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Several octaves - Euler's "gradus suavitatis" (degree of sweetness)

Leonhard Euler's formula for consonance of musical intervals:

 

Basically he says to multiply the numerator and denominator, then find the prime factors of the product, then subtract 1 from each prime, and then sum all the primes with 1.

 

So for 16/9, the product is 144, and the prime factors are [3, 3, 2, 2, 2, 2]. So Euler's number is 1+(2+2+1+1+1+1) = 9.

 

You can see that the octaves are not equivalent. In the third octave, the major third (5:1) ranks better than the perfect fourth (16:3), for instance, since it's part of the harmonic series.

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Uploaded on June 2, 2009