Strictly speaking a ("fully extended") major thirteenth chord looks like this, e.g. for CM13 (or CMA13 or CMaj13):
E (third or supertonic)
C (root or tonic)
But I read this today:
A major thirteenth chord (containing a major seventh) will nearly always feature a chromatically raised eleventh (C E G B D F♯ A), except for cases when the eleventh is omitted altogether. "It is customary to omit the eleventh on dominant or major thirteenth chords because the eleventh conflicts with the third," in these chords by a semitone.
The "conflicting" notes mentioned here are the F (eleventh) and the E (third or supertonic), i.e. there's only a semitone (half step) between an E and an F (well, technically they're an octave+a semitone apart, but it still can have a "disturbing" effect on the ears).
So in practice, to make the chord "easier on the ears", you either:
* raise the eleventh (in the case of CM13, from F to F#)
* remove the eleventh altogether
Here's an audio example comparing the sounds of the 3 different C-Major thirteenth chord versions discussed above: CM13 (full), CM13 (raised 11th), CM13 (omitted 11th).
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