12- & 24-note JI Cartesian Product Scales

I was playing around with the cartesian product scale concept I described
about a month ago in the tuning list and I found a few 12-note and
24-note scales which have particularly appealing structures. I looked for 12- and 24-note scales since obviously they map easily to conventional keyboards and are small enough to be actually playable.

I put up some scala files for those who want to try them out in practice.

http://magnus.smartelectronix.com/scales/

Here is the list of scales I found:

12 notes/oct:

[1,3] x [1,3,5,7,9,11,13]
[1,3,5] x [1,3,5] x ~[1,3,5]
[1,3,5] x [1,3,5,7,11]
[1,3,5,7,9] x ~[1,3,5]
[1,3,5,7] x [1,3,5,9]
[1,3] x [1,3] x [1,3,5] x [1,3,5]

24 notes/oct:

[1,3] x [1,3] x [1,3,5,7,9,11,13,15,17,19]
[1,3,5] x [1,3,5,7] x [1,3,5,7,9]
[1,3,5,7,9,11] x ~[1,3,5,7,9]
[1,3,5,7,9,11] x [1,3,5,7,9,11,15]

This list is not exhaustive and I'm sure there are other nice structures too. All of these scales can be inverted to create (more) utonal versions, but I didn't make scala files for that because I prefer otonalities to utonalities.

- Magnus Jonsson

>I put up some scala files for those who want to try them out in practice.
>
>http://magnus.smartelectronix.com/scales/
>
>Here is the list of scales I found:
>
>12 notes/oct:
>
>[1,3] x [1,3,5,7,9,11,13]
>[1,3,5] x [1,3,5] x ~[1,3,5]
>[1,3,5] x [1,3,5,7,11]
>[1,3,5,7,9] x ~[1,3,5]
>[1,3,5,7] x [1,3,5,9]
>[1,3] x [1,3] x [1,3,5] x [1,3,5]
>
>24 notes/oct:
>
>[1,3] x [1,3] x [1,3,5,7,9,11,13,15,17,19]
>[1,3,5] x [1,3,5,7] x [1,3,5,7,9]
>[1,3,5,7,9,11] x ~[1,3,5,7,9]
>[1,3,5,7,9,11] x [1,3,5,7,9,11,15]

Great! This is one of my favorite approaches to scale-making.

But say, it looks like B=11 and C=13 in your filenames. Is
that right?

-Carl

On Mon, 3 Oct 2005, Carl Lumma wrote:

> But say, it looks like B=11 and C=13 in your filenames. Is
> that right?

Yes, I used letters after I ran out of digits. So A = 10, B = 11, C = 13, etc.

I was playing a bit with the scale 1357x1357 and I noticed it has a very funny property. There are 10 unique pitches total in this scale, and four 1357 chords that have fundamentals 1357. I will assume octave equivalence and avoid all factors of 2.

harmonics
chord 1 3 5 7
chord 3 9 15 21
chord 5 15 25 35
chord 7 21 35 49

Now the interesting part is that modulation from any chord to any chord is possible in stuch as scale and it doesn't offend the ear (at least mine). The way to modulate from chord x to chord y is to emphasize the yth harmonic of chord x and then do the switch. The emphasized note will now be the xth harmonic of the y.

For example, when you modulate from 3 to 5, you emphasize the 5th harmonic, which will then become the third harmonic of the 5.

emph
3 9 15 21
5 15 25 35

Now to modulate from 5 to 7, emphasize 5's 7th harmonic

emph
5 15 25 35
7 21 35 49

Now what if we want to modulate from 7 to 7? Silly question but...
we emphasize the 7th harmonic, and as a result, we get a pitch 49 which can only be harmonized if we stay in the 7 chord.

So as long as we emphasize the xth harmonic of chord x, we more or less force ourselves to stay in the xth chord. Or more generally in a more optimistic viewpoint, emhasizing the xth harmonic gives us an opportunity to modulate to the xth chord.

If we add the next odd limit number, 9, we can use the same strategy to modulate, but also since 9 is a composite number we get additional possibilities too, and we get some more minor triads. We can add 11,13... and at 15 we again get more minor triads to play with.

These scales are more compact than Partch Diamond, and subjectively I think they have a happier sound to them... so they may be a nice complement to the Partch Diamond.

Odd-limit Unique pitches
square diamond
1 1 1
3 3 3
5 6 7
7 10 13
9 14 19
11 20 29
13 27 41
15 33 51
...
31 123 222
...
63 446 862

- Magnus Jonsson

I find happy music depress me, it is depressing music that makes me happy ( just jokin)

Magnus Jonsson wrote:

>
>These scales are more compact than Partch Diamond, and subjectively I >think they have a happier sound to them...
>
>
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

>I was playing a bit with the scale 1357x1357 and I noticed it has a very
>funny property. There are 10 unique pitches total in this scale, and
>four 1357 chords that have fundamentals 1357. I will assume octave
>equivalence and avoid all factors of 2.
>
> harmonics
>chord 1 3 5 7
>chord 3 9 15 21
>chord 5 15 25 35
>chord 7 21 35 49
>
>Now the interesting part is that modulation from any chord to any chord
>is possible in stuch as scale and it doesn't offend the ear (at least
>mine). The way to modulate from chord x to chord y is to emphasize the yth
>harmonic of chord x and then do the switch. The emphasized note will now
>be the xth harmonic of the y.
>
>For example, when you modulate from 3 to 5, you emphasize the 5th
>harmonic, which will then become the third harmonic of the 5.
>
> emph
>3 9 15 21
> 5 15 25 35
>
>Now to modulate from 5 to 7, emphasize 5's 7th harmonic
>
> emph
> 5 15 25 35
> 7 21 35 49

Yup. Fun. Such modulations make of the basis of evangelical
hymn chord progressions, from my point of view.

>These scales are more compact than Partch Diamond, and subjectively I
>think they have a happier sound to them...

Yes, and this was also the conclusion of Jules Siegel, who composed
some excellent music with [1 3 5 7 9 11 13 15] x [1 3 5 7 9 11 13 15].
His alboms were (are still?) available from the JI Network "store".

>so they may be a nice complement to the Partch Diamond.

It's a nice idea to have

[1 3 5 7 9 11] x [1 3 5 7 9 11] x u[1 3 5 7 9 11]

I think.

>Odd-limit Unique pitches
> square diamond
> 1 1 1
> 3 3 3
> 5 6 7
> 7 10 13
> 9 14 19
> 11 20 29
> 13 27 41
> 15 33 51
>...
>31 123 222
>...
>63 446 862

More generally...

square(n) = (pi(n) * (pi(n)+1)) / 2
and
diamond(n) = (pi(n) * (pi(n)-1)) + 1

...where pi(n) is the number of elements in a list of mutually-
prime factors n. If the factors are not mutually prime, you have
to subtract out the resulting doubled pitches.

Erv Wilson has done a great deal of work with both kinds of
scales.

-Carl

>I put up some scala files for those who want to try them out in practice.
>
>http://magnus.smartelectronix.com/scales/
>
>Here is the list of scales I found:
>
>12 notes/oct:
>
>[1,3] x [1,3,5,7,9,11,13] **1
>[1,3,5] x [1,3,5] x ~[1,3,5] **2
>[1,3,5] x [1,3,5,7,11]
>[1,3,5,7,9] x ~[1,3,5]
>[1,3,5,7] x [1,3,5,9]
>[1,3] x [1,3] x [1,3,5] x [1,3,5]

Great stuff. I see that **2 was known ("Major Wing").
**1 is a scale by David Canright, discussed recently
on the other list.

-Carl

there were people at oberlin working with a similar scale. possibly related or the very personage< do you know?

Carl Lumma wrote:

>
>
>Yes, and this was also the conclusion of Jules Siegel, who composed
>some excellent music with [1 3 5 7 9 11 13 15] x [1 3 5 7 9 11 13 15].
>His alboms were (are still?) available from the JI Network "store".
>
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

>>Yes, and this was also the conclusion of Jules Siegel, who composed
>>some excellent music with [1 3 5 7 9 11 13 15] x [1 3 5 7 9 11 13 15].
>>His alboms were (are still?) available from the JI Network "store".
>
>there were people at oberlin working with a similar scale. possibly
>related or the very personage< do you know?

I don't think Bob (Julse) was ever at Oberlin, so I'd like to know
who this was.

-Carl

Also Arnold Dreyblatt! www.dreyblatt.net. It seems he takes it out to
11 limit, which gives you 19 notes: (from
http://www.dreyblatt.net/Music.pdf)

The tuning system used in my music is calculated from the third,
fifth, seventh, ninth and eleventh
overtones and their multiples in the following pattern:
1 3 5 7 9 11
3 9 15 21 27 33
5 15 25 35 45 55
7 21 35 49 63 77
9 27 45 63 81 99
11 33 55 77 99 121

These mathematically related overones are heard as a tonal relation
when they are transposed
and sounded above a fundamental tone. In this process of transposition
from their position
in the natural overtone series, these tones fall in the span of one
octave in the following
order:
1, 33, 35, 9 77, 5, 81, 21, 45, 3, 49, 99, 25, 27, 55, 7, 15, 121, 63, (2)

this was in the mid 80's. the funny thing about it is one of the people was a person by the name of Kraig hill.
one of the few kraigs with a k i have ran accross.
He like the others changed their name , this one to Hougie, don't ask me why.
he was just in the group so don't know who headed it.
another kraig with k changed his name to ovid.
Carl Lumma wrote:

>>>Yes, and this was also the conclusion of Jules Siegel, who composed
>>>some excellent music with [1 3 5 7 9 11 13 15] x [1 3 5 7 9 11 13 15].
>>>His alboms were (are still?) available from the JI Network "store".
>>> >>>
>>there were people at oberlin working with a similar scale. possibly >>related or the very personage< do you know?
>> >>
>
>I don't think Bob (Julse) was ever at Oberlin, so I'd like to know
>who this was.
>
>-Carl
>
>
>
>
> >Yahoo! Groups Links
>
>
>
> >
>
>
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

On Mon, 3 Oct 2005, Carl Lumma wrote:

>> [1,3] x [1,3,5,7,9,11,13] **1
>> [1,3,5] x [1,3,5] x ~[1,3,5] **2
>> [1,3,5] x [1,3,5,7,11]
>> [1,3,5,7,9] x ~[1,3,5]
>> [1,3,5,7] x [1,3,5,9]
>> [1,3] x [1,3] x [1,3,5] x [1,3,5]
>
> Great stuff. I see that **2 was known ("Major Wing").
> **1 is a scale by David Canright, discussed recently
> on the other list.

Alright, added to http://magnus.smartelectronix.com/scales/notes.txt

On Mon, 3 Oct 2005, Carl Lumma wrote:

> Yes, and this was also the conclusion of Jules Siegel, who composed
> some excellent music with [1 3 5 7 9 11 13 15] x [1 3 5 7 9 11 13 15].
> His alboms were (are still?) available from the JI Network "store".

I'll check it out when I find time.

>> so they may be a nice complement to the Partch Diamond.
>
> It's a nice idea to have
>
> [1 3 5 7 9 11] x [1 3 5 7 9 11] x u[1 3 5 7 9 11]
>
> I think.
>

That has 87 pitches.

So, might we call this an 11-odd limit OOU scale? Similarly squares would be OO scales and partch diamonds would be OU scales.

Sizes of various odd-limit OOU scales:

Odd-limit Unique pitches
1 1
3 4
5 12
7 30
9 48
11 87
13 143
15 181

Obviously a lot of pitches, but for flexible pitch instruments that shouldn't matter, and since the structure is regular I'm sure some clever notation could be invented for it. As an example 579 might mean 5*7/9, 3AC could mean 3*11/13.

> square(n) = (pi(n) * (pi(n)+1)) / 2
> and
> diamond(n) = (pi(n) * (pi(n)-1)) + 1

Ah, so it appears squares are 2x smaller than diamonds asymptopically.

> Erv Wilson has done a great deal of work with both kinds of
> scales.

Oh, that's good to know.

- Magnus Jonsson

Minor correction: that should be 20 pitches, 11 is missing in the list.

His music is interesting...

On Tue, 4 Oct 2005, Jacob wrote:

> Also Arnold Dreyblatt! www.dreyblatt.net. It seems he takes it out to
> 11 limit, which gives you 19 notes: (from
> http://www.dreyblatt.net/Music.pdf)
>
> The tuning system used in my music is calculated from the third,
> fifth, seventh, ninth and eleventh
> overtones and their multiples in the following pattern:
> 1 3 5 7 9 11
> 3 9 15 21 27 33
> 5 15 25 35 45 55
> 7 21 35 49 63 77
> 9 27 45 63 81 99
> 11 33 55 77 99 121
>
> These mathematically related overones are heard as a tonal relation
> when they are transposed
> and sounded above a fundamental tone. In this process of transposition
> from their position
> in the natural overtone series, these tones fall in the span of one
> octave in the following
> order:
> 1, 33, 35, 9 77, 5, 81, 21, 45, 3, 49, 99, 25, 27, 55, 7, 15, 121, 63, (2)
>
>
>
>
>
>
>
> Yahoo! Groups Links
>
>
>
>
>
>
>

>>>>Yes, and this was also the conclusion of Jules Siegel, who composed
>>>>some excellent music with [1 3 5 7 9 11 13 15] x [1 3 5 7 9 11 13 15].
>>>>His alboms were (are still?) available from the JI Network "store".
>>>>
>>>>
>>>there were people at oberlin working with a similar scale. possibly
>>>related or the very personage< do you know?
>>
>>I don't think Bob (Jules) was ever at Oberlin, so I'd like to know
>>who this was.
>
>this was in the mid 80's. the funny thing about it is one of the
>people was a person by the name of Kraig hill.
> one of the few kraigs with a k i have ran accross.
> He like the others changed their name, this one to Hougie,
>don't ask me why.
> he was just in the group so don't know who headed it.
> another kraig with k changed his name to ovid.

Let us know if you think of it.

Oh, another person to work with modulating harmonics by harmonics
is, I think, Hans Andre-Stamm.

-Carl

>>> [1,3] x [1,3,5,7,9,11,13] **1
>>> [1,3,5] x [1,3,5] x ~[1,3,5] **2
>>> [1,3,5] x [1,3,5,7,11]
>>> [1,3,5,7,9] x ~[1,3,5]
>>> [1,3,5,7] x [1,3,5,9]
>>> [1,3] x [1,3] x [1,3,5] x [1,3,5]
>>
>> Great stuff. I see that **2 was known ("Major Wing").
>> **1 is a scale by David Canright, discussed recently
>> on the other list.
>
>Alright, added to http://magnus.smartelectronix.com/scales/notes.txt

Why not add these to the comments of the Scala files? I did
it this way:

!
[1 3 5] x [1 3 5] x u[1 3 5] cross set, Magnus Jonsson 2005.
12
!
25/24
9/8
6/5
5/4
4/3
3/2
25/16
8/5
5/3
9/5
15/8
2
!
! Also John Chalmer's "Major Wing".

-Carl

>So, might we call this an 11-odd limit OOU scale? Similarly squares
>would be OO scales and partch diamonds would be OU scales.

Sure, but I also like the [] x u[] notation. Maybe I would write

[1 3 5 7 9 11]^2 x u[1 3 5 7 9 11]

>Sizes of various odd-limit OOU scales:
>
>Odd-limit Unique pitches
>1 1
>3 4
>5 12
>7 30
>9 48
>11 87
>13 143
>15 181
>
>Obviously a lot of pitches, but for flexible pitch instruments that
>shouldn't matter, and since the structure is regular I'm sure some
>clever notation could be invented for it. As an example 579 might mean
>5*7/9, 3AC could mean 3*11/13.

The original way I thought of it was for an organ. 1 manual would
be the 'playing' manual, one manual would be the xU manual, and the
pedal could be xO. With a pair of MIDI keyboards, one could be
the 'playing' keyboard, and the other could be split to do the
modulations.

>> Erv Wilson has done a great deal of work with both kinds of
>> scales.
>
>Oh, that's good to know.

You can see his theory papers at

http://www.anaphoria.com

-Carl

I believe the scalatron in toranto has this capability between two bosanquet keyboards. the way to go.
actually has anyone done this with two 12 tone keyboards?

>
>The original way I thought of it was for an organ. 1 manual would
>be the 'playing' manual, one manual would be the xU manual, and the
>pedal could be xO. With a pair of MIDI keyboards, one could be
>the 'playing' keyboard, and the other could be split to do the
>modulations.
>
> >
>
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

>>The original way I thought of it was for an organ. 1 manual would
>>be the 'playing' manual, one manual would be the xU manual, and the
>>pedal could be xO. With a pair of MIDI keyboards, one could be
>>the 'playing' keyboard, and the other could be split to do the
>>modulations.
>
>I believe the scalatron in toranto has this capability between two
>bosanquet keyboards. the way to go.
> actually has anyone done this with two 12 tone keyboards?

Wendy Carlos did it with the Kurzweil K2000-series synths. Kurzweil
implemented the feature for her, and apparently if you buy one
off the shelf it can do it.

Werner Mohrlok (of Hermode Tuning) has software called Tadi that
can do it.

Kurt Bigler and I implemented a very flexible version of this,
described at

http://lumma.org/music/theory/xmw.txt

on his "digital pipes" 32-channel organ softsynth. Monz, Gene,
and Jonathan Glasier got a demo a few months ago.

And, as I just wrote, maybe Hans Andre-Stamm has something like
this.

-Carl

Sorry, I guess it was Peter Hulen at Wabash:

http://persweb.wabash.edu/facstaff/hulenp/musicwor/paper/paper2.html

--- In MakeMicroMusic@yahoogroups.com, "Paul Erlich" <paul@s...>
wrote:
> Also at Michigan State, the same scale again.
>
> --- In MakeMicroMusic@yahoogroups.com, Kraig Grady
<kraiggrady@a...>
> wrote:
> > there were people at oberlin working with a similar scale.
possibly
> > related or the very personage< do you know?
> >
> > Carl Lumma wrote:
> >
> > >
> > >
> > >Yes, and this was also the conclusion of Jules Siegel, who
composed
> > >some excellent music with [1 3 5 7 9 11 13 15] x [1 3 5 7 9 11
13
> 15].
> > >His alboms were (are still?) available from the JI
Network "store".
> > >
> > >
> > >
> >
> > --
> > Kraig Grady
> > North American Embassy of Anaphoria Island <http://anaphoria.com/>
> > The Wandering Medicine Show
> > KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los
Angeles

>Also Arnold Dreyblatt! www.dreyblatt.net.

Now here's a guy I've never heard of, with quite nice JI music
going back to the '80s. Wow!

>It seems he takes it out to 11 limit, which gives you
>19 notes: (from http://www.dreyblatt.net/Music.pdf)
>
>The tuning system used in my music is calculated from the third,
>fifth, seventh, ninth and eleventh overtones and their multiples
>in the following pattern:
>1 3 5 7 9 11
>3 9 15 21 27 33
>5 15 25 35 45 55
>7 21 35 49 63 77
>9 27 45 63 81 99
>11 33 55 77 99 121

This is 20 notes, not 19.

>These mathematically related overones are heard as a tonal
>relation when they are transposed and sounded above a fundamental
>tone. In this process of transposition from their position in
>the natural overtone series, these tones fall in the span of one
>octave in the following order:
>1 33 35 9 77 5 81 21 45 3 49 99 25 27 55 7 15 121 63 (2)

I checked the pdf, and this is right, he forgot 11!

-Carl

On Tue, 04 Oct 2005, Carl Lumma wrote:
>
> >So, might we call this an 11-odd limit OOU scale? Similarly squares
> >would be OO scales and partch diamonds would be OU scales.
>
> Sure, but I also like the [] x u[] notation. Maybe I would write
>
> [1 3 5 7 9 11]^2 x u[1 3 5 7 9 11]
...

Carl,

Why wouldn't you write:
[1 3 5 7 9 11]^2 / [1 3 5 7 9 11]
instead?

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.344 / Virus Database: 267.11.10/120 - Release Date: 5/10/05

I have no objection... let everyone use the notation they feel is best. I
still prefer u for some obscure reasons.

- Magnus

On Thu, 6 Oct 2005, Yahya Abdal-Aziz wrote:
Ë
>> [1 3 5 7 9 11]^2 x u[1 3 5 7 9 11]
>
> Why wouldn't you write:
> [1 3 5 7 9 11]^2 / [1 3 5 7 9 11]
> instead?

[Non-text portions of this message have been removed]

>> >So, might we call this an 11-odd limit OOU scale? Similarly squares
>> >would be OO scales and partch diamonds would be OU scales.
>>
>> Sure, but I also like the [] x u[] notation. Maybe I would write
>>
>> [1 3 5 7 9 11]^2 x u[1 3 5 7 9 11]
>...
>
>Carl,
>
>Why wouldn't you write:
> [1 3 5 7 9 11]^2 / [1 3 5 7 9 11]
>instead?
>
>Regards,
>Yahya

One could do that. I guess the only objection I could see is that
it looks very much like plain algebra, and thus equal to [1 3 5 7 9 11].
But I was just following Magnus' lead.

-Carl

--- In MakeMicroMusic@yahoogroups.com, "Yahya Abdal-Aziz"
<yahya@m...> wrote:
> On Tue, 04 Oct 2005, Carl Lumma wrote:
> >
> > >So, might we call this an 11-odd limit OOU scale? Similarly
squares
> > >would be OO scales and partch diamonds would be OU scales.
> >
> > Sure, but I also like the [] x u[] notation. Maybe I would write
> >
> > [1 3 5 7 9 11]^2 x u[1 3 5 7 9 11]
> ...
>
> Carl,
>
> Why wouldn't you write:
> [1 3 5 7 9 11]^2 / [1 3 5 7 9 11]
> instead?
>
> Regards,
> Yahya

Yikes. This notation would seem to imply that if you have a scale A
such that

A = [1 3 5 7 9 11]^2 / [1 3 5 7 9 11],

then you can "multiply" both sides by [1 3 5 7 9 11], yielding

A X [1 3 5 7 9 11] = [1 3 5 7 9 11]^2

But of course this is not the case.

Hi all,

On Thu, 06 Oct 2005, Paul Erlich wrote:
>
> --- In MakeMicroMusic@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
> > On Tue, 04 Oct 2005, Carl Lumma wrote:
> > >
> > > >So, might we call this an 11-odd limit OOU scale? Similarly
> > > > squares
> > > >would be OO scales and partch diamonds would be OU scales.
> > >
> > > Sure, but I also like the [] x u[] notation. Maybe I would write
> > >
> > > [1 3 5 7 9 11]^2 x u[1 3 5 7 9 11]
> > ...
> >
> > Carl,
> >
> > Why wouldn't you write:
> > [1 3 5 7 9 11]^2 / [1 3 5 7 9 11]
> > instead?
> >
> > Regards,
> > Yahya
>
> Yikes. This notation would seem to imply that if you have a scale A
> such that
>
> A = [1 3 5 7 9 11]^2 / [1 3 5 7 9 11],
>
> then you can "multiply" both sides by [1 3 5 7 9 11], yielding
>
> A X [1 3 5 7 9 11] = [1 3 5 7 9 11]^2
>
> But of course this is not the case.

Of course not! It's just a notational convenience to
separate the otonal and utonal components, with the
/ sign being a convenient mnemonic for the fact that
the utonal components involve taking reciprocals of
the listed numbers. It's a darn sight clearer and more
compact than writing it out in full, say as:
[1 3 5 7 9 11]^2 x [1/1 1/3 1/5 1/7 1/9 1/11]
which we could also possibly write as:
[1 3 5 7 9 11]^2 x 1/[1 3 5 7 9 11]

Either way, there's no implication that a "multiplicative
inverse" exists, or that we have a "cancellation law".
Just a shorthand.

Shorthand for what? The set of ratios obtained by
multiplying out the ordered sets:
[1 3 5 7 9 11], taken twice
and:
[1/1 1/3 1/5 1/7 1/9 1/11], taken once. That is:
[1 3 5 7 9 11] x [1 3 5 7 9 11] x [1/1 1/3 1/5 1/7 1/9 1/11].

Regards,
Yahya

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