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Message: 9050 - Contents - Hide Contents 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
Message: 9051 - Contents - Hide Contents 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 . . .
Message: 9052 - Contents - Hide Contents 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.
Message: 9053 - Contents - Hide Contents 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... * [with cont.]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.
Message: 9054 - Contents - Hide Contents 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
Message: 9055 - Contents - Hide Contents Date: Thu, 08 Jan 2004 08:46:00 Subject: Re: TOP and normed vector spaces From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> OK, why do we need a space and not a lattice?I want a metric on the dual space of linear functionals, so I start out with a normed vector space with which to define the dual. Let's put it another way. If vp is the p-adic valuation and T is a tuning map, then for a p-limit rational number q we have q = 2^v2(q) 3^v3(q) ... p^vp(q) A tuning map has numbers r2~2, r3~3, ... rp~p and is defined by T(q) = r2^v2(q) r3^v3(q) ... rp^vp(q) The norm ||T|| of the above tuning map (dual to the Tenney norm) is ||T|| = Max(|log2(r2)|, log3(r3), ..., logp(rp))|| However, we are not so much interested in the norm as in its distance from the JI tuning map which I've called SIZE, which is ||T - SIZE|| = Max(|log2(r2/2)|, |log3(r3/3)|, ..., |logp(rp/p)|) It is this which we want to minimize over the subspace of tuning maps such that T(c)=1 (writing it multiplicatively) for each comma c we are tempering out.
Message: 9056 - Contents - Hide Contents 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), > 22876792454961:19073486328125So 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.
Message: 9057 - Contents - Hide Contents 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 * [with cont.] (Wayb.) 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 ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
Message: 9058 - Contents - Hide Contents Date: Thu, 08 Jan 2004 23:23:11 Subject: Reply to Reply to Gene From: Gene Ward Smith It looks like TOP is in business; your tuning for meantone being the Tenney optimal one. I couldn't get code working using Maple's simplex routines, so I did something by hand and that screwed up. It's back to the drawing board for me, I want code which works for various limits and temperament dimensions. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
Message: 9059 - Contents - Hide Contents Date: Fri, 09 Jan 2004 01:55:34 Subject: Re: Reply to Reply to Gene From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> It looks like TOP is in business; your tuning for meantone being the > Tenney optimal one. I couldn't get code working using Maple's simplex > routines, so I did something by hand and that screwed up. It's back to > the drawing board for me, I want code which works for various limits > and temperament dimensions.I've got something working in the 5-limit which gets around Maple's stupid business of giving only one minimax solution when there are more than one. It may amuse Dave to lesrn I used the 1003611167-et in the code so as to work strictly in rational arithmetic. It all seems to check with Paul's results. Now for the higher limits code.
Message: 9060 - Contents - Hide Contents 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 1.7861The alaska tunings are essentially circulating versions of this tuning. -Carl
Message: 9061 - Contents - Hide Contents Date: Fri, 09 Jan 2004 02:03:53 Subject: Re: Comma(s) from 7-limit vectors From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> Hi, > > Could someone please show me the formula for extracting comma(s) from > 7-limit interval vectors? (I know how to do the cross-product to get > the comma from two 5-limit vectors.) Is this a wedge product? Does > it always generate two commas?It sounds almost like you are asking how to extract commas from commas. I can show how to extract commas from wedgies. If l is a 7-limit wedgie, then what I call the subgroup commas are these: 2^l[4]*3^(-l[2])*5^l[1], 2^(-l[5])*3^l[3]*7^(-l[1]), 2^l[6]*5^(-l[3])*7^l[2], 3^l[6]*5^(-l[5])*7^l[4] Each of these uses only three of the possible four primes. These commas are not linearly independent nor are they usually the simplest available; Hermite reducing them is one way to simply this. Do you have something available which handles linear algebra?
Message: 9062 - Contents - Hide Contents Date: Fri, 09 Jan 2004 21:55:45 Subject: Re: Temperament agreement From: Gene Ward Smith --- 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.> 81/80 > 32805/32768 > 2048/2025 > 15625/15552 > 128/125 > 3125/3072 > 250/243 > 78732/78125 > 20000/19683 > 25/24 > 648/625 > 135/128 > 256/243 > 393216/390625The 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.> 1600000/1594323 > 16875/16384 > 2109375/2097152 > 531441/524288 > > I'm afraid I disagree with Herman about including the temperament > where the apotome (2187/2048) vanishes.I'd like to see Herman's list too.
Message: 9063 - Contents - Hide Contents Date: Fri, 09 Jan 2004 02:47:36 Subject: Re: Temperament agreement From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> It looks like I'd be just as happy with straight lines on this chart.Could you enlighten the rest of us and give a comma list of commas you want?
Message: 9064 - Contents - Hide Contents Date: Fri, 09 Jan 2004 22:38:37 Subject: A peek at diaschisoid/ragatonic 6561/6250 From: Gene Ward Smith I proposed the name ragitone for 6561/6250, since it is a ragisma shy of the 21/20 minor septimal semitone. The diaschisoid name wouldn't work for the Fokker blocks since we had too many otheer similar names. You can get a good Ragitonic[12] from 5/59, but the top tuning might be interesting: [1203.324, 1896.686, 2794.033]. I may try some retuning experiments or see if I can get something interesting which circulates.
Message: 9066 - Contents - Hide Contents 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.
Message: 9067 - Contents - Hide Contents Date: Fri, 09 Jan 2004 03:59:03 Subject: Some seven limit TOP tunings From: Gene Ward Smith Comments? meantone [1201.69852049457, 1899.26290957479, 2790.25755632091, 3370.54832931857] miracle [1200.63101379502, 1900.95486766285, 2784.84854520592, 3368.45175676244] tertiathirds [1203.18730860812, 1907.00676548748, 2780.90050601280, 3359.87799995998] porcupine [1196.90596030328, 1906.85893776044, 2779.12957616631, 3367.71788629877] dominant seventh [1195.22895145839, 1894.57688783731, 2797.39174551566, 3382.21993307575] diminished [1194.12845965338, 1892.64882959452, 2788.24517433455, 3385.30940416124] pajara [1196.89342188995, 1901.90667876129, 2779.10046287214, 3377.54717381711] orwell [1199.53265690430, 1900.45553028115, 2784.11702916383, 3371.48183436847]
Message: 9068 - Contents - Hide Contents Date: Fri, 09 Jan 2004 04:01:14 Subject: Re: Comma(s) from 7-limit vectors From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:>> Do you have something available which handles linear algebra? >> Right now, just Excel. What do you recommend (without having to > spend too much?)I recommend you tell me what Excel can do for now. Can it row reduce a matrix, for example? Can it find a nullspace?
Message: 9069 - Contents - Hide Contents 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
Message: 9070 - Contents - Hide Contents 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
Message: 9071 - Contents - Hide Contents Date: Fri, 09 Jan 2004 08:37:33 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Gene Ward Smith --- 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.
Message: 9072 - Contents - Hide Contents 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 SchismicThis looks good to me. -Carl
Message: 9073 - Contents - Hide Contents 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)?
Message: 9074 - Contents - Hide Contents 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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