> some of the pitches in the harmonic series are approximated by the notes of the chromatic scale
is not the same thing as
> the chromatic scale is derived from the harmonic series
which is what the OP article claims.
You can find the chromatic scale in the harmonic series, yeah, if you ignore the majority of the notes in the harmonic series.
To find the chromatic scale in the harmonic series, you need to take the 2nd, 3rd, 4th, 5th, 9th, 15th, and 44th harmonics, and ignore of the rest. That's not a mathematically justified derivation, that's a post-hoc rationalization built on coincidence alone.
So, sure, fair question why these harmonics and not any of the others?
Well, powers of 2 are out because they are just higher octaves. Then we have a whole series of harmonics that are equivalent ratios to the fundamental when folded down into the octave range (3 (3:2),6 (6:4), 12 (12:8)), (5 (5:4), 10 (10:8)), (7,14,28), (13,26) and so on.
You'll notice the pattern: the harmonics the define the intervals in a 12T system are those that introduce new ratios into the list of intervals, so they lean toward being prime or only having factors not already introduced.
By the time you go through the list, it's easy to see that going up to the 31st harmonic really only leaves out a couple of possibilities from a 12T system: 25, 29, 31 and as far as I am aware this is because introducing them into the pitch class produces results extremely close to already existing members.
And sure, you could go higher, but the pattern will repeat: harmonics whose ratio folded into the octave range are identical, or which give rise to pitches extremely close to pitches defined already.
is not the same thing as
> the chromatic scale is derived from the harmonic series
which is what the OP article claims.
You can find the chromatic scale in the harmonic series, yeah, if you ignore the majority of the notes in the harmonic series.
To find the chromatic scale in the harmonic series, you need to take the 2nd, 3rd, 4th, 5th, 9th, 15th, and 44th harmonics, and ignore of the rest. That's not a mathematically justified derivation, that's a post-hoc rationalization built on coincidence alone.