I'm new to this forum so thanks for having me. I'm a musician and wave theorist currently working on a problem relating to tonality in music.
I need to find the extrema of the sum of 3 sine waves of whole-numbered frequencies A, B and C in lowest form i.e. f(t) = sin(2piAt) +sin(2piBt) +sin(2piCt) is periodic with a GCD of 1Hz.
The 1st derivative test for extrema gives
f'(t) = 2piAcos(2piAt) +2piBcos(2piBt) +2piCcos(2piCt) = 0
where I need to solve for t. Since the wave is periodic we can restrict the domain to 0 <= t <= 1. I've tried finding trig substitutions, reframing the problem in complex form etc but this one truly has me stumped. I would have pulled all my hair out by now if I had any.
If somebody could help me with this it would be greatly appreciated.