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Message: 5775 Date: Thu, 27 Dec 2001 23:57:16 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: dkeenanuqnetau It would be evidence that pelog actually is this 5-limit temperament if, when a pelog scale departs from 7-tET it does so by making all it's fifths but one, even narrower than the 7-tET fifth, i.e. even further from a 2:3. A 7-tET fifth is 685.7 c. The rms optimum "fifth" in this temperament is 677.1 c. A chain of 6 of these would leave a super-wide wolf of 737.2 c. Do pelog scales really tend to do this; have 6 fifths that are up to 25 c narrow and one that is up to 35 c wide (of 2:3)?
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Message: 5776 Date: Thu, 27 Dec 2001 04:24:16 Subject: Re: Paul's lattice math and my diagrams From: monz > From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, December 26, 2001 9:29 PM > Subject: [tuning-math] Re: Paul's lattice math and my diagrams > > > [me, monz] > > Right, of course... they continue infinitely in the direction > > of the meantone chain if you don't close the chain somewhere. > > I *am* interested in closing it so that I get a periodicity-block. > > No, sir, I'm afraid you're completely misunderstanding me. If 81:80 > is tempered out, then you can keep moving by as many 81:80s as you > want in the lattice, and you're still within the strip! In terms of > the cylinder, all you're doing is making a full circle around the > cylinder in the same direction over and over again. Oh, OK Paul, I've got you now. My description really is based on the planar representation, while you were talking about the cylindrical representation. > > > So you mean that on your ideal lattice you'd have long > > (or I probably should say wide) strips of cylinders, right? > > Wide strips, _or_ a single cylinder. Right... got it. > > My code transforms the prime-axes to a right-angled unit cube, > > transforms the primary lattice metrics along the 3 and 5 axes > > to the unit metrics along those new axes, then iterates thru > > the unit cube to fill it with coordinates x,y, always bouncing > > to the other side (i.e., modulo) when it goes beyond the > > floor or ceiling values (i.e., 1/2 > x,y > -1/2), then > > transforms back to the original lattice coordinates. > > > > This is exactly how I understood your paragraph. Please correct. > > You appear to have the correct picture of how to create periodicity > blocks. I was saying much more than that, but if that's all you were > looking for, then you're fine. Cool. But even tho it works, there still is something wrong with the mathematics in my spreadsheet. I'd appreciate some error correction. > If there's still any confusion, part 3 of the Gentle Introduction > should clear it up. Yes, I've since taken another look at that. When I have time I'll go over my spreadsheet with a fine-tooth comb and compare it to your webpage description. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
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Message: 5777 Date: Thu, 27 Dec 2001 01:14:39 Subject: Re: Keenan green Zometool struts From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > It might be better to get the creator kit with a different set of > greens more later. Umm . . . did you mean it might be better to get the creator kit first and then a kit with more greens later?
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Message: 5778 Date: Thu, 27 Dec 2001 01:18:22 Subject: Re: Keenan green Zometool struts From: dkeenanuqnetau --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > --- In tuning-math@y..., paul@s... wrote: > > Hey Dave, > > > > From Dave Keenan's Home Page * one might get the idea that the > Zome folks haven't implemented your green strut idea yet. Good point. I've fixed that now. > > But I recently saw a kit called "Advanced Mathematics" which did > contain green struts. > > > > Did your ideas in fact help this product to be developed? Yes. There's more detail about that now at the above URL > > Should I buy the [Advanced Math] kit? It's between 100 and 200 US$. I earlier wrote (in a hurry): > It might be better to get the creator kit with a different set of > greens more later. This is wrong. My more considered recommendations are now at the above URL.
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Message: 5779 Date: Thu, 27 Dec 2001 01:36:38 Subject: Re: My top 5--for Paul From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > --- 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: > > > There's nothing terribly personal about the fact that an error of > 0.5 > > c is imperceptible by humans. > > It sure is if you're playing with a loud or distorted sound system -- > my favorite! OK. So choose a lower number, it will still be higher than the 0.0002 c or whatever it was of that supposed number-one temperament that even Gene described as "absurd". > > Theorists have delved into systems such as 118, 171, 612-tET, but > has > > anything musical ever come of it? And if it has or does, surely we > > would be looking at subsets, not the entire 118 notes per octave > etc. > > i.e. we'd be looking at temperaments within these ETs where > consonant > > intervals are produced by considerably fewer than 50 notes in a > chain > > (or chains) of generators. > > Perhaps not yet . . . but what harm comes from _informing_ musicians > of these systems? I'd love it if a genius musician did make use of > not considerably fewer than 50 notes per octave -- oh, wait a minute, > my lips are a little partched today . . . Whether you take Partch's 41 to be schismic-41 plus 2 or miracle-45 minus 2, no consonant interval is produced by a generator-chain of more than 23 notes. I consider 23 to be considerably less than 50. > and when I make lattices > for these systems, you can be sure I'm going to start with the > simplest and work my way up until the impenetrable thickets of notes > make me decide a single line of data from Gene would be more > appropriate. OK. But don't some of the simplest ones have such large errors as to be absurd too? > Hey Dave, why not look at Gene's list of 5-limit temperaments and see > if he's missed anything? This would be a lot easier for me if he would deign to give the optimum generator (whether rms or max-absolute), in cents.
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Message: 5780 Date: Thu, 27 Dec 2001 01:44:30 Subject: Re: My top 5--for Paul From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > This is the 21st century--there is no particular obstacle to using 612 notes, For that matter there's no technical obstacle to using an essentially continuous spectrum. > other than that is lot of notes to get around to. That's the one! Namely the limitation is in human cognition. The composer can have computer assistance, but the listener can't.
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Message: 5783 Date: Fri, 28 Dec 2001 19:53:11 Subject: Re: Keenan green Zometool struts From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > > "The only problem with the Advanced Math kit is that it doesn't have > > any short whole greens (G0). I recommend adding 48 of these (at > > US$9.60)." > > > > I can't figure out how to order 48 short whole greens from the > > website. Can you help? > > You can't order them via the web. Just email your entire order to > sales@z... > > Are you getting the bundle of the Adv Math kit with George Hart and > Henri Picitto's Zome Geometry book? That's this one, yes? Advanced Math Creator Kit Bundle *
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Message: 5784 Date: Fri, 28 Dec 2001 21:12:01 Subject: Re: Paul's lattice math and my diagrams From: monz > From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, December 28, 2001 2:21 PM > Subject: [tuning-math] Re: Paul's lattice math and my diagrams > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > > So essentially what you're saying is that Chesnut, in his article > > on Mozart, *extrapolates* from Tosi's description of a 55-EDO > > conception, to Leopold Mozart's praise of Tosi, to W. A. Mozart, > > and that I have mistakenly accepted that as evidence? > > Nowhere does Chesnut claim that Mozart used 55-EDO or "9 commas per > whole-tone, 5 commas per diatonic semitone". He simply provides a > historical context in which Mozart's preferences can be understood. > Daniel Wold, for example, advocated 1/4-comma meantone for > Mozart . . . who can say? Hmmm... what about my ears telling me that my 55-EDO rendition of the beginning of Mozart's 40th Symphony, on my webpage Mozart's tuning: 55-EDO, (c) 2001 by Joseph L. Monzo * sounds so much like the great old recording of it from 78s that I loved as a kid? That was one of the most startling things that came out of this webpage, for me. OK, so that still doesn't weigh all that much... I haven't listened to versions in other meantones yet, and in any case it's only a few measures. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
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Message: 5785 Date: Fri, 28 Dec 2001 19:58:58 Subject: Re: more 2-D periodicity-block math (was: new 1/6-comma meantone lattice) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > One of the unison-vectors will tile the plane along > either of two parallel sides of the parallelogram, and the > other unison-vector will tile the plane along either > of the other two parallel sides. This is exactly what the "Gentle Introduction" shows.
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Message: 5786 Date: Fri, 28 Dec 2001 20:02:14 Subject: Re: Paul's lattice math and my diagrams From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > > > From: paulerlich <paul@s...> > > To: <tuning-math@y...> > > Sent: Thursday, December 27, 2001 1:51 PM > > Subject: [tuning-math] Re: Paul's lattice math and my diagrams > > > > > > > Oh, OK Paul, I've got you now. > > > Hope you didn't take that the wrong way... I meant that > I understand (I think...) > > > > > My description really is based on the planar representation, > > > > The wrong planar representation, in my opinion. > > > Even after emphasizing the equivalence of tiled > periodicity-blocks? I don't get it! We're talking about meantone, yes? > > > > P.S. How can you include W. A. Mozart under 55-EDO on your Equal > > Temperament definition page? I could understand if you wanted to put > > Mozart on a meantone page, but 55? Totally unjustified. Come on, > > let's not just make things up. > > > Well... his conception was clearly based on the > "9 commas per whole-tone, 5 commas per diatonic semitone" idea. There is no evidence for that. All we know is that he taught sharps lower than the "equivalent" flats.
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Message: 5787 Date: Fri, 28 Dec 2001 20:08:07 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > > > As a 5-limit approximation the 522.86c > > > generator is junk. It has an error of 25 c in the 2:3. > > > > On behalf of Herman Miller, Margo Schulter, Bill Sethares, and the > > entire island of Java, let me just say #(@*$& ?@#>$, and then let me > > just say, go play with this scale for a while. 'Junk' my &$$. > > Paul, I think you're severely distorting what I wrote. I didn't say > pelog is junk. I said "as a 5-limit approximation ..." But it's really the ratio of *3* that had the error you objected to. > Is there really any evidence that pelog is a 5-limit temperament? I think there's strong evidence it at least relates to the 3-limit, and that's the error you objected to. As far as 5-limit, it's definitely a matter of opinion, but I'm referring to Herman Miller's use of the scale, not necessarily the traditional one. I'm also referring to Margo and Bill's use of consonant sonorities where the departures from 5-limit JI are even larger than this. > > > So there are > > > plenty of other temperaments as good as this. > > > > By _as good as_, I mean having an equal or lower RMS error ANS and > > equal or lower 'gens' measure. > > Why can't I use my own criteria for "as good as"? Well, we're trying to find out if Gene is missing anything with his methods. But if you'd like to suggest a different measure of cents error and/or a different complexity measure, I'd hope Gene could be accomodating . . .
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Message: 5788 Date: Fri, 28 Dec 2001 20:11:34 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > > Do pelog scales really tend to do this; have 6 fifths that are up to > > 25 c narrow and one that is up to 35 c wide (of 2:3)? > > OK. I checked it out myself in the Scala archives and the answer is > yes! They really do. Well, this should be mentioned in our paper, I think. > > So, although pelog is well represented as a chain of very uneven (+- 25 > c) generators averaging 523 c +-15 c, I'm still waiting to learn > whether -1, 3 and -4 generators traditionally represent the > consonances of the system? Gamelan music doesn't operate with Western notions of "consonance" and "dissonance". There is lots of simultaneity though, so the issue wouldn't seem to be completely irrelevant . . . > Meanwhile, I'll asume this apparent 5-limit approximation is real and > will weight the gens by 1/log(max-odd-factor) and give you the list of > those with whole octave period that I consider equal or better than > this. Thanks! Hopefully, Gene can either locate all the ones you give in his own terms, or figure out why he missed them.
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Message: 5789 Date: Fri, 28 Dec 2001 20:19:14 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > I've modified my badness measure in ways that I hope take into account > the fact (assuming it is one) that pelog is some kind of 5-limit > temperament. I give the following possible ranking of 5-limit > temperaments having a whole octave period. > > Gen Gens in RMS err Name > (cents) 3 5 (cents) > ------------------------------------ > 503.8 [-1 -4] 4.2 meantone > 498.3 [-1 8] 0.3 schismic > 317.1 [ 6 5] 1.0 kleismic > 380.0 [ 5 1] 4.6 > 163.0 [-3 -5] 8.0 > 387.8 [ 8 1] 1.1 > 271.6 [ 7 -3] 0.8 orwell > 443.0 [ 7 9] 1.2 > 176.3 [ 4 9] 2.5 > 339.5 [-5 -13] 0.4 > 348.1 [ 2 8] 4.2 > 251.9 [-2 -8] 4.2 > 351.0 [ 2 1] 28.9 > 126.2 [-4 3] 6.0 > 522.9 [-1 3] 18.1 pelog? Dave, I was hoping that, instead of doing this, you would think in terms of two separate badness factors, an 'error' factor and a 'complexity' factor -- and let us know if you could find anything that was better on _both_ factors than _any_ of the temperaments I listed, but was not in the list anywhere . . . see?
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Message: 5790 Date: Fri, 28 Dec 2001 12:30:12 Subject: Re: Paul's lattice math and my diagrams From: monz > From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, December 28, 2001 12:02 PM > Subject: [tuning-math] Re: Paul's lattice math and my diagrams > > > > > [Paul] > > > P.S. How can you include W. A. Mozart under 55-EDO on your Equal > > > Temperament definition page? I could understand if you wanted to > > > put Mozart on a meantone page, but 55? Totally unjustified. > > > Come on, let's not just make things up. > > > > > > Well... his conception was clearly based on the > > "9 commas per whole-tone, 5 commas per diatonic semitone" idea. > > There is no evidence for that. All we know is that he taught sharps > lower than the "equivalent" flats. Hmmm... I'll have to find some time to dig back into this... too preoccupied with periodicity-block stuff math right now. So essentially what you're saying is that Chesnut, in his article on Mozart, *extrapolates* from Tosi's description of a 55-EDO conception, to Leopold Mozart's praise of Tosi, to W. A. Mozart, and that I have mistakenly accepted that as evidence? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
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Message: 5791 Date: Fri, 28 Dec 2001 21:14:21 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: clumma > I've modified my badness measure in ways that I hope take into > account the fact (assuming it is one) that pelog is some kind > of 5-limit temperament. Was there evidence for this, or is this just an assumption for further exploration? It strikes me as extremely unlikely that any Indonesian tuning is a 5-limit temperament. -Carl
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Message: 5792 Date: Fri, 28 Dec 2001 21:29:48 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: clumma >>I've modified my badness measure in ways that I hope take into >>account the fact (assuming it is one) that pelog is some kind >>of 5-limit temperament. > >Was there evidence for this, or is this just an assumption for >further exploration? It strikes me as extremely unlikely that >any Indonesian tuning is a 5-limit temperament. Posted this before I saw the bit on six narrow and one wide fifths. But: () The Scala scale archive is not a good source of actual pelogs, or any other ethnic tunings for that matter. () There may be many explanations for this pattern of fifths, including something like Sethares' treatment... have you seen his derivation of gamelan tunings in his book? While far from conclusive, it's the best treatment I've seen, and the approach strikes me as making sense... The long decay of gamelan instruments, the style of Indonesian music, the timbre of metalophones, and the ubiquity of the pythagorean scale suggest some chain of fifths over which the total sensory dissonance has been minimized. The 'experimental' way in which actual instances of these ensembles are tuned (as opposed to fixed tunings which are written down) fits this theory. () In any case, because Indonesian music doesn't use 5-limit consonances -- let alone modulate them -- I'd call it an abuse of terminology to say they use a 5-limit temperament, even if the data do match up. Since there is such wide variation in Indonesian tunings, it isn't very difficult to get the data to match, either... -Carl
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Message: 5793 Date: Fri, 28 Dec 2001 22:21:01 Subject: Re: Paul's lattice math and my diagrams From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > > > From: paulerlich <paul@s...> > > To: <tuning-math@y...> > > Sent: Friday, December 28, 2001 12:02 PM > > Subject: [tuning-math] Re: Paul's lattice math and my diagrams > > > > > > > > [Paul] > > > > P.S. How can you include W. A. Mozart under 55-EDO on your Equal > > > > Temperament definition page? I could understand if you wanted to > > > > put Mozart on a meantone page, but 55? Totally unjustified. > > > > Come on, let's not just make things up. > > > > > > > > > Well... his conception was clearly based on the > > > "9 commas per whole-tone, 5 commas per diatonic semitone" idea. > > > > There is no evidence for that. All we know is that he taught sharps > > lower than the "equivalent" flats. > > > Hmmm... I'll have to find some time to dig back into this... too > preoccupied with periodicity-block stuff math right now. So > essentially what you're saying is that Chesnut, in his article > on Mozart, *extrapolates* from Tosi's description of a 55-EDO > conception, to Leopold Mozart's praise of Tosi, to W. A. Mozart, > and that I have mistakenly accepted that as evidence? Nowhere does Chesnut claim that Mozart used 55-EDO or "9 commas per whole-tone, 5 commas per diatonic semitone". He simply provides a historical context in which Mozart's preferences can be understood. Daniel Wold, for example, advocated 1/4-comma meantone for Mozart . . . who can say?
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Message: 5794 Date: Fri, 28 Dec 2001 22:24:43 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote: > () There may be many explanations for this pattern of fifths, > including something like Sethares' treatment... have you seen > his derivation of gamelan tunings in his book? Looks totally contrived, and what about harmonic entropy? > () In any case, because Indonesian music doesn't use 5-limit > consonances -- let alone modulate them It modulates plenty, as we've recently discussed on the tuning list. And, listen to some Pelog-scale Indonesian music. It doesn't evoke 5- limit harmony to your ears? > I'd call it an abuse > of terminology to say they use a 5-limit temperament, even if > the data do match up. Since there is such wide variation in > Indonesian tunings, it isn't very difficult to get the data to > match, either... At least, we can call it a "creative interpretation" of Pelog, which Herman Miller has used effectively in his music, and the tuning of which is by no means precluded as a "statistical center" for actual Pelog tunings.
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Message: 5795 Date: Fri, 28 Dec 2001 00:03:46 Subject: Re: Keenan green Zometool struts From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > "The only problem with the Advanced Math kit is that it doesn't have > any short whole greens (G0). I recommend adding 48 of these (at > US$9.60)." > > I can't figure out how to order 48 short whole greens from the > website. Can you help? You can't order them via the web. Just email your entire order to sales@xxxxxxxx.xxx. Are you getting the bundle of the Adv Math kit with George Hart and Henri Picitto's Zome Geometry book?
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Message: 5797 Date: Fri, 28 Dec 2001 03:01:25 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: dkeenanuqnetau --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > Do pelog scales really tend to do this; have 6 fifths that are up to > 25 c narrow and one that is up to 35 c wide (of 2:3)? OK. I checked it out myself in the Scala archives and the answer is yes! They really do. I said I didn't know much about pelog, but I'm blowed if I know where I got that approx 7-tET idea. So, although pelog is well represented as a chain of very uneven (+-25 c) generators averaging 523 c +-15 c, I'm still waiting to learn whether -1, 3 and -4 generators traditionally represent the consonances of the system? Meanwhile, I'll asume this apparent 5-limit approximation is real and will weight the gens by 1/log(max-odd-factor) and give you the list of those with whole octave period that I consider equal or better than this.
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Message: 5798 Date: Fri, 28 Dec 2001 23:28:07 Subject: Re: Superparticular 5-limit scales From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > The three smallest 5-limit superparticulars are 81/80, 25/24, and 16/15. Putting these into the form of a matrix and inverting gives us > [h3 h5 h7], and hence (81/80)^3 (25/24)^5 (16/15)^7 = 2. We can arrange these 15 scale steps in a number of ways given by the multinomial coefficient 15!/(3! 5! 7!) = 360360, which rotations and inversions would reduce further. > > All of these scales are epimorphic, with defining val h15 = h3+h5+h7, so singling out the interesting ones means putting on additional contraints; convexity and connectedness suggest themselves, of course. > > Anyone care to take a shot at it? Sure -- let's adopt convexity and connectedness (via consonances).
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Message: 5799 Date: Fri, 28 Dec 2001 03:14:44 Subject: Re: My top 5--for Paul From: clumma I wrote... >The listener will sort out notes one way or the other, and >far fewer of them 171 or 612. Dave's point, I think, is >that he or she wouldn't be able to tell the difference. For ets, it probably craps out somewhere around 282-tET, where the 19-limit is consistently represented to within a cent rms, and no interval has more than 2 cents absolute error. -Carl
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