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.
G(12/5) = (1+ 2*1+1*(3-1)+1*(5-1)) = 9The 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: G[p_,q_]:=Module[{s=FactorInteger[LCM[p,q]]},1+Sum[s[[k,2]]*(s[[k,1]]-1),{k,Length[s]}]]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. |
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