Diaschismic family: Difference between revisions

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.

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.

=== No-19s 23-limit (Na"Naa') ===

<b>Na"Naa'</b> is a remarkable subgroup temperament of {{nowrap| 46 & 58 }} with a prime harmonic of 23.

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 keen, {{multival| 2 -4 18 -12 … }}, is really more interesting, adding 100/99 and 385/384 to the commas.