exchemist

I don't think the intervals of the octave are arbitrary, actually. If you take a fundamental and consider its harmonics (say a vibrating string with one, two, three, four etc standing waves), you start to get the other intervals that make up the musical scale. For example a C fundamental will have a 1st harmonic of C at the octave above, but the 3rd harmonic will be G at the octave plus a fifth above, the 4th will again be C but 2 octaves up, and the 5th harmonic will be E, a major third up from that (I think: it is hard to remember exactly how it works). So it is not coincidence that the major triad in harmony has the tonic, fifth and major third in it. These pitches are all part of the same family of overtones, from an implied lower fundamental. From what I recall, most of the notes of the Pythagorean scale were derived in this sort of way. The pentatonic scale, I think, still has the intervals of the third and the fifth in it, so is not inconsistent with the same physical principles - unsurprisingly.

Surely the point about octaves is that if you double the frequency you get something that resonates with the fundamental. And if you double it again, the same occurs. So doubling has a real significance, both from the point of view of physics and from the point of view of the hearer. In fact, when you play a certain pitch on any instrument, you also excite a whole series of so-called "overtones" at the same time, which are frequency multiples of 2, 4, 8 etc above the fundamental. You also get a sort of resonance at the fifth of the scale. It sounds, well, harmonious. Whereas if you play 2 pitches that are only a tone, or semitone, apart, you get no resonance but a harsh sounding beat frequency as the two pitches go in and out of phase with one another. So I do not think it is just a matter of convention. Doubling the frequency has a physical significance that the ear recognises.