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