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Diaschismic family: Difference between revisions

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atp just call it diaschismic
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reärranged some sections
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To get the 7-limit extensions, we add another comma:
To get the 7-limit extensions, we add another comma:


* Septimal diaschismic adds [[126/125]], the starling comma, to obtain 7-limit harmony by more complex methods than pajara, but with greater accuracy.
* Pajara derives from [[64/63]] and is a popular and well-known choice.  
* Pajara derives from [[64/63]] and is a popular and well-known choice.  
* Diaschismic adds [[126/125]], the starling comma, to obtain 7-limit harmony by more complex methods, but with greater accuracy.
* Srutal adds [[4375/4374]], the ragisma, which is about as accurate as septimal diaschismic but has a much more complex mapping of 7.  
* Srutal adds [[4375/4374]], the ragisma. It does no significant tuning damage, so we keep the 5-limit label srutal.  
* Keen adds [[875/864]].  
* Keen adds [[875/864]].  
* Bidia adds [[3136/3125]], the hemimean comma.
* Bidia adds [[3136/3125]], the hemimean comma.
Line 58: Line 58:
Pajara, diaschismic, srutal and keen keep the same half-octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as [[36/35]], the septimal quarter-tone) and echidna has a generator of 9/7. Bidia has a quarter-octave period and a fifth generator.
Pajara, diaschismic, srutal and keen keep the same half-octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as [[36/35]], the septimal quarter-tone) and echidna has a generator of 9/7. Bidia has a quarter-octave period and a fifth generator.


== Srutal ==
== Septimal diaschismic ==
{{Main| Diaschismic }}
{{See also| Srutal vs diaschismic }}
{{See also| Srutal vs diaschismic }}
A simpler characterization than the one given by the normal comma list is that diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called 46 & 58. However described, diaschismic has a 1/2-octave period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. [[58edo]] provides an excellent tuning, but an alternative is to make [[7/4]] just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58edo.
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-2 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.


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


[[Comma list]]: 2048/2025, 4375/4374
[[Comma list]]: 126/125, 2048/2025


{{Mapping|legend=1| 2 0 11 -42 | 0 1 -2 15 }}
{{Mapping|legend=1| 2 0 11 31 | 0 1 -2 -8 }}


{{Multival|legend=1| 2 -4 30 -11 42 81 }}
{{Multival|legend=1| 2 -4 -16 -11 -31 -26 }}


[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~3/2 = 704.814
[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~3/2 = 703.681


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [703.448, 705.882] (34\58 to 20\34)
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 705.882] (7\12 to 20\34)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]
* 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [703.448, 705.882]
* 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 705.882]


{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd, 332bcd }}
{{Optimal ET sequence|legend=1| 12, 46, 58, 104c, 162c }}


[[Badness]]: 0.091504
[[Badness]]: 0.037914


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


Comma list: 176/175, 896/891, 1331/1323
Comma list: 126/125, 176/175, 896/891


Mapping: {{mapping| 2 0 11 -42 -28 | 0 1 -2 15 11 }}
Mapping: {{mapping| 2 0 11 31 45 | 0 1 -2 -8 -12 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 704.856
Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 703.714


Tuning ranges:  
Tuning ranges:  
* 11-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 704.348] (7\12 to 27\46)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 704.348]


{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd }}
{{Optimal ET sequence|legend=1| 12, 46, 58, 104c, 162ce }}


Badness: 0.035315
Badness: 0.025034


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


Comma list: 169/168, 176/175, 325/324, 364/363
Comma list: 126/125, 176/175, 196/195, 364/363


Mapping: {{mapping| 2 0 11 -42 -28 -18 | 0 1 -2 15 11 8 }}
Mapping: {{mapping| 2 0 11 31 45 55 | 0 1 -2 -8 -12 -15 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 704.881
Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 703.704


Tuning ranges:  
Tuning ranges:  
* 13- and 15-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 13- and 15-odd-limit diamond monotone: ~3/2 = [703.448, 704.348] (34\58 to 27\46)
* 13-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 13-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [703.448, 704.348]


{{Optimal ET sequence|legend=1| 34d, 46, 80, 206cd, 286bcde }}
{{Optimal ET sequence|legend=1| 46, 58, 104c, 162cef }}


Badness: 0.025286
Badness: 0.018926


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 136/135, 169/168, 176/175, 221/220, 256/255
Comma list: 126/125, 136/135, 176/175, 196/195, 256/255


Mapping: {{mapping| 2 0 11 -42 -28 -18 5 | 0 1 -2 15 11 8 1 }}
Mapping: {{mapping| 2 0 11 31 45 55 5 | 0 1 -2 -8 -12 -15 1 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.840
Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 703.812


Tuning ranges:  
Tuning ranges:  
* 17-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 17-odd-limit diamond monotone: ~3/2 = [703.448, 704.348] (34\58 to 27\46)
* 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]
* 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]
* 17-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]
* 17-odd-limit diamond monotone and tradeoff: ~3/2 = [703.448, 704.348]


{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd }}
{{Optimal ET sequence|legend=1| 46, 58, 104c }}


Badness: 0.018594
Badness: 0.016425


=== 19-limit ===
=== No-19s 23-limit (Na"Naa') ===
Srutal, shrutar and bidia have similar 19-limit properties, tempering 190/189, related rank-3 [[julius]].
<b>Na"Naa'</b> is a remarkable subgroup temperament of 46&amp;58 with a prime harmonic of 23.


Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.23


Comma list: 136/135, 169/168, 176/175, 190/189, 221/220, 256/255
Comma list: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255


Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 | 0 1 -2 15 11 8 1 20 }}
Sval mapping: {{mapping| 2 0 11 31 45 55 5 63 | 0 1 -2 -8 -12 -15 1 -17 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.905
Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 703.870


{{Optimal ET sequence|legend=1| 34dh, 46, 80, 206cd }}
{{Optimal ET sequence|legend=1| 46, 58i, 104ci }}
 
Badness: 0.017063
 
==== Srutaloo ====
Srutaloo adds 576/575, 736/729 or 208/207, rhymes with [[Skidoo]].
 
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 256/255
 
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 | 0 1 -2 15 11 8 1 20 6 }}
 
Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.899
 
{{Optimal ET sequence|legend=1| 34dh, 46, 80, 206cd }}
 
Badness: 0.013555
 
===== 29-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23.29
 
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 232/231, 256/255
 
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 -76 | 0 1 -2 15 11 8 1 20 6 27 }}
 
Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.906
 
{{Optimal ET sequence|legend=1| 34dhj, 46, 80, 206cd }}
 
Badness: 0.013203
 
===== 31-limit =====
 
Subgroup: 2.3.5.7.11.13.17.19.23.29.31
 
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 217/216, 221/220, 232/231, 256/255
 
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 -76 48 | 0 1 -2 15 11 8 1 20 6 27 -12 }}
 
Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.817
 
{{Optimal ET sequence|legend=1| 46, 80, 126 }}
 
Badness: 0.015073


== Pajara ==
== Pajara ==
Line 477: Line 438:
Badness: 0.021790
Badness: 0.021790


== Septimal diaschismic ==
 
{{Main| Diaschismic }}
== Srutal ==
{{See also| Srutal vs diaschismic }}
{{See also| Srutal vs diaschismic }}
A simpler characterization than the one given by the normal comma list is that diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called 46 &amp; 58. However described, diaschismic has a 1/2-octave period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. [[58edo]] provides an excellent tuning, but an alternative is to make [[7/4]] just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58edo.
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-2 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.


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


[[Comma list]]: 126/125, 2048/2025
[[Comma list]]: 2048/2025, 4375/4374


{{Mapping|legend=1| 2 0 11 31 | 0 1 -2 -8 }}
{{Mapping|legend=1| 2 0 11 -42 | 0 1 -2 15 }}


{{Multival|legend=1| 2 -4 -16 -11 -31 -26 }}
{{Multival|legend=1| 2 -4 30 -11 42 81 }}


[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~3/2 = 703.681
[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~3/2 = 704.814


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 705.882] (7\12 to 20\34)
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [703.448, 705.882] (34\58 to 20\34)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]
* 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 705.882]
* 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [703.448, 705.882]


{{Optimal ET sequence|legend=1| 12, 46, 58, 104c, 162c }}
{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd, 332bcd }}


[[Badness]]: 0.037914
[[Badness]]: 0.091504


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


Comma list: 126/125, 176/175, 896/891
Comma list: 176/175, 896/891, 1331/1323


Mapping: {{mapping| 2 0 11 31 45 | 0 1 -2 -8 -12 }}
Mapping: {{mapping| 2 0 11 -42 -28 | 0 1 -2 15 11 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 703.714
Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 704.856


Tuning ranges:  
Tuning ranges:  
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 704.348] (7\12 to 27\46)
* 11-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 704.348]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]


{{Optimal ET sequence|legend=1| 12, 46, 58, 104c, 162ce }}
{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd }}


Badness: 0.025034
Badness: 0.035315


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


Comma list: 126/125, 176/175, 196/195, 364/363
Comma list: 169/168, 176/175, 325/324, 364/363


Mapping: {{mapping| 2 0 11 31 45 55 | 0 1 -2 -8 -12 -15 }}
Mapping: {{mapping| 2 0 11 -42 -28 -18 | 0 1 -2 15 11 8 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 703.704
Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 704.881


Tuning ranges:  
Tuning ranges:  
* 13- and 15-odd-limit diamond monotone: ~3/2 = [703.448, 704.348] (34\58 to 27\46)
* 13- and 15-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 13-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 13-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [703.448, 704.348]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]


{{Optimal ET sequence|legend=1| 46, 58, 104c, 162cef }}
{{Optimal ET sequence|legend=1| 34d, 46, 80, 206cd, 286bcde }}


Badness: 0.018926
Badness: 0.025286


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 126/125, 136/135, 176/175, 196/195, 256/255
Comma list: 136/135, 169/168, 176/175, 221/220, 256/255


Mapping: {{mapping| 2 0 11 31 45 55 5 | 0 1 -2 -8 -12 -15 1 }}
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 | 0 1 -2 15 11 8 1 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 703.812
Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.840


Tuning ranges:  
Tuning ranges:  
* 17-odd-limit diamond monotone: ~3/2 = [703.448, 704.348] (34\58 to 27\46)
* 17-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]
* 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]
* 17-odd-limit diamond monotone and tradeoff: ~3/2 = [703.448, 704.348]
* 17-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]
 
{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd }}
 
Badness: 0.018594
 
=== 19-limit ===
Srutal, shrutar and bidia have similar 19-limit properties, tempering 190/189, related rank-3 [[julius]].
 
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 136/135, 169/168, 176/175, 190/189, 221/220, 256/255
 
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 | 0 1 -2 15 11 8 1 20 }}
 
Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.905
 
{{Optimal ET sequence|legend=1| 34dh, 46, 80, 206cd }}
 
Badness: 0.017063
 
==== Srutaloo ====
Srutaloo adds 576/575, 736/729 or 208/207, rhymes with [[Skidoo]].
 
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 256/255
 
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 | 0 1 -2 15 11 8 1 20 6 }}
 
Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.899
 
{{Optimal ET sequence|legend=1| 34dh, 46, 80, 206cd }}
 
Badness: 0.013555
 
===== 29-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23.29
 
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 232/231, 256/255
 
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 -76 | 0 1 -2 15 11 8 1 20 6 27 }}
 
Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.906
 
{{Optimal ET sequence|legend=1| 34dhj, 46, 80, 206cd }}


{{Optimal ET sequence|legend=1| 46, 58, 104c }}
Badness: 0.013203


Badness: 0.016425
===== 31-limit =====


==== Na"Naa' ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31
<b>Na"Naa'</b> is a remarkable subgroup temperament of 46&amp;58 with a prime harmonic of 23.


Subgroup: 2.3.5.7.11.13.17.23
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 217/216, 221/220, 232/231, 256/255


Comma list: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 -76 48 | 0 1 -2 15 11 8 1 20 6 27 -12 }}


Sval mapping: {{mapping| 2 0 11 31 45 55 5 63 | 0 1 -2 -8 -12 -15 1 -17 }}
Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.817


Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 703.870
{{Optimal ET sequence|legend=1| 46, 80, 126 }}


{{Optimal ET sequence|legend=1| 46, 58i, 104ci }}
Badness: 0.015073


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