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The Pentagon Knot 
The TwelveNote ScaleIn normal Western music, an octave is divided into twelve notes. I never thought about it too much, but I always assumed that the number 12 was arbitrary. Well, guess what, it's not! In fact, it's almost completely determined by mathematics.I'll assume you know some basic facts about sound.
That takes care of ratio 2 … and also of ratio 4 (two octaves), ratio 1/2 (one octave the other way), and so on. But what about the next simplest case, ratio 3? Well, if we descend one octave to ratio 3/2, we find the pleasing sound that's known as a major fifth. That should definitely be part of any scale we construct! One way to accomplish that is just to declare that the scale contains an exact major fifth, but here I want to explore a different possibility. What if we divide an octave into n equallyspaced notes so that one of the notes is close to a major fifth? (A scale with equallyspaced notes is called equaltempered.) To be specific, suppose it's note #k that's close to a major fifth, where note #0 is the fundamental note and note #n is the note one octave higher. If we start at note #0 and take n steps of size k, that's roughly n major fifths, ratio (3/2)^{n}; if instead we take k steps of size n, that's exactly k octaves, ratio 2^{k}; and since we end at note #kn either way, those two ratios must be roughly the same: (3/2)^{n} ~ 2^{k}, or 3^{n} ~ 2^{k+n}. In other words, to obtain an equaltempered scale that has a decent major fifth, we need to find powers of 2 and 3 that are close together! If you want to try your hand, there are tables in some other essays.
Here I'll show you the systematic way to go about it. Starting from the equation above, we can take logarithms and simplify to get log_{2} 3 = (k+n)/n, which shows that instead of finding powers of 2 and 3 that are close together, we just need to find a good rational approximation to log_{2} 3. But that's easy … all we have to do is run the two continued fraction algorithms! One yields the continued fraction expansion, then the other spits out a series of good approximations. The only thing that's even remotely interesting about it is that the presence of the logarithm makes the calculation look like the Euclidean algorithm with subtraction replaced by division. In any case, here are the first few iterations of the first algorithm; you should know that log_{2} 3 = log 3 / log 2.
With a few more iterations included, the continued fraction expansion of log_{2} 3 turns out to be [1,1,1,2,2,3,1,5, … ]. Now we can run the second algorithm.
So, leaving out the uninteresting first few approximations, and adding the approximations that aren't convergents (that you get by varying the last coefficients in truncated expansions), here are the results … the first few equaltempered scales with decent major fifths.
The first, a fivenote scale, is based on the nice fact that 3^{5} = 243 is close to 2^{8} = 256. It's a pentatonic scale, since there are five notes, but when people talk about pentatonic scales, they usually mean different ones. The second, a sevennote scale, I don't know much to say about. You might think it would have something to do with the fact that we refer to notes using seven letters, plus sharps and flats, but I doubt it. Even if the seven letters are relics of a sevennote scale, it probably wasn't equaltempered. The third, the standard twelvenote chromatic scale, is based on the fact that 3^{12} = 531441 is close to 2^{19} = 524288. Beyond that, the numbers start to become inconvenient, so having twelve notes seems like just the right choice. By the way, here's an interesting fact. As the approximations to log_{2} 3 get better and better, the corresponding powers of 2 and 3 naturally get closer and closer … but only in terms of percentages. The numbers 8 and 9 (which can be used to construct a twonote scale) differ by 1, or about 12%; the numbers 243 and 256, by 13, or 5%; and the numbers 524288 and 531441, by 7153, or 1%. Does it ever happen that the numbers are adjacent again, like 8 and 9? Nope! In fact, if you consider the set of all powers (all numbers of the form a^{b} with both a and b at least 2), the numbers 8 and 9 are the only adjacent numbers in the whole set! That statement is Catalan's conjecture; it was apparently only proved a few years ago, in 2002. Isn't that amazing, that there are still things to learn even about the integers?

See AlsoDuodecimal @ November (2005) 