Conducting a fourier series given amplitude and frequency I'm currently writing an extended essay on why instruments sound different when they play the same notes, I have recorded a few instruments and plan on setting up some data but I have no Idea what data I should be looking at. I know I should be looking at fourier series but after watch videos I see that a fourier series is used to model a sinusoidal wave.
How does this connect to instruments other than the fact that they also produce sinusoidal waves, should I instead be looking at turning the initial sound wave into a frequency-amplitude graph. I understand all the mathematics involved but where do I start?
 A: You're on the right track, but you may have confused yourself with a belief that a Fourier series models a sinusoidal wave.  It does not; it models any arbitrary signal (such as the sound produced by a musical instrument) as a combination of sinusoidal waves of different frequencies. Each sin wave, which is identified by its frequency, will have a different amplitude.  A Fourier series describes the sound as a combination of all of these sin waves.
When you make a recording of an instrument, what you're really doing is periodically sampling the sound-wave intensity, or more accurately the variation in air pressure that our ears interpret as sound.  This sampling is typically performed at 44,100 times per second, so you'll get a recording of the intensity as a function of time as measured every 1/44100th of a second.  What you're interested in, however, is not the intensity as a function of time, but the intensity as a function of frequency.  In other words, you want to decompose the sound made by an instrument into different notes, all of which are produced simultaneously, and each of which has a different intensity.
The way to transform a signal from "intensity as a function of time" to "intensity as a function of frequency" is to take the Fourier transform of the signal (Wikipedia article: https://en.wikipedia.org/wiki/Fourier_transform).  A Fourier transform is able to resolve frequency components up to half the sampling frequency, so if you have a sampling frequency of 44100 Hz, you should be able to resolve frequencies up to 22050 Hz - which is just above the limit of human hearing.
If you have two instruments playing the same note, the strongest frequency component should be the same for both instruments.  For example, if you look at the Fourier-transformed signals of two instruments playing Middle C, for each instrument you should see the largest peak at a frequency of around 261.6 Hz (the frequency of Middle C).  What will be different is the relative amplitude of the other frequency components, particularly the various harmonics. For example, for the Middle C-playing instruments, you should see progressively smaller spikes at around 532.2 Hz, 784.8 Hz, 1046.4 Hz, etc.  The size of these peaks should be different for each instrument.  You will also see smaller amplitudes at various other frequencies; these amplitudes will also be different for each instrument.  This is what causes the instruments to sound different.
