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Message: 2300 - Contents - Hide Contents Date: Fri, 07 Dec 2001 03:12:24 Subject: Re: More lists From: dkeenanuqnetau --- In tuning-math@y..., graham@m... wrote:> I've updated the script at <Automatically generated temperaments * [with cont.] (Wayb.)> to > produce files using Dave Keenan's new figure of demerit. That is > > width**2 * math.exp((error/self.stdError*3)**2)Thanks for doing that Graham. I note that Graham is using maximum width and (optimised) maximum error where Gene is using rms width and (optimised) rms error. It will be interesting to see if this alone makes much difference to the rankings. I doubt it.> The stdError is from some complexity calculations we did before. I forget > what, but it's 17 cents.Actually that looks like the 1% std dev in frequency that came from some dude's experiments with actual live humans experiencing actual air vibrations. Paul can you remind us who it was and what s/he measured? So I see that while the gaussian with std error of 17 cents seems to do the right thing in eliminating temperaments with tiny errors but huge numbers of generators, it is too hard on those with larger errors. Notice that Ennealimmal is still in the 7-limit list (about number 22). The problem is that Paultone isn't there at all! It has 17.5 c error with 6 gens per tetrad. Those lists don't contain any temperament with errors greater than 10 cents. The 5-limit 163 cent neutral second temperament has the largest at 9.8 cents, with 5 generators per triad. So I have to agree with Paul that badness = num_gens^2 / gaussian(error/17c) doesn't work. I realised there's no need to have non-linear functions of _both_ num_gens and error (steps and cents) in this badness metric. e.g. This will give the same ranking as the above: badness = num_gens / gaussian(error/(17c * sqrt(2))) So all we really want to know is the relationship between error and num_gens. What shape is a line of constant badness (an isobad) on a plot of number of generators needed for a complete otonality (or diamond) against error in cents. The simplest badness, num_gens * error, would mean the isobads are hyperbolas, (and num_gens^2 * error or equivalently num_gens * sqrt(error) is of course very similar) but I think it is clear that, for constant badness, as error goes to zero, num-gens does _not_ go to infinity, but levels off. Even for zero error there is a limit to how many generators you can tolerate. I find it difficult to imagine anyone being seriously interested in using a temperament that needs 30 generators to get a single complete otonality, no matter how small the error is. And I think this limiting number of generators decreases as the odd-limit decreases. We can introduce this as a sudden limit as Gene suggested, or we can use some continuous function to make it come on gradually An isobad will also have a maximum number of cents error that can be tolerated even when everything is approximated by a single generator. Notice that the number of generators can't go below 1 (even for rms), so we don't care what an isobad does for num_gens < 1. What's a nice simple badness metric that will give us these effects?> Oh yes. Seeing as a 7-limit microtemperament is now causing something of > a storm, notice that the top 11-limit one is 26+46 (complexity of 30, > errors within 2.5 cents). And the simplest with all errors below a cent > is 118+152 (complexity of 74).Yes, even though we don't consider it a microtemperament at the 11-limit, Miracle temperament is really a serious 11-limit optimum, by any (reasonable) goodness measure. You have to pay an enormous cost in extra complexity to get the max error even _slightly_ lower than 11-limit-Miracle's 3.3 cents, or an enormous cost in cents to get the complexity down even slightly below 11-limit-Miracle's 22 generators. Is that what you are indicating?
Message: 2301 - Contents - Hide Contents Date: Fri, 07 Dec 2001 06:54:04 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>> I understand what the slope is in the HE case, but what slope are > you>> talking about re badness of linear temperament? Badness wrt what? >> What is the problem with a "flat" system and a cutoff?Dave is trying to understand why this _is_ a flat system.> It doesn't > commit to any particular theory about what humans are like and what > they should want, and I think that's a good plan. Thank you.
Message: 2302 - Contents - Hide Contents Date: Fri, 07 Dec 2001 03:23:14 Subject: Re: The grooviest linear temperaments for 7-limit music From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote: >>> Personally I'd feel much better if everyone could somehow agree what >> was the overall most sensible measure regardless of the results! >> Fat chance :) >>> In Gene's case, I would hope that it would be some elegant internal >> consistency that ties the whole deal together. I'd personally settle >> for that even if the results were a tad exotic. >> I feel the same way.It's nice to have pretty looking (i.e. simple) fomulae but we can hardly ignore the fact that we're trying to come up with a list of linear temperaments that will be of interest to the largest possible number of human beings. Unfortunately human perception and cognition is messy to model mathematically, not well established experimentally and highly variable between individuals. But I'm sure we can come up with something that is both reasonably elegant mathematically and that we (in this forum) can all agree isn't too bad. We certainly do it without trying some out and looking at the results! We should probably hone the badness metric using 5-limit, where the most experience exists.
Message: 2303 - Contents - Hide Contents Date: Fri, 07 Dec 2001 06:59:08 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> Possibly, though since gens and cents are two dimensions, we really >> need a shuf-off _curve_, don't we? >> If we bound one of them and gens^2 cents, we've bound the other; > that's what I'd do.Hmm . . . so if we simply put an upper bound on the RMS cents error, we'll have a closed search? That doesn't seem right . . .
Message: 2304 - Contents - Hide Contents Date: Fri, 07 Dec 2001 03:38:36 Subject: Re: More lists From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> Actually that looks like the 1% std dev in frequency that came from > some dude's experiments with actual live humans experiencing actual > air vibrations. Paul can you remind us who it was and what s/he > measured?It measured the typical uncertainties with which sine-wave partials in an optimal frequency range were heard, based on the uncertainties with which the virtual fundamentals were heard.
Message: 2305 - Contents - Hide Contents Date: Fri, 07 Dec 2001 07:00:21 Subject: Re: The grooviest linear temperaments for 7-limit music From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>> I understand what the slope is in the HE case, but what slope are > you>> talking about re badness of linear temperament? Badness wrt what? >> What is the problem with a "flat" system and a cutoff?I may be able to answer that when someone explains what is flat with respect to what. It doesn't> commit to any particular theory about what humans are like and what > they should want, and I think that's a good plan.Don't the cutoffs have to be based on a theory about what humans are like? If a "flat" system was miles from anything related what humans are like, would you still be interested in it? I don't think you can avoid this choice. You must publish a finite list. If you include more of certain extremes, you must omit more of the middle-of-the-road.
Message: 2306 - Contents - Hide Contents Date: Fri, 07 Dec 2001 03:47:20 Subject: Re: More lists From: dkeenanuqnetau --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> What's a nice simple badness metric that will give us these effects?Hey! What's wrong with simply badness = num_gens + error_in_cents (i.e. steps + cents) or if that seems too arbitrary, how about agreeing on some value of k in badness = k * num_gens + error_in_cents, where k ~= 1 or maybe even badness = k/odd_limit * num_gens + error_in_cents, where k ~= 5 Wanna give this one a spin Graham?
Message: 2307 - Contents - Hide Contents Date: Fri, 07 Dec 2001 07:03:27 Subject: Re: The grooviest linear temperaments for 7-limit music From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> We could search (16/15)^a (25/24)^b (81/80)^c to start out with, > and>> go to something more extreme if wanted. >> More extreme? I'm not getting this.(78732/78125)^a (32805/32768)^b (2109375/2097152)^c also gives the 5-limit, but is better for finding much smaller commas, to take a more or less random example.
Message: 2308 - Contents - Hide Contents Date: Fri, 07 Dec 2001 03:50:46 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> But I'm sure we can come up > with something that is both reasonably elegant mathematically and that > we (in this forum) can all agree isn't too bad.I felt that way about steps^3 cents, except where was 12+14?> We certainly do it > without trying some out and looking at the results!You mean a priori? The more arbitrary parameters we put into it, the more we'll have to rely on particular assumption on how someone is going to be making music, and this assumtion will be violated for the next person. The top 25 or 40 according to a very generalized criterion will best serve to present the _pattern_ of this whole endeavor, upon which any musician can base their _own_ evaluation, and if they don't want to, at least pick off one or two temperaments that interest them. But I have a nagging suspicion that there are even more "slippery" ones out there, especially on the ultra-simple end of things . . . I suspect we can use step^2 cents and cut it off at some point where there's a long gap in the step-cent plane. For example, the next point out after Ennealimmal is probably a long way out, so we can probably put a cutoff there. As for simple temperaments with large errors, I suspect there are more than Gene and Graham have found so far that would end up looking good on this criterion, so it may end up making sense to place another cutoff there . . . but I want to be sure we've caught all the slippery fish before we decide that. I would still like to see the "step" thing weighted -- there should be something very mathematically and acoustically elegant about doing it that way (if defined correctly) since we are using the Tenney lattice after all!> > We should probably hone the badness metric using 5-limit, where the > most experience exists.Yes, I was just going to say we should write the whole paper first in the 5-limit.
Message: 2309 - Contents - Hide Contents Date: Fri, 07 Dec 2001 07:05:05 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: >> It doesn't>> commit to any particular theory about what humans are like and what >> they should want, and I think that's a good plan. >> Don't the cutoffs have to be based on a theory about what humans are > like?I'm suggesting we place the cutoffs where we find big gaps, and comfortably outside any system that has been used to date.> > If a "flat" system was miles from anything related what humans are > like, would you still be interested in it?Again, any system that is "best" according to a "human" criterion will show up as "best in its neighborhood" under a flat criterion.
Message: 2310 - Contents - Hide Contents Date: Fri, 07 Dec 2001 04:18:34 Subject: Re: The grooviest linear temperaments for 7-limit music From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>> But I'm sure we can come up >> with something that is both reasonably elegant mathematically and > that>> we (in this forum) can all agree isn't too bad. >> I felt that way about steps^3 cents, except where was 12+14? >>> We certainly do it >> without trying some out and looking at the results!Oops! That should have been We certainly _can't_ do it without trying some out and looking at the results!> You mean a priori? The more arbitrary parameters we put into it, the > more we'll have to rely on particular assumption on how someone is > going to be making music, and this assumtion will be violated for the > next person."Not putting in" an arbitrary parameter is usually equivalent to putting it in but giving it an even more arbitrary value like 0 or 1.
Message: 2311 - Contents - Hide Contents Date: Fri, 07 Dec 2001 07:08:44 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>>> We could search (16/15)^a (25/24)^b (81/80)^c to start out with, >> and>>> go to something more extreme if wanted. >>>> More extreme? I'm not getting this. >> (78732/78125)^a (32805/32768)^b (2109375/2097152)^c also gives the > 5-limit, but is better for finding much smaller commas, to take a > more or less random example.Once a, b, and c are big enough, the original choice of commas will do little to induce any tendency of smallness or largeness in the result, correct?
Message: 2312 - Contents - Hide Contents Date: Fri, 07 Dec 2001 04:30:35 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> You mean a priori? The more arbitrary parameters we put into it, the >> more we'll have to rely on particular assumption on how someone is >> going to be making music, and this assumtion will be violated for > the >> next person. >> "Not putting in" an arbitrary parameter is usually equivalent to > putting it in but giving it an even more arbitrary value like 0 or 1.Well, I think Gene is saying that step^2 cents is clearly the right measure of "remarkability".
Message: 2313 - Contents - Hide Contents Date: Fri, 07 Dec 2001 07:11:53 Subject: Re: The grooviest linear temperaments for 7-limit music From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>>> Because those measures give an overall "slope" to the results, in >>> analogy to what the Farey series seeding does to harmonic entropy. >>>> What's objective about that? A certain slope may be _real_. i.e. >> humans on average may experience it that way, in which case > the "flat">> case will really be favouring one extreme. >> But I don't feel comfortable deciding that for anyone.But you _are_ deciding it. You can't help but decide it, unless you intend to publish an infinite list. No matter what you do there will be someone who thinks there's a lot of fluff in there and you missed out some others. They aren't going to be impressed by any argument that "our metric is 'objective' or 'flat'".>> I understand what the slope is in the HE case, but what slope are > you>> talking about re badness of linear temperament? Badness wrt what? >> Both step and cent.Huh? Obviously any badness metric _must_ slope down towards (0,0) on the (cents,gens) plain. If you make the gens and cents axes logarithmic then badness = gens^k * cents is simply a tilted plane. The only way you can decide on whether it should tilt more towards gens or cents (the exponent k) is through human considerations.
Message: 2314 - Contents - Hide Contents Date: Fri, 07 Dec 2001 04:35:25 Subject: Re: The grooviest linear temperaments for 7-limit music From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> The solutions represent?I take the 5-limit comma defined by the temperament, and then find another comma 2^p 3^q 5^r 7 such that the wedgie of this and the 5- limit comma is the correct wedgie, that means these two commas define the temperament.>> The pair of unisons >> returned in this way can be LLL reduced by the "com7" function, > which>> takes a pair of intervals and LLL reduces them. >> Why not TM-reduce them?I'd always LLL reduce them first.
Message: 2315 - Contents - Hide Contents Date: Fri, 07 Dec 2001 07:16:27 Subject: Re: The grooviest linear temperaments for 7-limit music From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>> If a "flat" system was miles from anything related what humans are >> like, would you still be interested in it? >> Again, any system that is "best" according to a "human" criterion > will show up as "best in its neighborhood" under a flat criterion.But some neighbourhoods may be so disadvantaged that their best doesn't even make it into the list.
Message: 2316 - Contents - Hide Contents Date: Fri, 07 Dec 2001 04:56:08 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> The solutions represent? >> I take the 5-limit comma defined by the temperament, and then find > another comma 2^p 3^q 5^r 7 such that the wedgie of this and the 5- > limit comma is the correct wedgie, that means these two commas define > the temperament. > >>>> The pair of unisons >>> returned in this way can be LLL reduced by the "com7" function, >> which>>> takes a pair of intervals and LLL reduces them. >>>> Why not TM-reduce them? >> I'd always LLL reduce them first. How come?
Message: 2317 - Contents - Hide Contents Date: Fri, 07 Dec 2001 07:25:50 Subject: Re: The grooviest linear temperaments for 7-limit music From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Hmm . . . so if we simply put an upper bound on the RMS cents error, > we'll have a closed search? That doesn't seem right . . .I was suggesting a *lower* bound on RMS cents as one possibility. If with all quantities positive we have g^2 c < A and c > B, then 1/c < 1/B, and so g^2 < A/B and g < sqrt(A/B). However, it probably makes more sense to use g>=1, so that if g^2 c <= A then c <= A.
Message: 2318 - Contents - Hide Contents Date: Fri, 07 Dec 2001 05:17:35 Subject: Re: The grooviest linear temperaments for 7-limit music From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> I'd always LLL reduce them first. > > How come?Because it makes the TM reduction dead easy.
Message: 2319 - Contents - Hide Contents Date: Fri, 07 Dec 2001 07:26:24 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>>>>> Because those measures give an overall "slope" to the results, > in>>>> analogy to what the Farey series seeding does to harmonic > entropy. >>>>>> What's objective about that? A certain slope may be _real_. i.e. >>> humans on average may experience it that way, in which case >> the "flat">>> case will really be favouring one extreme. >>>> But I don't feel comfortable deciding that for anyone. >> But you _are_ deciding it. You can't help but decide it, unless you > intend to publish an infinite list. No matter what you do there will > be someone who thinks there's a lot of fluff in there and you missed > out some others. They aren't going to be impressed by any argument > that "our metric is 'objective' or 'flat'".We won't be missing out on anyone's "best" (unless they are really far out on the plane, beyond the big gap where we will establish the cutoff). Then they can come up with their own criterion and get their own ranking. But at least we'll have something for everyone.>>> I understand what the slope is in the HE case, but what slope are >> you>>> talking about re badness of linear temperament? Badness wrt what? >>>> Both step and cent. >> Huh? Obviously any badness metric _must_ slope down towards (0,0) on > the (cents,gens) plain.The badness metric does, but the results don't. The results have a similar distribution everywhere on the plane, but only when gens^2 cents is the badness metric.
Message: 2320 - Contents - Hide Contents Date: Fri, 07 Dec 2001 05:18:59 Subject: Re: The grooviest linear temperaments for 7-limit music From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Well, I think Gene is saying that step^2 cents is clearly the right > measure of "remarkability".Huh? "Remarkability" sounds like a kind of goodness. Step^2 * cents is obviously a form of badness. I think I've already explained why no product of poynomials of these two things will ever be acceptable to me, at least not without cutoffs applied to them first. And I understand Gene to be saying that he wants at least an upper cutoff on "steps" (which seems like a bad name to me since it suggests scale steps, I prefer "num_gens" or just "gens"). gens^2 * cents gives exactly the same ranking as log(gens^2 * cents) [where the log base is arbitrary] because log(x) is monotonically increasing. Right? Now log(gens^2 * cents) = log(gens^2) + log(cents) = 2*log(gens) + log(cents) So this says that a doubling of the number of generators is twice as bad as a doubling of the error. And previously someone suggested it was 3 times as bad. You've arbitrarity decided that only the logarithms are comparable (when cents is already a logarithmic quantity) and you arbitrarily decided that the constant of proportionality between them must be an integer! So what's wrong with k*steps + cents? The basic idea here is that the unit of badness is the cent and we decide for a given odd-limit how many cents the error would need to be reduced for you to tolerate an extra generator in the width of your tetrads (or whatever), or how many generators you'd need to reduce the tetrad (or whatever) width by in order to tolerate another cent of error. Or maybe you think that a _doubling_ of the number of generators is worth a fixed number of cents. i.e. badness = k*log(gens) + cents But always you must decide a value for one parameter k that gives the proportionality between gens and cents because there is no relationship between their two units of measurement apart from the one that comes through human experience. Or at least I can't see any.
Message: 2321 - Contents - Hide Contents Date: Fri, 07 Dec 2001 07:28:22 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>>> If a "flat" system was miles from anything related what humans are >>> like, would you still be interested in it? >>>> Again, any system that is "best" according to a "human" criterion >> will show up as "best in its neighborhood" under a flat criterion. >> But some neighbourhoods may be so disadvantaged that their best > doesn't even make it into the list.That won't happen -- that's the point of the "flat" criterion. Only the neighborhoods outside our cutoff will be disadvantaged, but at least this will be explicit.
Message: 2322 - Contents - Hide Contents Date: Fri, 07 Dec 2001 05:22:38 Subject: Re: The grooviest linear temperaments for 7-limit music From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Yes, I was just going to say we should write the whole paper first in > the 5-limit.There's not much to the 5-limit--it basically is a mere comma search, and that can be done expeditiously using a decent 5-limit notation.
Message: 2323 - Contents - Hide Contents Date: Fri, 07 Dec 2001 07:34:36 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> Hmm . . . so if we simply put an upper bound on the RMS cents > error,>> we'll have a closed search? That doesn't seem right . . . >> I was suggesting a *lower* bound on RMS cents as one possibility.Oh . . . well I don't think we should frame it _that_ way!> If with all quantities positive we have g^2 c < A and c > B, then > 1/c < 1/B, and so g^2 < A/B and g < sqrt(A/B). However, it probably > makes more sense to use g>=1, so that if g^2 c <= A then c <= A.Are you saying that using g>=1 is enough to make this a closed search?
Message: 2324 - Contents - Hide Contents Date: Fri, 07 Dec 2001 05:34:10 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> Well, I think Gene is saying that step^2 cents is clearly the right >> measure of "remarkability". >> Huh? "Remarkability" sounds like a kind of goodness. Step^2 * cents is > obviously a form of badness.Right, but it's the _objective_ kind. Not the kind that has anything to do with any particular musician's desiderata. It's the only measure that doesn't favor a certain range of acceptable values for error or for complexity. It only favors the best examples within each range. The particular users of our findings can then decide what range suits them best. Within any narrow range, all reasonable measures will give the same ranking. This is kind of like using Tenney complexity to determine the seed set for harmonic entropy -- with different complexity measures the overall slope of the curve changes, changing the consonance ranking of intervals of different sizes, but the consonance ranking of nearby intervals remains the same regardless of how complexity is defined (as long as the 2-by-2 matrix formed by the numbers in adjacent seed fractions always has a determinant of 1).> I think I've already explained why no > product of poynomials of these two things will ever be acceptable to > me, at least not without cutoffs applied to them first. > And I > understand Gene to be saying that he wants at least an upper cutoffYes -- I discussed the situation a few messages back. We use an objective measure, and cut things off in a nice wide gap.