Yahoo Tuning Groups Ultimate Backup tuning Re; Dekany in 4D (Excel)

I've just put up an improved version, again at
http://dkeenan.com/Music/DekanyRotation.xls

I've changed the "gear-ratio" between successive rotation planes to phi
(1.618...) so the movie never repeats.

I've made the starting position one with 5-fold symmetry. I found this by
watching the animation until I saw it roughly right and then switching to
manual control with the sliders and painstakingly adjusting it until it was
(nearly) right. I didn't want to make it exactly right or else some edges
would dissapear behind others.

Paul, you will find the 4D coordinates of the vertices in the columns
marked "Triangulate". They do seem to be a bit "off center" and I'm not
sure why. You can look at the formulae to see how they were derived. It's
pretty straightforward except for Keenan Pepper's triangulation formulae.

-- Dave Keenan
http://dkeenan.com

🔗David J. Finnamore <daeron@bellsouth.net>

David K.,

This is one incredibly beautiful object! Thank you so much.

> I've just put up an improved version, again at
> http://dkeenan.com/Music/DekanyRotation.xls
>
> I've changed the "gear-ratio" between successive rotation planes to phi
> (1.618...) so the movie never repeats.

I think that also might now prevent "Reverse animation" from really reversing it. It seems to follow a different path backwards
than it came forwards. No matter, it's amazing from any angle.

Now, can someone point me to a place to find out how to translate this object's properties into a set of tunings? Do I need to go
to the archives and get a good grip on MOS/CPS first? I thought Anaphoria would be the natural place to find out about hexanies,
dekanies, etc. Unfortunately, the organization of the material is proving absolutely indecipherable to me, not to mention the
seemingly purposefully cryptic and esoteric nature of the presentation. Whoo! No disrespect meant, but that Mr. Wilson must be
one eccentric cat! I'm glad his material is available online in some form anyway. Thanks, Kraig!

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

--- In tuning@egroups.com, "David J. Finnamore" <daeron@b...> wrote:

> Now, can someone point me to a place to find out how to translate
this object's properties into a set of tunings?

Let's start with the hexany. As you can see in my movie, its six
elements are represented as

A*B
A*C
A*D
B*C
B*D
C*D

Usually, A, B, C, and D represent the numbers 1, 3, 5, and 7.

So calculating these products, arbitrarily choosing to divide them
all by 5 (i.e., using 1*5 as the tonic), and bringing to within one
octave yields:

1*3/5 = 3/5 -> 6/5
1*5/5 = 1/1 -> 1/1
1*7/5 = 7/5 -> 7/5
3*5/5 = 3/1 -> 3/2
3*7/5 = 21/5 -> 21/20
5*7/5 = 7/1 -> 7/4

There's your hexany.

Now for the 5-factor CPSs, A, B, C, D, and E usually represent 1, 3,
5, 7, and, 9. The 2)5 dekany, shown in dekany2.mpg, is
obtained by multiplying all possible _pairs_ out of the five factors;
and the 3)5 dekany, shown in dekany.mpg, is obtained by multiplying
all possible _triplets_ out of the five. Here's the 2)5 dekany with
1*5 as the tonic:

1*3/5 = 3/5 -> 6/5
1*5/5 = 1/1 -> 1/1
1*7/5 = 7/5 -> 7/5
1*9/5 = 9/5 -> 9/5
3*5/5 = 3/1 -> 3/2
3*7/5 = 21/5 -> 21/20
3*9/5 = 27/5 -> 27/20
5*7/5 = 7/1 -> 7/4
5*9/5 = 9/1 -> 9/8
7*9/5 = 63/5 -> 63/40

and here's the 3)5 dekany with 1*5*9 as the tonic:

1*3*5/(5*9) = 1/3 -> 4/3
1*3*7/(5*9) = 7/15 -> 28/15
1*3*9/(5*9) = 3/5 -> 6/5
1*5*7/(5*9) = 7/9 -> 14/9
1*5*9/(5*9) = 1/1 -> 1/1
1*7*9/(5*9) = 7/5 -> 7/5
3*5*7/(5*9) = 7/3 -> 7/6
3*5*9/(5*9) = 3/1 -> 3/2
3*7*9/(5*9) = 21/5 -> 21/20
5*7*9/(5*9) = 7/1 -> 7/4

Starting to become clear?

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

--- In tuning@egroups.com, "David J. Finnamore" <daeron@b...> wrote:

> Now, can someone point me to a place to find out how to translate
this object's properties into a set of tunings?

Let's start with the hexany. As you can see in my movie, its six
elements are represented as

A*B
A*C
A*D
B*C
B*D
C*D

Usually, A, B, C, and D represent the numbers 1, 3, 5, and 7.

So calculating these products, arbitrarily choosing to divide them
all by 5 (i.e., using 1*5 as the tonic), and bringing to within one
octave yields:

1*3/5 = 3/5 -> 6/5
1*5/5 = 1/1 -> 1/1
1*7/5 = 7/5 -> 7/5
3*5/5 = 3/1 -> 3/2
3*7/5 = 21/5 -> 21/20
5*7/5 = 7/1 -> 7/4

There's your hexany.

1*3/5 = 3/5 -> 6/5
1*5/5 = 1/1 -> 1/1
1*7/5 = 7/5 -> 7/5
1*9/5 = 9/5 -> 9/5
3*5/5 = 3/1 -> 3/2
3*7/5 = 21/5 -> 21/20
3*9/5 = 27/5 -> 27/20
5*7/5 = 7/1 -> 7/4
5*9/5 = 9/1 -> 9/8
7*9/5 = 63/5 -> 63/40

and here's the 3)5 dekany with 1*5*9 as the tonic:

Starting to become clear?

🔗Joseph Pehrson <pehrson@pubmedia.com>

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

Hi Dave,

>I've made the starting position one with 5-fold symmetry.

It doesn't look that way to me. That's odd. Maybe my system isn't updating
the file correctly?

Anyway, I'd like to keep the centered coordinates I'm using.

I had an idea.

The coordinates I have now are

(2,0,0)
(0,2,0)
(0,0,2)
(-2,0,0)
(0,-2,0)
(0,0,-2)
(-1,1,1)
(1,-1,1)
(1,1,-1)
(-1,-1,-1)

but in 4-D they should all be equidistant from the center. That could be the
case if the w-coordinate of the first six points is 0 and that of the last
four points has an absolute value of 1. The last four form a tetrany. The
only way to keep them all equidistant from one another is to make the sign
on all those 1s the same, so I'll choose 1.

OK -- the 4-D dekany movies are ready at
http://www.egroups.com/files/tuning/perlich/ and are named dekany_4d.mpg and
dekany2_4d.mpg. They just rotate along the ZW plane, for now. Enjoy!

🔗Kraig Grady <kraiggrady@anaphoria.com>

David!
Erv's teaching method is primarily Oral. documents are basic reminders and a sketch of the material he would cover in person. He
say he could elaborate more on any subject , all which could take a book. Instead he chooses to be highly condensed with the idea
that one picture equals a thousand words. He is always on to other things that would suffer if he spent all his time doing the work
the student could and should do himself. Anyway When The time permits a simple introduction to the CPS will be done! It is not for
beginners and those who don't understand it probably have little musical use for it.

"David J. Finnamore" wrote:

> David K.,
>
> This is one incredibly beautiful object! Thank you so much.
>
> > I've just put up an improved version, again at
> > http://dkeenan.com/Music/DekanyRotation.xls
> >
> > I've changed the "gear-ratio" between successive rotation planes to phi
> > (1.618...) so the movie never repeats.
>
> I think that also might now prevent "Reverse animation" from really reversing it. It seems to follow a different path backwards
> than it came forwards. No matter, it's amazing from any angle.
>
> Now, can someone point me to a place to find out how to translate this object's properties into a set of tunings? Do I need to go
> to the archives and get a good grip on MOS/CPS first? I thought Anaphoria would be the natural place to find out about hexanies,
> dekanies, etc. Unfortunately, the organization of the material is proving absolutely indecipherable to me, not to mention the
> seemingly purposefully cryptic and esoteric nature of the presentation. Whoo! No disrespect meant, but that Mr. Wilson must be
> one eccentric cat! I'm glad his material is available online in some form anyway. Thanks, Kraig!
>
> --
> David J. Finnamore
> Nashville, TN, USA
> http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html

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

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗David J. Finnamore <daeron@bellsouth.net>

Paul Erlich wrote:

> Starting to become clear?

Quite. I can't believe that's all there is to it. Thank you. I was picturing spending hours combing through the archives trying to
get a handle on it.

Just to be sure:
1) In the hexany if one were to use the 1*7, 3*5, or whatever as the tonic, the result would simply be a different mode of the scale
shown?
2) The "1, 3, 5, 7" is related to the "Euler genus" concept? (Or is Euler prime limit?)
3) Fractions or irrationals could be substituted for integers, requiring somewhat different usage of the term "dimension" as it
pertains harmonic lattices, but not its representation as a geometric figure?
4) The rotating dekany in 4D represents 2)5 and 3)5 dekanies equally well?

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

🔗Joseph Pehrson <josephpehrson@compuserve.com>

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

http://www.egroups.com/message/tuning/14984

> Joseph!
> I have to point out that Wilson would strongly object to using any
1/1 to define any of the
> CPS. The concept of a 1/1 is as far from any of these structures on
can get. Either you
> understand it as non-centered or you missing something.
>

Kraig!

YES. John Chalmers did go over this with me! That seems like a
*VERY* important concept, since that's what gives the CPS the
flexibility and multiplicity. Certainly I can hear that in *YOUR*
music...

Maybe Paul was just trying to simplify matters. I don't know why he
did it that way without further explanation...

Paul??

🔗David Finnamore <daeron@bellsouth.net>

Thanks, Kraig, that does help clear things up a bit. But it's
unrealistic and unnecesary in this society to require everyone to
travel to California and sit at the feet of the guru.

--- In tuning@egroups.com, Kraig Grady <kraiggrady@a...> wrote:
> Erv's teaching method is primarily Oral. documents are basic
reminders and a sketch of the material he would cover in person.

If his teaching is primarily oral, then the web presentation should
be primarily oral. Unless he does it for a living, I'd suggest that
someone tape record a trimester of lectures and lessons, or whatever,
encode them in Real Audio and upload them to your site. That way we
can use his documents as he intended. If it is his living, it should
be stated somewhere on your site that these documents are to be used
in conjunction with the classes, and where to register to take them.
Maybe it does and I haven't stumbled across it yet.

> he chooses to be highly condensed with the idea
> that one picture equals a thousand words. He is always on to other
things that would suffer if he spent all his time doing the work
> the student could and should do himself.

Smart man. I can respect that. But then why is he so adamant that
no one else try to explain these ideas to others? Why should each
student have to start from scratch?

> Anyway When The time permits a simple introduction to the CPS will
be done! It is not for
> beginners and those who don't understand it probably have little
musical use for it.

If all teachers had that attitude we'd still all be sitting around
camp fires sharpening stones. Progress takes teamwork. That's what
the Tuning List is all about.

David Finnamore

🔗Kraig Grady <kraiggrady@anaphoria.com>

Joseph!
There are some early diagrams where he shows all 6 "modes". I will try to put these up
soon. In this context it makes sense to have 1/1 but you need to do all six.
One of the other problems of "reducing" the hexany is the "functions" of each tone get
disguised. For instance, when you have 3*5, you know that within the hexany this tone will
function as a 3 and in another context 5. You will also know after it is pointed out to you
that it will function subharmonically in turn as 1 and 7. Once you see that each note has a
unique relationship and functional properties can one really enjoy the structure for what it
is worth.

Joseph Pehrson wrote:

> --- In tuning@egroups.com, Kraig Grady <kraiggrady@a...> wrote:
>
> http://www.egroups.com/message/tuning/14984
>
> > Joseph!
> > I have to point out that Wilson would strongly object to using any
> 1/1 to define any of the
> > CPS. The concept of a 1/1 is as far from any of these structures on
> can get. Either you
> > understand it as non-centered or you missing something.
> >
>
> Kraig!
>
> YES. John Chalmers did go over this with me! That seems like a
> *VERY* important concept, since that's what gives the CPS the
> flexibility and multiplicity. Certainly I can hear that in *YOUR*
> music...
>
> Maybe Paul was just trying to simplify matters. I don't know why he
> did it that way without further explanation...
>
> Paul??.

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

🔗Kraig Grady <kraiggrady@anaphoria.com>

David!
Erv is not your average person in some traditional or well defined position.
The idea of putting this stuff up on the web was my idea and not his.
He is also the last person who would ever want to be a guru. When he was growing up, he
had from an early age heard tones he could not play on his piano in the mountains of Mexico.
Only later in his late teens did he realize about microtones. On investigating further he
realized that much had to be done before he could do the music he wanted to do. He thought
that the best way to cause the music he wanted to hear was not in composing things that
couldn't be done, but by actualizing solution to the problems that prevent microtonal music
to come to fruition. He felt such an act could cause far more change in Music than anything
else. The bosanquet keyboard is something he has wanted for 50 years.
I have had the fortune to be present quite often when He first meets someone. The
conversation always end up centering on what this other person wants to do or is doing, not on
what he is doing. True he will pulls things out that have universal applications if they are
new, but more often than not, he will show them what he has that relates to them. He is always
the one asking the most questions. I have seen this go to the extreme where he says nothing
about what he is doing!
I would love to put up video of Erv, but i no longer have a camera. I have some video
though and someday i will figure out how to put it up. I spend quite a time getting him to let
me put up just these documents. I can be like pulling teeth!
I do what i can. Putting these Archives are my way of paying Erv back, by giving him some
of the credit he deserves. By the way he doesn't charge stating that "it would be unfair to
those who could not afford it". I can say out of all of my teachers in my life he is the best
model of a teacher regardless of subject.
The best idea might be for me to write a "My introduction to the Wilson Archives" with
probably one by John Chalmers " and one by Dan Wolf as these are the ones who know him best
over a longer period of time. I think sometimes a few guideposts are all that is needed. The
problem here is that his only request in putting this stuff up is that it "stand without
comment so that his work can speak for itself". More than anything else was a desire to
prevent himself from being "misquoted" as much as possible. Maybe he will consent to some
postludes someday.
Now if we get funding I will do a you suggest. At the moment I would like to redo
everything in PDF
and are not getting very far, although there are those trying to help!

David Finnamore wrote:

> Thanks, Kraig, that does help clear things up a bit. But it's
> unrealistic and unnecesary in this society to require everyone to
> travel to California and sit at the feet of the guru.
>
> --- In tuning@egroups.com, Kraig Grady <kraiggrady@a...> wrote:
> > Erv's teaching method is primarily Oral. documents are basic
> reminders and a sketch of the material he would cover in person.
>
> If his teaching is primarily oral, then the web presentation should
> be primarily oral. Unless he does it for a living, I'd suggest that
> someone tape record a trimester of lectures and lessons, or whatever,
> encode them in Real Audio and upload them to your site. That way we
> can use his documents as he intended. If it is his living, it should
> be stated somewhere on your site that these documents are to be used
> in conjunction with the classes, and where to register to take them.
> Maybe it does and I haven't stumbled across it yet.
>
> > he chooses to be highly condensed with the idea
> > that one picture equals a thousand words. He is always on to other
> things that would suffer if he spent all his time doing the work
> > the student could and should do himself.
>
> Smart man. I can respect that. But then why is he so adamant that
> no one else try to explain these ideas to others? Why should each
> student have to start from scratch?
>
> > Anyway When The time permits a simple introduction to the CPS will
> be done! It is not for
> > beginners and those who don't understand it probably have little
> musical use for it.
>
> If all teachers had that attitude we'd still all be sitting around
> camp fires sharpening stones. Progress takes teamwork. That's what
> the Tuning List is all about.
>
> David Finnamore

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

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

Fixed a bug, added a feature. Still at
http://dkeenan.com/Music/DekanyRotation.xls

Dave Finamore, thanks. I fixed reverse animation now.

I added a "speed" slider. It's much easier to follow stuff when it's going
real slow.

Paul Erlich, It's great to have the filled faces, wish I could do that in
Excel. But if you add other rotations to the movies they are gonna get
huge! Alresdy 10 times the size of the spreadsheet. I think there _is_ some
kind of 3D package for Excel. I'll have to investigate.

Robert Walker, can VRML handle 4 or 5 dimensions?

It seems we want a combination of the features of all 3 formats, Excel,
Matlab, VRML.

Paul, are you saying that when you open the speadsheet, you _don't_ see the
dekany as a little pentagram in a big pentagram in a pentagon? I don't
understand how that could be. Unless it's because your spreadsheet
application immediately starts iterating calculations (ignoring the
calculation-method settings in the document itself).

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

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

🔗Monz <MONZ@JUNO.COM>

--- In tuning@egroups.com, "Joseph Pehrson" wrote:
> http://www.egroups.com/message/tuning/14986
>
> --- In tuning@egroups.com, Kraig Grady <kraiggrady@a...> wrote:
>
> > http://www.egroups.com/message/tuning/14984
>
> > Joseph!
> > I have to point out that Wilson would strongly object to
> > using any 1/1 to define any of the CPS. The concept of a 1/1
> > is as far from any of these structures one can get. Either
> > you understand it as non-centered or you missing something.
> >
>
> Kraig!
>
> YES. John Chalmers did go over this with me! That seems like a
> *VERY* important concept, since that's what gives the CPS the
> flexibility and multiplicity. Certainly I can hear that in *YOUR*
> music...

Joe, I would say that this is *the* central concept (pardon the
pun) underlying CPS. It amounts to a very new definition of
tonality, wherein all intervals are consonant, but there is
no tonal center. Consonance without tonal centricity.

In a way, I think this is exactly what Schoenberg was trying to
do - remember that he referred to 'dissonances' as 'consonances
that are more remotely related to the keynote'. But he was
hampered by his insistent retention of 12-tET.

-monz
http://www.ixpres.com/interval/monzo/homepage.html

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

I think it is an exageration to say that any CPS has no preferred tonal
center. All consonant dyads are not equally consonant and otonal chords are
not equal to utonal. It seems as though the note corresponding to the
product of the smallest factors would be preferred. But still it's an
extraordinary acheivement to get as close to atonal as this, in JI.

I think that complete atonality can only be obtained by using all the notes
from some equal temperament (equal division of the octave). But that is
boring. So is the complete tonality of a diamond. The partial tonality of a
CPS is much more interesting.

Here's a dekany drawn as a standard 7-prime-limit lattice (3:9 same as 1:3,
no edges drawn for ratios of nine). This is the CPS formed by taking the
product of two at a time from {1,3,5,7,9}

5*7
,'/ \`.
1*5-/---\-3*5-------5*9
|\/ \/| /|\
|/\ /\| / | \
1*7-------3*7-------7*9 \
`.\ /,' /,' `.\
1*3-------1*9-------3*9

Here it is with meantone-ish names.

Fx
,'/ \`.
A -/---\- E ------- B
|\/ \/| /|\
|/\ /\| / | \
D#------- A#------- E# \
`.\ /,' /,' `.\
C ------- G ------- D

It seems clear to me that some notes will have a much stronger pull than
others because they are involved in more consonances that are simple and
otonal.

-- Dave Keenan
http://dkeenan.com

🔗Joseph Pehrson <pehrson@pubmedia.com>

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

http://www.egroups.com/message/tuning/14988

> Joseph!
> There are some early diagrams where he shows all 6 "modes". I
will try to put these up soon. In this context it makes sense to have
1/1 but you need to do all six.
> One of the other problems of "reducing" the hexany is the
"functions" of each tone get disguised. For instance, when you have
3*5, you know that within the hexany this tone will function as a 3
and in another context 5. You will also know after it is pointed out
to you that it will function subharmonically in turn as 1 and 7. Once
you see that each note has a unique relationship and functional
properties can one really enjoy the structure for what it is worth.
>

This is very interesting, Kraig! I can certainly hear "subharmonic"
relationships in your own music, and I'm anxious to learn more about
them!

Joseph

🔗Joseph Pehrson <pehrson@pubmedia.com>

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

http://www.egroups.com/message/tuning/14989

>
> The best idea might be for me to write a "My introduction to
the Wilson Archives" with probably one by John Chalmers " and one by
Dan Wolf as these are the ones who know him best over a longer period
of time.

This would really be fabulous and useful, and I assume that if Erv
Wilson approves it, everything would be "A-OK.."

>I think sometimes a few guideposts are all that is needed.

Perhaps. In some cases maybe a little more...

>The problem here is that his only request in putting this stuff up
is that it "stand without comment so that his work can speak for
>itself". More than anything else was a desire to prevent himself
from being "misquoted" as much as possible. Maybe he will consent to
some postludes someday.

Yeah... I see the problem...but maybe you guys who know him well
could run the results by him, and he would approve them.... (???)

> Now if we get funding I will do a you suggest. At the moment I
would like to redo everything in PDF and are not getting very far,
although there are those trying to
help!
>

It's an important service over in Anaphoria, and I would urge others
to become contributors as I have (small, alas)...

___________ ____ __ _
Joseph Pehrson

🔗Joseph Pehrson <pehrson@pubmedia.com>

🔗David J. Finnamore <daeron@bellsouth.net>

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

David Finnamore wrote,

>1) In the hexany if one were to use the 1*7, 3*5, or whatever as the
>tonic, the result would simply be a different mode of the scale
>shown?

Of course.

>2) The "1, 3, 5, 7" is related to the "Euler genus" concept? (Or is >Euler
prime limit?)

Yeah, Euler "is" prime limit, and Euler genuses are assymmetrical in the CPS
world, but you can make any Euler genus by simply combining _all_ the CPSs
with the same overall set of factors. So, for example, the 3·5·7 Euler genus
can be seen as a combination of the 1)4, 2)4, and 3)4 (1,3,5,7) CPSs.

>Fractions or irrationals could be substituted for integers, requiring
>somewhat different usage of the term "dimension" as it
>pertains harmonic lattices, but not its representation as a geometric
>figure?

Right, although the consonance/dissonance properties of the CPS may be lost.

>4) The rotating dekany in 4D represents 2)5 and 3)5 dekanies equally
>well?

I made two -- one for each!

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

Kraig wrote,

> There are some early diagrams where he shows all 6 "modes". I will >try
to put these up
>soon. In this context it makes sense to have 1/1 but you need to do >all
six.

Well, if you're just trying to get the scale on the keyboard, one is enough
-- the other five can come later.

> One of the other problems of "reducing" the hexany is the >"functions"
of each tone get
>disguised. For instance, when you have 3*5, you know that within the
>hexany this tone will
>function as a 3 and in another context 5. You will also know after it >is
pointed out to you
>that it will function subharmonically in turn as 1 and 7. Once you see
>that each note has a
>unique relationship and functional properties can one really enjoy the
>structure for what it
>is worth.

Very true! That's why I'd rather see 5*7 than 35/32, etc., on Robert
Walker's VRML.

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

David Finnamore wrote:

>Fractions or irrationals could be substituted for integers, requiring
>somewhat different usage of the term "dimension" as
>it pertains harmonic lattices, but not its representation as a geometric
>figure?

Hmm. This is where some actual text from Wilson would be helpful. If the
CPS idea is extended in this way it seems to me it could become almost
meaningless. But Wilson apparently tolerates a certain amount of degeneracy
in his CPS's, since the factors are not all relatively prime (or 1).

A picture isn't always worth a thousand words. Sometimes it can be
completely unintelligible without a few words to key one into what it's
about. If Erv Wilson doesn't write those words, others will. In fact we
are! If Wilson doesn't like the diverse (and sometimes bizarre) versions of
the idea, that folks are calling CPS, he has no-one to blame but himself.
Of course we are lucky to have the Apostle Kraig to guide our deliberations.
:-)

If a chain of 5 fifths, and Justin White's 3-at-a-time from
{1,2,3,4,5,7,9,11,13,15,21,27) can be considered CPS's, then the term
starts to seem useless.

>4) The rotating dekany in 4D represents 2)5 and 3)5 dekanies equally
>well?

If you'd said "the rotating _dispentachoron_ in 4d represents equally ...
", the answer would be an unqualified yes. Mine, without filled-in faces,
does. But Paul's, with otonal triads as filled faces, are different. I wish
I could do that in Excel, preferably using semi-transparent colours.

Paul Erlich wrote:

>Why did you leave out the line connecting 3*5 with 1*9?:

And what about the one between 3*7 and 1*9 too? It's because these edges
are "degeneracies". They are not edges of the dispentachoron. They break
its symmetry. They only occur because 3 and 9 are not relatively prime
(have common factors). I might have shown them dotted, but that's a little
difficult in ASCII-graphics.

I wrote:
> 5*7
> ,'/ \`.
>1*5-/---\-3*5-------5*9
> |\/ \/| /|\
> |/\ /\| / | \
>1*7-------3*7-------7*9 \
> `.\ /,' /,' `.\
> 1*3-------1*9-------3*9
>
>It seems clear to me that some notes will have a much stronger pull than
>others because they are involved in more consonances that are simple and
>otonal.

Carl Lumma asked:

>Which ones are those?

1*3, 1*5, 1*7, 1*9. When one recognises the extra "degenerate"
(non-dispentachoron) consonances, 1*3 looks like a clear winner to me,
which is why I called it "C" in:

Fx
,'/ \`.
A -/---\- E ------- B
|\/ \/|\ /|\
|/\ /\| \ / | \
D#------- A#------- E# \
`.\ /,' `.\ /,' `.\
C ------- G ------- D

The "degenerate" edges shown this time.

Is this the same as Kraig's usage of "degenerate". Is there a better term
for my usage?

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

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

David Keenan wrote,

>But Wilson apparently tolerates a certain amount of degeneracy
>in his CPS's, since the factors are not all relatively prime (or 1).

That doesn't necessarily imply any degeneracy -- for example, the (1 3 5 7
9) CPSs have no degeneracy.

>If a chain of 5 fifths, and Justin White's 3-at-a-time from
>{1,2,3,4,5,7,9,11,13,15,21,27) can be considered CPS's, then the term
>starts to seem useless.

I agree.

I wrote,

>>Why did you leave out the line connecting 3*5 with 1*9?:

Dave Keenan wrote,

>And what about the one between 3*7 and 1*9 too? It's because these edges
>are "degeneracies". They are not edges of the dispentachoron. They break
>its symmetry. They only occur because 3 and 9 are not relatively prime
>(have common factors). I might have shown them dotted, but that's a little
>difficult in ASCII-graphics.

Oh, _that's_ what you mean by degeneracy! I see, the structure has _more_
consonances that is should. OK! (in the past, we've used the term
"degeneracy" in connection with CPSs which have fewer than the expected
number of notes, for example if 1, 3, 11, and 33 are all factors, then 1*33
= 3*11 and we have degeneracy).

🔗David J. Finnamore <daeron@bellsouth.net>

David C Keenan wrote:

> David Finnamore wrote:
>
> >Fractions or irrationals could be substituted for integers, requiring
> >somewhat different usage of the term "dimension" as
> >it pertains harmonic lattices, but not its representation as a geometric
> >figure?
>
> Hmm. This is where some actual text from Wilson would be helpful.

You can say that again!!!

> Hmm. This is where some actual text from Wilson would be helpful.

Thank you.

> CPS idea is extended in this way it seems to me it could become almost
> meaningless.

I was purposefully pushing the envelope of my perception of the definition to provoke smart
people like you into helping me understand it. :-) Hope you don't mind.

> A picture isn't always worth a thousand words. Sometimes it can be
> completely unintelligible without a few words to key one into what it's
> about.

You can say that again!!! But I won't make you. You're exactly right; a picture without
context is meaningless. If the viewer happens to have sufficient context already in his mind,
then great. Otherwise, well, no wonder such "diverse (and sometimes bizarre) versions" of his
ideas are produced.

> Of course we are lucky to have the Apostle Kraig to guide our deliberations.
> :-)

Yes, indeed.

> >4) The rotating dekany in 4D represents 2)5 and 3)5 dekanies equally
> >well?
>
> If you'd said "the rotating _dispentachoron_ in 4d represents equally ...
> ", the answer would be an unqualified yes.

Good! That is what I meant. Sorry to you and Paul for the confusion. Perhaps the subject for
this thread should have been "Rotating Dispentachoron in 4D" all along. Or did we even know
that term when it started?

Paul, now that your solid faced rotating dekanies have been corrected, they have some of the
same magic as David K.'s tumbling dispentachoron. They're also a little easier to follow. But
there's something endlessly fascinating about the faceless version, the way it snaps together
into (nearly) symmetrical configurations of lines. I feel motivated to figure out how to use
music to show off the beauty of it. Judging by how long it's taking me just to plumb the
depths of a simple hexany, don't hold your breath on seeing anything like that from me any time
soon!

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--