[nix]

Supposedly there is a mathematical reason for the 2-2-1-2-2-2-1 spacing of the major scale. If you take all possible pairs of notes in the diatonic scale, you get a richer distribution of intervals than you can produce with any other seven-note selection from the twelve note scale. Similarly, the classic pentatonic scale provides the best set of intervals for any five-note selection from the twelve. A better set of intervals might lead to a better choice of chords too.

This is my somewhat fuzzy recollection from a paper I read a long time ago. Someone out there can check with three lines of R, right?

[manlon]

If you start with the 12 chromatic tones and start addding notes to a scale going up a circle of fifths, there are two natural stopping points where you have spanned the octave with a complete-sounding set of notes with relatively equal spacing and no gaps: five notes, which gives whole-step and minor-third intervals; and seven notes, which gives whole-step and half-step intervals. These two scales correspond to the spacing of the black notes and the white notes, which are mirror images of each other around the circle of fifths. Any other choice of scale size would have gaps, I believe.

[nix]

The drawback to this explanation is that the diatonic scale is 10,000 years older than the "circle of fifths". So it presumably had some appeal to musicians as well as to music theorists.

[dmoney]

Where do you find 10,000 year old music?

[nix]

You infer it from the existence of 10,000 year old musical instruments.

See http://en.wikipedia.org/wiki/Diatonic_scale#Prehistory - which also says that the circle of fifths was described much earlier than I thought.

[sampo]

So if we stard from C, and take steps of fifths, we get the notes in the order: C, G, D, A, E, B, F#, C#, G#, D#, A#, F (and then C again).

It is true that if you take the first 5, you get a scale with pattern of intervals C-<2>-D-<2>-E-<3>-G-<2>-A-<3>-C, and then it you'd add the sixth note (B), that would be only 1 step from C. So if we aim for a "nice" distribution of intervals, 5-note scale is a good stopping point.

But your explanation doesn't give any light on why the 7-note scale would be the next stopping point.