Bohlen scale as Fokker periodicity block (was: Re: open problem)

>I also wondered whether the Bohlen-Pierce scale is a periodicity
>block or not. I think it is.

It is (even by the strict parallelogram definition).

In http://members.aol.com/bpsite/scales.html, Bohlen gives his "chromatic"
JI scale as:

1/1, 27/25, 25/21, 9/7, 7/5, 75/49, 5/3, 9/5, 49/25, 15/7, 7/3, 63/25, 25/9,
3/1.

Constructing a parallelogram, centered on 1/1 using UVs (1 2) = 245:243 and
(5 -3) = 3125:3087 in the 5-7 tritave-equivalent lattice is:
. . . . . . .
. . . . . . .
. . . . . . .
| * | | | | |
| | | | | | |
| *| | | | | |
| | * | | | | |
...-147/125--*49/25--49/45---49/27--245/243-1225/729-6125/2187-..
| | | | | | |
| * | | * | | | |
| | | | | | |
| * | | | | | |
| | | *| | | |
| * | | | | | |
| | | | | | |
..189/125*---63/25----7/5-----7/3--*-35/27--175/81--875/729-...
| | | | | | |
|* | | | | | |
| | | | |* | |
* | | | | | |
| | | | | | |
| | | | | * | |
| * | | | | | |
...-243/125--27/25----9/5-----1/1-----5/3----25/9---125/81--...
| | | | | | * |
| | * | | | | |
| | | | | | |
| | | | | | *
| | *| | | | |
| | | | | | *|
| | | | | | |
...2187/875-243/175--81/35-*--9/7----15/7----25/21---*125/63...
| | | | | | |
| | | | | | * |
| | | |* | | |
| | | | | | * |
| | | | | | |
| | | | * | | * |
| | | | | | |
6561/2187-2187/1225-729/245--81/49--135/49--75/49*--125/49--...
| | | | | * | |
| | | | | |* |
| | | | | | |
| | | | | * |
. . . . . . .
. . . . . . .
. . . . . . .
So yes, it's just a matter of shifting the surrounding parallelogram. Can
Scala do that?

I would diagram Bohlen's "chromatic" JI scale using the consonant triads:

7 1/5-------1/3
/ \ \ /
/ \ and \ /
/ \ \ /
/ \ \ /
3---------5 1/7

(and 3=9 in tritave-equivalent land):

49/25
/ \
/ \
/ \
/ \
63/25------7/5-------7/3
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
27/25------9/5-------1/1-------5/3------25/9
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
9/7------15/7------25/21
\ /
\ /
\ /
\ /
75/49

Bohlen's "diatonic" JI scale (see
http://members.aol.com/bpsite/scales.html#anchor245013) is:

7/5-------7/3
/ \ / \
/ \ / \
/ \ / \
/ \ / \
9/5-------1/1-------5/3------25/9
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
9/7------15/7------25/21

and it is a periodicity block with UVs (1 2) = 245:243 and (4 -1) = 625:567:

. . . . . . .
. . . . . . .
. . . . . . .
| | | | | | |
| | | | | | |
| | | | | | |
| | | | | | |
...-147/125--49/25---49/45---49/27--245/243-1225/729-6125/2187-..
| | | | | | |
| | | | | | |
| | | | | | |
| | | | | | |
| | * | | | |
| | | * | | | |
| | *| * | | |
...-189/125--63/25----7/5-----7/3--*-35/27--175/81--875/729-...
| | * | | * | |
| | | | | * | |
| | * | | | * |
| | | | | | * |
| | * | | | | *
| | | | | | |
| | * | | | | *|
...-243/125--27/25----9/5-----1/1-----5/3----25/9---125/81--...
| | * | | | | * |
| | | | | | |
| |* | | | | * |
| | | | | | |
| * | | | | * |
| | * | | | | |
| | * | | | * |
...2187/875-243/175--81/35-*--9/7----15/7----25/21--125/63...
| | | * | | * |
| | | | * | | |
| | | | * |* |
| | | | | * | |
| | | | | * |
| | | | | | |
| | | | | | |
6561/2187-2187/1225-729/245--81/49--135/49---75/49--125/49--...
| | | | | | |
| | | | | | |
| | | | | | |
| | | | | | |
. . . . . . .
. . . . . . .
. . . . . . .

So I would be tempted to say that a chromatic alteration in Bohlen's system
corresponds to 625:567.

>Heinz Bohlen's "commas" are these:
>1) 245/243 minor BP diesis (1 2)
>2) 3125/3087 major BP diesis (5 -3)
>3) 15625/15309 great BP diesis (6 -1)
>4) 16875/16807 small BP diesis (4 -5)

You might also be interested in Kees van Prooijen's page,
http://www.kees.cc/tuning/s357.html, referred to from
http://www.kees.cc/tuning/perbl.html, which examines the "natural"
progression of PBs in the 5,7 tritave-equivalent lattice.