That doesn’t provide the full understanding you might think it does. For example: define ‘right now’.
Instantaneously, music is a single sound pressure measurement. That doesn’t have a Fourier transform. It doesn’t have a frequency. It’s just a single sample.
Fourier transforms work on functions. Typically functions in the time domain. And typically (but not always) on that function within a bounded range of time. And the result is another function, this one of frequency.
A spectrum analyzer, though, is showing the Fourier transform of a short snippet of some music. Then a moment later it’s showing you the transform for the next snippet.
Looking at a spectrum analyzer makes you think a Fourier transform is itself a function of time (to some vector of numbers perhaps?). That is not the case. So looking at a spectrum analyzer can give you an incorrect intuition for what Fourier does.
But you can do a Fourier transform on the whole of a piece of music. You’ll pick up frequency components like the overall beat, the bar structure, the verse/chorus alternation.
I think I get what you are trying to say... but the intuition about "this moment in time" is perfectly reasonable for a spectrum analyzer, since it's actually doing a DFT (not continuous from +-infinity) with the last sample (i.e. "now") defining the end of the window.
The thing that makes a DFT discrete is that it is over individual samples rather than a continuous function - not that it is over a finite domain.
A Fourier transform applied to a brief window of an underlying continuous function is called a ‘short-time Fourier transform’.
And the frequency information a STFT can pick up is bounded on the low end (think, like the opposite of the Nyquist limit) by the length of the window - this is called the ‘Rayleigh frequency’ - if your window is of length t, you can not detect frequencies lower than 1/t. Which is why your ‘instantaneous’ spectrum analyzer (looking at a short burst of maybe 0.05s of samples) for your 120bpm EDM doesn’t pick up a frequency component at 2Hz - even though that component is there in a Fourier analysis of the whole piece. It can only measure down to 20Hz. Which is fine because that’s also roughly the limit of the part of the song ‘function’ that we hear as ‘tone’ rather than ‘rhythm’.
Respectfully, your characterization of a DFT's infinite domain conflicts with the definition of the DFT -- it is defined as a finite sequence, and that's how it's used in common industry usage. Case in point:
> In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples [...]
In practice it’s hard to store an infinite number of discrete samples, let alone process them. So I assume that’s why people don’t try.
A spectrogram remains a visualization of a short time Fourier transform at a number of points in time. In practice usually produced using a DFT because discrete samples are what you have to work with.
Pedantics aside: Spectrum analyzers are computing DFTs over a finite window, and it's perfectly reasonable to think of these as (an approximation of) power spectral density changing over time.
Right. But if you think Fourier transforms produce a function of ‘power spectral density over time’ you are on a road to misunderstanding. Or even if you think that it makes sense to talk about the Fourier transform ‘at a moment in time’.
The thing I am railing against here is the idea that you can just look at a spectrogram to grasp Fourier. You can’t. It is an advanced application of Fourier transforms that creates a visualization of power spectral density over time but it is not a (simple) Fourier transform of the underlying data.
That’s a good place to start building intuition, but it can also distract you actual understanding.
What that visualization really is is a bunch of arbitrarily bounded Fourier transforms of little windowed slices of a piece of music. The true Fourier transform of a time bounded piece of music is a single unbounded function over an infinite frequency spectrum.
Well, yes. That part is obvious enough. The interesting part is how you build the mapping from time domain to frequency domain. That is the part that never clicked for me.
Maybe that's because a sinusoid of any frequence has its "roundness" and that is a matter of "hunting" which that formula performs every infinite-small moment to an x/y graph of your favourite music composition.
