1

The normal rule-of-thumb is that you can always omit the root and the fifth from seventh chords, but that seems to fall through with half-diminished chords. If you omit the fifth it looses its diminished quality.

So if I'm trying to construct three-note voicings of half-diminished chords, what can I leave out?

3

You can always omit a perfect 5th from a chord due to it being implied strongly by the root. You can sometimes omit the root of a chord, however this is only done when you know someone else is playing the root or is implied from the previous chord.

A half diminished 7th however does not have a perfect 5th and probably the most defining feature of the half diminished 7th is the tritone made between the root and the diminished 5th. The color of the chord is also brought through the 3rd and the 7th. In most situations you would want to just play all the chord tones to best reflect this chord.

2

As usually, it depends on the context. Without context, leaving out any note of a chord will often make it ambiguous. However, in context, you can usually get away with leaving out a note of a four-part chord. You're right that if the fifth isn't perfect, it is essential for the chord quality. However, I've often found that it can be left out, e.g., in a II-V-I progression in minor, the II chord can be played as a three-part chord (root, third, seventh), and the diminished fifth will be implied. Also the root is sometimes left out, which leaves you with a minor chord a minor third higher than the (left out) root of the half-diminished chord. In context, the root can be implied.

I haven't come across examples where the third or the seventh of a half-diminished chord are left out. Leaving out the seventh reduces the chord to a diminished triad, which can stand on its own, so the seventh can't be implied that easily. Note that the third and the seventh are usually important in voice leading, so it's less attractive to leave them out. E.g., in a II-V-I progression, the third of the II chord (half-diminished in a minor key) usually becomes the seventh of the V chord, and the seventh of the II chord (usually) leads to the third of the V chord.

0

You can omit root and 5th from a dominant 7th-shape chord and retain the function of the tritone. But you're losing the definition of the chord. Take G7, omit G and D leaving the tritone F - B. But is the chord now G7 or Db7? They may be functionally interchangable in jazz harmony, but they aren't the SAME chord.

A half-diminished chord doesn't have essential functional notes the way a dom7 does. Take away any one note, it will still function pretty well. But it won't be a half diminished chord any more.

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