audio2design
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You raise some interesting points, chiefly that the usual measurements of THD, IM and perhaps a few more esoteric forms such as TIM may not completely explain the story, insofar as it relies on the assumption that systems are mostly linear, and the most significant errors can be captured in frequency and phase deviations occurring in amplification and transduction, the latter being moving energy from electrical to mechanical forms in speakers and phonograph cartridges (IMO transduction could also embrace conversion not just in energy but also in forms of information, that is from digital to analog waveforms).
The non-linear errors that introduce new waveforms are assumed to consist entirely of IM and THD, which can be easily measured. That they may not tell the entire story is the consistent observation that we are not annoyed by levels as high as several percent in the case of THD, and even higher when the music is loud. But there are some irritating exceptions as postulated by Geddes in this paper and tested in this one. A forum discussion can be found here and more papers on his website at this link.
I'm not at all qualified to comment on the accuracy of these claims, but they are interesting and curiously largely ignored. Whether that is because subsequent research has shown them to be of trivial magnitude and therefore inconsequential, or whether they are an inconvenient truth I can't say. I just mention this in passing as your comments stimulated my recollection of the debate, but it may be worth pursuing in your quest to identify the "missing" measurements. So have a look at Geddes papers, in particular Part 2 which discusses the GedLee (the two authors, Geddes and Lee combined) metric.
While I agreed conceptually with Earl when he released the paper in 2003, I had issue with aspects of the hypothesis then and still do, but most of all, because his paper unfortunately didn't say a lot, other than there is a problem, it needs to be addressed, and here is a formula to discuss, but with little in the way of experimental data to back it up.
BUT, let's highlight one critical thing he wrote copied below. What he is saying is if the THD, IM, and multi-tone IM is low enough, then distortion is no longer relevant. What he is approaching is a practical discussion w.r.t. distortion audibility when costs are constrained, when dealing with things like transducers that are highly distorting, etc. and trying to achieve a metric for audibility. This is likely to be much more complex than even a fairly complex equation and may be best achieved through massive experimentation and deep learning, not an easy tasks since you need both human input in a controlled setting and accurate real time distortion measurement, hence a low distortion transducer. Hard to isolate audibility of distortion in an amplifier when you are playing into transducers of often much higher distortion. Even once you have that deep learning result, as pointed out by @tvrgeek (and you thought no one was paying attention), whether that distortion results in an audible difference is source material dependent (including level), because our sensitivity to distortion changes with level, and with masking which varies with the source material.