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Message: 1750 - Contents - Hide Contents Date: Fri, 5 Oct 2001 12:25 +01 Subject: Re: 3rd-best 11-limit temperament From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9pjjc9+5lu5@xxxxxxx.xxx> Gene wrote:> This is easy enough that I've been meaning to suggest that Manuel > consider putting into Scala a routine to calculate Gen(m, n, p) and > Mos(n,m,p) for two ets m and n and a prime limit p; in case m and n > are not relatively prime this needs to be adjusted by working inside > of the interval of repetition. Of course one can also think of this > in terms of the ets generated by linear combinations of hm and hn, as > for instance h53 = h22 + h31 and h72 = h31 + h41.That's roughly what my Python module does, and Manuel's welcome to take that code as inspiration. <# Temperament finding library -- definitions * [with cont.] (Wayb.)> The temperament's returned by temper.Temperament(m, n, temper.primes[:q]) where q is the number of prime intervals you're using, not including the 2:1 which is the interval of equivalence. So the temperament in question is temper.Temperament(31, 22, temper.primes[:4]) Graham
Message: 1751 - Contents - Hide Contents Date: Fri, 05 Oct 2001 19:22:32 Subject: Re: Tetrachordality and Scala From: Carl Lumma> 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 . . .Do we want to enforce scale order on the transposed pitches, or just let them fall where they are closest to the original pitches? Earlier in this thread, I claimed we don't care about scale order, and I still claim it. But I was enforcing order in the example in my last mail. I don't like your definition of omnitetrachordality because: () I can't tell if it's equivalent to What We Want^TM. That is, I can't tell what sort of semi-periodicity it enforces. This may be due to my utter lack of tools to eval. periodicity. A request for which spun this thread. () I doesn't punish approximate fifths -- only admits or rejects. -Carl
Message: 1752 - Contents - Hide Contents Date: Fri, 5 Oct 2001 12:25 +01 Subject: Re: 3rd-best 11-limit temperament From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9pibm5+jcmt@xxxxxxx.xxx> Paul wrote:> While the top two temperaments in Graham's 11-limit list are > essentially 31-out-of-72 and 46-out-of-72, the third one has the > lowest complexity measure of all in this list. Can anyone discuss > this, in terms of unison vectors, etc.? > > > 12/53, 271.1 cent generator > > basis: > (1.0, 0.22594789337911292) > > mapping by period and generator: > [(1, 0), (0, 7), (3, -3), (1, 8), (3, 2)] > > mapping by steps: > [(31, 22), (49, 35), (72, 51), (87, 62), (107, 76)] > > unison vectors: > [[7, 8, 0, -7, 0], [-21, 3, 7, 0, 0], [-21, -2, 0, 0, 7]] > > highest interval width: 17 > complexity measure: 17 (22 for smallest MOS) > highest error: 0.007764 (9.317 cents)This originally came out of Dave Keenan's spreadsheet. Note that it's compatible because the period is an octave. It was noted then that it was the simplest approximation -- better than meantone at 18 and schismic at 19. 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. [[7, 8, 0, -7, 0] + 2*[-21, 3, 7, 0, 0] ------------------- [-35 14 14 -7 0] = 7*[ -5 2 2 -1 0] [-21, 3, 7, 0, 0] - 2*[-21, -2, 0, 0, 7]] --------------------- [21 7 7 0 -14] = 7*[ 3 1 1 0 -2] [[7, 8, 0, -7, 0] - 3*[-21, -2, 0, 0, 7]] --------------------- [ 70 14 0 -7 -21] = 7*[ 10 2 0 -1 -3] so we have new unison vectors [-5 2 2 -1 0] [ 3 1 1 0 -2] [10 2 0 -1 -3] I don't know what you were planning to do with them. To check the determinants |3 -1 0 0| |2 2 -1 0| = -31 |1 1 0 -2| |2 0 -1 -3| |0 2 -2 0| |2 2 -1 0| = 22 |1 1 0 -2| |2 0 -1 -3| and the adjoint of | 1 0 2 -2 0| |-4 4 -1 0 0| |-5 2 2 -1 0| | 3 1 1 0 -2| |10 2 0 -1 -3| is | -31 -22 38 -36 24| | -49 -35 60 -57 38| | -72 -51 88 -84 56| | -87 -62 107 -102 68| |-107 -76 131 -124 83|> 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) mapping by period and generator: [(2, 0), (3, 2), (5, -4), (5, 7), (7, -1)] mapping by steps: [(46, 22), (73, 35), (107, 51), (129, 62), (159, 76)] unison vectors: [[-3, -1, -1, 0, 2], [4, 0, -2, -1, 1], [0, -5, 1, 2, 0]] highest interval width: 11 complexity measure: 22 (24 for smallest MOS) highest error: 0.007005 (8.406 cents) The increase in complexity is greater than the improved accuracy. It's as complex as Miracle, but more than double the error! It doesn't make limit11.key because it isn't unique. Ah! It is in limit11.mos. Note that the diaschismic temperament including 46 and 58 does make the 13-limit list (right at the bottom of mine) as does the 94+41 schismic. Graham
Message: 1753 - Contents - Hide Contents Date: Fri, 5 Oct 2001 21:05 +01 Subject: Re: 3rd-best 11-limit temperament From: graham@xxxxxxxxxx.xx.xx Paul Erlich wrote:>> highest interval width: 11 >> complexity measure: 22 (24 for smallest MOS) >> 24? What about the 22-tone MOS?The number of otonal (or utonal) complete chords is always the number of notes in the scale minus the complexity measure. So 24 notes gives you 2 otonalities. 22 notes would give you no complete chords. That follows from 22-22=0. Graham
Message: 1754 - Contents - Hide Contents Date: Fri, 5 Oct 2001 16:04:23 Subject: Re: 3rd-best 11-limit temperament From: manuel.op.de.coul@xxxxxxxxxxx.xxx Graham wrote:>That's roughly what my Python module does, and Manuel's welcome to take >that code as inspiration. <# Temperament finding library -- definitions * [with cont.] (Wayb.)>It would be a good addition, however it's quite a big piece of code. So probably I'd be quicker to rethink the algorithm myself. I already have the code for minimax temperament in Ada, although I'm not sure yet if it can be applied. I need to let this stuff sink in too. I'd also want to make the set of consonant p-limit intervals user definable. Manuel
Message: 1755 - Contents - Hide Contents Date: Fri, 05 Oct 2001 20:17:48 Subject: Re: Tetrachordality and Scala From: Paul Erlich --- In tuning-math@y..., "Carl Lumma" <carl@l...> 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 . . . >> Do we want to enforce scale order on the transposed pitches, or > just let them fall where they are closest to the original pitches?Don't know what you mean by "enforce scale order".> Earlier in this thread, I claimed we don't care about scale order, > and I still claim it. But I was enforcing order in the example > in my last mail. > > I don't like your definition of omnitetrachordality because: > > () I can't tell if it's equivalent to What We Want^TM. That is, > I can't tell what sort of semi-periodicity it enforces. This > may be due to my utter lack of tools to eval. periodicity. A > request for which spun this thread.Hmmm . . .> > () I doesn't punish approximate fifths -- only admits or rejects. > Same here.
Message: 1756 - Contents - Hide Contents Date: Fri, 05 Oct 2001 20:20:19 Subject: Re: 3rd-best 11-limit temperament From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> Paul Erlich wrote: >>>> highest interval width: 11 >>> complexity measure: 22 (24 for smallest MOS) >>>> 24? What about the 22-tone MOS? >> The number of otonal (or utonal) complete chords is always the number of > notes in the scale minus the complexity measure. So 24 notes gives you 2 > otonalities. 22 notes would give you no complete chords. That follows > from 22-22=0.So 22-out-of-46 MOS gives you no hexads? That's odd, since the 22-out- of-46 omnitetrachordal scale (which is very similar) does. This may be the first time we're seeing an omnitetrachordal scale look harmonically better than its MOS counterpart. Veddy veddy interresteeng.
Message: 1757 - Contents - Hide Contents Date: Fri, 05 Oct 2001 20:25:52 Subject: Re: Tetrachordality and Scala From: Carl Lumma>> >o we want to enforce scale order on the transposed pitches, or >> just let them fall where they are closest to the original pitches? >> Don't know what you mean by "enforce scale order".() Take the scale () Transpose it by 3:2. () Now compare the pitches of the two scales. 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. The question is, when comparing the scales, do we insist on lining up the pitches in order, for some rotation of the transposed scale. Or do allow a re-ordering of pitches to get the closest matches. For CS scales with a 3:2 in them, in order will always be the best order, I think.>> () I doesn't punish approximate fifths -- only admits or rejects. >> > Same here.That was supposed to be "it doesn't", referring to your measure. As you can see above, I want to punish approximate fifths, or at least formalize what we will allow as an approximate fifth. -Carl
Message: 1758 - Contents - Hide Contents Date: Fri, 5 Oct 2001 21:33 +01 Subject: Re: 3rd-best 11-limit temperament From: graham@xxxxxxxxxx.xx.xx Paul wrote:> What's limit11.mos?It's one of a series of files on my website that use the smallest MOS as the complexity measure in the figure of demerit. Check back through this forum and you'll find the discussion. It happens that the 46+22 temperament, with a smallest MOS of 24, performs much better here than with its standard complexity measure of 22. Graham
Message: 1759 - Contents - Hide Contents Date: Fri, 05 Oct 2001 20:35:23 Subject: Re: Tetrachordality and Scala From: Paul Erlich --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:>>> Do we want to enforce scale order on the transposed pitches, or >>> just let them fall where they are closest to the original pitches? >>>> Don't know what you mean by "enforce scale order". >> () Take the scale > () Transpose it by 3:2. > () Now compare the pitches of the two scales. > > 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. > > The question is, when comparing the scales, do we insist on lining > up the pitches in order, for some rotation of the transposed scale. > Or do allow a re-ordering of pitches to get the closest matches. > For CS scales with a 3:2 in them, in order will always be the best > order, I think. >>>> () I doesn't punish approximate fifths -- only admits or rejects. >>> >> Same here. >> That was supposed to be "it doesn't", referring to your measure. > As you can see above, I want to punish approximate fifths, or at > least formalize what we will allow as an approximate fifth. > > -CarlHow about a rule of thumb. 1.2% is what I'm currently using for both harmonic entropy and for these kinds of judgments. That's 20.775¢, or about 1 step in 58-tET.
Message: 1760 - Contents - Hide Contents Date: Fri, 05 Oct 2001 20:40:53 Subject: Question From: Paul Erlich What is the mapping from generators to primes in the 46-out-of-72 temperament?
Message: 1761 - Contents - Hide Contents Date: Fri, 05 Oct 2001 22:38:38 Subject: Re: Tetrachordality and Scala From: Carl Lumma>How about a rule of thumb. 1.2% is what I'm currently using for >both harmonic entropy and for these kinds of judgments. That's >20.775¢, or about 1 step in 58-tET.I'm perfectly happy to use a binary function like this, because badness doesn't continue to increase smoothly as you deviate from the 3:2 anyway -- you hit a wall before going towards 7:5, or whatever. How does it fit into the rest of the procedure, though? For cleanness I still like multiplying by a perfect 3:2 and seeing how much stuff changes. -Carl
Message: 1762 - Contents - Hide Contents Date: Sat, 06 Oct 2001 06:22:28 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: >>> Paul H., does this make any sense?Gene, I have no idea what your response has to do with whether this: "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." makes any sense or not, but I'll try to follow what you wrote anyway.> > 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?> 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.Can you give the "for musician dummies" version of this statement?> > 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).OK, I think I understand this, but what do tetrads have to do with what I was talking about?> This generalizes easily to any p-limit, with the caveat that treating > all odd primes the same makes progressively less sense.Well you know I agree with you on that!
Message: 1763 - Contents - Hide Contents Date: Sat, 6 Oct 2001 17:07 +01 Subject: Re: nugget from 4/21/00: first glimpse of "torsion"??? From: graham@xxxxxxxxxx.xx.xx> --- 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 0This whole matrix needs to be multiplied through by -1 so that the intervals are all small, instead of slightly smaller than an octave.> Instead of being an approximation of 20-tET, it's an double > approximation of > 10-tET with 539:540 and 880:881 pairs. ... > --- End forwarded message --- > > discuss . . .We know all about these now. It's like that 24 note periodicity block defining 12-equal. I've added it to the test cases at <#!/usr/local/bin/python * [with cont.] (Wayb.)> and <http.microtonal.co.uk/vectors.out> if you're interested. Is this "torsion" then? Graham
Message: 1764 - Contents - Hide Contents Date: Sat, 06 Oct 2001 06:28:32 Subject: Re: nugget from 4/21/00: first glimpse of "torsion"??? From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- 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,Wait a minute Gene. Although I did, for some strange reason, list six unison vectors, I only included five in the Fokker matrix. So what if I only had those five? Would I have to try all possible products and quotients of pairs out of that set of five?> 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.If you plug in indeterminants a, b, c, d, e, and f, how do you get something in terms of h to come out? I'd like to be able to do this calculation . . . but again, starting with only enough unison vectors to construct the Fokker matrix.
Message: 1765 - Contents - Hide Contents Date: Sat, 6 Oct 2001 17:07 +01 Subject: Re: 3rd-best 11-limit temperament From: graham@xxxxxxxxxx.xx.xx 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. Graham
Message: 1766 - Contents - Hide Contents Date: Sat, 06 Oct 2001 06:59:16 Subject: Re: 72 owns the 11-limit From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> I just finished an interesting calculation, where I took the nine > smallest superparticular ratios belonging to the 11-limit, namely > 225/224, 243/242, 385/384, 441/440, 540/539, 2401/2400, 3025/3024, > 4375/4374 and 9801/9800. I then checked all 126 4-element subsets, > finding 69 of rank 4. Astonisingly, all 69 had the 72 et as the > generator of its null space; in 59 of those cases the determinant of > the 5x5 matrix with 5 indeterminates for the first row was +-h72.So 10 had torsion . . . I get 144 as the Fokker determinant for these 10. Now . . . as to the justification of this calculation . . . I understand why superparticulars for unison vectors might be appealing in their own right, but when joining them together in triples, don't some of the triplets correspond to a basis that is more "skewed" than that of some triplet that includes at least one non-superparticular? What I mean by "skewed" is that the unison vectors all lie at a very small angle to some subspace (I'm thinking especially in my triangular lattice), so that certain consonant intervals and chords (small structures in the lattice) will require one to invoke many of the unison vectors to construct them . . . thus some effective unison vector, a product of some powers of some of the nominal unison vectors, will be more immediately relevant . . . and these won't necessarily be superparticular. Is there any way to determine the "canonical" set of unison vectors for a PB . . . perhaps this would be the set that minimizes the sizes of the numbers in the ratios representing the unison vectors . . . ? In the above I'm thinking of _all_ the unison vectors as commatic . . . but what if one is chromatic?
Message: 1767 - Contents - Hide Contents Date: Sat, 6 Oct 2001 17:07 +01 Subject: Re: Question From: graham@xxxxxxxxxx.xx.xx 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)] 25:24, 1029:1024 and 225:224 fail, apparently because it wants a half-octave generator, but doesn't give the usual clue. I think it should come out like this: 1/9, 66.7 cent generator basis: (0.5, 0.0555981053341) mapping by period and generator: [[2, 0], [4, -3], [5, 1], [6, 1]] mapping by steps: [(10, 8), (17, 13), (26, 21), (31, 25)] unison vectors: [[1, 0, 2, -2], [10, -1, -2, -1]] highest interval width: 4 complexity measure: 8 (10 for smallest MOS) highest error: 0.248243 (297.892 cents) Graham
Message: 1768 - Contents - Hide Contents Date: Sat, 06 Oct 2001 07:02:06 Subject: Re: Question From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Thanks a lot Bob.You're welcome. Who's 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 . . .I think you are asking for trouble in the form of torsion with this 20 business. Why 20? I could check all 7 choose 3 subsets of the commas you give below, but is there a reason to think this will work?
Message: 1769 - Contents - Hide Contents Date: Sat, 06 Oct 2001 18:33:02 Subject: Re: 72 owns the 11-limit From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Just for fun, I tried the 10 smallest, and I got the following Fokker > determinants: > > freq. determinant > 1 7 > 1 10 > 1 12 > 1 19 > 1 24 > 1 54 > 1 68 > 1 99 > 1 116 > 3 22 > 3 53 > 3 80 > 3 126 > 4 62 > 5 27 > 5 58 > 8 34 > 8 46 > 10 144 > 24 31 > 60 72 > 65 0 > > Which of these have torsion? (or how do I find out for myself)It's a pretty safe guess that all the ones with determinants 24, 54, 116, 126, 62 and 144 have torsion, but to really find out you should add the 2 column and take the gcd of the minors, or equivalently, add the 2 column and a row of indeterminants, if whatever you are using allows you to work with those, and take the gcd of the coefficients.> If I take all of the 11-limit superparticular ratios smaller than > 20.7¢ (thus those smaller than 17.6¢), I get > freq. determinant > 1 9 > 1 11 > 3 23 > 3 37Some fairly exotic scale possibilities in here!> So maybe 31 really owns the 11-limit? :)It makes a good run at it, but smaller is better when it comes to commas.
Message: 1770 - Contents - Hide Contents Date: Sat, 06 Oct 2001 07:20:32 Subject: Re: 72 owns the 11-limit From: Paul Erlich> --- In tuning-math@y..., genewardsmith@j... wrote:>> I just finished an interesting calculation, where I took the nine >> smallest superparticular ratios belonging to the 11-limit, namely >> 225/224, 243/242, 385/384, 441/440, 540/539, 2401/2400, 3025/3024, >> 4375/4374 and 9801/9800. I then checked all 126 4-element subsets, >> finding 69 of rank 4. Astonisingly, all 69 had the 72 et as the >> generator of its null space; in 59 of those cases the determinant > of>> the 5x5 matrix with 5 indeterminates for the first row was +-h72.Just for fun, I tried the 10 smallest, and I got the following Fokker determinants: freq. determinant 1 7 1 10 1 12 1 19 1 24 1 54 1 68 1 99 1 116 3 22 3 53 3 80 3 126 4 62 5 27 5 58 8 34 8 46 10 144 24 31 60 72 65 0 Which of these have torsion? (or how do I find out for myself) If I take all of the 11-limit superparticular ratios smaller than 20.7¢ (thus those smaller than 17.6¢), I get freq. determinant 1 1 1 9 1 11 1 29 1 49 1 51 1 60 1 64 1 65 1 79 1 91 1 96 2 90 2 116 3 20 3 23 3 24 3 37 3 48 3 50 3 68 3 80 3 99 3 126 5 28 5 82 8 10 8 18 11 4 11 144 13 58 14 45 14 54 17 15 17 53 19 7 19 19 20 38 21 12 21 26 24 41 24 62 34 14 40 22 40 46 41 27 45 8 46 34 63 72 187 0 188 31 If I take all of the 11-limit superparticular ratios smaller than 35¢, I get freq. determinant 1 30 1 59 1 63 1 65 1 66 1 76 1 79 1 81 1 86 1 91 1 92 1 93 2 56 2 78 2 96 2 116 3 13 3 21 3 32 3 40 3 47 3 49 3 61 3 80 3 99 3 126 4 1 4 29 4 33 4 51 4 60 5 11 5 42 5 82 5 88 6 108 7 90 9 64 11 6 11 39 11 48 11 50 11 144 14 58 16 37 16 52 17 53 18 23 18 36 20 2 20 9 27 3 27 41 32 68 33 45 39 18 40 44 40 46 40 62 43 5 47 28 53 38 54 16 54 54 65 72 68 17 81 26 83 20 90 4 91 24 92 19 131 15 133 27 145 34 164 7 175 14 176 22 198 8 218 10 251 12 298 31 582 0 Which of these have torsion? (or how do I find out for myself) So maybe 31 really owns the 11-limit? :)
Message: 1771 - Contents - Hide Contents Date: Sat, 06 Oct 2001 18:40:08 Subject: Re: 72 owns the 11-limit From: genewardsmith@xxxx.xxx --- In tuning-math@y..., genewardsmith@j... wrote:> It's a pretty safe guess that all the ones with determinants 24, 54, > 116, 126, 62 and 144 have torsion...Come to think of it, this is the 11-limit and the 24 may *not* have torsion; anyway, the only way to be sure is to check.
Message: 1772 - Contents - Hide Contents Date: Sat, 06 Oct 2001 07:23:42 Subject: Re: Question From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> Thanks a lot Bob. >> You're welcome. Who's Bob?Oops, sorry Gene. It's late!>>> 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 . . . >> I think you are asking for trouble in the form of torsion with this > 20 business. Why 20? I could check all 7 choose 3 subsets of the > commas you give below,8 choose 3?> but is there a reason to think this will work?If there is one without torsion, then we have something to look at that is potentially as interesting as Blackjack. Do you have a reason do think there won't be one?
Message: 1773 - Contents - Hide Contents Date: Sat, 06 Oct 2001 21:09:12 Subject: Re: nugget from 4/21/00: first glimpse of "torsion"??? From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:>> --- 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 >> This whole matrix needs to be multiplied through by -1 so that the > intervals are all small, instead of slightly smaller than an octave.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= t the determinant, or anything else significant . . . will it?> > Is this "torsion" then?Yes . . . Gene explained this a while back . . . I need to go out and get a= n abstract algebra textbook . . . I used to be a champion math student :(
Message: 1774 - Contents - Hide Contents Date: Sat, 06 Oct 2001 07:27:00 Subject: Re: Question From: Paul Erlich --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> --- In tuning-math@y..., genewardsmith@j... wrote:>> I could check all 7 choose 3 subsets of the >> commas you give below, >> 8 choose 3? >Wait a minute . . . there are only 8 cases to check . . . I gave 8 triplets with a Fokker determinant of 20.
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