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Message: 1800 - Contents - Hide Contents Date: Sun, 07 Oct 2001 03:45:47 Subject: Re: More from 4/21/00: does this make sense? From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Gene, I have no idea what your response has to do with whether this:You left out the most relevant portion, which talked about rhombic dodecahedra, the angles between 3/2, 5/4, and 7/4, and interlocking lattices. That was all by Paul Hahn.>> It certainly makes sense to look at the lattice-pair one gets by >> taking the lattice of utonal tetrads together with the lattice of >> otonal tetrads.> What does that mean?The tetrads form the dual honeycomb (or tesselation) to the lattice of 7-limit note classes, but it consists of two interlocking lattices of utonal and otonal tetrads. In the 5-limit, the dual tesselation to the triangular one of note classes is the hexagonal tiling, and also consists of two interlocking triangular lattices, one of major and one of minor triads. This situation generalizes to the p-limit.>> If we look at triples [a,b,c] with a quadradic form >> Q(a,b,c) = a^2+b^2+c^2+ab+ac+bc, we have the symmetric lattice of 7- >> limit note-classes.> Can you give the "for musician dummies" version of this statement?We can define a Euclidean metric on a real vector space in three equivalent ways: (1) A Euclidean metric; in three dimensions this might be d(u, v) = sqrt((u1-v1)^2 + (u2-v2)^2 + (u3-v3)^2) (2) A bilnear form, or dot product, for instance B(u, v) = u.v = u1 v1 + u2 v2 + u3 v3 (3) A quadratic form, such as Q(u) = u1^2 + u2^2 + u3^2. Each of the other two can be defined in terms of one of these, and in particular if Q is a positive definite quadratic form (meaning it is homogenous of the second order in the variables u1, u2, u3 in three dimensions, etc., and Q(u)>=0, with Q(u)=0 iff u=0) then we can define a corresponding bilinear form, or dot product, by B(u,v) = (Q(u+v)-Q(u)-Q(v))/2 We can also define the metric, by d(u,v) = sqrt(Q(u-v)) If you are going to draw the lattice diagrams you do, it could be helpful to realize that the Euclidean geometric structure is defined by the quadratic form. Hence for instance the distance between 4/3 and 15/8 is d([-1 0], [1 1]) = sqrt(Q([-2 -1]) = sqrt(7), and the cosine of the angle between the vector to 3/2 and the vector to 7/5 can be determined using the appropriate dot product, which would be u.v = ((u1+v1)^2 + (u2+v2)^2 + (u3+v3)^2 + (u1+v1)(u2+v2)+(u1+v1) (u3+v3) + (u2+v2)(u3+v3) - u1^2-u2^2-u3^2-u1u2-u1u3-u2u3-v1^2-v2^2- v3^2-v1v2-v1v3-v2v3)/2 = u1v1+u2v2+u3v3+(u1v2+u1v3+u2v1+u2v3+u3v1+u3v2)/2 We then have cos t = u.v/(||u||*||v||) = 0/(1*1) = 0, so that the classes defined by 3/2 and 7/5 are unit vectors at right angles, just as you've been drawing them.
Message: 1801 - Contents - Hide Contents Date: Sun, 07 Oct 2001 11:36:34 Subject: Re: More from 4/21/00: does this make sense? From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> Gene, I have no idea what your response has to do with whether this: >> You left out the most relevant portion, which talked about rhombic > dodecahedra, the angles between 3/2, 5/4, and 7/4, and interlocking > lattices. That was all by Paul Hahn.I think you're misunderstanding the context of that, as well as misunderstanding that the original "does this make sense" was not about any thing Paul Hahn wrote, but only about the other part.>>>> It certainly makes sense to look at the lattice-pair one gets by >>> taking the lattice of utonal tetrads together with the lattice of >>> otonal tetrads. >>> What does that mean? >> The tetrads form the dual honeycomb (or tesselation) to the lattice > of 7-limit note classes, but it consists of two interlocking lattices > of utonal and otonal tetrads. In the 5-limit, the dual tesselation to > the triangular one of note classes is the hexagonal tiling, and also > consists of two interlocking triangular lattices, one of major and > one of minor triads. This situation generalizes to the p-limit. >>>> If we look at triples [a,b,c] with a quadradic form >>> Q(a,b,c) = a^2+b^2+c^2+ab+ac+bc, we have the symmetric lattice of > 7- >>> limit note-classes. >>> Can you give the "for musician dummies" version of this statement? >> We can define a Euclidean metric on a real vector space in three > equivalent ways: > > (1) A Euclidean metric; in three dimensions this might be > > d(u, v) = sqrt((u1-v1)^2 + (u2-v2)^2 + (u3-v3)^2) > > (2) A bilnear form, or dot product, for instance > > B(u, v) = u.v = u1 v1 + u2 v2 + u3 v3 > > (3) A quadratic form, such as Q(u) = u1^2 + u2^2 + u3^2. > > Each of the other two can be defined in terms of one of these, and in > particular if Q is a positive definite quadratic form (meaning it is > homogenous of the second order in the variables u1, u2, u3 in three > dimensions, etc., and Q(u)>=0, with Q(u)=0 iff u=0) then we can > define a corresponding bilinear form, or dot product, by > > B(u,v) = (Q(u+v)-Q(u)-Q(v))/2 > > We can also define the metric, by > > d(u,v) = sqrt(Q(u-v)) > > If you are going to draw the lattice diagrams you do, it could be > helpful to realize that the Euclidean geometric structure is defined > by the quadratic form. Hence for instance the distance between 4/3 and > 15/8 is > > d([-1 0], [1 1]) = sqrt(Q([-2 -1]) = sqrt(7), > > and the cosine of the angle between the vector to 3/2 and the vector > to 7/5 can be determined using the appropriate dot product, which > would be > > u.v = ((u1+v1)^2 + (u2+v2)^2 + (u3+v3)^2 + (u1+v1)(u2+v2)+(u1+v1) > (u3+v3) + (u2+v2)(u3+v3) - u1^2-u2^2-u3^2-u1u2-u1u3-u2u3-v1^2-v2^2- > v3^2-v1v2-v1v3-v2v3)/2 = > u1v1+u2v2+u3v3+(u1v2+u1v3+u2v1+u2v3+u3v1+u3v2)/2 > > We then have > > cos t = u.v/(||u||*||v||) = 0/(1*1) = 0, > > so that the classes defined by 3/2 and 7/5 are unit vectors at right > angles, just as you've been drawing them.I may be coming back to this soon for other purposes, but I don't see the connection any of it has to the truncated octahedron question.
Message: 1802 - Contents - Hide Contents Date: Sun, 07 Oct 2001 12:05:10 Subject: torsion (was: Re: 72 owns the 11-limit) From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> I'd really like to be able to do this. The gcd of theminors . . .> can you explicate exactly what the>> procedure is . . . I have Matlab. >> Matlab actually has a lot of Maple stuff in it now, since they bought > some Maple functionality from the Maple people. Unfortunately, I > don't know Matlab so I don't exactly know how it works. > > The function igcd(n1, n2, ..., nk) in Maple returns the greatest > common divisor of k integers; this may be callable in Matlab, or > Matlab may have its own number theory functions. If you have a matrix > with k-1 rows and k columns, you can produce k different square > matricies by removing one of the columns; these (or the determinants > of these) are called minors. Since Matlab is strong on linear algebra > I assume it can do this, if not Maple can, and again that may be > callable. The proceedure would be to get the k different integers > det(min_j), 1<=j<=k, where min_j is the matrix you get by removing > the jth column, and take igcd(det(min_1), ..., det(min_k)). If this > is not 1, then you have torsion.OK! This time, eliminating all PBs with torsion: The set of 11-limit superparticular unison vectors smaller than 20.7 cents (thus smaller than 17.6 cents) yields the following, taking 4 at a time: freq. determinant 1 1 1 9 1 11 1 29 1 49 1 51 1 60 1 62 1 64 1 65 1 79 1 82 1 91 1 96 1 144 3 20 3 23 3 24 3 37 3 48 3 50 3 68 3 80 3 99 3 126 8 10 8 18 8 54 11 4 13 58 14 45 17 15 17 53 19 7 19 19 20 38 21 12 21 26 24 41 34 14 40 22 40 46 41 27 45 8 46 34 63 72 188 31 So there are three that legitimately have 126 notes, three that legitimately have 24, and one with 144. The set of 11-limit superparticular unison vectors smaller than 35 cents yields the following, taking 4 at a time: 1 30 1 40 1 59 1 63 1 65 1 66 1 76 1 79 1 82 1 86 1 91 1 92 1 96 1 144 2 78 3 13 3 21 3 32 3 47 3 49 3 61 3 62 3 80 3 88 3 90 3 99 3 126 4 1 4 29 4 33 4 42 4 51 4 52 4 60 5 11 5 28 6 68 9 64 10 48 11 6 11 39 11 50 13 36 14 58 15 44 16 37 17 53 18 16 18 23 20 2 20 9 21 54 27 3 27 41 33 45 39 18 40 46 43 5 48 38 56 20 65 72 67 24 68 17 81 26 90 4 92 19 131 15 133 27 137 34 164 7 169 14 176 22 188 8 218 10 251 12 298 31 A lot of legitimate 24's in this list!
Message: 1803 - Contents - Hide Contents
Date: Sun, 07 Oct 2001 14:08:00
Subject: Searching for interesting 7-limit MOS scales
From: Paul Erlich
Let's try chromatic unison vectors between 100 cents and 35 cents,
commatic unison vectors smaller than 35 cents but with numerator and
denominator less than 3000, disallow torsion, and select unison
vectors only from this list:
S2357 * [with cont.] (Wayb.)
The roundup:
freq. determinant
1 23
1 31
2 6
2 13
3 1
3 15
3 16
3 17
4 21
5 2
5 8
5 11
5 19
7 3
7 4
7 14
8 12
11 7
14 5
18 10
Most of the 10-tone scales correspond to the ones that carry my name
in Scala (10-out-of-22); true for the 14-tone ones as well (14-out-of-
26).
How can we try to eliminate "skewed" blocks from this list? Let's be
crude and ignore the difference between a taxicab metric on the Kees
van Prooijen lattice, and a Euclidean metric. So the "length" of a
unison vector is log(numerator), while the volume the three subtend
is simply the number of notes. A non-skewed block would have volume
equal to the product of the three lengths. Define the "straightness"
as the number of notes divided by this would-be volume. Here are the
rankings by straightness:
#1: 14 notes; commas 81:80, 50:49; chroma 25:24 (Erlich 14-of-~26)
#2: 31 notes; commas 1029:1024, 245:243; chroma 25:24 (v. improper!)
#3: 17 notes; commas 245:243, 64:63; chroma 25:24
#4: 19 notes; commas 126:125, 81:80; chroma 49:48
#5: 15 notes; commas 126:125, 64:63; chroma 28:27
#6: 21 notes; commas 1029:1024, 81:80; chroma 25:24 (not blackjack!)
#7: 14 notes; commas 81:80, 50:49; chroma 49:48 (Erlich 14-of-~26)
#8: 12 notes; commas 64:63, 50:49; chroma 36:35 (Erlich 12-of-22)
#9: 19 notes; commas 225:224, 81:80; chroma 49:48 (19-out-of-31???)
#10: 14 notes; commas 245:243, 50:49; chroma 25:24
#11: 12 notes; commas 81:80, 50:49; chroma 36:35
#12: 15 notes; commas 126:125, 64:63; chroma 49:48
#13: 10 notes; commas 64:63, 50:49; chroma 25:24 (Erlich 10-of-22)
#14: 19 notes; commas 225:224, 126:125, chroma 49:48
#15: 10 notes; commas 64:63, 50:49; chroma 28:27 (Erlich 10-of-22)
#16: 19 notes; commas 245:243, 126:125; chroma 49:48
#17: 23 notes; commas 2401:2400, 126:125; chroma 28:27
#18: 14 notes; commas 245:243, 81:80; chroma 25:24
#19: 16 notes; commas 245:243, 225:224; chroma 21:20
#20: 14 notes; commas 245:243, 50:49; chroma 49:48
#21: 12 notes; commas 126:125, 64:63; chroma 36:35
#22: 16 notes; commas 1029:1024, 50:49; chroma 36:35
#23: 19 notes; commas 245:243, 225:224; chroma 49:48
#24: 12 notes; commas 225:224, 50:49; chroma 36:35
#25: 10 notes; commas 64:63, 50:49; chroma 49:48
#26: 12 notes; commas 126:125, 81:80; chroma 36:35
#27: 21 notes; commas 1029:1024, 225:224; chroma 36:35 (Blackjack)
#28: 12 notes; commas 225:224, 64:63; chroma 36:35
#29: 10 notes; commas 225:224, 50:49; chroma 25:24 (Erlich 10-of-22)
#30: 17 notes; commas 2401:2400, 64:63; chroma 36:35
#31: 10 notes; commas 225:224, 50:49; chroma 28:27 (Erlich 10-of-22)
#32: 12 notes; commas 225:224, 81:80; chroma 36:35
#33: 21 notes; commas 2401:2400, 225:224; chroma 36:35 (Blackjack)
The list continues:
Straightness # of notes numerators of commas numeratr'chroma
0.13869 17 2401 81 36
0.13792 10 225 64 25
0.1358 11 245 126 21
0.13323 10 225 64 28
0.1331 16 1029 126 36
0.12844 11 245 126 25
0.12784 12 225 126 36
0.1269 8 126 50 28
0.12127 10 225 50 49
0.11899 7 81 64 25
0.11489 15 1029 126 49
0.11449 10 1029 50 25
0.11407 10 225 64 49
0.11156 8 245 50 28
0.10854 21 1029 2401 36
0.10818 7 126 81 21
0.10769 10 1029 64 25
0.10517 14 2401 81 49
0.10373 8 245 50 36
0.10232 7 126 81 25
0.098844 7 126 81 28
0.098556 10 2401 50 28
0.096602 7 225 81 21
0.096174 11 1029 225 21
0.094692 10 1029 50 49
0.092706 10 2401 64 28
0.091573 6 245 50 21
0.09137 7 225 81 25
0.09111 13 245 2401 28
0.090237 8 245 126 28
0.089861 5 81 64 21
0.088262 7 225 81 28
0.087777 7 225 126 21
0.085705 11 2401 225 21
0.084721 13 245 2401 36
0.08401 14 245 2401 49
0.083909 8 245 126 36
0.083022 7 225 126 25
0.082103 5 81 64 28
0.080199 7 225 126 28
0.079376 10 2401 64 49
0.072627 6 1029 50 21
0.071026 7 2401 64 21
0.070297 5 81 64 49
0.069443 4 126 50 21
0.068396 10 1029 225 49
0.067934 5 245 81 21
0.066921 11 1029 2401 21
0.065681 4 126 50 25
0.065584 5 245 64 28
0.062069 5 245 81 28
0.060985 5 245 64 36
0.060951 10 2401 225 49
0.057716 5 245 81 36
0.05693 5 1029 64 21
0.056154 5 245 64 49
0.054324 4 126 50 49
0.048991 3 126 64 21
0.048367 5 1029 64 36
0.047592 10 1029 2401 49
0.046337 3 126 64 25
0.045775 5 1029 81 36
0.043148 4 2401 50 21
0.042149 5 1029 81 49
0.040377 2 64 50 21
0.038212 2 81 50 21
0.036658 4 2401 50 36
0.034913 2 81 50 28
0.033669 5 1029 245 49
0.03128 3 245 225 25
0.031004 2 225 50 21
0.030216 3 245 225 28
0.029652 4 2401 126 36
0.029164 2 225 64 21
0.028808 3 2401 81 21
0.028097 3 245 225 36
0.027303 4 2401 126 49
0.026321 3 2401 81 28
0.0097913 1 1029 126 21
0.0092609 1 1029 126 25
0.0076707 1 245 2401 21
This shows that in general, the PBs with a lower number of notes
often tend to be more "skewed" and thus less interesting. But there
seem to be lots of interesting new scales that are quite "straight"!
Can someone please provide the generator, interval of repetition,
mapping from generators to primes (3,5,7), and maximum 7-limit error,
for at least the top 32 in the rankings, and preferably many more?
Message: 1805 - Contents - Hide Contents Date: Mon, 08 Oct 2001 06:20:19 Subject: Re: New file uploaded to tuning-math From: genewardsmith@xxxx.xxx If you take the fifteen 7-limit intervals 25/24, 28/27, 36/35, 49/48, 50/49, 64/63, 81/80, 126/125, 245/243, 1029/1024, 1728/1715, 2401/2400, 3136/3125, 4375/4374, 6144/6125 from the list S2357 * [with cont.] (Wayb.) which Paul used, and take the 15 choose 4 = 1365 four-element subsets, you will find that 642 of them give unimodular matricies, and hence give us notations. I uploaded all of these. The first list is a list of the four intervals used, and the second a list of the columns of the inverse matrix. If a number +-n appears on the list, then +-hn, the column vector consisting of n, round(n log_2(3)), round(n log_2(5)), round(n log_2(7)), is meant. If there is just a "v", it means it is not rounded off in this way, and the entire matrix is given in a compact form as a list of lists; the lists are horizontal but represent column vectors. There are vast numbers of blocks and tempered blocks which could be extracted from all of this, so it would be helpful to know what sort of thing people are looking for.
Message: 1806 - Contents - Hide Contents Date: Mon, 8 Oct 2001 11:32:23 Subject: Re: torsion (was: Re: 72 owns the 11-limit) From: manuel.op.de.coul@xxxxxxxxxxx.xxx By the way, taking the Breedsma, kalisma, ragisma and schisma as unison vectors gives a truly big 11-limit PB of 342 tones. Using the xenisma in addition gives the same PB. Manuel
Message: 1807 - Contents - Hide Contents Date: Mon, 08 Oct 2001 11:35:52 Subject: Re: 3rd-best 11-limit temperament From: Graham Breed Paul wrote:> Are you sure? This is the shrutar system, which you once said wasnot consistent with 22-equal.> So maybe Gene was right?The two systems are very different melodically. The diaschismic shruti scale in 46-equal looks like: 3 1 3 1 3 1 3 1 3 1 3 1 3 3 1 3 1 3 1 3 1 3 S r R g G M m P d D n N S' And the 22+46 temperament: 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 S r R g G M m P d D n N S' The former looks more like the canonical shruti scale, and so that's what I'd have expected if you mentioned tuning shrutis to 46-equal. The two are the same if you take only the named 12 notes. I half remember looking at this before. The straight diaschismic mapping is simpler in the 5- and 9-limits. They're both as complex in the 7-limit, and the shrutar is simpler in the 11-limit, but not so far as to be a compelling 11-limit temperament. Graham
Message: 1808 - Contents - Hide Contents Date: Mon, 8 Oct 2001 13:08 +01 Subject: Re: Question From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9pqocr+rc5u@xxxxxxx.xxx> Gene wrote:> --- In tuning-math@y..., graham@m... wrote: >>>>> 25:24, 1029:1024 and 225:224 fail, apparently because it wants > a>>>> half-octave generator, but doesn't give the usual clue. >>> >>> Gene? >>> The first column of the adjoint does have a common factor of 2.> But so do>> all the others. >> I don't know what your usual clue is, but mine is common factors. We > have > > [ a b c d] > det [ -5 2 2 -1] = -20a-32b-46c-56d = -2 h10 > [-10 1 0 3] > [ -3 -1 2 0] > > and therefore torsion.Oh yes, common factors in the left hand column are the definition of torsion. But I meant the clue for the period not being the octave. That's usually common factors in the second column, but not always. You can set 7:5 to be a unison vector with octave equivalent matrices, and the right mapping comes out, but no suggestion that the period is a half-octave. In the example above, all columns have a common factor of 2, so I divided through by it before checking for divisions of the octave. This is because of some inefficient unison vectors I have for the multiple-29 temperament that produce common factors of a power of 29 in all columns. Divide through by those common factors, and you get the right answer. It all depends on whether we take the adjoint matrix, or the inverse multiplied by the lowest common denominator. It now looks like the former, although I assumed the latter was simpler. I've tightened up the program so that it doesn't need this clue about the octave divisions. I also have better unison vectors for the multiple-29 case. I haven't studied the results in detail, but it's mostly working. <#!/usr/local/bin/python * [with cont.] (Wayb.)> <Unison vectors * [with cont.] (Wayb.)> <#!/usr/local/bin/python * [with cont.] (Wayb.)> <[(-1, 2, 0), (-2, 0, -1), (-1, -2, 4)] * [with cont.] (Wayb.)> I'm doing additional calculations without needing the chromatic unison vector. The problem is with octave equivalent matrices. It would be nice to do all the calculations with them. The rules are: 1) Form a matrix with the chromatic unison vector in the top row, and commatic unison vectors in the others. 2) The gcd of the left hand column of the adjoint is the number of equal divisions of the octave. 3) Divide through by this and you have the mapping by generators modulo the period. These work fine as long as the unison vectors are well behaved. It'd be nice if you could come up with a mathematical definition of this. But it looks like they really have to be approximate unisons, rather than approximate equal divisions of the octave (which don't work in the octave-specific case anyway). And it has to be possible to specify all consonant intervals as combinations of (commatic?) unison vectors. Once the mapping's chosen, it should be easy to find the ET for which the chromatic unison vector approximates to 0 steps, although I haven't worked out the algorithm for that yet. So, Paul's hypothesis can be made into a conjecture by saying "the above method always works". I'm fairly confident that it will this time. All we need is to convert it into mathematical language and prove it. Then we can try submitting a paper to Perspectives of New Music. Graham
Message: 1809 - Contents - Hide Contents Date: Mon, 8 Oct 2001 13:28 +01 Subject: Re: Question From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <memo.305965@xxx.xxxxxxxxx.xx.xx> Corrections already> These work fine as long as the unison vectors are well behaved. It'd > be nice if you could come up with a mathematical definition of this. > But it looks like they really have to be approximate unisons, rather > than approximate equal divisions of the octave (which don't work in the > octave-specific case anyway). And it has to be possible to specify all > consonant intervals as combinations of (commatic?) unison vectors.The second criterion isn't right. Firstly, the unison vectors need to be linearly independent. Secondly, taking each column independently, it must be possible to combine the unison vectors to get any number you want in there. Getting any consonant interval isn't possible unless you have a 1 note periodicity block. But this condition should remove torsion. Graham
Message: 1810 - Contents - Hide Contents Date: Mon, 08 Oct 2001 20:29:06 Subject: torsion (was: Re: 72 owns the 11-limit) From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> A lot of legitimate 24's in this list! >> Depending on what you mean by "legitimate". I think it would also be > interesting to see what you get after culling everything which does > not pass my validity test, by taking the product of all four > superparticular ratios, raising the result to the power of the number > of notes in the block (the number found by the absolute value of the > determinant, so that in the above case it would be 24), and removing > everything where the result is greater than 2.What does this test show?
Message: 1811 - Contents - Hide Contents Date: Mon, 08 Oct 2001 20:34:11 Subject: Re: Searching for interesting 7-limit MOS scales From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> Here are the >> rankings by straightness: >> Here are some other measures for the top ten on your list; the first > is a solid angle measure, the area on the unit sphere corresponding > to the three vectors.In a Cartesian lattice with 3, 5, and 7 axes? Also, doesn't this depend in an arbitrary way on the signs of the unison vectors?> The second is my validity condition; this is a > sufficient condition, not a necessary one, but one might well ask how > many of these correctly order the notes in the block--#2, the "very > improper" one, has a validity measure over 5.Can you explain what this validity condition is about?> The last measure is the > most like your measure; it is the volume (which is to say, the > determinant) divided by the product of the lengths of the sides.Lengths measured with Euclidean distance in the Cartesian lattice with 3, 5, and 7 axes?> Since a unit volume is the volume of the parallepiped with sides 3, > 5, and 7,A rectangular prism? Can you flesh this out for me please?> > > #1: 14 notes; commas 81:80, 50:49; chroma 25:24 (Erlich 14-of-~26) > > 1.337003903 2.796199310 1.120897076 > > #2: 31 notes; commas 1029:1024, 245:243; chroma 25:24 (v. improper!) > > 1.208253216 5.315252144 1.203940238 > > #3: 17 notes; commas 245:243, 64:63; chroma 25:24 > > 0.625778711 3.007295043 0.8997354106 > > #4: 19 notes; commas 126:125, 81:80; chroma 49:48 > > 1.036438116 2.179705030 1.149932312 > > #5: 15 notes; commas 126:125, 64:63; chroma 28:27 > > 1.397360786 2.462710473 0.899238708 > > #6: 21 notes; commas 1029:1024, 81:80; chroma 25:24 (not blackjack!) > > 0.907922503 3.388625357 0.9326427427 > > #7: 14 notes; commas 81:80, 50:49; chroma 49:48 (Erlich 14-of-~26) > > 2.405549309 2.107333077 1.120897076 > > #8: 12 notes; commas 64:63, 50:49; chroma 36:35 (Erlich 12-of-22) > > 1.714143895 2.158607408 1.309307341 > > #9: 19 notes; commas 225:224, 81:80; chroma 49:48 (19-out-of-31???) > > 0.420158312 2.038933737 1.014144974 > > #10: 14 notes; commas 245:243, 50:49; chroma 25:24 > > 0.71923786 2.635572871 0.9810960584
Message: 1812 - Contents - Hide Contents Date: Mon, 08 Oct 2001 20:58:41 Subject: Re: 3rd-best 11-limit temperament From: Paul Erlich --- In tuning-math@y..., "Graham Breed" <graham@m...> wrote:> Paul wrote: >>> Are you sure? This is the shrutar system, which you once said was> not consistent with 22-equal.>> So maybe Gene was right? >> The two systems are very different melodically. The diaschismic > shruti scale in 46-equal looks like: > > 3 1 3 1 3 1 3 1 3 1 3 1 3 3 1 3 1 3 1 3 1 3 > S r R g G M m P d D n N S' > > And the 22+46 temperament: > > 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 > S r R g G M m P d D n N S' >I meant the latter.> > The former looks more like the canonical shruti scale, and so that's > what I'd have expected if you mentioned tuning shrutis to 46-equal.Right, but this is a 7- and 11-limit adaptation of the sruti idea, keeping only the named 12 notes in their "canonical" tuning.> The two are the same if you take only the named 12 notes. I half > remember looking at this before. > > The straight diaschismic mapping is simpler in the 5- and 9- limits. > They're both as complex in the 7-limit, and the shrutar is simpler in > the 11-limit, but not so far as to be a compelling 11-limit > temperament.Well, I do get an 11-limit hexad that includes the open strings, which is probably the only one I'd be able to play anyway.
Message: 1813 - Contents - Hide Contents Date: Mon, 08 Oct 2001 21:07:19 Subject: ETs for 7-limit (was: Re: Searching for interesting 7-limit MOS scales) From: Paul Erlich --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Let's try chromatic unison vectors between 100 cents and 35 cents, > commatic unison vectors smaller than 35 cents but with numerator and > denominator less than 3000, disallow torsion, and select unison > vectors only from this list: > > S2357 * [with cont.] (Wayb.)I thought natural to see what ETs come up when we take three commatic unison vectors at a time, still defined as above, rather than the MOSs that come from one chromatic and two commatic. The results (ranked by straightness): Straighness ET numerators of unison vectors 0.24926 46 1029 245 126 0.2458 22 245 64 50 0.24401 27 245 126 64 0.24116 36 1029 245 50 0.21804 26 1029 81 50 0.21029 31 1029 126 81 0.19838 41 1029 245 225 0.18874 22 245 225 50 0.18778 31 1029 225 81 0.1874 31 2401 126 81 0.17754 22 245 225 64 0.17679 41 245 2401 225 0.17246 27 2401 126 64 0.17062 31 1029 225 126 0.16784 12 81 64 50 0.16734 31 2401 225 81 0.16251 19 245 126 81 0.15251 12 126 64 50 0.15205 31 2401 225 126 0.15162 27 245 2401 64 0.14803 14 245 81 50 0.14511 19 245 225 81 0.14433 12 126 81 50 0.13804 41 1029 245 2401 0.13577 12 126 81 64 0.13185 19 245 225 126 0.13066 31 1029 2401 81 0.13038 27 245 2401 126 0.12888 12 225 81 50 0.12192 16 1029 126 50 0.12123 12 225 81 64 0.1195 17 2401 81 64 0.11872 31 1029 2401 126 0.11711 12 225 126 50 0.11016 12 225 126 64 0.10752 15 1029 126 64 0.10463 14 2401 81 50 0.088612 10 1029 64 50 0.083576 14 245 2401 50 0.078966 10 2401 64 50 0.076863 8 245 126 50 0.074401 14 245 2401 81 0.068043 10 1029 225 50 0.064004 10 1029 225 64 0.060636 10 2401 225 50 0.057037 10 2401 225 64 0.049731 5 245 81 64 0.047346 10 1029 2401 50 0.044536 10 1029 2401 64 0.039442 5 1029 81 64 0.031507 5 1029 245 64 0.029818 5 1029 245 81 0.027162 4 2401 126 50 I'm shocked that 36 shows up so close to the top! Discuss.
Message: 1814 - Contents - Hide Contents Date: Mon, 08 Oct 2001 21:55:29 Subject: Re: Searching for interesting 7-limit MOS scales From: Paul Erlich No one calculated the information I requested (generators, mappings from primes to generators, minimax error). Gene, perhaps you can provide me with an algorithm to find the generator, and mapping from primes to generators, given the chromatic unison vector and set of commatic unison vectors? I'm not going to try to understand how it works now -- just want to get some results.
Message: 1815 - Contents - Hide Contents Date: Mon, 08 Oct 2001 22:54:30 Subject: torsion (was: Re: 72 owns the 11-limit) From: Paul Erlich --- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:> > By the way, taking the Breedsma, kalisma, ragisma and schisma > as unison vectors gives a truly big 11-limit PB of 342 tones.342-tET comes up in the list of the simplest ETs to achieve level-3 consistency in various odd limits: 3-limit -> 5-tET 5-limit -> 12-tET 7-limit -> 31-tET 9-limit -> 171-tET 11-limit -> 342-tET 13-limit -> 5585-tET 15-limit -> 5585-tET
Message: 1816 - Contents - Hide Contents Date: Tue, 09 Oct 2001 07:00:24 Subject: ETs for 7-limit (was: Re: Searching for interesting 7-limit MOS scales) From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> I'm shocked that 36 shows up so close to the top! Discuss.I also got a good number (1.2108) for my own straightness measure of 36 divided by the product of the lengths of 1029/1024, 245/243 and 50/49. However, my validity measure, which is the product of those three raised to the 36th power, was not very good: 3.3126. That measure may be more significant.
Message: 1817 - Contents - Hide Contents Date: Tue, 09 Oct 2001 19:56:12 Subject: Re: 7-limit PBs From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> There are "valid" sets with 1 step (see above). So what? This sort > of "validity" doesn't do much for me.The Paul Theorem applies, it just isn't very interesting. However, it's easy enough to cull out everything which does not have a certain minimal number of scale steps--what's the big deal?
Message: 1818 - Contents - Hide Contents Date: Tue, 9 Oct 2001 10:02 +01 Subject: Re: Question From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9ptp2v+3ilo@xxxxxxx.xxx> Gene wrote:>> In the example above, all columns have a common factor of 2, so I > divided>> through by it before checking for divisions of the octave. >> All columns of what have a common factor of 2? I get > > [ 1 0 0 0] [-20 0 0 0] > adj [ -5 2 2 -1] = [-32 -6 -2 -6] > [-10 1 0 3] [-46 -3 -1 -7] > [ -3 -1 2 0] [-56 2 -6 -2]No, not that one.> This has one column in the first matrix and one column in the second > divisible by 2. I don't see how a half-octave period even comes into > it.It has torsion, but not a half-octave period. I meant this one: [(-3, 0, 1), (0, 2, -2), (1, 0, 3)] Octave specifically, [ 1 0 0 0] [ 2 -3 0 1] [ 1 0 2 -2] [-10 1 0 3] adjoint [-20 0 0 0] [-32 6 0 -2] [-46 -2 -10 -6] [-56 -2 0 -6]>> 1) Form a matrix with the chromatic unison vector in the top row, > and>> commatic unison vectors in the others. > >> 2) The gcd of the left hand column of the adjoint is the number of > equal>> divisions of the octave. >> You've lost me completely here. Left hand column of the adjoint of > what matrix is supposed to do this? It can hardly be the matrix of > unison vectors, with or without 2 included in the picture. Do you > maybe mean you divide through by the gcd to get the number of > divisions?Without 2 (this method is all for octave equivalent matrices). It's the same as the gcd you take here: """ (4) 14/12 = 7/6 = 1+1/6, and the convergent to 7/6 is 1. We therefore want A + B = 93.651 cents as our generator; this happens to be very close to 1200/13 = 92.308 cents; not much of a surprise since g+h12=h26 and gcd(12,14)=2. We therefore can use 1200/13 as our generator, with a period of half an octave. """ GCD of 2, half octave period.> I don't see why you don't simply do what I do, and go for the whole > val, not just h(2). The number of divisions can be deceptive.Why don't I do what? I'm using octave equivalent matrices for the conjecture because one step using octave specific matrices involves solving what I think could be called a (very simple) system of Diophantine equations with only 1 unknown. Proving that that always works sounds harder than proving the octave-specific case. In fact, the number of notes in the periodicity block is the generator mapping of the chromatic unison vector. From the definition of the adjoint matrix, this will always be the same as the determinant. So we can ignore the determinant if we really want to. But the result we get is : The number of periods to the octave The mapping of generators to prime intervals The number of notes in the periodicity block Hopefully, we've defined away torsion. It'd be nice if we could get the number of steps to a generator in the equivalent ET. Did we have an algorithm for generating the periodicity block in order of pitch for octave equivalent vectors?>> These work fine as long as the unison vectors are well behaved. > It'd be>> nice if you could come up with a mathematical definition of this. >> I thought I did.Did you? Where? Graham
Message: 1819 - Contents - Hide Contents Date: Tue, 09 Oct 2001 20:01:37 Subject: Re: 7-limit PBs From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> There are "valid" sets with 1 step (see above). So what? This sort >> of "validity" doesn't do much for me. >> The Paul Theorem applies, it just isn't very interesting. However, > it's easy enough to cull out everything which does not have a certain > minimal number of scale steps--what's the big deal?It's not about the number of scale steps -- it's about the "skewness" of the block. Ultimately, this translates into greater errors with respect to the JI intervals -- which is what I really care about.
Message: 1820 - Contents - Hide Contents Date: Tue, 9 Oct 2001 10:02 +01 Subject: Re: Searching for interesting 7-limit MOS scales From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9pt7ch+ucrd@xxxxxxx.xxx> Paul wrote:> No one calculated the information I requested (generators, mappings > from primes to generators, minimax error). Gene, perhaps you can > provide me with an algorithm to find the generator, and mapping from > primes to generators, given the chromatic unison vector and set of > commatic unison vectors? I'm not going to try to understand how it > works now -- just want to get some results.If you look at <#!/usr/local/bin/python * [with cont.] (Wayb.)>, all you need to do is alter it to use different unison vectors (chromatic on top, all larger than a unison). If downloading ActivePython's too much trouble for you, I can run it at home sometime. Graham
Message: 1821 - Contents - Hide Contents Date: Tue, 09 Oct 2001 20:26:47 Subject: Re: 7-limit PBs From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> There _is_ a decent MOS for 7-limit with 11 steps: > > 11 note chain-of-minor-thirds scale * [with cont.] (Wayb.) > > But clearly not all the numbers in your list are to be taken > seriously.Aside from the ones with a low number of notes or a bad validity, which ones? It might be interesting to check on these 11 and 13 note PBs, and see if they make sense. 11 notes: [1.550683960, [3136/3125, 245/243, 36/35], 18*b+11*a+26*c+32*d] [1.696997541, [3136/3125, 245/243, 28/27], 18*b+11*a+26*c+32*d] [1.714645543, [4375/4374, 126/125, 25/24], 30*d+25*c+17*b+11*a] [1.718962570, [4375/4374, 245/243, 25/24], 30*d+25*c+17*b+11*a] [1.871717101, [126/125, 245/243, 25/24], 30*d+25*c+17*b+11*a] [2.113987015, [3136/3125, 36/35, 28/27], 18*b+11*a+26*c+32*d] [2.253538886, [1029/1024, 36/35, 25/24], 17*b+11*a+25*c+31*d] We are getting 11*a+17*b+25*c+30*d, 11*a+17*b+25*c+31*d, and 11*a+18*b+26*c+32*d. 13 notes: [1.509685048, [3136/3125, 1728/1715, 49/48], 20*b+13*a+30*c+36*d] [1.613159878, [2401/2400, 245/243, 36/35], 21*b+13*a+31*c+37*d] [1.665410631, [3136/3125, 1728/1715, 36/35], 20*b+13*a+30*c+36*d] [1.794547474, [2401/2400, 245/243, 28/27], 21*b+13*a+31*c+37*d] [1.842667322, [6144/6125, 1728/1715, 28/27], 37*d+30*c+21*b+13*a] [1.952519731, [6144/6125, 1728/1715, 25/24], 37*d+30*c+21*b+13*a] [1.973782564, [3136/3125, 49/48, 36/35], 20*b+13*a+30*c+36*d] [2.183860583, [3136/3125, 64/63, 25/24], 36*d+13*a+21*b+30*c] [2.326619401, [2401/2400, 36/35, 28/27], 21*b+13*a+31*c+37*d] [2.839788039, [6144/6125, 28/27, 25/24], 37*d+30*c+21*b+13*a] [3.009084772, [1728/1715, 28/27, 25/24], 37*d+30*c+21*b+13*a] With 13*a+20*b+30*c+36*d, 13*a+21*b+30*c+37*d, and 13*a+21*b+31*c+37*d.
Message: 1822 - Contents - Hide Contents Date: Tue, 9 Oct 2001 10:02 +01 Subject: Re: Searching for interesting 7-limit MOS scales From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9ptkum+fe30@xxxxxxx.xxx> Paul wrote:>> #19: 16 notes; commas 245:243, 225:224; chroma 21:20 >> 41-tET, 22-tET, 19-tET . . . is this Graham's MAGIC thing?Yes, that's Magic. Graham
Message: 1823 - Contents - Hide Contents Date: Tue, 09 Oct 2001 10:25:23 Subject: 7-limit PBs From: genewardsmith@xxxx.xxx I took the same 15 7-limit intervals as before, and this time took sets of three. After eliminating the ones with linear dependence or torsion, I obtained the following, which lists the validity measure, the intervals, and the dual val: [1.448457604, [4375/4374, 2401/2400, 6144/6125], 99*a+157*b+230*c+278*d] [1.509442953, [4375/4374, 2401/2400, 3136/3125], 99*a+157*b+230*c+278*d] [1.487593558, [4375/4374, 2401/2400, 1029/1024], 114*b+72*a+167*c+202*d] [1.247713548, [4375/4374, 2401/2400, 1728/1715], 43*b+27*a+63*c+76*d] [1.261826622, [4375/4374, 2401/2400, 126/125], 43*b+27*a+63*c+76*d] [1.269638870, [4375/4374, 2401/2400, 245/243], 43*b+27*a+63*c+76*d] [1.800467556, [4375/4374, 2401/2400, 81/80], 71*b+45*a+104*c+126*d] [1.455372564, [4375/4374, 2401/2400, 50/49], 28*b+18*a+41*c+50*d] [1.466326603, [4375/4374, 2401/2400, 49/48], 28*b+18*a+41*c+50*d] [1.296080214, [4375/4374, 2401/2400, 36/35], 22*c+26*d+15*b+9*a] [3.790362186, [4375/4374, 2401/2400, 28/27], 85*c+102*d+58*b+36*a] [1.968212894, [4375/4374, 6144/6125, 3136/3125], 99*a+157*b+230*c+278*d] [1.457979089, [4375/4374, 6144/6125, 1029/1024], 73*b+46*a+129*d+107*c] [1.779804660, [4375/4374, 6144/6125, 1728/1715], 84*b+53*a+123*c+149*d] [1.681227707, [4375/4374, 6144/6125, 126/125], 73*b+46*a+129*d+107*c] [1.698999914, [4375/4374, 6144/6125, 245/243], 73*b+46*a+129*d+107*c] [1.116544333, [4375/4374, 6144/6125, 81/80], 11*b+7*a+16*c+20*d] [1.142843389, [4375/4374, 6144/6125, 64/63], 11*b+7*a+16*c+20*d] [4.102976382, [4375/4374, 6144/6125, 50/49], 169*d+139*c+95*b+60*a] [2.544310889, [4375/4374, 6144/6125, 49/48], 62*b+39*a+91*c+109*d] [3.561534162, [4375/4374, 6144/6125, 28/27], 89*d+51*b+32*a+75*c] [2.763160012, [4375/4374, 3136/3125, 1029/1024], 331*d+187*b+118*a+274*c] [2.468277477, [4375/4374, 3136/3125, 1728/1715], 127*b+80*a+186*c+225*d] [1.249197248, [4375/4374, 3136/3125, 126/125], 19*a+30*b+44*c+53*d] [1.254634753, [4375/4374, 3136/3125, 245/243], 19*a+30*b+44*c+53*d] [3.283589564, [4375/4374, 3136/3125, 64/63], 142*c+97*b+61*a+172*d] [2.733807197, [4375/4374, 3136/3125, 50/49], 42*a+67*b+98*c+119*d] [2.083409549, [4375/4374, 3136/3125, 36/35], 54*c+37*b+23*a+66*d] [1.174027616, [4375/4374, 3136/3125, 28/27], 10*c+13*d+4*a+7*b] [1.951242152, [4375/4374, 3136/3125, 25/24], 23*b+15*a+34*c+40*d] [1.389479277, [4375/4374, 1029/1024, 1728/1715], 60*c+73*d+41*b+26*a] [1.824149922, [4375/4374, 1029/1024, 126/125], 73*b+46*a+129*d+107*c] [1.843432955, [4375/4374, 1029/1024, 245/243], 73*b+46*a+129*d+107*c] [1.577077172, [4375/4374, 1029/1024, 81/80], 60*c+73*d+41*b+26*a] [1.517338321, [4375/4374, 1029/1024, 64/63], 47*c+56*d+32*b+20*a] [1.672594979, [4375/4374, 1029/1024, 49/48], 47*c+56*d+32*b+20*a] [1.220941678, [4375/4374, 1029/1024, 36/35], 9*b+6*a+13*c+17*d] [4.347012006, [4375/4374, 1029/1024, 25/24], 90*d+50*b+32*a+73*c] [1.529904994, [4375/4374, 1728/1715, 126/125], 43*b+27*a+63*c+76*d] [1.539376974, [4375/4374, 1728/1715, 245/243], 43*b+27*a+63*c+76*d] [1.690915994, [4375/4374, 1728/1715, 81/80], 60*c+73*d+41*b+26*a] [1.887533019, [4375/4374, 1728/1715, 64/63], 43*b+27*a+63*c+76*d] [2.070013772, [4375/4374, 1728/1715, 50/49], 60*c+73*d+41*b+26*a] [1.028806584, [4375/4374, 1728/1715, 49/48], a+2*b+3*c+3*d] [1.036605118, [4375/4374, 1728/1715, 36/35], a+2*b+3*c+3*d] [3.442298035, [4375/4374, 1728/1715, 28/27], 45*b+28*a+66*c+79*d] [3.370477905, [4375/4374, 1728/1715, 25/24], 57*c+70*d+39*b+25*a] [1.479592122, [4375/4374, 126/125, 81/80], 19*a+30*b+44*c+53*d] [1.908883188, [4375/4374, 126/125, 64/63], 43*b+27*a+63*c+76*d] [1.255075583, [4375/4374, 126/125, 50/49], 13*b+8*a+19*c+23*d] [1.728934000, [4375/4374, 126/125, 49/48], 19*a+30*b+44*c+53*d] [1.337685863, [4375/4374, 126/125, 36/35], 13*b+8*a+19*c+23*d] [1.428343371, [4375/4374, 126/125, 28/27], 13*b+8*a+19*c+23*d] [1.714645543, [4375/4374, 126/125, 25/24], 30*d+25*c+17*b+11*a] [1.486032490, [4375/4374, 245/243, 81/80], 19*a+30*b+44*c+53*d] [1.920701505, [4375/4374, 245/243, 64/63], 43*b+27*a+63*c+76*d] [1.257372941, [4375/4374, 245/243, 50/49], 13*b+8*a+19*c+23*d] [1.736459703, [4375/4374, 245/243, 49/48], 19*a+30*b+44*c+53*d] [1.340134435, [4375/4374, 245/243, 36/35], 13*b+8*a+19*c+23*d] [1.430957888, [4375/4374, 245/243, 28/27], 13*b+8*a+19*c+23*d] [1.718962570, [4375/4374, 245/243, 25/24], 30*d+25*c+17*b+11*a] [1.219933467, [4375/4374, 81/80, 64/63], 11*b+7*a+16*c+20*d] [2.349492733, [4375/4374, 81/80, 50/49], 60*c+73*d+41*b+26*a] [1.881628622, [4375/4374, 81/80, 49/48], 19*a+30*b+44*c+53*d] [1.330764996, [4375/4374, 81/80, 36/35], 11*b+7*a+16*c+20*d] [1.800789429, [4375/4374, 81/80, 28/27], 28*c+33*d+19*b+12*a] [1.453990411, [4375/4374, 81/80, 25/24], 11*b+7*a+16*c+20*d] [3.421600769, [4375/4374, 64/63, 50/49], 79*c+96*d+54*b+34*a] [2.079078641, [4375/4374, 64/63, 49/48], 47*c+56*d+32*b+20*a] [1.362109799, [4375/4374, 64/63, 36/35], 11*b+7*a+16*c+20*d] [2.848783252, [4375/4374, 64/63, 28/27], 47*c+56*d+32*b+20*a] [2.093653566, [4375/4374, 50/49, 49/48], 28*b+18*a+41*c+50*d] [1.475235230, [4375/4374, 50/49, 36/35], 13*b+8*a+19*c+23*d] [3.011864701, [4375/4374, 50/49, 25/24], 28*b+18*a+41*c+50*d] [1.050240055, [4375/4374, 49/48, 36/35], a+2*b+3*c+3*d] [3.140275637, [4375/4374, 49/48, 28/27], 47*c+56*d+32*b+20*a] [3.034533867, [4375/4374, 49/48, 25/24], 28*b+18*a+41*c+50*d] [1.678896865, [4375/4374, 36/35, 28/27], 13*b+8*a+19*c+23*d] [1.623444754, [4375/4374, 36/35, 25/24], 11*b+7*a+16*c+20*d] [1.362979272, [4375/4374, 28/27, 25/24], 6*b+4*a+9*c+10*d] [1.296835720, [2401/2400, 6144/6125, 1029/1024], 49*b+31*a+87*d+72*c] [1.409207770, [2401/2400, 6144/6125, 1728/1715], 49*b+31*a+87*d+72*c] [1.427524260, [2401/2400, 6144/6125, 126/125], 49*b+31*a+87*d+72*c] [2.217365728, [2401/2400, 6144/6125, 245/243], 158*c+191*d+108*b+68*a] [2.039491520, [2401/2400, 6144/6125, 64/63], 104*d+59*b+37*a+86*c] [1.152921505, [2401/2400, 6144/6125, 50/49], 17*d+14*c+10*b+6*a] [1.155806812, [2401/2400, 6144/6125, 49/48], 17*d+14*c+10*b+6*a] [2.208066855, [2401/2400, 6144/6125, 36/35], 39*b+25*a+58*c+70*d] [5.556136148, [2401/2400, 6144/6125, 28/27], 69*b+43*a+100*c+121*d] [1.313691630, [2401/2400, 3136/3125, 1029/1024], 49*b+31*a+87*d+72*c] [1.427524260, [2401/2400, 3136/3125, 1728/1715], 49*b+31*a+87*d+72*c] [1.446078824, [2401/2400, 3136/3125, 126/125], 49*b+31*a+87*d+72*c] [2.281076121, [2401/2400, 3136/3125, 245/243], 158*c+191*d+108*b+68*a] [2.071170648, [2401/2400, 3136/3125, 64/63], 104*d+59*b+37*a+86*c] [1.155806812, [2401/2400, 3136/3125, 50/49], 17*d+14*c+10*b+6*a] [1.158699341, [2401/2400, 3136/3125, 49/48], 17*d+14*c+10*b+6*a] [2.231182923, [2401/2400, 3136/3125, 36/35], 39*b+25*a+58*c+70*d] [5.656559608, [2401/2400, 3136/3125, 28/27], 69*b+43*a+100*c+121*d] [1.488862040, [2401/2400, 1029/1024, 1728/1715], 49*b+31*a+87*d+72*c] [1.508213854, [2401/2400, 1029/1024, 126/125], 49*b+31*a+87*d+72*c] [1.738209169, [2401/2400, 1029/1024, 245/243], 65*b+41*a+115*d+95*c] [1.731539563, [2401/2400, 1029/1024, 81/80], 49*b+31*a+87*d+72*c] [1.234120624, [2401/2400, 1029/1024, 64/63], 16*b+10*a+23*c+28*d] [1.290335387, [2401/2400, 1029/1024, 50/49], 16*b+10*a+23*c+28*d] [1.295721876, [2401/2400, 1029/1024, 49/48], 16*b+10*a+23*c+28*d] [2.019048032, [2401/2400, 1029/1024, 36/35], 33*b+21*a+49*c+59*d] [1.547209978, [2401/2400, 1728/1715, 245/243], 43*b+27*a+63*c+76*d] [1.881579115, [2401/2400, 1728/1715, 81/80], 49*b+31*a+87*d+72*c] [1.897137590, [2401/2400, 1728/1715, 64/63], 43*b+27*a+63*c+76*d] [1.119277634, [2401/2400, 1728/1715, 50/49], 9*c+6*b+4*a+11*d] [1.121144263, [2401/2400, 1728/1715, 49/48], 9*c+6*b+4*a+11*d] [1.155526618, [2401/2400, 1728/1715, 36/35], 9*c+6*b+4*a+11*d] [2.772428686, [2401/2400, 1728/1715, 28/27], 37*b+23*a+65*d+54*c] [1.564710702, [2401/2400, 126/125, 245/243], 43*b+27*a+63*c+76*d] [1.906035357, [2401/2400, 126/125, 81/80], 49*b+31*a+87*d+72*c] [1.918596397, [2401/2400, 126/125, 64/63], 43*b+27*a+63*c+76*d] [1.121144263, [2401/2400, 126/125, 50/49], 9*c+6*b+4*a+11*d] [1.123014005, [2401/2400, 126/125, 49/48], 9*c+6*b+4*a+11*d] [1.157453700, [2401/2400, 126/125, 36/35], 9*c+6*b+4*a+11*d] [2.799119925, [2401/2400, 126/125, 28/27], 37*b+23*a+65*d+54*c] [1.342457835, [2401/2400, 245/243, 81/80], 22*b+14*a+32*c+39*d] [1.930474851, [2401/2400, 245/243, 64/63], 43*b+27*a+63*c+76*d] [1.496942601, [2401/2400, 245/243, 50/49], 22*b+14*a+32*c+39*d] [1.505698455, [2401/2400, 245/243, 49/48], 22*b+14*a+32*c+39*d] [1.613159878, [2401/2400, 245/243, 36/35], 21*b+13*a+31*c+37*d] [1.794547474, [2401/2400, 245/243, 28/27], 21*b+13*a+31*c+37*d] [1.625778481, [2401/2400, 81/80, 64/63], 27*b+17*a+40*c+48*d] [1.588174591, [2401/2400, 81/80, 50/49], 22*b+14*a+32*c+39*d] [1.597464076, [2401/2400, 81/80, 49/48], 22*b+14*a+32*c+39*d] [2.008061053, [2401/2400, 81/80, 36/35], 27*b+17*a+40*c+48*d] [1.159072634, [2401/2400, 81/80, 28/27], 8*c+9*d+5*b+3*a] [1.438608665, [2401/2400, 64/63, 50/49], 16*b+10*a+23*c+28*d] [1.444614119, [2401/2400, 64/63, 49/48], 16*b+10*a+23*c+28*d] [2.124866198, [2401/2400, 64/63, 36/35], 27*b+17*a+40*c+48*d] [1.691009707, [2401/2400, 64/63, 28/27], 16*b+10*a+23*c+28*d] [1.215506250, [2401/2400, 50/49, 36/35], 9*c+6*b+4*a+11*d] [1.768035978, [2401/2400, 50/49, 28/27], 16*b+10*a+23*c+28*d] [1.217533360, [2401/2400, 49/48, 36/35], 9*c+6*b+4*a+11*d] [1.775416623, [2401/2400, 49/48, 28/27], 16*b+10*a+23*c+28*d] [2.326619401, [2401/2400, 36/35, 28/27], 21*b+13*a+31*c+37*d] [1.427524260, [6144/6125, 3136/3125, 1029/1024], 49*b+31*a+87*d+72*c] [1.551220597, [6144/6125, 3136/3125, 1728/1715], 49*b+31*a+87*d+72*c] [1.571382931, [6144/6125, 3136/3125, 126/125], 49*b+31*a+87*d+72*c] [2.737190864, [6144/6125, 3136/3125, 245/243], 158*c+191*d+108*b+68*a] [2.287131427, [6144/6125, 3136/3125, 64/63], 104*d+59*b+37*a+86*c] [1.174547064, [6144/6125, 3136/3125, 50/49], 17*d+14*c+10*b+6*a] [1.177486492, [6144/6125, 3136/3125, 49/48], 17*d+14*c+10*b+6*a] [2.385833149, [6144/6125, 3136/3125, 36/35], 39*b+25*a+58*c+70*d] [6.347647197, [6144/6125, 3136/3125, 28/27], 69*b+43*a+100*c+121*d] [1.617873353, [6144/6125, 1029/1024, 1728/1715], 49*b+31*a+87*d+72*c] [2.103467335, [6144/6125, 1029/1024, 245/243], 73*b+46*a+129*d+107*c] [1.881579115, [6144/6125, 1029/1024, 81/80], 49*b+31*a+87*d+72*c] [1.427247693, [6144/6125, 1029/1024, 64/63], 15*a+24*b+42*d+35*c] [1.569463799, [6144/6125, 1029/1024, 50/49], 25*b+16*a+37*c+45*d] [1.535432120, [6144/6125, 1029/1024, 49/48], 15*a+24*b+42*d+35*c] [1.782870571, [6144/6125, 1029/1024, 36/35], 25*b+16*a+37*c+45*d] [1.050000000, [6144/6125, 1029/1024, 25/24], b+a+3*d+2*c] [1.780914442, [6144/6125, 1728/1715, 126/125], 49*b+31*a+87*d+72*c] [1.513769837, [6144/6125, 1728/1715, 245/243], 35*b+22*a+51*c+62*d] [2.044619737, [6144/6125, 1728/1715, 81/80], 49*b+31*a+87*d+72*c] [1.787357184, [6144/6125, 1728/1715, 64/63], 35*b+22*a+51*c+62*d] [1.971379996, [6144/6125, 1728/1715, 50/49], 35*b+22*a+51*c+62*d] [1.325000779, [6144/6125, 1728/1715, 49/48], 25*d+21*c+9*a+14*b] [1.418184249, [6144/6125, 1728/1715, 36/35], 25*d+21*c+9*a+14*b] [1.842667322, [6144/6125, 1728/1715, 28/27], 37*d+30*c+21*b+13*a] [1.952519731, [6144/6125, 1728/1715, 25/24], 37*d+30*c+21*b+13*a] [2.425554379, [6144/6125, 126/125, 245/243], 73*b+46*a+129*d+107*c] [2.071195136, [6144/6125, 126/125, 81/80], 49*b+31*a+87*d+72*c] [1.495120029, [6144/6125, 126/125, 64/63], 15*a+24*b+42*d+35*c] [1.649199242, [6144/6125, 126/125, 50/49], 25*b+16*a+37*c+45*d] [1.608449134, [6144/6125, 126/125, 49/48], 15*a+24*b+42*d+35*c] [1.873447987, [6144/6125, 126/125, 36/35], 25*b+16*a+37*c+45*d] [1.053257143, [6144/6125, 126/125, 25/24], b+a+3*d+2*c] [1.766847065, [6144/6125, 245/243, 81/80], 56*c+38*b+24*a+67*d] [1.812907661, [6144/6125, 245/243, 64/63], 35*b+22*a+51*c+62*d] [2.150970319, [6144/6125, 245/243, 49/48], 56*c+38*b+24*a+67*d] [1.082128318, [6144/6125, 245/243, 36/35], 3*b+2*a+5*c+5*d] [1.100014434, [6144/6125, 245/243, 28/27], 3*b+2*a+5*c+5*d] [2.835788482, [6144/6125, 245/243, 25/24], 46*c+57*d+32*b+20*a] [1.244677941, [6144/6125, 81/80, 64/63], 11*b+7*a+16*c+20*d] [3.886285184, [6144/6125, 81/80, 50/49], 107*d+88*c+60*b+38*a] [2.380563249, [6144/6125, 81/80, 49/48], 56*c+38*b+24*a+67*d] [1.357757518, [6144/6125, 81/80, 36/35], 11*b+7*a+16*c+20*d] [2.415933756, [6144/6125, 81/80, 28/27], 47*d+40*c+27*b+17*a] [2.360946697, [6144/6125, 64/63, 50/49], 35*b+22*a+51*c+62*d] [1.807551657, [6144/6125, 64/63, 49/48], 15*a+24*b+42*d+35*c] [1.389738102, [6144/6125, 64/63, 36/35], 11*b+7*a+16*c+20*d] [2.289191294, [6144/6125, 64/63, 28/27], 15*a+24*b+42*d+35*c] [1.518424276, [6144/6125, 64/63, 25/24], 11*b+7*a+16*c+20*d] [1.301497343, [6144/6125, 50/49, 49/48], 17*d+14*c+10*b+6*a] [2.278536109, [6144/6125, 50/49, 36/35], 25*b+16*a+37*c+45*d] [5.315847988, [6144/6125, 50/49, 28/27], 45*b+28*a+65*c+79*d] [1.469220000, [6144/6125, 50/49, 25/24], 17*d+14*c+10*b+6*a] [1.595180079, [6144/6125, 49/48, 36/35], 25*d+21*c+9*a+14*b] [2.462710473, [6144/6125, 49/48, 28/27], 15*a+24*b+42*d+35*c] [1.472896878, [6144/6125, 49/48, 25/24], 17*d+14*c+10*b+6*a] [1.144847592, [6144/6125, 36/35, 28/27], 3*b+2*a+5*c+5*d] [1.656373835, [6144/6125, 36/35, 25/24], 11*b+7*a+16*c+20*d] [2.839788039, [6144/6125, 28/27, 25/24], 37*d+30*c+21*b+13*a] [1.660204014, [3136/3125, 1029/1024, 126/125], 49*b+31*a+87*d+72*c] [4.231649866, [3136/3125, 1029/1024, 245/243], 87*a+138*b+202*c+244*d] [1.828192297, [3136/3125, 1029/1024, 64/63], 70*d+40*b+25*a+58*c] [1.187113513, [3136/3125, 1029/1024, 50/49], 14*c+17*d+9*b+6*a] [2.064939480, [3136/3125, 1029/1024, 49/48], 70*d+40*b+25*a+58*c] [3.867306154, [3136/3125, 1029/1024, 36/35], 58*b+37*a+86*c+104*d] [2.547030414, [3136/3125, 1029/1024, 25/24], 53*d+44*c+31*b+19*a] [1.804062272, [3136/3125, 1728/1715, 126/125], 49*b+31*a+87*d+72*c] [2.569847702, [3136/3125, 1728/1715, 245/243], 78*b+49*a+138*d+114*c] [3.720561269, [3136/3125, 1728/1715, 64/63], 78*b+49*a+138*d+114*c] [1.755627065, [3136/3125, 1728/1715, 50/49], 29*b+18*a+42*c+51*d] [1.509685048, [3136/3125, 1728/1715, 49/48], 20*b+13*a+30*c+36*d] [1.665410631, [3136/3125, 1728/1715, 36/35], 20*b+13*a+30*c+36*d] [1.296200144, [3136/3125, 1728/1715, 25/24], 9*b+5*a+12*c+15*d] [1.453386524, [3136/3125, 126/125, 245/243], 19*a+30*b+44*c+53*d] [1.386474440, [3136/3125, 126/125, 64/63], 34*d+28*c+19*b+12*a] [1.462600997, [3136/3125, 126/125, 50/49], 34*d+28*c+19*b+12*a] [1.840291986, [3136/3125, 126/125, 49/48], 19*a+30*b+44*c+53*d] [1.609356470, [3136/3125, 126/125, 36/35], 34*d+28*c+19*b+12*a] [1.397866900, [3136/3125, 126/125, 28/27], 7*a+11*b+16*c+19*d] [1.442139658, [3136/3125, 126/125, 25/24], 7*a+11*b+16*c+19*d] [3.840060916, [3136/3125, 245/243, 64/63], 78*b+49*a+138*d+114*c] [2.604911795, [3136/3125, 245/243, 50/49], 70*c+85*d+48*b+30*a] [1.550683960, [3136/3125, 245/243, 36/35], 18*b+11*a+26*c+32*d] [1.696997541, [3136/3125, 245/243, 28/27], 18*b+11*a+26*c+32*d] [1.522358304, [3136/3125, 245/243, 25/24], 12*b+8*a+18*c+21*d] [1.605730195, [3136/3125, 64/63, 50/49], 34*d+28*c+19*b+12*a] [2.710238360, [3136/3125, 64/63, 49/48], 70*d+40*b+25*a+58*c] [4.017844284, [3136/3125, 64/63, 28/27], 70*d+40*b+25*a+58*c] [2.183860583, [3136/3125, 64/63, 25/24], 36*d+13*a+21*b+30*c] [1.304754477, [3136/3125, 50/49, 49/48], 17*d+14*c+10*b+6*a] [1.863858578, [3136/3125, 50/49, 36/35], 34*d+28*c+19*b+12*a] [2.949144340, [3136/3125, 50/49, 28/27], 29*b+18*a+42*c+51*d] [1.472896878, [3136/3125, 50/49, 25/24], 17*d+14*c+10*b+6*a] [1.973782564, [3136/3125, 49/48, 36/35], 20*b+13*a+30*c+36*d] [4.538146945, [3136/3125, 49/48, 28/27], 70*d+40*b+25*a+58*c] [1.476582958, [3136/3125, 49/48, 25/24], 17*d+14*c+10*b+6*a] [2.113987015, [3136/3125, 36/35, 28/27], 18*b+11*a+26*c+32*d] [1.075200000, [3136/3125, 36/35, 25/24], 2*c+2*d+b+a] [1.759314432, [3136/3125, 28/27, 25/24], 7*a+11*b+16*c+19*d] [1.881579115, [1029/1024, 1728/1715, 126/125], 49*b+31*a+87*d+72*c] [1.108598313, [1029/1024, 1728/1715, 245/243], 8*b+5*a+12*c+14*d] [1.151256995, [1029/1024, 1728/1715, 64/63], 8*b+5*a+12*c+14*d] [2.335569854, [1029/1024, 1728/1715, 50/49], 60*c+73*d+41*b+26*a] [1.179639680, [1029/1024, 1728/1715, 49/48], 8*b+5*a+12*c+14*d] [1.225032023, [1029/1024, 1728/1715, 36/35], 8*b+5*a+12*c+14*d] [3.059142980, [1029/1024, 1728/1715, 25/24], 48*c+33*b+21*a+59*d] [2.631752268, [1029/1024, 126/125, 245/243], 73*b+46*a+129*d+107*c] [2.188267678, [1029/1024, 126/125, 81/80], 49*b+31*a+87*d+72*c] [1.535432120, [1029/1024, 126/125, 64/63], 15*a+24*b+42*d+35*c] [1.696672390, [1029/1024, 126/125, 50/49], 25*b+16*a+37*c+45*d] [1.651816855, [1029/1024, 126/125, 49/48], 15*a+24*b+42*d+35*c] [1.927376265, [1029/1024, 126/125, 36/35], 25*b+16*a+37*c+45*d] [1.055126953, [1029/1024, 126/125, 25/24], b+a+3*d+2*c] [1.135929305, [1029/1024, 245/243, 81/80], 8*b+5*a+12*c+14*d] [1.154976817, [1029/1024, 245/243, 64/63], 8*b+5*a+12*c+14*d] [3.312603948, [1029/1024, 245/243, 50/49], 57*b+36*a+83*c+101*d] [1.183451209, [1029/1024, 245/243, 49/48], 8*b+5*a+12*c+14*d] [5.315252144, [1029/1024, 245/243, 25/24], 31*a+87*d+71*c+49*b] [1.179639680, [1029/1024, 81/80, 64/63], 8*b+5*a+12*c+14*d] [2.650902363, [1029/1024, 81/80, 50/49], 60*c+73*d+41*b+26*a] [1.208722102, [1029/1024, 81/80, 49/48], 8*b+5*a+12*c+14*d] [1.255233531, [1029/1024, 81/80, 36/35], 8*b+5*a+12*c+14*d] [3.388625357, [1029/1024, 81/80, 25/24], 48*c+33*b+21*a+59*d] [1.504137953, [1029/1024, 64/63, 50/49], 16*b+10*a+23*c+28*d] [1.276281563, [1029/1024, 64/63, 36/35], 8*b+5*a+12*c+14*d] [1.848570832, [1029/1024, 64/63, 25/24], 16*b+10*a+23*c+28*d] [1.579217147, [1029/1024, 50/49, 49/48], 16*b+10*a+23*c+28*d] [2.344125081, [1029/1024, 50/49, 36/35], 25*b+16*a+37*c+45*d] [1.932774085, [1029/1024, 50/49, 25/24], 16*b+10*a+23*c+28*d] [1.307746559, [1029/1024, 49/48, 36/35], 8*b+5*a+12*c+14*d] [1.940842427, [1029/1024, 49/48, 25/24], 16*b+10*a+23*c+28*d] [2.253538886, [1029/1024, 36/35, 25/24], 17*b+11*a+25*c+31*d] [1.897137590, [1728/1715, 126/125, 245/243], 43*b+27*a+63*c+76*d] [2.377883156, [1728/1715, 126/125, 81/80], 49*b+31*a+87*d+72*c] [2.326207228, [1728/1715, 126/125, 64/63], 43*b+27*a+63*c+76*d] [1.153602745, [1728/1715, 126/125, 50/49], 9*c+6*b+4*a+11*d] [1.155526618, [1728/1715, 126/125, 49/48], 9*c+6*b+4*a+11*d] [1.190963383, [1728/1715, 126/125, 36/35], 9*c+6*b+4*a+11*d] [3.298310219, [1728/1715, 126/125, 28/27], 37*b+23*a+65*d+54*c] [1.151256995, [1728/1715, 245/243, 81/80], 8*b+5*a+12*c+14*d] [2.205432014, [1728/1715, 245/243, 50/49], 35*b+22*a+51*c+62*d] [1.199420137, [1728/1715, 245/243, 49/48], 8*b+5*a+12*c+14*d] [1.245573629, [1728/1715, 245/243, 36/35], 8*b+5*a+12*c+14*d] [1.297682532, [1728/1715, 245/243, 28/27], 8*b+5*a+12*c+14*d] [2.616129604, [1728/1715, 245/243, 25/24], 27*b+17*a+39*c+48*d] [1.195557177, [1728/1715, 81/80, 64/63], 8*b+5*a+12*c+14*d] [2.842253559, [1728/1715, 81/80, 50/49], 60*c+73*d+41*b+26*a] [1.225032023, [1728/1715, 81/80, 49/48], 8*b+5*a+12*c+14*d] [1.272171056, [1728/1715, 81/80, 36/35], 8*b+5*a+12*c+14*d] [3.584855584, [1728/1715, 81/80, 25/24], 48*c+33*b+21*a+59*d] [2.604025167, [1728/1715, 64/63, 50/49], 35*b+22*a+51*c+62*d] [1.245573629, [1728/1715, 64/63, 49/48], 8*b+5*a+12*c+14*d] [1.293503099, [1728/1715, 64/63, 36/35], 8*b+5*a+12*c+14*d] [1.347617144, [1728/1715, 64/63, 28/27], 8*b+5*a+12*c+14*d] [2.974491206, [1728/1715, 64/63, 25/24], 27*b+17*a+39*c+48*d] [1.213482515, [1728/1715, 50/49, 49/48], 9*c+6*b+4*a+11*d] [1.250696625, [1728/1715, 50/49, 36/35], 9*c+6*b+4*a+11*d] [3.171469175, [1728/1715, 50/49, 28/27], 29*b+18*a+42*c+51*d] [1.315616223, [1728/1715, 50/49, 25/24], 9*c+6*b+4*a+11*d] [1.380840823, [1728/1715, 49/48, 28/27], 8*b+5*a+12*c+14*d] [1.317810287, [1728/1715, 49/48, 25/24], 9*c+6*b+4*a+11*d] [1.433975353, [1728/1715, 36/35, 28/27], 8*b+5*a+12*c+14*d] [1.358223838, [1728/1715, 36/35, 25/24], 9*c+6*b+4*a+11*d] [3.009084772, [1728/1715, 28/27, 25/24], 37*d+30*c+21*b+13*a] [1.721440913, [126/125, 245/243, 81/80], 19*a+30*b+44*c+53*d] [2.367084273, [126/125, 245/243, 64/63], 43*b+27*a+63*c+76*d] [1.337685863, [126/125, 245/243, 50/49], 13*b+8*a+19*c+23*d] [2.011539314, [126/125, 245/243, 49/48], 19*a+30*b+44*c+53*d] [1.425733632, [126/125, 245/243, 36/35], 13*b+8*a+19*c+23*d] [1.522358304, [126/125, 245/243, 28/27], 13*b+8*a+19*c+23*d] [1.871717101, [126/125, 245/243, 25/24], 30*d+25*c+17*b+11*a] [1.542907401, [126/125, 81/80, 64/63], 34*d+28*c+19*b+12*a] [1.627623155, [126/125, 81/80, 50/49], 34*d+28*c+19*b+12*a] [2.179705030, [126/125, 81/80, 49/48], 19*a+30*b+44*c+53*d] [1.790936736, [126/125, 81/80, 36/35], 34*d+28*c+19*b+12*a] [1.487814607, [126/125, 81/80, 28/27], 7*a+11*b+16*c+19*d] [1.534936158, [126/125, 81/80, 25/24], 7*a+11*b+16*c+19*d] [1.693895333, [126/125, 64/63, 50/49], 34*d+28*c+19*b+12*a] [1.944562872, [126/125, 64/63, 49/48], 15*a+24*b+42*d+35*c] [1.863858578, [126/125, 64/63, 36/35], 34*d+28*c+19*b+12*a] [2.462710473, [126/125, 64/63, 28/27], 15*a+24*b+42*d+35*c] [1.213629630, [126/125, 64/63, 25/24], 3*a+5*b+8*d+7*c] [1.215506250, [126/125, 50/49, 49/48], 9*c+6*b+4*a+11*d] [1.675829337, [126/125, 50/49, 28/27], 13*b+8*a+19*c+23*d] [1.317810287, [126/125, 50/49, 25/24], 9*c+6*b+4*a+11*d] [1.254871699, [126/125, 49/48, 36/35], 9*c+6*b+4*a+11*d] [2.649382292, [126/125, 49/48, 28/27], 15*a+24*b+42*d+35*c] [1.320008011, [126/125, 49/48, 25/24], 9*c+6*b+4*a+11*d] [1.786134033, [126/125, 36/35, 28/27], 13*b+8*a+19*c+23*d] [1.360488960, [126/125, 36/35, 25/24], 9*c+6*b+4*a+11*d] [1.815034831, [126/125, 28/27, 25/24], 7*a+11*b+16*c+19*d] [1.199420137, [245/243, 81/80, 64/63], 8*b+5*a+12*c+14*d] [1.770935474, [245/243, 81/80, 50/49], 22*b+14*a+32*c+39*d] [1.276281563, [245/243, 81/80, 36/35], 8*b+5*a+12*c+14*d] [1.329675140, [245/243, 81/80, 28/27], 8*b+5*a+12*c+14*d] [2.363581242, [245/243, 81/80, 25/24], 22*b+14*a+32*c+39*d] [2.641250007, [245/243, 64/63, 50/49], 35*b+22*a+51*c+62*d] [1.249598197, [245/243, 64/63, 49/48], 8*b+5*a+12*c+14*d] [1.297682532, [245/243, 64/63, 36/35], 8*b+5*a+12*c+14*d] [1.351971425, [245/243, 64/63, 28/27], 8*b+5*a+12*c+14*d] [3.007295043, [245/243, 64/63, 25/24], 27*b+17*a+39*c+48*d] [1.986278255, [245/243, 50/49, 49/48], 22*b+14*a+32*c+39*d] [1.572336629, [245/243, 50/49, 36/35], 13*b+8*a+19*c+23*d] [1.678896865, [245/243, 50/49, 28/27], 13*b+8*a+19*c+23*d] [2.635572871, [245/243, 50/49, 25/24], 22*b+14*a+32*c+39*d] [1.329675140, [245/243, 49/48, 36/35], 8*b+5*a+12*c+14*d] [1.385302452, [245/243, 49/48, 28/27], 8*b+5*a+12*c+14*d] [2.650988754, [245/243, 49/48, 25/24], 22*b+14*a+32*c+39*d] [1.260576198, [245/243, 36/35, 25/24], 7*c+9*d+5*b+3*a] [1.291958414, [245/243, 28/27, 25/24], 7*c+9*d+5*b+3*a] [1.786901316, [81/80, 64/63, 50/49], 34*d+28*c+19*b+12*a] [1.276281563, [81/80, 64/63, 49/48], 8*b+5*a+12*c+14*d] [1.380840823, [81/80, 64/63, 28/27], 8*b+5*a+12*c+14*d] [1.620849023, [81/80, 64/63, 25/24], 11*b+7*a+16*c+20*d] [2.107333077, [81/80, 50/49, 49/48], 22*b+14*a+32*c+39*d] [2.074153775, [81/80, 50/49, 36/35], 34*d+28*c+19*b+12*a] [1.147959184, [81/80, 50/49, 28/27], 3*b+2*a+4*c+5*d] [2.796199310, [81/80, 50/49, 25/24], 22*b+14*a+32*c+39*d] [1.358068434, [81/80, 49/48, 36/35], 8*b+5*a+12*c+14*d] [1.414883587, [81/80, 49/48, 28/27], 8*b+5*a+12*c+14*d] [2.812554722, [81/80, 49/48, 25/24], 22*b+14*a+32*c+39*d] [1.469328077, [81/80, 36/35, 28/27], 8*b+5*a+12*c+14*d] [1.768103919, [81/80, 36/35, 25/24], 11*b+7*a+16*c+20*d] [1.872519988, [81/80, 28/27, 25/24], 7*a+11*b+16*c+19*d] [1.760686016, [64/63, 50/49, 49/48], 16*b+10*a+23*c+28*d] [2.158607408, [64/63, 50/49, 36/35], 34*d+28*c+19*b+12*a] [2.060991308, [64/63, 50/49, 28/27], 16*b+10*a+23*c+28*d] [2.154870412, [64/63, 50/49, 25/24], 16*b+10*a+23*c+28*d] [1.380840823, [64/63, 49/48, 36/35], 8*b+5*a+12*c+14*d] [2.163865892, [64/63, 49/48, 25/24], 16*b+10*a+23*c+28*d] [1.493966092, [64/63, 36/35, 28/27], 8*b+5*a+12*c+14*d] [1.809749792, [64/63, 36/35, 25/24], 11*b+7*a+16*c+20*d] [2.532938159, [64/63, 28/27, 25/24], 16*b+10*a+23*c+28*d] [1.317810287, [50/49, 49/48, 36/35], 9*c+6*b+4*a+11*d] [2.163865892, [50/49, 49/48, 28/27], 16*b+10*a+23*c+28*d] [1.969795692, [50/49, 36/35, 28/27], 13*b+8*a+19*c+23*d] [1.428724824, [50/49, 36/35, 25/24], 9*c+6*b+4*a+11*d] [2.648314659, [50/49, 28/27, 25/24], 16*b+10*a+23*c+28*d] [1.530797807, [49/48, 36/35, 28/27], 8*b+5*a+12*c+14*d] [1.431107521, [49/48, 36/35, 25/24], 9*c+6*b+4*a+11*d] [2.659370017, [49/48, 28/27, 25/24], 16*b+10*a+23*c+28*d] [1.371742112, [36/35, 28/27, 25/24], 7*c+9*d+5*b+3*a] As you can see if you sort through this, there are quite a few valid sets with rather exotic numbers of steps, such as 11 or 13. Perhaps we should learn to celebrate the fact?
Message: 1824 - Contents - Hide Contents Date: Tue, 09 Oct 2001 15:44:03 Subject: torsion (was: Re: 72 owns the 11-limit) From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> What does this test show? >> If something passes the test then the Paul Theorem will work; it > isn't a necessary condition, however.I suspect a necessary condition would have to distinguish between commatic and chromatic unison vectors. Does the Paul Theorem work for all the examples in my list?
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