Fourier Transform on a Sound

Unfortunately I don't think that Mathematica really knows anything about the frequency content of SoundNote objects. They are not audio files (recorded in the time domain), but more like MIDI objects. (In fact I think they're 'sonified' by sending them as MIDI events to the OS.)

You can, however, do a fourier transform on an audio file. Either import your own with Import, or use one of the example sounds:

sound = ExampleData[{"Sound", "ViolinScale"}]

We can extract the list of samples from the single SampledSoundList inside this Sound object:

samples = sound[[1, 1, 1]];

We can also extract the sample rate:

sampleRate = sound[[1, 2]];

From here we can compute the FFT and plot:

fft = Fourier[samples];
ListLogPlot[Abs[fft]^2, PlotRange -> All, 
 DataRange -> {0, (1 - 1/Length[samples]) sampleRate}]

There is a lot of other analysis you can do, but that should get you started.

This is easy to do now with Mathematica's Audio support introduced in v11.0 in 2016 - though note Spectrogram was available in 2012 v9.0:

au = Audio[Sound[SoundNote["G", 1, "Violin"]]];
Spectrogram[au]

(* ... or if you want an array ... *)
spectrogramData = SpectrogramArray[au];

Another option is to use the Periodogram function:

sound = ExampleData[{"Sound", "ViolinScale"}];
Periodogram[sound,
 Frame -> True,
 Axes -> None,
 BaseStyle -> {14, FontFamily -> "Helvetica", FontColor -> Black},
 FrameStyle -> Directive[Black, 14], 
 FrameLabel -> {"Freq (Hz)", "dB"}]