# Dwarf

A **dwarf** is a periodic scale obtained by sequentially mapping odd harmonics (1, 3, 5, 7, …) using a regular temperament. A dwarf is a kind of detempered scale. The name *dwarf* refers to the fact that you are choosing for each degree the smallest Tenney height. Dwarf scales often produce results which are rich harmonically, with a tendency to favor otonalities over utonalities.

## Construction

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*.

## Example

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.

Let us construct a JI dwarf by the patent val *V* = ⟨12 19 28 34] of 12et. As is shown above, 3/2 is mapped to 7\12, 5/4 to 4\12, and 7/4 to 10\12; we add these octave-reduced harmonics to the scale. We also have 9/8 mapped to 2\12. The next odd harmonic implied by the val is 15, which, after octave reduction, is mapped to 11\12. Follow the same process: 21/16 ~ 5\12, 25/16 ~ 7\12, and 27/16 ~ 8\12. Then there is 35/32 ~ 2\12, the same as ~9/8, so we reject it. Continuing on, we add 45/32 ~ 6\12, reject 49/32 ~ 8\12 which is the same as ~27/16, add 75/64 ~ 3\12, reject 81/64, 105/64, and 125/64, and add 135/128 ~ 1\12. With 135/128, we have added the last scale step of this 12-tone scale so the result is

- 135/128, 9/8, 75/64, 5/4, 21/16, 45/32, 3/2, 27/16, 7/4, 15/8, 2/1

And that is exactly Dwarf12 7, the dwarf of 12et in the 7-limit.