Tuning-Math Digests messages 8500 - 8524

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Message: 8500

Date: Sat, 22 Nov 2003 02:03:16

Subject: Re: Finding Generators to Primes etc

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > no, look at the parentheses. the complement of the wedge product is 
> > the cross product (when you're dealing with a 3 dimensional 
> problem).
> 
> Not exactly. A cross product takes vectors to vectors (or 
> pseudovectors, if you are a physicist) and in fact a three 
> dimensional real vector space with cross product is the real Lie 
> algebra o(3). The complement of a wedge product of bra vectors is a 
> ket vector, and conversely.

This is hilarious. I couldn't parody this any better than you're doing
it yourself. :-)

I thought Paul's statement was perfectly clear and correct.


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Message: 8501

Date: Sat, 22 Nov 2003 18:55:50

Subject: Re: Finding Generators to Primes etc

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >Not quite. The complement of the wedging is the same as crossing 
(but
> >crossing is only defined for 3D (5-limit)).
> 
> Is the cross product really only defined, for anything, for 3-item
> vectors?

yes. see

Cross Product -- from MathWorld *


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Message: 8503

Date: Sat, 22 Nov 2003 18:57:27

Subject: Re: Definition of val etc.

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Ah, so the "matrix product" is a pairwise dot product of sorts?

Matrix Multiplication -- from MathWorld *


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Message: 8505

Date: Sat, 22 Nov 2003 19:05:40

Subject: Re: Finding the complement

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:

> > You're probably doing something wrong, then. This is correct 
> > according to both the GABLE tutorial and the program itself.
> 
> Can you give a URL for this GABLE tutorial?

Again, it's http://carol.science.uva.nl/~leo/GABLE/tutorial.pdf - Ok * . . . 
and again, I'm looking at page 18

> Does it use alphabetical
> ordering of indices?

In the case of the trivector, yes -- and I would have thought that's 
all that matters here. Surely the question of whether 1 is the dual 
of its dual is independent of this ordering question? All the 
directly tuning-interpretable results of Grassmann algebra should be 
independent of this ordering question, and the cross-product is, 
thank goodness. Maybe the question of whether 1 is the dual of its 
dual is not intepretable in tuning terms.


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Message: 8507

Date: Sat, 22 Nov 2003 21:50:20

Subject: Re: Definition of val etc.

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> 
> wrote:
> 
> > > It says the cross product of two vectors is a pseudovector. Is 
it 
> > only
> > > 3D vals that are pseudovectors?
> > 
> > Argh. Let's leave pseudovectors out of it.
> 
> WHAT!!! Why pull the rug out from under me? I wish you had 
commented 
> when I posted this:
> 
> Yahoo groups: /tuning-math/message/7798 *
> 
> I took this as an important step in my learning about bra and ket 
> vectors. So I should forget about it? And if so, why?

I take it the answer is discrete vs. continuous spaces?


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Message: 8511

Date: Sat, 22 Nov 2003 22:43:45

Subject: Re: Definition of val etc.

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
Dave Keenan:
> >Well the transpose is only relevant if you're going to do it using
> >matrix operations in software like Mathematica, Maple, Matlab, Octave
> >(free) or Excel.
> 
> Then you need to say so.

Yes.

> >And if you're doing this you can read their help to
> >find out about transpose.
> 
> I typed "transpose" into Excel help and got back this single result:
> 
> >TRANSPOSE(array)
> >
> >Array   is an array or range of cells on a worksheet that you want to 
> >transpose. The transpose of an array is created by using the first row
> >of the array as the first column of the new array, the second row of
> >the array as the second column of the new array, and so on.
> 
> Is that right for matrices too?

Yes.

> >It's would be easy enough to explain transpose in this dictionary
> >entry if you really think I should. However, if I have to explain
> >transpose here, then presumably I have to explain "matrix product"
> >too? This would be more tedious.
> 
> I, for one, have no idea what a "matrix product" is.
> 
> >However, I suppose we could give the Excel formulae in a Monz
> >dictionary entry, considering it to be a sort of lowest common
> >denominator among math tools.
> 
> I'd prefer to actually know how to do these things by hand.

Me too. Only then do I feel I know what's happening when I use a
software calculator to do it.

> >Now lets look at doing it by hand, without using matrix multiplication.
...

> Ah, so the "matrix product" is a pairwise dot product of sorts?

Exactly. Each element of the resulting matrix is the dot product of a
row from the first matrix and a column from the second.


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Message: 8514

Date: Sat, 22 Nov 2003 22:50:35

Subject: Re: Finding the wedge product?

From: Dave Keenan

Oops. Something went missing near the end there. It should have been:

Now we sum the products with the same index.

product                   index
a1*b23 + a3*b12 - a2*b13  123
a1*b24 - a2*b14 + a4*b12  124
a1*b34 - a3*b14 + a4*b13  134
a2*b34 - a3*b24 + a4*b23  234

[but it was all there in the final result]

Now we list them in alphabetical order of index inside the right
number of brackets.

[[[a1*b23+a3*b12-a2*b13 a1*b24-a2*b14+a4*b12 a1*b34-a3*b14+a4*b13
a2*b34-a3*b24+a4*b23>>>


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Message: 8516

Date: Sat, 22 Nov 2003 22:55:56

Subject: Re: Definition of val etc.

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
> are we still hashing out what i should put into the Dictionary?
> i hope so ... unfortunately i'm understanding very little
> of what's been posted here in the last week.

Yeah. But I think it's close. Look for stuff in my posts between
--------------------------------------------------------------
...
--------------------------------------------------------------

> anyway, let's please be careful about using the word
> "transpose" in these definitions.  it already has a firmly
> established meaning to musicians, and you guys are using
> a different (mathematical) definition of it now.

A very good point which had completely slipped my mind in all this
heavy mathematics.


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Message: 8517

Date: Sat, 22 Nov 2003 23:03:26

Subject: Re: Finding the complement

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
> > Does it use alphabetical
> > ordering of indices?
> 
> In the case of the trivector, yes -- and I would have thought that's 
> all that matters here. Surely the question of whether 1 is the dual 
> of its dual is independent of this ordering question? 

You would expect so.

> All the 
> directly tuning-interpretable results of Grassmann algebra should be 
> independent of this ordering question, and the cross-product is, 
> thank goodness. Maybe the question of whether 1 is the dual of its 
> dual is not intepretable in tuning terms.

Hmmm. This is rather mystifying. I'm afraid I'm just going to go with
Browne's Euclidean complement and forget the GABLE "dual". Two reasons.
(a) GABLE is clearly not concerned with any dimension greater than 3.
(b) GABLE is "geometric algebra" which appears to be Grassman algebra
using "homogeneous" coordinates. This is where they add one more
component to the vector than there are dimensions in the space, so
they can distinguish points from vectors, or some such.


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Message: 8518

Date: Sun, 23 Nov 2003 04:16:28

Subject: Re: Finding the wedge product?

From: Dave Keenan

I think I've found one shortcut. But there may yet be a simpler one.

Given that the indexes of the two coefficients are respectively 
I = {i1 i2 i3 ...} and J = {j1 j2 j3 ...}.

The sign of the product is given by the parity (oddness) of
Min(Sum(I)-Card(I)*(Card(I)+1)/2, Card(J)*(2*Card(U)+1-Card(J))/2)-Sum(J))

where Sum({i1 i2 ... ig}) = i1 + i2 + ... ig
i.e. the sum of the digits in the coefficient's compound index.
and Card({i1 i2 ... ig}) = g
i.e. the grade of the thing that the coefficient came from.
and Card(U) is the cardinality of the universal set, in other words
the dimensionality n of the arguments, or their maximum simple index
number n.

I'll restate this in different terms.

We're talking about calculating a wedge product of two arguments. A
part of that process is finding the correct sign for each product of
coefficients.

Lets use 
n = the dimension of the arguments and result e.g. 3 for 5-limit, 4
for 7-limit, etc.
g1 = the grade of the first argument. e.g. 0 for a scalar, 1 for a
vector, 2 for a bivector, etc.
g2 = the grade of the second argument.
a = a coefficient of the first argument. i.e. one of the numbers
inside [ ... >, or [[ ... >>, etc.
b = a coefficient of the second argument.
s1 = the sum of the indices for the coefficient of the first argument (a).
s2 = the sum of the indices for the coefficient of the second argument
(b).

You negate the product a*b whenever the following expression is odd.

Min(s1 - g1*(g1+1)/2,  g2*(2*n+1-g2)/2) - s2)

Of course I'm not 100% certain of this.

The expression g*(2*n+1-g)/2) in the second argument to Min() is
intended to be equivalent to

n*(n+1)/2 - (n-g)*(n-g+1)/2


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Message: 8519

Date: Sun, 23 Nov 2003 19:21:43

Subject: Re: Finding Generators to Primes etc

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > *hands thrown up in air*
> > 
> > so why "Not exactly"???
> 
> The cross product of two bra vectors would be a bra vector, not a 
ket 
> vector, if we define things as usual and make the cross products of 
> two vectors be a vector in the same vector space. This is how it 
> manages to be an *algebra*.

hmm . . . but isn't that document claiming that, at least in physics, 
one should *not* define things as usual, and that things like force, 
which is often equal to the cross product of two legitimate vectors, 
are *pseudovectors*?


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Message: 8520

Date: Sun, 23 Nov 2003 06:10:35

Subject: Re: Finding the wedge product?

From: Dave Keenan

The more I think about it the less I think that index permutation
parity algorithm will work in general.

Here's one that does
Permutation Parity by Lou Piciullo *

It's very simple and designed for pencil and paper. It is much less
error-prone than trying to count how many index-swaps you need to get
to alphabetical/numerical order. It should be fairly easily adapted to
whatever data-structure is used for the compound indices.


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Message: 8521

Date: Sun, 23 Nov 2003 19:30:10

Subject: Re: Definition of val etc.

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
> <gwsmith@s...> 
> 
> > > Argh. Let's leave pseudovectors out of it.
> > 
> > WHAT!!! Why pull the rug out from under me? I wish you had 
> commented 
> > when I posted this:
> > 
> > Yahoo groups: /tuning-math/message/7798 *
> > 
> > I took this as an important step in my learning about bra and ket 
> > vectors. So I should forget about it? And if so, why?
> 
> It's a physics idea, is why. It seems to have no tuning theory 
> relevance to me. Mathematicians usually don't like the idea, 
because 
> to them a cross product either lies in the same vector space or it 
> doesn't, and if it does lie in the same vector space you get a Lie 
> algebra product with nothing to distinguish a pseudovector from a 
> vector, and if it doesn't we are in the realm of multilinear 
algebra. 
> Maybe I'm prejudiced; as a physics major it's up to you to make use 
> of the distinction for our purposes, perhaps, but to a 
mathematician 
> there isn't a lot of difference between pseudovector and bivector.

I don't get it. I thought what I was trying to do was exactly this -- 
identify a bivector with a pseudovector -- or was what you were 
objecting to in the correspondence with bra and ket vectors? It would 
really be nice if you could lay out all these terminologies for us in 
a slow, 'breathable' way. I'd like to gain a geometrical grasp on all 
this, and to help others do the same. The physics document above 
appeared to have notions with a striking resemblance to those I 
created in explaining the Hypothesis. If there's anything I can hang 
these mathematical terms on, there's a far greater chance I'll be 
able to understand what the equations mean in tuning terms, and to 
create my own . . .


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Message: 8522

Date: Sun, 23 Nov 2003 19:33:06

Subject: Re: contravariant vs. covariant vectors

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > 403 Forbidden *
> 
> I don't see anything about pseudovectors in this.

Whoops! So where did this term 'pseudovectors' come from? I remember 
something about 'axial' vectors somewhere . . . but was I the one 
that somehow slipped 'pseudovectors' into all this? I'M SORRY!!


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Message: 8523

Date: Sun, 23 Nov 2003 19:34:01

Subject: Re: Finding the complement

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
>  Maybe the question of whether 1 is the dual of its 
> > dual is not intepretable in tuning terms.
> 
> It's a part of the standard convention allowing us to define how 
the 
> complement is going to work--e1^e2^...^en is taken to define volume 
1.

So is GABLE just wrong?


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