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Message: 10800 - Contents - Hide Contents Date: Wed, 14 Apr 2004 19:19:56 Subject: Re: 32201 temps From: Paul Erlich Hi Gene.>> I would expect that, at high levels of complexity, one would see a >> large number of high-error temperament candidates. >> Why? The number of larger commas is limited on any list,Larger in terms of their sizes in cents? I don't see why that is necessarily limited on any list. I had no trouble finding myriad ridiculously large 'commas' in the 5-limit case I posted extensively on.> the number > of small ets, the same.It's true that there are a limited number of small ETs. But there aren't a limited number of large, garbage ET breeds (vals).> What method is going to produce for you a > large number of high-error garbage temperaments?Using ridiculously large commas, for example, or using garbage ET breeds (vals). I do realize that each of the items in your kitchen sink was 'limited' in the ways you seem to be claiming 'any' list would be (maybe you actually mean 'any reasonable' and not 'any'?), but I was still surprised as to the extremely small number of large-error temperaments in your list of 32201 -- and less so, by the vast that the vast majority of these had extremely large complexity. Thanks again for doing this work, Paul

Message: 10802 - Contents - Hide Contents Date: Wed, 14 Apr 2004 20:43:13 Subject: Re: 32201 temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Using ridiculously large commas, for example, or using garbage ET > breeds (vals).Using garbage commas or garbage vals will definately create garbage results, but why would anyone want garbage results? If you simply want all possibilities, those can be obtained by requiring the six integers of the wedgie to be in appropriate reduced form and satify a single algebraic conditions (wedgies are all rational points on a projective variety in mathspeak.)

Message: 10803 - Contents - Hide Contents Date: Wed, 14 Apr 2004 21:05:49 Subject: Re: 32201 temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> Using ridiculously large commas, for example, or using garbage ET >> breeds (vals). >> Using garbage commas or garbage vals will definately create garbage > results, but why would anyone want garbage results?No one would. That's why we're calling them "garbage". It just strikes me that near the logarithmic error and complexity of 7-limit blackwood, we hit the boundary of your search.> If you simply > want all possibilities, those can be obtained by requiring the six > integers of the wedgie to be in appropriate reduced form and satify a > single algebraic conditions (wedgies are all rational points on a > projective variety in mathspeak.)I would have thought the six integers would need to have a GCD of 1, and with that restriction alone you'd get two and only two occurrences of each temperament. But I take it it's more complicated than that?

Message: 10804 - Contents - Hide Contents Date: Wed, 14 Apr 2004 21:37:57 Subject: Circle for Carl From: Paul Erlich To refresh the memory, here's the plot of 7-limit "linear temperaments" in blue, with the moat (in red) that Dave and I favored: Yahoo groups: /tuning_files/files/Erlich/7lin2... * [with cont.] That used linear scaling on the axes; this version of the same plot used log scaling on both axes: Yahoo groups: /tuning_files/files/Erlich/7lin2... * [with cont.] if The numbers labeling the points refer to this list: Yahoo groups: /tuning-math/message/9317 * [with cont.] Carl expressed interest in using circular moats centered around the origin. This same moat becomes a circle around the origin if cube- root scaling is used on both of the axes: Yahoo groups: /tuning-math/files/Paul/carl.gif * [with cont.] I didn't label the points but it should be clear which point is which from comparison with the plots above. Ennealimmal is the point on the far lower right. This last plot has a lot more points, since it drew from Gene's fresh de-culling of his list of 32201, which had been culled to 126 when the earlier plots were made. In fact, this post is partially an excuse to immediately present something resulting from the de- cullation. Thanks again, Gene. I don't want this to contribute to or resume the argument about which scaling of the axes is appropriate, I just thought it might interest Carl, depending on what his predilection for circles was based on. I'd like to hear Carl's thoughts.

Message: 10805 - Contents - Hide Contents Date: Wed, 14 Apr 2004 22:27:13 Subject: Re: 32201 temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> I would have thought the six integers would need to have a GCD of 1, > and with that restriction alone you'd get two and only two > occurrences of each temperament. But I take it it's more complicated > than that?You can normalize by making the GCD 1, and then normalizing the sign, for instance by making the first nonzero coefficient positive. Another approach is simply to take any vector of six rational numbers equivalent to any other under multiplication by a nonzero scalar; this makes the five-dimensional projective space over the rationals. Either way, that is still not enough conditions, if <<x1 x2 x3 x4 x5 x6|| is our wedgie, we also must have that x1x6 - x2x5 + x3x4 = 0 This defines a 4D quadric in 5D (real or complex) projective space; it has an infinity of rational points, each of which corresponds to a wedgie. There's some math due to Klein connected to all of this.

Message: 10806 - Contents - Hide Contents Date: Wed, 14 Apr 2004 00:20:13 Subject: Re: 32201 temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> Gene, your zip file is corrupted. >> >> -C. >> Thanks, I uploaded again.Thanks, Gene. I finally got this into Matlab using 'importdata' and then for j=1:32201; s(j,:)=str2num(cell2mat(sy(j))); end I graphed it, and I see the same constellations as before, but there's "stuff" in the region where there is nothing. This is what I wanted. Thanks. I noticed that the great majority of the temperaments in the list had extremely high complexity, but virtually none -- a tiny, tiny number - - of them had very high error. Comments welcome. -Paul

Message: 10807 - Contents - Hide Contents Date: Wed, 14 Apr 2004 15:33:40 Subject: Re: Circle for Carl From: Carl Lumma>I don't want this to contribute to or resume the argument about which >scaling of the axes is appropriate, I just thought it might interest >Carl, depending on what his predilection for circles was based on. >I'd like to hear Carl's thoughts. Thanks Paul.My thoughts are hoping recorded in that thread. In any case I don't consider the "perfect list" so important. -Carl

Message: 10809 - Contents - Hide Contents Date: Wed, 14 Apr 2004 22:41:37 Subject: Re: 32201 temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Either way, that is still not enough conditions, if > <<x1 x2 x3 x4 x5 x6|| is our wedgie, we also must have that > > x1x6 - x2x5 + x3x4 = 0Aha! This is new.> This defines a 4D quadric in 5D (real or complex) projective space; > it has an infinity of rational points, each of which corresponds to a > wedgie. There's some math due to Klein connected to all of this.So the space of 7-limit "linear" temperaments is essentially only 4- dimensional? I was thinking 6-dimensional but then 1 dimension is redundant because of torsion considerations. So in musical terms, what consideration brings us down another dimension?

Message: 10810 - Contents - Hide Contents Date: Wed, 14 Apr 2004 16:34:23 Subject: Re: 32201 temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> I noticed that the great majority of the temperaments in the list had > extremely high complexity, but virtually none -- a tiny, tiny number - > - of them had very high error.One would not expect to get an infinite number of high-error temperament candidates, so why is this a surprise?

Message: 10811 - Contents - Hide Contents Date: Wed, 14 Apr 2004 22:44:34 Subject: Re: Circle for Carl From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> I don't want this to contribute to or resume the argument about which >> scaling of the axes is appropriate, I just thought it might interest >> Carl, depending on what his predilection for circles was based on. >> I'd like to hear Carl's thoughts. > > Thanks Paul. >> My thoughts are hoping recorded in that thread.Hoping recorded? You mean hopefully recorded? I've just re-read the entire contents of this list from the first 2+ months of this year, so I'm familiar with what you said. I was left with a big question mark as to your fundamentals, and was hoping that bouncing this off you would help clarify.> In any case I > don't consider the "perfect list" so important.Of course. Once you write music in all of them, then what? ;)

Message: 10812 - Contents - Hide Contents Date: Wed, 14 Apr 2004 18:01:38 Subject: Re: 32201 temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> I noticed that the great majority of the temperaments in the list > had>> extremely high complexity, but virtually none -- a tiny, tiny > number ->> - of them had very high error. >> One would not expect to get an infinite number of high-error > temperament candidates, so why is this a surprise?I would expect that, at high levels of complexity, one would see a large number of high-error temperament candidates.

Message: 10813 - Contents - Hide Contents Date: Wed, 14 Apr 2004 22:56:11 Subject: Re: 32201 temps 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: >>> Either way, that is still not enough conditions, if >> <<x1 x2 x3 x4 x5 x6|| is our wedgie, we also must have that >> >> x1x6 - x2x5 + x3x4 = 0 >> Aha! This is new. >>> This defines a 4D quadric in 5D (real or complex) projective space; >> it has an infinity of rational points, each of which corresponds to > a>> wedgie. There's some math due to Klein connected to all of this. >> So the space of 7-limit "linear" temperaments is essentially only 4- > dimensional?Correct; they correspond to what a number theorist or algebraic geometer would call the "Q-rational points" on the "Klein quadric", which is the x1x6-x2x5+x3x4 thingie; Q being the rational numbers. So we have a 4D quadric and an infinity of rational points on it, in 1-1 correspondence to 7-limit linear temperaments in some sense. I say in some sense because they can be arbitarily crappy. I was thinking 6-dimensional but then 1 dimension is> redundant because of torsion considerations. So in musical terms, > what consideration brings us down another dimension?In musical terms, you ask? That bears thinking about. The main thing is anything else doesn't give you an actual temperament.

Message: 10814 - Contents - Hide Contents Date: Wed, 14 Apr 2004 16:01:02 Subject: Re: Circle for Carl From: Carl Lumma>> >y thoughts are hoping recorded in that thread. >>Hoping recorded? You mean hopefully recorded? Yes. >I've just re-read the >entire contents of this list from the first 2+ months of this year, >so I'm familiar with what you said. I was left with a big question >mark as to your fundamentals, and was hoping that bouncing this off >you would help clarify.I thought I stated my fundamentals pretty clearly. I can't give msg. numbers, but I can copy and paste if you like. -Carl ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 10815 - Contents - Hide Contents Date: Wed, 14 Apr 2004 18:46:25 Subject: Re: 32201 temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> I would expect that, at high levels of complexity, one would see a > large number of high-error temperament candidates.Why? The number of larger commas is limited on any list, the number of small ets, the same. What method is going to produce for you a large number of high-error garbage temperaments? On the other hand, high complexity garbage temperaments I would expect, a priori, out the wazoo. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 10816 - Contents - Hide Contents Date: Thu, 15 Apr 2004 18:54:18 Subject: Re: 32201 temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> I was thinking 6-dimensional but then 1 dimension is>> redundant because of torsion considerations. So in musical terms, >> what consideration brings us down another dimension? >> In musical terms, you ask? That bears thinking about. The main thing > is anything else doesn't give you an actual temperament.In geometric terms?

Message: 10817 - Contents - Hide Contents Date: Thu, 15 Apr 2004 19:23:19 Subject: Re: 32201 temps 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: >>> I was thinking 6-dimensional but then 1 dimension is>>> redundant because of torsion considerations. So in musical terms, >>> what consideration brings us down another dimension? >>>> In musical terms, you ask? That bears thinking about. The main > thing>> is anything else doesn't give you an actual temperament. >> In geometric terms?Here's an on-line math book ("Projective and Polar Spaces", by Peter Cameron, revised on-line edition) with some relevant information: Projective and Polar Spaces * [with cont.] (Wayb.) Chapter eight is "The Klein quadric and triality", which discusses the Klein quadric (obviously) and the Klein correspondence. Chapter ten is "Exterior powers and Clifford algebras" which might be another place to find out about these.

Message: 10818 - Contents - Hide Contents Date: Thu, 15 Apr 2004 19:32:53 Subject: Re: 32201 temps 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: >>>>> I was thinking 6-dimensional but then 1 dimension is>>>> redundant because of torsion considerations. So in musical > terms,>>>> what consideration brings us down another dimension? >>>>>> In musical terms, you ask? That bears thinking about. The main >> thing>>> is anything else doesn't give you an actual temperament. >>>> In geometric terms? >> Here's an on-line math book ("Projective and Polar Spaces", by Peter > Cameron, revised on-line edition) with some relevant information: > > Projective and Polar Spaces * [with cont.] (Wayb.) > > Chapter eight is "The Klein quadric and triality", which discusses > the Klein quadric (obviously) and the Klein correspondence. Chapter > ten is "Exterior powers and Clifford algebras" which might be another > place to find out about these.Thanks. I've bookmarked this for later reference. Time to finish the paper.

Message: 10819 - Contents - Hide Contents Date: Thu, 15 Apr 2004 19:48:18 Subject: Re: 32201 temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Thanks. I've bookmarked this for later reference. Time to finish the > paper.It doesn't look that digestible; I may write a Wikipedia article.

Message: 10820 - Contents - Hide Contents Date: Thu, 15 Apr 2004 23:26:44 Subject: Commas of the form N/(N-2) From: Gene Ward Smith Here is something from the Lehmer article, all N/(N-2) through the 31-limit. As ratios of odd integers, I suppose these are particularly interesting to the non-octave school of thought. 3-limit 3 5-limit 5/3 27/25 7-limit 7/5 9/7 245/243 11-limit 11/9 35/33 77/75 13-limit 13/11 15/13 65/63 275/273 847/845 1575/1573 17-limit 17/15 51/49 119/117 121/119 189/187 19-limit 19/17 21/19 57/55 135/133 171/169 247/245 325/323 627/625 665/663 1617/1615 3213/3211 3971/3969 23-limit 23/21 25/23 117/115 209/207 255/253 299/297 345/343 1127/1125 1311/1309 2187/2185 2277/2275 2875/2873 3705/3703 6877/6875 8075/8073 9317/9315 18515/18513 41745/41743 57477/57475 1128127/1128125 1447875/1447873 29-limit 29/27 87/85 145/143 147/145 377/375 437/435 495/493 667/665 2873/2871 8381/8379 9947/9945 12675/12673 14877/14875 16445/16443 24565/24563 41327/41325 45619/45617 87725/87723 184877/184875 31-limit 31/29 33/31 93/91 95/93 155/153 343/341 405/403 527/525 529/527 715/713 899/897 1085/1083 1521/1519 1955/1953 2697/2695 3627/3625 4125/4123 5425/5423 7163/7161 19437/19435 22477/22475 86275/86273 130977/130975 203205/203203 2509047/2509045 3322055/3322053 287080367/287080365

Message: 10821 - Contents - Hide Contents Date: Fri, 16 Apr 2004 18:24:41 Subject: Request for Gene From: Paul Erlich Can you (would you) provide a nice list of commas that vanish in each of the following 7-limit wedgies? << 1 4 10 4 13 12 ]] << 2 -4 -4 -11 -12 2 ]] << 5 1 12 -10 5 25 ]] << 7 9 13 -2 1 5 ]] << 1 4 -2 4 -6 -16 ]] << 3 0 -6 -7 -18 -14 ]] << 4 -3 2 -14 -8 13 ]] << 2 8 1 8 -4 -20 ]] << 6 5 3 -6 -12 -7 ]] << 1 9 -2 12 -6 -30 ]] << 2 8 8 8 7 -4 ]] << 6 -7 -2 -25 -20 15 ]] << 6 10 10 2 -1 -5 ]] << 7 -3 8 -21 -7 27 ]] << 4 4 4 -3 -5 -2 ]] << 1 -8 -14 -15 -25 -10 ]] << 3 0 6 -7 1 14 ]] << 0 0 12 0 19 28 ]] << 1 4 -9 4 -17 -32 ]] << 0 5 0 8 0 -14 ]] << 3 12 -1 12 -10 -36 ]] << 10 9 7 -9 -17 -9 ]] << 3 5 -6 1 -18 -28 ]] For example, for the second wedgie, I'd at least want to see 50:49, 64:63, and 225:224, if not several more. Thanks, Paul

Message: 10822 - Contents - Hide Contents Date: Fri, 16 Apr 2004 20:23:47 Subject: Re: Request for Gene From: Gene Ward Smith --- 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: 10823 - Contents - Hide Contents Date: Fri, 16 Apr 2004 20:31:57 Subject: Re: Request for Gene From: Paul Erlich Thanks for the quick work, Gene! I appreciate it. (this is for the paper) --- 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: 10824 - Contents - Hide Contents Date: Fri, 16 Apr 2004 20:47:50 Subject: On From: Paul Erlich On <<0, 0, 12, 0, 19, 28]] . . . When I said 'we discussed this years ago', it turns out I had the wrong tuning. I was thinking of a tuning where 5120/5103 vanished, but 81/80 didn't. <<0, 0, 12, 0, 19, 28]], on the other hand, seems to be functionally the same as Jon Catler's '12-tone plus' tuning -- except that the offset is cleverly an eighthtone instead of a sixthtone, making for better 7:5s . . . right? So what's the wedgie, TOP primes, period, and generator for the tuning where 531441/524288 and 5120/5103 vanish? Waage is of course a *third* member of this 'family'. --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Thanks for the quick work, Gene! I appreciate it. (this is for the > paper) > > --- 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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