Augmented triad - Revision history

MisterShafXen at 11:04, 19 April 2025

2025-04-19T11:04:30Z

← Older revision		Revision as of 11:04, 19 April 2025
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	* (Augmented) 1 – 5/4 – 8/5		* (Augmented) 1 – 5/4 – 8/5

	The gap remaining to close the octave is another identical ~~fifth~~ (~5/4), giving rise to the temperament's name after this chord.		The gap remaining to close the octave is another identical third (~5/4), giving rise to the temperament's name after this chord.

	If [[225/224]] is tempered out, as in [[marvel]] and [[septimal meantone]], the ~25/16 augmented fifth is tempered together with ~[[14/9]]. The resulting intervals are ~5/4, ~5/4, and ~14/9, making the augmented triad a [[9-odd-limit]] essentially tempered chord (→ [[Marvel chords]]).		If [[225/224]] is tempered out, as in [[marvel]] and [[septimal meantone]], the ~25/16 augmented fifth is tempered together with ~[[14/9]]. The resulting intervals are ~5/4, ~5/4, and ~14/9, making the augmented triad a [[9-odd-limit]] essentially tempered chord (→ [[Marvel chords]]).

TallKite: /* In just intonation */ added ratios to the yaza chords

2024-09-24T21:35:35Z

In just intonation: added ratios to the yaza chords

← Older revision		Revision as of 21:35, 24 September 2024
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	In the [[7-limit]]:		In the [[7-limit]]:
	* [[36:45:56]] is the simplest JI tuning of the 7-limit marvel augmented chord.		* [[36:45:56]] (1/1 – 5/4 – 14/9) is the simplest JI tuning of the 7-limit marvel augmented chord.
	* [[35:45:56]]		* [[35:45:56]] (1/1 –  9/7 – 8/5)
	* [[28:35:45]]		* [[28:35:45]] (1/1 – 5/4 – [[45/28]])
	* [[28:36:45]]		* [[28:36:45]] (1/1 – 9/7 – [[45/28]])

	In the [[5-limit]]:		In the [[5-limit]]:

TallKite: /* In just intonation */ added more za chords

2024-09-24T21:30:30Z

In just intonation: added more za chords

← Older revision		Revision as of 21:30, 24 September 2024
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	In the [[7-limit]]:		In the [[7-limit]]:
	* [[36:45:56]] is the simplest JI tuning of the 7-limit marvel augmented chord.		* [[36:45:56]] is the simplest JI tuning of the 7-limit marvel augmented chord.
			* [[35:45:56]]
			* [[28:35:45]]
			* [[28:36:45]]

	In the [[5-limit]]:		In the [[5-limit]]:

Bcmills: The → A, since there are multiple augmented triads

2024-09-07T21:58:07Z

The → A, since there are multiple augmented triads

← Older revision		Revision as of 21:58, 7 September 2024
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	{{Wikipedia\|Augmented triad}}		{{Wikipedia\|Augmented triad}}

	~~The~~ '''augmented triad''' is a [[triad]] comprising a root, a [[major]] third, and an [[interval quality\|augmented]] fifth formed by stacking another major third.		An '''augmented triad''' is a [[triad]] comprising a root, a [[major]] third, and an [[interval quality\|augmented]] fifth formed by stacking another major third.

	== In temperaments ==		== In temperaments ==

Bcmills: Mention the comma earlier in the paragraph about augmented temperament. This makes the paragraph more parallel to the septimal one, and also allows the reader to more easily understand the paragraph without already knowing what "augmented temperament" is.

2024-08-24T03:38:55Z

Mention the comma earlier in the paragraph about augmented temperament. This makes the paragraph more parallel to the septimal one, and also allows the reader to more easily understand the paragraph without already knowing what "augmented temperament" is.

← Older revision		Revision as of 03:38, 24 August 2024
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	== In temperaments ==		== In temperaments ==

	In [[augmented (temperament)\|augmented temperament]], the ~[[25/16]] augmented fifth is [[enharmonic equivalence\|enharmonically equivalent]] to a minor sixth (~[[8/5]])~~, the difference of [[128/125]] being [[tempering out\|tempered out]]~~. Since the resulting internal intervals are ~5/4, ~5/4, and ~8/5, the augmented triad in augmented temperament is a [[5-odd-limit]] [[essentially tempered chord]]:		In [[augmented (temperament)\|augmented temperament]], which [[tempering out\|tempers out]] [[128/125]], the ~[[25/16]] augmented fifth is [[enharmonic equivalence\|enharmonically equivalent]] to a minor sixth (~[[8/5]]). Since the resulting internal intervals are ~5/4, ~5/4, and ~8/5, the augmented triad in augmented temperament is a [[5-odd-limit]] [[essentially tempered chord]]:

	* (Augmented) 1 – 5/4 – 8/5		* (Augmented) 1 – 5/4 – 8/5

Bcmills: de-emphasize marvel and emphasize septimal meantone in the temperaments section, to make the connection with common-practice tunings clearer

2024-08-24T03:19:10Z

de-emphasize marvel and emphasize septimal meantone in the temperaments section, to make the connection with common-practice tunings clearer

← Older revision		Revision as of 03:19, 24 August 2024
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	The gap remaining to close the octave is another identical fifth (~5/4), giving rise to the temperament's name after this chord.		The gap remaining to close the octave is another identical fifth (~5/4), giving rise to the temperament's name after this chord.

	In [[marvel~~\|marvel temperament~~]], the ~25/16 augmented fifth is tempered together with ~[[14/9]] ~~(tempering out the difference of [[225/224\|224/225]])~~. The resulting intervals are ~5/4, ~5/4, and ~14/9, making the augmented triad a [[9-odd-limit]] essentially tempered chord (→ [[Marvel chords]]).		If [[225/224]] is tempered out, as in [[marvel]] and [[septimal meantone]], the ~25/16 augmented fifth is tempered together with ~[[14/9]]. The resulting intervals are ~5/4, ~5/4, and ~14/9, making the augmented triad a [[9-odd-limit]] essentially tempered chord (→ [[Marvel chords]]).

	* (Marvel) 1 – 5/4 – 14/9		* (Marvel) 1 – 5/4 – 14/9
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	In that case, the gap remaining to close the octave is ~[[9/7]], a ''septimal major third''.		In that case, the gap remaining to close the octave is ~[[9/7]], a ''septimal major third''.

	Since [[12edo]] is a good tuning of augmented temperament and supports ~~marvel~~, and the historically prevalent [[quarter-comma meantone]] ~~also supports marvel~~, either or both of these essentially-tempered interpretations may be relevant for augmented chords found in common-practice and contemporary music.		Since [[12edo]] is a good tuning of augmented temperament and supports septimal meantone, and the historically prevalent [[quarter-comma meantone]] is a good tuning of septimal meantone, either or both of these essentially-tempered interpretations may be relevant for augmented chords found in common-practice and contemporary music.

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

FloraC: Saying meantone is a good tuning of marvel is a stretch. Explicitly link to marvel chords.

2024-08-23T07:07:19Z

Saying meantone is a good tuning of marvel is a stretch. Explicitly link to marvel chords.

← Older revision		Revision as of 07:07, 23 August 2024
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	The gap remaining to close the octave is another identical fifth (~5/4), giving rise to the temperament's name after this chord.		The gap remaining to close the octave is another identical fifth (~5/4), giving rise to the temperament's name after this chord.

	In [[marvel\|marvel temperament]], the ~25/16 augmented fifth is tempered together with ~[[14/9]] (tempering out the difference of [[225/224\|224/225]]). The resulting intervals are ~5/4, ~5/4, and ~14/9, making the augmented triad a [[9-odd-limit]] essentially tempered chord:		In [[marvel\|marvel temperament]], the ~25/16 augmented fifth is tempered together with ~[[14/9]] (tempering out the difference of [[225/224\|224/225]]). The resulting intervals are ~5/4, ~5/4, and ~14/9, making the augmented triad a [[9-odd-limit]] essentially tempered chord (→ [[Marvel chords]]).

	* ([[Marvel ~~chords\|Marvel]]~~) 1 – 5/4 – 14/9		* (Marvel) 1 – 5/4 – 14/9

	In that case, the gap remaining to close the octave is ~[[9/7]], a ''septimal major third''.		In that case, the gap remaining to close the octave is ~[[9/7]], a ''septimal major third''.

	Since [[12edo]] is a good tuning of augmented temperament and supports marvel, and the historically prevalent [[quarter-comma meantone]] ~~is a good tuning of~~ marvel, either or both of these essentially-tempered interpretations may be relevant for augmented chords found in common-practice and contemporary music.		Since [[12edo]] is a good tuning of augmented temperament and supports marvel, and the historically prevalent [[quarter-comma meantone]] also supports marvel, either or both of these essentially-tempered interpretations may be relevant for augmented chords found in common-practice and contemporary music.

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

Bcmills: associate 20:25:32 and 25:32:40 more closely with 16:20:25

2024-08-23T02:33:25Z

associate 20:25:32 and 25:32:40 more closely with 16:20:25

← Older revision		Revision as of 02:33, 23 August 2024
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	In the [[5-limit]]:		In the [[5-limit]]:
	* [[16:20:25]] is found on ♭II+ ({{Frac\|16\|15}}), ♭III+ ({{Frac\|6\|5}}) ♭VI+ ({{Frac\|8\|5}}), and ♭VII+ ({{Frac\|9\|5}}) in the [[duodene]].		* [[16:20:25]] is found on ♭II+ ({{Frac\|16\|15}}), ♭III+ ({{Frac\|6\|5}}) ♭VI+ ({{Frac\|8\|5}}), and ♭VII+ ({{Frac\|9\|5}}) in the [[duodene]]. Its [[rotation\|rotations]], [[20:25:32]] and [[25:32:40]], represent chords that would be enharmonically equivalent to itself in augmented temperament, substituting a minor sixth and a diminished fourth successively in place of the augmented fifth and major third.
	* [[20:25:32]] ~~is found on I+ ({{Frac\|1\|1}}), II+ ({{Frac\|9\|8}}), IV+ ({{Frac\|4\|3}}),~~ and ~~V+ ({{Frac\|3/2}}) in the duodene. It substitutes the 8/5 minor sixth found in augmented temperament in place of the 25/16 augmented fifth.~~
	* [[25:32:40]] ~~is found on III+ ({{Frac\|5\|4}})~~, ~~♯IV+ ({{Frac\|45\|32}}), VI+ ({{Frac\|5\|3}})~~, ~~VII+ ({{Frac\|15\|8}}) in the duodene. It substitutes an [[8/5]] ''~~minor sixth'' and ~~[[32/25]] ''~~diminished fourth'' in place of the augmented fifth and ~~lower~~ major third ~~respectively~~.

	In the [[3-limit]]:		In the [[3-limit]]:

Bcmills: add categories for Marvel chord

2024-08-22T03:09:18Z

add categories for Marvel chord

← Older revision		Revision as of 03:09, 22 August 2024
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	[[Category:Augmented]]		[[Category:Augmented]]
	[[Category:5-odd-limit chords]]		[[Category:5-odd-limit chords]]
			[[Category:Marvel]]
			[[Category:9-odd-limit chords]]
	[[Category:Essentially tempered chords]]		[[Category:Essentially tempered chords]]
	[[Category:Just intonation chords]]		[[Category:Just intonation chords]]

Bcmills at 03:08, 22 August 2024

2024-08-22T03:08:09Z

← Older revision		Revision as of 03:08, 22 August 2024
Line 11:		Line 11:
	The gap remaining to close the octave is another identical fifth (~5/4), giving rise to the temperament's name after this chord.		The gap remaining to close the octave is another identical fifth (~5/4), giving rise to the temperament's name after this chord.

	In [[marvel\|marvel temperament]], the ~25/16 augmented fifth is tempered together with ~[[14/9]] (tempering out the difference of [[224/225]]). The resulting intervals are ~5/4, ~5/4, and ~14/9, making the augmented triad a [[9-odd-limit]] essentially tempered chord:		In [[marvel\|marvel temperament]], the ~25/16 augmented fifth is tempered together with ~[[14/9]] (tempering out the difference of [[225/224\|224/225]]). The resulting intervals are ~5/4, ~5/4, and ~14/9, making the augmented triad a [[9-odd-limit]] essentially tempered chord:

	* ([[Marvel chords\|Marvel]]) 1 – 5/4 – 14/9		* ([[Marvel chords\|Marvel]]) 1 – 5/4 – 14/9