Diminished Seventh Root
Music theorists have struggled over the centuries to explain the meaning and function of diminished seventh chords. Currently, two approaches are generally used. The less complex method treats the leading tone as the root of the chord, and the other chord members as the third, fifth, and seventh of the chord, the same way other seventh chords are analyzed.
The other method is to analyze the chord as an "incomplete dominant ninth", that is a ninth chord with its root on the dominant, whose root is missing or implied. A vii°7 chord in the minor key (for example, in C minor, B♮, D, F, A♭) occurs naturally in the harmonic minor scale and is equivalent to the dominant 7(♭9) chord (G, B, D, F, A♭) without its root. Walter Piston has long been the champion of this analysis.
The dominant ninth theory has been questioned by Heinrich Schenker. He explained that although there is a kinship between all univalent chords rising out of the fifth degree, the dominant ninth chord is not a real chord formation.
Rameau explained the diminished seventh chord as a dominant seventh chord whose supposed fundamental bass is borrowed from the sixth degree in minor, raised a semitone producing a stack of minor thirds. Thus in C the dominant seventh is G7 (G-B-D-F) and the sixth degree borrowed from minor produces A♭-B-D-F. He observed in his Treatise on Harmony that three minor thirds and an augmented second make up a chord where the augmented second is such that "the ear is not offended" by it. He may have been talking of the augmented second in quarter-comma meantone, a tuning he favored, which is close to the just septimal minor third of 7/6.
Read more about this topic: Diminished Seventh Chord
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