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Message: 8550 - Contents - Hide Contents 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/b... * [with cont.] (Wayb.) ExpTheGeneralizedProduct.pdf Is this 'generalized product' of Grassman product nearly, but not quite, identical to the geometric product?

Message: 8551 - Contents - Hide Contents Date: Mon, 24 Nov 2003 05:43:54 Subject: a beautiful geometric algebra paper From: Paul Erlich %PDF-1.2 * [with cont.] (Wayb.) Should I ignore this one too? GABLE said it was using different conventions than Hestenes, so maybe this paper actually agrees with Grassmann Algebra?

Message: 8553 - Contents - Hide Contents 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? So, can we get a version of your Gospel with this rolled in? -Carl

Message: 8554 - Contents - Hide Contents 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? > > PaulWell, 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.

Message: 8555 - Contents - Hide Contents 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!

Message: 8557 - Contents - Hide Contents 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" >> >>> --- 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?

Message: 8558 - Contents - Hide Contents 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

Message: 8560 - Contents - Hide Contents 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/ * [with cont.] -Carl

Message: 8561 - Contents - Hide Contents 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 . . .

Message: 8562 - Contents - Hide Contents Date: Tue, 25 Nov 2003 01:29:33 Subject: Re: Finding Generators to Primes etc From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> 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.I don't believe in the 0-equal temperament.

Message: 8563 - Contents - Hide Contents 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 . . .

Message: 8564 - Contents - Hide Contents Date: Tue, 25 Nov 2003 22:35:13 Subject: Re: Finding the complement From: Gene Ward Smith --- 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.Browne has a certain point of view which isn't always what we need; for us the bracket between vals and monzos gives us what we want without a metric, or serves in place of one.> No, I don't think you need covariant vs contravariant.For our purposes this most certainly is what we want.> 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.You don't actually need a metric.> Yes, vectors represent both vals and monzos. Bad idea.If you don't like them

Message: 8566 - Contents - Hide Contents 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" >>> >>>> 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.

Message: 8568 - Contents - Hide Contents 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" >>>> >>> 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.

Message: 8569 - Contents - Hide Contents 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. ThanksNow *I* need to figure out how to get generators from commas!

Message: 8570 - Contents - Hide Contents Date: Wed, 26 Nov 2003 20:48:12 Subject: Re: Finding Generators to Primes etc From: Gene Ward Smith --- 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.

Message: 8571 - Contents - Hide Contents 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?

Message: 8573 - Contents - Hide Contents 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" >>> >>> 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 * [with cont.] (Wayb.)Isn't he *assuming* that the fifth is the generator here? Sorry, i'm having trouble following his reasoning . . .

Message: 8574 - Contents - Hide Contents 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" >>>>> >>>> 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 * [with cont.] (Wayb.) >> 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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