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Message: 8500 - Contents - Hide Contents 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.
Message: 8501 - Contents - Hide Contents 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 * [with cont.]
Message: 8502 - Contents - Hide Contents Date: Sat, 22 Nov 2003 08:13:54 Subject: Re: Finding Generators to Primes etc From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> This is hilarious. I couldn't parody this any better than you're doing > it yourself. :-)Your hilarity is noted. There is a way to relate the wedge product of two n-dimensional vectors to an n by n antisymmetric matrix in a natural way, and in concrete terms the Lie algebra product for o(n) can be thought of as the commutator for two such matricies. This gives us a product for pairs of linear temperaments in any prime limit, which would be nice if anyone could explain what the product meant and what use it was. If I find out, I'll be sure to tell you.
Message: 8503 - Contents - Hide Contents 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 * [with cont.]
Message: 8504 - Contents - Hide Contents Date: Sat, 22 Nov 2003 10:55:39 Subject: Re: Finding Generators to Primes etc From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Your hilarity is noted. There is a way to relate the wedge product of > two n-dimensional vectors to an n by n antisymmetric matrix in a > natural way, and in concrete terms the Lie algebra product for o(n) > can be thought of as the commutator for two such matricies. This > gives us a product for pairs of linear temperaments in any prime > limit, which would be nice if anyone could explain what the product > meant and what use it was. If I find out, I'll be sure to tell you.No doubt inevitably, I started fiddling with this and it seems there actually might be some uses. If you take the Lie bracket of two bivals, you get a bival which is orthogonal, using the usual inner product, to both. These means it really makes more sense for most purposes to consider it a bimonzo. This bracket seems to allow us to do in a more general way what I was doing in the 7-limit with pfaffians a while back--seeing whether two temperaments were related or not, and giving a measure of by how much. Taking wedgies of some important 7-limit temperaments for an example, if mean = <<1 4 10 4 13 12]] is the meantone wedgie, mir = <<6 -7 -2 -25 -20 15]] is the miracle wedgie, paj = <<2 -4 -4 -11 -12 2]] is the pajara wedgie, and orw = <<7 -3 8 -21 -7 27]] is the orwell wedgie, and if [a,b] is a modified Lie bracket, where the product is interpreted to be a bimonzo, then [mean,mir] = [mean,orw] = [orw,mir] = [[246 -223 46 -4 422 -307>> On the other hand, the brackets of mean, orw and mir with paj are all distinct. If we take the complement of these bimonzos, we get a bival again, which we can regard as a wedgie. What connection, if any, the resulting temperament has with the ones we started from I don't know. In the case of [mean,mir], the negative of the complement gives us <<307 422 404 -46 -223 -246]] This is certainly a wedgie for an honest temperament, with rms error (3.7 cents) in the vicinity of where we started from, but a much higher complexity, and so a high badness. An et which belongs to it is 31, and in 31 it is the same as the supermajor seconds temperament, with generator 6/31. We might note that 31 belongs to meantone, miracle and orwell, but not pajara.
Message: 8505 - Contents - Hide Contents 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 - Type Ok * [with cont.] (Wayb.) . . . 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.
Message: 8506 - Contents - Hide Contents Date: Sat, 22 Nov 2003 11:32:35 Subject: Re: Finding Generators to Primes etc From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> This is certainly a wedgie for an honest temperament, with rms error > (3.7 cents) in the vicinity of where we started from, but a much > higher complexity, and so a high badness.The MT basis is [[4 26 -17 -2>, [11 10 9 -17>] two not very distinguished commas. The generator mapping is, using a notation I'd like comments on, [<1 61 84 81], <0 -307 -422 -404]]
Message: 8507 - Contents - Hide Contents 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 * [with cont.] > > 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?
Message: 8508 - Contents - Hide Contents Date: Sat, 22 Nov 2003 22:23:53 Subject: Re: Finding Generators to Primes etc From: Gene Ward Smith --- 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?The orthogonal Lie algebras o(n), which concretely can be represented as commutator brackets for nxn antisymmetric matricies, is a generalization. My new thread was on how one could try to apply this to linear temperaments.
Message: 8509 - Contents - Hide Contents Date: Sat, 22 Nov 2003 22:25:36 Subject: Re: Finding the wedge product? From: Gene Ward Smith --- 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.
Message: 8510 - Contents - Hide Contents Date: Sat, 22 Nov 2003 22:31:41 Subject: Re: Finding Generators to Primes etc From: Gene Ward Smith --- 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*.
Message: 8511 - Contents - Hide Contents 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.
Message: 8512 - Contents - Hide Contents Date: Sat, 22 Nov 2003 22:44:22 Subject: Re: Definition of val etc. From: Gene Ward Smith --- 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 * [with cont.] > > 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.
Message: 8513 - Contents - Hide Contents Date: Sat, 22 Nov 2003 22:50:34 Subject: Re: contravariant vs. covariant vectors From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: I don't see anything about pseudovectors in this.
Message: 8514 - Contents - Hide Contents 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>>>
Message: 8515 - Contents - Hide Contents Date: Sat, 22 Nov 2003 22:54:16 Subject: Re: Finding the complement From: Gene Ward Smith --- 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.
Message: 8516 - Contents - Hide Contents 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.
Message: 8517 - Contents - Hide Contents 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.
Message: 8518 - Contents - Hide Contents 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
Message: 8519 - Contents - Hide Contents 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*?
Message: 8520 - Contents - Hide Contents 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 * [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.
Message: 8521 - Contents - Hide Contents 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 * [with cont.] >> >> 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 . . .
Message: 8522 - Contents - Hide Contents 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 * [with cont.] (Wayb.) >> 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!!
Message: 8523 - Contents - Hide Contents 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?
Message: 8524 - Contents - Hide Contents Date: Sun, 23 Nov 2003 07:26:31 Subject: Re: Finding the wedge product? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> We want a simple function of the (possibly compound) indexes of the > two coefficients, that gives us the sign of the result.Product of the differences might do.
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