spiraling into the vortex of ET approximations to 5-liimit JI -- for monz's page

successive zooms into the center:

/tuning/files/perlich/tree.gif
/tuning/files/perlich/treezuma.gif
/tuning/files/perlich/treezumb.gif
/tuning/files/perlich/treezumc.gif

if you can squint real hard, you can see 612 as 171+441, 118+494, and
289+323 near the center of the last graph . . . i was working on
another zoom but matlab crashed . . .

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
wrote:
> successive zooms into the center:
>
> /tuning/files/perlich/tree.gif
> /tuning/files/perlich/treezuma.gif
> /tuning/files/perlich/treezumb.gif
> /tuning/files/perlich/treezumc.gif
>
> if you can squint real hard, you can see 612 as 171+441, 118+494,
and
> 289+323 near the center of the last graph . . . i was working on
> another zoom but matlab crashed . . .

two more added, so as to reach monz's "magical" 4296:

/tuning/files/perlich/treezumd.gif
/tuning/files/perlich/treezume.gif

monz, (on your eqtemp page) you included 1432. maybe you really meant
1342? besides these graphs, marc jones also mentions 1342, not 1432,
at:

/tuning/files/dict/marc-edolist.htm

i also remember a post from marc that mentioned 3125 (=5^5) among
other such trivia. can anyone locate it?

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:
> successive zooms into the center:
>
> /tuning/files/perlich/tree.gif
> /tuning/files/perlich/treezuma.gif
> /tuning/files/perlich/treezumb.gif
> /tuning/files/perlich/treezumc.gif

You finally did it!

> if you can squint real hard, you can see 612 as 171+441, 118+494, and
> 289+323 near the center of the last graph . . . i was working on
> another zoom but matlab crashed . . .

The ennealimmal line (171+441=612) isn't bad even in the 5-limit.

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> i also remember a post from marc that mentioned 3125 (=5^5) among
> other such trivia. can anyone locate it?

I've always liked it because of the 5^5, so I might have mentioned that. It's really a 7 (and 9) limit system anyway--in terms of the 2401/2400 beat phenomenon, it allots two steps to 2401/2400 (and one to 4375/4374.)

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>

/tuning/topicId_39695.html#39695

wrote:
> successive zooms into the center:
>
> /tuning/files/perlich/tree.gif
> /tuning/files/perlich/treezuma.gif
> /tuning/files/perlich/treezumb.gif
> /tuning/files/perlich/treezumc.gif
>
> if you can squint real hard, you can see 612 as 171+441, 118+494,
and
> 289+323 near the center of the last graph . . . i was working on
> another zoom but matlab crashed . . .

***These are just great; they look like an Earle Brown piece! Maybe
we should try to play this stuff! A requisite would be a synth that
could instantaneously switch ETs! :)

JP

--- In tuning@y..., "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
>
> /tuning/topicId_39695.html#39695
>
> wrote:
> > successive zooms into the center:
> >
> > /tuning/files/perlich/tree.gif
> > /tuning/files/perlich/treezuma.gif
> > /tuning/files/perlich/treezumb.gif
> > /tuning/files/perlich/treezumc.gif
> >
> > if you can squint real hard, you can see 612 as 171+441, 118+494,
> and
> > 289+323 near the center of the last graph . . . i was working on
> > another zoom but matlab crashed . . .
>
>
> ***These are just great; they look like an Earle Brown piece!
Maybe
> we should try to play this stuff! A requisite would be a synth
that
> could instantaneously switch ETs! :)

note the close similarity of some of these plots with plots of just
triads such as:

/tuning/files/Erlich/closeup.jp
g

> Hello P&M

much of these numbers can be found in the following series
http://www.anaphoria.com/sieve.PDF

>
> --- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
> wrote:
> > successive zooms into the center:
> >
> > /tuning/files/perlich/tree.gif
> > /tuning/files/perlich/treezuma.gif
> > /tuning/files/perlich/treezumb.gif
> > /tuning/files/perlich/treezumc.gif
> >
> > if you can squint real hard, you can see 612 as 171+441, 118+494,
> and
> > 289+323 near the center of the last graph . . . i was working on
> > another zoom but matlab crashed . . .
>
> two more added, so as to reach monz's "magical" 4296:
>
> /tuning/files/perlich/treezumd.gif
> /tuning/files/perlich/treezume.gif
>
> monz, (on your eqtemp page) you included 1432. maybe you really meant
> 1342? besides these graphs, marc jones also mentions 1342, not 1432,
> at:
>
> /tuning/files/dict/marc-edolist.htm
>
> i also remember a post from marc that mentioned 3125 (=5^5) among
> other such trivia. can anyone locate it?
>
>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
KXLU 88.9 fm Wed. 8-9pm PST.
live stream kxlu.com

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> > Hello P&M
>
> much of these numbers can be found in the following series
> http://www.anaphoria.com/sieve.PDF

i see a lot of different numbers, too. thanks kraig. what i really
like is the patterns.

>/tuning/files/perlich/tree.gif
>/tuning/files/perlich/treezuma.gif
>/tuning/files/perlich/treezumb.gif
>/tuning/files/perlich/treezumc.gif

Rad!

>note the close similarity of some of these plots with plots of
>just triads such as:

I was just going to post on that. I wonder if Gene can tell us
why that is... (probably many different integer relations will
produce this kind of pattern?)...

-Carl

--- In tuning@y..., "Carl Lumma" <clumma@y...> wrote:

> I was just going to post on that. I wonder if Gene can tell us
> why that is... (probably many different integer relations will
> produce this kind of pattern?)...

I don't see any similarity beyond the fact that they both have lines. Here we have lines going through [6,8,10], [7,9,11], [8,10,12], etc. instead of ets.

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>

/tuning/topicId_39695.html#39722

>
> note the close similarity of some of these plots with plots of just
> triads such as:
>
>
/tuning/files/Erlich/closeup.jp
> g

***Yes, I believe we're close to "green pizza crust" ridges as well...

JP

> Hello Paul!

Likewise with Erv as he noticed that the same numbers would often re occur in the differences
between one number and the next.

>
> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
>
>
> --- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> > > Hello P&M
> >
> > much of these numbers can be found in the following series
> > http://www.anaphoria.com/sieve.PDF
>
> i see a lot of different numbers, too. thanks kraig. what i really
> like is the patterns.
>
>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
KXLU 88.9 fm Wed. 8-9pm PST.
live stream kxlu.com

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "Carl Lumma" <clumma@y...> wrote:
>
> > I was just going to post on that. I wonder if Gene can tell us
> > why that is... (probably many different integer relations will
> > produce this kind of pattern?)...
>
> I don't see any similarity beyond the fact that they both have
>lines. Here we have lines going through [6,8,10], [7,9,11],
[8,10,12], etc. instead of ets.

gene --

carl is correct. perhaps the situation would be clearer if each ET
were labeled, not by the total number of notes per octave, but by the
triad of numbers of notes per each of the consonant intervals. so
instead of 12 we'd have (3,4,5), instead of 19 we'd have (5,6,8), etc.

really, the two plots differ only in that one of them (any of the
sufficiently zoomed-in ET ones) shows all the triads whose intervals
are simply commensurable in log-frequency space, while the other (the
JI one) shows all the triads whose intervals are simply commesurable
in frequency space. many of the latter's lines are therefore slightly
curved on the graph in log-frequency space -- though the curvature is
barely visible at high zooms, it's easy to see here:

/tuning/files/triadp80000.jpg

btw -- john chalmers -- if you're reading this -- i never got a
response to my last e-mail to you. is this plot the one you wanted to
use for XH?

> really, the two plots differ only in that one of them (any of the
> sufficiently zoomed-in ET ones) shows all the triads whose
> intervals are simply commensurable in log-frequency space, while
> the other (the JI one) shows all the triads whose intervals are
> simply commesurable in frequency space.

Just like the 1-D scale tree. Say, how would you define "simply
commensurable"?

> groups.yahoo.com/group/harmonic_entropy/files/triadp80000.jpg
>
> btw -- john chalmers -- if you're reading this -- i never got a
> response to my last e-mail to you. is this plot the one you
> wanted to use for XH?

No, no! Use one of the new tree zooms. I insist!

-C.

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> carl is correct.

You lost me--what possible connection does a 7-9-11 triad have with 5-limit ets?

--- In tuning@y..., "Carl Lumma" <clumma@y...> wrote:
> > really, the two plots differ only in that one of them (any of the
> > sufficiently zoomed-in ET ones) shows all the triads whose
> > intervals are simply commensurable in log-frequency space, while
> > the other (the JI one) shows all the triads whose intervals are
> > simply commesurable in frequency space.
>
> Just like the 1-D scale tree. Say, how would you define "simply
> commensurable"?

expressible in integers up to whatever 'limit' -- for example, < 999-
tET in the former case, l*m*n < 1 million in the latter.

> > groups.yahoo.com/group/harmonic_entropy/files/triadp80000.jpg
> >
> > btw -- john chalmers -- if you're reading this -- i never got a
> > response to my last e-mail to you. is this plot the one you
> > wanted to use for XH?
>
> No, no! Use one of the new tree zooms. I insist!

the above is more symmetrical . . . but if john wants to use the
other one for the back cover . . .

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
wrote:
>
> > carl is correct.
>
> You lost me--what possible connection does a 7-9-11 triad have with
>5-limit ets?

the diagrams are very similar because they both show the distribution
of simple-integer triads up to some limit (at least within the area
zoomed in on). the only difference is that the proportions for one
are defined in log-frequency space, the other in linear frequency
space.

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:
> --- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> > --- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
> wrote:
> >
> > > carl is correct.
> >
> > You lost me--what possible connection does a 7-9-11 triad have with
> >5-limit ets?
>
> the diagrams are very similar because they both show the distribution
> of simple-integer triads up to some limit (at least within the area
> zoomed in on). the only difference is that the proportions for one
> are defined in log-frequency space, the other in linear frequency
> space.

How, exactly, does a plot of 5-limit ets show 7-9-11?

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
wrote:
> > --- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> > > --- In tuning@y..., "wallyesterpaulrus"
<wallyesterpaulrus@y...>
> > wrote:
> > >
> > > > carl is correct.
> > >
> > > You lost me--what possible connection does a 7-9-11 triad have
with
> > >5-limit ets?
> >
> > the diagrams are very similar because they both show the
distribution
> > of simple-integer triads up to some limit (at least within the
area
> > zoomed in on). the only difference is that the proportions for
one
> > are defined in log-frequency space, the other in linear frequency
> > space.
>
> How, exactly, does a plot of 5-limit ets show 7-9-11?

it doesn't. do you mean "what point in the et graph corresponds to
the point 7:9:11 in the just graph"? the answer depends how you map
things, but one way is to map 7:9:11 to the triad 0 7 9 in 11-equal.

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> > How, exactly, does a plot of 5-limit ets show 7-9-11?
>
> it doesn't. do you mean "what point in the et graph corresponds to
> the point 7:9:11 in the just graph"? the answer depends how you map
> things, but one way is to map 7:9:11 to the triad 0 7 9 in 11-equal.

I presumed that 7:9:11, considered as homogenous coordinates, was the same as the [7,9,11] et in the 5-limit, but either way we are back to my original answer that the similarity is simply that they both have lines, aren't we?

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

> I presumed that 7:9:11, considered as homogenous coordinates, was
>the same as the [7,9,11] et in the 5-limit, but either way we are
>back to my original answer that the similarity is simply that they
>both have lines, aren't we?

no, because the two graphs are almost identical, aside from a
slight "curve" in one of them.

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> no, because the two graphs are almost identical, aside from a
> slight "curve" in one of them.

Why don't you give a few examples of allegedly corresponding lines?

By the way, these sorts of graphs of points in homogenous coordinates can be thought of as picturing a part of a projective plane, which suggests the possibility of looking at the dual graph, where the points become lines and vice-versa. In the 5-limit case, the commas would now be represented by points, and the ets by lines.

>By the way, these sorts of graphs of points in homogenous
>coordinates can be thought of as picturing a part of a
>projective plane, which suggests the possibility of looking
>at the dual graph, where the points become lines and vice-
>versa. In the 5-limit case, the commas would now be represented
>by points, and the ets by lines.

A cool idea.

Also, these graphs appear to have scaling symmetry. In
the limit, are they black, or is there still whitespace?
Does their 'Lebesgue measure' tell us this?

-Carl

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
wrote:
>
> > no, because the two graphs are almost identical, aside from a
> > slight "curve" in one of them.
>
> Why don't you give a few examples of allegedly corresponding lines?

*any* line!!! don't you see, the two graphs are identical (up to a
logarithmic "squish"), they plot a point for every a:b:c that is in
lowest terms, up to some limit. it's the higher-dimensional analogue
of the farey series, etc. it appears to be some sort of fractal.

> By the way, these sorts of graphs of points in homogenous
>coordinates can be thought of as picturing a part of a projective
>plane, which suggests the possibility of looking at the dual graph,
>where the points become lines and vice-versa. In the 5-limit case,
>the commas would now be represented by points, and the ets by lines.

i would like very much to look at this dual graph, please give me
some information on how to create it, view it, etc. i'd especially
like it if it could have triangular symmetry like my latest "it's
getting better all the time" ET graph. thanks!

--- In tuning@y..., "Carl Lumma" <clumma@y...> wrote:

> Also, these graphs appear to have scaling symmetry. In
> the limit, are they black, or is there still whitespace?

since we're basically talking about the rationals in the reals
(except in a two-dimensional analogue), the density is always zero
(there's "infinitely" more white than black), but there are an
infinite number of black points in any region, no matter how small.

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>

/tuning/topicId_39695.html#39831

> *any* line!!! don't you see, the two graphs are identical (up to a
> logarithmic "squish"), they plot a point for every a:b:c that is in
> lowest terms, up to some limit. it's the higher-dimensional
analogue of the farey series, etc. it appears to be some sort of
fractal.
>

***You know, this is getting more amazing all the time. I'm just
getting a "glimmer" of that, but what I *am* getting is pretty
wild...

J. Pehrson

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> > Why don't you give a few examples of allegedly corresponding lines?
>
> *any* line!!! don't you see, the two graphs are identical (up to a
> logarithmic "squish"), they plot a point for every a:b:c that is in
> lowest terms, up to some limit. it's the higher-dimensional analogue
> of the farey series, etc. it appears to be some sort of fractal.

You seem to be saying they are the same becuase they are both the projective plane over Q, or a part of it.

> > By the way, these sorts of graphs of points in homogenous
> >coordinates can be thought of as picturing a part of a projective
> >plane, which suggests the possibility of looking at the dual graph,
> >where the points become lines and vice-versa. In the 5-limit case,
> >the commas would now be represented by points, and the ets by lines.
>
> i would like very much to look at this dual graph, please give me
> some information on how to create it, view it, etc. i'd especially
> like it if it could have triangular symmetry like my latest "it's
> getting better all the time" ET graph. thanks!

If you have a point [x, y, z] in homogenous coordinates, the equation of a line passing through it will be

Ax + By + Cz = 0

You can now represent the line as a point [A, B, C] in the dual plane, where now [x, y, z] becomes a line.

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
wrote:
>
> > > Why don't you give a few examples of allegedly corresponding
lines?
> >
> > *any* line!!! don't you see, the two graphs are identical (up to
a
> > logarithmic "squish"), they plot a point for every a:b:c that is
in
> > lowest terms, up to some limit. it's the higher-dimensional
analogue
> > of the farey series, etc. it appears to be some sort of fractal.
>
> You seem to be saying they are the same becuase they are both the
projective plane over Q, or a part of it.

unfortunately, i don't know what that means. i know that Q is the
rationals, but what's the name of the set that includes "4:5:6"
and "10:12:15" but not "8:10:12"? that's the set of points we're
plotting in both graphs, with three axes at 120-degree angles
representing the three intervals in the triad.

> > > By the way, these sorts of graphs of points in homogenous
> > >coordinates can be thought of as picturing a part of a
projective
> > >plane, which suggests the possibility of looking at the dual
graph,
> > >where the points become lines and vice-versa. In the 5-limit
case,
> > >the commas would now be represented by points, and the ets by
lines.
> >
> > i would like very much to look at this dual graph, please give me
> > some information on how to create it, view it, etc. i'd
especially
> > like it if it could have triangular symmetry like my latest "it's
> > getting better all the time" ET graph. thanks!
>
> If you have a point [x, y, z] in homogenous coordinates, the
equation of a line passing through it will be
>
> Ax + By + Cz = 0
>
> You can now represent the line as a point [A, B, C] in the dual
plane, where now [x, y, z] becomes a line.

i was hoping for something better, but i think i may see the answer
already -- for each linear temperament, i simply plot the *unison
vector*, right? let's see . . . 12-equal banishes [-4,1], [0,3],
[4,2], [8,1] . . . well the last three do lie on a straight line,
but . . . then there's the negatives of all of these . . . is it
unavoidable that we'll have considerably more than one line for each
ET? in the ETs-as-points graph, i cut down on the number by
insisisting that at least one of the just consonances be represented
by its best approximation . . . can i do something similar here?

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> > You seem to be saying they are the same becuase they are both the
> projective plane over Q, or a part of it.
>
> unfortunately, i don't know what that means. i know that Q is the
> rationals, but what's the name of the set that includes "4:5:6"
> and "10:12:15" but not "8:10:12"? that's the set of points we're
> plotting in both graphs, with three axes at 120-degree angles
> representing the three intervals in the triad.

The projective plane over Q are triples of rational numbers
[x,y,z] such that if q is any non-zero rational, the [qx,qy,qz]
represents the same point. This implies that up to sign, a point is
represented by [a,b,c] where a, b, and c are relatively prime integers.

This means that 4:5:6 and 8:10:12 are the same point; reducing
8:10:12 by the gcd gives us 4:5:6.

> i was hoping for something better, but i think i may see the answer
> already -- for each linear temperament, i simply plot the *unison
> vector*, right? let's see . . . 12-equal banishes [-4,1], [0,3],
> [4,2], [8,1] . . . well the last three do lie on a straight line,
> but . . . then there's the negatives of all of these . . .

Those are affine coordinates; you can projectivize them to
[-4,1,1], [0,3,1], etc. But remember, [-4,1,1] and [4,-1,-1] are the same point!

More natural would be to take [-4,4,-1], [-7,0,3] etc. as points on a projective plane, and [12,19,28] as the line through them. 3-limit commas would end up as points at infinity that way, though.

is it
> unavoidable that we'll have considerably more than one line for each
> ET?

There's one line for each val; how many vals an ET gets is up to you.

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

> More natural would be to take [-4,4,-1], [-7,0,3] etc. as points on a projective plane, and [12,19,28] as the line through them. 3-limit commas would end up as points at infinity that way, though.

I should add that there is no problem graphing points at infinity!

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
wrote:
>
> > > You seem to be saying they are the same becuase they are both
the
> > projective plane over Q, or a part of it.
> >
> > unfortunately, i don't know what that means. i know that Q is the
> > rationals, but what's the name of the set that includes "4:5:6"
> > and "10:12:15" but not "8:10:12"? that's the set of points we're
> > plotting in both graphs, with three axes at 120-degree angles
> > representing the three intervals in the triad.
>
> The projective plane over Q are triples of rational numbers
> [x,y,z] such that if q is any non-zero rational, the [qx,qy,qz]
> represents the same point. This implies that up to sign, a point is
> represented by [a,b,c] where a, b, and c are relatively prime
integers.
>
> This means that 4:5:6 and 8:10:12 are the same point; reducing
> 8:10:12 by the gcd gives us 4:5:6.

then you've got it! both graphs show this same structure. this
structure has baffled me for almost a decade. i learned many of the
wonderful properties of the farey series in one dimension, and
wondered if there were any analogous properties in this two-
dimensional structure. no one in the yale math faculty seemed to have
a clue. we've explored these questions mainly on harmonic_entropy, so
if you have any responses, please post them there.

> > i was hoping for something better, but i think i may see the
answer
> > already -- for each linear temperament, i simply plot the *unison
> > vector*, right? let's see . . . 12-equal banishes [-4,1], [0,3],
> > [4,2], [8,1] . . . well the last three do lie on a straight line,
> > but . . . then there's the negatives of all of these . . .
>
> Those are affine coordinates; you can projectivize them to
> [-4,1,1], [0,3,1], etc.

what does this mean? and how does this address their non-
collinearity? please reply to the rest of this on tuning-math, to
spare the usual poor souls . . .

> But remember, [-4,1,1] and [4,-1,-1] are the same point!

in what kind of coordinate system?

> More natural would be to take [-4,4,-1], [-7,0,3] etc. as points on
>a projective plane, and [12,19,28] as the line through them.

you lost me.

> is it
> > unavoidable that we'll have considerably more than one line for
each
> > ET?
>
> There's one line for each val;

that's exactly what i want. now i hope you can show me what it looks
like.

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

/tuning/topicId_39695.html#39916

> --- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
>
> > More natural would be to take [-4,4,-1], [-7,0,3] etc. as points
on a projective plane, and [12,19,28] as the line through them. 3-
limit commas would end up as points at infinity that way, though.
>
> I should add that there is no problem graphing points at infinity!

***I can see I'm going to need a newer ink jet...

JP