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Message: 10950 - Contents - Hide Contents Date: Thu, 13 May 2004 22:43:02 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. 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 12005:12096:12150I did something a little more straightforward. I calculated the wedgie for every triplet of consecutive 7-limit integers, through 99999. Then I just looked for the earliest occurence of each wedgie. I found a lot more, though a few wedgies apparently don't show up at all. There are a couple of improvements over the results I posted in Yahoo groups: /tuning-math/message/10408 * [with cont.] Meantone 1120:1125:1134 Magic 6048:6075:6125 Pajara 441:448:450 Semisixths 3375:3402:3430 Dominant Seventh 315:320:324 -- larger numbers than 245:252:256, but lcm much smaller at 181,440 Injera 392:400:405 OldKleismic 864:875:882 -- smaller numbers than 1000:1008:1029, lcm smaller at 5,292,000 Semifourths 240:243:245 Negri 672:675:686 Tripletone 125:126:128 Schismic 27783:28000:28125 Superpythagorean 1701:1715:1728 Orwell 12005:12096:12150 Augmented NOT FOUND Porcupine NOT FOUND <<6, 10, 10, 2, -1, -5]] 243:245:250 Supermajor seconds 5120:5145:5184 Flattone 2560:2592:2625 Diminished 245:250:252 <<6, 10, 3, 2, -12, -21]] NOT FOUND Catler NOT FOUND Gawel 10800:10935:10976 Nonkleismic 12000:12005:12096 Miracle 7168:7200:7203 Beatlemania 4725:4800:4802 <<6, -2, -2, -17, -20, 1]] 1024:1029:1050 <<8, 6, 6, -9, -13, -3]] 1715:1728:1750 Blackwood 189:192:196 Now for 5-limit ETs, one semi-improvement over Yahoo groups: /tuning-math/message/10404 * [with cont.] 3-equal 27:30:32 -- smaller numbers than 45:48:50 4-equal 24:25:27 5-equal 75:80:81 7-equal 240:243:250 9-equal 125:128:135 10-equal 16200:16384:16875 YUCK! 12-equal 625:640:648 15-equal 243:250:256 16-equal NOT FOUND 19-equal 15360:15552:15625 22-equal 6075:6144:6250 New ones: 27-equal 78125:78732:80000 lcm 196,830,000,000 29-equal 32768:32805:33750 lcm 134,369,280,000 ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 10951 - Contents - Hide Contents Date: Thu, 13 May 2004 04:19:47 Subject: Re: Adding wedgies? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> But how do you simply check whether a particular wedgie eats a > particular comma?Consider the wedgie in multimonzo form, and wedge with the monzo; if you get a zero multimonzo, you had a comma.

Message: 10952 - Contents - Hide Contents Date: Thu, 13 May 2004 04:22:00 Subject: Re: Adding wedgies? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> But how do you simply check whether a particular wedgie eats a >> particular comma? >> Consider the wedgie in multimonzo form, and wedge with the monzo; if > you get a zero multimonzo, you had a comma.I figured that. But how do you wedge a monzo with a multimonzo?

Message: 10953 - Contents - Hide Contents Date: Thu, 13 May 2004 04:24:19 Subject: Re: Adding wedgies? From: Paul Erlich Aha -- does the klein condition equate with the bivector being a simple bivector? --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> What's going on here? >> >> [5, 13, -17, 9, -41, -76], TOP error 0.27611 >> "plus" >> [13, 14, 35, -8, 19, 42], TOP error 0.26193 >> "equals" >> [18, 27, 18, 1, -22, -34], TOP error 0.036378 >> Parakleismic + Amity = Ennealimmal. Both parakleismic and amity have > 4375/4374 as a comma, and so does their sum (and difference, for that > matter.) > > I did talk about it before, though I can't recall what I said about > it. It is related to the Klein stuff. For 7-limit wedgies, define the > Pfaffian as follows: let > > X = <<x1 x2 x3 x4 x5 x6|| > Y = <<y1 y2 y3 y4 y5 y6|| > > Then > > Pf(X, Y) = y1x6 + x1y6 - y2x5 - x2y5 + y3x4 + x3y4 > > It is easily checked that we have the identity > > Pf(X+Y, X+Y) = Pf(X, X) + 2 Pf(X, Y) + Pf(Y, Y) > > The Klein condition for the wedgie X is Pf(X, X)=0. If X and Y both > satisfy the Klein condition, and if Pf(X, Y)=0, then X+Y also > satisfies the Klein condition, and hence is a wedgie. What > Pf(X, Y)=0 means is that X and Y are related; they share a comma. > > Probably, you will not want to talk about this in the paper. :)

Message: 10954 - Contents - Hide Contents Date: Thu, 13 May 2004 04:24:19 Subject: Re: Adding wedgies? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> Consider the wedgie in multimonzo form, and wedge with the monzo; if >> you get a zero multimonzo, you had a comma. >> I figured that. But how do you wedge a monzo with a multimonzo?It's just a special case of the general definition. If you mean me personally, I have Maple programs written to do it.

Message: 10955 - Contents - Hide Contents Date: Thu, 13 May 2004 04:26:00 Subject: Re: Adding wedgies? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>>> Consider the wedgie in multimonzo form, and wedge with the monzo; if >>> you get a zero multimonzo, you had a comma. >>>> I figured that. But how do you wedge a monzo with a multimonzo? >> It's just a special case of the general definition. If you mean me > personally, I have Maple programs written to do it.Is it a simple expression in terms of determinants?

Message: 10956 - Contents - Hide Contents Date: Thu, 13 May 2004 05:23:02 Subject: Re: tratios and yantras From: Paul Erlich I'm adding LCMs below, though i still don't know if the numbers and/or LCMs are smallest possible. It's interesting that the LCMs seem to give a much better ranking of complexity than the size of the numbers. Of course we're essentially using LCM (=n*d) already in the one-comma case. We should be able to visualize this, as well as the L1 (and maybe L-inf??) complexity measures, as specific sorts of "area" on the lattice, though 4-D is kind of hard. Need help, please . . . Of course if you divide the lcm by the numbers in the tratio you get another tratio with the same LCM, representing the same temperament, but I think always with larger numbers (if you start with one of these): 7-limit Blackwood 243:252:256 lcm 435,456 Dominant Sevenths 245:252:256 lcm 564,480 7-limit Diminished 343:350:360 lcm 617,400 Pajara 441:448:450 lcm 705,600 Semifourths 240:243:245 lcm 952,560 Tripletone 125:126:128 lcm 1,008,000 Injera 392:400:405 lcm 1,587,600 meantone 1120:1125:1134 lcm 2,268,000 7-limit Augmented: 1125:1152:1176 lcm 7,056,000 BUT ALSO 1568:1575:1620 lcm 3,175,200 (any simpler tratio?) OldKleismic 1000:1008:1029 lcm 6,174,000 Catler -- same as 12-equal -- 625:648:640 lcm 6,480,000 Negri 672:675:686 lcm 7,408,800 semisixths 3375:3402:3430 lcm 20,837,250 Superpythagorean 1701:1715:1728 lcm 26,671,680 magic 6048:6075:6125 lcm 47,628,000 miracle 7168:7200:7203 lcm 553,190,400 orwell 12005:12096:12150 lcm 933,508,800 (septimal) schismic 27783:28000:28125 lcm 2,778,300,000 ennealimmal 419904:420000:420175 lcm 4,410,829,080,000 What's bigger -- the lcm of ennealimmal or the national debt?

Message: 10957 - Contents - Hide Contents Date: Thu, 13 May 2004 05:25:40 Subject: Re: tratios and yantras From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> I'm adding LCMs below, though i still don't know if the numbers > and/or LCMs are smallest possible. It's interesting that the LCMs > seem to give a much better ranking of complexity than the size of the > numbers. Of course we're essentially using LCM (=n*d) already in the > one-comma case. We should be able to visualize this, as well as the > L1 (and maybe L-inf??) complexity measures, as specific sorts > of "area" on the lattice, though 4-D is kind of hard. Need help, > please . . . > > Of course if you divide the lcm by the numbers in the tratio you get > another tratio with the same LCM, representing the same temperament, > but I think always with larger numbers (if you start with one of > these): > > > 7-limit Blackwood 243:252:256 lcm 435,456 > Dominant Sevenths 245:252:256 lcm 564,480 > 7-limit Diminished 343:350:360 lcm 617,400 > Pajara 441:448:450 lcm 705,600 > Semifourths 240:243:245 lcm 952,560 > Tripletone 125:126:128 lcm 1,008,000 > Injera 392:400:405 lcm 1,587,600 > meantone 1120:1125:1134 lcm 2,268,000 > 7-limit Augmented: > 1125:1152:1176 lcm 7,056,000 > BUT ALSO > 1568:1575:1620 lcm 3,175,200 > (any simpler tratio?) > OldKleismic 1000:1008:1029 lcm 6,174,000 > Catler -- same as 12-equal -- 625:648:640 lcm 6,480,000 > Negri 672:675:686 lcm 7,408,800 > semisixths 3375:3402:3430 lcm 20,837,250 > Superpythagorean 1701:1715:1728 lcm 26,671,680 > magic 6048:6075:6125 lcm 47,628,000 > miracle 7168:7200:7203 lcm 553,190,400 > orwell 12005:12096:12150 lcm 933,508,800 > (septimal) schismic 27783:28000:28125 lcm 2,778,300,000 > ennealimmal 419904:420000:420175 lcm 4,410,829,080,000 > > What's bigger -- the lcm of ennealimmal or the national debt?. . . on Sept. 30th, 1993?

Message: 10958 - Contents - Hide Contents Date: Thu, 13 May 2004 06:01:24 Subject: Tratios vs. wedgies From: Paul Erlich I think in the codimension-2 case, the log of the tratio lcm complexity will be between 1 and 4 times the wedgie complexity (duly scaled). Proof or disproof?

Message: 10959 - Contents - Hide Contents Date: Thu, 13 May 2004 06:03:10 Subject: L1 multival norm complexity From: Paul Erlich I'm tired of writing this over and over again, so I'll call it 'genie' from now on (based on a suggestion from Aaron Johnston. It's the one with the funny coincidences with number of notes in the stereotypical scale.

Message: 10960 - Contents - Hide Contents Date: Thu, 13 May 2004 06:27:15 Subject: Re: Tratios vs. wedgies From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> I think in the codimension-2 case, the log of the tratio lcm > complexity will be between 1 and 4 times the wedgie complexity (duly > scaled). Proof or disproof?I no longer think it's true, but this sort of thing merits investigation. How do you express the lcm in terms of the wedgie? etc.

Message: 10961 - Contents - Hide Contents Date: Thu, 13 May 2004 07:13:23 Subject: Comparison of LOG-genie vs. tratio-LOG-lcm, 5-limit ETs From: Paul Erlich As we've seen, in this case genie comes out to about 3 times the number of notes per octave. So the comparison is essentially with number of notes. Again:> ET........tratio............lcm............. > 03-equal: 45:48:50......... 3,600........... > 04-equal: 24:25:27......... 5,400........... > 05-equal: 75:80:81......... 32,400.......... > 07-equal: 384:400:405...... 259,200......... > ("""""""""240:243:250...... 486,000).......( > 09-equal: 125:128:135...... 432,000......... > 10-equal: 729:768:800...... 4,665,600....... > 12-equal: 625:640:648...... 6,480,000....... > 15-equal: 243:250:256...... 7,776,000....... > 16-equal: 3072:3125:3240... 259,200,000..... > 19-equal: 15360:15552:15625 3,888,000,000 > 22-equal: 6075:6144:6250... 1,555,200,000 > > The monotonic pattern seems to break here. Did I miss any lower- weird > and/or simpler tratios?The next step is to draw a picture.

Message: 10962 - Contents - Hide Contents Date: Thu, 13 May 2004 07:23:49 Subject: 7-limit temperament and the algebra of spacetime? From: Paul Erlich The Algebra of Spacetime * [with cont.] (Wayb.) I guess I shouldn't have used the term "multival" -- multivector means a sum of elements of different grade . . .

Message: 10963 - Contents - Hide Contents Date: Thu, 13 May 2004 08:12:36 Subject: 41 "Hermanic" 7-limit linear temperaments (was: Re: 114 7-limit temperaments) From: Paul Erlich Now updating to include last 3 additions: --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Even though we were using a different complexity measure on January > 27 (L-inf instead of L1), my current list of 25 is quite close to > this one. So the method can't make that much difference -- maybe the > Klein constraint Gene was talking about has something to do with > this. Here's what made it in and what didn't: > > IN>> Number 1 Meantone >> >> [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]] >> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328] >> TOP generators [1201.698520, 504.1341314] >> bad: 6.5251 comp: 3.562072 err: 1.698521 > > IN>> Number 2 Magic >> >> [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]] >> TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877] >> TOP generators [1201.276744, 380.7957184] >> bad: 7.0687 comp: 4.274486 err: 1.276744 > > IN>> Number 3 Pajara >> >> [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]] >> TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174] >> TOP generators [598.4467109, 106.5665459] >> bad: 7.1567 comp: 2.988993 err: 3.106578 > > IN>> Number 4 Semisixths >> >> [7, 9, 13, -2, 1, 5] [[1, -1, -1, -2], [0, 7, 9, 13]] >> TOP tuning [1198.389531, 1903.732520, 2790.053107, 3364.304748] >> TOP generators [1198.389531, 443.1602931] >> bad: 7.8851 comp: 4.630693 err: 1.610469 > > IN>> Number 5 Dominant Seventh >> >> [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]] >> TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933] >> TOP generators [1195.228951, 495.8810151] >> bad: 8.0970 comp: 2.454561 err: 4.771049 > > IN>> Number 6 Injera >> >> [2, 8, 8, 8, 7, -4] [[2, 3, 4, 5], [0, 1, 4, 4]] >> TOP tuning [1201.777814, 1896.276546, 2777.994928, 3378.883835] >> TOP generators [600.8889070, 93.60982493] >> bad: 8.2512 comp: 3.445412 err: 3.582707 > > IN>> Number 7 Kleismic >> >> [6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]] >> TOP tuning [1203.187308, 1907.006766, 2792.359613, 3359.878000] >> TOP generators [1203.187309, 317.8344609] >> bad: 8.3168 comp: 3.785579 err: 3.187309 > > IN>> Number 8 Hemifourths >> >> [2, 8, 1, 8, -4, -20] [[1, 2, 4, 3], [0, -2, -8, -1]] >> TOP tuning [1203.668842, 1902.376967, 2794.832500, 3358.526166] >> TOP generators [1203.668841, 252.4803582] >> bad: 8.3374 comp: 3.445412 err: 3.66884 > > IN>> Number 9 Negri >> >> [4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]] >> TOP tuning [1203.187308, 1907.006766, 2780.900506, 3359.878000] >> TOP generators [1203.187309, 124.8419629] >> bad: 8.3420 comp: 3.804173 err: 3.187309 > > IN>> Number 10 Tripletone >> >> [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]] >> TOP tuning [1197.060039, 1902.640406, 2793.140092, 3377.079420] >> TOP generators [399.0200131, 92.45965769] >> bad: 8.4214 comp: 4.045351 err: 2.939961 > > IN>> Number 11 Schismic >> >> [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]] >> TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750] >> TOP generators [1200.760624, 498.1193303] >> bad: 8.5260 comp: 5.618543 err: .912904 > > IN>> Number 12 Superpythagorean >> >> [1, 9, -2, 12, -6, -30] [[1, 2, 6, 2], [0, -1, -9, 2]] >> TOP tuning [1197.596121, 1905.765059, 2780.732078, 3374.046608] >> TOP generators [1197.596121, 489.4271829] >> bad: 8.6400 comp: 4.602303 err: 2.403879 > > IN>> Number 13 Orwell >> >> [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]] >> TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834] >> TOP generators [1199.532657, 271.4936472] >> bad: 8.6780 comp: 5.706260 err: .946061 > > IN>> Number 14 Augmented >> >> [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]] >> TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546] >> TOP generators [399.9922103, 107.3111730] >> bad: 8.7811 comp: 2.147741 err: 5.870879 > > IN>> Number 15 Porcupine >> >> [3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]] >> TOP tuning [1196.905961, 1906.858938, 2779.129576, 3367.717888] >> TOP generators [1196.905960, 162.3176609] >> bad: 8.9144 comp: 4.295482 err: 3.094040 > > IN >> Number 16 >> >> [6, 10, 10, 2, -1, -5] [[2, 4, 6, 7], [0, -3, -5, -5]] >> TOP tuning [1196.893422, 1906.838962, 2779.100462, 3377.547174] >> TOP generators [598.4467109, 162.3159606] >> bad: 8.9422 comp: 4.306766 err: 3.106578 > > IN>> Number 17 Supermajor seconds >> >> [3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]] >> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.574099] >> TOP generators [1201.698520, 232.5214630] >> bad: 9.1819 comp: 5.522763 err: 1.698521 > > IN>> Number 18 Flattone >> >> [1, 4, -9, 4, -17, -32] [[1, 2, 4, -1], [0, -1, -4, 9]] >> TOP tuning [1202.536420, 1897.934872, 2781.593812, 3361.705278] >> TOP generators [1202.536419, 507.1379663] >> bad: 9.1883 comp: 4.909123 err: 2.536420 > > IN>> Number 19 Diminished >> >> [4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]] >> TOP tuning [1194.128460, 1892.648830, 2788.245174, 3385.309404] >> TOP generators [298.5321149, 101.4561401] >> bad: 9.2912 comp: 2.523719 err: 5.871540 Now IN!! > OUT >> Number 20 >> >> [6, 10, 3, 2, -12, -21] [[1, 2, 3, 3], [0, -6, -10, -3]] >> TOP tuning [1202.659696, 1907.471368, 2778.232381, 3359.055076] >> TOP generators [1202.659696, 82.97467050] >> bad: 9.3161 comp: 4.306766 err: 3.480440 > > IN >> Number 21 >> >> [0, 0, 12, 0, 19, 28] [[12, 19, 28, 34], [0, 0, 0, -1]] >> TOP tuning [1197.674070, 1896.317278, 2794.572829, 3368.825906] >> TOP generators [99.80617249, 24.58395811] >> bad: 9.3774 comp: 4.295482 err: 3.557008 > > OUT >> Number 22 >> >> [3, -7, -8, -18, -21, 1] [[1, 3, -1, -1], [0, -3, 7, 8]] >> TOP tuning [1202.900537, 1897.357759, 2790.235118, 3360.683070] >> TOP generators [1202.900537, 570.4479508] >> bad: 9.5280 comp: 4.891080 err: 2.900537Gawel -- IN!!!> OUT >> Number 23 >>>> [3, 12, 11, 12, 9, -8] [[1, 3, 8, 8], [0, -3, -12, -11]] >> TOP tuning [1202.624742, 1900.726787, 2792.408176, 3361.457323] >> TOP generators [1202.624742, 569.0491468] >> bad: 9.6275 comp: 5.168119 err: 2.624742 > > IN>> Number 24 Nonkleismic >> >> [10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]] >> TOP tuning [1198.828458, 1900.098151, 2789.033948, 3368.077085] >> TOP generators [1198.828458, 309.8926610] >> bad: 9.7206 comp: 6.309298 err: 1.171542 > > IN>> Number 25 Miracle >> >> [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]] >> TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756] >> TOP generators [1200.631014, 116.7206423] >> bad: 9.8358 comp: 6.793166 err: .631014 Now IN!!!!!Shelovesyouyeahyeahyeah > OUT>> Number 26 Beatles >> >> [2, -9, -4, -19, -12, 16] [[1, 1, 5, 4], [0, 2, -9, -4]] >> TOP tuning [1197.104145, 1906.544822, 2793.037680, 3369.535226] >> TOP generators [1197.104145, 354.7203384] >> bad: 9.8915 comp: 5.162806 err: 2.895855 > > IN>> Number 27 -- formerly Number 82 >> >> [6, -2, -2, -17, -20, 1] [[2, 2, 5, 6], [0, 3, -1, -1]] >> TOP tuning [1203.400986, 1896.025764, 2777.627538, 3379.328030] >> TOP generators [601.7004928, 230.8749260] >> bad: 10.0002 comp: 4.619353 err: 3.740932 > > OUT >> Number 28 >> >> [3, -5, -6, -15, -18, 0] [[1, 3, 0, 0], [0, -3, 5, 6]] >> TOP tuning [1195.486066, 1908.381352, 2796.794743, 3356.153692] >> TOP generators [1195.486066, 559.3589487] >> bad: 10.0368 comp: 4.075900 err: 4.513934 > > IN >> Number 29 >> >> [8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]] >> TOP tuning [1198.553882, 1907.135354, 2778.724633, 3378.001574] >> TOP generators [599.2769413, 272.3123381] >> bad: 10.1077 comp: 5.047438 err: 3.268439 > > IN>> Number 30 Blackwood >> >> [0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]] >> TOP tuning [1195.893464, 1913.429542, 2786.313713, 3348.501698] >> TOP generators [239.1786927, 83.83059859] >> bad: 10.1851 comp: 2.173813 err: 7.239629That's all, folks! All 28 I'm including were in the top 30 of the "Hermanic" list I posted 3 and a half months ago . . .

Message: 10964 - Contents - Hide Contents Date: Thu, 13 May 2004 08:19:24 Subject: Re: Comparison of LOG-genie vs. tratio-LOG-lcm, 5-limit ETs From: Paul Erlich Sorry, wrong subject line. --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> As we've seen, in this case genie comes out to about 3 times the > number of notes per octave. So the comparison is essentially with > number of notes. > > Again: > >> ET........tratio............lcm.............>> 03-equal: 45:48:50......... 3,600........... >> 04-equal: 24:25:27......... 5,400........... >> 05-equal: 75:80:81......... 32,400.......... >> 07-equal: 384:400:405...... 259,200......... >> ("""""""""240:243:250...... 486,000).......( >> 09-equal: 125:128:135...... 432,000......... >> 10-equal: 729:768:800...... 4,665,600....... >> 12-equal: 625:640:648...... 6,480,000....... >> 15-equal: 243:250:256...... 7,776,000....... >> 16-equal: 3072:3125:3240... 259,200,000..... >> 19-equal: 15360:15552:15625 3,888,000,000 >> 22-equal: 6075:6144:6250... 1,555,200,000 >> >> The monotonic pattern seems to break here. Did I miss any lower- > weird>> and/or simpler tratios? >

Message: 10965 - Contents - Hide Contents Date: Thu, 13 May 2004 09:44:11 Subject: Re: 22 7-limit temperaments in the upper uv quadrant From: Paul Erlich Gene posted this on Feb. 11th (click "up thread" or see below). I have no idea what you did here, Gene, but this list is awfully close to my Feb. 8th list of 23: Yahoo groups: /tuning-math/message/9317 * [with cont.] that led to my current list. Only #21 (SupermajorSeconds) and #22 (Nonkleismic) of the 23 in my list are omitted, and only Ennealimmal is added. We were so busy arguing that we didn't notice how very close our lists were. So what exactly led to your results here? I can't understand how you arrived at them. Your list has four distinct domains. All temperaments in domain n+1 are both more accurate and more complex than any temperament in domain n. Domain 1: Blackwood, Diminished, Augmented, DominantSevenths Domain 2: Everything on your list below that's not in other domains. The five temperaments I've just added, including Gawel, can be put in this domain too, without ruining the domains property above Domain 3: Orwell, Schismic, Miracle Domain 4: Ennealimmal --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> I found 22 of these; this is probably all that exist. I list the > temperament and the product uv. The list can be used as is, but if > Dave wants rounded corners, we could fix a value for uv and only > accept temperaments above it. Paul may dislike the unboundedness of > the upper quadrant; we can fix that by transforming coordinates back > to complexity and error. We have > > C = exp((8 + u - v)/3) > > E = exp((4 - 4u - v)/3) > > If we use the cuttoff hyperbola v = k/u, then we may plot that in the > C-E plane in parametric form as > > C(u) = exp((8 + u - k/u)) > E(u) = exp((4 - 4u - k/u)) > > This gives a curved line in an unbounded region, where zero error and > complexity (though not any temperaments exhibiting them) may be found. > > Other objections might be that you don't find your favorite > temperament, or you do find one you can't stand. I'm not too impressed > by the second, but for either, or if you think you see a "moat" > somewhere, you have a 5-parameter family to play with--two for the > origin, two for the slopes of the coordinate axes, and one for the > constant k in the hyperbola. That should accomodate anyone's desire to > fiddle, or even cook the books. > > Are there any remaining objections I have not answered above? > > Ennealimmal > 1 <18, 27, 18, 1, -22, -34| <3.629230331, .575465612| 2.088497 > > Meantone > 2 <1, 4, 10, 4, 13, 12| <1.005097996, 1.609665600| 1.617872 > > Magic > 3 <5, 1, 12, -10, 5, 25| <1.012524503, .7830372729| .792844 > > Pajara > 4 <2, -4, -4, -11, -12, 2| <.524469574, 1.498444023| .785888 > > Dominant seventh > 5 <1, 4, -2, 4, -6, -16| <.363511386, 2.141744135| .778548 > > Semisixths > 6 <7, 9, 13, -2, 1, 5| <.8521893702, .8383344292| .714420 > > Tripletone > 7 <3, 0, -6, -7, -18, -14| <.426316706, .940456192| .400932 > > Blackwood > 8 <0, 5, 0, 8, 0, -14| <.152491081, 2.548674345| .388650 > > Miracle > 9 <6, -7, -2, -25, -20, 15| <1.411065061, .2629785497| .371080 > > Diminished > 10 <4, 4, 4, -3, -5, -2| <.160875338, 1.953852436| .314327 > > Negri > 11 <4, -3, 2, -14, -8, 13| <.345586529, .859876933| .297162 > > Hemifourths > 12 <2, 8, 1, 8, -4, -20| <.283811920, 1.034875923| .293710 > > Kleismic > 13 <6, 5, 3, -6, -12, -7| <.322424097, .767227201| .247373 > > Superpythagorean > 14 <1, 9, -2, 12, -6, -30| <.4535440254, .4454275914| .202021 > > Injera > 15 <2, 8, 8, 8, 7, -4| <.245867248, .811824633| .199601 > > Augmented > 16 <3, 0, 6, -7, 1, 14| <.113185821, 1.762756409| .199519 > > "Number 43" {50/49, 245/243} Supermajor? > 17 <6, 10, 10, 2, -1, -5| <.287044794, .548744901| .157514This is "Biporky", not "Supermajor Seconds".> "Number 55" {81/80, 128/125} Duodecatonic? > 18 <0, 0, 12, 0, 19, 28| <.178510293, .520800106| .092968 > > Orwell > 19 <7, -3, 8, -21, -7, 27| <1.060730636, .7657743908e-1| .081228 > > Schismic > 20 <1, -8, -14, -15, -25, -10| <1.080908962, .5026039173e-1| .054327 > > Flattone > 21 <1, 4, -9, 4, -17, -32| <.3364294610, .1379787620| .046420 > > Porcupine > 22 <3, 5, -6, 1, -18, -28| <.176167209, .93101647e-1| .016401

Message: 10966 - Contents - Hide Contents Date: Thu, 13 May 2004 10:04:36 Subject: What's From: Paul Erlich For some reason, I can't find it on Gene's big list of 32,201 of these wedgies. Am I going crazy? Why isn't it there? Its LCM is comparable to the national debt on Sept. 30, 1994, only one year later than ennealimmal.

Message: 10967 - Contents - Hide Contents Date: Thu, 13 May 2004 10:19:35 Subject: Re: tratios and yantras From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> 7-limit Blackwood 243:252:256 lcm 435,456I just found this for Blackwood: 189:192:196 lcm 84,672 I guess "by inspection" doesn't cut it. I'll have to do this systematically. Here's my Matlab program to convert from tratios to wedgies: function out=trat2wedg(a1,a2,a3); j=factor(a1); a12=sum(j==2); a13=sum(j==3); a15=sum(j==5); a17=sum(j==7); j=factor(a2); a22=sum(j==2); a23=sum(j==3); a25=sum(j==5); a27=sum(j==7); j=factor(a3); a32=sum(j==2); a33=sum(j==3); a35=sum(j==5); a37=sum(j==7); w=([det([a22-a12 a23-a13;a22-a32 a23-a33]) det([a22-a12 a25-a15;a22- a32 a25-a35]) det([a22-a12 a27-a17;a22-a32 a27-a37]) det([a23-a13 a25- a15;a23-a33 a25-a35]) det([a23-a13 a27-a17;a23-a33 a27-a37]) det([a25- a15 a27-a17;a25-a35 a27-a37])]); g=gcd(w(1),gcd(w(2),gcd(w(3),gcd(w(4),gcd(w(5),w(6)))))); out=w/g;> Dominant Sevenths 245:252:256 lcm 564,480 > 7-limit Diminished 343:350:360 lcm 617,400 > Pajara 441:448:450 lcm 705,600 > Semifourths 240:243:245 lcm 952,560 > Tripletone 125:126:128 lcm 1,008,000 > Injera 392:400:405 lcm 1,587,600 > meantone 1120:1125:1134 lcm 2,268,000 > 7-limit Augmented: > 1125:1152:1176 lcm 7,056,000 > BUT ALSO > 1568:1575:1620 lcm 3,175,200 > (any simpler tratio?) > OldKleismic 1000:1008:1029 lcm 6,174,000 > Catler -- same as 12-equal -- 625:648:640 lcm 6,480,000 > Negri 672:675:686 lcm 7,408,800 > semisixths 3375:3402:3430 lcm 20,837,250 > Superpythagorean 1701:1715:1728 lcm 26,671,680 > magic 6048:6075:6125 lcm 47,628,000 > miracle 7168:7200:7203 lcm 553,190,400 > orwell 12005:12096:12150 lcm 933,508,800 > (septimal) schismic 27783:28000:28125 lcm 2,778,300,000 > ennealimmal 419904:420000:420175 lcm 4,410,829,080,000 > > What's bigger -- the lcm of ennealimmal or the national debt?

Message: 10968 - Contents - Hide Contents Date: Thu, 13 May 2004 10:20:59 Subject: Re: tratios and yantras From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> I'll have to do this systematically.But now I must sleep . . .

Message: 10969 - Contents - Hide Contents Date: Thu, 13 May 2004 11:14:33 Subject: Re: What's From: Graham Breed Paul Erlich wrote:> For some reason, I can't find it on Gene's big list of 32,201 of > these wedgies. Am I going crazy? Why isn't it there? Its LCM is > comparable to the national debt on Sept. 30, 1994, only one year > later than ennealimmal.should that be <<20, -15, 0, -70, -56, 42]]? 1/2, 113.7 cent generator basis: (0.20000000000000001, 0.094719200833399242) mapping by period and generator: [(5, 0), (6, 4), (13, -3), (14, 0)] mapping by steps: [(5, 5), (-2, 2), (19, 16), (14, 14)] highest interval width: 7 complexity measure: 35 (45 for smallest MOS) highest error: 0.007355 (8.826 cents) unique graham

Message: 10970 - Contents - Hide Contents Date: Thu, 13 May 2004 11:16:22 Subject: Re: How about From: Graham Breed <<4, 6, 6, 0, -2, -3]] gives 0/1, 277.1 cent generator basis: (0.5, 0.23088169264746539) mapping by period and generator: [(2, 0), (4, -2), (6, -3), (7, -3)] mapping by steps: [(2, 0), (-2, 2), (-3, 3), (-2, 3)] highest interval width: 3 complexity measure: 6 (10 for smallest MOS) highest error: 0.046726 (56.071 cents) Graham

Message: 10971 - Contents - Hide Contents Date: Thu, 13 May 2004 19:09:42 Subject: Re: 22 7-limit temperaments in the upper uv quadrant From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> We were so busy arguing that we didn't notice how very close our > lists were.It's hardly the case that I didn't notice that. Do you began to understand why I became so frustrated? I still can't figure out why it all blew up the way it did; normally, we communicate better than this.> So what exactly led to your results here? I can't understand how you > arrived at them.My idea was to get something closer to what people seemed to want, two exponents could be used; this could be smoothed out if we used a hyperbolic boundry in the log-log plane. In order to accomodate two exponents, making them the vertical and horizontal axis seemed like a good plan. This posting follows up a previous one which explains all of that. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 10972 - Contents - Hide Contents Date: Sat, 15 May 2004 17:55:14 Subject: Re: tratios and yantras From: monz hi Gene and Paul, definitions of tratio and yantra, please. thanks. -monz ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 10974 - Contents - Hide Contents Date: Wed, 19 May 2004 21:20:00 Subject: Re: tratios and yantras From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> hi Gene and Paul, > > > definitions of tratio and yantra, please. > thanks. > > > > -monz Hi Monz,Gene defined yantra here: Yahoo groups: /tuning-math/message/10095 * [with cont.] It's just the first N integers with no prime factors above P, for a given N and P. Tratio was something you posted to tell me I should use the terminology "proportion" for -- remember? If a temperament has codimension 1, it can be described by a vanishing ratio. If a temperament has codimension 2, it can be described by a vanishing tratio (a three-term proportion, like 625:640:648). Apparently, many, but not all, of the important codimension-1 temperaments can be described by vanishing tratios of three consective terms in the yantra (for which P is either 5 or 7, and N is infinite or simply "large enough").

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