Black-Scholes compression

New compression can be borrowed from other fields#

Most compressors exhibit the same small handful of gain-reduction curves (see compression). Tube, FET, VCA and opto are all classic compression at their core. So if you want a compressor that does something genuinely new, you need a new model of how gain reduction should move. Other disciplines already have compatible models. Finance has one. Quantum physics has one. Fluid dynamics has one. As far as I know, nobody had applied them to audio.

Finance is the obvious place to look. PCM audio and real-time stock price data are close to identical in form, and the finance world spends serious money on turning raw price data into something more useful. Meanwhile the conceptual framework inside audio compressors has changed only trivially since the 1940s.

An option is a price that depends on time and risk#

You do not need to know anything about trading for this. An option is a contract giving you the right to buy something at an agreed price, called the strike price, on an agreed expiry date. You pay a premium for that right. The premium goes up when the thing is more volatile, because the seller is taking on more risk. It goes up when the expiry is further away, because more can happen in two years than by tomorrow. If you hold the option while nothing moves, its value ticks away. If you buy more contracts, your exposure goes up.

The Black-Scholes-Merton model is the formula that turns all of that into a single price. Or stated technically: a parabolic partial differential equation, assuming the price follows a geometric Brownian motion.

I can't be the only person who has noticed this, but these concepts map onto a compressor very neatly:

• The underlying price is the signal amplitude.
• The strike price is the threshold.
• The volatility is the crest factor, or dynamic range.
• The time to maturity is the envelope time.
• The call option price is the scaled target level of the signal.

It is a proof of concept, and it works#

The paper below sets out the mappings and the architecture. The claim is that the model can be used directly for compression, and the plugin demonstrates it.

It is experimental, a bit unpredictable and CPU hungry by plugin standards. Even at 44.1 kHz the formula runs at least 44,100 times a second, which means over 132,000 expensive maths calls a second, and over a million with 4x oversampling. On an entry-level M4 Mac mini, Curvature's money algorithm used about 2 to 2.5% CPU at 44.1 kHz with 4x oversampling, and that was without SIMD. Which is a lot. Then again, finance runs hundreds of millions of these calls for a single serious backtest, so by its standards this isn't monstrous. It needs a sweet spot: overdo it and the gain reduction skyrockets and tanks the output; give it too little and you barely hear anything. It is probably not going to be your go-to mastering compressor. Where it earns its place is creative work on individual elements in a mix.

In the sweet spot it reacts violently to dynamic material and very gently to gentle material. To me it sounds more alive than ordinary compression, and, stupid as it sounds, almost more analogue than an analogue compressor, because an analogue compressor is still just doing normal, boring gain reduction most of the time.

Finance was only the first one#

The same approach works with quantum tunnelling, which is real, well documented physics and not new-age mysticism. A particle has a finite chance of crossing a barrier it does not have the energy to climb. The particle's energy is mapped to the level of a sample and the barrier to the threshold. Each tunnelling event, decided per sample by testing the real probability against a random number, adds a unit of gain reduction. The events compound like an attack phase and dissipate like a release. It does not sound like any other compression.

Both algorithms started in Curvature. Convexity replaces it, carries both, and adds others built the same way, including a simplified fluid-flow model.