One might ask “What do you mean “playing the rests”?
Utilizing periods of rests or pausing during an improvised solo is something many players never think of. As I listen to great improvisers, I realize that the short periods of “nothing” can be an oasis in the desert. These mini points of contemplation give the rest of the solo meaning and contrast. Often we are bombarded during a conversation with an over excited and sometimes recreationally medicated person who seems to be able to talk without stopping for air. The more the person prattles on, the less we tend to listen. Jazz solos can be the same.
To give a more contemporary example of this practice, compare the political meanderings of Donald Trump to the careful selection of utterances made by Dr. Ben Carson. Trumps style of speaking is measured by the pound and Dr. Carson’s is measured by the substance.
When playing a jazz solo, one needs to contrast action (notes) from contemplation (rests).
Another example of this approach would be the contrast of isis and snipers. isis’s approach is “cover your opponent with a barrage of fire power” and in contrast the snipers approach is “one bullet, one kill”. Much can be accomplished with just the right note in the right place during a jazz solo.
As an example of musical conservatism, enjoy Miles solo on this great standard. Notice his improvisation is more like a conversation than a solo. And in contrast to his sparse concept, listen to Trane demonstrate just the opposite. His contribution is much more aggressive and technical. Listen to one style for any length of time can get monotonous whereas the two contrasting styles complement each other.
This contrast is what I am suggesting as the execution of a perfect improvised solo. Technical virtuosity contrasted with thoughtful, pensive sensitivity.