--- 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