Revolutionary Approach To Music As A Math Formula

[make2c]

Currently, no mainstream audio codec, synthesizer, or analysis tool does exactly “one simple math function per detected time-domain peak/valley in a soundwave,” but my proposed novel approach called Analytic Lobe Hypothesis will do just that. To see the full write up google: Github Analytic-lobe-hypothesis

This approach to audio representation: detect every peak and valley, then fit each lobe with a single simple analytic function transforms any music to pure, differentiable math. Open invitation for musical hackers and DSP explorers to develop this idea further. Additive synthesis via FFT has served us brilliantly for decades, but it still forces us to work with millions of raw amplitude samples or frequency bins. What if we instead scanned a waveform once, located every local maximum and minimum, and for each tiny lobe between them fitted a single, compact analytic function—perhaps a Gaussian, a cubic polynomial, or a raised-cosine arc? The Analytic Lobe Hypothesis proposes exactly that: any complex musical signal, from a 2:45 classic like “Somewhere Over the Rainbow” down to its roughly 7 million samples and several million extrema, can be reconstructed as an ordered list of these micro-functions. The result is music expressed as pure mathematics—piecewise analytic, fully differentiable, and dramatically more amenable to symbolic computation, exact derivatives, and algebraic manipulation than any sample buffer or spectrogram has ever been. A laptop can do this in less than 1 second. Ultimately, AI could move from imitating audio to truly reasoning about acoustic shape—bridging the symbolic world of music theory and the continuous world of sound in a single, unified mathematical language.

[Tj Shredder]

Reading this
https://github.com/sfeltenberg/Analytic ... /README.md
I do not see any difference to Fouriers approach. It seems you are 200 years late…

[hugoderwolf]

Do you have any kind of background in the fields of DSP or mathematics?

[Music Engineer]

It reminds me of wavelet analysis or maybe a 1D version of radial basis function networks. Yes - one can reconstruct an arbitrary (smooth) function to arbitrary accuracy using a sum of appropriately scaled and shifted Gaussian bell curves, for example (and by scaled, I mean scaled in the x- and y-direction, where scaling in the x-direction would have the effect of narrowing or broadening the bell/bump). The question would be how useful such a representation would be. For a signal representation to be musically useful, it should reveal perceptually important features and make them available for editing in a straightforward way - and I'm not sure, if or how breaking down a function into Gaussian bumps would do that. But then, I did not yet think about this for very long, so this may not mean much.

Edit: A Gaussian bump is basically a smeared/smoothed impulse, so it would seem to me that breaking down a signal into a weighted sum of those (shifted in time and perhaps be adjusted in width) would amount to breaking down the signal into impulse transients with different bandwidths (via the widths, i.e. x-scale), locations/time-instants (via the x-shift) and amplitudes (via the y-scale). The broad bumps would represent strongly lowpassed transients that are more smeared out in time whereas the narrow bumps/spikes would represent transients with higher bandwidth. Such a time-centric representation may be seen as being somewhat complementary to the (frequency centric) Fourier representation via sinusoids. One might even argue that the raw signal samples themselves already constitute such a representation in terms of scaled and shifted bump functions. Each signal sample represents a sinc-shaped bump. It's not really a clean "bump" though because of all the pre- and post-ringing of the sinc shape.

[Surgo]

What in the AI psychosis is this.

[G-Spot]

Without delving into details, from a mathematical viewpoint there are three cases: the representation of audio by "micro-functions" requires less data than the ordinary representation with samples, or the same amount, or more.

The first case is noisy compression, and we already have efficient and principled algorithms for it. There might be hope for use in a synthesizer, which could be elegantly restricted to "nice" sounds.

The second case might be a starting point for fancy processing, but "truly reasoning about acoustic shape", or enabling something novel at all, remain to be demonstrated.

The clearly inefficient third case is very likely if perfect reconstruction is desired, due to effects like the increase of range from original values (between A and B) to sums (between 2A and 2B, one more bit of precision).

[MirkoVanHauten]

Doesn't FLAC work with breaking down segements to describe them with functions + error signal or at least very similar? I thought it was always described to me as the SVG of audio

[Music Engineer]

Doesn't FLAC work with breaking down segements to describe them with functions + error signal or at least very similar? I thought it was always described to me as the SVG of audio

FLAC is based on linear prediction. If I remember correctly (I wrote my final exam about this around 20 years ago) it computes for each block a bunch of LPC coeffs and stores them along with the reconstruction error that is produced when trying to predict the signal using these coeffs. The prediction error signal is typically a low amplitude white (noise...ish) signal, which is amenable to compression due to its low amplitude and the coeffs themselves are just a few handful of values. The details are a lot more messy but that's the general idea. The SVG of audio? I think, that would be something like midi files. Or maybe tracker files - which are basically like midi files that also store the instrument samples.

[mystran]

which is amenable to compression due to its low amplitude and the coeffs themselves are just a few handful of values.

Nitpick: "[...] low amplitude with high-probability [...]"

The reason I'm bringing this up is that it's truly fundamental to every lossless compression scheme (audio or otherwise): if you can massage the data (using an invertible scheme) in such a way that some "symbols" are much more likely than others, then you can use "entropy coding" to encode the likely symbols with fewer bits, with the trade-off that the unlikely symbols become longer.

[Cranky Man]

How about ATRAC? Each band is processed using a transform similar to an FFT (MDCT) And honestly I always liked how it sounded. Still have a bucket full of minidiscs

[Music Engineer]

Nitpick: "[...] low amplitude with high-probability [...]"

Yes, indeed. That's what the disclaimer "...The details are a lot more messy..." was there for.

[Music Engineer]

How about ATRAC? Each band is processed using a transform similar to an FFT (MDCT) And honestly I always liked how it sounded.

Really? A lossy audio compression scheme that sounds good? In which way does it sound good? Do you have a sound example? I don't really know how atrac artifacts sound like. I just know that I very much dislike the artifacts of mp3 and I would assume that most lossy audio codecs would sound qualitatively similar. Not so? If it does indeed sound good, maybe it could be turned into an effect. It totally wouldn't be the first time in the history of music technology that some artifact would later be used deliberately as effect.