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Message: 5426

Date: Thu, 13 Dec 2001 20:32 +0

Subject: Re: A hidden message (was: Re: Badness with gentle rolloff)

From: graham@xxxxxxxxxx.xx.xx

paulerlich wrote:

> 103168/62 = 1664 exactly!!
> 
> What is this magical mystical number 1664, and does the 62 suggest 
> that somehow 31-tET is making itself known across this vaster survey?

1664 is 128*13.  So 103169 is 13*256*31.  Interesting, don't know if it's 
meaningful, that it's lots of 2s and two prime numbers.  The obvious 
reason for it dividing by 31 is that it contains an interval taken from 
31-equal.  Well, I can't find any, but the best 7:5 in 2*10369-equal is 
15/31, so its influence can certainly be felt this far.


              Graham


top of page bottom of page up down Message: 5427 Date: Thu, 13 Dec 2001 18:50:48 Subject: Re: Badness with gentle rolloff From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > > What if you used n instead of log(n)? Would there still be this > same > > critical function? Or could a function with a different form be the > > critical one? > > This is what I was talking about in a previous posting; if we look at > |h(q)-n*log2(q)|^3, where q is in {3,5,7,5/3,7/3,7/5}, we can apply a > condition that |h(q)-n*log2(q)|^3 < f(n), where the integral of > f(n) or the sum of f(n) diverge--for instance, f(n) = 1/n, so > 1+1/2+1/3+..., the harmonic series, diverges, where int_1^n 1/x dx = > ln(n). The ln(n) means this is logarithmic; we can get other sorts of > density by changing it, but this is easiest and seems the best to me > anyway. But it's not really unique as a critical asymptotic function (?), is it?
top of page bottom of page up down Message: 5428 Date: Thu, 13 Dec 2001 20:39:32 Subject: A hidden message (was: Re: Badness with gentle rolloff) From: paulerlich --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > Actually, the Nyquist resolution (?) prevents me from saying whether > it's 1659.12658227848 (the nominal peak) or something plus or minus a > dozen or so. But clearly my visual estimate of 1664 has been > corroborated. In the 5-limit, one sees a similar pattern, and the "big peak" is at 612, predicably enough . . . spooky: 1664/612 = 2.718......
top of page bottom of page up down Message: 5430 Date: Thu, 13 Dec 2001 20:44:13 Subject: A hidden message (was: Re: Badness with gentle rolloff) From: paulerlich --- In tuning-math@y..., graham@m... wrote: > paulerlich wrote: > > > 103168/62 = 1664 exactly!! > > > > What is this magical mystical number 1664, and does the 62 suggest > > that somehow 31-tET is making itself known across this vaster survey? > > 1664 is 128*13. So 103169 is 13*256*31. No, but 103168 is. 103169 is 11*83*113. > Interesting, don't know if it's > meaningful, that it's lots of 2s and two prime numbers. The obvious > reason for it dividing by 31 is that it contains an interval taken from > 31-equal. Well, I can't find any, but the best 7:5 in 2*10369- equal is > 15/31, so its influence can certainly be felt this far. Confused . . . you mean 2*103168-equal? That's not consistent in the 7-limit . . .
top of page bottom of page up down Message: 5431 Date: Thu, 13 Dec 2001 18:55:43 Subject: A hidden message (was: Re: Badness with gentle rolloff) From: paulerlich Furthermore, noting a striking symmetry centered just above 50,000, I surmised that there must be an especially exceptional ET just above 100,000. And in fact there is -- 103169-tET, the new champion, only about 3/5 as bad as 171-tET. Now the periodicity we saw before appears to occur exactly 62 times from 1-tET to 103169-tET -- thus my current best estimate of the "wave period" is 103168/62 = 1664 exactly!! What is this magical mystical number 1664, and does the 62 suggest that somehow 31-tET is making itself known across this vaster survey?
top of page bottom of page up down Message: 5432 Date: Thu, 13 Dec 2001 20:50:49 Subject: Well . . . From: paulerlich I don't know what's going on here, but it sure reminds me of the Riemann zetafunction!
top of page bottom of page up down Message: 5433 Date: Thu, 13 Dec 2001 00:09:33 Subject: Re: Badness with gentle rolloff From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > > > But we are using 7-limit ETs as a trial run since we have much more > > collective experience of their subjective badness to draw on. > > > > So "steps" is the number of divisions in the octave and "cents" is > the > > 7-limit rms error. > > > > I understand that Paul and Gene favour a badness metric for these > that > > looks like this > > > > steps^2 * cents * if(min<=steps<=max, 1, infinity) > > The exponent would be 4/3, not 2, for ETs. Hey Paul, that's what I had originally but see what Gene wrote in Yahoo groups: /tuning-math/message/1833 * But as far as I can tell, the only flat one is steps * cents. I'll post my spreadsheet when I get it cleaned up. Or you can plot them for yourself.
top of page bottom of page up down Message: 5435 Date: Thu, 13 Dec 2001 21:46:21 Subject: Re: Badness with gentle rolloff From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > You'll be looking at the opposite extremes of the graph. So what? I was looking at the best in both cases. > Not really. At 612, you can't really see the difference yet. Go much > further and you'll see it. If you have to go much further than 612-tET it's hardly relevant to huan beings is it? Just how much further out were you planning to put your cutoff? How much further do you think I need to go to see it? Or to convince you that it doesn't exist? This reminds me of faiths regarding the second coming of Jesus. :-)
top of page bottom of page up down Message: 5436 Date: Thu, 13 Dec 2001 19:08:58 Subject: A hidden message (was: Re: Badness with gentle rolloff) From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > This partly makes sense to me and partly doesn't; it should have wave > frequencies corresponding to the good 7-limit ets, but why 1680? It > would be interesting to see a Fourier analysis of this. Matlab has fft. The FFT of the set of results up to 2^17 has a few extremely sharp peaks. With what formula should I interpret the results?
top of page bottom of page up down Message: 5438 Date: Thu, 13 Dec 2001 00:56:47 Subject: yahoo chokeup From: paulerlich I've replied to your last message twice, Dave, but the replies haven't shown up as yet . . . I hope they will!
top of page bottom of page up down Message: 5440 Date: Thu, 13 Dec 2001 21:59:53 Subject: Vitale 19 (was: Re: Temperament calculations online) From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > Hey Dave, > > Continuing our conversation from the tuning list, I plugged in the > unison vectors 243:245 and 224:225 into Graham's temperament finder, > and got Graham's MAGIC temperament. Graham gives a generator of > 380.39 cents. The 19-tone MOS would have 7 otonal and 7 utonal > tetrads, with a maximum error of 5+ cents. > > How many tetrads did your MIRACLE Vitale 19 have, Dave? (by which I > mean Rami Vitale's scale, without 21/16, 63/32, 8/7, 12/7, and > Miraclized.) It has 5 otonal and 5 utonal 7-limit tetrads with max error of 2.7 c. It's like this on a chain of secors. +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ Canasta +-+-+-+-----+-+-+-+-------+-+-+-------+-+-+-+-----+-+-+-+ MV19 5---------7---1-----------3-----------9----11 11-limit ratios
top of page bottom of page up down Message: 5441 Date: Thu, 13 Dec 2001 00:13:56 Subject: Re: Badness with gentle rolloff From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > > The exponent would be 4/3, not 2, for ETs. > > Hey Paul, that's what I had originally but see what Gene wrote in > Yahoo groups: /tuning-math/message/1833 * He was talking about linear temperaments there, not ETs (right, Gene?). > But as far as I can tell, the only flat one is steps * cents. That's "flat" for all ETs overall (though the wiggles aren't), but what we really care about is whether the goodness/badness values for the "very best" within each range show a flat pattern, or if their values go off to infinity or zero as "steps" increases.
top of page bottom of page up down Message: 5443 Date: Thu, 13 Dec 2001 23:10:04 Subject: Vitale 19 (was: Re: Temperament calculations online) From: clumma Dave, didn't you once show that the number of o- and u-tonal chords must be the same in any linear temp., of any number of notes? -Carl
top of page bottom of page up down Message: 5444 Date: Thu, 13 Dec 2001 00:53:22 Subject: Re: Badness with gentle rolloff From: paulerlich 2nd attempt at replying . . . > > The exponent would be 4/3, not 2, for ETs. > > Hey Paul, that's what I had originally but see what Gene wrote in > Yahoo groups: /tuning-math/message/1833 * I think Gene is referring to linear temperaments, not ETs, there. > But as far as I can tell, the only flat one is steps * cents. That's "flat" (but the wiggles aren't) if you look at each and every ET. But if you look at only the best ones in each range, or the best ones smaller than all better ones, or anything like that, you'll see that the "goodness" keeps increasing without bound. Gene was referring to the kind of "flatness" where it doesn't do that, nor does it drop toward zero after a certain point.
top of page bottom of page up down Message: 5445 Date: Thu, 13 Dec 2001 19:23:19 Subject: A hidden message (was: Re: Badness with gentle rolloff) From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > > > Matlab has fft. The FFT of the set of results up to 2^17 has a few > > extremely sharp peaks. With what formula should I interpret the > > results? > > I don't know what that means, but where are the spikes? I figured out how to get the power spectrum. Result: one big giant spike right at 1665-1666. I will upload the graph shortly.
top of page bottom of page up down Message: 5446 Date: Thu, 13 Dec 2001 23:12:25 Subject: Re: One way to block web advertising From: clumma --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > I'm using the Guidescope proxy service. It's working for me. > See http://www.guidescope.com * Doesn't seem to block the ads in the messages... any suggestions to get it to work? -Carl
top of page bottom of page up down Message: 5447 Date: Thu, 13 Dec 2001 02:17:39 Subject: Re: Badness with gentle rolloff From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > > But as far as I can tell, the only flat one is steps * cents. > > That's "flat" for all ETs overall (though the wiggles aren't), but > what we really care about is whether the goodness/badness values for > the "very best" within each range show a flat pattern, or if their > values go off to infinity or zero as "steps" increases. Well the size of wiggles and the best in each range look pretty damn flat to me for steps * cents (and not for steps^(4/3)*cents or steps^2*cents). Take a look for yourself. http://uq.net.au/~zzdkeena/Music/7LimitETBadness.xls.zip - Ok * 155 KB It comes set for steps*cents, so take a look at the "cut-off badness" chart, then change the yellow cell E6 to "=4/3" or "2" and look at the "cut-off badness" chart again.
top of page bottom of page up down Message: 5449 Date: Thu, 13 Dec 2001 23:27:10 Subject: Re: the 75 "best" 7-limit ETs below 100,000-tET From: dkeenanuqnetau Hey guys. This is _tuning_ math remember. It's a serious stretch of my imagination to think that ETs above 2000 have anything to do with tuning, let alone 100,000! --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > Up through this point in the list, most of the results tend to > be "small" ETs . . . hereafter, they don't. Doesn't that suggest your badness measure isn't flat? How about generating the corresponding charts for steps*cents for comparison with those you did for steps^(4/3)*cents? 1/badness seems to show it best and it still looks to me like steps^(4/3)*cents has gradually falling goodness. And when you give these "best of" lists, please quote your goodness or badness function.
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