> 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).
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).