The EarSpring paper showed that we can identify Sones measure for apparent loudness with the power ratio in the vibration amplitude of our “EarSprings”.
Whether or not that actually corresponds to basilar membrane vibrations or is just a convenient model, as simple as possible, to describe the whole system of hearing loudness perception, is unknown. And probably irrelevant.
But the basic truth uncovered by my research over the past 20 years is that the hearing correction gain, , needed to overcome an elevated threshold, , can be found by solving the Sones equation:
The equation states that in order to perceive a sound of level, , in the presence of an elevated threshold, one must simply supply enough additional gain, , so to overcome the elevated threshold Sones. In doing so, the sound will seem the same loudness to an impaired listener as it does to an unimpaired listener.
Each of the terms are power ratios against a common threshold power, and so they legitimately add as incoherent signal powers.
One might be further tempted to subtract a corrective term, , on the right hand side, in order to make the equation balance exactly when listening to absolute threshold levels of 0 dBHL. But , which is so tiny compared to elevated thresholds above 20 dB, that it may be neglected. It certainly falls below our ability to measure in the presence of elevated thresholds. You cannot discern any difference when corrections are developed with or without this corrective term.
What we know is that Sones grow, for everyday sound levels above 40 dBHL, as the cube root of the sound intensity. And that is a positive power, greater than zero, which means that Sones grow ever larger at loud levels. Hence at loud levels above the elevated threshold, the correction term, , itself becomes negligible, and sounds are perceived properly without much of any correction.
But the equation also hearkens back to our discussion about additive pseudo-noise fields. If we establish that a threshold is elevated by some amount, , as measured by the dBHL needed in the audiology signal, then we might be tempted to assign a pseudo-noise field as a causitive agent.
In order to achieve a 2 of 3 detection process, the actual threshold of hearing would be the signal power plus 18%, to account for same power random noise field enhancement, or 0.72 dB higher than shown by the audiology signal level. Again, this is too tiny to reliably detect at these levels. This is especially true with the optimistic estimate of +/- 5 dB accuracy in audiology measurements. (I believe the accuracy is worse than this in many cases.) So it is reasonable to count the threshold level as the same as the audiology signal level.
For sound levels above 37 dBHL, a reasonable approximation to Sones, from dBHL Phons, is:
This approximation is good to within about 1.5% relative error. All powers are measured in dBHL (Phons).
Finding the required gain for an impaired critical band can be done by first converting the incoming signal power from dBSPL to dBHL by subtracting the dBSPL absolute threshold of hearing, , for that frequency. Then substitute the signal dBHL power into the Sones approximation to form the right hand side of our equation.
The required gain is simply the inverse approximation of the right-hand sum, followed by subtracting the signal dBHL to leave in dB gain. It is really just that simple. And the result, in Crescendo, sounds fabulous!