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Message: 6000 - Contents - Hide Contents Date: Wed, 15 Jan 2003 05:43:31 Subject: Re: Nonoctave scales and linear temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>> Doesn't a map uniquely determine its generators? >>>> The problem is that the temperament does not uniquely >> determine the map. >> What is a temperament, then, if not a map?A regular temperament, when tuned, has values for the primes which map it to a group of lower rank. For instance, meantone might send 2 to 2, 3 to 5^(1/4) and 5 to 5. Starling, a 7-limit planar temperament, when tuned, might send 2 to 2, 3 to (375/14)^(1/3), 5 to 5 and 7 to (6125/18)^(1/3) (these are minimax tunings.) However, we want to define things more abstractly, and call the above 1/4-comma meantone, etc. not just "Meantone" or "Starling". In the case of a linear temperament, we can assume one generator is an octave or portion thereof, which means that while we have not defined the temperament as a mapping, we've at least come close. In the case of a planar temperament like Starling, this won't work. We could define standardized mappings, such as the Hermite reduced mapping, but it becomes more artificial. My answer has been to define the temperament by its associated wedgie; another option is to reduce a basis for the kernel (the commas of the temperament) in a standard way, such as Tenney-Minkowski. The more abstract creature we get in this way is what I think of as the temperament; it does not, for instance, presume octave equivalence.

Message: 6001 - Contents - Hide Contents Date: Thu, 16 Jan 2003 11:58:16 Subject: Re: Nonoctave scales and linear temperaments From: Graham Breed Gene Ward Smith wrote:> I think plusing is the same as taking the wedge product. If you want > n-dimensional temperaments (where linear is 2, planar 3, etc.) then > you can wedge n et maps. You may also wedge pi(p)-n commas together for the same result, where p is the prime limit and pi(x) is the number theory function counting primes less than or equal to x.The dual/complement of the wedge product. But the wedge product can't distinguish torsion from contorsion. I'm using the & operator for combining equal temperaments to get a linear temperament. The + operator can then be for adding equal temperaments to get another equal temperament. So h19&h31=meantone, but h19+h31=h50. Graham

Message: 6002 - Contents - Hide Contents Date: Thu, 16 Jan 2003 22:48:41 Subject: Re: Nonoctave scales and linear temperaments From: Carl Lumma>> >mm. How bad are such cases? Could we take them out at the >> end? >> i have no idea what you're asking. bad?Can you find them at the end of a map-space search, and take them out?>> That's what I call taxicab complexity, I think. >> not quite. for one thing, read this: > > lattice orientation * [with cont.] (Wayb.) > > including the link to my observations. //>>> if there's more than one comma being tempered out, we need >>> a notion of the "angle" between the commas . . . >> >> Please explain. >> search for "straightess" in these archives . . . Working...>> You assume there was an untweakable one in the 5-limit case? >> Bah! >> are you saying the octave should never be assumed to be > exactly 1200 cents?Yes, yes, and... yes.>> And in the 'now it's planar' case, you would no longer have an >> untweakable one. Something's got to give! >> i'm not following you.If you insist on there being an untweakable generator, you won't have it if you take a linear temperament, tweak the octave, and call it planar.>> Thanks. There is also, I gather, such a thing as a basis of >> commas? TM (TM stands for?) >> tenney-minkowski. tenney is the metric being minimized, and > minkowski provided a basis-reduction algorithm applicable to > such a case. >>> reduction applies to commas only, >> right? > > right.Thanks again. So if reduction is necc., it means that a temperament can be described by two different lists of commas, right? This means we'll have the same problem searching comma space as we did map space. So wedgies are our last hope.>>> no. you use a badness cutoff simply to define the list of >>> temperaments in the first place. >>>> That's the same as taking the 20 "best" temperaments. >> well, if your badness cutoff, extreme error cutoff, and extreme > complexity cutoff leave you with 20 inside, and if such a clunky > tripartite criterion is what you define as "best".Are you saying a badness cutoff is not sufficient to give a finite list of temperaments? -Carl

Message: 6003 - Contents - Hide Contents Date: Thu, 16 Jan 2003 12:00:38 Subject: Ultimate 5-limit again From: Gene Ward Smith 27/25 [[1, 2, 3], [0, -2, -3]] [1200., 268.056438833948093748427143263] 3.739252 35.609240 1861.731473 135/128 [[1, 2, 1], [0, -1, 3]] [1200., 522.862345874111793591855751693] 4.132031 18.077734 1275.365360 256/243 [[5, 8, 12], [0, 0, -1]] [240., 84.6637865678588914278600509674] 5.493061 12.759741 2114.877638 25/24 [[1, 1, 2], [0, 2, 1]] [1200., 350.977500432693708872243366367] 3.025593 28.851897 799.108711 648/625 [[4, 6, 9], [0, 1, 1]] [300., 94.1343573651111175944350240576] 6.437752 11.060060 2950.938432 16875/16384 [[1, 2, 2], [0, -4, 3]] [1200., 126.238272015257926746682149917] 8.172550 5.942563 3243.743713 250/243 [[1, 2, 3], [0, -3, -5]] [1200., 162.996026370546548951179738408] 5.948286 7.975801 1678.609846 128/125 [[3, 5, 7], [0, -1, 0]] [400., 91.2018560670299909777049249654] 4.828314 9.677666 1089.323984 3125/3072 [[1, 0, 2], [0, 5, 1]] [1200., 379.967949195094816842076920201] 7.741412 4.569472 2119.954990 20000/19683 [[1, 1, 1], [0, 4, 9]] [1200., 176.282270436412295298990817071] 9.785568 2.504205 2346.540676 531441/524288 [[12, 19, 28], [0, 0, -1]] [100., 14.6637865678588914278600509674] 13.183347 1.382394 3167.444999 81/80 [[1, 2, 4], [0, -1, -4]] [1200., 503.835154026035812053011163756] 4.132031 4.217731 297.556531 2048/2025 [[2, 3, 5], [0, 1, -2]] [600., 105.446531009812541696859310996] 6.271199 2.612822 644.408867 67108864/66430125 [[3, 5, 6], [0, -1, 4]] [400., 96.7879385616949726317268914802] 15.510107 .905187 3377.402314 78732/78125 [[1, -1, -1], [0, 7, 9]] [1200., 442.979297439105373735900374126] 12.192182 1.157498 2097.802867 393216/390625 [[1, -1, 2], [0, 8, 1]] [1200., 387.819673068349143521938606127] 12.543123 1.071950 2115.395301 2109375/2097152 [[1, 0, 3], [0, 7, -3]] [1200., 271.589599585245148575185388331] 12.772341 .800410 1667.723301 4294967296/4271484375 [[1, 2, 2], [0, -9, 7]] [1200., 55.2754932571412314963954609732] 18.573955 .483108 3095.692488 15625/15552 [[1, 0, 1], [0, 6, 5]] [1200., 317.079675185758890225628070818] 9.338935 1.029625 838.631548 1600000/1594323 [[1, 3, 6], [0, -5, -13]] [1200., 339.508825625715624367834924710] 13.794200 .383104 1005.555381 ``(2)^8*``(3)^14/``(5)^13 [[1, 5, 6], [0, -13, -14]] [1200., 315.250913337821936408197840098] 21.322672 .276603 2681.521263 ``(2)^24*``(5)^4/``(3)^21 [[1, 0, -6], [0, 4, 21]] [1200., 475.542233398945960632986914825] 21.733049 .153767 1578.433204 ``(2)^23*``(3)^6/``(5)^14 [[2, 4, 5], [0, -7, -3]] [600., 71.1460635722374759764193142621] 21.207625 .194018 1850.624306 ``(5)^19/``(2)^14/``(3)^19 [[19, 30, 44], [0, 1, 1]] [63.1578947368421052631578947368, 7.29225210195322285759291880280] 30.579320 .104784 2996.244873 ``(3)^18*``(5)^17/``(2)^68 [[1, 0, 4], [0, 17, -18]] [1200., 111.875426120872633513689333181] 38.845486 .058853 3449.774562 ``(2)^39*``(5)^3/``(3)^29 [[1, 3, 16], [0, -3, -29]] [1200., 565.988014913065527948022354197] 30.550812 .057500 1639.596150 ``(3)^8*``(5)/``(2)^15 [[1, 2, -1], [0, -1, 8]] [1200., 498.272487171563819993901705714] 9.459948 .161693 136.885775 ``(3)^45/``(2)^69/``(5) [[1, 2, 21], [0, -1, -45]] [1200., 498.082318148218414995068857757] 48.911647 .026391 3088.065497 ``(2)^38/``(3)^2/``(5)^15 [[1, 4, 2], [0, -15, 2]] [1200., 193.199614933859969427837273253] 24.977022 .060822 947.732642 ``(3)^35/``(2)^16/``(5)^17 [[1, -1, -3], [0, 17, 35]] [1200., 182.466089137089694182158775289] 38.845486 .025466 1492.763207 ``(2)*``(5)^18/``(3)^27 [[9, 15, 22], [0, -2, -3]] [133.333333333333333333333333333, 49.0088197863290461293795242156] 33.653272 .025593 975.428947 ``(2)^91/``(3)^12/``(5)^31 [[1, 5, 1], [0, -31, 12]] [1200., 132.194510561451335831533197063] 55.785793 .014993 2602.883149 ``(3)^10*``(5)^16/``(2)^53 [[2, 1, 6], [0, 8, -5]] [600., 162.741892126380267669129153916] 31.255737 .017725 541.228379 ``(2)^37*``(3)^25/``(5)^33 [[1, 17, 14], [0, -33, -25]] [1200., 560.546969532517954992081849041] 50.788153 .012388 1622.898233 ``(5)^51/``(2)^36/``(3)^52 [[1, 15, 16], [0, -51, -52]] [1200., 315.647874693157629813083932838] 82.462759 .004660 2613.109284 ``(2)^54*``(5)^2/``(3)^37 [[1, 2, 10], [0, -2, -37]] [1200., 249.018447894645757478665305415] 39.665603 .005738 358.125500 ``(3)^47*``(5)^14/``(2)^107 [[1, -1, 11], [0, 14, -47]] [1200., 221.567865486043404513033481373] 62.992219 .003542 885.454661 ``(2)^144/``(3)^22/``(5)^47 [[1, -2, 4], [0, 47, -22]] [1200., 91.5310212152783702805100052719] 86.914326 .002842 1866.076786 ``(3)^62/``(2)^17/``(5)^35 [[1, 11, 19], [0, -35, -62]] [1200., 322.801386591774163833303551406] 72.066208 .003022 1131.212237 ``(5)^86/``(2)^19/``(3)^114 [[2, 14, 19], [0, -43, -57]] [600., 151.117308517283810511306888810] 151.691690 .000751 2621.929721 ``(3)^54*``(5)^110/``(2)^341 [[2, -22, 17], [0, 55, -27]] [600., 274.580992585158530361667961540] 205.015253 .000385 3314.979642 ``(2)^232*``(5)^25/``(3)^183 [[1, 4, 20], [0, -25, -183]] [1200., 115.921782028626355899175361808] 191.093312 .000319 2223.857514 ``(2)^71*``(5)^37/``(3)^99 [[1, -9, -26], [0, 37, 99]] [1200., 343.296099168149974701274380276] 104.663080 .000511 586.422003 ``(5)^49/``(2)^90/``(3)^15 [[1, -6, 0], [0, 49, 15]] [1200., 185.754178875831648371588872508] 74.858154 .000761 319.341867 ``(3)^69*``(5)^61/``(2)^251 [[1, -22, 29], [0, 61, -69]] [1200., 463.966471361838581203914769766] 143.055244 .000194 566.898668 ``(3)^153*``(5)^73/``(2)^412 [[1, 38, -74], [0, -73, 153]] [1200., 598.603357209211686645273006535] 235.664038 .000052 683.835625 ``(2)^161/``(3)^84/``(5)^12 [[12, 19, 28], [0, 1, -7]] [100., 1.95516929663126752661956897955] 100.527798 .000120 121.841527 ``(2)^734/``(3)^321/``(5)^97 [[1, 18, -52], [0, -97, 321]] [1200., 203.072628472983063961058412679] 431.645735 .000032 2593.925421 ``(2)^21*``(3)^290/``(5)^207 [[1, 102, 143], [0, -207, -290]] [1200., 582.116159570683278092118297257] 374.222680 .000025 1307.744113 ``(2)^140*``(5)^195/``(3)^374 [[1, 40, 76], [0, -195, -374]] [1200., 236.400230712381991918098066318] 423.433817 .000023 1718.344823 ``(3)^237*``(5)^85/``(2)^573 [[1, 39, -102], [0, -85, 237]] [1200., 528.212294142246315995699039809] 332.899311 .000006 209.606840

Message: 6004 - Contents - Hide Contents Date: Thu, 16 Jan 2003 23:03:24 Subject: Re: Nonoctave scales and linear temperaments From: Graham Breed >> >Yes, it's the volume of a hypersphere. So it climbs dramatically with >the number of dimensions, which is one less the number of primes (or odd >numbers if you're being masochistic). For 8 dimensions, you still get >O((r**7)**7) = O(r**49) candidate wedgies. >Oops! That should be O((r**7)**6) or O(r**42) and 6 from 42 only gives 5 million or so combinations. I made the same mistake in the earlier post: there are n-2 commas needed to define a linear temperament using n primes. Graham

Message: 6005 - Contents - Hide Contents Date: Thu, 16 Jan 2003 13:48:33 Subject: Re: Nonoctave scales and linear temperaments From: Graham Breed Carl Lumma wrote:> If the above is true about commas, then complexity should be > defined in terms of commas, and we could search all sets of > simple commas...What sets of simple commas do you propose to take? The files I have here show equivalences between second-order tonality diamonds. If you have p odd numbers in your ratios, the tonality diamond will have of the order of p**2 (p squared) ratios. The second order diamond is made by combining these, so that gives O(p**4) ratios. You're then setting pairs of these ratios to be equivalent, giving O(p**8) commas. You then need to take combinations of pi(p)-1 commas. (That is one minus the number of primes in the ratios.) If you have many more commas than primes, that will go as the pi(p)-1st power. So the total number of wedgies you have to consider is O((p**8)**(pi(p)-1)) Lets simplify this by setting pi(p)~p. To get an understimate, I'll count it as pi(p) but still call it p. The complexity of finding p prime linear temperaments is then O(p**8(p-1)) In the 7-limit, there are only 4 primes, so the calculation is O(4**(8*3)) = O(4**24) = O(2.8e14) candidates (not that many, but that doesn't matter because we don't know how long each one will take). In the 19-limit, there are 8 primes. So we need O(8**(8*7)) = O(8**56) = O(3.7e50) If that really is 3.7e50 candidates, it's impossible. But even in comparison to the 7-limit case, it's huge. And (without much optimisation) I couldn't even do the 7-limit calculation! For display purposes, I'm only showing commas between the first n*3 ratios from the second-order diamond, where n is the size of the second order diamond. Using that would mean only O(p**4(p-1)) candidates. So only O(8**28) in the 19-limit, or O(2e25). But this isn't good enough -- most of my top 19-limit temperaments don't have 7 such equivalences. Whereas combing equal temperaments only gives O(n**2) calculations, where n is the number of ETs you consider. I find n=20 works well, requiring O(400) candidates. This is true in the 5-limit and also the 21-limit. I haven't heard of anybody doing a similar search with unison vectors in the 21-limit, or even suggesting ways to reduce the complexity. Graham

Message: 6006 - Contents - Hide Contents Date: Thu, 16 Jan 2003 23:06:02 Subject: Mega 11 and 13 limit ets From: Gene Ward Smith Here are lists from 50000 to 1000000 of ets with log-flat measure less than 1, together with its measure. Note that 258008 appears on both lists; it would also appear in the 15 and 17 limits. Eleven limit 54624 .687840 121524 .637140 218002 .933133 258008 .745125 312632 .988525 563949 .949222 903475 .565789 1161483 .555994 2064958 .531236 2968433 .912912 Thirteen limit 81860 .755085 87939 .756550 96478 .645002 161530 .656916 184417 .975711 258008 .614009 354486 .988733 442425 .925813 1024999 .971305 1106859 .997514 5033391 .592631 5291399 .984485 5475816 .931178 8192932 .817337

Message: 6007 - Contents - Hide Contents Date: Thu, 16 Jan 2003 19:43:14 Subject: Re: Ultimate 5-limit again From: wallyesterpaulrus thanks gene! look at this: Yahoo groups: /tuning/database? * [with cont.] method=reportRows&tbl=10&sortBy=5&sortDir=up the only thing i'd like is the numerator and denominator of each comma -- full digits. yahoo's sort doesn't recognize scientific notation. this way i can still sort by heuristic complexity. --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> 27/25 [[1, 2, 3], [0, -2, -3]] [1200., 268.056438833948093748427143263] > 3.739252 35.609240 1861.731473 > > > 135/128 [[1, 2, 1], [0, -1, 3]] [1200., 522.862345874111793591855751693] > 4.132031 18.077734 1275.365360 > > > 256/243 [[5, 8, 12], [0, 0, -1]] [240., 84.6637865678588914278600509674] > 5.493061 12.759741 2114.877638 > > > 25/24 [[1, 1, 2], [0, 2, 1]] [1200., 350.977500432693708872243366367] > 3.025593 28.851897 799.108711 > > > 648/625 [[4, 6, 9], [0, 1, 1]] [300., 94.1343573651111175944350240576] > 6.437752 11.060060 2950.938432 > > > 16875/16384 [[1, 2, 2], [0, -4, 3]] [1200., 126.238272015257926746682149917] > 8.172550 5.942563 3243.743713 > > > 250/243 [[1, 2, 3], [0, -3, -5]] [1200., 162.996026370546548951179738408] > 5.948286 7.975801 1678.609846 > > > 128/125 [[3, 5, 7], [0, -1, 0]] [400., 91.2018560670299909777049249654] > 4.828314 9.677666 1089.323984 > > > 3125/3072 [[1, 0, 2], [0, 5, 1]] [1200., 379.967949195094816842076920201] > 7.741412 4.569472 2119.954990 > > > 20000/19683 [[1, 1, 1], [0, 4, 9]] [1200., 176.282270436412295298990817071] > 9.785568 2.504205 2346.540676 > > > 531441/524288 [[12, 19, 28], [0, 0, -1]] [100., 14.6637865678588914278600509674] > 13.183347 1.382394 3167.444999 > > > 81/80 [[1, 2, 4], [0, -1, -4]] [1200., 503.835154026035812053011163756] > 4.132031 4.217731 297.556531 > > > 2048/2025 [[2, 3, 5], [0, 1, -2]] [600., 105.446531009812541696859310996] > 6.271199 2.612822 644.408867 > > > 67108864/66430125 [[3, 5, 6], [0, -1, 4]] [400., 96.7879385616949726317268914802] > 15.510107 .905187 3377.402314 > > > 78732/78125 [[1, -1, -1], [0, 7, 9]] [1200., 442.979297439105373735900374126] > 12.192182 1.157498 2097.802867 > > > 393216/390625 [[1, -1, 2], [0, 8, 1]] [1200., 387.819673068349143521938606127] > 12.543123 1.071950 2115.395301 > > > 2109375/2097152 [[1, 0, 3], [0, 7, -3]] [1200., 271.589599585245148575185388331] > 12.772341 .800410 1667.723301 > > > 4294967296/4271484375 [[1, 2, 2], [0, -9, 7]] [1200., 55.2754932571412314963954609732] > 18.573955 .483108 3095.692488 > > > 15625/15552 [[1, 0, 1], [0, 6, 5]] [1200., 317.079675185758890225628070818] > 9.338935 1.029625 838.631548 > > > 1600000/1594323 [[1, 3, 6], [0, -5, -13]] [1200., 339.508825625715624367834924710] > 13.794200 .383104 1005.555381 > > > ``(2)^8*``(3)^14/``(5)^13 [[1, 5, 6], [0, -13, -14]] [1200., 315.250913337821936408197840098] > 21.322672 .276603 2681.521263 > > > ``(2)^24*``(5)^4/``(3)^21 [[1, 0, -6], [0, 4, 21]] [1200., 475.542233398945960632986914825] > 21.733049 .153767 1578.433204 > > > ``(2)^23*``(3)^6/``(5)^14 [[2, 4, 5], [0, -7, -3]] [600., 71.1460635722374759764193142621] > 21.207625 .194018 1850.624306 > > > ``(5)^19/``(2)^14/``(3)^19 [[19, 30, 44], [0, 1, 1]] [63.1578947368421052631578947368, 7.29225210195322285759291880280] > 30.579320 .104784 2996.244873 > > > ``(3)^18*``(5)^17/``(2)^68 [[1, 0, 4], [0, 17, -18]] [1200., 111.875426120872633513689333181] > 38.845486 .058853 3449.774562 > > > ``(2)^39*``(5)^3/``(3)^29 [[1, 3, 16], [0, -3, -29]] [1200., 565.988014913065527948022354197] > 30.550812 .057500 1639.596150 > > > ``(3)^8*``(5)/``(2)^15 [[1, 2, -1], [0, -1, 8]] [1200., 498.272487171563819993901705714] > 9.459948 .161693 136.885775 > > > ``(3)^45/``(2)^69/``(5) [[1, 2, 21], [0, -1, -45]] [1200., 498.082318148218414995068857757] > 48.911647 .026391 3088.065497 > > > ``(2)^38/``(3)^2/``(5)^15 [[1, 4, 2], [0, -15, 2]] [1200., 193.199614933859969427837273253] > 24.977022 .060822 947.732642 > > > ``(3)^35/``(2)^16/``(5)^17 [[1, -1, -3], [0, 17, 35]] [1200., 182.466089137089694182158775289] > 38.845486 .025466 1492.763207 > > > ``(2)*``(5)^18/``(3)^27 [[9, 15, 22], [0, -2, -3]] [133.333333333333333333333333333, 49.0088197863290461293795242156] > 33.653272 .025593 975.428947 > > > ``(2)^91/``(3)^12/``(5)^31 [[1, 5, 1], [0, -31, 12]] [1200., 132.194510561451335831533197063] > 55.785793 .014993 2602.883149 > > > ``(3)^10*``(5)^16/``(2)^53 [[2, 1, 6], [0, 8, -5]] [600., 162.741892126380267669129153916] > 31.255737 .017725 541.228379 > > > ``(2)^37*``(3)^25/``(5)^33 [[1, 17, 14], [0, -33, -25]] [1200., 560.546969532517954992081849041] > 50.788153 .012388 1622.898233 > > > ``(5)^51/``(2)^36/``(3)^52 [[1, 15, 16], [0, -51, -52]] [1200., 315.647874693157629813083932838] > 82.462759 .004660 2613.109284 > > > ``(2)^54*``(5)^2/``(3)^37 [[1, 2, 10], [0, -2, -37]] [1200., 249.018447894645757478665305415] > 39.665603 .005738 358.125500 > > > ``(3)^47*``(5)^14/``(2)^107 [[1, -1, 11], [0, 14, -47]] [1200., 221.567865486043404513033481373] > 62.992219 .003542 885.454661 > > > ``(2)^144/``(3)^22/``(5)^47 [[1, -2, 4], [0, 47, -22]] [1200., 91.5310212152783702805100052719] > 86.914326 .002842 1866.076786 > > > ``(3)^62/``(2)^17/``(5)^35 [[1, 11, 19], [0, -35, -62]] [1200., 322.801386591774163833303551406] > 72.066208 .003022 1131.212237 > > > ``(5)^86/``(2)^19/``(3)^114 [[2, 14, 19], [0, -43, -57]] [600., 151.117308517283810511306888810] > 151.691690 .000751 2621.929721 > > > ``(3)^54*``(5)^110/``(2)^341 [[2, -22, 17], [0, 55, -27]] [600., 274.580992585158530361667961540] > 205.015253 .000385 3314.979642 > > > ``(2)^232*``(5)^25/``(3)^183 [[1, 4, 20], [0, -25, -183]] [1200., 115.921782028626355899175361808] > 191.093312 .000319 2223.857514 > > > ``(2)^71*``(5)^37/``(3)^99 [[1, -9, -26], [0, 37, 99]] [1200., 343.296099168149974701274380276] > 104.663080 .000511 586.422003 > > > ``(5)^49/``(2)^90/``(3)^15 [[1, -6, 0], [0, 49, 15]] [1200., 185.754178875831648371588872508] > 74.858154 .000761 319.341867 > > > ``(3)^69*``(5)^61/``(2)^251 [[1, -22, 29], [0, 61, -69]] [1200., 463.966471361838581203914769766] > 143.055244 .000194 566.898668 > > > ``(3)^153*``(5)^73/``(2)^412 [[1, 38, -74], [0, -73, 153]] [1200., 598.603357209211686645273006535] > 235.664038 .000052 683.835625 > > > ``(2)^161/``(3)^84/``(5)^12 [[12, 19, 28], [0, 1, -7]] [100., 1.95516929663126752661956897955] > 100.527798 .000120 121.841527 > > > ``(2)^734/``(3)^321/``(5)^97 [[1, 18, -52], [0, -97, 321]] [1200., 203.072628472983063961058412679] > 431.645735 .000032 2593.925421 > > > ``(2)^21*``(3)^290/``(5)^207 [[1, 102, 143], [0, -207, -290]] [1200., 582.116159570683278092118297257] > 374.222680 .000025 1307.744113 > > > ``(2)^140*``(5)^195/``(3)^374 [[1, 40, 76], [0, -195, -374]] [1200., 236.400230712381991918098066318] > 423.433817 .000023 1718.344823 > > > ``(3)^237*``(5)^85/``(2)^573 [[1, 39, -102], [0, -85, 237]] [1200., 528.212294142246315995699039809] > 332.899311 .000006 209.606840

Message: 6008 - Contents - Hide Contents Date: Thu, 16 Jan 2003 23:11:12 Subject: Re: Nonoctave scales and linear temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Carl Lumma wrote: > Lets simplify this by setting pi(p)~p.The prime number theorem says pi(x)~x/(ln(x)-1).> Whereas combing equal temperaments only gives O(n**2) calculations, > where n is the number of ETs you consider. I find n=20 works well, > requiring O(400) candidates.Once you take wedgies, you should have fewer candidates.

Message: 6009 - Contents - Hide Contents Date: Thu, 16 Jan 2003 20:35:43 Subject: meta tuning-math From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote:> Paul, > > I took your word for it when you wrote in > Yahoo groups: /tuning-math/message/5353 * [with cont.] > "... gene suggested using the entire range of p-norms, with p from 2 > to infinity ...". that's right. > But if it was actually you who first pointed it out, then thanks.i was the one who pointed out that the limit of the norm as p goes to infinity is the result that minimizes the maximum error amongst those errors which are not constant as a function of generator size. i.e., the kind of "minimax" value that manuel and kalle quoted.> I > can't find where Gene denied it,i think he said he deleted his claim (which was that you'd get to the center of the minimax range if you let p go to infinity) 10 seconds after posting it.

Message: 6010 - Contents - Hide Contents Date: Thu, 16 Jan 2003 23:15:31 Subject: Re: Ultimate 5-limit again From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> thanks gene! > > look at this: > > Yahoo groups: /tuning/database? * [with cont.] > method=reportRows&tbl=10&sortBy=5&sortDir=upNeat! What criterion did you use to select the ones you report on?

Message: 6011 - Contents - Hide Contents Date: Thu, 16 Jan 2003 20:37:35 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>> Eh? I go with Paul; this is the point of log-flat measures. >> Eh? Name your complexity range, and take the x most accurate > temperaments within it. Why do I need badness at all?it's a much easier, and prettier, way to acheive this. you get the distribution you want regardless of where you put the endpoints of your complexity ranges.

Message: 6012 - Contents - Hide Contents Date: Thu, 16 Jan 2003 23:25:13 Subject: Re: Nonoctave scales and linear temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> The wedge product of a set of commas is the complement of the wedge > product of a pair of equal mappings that define the same linear > temperament. The two are different, and it matters when you do > quantitative calculations, although Gene seems to get round this in a > way I don't understand.The two may be identified via Poincare duality, so I identify them. In practice, I reorder the terms of the wedge product I get from wedging commas so that it is the same list as the wedge product I get from wedging ets.

Message: 6013 - Contents - Hide Contents Date: Thu, 16 Jan 2003 20:38:26 Subject: Re: Ultimate 5-limit again From: Carl Lumma> look at this: > > Yahoo groups: /tuning/database? * [with cont.] > method=reportRows&tbl=10&sortBy=5&sortDir=upThis's great! I don't see the heuristic on it, though. -C.

Message: 6014 - Contents - Hide Contents Date: Thu, 16 Jan 2003 23:28:28 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>> Hmm. How bad are such cases? Could we take them out at the >>> end? >>>> i have no idea what you're asking. bad? >> Can you find them at the end of a map-space search, and take > them out?what do you want to take out? from what?>>> That's what I call taxicab complexity, I think. >>>> not quite. for one thing, read this: >> >> lattice orientation * [with cont.] (Wayb.) >> >> including the link to my observations. > //what does "//" mean?> If you insist on there being an untweakable generator, you won't > have it if you take a linear temperament, tweak the octave, and > call it planar.correct. the untweakable generator has been tweaked. is that all?>>> reduction applies to commas only, >>> right? >> >> right. >> Thanks again. So if reduction is necc., it means that a > temperament can be described by two different lists of commas, > right?right, although for single-comma temperaments, only one choice leaves you without torsion.> This means we'll have the same problem searching comma > space as we did map space. So wedgies are our last hope.the problem i was pointing out with map space, i think, was that the arbitrariness of the set of generators means your complexity ranking (if it's just based on the numbers of the map) will be meaningless.>>>> no. you use a badness cutoff simply to define the list of >>>> temperaments in the first place. >>>>>> That's the same as taking the 20 "best" temperaments. >>>> well, if your badness cutoff, extreme error cutoff, and extreme >> complexity cutoff leave you with 20 inside, and if such a clunky >> tripartite criterion is what you define as "best". >> Are you saying a badness cutoff is not sufficient to give a > finite list of temperaments?exactly. in *every* complexity range you have about the same number of temperaments with log-flat badness lower than some cutoff -- and there are an infinite number of non-overlapping complexity ranges.

Message: 6015 - Contents - Hide Contents Date: Thu, 16 Jan 2003 20:50:17 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:> Are you saying that the same set of commas could vanish > under two different maps, each with different gens? If > so, can you give an example? 81:80in terms of octave and fifth, the map is [1,0] [1,1] [4,0] in terms of octave and twelfth, the map is [1,0] [0,1] [4,-4]> If the above is true about commas, then complexity should be > defined in terms of commas,it's nice when there's only one comma. then the log of the numbers in the comma (say, the log of the odd limit) is an excellent estimate of complexity (it's what i call the heuristic complexity). if there's more than one comma being tempered out, we need a notion of the "angle" between the commas . . .>> that's not what i mean -- i mean, if you're dealing with a >> planar temperament (which might simply be a linear >> temperament with tweakable octaves) >> How can tweaking one of the generators of a linear temperament > turn it into a planar temperament? You need a third gen!it's a planar temperament in the sense that there are two independently tweakable generators, two independent dimensions needed in order to define each pitch of the tuning.>> or something with higher dimension, there's no unique choice >> of the basis of generators -- gene's used things such as >> hermite reduction to make this arbitrary choice for him. >> What's a "basis of generators"?for the case of meantone, one example would be octave and fifth. another example would be octave and twelfth. another example would be fifth and fourth. another example would be major second and minor second. each pair comprises a complete basis for the vector space of pitches in the tuning.>> the idea is that, if you sort by complexity, using a log- >> flat badness criterion guarantees that you'll have a similar >> number of temperaments to look at within each complexity >> range, so the complexity will increase rather smoothly in >> your list. >> You mean if I take the twenty "best" 5-limit temperaments > and sort by badness, the resulting list will alse be sorted > by complexity, then accuracy?no. you use a badness cutoff simply to define the list of temperaments in the first place. *then* you sort by complexity.>> though the mathematics of it is -- naturally -- heuristic >> in nature. > > ? >> AFAIK, a heuristic is an algorithm that attempts to search > only a fraction of a network yet still deliver results one > can have confidence in. from yourdictionary.com: heu·ris·tic(click to hear the word) (hy-rstk) adj. Of or relating to a usually speculative formulation serving as a guide in the investigation or solution of a problem: "The historian discovers the past by the judicious use of such a heuristic device as the 'ideal type'" (Karl J. Weintraub).

Message: 6016 - Contents - Hide Contents Date: Thu, 16 Jan 2003 23:29:04 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:>> >> >> Yes, it's the volume of a hypersphere. So it climbs dramatically with >> the number of dimensions, which is one less the number of primes (or odd >> numbers if you're being masochistic). For 8 dimensions, you still get >> O((r**7)**7) = O(r**49) candidate wedgies. >>> Oops! That should be O((r**7)**6) or O(r**42) and 6 from 42 only gives > 5 million or so combinations.still vastly redundant.

Message: 6017 - Contents - Hide Contents Date: Thu, 16 Jan 2003 20:53:12 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Gene Ward Smith wrote: >>> I think plusing is the same as taking the wedge product. If you want >> n-dimensional temperaments (where linear is 2, planar 3, etc.) then >> you can wedge n et maps. You may also wedge pi(p)-n commastogether for the same result, where p is the prime limit and pi(x) is the number theory function counting primes less than or equal to x.> > The dual/complement of the wedge product. But the wedge product can't > distinguish torsion from contorsion. > I'm using the & operator for combining equal temperaments to get a > linear temperament.that's the wedge product. i've never had any problems detecting when torsion is or isn't present with it. when the gcd isn't 1, you have torsion. right?

Message: 6018 - Contents - Hide Contents Date: Thu, 16 Jan 2003 23:31:03 Subject: Re: Mega 11 and 13 limit ets From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> Here are lists from 50000 to 1000000 of ets with log-flat measureless than 1, together with its measure. Note that 258008 appears on both lists; it would also appear in the 15 and 17 limits.> > > Eleven limit > > 54624 .687840 > 121524 .637140 > 218002 .933133 > 258008 .745125 > 312632 .988525 > 563949 .949222 > 903475 .565789 > 1161483 .555994 > 2064958 .531236 > 2968433 .912912what happened to 504504?

Message: 6019 - Contents - Hide Contents Date: Thu, 16 Jan 2003 20:57:38 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Whereas combing equal temperaments only gives O(n**2) calculations, > where n is the number of ETs you consider. I find n=20 works well, > requiring O(400) candidates. This is true in the 5-limit and also the > 21-limit. I haven't heard of anybody doing a similar search with unison > vectors in the 21-limit, or even suggesting ways to reduce the complexity. > > > Grahamhasn't gene done a 13-limit search . . . *not* by starting from ETs?

Message: 6020 - Contents - Hide Contents Date: Thu, 16 Jan 2003 23:32:32 Subject: Re: Ultimate 5-limit again From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:>> thanks gene! >> >> look at this: >> >> Yahoo groups: /tuning/database? * [with cont.] >> method=reportRows&tbl=10&sortBy=5&sortDir=up >> Neat! What criterion did you use to select the ones you report on?we went through this here on this list after you posted your big list. remember? i just cut things off after atomic because there was a big gap there.

Message: 6021 - Contents - Hide Contents Date: Thu, 16 Jan 2003 21:01:32 Subject: Re: Ultimate 5-limit again From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>> look at this: >> >> Yahoo groups: /tuning/database? * [with cont.] >> method=reportRows&tbl=10&sortBy=5&sortDir=up >> This's great! I don't see the heuristic on it, > though. > > -C.yahoo only supports 10 columns. i deleted the heuristic error since the RMS error has a visual correlate (distance from origin to corresponding line) on the corresponding graph. you're probably asking about heuristic complexity, though, since i wrote, "the only thing i'd like is the numerator and denominator of each comma -- full digits. yahoo's sort doesn't recognize scientific notation. this way i can still sort by heuristic complexity." by which i meant that sorting by denominator is the same as sorting by heuristic complexity. so you can now sort by heuristic complexity only until scientific notation comes in.

Message: 6022 - Contents - Hide Contents Date: Thu, 16 Jan 2003 23:39:40 Subject: Re: Ultimate 5-limit again From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith > <genewardsmith@j...>" <genewardsmith@j...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus> <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote: >>> thanks gene! >>>>>> look at this: >>> >>> Yahoo groups: /tuning/database? * [with cont.] >>> method=reportRows&tbl=10&sortBy=5&sortDir=up >>>> Neat! What criterion did you use to select the ones you report on? >> we went through this here on this list after you posted your big > list. remember? i just cut things off after atomic because there was > a big gap there.that is, i cut things off by complexity, not by your ordering, which was by the just size of the comma of all things.

Message: 6023 - Contents - Hide Contents Date: Thu, 16 Jan 2003 21:35:37 Subject: margo, manuel -- was it kirnberger? From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote: --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote: >>> Viking [161, -84, -12] .015361 cents >> this is the difference between 11 pythagorean commas and 12 syntonic > commas. i'm going to call it "atomic" instead, unless someone comes > up with a better name . . .was it kirnberger who proposed foreshortening each fifth by a schisma to approximate 12-equal? a chain of 12 such fifths would fail to close on itself by a mere "atom", or .015361 cents . . .

Message: 6024 - Contents - Hide Contents Date: Thu, 16 Jan 2003 23:40:04 Subject: Re: Ultimate 5-limit again From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith > <genewardsmith@j...>" <genewardsmith@j...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus> <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote: >>> thanks gene! >>>>>> look at this: >>> >>> Yahoo groups: /tuning/database? * [with cont.] >>> method=reportRows&tbl=10&sortBy=5&sortDir=up >>>> Neat! What criterion did you use to select the ones you report on? >> we went through this here on this list after you posted your big > list. remember? i just cut things off after atomic because there was > a big gap there.i would have liked ampersand in there, though . . .

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