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   John Carlos Baez (johncarlosbaez@mathstodon.xyz)'s status on Saturday, 28-Feb-2026 09:57:32 JST John Carlos Baez

   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 log₂(3/2) than any fraction with a smaller denominator.

   You get *some* of these by taking the continued fraction expansion of log₂(3/2), shown below, and truncating it at some point.

   This method gives you the 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

   and so on. But this method does not give *all* the fractions that come closer to log₂(3/2) than any fraction with a smaller denominator. For example, 2/3 and 4/7 are two you don't get by this method!

   These others show up whenever a number in bold is bigger than 1. The second time this happens is for

   1/(𝟭 + 1/(𝟭 + 1/(𝟮 + 1/𝟮)) = 7/12

   What we do then is look at the previous fraction (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 log₂(3/2) 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 log₂(3/2) than any fraction with a smaller denominator. But this procedure does give every fraction with that property.

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

   (1/n)

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     John Carlos Baez (johncarlosbaez@mathstodon.xyz)'s status on Saturday, 28-Feb-2026 09:58:05 JST John Carlos Baez

     in reply to

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

     But first, three useful buzzwords.

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

     • The other candidates for being best approximations, obtained by the weird procedure I described, are called 'semiconvergents'. These include convergents as a special case.

     • Given two fractions a/b and c/d their 'mediant' is (a + c)/(b + d). The weird 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'. See below!

     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".

     Or read this:

     https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree

     All this from trying to understand equal-tempered scales!

     (3/n, n = 3)

   • Embed this notice
     John Carlos Baez (johncarlosbaez@mathstodon.xyz)'s status on Saturday, 28-Feb-2026 09:58:06 JST John Carlos Baez

     in reply to

     Let's look at this wacky procedure in a more interesting case: when the number in boldface is even bigger, like 5. See below! This fraction is

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

     The previous fraction on the list is

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

     and before that is

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

     To look for good approximations to log₂(3/2) that aren't on our list, we 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, 179/306 (on our list).

     The four not on our list are new candidates for fractions closer to log₂(3/2) than any fraction with a smaller denominator! But only the last two 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 log₂(3/2) 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 log₂(3/2). And 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.

     (2/n)

     Blaise Pabón - controlpl4n3 repeated this.