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Message: 8975 - Contents - Hide Contents Date: Wed, 07 Jan 2004 22:31:54 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> Hiya Graham! Let me rephrase the above. Say I'm using unweighted > rms error over all the intervals in a given prime limit.As Paul pointed out, this is undefined.
Message: 8976 - Contents - Hide Contents Date: Wed, 07 Jan 2004 21:18:33 Subject: Re: Meantone reduced blocks From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> But doesn't that reduce to Augmented[12] when 128/125 is tempered > out>> and Diminished[12] when 648/625 is tempered out?? >> Ooops. Had my wiring crossed.The above follows from the Hypothesis, of course. So, what's your answer to my original question now? Or should I wait 'till you're done generating all these blocks?
Message: 8977 - Contents - Hide Contents Date: Wed, 07 Jan 2004 22:32:45 Subject: Re: non-1200: Tenney/heursitic meantone temperamentnt From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Carl Lumma wrote: >>> 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). > > Hello! >> How would an unweighted, unbounded RMS error work? The advantage of the > weighting is that more complex intervals get lower weights, and so the > weighted error stays roughly constant. Hence you can impose a weighted > minimax over all intervals within a given prime limit.And, in fact, over *all* intervals, if you're simply talking about the effect of tempering out a given comma in general -- or, if we really can generalize this, the effect of tempering out some "multi- comma" so that the codimension is specified but the dimension can be anything.
Message: 8978 - Contents - Hide Contents Date: Wed, 07 Jan 2004 00:25:26 Subject: Re: Meantone reduced blocks From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Have you found any that don't reduce to something other than: > > Meantone[12] > Diaschismic[12] > Augmented[12] > Diminished[12] > > when tempered accordingly? Thirds.scl qualifies.
Message: 8979 - Contents - Hide Contents Date: Wed, 07 Jan 2004 13:24:07 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Carl Lumma>> >hat if the generator isn't a just interval? Then isn't it still >> the same kind of multivariable optimization that you guys have been >> using all along? >>I didn't make any assumptions about what the generator was above. The >same formula works for any generator,You also didn't give the method for finding it, but I was assuming the TOP 3:2 and 2:1 are the meantone generators, for example.>and even when there is no >generator, as is the case for 7-limit and above.With a single comma there are two 5-limit generators, three 7-limit, and so on. Or so I suppose.>>>> And I don't understand your 'limitless' claim -- since p/q >>>> contains the factors it does and no others, //>> What I mean is, when extending meantone to a 7-limit mapping, it >> will naturally implicate different commas and change the optimal >> generator a bit, same as before. >>Yes, and there are different choices as to which commas to use to >extend meantone to a 7-limit linear temperament. But I wasn't talking >about that. I was talking about tempering out a single comma, which >would lead to a planar temperament in the 7-limit, etc.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? -Carl
Message: 8980 - Contents - Hide Contents Date: Wed, 07 Jan 2004 21:15:11 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Carl Lumma>>>>> >nd 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; are you saying the ^2 will turn it into a parabola? Oh man I'm braindead, I used to know this stuff. :(>>>> 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. Emphasis added. >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.Well I'm happy to learn the fixed-octave case first... or not. However.>>>> 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?One of my best friends studies rat brains (vision alas), and we're using matlab to model a potential business idea at the moment.>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. Tx. -Carl
Message: 8981 - Contents - Hide Contents Date: Wed, 07 Jan 2004 22:38:56 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Graham Breed Paul Erlich wrote:> Interesting! And is that truly the only one that matters?The size/complexity tells you the best value for the suitably weighted minimax in any temperament in which this comma vanishes. If a comma exists such that its size/complexity is equal to the optimim minimax error in a given linear temperament, and the comma is in the linear temperament's kernel, then the two temperaments must be identical. I'm not sure if such a comma will always exist, but provided it does it's the only one you need for TOPS. It doesn't even have to be made up of integers, so long as it's a linear combination of commas that are. Naturally, if such a comma does exist, there can be no other comma in the temperament that has a higher value of size/complexity.> So imposing octave-equivalence amounts to a uniform stretch/squish > of "Top", unless the octaves are already just? Bizarre!It's a generalization of the proof of the TOP meantone being stretched quarter-comma. All the factors that get tempered the same way as 2 will be stretched by the same amount. But if the octaves don't get tempered at all, some factors will be tempered in one method but not the other. Graham
Message: 8982 - Contents - Hide Contents Date: Wed, 07 Jan 2004 00:30:35 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> The main thing is to get goodness of an incomplete wedgie. Like we > could find out that 2401:2400 and 3025:3024 work well together, and so > keep looking for the next comma. But maybe looking at the planar > temperament would tell us that.Right; the geometric badness of a planar temperament will tell us that.
Message: 8983 - Contents - Hide Contents Date: Wed, 07 Jan 2004 14:47:06 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Carl Lumma>How would an unweighted, unbounded RMS error work?Hopefully by now you've seen the messages where I've seen the "error" of my ways, and settled on integer limit, or odd-limit with just octaves.>The advantage of the >weighting is that more complex intervals get lower weights, and so >the weighted error stays roughly constant.In fact, I once asked if the weighting couldn't be so steep that additionally specifying a limit would be unnecessary (since the contribution from the errors of higher-limit ratios would be vanishingly small). IOW, could the concept of choosing a "limit" be replaced entirely by that of choosing a steepness of the weighting function?>Hence you can impose a weighted >minimax over all intervals within a given prime limit.Aha! So why then isn't the prime limit also superfluous?>Still to the question. Yes, I think the answer is no.Cool, then you and Paul agree on that. -Carl
Message: 8984 - Contents - Hide Contents Date: Wed, 07 Jan 2004 21:36:42 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Graham Breed Paul Erlich wrote:> Hmm . . . by *all* the errors, I meant for lots and lots of > intervals, like I did.Oh, well, here's the 9-limit with a few bonuses: 3:1 0.002827 5:1 0.000000 5:3 0.001930 7:1 0.000903 7:3 0.000693 7:5 0.000903 9:1 0.002827 9:5 0.002827 9:7 0.002027 15:1 0.001147 27:1 0.002827 27:5 0.002827 The method for 7-limit temperaments is to find the worst possible comma that's in the basis, and temper that out. For meantone, it's the 81:80, so quarter comma meantone is still the 7-prime limit optimum (like it is for the unweighted 7-odd limit). I don't know if it always works, but it can. Also in miracle the worst 7-limit comma is the 5-limit one, [-25 7 6>. The comma size is 31.567 cents and the complexity 30032 cents (the size of [0 7 6>). The size of a fifth is then 1200*log2(3)*(1-31.567/30032)-1200 = 699.96 cents. A secor is a sixth or this, giving 116.66 cents. Here are the weighted 9-limit errors: 3:1 0.001051 5:1 0.001051 5:3 0.000334 7:1 0.000637 7:3 0.000043 7:5 0.000233 9:1 0.001051 9:5 0.000281 9:7 0.000487 The same tuning happens to work in the 11-limit as well, but it's no panacea because it fails beyond that. So here are some more numbers: 11:1 0.000344 11:3 0.000137 11:5 0.000361 11:7 0.000172 11:9 0.000619 13:1 0.002138 13:3 0.002588 13:5 0.002798 13:7 0.002621 13:9 0.003039 13:11 0.002460 15:1 0.001051 15:7 0.000594 15:11 0.000746 15:13 0.003076 17:1 0.002386 17:3 0.002794 17:5 0.002983 17:7 0.002823 17:9 0.003201 17:11 0.002677 17:13 0.000450 17:15 0.003391 19:1 0.000514 19:3 0.000906 19:5 0.001088 19:7 0.000935 19:9 0.001298 19:11 0.000794 19:13 0.001349 19:15 0.001481 19:17 0.001782 21:1 0.000786 21:5 0.000231 21:11 0.000515 21:13 0.002588 21:17 0.003007 21:19 0.001283 Oh yes, and this octave-equivalent method always gives the same result as your octave-specific one after you unstretch the scale so that octaves are just. Unless both the numerator and denominator of the comma are odd numbers, in which case you already get a just octave, and so only taking the larger one will give a different result. Graham
Message: 8985 - Contents - Hide Contents Date: Wed, 07 Jan 2004 14:47:35 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Carl Lumma>>>> > never understood this process, >>>>>> Solving a system of linear equations? >> >> Uh-huh. >>Well, the easiest way to understand is to solve one equation for one >variable, plug that solution into the other variables so that you've >eliminated one variable entirely, and repeat until you're done.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. What do these equations look like?>> Why are you assuming octave repetition, what does this assumption >> amount to? >>That you'll have the same pitches in each (possibly tempered) octave. >>> If 2 is in the map, one of the generators had better well generate >> it. If it isn't in the map, assuming octave repetition seems like >> a bad idea to me. >>Any recent cases where you'd prefer not to see 2 in the map?I'd always prefer to see it, but why assume?>> What's a Tenney limit? >>If the limit is L, it's all reduced ratios n/d such that n*d<=L (or >log(n*d)<=L, whatever). Ok. -Carl
Message: 8986 - Contents - Hide Contents Date: Wed, 07 Jan 2004 21:33:25 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Carl Lumma>> > 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?Traditionally a line, but in this case x is actually this other function, the summed errors from these arbitrary external just ratio things. As I move the generator size from 0-600 cents and pump it through say the meantone map, the errors could go up and down several times for all I know. Square that, and I'll just get sharper lumps. What am I missing? -Carl
Message: 8987 - Contents - Hide Contents Date: Wed, 07 Jan 2004 21:34:40 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> What if the generator isn't a just interval? Then isn't it still >>> the same kind of multivariable optimization that you guys have been >>> using all along? >>>> I didn't make any assumptions about what the generator was above. The >> same formula works for any generator, >> You also didn't give the method for finding it, but I was assuming the > TOP 3:2 and 2:1 are the meantone generators, for example.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. 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.>> and even when there is no >> generator, as is the case for 7-limit and above. >> With a single comma there are two 5-limit generators, three 7-limit, > and so on. Or so I suppose.Yes, but then there's even less uniqueness to the choice. Gene has proposed "hermite reduction", perhaps the issue is worth another look.>>>>> And I don't understand your 'limitless' claim -- since p/q >>>>> contains the factors it does and no others, > //>>> What I mean is, when extending meantone to a 7-limit mapping, it >>> will naturally implicate different commas and change the optimal >>> generator a bit, same as before. >>>> Yes, and there are different choices as to which commas to use to >> extend meantone to a 7-limit linear temperament. But I wasn't talking >> about that. I was talking about tempering out a single comma, which >> would lead to a planar temperament in the 7-limit, etc. >> 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.
Message: 8988 - Contents - Hide Contents Date: Wed, 07 Jan 2004 00:34:24 Subject: Re: Meantone reduced blocks From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> Have you found any that don't reduce to something other than: >> >> Meantone[12] >> Diaschismic[12] >> Augmented[12] >> Diminished[12] >> >> when tempered accordingly? > > Thirds.scl qualifies.What were the unison vectors for that again?
Message: 8989 - Contents - Hide Contents Date: Wed, 07 Jan 2004 21:40:17 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Paul Erlich wrote: >>> Hmm . . . by *all* the errors, I meant for lots and lots of >> intervals, like I did. >> Oh, well, here's the 9-limit with a few bonuses: > > 3:1 0.002827 > 5:1 0.000000 > 5:3 0.001930 > 7:1 0.000903 > 7:3 0.000693 > 7:5 0.000903 > 9:1 0.002827 > 9:5 0.002827 > 9:7 0.002027 > 15:1 0.001147 > 27:1 0.002827 > 27:5 0.002827So you're dividing by expressibility here? Interesting . . . !> The method for 7-limit temperaments is to find the worst possible comma > that's in the basis, and temper that out. For meantone, it's the 81:80, > so quarter comma meantone is still the 7-prime limit optimum (like it is > for the unweighted 7-odd limit). I don't know if it always works, but > it can.Hmm . . . are you talking about "septimal meantone", or are you talking about a 7-limit planar temperament?> > Also in miracle the worst 7-limit comma is the 5-limit one, [-25 7 6>. > The comma size is 31.567 cents and the complexity 30032 cents (the size > of [0 7 6>). The size of a fifth is then > 1200*log2(3)*(1-31.567/30032)-1200 = 699.96 cents. A secor is a sixth > or this, giving 116.66 cents. Here are the weighted 9-limit errors: > > 3:1 0.001051 > 5:1 0.001051 > 5:3 0.000334 > 7:1 0.000637 > 7:3 0.000043 > 7:5 0.000233 > 9:1 0.001051 > 9:5 0.000281 > 9:7 0.000487 > > The same tuning happens to work in the 11-limit as well, but it's no > panacea because it fails beyond that. So here are some more numbers: > > 11:1 0.000344 > 11:3 0.000137 > 11:5 0.000361 > 11:7 0.000172 > 11:9 0.000619 > 13:1 0.002138Hold on. Are you talking about some 13-limit linear extension of miracle, or a planar temperament that tempers out the same commas as miracle?> 13:3 0.002588 > 13:5 0.002798 > 13:7 0.002621 > 13:9 0.003039 > 13:11 0.002460 > 15:1 0.001051 > 15:7 0.000594 > 15:11 0.000746 > 15:13 0.003076 > 17:1 0.002386 > 17:3 0.002794 > 17:5 0.002983 > 17:7 0.002823 > 17:9 0.003201 > 17:11 0.002677 > 17:13 0.000450 > 17:15 0.003391 > 19:1 0.000514 > 19:3 0.000906 > 19:5 0.001088 > 19:7 0.000935 > 19:9 0.001298 > 19:11 0.000794 > 19:13 0.001349 > 19:15 0.001481 > 19:17 0.001782 > 21:1 0.000786 > 21:5 0.000231 > 21:11 0.000515 > 21:13 0.002588 > 21:17 0.003007 > 21:19 0.001283 > > Oh yes, and this octave-equivalent method always gives the same result > as your octave-specific one after you unstretch the scale so that > octaves are just. Unless both the numerator and denominator of the > comma are odd numbers, in which case you already get a just octave, and > so only taking the larger one will give a different result.Only taking the larger one?
Message: 8990 - Contents - Hide Contents Date: Wed, 07 Jan 2004 23:10:16 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Graham Breed I wrote:> The size/complexity tells you the best value for the suitably weighted > minimax in any temperament in which this comma vanishes. If a comma > exists such that its size/complexity is equal to the optimim minimax > error in a given linear temperament, and the comma is in the linear > temperament's kernel, then the two temperaments must be identical.Actually, it's more complicated than that. After finding the planar temperament, you then need to adjust intervals that didn't get tempered so that they work with the correct linear temperament family. And if you don't want to find the generator, that means solving simultaneous equations such that all the commas that are supposed to vanish do vanish. Graham
Message: 8991 - Contents - Hide Contents Date: Wed, 07 Jan 2004 00:35:37 Subject: Re: Meantone reduced blocks From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> Have you found any that don't reduce to something other than: >>> >>> Meantone[12] >>> Diaschismic[12] >>> Augmented[12] >>> Diminished[12] >>> >>> when tempered accordingly? >> >> Thirds.scl qualifies. >> Is there an extra negation in your sentence there, Paul? > > -CarlYup -- good catch, carl! I think my brain's taking a permanent vacation. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 8992 - Contents - Hide Contents Date: Wed, 07 Jan 2004 21:56:20 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Graham Breed Paul Erlich wrote:> So you're dividing by expressibility here? Interesting . . . !I'm dividing by the complexity. What's expressiblility?> Hmm . . . are you talking about "septimal meantone", or are you > talking about a 7-limit planar temperament?The usual 7-limit meantone.> Hold on. Are you talking about some 13-limit linear extension of > miracle, or a planar temperament that tempers out the same commas as > miracle?A 21-limit extension of miracle, based on the best approximations in 31- and 41-equal, or whatever else my program is doing.> Only taking the larger one?I don't now of any realistic commas with only odd numbers. So say you were tempering out 15/13. Using your method will work fine, it won't be very accurate, but you'll get a temperament out. And, because there are no factors of 2 in the commma, there's nothing to say the octaves should be stretched. So you already have an octave equivalent temperament! My method will take the complexity as log(15) rather than log(13). So it'll give some other result, and it may have to be like this for full generality. Graham
Message: 8993 - Contents - Hide Contents Date: Wed, 07 Jan 2004 21:57:40 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Carl Lumma>No, silly goose :), the squaring is done *before* the summing!Yes, well, I knew that. :)>The error of each interval will be a straight line. The errors >squared will be parabolas. The sum of a set of parabolas is a >parabola, since the sum of any number of functions of order 2 is a >function of order 2 -- since you're squaring some linear functions, >then adding, you'll have quadratic terms, linear terms, and constant >terms to add, and that's all.Great; this is what I was missing! Thanks! -Carl
Message: 8994 - Contents - Hide Contents Date: Wed, 07 Jan 2004 21:59:01 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Paul Erlich wrote: >>> So you're dividing by expressibility here? Interesting . . . ! >> I'm dividing by the complexity.Which complexity measure?> What's expressiblility?Kees's metric -- the log of the lowest odd limit the ratio belongs to.>> Hmm . . . are you talking about "septimal meantone", or are you >> talking about a 7-limit planar temperament? >> The usual 7-limit meantone.So how do you define and find "the worst comma"?>> Hold on. Are you talking about some 13-limit linear extension of >> miracle, or a planar temperament that tempers out the same commas as >> miracle? >> 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.>> Only taking the larger one? >> I don't now of any realistic commas with only odd numbers. 27/25. > So say you > were tempering out 15/13. Using your method will work fine, it won't be > very accurate, but you'll get a temperament out. And, because there are > no factors of 2 in the commma, there's nothing to say the octaves should > be stretched. So you already have an octave equivalent temperament! Yes. > My > method will take the complexity as log(15) rather than log(13).Who would use log(13)?> So > it'll give some other result, ???
Message: 8995 - Contents - Hide Contents Date: Wed, 07 Jan 2004 13:59:15 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Carl Lumma>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?>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, or what differentiates a period from a generator.>> With a single comma there are two 5-limit generators, three 7-limit, >> and so on. Or so I suppose. >>Yes, but then there's even less uniqueness to the choice. Gene has >proposed "hermite reduction", perhaps the issue is worth another look.Searching my new tuning-math archive for "hermite" yields 37 matches, most from March of 2002. The most recent bit (excluding the present thread) is from Gene from November of last year...>> You form a matrix with the octave (1 0 0 ...) at the top, then a >> chromatic unison vector (it doesn't matter which) and below them >> the commas. Take the adjoint (the inverse multiplied by the >> determinant). >>This proceedure only works if you have a linear temperament. >Something else you might try is finding a basis for the nullspace of >the matrix formed from the commas alone, without your additions, and >using this to obtain a reduced set of vals (which could involve some >extra work.) From there, one can put the vals in the form you like; I >am partial to Hermite reduction unless we are dealing with linear >temperaments, in which case we do period-generator and make the the >generator as small, greater than one, as possible, to get a standard >reduced form. > >Sometimes it suffices to simply find all standard vals which make all >of the commas zero and use this to start with. Finding the wedgie >from the commas, and the matrix from the wedgie, will also work; that >is how I would do this but I use Maple's Hermite reduction function >for it.From October 2003...>> I really hate to ask, but what do wedgies have to do with mapping >> generators to primes? >>I take the wedgie, and from it generate what I call the subgroup vals. >Then I hermite-reduce these, and apply a further reduction to make >the generators of the generator/period pair as small, greater than >one, as possible. This gives a standarized period/generator for the >temperament in question.As usual, however, nowhere on this list can I find any explanation of hermite reduction. Not what it is, not why we'd care, and not how to calculate it.>> 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, but somehow I think those doing rms around here were not including the 2s. -Carl
Message: 8996 - Contents - Hide Contents Date: Wed, 07 Jan 2004 22:02:56 Subject: Re: non-1200: Tenney/heursitic meantone temperamentt From: Graham Breed Carl Lumma 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"? Dave Keenan's original example was the 225:224 planar temperament, from what I remember. It gives different results, of course, because it's using a different complexity measure. I don't think anybody's done this before with continuous complexities. The 9-limit algorithm is a bit of a problem because 9 and 3 both have the same complexity. So at this point, it gets simpler if you ignore discrete limits. Still, with higher limits, it should be easier to calculate the RMS by finding the temperament mapping first than going straight to some kind of minimax by looking at the commas. Graham
Message: 8997 - Contents - Hide Contents Date: Wed, 07 Jan 2004 22:02:13 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:>> I don't now of any realistic commas with only odd numbers. > > 27/25.Another example is 245/243, but that's 7-limit, not 5-limit.
Message: 8998 - Contents - Hide Contents Date: Wed, 07 Jan 2004 17:27:28 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Carl Lumma>> > 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? -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: 8999 - Contents - Hide Contents Date: Wed, 07 Jan 2004 22:08:07 Subject: The Six Tertiadia Scales From: Gene Ward Smith ! tertiadia1.scl Tertiadia 2048/2025 262144/253125 scale 12 ! 16/15 256/225 75/64 5/4 4/3 64/45 3/2 8/5 3375/2048 225/128 15/8 2 ! tertiadia2.scl Tertiadia 2048/2025 262144/253125 scale 12 ! 135/128 9/8 6/5 32/25 675/512 45/32 3/2 8/5 128/75 2048/1125 15/8 2 ! tertiadia3.scl Tertiadia 2048/2025 262144/253125 scale 12 ! 16/15 256/225 4096/3375 5/4 675/512 45/32 3/2 8/5 128/75 225/128 15/8 2 ! tertiadia4.scl Tertiadia 2048/2025 262144/253125 scale 12 ! 16/15 256/225 6/5 10125/8192 675/512 45/32 3/2 8/5 128/75 225/128 15/8 2 ! tertiadia5.scl Tertiadia 2048/2025 262144/253125 scale 12 ! 16/15 256/225 75/64 5/4 4/3 45/32 3/2 8/5 3375/2048 225/128 15/8 2 ! tertiadia6.scl Tertiadia 2048/2025 262144/253125 scale 12 ! 16/15 256/225 75/64 5/4 4/3 64/45 1024/675 16384/10125 3375/2048 225/128 15/8 2
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