### The Circle of Fifths part 2 - Scales

When I rambled on about the circle of fifths, one of the things baked into it was the way there are twelve semi-tones in an octave. Various web pages discuss the golden ratio as an explanation for the number of notes, on the grounds that if you include the octave note there are 13 semi-tones, 8 white notes, and then 5 black notes. Hmmm.

So, what's the real thing that controls the number of notes in an octave? It basically comes down to having a note that's very close to a fifth. That is, given even-tempering, so that each semi-tone is the same ratio above the previous note, there is a note very close to 1.5x the base note. As it happens, with a twelve note scale, 2^(7/12) = 1.4983.

Put another way, what we're looking for is good rational approximations to log_2 1.5. And a really useful tool here is ... continued fractions! Continued fractions rock, and I'm rather disappointed they never appeared in the first year of undergrad maths I did. In the end, I learnt about them as a side-effect of learning some number theory for Project Euler.

Continued fractions provide a nice way of created truncated rational approximations to irrational numbers. So, log_2 1.5 is 1/(1 + (1/(1 + 1/(2 + 1/(2 + 1/(3 + ...)))))). According to the Wikipedia page algorithm, the approximations in increasing order of accuracy are 1, 1/2, 2/3, 3/5, 4/7, 7/12, 17/29, 24/41.

So, a 12-note scale is as good as it gets, unless you want a 29-note or 41-note scale. And that sounds excessively complicated.

Where did this Fibonacci/golden ratio business come from? Well, phi = 0.61803..., and log_2 1.5 = 0.58496..., so if you're looking for mystical numbers around 0.6, you might assume the golden ratio's involved. It comes up everywhere else, and 7/12 makes you think of 8/13. A case of mistaken identity!

Posted 2013-02-04.