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Message: 2100 - Contents - Hide Contents Date: Mon, 26 Nov 2001 08:58:23 Subject: Re: Wedge products and the torsion mess From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> 2048/2025 and 50/49 cannot be extended in a non-torsion way to three > 7-limit intervals, in other words, which would be suitable for a > block.Well that's a nice clarification. So perhaps it would have been better to focus on scales, rather than linear temperaments, after all!
Message: 2101 - Contents - Hide Contents Date: Mon, 26 Nov 2001 09:02:05 Subject: Re: Wedge products and the torsion mess From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> So perhaps it would have been better to focus on scales, rather than > linear temperaments, after all!Not really--we will still get uniqueness after booting out the torsion crud, and some of the things I am getting this way it would not have occured to look at. I still plan on seeing if we are missing something we shouldn't by looking at it from the other side also. Of course this is one more piece of weirdness it probably would be a pain to explain to a non-mathematical readership. :(
Message: 2102 - Contents - Hide Contents Date: Mon, 26 Nov 2001 09:03:51 Subject: Re: Survey From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:>> In either case, could you please explain the mathematical >> criterion that defines "Minkowski reduced", as you did for LLL? >> Let p/q be reduced to lowest terms; then T(p/q) = pq. A pair of > intervals {p/q, r/s} with p/q>1, r/s>1, T(p/q) < T(r/s) and p/q and > r/s independent is Minkowski reduced iff the only numbers in the set > {(p/q)^i (r/s)^j} such that T(t/u) < T(r/s) are powers of p/q.That's astoundingly simple! Wouldn't it be quite reasonable to further require that the only ratio t/u in the set {(p/q)^i (r/s)^j} such that T(t/u) < T(r/s), is p/q itself? The idea would be that otherwise, the two unison vectors are "mismatched".
Message: 2103 - Contents - Hide Contents Date: Mon, 26 Nov 2001 09:06:38 Subject: Re: Survey From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> That's astoundingly simple! Wouldn't it be quite reasonable to > further require that the only ratio t/u in the set {(p/q)^i (r/s) ^j} > such that T(t/u) < T(r/s), is p/q itself? The idea would be that > otherwise, the two unison vectors are "mismatched".I think it would be worthwhile to take a look and see if we miss anything which is really a keeper that way, and if not, go for it.
Message: 2104 - Contents - Hide Contents Date: Mon, 26 Nov 2001 09:20:52 Subject: Re: Survey From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> That's astoundingly simple! Wouldn't it be quite reasonable to >> further require that the only ratio t/u in the set {(p/q)^i (r/s) > ^j}>> such that T(t/u) < T(r/s), is p/q itself? The idea would be that >> otherwise, the two unison vectors are "mismatched". >> I think it would be worthwhile to take a look and see if we miss > anything which is really a keeper that way, and if not, go for it.I think this is priority #1. Simple, good, complicated, bad. The two shortest unison vectors. Nice. If we can narrow it down this way, we might really have a presentable paper together in time for publication. Offhand questions: (a) Can all, or a great majority, of the systems be expressed in terms of two superparticular unison vectors? (b) Are all systems expressible in terms of two superparticular unison vectors torsion-free?
Message: 2105 - Contents - Hide Contents Date: Mon, 26 Nov 2001 09:23:54 Subject: Re: Survey From: Paul Erlich> --- In tuning-math@y..., genewardsmith@j... wrote: >>>> In either case, could you please explain the mathematical >>> criterion that defines "Minkowski reduced", as you did for LLL? >>>> Let p/q be reduced to lowest terms; then T(p/q) = pq. A pair of >> intervals {p/q, r/s} with p/q>1, r/s>1, T(p/q) < T(r/s) and p/q and >> r/s independent is Minkowski reduced iff the only numbers in the > set>> {(p/q)^i (r/s)^j} such that T(t/u) < T(r/s) are powers of p/q.Rather than "Minkowski reduced", why don't we call this particular definition the definition of "Tenney-Minkowski" reduced; or, if Minkowski himself isn't really involved in any big way, "Tenney- Smith" reduced?
Message: 2106 - Contents - Hide Contents Date: Mon, 26 Nov 2001 09:39 +0 Subject: Re: Tormented by Torsion again From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9tstid+cipm@xxxxxxx.xxx>> I was just about to post using the above subject line when I saw > this. I'm bummed :) The reason is that 2048/2025 * 50/49 = (64/64)^2, > but no product of 2048/2025 and 50/49 makes 64/63. Neither LLL nor > Minkowski got rid of this problem, so I will need to check all the > results and see how to cure this disease.Gene! This is exactly the problem I've been trying to solve since before you arrived here! Now, if you look at the example you pointed out at the weekend, you'll see even the "right" pair of vectors seem to have torsion. Graham
Message: 2107 - Contents - Hide Contents Date: Mon, 26 Nov 2001 09:59:55 Subject: Re: Tormented by Torsion again From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> Gene! This is exactly the problem I've been trying to solve since before > you arrived here! Now, if you look at the example you pointed out at the > weekend, you'll see even the "right" pair of vectors seem to have torsion. > > > GrahamWhat example was that? I'm guessing that in that example, a power of p/q is shorter than r/s.
Message: 2108 - Contents - Hide Contents Date: Mon, 26 Nov 2001 10:48 +0 Subject: Re: Tormented by Torsion again From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9tt3qr+qem5@xxxxxxx.xxx> Me:>> Gene! This is exactly the problem I've been trying to solve since > before>> you arrived here! Now, if you look at the example you pointed out > at the>> weekend, you'll see even the "right" pair of vectors seem to have > torsion. Paul:> What example was that? I'm guessing that in that example, a power of > p/q is shorter than r/s.[-1, 0, -2, 2] [1, 7, -4, -1]. I can't get rid of the 7 in the second column. I can prove you can't get rid of the factor of 2 in the third column. For the second and fourth columns to be even, you have to add an even multiple of the lower vector to the higher. But in the first column, that means adding an even number to a multiple of -1, so you need to multiply the higher vector by an even number as well. Which means both are being multiplied by 2, and subsequently dividing by 2 doesn't simplify anything. I don't know what wedge products are or their relevance. Or what p, q, r and s are supposed to be. But we have two problems: to divide through by common factors and to know what common factors we don't have to divide through by. Graham
Message: 2109 - Contents - Hide Contents Date: Mon, 26 Nov 2001 11:55:14 Subject: Re: Tormented by Torsion again From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:>> What example was that? I'm guessing that in that example, a power of >> p/q is shorter than r/s. >> [-1, 0, -2, 2] 49/50 > [1, 7, -4, -1]. 4374/4375Sure enough, (49/50)^2 = 2401/2500 is simpler than 4374/4375. Hence we have a mismatch of unison vectors. I think torsion rules the realm outside the "strong Minkowski" condition I proposed.
Message: 2110 - Contents - Hide Contents Date: Mon, 26 Nov 2001 12:52 +0 Subject: Re: Tormented by Torsion again From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9ttaj2+539k@xxxxxxx.xxx> In article <9ttaj2+539k@xxxxxxx.xxx>, paul@xxxxxxxxxxxxx.xxx (Paul Erlich) wrote:> --- In tuning-math@y..., graham@m... wrote: >>>> What example was that? I'm guessing that in that example, a power > of>>> p/q is shorter than r/s. >>>> [-1, 0, -2, 2] > > 49/50 >>> [1, 7, -4, -1]. > > 4374/4375 >> Sure enough, (49/50)^2 = 2401/2500 is simpler than 4374/4375. Hence > we have a mismatch of unison vectors. I think torsion rules the realm > outside the "strong Minkowski" condition I proposed.Ah, but how about 64:63 and 3125:3087? [0, -2, 5, -3] [6, -2, 0, -1] Lots of common factors, but (64:63)^2 = 4096:3969. I'm not clear about the relevance of torsion here. Really, you should include the chromatic unison vector to see if the basis has torsion. So how do we know when we've gone far enough in removing those common factors if we don't have a chromatic UV? In case you haven't caught up with the simplification process, I'll work through your example of 2048:2025 and 50:49 [11 -4 -2 0] [ 1 0 2 -2] All you have to do is add them [11 -4 -2 0] +[1 0 2 -2] = [12 -4 0 -2] and divide the result through by 2 [6 -2 0 -1] which is 64:63. If [12 -4 0 -2] is a unison, so must [6 -2 0 -1] be. It also sounds like a cop-out to say the UVs are "mismatched". 49/50 and 4374/4375 remain the simplest way of producing the mapping [(2, 0), (2, 5), (3, 7), (4, 7)] does that mean the temperament itself is a mismatch? Incidentally, the bad vectors I got for this, 17496:16807 and 9765625:9565938, pass the square test. I'm having difficulty seeing any relevance to that. Graham
Message: 2111 - Contents - Hide Contents Date: Mon, 26 Nov 2001 13:12:01 Subject: Re: Tormented by Torsion again From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <9ttaj2+539k@e...> > In article <9ttaj2+539k@e...>, paul@s... (Paul Erlich) > wrote: >>> --- In tuning-math@y..., graham@m... wrote: >>>>>> What example was that? I'm guessing that in that example, a power >> of>>>> p/q is shorter than r/s. >>>>>> [-1, 0, -2, 2] >> >> 49/50 >>>>> [1, 7, -4, -1]. >> >> 4374/4375 >>>> Sure enough, (49/50)^2 = 2401/2500 is simpler than 4374/4375. Hence >> we have a mismatch of unison vectors. I think torsion rules the realm >> outside the "strong Minkowski" condition I proposed. >> Ah, but how about 64:63 and 3125:3087? > > [0, -2, 5, -3] > [6, -2, 0, -1] > > Lots of common factors, but (64:63)^2 = 4096:3969.So this is a good one. Why do you bring this example up? Isn't it just a normal example?> It also sounds like a cop-out to say the UVs are "mismatched". 49/50 > and 4374/4375 remain the simplest way of producing the mapping [(2, 0), > (2, 5), (3, 7), (4, 7)] does that mean the temperament itself is a > mismatch?That's what I'm suggesting we may wish to say. Look at it -- complexity of 14, worst error of 17.5 cents. The 4374/4375 is powerless to do any good in the presence of 49/50.
Message: 2112 - Contents - Hide Contents Date: Mon, 26 Nov 2001 13:15:10 Subject: Re: Survey From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., graham@m... wrote: >>> I can't even work out how to simplify the original vectors of >> [-1, 0, -2, 2] and [1, 7, -4, -1]. I'm worryingly close to > deciding that>> it can't be done. >> I get that <49/48, 4375/4374> is already Minkowski reduced,You mean 50/49, not 49/48, right?
Message: 2113 - Contents - Hide Contents Date: Mon, 26 Nov 2001 14:15 +0 Subject: Re: Tormented by Torsion again From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9ttf31+gk9@xxxxxxx.xxx> Me:>> Ah, but how about 64:63 and 3125:3087? >> >> [0, -2, 5, -3] >> [6, -2, 0, -1] >> >> Lots of common factors, but (64:63)^2 = 4096:3969. Paul:> So this is a good one. Why do you bring this example up? Isn't it > just a normal example?I don't know, what is "normal"? You can't combine these vectors to get a 1 in any column but the last. That has something to do with torsion, but I don't think it's the real problem.>> It also sounds like a cop-out to say the UVs are "mismatched". > 49/50>> and 4374/4375 remain the simplest way of producing the mapping [(2, > 0),>> (2, 5), (3, 7), (4, 7)] does that mean the temperament itself is a >> mismatch? >> That's what I'm suggesting we may wish to say. Look at it -- > complexity of 14, worst error of 17.5 cents. The 4374/4375 is > powerless to do any good in the presence of 49/50.50:49 and 245:243 give a complexity of 10 and a worst error of 17.5 cents. But there's no problem with them. Exactly how complex do you have to get to be a mismatch? The rule sounds arbitrary. I thought it was supposed to show that the vectors hadn't been reduced properly, but it doesn't work in all cases. Another example: 63:64 and 1024:1029 give 256:243 and 16807:15552 from my program. That is [-6, 2, 0, 1] [10, -1, 0, -3] comes out as [8, -5, 0, 0] [-6, -5, 0, 5] when we should have got [-6, 2, 0, 1] [ 4, 1, 0, -2] 49:48 and 63:64. But (256:243)^2=65536:59049 which is more complex than 16807:15552. So the test tells us nothing. Another example, 125:126 and 245:243 give 78732:78125 1647086:1594323. But 1647086:1594323 is simpler than 6198727824:6103515625. And again, 125:126 and 4374:4375 give 78732:78125 and 1647086:1594323. 225:224 and 245:243 give 3125:3072 and 537824:531441 225:224 and 1715:1728 give 839808:823543 and 2109375:2097152 245:243 and 4374:4375 give and 78732:78125 1647086:1594323 4374:4375 and 1715:1728 give 390625000:387420489 and 2038431744:1977326743 4374:4375 and 3125:3136 give 1224440064:1220703125 and 50797745488265216:50031545098999707 1715:1728 and 3125:3136 give 782757789696:762939453125 and 240734712102912:232630513987207 In all these cases, the bad vectors pass the square test. It looks like the special thing about the 2048:2025 and 50:49 example is that one of the vectors is correct. So does that mean Gene already has a way of handling bad pairs of vectors? As I don't know what algorithm he uses, I'm assuming it happens to get bad cases less often, but doesn't avoid them completely. Graham
Message: 2114 - Contents - Hide Contents Date: Mon, 26 Nov 2001 14:30 +0 Subject: Re: Survey From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9tt1hk+ua76@xxxxxxx.xxx> Paul asked:> (a) Can all, or a great majority, of the systems be expressed in > terms of two superparticular unison vectors?Well, I don't know of a pair for septimal schismic. The simplest are 225:224 and 5120:5103. With 5-limit temperaments, you don't have any choice. Graham
Message: 2115 - Contents - Hide Contents Date: Mon, 26 Nov 2001 14:36:31 Subject: Re: Tormented by Torsion again From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <9ttf31+gk9@e...> > Me:>>> Ah, but how about 64:63 and 3125:3087? >>> >>> [0, -2, 5, -3] >>> [6, -2, 0, -1] >>> >>> Lots of common factors, but (64:63)^2 = 4096:3969. > > Paul:>> So this is a good one. Why do you bring this example up? Isn't it >> just a normal example? >> I don't know, what is "normal"? Regular, ordinary. > You can't combine these vectors to get a > 1 in any column but the last.So? Why would you want to?> That has something to do with torsion, but > I don't think it's the real problem.So what exactly _is_ the problem with this system? Is there one?>>> It also sounds like a cop-out to say the UVs are "mismatched". >> 49/50>>> and 4374/4375 remain the simplest way of producing the mapping [(2, >> 0),>>> (2, 5), (3, 7), (4, 7)] does that mean the temperament itself is a >>> mismatch? >>>> That's what I'm suggesting we may wish to say. Look at it -- >> complexity of 14, worst error of 17.5 cents. The 4374/4375 is >> powerless to do any good in the presence of 49/50. >> 50:49 and 245:243 give a complexity of 10 and a worst error of 17.5 cents. > But there's no problem with them. Exactly how complex do you have to get > to be a mismatch? The rule sounds arbitrary.The idea is that if you were willing to temper 50:49, surely you'd also be willing to temper some interval in the lattice that you found sooner than 4374:4375.> I thought it was supposed > to show that the vectors hadn't been reduced properly,_If_ the rule is satisfied, _then_ the vectors are reduced properly. The converse doesn't hold, but the contrapositive does.> > 63:64 and 1024:1029 give 256:243 and 16807:15552 from my program. > > That is > > [-6, 2, 0, 1] > [10, -1, 0, -3] > > comes out as > > [8, -5, 0, 0] > [-6, -5, 0, 5] > > when we should have got > > [-6, 2, 0, 1] > [ 4, 1, 0, -2] > > 49:48 and 63:64.I don't know what your program is doing, or why. I wasn't talking about your program anyway.> But (256:243)^2=65536:59049 which is more complex than > 16807:15552. So the test tells us nothing.Graham, graham -- this is not a correct application of the test! 256:243 and 65536:59049 is _not_ a Minkowski-reduced basis for this system, so 256:243 and 65536:59049 don't pass "weak Minkowski", let alone "strong Minkowski"!! 49:48 and 63:64 is the Minkowski-reduced basis. _Those_ are the vectors that are supposed to satisfy the "strong Minkowski" condition.> Another example, 125:126 and 245:243 give 78732:78125 > 1647086:1594323.Forget about that. Your program is doing something weird. 125:126 and 245:243 is already Minkoski-reduced!
Message: 2116 - Contents - Hide Contents Date: Mon, 26 Nov 2001 15:05 +0 Subject: Re: Tormented by Torsion again From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9ttk1g+e71q@xxxxxxx.xxx> Paul wrote:> The idea is that if you were willing to temper 50:49, surely you'd > also be willing to temper some interval in the lattice that you found > sooner than 4374:4375.Well, that's up to you. I don't see what we gain by enforcing the square rule.>> I thought it was supposed >> to show that the vectors hadn't been reduced properly, >> _If_ the rule is satisfied, _then_ the vectors are reduced properly. > The converse doesn't hold, but the contrapositive does.What's a contrapositive? I've given an example that's fine when the rule isn't satisfied, and lots that are fine even though it isn't.>> But (256:243)^2=65536:59049 which is more complex than >> 16807:15552. So the test tells us nothing. >> Graham, graham -- this is not a correct application of the test! > 256:243 and 65536:59049 is _not_ a Minkowski-reduced basis for this > system, so 256:243 and 65536:59049 don't pass "weak Minkowski", let > alone "strong Minkowski"!! 49:48 and 63:64 is the Minkowski-reduced > basis. _Those_ are the vectors that are supposed to satisfy > the "strong Minkowski" condition.Why aren't they Minkowski reduced? Okay, my algorithm isn't perfect, but even if it didn't do the reduction perfectly, it'll never get to 49:48 and 63:64 using {(p/q)^i (r/s)^j} assuming i and j are integers. It's the cases where they aren't we're having trouble with. And if they're allowed, the test becomes redundant anyway. Gene did say in the original case "Neither LLL nor Minkowski got rid of this problem" so I assumed they did have to be integers. If i and j are allowed to be irrational, you can get anything you like, so shall we say they have to be rational?>> Another example, 125:126 and 245:243 give 78732:78125 >> 1647086:1594323. >> Forget about that. Your program is doing something weird. 125:126 and > 245:243 is already Minkoski-reduced!It doesn't matter where they came from. They aren't properly reduced but they pass your square test. Oh, while I'm posting, the 64:63 and 3125:3087 system is interesting because 3125:3136 is one of the original vectors. So there's a superparticular ratio that isn't one of the simplest pair. Graham
Message: 2117 - Contents - Hide Contents Date: Mon, 26 Nov 2001 16:20:03 Subject: Re: Tormented by Torsion again From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <9ttk1g+e71q@e...> > Paul wrote: >>> The idea is that if you were willing to temper 50:49, surely you'd >> also be willing to temper some interval in the lattice that you found >> sooner than 4374:4375. >> Well, that's up to you. I don't see what we gain by enforcing the square > rule.It's not a square rule. But what we gain is simplicity.>>> I thought it was supposed >>> to show that the vectors hadn't been reduced properly, >>>> _If_ the rule is satisfied, _then_ the vectors are reduced properly. >> The converse doesn't hold, but the contrapositive does. >> What's a contrapositive? I've given an example that's fine when the rule > isn't satisfied, and lots that are fine even though it isn't.Right -- but I'm claiming you won't find an example that's _not_ fine when the rule is satisfied.>>> But (256:243)^2=65536:59049 which is more complex than >>> 16807:15552. So the test tells us nothing. >>>> Graham, graham -- this is not a correct application of the test! >> 256:243 and 65536:59049 is _not_ a Minkowski-reduced basis for this >> system, so 256:243 and 65536:59049 don't pass "weak Minkowski", let >> alone "strong Minkowski"!! 49:48 and 63:64 is the Minkowski- reduced >> basis. _Those_ are the vectors that are supposed to satisfy >> the "strong Minkowski" condition. >> Why aren't they Minkowski reduced? Okay, my algorithm isn't perfect, but > even if it didn't do the reduction perfectly, it'll never get to 49:48 and > 63:64 using > > {(p/q)^i (r/s)^j} > > assuming i and j are integers.Yes, they are integers -- I thought you said 49:48 and 63:64 was equivalent to this, and I took "your" word for it. So what _do_ you get when you Minkowski-reduce this system? Gene?>>>> Another example, 125:126 and 245:243 give 78732:78125 >>> 1647086:1594323. >>>> Forget about that. Your program is doing something weird. 125:126 and >> 245:243 is already Minkoski-reduced! >> It doesn't matter where they came from. They aren't properly reduced but > they pass your square test.No, no, no! Again, my test _isn't_ a square test! It's a "strong Minkowski" condition, which includes Gene's Minkowski condition within it! My test, in case this wasn't clear, says that the shorter member of the reduced basis is the _only_ member of the kernel that is shorter than the longer member of the reduced basis. Is that clear? It's _not_ a square test -- though in some cases, the test may fail _because_ the square of the shorter vector is shorter than the longer vector. But that's not the only condition under which the test can fail!> Oh, while I'm posting, the 64:63 and 3125:3087 system is interesting > because 3125:3136 is one of the original vectors. So there's a > superparticular ratio that isn't one of the simplest pair.3125:3136 is not superparticular!> > > Graham
Message: 2118 - Contents - Hide Contents Date: Mon, 26 Nov 2001 17:01 +0 Subject: Re: Tormented by Torsion again From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9ttq3j+5tck@xxxxxxx.xxx> Me:>> Well, that's up to you. I don't see what we gain by enforcing the > square >> rule. Paul:> It's not a square rule. But what we gain is simplicity.How? And why is this worth losing perfectly valid results? Me:>> What's a contrapositive? I've given an example that's fine when > the rule>> isn't satisfied, and lots that are fine even though it isn't. Paul:> Right -- but I'm claiming you won't find an example that's _not_ fine > when the rule is satisfied. Sorry, typo. Me: >> {(p/q)^i (r/s)^j} >>>> assuming i and j are integers. Paul:> Yes, they are integers -- I thought you said 49:48 and 63:64 was > equivalent to this, and I took "your" word for it. So what _do_ you > get when you Minkowski-reduce this system? Gene?Okay, we'll assume 256:243 and 16807:15552 are Minkowski reduced then. 63:64 and 49:48 describe the same temperament. Incidentally, you did say 256:243 and 65536:59049 before, but my comments were for 256:243 and 16807:15552, although I didn't point that out. 65536:59049 is (256:243)^2. So 256:243 and 16807:15552 aren't Minkowski reduced, but there's no particular reason why they should be. The vectors are [8, -5, 0, 0] [-6, -5, 0, 5] How could you combine them to get anything simpler? My program's already checking the simplest cases. [14, 0, 0, -5] being the obvious one.>>> 16384*16807 275365888 >>> 16807*15552 261382464I'm certain these are Minkowski reduced, if I've got the definition right. Me:>> It doesn't matter where they came from. They aren't properly > reduced but>> they pass your square test. Paul:> No, no, no! Again, my test _isn't_ a square test! It's a "strong > Minkowski" condition, which includes Gene's Minkowski condition > within it!Well, whatever you call it, they pass.> My test, in case this wasn't clear, says that the shorter member of > the reduced basis is the _only_ member of the kernel that is shorter > than the longer member of the reduced basis. Is that clear? It's > _not_ a square test -- though in some cases, the test may fail > _because_ the square of the shorter vector is shorter than the longer > vector. But that's not the only condition under which the test can > fail!But without that condition, it becomes Gene's Minkowski, which these pairs pass. So how about Graham's Strong Minkowski test, where we start with Gene's definition, but allow i and j to be rationals? Keeping p/q, r/s and t/u as rationals. You could also replace (p/q) with (p/q)^i to make your test pass, which definitely isn't right.>> Oh, while I'm posting, the 64:63 and 3125:3087 system is > interesting>> because 3125:3136 is one of the original vectors. So there's a >> superparticular ratio that isn't one of the simplest pair. >> 3125:3136 is not superparticular! oops Graham
Message: 2119 - Contents - Hide Contents Date: Tue, 27 Nov 2001 19:53:37 Subject: Re: More fun with wedge products From: genewardsmith@xxxx.xxx --- In tuning-math@y..., graham@m... wrote:> What's a val?It's something which takes an interval and sends it to an integer. Examples would be: (1) An et; the val h12 for instance sends 2/1 to 12, 3/1 to 19, 5/4 to 4, etc. (2) A mapping of generators to primes, where the val tells you how many generator steps you need for a particular prime. From this you know how many generator steps to get to a p-limit interval, and that defines a val. (3) A p-adic valuation is a val. For instance, the valuation "v3" counts how many powers of 3, positive or negative, we find in a particular rational number, so that v3(9/8)=2, v3(5/4)=0, v3(5/3)=-1 and so forth. Vals are dual to intervals, since an interval takes a val and sends it to an integer; therefore, if intervals are represented by row vectors of integers, a val is a column vector of integers.> These wedge products seem to behave like the vector products we use in > physics. Are they similar? The same thing?The three dimensional wedge product sends a vector with basis e1, e2, e3 to one with basis e1^e2, e2^e3, e3^e1. Using the inner product, we can identify the two, which results in the vector product. Hence the vector product is just the 3-dimensional version of the wedge product.>> The {64/63, 49/48} system has 5, 10, and 15 for vals--in other words, >> these are common kernel elements. If we write them as vectors, we get >> >> h5: [ 5] h10: [10] h15: [15] >> [ 8] [16] [35] >> [12] [23] [35] >> [14] [28] [42] >> So these are the normal ET mappings. How did you get them?I checked to see what was dual to {64/63,49/48}, or in other words for what values of n I got hn(64/63) = hn(49/48) = 0. I notice that> h15 isn't the same as h5+h10. Is that right? Certainly. > Could the wedge product be used to remove the torsion, or only recognize > it? I'd still like a routing for working backwards from the mapping to > the unison vectors.You can recognize torsion easily with the wedge product. Removing torsion using it would also be possible, since you could look for vectors which when wedged with the wedge product give 0.> What algorithm are you using to find the temperaments? I don't remember > seeing it.I find two linearly independent vals, and LLL reduce them. I then convert this (which I call "map") to a version where one of the generators is an nth part of an octave, by column operations, and call this "reduced map". I usually put the second generator within the first, and always make them inside an octave. It might be useful to see your wedge product routine as well,> if they turn out to be important.I gave the definitions, that's about all the routine you need. But need to fix that again, I see!
Message: 2120 - Contents - Hide Contents Date: Tue, 27 Nov 2001 22:11:47 Subject: Re: Standardizing our wedge product From: genewardsmith@xxxx.xxx Graham is familiar with the cross-product, and chances are Paul and others are as well, so I think we should incorporate them as much as possible into a standard for the wedge product. Let's see if I can finally agree with myself on this! Definition of wedge product of 7-limit intervals Let [u1, u2, u3, u4] and [v1, v2, v3, v4] be two intervals r and s in the prime-power basis, so that r = 2^u1 3^u2 5^u3 7^u4 and s = 2^v1 3^v1 5^v3 7^v4. Then the wedge product r^s = [u1, u2, u3, u4]^[v1, v2, v3, v4] is defined as: [[u2,u3,u4] X [v2,v3,v4], u1*[v2,v3,v4]-v1*[u2,u3,u4]], where "X" is the cross-product and "*" is the scalar product. Written out in full, this would be r^s = [u3*v4-v3*u4,u4*v2-v4*u2,u2*v3-v2*u3,u1*v2-v1*u2, u1*v3-v1*u3,u1*v4-v1*u4] Definition of wedge product of 7-limit vals Let h = [u1,u2,u3,u4] and g = [v1,v2,v3,v4] be two vals in the prime-valuation basis, so that h(2) = u1, h(3)=u2, h(5)=u3, h(7)=u4 and g(2) = v1, g(3) = v2, g(5) = v3 and g(7) = v4. Then the wedge product h^g = [u1,u2,u3,u4]^[v1,v2,v3,v4] is defined as: [u1*[v2,v3,v4]-v1*[u2,u3,u4], [u2,u3,u4] X [v2,v3,v4]] Written out in full, this is h^g = [u1*v2-v1*u2,u1*v3-v1*u3,u1*v4-v1*u4,u3*v4-v3*u4, u4*v2-v4*u2,u2*v3-v2*u3] Note that the first three elements of the wedge invariant, defined from any starting place, gives the mapping to primes of the non- octave generator of the pair of generators. From the interval point of view, it is the cross-product of Graham's beloved octave equivalence classes. From the val point of view, it is the combination of vals I have been using for various purposes.
Message: 2121 - Contents - Hide Contents Date: Tue, 27 Nov 2001 22:40:21 Subject: Re: Standardizing our wedge product From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> Graham is familiar with the cross-product,Is that what you guys were referring to as the "vector product"? It seemed to be.> and chances are PaulYes . . .
Message: 2122 - Contents - Hide Contents Date: Tue, 27 Nov 2001 20:12:56 Subject: Unlocking the mysteries of the wedge invariant From: Gene W Smith This can be defined in multiple ways as cross-products involving intervals and vals, leading to multiple interpretations. We have for instance Interval product: [u1,u2,u3,u4]^[v1,v2,v3,v4] = [[u2,u3,u4]X[v2,v3,v4], [v3,v2,v1]X[u3,u2,u1]] These cross-products can be thought of either as involving interval classes, or intervals for which one of the prime powers are zero; from the latter point of view something perpendicular would be a val which mapped one of the primes to zero. We can exclude one of the primes in turn, and take the cross product, and compare it to the wedge invariant, and see how it can be interpreted in terms of vals. If the wedge invariant is [a,b,c,d,e,f], we find we get in this way four vals: [ 0] [ a] [ b] [ c] [ a] [ 0] [ f] [-e] [ b] [-f] [ 0] [ d] [ c] [ e] [-a] [ 0] Similarly, a cross-product of two vals represents an interval in the kernel of both, and so from the wedge invariant we get four commas: 2^f 3^-b 5^a, 3^d 5^e 7^f, 2^d 5^-c 7^b, 2^e 3^c 7^-a It seems therefore that going from the wedge invariant to the temperament it signifies is not too difficult. Here is paultone as an example: 64/63^50/49 = h12^h22 = [2,-4,-4,2,12,-11] We get the four vals: [ 0] [ 2] [ -4] [ -4] [ 2] [ 0] [-11] [-12] [-4] [11] [ 0] [ 2] [-4] [12] [ -2] [ 0] All these have 64/63 and 50/49 in the kernel. We also get the four commas: 2^-11 3^4 5^2 = 2025/2048 3^2 5^12 7^-11 = 2197265625/1977326743 2^2 5^4 7^-4 = (50/49)^2 2^12 3^-4 7^-2 = (64/63)^2 All these are commas of the temperament.
Message: 2123 - Contents - Hide Contents Date: Tue, 27 Nov 2001 04:49:04 Subject: Wedge products From: genewardsmith@xxxx.xxx The wedge product is associative and distributive, but the key thing to remember about it is that it is anticommunitive, which means that u^v = - v^u. This entails that u^u = 0. When we apply this to basis vectors, we see that if we start with four basis vectors {e2, e3, e5, e7} and take wedge products, we get six basis vectors for the vector space of the wedge product, ie {e2^e3, e2^e5, e2^e7, e3^e5, e3^e7, e5^e7}. Thus 4 choose 1 = four dimensions goes to 4 choose 2 = six dimensions. If we take a wedge product in this six-dimensional space with another 4-d vector, or in other words a triple wedge product, we get a basis {e3^e5^e7, e2^e5^e7, e2^e3^e7, e2^e3^e5} of 4 choose 3 = four dimensions. If we take a quadruple wedge product, we get a 4 choose 4 = one-demensional space, with basis element {e2^e3^e5^e7}; this can therefore be identified with a scalar, which makes the quadruple wedge product the determinant. The triple wedge product, being in effect a thing which takes an interval and sends it to an integer, can be identified with a val. The triple wedge product of three 7-limit intervals is the val we get by putting the three intervals in as rows of a 4x4 matrix and making the top row the four basis vectors. This same game can, of course, be played in any number of dimensions, where we get a Pascal's triangle arrangement to the dimensions. In the 11-limit, with five dimensions, it goes 5 10 10 5 1 and we get up to quintuple wedge products. The wedge product of two intervals will be in a space of dimension n choose 2 = n(n-1)/2, and the length of the vector in the ordinary norm will be the area of the paralleogram defined by the two intervals, and the direction will be defined by the orientation of the paralleogram. In four dimensions, this *isn't* simply a vector, because a vector is perpendicular to *three* dimensions, not two. The double wedge product is therefore an oriented area, the triple wedge product an oriented volume (defined by three vectors) and so forth--down finally to the determinant. The coefficients of the double wedge product can be thought of as the areas of the paralleogram of interval classes defined by taking only two of the four primes. If there is a common factor in each such reduction, it spells torsion. We may also see the torsion in comparing lenghts of wedge products, or their coefficients. We have 64/63^49/48 = [-14, 0, 8, 0, -5, 0] = u u.u = 3*5*19 16807/15552^256/243 = [70, 0, -40, 0, 25, 0] = v v = -5 u; and v.v = 3*5^3*19, showing it is 5 times longer. If someone wants to program this, the following might help: [u1, u2, u3, u4]^[v1, v2, v3, v4] = [u1*v2-v1*u2, u1*v3-v1*u3, u1*v4-v1*u4, u2*v3-v3*u2, u2*v4-v2*u4, u3*v4-v3*u4]
Message: 2124 - Contents - Hide Contents Date: Tue, 27 Nov 2001 06:07:16 Subject: More fun with wedge products From: genewardsmith@xxxx.xxx The triple wedge product of 7-limit intervals is basically a val. We can also take wedge products of vals, and the triple product can be identified with an interval. Taking a wedge product of two intervals or two vals gives us something in the Twilight Zone halfway between an interval and a val, and we can compare them directly. The {64/63, 49/48} system has 5, 10, and 15 for vals--in other words, these are common kernel elements. If we write them as vectors, we get h5: [ 5] h10: [10] h15: [15] [ 8] [16] [35] [12] [23] [35] [14] [28] [42] Taking wedge products gives us: h5^h10 = [0, -5, 0, -8, 0, 14] h15^h10 = [0, -5, 0, -8, 0, 14] h5^h15 = [0, -5, 0, -8, 0, 14] On the other hand, we also have 64/63^49/48 = [-14, 0, 8, 0, -5, 0] This is the the same as the previous wedge products read backwards, up to a change of sign, and this is no accident. The basis element v2^v3 for vals corresponds to e5^e7 for intervals, v2^v7 corresponds to e3^e5, and so forth. These wedge products tell us that neither the vals nor the intervals are giving us torsion, and that they are all giving us the same linear temperament. The wedge product, in fact, could be used to identify linear temperaments, rather than Minkowski reduced pairs of intervals.
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