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Message: 9000 - Contents - Hide Contents Date: Wed, 07 Jan 2004 22:10:01 Subject: Re: Meantone reduced blocks From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> So, what's your answer to my original question now? Or should I > wait 'till you're done generating all these blocks?The latter makes more sense to me.

Message: 9001 - Contents - Hide Contents Date: Wed, 07 Jan 2004 22:13:06 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Graham Breed Paul Erlich wrote:> Which complexity measure?The logarithm of the (larger) odd number in the comma ratio.>> What's expressiblility? >> Kees's metric -- the log of the lowest odd limit the ratio belongs to.I think that's the same, yes.> So how do you define and find "the worst comma"?It's defined as having the largest value for size/complexity. How you find it as an interesting question. I'm searching a range of combinations of the defining pair of commas and hoping I don't miss any. In general, it's probably easier to find the minimax by finding the temperament mapping first.>> A 21-limit extension of miracle, based on the best approximations > > in 31- >>> and 41-equal, or whatever else my program is doing. > > > Same question.I only know it's the worst comma in the 7-limit case, but as the 11-limit values are all within the weighted minimax it looks like it works there as well.>> My >> method will take the complexity as log(15) rather than log(13). >

Message: 9002 - Contents - Hide Contents Date: Wed, 07 Jan 2004 22:10:30 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> In truth, tempering a single comma (such as 81:80) from the 5- limit >> lattice yields a 2-dimensional tuning system, with no unique choice >> of generators. >> Aren't the units on either dimension the generators?The generators amount to a choice of basis for the tuning system. There is no unique choice of basis. I don't know what you mean by "units".>> But if we assume *octave-repetition*, then we're back >> to the usual period-generator mappings for the primes, which you can >> invert to find the generator. >> I never understood this process,Solving a system of linear equations?> or what differentiates a period > from a generator.In our parlance, when we assume *octave-repetition*, the 'period' will be the generator that generates the octave all by itself, while the 'generator' (usually the smallest possible is chosen, such as fourths for meantone) will produce all the other notes in the tuning in conjunction with the period -- they form a basis.>>> And does the old method give different results when going from >>> 5-limit linear to 7-limit planar? >>>> I believe so, though I can't remember the specifics. >>>>> Or are you claiming the answer >>> is "no" when "old method" was minimax, and "yes" when it was >>> anything else? >>>> If you mean Tenney-weighted minimax over all intervals, then this >> could very well be, though I don't think that was actually one of >> the "old" methods that were tried around here. >> I'm still partial to rms over all the intervals,How can you do that? Does it even converge? Or do you not really mean "all the intervals"?> but somehow I > think those doing rms around here were not including the 2s.If you don't include all the intervals, but don't want to assume octave-equivalence, you can use an integer limit, and I've posted some integer-limit rms results on the tuning list and elsewhere. But I don't like integer limit in comparison with Tenney limit, especially a Tenney limit that you don't even have to specify (as long as it's large enough to include all the primes you care about)!

Message: 9003 - Contents - Hide Contents Date: Wed, 07 Jan 2004 14:11:57 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Carl Lumma>> >nd does the old method give different results when going from >> 5-limit linear to 7-limit planar? Or are you claiming the answer >> is "no" when "old method" was minimax, and "yes" when it was >> anything else? >>What's "the old method"?Hiya Graham! Let me rephrase the above. Say I'm using unweighted rms error over all the intervals in a given prime limit. I want to find the 5-limit linear temperament that minimizes this error, call it Alex, and then I want to find the 7-limit planar temperament that does the same, call it Ben. Now, are the 5-limit intervals in Ben going to be different sizes than they are in Alex? In TOP temperament, the answer is no (I think). -Carl

Message: 9004 - Contents - Hide Contents Date: Wed, 07 Jan 2004 22:13:11 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> The method for 7-limit temperaments is to find the worst possible comma > that's in the basis, and temper that out.The definition of which is?

Message: 9005 - Contents - Hide Contents Date: Wed, 07 Jan 2004 22:14:38 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Paul Erlich wrote: >>> Which complexity measure? >> The logarithm of the (larger) odd number in the comma ratio. > >>> What's expressiblility? >>>> Kees's metric -- the log of the lowest odd limit the ratio belongs to. >> I think that's the same, yes. >>> So how do you define and find "the worst comma"? >> It's defined as having the largest value for size/complexity.Interesting! And is that truly the only one that matters?>>> My >>> method will take the complexity as log(15) rather than log(13). >>>> Who would use log(13)? > > Sorry, log(15*13)So imposing octave-equivalence amounts to a uniform stretch/squish of "Top", unless the octaves are already just? Bizarre!

Message: 9006 - Contents - Hide Contents Date: Wed, 07 Jan 2004 22:15:20 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> And does the old method give different results when going from >>> 5-limit linear to 7-limit planar? Or are you claiming the answer >>> is "no" when "old method" was minimax, and "yes" when it was >>> anything else? >>>> What's "the old method"? >> Hiya Graham! Let me rephrase the above. Say I'm using unweighted > rms error over all the intervals in a given prime limit.Wow. Surely that can't converge, can it?

Message: 9007 - Contents - Hide Contents Date: Wed, 07 Jan 2004 14:18:12 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Carl Lumma>Hiya Graham! Let me rephrase the above. Say I'm using unweighted >rms error over all the intervals in a given prime limit. I want to >find the 5-limit linear temperament that minimizes this error, call >it Alex, and then I want to find the 7-limit planar temperament >that does the same, call it Ben. Now, are the 5-limit intervals in >Ben going to be different sizes than they are in Alex? In TOP >temperament, the answer is no (I think).Drat! -- I meant "odd limit" above, not prime. And if the possibility of tempered 2s causes any problems, disallow it for now. -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: 9008 - Contents - Hide Contents Date: Thu, 08 Jan 2004 02:21:51 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> I remember this technique from Algebra, but I didn't think it would >>> be applicable here, since I assumed the variables wouldn't be >>> independent in that way. >>>> They're not, you actually have an extra equation. > > ? >>>> What do these equations look like? >> >> For meantone, >>>> prime2 = period; >> prime3 = period + generator; >> prime5 = 4*generator. >> >> You can throw out any equation -- say the first. >> >> so generator = .25*prime5, >> prime3 = period + .25*prime5, >> period = prime3 - .25*prime5. >> Sure, I've done these hundreds of times. But this is > just the map -- where are all the errors of all the > intervals? > > -CarlJust add up the primes that make them up!

Message: 9009 - Contents - Hide Contents Date: Thu, 08 Jan 2004 17:01:46 Subject: Tops and lattices From: Gene Ward Smith My vector space analysis was designed to find the optimal tuning in terms of the Tenney metric, so I thought it would turn out to be the same as TOP. However, I'm getting that 1/4-comma is optimal for meantone, and TOP had flattened octaves. If Paul would post again the TOP values for a few commas, making sure they are what he wants, I'll try to find out what the story is here. Anyway I'm happy with my approach, which seems to finally do what I was trying to do in the canonical tuning thread. Another issue is the naming of spaces and lattices. We have the following normed vector spaces and lattices living in them: Space A Space A for the p prime limit is the normed vector space with norm || |a2 a3 ... ap> || = |a2| + log2(3)|a3| + ... + log2(p)|ap| Living inside space A is the lattice whose coordinates are integers. Should Space A be the Tenney space and the lattice the Tenney lattice? Space B Dual to space A is Space B, of linear functionals on A. It is a normed vector space with norm || <b2 b3 ... bp> || = Max(|b2|, |b3/log2(3)|, ..., |bp/log2(p)|) Living inside of Space B is the lattice whose coordinates are integers, the lattice of vals. What is a good name for Space B and the lattice of vals? Also in Space B is a special point: <1 log2(3) ... log2(p)| I called this SIZE, but it could be called, for example, JIP for the JI point. What do people think? It might also be noted that if v is an equal temperament val, then ||v/v(2) - JIP|| is a Tenney-friendly measure of badness for v.

Message: 9011 - Contents - Hide Contents Date: Thu, 08 Jan 2004 19:40:13 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Me:>>> I'm not sure if such a comma will always exist, > > Paul:>> Wow -- now that's an interesting question to consider. > > Me:>>> but provided it does >>> it's the only one you need for TOPS. > > Paul:>> My intuition says it doesn't exist. >> If it doesn't, the worst one you get should get you close to the optimum > result.I thought "the worst one" is what you weren't sure existed.> > Me:>>> It doesn't even have to be made up >>> of integers, so long as it's a linear combination of commas that >>> are. > > Paul: >> ??? >> 81**2:80**2 or 6561:6400 has the double the size and double the > complexity of 81:80. That means its size/complexity ratio is the same, > and so tempering it out gives exactly the same result as tempering out > 81:80.Right, that's why I excluded such ratios from my charts.> So why stop there? The cube of 81:80 also gives the same > result. Any power will do, and it doesn't have to be an integer. The > square root of 81:80 works. Even the pith root works (provided you > express it as a monzo -- otherwise, it's an arbitrary number). > > That means instead of having two integers to define the resultant comma > of a 7-limit linear temperament, you only need one real number. For > example, the miracle kernel is (225:224)***i * (2401:2400)**j or i* [-5 2 > 2 -1> + j*[-5 -1 -2 4>. You can simplify that to x*[-5 2 2 -1> + > (1-x)*[-5 -1 -2 4> so that you only have 1 variable instead of 2.Hmm . . .>> There's something that seems strange about your octave-equivalent >> method. The comma is supposed to be distributed uniformly (per unit >> length, taxicabwise) among its constituent "rungs" in the lattice. >> But it seems that 81:80 = 81:5 would involve 1, not 0, rungs of 5 in >> the octave-equivalent lattice. But the octave-equivalent lattice >> can't be embedded in euclidean space, so this completely falls apart?? >> 81:5 involves three rungs of 3:1 and one rung of 3:5. Oh yeah. > For the 5-odd > limit, these rungs are of equal length, and so that error has to be > shared between them. That leaves 3:1 and 3:5 having an equal amount of > temperament, and so 1:5 must be untempered. Ha! > This is how Dave did it on his original web page: > > A method for optimally distributing any comma * [with cont.] (Wayb.) > > and we worked out at the time that it's correct. I didn't remember all > that stuff about octave-specific optimization. It looks like you've > cleared up the problems. But anyway, for the weighted, > octave-equivalent case: > > The complexity of 81:80 is entirely determined by the 81, and not at all > by the 80. The error can only be shared amongst those intervals that > contribute to the complexity. I don't see how this can be drawn on a > lattice, but it doesn't need to be.But you already cleared up the lattice situation above.> You can improve 3:1 at the expense of 3:5 and, because 3:5 has less > weight, the worst weighted error for 5 odd-limit ratios goes down. > However, the weighted error for 9:5 exceeds this, because it has the > same complexity as 9:8 but is more poorly tuned. The true minimax is > found by only tempering the 81.OK . . .> You certainly can embed an octave-equivalent lattice in Euclidian >space.Nope -- you need 'wormholes', i.e., non-euclidean features.> The triangular lattice is an example,It doesn't work -- 9:1 looks shorter than other ratios of 9.> but there's no reason to connect > up the 5:3s if you're measuring Euclidian distances.I certainly hope we're not measuring Euclidean distances!> In general, for > each prime interval you need to assign a weight (the unit lattice > distance) and also an angle.There should be no angles defined here, just as there are none in the Tenney lattice.> And you can probably do size/complexity > tempering with such a measure, but it'd be more complex, and I haven't > tried it. I'm guessing it wouldn't give the true weighted minimax, > because for that to work it looks like the complexity of a comma has to > be the sum of the complexities of its factors (one way or another).Right, but a non-euclidean version of the triangular lattice can respect this, can't it? Looks like that's similar to what Kees was geting at at the bottom of this page: lattice orientation * [with cont.] (Wayb.) except it doesn't look like he was thinking taxicab . . .

Message: 9012 - Contents - Hide Contents Date: Thu, 08 Jan 2004 23:54:25 Subject: Re: Temperament agreement From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> >> wrote:>>> Continued from the tuning list. >>> Paul:>>>> With my (Tenney) complexity and (all-interval-Tenney-minimax) > error >>>> measures? >>>>>> With these it seems I need to scale the parameters to >>> k=0.002 >>> p=0.5 >>> and max badness = 75 >>> >>> where badness = complexity * exp((error/k)**p) >>> >>> I'd be very interested to see how that compares with your other >> cutoff >>> lines. >> >> See >> >> Yahoo groups: /tuning_files/files/Erlich/dave3... * [with cont.] >>>> It looks like going back to logarithmic error scaling would be > better>> for seeing what is going on here . . .Yes. I like the log error scale.> Matlab is chugging . . . > > See Yahoo groups: /tuning_files/files/Erlich/dave4... * [with cont.] >Thanks very much for doing this Paul.> Your criterion (the cyan curve marked '75') allows in two commas of > about the complexity of those of amity and orwell, but giving about > the same error as diaschismic.Right. Well it was just an estimated modification to the original formula, to suit your new complexity and error measures. It runs very close to a lot of temps. I'd probably want to reduce the badness cutoff slightly so they all fall outside it.> I was hoping the numbers would be > readable on the latest graph; they aren't, but after all, they're > just numbers. But I don't know if anyone really wants to consider > adding these to their list at this point. I'd penalize complexity > more.I'll go along with that. A bit more penalty for complexity.> Anyway, it's not clear whether your function will allow in an > infinite number of commas. If we can find a similar function that is > b-e-p lower than a given epimericity contour (<1, right Gene?), then > we can be guaranteed that it gives a finite number, and perhaps find > them all.It looks like I'd be just as happy with straight lines on this chart. These should be easy enough to parameterise. e.g. the point (possibly off the chart) at which they all intersect. Then the badness is the inverse of the slope, or some such. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9013 - Contents - Hide Contents Date: Thu, 08 Jan 2004 02:48:15 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>> What do these equations look like? >>>> >>>> For meantone, >>>>>>>> prime2 = period; >>>> prime3 = period + generator; >>>> prime5 = 4*generator. >>>> >>>> You can throw out any equation -- say the first. >>>> >>>> so generator = .25*prime5, >>>> prime3 = period + .25*prime5, >>>> period = prime3 - .25*prime5. >>>>>> Sure, I've done these hundreds of times. But this is >>> just the map -- where are all the errors of all the >>> intervals? >>> >>> -Carl >>>> Just add up the primes that make them up! >> That sounds like TOP. I'm talking about the old way.Oh. The old way, you start with the mapping, and then solve for the period and generator, and then minimize your error function (which is over some finite set of intervals) by varying the period & generator, or in octave-equivalent cases, just the generator. You can use calculus and express the error function in terms of the generator size, take the derivative, set that equal to zero, and solve -- works great for sum-squared error (p=2), weighted or unweighted. Other methods work better for "harder" error functions like max-abs-error (p=inf) or sum-abs-error (p=1).

Message: 9014 - Contents - Hide Contents Date: Thu, 08 Jan 2004 19:48:07 Subject: Re: Tops and lattices From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> My vector space analysis was designed to find the optimal tuning in > terms of the Tenney metric, so I thought it would turn out to be the > same as TOP. However, I'm getting that 1/4-comma is optimal for > meantone, and TOP had flattened octaves. If Paul would post again the > TOP values for a few commas, making sure they are what he wants, I'll > try to find out what the story is here.You mean something like this? Yahoo groups: /tuning-math/message/8374 * [with cont.]> Anyway I'm happy with my > approach, which seems to finally do what I was trying to do in the > canonical tuning thread.But you just finished demonstrating that my method leads to Tenney- weighted minimax, and there can't be more than one solution for Tenney-weighted minimax -- right?> Another issue is the naming of spaces and lattices. We have the > following normed vector spaces and lattices living in them: > > Space A > > Space A for the p prime limit is the normed vector space with norm > > || |a2 a3 ... ap> || = |a2| + log2(3)|a3| + ... + log2(p)|ap| > > Living inside space A is the lattice whose coordinates are integers. > Should Space A be the Tenney space and the lattice the Tenney lattice? Sure. > Space B > > Dual to space A is Space B, of linear functionals on A. It is a normed > vector space with norm > > || <b2 b3 ... bp> || = Max(|b2|, |b3/log2(3)|, ..., |bp/log2(p)|) > > Living inside of Space B is the lattice whose coordinates are > integers, the lattice of vals. What is a good name for Space B and the > lattice of vals?Error space and the temperament lattice??????> Also in Space B is a special point: > > <1 log2(3) ... log2(p)| > > I called this SIZE, but it could be called, for example, JIP for the > JI point. What do people think?Seems to make a lot more sense than "SIZE"!

Message: 9015 - Contents - Hide Contents Date: Thu, 08 Jan 2004 20:38:00 Subject: Re: Temperament agreement From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:>> Continued from the tuning list. >> Paul:>>> With my (Tenney) complexity and (all-interval-Tenney-minimax) error >>> measures? >>>> With these it seems I need to scale the parameters to >> k=0.002 >> p=0.5 >> and max badness = 75 >> >> where badness = complexity * exp((error/k)**p) >> >> I'd be very interested to see how that compares with your other > cutoff >> lines. > > See > > Yahoo groups: /tuning_files/files/Erlich/dave3... * [with cont.] >> It looks like going back to logarithmic error scaling would be better > for seeing what is going on here . . . Matlab is chugging . . . See Yahoo groups: /tuning_files/files/Erlich/dave4... * [with cont.]Your criterion (the cyan curve marked '75') allows in two commas of about the complexity of those of amity and orwell, but giving about the same error as diaschismic. I was hoping the numbers would be readable on the latest graph; they aren't, but after all, they're just numbers. But I don't know if anyone really wants to consider adding these to their list at this point. I'd penalize complexity more. Anyway, it's not clear whether your function will allow in an infinite number of commas. If we can find a similar function that is b-e-p lower than a given epimericity contour (<1, right Gene?), then we can be guaranteed that it gives a finite number, and perhaps find them all.

Message: 9016 - Contents - Hide Contents Date: Thu, 08 Jan 2004 03:31:21 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> That sounds like TOP. I'm talking about the old way. >>>> Oh. The old way, you start with the mapping, and then solve for the >> period and generator, and then minimize your error function (which is >> over some finite set of intervals) by varying the period & generator, >> or in octave-equivalent cases, just the generator. You can use >> calculus >> Aha, I knew it! Calculus! :) >>> and express the error function in terms of the generator >> size, take the derivative, > > ok... >>> set that equal to zero, and solve >> Lost me here. The derivative itself is a curve,Right, and where it meets the x-axis is where it equals zero.> Oh, and if we're doing integer limit don't we need two > generators?We need two generators if we're talking about a 2D temperament -- either a planar temperament with octave-equivalence assumed, or a linear temperament where we can vary the octave (or period) as well as the generator.>> -- works great for sum-squared error (p=2), weighted or unweighted. >> Good, that's all I want. I've got enough software to put my eye > out with, I ought to be able to set this up. By the way, this now > includes Matlab, if you'd prefer to illustrate with code.Wow. Do you have the optimization toolbox?

Message: 9017 - Contents - Hide Contents Date: Thu, 08 Jan 2004 21:13:47 Subject: Re: Tops and lattices From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: Not exactly. What about a short list, giving your tuning for 2, 3, and 5 for various 5-limit commas?>> Anyway I'm happy with my >> approach, which seems to finally do what I was trying to do in the >> canonical tuning thread. >> But you just finished demonstrating that my method leads to Tenney- > weighted minimax, and there can't be more than one solution for > Tenney-weighted minimax -- right?No, I just finished finding out what Tenney optimized temperaments ought to be, under the assumption that that must be TOP. But it doesn't seem to be, and in fact is more like my canonical temperament business.>> Another issue is the naming of spaces and lattices. We have the >> following normed vector spaces and lattices living in them: >> >> Space A >> >> Space A for the p prime limit is the normed vector space with norm >> >> || |a2 a3 ... ap> || = |a2| + log2(3)|a3| + ... + log2(p)|ap| >> >> Living inside space A is the lattice whose coordinates are integers. >> Should Space A be the Tenney space and the lattice the Tenney > lattice? > > Sure.Are you and everyone else willing to stick with that? By that I mean are you really willing for the Tenney space and Tenney lattice to have a very specific definition, which will not dissolve into mush? I'm ready to adopt the names myself.>> Space B >> >> Dual to space A is Space B, of linear functionals on A. It is a > normed>> vector space with norm >> >> || <b2 b3 ... bp> || = Max(|b2|, |b3/log2(3)|, ..., |bp/log2(p)|) >> >> Living inside of Space B is the lattice whose coordinates are >> integers, the lattice of vals. What is a good name for Space B and > the>> lattice of vals? >> Error space and the temperament lattice??????Except that only a few vals are equal temperaments.>> Also in Space B is a special point: >> >> <1 log2(3) ... log2(p)| >> >> I called this SIZE, but it could be called, for example, JIP for > the>> JI point. What do people think? >> Seems to make a lot more sense than "SIZE"!I like it better too. The JIP point it is unless someone screams loudly at this point.

Message: 9018 - Contents - Hide Contents Date: Thu, 08 Jan 2004 21:16:42 Subject: Re: Temperament agreement From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Anyway, it's not clear whether your function will allow in an > infinite number of commas. If we can find a similar function that is > b-e-p lower than a given epimericity contour (<1, right Gene?), then > we can be guaranteed that it gives a finite number, and perhaps find > them all.Right. If Dave will give a list of commas he must have, we could simply find out the maximum epimericity. Then we could produce a list of commas with epimeriticy less than that, and see if there are any commas on it Dave won't tolerate.

Message: 9019 - Contents - Hide Contents Date: Thu, 08 Jan 2004 04:54:29 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> and express the error function in terms of the generator >>>> size, take the derivative, >>> >>> ok... >>>>>>> set that equal to zero, and solve >>>>>> Lost me here. The derivative itself is a curve, >>>> Right, and where it meets the x-axis is where it equals zero. >> That's where the original function is flat, but how do we > know the original function isn't flat at multiple places?If you're using least squares, the function is just a parabola, since there are no terms involving higher powers (than 2) of the generator.>>> Oh, and if we're doing integer limit don't we need two >>> generators? >>>> We need two generators if we're talking about a 2D temperament -- >> either a planar temperament with octave-equivalence assumed, or a >> linear temperament where we can vary the octave (or period) as well >> as the generator. >> I'm talking about linear temperaments now, strictly. And by > "integer limit", I mean variable octave,I didn't know you meant to use integer limit here. Then you have a function of two variables, so the calculus is a little harder, but the optimization toolbox has no problem -- and it's basically a paraboloid anyway.> and I've been calling > the period a generator for, oh, over a year.You were right.>>>> -- works great for sum-squared error (p=2), weighted or unweighted. >>>>>> Good, that's all I want. I've got enough software to put my eye >>> out with, I ought to be able to set this up. By the way, this now >>> includes Matlab, if you'd prefer to illustrate with code. >>>> Wow. Do you have the optimization toolbox? >> It looks like it. I just ran a "Large-scale unconstrained nonlinear > minimization" demo.How did you get so lucky? Anyway, for any temperament, write a function that computes your error function for a given choice of period and generator, then minimize it using fmin or fmins or whatever.

Message: 9020 - Contents - Hide Contents Date: Thu, 08 Jan 2004 21:21:22 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Graham Breed Paul Erlich wrote:> I thought "the worst one" is what you weren't sure existed.There'll always be a worst comma, but it might not be as bad as the linear temperament it belongs to.>> You certainly can embed an octave-equivalent lattice in Euclidian >> space. > >> Nope -- you need 'wormholes', i.e., non-euclidean features.Which kind of lattice are you talking about? I meant the crystallographic one.> It doesn't work -- 9:1 looks shorter than other ratios of 9.Or longer, indeed. But so what?> I certainly hope we're not measuring Euclidean distances!Yes, Euclidian distances. That's how it comes to be in Euclidian space.> There should be no angles defined here, just as there are none in the > Tenney lattice.Why "should"? They have to be, or it won't work.> Right, but a non-euclidean version of the triangular lattice can > respect this, can't it? Looks like that's similar to what Kees was > geting at at the bottom of this page: > > lattice orientation * [with cont.] (Wayb.) > > except it doesn't look like he was thinking taxicab . . .I don't know, I don't see what the point is. I don't think the odd limit counts as a norm, which I think is a problem, although I don't remember the details. Graham

Message: 9021 - Contents - Hide Contents Date: Thu, 08 Jan 2004 21:20:19 Subject: Re: Tops and lattices From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >> Not exactly. What about a short list, giving your tuning for 2, 3, > and 5 for various 5-limit commas? Like this? Yahoo groups: /tuning/message/51193 * [with cont.] (scroll down)>>> Anyway I'm happy with my >>> approach, which seems to finally do what I was trying to do in the >>> canonical tuning thread. >>>> But you just finished demonstrating that my method leads to Tenney- >> weighted minimax, and there can't be more than one solution for >> Tenney-weighted minimax -- right? >> No, I just finished finding out what Tenney optimized temperaments > ought to be, under the assumption that that must be TOP. But it > doesn't seem to be, and in fact is more like my canonical temperament > business.Well, i'd like to see your version of meantone that has a lower maximum Tenney-weighted error than mine!>>> Another issue is the naming of spaces and lattices. We have the >>> following normed vector spaces and lattices living in them: >>> >>> Space A >>> >>> Space A for the p prime limit is the normed vector space with norm >>> >>> || |a2 a3 ... ap> || = |a2| + log2(3)|a3| + ... + log2(p)|ap| >>> >>> Living inside space A is the lattice whose coordinates are > integers.>>> Should Space A be the Tenney space and the lattice the Tenney >> lattice? >> >> Sure. >> Are you and everyone else willing to stick with that? By that I mean > are you really willing for the Tenney space and Tenney lattice to > have a very specific definition, which will not dissolve into mush? > I'm ready to adopt the names myself.I'm afraid no one here is the mathematician you are, so you may be hoping for a bit too much. But I see nothing wrong with these, right now . . .>>> Space B >>> >>> Dual to space A is Space B, of linear functionals on A. It is a >> normed>>> vector space with norm >>> >>> || <b2 b3 ... bp> || = Max(|b2|, |b3/log2(3)|, ..., |bp/log2(p) |) >>> >>> Living inside of Space B is the lattice whose coordinates are >>> integers, the lattice of vals. What is a good name for Space B > and >> the>>> lattice of vals? >>>> Error space and the temperament lattice?????? >> Except that only a few vals are equal temperaments.I didn't say "equal", did I?>>> Also in Space B is a special point: >>> >>> <1 log2(3) ... log2(p)| >>> >>> I called this SIZE, but it could be called, for example, JIP for >> the>>> JI point. What do people think? >>>> Seems to make a lot more sense than "SIZE"! >> I like it better too. The JIP pointJI point point?

Message: 9022 - Contents - Hide Contents Date: Thu, 08 Jan 2004 05:21:15 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>>> and express the error function in terms of the generator >>>>>> size, take the derivative, >>>>> >>>>> ok... >>>>>>>>>>> set that equal to zero, and solve >>>>>>>>>> Lost me here. The derivative itself is a curve, >>>>>>>> Right, and where it meets the x-axis is where it equals zero. >>>>>> That's where the original function is flat, but how do we >>> know the original function isn't flat at multiple places? >>>> If you're using least squares, the function is just a parabola, since >> there are no terms involving higher powers (than 2) of the generator. >> I understand that functions of the type f(x) -> x^2 + c are shaped > like parabolas, but x isn't a generator size here, it's the sum of > errors resulting from a generator size. If I took out the ^2 it > might be shaped like anything;Huh? x + c is shaped like anything?

Message: 9023 - Contents - Hide Contents Date: Thu, 08 Jan 2004 21:23:06 Subject: Re: Temperament agreement From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> Anyway, it's not clear whether your function will allow in an >> infinite number of commas. If we can find a similar function that > is>> b-e-p lower than a given epimericity contour (<1, right Gene?), > then>> we can be guaranteed that it gives a finite number, and perhaps > find >> them all. >> Right. If Dave will give a list of commas he must have, we could > simply find out the maximum epimericity. Then we could produce a list > of commas with epimeriticy less than that, and see if there are any > commas on it Dave won't tolerate.Obviously there will be! 2/1, 3/2, 4/3, 5/4, 6/5, 9/8, 10/9, and 16/15 automatically qualify.

Message: 9024 - Contents - Hide Contents Date: Thu, 08 Jan 2004 21:27:40 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Paul Erlich wrote: >>> I thought "the worst one" is what you weren't sure existed. >> There'll always be a worst comma, but it might not be as bad as the > linear temperament it belongs to.What's the worst comma for 12-equal in the 5-limit?>>> You certainly can embed an octave-equivalent lattice in Euclidian >>> space. >> >>>> Nope -- you need 'wormholes', i.e., non-euclidean features. >> Which kind of lattice are you talking about? I meant the > crystallographic one.Me too -- but the lengths aren't compatible in a Euclidean space. Remember the whole big "wormholes" discussion from years ago?>> It doesn't work -- 9:1 looks shorter than other ratios of 9. >> Or longer, indeed. But so what? See above.>> I certainly hope we're not measuring Euclidean distances! >> Yes, Euclidian distances. That's how it comes to be in Euclidian space.We can either embed a lattice, with a taxicab distance, into Euclidean space, or we can't. But just because we can, doesn't mean we should use Euclidean distance! NONONONONONO!>> There should be no angles defined here, just as there are none in the >> Tenney lattice. >> Why "should"? They have to be, or it won't work.Why won't it? My Tenney, non-octave-equivalent way doesn't need angles defined. You can choose any set of angles you want, and still embed the result in Euclidean space, but that doesn't even matter -- what matters are the taxicab distances ONLY.

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