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Diaschismic family: Difference between revisions

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[[Tuning ranges]]:  
[[Tuning ranges]]:  
* valid range: [703.448, 705.882] (58 to 34d)
* valid range: ~3/2 = [703.448, 705.882] (34\58 to 20\34)
* nice range: [701.955, 706.843]
* nice range: ~3/2 = [701.955, 706.843]
* strict range: [703.448, 705.882]
* strict range: ~3/2 = [703.448, 705.882]


POTE generator: ~3/2 = 704.814
POTE generator: ~3/2 = 704.814
Line 44: Line 44:
Wedgie: <<2 -4 30 -11 42 81||
Wedgie: <<2 -4 30 -11 42 81||


{{Val list|legend=1| 46, 80, 126, 206cd, 332bcd }}
{{Val list|legend=1| 34d, 46, 80, 126, 206cd, 332bcd }}


[[Badness]]: 0.0915
[[Badness]]: 0.091504


== 11-limit ==
== 11-limit ==
Line 52: Line 52:


Tuning ranges:  
Tuning ranges:  
* valid range: [704.348, 705.882] (46 to 34d)
* valid range: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* nice range: [701.955, 706.843]
* nice range: ~3/2 = [701.955, 706.843]
* strict range: [704.348, 705.882]
* strict range: ~3/2 = [704.348, 705.882]


POTE generator: ~3/2 = 704.856
POTE generator: ~3/2 = 704.856
Line 60: Line 60:
Map: [<2 0 11 -42 -28|, <0 1 -2 15 11|]
Map: [<2 0 11 -42 -28|, <0 1 -2 15 11|]


Vals: {{Val list| 46, 80, 126, 206cd }}
Vals: {{Val list| 34d, 46, 80, 126, 206cd }}


Badness: 0.0353
Badness: 0.035315


== 13-limit ==
== 13-limit ==
Line 68: Line 68:


Tuning ranges:  
Tuning ranges:  
* valid range: [704.348, 705.882] (46 to 34d)
* valid range: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* nice range: [701.955, 706.843]
* nice range: ~3/2 = [701.955, 706.843]
* strict range: [704.348, 705.882]
* strict range: ~3/2 = [704.348, 705.882]


POTE generator: ~3/2 = 704.881
POTE generator: ~3/2 = 704.881
Line 78: Line 78:
Vals: {{Val list| 34d, 46, 80, 206cd, 286bcde }}
Vals: {{Val list| 34d, 46, 80, 206cd, 286bcde }}


Badness: 0.0253
Badness: 0.025286


= Pajara =
= Pajara =
Line 90: Line 90:


[[Tuning ranges]]:
[[Tuning ranges]]:
* valid range: [700.000, 720.000] (12 to 10)
* valid range: ~3/2 = [700.000, 720.000] (7\12 to 6\10)
* nice range: [701.955, 715.587]
* nice range: ~3/2 = [701.955, 715.587]
* strict range: [701.955, 715.587]
* strict range: ~3/2 = [701.955, 715.587]


[[POTE_tuning|POTE generator]]: 707.048
[[POTE_tuning|POTE generator]]: ~3/2 = 707.048


[[Mapping]]: [<2 0 11 12|, <0 1 -2 -2|]
[[Mapping]]: [<2 0 11 12|, <0 1 -2 -2|]
Line 100: Line 100:
{{Val list|legend=1| 10, 12, 22, 34d, 56d }}
{{Val list|legend=1| 10, 12, 22, 34d, 56d }}


[[Badness]]: 0.0200
[[Badness]]: 0.020033


== 11-limit ==
== 11-limit ==
Line 106: Line 106:


Tuning ranges:
Tuning ranges:
* valid range: [700.000, 709.091] (12 to 22)
* valid range: ~3/2 = [700.000, 709.091] (7\12 to 13\22)
* nice range: [701.955, 715.587]
* nice range: ~3/2 = [701.955, 715.587]
* strict range: [701.955, 709.091]
* strict range: ~3/2 = [701.955, 709.091]


[[POTE_tuning|POTE generator]]: 706.885
[[POTE_tuning|POTE generator]]: ~3/2 = 706.885


Map: [<2 0 11 12 26|, <0 1 -2 -2 -6|]
Mapping: [<2 0 11 12 26|, <0 1 -2 -2 -6|]


EDOs: 22, 34, 56, 146
Vals: {{Val list| 10e, 12, 22, 34d, 56d }}


Badness: 0.0203
Badness: 0.020343


===  13-limit ===
===  13-limit ===
Line 122: Line 122:


Tuning ranges:
Tuning ranges:
* valid range: [700.000, 709.091]
* valid range: ~3/2 = [700.000, 709.091] (7\12 to 13\22)
* nice range: [701.955, 738.573]
* nice range: ~3/2 = [701.955, 738.573]
* strict range: [701.955, 709.091]
* strict range: ~3/2 = [701.955, 709.091]


POTE generator: ~3/2 = 708.919
POTE generator: ~3/2 = 708.919


Map: [<2 0 11 12 26 1|, <0 1 -2 -2 -6 2|]
Mapping: [<2 0 11 12 26 1|, <0 1 -2 -2 -6 2|]


EDOs: 12, 22
Vals: {{Val list| 12, 22 }}


Badness: 0.0276
Badness: 0.027642


== Pajarous ==
== Pajarous ==
Commas: 50/49, 55/54, 64/63
Comma list: 50/49, 55/54, 64/63


valid range: 709.091 (22)
Tuning ranges:
 
* valid range: ~3/2 = 709.091 (13\22)
nice range: [701.955, 715.803]
* nice range: ~3/2 = [701.955, 715.803]
 
* strict range: ~3/2 = 709.091
strict range: 709.091


POTE generator: ~3/2 = 709.578
POTE generator: ~3/2 = 709.578


Map: [<2 0 11 12 -9|, <0 1 -2 -2 5|]
Mapping: [<2 0 11 12 -9|, <0 1 -2 -2 5|]


EDOs: 10, 12e, 22, 120bce, 142bce
Vals: {{Val list| 10, 12e, 22, 120bce, 142bce }}


Badness: 0.0283
Badness: 0.028349


=== 13-limit ===
=== 13-limit ===
Commas: 50/49, 55/54, 64/63, 65/63
Comma list: 50/49, 55/54, 64/63, 65/63
 
valid range: []
 
nice range: [701.955, 738.573]
 
strict range: []


POTE generator: ~3/2 = 710.240
POTE generator: ~3/2 = 710.240


Map: [<2 0 11 12 -9 1|, <0 1 -2 -2 5 2|]
Mapping: [<2 0 11 12 -9 1|, <0 1 -2 -2 5 2|]


EDOs: 10, 22, 54f, 76bdf
Vals: {{Val list| 10, 22, 54f, 76bdf }}


Badness: 0.0252
Badness: 0.025176


=== Pajaro ===
=== Pajaro ===
Commas: 40/39, 50/49, 55/54, 64/63
Comma list: 40/39, 50/49, 55/54, 64/63


POTE generator ~3/2 = 710.818
POTE generator ~3/2 = 710.818


Map: [<2 0 11 12 -9 17|, <0 1 -2 -2 5 -3|]
Mapping: [<2 0 11 12 -9 17|, <0 1 -2 -2 5 -3|]


EDOs: 10, 22f, 32f, 54f
Vals: {{Val list| 10, 22f, 32f, 54f }}


Badness: 0.0274
Badness: 0.0274


== Pajaric ==
== Pajaric ==
Commas: 45/44, 50/49, 56/55
Comma list: 45/44, 50/49, 56/55


POTE generator: ~3/2 = 705.524
POTE generator: ~3/2 = 705.524
Line 186: Line 179:
Map: [<2 0 11 12 7|, <0 1 -2 -2 0|]
Map: [<2 0 11 12 7|, <0 1 -2 -2 0|]


EDOs: 10, 12, 22e, 34de
Vals: {{Val list| 10, 12, 22e, 34de }}


Badness: 0.0238
Badness: 0.023798


=== 13-limit ===
=== 13-limit ===
Commas: 40/39, 45/44, 50/49, 56/55
Comma list: 40/39, 45/44, 50/49, 56/55


POTE generator: ~3/2 = 707.442
POTE generator: ~3/2 = 707.442
Line 197: Line 190:
Map: [<2 0 11 12 7 17|, <0 1 -2 -2 0 -3|]
Map: [<2 0 11 12 7 17|, <0 1 -2 -2 0 -3|]


EDOs: 10, 12f, 22ef, 34def
Vals: {{Val list| 10, 12f, 22ef, 34def }}


Badness: 0.0205
Badness: 0.0205
Line 208: Line 201:
Map: [<2 1 9 10 8|, <0 2 -4 -4 -1|]
Map: [<2 1 9 10 8|, <0 2 -4 -4 -1|]


EDOs: 20, 22, 68d, 90d
Vals: {{Val list| 20, 22, 68d, 90d }}


Badness: 0.0389
Badness: 0.038890


= Diaschismic =
= Diaschismic =
Line 217: Line 210:
Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher limit rank two temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363. The 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher limit harmonies, diaschismic is certainly one excellent way to do it; MOS of 34 notes and even more the 46 note MOS will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.
Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher limit rank two temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363. The 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher limit harmonies, diaschismic is certainly one excellent way to do it; MOS of 34 notes and even more the 46 note MOS will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.


Commas: 126/125, 2048/2025
[[Comma list]]: 126/125, 2048/2025


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


Map: [<2 0 11 31|, <0 1 -2 -8|]
[[Mapping]]: [<2 0 11 31|, <0 1 -2 -8|]


EDOs: 46, 58, 104c, 162c
{{Val list|legend=1| 12, 46, 58, 104c, 162c }}


Badness: 0.0379
[[Badness]]: 0.037914


== 11-limit ==
== 11-limit ==
Commas: 126/125, 176/175, 896/891
Comma list: 126/125, 176/175, 896/891


[[POTE_tuning|POTE generator]]: 703.714
POTE generator: ~3/2 = 703.714


Map: [<2 0 11 31 45|, <0 1 -2 -8 -12|]
Mapping: [<2 0 11 31 45|, <0 1 -2 -8 -12|]


EDOs: 46, 58, 104c, 162ce
Vals: {{Val list| 12, 46, 58, 104c, 162ce }}
 
Badness: 0.025034


== 13-limit ==
== 13-limit ==
Commas: 126/125, 196/195, 364/363, 2048/2025
Comma list: 126/125, 196/195, 364/363, 2048/2025


[[POTE_tuning|POTE generator]]: 703.704
POTE generator: ~3/2 = 703.704


Map: [<2 0 11 31 45 55|, <0 1 -2 -8 -12 -15|]
Mapping: [<2 0 11 31 45 55|, <0 1 -2 -8 -12 -15|]


EDOs: [[46edo|46]], [[58edo|58]], [[104edo|104c]], [[162edo|162cef]]
Vals: {{Val list| 46, 58, 104c, 162cef }}
 
Badness: 0.018926


== 17-limit ==
== 17-limit ==
Commas: 126/125, 136/135, 176/175, 196/195, 256/255
Comma list: 126/125, 136/135, 176/175, 196/195, 256/255


[[POTE_tuning|POTE generator]]: 703.812
POTE generator: ~3/2 = 703.812


Map: [<2 0 11 31 45 55 5|, <0 1 -2 -8 -12 -15 1|]
Mapping: [<2 0 11 31 45 55 5|, <0 1 -2 -8 -12 -15 1|]


EDOs: 46, 58, 104c
Vals: {{Val list| 46, 58, 104c }}
 
Badness: 0.016425


= Keen =
= Keen =
Line 264: Line 263:


EDOs: 22, 56, 78, 134b, 212b, 290b
EDOs: 22, 56, 78, 134b, 212b, 290b
Badness: 0.083971


== 11-limit ==
== 11-limit ==
Line 273: Line 274:


EDOs: 22, 56, 78, 212bf, 290bf
EDOs: 22, 56, 78, 212bf, 290bf
Badness: 0.045270


= Bidia =
= Bidia =
Retrieved from "https://en.xen.wiki/w/Diaschismic_family"