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Message: 2575 - Contents - Hide Contents Date: Wed, 19 Dec 2001 00:48:51 Subject: Re: Badness with gentle rolloff From: paulerlich --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "clumma" <carl@l...> wrote: >>> Yes. 404 Not Found * [with cont.] Search for http://uq.net.au/~zzdkeena/Music/7LimitETBadness.xls.zip in Wayback Machine >> >> Got it! >>>>>> I'd love to see consistency plotted against steps*rms. >>>>>> This has been added at your request. >>>> Thanks! I wish I could say I knew what I was looking at. >> Cut-off badness on x and "is consistent" on y? I was thinking >> steps on x and real-number (unrounded) Hahn consistency on y >> (it looks like you're using boolean consistency). >> If you're talking about the maximum real-number consistency _level_ > obeyed by an ET, this will be equal to 1/(max_error*1200*steps) > wherever it's greater than 1.5. So for the good ETs, you'll just be > plotting maximum error vs. rms error.Oops -- that's true if you plot maximum real-number consistency level against 1/(steps*rms), which is what I'm guessing you really had in mind anyway. . . ?
Message: 2576 - Contents - Hide Contents Date: Wed, 19 Dec 2001 00:50:04 Subject: Re: Flat 7 limit ET badness? (was: Badness with gentle rolloff) From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> You or someone might try graphing log steps vs badness for both of> these, or log steps vs number less than a cut-off value. > Gene, I'm disappointed. Dave has produced tons of graphs, in case you > didn't notice.Did he do the graphs I suggested? They might explain flatness better than some other sort of graph. Dave wants to understand how you use Diophantine> approximation theory here.Well, in case you haven't noticed, I did explain it. What is left to do?
Message: 2577 - Contents - Hide Contents Date: Wed, 19 Dec 2001 01:12:36 Subject: Re: Flat 7 limit ET badness? (was: Badness with gentle rolloff) From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>> You or someone might try graphing log steps vs badness for both of> these, or log steps vs number less than a cut-off value. > > Gene, I'm disappointed. Dave has produced tons of graphs, in case you > didn't notice. Dave wants to understand how you use Diophantine > approximation theory here.Not quite true. While a gentle introduction to Diophantine approximation theory would be welcome, I would also like to see it tested empirically. Paul, your criticism of my graphs was that they didn't go far enough. I only went to 612-tET. Going much further than 2000-tET is probably not practical for my spreadsheet approach, but you have plotted 1/(steps^(4/3)*cents) to 10^7-tET or some such. Could you maybe do the same for 1/(steps*cents)? However, plotting stuff and eyeballing it is one thing, but we should really be able to _calculate_ the flatness of a given badness measure out to large number ETs. Gene seems to be saying that for any given badness cutoff, the number less than that should be about the same in every decade (1 to 9-tET, 10 to 99-tET, 100 to 999-tET, etc). Could you check that with Matlab Paul, for both steps^(4/3)*cents and steps*cents? For various cutoffs?
Message: 2578 - Contents - Hide Contents Date: Wed, 19 Dec 2001 01:45:31 Subject: Re: formula for meantone implications? From: joemonz --- In tuning-math@y..., "unidala" <JGill99@i...> wrote: Yahoo groups: /tuning-math/message/1955 * [with cont.]> Joe, > > The more that I stared at my "correction" in message #1926, > the more I realized that I (erroneously) added something (2^) > to the solution for Z as a function of X. With the understanding > that the dependent variables Y and Z (as well as the independent > variable X), being in the *exponents" of your (nicely done) > diagram's algebraic identity [which was (3^(X/3))*(5^(X/6)) > ~= (3^Y)*(5^Z)] represent quantaties in the "logarithmic domain" > (and not the "linear domain"), the correction below should work > well for the such determinations of Y and Z as functions of X as > (it would seem) you have requested. >>> Z=(LN of(5^(X/2)-(3*(Z+(dY/dX)*X)+ X)/(LN of 3)))/(3*(LN of 5)))There's one parenthesis too many at the end. Can you go over this again and remove it? Thanks. I'm running your formulas thru Excel to see what the produce. -monz
Message: 2579 - Contents - Hide Contents Date: Wed, 19 Dec 2001 01:55:14 Subject: Re: Badness with gentle rolloff From: dkeenanuqnetau --- In tuning-math@y..., "clumma" <carl@l...> wrote:>> Yes. 404 Not Found * [with cont.] Search for http://uq.net.au/~zzdkeena/Music/7LimitETBadness.xls.zip in Wayback Machine > > Got it! >>>> I'd love to see consistency plotted against steps*rms. >>>> This has been added at your request. >> Thanks! I wish I could say I knew what I was looking at. > Cut-off badness on x and "is consistent" on y? I was thinking > steps on x and real-number (unrounded) Hahn consistency on y > (it looks like you're using boolean consistency).Yes. steps*rms on x and Boolean consistency on y. Tell me how to calculate real-number Hahn consistency. Do you mean you want to see both Hahn consistency and steps*cents badness (or do you want 1/(steps*cents) goodness? plotted against steps?
Message: 2581 - Contents - Hide Contents Date: Thu, 20 Dec 2001 03:00:15 Subject: Re: Two conditions on temperaments From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> It seems to me there are two conditions it would make sense to puton something intended as an odd-limit temperament:> > (1) Weak condition--no element of the tonality diamond is allowedto be a unison Fine. Then we won't need a lower bound on g anymore, will we?> > (2) Strong condition--all elements of the tonality diamond are distinctThat would make me cry, as it excludes twintone in the 7-limit.
Message: 2582 - Contents - Hide Contents Date: Thu, 20 Dec 2001 13:56 +0 Subject: Re: Meantone & co From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9vsgv8+d425@xxxxxxx.xxx> Gene wrote:> Graham is more likely to include things I wouldn't than vice-versa--his > methods can include more than one version of the same system, so that > you have for instance meantone, and another meantone with a seemingly > useless doubling of the generator steps. There are at least three > version of kleismic on his list--one the usual, one with doubled > generator steps, and one with a half-ocatave period.Yes. That's an interesting, uh, feature of my algorithm for going from ETs to linear temperaments. A normal 5-limit kleismic can be generated from 19- and 34-equal:>>> h19 = temper.PrimeET(19, temper.primes[:2]) >>> h34 = temper.PrimeET(34, temper.primes[:2]) >>> (h19&h34).mapping[(1, 0), (0, 6), (1, 5)] The defining wedge product is>>> (h19^h34).complement().flatten()(-6, -5, 6) One example where it goes wrong is 15- and 23-equal>>> h23 = temper.PrimeET(23, temper.primes[:2]) >>> h15 = temper.PrimeET(15, temper.primes[:2]) >>> (h23&h15).mapping[(1, 0), (0, 12), (1, 10)] the wedge product is>>> (h15^h23).complement().flatten()(12, 10, -12) and so simplifies to be equivalent to the above. In this case, it's obvious from looking at the generator mapping that something's wrong, because it has a common factor of 2. The other strange example is 4&34:>>> h4 = temper.PrimeET(4, temper.primes[:2]) >>> (h4&h34).mapping[(2, 0), (6, -6), (7, -5)] for the same reason>>> (h4^h34).complement().flatten()(-12, -10, 12) Here, though, there's nothing obviously wrong with the resulting temperament. So are these behaviours a problem? They follow from the generator and period being chosen ahead of the mapping. That's probably correct, because this is supposed to be an extension of the Scale Tree(s). They won't usually get to the top of the list, because they have a higher complexity than the equivalents. But it does mean temperaments will be underrated if they appear in this way. I'm hoping that any set of ETs containing a pair with torsion will also contain a pair without torsion. Also that it'll be far less common in the higher limits. If not, we can always get to those temperaments by using wedge products. With the current implementation, at least, I expect that would slow things down. Graham
Message: 2583 - Contents - Hide Contents Date: Thu, 20 Dec 2001 03:02:56 Subject: Re: Two conditions on temperaments From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> That would make me cry, as it excludes twintone in the 7-limit.That occurred to me also; I think the strong condition is too strong, but I see no problem with the weak condition.
Message: 2584 - Contents - Hide Contents Date: Thu, 20 Dec 2001 19:11:55 Subject: Re: Badness with gentle rolloff From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote:>> Tell me how to calculate real-number Hahn consistency. >> According to Paul, it's 1/(max_error*1200*steps)No, that's only correct if it's greater than 1.5.
Message: 2585 - Contents - Hide Contents Date: Thu, 20 Dec 2001 03:05:53 Subject: Re: Meantone & co From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> Here are four more: > > 81/80 > > Map: > > [ 0 1] > [-1 2] > [-4 4] > > Generators: a = 20.9931/50; b = 1 > > badness: 108 > rms: 4.22 > g: 2.944 > errors: [-5.79, -1.65, 4.14] > > Nothing left to say about this one. :) > > > 2048/2025 > > Map: > > [ 0 2] > [-1 4] > [ 2 3] > > Generators: 14.0123/34 (~4/3); b = 1/2 > > badness: 211 > rms: 2.613 > g: 4.32 > errors: [3.49, 2.79, -.70] > > A good way to take advantage of the 34-ets excellent 5-limit harmonies > is two gothish 17-et chains of fifths a sqrt(2) apart.This is my interpretation of the Indian scales.> > > 78732/78125 = 2^2 3^9 5^-7 > > Map: > > [ 0 1] > [ 7 -1] > [ 9 -1] > > Generators: 23.9947/65 (~9/7); b = 1 > > badness: 346 > rms: 1.157 > g: 6.68 > errors: [-1.1, 0.5, 1.6]Would you call this one a "unique facet of 65-tET"? Is this the kind of thing that Graham's searching pairs of ETs is likely to miss?> > > 393216/390625 = 2^17 3 5^-8 > > Map: > > [ 0 1] > [ 8 -1] > [ 1 2] > > Generators: a = 31.9951/99 (~5/4); b = 1 > Works with 31,34,65,99,164 > > badness: 251 > rms: 1.072 > g: 6.16 > error: [.602, 1.506, .904]Magic. Rather than continuing, Gene, could you start over going from lowest badness to highest badness?
Message: 2586 - Contents - Hide Contents Date: Thu, 20 Dec 2001 03:07:47 Subject: Re: 55-tET From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>>> It makes sense, but I don't think it defines a unique interval. >>>> Meaning you can always find smaller and smaller examples? Even if you >> disallow "potential torsion"? What if you fix all the commas except >> one, and just have to find the smallest candidate for the remaining >> comma. Isn't that choice unique? >> That should do it, though it seems a little arbitary. If you don't >fix all but one, I can prove easily enough you get arbitarily small >commas, so this may be your best shot. For the 55-et, or anything >else where there is a clear set of all-but-one keepers in the comma >department, it would make sense.Well? Did I find it? 3^15*5^10.
Message: 2587 - Contents - Hide Contents Date: Thu, 20 Dec 2001 03:16:11 Subject: Re: Meantone & co From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Magic.Not Magic; another 5/4 system. Rather than continuing, Gene, could you start over going from> lowest badness to highest badness?This way puts similar systems together; why do it the other way?
Message: 2588 - Contents - Hide Contents Date: Thu, 20 Dec 2001 03:17:35 Subject: Re: supernatural superparticulars? From: paulerlich --- In tuning-math@y..., "jpehrson2" <jpehrson@r...> wrote:>>> Curiously, J Gill >>>> No -- but what seems to be the case very often, is that when one >> comes up with such a scale in the form of a periodicity block, one >> has quite a few arbitrary choices to make as to which version of a >> particular scale degree one wants (the different versions differing >> by a unison vector), and then _one such set_ of arbitrary choices >> does lead to a scale with superparticular step sizes. > > Hi Paul... >> Well, that's pretty *mysterious* isn't it? Why does that happen that > the superparticular step sizes result?Usually they don't "result", but very often you can arbitrarily choose to use them.> Is it just the way the system > is set up.Well, superparticulars are not favored _by design_, if that's what you mean. You should ask Kraig Grady about superparticular step sizes too.> Spooky stuff! (If we can't believe in "magic primes" > that surely is something a little weird... yes?)One of the themes on the tuning-math list (busier than ever) is superparticulars . . . for example, we found that the graph of ET "goodness" has "waves" in it, and the most prominent visible wave by far for 7-limit rises and falls every 1664 ETs . . . and the superparticular ratio that Graham made famous in his Blackjack progression, 2401:2400, fits 1663.9 times in an octave.
Message: 2589 - Contents - Hide Contents Date: Thu, 20 Dec 2001 03:29:21 Subject: Re: Meantone & co From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > >> Magic. >> Not Magic; another 5/4 system. > > Rather than continuing, Gene, could you start over going from>> lowest badness to highest badness? >> This way puts similar systems together; why do it the other way?To get a sense of whether I can accept your kind of "flatness" in this context.
Message: 2590 - Contents - Hide Contents Date: Thu, 20 Dec 2001 03:30:40 Subject: Re: Meantone & co From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > >> Magic. >> Not Magic; another 5/4 system.Is this better or worse than Magic in the 5-limit?
Message: 2591 - Contents - Hide Contents Date: Thu, 20 Dec 2001 03:27:42 Subject: The Magic of Superparticulars From: paulerlich Basically, the magic of superparticulars is that they're the smallest possible intervals for a given complexity (or distance on the lattice). Which is kind of obvious if you think about it. If the two numbers in a ratio differ by 1, how can the interval be any smaller without increasing the two numbers in the ratio? Things fizzle out, though, once you get past the point where there are no superparticulars left. Think 5-limit; you can look at the page S235 * [with cont.] (Wayb.) and in the second column from the right, you'll see a bunch of ratios. Certain rows have the first few columns in parentheses, but those that don't correspond to the SMALLEST unison vectors for some given DISTANCE LIMIT in the lattice, or ODD-LIMIT, or INTEGER-LIMIT if you prefer. , you can see, are all superparticular, until you reach the last two, your old friends 25:24 and 81:80. Then there are no more superparticulars possible in the 5-limit. Now look at S2357 * [with cont.] (Wayb.) and again, all the non-parenthesized entries are superparticular, until you reach the last two, 2401:2400 and 4375:4374. The latter is the last superparticular possible in the 7-limit.
Message: 2592 - Contents - Hide Contents Date: Thu, 20 Dec 2001 03:33:18 Subject: Re: Two conditions on temperaments From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> That would make me cry, as it excludes twintone in the 7-limit. >> That occurred to me also; I think the strong condition is toostrong, but I see no problem with the weak condition. I agree. but we'd like to know when the strong condition applies to a temperament. This is shown as "unique" in Graham Breed's output, as in all elements of the diamond are represented by a unique interval in the temperament.
Message: 2593 - Contents - Hide Contents Date: Thu, 20 Dec 2001 03:40:38 Subject: Re: Flat 7 limit ET badness? (was: Badness with gentle rolloff) From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>> This is all sounding a little contrived. How low is low enough? How >> far from 1-tET is far enough? If we have to go out beyond what is >> musically relevant, then what's the point? >> Why not just exlude the inconsistent ets? The point is to get rid ofthe crap parade at the very beginning, and this looks like a use of consistency I could go for.>I understand this to be equivalent to putting a sharp cutoff at 600 on steps*cents. The usual objection to sharp cutoffs applies, namely people don't usually apply sharp cutoffs when making decisions about the usefulness of tunings.
Message: 2594 - Contents - Hide Contents Date: Thu, 20 Dec 2001 03:42:50 Subject: Re: Flat 7 limit ET badness? (was: Badness with gentle rolloff) From: dkeenanuqnetau --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> I understand this to be equivalent to putting a sharp cutoff at 600 on > steps*cents. The usual objection to sharp cutoffs applies, namely > people don't usually apply sharp cutoffs when making decisions about > the usefulness of tunings.Oops! Only when cents is max-absolute error, not rms.
Message: 2595 - Contents - Hide Contents Date: Thu, 20 Dec 2001 03:46:04 Subject: Re: Flat 7 limit ET badness? (was: Badness with gentle rolloff) From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>> I understand this to be equivalent to putting a sharp cutoff at 600 > on>> steps*cents. The usual objection to sharp cutoffs applies, namely >> people don't usually apply sharp cutoffs when making decisions about >> the usefulness of tunings. >> Oops! Only when cents is max-absolute error, not rms.Maybe it should be minimax. Maybe we should give a _range_ of optimal generators, rather than just one, when the same minimax is achieved for all within the range. Maybe we should also give the points at which the minimax is doubled. This would give an idea of the sensitivity of the tuning.
Message: 2596 - Contents - Hide Contents Date: Thu, 20 Dec 2001 03:50:55 Subject: Re: Meantone & co From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> This way puts similar systems together; why do it the other way? >> To get a sense of whether I can accept your kind of "flatness" in > this context.I don't get it--how will mixing the apples with the oranges do that?
Message: 2597 - Contents - Hide Contents Date: Thu, 20 Dec 2001 03:58:04 Subject: Re: Meantone & co From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>>> This way puts similar systems together; why do it the other way? >>>> To get a sense of whether I can accept your kind of "flatness" in >> this context. >> I don't get it--how will mixing the apples with the oranges do that?I just want to get the sense that some fixed value of badness is going to remove unimportant ones while retaining all the important ones. For example, I'd definitely set tbe badness cutoff high enough so that meantone, augmented, and diminished, even if they had errors as large as in 12-tET, make it in.
Message: 2599 - Contents - Hide Contents Date: Thu, 20 Dec 2001 04:44:35 Subject: [tuning] Re: great explanation [periodicity block] From: paulerlich --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:> <<Hmm? Those would be the stepsizes for the 12-tone scale in this > series, not the whole series, correct?>> > > Right, following the expansion or contraction rules will walk you up > and back in the series. So the stepsizes for the 22-tone scale in this > series would be 81/80 128/125 25/24, and the 2D UVs would be 3125/3072 > and 2048/2025.There's probably one more, if you take a look at the lattice. Why don't you make a quick-and-dirty periodicity-revealing lattice for the 12- and 22-tone scales that result in the "Tribonacci" system?
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