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Message: 10550 - Contents - Hide Contents Date: Sun, 07 Mar 2004 23:06:28 Subject: Re: Octave equivalent calculations (Was: Hanzos From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Gene Ward Smith wrote: >>> The two vals <21 33 49 59| and <41 65 95 115| should lead to miracle; >> if they don't something is wrong. The same goes in the 11-limit for >> <21 33 49 59 72| and <41 65 95 115 142|. >> Why should they? They never have before.If you get a different answer, I think you must be using an incorrect algorithm.>> This is the two vals <41 65 95 115 142| and <21 33 48 59 72|, which >> leads to the question why the second val for 21-equal; it doesn't seem >> like first choice. The temperament in question, with TM basis {100/99, >> 245/243, 1029/1024}, is associated firmly to 41 at any rate. >> That val is for the best equal temperament, isn't it? Can you find a > better one?That would depend on your definition of better; but the one with the TOP tuning closest to JI is the "standard" val, <21 33 48 59 73|. If we put that together with 41, we get an 11-limit temperament which adds 245/242 rather than 385/384 to 225/224 and 1029/1024, leading to a temperament with a half-secor generator. Incidentally, if we add 243/242 to 225/224 and 1029/1024, what does your program give?>> I think you are making a good case for the claim the best plan is to >> use wedgies. >> Wedgies give identical results to matrices, so what difference does it make?At this point it is by no means clear your matrix methods are giving correct answers, and in any case you seem to be working a lot harder for them.
Message: 10551 - Contents - Hide Contents Date: Sun, 07 Mar 2004 09:44:25 Subject: Re: Hanzos From: Graham Breed Paul Erlich wrote (2nd March):> Right, but that was a specific set of calculations, not a general > proof. You were just looking at 'linear' temperaments, if I recall > correctly, but torsion can afflict all types of temperaments. Plus it > seemed your method was far less elegant.What was inelegant about the torsion finder? I don't remember anything. I thought it was easier than for the octave-specific case because you don't have to check for a common divisor in the left hand column because the left hand column isn't there. So torsion should show up iff the adjoint matrix has a common divisor. That certainly should work, because if it gives different results to the octave-specific case it means the left hand (octave-specific) column must miss a common divisor that the rest of the adjoint has. For the simple case where the common divisor is 2, that means a power of each odd prime is equal to an odd number of octaves plus an even number of some comma. Divide it through, and you get a non-square rational number to be the square of another rational number, and so there's a formula for the exact solution of the square root of a rational number! This runs into problems with the fundamental theorem of arithmetic, so it can't be possible. Anyway, I'm less likely to miss something in the tests than that proof. And the general case doesn't matter, only the specific cases that refer to interesting temperaments. I don't even think all of those matter -- if the overwhelming majority of cases of torsion are discovered and dealt with, that's good enough. There are lots of sets of unison vectors that don't give sensible results, and there's always a set of unison vectors to give a specific temperament without torsion. The linear temperament issue doesn't matter. Torsion is a property of periodicity blocks, and it shouldn't matter how many chromatic unison vectors you choose (or which ones). I think I checked the whole adjoint matrix anyway. I thought we could prove that (for sensibly small unison vectors) the octave-equivalent adjoint is a subset of the octave-specific adjoint.> I care, and I hope Gene does too. I'd like to see this revisited.The problem is that you have to consider modulo arithmetic in the optimization problem. Let's take quarter comma meantone as the example. The rule is that four fifths have to add up to a 5:4. In other words, the generator is a quarter of a 5:4. Well, 5:4 is 386.3 cents so the generator must be 96.6 cents. You can see this is wrong because 96.6 is absurdly inaccurate for a 3:2 of 702.0 cents. There are four different results of division by 4 in modulo arithmetic. So a quarter of 5:4 could be 96.6, 396.6, 696.6 or 996.6 cents. Obviously you choose the best one. The old way of finding the minimax is to set the intervals between every consonant interval and every other consonant interval to be just. Then take whichever of these tunings gives the best result. The same should work for the octave equivalent case, so long as you take all the different generators that make each interval just. The newer way is to trace the worst error function downhill until you get to the bottom. I didn't expect this to work, but it does because the function only has one local minimum. I think it's that bit less likely to work in the octave-equivalent case, but you could give it a try. For the RMS, it gets more complicated because the total error can't be represented as a single quadratic function anymore. Maybe your numerical packages can solve this. The only way I can think is to find the minimax using the method above. The problem is that it would take a lot longer, whereas currently this is a simple calculation. Perhaps only solve for one consonance, and hope you get in the right basin of attraction. Still, it can certainly be done. Graham
Message: 10552 - Contents - Hide Contents Date: Sun, 07 Mar 2004 23:53:21 Subject: Re: Octave equivalent calculations (Was: Hanzos From: Graham Breed Gene Ward Smith wrote:> That would depend on your definition of better; but the one with the > TOP tuning closest to JI is the "standard" val, <21 33 48 59 73|. If > we put that together with 41, we get an 11-limit temperament which > adds 245/242 rather than 385/384 to 225/224 and 1029/1024, leading to > a temperament with a half-secor generator. Incidentally, if we add > 243/242 to 225/224 and 1029/1024, what does your program give?My definition is the closest to JI. <21 33 48 59 72| has a worst 11-limit error of 0.81 scale steps, but for <21 33 49 59 73| it's 0.92 steps. There isn't a standard val because the temperament is inconsistent. I get this temperament from the nearest prime approximations of 21 and 41 and also the unison vectors you gave: 3/62, 58.4 cent generator basis: (1.0, 0.048631497540919798) mapping by period and generator: [(1, 0), (1, 12), (3, -14), (3, -4), (4, -11)] mapping by steps: [(41, 21), (65, 33), (95, 49), (115, 59), (142, 73)] highest interval width: 38 complexity measure: 38 (41 for smallest MOS) highest error: 0.008391 (10.069 cents) unique Which looks like what you said, but not miracle. You get miracle from that other set.> At this point it is by no means clear your matrix methods are giving > correct answers, and in any case you seem to be working a lot harder > for them.They're giving exactly the same results as the wedgies. Graham
Message: 10553 - Contents - Hide Contents Date: Sun, 07 Mar 2004 10:14:57 Subject: Re: Hanzos From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> For the RMS, it gets more complicated because the total error can't be > represented as a single quadratic function anymore.There's nothing more complicated about it really if you do it in exact analogy to TOP; the question then is what are you taking to be the analog of the Tenney norm?
Message: 10554 - Contents - Hide Contents Date: Sun, 07 Mar 2004 10:23:39 Subject: Re: Hanzos From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> What was inelegant about the torsion finder? I don't remember anything. > I thought it was easier than for the octave-specific case because you > don't have to check for a common divisor in the left hand column because > the left hand column isn't there. So torsion should show up iff the > adjoint matrix has a common divisor.I think this would be clearer with some examples. Let's say you have 21 and 41. How do you get miracle out of the pair of them by your method? Then the same question for 1029/1024 and 16875/16807. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
Message: 10556 - Contents - Hide Contents Date: Mon, 08 Mar 2004 18:31:13 Subject: Re: Between Hahn and Euclid From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: > > If we put p=1 into it, I wonder if that is>> useful for anything? Paul liked the L1 error; this would be the >> corresponding norm on note classes. >> I looked at the p=1 norm around the unison.Does this lead to shells with cardinalities = rhombic dodecahedral numbers?
Message: 10557 - Contents - Hide Contents Date: Mon, 08 Mar 2004 20:15:51 Subject: Re: Between Hahn and Euclid From: Paul Erlich --- 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 "Gene Ward Smith" <gwsmith@s...> >> wrote: >> >> If we put p=1 into it, I wonder if that is>>> useful for anything? Paul liked the L1 error; this would be the >>> corresponding norm on note classes. >>>> I looked at the p=1 norm around the unison. >> Does this lead to shells with cardinalities = rhombic dodecahedral > numbers?Apparently the latter are customarily defined with respect to the cubic, not FCC, lattice :( Rhombic Dodecahedral Number -- from MathWorld * [with cont.]
Message: 10558 - Contents - Hide Contents Date: Mon, 08 Mar 2004 20:25:33 Subject: Re: Octave equivalent calculations (Was: Hanzos From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Me:>>> They're giving exactly the same results as the wedgies. > > Gene:>> They don't seem to be doing that. Are you saying you are getting >> results corresponding to my claims? >> I get the results I say I get, whether using wedgies, matrices (for > unison vectors) or the direct method for combining ETs. You keep > claiming I'm wrong, but don't say why.If you wedge <21 33 49 59| with <41 65 95 115| you get <12 -14 -4 -50 -40 30|; dividing this through by two gives you the wedgie for 7-limit miracle, <6 -7 -2 -25 -20 15|. If you wedge the monzo for 16805/16087 with that for 1029/1024 and take the compliment, once again you get twice the wedgie for miracle, and hence miracle. These are the results you should be getting by any correct method, but don't seem to be.
Message: 10559 - Contents - Hide Contents Date: Mon, 08 Mar 2004 20:36:20 Subject: Re: Canonical generators for 7-limit planar temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:>> Now I just need to understand how the lattice is formed from (in > this >> case) 3a^2+2ab+35b^2... >> Can someone tell me how this is used to form a lattice based on 3/2 > and 9/7? Thanks!If we define a norm by ||(3/2)^a (9/7)^b|| = sqrt(3a^2+2ab+35b^2) then we get a lattice, since 3a^2+2ab+35b^2 is positive-definite--the only way to get a zero distance from the unison is to be the unison.
Message: 10560 - Contents - Hide Contents Date: Mon, 08 Mar 2004 20:42:57 Subject: Re: Canonical generators for 7-limit planar temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> If we define a norm by > > ||(3/2)^a (9/7)^b|| = sqrt(3a^2+2ab+35b^2) > > then we get a lattice, since 3a^2+2ab+35b^2 is positive-definite--the > only way to get a zero distance from the unison is to be the unison.To see 3a^2+2ab+35b^2 is positive-definite, we can take the corresponding symmetric matrix [[3 1], [1 35]]. The characteristic polynomial for this is x^2 - 38x + 104, which has two positive real roots, 19+sqrt(257) and 19-sqrt(257), so it is positive-definite.
Message: 10561 - Contents - Hide Contents Date: Mon, 08 Mar 2004 07:56:19 Subject: Re: Octave equivalent calculations (Was: Hanzos From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Gene Ward Smith wrote: > They're giving exactly the same results as the wedgies.They don't seem to be doing that. Are you saying you are getting results corresponding to my claims?
Message: 10562 - Contents - Hide Contents Date: Mon, 08 Mar 2004 20:48:37 Subject: Re: Between Hahn and Euclid From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Apparently the latter are customarily defined with respect to the > cubic, not FCC, lattice :( > > Rhombic Dodecahedral Number -- from MathWorld * [with cont.]The handbook doesn't seem to know the sequence 1,13,19,55,79...
Message: 10563 - Contents - Hide Contents Date: Mon, 08 Mar 2004 09:04:11 Subject: Re: Octave equivalent calculations (Was: Hanzos From: Graham Breed Me:>> They're giving exactly the same results as the wedgies. Gene:> They don't seem to be doing that. Are you saying you are getting > results corresponding to my claims?I get the results I say I get, whether using wedgies, matrices (for unison vectors) or the direct method for combining ETs. You keep claiming I'm wrong, but don't say why. Except for torsion, the arithmetic is identical for matrices and wedge products, so I don't see what difference it should make. Graham
Message: 10564 - Contents - Hide Contents Date: Mon, 08 Mar 2004 21:36:12 Subject: Dual L1 norm deep hole scales From: Gene Ward Smith Here are the dual-L1 deep hole results Paul asked for. Shell 1 radius 3 6 notes [1, 21/20, 6/5, 7/5, 3/2, 7/4] Shell 2 radius 6 8 notes [7/6, 49/40, 5/4, 21/16, 8/5, 42/25, 12/7, 9/5] Shell 3 radius 7 24 notes [49/48, 36/35, 15/14, 35/32, 28/25, 9/8, 8/7, 63/50, 9/7, 4/3, 48/35, 10/7, 36/25, 35/24, 147/100, 49/32, 63/40, 49/30, 5/3, 147/80, 28/15, 15/8, 48/25, 49/25] Shell 4 radius 9 6 notes [441/400, 245/192, 98/75, 45/28, 288/175, 40/21] Shell 5 radius 10 24 notes [126/125, 25/24, 16/15, 343/320, 27/25, 147/128, 147/125, 60/49, 32/25, 168/125, 27/20, 49/36, 343/240, 72/49, 54/35, 14/9, 25/16, 105/64, 343/200, 25/14, 64/35, 35/18, 96/49, 63/32] Ball 1 radius 3 6 notes [1, 21/20, 6/5, 7/5, 3/2, 7/4] Ball 2 radius 6 14 notes [1, 21/20, 7/6, 6/5, 49/40, 5/4, 21/16, 7/5, 3/2, 8/5, 42/25, 12/7, 7/4,9/5] Ball 3 radius 7 38 notes [1, 49/48, 36/35, 21/20, 15/14, 35/32, 28/25, 9/8, 8/7, 7/6, 6/5, 49/40, 5/4, 63/50, 9/7, 21/16, 4/3, 48/35, 7/5, 10/7, 36/25, 35/24, 147/100, 3/2, 49/32, 63/40, 8/5, 49/30, 5/3, 42/25, 12/7, 7/4, 9/5, 147/80, 28/15, 15/8, 48/25, 49/25] Ball 4 radius 9 44 notes [1, 49/48, 36/35, 21/20, 15/14, 35/32, 441/400, 28/25, 9/8, 8/7, 7/6, 6/5, 49/40, 5/4, 63/50, 245/192, 9/7, 98/75, 21/16, 4/3, 48/35, 7/5, 10/7, 36/25, 35/24, 147/100, 3/2, 49/32, 63/40, 8/5, 45/28, 49/30, 288/175, 5/3, 42/25, 12/7, 7/4, 9/5, 147/80, 28/15, 15/8, 40/21, 48/25, 49/25] Ball 5 radius 10 68 notes [1, 126/125, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 343/320, 27/25, 35/32, 441/400, 28/25, 9/8, 8/7, 147/128, 7/6, 147/125, 6/5, 60/49, 49/40, 5/4, 63/50, 245/192, 32/25, 9/7, 98/75, 21/16, 4/3, 168/125, 27/20, 49/36, 48/35, 7/5, 10/7, 343/240, 36/25, 35/24, 72/49, 147/100, 3/2, 49/32, 54/35, 14/9, 25/16, 63/40, 8/5, 45/28, 49/30, 105/64, 288/175, 5/3, 42/25, 12/7, 343/200, 7/4, 25/14, 9/5, 64/35, 147/80, 28/15, 15/8, 40/21, 48/25, 35/18, 96/49, 49/25, 63/32]
Message: 10565 - Contents - Hide Contents
Date: Mon, 08 Mar 2004 03:29:42
Subject: inspiration for "jumping jacks"?
From: Carl Lumma
Jumping Champion -- from MathWorld * [with cont.]
-C.
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Message: 10566 - Contents - Hide Contents Date: Mon, 08 Mar 2004 21:54:47 Subject: Re: Dual L1 norm deep hole scales From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Ball 2 radius 6 14 notes > [1, 21/20, 7/6, 6/5, 49/40, 5/4, 21/16, 7/5, 3/2, 8/5, 42/25, > 12/7, 7/4,9/5]Just as I suspected. We've finally constructed the Stellated Hexany, aka Mandala!
Message: 10567 - Contents - Hide Contents Date: Mon, 08 Mar 2004 21:59:40 Subject: Re: Dual L1 norm deep hole scales From: Paul Erlich P.S. Why aren't they simply the "L1 deep hole" results? Why dual? I tend to think of the monzo space or lattice of notes itself as the 'standard' space, while the space of linear functionals on it (breeds) as its dual. Or am I misunderstanding?
Message: 10569 - Contents - Hide Contents Date: Mon, 08 Mar 2004 22:32:18 Subject: 7-limit 32805/32768-planar orthogonal projection scales From: Gene Ward Smith These are what you get from balls around the unison from the orthogonal projection of 32805/32768 onto the unison of the symmetrical lattice. The 32805/32768-planar tuning I used was just 3's and 7's, for three reasons--one, this gives a JI scale for JI affecianados, two it is easy to retune a scale given this way to the tuning you prefer, and three the tuning works just as it is. These scales or ones like them would be very suitable for Michael Harrison style music. I give first the radius and corresponding shell, and second the number of notes and scale, up to a 35 note scale. Shells 3 [4/3, 3/2] 6 [9/8, 16/9] 7 [32/27, 27/16] 9 [81/64, 128/81] 10 [256/243, 243/128] 11 [1024/729, 729/512] 13 [2187/2048, 4096/2187] 14 [8192/6561, 6561/4096] 15 [64/63, 63/32] 17 [256/189, 189/128] 18 [21/16, 32/21] 19 [567/512, 1024/567] 21 [8/7, 7/4] 22 [2048/1701, 1701/1024] 23 [7/6, 12/7] 25 [5103/4096, 8192/5103] 26 [19683/16384, 32768/19683] Scales 3 [1, 4/3, 3/2] 5 [1, 9/8, 4/3, 3/2, 16/9] 7 [1, 9/8, 32/27, 4/3, 3/2, 27/16, 16/9] 9 [1, 9/8, 32/27, 81/64, 4/3, 3/2, 128/81, 27/16, 16/9] 11 [1, 256/243, 9/8, 32/27, 81/64, 4/3, 3/2, 128/81, 27/16, 16/9, 243/128] 13 [1, 256/243, 9/8, 32/27, 81/64, 4/3, 1024/729, 729/512, 3/2, 128/81, 27/16, 16/9, 243/128] 15 [1, 256/243, 2187/2048, 9/8, 32/27, 81/64, 4/3, 1024/729, 729/512, 3/2, 128/81, 27/16, 16/9, 4096/2187, 243/128] 17 [1, 256/243, 2187/2048, 9/8, 32/27, 8192/6561, 81/64, 4/3, 1024/729, 729/512, 3/2, 128/81, 6561/4096, 27/16, 16/9, 4096/2187, 243/128] 19 [1, 64/63, 256/243, 2187/2048, 9/8, 32/27, 8192/6561, 81/64, 4/3, 1024/729, 729/512, 3/2, 128/81, 6561/4096, 27/16, 16/9, 4096/2187, 243/128, 63/32] 21 [1, 64/63, 256/243, 2187/2048, 9/8, 32/27, 8192/6561, 81/64, 4/3, 256/189, 1024/729, 729/512, 189/128, 3/2, 128/81, 6561/4096, 27/16, 16/9, 4096/2187, 243/128, 63/32] 23 [1, 64/63, 256/243, 2187/2048, 9/8, 32/27, 8192/6561, 81/64, 21/16, 4/3, 256/189, 1024/729, 729/512, 189/128, 3/2, 32/21, 128/81, 6561/4096, 27/16, 16/9, 4096/2187, 243/128, 63/32] 25 [1, 64/63, 256/243, 2187/2048, 567/512, 9/8, 32/27, 8192/6561, 81/64, 21/16, 4/3, 256/189, 1024/729, 729/512, 189/128, 3/2, 32/21, 128/81, 6561/4096, 27/16, 16/9, 1024/567, 4096/2187, 243/128, 63/32] 27 [1, 64/63, 256/243, 2187/2048, 567/512, 9/8, 8/7, 32/27, 8192/6561, 81/64, 21/16, 4/3, 256/189, 1024/729, 729/512, 189/128, 3/2, 32/21, 128/81, 6561/4096, 27/16, 7/4, 16/9, 1024/567, 4096/2187, 243/128, 63/32] 29 [1, 64/63, 256/243, 2187/2048, 567/512, 9/8, 8/7, 32/27, 2048/1701, 8192/6561, 81/64, 21/16, 4/3, 256/189, 1024/729, 729/512, 189/128, 3/2, 32/21, 128/81, 6561/4096, 1701/1024, 27/16, 7/4, 16/9, 1024/567, 4096/2187, 243/128, 63/32] 31 [1, 64/63, 256/243, 2187/2048, 567/512, 9/8, 8/7, 7/6, 32/27, 2048/1701, 8192/6561, 81/64, 21/16, 4/3, 256/189, 1024/729, 729/512, 189/128, 3/2, 32/21, 128/81, 6561/4096, 1701/1024, 27/16, 12/7, 7/4, 16/9, 1024/567, 4096/2187, 243/128, 63/32] 33 [1, 64/63, 256/243, 2187/2048, 567/512, 9/8, 8/7, 7/6, 32/27, 2048/1701, 5103/4096, 8192/6561, 81/64, 21/16, 4/3, 256/189, 1024/729, 729/512, 189/128, 3/2, 32/21, 128/81, 6561/4096, 8192/5103, 1701/1024, 27/16, 12/7, 7/4, 16/9, 1024/567, 4096/2187, 243/128, 63/32] 35 [1, 64/63, 256/243, 2187/2048, 567/512, 9/8, 8/7, 7/6, 32/27, 19683/16384, 2048/1701, 5103/4096, 8192/6561, 81/64, 21/16, 4/3, 256/189, 1024/729, 729/512, 189/128, 3/2, 32/21, 128/81, 6561/4096, 8192/5103, 1701/1024, 32768/19683, 27/16, 12/7, 7/4, 16/9, 1024/567, 4096/2187, 243/128, 63/32]
Message: 10571 - Contents - Hide Contents Date: Mon, 08 Mar 2004 22:33:25 Subject: Re: Dual L1 norm deep hole scales 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: >>> Ball 2 radius 6 14 notes >> [1, 21/20, 7/6, 6/5, 49/40, 5/4, 21/16, 7/5, 3/2, 8/5, 42/25, >> 12/7, 7/4,9/5] >> Just as I suspected. We've finally constructed the Stellated Hexany, > aka Mandala!Eh, I think that's turned up already. :)
Message: 10572 - Contents - Hide Contents Date: Mon, 08 Mar 2004 22:36:29 Subject: Re: Dual L1 norm deep hole scales From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> P.S. Why aren't they simply the "L1 deep hole" results? Why dual? I > tend to think of the monzo space or lattice of notes itself as > the 'standard' space, while the space of linear functionals on it > (breeds) as its dual. Or am I misunderstanding?Presumably L1 would mean ||3^a 5^b 7^c|| = |a|+|b|+|c| This has a different geometry from that, certainly. "Dual" comes in because if you use an L1 norm when measuring errors of tunings, dual to that is the "Dual L1 norm" on note-classes.
Message: 10573 - Contents - Hide Contents Date: Mon, 08 Mar 2004 22:38:19 Subject: Re: Canonical generators for 7-limit planar temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:>> To see 3a^2+2ab+35b^2 is positive-definite, we can take the >> corresponding symmetric matrix [[3 1], [1 35]]. The characteristic >> polynomial for this is x^2 - 38x + 104, which has two positive real >> roots, 19+sqrt(257) and 19-sqrt(257), so it is positive-definite. >> Thanks. Is the example lattice for this in Planarplots? I'd let to > look at it. It's p81.jpg
Message: 10574 - Contents - Hide Contents Date: Mon, 08 Mar 2004 22:46:50 Subject: Re: Dual L1 norm deep hole scales 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: >>>>> Ball 2 radius 6 14 notes >>> [1, 21/20, 7/6, 6/5, 49/40, 5/4, 21/16, 7/5, 3/2, 8/5, 42/25, >>> 12/7, 7/4,9/5] >>>> Just as I suspected. We've finally constructed the Stellated Hexany, >> aka Mandala! >