Is there a / what is the mathematic explanation of the warm analog sound?

I'm not talking about anything EQ related. Anyone who has played around with high end analog equipment understands that it transforms the sound in ways that aren't possible with simple EQ. I was wondering if there was a mathematical explanation for what is occurring when this happens, such as, how certain frequencies are shifting and by how much, etc. Plenty of VSTs exist that attempt to replicate that analog warmth sound, but I'm interested in knowing what sound/math transformation is actually taking place here. Is there any useful sources which discuss this topic from a mathematic stand point? Or perhaps some sort of fundamental/basic type of transformation that is acting on the sound?

Background: Musician for 5+ years, programmer for 15+ years.

• thermal/random noise vs quantization noise ? even harmonics vs odd harmonics ? Commented Sep 22, 2018 at 6:19
• I think your best bet is approaching from the programmer angle if you want to find a microscopic/component function is to search github for vst repos.Generally Warm sound is many things , mostly non-linearities harmonics and stuff... But it gets deeper once you start putting more components and wiring them. Commented Sep 27, 2018 at 18:33

Sadly "analog warmth" and "math transformation" are orthogonal. The former subjective and the latter precise. "Warmth" or any change to the audio implies distortion, in that the new audio is different than before. That said I would look at vacuum tube distortion vs. digital distortion (or mathematical distortion). I prefer and can hear the warmth of my guitar tube amps over solid state. To characterize distortion one looks at effects in the frequency domain as well as in the dynamics domain (compression, variation of frequency domain with amplitude etc). Modeling warmth may be possible to your ear, but not everyone's.

Here are papers on distortion:

https://phys.org/news/2017-02-physics-musicians-valve-amps.html

https://pdfs.semanticscholar.org/9903/44e66aa3851e90fc7a825f7481a33be21ecf.pdf

https://ccrma.stanford.edu/~dtyeh/papers/DavidYehThesissinglesided.pdf

What you need to do is look at Fourier transforms of you want to emulate analogue sounds in a digital environment.

Party of my degree focused specifically on this, so I have looked at mathematical explanations of overdrive, echo, phaser, flanger, detune, wah, and a few others.

There is no simple way to look at it though - each of these sites a body of different things. You are probably most interested in elements of overdrive - the non-linearity that analogue amplifiers bring in to a signal path. They sound "pleasant" to humans, in a way digital amplification sometimes doesn't.

But to be honest, the gap between high end analogue and high end digital is now all but gone. Sampling rates are now fast enough you can emulate analogue to a greater resolution than the human ear or brain.