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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♭).  
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♭).  


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


Line 1,087: Line 1,086:
Badness: 0.026288
Badness: 0.026288


==== 2.3.5.11 subgroup (Dequarter) ====
=== Bimeantone ===
[[Subgroup]]: 2.3.5.11
11/8 is mapped to half octave minus the [[128/125|meantone diesis]].  


[[Comma list]]: 33/32, 55/54
Subgroup: 2.3.5.7.11


{{Mapping|legend=2| 1 0 -4 5 | 0 1 4 -1 }}
Comma list: 81/80, 126/125, 245/242


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


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 696.0387
: mapping generators: ~63/44, ~3


{{Optimal ET sequence|legend=1| 5, 7, 19e, 26e }}
Optimal tunings:
* CTE: ~63/44 = 1\2, ~3/2 = 696.5199
* POTE: ~63/44 = 1\2, ~3/2 = 696.016


[[Badness]]: 0.0145
{{Optimal ET sequence|legend=1| 12, 26de, 38d, 50 }}


===== Dreamtone =====
Badness: 0.038122
Subgroup: 2.3.5.11.13


Comma list: 33/32, 55/54, 975/968
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Sval mapping: {{mapping| 1 0 -4 5 21 | 0 1 4 -1 -11 }}
Comma list: 81/80, 105/104, 126/125, 245/242


: sval mapping generators: ~2, ~3
Mapping: {{mapping| 2 0 -8 -26 -31 -40 | 0 1 4 10 12 15 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 689.6993
Optimal tunings:
* CTE: ~55/39 = 1\2, ~3/2 = 696.3410
* POTE: ~55/39 = 1\2, ~3/2 = 695.836


Optimal ET sequence: {{Optimal ET sequence| 7, 19eff, 26eff, 33ceeff, 40ceeff }}
{{Optimal ET sequence|legend=1| 12f, 26deff, 38df, 50 }}


Badness: 0.0353
Badness: 0.028817


=== Bimeantone ===
==== 17-limit ====
11/8 is mapped to half octave minus the [[128/125|meantone diesis]].  
Subgroup: 2.3.5.7.11.13.17


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


Comma list: 81/80, 126/125, 245/242
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 | 0 1 4 10 12 15 1 }}


Mapping: {{mapping| 2 0 -8 -26 -31 | 0 1 4 10 12 }}
Optimal tunings:  
* CTE: ~17/12 = 1\2, ~3/2 = 696.3526
* POTE: ~17/12 = 1\2, ~3/2 = 695.783


: mapping generators: ~63/44, ~3
{{Optimal ET sequence|legend=1| 12f, 38df, 50 }}


Optimal tunings:  
Badness: 0.022666
* CTE: ~63/44 = 1\2, ~3/2 = 696.5199
* POTE: ~63/44 = 1\2, ~3/2 = 696.016


{{Optimal ET sequence|legend=1| 12, 26de, 38d, 50 }}
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.038122
Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220


==== 13-limit ====
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 -1 | 0 1 4 10 12 15 1 3 }}
Subgroup: 2.3.5.7.11.13
 
Comma list: 81/80, 105/104, 126/125, 245/242
 
Mapping: {{mapping| 2 0 -8 -26 -31 -40 | 0 1 4 10 12 15 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/39 = 1\2, ~3/2 = 696.3410
* CTE: ~17/12 = 1\2, ~3/2 = 696.3837
* POTE: ~55/39 = 1\2, ~3/2 = 695.836
* POTE: ~17/12 = 1\2, ~3/2 = 695.752


{{Optimal ET sequence|legend=1| 12f, 26deff, 38df, 50 }}
{{Optimal ET sequence|legend=1| 12f, 26deff, 38df, 50 }}


Badness: 0.028817
Badness: 0.017785


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


Comma list: 81/80, 105/104, 126/125, 189/187, 221/220
Comma list: 81/80, 126/125, 1344/1331


Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 | 0 1 4 10 12 15 1 }}
Mapping: {{mapping| 1 2 4 7 5 | 0 -3 -12 -30 -11 }}
 
: mapping generators: ~2, ~11/10


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 1\2, ~3/2 = 696.3526
* CTE: ~2 = 1\1, ~11/10 = 167.7074
* POTE: ~17/12 = 1\2, ~3/2 = 695.783
* POTE: ~2 = 1\1, ~11/10 = 167.805


{{Optimal ET sequence|legend=1| 12f, 38df, 50 }}
{{Optimal ET sequence|legend=1| 7d, 36d, 43, 50, 93 }}


Badness: 0.022666
Badness: 0.050729


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


Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220
Comma list: 81/80, 126/125, 144/143, 364/363


Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 -1 | 0 1 4 10 12 15 1 3 }}
Mapping: {{mapping| 1 2 4 7 5 3 | 0 -3 -12 -30 -11 5 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 1\2, ~3/2 = 696.3837
* CTE: ~2 = 1\1, ~11/10 = 167.7121
* POTE: ~17/12 = 1\2, ~3/2 = 695.752
* POTE: ~2 = 1\1, ~11/10 = 167.790


{{Optimal ET sequence|legend=1| 12f, 26deff, 38df, 50 }}
{{Optimal ET sequence|legend=1| 7d, 43, 50, 93 }}


Badness: 0.017785
Badness: 0.035445


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


Comma list: 81/80, 126/125, 1344/1331
Comma list: 81/80, 126/125, 144/143, 189/187, 221/220


Mapping: {{mapping| 1 2 4 7 5 | 0 -3 -12 -30 -11 }}
Mapping: {{mapping| 1 2 4 7 5 3 8 | 0 -3 -12 -30 -11 5 -28 }}
 
: mapping generators: ~2, ~11/10


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/10 = 167.7074
* CTE: ~2 = 1\1, ~11/10 = 167.7047
* POTE: ~2 = 1\1, ~11/10 = 167.805
* POTE: ~2 = 1\1, ~11/10 = 167.786


{{Optimal ET sequence|legend=1| 7d, 36d, 43, 50, 93 }}
{{Optimal ET sequence|legend=1| 7dg, 43, 50, 93 }}


Badness: 0.050729
Badness: 0.025221


==== 13-limit ====
== Flattone ==
Subgroup: 2.3.5.7.11.13
{{Main| Flattone }}


Comma list: 81/80, 126/125, 144/143, 364/363
In flattone tunings, the fifth is typically even flatter than that of [[19edo]]. Here, 9 fourths get to the interval class for 7, so that [[7/4]] is a diminished seventh (C-B𝄫), [[7/6]] is a diminished third (C-E𝄫), and [[7/5]] is a doubly-diminished fifth (C-G𝄫). In general, septimal subminor intervals are diminished and septimal supermajor intervals are augmented, which makes it quite easy to learn flattone notation. Good tunings for flattone are [[45edo]], [[64edo]], and [[71edo]].


Mapping: {{mapping| 1 2 4 7 5 3 | 0 -3 -12 -30 -11 5 }}
[[Subgroup]]: 2.3.5.7


Optimal tunings:  
[[Comma list]]: 81/80, 525/512
* CTE: ~2 = 1\1, ~11/10 = 167.7121
* POTE: ~2 = 1\1, ~11/10 = 167.790


{{Optimal ET sequence|legend=1| 7d, 43, 50, 93 }}
{{Mapping|legend=1| 1 0 -4 17 | 0 1 4 -9 }}


Badness: 0.035445
{{Multival|legend=1| 1 4 -9 4 -17 -32 }}


==== 17-limit ====
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13.17
* [[CTE]]: ~2 = 1\1, ~3/2 = 693.5520
* [[POTE]]: ~2 = 1\1, ~3/2 = 693.779


Comma list: 81/80, 126/125, 144/143, 189/187, 221/220
[[Minimax tuning]]:  
* [[7-odd-limit]]: ~3/2 = {{monzo| 8/13 0 1/13 -1/13 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 21/13 0 1/13 -1/13 }}, {{monzo| 32/13 0 4/13 -4/13 }}, {{monzo| 32/13 0 -9/13 9/13 }}]
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5
* [[9-odd-limit]]: ~3/2 = {{monzo| 6/11 2/11 0 -1/11 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 17/11 2/11 0 -1/11 }}, {{monzo| 24/11 8/11 0 -4/11 }}, {{monzo| 34/11 -18/11 0 9/11 }}]
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7


Mapping: {{mapping| 1 2 4 7 5 3 8 | 0 -3 -12 -30 -11 5 -28 }}
[[Tuning ranges]]:  
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [692.353, 701.955]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]


Optimal tunings:  
[[Algebraic generator]]: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.
* CTE: ~2 = 1\1, ~11/10 = 167.7047
* POTE: ~2 = 1\1, ~11/10 = 167.786


{{Optimal ET sequence|legend=1| 7dg, 43, 50, 93 }}
{{Optimal ET sequence|legend=1| 7, 19, 26, 45 }}


Badness: 0.025221
[[Badness]]: 0.038553


== Flattone ==
=== 11-limit ===
{{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 doubly-diminished fifth (C-G𝄫). In general, septimal subminor intervals are diminished and septimal supermajor intervals are augmented, which makes it quite easy to learn flattone notation. Good tunings for flattone are [[45edo]], [[64edo]], and [[71edo]].
Comma list: 45/44, 81/80, 385/384


[[Subgroup]]: 2.3.5.7
Mapping: {{mapping| 1 0 -4 17 -6 | 0 1 4 -9 6 }}


[[Comma list]]: 81/80, 525/512
Optimal tuning:  
* CTE: ~2 = 1\1, ~3/2 = 693.2511
* POTE: ~2 = 1\1, ~3/2 = 693.126


{{Mapping|legend=1| 1 0 -4 17 | 0 1 4 -9 }}
Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]


{{Multival|legend=1| 1 4 -9 4 -17 -32 }}
{{Optimal ET sequence|legend=1| 7, 19, 26, 45, 71bc, 116bcde }}


[[Optimal tuning]]s:  
Badness: 0.033839
* [[CTE]]: ~2 = 1\1, ~3/2 = 693.5520
* [[POTE]]: ~2 = 1\1, ~3/2 = 693.779


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


[[Tuning ranges]]:  
Comma list: 45/44, 65/64, 78/77, 81/80
* 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 17 -6 10 | 0 1 4 -9 6 -4 }}


{{Optimal ET sequence|legend=1| 7, 19, 26, 45 }}
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 693.0293
* POTE: ~2 = 1\1, ~3/2 = 693.058


[[Badness]]: 0.038553
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]


=== Undecimal flattone ===
{{Optimal ET sequence|legend=1| 7, 19, 26, 45f, 71bcf, 116bcdef }}
==== Full 11-limit ====
Subgroup: 2.3.5.7.11


Comma list: 45/44, 81/80, 385/384
Badness: 0.022260


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


Optimal tuning:  
[[Subgroup]]: 2.3.5.7
* CTE: ~2 = 1\1, ~3/2 = 693.2511
* POTE: ~2 = 1\1, ~3/2 = 693.126


Tuning ranges:
[[Comma list]]: 81/80, 1875/1792
* 11-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]


{{Optimal ET sequence|legend=1| 7, 19, 26, 45, 71bc, 116bcde }}
{{Mapping|legend=1| 1 0 -4 -24 | 0 1 4 17 }}


Badness: 0.033839
: mapping generators: ~2, ~3


==== 13-limit ====
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13
* [[CTE]]: ~2 = 1\1, ~3/2 = 692.6984
* [[CWE]]: ~2 = 1\1, ~3/2 = 692.0479


Comma list: 45/44, 65/64, 78/77, 81/80
{{Optimal ET sequence|legend=1| 7d, 19d, 26, 59bcd, 85bccd }}


Mapping: {{mapping| 1 0 -4 17 -6 10 | 0 1 4 -9 6 -4 }}
[[Badness]]: 0.0961


Optimal tunings:
==== 11-limit ====
* CTE: ~2 = 1\1, ~3/2 = 693.0293
* POTE: ~2 = 1\1, ~3/2 = 693.058


Tuning ranges:  
[[Subgroup]]: 2.3.5.7
* 13- and 15-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]


{{Optimal ET sequence|legend=1| 7, 19, 26, 45f, 71bcf, 116bcdef }}
[[Comma list]]: 45/44, 81/80, 1375/1344


Badness: 0.022260
{{Mapping|legend=1| 1 0 -4 -24 0| 0 1 4 17 6 }}


==== 2.3.5.11 subgroup (Hypnotone) ====
: mapping generators: ~2, ~3


[[Subgroup]]: 2.3.5.11
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 692.642
* [[CWE]]: ~2 = 1\1, ~3/2 = 692.042


[[Comma list]]: 45/44, 81/80
{{Optimal ET sequence|legend=1| 7d, 19d, 26, 59bcd, 85bccd }}


{{Mapping|legend=2| 1 0 -4 -6 | 0 1 4 6 }}
== Dominant ==
{{See also| Archytas clan }}


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


[[Optimal tuning]] ([[CTE]]): ~2/1 = 1\1, ~3/2 = 694.6998
[[Subgroup]]: 2.3.5.7


{{Optimal ET sequence|legend=1| 7, 12, 19, 26, 45 }}
[[Comma list]]: 36/35, 64/63


[[Badness]]: 0.0104
{{Mapping|legend=1| 1 0 -4 6 | 0 1 4 -2 }}


===== 2.3.5.11.13 subgroup =====
{{Multival|legend=1| 1 4 -2 4 -6 -16 }}
Subgroup: 2.3.5.11.13


Comma list: 45/44, 65/64, 81/80
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.573


Sval mapping: {{mapping| 1 0 -4 -6 10 | 0 1 4 6 -4 }}
[[Tuning ranges]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[diamond monotone]]: ~3/2 = [700.000, 720.000] (7\12 to 3\5)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 715.587]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]


: sval mapping generators: ~2, ~3
{{Optimal ET sequence|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}


Optimal tuning (CTE): ~2/1 = 1\1, ~3/2 = 693.9513
[[Badness]]: 0.020690


Optimal ET sequence: {{Optimal ET sequence| 7, 12, 19, 26, 45f }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0141
Comma list: 36/35, 56/55, 64/63


== Flattertone ==
Mapping: {{mapping| 1 0 -4 6 13 | 0 1 4 -2 -6 }}
Flattertone tunings are typically at least as flat as [[26edo]]. Here, 17 fifths get to the interval class for 7, so that [[7/4]] is a double-augmented sixth (C-Ax). [[26edo]] and [[33edo|33cd-edo]] are the two primary flattertone tunings. [[1/2-comma meantone]] is also encompassed within flattertone's range. Any flatter than this, the meantone mapping for 5/4 is too inaccurate (it becomes more of a [[16/13]] or [[27/22]]), and [[deeptone]] temperament's mapping is more logical.


[[Subgroup]]: 2.3.5.7
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]


[[Comma list]]: 81/80, 1875/1792
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.254


{{Mapping|legend=1| 1 0 -4 -24 | 0 1 4 17 }}
{{Optimal ET sequence|legend=1| 5, 12, 17c, 29cde }}


: mapping generators: ~2, ~3
Badness: 0.024180


[[Optimal tuning]]s:
==== 13-limit ====
* [[CTE]]: ~2 = 1\1, ~3/2 = 692.6984
Subgroup: 2.3.5.7.11.13
* [[CWE]]: ~2 = 1\1, ~3/2 = 692.0479


{{Optimal ET sequence|legend=1| 7d, 19d, 26, 59bcd, 85bccd }}
Comma list: 36/35, 56/55, 64/63, 66/65


[[Badness]]: 0.0961
Mapping: {{mapping| 1 0 -4 6 13 18 | 0 1 4 -2 -6 -9 }}


==== 11-limit ====
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.636


[[Subgroup]]: 2.3.5.7
Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~3/2 = 705.882 (10\17)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]


[[Comma list]]: 45/44, 81/80, 1375/1344
{{Optimal ET sequence|legend=1| 12f, 17c, 29cdef }}


{{Mapping|legend=1| 1 0 -4 -24 0| 0 1 4 17 6 }}
Badness: 0.024108


: mapping generators: ~2, ~3
==== Dominion ====
Subgroup: 2.3.5.7.11.13


[[Optimal tuning]]s:  
Comma list: 26/25, 36/35, 56/55, 64/63
* [[CTE]]: ~2 = 1\1, ~3/2 = 692.642
* [[CWE]]: ~2 = 1\1, ~3/2 = 692.042


{{Optimal ET sequence|legend=1| 7d, 19d, 26, 59bcd, 85bccd }}
Mapping: {{mapping| 1 0 -4 6 13 -9 | 0 1 4 -2 -6 8 }}


== Dominant ==
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.905
{{See also| Archytas clan }}


The interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is [[12edo]], but it also works well with the Pythagorean tuning of pure [[3/2]] fifths, and with [[29edo]], [[41edo]], or [[53edo]].
{{Optimal ET sequence|legend=1| 5, 12, 17c, 46cde }}


[[Subgroup]]: 2.3.5.7
Badness: 0.027295


[[Comma list]]: 36/35, 64/63
=== Domineering ===
Subgroup: 2.3.5.7.11


{{Mapping|legend=1| 1 0 -4 6 | 0 1 4 -2 }}
Comma list: 36/35, 45/44, 64/63


{{Multival|legend=1| 1 4 -2 4 -6 -16 }}
Mapping: {{mapping| 1 0 -4 6 -6 | 0 1 4 -2 6 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.573
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.776


[[Tuning ranges]]:
{{Optimal ET sequence|legend=1| 5e, 7, 12, 19d, 43de }}
* [[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: 0.021978


[[Badness]]: 0.020690
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== 11-limit ===
Comma list: 36/35, 45/44, 52/49, 64/63
Subgroup: 2.3.5.7.11


Comma list: 36/35, 56/55, 64/63
Mapping: {{mapping| 1 0 -4 6 -6 10 | 0 1 4 -2 6 -4 }}


Mapping: {{mapping| 1 0 -4 6 13 | 0 1 4 -2 -6 }}
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 695.762


Tuning ranges:
{{Optimal ET sequence|legend=1| 5ef, 7, 12, 19d, 31def }}
* 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 tuning (POTE): ~2 = 1\1, ~3/2 = 703.254
Badness: 0.027039


{{Optimal ET sequence|legend=1| 5, 12, 17c, 29cde }}
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.024180
Comma list: 36/35, 45/44, 51/49, 52/49, 64/63


==== 13-limit ====
Mapping: {{mapping| 1 0 -4 6 -6 10 12 | 0 1 4 -2 6 -4 -5 }}
Subgroup: 2.3.5.7.11.13


Comma list: 36/35, 56/55, 64/63, 66/65
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.115


Mapping: {{mapping| 1 0 -4 6 13 18 | 0 1 4 -2 -6 -9 }}
{{Optimal ET sequence|legend=1| 5ef, 7, 12, 19d, 31def }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.636
Badness: 0.024539


Tuning ranges:
===== 19-limit =====
* 13- and 15-odd-limit diamond monotone: ~3/2 = 705.882 (10\17)
Subgroup: 2.3.5.7.11.13.17.19
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]


{{Optimal ET sequence|legend=1| 12f, 17c, 29cdef }}
Comma list: 36/35, 39/38, 45/44, 51/49, 52/49, 57/56


Badness: 0.024108
Mapping: {{mapping| 1 0 -4 6 -6 10 12 9 | 0 1 4 -2 6 -4 -5 -3 }}


==== Dominion ====
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.217
 
{{Optimal ET sequence|legend=1| 5ef, 7, 12, 19d, 31def }}
 
Badness: 0.020398
 
==== Dominatrix ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 26/25, 36/35, 56/55, 64/63
Comma list: 27/26, 36/35, 45/44, 64/63


Mapping: {{mapping| 1 0 -4 6 13 -9 | 0 1 4 -2 -6 8 }}
Mapping: {{mapping| 1 0 -4 6 -6 -1 | 0 1 4 -2 6 3 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.905
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.544


{{Optimal ET sequence|legend=1| 5, 12, 17c, 46cde }}
{{Optimal ET sequence|legend=1| 5e, 7, 12f, 19df }}


Badness: 0.027295
Badness: 0.018289


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


Comma list: 36/35, 45/44, 64/63
Comma list: 36/35, 64/63, 77/75


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


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.776
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.004


{{Optimal ET sequence|legend=1| 5e, 7, 12, 19d, 43de }}
{{Optimal ET sequence|legend=1| 5e, 12e, 17c, 46cd }}


Badness: 0.021978
Badness: 0.036562


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


Comma list: 36/35, 45/44, 52/49, 64/63
Comma list: 26/25, 36/35, 64/63, 66/65


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


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 695.762
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.496


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


Badness: 0.027039
Badness: 0.027435


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


Comma list: 36/35, 45/44, 51/49, 52/49, 64/63
Comma list: 22/21, 33/32, 36/35


Mapping: {{mapping| 1 0 -4 6 -6 10 12 | 0 1 4 -2 6 -4 -5 }}
Mapping: {{mapping| 1 0 -4 6 5 | 0 1 4 -2 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.115
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.491


{{Optimal ET sequence|legend=1| 5ef, 7, 12, 19d, 31def }}
{{Optimal ET sequence|legend=1| 5, 7, 12e }}


Badness: 0.024539
Badness: 0.026141


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


Comma list: 36/35, 39/38, 45/44, 51/49, 52/49, 57/56
Comma list: 22/21, 27/26, 33/32, 36/35


Mapping: {{mapping| 1 0 -4 6 -6 10 12 9 | 0 1 4 -2 6 -4 -5 -3 }}
Mapping: {{mapping| 1 0 -4 6 5 -1 | 0 1 4 -2 -1 3 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.217
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.743


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


Badness: 0.020398
Badness: 0.023300


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


Comma list: 27/26, 36/35, 45/44, 64/63
Comma list: 22/21, 27/26, 33/32, 36/35, 51/49


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


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.544
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.978


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


Badness: 0.018289
Badness: 0.024535


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


Comma list: 36/35, 64/63, 77/75
Comma list: 22/21, 27/26, 33/32, 36/35, 51/49, 57/56


Mapping: {{mapping| 1 0 -4 6 -14 | 0 1 4 -2 11 }}
Mapping: {{mapping| 1 0 -4 6 5 -1 12 9 | 0 1 4 -2 -1 3 -5 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.004
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 697.068


{{Optimal ET sequence|legend=1| 5e, 12e, 17c, 46cd }}
{{Optimal ET sequence|legend=1| 5, 7, 12ef, 19def }}


Badness: 0.036562
Badness: 0.021098


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


Comma list: 26/25, 36/35, 64/63, 66/65
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 0 -4 6 -14 -9 | 0 1 4 -2 11 8 }}
[[Comma list]]: 21/20, 28/27


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.496
{{Mapping|legend=1| 1 0 -4 -2 | 0 1 4 3 }}


{{Optimal ET sequence|legend=1| 5e, 12e, 17c }}
{{Multival|legend=1| 1 4 3 4 2 -4 }}


Badness: 0.027435
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 700.140


=== Arnold ===
{{Optimal ET sequence|legend=1| 5, 7d, 12d }}
Subgroup: 2.3.5.7.11


Comma list: 22/21, 33/32, 36/35
[[Badness]]: 0.024848


Mapping: {{mapping| 1 0 -4 6 5 | 0 1 4 -2 -1 }}
=== Meanertone ===
Subgroup: 2.3.5.7.11


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.491
Comma list: 21/20, 28/27, 33/32


{{Optimal ET sequence|legend=1| 5, 7, 12e }}
Mapping: {{mapping| 1 0 -4 -2 5 | 0 1 4 3 -1 }}


Badness: 0.026141
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.615


==== 13-limit ====
{{Optimal ET sequence|legend=1| 5, 7d, 12de }}
Subgroup: 2.3.5.7.11.13


Comma list: 22/21, 27/26, 33/32, 36/35
Badness: 0.025167


Mapping: {{mapping| 1 0 -4 6 5 -1 | 0 1 4 -2 -1 3 }}
== Supermean ==
[[Subgroup]]: 2.3.5.7


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.743
[[Comma list]]: 81/80, 672/625


{{Optimal ET sequence|legend=1| 5, 7, 12ef, 19def }}
{{Mapping|legend=1| 1 0 -4 -21 | 0 1 4 15 }}


Badness: 0.023300
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.889


==== 17-limit ====
{{Optimal ET sequence|legend=1| 5d, 12d, 17c, 29c }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 22/21, 27/26, 33/32, 36/35, 51/49
[[Badness]]: 0.134204


Mapping: {{mapping| 1 0 -4 6 5 -1 12 | 0 1 4 -2 -1 3 -5 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.978
Comma list: 56/55, 81/80, 132/125


{{Optimal ET sequence|legend=1| 5, 7, 12ef, 19def }}
Mapping: {{mapping| 1 0 -4 -21 -14 | 0 1 4 15 11 }}


Badness: 0.024535
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.096


==== 19-limit ====
{{Optimal ET sequence|legend=1| 5de, 12de, 17c, 29c }}
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 22/21, 27/26, 33/32, 36/35, 51/49, 57/56
Badness: 0.063262


Mapping: {{mapping| 1 0 -4 6 5 -1 12 9 | 0 1 4 -2 -1 3 -5 -3 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 697.068
Comma list: 26/25, 56/55, 66/65, 81/80


{{Optimal ET sequence|legend=1| 5, 7, 12ef, 19def }}
Mapping: {{mapping| 1 0 -4 -21 -14 -9 | 0 1 4 15 11 8 }}


Badness: 0.021098
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.094


== Sharptone ==
{{Optimal ET sequence|legend=1| 5de, 12de, 17c, 29c }}
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.


[[Subgroup]]: 2.3.5.7
Badness: 0.040324


[[Comma list]]: 21/20, 28/27
== Mohajira ==
{{Main| Mohajira }}


{{Mapping|legend=1| 1 0 -4 -2 | 0 1 4 3 }}
Mohajira can be viewed as derived from mohaha which maps the interval one quarter tone flat of 16/9 to 7/4, although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the porwell comma. [[31edo]] makes for an excellent (7-limit) mohajira tuning, with generator 9\31.


{{Multival|legend=1| 1 4 3 4 2 -4 }}
[[Subgroup]]: 2.3.5.7


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 700.140
[[Comma list]]: 81/80, 6144/6125


{{Optimal ET sequence|legend=1| 5, 7d, 12d }}
[[Mapping]]: [{{val| 1 1 0 6 }}, {{val| 0 2 8 -11 }}]


[[Badness]]: 0.024848
: mapping generators: ~2, ~128/105


=== Meanertone ===
{{Multival|legend=1| 2 8 -11 8 -23 -48 }}
Subgroup: 2.3.5.7.11


Comma list: 21/20, 28/27, 33/32
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~128/105 = 348.415


Mapping: {{mapping| 1 0 -4 -2 5 | 0 1 4 3 -1 }}
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~128/105 = {{monzo| 0 0 1/8 }}
: [{{Monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 6 0 -11/8 0 }}]
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.5


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.615
[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~128/105 = [347.368, 350.000] (11\38 to 7\24)
* 7-odd-limit [[diamond tradeoff]]: ~128/105 = [347.393, 350.978]
* 9-odd-limit diamond tradeoff: ~128/105 = [345.601, 350.978]


{{Optimal ET sequence|legend=1| 5, 7d, 12de }}
[[Algebraic generator]]: Mohabis, real root of 3''x''<sup>3</sup> - 3''x''<sup>2</sup> - 1, 348.6067 cents. Corresponding recurrence converges quickly.


Badness: 0.025167
{{Optimal ET sequence|legend=1| 7, 24, 31 }}


== Supermean ==
[[Badness]]: 0.055714
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 81/80, 672/625
Scales: [[mohaha7]], [[mohaha10]]


{{Mapping|legend=1|  1 0 -4 -21 | 0 1 4 15 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.889
Comma list: 81/80, 121/120, 176/175


{{Optimal ET sequence|legend=1| 5d, 12d, 17c, 29c }}
Mapping: [{{val| 1 1 0 6 2 }}, {{val| 0 2 8 -11 5 }}]


[[Badness]]: 0.134204
{{Multival|legend=1| 2 8 -11 5 8 -23 1 -48 -16 52 }}


=== 11-limit ===
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.477
Subgroup: 2.3.5.7.11


Comma list: 56/55, 81/80, 132/125
Minimax tuning:  
* 11-odd-limit: ~11/9 = {{monzo| 0 0 1/8 }}
: [{{Monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 6 0 -11/8 0 0 }}, {{monzo| 2 0 5/8 0 0 }}]
: Eigenmonzo (unchanged-interval) basis: 2.5


Mapping: {{mapping| 1 0 -4 -21 -14 | 0 1 4 15 11 }}
Tuning ranges:
* 11-odd-limit diamond monotone: ~11/9 = [348.387, 350.000] (9\31 to 7\24)
* 11-odd-limit diamond tradeoff: ~11/9 = [344.999, 350.978]


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.096
{{Optimal ET sequence|legend=1| 7, 24, 31 }}


{{Optimal ET sequence|legend=1| 5de, 12de, 17c, 29c }}
Badness: 0.026064


Badness: 0.063262
Scales: [[mohaha7]], [[mohaha10]]


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


Comma list: 26/25, 56/55, 66/65, 81/80
Comma list: 66/65, 81/80, 105/104, 121/120


Mapping: {{mapping| 1 0 -4 -21 -14 -9 | 0 1 4 15 11 8 }}
Mapping: [{{val| 1 1 0 6 2 4 }}, {{val| 0 2 8 -11 5 -1 }}]


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.094
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.558


{{Optimal ET sequence|legend=1| 5de, 12de, 17c, 29c }}
{{Optimal ET sequence|legend=1| 7, 24, 31 }}


Badness: 0.040324
Badness: 0.023388


== Mohajira ==
Scales: [[mohaha7]], [[mohaha10]]
{{Main| Mohajira }}


Mohajira can be viewed as derived from mohaha which maps the interval one quarter tone flat of 16/9 to 7/4, although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the porwell comma. [[31edo]] makes for an excellent (7-limit) mohajira tuning, with generator 9\31.  
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


[[Subgroup]]: 2.3.5.7
Comma list: 66/65, 81/80, 105/104, 121/120, 154/153


[[Comma list]]: 81/80, 6144/6125
Mapping: [{{val| 1 1 0 6 2 4 7 }}, {{val| 0 2 8 -11 5 -1 -10 }}]


[[Mapping]]: [{{val| 1 1 0 6 }}, {{val| 0 2 8 -11 }}]
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.736


: mapping generators: ~2, ~128/105
{{Optimal ET sequence|legend=1| 7, 24, 31, 86ef }}


{{Multival|legend=1| 2 8 -11 8 -23 -48 }}
Badness: 0.020576


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~128/105 = 348.415
Scales: [[mohaha7]], [[mohaha10]]
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


[[Minimax tuning]]:  
Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~128/105 = {{monzo| 0 0 1/8 }}
: [{{Monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 6 0 -11/8 0 }}]
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.5


[[Tuning ranges]]:
Mapping: [{{val| 1 1 0 6 2 4 7 6 }}, {{val| 0 2 8 -11 5 -1 -10 -6 }}]
* 7- and 9-odd-limit [[diamond monotone]]: ~128/105 = [347.368, 350.000] (11\38 to 7\24)
* 7-odd-limit [[diamond tradeoff]]: ~128/105 = [347.393, 350.978]
* 9-odd-limit diamond tradeoff: ~128/105 = [345.601, 350.978]


[[Algebraic generator]]: Mohabis, real root of 3''x''<sup>3</sup> - 3''x''<sup>2</sup> - 1, 348.6067 cents. Corresponding recurrence converges quickly.
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.810


{{Optimal ET sequence|legend=1| 7, 24, 31 }}
{{Optimal ET sequence|legend=1| 7, 24, 31, 55, 86efh }}


[[Badness]]: 0.055714
Badness: 0.017302


Scales: [[mohaha7]], [[mohaha10]]
Scales: [[mohaha7]], [[mohaha10]]


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


Comma list: 81/80, 121/120, 176/175
[[Comma list]]: 81/80, 392/375


Mapping: [{{val| 1 1 0 6 2 }}, {{val| 0 2 8 -11 5 }}]
[[Mapping]]: [{{val| 1 1 0 -1 }}, {{val| 0 2 8 13 }}]


{{Multival|legend=1| 2 8 -11 5 8 -23 1 -48 -16 52 }}
: mapping generators: ~2, ~25/21


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.477
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 350.586


Minimax tuning:
{{Optimal ET sequence|legend=1| 7d, 17c, 24, 65cc, 89ccd }}
* 11-odd-limit: ~11/9 = {{monzo| 0 0 1/8 }}
: [{{Monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 6 0 -11/8 0 0 }}, {{monzo| 2 0 5/8 0 0 }}]
: Eigenmonzo (unchanged-interval) basis: 2.5


Tuning ranges:
[[Badness]]: 0.077734
* 11-odd-limit diamond monotone: ~11/9 = [348.387, 350.000] (9\31 to 7\24)
* 11-odd-limit diamond tradeoff: ~11/9 = [344.999, 350.978]


{{Optimal ET sequence|legend=1| 7, 24, 31 }}
Scales: [[mohaha7]], [[mohaha10]]


Badness: 0.026064
=== 11-limit ===
Subgroup: 2.3.5.7.11


Scales: [[mohaha7]], [[mohaha10]]
Comma list: 56/55, 77/75, 243/242


=== 13-limit ===
Mapping: [{{val| 1 1 0 -1 2 }}, {{val| 0 2 8 13 5 }}]
Subgroup: 2.3.5.7.11.13


Comma list: 66/65, 81/80, 105/104, 121/120
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.565


Mapping: [{{val| 1 1 0 6 2 4 }}, {{val| 0 2 8 -11 5 -1 }}]
{{Optimal ET sequence|legend=1| 7d, 17c, 24, 65cc, 89ccd }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.558
Badness: 0.036207
 
{{Optimal ET sequence|legend=1| 7, 24, 31 }}
 
Badness: 0.023388


Scales: [[mohaha7]], [[mohaha10]]
Scales: [[mohaha7]], [[mohaha10]]


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


Comma list: 66/65, 81/80, 105/104, 121/120, 154/153
Comma list: 56/55, 66/65, 77/75, 243/242


Mapping: [{{val| 1 1 0 6 2 4 7 }}, {{val| 0 2 8 -11 5 -1 -10 }}]
Mapping: [{{val| 1 1 0 -1 2 4 }}, {{val| 0 2 8 13 5 -1 }}]


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.736
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.745


{{Optimal ET sequence|legend=1| 7, 24, 31, 86ef }}
{{Optimal ET sequence|legend=1| 7d, 17c, 24, 41c, 65cc }}


Badness: 0.020576
Badness: 0.028738


Scales: [[mohaha7]], [[mohaha10]]
Scales: [[mohaha7]], [[mohaha10]]


=== 19-limit ===
== Liese ==
Subgroup: 2.3.5.7.11.13.17.19
<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>


Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152
Liese splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. [[74edo]] makes for a good liese tuning, though [[19edo]] can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55.


Mapping: [{{val| 1 1 0 6 2 4 7 6 }}, {{val| 0 2 8 -11 5 -1 -10 -6 }}]
[[Subgroup]]: 2.3.5.7


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.810
[[Comma list]]: 81/80, 686/675


{{Optimal ET sequence|legend=1| 7, 24, 31, 55, 86efh }}
[[Mapping]]: [{{val| 1 0 -4 -3 }}, {{val| 0 3 12 11 }}]


Badness: 0.017302
: mapping generators: ~2, ~10/7


Scales: [[mohaha7]], [[mohaha10]]
{{Multival|legend=1| 3 12 11 12 9 -8 }}


== Mohamaq ==
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/7 = 632.406
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 81/80, 392/375
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/7 = {{monzo| 1/3 0 1/12 }}
: [{{Monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 2/3 0 11/12 0 }}]
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.5


[[Mapping]]: [{{val| 1 1 0 -1 }}, {{val| 0 2 8 13 }}]
[[Algebraic generator]]: Radix, the real root of ''x''<sup>5</sup> - 2''x''<sup>4</sup> + 2''x''<sup>3</sup> - 2''x''<sup>2</sup> + 2''x'' - 2, also a root of ''x''<sup>6</sup> - ''x''<sup>5</sup> - 2. The recurrence converges.


: mapping generators: ~2, ~25/21
{{Optimal ET sequence|legend=1| 17c, 19, 55, 74d }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 350.586
[[Badness]]: 0.046706


{{Optimal ET sequence|legend=1| 7d, 17c, 24, 65cc, 89ccd }}
=== Liesel ===
Subgroup: 2.3.5.7.11


[[Badness]]: 0.077734
Comma list: 56/55, 81/80, 540/539


Scales: [[mohaha7]], [[mohaha10]]
Mapping: [{{val| 1 0 -4 -3 4 }}, {{val| 0 3 12 11 -1 }}]


=== 11-limit ===
{{Multival|legend=1| 3 12 11 -1 12 9 -12 -8 -44 -41 }}
Subgroup: 2.3.5.7.11


Comma list: 56/55, 77/75, 243/242
Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 633.073


Mapping: [{{val| 1 1 0 -1 2 }}, {{val| 0 2 8 13 5 }}]
{{Optimal ET sequence|legend=1| 17c, 19, 36, 91cee }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.565
Badness: 0.040721


{{Optimal ET sequence|legend=1| 7d, 17c, 24, 65cc, 89ccd }}
==== 13-limit ====
Liesel is a very natural 13-limit tuning, given the generator is so near 13/9.


Badness: 0.036207
Subgroup: 2.3.5.7.11.13


Scales: [[mohaha7]], [[mohaha10]]
Comma list: 56/55, 78/77, 81/80, 91/90


=== 13-limit ===
Mapping: [{{val| 1 0 -4 -3 4 0 }}, {{val| 0 3 12 11 -1 7 }}]
Subgroup: 2.3.5.7.11.13


Comma list: 56/55, 66/65, 77/75, 243/242
Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 633.042


Mapping: [{{val| 1 1 0 -1 2 4 }}, {{val| 0 2 8 13 5 -1 }}]
{{Optimal ET sequence|legend=1| 17c, 19, 36, 91ceef }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.745
Badness: 0.027304


{{Optimal ET sequence|legend=1| 7d, 17c, 24, 41c, 65cc }}
=== Elisa ===
Subgroup: 2.3.5.7.11


Badness: 0.028738
Comma list: 77/75, 81/80, 99/98


Scales: [[mohaha7]], [[mohaha10]]
Mapping: [{{val| 1 0 -4 -3 -5 }}, {{val| 0 3 12 11 16 }}]


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


Liese splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. [[74edo]] makes for a good liese tuning, though [[19edo]] can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55.
{{Multival|legend=1| 3 12 11 16 12 9 15 -8 -4 7 }}


[[Subgroup]]: 2.3.5.7
Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 633.061


[[Comma list]]: 81/80, 686/675
{{Optimal ET sequence|legend=1| 17c, 19e, 36e }}


[[Mapping]]: [{{val| 1 0 -4 -3 }}, {{val| 0 3 12 11 }}]
Badness: 0.041592


: mapping generators: ~2, ~10/7
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Multival|legend=1| 3 12 11 12 9 -8 }}
Comma list: 66/65, 77/75, 81/80, 99/98


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/7 = 632.406
Mapping: [{{val| 1 0 -4 -3 -5 0 }}, {{val| 0 3 12 11 16 7 }}]


[[Minimax tuning]]:
Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 632.991
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/7 = {{monzo| 1/3 0 1/12 }}
: [{{Monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 2/3 0 11/12 0 }}]
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.5


[[Algebraic generator]]: Radix, the real root of ''x''<sup>5</sup> - 2''x''<sup>4</sup> + 2''x''<sup>3</sup> - 2''x''<sup>2</sup> + 2''x'' - 2, also a root of ''x''<sup>6</sup> - ''x''<sup>5</sup> - 2. The recurrence converges.
{{Optimal ET sequence|legend=1| 17c, 19e, 36e }}


{{Optimal ET sequence|legend=1| 17c, 19, 55, 74d }}
Badness: 0.026922


[[Badness]]: 0.046706
=== Lisa ===
 
=== Liesel ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 56/55, 81/80, 540/539
Comma list: 45/44, 81/80, 343/330


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


{{Multival|legend=1| 3 12 11 -1 12 9 -12 -8 -44 -41 }}
Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 631.370


Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 633.073
{{Optimal ET sequence|legend=1| 17cee, 19 }}


{{Optimal ET sequence|legend=1| 17c, 19, 36, 91cee }}
Badness: 0.054829
 
Badness: 0.040721


==== 13-limit ====
==== 13-limit ====
Liesel is a very natural 13-limit tuning, given the generator is so near 13/9.
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 56/55, 78/77, 81/80, 91/90
Comma list: 45/44, 81/80, 91/88, 147/143


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


Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 633.042
Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 631.221


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


Badness: 0.027304
Badness: 0.036144


=== Elisa ===
== Superpine ==
Subgroup: 2.3.5.7.11
The superpine temperament is generated by 1/3 of a fourth, represented by [[35/32]], which resembles [[porcupine]], but it favors flat fifths instead of sharp ones. Unlike in porcupine, the minor third reached by 2 generators up is strongly neutral-flavored and does not represent [[6/5]]–harmonics other than 3 all require the 15-tone mos to properly utilize. This temperament has an obvious 11-limit interpretation by treating the generator as [[11/10]] as in porcupine, which makes [[11/8]] high-[[complexity]] like the other harmonics, but in the 13-limit 5 generators up closely approximates [[13/8]]. [[43edo]] is a good tuning especially for the higher-limit extensions.


Comma list: 77/75, 81/80, 99/98
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 0 -4 -3 -5 }}, {{val| 0 3 12 11 16 }}]
[[Comma list]]: 81/80, 1119744/1071875


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


{{Multival|legend=1| 3 12 11 16 12 9 15 -8 -4 7 }}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~35/32 = 167.279


Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 633.061
{{Optimal ET sequence|legend=1| 7, 36, 43, 79c }}


{{Optimal ET sequence|legend=1| 17c, 19e, 36e }}
[[Badness]]: 0.137


Badness: 0.041592
=== 11-limit ===
Subgroup: 2.3.5.7.11


==== 13-limit ====
Comma list: 81/80, 176/175, 864/847
Subgroup: 2.3.5.7.11.13


Comma list: 66/65, 77/75, 81/80, 99/98
Mapping: {{mapping| 1 2 4 1 5 | 0 -3 -12 13 -11 }}


Mapping: [{{val| 1 0 -4 -3 -5 0 }}, {{val| 0 3 12 11 16 7 }}]
Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 167.407


Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 632.991
Optimal ET sequence: {{Optimal ET sequence| 7, 36, 43 }}


{{Optimal ET sequence|legend=1| 17c, 19e, 36e }}
Badness: 0.0576


Badness: 0.026922
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=== Lisa ===
Comma list: 78/77, 81/80, 144/143, 176/175
Subgroup: 2.3.5.7.11


Comma list: 45/44, 81/80, 343/330
Mapping: {{mapping| 1 2 4 1 5 3 | 0 -3 -12 13 -11 5 }}


Mapping: [{{val| 1 0 -4 -3 -6 }}, {{val| 0 3 12 11 18 }}]
Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 167.427


Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 631.370
Optimal ET sequence: {{Optimal ET sequence| 7, 36, 43 }}


{{Optimal ET sequence|legend=1| 17cee, 19 }}
Badness: 0.0368


Badness: 0.054829
== Lithium ==
Lithium is named after the 3rd element for being period-3, and also for lithium's molar mass of 6.9 g/mol since 69edo supports it. It supports a [[3L 6s]] scale and thus intuitively can be thought of as "tcherepnin meantone" in that context.


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


Comma list: 45/44, 81/80, 91/88, 147/143
[[Comma list]]: 81/80, 3125/3087


Mapping: [{{val| 1 0 -4 -3 -6 0 }}, {{val| 0 3 12 11 18 7 }}]
[[Mapping]]: [{{val| 3 0 -12 -20 }}, {{val| 0 1 4 6 }}]


Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 631.221
: mapping generators: ~56/45, ~3


{{Optimal ET sequence|legend=1| 17cee, 19 }}
[[Optimal tuning]] ([[CTE]]): ~56/45 = 1\3, ~3/2 = 695.827


Badness: 0.036144
{{Optimal ET sequence|legend=1| 12, 33cd, 45, 57 }}


== Superpine ==
[[Badness]]: 0.0692
The superpine temperament is generated by 1/3 of a fourth, represented by [[35/32]], which resembles [[porcupine]], but it favors flat fifths instead of sharp ones. Unlike in porcupine, the minor third reached by 2 generators up is strongly neutral-flavored and does not represent [[6/5]]–harmonics other than 3 all require the 15-tone mos to properly utilize. This temperament has an obvious 11-limit interpretation by treating the generator as [[11/10]] as in porcupine, which makes [[11/8]] high-[[complexity]] like the other harmonics, but in the 13-limit 5 generators up closely approximates [[13/8]]. [[43edo]] is a good tuning especially for the higher-limit extensions.
 
== Squares ==
{{Main| Squares }}
 
Squares splits the interval of an eleventh, or 8/3, into four supermajor third ([[9/7]]) intervals, and uses it for a generator. [[31edo]], with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out [[2401/2400]], the breedsma, as well as [[2430/2401]].


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


[[Comma list]]: 81/80, 1119744/1071875
[[Comma list]]: 81/80, 2401/2400


{{Mapping|legend=1| 1 2 4 1 | 0 -3 -12 13 }}
[[Mapping]]: [{{val| 1 3 8 6 }}, {{val| 0 -4 -16 -9 }}]


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~35/32 = 167.279
: mapping generators: ~2, ~9/7


{{Optimal ET sequence|legend=1| 7, 36, 43, 79c }}
{{Multival|legend=1| 4 16 9 16 3 -24 }}


[[Badness]]: 0.137
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 425.942


=== 11-limit ===
[[Minimax tuning]]:
Subgroup: 2.3.5.7.11
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~9/7 = {{monzo| 1/2 0 -1/16 }}
: [{{Monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 3/2 0 9/16 0 }}]
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.5


Comma list: 81/80, 176/175, 864/847
[[Algebraic generator]]: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.


Mapping: {{mapping| 1 2 4 1 5 | 0 -3 -12 13 -11 }}
{{Optimal ET sequence|legend=1| 14c, 17c, 31 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 167.407
[[Badness]]: 0.045993


Optimal ET sequence: {{Optimal ET sequence| 7, 36, 43 }}
Scales: [[skwares8]], [[skwares11]], [[skwares14]]


Badness: 0.0576
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 13-limit ===
Comma list: 81/80, 99/98, 121/120
Subgroup: 2.3.5.7.11.13


Comma list: 78/77, 81/80, 144/143, 176/175
Mapping: [{{val| 1 3 8 6 7 }}, {{val| 0 -4 -16 -9 -10 }}]


Mapping: {{mapping| 1 2 4 1 5 3 | 0 -3 -12 13 -11 5 }}
{{Multival|legend=1| 4 16 9 10 16 3 2 -24 -32 -3 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 167.427
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.957


Optimal ET sequence: {{Optimal ET sequence| 7, 36, 43 }}
{{Optimal ET sequence|legend=1| 14c, 17c, 31 }}


Badness: 0.0368
Badness: 0.021636


== Lithium ==
==== 13-limit ====
Lithium is named after the 3rd element for being period-3, and also for lithium's molar mass of 6.9 g/mol since 69edo supports it. It supports a [[3L 6s]] scale and thus intuitively can be thought of as "tcherepnin meantone" in that context.
Subgroup: 2.3.5.7.11.13


[[Subgroup]]: 2.3.5.7
Comma list: 66/65, 81/80, 99/98, 121/120


[[Comma list]]: 81/80, 3125/3087
Mapping: [{{val| 1 3 8 6 7 3 }}, {{val| 0 -4 -16 -9 -10 2 }}]


[[Mapping]]: [{{val| 3 0 -12 -20 }}, {{val| 0 1 4 6 }}]
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.550


: mapping generators: ~56/45, ~3
{{Optimal ET sequence|legend=1| 14c, 17c, 31, 79cf }}


[[Optimal tuning]] ([[CTE]]): ~56/45 = 1\3, ~3/2 = 695.827
Badness: 0.025514


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


[[Badness]]: 0.0692
Comma list: 78/77, 81/80, 91/90, 99/98


== Squares ==
Mapping: [{{val| 1 3 8 6 7 9 }}, {{val| 0 -4 -16 -9 -10 -15 }}]
{{Main| Squares }}


Squares splits the interval of an eleventh, or 8/3, into four supermajor third ([[9/7]]) intervals, and uses it for a generator. [[31edo]], with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out [[2401/2400]], the breedsma, as well as [[2430/2401]].
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.7516


[[Subgroup]]: 2.3.5.7
{{Optimal ET sequence|legend=1| 14cf, 17c, 31f }}


[[Comma list]]: 81/80, 2401/2400
Badness: 0.026877


[[Mapping]]: [{{val| 1 3 8 6 }}, {{val| 0 -4 -16 -9 }}]
==== Agora ====
Subgroup: 2.3.5.7.11.13


: mapping generators: ~2, ~9/7
Comma list: 81/80, 99/98, 105/104, 121/120


{{Multival|legend=1| 4 16 9 16 3 -24 }}
Mapping: [{{val| 1 3 8 6 7 14 }}, {{val| 0 -4 -16 -9 -10 -29 }}]


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 425.942
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 426.276


[[Minimax tuning]]:
{{Optimal ET sequence|legend=1| 14cf, 31, 45ef, 76e }}
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~9/7 = {{monzo| 1/2 0 -1/16 }}
: [{{Monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 3/2 0 9/16 0 }}]
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.5


[[Algebraic generator]]: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.
Badness: 0.024522


{{Optimal ET sequence|legend=1| 14c, 17c, 31 }}
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


[[Badness]]: 0.045993
Comma list: 81/80, 99/98, 105/104, 120/119, 121/119


Scales: [[skwares8]], [[skwares11]], [[skwares14]]
Mapping: [{{val| 1 3 8 6 7 14 8 }}, {{val| 0 -4 -16 -9 -10 -29 -11 }}]


=== 11-limit ===
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 426.187
Subgroup: 2.3.5.7.11


Comma list: 81/80, 99/98, 121/120
{{Optimal ET sequence|legend=1| 14cf, 31, 76e }}


Mapping: [{{val| 1 3 8 6 7 }}, {{val| 0 -4 -16 -9 -10 }}]
Badness: 0.022573


{{Multival|legend=1| 4 16 9 10 16 3 2 -24 -32 -3 }}
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.957
Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119


{{Optimal ET sequence|legend=1| 14c, 17c, 31 }}
Mapping: [{{val| 1 3 8 6 7 14 8 11 }}, {{val| 0 -4 -16 -9 -10 -29 -11 -19 }}]


Badness: 0.021636
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 426.225


==== 13-limit ====
{{Optimal ET sequence|legend=1| 14cf, 31, 76e }}
Subgroup: 2.3.5.7.11.13


Comma list: 66/65, 81/80, 99/98, 121/120
Badness: 0.018839


Mapping: [{{val| 1 3 8 6 7 3 }}, {{val| 0 -4 -16 -9 -10 2 }}]
=== Cuboctahedra ===
Subgroup: 2.3.5.7.11


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.550
Comma list: 81/80, 385/384, 1375/1372


{{Optimal ET sequence|legend=1| 14c, 17c, 31, 79cf }}
Mapping: [{{val| 1 3 8 6 -4 }}, {{val| 0 -4 -16 -9 21 }}]


Badness: 0.025514
{{Multival|legend=1| 4 16 9 -21 16 3 -47 -24 -104 -90 }}


==== Squad ====
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.993
Subgroup: 2.3.5.7.11.13


Comma list: 78/77, 81/80, 91/90, 99/98
{{Optimal ET sequence|legend=1| 14ce, 17ce, 31, 107b, 138b, 169be, 200be }}


Mapping: [{{val| 1 3 8 6 7 9 }}, {{val| 0 -4 -16 -9 -10 -15 }}]
Badness: 0.056826


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.7516
== Jerome ==
Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5<sup>1/20</sup>, or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size.


{{Optimal ET sequence|legend=1| 14cf, 17c, 31f }}
[[Subgroup]]: 2.3.5.7


Badness: 0.026877
[[Comma list]]: 81/80, 17280/16807


==== Agora ====
[[Mapping]]: [{{val| 1 1 0 2 }}, {{val| 0 5 20 7 }}]
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 99/98, 105/104, 121/120
: mapping generators: ~2, ~54/49


Mapping: [{{val| 1 3 8 6 7 14 }}, {{val| 0 -4 -16 -9 -10 -29 }}]
{{Multival|legend=1| 5 20 7 20 -3 -40 }}


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 426.276
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~54/49 = 139.343


{{Optimal ET sequence|legend=1| 14cf, 31, 45ef, 76e }}
{{Optimal ET sequence|legend=1| 17c, 26, 43, 69, 112bd }}


Badness: 0.024522
[[Badness]]: 0.108656


===== 17-limit =====
=== 11-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11


Comma list: 81/80, 99/98, 105/104, 120/119, 121/119
Comma list: 81/80, 99/98, 864/847


Mapping: [{{val| 1 3 8 6 7 14 8 }}, {{val| 0 -4 -16 -9 -10 -29 -11 }}]
Mapping: [{{val| 1 1 0 2 3 }}, {{val| 0 5 20 7 4 }}]


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 426.187
{{Multival|legend=1| 5 20 7 4 20 -3 -11 -40 -60 -13 }}


{{Optimal ET sequence|legend=1| 14cf, 31, 76e }}
Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 139.428


Badness: 0.022573
{{Optimal ET sequence|legend=1| 17c, 26, 43, 69 }}


===== 19-limit =====
Badness: 0.047914
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 1 3 8 6 7 14 8 11 }}, {{val| 0 -4 -16 -9 -10 -29 -11 -19 }}]
Comma list: 78/77, 81/80, 99/98, 144/143


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 426.225
Mapping: [{{val| 1 1 0 2 3 3 }}, {{val| 0 5 20 7 4 6 }}]


{{Optimal ET sequence|legend=1| 14cf, 31, 76e }}
Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 139.387


Badness: 0.018839
{{Optimal ET sequence|legend=1| 17c, 26, 43, 69 }}


=== Cuboctahedra ===
Badness: 0.029285
Subgroup: 2.3.5.7.11


Comma list: 81/80, 385/384, 1375/1372
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Mapping: [{{val| 1 3 8 6 -4 }}, {{val| 0 -4 -16 -9 21 }}]
Comma list: 78/77, 81/80, 99/98, 144/143, 189/187


{{Multival|legend=1| 4 16 9 -21 16 3 -47 -24 -104 -90 }}
Mapping: [{{val| 1 1 0 2 3 3 2 }}, {{val| 0 5 20 7 4 6 18 }}]


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.993
Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 139.362


{{Optimal ET sequence|legend=1| 14ce, 17ce, 31, 107b, 138b, 169be, 200be }}
{{Optimal ET sequence|legend=1| 17cg, 26, 43, 69 }}


Badness: 0.056826
Badness: 0.020878


== Jerome ==
=== 19-limit ===
Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5<sup>1/20</sup>, or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size.
Subgroup: 2.3.5.7.11.13.17.19


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


[[Comma list]]: 81/80, 17280/16807
Mapping: [{{val| 1 1 0 2 3 3 2 1 }}, {{val| 0 5 20 7 4 6 18 28 }}]


[[Mapping]]: [{{val| 1 1 0 2 }}, {{val| 0 5 20 7 }}]
Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 139.313


: mapping generators: ~2, ~54/49
{{Optimal ET sequence|legend=1| 17cgh, 26, 43, 69 }}


{{Multival|legend=1| 5 20 7 20 -3 -40 }}
Badness: 0.018229


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~54/49 = 139.343
== Meantritone ==
The ''meantritone'' temperament tempers out the mirkwai comma (16875/16807) and trimyna comma (50421/50000) in the 7-limit. In this temperament, three septimal tritones equals ~30/11 (an octave plus [[15/11]]-wide super-fourth) and five of them equals ~[[16/3]] (double-compound fourth). The name "meantritone" is a portmanteau of meantone and tritone, the latter is a generator of this temperament.


{{Optimal ET sequence|legend=1| 17c, 26, 43, 69, 112bd }}
[[Subgroup]]: 2.3.5.7


[[Badness]]: 0.108656
[[Comma list]]: 81/80, 16875/16807


=== 11-limit ===
[[Mapping]]: [{{val| 1 4 12 12 }}, {{val| 0 -5 -20 -19 }}]
Subgroup: 2.3.5.7.11


Comma list: 81/80, 99/98, 864/847
{{Multival|legend=1| 5 20 19 20 16 -12 }}


Mapping: [{{val| 1 1 0 2 3 }}, {{val| 0 5 20 7 4 }}]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 580.766


{{Multival|legend=1| 5 20 7 4 20 -3 -11 -40 -60 -13 }}
{{Optimal ET sequence|legend=1| 2cd, 29cd, 31 }}


Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 139.428
[[Badness]]: 0.082239


{{Optimal ET sequence|legend=1| 17c, 26, 43, 69 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.047914
Comma list: 81/80, 99/98, 2541/2500


=== 13-limit ===
Mapping: [{{val| 1 4 12 12 17 }}, {{val| 0 -5 -20 -19 -28 }}]
Subgroup: 2.3.5.7.11.13


Comma list: 78/77, 81/80, 99/98, 144/143
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.647


Mapping: [{{val| 1 1 0 2 3 3 }}, {{val| 0 5 20 7 4 6 }}]
{{Optimal ET sequence|legend=1| 2cde, 29cde, 31 }}


Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 139.387
Badness: 0.042869


{{Optimal ET sequence|legend=1| 17c, 26, 43, 69 }}
== Injera ==
Injera has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. [[38edo|38EDO]], which is two parallel [[19edo]]s, is an excellent tuning for injera.


Badness: 0.029285
[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name]


=== 17-limit ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17


Comma list: 78/77, 81/80, 99/98, 144/143, 189/187
[[Comma list]]: 50/49, 81/80


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


Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 139.362
: mapping generators: ~7/5, ~3


{{Optimal ET sequence|legend=1| 17cg, 26, 43, 69 }}
{{Multival|legend=1| 2 8 8 8 7 -4 }}


Badness: 0.020878
[[Optimal tuning]] ([[POTE]]): ~7/5 = 1\2, ~3/2 = 694.375


=== 19-limit ===
[[Tuning ranges]]:
Subgroup: 2.3.5.7.11.13.17.19
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [688.957, 701.955]
* 9-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]


Comma list: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143
{{Optimal ET sequence|legend=1| 12, 26, 38, 102bcd, 140bccd, 178bbccdd }}


Mapping: [{{val| 1 1 0 2 3 3 2 1 }}, {{val| 0 5 20 7 4 6 18 28 }}]
[[Badness]]: 0.031130


Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 139.313
; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 Two Pairs of Socks] (in [[26edo|26EDO]]) by [[Igliashon Jones]]


{{Optimal ET sequence|legend=1| 17cgh, 26, 43, 69 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.018229
Comma list: 45/44, 50/49, 81/80


== Meantritone ==
Mapping: [{{val| 2 0 -8 -7 -12 }}, {{val| 0 1 4 4 6 }}]
The ''meantritone'' temperament tempers out the mirkwai comma (16875/16807) and trimyna comma (50421/50000) in the 7-limit. In this temperament, three septimal tritones equals ~30/11 (an octave plus [[15/11]]-wide super-fourth) and five of them equals ~[[16/3]] (double-compound fourth). The name "meantritone" is a portmanteau of meantone and tritone, the latter is a generator of this temperament.


[[Subgroup]]: 2.3.5.7
{{Multival|legend=1| 2 8 8 12 8 7 12 -4 0 6 }}


[[Comma list]]: 81/80, 16875/16807
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 692.840


[[Mapping]]: [{{val| 1 4 12 12 }}, {{val| 0 -5 -20 -19 }}]
Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
* 11-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]


{{Multival|legend=1| 5 20 19 20 16 -12 }}
{{Optimal ET sequence|legend=1| 12, 14c, 26, 90bce, 116bcce }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 580.766
Badness: 0.023124


{{Optimal ET sequence|legend=1| 2cd, 29cd, 31 }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[Badness]]: 0.082239
Comma list: 45/44, 50/49, 78/77, 81/80


=== 11-limit ===
Mapping: [{{val| 2 0 -8 -7 -12 -21 }}, {{val| 0 1 4 4 6 9 }}]
Subgroup: 2.3.5.7.11


Comma list: 81/80, 99/98, 2541/2500
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 692.673


Mapping: [{{val| 1 4 12 12 17 }}, {{val| 0 -5 -20 -19 -28 }}]
Tuning ranges:  
* 13- and 15-odd-limit diamond monotone: ~3/2 = 692.308 (15\26)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.647
{{Optimal ET sequence|legend=1| 12f, 14cf, 26, 38e }}


{{Optimal ET sequence|legend=1| 2cde, 29cde, 31 }}
Badness: 0.021565
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.042869
Comma list: 45/44, 50/49, 78/77, 81/80, 85/84


== Injera ==
Mapping: [{{val| 2 0 -8 -7 -12 -21 5 }}, {{val| 0 1 4 4 6 9 1 }}]
Injera has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. [[38edo|38EDO]], which is two parallel [[19edo]]s, is an excellent tuning for injera.


[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name]
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 692.487


[[Subgroup]]: 2.3.5.7
{{Optimal ET sequence|legend=1| 12f, 14cf, 26 }}


[[Comma list]]: 50/49, 81/80
Badness: 0.018358


[[Mapping]]: [{{val| 2 0 -8 -7 }}, {{val| 0 1 4 4 }}]
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


: mapping generators: ~7/5, ~3
Comma list: 45/44, 50/49, 57/56, 78/77, 81/80, 85/84


{{Multival|legend=1| 2 8 8 8 7 -4 }}
Mapping: [{{val| 2 0 -8 -7 -12 -21 5 -1 }}, {{val| 0 1 4 4 6 9 1 3 }}]


[[Optimal tuning]] ([[POTE]]): ~7/5 = 1\2, ~3/2 = 694.375
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 692.299


[[Tuning ranges]]:
{{Optimal ET sequence|legend=1| 12f, 14cf, 26 }}
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
 
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [688.957, 701.955]
Badness: 0.015118
* 9-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]


{{Optimal ET sequence|legend=1| 12, 26, 38, 102bcd, 140bccd, 178bbccdd }}
==== Enjera ====
Subgroup: 2.3.5.7.11.13


[[Badness]]: 0.031130
Comma list: 27/26, 40/39, 45/44, 50/49


; Music
Mapping: [{{val| 2 0 -8 -7 -12 -2 }}, {{val| 0 1 4 4 6 3 }}]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 Two Pairs of Socks] (in [[26edo|26EDO]]) by [[Igliashon Jones]]


=== 11-limit ===
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 694.121
Subgroup: 2.3.5.7.11


Comma list: 45/44, 50/49, 81/80
{{Optimal ET sequence|legend=1| 12f, 14c, 26f, 38eff }}


Mapping: [{{val| 2 0 -8 -7 -12 }}, {{val| 0 1 4 4 6 }}]
Badness: 0.026542


{{Multival|legend=1| 2 8 8 12 8 7 12 -4 0 6 }}
=== Injerous ===
Subgroup: 2.3.5.7.11


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 692.840
Comma list: 33/32, 50/49, 55/54


Tuning ranges:  
Mapping: [{{val| 2 0 -8 -7 10 }}, {{val| 0 1 4 4 -1 }}]
* 11-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
* 11-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]


{{Optimal ET sequence|legend=1| 12, 14c, 26, 90bce, 116bcce }}
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 690.548


Badness: 0.023124
{{Optimal ET sequence|legend=1| 12e, 14c, 26e, 40cee }}


==== 13-limit ====
Badness: 0.038577
Subgroup: 2.3.5.7.11.13


Comma list: 45/44, 50/49, 78/77, 81/80
=== Lahoh ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 2 0 -8 -7 -12 -21 }}, {{val| 0 1 4 4 6 9 }}]
Comma list: 50/49, 56/55, 81/77


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 692.673
Mapping: [{{val| 2 0 -8 -7 7 }}, {{val| 0 1 4 4 0 }}]


Tuning ranges:
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 699.001
* 13- and 15-odd-limit diamond monotone: ~3/2 = 692.308 (15\26)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]


{{Optimal ET sequence|legend=1| 12f, 14cf, 26, 38e }}
{{Optimal ET sequence|legend=1| 2cd, 10cd, 12 }}


Badness: 0.021565
Badness: 0.043062


===== 17-limit =====
=== Teff ===
Subgroup: 2.3.5.7.11.13.17
{{Main| Teff }}


Comma list: 45/44, 50/49, 78/77, 81/80, 85/84
Teff (found by [[Mason Green]]) is to injera what mohajira is to meantone; it splits the generator in half in order to accommodate higher limit intervals, creating a half-octave quarter-tone temperament.


Mapping: [{{val| 2 0 -8 -7 -12 -21 5 }}, {{val| 0 1 4 4 6 9 1 }}]
Subgroup: 2.3.5.7.11


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 692.487
Comma list: 50/49, 81/80, 864/847


{{Optimal ET sequence|legend=1| 12f, 14cf, 26 }}
Mapping: [{{val| 2 1 -4 -3 8 }}, {{val| 0 2 8 8 -1 }}]


Badness: 0.018358
: mapping generators: ~7/5, ~16/11


===== 19-limit =====
Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 552.5303
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 45/44, 50/49, 57/56, 78/77, 81/80, 85/84
{{Optimal ET sequence|legend=1| 24d, 26, 50d }}


Mapping: [{{val| 2 0 -8 -7 -12 -21 5 -1 }}, {{val| 0 1 4 4 6 9 1 3 }}]
Badness: 0.070689


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 692.299
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Optimal ET sequence|legend=1| 12f, 14cf, 26 }}
Comma list: 50/49, 78/77, 81/80, 144/143


Badness: 0.015118
Mapping: [{{val| 2 1 -4 -3 8 2 }}, {{val| 0 2 8 8 -1 5 }}]


==== Enjera ====
Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 552.5324
Subgroup: 2.3.5.7.11.13


Comma list: 27/26, 40/39, 45/44, 50/49
{{Optimal ET sequence|legend=1| 24d, 26, 50d }}


Mapping: [{{val| 2 0 -8 -7 -12 -2 }}, {{val| 0 1 4 4 6 3 }}]
Badness: 0.040047


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 694.121
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


{{Optimal ET sequence|legend=1| 12f, 14c, 26f, 38eff }}
Comma list: 50/49, 78/77, 81/80, 85/84, 144/143


Badness: 0.026542
Mapping: [{{val| 2 1 -4 -3 8 2 6 }}, {{val| 0 2 8 8 -1 5 2 }}]


=== Injerous ===
Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 552.6558
Subgroup: 2.3.5.7.11


Comma list: 33/32, 50/49, 55/54
{{Optimal ET sequence|legend=1| 24d, 26 }}


Mapping: [{{val| 2 0 -8 -7 10 }}, {{val| 0 1 4 4 -1 }}]
Badness: 0.029499


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 690.548
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


{{Optimal ET sequence|legend=1| 12e, 14c, 26e, 40cee }}
Comma list: 50/49, 57/56, 78/77, 81/80, 85/84, 144/143


Badness: 0.038577
Mapping: [{{val| 2 1 -4 -3 8 2 6 2 }}, {{val| 0 2 8 8 -1 5 2 6 }}]


=== Lahoh ===
Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 552.6382
Subgroup: 2.3.5.7.11


Comma list: 50/49, 56/55, 81/77
{{Optimal ET sequence|legend=1| 24d, 26 }}


Mapping: [{{val| 2 0 -8 -7 7 }}, {{val| 0 1 4 4 0 }}]
Badness: 0.023133


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 699.001
== Pombe ==
Pombe (named after the African millet beer) is a variant of [[#Teff]] by [[User:Kaiveran|Kaiveran Lugheidh]] that eschews the tempering of 50/49 to attain more accuracy in the 7-limit. Oddly, the 7th harmonic has a lesser generator distance than in teff (-5 vs +8), but this combined with the fact that other harmonics are in the opposite direction means that the 7-limit diamond is more complex overall.


{{Optimal ET sequence|legend=1| 2cd, 10cd, 12 }}
[[Subgroup]]: 2.3.5.7


Badness: 0.043062
[[Comma list]]: 81/80, 300125/294912


=== Teff ===
[[Mapping]]: [{{val| 2 1 -4 11 }}, {{val| 0 2 8 -5 }}]
{{Main| Teff }}


Teff (found by [[Mason Green]]) is to injera what mohajira is to meantone; it splits the generator in half in order to accommodate higher limit intervals, creating a half-octave quarter-tone temperament.
: mapping generators: ~735/512, ~35/24


Subgroup: 2.3.5.7.11
{{Multival|legend=1| 4 16 -10 16 -27 -68 }}


Comma list: 50/49, 81/80, 864/847
[[Optimal tuning]] ([[POTE]]): ~735/512 = 1\2, ~48/35 = 552.2206


Mapping: [{{val| 2 1 -4 -3 8 }}, {{val| 0 2 8 8 -1 }}]
{{Optimal ET sequence|legend=1| 24, 26, 50, 126bcd, 176bcdd, 226bbcdd }}


: mapping generators: ~7/5, ~16/11
[[Badness]]: 0.116104


Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 552.5303
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Optimal ET sequence|legend=1| 24d, 26, 50d }}
Comma list: 81/80, 245/242, 385/384


Badness: 0.070689
Mapping: [{{val| 2 1 -4 11 8 }}, {{val| 0 2 8 -5 -1 }}]


==== 13-limit ====
Optimal tuning (POTE): ~99/70 = 1\2, ~11/8 = 552.0929
Subgroup: 2.3.5.7.11.13


Comma list: 50/49, 78/77, 81/80, 144/143
{{Optimal ET sequence|legend=1| 24, 26, 50 }}


Mapping: [{{val| 2 1 -4 -3 8 2 }}, {{val| 0 2 8 8 -1 5 }}]
Badness: 0.052099


Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 552.5324
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Optimal ET sequence|legend=1| 24d, 26, 50d }}
Comma list: 81/80, 105/104, 144/143, 245/242


Badness: 0.040047
Mapping: [{{val| 2 1 -4 11 8 2 }}, {{val| 0 2 8 -5 -1 5 }}]


==== 17-limit ====
Optimal tuning (POTE): ~99/70 = 1\2, ~11/8 = 552.1498
 
{{Optimal ET sequence|legend=1| 24, 26, 50 }}
 
Badness: 0.031039
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 50/49, 78/77, 81/80, 85/84, 144/143
Comma list: 81/80, 105/104, 144/143, 245/242, 273/272


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


Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 552.6558
Optimal tuning (POTE): ~17/12 = 1\2, ~11/8 = 552.1579


{{Optimal ET sequence|legend=1| 24d, 26 }}
{{Optimal ET sequence|legend=1| 24, 26, 50 }}


Badness: 0.029499
Badness: 0.021260


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


Comma list: 50/49, 57/56, 78/77, 81/80, 85/84, 144/143
Comma list: 81/80, 105/104, 133/132, 144/143, 171/170, 210/209


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


Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 552.6382
Optimal tuning (POTE): ~17/12 = 1\2, ~11/8 = 552.1196


{{Optimal ET sequence|legend=1| 24d, 26 }}
{{Optimal ET sequence|legend=1| 24, 26, 50 }}


Badness: 0.023133
Badness: 0.016548
 
== Pombe ==
Pombe (named after the African millet beer) is a variant of [[#Teff]] by [[User:Kaiveran|Kaiveran Lugheidh]] that eschews the tempering of 50/49 to attain more accuracy in the 7-limit. Oddly, the 7th harmonic has a lesser generator distance than in teff (-5 vs +8), but this combined with the fact that other harmonics are in the opposite direction means that the 7-limit diamond is more complex overall.


== Orphic ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 81/80, 300125/294912
[[Comma list]]: 81/80, 5898240/5764801


[[Mapping]]: [{{val| 2 1 -4 11 }}, {{val| 0 2 8 -5 }}]
[[Mapping]]: [{{val| 2 5 12 7 }}, {{val| 0 -4 -16 -3 }}]


: mapping generators: ~735/512, ~35/24
Mapping generators: ~2401/1728, ~7/6


{{Multival|legend=1| 4 16 -10 16 -27 -68 }}
{{Multival|legend=1| 8 32 6 32 -13 -76 }}


[[Optimal tuning]] ([[POTE]]): ~735/512 = 1\2, ~48/35 = 552.2206
[[Optimal tuning]] ([[POTE]]): ~2401/1728 = 1\2, ~7/6 = 275.794


{{Optimal ET sequence|legend=1| 24, 26, 50, 126bcd, 176bcdd, 226bbcdd }}
{{Optimal ET sequence|legend=1| 26, 48c, 74, 174bd, 248bbd }}


[[Badness]]: 0.116104
[[Badness]]: 0.258825


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


Comma list: 81/80, 245/242, 385/384
Comma list: 81/80, 99/98, 73728/73205


Mapping: [{{val| 2 1 -4 11 8 }}, {{val| 0 2 8 -5 -1 }}]
Mapping: [{{val| 2 5 12 7 6 }}, {{val| 0 -4 -16 -3 2 }}]


Optimal tuning (POTE): ~99/70 = 1\2, ~11/8 = 552.0929
Optimal tuning (POTE): ~363/256 = 1\2, ~7/6 = 275.762


{{Optimal ET sequence|legend=1| 24, 26, 50 }}
{{Optimal ET sequence|legend=1| 26, 48c, 74, 248bbd, 322bbdd }}


Badness: 0.052099
Badness: 0.101499


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


Comma list: 81/80, 105/104, 144/143, 245/242
Comma list: 81/80, 99/98, 144/143, 2200/2197


Mapping: [{{val| 2 1 -4 11 8 2 }}, {{val| 0 2 8 -5 -1 5 }}]
Mapping: [{{val| 2 5 12 7 6 12 }}, {{val| 0 -4 -16 -3 2 -10 }}]


Optimal tuning (POTE): ~99/70 = 1\2, ~11/8 = 552.1498
Optimal tuning (POTE): ~55/39 = 1\2, ~7/6 = 275.774


{{Optimal ET sequence|legend=1| 24, 26, 50 }}
{{Optimal ET sequence|legend=1| 26, 48c, 74, 174bd, 248bbd, 322bbdd }}


Badness: 0.031039
Badness: 0.053482


=== 17-limit ===
== Cloudtone ==
Subgroup: 2.3.5.7.11.13.17
The ''cloudtone'' temperament (5&amp;50) tempers out the [[cloudy comma]], 16807/16384 and the [[81/80|syntonic comma]], 81/80 in the 7-limit. It can be extended to the 11- and 13-limit by adding 385/384 and 105/104 to the comma list in this order.


Comma list: 81/80, 105/104, 144/143, 245/242, 273/272
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 2 1 -4 11 8 2 6 }}, {{val| 0 2 8 -5 -1 5 2 }}]
[[Comma list]]: 81/80, 16807/16384


Optimal tuning (POTE): ~17/12 = 1\2, ~11/8 = 552.1579
[[Mapping]]: [{{val| 5 0 -20 14 }}, {{val| 0 1 4 0 }}]


{{Optimal ET sequence|legend=1| 24, 26, 50 }}
: mapping generators: ~8/7, ~3


Badness: 0.021260
{{Multival|legend=1| 5 20 0 20 -14 -56 }}


=== 19-limit ===
[[Optimal tuning]] ([[POTE]]): ~8/7 = 1\5, ~3/2 = 695.720
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 81/80, 105/104, 133/132, 144/143, 171/170, 210/209
{{Optimal ET sequence|legend=1| 5, 45, 50 }}


Mapping: [{{val| 2 1 -4 11 8 2 6 2 }}, {{val| 0 2 8 -5 -1 5 2 6 }}]
[[Badness]]: 0.102256


Optimal tuning (POTE): ~17/12 = 1\2, ~11/8 = 552.1196
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Optimal ET sequence|legend=1| 24, 26, 50 }}
Comma list: 81/80, 385/384, 2401/2376


Badness: 0.016548
Mapping: [{{val| 5 0 -20 14 41 }}, {{val| 0 1 4 0 -3 }}]


== Orphic ==
Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 696.536
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 81/80, 5898240/5764801
{{Optimal ET sequence|legend=1| 5, 45, 50, 155bdd, 205bddd }}


[[Mapping]]: [{{val| 2 5 12 7 }}, {{val| 0 -4 -16 -3 }}]
Badness: 0.070378


Mapping generators: ~2401/1728, ~7/6
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Multival|legend=1| 8 32 6 32 -13 -76 }}
Comma list: 81/80, 105/104, 144/143, 2401/2376


[[Optimal tuning]] ([[POTE]]): ~2401/1728 = 1\2, ~7/6 = 275.794
Mapping: [{{val| 5 0 -20 14 41 -21 }}, {{val| 0 1 4 0 -3 5 }}]


{{Optimal ET sequence|legend=1| 26, 48c, 74, 174bd, 248bbd }}
Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 696.162


[[Badness]]: 0.258825
{{Optimal ET sequence|legend=1| 5, 45f, 50 }}


=== 11-limit ===
Badness: 0.048829
Subgroup: 2.3.5.7.11


Comma list: 81/80, 99/98, 73728/73205
== Subgroup extensions ==
=== Stützel (2.3.5.19) ===
Subgroup: 2.3.5.19


Mapping: [{{val| 2 5 12 7 6 }}, {{val| 0 -4 -16 -3 2 }}]
[[Comma list]]: 81/80, 96/95


Optimal tuning (POTE): ~363/256 = 1\2, ~7/6 = 275.762
[[Gencom]]: [2 4/3; 81/80 96/95]


{{Optimal ET sequence|legend=1| 26, 48c, 74, 248bbd, 322bbdd }}
[[Gencom|Gencom mapping]]: [{{val|1 2 4 0 0 0 0 3}}, {{val|0 -1 -4 0 0 0 0 3}}]


Badness: 0.101499
[[Mapping|Sval mapping]]: [{{val|1 2 4 3}}, {{val|0 -1 -4 3}}]


=== 13-limit ===
[[Tp tuning|POL2 generator]]: ~3/2 = 697.867
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 99/98, 144/143, 2200/2197
{{Optimal ET sequence|legend=1| 5, 7, 12, 31, 43 }}


Mapping: [{{val| 2 5 12 7 6 12 }}, {{val| 0 -4 -16 -3 2 -10 }}]
[[Tp tuning #T2 tuning|RMS error]]: 1.378 cents


Optimal tuning (POTE): ~55/39 = 1\2, ~7/6 = 275.774
=== Hypnotone ===
[[Subgroup]]: 2.3.5.11


{{Optimal ET sequence|legend=1| 26, 48c, 74, 174bd, 248bbd, 322bbdd }}
[[Comma list]]: 45/44, 81/80


Badness: 0.053482
{{Mapping|legend=2| 1 0 -4 -6 | 0 1 4 6 }}


== Cloudtone ==
: sval mapping generators: ~2, ~3
The ''cloudtone'' temperament (5&amp;50) tempers out the [[cloudy comma]], 16807/16384 and the [[81/80|syntonic comma]], 81/80 in the 7-limit. It can be extended to the 11- and 13-limit by adding 385/384 and 105/104 to the comma list in this order.


[[Subgroup]]: 2.3.5.7
[[Optimal tuning]] ([[CTE]]): ~2/1 = 1\1, ~3/2 = 694.6998


[[Comma list]]: 81/80, 16807/16384
{{Optimal ET sequence|legend=1| 7, 12, 19, 26, 45 }}


[[Mapping]]: [{{val| 5 0 -20 14 }}, {{val| 0 1 4 0 }}]
[[Badness]]: 0.0104


: mapping generators: ~8/7, ~3
==== 2.3.5.11.13 subgroup ====
Subgroup: 2.3.5.11.13


{{Multival|legend=1| 5 20 0 20 -14 -56 }}
Comma list: 45/44, 65/64, 81/80


[[Optimal tuning]] ([[POTE]]): ~8/7 = 1\5, ~3/2 = 695.720
Sval mapping: {{mapping| 1 0 -4 -6 10 | 0 1 4 6 -4 }}


{{Optimal ET sequence|legend=1| 5, 45, 50 }}
: sval mapping generators: ~2, ~3


[[Badness]]: 0.102256
Optimal tuning (CTE): ~2/1 = 1\1, ~3/2 = 693.9513


=== 11-limit ===
Optimal ET sequence: {{Optimal ET sequence| 7, 12, 19, 26, 45f }}
Subgroup: 2.3.5.7.11


Comma list: 81/80, 385/384, 2401/2376
Badness: 0.0141


Mapping: [{{val| 5 0 -20 14 41 }}, {{val| 0 1 4 0 -3 }}]
=== Dequarter ===
[[Subgroup]]: 2.3.5.11


Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 696.536
[[Comma list]]: 33/32, 55/54


{{Optimal ET sequence|legend=1| 5, 45, 50, 155bdd, 205bddd }}
{{Mapping|legend=2| 1 0 -4 5 | 0 1 4 -1 }}


Badness: 0.070378
: sval mapping generators: ~2, ~3


=== 13-limit ===
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 696.0387
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 105/104, 144/143, 2401/2376
{{Optimal ET sequence|legend=1| 5, 7, 19e, 26e }}


Mapping: [{{val| 5 0 -20 14 41 -21 }}, {{val| 0 1 4 0 -3 5 }}]
[[Badness]]: 0.0145


Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 696.162
==== Dreamtone ====
Subgroup: 2.3.5.11.13


{{Optimal ET sequence|legend=1| 5, 45f, 50 }}
Comma list: 33/32, 55/54, 975/968


Badness: 0.048829
Sval mapping: {{mapping| 1 0 -4 5 21 | 0 1 4 -1 -11 }}


== Subgroup extensions ==
: sval mapping generators: ~2, ~3
=== Stützel (2.3.5.19) ===
Subgroup: 2.3.5.19


[[Comma list]]: 81/80, 96/95
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 689.6993


[[Gencom]]: [2 4/3; 81/80 96/95]
Optimal ET sequence: {{Optimal ET sequence| 7, 19eff, 26eff, 33ceeff, 40ceeff }}


[[Gencom|Gencom mapping]]: [{{val|1 2 4 0 0 0 0 3}}, {{val|0 -1 -4 0 0 0 0 3}}]
Badness: 0.0353
 
[[Mapping|Sval mapping]]: [{{val|1 2 4 3}}, {{val|0 -1 -4 3}}]
 
[[Tp tuning|POL2 generator]]: ~3/2 = 697.867
 
{{Optimal ET sequence|legend=1| 5, 7, 12, 31, 43 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.378 cents


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