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Message: 6075 - Contents - Hide Contents Date: Fri, 17 Jan 2003 18:11:39 Subject: Re: A common notation for JI and ETs From: David C Keenan Continuing down the ratio popularity list, of those we don't yet have a symbol for: There are two 175 (5*5*7)commas of interest 175-diesis 512:525 43.41 c //| 0.40 c schisma 175'-diesis 127575:131072 46.82 c ./|) exact, no symbol without 5' two 245 (5*7*7) commas 245-comma 243:245 14.19 c ~|( 0.54 c schisma 245'-diesis 524288:535815 37.65 c /|~ 0.40 c schisma two 625 (5^4) commas 625-comma 4100625:4194304 39.11 c (|( 0.20 and 1.04 c schismas 625-comma 625:648 62.57 c '(|) 0.08 c sch, no sym w/o 5' You remember that you were concerned about symbols for 23 and 24 degrees of 494-ET if we eliminated the |\) and (/| symbols. Although I didn't notice at the time, we already had one for 23deg494 which is the 5:49'-diesis '|)) But we are left with the problem of 24deg494, or put another way: What should be the symbols for the apotome complements of |)) and '|)). The apotome complement of 56.48 c |)) is 59.16 c. The only possible two-flag symbol for that is |\) It seemed a lot tidier when we had (/| and |\) as complements, however this had a serious lateral confusability problem which we might now consider solved. So 24deg494 would be .|\) Not very nice to have to introduce 5' accents just for these two degrees, but there it is. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 6076 - Contents - Hide Contents Date: Fri, 17 Jan 2003 08:25:19 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>> "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.there are no blocks here, just temperaments.

Message: 6077 - Contents - Hide Contents Date: Fri, 17 Jan 2003 08:26:46 Subject: Re: Nonoctave scales and linear temperaments From: Graham Breed Gene Ward Smith wrote:> The prime number theorem says pi(x)~x/(ln(x)-1).Oh, I think I had that backwards.>> Whereas combing equal temperaments only gives O(n**2) calculations, >> where n is the number of ETs you consider. I find n=20 works well, >> requiring O(400) candidates. >> Once you take wedgies, you should have fewer candidates.Maybe, but isn't taking wedgies as hard as finding the maps? How do you avoid calculating a huge number of wedge products? (Oh, n=20 gives exactly 190 candidates) Graham

Message: 6078 - Contents - Hide Contents Date: Fri, 17 Jan 2003 00:05:07 Subject: Re: Mega 11 and 13 limit ets 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:>>> what happened to 504504? >> I checked from 3 to 23, and it's not nearly good enough to make the >cut in any of them. Are you sure this is the right number?i got it from: Definitions of tuning terms: list of EDOs used... * [with cont.] (Wayb.)

Message: 6079 - Contents - Hide Contents Date: Fri, 17 Jan 2003 08:31:53 Subject: Re: heuristic and straightness From: Carl Lumma>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.That's true; I just count the rungs. Any lattice that's 'topologically' (?) equivalent to the triangular lattice will do.>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.Why would it be terrible?>> Yahoo groups: /tuning-math/messages/2491?expand=1 * [with cont.] // >> 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.Thanks again! Now everything is clear. Except how you derrived the heuristics! Seriously man, one expository blurb would save you from having to do this for each person who's interested, and for me again when I've forgotten it in 6 months. :~) -Carl

Message: 6080 - Contents - Hide Contents Date: Fri, 17 Jan 2003 00:09:55 Subject: Re: Nonoctave scales and linear temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

Message: 6081 - Contents - Hide Contents Date: Fri, 17 Jan 2003 08:50:23 Subject: Re: heuristic and straightness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>> 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. >> That's true; I just count the rungs. Any lattice that's > 'topologically' (?) equivalent to the triangular lattice > will do.you mean approximately?>> 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. >> Why would it be terrible? heuristically speaking,in most cases the *difference* between the unison vectors will be of similar or greater magnitude in terms of JI comma interval size as the unison vectors themselves, but since the angle is very small, this *difference* vector will be very short (i.e., low complexity). a comma of a given JI interval size will lead to much higher error if tempered out over much fewer consonant rungs (i.e., if it's very short) than if it's tempered out over more consonant rungs (i.e., if it's long). therefore, you may end up with a temperament with much larger error than you would have expected given your original pair of unison vectors.> Thanks again! Now everything is clear. Except how you > derrived the heuristics!i did post the derivation a while back, probably before i even used the word "heuristic", but if you search for "heuristic", i think you'll find a post that links to the derivation post.> Seriously man, one expository blurb would save you from having > to do this for each person who's interested, and for me again > when I've forgotten it in 6 months. :~)i welcome suggestions or just make your own blurb, and let's put this on a webpage somewhere.

Message: 6082 - Contents - Hide Contents Date: Fri, 17 Jan 2003 00:13:49 Subject: Re: Nonoctave scales and linear temperaments From: Carl Lumma>> >an you find them at the end of a map-space search, and take >> them out? >>what do you want to take out?Can I identify the duplicate temperaments?>from what?A search of all possible maps.> what does "//" mean?Cut. I've been using it since I've been on these lists.>correct. the untweakable generator has been tweaked. is that all?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.>> Thanks again. So if reduction is necc., it means that a >> temperament can be described by two different lists of commas, >> right? >>right, although for single-comma temperaments, only one choice >leaves you without torsion.Thanks yet again.>> This means we'll have the same problem searching comma >> space as we did map space. So wedgies are our last hope. >>the problem i was pointing out with map space, i think, was >that the arbitrariness of the set of generators means your >complexity ranking (if it's just based on the numbers of the >map) will be meaningless.Oh. Now I get it! You're right. But doesn't the same problem occur with different commatic representations, when defining complexity off the commas?>> 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... -Carl

Message: 6084 - Contents - Hide Contents Date: Fri, 17 Jan 2003 00:18:38 Subject: Re: Mega 11 and 13 limit ets From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith > <genewardsmith@j...>" <genewardsmith@j...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus> <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote: >>>>> what happened to 504504? >>>> I checked from 3 to 23, and it's not nearly good enough to make the >> cut in any of them. Are you sure this is the right number? >I'd guess he chose it on the basis of its prime factorization: 504504 = 2^3 3^2 5^2 7 11 13 I'm not sure why he didn't choose the lcm of all numbers up to 15 (360360) or 16 (720720) instead.

Message: 6085 - Contents - Hide Contents Date: Fri, 17 Jan 2003 09:39:16 Subject: Re: heuristic and straightness From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:>> 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.The heuristic for complexity works particularly well as an estimate for geometric complexity. I did a regression analysis (which you could no doubt do better, if you have stat software) which also confirmed that your heuristic for error seems to have the right coefficients. The result is that your estimates look like an excellent way of quickly filtering a list of commas for geometric complexity, rms error, and geometic badness, which could then be used to compute wedgies. If we had estimates for the wedgies that would be even better.> don't forget to report standard error ranges for your coefficient > estimates!I don't have a program which gives me residuals or F-ratios or anything which one really wants to do this right. You can probably do it better; just remember to take the log of both sides.

Message: 6086 - Contents - Hide Contents Date: Fri, 17 Jan 2003 00:22:12 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>> Can you find them at the end of a map-space search, and take >>> them out? >>>> what do you want to take out? >> Can I identify the duplicate temperaments?duplicate? any temperament can be generated from an infinite number of possible basis vectors.>> correct. the untweakable generator has been tweaked. is that all? >> 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.>>> This means we'll have the same problem searching comma >>> space as we did map space. So wedgies are our last hope. >>>> the problem i was pointing out with map space, i think, was >> that the arbitrariness of the set of generators means your >> complexity ranking (if it's just based on the numbers of the >> map) will be meaningless. >> Oh. Now I get it! You're right. But doesn't the same > problem occur with different commatic representations, when > defining complexity off the commas?not if you define complexity right!>>> 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? gene has studied a few of them because they actually had low badness -- but the error is so high that the generators themselves sometimes provide better approximations to a consonance than the interval the consonance is mapped to!

Message: 6087 - Contents - Hide Contents Date: Fri, 17 Jan 2003 10:16:06 Subject: Re: heuristic and straightness From: Graham Breed --- wallyesterpaulrus wrote:> 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.Then that's what you need to reduce the complexity of the search. I can't find a quantitative definition of "straightness" in the archives. What is it? I presume it works for insufficient sets of unison vectors. Graham

Message: 6088 - Contents - Hide Contents Date: Fri, 17 Jan 2003 00:22:22 Subject: Re: Mega 11 and 13 limit ets From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith > > Definitions of tuning terms: list of EDOs used... * [with cont.] (Wayb.)> > I'd guess he chose it on the basis of its prime factorization:Sorry, that should be 504504 = 2^3 3^2 7^2 11 13 No 5s, obviously!

Message: 6089 - Contents - Hide Contents Date: Fri, 17 Jan 2003 10:18:52 Subject: Re: heuristic and straightness From: Carl Lumma>> >hat's true; I just count the rungs. Any lattice that's >> 'topologically' (?) equivalent to the triangular lattice >> will do. >>you mean approximately?? I originally meant that counting the rungs between a dyad and the origin would be approximately the same as the log of the odd limit of the dyad... lnoddlimit 7limittaxicab ratio 15:8 2.7 2 1.35 5:3 1.6 1 1.6 105:64 4.7 3 1.6 225:224 5.4 4 1.35>>> but in fact you may end up with a terrible temperament >>> if the unison vectors point in approximately the same >>> direction. >>>> Why would it be terrible? > >heuristically speaking, >>in most cases the *difference* between the unison vectors >will be of similar or greater magnitude in terms of JI comma >interval size as the unison vectors themselves, but since >the angle is very small, this *difference* vector will be >very short (i.e., low complexity). a comma of a given JI >interval size will lead to much higher error if tempered out >over much fewer consonant rungs (i.e., if it's very short) >than if it's tempered out over more consonant rungs (i.e., >if it's long). therefore, you may end up with a temperament >with much larger error than you would have expected given >your original pair of unison vectors.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? If so, this means that badness is not fixed for a given temperament... Also, can I change the straightness by transposing pitches by uvs? Finally, is "commatic basis" an acceptable synonym for "kernel"?>i did post the derivation a while back, probably before i >even used the word "heuristic", but if you search for >"heuristic", i think you'll find a post that links to the >derivation post.I think I remember it coming out, but I couldn't find it today in my searches. I did try.>i welcome suggestions or just make your own blurb, and >let's put this on a webpage somewhere.Maybe the original exposition can just be updated a bit, and then monz or I could host it, certainly. -C.

Message: 6090 - Contents - Hide Contents Date: Fri, 17 Jan 2003 00:24:05 Subject: Re: Mega 11 and 13 limit ets 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:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith >> >>> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus>>>> >>>> what happened to 504504? >>>>>> I checked from 3 to 23, and it's not nearly good enough to make the >>> cut in any of them. Are you sure this is the right number? >>> I'd guess he chose it on the basis of its prime factorization: > > 504504 = 2^3 3^2 5^2 7 11 13that wouldn't make sense, since he also chose 15601 as "most convenient" (this time for 3-limit), and it's prime!

Message: 6091 - Contents - Hide Contents Date: Fri, 17 Jan 2003 10:22:37 Subject: Re: heuristic and straightness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Graham Breed <graham@m...>" <graham@m...> wrote:> --- wallyesterpaulrus wrote: > Then that's what you need to reduce the complexity of the search. I > can't find a quantitative definition of "straightness" in the > archives.i haven't come up with one yet.> I presume it works for insufficient sets of > unison vectors.insufficient? how can a set of unison vectors be insufficient?

Message: 6092 - Contents - Hide Contents Date: Fri, 17 Jan 2003 00:25:26 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" > <clumma@y...> wrote:>>>> Can you find them at the end of a map-space search, and take >>>> them out? >>>>>> what do you want to take out? >>>> Can I identify the duplicate temperaments? >> duplicate? any temperament can be generated from an infinite number > of possible basis vectors.oops -- i meant sets of possible basis vectors, assuming the temperament is 2-d or more.

Message: 6093 - Contents - Hide Contents Date: Fri, 17 Jan 2003 10:26:40 Subject: Re: heuristic and straightness From: Graham Breed wallyesterpaulrus wrote:> insufficient? how can a set of unison vectors be insufficient?One unison vector is insufficient to define a 7-limit linear temperament, two or three unison vectors are insufficient to define a 21-limit linear temperament. For an arbitrary search to be practicable, there has to be a way of rejecting sets of three unison vectors because you know they can't give a good linear temperament. That would reduce the search to more like O(n**3) in the number of unison vectors instead of O(n**6). Graham

Message: 6094 - Contents - Hide Contents Date: Fri, 17 Jan 2003 00:45:24 Subject: Re: Nonoctave scales and linear temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:>> Oops! That should be O((r**7)**6) or O(r**42) and 6 from 42 only > gives>> 5 million or so combinations. >> still vastly redundant.One possibility would be to choose a comma list with a badness cutoff, where the badness was the badness of the corresponding codimension one, or one-comma, temperament. I think I'll try that unless someone has a better suggestion.

Message: 6095 - Contents - Hide Contents Date: Fri, 17 Jan 2003 10:28:00 Subject: Re: heuristic and straightness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"> 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. > 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.> 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.> 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).

Message: 6096 - Contents - Hide Contents Date: Sat, 18 Jan 2003 22:51:43 Subject: Re: heuristic and straightness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>> in most cases the *difference* between the unison vectors >>> will be of similar or greater magnitude in terms of JI comma >>> interval size as the unison vectors themselves, but since >>> the angle is very small, this *difference* vector will be >>> very short (i.e., low complexity). a comma of a given JI >>> interval size will lead to much higher error if tempered out >>> over much fewer consonant rungs (i.e., if it's very short) >>> than if it's tempered out over more consonant rungs (i.e., >>> if it's long). therefore, you may end up with a temperament >>> with much larger error than you would have expected given >>> your original pair of unison vectors. >>>> 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? >> 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 [the temperament] become "terrible"? >>>>> if you change to a "straighter" pair of unison vectors, >> Wait a minute -- straightness goes up or down with the angle > between the vectors? I thought up.yep -- until you reach 90 degrees (or whatever the equivalent is in the metric you're using).>>> one or both of them will have to be a lot shorter, thus less >>> distribution of error and a worse temperament. >> I don't follow the 'shorter' bit. The only thing I thought > straightness did was make the difference vector more complex.yes, or if you're decreasing from >90 degrees to 90 degrees, it's the sum vector that gets more complex.>> shortening the unison vectors makes the temperament worse, but >> in a given temperament, this would be counteracted by an >> increase in straighness, which makes the temperament better. >> You lost me.well, there are probably too many counterfactuals here. why don't we start again with any examples from the archives which came up in connection with straightness. your choice.>> overall, the measure must remain fixed for a given temperament, >> otherwise it's meaningless. >

Message: 6097 - Contents - Hide Contents Date: Sat, 18 Jan 2003 01:19:14 Subject: Re: heuristic and straightness From: Carl Lumma>> > was trying to point out that badness here has failed >> to reflect your opinion of the temperament. > > how so?You said the temperament got worse, but the badness remained constant. -C.

Message: 6098 - Contents - Hide Contents Date: Sat, 18 Jan 2003 01:53:18 Subject: Re: heuristic and straightness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>> I was trying to point out that badness here has failed >>> to reflect your opinion of the temperament. >> >> how so? >> You said the temperament got worse,shortening the unison vectors makes the temperament worse, but in a given temperament, this would be counteracted by an increase in straighness, which makes the temperament better. overall, the measure must remain fixed for a given temperament, otherwise it's meaningless.

Message: 6099 - Contents - Hide Contents Date: Sat, 18 Jan 2003 05:43:22 Subject: Re: heuristic and straightness From: Carl Lumma>> >n most cases the *difference* between the unison vectors >> will be of similar or greater magnitude in terms of JI comma >> interval size as the unison vectors themselves, but since >> the angle is very small, this *difference* vector will be >> very short (i.e., low complexity). a comma of a given JI >> interval size will lead to much higher error if tempered out >> over much fewer consonant rungs (i.e., if it's very short) >> than if it's tempered out over more consonant rungs (i.e., >> if it's long). therefore, you may end up with a temperament >> with much larger error than you would have expected given >> your original pair of unison vectors. >>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? >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 [the temperament] become "terrible"? >>> if you change to a "straighter" pair of unison vectors,Wait a minute -- straightness goes up or down with the angle between the vectors? I thought up.>> one or both of them will have to be a lot shorter, thus less >> distribution of error and a worse temperament.I don't follow the 'shorter' bit. The only thing I thought straightness did was make the difference vector more complex.>shortening the unison vectors makes the temperament worse, but >in a given temperament, this would be counteracted by an >increase in straighness, which makes the temperament better.You lost me.>overall, the measure must remain fixed for a given temperament, >otherwise it's meaningless.If by "the measure", you mean badness, I agree. -Carl

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