RE: Hello George!

In a message dated 10/21/00 2:53:57 AM EST, PErlich@Acadian-Asset.com
writes:

<< I am a musical theorist, and studying your page, I came to the
realization that the Dispentachoron is a musical structure known as a
"Dekany". There's an interesting musical structure known as an "Eikosany",
which would be a 5-dimensional uniform polytope with

0 Pentachora
10 Dispentachora
30 Octahedra
30 Tetrahedra
120 Triangles
90 Edges
20 Vertices

Do you know anything about this 5-dimension figure, particularly, any good
ways of depicting its structure? >>

George, I goofed. That should be 12 Dispentachora!

>As if there were any need for further proof of the deep connections between

>music and mathematics...!

>So in particular, your five-dimensional "eikosany" is the uniform
>dodecatetron. It has twelve (not 10--please note correction; I believe your

>other element counts are correct)

You got me!!! And yes, the others are correct.

>Its vertex figure is a triangular duoprism whose six
>cells are the vertex figures of the six dispentachora that come together at

>each vertex.

uh-oh . . .

>Whereas the ten vertices of a dispentachoron are located at the midpoints
of
>the ten edges of a regular pentachoron, the 20 vertices of the uniform
>dodecatetron are located at the centroids of the 20 triangles of a regular
>hexatetron (the 5D simplex, or tetrahedron-analog).

I'm falling in a 5-dimesional chasm!

>The Coxeter-Dynkin symbol
>for the dodecatetron is

>o-----o----(o)----o-----o

>and its symmetry group is the hexatetric group [3,3,3,3], of order 720.

720, huh? There are several musicians I know who will be interested in
knowing that.

>I was able to view your movie of the dispentachoron. Very pretty! Perhaps
>sometime you might contrive a movie of a projection of the same figure
>rotating about some arbitrary pivot plane in four-space.

I just asked the Tuning list (to which I'm cc-ing this e-mail, hope you
don't mind) if anyone could help with the mathematics of doing that. I bet
you can!!

>Then you could watch
>the octahedra and tetrahedra turn inside-out in the projection, as those
>cells rotated from the front of the figure to the back and vice versa.

Exactly -- by "front" and "back" you're of course referring to directions in
the unseen 4th dimension, correct?

>The 6D analog of the "eikosany" is the polytope

>o-----o----(o)----o-----o-----o

>It would have seven dodecatetra and seven dishexatetra (each of these is a
>uniform polytetron with six pentachora and six dispentachora as its
tetrons)
>as its pentons (so, with two different kinds of pentons, it is not an
>>isopenton<); one might call it a uniform disheptapenton. Its heptapentic
>symmetry group [3,3,3,3,3] has order 5040, and its 35 vertices are located
at
>the centroids of the 35 triangles (or 35 tetrahedra) of a regular
>heptapenton. If there is a musical analog of this, it perhaps would be
called
>a "triacontapentany."

By the way, these musical structures are an invention of Erv Wilson called
Combination Product Sets (CPSs). In a CPS, each vertex represents a musical
pitch and its frequency is determined by taking a product of k different
integers out of a master set of n different integers. Hence the scale has
n-choose-k notes. Each edge is a consonant dyad as it involves a frequency
ratio equal to the ratio of two integers. While you only recognize one type
of dispentachoron and one type of triacontapentany, the 2)5 and 3)5 CPSs are
quite different acoustically, as are the 3)7 and 4)7 CPSs. In fact, the
movie I sent you was in fact of the 3)5 CPS -- the 2)5 CPS would have a
different appearance, since I filled in only the triangles corresponding to
the "overtonal" or "harmonic" triads, of which the 3)5 CPS has 10 while the
2)5 CPS has 20. The numbers of "undertonal" or "subharmonic" triads are
reversed. The number of "overtonal" tetrads in the 3)5 CPS is 0, while the
2)5 CPS has, if I'm not mistaken again, 5 -- and the numbers are reversed
for the "undertonal" tetrads.

Anyway, we're all pretty clueless as to the mathematics of these things, so
any enlightenment would be much appreciated!

Paul

Paul!
Even when Erv showed his stuff to Steinhardt (sp), he felt they all got bogged down in
trying to plot this to a "solid.

Better yet would be to find this type of math already in existance-combination products

BTW one can just listen to a piece using the 1-3-5-7 hexany at
http://artists.mp3s.com/artists/72/the_tuning_punks.html

A Farewell Ring
down the page-if it ever loads up

"Paul H. Erlich" wrote:

> In a message dated 10/21/00 2:53:57 AM EST, PErlich@Acadian-Asset.com
> writes:
>
> << I am a musical theorist, and studying your page, I came to the
> realization that the Dispentachoron is a musical structure known as a
> "Dekany". There's an interesting musical structure known as an "Eikosany",
> which would be a 5-dimensional uniform polytope with
>
> 0 Pentachora
> 10 Dispentachora
> 30 Octahedra
> 30 Tetrahedra
> 120 Triangles
> 90 Edges
> 20 Vertices
>
> Do you know anything about this 5-dimension figure, particularly, any good
> ways of depicting its structure? >>
>
> George, I goofed. That should be 12 Dispentachora!
>
> >As if there were any need for further proof of the deep connections between
>
> >music and mathematics...!
>
> >So in particular, your five-dimensional "eikosany" is the uniform
> >dodecatetron. It has twelve (not 10--please note correction; I believe your
>
> >other element counts are correct)
>
> You got me!!! And yes, the others are correct.
>
> >Its vertex figure is a triangular duoprism whose six
> >cells are the vertex figures of the six dispentachora that come together at
>
> >each vertex.
>
> uh-oh . . .
>
> >Whereas the ten vertices of a dispentachoron are located at the midpoints
> of
> >the ten edges of a regular pentachoron, the 20 vertices of the uniform
> >dodecatetron are located at the centroids of the 20 triangles of a regular
> >hexatetron (the 5D simplex, or tetrahedron-analog).
>
> I'm falling in a 5-dimesional chasm!
>
> >The Coxeter-Dynkin symbol
> >for the dodecatetron is
>
> >o-----o----(o)----o-----o
>
> >and its symmetry group is the hexatetric group [3,3,3,3], of order 720.
>
> 720, huh? There are several musicians I know who will be interested in
> knowing that.
>
> >I was able to view your movie of the dispentachoron. Very pretty! Perhaps
> >sometime you might contrive a movie of a projection of the same figure
> >rotating about some arbitrary pivot plane in four-space.
>
> I just asked the Tuning list (to which I'm cc-ing this e-mail, hope you
> don't mind) if anyone could help with the mathematics of doing that. I bet
> you can!!
>
> >Then you could watch
> >the octahedra and tetrahedra turn inside-out in the projection, as those
> >cells rotated from the front of the figure to the back and vice versa.
>
> Exactly -- by "front" and "back" you're of course referring to directions in
> the unseen 4th dimension, correct?
>
> >The 6D analog of the "eikosany" is the polytope
>
> >o-----o----(o)----o-----o-----o
>
> >It would have seven dodecatetra and seven dishexatetra (each of these is a
> >uniform polytetron with six pentachora and six dispentachora as its
> tetrons)
> >as its pentons (so, with two different kinds of pentons, it is not an
> >>isopenton<); one might call it a uniform disheptapenton. Its heptapentic
> >symmetry group [3,3,3,3,3] has order 5040, and its 35 vertices are located
> at
> >the centroids of the 35 triangles (or 35 tetrahedra) of a regular
> >heptapenton. If there is a musical analog of this, it perhaps would be
> called
> >a "triacontapentany."
>
> By the way, these musical structures are an invention of Erv Wilson called
> Combination Product Sets (CPSs). In a CPS, each vertex represents a musical
> pitch and its frequency is determined by taking a product of k different
> integers out of a master set of n different integers. Hence the scale has
> n-choose-k notes. Each edge is a consonant dyad as it involves a frequency
> ratio equal to the ratio of two integers. While you only recognize one type
> of dispentachoron and one type of triacontapentany, the 2)5 and 3)5 CPSs are
> quite different acoustically, as are the 3)7 and 4)7 CPSs. In fact, the
> movie I sent you was in fact of the 3)5 CPS -- the 2)5 CPS would have a
> different appearance, since I filled in only the triangles corresponding to
> the "overtonal" or "harmonic" triads, of which the 3)5 CPS has 10 while the
> 2)5 CPS has 20. The numbers of "undertonal" or "subharmonic" triads are
> reversed. The number of "overtonal" tetrads in the 3)5 CPS is 0, while the
> 2)5 CPS has, if I'm not mistaken again, 5 -- and the numbers are reversed
> for the "undertonal" tetrads.
>
> Anyway, we're all pretty clueless as to the mathematics of these things, so
> any enlightenment would be much appreciated!
>
> Paul
>
>
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-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

>Even when Erv showed his stuff to Steinhardt (sp), he felt they all got
bogged down in
>trying to plot this to a "solid.

Well, once Dave Keenan fills me in on how to rotate the dekanies in 4-d, we
might have something even Erv would like.

>BTW one can just listen to a piece using the 1-3-5-7 hexany at
>http://artists.mp3s.com/artists/72/the_tuning_punks.html

>A Farewell Ring
>down the page-if it ever loads up

Beautiful, but don't you find the inharmonic timbres get in the way?

Paul!
Which ones, the ones in the background or the ones from the vibes. I find the vibes strong
enough to overcome either the delay (on the same pitches BTW) or the metered (mixtures of 13
and 8) gong in the bathtub. I take it you find it too much .

"Paul H. Erlich" wrote:

> Beautiful, but don't you find the inharmonic timbres get in the way?

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