The Twelve-Note Scale

I'll assume you know some basic facts about sound.

• Sound consists of pressure waves (in air, water, or whatever).
• A note corresponds to an oscillation at some frequency.
• Two notes sound nice together if the ratio of frequencies is close to a rational number with small numerator and denominator.
• If the ratio is 2, the corresponding notes are an octave apart.

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 equally-spaced notes so that one of the notes is close to a major fifth? (A scale with equally-spaced notes is called equal-tempered.)

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 3ⁿ ~ 2^k+n. In other words, to obtain an equal-tempered 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.

Powers of 2 and 3	up to	216	310
Powers of 2	up to	232
Powers of N	up to		320

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 log₂ 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 log₂ 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 log₂ 3 = log 3 / log 2.

					ai	rest
log	3	/	log	2	1	log	3/2	/	log	2
log	2	/	log	3/2	1	log	4/3	/	log	3/2
log	3/2	/	log	4/3	1	log	9/8	/	log	4/3
log	4/3	/	log	9/8	2	log	256/243	/	log	9/8

With a few more iterations included, the continued fraction expansion of log₂ 3 turns out to be [1,1,1,2,2,3,1,5, … ]. Now we can run the second algorithm.

		1	1	1	2	2	3	1
0	1	1	2	3	8	19	65	84
1	0	1	1	2	5	12	41	53

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 equal-tempered scales with decent major fifths.

k	3	4	7	17	24	31
k+n	8	11	19	46	65	84
n	5	7	12	29	41	53

The first, a five-note scale, is based on the nice fact that 3⁵ = 243 is close to 2⁸ = 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 seven-note 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 seven-note scale, it probably wasn't equal-tempered.

The third, the standard twelve-note chromatic scale, is based on the fact that 3¹² = 531441 is close to 2¹⁹ = 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 log₂ 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 two-note 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?