Tuning-Math Digests messages 2425 - 2449

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Message: 2425

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.


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Message: 2426

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


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Message: 2427

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.


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Message: 2428

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


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Message: 2429

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?


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Message: 2430

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......


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Message: 2431

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.


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Message: 2432

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 . . .


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Message: 2433

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?


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Message: 2434

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!


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Message: 2435

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.


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Message: 2436

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.


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Message: 2437

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. :-)


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Message: 2438

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?


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Message: 2439

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.


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Message: 2440

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!


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Message: 2441

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.


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Message: 2442

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


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Message: 2443

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.


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Message: 2444

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?


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Message: 2445

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


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Message: 2446

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.


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Message: 2447

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.


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Message: 2448

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


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Message: 2449

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.


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