Yahoo Tuning Groups Ultimate Backup tuning-math 3-d ("planar") temperaments request

Hi Gene,

I'd like to include a table in my paper which summarizes a bunch of 7-
limit, codimension-1 temperaments.

As you'd guess, for each, I want a set of three generators, one of
which generates 2:1 all by itself. And then the mappings from these
generators to primes.

The criteria for choosing the generators should be something I can
explain, like, "the second generator is constrained to be narrower
than the first generator, and given that constraint, is chosen so as
to map to the simplest ratio possible. The third generator is
constrained to be smaller than the second generator, and given that
constraint, is chosen so as to map to the simplest ratio possible."

This is the list of commas:

28 27
36 35
49 48
50 49
64 63
81 80
126 125
128 125
225 224
245 243
250 243
256 245
405 392
525 512
648 625
686 675
729 700
875 864
1029 1000
1029 1024
1323 1280
1728 1715
2048 2025
2240 2187
2401 2400
2430 2401
3125 3024
3125 3072
3125 3087
3136 3125
3645 3584
4000 3969
4375 4374
5103 5000
5120 5103
5625 5488
6144 6125
8748 8575
10976 10935
15625 15552
16875 16807
19683 19600
32805 32768

Thanks a bunch,
Paul

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> I'd like to include a table in my paper which summarizes a bunch of
7-
> limit, codimension-1 temperaments.

Fortunately for you, I now have a new computer which doesn't crash
all the time. What's the time frame here?

> As you'd guess, for each, I want a set of three generators, one of
> which generates 2:1 all by itself. And then the mappings from these
> generators to primes.

The easiest approach is simply to take the Hermite reduction of a set
of generators for the vals. This would mean if 2 is a generator, it
will give it, otherwise a fraction of an octave. If 2, 3, and 5 will
work, it will always give that; and so forth if you wanted a complete
description of the decision proceedure. This would be easiest for me,
and it seems to me it has a good claim to be the best choice.

By "give the generators", do you mean a TOP tuning plus a mapping?

> The criteria for choosing the generators should be something I can
> explain, like, "the second generator is constrained to be narrower
> than the first generator, and given that constraint, is chosen so
as
> to map to the simplest ratio possible. The third generator is
> constrained to be smaller than the second generator, and given that
> constraint, is chosen so as to map to the simplest ratio possible."

Hermite reduction would result in a criterion you could explain. Tell
me if that would be acceptable.

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> Hi Gene,
>
> I'd like to include a table in my paper which summarizes a bunch of
7-
> limit, codimension-1 temperaments.

Here's what I can do with programs I've already written; I give the
comma, the Hermite reduced mapping, and the rms tuning of the
corresponding generators.

28/27
[[1, 0, 0, -2], [0, 1, 0, 3], [0, 0, 1, 0]]
[1200., 1924.441037, 2795.308127]

36/35
[[1, 0, 0, 2], [0, 1, 0, 2], [0, 0, 1, -1]]
[1200., 1885.698206, 2794.442111]

49/48
[[1, 0, 0, 2], [0, 2, 0, 1], [0, 0, 1, 0]]
[1200., 950.9775006, 2780.364245]

50/49
[[2, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 1]]
[600.0000000, 1901.955001, 2777.569810]

64/63
[[1, 0, 0, 6], [0, 1, 0, -2], [0, 0, 1, 0]]
[1200., 1911.692177, 2792.156018]

81/80
[[1, 0, -4, 0], [0, 1, 4, 0], [0, 0, 0, 1]]
[1200., 1896.164845, 3366.344411]

126/125
[[1, 0, 0, -1], [0, 1, 0, -2], [0, 0, 1, 3]]
[1200., 1899.984322, 2789.269735]

128/125
[[3, 0, 7, 0], [0, 1, 0, 0], [0, 0, 0, 1]]
[400.0000000, 1908.798145, 3375.669050]

225/224
[[1, 0, 0, -5], [0, 1, 0, 2], [0, 0, 1, 2]]
[1200., 1899.812912, 2784.171625]

245/243
[[1, 0, 0, 0], [0, 1, 1, 2], [0, 0, 2, -1]]
[1200., 1904.876579, 440.9272508]

250/243
[[1, 2, 3, 0], [0, 3, 5, 0], [0, 0, 0, 1]]
[1200., -162.9960265, 3371.413597]

256/245
[[1, 0, 0, 4], [0, 1, 0, 0], [0, 0, 2, -1]]
[1200., 1918.248103, 1404.018924]

405/392
[[1, 0, 1, -1], [0, 1, 0, 2], [0, 0, 2, 1]]
[1200., 1888.775891, 789.3913963]

525/512
[[1, 0, 0, 9], [0, 1, 0, -1], [0, 0, 1, -2]]
[1200., 1892.089471, 2774.475077]

648/625
[[4, 0, 3, 0], [0, 1, 1, 0], [0, 0, 0, 1]]
[300.0000000, 1894.134355, 3368.825906]

686/675
[[1, 0, 2, 1], [0, 1, 0, 1], [0, 0, 3, 2]]
[1200., 1907.336788, 130.2063806]

729/700
[[1, 0, 0, -2], [0, 1, 0, 6], [0, 0, 1, -2]]
[1200., 1889.305200, 2784.908179]

875/864
[[1, 0, 0, 5], [0, 1, 0, 3], [0, 0, 1, -3]]
[1200., 1904.145204, 2781.933305]

1029/1000
[[1, 0, 0, 1], [0, 3, 0, -1], [0, 0, 1, 1]]
[1200., 632.3352811, 2791.262872]

1029/1024
[[1, 1, 0, 3], [0, 3, 0, -1], [0, 0, 1, 0]]
[1200., 233.4444415, 2785.016372]

1323/1280
[[1, 0, 0, 4], [0, 1, 1, -1], [0, 0, 2, 1]]
[1200., 1888.607610, 445.9928964]

1728/1715
[[1, 0, 0, 2], [0, 1, 0, 1], [0, 0, 3, -1]]
[1200., 1900.647644, 929.2070233]

2048/2025
[[2, 0, 11, 0], [0, 1, -2, 0], [0, 0, 0, 1]]
[600.0000000, 1905.446531, 3370.920823]

2240/2187
[[1, 0, 0, -6], [0, 1, 0, 7], [0, 0, 1, -1]]
[1200., 1908.500466, 2788.495535]

2401/2400
[[1, 1, 1, 2], [0, 2, 1, 1], [0, 0, 2, 1]]
[1200., 350.9775007, 617.6844971]

2430/2401
[[1, 0, 3, 1], [0, 1, 3, 2], [0, 0, 4, 1]]
[1200., 1898.792052, -1627.854498]

3125/3024
[[1, 0, 0, -4], [0, 1, 0, -3], [0, 0, 1, 5]]
[1200., 1905.114871, 2776.834104]

3125/3072
[[1, 0, 2, 0], [0, 5, 1, 0], [0, 0, 0, 1]]
[1200., 379.9679494, 3366.005567]

3125/3087
[[1, 0, 0, 0], [0, 1, 1, 1], [0, 0, 3, 5]]
[1200., 1903.069782, 293.4856589]

3136/3125
[[1, 0, 0, -3], [0, 1, 0, 0], [0, 0, 2, 5]]
[1200., 1902.435257, 1393.797198]

3645/3584
[[1, 0, 0, -9], [0, 1, 0, 6], [0, 0, 1, 1]]
[1200., 1897.216978, 2783.549864]

4000/3969
[[1, 0, 1, 4], [0, 1, 0, -2], [0, 0, 2, 3]]
[1200., 1904.436192, 793.1568564]

4375/4374
[[1, 0, 0, 1], [0, 1, 0, 7], [0, 0, 1, -4]]
[1200., 1902.005884, 2786.302405]

5103/5000
[[1, 0, 0, 3], [0, 1, 0, -6], [0, 0, 1, 4]]
[1200., 1896.830653, 2786.883085]

5120/5103
[[1, 0, 0, 10], [0, 1, 0, -6], [0, 0, 1, 1]]
[1200., 1902.888698, 2786.702752]

5625/5488
[[1, 0, 1, 0], [0, 1, 1, 2], [0, 0, 3, 4]]
[1200., 1896.338273, -105.9626599]

6144/6125
[[1, 0, 1, 4], [0, 1, 1, -1], [0, 0, 2, -3]]
[1200., 1902.491206, -157.4631742]

8748/8575
[[1, 0, 1, 0], [0, 1, 2, 1], [0, 0, 3, -2]]
[1200., 1897.239564, -736.0551389]

10976/10935
[[1, 0, 2, -1], [0, 1, 2, 3], [0, 0, 3, 1]]
[1200., 1902.880572, -1139.661549]

15625/15552
[[1, 0, 1, 0], [0, 6, 5, 0], [0, 0, 0, 1]]
[1200., 317.0796754, 3368.695142]

16875/16807
[[1, 0, 0, 0], [0, 1, 3, 3], [0, 0, 5, 4]]
[1200., 1901.307749, -583.6772476]

19683/19600
[[1, 0, 0, -2], [0, 2, 0, 9], [0, 0, 1, -1]]
[1200., 950.5282864, 2786.121192]

32805/32768
[[1, 0, 15, 0], [0, 1, -8, 0], [0, 0, 0, 1]]
[1200., 1901.727514, 3368.705472]

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > I'd like to include a table in my paper which summarizes a bunch
of
> 7-
> > limit, codimension-1 temperaments.
>
> Fortunately for you, I now have a new computer which doesn't crash
> all the time. What's the time frame here?

ASAP.

> > As you'd guess, for each, I want a set of three generators, one
of
> > which generates 2:1 all by itself. And then the mappings from
these
> > generators to primes.
>
> The easiest approach is simply to take the Hermite reduction of a
set
> of generators for the vals. This would mean if 2 is a generator, it
> will give it, otherwise a fraction of an octave. If 2, 3, and 5
will
> work, it will always give that; and so forth if you wanted a
complete
> description of the decision proceedure. This would be easiest for
me,
> and it seems to me it has a good claim to be the best choice.

Sure . . . just make sure that you do give a complete description of
the decision procedure, because I do want one.

> Hermite reduction would result in a criterion you could explain.
Tell
> me if that would be acceptable.

Sure -- as long as the explanation ends up being something I can
understand, then I should be able to explain it in the paper.