Tuning-Math Digests messages 1800 - 1824

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Message: 1800

Date: Sun, 07 Oct 2001 03:45:47

Subject: Re: More from 4/21/00: does this make sense?

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> Gene, I have no idea what your response has to do with whether this:

You left out the most relevant portion, which talked about rhombic 
dodecahedra, the angles between 3/2, 5/4, and 7/4, and interlocking 
lattices. That was all by Paul Hahn.

> > It certainly makes sense to look at the lattice-pair one gets by 
> > taking the lattice of utonal tetrads together with the lattice of 
> > otonal tetrads.

> What does that mean?

The tetrads form the dual honeycomb (or tesselation) to the lattice 
of 7-limit note classes, but it consists of two interlocking lattices 
of utonal and otonal tetrads. In the 5-limit, the dual tesselation to 
the triangular one of note classes is the hexagonal tiling, and also 
consists of two interlocking triangular lattices, one of major and 
one of minor triads. This situation generalizes to the p-limit.
 
> > If we look at triples [a,b,c] with a quadradic form
> > Q(a,b,c) = a^2+b^2+c^2+ab+ac+bc, we have the symmetric lattice of 
7-
> > limit note-classes. 

> Can you give the "for musician dummies" version of this statement?

We can define a Euclidean metric on a real vector space in three 
equivalent ways:

(1) A Euclidean metric; in three dimensions this might be

d(u, v) = sqrt((u1-v1)^2 + (u2-v2)^2 + (u3-v3)^2)

(2) A bilnear form, or dot product, for instance

B(u, v) = u.v = u1 v1 + u2 v2 + u3 v3

(3) A quadratic form, such as Q(u) = u1^2 + u2^2 + u3^2.

Each of the other two can be defined in terms of one of these, and in 
particular if Q is a positive definite quadratic form (meaning it is 
homogenous of the second order in the variables u1, u2, u3 in three 
dimensions, etc., and Q(u)>=0, with Q(u)=0 iff u=0) then we can 
define a corresponding bilinear form, or dot product, by

B(u,v) = (Q(u+v)-Q(u)-Q(v))/2

We can also define the metric, by

d(u,v) = sqrt(Q(u-v))

If you are going to draw the lattice diagrams you do, it could be 
helpful to realize that the Euclidean geometric structure is defined 
by the quadratic form. Hence for instance the distance between 4/3 and
15/8 is

d([-1 0], [1 1]) = sqrt(Q([-2 -1]) = sqrt(7),

and the cosine of the angle between the vector to 3/2 and the vector 
to 7/5 can be determined using the appropriate dot product, which 
would be

u.v = ((u1+v1)^2 + (u2+v2)^2 + (u3+v3)^2 + (u1+v1)(u2+v2)+(u1+v1)
(u3+v3) + (u2+v2)(u3+v3) - u1^2-u2^2-u3^2-u1u2-u1u3-u2u3-v1^2-v2^2-
v3^2-v1v2-v1v3-v2v3)/2 = 
u1v1+u2v2+u3v3+(u1v2+u1v3+u2v1+u2v3+u3v1+u3v2)/2

We then have 

cos t = u.v/(||u||*||v||) = 0/(1*1) = 0,

so that the classes defined by 3/2 and 7/5 are unit vectors at right 
angles, just as you've been drawing them.


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Message: 1801

Date: Sun, 07 Oct 2001 11:36:34

Subject: Re: More from 4/21/00: does this make sense?

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > Gene, I have no idea what your response has to do with whether 
this:
> 
> You left out the most relevant portion, which talked about rhombic 
> dodecahedra, the angles between 3/2, 5/4, and 7/4, and interlocking 
> lattices. That was all by Paul Hahn.

I think you're misunderstanding the context of that, as well as 
misunderstanding that the original "does this make sense" was not 
about any thing Paul Hahn wrote, but only about the other part.
> 
> > > It certainly makes sense to look at the lattice-pair one gets 
by 
> > > taking the lattice of utonal tetrads together with the lattice 
of 
> > > otonal tetrads.
> 
> > What does that mean?
> 
> The tetrads form the dual honeycomb (or tesselation) to the lattice 
> of 7-limit note classes, but it consists of two interlocking 
lattices 
> of utonal and otonal tetrads. In the 5-limit, the dual tesselation 
to 
> the triangular one of note classes is the hexagonal tiling, and 
also 
> consists of two interlocking triangular lattices, one of major and 
> one of minor triads. This situation generalizes to the p-limit.
>  
> > > If we look at triples [a,b,c] with a quadradic form
> > > Q(a,b,c) = a^2+b^2+c^2+ab+ac+bc, we have the symmetric lattice 
of 
> 7-
> > > limit note-classes. 
> 
> > Can you give the "for musician dummies" version of this statement?
> 
> We can define a Euclidean metric on a real vector space in three 
> equivalent ways:
> 
> (1) A Euclidean metric; in three dimensions this might be
> 
> d(u, v) = sqrt((u1-v1)^2 + (u2-v2)^2 + (u3-v3)^2)
> 
> (2) A bilnear form, or dot product, for instance
> 
> B(u, v) = u.v = u1 v1 + u2 v2 + u3 v3
> 
> (3) A quadratic form, such as Q(u) = u1^2 + u2^2 + u3^2.
> 
> Each of the other two can be defined in terms of one of these, and 
in 
> particular if Q is a positive definite quadratic form (meaning it 
is 
> homogenous of the second order in the variables u1, u2, u3 in three 
> dimensions, etc., and Q(u)>=0, with Q(u)=0 iff u=0) then we can 
> define a corresponding bilinear form, or dot product, by
> 
> B(u,v) = (Q(u+v)-Q(u)-Q(v))/2
> 
> We can also define the metric, by
> 
> d(u,v) = sqrt(Q(u-v))
> 
> If you are going to draw the lattice diagrams you do, it could be 
> helpful to realize that the Euclidean geometric structure is 
defined 
> by the quadratic form. Hence for instance the distance between 4/3 
and
> 15/8 is
> 
> d([-1 0], [1 1]) = sqrt(Q([-2 -1]) = sqrt(7),
> 
> and the cosine of the angle between the vector to 3/2 and the 
vector 
> to 7/5 can be determined using the appropriate dot product, which 
> would be
> 
> u.v = ((u1+v1)^2 + (u2+v2)^2 + (u3+v3)^2 + (u1+v1)(u2+v2)+(u1+v1)
> (u3+v3) + (u2+v2)(u3+v3) - u1^2-u2^2-u3^2-u1u2-u1u3-u2u3-v1^2-v2^2-
> v3^2-v1v2-v1v3-v2v3)/2 = 
> u1v1+u2v2+u3v3+(u1v2+u1v3+u2v1+u2v3+u3v1+u3v2)/2
> 
> We then have 
> 
> cos t = u.v/(||u||*||v||) = 0/(1*1) = 0,
> 
> so that the classes defined by 3/2 and 7/5 are unit vectors at 
right 
> angles, just as you've been drawing them.

I may be coming back to this soon for other purposes, but I don't see 
the connection any of it has to the truncated octahedron question.


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Message: 1802

Date: Sun, 07 Oct 2001 12:05:10

Subject: torsion (was: Re: 72 owns the 11-limit)

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > I'd really like to be able to do this. The gcd of the 
minors . . . 
> can you explicate exactly what the 
> > procedure is . . . I have Matlab.
> 
> Matlab actually has a lot of Maple stuff in it now, since they 
bought 
> some Maple functionality from the Maple people. Unfortunately, I 
> don't know Matlab so I don't exactly know how it works.
> 
> The function igcd(n1, n2, ..., nk) in Maple returns the greatest 
> common divisor of k integers; this may be callable in Matlab, or 
> Matlab may have its own number theory functions. If you have a 
matrix 
> with k-1 rows and k columns, you can produce k different square 
> matricies by removing one of the columns; these (or the 
determinants 
> of these) are called minors. Since Matlab is strong on linear 
algebra 
> I assume it can do this, if not Maple can, and again that may be 
> callable. The proceedure would be to get the k different integers 
> det(min_j), 1<=j<=k, where min_j is the matrix you get by removing 
> the jth column, and take igcd(det(min_1), ..., det(min_k)). If this 
> is not 1, then you have torsion.

OK! This time, eliminating all PBs with torsion:

The set of 11-limit superparticular unison vectors smaller than 20.7 
cents (thus smaller than 17.6 cents) yields the following, taking 4 
at a time:

freq. determinant
     1     1
     1     9
     1    11
     1    29
     1    49
     1    51
     1    60
     1    62
     1    64
     1    65
     1    79
     1    82
     1    91
     1    96
     1   144
     3    20
     3    23
     3    24
     3    37
     3    48
     3    50
     3    68
     3    80
     3    99
     3   126
     8    10
     8    18
     8    54
    11     4
    13    58
    14    45
    17    15
    17    53
    19     7
    19    19
    20    38
    21    12
    21    26
    24    41
    34    14
    40    22
    40    46
    41    27
    45     8
    46    34
    63    72
   188    31

So there are three that legitimately have 126 notes, three that 
legitimately have 24, and one with 144.

The set of 11-limit superparticular unison vectors smaller than 35 
cents yields the following, taking 4 at a time:

     1    30
     1    40
     1    59
     1    63
     1    65
     1    66
     1    76
     1    79
     1    82
     1    86
     1    91
     1    92
     1    96
     1   144
     2    78
     3    13
     3    21
     3    32
     3    47
     3    49
     3    61
     3    62
     3    80
     3    88
     3    90
     3    99
     3   126
     4     1
     4    29
     4    33
     4    42
     4    51
     4    52
     4    60
     5    11
     5    28
     6    68
     9    64
    10    48
    11     6
    11    39
    11    50
    13    36
    14    58
    15    44
    16    37
    17    53
    18    16
    18    23
    20     2
    20     9
    21    54
    27     3
    27    41
    33    45
    39    18
    40    46
    43     5
    48    38
    56    20
    65    72
    67    24
    68    17
    81    26
    90     4
    92    19
   131    15
   133    27
   137    34
   164     7
   169    14
   176    22
   188     8
   218    10
   251    12
   298    31

A lot of legitimate 24's in this list!


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Message: 1803

Date: Sun, 07 Oct 2001 14:08:00

Subject: Searching for interesting 7-limit MOS scales

From: Paul Erlich

Let's try chromatic unison vectors between 100 cents and 35 cents, 
commatic unison vectors smaller than 35 cents but with numerator and 
denominator less than 3000, disallow torsion, and select unison 
vectors only from this list:

S2357 *

The roundup:

freq.    determinant
     1    23
     1    31
     2     6
     2    13
     3     1
     3    15
     3    16
     3    17
     4    21
     5     2
     5     8
     5    11
     5    19
     7     3
     7     4
     7    14
     8    12
    11     7
    14     5
    18    10

Most of the 10-tone scales correspond to the ones that carry my name 
in Scala (10-out-of-22); true for the 14-tone ones as well (14-out-of-
26).

How can we try to eliminate "skewed" blocks from this list? Let's be 
crude and ignore the difference between a taxicab metric on the Kees 
van Prooijen lattice, and a Euclidean metric. So the "length" of a 
unison vector is log(numerator), while the volume the three subtend 
is simply the number of notes. A non-skewed block would have volume 
equal to the product of the three lengths. Define the "straightness" 
as the number of notes divided by this would-be volume. Here are the 
rankings by straightness:

#1: 14 notes; commas 81:80, 50:49; chroma 25:24 (Erlich 14-of-~26)
#2: 31 notes; commas 1029:1024, 245:243; chroma 25:24 (v. improper!)
#3: 17 notes; commas 245:243, 64:63; chroma 25:24
#4: 19 notes; commas 126:125, 81:80; chroma 49:48
#5: 15 notes; commas 126:125, 64:63; chroma 28:27
#6: 21 notes; commas 1029:1024, 81:80; chroma 25:24 (not blackjack!)
#7: 14 notes; commas 81:80, 50:49; chroma 49:48 (Erlich 14-of-~26) 
#8: 12 notes; commas 64:63, 50:49; chroma 36:35 (Erlich 12-of-22)
#9: 19 notes; commas 225:224, 81:80; chroma 49:48 (19-out-of-31???)
#10: 14 notes; commas 245:243, 50:49; chroma 25:24
#11: 12 notes; commas 81:80, 50:49; chroma 36:35
#12: 15 notes; commas 126:125, 64:63; chroma 49:48
#13: 10 notes; commas 64:63, 50:49; chroma 25:24 (Erlich 10-of-22)
#14: 19 notes; commas 225:224, 126:125, chroma 49:48
#15: 10 notes; commas 64:63, 50:49; chroma 28:27 (Erlich 10-of-22)
#16: 19 notes; commas 245:243, 126:125; chroma 49:48
#17: 23 notes; commas 2401:2400, 126:125; chroma 28:27
#18: 14 notes; commas 245:243, 81:80; chroma 25:24
#19: 16 notes; commas 245:243, 225:224; chroma 21:20
#20: 14 notes; commas 245:243, 50:49; chroma 49:48
#21: 12 notes; commas 126:125, 64:63; chroma 36:35
#22: 16 notes; commas 1029:1024, 50:49; chroma 36:35
#23: 19 notes; commas 245:243, 225:224; chroma 49:48
#24: 12 notes; commas 225:224, 50:49; chroma 36:35
#25: 10 notes; commas 64:63, 50:49; chroma 49:48
#26: 12 notes; commas 126:125, 81:80; chroma 36:35
#27: 21 notes; commas 1029:1024, 225:224; chroma 36:35 (Blackjack)
#28: 12 notes; commas 225:224, 64:63; chroma 36:35
#29: 10 notes; commas 225:224, 50:49; chroma 25:24 (Erlich 10-of-22)
#30: 17 notes; commas 2401:2400, 64:63; chroma 36:35
#31: 10 notes; commas 225:224, 50:49; chroma 28:27 (Erlich 10-of-22)
#32: 12 notes; commas 225:224, 81:80; chroma 36:35
#33: 21 notes; commas 2401:2400, 225:224; chroma 36:35 (Blackjack)

The list continues:

   Straightness    # of notes    numerators of commas  numeratr'chroma
      0.13869           17         2401           81           36
      0.13792           10          225           64           25
       0.1358           11          245          126           21
      0.13323           10          225           64           28
       0.1331           16         1029          126           36
      0.12844           11          245          126           25
      0.12784           12          225          126           36
       0.1269            8          126           50           28
      0.12127           10          225           50           49
      0.11899            7           81           64           25
      0.11489           15         1029          126           49
      0.11449           10         1029           50           25
      0.11407           10          225           64           49
      0.11156            8          245           50           28
      0.10854           21         1029         2401           36
      0.10818            7          126           81           21
      0.10769           10         1029           64           25
      0.10517           14         2401           81           49
      0.10373            8          245           50           36
      0.10232            7          126           81           25
     0.098844            7          126           81           28
     0.098556           10         2401           50           28
     0.096602            7          225           81           21
     0.096174           11         1029          225           21
     0.094692           10         1029           50           49
     0.092706           10         2401           64           28
     0.091573            6          245           50           21
      0.09137            7          225           81           25
      0.09111           13          245         2401           28
     0.090237            8          245          126           28
     0.089861            5           81           64           21
     0.088262            7          225           81           28
     0.087777            7          225          126           21
     0.085705           11         2401          225           21
     0.084721           13          245         2401           36
      0.08401           14          245         2401           49
     0.083909            8          245          126           36
     0.083022            7          225          126           25
     0.082103            5           81           64           28
     0.080199            7          225          126           28
     0.079376           10         2401           64           49
     0.072627            6         1029           50           21
     0.071026            7         2401           64           21
     0.070297            5           81           64           49
     0.069443            4          126           50           21
     0.068396           10         1029          225           49
     0.067934            5          245           81           21
     0.066921           11         1029         2401           21
     0.065681            4          126           50           25
     0.065584            5          245           64           28
     0.062069            5          245           81           28
     0.060985            5          245           64           36
     0.060951           10         2401          225           49
     0.057716            5          245           81           36
      0.05693            5         1029           64           21
     0.056154            5          245           64           49
     0.054324            4          126           50           49
     0.048991            3          126           64           21
     0.048367            5         1029           64           36
     0.047592           10         1029         2401           49
     0.046337            3          126           64           25
     0.045775            5         1029           81           36
     0.043148            4         2401           50           21
     0.042149            5         1029           81           49
     0.040377            2           64           50           21
     0.038212            2           81           50           21
     0.036658            4         2401           50           36
     0.034913            2           81           50           28
     0.033669            5         1029          245           49
      0.03128            3          245          225           25
     0.031004            2          225           50           21
     0.030216            3          245          225           28
     0.029652            4         2401          126           36
     0.029164            2          225           64           21
     0.028808            3         2401           81           21
     0.028097            3          245          225           36
     0.027303            4         2401          126           49
     0.026321            3         2401           81           28
    0.0097913            1         1029          126           21
    0.0092609            1         1029          126           25
    0.0076707            1          245         2401           21

This shows that in general, the PBs with a lower number of notes 
often tend to be more "skewed" and thus less interesting. But there 
seem to be lots of interesting new scales that are quite "straight"! 
Can someone please provide the generator, interval of repetition, 
mapping from generators to primes (3,5,7), and maximum 7-limit error, 
for at least the top 32 in the rankings, and preferably many more?


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Message: 1805

Date: Mon, 08 Oct 2001 06:20:19

Subject: Re: New file uploaded to tuning-math

From: genewardsmith@xxxx.xxx

If you take the fifteen 7-limit intervals 25/24, 28/27, 36/35, 49/48,
50/49, 64/63, 81/80, 126/125, 245/243, 1029/1024, 1728/1715, 
2401/2400, 3136/3125, 4375/4374, 6144/6125 from the list 
S2357 * which Paul used, and take the 
15 choose 4 = 1365 four-element subsets, you will find that 642 of 
them give unimodular matricies, and hence give us notations. I 
uploaded all of these. The first list is a list of the four intervals 
used, and the second a list of the columns of the inverse matrix. If 
a number +-n appears on the list, then +-hn, the column vector 
consisting of n, round(n log_2(3)), round(n log_2(5)), 
round(n log_2(7)), is meant. If there is just a "v", it means it is 
not rounded off in this way, and the entire matrix is given in a 
compact form as a list of lists; the lists are horizontal but 
represent column vectors.

There are vast numbers of blocks and tempered blocks which could be 
extracted from all of this, so it would be helpful to know what sort 
of thing people are looking for.


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Message: 1806

Date: Mon, 8 Oct 2001 11:32:23

Subject: Re: torsion (was: Re: 72 owns the 11-limit)

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

By the way, taking the Breedsma, kalisma, ragisma and schisma
as unison vectors gives a truly big 11-limit PB of 342 tones.
Using the xenisma in addition gives the same PB.

Manuel


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Message: 1807

Date: Mon, 08 Oct 2001 11:35:52

Subject: Re: 3rd-best 11-limit temperament

From: Graham Breed

Paul wrote:

> Are you sure? This is the shrutar system, which you once said was 
not consistent with 22-equal. 
> So maybe Gene was right?

The two systems are very different melodically.  The diaschismic 
shruti scale in 46-equal looks like:

 3 1 3 1 3 1 3 1 3 1 3 1 3 3 1 3 1 3 1 3 1 3
S   r   R   g G   M   m   P   d   D   n N   S'

And the 22+46 temperament:

 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2
S   r   R   g G   M   m   P   d   D   n N   S'


The former looks more like the canonical shruti scale, and so that's 
what I'd have expected if you mentioned tuning shrutis to 46-equal.  
The two are the same if you take only the named 12 notes.  I half 
remember looking at this before.

The straight diaschismic mapping is simpler in the 5- and 9-limits.  
They're both as complex in the 7-limit, and the shrutar is simpler in 
the 11-limit, but not so far as to be a compelling 11-limit 
temperament.


                     Graham


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Message: 1808

Date: Mon, 8 Oct 2001 13:08 +01

Subject: Re: Question

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9pqocr+rc5u@xxxxxxx.xxx>
Gene wrote:

> --- In tuning-math@y..., graham@m... wrote:
> 
> > > > 25:24, 1029:1024 and 225:224 fail, apparently because it wants 
> a 
> > > > half-octave generator, but doesn't give the usual clue.
> > > 
> > > Gene?
>  
> > The first column of the adjoint does have a common factor of 2.  
> But so do 
> > all the others. 
> 
> I don't know what your usual clue is, but mine is common factors. We 
> have
> 
>     [  a  b  c  d]
> det [ -5  2  2 -1] = -20a-32b-46c-56d = -2 h10
>     [-10  1  0  3]
>     [ -3 -1  2  0]
> 
> and therefore torsion.

Oh yes, common factors in the left hand column are the definition of 
torsion.  But I meant the clue for the period not being the octave.  
That's usually common factors in the second column, but not always.  You 
can set 7:5 to be a unison vector with octave equivalent matrices, and the 
right mapping comes out, but no suggestion that the period is a 
half-octave.

In the example above, all columns have a common factor of 2, so I divided 
through by it before checking for divisions of the octave.  This is 
because of some inefficient unison vectors I have for the multiple-29 
temperament that produce common factors of a power of 29 in all columns.  
Divide through by those common factors, and you get the right answer.

It all depends on whether we take the adjoint matrix, or the inverse 
multiplied by the lowest common denominator.  It now looks like the 
former, although I assumed the latter was simpler.

I've tightened up the program so that it doesn't need this clue about the 
octave divisions.  I also have better unison vectors for the multiple-29 
case.  I haven't studied the results in detail, but it's mostly working.

<#!/usr/local/bin/python *>
<Unison vectors *>
<#!/usr/local/bin/python *>
<[(-1, 2, 0), (-2, 0, -1), (-1, -2, 4)] *>

I'm doing additional calculations without needing the chromatic unison 
vector.  The problem is with octave equivalent matrices.  It would be nice 
to do all the calculations with them.  The rules are:


1) Form a matrix with the chromatic unison vector in the top row, and 
commatic unison vectors in the others.

2) The gcd of the left hand column of the adjoint is the number of equal 
divisions of the octave.

3) Divide through by this and you have the mapping by generators modulo 
the period.


These work fine as long as the unison vectors are well behaved.  It'd be 
nice if you could come up with a mathematical definition of this.  But it 
looks like they really have to be approximate unisons, rather than 
approximate equal divisions of the octave (which don't work in the 
octave-specific case anyway).  And it has to be possible to specify all 
consonant intervals as combinations of (commatic?) unison vectors.

Once the mapping's chosen, it should be easy to find the ET for which the 
chromatic unison vector approximates to 0 steps, although I haven't worked 
out the algorithm for that yet.

So, Paul's hypothesis can be made into a conjecture by saying "the above 
method always works".  I'm fairly confident that it will this time.  All 
we need is to convert it into mathematical language and prove it.  Then we 
can try submitting a paper to Perspectives of New Music.


                      Graham


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Message: 1809

Date: Mon, 8 Oct 2001 13:28 +01

Subject: Re: Question

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <memo.305965@xxx.xxxxxxxxx.xx.xx>
Corrections already

> These work fine as long as the unison vectors are well behaved.  It'd 
> be nice if you could come up with a mathematical definition of this.  
> But it looks like they really have to be approximate unisons, rather 
> than approximate equal divisions of the octave (which don't work in the 
> octave-specific case anyway).  And it has to be possible to specify all 
> consonant intervals as combinations of (commatic?) unison vectors.

The second criterion isn't right.  Firstly, the unison vectors need to be 
linearly independent.  Secondly, taking each column independently, it must 
be possible to combine the unison vectors to get any number you want in 
there.  Getting any consonant interval isn't possible unless you have a 1 
note periodicity block.  But this condition should remove torsion.


                Graham


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Message: 1810

Date: Mon, 08 Oct 2001 20:29:06

Subject: torsion (was: Re: 72 owns the 11-limit)

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > A lot of legitimate 24's in this list!
> 
> Depending on what you mean by "legitimate". I think it would also 
be 
> interesting to see what you get after culling everything which does 
> not pass my validity test, by taking the product of all four 
> superparticular ratios, raising the result to the power of the 
number 
> of notes in the block (the number found by the absolute value of 
the 
> determinant, so that in the above case it would be 24), and 
removing 
> everything where the result is greater than 2.

What does this test show?


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Message: 1811

Date: Mon, 08 Oct 2001 20:34:11

Subject: Re: Searching for interesting 7-limit MOS scales

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> >Here are the 
> > rankings by straightness:
> 
> Here are some other measures for the top ten on your list; the 
first 
> is a solid angle measure, the area on the unit sphere corresponding 
> to the three vectors.

In a Cartesian lattice with 3, 5, and 7 axes? Also, doesn't this 
depend in an arbitrary way on the signs of the unison vectors?

> The second is my validity condition; this is a 
> sufficient condition, not a necessary one, but one might well ask 
how 
> many of these correctly order the notes in the block--#2, the "very 
> improper" one, has a validity measure over 5.

Can you explain what this validity condition is about?

> The last measure is the 
> most like your measure; it is the volume (which is to say, the 
> determinant) divided by the product of the lengths of the sides.

Lengths measured with Euclidean distance in the Cartesian lattice 
with 3, 5, and 7 axes?
 
> Since a unit volume is the volume of the parallepiped with sides 3, 
> 5, and 7,

A rectangular prism? Can you flesh this out for me please?

> 
> 
> #1: 14 notes; commas 81:80, 50:49; chroma 25:24 (Erlich 14-of-~26)
> 
> 1.337003903  2.796199310  1.120897076
> 
> #2: 31 notes; commas 1029:1024, 245:243; chroma 25:24 (v. improper!)
> 
> 1.208253216  5.315252144 1.203940238
> 
> #3: 17 notes; commas 245:243, 64:63; chroma 25:24
> 
> 0.625778711  3.007295043  0.8997354106
> 
> #4: 19 notes; commas 126:125, 81:80; chroma 49:48
> 
> 1.036438116  2.179705030 1.149932312
> 
> #5: 15 notes; commas 126:125, 64:63; chroma 28:27
> 
> 1.397360786  2.462710473 0.899238708
> 
> #6: 21 notes; commas 1029:1024, 81:80; chroma 25:24 (not blackjack!)
> 
> 0.907922503  3.388625357  0.9326427427
> 
> #7: 14 notes; commas 81:80, 50:49; chroma 49:48 (Erlich 14-of-~26)
> 
> 2.405549309  2.107333077  1.120897076
>  
> #8: 12 notes; commas 64:63, 50:49; chroma 36:35 (Erlich 12-of-22)
> 
> 1.714143895  2.158607408  1.309307341
> 
> #9: 19 notes; commas 225:224, 81:80; chroma 49:48 (19-out-of-31???)
> 
> 0.420158312  2.038933737  1.014144974
> 
> #10: 14 notes; commas 245:243, 50:49; chroma 25:24
> 
> 0.71923786  2.635572871  0.9810960584


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Message: 1812

Date: Mon, 08 Oct 2001 20:58:41

Subject: Re: 3rd-best 11-limit temperament

From: Paul Erlich

--- In tuning-math@y..., "Graham Breed" <graham@m...> wrote:
> Paul wrote:
> 
> > Are you sure? This is the shrutar system, which you once said was 
> not consistent with 22-equal. 
> > So maybe Gene was right?
> 
> The two systems are very different melodically.  The diaschismic 
> shruti scale in 46-equal looks like:
> 
>  3 1 3 1 3 1 3 1 3 1 3 1 3 3 1 3 1 3 1 3 1 3
> S   r   R   g G   M   m   P   d   D   n N   S'
> 
> And the 22+46 temperament:
> 
>  2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2
> S   r   R   g G   M   m   P   d   D   n N   S'
> 
I meant the latter.
> 
> The former looks more like the canonical shruti scale, and so 
that's 
> what I'd have expected if you mentioned tuning shrutis to 46-equal.

Right, but this is a 7- and 11-limit adaptation of the sruti idea, 
keeping only the named 12 notes in their "canonical" tuning.
  
> The two are the same if you take only the named 12 notes.  I half 
> remember looking at this before.
> 
> The straight diaschismic mapping is simpler in the 5- and 9-
limits.  
> They're both as complex in the 7-limit, and the shrutar is simpler 
in 
> the 11-limit, but not so far as to be a compelling 11-limit 
> temperament.

Well, I do get an 11-limit hexad that includes the open strings, 
which is probably the only one I'd be able to play anyway.


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Message: 1813

Date: Mon, 08 Oct 2001 21:07:19

Subject: ETs for 7-limit (was: Re: Searching for interesting 7-limit MOS scales)

From: Paul Erlich

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> Let's try chromatic unison vectors between 100 cents and 35 cents, 
> commatic unison vectors smaller than 35 cents but with numerator 
and 
> denominator less than 3000, disallow torsion, and select unison 
> vectors only from this list:
> 
> S2357 *

I thought natural to see what ETs come up when we take three commatic 
unison vectors at a time, still defined as above, rather than the 
MOSs that come from one chromatic and two commatic. The results 
(ranked by straightness):

   Straighness          ET          numerators of unison vectors
      0.24926           46         1029          245          126
       0.2458           22          245           64           50
      0.24401           27          245          126           64
      0.24116           36         1029          245           50
      0.21804           26         1029           81           50
      0.21029           31         1029          126           81
      0.19838           41         1029          245          225
      0.18874           22          245          225           50
      0.18778           31         1029          225           81
       0.1874           31         2401          126           81
      0.17754           22          245          225           64
      0.17679           41          245         2401          225
      0.17246           27         2401          126           64
      0.17062           31         1029          225          126
      0.16784           12           81           64           50
      0.16734           31         2401          225           81
      0.16251           19          245          126           81
      0.15251           12          126           64           50
      0.15205           31         2401          225          126
      0.15162           27          245         2401           64
      0.14803           14          245           81           50
      0.14511           19          245          225           81
      0.14433           12          126           81           50
      0.13804           41         1029          245         2401
      0.13577           12          126           81           64
      0.13185           19          245          225          126
      0.13066           31         1029         2401           81
      0.13038           27          245         2401          126
      0.12888           12          225           81           50
      0.12192           16         1029          126           50
      0.12123           12          225           81           64
       0.1195           17         2401           81           64
      0.11872           31         1029         2401          126
      0.11711           12          225          126           50
      0.11016           12          225          126           64
      0.10752           15         1029          126           64
      0.10463           14         2401           81           50
     0.088612           10         1029           64           50
     0.083576           14          245         2401           50
     0.078966           10         2401           64           50
     0.076863            8          245          126           50
     0.074401           14          245         2401           81
     0.068043           10         1029          225           50
     0.064004           10         1029          225           64
     0.060636           10         2401          225           50
     0.057037           10         2401          225           64
     0.049731            5          245           81           64
     0.047346           10         1029         2401           50
     0.044536           10         1029         2401           64
     0.039442            5         1029           81           64
     0.031507            5         1029          245           64
     0.029818            5         1029          245           81
     0.027162            4         2401          126           50

I'm shocked that 36 shows up so close to the top! Discuss.


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Message: 1814

Date: Mon, 08 Oct 2001 21:55:29

Subject: Re: Searching for interesting 7-limit MOS scales

From: Paul Erlich

No one calculated the information I requested (generators, mappings 
from primes to generators, minimax error). Gene, perhaps you can 
provide me with an algorithm to find the generator, and mapping from 
primes to generators, given the chromatic unison vector and set of 
commatic unison vectors? I'm not going to try to understand how it 
works now -- just want to get some results.


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Message: 1815

Date: Mon, 08 Oct 2001 22:54:30

Subject: torsion (was: Re: 72 owns the 11-limit)

From: Paul Erlich

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
> 
> By the way, taking the Breedsma, kalisma, ragisma and schisma
> as unison vectors gives a truly big 11-limit PB of 342 tones.

342-tET comes up in the list of the simplest ETs to achieve level-3 
consistency in various odd limits:

3-limit -> 5-tET
5-limit -> 12-tET
7-limit -> 31-tET
9-limit -> 171-tET
11-limit -> 342-tET
13-limit -> 5585-tET
15-limit -> 5585-tET


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Message: 1816

Date: Tue, 09 Oct 2001 07:00:24

Subject: ETs for 7-limit (was: Re: Searching for interesting 7-limit MOS scales)

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> I'm shocked that 36 shows up so close to the top! Discuss.

I also got a good number (1.2108) for my own straightness measure of 
36 divided by the product of the lengths of 1029/1024, 245/243 and 
50/49. However, my validity measure, which is the product of those 
three raised to the 36th power, was not very good: 3.3126. That 
measure may be more significant.


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Message: 1817

Date: Tue, 09 Oct 2001 19:56:12

Subject: Re: 7-limit PBs

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> There are "valid" sets with 1 step (see above). So what? This sort 
> of "validity" doesn't do much for me.

The Paul Theorem applies, it just isn't very interesting. However, 
it's easy enough to cull out everything which does not have a certain 
minimal number of scale steps--what's the big deal?


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Message: 1818

Date: Tue, 9 Oct 2001 10:02 +01

Subject: Re: Question

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9ptp2v+3ilo@xxxxxxx.xxx>
Gene wrote:

> > In the example above, all columns have a common factor of 2, so I 
> divided 
> > through by it before checking for divisions of the octave. 
> 
> All columns of what have a common factor of 2? I get
> 
>     [  1  0  0  0]   [-20  0  0  0]
> adj [ -5  2  2 -1] = [-32 -6 -2 -6]
>     [-10  1  0  3]   [-46 -3 -1 -7]
>     [ -3 -1  2  0]   [-56  2 -6 -2]

No, not that one.

> This has one column in the first matrix and one column in the second 
> divisible by 2. I don't see how a half-octave period even comes into 
> it.

It has torsion, but not a half-octave period.

I meant this one:

[(-3, 0, 1),
 (0, 2, -2),
 (1, 0, 3)]

Octave specifically,

[  1  0  0  0]
[  2 -3  0  1]
[  1  0  2 -2]
[-10  1  0  3]

adjoint

[-20  0   0  0]
[-32  6   0 -2]
[-46 -2 -10 -6]
[-56 -2   0 -6]


> > 1) Form a matrix with the chromatic unison vector in the top row, 
> and 
> > commatic unison vectors in the others.
> 
> > 2) The gcd of the left hand column of the adjoint is the number of 
> equal 
> > divisions of the octave.
> 
> You've lost me completely here. Left hand column of the adjoint of 
> what matrix is supposed to do this? It can hardly be the matrix of 
> unison vectors, with or without 2 included in the picture. Do you 
> maybe mean you divide through by the gcd to get the number of 
> divisions?

Without 2 (this method is all for octave equivalent matrices).  It's the 
same as the gcd you take here:

"""
(4) 14/12 = 7/6 = 1+1/6, and the convergent to 7/6 is 1. We therefore 
want A + B = 93.651 cents as our generator; this happens to be very 
close to 1200/13 = 92.308 cents; not much of a surprise since 
g+h12=h26 and gcd(12,14)=2. We therefore can use 1200/13 as our 
generator, with a period of half an octave.
"""

GCD of 2, half octave period.

> I don't see why you don't simply do what I do, and go for the whole 
> val, not just h(2). The number of divisions can be deceptive.

Why don't I do what?  I'm using octave equivalent matrices for the 
conjecture because one step using octave specific matrices involves 
solving what I think could be called a (very simple) system of Diophantine 
equations with only 1 unknown.  Proving that that always works sounds 
harder than proving the octave-specific case.

In fact, the number of notes in the periodicity block is the generator 
mapping of the chromatic unison vector.  From the definition of the 
adjoint matrix, this will always be the same as the determinant.  So we 
can ignore the determinant if we really want to.  But the result we get is 
:

The number of periods to the octave
The mapping of generators to prime intervals
The number of notes in the periodicity block

Hopefully, we've defined away torsion.  It'd be nice if we could get the  
number of steps to a generator in the equivalent ET.  Did we have an 
algorithm for generating the periodicity block in order of pitch for 
octave equivalent vectors?

> > These work fine as long as the unison vectors are well behaved.  
> It'd be 
> > nice if you could come up with a mathematical definition of this.
> 
> I thought I did.

Did you?  Where?


                 Graham


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Message: 1819

Date: Tue, 09 Oct 2001 20:01:37

Subject: Re: 7-limit PBs

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > There are "valid" sets with 1 step (see above). So what? This 
sort 
> > of "validity" doesn't do much for me.
> 
> The Paul Theorem applies, it just isn't very interesting. However, 
> it's easy enough to cull out everything which does not have a 
certain 
> minimal number of scale steps--what's the big deal?

It's not about the number of scale steps -- it's about the "skewness" 
of the block. Ultimately, this translates into greater errors with 
respect to the JI intervals -- which is what I really care about.


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Message: 1820

Date: Tue, 9 Oct 2001 10:02 +01

Subject: Re: Searching for interesting 7-limit MOS scales

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9pt7ch+ucrd@xxxxxxx.xxx>
Paul wrote:

> No one calculated the information I requested (generators, mappings 
> from primes to generators, minimax error). Gene, perhaps you can 
> provide me with an algorithm to find the generator, and mapping from 
> primes to generators, given the chromatic unison vector and set of 
> commatic unison vectors? I'm not going to try to understand how it 
> works now -- just want to get some results.

If you look at <#!/usr/local/bin/python *>, all you 
need to do is alter it to use different unison vectors (chromatic on top, 
all larger than a unison).  If downloading ActivePython's too much trouble 
for you, I can run it at home sometime.


                    Graham


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Message: 1821

Date: Tue, 09 Oct 2001 20:26:47

Subject: Re: 7-limit PBs

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> There _is_ a decent MOS for 7-limit with 11 steps:
> 
> 11 note chain-of-minor-thirds scale *
> 
> But clearly not all the numbers in your list are to be taken 
> seriously.

Aside from the ones with a low number of notes or a bad validity, 
which ones? It might be interesting to check on these 11 and 13 note 
PBs, and see if they make sense.

11 notes:

[1.550683960, [3136/3125, 245/243, 36/35], 18*b+11*a+26*c+32*d]

[1.696997541, [3136/3125, 245/243, 28/27], 18*b+11*a+26*c+32*d]

[1.714645543, [4375/4374, 126/125, 25/24], 30*d+25*c+17*b+11*a]

[1.718962570, [4375/4374, 245/243, 25/24], 30*d+25*c+17*b+11*a]

[1.871717101, [126/125, 245/243, 25/24], 30*d+25*c+17*b+11*a]

[2.113987015, [3136/3125, 36/35, 28/27], 18*b+11*a+26*c+32*d]

[2.253538886, [1029/1024, 36/35, 25/24], 17*b+11*a+25*c+31*d]


We are getting 11*a+17*b+25*c+30*d, 11*a+17*b+25*c+31*d, and
11*a+18*b+26*c+32*d.


13 notes:

[1.509685048, [3136/3125, 1728/1715, 49/48], 20*b+13*a+30*c+36*d]

[1.613159878, [2401/2400, 245/243, 36/35], 21*b+13*a+31*c+37*d]

[1.665410631, [3136/3125, 1728/1715, 36/35], 20*b+13*a+30*c+36*d]

[1.794547474, [2401/2400, 245/243, 28/27], 21*b+13*a+31*c+37*d]

[1.842667322, [6144/6125, 1728/1715, 28/27], 37*d+30*c+21*b+13*a]

[1.952519731, [6144/6125, 1728/1715, 25/24], 37*d+30*c+21*b+13*a]

[1.973782564, [3136/3125, 49/48, 36/35], 20*b+13*a+30*c+36*d]

[2.183860583, [3136/3125, 64/63, 25/24], 36*d+13*a+21*b+30*c]

[2.326619401, [2401/2400, 36/35, 28/27], 21*b+13*a+31*c+37*d]

[2.839788039, [6144/6125, 28/27, 25/24], 37*d+30*c+21*b+13*a]

[3.009084772, [1728/1715, 28/27, 25/24], 37*d+30*c+21*b+13*a]


With 13*a+20*b+30*c+36*d, 13*a+21*b+30*c+37*d, and 
13*a+21*b+31*c+37*d.


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Message: 1822

Date: Tue, 9 Oct 2001 10:02 +01

Subject: Re: Searching for interesting 7-limit MOS scales

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9ptkum+fe30@xxxxxxx.xxx>
Paul wrote:

> > #19: 16 notes; commas 245:243, 225:224; chroma 21:20
> 
> 41-tET, 22-tET, 19-tET . . . is this Graham's MAGIC thing?

Yes, that's Magic.


                      Graham


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Message: 1823

Date: Tue, 09 Oct 2001 10:25:23

Subject: 7-limit PBs

From: genewardsmith@xxxx.xxx

I took the same 15 7-limit intervals as before, and this time took 
sets of three. After eliminating the ones with linear dependence or 
torsion, I obtained the following, which lists the validity measure, 
the intervals, and the dual val:

[1.448457604, [4375/4374, 2401/2400, 6144/6125], 
99*a+157*b+230*c+278*d]

[1.509442953, [4375/4374, 2401/2400, 3136/3125], 
99*a+157*b+230*c+278*d]

[1.487593558, [4375/4374, 2401/2400, 1029/1024], 
114*b+72*a+167*c+202*d]

[1.247713548, [4375/4374, 2401/2400, 1728/1715], 43*b+27*a+63*c+76*d]

[1.261826622, [4375/4374, 2401/2400, 126/125], 43*b+27*a+63*c+76*d]

[1.269638870, [4375/4374, 2401/2400, 245/243], 43*b+27*a+63*c+76*d]

[1.800467556, [4375/4374, 2401/2400, 81/80], 71*b+45*a+104*c+126*d]

[1.455372564, [4375/4374, 2401/2400, 50/49], 28*b+18*a+41*c+50*d]

[1.466326603, [4375/4374, 2401/2400, 49/48], 28*b+18*a+41*c+50*d]

[1.296080214, [4375/4374, 2401/2400, 36/35], 22*c+26*d+15*b+9*a]

[3.790362186, [4375/4374, 2401/2400, 28/27], 85*c+102*d+58*b+36*a]

[1.968212894, [4375/4374, 6144/6125, 3136/3125], 
99*a+157*b+230*c+278*d]

[1.457979089, [4375/4374, 6144/6125, 1029/1024], 
73*b+46*a+129*d+107*c]

[1.779804660, [4375/4374, 6144/6125, 1728/1715], 
84*b+53*a+123*c+149*d]

[1.681227707, [4375/4374, 6144/6125, 126/125], 73*b+46*a+129*d+107*c]

[1.698999914, [4375/4374, 6144/6125, 245/243], 73*b+46*a+129*d+107*c]

[1.116544333, [4375/4374, 6144/6125, 81/80], 11*b+7*a+16*c+20*d]

[1.142843389, [4375/4374, 6144/6125, 64/63], 11*b+7*a+16*c+20*d]

[4.102976382, [4375/4374, 6144/6125, 50/49], 169*d+139*c+95*b+60*a]

[2.544310889, [4375/4374, 6144/6125, 49/48], 62*b+39*a+91*c+109*d]

[3.561534162, [4375/4374, 6144/6125, 28/27], 89*d+51*b+32*a+75*c]

[2.763160012, [4375/4374, 3136/3125, 1029/1024], 
331*d+187*b+118*a+274*c]

[2.468277477, [4375/4374, 3136/3125, 1728/1715], 
127*b+80*a+186*c+225*d]

[1.249197248, [4375/4374, 3136/3125, 126/125], 19*a+30*b+44*c+53*d]

[1.254634753, [4375/4374, 3136/3125, 245/243], 19*a+30*b+44*c+53*d]

[3.283589564, [4375/4374, 3136/3125, 64/63], 142*c+97*b+61*a+172*d]

[2.733807197, [4375/4374, 3136/3125, 50/49], 42*a+67*b+98*c+119*d]

[2.083409549, [4375/4374, 3136/3125, 36/35], 54*c+37*b+23*a+66*d]

[1.174027616, [4375/4374, 3136/3125, 28/27], 10*c+13*d+4*a+7*b]

[1.951242152, [4375/4374, 3136/3125, 25/24], 23*b+15*a+34*c+40*d]

[1.389479277, [4375/4374, 1029/1024, 1728/1715], 60*c+73*d+41*b+26*a]

[1.824149922, [4375/4374, 1029/1024, 126/125], 73*b+46*a+129*d+107*c]

[1.843432955, [4375/4374, 1029/1024, 245/243], 73*b+46*a+129*d+107*c]

[1.577077172, [4375/4374, 1029/1024, 81/80], 60*c+73*d+41*b+26*a]

[1.517338321, [4375/4374, 1029/1024, 64/63], 47*c+56*d+32*b+20*a]

[1.672594979, [4375/4374, 1029/1024, 49/48], 47*c+56*d+32*b+20*a]

[1.220941678, [4375/4374, 1029/1024, 36/35], 9*b+6*a+13*c+17*d]

[4.347012006, [4375/4374, 1029/1024, 25/24], 90*d+50*b+32*a+73*c]

[1.529904994, [4375/4374, 1728/1715, 126/125], 43*b+27*a+63*c+76*d]

[1.539376974, [4375/4374, 1728/1715, 245/243], 43*b+27*a+63*c+76*d]

[1.690915994, [4375/4374, 1728/1715, 81/80], 60*c+73*d+41*b+26*a]

[1.887533019, [4375/4374, 1728/1715, 64/63], 43*b+27*a+63*c+76*d]

[2.070013772, [4375/4374, 1728/1715, 50/49], 60*c+73*d+41*b+26*a]

[1.028806584, [4375/4374, 1728/1715, 49/48], a+2*b+3*c+3*d]

[1.036605118, [4375/4374, 1728/1715, 36/35], a+2*b+3*c+3*d]

[3.442298035, [4375/4374, 1728/1715, 28/27], 45*b+28*a+66*c+79*d]

[3.370477905, [4375/4374, 1728/1715, 25/24], 57*c+70*d+39*b+25*a]

[1.479592122, [4375/4374, 126/125, 81/80], 19*a+30*b+44*c+53*d]

[1.908883188, [4375/4374, 126/125, 64/63], 43*b+27*a+63*c+76*d]

[1.255075583, [4375/4374, 126/125, 50/49], 13*b+8*a+19*c+23*d]

[1.728934000, [4375/4374, 126/125, 49/48], 19*a+30*b+44*c+53*d]

[1.337685863, [4375/4374, 126/125, 36/35], 13*b+8*a+19*c+23*d]

[1.428343371, [4375/4374, 126/125, 28/27], 13*b+8*a+19*c+23*d]

[1.714645543, [4375/4374, 126/125, 25/24], 30*d+25*c+17*b+11*a]

[1.486032490, [4375/4374, 245/243, 81/80], 19*a+30*b+44*c+53*d]

[1.920701505, [4375/4374, 245/243, 64/63], 43*b+27*a+63*c+76*d]

[1.257372941, [4375/4374, 245/243, 50/49], 13*b+8*a+19*c+23*d]

[1.736459703, [4375/4374, 245/243, 49/48], 19*a+30*b+44*c+53*d]

[1.340134435, [4375/4374, 245/243, 36/35], 13*b+8*a+19*c+23*d]

[1.430957888, [4375/4374, 245/243, 28/27], 13*b+8*a+19*c+23*d]

[1.718962570, [4375/4374, 245/243, 25/24], 30*d+25*c+17*b+11*a]

[1.219933467, [4375/4374, 81/80, 64/63], 11*b+7*a+16*c+20*d]

[2.349492733, [4375/4374, 81/80, 50/49], 60*c+73*d+41*b+26*a]

[1.881628622, [4375/4374, 81/80, 49/48], 19*a+30*b+44*c+53*d]

[1.330764996, [4375/4374, 81/80, 36/35], 11*b+7*a+16*c+20*d]

[1.800789429, [4375/4374, 81/80, 28/27], 28*c+33*d+19*b+12*a]

[1.453990411, [4375/4374, 81/80, 25/24], 11*b+7*a+16*c+20*d]

[3.421600769, [4375/4374, 64/63, 50/49], 79*c+96*d+54*b+34*a]

[2.079078641, [4375/4374, 64/63, 49/48], 47*c+56*d+32*b+20*a]

[1.362109799, [4375/4374, 64/63, 36/35], 11*b+7*a+16*c+20*d]

[2.848783252, [4375/4374, 64/63, 28/27], 47*c+56*d+32*b+20*a]

[2.093653566, [4375/4374, 50/49, 49/48], 28*b+18*a+41*c+50*d]

[1.475235230, [4375/4374, 50/49, 36/35], 13*b+8*a+19*c+23*d]

[3.011864701, [4375/4374, 50/49, 25/24], 28*b+18*a+41*c+50*d]

[1.050240055, [4375/4374, 49/48, 36/35], a+2*b+3*c+3*d]

[3.140275637, [4375/4374, 49/48, 28/27], 47*c+56*d+32*b+20*a]

[3.034533867, [4375/4374, 49/48, 25/24], 28*b+18*a+41*c+50*d]

[1.678896865, [4375/4374, 36/35, 28/27], 13*b+8*a+19*c+23*d]

[1.623444754, [4375/4374, 36/35, 25/24], 11*b+7*a+16*c+20*d]

[1.362979272, [4375/4374, 28/27, 25/24], 6*b+4*a+9*c+10*d]

[1.296835720, [2401/2400, 6144/6125, 1029/1024], 49*b+31*a+87*d+72*c]

[1.409207770, [2401/2400, 6144/6125, 1728/1715], 49*b+31*a+87*d+72*c]

[1.427524260, [2401/2400, 6144/6125, 126/125], 49*b+31*a+87*d+72*c]

[2.217365728, [2401/2400, 6144/6125, 245/243], 158*c+191*d+108*b+68*a]

[2.039491520, [2401/2400, 6144/6125, 64/63], 104*d+59*b+37*a+86*c]

[1.152921505, [2401/2400, 6144/6125, 50/49], 17*d+14*c+10*b+6*a]

[1.155806812, [2401/2400, 6144/6125, 49/48], 17*d+14*c+10*b+6*a]

[2.208066855, [2401/2400, 6144/6125, 36/35], 39*b+25*a+58*c+70*d]

[5.556136148, [2401/2400, 6144/6125, 28/27], 69*b+43*a+100*c+121*d]

[1.313691630, [2401/2400, 3136/3125, 1029/1024], 49*b+31*a+87*d+72*c]

[1.427524260, [2401/2400, 3136/3125, 1728/1715], 49*b+31*a+87*d+72*c]

[1.446078824, [2401/2400, 3136/3125, 126/125], 49*b+31*a+87*d+72*c]

[2.281076121, [2401/2400, 3136/3125, 245/243], 158*c+191*d+108*b+68*a]

[2.071170648, [2401/2400, 3136/3125, 64/63], 104*d+59*b+37*a+86*c]

[1.155806812, [2401/2400, 3136/3125, 50/49], 17*d+14*c+10*b+6*a]

[1.158699341, [2401/2400, 3136/3125, 49/48], 17*d+14*c+10*b+6*a]

[2.231182923, [2401/2400, 3136/3125, 36/35], 39*b+25*a+58*c+70*d]

[5.656559608, [2401/2400, 3136/3125, 28/27], 69*b+43*a+100*c+121*d]

[1.488862040, [2401/2400, 1029/1024, 1728/1715], 49*b+31*a+87*d+72*c]

[1.508213854, [2401/2400, 1029/1024, 126/125], 49*b+31*a+87*d+72*c]

[1.738209169, [2401/2400, 1029/1024, 245/243], 65*b+41*a+115*d+95*c]

[1.731539563, [2401/2400, 1029/1024, 81/80], 49*b+31*a+87*d+72*c]

[1.234120624, [2401/2400, 1029/1024, 64/63], 16*b+10*a+23*c+28*d]

[1.290335387, [2401/2400, 1029/1024, 50/49], 16*b+10*a+23*c+28*d]

[1.295721876, [2401/2400, 1029/1024, 49/48], 16*b+10*a+23*c+28*d]

[2.019048032, [2401/2400, 1029/1024, 36/35], 33*b+21*a+49*c+59*d]

[1.547209978, [2401/2400, 1728/1715, 245/243], 43*b+27*a+63*c+76*d]

[1.881579115, [2401/2400, 1728/1715, 81/80], 49*b+31*a+87*d+72*c]

[1.897137590, [2401/2400, 1728/1715, 64/63], 43*b+27*a+63*c+76*d]

[1.119277634, [2401/2400, 1728/1715, 50/49], 9*c+6*b+4*a+11*d]

[1.121144263, [2401/2400, 1728/1715, 49/48], 9*c+6*b+4*a+11*d]

[1.155526618, [2401/2400, 1728/1715, 36/35], 9*c+6*b+4*a+11*d]

[2.772428686, [2401/2400, 1728/1715, 28/27], 37*b+23*a+65*d+54*c]

[1.564710702, [2401/2400, 126/125, 245/243], 43*b+27*a+63*c+76*d]

[1.906035357, [2401/2400, 126/125, 81/80], 49*b+31*a+87*d+72*c]

[1.918596397, [2401/2400, 126/125, 64/63], 43*b+27*a+63*c+76*d]

[1.121144263, [2401/2400, 126/125, 50/49], 9*c+6*b+4*a+11*d]

[1.123014005, [2401/2400, 126/125, 49/48], 9*c+6*b+4*a+11*d]

[1.157453700, [2401/2400, 126/125, 36/35], 9*c+6*b+4*a+11*d]

[2.799119925, [2401/2400, 126/125, 28/27], 37*b+23*a+65*d+54*c]

[1.342457835, [2401/2400, 245/243, 81/80], 22*b+14*a+32*c+39*d]

[1.930474851, [2401/2400, 245/243, 64/63], 43*b+27*a+63*c+76*d]

[1.496942601, [2401/2400, 245/243, 50/49], 22*b+14*a+32*c+39*d]

[1.505698455, [2401/2400, 245/243, 49/48], 22*b+14*a+32*c+39*d]

[1.613159878, [2401/2400, 245/243, 36/35], 21*b+13*a+31*c+37*d]

[1.794547474, [2401/2400, 245/243, 28/27], 21*b+13*a+31*c+37*d]

[1.625778481, [2401/2400, 81/80, 64/63], 27*b+17*a+40*c+48*d]

[1.588174591, [2401/2400, 81/80, 50/49], 22*b+14*a+32*c+39*d]

[1.597464076, [2401/2400, 81/80, 49/48], 22*b+14*a+32*c+39*d]

[2.008061053, [2401/2400, 81/80, 36/35], 27*b+17*a+40*c+48*d]

[1.159072634, [2401/2400, 81/80, 28/27], 8*c+9*d+5*b+3*a]

[1.438608665, [2401/2400, 64/63, 50/49], 16*b+10*a+23*c+28*d]

[1.444614119, [2401/2400, 64/63, 49/48], 16*b+10*a+23*c+28*d]

[2.124866198, [2401/2400, 64/63, 36/35], 27*b+17*a+40*c+48*d]

[1.691009707, [2401/2400, 64/63, 28/27], 16*b+10*a+23*c+28*d]

[1.215506250, [2401/2400, 50/49, 36/35], 9*c+6*b+4*a+11*d]

[1.768035978, [2401/2400, 50/49, 28/27], 16*b+10*a+23*c+28*d]

[1.217533360, [2401/2400, 49/48, 36/35], 9*c+6*b+4*a+11*d]

[1.775416623, [2401/2400, 49/48, 28/27], 16*b+10*a+23*c+28*d]

[2.326619401, [2401/2400, 36/35, 28/27], 21*b+13*a+31*c+37*d]

[1.427524260, [6144/6125, 3136/3125, 1029/1024], 49*b+31*a+87*d+72*c]

[1.551220597, [6144/6125, 3136/3125, 1728/1715], 49*b+31*a+87*d+72*c]

[1.571382931, [6144/6125, 3136/3125, 126/125], 49*b+31*a+87*d+72*c]

[2.737190864, [6144/6125, 3136/3125, 245/243], 158*c+191*d+108*b+68*a]

[2.287131427, [6144/6125, 3136/3125, 64/63], 104*d+59*b+37*a+86*c]

[1.174547064, [6144/6125, 3136/3125, 50/49], 17*d+14*c+10*b+6*a]

[1.177486492, [6144/6125, 3136/3125, 49/48], 17*d+14*c+10*b+6*a]

[2.385833149, [6144/6125, 3136/3125, 36/35], 39*b+25*a+58*c+70*d]

[6.347647197, [6144/6125, 3136/3125, 28/27], 69*b+43*a+100*c+121*d]

[1.617873353, [6144/6125, 1029/1024, 1728/1715], 49*b+31*a+87*d+72*c]

[2.103467335, [6144/6125, 1029/1024, 245/243], 73*b+46*a+129*d+107*c]

[1.881579115, [6144/6125, 1029/1024, 81/80], 49*b+31*a+87*d+72*c]

[1.427247693, [6144/6125, 1029/1024, 64/63], 15*a+24*b+42*d+35*c]

[1.569463799, [6144/6125, 1029/1024, 50/49], 25*b+16*a+37*c+45*d]

[1.535432120, [6144/6125, 1029/1024, 49/48], 15*a+24*b+42*d+35*c]

[1.782870571, [6144/6125, 1029/1024, 36/35], 25*b+16*a+37*c+45*d]

[1.050000000, [6144/6125, 1029/1024, 25/24], b+a+3*d+2*c]

[1.780914442, [6144/6125, 1728/1715, 126/125], 49*b+31*a+87*d+72*c]

[1.513769837, [6144/6125, 1728/1715, 245/243], 35*b+22*a+51*c+62*d]

[2.044619737, [6144/6125, 1728/1715, 81/80], 49*b+31*a+87*d+72*c]

[1.787357184, [6144/6125, 1728/1715, 64/63], 35*b+22*a+51*c+62*d]

[1.971379996, [6144/6125, 1728/1715, 50/49], 35*b+22*a+51*c+62*d]

[1.325000779, [6144/6125, 1728/1715, 49/48], 25*d+21*c+9*a+14*b]

[1.418184249, [6144/6125, 1728/1715, 36/35], 25*d+21*c+9*a+14*b]

[1.842667322, [6144/6125, 1728/1715, 28/27], 37*d+30*c+21*b+13*a]

[1.952519731, [6144/6125, 1728/1715, 25/24], 37*d+30*c+21*b+13*a]

[2.425554379, [6144/6125, 126/125, 245/243], 73*b+46*a+129*d+107*c]

[2.071195136, [6144/6125, 126/125, 81/80], 49*b+31*a+87*d+72*c]

[1.495120029, [6144/6125, 126/125, 64/63], 15*a+24*b+42*d+35*c]

[1.649199242, [6144/6125, 126/125, 50/49], 25*b+16*a+37*c+45*d]

[1.608449134, [6144/6125, 126/125, 49/48], 15*a+24*b+42*d+35*c]

[1.873447987, [6144/6125, 126/125, 36/35], 25*b+16*a+37*c+45*d]

[1.053257143, [6144/6125, 126/125, 25/24], b+a+3*d+2*c]

[1.766847065, [6144/6125, 245/243, 81/80], 56*c+38*b+24*a+67*d]

[1.812907661, [6144/6125, 245/243, 64/63], 35*b+22*a+51*c+62*d]

[2.150970319, [6144/6125, 245/243, 49/48], 56*c+38*b+24*a+67*d]

[1.082128318, [6144/6125, 245/243, 36/35], 3*b+2*a+5*c+5*d]

[1.100014434, [6144/6125, 245/243, 28/27], 3*b+2*a+5*c+5*d]

[2.835788482, [6144/6125, 245/243, 25/24], 46*c+57*d+32*b+20*a]

[1.244677941, [6144/6125, 81/80, 64/63], 11*b+7*a+16*c+20*d]

[3.886285184, [6144/6125, 81/80, 50/49], 107*d+88*c+60*b+38*a]

[2.380563249, [6144/6125, 81/80, 49/48], 56*c+38*b+24*a+67*d]

[1.357757518, [6144/6125, 81/80, 36/35], 11*b+7*a+16*c+20*d]

[2.415933756, [6144/6125, 81/80, 28/27], 47*d+40*c+27*b+17*a]

[2.360946697, [6144/6125, 64/63, 50/49], 35*b+22*a+51*c+62*d]

[1.807551657, [6144/6125, 64/63, 49/48], 15*a+24*b+42*d+35*c]

[1.389738102, [6144/6125, 64/63, 36/35], 11*b+7*a+16*c+20*d]

[2.289191294, [6144/6125, 64/63, 28/27], 15*a+24*b+42*d+35*c]

[1.518424276, [6144/6125, 64/63, 25/24], 11*b+7*a+16*c+20*d]

[1.301497343, [6144/6125, 50/49, 49/48], 17*d+14*c+10*b+6*a]

[2.278536109, [6144/6125, 50/49, 36/35], 25*b+16*a+37*c+45*d]

[5.315847988, [6144/6125, 50/49, 28/27], 45*b+28*a+65*c+79*d]

[1.469220000, [6144/6125, 50/49, 25/24], 17*d+14*c+10*b+6*a]

[1.595180079, [6144/6125, 49/48, 36/35], 25*d+21*c+9*a+14*b]

[2.462710473, [6144/6125, 49/48, 28/27], 15*a+24*b+42*d+35*c]

[1.472896878, [6144/6125, 49/48, 25/24], 17*d+14*c+10*b+6*a]

[1.144847592, [6144/6125, 36/35, 28/27], 3*b+2*a+5*c+5*d]

[1.656373835, [6144/6125, 36/35, 25/24], 11*b+7*a+16*c+20*d]

[2.839788039, [6144/6125, 28/27, 25/24], 37*d+30*c+21*b+13*a]

[1.660204014, [3136/3125, 1029/1024, 126/125], 49*b+31*a+87*d+72*c]

[4.231649866, [3136/3125, 1029/1024, 245/243], 87*a+138*b+202*c+244*d]

[1.828192297, [3136/3125, 1029/1024, 64/63], 70*d+40*b+25*a+58*c]

[1.187113513, [3136/3125, 1029/1024, 50/49], 14*c+17*d+9*b+6*a]

[2.064939480, [3136/3125, 1029/1024, 49/48], 70*d+40*b+25*a+58*c]

[3.867306154, [3136/3125, 1029/1024, 36/35], 58*b+37*a+86*c+104*d]

[2.547030414, [3136/3125, 1029/1024, 25/24], 53*d+44*c+31*b+19*a]

[1.804062272, [3136/3125, 1728/1715, 126/125], 49*b+31*a+87*d+72*c]

[2.569847702, [3136/3125, 1728/1715, 245/243], 78*b+49*a+138*d+114*c]

[3.720561269, [3136/3125, 1728/1715, 64/63], 78*b+49*a+138*d+114*c]

[1.755627065, [3136/3125, 1728/1715, 50/49], 29*b+18*a+42*c+51*d]

[1.509685048, [3136/3125, 1728/1715, 49/48], 20*b+13*a+30*c+36*d]

[1.665410631, [3136/3125, 1728/1715, 36/35], 20*b+13*a+30*c+36*d]

[1.296200144, [3136/3125, 1728/1715, 25/24], 9*b+5*a+12*c+15*d]

[1.453386524, [3136/3125, 126/125, 245/243], 19*a+30*b+44*c+53*d]

[1.386474440, [3136/3125, 126/125, 64/63], 34*d+28*c+19*b+12*a]

[1.462600997, [3136/3125, 126/125, 50/49], 34*d+28*c+19*b+12*a]

[1.840291986, [3136/3125, 126/125, 49/48], 19*a+30*b+44*c+53*d]

[1.609356470, [3136/3125, 126/125, 36/35], 34*d+28*c+19*b+12*a]

[1.397866900, [3136/3125, 126/125, 28/27], 7*a+11*b+16*c+19*d]

[1.442139658, [3136/3125, 126/125, 25/24], 7*a+11*b+16*c+19*d]

[3.840060916, [3136/3125, 245/243, 64/63], 78*b+49*a+138*d+114*c]

[2.604911795, [3136/3125, 245/243, 50/49], 70*c+85*d+48*b+30*a]

[1.550683960, [3136/3125, 245/243, 36/35], 18*b+11*a+26*c+32*d]

[1.696997541, [3136/3125, 245/243, 28/27], 18*b+11*a+26*c+32*d]

[1.522358304, [3136/3125, 245/243, 25/24], 12*b+8*a+18*c+21*d]

[1.605730195, [3136/3125, 64/63, 50/49], 34*d+28*c+19*b+12*a]

[2.710238360, [3136/3125, 64/63, 49/48], 70*d+40*b+25*a+58*c]

[4.017844284, [3136/3125, 64/63, 28/27], 70*d+40*b+25*a+58*c]

[2.183860583, [3136/3125, 64/63, 25/24], 36*d+13*a+21*b+30*c]

[1.304754477, [3136/3125, 50/49, 49/48], 17*d+14*c+10*b+6*a]

[1.863858578, [3136/3125, 50/49, 36/35], 34*d+28*c+19*b+12*a]

[2.949144340, [3136/3125, 50/49, 28/27], 29*b+18*a+42*c+51*d]

[1.472896878, [3136/3125, 50/49, 25/24], 17*d+14*c+10*b+6*a]

[1.973782564, [3136/3125, 49/48, 36/35], 20*b+13*a+30*c+36*d]

[4.538146945, [3136/3125, 49/48, 28/27], 70*d+40*b+25*a+58*c]

[1.476582958, [3136/3125, 49/48, 25/24], 17*d+14*c+10*b+6*a]

[2.113987015, [3136/3125, 36/35, 28/27], 18*b+11*a+26*c+32*d]

[1.075200000, [3136/3125, 36/35, 25/24], 2*c+2*d+b+a]

[1.759314432, [3136/3125, 28/27, 25/24], 7*a+11*b+16*c+19*d]

[1.881579115, [1029/1024, 1728/1715, 126/125], 49*b+31*a+87*d+72*c]

[1.108598313, [1029/1024, 1728/1715, 245/243], 8*b+5*a+12*c+14*d]

[1.151256995, [1029/1024, 1728/1715, 64/63], 8*b+5*a+12*c+14*d]

[2.335569854, [1029/1024, 1728/1715, 50/49], 60*c+73*d+41*b+26*a]

[1.179639680, [1029/1024, 1728/1715, 49/48], 8*b+5*a+12*c+14*d]

[1.225032023, [1029/1024, 1728/1715, 36/35], 8*b+5*a+12*c+14*d]

[3.059142980, [1029/1024, 1728/1715, 25/24], 48*c+33*b+21*a+59*d]

[2.631752268, [1029/1024, 126/125, 245/243], 73*b+46*a+129*d+107*c]

[2.188267678, [1029/1024, 126/125, 81/80], 49*b+31*a+87*d+72*c]

[1.535432120, [1029/1024, 126/125, 64/63], 15*a+24*b+42*d+35*c]

[1.696672390, [1029/1024, 126/125, 50/49], 25*b+16*a+37*c+45*d]

[1.651816855, [1029/1024, 126/125, 49/48], 15*a+24*b+42*d+35*c]

[1.927376265, [1029/1024, 126/125, 36/35], 25*b+16*a+37*c+45*d]

[1.055126953, [1029/1024, 126/125, 25/24], b+a+3*d+2*c]

[1.135929305, [1029/1024, 245/243, 81/80], 8*b+5*a+12*c+14*d]

[1.154976817, [1029/1024, 245/243, 64/63], 8*b+5*a+12*c+14*d]

[3.312603948, [1029/1024, 245/243, 50/49], 57*b+36*a+83*c+101*d]

[1.183451209, [1029/1024, 245/243, 49/48], 8*b+5*a+12*c+14*d]

[5.315252144, [1029/1024, 245/243, 25/24], 31*a+87*d+71*c+49*b]

[1.179639680, [1029/1024, 81/80, 64/63], 8*b+5*a+12*c+14*d]

[2.650902363, [1029/1024, 81/80, 50/49], 60*c+73*d+41*b+26*a]

[1.208722102, [1029/1024, 81/80, 49/48], 8*b+5*a+12*c+14*d]

[1.255233531, [1029/1024, 81/80, 36/35], 8*b+5*a+12*c+14*d]

[3.388625357, [1029/1024, 81/80, 25/24], 48*c+33*b+21*a+59*d]

[1.504137953, [1029/1024, 64/63, 50/49], 16*b+10*a+23*c+28*d]

[1.276281563, [1029/1024, 64/63, 36/35], 8*b+5*a+12*c+14*d]

[1.848570832, [1029/1024, 64/63, 25/24], 16*b+10*a+23*c+28*d]

[1.579217147, [1029/1024, 50/49, 49/48], 16*b+10*a+23*c+28*d]

[2.344125081, [1029/1024, 50/49, 36/35], 25*b+16*a+37*c+45*d]

[1.932774085, [1029/1024, 50/49, 25/24], 16*b+10*a+23*c+28*d]

[1.307746559, [1029/1024, 49/48, 36/35], 8*b+5*a+12*c+14*d]

[1.940842427, [1029/1024, 49/48, 25/24], 16*b+10*a+23*c+28*d]

[2.253538886, [1029/1024, 36/35, 25/24], 17*b+11*a+25*c+31*d]

[1.897137590, [1728/1715, 126/125, 245/243], 43*b+27*a+63*c+76*d]

[2.377883156, [1728/1715, 126/125, 81/80], 49*b+31*a+87*d+72*c]

[2.326207228, [1728/1715, 126/125, 64/63], 43*b+27*a+63*c+76*d]

[1.153602745, [1728/1715, 126/125, 50/49], 9*c+6*b+4*a+11*d]

[1.155526618, [1728/1715, 126/125, 49/48], 9*c+6*b+4*a+11*d]

[1.190963383, [1728/1715, 126/125, 36/35], 9*c+6*b+4*a+11*d]

[3.298310219, [1728/1715, 126/125, 28/27], 37*b+23*a+65*d+54*c]

[1.151256995, [1728/1715, 245/243, 81/80], 8*b+5*a+12*c+14*d]

[2.205432014, [1728/1715, 245/243, 50/49], 35*b+22*a+51*c+62*d]

[1.199420137, [1728/1715, 245/243, 49/48], 8*b+5*a+12*c+14*d]

[1.245573629, [1728/1715, 245/243, 36/35], 8*b+5*a+12*c+14*d]

[1.297682532, [1728/1715, 245/243, 28/27], 8*b+5*a+12*c+14*d]

[2.616129604, [1728/1715, 245/243, 25/24], 27*b+17*a+39*c+48*d]

[1.195557177, [1728/1715, 81/80, 64/63], 8*b+5*a+12*c+14*d]

[2.842253559, [1728/1715, 81/80, 50/49], 60*c+73*d+41*b+26*a]

[1.225032023, [1728/1715, 81/80, 49/48], 8*b+5*a+12*c+14*d]

[1.272171056, [1728/1715, 81/80, 36/35], 8*b+5*a+12*c+14*d]

[3.584855584, [1728/1715, 81/80, 25/24], 48*c+33*b+21*a+59*d]

[2.604025167, [1728/1715, 64/63, 50/49], 35*b+22*a+51*c+62*d]

[1.245573629, [1728/1715, 64/63, 49/48], 8*b+5*a+12*c+14*d]

[1.293503099, [1728/1715, 64/63, 36/35], 8*b+5*a+12*c+14*d]

[1.347617144, [1728/1715, 64/63, 28/27], 8*b+5*a+12*c+14*d]

[2.974491206, [1728/1715, 64/63, 25/24], 27*b+17*a+39*c+48*d]

[1.213482515, [1728/1715, 50/49, 49/48], 9*c+6*b+4*a+11*d]

[1.250696625, [1728/1715, 50/49, 36/35], 9*c+6*b+4*a+11*d]

[3.171469175, [1728/1715, 50/49, 28/27], 29*b+18*a+42*c+51*d]

[1.315616223, [1728/1715, 50/49, 25/24], 9*c+6*b+4*a+11*d]

[1.380840823, [1728/1715, 49/48, 28/27], 8*b+5*a+12*c+14*d]

[1.317810287, [1728/1715, 49/48, 25/24], 9*c+6*b+4*a+11*d]

[1.433975353, [1728/1715, 36/35, 28/27], 8*b+5*a+12*c+14*d]

[1.358223838, [1728/1715, 36/35, 25/24], 9*c+6*b+4*a+11*d]

[3.009084772, [1728/1715, 28/27, 25/24], 37*d+30*c+21*b+13*a]

[1.721440913, [126/125, 245/243, 81/80], 19*a+30*b+44*c+53*d]

[2.367084273, [126/125, 245/243, 64/63], 43*b+27*a+63*c+76*d]

[1.337685863, [126/125, 245/243, 50/49], 13*b+8*a+19*c+23*d]

[2.011539314, [126/125, 245/243, 49/48], 19*a+30*b+44*c+53*d]

[1.425733632, [126/125, 245/243, 36/35], 13*b+8*a+19*c+23*d]

[1.522358304, [126/125, 245/243, 28/27], 13*b+8*a+19*c+23*d]

[1.871717101, [126/125, 245/243, 25/24], 30*d+25*c+17*b+11*a]

[1.542907401, [126/125, 81/80, 64/63], 34*d+28*c+19*b+12*a]

[1.627623155, [126/125, 81/80, 50/49], 34*d+28*c+19*b+12*a]

[2.179705030, [126/125, 81/80, 49/48], 19*a+30*b+44*c+53*d]

[1.790936736, [126/125, 81/80, 36/35], 34*d+28*c+19*b+12*a]

[1.487814607, [126/125, 81/80, 28/27], 7*a+11*b+16*c+19*d]

[1.534936158, [126/125, 81/80, 25/24], 7*a+11*b+16*c+19*d]

[1.693895333, [126/125, 64/63, 50/49], 34*d+28*c+19*b+12*a]

[1.944562872, [126/125, 64/63, 49/48], 15*a+24*b+42*d+35*c]

[1.863858578, [126/125, 64/63, 36/35], 34*d+28*c+19*b+12*a]

[2.462710473, [126/125, 64/63, 28/27], 15*a+24*b+42*d+35*c]

[1.213629630, [126/125, 64/63, 25/24], 3*a+5*b+8*d+7*c]

[1.215506250, [126/125, 50/49, 49/48], 9*c+6*b+4*a+11*d]

[1.675829337, [126/125, 50/49, 28/27], 13*b+8*a+19*c+23*d]

[1.317810287, [126/125, 50/49, 25/24], 9*c+6*b+4*a+11*d]

[1.254871699, [126/125, 49/48, 36/35], 9*c+6*b+4*a+11*d]

[2.649382292, [126/125, 49/48, 28/27], 15*a+24*b+42*d+35*c]

[1.320008011, [126/125, 49/48, 25/24], 9*c+6*b+4*a+11*d]

[1.786134033, [126/125, 36/35, 28/27], 13*b+8*a+19*c+23*d]

[1.360488960, [126/125, 36/35, 25/24], 9*c+6*b+4*a+11*d]

[1.815034831, [126/125, 28/27, 25/24], 7*a+11*b+16*c+19*d]

[1.199420137, [245/243, 81/80, 64/63], 8*b+5*a+12*c+14*d]

[1.770935474, [245/243, 81/80, 50/49], 22*b+14*a+32*c+39*d]

[1.276281563, [245/243, 81/80, 36/35], 8*b+5*a+12*c+14*d]

[1.329675140, [245/243, 81/80, 28/27], 8*b+5*a+12*c+14*d]

[2.363581242, [245/243, 81/80, 25/24], 22*b+14*a+32*c+39*d]

[2.641250007, [245/243, 64/63, 50/49], 35*b+22*a+51*c+62*d]

[1.249598197, [245/243, 64/63, 49/48], 8*b+5*a+12*c+14*d]

[1.297682532, [245/243, 64/63, 36/35], 8*b+5*a+12*c+14*d]

[1.351971425, [245/243, 64/63, 28/27], 8*b+5*a+12*c+14*d]

[3.007295043, [245/243, 64/63, 25/24], 27*b+17*a+39*c+48*d]

[1.986278255, [245/243, 50/49, 49/48], 22*b+14*a+32*c+39*d]

[1.572336629, [245/243, 50/49, 36/35], 13*b+8*a+19*c+23*d]

[1.678896865, [245/243, 50/49, 28/27], 13*b+8*a+19*c+23*d]

[2.635572871, [245/243, 50/49, 25/24], 22*b+14*a+32*c+39*d]

[1.329675140, [245/243, 49/48, 36/35], 8*b+5*a+12*c+14*d]

[1.385302452, [245/243, 49/48, 28/27], 8*b+5*a+12*c+14*d]

[2.650988754, [245/243, 49/48, 25/24], 22*b+14*a+32*c+39*d]

[1.260576198, [245/243, 36/35, 25/24], 7*c+9*d+5*b+3*a]

[1.291958414, [245/243, 28/27, 25/24], 7*c+9*d+5*b+3*a]

[1.786901316, [81/80, 64/63, 50/49], 34*d+28*c+19*b+12*a]

[1.276281563, [81/80, 64/63, 49/48], 8*b+5*a+12*c+14*d]

[1.380840823, [81/80, 64/63, 28/27], 8*b+5*a+12*c+14*d]

[1.620849023, [81/80, 64/63, 25/24], 11*b+7*a+16*c+20*d]

[2.107333077, [81/80, 50/49, 49/48], 22*b+14*a+32*c+39*d]

[2.074153775, [81/80, 50/49, 36/35], 34*d+28*c+19*b+12*a]

[1.147959184, [81/80, 50/49, 28/27], 3*b+2*a+4*c+5*d]

[2.796199310, [81/80, 50/49, 25/24], 22*b+14*a+32*c+39*d]

[1.358068434, [81/80, 49/48, 36/35], 8*b+5*a+12*c+14*d]

[1.414883587, [81/80, 49/48, 28/27], 8*b+5*a+12*c+14*d]

[2.812554722, [81/80, 49/48, 25/24], 22*b+14*a+32*c+39*d]

[1.469328077, [81/80, 36/35, 28/27], 8*b+5*a+12*c+14*d]

[1.768103919, [81/80, 36/35, 25/24], 11*b+7*a+16*c+20*d]

[1.872519988, [81/80, 28/27, 25/24], 7*a+11*b+16*c+19*d]

[1.760686016, [64/63, 50/49, 49/48], 16*b+10*a+23*c+28*d]

[2.158607408, [64/63, 50/49, 36/35], 34*d+28*c+19*b+12*a]

[2.060991308, [64/63, 50/49, 28/27], 16*b+10*a+23*c+28*d]

[2.154870412, [64/63, 50/49, 25/24], 16*b+10*a+23*c+28*d]

[1.380840823, [64/63, 49/48, 36/35], 8*b+5*a+12*c+14*d]

[2.163865892, [64/63, 49/48, 25/24], 16*b+10*a+23*c+28*d]

[1.493966092, [64/63, 36/35, 28/27], 8*b+5*a+12*c+14*d]

[1.809749792, [64/63, 36/35, 25/24], 11*b+7*a+16*c+20*d]

[2.532938159, [64/63, 28/27, 25/24], 16*b+10*a+23*c+28*d]

[1.317810287, [50/49, 49/48, 36/35], 9*c+6*b+4*a+11*d]

[2.163865892, [50/49, 49/48, 28/27], 16*b+10*a+23*c+28*d]

[1.969795692, [50/49, 36/35, 28/27], 13*b+8*a+19*c+23*d]

[1.428724824, [50/49, 36/35, 25/24], 9*c+6*b+4*a+11*d]

[2.648314659, [50/49, 28/27, 25/24], 16*b+10*a+23*c+28*d]

[1.530797807, [49/48, 36/35, 28/27], 8*b+5*a+12*c+14*d]

[1.431107521, [49/48, 36/35, 25/24], 9*c+6*b+4*a+11*d]

[2.659370017, [49/48, 28/27, 25/24], 16*b+10*a+23*c+28*d]

[1.371742112, [36/35, 28/27, 25/24], 7*c+9*d+5*b+3*a]


As you can see if you sort through this, there are quite a few valid 
sets with rather exotic numbers of steps, such as 11 or 13. Perhaps 
we should learn to celebrate the fact?


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Message: 1824

Date: Tue, 09 Oct 2001 15:44:03

Subject: torsion (was: Re: 72 owns the 11-limit)

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > What does this test show?
> 
> If something passes the test then the Paul Theorem will work; it 
> isn't a necessary condition, however.

I suspect a necessary condition would have to distinguish between 
commatic and chromatic unison vectors. Does the Paul Theorem work for 
all the examples in my list?


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