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Message: 2500 - Contents - Hide Contents Date: Sun, 16 Dec 2001 05:05:17 Subject: formula for meantone implications? From: monz Hello all, Please take a look at Yahoo groups: /tuning-math/files/monz/formula1... * [with cont.] This is an x-y plot of the numeric relationship between the pitches of 1/6-comma meantone their their acoustically closest implied 5-limit JI ratios, as illustrated on my lattice at Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) I suppose calculus is need to derive this numerically, since some values of x have two values for y and z, yes? In cases where there are two values for both y and z (i.e., -3 and +3 on my graph, and -3-6x and 3+6x if it were to be continued), the lower z goes with the top y and vice versa. If someone is interested in this, I would also appreciate formulas for the other meantone systems on my webpage, as well as a generalized formula if one can be derived. 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: 2501 - Contents - Hide Contents Date: Sun, 16 Dec 2001 22:42:03 Subject: Vitale 19 (was: Re: Temperament calculations online) From: dkeenanuqnetau --- In tuning-math@y..., "clumma" <carl@l...> wrote:> You said, "that doesn't quite do it".... > > Anyway, without the givens, one could read... "all linear > temperaments have the same number of o- and u-tonal chords", > as Paul seems to have done.You seem to have missed the important change I made. I put "same" in between "o" and "u", instead of after them. Instead of the number of o and u are the same in any ... I made it the number of o is the same as the number of u in any ...
Message: 2503 - Contents - Hide Contents Date: Sun, 16 Dec 2001 22:50:51 Subject: Re: Badness with gentle rolloff From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>>> It has to do with Diophantine approximation theory. Have you read >>> Dave Benson's course notes? >> Well, he does mention the Diophantine approximation exponent for > N-term ratios.Could you tell me what section this is in? I have searched all 8 pdf files for the word "diophantine" with no success.
Message: 2505 - Contents - Hide Contents Date: Sun, 16 Dec 2001 22:59:44 Subject: Vitale 19 (was: Re: Temperament calculations online) From: dkeenanuqnetau --- In tuning-math@y..., graham@m... wrote:> This is a useful thing to know for a temperament finder. When considering > unison vectors, you can check which ones can never produce a temperament > as accurate as the one you want. Do people have other rules of thumb for > filtering unison vectors, ETs or wedgies according to the simplest or most > accurate temperaments they can give rise to? It would make the search > less arbitrary. See A method for optimally distributing any comma * [with cont.] (Wayb.)
Message: 2507 - Contents - Hide Contents Date: Sun, 16 Dec 2001 17:18 +0 Subject: Re: Vitale 19 (was: Re: Temperament calculations online) From: graham@xxxxxxxxxx.xx.xx Dave Keenan:>> It has 5 otonal and 5 utonal 7-limit tetrads with max error of 2.7 > c. Paul Erlich> Now -- if you think of this as a planar temperament where _only_ > 224:225 is tempered out, I bet you can reduce that error even further.224:225 comes from 14:15 and 15:16 being equivalent. These are both second-order 7-limit intervals. So, the error has to be shared amongst 4 7-limit intervals. 224:225 is 7.7 cents, so any scale tempering it out can't be closer than 7.7/4=1.9 cents to 7-limit JI. So that's what the minimax for the planar temperament will be. This is a useful thing to know for a temperament finder. When considering unison vectors, you can check which ones can never produce a temperament as accurate as the one you want. Do people have other rules of thumb for filtering unison vectors, ETs or wedgies according to the simplest or most accurate temperaments they can give rise to? It would make the search less arbitrary. Graham
Message: 2509 - Contents - Hide Contents Date: Mon, 17 Dec 2001 19:37:22 Subject: Re: Badness with gentle rolloff From: clumma> You can search .pdf files for a particular word? Absolutely. -Carl
Message: 2510 - Contents - Hide Contents Date: Mon, 17 Dec 2001 20:18:31 Subject: Vitale 19 (was: Re: Temperament calculations online) From: paulerlich --- In tuning-math@y..., graham@m... wrote:> 224:225 comes from 14:15 and 15:16 being equivalent. These are both > second-order 7-limit intervals. So, the error has to be shared amongst 4 > 7-limit intervals.Or, the Hahn length of 224:225 in the 7-limit is 4. (Scala is supposedly able to compute this)
Message: 2511 - Contents - Hide Contents Date: Mon, 17 Dec 2001 12:41:01 Subject: Re: formula for meantone implications? From: monz Hi J,> From: unidala <JGill99@xxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 16, 2001 7:56 PM > Subject: [tuning-math] Re: formula for meantone implications? > > > J Gill: Monz, it sounds like you want to build a machine > than can "think" (like people do)! I guess if you can > define a set of JI ratios (which you like, or which meet > some "man-made" criteria for the numerical size of the > numerator/denominator involved, etc.), you could write > a program to "decide" which of those ratios your meantone > pitch value is "closest" to [by some pre-determined measure > such as RMS error in deviation from a function such as > 2^(pitch/reference)].Not at all! It's much simpler than that. I'm just looking for an elegant mathematical formula to explain what I'm showing on my lattices. The only measure I'm using is simple closeness in pitch-height. The only reason it gets complicated and requires two solutions sometimes is because some meantone pitches are exactly midway between the two closest implied ratios. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2512 - Contents - Hide Contents Date: Mon, 17 Dec 2001 21:01:20 Subject: Re: formula for meantone implications? From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> The only measure I'm using is simple closeness in pitch-height. > The only reason it gets complicated and requires two solutions > sometimes is because some meantone pitches are exactly midway > between the two closest implied ratios.Each meantone pitch implies an infinite number of ratios on the just 5-limit lattice. Restricting yourself to the two closest would be severely insufficient to describe a piece by, say, Mozart, where the tonic alone would have to imply several different 81:80 transpositions of itself over the course of the piece.
Message: 2513 - Contents - Hide Contents Date: Mon, 17 Dec 2001 22:24:41 Subject: Re: inverse of matrix --> for what? From: genewardsmith --- 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.> The next column is for 64:63 being the chromatic unison vector. As it has > a common factor of 2, you know the octave is divided into 2 equal parts. > You could set the generator as 434 cents. Then, 3 generators are a 3:2, > and 5 could be either 5:4 or 7:4 (with tritone reduction). Because 7:4 > and 5:4 are the same tritone-reduced, 7:5 must be a tritone. So 7:5 and > 10:7 are the same, and 50:49 is tempered out, as expected. I think this > one is Paultone.This is the chain-of-supermajor-thirds system, an interesting system with a unique association to the 22-et.> The last column is for 245:243 tempered out. I get a 109.4 cent > generator, with a 7-limit error of 17.5 cents.This is twintone, aka Paultone.> According to Gene, this: > > ( 1 -6 -2 ) > ( 9 -10 4 ) > (-2 -10 4 ) > > is the adjoint of the original matrix, and each column is the wedge > product of the relevant commatic unison vectors.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. From the full 7-limit point of view, we can do something equivalent by taking the odd part of the commas; we then have 25/49^1/63 = [-2,4,4,0,0,0] 25/49^245/243 = [6,10,10,0,0,0] 1/63^245/243 = [1,9,-2,0,0,0] Looking at this, we might think we have torsion in the first two examples; however 50/49^64/63 = [-2,4,4,-2,-12,11] 50/49^245/243 = [6,10,10,-5,1,2] 64/63^245/243 = [1,9,-2,-30,6,12] The above shows we do not have torsion, and tells us other things, such as how to calculate the second column of the period matrix.>> Do these integers tell us something about 22-EDO? >> Or about 22-EDO's representation of the prime-factors? >> >> ???? >> You should have left the factors of 2 in for that. Add the octave to the > matrix:Another way is to add a top row of basis vectors to the matrix; this is is the same as taking the triple wedge product: 50/49^64/63^245/243 = 22 i + 35 j + 51 k + 62 l; we see that the triple wedge product gives us the 22-et from the three commas.
Message: 2515 - Contents - Hide Contents Date: Mon, 17 Dec 2001 12:49 +0 Subject: Re: inverse of matrix --> for what? From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <002301c18678$e1e58f00$af48620c@xxx.xxx.xxx> monz wrote:> Excel's "mdeterm" function gives 22 as the determinant of > the original matrix. Multiplying the inverse of the matrix > by the determinant gives the inverse as fractional parts of 22: > > fractional inverse > | 1 -6 -2 | * 1 > | 9 -10 4 | -- > |-2 -10 4 | 22You shouldn't use | for the brackets. They're for determinants.> My questions: what does this inverse explain? > What purpose does it serve?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. The next column is for 64:63 being the chromatic unison vector. As it has a common factor of 2, you know the octave is divided into 2 equal parts. You could set the generator as 434 cents. Then, 3 generators are a 3:2, and 5 could be either 5:4 or 7:4 (with tritone reduction). Because 7:4 and 5:4 are the same tritone-reduced, 7:5 must be a tritone. So 7:5 and 10:7 are the same, and 50:49 is tempered out, as expected. I think this one is Paultone. The last column is for 245:243 tempered out. I get a 109.4 cent generator, with a 7-limit error of 17.5 cents. According to Gene, this: ( 1 -6 -2 ) ( 9 -10 4 ) (-2 -10 4 ) is the adjoint of the original matrix, and each column is the wedge product of the relevant commatic unison vectors.> Do these integers tell us something about 22-EDO? > Or about 22-EDO's representation of the prime-factors? > > ????You should have left the factors of 2 in for that. Add the octave to the matrix: ( 1 0 0 0) ( 1 0 2 -2) ( 6 -2 0 -1) ( 0 -5 1 2) then the adjoint is (22 0 0 0) (35 1 -6 -1) (51 9 -10 4) (62 -2 -10 4) so you now have an extra column that tells you the number of steps to each prime interval. It's also the wedge product of the three unison vectors. Graham
Message: 2516 - Contents - Hide Contents Date: Mon, 17 Dec 2001 08:34:56 Subject: Re: inverse of matrix --> for what? From: monz ----- Original Message ----- From: <graham@xxxxxxxxxx.xx.xx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Monday, December 17, 2001 4:49 AM Subject: [tuning-math] Re: inverse of matrix --> for what?> You shouldn't use | for the brackets. They're for determinants.Wow -- thanks for clearing that up!>>> My questions: what does this inverse explain? >> What purpose does it serve? >> Each column is a generator mapping. The left hand one corresponds to the > top row of the original, <snip...> > > According to Gene, this: > > ( 1 -6 -2 ) > ( 9 -10 4 ) > (-2 -10 4 ) > > is the adjoint of the original matrix, and each column is the wedge > product of the relevant commatic unison vectors.Thanks very much for explaining this, Graham. Now I'm at least beginning to hope that someday I'll understand Gene's work. Shouldn't I have Tuning Dictionary definitions for "wedge product" and "adjoint"? Please help. ... Gene? Paul?> then the adjoint is > > (22 0 0 0) > (35 1 -6 -1) > (51 9 -10 4) > (62 -2 -10 4)Looks like a typo... shouldn't the second row be (35 1 -6 -2) ? 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: 2517 - Contents - Hide Contents Date: Mon, 17 Dec 2001 16:50 +0 Subject: Re: inverse of matrix --> for what? From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <00c901c18718$c3a638a0$af48620c@xxx.xxx.xxx> monz wrote:> Shouldn't I have Tuning Dictionary definitions for "wedge product" > and "adjoint"? Please help. ... Gene? Paul?I don't know. It depends on how bloated you want it to get. They're both linear algebra terms that have a specialist application to tuning theory. And wedge products conceptually make the adjoint obsolete anyway. The adjoint's only useful because it can sometimes be calculated more efficiently if you already have a library that does inverses (or solves systems of linear equations, which comes to the same thing). Even then, it'll probably mean taking the inverse, multiplying by the determinant, and rounding off to integers. So knowing it's called an "adjoint" isn't much help.>> then the adjoint is >> >> (22 0 0 0) >> (35 1 -6 -1) >> (51 9 -10 4) >> (62 -2 -10 4) > >> Looks like a typo... shouldn't the second row be (35 1 -6 -2) ?Yes, looks like it, although I've lost the original calculation. Graham
Message: 2518 - Contents - Hide Contents Date: Mon, 17 Dec 2001 19:00:20 Subject: Re: formula for meantone implications? From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> > I suppose calculus is need to derive this numerically, > since some values of x have two values for y and z, yes? Calculus???
Message: 2519 - Contents - Hide Contents Date: Mon, 17 Dec 2001 19:03:04 Subject: Vitale 19 (was: Re: Temperament calculations online) From: paulerlich --- In tuning-math@y..., graham@m... wrote:> This is a useful thing to know for a temperament finder. When considering > unison vectors, you can check which ones can never produce a temperament > as accurate as the one you want. Do people have other rules of thumb for > filtering unison vectors, ETs or wedgies according to the simplest or most > accurate temperaments they can give rise to?Yes, I've talked about this before, but my version does not correspond to the minimax view of things.
Message: 2520 - Contents - Hide Contents Date: Mon, 17 Dec 2001 19:04:56 Subject: Re: Badness with gentle rolloff From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>>>> It has to do with Diophantine approximation theory. Have you > read>>>> Dave Benson's course notes? >>>> Well, he does mention the Diophantine approximation exponent for >> N-term ratios. >> Could you tell me what section this is in?I don't remember.> I have searched all 8 pdf > files for the word "diophantine" with no success.You can search .pdf files for a particular word? I've never heard of this ability. Try searching for "the".
Message: 2521 - Contents - Hide Contents Date: Mon, 17 Dec 2001 19:05:51 Subject: Vitale 19 (was: Re: Temperament calculations online) From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: Seems like a dead end. Time to redo this page with linear programming?
Message: 2522 - Contents - Hide Contents Date: Mon, 17 Dec 2001 19:10:18 Subject: Re: inverse of matrix --> for what? From: paulerlich --- In tuning-math@y..., graham@m... wrote:> The next column is for 64:63 being the chromatic unison vector. > As it has > a common factor of 2, you know the octave is divided into 2 equal parts. > You could set the generator as 434 cents. Then, 3 generators are a 3:2, > and 5 could be either 5:4 or 7:4 (with tritone reduction). Because 7:4 > and 5:4 are the same tritone-reduced, 7:5 must be a tritone. So 7:5 and > 10:7 are the same, and 50:49 is tempered out, as expected. I think this > one is Paultone.Generator of 434 cents? I don't think so!> > The last column is for 245:243 tempered out.You mean _not_ tempered out.> I get a 109.4 cent > generator, with a 7-limit error of 17.5 cents. _That's_ paultone!
Message: 2523 - Contents - Hide Contents Date: Tue, 18 Dec 2001 19:08:50 Subject: Re: 55-tET (was: Re: inverse of matrix --> for what?) From: monz> 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 ) <etc. -- snip>And of course, Yahoo's new space-removing "feature" ruined the careful formatting I put into that matrix, on the web-based version of the list. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2524 - Contents - Hide Contents Date: Tue, 18 Dec 2001 19:49:42 Subject: Vitale 19 (was: Re: Temperament calculations online) From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> 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.But it doesn't work right -- though of course if you could find a general fix, I'd be all for it . . . Linear programming can usually be done with pen and paper too.> If you just want an algorithm for computer, then numerical methods > (sucessive approximations) work just fine.You'd be surprised what a black-box minimization program can do with absolute value functions.
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