First, some definitions. An interval is a positive integer multiple of half steps. A scale is a sequence of intervals adding up to 12 half steps. Therefore, we can identify scales with (additive) compositions (ordered partitions) of the integer 12. We could take into account the playing of the same string on the same fret multiple times in a row by allowing an interval to be a non-negative integer multiple of half steps. Then scales could be identitified with weak compositions of 12. We could also take into account "backtracking" by allowing an interval to be any integer multiple of half steps. Finally, we could allow for scales that don't repeat every octave but instead repeat after some number of octaves by allowing a scale to be a sequence of intervals adding up to any integer multiple of 12. The concatenation of two scales is then again a scale. In this way the set of scales is given the structure of a non-commutative monoid (if we allow the empty scale). The subset of scales with only positive intervals, but that may not repeat every octave, would be a subsemigroup. What is this good for? Nothing but my mind runs rampant.