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 chromatic semitone / \ 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.