Tuning-Math Digests messages 9050 - 9074

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Message: 9050

Date: Thu, 08 Jan 2004 22:59:52

Subject: Re: non-1200: Tenney/heursitic meantone temperament

From: Graham Breed

Paul Erlich wrote:

> Wow. How did you find that?

Briefly (use the Reply thing so that indentation works),

Python 2.2.2 (#2, Feb  5 2003, 10:40:08)
[GCC 3.2.1 (Mandrake Linux 9.1 3.2.1-5mdk)] on linux-i386
Type "copyright", "credits" or "license" for more information.
IDLE 0.8 -- press F1 for help
 >>> import temper
 >>> h12 = temper.BestET(12,temper.limit5)
 >>> ~temper.Wedgie(h12)^temper.WedgableRatio((81,80))
{}
 >>> ~temper.Wedgie(h12)^temper.WedgableRatio((128,125))
{}
 >>> temper.factorizeRatio(128,125)
(7, 0, -3, 0, 0, 0, 0, 0)
 >>> def badness(a,b,c):
	return (a + b*temper.log2(3) + c*temper.log2(5)) / (
		abs(a) + abs(b)*temper.log2(3) + abs(c)*temper.log2(5))
 >>> for i in range(20):
	for j in range(-20,20):
		if i or j:
			print "%i %i %f" %(
				i, j, badness(
					7*j - 4*i,
					4*i,
					-i - 3*j))
0 -20 -0.002450
0 -19 -0.002450
0 -18 -0.002450
0 -17 -0.002450
0 -16 -0.002450
0 -15 -0.002450
0 -14 -0.002450
...
7 -2 0.000643
7 -1 0.001029
7 0 0.001415
7 1 0.001802
7 2 0.002189
7 3 0.002577
7 4 0.002964
7 5 0.002894
7 6 0.002841
...
19 15 0.002864
19 16 0.002846
19 17 0.002829
19 18 0.002813
19 19 0.002799
 >>> badness(0,28,-19)
0.0029641729677381511
 >>> psize = 28*temper.log2(3) - 19 * temper.log2(5)
 >>> worstbad = 0.0
 >>> for i in range(100):
	for j in range(-100,100):
		if i or j:
			worstbad=max(worstbad,
				abs(badness(
					7*j - 4*i,
					4*i,
					-i - 3*j)))
			
 >>> worstbad
0.0029641729677381628
 >>> worstbad/badness(0,28,-19)
1.000000000000004
 >>> (1+worstbad) * temper.log2(5)/28*12
0.99806172487682532
 >>> (1-worstbad) * temper.log2(3)/19*12
0.99806172487682532
 >>> print '%i:%i'%(3**28, 5**19)
22876792454961:19073486328125

>>TOPping it gives a narrow octave of 0.99806 2:1 octaves.
> 
> 
> Shall I proceed to calculate Tenney-weighted errors for all (well, a 
> bunch of) intervals? I hope you're onto something!

If you like.

> I thought that's what you were talking about in the thread where I 
> brought them up! Odd limit, right?

No, a lattice for octave-equivalent ratios.

> Did someone publish it before? It's currently not Gene's way, anyway.

How about Tenney?


                   Graham


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Message: 9051

Date: Thu, 08 Jan 2004 08:24:44

Subject: Re: TOP and normed vector spaces

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > > For a rational number, vp is the p-adic valuation of number q, 
that 
> > > is,the exponent in the factorization of q into primes. For 
other 
> > > points in the Tenney space it's just a coordinate.
> > 
> > There are no other points in the Tenney space. Anyway, I lost the 
> > train of thought.
> 
> Then why did you correct me to "Tenney space"?

What does that have to do with it? I just meant I lost where I was in 
my thinking when I stopped to ask you about the v's.

> Presumably, I knew what
> space I wanted even if I didn't have the right name for it. There
> *are* other points in my space, and what you seem to be talking 
about
> is a lattice.

OK, why do we need a space and not a lattice?

> > > > > and using the same proceedure we use to get 
> > > > > a unique minimax we can find a unique minimal distance 
point 
> > TOP 
> > > at 
> > > > > this minimum distance from SIZE
> > > > 
> > > > not following . . .
> > > 
> > > Remember, we have a way of measuring distance between tuning 
maps. 
> > 
> > In the dual space?
> 
> Correct. Tuning maps are points in the dual space, and that is a
> normed vector space, and hence a metric space, so we know the 
distance
> between two tuning maps. One, SIZE, is the JI tuning map. We want a
> tuning map in the subspace Null(C) as close as possible to SIZE.

OK . . .


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Message: 9052

Date: Thu, 08 Jan 2004 23:03:01

Subject: Re: non-1200: Tenney/heursitic meantone temperament

From: Graham Breed

> Graham, it sure doesn't look like you're using Euclidean distance 
> here!!!

No, I'm not.


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Message: 9053

Date: Thu, 08 Jan 2004 08:35:17

Subject: Re: Temperament agreement

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> Continued from the tuning list.
> Paul:
> >With my (Tenney) complexity and (all-interval-Tenney-minimax) error
> >measures?
> 
> With these it seems I need to scale the parameters to
> k=0.002
> p=0.5
> and max badness = 75
> 
> where badness = complexity * exp((error/k)**p)
> 
> I'd be very interested to see how that compares with your other 
cutoff
> lines.

See

Yahoo groups: /tuning_files/files/Erlich/dave3.gif *

It looks like going back to logarithmic error scaling would be better 
for seeing what is going on here . . . Matlab is chugging . . .

> These errors and complexities don't seem to have meaningful units.
> 
> Complexity used to have units of generators per diamond and error 
>used
> to have units of cents, both things you could relate to fairly 
>directly.

Here, complexity is length in the Tenney lattice = log2(n*d), and 
error is maximum over all intervals (or merely a simplest few, if you 
wish) of (cents error)/(interval complexity), where interval 
complexity is again log2(n*d). Most intervals achieve this maximum in 
the tuning in question. One advantage is that, if you choose to add 
more intervals into your optimization criterion, the optimum doesn't 
change.


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Message: 9054

Date: Thu, 08 Jan 2004 23:03:55

Subject: Re: non-1200: Tenney/heursitic meantone temperament

From: Graham Breed

Paul Erlich wrote:

> Well then we are talking about different things. I'm talking 
> about "expressibility" as the distance measure.

That's an entirely different part of the message you quoted, a few back.


                   Graham


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Message: 9056

Date: Thu, 08 Jan 2004 23:07:51

Subject: Re: non-1200: Tenney/heursitic meantone temperament

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
> 
> > Wow. How did you find that?
> 
> Briefly (use the Reply thing so that indentation works),

> 22876792454961:19073486328125

So it was a finite search? How do you know you won't keep finding 
worse and worse examples if you go farther out? You might be 
approaching a limit, but how do you know you'll ever reach it?

> >>TOPping it gives a narrow octave of 0.99806 2:1 octaves.
> > 
> > 
> > Shall I proceed to calculate Tenney-weighted errors for all 
(well, a 
> > bunch of) intervals? I hope you're onto something!
> 
> If you like.

OK, later -- gotta go perform now.

> > I thought that's what you were talking about in the thread where 
I 
> > brought them up! Odd limit, right?
> 
> No, a lattice for octave-equivalent ratios.

Yes, I meant that, with particular qualifications.

> > Did someone publish it before? It's currently not Gene's way, 
anyway.
> 
> How about Tenney?

No, I don't think he ever mentioned anything about comma-eating 
systems with tempered octaves or anything like that.


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Message: 9057

Date: Thu, 08 Jan 2004 13:05:22

Subject: Re: non-1200: Tenney/heursitic meantone temperament

From: Graham Breed

Me:
>>I'm not sure if such a comma will always exist,

Paul:
> Wow -- now that's an interesting question to consider.

Me:
>>but provided it does 
>>it's the only one you need for TOPS.

Paul:
> My intuition says it doesn't exist.

If it doesn't, the worst one you get should get you close to the optimum 
result.

Me:
>>It doesn't even have to be made up 
>>of integers, so long as it's a linear combination of commas that 
>>are.

Paul:
> ???

81**2:80**2 or 6561:6400 has the double the size and double the 
complexity of 81:80.  That means its size/complexity ratio is the same, 
and so tempering it out gives exactly the same result as tempering out 
81:80.  So why stop there?  The cube of 81:80 also gives the same 
result.  Any power will do, and it doesn't have to be an integer.  The 
square root of 81:80 works.  Even the pith root works (provided you 
express it as a monzo -- otherwise, it's an arbitrary number).

That means instead of having two integers to define the resultant comma 
of a 7-limit linear temperament, you only need one real number.  For 
example, the miracle kernel is (225:224)***i * (2401:2400)**j or i*[-5 2 
2 -1> + j*[-5 -1 -2 4>.  You can simplify that to x*[-5 2 2 -1> + 
(1-x)*[-5 -1 -2 4> so that you only have 1 variable instead of 2.

> There's something that seems strange about your octave-equivalent 
> method. The comma is supposed to be distributed uniformly (per unit 
> length, taxicabwise) among its constituent "rungs" in the lattice. 
> But it seems that 81:80 = 81:5 would involve 1, not 0, rungs of 5 in 
> the octave-equivalent lattice. But the octave-equivalent lattice 
> can't be embedded in euclidean space, so this completely falls apart??

81:5 involves three rungs of 3:1 and one rung of 3:5.  For the 5-odd 
limit, these rungs are of equal length, and so that error has to be 
shared between them.  That leaves 3:1 and 3:5 having an equal amount of 
temperament, and so 1:5 must be untempered.

This is how Dave did it on his original web page:

A method for optimally distributing any comma *

and we worked out at the time that it's correct.  I didn't remember all 
that stuff about octave-specific optimization.  It looks like you've 
cleared up the problems.  But anyway, for the weighted, 
octave-equivalent case:

The complexity of 81:80 is entirely determined by the 81, and not at all 
by the 80.  The error can only be shared amongst those intervals that 
contribute to the complexity.  I don't see how this can be drawn on a 
lattice, but it doesn't need to be.

You can improve 3:1 at the expense of 3:5 and, because 3:5 has less 
weight, the worst weighted error for 5 odd-limit ratios goes down. 
However, the weighted error for 9:5 exceeds this, because it has the 
same complexity as 9:8 but is more poorly tuned.  The true minimax is 
found by only tempering the 81.

You certainly can embed an octave-equivalent lattice in Euclidian space. 
  The triangular lattice is an example, but there's no reason to connect 
up the 5:3s if you're measuring Euclidian distances.  In general, for 
each prime interval you need to assign a weight (the unit lattice 
distance) and also an angle.  And you can probably do size/complexity 
tempering with such a measure, but it'd be more complex, and I haven't 
tried it.  I'm guessing it wouldn't give the true weighted minimax, 
because for that to work it looks like the complexity of a comma has to 
be the sum of the complexities of its factors (one way or another).


                   Graham



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Message: 9060

Date: Fri, 09 Jan 2004 13:49:56

Subject: Re: non-1200: Tenney/heursitic meantone temperament

From: Carl Lumma

>For an ET, just stretch so that the weighted errors of the most 
>upward-biased prime and most downward-biased prime are equal in 
>magnitude and opposite in sign. For 12-equal I take the mapping
>[12 19 28]
>divide (elementwise) by
>[1 log2(3) log2(5)]
>and get
>[12.00000000000000  11.98766531785769  12.05894362605501]
>Now we want to make the largest and smallest of these equidistant 
>from 12, so we divide [12 19 28] by their average
>[12.05894362605501+11.98766531785769 ]/2
>giving
>0.99806172487683   1.58026439772164   2.32881069137926

Awesome!

>So Graham had the all the digits right, I just needed more precision.
>Multiply by 12, and we get
>1197.67406985219          1896.31727726597          2794.57282965511
>Here it's clear we're hitting the maximum, 3.557, with both 3 and 5.
>
>
>           10            9       199.61       17.209       2.6508
>            9            8       199.61       4.2977      0.69655
>            6            5       299.42       16.223       3.3061
>            5            4       399.22       12.911       2.9873
>            4            3       499.03      0.98586        0.275
>            3            2       698.64       3.3118       1.2812
>            8            5       798.45       15.237        2.863
>            5            3       898.26       13.897        3.557
>            9            5       998.06       19.535        3.557
>            2            1       1197.7       2.3259       2.3259
>            9            4       1397.3       6.6236       1.2812
>           12            5       1497.1       18.549       3.1402
>            5            2       1596.9       10.585       3.1864
>            8            3       1696.7       1.3401      0.29227
>            3            1       1896.3       5.6377        3.557
>           16            5       1996.1       17.563       2.7781
>           10            3       2095.9       11.571       2.3581
>           18            5       2195.7        21.86       3.3674
>           15            4       2295.5       7.2733       1.2313
>            4            1       2395.3       4.6519       2.3259
>            9            2         2595       8.9495       2.1462
>            5            1       2794.6       8.2591        3.557
>           16            3       2894.4        3.666       0.6564
>            6            1         3094       7.9637       3.0808
>           25            4       3193.8        21.17       3.1864
>           20            3       3293.6        9.245       1.5651
>           15            2       3493.2       4.9473       1.0082
>            8            1         3593       6.9778       2.3259
>           25            3       3692.8       22.156        3.557
>            9            1       3792.6       11.275        3.557
>           10            1       3992.2       5.9332       1.7861
>           32            3       4092.1       5.9919      0.90994
>           12            1       4291.7        10.29       2.8702
>           25            2       4391.5       18.844       3.3389
>           27            2       4491.3       14.587       2.5348
>           15            1       4690.9       2.6214      0.67097
>           16            1       4790.7       9.3037       2.3259
>           18            1       4990.3       13.601       3.2618
>           20            1       5189.9       3.6073      0.83464
>           45            2       5389.5       0.6904      0.10635
>           24            1       5489.3       12.616       2.7515
>           25            1       5589.1       16.518        3.557
>           27            1         5689       16.913        3.557
>           30            1       5888.6      0.29546     0.060214
>           32            1       5988.4        11.63       2.3259
>           36            1         6188       15.927       3.0808
>           40            1       6387.6       1.2813      0.24076
>           45            1       6587.2       3.0163      0.54924
>           48            1         6687       14.941       2.6753
>           50            1       6786.8       14.192       2.5146
>           54            1       6886.6       19.239       3.3431
>           60            1       7086.2       2.0305      0.34375
>           64            1         7186       13.956       2.3259
>           72            1       7385.7       18.253       2.9584
>           75            1       7485.5       10.881       1.7468
>           80            1       7585.3       1.0446      0.16524
>           81            1       7585.3       22.551        3.557
>           90            1       7784.9       5.3423      0.82292
>           96            1       7884.7       17.267       2.6222
>          100            1       7984.5       11.866       1.7861

The alaska tunings are essentially circulating versions of this
tuning.

-Carl


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Message: 9066

Date: Fri, 09 Jan 2004 23:45:00

Subject: Re: Temperament agreement

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> 
> > I wouldn't want to include any outside the 5-limit linear temperaments
> > having the following 18 vanishing commas. And I wouldn't mind leaving
> > off the last four.
> 
> Your wishes can be accomodated by setting bounds for size and
> epimericity. For the short list, we have size < 93 cents and
> epimericity < 0.62, the only five limit comma which would be added to
> the list if we used these bounds would be 1600000/1594323. Presumably
> you have no objection to that, as it appears on your long list.

I could live with it, but I'd rather not.

> The long list has size < 93 and epimericity < 0.68. If we were to use
> these bounds, we would add 6561/6250 and 20480/19683. The second of
> these, 20480/19683, has epimericity 0.6757, which is a sliver higher
> than the actual maximum epimericity of your long list, 0.6739, and so
> setting the bound at 0.675 would leave it off. What do you make of the
> 6561/6250 comma? If you had no objection to letting it on to an
> amended long list, you'd be in business there as well.

I'd rather not. 

What I don't like about both of these proposals is the "corners" in
the cutoff line. I prefer straight or smoothly curved cutoffs.


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Message: 9069

Date: Fri, 09 Jan 2004 08:11:12

Subject: Re: Some seven limit TOP tunings

From: Paul Erlich

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

> pajara 
> 
> [1196.89342188995, 1901.90667876129, 2779.10046287214, 
3377.54717381711]

       numerator   denominator  temp.cents    error    error/comp.
           10            9       172.18       10.223       1.5748
            9            8       213.13       9.2231       1.4948
            8            7       213.13       18.041       3.1066
            7            6       278.75       11.876       2.2024
            6            5        319.7       4.0584      0.82707
            5            4       385.31       1.0001       0.2314
            9            7       426.27       8.8179       1.4752
            4            3       491.88       6.1648       1.7196
            7            5       598.45       15.935       3.1066
           10            7       598.45       19.041       3.1066
            3            2       705.01       3.0583       1.1831
            8            5       811.58       2.1065      0.39581
            5            3       877.19       7.1649       1.8339
           12            7       918.15       14.983       2.3439
            7            4       983.76       14.934       3.1066
            9            5       1024.7       7.1166       1.2958
            2            1       1196.9       3.1066       3.1066
           15            7       1303.5       15.983       2.3804
            9            4         1410       6.1165       1.1831
            7            3       1475.6       8.7696       1.9966
           12            5       1516.6      0.95177      0.16113
            5            2       1582.2       4.1067       1.2362
            8            3       1688.8       9.2714       2.0221
           14            5       1795.3       12.828       2.0929
            3            1       1901.9     0.048322     0.030488
           16            5       2008.5       5.2131       0.8246
           10            3       2074.1       10.272       2.0933
            7            2       2180.7       11.828       3.1066
           18            5       2221.6         4.01       0.6177
           15            4       2287.2       1.0484      0.17749
            4            1       2393.8       6.2132       3.1066
           21            5       2500.4       15.886        2.366
            9            2       2606.9       3.0099      0.72182
           14            3       2672.5        5.663       1.0502
            5            1       2779.1       7.2133       3.1066
           21            4       2885.7       14.886       2.3287
           16            3       2885.7       12.378       2.2163
            6            1       3098.8       3.1549       1.2205
           25            4       3164.4       8.2133       1.2362
           20            3         3271       13.378       2.2648
            7            1       3377.5       8.7213       3.1066
           15            2       3484.1        4.155      0.84677
            8            1       3590.7       9.3197       3.1066
           25            3       3656.3       14.378       2.3083
            9            1       3803.8     0.096644     0.030488
           28            3       3869.4       2.5564      0.39992
           10            1         3976        10.32       3.1066
           21            2       4082.6        11.78       2.1845
           32            3       4082.6       15.485       2.3515
           35            3       4254.7       1.5563       0.2318
           12            1       4295.7       6.2615       1.7466
           25            2       4361.3        11.32       2.0057
           27            2       4508.8       2.9616      0.51463
           14            1       4574.4       5.6147       1.4747
           15            1         4681       7.2616       1.8587
           16            1       4787.6       12.426       3.1066
           35            2       4959.8       4.6146      0.75288
           18            1       5000.7       3.2032      0.76817
           20            1       5172.9       13.426       3.1066
           21            1       5279.5       8.6729       1.9746
           45            2         5386       4.2033      0.64748
           24            1       5492.6       9.3681       2.0432
           49            2       5558.2       20.549       3.1066
           25            1       5558.2       14.427       3.1066
           27            1       5705.7      0.14497     0.030488
           28            1       5771.3       2.5081      0.52172
           30            1       5877.9       10.368        2.113
           32            1       5984.5       15.533       3.1066
           35            1       6156.6        1.508        0.294
           36            1       6197.6       6.3098       1.2205
           40            1       6369.8       16.533       3.1066
           42            1       6476.3       5.5664       1.0323
           45            1       6582.9       7.3099        1.331
           48            1       6689.5       12.475       2.2336
           49            1       6755.1       17.443       3.1066
           50            1       6755.1       17.533       3.1066
           54            1       6902.6       3.2515      0.56501
           56            1       6968.2      0.59847      0.10305
           60            1       7074.8       13.475       2.2812
           63            1       7181.4       8.6246       1.4429
           64            1       7181.4       18.639       3.1066
           70            1       7353.5       1.5986      0.26081
           72            1       7394.5       9.4164       1.5262
           75            1       7460.1       14.475       2.3238
           80            1       7566.7        19.64       3.1066
           81            1       7607.6      0.19329     0.030488
           84            1       7673.2       2.4598       0.3848
           90            1       7779.8       10.416       1.6045
           96            1       7886.4       15.581       2.3662
           98            1         7952       14.336       2.1673
          100            1         7952        20.64       3.1066
          105            1       8058.6       1.4597       0.2174
3.1066 looks like the max . . . compare with 22-equal:



       numerator   denominator  temp.cents    error    error/comp.
           10            9       163.64       18.767       2.8909
            9            8       218.18       14.272       2.3131
            8            7       218.18       12.992       2.2372
            7            6       272.73       5.8564       1.0861
            6            5       327.27       11.631       2.3704
            5            4       381.82       4.4955       1.0402
            9            7       436.36       1.2795      0.21407
            4            3       490.91       7.1359       1.9905
            7            5          600       17.488       3.4094
           10            7          600       17.488       2.8532
            3            2       709.09       7.1359       2.7605
            8            5       818.18       4.4955      0.84472
            5            3       872.73       11.631       2.9772
           12            7       927.27       5.8564      0.91616
            7            4       981.82       12.992       2.7026
            9            5       1036.4       18.767       3.4173
            2            1         1200            0            0
           15            7       1309.1       10.352       1.5418
            9            4       1418.2       14.272       2.7605
            7            3       1472.7       5.8564       1.3333
           12            5       1527.3       11.631       1.9691
            5            2       1581.8       4.4955       1.3533
            8            3       1690.9       7.1359       1.5564
           14            5         1800       17.488       2.8532
            3            1       1909.1       7.1359       4.5023
           16            5       2018.2       4.4955       0.7111
           10            3       2072.7       11.631       2.3704
            7            2       2181.8       12.992       3.4124
           18            5       2236.4       18.767       2.8909
           15            4       2290.9       2.6404        0.447
            4            1         2400            0            0
           21            5       2509.1       24.624       3.6674
            9            2       2618.2       14.272       3.4226
           14            3       2672.7       5.8564       1.0861
            5            1       2781.8       4.4955       1.9361
           21            4       2890.9       20.128       3.1488
           16            3       2890.9       7.1359       1.2777
            6            1       3109.1       7.1359       2.7605
           25            4       3163.6       8.9911       1.3533
           20            3       3272.7       11.631       1.9691
            7            1       3381.8       12.992       4.6279
           15            2       3490.9       2.6404       0.5381
            8            1         3600            0            0
           25            3       3654.5       16.127       2.5891
            9            1       3818.2       14.272       4.5023
           28            3       3872.7       5.8564      0.91616
           10            1       3981.8       4.4955       1.3533
           21            2       4090.9       20.128       3.7328
           32            3       4090.9       7.1359       1.0837
           35            3       4254.5       1.3608      0.20268
           12            1       4309.1       7.1359       1.9905
           25            2       4363.6       8.9911       1.5931
           27            2       4527.3       21.408       3.7199
           14            1       4581.8       12.992       3.4124
           15            1       4690.9       2.6404      0.67583
           16            1         4800            0            0
           35            2       4963.6       8.4967       1.3863
           18            1       5018.2       14.272       3.4226
           20            1       5181.8       4.4955       1.0402
           21            1       5290.9       20.128       4.5826
           45            2         5400       9.7763       1.5059
           24            1       5509.1       7.1359       1.5564
           49            2       5563.6       25.985       3.9283
           25            1       5563.6       8.9911       1.9361
           27            1       5727.3       21.408       4.5023
           28            1       5781.8       12.992       2.7026
           30            1       5890.9       2.6404       0.5381
           32            1         6000            0            0
           35            1       6163.6       8.4967       1.6565
           36            1       6218.2       14.272       2.7605
           40            1       6381.8       4.4955      0.84472
           42            1       6490.9       20.128       3.7328
           45            1         6600       9.7763       1.7801
           48            1       6709.1       7.1359       1.2777
           49            1       6763.6       25.985       4.6279
           50            1       6763.6       8.9911       1.5931
           54            1       6927.3       21.408       3.7199
           56            1       6981.8       12.992       2.2372
           60            1       7090.9       2.6404        0.447
           63            1         7200       27.264       4.5613
           64            1         7200            0            0
           70            1       7363.6       8.4967       1.3863
           72            1       7418.2       14.272       2.3131
           75            1       7472.7       1.8552      0.29783
           80            1       7581.8       4.4955       0.7111
           81            1       7636.4       28.544       4.5023
           84            1       7690.9       20.128       3.1488
           90            1         7800       9.7763       1.5059
           96            1       7909.1       7.1359       1.0837
           98            1       7963.6       25.985       3.9283
          100            1       7963.6       8.9911       1.3533
          105            1       8072.7       15.633       2.3283


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Message: 9070

Date: Fri, 09 Jan 2004 08:23:33

Subject: Re: non-1200: Tenney/heursitic meantone temperament

From: Paul Erlich

Gene, what do you get for the top system with the commas of 12-equal 
(in other words, some stretching or squashing of 12-equal)? Graham 
seems to gave gotten pretty close below, but no cigar . . .

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> 
wrote:
> > Paul Erlich wrote:
> > 
> > > Wow. How did you find that?
> > 
> > Briefly (use the Reply thing so that indentation works),
> 
> > 22876792454961:19073486328125
> 
> So it was a finite search? How do you know you won't keep finding 
> worse and worse examples if you go farther out? You might be 
> approaching a limit, but how do you know you'll ever reach it?
> 
> > >>TOPping it gives a narrow octave of 0.99806 2:1 octaves.
> > > 
> > > 
> > > Shall I proceed to calculate Tenney-weighted errors for all 
> (well, a 
> > > bunch of) intervals? I hope you're onto something!
> > 
> > If you like.
> 
> OK, later -- gotta go perform now.

I'm back . . . Looks like you might be off in the last digit or two 
(so maybe there is no worst comma?), but a lot of the Tenney-weighted 
errors are in the 3.5549 - 3.5591 range, so you're probably pretty 
close . . . 

           10            9       199.61       17.208       2.6508
            9            8       199.61        4.298      0.69661
            6            5       299.42       16.223       3.3062
            5            4       399.22        12.91       2.9872
            4            3       499.03        0.985      0.27476
            3            2       698.64        3.313       1.2816
            8            5       798.45       15.238       2.8633
            5            3       898.25       13.895       3.5566
            9            5       998.06       19.536       3.5573
            2            1       1197.7        2.328        2.328
            9            4       1397.3        6.626       1.2816
           12            5       1497.1       18.551       3.1406
            5            2       1596.9       10.582       3.1856
            8            3       1696.7        1.343      0.29291
            3            1       1896.3        5.641       3.5591
           16            5       1996.1       17.566       2.7786
           10            3       2095.9       11.567       2.3574
           18            5       2195.7       21.864        3.368
           15            4       2295.5       7.2693       1.2306
            4            1       2395.3        4.656        2.328
            9            2         2595        8.954       2.1473
            5            1       2794.6       8.2543       3.5549
           16            3       2894.4        3.671       0.6573
            6            1         3094        7.969       3.0828
           25            4       3193.8       21.165       3.1856
           20            3       3293.6       9.2393       1.5642
           15            2       3493.2       4.9413        1.007
            8            1         3593        6.984        2.328
           25            3       3692.8        22.15        3.556
            9            1       3792.6       11.282       3.5591
           10            1       3992.2       5.9263        1.784
           32            3         4092        5.999      0.91101
           12            1       4291.7       10.297       2.8723
           25            2       4391.5       18.837       3.3375
           27            2       4491.3       14.595       2.5361
           15            1       4690.9       2.6133      0.66889
           16            1       4790.7        9.312        2.328
           18            1       4990.3        13.61       3.2638
           20            1       5189.9       3.5983      0.83257
           45            2       5389.5      0.69972      0.10778
           24            1       5489.3       12.625       2.7536
           25            1       5589.1       16.509       3.5549
           27            1       5688.9       16.923       3.5591
           30            1       5888.6      0.28529      0.05814
           32            1       5988.4        11.64        2.328
           36            1         6188       15.938       3.0828
           40            1       6387.6       1.2703      0.23869
           45            1       6587.2       3.0277      0.55131
           48            1         6687       14.953       2.6774
           50            1       6786.8       14.181       2.5126
           54            1       6886.6       19.251       3.3452
           60            1       7086.2       2.0427      0.34582
           64            1         7186       13.968        2.328
           72            1       7385.6       18.266       2.9605
           75            1       7485.4       10.868       1.7447
           80            1       7585.3       1.0577      0.16731
           81            1       7585.3       22.564       3.5591
           90            1       7784.9       5.3557      0.82499
           96            1       7884.7       17.281       2.6243
          100            1       7984.5       11.853        1.784


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Message: 9072

Date: Fri, 09 Jan 2004 18:41:42

Subject: Re: 5-limit comma list

From: Carl Lumma

>These are all the 5-limit commas with size less than 100 cents and 
>epimericity less than 2/3; it looks reasonable to me.
>
>135/128 Pelogic
>256/243 Blackwood
>6561/6250 Ragitonic/Diaschizmoid
>25/24 Dicot
>648/625 Diminished
>16875/16384 Negri
>250/243 Porcupine
>128/125 Augmented 
>3125/3072 Magic
>20000/19683 Tetracot
>81/80 Meantone
>2048/2025 Diaschismic
>78732/78125 Hemisixths
>393216/390625 Wuerschmidt
>2109375/2097152 Orwell/Semicomma
>15625/15552 Kleismic
>1600000/1594323 Amity
>32805/32768 Schismic

This looks good to me.

-Carl


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Message: 9073

Date: Fri, 09 Jan 2004 08:48:09

Subject: Re: non-1200: Tenney/heursitic meantone temperament

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > Gene, what do you get for the top system with the commas of 12-
> equal 
> > (in other words, some stretching or squashing of 12-equal)? 
> 
> I don't know, but I plan on investigating TOP tunings of equal and 
> planar temperaments as well as linear ones. Presumably one gets a 
> squashing. The octaves of Dom7 are pretty short.

right, right . . . just wanted to do a cross-check between you and 
graham . . . so what's the formula for top linear in 7-limit (for 
pajara and/or in general)?


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Message: 9074

Date: Fri, 09 Jan 2004 00:53:19

Subject: Re: non-1200: Tenney/heursitic meantone temperament

From: Carl Lumma

>I don't know, but I plan on investigating TOP tunings of equal and 
>planar temperaments as well as linear ones. Presumably one gets a 
>squashing. The octaves of Dom7 are pretty short.

You have a way of combining commas, then?

-Carl


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