:
=== Ptolemy ===
* CTE: ~2 = 1\1, ~3/2 = 696.0585
See [[Rastmic clan #Ptolemy|Rastmic clan]].
* POTE: ~2 = 1\1, ~3/2 = 696.044


{{Optimal ET sequence|legend=1| 12, 19 }}
== Dominant ==
{{Main| Dominant (temperament) }}
{{See also| Archytas clan }}


Badness: 0.020139
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]].


====== 29-limit ======
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.  
Subgroup: 2.3.5.7.11.13.17.19.23.29


Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 14 8 | 0 1 4 10 6 -4 -5 -3 -6 -2 }}
[[Comma list]]: 36/35, 64/63


Optimal tunings:
{{Mapping|legend=1| 1 0 -4 6 | 0 1 4 -2 }}
* CTE: ~2 = 1\1, ~3/2 = 695.9824
* POTE: ~2 = 1\1, ~3/2 = 695.913


{{Optimal ET sequence|legend=1| 12, 19 }}
[[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: 0.018168
[[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]


====== 31-limit ======
{{Optimal ET sequence|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80, 93/92
[[Badness]] (Sintel): 0.524


Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 14 8 16 | 0 1 4 10 6 -4 -5 -3 -6 -2 -7 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Optimal tunings:  
Comma list: 36/35, 56/55, 64/63
* CTE: ~2 = 1\1, ~3/2 = 695.7980
* POTE: ~2 = 1\1, ~3/2 = 695.750


{{Optimal ET sequence|legend=1| 12, 19 }}
Mapping: {{mapping| 1 0 -4 6 13 | 0 1 4 -2 -6 }}


Badness: 0.017069
Tuning ranges:  
 
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 705.882] (7\12 to 10\17)
====== 37-limit ======
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37
 
Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92
 
Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 14 8 16 -9 | 0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 695.6746
* WE: ~2 = 1194.0169{{c}}, ~3/2 = 699.7473{{c}}
* POTE: ~2 = 1\1, ~3/2 = 695.603
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.2672{{c}}


{{Optimal ET sequence|legend=1| 12, 19 }}
{{Optimal ET sequence|legend=0| 5, 12, 17c, 29cde }}


Badness: 0.016129
Badness (Sintel): 0.799


====== 41-limit ======
==== 13-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41
Subgroup: 2.3.5.7.11.13


Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92, 124/123
Comma list: 36/35, 56/55, 64/63, 66/65


Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 | 0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 }}
Mapping: {{mapping| 1 0 -4 6 13 18 | 0 1 4 -2 -6 -9 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 695.7239
* WE: ~2 = 1193.8055{{c}}, ~3/2 = 700.0042{{c}}
* POTE: ~2 = 1\1, ~3/2 = 695.696
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.8254{{c}}


{{Optimal ET sequence|legend=1| 12, 19 }}
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]


Badness: 0.015356
{{Optimal ET sequence|legend=0| 12f, 17c, 29cdef }}


====== 43-limit ======
Badness (Sintel): 0.996
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41.43


Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 86/85, 93/92, 124/123
==== Dominion ====
Subgroup: 2.3.5.7.11.13


Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7 | 0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1 }}
Comma list: 26/25, 36/35, 56/55, 64/63
 
Mapping: {{mapping| 1 0 -4 6 13 -9 | 0 1 4 -2 -6 8 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 695.7160
* WE: ~2 = 1195.0293{{c}}, ~3/2 = 701.9847{{c}}
* POTE: ~2 = 1\1, ~3/2 = 695.688
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7698{{c}}


{{Optimal ET sequence|legend=1| 12, 19 }}
{{Optimal ET sequence|legend=0| 5, 12, 17c }}


Badness: 0.013906
Badness (Sintel): 1.13


====== 47-limit ======
=== Domination ===
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41.43.47
Subgroup: 2.3.5.7.11


Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 86/85, 93/92, 95/94, 124/123
Comma list: 36/35, 64/63, 77/75


Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7 4 | 0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1 1 }}
Mapping: {{mapping| 1 0 -4 6 -14 | 0 1 4 -2 11 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 695.6849
* WE: ~2 = 1194.8645{{c}}, ~3/2 = 701.9872{{c}}
* POTE: ~2 = 1\1, ~3/2 = 695.676
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5945{{c}}


{{Optimal ET sequence|legend=1| 12, 19 }}
{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}


Badness: 0.013818
Badness (Sintel): 1.21


===== Vincenzoid =====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13


Comma list: 34/33, 45/44, 51/50, 56/55, 65/64
Comma list: 26/25, 36/35, 64/63, 66/65


Mapping: {{mapping| 1 0 -4 -13 -6 10 -7 | 0 1 4 10 6 -4 7 }}
Mapping: {{mapping| 1 0 -4 6 -14 -9 | 0 1 4 -2 11 8 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 696.4125
* WE: ~2 = 1195.1324{{c}}, ~3/2 = 702.6343{{c}}
* POTE: ~2 = 1\1, ~3/2 = 695.358
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.0791{{c}}


{{Optimal ET sequence|legend=1| 7dg, 12, 19g }}
{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}


Badness: 0.022099
Badness (Sintel): 1.13


====== 19-limit ======
=== Domineering ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11


Comma list: 34/33, 45/44, 51/50, 56/55, 57/55, 65/64
Comma list: 36/35, 45/44, 64/63


Mapping: {{mapping| 1 0 -4 -13 -6 10 -7 -10 | 0 1 4 10 6 -4 7 9 }}
Mapping: {{mapping| 1 0 -4 6 -6 | 0 1 4 -2 6 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 696.9500
* WE: ~2 = 1194.7102{{c}}, ~3/2 = 695.6962{{c}}
* POTE: ~2 = 1\1, ~3/2 = 695.725
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1765{{c}}


{{Optimal ET sequence|legend=1| 7dgh, 12, 19gh }}
{{Optimal ET sequence|legend=0| 5e, 7, 12 }}


Badness: 0.019904
Badness (Sintel): 0.727


==== Meanundec ====
=== Arnold ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11


Comma list: 27/26, 40/39, 45/44, 56/55
Comma list: 22/21, 33/32, 36/35


Mapping: {{mapping| 1 0 -4 -13 -6 -1 | 0 1 4 10 6 3 }}
Mapping: {{mapping| 1 0 -4 6 5 | 0 1 4 -2 -1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 695.6202
* WE: ~2 = 1199.8507{{c}}, ~3/2 = 698.4045{{c}}
* POTE: ~2 = 1\1, ~3/2 = 697.254
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.4822{{c}}


{{Optimal ET sequence|legend=1| 7d, 12f, 19f }}
{{Optimal ET sequence|legend=0| 5, 7, 12e }}


Badness: 0.024243
Badness (Sintel): 0.864


===== 17-limit =====
=== Neutrominant ===
Subgroup: 2.3.5.7.11.13.17
See [[Rastmic clan #Neutrominant|Rastmic clan]].


Comma list: 27/26, 34/33, 40/39, 45/44, 56/55
== Flattertone ==
In flattertone, 17 fifths get to the interval class for 7, so that [[7/4]] is a double-augmented sixth (C–Ax). The fifth in flattertone is typically at least as flat as [[26edo]]. Here, 26edo and [[33edo|33cd-edo]] are the two primary flattertone tunings. [[1/2-comma meantone]] is also encompassed within flattertone's range. Any flatter than this, the meantone mapping for 5/4 is too inaccurate (it becomes more of a [[16/13]] or [[27/22]]), and [[deeptone]] temperament's mapping is more logical.


Mapping: {{mapping| 1 0 -4 -13 -6 -1 -7 | 0 1 4 10 6 3 7 }}
Flattertone was named by [[Flora Canou]] in 2024.


Optimal tunings:  
[[Subgroup]]: 2.3.5.7
* CTE: ~2 = 1\1, ~3/2 = 696.2789
* POTE: ~2 = 1\1, ~3/2 = 697.586


{{Optimal ET sequence|legend=1| 7dg, 12f }}
[[Comma list]]: 81/80, 1875/1792


Badness: 0.021400
{{Mapping|legend=1| 1 0 -4 -24 | 0 1 4 17 }}


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


Comma list: 27/26, 34/33, 40/39, 45/44, 56/55, 57/55
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1204.4511{{c}}, ~3/2 = 694.3258{{c}}
: [[error map]]: {{val| +4.451 -3.178 -9.011 +3.554 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 692.0479{{c}}
: error map: {{val| 0.000 -9.907 -18.122 -4.012 }}


Mapping: {{mapping| 1 0 -4 -13 -6 -1 -7 -10 | 0 1 4 10 6 3 7 9 }}
{{Optimal ET sequence|legend=1| 7d, 19d, 26, 59bcd, 85bccd }}


Optimal tunings:  
[[Badness]] (Sintel): 2.43
* CTE: ~2 = 1\1, ~3/2 = 696.8486
* POTE: ~2 = 1\1, ~3/2 = 698.118


{{Optimal ET sequence|legend=1| 7dgh, 12f }}
==== 11-limit ====
Subgroup: 2.3.5.7.11


Badness: 0.018996
Comma list: 45/44, 81/80, 1375/1344


=== Meanundeci ===
Mapping: {{mapping| 1 0 -4 -24 -6 | 0 1 4 17 6 }}
Meanundeci is a low-complexity low-accuracy entry that maps the 11/8 to the perfect fourth (C-F), and tridecimal meanundeci maps the 13/8 to the minor sixth (C-A♭).


Subgroup: 2.3.5.7.11
Optimal tunings:  
* WE: ~2 = 1203.4653{{c}}, ~3/2 = 693.8144{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 692.0422{{c}}


Comma list: 33/32, 55/54, 77/75
{{Optimal ET sequence|legend=0| 7d, 19d, 26 }}


Mapping: {{mapping| 1 0 -4 -13 5 | 0 1 4 10 -1 }}
Badness (Sintel): 1.53


Optimal tunings:
; Music
* CTE: ~2 = 1\1, ~3/2 = 696.7022
* [https://youtu.be/scCuGXnj5IY ''Music in 33EDO (33-Tone Equal Temperament) - Feb 2024''] by [[Budjarn Lambeth]] (2024)
* POTE: ~2 = 1\1, ~3/2 = 694.689


{{Optimal ET sequence|legend=1| 7d, 12e, 19e }}
== 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.


Badness: 0.031539
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).


==== 13-limit ====
The 11-limit extension was named by Gene Ward Smith in 2004<ref name="meantone & meanpop 2004"/>.  
Subgroup: 2.3.5.7.11.13


Comma list: 33/32, 55/54, 65/64, 77/75
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 0 -4 -13 5 10 | 0 1 4 10 -1 -4 }}
[[Comma list]]: 21/20, 28/27


Optimal tunings:
{{Mapping|legend=1| 1 0 -4 -2 | 0 1 4 3 }}
* CTE: ~2 = 1\1, ~3/2 = 696.2408
* POTE: ~2 = 1\1, ~3/2 = 694.764


{{Optimal ET sequence|legend=1| 7d, 12e, 19e }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1204.2961{{c}}, ~3/2 = 702.6463{{c}}
: [[error map]]: {{val| +4.296 +4.987 +24.271 -56.591 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4928{{c}}
: error map: {{val| 0.000 -0.462 +19.657 -64.347 }}


Badness: 0.026288
{{Optimal ET sequence|legend=1| 5, 7d, 12d }}


=== Bimeantone ===
[[Badness]] (Sintel): 0.629
11/8 is mapped to half octave minus the [[128/125|meantone diesis]].  


=== Meanertone ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 81/80, 126/125, 245/242
Comma list: 21/20, 28/27, 33/32


Mapping: {{mapping| 2 0 -8 -26 -31 | 0 1 4 10 12 }}
Mapping: {{mapping| 1 0 -4 -2 5 | 0 1 4 3 -1 }}
 
: mapping generators: ~63/44, ~3


Optimal tunings:  
Optimal tunings:  
* CTE: ~63/44 = 1\2, ~3/2 = 696.5199
* WE: ~2 = 1208.5304{{c}}, ~3/2 = 701.5669{{c}}
* POTE: ~63/44 = 1\2, ~3/2 = 696.016
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1117{{c}}


{{Optimal ET sequence|legend=1| 12, 26de, 38d, 50 }}
{{Optimal ET sequence|legend=0| 5, 7d, 12de }}


Badness: 0.038122
Badness (Sintel): 0.832


==== 13-limit ====
== Mildtone ==
Subgroup: 2.3.5.7.11.13
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.  


Comma list: 81/80, 105/104, 126/125, 245/242
Mildtone was named by [[User: Lucius Chiaraviglio|Lucius Chiaraviglio]] in 2024.


Mapping: {{mapping| 2 0 -8 -26 -31 -40 | 0 1 4 10 12 15 }}
[[Subgroup]]: 2.3.5.7


Optimal tunings:  
[[Comma list]]: 81/80, 16128/15625
* CTE: ~55/39 = 1\2, ~3/2 = 696.3410
* POTE: ~55/39 = 1\2, ~3/2 = 695.836


{{Optimal ET sequence|legend=1| 12f, 26deff, 38df, 50 }}
{{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 }}


Badness: 0.028817
[[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 }}


==== 17-limit ====
{{Optimal ET sequence|legend=1| 12, 43d, 55, 67 }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 81/80, 105/104, 126/125, 189/187, 221/220
[[Badness]] (Sintel): 2.67


Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 | 0 1 4 10 12 15 1 }}
=== 11-limit ===


Optimal tunings:  
[[Subgroup]]: 2.3.5.7.11
* CTE: ~17/12 = 1\2, ~3/2 = 696.3526
* POTE: ~17/12 = 1\2, ~3/2 = 695.783


{{Optimal ET sequence|legend=1| 12f, 38df, 50 }}
[[Comma list]]: 81/80, 176/175, 7058/6875


Badness: 0.022666
{{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 30}}


==== 19-limit ====
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13.17.19
* [[WE]]: ~2 = 1199.816{{c}}, ~3/2 = 698.355{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.455{{c}}


Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220
{{Optimal ET sequence|legend=1| 12, 43de, 55, 67 }}


Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 -1 | 0 1 4 10 12 15 1 3 }}
[[Badness]] (Sintel): 2.15


Optimal tunings:
=== 13-limit ===
* CTE: ~17/12 = 1\2, ~3/2 = 696.3837
* POTE: ~17/12 = 1\2, ~3/2 = 695.752


{{Optimal ET sequence|legend=1| 12f, 26deff, 38df, 50 }}
[[Subgroup]]: 2.3.5.7.11.13


Badness: 0.017785
[[Comma list]]: 81/80, 176/175, 196/195, 832/825


=== Trimean ===
{{Mapping|legend=1| 1 0 -4 -32 -44 | 0 1 4 22 30}}
Subgroup: 2.3.5.7.11


Comma list: 81/80, 126/125, 1344/1331
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.788{{c}}, ~3/2 = 698.355{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.471{{c}}


Mapping: {{mapping| 1 2 4 7 5 | 0 -3 -12 -30 -11 }}
{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}


: mapping generators: ~2, ~11/10
[[Badness]] (Sintel): 2.04


Optimal tunings:
=== 17-limit ===
* CTE: ~2 = 1\1, ~11/10 = 167.7074
* POTE: ~2 = 1\1, ~11/10 = 167.805


{{Optimal ET sequence|legend=1| 7d, 36d, 43, 50, 93 }}
[[Subgroup]]: 2.3.5.7.11.13.17


Badness: 0.050729
[[Comma list]]: 81/80, 176/175, 189/197, 196/195, 832/825


==== 13-limit ====
{{Mapping|legend=1| 1 0 -4 -32 -44 12| 0 1 4 22 30 -5}}
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 126/125, 144/143, 364/363
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.655{{c}}, ~3/2 = 698.295{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.488{{c}}


Mapping: {{mapping| 1 2 4 7 5 3 | 0 -3 -12 -30 -11 5 }}
{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}


Optimal tunings:  
[[Badness]] (Sintel): 1.98
* CTE: ~2 = 1\1, ~11/10 = 167.7121
* POTE: ~2 = 1\1, ~11/10 = 167.790


{{Optimal ET sequence|legend=1| 7d, 43, 50, 93 }}
=== 19-limit ===


Badness: 0.035445
[[Subgroup]]: 2.3.5.7.11.13.19
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Comma list: 81/80, 126/125, 144/143, 189/187, 221/220
[[Comma list]]: 81/80, 96/95, 176/175, 189/187, 196/195, 832/825


Mapping: {{mapping| 1 2 4 7 5 3 8 | 0 -3 -12 -30 -11 5 -28 }}
{{Mapping|legend=1| 1 0 -4 -32 -44 12 9| 0 1 4 22 30 -5 -3}}


Optimal tunings:  
[[Optimal tuning]]s:  
* CTE: ~2 = 1\1, ~11/10 = 167.7047
* [[WE]]: ~2 = 1199.371{{c}}, ~3/2 = 698.164{{c}}
* POTE: ~2 = 1\1, ~11/10 = 167.786
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.519{{c}}


{{Optimal ET sequence|legend=1| 7dg, 43, 50, 93 }}
{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}


Badness: 0.025221
[[Badness]] (Sintel): 1.95


== Flattone ==
{{Todo|unify precision|review}}
{{Main| Flattone }}


In flattone tunings, the fifth is typically even flatter than that of [[19edo]]. Here, 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 doubly-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. Good tunings for flattone are [[45edo]], [[64edo]], and [[71edo]].
== 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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 81/80, 525/512
[[Comma list]]: 81/80, 672/625
 
{{Mapping|legend=1| 1 0 -4 17 | 0 1 4 -9 }}


{{Multival|legend=1| 1 4 -9 4 -17 -32 }}
{{Mapping|legend=1| 1 0 -4 -21 | 0 1 4 15 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 693.5520
* [[WE]]: ~2 = 1195.4372{{c}}, ~3/2 = 702.2086{{c}}
* [[POTE]]: ~2 = 1\1, ~3/2 = 693.779
: [[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 }}


[[Minimax tuning]]:
{{Optimal ET sequence|legend=1| 5d, 12d, 17c }}
* [[7-odd-limit]]: ~3/2 = {{monzo| 8/13 0 1/13 -1/13 }}
: [{{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|eigenmonzo (unchanged-interval) basis]]: 2.7/5
* [[9-odd-limit]]: ~3/2 = {{monzo| 6/11 2/11 0 -1/11 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 17/11 2/11 0 -1/11 }}, {{monzo| 24/11 8/11 0 -4/11 }}, {{monzo| 34/11 -18/11 0 9/11 }}]
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7


[[Tuning ranges]]:
[[Badness]] (Sintel): 3.40
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [692.353, 701.955]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
 
[[Algebraic generator]]: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.
 
{{Optimal ET sequence|legend=1| 7, 19, 26, 45 }}
 
[[Badness]]: 0.038553


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


Comma list: 45/44, 81/80, 385/384
Comma list: 56/55, 81/80, 132/125


Mapping: {{mapping| 1 0 -4 17 -6 | 0 1 4 -9 6 }}
Mapping: {{mapping| 1 0 -4 -21 -14 | 0 1 4 15 11 }}


Optimal tuning:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 693.2511
* WE: ~2 = 1195.7270{{c}}, ~3/2 = 702.5848{{c}}
* POTE: ~2 = 1\1, ~3/2 = 693.126
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7471{{c}}


Tuning ranges:
{{Optimal ET sequence|legend=0| 5de, 12de, 17c }}
* 11-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]


{{Optimal ET sequence|legend=1| 7, 19, 26, 45, 71bc, 116bcde }}
Badness (Sintel): 2.09


Badness: 0.033839
=== 13-limit ===
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 45/44, 65/64, 78/77, 81/80
Comma list: 26/25, 56/55, 66/65, 81/80


Mapping: {{mapping| 1 0 -4 17 -6 10 | 0 1 4 -9 6 -4 }}
Mapping: {{mapping| 1 0 -4 -21 -14 -9 | 0 1 4 15 11 8 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 693.0293
* WE: ~2 = 1196.3958{{c}}, ~3/2 = 702.9766{{c}}
* POTE: ~2 = 1\1, ~3/2 = 693.058
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7940{{c}}


Tuning ranges:
{{Optimal ET sequence|legend=0| 5de, 12de, 17c, 29c }}
* 13- and 15-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]


{{Optimal ET sequence|legend=1| 7, 19, 26, 45f, 71bcf, 116bcdef }}
Badness (Sintel): 1.67


Badness: 0.022260
== Mohajira ==
{{Main| Mohajira }}


== Flattertone ==
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.
Flattertone tunings are typically at least as flat as [[26edo]]. Here, 17 fifths get to the interval class for 7, so that [[7/4]] is a double-augmented sixth (C-Ax). [[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.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 81/80, 1875/1792
[[Comma list]]: 81/80, 6144/6125


{{Mapping|legend=1| 1 0 -4 -24 | 0 1 4 17 }}
{{Mapping|legend=1| 1 1 0 6 | 0 2 8 -11 }}


: mapping generators: ~2, ~3
: mapping generators: ~2, ~128/105


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 692.6984
* [[WE]]: ~2 = 1200.8160{{c}}, ~128/105 = 348.6518{{c}}
* [[CWE]]: ~2 = 1\1, ~3/2 = 692.0479
: [[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 }}


{{Optimal ET sequence|legend=1| 7d, 19d, 26, 59bcd, 85bccd }}
[[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]]: 0.0961
[[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]


==== 11-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.


[[Subgroup]]: 2.3.5.7
{{Optimal ET sequence|legend=1| 7, 24, 31 }}


[[Comma list]]: 45/44, 81/80, 1375/1344
[[Badness]] (Sintel): 1.41


{{Mapping|legend=1| 1 0 -4 -24 0| 0 1 4 17 6 }}
Scales: [[mohaha7]], [[mohaha10]]


: mapping generators: ~2, ~3
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Optimal tuning]]s:  
Comma list: 81/80, 121/120, 176/175
* [[CTE]]: ~2 = 1\1, ~3/2 = 692.642
* [[CWE]]: ~2 = 1\1, ~3/2 = 692.042


{{Optimal ET sequence|legend=1| 7d, 19d, 26, 59bcd, 85bccd }}
Mapping: {{mapping| 1 1 0 6 2 | 0 2 8 -11 5 }}


== Dominant ==
Optimal tunings:
{{See also| Archytas clan }}
* WE: ~2 = 1201.1562{{c}}, ~11/9 = 348.8124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.4910{{c}}


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


[[Subgroup]]: 2.3.5.7
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]


[[Comma list]]: 36/35, 64/63
{{Optimal ET sequence|legend=0| 7, 24, 31 }}


{{Mapping|legend=1| 1 0 -4 6 | 0 1 4 -2 }}
Badness (Sintel): 0.862


{{Multival|legend=1| 1 4 -2 4 -6 -16 }}
Scales: [[mohaha7]], [[mohaha10]]


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.573
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Tuning ranges]]:  
Comma list: 66/65, 81/80, 105/104, 121/120
* [[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]


{{Optimal ET sequence|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}
Mapping: {{mapping| 1 1 0 6 2 4 | 0 2 8 -11 5 -1 }}


[[Badness]]: 0.020690
Optimal tunings:  
* WE: ~2 = 1200.4256{{c}}, ~11/9 = 348.6819{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.5622{{c}}


=== 11-limit ===
{{Optimal ET sequence|legend=0| 7, 24, 31 }}
Subgroup: 2.3.5.7.11


Comma list: 36/35, 56/55, 64/63
Badness (Sintel): 0.966


Mapping: {{mapping| 1 0 -4 6 13 | 0 1 4 -2 -6 }}
Scales: [[mohaha7]], [[mohaha10]]


Tuning ranges:
=== 17-limit ===
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 705.882] (7\12 to 10\17)
Subgroup: 2.3.5.7.11.13.17
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.254
Comma list: 66/65, 81/80, 105/104, 121/120, 154/153


{{Optimal ET sequence|legend=1| 5, 12, 17c, 29cde }}
Mapping: {{mapping| 1 1 0 6 2 4 7 | 0 2 8 -11 5 -1 -10 }}


Badness: 0.024180
Optimal tunings:  
* WE: ~2 = 1200.0382{{c}}, ~11/9 = 348.7471{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.7360{{c}}


==== 13-limit ====
{{Optimal ET sequence|legend=0| 7, 24, 31 }}
Subgroup: 2.3.5.7.11.13


Comma list: 36/35, 56/55, 64/63, 66/65
Badness (Sintel): 1.05


Mapping: {{mapping| 1 0 -4 6 13 18 | 0 1 4 -2 -6 -9 }}
Scales: [[mohaha7]], [[mohaha10]]


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.636
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Tuning ranges:  
Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152
* 13- and 15-odd-limit diamond monotone: ~3/2 = 705.882 (10\17)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]


{{Optimal ET sequence|legend=1| 12f, 17c, 29cdef }}
Mapping: {{mapping| 1 1 0 6 2 4 7 6 | 0 2 8 -11 5 -1 -10 -6 }}


Badness: 0.024108
Optimal tunings:  
* WE: ~2 = 1199.7469{{c}}, ~11/9 = 348.7367{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.8117{{c}}


==== Dominion ====
{{Optimal ET sequence|legend=0| 7, 24, 31, 55 }}
Subgroup: 2.3.5.7.11.13


Comma list: 26/25, 36/35, 56/55, 64/63
Badness (Sintel): 1.05


Mapping: {{mapping| 1 0 -4 6 13 -9 | 0 1 4 -2 -6 8 }}
Scales: [[mohaha7]], [[mohaha10]]


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.905
== 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.  


{{Optimal ET sequence|legend=1| 5, 12, 17c, 46cde }}
[[Subgroup]]: 2.3.5.7


Badness: 0.027295
[[Comma list]]: 81/80, 392/375


=== Domineering ===
{{Mapping|legend=1| 1 1 0 -1 | 0 2 8 13 }}
Subgroup: 2.3.5.7.11


Comma list: 36/35, 45/44, 64/63
: mapping generators: ~2, ~25/21


Mapping: {{mapping| 1 0 -4 6 -6 | 0 1 4 -2 6 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.0661{{c}}, ~25/21 = 350.3127{{c}}
: [[error map]]: {{val| -0.934 -2.264 +16.188 -13.827 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/21 = 350.4856{{c}}
: error map: {{val| 0.000 -0.984 +17.571 -12.513 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.776
{{Optimal ET sequence|legend=1| 7d, 17c, 24 }}


{{Optimal ET sequence|legend=1| 5e, 7, 12, 19d, 43de }}
[[Badness]] (Sintel): 1.97


Badness: 0.021978
Scales: [[mohaha7]], [[mohaha10]]


==== 13-limit ====
=== 11-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11


Comma list: 36/35, 45/44, 52/49, 64/63
Comma list: 56/55, 77/75, 243/242


Mapping: {{mapping| 1 0 -4 6 -6 10 | 0 1 4 -2 6 -4 }}
Mapping: {{mapping| 1 1 0 -1 2 | 0 2 8 13 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 695.762
Optimal tunings:
* WE: ~2 = 1199.1924{{c}}, ~11/9 = 350.3286{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.4821{{c}}


{{Optimal ET sequence|legend=1| 5ef, 7, 12, 19d, 31def }}
{{Optimal ET sequence|legend=0| 7d, 17c, 24 }}


Badness: 0.027039
Badness (Sintel): 1.20


===== 17-limit =====
Scales: [[mohaha7]], [[mohaha10]]
Subgroup: 2.3.5.7.11.13.17


Comma list: 36/35, 45/44, 51/49, 52/49, 64/63
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Mapping: {{mapping| 1 0 -4 6 -6 10 12 | 0 1 4 -2 6 -4 -5 }}
Comma list: 56/55, 66/65, 77/75, 243/242


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.115
Mapping: {{mapping| 1 1 0 -1 2 4 | 0 2 8 13 5 -1 }}


{{Optimal ET sequence|legend=1| 5ef, 7, 12, 19d, 31def }}
Optimal tunings:
* WE: ~2 = 1198.5986{{c}}, ~11/9 = 350.3353{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.6459{{c}}


Badness: 0.024539
{{Optimal ET sequence|legend=0| 7d, 17c, 24, 41c }}


===== 19-limit =====
Badness (Sintel): 1.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 36/35, 39/38, 45/44, 51/49, 52/49, 57/56
Scales: [[mohaha7]], [[mohaha10]]


Mapping: {{mapping| 1 0 -4 6 -6 10 12 9 | 0 1 4 -2 6 -4 -5 -3 }}
== Liese ==
<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.217
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.  


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


Badness: 0.020398
[[Comma list]]: 81/80, 686/675


==== Dominatrix ====
{{Mapping|legend=1| 1 0 -4 -3 | 0 3 12 11 }}
Subgroup: 2.3.5.7.11.13


Comma list: 27/26, 36/35, 45/44, 64/63
: mapping generators: ~2, ~10/7


Mapping: {{mapping| 1 0 -4 6 -6 -1 | 0 1 4 -2 6 3 }}
[[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 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.544
[[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


{{Optimal ET sequence|legend=1| 5e, 7, 12f, 19df }}
[[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.


Badness: 0.018289
{{Optimal ET sequence|legend=1| 17c, 19, 55, 74d }}
 
[[Badness]] (Sintel): 1.18


=== Domination ===
=== Liesel ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 36/35, 64/63, 77/75
Comma list: 56/55, 81/80, 540/539


Mapping: {{mapping| 1 0 -4 6 -14 | 0 1 4 -2 11 }}
Mapping: {{mapping| 1 0 -4 -3 4 | 0 3 12 11 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.004
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=1| 5e, 12e, 17c, 46cd }}
{{Optimal ET sequence|legend=0| 17c, 19, 36 }}


Badness: 0.036562
Badness (Sintel): 1.35


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


Comma list: 26/25, 36/35, 64/63, 66/65
Comma list: 56/55, 78/77, 81/80, 91/90


Mapping: {{mapping| 1 0 -4 6 -14 -9 | 0 1 4 -2 11 8 }}
Mapping: {{mapping| 1 0 -4 -3 4 0 | 0 3 12 11 -1 7 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.496
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=1| 5e, 12e, 17c }}
{{Optimal ET sequence|legend=0| 17c, 19, 36 }}


Badness: 0.027435
Badness (Sintel): 1.13


=== Arnold ===
=== Elisa ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 22/21, 33/32, 36/35
Comma list: 77/75, 81/80, 99/98


Mapping: {{mapping| 1 0 -4 6 5 | 0 1 4 -2 -1 }}
Mapping: {{mapping| 1 0 -4 -3 -5 | 0 3 12 11 16 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.491
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=1| 5, 7, 12e }}
{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}


Badness: 0.026141
Badness (Sintel): 1.37


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


Comma list: 22/21, 27/26, 33/32, 36/35
Comma list: 66/65, 77/75, 81/80, 99/98


Mapping: {{mapping| 1 0 -4 6 5 -1 | 0 1 4 -2 -1 3 }}
Mapping: {{mapping| 1 0 -4 -3 -5 0 | 0 3 12 11 16 7 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.743
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=1| 5, 7, 12ef, 19def }}
{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}


Badness: 0.023300
Badness (Sintel): 1.11


==== 17-limit ====
=== Lisa ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11


Comma list: 22/21, 27/26, 33/32, 36/35, 51/49
Comma list: 45/44, 81/80, 343/330


Mapping: {{mapping| 1 0 -4 6 5 -1 12 | 0 1 4 -2 -1 3 -5 }}
Mapping: {{mapping| 1 0 -4 -3 -6 | 0 3 12 11 18 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.978
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=1| 5, 7, 12ef, 19def }}
{{Optimal ET sequence|legend=0| 17cee, 19 }}


Badness: 0.024535
Badness (Sintel): 1.81


==== 19-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13


Comma list: 22/21, 27/26, 33/32, 36/35, 51/49, 57/56
Comma list: 45/44, 81/80, 91/88, 147/143


Mapping: {{mapping| 1 0 -4 6 5 -1 12 9 | 0 1 4 -2 -1 3 -5 -3 }}
Mapping: {{mapping| 1 0 -4 -3 -6 0 | 0 3 12 11 18 7 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 697.068
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=1| 5, 7, 12ef, 19def }}
{{Optimal ET sequence|legend=0| 17cee, 19 }}


Badness: 0.021098
Badness (Sintel): 1.49


== Sharptone ==
== Superpine ==
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.
{{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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 21/20, 28/27
[[Comma list]]: 81/80, 1119744/1071875


{{Mapping|legend=1| 1 0 -4 -2 | 0 1 4 3 }}
{{Mapping|legend=1| 1 2 4 1 | 0 -3 -12 13 }}


{{Multival|legend=1| 1 4 3 4 2 -4 }}
[[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 tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 700.140
{{Optimal ET sequence|legend=1| 7, 36, 43, 79c }}


{{Optimal ET sequence|legend=1| 5, 7d, 12d }}
[[Badness]] (Sintel): 3.46


[[Badness]]: 0.024848
=== 11-limit ===
 
=== Meanertone ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 21/20, 28/27, 33/32
Comma list: 81/80, 176/175, 864/847


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


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.615
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=1| 5, 7d, 12de }}
{{Optimal ET sequence|legend=0| 7, 36, 43 }}


Badness: 0.025167
Badness (Sintel): 1.90


== Supermean ==
=== 13-limit ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 81/80, 672/625
Comma list: 78/77, 81/80, 144/143, 176/175


{{Mapping|legend=1| 1 0 -4 -21 | 0 1 4 15 }}
Mapping: {{mapping| 1 2 4 1 5 3 | 0 -3 -12 13 -11 5 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.889
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=1| 5d, 12d, 17c, 29c }}
{{Optimal ET sequence|legend=0| 7, 36, 43 }}


[[Badness]]: 0.134204
Badness (Sintel): 1.52


=== 11-limit ===
== Lithium ==
Subgroup: 2.3.5.7.11
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.


Comma list: 56/55, 81/80, 132/125
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 0 -4 -21 -14 | 0 1 4 15 11 }}
[[Comma list]]: 81/80, 3125/3087


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.096
{{Mapping|legend=1| 3 0 -12 -20 | 0 1 4 6 }}


{{Optimal ET sequence|legend=1| 5de, 12de, 17c, 29c }}
: mapping generators: ~56/45, ~3


Badness: 0.063262
[[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 }}


=== 13-limit ===
{{Optimal ET sequence|legend=1| 12, 33cd, 45, 57 }}
Subgroup: 2.3.5.7.11.13


Comma list: 26/25, 56/55, 66/65, 81/80
[[Badness]] (Sintel): 1.75


Mapping: {{mapping| 1 0 -4 -21 -14 -9 | 0 1 4 15 11 8 }}
== Squares ==
{{Main| Squares }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.094
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]].


{{Optimal ET sequence|legend=1| 5de, 12de, 17c, 29c }}
[[Subgroup]]: 2.3.5.7


Badness: 0.040324
[[Comma list]]: 81/80, 2401/2400


== Mohajira ==
{{Mapping|legend=1| 1 -1 -8 -3 | 0 4 16 9 }}
{{Main| Mohajira }}


Mohajira can be viewed as derived from mohaha which maps the interval one quarter tone flat of 16/9 to 7/4, although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the porwell comma. [[31edo]] makes for an excellent (7-limit) mohajira tuning, with generator 9\31.
: mapping generators: ~2, ~14/9


[[Subgroup]]: 2.3.5.7
[[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 }}


[[Comma list]]: 81/80, 6144/6125
[[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


[[Mapping]]: [{{val| 1 1 0 6 }}, {{val| 0 2 8 -11 }}]
[[Algebraic generator]]: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.


: mapping generators: ~2, ~128/105
{{Optimal ET sequence|legend=1| 14c, 17c, 31, 169b, 200b }}


{{Multival|legend=1| 2 8 -11 8 -23 -48 }}
[[Badness]] (Sintel): 1.16


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~128/105 = 348.415
Scales: [[skwares8]], [[skwares11]], [[skwares14]]


[[Minimax tuning]]:
=== 11-limit ===
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~128/105 = {{monzo| 0 0 1/8 }}
Subgroup: 2.3.5.7.11
: [{{Monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 6 0 -11/8 0 }}]
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.5


[[Tuning ranges]]:
Comma list: 81/80, 99/98, 121/120
* 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 3''x''<sup>3</sup> - 3''x''<sup>2</sup> - 1, 348.6067 cents. Corresponding recurrence converges quickly.
Mapping: {{mapping| 1 -1 -8 -3 -3 | 0 4 16 9 10 }}


{{Optimal ET sequence|legend=1| 7, 24, 31 }}
Optimal tunings:
* WE: ~2 = 1201.6657{{c}}, ~11/7 = 775.1171{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.1754{{c}}


[[Badness]]: 0.055714
{{Optimal ET sequence|legend=0| 14c, 17c, 31, 130bee, 169beee }}


Scales: [[mohaha7]], [[mohaha10]]
Badness (Sintel): 0.715


=== 11-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 121/120, 176/175
Comma list: 66/65, 81/80, 99/98, 121/120


Mapping: [{{val| 1 1 0 6 2 }}, {{val| 0 2 8 -11 5 }}]
Mapping: {{mapping| 1 -1 -8 -3 -3 5 | 0 4 16 9 10 -2 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.477
Optimal tunings:
* WE: ~2 = 1199.8419{{c}}, ~11/7 = 774.3484{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4422{{c}}


Minimax tuning:
{{Optimal ET sequence|legend=0| 14c, 17c, 31, 79cf }}
* 11-odd-limit: ~11/9 = {{monzo| 0 0 1/8 }}
: [{{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 }}]
: Eigenmonzo (unchanged-interval) basis: 2.5


Tuning ranges:
Badness (Sintel): 1.05
* 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|legend=1| 7, 24, 31 }}
==== 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}}


Badness: 0.026064
{{Optimal ET sequence|legend=0| 14cf, 17c, 31f }}


Scales: [[mohaha7]], [[mohaha10]]
Badness (Sintel): 1.11


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


Comma list: 66/65, 81/80, 105/104, 121/120
Comma list: 81/80, 99/98, 105/104, 121/120


Mapping: [{{val| 1 1 0 6 2 4 }}, {{val| 0 2 8 -11 5 -1 }}]
Mapping: {{mapping| 1 -1 -8 -3 -3 -15 | 0 4 16 9 10 29 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.558
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=1| 7, 24, 31 }}
{{Optimal ET sequence|legend=0| 14cf, 31, 45ef, 76e }}


Badness: 0.023388
Badness (Sintel): 1.01


Scales: [[mohaha7]], [[mohaha10]]
===== 17-limit =====
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 66/65, 81/80, 105/104, 121/120, 154/153
Comma list: 81/80, 99/98, 105/104, 120/119, 121/119


Mapping: [{{val| 1 1 0 6 2 4 7 }}, {{val| 0 2 8 -11 5 -1 -10 }}]
Mapping: {{mapping| 1 -1 -8 -3 -3 -15 -3 | 0 4 16 9 10 29 11 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.736
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=1| 7, 24, 31, 86ef }}
{{Optimal ET sequence|legend=0| 14cf, 31 }}


Badness: 0.020576
Badness (Sintel): 1.15


Scales: [[mohaha7]], [[mohaha10]]
===== 19-limit =====
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152
Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119


Mapping: [{{val| 1 1 0 6 2 4 7 6 }}, {{val| 0 2 8 -11 5 -1 -10 -6 }}]
Mapping: {{mapping| 1 -1 -8 -3 -3 -15 -3 -8 | 0 4 16 9 10 29 11 19 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.810
Optimal tunings:
* WE: ~2 = 1201.2461{{c}}, ~11/7 = 774.5783{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8479{{c}}
 
{{Optimal ET sequence|legend=0| 14cf, 31 }}
 
Badness (Sintel): 1.15
 
=== Cuboctahedra ===
Subgroup: 2.3.5.7.11
 
Comma list: 81/80, 385/384, 1375/1372
 
Mapping: {{mapping| 1 -1 -8 -3 17 | 0 4 16 9 -21 }}
 
Optimal tunings:
* WE: ~2 = 1201.4436{{c}}, ~14/9 = 774.9386{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 774.0243{{c}}


{{Optimal ET sequence|legend=1| 7, 24, 31, 55, 86efh }}
{{Optimal ET sequence|legend=0| 31, 107b, 138b, 169be, 200be }}


Badness: 0.017302
Badness (Sintel): 1.88


Scales: [[mohaha7]], [[mohaha10]]
== 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.


== Mohamaq ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 81/80, 392/375
[[Comma list]]: 81/80, 17280/16807


[[Mapping]]: [{{val| 1 1 0 -1 }}, {{val| 0 2 8 13 }}]
{{Mapping|legend=1| 1 1 0 2 | 0 5 20 7 }}


: mapping generators: ~2, ~25/21
: mapping generators: ~2, ~54/49


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 350.586
[[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 }}


{{Optimal ET sequence|legend=1| 7d, 17c, 24, 65cc, 89ccd }}
{{Optimal ET sequence|legend=1| 17c, 26, 43 }}


[[Badness]]: 0.077734
[[Badness]] (Sintel): 2.75
 
Scales: [[mohaha7]], [[mohaha10]]


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


Comma list: 56/55, 77/75, 243/242
Comma list: 81/80, 99/98, 864/847


Mapping: [{{val| 1 1 0 -1 2 }}, {{val| 0 2 8 13 5 }}]
Mapping: {{mapping| 1 1 0 2 3 | 0 5 20 7 4 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.565
Optimal tunings:
* WE: ~2 = 1201.4436{{c}}, ~12/11 = 139.3714{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 139.4038{{c}}


{{Optimal ET sequence|legend=1| 7d, 17c, 24, 65cc, 89ccd }}
{{Optimal ET sequence|legend=0| 17c, 26, 43 }}


Badness: 0.036207
Badness (Sintel): 1.58
 
Scales: [[mohaha7]], [[mohaha10]]


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


Comma list: 56/55, 66/65, 77/75, 243/242
Comma list: 78/77, 81/80, 99/98, 144/143


Mapping: [{{val| 1 1 0 -1 2 4 }}, {{val| 0 2 8 13 5 -1 }}]
Mapping: {{mapping| 1 1 0 2 3 3 | 0 5 20 7 4 6 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.745
Optimal tunings:
* WE: ~2 = 1199.8860{{c}}, ~13/12 = 139.3737{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3817{{c}}


{{Optimal ET sequence|legend=1| 7d, 17c, 24, 41c, 65cc }}
{{Optimal ET sequence|legend=0| 17c, 26, 43 }}


Badness: 0.028738
Badness (Sintel): 1.21


Scales: [[mohaha7]], [[mohaha10]]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


== Liese ==
Comma list: 78/77, 81/80, 99/98, 144/143, 189/187
<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>


Liese splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. [[74edo]] makes for a good liese tuning, though [[19edo]] can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55.
Mapping: {{mapping| 1 1 0 2 3 3 2 | 0 5 20 7 4 6 18 }}


[[Subgroup]]: 2.3.5.7
Optimal tunings:  
* WE: ~2 = 1199.8346{{c}}, ~13/12 = 139.3431{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3544{{c}}


[[Comma list]]: 81/80, 686/675
{{Optimal ET sequence|legend=0| 17cg, 26, 43 }}


[[Mapping]]: [{{val| 1 0 -4 -3 }}, {{val| 0 3 12 11 }}]
Badness (Sintel): 1.06


: mapping generators: ~2, ~10/7
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


{{Multival|legend=1| 3 12 11 12 9 -8 }}
Comma list: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/7 = 632.406
Mapping: {{mapping| 1 1 0 2 3 3 2 1 | 0 5 20 7 4 6 18 28 }}


[[Minimax tuning]]:  
Optimal tunings:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/7 = {{monzo| 1/3 0 1/12 }}
* WE: ~2 = 1199.8891{{c}}, ~13/12 = 139.3001{{c}}
: [{{Monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 2/3 0 11/12 0 }}]
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3080{{c}}
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) 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=0| 17cgh, 26, 43, 69 }}


{{Optimal ET sequence|legend=1| 17c, 19, 55, 74d }}
Badness (Sintel): 1.11


[[Badness]]: 0.046706
== 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.


=== Liesel ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11


Comma list: 56/55, 81/80, 540/539
[[Comma list]]: 81/80, 16875/16807


Mapping: [{{val| 1 0 -4 -3 4 }}, {{val| 0 3 12 11 -1 }}]
{{Mapping|legend=1| 1 -1 -8 -7 | 0 5 20 19 }}


Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 633.073
: mapping generators: ~2, ~10/7


{{Optimal ET sequence|legend=1| 17c, 19, 36, 91cee }}
[[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 }}


Badness: 0.040721
{{Optimal ET sequence|legend=1| 29cd, 31, 188bcd, 219bbcd }}


==== 13-limit ====
[[Badness]] (Sintel): 2.08
Liesel is a very natural 13-limit tuning, given the generator is so near 13/9.


Subgroup: 2.3.5.7.11.13
=== 11-limit ===
Subgroup: 2.3.5.7.11


Comma list: 56/55, 78/77, 81/80, 91/90
Comma list: 81/80, 99/98, 2541/2500


Mapping: [{{val| 1 0 -4 -3 4 0 }}, {{val| 0 3 12 11 -1 7 }}]
Mapping: {{mapping| 1 -1 -8 -7 -11 | 0 5 20 19 28 }}


Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 633.042
Optimal tunings:
* WE: ~2 = 1201.2054{{c}}, ~10/7 = 619.9752{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.4223{{c}}


{{Optimal ET sequence|legend=1| 17c, 19, 36, 91ceef }}
{{Optimal ET sequence|legend=0| 29cde, 31 }}


Badness: 0.027304
Badness (Sintel): 1.42


=== Elisa ===
== Injera ==
Subgroup: 2.3.5.7.11
Injera has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a ~15/14 semitone difference between a half-octave and a perfect fifth. Injera 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.


Comma list: 77/75, 81/80, 99/98
[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name]


Mapping: [{{val| 1 0 -4 -3 -5 }}, {{val| 0 3 12 11 16 }}]
[[Subgroup]]: 2.3.5.7


Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 633.061
[[Comma list]]: 50/49, 81/80


{{Optimal ET sequence|legend=1| 17c, 19e, 36e }}
{{Mapping|legend=1| 2 0 -8 -7 | 0 1 4 4 }}


Badness: 0.041592
: mapping generators: ~7/5, ~3


==== 13-limit ====
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13
* [[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 }}


Comma list: 66/65, 77/75, 81/80, 99/98
[[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]


Mapping: [{{val| 1 0 -4 -3 -5 0 }}, {{val| 0 3 12 11 16 7 }}]
{{Optimal ET sequence|legend=1| 12, 26, 38 }}


Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 632.991
[[Badness]] (Sintel): 0.788


{{Optimal ET sequence|legend=1| 17c, 19e, 36e }}
; 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


Badness: 0.026922
=== 11-limit ===
 
=== Lisa ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 45/44, 81/80, 343/330
Comma list: 45/44, 50/49, 81/80


Mapping: [{{val| 1 0 -4 -3 -6 }}, {{val| 0 3 12 11 18 }}]
Mapping: {{mapping| 2 0 -8 -7 -12 | 0 1 4 4 6 }}


Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 631.370
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}})


{{Optimal ET sequence|legend=1| 17cee, 19 }}
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]


Badness: 0.054829
{{Optimal ET sequence|legend=0| 12, 26 }}
 
Badness (Sintel): 0.764


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


Comma list: 45/44, 81/80, 91/88, 147/143
Comma list: 45/44, 50/49, 78/77, 81/80


Mapping: [{{val| 1 0 -4 -3 -6 0 }}, {{val| 0 3 12 11 18 7 }}]
Mapping: {{mapping| 2 0 -8 -7 -12 -21 | 0 1 4 4 6 9 }}


Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 631.221
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}})


{{Optimal ET sequence|legend=1| 17cee, 19 }}
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]


Badness: 0.036144
{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}


== Superpine ==
Badness (Sintel): 0.891
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. 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 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
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


[[Comma list]]: 81/80, 1119744/1071875
Comma list: 45/44, 50/49, 78/77, 81/80, 85/84


{{Mapping|legend=1| 1 2 4 1 | 0 -3 -12 13 }}
Mapping: {{mapping| 2 0 -8 -7 -12 -21 5 | 0 1 4 4 6 9 1 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~35/32 = 167.279
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=1| 7, 36, 43, 79c }}
{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}


[[Badness]]: 0.137
Badness (Sintel): 0.935


=== 11-limit ===
===== 19-limit =====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 81/80, 176/175, 864/847
Comma list: 45/44, 50/49, 57/56, 78/77, 81/80, 85/84


Mapping: {{mapping| 1 2 4 1 5 | 0 -3 -12 13 -11 }}
Mapping: {{mapping| 2 0 -8 -7 -12 -21 5 -1 | 0 1 4 4 6 9 1 3 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 167.407
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: {{Optimal ET sequence| 7, 36, 43 }}
{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}


Badness: 0.0576
Badness (Sintel): 0.920


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


Comma list: 78/77, 81/80, 144/143, 176/175
Comma list: 27/26, 40/39, 45/44, 50/49


Mapping: {{mapping| 1 2 4 1 5 3 | 0 -3 -12 13 -11 5 }}
Mapping: {{mapping| 2 0 -8 -7 -12 -2 | 0 1 4 4 6 3 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 167.427
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: {{Optimal ET sequence| 7, 36, 43 }}
{{Optimal ET sequence|legend=0| 10cdeef, 12f }}


Badness: 0.0368
Badness (Sintel): 1.10


== Lithium ==
=== Injerous ===
Lithium is named after the 3rd element for being period-3, and also for lithium's molar mass of 6.9 g/mol since 69edo supports it. It supports a [[3L 6s]] scale and thus intuitively can be thought of as "tcherepnin meantone" in that context.
Subgroup: 2.3.5.7.11


[[Subgroup]]: 2.3.5.7
Comma list: 33/32, 50/49, 55/54


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


[[Mapping]]: [{{val| 3 0 -12 -20 }}, {{val| 0 1 4 6 }}]
Optimal tunings:  
* WE: ~7/5 = 603.1682{{c}}, ~3/2 = 694.1945{{c}} (~21/20 = 91.0264{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 691.6107{{c}} (~21/20 = 91.6107{{c}})


: mapping generators: ~56/45, ~3
{{Optimal ET sequence|legend=0| 12e, 14c, 26e, 40cee }}


[[Optimal tuning]] ([[CTE]]): ~56/45 = 1\3, ~3/2 = 695.827
Badness (Sintel): 1.28


{{Optimal ET sequence|legend=1| 12, 33cd, 45, 57 }}
=== Lahoh ===
Subgroup: 2.3.5.7.11


[[Badness]]: 0.0692
Comma list: 50/49, 56/55, 81/77


== Squares ==
Mapping: {{mapping| 2 0 -8 -7 7 | 0 1 4 4 0 }}
{{Main| Squares }}


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


[[Subgroup]]: 2.3.5.7
{{Optimal ET sequence|legend=0| 10cd, 12 }}


[[Comma list]]: 81/80, 2401/2400
Badness (Sintel): 1.42


[[Mapping]]: [{{val| 1 3 8 6 }}, {{val| 0 -4 -16 -9 }}]
=== Teff ===
{{Main| Teff }}


: mapping generators: ~2, ~9/7
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.


{{Multival|legend=1| 4 16 9 16 3 -24 }}
Subgroup: 2.3.5.7.11


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 425.942
Comma list: 50/49, 81/80, 864/847


[[Minimax tuning]]:  
Mapping: {{mapping| 2 1 -4 -3 8 | 0 2 8 8 -1 }}
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~9/7 = {{monzo| 1/2 0 -1/16 }}
: [{{Monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 3/2 0 9/16 0 }}]
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) 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.
: mapping generators: ~7/5, ~16/11


{{Optimal ET sequence|legend=1| 14c, 17c, 31 }}
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}})


[[Badness]]: 0.045993
{{Optimal ET sequence|legend=0| 24d, 26, 50d }}


Scales: [[skwares8]], [[skwares11]], [[skwares14]]
Badness (Sintel): 2.34


=== 11-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 99/98, 121/120
Comma list: 50/49, 78/77, 81/80, 144/143


Mapping: [{{val| 1 3 8 6 7 }}, {{val| 0 -4 -16 -9 -10 }}]
Mapping: {{mapping| 2 1 -4 -3 8 2 | 0 2 8 8 -1 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.957
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=1| 14c, 17c, 31 }}
{{Optimal ET sequence|legend=0| 24d, 26, 50d }}


Badness: 0.021636
Badness (Sintel): 1.65


==== 13-limit ====
==== 17-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17


Comma list: 66/65, 81/80, 99/98, 121/120
Comma list: 50/49, 78/77, 81/80, 85/84, 144/143


Mapping: [{{val| 1 3 8 6 7 3 }}, {{val| 0 -4 -16 -9 -10 2 }}]
Mapping: {{mapping| 2 1 -4 -3 8 2 6 | 0 2 8 8 -1 5 2 }}


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.550
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=1| 14c, 17c, 31, 79cf }}
{{Optimal ET sequence|legend=0| 24d, 26 }}


Badness: 0.025514
Badness (Sintel): 1.50


==== Squad ====
==== 19-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 78/77, 81/80, 91/90, 99/98
Comma list: 50/49, 57/56, 78/77, 81/80, 85/84, 144/143


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


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.7516
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=1| 14cf, 17c, 31f }}
{{Optimal ET sequence|legend=0| 24d, 26 }}


Badness: 0.026877
Badness (Sintel): 1.41


==== Agora ====
== Pombe ==
Subgroup: 2.3.5.7.11.13
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.


Comma list: 81/80, 99/98, 105/104, 121/120
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 3 8 6 7 14 }}, {{val| 0 -4 -16 -9 -10 -29 }}]
[[Comma list]]: 81/80, 300125/294912


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 426.276
{{Mapping|legend=1| 2 1 -4 11 | 0 2 8 -5 }}


{{Optimal ET sequence|legend=1| 14cf, 31, 45ef, 76e }}
: mapping generators: ~735/512, ~35/24


Badness: 0.024522
[[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 }}


===== 17-limit =====
{{Optimal ET sequence|legend=1| 24, 26, 50, 126bcd, 176bcdd, 226bbcdd }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 81/80, 99/98, 105/104, 120/119, 121/119
[[Badness]] (Sintel): 2.94


Mapping: [{{val| 1 3 8 6 7 14 8 }}, {{val| 0 -4 -16 -9 -10 -29 -11 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 426.187
Comma list: 81/80, 245/242, 385/384


{{Optimal ET sequence|legend=1| 14cf, 31, 76e }}
Mapping: {{mapping| 2 1 -4 11 8 | 0 2 8 -5 -1 }}


Badness: 0.022573
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}})


===== 19-limit =====
{{Optimal ET sequence|legend=0| 24, 26, 50 }}
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119
Badness (Sintel): 1.72


Mapping: [{{val| 1 3 8 6 7 14 8 11 }}, {{val| 0 -4 -16 -9 -10 -29 -11 -19 }}]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 426.225
Comma list: 81/80, 105/104, 144/143, 245/242


{{Optimal ET sequence|legend=1| 14cf, 31, 76e }}
Mapping: {{mapping| 2 1 -4 11 8 2 | 0 2 8 -5 -1 5 }}


Badness: 0.018839
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}})


=== Cuboctahedra ===
{{Optimal ET sequence|legend=0| 24, 26, 50 }}
Subgroup: 2.3.5.7.11


Comma list: 81/80, 385/384, 1375/1372
Badness (Sintel): 1.28


Mapping: [{{val| 1 3 8 6 -4 }}, {{val| 0 -4 -16 -9 21 }}]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.993
Comma list: 81/80, 105/104, 144/143, 245/242, 273/272


{{Optimal ET sequence|legend=1| 14ce, 17ce, 31, 107b, 138b, 169be, 200be }}
Mapping: {{mapping| 2 1 -4 11 8 2 6 | 0 2 8 -5 -1 5 2 }}


Badness: 0.056826
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}})


== Jerome ==
{{Optimal ET sequence|legend=0| 24, 26, 50 }}
Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5<sup>1/20</sup>, or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size.


[[Subgroup]]: 2.3.5.7
Badness (Sintel): 1.08


[[Comma list]]: 81/80, 17280/16807
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


[[Mapping]]: [{{val| 1 1 0 2 }}, {{val| 0 5 20 7 }}]
Comma list: 81/80, 105/104, 133/132, 144/143, 171/170, 210/209


: mapping generators: ~2, ~54/49
Mapping: {{mapping| 2 1 -4 11 8 2 6 2 | 0 2 8 -5 -1 5 2 6 }}


{{Multival|legend=1| 5 20 7 20 -3 -40 }}
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}})


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~54/49 = 139.343
{{Optimal ET sequence|legend=0| 24, 26, 50 }}


{{Optimal ET sequence|legend=1| 17c, 26, 43, 69, 112bd }}
Badness (Sintel): 1.01


[[Badness]]: 0.108656
== Orphic ==
Orphic has a semi-octave period and four generators plus a period gives the 3rd harmonic; its ploidacot is diploid alpha-tetracot.  


=== 11-limit ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11


Comma list: 81/80, 99/98, 864/847
[[Comma list]]: 81/80, 5898240/5764801


Mapping: [{{val| 1 1 0 2 3 }}, {{val| 0 5 20 7 4 }}]
{{Mapping|legend=1| 2 1 -4 4 | 0 4 16 3 }}


{{Multival|legend=1| 5 20 7 4 20 -3 -11 -40 -60 -13 }}
: mapping generators: ~2401/1728, ~343/288


Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 139.428
[[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 }}


{{Optimal ET sequence|legend=1| 17c, 26, 43, 69 }}
{{Optimal ET sequence|legend=1| 26, 48c, 74 }}


Badness: 0.047914
[[Badness]] (Sintel): 6.55


=== 13-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11


Comma list: 78/77, 81/80, 99/98, 144/143
Comma list: 81/80, 99/98, 73728/73205


Mapping: [{{val| 1 1 0 2 3 3 }}, {{val| 0 5 20 7 4 6 }}]
Mapping: {{mapping| 2 1 -4 4 8 | 0 4 16 3 -2 }}


Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 139.387
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}})


{{Optimal ET sequence|legend=1| 17c, 26, 43, 69 }}
{{Optimal ET sequence|legend=0| 26, 48c, 74 }}


Badness: 0.029285
Badness (Sintel): 3.36


=== 17-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13


Comma list: 78/77, 81/80, 99/98, 144/143, 189/187
Comma list: 81/80, 99/98, 144/143, 2200/2197


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


Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 139.362
Optimal tunings:
* WE: ~55/39 = 600.0540{{c}}, ~77/64 = 324.2551{{c}} (~7/6 = 275.7989{{c}})
* CWE: ~55/39 = 600.0000{{c}}, ~77/64 = 324.2307{{c}} (~7/6 = 275.7693{{c}})


{{Optimal ET sequence|legend=1| 17cg, 26, 43, 69 }}
{{Optimal ET sequence|legend=0| 26, 48c, 74 }}


Badness: 0.020878
Badness (Sintel): 2.21


=== 19-limit ===
== Cloudtone ==
Subgroup: 2.3.5.7.11.13.17.19
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.


Comma list: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 1 0 2 3 3 2 1 }}, {{val| 0 5 20 7 4 6 18 28 }}]
[[Comma list]]: 81/80, 16807/16384


Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 139.313
{{Mapping|legend=1| 5 0 -20 14 | 0 1 4 0 }}


{{Optimal ET sequence|legend=1| 17cgh, 26, 43, 69 }}
: mapping generators: ~8/7, ~3


Badness: 0.018229
[[Optimal tuning]]s:  
* [[WE]]: ~8/7 = 240.4267{{c}}, ~3/2 = 696.9566{{c}} (~49/48 = 24.3235{{c}})
: [[error map]]: {{val| +2.133 -2.865 +1.513 -2.852 }}
* [[CWE]]: ~8/7 = 240.0000{{c}}, ~3/2 = 696.1637{{c}} (~49/48 = 23.8373{{c}})
: error map: {{val| 0.000 -5.791 -1.659 -8.826 }}


== Meantritone ==
{{Optimal ET sequence|legend=1| 5, 40c, 45, 50 }}
The ''meantritone'' temperament tempers out the mirkwai comma (16875/16807) and trimyna comma (50421/50000) in the 7-limit. In this temperament, three septimal tritones equals ~30/11 (an octave plus [[15/11]]-wide super-fourth) and five of them equals ~[[16/3]] (double-compound fourth). The name "meantritone" is a portmanteau of meantone and tritone, the latter is a generator of this temperament.


[[Subgroup]]: 2.3.5.7
[[Badness]] (Sintel): 2.59


[[Comma list]]: 81/80, 16875/16807
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Mapping]]: [{{val| 1 4 12 12 }}, {{val| 0 -5 -20 -19 }}]
Comma list: 81/80, 385/384, 2401/2376


{{Multival|legend=1| 5 20 19 20 16 -12 }}
Mapping: {{mapping| 5 0 -20 14 41 | 0 1 4 0 -3 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 580.766
Optimal tunings:
* WE: ~8/7 = 240.2740{{c}}, ~3/2 = 697.3317{{c}} (~56/55 = 23.4904{{c}})
* CWE: ~8/7 = 240.0000{{c}}, ~3/2 = 696.6269{{c}} (~56/55 = 23.3731{{c}})


{{Optimal ET sequence|legend=1| 2cd, 29cd, 31 }}
{{Optimal ET sequence|legend=0| 5, 45, 50 }}


[[Badness]]: 0.082239
Badness (Sintel): 2.33


=== 11-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 99/98, 2541/2500
Comma list: 81/80, 105/104, 144/143, 2401/2376


Mapping: [{{val| 1 4 12 12 17 }}, {{val| 0 -5 -20 -19 -28 }}]
Mapping: {{mapping| 5 0 -20 14 41 -21 | 0 1 4 0 -3 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.647
Optimal tunings:
* WE: ~8/7 = 240.2435{{c}}, ~3/2 = 696.8686{{c}} (~91/90 = 23.8618{{c}})
* CWE: ~8/7 = 240.0000{{c}}, ~3/2 = 696.2653{{c}} (~91/90 = 23.7347{{c}})


{{Optimal ET sequence|legend=1| 2cde, 29cde, 31 }}
{{Optimal ET sequence|legend=0| 5, 45f, 50 }}


Badness: 0.042869
Badness (Sintel): 2.02


== Injera ==
== Subgroup extensions ==
Injera has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. [[38edo|38EDO]], which is two parallel [[19edo]]s, is an excellent tuning for injera.
=== Stützel (2.3.5.19) ===
[[Subgroup]]: 2.3.5.19


[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name]
[[Comma list]]: 81/80, 96/95


[[Subgroup]]: 2.3.5.7
{{Mapping|legend=2| 1 0 -4 9 | 0 1 4 -3 }}


[[Comma list]]: 50/49, 81/80
{{Mapping|legend=3| 1 0 -4 0 0 0 0 9 | 0 1 4 0 0 0 0 -3 }}


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


: mapping generators: ~7/5, ~3
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.5513{{c}}, ~3/2 = 697.6058{{c}}
: [[error map]]: {{val| -0.448 -4.798 +4.110 +6.977 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.8222{{c}}
: error map: {{val| 0.000 -4.133 +4.975 +9.020 }}


{{Multival|legend=1| 2 8 8 8 7 -4 }}
{{Optimal ET sequence|legend=1| 5, 7, 12, 31, 43, 98h }}


[[Optimal tuning]] ([[POTE]]): ~7/5 = 1\2, ~3/2 = 694.375
[[Badness]] (Sintel): 0.324


[[Tuning ranges]]:
=== Hypnotone ===
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
Hypnotone is no-sevens [[#Flattone|flattone]].
* 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, 102bcd, 140bccd, 178bbccdd }}
[[Subgroup]]: 2.3.5.11


[[Badness]]: 0.031130
[[Comma list]]: 45/44, 81/80


; Music
{{Mapping|legend=2| 1 0 -4 -6 | 0 1 4 6 }}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 Two Pairs of Socks] (in [[26edo|26EDO]]) by [[Igliashon Jones]]


=== 11-limit ===
{{Mapping|legend=3| 1 0 -4 0 -6 | 0 1 4 0 6 }}
Subgroup: 2.3.5.7.11


Comma list: 45/44, 50/49, 81/80
: mapping generators: ~2, ~3


Mapping: [{{val| 2 0 -8 -7 -12 }}, {{val| 0 1 4 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 }}


{{Multival|legend=1| 2 8 8 12 8 7 12 -4 0 6 }}
{{Optimal ET sequence|legend=1| 7, 12, 19, 26, 45 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 692.840
[[Badness]] (Sintel): 0.326


Tuning ranges:
==== 2.3.5.11.13 subgroup ====
* 11-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
Subgroup: 2.3.5.11.13
* 11-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]


{{Optimal ET sequence|legend=1| 12, 14c, 26, 90bce, 116bcce }}
Comma list: 45/44, 65/64, 81/80


Badness: 0.023124
Subgroup-val mapping: {{mapping| 1 0 -4 -6 10 | 0 1 4 6 -4 }}


==== 13-limit ====
Gencom mapping: {{mapping| 1 0 -4 0 -6 10 | 0 1 4 0 6 -4 }}
Subgroup: 2.3.5.7.11.13


Comma list: 45/44, 50/49, 78/77, 81/80
Optimal tunings:  
* WE: ~2 = 1202.6916{{c}}, ~3/2 = 694.4181{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0870{{c}}


Mapping: [{{val| 2 0 -8 -7 -12 -21 }}, {{val| 0 1 4 4 6 9 }}]
{{Optimal ET sequence|legend=0| 7, 12, 19, 26, 45f }}


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 692.673
Badness (Sintel): 0.561


Tuning ranges:
=== Dequarter ===
* 13- and 15-odd-limit diamond monotone: ~3/2 = 692.308 (15\26)
[[Subgroup]]: 2.3.5.11
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]


{{Optimal ET sequence|legend=1| 12f, 14cf, 26, 38e }}
[[Comma list]]: 33/32, 55/54


Badness: 0.021565
{{Mapping|legend=2| 1 0 -4 5 | 0 1 4 -1 }}


===== 17-limit =====
{{Mapping|legend=3| 1 0 -4 0 5 | 0 1 4 0 -1 }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 45/44, 50/49, 78/77, 81/80, 85/84
: mapping generators: ~2, ~3


Mapping: [{{val| 2 0 -8 -7 -12 -21 5 }}, {{val| 0 1 4 4 6 9 1 }}]
[[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 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 692.487
{{Optimal ET sequence|legend=1| 5, 7, 19e, 26e }}


{{Optimal ET sequence|legend=1| 12f, 14cf, 26 }}
[[Badness]] (Sintel): 0.451


Badness: 0.018358
==== Dreamtone ====
Subgroup: 2.3.5.11.13


===== 19-limit =====
Comma list: 33/32, 55/54, 975/968
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 45/44, 50/49, 57/56, 78/77, 81/80, 85/84
Subgroup-val mapping: {{mapping| 1 0 -4 5 21 | 0 1 4 -1 -11 }}


Mapping: [{{val| 2 0 -8 -7 -12 -21 5 -1 }}, {{val| 0 1 4 4 6 9 1 3 }}]
Gencom mapping: {{mapping| 1 0 -4 0 5 21 | 0 1 4 0 -1 -11 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 692.299
Optimal tunings:  
 
* WE: ~2 = 1207.8248{{c}}, ~3/2 = 694.7806{{c}}
{{Optimal ET sequence|legend=1| 12f, 14cf, 26 }}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 690.1826{{c}}
 
Badness: 0.015118
 
==== Enjera ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 27/26, 40/39, 45/44, 50/49
 
Mapping: [{{val| 2 0 -8 -7 -12 -2 }}, {{val| 0 1 4 4 6 3 }}]
 
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 694.121
 
{{Optimal ET sequence|legend=1| 12f, 14c, 26f, 38eff }}
 
Badness: 0.026542
 
=== Injerous ===
Subgroup: 2.3.5.7.11
 
Comma list: 33/32, 50/49, 55/54
 
Mapping: [{{val| 2 0 -8 -7 10 }}, {{val| 0 1 4 4 -1 }}]
 
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 690.548
 
{{Optimal ET sequence|legend=1| 12e, 14c, 26e, 40cee }}
 
Badness: 0.038577
 
=== Lahoh ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 56/55, 81/77
 
Mapping: [{{val| 2 0 -8 -7 7 }}, {{val| 0 1 4 4 0 }}]
 
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 699.001
 
{{Optimal ET sequence|legend=1| 2cd, 10cd, 12 }}
 
Badness: 0.043062
 
=== Teff ===
{{Main| Teff }}
 
Teff (found by [[Mason Green]]) is to injera what mohajira is to meantone; it splits the generator in half in order to accommodate higher limit intervals, creating a half-octave quarter-tone temperament.
 
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 81/80, 864/847
 
Mapping: [{{val| 2 1 -4 -3 8 }}, {{val| 0 2 8 8 -1 }}]
 
: mapping generators: ~7/5, ~16/11
 
Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 552.5303
 
{{Optimal ET sequence|legend=1| 24d, 26, 50d }}
 
Badness: 0.070689
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 50/49, 78/77, 81/80, 144/143
 
Mapping: [{{val| 2 1 -4 -3 8 2 }}, {{val| 0 2 8 8 -1 5 }}]
 
Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 552.5324
 
{{Optimal ET sequence|legend=1| 24d, 26, 50d }}
 
Badness: 0.040047
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 50/49, 78/77, 81/80, 85/84, 144/143
 
Mapping: [{{val| 2 1 -4 -3 8 2 6 }}, {{val| 0 2 8 8 -1 5 2 }}]
 
Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 552.6558
 
{{Optimal ET sequence|legend=1| 24d, 26 }}
 
Badness: 0.029499
 
==== 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: [{{val| 2 1 -4 -3 8 2 6 2 }}, {{val| 0 2 8 8 -1 5 2 6 }}]
 
Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 552.6382
 
{{Optimal ET sequence|legend=1| 24d, 26 }}
 
Badness: 0.023133
 
== 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. 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]]: [{{val| 2 1 -4 11 }}, {{val| 0 2 8 -5 }}]
 
: mapping generators: ~735/512, ~35/24
 
{{Multival|legend=1| 4 16 -10 16 -27 -68 }}
 
[[Optimal tuning]] ([[POTE]]): ~735/512 = 1\2, ~48/35 = 552.2206
 
{{Optimal ET sequence|legend=1| 24, 26, 50, 126bcd, 176bcdd, 226bbcdd }}
 
[[Badness]]: 0.116104
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 81/80, 245/242, 385/384
 
Mapping: [{{val| 2 1 -4 11 8 }}, {{val| 0 2 8 -5 -1 }}]
 
Optimal tuning (POTE): ~99/70 = 1\2, ~11/8 = 552.0929
 
{{Optimal ET sequence|legend=1| 24, 26, 50 }}
 
Badness: 0.052099
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 81/80, 105/104, 144/143, 245/242
 
Mapping: [{{val| 2 1 -4 11 8 2 }}, {{val| 0 2 8 -5 -1 5 }}]
 
Optimal tuning (POTE): ~99/70 = 1\2, ~11/8 = 552.1498
 
{{Optimal ET sequence|legend=1| 24, 26, 50 }}
 
Badness: 0.031039
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 81/80, 105/104, 144/143, 245/242, 273/272
 
Mapping: [{{val| 2 1 -4 11 8 2 6 }}, {{val| 0 2 8 -5 -1 5 2 }}]
 
Optimal tuning (POTE): ~17/12 = 1\2, ~11/8 = 552.1579
 
{{Optimal ET sequence|legend=1| 24, 26, 50 }}
 
Badness: 0.021260
 
=== 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: [{{val| 2 1 -4 11 8 2 6 2 }}, {{val| 0 2 8 -5 -1 5 2 6 }}]
 
Optimal tuning (POTE): ~17/12 = 1\2, ~11/8 = 552.1196
 
{{Optimal ET sequence|legend=1| 24, 26, 50 }}
 
Badness: 0.016548
 
== Orphic ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 81/80, 5898240/5764801
 
[[Mapping]]: [{{val| 2 5 12 7 }}, {{val| 0 -4 -16 -3 }}]
 
Mapping generators: ~2401/1728, ~7/6
 
{{Multival|legend=1| 8 32 6 32 -13 -76 }}
 
[[Optimal tuning]] ([[POTE]]): ~2401/1728 = 1\2, ~7/6 = 275.794
 
{{Optimal ET sequence|legend=1| 26, 48c, 74, 174bd, 248bbd }}
 
[[Badness]]: 0.258825
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 81/80, 99/98, 73728/73205
 
Mapping: [{{val| 2 5 12 7 6 }}, {{val| 0 -4 -16 -3 2 }}]
 
Optimal tuning (POTE): ~363/256 = 1\2, ~7/6 = 275.762
 
{{Optimal ET sequence|legend=1| 26, 48c, 74, 248bbd, 322bbdd }}
 
Badness: 0.101499
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 81/80, 99/98, 144/143, 2200/2197
 
Mapping: [{{val| 2 5 12 7 6 12 }}, {{val| 0 -4 -16 -3 2 -10 }}]
 
Optimal tuning (POTE): ~55/39 = 1\2, ~7/6 = 275.774
 
{{Optimal ET sequence|legend=1| 26, 48c, 74, 174bd, 248bbd, 322bbdd }}
 
Badness: 0.053482
 
== Cloudtone ==
The ''cloudtone'' temperament (5&amp;50) tempers out the [[cloudy comma]], 16807/16384 and the [[81/80|syntonic comma]], 81/80 in the 7-limit. 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]]: [{{val| 5 0 -20 14 }}, {{val| 0 1 4 0 }}]
 
: mapping generators: ~8/7, ~3
 
{{Multival|legend=1| 5 20 0 20 -14 -56 }}
 
[[Optimal tuning]] ([[POTE]]): ~8/7 = 1\5, ~3/2 = 695.720
 
{{Optimal ET sequence|legend=1| 5, 45, 50 }}
 
[[Badness]]: 0.102256
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 81/80, 385/384, 2401/2376
 
Mapping: [{{val| 5 0 -20 14 41 }}, {{val| 0 1 4 0 -3 }}]
 
Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 696.536
 
{{Optimal ET sequence|legend=1| 5, 45, 50, 155bdd, 205bddd }}
 
Badness: 0.070378
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 81/80, 105/104, 144/143, 2401/2376
 
Mapping: [{{val| 5 0 -20 14 41 -21 }}, {{val| 0 1 4 0 -3 5 }}]


Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 696.162
{{Optimal ET sequence|legend=0| 7, 19eff, 26eff, 33ceeff, 40ceeff }}


{{Optimal ET sequence|legend=1| 5, 45f, 50 }}
Badness (Sintel): 1.40


Badness: 0.048829
== References ==
<references/>


[[Category:Temperament families]]
[[Category:Temperament families]]

Latest revision as of 11:18, 9 May 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, 7058/6875

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, 189/197, 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: 2.3.5.7.11.13.19

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

Mapping: [⟨1 0 -4 -32 -44 12 9], ⟨0 1 4 22 30 -5 -3]]

Optimal tunings:

  • WE: ~2 = 1199.371 ¢, ~3/2 = 698.164 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 698.519 ¢

Optimal ET sequence: 12f, 55f, 67

Badness (Sintel): 1.95

Supermean

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

Subgroup: 2.3.5.7

Comma list: 81/80, 672/625

Mapping: [⟨1 0 -4 -21], ⟨0 1 4 15]]

Optimal tunings:

  • WE: ~2 = 1195.4372 ¢, ~3/2 = 702.2086 ¢
error map: ⟨-4.563 -4.309 +22.521 -8.319]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5375 ¢
error map: ⟨0.000 +2.583 +31.836 -0.763]

Optimal ET sequence: 5d, 12d, 17c

Badness (Sintel): 3.40

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 81/80, 132/125

Mapping: [⟨1 0 -4 -21 -14], ⟨0 1 4 15 11]]

Optimal tunings:

  • WE: ~2 = 1195.7270 ¢, ~3/2 = 702.5848 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.7471 ¢

Optimal ET sequence: 5de, 12de, 17c

Badness (Sintel): 2.09

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 56/55, 66/65, 81/80

Mapping: [⟨1 0 -4 -21 -14 -9], ⟨0 1 4 15 11 8]]

Optimal tunings:

  • WE: ~2 = 1196.3958 ¢, ~3/2 = 702.9766 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.7940 ¢

Optimal ET sequence: 5de, 12de, 17c, 29c

Badness (Sintel): 1.67

Mohajira

Main article: Mohajira

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

Subgroup: 2.3.5.7

Comma list: 81/80, 6144/6125

Mapping: [⟨1 1 0 6], ⟨0 2 8 -11]]

mapping generators: ~2, ~128/105

Optimal tunings:

  • WE: ~2 = 1200.8160 ¢, ~128/105 = 348.6518 ¢
error map: ⟨+0.816 -3.835 +2.901 +0.900]
  • CWE: ~2 = 1200.0000 ¢, ~128/105 = 348.4194 ¢
error map: ⟨0.000 -5.116 +1.041 -1.439]

Minimax tuning:

projection map: [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [6 0 -11/8 0⟩]
Unchanged-interval (eigenmonzo) basis: 2.5

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~128/105 = [347.368, 350.000] (11\38 to 7\24)
  • 7-odd-limit diamond tradeoff: ~128/105 = [347.393, 350.978]
  • 9-odd-limit diamond tradeoff: ~128/105 = [345.601, 350.978]

Algebraic generator: Mohabis, real root of 3x3 - 3x2 - 1, 348.6067 cents. Corresponding recurrence converges quickly.

Optimal ET sequence: 7, 24, 31

Badness (Sintel): 1.41

Scales: mohaha7, mohaha10

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 0 6 2], ⟨0 2 8 -11 5]]

Optimal tunings:

  • WE: ~2 = 1201.1562 ¢, ~11/9 = 348.8124 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.4910 ¢

Minimax tuning:

  • 11-odd-limit: ~11/9 = [0 0 1/8⟩
projection map: [[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [6 0 -11/8 0 0⟩, [2 0 5/8 0 0⟩]
unchanged-interval (eigenmonzo) basis: 2.5

Tuning ranges:

  • 11-odd-limit diamond monotone: ~11/9 = [348.387, 350.000] (9\31 to 7\24)
  • 11-odd-limit diamond tradeoff: ~11/9 = [344.999, 350.978]

Optimal ET sequence: 7, 24, 31

Badness (Sintel): 0.862

Scales: mohaha7, mohaha10

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 81/80, 105/104, 121/120

Mapping: [⟨1 1 0 6 2 4], ⟨0 2 8 -11 5 -1]]

Optimal tunings:

  • WE: ~2 = 1200.4256 ¢, ~11/9 = 348.6819 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.5622 ¢

Optimal ET sequence: 7, 24, 31

Badness (Sintel): 0.966

Scales: mohaha7, mohaha10

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 66/65, 81/80, 105/104, 121/120, 154/153

Mapping: [⟨1 1 0 6 2 4 7], ⟨0 2 8 -11 5 -1 -10]]

Optimal tunings:

  • WE: ~2 = 1200.0382 ¢, ~11/9 = 348.7471 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.7360 ¢

Optimal ET sequence: 7, 24, 31

Badness (Sintel): 1.05

Scales: mohaha7, mohaha10

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152

Mapping: [⟨1 1 0 6 2 4 7 6], ⟨0 2 8 -11 5 -1 -10 -6]]

Optimal tunings:

  • WE: ~2 = 1199.7469 ¢, ~11/9 = 348.7367 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.8117 ¢

Optimal ET sequence: 7, 24, 31, 55

Badness (Sintel): 1.05

Scales: mohaha7, mohaha10

Mohamaq

Mohamaq is a lower-accuracy alternative to mohajira that favors tunings sharp of 24edo. It may be described as 17c & 24; its ploidacot is dicot, the same as mohajira.

Subgroup: 2.3.5.7

Comma list: 81/80, 392/375

Mapping: [⟨1 1 0 -1], ⟨0 2 8 13]]

mapping generators: ~2, ~25/21

Optimal tunings:

  • WE: ~2 = 1199.0661 ¢, ~25/21 = 350.3127 ¢
error map: ⟨-0.934 -2.264 +16.188 -13.827]
  • CWE: ~2 = 1200.0000 ¢, ~25/21 = 350.4856 ¢
error map: ⟨0.000 -0.984 +17.571 -12.513]

Optimal ET sequence: 7d, 17c, 24

Badness (Sintel): 1.97

Scales: mohaha7, mohaha10

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 77/75, 243/242

Mapping: [⟨1 1 0 -1 2], ⟨0 2 8 13 5]]

Optimal tunings:

  • WE: ~2 = 1199.1924 ¢, ~11/9 = 350.3286 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.4821 ¢

Optimal ET sequence: 7d, 17c, 24

Badness (Sintel): 1.20

Scales: mohaha7, mohaha10

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 0 -1 2 4], ⟨0 2 8 13 5 -1]]

Optimal tunings:

  • WE: ~2 = 1198.5986 ¢, ~11/9 = 350.3353 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.6459 ¢

Optimal ET sequence: 7d, 17c, 24, 41c

Badness (Sintel): 1.19

Scales: mohaha7, mohaha10

Liese

Deutsch

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

Subgroup: 2.3.5.7

Comma list: 81/80, 686/675

Mapping: [⟨1 0 -4 -3], ⟨0 3 12 11]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 1201.5548 ¢, ~10/7 = 633.2251 ¢
error map: ⟨+1.555 -2.280 +6.168 -8.015]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 632.5640 ¢
error map: ⟨0.000 -4.263 +4.454 -10.622]

Minimax tuning:

projection map: [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [2/3 0 11/12 0⟩]
unchanged-interval (eigenmonzo) basis: 2.5

Algebraic generator: Radix, the real root of x5 - 2x4 + 2x3 - 2x2 + 2x - 2, also a root of x6 - x5 - 2. The recurrence converges.

Optimal ET sequence: 17c, 19, 55, 74d

Badness (Sintel): 1.18

Liesel

Subgroup: 2.3.5.7.11

Comma list: 56/55, 81/80, 540/539

Mapping: [⟨1 0 -4 -3 4], ⟨0 3 12 11 -1]]

Optimal tunings:

  • WE: ~2 = 1198.8507 ¢, ~10/7 = 632.4668 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 632.9963 ¢

Optimal ET sequence: 17c, 19, 36

Badness (Sintel): 1.35

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 78/77, 81/80, 91/90

Mapping: [⟨1 0 -4 -3 4 0], ⟨0 3 12 11 -1 7]]

Optimal tunings:

  • WE: ~2 = 1199.4968 ¢, ~10/7 = 632.7766 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 633.0082 ¢

Optimal ET sequence: 17c, 19, 36

Badness (Sintel): 1.13

Elisa

Subgroup: 2.3.5.7.11

Comma list: 77/75, 81/80, 99/98

Mapping: [⟨1 0 -4 -3 -5], ⟨0 3 12 11 16]]

Optimal tunings:

  • WE: ~2 = 1201.0489 ¢, ~10/7 = 633.6147 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 633.1644 ¢

Optimal ET sequence: 17c, 19e, 36e

Badness (Sintel): 1.37

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 -4 -3 -5 0], ⟨0 3 12 11 16 7]]

Optimal tunings:

  • WE: ~2 = 1201.4815 ¢, ~10/7 = 633.7720 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 633.1281 ¢

Optimal ET sequence: 17c, 19e, 36e

Badness (Sintel): 1.11

Lisa

Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80, 343/330

Mapping: [⟨1 0 -4 -3 -6], ⟨0 3 12 11 18]]

Optimal tunings:

  • WE: ~2 = 1202.6773 ¢, ~10/7 = 632.7783 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 631.6175 ¢

Optimal ET sequence: 17cee, 19

Badness (Sintel): 1.81

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 81/80, 91/88, 147/143

Mapping: [⟨1 0 -4 -3 -6 0], ⟨0 3 12 11 18 7]]

Optimal tunings:

  • WE: ~2 = 1203.6086 ¢, ~10/7 = 633.1193 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 631.5346 ¢

Optimal ET sequence: 17cee, 19

Badness (Sintel): 1.49

Superpine

See also: No-sevens subgroup temperaments § Superpine

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

Subgroup: 2.3.5.7

Comma list: 81/80, 1119744/1071875

Mapping: [⟨1 2 4 1], ⟨0 -3 -12 13]]

Optimal tunings:

  • WE: ~2 = 1199.3652 ¢, ~35/32 = 167.1615 ¢
error map: ⟨-0.635 -4.709 +5.209 +3.639]
  • CWE: ~2 = 1200.0000 ¢, ~35/32 = 167.2561 ¢
error map: ⟨0.000 -3.723 +6.613 +5.503]

Optimal ET sequence: 7, 36, 43, 79c

Badness (Sintel): 3.46

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 176/175, 864/847

Mapping: [⟨1 2 4 1 5], ⟨0 -3 -12 13 -11]]

Optimal tunings:

  • WE: ~2 = 1199.0522 ¢, ~11/10 = 167.1904 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 167.3382 ¢

Optimal ET sequence: 7, 36, 43

Badness (Sintel): 1.90

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 81/80, 144/143, 176/175

Mapping: [⟨1 2 4 1 5 3], ⟨0 -3 -12 13 -11 5]]

Optimal tunings:

  • WE: ~2 = 1199.4286 ¢, ~11/10 = 167.3105 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 167.3958 ¢

Optimal ET sequence: 7, 36, 43

Badness (Sintel): 1.52

Lithium

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

Subgroup: 2.3.5.7

Comma list: 81/80, 3125/3087

Mapping: [⟨3 0 -12 -20], ⟨0 1 4 6]]

mapping generators: ~56/45, ~3

Optimal tunings:

  • WE: ~56/45 = 400.6744 ¢, ~3/2 = 695.8474 ¢ {~15/14 = 105.5015 ¢)
error map: ⟨+2.023 -4.084 -2.924 +4.910]
  • CWE: ~56/45 = 400.0000 ¢, ~3/2 = 695.1413 ¢ {~15/14 = 104.8587 ¢)
error map: ⟨0.000 -6.814 -5.748 +2.022]

Optimal ET sequence: 12, 33cd, 45, 57

Badness (Sintel): 1.75

Squares

Main article: Squares

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

Subgroup: 2.3.5.7

Comma list: 81/80, 2401/2400

Mapping: [⟨1 -1 -8 -3], ⟨0 4 16 9]]

mapping generators: ~2, ~14/9

Optimal tunings:

  • WE: ~2 = 1201.2488 ¢, ~14/9 = 774.8640 ¢
error map: ⟨+1.249 -3.748 +1.520 +1.204]
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 774.1560 ¢
error map: ⟨0.000 -5.331 +0.183 -1.422]

Minimax tuning:

projection map: [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [3/2 0 9/16 0⟩]
unchanged-interval (eigenmonzo) basis: 2.5

Algebraic generator: Sceptre2, the positive root of 9x2 + x - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.

Optimal ET sequence: 14c, 17c, 31, 169b, 200b

Badness (Sintel): 1.16

Scales: skwares8, skwares11, skwares14

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -1 -8 -3 -3], ⟨0 4 16 9 10]]

Optimal tunings:

  • WE: ~2 = 1201.6657 ¢, ~11/7 = 775.1171 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 774.1754 ¢

Optimal ET sequence: 14c, 17c, 31, 130bee, 169beee

Badness (Sintel): 0.715

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -1 -8 -3 -3 5], ⟨0 4 16 9 10 -2]]

Optimal tunings:

  • WE: ~2 = 1199.8419 ¢, ~11/7 = 774.3484 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 774.4422 ¢

Optimal ET sequence: 14c, 17c, 31, 79cf

Badness (Sintel): 1.05

Squad

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 81/80, 91/90, 99/98

Mapping: [⟨1 -1 -8 -3 -3 -6], ⟨0 4 16 9 10 15]]

Optimal tunings:

  • WE: ~2 = 1202.0312 ¢, ~11/7 = 775.5589 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 774.4140 ¢

Optimal ET sequence: 14cf, 17c, 31f

Badness (Sintel): 1.11

Agora

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 105/104, 121/120

Mapping: [⟨1 -1 -8 -3 -3 -15], ⟨0 4 16 9 10 29]]

Optimal tunings:

  • WE: ~2 = 1202.3228 ¢, ~11/7 = 775.2214 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 773.8617 ¢

Optimal ET sequence: 14cf, 31, 45ef, 76e

Badness (Sintel): 1.01

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 99/98, 105/104, 120/119, 121/119

Mapping: [⟨1 -1 -8 -3 -3 -15 -3], ⟨0 4 16 9 10 29 11]]

Optimal tunings:

  • WE: ~2 = 1201.4340 ¢, ~11/7 = 774.7375 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 773.8955 ¢

Optimal ET sequence: 14cf, 31

Badness (Sintel): 1.15

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119

Mapping: [⟨1 -1 -8 -3 -3 -15 -3 -8], ⟨0 4 16 9 10 29 11 19]]

Optimal tunings:

  • WE: ~2 = 1201.2461 ¢, ~11/7 = 774.5783 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 773.8479 ¢

Optimal ET sequence: 14cf, 31

Badness (Sintel): 1.15

Cuboctahedra

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -1 -8 -3 17], ⟨0 4 16 9 -21]]

Optimal tunings:

  • WE: ~2 = 1201.4436 ¢, ~14/9 = 774.9386 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 774.0243 ¢

Optimal ET sequence: 31, 107b, 138b, 169be, 200be

Badness (Sintel): 1.88

Jerome

Jerome is related to Hieronymus' tuning; the Hieronymus generator is 51/20, or 139.316 cents. It may be described as 17c & 26; its ploidacot is pentacot. While the generator represents both 13/12 and 12/11, the CTE/CWE and Hieronymus generators are close to 13/12 in size.

Subgroup: 2.3.5.7

Comma list: 81/80, 17280/16807

Mapping: [⟨1 1 0 2], ⟨0 5 20 7]]

mapping generators: ~2, ~54/49

Optimal tunings:

  • WE: ~2 = 1200.1640 ¢, ~54/49 = 139.3624 ¢
error map: ⟨+0.164 -4.979 +0.934 +7.039]
  • CWE: ~2 = 1200.0000 ¢, ~54/49 = 139.3528 ¢
error map: ⟨0.000 -5.191 +0.741 +6.643]

Optimal ET sequence: 17c, 26, 43

Badness (Sintel): 2.75

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 864/847

Mapping: [⟨1 1 0 2 3], ⟨0 5 20 7 4]]

Optimal tunings:

  • WE: ~2 = 1201.4436 ¢, ~12/11 = 139.3714 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/11 = 139.4038 ¢

Optimal ET sequence: 17c, 26, 43

Badness (Sintel): 1.58

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 81/80, 99/98, 144/143

Mapping: [⟨1 1 0 2 3 3], ⟨0 5 20 7 4 6]]

Optimal tunings:

  • WE: ~2 = 1199.8860 ¢, ~13/12 = 139.3737 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.3817 ¢

Optimal ET sequence: 17c, 26, 43

Badness (Sintel): 1.21

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 78/77, 81/80, 99/98, 144/143, 189/187

Mapping: [⟨1 1 0 2 3 3 2], ⟨0 5 20 7 4 6 18]]

Optimal tunings:

  • WE: ~2 = 1199.8346 ¢, ~13/12 = 139.3431 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.3544 ¢

Optimal ET sequence: 17cg, 26, 43

Badness (Sintel): 1.06

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143

Mapping: [⟨1 1 0 2 3 3 2 1], ⟨0 5 20 7 4 6 18 28]]

Optimal tunings:

  • WE: ~2 = 1199.8891 ¢, ~13/12 = 139.3001 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.3080 ¢

Optimal ET sequence: 17cgh, 26, 43, 69

Badness (Sintel): 1.11

Meantritone

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

Subgroup: 2.3.5.7

Comma list: 81/80, 16875/16807

Mapping: [⟨1 -1 -8 -7], ⟨0 5 20 19]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 1201.3832 ¢, ~10/7 = 619.9478 ¢
error map: ⟨+1.383 -3.599 +1.576 +0.499]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.3176 ¢
error map: ⟨0.000 -5.367 +0.038 -1.791]

Optimal ET sequence: 29cd, 31, 188bcd, 219bbcd

Badness (Sintel): 2.08

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -1 -8 -7 -11], ⟨0 5 20 19 28]]

Optimal tunings:

  • WE: ~2 = 1201.2054 ¢, ~10/7 = 619.9752 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.4223 ¢

Optimal ET sequence: 29cde, 31

Badness (Sintel): 1.42

Injera

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

Origin of the name

Subgroup: 2.3.5.7

Comma list: 50/49, 81/80

Mapping: [⟨2 0 -8 -7], ⟨0 1 4 4]]

mapping generators: ~7/5, ~3

Optimal tunings:

  • WE: ~7/5 = 600.6662 ¢, ~3/2 = 695.1463 ¢ (~21/20 = 94.4801 ¢)
error map: ⟨+1.332 -5.476 -5.729 +12.425]
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 694.7712 ¢ (~21/20 = 94.7712 ¢)
error map: ⟨0.000 -7.184 -7.229 +10.259]

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
  • 7-odd-limit diamond tradeoff: ~3/2 = [688.957, 701.955]
  • 9-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]

Optimal ET sequence: 12, 26, 38

Badness (Sintel): 0.788

Music

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 0 -8 -7 -12], ⟨0 1 4 4 6]]

Optimal tunings:

  • WE: ~7/5 = 600.9350 ¢, ~3/2 = 693.9198 ¢ (~21/20 = 92.9848 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 693.3539 ¢ (~21/20 = 93.3539 ¢)

Tuning ranges:

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

Optimal ET sequence: 12, 26

Badness (Sintel): 0.764

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 50/49, 78/77, 81/80

Mapping: [⟨2 0 -8 -7 -12 -21], ⟨0 1 4 4 6 9]]

Optimal tunings:

  • WE: ~7/5 = 600.9982 ¢, ~3/2 = 693.8249 ¢ (~21/20 = 92.8267 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 693.0992 ¢ (~21/20 = 93.0992 ¢)

Tuning ranges:

  • 13-odd-limit diamond monotone: ~3/2 = 692.308 (15\26)
  • 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]

Optimal ET sequence: 12f, 14cf, 26

Badness (Sintel): 0.891

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 45/44, 50/49, 78/77, 81/80, 85/84

Mapping: [⟨2 0 -8 -7 -12 -21 5], ⟨0 1 4 4 6 9 1]]

Optimal tunings:

  • WE: ~7/5 = 601.1757 ¢, ~3/2 = 693.8441 ¢ (~21/20 = 92.6684 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 692.8879 ¢ (~21/20 = 92.8879 ¢)

Optimal ET sequence: 12f, 14cf, 26

Badness (Sintel): 0.935

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 45/44, 50/49, 57/56, 78/77, 81/80, 85/84

Mapping: [⟨2 0 -8 -7 -12 -21 5 -1], ⟨0 1 4 4 6 9 1 3]]

Optimal tunings:

  • WE: ~7/5 = 601.4245 ¢, ~3/2 = 693.9426 ¢ (~21/20 = 92.5181 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 692.7606 ¢ (~21/20 = 92.7606 ¢)

Optimal ET sequence: 12f, 14cf, 26

Badness (Sintel): 0.920

Enjera

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 0 -8 -7 -12 -2], ⟨0 1 4 4 6 3]]

Optimal tunings:

  • WE: ~7/5 = 599.1863 ¢, ~3/2 = 693.1791 ¢ (~21/20 = 93.9929 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 693.6809 ¢ (~21/20 = 93.6809 ¢)

Optimal ET sequence: 10cdeef, 12f

Badness (Sintel): 1.10

Injerous

Subgroup: 2.3.5.7.11

Comma list: 33/32, 50/49, 55/54

Mapping: [⟨2 0 -8 -7 10], ⟨0 1 4 4 -1]]

Optimal tunings:

  • WE: ~7/5 = 603.1682 ¢, ~3/2 = 694.1945 ¢ (~21/20 = 91.0264 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 691.6107 ¢ (~21/20 = 91.6107 ¢)

Optimal ET sequence: 12e, 14c, 26e, 40cee

Badness (Sintel): 1.28

Lahoh

Subgroup: 2.3.5.7.11

Comma list: 50/49, 56/55, 81/77

Mapping: [⟨2 0 -8 -7 7], ⟨0 1 4 4 0]]

Optimal tunings:

  • WE: ~7/5 = 597.3179 ¢, ~3/2 = 695.8759 ¢ (~21/20 = 98.5581 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 697.8757 ¢ (~21/20 = 97.8757 ¢)

Optimal ET sequence: 10cd, 12

Badness (Sintel): 1.42

Teff

Main article: Teff

Teff, found and named by Mason Green, is to injera what mohajira is to meantone; it splits the generator in halves in order to accommodate higher-limit intervals, creating a half-octave quartertone temperament. Its ploidacot is diploid alpha-dicot.

Subgroup: 2.3.5.7.11

Comma list: 50/49, 81/80, 864/847

Mapping: [⟨2 1 -4 -3 8], ⟨0 2 8 8 -1]]

mapping generators: ~7/5, ~16/11

Optimal tunings:

  • WE: ~7/5 = 600.2802 ¢, ~16/11 = 647.7720 ¢ (~33/32 = 47.4918 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~16/11 = 647.5224 ¢ (~33/32 = 47.5224 ¢)

Optimal ET sequence: 24d, 26, 50d

Badness (Sintel): 2.34

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 78/77, 81/80, 144/143

Mapping: [⟨2 1 -4 -3 8 2], ⟨0 2 8 8 -1 5]]

Optimal tunings:

  • WE: ~7/5 = 600.3037 ¢, ~16/11 = 647.7954 ¢ (~33/32 = 47.4917 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~16/11 = 647.5256 ¢ (~33/32 = 47.5256 ¢)

Optimal ET sequence: 24d, 26, 50d

Badness (Sintel): 1.65

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 78/77, 81/80, 85/84, 144/143

Mapping: [⟨2 1 -4 -3 8 2 6], ⟨0 2 8 8 -1 5 2]]

Optimal tunings:

  • WE: ~7/5 = 600.5123 ¢, ~16/11 = 647.8970 ¢ (~34/33 = 47.3846 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~16/11 = 647.4314 ¢ (~34/33 = 47.4314 ¢)

Optimal ET sequence: 24d, 26

Badness (Sintel): 1.50

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 50/49, 57/56, 78/77, 81/80, 85/84, 144/143

Mapping: [⟨2 1 -4 -3 8 2 6 2], ⟨0 2 8 8 -1 5 2 6]]

Optimal tunings:

  • WE: ~7/5 = 600.6308 ¢, ~16/11 = 648.0424 ¢ (~34/33 = 47.4116 ¢)
  • CWE: ~7/5 = 600.0000 ¢, ~16/11 = 647.4715 ¢ (~34/33 = 47.4715 ¢)

Optimal ET sequence: 24d, 26

Badness (Sintel): 1.41

Pombe

Pombe (named after the African millet beer) is a variant of #Teff by Kaiveran Lugheidh that eschews the tempering of 50/49 to attain more accuracy in the 7-limit. Its ploidacot is diploid alpha-dicot, the same as teff. Oddly, the 7th harmonic has a lesser generator distance than in teff (-5 vs +8), but this combined with the fact that other harmonics are in the opposite direction means that the 7-limit diamond is more complex overall.

Subgroup: 2.3.5.7

Comma list: 81/80, 300125/294912

Mapping: [⟨2 1 -4 11], ⟨0 2 8 -5]]

mapping generators: ~735/512, ~35/24

Optimal tunings:

  • WE: ~735/512 = 601.0652 ¢, ~35/24 = 648.9295 ¢ (~36/35 = 47.8642 ¢)
error map: ⟨+2.130 -3.031 +0.861 -1.756]
  • CWE: ~735/512 = 600.0000 ¢, ~35/24 = 647.8628 ¢ (~36/35 = 47.8628 ¢)
error map: ⟨0.000 -6.229 -3.411 -8.140]

Optimal ET sequence: 24, 26, 50, 126bcd, 176bcdd, 226bbcdd

Badness (Sintel): 2.94

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 245/242, 385/384

Mapping: [⟨2 1 -4 11 8], ⟨0 2 8 -5 -1]]

Optimal tunings:

  • WE: ~99/70 = 600.7890 ¢, ~16/11 = 648.7592 ¢ (~36/35 = 47.9701 ¢)
  • CWE: ~99/70 = 600.0000 ¢, ~16/11 = 647.9516 ¢ (~36/35 = 47.9516 ¢)

Optimal ET sequence: 24, 26, 50

Badness (Sintel): 1.72

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 1 -4 11 8 2], ⟨0 2 8 -5 -1 5]]

Optimal tunings:

  • WE: ~99/70 = 600.6971 ¢, ~16/11 = 648.6029 ¢ (~36/35 = 47.9058 ¢)
  • CWE: ~99/70 = 600.0000 ¢, ~16/11 = 647.8990 ¢ (~36/35 = 47.8990 ¢)

Optimal ET sequence: 24, 26, 50

Badness (Sintel): 1.28

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 105/104, 144/143, 245/242, 273/272

Mapping: [⟨2 1 -4 11 8 2 6], ⟨0 2 8 -5 -1 5 2]]

Optimal tunings:

  • WE: ~17/12 = 600.7610 ¢, ~16/11 = 648.6638 ¢ (~36/35 = 47.9028 ¢)
  • CWE: ~17/12 = 600.0000 ¢, ~16/11 = 647.8990 ¢ (~36/35 = 47.8990 ¢)

Optimal ET sequence: 24, 26, 50

Badness (Sintel): 1.08

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨2 1 -4 11 8 2 6 2], ⟨0 2 8 -5 -1 5 2 6]]

Optimal tunings:

  • WE: ~17/12 = 600.8048 ¢, ~16/11 = 648.7494 ¢ (~36/35 = 47.9446 ¢)
  • CWE: ~17/12 = 600.0000 ¢, ~16/11 = 647.9425 ¢ (~36/35 = 47.9425 ¢)

Optimal ET sequence: 24, 26, 50

Badness (Sintel): 1.01

Orphic

Orphic has a semi-octave period and four generators plus a period gives the 3rd harmonic; its ploidacot is diploid alpha-tetracot.

Subgroup: 2.3.5.7

Comma list: 81/80, 5898240/5764801

Mapping: [⟨2 1 -4 4], ⟨0 4 16 3]]

mapping generators: ~2401/1728, ~343/288

Optimal tunings:

  • WE: ~2401/1728 = 600.1767 ¢, ~343/288 = 324.3015 ¢ (~7/6 = 275.8751 ¢)
error map: ⟨+0.353 -4.572 +1.804 +4.785]
  • CWE: ~2401/1728 = 600.0000 ¢, ~343/288 = 324.2285 ¢ (~7/6 = 275.7715 ¢)
error map: ⟨0.000 -5.041 +1.342 +3.860]

Optimal ET sequence: 26, 48c, 74

Badness (Sintel): 6.55

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 73728/73205

Mapping: [⟨2 1 -4 4 8], ⟨0 4 16 3 -2]]

Optimal tunings:

  • WE: ~363/256 = 600.1011 ¢, ~77/64 = 324.2923 ¢ (~7/6 = 275.8088 ¢)
  • CWE: ~363/256 = 600.0000 ¢, ~77/64 = 324.2463 ¢ (~7/6 = 275.7537 ¢)

Optimal ET sequence: 26, 48c, 74

Badness (Sintel): 3.36

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 144/143, 2200/2197

Mapping: [⟨2 1 -4 4 8 2], ⟨0 4 16 3 -2 10]]

Optimal tunings:

  • WE: ~55/39 = 600.0540 ¢, ~77/64 = 324.2551 ¢ (~7/6 = 275.7989 ¢)
  • CWE: ~55/39 = 600.0000 ¢, ~77/64 = 324.2307 ¢ (~7/6 = 275.7693 ¢)

Optimal ET sequence: 26, 48c, 74

Badness (Sintel): 2.21

Cloudtone

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

Subgroup: 2.3.5.7

Comma list: 81/80, 16807/16384

Mapping: [⟨5 0 -20 14], ⟨0 1 4 0]]

mapping generators: ~8/7, ~3

Optimal tunings:

  • WE: ~8/7 = 240.4267 ¢, ~3/2 = 696.9566 ¢ (~49/48 = 24.3235 ¢)
error map: ⟨+2.133 -2.865 +1.513 -2.852]
  • CWE: ~8/7 = 240.0000 ¢, ~3/2 = 696.1637 ¢ (~49/48 = 23.8373 ¢)
error map: ⟨0.000 -5.791 -1.659 -8.826]

Optimal ET sequence: 5, 40c, 45, 50

Badness (Sintel): 2.59

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨5 0 -20 14 41], ⟨0 1 4 0 -3]]

Optimal tunings:

  • WE: ~8/7 = 240.2740 ¢, ~3/2 = 697.3317 ¢ (~56/55 = 23.4904 ¢)
  • CWE: ~8/7 = 240.0000 ¢, ~3/2 = 696.6269 ¢ (~56/55 = 23.3731 ¢)

Optimal ET sequence: 5, 45, 50

Badness (Sintel): 2.33

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨5 0 -20 14 41 -21], ⟨0 1 4 0 -3 5]]

Optimal tunings:

  • WE: ~8/7 = 240.2435 ¢, ~3/2 = 696.8686 ¢ (~91/90 = 23.8618 ¢)
  • CWE: ~8/7 = 240.0000 ¢, ~3/2 = 696.2653 ¢ (~91/90 = 23.7347 ¢)

Optimal ET sequence: 5, 45f, 50

Badness (Sintel): 2.02

Subgroup extensions

Stützel (2.3.5.19)

Subgroup: 2.3.5.19

Comma list: 81/80, 96/95

Subgroup-val mapping: [⟨1 0 -4 9], ⟨0 1 4 -3]]

Gencom mapping: [⟨1 0 -4 0 0 0 0 9], ⟨0 1 4 0 0 0 0 -3]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.5513 ¢, ~3/2 = 697.6058 ¢
error map: ⟨-0.448 -4.798 +4.110 +6.977]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.8222 ¢
error map: ⟨0.000 -4.133 +4.975 +9.020]

Optimal ET sequence: 5, 7, 12, 31, 43, 98h

Badness (Sintel): 0.324

Hypnotone

Hypnotone is no-sevens flattone.

Subgroup: 2.3.5.11

Comma list: 45/44, 81/80

Subgroup-val mapping: [⟨1 0 -4 -6], ⟨0 1 4 6]]

Gencom mapping: [⟨1 0 -4 0 -6], ⟨0 1 4 0 6]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1202.0621 ¢, ~3/2 = 694.5448 ¢
error map: ⟨+2.062 -5.348 -8.135 +15.951]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 693.9085 ¢
error map: ⟨0.000 -8.047 -10.680 +12.133]

Optimal ET sequence: 7, 12, 19, 26, 45

Badness (Sintel): 0.326

2.3.5.11.13 subgroup

Subgroup: 2.3.5.11.13

Comma list: 45/44, 65/64, 81/80

Subgroup-val mapping: [⟨1 0 -4 -6 10], ⟨0 1 4 6 -4]]

Gencom mapping: [⟨1 0 -4 0 -6 10], ⟨0 1 4 0 6 -4]]

Optimal tunings:

  • WE: ~2 = 1202.6916 ¢, ~3/2 = 694.4181 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 693.0870 ¢

Optimal ET sequence: 7, 12, 19, 26, 45f

Badness (Sintel): 0.561

Dequarter

Subgroup: 2.3.5.11

Comma list: 33/32, 55/54

Subgroup-val mapping: [⟨1 0 -4 5], ⟨0 1 4 -1]]

Gencom mapping: [⟨1 0 -4 0 5], ⟨0 1 4 0 -1]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1206.5832 ¢, ~3/2 = 695.8763 ¢
error map: ⟨+6.583 +0.504 -2.809 -20.862]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 693.1206 ¢
error map: ⟨0.000 -8.834 -13.831 -44.439]

Optimal ET sequence: 5, 7, 19e, 26e

Badness (Sintel): 0.451

Dreamtone

Subgroup: 2.3.5.11.13

Comma list: 33/32, 55/54, 975/968

Subgroup-val mapping: [⟨1 0 -4 5 21], ⟨0 1 4 -1 -11]]

Gencom mapping: [⟨1 0 -4 0 5 21], ⟨0 1 4 0 -1 -11]]

Optimal tunings:

  • WE: ~2 = 1207.8248 ¢, ~3/2 = 694.7806 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 690.1826 ¢

Optimal ET sequence: 7, 19eff, 26eff, 33ceeff, 40ceeff

Badness (Sintel): 1.40

References

  1. ↑ 1.0 1.1 Yahoo! Tuning Group | good 11-limit meantones
  2. ↑ Yahoo! Tuning Group | The meantone family
  3. ↑ 3.0 3.1 3.2 Yahoo! Tuning Group | names and definitions: meantone
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