a different example (was: coordinates from unison-vectors)

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, December 23, 2001 2:16 PM
> Subject: [tuning-math] Re: coordinates from unison-vectors (was: 55-tET)
>
>
> --- 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?

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

OK, here's a different example I've been playing around with.

Also note that it brings up another problem with my calculations:
my intention was to always have the periodicity-block centered
on 1/1, but it doesn't always work that way, even way I try
using reversed signs for the unison-vector exponent integers.

So anyway, I put in the matrix:

( 6 -14)
(-4 1)

and could see that the resulting periodicity-block had a strong
correlation (in the sense of my meantone-JI implied lattices)
with the neighborhood of 2/7- to 3/11-comma meantone.

The determinant is 50, so this agrees with my observation.

So then I made lattices comparing the JI periodicity-block
derived from that matrix with various fraction-of-a-comma
meantones, and can see by eye that the 7/25-comma meantone
axis goes right down the middle of this periodicity-block.

Therefore, my conclusion is that 7/25-comma meantone, 50-EDO,
and the (6 -14),(-4 1) periodicity-block are all intimately
related, and essentially identical. Other related meantones
are (in order of decreasing relatedness): 5/18-, 3/11-, and
2/7-comma.

This is getting closer to describing what I was asking about
a couple of weeks ago, for elegant mathematics which would
describe these kinds of relationships. Looking forward to
your reply.

-monz

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> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, December 23, 2001 3:31 PM
> Subject: [tuning-math] a different example (was: coordinates from
unison-vectors)
>
>
> So anyway, I put in the matrix:
>
> ( 6 -14)
> (-4 1)
>
> ... <snip>
>
> Therefore, my conclusion is that 7/25-comma meantone, 50-EDO,
> and the (6 -14),(-4 1) periodicity-block are all intimately
> related, and essentially identical. Other related meantones
> are (in order of decreasing relatedness): 5/18-, 3/11-, and
> 2/7-comma.

Oops... my bad. That should be the matrix:

( 6 -14)
( 4 -1)

and "the (6 -14),(4 -1) periodicity-block".

-monz

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

> Therefore, my conclusion is that 7/25-comma meantone, 50-EDO,
> and the (6 -14),(-4 1) periodicity-block are all intimately
> related, and essentially identical.

It would seem that you're totally ignoring the issue of how the
intervals are _tuned_, whether they're wolves or not, and focusing on
a rather superficial kind of similarity that results from sticking to
a 2-d JI lattice, and not using a cylinder to view 7/25-comma
meantone, and a torus to view 50-tET. In the latter cases, your
association of fixed JI pitches to each note in the tuning system is
at work, and Dave Keenan and I have been fighting to explain for
years now, it's the _intervals_ that matter in these lattices, not
the _pitches_. Think of the simple example of "where do you put the
1/1". C? D? The key of the piece? The key of the section? The current
chord? What if you have a chord like C-A-G-E-D?

As to the meantones in this range, I don't think your lattices
illuminate their similarity to any greater degree than is already
obvious from, say, the cents values of the fifths. The numeric data
on ratios you're pointing to in these studies has, I argue, no
musical or acoustical relevance to the perception of these meantones.

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

> So anyway, I put in the matrix:
>
> ( 6 -14)
> (-4 1)
>
> and could see that the resulting periodicity-block had a strong
> correlation (in the sense of my meantone-JI implied lattices)
> with the neighborhood of 2/7- to 3/11-comma meantone.
>
> The determinant is 50, so this agrees with my observation.

I'll work this out in another way; however, I'm afraid that while in some ways it is more elegant, it only becomes computationally easier when the number of scale steps is high, and I don't know that 50 is high enough for that to be the case.

First I put the 2 back into the above commas, and get

q1 = 2^23 3^6 5^-15 and q2 = 80/81. I have one et, h50, such that
h50(q1) = h50(q2) = 0. Now I search for something where h(q1)=0 and
h(q2)=1, obtainining h19, h69, -h31 and -h81. The simplest of these is h19, and I choose it.

Next, I look for something such that h(q1)=1 and h(q2)=0, and I get h16, -h34, -h84; I choose h16.

Now I form a 3x3 matrix from these, and invert it:

[ 50 19 16]
[ 79 30 25]^(-1) =
[116 44 37]

[-10 -1 5]
[ 23 6 -14]
[ 4 -4 1]

The rows of the inverted matrix correspond to the commas
q0 = 2^-10 3^-1 5 = 3125/3072 (small diesis), q1 = 2^23 3^6 5^14,
and q2=80/80.

Now I calculate the scale; the nth step is

scale[n] = q0^n * q1^round(19n/50) * q2^round(16n/50),

where "round" rounds to the nearest integer. It doesn't matter which paticular hn I selected when I do this, or where I start and end; though my definition of "nearest integer" does matter.

I got in this way:

1, 3125/3072, 128/125, 25/24, 82944/78125, 16/15, 625/576,
3456/3125, 10/9, 15625/13824, 144/125, 125/108, 18432/15625, 6/5, 625/512, 768/625, 5/4, 15625/12288, 32/25, 125/96, 20736/15625, 4/3, 3125/2304, 864/625, 25/18, 110592/78125, 36/25, 625/432, 4608/3125, 3/2, 15625/10368, 192/125, 25/16, 24576/15625, 8/5, 625/384, 1024/625, 5/3, 15625/9216, 216/125, 125/72, 27648/15625, 9/5, 3125/1728, 1152/625, 15/8, 78125/41472, 48/25, 125/64, 6144/3125

How does this compare with your results?

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

With 50 notes, some arbitrary decision has to be made -- no note can
be exactly in the center, since 50 is an even number. But you should
be getting the following block or its reflection through the origin:

p5's M3's
---- -----

3 -7
4 -7
1 -6
2 -6
3 -6
4 -6
1 -5
2 -5
3 -5
0 -4
1 -4
2 -4
3 -4
0 -3
1 -3
2 -3
3 -3
0 -2
1 -2
2 -2
-1 -1
0 -1
1 -1
2 -1
-1 0
0 0
1 0
-2 1
-1 1
0 1
1 1
-2 2
-1 2
0 2
-3 3
-2 3
-1 3
0 3
-3 4
-2 4
-1 4
0 4
-3 5
-2 5
-1 5
-4 6
-3 6
-2 6
-1 6
-4 7