basic CPS, stellation, hexany, etc. etc.

Hi, I've gotten on the list fairly recently, and I've been going through
the recent thread on "CPS" topics---stellations, hexanies, etc. etc.
Having, at first, no clue what these meant, I checked Joe Monzo's
tuning dictionary, which is usually totally helpful, however in the case
of this family terms, I still don't quite get it. In particular, I'm
looking for some simple examples along with the explanations.
Yes, I looked at the "Wilson archives" on Anaphoria, but they seem to
throw fairly complex examples at you right from the git-go. So I'm still
lost.

Does anyone know where I can find simple, clear examples that
gradually build in complexity along with clear explanations of those? (If
these might be found in the "archives" of this list, that's fine too of
course. . . . ) Maybe I missed something in the "Wilson Archives" or
wherever. . . .

ANOTHER RANDOM TOPIC:

Does anyone here use MAX/MSP on the Macintosh for tuning
experimentation? I've been makin' some patches recently. . .

CB

***From: Christopher Bailey******************

http://www.music.columbia.edu/chris

**********************************************

>***From: Christopher Bailey******************
>
>http://www.music.columbia.edu/chris
>
>**********************************************

Holy cow, dude! Kinda, Castrovalva? Did you write the tunes for
the Peter Davison whos?

Before you get into CPSs, are you familar with Partch's "tonality
diamond?" Have you read _Genesis of a Music_?

Tales is definitely some of my favorite music ever.

-Carl

> Christopher Bailey, TD 697.4]
>
> Hi, I've gotten on the list fairly recently, and I've been
> going through the recent thread on "CPS" topics---stellations,
> hexanies, etc. etc. Having, at first, no clue what these meant,
> I checked Joe Monzo's tuning dictionary, which is usually
> totally helpful, however in the case of this family terms,
> I still don't quite get it. In particular, I'm looking for
> some simple examples along with the explanations.
>
> Yes, I looked at the "Wilson archives" on Anaphoria, but they
> seem to throw fairly complex examples at you right from the
> git-go. So I'm still lost.

Hi Chris, and thanks for the good words about my Dictionary.

I must admit that, 'in the case of this family of terms',
I too 'still don't quite get it'. Thus the limited help
my Dictionary provides for the understanding of this stuff.

In face-to-face conversation with Erv Wilson, he's explained
to me that his published articles (i.e., those at the Wilson
Archives) are simply lecture-notes, and that without the
3-hour lecture that goes with them, there's a lot left unsaid.
That's why they 'seem to throw fairly complex examples at you
right from the git-go'.

The Dictionary is currently undergoing a heavy expansion.
I've asked in the past on this List if others could help
me with more detailed definitions for the Wilson terms, but
have only gotten a few enlightening responses. The offer
still stands. (Paul? Carl? Kraig? Manuel? Daniel Wolf?)

-monz

Joseph L. Monzo San Diego monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
| 'I had broken thru the lattice barrier...' |
| -Erv Wilson |
---------------------------------------------------

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[Monz wrote...]
>The Dictionary is currently undergoing a heavy expansion.
>I've asked in the past on this List if others could help
>me with more detailed definitions for the Wilson terms, but
>have only gotten a few enlightening responses. The offer
>still stands. (Paul? Carl? Kraig? Manuel? Daniel Wolf?)

Chris and Joe,

Be sure to see Paul Erlich's Gentle introduction to CPS!!
It looks like he did a great job. For the dictionary, a
link to the Gentle introduction may be the best bet.

I'll take a stab anyway:

We can take the pitches in a tuning, represent them as dots,
and arrange them in a space so that where any two pitches are
seperated by a consonant interval, that interval can be shown
as a straight line connecting the two dots (pitches) involved.

We can define any set of intervals we want as being consonant,
and we can always arrange the dots of any tuning so that _all_
our consonances are shown as straight lines, with none of the
lines crossing, if our space has enough dimensions. Sound like
Dr. Who yet? :)

Anyway, these spaces are called tonespaces -- usually, we
consider the intervals of just intonation (at some limit)
consonant, but really we can use any intervals we want.

Let's say we use JI. Consonant intervals are written as
fractions in JI. In order to keep our lines from crossing,
we need one dimension for every number appearing in a
numerator or denominator in our list of consonances.
Usually, we abide by octave-equivalence, where factors of
2 are ignored, so that a dot represents all octave copies
of its pitch.

One of the simplest and most common tonespaces is the two-
factor space of the 5-limit. Our list of consonances is:
(1:1, 5:4, 8:5, 3:2, 4:3, 6:5, 5:3). Ignoring the 2's, we
only need the numbers 3 and 5 to write all these fractions.
So we need two dimensions, x and y. Adding a 3 to a
numerator will move us + on x, adding a 3 to a denominator
will be a - move on x. Same for the 5's and y. So:

5/4-----15/8---45/32
/ \ / \ /
/ \ / \ /
/ \ / \/
1/1----3/2-----9/8

Every line on this diagram is in our list of consonances,
and no lines cross. The dots are named with a fraction
denoting the _pitch_ in the _tuning_ that's being shown.
In the above case, the tuning is [1/1, 9/8, 5/4, 45/32, 3/2,
15/8]. I wrote those fractions with /'s, to show that
they are _pitches_. Notice that I wrote the fractions in
our list of consonant intervals with :'s to show they are
_intervals_. That's a convention developed on this list,
to keep intervals straight from pitches. For example,
starting at 3/2, we move a 5:4 to 15/8. See?

Now say we wanted to add 15:8 to our list of consonances.
We'd add a dimension for 15, and then a step from 1/1 to
15/8 would be a single line on our 15-axis, rather than
two lines (one on the 3 and one on the 5). See?

Note!
Now look at this Wilson diagram:
http://www.anaphoria.com/dal12.html

It depicts two structures in 6-factor tonespace (1 3 5 7 9 11).
There appear to be lines crossing, but the lines only cross in
2-D (on the paper). The thing on the paper is only a shadow of
a 6-dimensional object. Cool, huh.

Anywho, the beat is that if we desire tunings with a good
number of consonances for a good economy of tones, they look
like connected structures in tonespace. One type of very
highly-connected structure is the tonality diamond (see
Partch -- the figure on the right of the above diagram is
the 11-limit diamond). Another type is the Combination
Product Set, or CPS.

To make a CPS tuning, start with your list of consonances, and
take, say, two of them at a time, follow them out on the lattice,
and mark a dot where you stop. For example, taking our 11-limit
consonance list: (1 3 5 7 9 11) two at-a-time, we get...

(1 3) (3 5) (5 7) (7 9) (9 11)
(1 5) (3 7) (5 9) (7 11)
(1 7) (3 9) (5 11)
(1 9) (3 11)
(1 11)

Order doesn't matter, since moving a 5:4 and then a 3:2 away
from 1/1 gets us to 15/8 the same as moving a 3:2 and then a
5:4. You can see that all this moving amounts to multiplying
the numbers. So our pitches are...

3/2 15/8 35/32 63/32 99/64
5/4 21/16 45/32 77/64
7/4 27/16 55/32
9/8 33/32
11/8

Notice there's no 1/1! All those pitches are measured from
1/1, but it's not in the tuning itself. That's one cool thing
about CPSs -- they have no "center" tonality (unlike tonality
diamonds).

Note also that while the numbers in these fractions are rather
large, the tuning has a lot of low-numbered consonant relationships.

Lastly, note that we could have taken the factors three at
a time (that would give the figure on the left of the Wilson
diagram above), four at a time, etc. One at a time gives a
tuning the same as the consonant list; six at a time gives
only one pitch.

You're ready to explore CPSs!

-Carl

>>Be sure to see Paul Erlich's Gentle introduction to CPS!!
>>It looks like he did a great job.
>
>I'm glad I did a great job, 'cause I haven't written it yet!

Whoa- I guess I was referring to the long post, which you recently
posted a corrected version of.

-Carl

Hi Paul,

>Partch, followed by Wilson, invented this whole language that we're
>speaking here.

Right, thanks, I look forward to reading his _Genesis of a Music_

Perhaps this may be helpful for anyone else interested in making the connection
between the pascal diagram and the geometrical shapes as named by mathematicians
- I have just today figured out how it works in mathematical terms.

http://www.anaphoria.com/dal16.html
as explained in
http://www.egroups.com/message/tuning/11031

=
point

pt, triangle, triangle, pt

pt, tetrahedron, octahedron, pt
Here one could show derivation of the hexany = octahedron by drawing a tetrahedron
with vertices 1, 3, 5, 7 so 1*3 = midpoint of the line joining the 1 to the 3, etc.

pt, 4-simplex, truncated 4-simplex, truncated 4-simplex, 4-simplex, pt
Here the truncated 4-simplex is obtained by putting points at the mid points of the edges
of the 4-simplex with vertices 1, 3, 5, 7, 9 (or 1,3, 5, 7, 11)

pt, 5-simplex, truncated 5-simplex, 2D face centred dual of 5-simplex, truncated 5-simplex, 5-simplex, pt
Here 2D face centred dual of 5-simplex = Eikosany as it has 10 vetrtices - also = two truncated
4-simplexes joined together

You can derive it from the 5-simplex with vertices 1, 3, 5, 7, 9, 11 (say) by putting points at
the middle of each of its triangular faces, e.g. at 1*3*5, etc.

(this is last row in the diagram at http://www.anaphoria.com/dal16.html)

Next row would be
pt, 6-simplex, truncated 6-simplex, 2D face centred dual of 6-simplex, 3D face centred dual of 6-simplex, 2D face
centred dual of 6-simplex, truncated 6-simplex, 6-simplex, pt

You could derive the middle figure of this row from the 6-simplex by putting points at the middle
of each of its tetrahedral 3-D faces, and joining them up.

Robert

Robert Walker!

Thanks for your VRML dekanies with sound. Awesome!

You wrote:

>Perhaps this may be helpful for anyone else interested in making the
connection between the pascal diagram and the geometrical shapes
>as named by mathematicians - I have just today figured out how it works in
mathematical terms.
[some editing]
>point
>pt, triangle, triangle, pt
>pt, tetrahedron, octahedron, pt
>pt, 4-simplex, truncated 4-simplex, truncated 4-simplex, 4-simplex, pt
>pt, 5-simplex, truncated 5-simplex, 2D face centred dual of 5-simplex,
truncated 5-simplex, 5-simplex, pt
>pt, 6-simplex, truncated 6-simplex, 2D face centred dual of 6-simplex, 3D
face centred dual of 6-simplex, 2D face centred dual of 6-simplex,
truncated 6-simplex, 6-simplex, pt

Yes!

You missed a tetrahedron in the 3rd row. And rather than "truncated" which
then needs to be qualified, why not say "edge-centred dual". Actually you
can drop the "centred" in all cases. Also a "3D face" is commonly called a
"cell". So we have:

point
pt, triangle, triangle, pt
pt, tetrahedron, octahedron, tetrahedron pt
pt, 4-simplex, edge-dual of 4-simplex, edge-dual of 4-simplex, 4-simplex, pt
pt, 5-simplex, edge-dual of 5-simplex, face-dual of 5-simplex, edge-dual of
5-simplex, 5-simplex, pt
pt, 6-simplex, edge-dual of 6-simplex, face-dual of 6-simplex, cell-dual of
6-simplex, face-dual of 6-simplex, edge-dual of 6-simplex, 6-simplex, pt

In place of "-hedron" for (2D) faces, we can use "-choron" for cells (3D
"faces) and "-tetron for 4D "faces", -penton for 5D etc. The prefix "dis-"
apparently means "edge-dual". So a 4-simplex is a pentachoron and the
dekanies are dispentachora.

I don't know a prefix for face-dual. Does anyone else? tris- ? Is the
eikosany a trishexatetron?

Regards,

-- Dave Keenan
http://dkeenan.com

"Edge-centered dual" is not a good term, because the dual of the dual is
supposed to give you the original object, which is true of the face-centered
dual but not of the edge-centered one.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> "Edge-centered dual" is not a good term, because the dual of the
dual is
> supposed to give you the original object, which is true of the face-
centered
> dual but not of the edge-centered one.

You are right of course,

Robert

Paul Erlich wrote:

> "Edge-centered dual" is not a good term, because the dual of the dual is
> supposed to give you the original object, which is true of the face-centered
> dual but not of the edge-centered one.

Ah! Good point. It's not necessarily a "dual". But notice that the
edge-center -> vertex transformation _is_ a dual in 2D, and the face-center
-> vertex transformation is _not_ a dual in 2D or 4D, or anything other
than 3D. In 4D, the cell-centered transformation gives the dual, etc.

Regards,
-- Dave Keenan
http://dkeenan.com