Diaschismic extensions: Difference between revisions

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~~'''Srutal''', known interchangeably as '''diaschismic''' in~~ the [[5-limit]], is a [[regular temperament]] 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]].

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]].

~~{{Tdlink~~|~~Diaschismic family #Srutal aka diaschismic}}~~

== 7-limit extensions ==

The two alternative names for this temperament are assigned to different strong extensions to the [[7-limit]]: srutal (34d&46) and diaschismic (46&58), though there are other mappings that are comparable in complexity and error: [[pajara]] (12&22) and keen (22&34).

Srutal tempers out [[4375/4374]] in addition to the diaschisma, and therefore [[7/4]] is represented by 15 semitones less a half octave, or five [[6/5]]s less a half octave. ~~Diaschismic sacrifices a slight amount of accuracy by tempering out [[126/125]], but slightly reduces complexity: [[8/~~7~~]] is represented by 8 semitones less a half~~-~~octave, or we can say 7/4 is equated to four~~ [[~~5/4~~]]~~s less a half octave~~.

=== Srutal ===

Srutal tempers out [[4375/4374]] in addition to the diaschisma, and therefore [[7/4]] is represented by 15 semitones less a half octave, or five [[6/5]]s less a half octave.

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

~~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~~.

=== Diaschismic ===

Diaschismic sacrifices a slight amount of accuracy by tempering out [[126/125]], but slightly reduces complexity: [[8/7]] is represented by 8 semitones less a half-octave, or we can say 7/4 is equated to four [[5/4]]s less a half octave.

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

For technical data on 7-limit and higher-limit ~~srutal:~~ see [[Diaschismic family #~~Srutal~~]].

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 ===

[[Pajara]] combines diaschismic with [[archy]], tempering the fifth to about 709 cents. The interval of two stacked fifths is equated to 16/7, and the harmonic seventh [[7/4]] and the just major third [[5/4]] are separated by a perfect semioctave.

For technical data on 7-limit and higher-limit pajara, see [[Diaschismic family #Pajara]].

=== Keen ===

== Intervals ==

=== Diaschismic (2.3.5.7.17) ===

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* [[Lumatone mapping for diaschismic]]

[[Category:~~Temperaments~~]]

[[Category:Temperament extensions]]

[[Category:~~Rank-2 temperaments]]~~

[[Category:Srutal]]

~~[[Category:Diaschismic family]]~~

[[Category:Diaschismic]]

~~[[Category:Srutal |~~Srutal ]]

[[Category:~~Diaschismic |~~Diaschismic ]]

@@ Line 1: / Line 1: @@
-{{URWTC}}
+{{Breadcrumb|Diaschismic}}
-'''Srutal''', known interchangeably as '''diaschismic''' in the [[5-limit]], is a [[regular temperament]] 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]].
+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]].
-{{Tdlink|Diaschismic family #Srutal aka diaschismic}}
 == 7-limit extensions ==
 The two alternative names for this temperament are assigned to different strong extensions to the [[7-limit]]: srutal (34d&amp;46) and diaschismic (46&amp;58), though there are other mappings that are comparable in complexity and error: [[pajara]] (12&amp;22) and keen (22&amp;34).
-Srutal tempers out [[4375/4374]] in addition to the diaschisma, and therefore [[7/4]] is represented by 15 semitones less a half octave, or five [[6/5]]s less a half octave. Diaschismic sacrifices a slight amount of accuracy by tempering out [[126/125]], but slightly reduces complexity: [[8/7]] is represented by 8 semitones less a half-octave, or we can say 7/4 is equated to four [[5/4]]s less a half octave.
+=== Srutal ===
+Srutal tempers out [[4375/4374]] in addition to the diaschisma, and therefore [[7/4]] is represented by 15 semitones less a half octave, or five [[6/5]]s less a half octave.
+For technical data on 7-limit and higher-limit srutal: see [[Diaschismic family #Srutal]].
-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.
+=== Diaschismic ===
+Diaschismic sacrifices a slight amount of accuracy by tempering out [[126/125]], but slightly reduces complexity: [[8/7]] is represented by 8 semitones less a half-octave, or we can say 7/4 is equated to four [[5/4]]s less a half octave.
 For technical data on 7-limit and higher-limit diaschismic: see [[Diaschismic family #Septimal diaschismic]].
-For technical data on 7-limit and higher-limit srutal: see [[Diaschismic family #Srutal]].
+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 ===
+[[Pajara]] combines diaschismic with [[archy]], tempering the fifth to about 709 cents. The interval of two stacked fifths is equated to 16/7, and the harmonic seventh [[7/4]] and the just major third [[5/4]] are separated by a perfect semioctave.
+For technical data on 7-limit and higher-limit pajara, see [[Diaschismic family #Pajara]].
+=== Keen ===
+{{todo|inline=1|complete section}}
 == Intervals ==
+{{todo|cleanup|complete section|inline=1}}
 === Diaschismic (2.3.5.7.17) ===
 <div><div style="display: inline-grid; margin-right: 25px;">
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 * [[Lumatone mapping for diaschismic]]
-[[Category:Temperaments]]
+[[Category:Temperament extensions]]
-[[Category:Rank-2 temperaments]]
+[[Category:Srutal]]
-[[Category:Diaschismic family]]
+[[Category:Diaschismic]]
-[[Category:Srutal |Srutal ]]
-[[Category:Diaschismic |Diaschismic ]]