Half-diminished seventh chord - Revision history

TallKite: /* In just intonation */ added 1/1 – 7/6 – 7/5 – 7/4

2024-09-27T23:42:44Z

In just intonation: added 1/1 – 7/6 – 7/5 – 7/4

← Older revision		Revision as of 23:42, 27 September 2024
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	In the [[7-limit]]:		In the [[7-limit]]:
	* [[5:6:7:9]] is a highly [[concordant]] 7-limit interpretation of the half-diminished seventh, found in the [[Ringer_scale\|Ringer 5]] scale and [[Euler-Fokker genus\|genus]] 3<sup>2</sup>{{dot}}5{{dot}}7, and tempered together with 45:54:64:81 in archytas temperament.		* [[5:6:7:9]] is a highly [[concordant]] 7-limit interpretation of the half-diminished seventh, found in the [[Ringer_scale\|Ringer 5]] scale and [[Euler-Fokker genus\|genus]] 3<sup>2</sup>{{dot}}5{{dot}}7, and tempered together with 45:54:64:81 in archytas temperament.
			* [[60:70:84:105]] (1/1 - 7/6 - 7/5 - 7/4) is a subharmonic chord.
	* [[20:24:28:35]] is a [[condissonant]] chord found in genus 3{{dot}}5{{dot}}7, and tempered together with 45:54:64:80 in archytas temperament.		* [[20:24:28:35]] is a [[condissonant]] chord found in genus 3{{dot}}5{{dot}}7, and tempered together with 45:54:64:80 in archytas temperament.
	* [[35:42:50:63]] is a [[condissonant]] chord found in genus 3<sup>2</sup>{{dot}}5<sup>2</sup>{{dot}}7, and tempered together with 25:30:36:45 in starling temperament.		* [[35:42:50:63]] is a [[condissonant]] chord found in genus 3<sup>2</sup>{{dot}}5<sup>2</sup>{{dot}}7, and tempered together with 25:30:36:45 in starling temperament.

Bcmills: /* In temperaments */ another -> a

2024-09-13T01:21:13Z

In temperaments: another -> a

← Older revision		Revision as of 01:21, 13 September 2024
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	[[Septimal meantone]], which is well-represented by the historically prevalent [[quarter-comma meantone]], tempers together all three of the tritones (10/7~36/25~64/45) and sevenths (9/5~16/9~25/14), so either of the above interpretations may be relevant for half-diminished seventh chords found in common-practice music. (→ [[Didymic chords #Dominant seventh chord]])		[[Septimal meantone]], which is well-represented by the historically prevalent [[quarter-comma meantone]], tempers together all three of the tritones (10/7~36/25~64/45) and sevenths (9/5~16/9~25/14), so either of the above interpretations may be relevant for half-diminished seventh chords found in common-practice music. (→ [[Didymic chords #Dominant seventh chord]])

	In [[archytas]] temperament, which tempers out [[64/63]], a ~[[16/9]] minor seventh is equated with ~[[7/4]], a ~64/45 diminished fifth is equated with ~[[7/5]], and the ~[[35/24]] ratio between ~7/4 and ~6/5 is equated with a ~[[3/2]] perfect fifth, yielding ~~another~~ 7-odd-limit essentially tempered chord:		In [[archytas]] temperament, which tempers out [[64/63]], a ~[[16/9]] minor seventh is equated with ~[[7/4]], a ~64/45 diminished fifth is equated with ~[[7/5]], and the ~[[35/24]] ratio between ~7/4 and ~6/5 is equated with a ~[[3/2]] perfect fifth, yielding a [[7-odd-limit]] essentially tempered chord:

	* (Archytas) 1/1 ‒ 6/5 ‒ 7/5 ‒ 7/4, with steps 6/5, 7/6, 5/4		* (Archytas) 1/1 ‒ 6/5 ‒ 7/5 ‒ 7/4, with steps 6/5, 7/6, 5/4

FloraC: Re-add septimal meantone

2024-09-12T15:34:26Z

Re-add septimal meantone

← Older revision		Revision as of 15:34, 12 September 2024
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	* (Starling) 1/1 ‒ [[6/5]] ‒ [[10/7]] ‒ [[9/5]], with steps 6/5, 6/5, 5/4		* (Starling) 1/1 ‒ [[6/5]] ‒ [[10/7]] ‒ [[9/5]], with steps 6/5, 6/5, 5/4

			[[Septimal meantone]], which is well-represented by the historically prevalent [[quarter-comma meantone]], tempers together all three of the tritones (10/7~36/25~64/45) and sevenths (9/5~16/9~25/14), so either of the above interpretations may be relevant for half-diminished seventh chords found in common-practice music. (→ [[Didymic chords #Dominant seventh chord]])

	In [[archytas]] temperament, which tempers out [[64/63]], a ~[[16/9]] minor seventh is equated with ~[[7/4]], a ~64/45 diminished fifth is equated with ~[[7/5]], and the ~[[35/24]] ratio between ~7/4 and ~6/5 is equated with a ~[[3/2]] perfect fifth, yielding another 7-odd-limit essentially tempered chord:		In [[archytas]] temperament, which tempers out [[64/63]], a ~[[16/9]] minor seventh is equated with ~[[7/4]], a ~64/45 diminished fifth is equated with ~[[7/5]], and the ~[[35/24]] ratio between ~7/4 and ~6/5 is equated with a ~[[3/2]] perfect fifth, yielding another 7-odd-limit essentially tempered chord:

FloraC: Partial undo

2024-09-12T15:23:22Z

Partial undo

← Older revision		Revision as of 15:23, 12 September 2024
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	* (Starling) 1/1 ‒ [[6/5]] ‒ [[10/7]] ‒ [[9/5]], with steps 6/5, 6/5, 5/4		* (Starling) 1/1 ‒ [[6/5]] ‒ [[10/7]] ‒ [[9/5]], with steps 6/5, 6/5, 5/4

	Similarly, in [[marvel]] temperament, which tempers out [[225/224]] and tempers together 16/9~25/14 (but ''not'' 9/5), the ~64/45 tritone is tempered together with ~10/7, again resulting in a 9-odd-limit essentially tempered chord:

	* (Marvel) 1/1 ‒ [[6/5]] ‒ [[10/7]] ‒ [[16/9]], with steps 6/5, 6/5, 5/4

	[[Septimal meantone]], which is well-represented by the historically prevalent [[quarter-comma meantone]]), tempers together all three of these sevenths (9/5~16/9~25/14), so any of the above interpretations may be relevant for half-diminished seventh chords found in common-practice music. (→ [[Didymic chords #Dominant seventh chord]])

	In [[archytas]] temperament, which tempers out [[64/63]], a ~[[16/9]] minor seventh is equated with ~[[7/4]], a ~64/45 diminished fifth is equated with ~[[7/5]], and the ~[[35/24]] ratio between ~7/4 and ~6/5 is equated with a ~[[3/2]] perfect fifth, yielding another 7-odd-limit essentially tempered chord:		In [[archytas]] temperament, which tempers out [[64/63]], a ~[[16/9]] minor seventh is equated with ~[[7/4]], a ~64/45 diminished fifth is equated with ~[[7/5]], and the ~[[35/24]] ratio between ~7/4 and ~6/5 is equated with a ~[[3/2]] perfect fifth, yielding another 7-odd-limit essentially tempered chord:

FloraC: Copypaste information from dominant seventh chord page. As this is the inverse of the dominant seventh chord, all that applies to the dominant seventh chord applies here too

2024-09-11T17:57:28Z

Copypaste information from dominant seventh chord page. As this is the inverse of the dominant seventh chord, all that applies to the dominant seventh chord applies here too

← Older revision		Revision as of 17:57, 11 September 2024
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	* (Meantone) 1/1 ‒ [[6/5]] ‒ [[36/25]] ‒ [[9/5]], with steps 6/5, 6/5, 5/4		* (Meantone) 1/1 ‒ [[6/5]] ‒ [[36/25]] ‒ [[9/5]], with steps 6/5, 6/5, 5/4

	In [[starling]] temperament, which [[tempering out\|tempers out]] [[126/125]], the ~36/25 diminished fifth is equated with ~[[10/7]], reducing the chord to the [[7-odd-limit]] as an [[essentially tempered chord]]:		In [[starling]] temperament, which [[tempering out\|tempers out]] [[126/125]], the ~36/25 diminished fifth is equated with ~[[10/7]], reducing the chord to the [[9-odd-limit]] as an [[essentially tempered chord]]:

	* (Starling) 1/1 ‒ [[6/5]] ‒ [[10/7]] ‒ [[9/5]], with steps 6/5, 6/5, 5/4		* (Starling) 1/1 ‒ [[6/5]] ‒ [[10/7]] ‒ [[9/5]], with steps 6/5, 6/5, 5/4

			Similarly, in [[marvel]] temperament, which tempers out [[225/224]] and tempers together 16/9~25/14 (but ''not'' 9/5), the ~64/45 tritone is tempered together with ~10/7, again resulting in a 9-odd-limit essentially tempered chord:

			* (Marvel) 1/1 ‒ [[6/5]] ‒ [[10/7]] ‒ [[16/9]], with steps 6/5, 6/5, 5/4

			[[Septimal meantone]], which is well-represented by the historically prevalent [[quarter-comma meantone]]), tempers together all three of these sevenths (9/5~16/9~25/14), so any of the above interpretations may be relevant for half-diminished seventh chords found in common-practice music. (→ [[Didymic chords #Dominant seventh chord]])

	In [[archytas]] temperament, which tempers out [[64/63]], a ~[[16/9]] minor seventh is equated with ~[[7/4]], a ~64/45 diminished fifth is equated with ~[[7/5]], and the ~[[35/24]] ratio between ~7/4 and ~6/5 is equated with a ~[[3/2]] perfect fifth, yielding another 7-odd-limit essentially tempered chord:		In [[archytas]] temperament, which tempers out [[64/63]], a ~[[16/9]] minor seventh is equated with ~[[7/4]], a ~64/45 diminished fifth is equated with ~[[7/5]], and the ~[[35/24]] ratio between ~7/4 and ~6/5 is equated with a ~[[3/2]] perfect fifth, yielding another 7-odd-limit essentially tempered chord:

Bcmills: /* In just intonation */

2024-09-11T03:08:59Z

In just intonation

← Older revision		Revision as of 03:08, 11 September 2024
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	In the [[7-limit]]:		In the [[7-limit]]:
	* [[5:6:7:9]] is a highly [[concordant]] 7-limit interpretation of the half-diminished seventh, found in the [[Ringer_scale\|Ringer 5]] scale and [[Euler-Fokker genus\|genus]]3<sup>2</sup>{{dot}}5{{dot}}7, and tempered together with 45:54:64:81 in archytas temperament.		* [[5:6:7:9]] is a highly [[concordant]] 7-limit interpretation of the half-diminished seventh, found in the [[Ringer_scale\|Ringer 5]] scale and [[Euler-Fokker genus\|genus]] 3<sup>2</sup>{{dot}}5{{dot}}7, and tempered together with 45:54:64:81 in archytas temperament.
	* [[20:24:28:35]] is a [[condissonant]] chord found in genus 3{{dot}}5{{dot}}7, and tempered together with 45:54:64:80 in archytas temperament.		* [[20:24:28:35]] is a [[condissonant]] chord found in genus 3{{dot}}5{{dot}}7, and tempered together with 45:54:64:80 in archytas temperament.
	* [[35:42:50:63]] is a [[condissonant]] chord found in genus 3<sup>2</sup>{{dot}}5<sup>2</sup>{{dot}}7, and tempered together with 25:30:36:45 in starling temperament.		* [[35:42:50:63]] is a [[condissonant]] chord found in genus 3<sup>2</sup>{{dot}}5<sup>2</sup>{{dot}}7, and tempered together with 25:30:36:45 in starling temperament.

Bcmills: fill in descriptions of JI interpretations

2024-09-11T03:08:43Z

fill in descriptions of JI interpretations

← Older revision		Revision as of 03:08, 11 September 2024
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	In the [[7-limit]]:		In the [[7-limit]]:
	* [[5:6:7:9]]		* [[5:6:7:9]] is a highly [[concordant]] 7-limit interpretation of the half-diminished seventh, found in the [[Ringer_scale\|Ringer 5]] scale and [[Euler-Fokker genus\|genus]]3<sup>2</sup>{{dot}}5{{dot}}7, and tempered together with 45:54:64:81 in archytas temperament.
	* [[20:24:28:35]]		* [[20:24:28:35]] is a [[condissonant]] chord found in genus 3{{dot}}5{{dot}}7, and tempered together with 45:54:64:80 in archytas temperament.
	* [[35:42:50:63]]		* [[35:42:50:63]] is a [[condissonant]] chord found in genus 3<sup>2</sup>{{dot}}5<sup>2</sup>{{dot}}7, and tempered together with 25:30:36:45 in starling temperament.

	In the [[5-limit]]:		In the [[5-limit]]:
	* [[25:30:36:45]]		* [[25:30:36:45]] is found on iii<sup>ø7</sup> ({{Frac\|5\|4}}) and vi<sup>ø7</sup> ({{Frac\|5\|3}}) in the [[duodene]].
	* [[45:54:64:80]]		* [[45:54:64:80]] found on vii<sup>ø7</sup> ({{Frac\|15\|8}}) in Ptolemy's intense diatonic scale ([[Zarlino]]), perhaps the most common 5-limit diatonic. It is also found on V<sup>ø7</sup> ({{Frac\|3\|2}}) in the [[duodene]].
	* [[45:54:64:81]], another [[preimage]] of the meantone half-diminished seventh chord, is found in [[Euler-Fokker genus\|genus]] 3<sup>4</sup>{{dot}}5, and tempered together with 5:6:7:9 in archytas temperament.		* [[45:54:64:81]], another [[preimage]] of the meantone half-diminished seventh chord, is found in [[Euler-Fokker genus\|genus]] 3<sup>4</sup>{{dot}}5, and tempered together with 5:6:7:9 in archytas temperament.
	* [[135:160:192:240]]		* [[135:160:192:240]] is found on ii<sup>ø7</sup> ({{Frac\|9\|8}}) and ♯iv<sup>ø7</sup> ({{Frac\|45\|32}}) in the [[duodene]].

	[[Category:Half-diminished seventh chords\| ]]		[[Category:Half-diminished seventh chords\| ]]
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	[[Category:9-odd-limit chords]]		[[Category:9-odd-limit chords]]
	[[Category:Essentially tempered chords]]		[[Category:Essentially tempered chords]]

	~~{{stub}}~~

Bcmills: add starling essentially-tempered chord; fix odd limit of archytas chord

2024-09-10T03:24:32Z

add starling essentially-tempered chord; fix odd limit of archytas chord

← Older revision		Revision as of 03:24, 10 September 2024
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	In meantone (including [[12edo]]), on which traditional tonal harmony is built, the interval of a diminished fifth [[tempering together\|tempers together]] [[36/25]]~[[64/45]]~[[1024/729]], and the minor seventh tempers together [[9/5]]~[[16/9]], giving a chord with an [[intervallic odd limit]] of 25:		In meantone (including [[12edo]]), on which traditional tonal harmony is built, the interval of a diminished fifth [[tempering together\|tempers together]] [[36/25]]~[[64/45]]~[[1024/729]], and the minor seventh tempers together [[9/5]]~[[16/9]], giving a chord with an [[intervallic odd limit]] of 25:

	* (Meantone) 1/1 ‒ [[6/5]] ‒ [[36/25]] ‒ [[9/5]]		* (Meantone) 1/1 ‒ [[6/5]] ‒ [[36/25]] ‒ [[9/5]], with steps 6/5, 6/5, 5/4

	In [[~~archytas~~]] temperament, which [[tempering out\|tempers out]] [[64/63]], a ~[[16/9]] minor seventh is equated with ~[[7/4]], a ~64/45 diminished fifth is equated with ~[[7/5]], and the ratio between ~7/4 and ~6/5 is equated with a ~[[3/2]] perfect fifth, yielding ~~a [[9~~-odd-limit~~]] [[~~essentially tempered chord]]:		In [[starling]] temperament, which [[tempering out\|tempers out]] [[126/125]], the ~36/25 diminished fifth is equated with ~[[10/7]], reducing the chord to the [[7-odd-limit]] as an [[essentially tempered chord]]:

			* (Starling) 1/1 ‒ [[6/5]] ‒ [[10/7]] ‒ [[9/5]], with steps 6/5, 6/5, 5/4

			In [[archytas]] temperament, which tempers out [[64/63]], a ~[[16/9]] minor seventh is equated with ~[[7/4]], a ~64/45 diminished fifth is equated with ~[[7/5]], and the ~[[35/24]] ratio between ~7/4 and ~6/5 is equated with a ~[[3/2]] perfect fifth, yielding another 7-odd-limit essentially tempered chord:

	* (Archytas) 1/1 ‒ 6/5 ‒ 7/5 ‒ 7/4, with steps 6/5, 7/6, 5/4		* (Archytas) 1/1 ‒ 6/5 ‒ 7/5 ‒ 7/4, with steps 6/5, 7/6, 5/4
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	* [[5:6:7:9]]		* [[5:6:7:9]]
	* [[20:24:28:35]]		* [[20:24:28:35]]
			* [[35:42:50:63]]

	In the [[5-limit]]:		In the [[5-limit]]:
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	[[Category:Half-diminished seventh chords\| ]]		[[Category:Half-diminished seventh chords\| ]]
	[[Category:Just intonation chords]]		[[Category:Just intonation chords]]
			[[Category:7-odd-limit chords]]
			[[Category:9-odd-limit chords]]
			[[Category:Essentially tempered chords]]

	{{stub}}		{{stub}}

Bcmills: add Archytas essentially-just chord

2024-09-10T03:08:14Z

add Archytas essentially-just chord

← Older revision		Revision as of 03:08, 10 September 2024
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	* (Archytas) 1/1 ‒ 6/5 ‒ 7/5 ‒ 7/4, with steps 6/5, 7/6, 5/4		* (Archytas) 1/1 ‒ 6/5 ‒ 7/5 ‒ 7/4, with steps 6/5, 7/6, 5/4

			Alternately, if we take the ~9/5 minor seventh instead, archytas gives a 9-odd-limit essentially just chord, tempering together [[5:6:7:9]] with [[45:54:64:81]]:

			* (Archytas) 1/1 ‒ 6/5 ‒ 7/5 ‒ 9/5, with steps 6/5, 7/6, 9/7

	== In just intonation ==		== In just intonation ==
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	* [[25:30:36:45]]		* [[25:30:36:45]]
	* [[45:54:64:80]]		* [[45:54:64:80]]
			* [[45:54:64:81]], another [[preimage]] of the meantone half-diminished seventh chord, is found in [[Euler-Fokker genus\|genus]] 3<sup>4</sup>{{dot}}5, and tempered together with 5:6:7:9 in archytas temperament.
	* [[135:160:192:240]]		* [[135:160:192:240]]

Bcmills: correct Archytas diminished fifth — it's based on 64/45, not 36/25.

2024-09-10T02:30:21Z

correct Archytas diminished fifth — it's based on 64/45, not 36/25.

← Older revision		Revision as of 02:30, 10 September 2024
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	* (Meantone) 1/1 ‒ [[6/5]] ‒ [[36/25]] ‒ [[9/5]]		* (Meantone) 1/1 ‒ [[6/5]] ‒ [[36/25]] ‒ [[9/5]]

	In [[archytas]] temperament, which [[tempering out\|tempers out]] [[64/63]], a ~[[16/9]] minor seventh is equated with ~[[7/4]], a ~36/25 diminished fifth is equated with ~[[7/5]], and the ratio between ~7/4 and ~6/5 is equated with a ~[[3/2]] perfect fifth, yielding a [[9-odd-limit]] [[essentially tempered chord]]:		In [[archytas]] temperament, which [[tempering out\|tempers out]] [[64/63]], a ~[[16/9]] minor seventh is equated with ~[[7/4]], a ~64/45 diminished fifth is equated with ~[[7/5]], and the ratio between ~7/4 and ~6/5 is equated with a ~[[3/2]] perfect fifth, yielding a [[9-odd-limit]] [[essentially tempered chord]]:

	* (Archytas) 1/1 ‒ 6/5 ‒ 7/5 ‒ 7/4, with steps 6/5, 6/5, 5/4		* (Archytas) 1/1 ‒ 6/5 ‒ 7/5 ‒ 7/4, with steps 6/5, 7/6, 5/4

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