>I'm interested in studying the historical conceptions of
>various shapes and sizes of periodicity blocks in music
>all over the world. I believe that this 'history of finity
>in tuning' (which, I now realize, is what my book(s?) attempts
>to be) can enrich our knowledge of many other aspects of our
>lives and histories, especially ancient religious beliefs,
>possibly even extending to modern scientific theories about
>the universe.

The Hindu system of 22 srutis, in its common JI form is essentially a Fokker
periodicity block, but not one of the good ones I'm looking for. Since this
is a 5-limit system, we need two unison vectors. The first one is the
diaschisma, ratio 2048/2025 or 3^-4 * 5^-2, as we've discussed before,
there's evidence that the same pitch (sruti #2) functioned as 135/128 and as
16/15 (or in ma-grama, as 45/32 and as 64/45). That's about 19.6 cents. The
other unison vector is large: the difference between sruti #1 in the two
gramas, 3^-9 * 5 or 68.7 cents. So the Fokker matrix is

-5 3
-9 1

The inverse of this matrix is

1/22 -3/22
9/22 -5/22

If we transform the set of lattice points with this matrix, we can define a
periodicity block within any 1 X 1 square of the transformed lattice. The
usual approach is to use the "unit square" from the origin to (1,1), but of
course we are free to translate this square wherever we want since the
result will still tile the plane with the unison vectors as generators. So
let us choose the square where the position along each dimension is greater
than -1/2 and less than or equal to 1/2, so the origin will serve as a
central note rather than a corner. Taking the resulting 22 points

-5/11 1/11
-9/22 2/11
-4/11 3/11
-7/22 4/11
-3/11 5/11
-5/22 -5/11
-2/11 -4/11
-3/22 -3/11
-1/11 -2/11
-1/22 -1/11
0 0
1/22 1/11
1/11 2/11
3/22 3/11
2/11 4/11
5/22 5/11
3/11 -5/11
7/22 -4/11
4/11 -3/11
9/22 -2/11
5/11 -1/11
1/2 0

and transforming them back to the lattice (using the original Fokker
matrix), we get

-1 -1 112
0 -1 814
1 -1 316
2 -1 1018
3 -1 520
-5 0 90
-4 0 792
-3 0 294
-2 0 996
-1 0 498
0 0 0
1 0 702
2 0 204
3 0 906
4 0 408
5 0 1110
-3 1 680
-2 1 182
-1 1 884
0 1 386
1 1 1088
2 1 590

in lattice form:

680---182---884---386---1088--590
/ \ / \ / \ / \ / \ / \
/ \ / \ / \ / \ / \ / \
90---792---294---996---498----0----702---204---906---408---1110
\ / \ / \ / \ / \ /
\ / \ / \ / \ / \ /
112---814---316---1018--520

Now of course all the properties of the block are preserved if we transpose
one (or more) note(s) by one of the unison vectors used to create the block.
This corresponds to distorting the edges of the block in parallel as I've
discussed before. Let's take (-3,1) (680.5 cents) and lower it by the unison
vector (-9,1) (68.7 cents) to obtain the note (6,0) (612 cents):

182---884---386---1088--590
/ \ / \ / \ / \ / \
/ \ / \ / \ / \ / \
90---792---294---996---498----0----702---204---906---408---1110--612
\ / \ / \ / \ / \ /
\ / \ / \ / \ / \ /
112---814---316---1018--520

This is the scale of srutis according to most reputable sources, such as S.
Ramanathan, Mathieu (see his _Harmonic Experience), etc. etc.

Clearly the scales distinguish notes a syntonic comma (22 cents) apart,
while one of the unison vectors is over three times larger. So this isn't
one of the blocks I'm looking for.

By the way, this process of transposing notes by unison vectors can in
general affect the analysis of whether the block is of the 'good' type or
not, as can the translation of the unit square mentioned earlier. Also one
could consider the diagonals of the block as representing unisons as well as
the edges. So my quest is complicated beyond belief.