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.