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The [[5-limit]] parent [[comma]] of the '''meantone family''' is the Didymus or [[Wikipedia: syntonic comma|syntonic comma]], [[81/80]]. This is the one they all temper out. The period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.
The [[5-limit]] parent [[comma]] of the '''meantone family''' is the syntonic comma, [[81/80]]. This is the one they all temper out. The [[period]] is an [[octave]], the [[generator]] is a [[3/2|fifth]], and four fifths go to make up a [[5/1]] interval.


== Meantone ==
== Meantone ==
Line 16: Line 16:
[[Mapping]]: [{{val| 1 0 -4 }}, {{val| 0 1 4 }}]
[[Mapping]]: [{{val| 1 0 -4 }}, {{val| 0 1 4 }}]


Mapping generators: ~2, ~3
: mapping generators: ~2, ~3


{{Multival|legend=1| 1 4 4 }}
{{Multival|legend=1| 1 4 4 }}
Line 24: Line 24:
[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[5-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }}
* [[5-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }}
: [[Eigenmonzo basis]] ([[unchanged-interval basis]]): 2.5
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.5


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 34: Line 34:


[[Badness]]: 0.007381
[[Badness]]: 0.007381
Scales: [[meantone5]], [[meantone7]], [[meantone12]]


=== Overview to extensions ===
=== Overview to extensions ===
The second comma of the normal comma list defines which [[7-limit]] family member we are looking at.
The second comma of the normal comma list defines which [[7-limit]] family member we are looking at.
* Septimal meantone adds [[Harrison's comma|{{Monzo| -13 10 0 -1 }}]],
* Septimal meantone adds [[Harrison's comma|{{monzo| -13 10 0 -1 }}]], finding the ~7/4 at the augmented sixth,  
* Flattone adds {{Monzo| -17 9 0 1 }},
* Flattone adds {{monzo| -17 9 0 1 }}, finding the ~7/4 at the diminished seventh,  
* Dominant adds [[64/63|{{Monzo| 6 -2 0 -1 }}]],
* Dominant adds [[64/63|{{monzo| 6 -2 0 -1 }}]], finding the ~7/4 at the minor seventh,  
* Sharptone adds [[28/27|{{Monzo| 2 -3 0 1 }}]],
* Sharptone adds [[28/27|{{monzo| 2 -3 0 1 }}]], finding the ~7/4 at the major sixth,  
Those all have a fifth as generator.
Those all have a fifth as generator.
* Injera adds {{Monzo| -7 8 0 -2 }} with a half-octave period.
* Injera adds {{monzo| -7 8 0 -2 }} with a half-octave period.
* Mohajira adds {{Monzo| -23 11 0 2 }} and splits the fifth in two.
* Mohajira adds {{monzo| -23 11 0 2 }} and splits the fifth in two.
* Godzilla adds [[49/48|{{Monzo| -4 -1 0 2 }}]] with an 8/7 generator, two of which give the fourth (4/3, an octave minus a fifth).
* Godzilla adds [[49/48|{{monzo| -4 -1 0 2 }}]] with an ~8/7 generator, two of which give the [[4/3|fourth]].
* Mothra adds [[1029/1024|{{Monzo| -10 1 0 3 }}]] with an 8/7 generator, three of which give the fifth.
* Mothra adds [[1029/1024|{{monzo| -10 1 0 3 }}]] with an ~8/7 generator, three of which give the fifth.
* Liese adds {{Monzo| -9 11 0 -3 }} with a 10/7 generator, three of which give the twelfth (3/1, an octave plus a fifth).
* Liese adds {{monzo| -9 11 0 -3 }} with a ~10/7 generator, three of which give the [[3/1|twelfth]].
* Squares adds {{Monzo| -3 9 0 -4 }} with a 9/7 generator, four of which give the eleventh (8/3, two octaves minus a fifth).
* Squares adds {{monzo| -3 9 0 -4 }} with a ~9/7 generator, four of which give the [[8/3|eleventh]].
* Jerome adds {{Monzo| 3 7 0 -5 }} and slices the fifth in five.
* Jerome adds {{monzo| 3 7 0 -5 }} and slices the fifth in five.
Temperaments discussed elsewhere include [[Very low accuracy temperaments #Plutus|plutus]].
Temperaments discussed elsewhere include [[Very low accuracy temperaments #Plutus|plutus]].


Line 64: Line 62:
[[Sval]] [[mapping]]: [{{val| 1 1 0 2 }}, {{val| 0 2 8 5 }}]
[[Sval]] [[mapping]]: [{{val| 1 1 0 2 }}, {{val| 0 2 8 5 }}]


Sval mapping generators: ~2, ~11/9
: sval mapping generators: ~2, ~11/9


Gencom mapping: [{{val| 1 1 0 0 2 }}, {{val| 0 2 8 0 5 }}]
[[Gencom]] [[mapping]]: [{{val| 1 1 0 0 2 }}, {{val| 0 2 8 0 5 }}]


[[Gencom]]: [2 11/9; 81/80 121/120]
: gencom: [2 11/9; 81/80 121/120]


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~11/9 = 348.8296
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~11/9 = 348.8296
Line 83: Line 81:
Sval mapping: [{{val| 1 1 0 2 4 }}, {{val| 0 2 8 5 -1 }}]
Sval mapping: [{{val| 1 1 0 2 4 }}, {{val| 0 2 8 5 -1 }}]


Sval mapping generators: ~2, ~11/9
: sval mapping generators: ~2, ~11/9


Gencom mapping: [{{val| 1 1 0 0 2 4 }}, {{val| 0 2 8 0 5 -1 }}]
Gencom mapping: [{{val| 1 1 0 0 2 4 }}, {{val| 0 2 8 0 5 -1 }}]


Gencom: [2 11/9; 66/65 81/80 121/120]
: gencom: [2 11/9; 66/65 81/80 121/120]


Optimal tuning (CTE): ~2 = 1\1, ~11/9 = 348.8794
Optimal tuning (CTE): ~2 = 1\1, ~11/9 = 348.8794
Line 115: Line 113:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }}
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }}
: [{{Monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| -3 0 5/2 0 }}]
: [{{Monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| -3 0 5/2 0 }}]
: [[Eigenmonzo basis]] ([[unchanged-interval basis]]): 2.5
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.5


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 129: Line 127:


[[Badness]]: 0.013707
[[Badness]]: 0.013707
Scales: [[meantone5]], [[meantone7]], [[meantone12]]


=== Unidecimal meantone aka Huygens ===
=== Unidecimal meantone aka Huygens ===
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* 11-odd-limit: ~3/2 = {{Monzo| 9/16 -1/8 0 0 1/16 }}
* 11-odd-limit: ~3/2 = {{Monzo| 9/16 -1/8 0 0 1/16 }}
: [{{Monzo| 1 0 0 0 0 }}, {{monzo| 25/16 -1/8 0 0 1/16 }}, {{monzo| 9/4 -1/2 0 0 1/4 }}, {{monzo| 21/8 -5/4 0 0 5/8 }}, {{monzo| 25/8 -9/4 0 0 9/8 }}]
: [{{Monzo| 1 0 0 0 0 }}, {{monzo| 25/16 -1/8 0 0 1/16 }}, {{monzo| 9/4 -1/2 0 0 1/4 }}, {{monzo| 21/8 -5/4 0 0 5/8 }}, {{monzo| 25/8 -9/4 0 0 9/8 }}]
: Eigenmonzo basis (unchanged-interval basis): 2.11/9
: Eigenmonzo (unchanged-interval) basis: 2.11/9


Tuning ranges:  
Tuning ranges:  
Line 173: Line 169:
Minimax tuning:  
Minimax tuning:  
* [[13-odd-limit|13-]] and [[15-odd-limit]]: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}
* [[13-odd-limit|13-]] and [[15-odd-limit]]: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}
: Eigenmonzo basis (unchanged-interval basis): 2.11/9
: Eigenmonzo (unchanged-interval) basis: 2.11/9


{{Optimal ET sequence|legend=1| 12f, 19e, 31 }}
{{Optimal ET sequence|legend=1| 12f, 19e, 31 }}
Line 316: Line 312:
Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/25 -2/25 0 0 0 1/25 }}
* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/25 -2/25 0 0 0 1/25 }}
: Eigenmonzo basis (unchanged-interval basis): 2.13/9
: Eigenmonzo (unchanged-interval) basis: 2.13/9


{{Optimal ET sequence|legend=1| 12f, 31f, 43 }}
{{Optimal ET sequence|legend=1| 12f, 31f, 43 }}
Line 407: Line 403:
Mapping: [{{val| 1 0 -4 -13 -25 -5 }}, {{val| 0 2 8 20 36 11 }}]
Mapping: [{{val| 1 0 -4 -13 -25 -5 }}, {{val| 0 2 8 20 36 11 }}]


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


Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 948.6109
Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 948.6109
Line 448: Line 444:
Mapping: [{{val| 2 0 -8 -26 -50 -59 }}, {{val| 0 1 4 10 18 21 }}]
Mapping: [{{val| 2 0 -8 -26 -50 -59 }}, {{val| 0 1 4 10 18 21 }}]


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


Optimal tuning (CTE): ~55/39 = 1\2, ~3/2 = 697.1678
Optimal tuning (CTE): ~55/39 = 1\2, ~3/2 = 697.1678
Line 491: Line 487:
Mapping: [{{val| 1 0 -4 -13 24 }}, {{val| 0 1 4 10 -13 }}]
Mapping: [{{val| 1 0 -4 -13 24 }}, {{val| 0 1 4 10 -13 }}]


Mapping generator: ~2, ~3
: mapping generator: ~2, ~3


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.5311
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.5311
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* 11-odd-limit: ~3/2 = {{monzo| 0 0 1/4 }}
* 11-odd-limit: ~3/2 = {{monzo| 0 0 1/4 }}
: [{{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 }}]
: [{{Monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| -3 0 5/2 0 0 }}, {{monzo| 11 0 -13/4 0 0 }}]
: Eigenmonzo basis (unchanged-interval basis): 2.5
: Eigenmonzo (unchanged-interval) basis: 2.5


Tuning ranges:  
Tuning ranges:  
Line 526: Line 522:
Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~3/2 = {{monzo| 4/7 0 0 0 -1/28 1/28 }}
* 13- and 15-odd-limit: ~3/2 = {{monzo| 4/7 0 0 0 -1/28 1/28 }}
: Eigenmonzo basis (unchanged-interval basis): 2.13/11
: Eigenmonzo (unchanged-interval) basis: 2.13/11


Tuning ranges:  
Tuning ranges:  
Line 600: Line 596:
Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~3/2 = {{monzo| 11/13 0 0 0 -1/13 }}
* 13- and 15-odd-limit: ~3/2 = {{monzo| 11/13 0 0 0 -1/13 }}
: Eigenmonzo basis (unchanged-interval basis): 2.11
: Eigenmonzo (unchanged-interval) basis: 2.11


{{Optimal ET sequence|legend=1| 12e, 19, 31f }}
{{Optimal ET sequence|legend=1| 12e, 19, 31f }}
Line 969: Line 965:
Mapping: [{{val| 1 1 0 -3 2 }}, {{val| 0 2 8 20 5 }}]
Mapping: [{{val| 1 1 0 -3 2 }}, {{val| 0 2 8 20 5 }}]


Mapping generators: ~2, ~11/9
: mapping generators: ~2, ~11/9


Optimal tuning (CTE): ~2 = 1\1, ~11/9 = 348.5324
Optimal tuning (CTE): ~2 = 1\1, ~11/9 = 348.5324
Line 999: Line 995:
Mapping: [{{val| 2 0 -8 -26 -31 }}, {{val| 0 1 4 10 12 }}]
Mapping: [{{val| 2 0 -8 -26 -31 }}, {{val| 0 1 4 10 12 }}]


Mapping generators: ~63/44, ~3
: mapping generators: ~63/44, ~3


Optimal tuning (CTE): ~63/44 = 1\2, ~3/2 = 696.5199
Optimal tuning (CTE): ~63/44 = 1\2, ~3/2 = 696.5199
Line 1,088: Line 1,084:
In flattone, 9 generator steps of 4/3 get to the interval class for 7, meaning that [[7/4]] is a diminished seventh interval (C-Bbb). Other intervals are [[7/6]], a diminished third (C-Ebb), and [[7/5]], a doubly diminshed fifth (C-Gbb). In general, most septimal subminor intervals are diminished and most septimal supermajor intervals are augmented, which makes it quite easy to learn flattone notation. Good tunings for flattone are [[26edo|26EDO]], [[45edo|45EDO]] and [[64edo|64EDO]].
In flattone, 9 generator steps of 4/3 get to the interval class for 7, meaning that [[7/4]] is a diminished seventh interval (C-Bbb). Other intervals are [[7/6]], a diminished third (C-Ebb), and [[7/5]], a doubly diminshed fifth (C-Gbb). In general, most septimal subminor intervals are diminished and most septimal supermajor intervals are augmented, which makes it quite easy to learn flattone notation. Good tunings for flattone are [[26edo|26EDO]], [[45edo|45EDO]] and [[64edo|64EDO]].


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 81/80, 525/512
[[Comma list]]: 81/80, 525/512
Line 1,096: Line 1,092:
{{Multival|legend=1| 1 4 -9 4 -17 -32 }}
{{Multival|legend=1| 1 4 -9 4 -17 -32 }}


[[POTE generator]]: ~3/2 = 693.779
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 693.779


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]: ~3/2 = {{Monzo| 8/13 0 1/13 -1/13 }}
* [[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 }}]
: [{{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]]s (unchanged-intervals): 2, 7/5
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.7/5
* [[9-odd-limit]]: ~3/2 = {{Monzo| 6/11 2/11 0 -1/11 }}
* [[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 }}]
: [{{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 }}]
: Eigenmonzos (unchanged-intervals): 2, 9/7
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.9/7


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 1,113: Line 1,109:
* 9-odd-limit diamond monotone and tradeoff: ~3/2 = [692.308, 694.737]
* 9-odd-limit diamond monotone and tradeoff: ~3/2 = [692.308, 694.737]


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.
[[Algebraic generator]]: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.


{{Optimal ET sequence|legend=1| 7, 19, 26, 45 }}
{{Optimal ET sequence|legend=1| 7, 19, 26, 45 }}
Line 1,128: Line 1,124:
Mapping: [{{val| 1 0 -4 17 -6 }}, {{val| 0 1 4 -9 6 }}]
Mapping: [{{val| 1 0 -4 17 -6 }}, {{val| 0 1 4 -9 6 }}]


POTE generator: ~3/2 = 693.126
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 693.126


Tuning ranges:  
Tuning ranges:  
Line 1,148: Line 1,144:
Mapping: [{{val| 1 0 -4 17 -6 10 }}, {{val| 0 1 4 -9 6 -4 }}]
Mapping: [{{val| 1 0 -4 17 -6 10 }}, {{val| 0 1 4 -9 6 -4 }}]


POTE generator: ~3/2 = 693.058
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 693.058


Tuning ranges:  
Tuning ranges:  
Line 1,168: Line 1,164:
Mapping: [{{val| 1 1 0 8 2 }}, {{val| 0 2 8 -18 5 }}]
Mapping: [{{val| 1 1 0 8 2 }}, {{val| 0 2 8 -18 5 }}]


POTE generator: ~11/9 = 346.922
: mapping generators: ~2, ~11/9
 
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 346.922


{{Optimal ET sequence|legend=1| 7, 31dd, 38d, 45e, 83bcddee }}
{{Optimal ET sequence|legend=1| 7, 31dd, 38d, 45e, 83bcddee }}
Line 1,181: Line 1,179:
Mapping: [{{val| 1 1 0 8 2 6 }}, {{val| 0 2 8 -18 5 -8 }}]
Mapping: [{{val| 1 1 0 8 2 6 }}, {{val| 0 2 8 -18 5 -8 }}]


POTE generator: ~11/9 = 346.910
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 346.910


{{Optimal ET sequence|legend=1| 7, 31ddf, 38df, 45ef, 83bcddeeff }}
{{Optimal ET sequence|legend=1| 7, 31ddf, 38df, 45ef, 83bcddeeff }}
Line 1,188: Line 1,186:


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 36/35, 64/63
[[Comma list]]: 36/35, 64/63
Line 1,198: Line 1,196:
{{Multival|legend=1| 1 4 -2 4 -6 -16 }}
{{Multival|legend=1| 1 4 -2 4 -6 -16 }}


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


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 1,222: Line 1,220:
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [700.000, 705.882]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [700.000, 705.882]


POTE generator: ~3/2 = 703.254
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.254


{{Optimal ET sequence|legend=1| 5, 12, 17c, 29cde }}
{{Optimal ET sequence|legend=1| 5, 12, 17c, 29cde }}
Line 1,235: Line 1,233:
Mapping: [{{val| 1 0 -4 6 13 18 }}, {{val| 0 1 4 -2 -6 -9 }}]
Mapping: [{{val| 1 0 -4 6 13 18 }}, {{val| 0 1 4 -2 -6 -9 }}]


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


Tuning ranges:  
Tuning ranges:  
Line 1,253: Line 1,251:
Mapping: [{{val| 1 0 -4 6 13 -9 }}, {{val| 0 1 4 -2 -6 8 }}]
Mapping: [{{val| 1 0 -4 6 13 -9 }}, {{val| 0 1 4 -2 -6 8 }}]


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


{{Optimal ET sequence|legend=1| 5, 12, 17c, 46cde }}
{{Optimal ET sequence|legend=1| 5, 12, 17c, 46cde }}
Line 1,266: Line 1,264:
Mapping: [{{val| 1 0 -4 6 -6 }}, {{val| 0 1 4 -2 6 }}]
Mapping: [{{val| 1 0 -4 6 -6 }}, {{val| 0 1 4 -2 6 }}]


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


{{Optimal ET sequence|legend=1| 5e, 7, 12, 19d, 43de }}
{{Optimal ET sequence|legend=1| 5e, 7, 12, 19d, 43de }}
Line 1,279: Line 1,277:
Mapping: [{{val| 1 0 -4 6 -6 10 }}, {{val| 0 1 4 -2 6 -4 }}]
Mapping: [{{val| 1 0 -4 6 -6 10 }}, {{val| 0 1 4 -2 6 -4 }}]


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


{{Optimal ET sequence|legend=1| 5ef, 7, 12, 19d, 31def }}
{{Optimal ET sequence|legend=1| 5ef, 7, 12, 19d, 31def }}
Line 1,292: Line 1,290:
Mapping: [{{val| 1 0 -4 6 -6 10 12 }}, {{val| 0 1 4 -2 6 -4 -5 }}]
Mapping: [{{val| 1 0 -4 6 -6 10 12 }}, {{val| 0 1 4 -2 6 -4 -5 }}]


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


{{Optimal ET sequence|legend=1| 5ef, 7, 12, 19d, 31def }}
{{Optimal ET sequence|legend=1| 5ef, 7, 12, 19d, 31def }}
Line 1,305: Line 1,303:
Mapping: [{{val| 1 0 -4 6 -6 10 12 9 }}, {{val| 0 1 4 -2 6 -4 -5 -3 }}]
Mapping: [{{val| 1 0 -4 6 -6 10 12 9 }}, {{val| 0 1 4 -2 6 -4 -5 -3 }}]


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


{{Optimal ET sequence|legend=1| 5ef, 7, 12, 19d, 31def }}
{{Optimal ET sequence|legend=1| 5ef, 7, 12, 19d, 31def }}
Line 1,318: Line 1,316:
Mapping: [{{val| 1 0 -4 6 -6 -1 }}, {{val| 0 1 4 -2 6 3 }}]
Mapping: [{{val| 1 0 -4 6 -6 -1 }}, {{val| 0 1 4 -2 6 3 }}]


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


{{Optimal ET sequence|legend=1| 5e, 7, 12f, 19df }}
{{Optimal ET sequence|legend=1| 5e, 7, 12f, 19df }}
Line 1,331: Line 1,329:
Mapping: [{{val| 1 0 -4 6 -14 }}, {{val| 0 1 4 -2 11 }}]
Mapping: [{{val| 1 0 -4 6 -14 }}, {{val| 0 1 4 -2 11 }}]


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


{{Optimal ET sequence|legend=1| 5e, 12e, 17c, 46cd }}
{{Optimal ET sequence|legend=1| 5e, 12e, 17c, 46cd }}
Line 1,344: Line 1,342:
Mapping: [{{val| 1 0 -4 6 -14 -9 }}, {{val| 0 1 4 -2 11 8 }}]
Mapping: [{{val| 1 0 -4 6 -14 -9 }}, {{val| 0 1 4 -2 11 8 }}]


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


{{Optimal ET sequence|legend=1| 5e, 12e, 17c }}
{{Optimal ET sequence|legend=1| 5e, 12e, 17c }}
Line 1,357: Line 1,355:
Mapping: [{{val| 1 0 -4 6 5 }}, {{val| 0 1 4 -2 -1 }}]
Mapping: [{{val| 1 0 -4 6 5 }}, {{val| 0 1 4 -2 -1 }}]


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


{{Optimal ET sequence|legend=1| 5, 7, 12e }}
{{Optimal ET sequence|legend=1| 5, 7, 12e }}
Line 1,370: Line 1,368:
Mapping: [{{val| 1 0 -4 6 5 -1 }}, {{val| 0 1 4 -2 -1 3 }}]
Mapping: [{{val| 1 0 -4 6 5 -1 }}, {{val| 0 1 4 -2 -1 3 }}]


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


{{Optimal ET sequence|legend=1| 5, 7, 12ef, 19def }}
{{Optimal ET sequence|legend=1| 5, 7, 12ef, 19def }}
Line 1,383: Line 1,381:
Mapping: [{{val| 1 0 -4 6 5 -1 12 }}, {{val| 0 1 4 -2 -1 3 -5 }}]
Mapping: [{{val| 1 0 -4 6 5 -1 12 }}, {{val| 0 1 4 -2 -1 3 -5 }}]


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


{{Optimal ET sequence|legend=1| 5, 7, 12ef, 19def }}
{{Optimal ET sequence|legend=1| 5, 7, 12ef, 19def }}
Line 1,396: Line 1,394:
Mapping: [{{val| 1 0 -4 6 5 -1 12 9 }}, {{val| 0 1 4 -2 -1 3 -5 -3 }}]
Mapping: [{{val| 1 0 -4 6 5 -1 12 9 }}, {{val| 0 1 4 -2 -1 3 -5 -3 }}]


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


{{Optimal ET sequence|legend=1| 5, 7, 12ef, 19def }}
{{Optimal ET sequence|legend=1| 5, 7, 12ef, 19def }}
Line 1,404: Line 1,402:
=== Neutrominant ===
=== Neutrominant ===
<span style="display: block; text-align: right;">[[:de:maqamisch|Deutsch]]</span>
<span style="display: block; text-align: right;">[[:de:maqamisch|Deutsch]]</span>
{{main| Neutrominant }}
{{Main| Neutrominant }}


The ''neutrominant'' temperament (formerly ''maqamic'' temperament) has a hemififth generator (~11/9) and tempers out 36/35 and 121/120. It makes the most sense if viewed as an adaptive temperament, whereby 7/4 and 9/5 simply share an equivalence class in the resulting scales, but don't need to share a particular tempered "middle-of-the-road" intonation.
The neutrominant temperament (formerly ''maqamic'' temperament) has a hemififth generator (~11/9) and tempers out 36/35 and 121/120. It makes the most sense if viewed as an adaptive temperament, whereby 7/4 and 9/5 simply share an equivalence class in the resulting scales, but don't need to share a particular tempered "middle-of-the-road" intonation.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 1,414: Line 1,412:
Mapping: [{{val| 1 1 0 4 2 }}, {{val| 0 2 8 -4 5 }}]
Mapping: [{{val| 1 1 0 4 2 }}, {{val| 0 2 8 -4 5 }}]


Mapping generators: ~2, ~11/9
: mapping generators: ~2, ~11/9


POTE generator: ~11/9 = 350.934
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.934


{{Optimal ET sequence|legend=1| 7, 17c, 24d, 41cd }}
{{Optimal ET sequence|legend=1| 7, 17c, 24d, 41cd }}
Line 1,429: Line 1,427:
Mapping: [{{val| 1 1 0 4 2 4 }}, {{val| 0 2 8 -4 5 -1 }}]
Mapping: [{{val| 1 1 0 4 2 4 }}, {{val| 0 2 8 -4 5 -1 }}]


Mapping generators: ~2, ~11/9
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.816
 
POTE generator: ~11/9 = 350.816


{{Optimal ET sequence|legend=1| 7, 17c, 24d, 41cd }}
{{Optimal ET sequence|legend=1| 7, 17c, 24d, 41cd }}
Line 1,438: Line 1,434:


== Sharptone ==
== Sharptone ==
Sharptone is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo|12EDO]] tuning does sharptone about as well as such a thing can be done, of course not in its patent val.
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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 21/20, 28/27
[[Comma list]]: 21/20, 28/27
Line 1,448: Line 1,444:
{{Multival|legend=1| 1 4 3 4 2 -4 }}
{{Multival|legend=1| 1 4 3 4 2 -4 }}


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


{{Optimal ET sequence|legend=1| 5, 7d, 12d }}
{{Optimal ET sequence|legend=1| 5, 7d, 12d }}
Line 1,461: Line 1,457:
Mapping: [{{val| 1 0 -4 -2 5 }}, {{val| 0 1 4 3 -1 }}]
Mapping: [{{val| 1 0 -4 -2 5 }}, {{val| 0 1 4 3 -1 }}]


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


{{Optimal ET sequence|legend=1| 5, 7d, 12de }}
{{Optimal ET sequence|legend=1| 5, 7d, 12de }}
Line 1,468: Line 1,464:


== Supermean ==
== Supermean ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 81/80, 672/625
[[Comma list]]: 81/80, 672/625
Line 1,474: Line 1,470:
[[Mapping]]: [{{val| 1 0 -4 -21 }}, {{val| 0 1 4 15 }}]
[[Mapping]]: [{{val| 1 0 -4 -21 }}, {{val| 0 1 4 15 }}]


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


{{Optimal ET sequence|legend=1| 5d, 12d, 17c, 29c }}
{{Optimal ET sequence|legend=1| 5d, 12d, 17c, 29c }}
Line 1,487: Line 1,483:
Mapping: [{{val| 1 0 -4 -21 -14 }}, {{val| 0 1 4 15 11 }}]
Mapping: [{{val| 1 0 -4 -21 -14 }}, {{val| 0 1 4 15 11 }}]


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


{{Optimal ET sequence|legend=1| 5de, 12de, 17c, 29c }}
{{Optimal ET sequence|legend=1| 5de, 12de, 17c, 29c }}
Line 1,500: Line 1,496:
Mapping: [{{val| 1 0 -4 -21 -14 -9 }}, {{val| 0 1 4 15 11 8 }}]
Mapping: [{{val| 1 0 -4 -21 -14 -9 }}, {{val| 0 1 4 15 11 8 }}]


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


{{Optimal ET sequence|legend=1| 5de, 12de, 17c, 29c }}
{{Optimal ET sequence|legend=1| 5de, 12de, 17c, 29c }}
Line 1,507: Line 1,503:


== Godzilla ==
== Godzilla ==
<span style="display: block; text-align: right;">[[:de:Semiphor,_Semaphor,_Godzilla|Deutsch]]</span>
<span style="display: block; text-align: right;">[[:de:Semiphor, Semaphor, Godzilla|Deutsch]]</span>
{{main| Semaphore and Godzilla }}
{{Main| Semaphore and godzilla }}
{{See also| Slendro clan }}


Godzilla tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-quarter intervals these represent give a fourth, and so step-and-a-quarter generators generate godzilla. [[19edo|19EDO]] is close to being the optimal generator tuning; hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes.
Godzilla tempers out [[49/48]], equating 8/7 with 7/6. Two of the step-and-a-quarter intervals these represent give a fourth, and so step-and-a-quarter generators generate godzilla. [[19edo]] is close to being the optimal generator tuning; hence it can be more or less equated with taking 4\19 as a generator. [[MOS scale]]s are of 5, 9, or 14 notes.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 49/48, 81/80
[[Comma list]]: 49/48, 81/80
Line 1,518: Line 1,515:
[[Mapping]]: [{{val| 1 0 -4 2 }}, {{val| 0 2 8 1 }}]
[[Mapping]]: [{{val| 1 0 -4 2 }}, {{val| 0 2 8 1 }}]


Mapping generators: ~2, ~7/4
: mapping generators: ~2, ~7/4


{{Multival|legend=1| 2 8 1 8 -4 -20 }}
{{Multival|legend=1| 2 8 1 8 -4 -20 }}


[[POTE generator]]: ~8/7 = 252.635
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 252.635


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 1,540: Line 1,537:
Mapping: [{{val| 1 0 -4 2 -6 }}, {{val| 0 2 8 1 12 }}]
Mapping: [{{val| 1 0 -4 2 -6 }}, {{val| 0 2 8 1 12 }}]


Mapping generators: ~2, ~7/4
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 254.027
 
POTE generator: ~8/7 = 254.027


Tuning ranges:  
Tuning ranges:  
Line 1,560: Line 1,555:
Mapping: [{{val| 1 0 -4 2 -6 -5 }}, {{val| 0 2 8 1 12 11 }}]
Mapping: [{{val| 1 0 -4 2 -6 -5 }}, {{val| 0 2 8 1 12 11 }}]


Mapping generators: ~2, ~7/4
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 253.603
 
POTE generator: ~8/7 = 253.603


Tuning ranges:  
Tuning ranges:  
Line 1,580: Line 1,573:
Mapping: [{{val| 1 0 -4 2 5 }}, {{val| 0 2 8 1 -2 }}]
Mapping: [{{val| 1 0 -4 2 5 }}, {{val| 0 2 8 1 -2 }}]


Mapping generators: ~2, ~7/4
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 254.042
 
POTE generator: ~8/7 = 254.042


{{Optimal ET sequence|legend=1| 14c, 19e, 33cdee }}
{{Optimal ET sequence|legend=1| 14c, 19e, 33cdee }}
Line 1,595: Line 1,586:
Mapping: [{{val| 1 0 -4 2 -10 }}, {{val| 0 2 8 1 17 }}]
Mapping: [{{val| 1 0 -4 2 -10 }}, {{val| 0 2 8 1 17 }}]


Mapping generators: ~2, ~7/4
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 251.079
 
POTE generator: ~8/7 = 251.079


{{Optimal ET sequence|legend=1| 19e, 24, 43de }}
{{Optimal ET sequence|legend=1| 19e, 24, 43de }}
Line 1,610: Line 1,599:
Mapping: [{{val| 1 0 -4 2 -10 -5 }}, {{val| 0 2 8 1 17 11 }}]
Mapping: [{{val| 1 0 -4 2 -10 -5 }}, {{val| 0 2 8 1 17 11 }}]


Mapping generators: ~2, ~7/4
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 251.165
 
POTE generator: ~8/7 = 251.165


{{Optimal ET sequence|legend=1| 19e, 24, 43de }}
{{Optimal ET sequence|legend=1| 19e, 24, 43de }}
Line 1,625: Line 1,612:
Mapping: [{{val| 1 0 -4 2 9 }}, {{val| 0 2 8 1 -7 }}]
Mapping: [{{val| 1 0 -4 2 9 }}, {{val| 0 2 8 1 -7 }}]


Mapping generators: ~2, ~7/4
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 251.173
 
POTE generator: ~8/7 = 251.173


{{Optimal ET sequence|legend=1| 5, 14ce, 19, 24, 43d }}
{{Optimal ET sequence|legend=1| 5, 14ce, 19, 24, 43d }}
Line 1,640: Line 1,625:
Mapping: [{{val| 1 0 -4 2 9 -5 }}, {{val| 0 2 8 1 -7 11 }}]
Mapping: [{{val| 1 0 -4 2 9 -5 }}, {{val| 0 2 8 1 -7 11 }}]


Mapping generators: ~2, ~7/4
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 251.198
 
POTE generator: ~8/7 = 251.198


{{Optimal ET sequence|legend=1| 5, 14cef, 19, 24, 43d }}
{{Optimal ET sequence|legend=1| 5, 14cef, 19, 24, 43d }}
Line 1,651: Line 1,634:
{{Main| Mohajira }}
{{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|31EDO]] makes for an excellent (7-limit) mohajira tuning, with generator 9/31.  
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.  


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 81/80, 6144/6125
[[Comma list]]: 81/80, 6144/6125
Line 1,659: Line 1,642:
[[Mapping]]: [{{val| 1 1 0 6 }}, {{val| 0 2 8 -11 }}]
[[Mapping]]: [{{val| 1 1 0 6 }}, {{val| 0 2 8 -11 }}]


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


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


[[POTE generator]]: ~128/105 = 348.415
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~128/105 = 348.415


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


[[Tuning ranges]]:
[[Tuning ranges]]:
Line 1,692: Line 1,675:
Mapping: [{{val| 1 1 0 6 2 }}, {{val| 0 2 8 -11 5 }}]
Mapping: [{{val| 1 1 0 6 2 }}, {{val| 0 2 8 -11 5 }}]


Mapping generators: ~2, ~11/9
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.477
 
POTE generator: ~11/9 = 348.477


Minimax tuning:  
Minimax tuning:  
* [[11-odd-limit]]: ~11/9 = {{Monzo| 0 0 1/8 }}
* 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 }}]
: [{{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 }}]
: Eigenmonzos (unchanged-intervals): 2, 5
: Eigenmonzo (unchanged-interval) basis: 2.5


[[Tuning ranges]]:
Tuning ranges:
* 11-odd-limit diamond monotone: ~11/9 = [348.387, 350.000] (9\31 to 7\24)
* 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]
* 11-odd-limit diamond tradeoff: ~11/9 = [344.999, 350.978]
Line 1,719: Line 1,700:
Mapping: [{{val| 1 1 0 6 2 4 }}, {{val| 0 2 8 -11 5 -1 }}]
Mapping: [{{val| 1 1 0 6 2 4 }}, {{val| 0 2 8 -11 5 -1 }}]


Mapping generators: ~2, ~11/9
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.558
 
POTE generator: ~11/9 = 348.558


{{Optimal ET sequence|legend=1| 7, 24, 31 }}
{{Optimal ET sequence|legend=1| 7, 24, 31 }}
Line 1,736: Line 1,715:
Mapping: [{{val| 1 1 0 6 2 4 7 }}, {{val| 0 2 8 -11 5 -1 -10 }}]
Mapping: [{{val| 1 1 0 6 2 4 7 }}, {{val| 0 2 8 -11 5 -1 -10 }}]


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


{{Optimal ET sequence|legend=1| 7, 24, 31, 86ef }}
{{Optimal ET sequence|legend=1| 7, 24, 31, 86ef }}
Line 1,753: Line 1,730:
Mapping: [{{val| 1 1 0 6 2 4 7 6 }}, {{val| 0 2 8 -11 5 -1 -10 -6 }}]
Mapping: [{{val| 1 1 0 6 2 4 7 6 }}, {{val| 0 2 8 -11 5 -1 -10 -6 }}]


Mapping generators: ~2, ~11/9
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.810
 
POTE generator: ~11/9 = 348.810


{{Optimal ET sequence|legend=1| 7, 24, 31, 55, 86efh }}
{{Optimal ET sequence|legend=1| 7, 24, 31, 55, 86efh }}
Line 1,764: Line 1,739:


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


[[Comma list]]: 81/80, 392/375
[[Comma list]]: 81/80, 392/375
Line 1,770: Line 1,745:
[[Mapping]]: [{{val| 1 1 0 -1 }}, {{val| 0 2 8 13 }}]
[[Mapping]]: [{{val| 1 1 0 -1 }}, {{val| 0 2 8 13 }}]


Mapping generators: ~2, ~25/21
: mapping generators: ~2, ~25/21


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


{{Optimal ET sequence|legend=1| 7d, 17c, 24, 65cc, 89ccd }}
{{Optimal ET sequence|legend=1| 7d, 17c, 24, 65cc, 89ccd }}
Line 1,787: Line 1,762:
Mapping: [{{val| 1 1 0 -1 2 }}, {{val| 0 2 8 13 5 }}]
Mapping: [{{val| 1 1 0 -1 2 }}, {{val| 0 2 8 13 5 }}]


Mapping generators: ~2, ~11/9
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.565
 
POTE generator: ~11/9 = 350.565


{{Optimal ET sequence|legend=1| 7d, 17c, 24, 65cc, 89ccd }}
{{Optimal ET sequence|legend=1| 7d, 17c, 24, 65cc, 89ccd }}
Line 1,804: Line 1,777:
Mapping: [{{val| 1 1 0 -1 2 4 }}, {{val| 0 2 8 13 5 -1 }}]
Mapping: [{{val| 1 1 0 -1 2 4 }}, {{val| 0 2 8 13 5 -1 }}]


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


{{Optimal ET sequence|legend=1| 7d, 17c, 24, 41c, 65cc }}
{{Optimal ET sequence|legend=1| 7d, 17c, 24, 41c, 65cc }}
Line 1,815: Line 1,786:


== Mothra ==
== Mothra ==
Mothra splits the fifth into three 8/7 generators. It uses [[1029/1024]], the gamelisma, to accomplish this deed and also tempers out [[1728/1715]], the orwell comma. Using [[31edo|31EDO]] with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra. In the 2.3.7 subgroup, mothra is identical to [[slendric]].
{{See also| Gamelismic clan }}
 
Mothra splits the fifth into three ~8/7 generators. It uses [[1029/1024]], the gamelisma, to accomplish this deed and also tempers out [[1728/1715]], the orwell comma. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra. In the 2.3.7 subgroup, mothra is identical to [[slendric]].


Note that mothra can also be called '''cynder''' in the 7-limit, which can be a little confusing sometimes.  
Note that mothra is also called '''cynder''' in the 7-limit, which can be a little confusing sometimes.  


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 81/80, 1029/1024
[[Comma list]]: 81/80, 1029/1024
Line 1,825: Line 1,798:
[[Mapping]]: [{{val| 1 1 0 3 }}, {{val| 0 3 12 -1 }}]
[[Mapping]]: [{{val| 1 1 0 3 }}, {{val| 0 3 12 -1 }}]


Mapping generators: ~2, ~8/7
: mapping generators: ~2, ~8/7


{{Multival|legend=1| 3 12 -1 12 -10 -36 }}
{{Multival|legend=1| 3 12 -1 12 -10 -36 }}


[[POTE generator]]: ~8/7 = 232.193
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 232.193


[[Algebraic generator]]: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.
[[Algebraic generator]]: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{Monzo| 0 0 1/12 }}
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 0 0 1/12 }}
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 3 0 -1/12 0 }}]
: [{{Monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 3 0 -1/12 0 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 5
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.5


{{Optimal ET sequence|legend=1| 5, 26, 31 }}
{{Optimal ET sequence|legend=1| 5, 26, 31 }}
Line 1,849: Line 1,822:
Mapping: [{{val| 1 1 0 3 5 }}, {{val| 0 3 12 -1 -8 }}]
Mapping: [{{val| 1 1 0 3 5 }}, {{val| 0 3 12 -1 -8 }}]


Mapping generators: ~2, ~8/7
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 232.031
 
POTE generator: ~8/7 = 232.031


{{Optimal ET sequence|legend=1| 5, 26, 31, 88, 150be, 181bee }}
{{Optimal ET sequence|legend=1| 5, 26, 31, 88, 150be, 181bee }}
Line 1,864: Line 1,835:
Mapping: [{{val| 1 1 0 3 5 1 }}, {{val| 0 3 12 -1 -8 14 }}]
Mapping: [{{val| 1 1 0 3 5 1 }}, {{val| 0 3 12 -1 -8 14 }}]


Mapping generators: ~2, ~8/7
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 231.811
 
POTE generator: ~8/7 = 231.811


{{Optimal ET sequence|legend=1| 5, 26, 31, 57, 88 }}
{{Optimal ET sequence|legend=1| 5, 26, 31, 57, 88 }}
Line 1,882: Line 1,851:
Mapping: [{{val| 1 1 0 3 0 }}, {{val| 0 3 12 -1 18 }}]
Mapping: [{{val| 1 1 0 3 0 }}, {{val| 0 3 12 -1 18 }}]


Mapping generators: ~2, ~8/7
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 231.317
 
POTE generator: ~8/7 = 231.317


{{Optimal ET sequence|legend=1| 5e, 26, 57e, 83bce }}
{{Optimal ET sequence|legend=1| 5e, 26, 57e, 83bce }}
Line 1,897: Line 1,864:
Mapping: [{{val| 1 1 0 3 0 1 }}, {{val| 0 3 12 -1 18 14 }}]
Mapping: [{{val| 1 1 0 3 0 1 }}, {{val| 0 3 12 -1 18 14 }}]


Mapping generators: ~2, ~8/7
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 231.293
 
POTE generator: ~8/7 = 231.293


{{Optimal ET sequence|legend=1| 5e, 26, 57e, 83bce }}
{{Optimal ET sequence|legend=1| 5e, 26, 57e, 83bce }}
Line 1,912: Line 1,877:
Mapping: [{{val| 1 1 0 3 -1 }}, {{val| 0 3 12 -1 23 }}]
Mapping: [{{val| 1 1 0 3 -1 }}, {{val| 0 3 12 -1 23 }}]


Mapping generators: ~2, ~8/7
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 232.419
 
POTE generator: ~8/7 = 232.419


{{Optimal ET sequence|legend=1| 31, 129, 160be, 191bce, 222bce, 253bcee }}
{{Optimal ET sequence|legend=1| 31, 129, 160be, 191bce, 222bce, 253bcee }}
Line 1,927: Line 1,890:
Mapping: [{{val| 1 1 0 3 -1 7 }}, {{val| 0 3 12 -1 23 -17 }}]
Mapping: [{{val| 1 1 0 3 -1 7 }}, {{val| 0 3 12 -1 23 -17 }}]


Mapping generators: ~2, ~8/7
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 232.640
 
POTE generator: ~8/7 = 232.640


{{Optimal ET sequence|legend=1| 31, 36, 67, 98 }}
{{Optimal ET sequence|legend=1| 31, 36, 67, 98 }}
Line 1,938: Line 1,899:
<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>
<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 81/80, 686/675
[[Comma list]]: 81/80, 686/675
Line 1,946: Line 1,907:
[[Mapping]]: [{{val| 1 0 -4 -3 }}, {{val| 0 3 12 11 }}]
[[Mapping]]: [{{val| 1 0 -4 -3 }}, {{val| 0 3 12 11 }}]


Mapping generators: ~2, ~10/7
: mapping generators: ~2, ~10/7


{{Multival|legend=1| 3 12 11 12 9 -8 }}
{{Multival|legend=1| 3 12 11 12 9 -8 }}


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


Minimax tuning:  
[[Minimax tuning]]:  
* 7- and 9-odd-limit: ~10/7 = {{Monzo| 1/3 0 1/12 }}
* [[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 }}]
: [{{Monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 2/3 0 11/12 0 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 5
: [[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.
[[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.
Line 1,970: Line 1,931:
Mapping: [{{val| 1 0 -4 -3 4 }}, {{val| 0 3 12 11 -1 }}]
Mapping: [{{val| 1 0 -4 -3 4 }}, {{val| 0 3 12 11 -1 }}]


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


{{Optimal ET sequence|legend=1| 17c, 19, 36, 91cee }}
{{Optimal ET sequence|legend=1| 17c, 19, 36, 91cee }}
Line 1,985: Line 1,946:
Mapping: [{{val| 1 0 -4 -3 4 0 }}, {{val| 0 3 12 11 -1 7 }}]
Mapping: [{{val| 1 0 -4 -3 4 0 }}, {{val| 0 3 12 11 -1 7 }}]


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


{{Optimal ET sequence|legend=1| 17c, 19, 36, 91ceef }}
{{Optimal ET sequence|legend=1| 17c, 19, 36, 91ceef }}
Line 1,998: Line 1,959:
Mapping: [{{val| 1 0 -4 -3 -5 }}, {{val| 0 3 12 11 16 }}]
Mapping: [{{val| 1 0 -4 -3 -5 }}, {{val| 0 3 12 11 16 }}]


POTE generator: ~10/7 = 633.061
Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 633.061


{{Optimal ET sequence|legend=1| 17c, 19e, 36e }}
{{Optimal ET sequence|legend=1| 17c, 19e, 36e }}
Line 2,011: Line 1,972:
Mapping: [{{val| 1 0 -4 -3 -5 0 }}, {{val| 0 3 12 11 16 7 }}]
Mapping: [{{val| 1 0 -4 -3 -5 0 }}, {{val| 0 3 12 11 16 7 }}]


POTE generator: ~10/7 = 632.991
Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 632.991


{{Optimal ET sequence|legend=1| 17c, 19e, 36e }}
{{Optimal ET sequence|legend=1| 17c, 19e, 36e }}
Line 2,024: Line 1,985:
Mapping: [{{val| 1 0 -4 -3 -6 }}, {{val| 0 3 12 11 18 }}]
Mapping: [{{val| 1 0 -4 -3 -6 }}, {{val| 0 3 12 11 18 }}]


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


{{Optimal ET sequence|legend=1| 17cee, 19 }}
{{Optimal ET sequence|legend=1| 17cee, 19 }}
Line 2,037: Line 1,998:
Mapping: [{{val| 1 0 -4 -3 -6 0 }}, {{val| 0 3 12 11 18 7 }}]
Mapping: [{{val| 1 0 -4 -3 -6 0 }}, {{val| 0 3 12 11 18 7 }}]


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


{{Optimal ET sequence|legend=1| 17cee, 19 }}
{{Optimal ET sequence|legend=1| 17cee, 19 }}
Line 2,052: Line 2,013:
[[Mapping]]: [{{val| 3 0 -12 -20 }}, {{val| 0 1 4 6 }}]
[[Mapping]]: [{{val| 3 0 -12 -20 }}, {{val| 0 1 4 6 }}]


Mapping generators: ~56/45, ~3
: mapping generators: ~56/45, ~3


[[Optimal tuning]] ([[CTE]]): ~56/45 = 1\3, ~3/2 = 695.827
[[Optimal tuning]] ([[CTE]]): ~56/45 = 1\3, ~3/2 = 695.827
Line 2,063: Line 2,024:
{{Main| 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|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]].
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, 2401/2400
[[Comma list]]: 81/80, 2401/2400
Line 2,071: Line 2,032:
[[Mapping]]: [{{val| 1 3 8 6 }}, {{val| 0 -4 -16 -9 }}]
[[Mapping]]: [{{val| 1 3 8 6 }}, {{val| 0 -4 -16 -9 }}]


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


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


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


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~9/7 = {{monzo| 1/2 0 -1/16 }}
* [[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 }}]
: [{{Monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 3/2 0 9/16 0 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 5
: [[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.
[[Algebraic generator]]: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.
Line 2,097: Line 2,058:
Mapping: [{{val| 1 3 8 6 7 }}, {{val| 0 -4 -16 -9 -10 }}]
Mapping: [{{val| 1 3 8 6 7 }}, {{val| 0 -4 -16 -9 -10 }}]


POTE generator: ~9/7 = 425.957
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.957


{{Optimal ET sequence|legend=1| 14c, 17c, 31 }}
{{Optimal ET sequence|legend=1| 14c, 17c, 31 }}
Line 2,110: Line 2,071:
Mapping: [{{val| 1 3 8 6 7 3 }}, {{val| 0 -4 -16 -9 -10 2 }}]
Mapping: [{{val| 1 3 8 6 7 3 }}, {{val| 0 -4 -16 -9 -10 2 }}]


POTE generator: ~9/7 = 425.550
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.550


{{Optimal ET sequence|legend=1| 14c, 17c, 31, 79cf }}
{{Optimal ET sequence|legend=1| 14c, 17c, 31, 79cf }}
Line 2,123: Line 2,084:
Mapping: [{{val| 1 3 8 6 7 9 }}, {{val| 0 -4 -16 -9 -10 -15 }}]
Mapping: [{{val| 1 3 8 6 7 9 }}, {{val| 0 -4 -16 -9 -10 -15 }}]


POTE generator: ~9/7 = 425.7516
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.7516


{{Optimal ET sequence|legend=1| 14cf, 17c, 31f }}
{{Optimal ET sequence|legend=1| 14cf, 17c, 31f }}
Line 2,136: Line 2,097:
Mapping: [{{val| 1 3 8 6 7 14 }}, {{val| 0 -4 -16 -9 -10 -29 }}]
Mapping: [{{val| 1 3 8 6 7 14 }}, {{val| 0 -4 -16 -9 -10 -29 }}]


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


{{Optimal ET sequence|legend=1| 14cf, 31, 45ef, 76e }}
{{Optimal ET sequence|legend=1| 14cf, 31, 45ef, 76e }}
Line 2,149: Line 2,110:
Mapping: [{{val| 1 3 8 6 7 14 8 }}, {{val| 0 -4 -16 -9 -10 -29 -11 }}]
Mapping: [{{val| 1 3 8 6 7 14 8 }}, {{val| 0 -4 -16 -9 -10 -29 -11 }}]


POTE generator: ~9/7 = 426.187
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 426.187


{{Optimal ET sequence|legend=1| 14cf, 31, 76e }}
{{Optimal ET sequence|legend=1| 14cf, 31, 76e }}
Line 2,162: Line 2,123:
Mapping: [{{val| 1 3 8 6 7 14 8 11 }}, {{val| 0 -4 -16 -9 -10 -29 -11 -19 }}]
Mapping: [{{val| 1 3 8 6 7 14 8 11 }}, {{val| 0 -4 -16 -9 -10 -29 -11 -19 }}]


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


{{Optimal ET sequence|legend=1| 14cf, 31, 76e }}
{{Optimal ET sequence|legend=1| 14cf, 31, 76e }}
Line 2,175: Line 2,136:
Mapping: [{{val| 1 3 8 6 -4 }}, {{val| 0 -4 -16 -9 21 }}]
Mapping: [{{val| 1 3 8 6 -4 }}, {{val| 0 -4 -16 -9 21 }}]


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


{{Optimal ET sequence|legend=1| 14ce, 17ce, 31, 107b, 138b, 169be, 200be }}
{{Optimal ET sequence|legend=1| 14ce, 17ce, 31, 107b, 138b, 169be, 200be }}
Line 2,184: Line 2,145:
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.
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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 81/80, 17280/16807
[[Comma list]]: 81/80, 17280/16807
Line 2,190: Line 2,151:
[[Mapping]]: [{{val| 1 1 0 2 }}, {{val| 0 5 20 7 }}]
[[Mapping]]: [{{val| 1 1 0 2 }}, {{val| 0 5 20 7 }}]


Mapping generators: ~2, ~54/49
: mapping generators: ~2, ~54/49


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


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


{{Optimal ET sequence|legend=1| 17c, 26, 43, 69, 112bd }}
{{Optimal ET sequence|legend=1| 17c, 26, 43, 69, 112bd }}
Line 2,207: Line 2,168:
Mapping: [{{val| 1 1 0 2 3 }}, {{val| 0 5 20 7 4 }}]
Mapping: [{{val| 1 1 0 2 3 }}, {{val| 0 5 20 7 4 }}]


Mapping generators: ~2, ~12/11
Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 139.428
 
POTE generator: ~12/11 = 139.428


{{Optimal ET sequence|legend=1| 17c, 26, 43, 69 }}
{{Optimal ET sequence|legend=1| 17c, 26, 43, 69 }}
Line 2,222: Line 2,181:
Mapping: [{{val| 1 1 0 2 3 3 }}, {{val| 0 5 20 7 4 6 }}]
Mapping: [{{val| 1 1 0 2 3 3 }}, {{val| 0 5 20 7 4 6 }}]


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


{{Optimal ET sequence|legend=1| 17c, 26, 43, 69 }}
{{Optimal ET sequence|legend=1| 17c, 26, 43, 69 }}
Line 2,237: Line 2,194:
Mapping: [{{val| 1 1 0 2 3 3 2 }}, {{val| 0 5 20 7 4 6 18 }}]
Mapping: [{{val| 1 1 0 2 3 3 2 }}, {{val| 0 5 20 7 4 6 18 }}]


Mapping generators: ~2, ~12/11
Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 139.362
 
POTE generator: ~12/11 = 139.362


{{Optimal ET sequence|legend=1| 17cg, 26, 43, 69 }}
{{Optimal ET sequence|legend=1| 17cg, 26, 43, 69 }}
Line 2,252: Line 2,207:
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 3 3 2 1 }}, {{val| 0 5 20 7 4 6 18 28 }}]


Mapping generators: ~2, ~12/11
Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 139.313
 
POTE generator: ~12/11 = 139.313


{{Optimal ET sequence|legend=1| 17cgh, 26, 43, 69 }}
{{Optimal ET sequence|legend=1| 17cgh, 26, 43, 69 }}
Line 2,263: Line 2,216:
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.
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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 81/80, 16875/16807
[[Comma list]]: 81/80, 16875/16807
Line 2,271: Line 2,224:
{{Multival|legend=1| 5 20 19 20 16 -12 }}
{{Multival|legend=1| 5 20 19 20 16 -12 }}


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


{{Optimal ET sequence|legend=1| 2cd, 29cd, 31 }}
{{Optimal ET sequence|legend=1| 2cd, 29cd, 31 }}
Line 2,284: Line 2,237:
Mapping: [{{val| 1 4 12 12 17 }}, {{val| 0 -5 -20 -19 -28 }}]
Mapping: [{{val| 1 4 12 12 17 }}, {{val| 0 -5 -20 -19 -28 }}]


POTE generator: ~7/5 = 580.647
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.647


{{Optimal ET sequence|legend=1| 2cde, 29cde, 31 }}
{{Optimal ET sequence|legend=1| 2cde, 29cde, 31 }}
Line 2,291: Line 2,244:


== Injera ==
== 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|19EDOs]], is an excellent tuning for 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.


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 50/49, 81/80
[[Comma list]]: 50/49, 81/80
Line 2,301: Line 2,254:
[[Mapping]]: [{{val| 2 0 -8 -7 }}, {{val| 0 1 4 4 }}]
[[Mapping]]: [{{val| 2 0 -8 -7 }}, {{val| 0 1 4 4 }}]


Mapping generators: ~7/5, ~3
: mapping generators: ~7/5, ~3


{{Multival|legend=1| 2 8 8 8 7 -4 }}
{{Multival|legend=1| 2 8 8 8 7 -4 }}


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


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 2,328: Line 2,281:
Mapping: [{{val| 2 0 -8 -7 -12 }}, {{val| 0 1 4 4 6 }}]
Mapping: [{{val| 2 0 -8 -7 -12 }}, {{val| 0 1 4 4 6 }}]


Mapping generators: ~7/5, ~3
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 692.840
 
POTE generator: ~3/2 = 692.840


Tuning ranges:  
Tuning ranges:  
Line 2,348: Line 2,299:
Mapping: [{{val| 2 0 -8 -7 -12 -21 }}, {{val| 0 1 4 4 6 9 }}]
Mapping: [{{val| 2 0 -8 -7 -12 -21 }}, {{val| 0 1 4 4 6 9 }}]


Mapping generators: ~7/5, ~3
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 692.673
 
POTE generator: ~3/2 = 692.673


Tuning ranges:  
Tuning ranges:  
Line 2,368: Line 2,317:
Mapping: [{{val| 2 0 -8 -7 -12 -21 5 }}, {{val| 0 1 4 4 6 9 1 }}]
Mapping: [{{val| 2 0 -8 -7 -12 -21 5 }}, {{val| 0 1 4 4 6 9 1 }}]


POTE generator: ~3/2 = 692.487
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 692.487


{{Optimal ET sequence|legend=1| 12f, 14cf, 26 }}
{{Optimal ET sequence|legend=1| 12f, 14cf, 26 }}
Line 2,381: Line 2,330:
Mapping: [{{val| 2 0 -8 -7 -12 -21 5 -1 }}, {{val| 0 1 4 4 6 9 1 3 }}]
Mapping: [{{val| 2 0 -8 -7 -12 -21 5 -1 }}, {{val| 0 1 4 4 6 9 1 3 }}]


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


{{Optimal ET sequence|legend=1| 12f, 14cf, 26 }}
{{Optimal ET sequence|legend=1| 12f, 14cf, 26 }}
Line 2,394: Line 2,343:
Mapping: [{{val| 2 0 -8 -7 -12 -2 }}, {{val| 0 1 4 4 6 3 }}]
Mapping: [{{val| 2 0 -8 -7 -12 -2 }}, {{val| 0 1 4 4 6 3 }}]


Mapping generators: ~7/5, ~3
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 694.121
 
POTE generator: ~3/2 = 694.121


{{Optimal ET sequence|legend=1| 12f, 14c, 26f, 38eff }}
{{Optimal ET sequence|legend=1| 12f, 14c, 26f, 38eff }}
Line 2,409: Line 2,356:
Mapping: [{{val| 2 0 -8 -7 10 }}, {{val| 0 1 4 4 -1 }}]
Mapping: [{{val| 2 0 -8 -7 10 }}, {{val| 0 1 4 4 -1 }}]


Mapping generators: ~7/5, ~3
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 690.548
 
POTE generator: ~3/2 = 690.548


{{Optimal ET sequence|legend=1| 12e, 14c, 26e, 40cee }}
{{Optimal ET sequence|legend=1| 12e, 14c, 26e, 40cee }}
Line 2,424: Line 2,369:
Mapping: [{{val| 2 0 -8 -7 7 }}, {{val| 0 1 4 4 0 }}]
Mapping: [{{val| 2 0 -8 -7 7 }}, {{val| 0 1 4 4 0 }}]


Mapping generators: ~7/5, ~3
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 699.001
 
POTE generator: ~3/2 = 699.001


{{Optimal ET sequence|legend=1| 2cd, 10cd, 12 }}
{{Optimal ET sequence|legend=1| 2cd, 10cd, 12 }}
Line 2,435: Line 2,378:
{{Main| Teff }}
{{Main| Teff }}


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


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 2,443: Line 2,386:
Mapping: [{{val| 2 1 -4 -3 8 }}, {{val| 0 2 8 8 -1 }}]
Mapping: [{{val| 2 1 -4 -3 8 }}, {{val| 0 2 8 8 -1 }}]


Mapping generators: ~7/5, ~16/11
: mapping generators: ~7/5, ~16/11


POTE generator: ~11/8 = 552.5303
Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 552.5303


{{Optimal ET sequence|legend=1| 24d, 26, 50d }}
{{Optimal ET sequence|legend=1| 24d, 26, 50d }}
Line 2,458: Line 2,401:
Mapping: [{{val| 2 1 -4 -3 8 2 }}, {{val| 0 2 8 8 -1 5 }}]
Mapping: [{{val| 2 1 -4 -3 8 2 }}, {{val| 0 2 8 8 -1 5 }}]


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


{{Optimal ET sequence|legend=1| 24d, 26, 50d }}
{{Optimal ET sequence|legend=1| 24d, 26, 50d }}
Line 2,471: Line 2,414:
Mapping: [{{val| 2 1 -4 -3 8 2 6 }}, {{val| 0 2 8 8 -1 5 2 }}]
Mapping: [{{val| 2 1 -4 -3 8 2 6 }}, {{val| 0 2 8 8 -1 5 2 }}]


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


{{Optimal ET sequence|legend=1| 24d, 26 }}
{{Optimal ET sequence|legend=1| 24d, 26 }}
Line 2,484: Line 2,427:
Mapping: [{{val| 2 1 -4 -3 8 2 6 2 }}, {{val| 0 2 8 8 -1 5 2 6 }}]
Mapping: [{{val| 2 1 -4 -3 8 2 6 2 }}, {{val| 0 2 8 8 -1 5 2 6 }}]


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


{{Optimal ET sequence|legend=1| 24d, 26 }}
{{Optimal ET sequence|legend=1| 24d, 26 }}
Line 2,491: Line 2,434:


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 81/80, 300125/294912
[[Comma list]]: 81/80, 300125/294912
Line 2,499: Line 2,442:
[[Mapping]]: [{{val| 2 1 -4 11 }}, {{val| 0 2 8 -5 }}]
[[Mapping]]: [{{val| 2 1 -4 11 }}, {{val| 0 2 8 -5 }}]


Mapping generators: ~735/512, ~35/24
: mapping generators: ~735/512, ~35/24


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


[[POTE generator]]: ~48/35 = 552.2206
[[Optimal tuning]] ([[POTE]]): ~735/512 = 1\2, ~48/35 = 552.2206


{{Optimal ET sequence|legend=1| 24, 26, 50, 126bcd, 176bcdd, 226bbcdd }}
{{Optimal ET sequence|legend=1| 24, 26, 50, 126bcd, 176bcdd, 226bbcdd }}
Line 2,516: Line 2,459:
Mapping: [{{val| 2 1 -4 11 8 }}, {{val| 0 2 8 -5 -1 }}]
Mapping: [{{val| 2 1 -4 11 8 }}, {{val| 0 2 8 -5 -1 }}]


POTE generator: ~11/8 = 552.0929
Optimal tuning (POTE): ~99/70 = 1\2, ~11/8 = 552.0929


{{Optimal ET sequence|legend=1| 24, 26, 50 }}
{{Optimal ET sequence|legend=1| 24, 26, 50 }}
Line 2,529: Line 2,472:
Mapping: [{{val| 2 1 -4 11 8 2 }}, {{val| 0 2 8 -5 -1 5 }}]
Mapping: [{{val| 2 1 -4 11 8 2 }}, {{val| 0 2 8 -5 -1 5 }}]


POTE generator: ~11/8 = 552.1498
Optimal tuning (POTE): ~99/70 = 1\2, ~11/8 = 552.1498


{{Optimal ET sequence|legend=1| 24, 26, 50 }}
{{Optimal ET sequence|legend=1| 24, 26, 50 }}
Line 2,542: Line 2,485:
Mapping: [{{val| 2 1 -4 11 8 2 6 }}, {{val| 0 2 8 -5 -1 5 2 }}]
Mapping: [{{val| 2 1 -4 11 8 2 6 }}, {{val| 0 2 8 -5 -1 5 2 }}]


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


{{Optimal ET sequence|legend=1| 24, 26, 50 }}
{{Optimal ET sequence|legend=1| 24, 26, 50 }}
Line 2,555: Line 2,498:
Mapping: [{{val| 2 1 -4 11 8 2 6 2 }}, {{val| 0 2 8 -5 -1 5 2 6 }}]
Mapping: [{{val| 2 1 -4 11 8 2 6 2 }}, {{val| 0 2 8 -5 -1 5 2 6 }}]


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


{{Optimal ET sequence|legend=1| 24, 26, 50 }}
{{Optimal ET sequence|legend=1| 24, 26, 50 }}
Line 2,562: Line 2,505:


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


[[Comma list]]: 81/80, 5898240/5764801
[[Comma list]]: 81/80, 5898240/5764801
Line 2,572: Line 2,515:
{{Multival|legend=1| 8 32 6 32 -13 -76 }}
{{Multival|legend=1| 8 32 6 32 -13 -76 }}


[[POTE generator]]: ~7/6 = 275.794
[[Optimal tuning]] ([[POTE]]): ~2401/1728 = 1\2, ~7/6 = 275.794


{{Optimal ET sequence|legend=1| 26, 48c, 74, 174bd, 248bbd }}
{{Optimal ET sequence|legend=1| 26, 48c, 74, 174bd, 248bbd }}
Line 2,585: Line 2,528:
Mapping: [{{val| 2 5 12 7 6 }}, {{val| 0 -4 -16 -3 2 }}]
Mapping: [{{val| 2 5 12 7 6 }}, {{val| 0 -4 -16 -3 2 }}]


Mapping generators: ~363/256, ~7/6
Optimal tuning (POTE): ~363/256 = 1\2, ~7/6 = 275.762
 
POTE generator: ~7/6 = 275.762


{{Optimal ET sequence|legend=1| 26, 48c, 74, 248bbd, 322bbdd }}
{{Optimal ET sequence|legend=1| 26, 48c, 74, 248bbd, 322bbdd }}
Line 2,600: Line 2,541:
Mapping: [{{val| 2 5 12 7 6 12 }}, {{val| 0 -4 -16 -3 2 -10 }}]
Mapping: [{{val| 2 5 12 7 6 12 }}, {{val| 0 -4 -16 -3 2 -10 }}]


Mapping generators: ~55/39, ~7/6
Optimal tuning (POTE): ~55/39 = 1\2, ~7/6 = 275.774
 
POTE generator: ~7/6 = 275.774


{{Optimal ET sequence|legend=1| 26, 48c, 74, 174bd, 248bbd, 322bbdd }}
{{Optimal ET sequence|legend=1| 26, 48c, 74, 174bd, 248bbd, 322bbdd }}
Line 2,611: Line 2,550:
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.
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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 81/80, 16807/16384
[[Comma list]]: 81/80, 16807/16384
Line 2,617: Line 2,556:
[[Mapping]]: [{{val| 5 0 -20 14 }}, {{val| 0 1 4 0 }}]
[[Mapping]]: [{{val| 5 0 -20 14 }}, {{val| 0 1 4 0 }}]


Mapping generators: ~8/7, ~3
: mapping generators: ~8/7, ~3


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


[[POTE generator]]: ~3/2 = 695.720
[[Optimal tuning]] ([[POTE]]): ~8/7 = 1\5, ~3/2 = 695.720


{{Optimal ET sequence|legend=1| 5, 45, 50 }}
{{Optimal ET sequence|legend=1| 5, 45, 50 }}
Line 2,634: Line 2,573:
Mapping: [{{val| 5 0 -20 14 41 }}, {{val| 0 1 4 0 -3 }}]
Mapping: [{{val| 5 0 -20 14 41 }}, {{val| 0 1 4 0 -3 }}]


Mapping generators: ~8/7, ~3
Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 696.536
 
POTE generator: ~3/2 = 696.536


{{Optimal ET sequence|legend=1| 5, 45, 50, 155bdd, 205bddd }}
{{Optimal ET sequence|legend=1| 5, 45, 50, 155bdd, 205bddd }}
Line 2,649: Line 2,586:
Mapping: [{{val| 5 0 -20 14 41 -21 }}, {{val| 0 1 4 0 -3 5 }}]
Mapping: [{{val| 5 0 -20 14 41 -21 }}, {{val| 0 1 4 0 -3 5 }}]


Mapping generators: ~8/7, ~3
Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 696.162
 
POTE generator: ~3/2 = 696.162


{{Optimal ET sequence|legend=1| 5, 45f, 50 }}
{{Optimal ET sequence|legend=1| 5, 45f, 50 }}
Line 2,658: Line 2,593:


== Meanmag ==
== Meanmag ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 81/80, 3125/3072
[[Comma list]]: 81/80, 3125/3072
Line 2,664: Line 2,599:
[[Mapping]]: [{{val| 19 30 44 0 }}, {{val| 0 0 0 1 }}]
[[Mapping]]: [{{val| 19 30 44 0 }}, {{val| 0 0 0 1 }}]


Mapping generators: ~25/24, ~7
: mapping generators: ~25/24, ~7


{{Multival|legend=1| 0 0 19 0 30 44 }}
{{Multival|legend=1| 0 0 19 0 30 44 }}


[[POTE generator]]: ~8/7 = 238.396
[[Optimal tuning]] ([[POTE]]): ~25/24 = 1\19, ~8/7 = 238.396


{{Optimal ET sequence|legend=1| 19, 38, 57, 76, 95bc }}
{{Optimal ET sequence|legend=1| 19, 38, 57, 76, 95bc }}
Line 2,681: Line 2,616:
Mapping: [{{val| 19 30 44 0 119 }}, {{val| 0 0 0 1 -1 }}]
Mapping: [{{val| 19 30 44 0 119 }}, {{val| 0 0 0 1 -1 }}]


Mapping generators: ~25/24, ~7
Optimal tuning (POTE): ~25/24 = 1\19, ~8/7 = 233.486
 
POTE generator: ~8/7 = 233.486


{{Optimal ET sequence|legend=1| 19, 38, 57, 76 }}
{{Optimal ET sequence|legend=1| 19, 38, 57, 76 }}
Line 2,696: Line 2,629:
Mapping: [{{val| 19 30 44 0 119 17 }}, {{val| 0 0 0 1 -1 1 }}]
Mapping: [{{val| 19 30 44 0 119 17 }}, {{val| 0 0 0 1 -1 1 }}]


Mapping generators: ~25/24, ~7
Optimal tuning (POTE): ~25/24 = 1\19, ~8/7 = 234.890
 
POTE generator: ~8/7 = 234.890


{{Optimal ET sequence|legend=1| 19, 38, 57, 76 }}
{{Optimal ET sequence|legend=1| 19, 38, 57, 76 }}
Line 2,705: Line 2,636:


== Undevigintone ==
== Undevigintone ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 49/48, 81/80, 126/125
[[Comma list]]: 49/48, 81/80, 126/125
Line 2,711: Line 2,642:
[[Mapping]]: [{{val| 19 30 44 53 0 }}, {{val| 0 0 0 0 1 }}]
[[Mapping]]: [{{val| 19 30 44 53 0 }}, {{val| 0 0 0 0 1 }}]


Mapping generators: ~21/20, ~11
: mapping generators: ~28/27, ~11


[[POTE generator]]: ~11/8 = 538.047
[[Optimal tuning]] ([[POTE]]): ~28/27 = 1\19, ~11/8 = 538.047


{{Optimal ET sequence|legend=1| 19, 38d }}
{{Optimal ET sequence|legend=1| 19, 38d }}
Line 2,726: Line 2,657:
Mapping: [{{val| 19 30 44 53 0 70 }}, {{val| 0 0 0 0 1 0 }}]
Mapping: [{{val| 19 30 44 53 0 70 }}, {{val| 0 0 0 0 1 0 }}]


Mapping generators: ~21/20, ~11
Optimal tuning (POTE): ~28/27 = 1\19, ~11/8 = 537.061
 
POTE generator: ~11/8 = 537.061


{{Optimal ET sequence|legend=1| 19, 38df }}
{{Optimal ET sequence|legend=1| 19, 38df }}
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