The overtone structures created by instruments have an enormous influence on their individual and signature timbres. Those overtones obviously don’t stop at 800Hz. The nature of the crossover plays a huge role in the loudspeaker’s ability to recreate both the fundamental tones and their overtones. Superior drivers make the design of the x/o an easier task.
In one of his Tech Talk Tuesday videos, Danny Richie (GR Research) shows the on and off axis response of individual drivers and complete loudspeakers, and demonstrates how improving the off axis response (often via the x/o, the design of which Danny is an expert) affects the loudspeaker’s total in room response, in both the amplitude and temporal domains.
Do yourself a huge favor and watch the videos!