Here's something that was leading me into a very modern, hiccupy, vaguely Chris Dave-like jazz concept. I was practicing along with the Coltrane recording of Big Nick, a medium-slow swing tune, and found myself doing this variation on my EZ Tony Williams-like method— double-timing it, so the rhythms are predominantly 16th notes and straight 8th notes. It's easy conceptually and technically, very modern in effect.
Read the original method, then proceed. We're using Progressive Steps To Syncopation, pp. 32-44— the “Syncopation” section. Here's the first line of the well-known p.37 exercise from that book, as played in the original method— in a fast 4, with straight 8th notes. Short notes (untied 8th notes) on the snare drum, and long notes (quarter notes, tied 8th notes) on the bass drum, with cymbal in unison with everything:
With today's thing, in a medium 4, but with the Syncopation part double-timed, it would be played like this— the dashed barlines show where the barlines were in the original part:
With 16th notes in jazz you want to think legato, but be sure your coordination is clean— it's easy for things to get mushy around those hihat notes on 2 and 4. So, legato, but not sloppily executed.
It's a good idea to also play the interpretation with a straight quarter note cymbal rhythm:
...and with a regular jazz beat on the cymbal— with a dotted-8th/16th rhythm on the skip note:
See the original method for ideas of what to do when you get into multiple 16th notes in a row on the snare drum— I prefer to play them alternating on the snare drum— without the cymbal, of course. Bring the cymbal back in on the first bass drum note after the run of 16ths on the snare. Or wherever you like.