Diaschismic: Difference between revisions

'''Diaschismic''', sometimes known as ''srutal'' in the [[5-limit]], is a half-octave [[regular temperament|temperament]] [[generator|generated]] by a [[3/2|perfect fifth]], ideally tuned slightly sharp, or that minus a half-octave [[period]], which is a semitone representing [[16/15]]. Two of these semitones give a whole tone of [[9/8]], so the diaschisma, [[2048/2025]] = 9/8 ÷ (16/15)², is [[tempering out|tempered out]], and a half-octave represents [[45/32]], and subtracting a whole tone from that gives [[5/4]]. Since the whole tone has been split in two equal halves of [[~]]16/15, it makes sense to equate that semitone with [[17/16]] and [[18/17]], by tempering out [[256/255]] ({{S|16}}), [[289/288]] ({{S|17}}), and their product [[136/135]], thus giving the [[2.3.5.17 subgroup|2.3.5.17-subgroup]] version of diaschismic, sometimes known as ''srutal archagall''.

The canonical [[extension]] to the [[7-limit]] reaches [[7/4]] at -8 fifths (a diminished fourth) [[octave-reduced]] plus a semioctave, tempering out the starling comma, [[126/125]], as well as the aberschisma, [[5120/5103]]. This equates the [[64/63|septimal comma]] with the [[81/80|syntonic comma]] and turns it into a generic comma step that can be used to bridge Pythagorean intervals with both classical and septimal intervals. This extension works best between [[46edo]] and [[58edo]].

In this tuning range, a stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)|diesis]]), or two comma steps, is close in size to a [[quartertone]], and that interval plus a half-octave can be used to represent [[16/11]]. Three more fifths on top of 16/11 gives [[16/13]] when octave-reduced, and thus [[13/11]] is equated with the Pythagorean minor third [[32/27]]. The mappings of primes [[11/1|11]] and [[13/1|13]] can also be characterized by [[parapyth]], where the major third at +4 fifths represents [[14/11]], and the minor third at -3 fifths represents [[13/11]], which makes sense as the fifth is tuned slightly sharp. Adding the mapping of 17/16 to the semitone, diaschismic is most naturally viewed as a full [[17-limit]] temperament, tempering out 126/125, 136/135, [[176/175]], [[196/195]], and 256/255. While this provides much more harmonic resources, the 2.3.5.17-subgroup version is simpler and has a more flexible tuning range.

An alternative extension to the full 17-limit is [[srutal]], which has a more complex mapping of prime 7 at +15 fifths, tempering out [[4375/4374]]. The mappings for primes 11 and 13 follow through parapyth, and 17 is again mapped to the semitone. This works best for a sharper tuning, between [[34edo]] (with the 34d [[val]]) and [[46edo]]. Another option for extension is [[pajara]], which equates the semioctave to [[7/5]] and [[10/7]]. This inherits the inaccuracy of [[archy]], while providing a much simpler (and arguably more elegant) representation of the 7-limit.