4000 4050 4100 4150 4200 4250 4300 4350 4400 4450 4500 4550 4600 4650 4700 4750 4800 4850 4900 4950 5000 5050 5100 5150 5200 5250 5300 5350 5400 5450 5500 5550 5600 5650 5700 5750 5800 5850 5900 5950 6000 6050 6100 6150 6200 6250 6300 6350 6400 6450 6500 6550
5650 - 5675 -
![]()
![]()
Message: 5650 Date: Sun, 23 Dec 2001 23:57:24 Subject: Re: a different example From: paulerlich > It's hard for me to visualize what happens on a cylinder > or torus since I'm dealing with planar graphs. Well then, the Hall article I'm sending you tomorrow may help.
![]()
![]()
![]()
Message: 5652 Date: Sun, 23 Dec 2001 21:22:39 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > lattice coordinates x, y : > > > x = ( (q*c) + (p*a) ) / n > > y = ( (q*d) + (p*b) ) / n There shouldn't be an "/ n" at the end of that. Maybe that's what's causing the weirdness, because sometimes n is negative.
![]()
![]()
![]()
Message: 5653 Date: Sun, 23 Dec 2001 00:28:19 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: monz > From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx>; <jhchalmers@xxxx.xxx>; <paul@xxxxxxxxxxxxx.xxx> > Sent: Saturday, December 22, 2001 7:35 PM > Subject: Re: [tuning-math] coordinates from unison-vectors (was: 55-tET) > > > By brute force, a bit of research into matrix transformations, > and a whole lot of luck, I figured out how to do it. Here's the pseudo-code for the formulas in my spreadsheet. Please feel free to correct any errors or to make the code more elegant. unison-vectors = (3^a) * (5^b) (3^c) * (5^d) unison-vector matrix = (a b) (c d) determinant n of the matrix : n = (a*d) - (c*b) inverse of the matrix = ( d -b) (-c a) ------- n inverse coordinates p, q : p = 0, q = 0 LOOP if ABS(p+d) > (ABS(n)/2) then p = MOD(p+d, ABS(n)) - ABS(n) else p = p + d end if if ABS(q-b) > (ABS(n)/2) then q = MOD(q-b, ABS(n)) - ABS(n) else q = q - b end if lattice coordinates x, y : x = ( (q*c) + (p*a) ) / n y = ( (q*d) + (p*b) ) / n END LOOP 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 *
![]()
![]()
![]()
Message: 5654 Date: Sun, 23 Dec 2001 19:40:42 Subject: Re: For Pierre, from tuning From: Pierre Lamothe Gene wrote: Judging by these examples, I would propose the following definitions: (1) We may write any positive rational number r in the form 2^n p/q, where p and q are odd integers and p/q is reduced to lowest form. The fraction p/q we call the *odd part* of r. (2) For any set S of rational numbers, we may take the odd parts of each element, and the least common multiple D of their denominators. The set of integers gammier(S) is defined as the set of the odd parts of the elements of S times D; gammier(S) is the *gammier of S*. Is there anything about this you want to accept, or to modify and then accept? There is no problem with (1). I would add only the terms I use for that. r = 2n p/q is the rational number r mod 2 = {2x p/q | all x in Z} is the corresponding octave class Ton ( r ) = 2k p/q | k such that r in [1,2[ is the tone representing the class Pivot ( r ) = p/q is the pivot representing the class : your odd part. There are few problems with (2). Gammier and generator First, the expression "gammier <a b c ...>" means the structure generated by <a b c ...>. The term <a b c ...> is refered as the generator g of the structure, while the elements of that structure g\g are a\a a\b a\c ... b\a b\b b\c ,,, c\a c\b c\c ... ,,, where u\v means here the interval between u and v. The symbol \ is independant of the composition law type, multiplicative or additive. So a.. 4\5 = 5/4 b.. (log4)\(log5) = log5 - log4 Gammier conditions More important, the gammier structure implies the existence of four conditions, the last axioms of the gammier structure, which are a.. regularity b.. contiguity c.. congruity d.. fertility so you refer simply to the harmoid structure if these conditions being unknow, you use only a.. this type of generation derived from the chordoid theory b.. giving a finite set of rational numbers (mod 2) implying implicitely a.. the appropriate restriction of the multiplication as the law b.. and the standard rational ordering all that being necessary to formulate the axioms. Minimal odd generator Any finite set of rational numbers may be considered as a chordic generator of an harmoid. Any line and any column of the matrix g\g, generated by a such chordic generator g, may generate the same harmoid. There exist also an infinity of odd set <a b c ...> generating the same harmoid. It is important to find the minimal odd generator of a given harmoid. The canonical order on the space of harmoids corresponds to the order of their minimal generator. Your definition may permit to find it but also may fail to find that minimal generator. For instance, <1 10/9 5/4 5/3 20/11> is a chordic generator of the gammier number 4 which contains the rast scale. Your definition gives the lcm D = 99 and then the following odd generator <99 110 165 180 495> while the minimal generator is <1 3 5 9 11> I imagine you have already understood the problem linked to the duality in chordoid structures: lines and columns of the chordic matrices being equigenerative. There exist two ways to reduce a set of "odd parts" and we have to compare them to find the minimal odd set. Pierre. [This message contained attachments]
![]()
![]()
![]()
Message: 5655 Date: Sun, 23 Dec 2001 21:28:54 Subject: Re: 55-tET & 1/6-comma meantone From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > Paul, I'm having a hard time understanding the difference between > these two conceptions, but I think I'm beginning to get it. > > The implied ratios on my lattices follow the general trend of > the meantone axis itself, which implies a handful of intervals > which can be stacked to build the entire scale. > > But some of these JI intervals are emphatically *not consonant*, > and are the "wolf intervals" which cause the displacement of > the trend-line of the periodicity-block to align it with the > meantone axis. > > Am I on the right track? No, not really. This doesn't have quite that much to do with the wolf intervals, though it's related to the fact that once 81/80 is tempered out, Gene (if I may speak for him) and I would view the operative lattice as a cylindrical one -- the planar 2-d JI lattice no longer applies to the tuning, musically, psychologically, or spiritually.
![]()
![]()
![]()
Message: 5656 Date: Sun, 23 Dec 2001 03:47:55 Subject: Re: For Pierre, from tuning From: Pierre Lamothe Gene wrote: The scales we get in this way are scale1: [1, 10/9, 6/5, 4/3, 3/2, 5/3, 9/5] scale2: [1, 9/8, 32/27, 4/3, 3/2, 27/16, 16/9] scale3: [1, 16/15, 5/4, 4/3, 3/2, 8/5, 15/8] There is overlap among these; their union is [1, 16/15, 10/9, 9/8, 32/27, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 27/16, 16/9, 9/5, 15/8]; Does this have anything to do with what you are saying? These scales and their union have certainly to do with what I am saying since there is confinement in a region near unison and your scales are true modes in the sense of the gammier theory. Moreover these modes are pretty well chosen to illustrate what I wrote in my recent posts about unison vectors, hexagonal region, S-matrix, etc. About the union I will say only that it misses only 27/20 and 40/27 to obtain the gammier generated by <1 3 5 9 15 27>. But none of the three scales is a suigeneris mode in that gammier, which is the Zarlino gammier <1 3 5 9 15> completed with the odd 27 and whose steps are <16/15 10/9 9/8>. I recall here what is a sui generis mode, that I would call "proper" modeif the term was not already used. On a finite set of intervals modulo 2 with a partial order defined by the partial composition law in the set, all totally ordered maximal subset are modes in that structure. Among the sui generis modes of a structure, those remarkable have a minimal transposition space, what is strongly correlated to the harmonic properties of such modes. I recall also that a gammier is obtained like a diamond with the differences between the elements of an appropriate odd generator which is necessarily non convex. I will add scale4: <1 9/8 5/4 4/3 3/2 5/3 15/8>, the Zarlino scale, for comparison. What are the steps of the scales? scale1: <27/25 10/9 9/8> scale2: <256/243 9/8> scale3: <16/15 9/8 75/64> scale4: <16/15 10/9 9/8> What are the transposition spaces of these scales, in other words, the intervals spaces spanned by the elements of these scales, in other words, the gammiers generated by these scales considered as chordic generators? scale1: gammier <15 25 27 45 75 81 135> scale2: gammier <1 3 9 27 81 243> scale3: gammier <1 3 5 15 45 75 225> scale4: gammier <1 3 5 9 15 27 45> What are the S-matrix associated with your scales? -------------------------------------------------------------------------------- HTML arrays scale1 27/25 10/9 9/8 27/25 1 250/243 25/24 10/9 243/250 1 81/80 9/8 24/25 80/81 1 scale2 256/243 9/8 256/243 1 2187/2048 9/8 2048/2187 1 scale3 16/15 9/8 75/64 16/15 1 135/128 1125/1024 9/8 128/135 1 25/24 75/64 1024/1125 24/25 1 scale4 16/15 10/9 9/8 16/15 1 25/24 135/128 10/9 24/25 1 81/80 9/8 128/135 80/81 1 -------------------------------------------------------------------------------- What are the hexagones associated with these scales? scale1: 81/80-25/24-250/243-80/81-24/25-243/250 scale2: no hexagone scale3: 25/24-1125/1024-135/128-24/25-1024/1125-128/35 scale4: 81/80-135/128-25/24-80/81-128/135-24/25 That corresponds to elements in the matrices in order of a "8" shape. We can look now at lattice representation in <2 3>Z2/ <2>Z for scale2 and <2 3 5>Z3 / <2>Z for the others scales and see that the contents of the segment or the hexagone is precisely the maximal gammier corresponding to the transposition space. (In blue: class 0 - in red: the scale). scale1 V . . . . . . 0 . . . . 2 6 3 0 . . . . . . . 0 4 1 5 2 . . . . . . . . 6 3 0 4 1 . . . . . . . . 5 2 6 3 U . . . . . . . 0 4 1 5 . . . . 0 . . . . . . 0 scale2 0 4 1 5 2 6 3 0 4 1 5 2 6 3 U scale3 . . . . . 0 . . . V 4 1 5 . . . 5 2 6 3 U . 6 3 0 4 1 . 0 4 1 5 2 . . . 2 6 3 0 . . . 0 . . . . . scale4 . . . V . . . . . 0 4 1 5 2 6 3 0 . . 2 6 3 0 4 1 5 . . 0 4 1 5 2 6 3 U . . . . . 0 . . . What are the periodicity blocks associated with these gammiers in segment or hexagone? scale1: <81/80 250/243> + <250/243 24/25> + <24/25 81/80> scale2: <2187/2048> + <2048/2187> scale3: <135/128 25/24> + <25/24 1024/1125> + <1024/1125 135/128> scale4: <81/80 25/24> + <25/24 128/135> + <128/135 81/80> As you may verified, the segment or the hexagone is obtained applying to the first block B a matrix M being a third root of the identity matrix [0 -1] [1 -1] in 3D and in the 2D case, at the reduced [-1] which is the square root of the unity. -------------------------------------------------------------------------------- Gene wrote: I need a definition before I get a statement or a proof — what do you mean by "simplest"? There exist for any "well-structured scale" a minimal gammier in which thatscale corresponds to a sui generis mode. And there exist a canonical order on the gammier space : the order of their minimal generator. What are the minimal generators of the minimal gammiers corresponding to our scales? scale1: <15 25 27 45 135> scale2: <1 3 27 243> scale3: <1 5 45 75> scale4: <1 5 9 15> For comparison, the three simplest generators in <2 3 5>Z3 are <1 3 5 9> <1 3 9 15> <1 5 9 15> Pierre [This message contained attachments]
![]()
![]()
![]()
Message: 5657 Date: Sun, 23 Dec 2001 13:38:40 Subject: Re: For Pierre, from tuning From: monz > From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 3:25 AM > Subject: [tuning-math] Re: For Pierre, from tuning > > > --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote: > > > What are the transposition spaces of these scales, in other words, the intervals spaces > > spanned by the elements of these scales, in other words, the gammiers generated by > > these scales considered as chordic generators? > > scale1: gammier <15 25 27 45 75 81 135> > > scale2: gammier <1 3 9 27 81 243> > > scale3: gammier <1 3 5 15 45 75 225> > > scale4: gammier <1 3 5 9 15 27 45> > > Judging by these examples, I would propose the following definitions: > > (1) We may write any positive rational number r in the form > 2^n p/q, where p and q are odd integers and p/q is reduced > to lowest form. The fraction p/q we call the *odd part* of r. I like this. What's interesting to me is to ponder the difference between thinking of a quantity in this form as opposed to the one I've preferred, which is simply 2^x 3^y 5^z... P^n, where P is the limiting prime-factor and x,y,z,n are integers (or often lately, fractions of integer terms). So is the general consensus that the former [2^n p/q] is best for describing dyads/intervals, and the latter [2^x 3^y 5^z... P^n] is best for describing larger entities such as entire tuning systems? What about those cases falling between, such as tri-, tetr-, pent-, hex-ads etc.? Most of you feel that the 2^n p/q notation is best for these, yes? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
![]()
![]()
![]()
Message: 5659 Date: Sun, 23 Dec 2001 13:41:40 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: monz > From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 1:18 PM > Subject: [tuning-math] Re: coordinates from unison-vectors (was: 55-tET) > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > > > I know that if the sign of the 3-exponent is changed, the sign > > for the 5-exponent must be reversed accordingly. But I find > > sometimes that using, for example, (4 -1) for the syntonic comma > > doesn't always give me the PB I expected, whereas making it > > (-4 1) does. > > As long as it's IN THE SAME FORM when you apply the inverse of the > matrix as well as when you apply the matrix itself, it won't matter -- > if you're centering around (0,0). My code does it automatically, so I guess it works correctly. All the user has to enter are the exponents of 3 and 5 for the two unison-vectors. Everything else is calculated from that. I've posted the Excel spreadsheet to the Files section. Yahoo groups: /tuning-math/files/monz/matrix math.xls * Also, I believe I've noticed that it makes a difference whether the larger or the smaller comma is listed first. Can someone check that and explain it? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
![]()
![]()
![]()
Message: 5660 Date: Sun, 23 Dec 2001 13:47:38 Subject: Re: 55-tET & 1/6-comma meantone From: monz > From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 1:28 PM > Subject: [tuning-math] Re: 55-tET & 1/6-comma meantone > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > > > Am I on the right track? > > No, not really. This doesn't have quite that much to do with the wolf > intervals, though it's related to the fact that once 81/80 is > tempered out, Gene (if I may speak for him) and I would view the > operative lattice as a cylindrical one -- the planar 2-d JI lattice > no longer applies to the tuning, musically, psychologically, or > spiritually. And I've agreed with both of you many times in the past, and wish to emphasize again that the only reason I'm using a planar lattice is because it's beyond my abilities to draw cylindrical ones. In fact, I'd very much appreciate someone posting the mathematics for converting my Excel lattices into cylindrical ones. What would emerge from my meantone/JI-implication lattices if they were to be "cylindrified", is that each flavor of meantone would slice the cylinder diagonally at a different angle, and would also impart a unique diameter to each cylinder. Right? C'mon, guys... I'm itching to draw this stuff now... -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
![]()
![]()
![]()
Message: 5661 Date: Sun, 23 Dec 2001 13:50:16 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: monz > From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 1:22 PM > Subject: [tuning-math] Re: coordinates from unison-vectors (was: 55-tET) > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > > > lattice coordinates x, y : > > > > > > x = ( (q*c) + (p*a) ) / n > > > > y = ( (q*d) + (p*b) ) / n > > There shouldn't be an "/ n" at the end of that. Why not? "n" is the determinant of the matrix, and plays a crucial role in the transformation from one perspective to another. > Maybe that's what's causing the weirdness, because > sometimes n is negative. Hmmm... so perhaps I need to keep "/ abs(n)"? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
![]()
![]()
![]()
Message: 5662 Date: Sun, 23 Dec 2001 14:00:01 Subject: Re: Temperament names From: monz > From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 1:20 PM > Subject: [tuning-math] Re: Temperament names > > > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > > > > > 393216/390625 = 2^17 3 5^-8 > > > > This is Wuerschmidt's comma, so obviously the temperament is the > > Wuerschmidt. Who the heck is Wuerschmidt? > > Do you read German? If so, you should seek out his research. He found > some very interesting stuff, essentially looking into periodicity > blocks before Fokker. This piqued my interest, so I took a look at the Tuning and Temperament Bibliography Tuning & temperament bibliography * and take the liberty of quoting the Joseph Würschmidt listings: "Logarithmische und graphische Darstellung der musikalischen Intervalle", Zeitschrift für Physik vol. 3, 1920, p. 89. "Viertel- und Sechsteltonmusik, eine kritische Studie", Neue Musikzeitung vol. 42, 1921, p. 183. "Über die neunzehnstufige Temperatur", Neue Musikzeitung vol. 42, 1921, p. 215. "Buchstabentonschrift und Von Oettingensches Tongewebe", Zeitschrift für Physik vol. 5, 1922, p. 111. "Die rationellen Tonsysteme in Quinten-Terzengewebe", Zeitschrift für Physik vol. 46, January 1928, p. 527. "Tonleitern, Tonarten und Tonsysteme. Eine historisch-theoretische Untersuchung", Sitzungsberichte der Physikalisch-medizinischen Sozietät zu Erlangen, Band 63, 1931, pp. 133-238. "Die neunzehn-stufige Skala; eine natürliche Erweiterung unseres Tonsystems" (The 19-tone scale; a natural expansion of our tonal system), Neue Musikzeitung vol. 14 no. 4, 1921, pp. 215-216. Which of these have you read, Paul? Can you summarize? I wonder if Tanaka wrote about periodicity-blocks before Würschmidt? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
![]()
![]()
![]()
Message: 5663 Date: Sun, 23 Dec 2001 14:04:33 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: monz > From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 1:41 PM > Subject: Re: [tuning-math] Re: coordinates from unison-vectors (was: 55-tET) > > > I've posted the Excel spreadsheet to the Files section. > Yahoo groups: /tuning-math/files/monz/matrix math.xls * I should have specified: the only worksheet in the file which draws the 5-limit periodicity-blocks is the one named "5-L PBs from UVs". The other worksheets illustrate 3-d examples from Graham's matrix tutorial webpage, and one of them was copied and currently has the unison-vectors for 22-EDO which were in my posts of a few days ago. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
![]()
![]()
![]()
Message: 5664 Date: 23 Dec 2001 19:15:55 -080 Subject: Keenan green Zometool struts From: paul@xxxxxxxxxxxxx.xxx 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. 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? Should I buy the kit? It's between 100 and 200 US$.
![]()
![]()
![]()
Message: 5666 Date: Sun, 23 Dec 2001 22:11:56 Subject: Re: For Pierre, from tuning From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > So is the general consensus that the former [2^n p/q] is > best for describing dyads/intervals, Yes, if octave-equivalence is assumed. > and the latter > [2^x 3^y 5^z... P^n] is best for describing larger entities > such as entire tuning systems? This doesn't make sense, as you'll only be describing a _single ratio_ here, and that single ratio can not correspond to more notes than a dyad. However, I would agree that if one is looking at JI tuning systems, the highest prime number P is one of the most important pieces of information you could want about a system . . . as well as which, if any, primes are not used.
![]()
![]()
![]()
Message: 5668 Date: Sun, 23 Dec 2001 22:18:06 Subject: Re: 55-tET & 1/6-comma meantone From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > And I've agreed with both of you many times in the past, and wish > to emphasize again that the only reason I'm using a planar lattice > is because it's beyond my abilities to draw cylindrical ones. It shouldn't be -- you can simply "ink" the cylinder and then "roll" it a bunch of times over a flat sheet. > In > fact, I'd very much appreciate someone posting the mathematics for > converting my Excel lattices into cylindrical ones. Your Excel lattices, though, are currently referring to JI ratios, including some rather complex ones -- we have to get rid of this feature first. Gene, any clever ideas? > > What would emerge from my meantone/JI-implication lattices if > they were to be "cylindrified", is that each flavor of meantone > would slice the cylinder diagonally at a different angle, and > would also impart a unique diameter to each cylinder. Right? Hmm . . . not really. It seems to me that the angle and the diameter would be fixed, and the _second_ unison vector tells you how _long_ the cylinder is before it meets itself, when bent into a torus representing the ET you're approximating. If there is no second unison vector, than the different flavors of meantone are functionally identical, their only salient difference being the level of beating in the consonant intervals. It *would* be nice, I admit, to see some actual cylindrical arrangements of notes, particularly in a VRML implementation or something. So far, I've simply printed out flat, repeating lattices and then rolled them up by hand.
![]()
![]()
![]()
Message: 5669 Date: Sun, 23 Dec 2001 22:19:39 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > > > From: paulerlich <paul@s...> > > To: <tuning-math@y...> > > Sent: Sunday, December 23, 2001 1:22 PM > > Subject: [tuning-math] Re: coordinates from unison-vectors (was: 55-tET) > > > > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > > > > > lattice coordinates x, y : > > > > > > > > > x = ( (q*c) + (p*a) ) / n > > > > > > y = ( (q*d) + (p*b) ) / n > > > > There shouldn't be an "/ n" at the end of that. > > > Why not? "n" is the determinant of the matrix, and plays > a crucial role in the transformation from one perspective > to another. All you need for the transformation in one direction is the matrix itself; and in the other direction, its inverse. You don't _additionally_ apply the determinant. > > Maybe that's what's causing the weirdness, because > > sometimes n is negative. > > Hmmm... so perhaps I need to keep "/ abs(n)"? Nope.
![]()
![]()
![]()
Message: 5670 Date: Sun, 23 Dec 2001 22:22:11 Subject: Re: Temperament names From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > Which of these have you read, Paul? None, but Mandelbaum touched on his work. A lot of it involves a conception of "rational tone-systems", which are essentially 5-limit periodicity blocks, expressed with only three step sizes, but then equally tempered anyway. W.'s "rational tone systems" included 12-, 19-, 22-, 31-, 34-, 41-, 53-, 65-, and 118-tET, and there weren't many more given the constrains he imposed.
![]()
![]()
![]()
Message: 5671 Date: Sun, 23 Dec 2001 14:26:29 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: monz > From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 2:19 PM > Subject: [tuning-math] Re: coordinates from unison-vectors (was: 55-tET) > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > > > > > From: paulerlich <paul@s...> > > > To: <tuning-math@y...> > > > Sent: Sunday, December 23, 2001 1:22 PM > > > Subject: [tuning-math] Re: coordinates from unison-vectors > > > > > > > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > > > > > > > lattice coordinates x, y : > > > > > > > > > > > > x = ( (q*c) + (p*a) ) / n > > > > > > > > y = ( (q*d) + (p*b) ) / n > > > > > > There shouldn't be an "/ n" at the end of that. > > > > > > Why not? "n" is the determinant of the matrix, and plays > > a crucial role in the transformation from one perspective > > to another. > > All you need for the transformation in one direction is the matrix > itself; and in the other direction, its inverse. You don't > _additionally_ apply the determinant. But when the inverse is described in integer terms, the determinant is part of it! Did you try my spreadsheet yet? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
![]()
![]()
![]()
Message: 5673 Date: Mon, 24 Dec 2001 04:19:44 Subject: Re: For Pierre, from tuning From: Pierre Lamothe Gene wote: > r mod 2 = {2^x p/q | all x in Z} is the corresponding octave class There's a problem with this--number theorists already mean something quite specific by r mod 2, and this isn't it. r mod 2 is 0 if x>0, is 1 if x=0, and is 'infinity' or undefined if x<0. How about <2^n r> for this? I made an inattention error here. Forget r mod 2 = in the line and read simply {2^x p/q | all x in Z} is the corresponding octave class I use normally r mod <2> as equivalent to Ton (r) or Pivot (r) implying only a multiplicative modulo. -------------------------------------------------------------------------------- Gene wrote: So if {a, b, c} is a set of odd integers, the structure it generates are the ratios greater than one between them? The structure has two levels. I don't have time to explain that in details. I will use simply an example. Let {a, b, c} = {1, 3, 5}. The ordered odd generator g is noted <1 5 3> and the corresponding harmoid structure is represented by the chordic matrix g\g .1. 5/4 3/2 8/5 .1. 6/5 4/3 5/3 .1. The ordered contents '(g\g) {1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3} is the class level of the corresponding harmoid structure. I use most often the tones of the first octave to represent the classes. In that case the ratios are greater than one. But there exist equivalent manners to represent the classes: the pivots and also the tones of the centered octave and then the ratios are not all greater than one. {1/5, 1/3, 3/5, 1, 5/3, 3/1, 5/1} {2/3, 4/5, 5/6, 1, 6/5, 5/4, 3/2} The harmoid structure is not only a class structure, it's first an interval structure: {... 3/10, 3/5, 6/5, 12/5, ...} for instance, belong to the structure at interval level. I add here that the three first axioms are respected: that harmoid is a gammoid. -------------------------------------------------------------------------------- > <99 110 165 180 495> Opps! I forgot to reduce. So <99 55 165 45 495>. -------------------------------------------------------------------------------- Gene wrote: However, taking 2 out of the picture gives me <1, 5/9, 5, 5/3, 5/11> which leads to <45,55,99,165,495> and finally to 1,3,5,9,5/3,9/5,11/3,11/5,11/9 which is not at all the same. > while the minimal generator is > <1 3 5 9 11> From this I get 1,3,5,9,5/3,9/5,11/3,11/5,11/9, with an extra 11 in there. Surely an error here. 495/45 = 11/1 Ton(11/1) = 11/8 -------------------------------------------------------------------------------- Gene wrote: > I imagine you have already understood the problem linked to the duality in chordoid > structures: lines and columns of the chordic matrices being equigenerative. I don't understand the above sentence; to follow you, I need clear, mathematical definitions. Ok, forget that for the moment. I thought you would have seen that there exist two ways to reduce a set of rational numbers to a set of odds. You have used only one. Using the precedent example, reduced first to pivots, I hope it will be now clear. These sets are strictly identical: {1, 5/9, 5/1, 5/3, 5/11} = {99/99, 55/99, 495/99, 165/99, 45/99} = {5/5, 5/9, 5/1, 5/3, 5/11} -------------------------------------------------------------------------------- I close here that session. I could only reply very shortly in these days. Merry Christmas! Pierre [This message contained attachments]
![]()
![]()
![]()
Message: 5674 Date: Mon, 24 Dec 2001 13:59:01 Subject: Re: For Pierre, from tuning From: Pierre Lamothe Gene wrote: This is Partch's "Tonality Diamond", but what does the order do for you? Partch called it arbitary, and it seems to me that he got that right. In any case, "tonality diamond" is the recognized name here. Shortly. May I conclude you didn't know that the Zarlino gammier corresponds to a matrix like the "Tonality Diamond"? I repeated often that the chordic matrix is like the "Tonality Diamond", but this matrix is justified by the chordoid theory, which reconstruct the abelian group ending, rather than starting, with the closure axiom. So, without the closure, that structure has well-defined properties like being generated by a chordic matrix. It's not only limited to convex generators of N-limit type. More, none of the N-limit diamond is a gammier, the non-convexity is essential to respect the fertility condition. Many are even not gammoid, like the 11-limit structure, having not the CS property. The order has no importance. The choice of an order is only to permit direct reading, fo instance, of a.. the treillis which exhibit modes, like <15 1 9 5> b.. the chords structure, like <1 5 3 15 9> I will give later the axioms definition with their sense. Now it's holidays. Pierre [This message contained attachments]
4000 4050 4100 4150 4200 4250 4300 4350 4400 4450 4500 4550 4600 4650 4700 4750 4800 4850 4900 4950 5000 5050 5100 5150 5200 5250 5300 5350 5400 5450 5500 5550 5600 5650 5700 5750 5800 5850 5900 5950 6000 6050 6100 6150 6200 6250 6300 6350 6400 6450 6500 6550
5650 - 5675 -