More measurement units

282 = 2 * 3 * 47
Strong through 35 limit, or maybe 41. Consistent
though 29 limit. One cent is 0.235 of these.

422 = 2 * 211
Strong through 35 limit, consistent to 27.

1578 = 2 * 3 * 263
Strong and consistent through the 29 limit. One
cent is 1.315 of these.

2460 = 2^2 * 3 * 5 * 41
The marvelous mina. Storng and consistent through
the 27 limit; one cent is 2.05 minas.

3178 = 2 * 7 * 227
Strong and consistent through the 27 limit.

29053 = 17 * 1709
Strong through the 27 limit, consistent to 29.

42165 = 3^2 * 5 * 937
Strong through the 27 limit, consistent to 29. One
cent is 35.1375 of these.

58973 = 17 * 3469
Strong through 27 limit, consistent to 47.

75781 prime
Strong and consistent through the 27 limit.

119365 = 5 * 23873
Pretty strong and consistent through 29.

178338 = 2 * 3 * 29723
Pretty strong and consistent through 27.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> 282 = 2 * 3 * 47
> Strong through 35 limit, or maybe 41.

BTW, what does "strong" mean?
That term slipped by me.

-monz
http://tonalsoft.com
Tonescape microtonal music software

Hi Gene,

I've only been looking up to the 41-limit ...
here are my observations on some of your candidates:

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:

> 282 = 2 * 3 * 47
> Strong through 35 limit, or maybe 41. Consistent
> though 29 limit. One cent is 0.235 of these.

Are you sure about that one? Doesn't look particularly
good to me, especially for 11 or 13.

> 1578 = 2 * 3 * 263
> Strong and consistent through the 29 limit. One
> cent is 1.315 of these.

Wow, really awesome thru the 11-limit!
Only drawback is that error for 3 is worse than
for 5/7/11.

> 3178 = 2 * 7 * 227
> Strong and consistent through the 27 limit.

Quite good thru the 23-limit ... except that 7 is
only so-so.

> 29053 = 17 * 1709
> Strong through the 27 limit, consistent to 29.

Excellent thru the 41-limit, with only 31 and 37
being so-so.

> 42165 = 3^2 * 5 * 937
> Strong through the 27 limit, consistent to 29. One
> cent is 35.1375 of these.

Fantastic thru the 23-limit, with 3/5/7 following
my criteria of decreasing closeness ... altho some
of the other ones near this cardinality do a better
job of approximating all three of those primes.

> 58973 = 17 * 3469
> Strong through 27 limit, consistent to 47.

Wow, this one is fantastic thru the 23-limit!
And quite good for 41-limit. Also follows criteria
of decreasing closeness for 3/5/7, and is extremely
good for 3.

> 119365 = 5 * 23873
> Pretty strong and consistent through 29.

Tha bomb for 29-limit, again with decreasing closeness
for 3/5/7.

> 178338 = 2 * 3 * 29723
> Pretty strong and consistent through 27.

Really sweet for 23-limit.

Is there some reason why so many of these are so good
up to the 23-limit, and then not as good after that?

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@>
> wrote:
> >
> > 282 = 2 * 3 * 47
> > Strong through 35 limit, or maybe 41.
>
>
> BTW, what does "strong" mean?
> That term slipped by me.

I didn't define it. However, I was calling a
logflat max error badness figure well under 1
"strong". Hence it is a description of how
something looks tuningwise in the q-odd-limit
compared to nearby ets in the same limit. These
"strong" systems stand out.

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Gene,
>
> I've only been looking up to the 41-limit ...
> here are my observations on some of your candidates:
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@>
> wrote:
>
> > 282 = 2 * 3 * 47
> > Strong through 35 limit, or maybe 41. Consistent
> > though 29 limit. One cent is 0.235 of these.
>
> Are you sure about that one? Doesn't look particularly
> good to me, especially for 11 or 13.

It's not particularly good, except by way of comparison.

Here's the kind of data I was looking at; I
should have said "pretty strong" though the
41 limit I suppose. This gives the odd limt,
the logflat badness at that limit, and whether
it is consistent or not at that limit.

3: 11.442093 true
5: 3.631912 true
7: 2.137224 true
9: 2.137224 true
11: 1.804242 true
13: 1.471148 true
15: 1.471148 true
17: 1.218937 true
19: 1.065717 true
21: 1.065717 true
23: .963580 true
25: .963580 true
27: .963580 true
29: .890956 true
31: .983367 false
33: .983367 false
35: .983367 false
37: 1.044747 false
39: 1.044747 false
41: 1.001033 false
43: 1.155781 false
45: 1.155781 false
47: 1.400901 false
49: 1.400901 false
51: 1.400901 false
53: 1.523463 false

>
> > 1578 = 2 * 3 * 263
> > Strong and consistent through the 29 limit. One
> > cent is 1.315 of these.
>
> Wow, really awesome thru the 11-limit!
> Only drawback is that error for 3 is worse than
> for 5/7/11.

Here's 1578; it's nice as it is hot stuff in low
limits also:

3: 111.763646 true
5: 2.813501 true
7: .824573 true
9: .824573 true
11: .999344 true
13: 1.355426 true
15: 1.355426 true
17: 1.060406 true
19: .889871 true
21: .889871 true
23: .780221 true
25: .780221 true
27: .780221 true
29: .906272 true
31: 1.194760 false
33: 1.194760 false
35: 1.194760 false
37: 1.516556 false
39: 1.516556 false
41: 1.565097 false
43: 1.492934 false
45: 1.492934 false
47: 1.433734 false
49: 1.433734 false
51: 1.433734 false
53: 1.384329 false

Here's 2460 to compare it to:

3: 19.069364 true
5: 3.205958 true
7: 2.024827 true
9: 2.024827 true
11: 1.821701 true
13: 1.232907 true
15: 1.232907 true
17: .950384 true
19: .915553 true
21: .915553 true
23: .796400 true
25: .796400 true
27: .796400 true
29: 1.353714 false
31: 1.505700 false
33: 1.505700 false
35: 1.505700 false
37: 1.402528 false
39: 1.402528 false
41: 1.428005 false
43: 1.358291 false
45: 1.358291 false
47: 1.301252 false
49: 1.301252 false
51: 1.301252 false
53: 1.253759 false

> > 42165 = 3^2 * 5 * 937
> > Strong through the 27 limit, consistent to 29. One
> > cent is 35.1375 of these.
>
> Fantastic thru the 23-limit, with 3/5/7 following
> my criteria of decreasing closeness ... altho some
> of the other ones near this cardinality do a better
> job of approximating all three of those primes.

Here's your criterion in action:

3: 2367.863803 true
5: 31.679630 true
7: 6.141319 true
9: 6.141319 true
11: 3.333135 true
13: 1.957065 true
15: 1.957065 true
17: 1.472474 true
19: 1.175186 true
21: 1.175186 true
23: .971669 true
25: .971669 true
27: .971669 true
29: 1.396057 true
31: 1.475892 false
33: 1.475892 false
35: 1.475892 false
37: 1.625211 false
39: 1.625211 false
41: 1.499244 false
43: 1.933381 false
45: 1.933381 false
47: 1.823499 false
49: 1.823499 false
51: 1.823499 false
53: 1.733333 false

>
> > 58973 = 17 * 3469
> > Strong through 27 limit, consistent to 47.
>
>
> Wow, this one is fantastic thru the 23-limit!
> And quite good for 41-limit. Also follows criteria
> of decreasing closeness for 3/5/7, and is extremely
> good for 3.

We can see how that differs from 42165 in these
respects:

3: 380.079290 true
5: 17.481037 true
7: 5.771011 true
9: 5.771011 true
11: 2.410890 true
13: 1.392017 true
15: 1.392017 true
17: 1.220839 true
19: .939878 true
21: .939878 true
23: .772469 true
25: .772469 true
27: .772469 true
29: 1.452505 true
31: 1.285613 true
33: 1.285613 true
35: 1.285613 true
37: 1.341341 true
39: 1.341341 true
41: 1.234235 true
43: 1.150315 true
45: 1.150315 true
47: 1.082940 true
49: 1.082940 false
51: 1.082940 false
53: 1.311734 false

> Is there some reason why so many of these are so good
> up to the 23-limit, and then not as good after that?

23 is the last prime before 29. How's about we look
at some of the ones I mentioned before?

1600

3: 95.998154 true
5: 5.798027 true
7: 3.708550 true
9: 3.708550 true
11: 2.041026 true
13: 1.692703 true
15: 1.692703 true
17: 1.323661 true
19: 1.166447 true
21: 1.166447 true
23: 1.022465 true
25: 1.022465 true
27: 1.022465 true
29: 1.102921 true
31: 1.016115 true
33: 1.016115 true
35: 1.016115 true
37: .950199 true
39: .950199 false
41: .973391 false
43: .928428 false
45: .928428 false
47: 1.183320 false
49: 1.183320 false
51: 1.183320 false
53: 1.318758 false

16808

3: 835.561333 true
5: 10.668964 true
7: 2.107995 true
9: 2.107995 true
11: 1.812993 true
13: 1.114604 true
15: 1.114604 true
17: .805879 true
19: .795808 true
21: .795808 true
23: .668887 true
25: .668887 true
27: .668887 true
29: .680382 true
31: .610665 true
33: .610665 true
35: .610665 true
37: 1.349745 false
39: 1.349745 false
41: 1.705450 false
43: 1.763175 false
45: 1.763175 false
47: 1.929018 false
49: 1.929018 false
51: 1.929018 false
53: 1.841683 false

31920

3: 96.494776 true
5: 10.404223 true
7: 7.420016 true
9: 7.420016 true
11: 3.851604 true
13: 2.293182 true
15: 2.293182 true
17: 1.622942 true
19: 1.267837 true
21: 1.267837 true
23: 1.201093 true
25: 1.201093 true
27: 1.201093 true
29: 1.089819 true
31: .971201 true
33: .971201 true
35: .971201 true
37: .985129 true
39: .985129 true
41: .910692 true
43: 1.459075 false
45: 1.459075 false
47: 1.598870 false
49: 1.598870 false
51: 1.598870 false
53: 1.872567 false

241200

3: 10812.044980 true
5: 49.756904 true
7: 6.306530 true
9: 6.306530 true
11: 3.325773 true
13: 1.831756 true
15: 1.831756 true
17: 1.211867 true
19: 1.232460 true
21: 1.232460 true
23: 1.117510 true
25: 1.117510 true
27: 1.117510 true
29: 1.102865 true
31: .982622 true
33: .982622 true
35: .982622 true
37: .895375 true
39: .895375 true
41: 1.708567 false
43: 1.578082 false
45: 1.578082 false
47: 1.474199 false
49: 1.474199 false
51: 1.474199 false
53: 1.987960 false

324296

3: 280.883122 true
5: 3.724566 true
7: 1.937199 true
9: 1.937199 true
11: 3.908910 true
13: 2.545319 true
15: 2.545319 true
17: 1.667417 true
19: 1.232626 true
21: 1.232626 true
23: 1.217679 true
25: 1.217679 true
27: 1.217679 true
29: 1.020921 true
31: .886664 true
33: .886664 true
35: .886664 true
37: .790059 true
39: .790059 true
41: 1.045695 true
43: .964003 true
45: .964003 true
47: .899081 true
49: .899081 true
51: .899081 true
53: .846362 true
55: .846362 true
57: .846362 true
59: .967555 true
61: 1.656537 false
63: 1.656537 false
65: 1.656537 false
67: 1.719754 false
69: 1.719754 false
71: 1.657114 false
73: 1.602691 false
75: 1.602691 false
77: 1.602691 false
79: 1.554994 false
81: 1.554994 false
83: 1.512865 false
85: 1.512865 false
87: 1.512865 false
89: 1.475397 false
91: 1.475397 false
93: 1.475397 false
95: 1.475397 false
97: 1.441868 false
99: 1.441868 false
101: 1.411694 false

monz wrote:

> I've only been looking up to the 41-limit ...

You want good 41-limit equal temperaments then? I had to modify my library both for such high prime limits and such large ETs. Some good octave sizes of 41-limit ETs by TOP-max error:

311, 1600, 2270, 4380, 4460, 5144, 5395, 6151, 6349, 9614, 12348, 14348, 14394, 15854, 17230, 18324, 18895, 18921, 20203, 20567, 22203, 27658, 28342, 29053, 30631, 31296, 31809, 31920, 33616, 34691

and some really good ones:

18895, 20567, 34691, 69702, 83096, 89023, 95524, 104253, 109590, 130215, 144281, 148418, 165226, 165879, 173958, 182687, 188496, 227377, 228955, 256119, 273580, 278435, 280028, 286350, 291745, 294277, 324296, 334553, 341757, 343757

TOP-max is probably the thing to go for here because if you want to measure JI intervals it's the worst approximation that causes trouble. But for comparison here's a TOP-RMS list:

311, 1395, 1920, 2742, 3041, 3084, 3395, 3992, 4349, 4501, 5144, 5638, 5809, 5927, 6349, 6501, 7361, 7527, 7847, 8539, 8893, 9204, 9717, 9934, 10589, 10729, 11353, 11534, 12245, 12348

311, 5144, 6349, and 12348 are also on the TOP-max list. (And 14348, 15854, 17230, 18324, 18895, 18921, 20203, 20567, 22203, 28342, 29053, 30631, 31809, 31920, 33616, and 34691 if you take the TOP-RMS list further.) So they have a good average error without the worst case getting too bad. A stricter TOP-RMS list:

34691, 58973, 65322, 67242, 81176, 83096, 95524, 109590, 112985, 148418, 165226, 201122, 227377, 228955, 256119, 286350, 324296, 343757, 345142, 360565, 369104, 377373, 381874, 385729, 389618, 398157, 418957, 421998, 430077, 437281

34691, 83096, 95224, 109590, 148418, 165226, 227377, 228955, 256119, 286350, 324296, and 343757 are in the cream of both crops.

I haven't checked if these are consistent but you should be able to work out the good mappings. I could post them to tuning-math but this is quite enough numbers for tuning.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> I haven't checked if these are consistent but you should be
> able to work out the good mappings. I could post them to
> tuning-math but this is quite enough numbers for tuning.

Of the 102 you give, 42 are 41-limit consistent.
Of the rest, many from my point of view are
fairly putrid; in fact, I wonder if 95224, one
of your cream of the crop numbers, isn't a typo.

One reason may be your use of TOP; and I can't
see how this makes any sense when trying to find
a measurement system. If you use max error, pure
octaves, the only ones to fall under the "1"
badness cutoff figure for logflat badness are
311, 1600, 20567, 31920, and 34691. Of these, all
except 1600 are consistent.

Some others I think look dubious are 6501, 7847,
22203, 31296, 31809, 67242, and 81176. I think
I'll run a 41-limit search and see what I get by
way of comparison; it would be interesting to
nail down why there seems to be such a big difference.

Gene Ward Smith wrote:
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> > >>I haven't checked if these are consistent but you should be >>able to work out the good mappings. I could post them to >>tuning-math but this is quite enough numbers for tuning.
> > Of the 102 you give, 42 are 41-limit consistent. > Of the rest, many from my point of view are > fairly putrid; in fact, I wonder if 95224, one
> of your cream of the crop numbers, isn't a typo.

Yes it's a typo. You'll notice it's 95524 in the original lists (with cut-and-paste accuracy).

> One reason may be your use of TOP; and I can't > see how this makes any sense when trying to find > a measurement system. If you use max error, pure
> octaves, the only ones to fall under the "1"
> badness cutoff figure for logflat badness are
> 311, 1600, 20567, 31920, and 34691. Of these, all
> except 1600 are consistent.

I use TOP with max error.

> Some others I think look dubious are 6501, 7847, > 22203, 31296, 31809, 67242, and 81176. I think
> I'll run a 41-limit search and see what I get by
> way of comparison; it would be interesting to > nail down why there seems to be such a big difference.

Yes, but may be better on tuning-math.

Graham

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:

> One reason may be your use of TOP; and I can't
> see how this makes any sense when trying to find
> a measurement system. If you use max error, pure
> octaves, the only ones to fall under the "1"
> badness cutoff figure for logflat badness are
> 311, 1600, 20567, 31920, and 34691. Of these, all
> except 1600 are consistent.

Going up to 400000, the only consistent ones with
a logflat figure less than 1 are 311, 2554, 20567,
31920 and 34691. Obviously, 1 is a little too strict
as a cuttoff. How did 2554 get missed, I wonder?

Our old friend 31920 remains very attractive because
of its divisibility properties. However, I want to
call attention to 20567, which is consistent--and
hot stuff--up to the 57 limit.

311 prime

3 23.841968 true
5 3.461784 true
7 1.329964 true
9 1.329964 true
11 1.146284 true
13 .891584 true
15 .891584 true
17 .947959 true
19 .826872 true
21 .826872 true
23 .764730 true
25 .764730 true
27 .764730 true
29 .887950 true
31 .859649 true
33 .859649 true
35 .859649 true
37 .815942 true
39 .815942 true
41 .781223 true
43 1.043355 false
45 1.043355 false
47 1.369367 false
49 1.369367 false
51 1.369367 false
53 1.332446 false
55 1.332446 false
57 1.332446 false
59 1.395939 false
61 1.366790 false
63 1.366790 false
65 1.366790 false
67 1.341392 false
69 1.341392 false

2554 = 2 * 1277

3 14.744646 true
5 10.619238 true
7 3.005611 true
9 3.005611 true
11 2.871183 true
13 2.234551 true
15 2.234551 true
17 1.720347 true
19 1.427223 true
21 1.427223 true
23 1.240648 true
25 1.240648 true
27 1.240648 true
29 1.161235 true
31 1.064295 true
33 1.064295 true
35 1.064295 true
37 .991031 true
39 .991031 true
41 .944946 true
43 1.332105 false
45 1.332105 false
47 1.295550 false
49 1.295550 false
51 1.295550 false
53 1.248042 false
55 1.248042 false
57 1.248042 false
59 1.490033 false
61 1.447670 false
63 1.447670 false
65 1.447670 false
67 1.411025 false
69 1.411025 false

20567 = 131 * 157

3 1568.185787 true
5 24.577261 true
7 6.204225 true
9 6.204225 true
11 3.384219 true
13 2.059681 true
15 2.059681 true
17 1.479202 true
19 1.167705 true
21 1.167705 true
23 1.168514 true
25 1.168514 true
27 1.168514 true
29 1.263145 true
31 1.131173 true
33 1.131173 true
35 1.131173 true
37 1.033519 true
39 1.033519 true
41 .958612 true
43 .899486 true
45 .899486 true
47 .930298 true
49 .930298 true
51 .930298 true
53 .953040 true
55 .953040 true
57 .953040 true
59 1.083102 false
61 1.322675 false
63 1.322675 false
65 1.322675 false
67 1.317205 false
69 1.317205 false

31920 = 3^4 * 3 * 5 * 7 * 19

3 96.494776 true
5 10.404223 true
7 7.420016 true
9 7.420016 true
11 3.851604 true
13 2.293182 true
15 2.293182 true
17 1.622942 true
19 1.267837 true
21 1.267837 true
23 1.201093 true
25 1.201093 true
27 1.201093 true
29 1.089819 true
31 .971201 true
33 .971201 true
35 .971201 true
37 .985129 true
39 .985129 true
41 .910692 true
43 1.459075 false
45 1.459075 false
47 1.598870 false
49 1.598870 false
51 1.598870 false
53 1.872567 false
55 1.872567 false
57 1.872567 false
59 1.793372 false
61 1.726280 false
63 1.726280 false
65 1.726280 false
67 1.668753 false
69 1.668753 false

34691 = 113 * 307

3 2285.702651 true
5 13.676202 true
7 2.394764 true
9 2.394764 true
11 3.740204 true
13 2.217606 true
15 2.217606 true
17 1.742964 true
19 1.425049 true
21 1.425049 true
23 1.182373 true
25 1.182373 true
27 1.182373 true
29 1.330277 true
31 1.184392 true
33 1.184392 true
35 1.184392 true
37 1.077012 true
39 1.077012 true
41 .995004 true
43 1.493092 false
45 1.493092 false
47 1.458597 false
49 1.458597 false
51 1.458597 false
53 1.481389 false
55 1.481389 false
57 1.481389 false
59 1.418246 false
61 1.364770 false
63 1.364770 false
65 1.364770 false
67 1.318932 false
69 1.318932 false

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Gene Ward Smith wrote:
> > --- In tuning@yahoogroups.com, Graham Breed <gbreed@> wrote:

> > Of the 102 you give, 42 are 41-limit consistent.
> > Of the rest, many from my point of view are
> > fairly putrid; in fact, I wonder if 95224, one
> > of your cream of the crop numbers, isn't a typo.
>
> Yes it's a typo. You'll notice it's 95524 in the original
> lists (with cut-and-paste accuracy).

It looks a lot better; in fact as a 41-limit system
it looks pretty damn good. In higher limits it
still looks good (using the patent tuning) even when
not consistent.

95524 = 2^2 * 11 * 13 * 167

3 4019.756169 true
5 43.474956 true
7 10.441955 true
9 10.441955 true
11 4.466493 true
13 2.517451 true
15 2.517451 true
17 1.717739 true
19 1.450344 true
21 1.450344 true
23 1.250744 true
25 1.250744 true
27 1.250744 true
29 1.105289 true
31 1.194415 true
33 1.194415 true
35 1.194415 true
37 1.125643 true
39 1.125643 true
41 1.031983 true
43 1.240260 false
45 1.240260 false
47 1.164527 false
49 1.164527 false
51 1.164527 false
53 1.102642 false
55 1.102642 false
57 1.102642 false
59 1.051197 false
61 1.007801 false
63 1.007801 false
65 1.007801 false
67 1.114510 false
69 1.114510 false