Is there a name for this operation?

Graham uses the "&" operator on his site to denote the combination of
two vals to make a mapping matrix, for example

7p&12p = <7 11 16] & <12 19 28] = [<7 11 16] <12 19 28]>

Note that on the right hand side of this equation, I've put the two
vals another ket. According to Graham, this is because we want to be
able to do

[<7 11 16] <12 19 28]>[-1 1 0> = [<7 11 16|-1 1 0> <12 19 28|-1 1 0>> = [4 7>

This tells us that 3/2 maps to the tempered monzo [4 7> in the basis
of 7p and 12p, which I believe works out to the meantone chromatic
half step and the diesis, respectively.

Another thing that results from this is the identity

<7 11 16] & <12 19 28] = [<7 11 16] <12 19 28]> = <[7 12> [11 19> [16 28>]

Which means that if you throw in a "tempered val," a normal val
results. In the basis of 7p and 12p, which again has steps of the
chromatic half step and diesis, let's see what happens if we use the
tempered vals <1 0], <0 1], <1 1], which by Hodge dual correspond to
the commas [0 -1>, [1 0>, [1 -1>

<[7 12> [11 19> [16 28>]<1 0] = <7 11 16]
<[7 12> [11 19> [16 28>]<0 1] = <12 19 28]
<[7 12> [11 19> [16 28>]<1 1] = <19 30 44]

It should make sense if you think about it why these tempered vals
spit out the real ones, because they correspond to dicot+meantone,
augmented+meantone, and magic+meantone respectively.

I've been spending the past week or so working out some axioms of this
& operator in order to formalize it. However, I'm very curious to know
if there's already a name for this sort of thing. It seems to
correspond somewhat to the Dirac outer product though, given by
[a><b]. For example, this would work out to

v1&v2 = [1 0><v1] + [0 1><v2]

Or something like that.

Does some analogue of this exist in multilinear algebra?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Graham uses the "&" operator on his site to denote the combination of
> two vals to make a mapping matrix, for example

Where does he say that?

> Which means that if you throw in a "tempered val," a normal val
> results. In the basis of 7p and 12p, which again has steps of the
> chromatic half step and diesis, let's see what happens if we use the
> tempered vals <1 0], <0 1], <1 1], which by Hodge dual correspond to
> the commas [0 -1>, [1 0>, [1 -1>
>
> <[7 12> [11 19> [16 28>]<1 0] = <7 11 16]
> <[7 12> [11 19> [16 28>]<0 1] = <12 19 28]
> <[7 12> [11 19> [16 28>]<1 1] = <19 30 44]

Can you explain this? What does the Hodge dual have to do with it?

On Mon, Nov 21, 2011 at 5:38 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > Graham uses the "&" operator on his site to denote the combination of
> > two vals to make a mapping matrix, for example
>
> Where does he say that?

For example

http://x31eq.com/cgi-bin/rt.cgi?ets=6p+11p&limit=5

This is a crappy temperament, but I picked it so that it clearly
displays 6p&11p on top. For most good temperaments which have names,
it displays the name instead. But you can see that 6p&11p is then
turned into

[<6 10 14]
<11 17 26]>

Which if displayed on one line is [<6 10 14] <11 17 26]>, two bras inside a ket.

We've been using this notation formally but in a self-consistent way,
and I'd like to axiomatize it as an operation that any jolly ol fellow
with a background in high school algebra can apply to polynomials,
given the axioms of the operator. I've spent a week on that so far,
but now I'm feeling as though I must be reinventing the wheel...

> > Which means that if you throw in a "tempered val," a normal val
> > results. In the basis of 7p and 12p, which again has steps of the
> > chromatic half step and diesis, let's see what happens if we use the
> > tempered vals <1 0], <0 1], <1 1], which by Hodge dual correspond to
> > the commas [0 -1>, [1 0>, [1 -1>
> >
> > <[7 12> [11 19> [16 28>]<1 0] = <7 11 16]
> > <[7 12> [11 19> [16 28>]<0 1] = <12 19 28]
> > <[7 12> [11 19> [16 28>]<1 1] = <19 30 44]
>
> Can you explain this? What does the Hodge dual have to do with it?

It's supposed to be tempered kets in a bra. If you feed in a tempered
bra, the <bra|ket> will be a scalar, and you'll have three scalars in
a bra, which is a val.

These "tempered vals" are just covectors acting on the temperamental
subspace, the basis of which is equal to that of the two vals that
you've used. For example, if our meantone basis is {octave, perfect
fifth}, a transversal of which is {2/1, 3/2}, the val <7 5] simply
means that 7 steps gets you to the meantone-tempered 2/1, and 5 steps
gets you to the meantone-tempered 3/2.

I'll work it out manually:

<[7 12> [11 19> [16 28>]<1 0] = <<1 0|7 12> <1 0|11 19> <1 0|16 28>] = <7 11 16]
<[7 12> [11 19> [16 28>]<0 1] = <<0 1|7 12> <0 1|11 19> <0 1|16 28>] =
<12 19 28]

As for the Hodge dual - in this case, since our vals are 7p&12p, the
tempered basis ends up being the diesis and the chromatic semitone. So
the covector <1 1] maps them to the same thing. We can consider this
to be a "1-map" by your terminology in a 2-dimensional space, and
apply the Hodge dual to it in the same way that we apply it to any
covector; if we do so, the sign of the second coefficient flips and we
get the "comma" [1 -1> expressed in coordinates in the temperamental
subspace. This is the difference between the meantone-tempered
chromatic semitone and the meantone-tempered diesis, which is the
meantone-tempered magic comma.

Not so coincidentally, here's what we get if we feed the <1 1]
tempered covector into <[7 12> [11 19> [16 28>]:

<[7 12> [11 19> [16 28>]<1 1] = <<1 1|7 12> <1 1|11 19> <1 1|16 28>] =
<19 30 44]

The result is 19p, as you'd expect.

-Mike

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

> > > Graham uses the "&" operator on his site to denote the combination of
> > > two vals to make a mapping matrix, for example
> >
> > Where does he say that?
>
> For example
>
> http://x31eq.com/cgi-bin/rt.cgi?ets=6p+11p&limit=5

How do you know he doesn't mean the same thing by it I mean, namely the temperament, not the matrix?

> > > <[7 12> [11 19> [16 28>]<1 0] = <7 11 16]
> > > <[7 12> [11 19> [16 28>]<0 1] = <12 19 28]
> > > <[7 12> [11 19> [16 28>]<1 1] = <19 30 44]
> >
> > Can you explain this? What does the Hodge dual have to do with it?
>
> It's supposed to be tempered kets in a bra. If you feed in a tempered
> bra, the <bra|ket> will be a scalar, and you'll have three scalars in
> a bra, which is a val.

I see. I think it would be clearer if you wrote it

<1 0| <|7 12> |11 19> |16 28>| = <7 11 16|

On Mon, Nov 21, 2011 at 9:42 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > For example
> >
> > http://x31eq.com/cgi-bin/rt.cgi?ets=6p+11p&limit=5
>
> How do you know he doesn't mean the same thing by it I mean, namely the temperament, not the matrix?

Maybe he does, but then I'm not sure how to define & formally, which
was my goal. A mapping matrix is certainly characteristic of a
temperament, so I thought that would be a good way to start.

My main point is that a^b takes in a vector and spits out a covector,
but a&b takes in a vector and spits out a tempered vector in a
different space. If that's not consistent with how others see the "&"
symbol being used, I'll pick a new symbol for it.

-Mike

On Mon, Nov 21, 2011 at 9:42 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> How do you know he doesn't mean the same thing by it I mean, namely the temperament, not the matrix?

But anyway, I don't want to get off track. If that's not what you want
& to mean, then okay, but I think the concept I laid out my original
post is useful. Do you know if this sort of operation has a name? Is
it the "direct sum" of the two vals?

-Mike

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

> Do you know if this sort of operation has a name? Is
> it the "direct sum" of the two vals?

I don't know a mathspeak name; it's sometimes called concatenating.