Tuning-Math Digests messages 10925 - 10949

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Message: 10925

Date: Sat, 08 May 2004 00:27:54

Subject: Re: 10+16 (continued from tuning)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> > 7-limit now . . .
> > 
> > val for 10:
> > 
> > <10 16 23 28]
> > 
> > val for 16:
> > 
> > <16 25 37 45]
> > 
> > wedgie:
> > 
> > <<-6 2 2 17 20 -1]]
> > 
> > According to Gene's file, this has TOP error of 3.740932 and L1 
> > complexity of 14.626943. Not too bad. It just barely, by a hair, 
> > falls outside the bound in my paper. Not too late to change that, 
> > though . . .
> 
> What is the bound of your paper at the moment?

It was going to be, for both 5-limit and 7-limit,

(error/16.6667)^(2/3) + (complexity/23.5)^(2/3) < 1

since the bound falls in a nice wide moat in both cases.

As of right now, though, I'm planning to use,

error/10 + complexity/23.5 < 1

partly due to Dave pleading against exponents smaller than 1. No more 
moats, which is OK since I don't really have the space for those 
graphs anyway.

This change only adds two to the 7-limit list and one to the 5-limit 
list, without taking any away.

The units are error = max. over all ratios of (cents error)/lg2(n*d),
complexity = L1 Tenney multival norm in the 7-limit case, but L1 
Tenney multival norm multiplied by 2 in the 5-limit case (I can't 
really justify that, but the multival has twice as many entries in 
the 7-limit case). This would appear to penalize complex 5-limit 
temperaments, but Orwell and Amity still make it in.


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Message: 10926

Date: Sat, 08 May 2004 00:40:14

Subject: Re: Request for Gene

From: Paul Erlich

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

> Are you giving names to all the temperaments you plan on tabulating,
> and if so, which names?

I guess this is all up for grabs at the moment. I'm 
using "semifourths" instead of "hemifourths" because generating 
by "semifourths" was already mentioned in an XH article. The 5-limit 
temperament where the Pythagorean comma vanishes will probably be 
called Compton since, thanks to Carl, Compton's patent is the 
earliest we know of . . . I'm planning to use Hanson only in the 5-
limit, to be true to Larry's intentions . . .


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Message: 10928

Date: Sat, 08 May 2004 05:13:54

Subject: Re: Request for Gene

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
> <gwsmith@s...> 
> > wrote:
> > 
> > > Are you giving names to all the temperaments you plan on 
> tabulating,
> > > and if so, which names?
> > 
> > I guess this is all up for grabs at the moment. I'm 
> > using "semifourths" instead of "hemifourths" because generating 
> > by "semifourths" was already mentioned in an XH article. The 5-
> limit 
> > temperament where the Pythagorean comma vanishes will probably be 
> > called Compton since, thanks to Carl, Compton's patent is the 
> > earliest we know of . . . I'm planning to use Hanson only in the 
5-
> > limit, to be true to Larry's intentions . . .
> 
> The hanson thing doesn't matter since catakleismic/hanson7 isn't 
> being discussed anyway. I could give what I presently have down as 
> names for these 25 temperaments, including switching to compton if 
> you like. 

Sure.

> What theory are you operating under regarding the point beyond 
>which 
> increasing accuracy of tuning no longer makes a practical 
>difference? 

None.

> Another question: are you stopping at the 7-limit?

Yes; you said you couldn't handle the 11-limit case, so I'll save 
that, as well as {2,3,7}, {3,5,7}, etc., for part 2 and future 
supplements. This paper is part 1.


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Message: 10930

Date: Mon, 10 May 2004 16:31:29

Subject: Re: Request for Gene

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote: 
>  
> > Yes; you said you couldn't handle the 11-limit case, so I'll 
save  
> > that, as well as {2,3,7}, {3,5,7}, etc., for part 2 and future  
> > supplements. This paper is part 1. 
>  
> My computer keeps crashing now every 1000 temperaments or so, but I 
> could finish doing 11 limit if your graph was truly crucial; I have 
> in mind buying a new one sometime and seeing if that helps, for 
that 
> matter. I think however that discussing 5 and 7 limit temperaments 
> is enough for one paper,

Yes.

> and that if you go to the 11 limit you 
> should probably continue on to 13.

Yes, in the next paper or two.

Math content is minimal, at John's request.



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Message: 10931

Date: Tue, 11 May 2004 18:37:27

Subject: 41 "Hermanic" 7-limit linear temperaments (was: Re: 114 7-limit temperaments)

From: Paul Erlich

Even though we were using a different complexity measure on January 
27 (L-inf instead of L1), my current list of 25 is quite close to 
this one. So the method can't make that much difference -- maybe the 
Klein constraint Gene was talking about has something to do with 
this. Here's what made it in and what didn't:

IN
>  Number 1 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: 6.5251 comp: 3.562072 err: 1.698521

IN
>  Number 2 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: 7.0687 comp: 4.274486 err: 1.276744

IN
>  Number 3 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: 7.1567 comp: 2.988993 err: 3.106578

IN
>  Number 4 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: 7.8851 comp: 4.630693 err: 1.610469

IN
>  Number 5 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: 8.0970 comp: 2.454561 err: 4.771049

IN
>  Number 6 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: 8.2512 comp: 3.445412 err: 3.582707

IN
>  Number 7 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: 8.3168 comp: 3.785579 err: 3.187309

IN
>  Number 8 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: 8.3374 comp: 3.445412 err: 3.66884

IN
>  Number 9 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: 8.3420 comp: 3.804173 err: 3.187309

IN
>  Number 10 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: 8.4214 comp: 4.045351 err: 2.939961

IN
>  Number 11 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: 8.5260 comp: 5.618543 err: .912904

IN
>  Number 12 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: 8.6400 comp: 4.602303 err: 2.403879

IN
>  Number 13 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: 8.6780 comp: 5.706260 err: .946061

IN
>  Number 14 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: 8.7811 comp: 2.147741 err: 5.870879

IN
>  Number 15 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: 8.9144 comp: 4.295482 err: 3.094040

IN
>  Number 16
>  
>  [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: 8.9422 comp: 4.306766 err: 3.106578

IN
>  Number 17 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: 9.1819 comp: 5.522763 err: 1.698521

IN
>  Number 18 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: 9.1883 comp: 4.909123 err: 2.536420

IN
>  Number 19 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: 9.2912 comp: 2.523719 err: 5.871540

OUT
>  Number 20
>  
>  [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: 9.3161 comp: 4.306766 err: 3.480440

IN
>  Number 21
>  
>  [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: 9.3774 comp: 4.295482 err: 3.557008

OUT
>  Number 22
>  
>  [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: 9.5280 comp: 4.891080 err: 2.900537

OUT
>  Number 23
>  
>  [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: 9.6275 comp: 5.168119 err: 2.624742

IN
>  Number 24 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: 9.7206 comp: 6.309298 err: 1.171542

IN
>  Number 25 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: 9.8358 comp: 6.793166 err: .631014

OUT
>  Number 26 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: 9.8915 comp: 5.162806 err: 2.895855

IN
>  Number 27 -- formerly 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: 10.0002 comp: 4.619353 err: 3.740932

OUT
>  Number 28
>  
>  [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: 10.0368 comp: 4.075900 err: 4.513934

IN
>  Number 29
>  
>  [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: 10.1077 comp: 5.047438 err: 3.268439

IN
>  Number 30 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: 10.1851 comp: 2.173813 err: 7.239629

OUT
> Number 31 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: 10.1855 comp: 6.742251 err: 1.049791

OUT
>  Number 32
>  
>  [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: 10.2131 comp: 6.411729 err: 1.525246

OUT
>  Number 33
>  
>  [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: 10.2154 comp: 5.369353 err: 2.939961

OUT
>  Number 34
>  
>  [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: 10.2188 comp: 5.168119 err: 3.215955

OUT
>  Number 35
>  
>  [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: 10.3332 comp: 3.445412 err: 5.664628

OUT
>  Number 36
>  
>  [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: 10.4461 comp: 3.804173 err: 5.291448

OUT
>  Number 37
>  
>  [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: 10.4972 comp: 4.075900 err: 4.974313

OUT
>  Number 38 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: 10.5077 comp: 6.742251 err: 1.371918

OUT
>  Number 39
>  
>  [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: 10.6719 comp: 5.706260 err: 2.939961

OUT
>  Number 40
>  
>  [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: 10.7036 comp: 3.820609 err: 5.526647

OUT
>  Number 41 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: 10.7079 comp: 6.966993 err: 1.267597


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Message: 10933

Date: Tue, 11 May 2004 19:29:33

Subject: 41 "Hermanic" 7-limit linear temperaments (was: Re: 114 7-limit temperaments)

From: Paul Erlich

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

> IN
> >  Number 17 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: 9.1819 comp: 5.522763 err: 1.698521

This appears, to the best of my fading recollection, to be the 
temperament behind Andrzej Gawel's 19-of-36-equal scale. Does anyone 
have the Mills tuning list archives? Robert Walker only made six or 
so members' posts public:

Mills messages - Contents *

If anyone has more, I'd love to see the results of a search 
for "Gawel".


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Message: 10935

Date: Tue, 11 May 2004 19:55:55

Subject: 41 "Hermanic" 7-limit linear temperaments (was: Re: 114 7-limit temperaments)

From: Carl Lumma

>>> Number 17 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: 9.1819 comp: 5.522763 err: 1.698521
>
>This appears, to the best of my fading recollection, to be the 
>temperament behind Andrzej Gawel's 19-of-36-equal scale. Does
>anyone have the Mills tuning list archives? Robert Walker only
>made six or so members' posts public:
>
>Mills messages - Contents *
>
>If anyone has more, I'd love to see the results of a search 
>for "Gawel".

Some months back I ganked all the stuff on the mills site I could
find.  It isn't much, and the string "gawel" apparently doesn't
appear within.  However, I did find this in my inbox...

"""
>>>One thing your example reminds me of is Andrzej Gawel's
>>>19-of-36-tET scale. Gawel ingeniously took the 7-of-12-tET
>>>diatonic scale and divided each of the six instances of the
>>>generator, 7/12 oct. = 19/12 oct., into a chain of three
>>>sub-generators, 19/36 oct., allowing all six of the ordinary
>>>diatonic triads to be completed as 7-limit tetrads,
>>>and in fact the scale has 14 7-limit tetrads.
>>
>>Wow.  Paul, is this right?
>>0 4 5 6 7 11 12 13 14 18 19 20 21 26 27 28 33 34 35
>
>No, it's
>
>0 2 4 6 8 10 12 14 16 18 19 21 23 25 27 29 31 33 35
"""

-Carl


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Message: 10936

Date: Tue, 11 May 2004 20:24:38

Subject: 41 "Hermanic" 7-limit linear temperaments (was: Re: 114 7-limit temperaments)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:
> >>> Number 17 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: 9.1819 comp: 5.522763 err: 1.698521
> >
> >This appears, to the best of my fading recollection, to be the 
> >temperament behind Andrzej Gawel's 19-of-36-equal scale. Does
> >anyone have the Mills tuning list archives? Robert Walker only
> >made six or so members' posts public:
> >
> >Mills messages - Contents *
> >
> >If anyone has more, I'd love to see the results of a search 
> >for "Gawel".
> 
> Some months back I ganked all the stuff on the mills site I could
> find.  It isn't much, and the string "gawel" apparently doesn't
> appear within.

No, it's too early.

> However, I did find this in my inbox...
> 
> """
> >>>One thing your example reminds me of is Andrzej Gawel's
> >>>19-of-36-tET scale. Gawel ingeniously took the 7-of-12-tET
> >>>diatonic scale and divided each of the six instances of the
> >>>generator, 7/12 oct. = 19/12 oct., into a chain of three
> >>>sub-generators, 19/36 oct., allowing all six of the ordinary

Hmm . . . so it's not quite the same. Gawel might be this one, though:

OUT
> Number 23
>
> [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: 9.6275 comp: 5.168119 err: 2.624742

Thanks, Carl!


> >>>diatonic triads to be completed as 7-limit tetrads,
> >>>and in fact the scale has 14 7-limit tetrads.
> >>
> >>Wow.  Paul, is this right?
> >>0 4 5 6 7 11 12 13 14 18 19 20 21 26 27 28 33 34 35
> >
> >No, it's
> >
> >0 2 4 6 8 10 12 14 16 18 19 21 23 25 27 29 31 33 35
> """
> 
> -Carl


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Message: 10937

Date: Tue, 11 May 2004 21:01:12

Subject: Re: Request for Gene

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> > Thanks again for this, Gene.
> > 
> > 
> > Would it be too much trouble to also do
> > 
> > [8, 6, 6, -9, -13, -3]
> 
> 390625/373248, 5971968/5764801, 50/49, 875/864, 1728/1715
> 
> > and
> > 
> > [6, -2, -2, -17, -20, 1]
> 
> 140625/131072, 525/512, 50/49, 1029/1024

Very interesting. Now, how about:

[3, 12, 11, 12, 9, -8] (Gawel?)
[6, 10, 3, 2, -12, -21]
[2, -9, -4, -19, -12, 16]

The idea is that I'm moving my boundary out to

error/10 + complexity/24 < 1

This adds these three 7-limit temperaments, for a total of 28. The 
number of 5-limit temperaments remains at 21. If I include 
ennealimmal as a "bonus" temperament, that's 50 altogether -- "Fifty 
Temperaments" will make a nice subtitle for the paper. 10, 24, and 50 
are all nice, round numbers. Finality approaches.



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Message: 10939

Date: Wed, 12 May 2004 05:22:41

Subject: Re: Request for Gene

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > Very interesting. Now, how about:
> > 
> > [3, 12, 11, 12, 9, -8] (Gawel?)
> 
> TM basis: {81/80, 686/675}
> commas: [1029/1000, 686/675, 81/80, 177147/175616, 10976/10935]
> 
> > [6, 10, 3, 2, -12, -21]
> 
> TM basis: {49/48, 250/243}
> commas: [282475249/262144000, 250/243, 49/48, 4000/3969]
> 
> > [2, -9, -4, -19, -12, 16]
> 
> TM basis: {64/63, 686/675}
> commas: [524288/492075, 6272/6075, 686/675, 64/63, 2401/2400}
> 
> > The idea is that I'm moving my boundary out to
> 
> > error/10 + complexity/24 < 1
> 
> Moving to a fixed exponent after all!

Huh? How does changing 23.5 to 24 make it a fixed exponent?

> > This adds these three 7-limit temperaments, for a total of 28. 
The 
> > number of 5-limit temperaments remains at 21. If I include 
> > ennealimmal as a "bonus" temperament, that's 50 altogether --
 "Fifty 
> > Temperaments" will make a nice subtitle for the paper. 10, 24, 
and 50 
> > are all nice, round numbers. Finality approaches.
> 
> I think ennealimmal is so striking that discussing it anyway makes
> sense--while your exponent favors low complexity over low error,

I have no idea what you're talking about. What do you mean?


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Message: 10941

Date: Wed, 12 May 2004 20:17:15

Subject: Re: Request for Gene

From: Paul Erlich

How do you calculate if a given comma vanishes in the temperament 
represented by a given wedgie?


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Message: 10942

Date: Wed, 12 May 2004 20:55:50

Subject: Adding wedgies?

From: Paul Erlich

What's going on here?

[5, 13, -17, 9, -41, -76], TOP error 0.27611
"plus"
[13, 14, 35, -8, 19, 42], TOP error 0.26193
"equals"
[18, 27, 18, 1, -22, -34], TOP error 0.036378

I think Gene talked about this before but I didn't quite catch on 
then.

BTW, these are the three most accurate 7-limit temperaments with L1 
multival complexity less than 40 (though it's over 38 for each of the 
three).


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Message: 10943

Date: Wed, 12 May 2004 21:09:24

Subject: Re: Adding wedgies?

From: Carl Lumma

>What's going on here?
>
>[5, 13, -17, 9, -41, -76], TOP error 0.27611
>"plus"
>[13, 14, 35, -8, 19, 42], TOP error 0.26193
>"equals"
>[18, 27, 18, 1, -22, -34], TOP error 0.036378
>
>I think Gene talked about this before but I didn't quite catch on 
>then.
>
>BTW, these are the three most accurate 7-limit temperaments with
>L1 multival complexity less than 40 (though it's over 38 for each
>of the three).

Boy, Paul, I am sure looking forward to your paper!!

This reminds me that my 'tuning-math forms' project is in limbo.
I have all the materials collected... just gotta find time to
do it... (if anybody wants to take a stab I'll make the materials
available...)

-Carl


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Message: 10944

Date: Wed, 12 May 2004 21:15:47

Subject: Re: Adding wedgies?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:
> >What's going on here?
> >
> >[5, 13, -17, 9, -41, -76], TOP error 0.27611
> >"plus"
> >[13, 14, 35, -8, 19, 42], TOP error 0.26193
> >"equals"
> >[18, 27, 18, 1, -22, -34], TOP error 0.036378
> >
> >I think Gene talked about this before but I didn't quite catch on 
> >then.
> >
> >BTW, these are the three most accurate 7-limit temperaments with
> >L1 multival complexity less than 40 (though it's over 38 for each
> >of the three).
> 
> Boy, Paul, I am sure looking forward to your paper!!

Other than ennealimmal, which is a "bonus" temperament, my paper 
stays below complexity < 24 -- and I doubt I'll be talking about 
adding wedgies. So don't get your hopes up.

> This reminds me that my 'tuning-math forms' project is in limbo.

What's that?


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Message: 10945

Date: Wed, 12 May 2004 22:01:06

Subject: Re: Adding wedgies?

From: Carl Lumma

>> Boy, Paul, I am sure looking forward to your paper!!
>
>Other than ennealimmal, which is a "bonus" temperament, my paper 
>stays below complexity < 24 -- and I doubt I'll be talking about 
>adding wedgies. So don't get your hopes up.

Oh, I suspected as much.  I'm looking forward to it for other
reasons!

>> This reminds me that my 'tuning-math forms' project is in limbo.
>
>What's that?

Remember, the one where I show how to calculate this stuff on
paper, 'long division' style?  By "materials" I just meant the
relevant posts from you, Gene, and Dave.  If Gene answers this
thread it will probably get included as well.

-Carl



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Message: 10946

Date: Thu, 13 May 2004 21:01:13

Subject: Re: 22 7-limit temperaments in the upper uv quadrant

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > We were so busy arguing that we didn't notice how very close our 
> > lists were.
> 
> It's hardly the case that I didn't notice that.

Well, that makes a lot more sense.

> Do you began to 
> understand why I became so frustrated? I still can't figure out why 
> it all blew up the way it did; normally, we communicate better than 
> this.
> 
> > So what exactly led to your results here? I can't understand how 
> you 
> > arrived at them.
> 
> My idea was to get something closer to what people seemed to want, 
> two exponents could be used; this could be smoothed out if we used 
a 
> hyperbolic boundry in the log-log plane. In order to accomodate two 
> exponents, making them the vertical and horizontal axis seemed like 
a 
> good plan. This posting follows up a previous one which explains 
all 
> of that.

I still don't see the intuition behind it. Could you draw some 
pictures to help?


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Message: 10947

Date: Thu, 13 May 2004 00:23:25

Subject: Re: Adding wedgies?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> > What's going on here?
> > 
> > [5, 13, -17, 9, -41, -76], TOP error 0.27611
> > "plus"
> > [13, 14, 35, -8, 19, 42], TOP error 0.26193
> > "equals"
> > [18, 27, 18, 1, -22, -34], TOP error 0.036378
> 
> Parakleismic + Amity = Ennealimmal. Both parakleismic and amity have
> 4375/4374 as a comma, and so does their sum (and difference, for 
that
> matter.)
> 
> I did talk about it before, though I can't recall what I said about
> it. It is related to the Klein stuff. For 7-limit wedgies, define 
the
> Pfaffian as follows: let
> 
> X = <<x1 x2 x3 x4 x5 x6||
> Y = <<y1 y2 y3 y4 y5 y6||
> 
> Then 
> 
> Pf(X, Y) = y1x6 + x1y6 - y2x5 - x2y5 + y3x4 + x3y4
> 
> It is easily checked that we have the identity
> 
> Pf(X+Y, X+Y) = Pf(X, X) + 2 Pf(X, Y) + Pf(Y, Y)
> 
> The Klein condition for the wedgie X is Pf(X, X)=0. If X and Y both
> satisfy the Klein condition, and if Pf(X, Y)=0, then X+Y also
> satisfies the Klein condition, and hence is a wedgie. What 
> Pf(X, Y)=0 means is that X and Y are related; they share a comma.

It seems below that the unshared commas, correspondingly, "add" 
(actually multiply).

But how do you simply check whether a particular wedgie eats a 
particular comma?


81:80 shared:

Meantone + DominantSevenths = Injera
126:125 "+" 35:36 "=" 49:50

Meantone + Catler = DominantSevenths 
126:125 "+" 125:128 "=" 63:64

Meantone + Flattone = Semifourths
225:224 "+" 512:525 "=" 48:49
126:125 "+" 4375:4374 "=" 245:243
126:125 "+" 875:846 "=" 49:48

Injera + Flattone = SupermajorSeconds
50:49 "+" 512:525 "=" 1024:1029
50:49 "+" 864:875 "=" 1728:1715
etc.


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Message: 10949

Date: Thu, 13 May 2004 03:40:28

Subject: Re: Vanishing tratios

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > Hmm . . . the two formulae seem to give the same result as long 
as gcd
> > (a,b,c) = 1, which was the case for all the tratios in question. 
> > Right?
> 
> Right.

So how can we express the tratio, or the lcm, in terms of the wedgie?


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