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Encyclopedia of Microtonal Music Theory

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augmented-5th / aug5 / #5 / +5

[Joe Monzo]

A diatonic interval which is one chromatic semitone larger than the perfect-5th of the diatonic scale. The augmented-5th is composed of 3 tones (i.e., whole-tones), one diatonic semitone, and one chromatic semitone. For example, in the key of A-minor:

                   --------- perfect-5th ---------
                  |                               |
A        B        C        D        E    F        G        A   =   A-natural-minor diatonic scale
    t        t        t        t       s      t        t
   M2       M2       M2       M2      m2     M2       M2

                                           chromatic semitone
                                                   / \
                                                  G   G#   A
                                                  (t-s)  s
                                                    +1  m2

   M2       M2       M2       M2      m2        +2      m2
    t        t        t        t       s      t+(t-s)    t
A        B        C        D        E    F            G#   A   =   A-harmonic-minor diatonic scale
                  |                                    |
                   ---------- augmented-5th -----------

augmented-5th  =  3t    + s  + (t-s)   =   4(t-s) + 4s
               =  3(M2) + m2 + (+1)    =   4(+1)  + 4(m2)

The augmented-5th thus contains 4 chromatic semitones and 4 diatonic semitones (or equivalently, 4 augmented-primes and 4 minor-2nds). In 12-edo, the augmented-5th encompasses 8 equal semitones, which makes it enharmonically equivalent to (i.e., exactly the same size as) the minor-6th -- but the augmented-5th is not properly the same as the minor-6th, and in all other tunings of the diatonic scale they are different intervals.

The augmented-5th arose primarily out of "common-practice" harmony as part of the harmonic-minor scale, where the 7th degree ("leading-tone") is raised a half-step so as to approach more closely to the tonic, the note to which it normally resolves.

The augmented-5th is the most prominent feature of the "augmented chord", which arises naturally as the triad built on the 3rd degree (III) of the harmonic-minor diatonic scale.

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