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Diaschismic family: Difference between revisions

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= Srutal =
= Srutal =
[[Comma list]]: 2048/2025, 4375/4374
[[Comma list]]: 2048/2025, 4375/4374
[[Tuning ranges]]:
* valid range: ~3/2 = [703.448, 705.882] (34\58 to 20\34)
* nice range: ~3/2 = [701.955, 706.843]
* strict range: ~3/2 = [703.448, 705.882]


[[Mapping]]: [{{val|2 0 11 -42}}, {{val|0 1 -2 15}}]
[[Mapping]]: [{{val|2 0 11 -42}}, {{val|0 1 -2 15}}]
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POTE generator: ~3/2 = 704.881
POTE generator: ~3/2 = 704.881
[[Tuning ranges]]:
* valid range: ~3/2 = [703.448, 705.882] (34\58 to 20\34)
* nice range: ~3/2 = [701.955, 706.843]
* strict range: ~3/2 = [703.448, 705.882]


Vals: {{Val list| 34d, 46, 80, 206cd, 286bcde }}
Vals: {{Val list| 34d, 46, 80, 206cd, 286bcde }}
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[[Comma list]]: 50/49, 64/63
[[Comma list]]: 50/49, 64/63
[[Mapping]]: [{{val|2 0 11 12}}, {{val|0 1 -2 -2}}]
[[POTE generator]]: ~3/2 = 707.048


[[Tuning ranges]]:
[[Tuning ranges]]:
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* nice range: ~3/2 = [701.955, 715.587]
* nice range: ~3/2 = [701.955, 715.587]
* strict range: ~3/2 = [701.955, 715.587]
* strict range: ~3/2 = [701.955, 715.587]
[[Mapping]]: [{{val|2 0 11 12}}, {{val|0 1 -2 -2}}]
[[POTE generator]]: ~3/2 = 707.048


{{Val list|legend=1| 10, 12, 22, 34d, 56d }}
{{Val list|legend=1| 10, 12, 22, 34d, 56d }}
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== 11-limit ==
== 11-limit ==
Comma list: 50/49, 64/63, 99/98
Comma list: 50/49, 64/63, 99/98
Mapping: [{{val|2 0 11 12 26}}, {{val|0 1 -2 -2 -6}}]
POTE generator: ~3/2 = 706.885


Tuning ranges:
Tuning ranges:
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* nice range: ~3/2 = [701.955, 715.587]
* nice range: ~3/2 = [701.955, 715.587]
* strict range: ~3/2 = [701.955, 709.091]
* strict range: ~3/2 = [701.955, 709.091]
Mapping: [{{val|2 0 11 12 26}}, {{val|0 1 -2 -2 -6}}]
POTE generator: ~3/2 = 706.885


Vals: {{Val list| 10e, 12, 22, 34d, 56d }}
Vals: {{Val list| 10e, 12, 22, 34d, 56d }}
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===  13-limit ===
===  13-limit ===
Comma list: 50/49, 64/63, 65/63, 99/98
Comma list: 50/49, 64/63, 65/63, 99/98
Mapping: [{{val|2 0 11 12 26 1}}, {{val|0 1 -2 -2 -6 2}}]
POTE generator: ~3/2 = 708.919


Tuning ranges:
Tuning ranges:
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* nice range: ~3/2 = [701.955, 738.573]
* nice range: ~3/2 = [701.955, 738.573]
* strict range: ~3/2 = [701.955, 709.091]
* strict range: ~3/2 = [701.955, 709.091]
Mapping: [{{val|2 0 11 12 26 1}}, {{val|0 1 -2 -2 -6 2}}]
POTE generator: ~3/2 = 708.919


Vals: {{Val list| 12, 22 }}
Vals: {{Val list| 12, 22 }}
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== Pajarous ==
== Pajarous ==
Comma list: 50/49, 55/54, 64/63
Comma list: 50/49, 55/54, 64/63
Mapping: [{{val|2 0 11 12 -9}}, {{val|0 1 -2 -2 5}}]
POTE generator: ~3/2 = 709.578


Tuning ranges:
Tuning ranges:
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* nice range: ~3/2 = [701.955, 715.803]
* nice range: ~3/2 = [701.955, 715.803]
* strict range: ~3/2 = 709.091
* strict range: ~3/2 = 709.091
Mapping: [{{val|2 0 11 12 -9}}, {{val|0 1 -2 -2 5}}]
POTE generator: ~3/2 = 709.578


Vals: {{Val list| 10, 12e, 22, 120bce, 142bce }}
Vals: {{Val list| 10, 12e, 22, 120bce, 142bce }}
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[[Mapping]]: [{{val|2 0 11 31}}, {{val|0 1 -2 -8}}]
[[Mapping]]: [{{val|2 0 11 31}}, {{val|0 1 -2 -8}}]


[[POTE_tuning|POTE generator]]: ~3/2 = 703.681
[[POTE generator]]: ~3/2 = 703.681


{{Val list|legend=1| 12, 46, 58, 104c, 162c }}
{{Val list|legend=1| 12, 46, 58, 104c, 162c }}
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== 11-limit ==
== 11-limit ==
Comma list: 176/175, 540/539, 896/891
Comma list: 176/175, 540/539, 896/891
Minimax tuning:
* 11-limit minimax
: [{{monzo|1 0 0 0 0}}, {{monzo|7/4 0 0 1/4 -1/4}}, {{monzo|2 0 0 -1/2 1/2}},
{{monzo|37/12 0 0 5/12 -5/12}}, {{monzo|37/12 0 0 -7/12 7/12}}]
* Eigenmonzos: 2, 11/7
* Minimax generator: (224/11)^(1/12) = 434.792


Mapping: [{{val|2 1 9 2 12}}, {{val|0 3 -6 5 -7}}]
Mapping: [{{val|2 1 9 2 12}}, {{val|0 3 -6 5 -7}}]


POTE generator: ~9/7 = 434.852
POTE generator: ~9/7 = 434.852
Minimax tuning:
* 11-odd-limit
: [{{monzo|1 0 0 0 0}}, {{monzo|7/4 0 0 1/4 -1/4}}, {{monzo|2 0 0 -1/2 1/2}}, {{monzo|37/12 0 0 5/12 -5/12}}, {{monzo|37/12 0 0 -7/12 7/12}}]
: Eigenmonzos: 2, 11/7
: Minimax generator: (224/11)<sup>(1/12)</sup> = 434.792


Vals: {{Val list| 22, 58, 80, 138cde, 218cde }}
Vals: {{Val list| 22, 58, 80, 138cde, 218cde }}
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= Shrutar =
= Shrutar =
Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. With wedgie &lt;&lt;4 -8 14 -22 11 55||, it can also be described as 22&amp;46. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. [[68edo]] makes for a good tuning, but another and excellent choice is a generator of 14^(1/7), making 7s just.
Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. With wedgie &lt;&lt;4 -8 14 -22 11 55||, it can also be described as 22&amp;46. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. [[68edo]] makes for a good tuning, but another and excellent choice is a generator of 14<sup>(1/7)</sup>, making 7s just.


By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14^(1/7) generator can again be used as tunings.
By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14<sup>(1/7)</sup> generator can again be used as tunings.


Commas: 245/243, 2048/2025
[[Comma list]]: 245/243, 2048/2025


[[POTE_tuning|POTE generator]]: 52.811
[[Mapping]]: [{{val| 2 1 9 -2 }}, {{val| 0 2 -4 7 }}]


Map: [&lt;2 1 9 -2|, &lt;0 2 -4 7|]
[[POTE generator]]: ~36/35 = 52.811


EDOs: 22, 46, 68, 182b, 250bc
{{Val list|legend=1| 22, 46, 68, 182b, 250bc }}


== 11-limit ==
== 11-limit ==
Commas: 121/120, 176/175, 245/243
Comma list: 121/120, 176/175, 245/243


[[POTE_tuning|POTE generator]]: 52.680
Mapping: [{{val| 2 1 9 -2 8 }}, {{val| 0 2 -4 7 -1 }}]


Map: [&lt;2 1 9 -2 8|, &lt;0 2 -4 7 -1|]
POTE generator: ~33/32 = 52.680


EDOs: 22, 46, 68, 114, 296bce, 410bce
Vals: {{Val list| 22, 46, 68, 114, 296bce, 410bce }}


== 13-limit ==
== 13-limit ==
Commas: 121/120, 176/175, 196/195, 245/243
Comma list: 121/120, 176/175, 196/195, 245/243


POTE generator: ~28/27 = 52.654
Mapping: [{{val| 2 1 9 -2 8 -10 }}, {{val| 0 2 -4 7 -1 16 }}]


Map: [&lt;2 1 9 -2 8 -10|, &lt;0 2 -4 7 -1 16|]
POTE generator: ~33/32 = 52.654


EDOs: 22, 24, 46, 68, 114
Vals: {{Val list| 22, 24, 46, 68, 114 }}


Badness: 0.0281
Badness: 0.0281


== 17-limit ==
== 17-limit ==
Commas: 121/120, 136/135, 154/153, 176/175, 196/195
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195


POTE generator: ~28/27 = 52.647
Mapping: [{{val| 2 1 9 -2 8 -10 6 }}, {{val| 0 2 -4 7 -1 16 2 }}]


Map: [&lt;2 1 9 -2 8 -10 6|, &lt;0 2 -4 7 -1 16 2|]
POTE generator: ~33/32 = 52.647


EDOs: 22, 24, 46, 68, 114
Vals: {{Val list| 22, 24, 46, 68, 114 }}


Badness: 0.0187
Badness: 0.0187


== 19-limit ==
== 19-limit ==
Commas: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342


POTE generator: ~28/27 = 52.730
Mapping: [{{val| 2 1 9 -2 8 -10 6 -10 }}, {{val| 0 2 -4 7 -1 16 2 17 }}]


Map: [&lt;2 1 9 -2 8 -10 6 -10|, &lt;0 2 -4 7 -1 16 2 17|]
POTE generator: ~33/32 = 52.730


EDOs: 22, 24, 46, 68, 114, 182bef
Vals: {{Val list| 22, 24, 46, 68, 114, 182bef }}


Badness: 0.0175
Badness: 0.0175


= Sruti =
= Sruti =
Commas: 2048/2025, 19683/19600
[[Comma list]]: 2048/2025, 19683/19600


POTE generator: ~175/144 = 351.876
[[Mapping]]: [{{val| 2 0 11 -15 }}, {{val| 0 2 -4 13 }}]


Map: [&lt;2 0 11 -15|, &lt;0 2 -4 13|]
{{Multival|legend=1| 4 -8 26 -22 30 83 }}


Wedgie: &lt;&lt;4 -8 26 -22 30 83||
POTE generator: ~175/144 = 351.876


EDOs: 24, 34d, 58, 150cd, 208cd, 266cd
{{Val list|legend=1| 24, 34d, 58, 150cd, 208cd, 266cd }}


Badness: 0.1174
[[Badness]]: 0.1174


== 11-limit ==
== 11-limit ==
Commas: 176/175, 243/242, 896/891
Comma list: 176/175, 243/242, 896/891
 
Mapping: [{{val| 2 0 11 -15 -1 }}, {{val| 0 2 -4 13 5 }}]


POTE generator: ~11/9 = 351.863
POTE generator: ~11/9 = 351.863


Map: [&lt;2 0 11 -15 -1|, &lt;0 2 -4 13 5|]
Vals: {{Val list| 24, 34d, 58, 150cde, 208cde }}
 
EDOs: 24, 34d, 58, 150cde, 208cde


Badness: 0.0415
Badness: 0.0415


== 13-limit ==
== 13-limit ==
Commas: 144/143, 176/175, 351/350, 676/675
Comma list: 144/143, 176/175, 351/350, 676/675
 
Mapping: [{{val| 2 0 11 -15 -1 9 }}, {{val| 0 2 -4 13 5 -1 }}]


POTE generator: ~11/9 = 351.886
POTE generator: ~11/9 = 351.886


Map: [&lt;2 0 11 -15 -1 9|, &lt;0 2 -4 13 5 -1|]
Vals: {{Val list| 24, 34d, 58, 150cdef, 208cdef }}
 
EDOs: 24, 34d, 58, 150cdef, 208cdef


Badness: 0.0238
Badness: 0.0238


= Anguirus =
= Anguirus =
Commas: 49/48, 2048/2025
[[Comma list]]: 49/48, 2048/2025


POTE generator: ~8/7 = 246.979
[[Mapping]]: [{{val| 2 0 11 4 }}, {{val| 0 2 -4 1 }}]


Map: [&lt;2 0 11 4|, &lt;0 2 -4 1|]
{{Multival|legend=1| 4 -8 2 -22 -8 27 }}


Wedgie: &lt;&lt;4 -8 2 -22 -8 27||
[[POTE generator]]: ~8/7 = 246.979


EDOs: 10, 24, 34
{{Val list|legend=1| 10, 24, 34 }}


Badness: 0.0780
[[Badness]]: 0.0780


== 11-limit ==
== 11-limit ==
Commas: 49/48, 56/55, 243/242
Comma list: 49/48, 56/55, 243/242
 
Mapping: [{{val| 2 0 11 4 -1 }}, {{val| 0 2 -4 1 5 }}]


POTE generator: ~8/7 = 247.816
POTE generator: ~8/7 = 247.816


Map: [&lt;2 0 11 4 -1|, &lt;0 2 -4 1 5|]
Vals: {{Val list| 10, 24, 34, 58d, 92de }}
 
EDOs: 10, 24, 34, 58d, 92de


Badness: 0.0493
Badness: 0.0493


== 13-limit ==
== 13-limit ==
Commas: 49/48, 56/55, 91/90, 352/351
Comma list: 49/48, 56/55, 91/90, 352/351
 
Mapping: [{{val| 2 0 11 4 -1 9 }}, {{val| 0 2 -4 1 5 -1 }}]


POTE generator: ~8/7 = 247.691
POTE generator: ~8/7 = 247.691


Map: [&lt;2 0 11 4 -1 9|, &lt;0 2 -4 1 5 -1|]
Vals: {{Val list| 10, 24, 34, 58d, 92def }}
 
EDOs: 10, 24, 34, 58d, 92def


Badness: 0.0308
Badness: 0.0308


= Shru =
= Shru =
Commas: 392/375, 1323/1280
[[Comma list]]: 392/375, 1323/1280


POTE generator: ~64/63 =  50.135
[[Mapping]]: [{{val| 2 1 9 11 }}, {{val| 0 2 -4 -5 }}]


Map: [&lt;2 1 9 11|, &lt;0 2 -4 -5|]
{{Multival|legend=1| 4 -8 -10 -22 -27 -1 }}


Wedgie: &lt;&lt;4 -8 -10 -22 -27 -1||
[[POTE generator]]: ~64/63 =  50.135


EDOs: 22d, 24
{{Val list|legend=1| 22d, 24 }}


Badness: 0.1576
[[Badness]]: 0.1576


== 11-limit ==
== 11-limit ==
Commas: 56/55, 77/75, 1323/1280
Comma list: 56/55, 77/75, 1323/1280


POTE generator: ~64/63 = 50.130
Mapping: [{{val| 2 1 9 11 8 }}, {{val| 0 2 -4 -5 -1 }}]


Map: [&lt;2 1 9 11 8|, &lt;0 2 -4 -5 -1|]
POTE generator: ~33/32 = 50.130


EDOs: 22d, 24
Vals: {{Val list| 22d, 24 }}


Badness: 0.0635
Badness: 0.0635


== 13-limit ==
== 13-limit ==
Commas: 56/55, 77/75, 105/104, 507/500
Comma list: 56/55, 77/75, 105/104, 507/500


POTE generator: ~64/63 = 50.535
Mapping: [{{val| 2 1 9 11 8 15 }}, {{val| 0 2 -4 -5 -1 -7 }}]


Map: [&lt;2 1 9 11 8 15|, &lt;0 2 -4 -5 -1 -7|]
POTE generator: ~33/32 = 50.535


EDOs: 24
Vals: {{Val list| 22df, 24 }}


Badness: 0.0457
Badness: 0.0457
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