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A very interesting collection of short essays about that m7b5 chord!
From here:
Invalid Link Removed
Invalid Link Removed, MM Music Composition, San Francisco Conservatory of Music
"Functioning as a half-diminished, though, this chord typically doesn't have a function on its own; it is commonly used as an upper structure
with an A in the bass, forming a chromatically altered
A7 chord (A7b13b9 to be precise). You'll note that A is also in the Bb melodic minor scale; in fact in jazz, it's common to treat the 7th mode of the melodic minor scale as the primary mode, called the "altered scale" or "diminished whole tone scale.The A7b13b9, as you might guess, tends to function as an altered dominant in D minor. If you again remove the A in the bass and look at the half-diminished seventh, you'll notice this leaves us with a ivø7, which is in neither the natural nor the melodic minor scale."
...
"The half-diminished seventh is a funny chord because its nature somehow tends to relax the pull of tonality. For instance, this same chord appears famously as a motto in Wagner's opera
Tristan und Isolde.You'll notice the famous "Tristan chord" appears in the very first phrase. Here the F half-diminished seventh is functioning as a French augmented 6th chord resolving to V7 in A. The G# appoggiatura leading into the A of the French +6 is given so much emphasis compared to the A itself that we almost forget it could ever resolve. Then after it does, we never get the full cadence, as the V is left unresolved before we move to another key to repeat the same phrase. By the end of the first minute of the prelude, the entire audience knows exactly the pining for unconsummated love that Tristan and Isolde will feel for the entire opera! Wagner has conjured a brilliant bit of dramatic poetry, and the half-diminished seventh chord becomes the very symbol of this feeling."
"The half-diminished seventh is what 19th century theorist Simon Sechter would have called a zwitterklang — literally a hermaphroditic chord..."
Invalid Link Removed, Lecturer in Music Theory and Piano at California State University, Channel Islands
"Most commonly, the half-diminished seventh chord functions as a
predominant, that is, a ii (supertonic chord) that comes before the dominant (V) which closes a progression in the minor mode. Most common is for the third of the chord to be in the bass.
any minor key: ii half-dim 6/5 - V7 - i
The half-diminished seventh chord can also function in the
dominant area, preceding the dominant (V) in the major mode.
any major key: vii half-dim 7 - V7 - I
In the 19th-century, composers started to take the predominant directly to the tonic (
plagal cadence), bypassing the dominant-tonic (V-I) convention because it began to feel too cliche and predictable. Its function is cadential, a sort of dominant-substitute although I wouldn't call it a dominant."
Invalid Link Removed, musician and teacher of music
"Diatonic vii7 chord in A flat major, functioning as a dominant area chord resolving to I
Diatonic ii7 chord in F minor, functioning as a pre-dominant area chord (some say subdominant area, same thing) about to go to V7.
Secondary vii half-diminished seventh chord, functioning as a vii7/x (some call it an incomplete V9/x), where x could be A flat major OR minor, so x could be a ii from major, III, iii, IV, iv, V, v, VI, vi, or VII in a major or a minor key. Notice I left out ii from a minor key, which doesn't get tonicised because it has a flat 5, and I also left out vii from a major key for the same reason. Go ahead and include ALL the minor key possibilities in the major keys as well (mode borrowing).
Secondary ii half-diminished seventh chord, functioning as the ii in a ii-V to a chord that is normally minor (so Fm in this case) OR to a major chord (F major)… oy, do I have to list all the keys where an F could happen?"