Eigenmonzo subgroups

On a Xenharmonic Alliance thread on Facebook, I had occasion to mention that the minimax tuning of a regular temperament mixes JI and tempered intervals. It occurs to me that for a rank r temperament in the q odd limit, the corresponding rank r minimax JI subgroup is interesting to consider. Here are a few examples.

126/125 7-limit: 2.5.7/3
126/125 9-limit: 2.3.7

225/224 7-limit: 2.5/3.7
225/224 9-limit: 2.9/5.7

1029/1024 7-limit: 2.5/3.7/3
1029/1024 9-limit: 2.9/5.9/7

2401/2400 7-limit: 2.3.5
2401/2400 11-limit: 2.3.5

{225/224, 385/384} 11-limit: 2.9/5.11/9

385/384 11-limit: 2.3.7/5.11/5

676/675 13-limit: 2.9/5.7.11.13
676/675 15-limit: 2.5/3.7.11.13

On Wed, May 18, 2011 at 2:44 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> On a Xenharmonic Alliance thread on Facebook, I had occasion to mention that the minimax tuning of a regular temperament mixes JI and tempered intervals. It occurs to me that for a rank r temperament in the q odd limit, the corresponding rank r minimax JI subgroup is interesting to consider. Here are a few examples.

So for 81/80 5-limit, if I understand what you're doing correctly, the
rank-2 minimax JI subgroup would be 2.5?

-Mike

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

> So for 81/80 5-limit, if I understand what you're doing correctly, the
> rank-2 minimax JI subgroup would be 2.5?

Right!

81/80 5-limit subgroup: 2.5

{81/80, 126/125} 7-limit subgroup: 2.5
{81/80, 126/125} 9-limit subgroup: 2.5

{81/80, 126/125, 385/384} 11-limit subgroup: 2.5

{81/80, 126/125, 99/98} 11-limit subgroup: 2.11/9

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
>
> > So for 81/80 5-limit, if I understand what you're doing correctly, the
> > rank-2 minimax JI subgroup would be 2.5?
>
> Right!
>
> 81/80 5-limit subgroup: 2.5
>
> {81/80, 126/125} 7-limit subgroup: 2.5
> {81/80, 126/125} 9-limit subgroup: 2.5
>
> {81/80, 126/125, 385/384} 11-limit subgroup: 2.5
>
> {81/80, 126/125, 99/98} 11-limit subgroup: 2.11/9
>

Interesting. BTW, I really like the Xenharmoic Wiki. Would someone
possibly be so kind as to show why 2.5 (for example, which I
take to mean 2 * 5 and not 2 1/2) is the rank-2 minimax JI subgroup
here; that is the Eigenmonzo in question (81/80 5-limit subgroup 2.5)
It's some kind of invariant in a matrix?

PGH

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:

> Interesting. BTW, I really like the Xenharmoic Wiki. Would someone
> possibly be so kind as to show why 2.5 (for example, which I
> take to mean 2 * 5 and not 2 1/2) is the rank-2 minimax JI subgroup
> here; that is the Eigenmonzo in question (81/80 5-limit subgroup 2.5)
> It's some kind of invariant in a matrix?

2.5 means {2, 5} are generators for the JI subgroup. If you look at 1/4 comma meantone, the 3 is flattened, and if you extend to the 7-limit the 7 is flattened also; but the 5 is just. The group generated by the just 2 and the just 5 is denoted "2.5". This accords with the usage of Graham's temperament finder.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
> --- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@> wrote:
>
> > Interesting. BTW, I really like the Xenharmoic Wiki. Would someone
> > possibly be so kind as to show why 2.5 (for example, which I
> > take to mean 2 * 5 and not 2 1/2) is the rank-2 minimax JI subgroup
> > here; that is the Eigenmonzo in question (81/80 5-limit subgroup 2.5)
> > It's some kind of invariant in a matrix?
>
> 2.5 means {2, 5} are generators for the JI subgroup. If you look at 1/4 comma meantone, the 3 is flattened, and if you extend to the 7-limit the 7 is flattened also; but the 5 is just. The group generated by the just 2 and the just 5 is denoted "2.5". This accords with the usage of Graham's temperament finder.

Thanks. I see, it's kind of a kernel, so to speak. Still not sure though what this has to do with your fractional monzos, which I guess
would make more sense to me when I study subgroups WITH fractions in their monzos...I guess it is what it is....

PGH

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:

> Thanks. I see, it's kind of a kernel, so to speak. Still not sure though what this has to do with your fractional monzos, which I guess
> would make more sense to me when I study subgroups WITH fractions in their monzos...I guess it is what it is....

If you compute the matrix with rows which are fractional monzos, and whose left eigenvectors with eigenvalue 1 are generated by generators of the eigenmonzo subgroup, and those with eigenvalue 0 are generated by generators of the comma subgroup, you end up with a projection matrix which projects JI onto the minimax tuning.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
> --- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@> wrote:
>
> > Thanks. I see, it's kind of a kernel, so to speak. Still not sure though what this has to do with your fractional monzos, which I guess
> > would make more sense to me when I study subgroups WITH fractions in their monzos...I guess it is what it is....
>
> If you compute the matrix with rows which are fractional monzos, and whose left eigenvectors with eigenvalue 1 are generated by generators of the eigenmonzo subgroup, and those with eigenvalue 0 are generated by generators of the comma subgroup, you end up with a projection matrix which projects JI onto the minimax tuning.
>

I see, are there any examples on xenharmonic wiki? I have read the article on eigenmonzos
I'd like to find a quick calculation to associate the minimax tuning with all this, for example
1/4 comma meantone for 81/80. I"m fuzzy on the distinction between eigenmonzo and comma subgroups, and which one(s) you use to make the projection matrix....

PGH

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:

> I see, are there any examples on xenharmonic wiki?

Tons of examples, in the form of all those matrices for "minimax tuning".

I have read the article on eigenmonzos
> I'd like to find a quick calculation to associate the minimax tuning with all this, for example
> 1/4 comma meantone for 81/80.

For 1/4 comma meantone, the eigenmonzo subgroup is generated by 2 and 5, and the comma subgroup by 81/80. Hence you just solve for the matrix which sends |1 0 0> and |0 0 1> to themselves and |-4 4 -1> to |0 0 0>, where I make these row vectors and multiply with them on the left.

> I"m fuzzy on the distinction between eigenmonzo and comma subgroups, and which one(s) you use to make the projection matrix....

You need both. One has eigenvalues of 1, the other eigenvalues of 0.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
> --- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@> wrote:
>
> > I see, are there any examples on xenharmonic wiki?
>
> Tons of examples, in the form of all those matrices for "minimax tuning".
>
> I have read the article on eigenmonzos
> > I'd like to find a quick calculation to associate the minimax tuning with all this, for example
> > 1/4 comma meantone for 81/80.
>
> For 1/4 comma meantone, the eigenmonzo subgroup is generated by 2 and 5, and the comma subgroup by 81/80. Hence you just solve for the matrix which sends |1 0 0> and |0 0 1> to themselves and |-4 4 -1> to |0 0 0>, where I make these row vectors and multiply with them on the left.
>
> > I"m fuzzy on the distinction between eigenmonzo and comma subgroups, and which one(s) you use to make the projection matrix....
>
> You need both. One has eigenvalues of 1, the other eigenvalues of 0.
>
Thanks. I'll try this in Excel with MINVERSE and/or MMULT functions.

PGH

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
> --- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@> wrote:
>
> > I see, are there any examples on xenharmonic wiki?
>
> Tons of examples, in the form of all those matrices for "minimax tuning".
>
> I have read the article on eigenmonzos
> > I'd like to find a quick calculation to associate the minimax tuning with all this, for example
> > 1/4 comma meantone for 81/80.
>
> For 1/4 comma meantone, the eigenmonzo subgroup is generated by 2 and 5, and the comma subgroup by 81/80. Hence you just solve for the matrix which sends |1 0 0> and |0 0 1> to themselves and |-4 4 -1> to |0 0 0>, where I make these row vectors and multiply with them on the left.
>
> > I"m fuzzy on the distinction between eigenmonzo and comma subgroups, and which one(s) you use to make the projection matrix....
>
> You need both. One has eigenvalues of 1, the other eigenvalues of 0.
>

I get ----

[[1,0,0],[1,0,.25],[0,0,1]] for the projection map ---

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:

> I get ----
>
> [[1,0,0],[1,0,.25],[0,0,1]] for the projection map ---

Correct, but it's best to do it in rational arithmetic rather than floats.