MusicAndNumbers

<i>The unreasonable effectiveness of number theory</i>, Proceedings of Symposia in Applied Mathematics vol. 46, pp. 4 and 5, AMS 1992.

String instruments produce simple frequency ratios when their strings are subdivided into equal lengths: shortening the string by one half produces the frequency ratio $2:1$, the octave; and making it a third shorter produces the frequency ratio $3:2$, the perfect fifth.

In perception, ratios of small integers avoid unpleasant beats between harmonics. Apart from the frequency ratio $1:1$ (unison), the octave is the most easily perceived interval. Next in importance comes the perfect fifth. However, musical scales exactly congruent modulo the octave cannot be constructed from the fifth alone because there are no positive integers $k$ and $m$ such that $$\left(\displaystyle\frac32\right)^m=\left(\displaystyle\frac21\right)^k.$$

A good approximation of the above equation can be obtained by writing $3^m=2^n$, $(n=m+k)$, and so $\displaystyle\frac nm\thickapprox\log_23$. We expand $\log_23$ into a continued fraction: $$\log_23=[1,1,1,1,2,2,\dots],$$ and obtain a close approximation $m=12$ and $n=19$ (hence $k=7$). Thus, the equal tempered fifth comes out as $2^{7/12}=1.498\dots$ with the basic interval $1:2^{1/12}$ of the semitone system.

(1636) Mersenne's Laws

(Pythagoras's law) When the tension on a string remains the same but the length $l$ is varied, the period of the vibration is proportional to $l$.
When the length of a string is held constant but the tension $T$ is varied, the frequency of oscillation is proportional to $\sqrt T$.
For different strings of the same length and tension, the period is proportional to $\sqrt w$, where $w$ is the weight of the string.

Just Intonation