Dominant seventh sharp five chord

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Augmented seventh chord

A dominant seventh sharp five chord, or augmented seventh chord, is a tetrad comprising a root, major third, augmented fifth, and minor seventh.

In temperaments

In meantone, on which traditional tonal harmony is built, the minor seventh tempers together 9/5~16/9, giving a chord with an intervallic odd limit of 125:

• (Meantone) 1/1 ‒ 5/4 ‒ 25/16 ‒ 9/5, with steps ~5/4, ~5/4, ~144/125.

In augmented temperament, which tempers out 128/125, a ~25/16 augmented fifth is equated with an ~8/5 minor sixth. The resulting chord is 25-odd-limit due to the 36/25~45/32 interval between the ~9/5 minor seventh and ~5/4 major third.

• (Augmented) 1/1 ‒ 5/4 ‒ 8/5 ‒ 9/5, with steps ~5/4, ~5/4, ~9/8.

In archytas temperament, which tempers out 64/63, ~16/9 is equated with ~7/4, giving another 25-odd chord:

• (Archytas) 1/1 ‒ 5/4 ‒ 25/16 ‒ 7/4, with steps ~5/4, ~5/4, ~28/25.

In marvel temperament, which tempers out 225/224, ~25/16 is equated with ~14/9, and the ~64/45 tritone between a ~5/4 major third a ~16/9 minor seventh is equated with ~10/7, yielding a 9-odd-limit essentially tempered chord:

• (Marvel) 1/1 ‒ 5/4 ‒ 14/9 ‒ 16/9, with steps ~5/4, ~5/4, ~8/7.

12edo is a good tuning of augmented temperament and supports all of the above temperaments to varying degrees, and septimal meantone — which is well-represented by the historically prevalent quarter-comma meantone — supports marvel, so any of the above interpretations may be relevant for seventh sharp five chords found in contemporary and common-practice music.

In just intonation

In the 7-limit:

• 16:20:25:28 is a condissonant chord found in genus 5² ⋅ 7. In archytas temperament, it is tempered together with 144:180:225:256.

• 36:45:56:64 is a condissonant chord found in genus 3² ⋅ 5 ⋅ 7. In marvel temperament, it is tempered together with 144:180:225:256.

In the 5-limit:

• 20:25:32:36, found on I+ (1⁄1) and VI+ (4⁄3) in the duodene, is a simple preimage of the 25-odd-limit chord found in augmented temperament, substituting an 8/5 minor sixth in place of the augmented fifth.

• 80:100:125:144, found in genus 3² ⋅ 5³, is a 125-odd-limit preimage of the chord found in meantone.

• 144:180:225:256, found on ♭III+ (6⁄5) and ♭VII+ (9⁄5) in the duodene, is also a preimage of the chords found in meantone, archytas, and marvel temperaments.

• 180:225:288:320, found on V+ (3⁄2) and II+ (9⁄8) in the duodene, is tempered together with 144:180:225:256, substituting an 8/5 minor sixth in place of the augmented fifth.

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