Some 12-tone, 2-step 46-et scales

I was facinated to discover that the 7,5 system did a little better than the completely symmetrical 6,6 system.

[0, 4, 8, 12, 16, 20, 23, 27, 31, 35, 39, 43]
[4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 3]
edges 24 24 40 connectivity 3 3 6

[0, 4, 8, 12, 16, 20, 24, 27, 31, 35, 39, 43]
[4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 3]
edges 24 25 41 connectivity 3 3 6

[0, 4, 8, 12, 16, 20, 24, 28, 31, 35, 39, 43]
[4, 4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 3]
edges 22 24 40 connectivity 2 3 6

[0, 4, 8, 12, 16, 20, 24, 28, 32, 35, 39, 43]
[4, 4, 4, 4, 4, 4, 4, 4, 3, 4, 4, 3]
edges 18 21 38 connectivity 1 2 5

[0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 39, 43]
[4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 4, 3]
edges 15 20 38 connectivity 1 2 5

[0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 43]
[4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3]
edges 12 19 38 connectivity 0 0 5

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> I was facinated to discover that the 7,5 system did a little better
than the completely symmetrical 6,6 system.
>
> [0, 4, 8, 12, 16, 20, 23, 27, 31, 35, 39, 43]
> [4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 3]
> edges 24 24 40 connectivity 3 3 6
>
> [0, 4, 8, 12, 16, 20, 24, 27, 31, 35, 39, 43]
> [4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 3]
> edges 24 25 41 connectivity 3 3 6

This is very neat and important stuff. It's been claimed that the
Indian scales derive from a second-order-maximally-even 7-out-of-12-
out-of-22 construction, which would imply the symmetrical 12-tone
system above. However, the actual evidence supports the
omnitetrachordal system. So you're saying one might explain this
using some ratio of 7? Or did I misread this? I'd like to see/make
lattices of these.

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

So you're saying one might explain this
> using some ratio of 7? Or did I misread this?

I'm not sure what your question is; what I was saying is that we get a little better count of 7-limit intervals with the 7,5 system than with the 6,6 system in the 46-et, which I did not expect.

I'd like to see/make
> lattices of these.

I could send you a gif file from Maple's graph-drawing program,of the sort I posted on the tuning list, but that would only be a starting point.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> So you're saying one might explain this
> > using some ratio of 7? Or did I misread this?
>
> I'm not sure what your question is; what I was saying is that we
>get a little better count of 7-limit intervals with the 7,5 system
>than with the 6,6 system in the 46-et, which I did not expect.

Right.

> I'd like to see/make
> > lattices of these.
>
> I could send you a gif file from Maple's graph-drawing program,of
>the sort I posted on the tuning list, but that would only be a
>starting point.

Sure. Now are there some 46-tET commas you did not take into account?
You didn't answer my "hyper-torus" point on the tuning list yet . . .

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

> Sure. Now are there some 46-tET commas you did not take into account?
> You didn't answer my "hyper-torus" point on the tuning list yet . . .

I wasn't clear what you meant, but there are topological considerations which come into graph theory. A graph can be a planar graph, for instance, or a graph on a quotient (cylinder or torus), so it can have a genus--it might be a graph on something with negative curvature. I plan on reading some graph theory and seeing if anything I run across suggests some application.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > Sure. Now are there some 46-tET commas you did not take into
account?
> > You didn't answer my "hyper-torus" point on the tuning list
yet . . .
>
> I wasn't clear what you meant, but there are topological
>considerations which come into graph theory. A graph can be a planar
>graph, for instance, or a graph on a quotient (cylinder or torus),
>so it can have a genus--it might be a graph on something with
>negative curvature. I plan on reading some graph theory and seeing
>if anything I run across suggests some application.

It shouldn't be so complicated. You can draw a big giant graph for
all of 72-tET, can't you?

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

> It shouldn't be so complicated. You can draw a big giant graph for
> all of 72-tET, can't you?

Certainly. What are you getting at, though?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > It shouldn't be so complicated. You can draw a big giant graph
for
> > all of 72-tET, can't you?
>
> Certainly. What are you getting at, though?

If you do that, you take _all_ the commas of 72-tET into account.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> I was facinated to discover that the 7,5 system did a little better than the completely symmetrical 6,6 system.

Here are the graphs. Looking at these, 12 might be a good place to center.

> [0, 4, 8, 12, 16, 20, 23, 27, 31, 35, 39, 43]
> [4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 3]
> edges 24 24 40 connectivity 3 3 6

/tuning-math/files/Gene/graph/g5_1.GIF
/tuning-math/files/Gene/graph/g7_1.GIF
/tuning-math/files/Gene/graph/g11_1.GIF

> [0, 4, 8, 12, 16, 20, 24, 27, 31, 35, 39, 43]
> [4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 3]
> edges 24 25 41 connectivity 3 3 6

/tuning-math/files/Gene/graph/g5_2.GIF
/tuning-math/files/Gene/graph/g7_2.GIF
/tuning-math/files/Gene/graph/g11_2.GIF

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

> If you do that, you take _all_ the commas of 72-tET into account.

I'm not getting you. I'm connecting things via intervals; the commas do not directly enter the picture.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > I was facinated to discover that the 7,5 system did a little
better than the completely symmetrical 6,6 system.
>
> Here are the graphs. Looking at these, 12 might be a good place to
center.
>
> > [0, 4, 8, 12, 16, 20, 23, 27, 31, 35, 39, 43]
> > [4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 3]
> > edges 24 24 40 connectivity 3 3 6
>
> /tuning-math/files/Gene/graph/g5_1.GIF
> /tuning-math/files/Gene/graph/g7_1.GIF
> /tuning-math/files/Gene/graph/g11_1.GIF
>
> > [0, 4, 8, 12, 16, 20, 24, 27, 31, 35, 39, 43]
> > [4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 3]
> > edges 24 25 41 connectivity 3 3 6
>
> /tuning-math/files/Gene/graph/g5_2.GIF
> /tuning-math/files/Gene/graph/g7_2.GIF
> /tuning-math/files/Gene/graph/g11_2.GIF

The note 31 would be the usual "tonic" of the Modern Indian Gamut
when equated with the 7,5 system.

Is there any way you could color the connecting lines, say, red for
ratios of 3, orange for ratios of 5, yellow for ratios of 7, green
for ratios of 9, and blue for ratios of 11?

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

> Is there any way you could color the connecting lines, say, red for
> ratios of 3, orange for ratios of 5, yellow for ratios of 7, green
> for ratios of 9, and blue for ratios of 11?

I've been wishing I could do that very thing. The output is the Maple graph drawing program output; if I could figure out a way of getting it to change color, and also of drawing more than one graph at a time, it could be done, but it doesn't seem to be implimented.