terms- plea for assistance

Hallo Joe Monzo and/or others,

I wonder if you'd be so kind to clarify:

(FROM THE DICTIONARY)

(which is a marvellous resource and thanks indeed for this Joe)

Combination Product Sets (CPS)

Harmonically symmetrical musical structures invented by Ervin M.
Wilson in the late 1960's. CPS's are generated by taking the
products of n harmonic factors m at a time and reducing the derived
tones to a common octave.

The best studied sets are the six tone Hexany (n=4, m=2) and the 20
element Eikosany (n=6, m=3).

CPS may be partitioned into sets of inversionally related chords,
triads in the case of the Hexany and tetrads in the Eikosany. CPS
are also characterized by intervallically symmetrical melodic
properties and may be defined in equal temperaments as well as Just
Intonation.

[from John Chalmers, Divisions of the Tetrachord]

see further comments by Paul Erlich and Carl Lumma

Could you please explain: what are:
Harmonically symmetrical musical structures invented by Ervin M. Wilson in
the late 1960's. CPS's are generated by taking the
products of n harmonic factors m at a time and reducing the derived
tones to a common octave.

for instance, does 4 harmonic factors 2 at a time mean 3/2 as a 3/2 and a
4/3 (or 9/8), then a 5/4 and an 8/5, then a 7/4 and an 8/7, and a 11/8 and a
16/11.........? Obviously not, but perhaps elaborate .........

and can you give an example of one of each of a hexany and the eikosany?

I understand lattice diagrams if that's any help.

help would not be forgotten........

best wishes

Lawrence Ball

Hi Lawrence.

Joseph Pehrson has suggested I do a web page of this in VRML.

As a first start towards it, maybe I can try explaining the idea of a
hexany to you.

Idea of a lattice taken for granted as you said you've understood that
(will need to explain that first in the web page).

Idea of a 1 3 5 7 hexany is that you take all pairs of numbers from 1 3 5 7 and
multiply them together:
1*3, 1*5, 1*7, 3*5, 3*7, 5*7

That gives six numbers. You can then place them on the vertices of an octahedron,
and if you wish, octave reduce them. The octahedron used this way is known as
a hexany because it has six vertices.

They will obviously be part of the 3 5 7 lattice.

Octave reduced:
3/2 5/4 7/4 15/8 21/16 35/32

In fact, it's made up of opposite pairs of triangles, such as 3 5 3*5 and 7 3*7 5*7
(3/2 5/4 15/8 and 7/4 21/16 35/32)
7 3*7 5*7 is a major chord 1 3 5
and 3 5 3*5 is a minor chord 1/5 1/3 1/1
The two triangles join together to make the octahedron.

1 3 5 is otonal (expressed most simply using overtone series)
and 1/5 1/3 1/1 is utonal (expressed most simply using undertone series).

Here it is in VRML, which you can get a plug in to view in p.c, and
there are some for the mac too, but they don't seem to work nearly as well
as they do for the p.c.

--------------------viewers-------------------------

I'd recommend Cosmo Player
http://www.cai.com/cosmo/home.htm

You could also try ticking Control Panel | Add Remove Programs | Windows Setup | Internet
tools | Microsoft VRML 2.0
viewer (which I think might be WorldView, another good one).

Another good one is Cortona:
http://www.parallelgraphics.com/products/cortona/download/

However, VRML causes lots of problems for the mac (especially, beware of the Cosmo Player
beta for the mac, which can erase your address book when you install it!!!
- while Cosmo Player is one of the best VRML viewers for the p.c.).

In fact, I haven't yet heard from anyone who has been able to view these
successfully on the mac, and if anyone has, would like to hear from them,
so I can recommend the viewer they use as one that works on at least some
macs!

------------------------------------------------------------

http://www.rcwalker.freeserve.co.uk/interactive_models_with_titles/hexany.wrl
Click on the ratios to hear the notes, and on the cyan or red spheres to hear the diads
or triads.

Octahedron has eight faces, so four "opposite pairs" of faces.

So lets look at the other three.

1*5 3*5 7*5 and 1*3 1*7 3*7
(5/4 15/8 35/32 and 3/2 7/4 21/16)
That is clearly
1 3 7 and 1/7 1/3 1/1

The others are going to be
1 5 7 and 1/7 1/5 1/1
and
3 5 7, 1/3 1/5 1/7.

So the 1 3 5 7 hexany notes can be used to make all the otonal triads that
are subsets of 1 3 5 7, and all the utonal triads that are subsets of 1/1 1/3 1/5 1/7
and the otonal triads are opposite the utonal ones.

That is a lot of consonant triads for a six note scale!

Now try five factors, say
1 3 5 7 11

You can make two different scales this time. One uses all pairs of factors,
and the other uses all triples, and both have 10 notes.

For instance, the one using pairs of factors has
1*3, 1*5, 1*7, 1*11, 3*5, 3*7, 3*11, 5*7, 5*11 and 7*11
as the ten vertices.

They are known as dekanies, because they have ten vertices, and so,
ten notes.

One of them is richer in otonal chords, and the other is richer in
utonal chords.

Now, with the extra factor, one can have otonal or utonal _tetrads_ (four note chords).

The one that you make using three factors at a time is the one rich
in utonal chords, and with utonal tetrads, and the other is especially
good for otonal chords, with otonal tetrads.

This time, three space dimensions aren't enough to show the
shape, as each time you add an extra factor, you need one extra
dimension.

1 3 = linear lattice - cycle of fifths, needs only one dimension
1 3 5 = the 3 5 lattice - needs two dimensions.
1 3 5 7 = needs three dimensions.

Add another, and you need a fourth space dimension at right angles to
the three we are familiar with.

It is _not_ the dimension of time though, just another space dimension, if
one can imagine such a thing, or more likely, fail to imagine it!

There are records of people who have said they can get an inkling of the idea
of a fourth space dimension, and even solve problems in 4D by imagining
the shapes in periods of concentration. But most are happy to view projections
of it, and just rely on the maths to get it right.

What one can do is to make a projection into a smaller number of dimensions.

We do this whenever we draw a 3D figure on a sheet of paper.
In the same way, one can draw a 4D figure in 2D, or indeed, in 3D.

So this is what one does for the dekany.

Here is one projection of each into 3D, using a particularly symmetrical
projection that I adapted from one that Paul did.

Taking the factors two at a time = known as the 2)5 dekany:
http://www.robertwalker.f9.co.uk/dekany/dekany2_1_3_5_7_11.wrl
http://www.robertwalker.f9.co.uk/dekany/dekany2_1_3_5_7_9.wrl

It will take a while to load as all the midi clips need to be transferred,
and it is best to wait for them all to load before you start clicking on them.

-----------------------------------------

Here is a zip with all the dekany files in it for off-line browsing
http://www.robertwalker.f9.co.uk/dekany/dekany.zip [200KB]
278 files Unpacks to 971 Kb.

Because of number of files in it, needs an extra Mb for disk partitions with 4 K clusters,
or up
to an extra 8-9 Mb for 32K clusters (= Fat 16, 1-2 Gb, or Fat 32, 32 Gb or more for
Wndows).
-----------------------------------------

This has otonal tetrads - click on any of the tetrahedra to hear them.

The tetrahedra are squashed because this is a 3D view on a 4D object.

It is a perspective view, and one is looking through the octahedron, which
is the nearest face, towards the inside tetrahedron, which is the furthest
away face in the fourth space dimension.

They are nested neatly within each other because one is looking at it
from directly opposite the outer octahedron in the fourth space dimension.
It is like the method of drawing a cube as two concentric squares.
Makes the 3D shape simpler to look at.

The faces you can see through are the utonal triads, and you will find
red or magenta spheres in the middle of each which you can click to hear the triads.

Taking the factors three at a time = known as the 3)5 dekany:
http://www.robertwalker.f9.co.uk/dekany/dekany3_tetrahedra_triads_1_3_5_7_11.wrl
http://www.robertwalker.f9.co.uk/dekany/dekany3_tetrahedra_triads_1_3_5_7_9.wrl
Click on the transparent tetrahedron outside the vacant faces to hear
the utonal tetrads.

This time, click on the faces to hear the otonal tetrads.

When I do a web page, I'll do these as gifs as well, (maybe do from
several angles for the dekany) for those who can't view the vrml.

Here is the Wilson diagram of a CPS set
http://www.anaphoria.com/dal16.html
and Paul's explanation
http://www.egroups.com/message/tuning/11031

If you have Excell, you can listen to the dekany, with notes fading and getting louder
again depending on distance, and watch as the 3D projections change, using Dave
Keenan's dekany spreadsheet.

N.B. if Monz or anyone else wants to use / adapt my VRML models for their web sites,
they are very welcome - can't remember if I said this before, or just took
it as understood.

It is very easy to change the notes for the vertices (both the text and the notes you hear
for them)
if one wants to use the same model, but for another combination of the factors.

I haven't tried doing a VRML of the complete Eikosany
which is a five dimensional shape made up of all the products of three factors from
a list of six, such as
1, 3, 5, 7, 11, 13
say.

It will have 6*5*4/6 = 20 notes.

However, one could do 3D projections for parts of it:

I've done that for the pentadekany, which consists of pairs of factors from
1, 3, 5, 7, 11, 13 (say)
(5*4/2 = 15 notes)

but not for the Eikosany, which will be a much more complex shape in 3D,
if one wants to be able to show all the tetrads, and pentads.

For the pentadekany, see:
http://www.robertwalker.f9.co.uk/pentadekany_c2-g4_br/Pentadekany.htm

There's another pentadekany which you get by taking the factors four at a
time.

Robert

Hi Lawrence --

I posted, some time ago, a nice introdution to Wilson CPS scales, that could
be considered a "Gentle Introduction". Does anyone have the appropriate
archive link handy?

-Paul

--- In tuning@y..., "Robert Walker" <robert_walker@r...> wrote:

/tuning/topicId_19157.html#19204

> Hi Lawrence.
>
> Joseph Pehrson has suggested I do a web page of this in VRML.
>
> As a first start towards it, maybe I can try explaining the idea of
a hexany to you.
>

This really is incredible work, Robert, and I love the way one can
rotate the hexany around to see it from all sides. Amazing.

I unzipped the Decany .zip file. It seems to view OK, but I can't
hear the MIDI files when viewing offline. Is there something I
should be doing? The files are in the same directory...

I vaguely remember we had some similar trouble before, but I can't
remember the solution!

Thanks again for your magnificent contributions!

________ _________ _______ _
Joseph Pehrson

List!
Wilson desire to have his work speak for itself was the main reason
it is up. I see no reason why it should not be referred to directly
since it is.

"Paul H. Erlich" wrote:

> Hi Lawrence --
>
> I posted, some time ago, a nice introdution to Wilson CPS scales, that
> could
> be considered a "Gentle Introduction". Does anyone have the
> appropriate
> archive link handy?
>
> -Paul

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

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:

/tuning/topicId_19157.html#19206

> Hi Lawrence --
>
> I posted, some time ago, a nice introdution to Wilson CPS scales,
that could be considered a "Gentle Introduction". Does anyone have
the
appropriate archive link handy?
>
> -Paul

One of these posts is archive #10566.

Is that the one you're looking for, Paul??

___________ _______ ______ _____
Joseph Pehrson

Robert!
The Wilson Archives have been converted to PDF files.
http://www.anaphoria.com/dal.PDF

Robert Walker wrote:

> Hi Kraig,
>
> Realised this link no longer works
> http://www.anaphoria.com/dal16.html
>
> What is the updated link?
>
> Robert
>
>
>
>
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-- Kraig Grady
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http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

Hi Kraig,

> The Wilson Archives have been converted to PDF files.
> http://www.anaphoria.com/dal.PDF

Thanks,

I'll be printing this out. I see it has the Penrose Tiling tonescape as well.

Robert

--- In tuning@y..., "Robert Walker" <robert_walker@r...> wrote:

/tuning/topicId_19157.html#19204

> As a first start towards it, maybe I can try explaining the idea of
a hexany to you.
>

This post by Robert Walker is one of the clearest and most
interesting explanations of the CPS that I have ever seen...

___________ ______ ______ _
Joseph Pehrson

Robert Walker wrote,

>The two triangles join together to make the octahedron.

Actually, four pairs of opposite triangles (one major and one minor triad)
join together to make the octahedron.

Each edge direction (there are 6 -- 2 edges in each direction) of the
octahedron corresponds to a particular interval -- in this case, one of the
6 7-limit consonances.

Yes, let's do that!

-----Original Message-----
From: Kraig Grady [mailto:kraiggrady@anaphoria.com]
Sent: Wednesday, February 21, 2001 8:38 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] terms- plea for assistance

List!
Wilson desire to have his work speak for itself was the main reason it
is up. I see no reason why it should not be referred to directly since it
is.

"Paul H. Erlich" wrote:

Hi Lawrence --

I posted, some time ago, a nice introdution to Wilson CPS scales, that could

be considered a "Gentle Introduction". Does anyone have the appropriate
archive link handy?

-Paul

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com <http://www.anaphoria.com>

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

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Hi Paul,

> Actually, four pairs of opposite triangles (one major and one minor triad)
> join together to make the octahedron.

> Each edge direction (there are 6 -- 2 edges in each direction) of the
> octahedron corresponds to a particular interval -- in this case, one of the
> 6 7-limit consonances.

Yes, and I mention the other three pairs of triangles later in the post.

However, I didn't say anything about the edge directions particularly,
or how it fits into the larger <3,5,7> lattice. This is something one
would need to do in a web page about it.

So here is a first attempt at that too.

The hexany is part of the tetrahedron + octahedron lattice, also known
as the face centred cubic, or the fruit stacking lattice.

Each face of the octahedron is one face of a 2D triangular sub-lattice.

So the 1, 3, 5 and 1/1, 1/3, 1/5 faces are in parallel <3,5> lattices.
(intervals 3, 5 and 5/3)

The 1 3 7 and 1/1 1/3 1/7 triangles are in parallel <3,7> lattices
(intervals 3, 7 and 7/3)

1, 5, 7 and 1/1 1/5 1/7 are in parallel <5,7> lattices.
(intervals 5, 7 and 7/5)

Then 3 5 7 and 1/3 1/5 1/7 are in parallel lattices generated by
the intervals 5/3 and 7/3.
(intervals 5/3, 7/3 and 7/5)

I'm not sure how one describes a lattice generated by 5/3 and 7/3,
except to use those words. Is there another way of looking at it that is
better?

The six edge directions are 3, 5, 7, 5/3, 7/3 and 7/5

If you look at a stack of fruit, you will see that each group of three
oranges (say) is positioned above another group of three in the previous
layer in a hexagonal star arrangement, and that is the hexany.

Each group of three has a single orange above and below it, and those
make tetrahedra, which are the tetrads of the lattice.

A nicely stacked pile of fruit makes a pyramid with four sides to it,
and those four sides are the four types of lattice.

The four edges of the pyramid, together with the two edge
directions that make its base, give the six edge directions of
the lattice.

Maybe for the web page, I'll do a vrml of a small stack of "fruit" showing
how this works, with tetrads, triads etc to click on.

If anyone has any comments about this, especially, anything left out that
one should say for a beginner, or anything that's unclear, please go ahead!

Robert

Sorry, got the stack of fruit wrong

It is a pyramid with three sides and a triangular base, so, a tetrahedron,
and the four faces are the four types of lattice, and the six edges of the
pyramid are the six lattice directions.

Robert

Sorry about the confusion about the fruit stacking lattice.

Of course, both descriptionsn are correct.

If you stack fruit, and start with the fruit as close together as
possible in the first layer, so that they make a triangular lattice,
you will end up with a tetrahedron - used for stacking cannon balls
sometimes.

If you stack them with the first layer in a regular square array
instead, you end up with a square based pyramid, or
the familiar stack of fruit with a rectangular base and a
ridge across the top.

Naturally, that one is more convenient for a rectangular fruit
stall!

Both are the same lattice, tilted at slightly different angles.
The tetrahedron has one of the triangular 2D lattices as
its base, and the other one has the square cross section of the
octahedron as its base.

The one used by greengrocers is particularly good for seeing
the octahedra.

Each orange is above a square of oranges in the previous row,
and then there is another orange beneath it too, making the octahedron.

Then each pair of fruit is above another pair in a little four orange
cluster, and those together make the tetrahedra.

Robert