Gene's notation & Schoenberg lattices

> From: <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Friday, August 17, 2001 7:06 PM
> Subject: [tuning-math] Re: Microtemperament and scale structure
>
>
> ... For another easy example, the
> rank 3 free group G can be sent to a rank 1 free group by a
> homomorphism h12(2) = 12, h12(3) = 19, h12(5) = 28.
> ...
> ... In the case of h12, the kernel is
> spanned by 81/80 (the diatonic comma) and 128/125 (the great diesis),
> where we have h12(81/80) = h12(128/125) = 0.
> ...
> Consider the system h72(2) = 72, h72(3) = 114, h72(5) = 167, h72(7) =
> 202, h72(11) = 249 ... h72(81/80) = 1 and h72(128/125) = 3...
> h31(2) = 31, h31(3) = 49, h31(5) = 72, h31(7) = 87 and h31(11) = 107
> *does* have the property that h31(81/80) = 0; and while h31(128/125) = 1
> we still find h31 is much closer in structre to h12 than is h72.

At last! I understand this!

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Wednesday, December 26, 2001 12:27 AM
> Subject: [tuning-math] Re: lattices of Schoenberg's rational implications
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > Can someone explain what's going on here, and what candidates
> > may be found for unison-vectors by extending the 11-limit system,
> > in order to define a 12-tone periodicity-block? Thanks.
>
> See if this helps;
>
> We can extend the set {33/32,64/63,81/80,45/44} to an 11-limit
> notation in various ways, for instance
>
> <56/55,33/32,65/63,81/80,45/44>^(-1) = [h7,h12,g7,-h2,h5]
>
> where g7 differs from h7 by g7(7)=19. Using this, we find
> the corresponding block is
>
> (56/55)^n (33/32)^round(12n/7) (64/63)^n (81/80)^round(-2n/12)
> (45/44)^round(5n/7), or 1-9/8-32/27-4/3-3/2-27/16-16/9;
> the Pythagorean scale. We don't need anything new to find a
> 12-note scale;

But alas! (pun intended) I have no idea what this means.

Gene (or anyone who knows), can you please explain this in
excruciating detail, illuminating every step and revealing
what all those cryptic letters represent? What's the business
with the "round" function? How do you "choose" your denominator?

If you can explain this in terms I understand, namely, the
matrix which lists the unison-vectors, then at least I can
begin to comprehend.

Can you tell me how the pseudo-code which I posted here
/tuning-math/messages/2069?expand=1
can be expanded to calculate higher-than-2-D periodicity-blocks?
(I know how to do the matrix stuff, but not the coordinates)

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> Gene (or anyone who knows), can you please explain this in
> excruciating detail, illuminating every step and revealing
> what all those cryptic letters represent? What's the business
> with the "round" function?

"Round" means round off to the nearest integer; sometimes, as with 1/2 or -1/2, there isn't a nearest integer. Then we can define it as round down, round up, or round towards, or round away--that is, round 1/2 down to 0 and -1/2 down to -1; or round 1/2 up to 1 and -1/2 up to 0; or round -1/2 to 0 and also 1/2 to 0, or round 1/2 to 1 and
-1/2 to -1. The last two systems are not good ones on mathematical groups, but get used anyway because they are easy to do and programmers are not always very sensative to good mathematical technique.

How do you "choose" your denominator?
>
> If you can explain this in terms I understand, namely, the
> matrix which lists the unison-vectors, then at least I can
> begin to comprehend.

If I start from <56/55,33/32,64/63,81/80,45/44> I can form the corresponding matrix:

[ 3 0 -1 1 -1]
[-5 1 0 1 1]
[ 6 -2 0 -1 0]
[-4 4 -1 0 0]
[-4 4 -1 0 0]
[-2 2 1 0 -1]

This matrix is unimodular, meaning it has determinant +-1. If I invert it, I get

[ 7 12 7 -2 5]
[11 19 11 -3 8]
[16 28 16 -5 12]
[20 34 19 -6 14]
[24 42 24 -7 17]

The columns of this are essentially a set of ets; they are maps from intervals to the integers, something I call a "val". The third column differs from the first by having a 19 instead of a 20 for the value that 7 gets mapped to. If I call the second column map "h12", I find that h12(56/55)=h12(64/63)=h12(81/80)=h12(45/44), but h12(33/32)=1; all of the columns are dual in this way to the rows of the previous matrix, by the definition of matrix inverse.

For any non-zero I can define a scale by calculating for 0<=n<d

step[n] = (56/55)^round(7n/d) (33/32)^round(12n/d)
(64/63)^round(7n/d) (81/80)^round(-2n/d) (45/44)^round(5n/d)

Choosing a value of d is what I meant by "chooing a denominator". If the value of d is not one of the values on the top row of the inverted matrix, which shows where 2 maps to, we may or may not get slightly goofy results such as repeated steps, but generally it seems to work reasonably well.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

>
> If I start from <56/55,33/32,64/63,81/80,45/44> I can form the
corresponding matrix:
>
> [ 3 0 -1 1 -1]
> [-5 1 0 1 1]
> [ 6 -2 0 -1 0]
> [-4 4 -1 0 0]
> [-4 4 -1 0 0]
> [-2 2 1 0 -1]

You put [-4 4 -1 0 0] in there twice.
>
> This matrix is unimodular, meaning it has determinant +-1.

Funny, if I take out [-4 4 -1 0 0], I get a determinant of 35!

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

> Funny, if I take out [-4 4 -1 0 0], I get a determinant of 35!

Hmmm...try

[ 3 0 -1 1 -1]
[-5 1 0 0 1]
[ 6 -2 0 -1 0]
[-4 4 -1 0 0]
[-2 2 1 0 -1]

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Wednesday, December 26, 2001 3:38 PM
> Subject: [tuning-math] Re: Gene's notation & Schoenberg lattices
>
>
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > Funny, if I take out [-4 4 -1 0 0], I get a determinant of 35!
>
> Hmmm...try
>
> [ 3 0 -1 1 -1]
> [-5 1 0 0 1]
> [ 6 -2 0 -1 0]
> [-4 4 -1 0 0]
> [-2 2 1 0 -1]

This gives me an adjoint of:

[ 7 12 7 -2 5]
[11 19 11 -3 8]
[16 28 16 -5 12]
[20 34 19 -6 14]
[24 42 24 -7 17]

So that:

h12(2) = 12
h12(3) = 19
h12(5) = 28
h12(7) = 34
h12(11) = 42

and

h5(2) = 5
h5(3) = 8
h5(5) = 12
h5(7) = 14
h5(11) = 17

Looks correct to me, altho the approximations of 5-EDO
to 5 and especially 11 are rather far off.

But I also see that:

h7(2) = 7
h7(3) = 11
h7(5) = 16
h7(7) = 19 and 20
h7(11) = 24

If I recall, there is some significance to the fact
that h7(7) = both 19 and 20. This is telling us that
7-EDO gives excellent approximations to 2, 3, and 5,
and a pretty good one to 11, but 7 lies approximately
midway between the two closest approximations. Correct?

(How am I doing on grokking your terminology, Gene?)

What about the column beginning with -2? h-2(2)=-2,
h-2(3)=-3, and h-2(11)=-7 all look OK, but -5 is closer
to h-2(6) than to h-2(5), and -6 is exactly h-2(8). ...???

And then... what does any of this have to do with finding
the 12-tone periodicity-block which Schoenberg had in mind?
I don't get it.

Also, I don't see any 7-limit ratios on the lattice...
that's because of the 56:55 and 64:63, right?

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> > [ 3 0 -1 1 -1]
> > [-5 1 0 0 1]
> > [ 6 -2 0 -1 0]
> > [-4 4 -1 0 0]
> > [-2 2 1 0 -1]
>
>
> This gives me an adjoint of:
>
> [ 7 12 7 -2 5]
> [11 19 11 -3 8]
> [16 28 16 -5 12]
> [20 34 19 -6 14]
> [24 42 24 -7 17]

In this case, that's also the inverse.

> If I recall, there is some significance to the fact
> that h7(7) = both 19 and 20.

h7(7)=20 by my definition, which says to round to the nearest integer. I called the map g7 which maps to 19: g7(7)=19.

This is telling us that
> 7-EDO gives excellent approximations to 2, 3, and 5,
> and a pretty good one to 11, but 7 lies approximately
> midway between the two closest approximations. Correct?

It's telling us that this particular set of commas does not work to give us something consistent so far as the mapping of 7 goes, since
h7 and g7 are the same except for how they map 7. Schoenberg's choice is a peculiar one, but interesting because of it.

> What about the column beginning with -2? h-2(2)=-2,
> h-2(3)=-3, and h-2(11)=-7 all look OK, but -5 is closer
> to h-2(6) than to h-2(5), and -6 is exactly h-2(8). ...???

It's h-2(7)=-6; only the maps of primes are shown, since everything else can be determined from them by addition.

> Also, I don't see any 7-limit ratios on the lattice...
> that's because of the 56:55 and 64:63, right?

Right; 56/55 64/63 = 512/495, which has no factor of 7. We can remove a row and column from the matrix of vals, and get

[ 7 12 -2 5]
[11 19 -3 8]
[16 28 -5 12]
[24 42 -7 17]

The inverse of this is

[ 9 -2 -2 -2]
[-5 1 0 1]
[-4 4 -1 0]
[-2 2 1 -1]

which corresponds to <512/495, 33/32, 81/80, 45/44>. Everything now is in a 2^a 3^b 5^c 11^d system without 7s, and it works equivalently for finding blocks.