I was puzzling over the same question. Some of these words are semantically overloaded, and as far as I can tell there's no single interpretation that makes it all "work out" the way you would require if this were math. There are two meanings to "adding 7":
1. adding 7 arithmetically to obtain the same note an octave higher;
2. adding in a 7th note to whatever chord you are playing.
But the way the chords are named isn't what you'd guess from those two meanings. If it were, you'd add a 6th to C major to get C6 (that part is true) and then a 7th to C6 to get C13 (that part is not true). Instead, there's yet a third way the notes are "added", namely cumulatively in the following sequence (I've cribbed most of this from the OP):
C C+E+G
C7 C+E+G+B♭
C9 C+E+G+B♭+D
C11 C+E+G+B♭+D+F
C13 C+E+G+B♭+D+F+A
(The author's argument is that while the chords are theoretically defined to include all these notes equally, some notes - especially B♭ - are more important than others. So even if you drop E,G,D,F and only play C+B♭+A, that's still arguably C13. He's saying that's the essence of the chord.)
I suppose some arbitrariness is inevitable, because there are only so many letters and numbers to go around, and more chords with a plausible claim to the "best" names -- which, if you know regular expressions, would be something like [A-G][1-9]+ -- than there are names available. One beautiful thing about music is that it isn't math (we have math for that). It seems to overlap with math and then mixes everything up in mind-blowing ways, at which point all you can do is feel. Well, memorize and feel.
Does anyone know historically how C13 got to mean "C+E+G+B♭+D+F+A" rather than "C+E+G+B♭+A", which if you consider things only symbolically, seems more likely? Obviously, it must be because that's what people were playing - but who? Presumably jazz musicians? When was this name settled upon?
Been out of Berklee a few years, but a few nuggets that might shed light:
In Jazz theory, the 7th scale degree is part of the chord. C7 isn't C (C+E+G) plus a B♭; C7 is its own flavor of C chord, C + E + G + B♭.
More generally, the ^3 and the ^7 are the two most important notes in the chord. The ^3 tells you its quality (minor, major) and the ^7 tells you its, uh, seventhness (dominant, diminished, maj7). It's common for players to leave off the 1 and the 5; they're low-information notes, and those fingers are better put to use playing the tensions, which are "chord extensions", the 9th, 11th and 13th. Critically, all three tensions are considered part of the chord, unlike Csus4, which means substitute a 4 for the 3, and don't play the 3.
C9, C11 and C13 can exist, but you won't often find C13 in particular like that. There's a whole set of memorizable rules to tell you which chord scales go with which chords performing which functions; if that C is a I chord in the measure, you're not likely to see a C11. You may see a C7(11), or C7(add11), which is IIRC the C chord plus an 11th note - but not the 9th. Then there's C7(#11), which you'll often find when C is the IV chord, 'cause the IV chord gets a Lydian scale.. but now I forget how that explains anything.
C+E+G+B♭+D+F+A can also be looked at as D-/C7, and often is. That whole "Upper Triad" polychord stuff is a whole semester to itself.
> Then there's C7(#11), which you'll often find when C is the IV chord, 'cause the IV chord gets a Lydian scale.. but now I forget how that explains anything.
I'd say it's usually the other way around: if you want to improvise in a Lydian mode, use a 7th(#11) chord on your IV, while improvising; if you're composing, use this chord to hint the interpret to use this scale. And so on.
> Does anyone know historically how C13 got to mean "C+E+G+B♭+D+F+A" rather than "C+E+G+B♭+A", which if you consider things only symbolically, seems more likely?
Changing the question to C11 to simplify: a possible answer (take it with a grain of salt) is that C11 got to mean "C+E+G+B♭+D+F" because in classical harmony you can't just stick a 11th there without the 7th and 9th. The 11th interval (C to upper F) is dissonant; not as much as the 4th, but still dissonant. However, it is consonant with the 7th and 9th. That is, B♭ to F is a 5th, and D to F is a 3rd, or simply: B♭-D-F is a triad.
The question now is, why do books say C11 means "C+E+G+B♭+D+F" but in practice it's often voiced as "C+E+G+B♭+F"? I think the simple answer is that in some traditions (like jazz) people are OK with that dissonance; and in some harmonical contexts (when the function of the chord is such and such; say jaylevvit's comment), we can omit such and such notes because the chord still "works". If you doubt, play several voicings. The "proof" is in the sound.
(but honestly I'm not fully convinced by that answer. In particular I dislike the notational mess).
1. adding 7 arithmetically to obtain the same note an octave higher;
2. adding in a 7th note to whatever chord you are playing.
But the way the chords are named isn't what you'd guess from those two meanings. If it were, you'd add a 6th to C major to get C6 (that part is true) and then a 7th to C6 to get C13 (that part is not true). Instead, there's yet a third way the notes are "added", namely cumulatively in the following sequence (I've cribbed most of this from the OP):
(The author's argument is that while the chords are theoretically defined to include all these notes equally, some notes - especially B♭ - are more important than others. So even if you drop E,G,D,F and only play C+B♭+A, that's still arguably C13. He's saying that's the essence of the chord.)I suppose some arbitrariness is inevitable, because there are only so many letters and numbers to go around, and more chords with a plausible claim to the "best" names -- which, if you know regular expressions, would be something like [A-G][1-9]+ -- than there are names available. One beautiful thing about music is that it isn't math (we have math for that). It seems to overlap with math and then mixes everything up in mind-blowing ways, at which point all you can do is feel. Well, memorize and feel.
Does anyone know historically how C13 got to mean "C+E+G+B♭+D+F+A" rather than "C+E+G+B♭+A", which if you consider things only symbolically, seems more likely? Obviously, it must be because that's what people were playing - but who? Presumably jazz musicians? When was this name settled upon?