I think you refere to Slew Rate.
If assertion was true then total frequency response was inversely proportional to SPL. Higher going level (SPL) demands higher Slew Rate. The tonal impact was a noticeable Low Pass filter affect which deletes top end with any blow up of loudness with master knob.
All kinds of transients are cracy stuff to comprehend. Although slew rate is very big in number at transients it does not (automatically) mean there is a very lot of high frequency content.
At the other hand side in filter theory slew rate can be used to calculate frequency bandwith.
In the meanwhile I claim that a very lot of tonal feel is captured at the very beginning of around 0.1-0.2 sec. any tone starts with. The burst of a branch lasts around 0.2 sec at most but is clearly identified by ears as burst of a branch. At the other hand side ears don't detect amplifier clipping of 0.2 sec duration. That's a "contradiction" we have to tolerate. And, clipped signals does not mean that "tonal" information is totally lost. It's still there but, somewhat different in composition, the ears do a distinct emphasis about what is "essential" and probably some amplifier clipping content is not in this regard, while a burst sound of a branch really is, cause it signals "danger to life" on to the brain!