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Message: 6900 - Contents - Hide Contents Date: Mon, 16 Jun 2003 21:41:01 Subject: Re: Interval Database Experiences From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >>> you were never very specific about what's wrong >> with it. >> What I mean by a "microtemperament" is something like this: one where > all of the relevant consonant intervals are within a cent of being > pure. What you've defined is a regular temperament of dimension > greater than or equal to one, so I think it should not be used as a > definition of microtemperament.agreed, though what monz (actually graham) defined in the first attempt is that the dimension must be *higher* than that of a linear temperament. but he appears to be willing to take that back further down on the page. this definition page is a huge mess! the final quote on the page should be attributed to graham, not to me. he's clearly not sticking to his original attempt at a definition, so why should it be on the top of the page? if this dictionary is to be of use to anyone, we have to eliminate the nonsense or at least relegate it to some back pages.
Message: 6901 - Contents - Hide Contents Date: Mon, 16 Jun 2003 15:16:44 Subject: Re: Fried Alaska From: Carl Lumma>>>> >nd for any given brat, you can get arbitrarily close to JI. >>>>>> Oh yeah? Then brats are nonsense, I say. >> >> wha??? >>At the least it means we'd need to consider the error as well >as the brat -- brats alone would not be reliable.Ironically, you use this sort of reasoning against prime limit all the time. -Carl
Message: 6902 - Contents - Hide Contents Date: Mon, 16 Jun 2003 22:24:07 Subject: Re: Fried Alaska From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>> and for any given brat, you can get arbitrarily close to JI. >>>>>>>> Oh yeah? Then brats are nonsense, I say. >>> >>> wha??? >>>> At the least it means we'd need to consider the error as well >> as the brat -- brats alone would not be reliable. >> Ironically, you use this sort of reasoning against prime limit > all the time.you're replying to yourself here. what sort of reasoning? let's see the analogy fleshed out.
Message: 6903 - Contents - Hide Contents Date: Mon, 16 Jun 2003 15:52:12 Subject: Re: Fried Alaska From: Carl Lumma>>> >t the least it means we'd need to consider the error as well >>> as the brat -- brats alone would not be reliable. >>>> Ironically, you use this sort of reasoning against prime limit >> all the time. >>you're replying to yourself here. Yeah... >what sort of reasoning? let's see the analogy fleshed out.There are ratios of low prime limit that are quite dissonant. There are chords indistinguishable from just with very high brats. I dunno, the sort of reasoning that looks at the function over the pitch continuum and sees if there are anomalies. -Carl
Message: 6904 - Contents - Hide Contents Date: Mon, 16 Jun 2003 23:06:12 Subject: Re: Fried Alaska From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> At the least it means we'd need to consider the error as well >>>> as the brat -- brats alone would not be reliable. >>>>>> Ironically, you use this sort of reasoning against prime limit >>> all the time. >>>> you're replying to yourself here. > > Yeah... >>> what sort of reasoning? let's see the analogy fleshed out. >> There are ratios of low prime limit that are quite dissonant. > There are chords indistinguishable from just with very high brats.i thought were were talking about well-temperaments with low, but rational, brats? you obviously don't care what the brat is for any chord close enough to just -- it's the significantly tempered ones where brats may or may not be meaningful. also, it's not the lowness of the brat, it's the simplicity of the ratio.
Message: 6905 - Contents - Hide Contents Date: Mon, 16 Jun 2003 16:24:36 Subject: Re: Fried Alaska From: Carl Lumma>i thought were were talking about well-temperaments with low, >but rational, brats?We were, but I suggested we compare bare chords instead of temperaments, because that's what the assumption about the larger tunings is based on.>you obviously don't care what the brat is for any chord close enough >to just -- it's the significantly tempered ones where brats may or >may not be meaningful. Apparently so. >also, it's not the lowness of the brat, it's the simplicity of tRight; I've just been saying lowness. By simplicity, don't we mean something like the Van Eck widths? -Carl
Message: 6906 - Contents - Hide Contents Date: Mon, 16 Jun 2003 16:27:16 Subject: Re: Fried Alaska From: Carl Lumma>> >ou obviously don't care what the brat is for any chord close enough >> to just -- it's the significantly tempered ones where brats may or >> may not be meaningful. > >Apparently so.Also, if we replace "just" with "interval x" in Gene's statement, brats are completely useless without some sort of tolerance adjustment, like we use for odd limit. -Carl
Message: 6907 - Contents - Hide Contents Date: Mon, 16 Jun 2003 23:48:20 Subject: Re: Fried Alaska From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> i thought were were talking about well-temperaments with low, but > rational, brats? you obviously don't care what the brat is for any > chord close enough to just -- it's the significantly tempered ones > where brats may or may not be meaningful. also, it's not the lowness > of the brat, it's the simplicity of the ratio.A brat of, say, -1 close to just will simply beat very slowly in sync. As for your second point, infinity happens to be a very nice brat.
Message: 6908 - Contents - Hide Contents Date: Mon, 16 Jun 2003 19:21:27 Subject: Re: Fried Alaska From: Carl Lumma>> >y simplicity, don't we mean something like the Van Eck widths? >>That's tough to answer without knowing what a Van Eck width is,I found gcdb by the way, but have no idea how it works. The Van Eck width of ratio Ri is log(R(i-1))-log(R(i+1)), where Ri is, say, the ith ratio in a Farey series of order n. Unfortunately, I guess these widths just shrink to nothing as n goes to infinity. The harmonic entropy based on them, however, converges to a finite value.>but the rules of simplicity are these: > >(1) q and -q, 1/q, and -1/q are equally simple. In particular, 0 and >infinity are equally simple. > >(2) To judge how simple q is, you need to look at how simple 5/(3-2q) >and (3-2q)/5q are as well. > >(3) Low numerators and denominators are better than high ones; >bearing in mind that infinity = 1/0 counts as low.Since we have no idea what's supposed to make one brat better than another, I suppose this is fine. -Carl
Message: 6909 - Contents - Hide Contents Date: Tue, 17 Jun 2003 18:48:31 Subject: harmonic entropy (was: Re: Fried Alaska) From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" > <wallyesterpaulrus@y...> wrote: > >> ok? >> No; how do you sum over row infinity of the Farey sequence?first of all, carl and i both had errors. carl's definition was wrong because the width goes from mediant to mediant. my probability definition was wrong because i only included the height term but forgot to multiply by the aforementioned width to get the area under the curve! now, row infinity? it's the farey, or mann, or tenney series of order n. n and s are the two parameters of the harmonic entropy function, plus you get to choose farey/mann/tenney/etc., and you get to choose bell/Vos.
Message: 6910 - Contents - Hide Contents Date: Tue, 17 Jun 2003 20:09:35 Subject: harmonic entropy (was: Re: Fried Alaska) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> > now, row infinity? it's the farey, or mann, or tenney series of order > n. n and s are the two parameters of the harmonic entropy function, > plus you get to choose farey/mann/tenney/etc., and you get to choose > bell/Vos.Does harmonic entropy seem to converge to a continuous function as n goes to infinity?
Message: 6911 - Contents - Hide Contents Date: Tue, 17 Jun 2003 20:17:32 Subject: harmonic entropy (was: Re: Fried Alaska) From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" > <wallyesterpaulrus@y...> wrote: >>>> now, row infinity? it's the farey, or mann, or tenney series of > order>> n. n and s are the two parameters of the harmonic entropy function, >> plus you get to choose farey/mann/tenney/etc., and you get to > choose >> bell/Vos. >> Does harmonic entropy seem to converge to a continuous function as n > goes to infinity?it's continuous for any n.
Message: 6912 - Contents - Hide Contents Date: Tue, 17 Jun 2003 20:19:04 Subject: harmonic entropy (was: Re: Fried Alaska) From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" > <wallyesterpaulrus@y...> wrote: >>>> now, row infinity? it's the farey, or mann, or tenney series of > order>> n. n and s are the two parameters of the harmonic entropy function, >> plus you get to choose farey/mann/tenney/etc., and you get to > choose >> bell/Vos. >> Does harmonic entropy seem to converge to a continuous function as n > goes to infinity?as n goes to infinity, the "shape" seems to converge, but it gets taller and flatter. if we had some suitable way to correct for the tallness and flatness as a function of n, we might be able to define a function which indeed converges to a limit as n goes to infinity. this is my hope, as i communicated to you some time ago.