The twelve notes per octave were derived from physics; they're not arbitrary. If you divide the sounding length of a string 3:2 it gives you an interval of a fifth. If you divide it 1:1 it gives you an octave above the original length. All the other notes in the octave have similar geometrical connections, which the brain recognises. In a chord, pulses occur at intervals, depending on the common factors of their vibration rates. Your brain recognises these. Every key has twelve notes in an octave in natural termperament. Unfortunately, the C in the key of C is not the same as the C in the key of E, so Equal Temperament was derived, which evens out the intervals slightly so that one fretboard or keyboard can play in every key. But that is minor compared with changing the number of notes in an octave.
12 is 2x2x3, so the number of recurrances of factors is vast. 13 is a prime number, and has no factors other than itself.
You just cannot decide on a different number of notes in a scale. It would be like deciding to alter the gravitational constant. Those non-European musicians who have more than 12 notes still have those same basic 12 notes, but they add fractions of those notes, which still are in accord with the laws of physics.
Western music also uses fractions of notes, particularly in jazz and blues. We as steel guitarists are renown for slurring notes.
I remember a few years ago some anthropologists wanted to know how an Egyptian flute from 3,000 years ago sounded, so they took a cast of it and made an identical replica. What they found was that the flute played in the key of C with the finger holes set up for a regular scale.
If you're going to change the number of notes in an octave, why 13? Why not 17? 23? 7?
http://bp.b0b.com/2013/06/fuse-blue/
I hope Bobby Lee won't mind me posting the above, which he recorded in the 13-note scale. To my hearing, it sounds flat, and gets more and more out-of-tune as he gets higher up the fretboard. With all due respect to b0b, and his enormous talent, I couldn't listen to music in this scale for very long.
B0b did mention that the 13-note scale is not an octave, but an octave plus a fifth. An octave plus a fifth is 17 tones, which is also a prime number.