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Message: 9550 - Contents - Hide Contents Date: Fri, 30 Jan 2004 15:24:40 Subject: Re: 114 7-limit temperaments From: Carl Lumma>> >hat I remember we gave you a hard time about, was not that linear >> temperaments are 2-dimensional without octave-equivalence, but that >> you wanted to call them "planar" (which would have been too >> confusing a departure from the historical usage). We wanted "Linear >> temperament" to be the constant name of the musical object which >> remains essentially the same while its mathematical >> models vary in dimensionality. >>Unfortunately for us, 'linear temperament' has probably never >referred to a multiple-chains-per-octave system (like pajara, >diminished, augmented, ennealimmal . . .) before we started using it >that way, and some of the original users of the term (say, Erv >Wilson) might be rather upset with this slight generalization.I can't remember Erv ever using the term, and if he had, I can't imagine him getting upset (by any stretch of the word) over something like this! -Carl
Message: 9551 - Contents - Hide Contents Date: Fri, 30 Jan 2004 23:59:43 Subject: Re: pelogic and kleismic/hanson From: Herman Miller On Sat, 31 Jan 2004 00:37:28 -0000, "Paul Erlich" <perlich@xxx.xxxx.xxx> wrote:>See > >http://www.anaphoria.com/keygrid.PDF - Type Ok * [with cont.] (Wayb.) > >page 7 seems to be using some pelog terminology; anyone familiar with >it?I'm not familiar with this terminology, but the keyboard is clearly based on a generator of 5 steps of 23-ET, while the generator of pelogic temperament is 10 steps of 23. In other words, this is the scale I've been calling "superpelog", with the basic 9-note MOS subset used as the basis for a system of notation. I'll add a reference to this paper on my superpelog page: Superpelog tuning * [with cont.] (Wayb.) -- see my music page ---> ---<The Music Page * [with cont.] (Wayb.)>-- hmiller (Herman Miller) "If all Printers were determin'd not to print any @io.com email password: thing till they were sure it would offend no body, \ "Subject: teamouse" / there would be very little printed." -Ben Franklin
Message: 9552 - Contents - Hide Contents Date: Fri, 30 Jan 2004 15:34:01 Subject: Re: 60 for Dave From: Carl Lumma>True. Even though it's out-Keenaning Keenan with respect to Smith, I >still think a straight line -- if not a *convex* curve, perish the >thought -- makes some sense. Both error and complexity are things >that we typically judge and compare in a *linear* fashion, so >performing various operations on them seems arbitrary at best. At >least, it seems that if there's zero error, doubling the complexity >should double the badness; and if there's zero complexity, doubling >the error should double the badness.John deLaubenfels seemed to feel that error is perceived quadratically, and this is the way he implemented error pain at one point in his software. Complexity, I should think, should definitely be punished more than 1:1. Using 8 notes instead of 7 notes would seem to demand more than eight 7ths the mental energy. -Carl
Message: 9553 - Contents - Hide Contents Date: Fri, 30 Jan 2004 22:24:45 Subject: Re: pelogic and kleismic/hanson From: Carl Lumma>> >ee >> >> http://www.anaphoria.com/keygrid.PDF - Type Ok * [with cont.] (Wayb.) >> >> page 7 seems to be using some pelog terminology; anyone familiar with >> it? >>I'm not familiar with this terminology, but the keyboard is clearly >based on a generator of 5 steps of 23-ET, while the generator of pelogic >temperament is 10 steps of 23. In other words, this is the scale I've >been calling "superpelog", with the basic 9-note MOS subset used as the >basis for a system of notation. > >I'll add a reference to this paper on my superpelog page: > >Superpelog tuning * [with cont.] (Wayb.)Be careful; Erv does not view these as ETs. He sees them as general- purpose cycles, which may be interpreted as ETs if you like, even though he doesn't usually like. -Carl
Message: 9554 - Contents - Hide Contents Date: Fri, 30 Jan 2004 23:41:50 Subject: Re: 114 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> What I remember we gave you a hard time about, was not that linear >>> temperaments are 2-dimensional without octave-equivalence, but that >>> you wanted to call them "planar" (which would have been too >>> confusing a departure from the historical usage). We wanted "Linear >>> temperament" to be the constant name of the musical object which >>> remains essentially the same while its mathematical >>> models vary in dimensionality. >>>> Unfortunately for us, 'linear temperament' has probably never >> referred to a multiple-chains-per-octave system (like pajara, >> diminished, augmented, ennealimmal . . .) before we started using it >> that way, and some of the original users of the term (say, Erv >> Wilson) might be rather upset with this slight generalization. >> I can't remember Erv ever using the term, and if he had, I can't > imagine him getting upset (by any stretch of the word) over something > like this! > > -CarlWell, a quick look shows that he used the terms "linear mapping", "linear scale", "linear notation", etc., in a way that almost certainly assumes *one* and only one chain of octave- equivalent pitch-classes. On page 5 of http://www.anaphoria.com/xen3a.PDF - Type Ok * [with cont.] (Wayb.) he mentions linear scales/notations of 5, 7, 8, 9, 11, 12, and 13 elements, generated by all the possible generators between 1/2 and 1/3 octave, and goes on to say that he has yet to consider linear systems which would be generated by a half-fifth or half-fourth, but the idea of *multiple* chains would not seem to fit into his rubric here. By the way, some of the simplest multiple-chain systems, augmented and diaschsimic/pajara, would immediately yield the "missing" numbers in the list -- 6 and 10, respectively. Given the clarification that I got on Wilson's MOS concept from Daniel, Kraig, and others recently, which prompted me to start using the "DE" terminology, and given his writings, I suspect (very strongly) that his linear temperament concept is similar.
Message: 9555 - Contents - Hide Contents Date: Fri, 30 Jan 2004 02:34:00 Subject: Re: 60 for Dave (was: 41 "Hermanic" 7-limit linear temperaments) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> I think Gene may be using the wrong norm to get his complexity > values. I'll wait until I'm sure they're right or corrected.What's the defintion of "right"?
Message: 9556 - Contents - Hide Contents Date: Fri, 30 Jan 2004 23:43:21 Subject: Crunch algorithm From: Gene Ward Smith Suppose we have a list of rational numbers greater than one, sorted in ascending order, the elements of which should be independent. Define the crunch function as follows: take the quotient of the last with the next-to-last, and then place it in its proper location in the list, and return the new sorted list. If we start from an independent set of numbers, and in particular a basis for the p-limit, we crunch down to a basis with smaller elements. Here is crunch, starting out on [2, 3, 5, 7]. The next column is the top row of the inverse matrix for the monzos, which is a val matrix; so these are scale divisions. I thought of this because Oldlyzko pointed me to an unpublished paper of his, which suggested the problem of finding very large scale divisions (in the context of the Riemann zeta function, not music!) is harder than I believe it to be. I think I'll send him something and mention Paul's discovery that 2401/2400 dominantes up to 100000 or so for 7-limit divisions, translated of course into zeta language. [2, 3, 5, 7] [1, 0, 0, 0] [7/5, 2, 3, 5] [0, 1, 0, 0] [7/5, 5/3, 2, 3] [0, 0, 1, 0] [7/5, 3/2, 5/3, 2] [0, 0, 0, 1] [6/5, 7/5, 3/2, 5/3] [1, 0, 0, 1] [10/9, 6/5, 7/5, 3/2] [1, 1, 0, 1] [15/14, 10/9, 6/5, 7/5] [1, 1, 1, 1] [15/14, 10/9, 7/6, 6/5] [1, 1, 1, 2] [36/35, 15/14, 10/9, 7/6] [2, 1, 1, 3] [36/35, 21/20, 15/14, 10/9] [2, 3, 1, 4] [36/35, 28/27, 21/20, 15/14] [2, 4, 3, 5] [50/49, 36/35, 28/27, 21/20] [5, 2, 4, 8] [81/80, 50/49, 36/35, 28/27] [8, 5, 2, 12] [245/243, 81/80, 50/49, 36/35] [12, 8, 5, 14] [126/125, 245/243, 81/80, 50/49] [14, 12, 8, 19] [4000/3969, 126/125, 245/243, 81/80] [19, 14, 12, 27] [19683/19600, 4000/3969, 126/125, 245/243] [27, 19, 14, 39] [4375/4374, 19683/19600, 4000/3969, 126/125] [39, 27, 19, 53] [250047/250000, 4375/4374, 19683/19600, 4000/3969] [53, 39, 27, 72] [250047/250000, 4375/4374, 1600000/1594323, 19683/19600] [53, 39, 72, 99] [53, 39, 99, 171] [53, 39, 270, 171] [53, 39, 441, 171] [53, 39, 612, 171] [53, 39, 783, 171] [53, 171, 39, 954] [53, 171, 993, 954] [53, 171, 954, 1947] [1947, 53, 171, 2901] [1947, 2901, 53, 3072] [3072, 1947, 2901, 3125] [3072, 1947, 6026, 3125] [3072, 1947, 9151, 3125] [3072, 1947, 12276, 3125] [3072, 1947, 15401, 3125] [3072, 1947, 18526, 3125] [3072, 1947, 21651, 3125] [3072, 1947, 24776, 3125] [3072, 1947, 27901, 3125] [3072, 1947, 31026, 3125]
Message: 9557 - Contents - Hide Contents Date: Fri, 30 Jan 2004 02:36:22 Subject: Re: 60 for Dave (was: 41 "Hermanic" 7-limit linear temperaments) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> Well something's wrong. Whether its the badness functions or only the > complexity I don't know. But Diaschismic shouldn't be so far down. I > don't think Miracle should be so far down either. Sure it gets a hit > for having 6 gens to the fifth, but not that much of a hit I would think.It gets an even harder hit for making the major sixth so complex, which seems fair enough.> And where's Shrutar?Far down the list somewhere.
Message: 9558 - Contents - Hide Contents Date: Fri, 30 Jan 2004 23:49:07 Subject: Re: 60 for Dave From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> True. Even though it's out-Keenaning Keenan with respect to Smith, I >> still think a straight line -- if not a *convex* curve, perish the >> thought -- makes some sense. Both error and complexity are things >> that we typically judge and compare in a *linear* fashion, so >> performing various operations on them seems arbitrary at best. At >> least, it seems that if there's zero error, doubling the complexity >> should double the badness; and if there's zero complexity, doubling >> the error should double the badness. >> John deLaubenfels seemed to feel that error is perceived >quadratically, > and this is the way he implemented error pain at one pointAt every point.> in his > software.That's mainly because the quadratic function was by far the easiest one to implement in software! But I am not at all adverse to assuming a quadratic penatly on error.> Complexity, I should think, should definitely be punished more than > 1:1. Using 8 notes instead of 7 notes would seem to demand more > than eight 7ths the mental energy.We could use a quadratic penalty on the complexity too. But now we're talking about *convex* badness contours, while Dave and especially Gene were proposing *concave* ones (Dave suggested using k*sqrt (error) + sqrt(complexity)). I think the difficulty with convex badness contours is that they imply a rather sudden cutoff for some maximum error and some maximum complexity. This is, indeed, probably a very accurate reflection of what's relevant for a particular musician working in a particular style. Against this, though, is the idea that different musicians may have different needs as regards lowness of complexity and lowness of error, so superimposing their individually-convex badness criteria could lead to a 'global' badness criterion that isn't convex. Taking this idea to its logical extreme leads to things like log-flat badness.
Message: 9559 - Contents - Hide Contents Date: Fri, 30 Jan 2004 02:42:07 Subject: Re: Characteristic polynomials and inverse matriies of interval matricies From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> I would be interested in learning how to derive characteristic > polynomials from interval matrices or scales. Thanks!It's just the usual characteristic polynomial of a matrix, but unfortunately does not seem to be telling us anything new.
Message: 9560 - Contents - Hide Contents Date: Fri, 30 Jan 2004 15:51:40 Subject: Re: 114 7-limit temperaments From: Carl Lumma>>> >nfortunately for us, 'linear temperament' has probably never >>> referred to a multiple-chains-per-octave system (like pajara, >>> diminished, augmented, ennealimmal . . .) before we started using >>> it that way, and some of the original users of the term (say, Erv >>> Wilson) might be rather upset with this slight generalization. >>>> I can't remember Erv ever using the term, and if he had, I can't >> imagine him getting upset (by any stretch of the word) over >> something like this! >>Well, a quick look shows that he used the terms "linear >mapping", "linear scale", "linear notation", etc., in a way that >almost certainly assumes *one* and only one chain of octave- >equivalent pitch-classes.But not a chain of one and only one generator.>he mentions linear scales/notations of 5, 7, 8, 9, 11, 12, and 13 >elements, generated by all the possible generators between 1/2 and >1/3 octave, and goes on to say that he has yet to consider linear >systems which would be generated by a half-fifth or half-fourth, but >the idea of *multiple* chains would not seem to fit into his rubric >here.But he doesn't think of these as temperaments.>Given the clarification that I got on Wilson's MOS concept from >Daniel, Kraig, and others recently, which prompted me to start using >the "DE" terminology, and given his writings, I suspect (very >strongly) that his linear temperament concept is similar.His concept of temperament is that "I would almost never do it". In any case, if you haven't met Erv, you might not realize that he is about as non-committal on terminology and "suspicious of language" as it's possible to get. -Carl
Message: 9561 - Contents - Hide Contents Date: Fri, 30 Jan 2004 23:01:20 Subject: Re: 60 for Dave From: Carl Lumma>>> >o that he could understand Gene's badness and my linear badness >>> in the same form, and propose a compromise. >>>> Ah. Is yours the one from the Attn: Gene post? >>No, it was the toy "Hermanic" example.This, I guess: "I thought I'd cull the list of 114 by applying a more stringent cutoff of 1.355*comp + error < 10.71. This is an arbitrary choice among the linear functions of complexity and error that could be chosen" You don't say what kind of comp and error you're using.>> That's good to know, but the above is just my value judgement, and >> as you point out log-flat badness frees us from those, in a sense. >>But it results in an infinite number of temperaments, or none at all, >depending on what level of badness you use as your cutoff....as I was trying to complain recently, when I said I'd be a lot more impressed if it didn't need cutoffs. -Carl
Message: 9562 - Contents - Hide Contents Date: Fri, 30 Jan 2004 02:45:08 Subject: Re: 114 7-limit temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> We've always said linear temperaments were actually 2-dimensional, so > equal temperaments are 1-dimensional, and are generated by their > step. Remember we've dropped octave-equivalence to get TOP.You boiled me in oil and rendered me down for lard when I first got here for wanting things this way, and now you claim we've always said it? :)> Could you take another look at the "Attn: Gene 2" post and explain > what's going on there, mathematically? I didn't use the maximum but > that was only 3-limit . . .I could but it might be one of those posts where I don't know what you are really asking for.
Message: 9563 - Contents - Hide Contents Date: Fri, 30 Jan 2004 23:59:08 Subject: Re: 114 7-limit temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> Unfortunately for us, 'linear temperament' has probably never >>>> referred to a multiple-chains-per-octave system (like pajara, >>>> diminished, augmented, ennealimmal . . .) before we started using >>>> it that way, and some of the original users of the term (say, Erv >>>> Wilson) might be rather upset with this slight generalization. >>>>>> I can't remember Erv ever using the term, and if he had, I can't >>> imagine him getting upset (by any stretch of the word) over >>> something like this! >>>> Well, a quick look shows that he used the terms "linear >> mapping", "linear scale", "linear notation", etc., in a way that >> almost certainly assumes *one* and only one chain of octave- >> equivalent pitch-classes. >> But not a chain of one and only one generator.Huh? How not??>> he mentions linear scales/notations of 5, 7, 8, 9, 11, 12, and 13 >> elements, generated by all the possible generators between 1/2 and >> 1/3 octave, and goes on to say that he has yet to consider linear >> systems which would be generated by a half-fifth or half-fourth, but >> the idea of *multiple* chains would not seem to fit into his rubric >> here. >> But he doesn't think of these as temperaments.When he does talk about some of these as temperaments (or the related just intonation structures with an undistributed commatic unison vector), though, they're all single-chain.
Message: 9564 - Contents - Hide Contents Date: Fri, 30 Jan 2004 02:46:28 Subject: Re: Graef article on rationalization of scales From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Pretty much. We've been through this before, whereupon Gene > defined "block" for convex PB and "semiblock" for something "not too > concave" or something, if my vague recollection is reliable.But we never settled anything, did we?
Message: 9565 - Contents - Hide Contents Date: Fri, 30 Jan 2004 16:04:43 Subject: Re: 114 7-limit temperaments From: Carl Lumma>> >ut not a chain of one and only one generator. >>Huh? How not??Because he doesn't temper, the generator varies in size depending on where you are in the map.>When he does talk about some of these as temperaments (or the related >just intonation structures with an undistributed commatic unison >vector), though, they're all single-chain...of a particular generator in scale steps, not interval size. -Carl
Message: 9566 - Contents - Hide Contents Date: Fri, 30 Jan 2004 02:53:06 Subject: Re: What the numbers mean From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> The complexity is defined by the weighted value for 5/3, which is > the>> worst case (as, in fact, it clearly is.) >> Meaning what, exactly?It's hard to find a reasonable take on septimal miracle which doesn't have it that 5/3 is the most complex consonance.> Can you show this process in action for the simpler, 3-limit and 5- > limit cases? And why do we take the worst case, instead of some sort > of product (which would appear to get you your spacial measure once > you've orthogonalized)?"Orthogonalized" is one of those words which is holding up communication, as I can only interpret that in terms of a L2 norm. The vals, if we assume the Tenney metric, have an L_inf norm, and I am regarding wedgies as multivals, hence the L_inf norm. It's logical from a mathematical point of view, but not something we are forced to beleive and adopt.
Message: 9567 - Contents - Hide Contents Date: Fri, 30 Jan 2004 02:55:59 Subject: Re: What the numbers mean From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> You are dividing by the _product_ of the two logs, shouldn't you be > dividing by their _sum_ (the log of the product of the primes)?No, not if we are using the dual norm induced on vals by the Tenney norm. We can define that as normalizing the vals by dividing through by log2(p); if we take products of pairs of these, which we do in finding the wedgie, we end up dividing by the products of pairs of primes.
Message: 9568 - Contents - Hide Contents Date: Fri, 30 Jan 2004 16:28:05 Subject: Re: 114 7-limit temperaments From: Carl Lumma>>>> >ut not a chain of one and only one generator. >>>>>> Huh? How not?? >>>> Because he doesn't temper, the generator varies in size >> depending on where you are in the map. >>That's what I meant by an undistributed commatic unison vector. But >there's still one and only one chain, in contradistinction with >pajara, augmented, diminished, ennealimmal, etc. . . . which was my >point.Yes, it's long been agreed that these have historically been missed. -Carl
Message: 9569 - Contents - Hide Contents Date: Fri, 30 Jan 2004 16:31:01 Subject: Re: 60 for Dave From: Carl Lumma>But I am not at all adverse to assuming a quadratic penatly on error. >>> Complexity, I should think, should definitely be punished more than >> 1:1. Using 8 notes instead of 7 notes would seem to demand more >> than eight 7ths the mental energy. >>We could use a quadratic penalty on the complexity too.If we square both terms, doesn't this give the same ranking? In March '02 I wrote, "I think I'd rather have a smooth pain function, like ms, and a stronger exponent on complexity.">But now we're talking about *convex* badness contours, while Dave >and especially Gene were proposing *concave* ones (Dave suggested >using k*sqrt (error) + sqrt(complexity)). I think the difficulty with >convex badness contours is that they imply a rather sudden cutoff >for some maximum error and some maximum complexity. This is, indeed, >probably a very accurate reflection of what's relevant for a >particular musician working in a particular style. Against this, >though, is the idea that different musicians may have different needs >as regards lowness of complexity and lowness of error, so >superimposing their individually-convex badness criteria could lead >to a 'global' badness criterion that isn't convex. Taking this idea >to its logical extreme leads to things like log-flat badness.Right. I can live with log-flat badness. By the way, when doing ms error, if an error is less than a cent it will get *smaller* when squared. Do you see this as a good thing, should we be ceilinging these to 1 before squarring, or...? -Carl
Message: 9570 - Contents - Hide Contents Date: Fri, 30 Jan 2004 08:26:13 Subject: Re: Attn: Gene 2 From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Again, it would really help if he went > through my original post and showed, step by step, how all the > results there generalize to a single formula.Which post, what results? ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9571 - Contents - Hide Contents Date: Fri, 30 Jan 2004 16:37:59 Subject: Re: 114 7-limit temperaments From: Carl Lumma>> > can't remember Erv ever using the term, >I didn't remember it. :) He seems to be arguing strongly here for temperament, but he either changed his mind before I met with him or he was using the term here to mean 'mapping to an MOS' rather than actual temperament. You can see from his later stuff that he lists both pitches when they coincide on a single key, and when I asked him about how he'd tune that, he said one or the other could be used in both contexts (a wolfish sort of temperament!) or an off-keyboard switch could be used to select between them, but never would he use a single averaged pitch. -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: 9572 - Contents - Hide Contents Date: Sat, 31 Jan 2004 00:42:53 Subject: Re: 60 for Dave From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> But I am not at all adverse to assuming a quadratic penatly on error. >>>>> Complexity, I should think, should definitely be punished more than >>> 1:1. Using 8 notes instead of 7 notes would seem to demand more >>> than eight 7ths the mental energy. >>>> We could use a quadratic penalty on the complexity too. >> If we square both terms, doesn't this give the same ranking?No, because following Dave, we're adding the terms, not multiplying them. Dave restated Gene's product as a sum of logs.> In March '02 I wrote, "I think I'd rather have a smooth pain function, > like ms, and a stronger exponent on complexity."In response to what?>> But now we're talking about *convex* badness contours, while Dave >> and especially Gene were proposing *concave* ones (Dave suggested >> using k*sqrt (error) + sqrt(complexity)). I think the difficulty with >> convex badness contours is that they imply a rather sudden cutoff >> for some maximum error and some maximum complexity. This is, indeed, >> probably a very accurate reflection of what's relevant for a >> particular musician working in a particular style. Against this, >> though, is the idea that different musicians may have different needs >> as regards lowness of complexity and lowness of error, so >> superimposing their individually-convex badness criteria could lead >> to a 'global' badness criterion that isn't convex. Taking this idea >> to its logical extreme leads to things like log-flat badness. >> Right. I can live with log-flat badness.Yecchhh . . .> By the way, when doing ms error, if an error is less than a cent > it will get *smaller* when squared.No, you can't compare cents to cents-squared. These quantities do not have the same dimension.> Do you see this as a good > thing, should we be ceilinging these to 1 before squarring, or...?To 1 cent? Definitely not -- there's no justification for treating 1 cent as a special error size. Besides, the errors Gene gave are only in units of cents if you're looking at the error of the octave -- other intervals have different units, since it's minimax Tenney-weighed error we're looking at.
Message: 9573 - Contents - Hide Contents Date: Sat, 31 Jan 2004 10:12:28 Subject: Re: the choice of wedgie-norm greatly impacts miracle's ranking From: Paul Erlich oops -- I may have done that all wrong. The scaling factors for the elements of the wedgie, the ones that you divide by to calculate the multival norm -- do you have to *multiply* by them when you calculate the multimonzo norm? --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> This time I'll try L_1 (multimonzo interpretation?) instead of > L_infinity (multival interpretation?) to get complexity from the > wedgie. Let's see how it affects the rankings -- we don't need to > worry about scaling because Gene's badness measure is > multiplicative . . . > > The top 10 get re-ordered as follows, though this is probably not the > new top 10 overall . . . > > 1.>> Number 1 Ennealimmal >> >> [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]] >> TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784] >> TOP generators [133.3373752, 49.02398564] >> bad: 4.918774 comp: 11.628267 err: .036377 >> 39.8287 -> bad = 57.7058 > > 2.>> Number 2 Meantone >> >> [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]] >> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328] >> TOP generators [1201.698520, 504.1341314] >> bad: 21.551439 comp: 3.562072 err: 1.698521 >> 11.7652 -> bad = 235.1092 > > 3.>> Number 9 Miracle >> >> [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]] >> TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756] >> TOP generators [1200.631014, 116.7206423] >> bad: 29.119472 comp: 6.793166 err: .631014 >> 21.1019 --> bad = 280.9843 > > 4.>> Number 7 Dominant Seventh >> >> [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]] >> TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933] >> TOP generators [1195.228951, 495.8810151] >> bad: 28.744957 comp: 2.454561 err: 4.771049 >> 7.9560 -> bad = 301.9952 > > 5.>> Number 3 Magic >> >> [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]] >> TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877] >> TOP generators [1201.276744, 380.7957184] >> bad: 23.327687 comp: 4.274486 err: 1.276744 >> 15.5360 -> bad = 308.1642 > > 6.>> Number 4 Beep >> >> [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]] >> TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548] >> TOP generators [1194.642673, 254.8994697] >> bad: 23.664749 comp: 1.292030 err: 14.176105 >> 4.7295 -> bad = 317.0935 > > 7.>> Number 6 Pajara >> >> [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]] >> TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174] >> TOP generators [598.4467109, 106.5665459] >> bad: 27.754421 comp: 2.988993 err: 3.106578 >> 10.4021 -> bad = 336.1437 > > 8.>> Number 10 Orwell >> >> [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]] >> TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834] >> TOP generators [1199.532657, 271.4936472] >> bad: 30.805067 comp: 5.706260 err: .946061 >> 19.9797 -> bad = 377.6573 > > 9.>> Number 8 Schismic >> >> [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]] >> TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750] >> TOP generators [1200.760624, 498.1193303] >> bad: 28.818558 comp: 5.618543 err: .912904 >> 20.2918 --> bad = 375.8947 > > 10.>> Number 5 Augmented >> >> [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]] >> TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546] >> TOP generators [399.9922103, 107.3111730] >> bad: 27.081145 comp: 2.147741 err: 5.870879 >> 8.3046 -> bad = 404.8933
Message: 9574 - Contents - Hide Contents Date: Sat, 31 Jan 2004 19:45:09 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: >>> So the below was wrong. I forgot that you reverse the order of the >> elements to convert a multival wedgie into a multimonzo wedgie! > Doing>> so would, indeed, give the same rankings as my original L_1 >> calculation. But that's gotta be the right norm. The Tenney lattice >> is set up to measure complexity, and the norm we always associate >> with it is the L_1 norm. Isn't that right? The L_1 norm on the > monzo>> is what I've been using all along to calculate complexity for the >> codimension-1 case, in my graphs and in the "Attn: Gene 2"> post . . . > > To me it seemed there was a reasonable case for either norm,What's the case for L_inf norm? It doesn't seem to agree with the purpose we're using the Tenney lattice in the first place . . .>Do you need me to redo the calculation using > the L1 norm?That would be excellent, and then I could make a graph for Dave . . .> I think it would be a good idea to stick to the normalization by > dividing, since for higher limits a linear temperament wedgie is > still a bival, so it's easier.OK . . .
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