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Message: 4100 - Contents - Hide Contents Date: Sun, 10 Mar 2002 04:54:04 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.Well it isn't. It is designed to fit what I've learned over the years by corresponding with people on the tuning lists, regarding the relative usefulness (or interest) in those 5-limit temperaments that have been known for a long time. For example, I think meantone must be in the top 3, or the badness measure is nonsense. I recently adjusted the cents parameter upwards to take into account Paul Erlich's suggestion that pelog just might exist in the real world because it is a MOS of a rough 5-limit temperament. I'd be happy to have lots of people play with those parameters to try to make the list come out with the temperaments they are familiar with, in the order they expect. 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. So you could think of the EXP((error/7.4_cents)^0.5) part as the "mistuning pain", except that would leave the "number-of-notes pain" as simply gens, which we both agree isn't right. If I take the "number-of-notes pain" as gens^2 then the "mistuning pain" is EXP((error/7.4_cents)^0.5) ^2 = EXP((error/1.85_cents)^0.5) I also designed it so that it would have a choice of parameters that gave a result very close to the "log flat" measure, so you could use that as a starting point.> Why not a non-fuzzy version, with a sharp cutoff?Because that's not how musicians or composers relate to temperaments. A temperament doesn't suddenly become of zero interest because it has overstepped some sharp boundary of harmonic error or number of generators. By the way, I think my 3 previous messages in this thread arrived on the list in the reverse order to the order I sent them. Thanks Yahoo.

Message: 4101 - Contents - Hide Contents Date: Sun, 10 Mar 2002 22:03:44 Subject: Re: Microtempered guitar design From: Carl Lumma>> >simpsons reference -- Answer: Yes, Kent.) >>And people let their children watch that crap.Oh, sorry Dave. I forgot you weren't a fan. I'll stop... it wasn't cool anyway. -Carl

Message: 4102 - Contents - Hide Contents Date: Sun, 10 Mar 2002 22:04:56 Subject: Re: 32 best 5-limit linear temperaments redux From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> 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 is trying to make an important subjective decision for musicians. gene, by insisting on a log-flat measure, best permits the musician to make this decision for him/herself. going too far in both directions doesn't hurt anyone. gene, you may note, has given his lists in order of complexity (or similar, but it should be in order of complexity), but not as an overall ranking. an overall ranking is pretty meaningless outside of a single musician's desiderata. so i'm completely with gene on this one.

Message: 4103 - Contents - Hide Contents Date: Sun, 10 Mar 2002 05:02:41 Subject: Re: 32 best 5-limit linear temperaments redux From: dkeenanuqnetau --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> 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? > > 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.

Message: 4104 - Contents - Hide Contents Date: Sun, 10 Mar 2002 22:07:25 Subject: Re: 32 best 5-limit linear temperaments redux From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>> 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.and it seems that the rms measure even agrees with the heuristic reasonably well -- though i'd prefer a weighted measure, as i've mentioned before. dave keenan has given the exact formula for the weighting i want.

Message: 4105 - Contents - Hide Contents Date: Sun, 10 Mar 2002 22:11:37 Subject: Re: 32 best 5-limit linear temperaments redux From: Carl Lumma>> >mmm. You once told me it had to be primes. Does this have anything >> to do with that? >>What had to be primes? The generators? Nope.The identities (column vectors of the map, I think). -Carl

Message: 4106 - Contents - Hide Contents Date: Sun, 10 Mar 2002 05:02:21 Subject: Re: 32 best 5-limit linear temperaments redux From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> 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.> What is different from what??Graham uses max error, and I use rms error.> 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. This only works for linear temperaments, so I'm not that happy with it.

Message: 4107 - Contents - Hide Contents Date: Sun, 10 Mar 2002 22:10:14 Subject: Re: 32 best 5-limit linear temperaments redux From: paulerlich --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> That's exactly what I _don't_ want. I want to know the size of > the chain I need to complete my map.why insist so vehemently on completeness? you can do wonderful musical things with an incomplete map.

Message: 4108 - Contents - Hide Contents Date: Sun, 10 Mar 2002 05:07:01 Subject: Re: 32 best 5-limit linear temperaments redux From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> 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.

Message: 4109 - Contents - Hide Contents Date: Sun, 10 Mar 2002 22:17:45 Subject: Re: 32 best 5-limit linear temperaments redux From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- 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.sounds like a terrible idea.

Message: 4110 - Contents - Hide Contents Date: Sun, 10 Mar 2002 22:24:15 Subject: Re: 32 best 5-limit linear temperaments redux From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> 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.i vote for just making a brief mention of the phenomenon, as anyone can easily calculate them (and their badness values) from the "real" ones if desired.

Message: 4111 - Contents - Hide Contents Date: Sun, 10 Mar 2002 05:17:04 Subject: Re: 32 best 5-limit linear temperaments redux From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> 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.Not clever enough to figure out why you want to use prefix notation.> 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.>> This only works for linear temperaments, so I'm not that happy with it. >

Message: 4112 - Contents - Hide Contents Date: Sun, 10 Mar 2002 22:12:59 Subject: Re: 32 best 5-limit linear temperaments redux From: paulerlich --- 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.hmm? listing in order of complexity? works wonderfully. no need to make separate lists, dave.

Message: 4113 - Contents - Hide Contents Date: Sun, 10 Mar 2002 23:28:09 Subject: Re: 32 best 5-limit linear temperaments redux From: Carl Lumma I wrote...>I show orwell with a 9-tone MOS, and this map: > >[0 7 -3 8 2] >[1 0 0 0 0] > >'zthat right?Whoops, I think it should be more something like this: [2] [0 1] [3] [7 0] [5] [-3 2] [7] [8 1] [11] [2 3] Is the column out front a way to write that? Certainly we wouldn't put the identities inside the map, as I first tried? Gene, when you showed me this, you gave: [0 1] [1 1] [4 0] for meantone. but shouldn't the last row be: [4 2] Graham's script gives: [0 1] [1 0] [4 -4] -Carl

Message: 4114 - Contents - Hide Contents Date: Sun, 10 Mar 2002 08:12:28 Subject: "Euclidean reduced" 8-tone scales From: genewardsmith By "Euclidean reduced" I mean I choose the least distance from the unison in the symmetric lattice of octave classes; if this is a tie, I try to use odd Tenney height (the Tenney height of the odd part) to break it, and if that doesn't work, Tenney height. If my thinking is right, the last should be good at tie-breaking. It would also be possible to take the distance from the middle of a fifth, or a triad, etc. h8, 5-limit 1, 10/9, 6/5, 4/3, 25/18, 3/2, 5/3, 9/5 h8, 7-limit (not epimorphic) 1, 7/6, 8/7, 4/3, 35/24, 3/2, 7/4, 12/7 h8, 9-limit (not epimorphic) 1, 7/6, 6/5, 4/3, 9/7, 3/2, 5/3, 12/7 h8+v7, 7-limit 1, 8/7, 6/5, 4/3, 7/5, 3/2, 5/3, 7/4 h8+v7, 9-limit (same as 7-limit) 1, 8/7, 6/5, 4/3, 7/5, 3/2, 5/3, 7/4

Message: 4115 - Contents - Hide Contents Date: Sun, 10 Mar 2002 08:15:44 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: > > 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.So you're gonna make different lists for different size ranges, are you? I think meantone is one of the 3 best overall.> 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.Of course I was meaning 5-limit for meantone. That's what this thread is about. I'd rather a single list that takes into account a typical tradeoff between accuracy number of notes. You can then go looking for your "size range" within that.

Message: 4116 - Contents - Hide Contents Date: Sun, 10 Mar 2002 08:27:22 Subject: Re: 32 best 5-limit linear temperaments redux From: genewardsmith --- 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?

Message: 4117 - Contents - Hide Contents Date: Sun, 10 Mar 2002 20:53:48 Subject: Re: 32 best 5-limit linear temperaments redux From: Carl Lumma>Supposing you saw a list which rated calculators, portable PCs, desktops, >workstations, servers, mainframes and supercomputers with a numerical >score, which combined price with performance. Would it make much sense to >find cheap $15 calculaors next to supercomputers?That depends on the price/performance function! You could be saying so yourself, in fact. There's nothing in your analogy keeping price to what you want it to be. Dave might think price is like size (complexity). -Carl

Message: 4118 - Contents - Hide Contents Date: Sun, 10 Mar 2002 10:08:29 Subject: Re: "Euclidean reduced" 8-tone scales From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> h8, 5-limit > > 1, 10/9, 6/5, 4/3, 25/18, 3/2, 5/3, 9/5If we look at commas for 5-limit h8, we find 16/15, 648/625 and 250/243, suggesting fourth-thirds, diminished and porcupine as possible temperaments. The above scale is in the class defined by (10/9)^4 (27/25)^3 (25/24); diminished equates 27/25~25/24, leading to L^4 s^4 systems, whereas porcupine equates 10/9~27/25, leading to L^7 s systems.> h8+v7, 7-limit > > 1, 8/7, 6/5, 4/3, 7/5, 3/2, 5/3, 7/4If we look at h8+v7, we add to our previous list of commas, in particular with 28/27, 36/35, 50/49, 126/125 and 245/243. Temperaments we get from these are 36/35^50/49 = [4,4,4,-2,5,-3], diminished; 50/49^245/243 = [6,10,10,-5,1,2], the "9/7" system of 22-et, with an 8/22 generator and a half-octave period; and 126/125^245/243 = [7,9,14,5,-1,-2], which if it ever had a name is one I've forgotten, but which has been mentioned in connection with 17/46 by both me and Graham, and which I find interesting. The steps for this scale are(8/7)^2 (10/9)^2 (15/14) (21/20)^3; this becomes 42322324 in 22-et or 93735739 in 46-et; permutations of this are of course entirely possible.

Message: 4119 - Contents - Hide Contents Date: Sun, 10 Mar 2002 11:15 +0 Subject: Re: 32 best 5-limit linear temperaments redux From: graham@xxxxxxxxxx.xx.xx genewardsmith wrote:> Graham uses max error, and I use rms error.I use RMS error, but max width for complexity. Graham

Message: 4120 - Contents - Hide Contents Date: Sun, 10 Mar 2002 18:38:56 Subject: Re: 32 best 5-limit linear temperaments redux From: Carl Lumma>> >hat's exactly what I _don't_ want. I want to know the size of >> the chain I need to complete my map. >>why insist so vehemently on completeness? you can do >wonderful musical things with an incomplete map.I agree, but then it's just a good temperament at a different limit. If we use the concept of limit, then we should. If a _few_ temperaments come up as really good except for one identity, they could be mentioned in a footnote or something. Otherwise, folks interested in stuff like 7:9:11 can go to Graham's site. -Carl

Message: 4121 - Contents - Hide Contents Date: Sun, 10 Mar 2002 20:55:19 Subject: Re: 32 best 5-limit linear temperaments redux From: Carl Lumma>If the analogy is say > >dollars <-> complexity >megaflop.gigabytes <-> 1/errorI swear I did not read this before I sent my reply. :) -Carl

Message: 4122 - Contents - Hide Contents Date: Sun, 10 Mar 2002 11:15 +0 Subject: Re: 32 best 5-limit linear temperaments redux From: graham@xxxxxxxxxx.xx.xx dkeenanuqnetau wrote:> 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.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. Graham

Message: 4123 - Contents - Hide Contents Date: Sun, 10 Mar 2002 11:15 +0 Subject: Re: 32 best 5-limit linear temperaments redux From: graham@xxxxxxxxxx.xx.xx Carl Lumma wrote:> 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? > > Carl's preferred map complexity measure: > (/ (- (max map) (min map)) (card map))That is ambiguous because you haven't defined what map or card mean.> Graham's preferred map complexity measure:This is the actual code: complexity = self.getWidth(self.getWidestInterval( consonances))*self.mapping[0][0] Depending on what your map is, it may give the same results up to the 7 limit and ignoring card. Beyond that, the widest interval may not be a column in the map. So I have a method for finding that, which is a bit uglier. It either loops over all the consonances and takes the largest absolute number of generator steps, or loops over the harmonics of a tonality diamond and takes the max-min. That mapping[0][0] is the number of periods to the equivalence interval because getWidth() doesn't take account of that. Graham

Message: 4124 - Contents - Hide Contents Date: Sun, 10 Mar 2002 18:54:18 Subject: Re: 32 best 5-limit linear temperaments redux From: Carl Lumma>> > agree, but then it's just a good temperament at a different >> limit. > >not necessarily. Example?>> If we use the concept of limit, then we should. > >?What it says.>i thought the paper was going to concern {2,3,5}, {2,3,7}, {2,5,7}, >{3,5,7}, and {2,3,5,7}. isn't that right?Oh- I didn't know. Okay then, you're not using limit. So you should really have no reason to list these temperaments, barring the example requested above. -Carl

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