Re: [tuning] Bizarre periodicity blocks found in 5 dimensions

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@ACADIAN-ASSET.COM>
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.
>
> ------------------------------------------------------------------------
> Get your money connected @ OnMoney.com - the first Web site that lets
> you see and manage all of your finances all in one place.
> http://click.egroups.com/1/3012/1/_/239029/_/956347882/
> ------------------------------------------------------------------------
>
> You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
> tuning-subscribe@egroups.com - join the tuning group.
> tuning-unsubscribe@egroups.com - unsubscribe from the tuning group.
> tuning-nomail@egroups.com - put your email message delivery on hold for
the tuning group.
> tuning-digest@egroups.com - change your subscription to daily digest
mode.
> tuning-normal@egroups.com - change your subscription to individual
emails.
>
>
>

Kees wrote,

>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.

Whoops, I didn't use that, please delete it.

>And the second enharmonic pair you found is 440:441, not 880:881.

Ah yes, thanks for the corrections.

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

Sorry, Kees, that doesn't work. Although this 1224-tone Fokker periodicity
block is very uneven, it does not form pairs of close approximations to
612-tone equal temperament

>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.

Thanks Kees, but I suspect there's a subtle mathematical issue that we're
missing here. Notice that in the 20-tone periodicity block the unison
vectors were far smaller than the steps of 20-tET or 10-tET. In the example
you gave, one of the unison vectors is about the same size as one step of
1224-tET, explaining the unevenness. In my example, the largest unison
vector is less that one eighth the size of a step of 20-tET, so a very even
scale would be expected. The doubling is what I didn't expect. I don't think
you can find this phenomenon in 2-d.

>The 'doubling' phenomenon is just a special case of the appearance of
linear
>combinations of 'good' ET case.

Not sure I understand.

Sorry, Kees, there was an error in my program, and you're right -- the
periodicity block you gave is a twin approximation of 612-tET. So in
general, when haphazardly combining unison vectors to form periodicity
blocks, one should check the ones with a non-prime number of notes to see if
they don't contain a periodicity within the periodicity. Thanks Kees.