Mathematical explanation of music and white/black notes in a piano

The short answer is: the diatonic scale in a given key consists of notes whose frequencies are those of the keynote times certain simple rational numbers; the white notes on the piano are those that belong to the diatonic scale based on C.

What's special about those simple rational-number ratios? Answer: on most musical instruments, notes whose frequencies are in simple rational ratios sound nice together. This turns out (surprisingly, at least to me) to be a fact about the instrument and not merely about the notes; when you play a given note on a given instrument, you get (kinda) sine waves whose frequencies are those of the given note, plus some higher frequencies; exactly what the higher frequencies are and how much of each you get depends on the instrument. For most instruments, the higher frequencies are integer multiples of the fundamental frequency, and that turns out to mean that the good-sounding combinations of notes are ones with simple rational frequency ratios; but there are instruments that behave differently (e.g., a tuned circular drum; or you can make a synthesized instrument that does anything you like) and different chords will sound good on them.

For much more on this, see William Sethares's book "Tuning, Timbre, Spectrum, Scale" and his web pages at http://eceserv0.ece.wisc.edu/~sethares/ttss.html where you can find, e.g., some music in unorthodox scales performed on (synthetic) instruments designed to make the harmony sound good.

For instance: listen to http://eceserv0.ece.wisc.edu/~sethares/mp3s/tenfingersX.mp3 and hear how out-of-tune it sounds. Now try http://eceserv0.ece.wisc.edu/~sethares/mp3s/Ten_Fingers.mp3 which has exactly the same notes but played on a synthetic instrument designed to make the harmonies work.

If you're just after a listen, there are lots of examples of alternative systems on youtube. Here's 19-tone, equal temperament (as opposed to the usual 12 TET): http://www.youtube.com/watch?v=1EP0KvbxW8o

I like the above example because it always starts by sounding "off" to me but seems okay by the end of the piece. It's a matter of what you're used to.

Years ago I watched a British documentary called Howard Goodall's Big Bangs. It's well worth watching. It explains how music theory developed from Pythagoras's (matching the 12 keys on the piano). It's an interesting exercise to "prove" these 12 steps based on this simple ratio.

What came much later was equal temperament. The 12 steps don't match up exactly. Equal temperament changes the notes slightly by changing the ratio slightly (to factors of the 12th root of 2). I believe it was Bach who first discovered this.

Not all cultures use equal temperament but it is overwhelmingly dominant in the West.

The series also explains chords, keys and so on. For someone like me who is more mathematically inclined it was fascinating. Give it a look if you can.

Oh also it wasn't the BBC as you might expect. It was Channel 4.

12 tones is not the only possibility. there is also a 19 tone scale:

http://en.wikipedia.org/wiki/19_equal_temperament

there are also some mp3 files using 19 tone scale: e.g. http://www.harrington.lunarpages.com/mp3/Jeff-Harrington_Pre...

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.