Diaschismic family: Difference between revisions
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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. | |||
==== 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'', documented right below. 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'', documented right below. The [[S-expression]]-based comma list of this temperament is {[[256/255|S16]], [[289/288|S17]]}. | ||
=== Srutal archagall === | === Srutal archagall === | ||
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{{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 | 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]] 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. 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. | ||
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{{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. | ||
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{{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 | ||
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== 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 & | 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 | ||
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== Bidia == | == Bidia == | ||
Bidia adds [[3136/3125]] to the commas, splitting the period into 1/4 octave. It may be called the {{nowrap| 12 & | 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. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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Badness (Sintel): 1.24 | Badness (Sintel): 1.24 | ||
== 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. | ||
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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. | |||
[[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: {{mapping| 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: {{mapping| 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 == | ||
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 | ||
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== 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 | ||
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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 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]]: | [[Comma list]]: 686/675, 1029/1024 | ||
{{Mapping|legend=1| 2 | {{Mapping|legend=1| 2 2 7 6 | 0 3 -6 -1 }} | ||
: mapping generators: ~45/32, ~ | : mapping generators: ~45/32, ~8/7 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[WE]]: ~45/32 = | * [[WE]]: ~45/32 = 599.7208{{c}}, ~8/7 = 234.8330{{c}} | ||
: [[error map]]: {{val| | : [[error map]]: {{val| -0.558 +1.986 +2.733 -5.334 }} | ||
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~ | * [[CWE]]: ~45/32 = 600.0000{{c}}, ~8/7 = 234.9539{{c}} | ||
: error map: {{val| 0.000 | : error map: {{val| 0.000 +2.907 +3.963 -3.780 }} | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 10, 26c, 36, 46 }} | ||
[[Badness]] (Sintel): | [[Badness]] (Sintel): 1.83 | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 385/384, 441/440, 686/675 | ||
Mapping: {{mapping| 2 | Mapping: {{mapping| 2 2 7 6 3 | 0 3 -6 -1 10 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~45/32 = 599.8022{{c}}, ~8/7 = 235.0185{{c}} | ||
* CWE: ~ | * CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0893{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148 }} | ||
Badness (Sintel): | Badness (Sintel): 1.49 | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | 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 | Mapping: {{mapping| 2 2 7 6 3 7 7 | 0 3 -6 -1 10 1 3 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~17/12 = 599.9571{{c}}, ~8/7 = 235.0709{{c}} | ||
* CWE: ~ | * 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) | |||
== 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 | ||