Tuning-Math Digests messages 9826 - 9850

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Message: 9826

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

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.


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Message: 9828

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.


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Message: 9830

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".


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Message: 9833

Date: Thu, 05 Feb 2004 23:38:45

Subject: Re: [tuning] Re: question about 24-tET

From: Carl Lumma

>> Can 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


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Message: 9834

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 *


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Message: 9838

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 . . .


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Message: 9839

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.


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Message: 9841

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


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Message: 9842

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" 
> > <paul.hjelmstad@u...> wrote:
> > > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
> > <gwsmith@s...> 
> > > wrote:
> > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
> > > > <paul.hjelmstad@u...> wrote:
> > > > 
> > > > > 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.


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Message: 9847

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.


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