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Message: 9075 Date: Fri, 09 Jan 2004 21:16:36 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Carl Lumma >>So Graham had the all the digits right, I just needed more precision. >>Multiply by 12, and we get >>1197.67406985219 1896.31727726597 2794.57282965511 >>Here it's clear we're hitting the maximum, 3.557, with both 3 and 5. >> >> >> 10 9 199.61 17.209 2.6508 >> 9 8 199.61 4.2977 0.69655 >> 6 5 299.42 16.223 3.3061 >> 5 4 399.22 12.911 2.9873 >> 4 3 499.03 0.98586 0.275 >> 3 2 698.64 3.3118 1.2812 >> 8 5 798.45 15.237 2.863 >> 5 3 898.26 13.897 3.557 >> 9 5 998.06 19.535 3.557 >> 2 1 1197.7 2.3259 2.3259 >> 9 4 1397.3 6.6236 1.2812 >> 12 5 1497.1 18.549 3.1402 >> 5 2 1596.9 10.585 3.1864 >> 8 3 1696.7 1.3401 0.29227 >> 3 1 1896.3 5.6377 3.557 >> 16 5 1996.1 17.563 2.7781 >> 10 3 2095.9 11.571 2.3581 >> 18 5 2195.7 21.86 3.3674 >> 15 4 2295.5 7.2733 1.2313 >> 4 1 2395.3 4.6519 2.3259 >> 9 2 2595 8.9495 2.1462 >> 5 1 2794.6 8.2591 3.557 >> 16 3 2894.4 3.666 0.6564 >> 6 1 3094 7.9637 3.0808 >> 25 4 3193.8 21.17 3.1864 >> 20 3 3293.6 9.245 1.5651 >> 15 2 3493.2 4.9473 1.0082 >> 8 1 3593 6.9778 2.3259 >> 25 3 3692.8 22.156 3.557 >> 9 1 3792.6 11.275 3.557 >> 10 1 3992.2 5.9332 1.7861 >> 32 3 4092.1 5.9919 0.90994 >> 12 1 4291.7 10.29 2.8702 >> 25 2 4391.5 18.844 3.3389 >> 27 2 4491.3 14.587 2.5348 >> 15 1 4690.9 2.6214 0.67097 >> 16 1 4790.7 9.3037 2.3259 >> 18 1 4990.3 13.601 3.2618 >> 20 1 5189.9 3.6073 0.83464 >> 45 2 5389.5 0.6904 0.10635 >> 24 1 5489.3 12.616 2.7515 >> 25 1 5589.1 16.518 3.557 >> 27 1 5689 16.913 3.557 >> 30 1 5888.6 0.29546 0.060214 >> 32 1 5988.4 11.63 2.3259 >> 36 1 6188 15.927 3.0808 >> 40 1 6387.6 1.2813 0.24076 >> 45 1 6587.2 3.0163 0.54924 >> 48 1 6687 14.941 2.6753 >> 50 1 6786.8 14.192 2.5146 >> 54 1 6886.6 19.239 3.3431 >> 60 1 7086.2 2.0305 0.34375 >> 64 1 7186 13.956 2.3259 >> 72 1 7385.7 18.253 2.9584 >> 75 1 7485.5 10.881 1.7468 >> 80 1 7585.3 1.0446 0.16524 >> 81 1 7585.3 22.551 3.557 >> 90 1 7784.9 5.3423 0.82292 >> 96 1 7884.7 17.267 2.6222 >> 100 1 7984.5 11.866 1.7861 > >The alaska tunings are essentially circulating versions of this >tuning. Which was based on... ! zeta12.scl ! 12 equal zeta tuning 12 ! 99.807 199.614 299.422 399.229 499.036 598.843 698.650 798.457 898.265 998.072 1097.879 1197.686 ...Notice the similarity... -Carl
Message: 9076 Date: Fri, 09 Jan 2004 08:55:36 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: > >I don't know, but I plan on investigating TOP tunings of equal and > >planar temperaments as well as linear ones. Presumably one gets a > >squashing. The octaves of Dom7 are pretty short. > > You have a way of combining commas, then? > > -Carl Seemingly -- Pajara uses two commas, after all. I'm trying to reproduce Gene's results, but I probably need to think about it away from the computer . . .
Message: 9077 Date: Fri, 09 Jan 2004 23:57:30 Subject: also... From: Carl Lumma >>> 10 9 199.61 17.209 2.6508 >>> 9 8 199.61 4.2977 0.69655 >>> 6 5 299.42 16.223 3.3061 >>> 5 4 399.22 12.911 2.9873 >>> 4 3 499.03 0.98586 0.275 >>> 3 2 698.64 3.3118 1.2812 >>> 8 5 798.45 15.237 2.863 >>> 5 3 898.26 13.897 3.557 >>> 9 5 998.06 19.535 3.557 >>> 2 1 1197.7 2.3259 2.3259 // >! zeta12.scl >! >12 equal zeta tuning > 12 >! >99.807 >199.614 >299.422 >399.229 >499.036 >598.843 >698.650 >798.457 >898.265 >998.072 >1097.879 >1197.686 > >...Notice the similarity... Also... Yahoo groups: /tuning-math/message/894 * >15 >Gram tuning = 15.052, 4.14 cents flat >Z tuning = 15.053, 4.26 cents flat Ok, so following Paul's method I'll take the 5-limit val < 15 24 35 ] and divide pairwise by log2(2 3 5), then find the average 15.07115704285749, divide the original val by it giving... < 0.9952785945594528 1.5924457512951244 2.322316720638723 ] ...and then * 1200... < 1194.3343134713434 1910.9349015541493 2786.7800647664676 ] (Is ket notation appropriate here? What is this, h1194.3343134713434 or h1200 or...?) This is not apparently the Gram or the Z tuning. To the 7-limit... < 1195.8934635210232 1913.4295416336372 2790.4180815490545 3348.5016978588646 ] ...this is close to the Gram tuning if I understand Gene's nomenclature there. Howabout the 17-limit... < 1197.365908554304 1915.7854536868863 2793.8537866267093 3352.6245439520508 4150.868482988254 4470.166058602735 4869.288028120835 ] ...whoops, we blew it. >19 >Gram tuning = 18.954, 2.93 cents sharp >Z tuning = 18.948, 3.29 cents sharp 5-limit... < 1202.2814046729093 1898.3390600098567 2784.2306213477896 ] 7-limit... < 1203.8338650199978 1900.7903131894698 2787.8257926778892 3358.06288663473 ] 11-limit... < 1201.3512212496696 1896.8703493415835 2782.076512367656 3351.137617170131 4173.114768551483 ] 31-limit... < 1201.2099597644576 // ...no cigar. >22 >Gram tuning = 22.025, 1.35 cents flat >Z tuning = 22.025, 1.37 cents flat 5-limit... < 1198.7183021467067 1907.051844324306 2778.846973158275 ] 7-limit... < 1198.6555970781733 1906.9520862607305 2778.7016114084927 3378.0294099475796 ] 11-limit... < 1198.6555970781733 // ...looks like we might have a winner here. By the way, I've decided I like the square bracket for ket notation, because it avoids confusion with abs. |n|, Euclidean distance or norm or whatever ||n|| is. -Carl
Message: 9078 Date: Fri, 09 Jan 2004 14:07:17 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich For an ET, just stretch so that the weighted errors of the most upward-biased prime and most downward-biased prime are equal in magnitude and opposite in sign. For 12-equal I take the mapping [12 19 28] divide (elementwise) by [1 log2(3) log2(5)] and get [12.00000000000000 11.98766531785769 12.05894362605501] Now we want to make the largest and smallest of these equidistant from 12, so we divide [12 19 28] by their average [12.05894362605501+11.98766531785769 ]/2 giving 0.99806172487683 1.58026439772164 2.32881069137926 So Graham had the all the digits right, I just needed more precision. Multiply by 12, and we get 1197.67406985219 1896.31727726597 2794.57282965511 Here it's clear we're hitting the maximum, 3.557, with both 3 and 5. 10 9 199.61 17.209 2.6508 9 8 199.61 4.2977 0.69655 6 5 299.42 16.223 3.3061 5 4 399.22 12.911 2.9873 4 3 499.03 0.98586 0.275 3 2 698.64 3.3118 1.2812 8 5 798.45 15.237 2.863 5 3 898.26 13.897 3.557 9 5 998.06 19.535 3.557 2 1 1197.7 2.3259 2.3259 9 4 1397.3 6.6236 1.2812 12 5 1497.1 18.549 3.1402 5 2 1596.9 10.585 3.1864 8 3 1696.7 1.3401 0.29227 3 1 1896.3 5.6377 3.557 16 5 1996.1 17.563 2.7781 10 3 2095.9 11.571 2.3581 18 5 2195.7 21.86 3.3674 15 4 2295.5 7.2733 1.2313 4 1 2395.3 4.6519 2.3259 9 2 2595 8.9495 2.1462 5 1 2794.6 8.2591 3.557 16 3 2894.4 3.666 0.6564 6 1 3094 7.9637 3.0808 25 4 3193.8 21.17 3.1864 20 3 3293.6 9.245 1.5651 15 2 3493.2 4.9473 1.0082 8 1 3593 6.9778 2.3259 25 3 3692.8 22.156 3.557 9 1 3792.6 11.275 3.557 10 1 3992.2 5.9332 1.7861 32 3 4092.1 5.9919 0.90994 12 1 4291.7 10.29 2.8702 25 2 4391.5 18.844 3.3389 27 2 4491.3 14.587 2.5348 15 1 4690.9 2.6214 0.67097 16 1 4790.7 9.3037 2.3259 18 1 4990.3 13.601 3.2618 20 1 5189.9 3.6073 0.83464 45 2 5389.5 0.6904 0.10635 24 1 5489.3 12.616 2.7515 25 1 5589.1 16.518 3.557 27 1 5689 16.913 3.557 30 1 5888.6 0.29546 0.060214 32 1 5988.4 11.63 2.3259 36 1 6188 15.927 3.0808 40 1 6387.6 1.2813 0.24076 45 1 6587.2 3.0163 0.54924 48 1 6687 14.941 2.6753 50 1 6786.8 14.192 2.5146 54 1 6886.6 19.239 3.3431 60 1 7086.2 2.0305 0.34375 64 1 7186 13.956 2.3259 72 1 7385.7 18.253 2.9584 75 1 7485.5 10.881 1.7468 80 1 7585.3 1.0446 0.16524 81 1 7585.3 22.551 3.557 90 1 7784.9 5.3423 0.82292 96 1 7884.7 17.267 2.6222 100 1 7984.5 11.866 1.7861 --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > Gene, what do you get for the top system with the commas of 12- equal > (in other words, some stretching or squashing of 12-equal)? Graham > seems to gave gotten pretty close below, but no cigar . . . > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: > > --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> > wrote: > > > Paul Erlich wrote: > > > > > > > Wow. How did you find that? > > > > > > Briefly (use the Reply thing so that indentation works), > > > > > 22876792454961:19073486328125 > > > > So it was a finite search? How do you know you won't keep finding > > worse and worse examples if you go farther out? You might be > > approaching a limit, but how do you know you'll ever reach it? > > > > > >>TOPping it gives a narrow octave of 0.99806 2:1 octaves. > > > > > > > > > > > > Shall I proceed to calculate Tenney-weighted errors for all > > (well, a > > > > bunch of) intervals? I hope you're onto something! > > > > > > If you like. > > > > OK, later -- gotta go perform now. > > I'm back . . . Looks like you might be off in the last digit or two > (so maybe there is no worst comma?), but a lot of the Tenney- weighted > errors are in the 3.5549 - 3.5591 range, so you're probably pretty > close . . . > > 10 9 199.61 17.208 2.6508 > 9 8 199.61 4.298 0.69661 > 6 5 299.42 16.223 3.3062 > 5 4 399.22 12.91 2.9872 > 4 3 499.03 0.985 0.27476 > 3 2 698.64 3.313 1.2816 > 8 5 798.45 15.238 2.8633 > 5 3 898.25 13.895 3.5566 > 9 5 998.06 19.536 3.5573 > 2 1 1197.7 2.328 2.328 > 9 4 1397.3 6.626 1.2816 > 12 5 1497.1 18.551 3.1406 > 5 2 1596.9 10.582 3.1856 > 8 3 1696.7 1.343 0.29291 > 3 1 1896.3 5.641 3.5591 > 16 5 1996.1 17.566 2.7786 > 10 3 2095.9 11.567 2.3574 > 18 5 2195.7 21.864 3.368 > 15 4 2295.5 7.2693 1.2306 > 4 1 2395.3 4.656 2.328 > 9 2 2595 8.954 2.1473 > 5 1 2794.6 8.2543 3.5549 > 16 3 2894.4 3.671 0.6573 > 6 1 3094 7.969 3.0828 > 25 4 3193.8 21.165 3.1856 > 20 3 3293.6 9.2393 1.5642 > 15 2 3493.2 4.9413 1.007 > 8 1 3593 6.984 2.328 > 25 3 3692.8 22.15 3.556 > 9 1 3792.6 11.282 3.5591 > 10 1 3992.2 5.9263 1.784 > 32 3 4092 5.999 0.91101 > 12 1 4291.7 10.297 2.8723 > 25 2 4391.5 18.837 3.3375 > 27 2 4491.3 14.595 2.5361 > 15 1 4690.9 2.6133 0.66889 > 16 1 4790.7 9.312 2.328 > 18 1 4990.3 13.61 3.2638 > 20 1 5189.9 3.5983 0.83257 > 45 2 5389.5 0.69972 0.10778 > 24 1 5489.3 12.625 2.7536 > 25 1 5589.1 16.509 3.5549 > 27 1 5688.9 16.923 3.5591 > 30 1 5888.6 0.28529 0.05814 > 32 1 5988.4 11.64 2.328 > 36 1 6188 15.938 3.0828 > 40 1 6387.6 1.2703 0.23869 > 45 1 6587.2 3.0277 0.55131 > 48 1 6687 14.953 2.6774 > 50 1 6786.8 14.181 2.5126 > 54 1 6886.6 19.251 3.3452 > 60 1 7086.2 2.0427 0.34582 > 64 1 7186 13.968 2.328 > 72 1 7385.6 18.266 2.9605 > 75 1 7485.4 10.868 1.7447 > 80 1 7585.3 1.0577 0.16731 > 81 1 7585.3 22.564 3.5591 > 90 1 7784.9 5.3557 0.82499 > 96 1 7884.7 17.281 2.6243 > 100 1 7984.5 11.853 1.784
Message: 9079 Date: Fri, 09 Jan 2004 14:25:49 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich So the stretch factor is 24/(19/log2(3) + 28/log2(5)). This looks related to the 'dual' of the comma Graham found, but I didn't have to go looking for it . . . --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > For an ET, just stretch so that the weighted errors of the most > upward-biased prime and most downward-biased prime are equal in > magnitude and opposite in sign. For 12-equal I take the mapping > [12 19 28] > divide (elementwise) by > [1 log2(3) log2(5)] > and get > [12.00000000000000 11.98766531785769 12.05894362605501] > Now we want to make the largest and smallest of these equidistant > from 12, so we divide [12 19 28] by their average > [12.05894362605501+11.98766531785769 ]/2 > giving > 0.99806172487683 1.58026439772164 2.32881069137926 > So Graham had the all the digits right, I just needed more precision. > Multiply by 12, and we get > 1197.67406985219 1896.31727726597 2794.57282965511 > Here it's clear we're hitting the maximum, 3.557, with both 3 and 5. > > > 10 9 199.61 17.209 2.6508 > 9 8 199.61 4.2977 0.69655 > 6 5 299.42 16.223 3.3061 > 5 4 399.22 12.911 2.9873 > 4 3 499.03 0.98586 0.275 > 3 2 698.64 3.3118 1.2812 > 8 5 798.45 15.237 2.863 > 5 3 898.26 13.897 3.557 > 9 5 998.06 19.535 3.557 > 2 1 1197.7 2.3259 2.3259 > 9 4 1397.3 6.6236 1.2812 > 12 5 1497.1 18.549 3.1402 > 5 2 1596.9 10.585 3.1864 > 8 3 1696.7 1.3401 0.29227 > 3 1 1896.3 5.6377 3.557 > 16 5 1996.1 17.563 2.7781 > 10 3 2095.9 11.571 2.3581 > 18 5 2195.7 21.86 3.3674 > 15 4 2295.5 7.2733 1.2313 > 4 1 2395.3 4.6519 2.3259 > 9 2 2595 8.9495 2.1462 > 5 1 2794.6 8.2591 3.557 > 16 3 2894.4 3.666 0.6564 > 6 1 3094 7.9637 3.0808 > 25 4 3193.8 21.17 3.1864 > 20 3 3293.6 9.245 1.5651 > 15 2 3493.2 4.9473 1.0082 > 8 1 3593 6.9778 2.3259 > 25 3 3692.8 22.156 3.557 > 9 1 3792.6 11.275 3.557 > 10 1 3992.2 5.9332 1.7861 > 32 3 4092.1 5.9919 0.90994 > 12 1 4291.7 10.29 2.8702 > 25 2 4391.5 18.844 3.3389 > 27 2 4491.3 14.587 2.5348 > 15 1 4690.9 2.6214 0.67097 > 16 1 4790.7 9.3037 2.3259 > 18 1 4990.3 13.601 3.2618 > 20 1 5189.9 3.6073 0.83464 > 45 2 5389.5 0.6904 0.10635 > 24 1 5489.3 12.616 2.7515 > 25 1 5589.1 16.518 3.557 > 27 1 5689 16.913 3.557 > 30 1 5888.6 0.29546 0.060214 > 32 1 5988.4 11.63 2.3259 > 36 1 6188 15.927 3.0808 > 40 1 6387.6 1.2813 0.24076 > 45 1 6587.2 3.0163 0.54924 > 48 1 6687 14.941 2.6753 > 50 1 6786.8 14.192 2.5146 > 54 1 6886.6 19.239 3.3431 > 60 1 7086.2 2.0305 0.34375 > 64 1 7186 13.956 2.3259 > 72 1 7385.7 18.253 2.9584 > 75 1 7485.5 10.881 1.7468 > 80 1 7585.3 1.0446 0.16524 > 81 1 7585.3 22.551 3.557 > 90 1 7784.9 5.3423 0.82292 > 96 1 7884.7 17.267 2.6222 > 100 1 7984.5 11.866 1.7861 > > > > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: > > Gene, what do you get for the top system with the commas of 12- > equal > > (in other words, some stretching or squashing of 12-equal)? Graham > > seems to gave gotten pretty close below, but no cigar . . . > > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > > wrote: > > > --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> > > wrote: > > > > Paul Erlich wrote: > > > > > > > > > Wow. How did you find that? > > > > > > > > Briefly (use the Reply thing so that indentation works), > > > > > > > 22876792454961:19073486328125 > > > > > > So it was a finite search? How do you know you won't keep finding > > > worse and worse examples if you go farther out? You might be > > > approaching a limit, but how do you know you'll ever reach it? > > > > > > > >>TOPping it gives a narrow octave of 0.99806 2:1 octaves. > > > > > > > > > > > > > > > Shall I proceed to calculate Tenney-weighted errors for all > > > (well, a > > > > > bunch of) intervals? I hope you're onto something! > > > > > > > > If you like. > > > > > > OK, later -- gotta go perform now. > > > > I'm back . . . Looks like you might be off in the last digit or two > > (so maybe there is no worst comma?), but a lot of the Tenney- > weighted > > errors are in the 3.5549 - 3.5591 range, so you're probably pretty > > close . . . > > > > 10 9 199.61 17.208 2.6508 > > 9 8 199.61 4.298 0.69661 > > 6 5 299.42 16.223 3.3062 > > 5 4 399.22 12.91 2.9872 > > 4 3 499.03 0.985 0.27476 > > 3 2 698.64 3.313 1.2816 > > 8 5 798.45 15.238 2.8633 > > 5 3 898.25 13.895 3.5566 > > 9 5 998.06 19.536 3.5573 > > 2 1 1197.7 2.328 2.328 > > 9 4 1397.3 6.626 1.2816 > > 12 5 1497.1 18.551 3.1406 > > 5 2 1596.9 10.582 3.1856 > > 8 3 1696.7 1.343 0.29291 > > 3 1 1896.3 5.641 3.5591 > > 16 5 1996.1 17.566 2.7786 > > 10 3 2095.9 11.567 2.3574 > > 18 5 2195.7 21.864 3.368 > > 15 4 2295.5 7.2693 1.2306 > > 4 1 2395.3 4.656 2.328 > > 9 2 2595 8.954 2.1473 > > 5 1 2794.6 8.2543 3.5549 > > 16 3 2894.4 3.671 0.6573 > > 6 1 3094 7.969 3.0828 > > 25 4 3193.8 21.165 3.1856 > > 20 3 3293.6 9.2393 1.5642 > > 15 2 3493.2 4.9413 1.007 > > 8 1 3593 6.984 2.328 > > 25 3 3692.8 22.15 3.556 > > 9 1 3792.6 11.282 3.5591 > > 10 1 3992.2 5.9263 1.784 > > 32 3 4092 5.999 0.91101 > > 12 1 4291.7 10.297 2.8723 > > 25 2 4391.5 18.837 3.3375 > > 27 2 4491.3 14.595 2.5361 > > 15 1 4690.9 2.6133 0.66889 > > 16 1 4790.7 9.312 2.328 > > 18 1 4990.3 13.61 3.2638 > > 20 1 5189.9 3.5983 0.83257 > > 45 2 5389.5 0.69972 0.10778 > > 24 1 5489.3 12.625 2.7536 > > 25 1 5589.1 16.509 3.5549 > > 27 1 5688.9 16.923 3.5591 > > 30 1 5888.6 0.28529 0.05814 > > 32 1 5988.4 11.64 2.328 > > 36 1 6188 15.938 3.0828 > > 40 1 6387.6 1.2703 0.23869 > > 45 1 6587.2 3.0277 0.55131 > > 48 1 6687 14.953 2.6774 > > 50 1 6786.8 14.181 2.5126 > > 54 1 6886.6 19.251 3.3452 > > 60 1 7086.2 2.0427 0.34582 > > 64 1 7186 13.968 2.328 > > 72 1 7385.6 18.266 2.9605 > > 75 1 7485.4 10.868 1.7447 > > 80 1 7585.3 1.0577 0.16731 > > 81 1 7585.3 22.564 3.5591 > > 90 1 7784.9 5.3557 0.82499 > > 96 1 7884.7 17.281 2.6243 > > 100 1 7984.5 11.853 1.784
Message: 9081 Date: Fri, 09 Jan 2004 14:30:54 Subject: Re: Temperament agreement From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: > > > It looks like I'd be just as happy with straight lines on this > chart. > > Could you enlighten the rest of us and give a comma list of commas > you want? I wouldn't want to include any outside the 5-limit linear temperaments having the following 18 vanishing commas. And I wouldn't mind leaving off the last four. 81/80 32805/32768 2048/2025 15625/15552 128/125 3125/3072 250/243 78732/78125 20000/19683 25/24 648/625 135/128 256/243 393216/390625 1600000/1594323 16875/16384 2109375/2097152 531441/524288 I'm afraid I disagree with Herman about including the temperament where the apotome (2187/2048) vanishes. I admit I haven't heard it. My rejection is based purely on the fact that it has errors of a similar size to others that I find marginal (as approximations of 5-limit JI) - pelogic (135/128) and quintuple-thirds (Blackwood's decatonic) (256/243) - while also having about 1.5 times their complexity. The only argument I've heard in favour of it is that Blackwood wrote something in 21-ET that sounds good. But does it sound good because it approximates 5-limit harmony, or despite not approximating it?
Message: 9085 Date: Sat, 10 Jan 2004 00:09:40 Subject: summary -- are these right? From: Carl Lumma TM reduction or LLL reduction -> canonical basis ...Which of TM, LLL is preferred these days, and is there a definition of "basis" somewhere? It's a list of commas, right? ---- Hermite normal form -> canonical map ...can someone give an algorithm for turning a basis (or whatever one needs) into a map in Hermite normal form by hand? ---- Standard val -> canonical val ...the standard val is just the best approximation of each identity in the ET, right? Are there any other contenders for canonical val? ---- TOP -> weighted minimax optimum tuning -> canonical temperament ...did Gene or Graham say there's a version of TOP equivalent to weighted rms? And Paul, have you looked at the non-weighted Tenney lattice? ---- Thanks, -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: 9088 Date: Sat, 10 Jan 2004 02:26:09 Subject: Re: Temperament agreement From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: > > > What I don't like about both of these proposals is the "corners" in > > the cutoff line. I prefer straight or smoothly curved cutoffs. > > It gives you the commas on your list, but you reject it anyway Read again. Yahoo groups: /tuning-math/message/8521 * I said I could live with it. > because it doesn't make use of your personal fetish about smooth > curves? You may be happy to known that the constant epimercity lines > *are* curved on Paul's graph. > > As for the rest, your obsession with curves is preposterous. It is neither fetish, obsession nor preposterous. (I guess I asked for that :-) And note that I said a single straight line would be fine. But rather, it comes from an understanding of how neural nets work (as in human perception).
Message: 9091 Date: Sat, 10 Jan 2004 00:44:36 Subject: Re: also... From: Carl Lumma [I wrote...] >Yahoo groups: /tuning-math/message/894 * > >>15 >>Gram tuning = 15.052, 4.14 cents flat >>Z tuning = 15.053, 4.26 cents flat > >Ok, so following Paul's method I'll take the 5-limit >val < 15 24 35 ] and // > >< 1194.3343134713434 1910.9349015541493 2786.7800647664676 ] > >(Is ket notation appropriate here? What is this, h1194.3343134713434 >or h1200 or...?) > >This is not apparently the Gram or the Z tuning. To the 7-limit... > >< 1195.8934635210232 1913.4295416336372 > 2790.4180815490545 3348.5016978588646 ] > >...this is close // > > Howabout the 17-limit... > >< 1197.365908554304 > 1915.7854536868863 > 2793.8537866267093 > 3352.6245439520508 > 4150.868482988254 > 4470.166058602735 > 4869.288028120835 ] > >...whoops, we blew it. Maybe I shouldn't be using the standard val at limits in which the ET is not consistent? >>19 >>Gram tuning = 18.954, 2.93 cents sharp >>Z tuning = 18.948, 3.29 cents sharp > >5-limit... > >< 1202.2814046729093 1898.3390600098567 2784.2306213477896 ] > >7-limit... > >< 1203.8338650199978 > 1900.7903131894698 > 2787.8257926778892 > 3358.06288663473 ] > >11-limit... // >...no cigar. ! >>22 >>Gram tuning = 22.025, 1.35 cents flat >>Z tuning = 22.025, 1.37 cents flat > >5-limit... > >< 1198.7183021467067 1907.051844324306 2778.846973158275 ] > >7-limit... > >< 1198.6555970781733 > 1906.9520862607305 > 2778.7016114084927 > 3378.0294099475796 ] > >11-limit... > >< 1198.6555970781733 // > >...looks like we might have a winner here. Going to the 13-limit... < 1200.7057937136167 1910.2137627262084 2783.454339972475 3383.8072368292833 4147.892741919766 4420.780422309225 ] ...the value seems to change sharply here too. -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: 9092 Date: Sun, 11 Jan 2004 21:57:09 Subject: Re: Temperament agreement From: Paul Erlich I don't like these two-curve boundaries when it's clear one simple curve could do. I personally could do without 78732/78125 and 20000/19683, but not without 531441/524288. --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote: > > > I wouldn't want to include any outside the 5-limit linear temperaments > > having the following 18 vanishing commas. And I wouldn't mind leaving > > off the last four. > > Your wishes can be accomodated by setting bounds for size and > epimericity. For the short list, we have size < 93 cents and > epimericity < 0.62, the only five limit comma which would be added to > the list if we used these bounds would be 1600000/1594323. Presumably > you have no objection to that, as it appears on your long list. > > > 81/80 > > 32805/32768 > > 2048/2025 > > 15625/15552 > > 128/125 > > 3125/3072 > > 250/243 > > 78732/78125 > > 20000/19683 > > 25/24 > > 648/625 > > 135/128 > > 256/243 > > 393216/390625 > > The long list has size < 93 and epimericity < 0.68. If we were to use > these bounds, we would add 6561/6250 and 20480/19683. The second of > these, 20480/19683, has epimericity 0.6757, which is a sliver higher > than the actual maximum epimericity of your long list, 0.6739, and so > setting the bound at 0.675 would leave it off. What do you make of the > 6561/6250 comma? If you had no objection to letting it on to an > amended long list, you'd be in business there as well. > > > 1600000/1594323 > > 16875/16384 > > 2109375/2097152 > > 531441/524288 > > > > I'm afraid I disagree with Herman about including the temperament > > where the apotome (2187/2048) vanishes. > > I'd like to see Herman's list too.
Message: 9094 Date: Sun, 11 Jan 2004 21:58:50 Subject: Re: Temperament agreement From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: > > > What I don't like about both of these proposals is the "corners" in > > the cutoff line. I prefer straight or smoothly curved cutoffs. > > It gives you the commas on your list, but you reject it anyway > because it doesn't make use of your personal fetish about smooth > curves? Uh-oh. > You may be happy to known that the constant epimercity lines > *are* curved on Paul's graph. > > As for the rest, your obsession with curves is preposterous. It may be time to run for the hills again :)
Message: 9095 Date: Sun, 11 Jan 2004 22:08:15 Subject: Re: The Two Diadiaschisma Scales From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > These are based on the diaschisma and the diaschisma-schisma (check > Manuel's list if you don't believe me) of 67108864/66430125. That's diaschisma *minus* schisma. I've been seeing to on all the latest charts. Note its appearance as the "misty" comma here, connecting 12, (51,) 63, 75, and the excellent 87 and 99: Tonalsoft Encyclopaedia of Tuning - equal-temperament, (c) 2004 Tonalsoft Inc. * > Scala > tells me the scale closest to diadiaschis1 in my scale archives is > bp12_17 "12-tET approximation with minimal order 17 beats". For > closest to diadiaschis2 I find that it is, according to Scala, > exactly equidistant from duoden12 "Almost equal 12-tone subset of > Duodenarium". The duodenarium is a huge Euler genus in the 5-limit lattice, with over 100 notes, I believe.
Message: 9096 Date: Sun, 11 Jan 2004 22:13:00 Subject: Re: summary -- are these right? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: > TM reduction or LLL reduction -> canonical basis > > ...Which of TM, LLL is preferred these days, LLL is just use to "set up" for TM, I believe. TM seems like one good option, but there are probably better or equally good ways to define things beyond 2 dimensions. > and is there > a definition of "basis" somewhere? You should hang it on your refrigerator. Once you do, you may be able to understand this: for the kernel of a temperament, it will be a list of linearly independent commas that don't lead to torsion; for a temperament, it will be a list of linearly independent intervals that generate the whole temperament. > ---- > Standard val -> canonical val > > ...the standard val is just the best approximation of each > identity in the ET, right? Are there any other contenders > for canonical val? Yes. (I'm in a hurry, my apologies)
Message: 9097 Date: Sun, 11 Jan 2004 22:38:32 Subject: Re: Temperament agreement From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: > > It gives you the commas on your list, but you reject it anyway > > because it doesn't make use of your personal fetish about smooth > > curves? > > Uh-oh. > > > You may be happy to known that the constant epimercity lines > > *are* curved on Paul's graph. > > > > As for the rest, your obsession with curves is preposterous. > > It may be time to run for the hills again :) Hee hee. Sorry to disappoint. :-)
Message: 9098 Date: Mon, 12 Jan 2004 17:52:32 Subject: Re: summary -- are these right? From: Carl Lumma >> > is there a definition of "basis" somewhere? // >Vector Space Basis -- from MathWorld * Ah, good. That's what I thought. >> >You should hang it on your refrigerator. Once you do, you may be >> >able to understand this: for the kernel of a temperament, it will >> >be a list of linearly independent commas that don't lead to >> >torsion; This is the only sense I've ever noticed it used around here, and it's what I meant by "TM reduction -> canonical basis". >> >for a temperament, it will be a list of linearly independent >> >intervals that generate the whole temperament. Generate the pitches in the temperament. One also needs the map. >> And did you see the posts where I compare zeta, gram, and TOP-et >> tunings? > >Yup . . . I've been wondering about working backwards from the technique to TOP for codimension > 1 temperaments. How would it apply to a pair of vals? Which commas is it tempering in the single-val case? etc. -Carl
Message: 9099 Date: Mon, 12 Jan 2004 18:12:05 Subject: Re: summary -- are these right? From: Carl Lumma >> >> And did you see the posts where I compare zeta, gram, and >> >> TOP-et tunings? >> > >> >Yup . . . >> >> I've been wondering about working backwards from the technique >> to TOP for codimension > 1 temperaments. How would it apply to >> a pair of vals? > >A pair of vals -> dimension = 2. How would what apply? We're looking for TOP for codimension 2, aren't we? >> Which commas is it tempering in the single-val case? > >Nothing new to TOP here. TOP is a single-comma technique last I heard. Yet ETs require more than a single comma in the 5-limit... Oh, and just in case these got lost... >...did Gene or Graham say there's a version of TOP equivalent >to weighted rms? And Paul, have you looked at the non-weighted >Tenney lattice? -Carl
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