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Message: 10950 - Contents - Hide Contents

Date: Thu, 13 May 2004 22:43:02

Subject: Re: tratios and yantras

From: Paul Erlich

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

> I took all the 7-limit integers less than 2^20 (recall this is a
> yantra when reduced to an octave) and found the smallest instance of
> three successive 7-limit integers which were mapped to the same val
> when wedged with an 7-limit wedgie, in order to get tratios for some
> of the most important 7-limit linear temperaments. Here's the 
results:
> 
> meantone 1120:1125:1134
> miracle 7168:7200:7203
> ennealimmal 419904:420000:420175
> magic 6048:6075:6125
> pajara 441:448:450
> (septimal) schismic 27783:28000:28125
> orwell 12005:12096:12150


I did something a little more straightforward. I calculated the 
wedgie for every triplet of consecutive 7-limit integers, through 
99999. Then I just looked for the earliest occurence of each wedgie. 
I found a lot more, though a few wedgies apparently don't show up at 
all. There are a couple of improvements over the results I posted in

Yahoo groups: /tuning-math/message/10408 * [with cont.] 

Meantone 1120:1125:1134 
Magic 6048:6075:6125  
Pajara 441:448:450 
Semisixths 3375:3402:3430 
Dominant Seventh 315:320:324
-- larger numbers than 245:252:256, but lcm much smaller at 181,440
Injera 392:400:405
OldKleismic 864:875:882
-- smaller numbers than 1000:1008:1029, lcm smaller at 5,292,000
Semifourths 240:243:245
Negri 672:675:686
Tripletone 125:126:128
Schismic 27783:28000:28125
Superpythagorean 1701:1715:1728
Orwell 12005:12096:12150
Augmented NOT FOUND
Porcupine NOT FOUND
<<6, 10, 10, 2, -1, -5]] 243:245:250 
Supermajor seconds 5120:5145:5184 
Flattone 2560:2592:2625 
Diminished 245:250:252
<<6, 10, 3, 2, -12, -21]] NOT FOUND
Catler NOT FOUND
Gawel 10800:10935:10976
Nonkleismic 12000:12005:12096 
Miracle 7168:7200:7203
Beatlemania 4725:4800:4802
<<6, -2, -2, -17, -20, 1]] 1024:1029:1050 
<<8, 6, 6, -9, -13, -3]] 1715:1728:1750
Blackwood 189:192:196

Now for 5-limit ETs, one semi-improvement over

Yahoo groups: /tuning-math/message/10404 * [with cont.] 

3-equal 27:30:32
-- smaller numbers than 45:48:50
4-equal 24:25:27
5-equal 75:80:81
7-equal 240:243:250
9-equal 125:128:135
10-equal 16200:16384:16875 YUCK!
12-equal 625:640:648
15-equal 243:250:256
16-equal NOT FOUND
19-equal 15360:15552:15625
22-equal 6075:6144:6250
New ones:
27-equal 78125:78732:80000 lcm 196,830,000,000
29-equal 32768:32805:33750 lcm 134,369,280,000



________________________________________________________________________
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Message: 10951 - Contents - Hide Contents

Date: Thu, 13 May 2004 04:19:47

Subject: Re: Adding wedgies?

From: Gene Ward Smith

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


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


Consider the wedgie in multimonzo form, and wedge with the monzo; if
you get a zero multimonzo, you had a comma.

Message: 10952 - Contents - Hide Contents

Date: Thu, 13 May 2004 04:22:00

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:
> 

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

> Consider the wedgie in multimonzo form, and wedge with the monzo; if
> you get a zero multimonzo, you had a comma.


I figured that. But how do you wedge a monzo with a multimonzo?

Message: 10953 - Contents - Hide Contents

Date: Thu, 13 May 2004 04:24:19

Subject: Re: Adding wedgies?

From: Paul Erlich

Aha -- does the klein condition equate with the bivector being a 
simple bivector?

--- 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.
> 
> Probably, you will not want to talk about this in the paper. :)

Message: 10954 - Contents - Hide Contents

Date: Thu, 13 May 2004 04:24:19

Subject: Re: Adding wedgies?

From: Gene Ward Smith

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


>> Consider the wedgie in multimonzo form, and wedge with the monzo; if
>> you get a zero multimonzo, you had a comma.
> 

> I figured that. But how do you wedge a monzo with a multimonzo?


It's just a special case of the general definition. If you mean me
personally, I have Maple programs written to do it.

Message: 10955 - Contents - Hide Contents

Date: Thu, 13 May 2004 04:26:00

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:
> 

>>> Consider the wedgie in multimonzo form, and wedge with the 
monzo; if
>>> you get a zero multimonzo, you had a comma.
>> 

>> I figured that. But how do you wedge a monzo with a multimonzo?
> 

> It's just a special case of the general definition. If you mean me
> personally, I have Maple programs written to do it.


Is it a simple expression in terms of determinants?

Message: 10956 - Contents - Hide Contents

Date: Thu, 13 May 2004 05:23:02

Subject: Re: tratios and yantras

From: Paul Erlich

I'm adding LCMs below, though i still don't know if the numbers 
and/or LCMs are smallest possible. It's interesting that the LCMs 
seem to give a much better ranking of complexity than the size of the 
numbers. Of course we're essentially using LCM (=n*d) already in the 
one-comma case. We should be able to visualize this, as well as the 
L1 (and maybe L-inf??) complexity measures, as specific sorts 
of "area" on the lattice, though 4-D is kind of hard. Need help, 
please . . .

Of course if you divide the lcm by the numbers in the tratio you get 
another tratio with the same LCM, representing the same temperament, 
but I think always with larger numbers (if you start with one of 
these):


7-limit Blackwood 243:252:256 lcm 435,456
Dominant Sevenths 245:252:256 lcm 564,480
7-limit Diminished 343:350:360 lcm 617,400
Pajara 441:448:450 lcm 705,600
Semifourths 240:243:245 lcm 952,560
Tripletone 125:126:128 lcm 1,008,000
Injera 392:400:405 lcm 1,587,600
meantone 1120:1125:1134 lcm 2,268,000
7-limit Augmented:
1125:1152:1176 lcm 7,056,000
BUT ALSO
1568:1575:1620 lcm 3,175,200
(any simpler tratio?)
OldKleismic 1000:1008:1029 lcm 6,174,000
Catler -- same as 12-equal -- 625:648:640 lcm 6,480,000
Negri 672:675:686 lcm 7,408,800
semisixths 3375:3402:3430 lcm 20,837,250
Superpythagorean 1701:1715:1728 lcm 26,671,680
magic 6048:6075:6125 lcm 47,628,000
miracle 7168:7200:7203 lcm 553,190,400
orwell 12005:12096:12150 lcm 933,508,800
(septimal) schismic 27783:28000:28125 lcm 2,778,300,000
ennealimmal 419904:420000:420175 lcm 4,410,829,080,000

What's bigger -- the lcm of ennealimmal or the national debt?

Message: 10957 - Contents - Hide Contents

Date: Thu, 13 May 2004 05:25:40

Subject: Re: tratios and yantras

From: Paul Erlich

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

> I'm adding LCMs below, though i still don't know if the numbers 
> and/or LCMs are smallest possible. It's interesting that the LCMs 
> seem to give a much better ranking of complexity than the size of 
the 
> numbers. Of course we're essentially using LCM (=n*d) already in 
the 
> one-comma case. We should be able to visualize this, as well as the 
> L1 (and maybe L-inf??) complexity measures, as specific sorts 
> of "area" on the lattice, though 4-D is kind of hard. Need help, 
> please . . .
> 
> Of course if you divide the lcm by the numbers in the tratio you 
get 
> another tratio with the same LCM, representing the same 
temperament, 
> but I think always with larger numbers (if you start with one of 
> these):
> 
> 
> 7-limit Blackwood 243:252:256 lcm 435,456
> Dominant Sevenths 245:252:256 lcm 564,480
> 7-limit Diminished 343:350:360 lcm 617,400
> Pajara 441:448:450 lcm 705,600
> Semifourths 240:243:245 lcm 952,560
> Tripletone 125:126:128 lcm 1,008,000
> Injera 392:400:405 lcm 1,587,600
> meantone 1120:1125:1134 lcm 2,268,000
> 7-limit Augmented:
> 1125:1152:1176 lcm 7,056,000
> BUT ALSO
> 1568:1575:1620 lcm 3,175,200
> (any simpler tratio?)
> OldKleismic 1000:1008:1029 lcm 6,174,000
> Catler -- same as 12-equal -- 625:648:640 lcm 6,480,000
> Negri 672:675:686 lcm 7,408,800
> semisixths 3375:3402:3430 lcm 20,837,250
> Superpythagorean 1701:1715:1728 lcm 26,671,680
> magic 6048:6075:6125 lcm 47,628,000
> miracle 7168:7200:7203 lcm 553,190,400
> orwell 12005:12096:12150 lcm 933,508,800
> (septimal) schismic 27783:28000:28125 lcm 2,778,300,000
> ennealimmal 419904:420000:420175 lcm 4,410,829,080,000
> 
> What's bigger -- the lcm of ennealimmal or the national debt?


. . . on Sept. 30th, 1993?

Message: 10958 - Contents - Hide Contents

Date: Thu, 13 May 2004 06:01:24

Subject: Tratios vs. wedgies

From: Paul Erlich

I think in the codimension-2 case, the log of the tratio lcm 
complexity will be between 1 and 4 times the wedgie complexity (duly 
scaled). Proof or disproof?

Message: 10959 - Contents - Hide Contents

Date: Thu, 13 May 2004 06:03:10

Subject: L1 multival norm complexity

From: Paul Erlich

I'm tired of writing this over and over again, so I'll call 
it 'genie' from now on (based on a suggestion from Aaron Johnston. 
It's the one with the funny coincidences with number of notes in the 
stereotypical scale.

Message: 10960 - Contents - Hide Contents

Date: Thu, 13 May 2004 06:27:15

Subject: Re: Tratios vs. wedgies

From: Paul Erlich

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

> I think in the codimension-2 case, the log of the tratio lcm 
> complexity will be between 1 and 4 times the wedgie complexity 
(duly 
> scaled). Proof or disproof?


I no longer think it's true, but this sort of thing merits 
investigation. How do you express the lcm in terms of the wedgie? etc.

Message: 10961 - Contents - Hide Contents

Date: Thu, 13 May 2004 07:13:23

Subject: Comparison of LOG-genie vs. tratio-LOG-lcm, 5-limit ETs

From: Paul Erlich

As we've seen, in this case genie comes out to about 3 times the 
number of notes per octave. So the comparison is essentially with 
number of notes.

Again:


> ET........tratio............lcm.............
> 03-equal: 45:48:50......... 3,600...........
> 04-equal: 24:25:27......... 5,400...........
> 05-equal: 75:80:81......... 32,400..........
> 07-equal: 384:400:405...... 259,200.........
> ("""""""""240:243:250...... 486,000).......(
> 09-equal: 125:128:135...... 432,000.........
> 10-equal: 729:768:800...... 4,665,600.......
> 12-equal: 625:640:648...... 6,480,000.......
> 15-equal: 243:250:256...... 7,776,000.......
> 16-equal: 3072:3125:3240... 259,200,000.....
> 19-equal: 15360:15552:15625 3,888,000,000  
> 22-equal: 6075:6144:6250... 1,555,200,000
> 
> The monotonic pattern seems to break here. Did I miss any lower-
weird 
> and/or simpler tratios?


The next step is to draw a picture.

Message: 10962 - Contents - Hide Contents

Date: Thu, 13 May 2004 07:23:49

Subject: 7-limit temperament and the algebra of spacetime?

From: Paul Erlich

The Algebra of Spacetime * [with cont.]  (Wayb.)

I guess I shouldn't have used the term "multival" -- multivector 
means a sum of elements of different grade . . .

Message: 10963 - Contents - Hide Contents

Date: Thu, 13 May 2004 08:12:36

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

From: Paul Erlich

Now updating to include last 3 additions:

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

> 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

Now IN!!
> 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


Gawel -- IN!!!

> 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

Now IN!!!!!Shelovesyouyeahyeahyeah
> 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


That's all, folks! All 28 I'm including were in the top 30 of 
the "Hermanic" list I posted 3 and a half months ago . . .

Message: 10964 - Contents - Hide Contents

Date: Thu, 13 May 2004 08:19:24

Subject: Re: Comparison of LOG-genie vs. tratio-LOG-lcm, 5-limit ETs

From: Paul Erlich

Sorry, wrong subject line.

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

> As we've seen, in this case genie comes out to about 3 times the 
> number of notes per octave. So the comparison is essentially with 
> number of notes.
> 
> Again:
> 
>> ET........tratio............lcm.............

>> 03-equal: 45:48:50......... 3,600...........
>> 04-equal: 24:25:27......... 5,400...........
>> 05-equal: 75:80:81......... 32,400..........
>> 07-equal: 384:400:405...... 259,200.........
>> ("""""""""240:243:250...... 486,000).......(
>> 09-equal: 125:128:135...... 432,000.........
>> 10-equal: 729:768:800...... 4,665,600.......
>> 12-equal: 625:640:648...... 6,480,000.......
>> 15-equal: 243:250:256...... 7,776,000.......
>> 16-equal: 3072:3125:3240... 259,200,000.....
>> 19-equal: 15360:15552:15625 3,888,000,000  
>> 22-equal: 6075:6144:6250... 1,555,200,000
>> 
>> The monotonic pattern seems to break here. Did I miss any lower-
> weird 

>> and/or simpler tratios?
> 

> The next step is to draw a picture.

Message: 10965 - Contents - Hide Contents

Date: Thu, 13 May 2004 09:44:11

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

From: Paul Erlich

Gene posted this on Feb. 11th (click "up thread" or see below).

I have no idea what you did here, Gene, but this list is awfully 
close to my Feb. 8th list of 23:

Yahoo groups: /tuning-math/message/9317 * [with cont.] 

that led to my current list.

Only #21 (SupermajorSeconds) and #22 (Nonkleismic) of the 23 in my 
list are omitted, and only Ennealimmal is added.

We were so busy arguing that we didn't notice how very close our 
lists were.

So what exactly led to your results here? I can't understand how you 
arrived at them.

Your list has four distinct domains. All temperaments in domain n+1 
are both more accurate and more complex than any temperament in 
domain n.

Domain 1: Blackwood, Diminished, Augmented, DominantSevenths
Domain 2: Everything on your list below that's not in other domains. 
The five temperaments I've just added, including Gawel, can be put in 
this domain too, without ruining the domains property above
Domain 3: Orwell, Schismic, Miracle
Domain 4: Ennealimmal


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

> I found 22 of these; this is probably all that exist. I list the
> temperament and the product uv. The list can be used as is, but if
> Dave wants rounded corners, we could fix a value for uv and only
> accept temperaments above it. Paul may dislike the unboundedness of
> the upper quadrant; we can fix that by transforming coordinates back
> to complexity and error. We have
> 
> C = exp((8 + u - v)/3)
> 
> E = exp((4 - 4u - v)/3)
> 
> If we use the cuttoff hyperbola v = k/u, then we may plot that in 
the
> C-E plane in parametric form as
> 
> C(u) = exp((8 + u - k/u))
> E(u) = exp((4 - 4u - k/u))
> 
> This gives a curved line in an unbounded region, where zero error 
and
> complexity (though not any temperaments exhibiting them) may be 
found.
> 
> Other objections might be that you don't find your favorite
> temperament, or you do find one you can't stand. I'm not too 
impressed
> by the second, but for either, or if you think you see a "moat"
> somewhere, you have a 5-parameter family to play with--two for the
> origin, two for the slopes of the coordinate axes, and one for the
> constant k in the hyperbola. That should accomodate anyone's desire 
to
> fiddle, or even cook the books.
> 
> Are there any remaining objections I have not answered above?
> 
> Ennealimmal
> 1 <18, 27, 18, 1, -22, -34| <3.629230331, .575465612| 2.088497
> 
> Meantone
> 2 <1, 4, 10, 4, 13, 12| <1.005097996, 1.609665600| 1.617872
> 
> Magic
> 3 <5, 1, 12, -10, 5, 25| <1.012524503, .7830372729| .792844
> 
> Pajara
> 4 <2, -4, -4, -11, -12, 2| <.524469574, 1.498444023| .785888
> 
> Dominant seventh
> 5 <1, 4, -2, 4, -6, -16| <.363511386, 2.141744135| .778548
> 
> Semisixths
> 6 <7, 9, 13, -2, 1, 5| <.8521893702, .8383344292| .714420
> 
> Tripletone
> 7 <3, 0, -6, -7, -18, -14| <.426316706, .940456192| .400932
> 
> Blackwood
> 8 <0, 5, 0, 8, 0, -14| <.152491081, 2.548674345| .388650
> 
> Miracle
> 9 <6, -7, -2, -25, -20, 15| <1.411065061, .2629785497| .371080
> 
> Diminished
> 10 <4, 4, 4, -3, -5, -2| <.160875338, 1.953852436| .314327
> 
> Negri
> 11 <4, -3, 2, -14, -8, 13| <.345586529, .859876933| .297162
> 
> Hemifourths
> 12 <2, 8, 1, 8, -4, -20| <.283811920, 1.034875923| .293710
> 
> Kleismic
> 13 <6, 5, 3, -6, -12, -7| <.322424097, .767227201| .247373
> 
> Superpythagorean
> 14 <1, 9, -2, 12, -6, -30| <.4535440254, .4454275914| .202021
> 
> Injera
> 15 <2, 8, 8, 8, 7, -4| <.245867248, .811824633| .199601
> 
> Augmented
> 16 <3, 0, 6, -7, 1, 14| <.113185821, 1.762756409| .199519
> 
> "Number 43" {50/49, 245/243} Supermajor?
> 17 <6, 10, 10, 2, -1, -5| <.287044794, .548744901| .157514


This is "Biporky", not "Supermajor Seconds".


> "Number 55" {81/80, 128/125} Duodecatonic?
> 18 <0, 0, 12, 0, 19, 28| <.178510293, .520800106| .092968
> 
> Orwell
> 19 <7, -3, 8, -21, -7, 27| <1.060730636, .7657743908e-1| .081228
> 
> Schismic
> 20 <1, -8, -14, -15, -25, -10| <1.080908962, .5026039173e-1| .054327
> 
> Flattone
> 21 <1, 4, -9, 4, -17, -32| <.3364294610, .1379787620| .046420
> 
> Porcupine
> 22 <3, 5, -6, 1, -18, -28| <.176167209, .93101647e-1| .016401

Message: 10966 - Contents - Hide Contents

Date: Thu, 13 May 2004 10:04:36

Subject: What's

From: Paul Erlich

For some reason, I can't find it on Gene's big list of 32,201 of 
these wedgies. Am I going crazy? Why isn't it there? Its LCM is 
comparable to the national debt on Sept. 30, 1994, only one year 
later than ennealimmal.

Message: 10967 - Contents - Hide Contents

Date: Thu, 13 May 2004 10:19:35

Subject: Re: tratios and yantras

From: Paul Erlich

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


> 7-limit Blackwood 243:252:256 lcm 435,456


I just found this for Blackwood:

189:192:196 lcm 84,672

I guess "by inspection" doesn't cut it.

I'll have to do this systematically.

Here's my Matlab program to convert from tratios to wedgies:

function out=trat2wedg(a1,a2,a3);
j=factor(a1);
a12=sum(j==2);
a13=sum(j==3);
a15=sum(j==5);
a17=sum(j==7);
j=factor(a2);
a22=sum(j==2);
a23=sum(j==3);
a25=sum(j==5);
a27=sum(j==7);
j=factor(a3);
a32=sum(j==2);
a33=sum(j==3);
a35=sum(j==5);
a37=sum(j==7);
w=([det([a22-a12 a23-a13;a22-a32 a23-a33]) det([a22-a12 a25-a15;a22-
a32 a25-a35]) det([a22-a12 a27-a17;a22-a32 a27-a37]) det([a23-a13 a25-
a15;a23-a33 a25-a35]) det([a23-a13 a27-a17;a23-a33 a27-a37]) det([a25-
a15 a27-a17;a25-a35 a27-a37])]);
g=gcd(w(1),gcd(w(2),gcd(w(3),gcd(w(4),gcd(w(5),w(6))))));
out=w/g;



> Dominant Sevenths 245:252:256 lcm 564,480
> 7-limit Diminished 343:350:360 lcm 617,400
> Pajara 441:448:450 lcm 705,600
> Semifourths 240:243:245 lcm 952,560
> Tripletone 125:126:128 lcm 1,008,000
> Injera 392:400:405 lcm 1,587,600
> meantone 1120:1125:1134 lcm 2,268,000
> 7-limit Augmented:
> 1125:1152:1176 lcm 7,056,000
> BUT ALSO
> 1568:1575:1620 lcm 3,175,200
> (any simpler tratio?)
> OldKleismic 1000:1008:1029 lcm 6,174,000
> Catler -- same as 12-equal -- 625:648:640 lcm 6,480,000
> Negri 672:675:686 lcm 7,408,800
> semisixths 3375:3402:3430 lcm 20,837,250
> Superpythagorean 1701:1715:1728 lcm 26,671,680
> magic 6048:6075:6125 lcm 47,628,000
> miracle 7168:7200:7203 lcm 553,190,400
> orwell 12005:12096:12150 lcm 933,508,800
> (septimal) schismic 27783:28000:28125 lcm 2,778,300,000
> ennealimmal 419904:420000:420175 lcm 4,410,829,080,000
> 
> What's bigger -- the lcm of ennealimmal or the national debt?

Message: 10968 - Contents - Hide Contents

Date: Thu, 13 May 2004 10:20:59

Subject: Re: tratios and yantras

From: Paul Erlich

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


> I'll have to do this systematically.


But now I must sleep . . .

Message: 10969 - Contents - Hide Contents

Date: Thu, 13 May 2004 11:14:33

Subject: Re: What's

From: Graham Breed

Paul Erlich wrote:

> For some reason, I can't find it on Gene's big list of 32,201 of 
> these wedgies. Am I going crazy? Why isn't it there? Its LCM is 
> comparable to the national debt on Sept. 30, 1994, only one year 
> later than ennealimmal.


should that be <<20, -15, 0, -70, -56, 42]]?

1/2, 113.7 cent generator

basis:
(0.20000000000000001, 0.094719200833399242)

mapping by period and generator:
[(5, 0), (6, 4), (13, -3), (14, 0)]

mapping by steps:
[(5, 5), (-2, 2), (19, 16), (14, 14)]

highest interval width: 7
complexity measure: 35  (45 for smallest MOS)
highest error: 0.007355  (8.826 cents)
unique

                 graham

Message: 10970 - Contents - Hide Contents

Date: Thu, 13 May 2004 11:16:22

Subject: Re: How about

From: Graham Breed

<<4, 6, 6, 0, -2, -3]] gives

0/1, 277.1 cent generator

basis:
(0.5, 0.23088169264746539)

mapping by period and generator:
[(2, 0), (4, -2), (6, -3), (7, -3)]

mapping by steps:
[(2, 0), (-2, 2), (-3, 3), (-2, 3)]

highest interval width: 3
complexity measure: 6  (10 for smallest MOS)
highest error: 0.046726  (56.071 cents)


                Graham

Message: 10971 - Contents - Hide Contents

Date: Thu, 13 May 2004 19:09:42

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

From: Gene Ward Smith

--- 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. 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.




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Message: 10972 - Contents - Hide Contents

Date: Sat, 15 May 2004 17:55:14

Subject: Re: tratios and yantras

From: monz

hi Gene and Paul,


definitions of tratio and yantra, please.
thanks.



-monz





 



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Message: 10974 - Contents - Hide Contents

Date: Wed, 19 May 2004 21:20:00

Subject: Re: tratios and yantras

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> hi Gene and Paul,
> 
> 
> definitions of tratio and yantra, please.
> thanks.
> 
> 
> 
> -monz

Hi Monz,


Gene defined yantra here:

Yahoo groups: /tuning-math/message/10095 * [with cont.] 

It's just the first N integers with no prime factors above P, for a 
given N and P.

Tratio was something you posted to tell me I should use the 
terminology "proportion" for -- remember? If a temperament has 
codimension 1, it can be described by a vanishing ratio. If a 
temperament has codimension 2, it can be described by a vanishing 
tratio (a three-term proportion, like 625:640:648).

Apparently, many, but not all, of the important codimension-1 
temperaments can be described by vanishing tratios of three 
consective terms in the yantra (for which P is either 5 or 7, and N 
is infinite or simply "large enough").

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