4000 4050 4100 4150 4200 4250 4300 4350 4400 4450 4500 4550 4600 4650 4700 4750 4800 4850 4900 4950 5000 5050 5100 5150 5200 5250 5300 5350 5400 5450 5500 5550 5600 5650 5700 5750 5800 5850 5900 5950 6000 6050 6100 6150 6200 6250 6300 6350 6400 6450 6500 6550
4800 - 4825 -
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Message: 4825 Date: Tue, 09 Oct 2001 16:10:46 Subject: Re: Searching for interesting 7-limit MOS scales From: Paul Erlich --- In tuning-math@y..., graham@m... wrote: > In-Reply-To: <9pt7ch+ucrd@e...> > 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 *>, 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. > If that's not too much trouble, I'd be in your debt!
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Message: 4826 Date: Tue, 09 Oct 2001 16:14:34 Subject: Re: validity measure From: Paul Erlich Using the validity measure _without_ a "straightness" measure is not recommended. All those 1-tone PBs look awfully good on the validity measure!
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Message: 4827 Date: Tue, 09 Oct 2001 16:27:10 Subject: Re: 7-limit PBs From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote: > 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: > [...] > > [1.055126953, [1029/1024, 126/125, 25/24], b+a+3*d+2*c] [...] > [2.253538886, [1029/1024, 36/35, 25/24], 17*b+11*a+25*c+31*d] > [...] > > [3.009084772, [1728/1715, 28/27, 25/24], 37*d+30*c+21*b+13*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? There are "valid" sets with 1 step (see above). So what? This sort of "validity" doesn't do much for me. There _is_ a decent MOS for 7-limit with 11 steps: 11 note chain-of-minor-thirds scale * But clearly not all the numbers in your list are to be taken seriously.
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Message: 4828 Date: Tue, 09 Oct 2001 05:26:14 Subject: Re: Searching for interesting 7-limit MOS scales From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: Here is the first triple on your list, 81/80, 50/49, 25/24 worked out as an example; let's see if you have any questions. (1) Step one is to find the val generating the dual group, and that means finding the minors or something equivalent, such as taking the determinant of [ a b c d] [-4 4 -1 0] [ 1 0 2 -2] [-3 -1 2 0] which is g = 14a + 22b + 32c + 39d. (2) Step two is finding a val v such that v(81/80) = v(50/49) = 0, and v(25/24) = 1. If we solve the linear system of equations -4a + 4b - c = 0 a + 2c - 2d = 0 -3a - b + 2c = 1 we get a = (7c-4)/16, b = (11c-4)/16, d = (39c-4)/32; since we want integter solutions we want 39c = 4 (mod 32), which means c = -4 will work. This gives us -2a-3b-4c-5d; we get the same linear span by adding this to our previous val, getting h12 = 12a + 19b + 28c + 34d. Note that while g+h12=h26, g is not h14, since g(5)=32, not 33. (3) We now want to find A and B such that A^g(q) B^h12(q) give good appoximations to 3,5,7,5/3,7/3 and 7/5. Since I don't want to mess around finding out how Maple's linear programming routines work and since least squares is so easy, I'll use that. Optimizing by least squares and assuming octaves are pure, I get A = 38.098 cents and B = 55.557 cents. (4) 14/12 = 7/6 = 1+1/6, and the convergent to 7/6 is 1. We therefore might choose A + B = 93.651 cents as our generator; this happens to be reasonably close to 1200/13 = 92.308 cents; not much of a surprise since g+h12=h26 and gcd(12,14)=2. If we use 1200/13 as our generator, with a period of half an octave, we have: (5) h26(3) = 41 = 2 (mod 13) h26(5) = 60 = -5 (mod 13) h26(7) = 73 = -5 (mod 13) The complexity is therefore 2*7 = 14. The example is actually a little too easy at this point, since the generator is 1/13 of an octave I don't need to do any mod 13 division.
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Message: 4829 Date: Tue, 09 Oct 2001 01:47:02 Subject: Re: Searching for interesting 7-limit MOS scales From: Paul Erlich From my ET posting today, I can try to understand what ETs some of these MOS scales (and others with the same generators) might be embeddable in: > > #1: 14 notes; commas 81:80, 50:49; chroma 25:24 (Erlich 14-of-~26) 26-tET of course! > #2: 31 notes; commas 1029:1024, 245:243; chroma 25:24 (v. improper!) 46-tET, 36-tET, 41-tET . . . if it's got more than 31 notes, try 31 anyway! > #3: 17 notes; commas 245:243, 64:63; chroma 25:24 22-tET, 27-tET. > #4: 19 notes; commas 126:125, 81:80; chroma 49:48 31-tET -- is this the famous 19-out-of-31? > #5: 15 notes; commas 126:125, 64:63; chroma 28:27 27-tET -- this must be the 15-out-of-27 Gene was talking about when he first joined this list. > #6: 21 notes; commas 1029:1024, 81:80; chroma 25:24 (not blackjack!) 26-tET, 31-tET > #7: 14 notes; commas 81:80, 50:49; chroma 49:48 (Erlich 14-of-~26) Same as #1. > #8: 12 notes; commas 64:63, 50:49; chroma 36:35 (Erlich 12-of-22) 22-tET, of course! > #9: 19 notes; commas 225:224, 81:80; chroma 49:48 (19-out-of-31???) 31-tET > #10: 14 notes; commas 245:243, 50:49; chroma 25:24 22-tET, 36-tET . . . what's this? > #11: 12 notes; commas 81:80, 50:49; chroma 36:35 26-tET, 14-tET . . . complement of Erlich 14-out-of-26 > #12: 15 notes; commas 126:125, 64:63; chroma 49:48 Same as #5 > #13: 10 notes; commas 64:63, 50:49; chroma 25:24 (Erlich 10-of-22) 22-tET, of course! > #14: 19 notes; commas 225:224, 126:125, chroma 49:48 31-tET > #15: 10 notes; commas 64:63, 50:49; chroma 28:27 (Erlich 10-of-22) Same as #13 > #16: 19 notes; commas 245:243, 126:125; chroma 49:48 46-tET, 27-tET . . . some kind of 8-tone scheme behind this . . . > #17: 23 notes; commas 2401:2400, 126:125; chroma 28:27 31-tET, 27-tET . . . some kind of 4-tone scheme behind this . . . kleismic??? > #18: 14 notes; commas 245:243, 81:80; chroma 25:24 NOTHING ON MY LIST! WHAT IS THIS? > #19: 16 notes; commas 245:243, 225:224; chroma 21:20 41-tET, 22-tET, 19-tET . . . is this Graham's MAGIC thing? > #20: 14 notes; commas 245:243, 50:49; chroma 49:48 Same as #10 > #21: 12 notes; commas 126:125, 64:63; chroma 36:35 Same as #5 > #22: 16 notes; commas 1029:1024, 50:49; chroma 36:35 36-tET, 26-tET . . . curious > #23: 19 notes; commas 245:243, 225:224; chroma 49:48 Same as #19 but longer chain > #24: 12 notes; commas 225:224, 50:49; chroma 36:35 22-tET -- Erlich 12-out-of-22 > #25: 10 notes; commas 64:63, 50:49; chroma 49:48 [Erlich 10-of-22] Same as #13 > #26: 12 notes; commas 126:125, 81:80; chroma 36:35 Complement of #4? > #27: 21 notes; commas 1029:1024, 225:224; chroma 36:35 (Blackjack) 41-tET, 31-tET > #28: 12 notes; commas 225:224, 64:63; chroma 36:35 22-tET > #29: 10 notes; commas 225:224, 50:49; chroma 25:24 (Erlich 10-of-22) 22-tET > #30: 17 notes; commas 2401:2400, 64:63; chroma 36:35 27-tET . . . some kind of 10-tone scheme behind this . . . > #31: 10 notes; commas 225:224, 50:49; chroma 28:27 (Erlich 10-of-22) 22-tET > #32: 12 notes; commas 225:224, 81:80; chroma 36:35 Complement of #9? > #33: 21 notes; commas 2401:2400, 225:224; chroma 36:35 (Blackjack) 41-tET, 31-tET Now, what _are_ the (optimal) generators?
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Message: 4830 Date: Tue, 09 Oct 2001 02:00:02 Subject: Re: New file uploaded to tuning-math From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote: > 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. The sort of thing described in my 22-tET paper <http://www- * math.cudenver.edu/~jstarret/22ALL.pdf>, things like blackjack, and everything in-between. Is there an 11-limit analogue to the decatonic system of my paper? What if we relax one or two of the rules in my paper?
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Message: 4831 Date: Tue, 09 Oct 2001 02:11:20 Subject: Breedsma, kalisma, ragisma, schisma From: genewardsmith@xxxx.xxx --- 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. > Using the xenisma in addition gives the same PB. One can extend this to an 11-limit notation by adding 385/384, 441/440 or 540/539; unfortunately, none of these seem to have names. We have: (32805/32768, 2401/2400, 4375/4374, 9801/9800, 385/384)^(-1) = [-h72, -h118, -h212, -h171, h342] (ditto, 441/440)^(-1) = [-h72, -h118, h130, h171, h342] (ditto, 540/539)^(-1) = [-h72, h224, h130, h171, h342]
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Message: 4832 Date: Tue, 09 Oct 2001 02:57:35 Subject: Re: Question From: genewardsmith@xxxx.xxx --- In tuning-math@y..., graham@m... wrote: > 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. 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] 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. > 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? 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. > 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.
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Message: 4833 Date: Tue, 09 Oct 2001 04:57:33 Subject: Re: Searching for interesting 7-limit MOS scales From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: > No one calculated the information I requested (generators, mappings > from primes to generators, minimax error). It seemed like the most reasonable way to do that would be to write a program, and I hadn't done that yet. 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. Let me work one example, and see if you have any questions. The first one on your list is 81/80, 50/49, 25/24 so I picked that one. (1) Step one is to find the val generating the dual group, and that means finding the minors or something equivalent, such as taking the determinant of [ a b c d] [-4 4 -1 0] [ 1 0 2 -2] [-3 -1 2 0] which is g = 14a + 22b + 32c + 39d. (2) Step two is finding a val v such that v(81/80) = v(50/49) = 0, and v(25/24) = 1. If we solve the linear system of equations -4a + 4b - c = 0 a + 2c - 2d = 0 -3a - b + 2c = 1 we get a = (7c-4)/16, b = (11c-4)/16, d = (39c-4)/32; since we want integter solutions we want 39c = 4 (mod 32), which means c = -4 will work. This gives us -2a-3b-4c-5d; we get the same linear span by adding this to our previous val, getting h12 = 12a + 19b + 28c + 34d. Note that while g+h12=h26, g is not h14, since g(5)=32, not 33. (3) We now want to find A and B such that A^g(q) B^h12(q) give good appoximations to 3,5,7,5/3,7/3 and 7/5. Since I don't want to mess around finding out how Maple's linear programming routines work and since least squares is so easy, I'll use that. Optimizing by least squares and assuming octaves are pure, I get A = 38.098 cents and B = 55.557 cents. (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. (5) We have h26(3) = 41 = 2 (mod 13) h26(5) = 60 = -5 (mod 13) h26(7) = 73 = -5 (mod 13) The complexity is therefore 2*7 = 14.
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Message: 4834 Date: Tue, 09 Oct 2001 05:26:14 Subject: Re: Searching for interesting 7-limit MOS scales From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: Here is the first triple on your list, 81/80, 50/49, 25/24 worked out as an example; let's see if you have any questions. (1) Step one is to find the val generating the dual group, and that means finding the minors or something equivalent, such as taking the determinant of [ a b c d] [-4 4 -1 0] [ 1 0 2 -2] [-3 -1 2 0] which is g = 14a + 22b + 32c + 39d. (2) Step two is finding a val v such that v(81/80) = v(50/49) = 0, and v(25/24) = 1. If we solve the linear system of equations -4a + 4b - c = 0 a + 2c - 2d = 0 -3a - b + 2c = 1 we get a = (7c-4)/16, b = (11c-4)/16, d = (39c-4)/32; since we want integter solutions we want 39c = 4 (mod 32), which means c = -4 will work. This gives us -2a-3b-4c-5d; we get the same linear span by adding this to our previous val, getting h12 = 12a + 19b + 28c + 34d. Note that while g+h12=h26, g is not h14, since g(5)=32, not 33. (3) We now want to find A and B such that A^g(q) B^h12(q) give good appoximations to 3,5,7,5/3,7/3 and 7/5. Since I don't want to mess around finding out how Maple's linear programming routines work and since least squares is so easy, I'll use that. Optimizing by least squares and assuming octaves are pure, I get A = 38.098 cents and B = 55.557 cents. (4) 14/12 = 7/6 = 1+1/6, and the convergent to 7/6 is 1. We therefore might choose A + B = 93.651 cents as our generator; this happens to be reasonably close to 1200/13 = 92.308 cents; not much of a surprise since g+h12=h26 and gcd(12,14)=2. If we use 1200/13 as our generator, with a period of half an octave, we have: (5) h26(3) = 41 = 2 (mod 13) h26(5) = 60 = -5 (mod 13) h26(7) = 73 = -5 (mod 13) The complexity is therefore 2*7 = 14. The example is actually a little too easy at this point, since the generator is 1/13 of an octave I don't need to do any mod 13 division.
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Message: 4835 Date: Tue, 09 Oct 2001 05:51:43 Subject: torsion (was: Re: 72 owns the 11-limit) From: genewardsmith@xxxx.xxx --- 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.
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Message: 4836 Date: Tue, 09 Oct 2001 06:01:22 Subject: Re: Searching for interesting 7-limit MOS scales From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: > 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? No, in the symmetrical lattice. However, I wouldn't pay much attention to this; for one thing there are six eight triangles for every three verticies, and which one to you pick? > > 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? With a high number like that, it seems likely that the val does not order the block linearly, and hence that "Paul" doesn't work. > > 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? No, lengths with the Euclidean distance where 1, 3 5 and 7 are the verticies of a regualar tetrahedron. > > 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? We want volume to correspond to number of lattice points in a region, so we want to make the vectors 3, 5, and 7, which give us the identity matrix, define a volume of 1; this means a cube with sides of length 1 has a measure of sqrt(2), but there is no reason to let that worry us. The 3, 5, 7 is not a rectangular prism, but a parallepidped; we have 3.5 = 3.7 = 5.7 = 1/2 so they are at 60 degree angles.
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Message: 4837 Date: Wed, 10 Oct 2001 18:37:34 Subject: Re: 7-limit PBs From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote: > --- In tuning-math@y..., genewardsmith@j... wrote: > > > 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] > > I decided to check on this one, with the best validity score of any > of these 11-note candidates. If I look at the list I uploaded, I have > two notations on it which complete the above: > > [3136/3125, 245/243, 50/49, 36/35] [v, 12, v, v] > [[-8, -13, -19, -23], [12, 19, 28, 34], [-11, -18, -26, -32], > [30, 48, 70, 85]] > > [3136/3125, 245/243, 36/35, 25/24] [v, v, v, v] > [[3, 5, 7, 9], [1, 1, 2, 2], [8, 12, 18, 21], [11, 18, 26, 32]] > > In the first case, we might look at the corresponding 11-30 > temperament, and in the second, at the 11-8 temperament; these > correspond to what look like reasonable steps rather than commas. > > Doing a least-squares for the 11-8 gives us generators of 70.06 cents > for the 11 val and 53.67 cents for the 8 val; this gives us a 3 which > is 3.13 cents sharp, a five 1.26 cents sharp and a 7 0.11 cents > sharp, so we are looking at something which might end up sounding > like music. To convert from the 11-8 basis to one of an octave plus a > generator, we find the convergent to 11/8, which is 4/3. The fact > that this is a convergent means that the transformation matrix > > [11 8] > [ 4 3] > > which transforms from the octave-generator basis to the 11-8 basis is > unimodular; we can therefore invert it, getting > > [ 3 -8] > [-4 11] > > which we may use to transform from the 11-8 basis to the octave- > generator basis. > > Our generator is therefore 4*70.06 + 3*53.67 = 441.24 cents; to a > good approximation this is 4*(4/68) + 3*(3/68) = 25/68 in terms of > octaves. Convergents to 25/68 are 3/8, 4/11 and 7/19, suggesting the > 8, 11 and 19 MOS with this generator in the 68-et as scale > possibilities; if we add the semiconvergents to the list we get 30 > and 49 as well. > > In the 68 division, we have 3, 5, 7, 5/3, 7/3 and 7/5 approximated by > 108, 158, 191, 50, 83 and 33 respectively; if we find n/25 mod 68 for > all of these we get -12, -10, -25, 2, -13, -15 for a complexity of > 27; however note the many 5/3 we are going to get--the generator is > approximately sqrt(5/3) = 442.18 cents. > > All of this tells me that this example at least is far from being > completely looney. If the complexity is 27 for an 11-note scale, this tells me the PB is very "skewed" in the lattice -- there isn't even enough room for "half a tetrad".
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Message: 4838 Date: Wed, 10 Oct 2001 18:40:18 Subject: Re: Searching for interesting 7-limit MOS scales From: Paul Erlich --- In tuning-math@y..., graham@m... wrote: > That produces the output file > paul.7limit.out which you can also find at my website. > > <404 Not Found *> This URL doesn't seem to work :(
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Message: 4839 Date: Wed, 10 Oct 2001 20:20 +0 Subject: Re: Searching for interesting 7-limit MOS scales From: graham@xxxxxxxxxx.xx.xx Paul wrote: > --- In tuning-math@y..., graham@m... wrote: > > > That produces the output file > > paul.7limit.out which you can also find at my website. > > > > <404 Not Found *> > > This URL doesn't seem to work :( Oops! Should be paul.limit7.out, etc. I'll get it all correct at <Unison vector to MOS script *>. Graham
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Message: 4840 Date: Wed, 10 Oct 2001 06:02:17 Subject: Re: 7-limit PBs From: genewardsmith@xxxx.xxx --- In tuning-math@y..., genewardsmith@j... wrote: > 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] I decided to check on this one, with the best validity score of any of these 11-note candidates. If I look at the list I uploaded, I have two notations on it which complete the above: [3136/3125, 245/243, 50/49, 36/35] [v, 12, v, v] [[-8, -13, -19, -23], [12, 19, 28, 34], [-11, -18, -26, -32], [30, 48, 70, 85]] [3136/3125, 245/243, 36/35, 25/24] [v, v, v, v] [[3, 5, 7, 9], [1, 1, 2, 2], [8, 12, 18, 21], [11, 18, 26, 32]] In the first case, we might look at the corresponding 11-30 temperament, and in the second, at the 11-8 temperament; these correspond to what look like reasonable steps rather than commas. Doing a least-squares for the 11-8 gives us generators of 70.06 cents for the 11 val and 53.67 cents for the 8 val; this gives us a 3 which is 3.13 cents sharp, a five 1.26 cents sharp and a 7 0.11 cents sharp, so we are looking at something which might end up sounding like music. To convert from the 11-8 basis to one of an octave plus a generator, we find the convergent to 11/8, which is 4/3. The fact that this is a convergent means that the transformation matrix [11 8] [ 4 3] which transforms from the octave-generator basis to the 11-8 basis is unimodular; we can therefore invert it, getting [ 3 -8] [-4 11] which we may use to transform from the 11-8 basis to the octave- generator basis. Our generator is therefore 4*70.06 + 3*53.67 = 441.24 cents; to a good approximation this is 4*(4/68) + 3*(3/68) = 25/68 in terms of octaves. Convergents to 25/68 are 3/8, 4/11 and 7/19, suggesting the 8, 11 and 19 MOS with this generator in the 68-et as scale possibilities; if we add the semiconvergents to the list we get 30 and 49 as well. In the 68 division, we have 3, 5, 7, 5/3, 7/3 and 7/5 approximated by 108, 158, 191, 50, 83 and 33 respectively; if we find n/25 mod 68 for all of these we get -12, -10, -25, 2, -13, -15 for a complexity of 27; however note the many 5/3 we are going to get--the generator is approximately sqrt(5/3) = 442.18 cents. All of this tells me that this example at least is far from being completely looney.
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Message: 4841 Date: Wed, 10 Oct 2001 20:03:30 Subject: Re: Searching for interesting 7-limit MOS scales From: Paul Erlich --- In tuning-math@y..., graham@m... wrote: > Paul wrote: > > > --- In tuning-math@y..., graham@m... wrote: > > > > > That produces the output file > > > paul.7limit.out which you can also find at my website. > > > > > > <404 Not Found *> > > > > This URL doesn't seem to work :( > > Oops! Should be paul.limit7.out, etc. I'll get it all correct at > <Unison vector to MOS script *>. > > > Graham Thanks a bunch Graham. You're my hero!
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Message: 4842 Date: Wed, 10 Oct 2001 11:03 +0 Subject: Re: Searching for interesting 7-limit MOS scales From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9pv7i6+6ddv@xxxxxxx.xxx> Okay, I've made it even easier to run my unison vectors to temperament program. You can feed it an input file containing the ratios as ratios. So I've run it over the examples Paul gave in <Yahoo groups: /tuning-math/message/1223 *>. Code is at <############################################################################### *> <#!/usr/local/bin/python *> <import sys, vectors *> You need Numeric Python. From the latest ActivePython release, you do "ppm install Numeric". But that is quite a download. If you can find where Numeric Python lives you only need that and the standard Python distribution from <Python Language Website *> (or a minimal distribution without TKInter, if you can find one). Here's the example input file: <chroma 25:24 (Erlich 14-of-~26) *> It assumes sets of ratios are separated by a line with no ratios in it. If it's always going to be one set per line, the program can be simplified. I'm taking the chromatic UV as the *last* on the list, as that's the way you seem to do it. To run this example, "python findTemperament.py paul.7limit" in the folder that has the scripts and input file. That produces the output file paul.7limit.out which you can also find at my website. I've also run this over the vectors Gene posted recently. <404 Not Found *> <404 Not Found *> <404 Not Found *> Sometime I'll get it to use the octave-specific vectors. For now, it converts them to octave-equivalent, and then back again. Paul did ask about this one before: [(-1, 2, 0), (0, 2, -2), (-5, 1, 2)] 5/7, 434.0 cent generator basis: (0.5, 0.3616541669070521) mapping by period and generator: [(2, 0), (1, 3), (1, 5), (2, 5)] mapping by steps: [[8, 6], [13, 9], [19, 14], [23, 17]] unison vectors: [[1, 0, 2, -2], [0, -5, 1, 2]] highest interval width: 5 complexity measure: 10 (14 for smallest MOS) highest error: 0.014573 (17.488 cents) Graham
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Message: 4843 Date: Wed, 10 Oct 2001 20:04:42 Subject: Does Miracle--11 count? From: genewardsmith@xxxx.xxx If the 11-note miracle scale counts in the quest for interesting 11- note 7-limit scales, there are things like it. 1/11 is a semiconvergent to 7/72, and similarly 3/11 is a semiconvergent to 15/58, for instance. We get 2,2,11,2,2,11,2,2,11,2,11 as the step pattern. Of course, for Paul's real quest, which is for interesting 7-limit scales, we can go to more steps; 7/27 is the penultimate convergent to 15/58, and gives us a 6*2, 3, 6*2, 3, 6*2, 3, 5*2, 3 pattern. The point of 15/58, (or 17/46, or 24/65 or 56/135 in 2^(1/2), etc.) is that it has a relatively low 7-limit complexity.
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Message: 4844 Date: Wed, 10 Oct 2001 20:38:13 Subject: Re: Does Miracle--11 count? From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote: > If the 11-note miracle scale counts in the quest for interesting 11- > note 7-limit scales, Not really . . . there's not even one tetrad. > Of course, for Paul's real quest, which is for interesting 7-limit > scales, we can go to more steps; 7/27 is the penultimate convergent > to 15/58, and gives us a 6*2, 3, 6*2, 3, 6*2, 3, 5*2, 3 pattern. > The point of 15/58, (or 17/46, This fraction doesn't seem to agree with the others. > or 24/65 or 56/135 in 2^(1/2), etc.) is > that it has a relatively low 7-limit complexity. What is the complexity, and how low can we get the errors? Or did this come up already?
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Message: 4845 Date: Wed, 10 Oct 2001 21:36:32 Subject: Re: Searching for interesting 7-limit MOS scales From: Paul Erlich Thanks to Graham, I'm now in a position to answer my questions: > > > #3: 17 notes; commas 245:243, 64:63; chroma 25:24 > > 22-tET, 27-tET. Generator 710.7 cents -- this is one of the three sizes of fifth that Dave Keenan recognizes as generating a good 7-limit scale with a single chain of fifths wrapped within the octave: Harmonic errors in single-chain-of-equal-fifths tunings * > > > #4: 19 notes; commas 126:125, 81:80; chroma 49:48 > > 31-tET -- is this the famous 19-out-of-31? Sure is! > > > #5: 15 notes; commas 126:125, 64:63; chroma 28:27 > > 27-tET -- this must be the 15-out-of-27 Gene was talking about when > he first joined this list. It sure is! > > > #6: 21 notes; commas 1029:1024, 81:80; chroma 25:24 (not blackjack!) > > 26-tET, 31-tET This one is fascinating. It's nothing like Blackjack, yet it has the same complexity measure and same number of notes per octave, and only slightly larger errors. The optimal generator achieves proper MOSs at 5, 26, and 31 notes -- and then again only at 584 notes per octave! Score another one for 31-tET. > > > #9: 19 notes; commas 225:224, 81:80; chroma 49:48 (19-out-of- 31???) Yup. > > > #10: 14 notes; commas 245:243, 50:49; chroma 25:24 > > 22-tET, 36-tET . . . what's this? Interesting . . . a 7:9 generator in a half-octave . . . Hey Graham . . . why does #11 open in your output with '5/6' while the otherwise identical #1 opens with '1/7'? > > > #14: 19 notes; commas 225:224, 126:125, chroma 49:48 > > 31-tET Good ol' meantone again. > > > #16: 19 notes; commas 245:243, 126:125; chroma 49:48 > > 46-tET, 27-tET . . . some kind of 8-tone scheme behind this . . . Curious one this! 27-tET doesn't really do it justice . . . but I suppose I could live with it . . . > > > #17: 23 notes; commas 2401:2400, 126:125; chroma 28:27 > > 31-tET, 27-tET . . . some kind of 4-tone scheme behind this . . . > kleismic??? This one actually looks really good . . . yes, a minor third generator here . . . score another one for 31-tET! > > #18: 14 notes; commas 245:243, 81:80; chroma 25:24 > > NOTHING ON MY LIST! WHAT IS THIS? This works in 19-tET . . . > > > #22: 16 notes; commas 1029:1024, 50:49; chroma 36:35 > > 36-tET, 26-tET . . . curious This is quite an interesting one . . . after 26, the next proper MOS is at 110 . . . score one for 26-tET! > > > #26: 12 notes; commas 126:125, 81:80; chroma 36:35 > > Complement of #4? Yup! > > #30: 17 notes; commas 2401:2400, 64:63; chroma 36:35 > > 27-tET . . . some kind of 10-tone scheme behind this . . . Yup . . . score another for 27-tET . . . > > > #32: 12 notes; commas 225:224, 81:80; chroma 36:35 > > Complement of #9? Yup. Alright, I think I'm getting a classical guitar (which tolerates greater mistuning in the fifths than other guitars) outfitted with Mark Rankin's fingerboards, two of which will be in 26-tET and 27- tET. I already have 22-tET and 31-tET electric guitars.
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Message: 4846 Date: Wed, 10 Oct 2001 22:07:04 Subject: Re: Does Miracle--11 count? From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: > > Of course, for Paul's real quest, which is for interesting 7- limit > > scales, we can go to more steps; 7/27 is the penultimate convergent > > to 15/58, and gives us a 6*2, 3, 6*2, 3, 6*2, 3, 5*2, 3 pattern. > > The point of 15/58, (or 17/46, > This fraction doesn't seem to agree with the others. 17/46 is a completely different generator than 15/58, if that's what's worrying you. > > or 24/65 or 56/135 in 2^(1/2), etc.) is > > that it has a relatively low 7-limit complexity. > What is the complexity, and how low can we get the errors? Or did > this come up already? Graham is presumably the one to ask about what has come up previously, but for the rest of it, we have: 3 10 1.49 5 9 6.79 7 7 3.59 5/3 -1 5.30 7/3 -3 2.09 7/5 -2 -3.20 The second column is the generator coordinate, and the third is sharpness in cents. We have a complexity of 13 and a maximum error of 6.79 cents; of course that can be reduced if we go to a linear temperament. A scale of 27 out of 58 would have a lot of complete tetrads, and if you go out farther you find 11 25 7.30 13 -5 7.75 to add to the fun.
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Message: 4847 Date: Wed, 10 Oct 2001 22:25:31 Subject: Re: Does Miracle--11 count? From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote: > --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: > > > > Of course, for Paul's real quest, which is for interesting 7- > limit > > > scales, we can go to more steps; 7/27 is the penultimate > convergent > > > to 15/58, and gives us a 6*2, 3, 6*2, 3, 6*2, 3, 5*2, 3 pattern. > > > The point of 15/58, (or 17/46, > > > This fraction doesn't seem to agree with the others. > > 17/46 is a completely different generator than 15/58, if that's > what's worrying you. > > > > or 24/65 or 56/135 in 2^(1/2), etc.) is > > > that it has a relatively low 7-limit complexity. > > > What is the complexity, and how low can we get the errors? Or did > > this come up already? > > Graham is presumably the one to ask about what has come up > previously, but for the rest of it, we have: > > 3 10 1.49 > 5 9 6.79 > 7 7 3.59 > 5/3 -1 5.30 > 7/3 -3 2.09 > 7/5 -2 -3.20 OK -- you're talking about the seventh temperament in 404 Not Found * i.e., "kleismic" (since the 5-limit unison vector that implies this generator is the kleisma). Gene -- the complexity is actually only 10, not 13 -- 11 notes in the chain are enough to give a complete tetrad. > A scale of 27 out of 58 would have a lot of complete > tetrads, Not significantly worse than 27 out of 31. Also, 11-, 15-, 19-, and 23-tone MOSs give tetrads. >and if you go out farther you find > > 11 25 7.30 Notice the seventh temperament in 22 26 29 31 41 46 58 72 80 87 89 94 111 113 118 121 130 145 149 152 * > 13 -5 7.75 Curiously, no kleismic temperament appears in 26 29 41 46 58 72 80 87 94 111 113 121 130 149 159 166 171 183 190 198 * or 29 41 58 72 80 87 94 111 121 130 149 159 183 190 198 212 217 224 241 253 * Graham found at least 10 ways to do "better" in 13-limit and 15-limit.
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Message: 4848 Date: Thu, 11 Oct 2001 18:16:09 Subject: Re: Does Miracle--11 count? From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote: > --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: > > > Gene -- the complexity is actually only 10, not 13 -- 11 notes in > the > > chain are enough to give a complete tetrad. > > Oops--so that's the definition people use? Well, that's Graham's complexity.
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Message: 4849 Date: Thu, 11 Oct 2001 18:19:27 Subject: Re: Searching for interesting 7-limit MOS scales From: Paul Erlich --- In tuning-math@y..., graham@m... wrote: > In-Reply-To: <9q2f10+h9nk@e...> > Paul wrote: > > > > > #22: 16 notes; commas 1029:1024, 50:49; chroma 36:35 > > > > > > 36-tET, 26-tET . . . curious > > > > This is quite an interesting one . . . after 26, the next proper MOS > > is at 110 . . . score one for 26-tET! > > You're very keen to take ETs here. Why don't you work out what ETs are > consistent with each unison vector, like at > <4 5 6 9 10 12 15 16 18 19 22 26 27 29 31 35 36 37 41 42 43 45 46 49 50 53 56 57 58 59 60 62 63 68 70 72 73 76 77 78 80 81 82 83 84 87 88 89 90 91 93 94 95 99 103 108 109 111 113 114 115 118 121 122 125 126 128 130 134 135 136 140 142 144 145 149 152 1 *> and see how often each comes > up? Well, if I can have a single guitar which will work for a lot of these good MOS scales, that saves money, plus I can switch from one to another withing a single composition. > > And in this case, you take the different sets > > (-10, 1, 0, 3), 1029:1024 > 5 10 15 16 26 31 36 41 46 56 57 62 72 77 82 87 ... > > (1, 0, 2, -2), 50:49 > 4 6 10 12 16 18 22 26 > > and check the intersection > > 5 10 15 16 26 31 41 ... > 4 6 10 12 16 18 22 26 > Not sure what that tells me.
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