Temperament integers

The positive integers form a commutative monoid under multiplication:

http://en.wikipedia.org/wiki/Monoid

That is, there is a product which is commutative and associative, with
an identity element. The p-limit positive integers for any p are also
a commutative monoid under multiplication, and so are the odd p-limit
integers. It is the latter one uses to contruct dwarf-type scales.

Given a temperament, one can define a monoid of "integers" and "odd
integers" relative to the temperament, where these "integers" are
actually rational numbers modulo the temperament, which me may view as
elements of the temperament group. We can define "positivity" by means
of vals which have natural number coefficients. For instance, for
225/224, <0 1 0 2| and <0 0 1 2| are the only reduced vals with only
two positive values, such that the 2-coefficient is zero. If an odd
ratio 7-limit rational number is non-negative according to these two
vals, it is a 225/224 odd marvel integer. Same with <0 4 0 1| and <0 0
2 1| for breed integers.

By using these "integers", monoid elements relative to a temperament,
we can define dwarf scales relative to the temperament. For rank one
temperaments this gives us generated scales such as MOS. For higher
rank temperaments, it gets more interesting, and you get something
intermediate between MOS and JI scales.