## Monday Morning Math: More Math and Music

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Good morning! This week we have a bit more info about music (thanks TwoPi!):   Last week I mentioned a “just fifth”, which is when the frequencies of two notes are in a 3:2 ratio: A is 440 Hz (on a piano at least) and the note E above that is at 660Hz.  These two notes are a perfect fifth, and the ratio 660:440 is a 3:2 ratio.

The frequencies in an octave, on the other hand, are in a 2:1 ratio:  The A above the one with 440 Hz has 880 Hz.  So the work that Jing Fang did, showing that 53 fifths was almost exactly 31 octaves, amounted to showing that $\left(\frac32\right)^{53}$ (which is approximately 2,151,972,563.22) is very close to  $2^{31}$ (which is 2,147,483,648).  If you want to avoid fractions, this means that $3^{53}$ is almost the same as $2^{84}$.  They won’t be exactly the same, but it’s pretty close.

But let’s go a little more in depth!  The savvy reader (that would be you) might realize that we can’t ever have a power of 3 exactly equal to a power of 2, because they are different prime factors.  This means we’ll never have a power of 3/2 equal to a power of 2.  This is a fundamental problem with piano tuning:  if you prioritize fifths then the octaves don’t exactly match up, and if you prioritize the octaves then the fifths don’t exactly match up.  Piano tuning today prioritizes the octaves, and uses the fact that 12 fifths is approximately the same as 7 octaves.  That is, $\left(\frac32\right)^{12}$ (which is approximately 129.7) is very close to $2^7$, which is 128.

Isn’t that cool?  If you want to hear some of it, here’s a 3-minute video of mathematician and musician Eugenia Cheng giving more detail and connecting it to Bach and to Category Theory:

And if you want even more music, here’s some music played on a Mobius strip, just for fun:

(And there are more videos on the AMS page on Math and Music.)