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Message: 10675 - Contents - Hide Contents Date: Tue, 23 Mar 2004 00:13:32 Subject: Re: 5-limit yantra commas From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> Yantras are naturally weighted by log(p), so it isn't surprising to >> find some connection. Are you saying the two will always lead to >> exactly identical results? > > Yes.Succinct but not convincing. The metric for yantras would give triangle shapes for balls. If we take 5 as perpendicular to 3, right triangle shapes, with the shorter sides in a log3:log5 proportion.
Message: 10676 - Contents - Hide Contents Date: Tue, 23 Mar 2004 00:34:54 Subject: Re: 5-limit yantra commas From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>>> Yantras are naturally weighted by log(p), so it isn't surprising to >>> find some connection. Are you saying the two will always lead to >>> exactly identical results? >> >> Yes. >> Succinct but not convincing. The metric for yantras would give > triangle shapes for balls.This may be the metric for yantra *pitches*, but you were constructing those ETs by combining yantra *intervals* (namely, by eating the smallest intervals (pi(limit) of them) in each yantra). So it would be more appropriate to look at the balls defined by yantra *intervals*.
Message: 10678 - Contents - Hide Contents Date: Tue, 23 Mar 2004 13:57:21 Subject: Re: some help with Pythagorean scale properties? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "jjensen142000" <jjensen14@h...> wrote:> Hi > > I am trying to understand meantone(s), and that led me to trying to > write a program to generate the Pythagorean scale which I have done: > > If I start with a string vibrating at a certain frequency which > I'll call the 1, and I repeatedly take 3rd harmonics up and down, > I generate > > { ..., 3^-2, 3^-1, 1, 3^1, 3^2, ... } > > then I rescale when necessary to put every note in the octave > [1,2] by multiplying by 2^k for some appropriate integer k. > > Questions: > > 1. So I have terms 2^k * 3^n where there seems to be no nice > formula to determine the k from the given n.If you allow yourself to use the "floor" function, there is.> But there is, > to me, a suprising amount of regularity between the distances > to the adjacent notes (its one of those commas, no doubt) > For example, Db to C# is 23.4600103... cents, and that > number occurs again from Gb to F# and Cb to B (and many > more places if I generate the sequence longer) > > Why should this be? It is very amazing to me that it should > be the same number...This number is simply 3^12 * 2^(-19), known as the pythagorean comma. Even ratio-fetishists often avoid the actual fraction, which, if you care, is 531441/524288.> 2. If I plot the log of my sequence of pure 3rd harmonics, I get > a line. I want to say that 1/4 comma meantone is another line > with a slightly lower slope to exactly hit a rescaled version > of the 5th harmonic.Correct -- if, instead of 3rd harmonics (3/1's), you use 3/2's.> There has been a substantial time lag between my working on > these calculations and my posting this note, but as I recall, > the big problem that I'm having is that meantone is defined as > deviating from pythagorean by some power of 81/80 but I > don't have a *formula* for pythagorean values, only an iterative > procedure for generating them, with the power of 2 found by > trial and error, and therefore I can't prove anything.You can use the floor function to compute the power of 2. What software are you using? ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 10679 - Contents - Hide Contents Date: Wed, 24 Mar 2004 17:39:14 Subject: Re: some help with Pythagorean scale properties? From: Paul Erlich Hi Jeff, --- In tuning-math@xxxxxxxxxxx.xxxx "jjensen142000" <jjensen14@h...> wrote:>>> 1. So I have terms 2^k * 3^n where there seems to be no nice >>> formula to determine the k from the given n. >>>> If you allow yourself to use the "floor" function, there is. >> >> You can use the floor function to compute the power of 2. What >> software are you using? > >> I'm using a Perl script that I wrote. I am familiar with the floor > function,Great. That's all you'll need. Let's say you have a ratio r which you determined as 3^n (or (3/2)^n, or whatever) for some given n. Define a new variable k as k=-floor(log(r)/log(2)) This is the k you wanted above, assuming you're reducing everything to be withing the octave from 1 to 2. Now recompute r from the old r and k thus: r=r/(2^-k). No trial and error involved.> and its great for computer programming, but I'm having > trouble using it in a mathematical proof.I didn't know you were trying to prove anything. What's this proof you're working on? Sorry if I misundertood your intention above.> To know that > floor(a*b) = floor(c*d) you have to actually compute the numbers, > you can't reason about it.Sometimes you can. Continued fractions can come into it. Let me know what you're trying to do. -Paul
Message: 10681 - Contents - Hide Contents Date: Wed, 24 Mar 2004 22:22:12 Subject: "On the Mathematical Structure of Tonal Harmony" From: Paul Erlich http://www.users.on.net/tpearce/0402204.pdf - Type Ok * [with cont.] (Wayb.) I'm too busy to read this now -- anything interesting here?
Message: 10683 - Contents - Hide Contents Date: Thu, 25 Mar 2004 01:02:35 Subject: Re: some help with Pythagorean scale properties? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "jjensen142000" <jjensen14@h...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >> Hi Jeff, >>>> --- In tuning-math@xxxxxxxxxxx.xxxx "jjensen142000" > <jjensen14@h...> >> wrote: >>>>>>> 1. So I have terms 2^k * 3^n where there seems to be no nice >>>>> formula to determine the k from the given n. >>>>>>>> If you allow yourself to use the "floor" function, there is. >>>> >>>> You can use the floor function to compute the power of 2. What >>>> software are you using? >>> >>>>>> I'm using a Perl script that I wrote. I am familiar with the > floor >>> function, >>>> Great. That's all you'll need. Let's say you have a ratio r which > you>> determined as 3^n (or (3/2)^n, or whatever) for some given n. > Define>> a new variable k as >> >> k=-floor(log(r)/log(2)) >> >> This is the k you wanted above, assuming you're reducing everything >> to be withing the octave from 1 to 2. Now recompute r from the old > r>> and k thus: >> >> r=r/(2^-k). >> >> No trial and error involved. >>>>> and its great for computer programming, but I'm having >>> trouble using it in a mathematical proof. >>>> I didn't know you were trying to prove anything. What's this proof >> you're working on? Sorry if I misundertood your intention above. >>>>> To know that >>> floor(a*b) = floor(c*d) you have to actually compute the numbers, >>> you can't reason about it. >>>> Sometimes you can. Continued fractions can come into it. Let me > know>> what you're trying to do. >> >> -Paul >> Ok, now I've got my notes in front of me so I can be a little more > specific. I think the Pythagorean note generating function is > > P(x) = 2^{k(x;3)} * 3^x > where 2^k(x;3) is the factor that rescales 3^x into the interval > [1,2] > > that means k(x;3) is the integer in [ x log3/log2 -1, x log3/log2], > which is floor( x log3/log2 ) > > Then we restrict x to be integer values only. (I guess I was hoping > to get some enlightenment by allowing x to be any real number...but > I don't think I did) > > Anyway, why did I write k(x;3)? Because for any meantone I think > the function would be > > M(x) = [ (81/80)^{-a} *3 ] * 2^k( x; (81/80)^{-a} *3 ) > > where a = 1/4 or any of the other values Benson mentions in > ch 5 of his Mathematics and Music online course notes. > > > Ok, so to answer your question, what I was trying to "prove" was > sort of based on something you said in "A Gentle Intro to Fokker > Peridodicity Blocks Part 1" where you more or less say that we > could generate the notes f,c,g,d,a and then identify e with f > and close the circle since they are only 90.22... cents apart, or > we could keep on generating notes until we get an even closer > match. So I'm trying to understand why b-c should be the SAME > 90.22... cents, and f#-g the same, and so on...Well, that should be fairly clear now. These intervals are all equal to P(5), where P(x) is as you've defined it above. Of course, if you're reducing everything to be within the traditional middle octave, which starts on C and ends just before the octave-higher c, then b-c will become b-C which is not 90.22 cents but rather -1109.78 cents. But then these, 90.22 and -1109.78 will be the only two possibilities for P(y)-P(z) where y-z = 5.> Maybe there is nothing more to understand, it is just a wierd > coincidence.There's nothing weird or coincidental about it.> But I would have expected 90.2 cents, then maybe 85.3 > cents, then 94 cents...I can't see why you would have expected that. If you have a clock, and mark off the point that your minute hand points to every 35.1 minutes, after a few hours go by, would you expect the pairs of nearby markings to have somewhat different distances between the two markings, or the same distances between the two markings?
Message: 10684 - Contents - Hide Contents Date: Thu, 25 Mar 2004 09:21:15 Subject: Re: "On the Mathematical Structure of Tonal Harmony" From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> http://www.users.on.net/tpearce/0402204.pdf - Type Ok * [with cont.] (Wayb.) > > I'm too busy to read this now -- anything interesting here?Skimming quickly shows there is a lot of BS in it, but that hardly precludes there being interesting things also. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 10686 - Contents - Hide Contents Date: Fri, 26 Mar 2004 22:17:47 Subject: Stormer's theorem From: Gene Ward Smith I got Haluska's book from the library, and already have learned one thing of great interest. I gave a proof here that for any p-limit, only a finite number m/n exist with m-n below a fixed bound. This, it turns out, is a theorem of Stormer dating to 1898, and is *constructive*. Rather than use Baker's theorem, which was unknown back then, Stormer showed how to constuct the entire list using Pell's equation. Lehmer has computed these up to the 41-limit; I have a huge box of old reprints of Lehmer's I got after he died, and I wonder if this was in it. Anyway, Lehmer's paper is "On a problem of Stormer", Illinois Journal of Mathematics 8(1964), 57-79, and I plan to dig it out if I can. In the preface we find: The author thanks composers of contemporary music and computer music theorists for their information-sharing and feedback. My thanks go to D. Benson, D. Canright, P. Erlich, K. Grady, M. Haluska, Y. Hellegouarch, B. Hero, D. Keenan, V. A. Lefebvre, H. L. Mittdendorf, J. L. Monzo, G. Morrison, G. Mazzola, E. Neuwirth, M. Op de Coul, R. Ruzicka, M. Schulter, J. Starrett, Ch. Stoddard, and W.Sethares.
Message: 10687 - Contents - Hide Contents Date: Fri, 26 Mar 2004 23:12:52 Subject: Re: Stormer's theorem From: Paul Erlich This should be communicated to John Chalmers post haste; in the most recent (6 years ago) issue of Xenharmonikon, he wrote an article called _The number of 23-Prime-Limit Superparticular Ratios Less Than 10,000,000_ -- since that's as high as his system would go. I'm sure he would be most interested to know that at least since 1964, the answer was in some sense known with the 10,000,000 relaxed to infinity. --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> I got Haluska's book from the library, and already have learned one > thing of great interest. I gave a proof here that for any p-limit, > only a finite number m/n exist with m-n below a fixed bound. This, it > turns out, is a theorem of Stormer dating to 1898, and is > *constructive*. Rather than use Baker's theorem, which was unknown > back then, Stormer showed how to constuct the entire list using Pell's > equation. Lehmer has computed these up to the 41-limit;For what fixed bound for m-n?> I have a huge > box of old reprints of Lehmer's I got after he died, and I wonder if > this was in it. Anyway, Lehmer's paper is "On a problem of Stormer", > Illinois Journal of Mathematics 8(1964), 57-79, and I plan to dig it > out if I can. > > In the preface we find: > > The author thanks composers of contemporary music and computer music > theorists for their information-sharing and feedback. My thanks go to > D. Benson, D. Canright, P. Erlich, K. Grady, M. Haluska, Y. > Hellegouarch, B. Hero, D. Keenan, V. A. Lefebvre, H. L. Mittdendorf, > J. L. Monzo, G. Morrison, G. Mazzola, E. Neuwirth, M. Op de Coul, R. > Ruzicka, M. Schulter, J. Starrett, Ch. Stoddard, and W.Sethares. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------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: 10688 - Contents - Hide Contents Date: Sat, 27 Mar 2004 07:34:25 Subject: Re: Digest Number 1011 From: John Chalmers Erv Wilson should be apprised of Stormer's theorem and Lehmer's result as it was he who asked me to compute the 23-limit up to 10 million. --John
Message: 10689 - Contents - Hide Contents Date: Sat, 27 Mar 2004 22:41:48 Subject: Stoermer From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx John Chalmers <JHCHALMERS@U...> wrote:> Erv Wilson should be apprised of Stormer's theorem and Lehmer's result > as it was he who asked me to compute the 23-limit up to 10 million.Does Erv have an email address? I have the Lehmer paper, which computes all 41-limit superparticular ratios. It also gives an improvement on Stoermer's result, which allows faster computation. Aside from the fact that the paper is by someone I knew, another thing that gets me is that I had heard that Stoermer had solved the 5-limit superparticular ratio problem, showing that the smallest such comma is 81/80. Apparently it was a mere minor detail his own computations went up to the 7-limit, and that his result was much more general! Lehmer proves the following: (Stoermer-Lehmer) Let 2=q1 < q2 < ... < qt be prime, let Q be {q1^a1 ... qt^at}, and let Q' be the set of all 2^t-1 square-free members of Q, excepting 2. Suppose (S+1)/S is a superparticular ratio belonging to Q. Then S = (xn-1)/2, where (xn, yn) is a solution of the Pell's equation x^2 - 2D y^2 = 1 where D is in Q', 1 <= n <= max(3, (1+qt)/2)), and yn is in Q. Conversely, if (xn, yn) is a solution to a Pell's equation with the above conditions, then (S+1)/S belongs to Q.
Message: 10690 - Contents - Hide Contents Date: Sat, 27 Mar 2004 23:56:06 Subject: Re: Stoermer From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: By the way, I misread the statement of Stoermer's theorem in Haluska; Stoermer only proves his theorem for (S+2)/S and (S+1)/S. Haluska, however, is aware that the more general result follows easily from Baker's theorem. I showed the stonger result that for any epimericity less than 1, we have only a finite list, in fact; also using Baker's theorem.
Message: 10691 - Contents - Hide Contents Date: Sun, 28 Mar 2004 07:55:35 Subject: Re: Wilson and Stormer From: John Chalmers Erv doesn't have email, but I sent him a copy of discussion yesterday, so he will know.
Message: 10692 - Contents - Hide Contents Date: Sun, 28 Mar 2004 16:34:35 Subject: Re: Stoermer From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx John Chalmers <JHCHALMERS@U...> wrote: >>> Erv Wilson should be apprised of Stormer's theorem and Lehmer's result >> as it was he who asked me to compute the 23-limit up to 10 million. >> Does Erv have an email address? > > I have the Lehmer paper, which computes all 41-limit superparticular > ratios.Great. Xenharmonikon readers might like to know: are there any 23- limit superparticulars above 10,000,000?
Message: 10693 - Contents - Hide Contents Date: Sun, 28 Mar 2004 21:11:40 Subject: Re: Stoermer From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Great. Xenharmonikon readers might like to know: are there any 23- > limit superparticulars above 10,000,000? Nope.
Message: 10694 - Contents - Hide Contents Date: Sun, 28 Mar 2004 00:03:57 Subject: Re: Stoermer From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: 4375/4374 is named "ragisma" on Manuel's list, but Stoermer was the first to know it was the largest 7-limit superparticular, and possibly the first to know about it at all, so naming it after him would seem logical. Dick Lehmer was the first to know about the whole lot of them up to the 41-limit, and I suppose a lehmerisma of 9801/9800 might be appropriate.
Message: 10695 - Contents - Hide Contents Date: Sun, 28 Mar 2004 21:27:28 Subject: Re: Stoermer From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> Great. Xenharmonikon readers might like to know: are there any 23- >> limit superparticulars above 10,000,000? > > Nope.I spoke too soon, after only looking at the strict 23 limit table. For some reason probably connected to the fact that 19 is the larger of a twin prime pair, there are two 19-limit commas smaller than any strictly 23-limit comma. They are: 5909761/5909760 |-8 -5 -1 0 2 2 2 -1> 11859211/11859210 |-1 -4 -1 1 -4 1 0 4>
Message: 10696 - Contents - Hide Contents Date: Sun, 28 Mar 2004 00:13:16 Subject: Re: Stoermer From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Dick Lehmer was the first to know about the whole lot of them > up to the 41-limit, and I suppose a lehmerisma of 9801/9800 might be > appropriate.Or perhaps 3025/3024 is a better lehmerisma, as Lehmer mentions it on the first page of his paper. In any case, 9801/9800 is called a "kalisma" for reasons unknown to me, but Manuel does not list 3025/3024, so attaching Dick's name to it seems like a good idea. He was a wonderful gentlemen and a hell of a mathematician, and deserves a comma as much as anyone.
Message: 10697 - Contents - Hide Contents Date: Sun, 28 Mar 2004 23:36:00 Subject: Re: Stoermer From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> >> wrote: >>>>> Great. Xenharmonikon readers might like to know: are there any 23- >>> limit superparticulars above 10,000,000? >> >> Nope. >> I spoke too soon, after only looking at the strict 23 limit table. For > some reason probably connected to the fact that 19 is the larger of a > twin prime pair, there are two 19-limit commas smallerYou mean larger?> than any > strictly 23-limit comma. They are: > > 5909761/5909760 > |-8 -5 -1 0 2 2 2 -1> > > 11859211/11859210 > |-1 -4 -1 1 -4 1 0 4>
Message: 10698 - Contents - Hide Contents Date: Sun, 28 Mar 2004 00:18:37 Subject: Re: Stoermer From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Or perhaps 3025/3024 is a better lehmerisma, as Lehmer mentions it on > the first page of his paper. In any case, 9801/9800 is called a > "kalisma" for reasons unknown to me, but Manuel does not list > 3025/3024, so attaching Dick's name to it seems like a good idea. He > was a wonderful gentlemen and a hell of a mathematician, and deserves > a comma as much as anyone.Lehmer unfortunately does not give a cite, but he claims Gauss mentioned 9801/9800, so I think I'll propose gaussisma for that, and lehmerisma for 3025/3024. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 10699 - Contents - Hide Contents Date: Sun, 28 Mar 2004 23:40:46 Subject: Re: Stoermer From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>>>> limit superparticulars above 10,000,000? >>> >>> Nope. >>>> I spoke too soon, after only looking at the strict 23 limit table. > For>> some reason probably connected to the fact that 19 is the larger of > a>> twin prime pair, there are two 19-limit commas smaller >> You mean larger?Never mind; larger numbers, smaller in cents.
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