Another multi-dimensional abstraction for music that's at a bit higher level is a musical lattice. The number of dimensions is limited by the largest prime you accept in the ratios you use. (Typical western music approximated by 12-tone equal temperament uses primes of 2, 3, and 5. Powers of 2 are often ignored because notes an octave apart are perceived to be sort of equivalent for harmonic purposes.)
Spectrogram is 2D (plot of amplitude given time and frequency).
Its interesting to think about it for a spectrogram because "similarity" is different in each dimension (freq vs. time).
Frequency is also perceived logarithmically, so you would probably want to convert to e.g. Mel scale before applying this algorithm (a 2000-2100Hz change is much subtler than a 200-300Hz change).
Isn't that 3 dimensions (amplitude, time, and frequency)? The plot of course fills 2 spatial dimensions and uses color to represent the 3rd dimension. But I don't know very much about this.
I don't have a mathematically rigorous understanding of it but the number of dimensions is basically the number of freely varying inputs to the corresponding functional representation.
In a 2d image, x position and y position are mapped to a color, e.g. I(x,y) = C.
In a spectrogram, freq and time are mapped to a color (amplitude) e.g. S(f,t) = A.
In neither case can you just pick an arbitrary color or amplitude and in general produce a singular x/y or f/t from that.
