I've got a multitracked choir recording with both tenor parts and soprano parts, both divided into melody + two harmony tracks. What I'd like to be able to do is (on-screen buttons for mixing, I'm talking about - I don't mean fader riding / automation or anything in the final track), say mute ALL the soprano parts (melodies AND harmonies) and hear ONLY the tenors... And then, for example, mute ONLY one harmony part (from both tenors and sopranos, or maybe hear only ONE harmony parts from the tenors.
 
Is there any way to set SONAR X3 up to do this without the use of lots of sends/busses? That'd be cool if someone knows how to do it!

Try grouping your faders and use control groups:
 
https://www.cakewalk.com/Documentation?product=SONAR%20X3&language=3&help=Mixing.41.html
 
 

Or grouping mute and solo buttons.

Not sure I fully understand the goal, but some combination of control-grouping mute and/or solo buttons (either in the same or opposite states) and possibly solo override (Shift+click Solo) should help minimize the number of clicks to change combinations. Right-click a control to add it to a group.