I already did, but I can do it again from a different perspective.
The saw blade cutting the lumber applies energy to the atoms. That makes them jiggle faster/harder which is heat. That energy imparted by the saw blade is not perfectly uniform over time, but pulses. For example as each saw tooth "hits" (see what I did there

) the wood, it imparts a burst of energy. The sound results from these pulses. If you take the number of saw teeth and multiply by the blade's rotational speed you can estimate the frequency of the sound being created. The sound results from the aggregate behavior of millions of atoms, while the heat results from their average "jiggle energy".
This all stems from the fact that 2 atoms cannot occupy the same space at the same time. As they near each other, electromagnetic forces push them apart. The closer they get to each other, the stronger that force gets. To be more precise the force increases with the inverse square of their distance. So whether the atoms actually touch each other is irrelevant. In fact, it's not even clear that it makes sense to say whether atoms "touch" because they are more like tiny clouds than tiny billiard balls. Atoms don't have precisely defined boundaries; their electron clouds are "fuzzy". For example two atoms in the same molecule having a covalent bond means they "share" one or more electrons. That means some electrons jump back and forth orbiting first one, then the other nucleus. But even then the electrons never hit each other, and the nucleuses are obviously separated. So no touching, at least not in the intuitive conventional sense of the word.