For deeper insight into the workings of Crescendo, I’m posting two of my early papers on the physics and models discovered through my lab measurements.
The papers remain largely correct, but some sections are probably nonsense.
The first paper writes about the model for unimpaired hearing, called the EarSpring paper. I call it that because I found that the entire system (cochlea, afferent 8th nerve, brain, efferent 8th nerve) behaved very much, in its loudness perception, like a classical spring, but with a nonlinear stiffness that increases as the power of vibration grows.
If anything is nonsense in this paper, it is probably the last section talking about a possible cause of hyper-recruitment. The equations found in this paper serve as my reference for how things ought to sound.
The next paper, called the Conductor paper, provides the equations I found to describe sensioneural hearing impairment. It describes how things do sound to us with hearing damage, and is the basis for those recruitment curves.
Knowing both, how things do sound with impairment (Conductor), and how things ought to sound (EarSpring), we can develop the required corrections at every frequency and every loudness presentation level.
The Conductor paper also developed an operator algebra for audio processing in the loudness domain. Operators can be used to represent Gain, Attenuation, Hearing Correction, and Hearing Perception. Equations can be written for the order of application of each operation (they are generally non-commutative), and equivalences can be found in other arrangements to produce the same results.
Later work, not written up in this presentable form, produced a final discovery that indeed the Pre-EQ used for correction of HyperRecruitment and Decruitment would work as intended, despite what appears to be a non-commutativity of some operations.
There is also an interesting final section in the Conductor paper that tries to reason about the nature of hearing impairment, as though produced through damaged hair cells. I no longer believe that our hair cells have been physically damaged (as in, broken off) but they do behave as though they were. (that’s another story for another post..)
The model derived there shows that only a very tiny amount of damage needs to occur before serious disruption of our perceptual abilities. Even profound hearing loss is attributable to fewer than 10% of hair cells becoming damaged.
I can tell you right now that when operating at the edge of available knowledge, everything you do and think seems quite uncertain. And while it seemed to work for me, back in 2006, I was very unsure about many of these things.
I recall that my first model of EarSpring had the exponent of nonlinearity at 1.5, instead of 2. That error was a result of the crudity of measurements of our hearing’s 3rd order IMD intercept. In retrospect, the refined value of 2 is supremely gratifying and reasonable from a physics perspective. (Why should / could the brain sense an oddball power like 1.5? But sensing acoustic power (exponent 2) seems much more reasonable).
It wasn’t until a substantial body of tests on other people before I gained the solid conviction I now hold. There is no doubt in my mind that this is largely correct.
The final puzzle for me was in seeing that everyone liked Crescendo, even when they have no impairment. Why should that be? It was only recently that I finally figured it out – it has to do with the attractiveness of NYC Compression. You and I, with damaged hearing, get our hearing restored. Everyone else hears NYC compression with slight amounts of Crescendo processing.
But now realize, as I told my students in the past, that nature does whatever it does, whether we understand it or not. Our models are not the way that nature is working. Our models merely serve as simple analogies for our puny brains to wrap around.
Our ears are not banks of nonlinear springs. But what we now know is that the behavior of our whole hearing system and a bank of nonlinear springs is in many ways very similar. The correspondence is close enough that predictions can be made on the basis of the model, and laboratory measurements agree with the predictions.
And the most significant prediction is that a Crescendo would work to restore proper musical hearing with all harmonics in proper proportion, preserving musical timbre. Oboes will continue to sound like oboes, only more fully for damaged hearing.