Augmented triad: Difference between revisions

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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.

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

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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 ==

@@ 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.
-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&thinsp;–&thinsp;5/4&thinsp;–&thinsp;14/9
@@ Line 17: / Line 17: @@
 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 ==