Re: [tuning] The CPS blues...[combination product sets]

Joseph!
Not my best written work and not well edited by another which kind of kills the whole
paper.
the terms like 3-7 mean 3 times 7 (3 x 7) if we take the hexany formed by 3 5 7 11 we need to
first find the combinations of 2 out of 4 cause that's what a hexany is!
we have ;
3x5
3x7
3x11
5x7
5x11
7x11.
now each one of these is a specific pitch if you have to (which I think is not simpler but
might make it clearer) factor these out where 3x5=15 3x7=21 etc. now if we take for instance
the 3 notes that all contain 3, 3x5 3x7 3x11 we should be able to see that the ratio between
the first two 3x5/3x7 = 10(5)/7. the 3 cancels out. from this you should be able to see that
the three mentioned above form a 5 7 11 triad (once again because the 3s cancel out. Where the
lattice helps is seeing the subharmonic triads not it so happens that the sub 5 7 11 is formed
by the opposite 3 tones 5x7 5x11 7x11 which illustrated what i showed peter the other day and
what Wolf was referring too. There are 8 triads with these 6 tones, 4 sets of opposite triads.
if you look at all of them you might notice that each tone will have a unique relationship to
the whole- where as the 3x5 will function in different triads as a 3 in one and 5 in another,
it will also function subharmonically as sub 7 in one and sub 11 in another. It represents one
of the tightest structures you can have next to a diamond. Here though any tone could be the
tonic or you can avoid a tonality altogether depending how you use it. Still the structure is
quite consonant as a whole. I don't know if this answers all your questions or not! please
fire away!

Joseph Pehrson wrote:

> I want to thank Kraig Grady and Paul Erlich for all the nice CPS
> mandalas I've been gazing at...
>
> I'm lost.
>
> I also read Kraig's 1/1 article from several years ago, with ROTATING
> CPSes... kind of like the way Stravinsky ROTATED hexachords when he
> wrote his "Variations, Aldous Huxley in Memoriam.."
>
> What I'm not getting is this... it looks like the terms 3-7, 7-11
> (there's a good one) are refering to LIMITS (??) not really ratios.
>
> These are then "rotated" and SOMEHOW create audible ratios. I don't
> have the foggiest idea how this is done. Are things just multiplied by
> 2 in some cases... Dunno.
>
> THEN we have the question of Scala. I know that Scala can create CPS
> scales rather "easily" if one knows what one is doing (ahem).
>
> There is the "Number of factors" (What is that?? the "limit" of the
> lattice??), and there is the "combination count"-- the "number of
> factors in each product." Huh??
>
> I'm not getting very far with this.
>
> Finally, I'm totally confused about the difference between the nice
> mandalas I've been staring at (no artificial stimulants, I promise you),
> and, supposedly, SQUARE (well, rather, RECTANGULAR) lattices produced
> somehow from the "Euler-Fokker" genera. What??
>
> Does this mean that there is the addition of ANOTHER FACTOR, a 1, that
> is giving another dimension and turning a triangle into a SQUARE
> lattice?
>
> Who wrote the "CPS for dummies" book?? Where is it??
>
> I'm singing the CPS blues....
> ________________ ________ ___ __ _
> Joseph Pehrson
>
>

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

Joseph -- I did promise a "Gentle Introduction to CPS scales" . . . but
first I want you to understand my last lattice post -- it's really not that
hard.

>Finally, I'm totally confused about the difference between the nice
>mandalas I've been staring at (no artificial stimulants, I promise you),
>and, supposedly, SQUARE (well, rather, RECTANGULAR) lattices produced
>somehow from the "Euler-Fokker" genera. What??

>Does this mean that there is the addition of ANOTHER FACTOR, a 1, that
>is giving another dimension and turning a triangle into a SQUARE
>lattice?

No. You can either use a triangular lattice for everything, or a square
lattice for everything.

For example, the hexany I drew:

,a#
,'/ \`.
C'-/---\-`G
|\/ \/| (triangular)
|/\ /\|
f#-------,c#
`.\ /,'
`Eb

in the rectangular lattice would look like this:

,a#
,' |
C'---|----G
| |
| | (rectangular)
| |
f#-------,c#
| ,'
Eb

And the Euler-Fokker genus, usually portrayed like

,a#-------,e#
,' | ,' |
C'---|----G' |
| | | |
| | | | (rectangular)
| | | |
| ,f#-------,c#
| ,' | ,'
Ab--------Eb

becomes, in the triangular lattice,

,a#-------,e#
,'/ \`. ,'/
C'-/---\-`G' /
/|\/ \/| / (triangular)
/ |/\ /\|/
/ ,f#-------,c#
/,' `.\ /,'
Ab-------`Eb

The difference is that the triangular lattice shows all the consonant
intervals explicitly:

3/2: 5/4: 5/3:
5 5
/ \
2---------3 / \
/ \
/ \
4 3

7/4: 7/5: 7/6:

,7 5 7.
,' | `.
4' | `6
7

while the rectangular lattice only shows those consonant intervals where one
side of the ratio is a power of 2:

3/2: 5/4: 7/4:

5
| ,7
2---------3 | ,'
| 4'
|
|
4

Hope this helps!

Joseph Pehrson wrote,

>Finally, I'm totally confused about the difference between the nice
>mandalas I've been staring at (no artificial stimulants, I promise you)

Strangely, John Chalmers states in the Tuning Dictionary that "Mandala" is
synonymous with "Stellated Hexany". If you take the hexany

,A#
,'/ \`.
C'-/---\-`G
|\/ \/|
|/\ /\|
F#-------,C#
`.\ /,'
`Eb

and complete the tetrad implied by each of the 8 triads, you get the 14-tone
Stellated Hexany or "The Mandala (?)":

E
/|\
/ | \
D#-------,A#-------,E#
\`. /,'/|\`.\ ,'/
\ `C'-/-|-\-`G' /
\/|\/.,Dx,\/|\/
/\|/,'Bbb`.\|/\
/ ,F#-------`C# \
/,' \`.\|/,'/ `.\
Ab-----\-`Eb-/-----`Bb
\ | /
\|/
A

whose 14 notes participate in a whopping 36 consonant intervals (hard to
beat in 7-limit JI). Incidentally, the Euler-Fokker genus mentioned earlier
can be found in this structure, as well as 3 other structures obtained by
rotating the genus in the triangular lattice.

Note that the interval between A and E above is not counted as a consonant
2:3 but rather as a dissonant 125:84 (in just intonation, 688¢).

Paul and Joseph!

the mandala can be seen at below link

http://www.anaphoria.com/dal06.html

"Paul H. Erlich" wrote:

> Joseph Pehrson wrote,
>
> >Finally, I'm totally confused about the difference between the nice
> >mandalas I've been staring at (no artificial stimulants, I promise you)
>
> Strangely, John Chalmers states in the Tuning Dictionary that "Mandala" is
> synonymous with "Stellated Hexany". If you take the hexany
>
> ,A#
> ,'/ \`.
> C'-/---\-`G
> |\/ \/|
> |/\ /\|
> F#-------,C#
> `.\ /,'
> `Eb
>
> and complete the tetrad implied by each of the 8 triads, you get the 14-tone
> Stellated Hexany or "The Mandala (?)":
>
> E
> /|\
> / | \
> D#-------,A#-------,E#
> \`. /,'/|\`.\ ,'/
> \ `C'-/-|-\-`G' /
> \/|\/.,Dx,\/|\/
> /\|/,'Bbb`.\|/\
> / ,F#-------`C# \
> /,' \`.\|/,'/ `.\
> Ab-----\-`Eb-/-----`Bb
> \ | /
> \|/
> A
>
> whose 14 notes participate in a whopping 36 consonant intervals (hard to
> beat in 7-limit JI). Incidentally, the Euler-Fokker genus mentioned earlier
> can be found in this structure, as well as 3 other structures obtained by
> rotating the genus in the triangular lattice.
>
> Note that the interval between A and E above is not counted as a consonant
> 2:3 but rather as a dissonant 125:84 (in just intonation, 688�).
>
> ------------------------------------------------------------------------
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-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

I wrote,

>and complete the tetrad implied by each of the 8 triads, you get
>the 14-tone Stellated Hexany or "The Mandala (?)":

> E
> /|\
> / | \
> D#-------,A#-------,E#
> \`. /,'/|\`.\ ,'/
> \ `C'-/-|-\-`G' /
> \/|\/.,Dx,\/|\/
> /\|/,'Bbb`.\|/\
> / ,F#-------`C# \
> /,' \`.\|/,'/ `.\
> Ab-----\-`Eb-/-----`Bb
> \ | /
> \|/
> A

>whose 14 notes participate in a whopping 36 consonant intervals (hard to
beat in 7-limit JI).

You can beat it with the good ol' 7-limit tonality diamond; it also has 36
consonant intervals, and also has 8 complete 7-limit consonant tetrads, but
has only 13 notes:

E---------B
/|\`. ,'/|\
/ | \ `Db / | \
/ ,A#-------,E# \
/,' \`.\|/.'/ `.\
C'-----\-`G'-/-----`D
\`. /,\/|\/.\ ,'/
\ Bbb-/\|/\--Fb /
\ | / ,C# \ | /
\|/,' `.\|/
Eb-------`Bb

These two scales are very similar in the number of consonant relationships
but very different in their interrelationships. In the mandala, there is no
central note (Dx sticks way out of the page, and Bbb sticks way into the
page). Each of the tetrads has an opposing tetrad with no notes in common.
In the diamond, there is a central note (in this case, G) which is consonant
with all other notes. Each of the tetrads has an opposing tetrad with 1 note
(the central note) in common.