This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).
- Contents - Hide Contents - Home - Section 54000 4050 4100 4150 4200 4250 4300 4350 4400 4450 4500 4550 4600 4650 4700 4750 4800 4850 4900 4950
4050 - 4075 -
Message: 4075 - Contents - Hide Contents Date: Sat, 09 Mar 2002 09:51:30 Subject: Re: 32 best 5-limit linear temperaments redux From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>> I don't think we are going to make any progress on this unless we can >> get beyond a badness measure that says the best 5-limit temperament is >> one that takes 49 generators before we get a single fifth (because it >> has such teensy weensy errors). >> This is just like saying we should not regard 2460 as a super-good > 5-limit scale because its errors are so small that it could make nopractical difference if they were larger, and that given the choice between 53 tones and 2460, 53 seems much more practical. This misses the point, which is that 2460 is very, very good compared to other things *in its size range*. If you compare wildly different values of "g", you are getting into apples and elephants.>>> This badness measure also says that meantone is only 7th best (or >> thereabouts) and thinks that a temperament whose perfect fifth is 758 >> cents and whose major third is 442 cents is only slightly worse than >> meantone (because it only needs 2 generators to get one of these >> supposed 1:3:5 chords). > >> Does anyone really believe this stuff? >> Paul has pointed out that ultra-funky scales may have morepossibilities than is at first apparent. Again, why compare apples with e coli? I assume we are making these lists of temperaments for people who are considering making practical musical use of them (as opposed to say theoretical mathematical, or practical engineering use). Such a person, searching in a list entitled "5-limit linear temperaments", can be presumed to want two things: 1. 5-limit harmony, and 2. temperament. The former implies that s/he wants intervals that _sound_ like some kind of approximation of ratios of 1, 3 and 5, and their octave equivalents and inversions. The latter implies that s/he doesn't want to have to deal with as many pitches as would be required in a 5-limit _rational_ tuning giving the same numbers of harmonies. The two temperaments I singled out would be of no interest to such a person (except as curiosities). The one, because it isn't 5-limit and the other because it isn't a temperament, for any practical purpose. The one has both its 4:5 approximation and its 3:4 approximation sounding exactly the same as each other. They both sound like 7:9s. The other requires _more_ notes than a 5-limit rational scale for any reasonable number of 5-limit harmonies, assuming we use a contiguous chain of generators.
Message: 4076 - Contents - Hide Contents Date: Sat, 09 Mar 2002 10:46:52 Subject: Re: 32 best 5-limit linear temperaments redux From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> The one has both its 4:5 approximation and its 3:4 approximation > sounding exactly the same as each other. They both sound like 7:9s. > The other requires _more_ notes than a 5-limit rational scale for any > reasonable number of 5-limit harmonies, assuming we use a contiguous > chain of generators.What you're saying is that the search was too broad, it seems. That could be rectified if there was general agreement it is so by the simple expedient of leaving off the extremes.
Message: 4077 - Contents - Hide Contents Date: Sat, 09 Mar 2002 21:12:06 Subject: Re: 32 best 5-limit linear temperaments redux From: Carl Lumma Dave wrote...>> Carl's preferred map complexity measure: >> (/ (- (max map) (min map)) (card map)) >>This isn't unambiguous mathematical notation. It's Lisp. Took me >a while to figure that out.It's generic prefix (Polish) notation with normal grouping by parens. Graham knows scheme now, and you have a paper on the lambda calculus on your web page, and Gene's a clever guy. Gene wrote...>> Meantone has a very compact 5-limit map. You only need 4 gens. In >> listening tests I've preferred a generator close to that of 69-et, >> though the rms optimum is closer to 31-et IIRC. In either case, why >> should we penalize meantone because it takes 31 or 69 gens to yield >> an et with the optimum generator? >>I don't; "g" has nothing to do with ets per se, and only measures >complexity. Good!>> Once again, I'll list my preferred map complexity measure in >> unambiguous mathematical notation. Why don't you and Graham >> give yours for the record, so Dave can tell us which one he >> likes best? >> >> Gene's preferred map complexity measure: >>rms generator steps, times the number of periods in an octave.What made you go to rms? Isn't it over-kill?>This only works for linear temperaments, so I'm not that happy with it.What else do you want it to work for? Planar temperaments? -Carl
Message: 4078 - Contents - Hide Contents Date: Sat, 09 Mar 2002 21:24:22 Subject: Re: 32 best 5-limit linear temperaments redux From: Carl Lumma Dave wrote...>Another reason it isn't ad hoc. The perceptual "pain" caused by >mistuning is not directly proportional to the error in cents. Even the >best microtonal ear on the planet apparently experiences essentially >zero pain with a 0.5 cent mistuning. Most people aren't significantly >bothered by a 3c mistuning (depending on the interval and how long it >is sustained). But a 30 cent mistuning is so bad that a 40 cent one >could hardly be much worse.Gene's "cents" are already rms, which we've long ago decided is the best single error measure. Chopping off anything less than .5 is a hack, and hopefully an un-necessary one. Gene wrote...>> For example, I think meantone must be in the top 3, or the badness >> measure is nonsense. >>This is like saying 12-et must be in the top three. It's great in its >size range-- if that happens to be the range you are interested in. If >it doesn't suit your requirements then it isn't great, whatever >number you come up with for it. What's top or not top depends on what >tone group you are looking at (5-limit, 7-limit?) and what sort of >accuracy you want.Dave's just saying that you're not weighting the span of the map enough, since he considers musical history to be a worthy badness measure in its own right -- one that selected meantone, diminished, augmented, over the infinity of temperaments bigger than schismic. You want something that exposes the pattern in the series of best temperaments. It would be nice to see a list with a much stronger penalty for size, but I can live with a flat measure with a sharp cutoff. -Carl
Message: 4079 - Contents - Hide Contents Date: Sat, 09 Mar 2002 21:30:56 Subject: Re: 32 best 5-limit linear temperaments redux From: Carl Lumma>> >t's generic prefix (Polish) notation with normal grouping by >> parens. Graham knows scheme now, and you have a paper on the >> lambda calculus on your web page, and Gene's a clever guy. >>Not clever enough to figure out why you want to use prefix notation.It's trivial one way or the other, though I personally find it easier to parse when dealing with ASCI.>> What made you go to rms? Isn't it over-kill? >>It's less sensitive to outliers; if a temperament does a lot of >things well and some badly, it still get credit for it.That's exactly what I _don't_ want. I want to know the size of the chain I need to complete my map. The only reason I divide by (card map) is so I can compare temperaments at different limits.>>> This only works for linear temperaments, so I'm not that happy >>> with it. >>>> What else do you want it to work for? Planar temperaments? > >Of course.There's nothing I want more than to get to planar temperaments, but we should finish with LTs first! I say this simply because I suspect if we can't handle LTs, we don't stand a ghost of a chance with PTs! -Carl
Message: 4080 - Contents - Hide Contents Date: Sun, 10 Mar 2002 12:35:14 Subject: Re: 32 best 5-limit linear temperaments redux From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > >> So you're gonna make different lists for different size ranges, are >> you? >> I've tried listing them in order of size, which makes sense to me,though I think Paul didn't like it. What do you think? I'd rather they were listed in increasing order of "badness", assuming "badness" actually means something, like badness. Then if I'm looking for the best temperament whose error is in a particular range of sizes I'll just go down the list until I find the first one _in_ that range. The same goes for any other property that might be important to me at the time, like having a half-octave period or having a single generator making the perfect fifth. If you think it's ok to have a badness measure that does not allow comparison between temperaments whose errors are in different size ranges, would you also think it ok to have one that didn't allow comparison between say temperaments whose generators were in different size ranges. i.e. you couldn't compare temperaments generated by approximate fourths with those generated by approximate thirds. If you agree that this would be a somewhat defective badness measure, then you should know that this is how I view the badness measure you are using.
Message: 4081 - Contents - Hide Contents Date: Sun, 10 Mar 2002 16:49:10 Subject: Re: 32 best 5-limit linear temperaments redux From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>> badness = wtd_rms_gens*EXP((rms_error/7.4_cents)^0.5) >> >> Where wtd_rms_gens are weighted by log of odd limit. >> This strikes me as completely ad hoc. Why not a non-fuzzy version,with a sharp cutoff? If you insist on sharp cutoffs (but why aren't _they_ too ad hoc?) I think you'd be quite safe to ignore temperaments whose rms complexity is more than 12 gens or whose rms error is more than 30 cents. That will get rid of all of what I consider junk from your list (all the unnamed ones, limmal, fourth-thirds and the two that end in "decal"). However, it won't solve the problem that the remaining temperaments will not be sensibly ranked by complexity^3*error, and it won't solve the problem that you are missing some temperaments that are IMHO at least as good as those that would remain. These temperaments are apparently rejected for the crime of having medium errors (around 5 cents) and medium complexity (around 6 gens). badness = complexity^3*error favours the extremes over these. The four I know of have an octave period. Two of these might never be generated by your algorithm, since you seemed to want to deny that they were anything but a repeat of meantone last time they were mentioned, despite the fact that they have different size MOS and work in different ETs. Here are the missing four. Mapping Gen to 3,5 (cents) Description MOS or ET sizes (improper) ------------------------------------------------------------------ [ 4, 9] 176.3 minimal diesic 6 7 (13 20 27) 34 (41 75 109 ...) [-4, 3] 126.2 16875/16384 9 10 19 (29 48 67 86 105 ...) [ 2, 8] 348.1 half meantone-fifth 7 (10 17 24) 31 (38 69 100) [-2,-8] 251.9 half meantone-fourth 5 (9 14) 19 (24 43 62) 81 (100 ...) It turns out that complexity*EXP((error/1_cent)^0.24) does not produce the same ranking as complexity^3*error. Even as gentle a rollof as that sends most of the junk to where it belongs and puts meantone on the top of the list where _it_ belongs. I've made a spreadsheet so you can play with sorting Gene's list (plus the four above) according to Gene's badness, my badness with variable parameters, or any badness you care to calculate from the given information. http://uq.net.au/~zzdkeena/Music/5LimitTemp.xls.zip - Type Ok * [with cont.] (Wayb.) 14KB zipped Excel spreadsheet with macros
Message: 4082 - Contents - Hide Contents Date: Sun, 10 Mar 2002 21:18:42 Subject: Re: 32 best 5-limit linear temperaments redux From: Carl Lumma>orwell with 9 or 13 notes. still 11-limit.I show orwell with a 9-tone MOS, and this map: [0 7 -3 8 2] [1 0 0 0 0] 'zthat right? That covers 10 gens in the 5-limit, 11-gens in the 7-limit, and 11 gens in the 11-limit. You're saying... [0 -3 2] [1 0 0] [1 5 11] ...only takes 5 notes? I know I said "limit", but I meant it figuratively, not literally. It's an abuse of terminology to call [1 5 11] the "[1 5 11]-limit", for example, but that's what I meant. So if your paper's really going to cover all these (by that I mean have a list for each one), I'd suggest ranking by complexity, with a sharp cutoff. Just show the most accurate three temperaments for each integer of complexity up to 15 or so. You could seed this with whatever badness measure you wanted, as long as you let it go up to 500. By then, even the kind that Dave can always add wouldn't make it up into my list You can tell I don't care about sharp cutoffs. :)>and what if the subset of 'primes' you're thinking about is not >really the primes, but is, say, {2, 3, 5/3} or {2, 3, 7/5}?Fine. (I don't understand what you thought I thought... Maybe I'm still thinking it! :) -Carl
Message: 4083 - Contents - Hide Contents Date: Sun, 10 Mar 2002 13:02:19 Subject: Re: 32 best 5-limit linear temperaments redux From: dkeenanuqnetau --- In tuning-math@y..., graham@m... wrote:> Have you tried running <Linear Temperament Finder * [with cont.] (Wayb.)> with > width*math.exp((error/1200.0)**0.24) as the figure of demerit? It might > be the wrong width, but it's worth a try.That's awesome! I had no idea it was that easy! You might mention on that page that "error" is in octaves, not cents. Or better still make "error" be in cents.
Message: 4084 - Contents - Hide Contents Date: Sun, 10 Mar 2002 17:37:59 Subject: Re: 32 best 5-limit linear temperaments redux From: dkeenanuqnetau Gene, Could you humor me and temporarily use badness = complexity*EXP((error/7.4_cents)^0.5) in your program and see if we get any others that come out better than pelogic by this measure but aren't in the list that started this thread (or my recent spreadsheet). There may be some with fractional-octave periods that I haven't found. Also these should just squeak in. Mapping Gen to 3,5 (cents) Description MOS ET sizes ------------------------------------------------------------------- [ 12, 10] 158.5 c half kleismic-minor-third 7 8 15 (23 38) 53 (68 121 ...) [-12,-10] 441.5 c half kleismic-major-sixth (5) 8 11 19 (30 49 68) 87 106
Message: 4085 - Contents - Hide Contents Date: Sun, 10 Mar 2002 21:19:45 Subject: Re: 32 best 5-limit linear temperaments redux From: Carl Lumma>e.g. Folks considering making a LT guitar and wanting to keep close to >standard open string tuning would favour a generator that was a >fourth.Ah, mere mention of the days of Keenan GuitarFrettings^TM make me shudder. In 6 months, will Dave be cooking up linear-tempered guitars... or maybe by then, planar-tempered guitars? Mmmmmmm.... Two things where I've always thought Dave Keenan kicks butt: notation systems, and guitar frettings. Well, I can't really speak to the latter. Paul- should the people at home start cracking eachother's skulls open and feasting on the sweet goo inside? (simpsons reference -- Answer: Yes, Kent.) Did you ever manage to automate the process at all, Dave? -Carl
Message: 4086 - Contents - Hide Contents Date: Sun, 10 Mar 2002 00:28:38 Subject: Re: 32 best 5-limit linear temperaments redux From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>> The one has both its 4:5 approximation and its 3:4 approximation >> sounding exactly the same as each other. They both sound like 7:9s. >> The other requires _more_ notes than a 5-limit rational scale for any >> reasonable number of 5-limit harmonies, assuming we use a contiguous >> chain of generators. >> What you're saying is that the search was too broad, it seems. Thatcould be rectified if there was general agreement it is so by the simple expedient of leaving off the extremes. We've been over this before. That's exactly what I'm trying to convince everyone to accept, not with sharp cutoffs, but gradual rolloffs. By choosing suitable values for the parameters k and p in badness = gens * EXP((cents/k)^p) we can start with a badness measure that gives exactly the same ranking as your current measure, and then by tweaking these parameters we can make those objectionable extreme cases fall off the bottom of any length best-of list we choose to make. I don't have time to check it at the moment, but I think if you set p = 0.24 and k = 1 cent you will get pretty much the same ranking as for gens^3 * cents. In fact if you divide it by e (~=2.718) and cube it, you will get a good approximation to your actual badness numbers. i.e. gens^3 * cents ~= (gens * EXP((cents/1)^0.24) / e)^3 but of course dividing by e and cubing doesn't change the ranking, so they can be omitted. Then gradually increase p and k until both of those objectionable temperaments (e-coli and elephant) just go off the bottom of your list of 32 best and see how the rest of them are then ranked. I guarantee it will make a lot more sense to most people, with meantone much closer to the top for one thing, and you probably won't need as many as 32 in the list to cover all the historical ones.
Message: 4087 - Contents - Hide Contents Date: Sun, 10 Mar 2002 10:43:15 Subject: Re: 32 best 5-limit linear temperaments redux From: Carl Lumma>Whether it is rms or max-absolute is irrelevant to my argument. I'm >happy to use minimum rms error as the input to badness (though I think >it's good to know the minimum max-absolute as well). If you're >suggesting that using rms somehow removes the need to apply a >nonlinear "pain" function, then you're mistaken. You've put "cents" in >scare quotes above as if you think the units aren't really cents when >it's rms. They are.That's true. If you want pain, it should just be ms.>> Chopping off anything less than .5 is a >> hack, and hopefully an un-necessary one. >>It is definitely unnecessary and I do not propose to chop anything >off. Gene's use of straight cents gives far too much credit to >temperaments that have very small sub-half-cent errors. It is allowed >to compensate so much for large gens that a temperament with a >complexity of 35.5 gens (rms) can be considered the best 5-limit >temperament! If you were to take a 0.5 cent threshold into account in >a discontinuous manner, you would treat any temperament whose rms >error was less than 0.5 c as if its error _was_ 0.5 c. But I'm not >proposing we do that. My pain function is non-linear but smooth. A >temperament still gets some credit for being sub-half-cent, just not >so much. ((rms_error/7.4_cents)^0.5)Well, this just has too many constants for my taste.>That's a good way of putting it, except I wouldn't say that Gene's not >weighting the complexity enough, I'd say he's failing to level off >(asymptote) with sub-cent errors and not weighting super-20-cent >errors enough.I think I'd rather have a smooth pain function, like ms, and a stronger exponent on complexity.>> You want something that exposes the pattern in the series of >> best temperaments. >> >> It would be nice to see a list with a much stronger penalty >> for size, but I can live with a flat measure with a sharp cutoff. >>You don't _have_ to live with it.True. But it will take more learning and more dissatisfaction with Gene and Graham than I'm currently experiencing to make me cook my own list. -Carl
Message: 4088 - Contents - Hide Contents Date: Sun, 10 Mar 2002 21:32:38 Subject: Re: 32 best 5-limit linear temperaments redux From: Carl Lumma>> >rote:>>> then the maximum interval width (which is also >>> the width of the complete chord - Carl's measure) seems more >>> relevant to me than the rms interval width. >>>> and shouldn't it be the maximum or rms *weighted* interval >> width? >>by the way, my now-famous heursitic for complexity would sort >the 5-limit temperaments by the size of the numerators (or >denominators, or n*d) of the commas.Sounds interesting, but I'll have to read up again on your heuristic a bit to even know. I have the original post here somewhere, I think.... -Carl
Message: 4089 - Contents - Hide Contents Date: Sun, 10 Mar 2002 10:51:43 Subject: Re: 32 best 5-limit linear temperaments redux From: Carl Lumma>I'd rather they were listed in increasing order of "badness", assuming >"badness" actually means something, like badness. Then if I'm looking >for the best temperament whose error is in a particular range of sizes >I'll just go down the list until I find the first one _in_ that range. >The same goes for any other property that might be important to me at >the time, like having a half-octave period or having a single >generator making the perfect fifth.I could be wrong, but I don't think you can have it both ways. If you want small temperaments to be better, your list will be finite. That's what happened with steps^3cents and ets (right, Gene?). I think I can feel a two-list paper cooking. First, we say, "in some meaningful sense, 2971 (or whatever) is as good as 31... there's a periodicity here... here's a list of the best 20 temperaments up to 10,000 (or something)...". Then, "but increasing the exponent on steps to yield a finite list meaningful for physical instruments such as guitars and pianos, we have ...". -Carl
Message: 4090 - Contents - Hide Contents Date: Sun, 10 Mar 2002 21:34:43 Subject: Re: 32 best 5-limit linear temperaments redux From: Carl Lumma>> >nd what if the subset of 'primes' you're thinking about is not >> really the primes, but is, say, {2, 3, 5/3} or {2, 3, 7/5}? >>{2,3,5/3} defines the same subgroup as {2,3,5}, but {2,3,7/5} is a >good example of a subgroup which doesn't fit the missing prime >paradigm. I posted a list of such subgroups a while back.Hmmm. You once told me it had to be primes. Does this have anything to do with that? -Carl
Message: 4091 - Contents - Hide Contents Date: Sun, 10 Mar 2002 10:54:54 Subject: Re: 32 best 5-limit linear temperaments redux From: Carl Lumma>http://uq.net.au/~zzdkeena/Music/5LimitTemp.xls.zip - Type Ok * [with cont.] (Wayb.) >14KB zipped Excel spreadsheet with macros Hooray! :) -Carl
Message: 4092 - Contents - Hide Contents Date: Sun, 10 Mar 2002 21:41:33 Subject: Re: 32 best 5-limit linear temperaments redux From: Carl Lumma>> >eight functions can be thought of as distance measures, but >> I'm not getting your point. >>a _lattice_ distance function.Does that mean a taxicab one? Whatever it is, if it can do n*d, who cares? -Carl
Message: 4093 - Contents - Hide Contents Date: Sun, 10 Mar 2002 11:25:16 Subject: Re: 32 best 5-limit linear temperaments redux From: Carl Lumma>> >arl's preferred map complexity measure: >> (/ (- (max map) (min map)) (card map)) >>That is ambiguous because you haven't defined what map or card mean.That's true. Actually, card is standard set theory stuff. I just need to define map. I was originally only referring to the map of the generator, but as Gene points out, you need to take into account if your other generator is a fraction of your ie. I'll see if I can't schlep something together. Now, it's off to brunch! -C.
Message: 4094 - Contents - Hide Contents Date: Sun, 10 Mar 2002 20:20:32 Subject: a crackpot idea From: Carl Lumma I wrote... <...Something agreeing with Dave that musical history is a sort of <badness measure (or at least, can be used to check badness measures). Also, we've been at this a couple of months, long enough that what we're talking about has a kind of ecosystem, so what we have names for might also be considered here -- what we have names for should all be in the 7-limit top 32, I'd wager. At least, for the musicians' list. -Carl
Message: 4095 - Contents - Hide Contents Date: Sun, 10 Mar 2002 03:53:37 Subject: Re: 32 best 5-limit linear temperaments redux From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> Okay, thanks, this does answer my question -- he's using the number of > gens in one instance of the map, as opposed to the number in the et > that provides a near-optimal generator size, or something.What's the difference?> Though my particular suggestion was not only this; g is the rms of > gens in a map, or something. I'm still not clear exactly how it's > calculated, or if it's different from what Graham calls complexity.It's a different measure of complexity.
Message: 4096 - Contents - Hide Contents Date: Sun, 10 Mar 2002 20:37:35 Subject: Re: 32 best 5-limit linear temperaments redux From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> If you insist on sharp cutoffs (but why aren't _they_ too ad hoc?)Because we know what they mean--we haven't hidden assumptions in a function; we've got it right out in the open. I> think you'd be quite safe to ignore temperaments whose rms complexity > is more than 12 gens or whose rms error is more than 30 cents.Well, I went up to 50 gens and 50 cents, which seems to be a lot of your complaint.> The four I know of have an octave period. Two of these might never be > generated by your algorithm, since you seemed to want to deny that > they were anything but a repeat of meantone last time they were > mentioned, despite the fact that they have different size MOS and work > in different ETs.And you can't get from one note to the other using 5-limit consonances, so that they aren't authentic 5-limit temperaments. Why complain about me introducing "junk" and then insist on this? Toss 'em, and consider them again at higher limits, where they make sense.> Here are the missing four. > > Mapping Gen > to 3,5 (cents) Description MOS or ET sizes (improper) > ------------------------------------------------------------------ > [ 4, 9] 176.3 minimal diesic 6 7 (13 20 27) 34 (41 75 109 ...) > [-4, 3] 126.2 16875/16384 9 10 19 (29 48 67 86 105 ...)Theese both have a badness under 1000; do you think a search in the range g < 12, rms < 30 and badness < 1000 would be a good idea? It seems you are saying that would be more relevant.
Message: 4097 - Contents - Hide Contents Date: Sun, 10 Mar 2002 03:51:23 Subject: Re: 32 best 5-limit linear temperaments redux From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> badness = wtd_rms_gens*EXP((rms_error/7.4_cents)^0.5) > > Where wtd_rms_gens are weighted by log of odd limit.This strikes me as completely ad hoc. Why not a non-fuzzy version, with a sharp cutoff?
Message: 4098 - Contents - Hide Contents Date: Sun, 10 Mar 2002 20:44:51 Subject: Re: 32 best 5-limit linear temperaments redux From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> I could be wrong, but I don't think you can have it both ways. If > you want small temperaments to be better, your list will be finite. > That's what happened with steps^3cents and ets (right, Gene?).You can tweak it a little and not get a finite list, but only a little, and log-flat seems like the right place for an infinite list. Dave's objections in good measure are that he doesn't *want* an infinite list, with or without microtemperaments on the actual list, because he doesn't want an infinity of micro-micro-micro-temperaments with no practical meaning being theoretically wonderful according to some measure. To me, that shows the measure is working, to Dave, that it's broken.
Message: 4099 - Contents - Hide Contents Date: Sun, 10 Mar 2002 21:39:30 Subject: Interesting 46-et, 8-tone scale From: genewardsmith I took the Euclidean reduced 8-tone scale 1--8/7--6/5--4/3--7/5--3/2--5/3--7/4 looked at its 46-et version in all 1680 permutations of its steps, and reduced this to 108 under equivalence under mode and inversion. Of these 108 classes, one stood out using counts of edges and 3-note chords, being vastly superior in the 3-note chord department. Moreover, it wasn't one that was particularly promising in its JI version! It is easy to find 3-note chords from the characteristic polynomial or the adjacency matrix, and it may be that looking at them will help a great deal in sorting out the gold; it certainly did in this case. Here it is: 39375739, plus all its modal forms and their inversions. It has 21 7-limit edges, and 20 7-limit 3-note chords. Its characteristic polynomial (the characteristic polynomial of the adjacency matrix, which has a 1 if two nodes are connected, and a 0 otherwise) is x^8-21*x^6-40*x^5+12*x^4+48*x^3; the -21*x^6 term means it has 21 7-limit intervals, and the -40*x^5 term means it has 20 7-limit three-note chords. The x^2, x, and constant term are all zero which means it has multiple zero eigenvalues, but I don't know what *that* means, at least as yet. The closest competition had only 14 3-note chords!
4000 4050 4100 4150 4200 4250 4300 4350 4400 4450 4500 4550 4600 4650 4700 4750 4800 4850 4900 4950
4050 - 4075 -