A website detailing piano chords uses the following pitch notation for A-flat 13 chords:
A♭13 → A♭-C-E♭-G♭-B♭-D♭-F (R, M3, P5, m7, M9, m11, m13)
A♭m13 → A♭-B-E♭-G♭-C♭♭-D♭-F (R, m3, P5, m7, M9, m11, M13)
My question is why use 'B♭' in the first case and 'C♭♭' in the second? They are the same.
Yeah, your site messed up the A♭m13 chord, which should be
A♭ C♭ E♭ G♭ B♭ D♭ F
. I think the problem was that they tried to make the minor third B rather than C♭, and they then wrote the 9th as C♭♭ in order to keep every letter represented in the 13th chord. The problem is, any third above any kind of A will be some kind of C. Here, the major third above A♭ is C, and the minor third above A♭ is C♭ (A♭-B is an augmented 2nd). Then the rest of the chord should follow, with the 9th being B♭.
Two notes that are the same but have different names are called 'enharmonic notes'. So, basically B♭ and C♭♭ are the same note. Which note name should be used, depends on the harmonic context of what you're doing.
Typical Western chords are build on thirds, so just ascend thirds from your root to find out which note you'll use. G♭ to C♭♭ is a diminished 4th, not a third. But, G♭ to B♭ is a major third, an interval that is used in the chord you're trying to build. *
So, the correct way to write A♭m13 would be:
A♭m13 → A♭-C♭-E♭-G♭-B♭-D♭-F (R, m3, P5, m7, M9, m11, M13)
*The same goes for B and C♭
O.k., the note names are incorrect, as already spotted. However, 13th chords are rarely played using 1,3,5,7,9,11,and 13. Think about it - that's all the notes from a scale. The 1,3,5 and m7 must be there, for reasons stated in many answers already, but often in this 13th case, 9 and 11 are missed out, but obviously the 13 itself is played. So, 1,3,5,m7,13 does the job, and in fact, if there's a root (1) played on bass or piano, the chord can leave that out, and 5 is often left out for other reasons. At that point, there's room for the 9 and/or 11, with consideration to how all the notes are voiced.
