Different Scales:

You're basically correct with the formation of new
scales by selecting the exponential base and the 
number of subdivisiions.  One thing to do might be
to pick an exponential base, say 5, and subdivide
it in various ways (25 notes pops into my mind for
some cool squares reason), and see if you can get
intervals to sound better to you by tweaking the
notes slightly in either direction (deviate from 
"equal temperament").  You could then go back and
try to explain why the tweaked notes sound nicer.
Then you'd have a scale and a tuning-system (temperament) 
that you could write a piece in.  

Another cool thing to do is to find intervals that
sound bad, and devise new timbres (different frequency
components) that make them sound better.

Physical Model, Bowed Wineglass: 
See Banded Waveguides in ChucK and STK, and Georg Essl's 
PhD Thesis, and our joint papers on the topic of bowed
stiff modal systems (bars, prayer bowls, wine glasses,
etc.)