This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).
- Contents - Hide Contents - Home - Section 10 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950
350 - 375 -
Message: 353 - Contents - Hide Contents Date: Sun, 25 Jun 2000 08:41:33 Subject: [tuning-math] better than ratios (was: Re: Hypothesis revisited) From: monz> ----- Original Message ----- > From: Paul Erlich <paul@s...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, June 25, 2001 4:27 AM > Subject: [tuning-math] Re: Hypothesis revisited > > > --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote: >>> (Incidentally, I think the 135/128 "major chroma" is a >> chromatic unison vector of the so-called miracle generator at >> two-dimensions; with 34171875/33554432 being the commatic unison >> vector if the generator is taken to a 10- or 11-tone MOS.) >> I see the MIRACLE scales as needing three or four unison vectors > each, since they live in a 7- or 11-limit lattice (i.e., they're > 3D or 4D).For the benefit of anyone else who (like me) got lost in this thread, Dan's commatic unison vector is a 5-limit one, expressible in prime-factor notation (much better, I think) as 2^-25 * 3^7 * 5^6, or [-25 7 6] (which is how Graham writes it). Dan's chromatic unison vector (also 5-limit) is 2^-7 * 3^2 * 5^1 = [-7 2 1]. So much better than ratios... -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 354 - Contents - Hide Contents Date: Sun, 25 Jun 2000 08:45:13 Subject: Re: 41 "miracle" and 43 tone scales From: monz> ----- Original Message ----- > From: Paul Erlich <paul@s...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, June 25, 2001 4:38 AM > Subject: [tuning-math] Re: 41 "miracle" and 43 tone scales > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> It was Erv Wilson who hypothesized that Partch was intuitively >> "feeling out" a version of 41-EDO where two of the pitches could >> imply either of a pair of ratios (12/11 and 11/10, and their >> "octave"-complements). >> Actually, the pair was 11/10 and 10/9 . . . you don't get a > PB or CS the other way.OK, I understand that *theoretically* this is the elegant comparison. But we had a discussion about this around two years ago... Didn't Daniel Wolf present cases in Partch's actual compositions where either pair could be interchangeable? That's what I remember. (I should have mentioned it the first time around... my bad.) -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 355 - Contents - Hide Contents Date: Sun, 25 Jun 2000 08:49:31 Subject: Re: pairwise entropy minimizer From: monz> ----- Original Message ----- > From: Paul Erlich <paul@s...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, June 25, 2001 4:35 AM > Subject: [tuning-math] Re: pairwise entropy minimizer > > > --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:>> Back in the day, Paul Erlich was working on finding scales for >> which the sum of the harmonic entropy of their dyads was low. >> ... >> (1) There were ever results for other cardinalities. >> Oh yes . . . by the time I got to 12 notes, I was finding that > the program was getting "stuck" in some kind of higher-dimensional > "crevices" leading to curious 12-tone well-temperaments which > were not even local minima . . . they could be nudged closer > to 12-tET without ever increasing the total dyadic harmonic > entropy at any stage. Monz made a webpage of these well-temperaments. > This was all posted to the tuning list . . . you'll have to dig > through the archives.Uh-oh... apparently my webpages must be getting "stuck in some kind of higher-dimensional crevices"!!! This seems to be another of the "lost Monzo webpages". Again, Paul, if you can remember any text from this page I can probably find it and make a prominent link. -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 357 - Contents - Hide Contents Date: Mon, 25 Jun 2001 16:39:33 Subject: Re: Hypothesis revisited From: carl@l... --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Progress seems to have halted on the paper that was to introduce > MIRACLE . . . /.../ > If we can do the following math problem, we'll be fine: > > Given a k-by-k matrix, containing k-1 commatic unison vectors and 1 > chromatic unison vector, delimiting a periodicity block, find: > > (a) the generator of the resulting WF (MOS) scale; > > (b) the integer N such that the interval of repetition is 1/N > octaves.Can somebody fill me in on what is meant by "interval of repetition" here? -C.
Message: 358 - Contents - Hide Contents Date: Mon, 25 Jun 2001 16:45:16 Subject: Re: pairwise entropy minimizer From: carl@l... --- In tuning-math@y..., "M. Edward Borasky" <znmeb@a...> wrote:> Hmmm ... multi-dimensional optimization isn't a particularly > difficult problem, as long as the function to be optimized is > reasonably well behaved.IIRC, that's the problem with harmonic entropy. -Carl
Message: 359 - Contents - Hide Contents Date: Mon, 25 Jun 2001 16:50:30 Subject: Re: Hypothesis revisited From: Dave Keenan --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:>>>> 2. Masses of people over centuries have effectively given us a >>> short>>>> list of those they found useful. ...> I am [objecting to the above sentence].I mean the ancient scales that are still in popular use today in various cultures. eg. "meantone" diatonic. Arabic scales. Various pentatonics. Gamelan scales.>> I'll adress the >> second. Graham Breed (and George Secor) have shown that MIRACLE_41 is >> almost identical to several of Partch's scales. >> Eh . . . not quite.Err Paul, "almost" is a synonym for "not quite". See my post to the tuning list entitled "Partch's scales on the Miracle keyboard".>> I can't help seeing >> Partch's various scales as gropings towards either Canasta >> Don't see it.No. I was wrong there.>> or >> MIRACLE-41. >> Toward modulus-41, yes . . . with many other generators functioningas well as, if not better than,> the 4/41 (MIRACLE) generator.No. I'm talking about Miracle-41 and the 7/72 oct generator. 4/41 oct is only borderline Miracle.> Yes, Dave, we both want to "rule out" the MOSs with noapproximations to any JI> intervals/chords (if such a thing is possible). That is where we(the originators of "MIRACLE")> differ from Dan Stearns (at least in the viewpoint that goes behindthis paper we're> contemplating). But that still leaves a great number ofpossibilities, as Robert Valentine, for> example, has been finding.Oh sure. I was assuming you had read Dan's post and my response to it, and were referring to that. Sorry. -- Dave Keenan
Message: 361 - Contents - Hide Contents Date: Mon, 25 Jun 2001 16:58:55 Subject: Re: 41 "miracle" and 43 tone scales From: Dave Keenan --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> He stopped at 43 in order to make a melodically fairly even scale.With 10/9 and 11/10 seen as> a commatic pair (the unison vector involved is 100:99), and theiroctave complements another> such pair, Partch's scale is a 41-tone periodicity block -- or whatWilson calls a "Constant> Structure".I think George Secor, Graham Breed and Dave Keenan disagree with this analysis, preferring one based on filling in the the diamond gaps using rationalised Miracle generators. See Yahoo groups: /tuning/message/25575 * [with cont.] Does anyone know if Partch regularly used any of the many approximate JI intervals in his scale such as those with only a 224:225 or 384:385 error (less than 8 cents)? -- Dave Keenan
Message: 362 - Contents - Hide Contents Date: Mon, 25 Jun 2001 17:09:27 Subject: Re: Hypothesis revisited From: Dave Keenan --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:>> Actually, with the 10-note 369c MOS, I was looking for a MOS scale >> that Paul would have difficulty finding unison-vectors for, that are >> anything like unisons. i.e. This one was meant to have _big_ UVs, and >> not to contain any good approximations to SWNRs. >> >> Are you asking us to find a linear temperament that treats those >> unison vectors (49/40 and 4375/4096) as commas, and to tell you how >> "good" it is relative to the usual JI criteria. >> I think Dan just found unison vectors for your example, Dave!If that's the case, then it makes my point quite well. Isn't it just a little ridiculous to refer to intervals of 351c and 114c as "unison" vectors or "commas"? -- Dave Keenan
Message: 363 - Contents - Hide Contents Date: Mon, 25 Jun 2001 17:16:05 Subject: Re: 41 "miracle" and 43 tone scales From: Dave Keenan --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Graham and Dave, Wilson knew Partch, and his mappings for the[43-tone scale] to Modulus-41 and> Modulus-72 keyboards did not use the MIRACLE generator, but rather other generators. Which ones? > So I > don't see how one could say that Partch was using, or implyingMIRACLE, in any way> whatsoever.All that means is that Partch wasn't intentionally using Miracle and that Wilson missed the fact that Partch's scales imply it. -- Dave Keenan
Message: 364 - Contents - Hide Contents Date: Mon, 25 Jun 2001 17:25:33 Subject: Re: 41 "miracle" and 43 tone scales From: Dave Keenan --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> It was Erv Wilson who hypothesized that Partch was intuitively >>> "feeling out" a version of 41-EDO where two of the pitches could >>> imply either of a pair of ratios (12/11 and 11/10, and their >>> "octave"-complements). >>>> Actually, the pair was 11/10 and 10/9 . . . you don't get a >> PB or CS the other way. > >> OK, I understand that *theoretically* this is the elegant comparison. > > But we had a discussion about this around two years ago... > > Didn't Daniel Wolf present cases in Partch's actual compositions > where either pair could be interchangeable? That's what I remember.It's interesting that Miracle distinguishes all three of these ratios, as Partch did. 11:12 is -9 generators 10:11 is 22 generators 9:10 is -19 generators -- Dave Keenan
Message: 365 - Contents - Hide Contents Date: Mon, 25 Jun 2001 17:38:34 Subject: Re: Hypothesis revisited From: Dave Keenan --- In tuning-math@y..., carl@l... wrote:> Can somebody fill me in on what is meant by "interval of > repetition" here?It's just Paul inventing yet another term for what has been called (ill advisedly when relating to MOS) formal octave interval of equivalence and more sensibly called period interval of periodicity It gets a little ridiculous referring to 1/29 octave as a formal octave or an interval of equivalence, as in Graham's 15-limit temperament. -- Dave Keenan
Message: 366 - Contents - Hide Contents Date: Sun, 25 Jun 2000 12:58:27 Subject: Re: 41 "miracle" and 43 tone scales From: monz I'm replying here to two of Graham's posts about Partch and MIRACLE.> ----- Original Message ----- > From: <graham@m...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, June 25, 2001 2:53 AM > Subject: [tuning-math] Re: 41 "miracle" and 43 tone scales > > > One question is, how much did Partch know about Miracle when he drew up > that original, unpublished scale? It may be stretching credulity to > suggest he worked it all out, and then pretended it was pure JI. But the > criteria he was using may well have matched those that are enshrined in > Miracle. Roughly equal melodic steps will of course favour an MOS. And > he would have been able to hear the intervals that were almost just by > Miracle approximations. And so he could have chosen the extra notes to > maximise these consonances. > > In which case, why did he change his mind later? I think it was to get > more modulation by fifths in the 5-limit plane. With experience, he > decided this was more important than matching the consonances. > > The limitations on modulation by fifths is one of the problems with > Miracle, at least in a traditional context. Boomsliter and Creel's > theories work very well with schismic, but not at all well with Miracle, > temperament. > > ----- Original Message ----- > From: <graham@m...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Cc: <gbreed@c...> > Sent: Monday, June 25, 2001 7:03 AM > Subject: [tuning-math] Re: 41 "miracle" and 43 tone scales > > > Oh, come come. If Partch was ever feeling towards Miracle he would have > stopped doing so long before Wilson came up with his Modulus-41 ideas. > That the scale works so well with 41 and 72 does imply Miracle. Then > again, simply using 11-limit JI implies Miracle. > > It is interesting that 31, 41 and 72 don't get a mention in Genesis. > Deliberate avoidance of temperaments he can't dismiss so lightly? You > decide!Graham, you know that I also love speculation! I'm very impressed by yours here. John Chalmers is the subscriber on this list who can really document the relationship between Secor and Partch. (Perhaps we should also post a query on another list for Kraig Grady?) I do know, however, that their meeting ocurred quite late in Partch's life. Partch lamented that Secor's Scalatron was the instrument he had always wanted, but it came along too late to do him any good. This was probably early 1970s, possibly late 1960s. _Genesis_ was published in 1947 or 1949 [1] (1st ed.) and 1974 (2nd ed.), and the only substantial changes in the 2nd edition concerned Partch's new instruments. The theoretical and historical sections of the book remained virtually intact. So I'm certain beyond any doubt that Partch was not *consciously* aware of MIRACLE before the late 1960s. (note my emphasis) But Graham's speculations are intriguing, and I'm fairly convinced by them that Partch *intuitively* understood the MIRACLE concept and perhaps was indeed guided in constructing his 43-tone scale by some of the additional "senses" in which the 14 new (and original 29) pitches could be taken in MIRACLE. Daniel Wolf, who has had the opportunity to study Partch's scores in *much* greater depth than I have, has remarked on how Partch did not always construct his harmonies according to the lowest-odd-integer hexadic theory presented in _Genesis_. So perhaps some of these "nonstandard" usages *do* conform to MIRACLE-like approximations. Partch's 14 additional pitches are, as Graham correctly states, primarily an expansion of the Tonality Diamond in the prime-factor-3 dimension, which Graham notes is *not* a feature of MIRACLE. I've noted before how I thought it was a paradox that for all his vitriolic abrogation of Pythagoreanism, Partch took exactly this route in expanding his pitch gamut. It seems that he valued *something* about traditional music-theory after all, and that "something" is, again as Graham points out, modulation or root-movement by 3:2s. About the equal temperaments discussed in _Genesis_: First of all, I should say that I was simply writing from memory before. Now I have the book in front of me, and there are indeed some ETs that I left out. I'll correct that omission abundantly now. Partch (1974, p 417) does make this interesting general observation:> Fundamentally, equal temperaments are based upon and deduced > from Pythagorean "cycles," in whole or part.He opens his chapter on equal-temperaments with a long and scathing diatribe against 12-EDO, which, by this point in the book, should not surprise the reader. Then he discusses the 'First Result of Expansion - "Quartertones"'. Upon mentioning Carillo, Partch also thus mentions 48- and 96-EDO. But he actually does go into a little detail about 24-EDO, and he's even generous enough about its potential to say that 'As a temporary expedient, as an immediately feasible method of creating new musical resources, "quartertones" are valuable'. He mentions Haba [which should be spelled Hába], Hans Barth, and Mildred Couper and their use of dual regular keyboards, and Meyer and Moellendorf and their new keyboards. Then Partch breifly discusses Busoni and 36-EDO, which he characterizes as "another Polypythagoreanism in tempered expression". In the middle of this text, on p 430, is Partch's comparative table of tunings. I will come back to say more about this table after describing the rest of the text. Next comes the discussion of Yasser's 19-EDO, then finally 53-EDO. About Yasser's proposal, Partch emphasizes that its goal is not the betterment of intonation, but simply an expansion of scalar resources. He notes the improved approximations to 5- and 7-limit ratios, and also that "The ratios of 7 are somewhat better also, but still with a maximum falsity of 21.4 cents (33.1 cents in twelve-tone temperament). The ratios of 11 are not represented at all". Actually, 19-EDO's closest approximations to the 11-limit ratios are all between +/- 17.1 and 31.5 cents, significantly better than 12-EDO's. Partch had mentioned in "Chapter 15: A Thumbnail Sketch of the History of Intonation" that King Fang (in China) and Mersenne, Kircher, and Mercator (in Europe) all proposed this tuning. In the middle of the discussion of 53-EDO is a digression "On the Matter of Hearing a 2-Cents Falsity". Partch notes that 53-EDO is indeed extremely close to 3- and 5-limit JI, but does not consider it suitable for his own use as it offers little improvement in approximating the 7- and 11-limit ratios he wanted to use. Finally he examines the keyboard proposals of Nicolaus Ramarinus (1640) [2], Bosanquet (no date given by Partch, c. 1875?), and Jas. Paul White (1883) [3]. And that wraps up Partch's "Chapter 17: Equal Temperaments". Now, back to that comparative table... Partch's table on p 430 compares his Monophonic 43-tone scale with, in order: - 12-EDO, - 12-tone Pythagorean: a 3^(-6...+5) system, - 16-tone Meantone: a cycle of implied "5ths" 3^(-5...10) tuned in 1/4-comma meantone, the pair of notes at either end of the cycle being the additional notes on Handel's organ (according to Partch), - 17-tone Arabic: a Pythagorean 3^(-12...+4) system, - 19-EDO, - 24-EDO, - 31-EDO, - 36-EDO, - 53-EDO. First, I should note that there are obviously tunings here (the second, third, and fourth) which are not ETs. Partch had already discussed these in his "Chapter 16: Polypythagoreanism". But - SURPRISE! - there's 31-EDO in the table, but WITH NO MENTION WHATSOEVER IN THE TEXT!! And I checked all the other chapters in _Genesis_... there's no mention at all of Huyghens, Fokker, or anything else concerning 31-EDO. Now THAT'S interesting! ... And I never noticed it before, having been duped by 31-EDO's appearance in that table into thinking that Partch said something about it somewhere. So Graham is right that, except for this inconspicuous little tabulation, Partch does not mention 31-, 41- or 72-EDO. Good detective work, Graham!!! NOTES [1] I asked before (on the main list) about the actual publication date. I don't remember now what the outcome was, but I've seen it listed in catalogs under both dates. The original Preface is dated April 1947, but the copyright date is 1949. [2] About Ramarinus, Partch says:> the "tone" (9/8) was divided into nine "commas", > according to Hawkins [_History of the Science and > Practice of Music, vol 1, p 396]. The fifty-third part > of 2/1 is approximately the width of the "comma" of > Didymus, 81/80 (21.5 cents; see table above), and since > six 9/8's are larger than a 2/1 by approximately this > interval (the "comma" of Pythagoras, 23.5 cents), this > procedure would result in a fifty-three-tone scale.Of course, we are well aware that the 9-commas-per-tone temperament works out to exactly 55-EDO, which is a meantone, whereas 53-EDO is quasi-just. This choice probably reflects Partch's own bias; I'd bet that Ramarinus most likely meant something more like 55-EDO. [3] Paul (or anyone else in Boston): It still says in the 1974 edition of _Genesis_ that White's harmonium was housed in a practice room at New England Conservatory, and that Partch examined it in 1943. I've found page references in _Genesis_ that should have been renumbered from the 1st edition and weren't, so perhaps this is a story that also should have been updated. Please... go take a look and let us know! -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 368 - Contents - Hide Contents Date: Mon, 25 Jun 2001 21:46:57 Subject: Re: Hypothesis revisited From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> The unison vectors I used for 31+41n are: > > [[ 2 -2 2 0 -1] > [-7 -1 1 1 1] > [-1 5 0 0 -2] > [-5 2 2 -1 0]] >Getting rid of the first column: [-2 2 0 -1] [-1 1 1 1] [ 5 0 0 -2] [ 2 2 -1 0] the resulting FPB is cents numerator denominator 38.906 45 44 70.672 25 24 79.965 288 275 111.73 16 15 150.64 12 11 182.4 10 9 203.91 9 8 235.68 55 48 262.37 64 55 294.13 32 27 315.64 6 5 347.41 11 9 386.31 5 4 425.22 225 176 427.37 32 25 466.28 72 55 498.04 4 3 536.95 15 11 551.32 11 8 590.22 45 32 609.78 64 45 648.68 16 11 663.05 22 15 701.96 3 2 733.72 55 36 772.63 25 16 774.78 352 225 813.69 8 5 852.59 18 11 884.36 5 3 905.87 27 16 937.63 55 32 964.32 96 55 996.09 16 9 1017.6 9 5 1049.4 11 6 1088.3 15 8 1120 275 144 1129.3 48 25 1161.1 88 45 1200 1 1> > That uses 100:99 as the chromatic UV. The more obvious choicewould be a> schisma, so that > > [[-15 8 1 0 0] > [-7 -1 1 1 1] > [-1 5 0 0 -2] > [-5 2 2 -1 0]] > > would give the same results.Again getting rid of the first column, this is [ 8 1 0 0] [-1 1 1 1] [ 5 0 0 -2] [ 2 2 -1 0] giving the FPB cents numerator denominator 31.767 55 54 60.412 729 704 92.179 135 128 111.73 16 15 143.5 88 81 172.14 243 220 203.91 9 8 235.68 55 48 262.37 64 55 296.09 1215 1024 315.64 6 5 347.41 11 9 386.31 5 4 407.82 81 64 439.59 165 128 466.28 72 55 498.04 4 3 519.55 27 20 558.46 243 176 590.22 45 32 609.78 64 45 643.5 1485 1024 670.19 81 55 701.96 3 2 733.72 55 36 760.41 256 165 794.13 405 256 813.69 8 5 845.45 44 27 884.36 5 3 905.87 27 16 937.63 55 32 964.32 96 55 996.09 16 9 1017.6 9 5 1056.5 81 44 1088.3 15 8 1107.8 256 135 1141.5 495 256 1168.2 108 55 1200 2 1 The difference between these two scales is numerator denominator 242 243 2187 2200 4125 4096 1 1 242 243 2187 2200 1 1 1 1 1 1 32805 32768 1 1 1 1 1 1 99 100 4125 4096 1 1 1 1 99 100 243 242 1 1 1 1 16335 16384 243 242 1 1 1 1 4096 4125 91125 90112 1 1 242 243 1 1 1 1 1 1 1 1 1 1 1 1 243 242 1 1 4096 4125 4125 4096 243 242 2 1 So if the schisma (32805:32768) is the _chromatic_ unison vector of one of these scales, the two scales are _not_ equivalent, even up to arbitrary transpositions by _commatic_ unison vectors.> I can't check this now, as I don't have > Numerical Python installed, or even Excel. But you may be able to. Try > inverting this matrix, and multiplying it by its determinant: >[[ 1 0 0 0 0] [-15 8 1 0 0] [-7 -1 1 1 1] [-1 5 0 0 -2] [-5 2 2 -1 0]] The determinant is -41, and the inverse is [ 1 0 0 0 0 ] [ 65/41 6/41 -2/41 -1/41 -2/41] [ 95/41 -7/41 16/41 8/41 16/41] [ 115/41 -2/41 28/41 14/41 -13/41] [ 142/41 15/41 -5/41 -23/41 -5/41]> The left hand two columns should be > > [[ 41 0] > [ 65 -6] > [ 95 7] > [115 2] > [142 -15]]Up to a minus sign, yes.> > If they are, the two sets of unison vectors give exactly the same > results. They don't! > I think they must be, because I remember checking the > determinant before, and any chroma that gives a determinant of 41 when > placed with Miracle commas should give this result.Something must be wrong with one of your assumptions.> You most certainly do need octave-specific matrices. Otherwise, that > left-hand column won't be there.I see that as a good thing . . . don't you?> There may be an algorithm that works with octave > invariant matrices, but it's easier to upgrade them to be > octave-specific, and use a common or garden inverse. ?
Message: 369 - Contents - Hide Contents Date: Mon, 25 Jun 2001 21:50:22 Subject: Re: 41 "miracle" and 43 tone scales From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <9h7845+e2fi@e...> > Paul wrote: >>> Graham and Dave, Wilson knew Partch, and his mappings for the Diamond >> to Modulus-41 and Modulus-72 keyboards did not use the MIRACLE >> generator, but rather other generators. So I don't see how one could >> say that Partch was using, or implying MIRACLE, in any way whatsoever. >> Oh, come come. If Partch was ever feeling towards Miracle he would have > stopped doing so long before Wilson came up with his Modulus-41 ideas. ??? > That the scale works so well with 41 and 72 does imply Miracle.Now you're stretching the meaning of the word "imply".> Then > again, simply using 11-limit JI implies Miracle.Now you're _really_ stretching the meaning of the word "imply"!!! :)> It is interesting that 31, 41 and 72 don't get a mention in Genesis. > Deliberate avoidance of temperaments he can't dismiss so lightly? You > decide!I think he was simply ignorant of these temperaments, in the literature he was familiar with (which concentrated on 19, 24, and 53). Actually, 31 _is_ in his ET comparison table, isn't it?
Message: 370 - Contents - Hide Contents Date: Mon, 25 Jun 2001 21:55:13 Subject: Re: pairwise entropy minimizer From: Paul Erlich --- In tuning-math@y..., carl@l... wrote:> --- In tuning-math@y..., "M. Edward Borasky" <znmeb@a...> wrote:>> Hmmm ... multi-dimensional optimization isn't a particularly >> difficult problem, as long as the function to be optimized is >> reasonably well behaved. >> IIRC, that's the problem with harmonic entropy. > Huh?
Message: 371 - Contents - Hide Contents Date: Mon, 25 Jun 2001 22:46:03 Subject: Unison vectors and MOS From: Graham Breed My temperament finding program has been updated to find commatic unison vectors as well. I tried to get it to find the simplest ones, but it still needs some work there. See <Unison vector to MOS script * [with cont.] (Wayb.)> and <Automatically generated temperaments * [with cont.] (Wayb.)> From plugging in some of the unison vectors mentioned before, it's apparent that we don't have as much choice in the chroma as I thought. But it still isn't unique for a temperament. -- Graham "I toss therefore I am" -- Sartre
Message: 372 - Contents - Hide Contents Date: Mon, 25 Jun 2001 22:36:28 Subject: Re: 41 "miracle" and 43 tone scales From: Graham Breed Monz wrote:> _Genesis_ was published in 1947 or 1949 [1] (1st ed.) and > 1974 (2nd ed.), and the only substantial changes in the 2nd edition > concerned Partch's new instruments. The theoretical and historical > sections of the book remained virtually intact.So, if "Exposition on Monophony" was1933, that's well in advance.> But Graham's speculations are intriguing, and I'm fairly convinced > by them that Partch *intuitively* understood the MIRACLE concept > and perhaps was indeed guided in constructing his 43-tone scale > by some of the additional "senses" in which the 14 new (and > original 29) pitches could be taken in MIRACLE.Be careful you don't get carried away with these speculations. It seems plausible that he was feeling for something like 41-equal but with improved 11-limit harmony. In that case, you'd expect the result to look something like a 41-note MOS of a good 11-limit temperament. The scale he ends up with does fit schismic better than Miracle. As mathematicians, we should be aware of the dangers of imposing patterns on data. For the rest, I think the discussion should be taken to the main list if you think you have a case. Dave Keenan has already come up with some new arguments.> Partch's 14 additional pitches are, as Graham correctly states, > primarily an expansion of the Tonality Diamond in the prime-factor-3 > dimension, which Graham notes is *not* a feature of MIRACLE. > > I've noted before how I thought it was a paradox that for all > his vitriolic abrogation of Pythagoreanism, Partch took exactly > this route in expanding his pitch gamut. It seems that he valued > *something* about traditional music-theory after all, and that > "something" is, again as Graham points out, modulation or > root-movement by 3:2s.D'alessandro also ends up with a long chain of 3:2s, and so doesn't work so well as Miracle.> And I checked all the other chapters in _Genesis_... there's no > mention at all of Huyghens, Fokker, or anything else concerning > 31-EDO.I thought Fokker did his music theory during the Nazi occupation, hence after the original publication of Genesis. And Huygens' music theory wouldn't have been known until then either. Yasser still suggested eventual evolution to 31 though.> Now THAT'S interesting! ... And I never noticed it before, > having been duped by 31-EDO's appearance in that table into > thinking that Partch said something about it somewhere. > > So Graham is right that, except for this inconspicuous little > tabulation, Partch does not mention 31-, 41- or 72-EDO. > Good detective work, Graham!!!With you're detective work we can now say that he avoided *all* consistent 11-limit temperaments! Graham "I toss therefore I am" -- Sartre
Message: 373 - Contents - Hide Contents Date: Mon, 25 Jun 2001 22:00:13 Subject: Re: Hypothesis revisited From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:>>>>> 2. Masses of people over centuries have effectively given us a >>>> short>>>>> list of those they found useful. > ...>> I am [objecting to the above sentence]. >> I mean the ancient scales that are still in popular use today in > various cultures. eg. "meantone" diatonic. Arabic scales. Various > pentatonics. Gamelan scales.There are a lot of cultural accidents that lead to "popular use". And those Gamelan scales . . . you'd need some large unison vectors for those, wouldn't you?>>>> I can't help seeing >>> Partch's various scales as gropings towards either Canasta >>>> Don't see it. >> No. I was wrong there. > >>> or >>> MIRACLE-41. >>>> Toward modulus-41, yes . . . with many other generators functioning> as well as, if not better than,>> the 4/41 (MIRACLE) generator. >> No. I'm talking about Miracle-41 and the 7/72 oct generator. 4/41 oct > is only borderline Miracle.I meant 4/41 in a modulus-41, not 41-tET, sense. Doesn't the 19/72 generator work as well for Partch's scale as the 7/72 generator?
Message: 374 - Contents - Hide Contents Date: Mon, 25 Jun 2001 22:02:47 Subject: Re: 41 "miracle" and 43 tone scales From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:>> He stopped at 43 in order to make a melodically fairly even scale.> With 10/9 and 11/10 seen as>> a commatic pair (the unison vector involved is 100:99), and their> octave complements another>> such pair, Partch's scale is a 41-tone periodicity block -- or what> Wilson calls a "Constant >> Structure". >> I think George Secor, Graham Breed and Dave Keenan disagree with this > analysis, preferring one based on filling in the the diamond gaps > using rationalised Miracle generators. See > Yahoo groups: /tuning/message/25575 * [with cont.]The analyses are not necessarily incompatible!!!
0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950
350 - 375 -