Em #5 would have a B#, not a C. This isn't a pedantic point; it very much matters if we're trying to be accurate with chord naming.
Another example to illustrate how identifying the root is important: C E G A. It's either a C6 or an amin7.
Another example to illustrate how identifying the root is important: C E G A. It's either a C6 or an amin7.
I think the point that root and bass should not be conflated is also valid.
Inverted chords have, in my ears, never sounded anything like the chords they are supposedly an inversion of. It's like saying "10 is just really 3+7". It's true, and it's also useless. Let 10 be 10. Let 3 be 3. Etc.
Most composers don't even use this system as they compose, it's just for after the fact analysis. Composers hear a tune in their head and then transcribe it. Unless they're writing cookie cutter shit without inspiration, then they might synthesize some soulless crap that intentionally follows some well known chord progression as picked by a dart thrown at a poster of the circle of fifths.
Ok. Then what do you call it?
Known in western classical harmonization theory as "Em+" (E chord with minor third, augmented 5th). C first inv is actually Gsus4add6no5 or as I like to call it G+5+4. It has an incredibly warm and rich tone, nothing like C+4+3 (C major triad).
It works in C because all of its notes are also present in the C major scale, but that's just a coincidence. Calling it a C inverted chord is like making astrological or numerological conclusions.
And no, shifting up or down by octaves is not the same sound. It's close but not equal. You can probably get away with octave substitutions in a busy song but you can not say it sounds exactly the same. A 9th is not a 2nd. 11 not 4, 13 not 6. When you play La Campanella, is that your excuse for skipping those octave jumps? :)
As I said, unpopular opinion :)
If you are already ignoring the actual chord "voicing" as written in the "classical" system, then just keep doing that in my system??
If you are already able to intuit that a note should go elsewhere, then keep doing that for your jazz sets? What's the issue again?
"Gsus4x5" is just as unreadable and ignorable and malleable as "G+5+4".