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Message: 2525 - Contents - Hide Contents Date: Tue, 18 Dec 2001 19:47:26 Subject: Re: 55-tET From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, December 18, 2001 6:48 PM > Subject: [tuning-math] 55-tET (was: Re: inverse of matrix --> for what?) > > > What I had in mind was that there should be a pair of > unison-vectors which defines the set of acoustically implied > ratios which I put on my lattice at > <Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.)>? > ... assuming, of course, that in the places where two > ratios are implied equally well/badly, only one can be chosen. > > I find that if I continue my diagram, the unison-vector that > "works" together with the syntonic comma (-4 4 -1 ) to close > the system at 55 tones, is the (-51 19 9 ). The 8ve-invariant > tuning of the 55th quasi-meantone pitch would be 3^19 * 5^(55/6), > which is ~10.38405963 cents higher than the starting pitch, and > the ratio it implies most closely is 3^19 * 5^9.Oops... my bad. Two errors here. That should say "The 8ve-invariant tuning of the 55th quasi-meantone *generator*...", calling the starting pitch the zero-th generator. And the tuning itself is wrong: it should be 3^(55/3) * 5^(55/6). (55/3 = 18 & 1/3, and 55/6 = 9 & 1/6. A simple foul-up: the 54 generator is exactly the ratio 3^18 * 5^9; for the next one I accidentally added 1 instead of 1/3 to 18.) The next "closure" size for 1/6-comma meantone is a 67-note set. The 8ve-invariant 67th generator is ~9.168509182 lower (narrower) than the starting pitch, and its tuning is 3^(67/3) * 5^(67/6). The ratio it implies acoustically most closely is 3^23 * 5^11. The unison-vector would therefore be described, in my matrix notation, as (-61 23 11). Gene, does this agree with your program's output? love / peace / harmony ... -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: 2526 - Contents - Hide Contents Date: Tue, 18 Dec 2001 19:52:26 Subject: Vitale 19 (was: Re: Temperament calculations online) From: paulerlich --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <9vlfh8+9o62@e...> > Paul wrote: >>> Yes, I've talked about this before, but my version does not >> correspond to the minimax view of things. >> About what? When? Where? > > > Graham Yahoo groups: /tuning-math/message/1437 * [with cont.]
Message: 2527 - Contents - Hide Contents Date: Tue, 18 Dec 2001 22:02:23 Subject: Re: inverse of matrix --> for what? From: genewardsmith --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <9vlrb9+8s4b@e...> > genewardsmith wrote: > Is the information we lose really that valuable? Ignoring torsion, the > commatic, octave-equivalent unison vectors still give us the mapping by > generators and the number of periods to an octave.We can't ignore torsion, for starters, and if we simply use the information that the cross-product supplies, we don't have enough to define the temperament, which to my mind is one of the main points of it all. Can you go from that> to an optimum generator, and reconstruct the period mapping?No, although usually there will be an obvious "best" choice, I presume. What is your objection to an invariant which actually does the job, instead of only some of it? It seems a little perverse to me. If the cross-product gives us [-1,-4,-10] it is a fair bet that we have meantone, but if the wedgie is [-1,-4,-10,-12,13,-4] then we *know* we have meantone; and not [-1,-4,-10,-8,17,-6] or [-1,-4,-10,-10,15,-5] or something.
Message: 2528 - Contents - Hide Contents Date: Tue, 18 Dec 2001 14:13:48 Subject: Re: inverse of matrix --> for what? From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, December 17, 2001 2:24 PM > Subject: [tuning-math] Re: inverse of matrix --> for what? > > > --- In tuning-math@y..., graham@m... wrote: >>> Each column is a generator mapping. The left hand one corresponds to the >> top row of the original, 50:49, being the chromatic unison vector. That >> gives a 710 cent generator that approximates 3;2, with 9 octave reduced >> fifths approximating 5:4 and 2 octave reduced fourths approximating 7:4. >> It can be done as 9/22, better as 11/27, and best of all as 20/49, where it > is the 27+22 system.<etc. -- snip> Could you (or someone else?) please give an analysis similar to the one in this post, but for 55-EDO? Thanks. I'm especially interested in all the 5-limit unison-vectors which can define 55-EDO. love / peace / harmony ... -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: 2529 - Contents - Hide Contents Date: Tue, 18 Dec 2001 22:33:55 Subject: Re: inverse of matrix --> for what? From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> I'm especially interested in all the 5-limit unison-vectors which can > define 55-EDO.I just wrote a program yesterday which finds the 5-limit comma associated to (n, g) for the et n and generator g; here is what I get for 55: 2^90/3^26/5^21, 2^82/3^18/5^23, 2^7*3^25/5^20, 2^31*3/5^14, 2^27*3^5/5^15, 2^39/3^7/5^12, 3^27*5^7/2^59, 2^35/3^3/5^13, 2^74/3^10/5^25, 2^23*3^9/5^16, 2^19*3^13/5^17, 2^66/3^2/5^27, 2^47/3^15/5^10, 2^15*3^17/5^18, 3^19*5^9/2^51, 2^43/3^11/5^11, 3^4/2^4/5, 2^11*3^21/5^19, 3^23*5^8/2^55 Each of these 19 commas defines a linear temperament associated to 55; one, of course (3^4/2^4/5 in the notation my computer used) is 81/80.
Message: 2530 - Contents - Hide Contents Date: Tue, 18 Dec 2001 00:54:06 Subject: Re: Badness with gentle rolloff From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> I have searched all 8 pdf >> files for the word "diophantine" with no success. >> You can search .pdf files for a particular word?Sure. Type Ctrl-F or choose Edit/Find or click the binoculars icon. I believe it is possible to generate a PDF which is essentially just an image and searching in those is impossible. But that is not the case for Dave Benson's files.> I've never heard of this ability. Try searching for "the".Just in case, I tried searching for "the" in all 8 files. It works fine.
Message: 2531 - Contents - Hide Contents Date: Tue, 18 Dec 2001 22:42:21 Subject: Re: inverse of matrix --> for what? From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> Could you (or someone else?) please give an analysis similar to the one in > this post, but for 55-EDO? Thanks. > > I'm especially interested in all the 5-limit unison-vectors which can > define 55-EDO. Hi Monz.The three unison vectors we came up with for 22-tET were a "Minkowski- reduced trio" . . . that is, they're essentially the three simplest unison vectors which define 22-tET in the 7-limit. But there are many other ways to define 22-tET in the 7-limit with three unison vectors - - this was just the simplest way. For 5-limit, we will only need two unison vectors to define an ET, in this case 55-tET. One of these unison vectors should of course 81:80, the unison vector that defines meantone. I don't know what the other pair of the "Minkowski-reduced duo" for 55-tET is, but the choice is essentially immaterial -- any such choice will be equivalent when 81:80 is tempered out, and when 81:80 is not tempered out, you essentially have garbage, since 55-tET only makes sense as a particular meantone and not as some other kind of 5-limit linear temperament. If you're still interested, it should be easy for you to find candidates for the second unison vector, Monz. Just look at one of the enharmonic equivalencies in 55-tET and express both of the notes comprising the equivalency as a JI ratio in several different ways. Then take the quotient of various pairs of ratios representing the two notes, and voila -- various unison vector candidates.
Message: 2532 - Contents - Hide Contents Date: Tue, 18 Dec 2001 03:43:21 Subject: Vitale 19 (was: Re: Temperament calculations online) From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > >> See A method for optimally distributing any comma * [with cont.] (Wayb.) >> Seems like a dead end. Time to redo this page with linear programming?Be my guest. It's not something I'm interested in doing, and it's a very long time since I learnt about linear programming, and I've never used it for anything real since.
Message: 2533 - Contents - Hide Contents Date: Tue, 18 Dec 2001 22:47:35 Subject: Re: inverse of matrix --> for what? From: paulerlich I wrote,> 55-tET only makes sense as a > particular meantone and not as some other kind of 5-limit linear > temperament.By "makes sense", I of course meant "has reasonable complexity", not "is mathematically correct". Sure, the other unison vectors Gene described corresponds to a way of generating 55-tET from some interval other than the fifth (fourth), but these ways entail great complexity and are irrelevant to the historical use of 55-tET, which was as a measuring system for a particular flavor of meantone. (Just being the voice of grounding in reality -- there's nothing wrong with pursuing these curiosities for their own sake, of course.)
Message: 2534 - Contents - Hide Contents Date: Tue, 18 Dec 2001 03:48:04 Subject: Vitale 19 (was: Re: Temperament calculations online) From: dkeenanuqnetau --- 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: >> >>> See A method for optimally distributing any comma * [with cont.] (Wayb.) >>>> Seems like a dead end. Time to redo this page with linear > programming? >> Be my guest. It's not something I'm interested in doing, and it's a > very long time since I learnt about linear programming, and I've never > used it for anything real since.Actually, what would be the point. The point of my attempt on that page, is that you can do it with nothing more than pen and paper and you can follow what and why. If you just want an algorithm for computer, then numerical methods (sucessive approximations) work just fine.
Message: 2535 - Contents - Hide Contents Date: Tue, 18 Dec 2001 22:53:51 Subject: Vitale 19 (was: Re: Temperament calculations online) From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Linear programming can usually > be done with pen and paper too.Why so it can. I'd love to see such a method explained.> You'd be surprised what a black-box minimization program can do with > absolute value functions.I don't think it is hard to make it "absolute-value aware".
Message: 2536 - Contents - Hide Contents Date: Tue, 18 Dec 2001 08:42:34 Subject: Re: Badness with gentle rolloff From: clumma> I believe it is possible to generate a PDF which is essentially > just an image and searching in those is impossible. But that is > not the case for Dave Benson's files.Correct, and it's possible to have a file containing mixed data types -- you can embed images right along with postscript stuff, so if "diophantine" (or whatever) is in an image with a bunch of math symbols, then you won't find it. Dave- do you have a spreadsheet for badness measures? I'd love to see consistency plotted against steps*rms. -Carl
Message: 2537 - Contents - Hide Contents Date: Tue, 18 Dec 2001 23:05:12 Subject: Vitale 19 (was: Re: Temperament calculations online) From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> Linear programming can usually >> be done with pen and paper too. >> Why so it can. I'd love to see such a method explained.Well, above the 5-limit, you might need a 3-dimensional "paper" :)>> You'd be surprised what a black-box minimization program can do with >> absolute value functions. >> I don't think it is hard to make it "absolute-value aware".Well, I wish the Matlab people had thought of that!
Message: 2538 - Contents - Hide Contents Date: Tue, 18 Dec 2001 10:57 +0 Subject: Re: inverse of matrix --> for what? From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9vlrb9+8s4b@xxxxxxx.xxx> genewardsmith wrote:> It's the adjoint matrix; the columns are wedge products in a 3D space > of octave equivalence classes, where the wedge product becomes a > cross-product. I don't recommend this point of view, which throws away > some valuable information.Is the information we lose really that valuable? Ignoring torsion, the commatic, octave-equivalent unison vectors still give us the mapping by generators and the number of periods to an octave. Can you go from that to an optimum generator, and reconstruct the period mapping? Including the chromatic unison vector gives us the number of notes in a given MOS, and don't we have a way of getting the periodicity block as monotonically increasing pitches? That should make it even easier. Graham
Message: 2539 - Contents - Hide Contents Date: Tue, 18 Dec 2001 23:17:48 Subject: Re: inverse of matrix --> for what? From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> For 5-limit, we will only need two unison vectors to define an ET, in > this case 55-tET. One of these unison vectors should of course 81:80, > the unison vector that defines meantone.I got two of the commas on my list--one, of course, 81/80, and the other 6442450944/6103515625 = 2^31*3*5^(-14). My badness score for the associated temperament is 6590, but some of the other commas do better--in particular, 2^47 3^(-15) 5^(-10) scores 1378; which hardly compares with the score of 108 for meantone and would not make my best list, where I have a cutoff of 500, but it isn't garbage. The period matrix is [ 0 5] [ -2 11] [ 3 7] and the generators are a = 19.98/65 and b = 1/5; it really is more of a 65-et system than a 55-et system, and scores as well as it does since it is in much better tune than the 55-et itself, with errors: 3: .317 5: .228 5/3: -.040
Message: 2540 - Contents - Hide Contents Date: Tue, 18 Dec 2001 12:24 +0 Subject: Re: Vitale 19 (was: Re: Temperament calculations online) From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9vlfh8+9o62@xxxxxxx.xxx> Paul wrote:> Yes, I've talked about this before, but my version does not > correspond to the minimax view of things.About what? When? Where? Graham
Message: 2541 - Contents - Hide Contents Date: Tue, 18 Dec 2001 23:36:36 Subject: Re: Badness with gentle rolloff From: dkeenanuqnetau --- In tuning-math@y..., "clumma" <carl@l...> wrote:> Dave- do you have a spreadsheet for badness measures? Yes. 404 Not Found * [with cont.] Search for http://uq.net.au/~zzdkeena/Music/7LimitETBadness.xls.zip in Wayback Machine > I'd love to see consistency plotted against steps*rms.This has been added at your request.
Message: 2543 - Contents - Hide Contents Date: Tue, 18 Dec 2001 23:53:17 Subject: 55-tET (was: Re: inverse of matrix --> for what?) From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> For 5-limit, we will only need two unison vectors to define an ET, in >> this case 55-tET. One of these unison vectors should of course 81:80, >> the unison vector that defines meantone. >> I got two of the commas on my list--one, of course, 81/80, and theother 6442450944/6103515625 = 2^31*3*5^(-14). My badness score for the associated temperament is 6590, but some of the other commas do better--in particular, 2^47 3^(-15) 5^(-10) scores 1378; which hardly compares with the score of 108 for meantone and would not make my best list, where I have a cutoff of 500, but it isn't garbage. The period matrix is> > [ 0 5] > [ -2 11] > [ 3 7] > > and the generators are a = 19.98/65 and b = 1/5; it really is moreof a 65-et system than a 55-et system, and scores as well as it does since it is in much better tune than the 55-et itself, with errors:> > 3: .317 > 5: .228 > 5/3: -.040So, in an evaluation of 55-tET generators, it's pretty much garbage. So, Monz, according to Gene the simplest pair of unison vectors for defining 55-tET is 81:80 and 6442450944:6103515625 The latter is the result of going 14 major thirds down and one perfect fifth up. In JI, it's about 93.563 cents; and in 55-tET, it's of course 0.000 ;)
Message: 2545 - Contents - Hide Contents Date: Tue, 18 Dec 2001 18:48:00 Subject: 55-tET (was: Re: inverse of matrix --> for what?) From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, December 18, 2001 2:33 PM > Subject: [tuning-math] Re: inverse of matrix --> for what? > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> I'm especially interested in all the 5-limit unison-vectors >> which can define 55-EDO. >> I just wrote a program yesterday which finds the 5-limit comma > associated to (n, g) for the et n and generator g; here is > what I get for 55: > > 2^90/3^26/5^21, 2^82/3^18/5^23, 2^7*3^25/5^20, 2^31*3/5^14, > 2^27*3^5/5^15, 2^39/3^7/5^12, 3^27*5^7/2^59, 2^35/3^3/5^13, > 2^74/3^10/5^25, 2^23*3^9/5^16, 2^19*3^13/5^17, 2^66/3^2/5^27, > 2^47/3^15/5^10, 2^15*3^17/5^18, 3^19*5^9/2^51, 2^43/3^11/5^11, > 3^4/2^4/5, 2^11*3^21/5^19, 3^23*5^8/2^55 > > Each of these 19 commas defines a linear temperament associated > to 55; one, of course (3^4/2^4/5 in the notation my computer used) > is 81/80.So, rewritten in a form that I'm more familiar with, that's: where unison-vector = 2^x * 3^y * 5^z, x y z ( 90 -26 -21 ) ( 82 -18 -23 ) ( 7 25 -20 ) ( 31 1 -14 ) ( 27 5 -15 ) ( 39 -7 -12 ) (-59 27 7 ) ( 35 -3 -13 ) ( 74 -10 -25 ) ( 23 9 -16 ) ( 19 13 -17 ) ( 66 -2 -27 ) ( 47 -15 -10 ) ( 15 17 -18 ) (-51 19 9 ) ( 43 -11 -11 ) ( -4 4 -1 ) ( 11 21 -19 ) (-55 23 8 ) Thanks to Paul for the invaluable subsequent comments.> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, December 18, 2001 3:53 PM > Subject: [tuning-math] 55-tET (was: Re: inverse of matrix --> for what?) > > > So, Monz, according to Gene the simplest pair of unison vectors for > defining 55-tET is > > 81:80 > and > 6442450944:6103515625 > > The latter is the result of going 14 major thirds down and one > perfect fifth up. In JI, it's about 93.563 cents; and in 55-tET, it's > of course 0.000 ;)What I had in mind was that there should be a pair of unison-vectors which defines the set of acoustically implied ratios which I put on my lattice at <Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.)>? ... assuming, of course, that in the places where two ratios are implied equally well/badly, only one can be chosen. I find that if I continue my diagram, the unison-vector that "works" together with the syntonic comma (-4 4 -1 ) to close the system at 55 tones, is the (-51 19 9 ). The 8ve-invariant tuning of the 55th quasi-meantone pitch would be 3^19 * 5^(55/6), which is ~10.38405963 cents higher than the starting pitch, and the ratio it implies most closely is 3^19 * 5^9. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2546 - Contents - Hide Contents Date: Wed, 19 Dec 2001 19:11:54 Subject: Re: 55-tET From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: monz <joemonz@y...> >> To: <tuning-math@y...> >> Sent: Tuesday, December 18, 2001 6:48 PM >> Subject: [tuning-math] 55-tET (was: Re: inverse of matrix --> for what?) >> >> >> What I had in mind was that there should be a pair of >> unison-vectors which defines the set of acoustically implied >> ratios which I put on my lattice at >> <Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.)>? >> ... assuming, of course, that in the places where two >> ratios are implied equally well/badly, only one can be chosen. >> >> I find that if I continue my diagram, the unison-vector that >> "works" together with the syntonic comma (-4 4 -1 ) to close >> the system at 55 tones, is the (-51 19 9 ). The 8ve-invariant >> tuning of the 55th quasi-meantone pitch would be 3^19 * 5^(55/6), >> which is ~10.38405963 cents higher than the starting pitch, and >> the ratio it implies most closely is 3^19 * 5^9. > >> Oops... my bad. Two errors here. > > That should say "The 8ve-invariant tuning of the 55th > quasi-meantone *generator*...", calling the starting pitch > the zero-th generator. And the tuning itself is wrong: > it should be 3^(55/3) * 5^(55/6).What you completely fail to mention is that you're actually using 1/6- comma meantone. This was nowhere implied in your original question to us. But you can indeed define 55-tET with the two unison vectors (-4 4 - 1) and (-51 19 9), and in fact it's quite logical to do so, since these intervals are only 21.51 cents and 13.97 cents in JI. In fact, I've been wondering if there's an analogue to Minkowski reduction which, instead of finding the simplest commatic unison vectors, finds the _smallest_ ones that work, without torsion (or "potential torsion"). This could be valuable to JI-oriented theorists like Monz and Kraig. For example, it seems that for Blackjack in the 7-limit, the answer might be 2401:2400 and 16875:16807 . . . Gene, does this make any sense?
Message: 2547 - Contents - Hide Contents Date: Wed, 19 Dec 2001 19:17:00 Subject: Re: Flat 7 limit ET badness? (was: Badness with gentle rolloff) From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> Gene seems to be saying that for any given badness cutoff,Only if it's low enough, or if you start far enough away from zero.> the number > less than that should be about the same in every decade (1 to 9- tET, > 10 to 99-tET, 100 to 999-tET, etc). Could you check that with Matlab > Paul, for both steps^(4/3)*cents and steps*cents? For various cutoffs?Sure -- just tell me how far out I should go, what cutoffs to use -- perhaps Gene would like to weigh in on these decisions to help guide us toward something that will make the distinction more clear . . .
Message: 2548 - Contents - Hide Contents Date: Wed, 19 Dec 2001 04:25:37 Subject: Re: 55-tET From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> The next "closure" size for 1/6-comma meantone is a 67-note set. > The 8ve-invariant 67th generator is ~9.168509182 lower (narrower) > than the starting pitch, and its tuning is 3^(67/3) * 5^(67/6). > The ratio it implies acoustically most closely is 3^23 * 5^11. > The unison-vector would therefore be described, in my matrix > notation, as (-61 23 11). > > Gene, does this agree with your program's output?I'm not sure what your question means; however I can make the following comments: (1) Presumably you meant the comma 2^62 3^(-23) 5^(-11) (2) This is a 67-et comma; however, and much more significantly, it is a 65-et comma. It really doesn't work very well for anything *but* 65-et, in fact. (3) For the associated linear temperament, we have a map [ 0 1] [-11 7] [ 23 9] The generator is 31.997/65, so this can be more or less equated with 32/65.
Message: 2549 - Contents - Hide Contents Date: Wed, 19 Dec 2001 12:07:44 Subject: Re: 55-tET From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, December 19, 2001 11:11 AM > Subject: [tuning-math] Re: 55-tET > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> Oops... my bad. Two errors here. >> >> That should say "The 8ve-invariant tuning of the 55th >> quasi-meantone *generator*...", calling the starting pitch >> the zero-th generator. And the tuning itself is wrong: >> it should be 3^(55/3) * 5^(55/6). >> What you completely fail to mention is that you're actually using 1/6- > comma meantone. This was nowhere implied in your original question to > us.Wow -- my bad again. Thanks for catching that, Paul. I made reference to my webpage, so it should have been pretty clear that I was referring to 1/6-comma meantone specifically, as well as its analogue in 55-EDO... but you're right that I should have made that clear in my post.> But you can indeed define 55-tET with the two unison vectors > (-4 4 -1) and (-51 19 9), and in fact it's quite logical to do so, > since these intervals are only 21.51 cents and 13.97 cents in JI.Paul, I've been surprised all along that you disagree so strongly with my interpretations of meantone systems. Is this perhaps bringing our individual conceptions a bit closer together?> In fact, I've been wondering if there's an analogue to Minkowski > reduction which, instead of finding the simplest commatic unison > vectors, finds the _smallest_ ones that work, without torsion > (or "potential torsion"). This could be valuable to JI-oriented > theorists like Monz and Kraig. For example, it seems that for > Blackjack in the 7-limit, the answer might be 2401:2400 and > 16875:16807 . . . Gene, does this make any sense?Yes, this does sound interesting to me... but I really don't know what "Minkowski reduction" is... -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
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