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Message: 9700 - Contents - Hide Contents Date: Mon, 02 Feb 2004 03:03:02 Subject: Re: Back to the 5-limit cutoff (was: 60 for Dave) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> It'll generally be different people, though. For a given > individual,>> very low error doesn't make complexity any easier to tolerate than >> merely tolerable error. Nevertheless, I think we should use > something>> close to a straight line, slightly convex or concave to best fit >> a "moat". >> Recall that any goofy, ad-hoc weirdness may need to be both explained > and justified.What did you have in mind?

Message: 9701 - Contents - Hide Contents Date: Mon, 02 Feb 2004 05:50:33 Subject: Re: Back to the 5-limit cutoff From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> But Yes, true. Increasing my tolerance for complexity >>> simultaneously increases my tolerance for error, since this is Max(). >>>> I have no idea why you say that. However, when I said "more of one", >> I didn't mean "more tolerance for one", I simply meant "higher values >> of one". >> If I have a certain expectation of max error and a separate > expectation of max complexity, but I can't measure them directly, > I have to use Dave's formula, I wind up with more of whatever I > happened to expect less of.More of whatever you happened to expect less of? What do you mean? Can you explain with an example?> Dave's function is thus a badness > function, since it represents both error and complexity.A badness function has to take error and complexity as inputs, and give a number as output.

Message: 9702 - Contents - Hide Contents Date: Mon, 02 Feb 2004 09:26:03 Subject: Re: Back to the 5-limit cutoff From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>>> I'm arguing that, along this particular line of thinking, >>>>>> complexity does one thing to music, and error another, but >>>>>> there's no urgent reason more of one should limit your >>>>>> tolerance for the other . . . > // >>>>>> If I'm bounding a list of temperaments with Dave's formula only, >>> and I desire that error not exceed 10 cents rms and complexity not >>> exceed 20 notes (and a and b somehow put cents and notes into the >>> same units), what bound on Dave's formula should I use? >>>> You'd pick a and b such that max(cents/10,complexity/20) < 1. >

Message: 9703 - Contents - Hide Contents Date: Mon, 02 Feb 2004 03:05:36 Subject: Re: 114 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:>> You're using temperaments to construct scales, aren't you? >> Not me, for the most part. I think the non-keyboard composer is > simply being ignored in these discussions, and I think I'll stand up > for him.And the new-keyboard composer!>> Oh, yes, I think the 9-limit calculation can be done by giving 3 a >> weight of a half. That places 9 on an equal footing with 5 and 7, > and I>> think it works better than vaguely talking about the number of >> consonances. >> You also get a lattice which is two symmetrical lattices glued > together that way.Really? Can you elaborate? Do you still get an infinite number of images of each ratio, times 9/3^2, 3^2/9, 9^2/3^4, etc.?

Message: 9704 - Contents - Hide Contents Date: Mon, 02 Feb 2004 05:03:26 Subject: Re: Back to the 5-limit cutoff From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:>> My favourite cutoff for 5-limit temperaments is now. >> >> (error/8.13)^2 + (complexity/30.01)^2 < 1 >> >> This has an 8.5% moat, in the sense that we must go out to >> >> (error/8.13)^2 + (complexity/30.01)^2 < 1.085 >> >> before we will include another temperament (semisixths).That was wrong. I forgot to square the radius. It has an 8.5% moat in the sense that we must go out to (error/8.13)**2 + (complexity/30.01)**2 < 1.085**2 before we will include another temperament (semisixths). I'm trying to remember to use "**" for power now that "^" is wedge product.>> It includes the following 17 temperaments. >> is this in order of (error/8.13)^2 + (complexity/30.01)^2 ?Yes. Or if it isn't, it's pretty close to it. The last four are essentially _on_ the curve, so their order is irrelevant.>> meantone 80:81 >> augmented 125:128 >> porcupine 243:250 >> diaschismic 2025:2048 >> diminished 625:648 >> magic 3072:3125 >> blackwood 243:256 >> kleismic 15552:15625 >> pelogic 128:135 >> 6561/6250 6250:6561 >> quartafifths (tetracot) 19683:20000 >> negri 16384:16875 >> 2187/2048 2048:2187 >> neutral thirds (dicot) 24:25 >> superpythag 19683:20480 >> schismic 32768:32805 >> 3125/2916 2916:3125 >> >> Does this leave out anybody's "must-have"s? >> >> Or include anybody's "no-way!"s? >> I suspect you could find a better moat if you included semisixths > too -- but you might need to hit at least one axis at less than a 90- > degree angle. Then again, you might not.If you keep the power at 2, there is no better moat that includes semisixths. The best such only has a 6.7% moat. This is (error/8.04)**2 + (complexity/32.57)**2 = 1**2 However if the power is reduced to 1.75 then we get a 9.3% moat outside of (error/8.25)**1.75 + (complexity/32.62)**1.75 = 1**1.75 which adds only semisixths to the above list. However I'm coming around to thinking that the power of 2 (and the resultant meeting the axes at right angles) is far easier to justify than anything else. Has anyone (e.g. Herman) expressed any particular interest in semisixths?

Message: 9705 - Contents - Hide Contents Date: Mon, 02 Feb 2004 05:48:12 Subject: Re: Weighting From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> I'm equating "field of attraction" with the width from the top of one > hump (maximum) to the next. I guess what I'm adding to Partch is that > once an interval is mistuned so much that it's outside of the original > field of attraction and into that of another ratio, then it is no > longer meaningful to call it an approximation of the original ratio. > > And so, with TOP weighting you will more easily do this with the more > complex ratios.We don't yet know what harmonic entropy says about the tolerance of the tuning of individual intervals in a consonant chord. And in the past, complexity computations have often been geared around complete consonant chords. They're definitely an important consideration . . . For dyads, you have more of a point. As I mentioned before, TOP can be viewed as an optimization over *only* a set of equally-complex, fairly complex ratios, all containing the largest prime in your lattice as one term, and a number within a factor of sqrt(2) or so of it as the other. So as long as these ratios have a standard of error applied to them which keeps them "meaningful", you should have no objection. Otherwise, you had no business including that prime in your lattice in the first place, something I've used harmonic entropy to argue before. But clearly you are correct in implying we'll need to tighten our error tolerance when we do 13-limit "moats", etc. I think that's true but really just tells us that with the kinds of timbres and other musical factors that high-error low-limit timbres are useful for, you simply won't have access to any 13-limit effects - - from dyads alone.

Message: 9706 - Contents - Hide Contents Date: Mon, 02 Feb 2004 09:28:57 Subject: Re: Back to the 5-limit cutoff From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> Even if you accept this (which I don't), wouldn't it merely tell you >> that the power should be *at least 2* or something, rather than >> *exactly 2*? >> Yes. I was playing with things like comp**5(err**2) back in the day. > But I may have been missing out on the value of adding...Taking the log of the whole thing doesn't change anything since you can just take the log of the cutoff value too, and the log of a constant is still a constant. So the above is equivalent to 5*log(comp) + 2*log(err)

Message: 9707 - Contents - Hide Contents Date: Mon, 02 Feb 2004 01:35:14 Subject: Re: Back to the 5-limit cutoff From: Carl Lumma>> >k, I walked into that one by giving fixed bounds on what I wanted. >> But re. your original suggestion (above), for any fixed version of >> the formula, more of one *increases* my tolerance for the other. > >Nonsense.Nonsense, eh? This is pretty much the definition of Max(). It throws away information on the smaller thing. You can tweak your precious constants after the fact to fix it, but not before the fact. -Carl

Message: 9708 - Contents - Hide Contents Date: Mon, 02 Feb 2004 03:09:35 Subject: Re: 114 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> I don't think that's quite what Partch says. Manuel, at least, has >>> always insisted that simpler ratios need to be tuned more accurately, >>> and harmonic entropy and all the other discordance functions I've >>> seen show that the increase in discordance for a given amount of >>> mistuning is greatest for the simplest intervals. >>>> Did you ever track down what Partch said? >> Observation One: The extent and intensity of the influence of a > magnet is in inverse proportion to its ratio to 1. > > "To be taken in conjunction with the following" > > Observation Two: The intensity of the urge for resolution is in > direct proportion to the proximity of the temporarily magnetized > tone to the magnet. Thanks, Carl.>> It also shows that, if all intervals are equally mistuned, the more >> complex ones will have the highest entropy. >> ? The more complex ones already have the highest entropy. You mean > they gain the most entropy from the mistuning? I think Paul's saying > the entropy gain is about constant per mistuning of either complex > or simple putative ratios.It's a lot greater for the sinple ones.> Thus if consonance really *does* > deteriorate at the same rate for all ratios as Paul claims,Where did I claim that?

Message: 9709 - Contents - Hide Contents Date: Mon, 02 Feb 2004 05:05:34 Subject: Harmonic Entropy (was: Re: Question for Dave Keenan) From: Paul Erlich Action on the harmonic entropy list, where I just explained the calculation again, hopefully more clearly . . .

Message: 9710 - Contents - Hide Contents Date: Mon, 02 Feb 2004 05:54:01 Subject: Re: Back to the 5-limit cutoff From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> Also try including semisixths *and* wuerschmidt -- for a list of 19 -- >> particularly if you're willing to try a straighter curve. >> No. There's no way to get a better moat by adding wuerschmidt. It's > too close to aristoxenean, and if you also add aristoxenean it's too > close to ... etc.Sorry -- I was looking at an unlabeled graph and thought semisixths was wuerschmidt for a second . . .

Message: 9711 - Contents - Hide Contents Date: Mon, 02 Feb 2004 03:11:53 Subject: Re: Weighting (was: 114 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Oh no, the simple intervals gain the most entropy. That's Paul's > argument for them being well tuned. After a while, the complex > intervals stop gaining entropy altogether, and even start losing it. At > that point I'd say they should be ignored altogether, rather than > included with a weighting that ensures they can never be important. > Some of the temperaments being bandied around here must get way beyond > that point. Examples? > Actually, any non-unique temperament will be a problem. ?

Message: 9712 - Contents - Hide Contents Date: Mon, 02 Feb 2004 05:10:50 Subject: Re: Weighting From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> To me, that's an argument for why TOP isn't necessarily what you >> want. >> The entropy minima are wider for simple ratios, but that doesn't > mean that error is less damaging to them. What it does mean is > that you're less likely to run afoul of extra-JI effects when > measuring error from a rational interval when that interval is > simple.How does measuring error run you afoul of extra-JI effects, and what are these extra-JI effects?

Message: 9713 - Contents - Hide Contents Date: Mon, 02 Feb 2004 05:58:58 Subject: Re: 7-limit horagrams From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> I'm ssorry. >> >> Green-black-green-black-green-black-green-black-green-black-green- >> black . . . >> >> Wasn't that your idea in the first place? >> I think so, and this is one of the patterns (including even, odd, > prime, proper ...) I ruled out. Look at decimal.bmp. 10 & 14 > are both green. > > -CarlI guess my debugging is at fault, then! Wow, I sure win the ass award for suggesting you needed to look more!

Message: 9714 - Contents - Hide Contents Date: Mon, 02 Feb 2004 09:39:11 Subject: Re: Back to the 5-limit cutoff From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> Ok, I walked into that one by giving fixed bounds on what I wanted. >>> But re. your original suggestion (above), for any fixed version of >>> the formula, more of one *increases* my tolerance for the other. >> >> Nonsense. >> Nonsense, eh? This is pretty much the definition of Max(). It > throws away information on the smaller thing.Yes -- thus more of one has no effect on the tolerance for the other - - it's either the bigger thing, making the tolerance for the other irrelevant anyway, or it's the smaller thing, it which case the tolerance for the other is a constant.> You can tweak your > precious constants after the fact to fix it, but not before the > fact.Isn't this true of any badness criterion? I don't see how one with rectangular contours is suddently any different.

Message: 9715 - Contents - Hide Contents Date: Mon, 02 Feb 2004 03:11:28 Subject: Re: Back to the 5-limit cutoff From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> You should know how to calculate them by now: log(n/d)*log(n*d) > and log(n*d) respectively. You mean log(n/d)/log(n*d)where n:d is the comma that vanishes. I prefer these scalings complexity = lg2(n*d) error = comma_size_in_cents / complexity = 1200 * log(n/d) / log(n*d) My favourite cutoff for 5-limit temperaments is now. (error/8.13)^2 + (complexity/30.01)^2 < 1 This has an 8.5% moat, in the sense that we must go out to (error/8.13)^2 + (complexity/30.01)^2 < 1.085 before we will include another temperament (semisixths). Note that I haven't called it a "badness" function, but rather a "cutoff" function. So there's no need to see it as competing with log-flat badness. What it is competing with is log-flat badness plus cutoffs on error and complexity (or epimericity). Yes it's arbitrary, but at least it's not capricious, thanks to the existence of a reasonable-sized moat around it. It includes the following 17 temperaments. meantone 80:81 augmented 125:128 porcupine 243:250 diaschismic 2025:2048 diminished 625:648 magic 3072:3125 blackwood 243:256 kleismic 15552:15625 pelogic 128:135 6561/6250 6250:6561 quartafifths (tetracot) 19683:20000 negri 16384:16875 2187/2048 2048:2187 neutral thirds (dicot) 24:25 superpythag 19683:20480 schismic 32768:32805 3125/2916 2916:3125 Does this leave out anybody's "must-have"s? Or include anybody's "no-way!"s? The more I think about this sum-of-squares type of cutoff function, the more I think it is the sort of thing I might have suggested a very long time ago if log-flat badness (with error and complexity cutoffs) wasn't being pushed so hard by a certain Erlich (who shall remain nameless). ;-)

Message: 9716 - Contents - Hide Contents Date: Mon, 02 Feb 2004 05:12:56 Subject: Re: Back to the 5-limit cutoff From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>>> I'm arguing that, along this particular line of thinking, >>>>>> complexity does one thing to music, and error another, but >>>>>> there's no urgent reason more of one should limit your >>>>>> tolerance for the other . . . >>>>>>>>>> Taking this to its logical extreme, wouldn't we abandon badness >>>>> alltogether? >>>>> >>>>> -Carl >>>>>>>> No, it would just become 'rectangular', as Dave noted. >>>>>> I didn't follow that. >>>> Your badness function would become max(a*complexity, b*error), thus >> having rectangular contours. >> More of one can here influence the tolerance for the other. Not true.>>> Maybe you could explain how it explains >>> how someone who sees no relation between error and complexity >>> could possibly be interested in badness. >>>> Dave and I abandoned badness in favor of a "moat". >> "Badness" to me is any combination of complexity and error, which > I took your moat to be.No, it's just a single in/out dividing line, which can be taken as a single arbitrary contour of some badness function, but doesn't have to be.

Message: 9717 - Contents - Hide Contents Date: Mon, 02 Feb 2004 06:01:21 Subject: Re: Back to the 5-limit cutoff From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:>> However I'm coming around to thinking that the power of 2 (and the >> resultant meeting the axes at right angles) is far easier to justify >> than anything else. > > How so?It's what you said yesterday (I think). At some point (1 cent, 0.5 cent?) the error is so low and the complexity so high, that any further reduction in error is irrelevant and will not cause you to allow any further complexity. So it should be straight down to the complexity axis from there. Similarly, at some point (10 notes per whatever, 5?) the complexity is so low and the error so high, that any further reduction will not cause you to allow any further error. So it should be straight across to the error axis from there. It also corresponds to mistuning-pain being the square of the error. As you pointed out, that may have just been used by JdL as it is convenient, but don't the bottoms of your HE notches look parabolic? To justify using the square of complexity (as I think Carl suggested) we also have the fact that the number of intervals is O(comp**2).

Message: 9718 - Contents - Hide Contents Date: Mon, 02 Feb 2004 09:48:47 Subject: Re: Back to the 5-limit cutoff From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> Ok, I walked into that one by giving fixed bounds on what I > wanted.>>>> But re. your original suggestion (above), for any fixed version > of>>>> the formula, more of one *increases* my tolerance for the other. >>> >>> Nonsense. >>>> Nonsense, eh? This is pretty much the definition of Max(). It >> throws away information on the smaller thing. >> Yes -- thus more of one has no effect on the tolerance for the other - > - it's either the bigger thing, making the tolerance for the other > irrelevant anyway, or it's the smaller thing, it which case the > tolerance for the other is a constant. >>> You can tweak your >> precious constants after the fact to fix it, but not before the >> fact. >> Isn't this true of any badness criterion? I don't see how one with > rectangular contours is suddently any different.Paul and Carl, I think you're both right. You're just talking about slightly different things. As a function, max(x,y) "depends on" both x and y but at any given point on the "curve" it only "depends on" one of them in the sense that if you take the partial derivatives wrt x and y, one of them will always be zero.

Message: 9719 - Contents - Hide Contents Date: Mon, 02 Feb 2004 05:14:29 Subject: Re: The true top 32 in log-flat? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>>>>> TOP generators [1201.698520, 504.1341314] >>>>>>>>>> So how are these generators being chosen? Hermite? >>>>>>>> No, just assume octave repetition, find the period (easy) and then >>>> the unique generator that is between 0 and 1/2 period. >>>> >>>>> I confess>>>>> I don't know how to 'refactor' a generator basis. >>>>>>>> What do you have in mind? >>>>>> Isn't it possible to find alternate generator pairs that give >>> the same temperament when carried out to infinity? >>>> Yup! You can assume tritave-equivalence instead of octave- >> equivalence, for one thing . . . >> And can doing so change the DES series? > > -CarlWell of course . . . can you think of any octave-repeating DESs that are also tritave-repeating?

Message: 9720 - Contents - Hide Contents Date: Mon, 02 Feb 2004 06:03:45 Subject: Re: The true top 32 in log-flat? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> I'm using your formula from >> >> Yahoo groups: /tuning-math/message/8806 * [with cont.] >> >> but instead of "max", I'm using "sum" . . . >> So these cosmically great answers are coming from the L1 norm applied > to the scaling we got from vals, where we divide by log2(p)'s. What > does that mean, I wonder?The formulas, I think, are basically the same ones being used in the "cross-check" post -- did you have a chance to think about it? Yahoo groups: /tuning-math/message/9052 * [with cont.]

Message: 9721 - Contents - Hide Contents Date: Mon, 02 Feb 2004 09:52:44 Subject: Re: Back to the 5-limit cutoff 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:>>>>> Ok, I walked into that one by giving fixed bounds on what I >> wanted.>>>>> But re. your original suggestion (above), for any fixed version >> of>>>>> the formula, more of one *increases* my tolerance for the other. >>>> >>>> Nonsense. >>>>>> Nonsense, eh? This is pretty much the definition of Max(). It >>> throws away information on the smaller thing. >>>> Yes -- thus more of one has no effect on the tolerance for the other - >> - it's either the bigger thing, making the tolerance for the other >> irrelevant anyway, or it's the smaller thing, it which case the >> tolerance for the other is a constant. >>>>> You can tweak your >>> precious constants after the fact to fix it, but not before the >>> fact. >>>> Isn't this true of any badness criterion? I don't see how one with >> rectangular contours is suddently any different. >> Paul and Carl, > > I think you're both right. You're just talking about slightly > different things. > > As a function, max(x,y) "depends on" both x and y but at any given > point on the "curve" it only "depends on" one of them in the sense > that if you take the partial derivatives wrt x and y, one of them will > always be zero.That's what I was saying. So what was Carl saying?

Message: 9723 - Contents - Hide Contents Date: Mon, 02 Feb 2004 01:54:36 Subject: Re: Back to the 5-limit cutoff From: Carl Lumma>Yes -- thus more of one has no effect on the tolerance for the >other -- it's either the bigger thing, making the tolerance for >the other irrelevant anyway, or it's the smaller thing, it which >case the tolerance for the other is a constant.If you make the bigger one bigger, you're also allowing the smaller one to get bigger without knowing about it. Or maybe I'm misunderstanding "tolerance" here, or the setup of the procedure.>> You can tweak your >> precious constants after the fact to fix it, but not before >> the fact. >>Isn't this true of any badness criterion?Yes, that's why I said someone who wants to change his expectations of error without changing his expectations of complexity shouldn't use badness. -Carl

Message: 9724 - Contents - Hide Contents Date: Mon, 02 Feb 2004 05:15:22 Subject: Re: 7-limit horagrams From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>> Beautiful! I take it the green lines are proper scales? >>>>> >>>>> -C. >>>>>>>> Guess again (it's easy)! >>>>>> Obviously not easy enough if we've had to exchange three >>> messages about it. >>> >>> -Carl >>>> Then you can't actually be looking at the horagrams ;) >> Why not just explain things rather than riddling your users?Because I'm trying to encourage some looking.

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