Re: [tuning] Re: stellation continued

Paul wrote:
>. . . 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. . . .

No, the symmetry is not ruined. You forget the utonal chords. The
superstellated 1)4 CPS has the same number of utonal tones than
the 3)4 has otonal tones. Both sides together, they have the same number
of tones. To illustrate, these are the rectangular lattices.

hor > 3/1 - ver ^ 5/1 - hor >> 7/1

Superstellated 1 out of 1 3 5 7 CPS:

* | * * | * * * | * * * * |
. | . * | 0 * | * * * |
| | * | * * |
| | | * |

Superstellated 2 out of 1 3 5 7 CPS (identical to stellated one):

| * | | |
* | * * | * * * | |
. | 0 * * | * * | * |
| | * | |

Superstellated 3 out of 1 3 5 7 CPS:

* | | | |
* * | * | | |
* * * | * * | * | |
0 * * * | * * * | * * | * |

And, for comparison, the "first-order" stellated lattices.

Stellated 1 out of 1 3 5 7 CPS:

* * | * * * | * * |
. * | 0 * | * * * |
| * | * * |

Stellated 2 out of 1 3 5 7 CPS:

| * | | |
* | * * | * * * | |
. | 0 * * | * * | * |
| | * | |

Stellated 3 out of 1 3 5 7 CPS:

* * | * | |
* 0 * | * * | * . |
* * | * * * | * * |

Here's the difference for an odd number of factors. The dimension of
prime 11 is not shown. An "x" instead of an "*" is drawn when there's
more than one pitch projected onto that point.

hor > 3/1 - ver ^ 5/1 - hor >> 7/1

Stellated 2 out of 1 3 5 7 11 CPS:

| * | | |
* x | * x x | x x x | |
. * | O x * | * O x | * |
| * | * x | |

Superstellated 2 out of 1 3 5 7 11 CPS:

| | * | | |
* | * x | * x x | * x x x | |
. | . * | O x * | * O x | * |
| | * | * x | |
| | | * | |

Manuel Op de Coul coul@ezh.nl

--- In tuning@egroups.com, <MANUEL.OP.DE.COUL@E...> wrote:
>
> Paul wrote:
> >. . . 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. . . .
>
> No, the symmetry is not ruined . . . [text and diagrams deleted]

Manuel, all your examples seem correct, but once again the "drawing
with replacement" definition is what I was objecting to, and what
appears problematic. What am I missing? How does "drawing with
replacement" lead to the scales you diagrammed?

Paul wrote in the Sears Tower:
>What am I missing? How does "drawing with replacement"
>lead to the scales you diagrammed?

Sorry, I thought the formula for the number of tones would make
it clear. You need to "draw with replacement" two times, another time
for the utonal tones.

| N | | N | ( N )
Ts = | | + | | - ( )
| M | | N-M | ( M )

where

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

The second term represents drawing the N-M out of N utonal tones
with replacement. This will add the missing complements of the
original CPS combinations. Then the utonal tones are the product of
all factors divided by all N-M sized combinations. So for the 2)4 CPS:

complements:
ab cd
ac bd
ad bc
bc ad
bd ac
cd ab

missing complements:
aa => abcd/aa = bcd/a
bb => abcd/bb = acd/b
cc => abcd/cc = abd/c
dd => abcd/dd = abc/d

You see that "drawing with replacement" two times gives the tones for
the original CPS twice, hence N-above-M must be subtracted once.
Have fun on your trip.

Manuel Op de Coul coul@ezh.nl

Manuel wrote,

>hor > 3/1 - ver ^ 5/1 - hor >> 7/1

>Superstellated 1 out of 1 3 5 7 CPS:

> * | * * | * * * | * * * * |
> . | . * | 0 * | * * * |
> | | * | * * |
> | | | * |

>Superstellated 3 out of 1 3 5 7 CPS:

> * | | | |
> * * | * | | |
> * * * | * * | * | |
> 0 * * * | * * * | * * | * |

Could you motivate these for us using some definitions, etc.? Also, what's
with the "."s?

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Paul wrote:
>Could you motivate these for us using some definitions, etc.?

Well, those may look a bit pathological, as may all 1)N and N-1)N
superstellations but it's the most elegant generalisation that I can
think of. The definition is no different from Carl's first attempt at
a formula on 13-6-2000, the relevant part being:

Co: cardinality of the basic otonal chord, 1+(N-M)
Cu: cardinality of the basic utonal chord, 1+M
So: number of tones needed to saturate the basic otonal chord, N-Co
Su: number of tones needed to saturate the basic utonal chord, N-Cu

...with the addition that the basic chords formed by the added tones
are also saturated by new tones and if they form new basic chords,
they are also saturated, etc.
So, "first-order" stellation and "superstellation" are the same if
M and N-M both are less than or equal to 2.
I'll explain the process for the 1)4 CPS:
We have 4 tones, A, B, C and D. These form the basic otonal chord
already so there's nothing to be saturated there.
Then we write the tones as
ABCD/BCD, ABCD/ACD, ABCD/ABD and ABCD/ABC.
We'll write this for short leaving the ABCD numerator implicit:
/BCD /ACD /ABD /ABC
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.
Then if we inspect all subharmonic combinations obtained in this way
from the first four (being a 3)4 draw), we see that's it's a 3)4 draw
with replacement.

>Also, what's with the "."s?

They symbolise the point where the vertical axis through the origin
goes through the plane, as an aid to imagining the planes stacked
on top of eachother to form a 3-D lattice.

Manuel Op de Coul coul@ezh.nl

Paul!
I am not sure what exactly superstellations are at this point, but if you are referring to
the completion of every dyad into a tetrad, I doubt the it would have much interest to him.

Paul Erlich wrote:

> --- 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.
>
>

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

Paul!
Even though Erv has commented on the possibilities of modulating diamonds around
eikosanies and vice versa, it is not always desirable. From working with a particular Eikosany
at a fixed pitch level, each pitch will acquire a particular "personality" that becomes more
and more perceptible over time. For instance, the 1-3-7 in the 1-3-7-9-11-15 eikosany will
functions harmonically as the 1, the 3 ,and the 7, and subharmonically, as the 9, the 1, and
the 15. This all becomes perceptible with extended use. Tetrad become especially this way, i
just hear the 1-3-7-9 tetrad and i "know" where every thing else is.

Paul Erlich wrote:

> --- 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?

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

--- In tuning@egroups.com, Kraig Grady <kraiggrady@a...> wrote:
>
> Paul!
> I am not sure what exactly superstellations are at this point,
but if you are referring to
> the completion of every dyad into a tetrad, I doubt the it would
have much interest to him.

Read the recent posts again, Kraig. What we're calling
"superstellation" is exactly what Wilson and Chalmers called
"stellation".

>
> Paul Erlich wrote:
>
> > --- 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.

then lets call them "stellation"!!

Paul Erlich wrote:

>
> Read the recent posts again, Kraig. What we're calling
> "superstellation" is exactly what Wilson and Chalmers called
> "stellation".
>

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

--- In tuning@egroups.com, Kraig Grady <kraiggrady@a...> wrote:

> then lets call them "stellation"!!

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. For
example, Carl's version of stellation, applied to the eikosany,
completes all the incomplete tetrads to hexads. Since there are 30 of
these tetrads, 60 tones are added, bringing the total to 80. Wilson's
formulation proceeds to find the incomplete pentads within these 60
tones, of which there are apparently 12, and complete those into
hexads, bringing the total number of tones to 92.

Manuel followed Carl's lead, and introduced the term
"superstellation"
for the case which, at least for (n/2))n CPSs, agrees with the Wilson/
Chalmers result. This type of stellation is easy to calculate, using
either Manuel's method of "drawing with replacement" (twice), or the
Wilson method quotes by Daniel Wolf, where the intersection of the
harmonic cross-set with the subharmonic cross-set is taken. If
nothing
else, might we call this "Wilson stellation" instead of
superstellation?

Carl wrote,

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

I wrote,

>>Why appeal to these structures of lower symmetry???

Carl wrote,

>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.
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?

>Manuel's
>procedure takes the cake for ease -- maybe we should call these things
>after it.

Huh?

>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.

_________________________________________________

We have moved!

As of June 26, 2000, Acadian Asset Management will be at a
new location in Boston's financial district.

Please contact us at:
Acadian Asset Management
Ten Post Office Square, 8th Floor
Boston, MA 02109.

All phone, fax and email remain the same.