A Dekany graph with 3-fold symmetry

If you are interested in dekanies check out
http://dkeenan.com/Music/DekanyGraph.gif (14 kB)

It shows all dyads in the 2 of {1,3,5,7,9} dekany, using a different colour
for each ratio, similar to my earlier lattices. The colour of a 3:9 is even
slightly different to a 1:3. It shows all three 5-limit triads as centered
equilateral triangles. A small one for the minor and two large ones for the
majors.

I discovered it by watching the dekany tumbling in 4D.

Regards,

-- Dave Keenan
http://dkeenan.com

Cool, Dave!

Why not put the E# in the center?

--- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:
> If you are interested in dekanies check out
> http://dkeenan.com/Music/DekanyGraph.gif (14 kB)
>
> It shows all dyads in the 2 of {1,3,5,7,9} dekany, using a
different
colour
> for each ratio, similar to my earlier lattices. The colour of a 3:9
is even
> slightly different to a 1:3.

Yes, and the 1*3 - 1*9 is the same interval as 1*9 - 3*9 but they're
different in both angle and length in the diagram. I thought that
was a no-no.

David Finnamore

>Yes, and the 1*3 - 1*9 is the same interval as 1*9 - 3*9 but they're
>different in both angle and length in the diagram. I thought that
>was a no-no.

>David Finnamore

David F., what Dave K. created here is a graph (in the graph-theoretic
sense), not a lattice. The actual lattice resides in 4-dimensional space, as
Dave and I have been trying to depict in various ways. What Dave did here
was a topological transformation (like the one that makes a donut out of a
coffee cup) of the 4-D dekany into 2-D space, distorting positions and
directions, but preserving all the connections betweeen points so that one
can more easily see them.

As another example, I can make a graph of the hexany as follows:

C*D---------------------B*D
|\`--.__ __,--'/|
| \ `--A*D--' / |
\ \ / \ / /
\ \ / \ / /
| \ / \ / |
| \ / \ / |
\ A*C-------A*B /
\ \ / /
\_ \ / _/
\ \ / /
\_ \ / _/
`--C*B--'

which you can imagine as the result of stretching out an octahedron and
flattening it out onto a flat surface.

Again, it's not a lattice, it's a graph. Only in the (possibly
higher-dimensional) lattice can you see the full symmetry of the CPS.

David Finnamore wrote,

>Yes, and the 1*3 - 1*9 is the same interval as 1*9 - 3*9 but they're
>different in both angle and length in the diagram. I thought that
>was a no-no.

Actually, Erv Wilson did often use an independent 9-axis when mapping the
dekany and other structures. This would result in certain identical notes
appearing in more than one place, and thus certain identical intervals
appearing in more than one way. However, it was still true that the direct
connections in the lattice always represented the most consonant intervals
in the system, and even if one of these intervals could also be found in
another way as not a direct connection, it was still true that any interval
represented by a direct connection would be _at least_ as consonant as any
interval represented by not a direct connection.

I wrote,

>Cool, Dave!

>Why not put the E# in the center?

An additional advantage of doing this would be that one could see the 5
hexanies much more easily.

Paul Erlich wrote:

>Cool, Dave!
>Why not put the E# in the center?
>An additional advantage of doing this would be that one could see the 5
>hexanies much more easily.

Of course that's how I originally saw it (but sort of side-on). It's just
that all the hexanies (and triads) are not equally interesting and it was
kind of cluttered, so I exploded the note with the least consonant
relationships (the 7*9 E#) to infinity. You can imagine it mapped to the
surface of a sphere with the E# diametrically opposite from the center of
the central triangle. 3 around the north pole (A,C,E), 6 around the
equator, 1 at the south pole (E#). Actually the 6 at the equator could
instead be every second one at a different tropic. The arctic circle and
the tropics would be the three 5-limit triads.

Now I want a dekany ball that I can play, as well as an eikosany donut.

Here's it is with the E# in the center.
http://dkeenan.com/Music/DekanyGraph2.gif

I guess you learn different things from both of them. Here's the previous
one again.
http://dkeenan.com/Music/DekanyGraph.gif

I also updated the chord chart for your decatonics, to show that fully
dyadically-consonant hexad (not hexany) you found way back.
http://dkeenan.com/Music/ErlichDecChords.gif

Regards,
-- Dave Keenan
http://dkeenan.com

Dave Keenan wrote,

>so I exploded the note with the least consonant
>relationships (the 7*9 E#)

? It's part of 6 consonant intervals, just like all of the other notes
(well, a few of the notes are part of 7 consonant intervals --- you
occasionally get an extra one since 3*3=9, as you pointed out).

>I also updated the chord chart for your decatonics, to show that fully
>dyadically-consonant hexad (not hexany) you found way back.
>http://dkeenan.com/Music/ErlichDecChords.gif

Thanks!

Paul Erlich wrote:

>Dave Keenan wrote,
>
>>so I exploded the note with the least consonant
>>relationships (the 7*9 E#)
>
>? It's part of 6 consonant intervals, just like all of the other notes
>(well, a few of the notes are part of 7 consonant intervals --- you
>occasionally get an extra one since 3*3=9, as you pointed out).

That was "(least consonant) relationships", not "least (consonant
relationships)".

-- Dave Keenan
http://dkeenan.com

Paul Erlich wrote:

>Is this a dekany?
>
>Go to this site, http://members.aol.com/jmtsgibbs/draw4d.htm, which Carl
>pointed me to. Select "Truncated Simplex". Is this the
>dispentachoron/dekany? It sure looks like it!

Yes it is.
-- Dave Keenan
http://dkeenan.com

> Paul Erlich wrote:
>
> >Is this a dekany?
> >
> >Go to this site, http://members.aol.com/jmtsgibbs/draw4d.htm, which Carl
> >pointed me to. Select "Truncated Simplex". Is this the
> >dispentachoron/dekany? It sure looks like it!
>
> Yes it is.
> -- Dave Keenan

Great, now I know what a dekany is!

4D tetrahedron has 5 vertices.

It can be shown as pentagonal star in pentagon:

http://w3.one.net/~monkey/mathematics/simplex/

Now, each vertex is met by all except two of the triangles of the simplex. So 8 triangles meet
at every vertex of simplex.

If you join mid-points of the edges together, these 8 triangles truncate to triangles again,
and each is a triangular face of the 3-d figure that replaces the original vertex on truncation.

So you end up with five octahedra.

Joining mid-points of the edges will also truncate the original five tetrahedra to make
five new tetrahedra.

(
You _don't_ get the "truncated tetrahedra
http://www.georgehart.com/virtual-polyhedra/vrml/truncated_tetrahedron.wrl"
with triangles and hexagons faces, because you are truncating all the way
to the mid-point of the edge, rather than truncating a third of the way in only
)

So truncated simplex has five octahedral faces, and five tetrahedral ones.

Obtained by joining mid-points of edges of 4D equiv. of tetrahedron.

Robert

--- In tuning@egroups.com, "Robert Walker" <robert_walker@r...> wrote:

http://www.egroups.com/message/tuning/15225

>
http://www.georgehart.com/virtual-polyhedra/vrml/truncated_tetrahedron
.wrl"

This rotating thing is truly incredible...

Too bad it doesn't make any kind of sound. Could it illustrate some
kind of sound?? Dunno.

Joseph

--- In tuning@egroups.com, "Robert Walker" <robert_walker@r...> wrote:

> So you end up with five octahedra.

Which you should be able to see in my movies dekany.mpg and
dekany_4d.mpg.

> Joining mid-points of the edges will also truncate the original
five tetrahedra to make
> five new tetrahedra.

Which you should be able to see in my movies dekany2.mpg and
dekany2_4d.mpg.

>> So you end up with five octahedra.

>Which you should be able to see in my movies dekany.mpg and
>dekany_4d.mpg.

>> Joining mid-points of the edges will also truncate the original
>five tetrahedra to make
> five new tetrahedra.

>Which you should be able to see in my movies dekany2.mpg and
>dekany2_4d.mpg.

Yes, indeed.

I find the octahedra easiest to see in the dekany.mpg as there
are four, one for each of the solid faces (kind of squashed, inside),
+ the octahedron that you use as the framework for the model.

Then you can see the tetrahedra pretty clearly in this model too once
one knows they are there - one for each of the vacant faces of the
framing octahedron, and one in the centre.

Robert