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Message: 1775 - Contents - Hide Contents Date: Sat, 06 Oct 2001 21:13:05 Subject: Re: Question From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> Paul wrote: >>> How many distinct MOS scales are represented here? What are the >> generators, and mapping from generators to primes, in each? >> I've written a test script. See > <#!/usr/local/bin/python * [with cont.] (Wayb.)> and the results at > <[(-1, 2, 0), (-2, 0, -1), (-1, -2, 4)] * [with cont.] (Wayb.)>. In summary > > > > 1/4, 356.0 cent generator > [(1, 0), (1, 2), (5, -9), (4, -4)] > > 1/20, 116.6 cent generator > [(1, 0), (1, 6), (3, -7), (3, -2)] > > 1/10, 116.6 cent generator > [(1, 0), (1, 6), (3, -7), (3, -2)] > > > 1/10, 55.2 cent generator > [(1, 0), (2, -6), (2, 7), (3, 2)] > > > ? > [(?, 0), (?, -3), (?, 1), (?, 1)] > > > > 1/20, 55.2 cent generator > [(1, 0), (2, -6), (2, 7), (3, 2)] > > 9/20, 578.1 cent generator > [(1, 0), (-1, 6), (6, -7), (4, -2)] > > 9/20, 578.1 cent generator > [(1, 0), (-1, 6), (6, -7), (4, -2)] >So some of them _don't_ have torsion?> > > 25:24, 1029:1024 and 225:224 fail, apparently because it wants a > half-octave generator, but doesn't give the usual clue. Gene?
Message: 1776 - Contents - Hide Contents Date: Sat, 06 Oct 2001 21:10:34 Subject: Re: 3rd-best 11-limit temperament From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> Me: >>>>> 3/34, 52.2 cent generator >>>> >>>> basis: >>>> (0.5, 0.043499613319368802) > > Gene: >>>> It seems to me he probably means the 22-24 system, with 22+24=46, >> and>>> not the 22-46 system, with 22+46=68. Paul? > > Paul: >>> I don't know. Graham got the right generator for the system I meant. >> Does that mean you're wrong, Gene? I don't know. I do find it >> interesting that though the 22-tone MOS has no 11-limit hexads, the >> corresponding 22-tone omnitetrachordal scale has some. >> This is the system generated from the consistent mappings of 46- and > 22-equal. There's also a system consistent with 46 -and 58-equal which I > called "diaschismic". It does have a 22 note MOS, but it's too complex to > be the 22 from 46 that Paul asked for. >Are you sure? This is the shrutar system, which you once said was not consistent with 22-equal. So maybe Gene was right?
Message: 1777 - Contents - Hide Contents Date: Sat, 06 Oct 2001 21:14:41 Subject: Re: 72 owns the 11-limit From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:>>> So maybe 31 really owns the 11-limit? :) >> It makes a good run at it, but smaller is better when it comes to > commas.Well if that's so, wouldn't 250000:250047 be better than all the superparticular commas? No, I think larger commas with smaller numbers can often be better because they portend simpler systems.
Message: 1778 - Contents - Hide Contents Date: Sat, 06 Oct 2001 21:17:10 Subject: torsion (was: Re: 72 owns the 11-limit) From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> but to really find out you should > add the 2 column and take the gcd of the minors,I'd really like to be able to do this. The gcd of the minors . . . can you explicate exactly what the procedure is . . . I have Matlab.
Message: 1779 - Contents - Hide Contents Date: Sat, 06 Oct 2001 21:46:10 Subject: Re: 72 owns the 11-limit From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:>> freq. determinant > >> 1 9 >> 1 11 >> 3 23 >> 3 37 >> Some fairly exotic scale possibilities in here!Well, the 11- and 23-tone ones especially would correspond to "skewed" periodicity blocks, a concept I was trying to explain early this morning (which means that some of the simple consonant intervals will involve many unison vectors and thus be quite out-of-tune). Did that make sense to you? Is there any way to measure this "skewness", say with dot products of the vectors, but modified to conform with the tetrahedral-octahedral lattice instead of the Cartesian one? What about the idea of a "canonical" basis? Perhaps I need to go off and get some math textbooks and figure this all out for myself . . .
Message: 1780 - Contents - Hide Contents Date: Sat, 06 Oct 2001 15:50:25 Subject: directional sound From: Carl Lumma The Audio Spotlight * [with cont.] (Wayb.) -Carl
Message: 1781 - Contents - Hide Contents Date: Sat, 06 Oct 2001 01:15:12 Subject: Re: 3rd-best 11-limit temperament From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: If you wrote such a paper in the next> couple of months, you could probably get it published in > Xenharmonikon 18 . . .Thanks for the suggestion; I've been meaning to ask about how much of this stuff you think is publishable and how much might suffer from the "too mathematical" virus. I hope to clean things up, put up a web page, and then think about any further possibilities.
Message: 1782 - Contents - Hide Contents Date: Sat, 06 Oct 2001 01:17:08 Subject: Re: Tetrachordality and Scala From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> I'm afraid I've lost you on this. I defined omnitetrachordality on > the tuning list, and Gene said he was going to come up with some > theorems about it . . .I asked if it would be a good idea, since I think it should be possible. I need to take projects a few at a time, though.
Message: 1783 - Contents - Hide Contents Date: Sat, 06 Oct 2001 01:19:12 Subject: Re: Tetrachordality and Scala From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:> That's where we left off. We were counting the number of changing > pitches. But what if they all change slightly. I suggest comparing > them statistically in log-freq. space. Maybe just mean-deviation is > okay here.I thought you'd rejected the idea of caring about the sizes of intervals?
Message: 1784 - Contents - Hide Contents Date: Sat, 06 Oct 2001 04:16:57 Subject: nugget from 4/21/00: first glimpse of "torsion"??? From: Paul Erlich --- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote: I've been exploring some 13-limit periodicity blocks due to Polychroni's questions, and I've found some which seem to contradict my conceptions periodicity blocks so far. For example, using the unison vectors 243:242 (7.1¢), 352:351 (4.9¢), 385:384 (4.5¢), 676:675 (2.6¢), 2401:2400 (0.7¢), and 3025:3024 (0.6¢), so that the Fokker matrix is -5 0 0 2 0 3 0 0 -1 1 3 2 0 0 -2 1 2 -4 0 0 3 -2 1 -2 0 (whose determinant is 20), the 5-d "parallelogram" contains the following pitches: cents numerator denominator 0 1 1 116.23 77 72 119.44 15 14 235.68 55 48 238.89 225 196 359.47 16 13 363.4 882 715 478.92 120 91 482.85 189 143 595.15 55 39 598.36 900 637 717.15 286 189 721.08 91 60 836.6 715 441 840.53 13 8 961.11 392 225 964.32 96 55 1080.6 28 15 1083.8 144 77 1196.1 880 441 Instead of being an approximation of 20-tET, it's an double approximation of 10-tET with 539:540 and 880:881 pairs. What's going on here??? Can anyone come up with a mathematical explanation for this phenomenon? Does it only occur in higher dimensions? Certainly my belief, which I think came from Paul Hahn, that an N-tone periodicity block with small unison vectors will always be a good approximation of N-tET, turned out to be wrong. --- End forwarded message --- discuss . . .
Message: 1785 - Contents - Hide Contents Date: Sat, 06 Oct 2001 04:19:00 Subject: More from 4/21/00: does this make sense? From: Paul Erlich --- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote: On October 29, 1999, Paul Hahn wrote,>If you look at the way the truncated >octa fills space, you'll find that its symmetry group is that of a >body-centered cubic lattice. The rhombic dodec, OTOH, fills space such >that is _its_ symmetry group is that of a _face_-centered cubic >lattice. >Why is this important? Well, let's think about why we use FCC (= oc- tet >or triangulated lattice) for pitch diagrams in the first place. If you >ignore the edges and just look at the lattice points, it's equivalent to >the cubic lattice--it's just been subjected to a couple of affine >(shear) transformations. (Translation: we squished it a bit so that it >slants.) This makes sense because the lattice is actually a space whose >basis is the three vectors representing the 3/2, the 5/4, and the 7/4. >Right? >Now look at the way the shapes you want to use to tesselate space with. >They _also_ are related to each other by three basic vectors, it's just >that this time, it's the three unison vectors. Other than that, the >relationship is the same. But there's no way you can map the BCC to a >straightforward cubic lattice--you either have to leave some points out >of the cubic lattice, or interlock two cubic lattices together.Conjecture: the bizarre, double-vision periodicity block I found could even happen in 3 dimensions, if there are 4 unison vectors defining truncated-octahedron equivalence regions, but due to the parallelopiped basis of the periodicity block construction from three unison vectors, these truncated-octahedron regions could only come up two at a time. What I found may be some sort of higher-dimensional analogue with 5 dimensions and 7(?) operative unison vectors. Paul H., does this make any sense? --- End forwarded message ---
Message: 1786 - Contents - Hide Contents Date: Sat, 06 Oct 2001 04:47:13 Subject: Re: Question From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> What is the mapping from generators to primes in the 46-out-of-72 > temperament?I get that r = 2^(11/72) is the generator within a sqrt(2); we have 2 r^(-6) = 2^(42/72) ~ 3/2 2 r^(-11) = 2^(23/72) ~ 5/4 2^(1/2) r^2 = 2^(58/72) ~ 7/4 r^3 = 2^(33/72) ~ 11/8. The complexity is 2*(3-(-11)) = 28.
Message: 1787 - Contents - Hide Contents Date: Sat, 06 Oct 2001 05:11:40 Subject: Re: 3rd-best 11-limit temperament From: genewardsmith@xxxx.xxx --- In tuning-math@y..., graham@m... wrote:> The unison vectors are not in their simplest terms. Dan Stearns claimed > before to have an algorithm for finding unison vectors, so I'd still like > to see it.There are such things as lattice basis reduction algorithms, but this isn't even a lattice basis.>> Why is this better than an ME 22-out-of-46, which has a maximum error >> of 8.6 cents in the 11-limit, probably reducable further in a non- ET >> setting?> Presumably, you mean >>>>> temper.Temperament(46, 22, temper.primes[:4]) >> 3/34, 52.2 cent generator > > basis: > (0.5, 0.043499613319368802)It seems to me he probably means the 22-24 system, with 22+24=46, and not the 22-46 system, with 22+46=68. Paul?
Message: 1788 - Contents - Hide Contents Date: Sat, 06 Oct 2001 05:44:45 Subject: Re: More from 4/21/00: does this make sense? From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Paul H., does this make any sense?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. If we look at triples [a,b,c] with a quadradic form Q(a,b,c) = a^2+b^2+c62+ab+ac+bc, we have the symmetric lattice of 7- limit note-classes. The tetrads are defined as four lattice points, each of which is at unit distance from the other three. The tetrad centroids are simply the means of the lattice points, if we omit the division by 4, we get for the 1-3-5-7 otonal tetrad [0 0 0]+[1 0 0]+[0 1 0]+[0 0 1] = [1 1 1], where 1+1+1=(-1) (mod 4). For the utonal tetrad which is its inversion, we get [0 0 0]+[-1 0 0]+[0 -1 0]+[0 0 -1] = [-1 -1 -1], and -1-1-1=1 (mod 4). If we translate either of these tetrads by an arbitary [a b c] we end up with the same result mod 4 (since we add four each of a, b, and c.) Hence the otonal tetrads can be considered as a lattice with base point [1 1 1], defined as [4a+1, 4b+1, 4c+1], or equivalently as those triples [u v w] such that u+v+w=-1 (mod 4); the same goes for the utonal lattice, with base point [-1 -1 -1] and u+v+w=1 (mod 4). This generalizes easily to any p-limit, with the caveat that treating all odd primes the same makes progressively less sense. I've been intending to bring this up, partly because it allows us to do strange things of the kind Robert seems fond of doing.
Message: 1789 - Contents - Hide Contents Date: Sat, 06 Oct 2001 06:12:05 Subject: Re: Question From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> What is the mapping from generators to primes in the 46-out-of-72 >> temperament? >> I get that r = 2^(11/72) is the generator within a sqrt(2);Saw that on Graham's 11-limit and 13-limit results . . .> we have > > 2 r^(-6) = 2^(42/72) ~ 3/2 > > 2 r^(-11) = 2^(23/72) ~ 5/4 > > 2^(1/2) r^2 = 2^(58/72) ~ 7/4 > > r^3 = 2^(33/72) ~ 11/8. >Thanks a lot Bob. So the 20-tone-per-octave MOS scale will have three 1:3:7:11 and three 1/(1:3:7:11) chords (right?). Not enough to warrant too much interest in this 20-tone-per-octave MOS at this point . . . BUT: merely within the 7-limit, the following 20-tone MOS scales (unless some of them are double image) suggest themselves from PB considerations: 2401 : 2400 commatic 64 : 63 commatic 25 : 24 chromatic 2401 : 2400 commatic 225 : 224 commatic 25 : 24 chromatic 2401 : 2400 commatic 225 : 224 commatic 28 : 27 chromatic 1029 : 1024 commatic 50 : 49 commatic 28 : 27 chromatic 225 : 224 commatic 1029 : 1024 commatic 25 : 24 chromatic 225 : 224 commatic 1029 : 1024 commatic 28 : 27 chromatic 2401 : 2400 commatic 1029 : 1024 commatic 25 : 24 chromatic 2401 : 2400 commatic 1029 : 1024 commatic 28 : 27 chromatic How many distinct MOS scales are represented here? What are the generators, and mapping from generators to primes, in each?
Message: 1790 - Contents - Hide Contents Date: Sat, 06 Oct 2001 06:12:55 Subject: Re: nugget from 4/21/00: first glimpse of "torsion"??? From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> discuss . . .You can do the same calculation with these six commas as I did in the "72 owns the 11-limit" article, with a similar result. Taking the six combinations of these five at a time, and adding an indeterminant row [a b c d e f], we get a determinant of zero in two cases (linear dependency), a determinant of h10 and of -h10, meaning we can construct a 10-block, and two determinants of -2 h10, meaning we have torsion. I'm afraid this sort of thing will happen a lot.
Message: 1791 - Contents - Hide Contents Date: Sat, 06 Oct 2001 06:15:09 Subject: Re: 3rd-best 11-limit temperament From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:>> 3/34, 52.2 cent generator >> >> basis: >> (0.5, 0.043499613319368802) >> It seems to me he probably means the 22-24 system, with 22+24=46, and > not the 22-46 system, with 22+46=68. Paul?I don't know. Graham got the right generator for the system I meant. Does that mean you're wrong, Gene? I don't know. I do find it interesting that though the 22-tone MOS has no 11-limit hexads, the corresponding 22-tone omnitetrachordal scale has some.
Message: 1792 - Contents - Hide Contents Date: Sun, 7 Oct 2001 16:44 +01 Subject: Re: nugget from 4/21/00: first glimpse of "torsion"??? From: graham@xxxxxxxxxx.xx.xx Paul wrote:> What are you talking about? There's no 2 column here, and there's no > reason= > to define the > intervals downward instead of upward. Anyway, multiplying by -1 won't > affec=Once again, I don't know how to find the temeperaments from octave equivalent matrices. My conversion to octave specificity assumes the intervals are between a unison and an octave. Graham
Message: 1793 - Contents - Hide Contents Date: Sun, 7 Oct 2001 16:44 +01 Subject: Re: Question From: graham@xxxxxxxxxx.xx.xx Paul wrote:> So some of them _don't_ have torsion?The last figures were wrong (some of my unison vectors were wrong) so here's what they should be: [(-1, 2, 0), (-2, 0, -1), (-1, -2, 4)] 3/10, 356.0 cent generator complexity measure: 11 (17 for smallest MOS) highest error: 0.008316 (9.980 cents) unique [(-1, 2, 0), (2, 2, -1), (-1, -2, 4)] 1/10, 116.6 cent generator complexity measure: 13 (21 for smallest MOS) highest error: 0.002024 (2.428 cents) unique [(-3, 0, 1), (2, 2, -1), (-1, -2, 4)] 1/10, 116.6 cent generator complexity measure: 13 (21 for smallest MOS) highest error: 0.002024 (2.428 cents) unique [(-3, 0, 1), (0, 2, -2), (1, 0, 3)] 2/5, 231.2 cent generator complexity measure: 8 (10 for smallest MOS) highest error: 0.014573 (17.488 cents) [(-1, 2, 0), (1, 0, 3), (2, 2, -1)] 1/10, 116.6 cent generator complexity measure: 13 (21 for smallest MOS) highest error: 0.002024 (2.428 cents) unique [(-3, 0, 1), (1, 0, 3), (2, 2, -1)] 1/10, 116.6 cent generator complexity measure: 13 (21 for smallest MOS) highest error: 0.002024 (2.428 cents) unique [(-1, 2, 0), (1, 0, 3), (-1, -2, 4)] 1/10, 116.6 cent generator complexity measure: 13 (21 for smallest MOS) highest error: 0.002024 (2.428 cents) unique [(-3, 0, 1), (1, 0, 3), (-1, -2, 4)] 1/10, 116.6 cent generator complexity measure: 13 (21 for smallest MOS) highest error: 0.002024 (2.428 cents) unique This one doesn't have torsion: [(-3, 0, 1), (0, 2, -2), (1, 0, 3)] conversion [[10 0] [16 -3] [23 1] [28 1]] 2/5, 231.2 cent generator basis: (0.5, 0.19264507794239583) mapping by period and generator: [(2, 0), (2, 3), (5, -1), (6, -1)] mapping by steps: [[6, 4], [9, 5], [14, 9], [17, 11]] unison vectors: [[1, 0, 2, -2], [-9, 1, 2, 1]] highest interval width: 4 complexity measure: 8 (10 for smallest MOS) highest error: 0.014573 (17.488 cents) No, wait, it does give a 20 note periodicity block, but the whole adjoint matrix divides through by 2. So this was falling victim to the trap I set for the bad multiple-29 unison vectors. I've worked out some better ones for that now, that I can try out next time I'm in Linux.>> 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. Still, I changed the program to find the period a different way. If we want to use only octave-equivalent matrices, I think we'll still have to make sure we start with a good set of unison vectors, and trust that left hand column. Graham
Message: 1794 - Contents - Hide Contents Date: Sun, 07 Oct 2001 19:07:26 Subject: torsion (was: Re: 72 owns the 11-limit) From: genewardsmith@xxxx.xxx --- 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.
Message: 1795 - Contents - Hide Contents Date: Sun, 07 Oct 2001 23:15:03 Subject: Re: Searching for interesting 7-limit MOS scales From: genewardsmith@xxxx.xxx --- 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. 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. 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. Since a unit volume is the volume of the parallepiped with sides 3, 5, and 7, and not a unit cube, the maximum this can attain is actually sqrt(2), which is what a cube would give. #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: 1796 - Contents - Hide Contents Date: Sun, 07 Oct 2001 23:27:23 Subject: Re: Question From: genewardsmith@xxxx.xxx --- 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.
Message: 1797 - Contents - Hide Contents Date: Sun, 07 Oct 2001 00:04:30 Subject: Re: Tetrachordality and Scala From: Carl Lumma>> >hat's where we left off. We were counting the number of changing >> pitches. But what if they all change slightly. I suggest >> comparing them statistically in log-freq. space. Maybe just mean- >> deviation is okay here. >> I thought you'd rejected the idea of caring about the sizes of > intervals?While I'm not sure which intervals you're referring to here, it's safe to say I haven't rejected anything yet. :) What I care about is modeling an effect that Paul cooked up way long ago -- listeners use the 3:2 to chunk together pitches they here, much the same way they use the 2:1 to do this. So what I proposed in the mail you're replying to, was to transpose the scale by a 3:2, and call that the thing the listener has in his/her brain when hearing the original scale. Then ask how well the original scale fits this mental space. The only things I'm not sure of are: 1. We don't want to acknowledge the existence of the "scale" as such. We want the measure to represent the average sequence of music made with the scale. But for ease of calculation, this won't due. I'm reasonably convinced that the entire scale represents the set of all subsets of itself fairly well for most purposes, but I really have no idea. 2. What does it mean to allow or forbid a re-arrangement of the transposed scale to best fit the original scale? For example, transposing the diatonic scale by a tritone, the 4th and 7th degrees of the original scale will appear in the transposed scale, _but in no mode of the transposed scale will these pitches be a 4th and a 7th_. Should we just do a 'picture pages' match-up? My feeling is yes, we should, since our perceptual model (equivalence at the 3:2) doesn't assume our listener will be keeping track of scale degrees. But I'm open to suggestions here. Paul would like to forget these details, and transpose by an interval that appears in the scale. But Paul, how does this work if the scale contains more than one size of acceptable 3:2? And this measure isn't defined on scales without an acceptable 3:2 (granted such scales will all be "bad" on my measure, anyway). Does this help clear anything up for you, Gene? -Carl
Message: 1798 - Contents - Hide Contents Date: Sun, 07 Oct 2001 00:54:38 Subject: torsion (was: Re: 72 owns the 11-limit) From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> I'd really like to be able to do this. The gcd of the minors . . .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.
Message: 1799 - Contents - Hide Contents Date: Sun, 07 Oct 2001 02:23:58 Subject: Re: torsion From: Carl Lumma I confess I haven't been following this thread, but gcd has a nice recursive definition based on the Euclidean algorithm. Or something. Here's lisp that'll do it: (define gcd (lambda (ls) (let ((y (car (cdr ls)))) (cond [(zero? y) (car ls)] [else (gcd (cons y (cons (modulo (car ls) y) '())))])))) -Carl
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