Dominant seventh sharp five 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:
In augmented temperament, which tempers out 128/125, a ~25/16 augmented fifth is equated with an ~8/5 minor sixth. However, the chord is still 25-odd-limit due to the 36/25~45/32 interval between the ~9/5 minor seventh and ~5/4 major third.
In archytas temperament, which tempers out 64/63, ~16/9 is equated with ~7/4, giving another 25-odd essentially tempered chord:
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:
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
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