Re: [tuning] RE: Re: The CPS blues... A twofold stellated apology

My apologies for my delayed answer -- an unplanned trip out of the country and a large translation project have tied me up for the past week.

My apologies also for too rapidly glancing over Erv Wilson's _Letter to John Chalmers_. I was in error when I said that the CPS other than 1(2, 2(4, 3(6, 4(8 etc had more than one stellation solution, it is rather the combined figures like the dekateserany for which this is true. I do, however, stand by my opinion that stellated CPS outside of the above mentioned are trivial, as their members are subsets of those located in the central column of Pascal's triangle

Wilson's actual presentation of stellation deserves to be summarized, for his approach is surprising. By going at the problem from this other direction, it's not really necessary to use brute force.

Given the set of factors 1,3,5,7, Wilson makes matrices and graphs the two cross sets (1:3:5:7)^2 and (/1:/3:/5:/7)^2. The intersection of the two sets -- the tones common to both -- is the 2(4 hexany (or octahedron) and the union of the two is the stellated hexany (also called a "mandala" or stellate octahedron).

Wilson notes that the same construction procedure can be followed for the stellate Eikosany, but that a three dimensional matrix is required. In other words,(1:3:5:7:9:11)^3 will intersect point-for-point with the complementary (/1:/3:/5:/7:/9:/11)^3 at the 3(6 eikosany, the union of the two forming the stellate eokosany.

Wilson's final example, is a stellation of the union of a 2(5 CPS and a 3(5 CPS. He takes a five element cross set, and combines it with the complementary set divided by one of the five elements, thus the five possible solutions. From the text and my notes to Wilson's presentation to me 20-some years ago, I have the impression that he arrived at this solution by constructing a styrofoam-ball-and-dowell model.

Daniel Wolf

--- In tuning@egroups.com, "Daniel Wolf" <djwolf@s...> wrote:
> My apologies for my delayed answer -- an unplanned trip out of the
>country and a large translation project have tied me up for the past
>week.
>
> My apologies also for too rapidly glancing over Erv Wilson's
>_Letter to John Chalmers_. I was in error when I said that the CPS
>other than 1(2, 2(4, 3(6, 4(8 etc had more than one stellation
>solution, it is rather the combined figures like the dekateserany
>for which this is true.

So you're referring to a "dekateserany" other than one which is the
stellated hexany? We need to modify the Tuning Dictionary entry.

>I do, however, stand by my opinion that stellated CPS outside of the
>above mentioned are trivial, as their members are subsets of those
>located in the central column of Pascal's triangle

Assuming that the stellation of the 1(4 CPS that I illustrated, and
that Manuel referred to (it has 16 notes) agrees with what Wilson
intended, I see nothing trivial about it. It is certainly not a
subset of the stellated 2(4 CPS. If it happens to be a subset of a
CPS with 2, 4, or 6 more factors, that really wouldn't be very
helpful to a composer who's working only within the 7-limit, now
would it?
>
> Wilson's actual presentation of stellation deserves to be
>summarized, for his approach is surprising. By going at the problem
>from this other direction, it's not really necessary to use brute
>force.
>
> Given the set of factors 1,3,5,7, Wilson makes matrices and graphs
>the two cross sets (1:3:5:7)^2 and (/1:/3:/5:/7)^2. The
>intersection of the two sets -- the tones common to both -- is the 2>
(4 hexany (or octahedron) and the union of the two is the stellated
>hexany (also called a "mandala" or stellate octahedron).

On the face of it it would seem that there is only one note common to
both -- 1. What are you leaving out? Are the cross sets transposed in
some way to create more of an overlap?

> Wilson's final example, is a stellation of the union of a 2(5 CPS
>and a 3(5 CPS. He takes a five element cross set, and combines it
>with the complementary set divided by one of the five elements, thus
>the five possible solutions.

My guess at this point (with extreme ignorance) is that this results
from the fact that there are five "most compact ways" to construct
the initial union itself -- the "naive" or "Eulerian" way that takes
the products in an absolute sense; and then the results of
multiplying the 2(5 CPS by one of the factors other than 1. If this
is wrong, then are there 25 possible stellations resulting from 5
ways to construct the union and 5 ways to stellate each one; or do
some of these come out being identical?

>From the text and my notes to Wilson's presentation to me 20-some
>years ago, I have the impression that he arrived at this solution by
>constructing a styrofoam-ball-and-dowell model.

These would undoubtedly help greatly in grasping (literally and
figuratively) these structures. It's too bad e-mail doesn't allow you
to portray these things very well.