Epimoric epilogue

I previously derived the PB scale

1-10/9-6/5-4/3-3/2-5/3-9/5-(2)

(and once again wrote it down incorrectly) from the notation I called
N(4), with basis <16/15, 25/24, 81/80>, which in turn was dervied
from the basis <S(3), T(4), S(4)> = <9/8, 10/9, 16/15> by dividing
adjacent elements. This second basis, consisting of epimoric
elements, is well-suited to producing scales with epimoric steps, and
so one might wonder if we can obtain such scales as PBs. The scale
above was calculated as the one whose nth degree is

(16/15)^n * (25/24)^ceil(b1 + 5n/7) * (81/80)^ceil(b2 + 3n/7),

where the base points were b1 = b2 = -1/2. However, we may pick any
base points we like in the square region -1 < b1, b2 <= 0 and obtain
a PB, and this region is chopped up by lines into subregions which
produce a variety of scales and scale modes. If we consider the JI
diatonic scale written in the notation, defined by [h_7, h_5, h_3]
we get

9/8 --> [1, 1, 1]
5/4 --> [2, 1, 1]
4/3 --> [3, 2, 1]
3/2 --> [4, 3, 2]
5/3 --> [5, 4, 2]
15/8--> [6, 5, 3]

If we add (or on the left, multiply) we get [21, 17, 10]; subtracting
3*[7, 5, 3] we get [0, 2, 1]; if therefore we adjust the base point
by setting b1 = -1/2 + 2/7 = -3/14 and b2 = -1/2 + 1/7 = -5/14, we
get something which should be in the middle of the region producing
the diatonic JI if there is one. Calculating

(16/15)^n * (25/24)^ceil(-3/14 + 5n/7) * (81/80)^ceil(-5/14 + 3n/7)

we find that we do, in fact, obtain the diatonic JI as a PB scale. If
we want to obtain an et suitable to the nature of the scale, we can
look at the group dual to the smallest basis element, 81/80, giving
us p h_7 + q h_5; for p and q both positive we obtain
h_7 + h_5 = h_12, 2h_7 + h_5 = h_19 and so forth. It should be noted
that we are speaking of vals, not simply ets here; we have

[17]
h_7 + 2h_5 = [27]
[40]

in null(81/80), but

[17]
2h_7 + h_3 = [27]
[39]

in null(25/24).

The same sort of considerations apply to the other notations
discussed, giving rise to many PBs with epimoric scale steps.