Diaschismic family: Difference between revisions

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Line 40:

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.

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

==== 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 ==

Line 78:

Line 82:

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

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

Optimal tunings:

Line 97:

Line 101:

Comma list: 126/125, 176/175, 196/195, 364/363

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

Optimal tunings:

Line 117:

Line 121:

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 }}

Optimal tunings:

Line 138:

Line 142:

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:

Line 180:

Line 184:

Comma list: 50/49, 64/63, 85/84

~~Mapping:~~ {{~~mapping~~| 2 0 11 12 5 | 0 1 -2 -2 1 }}

Optimal tunings:

Line 195:

Line 199:

Comma list: 50/49, 64/63, 99/98

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

Optimal tunings:

Line 214:

Line 218:

Comma list: 50/49, 64/63, 85/84, 99/98

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

Optimal tunings:

Line 229:

Line 233:

Comma list: 50/49, 64/63, 65/63, 99/98

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

Optimal tunings:

Line 244:

Line 248:

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 }}

Optimal tunings:

Line 259:

Line 263:

Comma list: 50/49, 64/63, 78/77, 99/98

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

Optimal tunings:

Line 274:

Line 278:

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 }}

Optimal tunings:

Line 289:

Line 293:

Comma list: 40/39, 50/49, 64/63, 66/65

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

Optimal tunings:

Line 304:

Line 308:

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 }}

Optimal tunings:

Line 319:

Line 323:

Comma list: 50/49, 55/54, 64/63

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

Optimal tunings:

Line 338:

Line 342:

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 }}

Optimal tunings:

Line 353:

Line 357:

Comma list: 50/49, 55/54, 64/63, 65/63

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

Optimal tunings:

Line 368:

Line 372:

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 }}

Optimal tunings:

Line 383:

Line 387:

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

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

Optimal tunings:

Line 398:

Line 402:

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 }}

Optimal tunings:

Line 413:

Line 417:

Comma list: 45/44, 50/49, 56/55

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

Optimal tunings:

Line 428:

Line 432:

Comma list: 40/39, 45/44, 50/49, 56/55

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

Optimal tunings:

Line 443:

Line 447:

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 }}

Optimal tunings:

Line 458:

Line 462:

Comma list: 50/49, 64/63, 121/120

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

: mapping generators: ~2, ~16/11

Line 475:

Line 479:

Comma list: 50/49, 64/63, 243/242

~~Mapping:~~ {{~~mapping~~| 2 0 11 12 -1 | 0 2 -4 -4 5 }}

: mapping generators: ~2, ~55/32

Line 491:

Line 495:

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

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

Optimal tunings:

Line 506:

Line 510:

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 }}

Optimal tunings:

Line 546:

Line 550:

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

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

Optimal tunings:

Line 565:

Line 569:

Comma list: 169/168, 176/175, 325/324, 364/363

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

Optimal tunings:

Line 585:

Line 589:

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 }}

Optimal tunings:

Line 604:

Line 608:

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:

Line 621:

Line 625:

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:

Line 636:

Line 640:

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:

Line 651:

Line 655:

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:

Line 685:

Line 689:

Comma list: 100/99, 385/384, 1232/1215

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

Optimal tunings:

Line 700:

Line 704:

Comma list: 100/99, 105/104, 144/143, 1078/1053

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

Optimal tunings:

Line 715:

Line 719:

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}}

Optimal tunings:

Line 730:

Line 734:

Comma list: 91/90, 100/99, 352/351, 385/384

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

Optimal tunings:

Line 745:

Line 749:

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 }}

Optimal tunings:

Line 780:

Line 784:

Comma list: 176/175, 896/891, 1375/1372

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

Optimal tunings:

Line 795:

Line 799:

Comma list: 176/175, 325/324, 640/637, 896/891

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

Optimal tunings:

Line 810:

Line 814:

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 }}

Optimal tunings:

Line 825:

Line 829:

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:

Line 840:

Line 844:

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:

Line 879:

Line 883:

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

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

Optimal tunings:

Line 890:

Line 894:

=== 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

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

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

Optimal tunings:

Line 909:

Line 915:

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 }}

Optimal tunings:

Line 924:

Line 930:

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:

Line 939:

Line 945:

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:

Line 950:

Line 956:

== 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

Line 974:

Line 982:

Comma list: 56/55, 77/75, 1323/1280

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

Optimal tunings:

Line 989:

Line 997:

Comma list: 56/55, 77/75, 105/104, 507/500

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

Optimal tunings:

Line 1,000:

Line 1,008:

== Sruti ==

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 1,024:

Line 1,034:

Comma list: 176/175, 243/242, 896/891

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

Optimal tunings:

Line 1,039:

Line 1,049:

Comma list: 144/143, 176/175, 351/350, 676/675

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

Optimal tunings:

Line 1,054:

Line 1,064:

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 }}

Optimal tunings:

Line 1,089:

Line 1,099:

Comma list: 49/48, 56/55, 243/242

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

Optimal tunings:

Line 1,104:

Line 1,114:

Comma list: 49/48, 56/55, 91/90, 243/242

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

Optimal tunings:

Line 1,119:

Line 1,129:

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 }}

Optimal tunings:

Line 1,128:

Line 1,138:

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 ==

Line 1,230:

Line 1,164:

Comma list: 385/384, 441/440, 686/675

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

Optimal tunings:

Line 1,245:

Line 1,179:

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:

Line 1,260:

Line 1,194:

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:

Line 1,299:

Line 1,233:

Comma list: 176/175, 896/891, 2401/2400

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

Optimal tunings:

Line 1,314:

Line 1,248:

Comma list: 176/175, 196/195, 512/507, 676/675

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

Optimal tunings:

Line 1,329:

Line 1,263:

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 }}

Optimal tunings:

Line 1,344:

Line 1,278:

Comma list: 243/242, 441/440, 2048/2025

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

Optimal tunings:

Line 1,359:

Line 1,293:

Comma list: 144/143, 196/195, 243/242, 676/675

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

Optimal tunings:

Line 1,374:

Line 1,308:

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 }}

Optimal tunings:

Line 1,387:

Line 1,321:

=== Srutal archagall (2.3.5.17) ===

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

Line 1,392:

Line 1,328:

Comma list: 136/135, 256/255

~~Subgroup-val mapping:~~ {{~~mapping~~| 2 0 11 5 | 0 1 -2 1 }}

: mapping generators: ~17/12, ~3

Line 1,403:

Line 1,339:

Badness (Sintel): 0.212

[[Category:Diaschismic family| ]]

[[Category:Temperament families]]

[[Category:~~Diaschismic family| ]] <!~~-~~- main article -->~~

[[Category:Catalogs of rank-2 temperaments]]

~~[[Category:Rank~~ 2]]

@@ Line 40: / Line 40: @@
 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 ====
-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 ==
@@ Line 78: / Line 82: @@
 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:
@@ Line 97: / Line 101: @@
 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:
@@ Line 117: / Line 121: @@
 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:
@@ Line 138: / Line 142: @@
 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:
@@ Line 180: / Line 184: @@
 Comma list: 50/49, 64/63, 85/84
-Mapping: {{mapping| 2 0 11 12 5 | 0 1 -2 -2 1 }}
+{{Mapping|legend=0| 2 0 11 12 5 | 0 1 -2 -2 1 }}
 Optimal tunings:
@@ Line 195: / Line 199: @@
 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:
@@ Line 214: / Line 218: @@
 Comma list: 50/49, 64/63, 85/84, 99/98
-Mapping: {{mapping| 2 0 11 12 26 5 | 0 1 -2 -2 -6 1 }}
+{{Mapping|legend=0| 2 0 11 12 26 5 | 0 1 -2 -2 -6 1 }}
 Optimal tunings:
@@ Line 229: / Line 233: @@
 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:
@@ Line 244: / Line 248: @@
 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:
@@ Line 259: / Line 263: @@
 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:
@@ Line 274: / Line 278: @@
 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:
@@ Line 289: / Line 293: @@
 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:
@@ Line 304: / Line 308: @@
 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:
@@ Line 319: / Line 323: @@
 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:
@@ Line 338: / Line 342: @@
 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:
@@ Line 353: / Line 357: @@
 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:
@@ Line 368: / Line 372: @@
 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:
@@ Line 383: / Line 387: @@
 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:
@@ Line 398: / Line 402: @@
 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:
@@ Line 413: / Line 417: @@
 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:
@@ Line 428: / Line 432: @@
 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:
@@ Line 443: / Line 447: @@
 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:
@@ Line 458: / Line 462: @@
 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
@@ Line 475: / Line 479: @@
 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
@@ Line 491: / Line 495: @@
 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:
@@ Line 506: / Line 510: @@
 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:
@@ Line 546: / Line 550: @@
 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:
@@ Line 565: / Line 569: @@
 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:
@@ Line 585: / Line 589: @@
 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:
@@ Line 604: / Line 608: @@
 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:
@@ Line 621: / Line 625: @@
 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:
@@ Line 636: / Line 640: @@
 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:
@@ Line 651: / Line 655: @@
 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:
@@ Line 685: / Line 689: @@
 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:
@@ Line 700: / Line 704: @@
 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:
@@ Line 715: / Line 719: @@
 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:
@@ Line 730: / Line 734: @@
 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:
@@ Line 745: / Line 749: @@
 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:
@@ Line 780: / Line 784: @@
 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:
@@ Line 795: / Line 799: @@
 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:
@@ Line 810: / Line 814: @@
 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:
@@ Line 825: / Line 829: @@
 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:
@@ Line 840: / Line 844: @@
 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:
@@ Line 879: / Line 883: @@
 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:
@@ Line 890: / Line 894: @@
 === 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
 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:
@@ Line 909: / Line 915: @@
 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:
@@ Line 924: / Line 930: @@
 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:
@@ Line 939: / Line 945: @@
 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:
@@ Line 950: / Line 956: @@
 == 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
@@ Line 974: / Line 982: @@
 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:
@@ Line 989: / Line 997: @@
 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:
@@ Line 1,000: / Line 1,008: @@
 == 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.
@@ Line 1,024: / Line 1,034: @@
 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:
@@ Line 1,039: / Line 1,049: @@
 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:
@@ Line 1,054: / Line 1,064: @@
 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:
@@ Line 1,089: / Line 1,099: @@
 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:
@@ Line 1,104: / Line 1,114: @@
 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:
@@ Line 1,119: / Line 1,129: @@
 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:
@@ Line 1,128: / Line 1,138: @@
 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 ==
@@ Line 1,230: / Line 1,164: @@
 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:
@@ Line 1,245: / Line 1,179: @@
 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:
@@ Line 1,260: / Line 1,194: @@
 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:
@@ Line 1,299: / Line 1,233: @@
 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:
@@ Line 1,314: / Line 1,248: @@
 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:
@@ Line 1,329: / Line 1,263: @@
 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:
@@ Line 1,344: / Line 1,278: @@
 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:
@@ Line 1,359: / Line 1,293: @@
 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:
@@ Line 1,374: / Line 1,308: @@
 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:
@@ Line 1,387: / Line 1,321: @@
 === 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
@@ Line 1,392: / Line 1,328: @@
 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
@@ Line 1,403: / Line 1,339: @@
 Badness (Sintel): 0.212
+[[Category:Diaschismic family| ]] <!-- main article -->
 [[Category:Temperament families]]
-[[Category:Diaschismic family| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]