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Message: 10025 - Contents - Hide Contents Date: Tue, 10 Feb 2004 02:40:43 Subject: Re: The same page From: Carl Lumma> Complexity is the Erlich magic L1 norm; if > <<a1 a2 a3 a4 a5 a6|| is the wedgie, then complexity is > |a1/p3|+|a2/p5|+|a3/p7|+|a4/p3p5|+|a5/p3p7|+|a6/p5p7|. > Log complexity is log of this. Error is the distance from > the JIP of the 7-limit TOP tuning for the temperament; log > (complexity) and log(error) are logs of complexity and > error, so defined.What are p3, p5, etc.? -Carl
Message: 10026 - Contents - Hide Contents Date: Tue, 10 Feb 2004 20:11:21 Subject: Re: The same page 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:>>> Just to be sure we are on it, in terms of defintions of compexity >> and>>> error, here is my page. >>>> Where are ETs? >> Forgot 'em, but you seem to have them figured out. Modulo some slight > fiddling if you must fiddle,I'd like to understand this slight fiddling, and apply this understanding to the 7-limit linear case (and elsewhere).> complexity is n for the n-et, so log > complexity is log(n). >>>> so log(complexity) is loglog(n*d). Error is >>> distance from the TOP tuning to the JIP, or in other words the > max >> of>>> the absolute values of the errors for 2, 3 and 5 in TOP tuning, >>> divided by log2(2), log2(3) and log2(5) respectively. >>>> It also can be expressed as log(n/d)/log(n*d) (*1200). >> How can either log(n*d) or loglog(n*d) also be expressed as > epimericity, which this is very close to being? Not following. >>> Log(error) is>>> the log of this. Loglog plots compare loglog(n*d) with log (error). >>>> i.e., log(log(n*d)) with log(log(n/d)/log(n*d)). >>> >>> 7-limit linear >>>>>> Complexity is the Erlich magic L1 norm; if <<a1 a2 a3 a4 a5 a6|| > is >>> the wedgie, >> >> val-wedgie, yes. >> That's how "wedgie" is defined.They're merely duals of one another, but why this definition? I can understand taking the wedge product of monzos much better than I understand the wedge product of vals.
Message: 10027 - Contents - Hide Contents Date: Tue, 10 Feb 2004 21:20:23 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> Thus it's great for a paper for mathematicians. Not for musicians. >>>>>> The *contents* of the list is what's great for musicians, not >>> how it was generated. >>>> No; I agree with Graham that we should "teach a man to fish". >> I disagree. It's just too hard for non-mathematicians. Unless by > "fish" you mean "go to Graham's web site and use the temperament > finder there" in which case I'm all for it! And this would let us not > worry too much that we may have left some temperament out of the paper > that someone someday may find useful.This is a music *theory* paper, so presenting the bare minimum of math to actually *derive* our results is appropriate. True, only heavy theorists will probably want to reproduce the calculations. But we want to leave referees, at least, with fairly complete confidence that what we're doing is correct.
Message: 10028 - Contents - Hide Contents Date: Tue, 10 Feb 2004 22:51:06 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>>>> >ould you do me a favor and attempt to speak to me as a human being, >>>> and not deal with me like a chess opponent, trying to look several >>>> moves ahead so that you can defeat me? >>>>>> I washed out of the first round of the US correspondence championship. >>> It's my brother who is the grandmaster. >>>> Is he really a grandmaster? >>He's not an OTB grandmaster, he is a correspondence grandmaster and >several times US champion.Wow, cool! I don't follow the correspondence scene, but I always get a kick out of the name of the column in No Life magazine: The Check Is In The Mail. :) -Carl
Message: 10029 - Contents - Hide Contents Date: Tue, 10 Feb 2004 23:58:04 Subject: Hyperbolic cuttoff From: Gene Ward Smith The -1 slope seems to be what is wanted on the high-complexity side; and arctan(-1) is -pi/4. If I take a -pi/3 from that, I get a slope for the line at and angle of 120 degrees, which is tan(-5pi/12) = -2-sqrt(3) = -3.732 which should do nicely as a high-complexity cuttoff. A hyperbola in the loglog plane suggests itself, and I'll find out what list I can get in this way if it won't violate some sacred principle--in other words, if there is any interest, and minds are still open.
Message: 10031 - Contents - Hide Contents Date: Tue, 10 Feb 2004 06:43:29 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> I seem to have been doing nothing but that for the past two days. The > fact that you haven't recognised it as such says to me that we're > somehow talking past each other much worse than I thought.Then first loglog plots I've seen were just now posted by Paul; they make a *very* strong case for loglog, not at all to my surprise. It would be interesting now to see linears.> 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.There's only one way to do it, which is to do it. I don't see why this is any kind of argument. Something on the very edge of your criterion is by definition marginal, whereever your margin lies. You can try to avoid this by moats, but that's only going to take you so far, and if you are not careful (and I've seen no signs of care) into regions where the justification is dubious. If you want a list, why not just pick your favorites and put them on it?> > 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".If the moat is gerrymandered, you get that question anyway, don't you?> The way to find a useful moat is to start with the temperaments you > know everyone will want included, and those that almost no one will > care about, and check out the space between the two.Right. Then you put them on a loglog plot, and try to draw a straight line between them, and find to your amazement that it works. Now you only have the corners to worry about, and what you are doing is easier to justify. Is this so bad? Why the opposition to even trying? When the response is "this isn't helping" my impression is that I am not being listened to at all, hence I started shouting. Now I think I may have gotten through a little, so let's talk.
Message: 10032 - Contents - Hide Contents Date: Tue, 10 Feb 2004 20:15:01 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:>> --- 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.I'm not following you, and I'm at a loss to understand why the above definition of TOP error is suddenly a problem for you.>>> 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.Yes; TOP error is always defined as above (though there are plenty of equivalent definitions, such as over the primes alone).>>>> 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.Sure, but Dave's talking about measuring the errors directly (which may be less useful for finding a solution, but may be more useful for understanding what the solution means).
Message: 10033 - Contents - Hide Contents Date: Tue, 10 Feb 2004 21:24:59 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> The rectangle enclosed by error and complexity bounds. You answered > that the axes were infinitely far away, but the badness line AB > doesn't seem to be helping that.If you simply bound complexity alone, you get a finite number of temperaments. Most are complete crap.
Message: 10034 - Contents - Hide Contents Date: Tue, 10 Feb 2004 00:01:26 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> I'm sorry it came across that way. But the fact is we had already >>> thought about it and found it too extreme, not possible to match up >>> with the historical data (vague though that is). Sorry we didn't >>> spell that out. >>>> It would be nice if some attempt was made to bring the rest of us on >> board. I don't know what Carl or Graham think, but I have not been >> convinced. >> My latest position is that I can live with log-flat badness with > appropriate cutoffs. The problem with anything more trickyMore tricky? Log-flat is tricky enough to be interesting for mathematicians and mind-boggling for musicians.> is that > we have no data. Not vague historical data, actually no data.Less data than in the log-flat case?> By > putting all this energy into the list of temperaments, we're loosing > 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.Seems to be a creepy coincidence, since it's an affine-geometrical measure of area in the Tenney lattice, not something with units of number of notes. But I'm not surprised that it gives more "notes" for more complex temperaments, and fewer for less complex temperaments. ;)
Message: 10036 - Contents - Hide Contents Date: Tue, 10 Feb 2004 12:18:09 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>>> >hus it's great for a paper for mathematicians. Not for musicians. >>>> The *contents* of the list is what's great for musicians, not >> how it was generated. >>No; I agree with Graham that we should "teach a man to fish".That's all the more reason particulars like moats v. logflat don't matter -- if I can fish, I can implement my own list. But I still don't think the hypothetical "musicians" you're battering Gene with will be able to do it.>>>>> Log-flat badness with cutoffs >>>>>>>> The cutoffs are of course completely arbitrary, but can be easily >>>> justified and explained in the context of a paper. >>>>>> But there are *three* of them! >>>> ...still trying to understand why the rectangle doesn't enclose >> a finite number of temperaments... > >Which rectangle?The rectangle enclosed by error and complexity bounds. You answered that the axes were infinitely far away, but the badness line AB doesn't seem to be helping that.>> My thoughts are that in the 5-limit, we might reasonably have a >> chance of guessing a good list. But beyond that, I would cry >> Judas if anyone here claimed they could hand-pick anything. So, >> my question to you is: can a 5-limit moat be extrapolated upwards >> nicely? >>Not sure what you mean by that.Which part? Can the equation/coordinates that defines your fav. moat be taken from a 5-limit plot and slapped onto a 7-limit one? -Carl
Message: 10037 - Contents - Hide Contents Date: Tue, 10 Feb 2004 21:27:12 Subject: Re: ! From: Paul Erlich --- 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 "Paul Erlich" <perlich@a...> > wrote: >>>>> Did either of you guys look at the loglog version of the moat- of- > 23 7->>> limit linear temperaments? >>>> I have a plot with unlabled axes and a curved red line on it. >> Obviously, since I don't know what is being plotted, I draw no >> conclusion. >> I've already clarified this for you!!Again: The horizontal axis, as always, is *complexity*. The vertical axis, as always, is *error*. We've already established we're on the same page on those. It's easy to see, by the tick marks, if either or both of the axes is scaled logarithmically. The red line is our proposed moat. And again, the 7-limit 'linear' temperaments are indexed as follows (I show the first three numbers in the val-wedgie, since you feel they are the most important): 1) 1 4 10 2) 2 -4 -4 3) 5 1 12 4) 7 9 13 5) 1 4 -2 6) 3 0 -6 7) 4 -3 2 8) 2 8 1 9) 6 5 3 10) 1 9 -2 11) 2 8 8 12) 6 -7 -2 13) 6 10 10 14) 7 -3 8 15) 4 4 4 16) 1 -8 -14 17) 3 0 6 18) 0 0 12 19) 1 4 -9 20) 0 5 0 21) 3 12 -1 22) 10 9 7 23) 3 5 -6 24) 9 5 -3 25) 8 6 6 26) 6 -2 -2 27) 6 5 22 28) 3 12 11 29) 2 -9 -4 30) 11 13 17 31) 6 10 3 32) 4 2 2
Message: 10038 - Contents - Hide Contents Date: Tue, 10 Feb 2004 00:09:02 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> Complexity is horizontal, error is vertical, >> >> Aha. >>>>> labels are the notes per octave of the ET. >>>> How can error be in notes? >> Sorry. I was referring to the labels on the points. i.e. each point is > labelled with the n of the n-tET that it is. > > The error is minimax error in cents where the weighting is log_2 (n*d) > for the ratio n/d in lowest terms. > > The complexity I'm not sure about. Paul?This was explained in the post itself, though it's obviously giving something extremely close to the number of notes per octave.
Message: 10040 - Contents - Hide Contents Date: Tue, 10 Feb 2004 06:49:58 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> 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.Ridiculous. I've *composed* in about that many temperaments.
Message: 10041 - Contents - Hide Contents Date: Tue, 10 Feb 2004 20:17:42 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 "Dave Keenan" <d.keenan@b...> > wrote: >>> I don't have a problem with that. I still think the simplest curves >> through moats that are in the right ballpark will be of the form >> >> (err/k1)^p + (comp/k2)^p < x where p is 1 or slightly less than 1. >> The ets Paul just got through plotting are lying more or less along > straight lines. I don't see any way to make a sensible moat unless > your line follows the lay of the land, so to speak.Except you suddenly depart from the lay of the land in two places? Why this suddenness? What powerful psychological force operates at these two points?>> Is there a simpler function of log(err) and log(comp) that gives >> similar shaped curves in the region of interest? > > Lines.As I showed, the curve in question, on a log-log plot, looks clearly unlike a line.
Message: 10042 - Contents - Hide Contents Date: Tue, 10 Feb 2004 21:27:57 Subject: Re: ! From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> >> wrote:>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> >>> wrote: Dave wrote:>> I don't much care how it's plotted, so long as we zoom in on the >> interesting bit. So, on these plots, what shape would you make a >> smooth curve that encloses only (or mostly) those ETs that musicians >> have actually found useful (or that you think are likely to be found >> useful) for approximating JI to the relevant limit? Having regard > for>> the difficulty caused by complexity as well as error. >> Did either of you guys look at the loglog version of the moat-of-23 7- > limit linear temperaments?Sure. I looked at it and agree with it just fine. That should be obvious since I agreed just fine with it on a linear-linear plot. I was asking Gene what shape _he_ thought it should be, and particularly in regard to 5-limit ETs. He says "a straight line", so I think we're doomed to disagree.> I'd say that if you planned to use any set of commas that generate > the ET's kernel (for chord-pump progressions, say), we're justified > to consider that you're planning to use the ET itself. Fair enough.
Message: 10043 - Contents - Hide Contents Date: Tue, 10 Feb 2004 00:10:19 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> ... the dreaded error and complexity bounds. >>>> My 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.>> 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?Same reason as above.
Message: 10045 - Contents - Hide Contents Date: Tue, 10 Feb 2004 06:49:50 Subject: Re: ! From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:>> Why should we want to multiply instead of add? >> Oh, for God's sake Paul-have you looked at your own plots? Did you > notice how straight the thing looks in loglog coordinates? Your plots > make it clear that loglog is the right approach. Look at them!I don't much care how it's plotted, so long as we zoom in on the interesting bit. So, on these plots, what shape would you make a smooth curve that encloses only (or mostly) those ETs that musicians have actually found useful (or that you think are likely to be found useful) for approximating JI to the relevant limit? Having regard for the difficulty caused by complexity as well as error. I wonder if, when you say that there is no particular problem with complexity you are thinking of cases where you may use a subset of an ET, in the way that Joseph Pehrson is using a 21 note subset of 72-ET. In that case you are really using a linear temperament, not the ET itself. I think the complexity of an ET should be considered as if you planned to use _all_ its notes.
Message: 10046 - Contents - Hide Contents Date: Tue, 10 Feb 2004 16:11:27 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>> > haven't seen any cluster analysis yet! >>It was principal components analysis, but the reasoning behind the >implementation was obscure.I have no idea what you're talking about.>>> 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. >>OK, Carl, so everyone's been sorely underestimating the true >usefulness of 665-equal and 612-equal, yes? No. -Carl
Message: 10047 - Contents - Hide Contents Date: Tue, 10 Feb 2004 20:21:04 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> log-flat is natural, in a way. And it should be one of the easier >>> concepts around here to explain to musicians. >>>> I don't recall even Dave understanding its derivation, let along any >> full-time musicians. >> I recall that Dave rejected the idea of a critical exponent. But I > didn't understand it until I coded it. But anyway, it's no big deal. > At the level I'd imagine this stuff being explained to a full-time > musician, it wouldn't be any harder to explain than a moat.I think the regular plot will be easier to explain than the log-log plot. After that, the moat will be quite easy to demonstrate. It seems you just grokked it yourself.>>> or why we should >>> want to add instead of multiply to get badness. >>>> Why should we want to multiply instead of add? >> Gene multiplies logs, and you and Dave are adding them. > Or so I thought...No, Gene adds logs, while Dave and I add without taking logs.
Message: 10048 - Contents - Hide Contents Date: Tue, 10 Feb 2004 21:28:45 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Except you suddenly depart from the lay of the land in two places? > Why this suddenness? What powerful psychological force operates at > these two points?Lack of interest in tempering out 3/2. However, you don't need to set any error bound if this doesn't worry you, and don't need any complexity bound at all. We discussed this extensively in the past, back in the days when you didn't like the idea of getting a finite list, which this would do for the right badness exponent or slope.> As I showed, the curve in question, on a log-log plot, looks clearly > unlike a line.Can you tell me where this plot is, with axes clearly labeled, so the rest of us have a clue?
Message: 10049 - Contents - Hide Contents Date: Tue, 10 Feb 2004 00:12:09 Subject: Re: Loglog From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> I checked the files I saved of the graphs being posted, and found > no>> loglog examples. I then went over to tuning-files, and found one >> example, >> You missed quite a few then,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.
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