nugget from 4/21/00: first glimpse of "torsion"???

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
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.
--- End forwarded message ---

discuss . . .

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> discuss . . .

You can do the same calculation with these six commas as I did in
the "72 owns the 11-limit" article, with a similar result. Taking the
six combinations of these five at a time, and adding an indeterminant
row [a b c d e f], we get a determinant of zero in two cases (linear
dependency), a determinant of h10 and of -h10, meaning we can
construct a 10-block, and two determinants of -2 h10, meaning we have
torsion. I'm afraid this sort of thing will happen a lot.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > discuss . . .
>
> You can do the same calculation with these six commas as I did in
> the "72 owns the 11-limit" article, with a similar result. Taking
the
> six combinations of these five at a time,

Wait a minute Gene. Although I did, for some strange reason, list six
unison vectors, I only included five in the Fokker matrix. So what if
I only had those five? Would I have to try all possible products and
quotients of pairs out of that set of five?

> and adding an indeterminant
> row [a b c d e f], we get a determinant of zero in two cases
(linear
> dependency), a determinant of h10 and of -h10, meaning we can
> construct a 10-block, and two determinants of -2 h10, meaning we
have
> torsion. I'm afraid this sort of thing will happen a lot.

If you plug in indeterminants a, b, c, d, e, and f, how do you get
something in terms of h to come out? I'd like to be able to do this
calculation . . . but again, starting with only enough unison vectors
to construct the Fokker matrix.

> --- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
> 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

This whole matrix needs to be multiplied through by -1 so that the
intervals are all small, instead of slightly smaller than an octave.

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

...

> --- End forwarded message ---
>
> discuss . . .

We know all about these now. It's like that 24 note periodicity block
defining 12-equal. I've added it to the test cases at
<http://x31eq.com/vectors.py> and
<http.microtonal.co.uk/vectors.out> if you're interested.

Is this "torsion" then?

Graham

--- In tuning-math@y..., graham@m... wrote:
> > --- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
> > 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
>
> This whole matrix needs to be multiplied through by -1 so that the
> intervals are all small, instead of slightly smaller than an octave.

What are you talking about? There's no 2 column here, and there's no reason=
to define the
intervals downward instead of upward. Anyway, multiplying by -1 won't affec=
t the determinant,
or anything else significant . . . will it?
>
> Is this "torsion" then?

Yes . . . Gene explained this a while back . . . I need to go out and get a=
n abstract algebra
textbook . . . I used to be a champion math student :(

Paul wrote:

> What are you talking about? There's no 2 column here, and there's no
> reason=
> to define the
> intervals downward instead of upward. Anyway, multiplying by -1 won't
> affec=

Once again, I don't know how to find the temeperaments from octave
equivalent matrices. My conversion to octave specificity assumes the
intervals are between a unison and an octave.

Graham