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Message: 6025 - Contents - Hide Contents Date: Thu, 16 Jan 2003 21:45:18 Subject: Re: Nonoctave scales and linear temperaments From: Graham Breed wallyesterpaulrus wrote:> 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?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. Graham

Message: 6026 - Contents - Hide Contents Date: Thu, 16 Jan 2003 23:41:33 Subject: Re: Nonoctave scales and linear temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> > tenney-minkowski. tenney is the metric being minimized, and minkowski > provided a basis-reduction algorithm applicable to such a case.He supplied a criterion; there seem to be no algorithms better than brute force.

Message: 6027 - Contents - Hide Contents Date: Thu, 16 Jan 2003 21:48:28 Subject: Re: Nonoctave scales and linear temperaments From: Carl Lumma>> >f 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?All of them within taxicab radius r on the triangular harmonic lattice. T(r), the number of tones within r, then seems to be 6r + T(r-1); T(0)=1 in the 5-limit. That seems to be roughly O(r**2). The number of possible commas is than the 2-combin. of this, which is again is roughly **2. So O(r**4) at the end of the day. Perhaps that first 2 is the number of dimensions on the lattice, in which case the 15-limit would be O(r**14).>The files I have here show equivalences between second-order >tonality diamonds.Which are? Tonality diamonds of the pitches in a tonality diamond?>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.Ah yeah, that's a lot of commas, and it seems a rather ad hoc way to select them.>You then need to take combinations of pi(p)-1 commas. >(That is one minus the number of primes in the ratios.) Right. >If you have many more commas than primes, that will go as >the pi(p)-1st power.I'll take your word for it.>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 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. :( >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.There must be many ways to reduce the complexity of the method I suggest. For example, rather than finding all the pitches and taking combinations, we might set a max comma size (as an interval) and step through ratios involving up to r compoundings of the allowed factors that are smaller than that bound. -Carl

Message: 6028 - Contents - Hide Contents Date: Thu, 16 Jan 2003 23:42:30 Subject: Re: Nonoctave scales and linear temperaments 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:>> >> tenney-minkowski. tenney is the metric being minimized, and minkowski >> provided a basis-reduction algorithm applicable to such a case. >> He supplied a criterion; there seem to be no algorithms better than brute force.but it's a criterion guaranteed to give a unique answer, right?

Message: 6029 - Contents - Hide Contents Date: Thu, 16 Jan 2003 21:56:23 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> wallyesterpaulrus wrote: >>> 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? >> 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.well let's get these things understood!

Message: 6030 - Contents - Hide Contents Date: Thu, 16 Jan 2003 23:43:56 Subject: Re: Ultimate 5-limit again From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus > <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith >> >>> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus>>>>> 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.which is the same sort implemented in the url above -- maybe you just forgot to click on "next"?

Message: 6031 - Contents - Hide Contents Date: Thu, 16 Jan 2003 04:46:57 Subject: Re: Notating Pajara From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> --- 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 infact 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? Gene,As evidence, I offer the minimax generators for Pajara quoted by a number of different people earlier in this thread. I believe the same value was supplied independently by 2 or 3 people, and I think you will find that it is the p-->infinity value. I think you'd better include p=1 before Paul starts campaigning for the limit as p-->0 or p negative. ;-) Actually I think that the absolutely and ideally perfect optimisation will probably occur when p = 2*pi. ;-) --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> --- 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. 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 ...". But if it was actually you who first pointed it out, then thanks. I can't find where Gene denied it, but if he did, I suspect it was only because his algorithm was apparently producing counterexamples. We now know it is was the algorithm that was the problem. Regards, -- Dave Keenan

Message: 6032 - Contents - Hide Contents Date: Thu, 16 Jan 2003 21:59:34 Subject: Re: Ultimate 5-limit again From: Carl Lumma> 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.Yep; another nice thing about RMS. But how does heuristic error differ from RMS?

Message: 6033 - Contents - Hide Contents Date: Thu, 16 Jan 2003 23:58:28 Subject: Re: Mega 11 and 13 limit ets From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> what happened to 504504?I checked from 3 to 23, and it's not nearly good enough to make the cut in any of them. Are you sure this is the right number?

Message: 6034 - Contents - Hide Contents Date: Thu, 16 Jan 2003 07:26:20 Subject: Re: Notating Pajara From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote:> As evidence, I offer the minimax generators for Pajara quoted by a > number of different people earlier in this thread. I believe the same > value was supplied independently by 2 or 3 people, and I think you > will find that it is the p-->infinity value.I've decided to cave, and re-wrote my program to get it to correspond to what other people are doing.> I think you'd better include p=1 before Paul starts campaigning for > the limit as p-->0 or p negative. ;-)So it seems, but I really think we'd better draw a line there.> Actually I think that the absolutely and ideally perfect optimisation > will probably occur when p = 2*pi. ;-)Obviously we need to work pi in somehow.

Message: 6035 - Contents - Hide Contents Date: Thu, 16 Jan 2003 22:00:05 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> 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? >> All of them within taxicab radius r on the triangular harmonic > lattice. T(r), the number of tones within r, then seems to be > 6r + T(r-1); T(0)=1 in the 5-limit. That seems to be roughly > O(r**2). The number of possible commas is than the 2-combin. > of this, which is again is roughly **2. So O(r**4) at the end > of the day.this is silly. all you need to do to get all these commas to come up as *tones* is to double your radius. so 2*O(r**2) will do -- the O (r**4) method is extremely redundant.> Perhaps that first 2 is the number of dimensions on the lattice, > in which case the 15-limit would be O(r**14).if we're looking for an efficient search, we'd definitely use a prime- limit lattice, not one with redundant points!

Message: 6036 - Contents - Hide Contents Date: Thu, 16 Jan 2003 08:19:28 Subject: Re: Nonoctave scales and linear temperaments From: Carl Lumma>> >he 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.Well, I don't know what a basis is, let alone a TM reduced one, let alone a Hermite normal form.>>>> 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.Eh? Name your complexity range, and take the x most accurate temperaments within it. Why do I need badness at all? -Carl

Message: 6037 - Contents - Hide Contents Date: Thu, 16 Jan 2003 22:01:24 Subject: Re: Ultimate 5-limit again From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>> 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. >> Yep; another nice thing about RMS. > > But how does heuristic error differ from RMS?not too much -- less than a factor of 2 in every case that i've seen.

Message: 6038 - Contents - Hide Contents Date: Thu, 16 Jan 2003 08:59:23 Subject: Re: Nonoctave scales and linear temperaments From: Carl Lumma>hmm . . . aren't you then just talking about a finite limit >of some sort?With a steep weighting and a promise that what we've left out affects things less than x, sure. Maybe the Zeta function stuff can do even better...>>>> 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 >>not sure if this concept has been pinned down . . .Certainly it hasn't. For linear 5-limit temps, if I take something like this... 2 3 5 gen1 gen2 ...and start filling in numbers, either every choice of numbers converges on a pair of exact sizes for gen1 and gen2 under an error function of a certain class (say, one to which RMS, and apparently not minimax, belongs), or it doesn't. Am I to understand it doesn't?>> (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.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?>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 remember Graham using the +ing ets method. He claimed it was pretty fast and comprehensive. I remember thinking it was anything but pretty. I don't understand wedgies, so... 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...>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!>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"?>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? There didn't seem to be any exponents that would do this on Dave's spreadsheet, and I thought I tried the critical exponent for log-flat temps.>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. -Carl

Message: 6039 - Contents - Hide Contents Date: Thu, 16 Jan 2003 22:03:45 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" > <clumma@y...> 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? >>>> All of them within taxicab radius r on the triangular harmonic >> lattice. T(r), the number of tones within r, then seems to be >> 6r + T(r-1); T(0)=1 in the 5-limit. That seems to be roughly >> O(r**2). The number of possible commas is than the 2-combin. >> of this, which is again is roughly **2. So O(r**4) at the end >> of the day. >> this is silly. all you need to do to get all these commas to come up > as *tones* is to double your radius. so 2*O(r**2) will do -- the O > (r**4) method is extremely redundant.and, in fact, kees van prooijen does exactly this search on tones, here: Searching Small Intervals * [with cont.] (Wayb.) and keeps the smallest, second-smallest, and third-smallest commas for each possible r, and comes up with this list: S235 * [with cont.] (Wayb.)

Message: 6040 - Contents - Hide Contents Date: Thu, 16 Jan 2003 22:13:29 Subject: Re: Nonoctave scales and linear temperaments From: Carl Lumma>> >re 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? > >81:80 >>in 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]Hmm. How bad are such cases? Could we take them out at the end?>> 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).That's what I call taxicab complexity, I think. Isn't it also the case that the same temperament may be defined by different lists of commas?>if there's more than one comma being tempered out, we need >a notion of the "angle" between the commas . . . Please explain.>> 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.You assume there was an untweakable one in the 5-limit case? Bah! And in the 'now it's planar' case, you would no longer have an untweakable one. Something's got to give!>> 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.Thanks. There is also, I gather, such a thing as a basis of commas? TM (TM stands for?) reduction applies to commas only, right?>> 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.That's the same as taking the 20 "best" temperaments.>*then* you sort by complexity.Aha! This isn't better than taking the 20 simplest temperaments and sorting by accuracy. -Carl

Message: 6041 - Contents - Hide Contents Date: Thu, 16 Jan 2003 22:18:01 Subject: Re: Nonoctave scales and linear temperaments From: Carl Lumma>and, in fact, kees van prooijen does exactly this search >on tones, here: > > Searching Small Intervals * [with cont.] (Wayb.)Man, it's been a while since I looked at Kees' site. This is a goldmine!>and keeps the smallest, second-smallest, and third-smallest >commas for each possible r, and comes up with this list: > > S235 * [with cont.] (Wayb.) That's badass. -C.

Message: 6042 - Contents - Hide Contents Date: Thu, 16 Jan 2003 22:28:25 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? >> >> 81:80 >>>> in 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] >> Hmm. How bad are such cases? Could we take them out at the > end?i have no idea what you're asking. bad?>>> 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). >> 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.> Isn't it also the case that the same temperament may be defined > by different lists of commas? >>> 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 . . .>>> 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. >> 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?> 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.>>> 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. >> 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.>>> 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. >> 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".>> *then* you sort by complexity. >> Aha! This isn't better than taking the 20 simplest temperaments > and sorting by accuracy.of course it is. most of the 20 simplest temperaments are garbage. don't you want the several best temperaments in each complexity range, going up to complexity higher than you could ever use?

Message: 6043 - Contents - Hide Contents Date: Thu, 16 Jan 2003 22:31:24 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>> and, in fact, kees van prooijen does exactly this search >> on tones, here: >> >> Searching Small Intervals * [with cont.] (Wayb.) >> Man, it's been a while since I looked at Kees' site. This > is a goldmine! >>> and keeps the smallest, second-smallest, and third-smallest >> commas for each possible r, and comes up with this list: >> >> S235 * [with cont.] (Wayb.) > > That's badass. > > -C.and, since there are only two coordinates in the 5-limit, the comma search (and thus the linear temperament search) requires only O(r^2) operations -- kees apparently took r to be on the order of 10^5, much more than anyone could ever possibly need :)

Message: 6044 - Contents - Hide Contents Date: Thu, 16 Jan 2003 22:32:41 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> search for "straightess" in these archives . . .oops -- i meant "straightness"!

Message: 6045 - Contents - Hide Contents Date: Thu, 16 Jan 2003 22:46:08 Subject: Re: Nonoctave scales and linear temperaments From: Graham Breed Carl Lumma wrote:> Perhaps that first 2 is the number of dimensions on the lattice, > in which case the 15-limit would be O(r**14).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.>> The files I have here show equivalences between second-order >> tonality diamonds. >> Which are? Tonality diamonds of the pitches in a tonality > diamond? Yes > Ah yeah, that's a lot of commas, and it seems a rather ad hoc > way to select them.The advantage of commas taken from nth order tonality diamond is that you can predict how good a temperament can be involved in. The size of the comma in cents divided by the order of the tonality diamond is the best possible minimax error.> There must be many ways to reduce the complexity of the > method I suggest. For example, rather than finding all > the pitches and taking combinations, we might set a max > comma size (as an interval) and step through ratios > involving up to r compoundings of the allowed factors > that are smaller than that bound.Yes, you could, for example, take only commas less than 10 cents. That means only 10/1200 or 1/120 of the commas stay. So you save two orders of magnitude from you set of commas, but no more. I've found the code for my search, and I was using this kind of optimization. The big reduction in complexity would come from reducing the number of combinations of commas you need to take. Ideally some way of predicting that a small set of commas (all good on their own) will never give a good temperament by adding more. There are certainly a lot of commas that can be relevant. Below are all 42 the equivalences from my list of 19-limit temperaments (generated from 20 equal temperaments). Some may not be unique unison vectors, but whatever, they aren't sufficient to generate most of the linear temperaments anyway. There are 27 million combinations of 7 from 42. 7 from 28 already gives over a million combinations. 22:19 =~ 15:13 16:15 =~ 17:16 15:11 =~ 26:19 135:128 =~ 19:18 65:48 =~ 19:14 20:19 =~ 19:18 24:19 =~ 19:15 39:32 =~ 128:105 14:13 =~ 128:119 45:32 =~ 128:91 256:195 =~ 21:16 256:221 =~ 22:19 65:64 =~ 64:63 128:117 =~ 35:32 17:13 =~ 64:49 13:10 =~ 64:49 21:20 =~ 20:19 40:33 =~ 17:14 48:35 =~ 256:187 18:17 =~ 128:121 40:39 =~ 49:48 128:117 =~ 12:11 11:9 =~ 39:32 256:221 =~ 22:19 25:24 =~ 133:128 22:17 =~ 128:99 128:91 =~ 7:5 64:57 =~ 28:25 32:27 =~ 13:11 256:209 =~ 11:9 19:18 =~ 128:121 28:25 =~ 143:128 18:17 =~ 17:16 9:8 =~ 64:57 171:128 =~ 4:3 28:27 =~ 133:128 128:95 =~ 27:20 32:27 =~ 19:16 24:19 =~ 81:64 135:128 =~ 20:19 165:128 =~ 128:99 135:128 =~ 19:18 =~ 128:121 Graham

Message: 6046 - Contents - Hide Contents Date: Fri, 17 Jan 2003 00:59:54 Subject: Re: Nonoctave scales and linear temperaments 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:>> --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote: >>>> 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. >> One possibility would be to choose a comma list with a badness >cutoff, where the badness was the badness of the corresponding >codimension one, or one-comma, temperament.you'd get a lot of temperaments that are worse than a lot you woudn't get, because of the straightness thing.

Message: 6047 - Contents - Hide Contents Date: Fri, 17 Jan 2003 10:30:03 Subject: Re: heuristic and straightness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> wallyesterpaulrus wrote: >>> insufficient? how can a set of unison vectors be insufficient? >> For an arbitrary search to be practicable, > there has to be a way of rejecting sets of three unison vectors because > you know they can't give a good linear temperament.then you need a 3-d generalization of "straightness". i bet if i went and learned grassmann algebra i'd be able to get a better grasp on all this.

Message: 6048 - Contents - Hide Contents Date: Fri, 17 Jan 2003 11:31:56 Subject: Re: heuristic and straightness From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith > <genewardsmith@j...>" <genewardsmith@j...> wrote: >>> I'm not sure what you want here--if all of the commas point in >> about the same direction, do you mean with or without 2? >> i'm thinking without.One method which might come to the same thing as "straightness" in effect is to take two commas, and combine to get a codimension 2 wedgie. Produce a list of these by taking the best (here you run into geometric badness) and then wedge these with another comma, and so forth.

Message: 6049 - Contents - Hide Contents Date: Fri, 17 Jan 2003 01:37:37 Subject: Re: Nonoctave scales and linear temperaments From: Carl Lumma>> >an I identify the duplicate temperaments? >> duplicate? any temperament can be generated from an infinite > number of possible basis vectors."The results of a search of all possible maps is bound to return pairs, trios, etc. of maps that represent the same temperament. Can we find them in the mess of results?" Was all I was asking!>> Assuming 2:1 reduction makes me squirm in my chair, is all. >> Plentiful near-2:1s should emerge from the search if the criteria >> are right. >>if the criteria include mapping to, and minimizing error from, >2:1, then of course a near 2:1 will emerge in each temperament.Yup, that's what I've been saying alright.>> Oh. Now I get it! You're right. But doesn't the same >> problem occur with different commatic representations, when >> defining complexity off the commas? >> not if you define complexity right! Aha!>>>> 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. >>>> Oh. I guess I need some examples, then, of most of the simple >> temperaments that are garbage... >> what are the 20 simplest 5-limit intervals? now set each of > these to be the commatic unison vector, and what temperaments > do you get?Perhaps we could enforce "validity", and maybe also Kees' 'complexity validity'. -Carl

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