Augmented triad: Difference between revisions

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~~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 [[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

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.

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:

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 ~~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''.

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 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 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]] (1/1 –  9/7 – 8/5)

* [[28:35:45]] (1/1 – 5/4 – [[45/28]])

* [[28:36:45]] (1/1 – 9/7 – [[45/28]])

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]]:

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 {{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 [[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&thinsp;–&thinsp;5/4&thinsp;–&thinsp;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.
-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:
+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 chords|Marvel]]) 1&thinsp;–&thinsp;5/4&thinsp;–&thinsp;14/9
+* (Marvel) 1&thinsp;–&thinsp;5/4&thinsp;–&thinsp;14/9
 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 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 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&thinsp;– 14/9) is the simplest JI tuning of the 7-limit marvel augmented chord.
+* [[35:45:56]] (1/1&thinsp;–&thinsp; 9/7&thinsp;– 8/5)
+* [[28:35:45]] (1/1 – 5/4 – [[45/28]])
+* [[28:36:45]] (1/1 – 9/7 –  [[45/28]])
 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]]: