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Message: 10901 Date: Tue, 27 Apr 2004 20:59:09 Subject: Re: Vanishing tratios From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > > Speaking of vanishing, I was going to respond to a post of yours about > tratios and yantras, and it seems to have vanished! I figured out what you meant.
Message: 10902 Date: Tue, 27 Apr 2004 21:04:16 Subject: Re: Vanishing tratios From: Carl Lumma >>Speaking of vanishing, I was going to respond to a post >>of yours about tratios and yantras, and it seems to have >>vanished! > >I figured out what you meant. I appreciate the desire to focus the discussion, but thought I'd mention that your readers might not have figured it out... -Carl
Message: 10903 Date: Tue, 27 Apr 2004 21:05:56 Subject: Re: Vanishing tratios 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 "Paul Erlich" <perlich@a...> > > wrote: > > > > > 5-limit 12-equal -- 625:640:648? > > > > > > I'm going tratio-wild, but I have to go! :( > > > > Let's define the function > > > > weird(a,b,c) = a*b*c/gcd(a,b)/gcd(a,c)/gcd(a,b) > > What's wrong with lcm(a,b,c) = abc/gcd(ab,ac,bc)? You're right -- all these formulae seem to produce the same result (that's what you meant, right?) . . .
Message: 10905 Date: Tue, 27 Apr 2004 21:22:08 Subject: Re: Vanishing tratios 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: > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > > wrote: > > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > > > > wrote: > > > > > > > > > 5-limit 12-equal -- 625:640:648? > > > > > > > > > > I'm going tratio-wild, but I have to go! :( > > > > > > > > Let's define the function > > > > > > > > weird(a,b,c) = a*b*c/gcd(a,b)/gcd(a,c)/gcd(a,b) > > > > > > What's wrong with lcm(a,b,c) = abc/gcd(ab,ac,bc)? > > > > You're right -- all these formulae seem to produce the same result > > (that's what you meant, right?) . . . > > What I actually meant was that I thought what you probably wanted is > the least common multiple, and that abc/gcd(ab,ac,bc) is a formula for > it in terms of the gcd. Weird(a,b,c) is a different arthmetic > function, unknown to me. > > weird(2,4,6) = 2*4*6/gcd(2,4)/gcd(2,6)/gcd(4,6) = 48/8 = 6 > lcm(2,4,6) = 2*4*6/gcd(8,12,24) = 48/4 = 12 Hmm . . . the two formulae seem to give the same result as long as gcd (a,b,c) = 1, which was the case for all the tratios in question. Right?
Message: 10906 Date: Tue, 27 Apr 2004 06:21:07 Subject: Re: Vanishing tratios From: monz hi paul, --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > Before I finalize my paper, I'd like to explore the > following idea. > > What if I take a 3-term ratio ("tratio"?) and have it vanish? > > Let's say 125:126:128. > > So 128:125 vanishes, 128:126 = 64:63 vanishes, and >126:125 vanishes. traditionally, "ratio" has been used for 2-term comparisons, and for >2-term, "proportion". just nit-picking on terminology. but that's a cool idea you posted about. -monz
Message: 10910 Date: Tue, 27 Apr 2004 14:51:11 Subject: Re: tratios and yantras From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > I took all the 7-limit integers less than 2^20 (recall this is a > yantra when reduced to an octave) and found the smallest instance of > three successive 7-limit integers which were mapped to the same val > when wedged with an 7-limit wedgie, in order to get tratios for some > of the most important 7-limit linear temperaments. You lost me. Can you break down this process, step-by-step, with examples? Here's the results: > > meantone 1120:1125:1134 > miracle 7168:7200:7203 > ennealimmal 419904:420000:420175 > magic 6048:6075:6125 > pajara 441:448:450 > (septimal) schismic 27783:28000:28125 > orwell 12005:12096:12150 ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * <*> 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 *
Message: 10911 Date: Tue, 27 Apr 2004 23:50:14 Subject: Re: notation of monzos From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote: > I think there was only one person who disliked the idea (I'm pretty > sure it wasn't Dave, because he's a stickler for perceptual > improvements in notation), IIRC on the grounds that the commas are a > convenience rather than a necessity. I responded that commas are > widely accepted as serving a similar purpose as place-markers in > large decimal numbers. However, not enough people expressed any > further opinions one way or another for my suggestion to be either > adopted or rejected. > > Now's the time for others to speak up so the issue can be resolved. Yes. I wholeheartedly support putting a comma after the 3-exponent and then after every third exponent beyond that. Not only because of the improvement in readability (not having to count all the way from the end to figure which exponent you're looking at), but also because it allows one to distinguish octave-equivalent vectors (and even the 2,3-reduced ones George and I sometimes use in describing notational commas relative to a chain of fifths) from complete vectors. [2 3, 5 7 11, 13 17 19> ordinary, complete [3, 5 7 11, 13 17 19> octave-equivalent, 2-reduced [, 5 7 11, 13 17 19> 2,3-reduced And I agree that 11 and 19 are two natural stopping places, for various reasons. I don't remember anyone objecting to these commas, but maybe some wouldn't bother using them that way themselves. I don't like the pipe or vertical bar since it is too easily confused with a digit one. You folks who still have perfect eyesight may not think so, but wait 'til you get a bit older. Even for the inner product I prefer <2 3, 5][2 3, 5> or <2 3, 5].[2 3, 5> ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * <*> 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 *
Message: 10912 Date: Thu, 29 Apr 2004 05:17:26 Subject: lattice definitions and geometries From: Paul Erlich Gene, would you comment on this: Lattice Geometries * in regard to tuning, and what they have been / might be useful for (see concurrent tuning list thread) . . . ? -p
Message: 10914 Date: Fri, 30 Apr 2004 00:09:27 Subject: Re: lattice definitions and geometries From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: > > Gene, would you comment on this: > > > > Lattice Geometries * > > > > in regard to tuning, and what they have been / might be useful for > > (see concurrent tuning list thread) . . . > > I don't see the concurrent thread. On "cubic ice", the diamond form of carbon, etc.
Message: 10915 Date: Fri, 30 Apr 2004 01:25:39 Subject: book: Harmonic Analysis and Hypergroups From: Carl Lumma just something i came across: Harmonic Analysis and Hypergroups * -Carl ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * <*> 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 *
Message: 10919 Date: Wed, 05 May 2004 22:27:04 Subject: 10+16 (continued from tuning) From: Paul Erlich 7-limit now . . . val for 10: <10 16 23 28] val for 16: <16 25 37 45] wedgie: <<-6 2 2 17 20 -1]] According to Gene's file, this has TOP error of 3.740932 and L1 complexity of 14.626943. Not too bad. It just barely, by a hair, falls outside the bound in my paper. Not too late to change that, though . . .
Message: 10921 Date: Thu, 06 May 2004 00:50:48 Subject: Re: Request for Gene From: Paul Erlich Thanks again for this, Gene. Would it be too much trouble to also do [8, 6, 6, -9, -13, -3] and [6, -2, -2, -17, -20, 1] ? Those would be great. Also add these to the tratio request . . . --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > > > Can you (would you) provide a nice list of commas that vanish in each > > of the following 7-limit wedgies? > > Below I give a list in descending size order of the subgroup commas > for each temperament, plus any on a 7-limit comma list which I either > cooked up for you or which you computed, of commas with relative error > less than 0.06 and epimericity less than 0.5. > > > [1, 4, 10, 4, 13, 12] > [59049/57344, 81/80, 126/125, 225/224, 3136/3125, 703125/702464] > > [2, -4, -4, -11, -12, 2] > [50/49, 64/63, 2048/2025, 225/224] > > [5, 1, 12, -10, 5, 25] > [3125/3072, 875/864, 245/243, 225/224, 10976/10935] > > [7, 9, 13, -2, 1, 5] > [686/675, 245/243, 126/125, 78732/78125, 4375/4374] > > [1, 4, -2, 4, -6, -16] > [256/245, 36/35, 64/63, 81/80, 5120/5103] > > [3, 0, -6, -7, -18, -14] > [128/125, 64/63, 126/125, 4000/3969, 250047/250000] > > [4, -3, 2, -14, -8, 13] > [16875/16384, 525/512, 49/48, 686/675, 225/224] > > [2, 8, 1, 8, -4, -20] > [49/48, 81/80, 245/243, 19683/19600] > > [6, 5, 3, -6, -12, -7] > [1029/1000, 49/48, 875/864, 126/125, 15625/15552] > > [1, 9, -2, 12, -6, -30] > [20480/19683, 64/63, 245/243, 1728/1715, 420175/419904] > > [2, 8, 8, 8, 7, -4] > [6561/6272, 405/392, 50/49, 81/80, 4000/3969] > > [6, -7, -2, -25, -20, 15] > [34171875/33554432, 1063125/1048576, 1029/1024, 225/224, 16875/16807, > 2401/2400] > > [6, 10, 10, 2, -1, -5] > [250/243, 50/49, 2430/2401, 245/243] > > [7, -3, 8, -21, -7, 27] > [2430/2401, 1728/1715, 2109375/2097152, 225/224, 6144/6125, 65625/65536] > > [4, 4, 4, -3, -5, -2] > [360/343, 648/625, 36/35, 50/49, 3125/3087, 126/125] > > [1, -8, -14, -15, -25, -10] > [3125/3087, 4000/3969, 225/224, 5120/5103, 33554432/33480783, 32805/32768] > > [3, 0, 6, -7, 1, 14] > [405/392, 36/35, 128/125, 225/224] > > [0, 0, 12, 0, 19, 28] > [648/625, 128/125, 531441/524288, 81/80, 2048/2025, 32805/32768] > > [1, 4, -9, 4, -17, -32] > [137781/131072, 525/512, 875/864, 81/80, 4375/4374] > > [0, 5, 0, 8, 0, -14] > [256/243, 28/27, 49/48, 64/63, 1029/1024] > > [3, 12, -1, 12, -10, -36] > [81/80, 1728/1715, 1029/1024] > > [10, 9, 7, -9, -17, -9] > [10077696/9765625, 559872/546875, 126/125, 1728/1715, 2401/2400] > > [3, 5, -6, 1, -18, -28] > [250/243, 64/63, 875/864, 6144/6125]
Message: 10924 Date: Sat, 08 May 2004 00:16:34 Subject: Re: Request for Gene From: Paul Erlich The tratio request I referred to would involve finding the smallest tratios, and/or the tratios with smallest LCM, for the 23 (now 25) wedgies listed below. --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > Thanks again for this, Gene. > > > Would it be too much trouble to also do > > [8, 6, 6, -9, -13, -3] > > and > > [6, -2, -2, -17, -20, 1] > > ? > > Those would be great. > > > Also add these to the tratio request . . . > > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: > > > > > Can you (would you) provide a nice list of commas that vanish in > each > > > of the following 7-limit wedgies? > > > > Below I give a list in descending size order of the subgroup commas > > for each temperament, plus any on a 7-limit comma list which I > either > > cooked up for you or which you computed, of commas with relative > error > > less than 0.06 and epimericity less than 0.5. > > > > > > [1, 4, 10, 4, 13, 12] > > [59049/57344, 81/80, 126/125, 225/224, 3136/3125, 703125/702464] > > > > [2, -4, -4, -11, -12, 2] > > [50/49, 64/63, 2048/2025, 225/224] > > > > [5, 1, 12, -10, 5, 25] > > [3125/3072, 875/864, 245/243, 225/224, 10976/10935] > > > > [7, 9, 13, -2, 1, 5] > > [686/675, 245/243, 126/125, 78732/78125, 4375/4374] > > > > [1, 4, -2, 4, -6, -16] > > [256/245, 36/35, 64/63, 81/80, 5120/5103] > > > > [3, 0, -6, -7, -18, -14] > > [128/125, 64/63, 126/125, 4000/3969, 250047/250000] > > > > [4, -3, 2, -14, -8, 13] > > [16875/16384, 525/512, 49/48, 686/675, 225/224] > > > > [2, 8, 1, 8, -4, -20] > > [49/48, 81/80, 245/243, 19683/19600] > > > > [6, 5, 3, -6, -12, -7] > > [1029/1000, 49/48, 875/864, 126/125, 15625/15552] > > > > [1, 9, -2, 12, -6, -30] > > [20480/19683, 64/63, 245/243, 1728/1715, 420175/419904] > > > > [2, 8, 8, 8, 7, -4] > > [6561/6272, 405/392, 50/49, 81/80, 4000/3969] > > > > [6, -7, -2, -25, -20, 15] > > [34171875/33554432, 1063125/1048576, 1029/1024, 225/224, > 16875/16807, > > 2401/2400] > > > > [6, 10, 10, 2, -1, -5] > > [250/243, 50/49, 2430/2401, 245/243] > > > > [7, -3, 8, -21, -7, 27] > > [2430/2401, 1728/1715, 2109375/2097152, 225/224, 6144/6125, > 65625/65536] > > > > [4, 4, 4, -3, -5, -2] > > [360/343, 648/625, 36/35, 50/49, 3125/3087, 126/125] > > > > [1, -8, -14, -15, -25, -10] > > [3125/3087, 4000/3969, 225/224, 5120/5103, 33554432/33480783, > 32805/32768] > > > > [3, 0, 6, -7, 1, 14] > > [405/392, 36/35, 128/125, 225/224] > > > > [0, 0, 12, 0, 19, 28] > > [648/625, 128/125, 531441/524288, 81/80, 2048/2025, 32805/32768] > > > > [1, 4, -9, 4, -17, -32] > > [137781/131072, 525/512, 875/864, 81/80, 4375/4374] > > > > [0, 5, 0, 8, 0, -14] > > [256/243, 28/27, 49/48, 64/63, 1029/1024] > > > > [3, 12, -1, 12, -10, -36] > > [81/80, 1728/1715, 1029/1024] > > > > [10, 9, 7, -9, -17, -9] > > [10077696/9765625, 559872/546875, 126/125, 1728/1715, 2401/2400] > > > > [3, 5, -6, 1, -18, -28] > > [250/243, 64/63, 875/864, 6144/6125]
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