German sixth chord - Revision history

Bcmills: clarify meantone vs. septimal meantone in chord summaries

2024-08-24T05:31:39Z

clarify meantone vs. septimal meantone in chord summaries

← Older revision		Revision as of 05:31, 24 August 2024
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	However, in the historically prevalent [[quarter-comma meantone]], the augmented sixth is tuned only a few cents shy of a just [[7/4]], so the meantone German sixth chord can be considered to approximate the ''harmonic seventh chord'' as in [[septimal meantone]]:		However, in the historically prevalent [[quarter-comma meantone]], the augmented sixth is tuned only a few cents shy of a just [[7/4]], so the meantone German sixth chord can be considered to approximate the ''harmonic seventh chord'' as in [[septimal meantone]]:

	* (~~Meantone~~) 1 – 5/4 – 3/2 – 7/4		* (Septimal meantone) 1 – 5/4 – 3/2 – 7/4

	representing [[4:5:6:7]] (with steps 5/4, 6/5, and 7/6).		representing [[4:5:6:7]] (with steps 5/4, 6/5, and 7/6).

Bcmills at 04:42, 20 August 2024

2024-08-20T04:42:17Z

← Older revision		Revision as of 04:42, 20 August 2024
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	simultaneously represents both [[72:90:108:125]] (with steps 5/4, 6/5, and [[125/108]]) and [[128:160:192:225]] (with steps 5/4, 6/5, and [[75/64]]).		simultaneously represents both [[72:90:108:125]] (with steps 5/4, 6/5, and [[125/108]]) and [[128:160:192:225]] (with steps 5/4, 6/5, and [[75/64]]).

	However, in the historically prevalent [[quarter-comma meantone]], the augmented ~~seventh~~ is tuned only a few cents shy of a just [[7/4]], so the meantone German sixth chord can be considered to approximate the ''harmonic seventh chord'' as in [[septimal meantone]]:		However, in the historically prevalent [[quarter-comma meantone]], the augmented sixth is tuned only a few cents shy of a just [[7/4]], so the meantone German sixth chord can be considered to approximate the ''harmonic seventh chord'' as in [[septimal meantone]]:

	* (Meantone) 1 – 5/4 – 3/2 – 7/4		* (Meantone) 1 – 5/4 – 3/2 – 7/4

Bcmills: factor out category

2024-08-20T04:22:58Z

factor out category

← Older revision		Revision as of 04:22, 20 August 2024
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	[[Category:Just intonation chords]]		[[Category:Just intonation chords]]
	[[Category:~~Tetrads~~]]		[[Category:German sixth chords\| ]] <!-- main article -->

Bcmills: mention both occurrences of the JI chord in the duodone

2024-08-20T04:17:14Z

mention both occurrences of the JI chord in the duodone

← Older revision		Revision as of 04:17, 20 August 2024
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	In the [[5-limit]]:		In the [[5-limit]]:

	* [[128:160:192:225]], one of the possible 5-limit interpretations of the meantone chord, is found rooted on ♭VI in the [[duodene]].		* [[128:160:192:225]], one of the possible 5-limit interpretations of the meantone chord, is found rooted at the ♭II ({{Frac\|16\|15}}) and ♭VI ({{Frac\|8\|5}}) of the [[duodene]].

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

Bcmills: link 7/4

2024-08-20T04:10:36Z

link 7/4

← Older revision		Revision as of 04:10, 20 August 2024
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	simultaneously represents both [[72:90:108:125]] (with steps 5/4, 6/5, and [[125/108]]) and [[128:160:192:225]] (with steps 5/4, 6/5, and [[75/64]]).		simultaneously represents both [[72:90:108:125]] (with steps 5/4, 6/5, and [[125/108]]) and [[128:160:192:225]] (with steps 5/4, 6/5, and [[75/64]]).

	However, in the historically prevalent [[quarter-comma meantone]], the augmented seventh is tuned only a few cents shy of a just 7/4, so the meantone German sixth chord can be considered to approximate the ''harmonic seventh chord'' as in [[septimal meantone]]:		However, in the historically prevalent [[quarter-comma meantone]], the augmented seventh is tuned only a few cents shy of a just [[7/4]], so the meantone German sixth chord can be considered to approximate the ''harmonic seventh chord'' as in [[septimal meantone]]:

	* (Meantone) 1 – 5/4 – 3/2 – 7/4		* (Meantone) 1 – 5/4 – 3/2 – 7/4

Bcmills: Streamline prose in meantone section; emphasize that the closeness to 7/4 is specifically in quarter-comma meantone, not “most meantones” in general (which would seem to imply 12edo and 19edo); link to chord pages for meantone alternatives (especially for infobox data); emphasize the connection between the duodene chord and the possible meantone interpretations.

2024-08-20T04:04:30Z

Streamline prose in meantone section; emphasize that the closeness to 7/4 is specifically in quarter-comma meantone, not “most meantones” in general (which would seem to imply 12edo and 19edo); link to chord pages for meantone alternatives (especially for infobox data); emphasize the connection between the duodene chord and the possible meantone interpretations.

← Older revision		Revision as of 04:04, 20 August 2024
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	== In meantone ==		== In meantone ==
	If we stick to the [[5-limit]], which traditional tonal harmony mainly concerns, the augmented sixth represents both [[125/72]] and [[225/128]]. It is treated as a rare and special dissonance, especially when used on the sixth degree. The chord is

	* (Meantone) 1 – 5/~~4 – 3/2 – ~~125/72 ~~with steps 5/4~~, 6/5, ~~75/64.~~		In 5-limit meantone, the traditional foundation of tonal harmony, the augmented sixth represents both [[225/128]] and [[125/72]]. It is treated as a rare and special dissonance, especially when used on the sixth degree. The 5-limit meantone German sixth chord,

	~~Note the ~125/72 is simultaneously ~225~~/~~128, and the ~[[75~~/~~64]] is simultaneously ~[[~~125/~~108]].~~		* (Meantone) 1 – 5/4 – 3/2 – 125/72

	~~However, most~~ [[~~meantone~~]] ~~tunings have the augmented sixth placed near~~ [[7/4]]~~. In the historically-prevalent~~ [[~~quarter-comma meantone~~]], ~~for example, it is tuned only a few cents shy of a just 7~~/4, ~~so the meantone German sixth chord can be considered to approximate the~~ [[~~harmonic seventh chord~~]]:		simultaneously represents both [[72:90:108:125]] (with steps 5/4, 6/5, and [[125/108]]) and [[128:160:192:225]] (with steps 5/4, 6/5, and [[75/64]]).

	* (Meantone) 1 – 5/4 – 3/2 – 7/4 with steps 5/4, 6/5, 7/6.		However, in the historically prevalent [[quarter-comma meantone]], the augmented seventh is tuned only a few cents shy of a just 7/4, so the meantone German sixth chord can be considered to approximate the ''harmonic seventh chord'' as in [[septimal meantone]]:

			* (Meantone) 1 – 5/4 – 3/2 – 7/4

			representing [[4:5:6:7]] (with steps 5/4, 6/5, and 7/6).
			<!-- Note: The close approximation of 7/4 is specific to quarter-comma meantone. Many other meantones, such as 12edo and 19edo, do not approximate it nearly as closely, although they may nonetheless support the septimal meantone interpretation that unifies it with the German sixth. -->

	== In just intonation ==		== In just intonation ==
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	In the [[5-limit]]:		In the [[5-limit]]:

	* ~~Because the German sixth chord is often built on the sixth degree of a minor scale, the~~ [[128:160:192:225]] found on ~~the~~ ♭VI of the [[duodene]] ~~is a good candidate for its interpretation~~.		* [[128:160:192:225]], one of the possible 5-limit interpretations of the meantone chord, is found rooted on ♭VI in the [[duodene]].

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

Bcmills at 21:37, 19 August 2024

2024-08-19T21:37:22Z

← Older revision		Revision as of 21:37, 19 August 2024
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	[[Category:Just intonation chords]]		[[Category:Just intonation chords]]
			[[Category:Tetrads]]

Bcmills: add to JI Chords category; adjust formatting for consistency with other pages in that category; add links to mentioned intervals

2024-08-19T20:51:30Z

add to JI Chords category; adjust formatting for consistency with other pages in that category; add links to mentioned intervals

← Older revision		Revision as of 20:51, 19 August 2024
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	* (Meantone) 1 – 5/4 – 3/2 – 125/72 with steps 5/4, 6/5, 75/64.		* (Meantone) 1 – 5/4 – 3/2 – 125/72 with steps 5/4, 6/5, 75/64.

	Note the ~125/72 is simultaneously ~225/128, and the ~75/64 is simultaneously ~125/108.		Note the ~125/72 is simultaneously ~225/128, and the ~[[75/64]] is simultaneously ~[[125/108]].

	However, most [[meantone]] tunings have the augmented sixth placed near [[7/4]]. In the historically-prevalent [[quarter-comma meantone]], for example, it is tuned only a few cents shy of a just 7/4, so the meantone German sixth chord can be considered to approximate the [[harmonic seventh chord]]:		However, most [[meantone]] tunings have the augmented sixth placed near [[7/4]]. In the historically-prevalent [[quarter-comma meantone]], for example, it is tuned only a few cents shy of a just 7/4, so the meantone German sixth chord can be considered to approximate the [[harmonic seventh chord]]:
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	== In just intonation ==		== In just intonation ==
	Because the German sixth chord is often built on the sixth degree of a minor scale, the [[128:160:192:225]] found on the ♭VI of the [[duodene]] is a good candidate for its interpretation ~~in the 5-limit~~.
			In the [[5-limit]]:

			* Because the German sixth chord is often built on the sixth degree of a minor scale, the [[128:160:192:225]] found on the ♭VI of the [[duodene]] is a good candidate for its interpretation.

			[[Category:Just intonation chords]]

Bcmills: /* In meantone */Typo fix (has -> have)

2024-08-18T02:26:21Z

In meantone: Typo fix (has -> have)

← Older revision		Revision as of 02:26, 18 August 2024
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	Note the ~125/72 is simultaneously ~225/128, and the ~75/64 is simultaneously ~125/108.		Note the ~125/72 is simultaneously ~225/128, and the ~75/64 is simultaneously ~125/108.

	However, most [[meantone]] tunings ~~has~~ the augmented sixth placed near [[7/4]]. In the historically-prevalent [[quarter-comma meantone]], for example, it is tuned only a few cents shy of a just 7/4, so the meantone German sixth chord can be considered to approximate the [[harmonic seventh chord]]:		However, most [[meantone]] tunings have the augmented sixth placed near [[7/4]]. In the historically-prevalent [[quarter-comma meantone]], for example, it is tuned only a few cents shy of a just 7/4, so the meantone German sixth chord can be considered to approximate the [[harmonic seventh chord]]:

	* (Meantone) 1 – 5/4 – 3/2 – 7/4 with steps 5/4, 6/5, 7/6.		* (Meantone) 1 – 5/4 – 3/2 – 7/4 with steps 5/4, 6/5, 7/6.

FloraC: /* In meantone */ 75/64~125/108 could be useful too

2024-08-16T16:03:55Z

In meantone: 75/64~125/108 could be useful too

← Older revision		Revision as of 16:03, 16 August 2024
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	If we stick to the [[5-limit]], which traditional tonal harmony mainly concerns, the augmented sixth represents both [[125/72]] and [[225/128]]. It is treated as a rare and special dissonance, especially when used on the sixth degree. The chord is		If we stick to the [[5-limit]], which traditional tonal harmony mainly concerns, the augmented sixth represents both [[125/72]] and [[225/128]]. It is treated as a rare and special dissonance, especially when used on the sixth degree. The chord is

	* (Meantone) 1 – 5/4 – 3/2 – 125/72 with steps 5/4, 6/5, 75/64		* (Meantone) 1 – 5/4 – 3/2 – 125/72 with steps 5/4, 6/5, 75/64.

	Note the ~125/72 is simultaneously ~225/128.		Note the ~125/72 is simultaneously ~225/128, and the ~75/64 is simultaneously ~125/108.

	However, most [[meantone]] tunings has the augmented sixth placed near [[7/4]]. In the historically-prevalent [[quarter-comma meantone]], for example, it is tuned only a few cents shy of a just 7/4, so the meantone German sixth chord can be considered to approximate the [[harmonic seventh chord]]:		However, most [[meantone]] tunings has the augmented sixth placed near [[7/4]]. In the historically-prevalent [[quarter-comma meantone]], for example, it is tuned only a few cents shy of a just 7/4, so the meantone German sixth chord can be considered to approximate the [[harmonic seventh chord]]: