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Message: 2425 - Contents - Hide Contents Date: Thu, 13 Dec 2001 16:40:43 Subject: A hidden message (was: Re: Badness with gentle rolloff) From: paulerlich --- In tuning-math@y..., graham@m... wrote:> > I don't know either, but I'll register an interest in finding out. I've > thought for a while that the set of consistent ETs may have properties > similar to the set of prime numbers.Well, this pattern I found shows up regardless of whether you look at consistent ETs only, or fail to enforce consistency at all.> It really gets down to details of > the distribution of rational numbers. One thing I noticed is that you > seem to get roughly the same number of consistent ETs within any linear > range. Is that correct?Yup -- in the 7-limit, it's always half! You know how to view this table: range #inconsistent 1-10000 5006 10001-20000 4996 20001-30000 5004 30001-40000 5002 40001-50000 4996 50001-60000 4996 60001-70000 4996 70001-80000 5002 80001-90000 5006 90001-100000 4999 (the first odd number so far)> > As to these diagrams, one thing I notice is that the resolution is way > below the number of ETs being considered. So could this be some sort of > aliasing problem?No, because the same exact behavior showed up in the Excel chart, no matter how I stretched it out . . .> Best way of checking is to be sure each bin contains > the same *number* of ETs, not merely that the x axis is divided into > near-enough equal parts.Hmm . . . all the maxima are visible, so I'm not sure this is relevant anyway.
Message: 2426 - Contents - Hide Contents
Date: Thu, 13 Dec 2001 20:30:48
Subject: the 75 "best" 5-limit ETs below 2^17-tET
From: paulerlich
Assuming a "critical exponent" of 3/2 for this case (is that right?)
:
Out of the consistent ones:
rank ET "badness"
1 4296
0.20153554902775
2 78005
0.253840852090173
3 118
0.298051576414275
4 3
0.325158891374691
5 53
0.361042754847595
6 1783
0.376704792560154
7 2513
0.38157807050998
8 25164
0.410594002644579
9 19
0.410991123902702
10 12
0.417509911542676
11 612
0.436708226862349
12 730
0.440328484445999
13 34
0.458833616575689
14 171
0.461323498406156
15 20868
0.462440101460723
16 7
0.479263869467813
17 4
0.517680428544775
18 441
0.525786933473794
19 1171
0.54066707734392
20 8592
0.570028613470703
21 65
0.580261609859836
22 52841
0.584468600555837
23 73709
0.592105848504379
24 6809
0.613067695688349
25 15
0.644650341848039
26 5
0.654939089766412
27 31
0.659243117645396
28 289
0.666113665527379
29 22
0.713295533690924
30 1342
0.734143972117584
31 16572
0.736198397866562
32 323
0.744599492497238
33 559
0.763541910323762
34 152
0.785452598431966
35 9
0.804050483021927
36 29460
0.806162085936717
37 98873
0.808456458619207
38 1053
0.816063953343609
39 10
0.831348880236421
40 27677
0.840139252565266
41 236
0.843017163303497
42 6079
0.854618300478436
43 87
0.855517482964681
44 1901
0.875919286932322
45 8
0.885030392786763
46 3684
0.886931822414785
47 48545
0.889578653724097
48 11105
0.89024911748373
49 84
0.908733078006219
50 46
0.91773712251282
51 6
0.919688228216577
52 99
0.929380204600093
53 205
0.94016876679068
54 103169
0.941197892471069
55 57137
0.942202383987665
56 12276
0.953572058507306
57 31973
0.95740445574325
58 494
0.959416685644995
59 270
0.962515005704479
60 23381
0.982018213968414
61 5467
0.999256787729657
62 2954
1.01217901495476
63 130846
1.01554445408754
64 16
1.02089438881021
65 106
1.02118312100403
66 3125
1.02398156287357
67 7980
1.03541593154556
68 12888
1.04720943128646
69 46032
1.06067411398872
70 3566
1.06548205329903
71 5026
1.07926576483874
72 41
1.08648217274233
73 72
1.09019587056164
74 2395
1.12961831514844
75 82301
1.14050108729716
Message: 2427 - Contents - Hide Contents Date: Thu, 13 Dec 2001 18:45:44 Subject: Re: Badness with gentle rolloff From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> When you measure the size of an et n by log(n), and are at the >> critical exponent, the ets less than a certain fixed badness are >> evenly distributed on average; >> This is only true if you choose a very low value for your "certain > fixed badness", right?Or start a bit away from 0.> 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.
Message: 2428 - Contents - Hide Contents 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
Message: 2429 - Contents - Hide Contents 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?
Message: 2430 - Contents - Hide Contents 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......
Message: 2431 - Contents - Hide Contents Date: Thu, 13 Dec 2001 18:52:25 Subject: A hidden message (was: Re: Badness with gentle rolloff) From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Rather than looking like random "noise", the pattern of "best local > ETs" seems to have a definite "wave" to it, with a frequency of about > 1680 -- that is, the "wave" repeats itself about 19 1/2 times within > the first 32768 ETs, seemingly with quite a bit of regularity.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.
Message: 2432 - Contents - Hide Contents 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 . . .
Message: 2433 - Contents - Hide Contents 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?
Message: 2434 - Contents - Hide Contents 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!
Message: 2435 - Contents - Hide Contents 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 * [with cont.] 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.
Message: 2436 - Contents - Hide Contents Date: Thu, 13 Dec 2001 19:04:14 Subject: Re: Badness with gentle rolloff From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> But it's not really unique as a critical asymptotic function (?), is > it?It is among functions n^e, for some fixed e; the value e=-1 is the critical exponent where n^(e+1)/(e+1) no longer works as an antiderivative, and going to smaller values of e leads to convergent series and integrals. You can get cute at the critical exponent, by looking at things like 1/(n ln n), 1/(n ln n ln ln n) and so forth. These diverge even more slowly than 1/n.
Message: 2437 - Contents - Hide Contents 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. :-)
Message: 2438 - Contents - Hide Contents 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?
Message: 2439 - Contents - Hide Contents Date: Thu, 13 Dec 2001 21:54:48 Subject: Re: Well . . . From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> I don't know what's going on here, but it sure reminds me of the > Riemann zetafunction!It implies things about the zeta function, and I want to post about it to sci.math.research; am I correct in thinking that you are the one who discovered this? I also am wondering if you are going to sic Matlab's FFT on the 5-limit also.
Message: 2440 - Contents - Hide Contents 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!
Message: 2441 - Contents - Hide Contents Date: Thu, 13 Dec 2001 19:11:11 Subject: Re: Badness with gentle rolloff From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> You can get cute at the critical exponent, by looking at things like > 1/(n ln n), 1/(n ln n ln ln n) and so forth. These diverge even more > slowly than 1/n.It should be noted that these work only "almost always", whereas 1/n works without exception, giving us an infinite set. It is highly probable that the badness of the very best systems using the critical expondent goes to zero, and goes fast enough that we could work in an extra log factor, but proving it would be another matter.
Message: 2442 - Contents - Hide Contents 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
Message: 2443 - Contents - Hide Contents 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 * [with cont.]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.
Message: 2444 - Contents - Hide Contents Date: Thu, 13 Dec 2001 19:12:27 Subject: A hidden message (was: Re: Badness with gentle rolloff) From: genewardsmith --- 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?
Message: 2445 - Contents - Hide Contents 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
Message: 2446 - Contents - Hide Contents 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 * [with cont.]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.
Message: 2447 - Contents - Hide Contents 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.
Message: 2448 - Contents - Hide Contents 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: Doesn't seem to block the ads in the messages... any suggestions to get it to work? -Carl
Message: 2449 - Contents - Hide Contents 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. 404 Not Found * [with cont.] Search for http://uq.net.au/~zzdkeena/Music/7LimitETBadness.xls.zip in Wayback Machine 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.
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