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Message: 5501

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.


top of page bottom of page up down Message: 5503 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 *
top of page bottom of page up down Message: 5505 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
top of page bottom of page up down Message: 5507 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
top of page bottom of page up down Message: 5508 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)
top of page bottom of page up down Message: 5509 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 Setup *
top of page bottom of page up down Message: 5510 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.
top of page bottom of page up down Message: 5513 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 | 22 You 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
top of page bottom of page up down Message: 5514 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 http://www.monz.org * "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
top of page bottom of page up down Message: 5515 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
top of page bottom of page up down Message: 5516 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???
top of page bottom of page up down Message: 5517 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.
top of page bottom of page up down Message: 5518 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".
top of page bottom of page up down Message: 5519 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: > See A method for optimally distributing any comma * Seems like a dead end. Time to redo this page with linear programming?
top of page bottom of page up down Message: 5520 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!
top of page bottom of page up down Message: 5521 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 Setup *
top of page bottom of page up down Message: 5522 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.
top of page bottom of page up down Message: 5523 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 > <lattices comparing various Meantone Cycles, (c)2001 by Joseph L. Monzo *>? > ... 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 http://www.monz.org * "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
top of page bottom of page up down Message: 5524 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 *
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