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Note that the "Lokki" Griesinger quotes is Finnish acoustics researcher Tapio Lokki, rather than the step-brother and nemesis of Thor.
My understanding is that one area of importance is the phase relationship between fundamentals and overtones. When the overtones arrive at the same instant as the fundamentals, there is a brief spike in SPL. When the overtones are smeared out in time, those spikes disappear.
The reason those spikes matter has to do with signal-to-noise ratio, which in turn has to do with clarity and intelligibility.
Woe is me to disagree with someone of Griesinger's status, but I'm aware of a lot of psychoacoustics literature that appears to contradict this position (if indeed I've understood correctly what his position is).
Of all the group delay audibility studies of which I'm aware, this is the study that found the lowest thresholds for group delay audibility. In this study, all-pass phase characteristics of numerous loudspeakers were convolved with impulses that were then played back to subjects over headphones. Subjects were tested to determine whether they were able to discern a difference between the convolved impulse and the un-convolved original (note that some low-pass filtering was applied to increase subjects' sensitivity - I won't go into the details here but it's in the linked paper).
Previous studies (e.g. the classic by Blauert) had already determined that subjects are far more sensitive to phase distortion when listening on headphones (vs. loudspeakers). Note also that previous studies had found that subjects are more sensitive to phase distortions when the stimuli are impulses than when the stimuli are music signals.
Keeping all that in mind, these graphs summarise the study's findings:
Even under highly contrived conditions (in comparison with normal loudspeaker listening and music), subjects were unable to detect phase differences of up to 1ms above 300Hz, and were unable to detect phase differences of well beyond 10ms across the entire audio band.
To put those results into perspective, a 4th order crossover at 500Hz or above will not exceed the threshold established in Graph (a). Even an 8th-order crossover at (for example) 2kHz will not exceed this threshold.
In other words, a typical loudspeaker is quite unlikely to produce enough phase distortion to be audible, even under the most contrived conditions (i.e. taking the speaker's phase characteristic, convolving it with an impulse, filtering it in such a way that sensitivity to phase distortion is increased, then playing it back through headphones).
Thus the idea that the group delay generated by normal loudspeaker crossovers could be a relevant factor in subjective sound quality does not seem plausible to me.
Anyway, I'm really not an expert on this (@j_j is who we would want to ask). But my understanding is that gradual shifts in phase such as those generally introduced by typical crossovers cannot result in audible group delay.
(IIUC, sharp changes in phase within a single critical band can be audible. This does not occur under normal circumstances in loudspeakers. It could occur with atypically high-order crossover filters, but we haven't seen that so far in the speakers measured.)
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