coordinates from unison-vectors (was: 55-tET)

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Wednesday, December 19, 2001 1:36 PM
> Subject: [tuning-math] Re: 55-tET
>
>
> ... So in your [monz's] view, the 55 tones would be much
> better understood as the Fokker periodicity block defined
> by the two unison vectors (-4 4 -1) and (-51 19 9). Since
> I'm sure you're interested, here are the coordinates of
> these 55 tones in the (3,5) lattice:
>
> 3 5
> --- ----
>
> -11 -4
> -10 -4
> -9 -4
> -8 -4
> -7 -4 <etc. -- snip>

Paul, can you please explain the procedure you use to find
coordinates from a given set of unison-vectors, as you did
here? Thanks.

-monz

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> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, December 22, 2001 12:21 PM
> Subject: [tuning-math] coordinates from unison-vectors (was: 55-tET)
>
>
> Paul, can you please explain the procedure you use to find
> coordinates from a given set of unison-vectors, as you did
> here? Thanks.

I've figured out how to use Excel to calculate the coordinates
within the unit square of the inverse of a 2-dimensional matrix,
and even how to have it centered on 0,0... I think. Here are
my results for the periodicity-block [3^x * 5^y] with
unison-vectors (x y) of (4 -1) and (19 9):

0/55 0/55
9/55 1/55
18/55 2/55
27/55 3/55
-19/55 4/55
-10/55 5/55
-1/55 6/55
8/55 7/55
17/55 8/55
26/55 9/55
-20/55 10/55
-11/55 11/55
-2/55 12/55
7/55 13/55
16/55 14/55
25/55 15/55
-21/55 16/55
-12/55 17/55
-3/55 18/55
6/55 19/55
15/55 20/55
24/55 21/55
-22/55 22/55
-13/55 23/55
-4/55 24/55
5/55 25/55
14/55 26/55
23/55 27/55
-23/55 -27/55
-14/55 -26/55
-5/55 -25/55
4/55 -24/55
13/55 -23/55
22/55 -22/55
-24/55 -21/55
-15/55 -20/55
-6/55 -19/55
3/55 -18/55
12/55 -17/55
21/55 -16/55
-25/55 -15/55
-16/55 -14/55
-7/55 -13/55
2/55 -12/55
11/55 -11/55
20/55 -10/55
-26/55 -9/55
-17/55 -8/55
-8/55 -7/55
1/55 -6/55
10/55 -5/55
19/55 -4/55
-27/55 -3/55
-18/55 -2/55
-9/55 -1/55

Right?

The graph of this is at
/tuning-math/files/monz/inv-matrix.gif

(I think my axis labels may be wrong, but the graph appears to be
showing the correct periodicity-block shape.)

But now how do I go about "transforming them back to the lattice
(using the original Fokker matrix)" as described at
<http://www.ixpres.com/interval/td/erlich/srutipblock.htm>,
to get the actual lattice coordinates?

-monz

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> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, December 22, 2001 3:29 PM
> Subject: Re: [tuning-math] coordinates from unison-vectors (was: 55-tET)
>
>
>
> > From: monz <joemonz@yahoo.com>
> > To: <tuning-math@yahoogroups.com>
> > Sent: Saturday, December 22, 2001 12:21 PM
> > Subject: [tuning-math] coordinates from unison-vectors (was: 55-tET)
> >
> >
> > Paul, can you please explain the procedure you use to find
> > coordinates from a given set of unison-vectors, as you did
> > here? Thanks.

Never mind! Trial and error wins again!

By brute force, a bit of research into matrix transformations,
and a whole lot of luck, I figured out how to do it.

-monz

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> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, December 22, 2001 7:35 PM
> Subject: Re: [tuning-math] coordinates from unison-vectors (was: 55-tET)
>
>
> By brute force, a bit of research into matrix transformations,
> and a whole lot of luck, I figured out how to do it.

I've figured out how to do what I was trying to do, but I'm
still puzzled over the way changing the sign of the exponents
in the unison-vector changes the shape of the periodicity-block.

I know that if the sign of the 3-exponent is changed, the sign
for the 5-exponent must be reversed accordingly. But I find
sometimes that using, for example, (4 -1) for the syntonic comma
doesn't always give me the PB I expected, whereas making it
(-4 1) does.

Now that I have an Excel spreadsheet set up to graph the
periodicity-blocks, I can simply play with the signs until
I get the block I'm looking for. But I'm curious as to why
the shape changes as it does. Can anyone explain this?

-monz

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> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>; <jhchalmers@ucsd.edu>;
<paul@stretch-music.com>
> Sent: Saturday, December 22, 2001 7:35 PM
> Subject: Re: [tuning-math] coordinates from unison-vectors (was: 55-tET)
>
>
> By brute force, a bit of research into matrix transformations,
> and a whole lot of luck, I figured out how to do it.

Here's the pseudo-code for the formulas in my spreadsheet.
Please feel free to correct any errors or to make the code
more elegant.

unison-vectors =

(3^a) * (5^b)
(3^c) * (5^d)

unison-vector matrix =

(a b)
(c d)

determinant n of the matrix :

n = (a*d) - (c*b)

inverse of the matrix =

( d -b)
(-c a)
-------
n

inverse coordinates p, q :

p = 0, q = 0

LOOP

if ABS(p+d) > (ABS(n)/2)

then p = MOD(p+d, ABS(n)) - ABS(n)

else p = p + d

end if

if ABS(q-b) > (ABS(n)/2)

then q = MOD(q-b, ABS(n)) - ABS(n)

else q = q - b

end if

lattice coordinates x, y :

x = ( (q*c) + (p*a) ) / n

y = ( (q*d) + (p*b) ) / n

END LOOP

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Wednesday, December 19, 2001 1:36 PM
> > Subject: [tuning-math] Re: 55-tET
> >
> >
> > ... So in your [monz's] view, the 55 tones would be much
> > better understood as the Fokker periodicity block defined
> > by the two unison vectors (-4 4 -1) and (-51 19 9). Since
> > I'm sure you're interested, here are the coordinates of
> > these 55 tones in the (3,5) lattice:
> >
> > 3 5
> > --- ----
> >
> > -11 -4
> > -10 -4
> > -9 -4
> > -8 -4
> > -7 -4 <etc. -- snip>
>
>
>
> Paul, can you please explain the procedure you use to find
> coordinates from a given set of unison-vectors, as you did
> here? Thanks.
>
>
>
> -monz

This is explained in the _Gentle Introduction, part 3. Though there
I'm dealing with a 3-d case, here it's only a 2-d case.

Gene has a better way, though, where he can give a single formula to
produce _all_ the points and _only_ those points in one fell swoop.

----- Original Message -----
From: paulerlich <paul@stretch-music.com>
To: <tuning-math@yahoogroups.com>
Sent: Sunday, December 23, 2001 1:03 PM
Subject: [tuning-math] Re: coordinates from unison-vectors (was: 55-tET)

> This is explained in the _Gentle Introduction, part 3. Though there
> I'm dealing with a 3-d case, here it's only a 2-d case.

Hmmm... I studied all the pages in your _Gentle Introduction_ series
*except* the 3-d (7-limit) one!

> Gene has a better way, though, where he can give a single formula to
> produce _all_ the points and _only_ those points in one fell swoop.

I'm interested in seeing the differences between his formula and
the method I jury-rigged. :)

... always searching for greater elegance ...

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > From: monz <joemonz@y...>
> > To: <tuning-math@y...>
> > Sent: Saturday, December 22, 2001 12:21 PM
> > Subject: [tuning-math] coordinates from unison-vectors (was: 55-
tET)
> >
> >
> > Paul, can you please explain the procedure you use to find
> > coordinates from a given set of unison-vectors, as you did
> > here? Thanks.
>
>
>
> I've figured out how to use Excel to calculate the coordinates
> within the unit square of the inverse of a 2-dimensional matrix,
> and even how to have it centered on 0,0... I think.

Good for you! So you read part 3 already?

> But now how do I go about "transforming them back to the lattice
> (using the original Fokker matrix)" as described at
> <http://www.ixpres.com/interval/td/erlich/srutipblock.htm>,
> to get the actual lattice coordinates?

Just multiply each pair of coordinates _inside the box_ by the
original, non-inverted, Fokker matrix.

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

> I know that if the sign of the 3-exponent is changed, the sign
> for the 5-exponent must be reversed accordingly. But I find
> sometimes that using, for example, (4 -1) for the syntonic comma
> doesn't always give me the PB I expected, whereas making it
> (-4 1) does.

As long as it's IN THE SAME FORM when you apply the inverse of the
matrix as well as when you apply the matrix itself, it won't matter --
if you're centering around (0,0).

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

> lattice coordinates x, y :
>
>
> x = ( (q*c) + (p*a) ) / n
>
> y = ( (q*d) + (p*b) ) / n

There shouldn't be an "/ n" at the end of that. Maybe that's what's
causing the weirdness, because sometimes n is negative.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, December 23, 2001 1:18 PM
> Subject: [tuning-math] Re: coordinates from unison-vectors (was: 55-tET)
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > I know that if the sign of the 3-exponent is changed, the sign
> > for the 5-exponent must be reversed accordingly. But I find
> > sometimes that using, for example, (4 -1) for the syntonic comma
> > doesn't always give me the PB I expected, whereas making it
> > (-4 1) does.
>
> As long as it's IN THE SAME FORM when you apply the inverse of the
> matrix as well as when you apply the matrix itself, it won't matter --
> if you're centering around (0,0).

My code does it automatically, so I guess it works correctly.
All the user has to enter are the exponents of 3 and 5 for the
two unison-vectors. Everything else is calculated from that.

I've posted the Excel spreadsheet to the Files section.
/tuning-math/files/monz/matrix%20math.xls

Also, I believe I've noticed that it makes a difference whether
the larger or the smaller comma is listed first. Can someone
check that and explain it?

-monz

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> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, December 23, 2001 1:22 PM
> Subject: [tuning-math] Re: coordinates from unison-vectors (was: 55-tET)
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > lattice coordinates x, y :
> >
> >
> > x = ( (q*c) + (p*a) ) / n
> >
> > y = ( (q*d) + (p*b) ) / n
>
> There shouldn't be an "/ n" at the end of that.

Why not? "n" is the determinant of the matrix, and plays
a crucial role in the transformation from one perspective
to another.

> Maybe that's what's causing the weirdness, because
> sometimes n is negative.

Hmmm... so perhaps I need to keep "/ abs(n)"?

-monz

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> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, December 23, 2001 1:41 PM
> Subject: Re: [tuning-math] Re: coordinates from unison-vectors (was:
55-tET)
>
>
> I've posted the Excel spreadsheet to the Files section.
> /tuning-math/files/monz/matrix%20math.xls

I should have specified: the only worksheet in the file which
draws the 5-limit periodicity-blocks is the one named
"5-L PBs from UVs".

The other worksheets illustrate 3-d examples from Graham's matrix
tutorial webpage, and one of them was copied and currently has the
unison-vectors for 22-EDO which were in my posts of a few days ago.

-monz

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

> I'm interested in seeing the differences between his formula and
> the method I jury-rigged. :)
>
> ... always searching for greater elegance ...

Why don't you give an example and I'll work it out in a different way?

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Sunday, December 23, 2001 1:22 PM
> > Subject: [tuning-math] Re: coordinates from unison-vectors (was:
55-tET)
> >
> >
> > --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> >
> > > lattice coordinates x, y :
> > >
> > >
> > > x = ( (q*c) + (p*a) ) / n
> > >
> > > y = ( (q*d) + (p*b) ) / n
> >
> > There shouldn't be an "/ n" at the end of that.
>
>
> Why not? "n" is the determinant of the matrix, and plays
> a crucial role in the transformation from one perspective
> to another.

All you need for the transformation in one direction is the matrix
itself; and in the other direction, its inverse. You don't
_additionally_ apply the determinant.

> > Maybe that's what's causing the weirdness, because
> > sometimes n is negative.
>
> Hmmm... so perhaps I need to keep "/ abs(n)"?

Nope.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, December 23, 2001 2:19 PM
> Subject: [tuning-math] Re: coordinates from unison-vectors (was: 55-tET)
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> >
> > > From: paulerlich <paul@s...>
> > > To: <tuning-math@y...>
> > > Sent: Sunday, December 23, 2001 1:22 PM
> > > Subject: [tuning-math] Re: coordinates from unison-vectors
> > >
> > >
> > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > >
> > > > lattice coordinates x, y :
> > > >
> > > >
> > > > x = ( (q*c) + (p*a) ) / n
> > > >
> > > > y = ( (q*d) + (p*b) ) / n
> > >
> > > There shouldn't be an "/ n" at the end of that.
> >
> >
> > Why not? "n" is the determinant of the matrix, and plays
> > a crucial role in the transformation from one perspective
> > to another.
>
> All you need for the transformation in one direction is the matrix
> itself; and in the other direction, its inverse. You don't
> _additionally_ apply the determinant.

But when the inverse is described in integer terms, the
determinant is part of it!

Did you try my spreadsheet yet?

-monz

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OK, there's something in the way you're calculating things that
forces you to use the determinant again at the end. So, for now, I
would recommend dividing by abs(n).

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

> > All you need for the transformation in one direction is the
matrix
> > itself; and in the other direction, its inverse. You don't
> > _additionally_ apply the determinant.
>
>
> But when the inverse is described in integer terms, the
> determinant is part of it!

Yes, it's part of the usual calculation of the inverse. But it's
certainly not part of the usual application of the matrix itself at
the end of the process.

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

> Yes, it's part of the usual calculation of the inverse. But it's
> certainly not part of the usual application of the matrix itself at
> the end of the process.

Maybe he would be better off calculating the adjoint instead.