Diaschismic extensions - Revision history

Sintel: -TempClean

2026-05-28T14:41:53Z

-TempClean

← Older revision		Revision as of 14:41, 28 May 2026
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	{{Breadcrumb\|Diaschismic}}		{{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]].		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]].

Overthink: rework intro paragraph, note parapyth

2026-01-10T20:54:29Z

rework intro paragraph, note parapyth

← Older revision		Revision as of 20:54, 10 January 2026
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	{{Breadcrumb\|Diaschismic}}		{{Breadcrumb\|Diaschismic}}
	{{URWTC}}		{{URWTC}}
	~~'''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]].		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 #Diaschismic}}~~

	== 7-limit extensions ==		== 7-limit extensions ==
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	For technical data on 7-limit and higher-limit diaschismic: see [[Diaschismic family #Septimal diaschismic]].		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 ===		=== Pajara ===
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	== Intervals ==		== Intervals ==
			{{todo\|cleanup\|complete section\|inline=1}}
	=== Diaschismic (2.3.5.7.17) ===		=== Diaschismic (2.3.5.7.17) ===
	<div><div style="display: inline-grid; margin-right: 25px;">		<div><div style="display: inline-grid; margin-right: 25px;">

Overthink: fix todo

2025-12-27T06:59:08Z

fix todo

← Older revision		Revision as of 06:59, 27 December 2025
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	=== Keen ===		=== Keen ===
	{{todo\|inline=1\|~~expand~~}}		{{todo\|inline=1\|complete section}}

	== Intervals ==		== Intervals ==

Overthink: /* 7-limit extensions */ todo:expand on Keen

2025-12-02T07:20:17Z

7-limit extensions: todo:expand on Keen

← Older revision		Revision as of 07:20, 2 December 2025
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	=== Keen ===		=== Keen ===
			{{todo\|inline=1\|expand}}

	== Intervals ==		== Intervals ==

VectorGraphics at 06:26, 23 October 2025

2025-10-23T06:26:09Z

← Older revision		Revision as of 06:26, 23 October 2025
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	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).		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]].

			=== 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]].

	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.

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

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

	== Intervals ==		== Intervals ==

Overthink at 23:20, 22 October 2025

2025-10-22T23:20:05Z

← Older revision		Revision as of 23:20, 22 October 2025
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			{{Breadcrumb\|Diaschismic}}
	{{URWTC}}		{{URWTC}}
	~~{{Breadcrumb\|Diaschismic}}~~
	'''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]].		'''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]].

Overthink: + Breadcrumb

2025-10-22T23:19:40Z

+ Breadcrumb

← Older revision		Revision as of 23:19, 22 October 2025
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	{{URWTC}}		{{URWTC}}
			{{Breadcrumb\|Diaschismic}}
	'''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]].		'''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]].

VectorGraphics at 03:11, 9 September 2025

2025-09-09T03:11:11Z

← Older revision		Revision as of 03:11, 9 September 2025
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	'''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]].		'''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 #~~Srutal aka diaschismic~~}}		{{Tdlink\|Diaschismic family #Diaschismic}}

	== 7-limit extensions ==		== 7-limit extensions ==

VectorGraphics at 03:10, 9 September 2025

2025-09-09T03:10:46Z

← Older revision		Revision as of 03:10, 9 September 2025
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	{{URWTC}}		{{URWTC}}
	'''~~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]].		'''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 #Srutal aka diaschismic}}		{{Tdlink\|Diaschismic family #Srutal aka diaschismic}}

FloraC: Recategorize

2025-04-28T14:02:13Z

Recategorize

← Older revision		Revision as of 14:02, 28 April 2025
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	* [[Lumatone mapping for diaschismic]]		* [[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 ]]