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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.
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 ====
Line 86: 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 120: Line 120:
{{See also| Huygens vs meanpop }}
{{See also| Huygens vs meanpop }}


Undecimal meantone<ref name="meantone & meanpop 2003">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_6048.html#6052 Yahoo! Tuning Group | ''good 11-limit meantones'']</ref> a.k.a. huygens<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10437.html Yahoo! Tuning Group | ''The meantone family'']</ref><ref name="meantone & meanpop 2004">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10864.html#10870 Yahoo! Tuning Group | ''names and definitions: meantone'']</ref> maps the [[11/8]] to the double-augmented third (C–E𝄪), 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 a 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 150: Line 150:
* [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:  
* WE: ~2 = 1200.8149{{c}}, ~3/2 = 697.1155{{c}}
* WE: ~2 = 1199.9389{{c}}, ~3/2 = 697.2282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.7085{{c}}
* 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 }}
: unchanged-interval (eigenmonzo) 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=0| 12f, 19e, 31 }}
{{Optimal ET sequence|legend=0| 12, 31, 43, 74 }}


Badness (Sintel): 0.746
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:  
* WE: ~2 = 1201.2376{{c}}, ~3/2 = 697.0954{{c}}
* WE: ~2 = 1199.5811{{c}}, ~3/2 = 697.0918{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4563{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3303{{c}}


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


Badness (Sintel): 0.970
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:
* WE: ~2 = 1199.2931{{c}}, ~3/2 = 696.9690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3736{{c}}
 
{{Optimal ET sequence|legend=0| 12, 31, 43, 74gh }}
 
Badness (Sintel): 1.07
 
==== Fokkertone ====
Fokkertone maps the [[13/8]] to the double-augmented fifth (C–G𝄪). Note that the minor third conflates 13/11 with 6/5, and that 11/10~13/12 is a double-augmented unison; 12/11 is a double-diminished third; and 14/13 is a minor second. 31edo can be recommended as a tuning since it is the only 13-odd-limit diamond monotone tuning.
 
This extension used to be known as ''tridecimal meantone'', but was decanonicalized in 2025.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 66/65, 81/80, 99/98, 105/104
 
Mapping: {{mapping| 1 0 -4 -13 -25 -20 | 0 1 4 10 18 15 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.4134{{c}}, ~3/2 = 697.0933{{c}}
* WE: ~2 = 1200.8149{{c}}, ~3/2 = 697.1155{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.3526{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.7085{{c}}


{{Optimal ET sequence|legend=0| 12fghh, 19egh, 31, 50e }}
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 (Sintel): 1.09
{{Optimal ET sequence|legend=0| 12f, 19e, 31 }}


===== Huygens =====
Badness (Sintel): 0.746
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


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Badness (Sintel): 1.02
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 233: Line 260:
Badness (Sintel): 1.10
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:  
* WE: ~2 = 1199.9389{{c}}, ~3/2 = 697.2282{{c}}
* WE: ~2 = 1199.9122{{c}}, ~3/2 = 697.4779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.2627{{c}}
* 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=0| 12, 31, 43, 74 }}
Badness (Sintel): 1.09
 
Badness (Sintel): 1.07


===== 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:  
* WE: ~2 = 1199.5811{{c}}, ~3/2 = 697.0918{{c}}
* WE: ~2 = 1199.3793{{c}}, ~3/2 = 697.2833{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3303{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6222{{c}}


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


Badness (Sintel): 1.06
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:  
* WE: ~2 = 1199.2931{{c}}, ~3/2 = 696.9690{{c}}
* WE: ~2 = 1199.0260{{c}}, ~3/2 = 697.1486{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3736{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6887{{c}}


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


Badness (Sintel): 1.07
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:  
* WE: ~2 = 1199.9122{{c}}, ~3/2 = 697.4779{{c}}
* WE: ~2 = 1201.0387{{c}}, ~26/15 = 949.2863{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.5241{{c}}
* 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 }}
: unchanged-interval (eigenmonzo) basis: 2.13/9


{{Optimal ET sequence|legend=0| 12f, 31f, 43 }}
Badness (Sintel): 1.30


Badness (Sintel): 1.09
===== 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:  
* WE: ~2 = 1199.9428{{c}}, ~3/2 = 697.4804{{c}}
* WE: ~2 = 1201.0270{{c}}, ~26/15 = 949.2892{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.5113{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5169{{c}}


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


Badness (Sintel): 1.41
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:  
* WE: ~2 = 1200.0089{{c}}, ~3/2 = 697.4864{{c}}
* WE: ~2 = 1201.0339{{c}}, ~19/11 = 949.2902{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.4815{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 948.5111{{c}}
 
{{Optimal ET sequence|legend=0| 19egh, 43, 62 }}


{{Optimal ET sequence|legend=0| 12fghh, 31fgh, 43 }}
Badness (Sintel): 1.15


Badness (Sintel): 1.54
==== Semimeantone ====
Subgroup: 2.3.5.7.11.13


===== Sauveuric =====
Comma list: 81/80, 99/98, 126/125, 847/845
Subgroup: 2.3.5.7.11.13.17


Comma list: 78/77, 81/80, 99/98, 120/119, 126/125
Mapping: {{mapping| 2 0 -8 -26 -50 -59 | 0 1 4 10 18 21 }}


Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 | 0 1 4 10 18 27 -5 }}
: mapping generators: ~55/39, ~3


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.3793{{c}}, ~3/2 = 697.2833{{c}}
* WE: ~55/39 = 600.3606{{c}}, ~3/2 = 697.4241{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6222{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 697.0545{{c}}


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


Badness (Sintel): 1.22
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: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125
Comma list: 81/80, 99/98, 126/125, 221/220, 289/288


Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 9 | 0 1 4 10 18 27 -5 -3 }}
Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 | 0 1 4 10 18 21 1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.0260{{c}}, ~3/2 = 697.1486{{c}}
* WE: ~17/12 = 600.5426{{c}}, ~3/2 = 697.5571{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6887{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9858{{c}}


{{Optimal ET sequence|legend=0| 12f, 43 }}
{{Optimal ET sequence|legend=0| 12f, 50eff, 62, 136bg }}


Badness (Sintel): 1.25
Badness (Sintel): 1.60


==== Hemimeantone ====
===== 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, 169/168
Comma list: 81/80, 99/98, 126/125, 153/152, 209/208, 221/220


Mapping: {{mapping| 1 0 -4 -13 -25 -5 | 0 2 8 20 36 11 }}
Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 -1 | 0 1 4 10 18 21 1 3 }}
 
: mapping generators: ~2, ~26/15


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.0387{{c}}, ~26/15 = 949.2863{{c}}
* WE: ~17/12 = 600.5959{{c}}, ~3/2 = 697.5985{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5065{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9638{{c}}


{{Optimal ET sequence|legend=0| 19e, 43, 62 }}
{{Optimal ET sequence|legend=0| 12f, 50eff, 62 }}


Badness (Sintel): 1.30
Badness (Sintel): 1.47


===== 17-limit =====
=== Meanpop ===
Subgroup: 2.3.5.7.11.13.17
{{See also| Huygens vs meanpop }}


Comma list: 81/80, 99/98, 126/125, 169/168, 221/220
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 -5 -22 | 0 2 8 20 36 11 33 }}
Subgroup: 2.3.5.7.11


Optimal tunings:  
Comma list: 81/80, 126/125, 385/384
* WE: ~2 = 1201.0270{{c}}, ~26/15 = 949.2892{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5169{{c}}


{{Optimal ET sequence|legend=0| 19eg, 43, 62 }}
Mapping: {{mapping| 1 0 -4 -13 24 | 0 1 4 10 -13 }}


Badness (Sintel): 1.19
: mapping generator: ~2, ~3


===== 19-limit =====
Optimal tunings:
Subgroup: 2.3.5.7.11.13.17.19
* 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, 153/152, 169/168, 221/220
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 -22 -25 | 0 2 8 20 36 11 33 37 }}
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.)


Optimal tunings:  
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.
* WE: ~2 = 1201.0339{{c}}, ~19/11 = 949.2902{{c}}
 
* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 948.5111{{c}}
{{Optimal ET sequence|legend=0| 12e, 19, 31, 81, 112b }}


{{Optimal ET sequence|legend=0| 19egh, 43, 62 }}
Badness (Sintel): 0.712


Badness (Sintel): 1.15
; 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]


==== Semimeantone ====
==== Tridecimal meanpop ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


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


Optimal tunings:  
Optimal tunings:  
* WE: ~55/39 = 600.3606{{c}}, ~3/2 = 697.4241{{c}}
* WE: ~2 = 1201.0765{{c}}, ~3/2 = 696.8361{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 697.0545{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2347{{c}}


{{Optimal ET sequence|legend=0| 12f, …, 50eff, 62, 136b }}
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 (Sintel): 1.68
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.)


===== 17-limit =====
{{Optimal ET sequence|legend=0| 19, 31, 50, 81 }}
 
Badness (Sintel): 0.863
 
===== Meanpoppic =====
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, 126/125, 144/143, 273/272


Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 | 0 1 4 10 18 21 1 }}
Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 | 0 1 4 10 -13 15 26 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~17/12 = 600.5426{{c}}, ~3/2 = 697.5571{{c}}
* WE: ~2 = 1201.0727{{c}}, ~3/2 = 696.8168{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9858{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2195{{c}}


{{Optimal ET sequence|legend=0| 12f, 50eff, 62, 136bg }}
{{Optimal ET sequence|legend=0| 19g, 31, 50, 81, 131bd }}


Badness (Sintel): 1.60
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


Comma list: 81/80, 99/98, 126/125, 153/152, 209/208, 221/220
Comma list: 81/80, 105/104, 126/125, 144/143, 153/152, 273/272


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 -37 -40 | 0 1 4 10 -13 15 26 28 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~17/12 = 600.5959{{c}}, ~3/2 = 697.5985{{c}}
* WE: ~2 = 1201.0719{{c}}, ~3/2 = 696.8101{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9638{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2137{{c}}


{{Optimal ET sequence|legend=0| 12f, 50eff, 62 }}
{{Optimal ET sequence|legend=0| 19gh, 31, 50, 81 }}


Badness (Sintel): 1.47
Badness (Sintel): 1.08


=== Meanpop ===
===== Meanpoid =====
{{See also| Meantone vs meanpop }}
Subgroup: 2.3.5.7.11.13.17


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 still maps the 13/8 to the double-augmented fifth (C–G𝄪). Note also 11/10 is a double-diminished third; 12/11~13/12, double-augmented unison; and 14/13, minor second.
Comma list: 81/80, 105/104, 120/119, 126/125, 144/143


Subgroup: 2.3.5.7.11
Mapping: {{mapping| 1 0 -4 -13 24 -20 12 | 0 1 4 10 -13 15 -5 }}


Comma list: 81/80, 126/125, 385/384
Optimal tunings:  
* WE: ~2 = 1200.2768{{c}}, ~3/2 = 696.5683{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4114{{c}}


Mapping: {{mapping| 1 0 -4 -13 24 | 0 1 4 10 -13 }}
{{Optimal ET sequence|legend=0| 19, 31 }}


: mapping generator: ~2, ~3
Badness (Sintel): 1.17


Optimal tunings:
====== 19-limit ======
* WE: ~2 = 1201.3464{{c}}, ~3/2 = 697.2159{{c}}
Subgroup: 2.3.5.7.11.13.17.19
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4509{{c}}


Minimax tuning:  
Comma list: 77/76, 81/80, 96/95, 105/104, 120/119, 126/125
* 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


Tuning ranges:  
Mapping: {{mapping| 1 0 -4 -13 24 -20 12 9 | 0 1 4 10 -13 15 -5 -3 }}
* 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.
Optimal tunings:  
* WE: ~2 = 1199.7905{{c}}, ~3/2 = 696.3779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4973{{c}}


{{Optimal ET sequence|legend=0| 12e, 19, 31, 81, 112b }}
{{Optimal ET sequence|legend=0| 19, 31 }}


Badness (Sintel): 0.712
Badness (Sintel): 1.25


; Music
==== Semimeanpop ====
* [http://soonlabel.com/xenharmonic/archives/607 Scott Joplin's "The Entertainer" tuned into meanpop]{{dead link}}
Subgroup: 2.3.5.7.11.13
* [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 ====
Comma list: 81/80, 126/125, 385/384, 847/845
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 105/104, 126/125, 144/143
Mapping: {{mapping| 2 0 -8 -26 48 39 | 0 1 4 10 -13 -10 }}


Mapping: {{mapping| 1 0 -4 -13 24 -20 | 0 1 4 10 -13 15 }}
: mapping generators: ~55/39, ~3


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.0765{{c}}, ~3/2 = 696.8361{{c}}
* WE: ~55/39 = 600.6704{{c}}, ~3/2 = 697.2151{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2347{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.4341{{c}}
 
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


Tuning ranges:
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112b }}
* 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=0| 19, 31, 50, 81 }}
Badness (Sintel): 1.78


Badness (Sintel): 0.863
===== 17-limit =====
 
===== Meanpoppic =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 81/80, 105/104, 126/125, 144/143, 273/272
Comma list: 81/80, 126/125, 221/220, 273/272, 289/288


Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 | 0 1 4 10 -13 15 26 }}
Mapping: {{mapping| 2 0 -8 -26 48 39 5 | 0 1 4 10 -13 -10 1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.0727{{c}}, ~3/2 = 696.8168{{c}}
* WE: ~17/12 = 600.7232{{c}}, ~3/2 = 697.2820{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2195{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4411{{c}}


{{Optimal ET sequence|legend=0| 19g, 31, 50, 81, 131bd }}
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bg }}


Badness (Sintel): 1.02
Badness (Sintel): 1.45


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


Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 -40 | 0 1 4 10 -13 15 26 28 }}
Mapping: {{mapping| 2 0 -8 -26 48 39 5 -1 | 0 1 4 10 -13 -10 1 3 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.0719{{c}}, ~3/2 = 696.8101{{c}}
* WE: ~17/12 = 600.7527{{c}}, ~3/2 = 697.3244{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2137{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4525{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bgh }}


{{Optimal ET sequence|legend=0| 19gh, 31, 50, 81 }}
Badness (Sintel): 1.28


Badness (Sintel): 1.08
=== 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.  


===== Meanpoid =====
Subgroup: 2.3.5.7.11
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, 81/80


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.2768{{c}}, ~3/2 = 696.5683{{c}}
* WE: ~2 = 1199.6946{{c}}, ~3/2 = 696.0729{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4114{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2083{{c}}


{{Optimal ET sequence|legend=0| 19, 31 }}
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 (Sintel): 1.17
{{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: 77/76, 81/80, 96/95, 105/104, 120/119, 126/125
==== 13-limit ====
 
Mapping: {{mapping| 1 0 -4 -13 24 -20 12 9 | 0 1 4 10 -13 15 -5 -3 }}
 
Optimal tunings:
* WE: ~2 = 1199.7905{{c}}, ~3/2 = 696.3779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4973{{c}}
 
{{Optimal ET sequence|legend=0| 19, 31 }}
 
Badness (Sintel): 1.25
 
==== Meanplop ====
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, 78/77, 81/80


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1202.3237{{c}}, ~3/2 = 697.5502{{c}}
* WE: ~2 = 1199.7931{{c}}, ~3/2 = 696.0258{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2135{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1241{{c}}


Minimax tuning:
{{Optimal ET sequence|legend=0| 7df, 12f, 19, 31e }}
* 13- and 15-odd-limit: ~3/2 = {{monzo| 11/13 0 0 0 -1/13 }}
: unchanged-interval (eigenmonzo) basis: 2.11


{{Optimal ET sequence|legend=0| 12e, 19, 31f }}
Badness (Sintel): 0.875
 
Badness (Sintel): 1.14


===== 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, 56/55, 78/77, 81/80, 120/119


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.4737{{c}}, ~3/2 = 697.2690{{c}}
* WE: ~2 = 1198.6665{{c}}, ~3/2 = 695.8010{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4129{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4998{{c}}


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


Badness (Sintel): 1.37
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: 39/38, 52/51, 65/64, 77/76, 81/80, 91/90
Comma list: 45/44, 56/55, 78/77, 81/80, 96/95, 120/119


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 -20 12 9 | 0 1 4 10 6 15 -5 -3 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.8839{{c}}, ~3/2 = 697.0104{{c}}
* WE: ~2 = 1198.2880{{c}}, ~3/2 = 695.7123{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4949{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.6370{{c}}


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


Badness (Sintel): 1.43
Badness (Sintel): 1.23


=== Meanenneadecal ===
==== Vincenzo ====
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 a major second; 12/11~14/13, minor second; and 13/12, double-augmented unison.
Subgroup: 2.3.5.7.11.13


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


Comma list: 45/44, 56/55, 81/80
Mapping: {{mapping| 1 0 -4 -13 -6 10 | 0 1 4 10 6 -4 }}
 
Mapping: {{mapping| 1 0 -4 -13 -6 | 0 1 4 10 6 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.6946{{c}}, ~3/2 = 696.0729{{c}}
* WE: ~2 = 1202.1684{{c}}, ~3/2 = 696.3160{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2083{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.2045{{c}}


Tuning ranges:
{{Optimal ET sequence|legend=0| 7d, 12, 19 }}
* 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=0| 7d, 12, 19, 31e }}
Badness (Sintel): 1.02


Badness (Sintel): 0.708
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


==== 13-limit ====
Comma list: 45/44, 52/51, 56/55, 65/64, 81/80
Subgroup: 2.3.5.7.11.13


Comma list: 45/44, 56/55, 78/77, 81/80
Mapping: {{mapping| 1 0 -4 -13 -6 10 12 | 0 1 4 10 6 -4 -5 }}
 
Mapping: {{mapping| 1 0 -4 -13 -6 -20 | 0 1 4 10 6 15 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.7931{{c}}, ~3/2 = 696.0258{{c}}
* WE: ~2 = 1200.5137{{c}}, ~3/2 = 696.1561{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1241{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.8771{{c}}


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


Badness (Sintel): 0.875
Badness (Sintel): 1.30


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


Comma list: 45/44, 56/55, 78/77, 81/80, 120/119
Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1198.6665{{c}}, ~3/2 = 695.8010{{c}}
* WE: ~2 = 1199.8261{{c}}, ~3/2 = 696.0298{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4998{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1262{{c}}


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


Badness (Sintel): 1.17
Badness (Sintel): 1.36


===== 19-limit =====
=== Bimeantone ===
Subgroup: 2.3.5.7.11.13.17.19
11/8 is mapped to half octave minus the [[128/125|meantone diesis]].  


Comma list: 45/44, 56/55, 78/77, 81/80, 96/95, 120/119
Subgroup: 2.3.5.7.11


Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 9 | 0 1 4 10 6 15 -5 -3 }}
Comma list: 81/80, 126/125, 245/242


Optimal tunings:  
Mapping: {{mapping| 2 0 -8 -26 -31 | 0 1 4 10 12 }}
* WE: ~2 = 1198.2880{{c}}, ~3/2 = 695.7123{{c}}
 
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.6370{{c}}
: mapping generators: ~63/44, ~3
 
Optimal tunings:  
* WE: ~63/44 = 600.7492{{c}}, ~3/2 = 696.8853{{c}}
* CWE: ~63/44 = 600.0000{{c}}, ~3/2 = 696.1908{{c}}


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


Badness (Sintel): 1.23
Badness (Sintel): 1.26


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


Comma list: 45/44, 56/55, 65/64, 81/80
Comma list: 81/80, 105/104, 126/125, 245/242


Mapping: {{mapping| 1 0 -4 -13 -6 10 | 0 1 4 10 6 -4 }}
Mapping: {{mapping| 2 0 -8 -26 -31 -40 | 0 1 4 10 12 15 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1202.1684{{c}}, ~3/2 = 696.3160{{c}}
* WE: ~55/39 = 600.8309{{c}}, ~3/2 = 696.8000{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.2045{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.0066{{c}}


{{Optimal ET sequence|legend=0| 7d, 12, 19 }}
{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}


Badness (Sintel): 1.02
Badness (Sintel): 1.19


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


Comma list: 45/44, 52/51, 56/55, 65/64, 81/80
Comma list: 81/80, 105/104, 126/125, 189/187, 221/220


Mapping: {{mapping| 1 0 -4 -13 -6 10 12 | 0 1 4 10 6 -4 -5 }}
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 | 0 1 4 10 12 15 1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.5137{{c}}, ~3/2 = 696.1561{{c}}
* WE: ~17/12 = 600.9234{{c}}, ~3/2 = 696.8536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.8771{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.9317{{c}}


{{Optimal ET sequence|legend=0| 12, 19 }}
{{Optimal ET sequence|legend=0| 12f, 38df, 50 }}


Badness (Sintel): 1.30
Badness (Sintel): 1.15


===== 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, 45/44, 52/51, 56/55, 65/64, 81/80
Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220


Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 | 0 1 4 10 6 -4 -5 -3 }}
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 -1 | 0 1 4 10 12 15 1 3 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.8261{{c}}, ~3/2 = 696.0298{{c}}
* WE: ~17/12 = 600.9845{{c}}, ~3/2 = 696.8939{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1262{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.8947{{c}}


{{Optimal ET sequence|legend=0| 12, 19 }}
{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}


Badness (Sintel): 1.36
Badness (Sintel): 1.08


==== Meanundec ====
=== Trimean ===
Subgroup: 2.3.5.7.11.13
{{See also| No-sevens subgroup temperaments #Superpine }}


Comma list: 27/26, 40/39, 45/44, 56/55
Subgroup: 2.3.5.7.11


Mapping: {{mapping| 1 0 -4 -13 -6 -1 | 0 1 4 10 6 3 }}
Comma list: 81/80, 126/125, 1344/1331


Optimal tunings:  
Mapping: {{mapping| 1 2 4 7 5 | 0 -3 -12 -30 -11 }}
* WE: ~2 = 1196.0359{{c}}, ~3/2 = 694.9504{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.7474{{c}}


{{Optimal ET sequence|legend=0| 7d, 12f, 19f }}
: mapping generators: ~2, ~11/10


Badness (Sintel): 1.00
Optimal tunings:
 
* WE: ~2 = 1200.7155{{c}}, ~11/10 = 167.9055{{c}}
===== 17-limit =====
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7749{{c}}
Subgroup: 2.3.5.7.11.13.17
 
{{Optimal ET sequence|legend=0| 7d, 36d, 43, 50, 93 }}
 
Badness (Sintel): 1.68
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Comma list: 27/26, 34/33, 40/39, 45/44, 56/55
Comma list: 81/80, 126/125, 144/143, 364/363


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1196.8604{{c}}, ~3/2 = 695.7613{{c}}
* WE: ~2 = 1200.6104{{c}}, ~11/10 = 167.8749{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.1744{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7728{{c}}


{{Optimal ET sequence|legend=0| 7dg, 12f }}
{{Optimal ET sequence|legend=0| 7d, 43, 50, 93 }}


Badness (Sintel): 1.09
Badness (Sintel): 1.46


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


Comma list: 27/26, 34/33, 40/39, 45/44, 56/55, 57/55
Comma list: 81/80, 126/125, 144/143, 189/187, 221/220


Mapping: {{mapping| 1 0 -4 -13 -6 -1 -7 -10 | 0 1 4 10 6 3 7 9 }}
Mapping: {{mapping| 1 2 4 7 5 3 8 | 0 -3 -12 -30 -11 5 -28 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1196.9296{{c}}, ~3/2 = 696.3321{{c}}
* WE: ~2 = 1200.6144{{c}}, ~11/10 = 167.8716{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.7122{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7682{{c}}


{{Optimal ET sequence|legend=0| 7dgh, 12f }}
{{Optimal ET sequence|legend=0| 7dg, 43, 50, 93 }}


Badness (Sintel): 1.16
Badness (Sintel): 1.28


=== Meanundeci ===
=== Migration ===
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♭).  
See [[Rastmic clan #Migration|Rastmic clan]].


Subgroup: 2.3.5.7.11
== Flattone ==
{{Main| Flattone }}


Comma list: 33/32, 55/54, 77/75
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]].


Mapping: {{mapping| 1 0 -4 -13 5 | 0 1 4 10 -1 }}
[[Subgroup]]: 2.3.5.7


Optimal tunings:  
[[Comma list]]: 81/80, 525/512
* WE: ~2 = 1205.7146{{c}}, ~3/2 = 697.9977{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.1805{{c}}


{{Optimal ET sequence|legend=0| 7d, 12e, 19e }}
{{Mapping|legend=1| 1 0 -4 17 | 0 1 4 -9 }}


Badness (Sintel): 1.04
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1203.6308{{c}}, ~3/2 = 695.8782{{c}}
: [[error map]]: {{val| +3.631 -2.446 -2.801 -2.684 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.7334{{c}}
: error map: {{val| 0.000 -8.222 -11.380 -12.426 }}


==== 13-limit ====
[[Minimax tuning]]:
Subgroup: 2.3.5.7.11.13
* [[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


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


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


Optimal tunings:
{{Optimal ET sequence|legend=1| 7, 19, 26, 45 }}
* WE: ~2 = 1205.5631{{c}}, ~3/2 = 697.9847{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.0144{{c}}


{{Optimal ET sequence|legend=0| 7d, 12e, 19e }}
[[Badness]] (Sintel): 0.976


Badness (Sintel): 1.09
=== 11-limit ===
 
This can also be considered a no-sevens temperament: [[#Hypnotone|hypnotone]].
=== Bimeantone ===
11/8 is mapped to half octave minus the [[128/125|meantone diesis]].  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 81/80, 126/125, 245/242
Comma list: 45/44, 81/80, 385/384


Mapping: {{mapping| 2 0 -8 -26 -31 | 0 1 4 10 12 }}
Mapping: {{mapping| 1 0 -4 17 -6 | 0 1 4 -9 6 }}


: mapping generators: ~63/44, ~3
Optimal tuning:  
* WE: ~2 = 1202.3247{{c}}, ~3/2 = 694.4688{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.1467{{c}}


Optimal tunings:  
Tuning ranges:  
* WE: ~63/44 = 600.7492{{c}}, ~3/2 = 696.8853{{c}}
* 11-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* CWE: ~63/44 = 600.0000{{c}}, ~3/2 = 696.1908{{c}}
* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]


{{Optimal ET sequence|legend=0| 12, 26de, 38d, 50 }}
{{Optimal ET sequence|legend=0| 7, 19, 26, 45, 71bc, 116bcde }}


Badness (Sintel): 1.26
Badness (Sintel): 1.12


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


Comma list: 81/80, 105/104, 126/125, 245/242
Comma list: 45/44, 65/64, 78/77, 81/80


Mapping: {{mapping| 2 0 -8 -26 -31 -40 | 0 1 4 10 12 15 }}
Mapping: {{mapping| 1 0 -4 17 -6 10 | 0 1 4 -9 6 -4 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~55/39 = 600.8309{{c}}, ~3/2 = 696.8000{{c}}
* WE: ~2 = 1202.5156{{c}}, ~3/2 = 694.5107{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.0066{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0538{{c}}


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


Badness (Sintel): 1.19
{{Optimal ET sequence|legend=0| 7, 19, 26, 45f, 71bcf, 116bcdef }}


==== 17-limit ====
Badness (Sintel): 0.920
Subgroup: 2.3.5.7.11.13.17


Comma list: 81/80, 105/104, 126/125, 189/187, 221/220
=== Ptolemy ===
See [[Rastmic clan #Ptolemy|Rastmic clan]].


Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 | 0 1 4 10 12 15 1 }}
== Dominant ==
{{Main| Dominant (temperament) }}
{{See also| Archytas clan }}


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


{{Optimal ET sequence|legend=0| 12f, 38df, 50 }}
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.


Badness (Sintel): 1.15
[[Subgroup]]: 2.3.5.7


==== 19-limit ====
[[Comma list]]: 36/35, 64/63
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220
{{Mapping|legend=1| 1 0 -4 6 | 0 1 4 -2 }}


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


Optimal tunings:  
[[Tuning ranges]]:  
* WE: ~17/12 = 600.9845{{c}}, ~3/2 = 696.8939{{c}}
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[diamond monotone]]: ~3/2 = [700.000, 720.000] (7\12 to 3\5)
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.8947{{c}}
* 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=0| 12f, 26deff, 38df, 50 }}
{{Optimal ET sequence|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}


Badness (Sintel): 1.08
[[Badness]] (Sintel): 0.524
 
=== Trimean ===
{{See also| No-sevens subgroup temperaments #Superpine }}


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


Comma list: 81/80, 126/125, 1344/1331
Comma list: 36/35, 56/55, 64/63


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


: mapping generators: ~2, ~11/10
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:  
Optimal tunings:  
* WE: ~2 = 1200.7155{{c}}, ~11/10 = 167.9055{{c}}
* WE: ~2 = 1194.0169{{c}}, ~3/2 = 699.7473{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7749{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.2672{{c}}


{{Optimal ET sequence|legend=0| 7d, 36d, 43, 50, 93 }}
{{Optimal ET sequence|legend=0| 5, 12, 17c, 29cde }}


Badness (Sintel): 1.68
Badness (Sintel): 0.799


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


Comma list: 81/80, 126/125, 144/143, 364/363
Comma list: 36/35, 56/55, 64/63, 66/65


Mapping: {{mapping| 1 2 4 7 5 3 | 0 -3 -12 -30 -11 5 }}
Mapping: {{mapping| 1 0 -4 6 13 18 | 0 1 4 -2 -6 -9 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.6104{{c}}, ~11/10 = 167.8749{{c}}
* WE: ~2 = 1193.8055{{c}}, ~3/2 = 700.0042{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7728{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.8254{{c}}


{{Optimal ET sequence|legend=0| 7d, 43, 50, 93 }}
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|legend=0| 12f, 17c, 29cdef }}


Badness (Sintel): 1.46
Badness (Sintel): 0.996


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


Comma list: 81/80, 126/125, 144/143, 189/187, 221/220
Comma list: 26/25, 36/35, 56/55, 64/63


Mapping: {{mapping| 1 2 4 7 5 3 8 | 0 -3 -12 -30 -11 5 -28 }}
Mapping: {{mapping| 1 0 -4 6 13 -9 | 0 1 4 -2 -6 8 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.6144{{c}}, ~11/10 = 167.8716{{c}}
* WE: ~2 = 1195.0293{{c}}, ~3/2 = 701.9847{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7682{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7698{{c}}


{{Optimal ET sequence|legend=0| 7dg, 43, 50, 93 }}
{{Optimal ET sequence|legend=0| 5, 12, 17c }}


Badness (Sintel): 1.28
Badness (Sintel): 1.13


== Flattone ==
=== Domination ===
{{Main| Flattone }}
Subgroup: 2.3.5.7.11


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 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. Good tunings for flattone are [[45edo]], [[64edo]], and [[71edo]].
Comma list: 36/35, 64/63, 77/75


[[Subgroup]]: 2.3.5.7
Mapping: {{mapping| 1 0 -4 6 -14 | 0 1 4 -2 11 }}


[[Comma list]]: 81/80, 525/512
Optimal tunings:  
* WE: ~2 = 1194.8645{{c}}, ~3/2 = 701.9872{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5945{{c}}


{{Mapping|legend=1| 1 0 -4 17 | 0 1 4 -9 }}
{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}


[[Optimal tuning]]s:  
Badness (Sintel): 1.21
* [[WE]]: ~2 = 1203.6308{{c}}, ~3/2 = 695.8782{{c}}
: [[error map]]: {{val| +3.631 -2.446 -2.801 -2.684 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.7334{{c}}
: error map: {{val| 0.000 -8.222 -11.380 -12.426 }}


[[Minimax tuning]]:
==== 13-limit ====
* [[7-odd-limit]]: ~3/2 = {{monzo| 8/13 0 1/13 -1/13 }}
Subgroup: 2.3.5.7.11.13
: [[projection map]]: [{{monzo| 1 0 0 0 }}, {{monzo| 21/13 0 1/13 -1/13 }}, {{monzo| 32/13 0 4/13 -4/13 }}, {{monzo| 32/13 0 -9/13 9/13 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
* [[9-odd-limit]]: ~3/2 = {{monzo| 6/11 2/11 0 -1/11 }}
: [[projection map]]: [{{monzo| 1 0 0 0 }}, {{monzo| 17/11 2/11 0 -1/11 }}, {{monzo| 24/11 8/11 0 -4/11 }}, {{monzo| 34/11 -18/11 0 9/11 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7


[[Tuning ranges]]:  
Comma list: 26/25, 36/35, 64/63, 66/65
* 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.
Mapping: {{mapping| 1 0 -4 6 -14 -9 | 0 1 4 -2 11 8 }}


{{Optimal ET sequence|legend=1| 7, 19, 26, 45 }}
Optimal tunings:
* WE: ~2 = 1195.1324{{c}}, ~3/2 = 702.6343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.0791{{c}}


[[Badness]] (Sintel): 0.976
{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}


=== 11-limit ===
Badness (Sintel): 1.13
This can also be considered a no-sevens temperament: [[#Hypnotone|hypnotone]].


=== Domineering ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 45/44, 81/80, 385/384
Comma list: 36/35, 45/44, 64/63


Mapping: {{mapping| 1 0 -4 17 -6 | 0 1 4 -9 6 }}
Mapping: {{mapping| 1 0 -4 6 -6 | 0 1 4 -2 6 }}


Optimal tuning:  
Optimal tunings:  
* WE: ~2 = 1202.3247{{c}}, ~3/2 = 694.4688{{c}}
* WE: ~2 = 1194.7102{{c}}, ~3/2 = 695.6962{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.1467{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1765{{c}}


Tuning ranges:
{{Optimal ET sequence|legend=0| 5e, 7, 12 }}
* 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=0| 7, 19, 26, 45, 71bc, 116bcde }}
Badness (Sintel): 0.727


Badness (Sintel): 1.12
=== Arnold ===
Subgroup: 2.3.5.7.11


=== 13-limit ===
Comma list: 22/21, 33/32, 36/35
Subgroup: 2.3.5.7.11.13


Comma list: 45/44, 65/64, 78/77, 81/80
Mapping: {{mapping| 1 0 -4 6 5 | 0 1 4 -2 -1 }}
 
Mapping: {{mapping| 1 0 -4 17 -6 10 | 0 1 4 -9 6 -4 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1202.5156{{c}}, ~3/2 = 694.5107{{c}}
* WE: ~2 = 1199.8507{{c}}, ~3/2 = 698.4045{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0538{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.4822{{c}}


Tuning ranges:
{{Optimal ET sequence|legend=0| 5, 7, 12e }}
* 13- and 15-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]


{{Optimal ET sequence|legend=0| 7, 19, 26, 45f, 71bcf, 116bcdef }}
Badness (Sintel): 0.864


Badness (Sintel): 0.920
=== Neutrominant ===
See [[Rastmic clan #Neutrominant|Rastmic clan]].


== Flattertone ==
== Flattertone ==
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.
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.
 
Flattertone was named by [[Flora Canou]] in 2024.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,047: Line 1,063:
* [https://youtu.be/scCuGXnj5IY ''Music in 33EDO (33-Tone Equal Temperament) - Feb 2024''] by [[Budjarn Lambeth]] (2024)
* [https://youtu.be/scCuGXnj5IY ''Music in 33EDO (33-Tone Equal Temperament) - Feb 2024''] by [[Budjarn Lambeth]] (2024)


== Dominant ==
== Sharptone ==
{{Main| Dominant (temperament) }}
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| 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]].
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).


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 takes the tritone as 16/11, which it barely sounds like. The first alternative, domineering, takes the same step as 11/8, which it barely sounds like either. Domination tempers out 77/75 and identifies 11/8 with the augmented third; 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.  
The 11-limit extension was named by Gene Ward Smith in 2004<ref name="meantone & meanpop 2004"/>.  


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


[[Comma list]]: 36/35, 64/63
[[Comma list]]: 21/20, 28/27


{{Mapping|legend=1| 1 0 -4 6 | 0 1 4 -2 }}
{{Mapping|legend=1| 1 0 -4 -2 | 0 1 4 3 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1195.3384{{c}}, ~3/2 = 698.8478{{c}}
* [[WE]]: ~2 = 1204.2961{{c}}, ~3/2 = 702.6463{{c}}
: [[error map]]: {{val| -4.662 -7.769 +9.077 +14.832 }}
: [[error map]]: {{val| +4.296 +4.987 +24.271 -56.591 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.1125{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4928{{c}}
: error map: {{val| 0.000 -0.842 +18.136 +28.949 }}
: error map: {{val| 0.000 -0.462 +19.657 -64.347 }}


[[Tuning ranges]]:
{{Optimal ET sequence|legend=1| 5, 7d, 12d }}
* [[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 }}
[[Badness]] (Sintel): 0.629


[[Badness]] (Sintel): 0.524
=== Meanertone ===
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 36/35, 56/55, 64/63
Comma list: 21/20, 28/27, 33/32


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


Tuning ranges:  
Optimal tunings:  
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 705.882] (7\12 to 10\17)
* WE: ~2 = 1208.5304{{c}}, ~3/2 = 701.5669{{c}}
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1117{{c}}


Optimal tunings:
{{Optimal ET sequence|legend=0| 5, 7d, 12de }}
* WE: ~2 = 1194.0169{{c}}, ~3/2 = 699.7473{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.2672{{c}}


{{Optimal ET sequence|legend=0| 5, 12, 17c, 29cde }}
Badness (Sintel): 0.832


Badness (Sintel): 0.799
== 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.


==== 13-limit ====
Mildtone was named by [[User: Lucius Chiaraviglio|Lucius Chiaraviglio]] in 2024.  
Subgroup: 2.3.5.7.11.13


Comma list: 36/35, 56/55, 64/63, 66/65
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 0 -4 6 13 18 | 0 1 4 -2 -6 -9 }}
[[Comma list]]: 81/80, 16128/15625


Optimal tunings:  
{{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 }}
* WE: ~2 = 1193.8055{{c}}, ~3/2 = 700.0042{{c}}
 
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.8254{{c}}
[[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 }}


Tuning ranges:
{{Optimal ET sequence|legend=1| 12, 43d, 55, 67 }}
* 13- and 15-odd-limit diamond monotone: ~3/2 = 705.882 (10\17)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]


{{Optimal ET sequence|legend=0| 12f, 17c, 29cdef }}
[[Badness]] (Sintel): 2.67


Badness (Sintel): 0.996
=== 11-limit ===


==== Dominion ====
[[Subgroup]]: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 26/25, 36/35, 56/55, 64/63
[[Comma list]]: 81/80, 176/175, 7056/6875


Mapping: {{mapping| 1 0 -4 6 13 -9 | 0 1 4 -2 -6 8 }}
{{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 30}}


Optimal tunings:  
[[Optimal tuning]]s:  
* WE: ~2 = 1195.0293{{c}}, ~3/2 = 701.9847{{c}}
* [[WE]]: ~2 = 1199.816{{c}}, ~3/2 = 698.355{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7698{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.455{{c}}


{{Optimal ET sequence|legend=0| 5, 12, 17c }}
{{Optimal ET sequence|legend=1| 12, 43de, 55, 67 }}


Badness (Sintel): 1.13
[[Badness]] (Sintel): 2.15


=== Domineering ===
=== 13-limit ===
Subgroup: 2.3.5.7.11


Comma list: 36/35, 45/44, 64/63
[[Subgroup]]: 2.3.5.7.11.13


Mapping: {{mapping| 1 0 -4 6 -6 | 0 1 4 -2 6 }}
[[Comma list]]: 81/80, 176/175, 196/195, 832/825


Optimal tunings:
{{Mapping|legend=1| 1 0 -4 -32 -44 | 0 1 4 22 30}}
* WE: ~2 = 1194.7102{{c}}, ~3/2 = 695.6962{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1765{{c}}


{{Optimal ET sequence|legend=0| 5e, 7, 12 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.788{{c}}, ~3/2 = 698.355{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.471{{c}}


Badness (Sintel): 0.727
{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}


==== 13-limit ====
[[Badness]] (Sintel): 2.04
Subgroup: 2.3.5.7.11.13


Comma list: 36/35, 45/44, 52/49, 64/63
=== 17-limit ===
 
Mapping: {{mapping| 1 0 -4 6 -6 10 | 0 1 4 -2 6 -4 }}


Optimal tunings:  
[[Subgroup]]: 2.3.5.7.11.13.17
* WE: ~2 = 1198.1958{{c}}, ~3/2 = 694.7159{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.6809{{c}}


{{Optimal ET sequence|legend=0| 7, 12 }}
[[Comma list]]: 81/80, 176/175, 189/187, 196/195, 832/825


Badness (Sintel): 1.12
{{Mapping|legend=1| 1 0 -4 -32 -44 12| 0 1 4 22 30 -5}}


===== 17-limit =====
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13.17
* [[WE]]: ~2 = 1199.655{{c}}, ~3/2 = 698.295{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.488{{c}}


Comma list: 36/35, 45/44, 51/49, 52/49, 64/63
{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}


Mapping: {{mapping| 1 0 -4 6 -6 10 12 | 0 1 4 -2 6 -4 -5 }}
[[Badness]] (Sintel): 1.98


Optimal tunings:
=== 19-limit ===
* WE: ~2 = 1197.7959{{c}}, ~3/2 = 694.8362{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.0834{{c}}
 
{{Optimal ET sequence|legend=0| 7, 12 }}
 
Badness (Sintel): 1.25
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 36/35, 39/38, 45/44, 51/49, 52/49, 57/56
 
Mapping: {{mapping| 1 0 -4 6 -6 10 12 9 | 0 1 4 -2 6 -4 -5 -3 }}
 
Optimal tunings:
* WE: ~2 = 1197.6198{{c}}, ~3/2 = 694.8362{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2075{{c}}
 
{{Optimal ET sequence|legend=0| 5ef, 7, 12, 19d, 31def }}
 
Badness (Sintel): 1.24
 
==== Dominatrix ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 27/26, 36/35, 45/44, 64/63
 
Mapping: {{mapping| 1 0 -4 6 -6 -1 | 0 1 4 -2 6 3 }}
 
Optimal tunings:
* WE: ~2 = 1193.1574{{c}}, ~3/2 = 694.5610{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.7268{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 7, 12f }}
 
Badness (Sintel): 0.756
 
=== Domination ===
Subgroup: 2.3.5.7.11
 
Comma list: 36/35, 64/63, 77/75
 
Mapping: {{mapping| 1 0 -4 6 -14 | 0 1 4 -2 11 }}
 
Optimal tunings:
* WE: ~2 = 1194.8645{{c}}, ~3/2 = 701.9872{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5945{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 0 -4 6 -14 -9 | 0 1 4 -2 11 8 }}
 
Optimal tunings:
* WE: ~2 = 1195.1324{{c}}, ~3/2 = 702.6343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.0791{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}
 
Badness (Sintel): 1.13
 
=== Arnold ===
Subgroup: 2.3.5.7.11
 
Comma list: 22/21, 33/32, 36/35
 
Mapping: {{mapping| 1 0 -4 6 5 | 0 1 4 -2 -1 }}
 
Optimal tunings:
* WE: ~2 = 1199.8507{{c}}, ~3/2 = 698.4045{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.4822{{c}}
 
{{Optimal ET sequence|legend=0| 5, 7, 12e }}
 
Badness (Sintel): 0.864
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 22/21, 27/26, 33/32, 36/35
 
Mapping: {{mapping| 1 0 -4 6 5 -1 | 0 1 4 -2 -1 3 }}
 
Optimal tunings:
* WE: ~2 = 1197.8123{{c}}, ~3/2 = 695.4727{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.5713{{c}}
 
{{Optimal ET sequence|legend=0| 5, 7 }}
 
Badness (Sintel): 0.963
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 22/21, 27/26, 33/32, 36/35, 51/49
 
Mapping: {{mapping| 1 0 -4 6 5 -1 12 | 0 1 4 -2 -1 3 -5 }}
 
Optimal tunings:
* WE: ~2 = 1197.6327{{c}}, ~3/2 = 695.6030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.9316{{c}}
 
{{Optimal ET sequence|legend=0| 5, 7 }}
 
Badness (Sintel): 1.25
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 22/21, 27/26, 33/32, 36/35, 51/49, 57/56
 
Mapping: {{mapping| 1 0 -4 6 5 -1 12 9 | 0 1 4 -2 -1 3 -5 -3 }}
 
Optimal tunings:
* WE: ~2 = 1197.5295{{c}}, ~3/2 = 695.6325{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.0579{{c}}
 
{{Optimal ET sequence|legend=0| 5, 7, 12ef, 19def }}
 
Badness (Sintel): 1.28
 
== 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<ref name="meantone & meanpop 2004"/>.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7.11.13.17.19


[[Comma list]]: 21/20, 28/27
[[Comma list]]: 81/80, 96/95, 176/175, 189/187, 196/195, 832/825


{{Mapping|legend=1| 1 0 -4 -2 | 0 1 4 3 }}
{{Mapping|legend=1| 1 0 -4 -32 -44 12 9| 0 1 4 22 30 -5 -3}}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1204.2961{{c}}, ~3/2 = 702.6463{{c}}
* [[WE]]: ~2 = 1199.371{{c}}, ~3/2 = 698.164{{c}}
: [[error map]]: {{val| +4.296 +4.987 +24.271 -56.591 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.519{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4928{{c}}
: error map: {{val| 0.000 -0.462 +19.657 -64.347 }}
 
{{Optimal ET sequence|legend=1| 5, 7d, 12d }}
 
[[Badness]] (Sintel): 0.629
 
=== Meanertone ===
Subgroup: 2.3.5.7.11
 
Comma list: 21/20, 28/27, 33/32
 
Mapping: {{mapping| 1 0 -4 -2 5 | 0 1 4 3 -1 }}


Optimal tunings:
{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}
* WE: ~2 = 1208.5304{{c}}, ~3/2 = 701.5669{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1117{{c}}


{{Optimal ET sequence|legend=0| 5, 7d, 12de }}
[[Badness]] (Sintel): 1.95


Badness (Sintel): 0.832
{{Todo|unify precision|review}}


== Supermean ==
== Supermean ==
Line 1,384: Line 1,240:
{{Main| Mohajira }}
{{Main| Mohajira }}


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.
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.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,555: Line 1,411:
<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>
<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 2,060: Line 1,916:


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


Line 2,495: Line 2,351:


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Meantone family| ]] <!-- main article -->
[[Category:Meantone family| ]] <!-- main article -->
[[Category:Meantone| ]] <!-- key article -->
[[Category:Meantone| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/Meantone_family"