Re: Octony Lullaby

Hi Dave,

>Very nice. But what's an octony? And in particular what is the 7 limit
octony?

>It sounds like it should be a Wilson CPS with 8 notes, but the number
>8 doesn't appear in Pascal's triangle except as 1 of 8 or 7 of 8 and
>those would be octads, not octanys, as in the difference between a
>hexad and a hexany. And you spell it octony not octany.

The 7-limit octony consists of all products of any of 1, 3, 5, 7

As a result, it is complete row of Pascal triangle for 3,5,7 instead of the usual 1,3,5 , as cube.

(relates to the result that sum of numbers in nth row of tri. is 2^n)

You can find the two triangles 3, 5, 7 and 3*5, 5*7, 7*3 in it, as well as 1 and 1*3*5*7.

Similarly you can find the tetrads and hexany in the hypercube.

Since hexany is combinatin of the two triangles, you can also find the 1,3,5,7 hexany in the
7-limit octony, as 1*3, 1*5, 1*7, 3*5, 5*7, 7*3.

Since combination of tetrad and hexany is a dekany, you can also find both dekanies in the
_4-dim_ hypercube (they share six vertices in common) as

1
( one corner of hypercube)

3 5 7 11
(= tetrahedron - consisting of all vertices one vertex in from corner)

3*5 3*7 3*11 5*7 5*11 7*11
(= octahedron, 2 vertices in from corner)

3*5*7 3*7*11 3*5*11 3*7*11
(= tetrahedron - 3 vertices in)

1*3*5*11
(= opposite corner of hypercube)

The first tetrahedron + octahedron = 1,3,5,7,11 2)5 Dekany.

The second tetrahedron + octahedron = 1,3,5,7,11 3)5 Dekany.

I'd like to do model of this some time, or as much of it as one can do
without being confusing if one can't do it all.

I've already done a model of Octony as cube, with octahedron inside
- see post day or two ago:

http://www.egroups.com/message/tuning/15879

Only realised this connection of the Pascal triangle with the n-dim cube recently,
but Manuel Op de Coul tells me it's been mentioned before on TL.

Robert