Paul,

First of all, you slipped 385:384 (1,-1,-1,-1,0) in your list as a sixth
interval that's not in the matrix.
And the second enharmonic pair you found is 440:441, not 880:881.

And here's an example in 2 dimensional 5-limit:
27 -18
47 14
With determinant 1224, doubling 612

I think the problem is that the periodicity block method to get ET fails
when the range of 'building' intervals are to widely spread. Intervals that
are smaller than those spanning the block, fall inside the parallelogram and
cause periodicity inside the periodicity. My method was completely based on
avoiding this situation.
The 'doubling' phenomenon is just a special case of the appearance of linear
combinations of 'good' ET case.

Kees

----- Original Message -----
From: Paul H. Erlich <PERLICH@...>
To: 'tuning@onelist.com' <tuning@egroups.com>
Sent: Friday, April 21, 2000 1:10 PM
Subject: [tuning] 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.
>
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