re: stellation continued

>Carl: My formula gave 92 tones for the stellated 20-any and I proposed
>adding 2 more to get 94, not 96. Also, the stellated 70-any appears to
>have 590 tones, IIRC.

D'oh! My bad.

>I had to write Erv about another matter, so I put the question to him
>yesterday. Will report his reply.

Cool!

>I have tried to use his conception of stellation, not my own. Erv is
>extremely conscious of structure and relations, so I imagine he quickly
>saw that adding 12 more tones to the first level stellation of the
>20-any produced a more symmetrical, connected, and efficient structure.

Bleck!

-Carl

>Well, I guess you could call that "partially stellated", but of course it's
>not _the_ stellated hexany.

Ah, huh. I said that the stellated hexany is a stellated E.F. genus.
Not that the E.F. genus is a stellated hexany!

>Just showimg that the _fully_ stellated CPS has to be distinct from the E.F.
>genus.

Of course, yes. My famous half-delirious posts must be closer to fully-
delirious than I suspected -- did I say something debating that?

What I meant to say was that maybe the _fully_ stellated CPS was a
_stellated_ E.F. genus.

>>>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
>
>No, it only exploits a few of these resources.

Could you give an example?

>>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?
>
>I'm betting it's the latter.

My bet too. What I may now try to find is x, given the structure. Of
course, I also bet that x can be defined for classes of structures, of
which there are relatively few.

>>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, they're all 6-D projections.

My own naive belief (based on what Wilson said to me,^1 and what little
experience I have with these matters^2) is that the 5-fold projections
are 2-D shadows of 6-D structures, with the 6-D structures rotated so
they look the same as 5-D structures in 2-D. For example, you can rotate
a 4-D cube so it looks like the shadow of 3-D cube in 2-D.

^1 Wilson called the 6-factor stuff, with the 1 in the center of the
pentagon, "cheating". If you look at the Pascal's triangle of CPSs,
he uses the same projection for the x)5 CPSs as the x)6 ones.

^2 The centered-pentagon stuff shares the 5-fold symmetry of the
triakontahedron mapping and the Penrose tilings, which can all be
projected, I believe I've read, from polytopes of no higher dimension
than five.

-Carl

>Just find a couple of neighboring hexanies in the oct-tet lattice and see
>for yourself how the E.F. genuses only include a few of the connecting
>tetrads.

IIRC, I was referring to stellated E.F. geni, but no matter...

>>^1 Wilson called the 6-factor stuff, with the 1 in the center of the
>>pentagon, "cheating". If you look at the Pascal's triangle of CPSs,
>>he uses the same projection for the x)5 CPSs as the x)6 ones.
>
>But in the Wilson lattices I've seen, he doesn't put 1 in the center of
>the pentagon, but at a vertex.

You mean in the 5-factor lattices, the 1 is on a vertex (instead of in the
center, as in the 6-factor pentagonal lattice)? That true. I think
that's why he called the centered-pentagon stuff "cheating".

I just got a letter from Erv today, stating that he modeled the stellated
eikosany in 3-D with the Zometool I gave him. He says it looks like a
snowflake. 'd Love to see that (even tho I still don't approve of him
stellating the stellation!)!

-Carl

>>Just find a couple of neighboring hexanies in the oct-tet lattice and see
>>for yourself how the E.F. genuses only include a few of the connecting
>>tetrads.
>
>IIRC, I was referring to stellated E.F. geni, but no matter...

No, no, very sorry, I _was_ referring to the plain E.F. genus this time,
but I thought you meant modulation between different CPSs, rather than
between different instances of the same CPS on the lattice. That's a
kind of modulation I havn't thought about before. Thanks- I'll look into
it.

-Carl

Hello to all from lovely Madison, WI!

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

> >>^1 Wilson called the 6-factor stuff, with the 1 in the center of
the
> >>pentagon, "cheating". If you look at the Pascal's triangle of
CPSs,
> >>he uses the same projection for the x)5 CPSs as the x)6 ones.
> >
> >But in the Wilson lattices I've seen, he doesn't put 1 in the
center of
> >the pentagon, but at a vertex.
>
> You mean in the 5-factor lattices, the 1 is on a vertex (instead of
in the
> center, as in the 6-factor pentagonal lattice)?

I mean that it's on a vertex in the 6-factor, centered pentagonal
lattice.

> (even tho I still don't approve of him
> stellating the stellation!)!

I don't know Carl . . . If I were you, I'd spend more time studying
the hebdomekontany before being so dismissive of Wilson's ideas . . .

>>You mean in the 5-factor lattices, the 1 is on a vertex (instead of
>>in the center, as in the 6-factor pentagonal lattice)?
>
>I mean that it's on a vertex in the 6-factor, centered pentagonal
>lattice.

The 1 identity is usually at the center of the centered pentagon in
the 6-factor stuff, although "A" is on a vertex in figure 19. Not
that it matters which identity is where...

>>(even tho I still don't approve of him stellating the stellation!)!
>
>I don't know Carl . . . If I were you, I'd spend more time studying
>the hebdomekontany before being so dismissive of Wilson's ideas . . .

Why, what's up with the hebdomekontany? I have a diagram of it
right here.

-Carl

A while ago, when Carl presented his formula for the number of tones
in a stellated CPS, I had implemented the stellation process in Scala,
and called the command CPS/STELLATED.
Then I had to try very hard to figure out how to enhance the routine
for adding the extra tones necessary to complete the extra incomplete
chords, for the general M-out-of-N case.
In a small revelation I found the solution which was very simple.
A completely stellated CPS is made by drawing with putting back!
Why did nobody tell this before?
(A normal CPS is made by drawing without putting back.)
I had implemented this a long time ago, but only halfway, namely only the
otonal side. That command is CPS/CORNER.
Now the routine is written, I'm thinking about calling it
CPS/SUPERSTELLATED and keep the name /STELLATED for the "first-order"
stellation.
So the cases where M not equals N/2 are also solved.
The number of tones I find for M=N/2 agree with John Chalmers' formula.
His formula is not correct for the other cases though. But it can be
adapted to make it correct.
The number of combinations of M out of N with putting back is (if we
adopt vertical lines for the symbol):

| N | (N + M - 1) (N + M - 1)!
| | = ( ) = ------------
| M | ( M ) M! (N - 1)!

Then John's formula is

| N | ( N )
2 | | - ( )
| M | ( M )

which should be changed to

| N | | N | ( N )
| | + | | - ( )
| M | | N-M | ( M )

The first term is the number of tones in the otonal part, the second
term in the utonal part, and the third one subtracted is the number of
tones in common between them, being the bare CPS.
After writing this formula out I don't see how to simplify it:

(N+M-1)!(N-M)! + (2N-M-1)!M! - (N-1)!N!
---------------------------------------
M!(N-M)!(N-1)!

But there you have it.

Manuel Op de Coul coul@ezh.nl

Hello from beautiful Marquette, MI!
--- In tuning@egroups.com, Carl Lumma <CLUMMA@N...> wrote:
> >>You mean in the 5-factor lattices, the 1 is on a vertex (instead
of
> >>in the center, as in the 6-factor pentagonal lattice)?
> >
> >I mean that it's on a vertex in the 6-factor, centered pentagonal
> >lattice.
>
> The 1 identity is usually at the center of the centered pentagon in
> the 6-factor stuff, although "A" is on a vertex in figure 19. Not
> that it matters which identity is where...

It matters because putting the 1 identity at the center can give a
false sense of symmetry when it's not really there. In Wilson's
diagrams of the 6-factor Grand Slam, he does not put the 1 identity
in the center, allowing the asymmetry to be clearly seen.
>
> >>(even tho I still don't approve of him stellating the
stellation!)!
> >
> >I don't know Carl . . . If I were you, I'd spend more time
studying
> >the hebdomekontany before being so dismissive of Wilson's
ideas . . .
>
> Why, what's up with the hebdomekontany? I have a diagram of it
> right here.
>
Whoops, I guess I meant the eikosany, and its relations with other
eikosanies in hexadic tone space.

--- In tuning@egroups.com, <MANUEL.OP.DE.COUL@E...> wrote:

> The number of tones I find for M=N/2 agree with John Chalmers'
formula.
> His formula is not correct for the other cases though.

I'm not sure that the other cases are well-defined -- if you think
about the 1)4 CPS, the "superstellation" would never be complete --
it would extend to infinity . . .

I wrote,
> --- In tuning@egroups.com, <MANUEL.OP.DE.COUL@E...> wrote:
>
> > The number of tones I find for M=N/2 agree with John Chalmers'
> formula.
> > His formula is not correct for the other cases though.
>
> I'm not sure that the other cases are well-defined -- if you think
> about the 1)4 CPS, the "superstellation" would never be complete --
> it would extend to infinity . . .

. . . while if you use the "drawing with replacement" definition, it
would only contain the 4 notes of the CPS itself, while the
superstellated 3)4 CPS would have far more notes, ruining the
symmetry between m)n and (n-m))n CPSs. . . .

>I'm not sure that the other cases are well-defined -- if you think
>about the 1)4 CPS, the "superstellation" would never be complete --
>it would extend to infinity . . .

Aha!

-Carl

>>I'm not sure that the other cases are well-defined -- if you think
>>about the 1)4 CPS, the "superstellation" would never be complete --
>>it would extend to infinity . . .
>
>Aha!

Fancy, and a little deceiving. Things depend somewhat on the cardinality
of the "deficient" chord. In this case, we've declared all dyads deficient
from the start, so stellation will never terminate, regardless of the
initial structure. The same is obviously true when we consider single
points deficient, or any range of cardinalities that includes 1 or 2, since
three notes are required to distinguish o- and u-tonal structures in JI;
the extra symmetry between them means you can't keep your points from
participating in new incomplete chords.

Considering nothing smaller than a triad deficient, the shape of the
structure may come into play. On the oct. lattice, however, I believe
all stellations terminate after a single interation when triads are
considered deficient.

Let's explore. We have, 3->4 in 1.

Now the question- has Wilson's structure terminated after 2 iterations?
Let's say it has. For the first iteration, 4->6. But what was the
cardinality of the incomplete chords at the second iteration?

[Chalmers]
>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.

As I've been saying, we're running up against adjacent pentadekanies
here, and completing the pentads to hexands (while ignoring the
incomplete triads, I might add). So if, as I suggested way back in this
thread, we add 1 to m at each iteration, stellation terminates when m=n.
For the hebdomekontany, we must add three tones to each chord at the
first iteration, then two, and then 1 (again, ignoring the "lesser"
deficient chords in the adjacent CPSs -- else, three, two and six, one
and seven).

All this isn't worth the trouble. Might as well just say, "I'm going
to use the entire lattice". The idea of the CPS as a structure providing
"special" access to the relations in a tonespace looses all meaning.

-Carl

Hello from the Sears Tower, Chicago, IL!!!
--- In tuning@egroups.com, <MANUEL.OP.DE.COUL@E...> wrote:

> A completely stellated CPS is made by drawing with putting back!
> Why did nobody tell this before?
> (A normal CPS is made by drawing without putting back.)

Manuel, this doesn't seem to work. For example, for the 2-out-of-4
case, drawing with putting back gets you

aa
bb
cc
dd
ab
ac
ad
bc
bd
cb,

only 10 tones. You've got the hexany, and you've completed the otonal
tetrads, but you also need

abc/d
abd/c
acd/b
bcd/a

to complete the utonal tetrads and get the full 14-note stellated
hexany.

Am I misunderstanding you?

Manuel wrote...

>If we take the first two, /ACD and /BCD, we see they form a
>subharmonic dyad which can be completed to a tetrad by adding
>/CCD and /DCD making: /ACD /BCD /CCD /DCD. We do this for all
>combinations of the two out of 4. Then the new tones form more
>incomplete chords (now triads), such as /AAB /AAC /AAD so we
>have to add /AAA, and likewise /BBB, /CCC and /DDD. At each
>iteration the number of tones to be added is one less, as is
>each incomplete chord one larger.

Did you read my post from 2/7/00? I have a problem with the
above procedure -- considering a dyad an incomplete tetrad at one
iteration but not at the others. Yes, the cardinality of the
incomplete chord increases with each iteration, but you need to
start with at least triads -- if you start with dyads the first
iteration never really stops, because there will always be
incomplete dyads.

It's nice to see what's happening with these superstellations, but
it doesn't make sense to call them stellated CPSs. I think
they're stellated Euler-Fokker genera. Anyway, a CPS is supposed
to be a fancy subset of a tonespace, not a gross chunk of it. I
find it much more natural to think of stellation as simply
completing all the chords I had in my original structure. Using
them all is enough of a challenge, without extras besides.

-Carl

I wrote...
>It's nice to see what's happening with these superstellations, but
>it doesn't make sense to call them stellated CPSs. I think
>they're stellated Euler-Fokker genera.

I should have qualified that: the superstellated m/n=2/1 CPSs are.

Paul Erlich wrote...
>Manuel, now that I think about it, your construction appears eminently
>logical. Once again, I'd like to state my conjecture (which I mentioned in
>a previous response to Carl) as to the "why" of Wilson stellation, or what
>we're calling "superstellation". It seems to get you the biggest fully
>connected region in the lattice that includes the initial CPS but no
>additional copies of that CPS.

It's nice to know that drawing with putting back gives us this structure,
but as I was going to reply before, the CPSs aren't usually gestalts that
one would transpose -- its chords are. The only possible exceptions being
the hexany, whose 1st stellation is already super, and the x)5 dekanies.

-Carl

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

> It's nice to see what's happening with these superstellations, but
> it doesn't make sense to call them stellated CPSs.

I wouldn't so cavalierly dismiss a concept that Wilson put years of
thought into.

> I think
> they're stellated Euler-Fokker genera.

Why appeal to these structures of lower symmetry???

> Anyway, a CPS is supposed
> to be a fancy subset of a tonespace, not a gross chunk of it. I
> find it much more natural to think of stellation as simply
> completing all the chords I had in my original structure. Using
> them all is enough of a challenge, without extras besides.

That's fine, but the (super)stellation gives you a very naturally-
defined "chunk": the largest one that includes no additional copies
of
the original CPS.

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

>the CPSs aren't usually gestalts that
> one would transpose -- its chords are. The only possible
exceptions being
> the hexany, whose 1st stellation is already super, and the x)5
dekanies.

Gee . . . why wouldn't you want to transpose CPSs, and why would you
make those particular exceptions?

>>I think they're stellated Euler-Fokker genera.
>
>Why appeal to these structures of lower symmetry???

What does symmetry have to do with it? I thought it might be easier
to think of a single stellation, rather than having to find a bunch of
incomplete chords at each iteration (though a good excercise). Manuel's
procedure takes the cake for ease -- maybe we should call these things
after it.

>That's fine, but the (super)stellation gives you a very naturally-
>defined "chunk": the largest one that includes no additional copies
>of the original CPS.

Agree.

>>the CPSs aren't usually gestalts that one would transpose -- its
>>chords are. The only possible exceptions being the hexany, whose
>>1st stellation is already super, and the x)5 dekanies.
>
>Gee . . . why wouldn't you want to transpose CPSs, and why would you
>make those particular exceptions?

It's not that I wouldn't want to, just that the listener probably
wouldn't be able to tell. It's all just a bunch of chord modulations;
I don't imagine perceiving an entire eikosany as a scale. The noted
exceptions are the only CPSs with few enough tones to be treated as
scales (in my opinion), not counting the raw o- and u-tonalities along
the edges of the triangle.

>I have no problem with that -- it's just that Carl Lumma seems to
>think he is wiser than Wilson, and prefers to use the term "stellation"
>for the result of only the first iteration of completing chords.

Whoa dude! I may prefer the term stellation for the first stellation,
but where do you get the wiser than Erv bit? The only thing I've heard
from Erv on this topic is that he built a model of the "stellate Eikosany".
Have you heard something I haven't? In my experience, Erv is less
concerned with making rigid definitions than he is in telling people
about stuff he's interested in. That's why all this stuff has like six
names (i.e. stellate hexany, mandala, dekatesserany, etc.). I doubt
Erv thinks of stellation as anything more than a process of adding points.
On the other hand, he may be adamant. In which case, I would fold as
on laundry day.

-Carl

>>>>I think they're stellated Euler-Fokker genera.
>>>
>>>Why appeal to these structures of lower symmetry???
>>
>>What does symmetry have to do with it?
>
>I guess I've been trained to think like a physicist, and to me it makes
>little sense to construct something symmetrical from something asymmetrical.

Huh. I appealed to the EF genera in case they would eliminate the need for
multiple iterations. I was never sure they did that, and I'm even less
sure now.

>As to the specifics, first of all this only works for n=4 when one of the
>factors is 1; and I doubt it would continue to work for higher n. Care to
>prove me wrong?

Not especially, and I'm not sure how. But if you look at Erv's diagrams
of the 6-factor grand slams, you can see that all the points shown will
occur in the superstellated eikosany.

>>Whoa dude! I may prefer the term stellation for the first stellation,
>>but where do you get the wiser than Erv bit? The only thing I've heard
>>from Erv on this topic is that he built a model of the "stellate Eikosany".
>>Have you heard something I haven't?
>
>John Chalmers recently asked Erv about this and Erv said the stellated
>eikosany has 92 tones.

Right. And Erv certainly should have dibs on the name. In the meantime,
I have no reason to believe that he wouldn't call both the 80 and 92-tone
structures 'stellated eikosanies'.

-Carl

In July, Carl Lumma wrote:
>
> Did you read my post from 2/7/00? I have a problem with the
> above procedure -- considering a dyad an incomplete tetrad at one
> iteration but not at the others. [ . . . ]
> It's nice to see what's happening with these superstellations, but
> it doesn't make sense to call them stellated CPSs. [ . . . ]
> Anyway, a CPS is supposed
> to be a fancy subset of a tonespace, not a gross chunk of it. I
> find it much more natural to think of stellation as simply
> completing all the chords I had in my original structure. Using
> them all is enough of a challenge, without extras besides.

Hi Carl.

Besides the other reasons I brought up to explain why Erv Wilson may have chosen the
92-tone definition of the Stellated Eikosany rather than your 80-tone one, I'm pretty sure
there's an additional reason -- the geometrical definition of stellation. You'll find much
about that on George Hart's pages. Basically, the planes (or hyperplanes) of the figure
being stellated have to be extended until they meet at a point. So that's why the
stellation of the Eikosany doesn't stop when the incomplete triads are completed to
tetrads -- the same geometrical extensions that allowed for those completions would
leave "holes" unless they were continued out to where they met, namely, to the tones
that complete the new incomplete tetrads into pentads. (I'm sure George Olshevsky
could give us a better geometrical explanation of this, but I think I scared him away with
too many questions already.)

-Paul

>Besides the other reasons I brought up to explain why Erv Wilson may have
>chosen the 92-tone definition of the Stellated Eikosany rather than your
>80-tone one, I'm pretty sure there's an additional reason -- the
>geometrical definition of stellation. You'll find much about that on
>George Hart's pages. Basically, the planes (or hyperplanes) of the figure
>being stellated have to be extended until they meet at a point. So that's
>why the stellation of the Eikosany doesn't stop when the incomplete triads
>are completed to tetrads -- the same geometrical extensions that allowed
>for those completions would leave "holes" unless they were continued out
>to where they met, namely, to the tones that complete the new incomplete
>tetrads into pentads. (I'm sure George Olshevsky could give us a better
>geometrical explanation of this, but I think I scared him away with too
>many questions already.)

Hmm... my 5-D vissi isn't quite there yet... I understand what you're
saying, though. I had always opperated under a (perhaps different, perhaps
incorrect) definition of stellation, which was to create the compound of a
figure and its dual. That should be accomplished by adding a single point
over each face, holes or no, eh?

Anyway, by the definition above, where does the stellation process end in
a given case? Is the given Euler-Fokker genus always a subset of the final
product? I think musically, I'd still prefer to think of the 80-tone
structure as a stellated eikosany, and the 92-tone one as a stellated
6-factor Euler-Fokker genus.

-Carl

Carl Lumma wrote,

>I had always opperated under a (perhaps different, perhaps
>incorrect) definition of stellation, which was to create the compound of a
>figure and its dual.

That's incorrect, as you can see by simply considering the tetrad [the 1)4
or 3)4 CPS].

>Anyway, by the definition above, where does the stellation process end in
>a given case? Is the given Euler-Fokker genus always a subset of the final
>product?

The 64-tone B*C*D*E*F Euler-Fokker genus is not a subset of the 92-tone
stellated A,B,C,D,E,F Eikosany.

>I think musically, I'd still prefer to think of the 80-tone
>structure as a stellated eikosany, and the 92-tone one as a stellated
>6-factor Euler-Fokker genus.

Besides not including it, the 92-tone Stellated Eikosany also contains
different symmetries from the Euler-Fokker genus -- quite against the spirit
of stellation.

>>I had always opperated under a (perhaps different, perhaps
>>incorrect) definition of stellation, which was to create the compound of a
>>figure and its dual.
>
>That's incorrect, as you can see by simply considering the tetrad [the 1)4
>or 3)4 CPS].

Ah, it has to on on the lattice, huh? :)

>>Anyway, by the definition above, where does the stellation process end in
>>a given case? Is the given Euler-Fokker genus always a subset of the final
>>product?
>
>The 64-tone B*C*D*E*F Euler-Fokker genus is not a subset of the 92-tone
>stellated A,B,C,D,E,F Eikosany.

Herm... Why'd you leave out the "A" from the Euler-Fokker genus in
question?

-Carl

I wrote,

>>The 64-tone B*C*D*E*F Euler-Fokker genus is not a subset of the 92-tone
>>stellated A,B,C,D,E,F Eikosany.

Carl wrote,

>Herm... Why'd you leave out the "A" from the Euler-Fokker genus in
>question?

Because it's 1. I should have said the 1,B,C,D,E,F Eikosany.