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{{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 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"