Is $\sin(440\cdot2\pi x)+\sin(440\cdot 2^{1/4}\cdot2\pi x)+\sin(440\cdot2^{1/2}\cdot2\pi x)$ periodic?

Question

Can anyone help me to determine whether the function below is periodic or not? If it is periodic, can anyone tell me how to find the period.

$$y=\sin(440\cdot2\pi x)+\sin(440\cdot 2^{1/4}\cdot2\pi x)+\sin(440\cdot2^{1/2}\cdot2\pi x)$$

Answer 1

To expand a bit on Ajay's quick comment: that function is not periodic due to being the sum of three continuous periodic functions with incommensurate periods (i.e. at least one of the ratios between the different periods is irrational) such that all of the possible sums of these functions are non-constant (i.e. $f_1$, $f_2$, $f_3$, $\lambda f_1 + \mu f_2$, $\lambda f_1 + \mu f_3$, $\lambda f_2 + \mu f_3$ and $\lambda f_1 + \mu f_2 + \nu f_3$ are all non-constant; without this condition, you can easily create "degenerate" counterexamples, for example by picking $f_2 := -f_1$, in which case $f_1 + f_2 + f_3$ is periodic iff $f_3$ is periodic). See Theorem $1$ in this paper for reference.

There are quite a few posts on MSE for the particular case of a sum of two continuous periodic functions but I couldn't seem to find a satisfying one for the generalisation to multiple functions, to my dismay. But oh well.