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

--- In tuning@yahoogroups.com, Magnus Jonsson <magnus@s...> wrote:
>
> 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.

Does the tilde mean inversion?

I think it would make more sense to simply call these "products" and
drop the "cartesian", by the way.

> [1,3,5] x [1,3,5] x ~[1,3,5]

This is John Chalmer's "Major Wing" scale if, as I presume, I am
interpreting the "~" correctly. I'll see what else comes up with
these; this one is nice if constant structures and so forth are not a
concern.

On Mon, 3 Oct 2005, Gene Ward Smith wrote:

> Does the tilde mean inversion?

Yes. ~[x,y,...,z] = [1/x,1/y,...,1/z]

> I think it would make more sense to simply call these "products" and
> drop the "cartesian", by the way.

That is fine by me, although I think it has been called cartesian in the past by others.

>> [1,3,5] x [1,3,5] x ~[1,3,5]
>
> This is John Chalmer's "Major Wing" scale if, as I presume, I am
> interpreting the "~" correctly. I'll see what else comes up with
> these; this one is nice if constant structures and so forth are not a
> concern.

Cool. I added a notes.txt into my scales directory with this information.

--- In tuning@yahoogroups.com, Magnus Jonsson <magnus@s...> wrote:
>
> On Mon, 3 Oct 2005, Gene Ward Smith wrote:
>
> > Does the tilde mean inversion?
>
> Yes. ~[x,y,...,z] = [1/x,1/y,...,1/z]
>
> > I think it would make more sense to simply call these "products" and
> > drop the "cartesian", by the way.
>
> That is fine by me, although I think it has been called cartesian in
the
> past by others.

Simply calling these "products" is ambiguous and could lead to
confusion, couldn't it? Does everyone know there's one and only one way
to multiply sets? "Cartesian product" or something similar seems
necessary in order to be clear.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> Simply calling these "products" is ambiguous and could lead to
> confusion, couldn't it? Does everyone know there's one and only one way
> to multiply sets? "Cartesian product" or something similar seems
> necessary in order to be clear.

The trouble with calling it a Cartesian product is that it isn't a
Cartesian product, and since that is a math term with a very
well-established meaning, it seems to me best not to mess with it.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > Simply calling these "products" is ambiguous and could lead to
> > confusion, couldn't it? Does everyone know there's one and only one
way
> > to multiply sets? "Cartesian product" or something similar seems
> > necessary in order to be clear.
>
> The trouble with calling it a Cartesian product is that it isn't a
> Cartesian product, and since that is a math term with a very
> well-established meaning, it seems to me best not to mess with it.

What's a better term, then, for the set of all possible pairwise
products of one element from set 1 with one element from set 2?

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> What's a better term, then, for the set of all possible pairwise
> products of one element from set 1 with one element from set 2?

I think I'll ask on sci.math if someone knows of a term for this.
However, since "combination product set" is already in use here, what
about calling it the combination product?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > What's a better term, then, for the set of all possible pairwise
> > products of one element from set 1 with one element from set 2?
>
> I think I'll ask on sci.math if someone knows of a term for this.
> However, since "combination product set" is already in use here, what
> about calling it the combination product?

() That's when there's only 1 set, not 2.

() And even if the 2 sets are exactly the same, the result will still
be different from the relevant combination product set, since the
latter won't include the product of an element with itself. For
example, the 2)4 {1,3,5,7} CPS has six notes, while {1,3,5,7} X
{1,3,5,7} has ten.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> > I think I'll ask on sci.math if someone knows of a term for this.
> > However, since "combination product set" is already in use here, what
> > about calling it the combination product?
>
> () That's when there's only 1 set, not 2.

I've gotten two replies already on sci.math, both of which amount to
suggesting one should simply define, in the course of one's paper,
something like "A*B" for sets of numbers A and B. Not very encouraging
for the hope that there's already a term for it.