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Message: 5975 - Contents - Hide Contents Date: Tue, 14 Jan 2003 11:13:07 Subject: Re: Notating Pajara From: Graham Breed Gene:>> minimax 706.8431431 Manuel:> Could this be wrong? I have 709.363 in the scale archive.I agree with Manuel Graham

Message: 5976 - Contents - Hide Contents Date: Tue, 14 Jan 2003 22:34:36 Subject: Re: Nonoctave scales and linear temperaments From: Carl Lumma>> >ecause complexity is comparitive. The idea of a complete >> otonal chord is not less arbitrary. >>right, but whether a particular mapping is more complex >than another shouldn't be this arbitrary!I'm lost. If you agree with that, then what's arbitrary?>> Oh, crap. What is it you've been pushing, then? >> Weighted complexity? >>that's been much more common, yes.Ok. Here's my latest thinking, as promised. Ideally we'd base everything on complete n-ads, with harmonic entropy. Since that's not available, we'll look at dyadic breakdowns. If you use the concept of odd limits, and your best way of measuring the error of an n-ad is to break it down into dyads, you're basically saying that ratio containing n is much different than any ratio containing at most n-2. Thus, I suspect that my sum of abs-errors for each odd identity up to the limit would make sense despite the fact that for dyads like 5:3 the errors may cancel. If we throw out odd-limit, however, we might be better off. If there were a weighting that followed Tenney limit but was steep enough to make near-perfect 2:1s a fact of life and anything much beyond the 17-limit go away, we could have individually-weighted errors and 'limit infinity'. We should be able to search map space and assign generator values from scratch. Pure 2:1 generators should definitely not be assumed. Instead, we might use the appearence of many near-octave generators as evidence the weighting is right. As far as my combining error and complexity before optimizing generators, that was wrong. Moreover, combining them at all is not for me. I'm not bound to ask, "What's the 'best' temp. in size range x?". Rather, I might ask, "What's the most accurate temperament in complexity range x?". Which is just a sort on all possible temperaments, first by complexity, then by accuracy. Which is how I set up Dave's 5-limit spreadsheet after endlessly trying exponents in the badness calc. without being able to get a sensicle ranking. As for which complexity to use, we have the question of how to define a map at 'limit infinity'. . . In the meantime, what about standard n-limit complexity? () Gene's geometric complexity sounds interesting (assuming it's limit-specific...). () The number of notes of the temperament needed to get all of the n-limit dyads. () The taxicab complexity of the n-limit commas on the harmonic lattice. Or something that measured how much smaller the average harmonic structure would be in the temperament than in JI. This sort of formulation is probably best, and may in fact be what Gene's geometric complexity does...>>> f(w3*error(3),w5*error(5),w5*error(5:3)) >>> >>> where f is either RMS or MAD or MAX or whatever, and w3 is >>> your weight on ratios of 3, and w5 is your weight on ratios >>> of 5. >>>> Thanks. So my f is +, >>are you sure? aren't there absolute values, in which case it's >equivalent to MAD? (or p=1, which gene doesn't want to consider)Yep, you're right. Though its choice was just an expedient here, and it would be the MAD of just the identities, not of all the dyads in the limit. Last time we tested these things for all-the-dyads-in-a-chord, I believe I preferred RMS. Which is not to say that MAD shouldn't be included in the poptimal series. -Carl

Message: 5977 - Contents - Hide Contents Date: Tue, 14 Jan 2003 22:53:47 Subject: Re: Notating Pajara From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> though i suppose if you were taking the limit as p approached > infinity, you'd essentially take the minimax without considering 7:5.No, you'd get the value half-way between the endpoints of the range of minimax values.

Message: 5978 - Contents - Hide Contents Date: Tue, 14 Jan 2003 22:57:16 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>> Because complexity is comparitive. The idea of a complete >>> otonal chord is not less arbitrary. >>>> right, but whether a particular mapping is more complex >> than another shouldn't be this arbitrary! >> I'm lost. If you agree with that, then what's arbitrary?if you choose a different set of generators, you'll get a different ranking for which mapping is more complex then which! on the rest of this message, i'll have to get back to you later . . .

Message: 5979 - Contents - Hide Contents Date: Wed, 15 Jan 2003 06:04:54 Subject: 103169 etc. From: Gene Ward Smith Part of my plan for fixing my minimax calculations involves using rational arithmetic, so I'm interested in ets in other limits which correspond, more or less, to 103169 in the 7-limit. This isn't an actual requirement, but it seems better style. If anyone has done similar computations this far out in the 11 and 13 limits, could you post the results here? I'll defend you from the charge that such computations are useless by way of payment.

Message: 5980 - Contents - Hide Contents Date: Wed, 15 Jan 2003 16:48:02 Subject: Re: 103169 etc. From: manuel.op.de.coul@xxxxxxxxxxx.xxx Probably only of marginal help, but it took quite some time to run on my PC: 1200.0000 cents divided by 103169, step = 0.0116 cents Nearest to 5/4 : 33213, 386.3137 cents, diff. 0.000378 steps, 0.0000 cents Nearest to 3/2 : 60350, 701.9550 cents, diff. 0.003763 steps, 0.0000 cents Nearest to 7/4 : 83294, 968.8259 cents, diff. 0.000046 steps, 0.0000 cents Nearest to 11/8 : 47399, 551.3168 cents, diff. -0.100663 steps, -0.0012 cents Nearest to 13/8 : 72264, 840.5316 cents, diff. 0.334719 steps, 0.0039 cents Misfit numbers M1-M5 : 0.0000 0.0000 0.0000 0.0000 0.0000 Relative errors R1-R5 : 1.505 0.828 0.558 10.485 35.166 % of average Combined error factor : 0.0000 (3 - 7) Combined error factor : 0.0028 (3 - 15) Highest harmonic represented consistently : 16 highest error 13/11 : 24865, diff. 0.435382 steps, 0.0051 cents, level 1 Highest harmonic represented uniquely : 398 highest error 109/86 : 35276, diff. 0.499875 steps, 0.0058 cents, level 1 Highest harm. represented uniquely inv. equiv.: 396 highest error 109/86 : 35276, diff. 0.499875 steps, 0.0058 cents, level 1 Consistency levels : 3: 132 5: 132 7: 132 9: 66 11: 4 13: 1 15: 1 Diameters : 3: 51584 5: 229 7: 44 9: 31 11: 15 13: 11 15: 11 Consistency 16 region : 103168.89683 - 103169.15475 tones/octave Number of possible Pythagorean generators : 91840 Pyth. maj. third : 35062, 407.8202 cents, diff. 1849.000 steps, 21.5065 cents Pyth. dim. fourth : 33045, 384.3596 cents, diff. -167.9996 steps, -1.9541 cents Basic fifth : 60182, 700.0010 cents, diff. -167.9962 steps, -1.9540 cents, -0.09086 syntonic commas ( 1/11 ), -0.08329 Pythagorean commas ( 1/12 ) Basic Pyth. third : 34390, 400.0039 cents, diff. 1177.000 steps, 13.6902 cents Number of recognisable fifths : 2948 Number of recognisable thirds : 8597 Best Pyth. comma : 2017, Basic Pyth. comma : 1 Best diesis : 3530, Basic diesis : 2 Syntonic comma : 1849, Basic syntonic comma : 1177 Diaschisma : 1681, Schisma : 168 Diatonic semitone : 9606, Chromatic semitone : 7925 Minor tone : 15682,Minor chroma : 6076 Pythagorean limma : 7757, Apotome : 9774, Pyth. whole tone : 17531 R = W / H : 2.260023 = 17531/7757 Basic R : 2.000116 = 17195/8597 Hyperoche : 11791,Eschatum :-13808 Kleisma : 697, Wuerschmidt comma : 984 Septimal kleisma : 663, Septimal comma : 2344 Septimal diesis : 4193, Harrison comma : 4361 Undecimal comma : 4580, Tridecimal comma : 5617 Basic third : 34389, 399.9922 cents, diff. 13.6785 cents Number of cycles of basic fifths : 1 of 103169 tones Number of cycles of best fifths : 1 of 103169 tones Number of basic fifths in basic third : 103161 or -8 Number of basic fifths in best third : 89049 or -14120 Number of best fifths in basic third : 64909 or -38260 Number of best fifths in best third : 46795 or -56374 Number of basic fifths in best seventh : 71007 or -32162 Number of best fifths in best seventh : 10214 or -92955 Number of basic fifths in best sixth : 87032 or -16137 Number of best fifths in best sixth : 46794 or -56375 Number of basic thirds in best seventh : 81397 or -21772 Number of best thirds in best seventh : 81120 or -22049 Number of cycles of basic thirds : 1 of 103169 tones Number of cycles of best thirds : 1 of 103169 tones Number of basic thirds in basic fifth : 12896 or -90273 Number of basic thirds in best fifth : 12644 or -90525 Number of best thirds in basic fifth : 81315 or -21854 Number of best thirds in best fifth : 76814 or -26355 Number of cycles of best sevenths : 1 of 103169 tones Number of best sevenths in basic fifth : 102922 or -247 Number of best sevenths in best fifth : 17646 or -85523 Number of best sevenths in basic third : 1976 or -101193 Number of best sevenths in best third : 83063 or -20106 Embedded divisions: 11 83 113 913 1243 9379 Manuel

Message: 5981 - Contents - Hide Contents Date: Wed, 15 Jan 2003 08:45:14 Subject: Re: Notating Pajara From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> I'm afraid minimax can sometimes involve a range of values, andthat's what's happened here. I was simply feeding it to Maple's simplex routine and turning the crank, but I need to redo the code so that the range is returned when there is one.>I've been Internet-challenged for the past few days. I'm glad Paul remembered this common pitfall with max-absolute calculations. Standard practice (among tuning folk) when calculating generators giving the least max-absolute error (minimax) is not to return a range (musically we don't really care about such ranges) but simply to eliminate from the optimisation any consonance whose size is independent of the generator size (like the 5:7 in Pajara). Of course one must still take the errors of such consonances into account when quoting the least max-absolute error for the tuning in cents, as required for comparison with other temperaments. Previously, in my case at least, the justifaction for this procedure was purely musical, and so it appeared somewhat ad hoc mathematically. Thanks Gene for pointing out that least max-absolute corresponds to least sum-of-p_th-powers-of-absolutes as p -> oo. As Paul pointed out, this now provides a purely mathematical justification for the standard procedure. So Gene, does this now throw into doubt all your previous p-optimal calculations?

Message: 5982 - Contents - Hide Contents Date: Wed, 15 Jan 2003 17:03:44 Subject: Re: Notating Pajara 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:>>> i get 709.051244958997 -- we have some discrepancy!! >> i get a p=4 norm of 0.0162470021271903 for your value, and >> 0.0162469162102062 for mine, so clearly something's wrong with your >> algorithm! >> Nothing is wrong with my algorithm; I reran it and got your value.I have no idea where mine came from.> >>> minimax 706.8431431 >>>> i'm not sure, but i do get this same figure for MAD. >>>>> From this we can't prove that 22-equal is poptimal, though it at >>> least comes close and might be. >>>> according to my calculations, for p=5 we get 709.112411004975, so 22- >> equal is in there! >> And p=6 gives 709.1585, better yet; it's simply headed off where >you suggested, to 709.363.when did i suggest that? and isn't it true that this value, the limit as p goes to infinity, is *not* the midpoint of the minimax range, as you stated it would be?>> The best choice might still be 22, but it isn't a walk. >> >> on my keyboard it is! >> Depends on how much you favor triads, I suppose. Compromising >between 22 and 34 does seem like an interesting possibility.well, with 22 i have more triads and more of everything, since the chain closes on itself. plus the guitar, by its nature with frets across differently-tuned strings, *really* calls strongly for a closed chain.

Message: 5983 - Contents - Hide Contents Date: Wed, 15 Jan 2003 17:08:50 Subject: Re: Notating Pajara From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote:> Thanks Gene for pointing out that least max-absolute corresponds to > least sum-of-p_th-powers-of-absolutes as p -> oo.huh? *i* suggested that this was the case, and gene denied it.> So Gene, does this now throw into doubt all your previous p-optimal > calculations?something else seems to have thrown a large number of them into doubt -- copying error perhaps, but since the conclusions were wrong about 22 and pajara, i don't feel too comfortable about any others. and i think it's insanity not to allow p to go as low as 1, or perhaps even lower, when talking this poptimal stuff.

Message: 5984 - Contents - Hide Contents Date: Wed, 15 Jan 2003 17:39:53 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>> Because complexity is comparitive. The idea of a complete >>> otonal chord is not less arbitrary. >>>> right, but whether a particular mapping is more complex >> than another shouldn't be this arbitrary! >> I'm lost. If you agree with that, then what's arbitrary? >>>> Oh, crap. What is it you've been pushing, then? >>> Weighted complexity? >>>> that's been much more common, yes. >> Ok. Here's my latest thinking, as promised. > > Ideally we'd base everything on complete n-ads, with > harmonic entropy. Since that's not available, we'll look > at dyadic breakdowns. > > If you use the concept of odd limits, and your best way > of measuring the error of an n-ad is to break it down > into dyads, you're basically saying that ratio containing > n is much different than any ratio containing at most n-2. > Thus, I suspect that my sum of abs-errors for each odd > identity up to the limit would make sense despite the fact > that for dyads like 5:3 the errors may cancel.i don't understand this reasoning at all. completely baffled, i am.> If we throw out odd-limit, however, we might be better off. > If there were a weighting that followed Tenney limit but > was steep enough to make near-perfect 2:1s a fact of life > and anything much beyond the 17-limit go away, we could > have individually-weighted errors and 'limit infinity'.but there would have to be an infinitely long map, or wedgie, or list of unison vectors in order to define the temperament family.> We should be able to search map space and assign generator > values from scratch.i don't understand this.> Pure 2:1 generators should definitely > not be assumed. Instead, we might use the appearence of > many near-octave generators as evidence the weighting is > right.as gene expained, we can let the 2:1s fall as they may even with the current framework. though the choice of what gets defined as "generator" becomes arbitrary when the tuning actually has more than one dimension of freedom to it, the actual parameters describing the temperament's tuning remain perfectly well-defined.> As far as my combining error and complexity before optimizing > generators, that was wrong. Moreover, combining them at all > is not for me. I'm not bound to ask, "What's the 'best' temp. > in size range x?". Rather, I might ask, "What's the most > accurate temperament in complexity range x?".that's exactly how i've been looking at all this for the entire history of this list -- witness my comments to dave defending gene's log-flat badness measures; i took exactly this tack!> Which is just > a sort on all possible temperaments, first by complexity,this is exactly how i proposed that we present the results in our paper . . .> then > by accuracy.well, you'll rarely have two temperaments with the same complexity, though many complexity measures do put meantone and pelogic as identical complexity, and my heuristic puts blackwood and porcupine as identical complexity, for instance . . .> Which is how I set up Dave's 5-limit spreadsheet > after endlessly trying exponents in the badness calc. without > being able to get a sensicle ranking.well, at this point, it's easy enough to sort the 5-limit database by complexity, at least complexity as defined by my heuristic: Yahoo groups: /tuning/database? * [with cont.] method=reportRows&tbl=10&sortBy=3 or Yahoo groups: /tuning/database? * [with cont.] method=reportRows&tbl=10&sortBy=8>>>> f(w3*error(3),w5*error(5),w5*error(5:3)) >>>> >>>> where f is either RMS or MAD or MAX or whatever, and w3 is >>>> your weight on ratios of 3, and w5 is your weight on ratios >>>> of 5. >>>>>> Thanks. So my f is +, >>>> are you sure? aren't there absolute values, in which case it's >> equivalent to MAD? (or p=1, which gene doesn't want to consider) >> Yep, you're right. Though its choice was just an expedient > here, and it would be the MAD of just the identities, not of > all the dyads in the limit.this again. how strange since the original context in which you proposed MAD was, iirc, 15-equal. 15-equal has a minor third which is less that 4 cents off just, giving its triads a "locked" quality which you liked. if you keep the magnitudes of the errors on the identities, but make them disagree in sign, the minor third will be 32 cents off just. is such a tuning really just as good as 15-equal? try it!

Message: 5985 - Contents - Hide Contents Date: Wed, 15 Jan 2003 18:37:51 Subject: Re: Ultimate 5-limit comma list From: wallyesterpaulrus well, could i? --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> could i ask for periods and rms-optimal generators for each? > > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" > <genewardsmith@j...> wrote:>> Not that any list is really ultimate, but with rms error < 40,> geometric complexity < 500, and badness < 3500, it covers a lot of > ground. >>>> 27/25 3.739252 35.60924 1861.731473 >> >> 135/128 4.132031 18.077734 1275.36536 >> >> 256/243 5.493061 12.759741 2114.877638 >> >> 25/24 3.025593 28.851897 799.108711 >> >> 648/625 6.437752 11.06006 2950.938432 >> >> 16875/16384 8.17255 5.942563 3243.743713 >> >> 250/243 5.948286 7.975801 1678.609846 >> >> 128/125 4.828314 9.677666 1089.323984 >> >> 3125/3072 7.741412 4.569472 2119.95499 >> >> 20000/19683 9.785568 2.504205 2346.540676 >> >> 531441/524288 13.183347 1.382394 3167.444999 >> >> 81/80 4.132031 4.217731 297.556531 >> >> 2048/2025 6.271199 2.612822 644.408867 >> >> 67108864/66430125 15.510107 .905187 3377.402314 >> >> 78732/78125 12.192182 1.157498 2097.802867 >> >> 393216/390625 12.543123 1.07195 2115.395301 >> >> 2109375/2097152 12.772341 .80041 1667.723301 >> >> 4294967296/4271484375 18.573955 .483108 3095.692488 >> >> 15625/15552 9.338935 1.029625 838.631548 >> >> 1600000/1594323 13.7942 .383104 1005.555381 >> >> (2)^8*(3)^14/(5)^13 21.322672 .276603 2681.521263 >> >> (2)^24*(5)^4/(3)^21 21.733049 .153767 1578.433204 >> >> (2)^23*(3)^6/(5)^14 21.207625 .194018 1850.624306 >> >> (5)^19/(2)^14/(3)^19 30.57932 .104784 2996.244873 >> >> (3)^18*(5)^17/(2)^68 38.845486 .058853 3449.774562 >> >> (2)^39*(5)^3/(3)^29 30.550812 .057500 1639.59615 >> >> (3)^8*(5)/(2)^15 9.459948 .161693 136.885775 >> >> (3)^45/(2)^69/(5) 48.911647 .026391 3088.065497 >> >> (2)^38/(3)^2/(5)^15 24.977022 .060822 947.732642 >> >> (3)^35/(2)^16/(5)^17 38.845486 .025466 1492.763207 >> >> (2)*(5)^18/(3)^27 33.653272 .025593 975.428947 >> >> (2)^91/(3)^12/(5)^31 55.785793 .014993 2602.883149 >> >> (3)^10*(5)^16/(2)^53 31.255737 .017725 541.228379 >> >> (2)^37*(3)^25/(5)^33 50.788153 .012388 1622.898233 >> >> (5)^51/(2)^36/(3)^52 82.462759 .004660 2613.109284 >> >> (2)^54*(5)^2/(3)^37 39.665603 .005738 358.1255 >> >> (3)^47*(5)^14/(2)^107 62.992219 .003542 885.454661 >> >> (2)^144/(3)^22/(5)^47 86.914326 .002842 1866.076786 >> >> (3)^62/(2)^17/(5)^35 72.066208 .003022 1131.212237 >> >> (5)^86/(2)^19/(3)^114 151.69169 .000751 2621.929721 >> >> (3)^54*(5)^110/(2)^341 205.015253 .000385 3314.979642 >> >> (2)^232*(5)^25/(3)^183 191.093312 .000319 2223.857514 >> >> (2)^71*(5)^37/(3)^99 104.66308 .000511 586.422003 >> >> (5)^49/(2)^90/(3)^15 74.858154 .000761 319.341867 >> >> (3)^69*(5)^61/(2)^251 143.055244 .000194 566.898668 >> >> (3)^153*(5)^73/(2)^412 235.664038 5.224825e-05 683.835625 >> >> (2)^161/(3)^84/(5)^12 100.527798 .000120 121.841527 >> >> (2)^734/(3)^321/(5)^97 431.645735 3.225337e-05 2593.925421 >> >> (2)^21*(3)^290/(5)^207 374.22268 2.495356e-05 1307.744113 >> >> (2)^140*(5)^195/(3)^374 423.433817 2.263360e-05 1718.344823 >> >> (3)^237*(5)^85/(2)^573 332.899311 5.681549e-06 209.60684

Message: 5986 - Contents - Hide Contents Date: Wed, 15 Jan 2003 18:48:32 Subject: Re: Nonoctave scales and linear temperaments From: Carl Lumma>> >f we throw out odd-limit, however, we might be better off. >> If there were a weighting that followed Tenney limit but >> was steep enough to make near-perfect 2:1s a fact of life >> and anything much beyond the 17-limit go away, we could >> have individually-weighted errors and 'limit infinity'. >>but there would have to be an infinitely long map, or wedgie, >or list of unison vectors in order to define the temperament >family.Yeah, but we could approximate it with a finite map.>> We should be able to search map space and assign generator >> values from scratch. >>i don't understand this.The number of possible maps I'm interested in isn't that large. Gene didn't deny that a map uniquely defined its generators (still working through how a list of commas could be more fundamental than a map...).>as gene expained, we can let the 2:1s fall as they may even >with the current framework.What is the current framework? How have we been searching for new systems?>though the choice of what gets defined as "generator" becomes >arbitraryI never gave any special significance to the generator vs. the ie.>> As far as my combining error and complexity before optimizing >> generators, that was wrong. Moreover, combining them at all >> is not for me. I'm not bound to ask, "What's the 'best' temp. >> in size range x?". Rather, I might ask, "What's the most >> accurate temperament in complexity range x?". >>that's exactly how i've been looking at all this for the entire >history of this list -- witness my comments to dave defending >gene's log-flat badness measures; i took exactly this tack!How could it defend Gene's log-flat badness? It's utterly opposed to it!>> Which is just a sort on all possible temperaments, first by >> complexity, >>this is exactly how i proposed that we present the results in >our paper . . . Cool.>> then by accuracy. >>well, you'll rarely have two temperaments with the same >complexity,Funny, I don't see Dave's 5-limitTemp spreadsheet on his website, but with Graham complexity, you do get a fair number of collisions IIRC.>well, at this point, it's easy enough to sort the 5-limit >database by complexity, at least complexity as defined by >my heuristic:Too bad there's nothing explaining the heuristic. :( -C.

Message: 5987 - Contents - Hide Contents Date: Wed, 15 Jan 2003 18:50:06 Subject: Re: Ultimate 5-limit comma list From: wallyesterpaulrus --- 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 0.015 cents . . .

Message: 5988 - Contents - Hide Contents Date: Wed, 15 Jan 2003 19:00:05 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>> If we throw out odd-limit, however, we might be better off. >>> If there were a weighting that followed Tenney limit but >>> was steep enough to make near-perfect 2:1s a fact of life >>> and anything much beyond the 17-limit go away, we could >>> have individually-weighted errors and 'limit infinity'. >>>> but there would have to be an infinitely long map, or wedgie, >> or list of unison vectors in order to define the temperament >> family. >> Yeah, but we could approximate it with a finite map.hmm . . . aren't you then just talking about a finite limit of some sort?>>> We should be able to search map space and assign generator >>> values from scratch. >>>> i don't understand this. >> The number of possible maps I'm interested in isn't that > large. Gene didn't deny that a map uniquely defined its > generatorsnot sure if this concept has been pinned down . . .> (still working through how a list of commas > could be more fundamental than a map...).given a list of commas, you can determine the mapping, no matter how you define your generators.>> as gene expained, we can let the 2:1s fall as they may even >> with the current framework. >> What is the current framework? How have we been searching > for new systems?well, i guess i really meant a *generalization* of the current framework, using say an integer limit or product limit instead of an odd limit to optimize (there, we've come full circle). gene has pulled out a few examples, iirc. we've been searching by commas, i believe -- i'll let gene answer this more fully. graham has been searching by "+"ing ET maps to get linear temperament maps -- something i'm not sure i can explain right now.>> though the choice of what gets defined as "generator" becomes >> arbitrary >> I never gave any special significance to the generator > vs. the ie.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) 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.>>> As far as my combining error and complexity before optimizing >>> generators, that was wrong. Moreover, combining them at all >>> is not for me. I'm not bound to ask, "What's the 'best' temp. >>> in size range x?". Rather, I might ask, "What's the most >>> accurate temperament in complexity range x?". >>>> that's exactly how i've been looking at all this for the entire >> history of this list -- witness my comments to dave defending >> gene's log-flat badness measures; i took exactly this tack! >> How could it defend Gene's log-flat badness? It's utterly > opposed to it!hardly!! 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.> Too bad there's nothing explaining the heuristic. :(you can find old posts on this list explaining it, and other posts which link to that explanation. the logic is complete, though the mathematics of it is -- naturally -- heuristic in nature.

Message: 5989 - Contents - Hide Contents Date: Wed, 15 Jan 2003 21:14:05 Subject: Re: Nonoctave scales and linear temperaments From: Graham Breed wallyesterpaulrus wrote:> we've been searching by commas, i believe -- i'll let gene answer > this more fully. graham has been searching by "+"ing ET maps to get > linear temperament maps -- something i'm not sure i can explain right > now.I wrote my method up a while back: How to find linear temperaments * [with cont.] (Wayb.) I've implemented a search by unison vectors as well. But it isn't as efficient if you use an arbitrarily large set of unison vectors, as you really should for an exhaustive search.> 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) 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.Even with linear temperaments, it can be difficult to make the choice unique. I had a lot of trouble with scales with a small period generated by unison vectors. The relative sizes of the possible generators can change after you optimize. Graham

Message: 5990 - Contents - Hide Contents Date: Wed, 15 Jan 2003 21:46:51 Subject: Re: Notating Pajara From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote:> Standard practice (among tuning folk) when calculating generators > giving the least max-absolute error (minimax) is not to return a range > (musically we don't really care about such ranges)It seems to me that for this business that musically we might in fact care about them. Of course it would be easy enough to code so that the p-->infinity limit minimax was returned, but is this what we want?> So Gene, does this now throw into doubt all your previous p-optimal > calculations?Fraid so. :(

Message: 5991 - Contents - Hide Contents Date: Wed, 15 Jan 2003 21:51:35 Subject: Re: Notating Pajara From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:>> And p=6 gives 709.1585, better yet; it's simply headed off where >> you suggested, to 709.363. >> when did i suggest that? and isn't it true that this value, the limit > as p goes to infinity, is *not* the midpoint of the minimax range, as > you stated it would be?I cancel that 10 seconds after posting it and you want to hold me to it. :)

Message: 5992 - Contents - Hide Contents Date: Wed, 15 Jan 2003 22:03:30 Subject: Re: Nonoctave scales and linear temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:> The number of possible maps I'm interested in isn't that > large. Gene didn't deny that a map uniquely defined its > generators (still working through how a list of commas > could be more fundamental than a map...).It is easy to define a canonical set of commas by insisting it be TM reduced; and thereby associate something (a set of commas) as a unique marker for the temperament. We can also do this with maps by for instance Hermite reducing the map. No one seems to like Hermite reduction much, so if you want to define a canonical map which is better the field of opportunity is open.>> as gene expained, we can let the 2:1s fall as they may even >> with the current framework. >> What is the current framework? How have we been searching > for new systems?I've been coming up with a set of wedgies, setting limits to error, complexity and log-flat badness, and filtering the list. This is, obviously, not the only way to do things, but it works, and has advantages.>>> As far as my combining error and complexity before optimizing >>> generators, that was wrong. Moreover, combining them at all >>> is not for me. I'm not bound to ask, "What's the 'best' temp. >>> in size range x?". Rather, I might ask, "What's the most >>> accurate temperament in complexity range x?". >>>> that's exactly how i've been looking at all this for the entire >> history of this list -- witness my comments to dave defending >> gene's log-flat badness measures; i took exactly this tack! >> How could it defend Gene's log-flat badness? It's utterly > opposed to it!Eh? I go with Paul; this is the point of log-flat measures.

Message: 5993 - Contents - Hide Contents Date: Wed, 15 Jan 2003 22:15:10 Subject: Re: Nonoctave scales and linear temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" > <clumma@y...> wrote:>>>> If we throw out odd-limit, however, we might be better off. >>>> If there were a weighting that followed Tenney limit but >>>> was steep enough to make near-perfect 2:1s a fact of life >>>> and anything much beyond the 17-limit go away, we could >>>> have individually-weighted errors and 'limit infinity'. >>>>>> but there would have to be an infinitely long map, or wedgie, >>> or list of unison vectors in order to define the temperament >>> family. >>>> Yeah, but we could approximate it with a finite map. >> hmm . . . aren't you then just talking about a finite limit of some > sort?My Zeta function method in theory goes out to limit infinity, but gives much greater weight to smaller primes, 2 in particular.> we've been searching by commas, i believe -- i'll let gene answer > this more fully.One can use various methods to produce a list of wedgies to test, and merge the lists. graham has been searching by "+"ing ET maps to get> linear temperament maps -- something i'm not sure i can explain right > now.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.

Message: 5994 - Contents - Hide Contents Date: Wed, 15 Jan 2003 22:24:08 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> wrote:> At 11:13 PM 10/01/2003 +0000, Dave Keenan <d.keenan@u...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" >>> Now that we've agreed on the 5' comma symbols, may I suggest that the >> ascii symbols for -5' and +5' be ' and ` respectively, > > I assume you meant to write ` and ' respectively?Yes -- just goes to show you how confusable those two ascii symbols are when there's no vertical distinction.>> regardless of >> the direction of alteration of the main symbol (particularly since >> the actual accents don't appear aligned with the point of the arrow >> in the actual symbols)? I think that the period and comma are too >> difficult to remember, especially the way you've done the 125- diesis >> above (which is different than before), and I think `//| and '\\! >> should be clear enough for a 125-diesis up and down, respectively.Theoretically clear, but in practice confusable, as I noted above.> > Yes it's different than before. I find that `//| and '\\! don't look like > inverses of each other. Right! > My thinking is that, with these tiny ASCII symbols, > the vertical position is a much stronger cue than the slope, particularly > since neither ' nor . have any slope. Right again! > I find that `//| and ,\\! look like > inverses, but unfortunately position and slope cues conflict with each > other in these two symbols.And right again!> That only leaves .//| and '\\! > > So I'm proposing that the ascii symbols for -5' and +5' be . and ' > respectively, regardless of the direction of alteration of the main symbol. Agreed! > Consider distinguishing the Pythagorean comma from the diaschisma. Which > pair makes it clearer which is which. > '/| `/| > or > '/| ./| > and in the other direction > `\! '/! > or > .\! '\! > > I have to say both options are pretty unsatisfactory.So we use the one that's least unsatisfactory. :-)>>> ... >>> Perhaps we should ditch the (/| symbol entirely and use |)) for the 31' >>> comma since |)) is the more obvious symbol for the 49'-diesis. >>>> For the 31' comma only the divisions that have any semblance of >> consistency up to the 31 limit would have any practical bearing on >> this decision. For 270 and 311 |)) is required, while for 217, 388, >> and 653 either one is valid; 494 requires (/|, but is not 1,7,31,49- >> consistent. It looks like |)) takes it. But this would require >> other symbols for 23 and 24deg494; any ideas? >> I don't think that notating 494-ET is a high enough priority to delay the > adoption of |)) as both the 49' and 31' diesis symbol. I can only think > that we might be forced to use some symbols involving 5' for 494- ET. We > could wait and see if suitable symbols come up as we work our way down the > ratio popularity list.Sounds like a good idea. (Also looks like you've been making more progress on that list lately than I have.) --George

Message: 5995 - Contents - Hide Contents Date: Wed, 15 Jan 2003 22:29:45 Subject: Re: Notating Pajara 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:>>>> And p=6 gives 709.1585, better yet; it's simply headed off where >>> you suggested, to 709.363. >>>> when did i suggest that? and isn't it true that this value, the limit >> as p goes to infinity, is *not* the midpoint of the minimax range, as >> you stated it would be? >> I cancel that 10 seconds after posting it and you want to hold me >to it. :)sorry, i wasn't aware of the cancellation! did you delete the post? wow, i must have really had my trigger finger on the "next" button! glad to know my intuition isn't failing me at least on *some* things . . .

Message: 5996 - Contents - Hide Contents Date: Wed, 15 Jan 2003 01:36:49 Subject: Re: Nonoctave scales and linear temperaments From: Carl Lumma>if you choose a different set of generators, you'll get >a different ranking for which mapping is more complex >then which!Doesn't a map uniquely determine its generators? -Carl

Message: 5997 - Contents - Hide Contents Date: Wed, 15 Jan 2003 02:08:16 Subject: Re: Notating Pajara From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> i get 709.051244958997 -- we have some discrepancy!! > i get a p=4 norm of 0.0162470021271903 for your value, and > 0.0162469162102062 for mine, so clearly something's wrong with your > algorithm!Nothing is wrong with my algorithm; I reran it and got your value. I have no idea where mine came from.>> minimax 706.8431431 >> i'm not sure, but i do get this same figure for MAD. >>> From this we can't prove that 22-equal is poptimal, though it at >> least comes close and might be. >> according to my calculations, for p=5 we get 709.112411004975, so 22- > equal is in there!And p=6 gives 709.1585, better yet; it's simply headed off where you suggested, to 709.363.>The best choice might still be 22, but it isn't a walk. > > on my keyboard it is!Depends on how much you favor triads, I suppose. Compromising between 22 and 34 does seem like an interesting possibility.

Message: 5998 - Contents - Hide Contents Date: Wed, 15 Jan 2003 03:55:06 Subject: Re: Nonoctave scales and linear temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>> if you choose a different set of generators, you'll get >> a different ranking for which mapping is more complex >> then which!> Doesn't a map uniquely determine its generators?The problem is that the temperament does not uniquely determine the map.

Message: 5999 - Contents - Hide Contents Date: Wed, 15 Jan 2003 04:44:36 Subject: Re: Nonoctave scales and linear temperaments From: Carl Lumma>> >oesn'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? -Carl

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