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Message: 1825 - Contents - Hide Contents Date: Tue, 09 Oct 2001 15:51:12 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: >>> 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. Oh, goody! > However, I wouldn't pay much > attention to this; for one thing there are six eight triangles for > every three verticies,Six eight triangles?> and which one to you pick?Well, once you're tempering out the commatic unison vectors, much of the choice becomes irrelevant -- right?>>>> 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.Well, I guess I'll have to try programming the algorithm you just gave me and see what comes out as the generator, etc.>>>> 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.I'd prefer an "isosceles tetrahedron", but this is good too . . .>>>> 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.OK -- when you said "sides 3, 5, and 7", I thought you meant lengths 3, 5, and 7. Sorry. Have you given any thought to the idea of a "canonical basis" for the case where all the unison vectors are chromatic, and the case where one is commatic? A lot of the measures we've come up here depend on the specific unison vectors we name as the basis, even though many other sets yield exactly the same temperament.
Message: 1826 - Contents - Hide Contents Date: Tue, 09 Oct 2001 16:08:56 Subject: validity measure From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- 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.Your validity measure is equivalent to something I proposed a couple of years ago on the tuning list. If you're constructing the parallelepiped PB in JI (i.e. a hyper-MOS with no unison vectors tempered out), then the largest interval functioning as a unison, and directly affecting what notes do and do not appear in the block, should be smaller than the smallest step in the block. Clearly if that interval greater than 1/N octave, it can't be smaller than the smallest step. The largest interval functioning as a unison, etc., spans a diagonal of the block -- specifically that diagonal which is the result of adding the unison vectors when all are taken as ascending intervals. So in this case, we have 1029/1024 = 8.4327¢ 245/243 = 14.1905¢ 50/49 = 34.9756¢ sum = 57.5989¢ > 33.3333¢ HOWEVER, when you're tempering out unison vectors, this validity measure ceases to mean very much. For example, 12-tET can be defined using the minor diesis and the major diesis, but clearly fails the validity measure with respect to these unison vectors. So what? You see how a "canonical basis" might be useful. Now, 36-tET only showed up once in our list -- since we stuck to the smallest unison vectors for their lengths, we probably found the "canonical" basis for 36-tET, and it turned out invalid. OK. But ultimately I'd like to drop the restriction on taking unison vectors from Kees' list -- Herman Miller has made wonderful music using the maximal diesis (250:243) as a commatic unison vector!
Message: 1827 - Contents - Hide Contents 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 * [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. >If that's not too much trouble, I'd be in your debt!
Message: 1828 - Contents - Hide Contents 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!
Message: 1829 - Contents - Hide Contents 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 * [with cont.] (Wayb.) But clearly not all the numbers in your list are to be taken seriously.
Message: 1830 - Contents - Hide Contents
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.
Message: 1831 - Contents - Hide Contents 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 49:4831-tET -- is this the famous 19-out-of-31?> #5: 15 notes; commas 126:125, 64:63; chroma 28:2727-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 25:2422-tET, 36-tET . . . what's this?> #11: 12 notes; commas 81:80, 50:49; chroma 36:3526-tET, 14-tET . . . complement of Erlich 14-out-of-26> #12: 15 notes; commas 126:125, 64:63; chroma 49:48Same 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 49:4846-tET, 27-tET . . . some kind of 8-tone scheme behind this . . .> #17: 23 notes; commas 2401:2400, 126:125; chroma 28:2731-tET, 27-tET . . . some kind of 4-tone scheme behind this . . . kleismic???> #18: 14 notes; commas 245:243, 81:80; chroma 25:24NOTHING ON MY LIST! WHAT IS THIS?> #19: 16 notes; commas 245:243, 225:224; chroma 21:2041-tET, 22-tET, 19-tET . . . is this Graham's MAGIC thing?> #20: 14 notes; commas 245:243, 50:49; chroma 49:48Same as #10> #21: 12 notes; commas 126:125, 64:63; chroma 36:35Same as #5> #22: 16 notes; commas 1029:1024, 50:49; chroma 36:3536-tET, 26-tET . . . curious> #23: 19 notes; commas 245:243, 225:224; chroma 49:48Same as #19 but longer chain> #24: 12 notes; commas 225:224, 50:49; chroma 36:3522-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 36:35Complement 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 36:3527-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 36:35Complement of #9?> #33: 21 notes; commas 2401:2400, 225:224; chroma 36:35 (Blackjack) 41-tET, 31-tETNow, what _are_ the (optimal) generators?
Message: 1832 - Contents - Hide Contents 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- * [with cont.] (Wayb.) 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?
Message: 1833 - Contents - Hide Contents 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]
Message: 1834 - Contents - Hide Contents 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.
Message: 1835 - Contents - Hide Contents 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.
Message: 1836 - Contents - Hide Contents
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.
Message: 1837 - Contents - Hide Contents 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.
Message: 1838 - Contents - Hide Contents 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.
Message: 1839 - Contents - Hide Contents 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".
Message: 1840 - Contents - Hide Contents 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 * [with cont.] Search for http://www.microtonal.co.uk/paul.7limit.out in Wayback Machine>This URL doesn't seem to work :(
Message: 1841 - Contents - Hide Contents 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 * [with cont.] Search for http://www.microtonal.co.uk/paul.7limit.out in Wayback Machine> >> 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 * [with cont.] (Wayb.)>. Graham
Message: 1842 - Contents - Hide Contents 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.
Message: 1843 - Contents - Hide Contents 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 * [with cont.] Search for http://www.microtonal.co.uk/paul.7limit.out in Wayback Machine> >>>> 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 * [with cont.] (Wayb.)>. > > > GrahamThanks a bunch Graham. You're my hero!
Message: 1844 - Contents - Hide Contents
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 * [with cont.] >.
Code is at
<# Temperament finding library -- definitions * [with cont.] (Wayb.)>
<#!/usr/local/bin/python * [with cont.] (Wayb.)>
<import sys, vectors * [with cont.] (Wayb.)>
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 * [with cont.] (Wayb.)> (or a minimal distribution
without TKInter, if you can find one).
Here's the example input file:
<chroma 25:24 (Erlich 14-of-~26) * [with cont.] (Wayb.)>
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 * [with cont.] Search for http://www.microtonal.co.uk/paul.7limit.out in Wayback Machine>
<404 Not Found * [with cont.] Search for http://www.microtonal.co.uk/gene.7limit.vectors in Wayback Machine>
<404 Not Found * [with cont.] Search for http://www.microtonal.co.uk/gene.7limit.out in Wayback Machine>
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
Message: 1845 - Contents - Hide Contents 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.
Message: 1846 - Contents - Hide Contents 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?
Message: 1847 - Contents - Hide Contents 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-fifth... * [with cont.] (Wayb.)>>> #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-tETThis 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-tETGood 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 . . . curiousThis 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.
Message: 1848 - Contents - Hide Contents 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.
Message: 1849 - Contents - Hide Contents 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 -3.20OK -- you're talking about the seventh temperament in 4 5 6 9 10 12 15 16 18 19 22 26 27 29 31 35 36... * [with cont.] (Wayb.) 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 7.30Notice the seventh temperament in 22 26 29 31 41 46 58 72 80 87 89 94 111 113 11... * [with cont.] (Wayb.)> 13 -5 7.75Curiously, no kleismic temperament appears in 26 29 41 46 58 72 80 87 94 111 113 121 130 149... * [with cont.] (Wayb.) or 29 41 58 72 80 87 94 111 121 130 149 159 183 1... * [with cont.] (Wayb.) Graham found at least 10 ways to do "better" in 13-limit and 15-limit.
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