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Message: 9375 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 13:02:37

Subject: Re: Maple code for ?(x)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:


> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 


>>> quest := proc(x)
>>> # Minkowsky question mark function
>>> local i, j, d, l, s, t;
>>> l := convert(x, confrac);
>>> d := nops(l);
>>> s := l[1];
>>> for i from 2 to d do
>>> t := 1;
>>> for j from 2 to i do
>>> t := t - l[j] od;
>>> s := s + (-1)^i * 2^t od;
>>> s end:
>> 


>> what is nops?
> 


> Nops gives the number of operands for a variety of expressions.
> Convert(x, confrac) returns a list of integers, and for a list, nops
> gives the number of elements of the list.



Aren't 'most' continued fractions infinite?
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Message: 9376 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 13:54:40

Subject: *Correction01* Re: 114 7-limit temperaments

From: Carl Lumma


>>> >umber 8 Schismic
>>> 
>>> [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
>>> TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
>>> TOP generators [1200.760624, 498.1193303]
>>> bad: 28.818558 comp: 5.618543 err: .912904
>>> 
>>> 
>>> Number 9 Miracle
>>> 
>>> [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
>>> TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
>>> TOP generators [1200.631014, 116.7206423]
>>> bad: 29.119472 comp: 6.793166 err: .631014
//


>> And I don't see how you figure schismic is less complex than
>> miracle in light of the maps given.
>


>Probably the shortness of the fifths in the lattice wins it for 
>schismic . . .



After I wrote that I reflected a bit on comma complexity vs. map
complexity.  Comma complexity gives you the number of notes you'd
have to search to find the comma, on average (Kees points out that
the symmetry of the lattice allows you to search 1/4 this numeber
in the 5-limit, or something, but anyway...).  Map complexity is
the number of notes you need to complete the map *with contiguous
chains of generators*.  It's this contiguous-chain restriction
that makes me wonder -- what good is it?  I suppose it helps keep
the number of step sizes (mean variety) low in the resulting scales.
But it implies a generator-stacking process that produces linear
temperaments in some (DE/MOS) cases and I'm guessing planar
temperaments otherwise (when there are 3 step sizes)... if so what
relation do these planar temperaments bear to the linear temperaments
arrived at with the same-sized generators?... in the case of Marvel,
Gene says 384:whatever always goes with 225:224, and this notion of
natural planar extensions seems highly interesting...  Aside from
the Hypothesis, the link between these two ways of approaching
temperament (chains vs. commas) seems little-explored.

-Carl
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Message: 9377 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 13:08:02

Subject: Re: TOP take on 7-limit temperaments

From: Paul Erlich

I appreciate this work, Gene.

How about a worked-out, hand-holding example for one of these error 
and complexity calculations?

P.S. Instead of using log-flat badness, why don't we use the same 
function of error and complexity that yielded epimericity in the 
codimension-1 case?


--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:


> Here is my old list of 45 7-limit linear temperaments, this time 
using
> top to sort it all out. The goodness winner is ennealimmal by a huge
> margin, with meantone in second and magic coming up third. Ones with
> badness under 30 are dominant seventh, augmented, pajara, meantone,
> magic, miracle, schismic, and ennealimmal.
> 
> Decimal <4 2 2 -6 -8 -1]
> [1207.657798 1914.092323 2768.532858 3372.361757]
> [[2 4 5 6] [0 -2 -1 -1]] [600.0000000 249.0224992]
> err: 7.657798 comp: 2.523719 bad: 48.773723
> 
> 
> Dominant seventh <1 4 -2 4 -6 -16]
> [1195.228951 1894.576888 2797.391744 3382.219933]
> [[1 2 4 2] [0 -1 -4 2]] [1200. 497.7740225]
> err: 4.771049 comp: 2.454561 bad: 28.744957
> 
> 
> Diminished <4 4 4 -3 -5 -2]
> [1194.128460 1892.648830 2788.245174 3385.309404]
> [[4 6 9 11] [0 1 1 1]] [300.0000000 85.69820677]
> err: 5.871540 comp: 2.523719 bad: 37.396767
> 
> 
> Blackwood <0 5 0 8 0 -14]
> [1195.893464 1913.429542 2786.313713 3348.501698]
> [[5 8 12 14] [0 0 -1 0]] [240.0000000 90.61325640]
> err: 7.239629 comp: 2.173813 bad: 34.210608
> 
> 
> Augmented <3 0 6 -7 1 14]
> [1199.976630 1892.649878 2799.945472 3385.307546]
> [[3 5 7 9] [0 -1 0 -2]] [400.0000000 110.2596913]
> err: 5.870879 comp: 2.147741 bad: 27.081145
> 
> 
> Pajara <2 -4 -4 -11 -12 2]
> [1196.893422 1901.906680 2779.100462 3377.547174]
> [[2 3 5 6] [0 1 -2 -2]] [600.0000000 108.8143299]
> err: 3.106578 comp: 2.988993 bad: 27.754421
> 
> 
> Hexadecimal <1 -3 5 -7 5 20]
> [1208.959294 1887.754858 2799.450479 3393.977822]
> [[1 2 1 5] [0 -1 3 -5]] [1200. 526.8909182]
> err: 8.959294 comp: 3.068202 bad: 84.341555
> 
> 
> Negri <4 -3 2 -14 -8 13]
> [1203.187308 1907.006766 2780.900506 3359.878000]
> [[1 2 2 3] [0 -4 3 -2]] [1200. 125.4687958]
> err: 3.187309 comp: 3.804173 bad: 46.125884
> 
> 
> Kleismic <6 5 3 -6 -12 -7]
> [1203.187308 1907.006766 2792.359613 3359.878000]
> [[1 0 1 2] [0 6 5 3]] [1200. 316.6640534]
> err: 3.187309 comp: 3.785579 bad: 45.676063
> 
> 
> Tripletone <3 0 -6 -7 -18 -14]
> [1197.060039 1902.640406 2793.140092 3377.079420]
> [[3 5 7 8] [0 -1 0 2]] [400.0000000 88.72066409]
> err: 2.939961 comp: 4.045351 bad: 48.112067
> 
> 
> Hemifourth <2 8 1 8 -4 -20]
> [1203.668842 1902.376967 2794.832500 3358.526166]
> [[1 2 4 3] [0 -2 -8 -1]] [1200. 252.7423121]
> err: 3.668842 comp: 3.445412 bad: 43.552336
> 
> 
> Meantone <1 4 10 4 13 12]
> [1201.698521 1899.262909 2790.257556 3370.548328]
> [[1 2 4 7] [0 -1 -4 -10]] [1200. 503.3520320]
> err: 1.698521 comp: 3.562072 bad: 21.551439
> 
> 
> Injera <2 8 8 8 7 -4]
> [1201.777814 1896.276546 2777.994928 3378.883835]
> [[2 3 4 5] [0 1 4 4]] [600.0000000 93.65102578]
> err: 3.582707 comp: 3.445412 bad: 42.529834
> 
> 
> Double wide <8 6 6 -9 -13 -3]
> [1198.553882 1907.135354 2778.724633 3378.001574]
> [[2 5 6 7] [0 -4 -3 -3]] [600.0000000 274.3886321]
> err: 3.268439 comp: 5.047438 bad: 83.268810
> 
> 
> Porcupine <3 5 -6 1 -18 -28]
> [1196.905961 1906.858938 2779.129576 3367.717888]
> [[1 2 3 2] [0 -3 -5 6]] [1200. 162.3778142]
> err: 3.094040 comp: 4.295482 bad: 57.088650
> 
> 
> Superpythagorean <1 9 -2 12 -6 -30]
> [1197.596121 1905.765059 2780.732078 3374.046608]
> [[1 2 6 2] [0 -1 -9 2]] [1200. 489.6151808]
> err: 2.403879 comp: 4.602303 bad: 50.917015
> 
> 
> Muggles <5 1 -7 -10 -25 -19]
> [1203.148010 1896.965522 2785.689126 3359.988323]
> [[1 0 2 5] [0 5 1 -7]] [1200. 377.6398800]
> err: 3.148011 comp: 5.618543 bad: 99.376477
> 
> 
> Beatles <2 -9 -4 -19 -12 16]
> [1197.104145 1906.544822 2793.037680 3369.535226]
> [[1 1 5 4] [0 2 -9 -4]] [1200. 356.3080304]
> err: 2.895855 comp: 5.162806 bad: 77.187771
> 
> 
> Flattone <1 4 -9 4 -17 -32]
> [1202.536420 1897.934872 2781.593812 3361.705278]
> [[1 2 4 -1] [0 -1 -4 9]] [1200. 506.5439220]
> err: 2.536420 comp: 4.909123 bad: 61.126418
> 
> 
> Magic <5 1 12 -10 5 25]
> [1201.276744 1903.978592 2783.349206 3368.271877]
> [[1 0 2 -1] [0 5 1 12]] [1200. 380.5064473]
> err: 1.276744 comp: 4.274486 bad: 23.327687
> 
> 
> Nonkleismic <10 9 7 -9 -17 -9]
> [1198.828458 1900.098151 2789.033948 3368.077085]
> [[1 -1 0 1] [0 10 9 7]] [1200. 309.9514712]
> err: 1.171542 comp: 6.309298 bad: 46.635848
> 
> 
> Semisixths <7 9 13 -2 1 5]
> [1198.389531 1903.732520 2790.053106 3364.304748]
> [[1 -1 -1 -2] [0 7 9 13]] [1200. 443.6203855]
> err: 1.610469 comp: 4.630693 bad: 34.533812
> 
> 
> Orwell <7 -3 8 -21 -7 27]
> [1199.532657 1900.455530 2784.117029 3371.481834]
> [[1 0 3 1] [0 7 -3 8]] [1200. 271.3263635]
> err: .946061 comp: 5.706260 bad: 30.805067
> 
> 
> Miracle <6 -7 -2 -25 -20 15]
> [1200.631014 1900.954868 2784.848544 3368.451756]
> [[1 1 3 3] [0 6 -7 -2]] [1200. 116.5729472]
> err: .631014 comp: 6.793166 bad: 29.119472
> 
> 
> Quartaminorthirds <9 5 -3 -13 -30 -21]
> [1199.792743 1900.291122 2788.751252 3365.878770]
> [[1 1 2 3] [0 9 5 -3]] [1200. 77.70708732]
> err: 1.049791 comp: 6.742251 bad: 47.721346
> 
> 
> Supermajor seconds <3 12 -1 12 -10 -36]
> [1201.698521 1899.262909 2790.257556 3372.574099]
> [[1 1 0 3] [0 3 12 -1]] [1200. 232.1235474]
> err: 1.698521 comp: 5.522763 bad: 51.806440
> 
> 
> Schismic <1 -8 -14 -15 -25 -10]
> [1200.760625 1903.401919 2784.194017 3371.388750]
> [[1 2 -1 -3] [0 -1 8 14]] [1200. 497.8598384]
> err: .912904 comp: 5.618543 bad: 28.818563
> 
> 
> Superkleismic <9 10 -3 -5 -30 -35]
> [1201.371918 1904.129438 2783.128219 3369.863245]
> [[1 4 5 2] [0 -9 -10 3]] [1200. 321.8581276]
> err: 1.371918 comp: 6.742251 bad: 62.364566
> 
> 
> Squares <4 16 9 16 3 -24]
> [1201.698521 1899.262909 2790.257556 3372.067656]
> [[1 3 8 6] [0 -4 -16 -9]] [1200. 425.9591136]
> err: 1.698521 comp: 6.890825 bad: 80.651668
> 
> 
> Semififth <2 8 -11 8 -23 -48]
> [1201.698521 1899.262909 2790.257556 3373.586984]
> [[1 1 0 6] [0 2 8 -11]] [1200. 348.3528922]
> err: 1.698521 comp: 7.363684 bad: 92.100337
> 
> 
> Diaschismic <2 -4 -16 -11 -31 -26]
> [1198.732403 1901.885616 2789.256983 3365.267311]
> [[2 3 5 7] [0 1 -2 -8]] [600.0000000 103.7370914]
> err: 1.267597 comp: 6.966993 bad: 61.527901
> 
> 
> Octacot <8 18 11 10 -5 -25]
> [1199.031259 1903.490418 2784.064367 3366.693863]
> [[1 1 1 2] [0 8 18 11]] [1200. 88.14540671]
> err: .968741 comp: 7.752178 bad: 58.217715
> 
> 
> Tritonic <5 -11 -12 -29 -33 3]
> [1201.023211 1900.333250 2785.201472 3365.953391]
> [[1 4 -3 -3] [0 -5 11 12]] [1200. 580.4242150]
> err: 1.023211 comp: 7.880073 bad: 63.536850
> 
> 
> Supersupermajor <3 17 -1 20 -10 -50]
> [1200.231588 1903.372996 2784.236389 3366.314293]
> [[1 1 -1 3] [0 3 17 -1]] [1200. 234.4104084]
> err: .894655 comp: 7.670504 bad: 52.638504
> 
> 
> Shrutar <4 -8 14 -22 11 55]
> [1198.920873 1903.665377 2786.734051 3365.796415]
> [[2 3 5 5] [0 2 -4 7]] [600.0000000 52.89351739]
> err: 1.079127 comp: 8.437555 bad: 76.825572
> 
> 
> Catakleismic <6 5 22 -6 18 37]
> [1200.536356 1901.438376 2785.068335 3370.331646]
> [[1 0 1 -3] [0 6 5 22]] [1200. 316.7238784]
> err: .536356 comp: 7.836558 bad: 32.938503
> 
> 
> Hemiwuerschmidt <16 2 5 -34 -37 6]
> [1199.692003 1901.466838 2787.028860 3368.496143]
> [[1 -1 2 2] [0 16 2 5]] [1200. 193.9099372]
> err: .307997 comp: 10.094876 bad: 31.386987
> 
> 
> Hemikleismic <12 10 -9 -12 -48 -49]
> [1199.411231 1902.888178 2785.151380 3370.478790]
> [[1 0 1 4] [0 12 10 -9]] [1200. 158.7324720]
> err: .588769 comp: 10.787602 bad: 68.516458
> 
> 
> Hemithird <15 -2 -5 -38 -50 -6]
> [1200.363229 1901.194685 2787.427555 3367.479202]
> [[1 4 2 2] [0 -15 2 5]] [1200. 193.2841225]
> err: .479706 comp: 11.237086 bad: 60.573479
> 
> 
> Wizard <12 -2 20 -31 -2 52]
> [1200.639571 1900.941305 2784.828674 3368.342104]
> [[2 1 5 2] [0 6 -1 10]] [600.0000000 216.7129477]
> err: .639571 comp: 8.423526 bad: 45.381303
> 
> 
> Duodecimal <0 12 24 19 38 22]
> [1200.617051 1900.976998 2785.844725 3370.558188]
> [[12 19 28 34] [0 0 -1 -2]] [100.0000000 15.94743281]
> err: .617051 comp: 8.548972 bad: 45.097159
> 
> 
> Slender <13 -10 6 -46 -27 42]
> [1200.337238 1901.055858 2784.996493 3370.418508]
> [[1 2 2 3] [0 -13 10 -6]] [1200. 38.46612667]
> err: .567296 comp: 12.499426 bad: 88.631905
> 
> 
> Amity <5 13 -17 9 -41 -76]
> [1199.723894 1902.392618 2786.717797 3369.601033]
> [[1 3 6 -2] [0 -5 -13 17]] [1200. 339.4147297]
> err: .276106 comp: 11.659166 bad: 37.532790
> 
> 
> Hemififth <2 25 13 35 15 -40]
> [1199.700353 1902.429930 2785.617954 3368.041901]
> [[1 1 -5 -1] [0 2 25 13]] [1200. 351.4712147]
> err: .299647 comp: 10.766914 bad: 34.737019
> 
> 
> Ennealimmal <18 27 18 1 -22 -34]
> [1200.036377 1902.012658 2786.350298 3368.723784]
> [[9 15 22 26] [0 -2 -3 -2]] [133.3333333 48.99915090]
> err: .036377 comp: 11.628267 bad: 4.918774
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Message: 9378 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 21:59:25

Subject: Re: TOP take on 7-limit temperaments

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:


> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 


>> It was the *badness* function you used that I didn't like, which 
is 
>> why I was suggesting this *other* function of complexity and 
error 
>> which, as well, *you yourself* posted.
> 


> That was in a discussion of commas, not wedgies.



You gave a function of complexity and error, in which you managed to 
eliminate all reference to the original comma, so that I could plot 
the contours over the whole graph regardless of how few commas were 
actually in the vicinity.



> How do you apply it 
> to wedgies?



Compute the complexity and error from the wedgie, and the compute 
that same function of complexity and error.
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Message: 9379 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 13:09:53

Subject: Re: 114 7-limit temperaments

From: Paul Erlich

Gene, you rock!

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:


> This is a list of linear temperaments with top complexity < 15, top
> error < 15, and top badness < 100. I searched extensively without
> adding to the list, which is probably complete. Most of the names 
are
> old ones. In some cases I extended a 5-limit name to what seemed 
like
> the appropriate 7-limit temperament, and in the case of The
> Temperament Formerly Known as Duodecimal, am suggesting Waage or
> Compton if one of these gentlemen invented it. There are a few new
> names being suggested, none of which are yet etched in stone--not 
even
> when the name is Bond, James Bond.
> 
> Number 1 Ennealimmal
> 
> [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]]
> TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784]
> TOP generators [133.3373752, 49.02398564]
> bad: 4.918774 comp: 11.628267 err: .036377
> 
> 
> Number 2 Meantone
> 
> [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
> TOP generators [1201.698520, 504.1341314]
> bad: 21.551439 comp: 3.562072 err: 1.698521
> 
> 
> Number 3 Magic
> 
> [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
> TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
> TOP generators [1201.276744, 380.7957184]
> bad: 23.327687 comp: 4.274486 err: 1.276744
> 
> 
> Number 4 Beep
> 
> [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
> TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]
> TOP generators [1194.642673, 254.8994697]
> bad: 23.664749 comp: 1.292030 err: 14.176105
> 
> 
> Number 5 Augmented
> 
> [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
> TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]
> TOP generators [399.9922103, 107.3111730]
> bad: 27.081145 comp: 2.147741 err: 5.870879
> 
> 
> Number 6 Pajara
> 
> [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
> TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
> TOP generators [598.4467109, 106.5665459]
> bad: 27.754421 comp: 2.988993 err: 3.106578
> 
> 
> Number 7 Dominant Seventh
> 
> [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
> TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933]
> TOP generators [1195.228951, 495.8810151]
> bad: 28.744957 comp: 2.454561 err: 4.771049
> 
> 
> Number 8 Schismic
> 
> [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
> TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
> TOP generators [1200.760624, 498.1193303]
> bad: 28.818558 comp: 5.618543 err: .912904
> 
> 
> Number 9 Miracle
> 
> [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
> TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
> TOP generators [1200.631014, 116.7206423]
> bad: 29.119472 comp: 6.793166 err: .631014
> 
> 
> Number 10 Orwell
> 
> [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
> TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834]
> TOP generators [1199.532657, 271.4936472]
> bad: 30.805067 comp: 5.706260 err: .946061
> 
> 
> Number 11 Hemiwuerschmidt
> 
> [16, 2, 5, -34, -37, 6] [[1, -1, 2, 2], [0, 16, 2, 5]]
> TOP tuning [1199.692003, 1901.466838, 2787.028860, 3368.496143]
> TOP generators [1199.692003, 193.8224275]
> bad: 31.386908 comp: 10.094876 err: .307997
> 
> 
> Number 12 Catakleismic
> 
> [6, 5, 22, -6, 18, 37] [[1, 0, 1, -3], [0, 6, 5, 22]]
> TOP tuning [1200.536356, 1901.438376, 2785.068335, 3370.331646]
> TOP generators [1200.536355, 316.9063960]
> bad: 32.938503 comp: 7.836558 err: .536356
> 
> 
> Number 13 Father
> 
> [1, -1, 3, -4, 2, 10] [[1, 2, 2, 4], [0, -1, 1, -3]]
> TOP tuning [1185.869125, 1924.351908, 2819.124589, 3401.317477]
> TOP generators [1185.869125, 447.3863410]
> bad: 33.256527 comp: 1.534101 err: 14.130876
> 
> 
> Number 14 Blackwood
> 
> [0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]]
> TOP tuning [1195.893464, 1913.429542, 2786.313713, 3348.501698]
> TOP generators [239.1786927, 83.83059859]
> bad: 34.210608 comp: 2.173813 err: 7.239629
> 
> 
> Number 15 Semisixths
> 
> [7, 9, 13, -2, 1, 5] [[1, -1, -1, -2], [0, 7, 9, 13]]
> TOP tuning [1198.389531, 1903.732520, 2790.053107, 3364.304748]
> TOP generators [1198.389531, 443.1602931]
> bad: 34.533812 comp: 4.630693 err: 1.610469
> 
> 
> Number 16 Hemififths
> 
> [2, 25, 13, 35, 15, -40] [[1, 1, -5, -1], [0, 2, 25, 13]]
> TOP tuning [1199.700353, 1902.429930, 2785.617954, 3368.041901]
> TOP generators [1199.700353, 351.3647888]
> bad: 34.737019 comp: 10.766914 err: .299647
> 
> 
> Number 17 Diminished
> 
> [4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]]
> TOP tuning [1194.128460, 1892.648830, 2788.245174, 3385.309404]
> TOP generators [298.5321149, 101.4561401]
> bad: 37.396767 comp: 2.523719 err: 5.871540
> 
> 
> Number 18 Amity
> 
> [5, 13, -17, 9, -41, -76] [[1, 3, 6, -2], [0, -5, -13, 17]]
> TOP tuning [1199.723894, 1902.392618, 2786.717797, 3369.601033]
> TOP generators [1199.723894, 339.3558130]
> bad: 37.532790 comp: 11.659166 err: .276106
> 
> 
> Number 19 Pelogic
> 
> [1, -3, -4, -7, -9, -1] [[1, 2, 1, 1], [0, -1, 3, 4]]
> TOP tuning [1209.734056, 1886.526887, 2808.557731, 3341.498957]
> TOP generators [1209.734056, 532.9412251]
> bad: 39.824125 comp: 2.022675 err: 9.734056
> 
> 
> Number 20 Parakleismic
> 
> [13, 14, 35, -8, 19, 42] [[1, 5, 6, 12], [0, -13, -14, -35]]
> TOP tuning [1199.738066, 1902.291445, 2786.921905, 3368.090564]
> TOP generators [1199.738066, 315.1076065]
> bad: 40.713036 comp: 12.467252 err: .261934
> 
> 
> Number 21 {21/20, 28/27}
> 
> [1, 4, 3, 4, 2, -4] [[1, 2, 4, 4], [0, -1, -4, -3]]
> TOP tuning [1214.253642, 1919.106053, 2819.409644, 3328.810876]
> TOP generators [1214.253642, 509.4012304]
> bad: 42.300772 comp: 1.722706 err: 14.253642
> 
> 
> Number 22 Injera
> 
> [2, 8, 8, 8, 7, -4] [[2, 3, 4, 5], [0, 1, 4, 4]]
> TOP tuning [1201.777814, 1896.276546, 2777.994928, 3378.883835]
> TOP generators [600.8889070, 93.60982493]
> bad: 42.529834 comp: 3.445412 err: 3.582707
> 
> 
> Number 23 Dicot
> 
> [2, 1, 6, -3, 4, 11] [[1, 1, 2, 1], [0, 2, 1, 6]]
> TOP tuning [1204.048158, 1916.847810, 2764.496143, 3342.447113]
> TOP generators [1204.048159, 356.3998255]
> bad: 42.920570 comp: 2.137243 err: 9.396316
> 
> 
> Number 24 Hemifourths
> 
> [2, 8, 1, 8, -4, -20] [[1, 2, 4, 3], [0, -2, -8, -1]]
> TOP tuning [1203.668842, 1902.376967, 2794.832500, 3358.526166]
> TOP generators [1203.668841, 252.4803582]
> bad: 43.552336 comp: 3.445412 err: 3.668842
> 
> 
> Number 25 Waage? Compton? Duodecimal?
> 
> [0, 12, 24, 19, 38, 22] [[12, 19, 28, 34], [0, 0, -1, -2]]
> TOP tuning [1200.617051, 1900.976998, 2785.844725, 3370.558188]
> TOP generators [100.0514209, 16.55882096]
> bad: 45.097159 comp: 8.548972 err: .617051
> 
> 
> Number 26 Wizard
> 
> [12, -2, 20, -31, -2, 52] [[2, 1, 5, 2], [0, 6, -1, 10]]
> TOP tuning [1200.639571, 1900.941305, 2784.828674, 3368.342104]
> TOP generators [600.3197857, 216.7702531]
> bad: 45.381303 comp: 8.423526 err: .639571
> 
> 
> Number 27 Kleismic
> 
> [6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]]
> TOP tuning [1203.187308, 1907.006766, 2792.359613, 3359.878000]
> TOP generators [1203.187309, 317.8344609]
> bad: 45.676063 comp: 3.785579 err: 3.187309
> 
> 
> Number 28 Negri
> 
> [4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]]
> TOP tuning [1203.187308, 1907.006766, 2780.900506, 3359.878000]
> TOP generators [1203.187309, 124.8419629]
> bad: 46.125886 comp: 3.804173 err: 3.187309
> 
> 
> Number 29 Nonkleismic
> 
> [10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]]
> TOP tuning [1198.828458, 1900.098151, 2789.033948, 3368.077085]
> TOP generators [1198.828458, 309.8926610]
> bad: 46.635848 comp: 6.309298 err: 1.171542
> 
> 
> Number 30 Quartaminorthirds
> 
> [9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]]
> TOP tuning [1199.792743, 1900.291122, 2788.751252, 3365.878770]
> TOP generators [1199.792743, 77.83315314]
> bad: 47.721352 comp: 6.742251 err: 1.049791
> 
> 
> 
> Number 31 Tripletone
> 
> [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]]
> TOP tuning [1197.060039, 1902.640406, 2793.140092, 3377.079420]
> TOP generators [399.0200131, 92.45965769]
> bad: 48.112067 comp: 4.045351 err: 2.939961
> 
> 
> Number 32 Decimal
> 
> [4, 2, 2, -6, -8, -1] [[2, 4, 5, 6], [0, -2, -1, -1]]
> TOP tuning [1207.657798, 1914.092323, 2768.532858, 3372.361757]
> TOP generators [603.8288989, 250.6116362]
> bad: 48.773723 comp: 2.523719 err: 7.657798
> 
> 
> Number 33 {1029/1024, 4375/4374}
> 
> [12, 22, -4, 7, -40, -71] [[2, 5, 8, 5], [0, -6, -11, 2]]
> TOP tuning [1200.421488, 1901.286959, 2785.446889, 3367.642640]
> TOP generators [600.2107440, 183.2944602]
> bad: 50.004574 comp: 10.892116 err: .421488
> 
> 
> Number 34 Superpythagorean
> 
> [1, 9, -2, 12, -6, -30] [[1, 2, 6, 2], [0, -1, -9, 2]]
> TOP tuning [1197.596121, 1905.765059, 2780.732078, 3374.046608]
> TOP generators [1197.596121, 489.4271829]
> bad: 50.917015 comp: 4.602303 err: 2.403879
> 
> 
> Number 35 Supermajor seconds
> 
> [3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]]
> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.574099]
> TOP generators [1201.698520, 232.5214630]
> bad: 51.806440 comp: 5.522763 err: 1.698521
> 
> 
> Number 36 Supersupermajor
> 
> [3, 17, -1, 20, -10, -50] [[1, 1, -1, 3], [0, 3, 17, -1]]
> TOP tuning [1200.231588, 1903.372996, 2784.236389, 3366.314293]
> TOP generators [1200.231587, 234.3804692]
> bad: 52.638504 comp: 7.670504 err: .894655
> 
> 
> Number 37 {6144/6125, 10976/10935} Hendecatonic?
> 
> [11, -11, 22, -43, 4, 82] [[11, 17, 26, 30], [0, 1, -1, 2]]
> TOP tuning [1199.662182, 1902.490429, 2787.098101, 3368.740066]
> TOP generators [109.0601984, 48.46705632]
> bad: 53.458690 comp: 12.579627 err: .337818
> 
> 
> Number 38 {3136/3125, 5120/5103} Misty
> 
> [3, -12, -30, -26, -56, -36] [[3, 5, 6, 6], [0, -1, 4, 10]]
> TOP tuning [1199.661465, 1902.491566, 2787.099767, 3368.765021]
> TOP generators [399.8871550, 96.94420930]
> bad: 53.622498 comp: 12.585536 err: .338535
> 
> 
> Number 39 {1728/1715, 4000/3993}
> 
> [11, 18, 5, 3, -23, -39] [[1, 2, 3, 3], [0, -11, -18, -5]]
> TOP tuning [1199.083445, 1901.293958, 2784.185538, 3371.399002]
> TOP generators [1199.083445, 45.17026643]
> bad: 55.081549 comp: 7.752178 err: .916555
> 
> 
> Number 40 {36/35, 160/147} Hystrix?
> 
> [3, 5, 1, 1, -7, -12] [[1, 2, 3, 3], [0, -3, -5, -1]]
> TOP tuning [1187.933715, 1892.564743, 2758.296667, 3402.700250]
> TOP generators [1187.933715, 161.1008955]
> bad: 55.952057 comp: 2.153383 err: 12.066285
> 
> 
> Number 41 {28/27, 50/49}
> 
> [2, 6, 6, 5, 4, -3] [[2, 3, 4, 5], [0, 1, 3, 3]]
> TOP tuning [1191.599639, 1915.269258, 2766.808679, 3362.608498]
> TOP generators [595.7998193, 127.8698005]
> bad: 56.092257 comp: 2.584059 err: 8.400361
> 
> 
> Number 42 Porcupine
> 
> [3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]]
> TOP tuning [1196.905961, 1906.858938, 2779.129576, 3367.717888]
> TOP generators [1196.905960, 162.3176609]
> bad: 57.088650 comp: 4.295482 err: 3.094040
> 
> 
> Number 43
> 
> [6, 10, 10, 2, -1, -5] [[2, 4, 6, 7], [0, -3, -5, -5]]
> TOP tuning [1196.893422, 1906.838962, 2779.100462, 3377.547174]
> TOP generators [598.4467109, 162.3159606]
> bad: 57.621529 comp: 4.306766 err: 3.106578
> 
> 
> Number 44 Octacot
> 
> [8, 18, 11, 10, -5, -25] [[1, 1, 1, 2], [0, 8, 18, 11]]
> TOP tuning [1199.031259, 1903.490418, 2784.064367, 3366.693863]
> TOP generators [1199.031259, 88.05739491]
> bad: 58.217715 comp: 7.752178 err: .968741
> 
> 
> Number 45 {25/24, 81/80} Jamesbond?
> 
> [0, 0, 7, 0, 11, 16] [[7, 11, 16, 20], [0, 0, 0, -1]]
> TOP tuning [1209.431411, 1900.535075, 2764.414655, 3368.825906]
> TOP generators [172.7759159, 86.69241190]
> bad: 58.637859 comp: 2.493450 err: 9.431411
> 
> 
> Number 46 Hemithirds
> 
> [15, -2, -5, -38, -50, -6] [[1, 4, 2, 2], [0, -15, 2, 5]]
> TOP tuning [1200.363229, 1901.194685, 2787.427555, 3367.479202]
> TOP generators [1200.363229, 193.3505488]
> bad: 60.573479 comp: 11.237086 err: .479706
> 
> 
> Number 47
> 
> [12, 34, 20, 26, -2, -49] [[2, 4, 7, 7], [0, -6, -17, -10]]
> TOP tuning [1200.284965, 1901.503343, 2786.975381, 3369.219732]
> TOP generators [600.1424823, 83.17776441]
> bad: 61.101493 comp: 14.643003 err: .284965
> 
> 
> Number 48 Flattone
> 
> [1, 4, -9, 4, -17, -32] [[1, 2, 4, -1], [0, -1, -4, 9]]
> TOP tuning [1202.536420, 1897.934872, 2781.593812, 3361.705278]
> TOP generators [1202.536419, 507.1379663]
> bad: 61.126418 comp: 4.909123 err: 2.536420
> 
> 
> Number 49 Diaschismic
> 
> [2, -4, -16, -11, -31, -26] [[2, 3, 5, 7], [0, 1, -2, -8]]
> TOP tuning [1198.732403, 1901.885616, 2789.256983, 3365.267311]
> TOP generators [599.3662015, 103.7870123]
> bad: 61.527901 comp: 6.966993 err: 1.267597
> 
> 
> Number 50 Superkleismic
> 
> [9, 10, -3, -5, -30, -35] [[1, 4, 5, 2], [0, -9, -10, 3]]
> TOP tuning [1201.371917, 1904.129438, 2783.128219, 3369.863245]
> TOP generators [1201.371918, 322.3731369]
> bad: 62.364585 comp: 6.742251 err: 1.371918
> 
> 
> Number 51
> 
> [8, 1, 18, -17, 6, 39] [[1, -1, 2, -3], [0, 8, 1, 18]]
> TOP tuning [1201.135544, 1899.537544, 2789.855225, 3373.107814]
> TOP generators [1201.135545, 387.5841360]
> bad: 62.703297 comp: 6.411729 err: 1.525246
> 
> 
> Number 52 Tritonic
> 
> [5, -11, -12, -29, -33, 3] [[1, 4, -3, -3], [0, -5, 11, 12]]
> TOP tuning [1201.023211, 1900.333250, 2785.201472, 3365.953391]
> TOP generators [1201.023211, 580.7519186]
> bad: 63.536850 comp: 7.880073 err: 1.023211
> 
> 
> Number 53
> 
> [1, 33, 27, 50, 40, -30] [[1, 2, 16, 14], [0, -1, -33, -27]]
> TOP tuning [1199.680495, 1902.108988, 2785.571846, 3369.722869]
> TOP generators [1199.680495, 497.2520023]
> bad: 64.536886 comp: 14.212326 err: .319505
> 
> 
> Number 54
> 
> [6, 10, 3, 2, -12, -21] [[1, 2, 3, 3], [0, -6, -10, -3]]
> TOP tuning [1202.659696, 1907.471368, 2778.232381, 3359.055076]
> TOP generators [1202.659696, 82.97467050]
> bad: 64.556006 comp: 4.306766 err: 3.480440
> 
> 
> Number 55
> 
> [0, 0, 12, 0, 19, 28] [[12, 19, 28, 34], [0, 0, 0, -1]]
> TOP tuning [1197.674070, 1896.317278, 2794.572829, 3368.825906]
> TOP generators [99.80617249, 24.58395811]
> bad: 65.630949 comp: 4.295482 err: 3.557008
> 
> 
> Number 56
> 
> [2, 1, -4, -3, -12, -12] [[1, 1, 2, 4], [0, 2, 1, -4]]
> TOP tuning [1204.567524, 1916.451342, 2765.076958, 3394.502460]
> TOP generators [1204.567524, 355.9419091]
> bad: 66.522610 comp: 2.696901 err: 9.146173
> 
> 
> Number 57
> 
> [2, -2, 1, -8, -4, 8] [[1, 2, 2, 3], [0, -2, 2, -1]]
> TOP tuning [1185.869125, 1924.351909, 2819.124589, 3333.914203]
> TOP generators [1185.869125, 223.6931705]
> bad: 66.774944 comp: 2.173813 err: 14.130876
> 
> 
> Number 58
> 
> [5, 8, 2, 1, -11, -18] [[1, 2, 3, 3], [0, -5, -8, -2]]
> TOP tuning [1194.335372, 1892.976778, 2789.895770, 3384.728528]
> TOP generators [1194.335372, 99.13879319]
> bad: 67.244049 comp: 3.445412 err: 5.664628
> 
> 
> Number 59
> 
> [3, 5, 9, 1, 6, 7] [[1, 2, 3, 4], [0, -3, -5, -9]]
> TOP tuning [1193.415676, 1912.390908, 2789.512955, 3350.341372]
> TOP generators [1193.415676, 158.1468146]
> bad: 67.670842 comp: 3.205865 err: 6.584324
> 
> 
> Number 60
> 
> [3, 0, 9, -7, 6, 21] [[3, 5, 7, 9], [0, -1, 0, -3]]
> TOP tuning [1193.415676, 1912.390908, 2784.636577, 3350.341372]
> TOP generators [397.8052253, 76.63521863]
> bad: 68.337269 comp: 3.221612 err: 6.584324
> 
> 
> Number 61 Hemikleismic
> 
> [12, 10, -9, -12, -48, -49] [[1, 0, 1, 4], [0, 12, 10, -9]]
> TOP tuning [1199.411231, 1902.888178, 2785.151380, 3370.478790]
> TOP generators [1199.411231, 158.5740148]
> bad: 68.516458 comp: 10.787602 err: .588769
> 
> 
> Number 62
> 
> [2, -2, -2, -8, -9, 1] [[2, 3, 5, 6], [0, 1, -1, -1]]
> TOP tuning [1185.468457, 1924.986952, 2816.886876, 3409.621105]
> TOP generators [592.7342285, 146.7842660]
> bad: 68.668284 comp: 2.173813 err: 14.531543
> 
> 
> Number 63
> 
> [8, 13, 23, 2, 14, 17] [[1, 2, 3, 4], [0, -8, -13, -23]]
> TOP tuning [1198.975478, 1900.576277, 2788.692580, 3365.949709]
> TOP generators [1198.975478, 62.17183489]
> bad: 68.767371 comp: 8.192765 err: 1.024522
> 
> 
> Number 64
> 
> [3, -7, -8, -18, -21, 1] [[1, 3, -1, -1], [0, -3, 7, 8]]
> TOP tuning [1202.900537, 1897.357759, 2790.235118, 3360.683070]
> TOP generators [1202.900537, 570.4479508]
> bad: 69.388565 comp: 4.891080 err: 2.900537
> 
> 
> Number 65
> 
> [3, 12, 11, 12, 9, -8] [[1, 3, 8, 8], [0, -3, -12, -11]]
> TOP tuning [1202.624742, 1900.726787, 2792.408176, 3361.457323]
> TOP generators [1202.624742, 569.0491468]
> bad: 70.105427 comp: 5.168119 err: 2.624742
> 
> 
> Number 66
> 
> [17, 6, 15, -30, -24, 18] [[1, -5, 0, -3], [0, 17, 6, 15]]
> TOP tuning [1199.379215, 1900.971080, 2787.482526, 3370.568669]
> TOP generators [1199.379215, 464.5804210]
> bad: 71.416917 comp: 10.725806 err: .620785
> 
> 
> Number 67
> 
> [11, 13, 17, -5, -4, 3] [[1, 3, 4, 5], [0, -11, -13, -17]]
> TOP tuning [1198.514750, 1899.600936, 2789.762356, 3371.570447]
> TOP generators [1198.514750, 154.1766650]
> bad: 71.539673 comp: 6.940227 err: 1.485250
> 
> 
> Number 68
> 
> [3, -24, -1, -45, -10, 65] [[1, 1, 7, 3], [0, 3, -24, -1]]
> TOP tuning [1200.486331, 1902.481504, 2787.442939, 3367.460603]
> TOP generators [1200.486331, 233.9983907]
> bad: 72.714599 comp: 12.227699 err: .486331
> 
> 
> Number 69
> 
> [23, -1, 13, -55, -44, 33] [[1, 9, 2, 7], [0, -23, 1, -13]]
> TOP tuning [1199.671611, 1901.434518, 2786.108874, 3369.747810]
> TOP generators [1199.671611, 386.7656515]
> bad: 73.346343 comp: 14.944966 err: .328389
> 
> 
> Number 70
> 
> [6, 29, -2, 32, -20, -86] [[1, 4, 14, 2], [0, -6, -29, 2]]
> TOP tuning [1200.422358, 1901.285580, 2787.294397, 3367.645998]
> TOP generators [1200.422357, 483.4006416]
> bad: 73.516606 comp: 13.193267 err: .422358
> 
> 
> Number 71
> 
> [7, -15, -16, -40, -45, 5] [[1, 5, -5, -5], [0, -7, 15, 16]]
> TOP tuning [1200.210742, 1900.961474, 2784.858222, 3370.585685]
> TOP generators [1200.210742, 585.7274621]
> bad: 74.053446 comp: 10.869066 err: .626846
> 
> 
> Number 72
> 
> [5, 3, 7, -7, -3, 8] [[1, 1, 2, 2], [0, 5, 3, 7]]
> TOP tuning [1192.540126, 1890.131381, 2803.635005, 3361.708008]
> TOP generators [1192.540126, 139.5182509]
> bad: 74.239244 comp: 3.154649 err: 7.459874
> 
> 
> Number 73
> 
> [4, 21, -3, 24, -16, -66] [[1, 0, -6, 4], [0, 4, 21, -3]]
> TOP tuning [1199.274449, 1901.646683, 2787.998389, 3370.862785]
> TOP generators [1199.274449, 475.4116708]
> bad: 74.381278 comp: 10.125066 err: .725551
> 
> 
> Number 74
> 
> [3, -5, -6, -15, -18, 0] [[1, 3, 0, 0], [0, -3, 5, 6]]
> TOP tuning [1195.486066, 1908.381352, 2796.794743, 3356.153692]
> TOP generators [1195.486066, 559.3589487]
> bad: 74.989802 comp: 4.075900 err: 4.513934
> 
> 
> Number 75
> 
> [6, 0, 3, -14, -12, 7] [[3, 4, 7, 8], [0, 2, 0, 1]]
> TOP tuning [1199.400031, 1910.341746, 2798.600074, 3353.970936]
> TOP generators [399.8000105, 155.5708520]
> bad: 76.576420 comp: 3.804173 err: 5.291448
> 
> 
> Number 76
> 
> [13, 2, 30, -27, 11, 64] [[1, 6, 3, 13], [0, -13, -2, -30]]
> TOP tuning [1200.672456, 1900.889183, 2786.148822, 3370.713730]
> TOP generators [1200.672456, 407.9342733]
> bad: 76.791305 comp: 10.686216 err: .672456
> 
> 
> Number 77 Shrutar
> 
> [4, -8, 14, -22, 11, 55] [[2, 3, 5, 5], [0, 2, -4, 7]]
> TOP tuning [1198.920873, 1903.665377, 2786.734051, 3365.796415]
> TOP generators [599.4604367, 52.64203308]
> bad: 76.825572 comp: 8.437555 err: 1.079127
> 
> 
> Number 78
> 
> [12, 10, 25, -12, 6, 30] [[1, 6, 6, 12], [0, -12, -10, -25]]
> TOP tuning [1199.028703, 1903.494472, 2785.274095, 3366.099130]
> TOP generators [1199.028703, 440.8898120]
> bad: 77.026097 comp: 8.905180 err: .971298
> 
> 
> Number 79 Beatles
> 
> [2, -9, -4, -19, -12, 16] [[1, 1, 5, 4], [0, 2, -9, -4]]
> TOP tuning [1197.104145, 1906.544822, 2793.037680, 3369.535226]
> TOP generators [1197.104145, 354.7203384]
> bad: 77.187771 comp: 5.162806 err: 2.895855
> 
> 
> Number 80
> 
> [6, -12, 10, -33, -1, 57] [[2, 4, 3, 7], [0, -3, 6, -5]]
> TOP tuning [1199.025947, 1903.033657, 2788.575394, 3371.560420]
> TOP generators [599.5129735, 165.0060791]
> bad: 78.320453 comp: 8.966980 err: .974054
> 
> 
> Number 81
> 
> [4, 4, 0, -3, -11, -11] [[4, 6, 9, 11], [0, 1, 1, 0]]
> TOP tuning [1212.384652, 1905.781495, 2815.069985, 3334.057793]
> TOP generators [303.0961630, 63.74881402]
> bad: 78.879803 comp: 2.523719 err: 12.384652
> 
> 
> Number 82
> 
> [6, -2, -2, -17, -20, 1] [[2, 2, 5, 6], [0, 3, -1, -1]]
> TOP tuning [1203.400986, 1896.025764, 2777.627538, 3379.328030]
> TOP generators [601.7004928, 230.8749260]
> bad: 79.825592 comp: 4.619353 err: 3.740932
> 
> 
> Number 83
> 
> [1, 6, 5, 7, 5, -5] [[1, 2, 5, 5], [0, -1, -6, -5]]
> TOP tuning [1211.970043, 1882.982932, 2814.107292, 3355.064446]
> TOP generators [1211.970043, 540.9571536]
> bad: 79.928319 comp: 2.584059 err: 11.970043
> 
> 
> Number 84 Squares
> 
> [4, 16, 9, 16, 3, -24] [[1, 3, 8, 6], [0, -4, -16, -9]]
> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.067656]
> TOP generators [1201.698520, 426.4581630]
> bad: 80.651668 comp: 6.890825 err: 1.698521
> 
> 
> Number 85
> 
> [6, 0, 0, -14, -17, 0] [[6, 10, 14, 17], [0, -1, 0, 0]]
> TOP tuning [1194.473353, 1901.955001, 2787.104490, 3384.341166]
> TOP generators [199.0788921, 88.83392059]
> bad: 80.672767 comp: 3.820609 err: 5.526647
> 
> 
> Number 86
> 
> [7, 26, 25, 25, 20, -15] [[1, 5, 15, 15], [0, -7, -26, -25]]
> TOP tuning [1199.352846, 1902.980716, 2784.811068, 3369.637284]
> TOP generators [1199.352846, 584.8262161]
> bad: 81.144087 comp: 11.197591 err: .647154
> 
> 
> Number 87
> 
> [18, 15, -6, -18, -60, -56] [[3, 6, 8, 8], [0, -6, -5, 2]]
> TOP tuning [1200.448679, 1901.787880, 2785.271912, 3367.566305]
> TOP generators [400.1495598, 83.18491309]
> bad: 81.584166 comp: 13.484503 err: .448679
> 
> 
> Number 88
> 
> [9, -2, 14, -24, -3, 38] [[1, 3, 2, 5], [0, -9, 2, -14]]
> TOP tuning [1201.918556, 1904.657347, 2781.858962, 3363.439837]
> TOP generators [1201.918557, 189.0109248]
> bad: 81.594641 comp: 6.521440 err: 1.918557
> 
> 
> Number 89
> 
> [1, -8, -2, -15, -6, 18] [[1, 2, -1, 2], [0, -1, 8, 2]]
> TOP tuning [1195.155395, 1894.070902, 2774.763716, 3382.790568]
> TOP generators [1195.155395, 496.2398890]
> bad: 82.638059 comp: 4.075900 err: 4.974313
> 
> 
> Number 90
> 
> [3, 7, -1, 4, -10, -22] [[1, 1, 1, 3], [0, 3, 7, -1]]
> TOP tuning [1205.820043, 1890.417958, 2803.215176, 3389.260823]
> TOP generators [1205.820043, 228.1993049]
> bad: 82.914167 comp: 3.375022 err: 7.279064
> 
> 
> 
> Number 91
> 
> [6, 5, -31, -6, -66, -86] [[1, 0, 1, 11], [0, 6, 5, -31]]
> TOP tuning [1199.976626, 1902.553087, 2785.437532, 3369.885264]
> TOP generators [1199.976626, 317.0921813]
> bad: 83.023430 comp: 14.832953 err: .377351
> 
> 
> Number 92
> 
> [8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]]
> TOP tuning [1198.553882, 1907.135354, 2778.724633, 3378.001574]
> TOP generators [599.2769413, 272.3123381]
> bad: 83.268810 comp: 5.047438 err: 3.268439
> 
> 
> Number 93
> 
> [4, 2, 9, -6, 3, 15] [[1, 3, 3, 6], [0, -4, -2, -9]]
> TOP tuning [1208.170435, 1910.173796, 2767.342550, 3391.763218]
> TOP generators [1208.170435, 428.5843770]
> bad: 83.972208 comp: 3.205865 err: 8.170435
> 
> 
> Number 94 Hexidecimal
> 
> [1, -3, 5, -7, 5, 20] [[1, 2, 1, 5], [0, -1, 3, -5]]
> TOP tuning [1208.959294, 1887.754858, 2799.450479, 3393.977822]
> TOP generators [1208.959293, 530.1637287]
> bad: 84.341555 comp: 3.068202 err: 8.959294
> 
> 
> Number 95
> 
> [6, 0, 15, -14, 7, 35] [[3, 5, 7, 9], [0, -2, 0, -5]]
> TOP tuning [1197.060039, 1902.856975, 2793.140092, 3360.572393]
> TOP generators [399.0200131, 46.12154491]
> bad: 84.758945 comp: 5.369353 err: 2.939961
> 
> 
> Number 96
> 
> [0, 12, 12, 19, 19, -6] [[12, 19, 28, 34], [0, 0, -1, -1]]
> TOP tuning [1198.015473, 1896.857833, 2778.846497, 3377.854234]
> TOP generators [99.83462277, 16.52294019]
> bad: 85.896401 comp: 5.168119 err: 3.215955
> 
> 
> Number 97
> 
> [11, -6, 10, -35, -15, 40] [[1, 4, 1, 5], [0, -11, 6, -10]]
> TOP tuning [1200.950404, 1901.347958, 2784.106944, 3366.157786]
> TOP generators [1200.950404, 263.8594234]
> bad: 85.962459 comp: 9.510433 err: .950404
> 
> 
> Number 98 Slender
> 
> [13, -10, 6, -46, -27, 42] [[1, 2, 2, 3], [0, -13, 10, -6]]
> TOP tuning [1200.337238, 1901.055858, 2784.996493, 3370.418508]
> TOP generators [1200.337239, 38.43220154]
> bad: 88.631905 comp: 12.499426 err: .567296
> 
> 
> Number 99
> 
> [0, 5, 10, 8, 16, 9] [[5, 8, 12, 15], [0, 0, -1, -2]]
> TOP tuning [1195.598382, 1912.957411, 2770.195472, 3388.313857]
> TOP generators [239.1196765, 99.24064453]
> bad: 89.758630 comp: 3.595867 err: 6.941749
> 
> 
> Number 100
> 
> [1, -1, -5, -4, -11, -9] [[1, 2, 2, 1], [0, -1, 1, 5]]
> TOP tuning [1185.210905, 1925.395162, 2815.448458, 3410.344145]
> TOP generators [1185.210905, 445.0266480]
> bad: 90.384580 comp: 2.472159 err: 14.789095
> 
> 
> Number 101
> 
> [2, 8, -11, 8, -23, -48] [[1, 1, 0, 6], [0, 2, 8, -11]]
> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3373.586984]
> TOP generators [1201.698520, 348.7821945]
> bad: 92.100337 comp: 7.363684 err: 1.698521
> 
> 
> Number 102
> 
> [3, 12, 18, 12, 20, 8] [[3, 5, 8, 10], [0, -1, -4, -6]]
> TOP tuning [1202.260038, 1898.372926, 2784.451552, 3375.170635]
> TOP generators [400.7533459, 105.3938041]
> bad: 92.910783 comp: 6.411729 err: 2.260038
> 
> 
> Number 103
> 
> [4, -8, -20, -22, -43, -24] [[4, 6, 10, 13], [0, 1, -2, -5]]
> TOP tuning [1199.003867, 1903.533834, 2787.453602, 3371.622404]
> TOP generators [299.7509668, 105.0280329]
> bad: 93.029698 comp: 9.663894 err: .996133
> 
> 
> Number 104
> 
> [3, 0, -3, -7, -13, -7] [[3, 5, 7, 8], [0, -1, 0, 1]]
> TOP tuning [1205.132027, 1884.438632, 2811.974729, 3337.800149]
> TOP generators [401.7106756, 124.1147448]
> bad: 94.336372 comp: 2.921642 err: 11.051598
> 
> 
> Number 105
> 
> [4, 7, 2, 2, -8, -15] [[1, 2, 3, 3], [0, -4, -7, -2]]
> TOP tuning [1190.204869, 1918.438775, 2762.165422, 3339.629125]
> TOP generators [1190.204869, 115.4927407]
> bad: 94.522719 comp: 3.014736 err: 10.400103
> 
> 
> 
> Number 106
> 
> [13, 19, 23, 0, 0, 0] [[1, 0, 0, 0], [0, 13, 19, 23]]
> TOP tuning [1200.0, 1904.187463, 2783.043215, 3368.947050]
> TOP generators [1200., 146.4759587]
> bad: 94.757554 comp: 8.202087 err: 1.408527
> 
> 
> Number 107
> 
> [2, -6, -6, -14, -15, 3] [[2, 3, 5, 6], [0, 1, -3, -3]]
> TOP tuning [1206.548264, 1891.576247, 2771.109113, 3374.383246]
> TOP generators [603.2741324, 81.75384943]
> bad: 94.764743 comp: 3.804173 err: 6.548265
> 
> 
> Number 108
> 
> [2, -6, -6, -14, -15, 3] [[2, 3, 5, 6], [0, 1, -3, -3]]
> TOP tuning [1206.548264, 1891.576247, 2771.109113, 3374.383246]
> TOP generators [603.2741324, 81.75384943]
> bad: 94.764743 comp: 3.804173 err: 6.548265
> 
> 
> Number 109
> 
> [1, -13, -2, -23, -6, 32] [[1, 2, -3, 2], [0, -1, 13, 2]]
> TOP tuning [1197.567789, 1904.876372, 2780.666293, 3375.653987]
> TOP generators [1197.567789, 490.2592046]
> bad: 94.999539 comp: 6.249713 err: 2.432212
> 
> 
> Number 110
> 
> [9, 0, 9, -21, -11, 21] [[9, 14, 21, 25], [0, 1, 0, 1]]
> TOP tuning [1197.060039, 1897.499011, 2793.140092, 3360.572393]
> TOP generators [133.0066710, 35.40561749]
> bad: 95.729260 comp: 5.706260 err: 2.939961
> 
> 
> Number 111
> 
> [5, 1, 9, -10, 0, 18] [[1, 0, 2, 0], [0, 5, 1, 9]]
> TOP tuning [1193.274911, 1886.640142, 2763.877849, 3395.952256]
> TOP generators [1193.274911, 377.3280283]
> bad: 99.308041 comp: 3.205865 err: 9.662601
> 
> 
> Number 112 Muggles
> 
> [5, 1, -7, -10, -25, -19] [[1, 0, 2, 5], [0, 5, 1, -7]]
> TOP tuning [1203.148010, 1896.965522, 2785.689126, 3359.988323]
> TOP generators [1203.148011, 379.3931044]
> bad: 99.376477 comp: 5.618543 err: 3.148011
> 
> 
> Number 113
> 
> [11, 6, 15, -16, -7, 18] [[1, 1, 2, 2], [0, 11, 6, 15]]
> TOP tuning [1202.072164, 1905.239303, 2787.690040, 3363.008608]
> TOP generators [1202.072164, 63.92428535]
> bad: 99.809415 comp: 6.940227 err: 2.072164
> 
> 
> Number 114
> 
> [1, -8, -26, -15, -44, -38] [[1, 2, -1, -8], [0, -1, 8, 26]]
> TOP tuning [1199.424969, 1900.336158, 2788.685275, 3365.958541]
> TOP generators [1199.424969, 498.5137806]
> bad: 99.875385 comp: 9.888635 err: 1.021376
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Message: 9380 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 22:00:24

Subject: Re: Maple code for ?(x)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:


>>> Obviously, Maple is not going to compute an infinte number of
>>> convergents. What it does depends on the setting of Digits.
>> 


>> So we're getting some *approximation* to the ? function, yes? 
Matlab 
>> does continued fractions, but they're strings not vectors, they 
use 
>> an error tolerance to determine when to stop, and they allow 
negative 
>> entries. I've written code to get around the last limitation; the 
>> other two shouldn't be that hard . . .
> 


> By the way, somebody once told me that maple code can be executed
> in Matlab.  Dunno if that's true...



They must have had an intepreter or something . . .
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Message: 9381 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 13:11:31

Subject: Re: 114 7-limit temperaments

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:


> This list is attractive, but Meantone, Magic, Pajara, maybe
> Injera to name a few are too low for my taste, if I'm reading
> these errors right (they're weighted here, I take it).



I think log-flat badness has outlived its popularity :)



> And I don't see how you figure schismic is less complex than
> miracle in light of the maps given.



Probably the shortness of the fifths in the lattice wins it for 
schismic . . .




> 
> -Carl
> 


>> Number 1 Ennealimmal
>> 
>> [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]]
>> TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784]
>> TOP generators [133.3373752, 49.02398564]
>> bad: 4.918774 comp: 11.628267 err: .036377
>> 
>> 
>> Number 2 Meantone
>> 
>> [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
>> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
>> TOP generators [1201.698520, 504.1341314]
>> bad: 21.551439 comp: 3.562072 err: 1.698521
>> 
>> 
>> Number 3 Magic
>> 
>> [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
>> TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
>> TOP generators [1201.276744, 380.7957184]
>> bad: 23.327687 comp: 4.274486 err: 1.276744
>> 
>> 
>> Number 4 Beep
>> 
>> [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
>> TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]
>> TOP generators [1194.642673, 254.8994697]
>> bad: 23.664749 comp: 1.292030 err: 14.176105
>> 
>> 
>> Number 5 Augmented
>> 
>> [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
>> TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]
>> TOP generators [399.9922103, 107.3111730]
>> bad: 27.081145 comp: 2.147741 err: 5.870879
>> 
>> 
>> Number 6 Pajara
>> 
>> [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
>> TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
>> TOP generators [598.4467109, 106.5665459]
>> bad: 27.754421 comp: 2.988993 err: 3.106578
>> 
>> 
>> Number 7 Dominant Seventh
>> 
>> [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
>> TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933]
>> TOP generators [1195.228951, 495.8810151]
>> bad: 28.744957 comp: 2.454561 err: 4.771049
>> 
>> 
>> Number 8 Schismic
>> 
>> [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
>> TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
>> TOP generators [1200.760624, 498.1193303]
>> bad: 28.818558 comp: 5.618543 err: .912904
>> 
>> 
>> Number 9 Miracle
>> 
>> [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
>> TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
>> TOP generators [1200.631014, 116.7206423]
>> bad: 29.119472 comp: 6.793166 err: .631014
>> 
>> 
>> Number 10 Orwell
>> 
>> [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
>> TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834]
>> TOP generators [1199.532657, 271.4936472]
>> bad: 30.805067 comp: 5.706260 err: .946061
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Message: 9382 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 22:01:30

Subject: Re: 114 7-limit temperaments

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:


> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> 


>> What kind of complexity is this? 
> 


> It's the complexity which arises naturally out of the Tenney space 
> and dual val space point of view, as the norm on a bival. It 
> therefore gives more weight to lower primes such as 2 and 3 as 
> opposed to higher ones such as 5 and 7.
> 


>> Do you always use the same kind?
> 


> I wanted to do things from a TOP point of view, so I used something 
> consistent with that.



Carl, did you read the message that you got 3 copies of? This is what 
Gene was addressing with his list.
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Message: 9383 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 14:04:25

Subject: Re: 114 7-limit temperaments

From: Carl Lumma


>> >ap complexity is
>> the number of notes you need to complete the map *with contiguous
>> chains of generators*. 
>


>Thus it will depend on the choice of generators. For so-called linear 
>temperaments, this is only made definite by fixing one of them to be 
>1/N octaves.



I don't get thus.  The map contains the consonances you want, and a
way to get the generators to hit them.  Why does one of the generators
have to generate an octave all by itself?



>For planar and higher-dimensional temperaments, the 
>choice is even more arbitrary. Comma complexity, or wedgie complexity 
>for higher codimensions, is well-defined, and is (according to Gene) 
>the natural generalization of the complexity measures we all agree on 
>for the simplest cases.



I agree that comma complexity seems more desirable.  I don't have
the commas Gene used for schismic and miracle handy (and I don't
know how to compute complexity from more than a single comma) but
what's all this talk about generator steps in a 3/2 then?

-Carl
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Message: 9384 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 15:00:07

Subject: Re: A potentially informative property of tunings

From: George D. Secor

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:


> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" 
<gdsecor@y...> 
> wrote:


>> --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> 
wrote:


>>> Take a generator of 260.76 cents and a period of 1206.55 cents. 
This
>>> defines a linear tuning which belongs to a family of related 
linear
>>> temperaments. The simplest mapping is the "beep" mapping, which 
distributes
>>> the 27;25 interval:
>>> 
>>> [(1, 0), (2, -2), (3, -3)]
>>> 
>>> but after 6 iterations of the generator, there's a better 5:1 


at (1, 6),


>>> about 15 cents flat (compared with the 51 cent sharp "beep" 


version of the


>>> interval). That means this particular tuning is consistent 
with "beep"
>>> temperament only up to a range of 5 generators -- or to coin a 
phrase, its
>>> "consistency range" with respect to "beep" is 5. In comparison, 
top
>>> meantone has a "consistency range" of 34: its (17, -35) version 


of 5:1 is


>>> only 2 cents flat, compared with the 4-cent sharp (4, -4). 
Quarter-comma
>>> meantone has a "consistency range" of 29, since it has a better 
3:1 at
>>> (-11, 30).
>>> 
>>> First of all, I don't like the term "consistency range", but I 
couldn't
>>> think of anything better. I'd appreciate ideas for what to call 
this
>>> property. 
>> 


>> Since you're describing a relationship or comparison between two 
>> temperaments, I would suggest "compatibility range".  The 
>> term "consistency" is usually used to describe only relationships 
>> within a single temperament.
> 


> I don't see any comparison between two temperaments in what Herman 
is 
> proposing! It all looks "within temperament" to me.



I haven't really been following this discussion, so I may be 
misinterpreting something here, but it sure looks to me like Herman 
is comparing or relating "this particular tuning" with "'beep' 
temperament" in the first sentence of the following excerpt (which I 
am requoting from above):



>>> ...  That means this particular tuning is consistent with "beep"
>>> temperament only up to a range of 5 generators -- or to coin a 
phrase, its
>>> "consistency range" with respect to "beep" is 5.



So if "this particular tuning" and "'beep' temperament" are both 
temperaments, is he not comparing two temperaments?

--George
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Message: 9385 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 14:05:18

Subject: Re: Maple code for ?(x)

From: Carl Lumma


>> >y the way, somebody once told me that maple code can be executed
>> in Matlab.  Dunno if that's true...
>


>They must have had an intepreter or something . . .



They claimed the maple kernel is included with matlab.  Though I
suspect they were wrong.  :)

-Carl
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Message: 9386 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 15:19:05

Subject: Re: A potentially informative property of tunings

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> 
wrote:


> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:


>> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" 
> <gdsecor@y...> 
>> wrote:


>>> --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller 
<hmiller@I...> 
> wrote:


>>>> Take a generator of 260.76 cents and a period of 1206.55 
cents. 
> This


>>>> defines a linear tuning which belongs to a family of related 
> linear


>>>> temperaments. The simplest mapping is the "beep" mapping, 
which 
> distributes


>>>> the 27;25 interval:
>>>> 
>>>> [(1, 0), (2, -2), (3, -3)]
>>>> 
>>>> but after 6 iterations of the generator, there's a better 5:1 


> at (1, 6),


>>>> about 15 cents flat (compared with the 51 cent sharp "beep" 


> version of the


>>>> interval). That means this particular tuning is consistent 
> with "beep"


>>>> temperament only up to a range of 5 generators -- or to coin 
a 
> phrase, its


>>>> "consistency range" with respect to "beep" is 5. In 
comparison, 
> top


>>>> meantone has a "consistency range" of 34: its (17, -35) 
version 


> of 5:1 is


>>>> only 2 cents flat, compared with the 4-cent sharp (4, -4). 
> Quarter-comma


>>>> meantone has a "consistency range" of 29, since it has a 
better 
> 3:1 at
>>>> (-11, 30).
>>>> 
>>>> First of all, I don't like the term "consistency range", but 
I 
> couldn't


>>>> think of anything better. I'd appreciate ideas for what to 
call 
> this
>>>> property. 
>>> 


>>> Since you're describing a relationship or comparison between 
two 
>>> temperaments, I would suggest "compatibility range".  The 
>>> term "consistency" is usually used to describe only 
relationships 
>>> within a single temperament.
>> 


>> I don't see any comparison between two temperaments in what 
Herman 
> is 


>> proposing! It all looks "within temperament" to me.
> 


> I haven't really been following this discussion, so I may be 
> misinterpreting something here, but it sure looks to me like Herman 
> is comparing or relating "this particular tuning" with "'beep' 
> temperament" in the first sentence of the following excerpt (which 
I 
> am requoting from above):
> 


>>>> ...  That means this particular tuning is consistent 
with "beep"
>>>> temperament only up to a range of 5 generators -- or to coin 
a 
> phrase, its


>>>> "consistency range" with respect to "beep" is 5.
> 


> So if "this particular tuning" and "'beep' temperament" are both 
> temperaments, is he not comparing two temperaments?



George, as explained at the top of this message, "this particular 
tuning" does result from applying "'beep' temperament", so no.
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Message: 9387 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 14:11:42

Subject: Re: Maple code for ?(x)

From: Carl Lumma


>>> >y the way, somebody once told me that maple code can be executed
>>> in Matlab.  Dunno if that's true...
>> 


>> They must have had an intepreter or something . . .
>


>They claimed the maple kernel is included with matlab.  Though I
>suspect they were wrong.  :)



From Matlab help...



>Using Maple Functions
>The maple function lets you access Maple functions directly.
>This function takes sym objects, strings, and doubles as inputs.
>It returns a symbolic object, character string, or double
>corresponding to the class of the input. You can also use the
>maple function to debug symbolic math programs that you develop.

>Precompiled Maple Procedures
>When Maple loads a source (ASCII text) procedure into its workspace,
>it compiles (translates) the procedure into an internal format. You
>can subsequently use the maple function to save the procedures in
>the internal format. The advantage is you avoid recompiling the
>procedure the next time you load it, thereby speeding up the process.

>maple
>Access Maple kernel.
>
>Syntax
>r = maple('statement')
>r = maple('function',arg1,arg2,...)
>[r, status] = maple(...)
>maple('traceon') or maple trace on
>maple('traceoff') or maple trace off

>The computational engine underlying the [symbolic math] toolboxes is
>the kernel of Maple, a system developed primarily at the University
>of Waterloo, Canada, and, more recently, at the Eidgenössiche
>Technische Hochschule, Zürich, Switzerland. Maple is marketed and
>supported by Waterloo Maple, Inc.



Once again I learn not to suspect Dan of being wrong.

-Carl
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Message: 9388 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 15:31:05

Subject: Re: A potentially informative property of tunings

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> 
wrote:


> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:


>> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" 
> <gdsecor@y...> 
>> wrote:


>>> --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller 
<hmiller@I...> 
> wrote:


>>>> Take a generator of 260.76 cents and a period of 1206.55 
cents. 
> This


>>>> defines a linear tuning which belongs to a family of related 
> linear


>>>> temperaments. The simplest mapping is the "beep" mapping, 
which 
> distributes


>>>> the 27;25 interval:
>>>> 
>>>> [(1, 0), (2, -2), (3, -3)]
>>>> 
>>>> but after 6 iterations of the generator, there's a better 5:1 


> at (1, 6),


>>>> about 15 cents flat (compared with the 51 cent sharp "beep" 


> version of the


>>>> interval). That means this particular tuning is consistent 
> with "beep"


>>>> temperament only up to a range of 5 generators -- or to coin 
a 
> phrase, its


>>>> "consistency range" with respect to "beep" is 5. In 
comparison, 
> top


>>>> meantone has a "consistency range" of 34: its (17, -35) 
version 


> of 5:1 is


>>>> only 2 cents flat, compared with the 4-cent sharp (4, -4). 
> Quarter-comma


>>>> meantone has a "consistency range" of 29, since it has a 
better 
> 3:1 at
>>>> (-11, 30).
>>>> 
>>>> First of all, I don't like the term "consistency range", but 
I 
> couldn't


>>>> think of anything better. I'd appreciate ideas for what to 
call 
> this
>>>> property. 
>>> 


>>> Since you're describing a relationship or comparison between 
two 
>>> temperaments, I would suggest "compatibility range".  The 
>>> term "consistency" is usually used to describe only 
relationships 
>>> within a single temperament.
>> 


>> I don't see any comparison between two temperaments in what 
Herman 
> is 


>> proposing! It all looks "within temperament" to me.
> 


> I haven't really been following this discussion, so I may be 
> misinterpreting something here, but it sure looks to me like Herman 
> is comparing or relating "this particular tuning" with "'beep' 
> temperament" in the first sentence of the following excerpt (which 
I 
> am requoting from above):
> 


>>>> ...  That means this particular tuning is consistent 
with "beep"
>>>> temperament only up to a range of 5 generators -- or to coin 
a 
> phrase, its


>>>> "consistency range" with respect to "beep" is 5.
> 


> So if "this particular tuning" and "'beep' temperament" are both 
> temperaments, is he not comparing two temperaments?
> 
> --George



Note that Herman also says, above, 'Quarter-comma meantone has 
a "consistency range" of 29, since it has a better 3:1 at (-11, 30)'. 
This is an example which concerns a temperament you already 
understand, and clearly only one, not two.
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Message: 9389 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 15:54:52

Subject: Re: A potentially informative property of tunings

From: George D. Secor

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:


> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" 
<gdsecor@y...> 
>> ...


>> So if "this particular tuning" and "'beep' temperament" are both 
>> temperaments, is he not comparing two temperaments?
> 


> George, as explained at the top of this message, "this particular 
> tuning" does result from applying "'beep' temperament", so no.
> 
> Note that Herman also says, above, 'Quarter-comma meantone has 
> a "consistency range" of 29, since it has a better 3:1 at (-11, 
30)'. 
> This is an example which concerns a temperament you already 
> understand, and clearly only one, not two.



Okay, I get it.  Thanks.

--George
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Message: 9390 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 22:15:08

Subject: Re: 114 7-limit temperaments

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:


>>> Map complexity is
>>> the number of notes you need to complete the map *with contiguous
>>> chains of generators*. 
>> 


>> Thus it will depend on the choice of generators. For so-called 
linear 
>> temperaments, this is only made definite by fixing one of them to 
be 
>> 1/N octaves.
> 


> I don't get thus.  The map contains the consonances you want, and a
> way to get the generators to hit them.  Why does one of the 
generators
> have to generate an octave all by itself?



It doesn't, but that's how just about everyone has always described 
just about all of them.



>> For planar and higher-dimensional temperaments, the 
>> choice is even more arbitrary. Comma complexity, or wedgie 
complexity 
>> for higher codimensions, is well-defined, and is (according to 
Gene) 
>> the natural generalization of the complexity measures we all agree 
on 
>> for the simplest cases.
> 


> I agree that comma complexity seems more desirable.  I don't have
> the commas Gene used for schismic and miracle handy



He didn't (as he keeps insisting in another thread here now).



> (and I don't
> know how to compute complexity from more than a single comma)



You have to compute the wedgie, and then use the formula for wedgie 
(top) complexity that he just posted.



> but
> what's all this talk about generator steps in a 3/2 then?



Complexity (if reasonably defined) is complexity, and the various 
ways of looking at it are essentially equivalent.
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Message: 9391 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 18:39:15

Subject: Re: 114 7-limit temperaments

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:


> This list is attractive, but Meantone, Magic, Pajara, maybe
> Injera to name a few are too low for my taste, if I'm reading
> these errors right (they're weighted here, I take it).
> 
> If you could make this list finite with badness bounds only,
> I'd be more impressed by claims that log-flat badness is
> desirable (allows the comparison of ennealimmal with all
> temperaments in a sense, not just the others on the list, or
> whatever).



Log flat badness is deliberately designed not to be finite. and it
seems to me your objection is strange--do you think epimericity allows
comparison of one comma with another, while a log flat badness does not?

As for meantone, magic and pajara being too low, they are all near tht
top of the list. It would seem the list is doing exactly what you want
it to do.

You can make the list finite by bounding complexity, which is what
I've done.



> And I don't see how you figure schismic is less complex than
> miracle in light of the maps given.



Schismic gets to 3/2 in one generator step, and miracle takes six.
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Message: 9392 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 22:16:45

Subject: Re: Maple code for ?(x)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:


>>>> By the way, somebody once told me that maple code can be 
executed
>>>> in Matlab.  Dunno if that's true...
>>> 


>>> They must have had an intepreter or something . . .
>> 


>> They claimed the maple kernel is included with matlab.  Though I
>> suspect they were wrong.  :)
> 


> From Matlab help...
> 


>> Using Maple Functions
>> The maple function lets you access Maple functions directly.
>> This function takes sym objects, strings, and doubles as inputs.
>> It returns a symbolic object, character string, or double
>> corresponding to the class of the input. You can also use the
>> maple function to debug symbolic math programs that you develop.
> 
>> Precompiled Maple Procedures
>> When Maple loads a source (ASCII text) procedure into its 
workspace,
>> it compiles (translates) the procedure into an internal format. You
>> can subsequently use the maple function to save the procedures in
>> the internal format. The advantage is you avoid recompiling the
>> procedure the next time you load it, thereby speeding up the 
process.
> 
>> maple
>> Access Maple kernel.
>> 
>> Syntax
>> r = maple('statement')
>> r = maple('function',arg1,arg2,...)
>> [r, status] = maple(...)
>> maple('traceon') or maple trace on
>> maple('traceoff') or maple trace off
> 
>> The computational engine underlying the [symbolic math] toolboxes 
is
>> the kernel of Maple, a system developed primarily at the University
>> of Waterloo, Canada, and, more recently, at the Eidgenössiche
>> Technische Hochschule, Zürich, Switzerland. Maple is marketed and
>> supported by Waterloo Maple, Inc.
> 


> Once again I learn not to suspect Dan of being wrong.
> 
> -Carl



I don't have the symbolic math toolbox. I have the optimization 
toolbox and the statistics toolbox, that's it. Each toolbox is like 
$1000!
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Message: 9393 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 18:40:44

Subject: Re: Maple code for ?(x)

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:



> Aren't 'most' continued fractions infinite?



Obviously, Maple is not going to compute an infinte number of
convergents. What it does depends on the setting of Digits.
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Message: 9394 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 22:28:19

Subject: Poor man's harmonic entropy graphs uploaded

From: Gene Ward Smith

I've put these in the photos section.

Recall that "standard" ? HE is a Gaussian smoothing of ?(2^(x/1200))',
whereas the poor man simply contents himself with a central difference
operator on ?(2^(x/1200)). I've put up graphs of poor man for Del_s,
with s 5, 10, 25 and 50 cents. These are very fast and easy to
compute, and might be useful for that reason.
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Message: 9395 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 18:42:31

Subject: Re: TOP take on 7-limit temperaments

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:


> I appreciate this work, Gene.
> 
> How about a worked-out, hand-holding example for one of these error 
> and complexity calculations?
> 
> P.S. Instead of using log-flat badness, why don't we use the same 
> function of error and complexity that yielded epimericity in the 
> codimension-1 case?



What's your proposal specifically?
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Message: 9396 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 22:33:30

Subject: Re: Poor man's harmonic entropy graphs uploaded

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:


> I've put these in the photos section.
> 
> Recall that "standard" ? HE is a Gaussian smoothing of ?(2^
(x/1200))',
> whereas the poor man simply contents himself with a central 
difference
> operator on ?(2^(x/1200)). I've put up graphs of poor man for Del_s,
> with s 5, 10, 25 and 50 cents. These are very fast and easy to
> compute, and might be useful for that reason.



Unfortunately, their features bear little resemblance to those of 
harmonic entropy curves. Yet they are interesting in their own right. 
Are those global maxima at the golden ratio or something?
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Message: 9397 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 18:44:24

Subject: Re: 114 7-limit temperaments

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:


> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:


>> This list is attractive, but Meantone, Magic, Pajara, maybe
>> Injera to name a few are too low for my taste, if I'm reading
>> these errors right (they're weighted here, I take it).
> 


> I think log-flat badness has outlived its popularity :)



Not with me. However, an alternative which isn't simply ad-hoc
randomness would be nice.
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Message: 9399 - Contents - Hide Contents

Date: Wed, 21 Jan 2004 17:54:09

Subject: Re: Maple code for ?(x)

From: Carl Lumma


>>> >he computational engine underlying the [symbolic math] toolboxes 
>>> is the kernel of Maple, a system developed primarily at the
>>> University of Waterloo, Canada, and, more recently, at the
>>> Eidgenössiche Technische Hochschule, Zürich, Switzerland. Maple
>>> is marketed and supported by Waterloo Maple, Inc.
>> 


>> Once again I learn not to suspect Dan of being wrong.
>


>I don't have the symbolic math toolbox. I have the optimization 
>toolbox and the statistics toolbox, that's it. Each toolbox is like 
>$1000!



How can one tell what toolboxes are installed?  I have 27 toolboxes
in my "launch pad", whatever that means.

-Carl
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