Why no 8th, 10th, 12th, & 14th chords?

[matt.guitarteacher]

A lot of this discussion about chord voicing seems to imply that you would, for some reason, account for every note in the voicing by naming it.

A C710, for example, is naming a note which is already in the chord. You could voice that 3rd anywhere in the chord and it's still a 3rd. So you don't name it as a 10th.

However, C7b10 (usually named C7#9) provides information about additional pitches in the chord. The chord already contains an E natural and the name of the chord tells you that there's also an Eb. That chord's name tells you about something that isn't included in the original chord symbol.

What if a triad is voiced as follows?

C
E
G

With the C on top would you call the E a 3rd or 10th? The whole 10th, 12th, 14th thing as a way of trying to meticulously spell a chord as it is actually voiced kind of falls apart if it's not in root position anyway. You're back to figured bass. E natural is the third of a C chord regardless of where it's voiced.