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

From: Paul Erlich <PERLICH@ACADIAN-ASSET.COM>
>
> So you're referring to a "dekateserany" other than one which is the
> stellated hexany? We need to modify the Tuning Dictionary entry.
>

Evidently

> 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?
> >

In Pascal's triangle, the 1(4 CPS appears in the same row as the hexany,
therefore it cannot be a subset of the hexany, but it sure is a subset of
the Eikosany. The limit remark doesn't make much sense -- the factors in
the CPS are entirely free (i.e. I could make an Eikosany with factors
1,3,5,7,9,15 or 1,3,5,7,9,21 etc.) and one might very well use the 1(4 CPSs
that are subsets of the 2(5 dekany subset of the eikosany without factors
above 7.

> 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?
>

Yep, I left something out -- Wilson transposes the complementary cross set
by the product of all the factors.

> > 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.

that's right.

> >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.
>
Yes, for once, Wilson doesn't even bother to try to graph the thing on
paper...

> From: Paul Erlich <PERLICH@ACADIAN-ASSET.COM>
> >
> > 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?

In all the other cases of stellation, the factors that are being added to are not ambiguous.
In this case ( dyads ) are treated as two different things without any acoustical support. I
don't think that when we hear a harmonic tetrad anyone perceives it as combination of
subharmonic dyads. Also once we map out it is not "seen" as a stellated tetrad but as a
structure where each triad in the tetrad is treated as a member of a hexany (there being
four). I will have to say seen from this viewpoint it is not a trivial structure but not
expressed in its simplest means.

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

--- In tuning@egroups.com, Kraig Grady <kraiggrady@a...> wrote:
>
>
>
> > From: Paul Erlich <PERLICH@A...>
> > >
> > > 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?
>
> In all the other cases of stellation, the factors that are being
>added to are not ambiguous.
> In this case ( dyads ) are treated as two different things
>without any acoustical support.

Support?

>I
> don't think that when we hear a harmonic tetrad anyone perceives it
>as combination of
> subharmonic dyads.

But the mirror structure, the stellated 3(4 CPS, could be thought of
as related to the fact that when we hear a subharmonic tetrad we may
in fact perceive it as a combination of harmonic dyads.

>Also once we map out it is not "seen" as a stellated tetrad but as a
> structure where each triad in the tetrad is treated as a member of
>a hexany (there being
> four).

That's just an alternate way of seeing it, as I mentioned before. In
my lattice diagram I only drew lines if they belonged to a complete
tetrad, thus the hexanies were not as evident.

>I will have to say seen from this viewpoint it is not a
>trivial structure but not
>expressed in its simplest means.

?