43 7-limit planar temperaments

Below I give the comma, mapping, and TOP generators for Paul's list
of 43 planar temperaments. The generator and mapping result from a
modified Hermite reduction, the modification being to change signs
when needed to ensure the generators are all positive.

28/27
[[1, 0, 0, -2], [0, 1, 0, 3], [0, 0, 1, 0]]
[1193.415676, 1912.390908, 2786.313714]

36/35
[[1, 0, 0, 2], [0, 1, 0, 2], [0, 0, 1, -1]]
[1195.264647, 1894.449645, 2797.308862]

49/48
[[1, 0, 0, 2], [0, 2, 0, 1], [0, 0, 1, 0]]
[1203.187309, 953.5033827, 2786.313714]

50/49
[[2, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 1]]
[598.4467109, 1901.955001, 2779.100463]

64/63
[[1, 0, 0, 6], [0, 1, 0, -2], [0, 0, 1, 0]]
[1197.723683, 1905.562879, 2786.313714]

81/80
[[1, 0, -4, 0], [0, 1, 4, 0], [0, 0, 0, 1]]
[1201.698520, 1899.262910, 3368.825906]

126/125
[[1, 0, 0, -1], [0, 1, 0, -2], [0, 0, 1, 3]]
[1199.010636, 1900.386896, 2788.610946]

128/125
[[3, 0, 7, 0], [0, 1, 0, 0], [0, 0, 0, 1]]
[399.0200131, 1901.955001, 3368.825906]

225/224
[[1, 0, 0, -5], [0, 1, 0, 2], [0, 0, 1, 2]]
[1200.493660, 1901.172569, 2785.167472]

245/243
[[1, 0, 0, 0], [0, 1, 1, 2], [0, 0, 2, -1]]
[1200., 1903.372995, 440.4316973]

250/243
[[1, 2, 3, 0], [0, -3, -5, 0], [0, 0, 0, 1]]
[1196.905960, 162.3176609, 3368.825906]

256/245
[[1, 0, 0, 4], [0, 1, 0, 0], [0, 0, 2, -1]]
[1195.228951, 1901.955001, 1398.695873]

405/392
[[1, 0, 1, -1], [0, 1, 0, 2], [0, 0, 2, 1]]
[1203.269293, 1896.773294, 787.7266785]

525/512
[[1, 0, 0, 9], [0, 1, 0, -1], [0, 0, 1, -2]]
[1202.406737, 1898.140412, 2780.725442]

648/625
[[4, 0, 3, 0], [0, 1, 1, 0], [0, 0, 0, 1]]
[299.1603149, 1896.631523, 3368.825906]

686/675
[[1, 0, 2, 1], [0, 1, 0, 1], [0, 0, 3, 2]]
[1198.513067, 1904.311735, 130.9133777]

729/700
[[1, 0, 0, -2], [0, 1, 0, 6], [0, 0, 1, -2]]
[1203.706383, 1896.080523, 2794.919668]

875/864
[[1, 0, 0, 5], [0, 1, 0, 3], [0, 0, 1, -3]]
[1201.121570, 1903.732647, 2783.709509]

1029/1000
[[1, 0, 0, 1], [0, 3, 0, -1], [0, 0, 1, 1]]
[1202.477948, 632.6758490, 2792.067330]

1029/1024
[[1, 1, 0, 3], [0, 3, 0, -1], [0, 0, 1, 0]]
[1200.421488, 233.6218235, 2786.313714]

1323/1280
[[1, 0, 0, 4], [0, 1, 1, -1], [0, 0, 2, 1]]
[1202.764567, 1897.573266, 447.5797863]

1728/1715
[[1, 0, 0, 2], [0, 1, 0, 1], [0, 0, 3, -1]]
[1199.391895, 1900.991178, 929.2418964]

2048/2025
[[2, 0, 11, 0], [0, 1, -2, 0], [0, 0, 0, 1]]
[599.5552941, 1903.364685, 3368.825906]

2240/2187
[[1, 0, 0, -6], [0, 1, 0, 7], [0, 0, 1, -1]]
[1198.134693, 1904.911442, 2781.982606]

2401/2400
[[1, 1, 1, 2], [0, 2, 1, 1], [0, 0, 2, 1]]
[1200.032113, 350.9868928, 617.6846359]

2430/2401
[[1, 0, 3, 1], [0, 1, 3, 2], [0, 0, -4, -1]]
[1199.075238, 1900.489288, 1628.631774]

3125/3024
[[1, 0, 0, -4], [0, 1, 0, -3], [0, 0, 1, 5]]
[1202.454598, 1905.845447, 2780.614314]

3125/3072
[[1, 0, 2, 0], [0, 5, 1, 0], [0, 0, 0, 1]]
[1201.276744, 380.7957184, 3368.825906]

3125/3087
[[1, 0, 0, 0], [0, 1, 1, 1], [0, 0, 3, 5]]
[1200., 1903.401919, 293.5973664]

3136/3125
[[1, 0, 0, -3], [0, 1, 0, 0], [0, 0, 2, 5]]
[1199.738066, 1901.955001, 1393.460953]

3645/3584
[[1, 0, 0, -9], [0, 1, 0, 6], [0, 0, 1, 1]]
[1201.235997, 1899.995991, 2783.443817]

4000/3969
[[1, 0, 1, 4], [0, 1, 0, -2], [0, 0, 2, 3]]
[1199.436909, 1902.847479, 792.7846742]

4375/4374
[[1, 0, 0, 1], [0, 1, 0, 7], [0, 0, 1, -4]]
[1200.016360, 1901.980932, 2786.275726]

5103/5000
[[1, 0, 0, 3], [0, 1, 0, -6], [0, 0, 1, 4]]
[1201.434720, 1899.681024, 2789.645030]

5120/5103
[[1, 0, 0, 10], [0, 1, 0, -6], [0, 0, 1, 1]]
[1199.766314, 1902.325384, 2785.771112]

5625/5488
[[1, 0, 1, 0], [0, 1, 1, 2], [0, 0, -3, -4]]
[1201.715742, 1899.235615, 106.2071570]

6144/6125
[[1, 0, 1, 4], [0, 1, 1, -1], [0, 0, -2, 3]]
[1199.786928, 1901.617290, 157.2978838]

8748/8575
[[1, 0, 1, 0], [0, 1, 2, 1], [0, 0, -3, 2]]
[1198.678173, 1899.859955, 736.3383942]

10976/10935
[[1, 0, 2, -1], [0, 1, 2, 3], [0, 0, -3, -1]]
[1199.758595, 1902.337618, 1139.106063]

15625/15552
[[1, 0, 1, 0], [0, 6, 5, 0], [0, 0, 0, 1]]
[1200.291038, 317.0693810, 3368.825906]

16875/16807
[[1, 0, 0, 0], [0, 1, 3, 3], [0, 0, -5, -4]]
[1200., 1901.560426, 583.7891213]

19683/19600
[[1, 0, 0, -2], [0, 2, 0, 9], [0, 0, 1, -1]]
[1200.256485, 950.7742412, 2786.909253]

32805/32768
[[1, 0, 15, 0], [0, 1, -8, 0], [0, 0, 0, 1]]
[1200.065120, 1901.851787, 3368.825906]

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Below I give the comma, mapping, and TOP generators for Paul's list
> of 43 planar temperaments. The generator and mapping result from a
> modified Hermite reduction, the modification being to change signs
> when needed to ensure the generators are all positive.

Thanks so much, Gene. If you could describe in common, non-technical
language the criteria you used to choose the set of generators, I'll
be able to explain it in my paper.

>
> 28/27
> [[1, 0, 0, -2], [0, 1, 0, 3], [0, 0, 1, 0]]
> [1193.415676, 1912.390908, 2786.313714]
>
> 36/35
> [[1, 0, 0, 2], [0, 1, 0, 2], [0, 0, 1, -1]]
> [1195.264647, 1894.449645, 2797.308862]
>
> 49/48
> [[1, 0, 0, 2], [0, 2, 0, 1], [0, 0, 1, 0]]
> [1203.187309, 953.5033827, 2786.313714]
>
> 50/49
> [[2, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 1]]
> [598.4467109, 1901.955001, 2779.100463]
>
> 64/63
> [[1, 0, 0, 6], [0, 1, 0, -2], [0, 0, 1, 0]]
> [1197.723683, 1905.562879, 2786.313714]
>
> 81/80
> [[1, 0, -4, 0], [0, 1, 4, 0], [0, 0, 0, 1]]
> [1201.698520, 1899.262910, 3368.825906]
>
> 126/125
> [[1, 0, 0, -1], [0, 1, 0, -2], [0, 0, 1, 3]]
> [1199.010636, 1900.386896, 2788.610946]
>
> 128/125
> [[3, 0, 7, 0], [0, 1, 0, 0], [0, 0, 0, 1]]
> [399.0200131, 1901.955001, 3368.825906]
>
> 225/224
> [[1, 0, 0, -5], [0, 1, 0, 2], [0, 0, 1, 2]]
> [1200.493660, 1901.172569, 2785.167472]
>
> 245/243
> [[1, 0, 0, 0], [0, 1, 1, 2], [0, 0, 2, -1]]
> [1200., 1903.372995, 440.4316973]
>
> 250/243
> [[1, 2, 3, 0], [0, -3, -5, 0], [0, 0, 0, 1]]
> [1196.905960, 162.3176609, 3368.825906]
>
> 256/245
> [[1, 0, 0, 4], [0, 1, 0, 0], [0, 0, 2, -1]]
> [1195.228951, 1901.955001, 1398.695873]
>
> 405/392
> [[1, 0, 1, -1], [0, 1, 0, 2], [0, 0, 2, 1]]
> [1203.269293, 1896.773294, 787.7266785]
>
> 525/512
> [[1, 0, 0, 9], [0, 1, 0, -1], [0, 0, 1, -2]]
> [1202.406737, 1898.140412, 2780.725442]
>
> 648/625
> [[4, 0, 3, 0], [0, 1, 1, 0], [0, 0, 0, 1]]
> [299.1603149, 1896.631523, 3368.825906]
>
> 686/675
> [[1, 0, 2, 1], [0, 1, 0, 1], [0, 0, 3, 2]]
> [1198.513067, 1904.311735, 130.9133777]
>
> 729/700
> [[1, 0, 0, -2], [0, 1, 0, 6], [0, 0, 1, -2]]
> [1203.706383, 1896.080523, 2794.919668]
>
> 875/864
> [[1, 0, 0, 5], [0, 1, 0, 3], [0, 0, 1, -3]]
> [1201.121570, 1903.732647, 2783.709509]
>
> 1029/1000
> [[1, 0, 0, 1], [0, 3, 0, -1], [0, 0, 1, 1]]
> [1202.477948, 632.6758490, 2792.067330]
>
> 1029/1024
> [[1, 1, 0, 3], [0, 3, 0, -1], [0, 0, 1, 0]]
> [1200.421488, 233.6218235, 2786.313714]
>
> 1323/1280
> [[1, 0, 0, 4], [0, 1, 1, -1], [0, 0, 2, 1]]
> [1202.764567, 1897.573266, 447.5797863]
>
> 1728/1715
> [[1, 0, 0, 2], [0, 1, 0, 1], [0, 0, 3, -1]]
> [1199.391895, 1900.991178, 929.2418964]
>
> 2048/2025
> [[2, 0, 11, 0], [0, 1, -2, 0], [0, 0, 0, 1]]
> [599.5552941, 1903.364685, 3368.825906]
>
> 2240/2187
> [[1, 0, 0, -6], [0, 1, 0, 7], [0, 0, 1, -1]]
> [1198.134693, 1904.911442, 2781.982606]
>
> 2401/2400
> [[1, 1, 1, 2], [0, 2, 1, 1], [0, 0, 2, 1]]
> [1200.032113, 350.9868928, 617.6846359]
>
> 2430/2401
> [[1, 0, 3, 1], [0, 1, 3, 2], [0, 0, -4, -1]]
> [1199.075238, 1900.489288, 1628.631774]
>
> 3125/3024
> [[1, 0, 0, -4], [0, 1, 0, -3], [0, 0, 1, 5]]
> [1202.454598, 1905.845447, 2780.614314]
>
> 3125/3072
> [[1, 0, 2, 0], [0, 5, 1, 0], [0, 0, 0, 1]]
> [1201.276744, 380.7957184, 3368.825906]
>
> 3125/3087
> [[1, 0, 0, 0], [0, 1, 1, 1], [0, 0, 3, 5]]
> [1200., 1903.401919, 293.5973664]
>
> 3136/3125
> [[1, 0, 0, -3], [0, 1, 0, 0], [0, 0, 2, 5]]
> [1199.738066, 1901.955001, 1393.460953]
>
> 3645/3584
> [[1, 0, 0, -9], [0, 1, 0, 6], [0, 0, 1, 1]]
> [1201.235997, 1899.995991, 2783.443817]
>
> 4000/3969
> [[1, 0, 1, 4], [0, 1, 0, -2], [0, 0, 2, 3]]
> [1199.436909, 1902.847479, 792.7846742]
>
> 4375/4374
> [[1, 0, 0, 1], [0, 1, 0, 7], [0, 0, 1, -4]]
> [1200.016360, 1901.980932, 2786.275726]
>
> 5103/5000
> [[1, 0, 0, 3], [0, 1, 0, -6], [0, 0, 1, 4]]
> [1201.434720, 1899.681024, 2789.645030]
>
> 5120/5103
> [[1, 0, 0, 10], [0, 1, 0, -6], [0, 0, 1, 1]]
> [1199.766314, 1902.325384, 2785.771112]
>
> 5625/5488
> [[1, 0, 1, 0], [0, 1, 1, 2], [0, 0, -3, -4]]
> [1201.715742, 1899.235615, 106.2071570]
>
> 6144/6125
> [[1, 0, 1, 4], [0, 1, 1, -1], [0, 0, -2, 3]]
> [1199.786928, 1901.617290, 157.2978838]
>
> 8748/8575
> [[1, 0, 1, 0], [0, 1, 2, 1], [0, 0, -3, 2]]
> [1198.678173, 1899.859955, 736.3383942]
>
> 10976/10935
> [[1, 0, 2, -1], [0, 1, 2, 3], [0, 0, -3, -1]]
> [1199.758595, 1902.337618, 1139.106063]
>
> 15625/15552
> [[1, 0, 1, 0], [0, 6, 5, 0], [0, 0, 0, 1]]
> [1200.291038, 317.0693810, 3368.825906]
>
> 16875/16807
> [[1, 0, 0, 0], [0, 1, 3, 3], [0, 0, -5, -4]]
> [1200., 1901.560426, 583.7891213]
>
> 19683/19600
> [[1, 0, 0, -2], [0, 2, 0, 9], [0, 0, 1, -1]]
> [1200.256485, 950.7742412, 2786.909253]
>
> 32805/32768
> [[1, 0, 15, 0], [0, 1, -8, 0], [0, 0, 0, 1]]
> [1200.065120, 1901.851787, 3368.825906]

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> Thanks so much, Gene. If you could describe in common, non-
technical
> language the criteria you used to choose the set of generators,
I'll
> be able to explain it in my paper.

See if this will work for you:

Rules

1. The mapping matrix is a M[i,j] 4x3 integral matrix, meaning four
rows and three columns with integer values.

For i from 1 to 3 (the first three rows) we have the following rules
if the comma is 7-limit; in case it is a 5-limit comma the rules
apply only to the first two rows. Then the last row is [0,0,1] and
the last column is [0,0,0,1], giving a 7 as generator. Also only the
2x2 matrix obtained by deleting the last column and the last two rows
needs to have a determinant of +-1.

2. If j>i then M[i,j] = 0.

3. If j=i then M[i,j] is not equal to 0.

4. If j<i then |M[i,j]| < |M[i,i]|

5. The 3x3 matrix obtained by leaving off the last row has
determinant +-1.

6. The generators are three real numbers A, B, C greater than 1 (ie
in terms of cents, such that their value in cents is positive.)

7. A^M[i,1] * B^M[i,2] * C^M[i,3] gives the TOP tuning of the ith
prime, 1<=i<=4.

A, B, and C are associated to 2, 3, and 5 respectively, and these
rules result in A being an approximate octave if possible, B being
an approximate 3 if that is consistent with what we have for A, and C
being an approximate 5 if consistent with A and B.

That's not quite what I was asking for, but thanks! That's awesome.

What I was asking for was the simplest possible criterion for
determining that it's *this* set of generators and not any of the
other equivalent sets. *That's* what I feel I owe the readers, even
more than a method for the mathematical ones to be able to do it
themselves.

Of course, I could just give the readers the results without
explaining the criterion, but that's a last resort.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> What I was asking for was the simplest possible criterion for
> determining that it's *this* set of generators and not any of the
> other equivalent sets. *That's* what I feel I owe the readers, even
> more than a method for the mathematical ones to be able to do it
> themselves.

Don't the rules I give do that job? They force the result.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > What I was asking for was the simplest possible criterion for
> > determining that it's *this* set of generators and not any of the
> > other equivalent sets. *That's* what I feel I owe the readers,
even
> > more than a method for the mathematical ones to be able to do it
> > themselves.
>
> Don't the rules I give do that job? They force the result.

Yes, but it requires a lot more math than I've introduced in the
paper. If you were able to implement a criterion like the one I
sketched out, then I could provide a complete explanation for the
reader. But don't worry if it's not easy . . . I'll just use what you
did, with your rules in a footnote.