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Diaschismic family: Difference between revisions

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{{Main| Diaschismic }}
{{Main| Diaschismic }}


The [[period]] of diaschismic is half an [[2/1|octave]], and the [[generator]] is a fifth; the [[ploidacot]] is diploid monocot. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]], or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a [[mos]] of diaschismic gives two scale possibilities.
The [[period]] of diaschismic is half an [[octave]], and the [[generator]] is a fifth; the [[ploidacot]] is diploid monocot. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]], or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a [[mos]] of diaschismic gives two scale possibilities.


This temperament is also known as '''srutal''' in the 5-limit, but that name more strictly speaking refers to the [[#Srutal|34d & 46 extension]] to the [[7-limit]] that adds [[4375/4374]] to the comma list.
This temperament is also known as '''srutal''' in the 5-limit, but that name more strictly speaking refers to the [[#Srutal|34d & 46 extension]] to the [[7-limit]] that adds [[4375/4374]] to the comma list.
Line 40: Line 40:
Those all keep the same half-octave period and fifth generator.  
Those all keep the same half-octave period and fifth generator.  


Bidia adds [[3136/3125]], the hemimean comma, with a 1/4-octave period. Shrutar adds [[245/243]] and shru adds [[392/375]], with a quartertone generator. Sruti adds [[19683/19600]] and anguirus adds [[49/48]], with a neutral third or hemitwelfth generator. Those split the original generator in two. Echidna adds [[1728/1715]], the orwellisma, with a ~9/7 generator. Echidnic adds [[686/675]], the senga, with a ~8/7 generator. Those split the original generator in three. Finally, quadrasruta adds [[2401/2400]] and splits the original generator in four.
Bidia adds [[3136/3125]], the hemimean comma, with a 1/4-octave period. Shrutar adds [[245/243]] and shru adds [[392/375]], with a quartertone generator. Sruti adds [[19683/19600]] and anguirus adds [[49/48]], with a neutral third or hemitwelfth generator. Those split the original generator in two. Echidna adds [[1728/1715]], the orwellisma, with a ~9/7 generator. Echidnic adds [[686/675]], the senga, with a ~8/7 generator. Those split the original generator in three. Finally, quadrasruta adds [[2401/2400]] and splits the original generator in four.
 
Temperaments discussed elsewhere include [[stearnsmic clan #Echidna|echidna]]. The rest are considered below.  


==== Subgroup extensions ====
==== Subgroup extensions ====
Since the diaschisma factors into ([[256/255]])<sup>2</sup>([[289/288]]) in the 17-limit, it extends naturally to the 2.3.5.17 subgroup as ''srutal archagall'', considered in [[#Subgroup extensions]]. The [[S-expression]]-based comma list of this temperament is {[[256/255|S16]], [[289/288|S17]]}.
Since the diaschisma factors into ([[256/255]])<sup>2</sup>([[289/288]]) in the 17-limit, it extends naturally to the 2.3.5.17 subgroup as ''srutal archagall'', considered in [[#Subgroup extensions_2|#Subgroup extensions]]. The [[S-expression]]-based comma list of this temperament is {[[256/255|S16]], [[289/288|S17]]}.
 
More generally, one may note that since the fifth of diaschismic is sharp in any good tuning, it is compatible with the logic of [[parapyth]] in making ~13/11 a minor third and ~14/11 a major third, leading to the rank-3 temperament [[varda]], which multiple temps here are rank-2 tunings of, such as 17-limit diaschismic and srutal (and their obvious extensions to prime 23, na"naa' and srutaloo, both by mapping ~23/16 as an augmented fourth aka. tritone).


== Septimal diaschismic ==
== Septimal diaschismic ==
Line 78: Line 82:
Comma list: 126/125, 176/175, 896/891
Comma list: 126/125, 176/175, 896/891


Mapping: {{mapping| 2 0 11 31 45 | 0 1 -2 -8 -12 }}
{{Mapping|legend=0| 2 0 11 31 45 | 0 1 -2 -8 -12 }}


Optimal tunings:  
Optimal tunings:  
Line 97: Line 101:
Comma list: 126/125, 176/175, 196/195, 364/363
Comma list: 126/125, 176/175, 196/195, 364/363


Mapping: {{mapping| 2 0 11 31 45 55 | 0 1 -2 -8 -12 -15 }}
{{Mapping|legend=0| 2 0 11 31 45 55 | 0 1 -2 -8 -12 -15 }}


Optimal tunings:  
Optimal tunings:  
Line 117: Line 121:
Comma list: 126/125, 136/135, 176/175, 196/195, 256/255
Comma list: 126/125, 136/135, 176/175, 196/195, 256/255


Mapping: {{mapping| 2 0 11 31 45 55 5 | 0 1 -2 -8 -12 -15 1 }}
{{Mapping|legend=0| 2 0 11 31 45 55 5 | 0 1 -2 -8 -12 -15 1 }}


Optimal tunings:  
Optimal tunings:  
Line 138: Line 142:
Comma list: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255
Comma list: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255


Subgroup-val mapping: {{mapping| 2 0 11 31 45 55 5 63 | 0 1 -2 -8 -12 -15 1 -17 }}
{{Mapping|legend=2| 2 0 11 31 45 55 5 63 | 0 1 -2 -8 -12 -15 1 -17 }}


Optimal tunings:  
Optimal tunings:  
Line 174: Line 178:


[[Badness]] (Sintel): 0.507
[[Badness]] (Sintel): 0.507
=== 2.3.5.7.17 subgroup ===
Subgroup: 2.3.5.7.17
Comma list: 50/49, 64/63, 85/84
{{Mapping|legend=0| 2 0 11 12 5 | 0 1 -2 -2 1 }}
Optimal tunings:
* WE: ~7/5 = 599.053{{c}}, ~3/2 = 706.355{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.607{{c}}
{{Optimal ET sequence|legend=0| 10, 12, 22, 56d }}
Badness (Sintel): 0.438


=== 11-limit ===
=== 11-limit ===
Line 180: Line 199:
Comma list: 50/49, 64/63, 99/98
Comma list: 50/49, 64/63, 99/98


Mapping: {{mapping| 2 0 11 12 26 | 0 1 -2 -2 -6 }}
{{Mapping|legend=0| 2 0 11 12 26 | 0 1 -2 -2 -6 }}


Optimal tunings:  
Optimal tunings:  
Line 193: Line 212:


Badness (Sintel): 0.673
Badness (Sintel): 0.673
==== 2.3.5.7.11.17 subgroup ====
Subgroup: 2.3.5.7.11.17
Comma list: 50/49, 64/63, 85/84, 99/98
{{Mapping|legend=0| 2 0 11 12 26 5 | 0 1 -2 -2 -6 1 }}
Optimal tunings:
* WE: ~7/5 = 599.062{{c}}, ~3/2 = 706.095{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.370{{c}}
{{Optimal ET sequence|legend=0| 10e, 12, 22, 34d, 56d }}
Badness (Sintel): 0.645


==== 13-limit ====
==== 13-limit ====
Line 199: Line 233:
Comma list: 50/49, 64/63, 65/63, 99/98
Comma list: 50/49, 64/63, 65/63, 99/98


Mapping: {{mapping| 2 0 11 12 26 1 | 0 1 -2 -2 -6 2 }}
{{Mapping|legend=0| 2 0 11 12 26 1 | 0 1 -2 -2 -6 2 }}


Optimal tunings:  
Optimal tunings:  
Line 214: Line 248:
Comma list: 50/49, 52/51, 64/63, 65/63, 99/98
Comma list: 50/49, 52/51, 64/63, 65/63, 99/98


Mapping: {{mapping| 2 0 11 12 26 1 5 | 0 1 -2 -2 -6 2 1 }}
{{Mapping|legend=0| 2 0 11 12 26 1 5 | 0 1 -2 -2 -6 2 1 }}


Optimal tunings:  
Optimal tunings:  
Line 229: Line 263:
Comma list: 50/49, 64/63, 78/77, 99/98
Comma list: 50/49, 64/63, 78/77, 99/98


Mapping: {{mapping| 2 0 11 12 26 36 | 0 1 -2 -2 -6 -9 }}
{{Mapping|legend=0| 2 0 11 12 26 36 | 0 1 -2 -2 -6 -9 }}


Optimal tunings:  
Optimal tunings:  
Line 244: Line 278:
Comma list: 50/49, 64/63, 78/77, 85/84, 99/98
Comma list: 50/49, 64/63, 78/77, 85/84, 99/98


Mapping: {{mapping| 2 0 11 12 26 36 5 | 0 1 -2 -2 -6 -9 1 }}
{{Mapping|legend=0| 2 0 11 12 26 36 5 | 0 1 -2 -2 -6 -9 1 }}


Optimal tunings:  
Optimal tunings:  
Line 259: Line 293:
Comma list: 40/39, 50/49, 64/63, 66/65
Comma list: 40/39, 50/49, 64/63, 66/65


Mapping: {{mapping| 2 0 11 12 26 17 | 0 1 -2 -2 -6 -3 }}
{{Mapping|legend=0| 2 0 11 12 26 17 | 0 1 -2 -2 -6 -3 }}


Optimal tunings:  
Optimal tunings:  
Line 274: Line 308:
Comma list: 40/39, 50/49, 64/63, 66/65, 85/84
Comma list: 40/39, 50/49, 64/63, 66/65, 85/84


Mapping: {{mapping| 2 0 11 12 26 17 5 | 0 1 -2 -2 -6 -3 1 }}
{{Mapping|legend=0| 2 0 11 12 26 17 5 | 0 1 -2 -2 -6 -3 1 }}


Optimal tunings:  
Optimal tunings:  
Line 289: Line 323:
Comma list: 50/49, 55/54, 64/63
Comma list: 50/49, 55/54, 64/63


Mapping: {{mapping| 2 0 11 12 -9 | 0 1 -2 -2 5 }}
{{Mapping|legend=0| 2 0 11 12 -9 | 0 1 -2 -2 5 }}


Optimal tunings:  
Optimal tunings:  
Line 302: Line 336:


Badness (Sintel): 0.937
Badness (Sintel): 0.937
==== 2.3.5.7.11.17 subgroup ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 50/49, 52/51, 55/54, 64/63, 65/63
{{Mapping|legend=0| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}
Optimal tunings:
* WE: ~7/5 = 599.408{{c}}, ~3/2 = 708.878{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 709.544{{c}}
{{Optimal ET sequence|legend=0| 10, 12e, 22 }}
Badness (Sintel): 0.766


==== 13-limit ====
==== 13-limit ====
Line 308: Line 357:
Comma list: 50/49, 55/54, 64/63, 65/63
Comma list: 50/49, 55/54, 64/63, 65/63


Mapping: {{mapping| 2 0 11 12 -9 1 | 0 1 -2 -2 5 2 }}
{{Mapping|legend=0| 2 0 11 12 -9 1 | 0 1 -2 -2 5 2 }}


Optimal tunings:  
Optimal tunings:  
Line 323: Line 372:
Comma list: 50/49, 52/51, 55/54, 64/63, 65/63
Comma list: 50/49, 52/51, 55/54, 64/63, 65/63


Mapping: {{mapping| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}
{{Mapping|legend=0| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}


Optimal tunings:  
Optimal tunings:  
Line 338: Line 387:
Comma list: 40/39, 50/49, 55/54, 64/63
Comma list: 40/39, 50/49, 55/54, 64/63


Mapping: {{mapping| 2 0 11 12 -9 17 | 0 1 -2 -2 5 -3 }}
{{Mapping|legend=0| 2 0 11 12 -9 17 | 0 1 -2 -2 5 -3 }}


Optimal tunings:  
Optimal tunings:  
Line 353: Line 402:
Comma list: 40/39, 50/49, 55/54, 64/63, 85/84
Comma list: 40/39, 50/49, 55/54, 64/63, 85/84


Mapping: {{mapping| 2 0 11 12 -9 17 5 | 0 1 -2 -2 5 -3 1 }}
{{Mapping|legend=0| 2 0 11 12 -9 17 5 | 0 1 -2 -2 5 -3 1 }}


Optimal tunings:  
Optimal tunings:  
Line 368: Line 417:
Comma list: 45/44, 50/49, 56/55
Comma list: 45/44, 50/49, 56/55


Mapping: {{mapping| 2 0 11 12 7 | 0 1 -2 -2 0 }}
{{Mapping|legend=0| 2 0 11 12 7 | 0 1 -2 -2 0 }}


Optimal tunings:  
Optimal tunings:  
Line 383: Line 432:
Comma list: 40/39, 45/44, 50/49, 56/55
Comma list: 40/39, 45/44, 50/49, 56/55


Mapping: {{mapping| 2 0 11 12 7 17 | 0 1 -2 -2 0 -3 }}
{{Mapping|legend=0| 2 0 11 12 7 17 | 0 1 -2 -2 0 -3 }}


Optimal tunings:  
Optimal tunings:  
Line 398: Line 447:
Comma list: 34/33, 40/39, 45/44, 50/49, 56/55
Comma list: 34/33, 40/39, 45/44, 50/49, 56/55


Mapping: {{mapping| 2 0 11 12 7 17 5 | 0 1 -2 -2 0 -3 1 }}
{{Mapping|legend=0| 2 0 11 12 7 17 5 | 0 1 -2 -2 0 -3 1 }}


Optimal tunings:  
Optimal tunings:  
Line 413: Line 462:
Comma list: 50/49, 64/63, 121/120  
Comma list: 50/49, 64/63, 121/120  


Mapping: {{mapping| 2 1 9 10 8 | 0 2 -4 -4 -1 }}
{{Mapping|legend=0| 2 1 9 10 8 | 0 2 -4 -4 -1 }}


: mapping generators: ~2, ~16/11
: mapping generators: ~2, ~16/11
Line 430: Line 479:
Comma list: 50/49, 64/63, 243/242
Comma list: 50/49, 64/63, 243/242


Mapping: {{mapping| 2 0 11 12 -1 | 0 2 -4 -4 5 }}
{{Mapping|legend=0| 2 0 11 12 -1 | 0 2 -4 -4 5 }}
: mapping generators: ~2, ~55/32
: mapping generators: ~2, ~55/32


Line 446: Line 495:
Comma list: 50/49, 64/63, 78/77, 144/143
Comma list: 50/49, 64/63, 78/77, 144/143


Mapping: {{mapping| 2 0 11 12 -1 9 | 0 2 -4 -4 5 -1 }}
{{Mapping|legend=0| 2 0 11 12 -1 9 | 0 2 -4 -4 5 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 461: Line 510:
Comma list: 50/49, 64/63, 78/77, 85/84, 144/143
Comma list: 50/49, 64/63, 78/77, 85/84, 144/143


Mapping: {{mapping| 2 0 11 12 -1 9 5 | 0 2 -4 -4 5 -1 2 }}
{{Mapping|legend=0| 2 0 11 12 -1 9 5 | 0 2 -4 -4 5 -1 2 }}


Optimal tunings:  
Optimal tunings:  
Line 501: Line 550:
Comma list: 176/175, 896/891, 1331/1323
Comma list: 176/175, 896/891, 1331/1323


Mapping: {{mapping| 2 0 11 -42 -28 | 0 1 -2 15 11 }}
{{Mapping|legend=0| 2 0 11 -42 -28 | 0 1 -2 15 11 }}


Optimal tunings:  
Optimal tunings:  
Line 520: Line 569:
Comma list: 169/168, 176/175, 325/324, 364/363
Comma list: 169/168, 176/175, 325/324, 364/363


Mapping: {{mapping| 2 0 11 -42 -28 -18 | 0 1 -2 15 11 8 }}
{{Mapping|legend=0| 2 0 11 -42 -28 -18 | 0 1 -2 15 11 8 }}


Optimal tunings:  
Optimal tunings:  
Line 540: Line 589:
Comma list: 136/135, 169/168, 176/175, 221/220, 256/255
Comma list: 136/135, 169/168, 176/175, 221/220, 256/255


Mapping: {{mapping| 2 0 11 -42 -28 -18 5 | 0 1 -2 15 11 8 1 }}
{{Mapping|legend=0| 2 0 11 -42 -28 -18 5 | 0 1 -2 15 11 8 1 }}


Optimal tunings:  
Optimal tunings:  
Line 559: Line 608:
Comma list: 136/135, 169/168, 176/175, 190/189, 221/220, 256/255
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 }}
{{Mapping|legend=0| 2 0 11 -42 -28 -18 5 -55 | 0 1 -2 15 11 8 1 20 }}


Optimal tunings:  
Optimal tunings:  
Line 576: Line 625:
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 256/255
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 }}
{{Mapping|legend=0| 2 0 11 -42 -28 -18 5 -55 -10 | 0 1 -2 15 11 8 1 20 6 }}


Optimal tunings:  
Optimal tunings:  
Line 591: Line 640:
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 232/231, 256/255
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 }}
{{Mapping|legend=0| 2 0 11 -42 -28 -18 5 -55 -10 -76 | 0 1 -2 15 11 8 1 20 6 27 }}


Optimal tunings:  
Optimal tunings:  
Line 606: Line 655:
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 217/216, 221/220, 232/231, 256/255
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 }}
{{Mapping|legend=0| 2 0 11 -42 -28 -18 5 -55 -10 -76 48 | 0 1 -2 15 11 8 1 20 6 27 -12 }}


Optimal tunings:  
Optimal tunings:  
Line 640: Line 689:
Comma list: 100/99, 385/384, 1232/1215
Comma list: 100/99, 385/384, 1232/1215


Mapping: {{mapping| 2 0 11 -23 26 | 0 1 -2 9 -6 }}
{{Mapping|legend=0| 2 0 11 -23 26 | 0 1 -2 9 -6 }}


Optimal tunings:  
Optimal tunings:  
Line 655: Line 704:
Comma list: 100/99, 105/104, 144/143, 1078/1053
Comma list: 100/99, 105/104, 144/143, 1078/1053


Mapping: {{mapping| 2 0 11 -23 26 -18 | 0 1 -2 9 -6 8 }}
{{Mapping|legend=0| 2 0 11 -23 26 -18 | 0 1 -2 9 -6 8 }}


Optimal tunings:  
Optimal tunings:  
Line 670: Line 719:
Comma list: 100/99, 105/104, 119/117, 144/143, 154/153
Comma list: 100/99, 105/104, 119/117, 144/143, 154/153


Mapping: {{mapping| 2 0 11 -23 26 -18 5 | 0 1 -2 9 -6 8 1}}
{{Mapping|legend=0| 2 0 11 -23 26 -18 5 | 0 1 -2 9 -6 8 1}}


Optimal tunings:  
Optimal tunings:  
Line 685: Line 734:
Comma list: 91/90, 100/99, 352/351, 385/384
Comma list: 91/90, 100/99, 352/351, 385/384


Mapping: {{mapping| 2 0 11 -23 26 36 | 0 1 -2 9 -6 -9 }}
{{Mapping|legend=0| 2 0 11 -23 26 36 | 0 1 -2 9 -6 -9 }}


Optimal tunings:  
Optimal tunings:  
Line 700: Line 749:
Comma list: 91/90, 100/99, 136/135, 154/153, 256/255
Comma list: 91/90, 100/99, 136/135, 154/153, 256/255


Mapping: {{mapping| 2 0 11 -23 26 36 5 | 0 1 -2 9 -6 -9 1 }}
{{Mapping|legend=0| 2 0 11 -23 26 36 5 | 0 1 -2 9 -6 -9 1 }}


Optimal tunings:  
Optimal tunings:  
Line 735: Line 784:
Comma list: 176/175, 896/891, 1375/1372
Comma list: 176/175, 896/891, 1375/1372


Mapping: {{mapping| 4 0 22 43 71 | 0 1 -2 -5 -9 }}
{{Mapping|legend=0| 4 0 22 43 71 | 0 1 -2 -5 -9 }}


Optimal tunings:  
Optimal tunings:  
Line 750: Line 799:
Comma list: 176/175, 325/324, 640/637, 896/891
Comma list: 176/175, 325/324, 640/637, 896/891


Mapping: {{mapping| 4 0 22 43 71 -36 | 0 1 -2 -5 -9 8 }}
{{Mapping|legend=0| 4 0 22 43 71 -36 | 0 1 -2 -5 -9 8 }}


Optimal tunings:  
Optimal tunings:  
Line 765: Line 814:
Comma list: 136/135, 176/175, 256/255, 325/324, 640/637
Comma list: 136/135, 176/175, 256/255, 325/324, 640/637


Mapping: {{mapping| 4 0 22 43 71 -36 10 | 0 1 -2 -5 -9 8 1 }}
{{Mapping|legend=0| 4 0 22 43 71 -36 10 | 0 1 -2 -5 -9 8 1 }}


Optimal tunings:  
Optimal tunings:  
Line 780: Line 829:
Comma list: 136/135, 176/175, 190/189, 256/255, 325/324, 640/637
Comma list: 136/135, 176/175, 190/189, 256/255, 325/324, 640/637


Mapping: {{mapping| 4 0 22 43 71 -36 10 17 | 0 1 -2 -5 -9 8 1 0 }}
{{Mapping|legend=0| 4 0 22 43 71 -36 10 17 | 0 1 -2 -5 -9 8 1 0 }}


Optimal tunings:  
Optimal tunings:  
Line 795: Line 844:
Comma list: 136/135, 176/175, 190/189, 253/252, 256/255, 325/324, 640/637
Comma list: 136/135, 176/175, 190/189, 253/252, 256/255, 325/324, 640/637


Mapping: {{mapping| 4 0 22 43 71 -36 10 17 -20 | 0 1 -2 -5 -9 8 1 0 6 }}
{{Mapping|legend=0| 4 0 22 43 71 -36 10 17 -20 | 0 1 -2 -5 -9 8 1 0 6 }}


Optimal tunings:
Optimal tunings:
Line 834: Line 883:
Comma list: 121/120, 176/175, 245/243
Comma list: 121/120, 176/175, 245/243


Mapping: {{mapping| 2 1 9 -2 8 | 0 2 -4 7 -1 }}
{{Mapping|legend=0| 2 1 9 -2 8 | 0 2 -4 7 -1 }}


Optimal tunings:
Optimal tunings:
Line 845: Line 894:


=== 13-limit ===
=== 13-limit ===
Note that an alternative, simpler mapping for prime 13 exists at -7 gens, as in [[#Shru]]. This mapping is especially valuable when considering shrutar as a no-7's 17-limit temperament (22 & 24, supported by 46, 68f and 70), as the complexity from mapping 7 at +7 gens is one of the motivations for picking a +16 gen mapping for 13. This temperament isn't as accurate, but it is simple, and makes many [[MOS]]es, specifically 2L 2''n''s for ''n'' up to [[2L 20s|10]], followed by [[22L 2s]] for the 24-note MOS, with a quarter-tone generator equivalent to ~11/8.
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 176/175, 196/195, 245/243
Comma list: 121/120, 176/175, 196/195, 245/243


Mapping: {{mapping| 2 1 9 -2 8 -10 | 0 2 -4 7 -1 16 }}
{{Mapping|legend=0| 2 1 9 -2 8 -10 | 0 2 -4 7 -1 16 }}


Optimal tunings:
Optimal tunings:
Line 864: Line 915:
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195


Mapping: {{mapping| 2 1 9 -2 8 -10 6 | 0 2 -4 7 -1 16 2 }}
{{Mapping|legend=0| 2 1 9 -2 8 -10 6 | 0 2 -4 7 -1 16 2 }}


Optimal tunings:
Optimal tunings:
Line 879: Line 930:
Comma list: 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


Mapping: {{mapping| 2 1 9 -2 8 -10 6 -10 | 0 2 -4 7 -1 16 2 17 }}
{{Mapping|legend=0| 2 1 9 -2 8 -10 6 -10 | 0 2 -4 7 -1 16 2 17 }}


Optimal tunings:
Optimal tunings:
Line 894: Line 945:
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 253/252, 343/342
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 253/252, 343/342


Mapping: {{mapping| 2 1 9 -2 8 -10 6 -10 -4 | 0 2 -4 7 -1 16 2 17 12 }}
{{Mapping|legend=0| 2 1 9 -2 8 -10 6 -10 -4 | 0 2 -4 7 -1 16 2 17 12 }}


Optimal tunings:
Optimal tunings:
Line 905: Line 956:


== Shru ==
== Shru ==
Shru tempers out 392/375 and slices the compound semitone into two generators of ~10/7. Its ploidacot is diploid alpha-dicot, the same as shrutar.  
Shru tempers out 392/375 and slices the compound semitone into two generators of ~10/7. Its ploidacot is diploid alpha-dicot, the same as shrutar. It can be seen as one way of merging [[parapyth]] with [[diaschismic]] (and hence as a lower-accuracy rank 2 tuning of [[varda]]).
 
Note that shru is especially efficient as a no-7's 17-limit temperament, as its 7 is the highest damage, and without it, the other primes can be tuned more accurately.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 929: Line 982:
Comma list: 56/55, 77/75, 1323/1280
Comma list: 56/55, 77/75, 1323/1280


Mapping: {{mapping| 2 1 9 11 8 | 0 2 -4 -5 -1 }}
{{Mapping|legend=0| 2 1 9 11 8 | 0 2 -4 -5 -1 }}


Optimal tunings:
Optimal tunings:
Line 944: Line 997:
Comma list: 56/55, 77/75, 105/104, 507/500
Comma list: 56/55, 77/75, 105/104, 507/500


Mapping: {{mapping| 2 1 9 11 8 15 | 0 2 -4 -5 -1 -7 }}
{{Mapping|legend=0| 2 1 9 11 8 15 | 0 2 -4 -5 -1 -7 }}


Optimal tunings:
Optimal tunings:
Line 955: Line 1,008:


== Sruti ==
== Sruti ==
{{Redirect|Sruti|the tuning concept in Indian music|Shruti}}
Sruti tempers out 19683/19600, setting itself up as a [[hemipyth]] temperament. It has the same semi-octave period as diaschismic, but the generator can be taken as a neutral third or a hemitwelfth. The temperament can be described as {{nowrap| 24 & 34d }}; its ploidacot is diploid dicot. [[58edo]] may be recommended as a tuning.  
Sruti tempers out 19683/19600, setting itself up as a [[hemipyth]] temperament. It has the same semi-octave period as diaschismic, but the generator can be taken as a neutral third or a hemitwelfth. The temperament can be described as {{nowrap| 24 & 34d }}; its ploidacot is diploid dicot. [[58edo]] may be recommended as a tuning.  


Line 979: Line 1,034:
Comma list: 176/175, 243/242, 896/891
Comma list: 176/175, 243/242, 896/891


Mapping: {{mapping| 2 0 11 -15 -1 | 0 2 -4 13 5 }}
{{Mapping|legend=0| 2 0 11 -15 -1 | 0 2 -4 13 5 }}


Optimal tunings:
Optimal tunings:
Line 994: Line 1,049:
Comma list: 144/143, 176/175, 351/350, 676/675
Comma list: 144/143, 176/175, 351/350, 676/675


Mapping: {{mapping| 2 0 11 -15 -1 9 | 0 2 -4 13 5 -1 }}
{{Mapping|legend=0| 2 0 11 -15 -1 9 | 0 2 -4 13 5 -1 }}


Optimal tunings:
Optimal tunings:
Line 1,009: Line 1,064:
Comma list: 136/135, 144/143, 170/169, 176/175, 221/220
Comma list: 136/135, 144/143, 170/169, 176/175, 221/220


Mapping: {{mapping| 2 0 11 -15 -1 9 5 | 0 2 -4 13 5 -1 2 }}
{{Mapping|legend=0| 2 0 11 -15 -1 9 5 | 0 2 -4 13 5 -1 2 }}


Optimal tunings:
Optimal tunings:
Line 1,044: Line 1,099:
Comma list: 49/48, 56/55, 243/242
Comma list: 49/48, 56/55, 243/242


Mapping: {{mapping| 2 0 11 4 -1 | 0 2 -4 1 5 }}
{{Mapping|legend=0| 2 0 11 4 -1 | 0 2 -4 1 5 }}


Optimal tunings:
Optimal tunings:
Line 1,059: Line 1,114:
Comma list: 49/48, 56/55, 91/90, 243/242
Comma list: 49/48, 56/55, 91/90, 243/242


Mapping: {{mapping| 2 0 11 4 -1 9 | 0 2 -4 1 5 -1 }}
{{Mapping|legend=0| 2 0 11 4 -1 9 | 0 2 -4 1 5 -1 }}


Optimal tunings:
Optimal tunings:
Line 1,074: Line 1,129:
Comma list: 49/48, 56/55, 91/90, 119/117, 154/153
Comma list: 49/48, 56/55, 91/90, 119/117, 154/153


Mapping: {{mapping| 2 0 11 4 -1 9 5 | 0 2 -4 1 5 -1 2 }}
{{Mapping|legend=0| 2 0 11 4 -1 9 5 | 0 2 -4 1 5 -1 2 }}


Optimal tunings:
Optimal tunings:
Line 1,083: Line 1,138:


Badness (Sintel): 1.10
Badness (Sintel): 1.10
== Echidna ==
Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the {{nowrap| 22 & 58 }} temperament; its ploidacot is diploid alpha-tricot. [[58edo]] or [[80edo]] make for good tunings, or their vals can be added to {{val| 138 219 321 388 }} (138cde). In most of the tunings it has a significantly sharp 7/4 which some prefer.
Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 540/539 or 896/891 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-odd-limit diamond to within about six cents of error, within a compass of 24 notes. The 22-note 2mos gives scope for this, and the 36-note mos much more. Better yet, it is related to three important 11-limit edos: 22edo, a trivial tuning, is the smallest consistent in the 11-odd-limit, corresponding to the merge of this temperament with [[hedgehog]]; [[58edo]] is the smallest tuning that is distinctly consistent in the 11-odd-limit and [[80edo]] is the third smallest distinctly consistent in the 11-odd-limit.
The generator can be interpreted as 11/10, the period complement of 9/7, as a stack of 11/10 and 9/7 makes [[99/70]] which is extremely close to 600{{cent}} and is equal to it if we temper out [[9801/9800|S99]]. Three 11/10's then make a 4/3 (tempering out [[4000/3993|S10/S11]] thus making 10/9 and 12/11 equidistant from 11/10), implying a flat tuning of 4/3.
Like most srutal extensions, the 13- and 17-limit interpretations are possible by observing that since we have tempered out [[176/175]], tempering out [[351/350]] and [[352/351]] which sum to 176/175 is very elegant. In the 17-limit we can equate the half-octave with 17/12 and 24/17 and we can take advantage of the sharp fifth by combining echidna with [[srutal archagall]], leading to a particularly beautiful temperament (one that prefers a very slightly less sharp fifth than srutal archagall). This mapping of 13 and 17 is supported by the patent vals of the three main echidna edos of 22, 58 and 80, of which all except 22 are consistent in the [[17-odd-limit]].
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 1728/1715, 2048/2025
{{Mapping|legend=1| 2 1 9 2 | 0 3 -6 5 }}
: mapping generators: ~45/32, ~9/7
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.3056{{c}}, ~9/7 = 434.3524{{c}}
: [[error map]]: {{val| -1.389 +0.408 +1.322 +1.547 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8327{{c}}
: error map: {{val| 0.000 +2.543 +4.690 +5.338 }}
{{Optimal ET sequence|legend=1| 22, 58, 80, 138cd, 218cd }}
[[Badness]] (Sintel): 1.47
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 176/175, 540/539, 896/891
Mapping: {{mapping| 2 1 9 2 12 | 0 3 -6 5 -7 }}
Optimal tunings:
* WE: ~45/32 = 599.3085{{c}}, ~9/7 = 434.3511{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8647{{c}}
Minimax tuning:
* 11-odd-limit: ~9/7 = {{monzo| 5/12 0 0 1/12 -1/12 }}
: [{{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 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/7
{{Optimal ET sequence|legend=0| 22, 58, 80, 138cde, 218cde }}
Badness (Sintel): 0.859
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 351/350, 364/363, 540/539
Mapping: {{mapping| 2 1 9 2 12 19 | 0 3 -6 5 -7 -16 }}
Optimal tunings:
* WE: ~45/32 = 599.3397{{c}}, ~9/7 = 434.2772{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.7864{{c}}
{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}
Badness (Sintel): 0.978
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 136/135, 176/175, 221/220, 256/255, 540/539
Mapping: {{mapping| 2 1 9 2 12 19 6 | 0 3 -6 5 -7 -16 3 }}
Optimal tunings:
* WE: ~45/32 = 599.4645{{c}}, ~9/7 = 434.4282{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8340{{c}}
{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}
Badness (Sintel): 1.03


== Echidnic ==
== Echidnic ==
Line 1,185: Line 1,164:
Comma list: 385/384, 441/440, 686/675
Comma list: 385/384, 441/440, 686/675


Mapping: {{mapping| 2 2 7 6 3 | 0 3 -6 -1 10 }}
{{Mapping|legend=0| 2 2 7 6 3 | 0 3 -6 -1 10 }}


Optimal tunings:
Optimal tunings:
Line 1,200: Line 1,179:
Comma list: 91/90, 169/168, 385/384, 441/440
Comma list: 91/90, 169/168, 385/384, 441/440


Mapping: {{mapping| 2 2 7 6 3 7 | 0 3 -6 -1 10 1 }}
{{Mapping|legend=0| 2 2 7 6 3 7 | 0 3 -6 -1 10 1 }}


Optimal tunings:
Optimal tunings:
Line 1,215: Line 1,194:
Comma list: 91/90, 136/135, 154/153, 169/168, 256/255
Comma list: 91/90, 136/135, 154/153, 169/168, 256/255


Mapping: {{mapping| 2 2 7 6 3 7 7 | 0 3 -6 -1 10 1 3 }}
{{Mapping|legend=0| 2 2 7 6 3 7 7 | 0 3 -6 -1 10 1 3 }}


Optimal tunings:
Optimal tunings:
Line 1,254: Line 1,233:
Comma list: 176/175, 896/891, 2401/2400
Comma list: 176/175, 896/891, 2401/2400


Mapping: {{mapping| 2 0 11 8 22 | 0 4 -8 -3 -19 }}
{{Mapping|legend=0| 2 0 11 8 22 | 0 4 -8 -3 -19 }}


Optimal tunings:
Optimal tunings:
Line 1,269: Line 1,248:
Comma list: 176/175, 196/195, 512/507, 676/675
Comma list: 176/175, 196/195, 512/507, 676/675


Mapping: {{mapping| 2 0 11 8 22 9 | 0 4 -8 -3 -19 -2 }}
{{Mapping|legend=0| 2 0 11 8 22 9 | 0 4 -8 -3 -19 -2 }}


Optimal tunings:
Optimal tunings:
Line 1,284: Line 1,263:
Comma list: 136/135, 170/169, 176/175, 196/195, 256/255
Comma list: 136/135, 170/169, 176/175, 196/195, 256/255


Mapping: {{mapping| 2 0 11 8 22 9 5 | 0 4 -8 -3 -19 -2 4 }}
{{Mapping|legend=0| 2 0 11 8 22 9 5 | 0 4 -8 -3 -19 -2 4 }}


Optimal tunings:
Optimal tunings:
Line 1,299: Line 1,278:
Comma list: 243/242, 441/440, 2048/2025
Comma list: 243/242, 441/440, 2048/2025


Mapping: {{mapping| 2 0 11 8 -1 | 0 4 -8 -3 10 }}
{{Mapping|legend=0| 2 0 11 8 -1 | 0 4 -8 -3 10 }}


Optimal tunings:
Optimal tunings:
Line 1,314: Line 1,293:
Comma list: 144/143, 196/195, 243/242, 676/675
Comma list: 144/143, 196/195, 243/242, 676/675


Mapping: {{mapping| 2 0 11 8 -1 9 | 0 4 -8 -3 10 -2 }}
{{Mapping|legend=0| 2 0 11 8 -1 9 | 0 4 -8 -3 10 -2 }}


Optimal tunings:
Optimal tunings:
Line 1,329: Line 1,308:
Comma list: 136/135, 144/143, 170/169, 196/195, 221/220
Comma list: 136/135, 144/143, 170/169, 196/195, 221/220


Mapping: {{mapping| 2 0 11 8 -1 9 5 | 0 4 -8 -3 10 -2 4 }}
{{Mapping|legend=0| 2 0 11 8 -1 9 5 | 0 4 -8 -3 10 -2 4 }}


Optimal tunings:
Optimal tunings:
Line 1,342: Line 1,321:
=== Srutal archagall (2.3.5.17) ===
=== Srutal archagall (2.3.5.17) ===
{{See also | Fiventeen }}
{{See also | Fiventeen }}
This extension of 5-limit diaschismic adds prime 17 and which with respect to [[MVP archagall]] is able to express the harmonics [[75/1|75]] and [[85/1|85]] in their appropriate prime subgroup. It achieves this by equating [[85/64]] with [[4/3]] by tempering out their difference of [[256/255]] (S16). Therefore it also tempers out [[289/288]] (S17) and thus equates [[17/15]] with [[9/8]] due to tempering out [[136/135]] (S16⋅S17). It could be described as the 10 & 12 temperament with strong emphasis on 12edo being the better tuning on the 2.3.5.17 subgroup, implying ideal tunings of 34edo, 46edo or 80edo.


Subgroup: 2.3.5.17
Subgroup: 2.3.5.17
Line 1,347: Line 1,328:
Comma list: 136/135, 256/255
Comma list: 136/135, 256/255


Subgroup-val mapping: {{mapping| 2 0 11 5 | 0 1 -2 1 }}
{{Mapping|legend=2| 2 0 11 5 | 0 1 -2 1 }}
: mapping generators: ~17/12, ~3
: mapping generators: ~17/12, ~3


Line 1,358: Line 1,339:
Badness (Sintel): 0.212
Badness (Sintel): 0.212


[[Category:Diaschismic family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Diaschismic family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]
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