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Message: 1950 - Contents - Hide Contents Date: Mon, 05 Nov 2001 04:30:59 Subject: Re: A non-Tribonacci example From: genewardsmith@xxxx.xxx --- In tuning-math@y..., genewardsmith@j... wrote:> Let's see what can be done with the following recurrence: > > 6 1 2 7 3 9 10 12 19 22 31 41 53 72 94 ... > 9 2 3 11 5 14 16 19 30 35 49 65 84 114 149 ... > 14 2 5 16 7 21 23 28 44 51 72 95 123 167 218 ...Let's revisit this, and add rows for 7 and 11: 6 1 2 7 3 9 10 12 19 22 31 41 ... 9 2 3 11 5 14 16 19 30 35 49 65 ... 14 2 5 16 7 21 23 28 44 51 72 95 ... 16 3 6 19 9 25 28 34 53 62 87 115 ... 21 4 6 25 10 31 35 41 66 76 107 142 ... The rows are no longer linearly independent, and we can find the dependency by inverting the matrix of the first three columns, getting the matrix of step sizes, and multiplying this by the first three elements of our new rows. The matrix is [ 4 -1 -1] [-3 2 0] [10 2 3] Multiplying [16 3 6] by this gives [-5 2 3], and 2^(-5)*3^2*5^2/7 = 225/224, which tells us that this is a kernel element for this system. Similarly, multiplying by [21 4 6] gives us [12 -1 -3], and this divided by 11 is 4096/4125, which is therefore also in the kernel. If we take (4125/4096)/(225/224) = 385/384, we find another kernel element. Calculating the approximations to 7 and 11 shows that 7 is 1/2 cent sharp and 11 2.9 cents flat. We therefore have 3,5,7, and 11 all quite well approximated.
Message: 1951 - Contents - Hide Contents Date: Mon, 05 Nov 2001 05:20:09 Subject: (unknown) From: genewardsmith@xxxx.xxx --- In tuning-math@y..., h_cahyadi@y... wrote:> hi i have problem with understanding math probability > anybody can give suggestions what i should do to ahve better > understandingThe Usenet newsgroups sci.math and sci.stat.math would be better places to ask.
Message: 1952 - Contents - Hide Contents Date: Mon, 05 Nov 2001 06:32:07 Subject: Osmium-Orwell-Secor From: genewardsmith@xxxx.xxx One way of doing something along the lines Dan suggested would be to use the Osmium versions of the orwell and the secor, together with the octave, to express notes. These are pretty far off their usual values, but looking at the 9-note scale I gave we find a lot of 16/15 (secor) and some 75/64 (orwell) relationships, so it might be just the ticket. If we invert the matrix for <2,16/15,75/64> we get [1 0 0] [2 -1 -2] [2 1 1] From this we see we can express notes in the Osmium system by q ~ 2^f1(q) * o^f2(q) * s^f3(q), where f1=v2+2v3+2v5,f2=-v3+v5, f3=-2v3+v5. Here o is the Osmium orwell, (40-11z-14z^2)/241, of 268.1254776 cents, and s is the Osmium secor, (43-11z-3z^2)/241, of 115.3367774 cents. Of course given our 225/224~1 and 385/384~1 we have the 11-limit expressed in these terms. In these terms, we have 3 ~ 2^2 o^(-1) s^(-2) 5 ~ 2^2 o s 7 ~ 2^3 s^(-2) 11 ~ 2^4 o^(-2) s^(-1)
Message: 1953 - Contents - Hide Contents Date: Mon, 05 Nov 2001 07:20:04 Subject: Re: Osmium-Orwell-Secor From: genewardsmith@xxxx.xxx --- In tuning-math@y..., genewardsmith@j... wrote:> Here o is the Osmium orwell, (40-11z-14z^2)/241, of > 268.1254776 cents, and s is the Osmium secor, (43-11z-3z^2)/241, of > 115.3367774 cents.I calculated the 11-limit least squares versions of these, and got the following temperament: o = 267.1445284 cents, s = 116.0783521 cents. The same map to the primes obtains, of course:> 3 ~ 2^2 o^(-1) s^(-2) > 5 ~ 2^2 o s > 7 ~ 2^3 s^(-2) > 11 ~ 2^4 o^(-2) s^(-1)
Message: 1954 - Contents - Hide Contents Date: Mon, 05 Nov 2001 08:33:49 Subject: Re: Blocks for my example From: genewardsmith@xxxx.xxx --- In tuning-math@y..., genewardsmith@j... wrote:> Here are 9 and 10 note Fokker blocks to go with my example: > > 1-16/15-75/64-5/4-4/3-3/2-8/5-128/75-15/8-(2),Here is a picture of the above in orwell-secor coordinates: 4/3 16/15 5/4 12/7 1 7/6 8/5 15/8 3/2 I've adjusted using some 7-limit approximations based on 225/224, which belongs to the system.
Message: 1955 - Contents - Hide Contents Date: Mon, 05 Nov 2001 08:49:23 Subject: Re: Blocks for my example From: genewardsmith@xxxx.xxx --- In tuning-math@y..., genewardsmith@j... wrote: The block example suggests that the equivalent Magic-Miracle coordinates might be a better choice than Orwell-Miracle; if we call the Osmium magic generator m, it is m = os, so that o = m/s. Substituting this gives 3 ~ 2^2 m s^(-2) 5 ~ 2^2 m 7 ~ 2^3 s^(-2) 11 ~ 2^4 m^(-2) s
Message: 1956 - Contents - Hide Contents Date: Tue, 06 Nov 2001 03:20:41 Subject: Semiconvergent stuff From: genewardsmith@xxxx.xxx --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote: > Gene, if there's a simple algorithm that gets us only these > semiconvergents and not the others (simpler, that is, than > calculating all the semiconvergents and then throwing some out), let > us know on tuning-math.Here's another way to do it: the convergents to 2^(7/24) are 5/4,11/9,71/58,224/183... which tells us that before 71/58 we have semiconvergents of the form (11n+5)/(9n+4). We can solve the linear equation for n which finds the value for which this is exactly as far from 2^(7/24) as is 11/9, but on the other side, which gives n=2.926... . This tells us we can toss out the values for n=1 and n=2. If you were going to code this this would be a pretty good system, but I doubt it's any kind of improvement by hand except in the cases where we have a very good convergent, leading to a long line of semiconvergents.
Message: 1959 - Contents - Hide Contents Date: Sat, 10 Nov 2001 02:03:30 Subject: we have 50 members! From: Paul Erlich We're officially a community now :) Many of you have never posted here or on the tuning list. Let this be your opportunity to introduce yourselves, describe your interests, and ask questions (no matter how strange they may seem). Nothing spurs interesting new research like new voices and new ideas! -Paul
Message: 1960 - Contents - Hide Contents Date: Sat, 10 Nov 2001 20:39:52 Subject: Temperament bases and lattice basis reduction From: genewardsmith@xxxx.xxx I've been finding the LLL lattice basis reduction algorithm very useful for finding a basis for a planar temperament, and it probably would be even more useful for higher dimensions. The notation [h19,h22,h27,h31] is a useful one for looking at some of these, since [h19,h22,h27,h31]^(-1) = <1728/1715,126/125,225/224,245/243>. This gives us four planar temperaments, each of which is interesting. The above means, for instance, that the 19,22, and 31 ets each set 225/224 to a unison, whereas h27(225/224)=1, a single step. To get a basis for the 225/224 planar temperament, we therefore eliminate h27. We now may perform lattice basis reduction on the columns of the remaining three vals, getting: [19 22 31] [ 0 1 0] [30 35 49] [-1 1 -1] [44 51 72] ==> [ 1 1 0] [53 62 87] [ 0 -1 -2] Similar calculations with the other commas gives us the following: 1728/1715: basis 6,1/2,12/7 equivalent to 2,3/2,7/6 126/125: basis 6/25,5,5/3 equivalent to 2,5,5/3 or 2,3,5/3 245/243: basis 1/2,9/7,3 equivalent to 2,3/2,9/7 The LLL lattice basis reduction algorithm is in Maple, which may mean it is in Matlab also. We keep the top row, since we want octave equivalence, and by preference the second row, since we like 3. We then choose which of the last two rows will give us a unimodular 3x3 matrix; in this case it is the third row, so 2,3 and 5 are the remaining primes. We then invert: [ 0 1 0]^(-1) [-1 0 1] [-1 1 -1] = [ 1 0 0] [ 1 1 0] [ 2 -1 -1] The three rows of the inverted matrix represent 5/2, 2, and 4/15; a minor adjustment gives us the equivalent basis of 2,5/4, and 16/15, which is the Miracle-Magic basis.
Message: 1961 - Contents - Hide Contents Date: Sat, 10 Nov 2001 21:41:25 Subject: meantone puzzles From: monz Hi folks, I've been immersed in the world of meantone all day today, first because I'm trying to understand what Telemann wrote on microtonality, and secondly because I wrote a short piece on my broken 19-tone guitar, and only realized after it was written that the two lowest strings were out of tune (more than ~60 cents sharp), and I've been going crazy trying to figure out what I played that sounded so good. Anyway, been making lots of calculations on various meantones, and have some puzzling observations that I'd like to understand better. The EDOs which approximate basic common meantones have L/s (Large/small) relationships as follows: EDO degrees meantone EDO L s 1/3 19 3 2 1/4 31 5 3 1/5 43 7 4 1/6 55 9 5 In all cases I name the notes according to a meantone cycle with the "5th" a few cents narrow, and use L to represent the diatonic whole-tone, and s for the diatonic semitone. There are linear progressions in all four of these columns: - the denominator of the fraction of a comma increases by 1, - the number of EDO degrees in the 8ve increases by 12, - the number of EDO degrees in L increases by 2, - and the number of EDO degrees in s increases by 1. I'm sure these relationships have been noticed before, but I'm rather inexperienced with meantones, most of my research having been done on JIs. Can anyone explain this? 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: 1962 - Contents - Hide Contents Date: Sun, 11 Nov 2001 22:31:50 Subject: Harmonic entropy From: genewardsmith@xxxx.xxx Is someone ever going to give a precise definition? One uploaded to the files area of the entropy group would be nice. If you can calculate it, you can define it--if nothing else works, say exactly what you are calculating.
Message: 1963 - Contents - Hide Contents Date: Sun, 11 Nov 2001 08:52:52 Subject: Re: meantone puzzles From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> EDO degrees > meantone EDO L s > > 1/3 19 3 2 > 1/4 31 5 3 > 1/5 43 7 4 > 1/6 55 9 5 > I'm sure these relationships have been noticed before, but > I'm rather inexperienced with meantones, most of my research > having been done on JIs. Can anyone explain this?The 7 et has a fifth 9.47 relative cents flat, and the 12 et is 1.96 rc flat. Your sequence of ets is 7+12n, leading to fifths which are 12(9.47+1.96n)/(7+12n) cents flat. Taking the reciprocal, converting to commas, and expanding as a power series gives 1.32322 + 1.99683 n - 0.412064 n^2 + 0.0850336 n^3 - ... comma temperament, so it is not really linear, but it is linear to a first approximation. The 12 et has L 2 and s 1, and the 7 et has L 1 and s 1. Hence h7 + n h12 has L 2n+1 and s n+1. That's both parts of your question.
Message: 1965 - Contents - Hide Contents Date: Mon, 12 Nov 2001 09:17:38 Subject: Re: meantones From: genewardsmith@xxxx.xxx --- In tuning-math@y..., Robert C Valentine <BVAL@I...> wrote:> So higher meantones are successively > better approximations to 12tet!These ones are, since we add 12 to get the sequence. Other sequences approach other things, of course.
Message: 1966 - Contents - Hide Contents Date: Mon, 12 Nov 2001 11:15:37 Subject: Re: meantones From: manuel.op.de.coul@xxxxxxxxxxx.xxx>You can also say that >L 2n+1 >- = ---- >s n+1 >which, as n goes to infinity, leads to L/s = 2. >So higher meantones are successively >better approximations to 12tet!Only until 1/11-comma MT, after that the fifth becomes larger than 700 cents. Manuel
Message: 1967 - Contents - Hide Contents Date: Mon, 12 Nov 2001 10:47:02 Subject: Re: meantones From: genewardsmith@xxxx.xxx --- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:> Only until 1/11-comma MT, after that the fifth becomes > larger than 700 cents.The best fifth does, not the one from this sequence.
Message: 1968 - Contents - Hide Contents Date: Tue, 13 Nov 2001 00:39:03 Subject: Re: Temperament bases and lattice basis reduction From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> I've been finding the LLL lattice basis reduction algorithm very > useful for finding a basis for a planar temperament, and it probably > would be even more useful for higher dimensions. > > The notation [h19,h22,h27,h31] is a useful one for looking at some of > these, since [h19,h22,h27,h31]^(-1) = > <1728/1715,126/125,225/224,245/243>. This gives us four planar > temperaments, each of which is interesting. > > The above means, for instance, that the 19,22, and 31 ets each set > 225/224 to a unison, whereas h27(225/224)=1, a single step. To get a > basis for the 225/224 planar temperament, we therefore eliminate h27. > We now may perform lattice basis reduction on the columns of the > remaining three vals, getting: > > [19 22 31] [ 0 1 0] > [30 35 49] [-1 1 -1] > [44 51 72] ==> [ 1 1 0] > [53 62 87] [ 0 -1 -2] > > Similar calculations with the other commas gives us the following: > > 1728/1715: basis 6,1/2,12/7 equivalent to 2,3/2,7/6 > > 126/125: basis 6/25,5,5/3 equivalent to 2,5,5/3 or 2,3,5/3 > > 245/243: basis 1/2,9/7,3 equivalent to 2,3/2,9/7Not sure what this is telling me.> > The LLL lattice basis reduction algorithm is in Maple, which may mean > it is in Matlab also.I don't see it: MAERESIZE Change the size of a matrix to be [m n]. mldivide.m: %\ Backslash or left matrix divide. mpower.m: %^ Matrix power. mrdivide.m: %/ Slash or right matrix divide. mtimes.m: %* Matrix multiply. slash.m: %Matrix division. COMPAN Companion matrix. DIAG Diagonal matrices and diagonals of a matrix. EYE Identity matrix. FLIPDIM Flip matrix along specified dimension. FLIPLR Flip matrix in left/right direction. FLIPUD Flip matrix in up/down direction. HADAMARD Hadamard matrix. HANKEL Hankel matrix. HILB Hilbert matrix. INVHILB Inverse Hilbert matrix. ISEMPTY True for empty matrix. PASCAL Pascal matrix. ROT90 Rotate matrix 90 degrees. SIZE Size of matrix. TOEPLITZ Toeplitz matrix. VANDER Vandermonde matrix. WILKINSON Wilkinson's eigenvalue test matrix. EXPM Matrix exponential. EXPM1 Matrix exponential via Pade approximation. EXPM2 Matrix exponential via Taylor series. EXPM3 Matrix exponential via eigenvalues and eigenvectors. FUNM Evaluate general matrix function. INV Matrix inverse. LOGM Matrix logarithm. NORM Matrix or vector norm. NORMEST Estimate the matrix 2-norm. RANK Matrix rank. SQRTM Matrix square root. COV Covariance matrix. POLYVALM Evaluate polynomial with matrix argument. FULL Convert sparse matrix to full matrix. ISSPARSE True for sparse matrix. NNZ Number of nonzero matrix elements. NONZEROS Nonzero matrix elements. NZMAX Amount of storage allocated for nonzero matrix elements. SPALLOC Allocate space for sparse matrix. SPARSE Create sparse matrix. SPCONVERT Import from sparse matrix external format. SPDIAGS Sparse matrix formed from diagonals. SPEYE Sparse identity matrix. SPFUN Apply function to nonzero matrix elements. SPONES Replace nonzero sparse matrix elements with ones. SPPARMS Set parameters for sparse matrix routines. SPRAND Sparse uniformly distributed random matrix. SPRANDN Sparse normally distributed random matrix. SPRANDSYM Sparse random symmetric matrix. UNMESH Convert a list of bedges to a graph or matrix. VIEWMTX View transformation matrix. PLOTMATRIX Scatter plot matrix. MATQUEUE Creates and manipulates a figure-based matrix queue. TEXTWRAP Return wrapped string matrix for given UI Control. MAT2STR Convert matrix to eval'able string. STR2MAT Form blank padded character matrix from strings. STR2NUM Convert string matrix to numeric array. AIRFOIL Display sparse matrix from NASA airfoil. FEM1ODE Stiff problem with a time-dependent mass matrix, M(t)*y' = f (t,y). FEM2ODE Stiff problem with a constant mass matrix, M*y' = f(t,y). MATDEMS For setting up matrix computation demos from the MATLAB DEMO. PLTMAT Display a matrix in a figure window. SPIRAL SPIRAL(n) is an n-by-n matrix with elements ranging CASEWRITE Writes casenames from a string matrix to a file. PCACOV Principal Component Analysis using the covariance matrix. SQUAREFORM Square matrix formatted distance. X2FX Factor settings matrix (x) to design matrix (fx). ATAMULT Example Jacobian-matrix multiply FINDMAX2 Interpolates the maxima in a matrix of data. HMULT Hessian-matrix product normal.m: % NORMAL.M - function to normalize a matrix. standardize.m: % STANDARDIZE.M - function to standardize a matrix. DISP3D Display 3D matrix ELEM3D Element positions of 3-D matrix packed in a 2-D matrix. FIND3D Return position of non-zero elements in 3-D matrix. NDX3D Index into 3-D matrix packed in a 2-D matrix. SIZE3D Size of 2-D matrix to hold 3-D matrix. CAUCHY Cauchy matrix. CHEBSPEC Chebyshev spectral differentiation matrix. CHEBVAND Vandermonde-like matrix for the Chebyshev polynomials. CHOW Chow matrix (singular Toeplitz lower Hessenberg matrix). CIRCUL Circulant matrix. CLEMENT Clement matrix. CONDEX "Counter-examples" to matrix condition number estimators. CYCOL Matrix whose columns repeat cyclically. DORR Dorr matrix. DRAMADAH Matrix of zeros and ones whose inverse has large integer entries. FIEDLER Fiedler matrix. FORSYTHE Forsythe matrix (perturbed Jordan block). FRANK Frank matrix. GEARMAT Gear matrix. GRCAR Grcar matrix. HANOWA Matrix whose eigenvalues lie on a vertical line. HOUSE Householder matrix. INVHESS Inverse of an upper Hessenberg matrix. INVOL Involutory matrix. IPJFACT Hankel matrix with factorial elements. KAHAN Kahan matrix. KMS Kac-Murdock-Szego Toeplitz matrix. KRYLOV Krylov matrix. LAUCHLI Lauchli matrix. LEHMER Lehmer matrix. LESP Tridiagonal matrix with real, sensitive eigenvalues. LOTKIN Lotkin matrix. MINIJ Symmetric positive definite matrix MIN(i,j). MOLER Moler matrix (symmetric positive definite). NEUMANN Singular matrix from the discrete Neumann problem. PARTER Parter matrix (Toeplitz with singular values near pi). PEI Pei matrix. POISSON Block tridiagonal matrix from Poisson's equation. PROLATE Prolate matrix (symmetric, ill-conditioned Toeplitz matrix). RANDHESS Random, orthogonal upper Hessenberg matrix. RANDO Random matrix with elements -1, 0 or 1. RANDSVD Random matrix with pre-assigned singular values. REDHEFF Redheffer matrix. RIEMANN Matrix associated with the Riemann hypothesis. RIS Symmetric Hankel matrix. SMOKE Complex matrix with a "smoke ring" pseudospectrum. TOEPPD Symmetric positive definite Toeplitz matrix. TOEPPEN Pentadiagonal Toeplitz matrix. TRIDIAG Tridiagonal matrix (sparse). TRIW Upper triangular matrix discussed by Wilkinson and others. WATHEN Wathen matrix. ITERAPP Apply matrix operator to vector and error gracefully. FINDP Nonsingular basis permutation matrix.> We keep the top row, since we want octave equivalence, and by > preference the second row, since we like 3. We then choose which of > the last two rows will give us a unimodular 3x3 matrix; in this case > it is the third row, so 2,3 and 5 are the remaining primes. We then > invert: > > [ 0 1 0]^(-1) [-1 0 1] > [-1 1 -1] = [ 1 0 0] > [ 1 1 0] [ 2 -1 -1] > > The three rows of the inverted matrix represent 5/2, 2, and 4/15; a > minor adjustment gives us the equivalent basis of 2,5/4, and 16/15, > which is the Miracle-Magic basis.Wish I followed this post.
Message: 1969 - Contents - Hide Contents Date: Tue, 13 Nov 2001 21:01:39 Subject: Steps from ets using lattice basis reduction From: genewardsmith@xxxx.xxx Another game one can play with lattice basis reduction finds p-limit elements which are represented by a small number of steps in a given et. Ratios of these can then produce kernel elements. For instance, h72 in the 11-limit is represented by [72,114,167,202,249]. From this we obtain the five lattice vectors [72,1,0,0,0,0,0], [114,0,1,0,0,0], [167,0,0,1,0,0], [202,0,0,0,1,0], and [249,0,0,0,0,1]. The first entry is the number of steps in the 72- et for the given element, and the rest are that element in standard prime-power notation--the ones I give above being the primes themselves. We now reduce the above using LLL, and obtain [1,1,2,-3,1,0], meaning 126/125 represented by 1 step of the 72 et, [2,1,0,2,-2,0], meaning 50/49 represented by 2 steps of the 72 et, and so forth for 1 and 99/98; -1 and 242/245; and 2 and 55/54. We then may find (99/98)/(126/125) = 1375/1372, (50/49)/(55/54) = 540/539, (245/242)/(126/125) = 4375/4356, etc as commas for the 72 et.
Message: 1970 - Contents - Hide Contents Date: Tue, 13 Nov 2001 01:12:22 Subject: Re: Harmonic entropy From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> Is someone ever going to give a precise definition? One uploaded to > the files area of the entropy group would be nice. If you can > calculate it, you can define it--if nothing else works, say exactly > what you are calculating.Well, there are variations. But it's always a function (meant to reflect one component of dissonance) of a precise specified input interval or chord. And entropy is always defined as in information theory: sum(p*log(p)) where the sum is over all possible states and p is the probability of each state. For harmonic entropy, each "state" is a just dyad (or n-ad in future versions), i.e., a ratio. The universe of possible ratios is determined by a rule (such as max(p,q)<N or p*q<N, generally any rule such that p(j)*q(i) - p(i)*q(j) = 1 for any pair of adjacent ratios p (i)/q(i), p(j)/q(j), and with N tending toward infinity). The probability of each dyad is determined by determining the corresponding area under a normal curve (whose s.d. is an input parameter) standard, centered around the actual input value. The width of the "slice" corresponding to each dyad is determined assuming that it occupies the full "range" between the adjacent mediants. My Matlab entropy function is of the form output = entropy(cents,s,N) where s is a cents value for the input interval, s is the s.d. of the normal curve, and N is (these days) typically used for the rule p*q<N. Why, because: It was found that the local minima P/Q tend to satisfy P*Q<C no matter what "rule" was chosen. The curve as a whole has an overall "slope" unless the Tenney rule (p*q<N) is chosen. Then it is often found that the entropy for the simpler local minima is proportional to the Tenney Harmonic Distance, log(P*Q). It is also found that using 1/sqrt(p*q) as a proxy for each dyad's "range", and simply multiplying this by the height of the bell curve _exactly_ at p/q, leads to a nearly identical functional appearance, except that there is less sensitivity to tiny changes in N. Open questions: Is there a function, F(x,y), such that F(entropy,s) is invariant to changes in N? For s=1%, F(entropy,1%) = exp(entropy/2.3) seemed to work. Can we explicitly calculate what this function converges to for N->infinity? Or at least prove that is does converge, and calculate thelimit to some computational error? Can we prove that the observations mentioned above (about the local minima and about proxying for the width being OK) are in some sense true? I prepared a full plan for calculating triadic harmonic entropy. See the harmonic_entropy@xxxxxxxxxxx.xxx archives. How can we optimize the calculation?
Message: 1971 - Contents - Hide Contents Date: Tue, 13 Nov 2001 21:38:40 Subject: Re: Steps from ets using lattice basis reduction From: genewardsmith@xxxx.xxx I forgot to carry out the second part of my example; I've just now done it, and find the results a little surprising, but interesting. Since my original lattice was unimodular, the steps I get from the lattice basis reduction are unimodular also, forming a basis for a notation. We have [ 1 2 -3 1 0] [14 11 4 -8 12] [ 1 0 2 -2 0]^(-1) [22 17 7 -13 19] [-1 2 0 -2 1] = [32 25 10 -19 28] [ 1 0 -1 -2 2] [39 30 12 -23 34] [-1 -3 1 0 1] [48 37 15 -28 42] This is a notation, but I was suprised at the somewhat exotic 14, 11 and 8 entries; which, however seem fruitful scale possibilities. I expect it is due in good part to the fact that I went all the way to the 11-limit, and it makes me wonder what would happen if I went totally overboard and looked at 311.
Message: 1972 - Contents - Hide Contents Date: Tue, 13 Nov 2001 03:06:19 Subject: Re: Temperament bases and lattice basis reduction From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:>> The LLL lattice basis reduction algorithm is in Maple, which may > mean>> it is in Matlab also.> I don't see it:In Maple the function is called "lattice" and requires a "readlib(lattice)" before it can be used; however it's clear that Matlab mostly has different names. Since it had a lot of functionality before the Maple addition, that's to be expected. Does it have a help function? Can you put in "lattice", "lattice basis" "lattice basis reduction" or "LLL" and see what happens? Thanks for the entropy post, I'll see if I can finally figure out what you are doing.
Message: 1973 - Contents - Hide Contents Date: Tue, 13 Nov 2001 03:11:24 Subject: Re: Temperament bases and lattice basis reduction From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> however it's clear that > Matlab mostly has different names. Since it had a lot of > functionality before the Maple addition, that's to be expected. Does > it have a help function? Can you put in "lattice", "lattice > basis" "lattice basis reduction" or "LLL" and see what happens? Nothing found.
Message: 1974 - Contents - Hide Contents Date: Tue, 13 Nov 2001 10:21:24 Subject: Re: Temperament bases and lattice basis reduction From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Nothing found.If you are very adventerous, of if by some chance you are on a Linux box, you could try the freeware mathematicians have cooked up--Pari and Lydia both have lattice basis reduction. Alas, they come from a Unix/academic environment, and getting them up and running on a PC is a pain.
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