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Message: 5850 - Contents - Hide Contents Date: Mon, 06 Jan 2003 19:44:54 Subject: We win From: Gene Ward Smith We made it to 72 members before tuning got up to 612.
Message: 5851 - Contents - Hide Contents Date: Mon, 06 Jan 2003 08:04:30 Subject: Re: Scale theory resources From: Carl Lumma>By the way, Carl, did you see my question about scale theory resources?I had, but I don't really know any good sources for this type of stuff. Might be a good question to ask on specmus.>Does anyone know of a good source for these, especially >on-line? I was browsing around and found that Clampitt >has been using Ramsey theory, which sounds as if people >are wading out past the shallow end of the pond, and >I'd like to catch up.If you can get a hold of any of these... Rothenberg, David. "A Mathematical Model for Perception Applied to the Perception of Pitch", Lecture Notes in Computer Science: Formal Aspects of the Cognitive Process, G. Goos and J. Harmanis (eds.), chapter 22, 1975, pp. 127-141. Rothenberg, David. "A Model for Pattern Perception with Musical Applications. Part I: Pitch Structures as Order-Preserving Maps", Mathematical Systems Theory vol. 11, 1978, pp. 199-234. Rothenberg, David. "A Model for Pattern Perception with Musical Applications Part II: The Information Content of Pitch structures", Mathematical Systems Theory vol. 11, 1978, pp. 353-372. Rothenberg, David. "A Model for Pattern Perception with Musical Applications Part III: The Graph Embedding of Pitch Structures", Mathematical Systems Theory vol. 12, 1978, pp. 73-101. (from Manuel's Tuning & Temperament Bib.) -Carl
Message: 5852 - Contents - Hide Contents Date: Mon, 06 Jan 2003 20:00:42 Subject: Re: Poptimal generators From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> "Poptimal" is short for "p-optimal". The p here is a real variable > p>=2, which is what analysts normally use when discussing these Holder > type normed linear spaces. > > A pair of generators [1/n, x] for a linear temperament is >*poptimal* if there is some p, 2 <= p <= infinity,why not go all the way to 1? MAD, or p-1, error certainly seems most appropriate for dissonance curves such as vos's or secor's -- which are in fact even pointier at the local minima (resembling exp (|error|)) . . .> "Muggles" [5, 1, -7, -19, 25, -10] [1, 62/197] > > "Beatles" [2, -9, -4, 16, 12, -19] [1, 19/64]what did i miss?
Message: 5853 - Contents - Hide Contents Date: Mon, 06 Jan 2003 20:05:53 Subject: Re: Poptimal generators From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:>MAD, or p-1, errori meant p=1, not p-1.
Message: 5854 - Contents - Hide Contents Date: Mon, 06 Jan 2003 20:13:20 Subject: thanks manuel From: wallyesterpaulrus thanks to manuel for putting me up for the night, playing lots of cds for me, helping me with train tickets, showing me scala, and everything else. regarding complexity attributes in scala -- i don't think i would say that the "erlich attribute" or "tenney attribute" of the notes of a scale should be based on *pitch ratios*. they should be based on *interval ratios*. so one might either be interested in the *average* complexity of the intervals formed by the note in question from all the other notes in the scale, or, in special cases, the complexity of the interval formed by the note from the tonic (1/1). the latter would be the current value of the attribute, but i don't think that is of as general interest or the natural application of these interval complexity measures to a scale . . . thanks again!! (and so sorry graham i missed you -- i assumed incorrectly which friday and saturday you meant!!!)
Message: 5855 - Contents - Hide Contents Date: Mon, 06 Jan 2003 20:28:26 Subject: 31, 112 and 11-limit Meantone From: Gene Ward Smith We get a good version of 11-limit Meantone by using the mapping the 31-et gives us. This has [1, 4, 10, -13, 4, 13, -24, 12, -44, -71] for a wedgie and [[1, 1, 0, -3, 11], [0, 1, 4, 10, -13]] for a mapping to primes. I *still* cannot get the 31-et to be poptimal using this! In fact, the best poptimal for it seems to be our old pal, 112-equal Meantone, a hitherto unknown star in the Meantone firmament.
Message: 5856 - Contents - Hide Contents Date: Mon, 06 Jan 2003 20:35:18 Subject: Re: Poptimal generators From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> why not go all the way to 1? MAD, or p-1, error certainly seems most > appropriate for dissonance curves such as vos's or secor's -- which > are in fact even pointier at the local minima (resembling exp > (|error|)) . . .I didn't trust a p as low as 1 to give good results, since it tends to ignore outliers, and the argument that we get continuous functions for p as a function of generator size is easier to make if we have p>=2; not that that really matters, I suppose. Maybe I should give it a shot and see if I can convince 31 to be a poptimal meantone in this way.>> "Muggles" [5, 1, -7, -19, 25, -10] [1, 62/197] >> >> "Beatles" [2, -9, -4, 16, 12, -19] [1, 19/64] >> what did i miss?They've been around forever, I just decided to name them. "Muggles" is an inferior sort of Magic, and 19/64 goes with Beatles for those of us who were around back then. It certainly seems better than "Gulf of Tonkin Incident" or "Lyndon Johnson Elected in a Landslide", which are the only other public events I recall from that year.
Message: 5857 - Contents - Hide Contents Date: Mon, 06 Jan 2003 21:12:09 Subject: Re: Poptimal generators From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith > <genewardsmith@j...>" <genewardsmith@j...> wrote:>> "Poptimal" is short for "p-optimal". The p here is a real variable >> p>=2, which is what analysts normally use when discussing these > Holder>> type normed linear spaces. >> >> A pair of generators [1/n, x] for a linear temperament is >> *poptimal* if there is some p, 2 <= p <= infinity, >> why not go all the way to 1? MAD, or p=1, error certainly seems most > appropriate for dissonance curves such as vos's or secor's -- which > are in fact even pointier at the local minima (resembling exp > (|error|)) . . .Possibly because no one in the history of this endeavour has ever before now suggested that mean-absolute error corresponds in any way to the human perception of these things. I think everyone agrees that the worst errors dominate; either totally as in max-absolute (p=oo) or partially as in RMS (p=2). However, I strongly object to this statement of Gene's in the tuning list: "A "poptimal" generator can lay claim to being absolutely and ideally perfect as a generator for a given temperament ..." When we're talking about human perception, as we are, it should be obvious that nothing can be absolutely and ideally perfect for everyone. Even a single person might prefer slightly different generators for different purposes. To validate such a claim of "perfection" you would at least need to produce statistics on the opinions of many listeners. Gene, you seem to be confusing beautiful mathematics with accurate modelling of a psychoacoustic property (yet to be established).>> "Muggles" [5, 1, -7, -19, 25, -10] [1, 62/197]I think the names "Muggles" and "Wizard" could be swapped for these two temperaments. The one above, also called "Narrow major thirds", is significantly better than the Twin major thirds temperament by anyone's badness. The twin major thirds temperament is so bad (due mostly to its complexity) that it's barely worth mentioning (at least at the 7-limit).>> "Beatles" [2, -9, -4, 16, 12, -19] [1, 19/64]Previously called "Neutral thirds with complex 5s". I guess the name's based of the "64" denominator?
Message: 5858 - Contents - Hide Contents Date: Mon, 06 Jan 2003 21:20:25 Subject: Re: 31, 112 and 11-limit Meantone From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> We get a good version of 11-limit Meantone by using the mapping the31-et gives us. This has> > [1, 4, 10, -13, 4, 13, -24, 12, -44, -71] > > for a wedgie and > > [[1, 1, 0, -3, 11], [0, 1, 4, 10, -13]] > > for a mapping to primes. I *still* cannot get the 31-et to bepoptimal using this! In fact, the best poptimal for it seems to be our old pal, 112-equal Meantone, a hitherto unknown star in the Meantone firmament. What this says to me is that p-optimality is an interesting guide in many cases but we shouldn't let it take over from commonsense. In the case of meantone it says to me that the range of generators is too wide and we may need two ETs to cover it.
Message: 5859 - Contents - Hide Contents Date: Mon, 06 Jan 2003 22:59:27 Subject: Re: 31, 112 and 11-limit Meantone From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote:> What this says to me is that p-optimality is an interesting guide in > many cases but we shouldn't let it take over from commonsense.Sounds right, but I think it should be considered. In the> case of meantone it says to me that the range of generators is too > wide and we may need two ETs to cover it.Too wide? The problem has been that it is too damned narrow, being stuck in the neighborhood of 1.53 to 1.58 or thereabouts in terms of Blackwood's constant. We can't seem to get down to 1.5, for 31, and only for 5-limit do we get up to 1.6, for 81.
Message: 5860 - Contents - Hide Contents Date: Mon, 06 Jan 2003 23:17:39 Subject: Re: 31, 112 and 11-limit Meantone From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> Too wide? The problem has been that it is too damned narrow, being > stuck in the neighborhood of 1.53 to 1.58 or thereabouts in terms of Blackwood's constant. We can't seem to get down to 1.5, for 31, and only for 5-limit do we get up to 1.6, for 81.At least t = pi/2 is in the range, so if you are bored with Lucy tuning (t = (2*pi-5)/(7-2*pi)) you could try this instead, and tune to fifths of 2400*((2*pi+3)/(7*pi+10)) cents. It's poptimal.
Message: 5861 - Contents - Hide Contents Date: Tue, 07 Jan 2003 11:46:51 Subject: Re: Nonoctave scales and linear temperaments From: Carl Lumma>I also looked at beta and gamma, but I didn't find much. Beta >can be related to a cheeseball system obtainable as h75&h94, >and it is possible to regard gamma as 5/171. Maybe Graham or >Dave can point out something I am missing here.Would you expect that tempering the "octave" instead of the "generator" at a given limit would lead to a different list of optimal temperaments or optimal generator for a temperament? Perhaps we should, since our lists tend to keep 2:1 on both axes of the map (or at least with a very short period of ie's) while not considering its temperament. That is, we often discuss temperaments with 2:1 "octaves" and irrational or dissonant "generators". Perhaps we should instead allow the "octave" to be any size while applying a single weighted error function to all consonances, including the 2:1. Most weighting schemes would probably attract the good generators of good temperaments to consonant intervals. It might even be possible to always force one of the generators to a pure 2:1 with a steep enough weighting. Rather than guessing, it seems like a good place to plug in harmonic entropy. But why average (max, rms, etc.) complexity and error across a map before weighting and calculating badness? Why not weight per harmonic identity, then just sum to find badness at the given limit? If you weight the error right, you shouldn't have to weight the complexity. Let g(x) be the graham complexity of identity x, and e(x) be the weighted error of that identity. Then minimize Sum [g(r) * e(r)] where r goes over all the identities in the given limit. If a minimum could be found, we would know the optimal generator of the optimal temperament. Assuming a perfect error function, which can't exist outside of perfect, deterministic neuroscience. Given the way the error and graham complexity of identities compound when covering a limit (some sort of consistency is required by Gene's def. for linear temperaments, IIRC), per- identity consideration should be enough. But one could imagine weighting per-dyad, or even per-chord. Or maybe I'm missing something? -Carl
Message: 5862 - Contents - Hide Contents Date: Tue, 07 Jan 2003 19:41:10 Subject: Re: Nonoctave scales and linear temperaments From: Carl Lumma>Let g(x) be the graham complexity of identity x, and e(x) be >the weighted error of that identity. Then minimize > >Sum [g(r) * e(r)] > >where r goes over all the identities in the given limit.That's supposed to be... Let g(x) be the graham complexity of identity x, and e(x) be the weighted error of that identity. Then minimize Sum [g(r) * e(r)] over all maps, where r goes over the identities of the given map. -Carl
Message: 5863 - Contents - Hide Contents Date: Tue, 07 Jan 2003 20:01:48 Subject: Re: Improved generators for 7-limit linear temperaments From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> Since 9-odd-limit and 7-odd-limit involve the same set of primes,we can justify using either. If we regard them both as important, we can justify using anything between the ranges of the poptimal generators even when these are disjoint. To be sure, when 3 has high complexity we might be interested in 7 but not 9, but I think this is pretty much of a quibble. I calculated a 9-limit poptimal, (the second listed below) and then found the best which can be had by putting 7 and 9 together. The "univeral" generators listed below are where all the p values give the same result; since this didn't happen for both 7 and 9 I still got an et.> > At this point I would say things are generally looking good. We nowhave 72 for Miracle, 84 for Orwell, 12 for Diminished and Dominant Seventh, 22 for Porcupine, 31 for Semififths and Supermajor Seconds,> 41 for Superkleismic and 46 for Semisixths. The 112-et for Meantoneis still a little over the top, but is at least worth considering; the tendency for Meantone to want a generator in this very small interval is rather striking.> > ... > "Kleismic", > [6, 5, 3, -7, 12, -6], [[1, 0, 1, 2], [0, 6, 5, 3]], > > [1, 14/53], [1, 14/53], [1, 14/53],I'm a little puzzled how 53-ET popped up (instead of 72-ET) for this one. In this regard, please refer to: Yahoo groups: /tuning-math/message/5253 * [with cont.] There I evaluated by minimax only, but I thought that it seemed pretty clear that 53 was not our choice to notate kleismic. (I haven't waded through all of the details of your explanation, because I'm just interested in the results, and this one puzzles me.) --George
Message: 5864 - Contents - Hide Contents Date: Tue, 07 Jan 2003 20:51:22 Subject: Re: Nonoctave scales and linear temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:> Would you expect that tempering the "octave" instead of the > "generator" at a given limit would lead to a different list > of optimal temperaments or optimal generator for a temperament?The problem with this is that it makes the question of what the consonancesof the temperament are murky--we can't simply use everything in the odd limit for some odd n. Of course, we could simply create a set of intervals and then temper, or use n-limit including evens. Perhaps we should instead allow the> "octave" to be any size while applying a single weighted error > function to all consonances, including the 2:1.Should we have 8-limit as well as 7-limit and 9-limit?> But why average (max, rms, etc.) complexity and error across > a map before weighting and calculating badness? Why not > weight per harmonic identity, then just sum to find badness at > the given limit? If you weight the error right, you shouldn't > have to weight the complexity.I'll return to this after I've had my breakfast coffee. :)
Message: 5865 - Contents - Hide Contents Date: Tue, 7 Jan 2003 12:57:28 Subject: Fw: FW: PLEEEEEASE READ!!!! It was on the news! From: monz ----- Original Message ----- From: "Can Akkoc" <can193849@xxxxx.xxx> To: "Gonca Tokuz" <gensek1@xxxxxx.xxx.xx>; "Mustafa Tokyay" <mtokyay@xx.xxxx.xxx.xx>; "Oktay Yagiz" <obyagiz@xxx.xxx>; "Ali Yuksel Selcukoglu" <selcukya@xxx.xxx>; "Ayhan Sicimoglu" <ayhan@xxxxxxxxxxxx.xxx>; "Zerrin Soylemez" <soylemez@xxxxxx.xxx.xx>; "Mete Soyoguz" <soyoguzm@xxx.xxx>; "Necil Toktay" <ntoktay@xxxxxx.xxx.xx>; "Aydin Nurhan" <anurhan@xxxxx.xxx>; "Mahmut Esat Ozan" <meozan@xxx.xxx>; "Asli Ozel" <asliozel73@xxxxxxx.xxx>; "Virginia Parks" <vparks@xxxxxxxx.xxx>; "Keenan Pars" <keenanpars@xxxxx.xxx>; "Julio Ruiz" <ruizjulioc@xxxxx.xxx.xx>; "Joe Monzo" <monz@xxxxxxxxx.xxx>; "Filiz Hosukoglu" <flzchsk@xxxxx.xxx>; "Rifat Karakimseli" <rifatk@xxxxxxxxxxx.xxx>; "Zeki Karakimseli" <zkarakimseli@xxxx.xxx>; "M. Kemal Karaosmanoğlu" <mkemal@xxxxxx.xxx.xx>; "Beyhan Karsligil" <beyhan_karsligil@xxxxx.xxx>; "Murat Karsligil" <mkarsligil@xxxxx.xxx>; "Necdet Kesmez" <kesmez@xxxxx.xxx.xx>; "Suzan Konar" <sosbil@xxxxxx.xxx.xx>; "Hayri Korezlioglu" <hayri@xxxx.xxx.xx>; "Dogan Erbahar" <doganerbahar@xxxxxxxxxxx.xxx>; "Cenap Erenben" <erenben@xxxxxx.xxx>; "Yener Erguven" <erguven@xxxxxxx.xxxxxxx.xxx.xx>; "Özkan Esmer" <oesmer@xxxxxxxxxxx.xxx>; "Jon Garcia" <jgarcia@xxxxxxxx.xxx>; "Hasan Gokpinar" <hasan_gokpinar@xxxxxxx.xxx>; "Metin Guney" <guney_m@xxxxxxx.xxx>; "John van der Hoek" <jvanderh@xxxxx.xxxxxxxx.xxx.xx>; "Ergun Cagatay" <tetragon@xxxx.xxx>; "Gunsan Cetin" <gunsan_cetin@xxx.xxx>; "John Chalmers" <jhchalmers@xxxx.xxx>; "Christopher Chapman" <christopher.chapman@xxxxxxxx.xxx>; "Ryan Culpepper" <ryanc@xxxxxx.xxxx.xxx>; "Ewa Dahlig" <eda@xxxxxxx.xx.xx.xxx.xx>; "Metin Atamer" <interwind@xxx.xxx.xx>; "Yusuf Atmaca" <yusuf@xxxxx.xxx.xx>; "Erdal Atrek" <erdalatrek@xxx.xxx>; "Ruhi Ayangil" <ayangil@xxxxxx.xxx.xx>; "Cengiz Aydin" <cengiz.aydin@xxx.xxx>; "Tahir Aydogdu" <tahiraydogdu@xxxxxxxxxxxx.xxx>; "Aydan Bakan" <aydanbakan@xxxxx.xxx>; "Baris Baraz" <bbaraz@xxxxxxx.xxx.xx>; "Kaya Buyukataman" <grassroots@xxxxxxxxxxxx.xxx>; "Gul Abut" <gul@xxxxxxxxxxx.xxxx.xxx>; "Huseyin Abut" <abut@xxxxxxx.xxxx.xxx>; "Gulten Akkoc" <gakkoc@xxxxx.xxx>; "Nevra Akkoc" <nakkoc@xxxxx.xxx>; "Joseph Albree" <joe2@xxxxxxx.xxx.xxx>; "Server Acim" <serveracim@xxxxx.xxx>; "Asim Addemir" <addemira@xxxxxxxx.xxx>; "Selim Akkoc" <selim@xxxxxxxxxx.xxx>; "Ali Askin" <aaskin@xxxxxx.xxx.xx>; "Cem Baysal" <shrama@xxxxxxxxxxx.xxx>; "Aykut Berk" <ayberk@xxxxx.xxx>; "Sehvar Besiroglu" <besir@xxx.xxx.xx>; "Hakan Cevher" <cevher@xxxxxxx.xxx.xxx.xx>; "Cullen Duke" <dukecul@xxxxxx.xxx>; "Ibrahim Halil Guzelbey" <guzelbey@xxxxxx.xxx.xx>; "Yuruk Iyriboz" <iyri@xxx.xxx>; "Muammer Karabey" <mmkarabey@xxxx.xxx>; "Zihni Kutlar" <zihni.kutlar@xxxxxx.xxx.xx>; "Tolga Larlar" <tolgalarlar@xxxxx.xxx>; "Sevgin Oktay" <sevgin@xxxxxxxxxxxxxxxx.xxx>; "Yuksel Oktay" <yoktay@xxx.xxx.xx>; "Ceylan Orhun" <herana@xxxxxxxxxxx.xxx>; "Gamze Sisman" <gamzesisman@xxxxx.xxx.xx>; "Erol Tasdemiroglu" <siberasertas@xxxxxxxxxxx.xxx>; "Ufuk Tezer" <ufuk_tezer@xxxxxxx.xxx>; "Hasan Toy" <hasantoy@xxxxxxx.xxx>; "Tacettin Yuksel" <t_yuksel@xxxxxxx.xxx> Sent: Tuesday, January 07, 2003 12:24 PM Subject: Fwd: FW: PLEEEEEASE READ!!!! It was on the news!> > > Sehvar Besiroglu <besir@xxx.xxx.xx> wrote:Date: Wed, 08 Jan 2003 05:53:23 -0800 > Subject: FW: PLEEEEEASE READ!!!! It was on the news! > From: "Sehvar Besiroglu" > To: "Aykanat, Acar" , > Can Akkoc , Cihat Askin , > "Murat AYDEMĞR" , > "n.serhan aytan" , > "Münir Nurettin Beken" , > Sibel Bozdogan , > ezgi CARDAKTAN , > hedia chaffai , > "Derya TÜRKAN" > , > nilgun dogrusoz , > Sedat Eren , > "Tolga Gülen" , > Kamran Hooshmand , > "Ciğdem Kafescioğlu" , > Sevgi KANTAR , > Belma Kurtisoglu , > kurtisoglu , > oozbilen , Siir Ozbilge , > "Hadass Pal-Yarden" > > > > ---------- > From: "Feridun Ozgoren" <feridun.ozgoren@xxxxxxx.xxx> > To: <VTunaligil@xxx.xxx>, "Semin Cagin" <gzaimler@xxxxxxx.xxx>, "SelisOnel" <selis@xxx.xxx.xxx>, "Selim Alptekin" <alptekin@xxxxxxx.xxx>, "Sehvar Besiroglu" <besir@xxx.xxx.xx>, "Sandra Layman" <sandralayman@xxxxxxxxx.xxx>, "Nan Freeman" <nan.freeman@xxxxxxx.xxx>, "Muzaffer Kanaan" <muzafferkanaan@xxxxxxx.xxx>, "Moshahida Sultana" <sultana.m@xxx.xxx>, "Metin Sezgin" <mtsezgin@xxx.xxx>, "Ibrahim Kalin" <ikalin@xxxxxxxxx.xxx>, "ibrahim hakki yigit" <ibrahimhakkiyigit@xxxxxxx.xxx>, Hüseyin Adalar <hadalar4@xxxxx.xxx>, "Himmet, Hesna Taskomur" <taskomur@xxx.xxxxxxx.xxx>, "Hazal Selcuk" <hazalselcuk@xxxxxxx.xxx>, "Harun" <harunmusic@xxxxxxx.xxx>, "Hamza Zeytinoglu" <hamza_zeytinoglu@xxx.xxxxxxx.xxx>, "Hakan Talu" <refikhakan@xxxxxxxxxxx.xxx>, "Hakan Partal" <hpartal@xxxxx.xxx>, "Gunduz Saner" <gsaner2003@xxxxx.xxx>, "Guliz Pamukoglu" <gupam@xxx.xxx>, "Gulhan G. Ayyildiz" <gulhangayyildiz@xxxxxxx.xxx>, "gulden ayboga" <ayboga@xxxxx.xxx>, "Ferhan Ozgoren" <grafik@xxxxxx.xxx.xx>, "Fatma" <fazomers@xxx.xxx>, "Emre Yildirim" <emeyil@xxxxx.xxx>, "Derya Turkan" <deryaturkan@xxxxxx.xxx>, "Bora Pervane" <bora@xxxxxxx.xxx>, "Birol Yesilada" <BYesilada@xxx.xxx>, "Barihuda Tanrikorur" <barihudatanrikorur@xxxxxxx.xxx>, "Ahmet,Nurten,Zafer Sahin" <ab_ce_43@xxxxx.xxx>, "Ahmet Ersoy" <ersoya@xxxx.xxx.xx>, "Ahmet Erdogdular" <ahmeterdogdular@xxxxxxx.xxx>, "Ahmed Husrev Isbilir" <ahmedhusrevi@xxxxx.xxx>> Subject: Fw: PLEEEEEASE READ!!!! It was on the news! > Date: Mon, Jan 6, 2003, 7:22 AM > > > > ----- Original Message ----- > From: Muzaffer Kanaan <mailto:muzaffer.kanaan@xxxxxxx.xxx> > To: feridun.ozgoren@xxxxxxx.xxx <mailto:feridun.ozgoren@xxxxxxx.xxx> ; gupam@xxx.xxx <mailto:gupam@xxx.xxx> > Sent: Tuesday, January 07, 2003 9:58 AM > Subject: FW: PLEEEEEASE READ!!!! It was on the news! > > Boyle seylere de hic inanmam ama.,.....bir deneyelim bakalim. > > Muzaffer > > -----Original Message----- > From: Burak KALAC [mailto:burak.kalac@xxxxxxx.xxx.xxx > Sent: Tuesday, January 07, 2003 7:44 AM > Subject: PLEEEEEASE READ!!!! It was on the news! > > BLL ABMLE ANLATIK DA........... > > > > Subject: FW: PLEEEEEASE READ!!!! It was on the news! > > To all of my friends, I do not usually forward messages, but this is frommy good friend Pearlas Sanborn and she really is an attorney.> > If she says that this will work - it WILL work. After all, what have yougot to lose?> SORRY EVERYBODY..JUST HAD TO TAKE THE CHANCE!!! > I'm an attorney, and I know the law. This thing is for real. > Rest assured AOL and Intel will follow through with their promises forfear of facing a multimillion-dollar class action suit similar to the one filed by PepsiCo against General Electric not too long ago.> > > > Dear Friends, > Please do not take this for a junk letter. Bill Gates is sharing hisfortune. If you ignore this you will repent later. Microsoft and AOL are now the largest Internet companies and in an effort to make sure that Internet Explorer remains the most widely used program, Microsoft and AOL are running an e-mail beta test.> > When you forward this e-mail to friends, Microsoft can and will track it(if you are a Microsoft Windows user) for a two week time period.> > For every person that you forward this e-mail to, Microsoft will pay you$245.00, for every person that you sent it to that forwards it on, Microsoft will pay you $243.00 and for every third person that receives it, you will be paid $241.00. Within two weeks, Microsoft will contact you for your address and then send you a cheque.> > Regards. > Charles S. Bailey > General Manager Field Operations > 1-800-842-2332 Ext. 1085 or 904-245-1085 or RNX 292-1085 > Charles_Bailey@xxx.xxx <mailto:Charles_Bailey@xxx.xxx> > > > > I thought this was a scam myself, but two weeks after receiving thise-mail and forwarding it on, Microsoft contacted me for my address and within days, I received a cheque for US$24,800.00. You need to respond before the beta testing is over. If anyone can afford this Bill Gates is the man.> > It's all marketing expense to him. Please forward this to as many people as possible. > You are bound to get at least US$10,000.00. > We're not going to help them out with their e-mail beta test withoutgetting a little something for our time. My brother's girlfriend got in on this a few months ago. When I went to visit him for the Baylor/UT game.> > > > She showed me her check. 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Message: 5866 - Contents - Hide Contents Date: Tue, 07 Jan 2003 21:12:28 Subject: Re: Improved generators for 7-limit linear temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" <gdsecor@y...>wrote:> I'm a little puzzled how 53-ET popped up (instead of 72-ET) for this > one. In this regard, please refer to: > > Yahoo groups: /tuning-math/message/5253 * [with cont.] > > There I evaluated by minimax only, but I thought that it seemed > pretty clear that 53 was not our choice to notate kleismic.In a nutshell, 53 is smaller than 72; it is very close to the minimax valuewhile 72 is very close to the rms value, but either will work. We have: Least squares 316.664 Least cubes 316.466 Least fourth powers 316.566 Minimax 316.993 Both the 53-et value of 316.981 cents and the 72-et value of 316.667 cents are between the rms and minimax optimal generators. How differently would the notation look in 53 as opposed to 72?
Message: 5867 - Contents - Hide Contents Date: Tue, 07 Jan 2003 21:17:42 Subject: Re: Nonoctave scales and linear temperaments From: Carl Lumma>The problem with this is that it makes the question of what the >consonances of the temperament are murky--we can't simply use >everything in the odd limit for some odd n. Of course, we could >simply create a set of intervals and then temper, or use n-limit >including evens.Perhaps I'm not seeing it, but I don't think we need to change our concept of limit.>> But why average (max, rms, etc.) complexity and error across >> a map before weighting and calculating badness? Why not >> weight per harmonic identity, then just sum to find badness at >> the given limit? If you weight the error right, you shouldn't >> have to weight the complexity. >>I'll return to this after I've had my breakfast coffee. :)That was a seriously late-night post, and hopefully it made sense. For all I know you could already be calculating badness this way. For linear temperaments, we have a 2-D lattice of generators. The map turns points on this lattice into points on the (weighted, if you like) harmonic lattice, and back again. The complexity of a pair of such points is the taxicab distance on the lattice of generators, and the error is the taxicab distance on the harmonic lattice. You can define as many mapping points as you like. But if you stick to consistent maps of the identities only, you should get reasonable results for all the members of what we normally call a limit. -Carl
Message: 5868 - Contents - Hide Contents Date: Tue, 07 Jan 2003 21:18:15 Subject: Re: Temperament notation From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>> That was a bad choice of terminology. The issue that Gene and >>> I are raising is: should notation be based on the melodic scale >>> being used, or should they be based on the 7-tone meantone >>> diatonic scale musicians are already familiar with? >>>> The issue that I raise is: how many different notations can you >> expect a person to learn? >> As many as he needs to play the music he wants to.I'm talking about microtonal performers, who (I imagine) would want to learn as few systems of notation as possible -- I think most would settle for one, if it will handle most everything.> Since learning septimal notation gives access to 300 years of > Western music, the potential payoff for each new one shouldn't > be grounds for complaint.What, exactly, do you mean by "septimal notation"? The notation Dave and I have been working on includes septimal intervals -- and a lot more besides. One major problem with learning different notations is that a given symbol could mean one thing in one notation and something else in another, and even without this you start almost from scratch with each new notation. Better to have a single versatile (albeit complicated) notation for the performer that requires knowledge of only a small subset of symbols for a given application; another application may include other symbols, but there would be commonality (and no conflict) between them.>>>> But what value is the decimal notation to me, and what incentive >>>> would I have to learn it without a decimal keyboard? >>>>>> () It gives you an invaluable tool for understanding the music. >>>> Okay, but only as long as the music is written in the Miracle >> temperament, yes? (More about this below.) >> Only if the music is written in a decatonic scale, yes. There > are many ways to supply the chromatic pitches outside of Miracle > temperament. >[GS, re mapping the Miracle temperament on a Bosanquet generalized keyboard:]>> The tones would be on a diagonal row of keys that (ascending in >> pitch) would go off the near edge of the keyboard; but they could >> be picked up at the far edge,This would be true of a 31 mapping; a 41 mapping (ascending) would go off the far edge. But in either case my conclusion holds:> so yes, it can be done without >> extraordinary effort. >> I'd say there's nothing far worse about such a setup than in > using the Halberstadt for 12-equal.I'd be the first to say that it's far better than the Halberstadt for 12-ET, because like intervals occur in like patterns, so you need only a single fingering pattern for a given interval, chord, scale, etc., regardless of the key. (However, like patterns do not guarantee like intervals, as would be the case with a decimal keyboard, so you must take care to observe in which keys this will work, just as you would need to avoid the wolf of meantone temperament on a Halberstadt keyboard.)>>>> Now give me the same decimal keyboard with Partch's 11-limit >>>> JI mapped onto it (observe that this was the reason that I >>>> originally came up with the layout). Again, I should do just >>>> fine with 72-ET sagittal notation, assuming that I am proficient >>>> with the keyboard. >>>>>> You seem to be saying that it's easier to learn to find pitches >>> on a keyboard than it is to learn to find pitches in a notation... >>> For me, it's the opposite. >>>> Actually I do find it easier to perceive the pitch relationships >> on a generalized keyboard (of whatever sort) than from a notation, >> Well, that's an important statement. Since I don't have any > experience playing a generalized keyboard of any sort, I have > nothing to offer. I can say that learning to read music was > easier for me than learning to finger the piano. I really have > no idea if this relationship would remain when learning a new > keyboard/notation pair.That involved acquiring a certain amount of manual dexterity (including independent movement of hands and fingers) that you already possess when learning a new keyboard. I could play simple things immediately the first time I sat down at a Bosanquet keyboard. A decimal keyboard would not immediately be as familiar, but things come quickly if you learn to do things by learning interval patterns.>> The broader point that I was trying to make seems to have gotten >> lost in all of the details of the discussion. I was trying to >> show that there is no particular advantage in using decimal >> notation to notate music that is *not* based on the Miracle >> geometry (e.g., Partch's music, which is better understood in >> reference to an 11-limit tonality diamond), but for which the >> tones may still be very suitably mapped onto a decimal keyboard. >> The advantage of decimal notation comes into effect only when and >> if you are using the Miracle temperament itself, i.e., exploiting >> the tonal relationships that are unique to Miracle. Likewise, if >> I play something in 31, 41, or 72-ET on a decimal keyboard that >> was composed by someone utterly ignorant of Miracle as an >> organizing principle for tonality (as I believe *all* of us were >> up until a couple of years ago -- myself included), is the decimal >> notation going to benefit me in any way if the composition which >> I am playing was not conceived as being decatonic? >> No! But there are other temperaments with interesting decatonic > scales besides 31, 41, and 72, and all of them would get ten > nominals under my pen, and I suspect the differences in > accidentals would be easy for performers to learn. >>> But it would be unrealistic to expect anyone to learn three >> different notations for one tonal system, according to which >> tonal relationships are exploited in a given piece. >> Perhaps we'll have to agree to disagree. >>> (Or suppose that a piece is heptatonic in one place and decatonic >> in another. Do we switch notations in the middle of the page?) >> Absolutely! Just like switching clefs or key signatures.Hey, you were supposed to say "no, of course not!" Suppose it goes back and forth from heptatonic to decatonic every measure, or the right hand does something decatonic, while the left hand is heptatonic, etc. Or there may come a point where the music can get so sophisticated that it would be difficult to tell what-tonic the composer had in mind. Let's try not to confuse the poor performer (or the *good* performer either).> ... > You mention Partch's music, which doesn't really use any fixed > melodic scale. I would think transpositionally invariant > notation would be optimal, for the scores at least. > > But Partch had the right idea... since his instruments played > different scales (tonality diamond, microchromatic scales, > ancient melodic scales, etc.), he notated differently for each > of them.Help! That's what I'm trying to avoid -- insofar as possible, I want the pitch notation to be independent of the instrument. How can you understand vertical harmony in a score if the notation is different in different lines?>> Since we are already acquainted with a notation that uses 7 >> nominals, and if that works reasonably well for many alternative >> tunings, then why not have a generalized notation that builds on >> that? >> I am genuinely interested to see how it looks. You should debut > it with sample music, both original and classical.My paper has a couple of samples, plus examples of the 17-limit consonances in the key of C. These at least give you an idea of how things look on paper. This, along with the explanation of the notation, would be enough to get you started. (I'll have to get part of that posted soon, after I go through it and make a few small changes.)> ...>>> Let's ask it this way: take the well-tempered clavier and re- >>> write it with 6 nominals. Is that only a slight disadvantage? >>>> I was about to say no, but only because 6 nominals will hardly >> work well with anything, >> They'd work spledidly for wholetone music. It's fortunate that > 12-equal supports the wholetone scale without collisions. But > if you look at octatonic music, it would be much better notated > with 8 nominals. I would go so far as to suggest that this held > back octatonic music in the last century. >>> even for Partch (since 6:7:8:9:10:11 isn't a constant structure). >> The diatonic scale in 12-equal isn't a constant structure either.That's true only as a technicality -- the augmented fourth and diminished fifth in the diatonic scale just happen to be the same size in 12-ET, but since they are functionally different, this isn't a liability. Partch's hexads make more sense if you consider them heptatonically (with a vacanct position between 6 and 7).> Actually, it's strict propriety that's important, and the > violations aren't too bad here, and I think 6 nominals would be > ideal. But Partch doesn't really stick to this scale, so... >>> I'm not really arguing against specialized notations with other >> than 7 nominals, but I don't think that we can expect very many >> players to learn them. >> They'll learn them if there's cool music written in them.This year I intend to spend more time composing music than writing papers! -- now that I have a notation that will do everything I need it to.> ... > Did you see any of the virtual keyboard projector posts I've > made to the main list over the past year?No, I haven't. I've been spending so much time on the notation project that I usually only have time to read quickly (or scan) through postings on the main list, so there are a lot of things I have spent less time on than I would have liked. Would you refer me to an example or two? --George
Message: 5869 - Contents - Hide Contents Date: Tue, 07 Jan 2003 21:51:17 Subject: Re: Temperament notation From: Carl Lumma>> >ince learning septimal notation gives access to 300 years of >> Western music, the potential payoff for each new one shouldn't >> be grounds for complaint. >>What, exactly, do you mean by "septimal notation"?A notation with 7 nominals!>But in either case my conclusion holds: >>>> so yes, it can be done without >>> extraordinary effort. >>>> I'd say there's nothing far worse about such a setup than >> in using the Halberstadt for 12-equal. >>I'd be the first to say that it's far better than the >Halberstadt for 12-ET, because like intervals occur in like >patterns, so you need only a single fingering pattern for a >given interval, chord, scale, etc., regardless of the key.We agree, then.>(However, like patterns do not guarantee like intervals, as >would be the case with a decimal keyboard, so you must take >care to observe in which keys this will work, just as you >would need to avoid the wolf of meantone temperament on a >Halberstadt keyboard.)I don't follow this. It is possible to make a transpositionally invariant decimal keyboard, so that like patterns do guarantee like intervals.>>> (Or suppose that a piece is heptatonic in one place and >>> decatonic in another. Do we switch notations in the middle >>> of the page?) >>>> Absolutely! Just like switching clefs or key signatures. >>Hey, you were supposed to say "no, of course not!" Suppose it >goes back and forth from heptatonic to decatonic every measure, >or the right hand does something decatonic, while the left hand >is heptatonic, etc.It's just a judgement call on the part of the composer, as it is with clefs and key signatures.>> You mention Partch's music, which doesn't really use any fixed >> melodic scale. I would think transpositionally invariant >> notation would be optimal, for the scores at least. >> >> But Partch had the right idea... since his instruments played >> different scales (tonality diamond, microchromatic scales, >> ancient melodic scales, etc.), he notated differently for each >> of them. >>Help! That's what I'm trying to avoid -- insofar as possible, I >want the pitch notation to be independent of the instrument. How >can you understand vertical harmony in a score if the notation is >different in different lines?Oh, you have to have unified notation for a score, as I say, "for scores at least". But parts are a different matter.>> I am genuinely interested to see how it looks. You should debut >> it with sample music, both original and classical. >>My paper has a couple of samples, plus examples of the 17-limit >consonances in the key of C. These at least give you an idea of >how things look on paper. This, along with the explanation of >the notation, would be enough to get you started. (I'll have to >get part of that posted soon, after I go through it and make a >few small changes.)Great! Please let me know with XH 18 (or is it 19?) comes out.>>> ... 6:7:8:9:10:11 isn't a constant structure). >>>> The diatonic scale in 12-equal isn't a constant structure either. >>That's true only as a technicality -- the augmented fourth and >diminished fifth in the diatonic scale just happen to be the same >size in 12-ET, but since they are functionally different, this >isn't a liability.And in harmonics 6-12, the aug 3rd and dim 4th don't function differently?>Partch's hexads make more sense if you consider them >heptatonically (with a vacanct position between 6 and 7).I'd imagine I'd mainly stick to triads if I were to write 'diatonic' music with harmonics 6-12.>> Did you see any of the virtual keyboard projector posts I've >> made to the main list over the past year? >>No, I haven't. I've been spending so much time on the notation >project that I usually only have time to read quickly (or scan) >through postings on the main list, so there are a lot of things >I have spent less time on than I would have liked. Would you >refer me to an example or two?Msg. #s 35809 and 41680. -Carl
Message: 5870 - Contents - Hide Contents Date: Tue, 07 Jan 2003 21:55:43 Subject: Re: Improved generators for 7-limit linear temperaments From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" <gdsecor@y...> wrote: >>> I'm a little puzzled how 53-ET popped up (instead of 72-ET) for this >> one. In this regard, please refer to: >> >> Yahoo groups: /tuning-math/message/5253 * [with cont.] >> >> There I evaluated by minimax only, but I thought that it seemed >> pretty clear that 53 was not our choice to notate kleismic. >> In a nutshell, 53 is smaller than 72; it is very close to theminimax value while 72 is very close to the rms value, but either will work.> > We have: > > Least squares 316.664 > Least cubes 316.466 > Least fourth powers 316.566 > Minimax 316.993 > > > Both the 53-et value of 316.981 cents and the 72-et value of316.667 cents are between the rms and minimax optimal generators.> > How differently would the notation look in 53 as opposed to 72?To illustrate, I will repeat something from my earlier posting (#5253): << Something else I noticed about the choice of 72 as the notation for both the Miracle and kleismic temperaments: the progression of sagittal symbols for a 72-ET panchromatic scale (one passing through all the tones) is the same as that for a sequence of tones differing by the generating interval in both temperaments. To illustrate: 72-ET: C C\! C!) C\!/ B|) B/| B Miracle: C B\! Bb!) A\!/ G|) F#/| F kleismic: C A\! F#!) Eb\!/ B|) G#/| F The pattern then repeats. I believe that this is a useful property that provides a further justification for basing the kleismic notation on 72. >> In 53-ET sagittal notation the last line of the above table would be: kleismic: C A\! F#\! Eb\!/ B/| G# F or Ab/| The Ab/| respelling in 53 is the same in 72 (since G#=Ab in 72). The essential differences between the 53 and 72 symbols are: 1) The 7 comma symbol |) occurs only in the 72 standard set; it is the same number of degrees as the 5 comma /| in 53; and 2) The apotome minus the 11 diesis is the same number of degrees as the 11 diesis in 72 -- so A\!/ equals G#/|\ -- but in 53 it is different, so A\!/ -- A lowered by ~32:33, the 11 diesis -- would be respelled with a different symbol, G#(|) -- G# raised by ~704:729, the 11' diesis. --George
Message: 5871 - Contents - Hide Contents Date: Tue, 07 Jan 2003 22:01:01 Subject: Quick summary of thoughts on notation From: Carl Lumma () I would like to distinguish between two types of notation. ....() Systems in which a given acoustic interval always covers the same distance on the staff, in terms of lines and spaces. ("trans. invariant notations") ....() Systems in which a given scale interval (2nd, 3rd, etc.) always covers the same distance on the staff. ("diatonic" notations) () For non-diatonic music, the former type of notation is preferable. () Regarding the former type, the number of nominals is more-or-less irrelevant. () For diatonic music, the latter type of notation is preferable. () Regarding the latter type, the number of nominals is crucial. () A single notation of the latter type has occupied generations of musicians in our culture. -Carl
Message: 5872 - Contents - Hide Contents Date: Tue, 07 Jan 2003 00:32:30 Subject: Re: 31, 112 and 11-limit Meantone From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> Too wide? The problem has been that it is too damned narrow, being > stuck in the neighborhood of 1.53 to 1.58 or thereabouts in termsof Blackwood's constant. We can't seem to get down to 1.5, for 31, and only for 5-limit do we get up to 1.6, for 81. I guess I don't really understand this poptimal stuff after all. Isn't there rather a big difference between the RMS and the max- absolute optimum generators for meantone, at least at the 5 limit. I'm sorry I don't know what Blackwood's constant is.
Message: 5873 - Contents - Hide Contents Date: Tue, 07 Jan 2003 22:38:09 Subject: Re: Temperament notation From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>> Since learning septimal notation gives access to 300 years of >>> Western music, the potential payoff for each new one shouldn't >>> be grounds for complaint. >>>> What, exactly, do you mean by "septimal notation"? >> A notation with 7 nominals!What I would prefer to call a heptatonic notation.>> (However, like patterns do not guarantee like intervals, as >> would be the case with a decimal keyboard, so you must take >> care to observe in which keys this will work, just as you >> would need to avoid the wolf of meantone temperament on a >> Halberstadt keyboard.) >> I don't follow this. It is possible to make a transpositionally > invariant decimal keyboard, so that like patterns do guarantee > like intervals.I'm sorry, but evidently I didn't make myself clear. I was speaking of what occurs with either a 31 or 41 mapping of Miracle onto a *Bosanquet* generalized keyboard. For Miracle on a decimal keyboard you have an ideal situation, such as you describe.> ...>>> You mention Partch's music, which doesn't really use any fixed >>> melodic scale. I would think transpositionally invariant >>> notation would be optimal, for the scores at least. >>> >>> But Partch had the right idea... since his instruments played >>> different scales (tonality diamond, microchromatic scales, >>> ancient melodic scales, etc.), he notated differently for each >>> of them. >>>> Help! That's what I'm trying to avoid -- insofar as possible, I >> want the pitch notation to be independent of the instrument. How >> can you understand vertical harmony in a score if the notation is >> different in different lines? >> Oh, you have to have unified notation for a score, as I say, > "for scores at least". But parts are a different matter.As a practical expedient, a part will have to be in whatever notation the player is comfortable with. This might make communication between a conductor or music director (reading the score) and a player more difficult, but conductors will need to become astute and versatile in these situations.>>> I am genuinely interested to see how it looks. You should debut >>> it with sample music, both original and classical. >>>> My paper has a couple of samples, plus examples of the 17-limit >> consonances in the key of C. These at least give you an idea of >> how things look on paper. This, along with the explanation of >> the notation, would be enough to get you started. (I'll have to >> get part of that posted soon, after I go through it and make a >> few small changes.) >> Great! Please let me know with XH 18 (or is it 19?) comes out.It will be XH18, and I have no idea how long it will be (which is why I'll try to put some of the explanation about the notation in the tuning-math files soon).>>>> ... 6:7:8:9:10:11 isn't a constant structure). >>>>>> The diatonic scale in 12-equal isn't a constant structure either. >>>> That's true only as a technicality -- the augmented fourth and >> diminished fifth in the diatonic scale just happen to be the same >> size in 12-ET, but since they are functionally different, this >> isn't a liability. >> And in harmonics 6-12, the aug 3rd and dim 4th don't function > differently?I was interpreting the phrase "diatonic scale in 12-equal" to mean 5- limit with no chromatic intervals, but the harmonic minor scale slipped my mind. To address your question, the aug 3rd and dim 4th *do* function differently in the scale, and the larger interval subtends fewer diatonic degrees -- yet we are not disoriented! It is interesting to contemplate that if we used a notation with 12 nominals for 12-ET that we would be unable to observe a distinction on the printed page. So you have a point.>>> Did you see any of the virtual keyboard projector posts I've >>> made to the main list over the past year? >>>> No, I haven't. I've been spending so much time on the notation >> project that I usually only have time to read quickly (or scan) >> through postings on the main list, so there are a lot of things >> I have spent less time on than I would have liked. Would you >> refer me to an example or two? >> Msg. #s 35809 and 41680.Thanks! I'll have to take a look. --George
Message: 5874 - Contents - Hide Contents Date: Tue, 07 Jan 2003 02:14:53 Subject: Blackwood transformations From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>"> I'm sorry I don't know what Blackwood's constant is.I didn't use the right one anyway, but let me define something more general. Let 0 < p/q < 1 be a fraction in lowest terms representing a generator as a fractional part of an interval of repetition. Let p1/q1 and p2/q2 be the fractions on either side of p/q in the qth row of the Farey sequence, with p1/q1 on the side with the better values of the generator, and p2/q2 on the side with the values going in the wrong direction, so to speak. We now define the linear fractional transformation BI(z, p/q) = (p1*z + p2)/(q1*z + q2) This has the property that BI(0,p/q)=p2/q2, BI(1, p/q)=(p1+p2)/(q1+q2)=p/q, BI(infinity,p/q)=p1/q1. The value of x=BI(z,p/q) for z>=1 is the generator which gives a ratio between the large and small steps of the q-step MOS; we may call BI the inverse Blackwood transformation. If we solve for z in terms of x, we get BW(x, p/q) = -(q2*x - p2)/(q1*x - p1), the Blackwood transformation which takes a generator x and associates it to a Blackwood-style constant z. The transformation I used was BW(x, 7/12), whereas Blackwood used BW(x, 4/7). (If there is confusion, one could use BI(z, p/q, +) for the case where p1/q1>p/q and BI(z, p/q, -) if p1/q1<p/q.)
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