a traditional "lossless" audio format such as wav is comprised of a 1 dimensional sample (volume) refreshed thousands of time per second (up to 192.1) . Currently 24 bit audio allows 16 million possible loudness levels which generally gives more than enough dynamic range.

The problem is if you zoom
in extremely close it looks like a stair step instead of a fluid sine shape. This also creates large files as you as 24 bit 96khz sample rate wav file will be 4 mb/sec of data.

I suggest using trigonometry and physics to model sound as a sine based function. physics modeling already exists for drums and other instruments and the effects can be eerily life like. These programs model how the an object vibrates a surface to make sound. Drums are "easy" (but still hard to figure out the math)

Basically one would use a regression analysis to find the best fit curve of an audio signal. You would use a variable "function rate" instead of a variable bit rate, to determine how long each sine wave function would be. You would also use addition to add multiple sine waves together instead of using many shorter sine waves to describe the sound.

The payoff? Extremely small bit rate for lossless audio and high sound quality, plus dynamic range and frequency response can be arbitrary as you can define a waveform as high frequency or as loud as you want (limited by hardware). Using a raster based approach is a brute force method of describing sound.

the savings come because a single waveform function on general would be much longer than a .wav bit rate. I'm guessing you would only need a function rate of around 100-1Khz to successfully design an audio signal. The sine wave function could probably be parsed down to a few hundred bits, it would comprise of a sine function or linear function of say 3 digits of say 2 digits of precision then a multiplier to define loudness a multiplier to determine frequency and a marker to define the stop. As long as the sine wave function size x sample rate was lower than a similar lossless audio format file size could be remarkbly smaller.

The trade off would be increased complexity in encoding. (however sound processing is no longer considered computationally intense)

The alternative method would be to describe the sound at a fixed sample rate some fraction of the maximum desired hearable frequency (22khz). For example if you picked a 11khz frequency rate. Each function would describe 1/2 of a sine wave at 22 hz. Which might allow for less information.

Stair-steps in digital audiohttp://www.tomshard...nyl-digital#t351127 A good post about the stair-step myth. The rest of the thread has things to say about it too. [Wrongfellow, Jan 19 2011]

Waveletshttp://en.wikipedia.org/wiki/Wavelet [Jinbish, Jan 19 2011]

Audiophileshttp://xkcd.com/841/ //The problem is if you zoom in extremely close it looks like a stair step instead of a fluid sine shape.// Well don't look so bloody close! [Jinbish, Jan 19 2011]

//The problem is if you zoom in extremely close it looks like a stair step instead of a fluid sine shape.//

Only if the reconstruction filter is broken (or if you're looking at the signal prior to filtering).

The idea that sampled audio is made up of "stair-steps" is one of the simplifying myths that gets used to avoid exposing people to more mathematical detail than they need.

Did you get taught in school that electrons moved around the nucleus like planets orbiting the sun? Did they then reveal to you a couple of years later that this was actually a complete lie?

Stair-steps in sampled waveforms are the same kind of lie.

The usual challenge for lossless codecs is pink noise (equal energy per octave.) As described, it's highly unlikely that "regression" would reduce bit rate. [-]

Would there not be gains to reprsenting sound (waveforms) mathmatically as wave forms. instead of representing them as raster images of millions of discrete points? at the very least I think there would be file size savings.

Well, one issue is periodicity. For example, any periodic
waveform can by represented by a series of sinusoids. As
soon as you lose the periodicity, you (theoretically) need
infinite sinusoids.

There are entire fields of mathematics dedicated to baking this. Look up compression (lossless), compression (lossy), filter theory, sampling theory, signal processing, and psychoacoustics.