Simple "double" periodicity block

Using the (3,5) unison vectors

(12 0) = Pythagorean comma
(4 2) = Diaschisma

the Fokker periodicity block has 24 notes (cents shown below with fifths
vertical, major thirds horizontal):

2
|
|
500
|
|
612-------998
| |
| |
1110------296
| |
| |
408-------794
| |
| |
906--------92
| |
|
204------ 590
| |
| |
702 ------1088
|
| |
0--------386
| |
| |
498-------884
| |
| |
996-------182
| |
| |
294-------680
|
|
792
|
|
90

This is a double approximation to 12-tET, with syntonic comma and schisma
pairs.

What seems to be happening in all these "double" periodicity blocks is that
there is an approximate half-octave interval that is some combination of the
unison vectors divided by two (in this case, (6 0)) and the periodicity
block can be broken into two similar portions a half-octave apart (breaks
shown above). If each portion already contains a half-octave, you get this
"doubling" effect.

So I conjecture that this doubling only happens when the determinant is a
multiple of 4. Similarly, tripling probably can only occur if the
determinant is a multiple of 9.

I wrote,

>So I conjecture that this doubling only happens when the determinant is a
multiple of >4. Similarly, tripling probably can only occur if the
determinant is a multiple of 9.

That conjecture was incorrect. A look at the 66-tone Fokker periodicity
block formed from unison vectors

3 7 0 (=10.06¢),
-8 -6 2 (=4.13¢), and
-8 2 5 (=1.12¢),

shows a triple-22-tone character, with small "seconds" of 65625:65536
(2.35¢), 6144:6125 (5.36¢), 2460375:2458624 (1.23¢), and 10976:10935
(6.48¢).

However, another 66-toner, formed from

-5 -2 -3 (=11.12¢),
3 7 0 (=10.06¢), and
1 0 3 (=8.43¢),

looks more like a 53-tone scale with 9 notes doubled, by either 225:224
(7.71¢) or 3136:3125 (6.08¢), and 2 notes tripled, by both of these
intervals.

So a small interval within the periodicity block is not sufficient (but of
course it is necessary) to establish a periodicity within the periodicity. I
wonder, is there a necessary condition that we can establish?