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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. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links To visit your group on the web, go to: Yahoo groups: /tuning-math/ * To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service *
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.
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.
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".
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
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 *
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 . . .
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.
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
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.
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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