Diaschismic: Difference between revisions

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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]]~~. We~~ also have a whole tone plus a period represent [[8/5]]~~, and~~ in ~~the canonical~~ [[~~extension~~]] to ~~the~~ [[7-~~limit~~]], ~~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~~]].

'''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'''.

A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)|diesis]]) 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]].

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.

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<nowiki/>* In 17-limit CWE tuning, octave-reduced

=== As a ~~detempereament~~ of 12et ===

=== As a detemperament of 12et ===

[[File: Diaschismic 12et Detempering.png|thumb|Diaschismic as a 58-tone 12et detempering]]

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The 13th harmonic is just beyond the specified generator range, so the diagram does not show it.

Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 46 generator steps, so it vanishes in 46edo, but is tuned to the same size as the syntonic comma in 58edo. 104edo tunes it to one half the size of the syntonic comma, which may be seen as a good compromise.

== Chords ==

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|-

! Edo<br>generator

! [[Eigenmonzo|~~Eigenmonzo~~<br>(~~unchanged-interval~~)]]*

! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*

! Generator (¢)

! Comments

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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]]. We also have a whole tone plus a period represent [[8/5]], and in the canonical [[extension]] to the [[7-limit]], 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]].
+'''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'''.
-A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)|diesis]]) 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]].
+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.
@@ Line 143: / Line 145: @@
 <nowiki/>* In 17-limit CWE tuning, octave-reduced
-=== As a detempereament of 12et ===
+=== As a detemperament of 12et ===
 [[File: Diaschismic 12et Detempering.png|thumb|Diaschismic as a 58-tone 12et detempering]]
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 The 13th harmonic is just beyond the specified generator range, so the diagram does not show it.
 Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 46 generator steps, so it vanishes in 46edo, but is tuned to the same size as the syntonic comma in 58edo. 104edo tunes it to one half the size of the syntonic comma, which may be seen as a good compromise.
 == Chords ==
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 |-
 ! Edo<br>generator
-! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
+! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
 ! Generator (¢)
 ! Comments