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Message: 9650 - Contents - Hide Contents Date: Sun, 01 Feb 2004 05:47:06 Subject: Re: Back to the 5-limit cutoff (was: 60 for Dave) From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: >>> Can you easily re-plot dave3.jpg showing a quarter-ellipse >> >> (k*err)^2 + comp^2 = x^2 >> >> that passes thru 24;25 (neutral thirds) and 78125;78732 > (semisixths)? >>>> This is not intended to represent the centre of the moat but one > edge >> of it. >> Tough assignment . . . you?I don't have the data to plot them all. If you just want me to give you k and x I'll only need the error and complexity figures for these two temperaments. Are they still the same as used on dave3.jpg, or are you about to change them?>> I disagree. I think people will only tolerate more complexity if it >> gives them less errolr and vice versa. >> 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.Very true. So that makes the quadratic (elliptical) case look good> Nevertheless, I think we should use something > close to a straight line, slightly convex or concave to best fit > a "moat".Sounds good to me. So we can parameterise it as (err/max_err)^p + (comp/max_comp)^p < 1, where 0.5<=p<=2.
Message: 9651 - Contents - Hide Contents Date: Sun, 01 Feb 2004 19:17:34 Subject: Re: Back to the 5-limit cutoff From: Carl Lumma>>> >'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. Maybe you could explain how it explains how someone who sees no relation between error and complexity could possibly be interested in badness. -Carl
Message: 9652 - Contents - Hide Contents Date: Sun, 01 Feb 2004 05:57:51 Subject: Re: The true top 32 in log-flat? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> There's something VERY CREEPY about my complexity values. I'm going >> to have to accept this as *the* correct scaling for complexity (I'm >> already convinced this is the correct formulation too, i.e. L_1 norm, >> for the time being) . . . > ...>> Creepy, isn't it? > > Woah! Yes.So when I explained this as the number of notes in the bivector, I must have been onto something. It's interesting how the *proper* scales are "attractors" for this measure, but improper DEs don't seem to be at all -- horagrams about to be uploaded . . .
Message: 9653 - Contents - Hide Contents Date: Sun, 01 Feb 2004 19:18:43 Subject: Re: The true top 32 in log-flat? From: Carl Lumma>>>>> >OP 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? -Carl
Message: 9654 - Contents - Hide Contents Date: Sun, 01 Feb 2004 05:59:56 Subject: Re: Back to the 5-limit cutoff (was: 60 for Dave) 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 "Dave Keenan" <d.keenan@b...> >> wrote: >>>>> Can you easily re-plot dave3.jpg showing a quarter-ellipse >>> >>> (k*err)^2 + comp^2 = x^2 >>> >>> that passes thru 24;25 (neutral thirds) and 78125;78732 >> (semisixths)? >>>>>> This is not intended to represent the centre of the moat but one >> edge >>> of it. >>>> Tough assignment . . . you? >> I don't have the data to plot them all. If you just want me to give > you k and x I'll only need the error and complexity figures for these > two temperaments. > > Are they still the same as used on dave3.jpg, Yes. > or are you about to > change them?No. You should know how to calculate them by now: log(n/d)*log(n*d) and log(n*d) respectively.
Message: 9655 - Contents - Hide Contents Date: Sun, 01 Feb 2004 19:20:18 Subject: Re: 7-limit horagrams From: Carl Lumma>> >eautiful! 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
Message: 9656 - Contents - Hide Contents Date: Sun, 01 Feb 2004 06:12:31 Subject: 7-limit horagrams From: Paul Erlich Yahoo groups: /tuning_files/files/Erlich/seven... * [with cont.]
Message: 9657 - Contents - Hide Contents Date: Sun, 01 Feb 2004 19:23:15 Subject: Re: 114 7-limit temperaments From: Carl Lumma>>> >uch distinctions may be important for *scales*, but for >>> temperaments, I'm perfectly happy not to have to worry about >>> them. Any reasons I shouldn't be? >>>> You're using temperaments to construct scales, aren't you? >>Not necessarily -- they can be used directly to construct music, >mapped say to a MicroZone or a Z-Board. > > * [with cont.] (Wayb.)??? Doing so creates a scale. -Carl
Message: 9658 - Contents - Hide Contents Date: Sun, 01 Feb 2004 06:17:17 Subject: Re: I guess Pajara's not #2 From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Please remember to post the results as a delimited table, with one > temperament per row. That way, most of us will be able to graph them > easily.You'd better give a sample of what you want; I'll see if Maple can accomodate you.
Message: 9659 - Contents - Hide Contents Date: Sun, 01 Feb 2004 19:26:27 Subject: Re: 114 7-limit temperaments From: Carl Lumma>>> >t 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. Ok.>> Thus if consonance really *does* >> deteriorate at the same rate for all ratios as Paul claims, >>Where did I claim that?In your decatonic paper you say the consonance deteriorates 'at least as fast', and opt to go sans weighting, IIRC. -Carl
Message: 9660 - Contents - Hide Contents Date: Sun, 01 Feb 2004 21:21:54 Subject: Re: Weighting From: Carl Lumma>>> >o 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,You may leave the minimum of the putative ratio.>and what are these extra-JI effects?Even in "pure JI" you can hit entropy maxima. -C.
Message: 9661 - Contents - Hide Contents Date: Sun, 01 Feb 2004 06:16:34 Subject: Re: The true top 32 in log-flat? From: Paul Erlich --- 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 "Paul Erlich" <perlich@a...> > wrote:>>> There's something VERY CREEPY about my complexity values. I'm > going>>> to have to accept this as *the* correct scaling for complexity > (I'm>>> already convinced this is the correct formulation too, i.e. L_1 > norm,>>> for the time being) . . . >> ...>>> Creepy, isn't it? >> >> Woah! Yes. >> So when I explained this as the number of notes in the bivector, I > must have been onto something. It's interesting how the *proper* > scales are "attractors" for this measure, but improper DEs don't seem > to be at allExcept Blackjack . . .
Message: 9662 - Contents - Hide Contents Date: Sun, 01 Feb 2004 21:23:56 Subject: Re: Back to the 5-limit cutoff From: Carl Lumma>>>>>>> >'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? >>>>>>>>> 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.Actually what are a and b? But Yes, true. Increasing my tolerance for complexity simultaneously increases my tolerance for error, since this is Max(). -Carl
Message: 9663 - Contents - Hide Contents Date: Sun, 01 Feb 2004 06:23:23 Subject: Re: 7-limit horagrams From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: I dumped the awful name "quartiminorthirds" in favor of "valentine", which you may recall from over a year ago: Yahoo groups: /tuning-math/files/Paul/dualzoom... * [with cont.]
Message: 9664 - Contents - Hide Contents Date: Sun, 01 Feb 2004 21:25:59 Subject: Re: The true top 32 in log-flat? From: Carl Lumma>>>>>>>>> >OP 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? >>Well of course . . . can you think of any octave-repeating DESs >that are also tritave-repeating?Right, so when trying to explain a creepy coincidence between complexity and DES cardinalities, might not we take this into account? -Carl
Message: 9665 - Contents - Hide Contents Date: Sun, 01 Feb 2004 07:11:51 Subject: Re: I guess Pajara's not #2 From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> Please remember to post the results as a delimited table, with one >> temperament per row. That way, most of us will be able to graph them >> easily. >> You'd better give a sample of what you want; I'll see if Maple can > accomodate you.[[w, e, d, g, i, e] error] works fine for me, I can calculate the complexity . . . [[6, -7, -2, -25, -20, 15] .630134] [[1, 4, 10, 4, 13, 12] 1.698521] [[5, 1, 12, -10, 5, 25] 1.276744] [[2, -4, -4, -11, -12, 2] 3.106578] [[4, 4, 4, -3, -5, -2] 5.871540]
Message: 9666 - Contents - Hide Contents Date: Sun, 01 Feb 2004 05:25:03 Subject: Re: Back to the 5-limit cutoff From: Carl Lumma>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 ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9667 - Contents - Hide Contents Date: Sun, 01 Feb 2004 21:28:20 Subject: Re: 7-limit horagrams From: Carl Lumma>>>>>> >eautiful! I take it the green lines are proper scales? >>>>>>>>>> Guess again (it's easy)! >>>>>>>> Obviously not easy enough if we've had to exchange three >>>> messages about it. >>>>>> 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.I've tested several possibilities about what the green could mean, and your continued refusal to simply provide the answer is assinine, with a double s. -Carl ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9668 - Contents - Hide Contents Date: Mon, 02 Feb 2004 00:46:56 Subject: Re: Back to the 5-limit cutoff From: Carl Lumma>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.Picking a single point is hard. It should be asymptotic.>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.I'd say 5, and yes both of these suggestions make sense.>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). Right on! -Carl
Message: 9669 - Contents - Hide Contents Date: Mon, 02 Feb 2004 03:55:08 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? > > -CarlYup! You can assume tritave-equivalence instead of octave- equivalence, for one thing . . .
Message: 9670 - Contents - Hide Contents Date: Mon, 02 Feb 2004 23:36:48 Subject: Re: finding a moat in 7-limit commas a bit tougher . . . From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: And you can safely cut it off above 25/24. Since this was marginal as a 5-limit linear temperament it isn't going to fare any better as a 7-limit planar merely by adding some just ratio of 7 as a second generator.
Message: 9671 - Contents - Hide Contents Date: Mon, 02 Feb 2004 23:54:27 Subject: Re: 114 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Me:>>> If you don't >>> want more than 18 notes in your scale, miracle is a contender in >> the>>> 7-limit but not the 9-limit. And if you don't want errors more >> than 6>>> cents, you can use meantone in the 7-limit but not the 9-limit. > > Paul E:>> What if you don't assume total octave-equivalence? >> I don't think it matters.What do you mean? The 7-limit and 9-limit you refer to assume total octave-equivalence in the way you evaluate intervals.>> In the Tenney-lattice view of harmony, 'limit' and chord structure is >> a more fluid concept. >> Yes, that's the problem.Seems like a feature, not a bug, to me.>>> 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. >>>> Number of consonances? >> For minimax error with equal weighting, you share the comma among all > the constituent intervals in the given limit. But it's ambiguous how > you do the counting. If you take 28:27, that has odd factors > > 7:3*3*3 > > So in the 9-limit that gives two intervals, which could be 7:6 and 8:9 > or 7:9 and 2:3. > > If you give 9 half the error, that means 3 has 1/4 of it. For 7:6 to > have half, 7 must take the other quarter in the other direction. That > means 7:9 has three quarters of the comma, but no 9-limit interval is > supposed to have more than half of it. > > If you give 2:3 half the error, then 9:8 must take the whole error, > which is already wrong. > > So the right thing is to treat 3 as having a weighting of half. Then, > the complexity is 1.5 and the minimax should be 2/3 of the original > comma. And that worst interval will be 9:8, or anything else with a 9 > in it.We've come up with slight refinements of this idea, if you followed the posts where I gave Gene some error weightings and he described the geometry of the corresponding dual lattices.>>> I'm expecting the limit of this calculation as the odd limit tends >> to>>> infinity will be the same as this Kees metric. >> >>>> Can you clarify which calculation and which Kees metric you're >> talking about? >> The calculation of sharing the comma to give an odd-limit minimax, and > the metric taking the logarithm of the (larger) odd number in the ratio.I wish Gene would turn his attention to this . . .>> I still remain unclear on what you were doing with your octave- >> equivalent TOP stuff. Gene ended up interested in the topic later but >> you missed each other. I rediscovered your 'worst comma in 12- equal' >> when playing around with "orthogonalization" and now figure I must >> have misunderstood your code. You weren't searching an infinite >> number of commas, but just three, right? >> I was searching a large number but not infinite. And this was to solve > a lower dimensioned temperament, nothing to do with octave equivalence.I know, I was talking about two different things above.> I don't know how Gene's doing it, but I thought it was some numerical > method to calculate the weighted minimax directly.Yes, linear programming, which searches 2^n possibilities. I gave my method here: Yahoo groups: /tuning-math/message/8512 * [with cont.]> The octave-equivalent TOP is easy -- you use the same weighting formula, > but using the log of the larger odd number of n and d, instead of n*d.I know this is what you suggested, but I'm not sure how it's justified. Certainly, it's far less clear than TOP itself. Gene, if you can, would you look at this?> I don't think TOP really favors simple ratios at all. The weighting is > only reflecting the mathematics of composite numbers, which ensure that > simple intervals will have a lower error. The result will be close to > some average of the odd (or Farey) limits that could apply to that prime > limit. So it isn't a fully general case, but is a convenient > approximation where it's easier to calculate than the true odd limit > optimum.It does better than any odd-limit formulation since, in truth, the fifth is more consonant than the fourth, etc. etc.
Message: 9672 - Contents - Hide Contents Date: Mon, 02 Feb 2004 00:52:55 Subject: Re: Weighting From: Carl Lumma>> >e 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. >>Yes. I can agree to all that.Yes, it's unfortunately very hard to see how my recent suggestion of using harmonic entropy directly (instead of error) could work. The entropies are unlikely to change in the same direction as you resize a given axis. Therefore I retract the suggestion. -Carl
Message: 9673 - Contents - Hide Contents Date: Mon, 02 Feb 2004 03:56:15 Subject: Re: 114 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> Such distinctions may be important for *scales*, but for >>>> temperaments, I'm perfectly happy not to have to worry about >>>> them. Any reasons I shouldn't be? >>>>>> You're using temperaments to construct scales, aren't you? >>>> Not necessarily -- they can be used directly to construct music, >> mapped say to a MicroZone or a Z-Board. >> >> * [with cont.] (Wayb.) >> ??? Doing so creates a scale. > > -CarlA 108-tone scale?
Message: 9674 - Contents - Hide Contents Date: Mon, 02 Feb 2004 05:31:51 Subject: Re: Weighting From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:>> So surely he means that as >> the numbers in the ratio get larger, the width of the field of >> attraction gets smaller. > > Yes. >>> To me, that's an argument for why TOP isn't necessarily what you >> want. >> Why, if this only addresses complexity and ignores tolerance? Partch > isn't expressing his views on tolerance/mistuning here. > > And while the Farey or whatever series that are used to calculate > harmonic entropy follow this same observation if one equates "field > of attraction" with "interval between it and adjacent ratios", the > harmonic entropy that comes out of this shows that simpler ratios are > most sensitive to mistuning, precisely because their great consonance > arises from this very remoteness from neighbors, a unique property > that rapidly subsides as one shifts away from the correct tuning.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.
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