Last night I went back to some Lisp models I wrote several years ago, searching for the best parameters to use for NYC parallel compression as an approximation to what EarSpring and Conductor tell us about the recruitment compensation needed for each given level of impairment at each frequency.
The best psychoacoustic compression ratio seems to be 3.7:1, with a makeup gain of about 21 dB using a threshold of compression at 20 dB below the threshold elevation at each frequency. There is something magic about NYC compression, as we all know. And the best compression ratio seems to be that 3.7:1.
So I happen to use this for entirely musical reasons too, well ahead of any Crescendo processing. I split my final mix channel, sending some to a side chain containing an EQ with a low shelf filter and a compressor, mixing that back into the final mix output. The EQ de-emphasizes the bass frequencies below 1 kHz by 24 dB, and the compressor has a low threshold. The intention is to enhance the higher frequencies, but only at low levels. The processing mirrors exactly the sort of approximate hearing corrections used by a simple Crescendo engine. This is psychoacoustically pleasing to every listener, hearing impaired or not.
But… in that code I found a section describing the use of an 8 kHz 2-pole High-Pass filter as being used to derive the required corrections at all lower frequencies. It turns out that an 8 kHz 2-pole HPF has a roll-off of around 12 dB/octave. And this roll off is almost exactly what we see in the damage levels among impaired listeners.
And the damning thing about this is that these two aspects – the roll off of damage, and the roll off of a HPF have nothing to do with each other. They are from incompatible dimensions. It seems to be a mere coincidence between them.
The damage caused along the cochlea should arise from power dissipation of the acoustic waves in the cochlear fluid. And the damage rate should be a result of the efficiency of coupling between these acoustic fluid waves and the inner hair cells, which in turn is a byproduct of the cochlear fluid viscosity. A unit of 1 Bark bandwidth corresponds to some linear distance along the basilar membrane, where highest frequencies are sensed at the oval window end, and bass frequencies are sensed at the apical end.
That this damage occurs with a coincidental rate equivalent to 12 dB/octave is curious. But this gives us a chance to verify the relationship of Bark bandwidths to Hz frequency. We know the damage rate is on average about 3.5 dB/Bark among nearly everyone. From a multitude of measurements on different individuals, we found damage slopes ranging from 3.1 to 4.0 dB/Bark, with an average around 3.5 dB/Bark.
For this to be closely equivalent to 12 dB/octave – and the equivalence is very close indeed – this means that at these higher frequencies, near the oval window end of the basilar membrane, we must have 1 Bark width = 0.29 octave. And that too is very nearly correct, based on other kinds of measurements.
But our confused use of an 8 kHz HPF aroused some suspicion when I reexamined it. It seems daft to consider such a device in this context. The damage rate probably also matches the roll off from a 10 kHz or higher HPF. It is the 12 dB/octave roll off that we match. And that seems just a point of curiosity.
I’d love for there to be some deeper physics reason for such a damage roll off. But I cannot think of anything.
Anyway, this is an indication of the circuitous route we all follow on the way to major discoveries. A lot of tangents, nonsense, constantly reconsidering, and coincidences, along the way. It seems daft, but this one also gave us another method for estimating Bark bandwidths. They are a little bit narrower than 1/3 octave filter banks found in graphic equalizers.
- DM