Yahoo Tuning Groups Ultimate Backup tuning dekany movie

Using the projection shown at http://www.anaphoria.com/dal24.html, I made a
movie of a rotating 3)5 dekany. Again, the otonal triads are shown as solid
triangles.

Go to http://www.egroups.com/files/tuning/perlich/ and download dekany.mpg.

Can you see the dekany as five hexanies? If not, look at hexany.mpg again.
Now go back to dekany.mpg. You should see the same old hexany hiding among
the four new "squashed" hexanies.

Remember, though, that the dekany only has 10 notes -- not the 30 you might
expect for five hexanies.

I would have used Jon Szanto's recommended Alchemy GIF Animator program to
produce a movie that looked cleaner, took up less memory, and repeated
infinitely, but today Matlab refuses to save more than the first frame of
the movie as a valid .bmp. I've contacted Matlab support . . .

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

Paul Erlich wrote:
>So, Dave, what did you think of my 3)5 dekany movie?

Great! I love the idea of the filled triangles for otonal triads.

>Can you help me
>make it rotate
>through the fourth dimension, such that one of the "squashed"
>hexanies becomes the
>"big" one, and vice versa?

I know nothing about Matlab. I'm sure you know what a rotation matrix looks
like. What do you want to know?

>Or can you implement that yourself in
>Excel?

I'll get to it eventually.

>Now if only we could add a soundtrack to the file that makes each
>otonal triad actually
>play as it lights up . . .

Looks like Robert Walker is working on it! Sort of.

Good detective work re Dekany = Rectified Pentachoron. I take it that
"rectified" means vertices in place of edge-midpoints and edges between
what were midpoints of edges that shared a vertex.

Notice that an octagon is a rectified tetrahedron.

Here's a bit of a recap for readers who may have just tuned in.

Take good ol' Pascal's triangle.

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1

This gives us the number of ways of choosing N things from M. e.g. 5choose2
and 5choose3 are both 10.

As Erv Wilson realised, the musical application comes when the things we
are choosing are small integers to be multiplied together to give the
frequencies of the notes of a scale. Musically, the disadvantages probably
outweigh the advantages of going beyond 6 factors, given the limitations of
the human auditory system.

Leaving off all the 1's around the outside and translating into musical
terms (Wilson's) we get.
dyad
triad triad
tetrad hexany tetrad
pentad dekany dekany pentad
hexad pentadekany eikosany pentadekany hexad

To better understand the symmetries of each we want to translate them into
regular geometrical objects. Leaving off the right half of the table (which
is just a mirror image of the left) we get:

dimensionality
line-segment 1D
triangle 2D
tetrahedron octahedron 3D
pentachoron rectified-pentachoron 4D
hexa4ron ?pentadekany? ?eikosany? 5D

Those on the left side of each row are also called "simplexes". e.g. the
tetrahedron is the 3D simplex, the pentachoron is the 4D simplex.

My guess is that the pentadekany is the rectified-hexa4ron. Where
"rectified" means vertices in place of edge-midpoints, and edges between
what were midpoints of edges that shared a vertex.

And I guess the eikosany is the <something-else>-ified hexa4ron, where
<something-else>-ified means vertices in place of face-midpoints, and edges
between what were midpoints of faces that shared an edge.

Can anyone fill in the question marks below?

Musical Number of Geometrical name(s)
name vert edge face cell 4ron 5ron
-----------------------------------------------------------------------
Dyad 2 1 Line-segment, 1D-simplex
Triad 3 3 1 Triangle, 2D-simplex
Tetrad 4 6 4 1 Tetrahedron, 3D-simplex
Hexany 6 12 8 1 Octahedron, rect-3D-simplex
Pentad 5 10 10 5 1 Pentachoron, 4D-simplex
Dekany 10 30 30 10 1 Rect-Pentachoron,
rect-4D-simplex
Hexad 6 15 20 15 6 1 Hexa4ron, 5D-simplex
Pentadekany 15 ? ? ? ? 1 ?
Eikosany 20 90 120 60? 10? 1 ?

Notice how Pascal's triangle reappears in the table above for the
simplexes. It suggests there should be a column for the number of
"negative-one dimensional" objects! Which would always contain "1". I can
only interpret it as the mystical Tao or void. :-)

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

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

George, I notice that your Coxeter-Dynkin symbol corresponds to the CPS's
position on Pascal's triangle in the row with the indicated number of
elements!

David Keenan wrote,

>I know nothing about Matlab. I'm sure you know what a rotation matrix looks
>like. What do you want to know?

I think I need a 4-d rotation matrix -- is it that obvious?

>Notice that an octagon is a rectified tetrahedron.

You mean octahedron.

>My guess is that the pentadekany is the rectified-hexa4ron. Where
>"rectified" means vertices in place of edge-midpoints, and edges between
>what were midpoints of edges that shared a vertex.

I bet George knows!

>And I guess the eikosany is the <something-else>-ified hexa4ron, where
><something-else>-ified means vertices in place of face-midpoints, and edges
>between what were midpoints of faces that shared an edge.

According to George, you're exactly right! Whoa, my 5-dimensional
imagination is no match for you guys'! But David, did you know about the
symmetry order being 720?

>Can anyone fill in the question marks below?

I'll fill in some and let Geroge fill in the rest.

If I counted right, the 2)6 Pentadekany has 60 otonal and 20 utonal tetrads,
60 otonal tetrads and 15 hexanies, 6 pentads and 6 dekanies. Otonal and
utonal switch for the 4)6 Pentadekany.

I can add something to your list, Dave, thanks to George:

Musical Number of Geometrical name(s)
name vert edge face cell 4ron 5ron 6ron
-----------------------------------------------------------------------
Triacontapentany 35 1 Uniform Disheptapenton

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

Paul Erlich wrote:

>But David, did you know about the symmetry order being 720?

No. I don't even know what that means! I'm guessing it means it has 720
pivot planes having n-fold rotational symmetry (n>=2). Seems too many.
Maybe 720 is the product of all the symmetries?
Number of pivot planes with symmetry
2-fold 3-fold 4-fold 5-fold ?-fold
---------------------------------------------------------
line-segment 1
triangle 1
tetrahedron 3 4
octahedron 6 4 3
pentachoron
dispentachoron
hexatetron ?
dishexatetron? or
pentadecatetron?
dodecatetron

I think we can leave the word "uniform" off the polytope names, on the
understanding that we want to draw them _all_ as uniform, although of
course musically, none of them are.

Paul, do you have a zometool kit? See http://www.zometool.com.

I can't say I'm terribly excited by the prospect of a triacontapentany
(35-any, 7choose3 or 7choose4). 6 factors seems enough to keep one busy for
a very long time. I guess that was Wilson's (and Partch's?) conclusion too.

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

🔗Kraig Grady <kraiggrady@anaphoria.com>

I don't know if this is relevant but you have the centered pentagram lattice (one tone in the
center with five others forming a pentagram) now you have 144 permutations half of which are
"mirrors". this leave 72 and if you rotate the ten sided figure that this generates you have
720.

David C Keenan wrote:

> Paul Erlich wrote:
>
> >But David, did you know about the symmetry order being 720?
>
> No. I don't even know what that means! I'm guessing it means it has 720
> pivot planes having n-fold rotational symmetry (n>=2). Seems too many.
> Maybe 720 is the product of all the symmetries?
> Number of pivot planes with symmetry
> 2-fold 3-fold 4-fold 5-fold ?-fold
> ---------------------------------------------------------
> line-segment 1
> triangle 1
> tetrahedron 3 4
> octahedron 6 4 3
> pentachoron
> dispentachoron
> hexatetron ?
> dishexatetron? or
> pentadecatetron?
> dodecatetron
>
> I think we can leave the word "uniform" off the polytope names, on the
> understanding that we want to draw them _all_ as uniform, although of
> course musically, none of them are.
>
> Paul, do you have a zometool kit? See http://www.zometool.com.
>
> I can't say I'm terribly excited by the prospect of a triacontapentany
> (35-any, 7choose3 or 7choose4). 6 factors seems enough to keep one busy for
> a very long time. I guess that was Wilson's (and Partch's?) conclusion too.
>
> Regards,
> -- Dave Keenan
> http://dkeenan.com
>
>
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-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com