Function
The half-diminished chord has three functions in contemporary harmony: predominant function, diminished, and dominant function. The vast majority of its occurrence is on the II chord in the minor mode, wherein it takes a predominant function, leading naturally to the dominant V chord. Not including the root motion, there is only a one note difference between a half diminished chord and a V chord with a 9th. Since it is built on the diatonic II chord of the minor scale, most of the time the II-V pattern resolves to a minor tonic (such as in the progression Dm7(♭5) - G7(♭9) - Cm)., but there are instances where there is a major tonic resolution.
Diminished chord function is rarer, but it still exists. Half-diminished chords can function in the same way as fully diminished chords do, such as in the chord progression Cmaj7 - C♯dim7 - Dm7, or Em7 - E♭dim7 - Dm7, where the diminished chord serves as a chromatic passing chord preceding a chord with a diatonic root. A typical example of this is when ♯IVm7(♭5) progresses to IVm7, such as in the Cole Porter song "Night and Day", where you have the progression F♯m7(♭5) - Fm7 - Em7 - E♭dim7 - Dm7 - G7 - Cmaj7. If analyzed in its predominant function, it wouldn't sufficiently explain how it functions preceding the Fm7 chord.
In dominant function, the VII half diminished chord, like its fully diminished counterpart, can take the place of the dominant V chord at a point of cadential motion. This generally occurs in a major key, since the flattening of the sixth degree in the natural minor scale renders a dominant diminished seventh chord fully diminished if played within the scale. Indeed, the VII half diminished chord in a major key is identical to a dominant ninth chord (a dominant seventh with an added ninth) but with its root omitted.
The dominant function of the half-diminished seventh chord may also occur in a secondary dominant context, i.e., as part of a progression where the chord performs the dominant function with respect to the overall key's dominant chord. In this scenario, the half-diminished seventh chord is built on the tritone of the overall key and is equivalent to a secondary dominant seventh chord with added ninth and omitted root. If written with respect to the overall key, this chord is styled "#ivø7," but in terms of its function in the progression, the styling "viiø7/V" is more descriptive. Examples of the #ivø7-to-V or viiø7/V-to-V transition include the half-line "know when to run" in the Kenny Rogers song "The Gambler", and measures 11-12 of Scott Joplin's "The Entertainer".
Read more about this topic: Half-diminished Seventh Chord
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