posted 25 March 2002 05:13            

linear_crescendo.zip

Behringer has a 6 band stereo compressor that can be had now for $180 USD, using 24 bit A/D and D/A. It is a linear compressor, which is not optimal for hearing correction, but it can work respectably well enough. So can Kyma, but Kyma is an expensive way to do things in the long run. There is a lot of disagreement on the number of bands needed for proper hearing correction. I have finally settled on 11 bands of nonlinear compression in my personal Crescendo, but experiments with Kyma show that 6 1-octave bands with linear compression works well enough.

So now that such an affordable unit is readily available, here is how to use it to correct one's hearing, and even if you have perfect hearing, how to correct for your headphone excesses...

I have successfully modeled human hearing impairment as a depression in the mechanism that conveys loudness information to the brain. We perceive loudness in a measure called Sones, for complex sounds. Phones are used only for pure sinewaves. The expression relating Sones to dBSPL or sound pressure is

Sones = 10^(alpha/20*(P - Pt - 40)), valid for P - Pt > 40 dB

(from Benade)

where P is the sound pressure in dBSPL, and Pt is the threshold pressure level. Impaired hearing is very closely approximated as a depression in this level, such that one's hearing thresholds are elevated above Pt in a frequency dependent manner. [By definition 1 Sone = 40 dBSPL above the 0 dB isophone -- similar to the Fletcher-Munson loudness curves].

An approximation of the needed compression gain based on this equation is

G = 20/alpha * Log10[1 + 10^(-alpha/20*(P - Pt)]

where now P is the incoming sound level in dB above the 0 dB isophone, Pt is one's measured threshold with a particular set of headphones. These two equations are frequency dependent and will vary from band to band as both the sound level P and perceived threshold Pt both vary. [This approximation neglects a small contribution at the faint end of loudness of magnitude 0.08 dB, so it is quite good at normal listening levels.]

[BTW -- this release is partly in celebration of this equation, found just this weekend. Several months ago I found that, to a very good approximation, dG(P,Pt)/dP = -dG(P,Pt)/dPt, and hence, this indicated that the gain equation depends almost only on the difference (P-Pt). This means that headphone corrections or calibrated headphones were unnecessary. This weekend I finally found my expression involving only (P-Pt) and I could do away with banks of cubic and quartic best fits to the individual compression curves. One equation handles all of the bands identically!!!! Hoorray!!!]

There is no need for headphone compensation since our equation depends on the difference (P-Pt). That means that if you measure your threshold hearing with a particular set of headphones then the required headphone compensation, whatever that is, is automatically accommodated by this equation. Headphones are wildly variable in response, and characterizing them is quite difficult. But it is very easy to characterize the combination of headphone + listener. That's what a simple self-administered audiology test provides. One of the Sounds in the attachment turns Kyma into an audiology test instrument for just this purpose.

The constant alpha = 0.6. Some investigators disagree, but this value corresponds to the commonly held belief that +10 dB sound increase corresponds to "twice as loud". My own tests over the past 2 years indcate that Benade is closer to being correct than many researchers in Germany... (Of course, humans are highly variable and this value is for the hypothetical "average" human).

The compression gain is a nonlinear function of sound level. But one can ask for the "best fitting" linear approximation of it over some musically relevant range of loudnesses. Using a technique known as Remez's method, one can derive straight line fits to this nonlinear equation over the range -30 dBVU to +10 dBVU such that the maximum absolute error is minimized in this interval. I have done just this and you will find the results in the attached text file. The results are shown in terms of required compressor makeup gain at 0 dbVU, and compression ratio, as a function of threshold elevation. Most of the errors are less than 1.5 dB, and so should be hardly discernable in typical musical listening sessions.

The Kyma compressors are idiosyncratic (aren't they all?) in that the sidechain envelope follower works on the sidechain signal directly, while the input signal is first attenuated by 10 dB before going through the compressor. Makeup gain at 0 dBVU is obtained by realizing that 0 dBVU = -10 dBFS in the Capybara, and so the actual makeup gain to be applied to a Kyma compressor is (G0 + 10/R + attenuation) where G0 is the gain shown in the text file, R is the compression ratio, and attenuation is additional attenuation applied to prevent the output from clipping.

The amount of this attenuation will depend on program material and the strength of your compressive gains. Since your threshold elevation will be frequency dependent, a bank of 6 compressors is utilized following each of a bank of Kyma FIR graphic equalizer filters. A script at the front of the Crescendo Sound contains the individual compressor parameters. The threshold of each compressor is set at -40 dB which corresponds to -30 dBVU. Hence the range of compression covers -30 dBVU to +10 dBVU.

For use with the Behringer DSP9024 you will use a slightly different makeup gain. As soon as my unit arrives I will let you know how to set it up.

The results of a 6-band linear multiband compression with these particular settings is quite a good listening enhancer. It isn't a real Crescendo, but it is pretty good nonetheless. At $4K for a Crescendo, versus $180 for a Behringer DSP9024, I think the compromise is quite reasonable.

While some of us have what look like severe hearing impairment, those severe impairment levels only apply to threshold level sounds. Normal musical levels are much less impaired than one's audiology report might indicate. For example, with 60 dB threshold elevation I might only need 4 to 10 dB of compression gain to hear tonally balanced loudness levels -- not 60 dB! This seems to be a little known fact, and its ignorance might possibly scare some individuals into purchasing hearing aids...

- DM