Some facts about the Hofstadter Butterfly: 

For rational values of a = p/q all the gaps are open, except the middle gap for even q.  
For a = p/q, the Hall conductance s corresponding to the gap number r satisfies the diophantine equation p s + q m = r. The unique solution is chosen such that s < q / 2.  
The number of disjoint regions in the colored figure having the (same) color corresponding to the
Hall conductance s is given by: where the function is the Euler function which counts the number of positive integers not greater than j and relatively prime to j. This result assumes that for irrational values of a all the gaps are open (see open questions below). 

The edges of gap number r at rational value of a = p/q, are common points for infinitely many regions. The set of Hall conductances s of these regions is exactly the set of solutions to the diophantine equation p s + q m = r. 
Still open questions: 

Are all the gaps open for irrational values of a ?  
For almost all values of a (having full Lebesgue measure), the spectrum is a cantor set of zero Lebesgue measure. Is this true for all irrational values of a ? 