Diaschismic extensions: Difference between revisions

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~~{{URWTC}}~~

In the [[5-limit]], '''diaschismic''' is a [[regular temperament]] (also known as ''srutal'', though they refer to different extensions in higher limits) defined by [[tempering out]] the comma [[2048/2025]] = [11 -4 -2⟩, the diaschisma. The octave is split into two periods, each representing [[~]][[45/32]]~[[64/45]]; and the [[generator]] can be considered to be a perfect fifth (~[[3/2]]), or a perfect fifth less a period, which is a diatonic semitone of ~[[16/15]]. Tempering out the diaschisma implies that two of these semitones are equated to [[9/8]], and as [[9/8]] = ([[18/17]])([[17/16]]), ~[[16/15]] can very naturally be equated to 17/16 and 18/17 as well, producing a 2.3.5.17 [[subgroup]] extension known as '''srutal archagall''', whose commas are [[136/135]] and [[256/255]]. There are multiple ways to extend diaschismic to primes [[7/1|7]], [[11/1|11]], and [[13/1|13]].

~~'''Diaschismic''' in~~ the [[5-limit]] is a [[regular temperament]] (also known as ''srutal'') defined by [[tempering out]] the comma [[2048/2025]] = [11 -4 -2⟩, the diaschisma. The octave is split into two periods, each representing [[~]][[45/32]]~[[64/45]]; and the [[generator]] can be considered to be a perfect fifth (~[[3/2]]), or a perfect fifth less a period, which is a diatonic semitone of ~[[16/15]]. Tempering out the diaschisma implies that two of these semitones are equated to [[9/8]], and ~~therefore~~ as [[9/8]] = ([[18/17]])([[17/16]]), ~[[16/15]] can very naturally be equated to 17/16 and 18/17 as well, producing a 2.3.5.17 [[subgroup]] extension known as '''srutal archagall''', whose commas are [[136/135]] and [[256/255]].

~~{{Tdlink~~|~~Diaschismic family #Diaschismic}}~~

== 7-limit extensions ==

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For technical data on 7-limit and higher-limit diaschismic: see [[Diaschismic family #Septimal diaschismic]].

Both of these can be extended straightforwardly to the [[11-limit|11-]], [[13-limit|13-]], and [[17-limit]] by adding 176/175, 352/351, and 221/220 to the comma list in this order.

Both of these can be extended straightforwardly to the [[11-limit|11-]], [[13-limit|13-]], and [[17-limit]] by adding 176/175, 352/351, and 221/220 to the comma list in this order. The extensions to prime [[11/1|11]] and [[13/1|13]] can be characterized by [[parapyth]], which makes sense as the fifth is tuned slightly sharp, and prime 17 is found via srutal archagall.

=== Pajara ===

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=== Keen ===

== Intervals ==

=== Diaschismic (2.3.5.7.17) ===

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 {{Breadcrumb|Diaschismic}}
-{{URWTC}}
+In the [[5-limit]], '''diaschismic''' is a [[regular temperament]] (also known as ''srutal'', though they refer to different extensions in higher limits) defined by [[tempering out]] the comma [[2048/2025]] = [11 -4 -2⟩, the diaschisma. The octave is split into two periods, each representing [[~]][[45/32]]~[[64/45]]; and the [[generator]] can be considered to be a perfect fifth (~[[3/2]]), or a perfect fifth less a period, which is a diatonic semitone of ~[[16/15]]. Tempering out the diaschisma implies that two of these semitones are equated to [[9/8]], and as [[9/8]] = ([[18/17]])([[17/16]]), ~[[16/15]] can very naturally be equated to 17/16 and 18/17 as well, producing a 2.3.5.17 [[subgroup]] extension known as '''srutal archagall''', whose commas are [[136/135]] and [[256/255]]. There are multiple ways to extend diaschismic to primes [[7/1|7]], [[11/1|11]], and [[13/1|13]].
-'''Diaschismic''' in the [[5-limit]] is a [[regular temperament]] (also known as ''srutal'') defined by [[tempering out]] the comma [[2048/2025]] = [11 -4 -2⟩, the diaschisma. The octave is split into two periods, each representing [[~]][[45/32]]~[[64/45]]; and the [[generator]] can be considered to be a perfect fifth (~[[3/2]]), or a perfect fifth less a period, which is a diatonic semitone of ~[[16/15]]. Tempering out the diaschisma implies that two of these semitones are equated to [[9/8]], and therefore as [[9/8]] = ([[18/17]])([[17/16]]), ~[[16/15]] can very naturally be equated to 17/16 and 18/17 as well, producing a 2.3.5.17 [[subgroup]] extension known as '''srutal archagall''', whose commas are [[136/135]] and [[256/255]].
-{{Tdlink|Diaschismic family #Diaschismic}}
 == 7-limit extensions ==
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 For technical data on 7-limit and higher-limit diaschismic: see [[Diaschismic family #Septimal diaschismic]].
-Both of these can be extended straightforwardly to the [[11-limit|11-]], [[13-limit|13-]], and [[17-limit]] by adding 176/175, 352/351, and 221/220 to the comma list in this order.
+Both of these can be extended straightforwardly to the [[11-limit|11-]], [[13-limit|13-]], and [[17-limit]] by adding 176/175, 352/351, and 221/220 to the comma list in this order. The extensions to prime [[11/1|11]] and [[13/1|13]] can be characterized by [[parapyth]], which makes sense as the fifth is tuned slightly sharp, and prime 17 is found via srutal archagall.
 === Pajara ===
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 === Keen ===
-{{todo|inline=1|expand}}
+{{todo|inline=1|complete section}}
 == Intervals ==
+{{todo|cleanup|complete section|inline=1}}
 === Diaschismic (2.3.5.7.17) ===
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