Bird's eye view of temperaments by accuracy: Difference between revisions
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== 11-limit focus == | == 11-limit focus == | ||
== ~17-limit focus == | == ~17-limit focus == | ||
=== [[Echidna]] === | |||
Note counts: | |||
* 24 for {3, 5, 7, 11, 15, 17, 33} ([[14L 8s]]) | |||
* 26 for adding {9} (14L 8s or 22L 14s) | |||
* 46 for adding {13} (22L 14s) | |||
Echidna has a generator of [[11/10]] or equivalently [[9/7]] because 11/10 * 9/7 = [[99/70]] is its period of half an octave, and can be seen as splitting the fourth of [[srutal archagall]] into three [[11/10]]'s by tempering [[4000/3993|S10/S11]] = (12/9)/(11/10)<sup>3</sup> = (4/3)/(11/10)<sup>3</sup> so that [[12/11]] and [[10/9]] are made equidistant from 11/10. It can be seen as a high-accuracy version of [[hedgehog]] and as a mild detempering of [[22edo]] that achieves an accurate and distinctly consistent [[11-odd-limit]]. In fact, of the three smallest edos that are distinctly consistent in the 11-odd-limit, which are [[58edo]], [[72edo]] and [[80edo]], echidna is supported by the smallest and third-smallest (so 72edo is in a sense the odd one out, being the one that ''doesn't'' support echidna). The smallest edo consistent in the 11-odd-limit, 22edo, is in fact a trivial tuning of echidna, where the generator is conflated with 12/11 and 10/9. 58edo and 80edo are both interesting options, being the merge of echidna and a variety of other notable temperaments, so depending on preference and tuning needs, though 80edo is the more optimal tuning for it (especially in the full 17-limit). | |||
Echidna is notable as achieving no-13's [[17-limit]] harmony with accuracy in a surprisingly small number of notes. [[13/8]] can be found too but is the most complex, being found at (11/10)<sup>16</sup> plus a half-octave period, octave-reduced. However, as primes 5 and 11 are also found in the same direction, intervals of 13 are common even in the 22-note MOS, so the 36-note MOS is more useful than might be suspected, despite not finding every odd from the same position. | |||
=== [[Diaschismic]] === | === [[Diaschismic]] === | ||
Note count: 36 for { | Note count: 36 for {3, 5, 7, 9, 11, 13, 15, 17, 21, 25, 33, 35, 39, 51} (12L 22s) | ||
Diaschismic is an extension of [[srutal archagall]] to the full [[17-limit]] of similar complexity to [[srutal]] (with which it merges in [[46edo]] so that they're complimentary) but which damages the 2.3.5.17 subgroup slightly more. Familiarizing oneself with the structure of srutal archagall is recommendable, even if the ideal tunings differ slightly, as navigation will be similar. | Diaschismic is an extension of [[srutal archagall]] to the full [[17-limit]] of similar complexity to [[srutal]] (with which it merges in [[46edo]] so that they're complimentary) but which damages the 2.3.5.17 subgroup slightly more. Familiarizing oneself with the structure of srutal archagall is recommendable, even if the ideal tunings differ slightly, as navigation will be similar. | ||