chromablocks

Give me an abelian group G*, where G is a collection of positive
odd integers not exceeding 17 (with 2 <= |G| <= 9). A "block" B
in G* is a finite subset of G* such that elements of U, another
finite subset of G*, cannot be recovered from the factors of any
element of B, with B n U = [null].

Require every element of U be =< 2^( 1/|G| ).

Now find all B such that 5 <= |B| <= 10 and all elements of
the corresponding U are squares of G.

Maybe somebody can tell me what problems I'll encounter by
allowing G to contain composite odds. I still don't understand
why adding a 'factor-backwards' rule won't preserve something
equivalent to the fundamental theorem of arithmetic.

Anyway, so far I'd call these untempered periodicity blocks of
5-10 elements whose chroma are square rationals no bigger than
240 cents. The whole point of the obtuse language is that the
bases for defining JI and "square" are not typical prime or
odd limits.

I would like to extend this regime to further include rank 2
temperaments (again on a group based on potentially non-
consecutive odd- with a factor-backwards rule) whose commas are
all *ratios* of two squares (of this group), and who have chains
of 5-10 terms that give chroma which are all square under the
kernel.

Hopefully this makes some sense.

-Carl

At 02:06 PM 7/24/2006, you wrote:
>Give me an abelian group G*, where G is a collection of positive
>odd integers not exceeding 17 (with 2 <= |G| <= 9). A "block" B
>in G* is a finite subset of G* such that elements of U, another
>finite subset of G*, cannot be recovered from the factors of any
>element of B, with B n U = [null].
>
>Require every element of U be =< 2^( 1/|G| ).
>
>Now find all B such that 5 <= |B| <= 10 and all elements of
>the corresponding U are squares of G.

Whoops, not squares of G, but of the consonances. It isn't
clear to me what formalism to use here, without mixing metaphors
in an unsightly way.

-C.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> At 02:06 PM 7/24/2006, you wrote:
> >Give me an abelian group G*, where G is a collection of positive
> >odd integers not exceeding 17 (with 2 <= |G| <= 9). A "block" B
> >in G* is a finite subset of G* such that elements of U, another
> >finite subset of G*, cannot be recovered from the factors of any
> >element of B, with B n U = [null].
> >
> >Require every element of U be =< 2^( 1/|G| ).
> >
> >Now find all B such that 5 <= |B| <= 10 and all elements of
> >the corresponding U are squares of G.
>
> Whoops, not squares of G, but of the consonances. It isn't
> clear to me what formalism to use here, without mixing metaphors
> in an unsightly way.

I'm not clear what you are saying, so saying it another way wouldn't
hurt. What does B n U mean? What does "not recovered" mean? Are you
just saying the groups they generate are such that the group
generated by U is not contained in the group generated by B, or what?

Why not give some examples, and tell us what the point of it all is?

At 04:29 PM 7/24/2006, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>>
>> At 02:06 PM 7/24/2006, you wrote:
>> >Give me an abelian group G*, where G is a collection of positive
>> >odd integers not exceeding 17 (with 2 <= |G| <= 9). A "block" B
>> >in G* is a finite subset of G* such that elements of U, another
>> >finite subset of G*, cannot be recovered from the factors of any
>> >element of B, with B n U = [null].
>> >
>> >Require every element of U be =< 2^( 1/|G| ).
>> >
>> >Now find all B such that 5 <= |B| <= 10 and all elements of
>> >the corresponding U are squares of G.
>>
>> Whoops, not squares of G, but of the consonances. It isn't
>> clear to me what formalism to use here, without mixing metaphors
>> in an unsightly way.
>
>I'm not clear what you are saying, so saying it another way wouldn't
>hurt. What does B n U mean?

That was supposed to be intersection (no elements in U are in B).

>What does "not recovered" mean?

No element in U is a factors of an element in B.

>Why not give some examples, and tell us what the point of it all is?

I'm trying to describe periodicity blocks without typical
harmonic limits.

-Carl

Hi Carl,

[Gene Ward Smith]
>Why not give some examples, and tell us what the point of it all is?

[Carl Lumma]
> I'm trying to describe periodicity blocks without typical
> harmonic limits.

Do you have an example of this? It might make all the difference.

Regards,
Yahya

>[Gene Ward Smith]
>>Why not give some examples, and tell us what the point of it all is?
>
>[Carl Lumma]
>> I'm trying to describe periodicity blocks without typical
>> harmonic limits.
>
>Do you have an example of this? It might make all the difference.

Gene's cooked up no-fives blocks for Aaron, I think.

-Carl