Fourier Transform: Musical instruments How do I Fourier Analyse the music produced by a musical instrument? What I mean is that what tools/applications are best suited to Fourier Analyse waves from musical instruments?
 A: Since I'm a Matlab user, here's how I would do it.  A lot of professional audio suites have Fourier analysis built in, they just call it "spectrum analysis".


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*Record the instrument using an audio recording software such as Audacity.  Or, you could just record a cell phone video, then use one of many applications to extract audio from cell phone videos (do a Google search to find one).

*Export the audio as a .wav or .mp3 file (at least, these are the most common).

*Import the .wav file into Matlab using the audioread function.  This will represent your (digitized) music as a vector, where each element in the vector is a single digital sample.  For example, a .wav file will have 44,100 samples for every second of audio - i.e. your vector will live in $\Bbb{R}^{44,100\cdot T}$ if your audio is mono (as opposed to stereo) and $T$ seconds.  Actually, digital audio is also quantized, meaning you don't get an actual real number for each sample, but rather an integer.

*Once you have this vector, you can then run whatever Fourier analysis you want - fft, wavelet transforms, etc. 
If you don't have Matlab, octave and python have similar packages for working with audio files (a quick Google will let you know what's out there).
Here's a quick example using Radiohead's song "Burn the Witch".  I imported it using audioread('file.mp3').  This gives me a matrix $x$ that is $9728781\times 2$ (two columns because it's a stereo recording).  Here's a plot of the first second of the song - one channel is blue, the other is red.

I can do an FFT on this vector by calling $X = fft(x)$.  Here's a plot of the modulus of $X$: 

Now that your song is just a vector, you can do all sorts of fun things like filter it, use it as the initial condition for a heat equation and see what the result sounds like, you dream it!
