Trying to pin this terminology down right about mapping matrices

A rather mundane operation: here's a sample mapping matrix: [<1 1 0|,
<0 1 4|]. We know that applying either of these homomorphisms to |4 -4
1> will get you 0.

All of the above are operations we're performing on a 3d vector space,
hence on sets of 3 integers at a time. No ratios or rationals or
limits or anything is involved at this point.

Now, on top of that layer, you have a subgroup identifier like [2 3
5], which specifies a way for you to map every one of these sets of 3
integers into the rationals. In this case, if your identifier is [2 3
5], you get meantone, but if it's [2 3 24/5], you get mavila.

Graham suggested that this operation is a monomorphism from sets of 3
integers into the rationals. I suggested also that it's an isomorphism
from sets of 3 integers onto the subgroup of the rationals containing
only the factors present in the mapping, which is probably the same
thing.

Is the above right? That the mapping matrix represents some sort of
group homomorphism from sets of integers into the rationals? What's
the terminologically correct way to precisely state this? To pin this
down strictly would help a lot with the library I'm trying to write.

Thanks,
Mike

Mike Battaglia <battaglia01@gmail.com> wrote:

> Is the above right? That the mapping matrix represents
> some sort of group homomorphism from sets of integers
> into the rationals? What's the terminologically correct
> way to precisely state this? To pin this down strictly
> would help a lot with the library I'm trying to write.

The mapping matrix for a regular temperament defines an
epimorphism, give or take niggles about the definitions of
terms.

Graham

On Tue, Sep 13, 2011 at 8:24 AM, Graham Breed <gbreed@gmail.com> wrote:
>
> Mike Battaglia <battaglia01@gmail.com> wrote:
>
> > Is the above right? That the mapping matrix represents
> > some sort of group homomorphism from sets of integers
> > into the rationals? What's the terminologically correct
> > way to precisely state this? To pin this down strictly
> > would help a lot with the library I'm trying to write.
>
> The mapping matrix for a regular temperament defines an
> epimorphism, give or take niggles about the definitions of
> terms.

I shouldn't have said "mapping matrix" above. I was talking
specifically about the [2 3 5] thing, which as far as I know has no
name.

I specifically want to know what it's called when [2 3 5] transforms
|-4 4 -1> into 81/80. Sounds like it's a second homomorphism.

-Mike

On Tue, Sep 13, 2011 at 8:27 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> I shouldn't have said "mapping matrix" above. I was talking
> specifically about the [2 3 5] thing, which as far as I know has no
> name.
>
> I specifically want to know what it's called when [2 3 5] transforms
> |-4 4 -1> into 81/80. Sounds like it's a second homomorphism.

Gene posted on Facebook that the [2 3 5] thing, which has no name,
actually does have a name, and that that name is "icon." I also
inferred an answer to my original question from his reply, which is
that the icon represents a homomorphism from JI onto the abstract
regular temperament. It's the same situation I ran into with vals.

So if I understand correctly then, everything we were saying above is
wrong, because every element in the tempered space doesn't map to a
unique element in JI. The problem isn't that some rationals aren't
covered, it's that each element maps to multiple rationals. For
example, given the val <12 19 28| and the icon [2 3 5], 4\12 maps onto
all of 5/4, 81/64, 32/25. So an icon isn't a monomorphism from sets of
integers into the rationals.

However, we can say that an icon is a homomorphism from sets of
integers to groups of equivalence classes in JI, the elements of which
differ by the kernel of the homomorphism, which is the first
isomorphism theorem. But it's probably easier to do it the other way
around.

-Mike

Mike wrote:

>> I specifically want to know what it's called when [2 3 5] transforms
>> |-4 4 -1> into 81/80. Sounds like it's a second homomorphism.
>
>Gene posted on Facebook that the [2 3 5] thing, which has no name,
>actually does have a name, and that that name is "icon."

[2 3 5] is a basis. An icon is what Graham calls a mapping.

>So an icon isn't a monomorphism from sets of
>integers into the rationals.

Hence the problem recovering JI from tempered scores.

>However, we can say that an icon is a homomorphism

A morphism is a relationship, not a thing. You can say
that a mapping describes a morphism.

-Carl

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Mike Battaglia <battaglia01@...> wrote:
>
> > Is the above right? That the mapping matrix represents
> > some sort of group homomorphism from sets of integers
> > into the rationals? What's the terminologically correct
> > way to precisely state this? To pin this down strictly
> > would help a lot with the library I'm trying to write.
>
> The mapping matrix for a regular temperament defines an
> epimorphism, give or take niggles about the definitions of
> terms.

This is what I called an "icon".

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> I shouldn't have said "mapping matrix" above. I was talking
> specifically about the [2 3 5] thing, which as far as I know has no
> name.

You could always call it a list of basis elements.

On Tue, Sep 13, 2011 at 1:41 PM, Carl Lumma <carl@lumma.org> wrote:
>
> Mike wrote:
>
> >> I specifically want to know what it's called when [2 3 5] transforms
> >> |-4 4 -1> into 81/80. Sounds like it's a second homomorphism.
> >
> >Gene posted on Facebook that the [2 3 5] thing, which has no name,
> >actually does have a name, and that that name is "icon."
>
> [2 3 5] is a basis. An icon is what Graham calls a mapping.

Gene's telling me in XA chat that an icon is the name for the actual
homomorphic mapping that maps JI onto some abstract regular
temperament. A mapping or a wedgie or a comma kernel are all ways of
notating a particular icon.

> >So an icon isn't a monomorphism from sets of
> >integers into the rationals.
>
> Hence the problem recovering JI from tempered scores.

I think you're talking about the fact that temperaments, when viewed
as homomorphisms, from JI aren't injective. What I'm talking about is
more basic and probably more useless, which is how to describe what
happens when you apply the [2, 3, 5] basis to [<1 0 0], <0 1 0], <0 0
1]>, hence mapping JI onto something like |-3 1 1>.

> >However, we can say that an icon is a homomorphism
>
> A morphism is a relationship, not a thing. You can say
> that a mapping describes a morphism.

That's not consistent with how Gene describes it here:
http://lumma.org/tuning/gws/intval.html

"A val is a homomorphism of Np to the integers Z."

Although it was said offlist that a val describes an icon too.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> That's not consistent with how Gene describes it here:
> http://lumma.org/tuning/gws/intval.html
>
> "A val is a homomorphism of Np to the integers Z."
>
> Although it was said offlist that a val describes an icon too.

If the integers denote notes of a rank one temperament then the val is an icon.