Which equation? Sin, math and music

Question

I found it inside an italian music book:

It [artistically] represents the music scale. I think it is a series of trigonometry sin equation. With some difference in the amplitude and in the frequency.

What are the exact parameters of the curves?

Answer 1

The figure appears to shows 12 superposed sine curves with wavelenghts $1$, $\frac12$, $\frac13$, down to $\frac1{12}$, and amplitude proportional to the wavelenghts, such that the slopes at the nodes are equal. In more elementary mathematical terms, it is a plot of the 12 functions $$ f_n(x) = \frac1n \sin(nx), \qquad\qquad 1\le n\le 12 $$ on the interval $x\in [0,2\pi]$ (and then everything turned 90­° such that the $x$-axis is vertical).

This is not directly related to the diatonic scale alluded to on the left.

If the vertical (on the page) axis represents time, the notes corresponding to the sine waves are the first 11 harmonics of the one represented by the largest wave -- many of these do correspond to "pure" intervals in music, but don't directly map to the seven steps of a diatonic scale.

Assuming that the base tone is a C ("do"), the 12 tones depicted are $$ C, C, G, C, E, G, A^{(\sharp)}, C, D, E, F^{(\sharp)}, G $$ spanning $3\frac12$ octaves, where the bracketed sharp signs indicate in-between notes that are not usually used in Western music.

Answer 2

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These are the 12 harmonics of a given frequency, as Henning Makholm points out in his answer. All the sine waves shown are integer multiples of some base frequency, one cycle of which is shown. Many of these correspond to pitches of the just intonation scale, which is not the scale to which modern Western instruments e.g. keyboards are tuned. In that tuning, the "tempered scale", pitches are adjusted ("tempered") so that all keys are equally, subtly out of tune :) and so that modulations become possible and none sound sour.

The diagram shows one cycle of "do", whatever frequency that might be, and all the other harmonics plotted against it (plus the octave above "do"). No base frequency is given, so the diagram is relative to some/any chosen reference. If $\theta_{do}$ is that frequency in Hz (cycles per second), then this shows one cycle of $\sin \theta_{do}$, plotted against $\sin$ of various multiples of $\theta_{do}$. The amplitudes seem a little arbitrary — happened to make for a legible diagram. (They're not entirely arbitrary, of course: sine waves with more cycles per the period shown have smaller amplitude.)

The octave above is twice the frequency, and that's represented in the diagram by the sine wave which completes 2 cycles. The sine wave of the octave is $\sin 2\theta_{do}$.

"so", the 5th, is $\frac 3 2$ times the frequency of "do" in just intonation, so that's represented in the diagram by the sine wave in which completes $3$ cycles. The pitch of this is actually "so" of the next octave.

"mi", the natural $3$rd, has ratio $\frac 5 4$ with "do". It's represented by the sine wave which completes 5 cycles. Note that the frequency of the sine wave shown is thus $2$ octaves above "mi".

The ratios of all notes of the just intonation scale are as follows:

$$\begin{align} &\text{do} \quad&1/1 \\ &\text{♭ re} \quad&16/15 \\ &\text{re} \quad&9/8 \\ &\text{♭ mi} \quad&6/5 \\ &\text{mi} \quad&5/4 \\ &\text{fa} \quad&4/3 \\ &\text{#fa} \quad&45/32 \\ &\text{so} \quad&3/2 \\ &\text{♭ la} \quad&8/5 \\ &\text{la} \quad&5/3 \\ &\text{♭ ti} \quad&9/5 \\ &\text{ti} \quad &15/8 \\ &\text{do'} \quad & 2/1 &\quad \text{(do, an octave up)} \end{align}$$

Some of these can't be octave-adjusted (multiplied or divided by some power of $2$) to fit within a single cycle of "do". This is why tempered tuning was invented! Tempered tuning provides closure; for any interval, some integer multiple of that interval gets you to an octave of "do". For example, three rising Major 3rds brings you to the octave of your starting note, whereas no integer multiple of the just-intonation "mi" is an octave of "do".