You cannot omit an altered fifth.
In other words, you cannot omit the b5 from either a half-diminished chord or a diminished chord, or the #5 from an augmented chord.
The omission of the 5th only works for unaltered fifths that are implied by the harmonic overtones of the root.
music.stackexchange.com/users/6780/Kirk A
Quite often the most important note of a "m7-5" chord is the "-5", and leaving that out spoils its harmonic effect more than leaving out some other note.
If you have to leave something out, ditch the seventh or maybe even the third.
This is one reason why it's better to look at "Bm7-5" as Dm6/B.
In the contexts where it's commonly used, there are two important notes: F and B.
It's like, F is for the minor character (like in Dm) and B is for the bass movement, like in Bm7-5 - E7 - Am.
Or what do you say, isn't the fact that the seventh can be left out rather than the "-5", a telltale sign that it's more of a Dm than a Bm? :)
music.stackexchange.com/users/51766/piiperi Reinstate Monica
The P5 from the root of a major or minor chord can be omitted because that sound is already heard in a harmonic from the root note, albeit more quietly than if it was played as a saparate note.
The diminished chord has 1, &flat;3 and diminished 5.
That dim.5 is not contained in the harmonics of ,well, either note from that chord.
So leaving it out means just that - it's not going to be there at all.
So, no proper chord!
So, the 'rule' that you can leave out the 5th won't apply to diminished (or augmented) chords, only majors and minors.
In fact, the 'rule' ought to be you can leave out P5 .
music.stackexchange.com/users/2808/Tim
In Levine's Piano Jazz book he gives an example of Bud Powell voicings (shells) where the minor seventh and half diminished are the same - a minor seventh over root, all other tones omitted... ...
it isn't stated in the text around that figure, but I think the assumption is the flat fifth will be supplied in the right hand.
...
what's the difference between the the two chords if we have the removed the 5th which is not essential?
The diminished fifth is essential to define the half diminished chord, but apparently it isn't required to be in the shell.
Supply it in another voice (or possibly it will be implied by the surrounding harmony.)
music.stackexchange.com/users/23919/Michael Curtis
The whole point of shell voicings is to provide some "flavour", often for the purpose of supporting a soloist (maybe yourself), rather than to completely define every single chord, or to tell the soloist what they should play.
Similarly, the lead sheet may say "G13" but you might well choose to play a G7 - and leave a bit of space.
So although the answers above may be technically correct in terms of harmonic theory - from the point of view of a musician on a gig - they should not be your primary consideration.
Which actual notes you choose should be determined by what you hear the other musicians do, making nice voice leading, and so on.
music.stackexchange.com/users/62227/danmcb
I am learning about shell voicings for 7th chords. I understand that you you remove the 5th because it makes no difference in the quality of the chord.That leaves me with Cmin7 being C Eb BbC half diminished being C Eb Bb  So because we removed the 5th, both chords have the same notes and therefore the same sound. In that case, what's the difference between the the two chords if we have the removed the 5th which is not essential?
What's the difference in sound between the shell voicing of a minor 7th chord vs a half diminished 7th chord?
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<chords><jazz><chord-voicings>
music.stackexchange.com/questions/94607
music.stackexchange.com/users/65956/JFarobek
The P5 from the root of a major or minor chord can be omitted because that sound is already heard in a harmonic from the root note, albeit more quietly than if it was played as a saparate note. The whole point of shell voicings is to provide some "flavour", often for the purpose of supporting a soloist (maybe yourself), rather than to completely define every single chord, or to tell the soloist what they should play. Actual notes you choose should be determined by what you hear the other musicians do, making nice voice leading, and so on.
The P5 from the root of a major or minor chord can be omitted because that sound is already heard in a harmonic from the root note Shell voicings are often used for the purpose of supporting a soloist, rather than to define every single chord. Similarly, the lead sheet may say "G13" but you might well choose to play a G7 - and leave a bit of space. Which actual notes you choose should be determined by what you hear the other musicians do, making nice voice leading, and so on.