Robert,
It's possible to measure time with high accuracy - via performance.now() - but you cannot get a callback with that kind of precision. In fact, in light of layout passes and JavaScript execution in the main thread, and the ever-looming threat of garbage collection happening in the main thread, you can't get anywhere NEAR even millisecond precision; you generally ought to be planning on potential interruptions in the tens of milliseconds for robustness.
The answer to this is to use scheduling, particularly in the Web Audio API - I see that you saw the article I wrote about this a year ago on HTML5Rocks (http://www.html5rocks.com/en/tutorials/audio/scheduling/), but you missed the significant piece - you shouldn't be calling
audioSource2.noteOn(0, 0.1190, 1.875);
you need the time offset to schedule it ahead appropriately:
audioSource2.noteOn(time, 0.1190, 1.875);
If you look at my original code, that's how I'm scheduling the oscillator ahead of time. The scheduler runs in a "slow" callback loop - being called only every 100ms or so - but schedules ahead a few beats. If you truly need to mute notes that may already be scheduled in the next 1/10th of a second, then you can keep a node in the middle to disconnect().