Dominant seventh sharp five chord - Revision history

Bcmills: more categories

2024-09-17T03:47:32Z

more categories

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	[[Category:Just intonation chords]]		[[Category:Just intonation chords]]
	[[Category:Dominant seventh sharp five chords\| ]]		[[Category:Dominant seventh sharp five chords\| ]]
			[[Category:Essentially tempered chords]]
			[[Category:9-odd-limit chords]]

Bcmills: unstub

2024-09-17T01:12:41Z

unstub

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	[[Category:Just intonation chords]]		[[Category:Just intonation chords]]
	[[Category:Dominant seventh sharp five chords\| ]]		[[Category:Dominant seventh sharp five chords\| ]]

	~~{{stub}}~~

Bcmills: /* In just intonation */

2024-09-17T01:12:05Z

In just intonation

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	* [[80:100:125:144]], found in genus 3<sup>2</sup>{{dot}}5<sup>3</sup>, is a 125-odd-limit preimage of the chord found in meantone.		* [[80:100:125:144]], found in genus 3<sup>2</sup>{{dot}}5<sup>3</sup>, is a 125-odd-limit preimage of the chord found in meantone.

	* [[144:180:225:256]], found on ♭III+ ({{Frac\|6\|5}}) and ♭VII+ ({{Frac\|9\|5}}) in the [[duodene]], is ~~also~~ a preimage of the chords found in ~~meantone,~~ archytas, and marvel temperaments.		* [[144:180:225:256]], found on ♭III+ ({{Frac\|6\|5}}) and ♭VII+ ({{Frac\|9\|5}}) in the [[duodene]], is a preimage of both the 5-limit meantone chord and the simpler 7-limit chords found in archytas and marvel temperaments.

	* [[180:225:288:320]], found on V+ ({{Frac\|3\|2}}) and II+ ({{Frac\|9\|8}}) in the [[duodene]], is tempered together with 144:180:225:256 in augmented temperament, substituting an 8/5 minor sixth in place of the augmented fifth.		* [[180:225:288:320]], found on V+ ({{Frac\|3\|2}}) and II+ ({{Frac\|9\|8}}) in the [[duodene]], is tempered together with 144:180:225:256 in augmented temperament, substituting an 8/5 minor sixth in place of the augmented fifth.

Bcmills: /* In just intonation */

2024-09-17T01:08:25Z

In just intonation

← Older revision		Revision as of 01:08, 17 September 2024
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	* [[144:180:225:256]], found on ♭III+ ({{Frac\|6\|5}}) and ♭VII+ ({{Frac\|9\|5}}) in the [[duodene]], is also a preimage of the chords found in meantone, archytas, and marvel temperaments.		* [[144:180:225:256]], found on ♭III+ ({{Frac\|6\|5}}) and ♭VII+ ({{Frac\|9\|5}}) in the [[duodene]], is also a preimage of the chords found in meantone, archytas, and marvel temperaments.

	* [[180:225:288:320]], found on V+ ({{Frac\|3\|2}}) and II+ ({{Frac\|9\|8}}) in the [[duodene]], is tempered together with 144:180:225:256, substituting an 8/5 minor sixth in place of the augmented fifth.		* [[180:225:288:320]], found on V+ ({{Frac\|3\|2}}) and II+ ({{Frac\|9\|8}}) in the [[duodene]], is tempered together with 144:180:225:256 in augmented temperament, substituting an 8/5 minor sixth in place of the augmented fifth.

	[[Category:Just intonation chords]]		[[Category:Just intonation chords]]

Bcmills: add several JI chords

2024-09-17T01:08:01Z

add several JI chords

← Older revision		Revision as of 01:08, 17 September 2024
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	== In just intonation ==		== In just intonation ==

			In the [[7-limit]]:

			* [[16:20:25:28]] is a [[condissonant]] chord found in [[Euler-Fokker genus\|genus]] 5<sup>2</sup>{{dot}}7. In archytas temperament, it is [[tempered together]] with 144:180:225:256.

			* [[36:45:56:64]] is a condissonant chord found in genus 3<sup>2</sup>{{dot}}5{{dot}}7. In marvel temperament, it is tempered together with 144:180:225:256.

			In the [[5-limit]]:

			* [[20:25:32:36]], found on I+ ({{Frac\|1\|1}}) and VI+ ({{Frac\|4\|3}}) in the [[duodene]], is a simple [[preimage]] of the 25-odd-limit chord found in augmented temperament, substituting an 8/5 minor sixth in place of the augmented fifth.

			* [[80:100:125:144]], found in genus 3<sup>2</sup>{{dot}}5<sup>3</sup>, is a 125-odd-limit preimage of the chord found in meantone.

			* [[144:180:225:256]], found on ♭III+ ({{Frac\|6\|5}}) and ♭VII+ ({{Frac\|9\|5}}) in the [[duodene]], is also a preimage of the chords found in meantone, archytas, and marvel temperaments.

			* [[180:225:288:320]], found on V+ ({{Frac\|3\|2}}) and II+ ({{Frac\|9\|8}}) in the [[duodene]], is tempered together with 144:180:225:256, substituting an 8/5 minor sixth in place of the augmented fifth.

	[[Category:Just intonation chords]]		[[Category:Just intonation chords]]

Bcmills: /* In temperaments */

2024-09-16T04:04:16Z

In temperaments

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	* (Meantone) 1/1 ‒ [[5/4]] ‒ [[25/16]] ‒ [[9/5]], with steps ~5/4, ~5/4, ~144/125.		* (Meantone) 1/1 ‒ [[5/4]] ‒ [[25/16]] ‒ [[9/5]], with steps ~5/4, ~5/4, ~144/125.

	In [[augmented (temperament)\|augmented temperament]], which tempers out [[128/125]], a ~[[25/16]] augmented fifth is equated with an ~[[8/5]] minor sixth. ~~However, the~~ chord is ~~still~~ [[25-odd-limit]] due to the [[36/25]]~[[45/32]] interval between the ~[[9/5]] minor seventh and ~[[5/4]] major third.		In [[augmented (temperament)\|augmented temperament]], which tempers out [[128/125]], a ~[[25/16]] augmented fifth is equated with an ~[[8/5]] minor sixth. The resulting chord is [[25-odd-limit]] due to the [[36/25]]~[[45/32]] interval between the ~[[9/5]] minor seventh and ~[[5/4]] major third.

	* (Augmented) 1/1 ‒ [[5/4]] ‒ [[8/5]] ‒ [[9/5]], with steps ~5/4, ~5/4, ~9/8.		* (Augmented) 1/1 ‒ [[5/4]] ‒ [[8/5]] ‒ [[9/5]], with steps ~5/4, ~5/4, ~9/8.

	In [[archytas]] temperament, which tempers out [[64/63]], ~16/9 is equated with ~[[7/4]], giving another 25-odd ~~essentially tempered~~ chord:		In [[archytas]] temperament, which tempers out [[64/63]], ~16/9 is equated with ~[[7/4]], giving another 25-odd chord:

	* (Archytas) 1/1 ‒ [[5/4]] ‒ [[25/16]] ‒ [[7/4]], with steps ~5/4, ~5/4, ~28/25.		* (Archytas) 1/1 ‒ [[5/4]] ‒ [[25/16]] ‒ [[7/4]], with steps ~5/4, ~5/4, ~28/25.

Bcmills: fix Archytas steps

2024-09-16T03:49:47Z

fix Archytas steps

← Older revision		Revision as of 03:49, 16 September 2024
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	In [[archytas]] temperament, which tempers out [[64/63]], ~16/9 is equated with ~[[7/4]], giving another 25-odd essentially tempered chord:		In [[archytas]] temperament, which tempers out [[64/63]], ~16/9 is equated with ~[[7/4]], giving another 25-odd essentially tempered chord:

	* (Archytas) 1/1 ‒ [[5/4]] ‒ [[25/16]] ‒ [[7/4]], with steps ~5/4, ~5/4, ~8/7.		* (Archytas) 1/1 ‒ [[5/4]] ‒ [[25/16]] ‒ [[7/4]], with steps ~5/4, ~5/4, ~28/25.

	In [[marvel]] temperament, which tempers out [[225/224]], ~25/16 is equated with ~[[14/9]], and the ~[[64/45]] tritone between a ~5/4 major third a ~[[16/9]] minor seventh is equated with ~[[10/7]], yielding a [[9-odd-limit]] [[essentially tempered chord]]:		In [[marvel]] temperament, which tempers out [[225/224]], ~25/16 is equated with ~[[14/9]], and the ~[[64/45]] tritone between a ~5/4 major third a ~[[16/9]] minor seventh is equated with ~[[10/7]], yielding a [[9-odd-limit]] [[essentially tempered chord]]:

Bcmills at 03:44, 16 September 2024

2024-09-16T03:44:37Z

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	* (Augmented) 1/1 ‒ [[5/4]] ‒ [[8/5]] ‒ [[9/5]], with steps ~5/4, ~5/4, ~9/8.		* (Augmented) 1/1 ‒ [[5/4]] ‒ [[8/5]] ‒ [[9/5]], with steps ~5/4, ~5/4, ~9/8.

			In [[archytas]] temperament, which tempers out [[64/63]], ~16/9 is equated with ~[[7/4]], giving another 25-odd essentially tempered chord:

			* (Archytas) 1/1 ‒ [[5/4]] ‒ [[25/16]] ‒ [[7/4]], with steps ~5/4, ~5/4, ~8/7.

	In [[marvel]] temperament, which tempers out [[225/224]], ~25/16 is equated with ~[[14/9]], and the ~[[64/45]] tritone between a ~5/4 major third a ~[[16/9]] minor seventh is equated with ~[[10/7]], yielding a [[9-odd-limit]] [[essentially tempered chord]]:		In [[marvel]] temperament, which tempers out [[225/224]], ~25/16 is equated with ~[[14/9]], and the ~[[64/45]] tritone between a ~5/4 major third a ~[[16/9]] minor seventh is equated with ~[[10/7]], yielding a [[9-odd-limit]] [[essentially tempered chord]]:
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	* (Marvel) 1/1 ‒ [[5/4]] ‒ [[14/9]] ‒ [[16/9]], with steps ~5/4, ~5/4, ~8/7.		* (Marvel) 1/1 ‒ [[5/4]] ‒ [[14/9]] ‒ [[16/9]], with steps ~5/4, ~5/4, ~8/7.

	[[12edo]] is a good tuning of augmented temperament, and [[septimal meantone]], which is well-represented by the historically prevalent [[quarter-comma meantone]], ~~tempers out the marvel comma;~~ so any of the above interpretations may be relevant for seventh sharp five chords found in contemporary and common-practice music.		[[12edo]] is a good tuning of augmented temperament and supports all of the above temperaments to varying degrees, and [[septimal meantone]] — which is well-represented by the historically prevalent [[quarter-comma meantone]] — supports marvel, so any of the above interpretations may be relevant for seventh sharp five chords found in contemporary and common-practice music.

	== In just intonation ==		== In just intonation ==

Bcmills: add conclusion to temperaments section

2024-09-16T01:23:54Z

add conclusion to temperaments section

← Older revision		Revision as of 01:23, 16 September 2024
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	* (Meantone) 1/1 ‒ [[5/4]] ‒ [[25/16]] ‒ [[9/5]], with steps ~5/4, ~5/4, ~144/125.		* (Meantone) 1/1 ‒ [[5/4]] ‒ [[25/16]] ‒ [[9/5]], with steps ~5/4, ~5/4, ~144/125.

	In [[augmented (temperament)\|augmented temperament]] ~~(including~~ [[~~12edo~~]]), a ~[[25/16]] augmented fifth is equated with an ~[[8/5]] minor sixth. However, the chord is still [[25-odd-limit]] due to the [[36/25]]~[[45/32]] interval between the ~[[9/5]] minor seventh and ~[[5/4]] major third.		In [[augmented (temperament)\|augmented temperament]], which tempers out [[128/125]], a ~[[25/16]] augmented fifth is equated with an ~[[8/5]] minor sixth. However, the chord is still [[25-odd-limit]] due to the [[36/25]]~[[45/32]] interval between the ~[[9/5]] minor seventh and ~[[5/4]] major third.

	* (Augmented) 1/1 ‒ [[5/4]] ‒ [[8/5]] ‒ [[9/5]], with steps ~5/4, ~5/4, ~9/8.		* (Augmented) 1/1 ‒ [[5/4]] ‒ [[8/5]] ‒ [[9/5]], with steps ~5/4, ~5/4, ~9/8.
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	* (Marvel) 1/1 ‒ [[5/4]] ‒ [[14/9]] ‒ [[16/9]], with steps ~5/4, ~5/4, ~8/7.		* (Marvel) 1/1 ‒ [[5/4]] ‒ [[14/9]] ‒ [[16/9]], with steps ~5/4, ~5/4, ~8/7.
	<!-- ~~TODO: starling~~, and~~/or archytas? -~~->
			[[12edo]] is a good tuning of augmented temperament, and [[septimal meantone]], which is well-represented by the historically prevalent [[quarter-comma meantone]], tempers out the marvel comma; so any of the above interpretations may be relevant for seventh sharp five chords found in contemporary and common-practice music.

	== In just intonation ==		== In just intonation ==

Bcmills at 00:46, 16 September 2024

2024-09-16T00:46:49Z

← Older revision		Revision as of 00:46, 16 September 2024
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	* (Augmented) 1/1 ‒ [[5/4]] ‒ [[8/5]] ‒ [[9/5]], with steps ~5/4, ~5/4, ~9/8.		* (Augmented) 1/1 ‒ [[5/4]] ‒ [[8/5]] ‒ [[9/5]], with steps ~5/4, ~5/4, ~9/8.

	In [[marvel]] temperament, which tempers out [[225/224]], ~25/16 is equated with ~[[14/9]], and the ~[[64/45]] tritone between ~~the~~ 5/4 major third ~~and~~ 16/9 minor seventh is equated with ~[[10/7]], yielding a [[9-odd-limit]] [[essentially tempered chord]]:		In [[marvel]] temperament, which tempers out [[225/224]], ~25/16 is equated with ~[[14/9]], and the ~[[64/45]] tritone between a ~5/4 major third a ~[[16/9]] minor seventh is equated with ~[[10/7]], yielding a [[9-odd-limit]] [[essentially tempered chord]]:

	* (Marvel) 1/1 ‒ [[5/4]] ‒ [[14/9]] ‒ [[16/9]], with steps ~5/4, ~5/4, ~8/7.		* (Marvel) 1/1 ‒ [[5/4]] ‒ [[14/9]] ‒ [[16/9]], with steps ~5/4, ~5/4, ~8/7.