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Message: 2451 - Contents - Hide Contents 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.
Message: 2452 - Contents - Hide Contents Date: Thu, 13 Dec 2001 03:23:42 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:>> --- 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 KBDave, you have to plot "goodness", not "badness".
Message: 2453 - Contents - Hide Contents Date: Thu, 13 Dec 2001 03:40:36 Subject: Re: Badness with gentle rolloff From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> 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.]I seem to have confused the issue, since I thought it was about temperaments; it would be 4/3=1+1/3 for the 7-limit ets, and 2=(1+1/3)/(1-1/3) for 7-limit temperaments, but 5/4 = 1+1/4 for 11-limit ets, and 5/3 = (1+1/4)/(1-1/4) for 11-limit temperaments.
Message: 2454 - Contents - Hide Contents Date: Thu, 13 Dec 2001 04:38:13 Subject: Re: Badness with gentle rolloff From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> My (incomplete) understanding is that flatness is flatness. It's what > you acheive when you hit the critical exponent. The "logarithmic" > character that we see is simply a by-product of the criticality. > > I look forward to a fuller and more accurate reply from Gene.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; if you plotted numbers of ets less than the limit up to n versus log(n), it should be a rough line. If you go over the critical exponent, you should get a finite list. If you go under, it is weighted in favor of large ets, in terms of the log of the size.
Message: 2455 - Contents - Hide Contents Date: Thu, 13 Dec 2001 04:51:26 Subject: Re: Badness with gentle rolloff From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> My (incomplete) understanding is that flatness is flatness. It's > what>> you acheive when you hit the critical exponent. The "logarithmic" >> character that we see is simply a by-product of the criticality. >> >> I look forward to a fuller and more accurate reply from Gene. >> 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?> if you plotted numbers of ets less > than the limit up to n versus log(n), it should be a rough line. If > you go over the critical exponent, you should get a finite list. If > you go under, it is weighted in favor of large ets, in terms of the > log of the size.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?
Message: 2456 - Contents - Hide Contents Date: Thu, 13 Dec 2001 05:45:53 Subject: Re: Badness with gentle rolloff From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Dave, you have to plot "goodness", not "badness".Paul, I assume goodness = 1/badness? How could any reasonable transformation from badness to goodness change whether it looks flat or not? I've added goodness plots below the badness plots in 404 Not Found * [with cont.] Search for http://uq.net.au/~zzdkeena/Music/7LimitETBadness.xls.zip in Wayback Machine I agree that goodness lets you see the trends in the best more easily. But with the limited sample we have, up to 612-tET, it looks like the goodness of the best in any range is already falling off with increasing number of steps, even with steps*cents. Going to steps^(4/3)*cents just makes it fall off faster. So I still think steps*cents is the flat one. Yahoo's advertising has sure taken a quantum leap in obtrusiveness, if not badness! Yikes.
Message: 2457 - Contents - Hide Contents Date: Thu, 13 Dec 2001 05:51:09 Subject: Re: Temperament calculations online From: dkeenanuqnetau --- In tuning-math@y..., graham@m... wrote:> <Linear Temperament Finding Home * [with cont.] (Wayb.)> > > Early days yet, but it is working. > > > Graham That's awesome!
Message: 2458 - Contents - Hide Contents Date: Thu, 13 Dec 2001 06:08:21 Subject: One way to block web advertising From: dkeenanuqnetau I'm using the Guidescope proxy service. It's working for me. See * [with cont.] (Wayb.)
Message: 2459 - Contents - Hide Contents Date: Thu, 13 Dec 2001 11:47 +0 Subject: Re: More lists From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9v8gp5+lv0o@xxxxxxx.xxx> Me:>> so if you'd like to check this should be >> Paultone minimax: >> >> >> 2/11, 106.8 cent generator Paul:> That's clearly wrong, as the 7:4 is off by 17.5 cents!Well, that is interesting. It turns out the minimax temperament corresponds to a just 35:24. But I'd previously assumed that the minimax had to have a just interval within the consonance limit. Well, I've added some sticking tape to the algorithm, but I'm not sure it'll hold. This is what I get now 2/11, 109.4 cent generator basis: (0.5, 0.09113589675523795) mapping by period and generator: [(2, 0), (3, 1), (5, -2), (6, -2)] mapping by steps: [(12, 10), (19, 16), (28, 23), (34, 28)] highest interval width: 3 complexity measure: 6 (8 for smallest MOS) highest error: 0.014573 (17.488 cents) 7:5 =~ 10:7 consistent with: 10, 12, 22> I don't think it should count as unique since >>> 7:5 =~ 10:7Yes, that was a different problem. I wasn't including tritone-equivalences as duplicates. Graham
Message: 2460 - Contents - Hide Contents Date: Thu, 13 Dec 2001 12:05:50 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:>> Dave, you have to plot "goodness", not "badness". >> Paul, I assume goodness = 1/badness? How could any reasonable > transformation from badness to goodness change whether it looks flat > or not?You'll be looking at the opposite extremes of the graph.> I've added goodness plots below the badness plots in > 404 Not Found * [with cont.] Search for http://uq.net.au/~zzdkeena/Music/7LimitETBadness.xls.zip in Wayback Machine > > I agree that goodness lets you see the trends in the best more easily. > But with the limited sample we have, up to 612-tET, it looks like the > goodness of the best in any range is already falling off with > increasing number of steps, even with steps*cents. Going to > steps^(4/3)*cents just makes it fall off faster.Not really. At 612, you can't really see the difference yet. Go much further and you'll see it.
Message: 2461 - Contents - Hide Contents Date: Thu, 13 Dec 2001 12:23:18 Subject: Vitale 19 (was: Re: Temperament calculations online) From: paulerlich --- In tuning-math@y..., graham@m... wrote:> <Linear Temperament Finding Home * [with cont.] (Wayb.)> > > Early days yet, but it is working. > > > GrahamNice work, Graham! 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.)
Message: 2462 - Contents - Hide Contents Date: Thu, 13 Dec 2001 15:28:46 Subject: A hidden message (was: Re: Badness with gentle rolloff) From: paulerlich I wrote,> Not really. At 612, you can't really see the difference yet. Go much > further and you'll see it.Well I extended the graph out to 32768, and 4/3 starts to make more sense as an exponent. But I noticed something else -- something totally unexpected. 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. Are my eyes decieving me here, or is something going on? Gene? I'll try Matlab next . . .
Message: 2463 - Contents - Hide Contents Date: Thu, 13 Dec 2001 16:02:12 Subject: A hidden message (was: Re: Badness with gentle rolloff) From: paulerlich I 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. > > Are my eyes decieving me here, or is something going on? Gene? > > I'll try Matlab next . . .Take a look at the two pictures in Yahoo groups: /tuning-math/files/Paul/ * [with cont.] (I didn't enforce consistency, but we're only focusing on the "goodest" ones, which are consistent anyway). In both of them, you can spot the same periodicity, occuring 60 times with regular frequency among the first 100,000 ETs. Thus we see a frequency of about 1670 in the wave, agreeing closely with the previous estimate? What the heck is going on here? Riemann zetafunction weirdness?
Message: 2464 - Contents - Hide Contents Date: Fri, 14 Dec 2001 18:39:35 Subject: Re: the 75 "best" 7-limit ETs below 100,000-tET From: clumma --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > > > > > > > > > > > > > > > > >>> 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! >> These high numbers have their uses, and Paul seems to have > discovered something of considerable number-theoretic interest, > so give us a break, please.It's a fine thread, but it's also the first one on this list that I can remember for which I can see no musical application, and it's probably good somebody pointed this out. If it leads to more powerful methods for finding good temperaments, that could be used in something like Graham's temperament finder, then I'll take this back.
Message: 2465 - Contents - Hide Contents Date: Fri, 14 Dec 2001 20:03:19 Subject: Vitale 19 (was: Re: Temperament calculations online) From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote:>> If the tuning has a point of reflective symmetry (not necessarily >> at a note) then there will be the same number of otonal as utonal. >> Symmetry with respect to what? If it doesn't have to be a note, > any continuous single-generator chain will have it. > > -CarlYou bet! But if it isn't a continuous single-generator chain, then . . .
Message: 2466 - Contents - Hide Contents Date: Fri, 14 Dec 2001 18:44:23 Subject: Re: Badness with gentle rolloff From: clumma [Dave Keenan wrote...]> As has been done many times before, we are looking for a single > figure that combines the error in cents with the number of notes > in the tuning to give a single figure-of-demerit with which to > rank tunings for the purpose of deciding what to leave out of a > published list or catalog for which limited space is available. > One simply lowers the maximum badness bar until the right number > of tunings get under it. Thanks.Has consistency been considered? It is an error per note measure. [Gene Ward Smith wrote...]>it would be 4/3=1+1/3 for the 7-limit ets, and 2=(1+1/3)/(1-1/3) >for 7-limit temperaments, but 5/4 = 1+1/4 for 11-limit ets, and >5/3 = (1+1/4)/(1-1/4) for 11-limit temperamentsSorry, Gene, but I'm not following where you're getting these exponents. Is there a simple rule or reason I'm missing? -Carl
Message: 2467 - Contents - Hide Contents Date: Fri, 14 Dec 2001 20:04:41 Subject: Re: Badness with gentle rolloff From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote:> Sorry, Gene, but I'm not following where you're getting these > exponents. Is there a simple rule or reason I'm missing? > > -CarlIt has to do with Diophantine approximation theory. Have you read Dave Benson's course notes?
Message: 2468 - Contents - Hide Contents Date: Fri, 14 Dec 2001 20:23:55 Subject: Re: Proxomitron - freeware "proxy" alternative? From: genewardsmith --- In tuning-math@y..., "unidala" <JGill99@i...> wrote:> J Gill: This (freeware) item Now that's what I call a dead Proxomitron! * [with cont.] (Wayb.) (earlier version > 2.0, anyway) worked well as a software add-stripper in "harmony" with > Zonelabs' "ZoneAlarm" software firewall (freeware version > downloadable at: Zone Labs * [with cont.] (Wayb.) ).Same question--I installed Proxomitron and ZoneAlarm; now how do I exterminate WalMart?
Message: 2469 - Contents - Hide Contents Date: Fri, 14 Dec 2001 20:27:58 Subject: Re: A theory From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Since 612 is the "tuning of schismas", any near-multiple of 612 will > be more likely to have the schisma vanish, and thus to do 5-limit > well, than non-near-multiple of 612.It's not likely to have the schisma vanish; quite the reverse. What it *is* going to do is to represent the schisma as a certain number of steps with great accuracy.> So . . . there must be some very significant 7-limit comma lurking at > about 1/1664 octave. This is the Breedsma, which equals exactly one > step of 1663.89978-tET.You seem to be on to something, but why just the breedsma? This still seems to require more explanation.
Message: 2470 - Contents - Hide Contents Date: Fri, 14 Dec 2001 20:31:36 Subject: Re: Badness with gentle rolloff From: clumma>> >orry, Gene, but I'm not following where you're getting these >> exponents. Is there a simple rule or reason I'm missing? >> >> -Carl >> It has to do with Diophantine approximation theory. Have you read > Dave Benson's course notes?I've looked at them. What I could understand looked mundane, and what I couldn't looked like it required quite a bit more math than I know. Is there a section of the Benson which is particularly helpful here? -Carl
Message: 2471 - Contents - Hide Contents Date: Fri, 14 Dec 2001 20:53:41 Subject: Re: the 75 "best" 5-limit ETs below 2^17-tET From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Assuming a "critical exponent" of 3/2 for this case (is that right?)Correct, but could you give the exact definition of badness you are using? I also could not find the FFT data you said you posted.
Message: 2472 - Contents - Hide Contents Date: Fri, 14 Dec 2001 21:21:08 Subject: Some commas and Paul's chart From: genewardsmith I was wondering whether the secondary peak on Paul's chart had something to do with the ragisma, but that does not seem to be the case: Comma 1/log2(comma) 25/24 16.97974789 28/27 19.05944685 36/35 24.60509736 49/48 33.61644777 50/49 34.30961893 64/63 44.01393598 81/80 55.79763048 2048/2025 61.37300833 245/243 84.56347925 126/125 86.98951088 4000/3969 89.09131854 1728/1715 91.78824531 1029/1024 142.3028203 225/224 155.6112748 3136/3125 197.2631833 5120/5103 208.4128371 6144/6125 223.7951436 2401/2400 1663.898452 4375/4374 3032.168156 It would be interesting to see the numeric values of the other peaks. No apologies--deciding in advance that something is "useless" is *not* the way research should be done, and since our resources really don't require conservation, there is no point in acting like congresscritters deciding what to cut out of next year's NSF budget. There is something here we don't understand, and it could make a difference even to the relentlessly and prosaically "practical"--if that is what music is supposed to be.
Message: 2473 - Contents - Hide Contents Date: Fri, 14 Dec 2001 22:38:17 Subject: Vitale 19 (was: Re: Temperament calculations online) From: dkeenanuqnetau --- In tuning-math@y..., "clumma" <carl@l...> wrote:>>> 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 >>>> Huh? Compare 7-meantone with 7-chain-of-minor thirds. >> I realized this ambiguity after I posted. I meant, > any given linear temp. taken to any given number of > notes.That doesn't quite do it. You meant: .. that the number of o-tonal chords is the same as the number of u-tonal chords in any linear temp. taken to any number of notes.
Message: 2474 - Contents - Hide Contents Date: Fri, 14 Dec 2001 22:58:14 Subject: Vitale 19 (was: Re: Temperament calculations online) From: clumma>> > realized this ambiguity after I posted. I meant, >> any given linear temp. taken to any given number of >> notes. >> That doesn't quite do it. You meant: > > .. that the number of o-tonal chords is the same as the number > of u-tonal chords in any linear temp. taken to any number of notes.You took out the "given"?? -Carl
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