Dwarves

Suppose v is a val such that the gcd of the coefficients is 1, v(2)>0 and for every odd prime p, v(p)>=0. According to my original definition of val at any rate, vals are defined for all positive rational numbers, but for all but a finite number of primes they map the prime to 0. Bearing the precise definition in mind, we can define the dwarf scale for v, dwarf(v), as the reduction to the octave of the integers n of minimal Tenney height which form a complete set of residues v(n) mod v(2). The reason for the name "dwarf" is that height is as small as possible. The results are not extremely sensitive to this exact definition, as sorting by Hahn distance from unity and using Tenney height to break ties seems to usually lead to the same result.

Because v(2n) mod v(2) = v(n) mod v(2), the integers n will always be odd. Because if v(q) = 0 then v(qn) = v(n), the integers n will always be in the p-limit, where p is the largest prime for which v(p)>0. If r is any prime for which v(r)=0, then r does not appear in the factorization of the integers n, so this definition also covers subgroup situations, such as {2,3,7}-scales, so long as 2 is in the picture and we are using octave equivalence.

Because of the way they are constructed, dwarves are always permutation epimorphic and have a bias towards otonality over utonality. Here are some examples:

<7 11 16| 1, 9/8, 5/4, 45/32, 3/2, 27/16, 15/8
transposes to: 1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8
<7 11 16 20| 1, 9/8, 5/4, 21/16, 3/2, 27/16, 7/4
transposes to: 1, 9/8, 7/6, 4/3, 3/2, 5/3, 7/4
<10 16 23 28| 1, 35/32, 9/8, 5/4, 21/16 45/32, 3/2, 105/64, 7/4, 15/8
transposes to: 1, 16/15, 7/6, 6/5, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8

Marvelous dwarves

A marvelous dwarf is a scale with the following attributes:

(1) It is a marvel tempering of a 5-limit dwarf.
(2) It has the same number n of otonal tetrads, otonal pentads, utonal tetrads and utonal pentads. As a consequence of this it also has n subminor and n supermajor tetrads.
(3) It is covered by its pentads--that is, every note is harmonized by a pentad, and the scale is the union of its pentads.
(4) It has more 5-limit triads than pentads.
(5) It has no approximate tetrads deriving from anything but marvel; in the 5-limit scale which is tempered the smallest comma which produces approximate tetrads is 225/224.

If every condition but the third--the covering condition--is satisfied, I call it a semimarvelous dwarf. Why there are these scales exhibiting such regularity as a result of finding the 5-limit dwarf is an interesting question. Whatever the reason for it, the marvelous dwarves--of size 12, 15, 18, 19, 20, 21, and 25--seem like excellent scales for instrumentalists and composers interested in 9-limit harmony and scales in this size range. The 25-note scale, whose 5-limit preimage is the Euler genus of 15^4, is particularly striking from the point of view of the quantity of pentads it supplies.

There is a marvelous or semimarvelous dwarf for each scale size from 11 to 22, and then the 25 note scale. So far as I know this is the complete list.

Here is a brief description; the numbers are pentad number, numbers of major-minor triads, and size/pentad ratio.

<11 17 26| 1 6-6 11 semimarvelous
<12 19 28| 2 6-6 6 marvelous
<13 20 30| 1 7-6 13 semimarvelous
<14 22 33| 2 7-6 7 semimarvelous
<15 24 35| 3 8-8 5 marvelous
<16 25 37| 2 7-6 8 semimarvelous
<17 27 40| 4 10-9 4.25 semimarvelous
<18 29 42| 4 10-10 4.5 marvelous
<19 30 44| 5 12-11 3.8 marvelous
<20 32 47| 6 12-12 3.333 marvelous
<21 33 49| 5 12-12 4.2 marvelous
<22 35 51| 6 14-13 3.667 semimarvelous
<25 40 58| 9 16-16 2.778 marvelous