Tuning-Math Digests messages 8550 - 8574

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Message: 8550

Date: Mon, 24 Nov 2003 04:34:50

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:
> 
> > 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.

I've come to section 10:

Index of /homes/browne/grassmannalgebra/book/bookpdf *
ExpTheGeneralizedProduct.pdf

Is this 'generalized product' of Grassman product nearly, but not 
quite, identical to the geometric product?


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Message: 8551

Date: Mon, 24 Nov 2003 05:43:54

Subject: a beautiful geometric algebra paper

From: Paul Erlich

http://modelingnts.la.asu.edu/pdf/OerstedMedalLecture.pdf - Ok *

Should I ignore this one too? GABLE said it was using different 
conventions than Hestenes, so maybe this paper actually agrees with 
Grassmann Algebra?


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Message: 8553

Date: Tue, 25 Nov 2003 11:49:34

Subject: Re: Finding the wedge product?

From: Carl Lumma

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

You mean I shouldn't take your thing as Gospel just yet?

>Here's one that does //

Permutation Parity by Lou Piciullo *

So, can we get a version of your Gospel with this rolled in?

-Carl


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Message: 8554

Date: Tue, 25 Nov 2003 20:19:27

Subject: Re: Finding Generators to Primes etc

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > The mapping from *this* period and 
> > generator to primes should be identical for both ETs, and this 
> > carries over to the '&'.
> > 
> Interesting. So is rms or minimax applied, to say, 5/12 and 8/19
> for 12&19 together?
> 
> Paul

Well, what I was thinking was that, once you've found the period and 
generator, you can determine the mapping between primes and 
generators using either ET. But that's not quite right, you need 
both. For example, prime 3 is -1 generator but also 11 generators in 
12-equal, and prime 3 is -1 generator but also 18 generators in 19-
equal. So you need to pick the one that's in common between the two 
ETs. Once you've done that, you can drop any reference to the ETs 
themselves, and just optimize, rms or minimaxm, using the mapping 
itself.


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Message: 8555

Date: Tue, 25 Nov 2003 20:27:10

Subject: Re: Finding the complement

From: Paul Erlich

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

> 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.

I actually found this not in GABLE, but in Browne's book, right in 
chapter 1!!

Chapter 1 is going quite well so far, but with no mention of 
contravariant vs. covariant, the analogy with what we're doing here 
is not clear yet. What's disturbing to me so far is that the 
complement defines a metric, but I would have hoped all our 
operations could be done without any choice of metric. And then, why 
we would need to take the *complement* of the wedge product to get 
the val -- in 3D, the wedge product is a bivector, and the complement 
of that (given a metric) would be a vector, but don't vectors 
represent monzos? But onwards!


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Message: 8557

Date: Tue, 25 Nov 2003 20:49:12

Subject: Re: Finding Generators to Primes etc

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
> > <paul.hjelmstad@u...> wrote:
> > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" 
<perlich@a...> 
> > > wrote:
> > > > The mapping from *this* period and 
> > > > generator to primes should be identical for both ETs, and 
this 
> > > > carries over to the '&'.
> > > > 
> > > Interesting. So is rms or minimax applied, to say, 5/12 and 8/19
> > > for 12&19 together?
> > > 
> > > Paul
> > 
> > Well, what I was thinking was that, once you've found the period 
> and 
> > generator, you can determine the mapping between primes and 
> > generators using either ET. But that's not quite right, you need 
> > both. For example, prime 3 is -1 generator but also 11 generators 
> in 
> > 12-equal, and prime 3 is -1 generator but also 18 generators in 
19-
> > equal. So you need to pick the one that's in common between the 
two 
> > ETs. Once you've done that, you can drop any reference to the ETs 
> > themselves, and just optimize, rms or minimaxm, using the mapping 
> > itself.
> 
> Thanks. Could you give an example?

Hmm, I thought this *was* an example . . . you already know how to do 
the optimization using the mapping between primes and generators, 
right?


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Message: 8558

Date: Tue, 25 Nov 2003 21:10:19

Subject: Re: Finding the complement

From: Graham Breed

Paul Erlich wrote:

> Chapter 1 is going quite well so far, but with no mention of 
> contravariant vs. covariant, the analogy with what we're doing here 
> is not clear yet. What's disturbing to me so far is that the 
> complement defines a metric, but I would have hoped all our 
> operations could be done without any choice of metric. And then, why 
> we would need to take the *complement* of the wedge product to get 
> the val -- in 3D, the wedge product is a bivector, and the complement 
> of that (given a metric) would be a vector, but don't vectors 
> represent monzos? But onwards!

No, I don't think you need covariant vs contravariant.  It's only a way 
of labelling the elements so that you know when you have to take the 
complement.  If you think of the complement as a way of transforming one 
exterior element into another it still works.

The number of steps to the octave of an equal temperament is the same as 
the area of an octave-equivalent periodicity block.  To calculate an 
area, you need a metric, and we're implying one by relating unison 
vectors to mappings.  Geometric complexity might make sense in terms of 
a different metric, but I don't understand that yet.

Yes, vectors represent both vals and monzos.  If you don't like them 
both to be plain vectors, you can call one "covariant" and the other 
"contravariant".


                             Graham


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Message: 8560

Date: Tue, 25 Nov 2003 13:58:04

Subject: Re: Finding Generators to Primes etc

From: Carl Lumma

>Yes. I guess what I meant was, could you give me the pre-optimized 
>generator in this case. Thanx!

Many examples are given here in the linear temperament database.

Yahoo groups: /tuning/database/ *

-Carl


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Message: 8561

Date: Tue, 25 Nov 2003 22:21:00

Subject: Re: Finding Generators to Primes etc

From: Paul Erlich

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

> > Hmm, I thought this *was* an example . . . you already know how 
to 
> do 
> > the optimization using the mapping between primes and generators, 
> > right?
> 
> Yes. I guess what I meant was, could you give me the pre-optimized 
> generator in this case. Thanx!

I don't know what you mean by 'pre-optimized' generator! Sorry, I 
really do want to help . . .


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Message: 8563

Date: Tue, 25 Nov 2003 22:24:17

Subject: Re: Finding the complement

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
> 
> > Chapter 1 is going quite well so far, but with no mention of 
> > contravariant vs. covariant, the analogy with what we're doing 
here 
> > is not clear yet. What's disturbing to me so far is that the 
> > complement defines a metric, but I would have hoped all our 
> > operations could be done without any choice of metric. And then, 
why 
> > we would need to take the *complement* of the wedge product to 
get 
> > the val -- in 3D, the wedge product is a bivector, and the 
complement 
> > of that (given a metric) would be a vector, but don't vectors 
> > represent monzos? But onwards!
> 
> No, I don't think you need covariant vs contravariant.  It's only a 
way 
> of labelling the elements so that you know when you have to take 
the 
> complement.  If you think of the complement as a way of 
transforming one 
> exterior element into another it still works.
> 
> The number of steps to the octave of an equal temperament is the 
same as 
> the area of an octave-equivalent periodicity block.  To calculate 
an 
> area, you need a metric,

I don't believe you need a metric to calculate the number of steps 
per octave. The exterior product gives you the right answer, without 
assuming any metric -- just linearity.

> and we're implying one by relating unison 
> vectors to mappings.

I don't think that's necessarily true either.

> Yes, vectors represent both vals and monzos.  If you don't like 
them 
> both to be plain vectors, you can call one "covariant" and the 
other 
> "contravariant".

I guess I'll have to wait 'till I get to the relevant section in 
Browne's book . . .


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Message: 8566

Date: Tue, 25 Nov 2003 23:25:14

Subject: Re: Finding Generators to Primes etc

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
> > <paul.hjelmstad@u...> wrote:
> > 
> > > > Hmm, I thought this *was* an example . . . you already know 
how 
> > to 
> > > do 
> > > > the optimization using the mapping between primes and 
> generators, 
> > > > right?
> > > 
> > > Yes. I guess what I meant was, could you give me the pre-
> optimized 
> > > generator in this case. Thanx!
> > 
> > I don't know what you mean by 'pre-optimized' generator! Sorry, I 
> > really do want to help . . .
> 
> Hmm. Sorry, let me approach it this way. in 12&19, you have 5/12 for
> the one and 8/19 for the other. How do you come up with one raw
> generator-to-prime mapping.

As I was saying, for each prime, use the mapping common to both 5/12 
and 8/19, which is

prime 2 = 1 period
prime 3 = 2 periods - 1 generator
prime 5 = 4 periods - 4 generators

> And then is rms applied after
> that?

yes, you want to 'solve' the above system of equations for the 
generator, to minimize your desired error function.


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Message: 8568

Date: Wed, 26 Nov 2003 20:24:00

Subject: Re: Finding Generators to Primes etc

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
> > <paul.hjelmstad@u...> wrote:
> > >
> > > Hmm. Sorry, let me approach it this way. in 12&19,
> > > you have 5/12 for the one and 8/19 for the other.
> > > How do you come up with one raw generator-to-prime mapping.
> > 
> > As I was saying, for each prime, use the mapping 
> > common to both 5/12 and 8/19, which is
> > 
> > prime 2 = 1 period
> > prime 3 = 2 periods - 1 generator
> > prime 5 = 4 periods - 4 generators
> 
> 
> i'm not understanding a lot of this ...

This should be very familiar territory for you -- it's just meantone 
we're talking about here.

> but i am curious
> about this: why are you overshooting the prime with the
> periods and then subtracting generators (instead of
> coming as close under the prime as you can with the periods,
> then adding generators)?

Because that's how meantone approximates the primes.

> the latter is the way i've always thought of prime-mapping.
> is there some special reason to do it "backwards" like this?

Monz, sometimes you have to use minus signs. This is one of those 
cases. You might think that all you have to do is restate the 
generator to be the fifth (7/12 oct. or 11/19 oct.) instead of the 
fourth, and then you won't have any minus signs. But that wouldn't 
help in other cases, for example schismic:

If we state the generator of schismic as the fourth (~498.3 cents), 
we have

prime 2 = 1 period
prime 3 = 2 periods - 1 generator
prime 5 = -1 period + 8 generators

If we state the generator of schismic as the fifth (~701.7 cents), we 
have

prime 2 = 1 period
prime 3 = 1 periods + 1 generator
prime 5 = 7 periods - 8 generators

So either way, we have minus signs to contend with. In general, they 
will be needed, and one should not be afraid of them.

It's fairly conventional around here to state the generator in 
smallest possible (cents) terms.


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Message: 8569

Date: Wed, 26 Nov 2003 20:24:55

Subject: Re: Finding Generators to Primes etc

From: Paul Erlich

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

> I see, finally. Now my triangle is complete, Generators - Commas -
> Temperaments. Thanks

Now *I* need to figure out how to get generators from commas!


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Message: 8571

Date: Wed, 26 Nov 2003 20:59:05

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:
> 
> > Now *I* need to figure out how to get generators from commas!
> 
> My personal approach is to turn everything into wedgies, and then 
> derive everything *from* wedgies.

if there's only one comma, is that the wedgie?


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Message: 8573

Date: Wed, 26 Nov 2003 21:11:16

Subject: Re: Finding Generators to Primes etc

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
> > <paul.hjelmstad@u...> wrote:
> > 
> > > I see, finally. Now my triangle is complete, Generators - 
Commas -
> > > Temperaments. Thanks
> > 
> > Now *I* need to figure out how to get generators from commas!
> 
> Graham Breed has a website that shows how to do this with matrices.
> (I also want to learnt the wedgie way...) It's pretty cool ...
> 
> Linear temperaments from matrix formalism *

Isn't he *assuming* that the fifth is the generator here? Sorry, i'm 
having trouble following his reasoning . . .


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Message: 8574

Date: Wed, 26 Nov 2003 21:20:06

Subject: Re: Finding Generators to Primes etc

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
> <paul.hjelmstad@u...> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> > wrote:
> > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
> > > <paul.hjelmstad@u...> wrote:
> > > 
> > > > I see, finally. Now my triangle is complete, Generators - 
> Commas -
> > > > Temperaments. Thanks
> > > 
> > > Now *I* need to figure out how to get generators from commas!
> > 
> > Graham Breed has a website that shows how to do this with 
matrices.
> > (I also want to learnt the wedgie way...) It's pretty cool ...
> > 
> > Linear temperaments from matrix formalism *
> 
> Isn't he *assuming* that the fifth is the generator here? Sorry, 
i'm 
> having trouble following his reasoning . . .

But what I really want is a way of getting it through geometrical 
understanding, yet without assuming a metric . . .


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