Bizarre periodicity blocks found in 5 dimensions

I've been exploring some 13-limit periodicity blocks due to Polychroni's
questions, and I've found some which seem to contradict my conceptions
periodicity blocks so far. For example, using the unison vectors 243:242
(7.1¢), 352:351 (4.9¢), 385:384 (4.5¢), 676:675 (2.6¢), 2401:2400 (0.7¢),
and 3025:3024 (0.6¢), so that the Fokker matrix is

-5 0 0 2 0
3 0 0 -1 1
3 2 0 0 -2
1 2 -4 0 0
3 -2 1 -2 0

(whose determinant is 20), the 5-d "parallelogram" contains the following
pitches:

cents numerator denominator
0 1 1
116.23 77 72
119.44 15 14
235.68 55 48
238.89 225 196
359.47 16 13
363.4 882 715
478.92 120 91
482.85 189 143
595.15 55 39
598.36 900 637
717.15 286 189
721.08 91 60
836.6 715 441
840.53 13 8
961.11 392 225
964.32 96 55
1080.6 28 15
1083.8 144 77
1196.1 880 441

Instead of being an approximation of 20-tET, it's an double approximation of
10-tET with 539:540 and 880:881 pairs.

What's going on here??? Can anyone come up with a mathematical explanation
for this phenomenon? Does it only occur in higher dimensions?

Certainly my belief, which I think came from Paul Hahn, that an N-tone
periodicity block with small unison vectors will always be a good
approximation of N-tET, turned out to be wrong.

On October 29, 1999, Paul Hahn wrote,

>If you look at the way the truncated
>octa fills space, you'll find that its symmetry group is that of a
>body-centered cubic lattice. The rhombic dodec, OTOH, fills space such
>that is _its_ symmetry group is that of a _face_-centered cubic
>lattice.

>Why is this important? Well, let's think about why we use FCC (= oc-tet
>or triangulated lattice) for pitch diagrams in the first place. If you
>ignore the edges and just look at the lattice points, it's equivalent to
>the cubic lattice--it's just been subjected to a couple of affine
>(shear) transformations. (Translation: we squished it a bit so that it
>slants.) This makes sense because the lattice is actually a space whose
>basis is the three vectors representing the 3/2, the 5/4, and the 7/4.
>Right?

>Now look at the way the shapes you want to use to tesselate space with.
>They _also_ are related to each other by three basic vectors, it's just
>that this time, it's the three unison vectors. Other than that, the
>relationship is the same. But there's no way you can map the BCC to a
>straightforward cubic lattice--you either have to leave some points out
>of the cubic lattice, or interlock two cubic lattices together.

Conjecture: the bizarre, double-vision periodicity block I found could even
happen in 3 dimensions, if there are 4 unison vectors defining
truncated-octahedron equivalence regions, but due to the parallelopiped
basis of the periodicity block construction from three unison vectors, these
truncated-octahedron regions could only come up two at a time.

What I found may be some sort of higher-dimensional analogue with 5
dimensions and 7(?) operative unison vectors.

Paul H., does this make any sense?

On Fri, 21 Apr 2000, Paul H. Erlich wrote:
[lotsa fascinating stuff snipped]
> Paul H., does this make any sense?

I'm not sure. I'll have to take some time to study this, and get back
to you.

--pH <manynote@library.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "Play it over? When I need one and you need seven?"
-\-\-- o

Meanwhile, I'll try to find a 3-d (7-limit) example.

-----Original Message-----
From: Paul Hahn [mailto:PAUL-HAHN@LIBRARY.WUSTL.EDU]
Sent: Friday, April 21, 2000 4:18 PM
To: 'tuning@onelist.com'
Subject: Re: [tuning] RE: Bizarre periodicity blocks found in 5
dimensions

On Fri, 21 Apr 2000, Paul H. Erlich wrote:
[lotsa fascinating stuff snipped]
> Paul H., does this make any sense?

I'm not sure. I'll have to take some time to study this, and get back
to you.

--pH <manynote@library.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "Play it over? When I need one and you need seven?"
-\-\-- o

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>Instead of being an approximation of 20-tET, it's an double approximation of
>10-tET with 539:540 and 880:881 pairs.

Paul, could you explain what you mean by "539:540 and 880:881 pairs" here?
Are you saying you found a 20-tone periodicity block that can be viewed as two
smaller blocks connected by some sort of meta unison vectors?

Do I understand you to imply that there are a number of different smaller
blocks (each approximating 10-tet) that could be used here, which, taken all
together, outline a single 7-D block?

-Carl

>>>Instead of being an approximation of 20-tET, it's an double approximation
>>>of 10-tET with 539:540 and 880:881 pairs.
>
>>Paul, could you explain what you mean by "539:540 and 880:881 pairs" here?
>>Are you saying you found a 20-tone periodicity block that can be viewed as
>>two smaller blocks connected by some sort of meta unison vectors?
>
>I just meant that of the pairs of notes associated with each 10-tET degree,
>five were 539:540 apart, and five were 881:880 apart.

That's what I thought, but 881:880 was too small. Kees' post cleared that
up.

>>Do I understand you to imply that there are a number of different smaller
>>blocks (each approximating 10-tet) that could be used here, which, taken
>>all together, outline a single 7-D block?
>
>No. There are only 5 dimensions in 13-limit tuning.

I was just trying to figure out what you meant by...

>What I found may be some sort of higher-dimensional analogue with 5
>dimensions and 7(?) operative unison vectors.

>Although the 20-tone 13-limit periodicity blocks turned out to be double
>10-tone ones, this 11-limit one is not:

But that one does seem to approximate 20-tET. Question is, is the original
idea correct -- that all periodicity blocks (or perhaps only those with
unison vectors smaller than their smallest 2nd) approximate N-tET? Or
don't we know yet?

-Carl