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Message: 9925 - Contents - Hide Contents Date: Mon, 09 Feb 2004 22:31:05 Subject: Re: The same page From: Carl Lumma>log2(3), log2(5), etc.Thanks. I think that's the same as Gene was using before, then. -Carl

Message: 9926 - Contents - Hide Contents Date: Mon, 09 Feb 2004 23:49:59 Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> Yes. That's a good point (about e.g. 5-limit JI having infinite > complexity as a linear temperament), but obviously there's another > point of view available where 5-limit JI has finite complexity as a > planar temperament.But there are no other 5-limit planar temperaments to compare it to, so this is irrelevant.> Psychologically it would seem that there is some point in the > complexity of low-error 5-limit linear temperaments where one would > rather have the planar complexity of 5-limit JI than bother with the > linear complexity of a temperament.Rather, it seems to me, one wouldn't care. Certainly the linear temperament can never become *more* complex than a planar -- it can merely become *equally* complex for all intents and purposes.> I suggest that occurs somewhere > between the complexities of schismic and the least complex temperament > with error less than schismic.I suggest much more complex temperaments belong in a math paper, not a music paper.

Message: 9927 - Contents - Hide Contents Date: Mon, 09 Feb 2004 22:38:33 Subject: Re: The same page From: Carl Lumma>>> >-limit, comma = n/d >>> >>> Complexity is log2(n*d), >>>> Yes, but this can also be expressed in other ways, for example if >> >> <<a1 a2 a3|| >> >> is the val-wedgie (dual to the comma),I thought val ^ val -> comma, so val ^ val must not be a val-wedgie. What's a val-wedgie? Anybody have a handy asci 'units' table for popular wedge products in ket notation? ie, [ val > ^ [ val > -> [[ wedgie >> < monzo ] ^ < monzo ] -> ? ...etc.>>> Error is the distance from the JIP of the 7-limit TOP >>> tuning for the temperament; >>>> Or same as 5-limit linear error but with an additional term for 7.What's linear error? -Carl

Message: 9928 - Contents - Hide Contents Date: Mon, 09 Feb 2004 23:54:26 Subject: Re: Beep isn't useless.... From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> but finding something that > doesn't at the same time exploit the meantone comma is going to be > tricky.Yes, it's quite rare. Even the Canon has some rough spots where, for example, scale degree 2 appears over the IV chord and vi chord.

Message: 9929 - Contents - Hide Contents Date: Mon, 09 Feb 2004 11:00:32 Subject: The seven-limit lattices From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> Hate to look dumb, but need to ask how the numbers in brackets are > calculated. [0,1,0] sends major to minor, but how?Instead of writing a response to this, I made a new web page: Seven limit lattices * [with cont.] (Wayb.) ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 9930 - Contents - Hide Contents Date: Mon, 09 Feb 2004 23:56:53 Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> I'm hoping paul can easily replot those ET plots loglog.When I do so, at least keep in mind that rather than log(complexity), 2^complexity has actually been proposed as a criterion (i.e., by Fokker), and that error^2, at least, has gotten much attention as a measure of pain, while log(error) has gotten none.

Message: 9931 - Contents - Hide Contents Date: Mon, 09 Feb 2004 22:44:01 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>We're trying to come up with some reasonable way to decide on which >temperaments of each type to include in a paper on temperaments, given >that space is always limited. We want to include those few (maybe only >about 20 of each type)For musicians, I'd make the list 5 for each limit; 10 tops. For people reading a theory paper, 20 would be interesting.>which we feel are most likely to actually be >found useful by musicians, and we want to be able to answer questions >of the kind: "since you included this and this, then why didn't you >included this". So Gene may have a point when he talks about cluster >analysis, I just don't find his applications of it so far to be >producing useful results.I haven't seen any cluster analysis yet!>Our starting point (but _only_ a starting point) is the knowledge >we've built up, over many years spent on the tuning list, regarding >what people find musically useful, with 5-limit ETs having had the >greatest coverage.You're gravely mistaken about the pertinence of this 'data source'. Even worse than culling intervals from the Scala archive.>It may be an objective mathematical fact that log-flat badness gives >uniform distribution, but you don't need a multiple-choice survey to >know it is a psychological fact that musicians aren't terribly >interested in availing themselves of the full resources of 4276-ET >()or whatever it was.So far this can be addressed with a complexity bound.>So we add complexity and error cutoffs which >utterly violate log-flat badness in their region of application (so >why violate log-flat badness elsewhere and make the transition to >non-violatedness as smooth as possible. ? >Corners in the cutoff line are bad because there are too many ways for >a temperament to be close to the outside of a corner. Agreed. >A moat is a wide and straight (or smoothly curved) band of white space >on the complexity-error chart, surrounding your included temperaments. >It is good to have a moat so that you can answer questions like "since >you included this and this, then why didn't you included this", by at >least offering that "it's a long way from any of the included >temperaments, on an error complexity plot".Okay, now I have a definition of moat. How do they compare to Gene's "acceptance regions"? -Carl

Message: 9932 - Contents - Hide Contents Date: Mon, 09 Feb 2004 22:51:29 Subject: loglog! From: Carl Lumma Yahoo groups: /tuning-math/files/Paul/et5loglo... * [with cont.] Ok, easy! No moat needed, at least for ETs. Just draw a circle around the origin and grow the radius until it would include something that exceeds a single bound -- a "TOP notes per 1200 cents" bound. For ETs at least. Choose a bound according to sensibilities in the 5-limit, round it to the nearest ten, and use it for all limits. -Carl

Message: 9933 - Contents - Hide Contents Date: Mon, 09 Feb 2004 00:40:53 Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears) From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Some evidence you've actually considered it would be nice. A plot > would be grand. Some attempt to theoretically justify what you two are > doing would be appreciated.I'm not sure what "it" is that you think we haven't considered. If it's log-flat badness then that seems to have been the only such measure being considered on this list for the past several years, despite the objections (from psychology) that I thought I spelled out in great detail when it was first mooted. And by "theoretically justify" do you mean justify purely from mathematical considerations? I believe that to be futile. It eventually needs to be grounded in human psychology, both perceptual and cognitive. I understand you're still in favour of log-flat cutoffs which can be written in the form log(err) + k * log(complexity) < x Paul and I have been considering those of the form err^p + k * comp^p < x which can be made to look a lot like the previous one when 0<p<0.5. Paul and I have not so much been trying to theoretically justify, but rather empirically determine, appropriate values for p, admittedly based on some pretty sketchy and anecdotal evidence. But that's all we have. By far the greatest body of evidence, about which temperaments people consider musically interesting or useful, relates to equal temperaments, particularly at the 5-limit. And we find that what works best is a value of p that's slightly less than one, i.e. the cutoff functions that we construct based on our knowledge of which ETs have been popular historically, are somewhere between log and linear, but much closer to linear. Since you and Paul seem to have done a marvelous job of giving us error and complexity measures that generalise from equal temps to linear temps and beyond, then it seems likely that the general shape of equal-interest contours we find for equal temps will be repeated for higher dimensions. I suppose you could say this is the theoretical part of the justification. But rather than trying to come up with precise values for p and the scaling constants for cutoffs, we are looking for what we call "moats". These are places where moderate changes in these constants will make no difference to which temperaments are included. They would ideally look like a band of whitespace on the graph shaped like a pair of back-to-back horns (something I hadn't realised before). In other words it doesn't matter so much if a moat has a narrow waist. What is most important is that it is wide near the axes. But we can't just use any old moat. There are bound to be some very wide moats that are unusable because they bear no resemblance to an equal-interest cutoff. The idea is that they should agree with the subjective cutoff functions (implicit or otherwise) of as many different people as possible.

Message: 9934 - Contents - Hide Contents Date: Mon, 09 Feb 2004 00:57:22 Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears) From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> A bit more concavity still and we include >> >> 45. Blackwood >> Following what Dave did for the 5-limit and ET cases, I found that an > exponent of 2/3 produces the desired moat, for example when > > err^(2/3)/6.3+complexity^(2/3)/9.35 < 1.I prefer to put the scaling constants inside the exponentiation like this (err/15.8)^(2/3) + (complexity/28.6)^(2/3) < 1. Then you can see at a glance what maximum error and complexity are allowed by this cutoff. For similar reasons I prefer to show the chart with both axes starting from zero.> Please look at the resulting graph: > > Yahoo groups: /tuning_files/files/Erlich/7lin2... * [with cont.] > > The temperaments in thie graph are identified by their ranking > according to the badness measure implied above: > > 1. Huygens meantone > 2. Pajara > 3. Magic > 4. Semisixths > 5. Dominant Seventh > 6. Tripletone > 7. Negri > 8. Hemifourths > 9. Kleismic/Hanson > 10. Superpythagorean > 11. Injera > 12. Miracle > 13. Biporky > 14. Orwell > 15. Diminished > 16. Schismic > 17. Augmented > 18. 1/12 oct. period, 25 cent generator (we discussed this years ago) > 19. Flattone > 20. Blackwood > 21. Supermajor seconds > 22. Nonkleismic > 23. PorcupineThis looks reasonable to me as a cutoff, although maybe still too many, but making a badness measure out of it may be going too far. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 9935 - Contents - Hide Contents Date: Tue, 10 Feb 2004 00:14:08 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>>> >ur starting point (but _only_ a starting point) is the knowledge >>> we've built up, over many years spent on the tuning list, regarding >>> what people find musically useful, with 5-limit ETs having had the >>> greatest coverage. >>>> You're gravely mistaken about the pertinence of this 'data source'. >> Even worse than culling intervals from the Scala archive. >>How do you know this?Assuming a system is never exhausted, how close do you think we've come to where schismic, meantone, dominant 7ths, augmented, and diminshed are today with any other system? If you had gone to apply your program in Bach's time, would you have included augmented and diminished? "Oh, nobody's ever expressed interest about them on a particular mailing list with about enough aggregate musical talent to dimly light a pantry, so they must not be worth mentioning." It is said the musicians of Bach's time did not accept the errors of 12-tET. 5-limit ETs being shown musically useful on the tuning list? Exactly what music are you thinking of? We're fortunate to have had some great musicians working with new systems -- Haverstick, Catler, Hobbs, Grady -- but we've chased all of them off the list, and only Haverstick could be said to have worked in a "5-limit ET" (and it's a stretch). We've got Miller, Smith and Pehrson left, with the promising Erlich and monz stuck in theory and/or 12-tET land. We're so far from any kind of form that would allow us to make statements about musical utility that it's laughable. -Carl

Message: 9936 - Contents - Hide Contents Date: Tue, 10 Feb 2004 00:43:33 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>>> >y objection was not to limits on them per se, but to acceptance >>> regions shaped like this (on a log-log plot). >>> >>> err >>> | >>> | (a) >>> |---\ >>> | \ >>> | \ >>> | \ (b) >>> | | >>> | | >>> ------------ comp >>> >>> as opposed to a smooth curve that rounds off those corners marked >>> (a) and (b). >>>> Aha, now I understand your objection. But wait, what's stopping >> this from being a rectangle? Is the badness bound giving the >> line AB? > >Yes. >>> If so, it looks like a badness cutoff alone would give a >> finite region... >>No, because the zero-error line is infinitely far away on a loglog >plot.Can you illustrate this? It looks like the zero-error line is three dashes away on the above loglog plot. :)>>> It turns out that the simplest way to round off those corners >>> is to do the following on a linear-linear plot. >>> >>> err >>> | >>> | >>> |\ >>> | \ >>> | \ >>> | \ >>> | \ >>> ------------ comp >>>> Why not this on a loglog plot? >

Message: 9937 - Contents - Hide Contents Date: Tue, 10 Feb 2004 04:18:36 Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> >> wrote: >>>>> I'm hoping paul can easily replot those ET plots loglog. >>>> When I do so, at least keep in mind that rather than log > (complexity),>> 2^complexity has actually been proposed as a criterion (i.e., by >> Fokker), and that error^2, at least, has gotten much attention as a >> measure of pain, while log(error) has gotten none. >> That it's gotten none is what I'm complaining about.No one creates a psychological model where one of the response variables goes to minus infinity!> That > 2^complexity has been discussed bores me to tears, unless you can > explain *why*.One reason might be because, for an ET, the number of possible chords goes as 2^complexity.

Message: 9938 - Contents - Hide Contents Date: Tue, 10 Feb 2004 07:12:58 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> >> wrote: >>>>> The error is minimax error in cents where the weighting is log_2 >> (n*d)>>> for the ratio n/d in lowest terms. >> The weighting is actually ONE OVER log2(n*d).It's not either one when I'm doing it, to me log(n/d)/log(n*d) is just a variant on epimericity.>> What in the world does this mean? >> Just because he's off by a multiplicate inverse, you suddenly have no > idea what he's talking about?No, and you aren't making much sense to me either; we seem to have differing ideas of what the topic under discussion is. Maybe I'm not tracking it, but I thought we were talking about TOP error and complexity.>> Do you mean TOP error for an equal >> temperament, >> Of course that's what he means. >>> which is dual to the above? >> Dual? How does duality come into play here?The dual to Tenney distance is how the error is measured.

Message: 9939 - Contents - Hide Contents Date: Tue, 10 Feb 2004 16:20:12 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>Years ago, when you first made be aware of this fact, I was seduced >by it, to Dave's dismay. Did you forget? Now, I'm thinking about it >from a musician's point of view. Simply put, music based on >constructs requiring large numbers of pitches doesn't seem to be able >to cohere in the way almost all the world's music does. Of all >people, I'm suprised Carl is now throwing his investigations along >these lines by the wayside.I'm not. It is well known that Dave, for example, is far more micro-biased than I! I'm just exploring possibilities. -Carl

Message: 9940 - Contents - Hide Contents Date: Tue, 10 Feb 2004 20:36:47 Subject: Re: ! From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> >> wrote: > >>> Your plots>>> make it clear that loglog is the right approach. Look at them! >>>> Geez, you must really be thinking like a mathematician and not a >> musician. >> A musician is going to look at these plots, see that they show a > slantwise arrangement of ets, and conclude circles are the way to > analyze them,I wasn't one of those who brought up or discussed circles, but I certainly wouldn't want to seduce musicians with a plot that is not likely to correspond with musically meaningful pain measures -- not by a long shot!

Message: 9941 - Contents - Hide Contents Date: Tue, 10 Feb 2004 21:40:13 Subject: Re: The same page From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> Anybody have a handy asci 'units' table for popular wedge products >>> in ket notation? ie, >>> >>> [ val > ^ [ val > -> [[ wedgie >><val| ^ <val| = || wedgie >>>> In 4D (e.g., 7-limit), for linear temperaments the bival is dual to >> the bimonzo, and both are referred to as the "wedgie" (though Gene >> uses the bival form).Both are referred to as the "wedgie" by whom?> Ok great. But what's all about this algebraic dual? Is this > something I can do to matrices, like complement and transpose?It's the complement.

Message: 9942 - Contents - Hide Contents Date: Tue, 10 Feb 2004 00:44:19 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> My latest position is that I can live with log-flat badness with > appropriate cutoffs. The problem with anything more tricky is that > we have no data. Not vague historical data, actually no data.Three questions regarding this statement. 1. Why is log-flat badness with cutoffs (on error and complexity) less tricky than the cutoff functions Paul and I have been looking at. Log-flat badness with cutoffs looks like this max(err/k1, comp/k2, err * comp^k3) < x or equivalently this (with different choices of k1, k2 and x) max(err/k1, comp/k2, log(err) + k3*log(comp)) < x where k3 is the number of primes divided by the number of primes less the number of degrees of freedom. This has two discontinuities in the cutoff curve. Is that less tricky than the single straight line err/k1 + comp/k2 < x ? or the slightly curved line (err/k1)^(2/3) + comp^(2/3) < x ? If so, why? 2. Assuming for the moment that we have no data, why isn't that just as much of a problem for log-flat badness with e&c cutoffs as for any other proposed cutoff relation? i.e. How should we decide what cutoffs to use on error, complexity and log-flat badness? 3. Why don't discussions of the value of various temperaments in the archives of the tuning list constitute data on this, or at least evidence? I assume "evidence" is what you mean by "data" here. It's what I meant. If, by "data", you mean something already organised as lists of relevant numbers then I agree we don't have it, but what could possibly be meant by "vague historical" lists of numbers.> By > putting all this energy into the list of temperaments, we're loosing > touch with reality.Well Paul and I see it as bringing it in closer touch with reality.> Rather than worry about what is and isn't on > the list, I'd like to figure out why Paul's creepy complexity gives > the numbers it does.Well sure. That would be a good thing to do. But I don't have a handle on it. I think that's Paul and Gene's department. I'm happy just to take it as evidence that Paul has hit on a very good complexity measure and we should use it.> But as long as Dave and Paul were having fun I > didn't want to say anything. They have a way of coming up with neat > stuff, though so far their conversation has been impenetrable to me.Thanks and sorry. Did this one help? Yahoo groups: /tuning-math/message/9330 * [with cont.]

Message: 9943 - Contents - Hide Contents Date: Tue, 10 Feb 2004 04:20:25 Subject: Re: Loglog From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> My apologies again, these used log of error, but not log of >> complexity. Using log of complexity crammed all the interesting > stuff>> to the far left to the point of illegibility, in the cases I >> originally tried. >> Sounds like a reason to get rid of most of your points, which are > gumming up the works anyway, and look at the good stuff.I actually meant the right, not the left -- but this isn't so much of a problem for the loglog graph I made for you before and for the current batch, is it?

Message: 9944 - Contents - Hide Contents Date: Tue, 10 Feb 2004 16:22:50 Subject: Re: ! From: Carl Lumma>> > musician is going to look at these plots, see that they show a >> slantwise arrangement of ets, and conclude circles are the way to >> analyze them, >>I wasn't one of those who brought up or discussed circles, but I >certainly wouldn't want to seduce musicians with a plot that is not >likely to correspond with musically meaningful pain measures -- not >by a long shot!The circle rocks, dude. It penalizes temperaments equally for trading too much of their error for complexity, or complexity for error. Look at the plots, and the first things you hit are 19, 12, and 53. And 22 in the 7-limit. Further, my suggestion that 1cents = zero should satisfy Dave's micro fears. Or make 0 cents = zero. It works either way. No origin; pfff. -Carl

Message: 9946 - Contents - Hide Contents Date: Tue, 10 Feb 2004 20:39:14 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> No way, dude! The decision is virtually made for us. >> Prove it. Give log-log plots for your proposed moats,I already did that for one case, pointed it out twice, and asked for your comments.> It's possible we could come to some kind of consensus > if you would attempt to treat people with something better than the > contempt you have shown lately.I can take your attitude in no other way, unless you either ignored completely or have an abominably low level of respect for the discussions Dave and I posted on the topic. Let's start over. If I'm willing to tolerate a certain level of error, and a certain level of complexity, why wouldn't I be willing to tolerate both together?

Message: 9947 - Contents - Hide Contents Date: Tue, 10 Feb 2004 21:42:02 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Years ago, when you first made be aware of this fact, I was seduced > by it, to Dave's dismay. Did you forget? Now, I'm thinking about it > from a musician's point of view. Simply put, music based on > constructs requiring large numbers of pitches doesn't seem to be able > to cohere in the way almost all the world's music does.You've gotten all the way up to 22 notes to the octave. I suggest you have zero experience along these lines.

Message: 9948 - Contents - Hide Contents Date: Tue, 10 Feb 2004 00:44:15 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>we are left with trying to cook up some >scheme which doesn't look as if we are simply cooking up some scheme >to get rid of them. If this isn't basically just a shell game, I >think the thing should be defined in a way where the definition gives >us the list, and not the list the definition. Some kind of cluster >analysis or something.Agreed completely, but let's hear Paul and Dave out. They may already have something! -C.

Message: 9949 - Contents - Hide Contents Date: Tue, 10 Feb 2004 04:28:02 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: >>> The error is minimax error in cents where the weighting is log_2 > (n*d)>> for the ratio n/d in lowest terms. >> What in the world does this mean? Do you mean TOP error for an equal > temperament, which is dual to the above?Yes, I should have said the weights were 1/log_2(n*d) or that the errors were _divided_ by the weights I gave. Yes. I mean TOP error but didn't want to assume all readers would know what that meant.

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