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Message: 8525 - Contents - Hide Contents

Date: Sun, 23 Nov 2003 19:38:13

Subject: Re: Finding the complement

From: Paul Erlich

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


> --- 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.



Hmm? In GABLE there are eight components to every object, the basis 
being (to within signs){1,e1,e2,e3,e1^e2,e2^e3,e1^e3,e1^e2^e3}.
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Message: 8526 - Contents - Hide Contents

Date: Sun, 23 Nov 2003 07:26:27

Subject: Re: Finding the wedge product?

From: monz

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


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


> Can you post the entire corrected thing?
> 
> -C.





i've already done it here:

Definitions of tuning terms: val, (c) 2001 by ... * [with cont.]  (Wayb.)


Gene, can you give me the rigorous mathematical definition
as well?


-monz
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Message: 8527 - Contents - Hide Contents

Date: Sun, 23 Nov 2003 20:37:57

Subject: Re: Finding the complement

From: Graham Breed

Dave Keenan wrote:



> (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.



Oh, geometric algebra.  Here are some pages about it:

Introduction to Geometric Algebra || kuro5hin.org * [with cont.]  (Wayb.)

Introduction to Geometric Algebra (part two) |... * [with cont.]  (Wayb.)

It's similar to Grassman algebra, but not the same.


                     Graham
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Message: 8528 - Contents - Hide Contents

Date: Sun, 23 Nov 2003 07:28:31

Subject: Re: Definition of val etc.

From: monz

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


>>> transpose
> 


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


> It isn't usually used as a noun in music.  And where there are
> collisions the math terminology should probably be favoured because
> it's more precise.
> 
> -Carl





Carl, that's an extremely good point ... in fact, musicians
*never* use "transpose" as a noun, calling a transposed part
either a "transposed part" or (most often) a "transposition".

"transpose" is strictly a verb in musical usage.

still, i want to put both definitions (the musical verb and
the mathematical noun) into the Dictionary, since both have
something to do with tuning.



-monz
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Message: 8529 - Contents - Hide Contents

Date: Sun, 23 Nov 2003 21:02:19

Subject: Re: Definition of val etc.

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:



> 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? 



I don't recall trying to identify anything with a pseudovector; the 
complement allows the identification of m-kets with (n-m)-bras, which 
is useful for tuning purposes.

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. 



That looked fine to me, and had the usual physicist's explanation of 
a linear form/functional (aka dual vector, bra vector, etc.) in terms 
of parallel spaces the vector pierces. I saw nothing in it about 
pseudovectors.
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Message: 8530 - Contents - Hide Contents

Date: Sun, 23 Nov 2003 08:57:15

Subject: Re: Finding the wedge product?

From: Graham Breed

Dave Keenan wrote:



> Here's one that does
> Permutation Parity by Lou Piciullo * [with cont.]  (Wayb.)
> 
> 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.



I think that's equivalent to the one I'm using, which I got from a group 
theory book.  (Actually, the book gave a slightly different algorithm, 
presumably what Gene meant by "product of differences".  But that's less 
efficient because multiplication's harder than addition.)

def equivalentValue(base, value):
     """alternative algorithm that avoids the evil bubblesort"""
     for i in range(len(base)-1):
         for j in range(i+1, len(base)):
             if base[j]<base[i]:
                 value = -value
     return value

And here's the same kind of thing in C, from another program I happen to 
be working on:

int odd_permutation(const int board[N_CELLS]) {
     int i, j, total=0;

     for (i=0; i<N_CELLS-1; i++) {
             for (j=i+1; j<N_CELLS; j++) {
                     total += board[i] > board[j];
             }
     }
     return total % 2;
}

It's still an O(n**2) algorighm, so there's no huge saving over bubble 
sorting and counting swaps, especially if you want the result sorted anyway.


                      Graham
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Message: 8531 - Contents - Hide Contents

Date: Sun, 23 Nov 2003 21:03:47

Subject: Re: Finding the complement

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:


> --- 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?



If it isn't calling e1^...^en a 1, it isn't right for our purposes.
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Message: 8532 - Contents - Hide Contents

Date: Sun, 23 Nov 2003 21:18:21

Subject: Re: Finding the complement

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:



> If it isn't calling e1^...^en a 1, it isn't right for our purposes.



Calling it a 1? Isn't it an n-vector, not a scalar?
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Message: 8533 - Contents - Hide Contents

Date: Sun, 23 Nov 2003 22:18:32

Subject: Re: Finding the complement

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:


> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> 
> wrote:
> 


>> If it isn't calling e1^...^en a 1, it isn't right for our 
purposes.
> 


> Calling it a 1? Isn't it an n-vector, not a scalar?



Right, but the complement is a 1.
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Message: 8534 - Contents - Hide Contents

Date: Sun, 23 Nov 2003 22:37:47

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:


>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
> <gwsmith@s...> 
>> wrote:
>> 


>>> If it isn't calling e1^...^en a 1, it isn't right for our 
> purposes.
>> 


>> Calling it a 1? Isn't it an n-vector, not a scalar?
> 


> Right, but the complement is a 1.



As you may have seen, GABLE defines the dual as division by e1^e2^e3. 
division is in the sense of the so-called 'geometric product'. Is the 
geometric product non-existent and/or meaningless in Grassmann 
algebra?
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Message: 8535 - Contents - Hide Contents

Date: Sun, 23 Nov 2003 00:40:08

Subject: Re: Finding the wedge product?

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:


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


>>> That's great Graham. I think I get it now. Let me try feeding it 
> back


>>> in a different way so you can tell me if I've got it right, and so
>>> others may have another chance at following it.
>> 


>> Thanks Dave.  Can someone confirm this?  I'm about to take it as
>> Gospel.
> 


> It looked good to me.



Thanks Gene. I'll bet there were lots of mathematical nits you could
have picked, with my presentation, so I really appreciate a simple
response like this. And I expect Carl does too. Thanks for all your
help in getting us to this point.
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Message: 8536 - Contents - Hide Contents

Date: Sun, 23 Nov 2003 02:15:37

Subject: Re: Finding the wedge product?

From: Dave Keenan

It would be nice to have a shortcut to avoid actually doing all those
index position swaps to find the sign of each product of scalars. 

We already have such a shortcut rule when calculating the complement,
namely sum the indexes plus Ceiling(grade/2) and negate if odd. But
that's no use for the wedge product.

We want a simple function of the (possibly compound) indexes of the
two coefficients, that gives us the sign of the result.

We can write the two indices as I = {i1 i2 i3 ...} and J = {j1 j2 j3
...} where the i's and j's are integers (simple indices). Note that
the compound-indices are rightly represented as sets since the order
of their elements doesn't matter, and we never have duplicates,
_within_ each compound-index. 

These sets could be implemented efficiently on a computer as a
bit-array or bit-set with a one-bit for each simple index that's in
the set. 16 bits would be plenty for any conceivable tuning
application. These could either be carried around with the
coefficients or computed as needed, given the grade and the number of
coefficients, or grade and dimension.

The condition for throwing away the product (of coefficients), can
then be expressed as I * J =/= {}, where "*" is set-intersection. i.e.
Throw it away if there is any simple index common to both sets.

And the index of the product is I + J, where "+" is set-union.

Now what's the simplest algorithm to give us the sign of the product
given those two index sets, in the right order?
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Message: 8539 - Contents - Hide Contents

Date: Mon, 24 Nov 2003 19:05:25

Subject: Re: a beautiful geometric algebra paper

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Kees van Prooijen" <lists@k...> 
wrote:



> Also, I have collected most of all geometric algebra papers 
available 
> on the net (aprx. 27 Mb) which I will gladly put on my website for 
a 
> while if there's interest.



What would be nice to find would be a simple, straightforward 
exposition of the basics of exterior algebra, without all the bells 
and whistles. The heavily mathematical books I use won't cut it, 
Browne has far more complexity than we need, and geometric algebra 
intoduces stuff I don't think we can use.
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Message: 8540 - Contents - Hide Contents

Date: Mon, 24 Nov 2003 20:07:38

Subject: Re: Finding Generators to Primes etc

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
<paul.hjelmstad@u...> wrote:



> Another question - is it possible to go from 3 to 1, without going 
> through 2?



Yes -- in the two-ET case, the period will be 1/n octaves where n is 
the greatest common divisor of the two ET cardinalities, and the 
generator will be in the vicinity of the best-matching other interval 
between the two ETs. For example, for 12&19, the period is 1 octave, 
while the best other match is between 5/12 oct. and 8/19 oct., so the 
generator is near these values. For 12&22, the period is 1/2 octave, 
and the best other match is between 1/12 oct. and 2/22 oct., so the 
generator is near these values. The mapping from *this* period and 
generator to primes should be identical for both ETs, and this 
carries over to the '&'.



> Also, do you always need 2 commas (in the 5-limit) to get
> Generators to Primes?



No, with one comma you get a so-called 'linear temperament' (or 2D 
temperament -- one period and one generator), just as you get from 
12&19 or 12&22 above.

I know you've seen this before, but on each row in this table:

Yahoo groups: /tuning/database? * [with cont.] 
method=reportRows&tbl=10&sortBy=3

The single comma is described in the first five columns, the period 
and optimal generator are shown in the eighth column, and the mapping 
from period and generator to primes is shown in the ninth column.

81:80 corresponds to the 12&19 case above, and 2048:2025 corresponds 
to the 12&22 case above.

The comma is more unique, since 12&19 might be written 19&31, etc., 
12&22 might be written 22&34, etc. . . .
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Message: 8541 - Contents - Hide Contents

Date: Mon, 24 Nov 2003 20:12:12

Subject: Re: a beautiful geometric algebra paper

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:


> --- In tuning-math@xxxxxxxxxxx.xxxx "Kees van Prooijen" 
<lists@k...> 
> wrote:
> 


>> Also, I have collected most of all geometric algebra papers 
> available 


>> on the net (aprx. 27 Mb) which I will gladly put on my website 
for 
> a 


>> while if there's interest.
> 


> What would be nice to find would be a simple, straightforward 
> exposition of the basics of exterior algebra, without all the bells 
> and whistles. The heavily mathematical books I use won't cut it, 
> Browne has far more complexity than we need, and geometric algebra 
> intoduces stuff I don't think we can use.



the fact that different references appear to contradict one another 
is the worst part of all.
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Message: 8542 - Contents - Hide Contents

Date: Mon, 24 Nov 2003 20:33:12

Subject: Re: Finding Generators to Primes etc

From: Graham Breed

Paul G Hjelmstad wrote:



> Another question - is it possible to go from 3 to 1, without going 
> through 2? Also, do you always need 2 commas (in the 5-limit) to get
> Generators to Primes?



It's possible to do anything you like without commas, unless it 
explicitly involves commas.  If you're using commas, you need 2 of them 
to get a 5-limit equal temperament.

What's 3 to 1, a linear temperament to a pair of equal temperaments? 
The usual mapping of primes to generators already gives you this.  They 
may be bizarre equal temperments, with 1 or 0 notes to the octave.  But 
you can add them together to get a different 1 note equal temperment. 
And then add the two 1s to get a 2 note temperament.  And so on, as far 
as you like.


                    Graham
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Message: 8543 - Contents - Hide Contents

Date: Mon, 24 Nov 2003 20:33:29

Subject: Re: Finding Generators to Primes etc

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:



> The usual mapping of primes to generators 



generators to primes?
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Message: 8544 - Contents - Hide Contents

Date: Mon, 24 Nov 2003 21:40:01

Subject: Re: Finding Generators to Primes etc

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:



> may be bizarre equal temperments, with 1 or 0 notes to the octave.



A good reason to call them vals.
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Message: 8545 - Contents - Hide Contents

Date: Mon, 24 Nov 2003 21:42:13

Subject: Re: Finding Generators to Primes etc

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:


> --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> 
wrote:
> 


>> The usual mapping of primes to generators 
> 


> generators to primes?



Graham's way of saying it really makes more sense. We have a mapping 
h, for which h(2) gives the number of generator steps to 2, h(3) to 
3, and so forth.
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Message: 8546 - Contents - Hide Contents

Date: Mon, 24 Nov 2003 23:37:29

Subject: Re: Finding Generators to Primes etc

From: Dave Keenan

Graham:
...


>> may be bizarre equal temperments, with 1 or 0 notes to the octave.

Gene: 


> A good reason to call them vals.



Or single-generator mappings.

But whether he called them mappings or vals he would still have wanted
to say that they can be considered as mappings or vals _for_ equal
temperaments, albeit bizarre ones. And that linear combinations of
them are all that is needed to give you mappings for more reasonable ETs.

I had not realised that. Thanks Graham.
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Message: 8547 - Contents - Hide Contents

Date: Mon, 24 Nov 2003 01:38:26

Subject: Re: Finding the wedge product?

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:


> It's still an O(n**2) algorighm, so there's no huge saving over bubble 
> sorting and counting swaps, especially if you want the result sorted 
> anyway.



Right, but if you use a fixed-length bitset implementation of the
compound indices (e.g. 32 bits) then there's no requirement for
sorting. You just OR them together to get the compound index of the
result. And the parity algorithm is then only O(n) and consists of
shifts and ANDs followed by XORing the parity of the number of 1 bits
in the results. I guess this is still O(n*log(n)) unless you have an
O(1) bitwise parity operation. Otherwise bitwise parity will need to
be implemented as log(n) shifts and xors. Here's some pseudo-C-code.

long function Parity(long i, j)
  /* Where i and j representing the two compound indices as bitsets */
  result = 0
  for n = 1 to 31
    i >>= 1
    result ^= bitParity(i & j) 
  next n
  return result

long function bitParity(long i)
  i ^= i>>1
  i ^= i>>2
  i ^= i>>4
  i ^= i>>8
  i ^= i>>16
  i &= 1
  return i
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Message: 8548 - Contents - Hide Contents

Date: Mon, 24 Nov 2003 03:40:57

Subject: Re: Finding the complement

From: Paul Erlich

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


> --- 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.



In any case, could such considerations argue that we should make 
standard use of, as defined in section 6.9 or pp. 32-37 and motivated 
further down here,

Index of /homes/browne/grassmannalgebra/book/b... * [with cont.]  (Wayb.)
TheInteriorProduct.pdf

the (generalized) cross-product?
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Message: 8549 - Contents - Hide Contents

Date: Mon, 24 Nov 2003 04:28:34

Subject: Re: Finding the complement

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:



> As you may have seen, GABLE defines the dual as division by 
e1^e2^e3. 
> division is in the sense of the so-called 'geometric product'. Is 
the 
> geometric product non-existent and/or meaningless in Grassmann 
> algebra?



Geometric algebra in some sense encompasses Grassmann algebra, but 
not in a way that strikes me as useful for our purposes. Even more 
general are Clifford algebras, which is what algebraists are most 
interested in; but again, I don't see a payoff.
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