The ennealimmal band

We can use the three layer comma, 250047/250000, to reduce the 7-limit
to just three layers of the 5-limit lattice. Similarly, we can use the
ennealimma to reduce the 5-limit to a band of values; any 5-limit
comma will roll up the 5-limit, but only some, such as 128/125,
648/625, or the ennealimma, roll it up with even edges.

The ennealimma, 2/(27/25)^9 = |1 -27 18>, can be written in terms of
5/3 and 3 instead, giving a 9 in the exponent. 27/25 = 3 (5/3)^(-2),
so the ennealimma is 2 3^(-9) (5/3)^18. The exponent of the 3 can
therefore be reduced mod 9 to the range -4 to 4, giving a band picture
of the 5-limit, consisting of nine chains of minor thirds. Not only
can the entire 5-limit be reduced to this band of chains of minor
thirds, the 7-limit, when expressed in ennealimmal, reduces as well,
using the 4375/4374 approximation to 7 as 7 ~ 4374/625 = 2 * 3^7 *
5^(-4) = 2 * 3^3 * (5/3)^(-7), showing it is in the +3 band. What we
have now done is simply to use the minor third as the generator for
ennealimmal. If now we close the chains of minor thirds to form a
circle of 19, we get 171-et, a rather unique feature to 171.