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Message: 6050 - Contents - Hide Contents Date: Fri, 17 Jan 2003 10:32:34 Subject: Re: heuristic and straightness From: Carl Lumma>> >ight, the difference vector has to vanish, too. Ok. What >> I don't get is, for a given temperament, can I change the >> straightness by changing the unison vector representation? > > yes. Ok.>> If so, this means that badness is not fixed for a given >> temperament... >> that's not true. since both the defining unison vectors *and* > the straightness change, the badness can (and will) remain > constant.Then how can it become "terrible"?>> Also, can I change the straightness by transposing pitches >> by uvs? >> this is meaningless, as we're talking about temperaments, not > irregular finite periodicity blocks. we're talking either equal > temperaments or infinite regular tunings.Straightness certainly sounds like it can be defined on an untempered block.>> Finally, is "commatic basis" an acceptable synonym for >> "kernel"? >> no. every vector that can be generated from the basis belongs > to the kernel, but not every set of n members of the kernel > (even if they're linearly independent) is a basis (since you may > get torsion).So is it possible to have more than one kernel representation for a temperament? -Carl
Message: 6051 - Contents - Hide Contents Date: Fri, 17 Jan 2003 11:38:47 Subject: Re: heuristic and straightness From: Graham Breed Gene Ward Smith wrote:> One method which might come to the same thing as "straightness" in effect is to take two commas, and combine to get a codimension 2 wedgie. Produce a list of these by taking the best (here you run into geometric badness) and then wedge these with another comma, and so forth.So is "geometric badness" simplicity or complexity of the exterior element? Graham
Message: 6052 - Contents - Hide Contents Date: Fri, 17 Jan 2003 01:41:31 Subject: Re: Nonoctave scales and linear temperaments From: Carl Lumma [I wrote...]>Perhaps we could enforce "validity", and maybe also Kees' >'complexity validity'.I guess I should have called that 'expressibility validity'. -Carl
Message: 6053 - Contents - Hide Contents Date: Fri, 17 Jan 2003 10:33:29 Subject: Re: heuristic and straightness From: Graham Breed wallyesterpaulrus wrote:> then you need a 3-d generalization of "straightness". i bet if i went > and learned grassmann algebra i'd be able to get a better grasp on > all this.Yes, probably. Vectors are straighter the smaller the area of the parallelogram they describe. The wedge product is a generalisation of area, as it tends towards the determinant. So hopefully we can do something with the intermediate wedge products. But I don't know what :( Graham
Message: 6054 - Contents - Hide Contents Date: Fri, 17 Jan 2003 15:16:52 Subject: Re: [tuning] margo, manuel -- was it kirnberger? From: manuel.op.de.coul@xxxxxxxxxxx.xxx Paul asked:>was it kirnberger who proposed foreshortening each fifth by a schisma >to approximate 12-equal? a chain of 12 such fifths would fail to >close on itself by a mere "atom", or .015361 cents . . .That I don't know, but he was at least the first to publish a temperament with a fifth flattened by a schisma, in the 1780's. Then Prinz did so around 1810. Found this on Nigel Taylor's tuning page * [with cont.] (Wayb.) Manuel
Message: 6055 - Contents - Hide Contents Date: Fri, 17 Jan 2003 01:48:49 Subject: heuristic and straightness From: Carl Lumma>>> >t's nice when there's only one comma. then the log of the >>> numbers in the comma (say, the log of the odd limit) is an >>> excellent estimate of complexity (it's what i call the >>> heuristic complexity). >>>> That's what I call taxicab complexity, I think. >> not quite. for one thing, read this: > > lattice orientation * [with cont.] (Wayb.) > > including the link to my observations.All I get from this is that it depends whether one uses a triangular or rectangular lattice. I must be missing something... ----->>> if there's more than one comma being tempered out, we need >>> a notion of the "angle" between the commas . . . >> >> Please explain. >> search for "straightess" in these archives . . .For a given block, notes can be transposed by unison vectors. This changes the shape of the block. Does it change its straightness? Don't follow why the less straight blocks are supposed to be less interesting. ----- Here's what I found on the heuristic. Last time I asked, you referred me to this message: Yahoo groups: /tuning-math/messages/2491?expand=1 * [with cont.] Which I can't follow at all. Which column is the heuristic, what are the other columns, and what are their values expected to do (go down or up...)? I also found this blurb:>the heuristics are only formulated for the one-unison-vector >case (e.g., 5-limit linear temperaments), and no one has bothered >to figure out the metric that makes it work exactly (though it >seems like a tractable math problem). but they do seem to work >within a factor of two for the current "step" and "cent" >functions. "step" is approximately proportional to log(d), >and "cent" is approximately proportional to (n-d)/(d*log(d)).Why are they called "step" and "cent"? How were they derrived? -Carl
Message: 6056 - Contents - Hide Contents Date: Fri, 17 Jan 2003 10:49:05 Subject: Re: heuristic and straightness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>> Right, the difference vector has to vanish, too. Ok. What >>> I don't get is, for a given temperament, can I change the >>> straightness by changing the unison vector representation? >> >> yes. > > Ok. >>>> If so, this means that badness is not fixed for a given >>> temperament... >>>> that's not true. since both the defining unison vectors *and* >> the straightness change, the badness can (and will) remain >> constant. >> Then how can it become "terrible"?if you change to a "straighter" pair of unison vectors, one or both of them will have to be a lot shorter, thus less distribution of error and a worse temperament.>>> Also, can I change the straightness by transposing pitches >>> by uvs? >>>> this is meaningless, as we're talking about temperaments, not >> irregular finite periodicity blocks. we're talking either equal >> temperaments or infinite regular tunings. >> Straightness certainly sounds like it can be defined on an > untempered block.well, that's not the context in which it's been discussed, and thus not the context into which your questions about it were posed.>>> Finally, is "commatic basis" an acceptable synonym for >>> "kernel"? >>>> no. every vector that can be generated from the basis belongs >> to the kernel, but not every set of n members of the kernel >> (even if they're linearly independent) is a basis (since you may >> get torsion). >> So is it possible to have more than one kernel representation > for a temperament?no. the kernel is an infinite set of vectors, and is unique to the temperament.
Message: 6057 - Contents - Hide Contents Date: Fri, 17 Jan 2003 18:38:30 Subject: Re: heuristic and straightness From: Carl Lumma>>> >hat's not true. since both the defining unison vectors *and* >>> the straightness change, the badness can (and will) remain >>> constant. >>>> Then how can it become "terrible"? >> if you change to a "straighter" pair of unison vectors, one or > both of them will have to be a lot shorter, thus less distribution > of error and a worse temperament.I was trying to point out that badness here has failed to reflect your opinion of the temperament.>>>> Also, can I change the straightness by transposing pitches >>>> by uvs? >>>>>> this is meaningless, as we're talking about temperaments, not >>> irregular finite periodicity blocks. we're talking either equal >>> temperaments or infinite regular tunings. >>>> Straightness certainly sounds like it can be defined on an >> untempered block. >>well, that's not the context in which it's been discussed, and >thus not the context into which your questions about it were posed.That's true, but I couldn't see making a separate post. I'm trying to understand your concept of straightness.>no. the kernel is an infinite set of vectors, and is unique to the >temperament.Whew. Got it! -C.
Message: 6058 - Contents - Hide Contents Date: Fri, 17 Jan 2003 01:55:47 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>> Can I identify the duplicate temperaments? >>>> duplicate? any temperament can be generated from an infinite >> number of possible basis vectors. >> "The results of a search of all possible maps is bound to return > pairs, trios, etc. of maps that represent the same temperament. > Can we find them in the mess of results?" Was all I was asking!yes, since you can calculate the wedgie of each. but clearly this is a terrible way to go about the search. how would you even delimit it?>>> Assuming 2:1 reduction makes me squirm in my chair, is all. >>> Plentiful near-2:1s should emerge from the search if the criteria >>> are right. >>>> if the criteria include mapping to, and minimizing error from, >> 2:1, then of course a near 2:1 will emerge in each temperament. >> Yup, that's what I've been saying alright.and this kind of optimization has been done a few times, for example by gene.>>>>> Are you saying a badness cutoff is not sufficient to give a >>>>> finite list of temperaments? >>>>>>>> exactly. in *every* complexity range you have about the same >>>> number of temperaments with log-flat badness lower than some >>>> cutoff -- and there are an infinite number of non-overlapping >>>> complexity ranges. >>>>>> Oh. I guess I need some examples, then, of most of the simple >>> temperaments that are garbage... >>>> what are the 20 simplest 5-limit intervals? now set each of >> these to be the commatic unison vector, and what temperaments >> do you get? >> Perhaps we could enforce "validity", ? > and maybe also Kees' > 'complexity validity'. ??
Message: 6059 - Contents - Hide Contents Date: Fri, 17 Jan 2003 10:52:41 Subject: Re: heuristic and straightness From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:> Maybe the original exposition can just be updated a bit, and > then monz or I could host it, certainly.You might want to add to complexity ~ log(d) error ~ log(n-d)/(d log(d)) a badness heursitic of badness ~ log(n-d) log(d)^e / d where e = pi(prime limit)-1 = number of odd primes in limit.
Message: 6060 - Contents - Hide Contents Date: Fri, 17 Jan 2003 23:49:59 Subject: Re: heuristic and straightness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>>> that's not true. since both the defining unison vectors *and* >>>> the straightness change, the badness can (and will) remain >>>> constant. >>>>>> Then how can it become "terrible"? >>>> if you change to a "straighter" pair of unison vectors, one or >> both of them will have to be a lot shorter, thus less distribution >> of error and a worse temperament. >> I was trying to point out that badness here has failed to reflect > your opinion of the temperament. how so?
Message: 6061 - Contents - Hide Contents Date: Fri, 17 Jan 2003 02:06:51 Subject: Re: heuristic and straightness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>>> it's nice when there's only one comma. then the log of the >>>> numbers in the comma (say, the log of the odd limit) is an >>>> excellent estimate of complexity (it's what i call the >>>> heuristic complexity). >>>>>> That's what I call taxicab complexity, I think. >>>> not quite. for one thing, read this: >> >> lattice orientation * [with cont.] (Wayb.) >> >> including the link to my observations. >> All I get from this is that it depends whether one uses a > triangular or rectangular lattice. I must be missing > something...it shows that the 'expressibility' metric is not quite the same as the taxicab metric on the isosceles-triangular lattice. you need to use scalene triangles of a certain type, which i don't think is what you were thinking when you wrote "taxicab complexity" above.>>>> if there's more than one comma being tempered out, we need >>>> a notion of the "angle" between the commas . . . >>> >>> Please explain. >>>> search for "straightess" in these archives . . . >> For a given block, notes can be transposed by unison vectors. > This changes the shape of the block. Does it change its > straightness?straightness applies to a set of unison vectors. different sets of unison vectors can define the same temperament. a temperament may look good on the basis of being defined by good unison vectors. but in fact you may end up with a terrible temperament if the unison vectors point in approximately the same direction.> Here's what I found on the heuristic. Last time I asked, > you referred me to this message: > > Yahoo groups: /tuning-math/messages/2491?expand=1 * [with cont.] > > Which I can't follow at all.well, let me help you then.> Which column is the heuristic,column V is proportional to the heuristic error, and Y is proportional to the heuristic complexity.> what are the other columns,U is the rms error, W is the ratio of the two error measures. X is the complexity (weighted rms of generators-per-consonance at that point i believe), and Z is the ratio of the two complexity measures.> and what are their values expected > to do (go down or up...)?W and Z are expected to remain relatively constant.> I also found this blurb: >>> the heuristics are only formulated for the one-unison-vector >> case (e.g., 5-limit linear temperaments), and no one has bothered >> to figure out the metric that makes it work exactly (though it >> seems like a tractable math problem). but they do seem to work >> within a factor of two for the current "step" and "cent" >> functions. "step" is approximately proportional to log(d), >> and "cent" is approximately proportional to (n-d)/(d*log(d)). >> Why are they called "step" and "cent"? How were they derrived?that's what gene used to call them. "step" is simply complexity, and "cent" is simply rms error.
Message: 6062 - Contents - Hide Contents Date: Fri, 17 Jan 2003 10:56:01 Subject: Re: heuristic and straightness From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> wallyesterpaulrus wrote: >>> then you need a 3-d generalization of "straightness". i bet if i went >> and learned grassmann algebra i'd be able to get a better grasp on >> all this. >> Yes, probably. Vectors are straighter the smaller the area of the > parallelogram they describe. The wedge product is a generalisation of > area, as it tends towards the determinant. So hopefully we can do > something with the intermediate wedge products. But I don't know what :(The geometic complexity is a measure of the length of the wedge product of the commas.
Message: 6063 - Contents - Hide Contents Date: Fri, 17 Jan 2003 23:47:34 Subject: Re: heuristic and straightness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> wallyesterpaulrus wrote: >>> so the main difference is that you're using a euclidean metric (for >> geometric complexity), while i'm using a taxicab one (for heuristic >> complexity). >> I take it you're adding either the moduli or squares of the coefficients > of the exterior element (wedge product of vector)?i don't even know what that means.> Well, that's easy > enough. And the more complex the better, because that covers the > complexity of the original unison vectors and their straightness. If > we've already chosen unison vectors that are small pitch intervals, do > we have a badness measure? not following.
Message: 6064 - Contents - Hide Contents Date: Fri, 17 Jan 2003 03:13:30 Subject: Re: heuristic and straightness From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote: but they do seem to work>>> within a factor of two for the current "step" and "cent" >>> functions. "step" is approximately proportional to log(d), >>> and "cent" is approximately proportional to (n-d)/(d*log(d)). >>>> Why are they called "step" and "cent"? How were they derrived? >> that's what gene used to call them. "step" is simply complexity, > and "cent" is simply rms error.We could put this together, and get "bad" as heuristically as bad = (n-d)*log(d)^2/d We might also do multiple linear regression analysis on log badness vs log(n-d), loglog(d) and log(d) and see what we got.
Message: 6065 - Contents - Hide Contents Date: Fri, 17 Jan 2003 10:59:34 Subject: Re: heuristic and straightness From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> The geometic complexity is a measure of the length of the wedge product of the commas.Of commas, taken mod 2 (note classes or whatever you want to call it.) I'm not sure what you want here--if all of the commas point in about the same direction, do you mean with or without 2?
Message: 6066 - Contents - Hide Contents Date: Fri, 17 Jan 2003 05:40:49 Subject: Re: heuristic and straightness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> > but they do seem to work>>>> within a factor of two for the current "step" and "cent" >>>> functions. "step" is approximately proportional to log(d), >>>> and "cent" is approximately proportional to (n-d)/(d*log(d)). >>>>>> Why are they called "step" and "cent"? How were they derrived? >>>> that's what gene used to call them. "step" is simply complexity, >> and "cent" is simply rms error. >> We could put this together, and get "bad" as heuristically as > > bad = (n-d)*log(d)^2/dof course -- where the second "d" should really be odd limit.> We might also do multiple linear regression analysis on log badness >vs log(n-d), loglog(d) and log(d) and see what we got.i'm not following. i do multiple linear regression analysis every day, so please clarify!
Message: 6067 - Contents - Hide Contents Date: Fri, 17 Jan 2003 11:07:42 Subject: Re: heuristic and straightness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote: >> wallyesterpaulrus wrote: >>>>> then you need a 3-d generalization of "straightness". i bet if i went >>> and learned grassmann algebra i'd be able to get a better grasp on >>> all this. >>>> Yes, probably. Vectors are straighter the smaller the area of the >> parallelogram they describe. The wedge product is a generalisation of >> area, as it tends towards the determinant. So hopefully we can do >> something with the intermediate wedge products. But I don't know what :( >> The geometic complexity is a measure of the length of the wedge >product of the commas.or the length of the comma, if there's only one. so the main difference is that you're using a euclidean metric (for geometric complexity), while i'm using a taxicab one (for heuristic complexity).
Message: 6068 - Contents - Hide Contents Date: Fri, 17 Jan 2003 06:17:50 Subject: Re: heuristic and straightness From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:>> We could put this together, and get "bad" as heuristically as >> >> bad = (n-d)*log(d)^2/d >> of course -- where the second "d" should really be odd limit."d" does not mean denominator?>> We might also do multiple linear regression analysis on log badness >> vs log(n-d), loglog(d) and log(d) and see what we got.> i'm not following. i do multiple linear regression analysis every > day, so please clarify!I'm not following what you're not following. I find that log(d) is a good heuristic for geometric 5-limit complexity, at least in the comma range of sizes. A linear regression of log(complexity) vs log(d) gives c ~ .991685*log(d)^.986763, which is awfully close to log(d); in fact some of the time the geometric complexity is exactly log(d). A model using log|n-d|, loglog(d), and log(d) (each d being the denominator, and n the numerator) gives log(c) ~ -.009541*log|n-d|+.009534*log(d)+.967642*loglog(d)-.0090687
Message: 6069 - Contents - Hide Contents Date: Fri, 17 Jan 2003 11:08:35 Subject: Re: heuristic and straightness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> I'm not sure what you want here--if all of the commas point in >about the same direction, do you mean with or without 2?i'm thinking without.
Message: 6070 - Contents - Hide Contents Date: Fri, 17 Jan 2003 07:06:09 Subject: Re: heuristic and straightness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:>>>> We could put this together, and get "bad" as heuristically as >>> >>> bad = (n-d)*log(d)^2/d >>>> of course -- where the second "d" should really be odd limit. >> "d" does not mean denominator?it can, or it can mean odd limit of the ratio (in other words, the ratio is a "Ratio Of d") -- the difference is slight. the second and third "d", i meant to say.>>> We might also do multiple linear regression analysis on log badness >>> vs log(n-d), loglog(d) and log(d) and see what we got. >>> i'm not following. i do multiple linear regression analysis every >> day, so please clarify! >> I'm not following what you're not following. > > I find that log(d) is a good heuristic for geometric 5-limit >complexity, at least in the comma range of sizes.in all ranges, i think you'll find.> A linear regression > of log(complexity) vs log(d) gives c ~ .991685*log(d)^.986763,oh, so you mean to use *your* complexity/badness/whatever as the dependent variable in the regression! clearly, though, you understand the heuristic.>which is awfully close to log(d); in fact some of the time the >geometric complexity is exactly log(d). A model using log|n-d|, >loglog(d), and log(d) (each d being the denominator, and n the >numerator) gives > > log(c) ~ -.009541*log|n-d|+.009534*log(d)+.967642*loglog(d)-.0090687where c is complexity . . . don't forget to report standard error ranges for your coefficient estimates!
Message: 6071 - Contents - Hide Contents Date: Fri, 17 Jan 2003 11:14:25 Subject: Re: heuristic and straightness From: Graham Breed wallyesterpaulrus wrote:> so the main difference is that you're using a euclidean metric (for > geometric complexity), while i'm using a taxicab one (for heuristic > complexity).I take it you're adding either the moduli or squares of the coefficients of the exterior element (wedge product of vector)? Well, that's easy enough. And the more complex the better, because that covers the complexity of the original unison vectors and their straightness. If we've already chosen unison vectors that are small pitch intervals, do we have a badness measure? Graham
Message: 6072 - Contents - Hide Contents Date: Fri, 17 Jan 2003 07:07:35 Subject: kewl math diversion for anyone bored/disinterested/lost From: wallyesterpaulrus 404 Not Found * [with cont.] Search for http://www.maths.bris.ac.uk/~maadb/research/seminars/online/fgfut/fgfu in Wayback Machine t21.html
Message: 6073 - Contents - Hide Contents Date: Fri, 17 Jan 2003 07:08:17 Subject: kewl math diversion for anyone bored/disinterested/lost From: wallyesterpaulrus 404 Not Found * [with cont.] Search for http://www.maths.bris.ac.uk/~maadb/research/seminars/online/fgfut/fgfu in Wayback Machine t21.html
Message: 6074 - Contents - Hide Contents Date: Fri, 17 Jan 2003 07:53:09 Subject: Re: Nonoctave scales and linear temperaments From: Carl Lumma>> >The results of a search of all possible maps is bound to return >> pairs, trios, etc. of maps that represent the same temperament. >> Can we find them in the mess of results?" Was all I was asking! >>yes, since you can calculate the wedgie of each. but clearly this >is a terrible way to go about the search.Well, well by now I of course agree.>how would you even delimit it?I won't ask...>> Perhaps we could enforce "validity", > > ?That's Gene's name for a concept you said was equivalent to the condition that all steps of a block be larger than its unison vectors.>> and maybe also Kees' 'complexity validity'. > > ??See later message. -Carl
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