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Diaschismic family: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
The [[5-limit]] parent [[comma]] for the '''diaschismic family''' of [[regular temperament|temperaments]] is 2048/2025, the [[diaschisma]]. The [[period]] is half an [[2/1|octave]], and the [[generator]] is a fifth. 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 '''diaschismic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the diaschisma, [[2048/2025]].  


== Diaschismic ==
== Diaschismic ==
{{Main| Diaschismic }}
{{Main| Diaschismic }}
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 12: Line 14:


{{Mapping|legend=1| 2 0 11 | 0 1 -2 }}
{{Mapping|legend=1| 2 0 11 | 0 1 -2 }}
: mapping generators: ~45/32, ~3
: mapping generators: ~45/32, ~3


Line 32: Line 33:
==== 7-limit extensions ====
==== 7-limit extensions ====
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.  
* 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 adds [[50/49]] or [[64/63]] and is a popular and well-known choice.  
* 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, which is about as accurate as septimal diaschismic but has a much more complex mapping of 7.  
* Keen adds [[875/864]].  
* Keen adds [[875/864]].  
* Bidia adds [[3136/3125]], the hemimean comma.
* Echidna adds [[1728/1715]], the orwellisma.
* Shrutar adds [[245/243]], the sensamagic comma.


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.
Those all keep the same half-octave period and fifth generator.  
 
==== 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'', documented right below. The [[S-expression]]-based comma list of this temperament is {[[256/255|S16]], [[289/288|S17]]}.  


=== Srutal archagall ===
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.
{{See also | Fiventeen }}


Subgroup: 2.3.5.17
Temperaments discussed elsewhere include [[stearnsmic clan #Echidna|echidna]]. The rest are considered below.  


Comma list: 136/135, 256/255
==== 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_2|#Subgroup extensions]]. The [[S-expression]]-based comma list of this temperament is {[[256/255|S16]], [[289/288|S17]]}.


Subgroup-val mapping: {{mapping| 2 0 11 5 | 0 1 -2 1 }}
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).
 
: mapping generators: ~17/12, ~3
 
Optimal tunings:
* WE: ~45/32 = 599.5585{{c}}, ~3/2 = 704.6188{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 705.1356{{c}}
 
{{Optimal ET sequence|legend=0| 10, 12, 22, 34, 80, 114, 194bc }}
 
Badness (Sintel): 0.212


== Septimal diaschismic ==
== Septimal diaschismic ==
Line 69: Line 53:
{{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 {{nowrap| 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.
A simpler characterization than the one given by the normal comma list is that septimal diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called {{nowrap| 46 & 58 }}. However described, septimal diaschismic has a 1/2-octave period and a sharp fifth generator like the 5-limit version, but not so sharp, giving a more accurate but more complex temperament. [[104edo]] with the 104c [[val]] provides an excellent tuning, which is close to tuning [[7/4]] just by making the fifth 703.897 cents.  


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.
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. This mapping can also be rationalized by [[parapyth]], which makes sense due to the sharp fifth, and prime 17 is found as in srutal archagall. If you want to explore higher-limit harmonies, diaschismic is certainly one excellent way to do it; [[mos]] scales 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
Line 98: 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 117: 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 137: 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 158: 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 171: Line 155:
{{Main| Pajara }}
{{Main| Pajara }}


Pajara is closely associated with 22edo (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2-octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22edo, 34 with the val {{val| 34 54 79 96 }} and 56 with the val {{val| 56 89 130 158 }} are interesting alternatives, with more acceptable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12edo and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.
Pajara is closely associated with 22edo (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2-octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22edo, 34 with the val {{val| 34 54 79 96 }} (34d) and 56 with the val {{val| 56 89 130 158 }} (56d) are interesting alternatives, with more acceptable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12edo and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.


Pajara extends nicely to an 11-limit version, for which the 56edo tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.
Pajara extends nicely to an 11-limit version, for which the 56edo tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.
Line 194: 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 200: 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 213: 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 219: 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 234: 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 249: 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 264: 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 279: 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 294: 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 309: 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 322: 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 328: 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 343: 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 358: 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 373: 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 388: 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 403: 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 418: 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 433: 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 450: 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 467: 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 482: 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 495: Line 523:
{{See also| Srutal vs diaschismic }}
{{See also| Srutal vs diaschismic }}


Srutal can be described as the 34d & 46 temperament, where 7/4 is located at 15 generator steps, or the double-augmented fifth (C–Gx). 80edo and 126edo are among the possible tunings. Srutal, shrutar and bidia have similar 19-limit properties, tempering out 190/189, related to rank-3 [[julius]].
Srutal can be described as the {{nowrap| 34d & 46 }} temperament, where 7/4 is located at 15 generator steps, or the double-augmented fifth (C–Gx). As such, it weakly extends [[leapfrog]]. 80edo and [[126edo]] are among the possible tunings. Srutal, shrutar and bidia have similar 19-limit properties, tempering out 190/189, related to rank-3 [[julius]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 522: 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 541: 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 561: 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 580: 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 597: 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 612: 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 627: 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 638: Line 666:


== Keen ==
== Keen ==
Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the {{nowrap| 22 & 56 }} temperament. [[78edo]] is a good tuning choice, and remains a good one in the 11-limit, where the temperament is really more interesting, adding 100/99 and 385/384 to the list of commas.
Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the {{nowrap| 22 & 34 }} temperament. [[78edo]] is a good tuning choice, and remains a good one in the 11-limit, where the temperament is really more interesting, adding 100/99 and 385/384 to the list of commas.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 661: 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 676: 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 691: 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 706: 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 721: 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 732: Line 760:


== Bidia ==
== Bidia ==
Bidia adds [[3136/3125]] to the commas, splitting the period into 1/4 octave. It may be called the {{nowrap| 12 & 56 }} temperament.
Bidia adds [[3136/3125]] to the commas, splitting the period into 1/4 octave. It may be called the {{nowrap| 12 & 68 }} temperament; its ploidacot is tetraploid monocot. Scales of bidia [[cluster temperament|cluster]] around [[12edo]], with a small residue left behind when three semitones exceed the quarter-octave period. This residue represents [[64/63]], and somewhat peculiarly, [[81/80]] is represented by ''two'' of these intervals.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 739: Line 767:


{{Mapping|legend=1| 4 0 22 43 | 0 1 -2 -5 }}
{{Mapping|legend=1| 4 0 22 43 | 0 1 -2 -5 }}
: mapping generators: ~25/21, ~3
: mapping generators: ~25/21, ~3


Line 757: 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:  
* WE: ~25/21 = 299.6809{{c}}, ~3/2 = 704.3367{{c}}
* WE: ~25/21 = 299.6809{{c}}, ~3/2 = 704.3367{{c}}
* CWE: ~25/21 = 600.0000{{c}}, ~3/2 = 705.2170{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 705.2170{{c}}


{{Optimal ET sequence|legend=0| 12, 56e, 68, 80 }}
{{Optimal ET sequence|legend=0| 12, 56e, 68, 80 }}
Line 772: 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:  
* WE: ~25/21 = 299.7538{{c}}, ~3/2 = 704.7222{{c}}
* WE: ~25/21 = 299.7538{{c}}, ~3/2 = 704.7222{{c}}
* CWE: ~25/21 = 600.0000{{c}}, ~3/2 = 705.3241{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 705.3241{{c}}


{{Optimal ET sequence|legend=0| 12, 68, 80, 148d, 228bcd, 376bbcddf }}
{{Optimal ET sequence|legend=0| 12, 68, 80, 148d, 228bcd, 376bbcddf }}
Line 787: 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:  
* WE: ~25/21 = 299.7883{{c}}, ~3/2 = 704.8365{{c}}
* WE: ~25/21 = 299.7883{{c}}, ~3/2 = 704.8365{{c}}
* CWE: ~25/21 = 600.0000{{c}}, ~3/2 = 705.3496{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 705.3496{{c}}


{{Optimal ET sequence|legend=0| 12, 68, 80, 148d }}
{{Optimal ET sequence|legend=0| 12, 68, 80, 148d }}
Line 802: 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:  
* WE: ~19/16 = 299.7967{{c}}, ~3/2 = 704.8609{{c}}
* WE: ~19/16 = 299.7967{{c}}, ~3/2 = 704.8609{{c}}
* CWE: ~19/16 = 600.0000{{c}}, ~3/2 = 705.3519{{c}}
* CWE: ~19/16 = 300.0000{{c}}, ~3/2 = 705.3519{{c}}


{{Optimal ET sequence|legend=0| 12, 68, 80, 148d }}
{{Optimal ET sequence|legend=0| 12, 68, 80, 148d }}
Line 817: 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 826: Line 853:


Badness (Sintel): 1.24
Badness (Sintel): 1.24
== Echidna ==
Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the {{nowrap| 22 & 58 }} temperament. [[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 ==
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 686/675, 1029/1024
{{Mapping|legend=1| 2 2 7 6 | 0 3 -6 -1 }}
: mapping generators: ~45/32, ~8/7
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.7208{{c}}, ~8/7 = 234.8330{{c}}
: [[error map]]: {{val| -0.558 +1.986 +2.733 -5.334 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~8/7 = 234.9539{{c}}
: error map: {{val| 0.000 +2.907 +3.963 -3.780 }}
{{Optimal ET sequence|legend=1| 10, 26c, 36, 46 }}
[[Badness]] (Sintel): 1.83
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 385/384, 441/440, 686/675
Mapping: {{mapping| 2 2 7 6 3 | 0 3 -6 -1 10 }}
Optimal tunings:
* WE: ~45/32 = 599.8022{{c}}, ~8/7 = 235.0185{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0893{{c}}
{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148 }}
Badness (Sintel): 1.49
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 169/168, 385/384, 441/440
Mapping: {{mapping| 2 2 7 6 3 7 | 0 3 -6 -1 10 1 }}
Optimal tunings:
* WE: ~45/32 = 599.9570{{c}}, ~8/7 = 235.0708{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0862{{c}}
{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}
Badness (Sintel): 1.19
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
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 }}
Optimal tunings:
* WE: ~17/12 = 599.9571{{c}}, ~8/7 = 235.0709{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~8/7 = 235.0860{{c}}
{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}
Badness (Sintel): 0.983
; Music
* [https://untwelve.org/competition/2011 ''A Stiff Shot of Turpentine''] [https://untwelve.org/static/audio/competition/2011/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3 play] by [[Peter Kosmorsky]]
* [https://www.youtube.com/watch?v=VsBXIvBZY6A ''56edo Track (Echidnic16 Scale)''] by [[Budjarn Lambeth]] (2025)


== Shrutar ==
== Shrutar ==
Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. It can also be described as {{nowrap| 22 & 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 excellent choice is a generator of 14<sup>(1/7)</sup>, making 7's just.
Shrutar adds 245/243 to the commas, and also tempers out [[6144/6125]]. It can also be described as {{nowrap| 22 & 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. Its ploidacot is diploid alpha-dicot. [[68edo]] makes for a good tuning, but another excellent choice is a generator of 14<sup>(1/7)</sup>, making 7's 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<sup>(1/7)</sup> 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.
Additionally, shrutar can employ the standard diaschismic mapping of prime 17, and most naturally represents the 2.3.5.7.11.17 subgroup temperament where 15:16:17:18 and 32:33:34:35:36 are equalized. Shrutar canonically maps primes 13, 19, and 23 as the 46 & 68 temperament; these mappings are significantly more complex and need finer tuning, however.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 982: Line 866:


{{Mapping|legend=1| 2 1 9 -2 | 0 2 -4 7 }}
{{Mapping|legend=1| 2 1 9 -2 | 0 2 -4 7 }}
: mapping generators: ~45/32, ~35/24
: mapping generators: ~45/32, ~35/24


Line 1,000: 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 1,011: 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 1,025: Line 910:
Badness (Sintel): 1.16
Badness (Sintel): 1.16


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


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 1,040: Line 925:
Badness (Sintel): 0.953
Badness (Sintel): 0.953


=== 19-limit ===
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


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 1,055: Line 940:
Badness (Sintel): 1.07
Badness (Sintel): 1.07


=== 23-limit ===
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13.17.19.23


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 1,069: Line 954:


Badness (Sintel): 1.03
Badness (Sintel): 1.03
== 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. 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
[[Comma list]]: 392/375, 1323/1280
{{Mapping|legend=1| 2 1 9 11 | 0 2 -4 -5 }}
: mapping generators: ~45/32, ~10/7
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 600.2519{{c}}, ~10/7 = 650.4083{{c}}
: [[error map]]: {{val| +0.504 -0.887 +14.321 -18.096 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~10/7 = 650.1017{{c}}
: error map: {{val| 0.000 -1.752 +13.279 -19.334 }}
{{Optimal ET sequence|legend=1| 2, 22d, 24 }}
[[Badness]] (Sintel): 3.99
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 56/55, 77/75, 1323/1280
{{Mapping|legend=0| 2 1 9 11 8 | 0 2 -4 -5 -1 }}
Optimal tunings:
* WE: ~17/12 = 600.2356{{c}}, ~10/7 = 650.3856{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~10/7 = 650.1008{{c}}
{{Optimal ET sequence|legend=0| 2, 22d, 24 }}
Badness (Sintel): 2.10
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 77/75, 105/104, 507/500
{{Mapping|legend=0| 2 1 9 11 8 15 | 0 2 -4 -5 -1 -7 }}
Optimal tunings:
* WE: ~45/32 = 599.9067{{c}}, ~10/7 = 649.4907{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~10/7 = 649.5950{{c}}
{{Optimal ET sequence|legend=0| 2, 24 }}
Badness (Sintel): 2.12


== 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.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 1,076: Line 1,017:


{{Mapping|legend=1| 2 0 11 -15 | 0 2 -4 13 }}
{{Mapping|legend=1| 2 0 11 -15 | 0 2 -4 13 }}
: mapping generators: ~45/32, ~140/81
: mapping generators: ~45/32, ~140/81


Line 1,094: 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 1,109: 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 1,124: 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 1,135: Line 1,075:


== Anguirus ==
== Anguirus ==
As another hemipyth temperament, anguirus tempers out 49/48. It can be described as the {{nowrap| 10 & 24 }} temperament; its ploidacot is diploid dicot, the same as sruti.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 1,140: Line 1,082:


{{Mapping|legend=1| 2 0 11 4 | 0 2 -4 1 }}
{{Mapping|legend=1| 2 0 11 4 | 0 2 -4 1 }}
: mapping generators: ~45/32, ~7/4
: mapping generators: ~45/32, ~7/4


Line 1,158: 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 1,173: 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 1,188: 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 1,198: Line 1,139:
Badness (Sintel): 1.10
Badness (Sintel): 1.10


== Shru ==
== Echidnic ==
Echidnic tempers out 686/675 and [[1029/1024]]. It has the same semi-octave period as diaschismic, but slices the generator of a fifth into three ~8/7's. It can be described as the {{nowrap| 10 & 46 }} temperament; its ploidacot is diploid tricot.
 
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 392/375, 1323/1280
[[Comma list]]: 686/675, 1029/1024


{{Mapping|legend=1| 2 1 9 11 | 0 2 -4 -5 }}
{{Mapping|legend=1| 2 2 7 6 | 0 3 -6 -1 }}
 
: mapping generators: ~45/32, ~8/7
: mapping generators: ~45/32, ~10/7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~45/32 = 600.2519{{c}}, ~10/7 = 650.4083{{c}}
* [[WE]]: ~45/32 = 599.7208{{c}}, ~8/7 = 234.8330{{c}}
: [[error map]]: {{val| +0.504 -0.887 +14.321 -18.096 }}
: [[error map]]: {{val| -0.558 +1.986 +2.733 -5.334 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~10/7 = 650.1017{{c}}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~8/7 = 234.9539{{c}}
: error map: {{val| 0.000 -1.752 +13.279 -19.334 }}
: error map: {{val| 0.000 +2.907 +3.963 -3.780 }}


{{Optimal ET sequence|legend=1| 2, 22d, 24 }}
{{Optimal ET sequence|legend=1| 10, 26c, 36, 46 }}


[[Badness]] (Sintel): 3.99
[[Badness]] (Sintel): 1.83


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


Comma list: 56/55, 77/75, 1323/1280
Comma list: 385/384, 441/440, 686/675


Mapping: {{mapping| 2 1 9 11 8 | 0 2 -4 -5 -1 }}
{{Mapping|legend=0| 2 2 7 6 3 | 0 3 -6 -1 10 }}


Optimal tunings:
Optimal tunings:
* WE: ~17/12 = 600.2356{{c}}, ~10/7 = 650.3856{{c}}
* WE: ~45/32 = 599.8022{{c}}, ~8/7 = 235.0185{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~10/7 = 650.1008{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0893{{c}}


{{Optimal ET sequence|legend=0| 2, 22d, 24 }}
{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148 }}


Badness (Sintel): 2.10
Badness (Sintel): 1.49


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


Comma list: 56/55, 77/75, 105/104, 507/500
Comma list: 91/90, 169/168, 385/384, 441/440
 
{{Mapping|legend=0| 2 2 7 6 3 7 | 0 3 -6 -1 10 1 }}
 
Optimal tunings:
* WE: ~45/32 = 599.9570{{c}}, ~8/7 = 235.0708{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0862{{c}}
 
{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}
 
Badness (Sintel): 1.19
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 91/90, 136/135, 154/153, 169/168, 256/255


Mapping: {{mapping| 2 1 9 11 8 15 | 0 2 -4 -5 -1 -7 }}
{{Mapping|legend=0| 2 2 7 6 3 7 7 | 0 3 -6 -1 10 1 3 }}


Optimal tunings:
Optimal tunings:
* WE: ~45/32 = 599.9067{{c}}, ~10/7 = 649.4907{{c}}
* WE: ~17/12 = 599.9571{{c}}, ~8/7 = 235.0709{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~10/7 = 649.5950{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~8/7 = 235.0860{{c}}
 
{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}


{{Optimal ET sequence|legend=0| 2, 24 }}
Badness (Sintel): 0.983


Badness (Sintel): 2.12
; Music
* [https://untwelve.org/competition/2011 ''A Stiff Shot of Turpentine''] [https://untwelve.org/static/audio/competition/2011/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3 play] by [[Peter Kosmorsky]]
* [https://www.youtube.com/watch?v=VsBXIvBZY6A ''56edo Track (Echidnic16 Scale)''] by [[Budjarn Lambeth]] (2025)


== Quadrasruta ==
== Quadrasruta ==
Named by [[Xenllium]] in 2022, quadrasruta tempers out 2401/2400, the breedsma, and extends [[buzzard]]. It may be described as {{nowrap| 58 & 68 }}; its ploidacot is diploid alpha-tetracot. 126edo may be recommended as a tuning.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 1,253: Line 1,216:


{{Mapping|legend=1| 2 0 11 8 | 0 4 -8 -3 }}
{{Mapping|legend=1| 2 0 11 8 | 0 4 -8 -3 }}
: mapping generators: ~45/32, ~21/16
: mapping generators: ~45/32, ~21/16


Line 1,271: 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 1,286: 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 1,301: 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 1,316: 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 1,331: 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 1,346: 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 1,356: Line 1,318:
Badness (Sintel): 1.13
Badness (Sintel): 1.13


== Subgroup extensions ==
=== Srutal archagall (2.3.5.17) ===
{{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
Comma list: 136/135, 256/255
{{Mapping|legend=2| 2 0 11 5 | 0 1 -2 1 }}
: mapping generators: ~17/12, ~3
Optimal tunings:
* WE: ~45/32 = 599.5585{{c}}, ~3/2 = 704.6188{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 705.1356{{c}}
{{Optimal ET sequence|legend=0| 10, 12, 22, 34, 80, 114, 194bc }}
Badness (Sintel): 0.212
[[Category:Diaschismic family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Diaschismic family| ]] <!-- main article -->
[[Category:Diaschismic| ]] <!-- key article -->
[[Category:Rank 2]]
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