Diaschismic: Difference between revisions
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There's only one canonical extension. Also mention the hemifamity comma |
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'''Diaschismic''', sometimes known as [[srutal vs diaschismic|srutal]] in the [[5-limit]], is a half-octave [[regular temperament|temperament]] [[generator|generated]] by a [[3/2|perfect fifth]] 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]], is [[tempering out|tempered out]], and we also have a whole tone plus a period represent [[8/5]]. 9/8 splits in two very naturally into [[17/16]] × [[18/17]], and since we are equating half 9/8 to 16/15, it makes good sense to equate that interval to 17/16 and 18/17 as well, by tempering out [[S-expression|S16]] = [[256/255]], S17 = [[289/288]], and their product [[136/135]], leading to a 2.3.5.17 [[subgroup]] extension called '''srutal archagall'''. | '''Diaschismic''', sometimes known as [[srutal vs diaschismic|srutal]] in the [[5-limit]], is a half-octave [[regular temperament|temperament]] [[generator|generated]] by a [[3/2|perfect fifth]] 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]], is [[tempering out|tempered out]], and we also have a whole tone plus a period represent [[8/5]]. 9/8 splits in two very naturally into [[17/16]] × [[18/17]], and since we are equating half 9/8 to 16/15, it makes good sense to equate that interval to 17/16 and 18/17 as well, by tempering out [[S-expression|S16]] = [[256/255]], S17 = [[289/288]], and their product [[136/135]], leading to a 2.3.5.17 [[subgroup]] extension called '''srutal archagall'''. | ||
The canonical [[extension]] to the [[7-limit]] lies where the fifth is tuned a little sharp such that eight of them octave reduced (an augmented fifth) minus a period approximate [[8/7]], tempering out the starling comma, [[126/125]], as well as the hemifamity comma, [[5120/5103]]. | |||
A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)|diesis]]), in this tuning range, is close in size to a quartertone, and that plus a period can be used to represent [[16/11]]. Three more fifths on top of 16/11 give [[16/13]]. Finally, since the whole tone has been split in two, each can be used to represent [[17/16]]~[[18/17]]. Therefore, diaschismic is most naturally viewed as a full 17-limit temperament, tempering out 126/125, 136/135, [[176/175]], [[196/195]], and 256/255. | |||
See [[Diaschismic family #Diaschismic]] and [[Diaschismic family #Septimal diaschismic]] for technical data. | See [[Diaschismic family #Diaschismic]] and [[Diaschismic family #Septimal diaschismic]] for technical data. | ||