# period of a sine function in the form of $F(x) = \sin(ax) + \sin(bx)$. where the amplitudes of the two individual sine waves are the same? [duplicate]

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!

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.

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.

• 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. Jan 30, 2023 at 8:28