A Dictionary of Music and Musicians/Diminished Intervals

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DIMINISHED INTERVALS are such as are either less than perfect or less than minor by one semitone. Thus (a) being a perfect fifth, (b) is a diminished fifth; and (c) being a perfect fourth, (d) is a diminished fourth:—

{ \override Score.TimeSignature #'stencil = ##f \relative g' { \cadenzaOn <g d'>1^"(a)" <g des'>^"(b)" <a e>^"(c)" <aes e>^"(d)" } }
These are both of discordant nature, the diminished fourth always so; but if a major sixth be added below the bass note of the diminished fifth it is considered to modify the discordance so far as to admit of its being used as a concord. This rule is of old standing, especially in regard to the occurrence of the chord diatonically, as (e) in the key of C, which was admitted in the strict old style where discords were excluded. Of intervals which are changeable into major or minor the diminished seventh is the commonest, (f), which is a semitone less than the ordinary minor seventh (g), according to the rule above given. The complete chord, which is commonly known as that of the 'diminished seventh,' (h), is properly speaking an inversion of a chord of the minor ninth, (i). It occurs with remarkable
{ \override Score.TimeSignature #'stencil = ##f \relative d' { \cadenzaOn <d b' f'>1^"(e)" <cis bes'>^"(f)" <c bes!>^"(g)" <cis e g bes!>^"(h)" <a cis e g bes!>^"(i)" } }
frequency in modern music, part of its popularity no doubt arising from the singular facilities for modulation which it affords. For the notes of which it is composed being at equal distances from one another, any one of them can be chosen at will to stand as minor ninth to the root which is understood. Thus the above chord might be written in either of the following ways—
{ \override Score.TimeSignature #'stencil = ##f \relative d' { \cadenzaOn <des e g bes>1 <des fes g bes> <cis e g ais> } }
in which D♭, F♭, and G are respectively the minor ninths to C, E♭, and F♯, the absent root notes, and could pass into as many different keys as those root notes could serve, either as dominant, tonic, or supertonic. [See Change, Modulation.] The chord of the diminished third, as (k), occurs in music as the inversion of the chord of the augmented sixth, as (l). It has such a strongly
{ \override Score.TimeSignature #'stencil = ##f \relative b' { \cadenzaOn <bes gis>1^\markup { \halign #-3 (k) } <bes gis,> <gis bes,>^"(l)" } }
marked character of its own that great composers seem agreed to reserve it for special occasions. Bach uses it with powerful effect at the end of the 'Crucifixus' in his B minor Mass, and Beethoven in the chorus to the same words in his 'Missa Solennis.'