# Dwarves

Suppose v is a val <n ... | whose first coordinate is a positive integer n, and suppose the coordinates of the val, reduced modulo n, are distinct. An example would be <12 19 28 34|; reduced mod 12 this is <0 7 4 10| and 0, 7, 4, and 10 are all distinct. Starting from 1, take the odd positive integers in order of increasing size, 1, 3, 5, 7, ... and map them by the val v, reducing the result mod n. If this number (from 0 to n-1) has not appeared before, add the odd positive integer to a set. When n values have been obtained and no further additions are possible, take the resulting set and reduce its elements to an octave. The result is Dwarf(v), the dwarf scale resulting from the val v. Examples may be found on the Scalesmith page.

Of particular interest are dwarf scales resulting from equal temperament vals which are epimorphic for the val v, but even vals far removed from an equal temperament will produce a scale. Dwarf scales often produce results which are rich harmonically, with a tendency to favor otonalities over utonalities. The name "dwarf" refers to the fact that you are choosing for each degree the smallest Benedetti height.