A uniform way of denoting temperaments

The idea is to denote temperaments in a way which includes both full p-limit groups and subgroups, treating them in the same way, and explicitly giving generators, commas and mappings. In a nutshell, the idea is to denote a temperament by giving generator transversals, in a specific order, followed by commas, also listed in a specific order. This has the advantage of explicitness, so there's no more arguing about what the generators or mappings should look like. It has the disadvantage of being noncanonical, so that it can't be decided if two temperments are the same without further work. The format I'm suggesting, for which I have not provided a name as yet, is [generator list; comma list].

For example, consider [2, 9/8; 81/80, 99/98, 126/125]. Finding the corresponding normal interval list yields 2.9.5.7.11 as the subgroup in canonical form. Note that there is only one subgroup at issue, 2.9.5.7.11, not a multitude. Converting the generators-commas list to a matrix with rows of monzos yields a matrix for which the pseudoinverse can be found and transposed, yielding an extended mapping matrix with fractional val rows:

[<1 3/2 2 2 2|, <0 1/2 2 5 9|, <0 0 -1 -3 -6|, <0 0 0 0 1|, <0 0 0 1 2|]

Since there are two generators, the first two rows of this matrix gives the actual mapping; the others map the commas. The inverse transpose of the mapping matrix gives the generators-commas list back.

Applying the mapping matrix, we can test if an interval belongs to the subgroup or not. Hence it would be possible to obtain, for example, least squares values for the temperament for that part of a tonality diamond which belongs to the subgroup.

On Tue, Jun 12, 2012 at 2:31 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> For example, consider [2, 9/8; 81/80, 99/98, 126/125]. Finding the
> corresponding normal interval list yields 2.9.5.7.11 as the subgroup in
> canonical form. Note that there is only one subgroup at issue, 2.9.5.7.11,
> not a multitude.

OK, how does your method handle the 2.3.7/5 subgroup, for instance?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Jun 12, 2012 at 2:31 PM, genewardsmith <genewardsmith@...>
> wrote:
> >
> > For example, consider [2, 9/8; 81/80, 99/98, 126/125]. Finding the
> > corresponding normal interval list yields 2.9.5.7.11 as the subgroup in
> > canonical form. Note that there is only one subgroup at issue, 2.9.5.7.11,
> > not a multitude.
>
> OK, how does your method handle the 2.3.7/5 subgroup, for instance?
>
> -Mike

Denoted [2, 3, 7/5;]. Mapping matrix is
[<1 0 0 0|, <0 1 0 0|, <0 0 -1/2 1/2|]

On Tue, Jun 12, 2012 at 5:46 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> Denoted [2, 3, 7/5;]. Mapping matrix is
> [<1 0 0 0|, <0 1 0 0|, <0 0 -1/2 1/2|]

OK, so what I'm saying is that this matrix does indeed have the
property that monzos of the form a*|1 0 0 0> + b*|0 1 0 0> + c*|0 0 -1
1> will map to |a b c>, so it could be said to represent the 2.3.7/5
subgroup in that way.

The only thing is that there are an infinite amount of other subgroups
of the form x.y.z which also have the property that a*x + b*y + c*z
maps to |a b c>, in addition to 2.3.7/5.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Jun 12, 2012 at 5:46 PM, genewardsmith <genewardsmith@...>
> wrote:
> >
> > Denoted [2, 3, 7/5;]. Mapping matrix is
> > [<1 0 0 0|, <0 1 0 0|, <0 0 -1/2 1/2|]
>
> OK, so what I'm saying is that this matrix does indeed have the
> property that monzos of the form a*|1 0 0 0> + b*|0 1 0 0> + c*|0 0 -1
> 1> will map to |a b c>, so it could be said to represent the 2.3.7/5
> subgroup in that way.
>
> The only thing is that there are an infinite amount of other subgroups
> of the form x.y.z which also have the property that a*x + b*y + c*z
> maps to |a b c>, in addition to 2.3.7/5.

OK. 7/5 maps to a triple of integers, but so does 49. Describing this in terms of equivalence classes of subgroups I think is confusing. The set of 7-limit intervals mapped to a triple of integers is in fact a group, a subgroup of the 7-limit. You could call it 2.3.25.35. None of this, so far as I can see, does any damage to the system of temperament representation I described.

On Wed, Jun 13, 2012 at 12:12 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> OK. 7/5 maps to a triple of integers, but so does 49. Describing this in
> terms of equivalence classes of subgroups I think is confusing. The set of
> 7-limit intervals mapped to a triple of integers is in fact a group, a
> subgroup of the 7-limit. You could call it 2.3.25.35. None of this, so far
> as I can see, does any damage to the system of temperament representation I
> described.

The reason I was saying it's an equivalence class of subgroups instead
of a larger subgroup is because intervals in the 2.3.25.35 subgroup -
a 4D subgroup - go down a dimension to -triples- of integers.
Furthermore, 35/1 maps to |0 0 0>.

Contrast this with the matrix [<1 0 0 0|, <0 1 0 0|, <0 0 0.5 0|, <0 0
0.5 0.5|], in which 2.3.25.35 also maps to a set of integers, this
time a quadruple.

I'm not sure if there's any damage; I'm just trying to figure out what
significance you place on the group that maps to a tuple of integers
under the matrix in question.

-Mike

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

> I'm not sure if there's any damage; I'm just trying to figure out what
> significance you place on the group that maps to a tuple of integers
> under the matrix in question.

It's a group which contains the group I'm really interested in, which is the group generated by the generators-commas list. That you can obtain from the extended mapping via the pseudoinverse.

On Wed, Jun 13, 2012 at 3:29 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
>
> > I'm not sure if there's any damage; I'm just trying to figure out what
> > significance you place on the group that maps to a tuple of integers
> > under the matrix in question.
>
> It's a group which contains the group I'm really interested in, which is
> the group generated by the generators-commas list. That you can obtain from
> the extended mapping via the pseudoinverse.

It occurred to me, given my recent post about the pseudoinverse, that
the pseudoinverse of a subgroup matrix is a mapping matrix in which
the kernel of the mapping is the complementary subgroup to the
original subgroup matrix. That is, if you have a JI group G and
subgroup matrix S, and S° is the pseudoinverse of S, then ker(S°) =
complement(S).

This means that

1) for any subgroup matrix, there is exactly one unique mapping matrix
which tempers out the subgroup's complement, and for which the
subgroup generators are a transversal of the mapping generators
2) for any mapping matrix, there is exactly one subgroup matrix whose
complement is tempered out by the mapping, and for which the subgroup
matrix generators are a transversal of the mapping generators
3) for any subgroup matrix, there is exactly one mapping matrix for
which the set of all vals NOT supporting the mapping in question are
also the set of all vals which are projected to the zero sval on the
subgroup
4) for any mapping matrix, there is exactly one subgroup matrix which
projects the set of all vals not supported by the mapping to the zero
sval on teh subgroup

and in all these cases, the unique dual object is the pseudoinverse of
the matrix in question. I'm not sure what use this might have yet, but
it's certainly interesting.

Keep in mind again that I'm treating "subgroup matrices" as matrices
with linearly independent columns representing monzos.

-Mike