Interesting thread! I'm not a mathematician, just an IT guy with barely enough math knowledge to be dangerous, but hear me out.

It always seemed to me that a waveform is something that can only be approximated. Unless it's generated by a function, that is (like a sine wave.) In music, the waveforms can only be approximated since the underlying sounds are very complex and chaotic. The actual complexity of a waveform makes it a very fractal-like structure. As such, it might make more sense to assign a fractal dimensionality to waveforms, which probably lies somewhere between 1 and 2?

No.

As you said, a simple sine can be accurately expressed through it's coefficients, like frequency, amplitude, phase-shift and bias. The same is true for any higher order signals, you just add more and more of the sinusoidal components (all signals can be expressed as a sum of sinusoids). For some signals, that number of components is infinite, those can only be approximated at a certain cut-off.

However, this is not a problem in practical applications, since prior to sampling, you filter the signal below the max frequency, according to the Shannon-Nyquist sampling theorem, to avoid artifacts through under-sampling.

Assuming your sampling frequency is 48kHz (as most sound cards do), the maximum frequency is 24kHz, according to the Shannon-Nyquist theorem. All signals sampled by that device are accurately reproduce-able, up to the frequency of 24kHz. In practice, the sound card must therefore low-pass the analog sound at 24kHz. This low-passed signal can now be exactly reproduced, from the sampled data. Keep in mind, that we're discussing sound in the human aural spectrum here. Anything above 20kHz is inaudible, anyhow. The higher frequency cut-off is mainly to allow the filter some headroom, as no filter is abrupt. Also note, this is us discussing the ADCs ability to sample sounds. As an ADC has a technical limit at which it can reproduce a signal, it will simply refuse to output signals of higher sampling rates.

Also, keep in mind that this is also exactly true in the RF spectrum. you can easily build a device to sample in the 10GHz range and, reproduce whatever signal at ½f of that (<5GHz, in this example).

The idea of dimensionality as you tried to infer, is a non-sequitur, and is kinda in-line with what Saratoga mentions, how dimensionality is misunderstood in this context. I must admit, me explaining complex planes as being two dimensional was kinda misleading, too. I should've kept this out of the discussion.

It is also exactly not a fractal, or fractal-like! Fractals are also by definition functions, which are not differentiable at any point however signals are very much differentiable at allmost all points. Fractals are highly structured, the exact opposite of noise! However in analog sampled signals (sound waves in particular) we're dealing specifically with noise, which is by definition, un-structured. Fractals are the very essence of patters, while noise is the absence of patterns. Especially in music the wave forms are extremely well-formed and highly harmonic, compared to fractals. Fractals are essentially a form of symmetry.

However, there is fractal-like music or perhaps less so music, but fractal-like tones: The Shepard Tone is an Example that comes to mind: https://www.youtube.com/watch?v=BzNzgsAE4F0

One attribute of fractals is self-similarity, the Shepard Tone essentially implements that, but obviously the non-differentiability isn't there.

For some reason I just forgot the rest of my ideas I should go to bed.

Last post by loft -
I know this will not be directly helpful but... I've been searching for something similar a few months back without much success: most (if not all) solutions were asking for money. I remember one letting non-paying users scrobble some 10 tracks per day, but I considered it waaay to little (like probably you would too).

More importantly, being complex valued in this case is distinct from defining a plane as you are assuming. Recall that the FFT of a real valued function (such as a series of sampled points of a transducer in time) must be symmetric about zero. This means N unique values would seem to map to N/2 values - which would make the transform non invertible. Instead it actually maps to N/2 complex value pairs, or more simply just N values. Thus it's actually a mapping from N to N and therefore invertible.

Ah right, this is actually a good point. For some reason I ended up going more into complex numbers as composite values.

A two-dimensional DFT also results in a two-dimensional frequency domain function which is also complex-valued, as it is with image processing. I was probably focusing too much on explaining that a complex number has two components (which is more of a consideration in programming).

Btw. FYE: MRI machines return a (2D) frequency domain, the picture has to be IFFT'd to reconstruct the image.

The two pictures I've attached is Mr. Joseph Fourier, and Mr. Fourier, fourier-transformed. Should be noted, that's the combined image with both magnitude and phase merged, and centered around 50% gray for 0.

Last post by Nikaki -
Interesting thread! I'm not a mathematician, just an IT guy with barely enough math knowledge to be dangerous, but hear me out.

It always seemed to me that a waveform is something that can only be approximated. Unless it's generated by a function, that is (like a sine wave.) In music, the waveforms can only be approximated since the underlying sounds are very complex and chaotic. The actual complexity of a waveform makes it a very fractal-like structure. As such, it might make more sense to assign a fractal dimensionality to waveforms, which probably lies somewhere between 1 and 2?

Last post by Peter -
Thanks for the bug report. The new version also includes latest FFmpeg code, which seems to have backfired for some older devices. The change will be reverted shortly.