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. They exclusively contain otonal intervals, expressible as octave-reduced forms of harmonics, and may be flipped to provide utonal versions of the same scales.
Construction
For an equal temperament n-ET, starting from 1, take the odd positive integers (within the subgroup of the equal temperament) in order of increasing size - 1, 3, 5, 7, … - and temper them to the equal temperament, octave-reducing the result. 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.
The dwarf will ultimately end up being in a subgroup where no two primes map to the same interval when octave-reduced, as ultimately the simpler prime would always be chosen due to the method of the scale's construction.
For example, let us consider 7edo in the 5-limit: ⟨7 11 16], reduced to ⟨0 4 2].
- 1 is mapped to 0 steps (or 2/1 to the octave)
- 3 is mapped to 4 steps
- 5 is mapped to 2 steps
- 9 is mapped to a single step
- 15 is mapped to 6 steps
- 25 is mapped to 4 steps (but 3 is already chosen for 4 steps, so 25 can be ignored)
- 27 is mapped to 5 steps
- 45 is mapped to 3 steps
At this point, we have have 7 intervals, so the dwarf is complete: [9/8, 5/4, 45/32, 3/2, 27/16, 15/8, 2/1]. In fact, this is the Lydian mode of the zarlino diatonic scale, which exposes the fact that all the intervals of Lydian can be expressed as otonalities.
Symmetrical dwarf
For a symmetrical scale, we may consider for each odd both the reduced interval and its octave complement, with the same priority (in the case of a tie, like in an even edo, either may be chosen). Unlike standard dwarves, which contain exclusively otonalities or utonalities, these contain a balanced number of both. Again taking 7edo in the 5-limit as an example:
- 1 is mapped to 0 steps (or 2/1 to the octave)
- 3 is mapped to 4 steps
- 4/3 is mapped to 3 steps
- 5 is mapped to 2 steps
- 8/5 is mapped to 5 steps
- 9 is mapped to a single step
- 16/9 is mapped to 6 steps
So the result is [9/8, 5/4, 4/3, 3/2, 8/5, 16/9, 2/1].