David,
Do all your appointments start at increments of 0:15? Or do you have
appointments with arbitrary start/end times?
For example, could you have a schedule like this?

0001 9:00 9:45
0002 9:15 9:50
0003 9:20 9:40

And if you have a schedule like this, how do you propose to encode it
in the new chart? Like this?

0001 9:00 9:15
0001 9:15 9:30
0001 9:30 9:45
0002 9:15 9:30
0002 9:30 9:45
0002 9:45 9:50
0003 9:20 9:30
0003 9:30 9:40

Kostis Vezerides