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diminished-4th / dim4 / b4 / -4

[Joe Monzo]

A diatonic interval which is one chromatic semitone smaller than the perfect-4th of the diatonic scale. The diminished-4th is composed of 1 tone (i.e., whole-tone) and 2 diatonic-semitones.

The diminished-4th 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 diminished 4th occurs naturally in every diatonic harmonic-minor scale, between the "leading-tone" (7th degree or "VII") and the "mediant" (3rd degree or "III"). Example, in the key of C-major ("t" = tone, "s" = semitone) :

 ---- perfect-4th -----
|                      |
G        A        B    C        D        E    F        G    =   A-natural-minor diatonic scale
    t        t      s      t        t      s      t
   M2       M2     m2     M2       M2     m2     M2

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

      m2    M2     m2     M2       M2     m2        +2
       s     t      s      t        t      s      t+(t-s)
    G#   A        B    C        D        E    F            G#   =   A-harmonic-minor diatonic scale
    |                  |
     - diminished-4th -

diminished-4th  =  t + 2s     =  (t-s) + 3s
                = M2 + 2(m2)  =   +1   + 3(m2)

Thus, the diminished-4th contains 1 chromatic semitone and 3 diatonic semitones (or equivalently, 1 augmented-prime and 3 minor-2nds). In 12-edo, the diminished-4th encompasses 4 equal semitones and is enharmonically equivalent to the major-3rd.

The diminished-4th is particularly important in medieval theory because in the pythagorean tuning then prevalent, the diminished-4th is only a skhisma (~ 2 cents) smaller than the just major-3rd of ratio 5:4. Composers of the 1400s began using the diminished-4th in place of the much more discordant pythagorean major-3rd, to take advantage of its more concordant quality. This led to the general recognition of true 5-limit tuning beginning with Ramos in 1480. In more modern times, the diminished-4th became a feature of schismic tunings such as those of Helmholtz and Groven.

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