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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.