Adding wedgies?

What's going on here?

[5, 13, -17, 9, -41, -76], TOP error 0.27611
"plus"
[13, 14, 35, -8, 19, 42], TOP error 0.26193
"equals"
[18, 27, 18, 1, -22, -34], TOP error 0.036378

I think Gene talked about this before but I didn't quite catch on
then.

BTW, these are the three most accurate 7-limit temperaments with L1
multival complexity less than 40 (though it's over 38 for each of the
three).

>What's going on here?
>
>[5, 13, -17, 9, -41, -76], TOP error 0.27611
>"plus"
>[13, 14, 35, -8, 19, 42], TOP error 0.26193
>"equals"
>[18, 27, 18, 1, -22, -34], TOP error 0.036378
>
>I think Gene talked about this before but I didn't quite catch on
>then.
>
>BTW, these are the three most accurate 7-limit temperaments with
>L1 multival complexity less than 40 (though it's over 38 for each
>of the three).

Boy, Paul, I am sure looking forward to your paper!!

This reminds me that my 'tuning-math forms' project is in limbo.
I have all the materials collected... just gotta find time to
do it... (if anybody wants to take a stab I'll make the materials
available...)

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> >What's going on here?
> >
> >[5, 13, -17, 9, -41, -76], TOP error 0.27611
> >"plus"
> >[13, 14, 35, -8, 19, 42], TOP error 0.26193
> >"equals"
> >[18, 27, 18, 1, -22, -34], TOP error 0.036378
> >
> >I think Gene talked about this before but I didn't quite catch on
> >then.
> >
> >BTW, these are the three most accurate 7-limit temperaments with
> >L1 multival complexity less than 40 (though it's over 38 for each
> >of the three).
>
> Boy, Paul, I am sure looking forward to your paper!!

Other than ennealimmal, which is a "bonus" temperament, my paper
stays below complexity < 24 -- and I doubt I'll be talking about
adding wedgies. So don't get your hopes up.

> This reminds me that my 'tuning-math forms' project is in limbo.

What's that?

>> Boy, Paul, I am sure looking forward to your paper!!
>
>Other than ennealimmal, which is a "bonus" temperament, my paper
>stays below complexity < 24 -- and I doubt I'll be talking about
>adding wedgies. So don't get your hopes up.

Oh, I suspected as much. I'm looking forward to it for other
reasons!

>> This reminds me that my 'tuning-math forms' project is in limbo.
>
>What's that?

Remember, the one where I show how to calculate this stuff on
paper, 'long division' style? By "materials" I just meant the
relevant posts from you, Gene, and Dave. If Gene answers this
thread it will probably get included as well.

-Carl

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> What's going on here?
>
> [5, 13, -17, 9, -41, -76], TOP error 0.27611
> "plus"
> [13, 14, 35, -8, 19, 42], TOP error 0.26193
> "equals"
> [18, 27, 18, 1, -22, -34], TOP error 0.036378

Parakleismic + Amity = Ennealimmal. Both parakleismic and amity have
4375/4374 as a comma, and so does their sum (and difference, for that
matter.)

I did talk about it before, though I can't recall what I said about
it. It is related to the Klein stuff. For 7-limit wedgies, define the
Pfaffian as follows: let

X = <<x1 x2 x3 x4 x5 x6||
Y = <<y1 y2 y3 y4 y5 y6||

Then

Pf(X, Y) = y1x6 + x1y6 - y2x5 - x2y5 + y3x4 + x3y4

It is easily checked that we have the identity

Pf(X+Y, X+Y) = Pf(X, X) + 2 Pf(X, Y) + Pf(Y, Y)

The Klein condition for the wedgie X is Pf(X, X)=0. If X and Y both
satisfy the Klein condition, and if Pf(X, Y)=0, then X+Y also
satisfies the Klein condition, and hence is a wedgie. What
Pf(X, Y)=0 means is that X and Y are related; they share a comma.

Probably, you will not want to talk about this in the paper. :)

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > What's going on here?
> >
> > [5, 13, -17, 9, -41, -76], TOP error 0.27611
> > "plus"
> > [13, 14, 35, -8, 19, 42], TOP error 0.26193
> > "equals"
> > [18, 27, 18, 1, -22, -34], TOP error 0.036378
>
> Parakleismic + Amity = Ennealimmal. Both parakleismic and amity have
> 4375/4374 as a comma, and so does their sum (and difference, for
that
> matter.)
>
> I did talk about it before, though I can't recall what I said about
> it. It is related to the Klein stuff. For 7-limit wedgies, define
the
> Pfaffian as follows: let
>
> X = <<x1 x2 x3 x4 x5 x6||
> Y = <<y1 y2 y3 y4 y5 y6||
>
> Then
>
> Pf(X, Y) = y1x6 + x1y6 - y2x5 - x2y5 + y3x4 + x3y4
>
> It is easily checked that we have the identity
>
> Pf(X+Y, X+Y) = Pf(X, X) + 2 Pf(X, Y) + Pf(Y, Y)
>
> The Klein condition for the wedgie X is Pf(X, X)=0. If X and Y both
> satisfy the Klein condition, and if Pf(X, Y)=0, then X+Y also
> satisfies the Klein condition, and hence is a wedgie. What
> Pf(X, Y)=0 means is that X and Y are related; they share a comma.

It seems below that the unshared commas, correspondingly, "add"
(actually multiply).

But how do you simply check whether a particular wedgie eats a
particular comma?

81:80 shared:

Meantone + DominantSevenths = Injera
126:125 "+" 35:36 "=" 49:50

Meantone + Catler = DominantSevenths
126:125 "+" 125:128 "=" 63:64

Meantone + Flattone = Semifourths
225:224 "+" 512:525 "=" 48:49
126:125 "+" 4375:4374 "=" 245:243
126:125 "+" 875:846 "=" 49:48

Injera + Flattone = SupermajorSeconds
50:49 "+" 512:525 "=" 1024:1029
50:49 "+" 864:875 "=" 1728:1715
etc.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> But how do you simply check whether a particular wedgie eats a
> particular comma?

Consider the wedgie in multimonzo form, and wedge with the monzo; if
you get a zero multimonzo, you had a comma.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > But how do you simply check whether a particular wedgie eats a
> > particular comma?
>
> Consider the wedgie in multimonzo form, and wedge with the monzo; if
> you get a zero multimonzo, you had a comma.

I figured that. But how do you wedge a monzo with a multimonzo?

Aha -- does the klein condition equate with the bivector being a
simple bivector?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > What's going on here?
> >
> > [5, 13, -17, 9, -41, -76], TOP error 0.27611
> > "plus"
> > [13, 14, 35, -8, 19, 42], TOP error 0.26193
> > "equals"
> > [18, 27, 18, 1, -22, -34], TOP error 0.036378
>
> Parakleismic + Amity = Ennealimmal. Both parakleismic and amity have
> 4375/4374 as a comma, and so does their sum (and difference, for
that
> matter.)
>
> I did talk about it before, though I can't recall what I said about
> it. It is related to the Klein stuff. For 7-limit wedgies, define
the
> Pfaffian as follows: let
>
> X = <<x1 x2 x3 x4 x5 x6||
> Y = <<y1 y2 y3 y4 y5 y6||
>
> Then
>
> Pf(X, Y) = y1x6 + x1y6 - y2x5 - x2y5 + y3x4 + x3y4
>
> It is easily checked that we have the identity
>
> Pf(X+Y, X+Y) = Pf(X, X) + 2 Pf(X, Y) + Pf(Y, Y)
>
> The Klein condition for the wedgie X is Pf(X, X)=0. If X and Y both
> satisfy the Klein condition, and if Pf(X, Y)=0, then X+Y also
> satisfies the Klein condition, and hence is a wedgie. What
> Pf(X, Y)=0 means is that X and Y are related; they share a comma.
>
> Probably, you will not want to talk about this in the paper. :)

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> > Consider the wedgie in multimonzo form, and wedge with the monzo; if
> > you get a zero multimonzo, you had a comma.
>
> I figured that. But how do you wedge a monzo with a multimonzo?

It's just a special case of the general definition. If you mean me
personally, I have Maple programs written to do it.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > > Consider the wedgie in multimonzo form, and wedge with the
monzo; if
> > > you get a zero multimonzo, you had a comma.
> >
> > I figured that. But how do you wedge a monzo with a multimonzo?
>
> It's just a special case of the general definition. If you mean me
> personally, I have Maple programs written to do it.

Is it a simple expression in terms of determinants?

Though I'd still like to see this answered, it would appear to be
true according to pages 7-8 of this paper:

http://arxiv.org/PS_cache/hep-th/pdf/0311/0311162.pdf

where it seems that the Klein correspondence does indeed equate to
the condition that the bivector be simple.

But I could be reading it wrong.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> Aha -- does the klein condition equate with the bivector being a
> simple bivector?
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > > What's going on here?
> > >
> > > [5, 13, -17, 9, -41, -76], TOP error 0.27611
> > > "plus"
> > > [13, 14, 35, -8, 19, 42], TOP error 0.26193
> > > "equals"
> > > [18, 27, 18, 1, -22, -34], TOP error 0.036378
> >
> > Parakleismic + Amity = Ennealimmal. Both parakleismic and amity
have
> > 4375/4374 as a comma, and so does their sum (and difference, for
> that
> > matter.)
> >
> > I did talk about it before, though I can't recall what I said
about
> > it. It is related to the Klein stuff. For 7-limit wedgies, define
> the
> > Pfaffian as follows: let
> >
> > X = <<x1 x2 x3 x4 x5 x6||
> > Y = <<y1 y2 y3 y4 y5 y6||
> >
> > Then
> >
> > Pf(X, Y) = y1x6 + x1y6 - y2x5 - x2y5 + y3x4 + x3y4
> >
> > It is easily checked that we have the identity
> >
> > Pf(X+Y, X+Y) = Pf(X, X) + 2 Pf(X, Y) + Pf(Y, Y)
> >
> > The Klein condition for the wedgie X is Pf(X, X)=0. If X and Y
both
> > satisfy the Klein condition, and if Pf(X, Y)=0, then X+Y also
> > satisfies the Klein condition, and hence is a wedgie. What
> > Pf(X, Y)=0 means is that X and Y are related; they share a comma.
> >
> > Probably, you will not want to talk about this in the paper. :)

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> Though I'd still like to see this answered, it would appear to be
> true according to pages 7-8 of this paper:
>
> http://arxiv.org/PS_cache/hep-th/pdf/0311/0311162.pdf
>
> where it seems that the Klein correspondence does indeed equate to
> the condition that the bivector be simple.

I thought I did answer that; in any case, it's true.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > Though I'd still like to see this answered, it would appear to be
> > true according to pages 7-8 of this paper:
> >
> > http://arxiv.org/PS_cache/hep-th/pdf/0311/0311162.pdf
> >
> > where it seems that the Klein correspondence does indeed equate
to
> > the condition that the bivector be simple.
>
> I thought I did answer that; in any case, it's true.

You didn't answer that or any of the other approximately 10 questions
I posted that day.