are such as are either less than perfect or less than minor by one semitone. Thus (a) being a perfect fifth, (4) is a diminished fifth ; and (c) being a perfect fourth, (d) is a diminished fourth : 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 i t 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 k ey 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 (/), 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 t occurs (/) fe) (ft) (") - o fe- .... bg - with remarkable 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 a t will to stand as minor ninth to the root which is understood. (See H armony, subsection t h e Clas s if icat io n o f Ch o r d s ; and Mo dulat ion.) The chord of the diminished third, as (k), occurs in music as the inversion of the chord of the augmented sixth, as ^ (k) (Q (I). Bach uses it with power- z=i ful effect a t the end of the - tcJ~" 4 Crucifixus ' in his B minor W=r Mass, and Beethoven in the chorus to the same words in his ' Missa Solennis.' c. H. h . p.