Applying M7 Transform to Wedge Products

So, you know, my D4 X S3 stuff (Symmetry of Square and Triangle, which reduces for the "M7" transform...)
Anyway FINALLY I have a way to bring it into Exterior Algebra...Here's a trivial example

Let's use 2, 3/2*sqrt(2) and 5 instead of 2, 3, 5 as a basis, this applies M7 (Move all odd steps by a tritone, even ones are fixed,
so that now scales based on the circle of fifths are based on semitones...)

Now wedge (-4, 2, -1) and (7, 0, -3) instead of (-4, 4, -1) and (7, 0, 3) for 5-limit meantone. This is of course
81/20 and 128/125 instead of 81/80 and 128/125 due to 3/2*sqrt(2) instead of 3. Wedging out 81/20 is a bit odd...

I get 12, 13, 28! So now 19 -> 13 in the way that 19 steps goes to 13, which is a semitone, really.

I know this is quite trivial, but I was happy to see it works. I want to try to wedge other things, like sqrt(3)/2 which splits the perfect forth, and equals sin(pi/3)...

On Sat, Feb 26, 2011 at 1:29 PM, Paul <phjelmstad@msn.com> wrote:
>
> So, you know, my D4 X S3 stuff (Symmetry of Square and Triangle, which reduces for the "M7" transform...)
> Anyway FINALLY I have a way to bring it into Exterior Algebra...Here's a trivial example
>
> Let's use 2, 3/2*sqrt(2) and 5 instead of 2, 3, 5 as a basis, this applies M7 (Move all odd steps by a tritone, even ones are fixed,
> so that now scales based on the circle of fifths are based on semitones...)
>
> Now wedge (-4, 2, -1) and (7, 0, -3) instead of (-4, 4, -1) and (7, 0, 3) for 5-limit meantone. This is of course
> 81/20 and 128/125 instead of 81/80 and 128/125 due to 3/2*sqrt(2) instead of 3. Wedging out 81/20 is a bit odd...
>
> I get 12, 13, 28! So now 19 -> 13 in the way that 19 steps goes to 13, which is a semitone, really.
>
> I know this is quite trivial, but I was happy to see it works. I want to try to wedge other things, like sqrt(3)/2 which splits the perfect forth, and equals sin(pi/3)...

This is fascinating, but I'm not sure I completely understand it.
Probably it'll make more sense to me once I finish reading up on the
linear algebra stuff. So what you're trying to do here is to build in
a tritone substitution for 5 to the basis vectors for the lattice? If
so, maybe it also more sense to view the basis vectors as sqrt(2), 3,
and 5, which I believe is a superset of what you posted.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, Feb 26, 2011 at 1:29 PM, Paul <phjelmstad@...> wrote:
> >
> > So, you know, my D4 X S3 stuff (Symmetry of Square and Triangle, which reduces for the "M7" transform...)
> > Anyway FINALLY I have a way to bring it into Exterior Algebra...Here's a trivial example
> >
> > Let's use 2, 3/2*sqrt(2) and 5 instead of 2, 3, 5 as a basis, this applies M7 (Move all odd steps by a tritone, even ones are fixed,
> > so that now scales based on the circle of fifths are based on semitones...)
> >
> > Now wedge (-4, 2, -1) and (7, 0, -3) instead of (-4, 4, -1) and (7, 0, 3) for 5-limit meantone. This is of course
> > 81/20 and 128/125 instead of 81/80 and 128/125 due to 3/2*sqrt(2) instead of 3. Wedging out 81/20 is a bit odd...
> >
> > I get 12, 13, 28! So now 19 -> 13 in the way that 19 steps goes to 13, which is a semitone, really.
> >
> > I know this is quite trivial, but I was happy to see it works. I want to try to wedge other things, like sqrt(3)/2 which splits the perfect forth, and equals sin(pi/3)...
>
> This is fascinating, but I'm not sure I completely understand it.
> Probably it'll make more sense to me once I finish reading up on the
> linear algebra stuff. So what you're trying to do here is to build in
> a tritone substitution for 5 to the basis vectors for the lattice? If
> so, maybe it also more sense to view the basis vectors as sqrt(2), 3,
> and 5, which I believe is a superset of what you posted.
>
> -Mike

Almost, tritone substitution for 3 actually. And the vectors should
be (-2, 4, -1) and (7, 0, -3). You wedge these to find 12,13,28 for
2,3/2(sqrt(2),5. The M7 affine action (based on D4 X S3) sends
1->7, 2->2, 3->9, 4->4, 5->11, 6->6, 7->1, 8->8, 9->3, 10->10 11->5.

This is based on the group of units on the Z12 ring, which are the numbers less than 12 relatively prime to it 1,5,7,11. These numbers
can be used for M1, M5, M7, M11, which multiply the elements in Z12.

Mostly this group discusses Multilinear Algebra (Grassman etc) but
sometimes Group Theory so I was trying to mix that in but I don't
think my approach is entirely orthodox, for example, 81/20 is not
well-formed (I think this is related to torsion) --- in the sense
that four semitones fit into 5/4 (instead of 3/2 four times fitting
into 5). 81/64 actually will produce 12,25,28 where you can see
that 19->25 is also a tritone substitution. Now four semitones
fitting into 5/4 still gives 81/80, but that will go to 12,19,28
so I needed to tweak things a little.

PGH

On Sat, Feb 26, 2011 at 3:18 PM, Paul <phjelmstad@msn.com> wrote:
>
> Almost, tritone substitution for 3 actually. And the vectors should
> be (-2, 4, -1) and (7, 0, -3). You wedge these to find 12,13,28 for
> 2,3/2(sqrt(2),5. The M7 affine action (based on D4 X S3) sends
> 1->7, 2->2, 3->9, 4->4, 5->11, 6->6, 7->1, 8->8, 9->3, 10->10 11->5.

Sorry, for 3, that's what I meant. Not 5.

> This is based on the group of units on the Z12 ring, which are the numbers less than 12 relatively prime to it 1,5,7,11. These numbers
> can be used for M1, M5, M7, M11, which multiply the elements in Z12.
>
> Mostly this group discusses Multilinear Algebra (Grassman etc) but
> sometimes Group Theory so I was trying to mix that in but I don't
> think my approach is entirely orthodox, for example, 81/20 is not
> well-formed (I think this is related to torsion) --- in the sense
> that four semitones fit into 5/4 (instead of 3/2 four times fitting
> into 5). 81/64 actually will produce 12,25,28 where you can see
> that 19->25 is also a tritone substitution. Now four semitones
> fitting into 5/4 still gives 81/80, but that will go to 12,19,28
> so I needed to tweak things a little.

Doesn't ripple temperament do this? Or perhaps it's passion.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, Feb 26, 2011 at 3:18 PM, Paul <phjelmstad@...> wrote:
> >
> > Almost, tritone substitution for 3 actually. And the vectors should
> > be (-2, 4, -1) and (7, 0, -3). You wedge these to find 12,13,28 for
> > 2,3/2(sqrt(2),5. The M7 affine action (based on D4 X S3) sends
> > 1->7, 2->2, 3->9, 4->4, 5->11, 6->6, 7->1, 8->8, 9->3, 10->10 11->5.
>
> Sorry, for 3, that's what I meant. Not 5.
>
> > This is based on the group of units on the Z12 ring, which are the numbers less than 12 relatively prime to it 1,5,7,11. These numbers
> > can be used for M1, M5, M7, M11, which multiply the elements in Z12.
> >
> > Mostly this group discusses Multilinear Algebra (Grassman etc) but
> > sometimes Group Theory so I was trying to mix that in but I don't
> > think my approach is entirely orthodox, for example, 81/20 is not
> > well-formed (I think this is related to torsion) --- in the sense
> > that four semitones fit into 5/4 (instead of 3/2 four times fitting
> > into 5). 81/64 actually will produce 12,25,28 where you can see
> > that 19->25 is also a tritone substitution. Now four semitones
> > fitting into 5/4 still gives 81/80, but that will go to 12,19,28
> > so I needed to tweak things a little.
>
> Doesn't ripple temperament do this? Or perhaps it's passion.
>
> -Mike

Not sure --- I have one more correction, actually you always get 81/80
because of the change in the denominator --- for example with 2, 3/2*sqrt(2) you pick up 4 in the denominator --- so you get 81/80 anyway. Augh. Sometimes I make dumb mistakes:)