Stellation continued

Carl, Manuel, Paul E. Dan, Kraig, et al., :

I looked in my notes last night and was quite surprised with what I
found. I have notes in several places to the effect that Erv said that
the stellated eikosany has 92 tones. I also found a note saying that my
1981 formula is correct at least for the stellated 2)4 hexany and the
stellated 3)6 eiksosany. Alas, I didn't find any suggestions as to how
I derived it.

My formula may apply only to R)N CPS when R= N/2. In these cases, the
number of tones
S= 2*P-C where P and C are computed according to the following formulae:

P = (N+R-1)!
----------
R!(N-1)!

and

C= N1
------
R!(N-R)!

Using these formulas I get 14 for the stellated 2)4 hexany and 92 for
the stellated 3)6
20-any. The stellated 4)8 70-any would then have 590 tones.

I can account for the 14 tones of the stellated hexany and the 92 of the
20-any by brute force and symmetry. The hexany case is well-known, so I
won't discuss it here, but the 3)6 20-any is not obvious, at least to me
this AM (I still have mild hypocaffeinemia).

For the general generating hexad A.B.C.D.E.F, each tetrad "facet" is a
set of four products of three factors of the form A*E*F, B*E*F, C*E*F,
D*E*F. To complete the hexad, we must add
two extra tones E*E*F and F*E*F to each tetrad. There are 15 such
harmonic tetrads (six items, 4 at a time) and hence 30 extra tones must
be added. The subharmonic facets, of which there are also 15, have the
form A*B*C, A*B*D, A*C*D, and B*C*D, corresponding to sub-D, Sub-C,
Sub-B and Sub-A respectively. To these we add A*B*C*D/E and A*B*C*D/F.
Since there are 15 subharmonic tetrads, this accounts for an additional
30 tones or 80 tones in all.

However, six new incomplete harmonic hexads are created by the addition
of the first 30 tones above. These are formed by the notes E*E*F and
F*F*E for all selections of 2 factors from the 6 of the generator. To
complete these hexads, we must add 6 more tones of the form E*E*E to
generate chordal sets such as A*E*E, B*E*E, C*E*E, D*E*E, F*E*E and
E*E*E for all such combinations of the factors. This step adds 6 more
tones and makes 86 in altogether.

By symmetry, six more tones must be added to the 86 tones above to
complete the subharmonic facets. I seem to be dense this AM and am not
sure exactly what form these 6 tones must have. My guess is that they
are of the form A*B*C*D*E, A*B*C*D*F, corresponding to Sub-F, Sub-E,
etc. These tones raise the total to 92, in agreement with Erv's number.

(One might add two more tones, A*B*C*D*E*F and A*B*C*D*E*F/A*B*C*D*E*F
or 1/1 as well and I have a vague recollection of seeing them on a
diagram some time in the past.)

We might just decide to punt and ask the master himself.

--John

[John Chalmers wrote...]
>However, six new incomplete harmonic hexads are created by the addition
>of the first 30 tones above.

Right. Perhaps we should call the 96-tone structure some sort of "2nd-
iteration stellation". :) The 1st stellation might be of interest to
somebody, no?

>These are formed by the notes E*E*F and F*F*E for all selections of 2
>factors from the 6 of the generator.

Remembering that the x)6 CPSs mate to form the 6-factor Euler-Fokker
genus, it's no suprise that we run up against Pentadekanies when
stellating the Eikosany. As above, stellating these adjacent
Pentadekanies doesn't really strike me as the thing to do.

I believe your 96-tone structure is actually a stellated E.F. genus,
and I wonder if your 570-tone is? The 2)4 is adjacent to the basic,
completed 4-factor chords, and thus requires only a single iteration
of stellation to become a stellated E.F. genus. The 3)6 must cross a
pair of neighboring m-1)n, m+1)n CPSs to reach the basic hexads, and
thus it takes two interations to become a stellated EFG. The 4)8 will
have more than one pair of neighbors -- how many "iterations" does
your formula perform?

-Carl

--- In tuning@egroups.com, Carl Lumma <CLUMMA@N...> wrote:

> I believe your 96-tone structure is actually a stellated E.F. genus,

Carl, I doubt that -- as I've noted, the E.F. genus is symmetrical in
the square lattice, while the CPSs and their stellations are
symmetrical in the triangular lattice. Don't forget, also, that the
E.F. genus is only one of many equally symmetrical ways of combining
all the CPS scales with a given number of factors. From the
triangular viewpoint, the E.F. genus is only one of a family of 2^(n-
1) equivalent (by rotation) structures, where n is the number of
factors.

>>I believe your 96-tone structure is actually a stellated E.F. genus,
>
>Carl, I doubt that --

Yeah, it was a shot in the dark, but... the 4-factor stellated 1:2 CPS
is a stellated E.F. genus, and it does take 6 tones to stellate the
large chords in each of the two Pentadekanies, accounting for the 12
extra tones in the Chalmers structure.

>as I've noted, the E.F. genus is symmetrical in the square lattice,
>while the CPSs and their stellations are symmetrical in the
>triangular lattice.

And?

>Don't forget, also, that the E.F. genus is only one of many equally
>symmetrical ways of combining all the CPS scales with a given number
>of factors.

True.

I guess what I was really trying to establish as important:

1. Do we always want to continue stellating until all tones participate
only in complete chords?

2. Even if we do, are there structures which never reach this fully-
stellated point? Is there a way to tell how many iterations are required
to fully-stellate a given structure?

3. What exactly is Chalmers' formula finding?

-Carl

--- In tuning@egroups.com, Carl Lumma <CLUMMA@N...> wrote:
> >>I believe your 96-tone structure is actually a stellated E.F.
genus,
> >
> >Carl, I doubt that --
>
> Yeah, it was a shot in the dark, but... the 4-factor stellated 1:2
CPS

Huh?

> is a stellated E.F. genus,

Can you show me?

> and it does take 6 tones to stellate the
> large chords in each of the two Pentadekanies, accounting for the 12
> extra tones in the Chalmers structure.
>
> >as I've noted, the E.F. genus is symmetrical in the square lattice,
> >while the CPSs and their stellations are symmetrical in the
> >triangular lattice.
>
> And?

By symmetrical I mean "maximally symmetrical". Other than a single
point, a structure can't be maximally symmetrical in both lattices.
>
> >Don't forget, also, that the E.F. genus is only one of many equally
> >symmetrical ways of combining all the CPS scales with a given
number
> >of factors.
>
> True.
>
>
> I guess what I was really trying to establish as important:
>
> 1. Do we always want to continue stellating until all tones
participate
> only in complete chords?
>
I guess it could be interesting to work with the intermediate
stellations, like the 80-tone one . . . if we were aliens capable of
conceptualizing 80- and 92-tone scales.

> 2. Even if we do, are there structures which never reach this fully-
> stellated point? Is there a way to tell how many iterations are
required
> to fully-stellate a given structure?
>
> 3. What exactly is Chalmers' formula finding?

I think the idea is that you always reach a point where you are up
against other copies of the CPS you started with in all directions,
so with stellation you've essentially included all the resources you
could possibly use to move from one CPS to another.

Carl, take a close look at http://www.anaphoria.com/dal25.html. It
shows the 1.3.5.7.9.11 Euler-Fokker genus (which Wilson calls
the "Grand Slam") in four different lattice orientations. The Figures
37 and 36 should be familiar from our previous discussions -- the
symmetrical figure at the center of each is of course the Eikosany,
using the centered pentagon lattice, with different lines made
visible in each case. The Stellated Eikosany would be as symmetrical
as the Eikosany itself, while the E.F. genus, as you can see, is
elongated in one direction. Figure 35 is how Fokker would probably
lattice the thing -- using four directions for the four primes 3, 5,
7, and 11. The factor of 9 accounts for the basic hypercube structure
being repeated two extra times along the 3 axis. Figure 34 is a
Wilson variation that can be better understood by looking at
http://www.anaphoria.com/dal23.html.

>>Yeah, it was a shot in the dark, but... the 4-factor stellated 1:2
>>CPS
>
>Huh?

I just meant the stellated hexany. I phrased it that way because Chalmers
said his formula might only apply to m/n=1/2 CPSs.

>> is a stellated E.F. genus,
>
>Can you show me?

The 4-factor E.F. genus is an octahedron with two of its faces stellated.

>>>as I've noted, the E.F. genus is symmetrical in the square lattice,
>>>while the CPSs and their stellations are symmetrical in the
>>>triangular lattice.
>>
>>And?
>
>By symmetrical I mean "maximally symmetrical". Other than a single
>point, a structure can't be maximally symmetrical in both lattices.

Yes- could you explain how that fits into the discussion?

>> I guess what I was really trying to establish as important:
>>
>> 1. Do we always want to continue stellating until all tones
>> participate only in complete chords?
>
>I guess it could be interesting to work with the intermediate
>stellations, like the 80-tone one . . . if we were aliens capable of
>conceptualizing 80- and 92-tone scales.

Certainly these are not meant as melodic scales in the sense you
imply. But I certainly find the "intermediate" (I would call it
the "true") stellated eikosany easier to conceptualize, since the
original tetrads are all stellated, but without bothersome added
tones serving auxillary purpose. There's no point in a "stellated
eikosany" if you're going to use the whole lattice.

>> 2. Even if we do, are there structures which never reach this fully-
>> stellated point? Is there a way to tell how many iterations are
>> required to fully-stellate a given structure?
>>
>> 3. What exactly is Chalmers' formula finding?
>
>I think the idea is that you always reach a point where you are up
>against other copies of the CPS you started with in all directions,
>so with stellation you've essentially included all the resources you
>could possibly use to move from one CPS to another.

The plain E.F. genus should do that. But my second question is a
little more arcane; is there any structure, containing incomplete
chords, such that every time you complete all the chords you create
new structure with some incomplete chords? Or do all structures
become fully stellated after x iterations?

>Carl, take a close look at http://www.anaphoria.com/dal25.html. It
>shows the 1.3.5.7.9.11 Euler-Fokker genus (which Wilson calls
>the "Grand Slam") in four different lattice orientations.

I used to call it a grand slam too, until you pointed out that it
was an E.F. genus.

>while the E.F. genus, as you can see, is elongated in one direction.
>Figure 35 is how Fokker would probably lattice the thing -- using four
>directions for the four primes 3, 5, 7, and 11. The factor of 9
>accounts for the basic hypercube structure being repeated two extra
>times along the 3 axis.

More about being symmetrical in one lattice and not another? Again,
I'm missing the tie-in.

>Figure 34 is a Wilson variation that can be better understood by looking
>at http://www.anaphoria.com/dal23.html.

Yes- I believe those are true 6-D projections, but I find the 5-fold stuff
of the "Treetoad" and "Pascal's Triangle of CPSs" easier to use.

-Carl