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How does this make sense? Yes, I know jazz players use this term for the chord, but why?
Chords are named for their main notes in order:
* 1st: the main pat of the name (Eb or A). This is so important that even where there is no root played we still say something like "C#7 (no root)".
* 3rd: second in importance, this determines whether the chord is major or minor. We assume major unless otherwise stated, or where that is clearly implied another way (as discussed below).
* 5th: third in importance. We don't generally mention the 5th, we assume that it is perfect, but in those (moderately uncommon) instances where the 5th is not perfect, we name the chord "augmented" or "diminished". These terms also imply the appropriate 3rd - e.g., a diminished chord always has a flat 3rd and we don't need to say "C minor diminished", just "C diminished".
* 7th : last of the four main determinants of a chord name but always mentioned where present. We assume a minor (flat) 7th unless specified otherwise.
* Others: last of all, if there are any other notes we want to indicate, we add them on to the end of the chord name. Hence "D add 4" or "Cm6" or (where a 7th is present) "E9".
According to the standard rules of chord naming (which I think I've done reasonable justice to above) we can certainly have an "Eb augmented Major 7th". But Eb Major 7th #5? Give me a break! It's an augmented chord all day long. Why not call it that?
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Slightly less flippantly though (and I can only speak from my perspective), if I was given a chart with lots of chords on - say 2 chords per bar at a pretty fast tempo, I would certainly appreciate seeing Eb∆7#5 (or possibly better still G/Eb) as apposed to Eb Augmented Major 7th. Even reducing that down to Eb+∆7, from a visual perspective feels clunky and not particularly intuitive from my perspective.
You've got good grounds for calling it that, but for me Eb∆7#5 is much more descriptive, particularly in the heat of the moment. I just raise the 5th of any give ∆7 voicing I'm playing at a given moment, less mental gymnastics. That's my reasoning behind your question of "why?"
I’d wager the other aspect is from an improv perspective where it’d probably be thought more as from Melodic rather than Harmonic minor.
The #4 is a safer note choice over a ∆7 type chord rather than the P4 of the Harmonic Minor mode (Major/Ionian#5?) Eb F G Ab B C D. Granted it completely works with the way you want prefer to name it as well. But if it’s a substitute for a plain old maj7 chord?
I dunno, just my take on it.
Also as @Keefy says, it’s an Americanism and Americanisms tend to stick whether will like them or not