In western music an octave is devided into 12 equal steps. The frequencies form a geometric sequence with factor 2 1/12 . This multiplicative structure comes from the fact that the human ear listens logarithmically. Rational Frequency relations are considered "harmonic" . A geometric sequence is translational invariant but can only approximate rational numbers. With 12 steps, this can be achieved well, better for example than in Stockhausen's 5 1/5 scale.
Euler's music theory assigns to a frequency ratio p/q a number G(p,q)=G(p/q) called "gradus suavitatis" which could be translated as "degree of pleasure". G(p/q) is defined as 1+ sum ei (p i-1), where p i are the prime factors with multiplicity ei of the least common multiple of p and q. Lets look at the little decime 12/5=p/q. Since lcm(12,5)= 60=22+3+5, we have
 G(12/5) = (1+ 2*1+1*(3-1)+1*(5-1)) = 9
The geometric scale with 12 steps interpolating the frequency doubling 1:2 allows to approximate some rational numbers "with pleasure". For example, 2 2/12 = 1.122462048 ... is close to 9/8=1.125. The function G(n,m) can be determined with the online-calculator to the left (look at the source to see the implementation in Javascript). Here is a Mathematica implementation of the Gradus Suavitatis:
which was used to plot the function G(n,m) on the positive quadrant in the integer plane. The color encodes the value of G(n,m).
References: Wille: Mathematische Musiktheorie, 1983 [PDF], in "Musik and Mathematik" (in german), editors Heinz Götze and Rudolf Wille, Springer. Acknowledgement: Thanks to Vicente Liern for pointing out an error about the Euler Suavis function. This got only corrected in April 28, 2012.
© 2000-2012