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Message: 9825 - Contents - Hide Contents
Date: Wed, 04 Feb 2004 08:25:40
Subject: Re: Acceptance regions
From: Gene Ward Smith
It occurs to me we aren't getting a triangular region, we are getting
a quadrant and then sawing off a corner using the badness cutoff
line, so we have an unbounded region. It acts like it's bounded
because we can only get so far when trying simultaneously for small
error and small compexity, but no error and no complexity is down at
-infinity, -infinity. This makes the idea of using an ellipse pretty
dubious; a parabolic region might make more sense.
Message: 9826 - Contents - Hide Contents
Date: Wed, 04 Feb 2004 09:00:32
Subject: Re: finding a moat in 7-limit commas a bit tougher . . .
From: Dave Keenan
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...>
> wrote:
>> Perhaps we should limit such tests to otonalities having at most one
>> note per prime (or odd) in the limit. e.g. If you can't make a
>> convincing major triad then it aint 5-limit. And you can't use
>> scale-spectrum timbres although you can use inharmonics that have no
>> relation to the scale.
>
> yes, mastuuuhhhhh . . . =(
It was just a suggestion. I wrote "perhaps we should" and "e.g.".
What does "=(" mean?
I'm guessing you think it's a bad idea.
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Message: 9827 - Contents - Hide Contents
Date: Thu, 05 Feb 2004 21:37:32
Subject: Re: Some convex hull badness measures
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> Hi Gene,
>
> To be able to comment on any of this, I really need to see them
> plotted in the (linear) error vs. complexity plane.
>
> Could you just post something like that list of 114 TOP 7-limit
linear
> temps again, but with Paul's latest favourite complexity measure
It should be a new list based on that complexity measure. The list
should agree with the complexity measure. Otherwise things will be
missing.
Message: 9828 - Contents - Hide Contents
Date: Thu, 05 Feb 2004 21:41:20
Subject: Re: Acceptance regions
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
> wrote:
>
>> No; the idea was to do a complete search within an extra-large
> region
>> and then look for the widest moats. Dave and I have done this for
>> equal temperaments, 5-limit linear temperaments, 7-limit planar
>> temperaments. Now we're asking for your help.
>
> And the reason why we care about moats is?
To come up with a list of temperaments which would not change even if
our cutoff criterion were to be altered by a fair amount.
Message: 9829 - Contents - Hide Contents
Date: Thu, 05 Feb 2004 22:24:40
Subject: Re: Acceptance regions
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
>> And the reason why we care about moats is?
>
> To come up with a list of temperaments which would not change even if
> our cutoff criterion were to be altered by a fair amount.
I thought these moats were gerrymandered, so how is that going to
work? Anyway, isn't it more important to have a list with the good
stuff on it, moat or no moat?
Message: 9830 - Contents - Hide Contents
Date: Thu, 05 Feb 2004 22:28:39
Subject: Re: Acceptance regions
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
wrote:
>
>>> And the reason why we care about moats is?
>>
>> To come up with a list of temperaments which would not change
even if
>> our cutoff criterion were to be altered by a fair amount.
>
> I thought these moats were gerrymandered, so how is that going to
> work?
Unclear on your question . . .
> Anyway, isn't it more important to have a list with the good
> stuff on it,
That's obviously the starting point.
> moat or no moat?
Without a moat, there would be questionable cases, of "if those are
in, why isn't this in" and "if those are out, why isn't this out".
Message: 9831 - Contents - Hide Contents
Date: Thu, 05 Feb 2004 23:29:18
Subject: A 41-limit temperament
From: Gene Ward Smith
The 140&171 7-limit temperament has a 311-et generator of 20/311;
extending that to the 41-limit gives a mapping of
[<1, 3, 2, 3, -4, 1, 1, 11, -3, 1, 11, 0, 6|,
<0, -22, 5, -3, 116, 42, 48, -105, 117, 60, -94, 81, -10|]
An LLL-reduced basis is
{703/702, 784/783, 875/874, 1000/999, 1625/1624, 1729/1728, 8092/8073,
10557/10556, 68921/68894, 177023/177008, 11670750/11648281}
Message: 9832 - Contents - Hide Contents
Date: Thu, 05 Feb 2004 23:31:55
Subject: Re: Acceptance regions
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> Without a moat, there would be questionable cases, of "if those are
> in, why isn't this in" and "if those are out, why isn't this out".
With a moat, there might be a question of why you are using a
seemingly unmotivated, ad hoc criterion. Maybe we could formalize it
to a similarity circle or something that could be justified?
Message: 9833 - Contents - Hide Contents
Date: Thu, 05 Feb 2004 23:38:45
Subject: Re: [tuning] Re: question about 24-tET
From: Carl Lumma
>> >an we get generators for 5-limit meantone, 7-limit schismic,
>> and 11-limit miracle for each of:
>>
>> (1) TOP
>> (2) odd-limit TOP
>> (3) rms TOP (or can you only do integer-limit rms TOP?)
>> (4) rms odd-limit TOP
>
>I can do the TOP. What's the definition for the others?
You know what (1) is.
I thought you just posted something about doing (2) & (4) by
leaving out the 2-terms in a certain formula. Here:
>For any set of consonances C we want to do an rms optimization for,
>we can find a corresponding Euclidean norm on the val space (or
>octave-excluding subspace if we are interested in the odd limit) by
>taking the sum of terms
>
>(c2 x2 + c3 x3 + ... + cp xp)^2
>
>for each monzo |c2 c3 ... cp> in C. If we want something corresponded
>to weighted optimization we would add weights, and if we wanted the
>odd limit, the consonances in C can be restricted to quotients of odd
>integers,
In (2) I mean the tuning that gives minimax error over all odd-limit
consonances (try the 9-limit). As far as weighting for this, I'd try
the usual Tenney weighting as in (1), and Paul's odd-limit weighting
suggestions:
>>> Now what if we apply 'odd-limit-weighting' to each of the intervals,
>>> including 9:3 which is treated as having an odd-limit of 9? Try
>>> using 'odd-limit' plus-or-minus 1 or 1/2 too.
>>
>> Is the weighting by multiplying or dividing by the log of the odd
>> limit? Presumably mutliplying will make more sense. Do we square and
>> then multiply, since we will be taking square roots?
>
>Divide. As in TOP, errors of more complex intervals are divided by
>larger numbers.
For (4) it's the tuning that gives minimum rms error over the 9-limit
consonances. All weighting suggestions apply.
For (3) it's the tuning that gives minimum rms over all intervals
with Tenney weighting as in (1).
>If I'm doing rms analogs of TOP, don't I need a list of intervals
>and maybe weights for them in order to cook up a Euclidean metric?
>I think Paul wanted something like that, and I could do it if I
>could remember exactly what it was.
See above.
-Carl
Message: 9834 - Contents - Hide Contents
Date: Thu, 05 Feb 2004 00:04:46
Subject: Off topic - Emoticon humor
From: Dave Keenan
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
>>> yes, mastuuuhhhhh . . . =(
> It's a picture of me succumbing to your authority.
I can't see it. While searching for any precedent for this emoticon I
came across the following, which cracked me up.
oops * [with cont.] (Wayb.)
Message: 9835 - Contents - Hide Contents
Date: Thu, 05 Feb 2004 12:08:21
Subject: 126 7-limit linears
From: Gene Ward Smith
I first made a candidate list by the kitchen sink method:
(1) All pairs n,m<=200 of standard vals
(2) All pairs n,m<=200 of TOP vals
(3) All pairs 100<=n,m<400 of standard vals
(4) All pairs 100<=n,m<=400 of TOP vals
(5) Generators of standard vals up to 100
(6) Generators of certain nonstandard vals up to 100
(7) Pairs of commas from Paul's list of relative error < 0.06,
epimericity < 0.5
(8) Pairs of vals with consistent badness figure < 1.5 up to 5000
This lead to a list of 32201 candidate wedgies, most of which of
course were incredible garbage. I then accepted everything with a 2.8
exponent badness less than 10000, where error is TOP error and
complexity is our mysterious L1 TOP complexity. I did not do any
cutting off for either error or complexity, figuring people could
decide how to do that for themselves. The first six systems are
macrotemperaments of dubious utility, number 7 is the {15/14, 25/24}
temperament, and 8 and 9 are the beep-ennealimmal pair, and number 13
is father. After ennealimmal, we don't get back into the micros until
number 46; if we wanted to avoid going there we can cutoff at 4000.
Number 46, incidentally, has TM basis {2401/2400, 65625/65536} and is
covered by 140, 171, 202 and 311; the last is interesting because of
the peculiar talents of 311.
1 [0, 0, 2, 0, 3, 5] 662.236987 77.285947 2.153690
2 [1, 1, 0, -1, -3, -3] 806.955502 64.326132 2.467788
3 [0, 0, 3, 0, 5, 7] 829.171704 30.152577 3.266201
4 [0, 2, 2, 3, 3, -1] 870.690617 33.049025 3.216583
5 [1, 2, 1, 1, -1, -3] 888.831828 49.490949 2.805189
6 [1, 2, 3, 1, 2, 1] 1058.235145 33.404241 3.435525
7 [2, 1, 3, -3, -1, 4] 1076.506437 16.837898 4.414720
8 [2, 3, 1, 0, -4, -6] 1099.121425 14.176105 4.729524
9 [18, 27, 18, 1, -22, -34] 1099.959348 .036377 39.828719
10 [1, -1, 0, -4, -3, 3] 1110.471803 39.807123 3.282968
11 [0, 5, 0, 8, 0, -14] 1352.620311 7.239629 6.474937
12 [1, -1, -2, -4, -6, -2] 1414.400610 20.759083 4.516198
13 [1, -1, 3, -4, 2, 10] 1429.376082 14.130876 5.200719
14 [1, 4, -2, 4, -6, -16] 1586.917865 4.771049 7.955969
15 [1, 4, 10, 4, 13, 12] 1689.455290 1.698521 11.765178
16 [2, 1, -1, -3, -7, -5] 1710.030839 16.874108 5.204166
17 [1, 4, 3, 4, 2, -4] 1749.120722 14.253642 5.572288
18 [0, 0, 4, 0, 6, 9] 1781.787825 33.049025 4.153970
19 [1, -1, 1, -4, -1, 5] 1827.319456 54.908088 3.496512
20 [4, 4, 4, -3, -5, -2] 1926.265442 5.871540 7.916963
21 [2, -4, -4, -11, -12, 2] 2188.881053 3.106578 10.402108
22 [3, 0, 6, -7, 1, 14] 2201.891023 5.870879 8.304602
23 [0, 0, 5, 0, 8, 12] 2252.838883 19.840685 5.419891
24 [4, 2, 2, -6, -8, -1] 2306.678659 7.657798 7.679190
25 [2, 1, 6, -3, 4, 11] 2392.139586 9.396316 7.231437
26 [2, -1, 1, -6, -4, 5] 2452.275337 22.453717 5.345120
27 [0, 0, 7, 0, 11, 16] 2580.688285 9.431411 7.420171
28 [1, -3, -4, -7, -9, -1] 2669.323351 9.734056 7.425960
29 [5, 1, 12, -10, 5, 25] 2766.028555 1.276744 15.536039
30 [7, 9, 13, -2, 1, 5] 2852.991531 1.610469 14.458536
31 [2, -2, 1, -8, -4, 8] 3002.749158 14.130876 6.779481
32 [3, 0, -6, -7, -18, -14] 3181.791246 2.939961 12.125211
33 [2, 8, 1, 8, -4, -20] 3182.905310 3.668842 11.204461
34 [6, -7, -2, -25, -20, 15] 3222.094343 .631014 21.101881
35 [4, -3, 2, -14, -8, 13] 3448.998676 3.187309 12.124601
36 [1, -3, -2, -7, -6, 4] 3518.666155 18.633939 6.499551
37 [1, 4, 5, 4, 5, 0] 3526.975600 19.977396 6.345287
38 [2, 6, 6, 5, 4, -3] 3589.967809 8.400361 8.700992
39 [2, 1, -4, -3, -12, -12] 3625.480387 9.146173 8.470366
40 [2, -2, -2, -8, -9, 1] 3634.089963 14.531543 7.185526
41 [3, 2, 4, -4, -2, 4] 3638.704033 20.759083 6.329002
42 [6, 5, 3, -6, -12, -7] 3680.095702 3.187309 12.408714
43 [2, 8, 8, 8, 7, -4] 3694.344150 3.582707 11.917575
44 [2, 3, 6, 0, 4, 6] 3938.578264 20.759083 6.510560
45 [0, 0, 5, 0, 8, 11] 3983.263457 38.017335 5.266481
46 [22, -5, 3, -59, -57, 21] 4009.709706 .073527 49.166221
47 [3, 5, 9, 1, 6, 7] 4092.014696 6.584324 9.946084
48 [7, -3, 8, -21, -7, 27] 4145.427852 .946061 19.979719
49 [1, -8, -14, -15, -25, -10] 4177.550548 .912904 20.291786
50 [3, 5, 1, 1, -7, -12] 4203.022260 12.066285 8.088219
51 [1, 9, -2, 12, -6, -30] 4235.792998 2.403879 14.430906
52 [6, 10, 10, 2, -1, -5] 4255.362112 3.106578 13.189661
53 [2, 5, 3, 3, -1, -7] 4264.417050 21.655518 6.597656
54 [6, 5, 22, -6, 18, 37] 4465.462582 .536356 25.127403
55 [0, 0, 12, 0, 19, 28] 4519.315488 3.557008 12.840061
56 [1, -3, 3, -7, 2, 15] 4555.017089 15.315953 7.644302
57 [1, -1, -5, -4, -11, -9] 4624.441621 14.789095 7.782398
58 [16, 2, 5, -34, -37, 6] 4705.894319 .307997 31.211875
59 [4, -32, -15, -60, -35, 55] 4750.916876 .066120 54.255591
60 [1, -8, 39, -15, 59, 113] 4919.628715 .074518 52.639423
61 [3, 0, -3, -7, -13, -7] 4967.108742 11.051598 8.859010
62 [6, 0, 0, -14, -17, 0] 5045.450988 5.526647 11.410361
63 [37, 46, 75, -13, 15, 45] 5230.896745 .021640 83.678088
64 [1, 6, 5, 7, 5, -5] 5261.484667 11.970043 8.788871
65 [3, 2, -1, -4, -10, -8] 5276.949135 17.564918 7.671954
66 [1, 4, -9, 4, -17, -32] 5338.184867 2.536420 15.376139
67 [1, -3, 5, -7, 5, 20] 5338.971970 8.959294 9.797992
68 [10, 9, 7, -9, -17, -9] 5386.217633 1.171542 20.325677
69 [19, 19, 57, -14, 37, 79] 5420.385757 .046052 64.713343
70 [5, 3, 7, -7, -3, 8] 5753.932407 7.459874 10.743721
71 [3, 5, -6, 1, -18, -28] 5846.930660 3.094040 14.795975
72 [3, 12, -1, 12, -10, -36] 5952.918469 1.698521 18.448015
73 [6, 0, 3, -14, -12, 7] 6137.760804 5.291448 12.429144
74 [4, 4, 0, -3, -11, -11] 6227.282004 12.384652 9.221275
75 [3, 0, 9, -7, 6, 21] 6250.704457 6.584324 11.570803
76 [9, 5, -3, -13, -30, -21] 6333.111158 1.049791 22.396682
77 [0, 0, 8, 0, 13, 19] 6365.852053 14.967465 8.686091
78 [4, 2, 5, -6, -3, 6] 6370.380556 16.499269 8.391154
79 [1, -8, -2, -15, -6, 18] 6507.074340 4.974313 12.974488
80 [2, -6, 1, -14, -4, 19] 6598.741284 6.548265 11.820058
81 [2, 25, 13, 35, 15, -40] 6657.512727 .299647 35.677429
82 [6, -2, -2, -17, -20, 1] 6845.573750 3.740932 14.626943
83 [1, 7, 3, 9, 2, -13] 6852.061008 12.161876 9.603642
84 [0, 5, 5, 8, 8, -2] 7042.202107 19.368923 8.212986
85 [4, 2, 9, -6, 3, 15] 7074.478038 8.170435 11.196673
86 [8, 6, 6, -9, -13, -3] 7157.960980 3.268439 15.596153
87 [5, 8, 2, 1, -11, -18] 7162.155511 5.664628 12.817743
88 [3, 17, -1, 20, -10, -50] 7280.048554 .894655 24.922952
89 [4, 2, -1, -6, -13, -8] 7307.246603 13.289190 9.520562
90 [5, 13, -17, 9, -41, -76] 7388.593186 .276106 38.128083
91 [8, 18, 11, 10, -5, -25] 7423.457669 .968741 24.394122
92 [3, -2, 1, -10, -7, 8] 7553.291925 18.095699 8.628089
93 [3, 7, -1, 4, -10, -22] 7604.170165 7.279064 11.973078
94 [6, 10, 3, 2, -12, -21] 7658.950254 3.480440 15.622931
95 [14, 59, 33, 61, 13, -89] 7727.766150 .037361 79.148236
96 [3, -5, -6, -15, -18, 0] 7760.555544 4.513934 14.304666
97 [13, 14, 35, -8, 19, 42] 7785.862490 .261934 39.585940
98 [11, 13, 17, -5, -4, 3] 7797.739891 1.485250 21.312375
99 [2, -4, -16, -11, -31, -26] 7870.803242 1.267597 22.628529
100 [2, -9, -4, -19, -12, 16] 7910.552221 2.895855 16.877046
101 [0, 0, 9, 0, 14, 21] 7917.731843 14.176105 9.573860
102 [3, 12, 11, 12, 9, -8] 7922.981072 2.624742 17.489863
103 [1, -6, 3, -12, 2, 24] 8250.683192 8.474270 11.675656
104 [55, 73, 93, -12, -7, 11] 8282.844862 .017772 105.789216
105 [4, 7, 2, 2, -8, -15] 8338.658153 10.400103 10.893408
106 [0, 5, -5, 8, -8, -26] 8426.314560 8.215515 11.894828
107 [5, 8, 14, 1, 8, 10] 8428.707855 4.143252 15.190723
108 [6, 7, 5, -3, -9, -8] 8506.845926 6.986391 12.646486
109 [8, 13, 23, 2, 14, 17] 8538.660000 1.024522 25.136807
110 [0, 0, 10, 0, 16, 23] 8630.819015 11.358665 10.686371
111 [3, -7, -8, -18, -21, 1] 8799.551719 2.900537 17.521249
112 [0, 5, 10, 8, 16, 9] 8869.402675 6.941749 12.865826
113 [4, 16, 9, 16, 3, -24] 8931.184092 1.698521 21.324102
114 [6, 5, 7, -6, -6, 2] 8948.277847 9.097987 11.718042
115 [3, -3, 1, -12, -7, 11] 9072.759561 14.130876 10.062449
116 [0, 12, 24, 19, 38, 22] 9079.668325 .617051 30.795105
117 [33, 78, 90, 47, 50, -10] 9153.275887 .016734 112.014440
118 [5, 1, -7, -10, -25, -19] 9260.372155 3.148011 17.329377
119 [1, -6, -2, -12, -6, 12] 9290.939644 13.273963 10.377495
120 [2, -2, 4, -8, 1, 15] 9367.180611 25.460673 8.247748
121 [3, 5, 16, 1, 17, 23] 9529.360455 3.220227 17.366255
122 [6, 3, 5, -9, -9, 3] 9771.701969 9.773087 11.787090
123 [15, -2, -5, -38, -50, -6] 9772.798330 .479706 34.589494
124 [2, -6, -6, -14, -15, 3] 9810.819078 6.548265 13.618691
125 [1, 9, 3, 12, 2, -18] 9825.667878 9.244393 12.047225
126 [1, -13, -2, -23, -6, 32] 9884.172505 2.432212 19.449425
Message: 9836 - Contents - Hide Contents
Date: Thu, 05 Feb 2004 12:32:54
Subject: Re: Acceptance regions
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
wrote:
> No; the idea was to do a complete search within an extra-large
region
> and then look for the widest moats. Dave and I have done this for
> equal temperaments, 5-limit linear temperaments, 7-limit planar
> temperaments. Now we're asking for your help.
And the reason why we care about moats is?
________________________________________________________________________
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Message: 9837 - Contents - Hide Contents
Date: Fri, 06 Feb 2004 12:37:36
Subject: [tuning] Re: question about 24-tET
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
I used the 45 (counting multiplicities) 10-limit intervals to define a
norm, and the result clearly did not make sense as a way of ranking
musical intervals. I could add weighting, but there already is heavy
weighting for the lower primes automatically.
I think Paul's theory about this is wrong, and mine was right--we are
better off starting from a norm we know works reasonably well, like the
sqrt(sum log(p)log(q)x_p x_q) norm.
Message: 9838 - Contents - Hide Contents
Date: Fri, 06 Feb 2004 16:44:34
Subject: Re: 126 7-limit linears
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
> wrote:
> After all the complaints, no response. :(
Some of us have to sleep sometimes . . . patience . . .
Message: 9839 - Contents - Hide Contents
Date: Fri, 06 Feb 2004 16:47:25
Subject: [tuning] Re: question about 24-tET
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>
> I used the 45 (counting multiplicities) 10-limit intervals to
define a
> norm, and the result clearly did not make sense as a way of ranking
> musical intervals. I could add weighting, but there already is heavy
> weighting for the lower primes automatically.
>
> I think Paul's theory about this is wrong, and mine was right--we
are
> better off starting from a norm we know works reasonably well, like
the
> sqrt(sum log(p)log(q)x_p x_q) norm.
I wish I knew what you were talking about.
Message: 9841 - Contents - Hide Contents
Date: Fri, 06 Feb 2004 18:47:29
Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears)
From: Paul Erlich
Since there's a huge empty gap between complexity ~25+ and ~31, I was
forced to look for a lower-complexity moat (probably a good thing
anyway). I'll upload a graph showing the temperaments indicated by
their ranking according to error/8.125 + complexity/25, since I saw a
reasonable linear moat where this measure equals 1. Twenty
temperaments make it in:
1. Huygens meantone
2. Semisixths
3. Magic
4. Pajara
5. Tripletone
6. Superpythagorean
7. Negri
8. Kleismic
9. Hemifourths
10. Dominant Seventh
11. [598.4467109, 162.3159606],[[2, 4, 6, 7], [0, -3, -5, -5]]
12. Orwell
13. Injera
14. Miracle
15. Schismic
16. Flattone
17. Supermajor seconds
18. 1/12 oct. period, 25 cent generator (we discussed this years ago)
19. Nonkleismic
20. Porcupine
If we allow the moat to be slightly concave, we would include:
26. Diminished
29. Augmented
A bit more concavity still and we include
45. Blackwood
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> I first made a candidate list by the kitchen sink method:
>
> (1) All pairs n,m<=200 of standard vals
>
> (2) All pairs n,m<=200 of TOP vals
>
> (3) All pairs 100<=n,m<400 of standard vals
>
> (4) All pairs 100<=n,m<=400 of TOP vals
>
> (5) Generators of standard vals up to 100
>
> (6) Generators of certain nonstandard vals up to 100
>
> (7) Pairs of commas from Paul's list of relative error < 0.06,
> epimericity < 0.5
>
> (8) Pairs of vals with consistent badness figure < 1.5 up to 5000
>
> This lead to a list of 32201 candidate wedgies, most of which of
> course were incredible garbage. I then accepted everything with a
2.8
> exponent badness less than 10000, where error is TOP error and
> complexity is our mysterious L1 TOP complexity. I did not do any
> cutting off for either error or complexity, figuring people could
> decide how to do that for themselves. The first six systems are
> macrotemperaments of dubious utility, number 7 is the {15/14, 25/24}
> temperament, and 8 and 9 are the beep-ennealimmal pair, and number
13
> is father. After ennealimmal, we don't get back into the micros
until
> number 46; if we wanted to avoid going there we can cutoff at 4000.
> Number 46, incidentally, has TM basis {2401/2400, 65625/65536} and
is
> covered by 140, 171, 202 and 311; the last is interesting because of
> the peculiar talents of 311.
>
>
>
> 1 [0, 0, 2, 0, 3, 5] 662.236987 77.285947 2.153690
> 2 [1, 1, 0, -1, -3, -3] 806.955502 64.326132 2.467788
> 3 [0, 0, 3, 0, 5, 7] 829.171704 30.152577 3.266201
> 4 [0, 2, 2, 3, 3, -1] 870.690617 33.049025 3.216583
> 5 [1, 2, 1, 1, -1, -3] 888.831828 49.490949 2.805189
> 6 [1, 2, 3, 1, 2, 1] 1058.235145 33.404241 3.435525
> 7 [2, 1, 3, -3, -1, 4] 1076.506437 16.837898 4.414720
> 8 [2, 3, 1, 0, -4, -6] 1099.121425 14.176105 4.729524
> 9 [18, 27, 18, 1, -22, -34] 1099.959348 .036377 39.828719
> 10 [1, -1, 0, -4, -3, 3] 1110.471803 39.807123 3.282968
> 11 [0, 5, 0, 8, 0, -14] 1352.620311 7.239629 6.474937
> 12 [1, -1, -2, -4, -6, -2] 1414.400610 20.759083 4.516198
> 13 [1, -1, 3, -4, 2, 10] 1429.376082 14.130876 5.200719
> 14 [1, 4, -2, 4, -6, -16] 1586.917865 4.771049 7.955969
> 15 [1, 4, 10, 4, 13, 12] 1689.455290 1.698521 11.765178
> 16 [2, 1, -1, -3, -7, -5] 1710.030839 16.874108 5.204166
> 17 [1, 4, 3, 4, 2, -4] 1749.120722 14.253642 5.572288
> 18 [0, 0, 4, 0, 6, 9] 1781.787825 33.049025 4.153970
> 19 [1, -1, 1, -4, -1, 5] 1827.319456 54.908088 3.496512
> 20 [4, 4, 4, -3, -5, -2] 1926.265442 5.871540 7.916963
> 21 [2, -4, -4, -11, -12, 2] 2188.881053 3.106578 10.402108
> 22 [3, 0, 6, -7, 1, 14] 2201.891023 5.870879 8.304602
> 23 [0, 0, 5, 0, 8, 12] 2252.838883 19.840685 5.419891
> 24 [4, 2, 2, -6, -8, -1] 2306.678659 7.657798 7.679190
> 25 [2, 1, 6, -3, 4, 11] 2392.139586 9.396316 7.231437
> 26 [2, -1, 1, -6, -4, 5] 2452.275337 22.453717 5.345120
> 27 [0, 0, 7, 0, 11, 16] 2580.688285 9.431411 7.420171
> 28 [1, -3, -4, -7, -9, -1] 2669.323351 9.734056 7.425960
> 29 [5, 1, 12, -10, 5, 25] 2766.028555 1.276744 15.536039
> 30 [7, 9, 13, -2, 1, 5] 2852.991531 1.610469 14.458536
> 31 [2, -2, 1, -8, -4, 8] 3002.749158 14.130876 6.779481
> 32 [3, 0, -6, -7, -18, -14] 3181.791246 2.939961 12.125211
> 33 [2, 8, 1, 8, -4, -20] 3182.905310 3.668842 11.204461
> 34 [6, -7, -2, -25, -20, 15] 3222.094343 .631014 21.101881
> 35 [4, -3, 2, -14, -8, 13] 3448.998676 3.187309 12.124601
> 36 [1, -3, -2, -7, -6, 4] 3518.666155 18.633939 6.499551
> 37 [1, 4, 5, 4, 5, 0] 3526.975600 19.977396 6.345287
> 38 [2, 6, 6, 5, 4, -3] 3589.967809 8.400361 8.700992
> 39 [2, 1, -4, -3, -12, -12] 3625.480387 9.146173 8.470366
> 40 [2, -2, -2, -8, -9, 1] 3634.089963 14.531543 7.185526
> 41 [3, 2, 4, -4, -2, 4] 3638.704033 20.759083 6.329002
> 42 [6, 5, 3, -6, -12, -7] 3680.095702 3.187309 12.408714
> 43 [2, 8, 8, 8, 7, -4] 3694.344150 3.582707 11.917575
> 44 [2, 3, 6, 0, 4, 6] 3938.578264 20.759083 6.510560
> 45 [0, 0, 5, 0, 8, 11] 3983.263457 38.017335 5.266481
> 46 [22, -5, 3, -59, -57, 21] 4009.709706 .073527 49.166221
> 47 [3, 5, 9, 1, 6, 7] 4092.014696 6.584324 9.946084
> 48 [7, -3, 8, -21, -7, 27] 4145.427852 .946061 19.979719
> 49 [1, -8, -14, -15, -25, -10] 4177.550548 .912904 20.291786
> 50 [3, 5, 1, 1, -7, -12] 4203.022260 12.066285 8.088219
> 51 [1, 9, -2, 12, -6, -30] 4235.792998 2.403879 14.430906
> 52 [6, 10, 10, 2, -1, -5] 4255.362112 3.106578 13.189661
> 53 [2, 5, 3, 3, -1, -7] 4264.417050 21.655518 6.597656
> 54 [6, 5, 22, -6, 18, 37] 4465.462582 .536356 25.127403
> 55 [0, 0, 12, 0, 19, 28] 4519.315488 3.557008 12.840061
> 56 [1, -3, 3, -7, 2, 15] 4555.017089 15.315953 7.644302
> 57 [1, -1, -5, -4, -11, -9] 4624.441621 14.789095 7.782398
> 58 [16, 2, 5, -34, -37, 6] 4705.894319 .307997 31.211875
> 59 [4, -32, -15, -60, -35, 55] 4750.916876 .066120 54.255591
> 60 [1, -8, 39, -15, 59, 113] 4919.628715 .074518 52.639423
> 61 [3, 0, -3, -7, -13, -7] 4967.108742 11.051598 8.859010
> 62 [6, 0, 0, -14, -17, 0] 5045.450988 5.526647 11.410361
> 63 [37, 46, 75, -13, 15, 45] 5230.896745 .021640 83.678088
> 64 [1, 6, 5, 7, 5, -5] 5261.484667 11.970043 8.788871
> 65 [3, 2, -1, -4, -10, -8] 5276.949135 17.564918 7.671954
> 66 [1, 4, -9, 4, -17, -32] 5338.184867 2.536420 15.376139
> 67 [1, -3, 5, -7, 5, 20] 5338.971970 8.959294 9.797992
> 68 [10, 9, 7, -9, -17, -9] 5386.217633 1.171542 20.325677
> 69 [19, 19, 57, -14, 37, 79] 5420.385757 .046052 64.713343
> 70 [5, 3, 7, -7, -3, 8] 5753.932407 7.459874 10.743721
> 71 [3, 5, -6, 1, -18, -28] 5846.930660 3.094040 14.795975
> 72 [3, 12, -1, 12, -10, -36] 5952.918469 1.698521 18.448015
> 73 [6, 0, 3, -14, -12, 7] 6137.760804 5.291448 12.429144
> 74 [4, 4, 0, -3, -11, -11] 6227.282004 12.384652 9.221275
> 75 [3, 0, 9, -7, 6, 21] 6250.704457 6.584324 11.570803
> 76 [9, 5, -3, -13, -30, -21] 6333.111158 1.049791 22.396682
> 77 [0, 0, 8, 0, 13, 19] 6365.852053 14.967465 8.686091
> 78 [4, 2, 5, -6, -3, 6] 6370.380556 16.499269 8.391154
> 79 [1, -8, -2, -15, -6, 18] 6507.074340 4.974313 12.974488
> 80 [2, -6, 1, -14, -4, 19] 6598.741284 6.548265 11.820058
> 81 [2, 25, 13, 35, 15, -40] 6657.512727 .299647 35.677429
> 82 [6, -2, -2, -17, -20, 1] 6845.573750 3.740932 14.626943
> 83 [1, 7, 3, 9, 2, -13] 6852.061008 12.161876 9.603642
> 84 [0, 5, 5, 8, 8, -2] 7042.202107 19.368923 8.212986
> 85 [4, 2, 9, -6, 3, 15] 7074.478038 8.170435 11.196673
> 86 [8, 6, 6, -9, -13, -3] 7157.960980 3.268439 15.596153
> 87 [5, 8, 2, 1, -11, -18] 7162.155511 5.664628 12.817743
> 88 [3, 17, -1, 20, -10, -50] 7280.048554 .894655 24.922952
> 89 [4, 2, -1, -6, -13, -8] 7307.246603 13.289190 9.520562
> 90 [5, 13, -17, 9, -41, -76] 7388.593186 .276106 38.128083
> 91 [8, 18, 11, 10, -5, -25] 7423.457669 .968741 24.394122
> 92 [3, -2, 1, -10, -7, 8] 7553.291925 18.095699 8.628089
> 93 [3, 7, -1, 4, -10, -22] 7604.170165 7.279064 11.973078
> 94 [6, 10, 3, 2, -12, -21] 7658.950254 3.480440 15.622931
> 95 [14, 59, 33, 61, 13, -89] 7727.766150 .037361 79.148236
> 96 [3, -5, -6, -15, -18, 0] 7760.555544 4.513934 14.304666
> 97 [13, 14, 35, -8, 19, 42] 7785.862490 .261934 39.585940
> 98 [11, 13, 17, -5, -4, 3] 7797.739891 1.485250 21.312375
> 99 [2, -4, -16, -11, -31, -26] 7870.803242 1.267597 22.628529
> 100 [2, -9, -4, -19, -12, 16] 7910.552221 2.895855 16.877046
> 101 [0, 0, 9, 0, 14, 21] 7917.731843 14.176105 9.573860
> 102 [3, 12, 11, 12, 9, -8] 7922.981072 2.624742 17.489863
> 103 [1, -6, 3, -12, 2, 24] 8250.683192 8.474270 11.675656
> 104 [55, 73, 93, -12, -7, 11] 8282.844862 .017772 105.789216
> 105 [4, 7, 2, 2, -8, -15] 8338.658153 10.400103 10.893408
> 106 [0, 5, -5, 8, -8, -26] 8426.314560 8.215515 11.894828
> 107 [5, 8, 14, 1, 8, 10] 8428.707855 4.143252 15.190723
> 108 [6, 7, 5, -3, -9, -8] 8506.845926 6.986391 12.646486
> 109 [8, 13, 23, 2, 14, 17] 8538.660000 1.024522 25.136807
> 110 [0, 0, 10, 0, 16, 23] 8630.819015 11.358665 10.686371
> 111 [3, -7, -8, -18, -21, 1] 8799.551719 2.900537 17.521249
> 112 [0, 5, 10, 8, 16, 9] 8869.402675 6.941749 12.865826
> 113 [4, 16, 9, 16, 3, -24] 8931.184092 1.698521 21.324102
> 114 [6, 5, 7, -6, -6, 2] 8948.277847 9.097987 11.718042
> 115 [3, -3, 1, -12, -7, 11] 9072.759561 14.130876 10.062449
> 116 [0, 12, 24, 19, 38, 22] 9079.668325 .617051 30.795105
> 117 [33, 78, 90, 47, 50, -10] 9153.275887 .016734 112.014440
> 118 [5, 1, -7, -10, -25, -19] 9260.372155 3.148011 17.329377
> 119 [1, -6, -2, -12, -6, 12] 9290.939644 13.273963 10.377495
> 120 [2, -2, 4, -8, 1, 15] 9367.180611 25.460673 8.247748
> 121 [3, 5, 16, 1, 17, 23] 9529.360455 3.220227 17.366255
> 122 [6, 3, 5, -9, -9, 3] 9771.701969 9.773087 11.787090
> 123 [15, -2, -5, -38, -50, -6] 9772.798330 .479706 34.589494
> 124 [2, -6, -6, -14, -15, 3] 9810.819078 6.548265 13.618691
> 125 [1, 9, 3, 12, 2, -18] 9825.667878 9.244393 12.047225
> 126 [1, -13, -2, -23, -6, 32] 9884.172505 2.432212 19.449425
Message: 9842 - Contents - Hide Contents
Date: Fri, 06 Feb 2004 18:57:59
Subject: Re: Comma reduction?
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
> wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
>>
>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith"
>> >> wrote:
>>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
>>>>
>>>
>>>>> Thanks. Are they called 2-val and 2-monzo because they
>>> are "linear"
>>>>> or is there some other reason?
>>>>
>>>> 2-vals are two vals wedged, 2-monzos are two monzos wedged.
The
>>> former
>>>> is linear unless it reduces to the zero wedgie, the latter is
>> linear
>>>> only in the 7-limit.
>>>
>>> Thanks! So the latter is linear in the 7-limit because the 7-
> limit
>> is
>>> formed from two commas...I see.
>>
>> The 7-limit is 4-dimensional, so if you temper out 2 commas
you're
>> left with a 2-dimensional system, which is what we usually refer
to
>> as "linear". Is that what you meant?
>
> Yes, I guess so. Why does tempering out two commas in a 4-
dimensional
> system leave a 2-dimensional system?
Roughly: the two commas in addition to two other basis vectors will
span the 4-dimensional system (only if the four vectors are linearly
independent). If you temper out the two commas, the remaining two
basis vectors will form a basis for the entire resulting system of
pitches, which we therefore regard as two-dimensional.
Message: 9843 - Contents - Hide Contents
Date: Fri, 06 Feb 2004 01:28:29
Subject: Re: A 41-limit temperament
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> The 140&171 7-limit temperament has a 311-et generator of 20/311;
> extending that to the 41-limit gives a mapping of
>
>
> [<1, 3, 2, 3, -4, 1, 1, 11, -3, 1, 11, 0, 6|,
> <0, -22, 5, -3, 116, 42, 48, -105, 117, 60, -94, 81, -10|]
>
> An LLL-reduced basis is
>
> {703/702, 784/783, 875/874, 1000/999, 1625/1624, 1729/1728, 8092/8073,
> 10557/10556, 68921/68894, 177023/177008, 11670750/11648281}
It seems this isn't a basis for the whole temperament; it has
3-torsion. I need to find non-cube products which are cubes, and take
the cube root
Message: 9845 - Contents - Hide Contents
Date: Fri, 06 Feb 2004 05:07:48
Subject: Re: A 41-limit temperament
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> It seems this isn't a basis for the whole temperament; it has
> 3-torsion. I need to find non-cube products which are cubes, and take
> the cube root
A correct LLL reduced basis for the temperament is
{6290/6279, 714/713, 820/819, 1015/1014, 1105/1104, 1365/1364,
2002/2001, 2146/2145, 2185/2184, 16530/16523, 4060/4059}
4000/3993 is zero steps of 311, one step of 140, and 385/384 is one
step of 311, zero steps of 140; adding the correct one of these gives
us a basis for 311 or 140, and adding both gives us a "notation". This
one has three different versions of 87-et, which seems excessive, but
it does easily allow us to compute a 311 Fokker block (I just did it,
so I know its easy) in case anyone is tired of piddling around with
tiny 11-limit scales of size 43 or so. That 6290/6279 is not very
nice, so maybe that could be reduced. An ambitious person might want
to TM reduce the whole thing. Booting it would give us a planar
temperament, in case anyone knows what to do with a 41-limit planar.
Message: 9846 - Contents - Hide Contents
Date: Fri, 06 Feb 2004 20:52:30
Subject: Ennealimmal[45] as a chord block
From: Gene Ward Smith
The major chord with root 21/20 is [0,2,0] in the 7-limit chord
lattice, that with root 2401/2400 is [2,-3,-3], and that with
4375/4374 is [6,-6,-3]. If I form the block in the lattice centered at
[0,0,0] and with the inverse matrix coordinates running -1 < coordinat
<= 1, I get a block with 127 chords, consisting of 191 notes. Reducing
this by ennealimmal gives Ennealimmal[45]. Below is a TM reduced JI
scale corresponding to Ennealimmal[45] which people with an aversion
to tempering could use instead, not to mention people who just plain
like the idea. Sticking it in Scala shows 7 of the 18 major tetrads
are very slightly tempered, and the other 11 are pure JI. In the case
of minor tetrads, we have eight tempered ones, and ten untempered. The
maximum error is 2401/2400, or 0.721 cents, in both cases.
We also have 27 supermajor and 18 subminor tetrads, defined as
1--9/7--3/2--9/5 and 1--7/6--3/2--5/3 (or 6:7:9:10 for those who
prefer.) Twelve supermajor and eight subminor tetrads are tempered.
Finding ways of harmonizing things is apparently not a problem.
! enn45.scl
Detempered Ennealimmal[45], TM reduced
45
!
49/48
25/24
21/20
15/14
27/25
54/49
9/8
245/216
81/70
7/6
25/21
175/144
49/40
5/4
63/50
9/7
21/16
250/189
27/20
49/36
25/18
486/343
10/7
35/24
72/49
3/2
49/32
54/35
63/40
100/63
81/50
81/49
5/3
245/144
12/7
7/4
25/14
9/5
90/49
50/27
189/100
27/14
35/18
125/63
2
Message: 9847 - Contents - Hide Contents
Date: Fri, 06 Feb 2004 05:55:46
Subject: Re: Acceptance regions
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
wrote:
>
>> Without a moat, there would be questionable cases, of "if those
are
>> in, why isn't this in" and "if those are out, why isn't this out".
>
> With a moat, there might be a question of why you are using a
> seemingly unmotivated, ad hoc criterion. Maybe we could formalize it
> to a similarity circle or something that could be justified?
If the two agree, all the better.
Message: 9848 - Contents - Hide Contents
Date: Fri, 06 Feb 2004 20:56:41
Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears)
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> Since there's a huge empty gap between complexity ~25+ and ~31, I was
> forced to look for a lower-complexity moat (probably a good thing
> anyway). I'll upload a graph showing the temperaments indicated by
> their ranking according to error/8.125 + complexity/25, since I saw a
> reasonable linear moat where this measure equals 1. Twenty
> temperaments make it in:
Given that we normally relate error and complexity multiplicitively, I
think using log(err) and log(complexity) makes far more sense. Can you
justify using them additively?
I might be more willing to believe in this stuff if it made some
logical sense to me, but maybe I am just strange.
Message: 9849 - Contents - Hide Contents
Date: Fri, 06 Feb 2004 21:29:38
Subject: Detempered Ennealimmal[36]
From: Gene Ward Smith
This one has nine major tetrads, three tempered, nine minor tetrads,
four tempered, eighteen supermajor tetrads, seven tempered, and
eighteen subminor tetrads, eight tempered.
! enn36.scl
TM reduced detempering of Ennealimmal[36]
36
!
49/48
21/20
15/14
27/25
54/49
245/216
81/70
7/6
25/21
49/40
5/4
63/50
9/7
250/189
27/20
49/36
25/18
10/7
35/24
72/49
3/2
54/35
63/40
100/63
81/50
5/3
245/144
12/7
7/4
9/5
90/49
50/27
189/100
35/18
125/63
2
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