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Message: 10525 - Contents - Hide Contents Date: Fri, 05 Mar 2004 20:57:32 Subject: Re: Hahn norm formula From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> Despite its apparent theoretical interest, Crystal Ball Two should >> suffice as a name, >> Paul Hahn might have called it the 7-limit radius 2 scale or perhaps > more likely, the Level 2 7-limit Diamond.Since apparently it is called a cystal ball in some nonmusical connection already, it seems to me better to stick with that, on your own principle of established names.

Message: 10526 - Contents - Hide Contents Date: Fri, 05 Mar 2004 21:12:11 Subject: Re: Hahn norm formula From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>>> Despite its apparent theoretical interest, Crystal Ball Two should >>> suffice as a name, >>>> Paul Hahn might have called it the 7-limit radius 2 scale or perhaps >> more likely, the Level 2 7-limit Diamond. >> Since apparently it is called a cystal ball in some nonmusical > connection already, it seems to me better to stick with that, on your > own principle of established names.In nonmusical connections?

Message: 10527 - Contents - Hide Contents Date: Fri, 05 Mar 2004 13:21:37 Subject: Re: Hahn norm formula From: Carl Lumma Can someone briefly explain shell and hull, and the difference? The mathworld definitions are, as usual, obtuse, and it isn't clear which shell definition is in use here. And has anyone noticed how mathworld is slowly becoming a Mathematica help file? My guess (based on the "generalization of an annulus" definition of shell) is that in 3-D, a hull is a surface while a shell is a volume. Is this correct? -Carl

Message: 10528 - Contents - Hide Contents Date: Fri, 05 Mar 2004 21:25:31 Subject: Re: Hahn norm formula From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> Can someone briefly explain shell and hull, and the difference?Gene uses "shell" to mean the set of notes at a given "distance" from the origin. I know what a "convex hull" is but don't know what "hull" in general means, if anything.

Message: 10529 - Contents - Hide Contents Date: Fri, 05 Mar 2004 22:08:43 Subject: Re: Hahn norm formula From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Now what do you get if you center around a major tetrad (shallow > hole)?At first you get exactly the same result as using the Euclidean norm. It is only when we get to the fourth populated shell that we see a difference--this is a funny one, with only four notes in it. The Euclidean norm would put these notes together with some from shell 5. It would seem that the Hahn norm is not always drawing less refined distinctions. Shells Radius 3/4 Notes 4 [1, 5/4, 3/2, 7/4] Radius 3/2 Notes 12 [21/20, 15/14, 35/32, 7/6, 6/5, 21/16, 7/5, 10/7, 35/24, 5/3, 12/7,15/8] Radius 7/4 Notes 12 [49/48, 25/24, 9/8, 8/7, 49/40, 9/7, 4/3, 49/32, 25/16, 8/5, 25/14, 9/5] Radius 9/4 Notes 4 [60/49, 105/64, 42/25, 35/18] Radius 5/2 Notes 36 [50/49, 36/35, 10/9, 28/25, 147/128, 75/64, 25/21, 175/144, 63/50, 245/192, 75/56, 49/36, 175/128, 48/35, 25/18, 45/32, 36/25, 72/49, 147/100, 75/49,14/9, 63/40, 45/28, 80/49, 49/30, 245/144, 175/96, 90/49, 147/80, 28/15, 40/21, 245/128, 48/25, 96/49, 49/25, 63/32] Radius 11/4 Notes 24 [16/15, 343/320, 27/25, 54/49, 125/112, 343/288, 125/98, 32/25, 125/96,64/49, 343/256, 27/20, 343/240, 125/84, 32/21, 54/35, 27/16, 343/200, 125/72,16/9, 343/192, 64/35, 27/14, 125/64] Scales Radius 3/4 Notes 4 [1, 5/4, 3/2, 7/4] Radius 3/2 Notes 16 [1, 21/20, 15/14, 35/32, 7/6, 6/5, 5/4, 21/16, 7/5, 10/7, 35/24, 3/2, 5/3, 12/7, 7/4, 15/8] Radius 7/4 Notes 28 [1, 49/48, 25/24, 21/20, 15/14, 35/32, 9/8, 8/7, 7/6, 6/5, 49/40, 5/4, 9/7, 21/16, 4/3, 7/5, 10/7, 35/24, 3/2, 49/32, 25/16, 8/5, 5/3, 12/7, 7/4, 25/14, 9/5, 15/8] Radius 9/4 Notes 32 [1, 49/48, 25/24, 21/20, 15/14, 35/32, 9/8, 8/7, 7/6, 6/5, 60/49, 49/40, 5/4, 9/7, 21/16, 4/3, 7/5, 10/7, 35/24, 3/2, 49/32, 25/16, 8/5, 105/64, 5/3, 42/25, 12/7, 7/4, 25/14, 9/5, 15/8, 35/18] Radius 5/2 Notes 68 [1, 50/49, 49/48, 36/35, 25/24, 21/20, 15/14, 35/32, 10/9, 28/25, 9/8, 8/7, 147/128, 7/6, 75/64, 25/21, 6/5, 175/144, 60/49, 49/40, 5/4, 63/50, 245/192, 9/7, 21/16, 4/3, 75/56, 49/36, 175/128, 48/35, 25/18, 7/5, 45/32, 10/7, 36/25, 35/24, 72/49, 147/100, 3/2, 75/49, 49/32, 14/9, 25/16, 63/40, 8/5, 45/28, 80/49, 49/30, 105/64, 5/3, 42/25, 245/144, 12/7, 7/4, 25/14, 9/5, 175/96,90/49, 147/80, 28/15, 15/8, 40/21, 245/128, 48/25, 35/18, 96/49, 49/25, 63/32] Radius 11/4 Notes 92 [1, 50/49, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 343/320, 27/25, 35/32, 54/49, 10/9, 125/112, 28/25, 9/8, 8/7, 147/128, 7/6, 75/64, 25/21, 343/288, 6/5, 175/144, 60/49, 49/40, 5/4, 63/50, 125/98, 245/192, 32/25, 9/7, 125/96, 64/49, 21/16, 4/3, 75/56, 343/256, 27/20, 49/36, 175/128, 48/35, 25/18, 7/5, 45/32, 10/7, 343/240, 36/25, 35/24, 72/49, 147/100, 125/84, 3/2, 32/21, 75/49,49/32, 54/35, 14/9, 25/16, 63/40, 8/5, 45/28, 80/49, 49/30, 105/64, 5/3, 42/25, 27/16, 245/144, 12/7, 343/200, 125/72, 7/4, 16/9, 25/14, 343/192, 9/5, 175/96, 64/35, 90/49, 147/80, 28/15, 15/8, 40/21, 245/128, 48/25, 27/14, 35/18, 125/64, 96/49, 49/25, 63/32]

Message: 10530 - Contents - Hide Contents Date: Fri, 05 Mar 2004 22:18:17 Subject: Re: Hahn norm formula From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> Can someone briefly explain shell and hull, and the difference? > The mathworld definitions are, as usual, obtuse, and it isn't > clear which shell definition is in use here.The relevant mathworld definition for hull is the one for convex hull; at least, I haven't noticed anyone using another. "Shell" is used in connection with lattices to mean sets of points all at the same distance from a given point (which need not be a lattice point.) A spherical shell could then be found that only the shell points lie in.

Message: 10531 - Contents - Hide Contents Date: Fri, 05 Mar 2004 22:24:10 Subject: Re: Hahn norm formula From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> At first you get exactly the same result as using the Euclidean norm. > It is only when we get to the fourth populated shell that we see a > difference--this is a funny one, with only four notes in it. The > Euclidean norm would put these notes together with some from shell 5. > It would seem that the Hahn norm is not always drawing less refined > distinctions.Message 9756 is relevant to this; adding those four extra notes gives us the 3x3x3 chord cube.

Message: 10532 - Contents - Hide Contents Date: Fri, 05 Mar 2004 00:10:11 Subject: Re: Hahn norm formula From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Ball 2 30 notes > [1, 49/48, 36/35, 21/20, 15/14, 35/32, 28/25, 9/8, 8/7, 6/5, 63/50, > 9/7,4/3, 48/35, 7/5, 10/7, 36/25, 35/24, 147/100, 3/2, 49/32, 63/40, > 49/30, 5/3, 7/4, 147/80, 28/15, 15/8, 48/25, 49/25]If we take the aproximate 7-limit consonances of this with commas less than 10 cents, we get 2401/2400, 6144/6125, 225/224 and 1029/1024; once again hemiwuerschmidt or miracle suggests itself.> Ball 3 84 notes > [1, 50/49, 49/48, 36/35, 1029/1000, 21/20, 15/14, 49/45, 35/32, > 192/175,54/49, 441/400, 10/9, 28/25, 9/8, 8/7, 343/300, 144/125, > 75/64, 288/245, 189/160, 25/21, 343/288, 6/5, 175/144, 216/175, 56/45, > 63/50, 245/192, 9/7, 1029/800, 64/49, 98/75, 4/3, 75/56, 343/256, > 175/128, 48/35, 343/250, 441/320, 25/18, 7/5, 45/32, 10/7, 36/25, > 35/24, 147/100, 112/75, 3/2, 189/125, 32/21, 75/49, 49/32, 192/125, > 384/245, 196/125, 63/40, 45/28, 1029/640, 80/49, 49/30, 288/175, 5/3, > 27/16, 245/144, 216/125, 7/4, 432/245, 441/250, 16/9, 343/192, > 224/125, 175/96, 90/49, 147/80, 28/15, 15/8, 189/100, 40/21, 343/180, > 245/128,48/25, 27/14, 49/25]For these we get 2401/2400, 589824/588245, 6144/6125, 5120/5103, 3136/3125, 16875/16807, 225/224, 15625/15552 and 1029/1024. The first, second third and fifth commas are commas of hemiwuerschimidt; 5120/5103 isn't, but putting it together with the other four among the smallest five commas leads to 99-et. Putting it all into 99-et shrinks the scale down to 62 notes with steps of sizes 1, 2, and 3 99-equal steps.

Message: 10534 - Contents - Hide Contents Date: Sat, 06 Mar 2004 18:40:37 Subject: Between Hahn and Euclid From: Gene Ward Smith The symmetric Euclidean metric on note-classes corresponds to rms error when evaluating temperaments; the Hahn metric, it turns out, corresponds to minimax error. This suggests we might want to look at metrics between Euclid and Hahn, corresponding to p-th roots of sums of p-th powers of absolute values. What we get is this: ||3^a 5^b 7^c||_p = (|a|^p + |b|^p + |c|^p + |b+c|^p + |a+c|^p + |a+b|^p + |a+b+c|^p)^(1/p) ||(a,b,c)||_2 using this definition is twice what I've been using, but that changes nothing. 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.

Message: 10535 - Contents - Hide Contents Date: Sat, 06 Mar 2004 19:36:55 Subject: Re: Between Hahn and Euclid From: Gene Ward Smith --- 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. The first two shells correspond to the first two shells of Euclidean, leading to this 19-note scale once again: [1, 21/20, 15/14, 8/7, 7/6, 6/5, 5/4, 4/3, 48/35, 7/5, 10/7, 35/24, 3/2, 8/5, 5/3, 12/7, 7/4, 28/15, 40/21] The next shell, with 36 elements, is the union of the third and fourth Euclidean shells, and the one after that, with 24 elements, is the Euclidean fifth shell. Beyond this point the shells don't correspond.

Message: 10536 - Contents - Hide Contents Date: Sat, 06 Mar 2004 23:44:11 Subject: Re: Canonical generators for 7-limit planar temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" >>> >>>> I gave a semidefinite form in terms of 3, 5 and 9; you feed 3/2 >> and>>>> 9/7 into that, and get a definite form in terms of those two. >>>>>> Okay. I have to figure out how you killed off "c", and how "b" > is>>> based on 9/7... >>> on 9/7, and so forth >>>> The new a and b are different from the old a and b; I killed off c >> because I never fed a c in in the first place. >> True, I should know better. I just need to figure out how it is > *calculated*. I certainly don't expect you to crunch numbers for me... > that I can do myself (hopefully) once I understand things > theoretically. Here's a more theoretical question: Why does doing > an orthogonal projection make the comma vanish?I think I can jump in with the answer here -- because it makes all comma-separated pairs of notes overlap exactly, thus representing the comma by a zero length in the lattice.

Message: 10537 - Contents - Hide Contents Date: Sat, 06 Mar 2004 23:46:12 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. The first two shells > correspond to the first two shells of Euclidean, leading to this > 19-note scale once again: > > [1, 21/20, 15/14, 8/7, 7/6, 6/5, 5/4, 4/3, 48/35, 7/5, 10/7, 35/24, > 3/2, 8/5, 5/3, 12/7, 7/4, 28/15, 40/21] > > The next shell, with 36 elements, is the union of the third and fourth > Euclidean shells, and the one after that, with 24 elements, is the > Euclidean fifth shell. Beyond this point the shells don't correspond.Cool. What if you center around a deep hole? Thanks.

Message: 10539 - Contents - Hide Contents Date: Sat, 06 Mar 2004 01:32:02 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:>> Yes. I still need to figure out how you get 3a^2 + 2ab +35b^2. Give >> me a couple days... >>>> Hmm. Do you feed a unit vector in the 81/80 direction (based on 3/2 > and 9/7) into an orthogonal projection, and then feed that back > into Q? (Also based on 3/2 and 9/7) or I am going back too far? > > Thanx, I know it hasn't been a couple days, but its Friday...I gave a semidefinite form in terms of 3, 5 and 9; you feed 3/2 and 9/7 into that, and get a definite form in terms of those two.

Message: 10541 - Contents - Hide Contents Date: Sat, 06 Mar 2004 01:57:56 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:>> I gave a semidefinite form in terms of 3, 5 and 9; you feed 3/2 and >> 9/7 into that, and get a definite form in terms of those two. >> Okay. I have to figure out how you killed off "c", and how "b" is > based on 9/7... > on 9/7, and so forthThe new a and b are different from the old a and b; I killed off c because I never fed a c in in the first place.

Message: 10543 - Contents - Hide Contents Date: Sat, 06 Mar 2004 02:51:36 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:> True, I should know better. I just need to figure out how it is > *calculated*. I certainly don't expect you to crunch numbers for me... > that I can do myself (hopefully) once I understand things > theoretically. Here's a more theoretical question: Why does doing > an orthogonal projection make the comma vanish?Because you are projecting orthogonally from the comma.

Message: 10545 - Contents - Hide Contents Date: Sun, 07 Mar 2004 15:00:58 Subject: Re: Hanzos From: Graham Breed Gene Ward Smith wrote:> 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?You can do it the old pre-TOP way by taking the number of n-limit intervals to the comma. Or use the Kees metric, being like the Tenney one, but only considering the larger odd number in the ratio. Either way, it could well give the right tuning (depending on what you're doing for multiple commas) but you still have to be able to extract the right generator from it. Graham

Message: 10546 - Contents - Hide Contents Date: Sun, 07 Mar 2004 15:50:43 Subject: Re: Hanzos From: Graham Breed Gene Ward Smith wrote:> 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.Even with the octave specific method, 21 and 41 don't give miracle, but: 3/62, 58.4 cent generator basis: (1.0, 0.048647720621243257) mapping by period and generator: [(1, 0), (1, 12), (1, 27), (3, -4), (2, 30)] mapping by steps: [(41, 21), (65, 33), (95, 48), (115, 59), (142, 72)] highest interval width: 34 complexity measure: 34 (41 for smallest MOS) highest error: 0.008440 (10.128 cents) 1029/1024 and 16875/16807 confuse the program, which suggests that whatever I was doing before won't work here. Hmm. Anyway, using 3:1 as the chroma gives [[ 1, 0, 0, 0], [ 0, 1, 0, 0], [-10, 1, 0, 3], [ 0, 3, 4, -5]] and an adjoint of [[ 12, 0, 0, 0], [ 24, 12, 0, 0], [ 22, -14, 5, 3], [ 32, -4, 4, 0]] So there is some spurious torsion in the second column. I wonder how I got rid of it before. The way the algorithm actually works, it discovers that a period of 6 will work before it tries the period of 3 that it thinks will work. Well, if you need to find torsion with octave-equivalent vectors, you can always work out the periodicity block I suppose. Graham

Message: 10547 - Contents - Hide Contents Date: Sun, 07 Mar 2004 21:31:22 Subject: Re: Hanzos From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Gene Ward Smith wrote: >>> 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. >> Even with the octave specific method, 21 and 41 don't give miracle, but: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|.> 3/62, 58.4 cent generator > > basis: > (1.0, 0.048647720621243257) > > mapping by period and generator: > [(1, 0), (1, 12), (1, 27), (3, -4), (2, 30)] > > mapping by steps: > [(41, 21), (65, 33), (95, 48), (115, 59), (142, 72)]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.> So there is some spurious torsion in the second column. I wonder how I > got rid of it before. The way the algorithm actually works, it > discovers that a period of 6 will work before it tries the period of 3 > that it thinks will work. > > Well, if you need to find torsion with octave-equivalent vectors, you > can always work out the periodicity block I suppose.I think you are making a good case for the claim the best plan is to use wedgies.

Message: 10548 - Contents - Hide Contents Date: Sun, 07 Mar 2004 22:01:28 Subject: Octave equivalent calculations (Was: Hanzos From: Graham Breed 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.> 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?> 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? Graham

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