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As the title says, I'm working on a research paper for my IB diploma, that is trying to find a link between patterns in music and their waves. but I've encountered a problem when trying to mathematically calculate the period of a function that is a sum of two different sin functions with the same amplitude but different frequencies. What I have been doing is plotting the function on Desmos and finding it manually, which doesn't help me with what i am currently trying to do. I looked far and wide on the internet but I found nothing.

I'm doing this to see how the change in the difference between the frequencies (b/a) effect the period of a function. and would appreciate any help!

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If $a/b$ is real rational, then the period of $f(x)$ is LCM$\left(\frac{2 \pi}{a}, \frac{2 \pi}{b}\right)$.

If $a/b$ is irrational then $f(x)$ is not periodic.

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If the ratio $b/a$ is not rational then the resulting function will not be periodic. For example, the tritone (e.g. C to F#) in well tempered tuning has a ratio of $\sqrt 2$.

If the ratio is rational, $\frac{b}{a} = \frac{p}{q}$ where $p$ and $q$ are integers with no common factor then $p$ cycles of the $a$ wave will match $q$ cycles of the $b$ wave. The sum will be periodic. Generally, this will the period but they may be exceptional cases, e.g. if $a = -b$ then the sum will be $0$.

E.g. a perfect fifth in just tempered tuning has the ratio $\frac{3}{2}$ so $2$ cycles of the lower frequency will match 3 cycles of the higher.

As well as assuming the same amplitude, you are assuming that the waves start together. This won't be true in general.

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  • $\begingroup$ Yes, i have encountered a problem with well-tempered tuning. and thus I have done all calculations based on just tempered tuning. you'll have to give me a moment for me to check it out. $\endgroup$
    – Bouncyy
    Jan 30, 2023 at 8:28

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