Bandgap of 'white' solids?

Under 'white visible light' illumination, my understanding is that
colour is percieved as the combined effect of photons that are
reflected instead of absorbed. So an ideal matt black surface absorbes
everything, whereas a matt white surface reflects everything. A
transparent surface transmits most of the light, because the band gap
is so large that only high-energy photons (UV range) have sufficient
energy to excite electrons from the valance band to the conduction band
for absorption. Conversely, the band gap of a 'black surface' is so
small that virtually all photons have sufficient energy for absorption
to occur. Therefore, what type of band gap does a 'white' surface have,
or does it matter? Is it more of a specular surface phenomenon related
to light scattering?
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