The dwarf

Suppose v is a val such that v(2)>0 and for every odd prime p,
v(p)>=0. According to my orginal 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.

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 contructed, 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