Equal Temperament (Part 3)

Say you want to find all the N-tone equal tempered scales that have better fifths than any scale with fewer notes. Mathematically this means you want to find all fractions that come closer to

than any fraction with a smaller denominator.

Let me show you what they are, and then talk about how you can efficiently find them:

You get some of these by taking the continued fraction expansion of shown below, and truncating it at some point.

This method gives you a list of fractions

1/𝟭 = 1 1/(𝟭+ 1/𝟭)) = 1/2 1/(𝟭 + 1/(𝟭 + 1/𝟮)) = 3/5 1/(𝟭 + 1/(𝟭 + 1/(𝟮 + 1/𝟮)) = 7/12 1/(𝟭 + 1/(𝟭 + 1/(𝟮 + 1/(𝟮 + 1/𝟯)) = 24/41

and so on. These are all closer to than any fraction with a smaller denominator. But this method does not give all the fractions with that property. For example, 2/3 and 4/7 are two you don’t get by this method!

The others show up whenever a number in bold is bigger than 1. For example, the second time this happens is for

1/(𝟭 + 1/(𝟭 + 1/(𝟮 + 1/𝟮)) = 7/12

What we do in this case is look at the previous fraction in the list (3/5) and the one before that (1/2), and write down this funny thing built from those two:

(3n + 1)/(5n + 2)

When n = 0 this is 1/2 (already on our list), when n = 1 this is 4/7 (new), and when n = 2 this is 7/12 (already on our list).

The number 4/7 is not on our list, so it’s a new candidate for being closer to than any fraction with a smaller denominator. And it is!

You may get more than one new number using this procedure. Alas, they aren’t always closer to than any fraction with a smaller denominator. But this procedure does give every fraction with that property.

Even better, this algorithm is a completely general procedure for finding the best rational approximations to irrational numbers!—where by ‘best’ I mean: closer than any fraction with a smaller denominator.

Let’s look at this wacky procedure in a more interesting case: when the number in boldface is even bigger, like 5.

This fraction is

1/(𝟭 + 1/(𝟭 + 1/(𝟮 + 1/(𝟮 + 1/(𝟯 + 1/(𝟭 + 1/𝟱)))))) = 179/306

We do the same thing as before. We look at the previous fraction in the list:

1/(𝟭 + 1/(𝟭 + 1/(𝟮 + 1/(𝟮 + 1/(𝟯 + 1/𝟭))))) = 31/53

and the one before that:

1/(𝟭 + 1/(𝟭 + 1/(𝟮 + 1/(𝟮 + 1/𝟯)))) = 24/41

We use these to write down this funny thing:

(31n + 24)/(53n + 41)

Taking n = 0,1,2,3,4,5 this gives 24/41 (on our list), 55/94, 86/147, 117/200, 148/253, and 179/306 (on our list).

The four not on our list are new candidates for fractions closer to than any fraction with a smaller denominator! But only the last two of these actually have this good property: 117/200 and 148/253.

In general, when we get an even number of new candidates this way, the last half have this good property. The first half do not.

When we get an odd number of new candidates, it becomes more tricky. The middle one can go either way—but all those after it are closer to than any fraction with a smaller denominator, and none before are.

There is a rule to decide this tricky middle case, but you’ve probably had enough by now!

Again: what makes all this stuff worth knowing is that it gives the best rational approximations of any positive irrational number, not just This is relevant to resonance problems like the rings of Saturn, which have gaps at orbital periods that are close to simple rational multiples of the periods of the big moons. But more importantly, it’s a basic part of number theory.

There are various places to read more about this stuff. I haven’t read them yet, I’m ashamed to say!

But first, some useful buzzwords.

• The best approximations to an irrational number coming from truncating its continued fraction are called ‘convergents’.

• The other candidates for best approximations, obtained by the wacky procedure I described, are called ‘semiconvergents’. These include convergents as a special case. Here are the first 14 fractions that are closer to than any fraction with a smaller denominator:

9 are convergents, and the rest are only semiconvergents.

• Given two fractions a/b and c/d their ‘mediant’ is (a + c)/(b + d). The procedure I described is based on mediants. Starting from the numbers 0/1 and 1/0 you can build a tree of numbers by taking mediants, called the ‘Stern-Brocot tree’. It looks like this:

Here are some books:

• Khinchin’s Continued Fractions covers best approximations and semiconvergents carefully, including the delicate middle case.

• Rockett and Szüsz’s Continued Fractions goes into the best-approximation theory in lots of detail.

• If you like the Stern–Brocot tree, you may like to think about how semiconvergents are connected to that. For this, see Conway and Guy’s The Book of Numbers, and Graham, Knuth, and Patashnik’s Concrete Mathematics. Both these books are packed with fun.

Or read this:

• Wikipedia, Stern–Brocot tree.

All this from trying to understand equal-tempered scales!

For more on equal temperament, read these:

• Part 1: Some equal-tempered scales with better perfect fifths than all equal-tempered scales with fewer notes: 1-TET, 2-TET, 3-TET, 5-TET, 7-TET, 12-TET, 29-TET, 41-TET and 53-TET.

• Part 2: Patterns that emerge when we study which equal-tempered scales have the best perfect fifths, major thirds or minor thirds.

For more on Pythagorean tuning, read this series:

• Pythagorean tuning.

For more on just intonation, read this series:

• Just intonation.

For more on quarter-comma meantone tuning, read this series:

• Quarter-comma meantone.

For more on well-tempered scales, read this series:

• Well temperaments.