A new graph for Paul?

Dual to the 5-limit symmetrical lattice of intervals is a 5-limit
symmetrical lattice of vals whose first component is zero--which
includes the generators in the period-generator of linear temperaments.
The 3 axis and the 5 axis for intervals is 60 degrees apart; for a
graph of the lattice of generators, |0 1> and |1 0> should be 120
degrees apart. There are interesting lines to draw on such a graph;
the |-3 -5> of porcupine, |1 4> of meantone and |5 13> of amity lie
along a line, for instance. Each generator can be graphed twice, by
graphing +-|1 4>, etc. This would give us additional lines; the line
between |-1 -4> and |-3 -5> includes pelogic at |1 -3>.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Dual to the 5-limit symmetrical lattice of intervals is a 5-limit
> symmetrical lattice of vals whose first component is zero--which
> includes the generators in the period-generator of linear
>temperaments.

Don't get it.

> The 3 axis and the 5 axis for intervals is 60 degrees apart; for a
> graph of the lattice of generators, |0 1> and |1 0> should be 120
> degrees apart. There are interesting lines to draw on such a graph;
> the |-3 -5> of porcupine, |1 4> of meantone and |5 13> of amity lie
> along a line, for instance.

What does that line mean? The dual graph I recently produced
puts 'linear temperaments' that share an ET on straight lines . . .

> Each generator can be graphed twice, by
> graphing +-|1 4>, etc. This would give us additional lines; the line
> between |-1 -4> and |-3 -5> includes pelogic at |1 -3>.

Meaning?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > Dual to the 5-limit symmetrical lattice of intervals is a 5-limit
> > symmetrical lattice of vals whose first component is zero--which
> > includes the generators in the period-generator of linear
> >temperaments.
>
> Don't get it.

What's the hang-up? Do you understand the part about pairs of integers
representing generators for 5-limit linear temperaments? Do you
understand the norm I've placed on these? Do you understand this makes
it a lattice, and what that lattice looks like?

> > The 3 axis and the 5 axis for intervals is 60 degrees apart; for a
> > graph of the lattice of generators, |0 1> and |1 0> should be 120
> > degrees apart. There are interesting lines to draw on such a graph;
> > the |-3 -5> of porcupine, |1 4> of meantone and |5 13> of amity lie
> > along a line, for instance.
>
> What does that line mean?

That's to be explored. We have, along a line,

porcupine-->meantone-->tetracot-->amity

In terms of generator mapping, but not the period part of the map, and
therefore not in terms of the generators in cents, we can transform
one to the next continuously. That's the sort of thing I think is
worth thiking about from a compositional point of view, for starters.
I suggest we might find out more about what it might mean with a graph
in front of us.

> > Each generator can be graphed twice, by
> > graphing +-|1 4>, etc. This would give us additional lines; the line
> > between |-1 -4> and |-3 -5> includes pelogic at |1 -3>.
>
> Meaning?

Let's find out.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> > > Dual to the 5-limit symmetrical lattice of intervals is a 5-
limit
> > > symmetrical lattice of vals whose first component is zero--which
> > > includes the generators in the period-generator of linear
> > >temperaments.
> >
> > Don't get it.
>
> What's the hang-up? Do you understand the part about pairs of
integers
> representing generators for 5-limit linear temperaments?

No, it would seem you need 4 integers to specify the mapping. Oh,
you're throwing out the period. Seems dirty to do that.

> > What does that line mean?
>
> That's to be explored. We have, along a line,
>
> porcupine-->meantone-->tetracot-->amity
>
> In terms of generator mapping, but not the period part of the map,
and
> therefore not in terms of the generators in cents, we can transform
> one to the next continuously. That's the sort of thing I think is
> worth thiking about from a compositional point of view, for
>starters.

What can you do with it?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> No, it would seem you need 4 integers to specify the mapping. Oh,
> you're throwing out the period. Seems dirty to do that.

Dirty??

> > > What does that line mean?
> >
> > That's to be explored. We have, along a line,
> >
> > porcupine-->meantone-->tetracot-->amity
> >
> > In terms of generator mapping, but not the period part of the map,
> and
> > therefore not in terms of the generators in cents, we can transform
> > one to the next continuously. That's the sort of thing I think is
> > worth thiking about from a compositional point of view, for
> >starters.
>
> What can you do with it?

I just said.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > No, it would seem you need 4 integers to specify the mapping. Oh,
> > you're throwing out the period. Seems dirty to do that.
>
> Dirty??

For one thing, the generator is not unique, and its multiplicity is
proportional to periods per octave. For example, diaschismic can be
understood as having, like meantone, a generator of a fourth, but
its 'canonical' generator is sort of a minor second. Which do you
use, and what's the rule to determine which?

> > > > What does that line mean?
> > >
> > > That's to be explored. We have, along a line,
> > >
> > > porcupine-->meantone-->tetracot-->amity
> > >
> > > In terms of generator mapping, but not the period part of the
map,
> > and
> > > therefore not in terms of the generators in cents, we can
transform
> > > one to the next continuously. That's the sort of thing I think
is
> > > worth thiking about from a compositional point of view, for
> > >starters.
> >
> > What can you do with it?
>
> I just said.

If you're not transforming continuously in terms of the generator in
cents, what are you transforming continuously in terms of?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> For one thing, the generator is not unique, and its multiplicity is
> proportional to periods per octave.

Let's stick with octave periods for now.

> If you're not transforming continuously in terms of the generator in
> cents, what are you transforming continuously in terms of?

It might be something you can do by using 7-equal as a way to pass
from one temperament to another. The lines in question are 7-et lines;
if you take the generator val <0 a b| and wedge/cross-product it with
<1 11/7 16/7| you get the monzos for the various temperaments.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > For one thing, the generator is not unique, and its multiplicity
is
> > proportional to periods per octave.
>
> Let's stick with octave periods for now.
>
> > If you're not transforming continuously in terms of the generator
in
> > cents, what are you transforming continuously in terms of?
>
> It might be something you can do by using 7-equal as a way to pass
> from one temperament to another. The lines in question are 7-et
lines;
> if you take the generator val <0 a b| and wedge/cross-product it
with
> <1 11/7 16/7| you get the monzos for the various temperaments.

Let me get back to this after we're done talking about error
functions and the metrics of their duals.

>> <1 -3| 125/128
>
>135/128

Gene and all,

What if, instead of issuing a correction post like this,
we were to post a full corrected version and delete the
original from the archives? Posterity may thank us...

Just a thought.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> <1 -3| 125/128
> >
> >135/128
>
> Gene and all,
>
> What if, instead of issuing a correction post like this,
> we were to post a full corrected version and delete the
> original from the archives? Posterity may thank us...

Not a bad plan; I do wonder how it works for people who read via email.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> Let me get back to this after we're done talking about error
> functions and the metrics of their duals.

OK. Here's something to note when you do--these lines can go on for
quite a ways including temperaments of reasonably low badness. Here is
my line, each temperament of which differs from the next by meantone,
meaning the successive differences are <1 4|:

<0 -7| 2187/2048
<1 -3| 135/128
<2 1| 25/24
<3 5| 250/243
<4 9| 20000/19683
<5 13| 1600000/1594323
<6 17| 129140163/128000000

>Not a bad plan; I do wonder how it works for people who read
>via email.

I just got your correction to the message in question by
e-mail. It would help to have a standard header for the
subject line. Maybe...

Correction01: [old subject]

...to be followed by Correction02 and so forth.

-Carl