Difference set modmos structures

Given a mod n perfect difference set, we can take the differences and
use the set so formed as a modmos for some temperament. In particular,
we might pick a temperament with a MOS for that number of notes. For
13, for instance, we might pick on orwell, magic, hemithirds or
wuerschmidt.

A perfect difference set mod 13 is {0,1,3,9}; if we substract off 1,
3, or 9 mod 13, we get another difference set containing 0. Taking the
corresponding sets of 13 integers gives us the following modmos
structures:

difference set: {0, 1, 3, 9}
modmos structure: [-9, -8, -6, -3, -2, -1, 0, 1, 2, 3, 6, 8, 9]

difference set: {0, 2, 8, 12}
modmos structure: [-12, -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10, 12]

difference set: {0, 6, 10, 11}
modmos structure: [-11, -10, -6, -5, -4, -1, 0, 1, 4, 5, 6, 10, 11]

difference set: {0, 4, 5, 7}
modmos structure: [-7, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 7]

In order to get an actual modmos, we have to pick a generator. This
proceedure can lead to interesting results; the {0, 6, 10, 11}
difference set structure, applied to orwell, gives six major and six
minor tetrads, which means the 13-note orwell difference diamond has
the same number of tetrads as Orwell[16], using three fewer notes. I
haven't taken a look at any other temperaments as yet but this is
clearly a strong start!

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> In order to get an actual modmos, we have to pick a generator. This
> proceedure can lead to interesting results; the {0, 6, 10, 11}
> difference set structure, applied to orwell, gives six major and six
> minor tetrads, which means the 13-note orwell difference diamond has
> the same number of tetrads as Orwell[16], using three fewer notes.

Miscounted--rats! We get something equaling the MOS, anyway.