Diaschismic family: Difference between revisions

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==== 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]]}.

== Septimal diaschismic ==

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=== 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

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 ==== 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]]}.
 == Septimal diaschismic ==
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 === 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