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Message: 6100 - Contents - Hide Contents Date: Sun, 19 Jan 2003 14:48:04 Subject: Calculating geometric complexity and badness From: Gene Ward Smith I've just finished computing the coefficients to calculate geometric complexity directly from the wedgie up to the 11 limit; an aggravating but potentially very useful chore. I'm sorry for the variation in notation, but I took this from various Maple routines which did not have a consistent internal system, having been written at various times. Here are the results. 5 limit In the 5 limit, if q = 2^u0 3^u1 5^u2 is a comma, then [u0, u1, u2] is the wedgie and sqrt(log(3)^2 u1^2 + log(3)^2 u1 u2 + log(5)^2 u2^2) is the geometric complexity. 7 limit If p = 2^u1 3^u2 5^u3 7^u4 and q = 2^v1 3^v2 5^v3 7^v4 are two commas, then [u3*v4-v3*u4,u4*v2-v4*u2,u2*v3-v2*u3,u1*v2-v1*u2, u1*v3-v1*u3,u1*v4-v1*u4] is their wedge product p^q. If u=[u1,u2,u3,u4] and v=[v1,v2,v3,v4] instead are vals, then [u1*v2-v1*u2,u1*v3-v1*u3,u1*v4-v1*u4,u3*v4-v3*u4, u4*v2-u2*v4,u2*v3-v2*u3] is their wedge product, u^v. These two types of wedge product can be indentified with each other, by "Poincare duality". It does not matter whether the wedgie comes from commas or vals, therefore. If [l[1], l[2], l[3], l[4], l[5], l[6]] a wedgie, then sqrt(8.13090525300395*l[1]^2+4.20601082222396*l[2]^2+ 2.76216685071565*l[3]^2-3.00701812111698*l[1]*l[2]- 1.56317414960864*l[1]*l[3]-2.39798540221393*l[2]*l[3]) is its geometric complexity. This depends only on l[1], l[2], l[3], which are the number of generator steps times how many intervals of repetition there are in the octave. Hence it can also be computed directly from the mapping. Planar If q = 2^u0 3^u1 5^u2 7^u3 11^u4 is a comma, [u0,u1,u2,u3,u4] is the corresponding planar temperament wedgie, and sqrt(log(3)^2*u1^2+log(5)^2*u2^2+log(7)^2*u3^2 +log(3)^2*u1*u2+log(3)^2*u1*u3+log(5)^2*u2*u3) is the geometric complexity. 11 limit Linear If p = 2^u0 3^u1 5^u2 7^u3 11^u4 q = 2^v0 3^v1 5^v2 7^v3 11^v4 r = 2^w0 3^w1 5^w2 7^w3 11^w4 are three intervals, then [u2*v3*w4-u2*v4*w3-v2*u3*w4+v2*u4*w3+w2*u3*v4-w2*u4*v3, -u1*v3*w4+u1*v4*w3+v1*u3*w4-v1*u4*w3-w1*u3*v4+w1*u4*v3, u1*v2*w4-u1*v4*w2-v1*u2*w4+v1*u4*w2+w1*u2*v4-w1*u4*v2, -u1*v2*w3+u1*v3*w2+v1*u2*w3-v1*u3*w2-w1*u2*v3+w1*u3*v2, u0*v3*w4-u0*v4*w3-v0*u3*w4+v0*u4*w3+w0*u3*v4-w0*u4*v3, -u0*v2*w4+u0*v4*w2+v0*u2*w4-v0*u4*w2-w0*u2*v4+w0*u4*v2, u0*v2*w3-u0*v3*w2-v0*u2*w3+v0*u3*w2+w0*u2*v3-w0*u3*v2, u0*v1*w4-u0*v4*w1-v0*u1*w4+v0*u4*w1+w0*u1*v4-w0*u4*v1, -u0*v1*w3+u0*v3*w1+v0*u1*w3-v0*u3*w1-w0*u1*v3+w0*u3*v1, u0*v1*w2-u0*v2*w1-v0*u1*w2+v0*u2*w1+w0*u1*v2-w0*u2*v1] is their wedge product p^q^r. Similarly, if u=[u1,u2,u3,u4,u5] and v=[v1,v2,v3,v4,v5] are two vals, then [u1*v2-v1*u2,u1*v3-v1*u3,u1*v4-v1*u4,u1*v5-v1*u5, u2*v3-v2*u3,u2*v4-v2*u4, u2*v5-v2*u5,u3*v4-v3*u4,u3*v5-v3*u5,u4*v5-v4*u5] is their wedge product, u^v. Both methods can be used to calculate the wedgie of a linear temperament; again by Poincare duality we can identify these two types of wedge products. If [l[1],l[2],l[3],l[4],l[5],l[6],l[7],l[8],l[9],l[10]] is a linear temperament wedgie, then its geometric complexity is given by sqrt(37.4669556799257*l[1]^2+19.8578149236787*l[2^2 +13.8576504872655*l[3]^2+8.43459044261178*l[4]^2 -9.24811506924099*l[2]*l[3]-12.9637247061659*l[1]*l[2] -6.02856647730047*l[1]*l[3]-6.41005295068753*l[3]*l[4] -2.95953128437572*l[1]*l[4]-4.54006536578504*l[2]*l[4]) Again, this depends only on l[1] through l[4] which are the number of generator steps times the number of intervals of equivalence in an octave, so this is also computable directly from the mapping. Planar If p = 2^u0 3^u1 5^u2 7^u3 11^u4 and q = 2^v0 3^v1 5^v2 7^v3 11^v4 are two intervals, then [u3*v4-u4*v3, u2*v4-u4*v2, u2*v3-u3*v2, u1*v4-u4*v1, u1*v3-u3*v1, u1*v2-u2*v1, u0*v4-v0*u4, u0*v3-u3*v0, u0*v2-v0*u2, u0*v1-u1*v0] is their wedge product. If u=[u0,u1,u2,u3,u4], v=[v0,v1,v2,v3,v4], and w=[w0,w1,w2,w3,w4] are three vals, then [-u0*v1*w2+u0*v2*w1+v0*u1*w2-v0*u2*w1-w0*u1*v2+w0*u2*v1, u0*v1*w3-u0*v3*w1-v0*u1*w3+v0*u3*w1+w0*u1*v3-w0*u3*v1, -u0*v1*w4+u0*v4*w1+v0*u1*w4-v0*u4*w1-w0*u1*v4+w0*u4*v1, -u0*v2*w3+u0*v3*w2+v0*u2*w3-v0*u3*w2-w0*u2*v3+w0*u3*v2, u0*v2*w4-u0*v4*w2-v0*u2*w4+v0*u4*w2+w0*u2*v4-w0*u4*v2, -u0*v3*w4+u0*v4*w3+v0*u3*w4-v0*u4*w3-w0*u3*v4+w0*u4*v3, u1*v2*w3-u1*v3*w2-v1*u2*w3+v1*u3*w2+w1*u2*v3-w1*u3*v2, -u1*v2*w4+u1*v4*w2+v1*u2*w4-v1*u4*w2-w1*u2*v4+w1*u4*v2, u1*v3*w4-u1*v4*w3-v1*u3*w4+v1*u4*w3+w1*u3*v4-w1*u4*v3, -u2*v3*w4+u2*v4*w3+v2*u3*w4-v2*u4*w3-w2*u3*v4+w2*u4*v3] is their wedge product. Again by Poincare duality, we identify these two types of wedge product, and an 11-limit planar temperament wedgie may be calculated from either. If [l[1],l[2],l[3],l[4],l[5],l[6],l[7],l[8],l[9],l[10]] is a planar temperament wedgie, then its geometric complexity is given by sqrt(18.1878630999135*l[1]^2 +13.2165141603747*l[2]^2 +8.13090525300357*l[3]^2 +6.57565648053141*l[4]^2 +4.20601082222387*l[5]^2 +2.76216685071556*l[6]^2 +9.98976207451545*l[1]*l[2] -4.90415316714511*l[1]*l[3] +4.65474179366984*l[1]*l[4] -2.28509613536279*l[1]*l[5] -.721921985754273*l[1]*l[6] +6.45350417171744*l[2]*l[3] +5.37666377942441*l[2]*l[4] +1.44384397150855*l[2]*l[5] -1.56317414960855*l[2]*l[6] +3.00701812111700*l[3]*l[5] -1.56317414960855*l[3]*l[6] +3.84182937372243*l[4]*l[5] +2.39798540221390*l[4]*l[6] +2.39798540221390*l[5]*l[6]) This depends only on l[1] through l[6]. Spacial If q = 2^u0 3^u1 5^u2 7^u3 11^u4 is a comma, [u0,u1,u2,u3,u4] is the corresponding spacial temperament wedgie, and sqrt(log(3)^2*u1^2+log(5)^2*u2^2+log(7)^2*u3^2+log(11)^2*u4^2+ log(3)^2*u1*u2+log(3)^2*u1*u3+log(3)^2*u1*u4 +log(5)^2*u2*u3+log(5)^2*u2*u4+log(7)^2*u3*u4) is the geometric complexity. Badness If r is error (usually rms error), c is complexity, n is the number of primes in the p-limit (n=pi(p)) and d is the codimension, which is to say the number of commas used to define the temperament, then badness = r c^(n/d) is the log-flat badness measure.
Message: 6101 - Contents - Hide Contents Date: Sun, 19 Jan 2003 00:16:03 Subject: A comma list example From: Gene Ward Smith Using heuristics, I came up with the following list of 11-limit commas: slz := [36/35, 77/75, 128/125, 45/44, 49/48, 50/49, 55/54, 56/55, 64/63, 81/80, 245/242, 99/98, 100/99, 121/120, 245/243, 126/125, 1331/1323, 176/175, 896/891, 1029/1024, 225/224, 243/242, 3388/3375, 3136/3125, 5120/5103, 6144/6125,385/384, 8019/8000, 441/440, 1375/1372, 12005/11979, 6250/6237, 540/539, 4000/3993, 19712/19683, 5632/5625, 32805/32768, 172032/171875, 102487/102400, 180224/180075, 41503/41472, 131072/130977, 420175/419904, 160083/160000, 234375/234256, 43923/43904, 2401/2400, 2097152/2096325,2657205/2656192, 3025/3024, 117649/117612, 820125/819896, 4375/4374, 56953125/56942116, 250047/250000, 1771875/1771561, 47265625/47258883, 246071287/246037500, 21437500/21434787, 184549376/184528125, 369140625/369098752, 9801/9800, 35156250/35153041, 151263/151250, 226492416/226474325, 214375000/214358881, 645700815/645657712, 100663296/100656875, 15625959723/15625000000, 1771561/1771470, 14348907/14348180, 78125000/78121827, 13153840425/13153337344, 199297406/199290375, 26795786661/26794860125, 44660948992/44659644435, 40283203125/40282095616, 210087500000/210082347387,67110351/67108864, 3294225/3294172, 6576668672/6576582375, 16748046875/16747855872, 781258401/781250000, 13841287201/13841203200, 4882812500/4882786447, 2541867610898/2541865828329] There are 86 of these, and hence 102340 combinations taken three at a time. This leads to 44581 wedgies. If we limit it by filtering out all complexities greater than 300, we can reduce this to 15382 candidates, although this removes most of the rationale for including so many very small commas in the original list. Possibly the thing to do is to band-limit the wedge products, so that the smallest comma is not too much smaller than the largest comma. Of course, the more filters we introduce before testing for badness, the more likely it becomes we are missing something of interest.
Message: 6103 - Contents - Hide Contents Date: Sun, 19 Jan 2003 12:23:08 Subject: Re: A common notation for JI and ETs From: David C Keenan I earlier referred to a "uselessnes" figure-of-demerit for notational commas. This was obtained by multiplying the power of 3 in the comma (number of fifths) by the size of the comma in cents. I have since realised that for the purpose of our notation (or any notation based on a chain of best fifths) the ideal power of 3 in a comma is not zero, but ideal_power_of_three = 7 * size_of_comma / size_of_apotome or 7 * comma_in_cents / 113.685 This ideal value would ensure that the comma always represented the same fraction of an apotome irrespective of the size of the fifth. This can be used to show that a near-ideal half-apotome comma cannot exist. It would need to have 3 to the power of 3.5, but the power of 3 must be an integer. The best we could do is find a near 57 cent comma with 3^3 or 3^4, which is what we have in '|)) the 5:49 comma 392:405 (56.5 c). We can then define the "slope" of a comma as slope = actual_power_of_three - ideal_power_of_three For example (|( as the 5:11' comma 44:45 (38.9 c) has a slope of 2 - 7 * 38.9/113.7 = -0.4 which tells us that it should be extremely good at representing 2/7 of an apotome across a wide range of fifth sizes. In fact it works for ETs 49,56,63,70,77,84 and 91. I believe we should not use a comma for notating temperaments if it has an absolute value of slope greater than 8. The following have slopes between 7 and 8 and so I think they should also be avoided if possible '| as 5' comma ~|( as 7:13 comma |~ as 23 comma )|~ as 19' comma |)) as 49' diesis This also says that we should equally allow both '/|) as 7' diesis 57344:59049 and its apotome complement .(|\ as 7" diesis 27:28 as you first suggested, since they have slopes of 6.9 and -6.9. In fact true apotome complements always have the same absolute value of slope, but opposite signs. This suggests we modify "uselessness" to use slope instead of number of fifths. uselessness = ABS(slope) * size_of_comma but in fact I find uselessness fairly useless now, and I'm happy to simply limit the absolute slope to 8 and the comma size to 70.17 cents, the largest that could conceivably be notated as '((|
Message: 6105 - Contents - Hide Contents Date: Sun, 19 Jan 2003 05:50:09 Subject: Re: heuristic and straightness From: Carl Lumma>>> >hortening the unison vectors makes the temperament worse, but >>> in a given temperament, this would be counteracted by an >>> increase in straighness, which makes the temperament better. >>>> You lost me. >>well, there are probably too many counterfactuals here. why don't >we start again with any examples from the archives which came up >in connection with straightness. your choice.Good idea. I did find some talk between you and Gene, referring to results I couldn't find. :( At one point, you mention low- badness blocks of one note... If you pick the example, I'm happy to attempt to build the block and find the alternate versions... no promises, though, since I'm new to this. If we get a good example or two, I'll collect everything and give you a URL. Bless you, monz, for doing this sort of thing. -Carl
Message: 6107 - Contents - Hide Contents Date: Sun, 19 Jan 2003 08:23:55 Subject: Re: heuristic and straightness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>>> shortening the unison vectors makes the temperament worse, but >>>> in a given temperament, this would be counteracted by an >>>> increase in straighness, which makes the temperament better. >>>>>> You lost me. >>>> well, there are probably too many counterfactuals here. why don't >> we start again with any examples from the archives which came up >> in connection with straightness. your choice. >> Good idea. I did find some talk between you and Gene, referring > to results I couldn't find. :( At one point, you mention low- > badness blocks of one note... If you pick the example, I'm happy > to attempt to build the block and find the alternate versions... > no promises, though, since I'm new to this.what are we talking about, anyway? (no offence)
Message: 6108 - Contents - Hide Contents Date: Sun, 19 Jan 2003 20:16:25 Subject: Re: Calculating geometric complexity and badness From: Carl Lumma> 5 limit > > In the 5 limit, if q = 2^u0 3^u1 5^u2 is a comma, > then [u0, u1, u2] is the wedgieThat's all the wedgie is for a single comma? I take it this would be the chromatic uv in the linear case?> sqrt(log(3)^2 u1^2 + log(3)^2 u1 u2 + log(5)^2 u2^2) > > is the geometric complexity.Cool. I can do that. ;)> 11 limit > > Linear > > If > > p = 2^u0 3^u1 5^u2 7^u3 11^u4 > q = 2^v0 3^v1 5^v2 7^v3 11^v4 > r = 2^w0 3^w1 5^w2 7^w3 11^w4 > > are three intervals,You mean commas, right?> SpacialYou the man, Gene!> Badness > > If r is error (usually rms error), c is complexity, n is the > number of primes in the p-limit (n=pi(p)) and d is the > codimension, which is to say the number of commas used to > define the temperament,Does it matter how many we temper out here?> then > > badness = r c^(n/d) > > is the log-flat badness measure.Great! I always thought the critical exponent had to be determined empirically. -Carl
Message: 6109 - Contents - Hide Contents Date: Sun, 19 Jan 2003 20:24:13 Subject: Re: heuristic and straightness From: Carl Lumma>>>>> >hortening the unison vectors makes the temperament worse, >>>>> but in a given temperament, this would be counteracted by >>>>> an increase in straighness, which makes the temperament >>>>> better. >>>>>>>> You lost me. >>>>>> well, there are probably too many counterfactuals here. why >>> don't we start again with any examples from the archives >>> which came up in connection with straightness. your choice. >>>> Good idea. I did find some talk between you and Gene, >> referring to results I couldn't find. :( At one point, you >> mention low-badness blocks of one note... If you pick the >> example, I'm happy to attempt to build the block and find the >> alternate versions... no promises, though, since I'm new to >> this. >>what are we talking about, anyway? (no offence)I'm trying to figure out what you meant in the top paragraph, there...>>>>> shortening the unison vectors makes the temperament worse,If we heuristically ignore the sizes, then yes.>>>>> but in a given temperament, this would be counteracted by >>>>> an increase in straighness, which makes the temperament >>>>> better.??? If straightness is maximal when the uvs are maximally orthogonal, how does this mean the uvs have gotten shorter? I thought it was a *decrease* in straightness that made the difference/sum vector shorten, making the temperament *worse*. Ultimately, I'm trying to figure out how changing the straigtness can make a temperament "worse" but keep badness constant. Either badness is broken or it doesn't! -Carl
Message: 6110 - Contents - Hide Contents Date: Sun, 19 Jan 2003 20:31:10 Subject: Re: New file uploaded to tuning-math From: Carl Lumma Paul, These are tha bomb! -Carl
Message: 6111 - Contents - Hide Contents Date: Mon, 20 Jan 2003 13:32:11 Subject: An 11-limit linear temperament top 100 list From: Gene Ward Smith This isn't *actually* top 100, since I left in some temperaments which were plainly too inaccurate for study purposes, but 100 is a lot of temperaments so I'm not worried. If we define "epimericity" for p/q > 1 reduced to its lowest form as log(p-q)/log(q), then as I suggested in Message 4458, we can use a list with bounded epimericity as a comma list. While Carl thought my consideration of superparticular temperaments was a dud, it inspired me to think this idea is a good plan, especially now that I have a better computer and version of Maple in hand. The extra commas I suggested were all that was needed in the 7-limit all had epimericity less than .46. I suggested .5 as a cutoff for the 7-limit and .3 for the 11-limit; I boosted this to .35, with a 50 cent cutoff for size. This gave me the following list of 51 commas, in order of badness of the corresponding planar temperament: [9801/9800, 3025/3024, 3294225/3294172, 151263/151250, 441/440, 385/384, 225/224, 2401/2400, 56/55, 176/175, 4375/4374, 540/539, 64/63, 100/99, 250047/250000, 5632/5625, 36/35, 1375/1372, 126/125, 45/44, 99/98, 43923/43904, 896/891, 81/80, 49/48, 50/49, 121/120, 117649/117612, 55/54, 41503/41472, 1771561/1771470, 77/75, 4000/3993, 6250/6237, 8019/8000, 6144/6125, 1029/1024,5120/5103, 3388/3375, 3136/3125, 32805/32768, 245/242, 243/242, 128/125, 12005/11979, 245/243, 1728/1715, 19712/19683, 625/616, 1331/1323, 2200/2187] Wedging these three at a time led to 6135 wedgies. Taking the best 100 of these by geometric badness gave me my list. As I remarked, some of the resulting temperaments should be tossed as too inaccurate, but I left them as examples to consider in connection with the straightness discussion. We now have so many versions of Meantone we might need to beat them off with a stick; the one I called "Meanpop" I gave that name to since it has a fifth which is poptimal for seven-limit Meantone. Many names familiar from the seven-limit or named before appear here. Hemiennealimmal [36, 54, 36, 18, 2, -44, -96, -68, -145, -74] [[18, 28, 41, 50, 62], [0, 2, 3, 2, 1]] generators [66.6666666667, 17.6128210979] bad 2055.541669 rms .198798 comp 256.276437 Miracle [6, -7, -2, 15, -25, -20, 3, 15, 59, 49] [[1, 1, 3, 3, 2], [0, 6, -7, -2, 15]] generators [1200., 116.672264296] bad 2362.204792 rms 1.901466 comp 71.868304 [38, 38, 114, 76, -28, 74, -11, 158, 45, -181] [[38, 60, 88, 106, 131], [0, 1, 1, 3, 2]] generators [31.5789473684, 7.15256754222] bad 2665.433369 rms .129891 comp 386.646417 [0, 0, 0, 1, 0, 0, 2, 0, 3, 3] [[1, 2, 3, 3, 4], [0, 0, 0, 0, -1]] generators [1200., 140.882982221] bad 3059.967157 rms 517.602977 comp 2.904237 [2, -16, 78, 58, -30, 118, 85, 226, 190, -107] [[2, 3, 6, -1, 2], [0, 1, -8, 39, 29]] generators [600.000000000, 101.758845067] bad 3156.846100 rms .208485 comp 322.186595 [20, -30, -10, -80, -94, -72, -196, 61, -82, -190] [[10, 16, 23, 28, 34], [0, -2, 3, 1, 8]] generators [120.000000000, 8.93614137404] bad 3203.569897 rms .254545 comp 288.351390 [102, 210, 216, 222, 96, 56, -1, -88, -211, -124] [[6, 11, 17, 20, 24], [0, -17, -35, -36, -37]] generators [200.000000000, 17.5327039657] bad 3334.010974 rms .036009 comp 954.841112 Unidec [12, 22, -4, -6, 7, -40, -51, -71, -90, -3] [[2, 5, 8, 5, 6], [0, -6, -11, 2, 3]] generators [600.000000000, 183.182783130] bad 3535.629462 rms 1.249417 comp 117.775665 [14, 6, 74, 52, -23, 78, 34, 155, 100, -110] [[2, 4, 5, 10, 10], [0, -7, -3, -37, -26]] generators [600.000000000, 71.1037242400] bad 3709.724174 rms .352551 comp 258.983392 [44, -10, 6, 79, -118, -114, -27, 42, 218, 201] [[1, 3, 2, 3, 6], [0, -44, 10, -6, -79]] generators [1200., 38.5948091514] bad 3815.546980 rms .219371 comp 350.127551 Wizard [12, -2, 20, -6, -31, -2, -51, 52, -7, -86] [[2, 1, 5, 2, 8], [0, 6, -1, 10, -3]] generators [600.000000000, 216.784022808] bad 3830.786400 rms 1.584515 comp 107.160572 [8, -64, -30, -110, -120, -70, -202, 110, -34, -205] [[2, 2, 14, 10, 23], [0, 4, -32, -15, -55]] generators [600.000000000, 175.435046942] bad 3951.940564 rms .205760 comp 371.595182 Quartaminorthirds [9, 5, -3, 7, -13, -30, -20, -21, -1, 30] [[1, 1, 2, 3, 3], [0, 9, 5, -3, 7]] generators [1200., 77.9320061602] bad 4041.237407 rms 4.418576 comp 59.805237 [18, 15, -6, 9, -18, -60, -48, -56, -31, 46] [[3, 6, 8, 8, 11], [0, -6, -5, 2, -3]] generators [400.000000000, 83.1441780398] bad 4104.955169 rms 1.357052 comp 122.582132 Pajara [2, -4, -4, -12, -11, -12, -26, 2, -14, -20] [[2, 3, 5, 6, 8], [0, 1, -2, -2, -6]] generators [600.000000000, 107.105867271] bad 4125.050690 rms 9.552922 comp 38.122013 Octoid [24, 32, 40, 24, -5, -4, -45, 3, -55, -71] [[8, 13, 19, 23, 28], [0, -3, -4, -5, -3]] generators [150.000000000, 16.0721625528] bad 4139.349018 rms .768706 comp 173.261857 Tripletone [3, 0, -6, -6, -7, -18, -20, -14, -14, 4] [[3, 5, 7, 8, 10], [0, -1, 0, 2, 2]] generators [400.000000000, 87.7987973153] bad 4275.784995 rms 12.772525 comp 32.722273 Orwell [7, -3, 8, 2, -21, -7, -21, 27, 15, -22] [[1, 0, 3, 1, 3], [0, 7, -3, 8, 2]] generators [1200., 271.444627222] bad 4352.766535 rms 5.548615 comp 54.544189 [2, -4, -16, -24, -11, -31, -45, -26, -42, -12] [[2, 3, 5, 7, 9], [0, 1, -2, -8, -12]] generators [600.000000000, 103.783535999] bad 4368.478166 rms 3.182069 comp 76.308394 Meantone [1, 4, 10, 18, 4, 13, 25, 12, 28, 16] [[1, 2, 4, 7, 11], [0, -1, -4, -10, -18]] generators [1200., 502.999427608] bad 4405.132983 rms 6.584357 comp 49.575532 Magic [5, 1, 12, -8, -10, 5, -30, 25, -22, -64] [[1, 0, 2, -1, 6], [0, 5, 1, 12, -8]] generators [1200., 380.713812625] bad 4474.854562 rms 4.730404 comp 61.027896 [0, 0, 0, 5, 0, 0, 8, 0, 12, 14] [[5, 8, 12, 14, 17], [0, 0, 0, 0, 1]] generators [240.000000000, 99.1170177792] bad 4584.543716 rms 53.042760 comp 14.521183 Duodecimal [0, 12, 24, 36, 19, 38, 57, 22, 42, 18] [[12, 19, 28, 34, 42], [0, 0, -1, -2, -3]] generators [100.000000000, 16.8004014251] bad 4602.747844 rms 1.886540 comp 107.748365 [38, -3, 8, 64, -93, -94, -30, 27, 159, 152] [[1, -7, 3, 1, -11], [0, 38, -3, 8, 64]] generators [1200., 271.110310111] bad 4668.827965 rms .383188 comp 282.799004 [1, -1, 3, 4, -4, 2, 3, 10, 13, 1] [[1, 2, 2, 4, 5], [0, -1, 1, -3, -4]] generators [1200., 455.784552041] bad 4694.936246 rms 44.341252 comp 16.401858 Pentoid [2, 3, 1, -2, 0, -4, -10, -6, -15, -9] [[1, 2, 3, 3, 3], [0, -2, -3, -1, 2]] generators [1200., 262.017672737] bad 4792.041040 rms 40.160927 comp 17.620981 Catakleismic [6, 5, 22, -21, -6, 18, -54, 37, -66, -135] [[1, 0, 1, -3, 9], [0, 6, 5, 22, -21]] generators [1200., 316.707652223] bad 4805.477920 rms 1.697137 comp 117.818038 Nonkleismic [10, 9, 7, 25, -9, -17, 5, -9, 27, 46] [[1, -1, 0, 1, -3], [0, 10, 9, 7, 25]] generators [1200., 310.147077476] bad 4830.505211 rms 3.316530 comp 79.065160 [8, -64, -30, 61, -120, -70, 69, 110, 363, 275] [[1, 1, 7, 5, -1], [0, 8, -64, -30, 61]] generators [1200., 87.7196164898] bad 4849.969370 rms .241148 comp 382.009103 [2, -57, -28, 46, -95, -50, 66, 95, 304, 226] [[1, 1, 19, 11, -10], [0, 2, -57, -28, 46]] generators [1200., 351.114995251] bad 4873.582183 rms .331141 comp 316.738528 Hemithird [15, -2, -5, 22, -38, -50, -17, -6, 58, 79] [[1, 4, 2, 2, 7], [0, -15, 2, 5, -22]] generators [1200., 193.222638998] bad 4937.466578 rms 1.772097 comp 116.683657 [0, 0, 0, 3, 0, 0, 5, 0, 7, 9] [[3, 5, 7, 9, 11], [0, 0, 0, 0, -1]] generators [400.000000000, 140.882982221] bad 4948.732044 rms 134.144184 comp 8.712710 Dominant Seventh [1, 4, -2, -6, 4, -6, -13, -16, -28, -10] [[1, 2, 4, 2, 1], [0, -1, -4, 2, 6]] generators [1200., 495.145082634] bad 4962.157739 rms 18.933026 comp 28.253374 [1, -3, -4, -1, -7, -9, -5, -1, 8, 11] [[1, 2, 1, 1, 3], [0, -1, 3, 4, 1]] generators [1200., 526.844177545] bad 4967.542223 rms 36.627981 comp 19.028176 Schismic [1, -8, -14, -18, -15, -25, -32, -10, -14, -2] [[1, 2, -1, -3, -4], [0, -1, 8, 14, 18]] generators [1200., 497.529640592] bad 4970.055833 rms 5.290179 comp 60.776340 [24, -9, -66, 12, -70, -172, -64, -128, 59, 262] [[3, 2, 8, 16, 9], [0, 8, -3, -22, 4]] generators [400.000000000, 137.769752422] bad 5047.036500 rms .354706 comp 310.384815 [1, -1, 0, 1, -4, -3, -2, 3, 6, 3] [[1, 2, 2, 3, 4], [0, -1, 1, 0, -1]] generators [1200., 459.070228117] bad 5056.794051 rms 130.796462 comp 8.961229 Septimal [0, 0, 7, 0, 0, 11, 0, 16, 0, -24] [[7, 11, 16, 20, 24], [0, 0, 0, -1, 0]] generators [171.428571428, 85.5877110034] bad 5184.550217 rms 22.636347 comp 26.058106 [24, 20, 16, -12, -24, -42, -102, -19, -97, -89] [[4, 6, 9, 11, 14], [0, 6, 5, 4, -3]] generators [300.000000000, 16.8775244692] bad 5185.657721 rms 1.086899 comp 161.127182 [6, -48, -108, 3, -90, -188, -16, -116, 173, 382] [[3, 4, 13, 22, 10], [0, 2, -16, -36, 1]] generators [400.000000000, 150.873707764] bad 5244.564671 rms .228287 comp 413.748659 [0, 0, 0, 2, 0, 0, 3, 0, 5, 5] [[2, 3, 5, 5, 7], [0, 0, 0, 0, -1]] generators [600.000000000, 140.882982221] bad 5347.782721 rms 284.929289 comp 5.808473 [12, 34, 20, 30, 26, -2, 6, -49, -48, 15] [[2, 4, 7, 7, 9], [0, -6, -17, -10, -15]] generators [600.000000000, 83.1977090004] bad 5359.187204 rms 1.462302 comp 137.542589 [23, -1, 13, 42, -55, -44, -13, 33, 101, 73] [[1, 9, 2, 7, 17], [0, -23, 1, -13, -42]] generators [1200., 386.859261176] bad 5380.665034 rms 1.013641 comp 171.779435 [0, 2, 2, 2, 3, 3, 3, -1, -2, -1] [[2, 3, 5, 6, 7], [0, 0, -1, -1, -1]] generators [600.000000000, 266.469146634] bad 5407.715441 rms 129.838957 comp 9.370554 Supersupermajor [3, 17, -1, -13, 20, -10, -31, -50, -89, -33] [[1, 1, -1, 3, 6], [0, 3, 17, -1, -13]] generators [1200., 234.451570240] bad 5412.588259 rms 3.005389 comp 89.805421 Meanpop [1, 4, 10, -13, 4, 13, -24, 12, -44, -71] [[1, 2, 4, 7, -2], [0, -1, -4, -10, 13]] generators [1200., 503.595073256] bad 5420.225629 rms 5.644271 comp 61.580856 [3, -5, -6, -1, -15, -18, -12, 0, 15, 18] [[1, 3, 0, 0, 3], [0, -3, 5, 6, 1]] generators [1200., 562.608972648] bad 5472.363478 rms 13.781284 comp 36.251932 Schismatic [1, -8, -14, 23, -15, -25, 33, -10, 81, 113] [[1, 2, -1, -3, 13], [0, -1, 8, 14, -23]] generators [1200., 497.816548050] bad 5478.851624 rms 2.447559 comp 102.323143 [0, 0, 0, 4, 0, 0, 6, 0, 9, 11] [[4, 6, 9, 11, 14], [0, 0, 0, 0, -1]] generators [300.000000000, 140.882982221] bad 5506.167087 rms 92.405127 comp 11.616947 [1, -3, -2, -1, -7, -6, -5, 4, 8, 4] [[1, 2, 1, 2, 3], [0, -1, 3, 2, 1]] generators [1200., 510.517333268] bad 5513.617840 rms 54.982509 comp 15.875495 [6, -19, -26, -21, -44, -58, -54, -7, 17, 31] [[1, 2, 1, 1, 2], [0, -6, 19, 26, 21]] generators [1200., 83.3216367302] bad 5534.523131 rms 1.883084 comp 120.483135 [6, -36, -84, -132, -71, -150, -230, -94, -182, -80] [[6, 10, 11, 10, 10], [0, -1, 6, 14, 22]] generators [200.000000000, 97.7979183888] bad 5608.137311 rms .259073 comp 399.243757 Superkleismic [9, 10, -3, 2, -5, -30, -28, -35, -30, 16] [[1, 4, 5, 2, 4], [0, -9, -10, 3, -2]] generators [1200., 321.939679550] bad 5706.061896 rms 5.302952 comp 65.931245 [2, -2, -2, 0, -8, -9, -7, 1, 7, 7] [[2, 3, 5, 6, 7], [0, 1, -1, -1, 0]] generators [600.000000000, 145.338368448] bad 5731.615600 rms 46.396405 comp 17.991842 [2, -16, -40, -60, -30, -69, -102, -48, -84, -30] [[2, 3, 6, 9, 12], [0, 1, -8, -20, -30]] generators [600.000000000, 101.618164890] bad 5760.166908 rms .979643 comp 182.650266 Porcupine [3, 5, -6, 4, 1, -18, -4, -28, -8, 32] [[1, 2, 3, 2, 4], [0, -3, -5, 6, -4]] generators [1200., 162.926556665] bad 5765.207416 rms 11.793935 comp 41.067070 [6, 0, 3, 3, -14, -12, -16, 7, 7, -2] [[3, 4, 7, 8, 10], [0, 2, 0, 1, 1]] generators [400.000000000, 152.119884703] bad 5804.051786 rms 14.472091 comp 36.468676 [0, 1, 0, 0, 2, 0, 0, -3, -4, 0] [[1, 2, 3, 3, 4], [0, 0, -1, 0, 0]] generators [1200., 338.888056433] bad 5837.190159 rms 483.720277 comp 4.456211 [2, 1, 3, 5, -3, -1, 1, 4, 8, 4] [[1, 1, 2, 2, 2], [0, 2, 1, 3, 5]] generators [1200., 344.800380608] bad 5859.936960 rms 55.689539 comp 16.340735 [0, 5, 0, -5, 8, 0, -8, -14, -29, -14] [[5, 8, 12, 14, 17], [0, 0, -1, 0, 1]] generators [240.000000000, 82.5021142548] bad 5925.447275 rms 22.089446 comp 28.649813 Injera [2, 8, 8, 12, 8, 7, 12, -4, 0, 6] [[2, 3, 4, 5, 6], [0, 1, 4, 4, 6]] generators [600.000000000, 91.3378934315] bad 5930.390624 rms 13.344995 comp 38.784565 [1, -1, -2, 1, -4, -6, -2, -2, 6, 10] [[1, 2, 2, 2, 4], [0, -1, 1, 2, -1]] generators [1200., 492.054891356] bad 5979.311922 rms 96.113418 comp 11.921209 [2, 1, 3, -2, -3, -1, -10, 4, -8, -16] [[1, 1, 2, 2, 4], [0, 2, 1, 3, -2]] generators [1200., 336.439285435] bad 6018.139480 rms 52.234534 comp 17.254513 [4, 2, 2, -4, -6, -8, -20, -1, -16, -18] [[2, 4, 5, 6, 6], [0, -2, -1, -1, 2]] generators [600.000000000, 257.288990758] bad 6037.202663 rms 22.632565 comp 28.553621 [6, -12, 10, -14, -33, -1, -43, 57, 9, -74] [[2, 4, 3, 7, 5], [0, -3, 6, -5, 7]] generators [600.000000000, 165.152290841] bad 6063.419880 rms 2.986631 comp 96.498734 [6, -48, -108, -168, -90, -188, -287, -116, -224, -98] [[6, 10, 10, 8, 7], [0, -1, 8, 18, 28]] generators [200.000000000, 98.2539292888] bad 6150.370425 rms .191092 comp 506.518200 [42, 47, 34, 33, -23, -64, -93, -53, -86, -25] [[1, -13, -14, -9, -8], [0, 42, 47, 34, 33]] generators [1200., 416.714284033] bad 6156.791026 rms .579519 comp 260.478466 Tritonic [5, -11, -12, -3, -29, -33, -22, 3, 31, 33] [[1, 4, -3, -3, 2], [0, -5, 11, 12, 3]] generators [1200., 580.274408364] bad 6158.168745 rms 5.154394 comp 70.204409 Double wide [8, 6, 6, -4, -9, -13, -34, -3, -30, -32] [[2, 5, 6, 7, 6], [0, -4, -3, -3, 2]] generators [600.000000000, 274.687303196] bad 6195.802215 rms 8.163015 comp 53.474677 [17, 6, 15, 27, -30, -24, -16, 18, 42, 24] [[1, -5, 0, -3, -7], [0, 17, 6, 15, 27]] generators [1200., 464.880312701] bad 6229.034828 rms 2.412281 comp 111.479674 Meanertone [1, 4, 3, -1, 4, 2, -5, -4, -16, -13] [[1, 2, 4, 4, 3], [0, -1, -4, -3, 1]] generators [1200., 503.381925652] bad 6235.745072 rms 47.548543 comp 18.648791 Meanenneadecal [1, 4, 10, 6, 4, 13, 6, 12, 0, -18] [[1, 2, 4, 7, 6], [0, -1, -4, -10, -6]] generators [1200., 504.558724590] bad 6252.411315 rms 18.965801 comp 32.422449 [64, 172, 102, 146, 124, -18, 10, -246, -256, 57] [[2, -6, -20, -9, -14], [0, 32, 86, 51, 73]] generators [600.000000000, 171.934587317] bad 6258.003982 rms .114672 comp 695.323226 [1, 33, 27, -18, 50, 40, -32, -30, -156, -144] [[1, 2, 16, 14, -4], [0, -1, -33, -27, 18]] generators [1200., 497.374746314] bad 6259.261024 rms 1.115730 comp 177.573572 [3, -24, -1, 28, -45, -10, 34, 65, 148, 82] [[1, 1, 7, 3, -2], [0, 3, -24, -1, 28]] generators [1200., 233.937160254] bad 6259.999412 rms 1.499793 comp 148.705092 Pajaric [2, -4, -4, 0, -11, -12, -7, 2, 14, 14] [[2, 3, 5, 6, 7], [0, 1, -2, -2, 0]] generators [600.000000000, 106.675554557] bad 6293.616955 rms 27.006882 comp 26.330282 [18, 39, 42, 9, 20, 16, -48, -12, -114, -120] [[3, 2, 1, 2, 9], [0, 6, 13, 14, 3]] generators [400.000000000, 183.522617689] bad 6297.037343 rms 1.049818 comp 184.847442 Supermajor seconds [3, 12, -1, -8, 12, -10, -23, -36, -60, -19] [[1, 1, 0, 3, 5], [0, 3, 12, -1, -8]] generators [1200., 231.991673974] bad 6304.096122 rms 6.456792 comp 62.196165 Diminished [4, 4, 4, 0, -3, -5, -14, -2, -14, -14] [[4, 6, 9, 11, 14], [0, 1, 1, 1, 0]] generators [300.000000000, 114.119995044] bad 6306.500152 rms 27.265894 comp 26.212063 Slender [13, -10, 6, 17, -46, -27, -18, 42, 74, 27] [[1, 2, 2, 3, 4], [0, -13, 10, -6, -17]] generators [1200., 38.3548342416] bad 6321.956492 rms 2.438407 comp 111.749905 [18, -9, 18, 9, -56, -22, -48, 67, 52, -37] [[9, 14, 21, 25, 31], [0, 2, -1, 2, 1]] generators [133.333333333, 17.0160504962] bad 6325.840259 rms 1.689101 comp 139.340563 [30, 13, 14, 3, -49, -62, -99, -4, -38, -40] [[1, -13, -4, -4, 2], [0, 30, 13, 14, 3]] generators [1200., 583.380644845] bad 6326.911371 rms 1.179376 comp 172.871604 Kleismic [6, 5, 3, -2, -6, -12, -24, -7, -22, -16] [[1, 0, 1, 2, 4], [0, 6, 5, 3, -2]] generators [1200., 317.610475585] bad 6369.686860 rms 14.472383 comp 38.560870 [5, 3, 7, 4, -7, -3, -11, 8, -1, -13] [[1, 1, 2, 2, 3], [0, 5, 3, 7, 4]] generators [1200., 141.164897166] bad 6400.766041 rms 19.644403 comp 32.195552 [4, 2, 2, 10, -6, -8, 2, -1, 16, 21] [[2, 4, 5, 6, 9], [0, -2, -1, -1, -5]] generators [600.000000000, 252.994745924] bad 6414.557575 rms 19.453599 comp 32.426498 [2, 1, 6, 5, -3, 4, 1, 11, 8, -7] [[1, 1, 2, 1, 2], [0, 2, 1, 6, 5]] generators [1200., 355.041079645] bad 6442.538585 rms 37.484665 comp 21.934212 [1, -1, 3, -4, -4, 2, -10, 10, -6, -22] [[1, 2, 2, 4, 2], [0, -1, 1, -3, 4]] generators [1200., 455.251489802] bad 6476.838112 rms 43.037876 comp 20.253838 Hemiwuerschmidt [16, 2, 5, 40, -34, -37, 8, 6, 86, 95] [[1, -1, 2, 2, -3], [0, 16, 2, 5, 40]] generators [1200., 193.827642803] bad 6485.787554 rms 1.764586 comp 137.781843 [1, 2, 0, 1, 1, -3, -2, -6, -5, 3] [[1, 2, 3, 3, 4], [0, -1, -2, 0, -1]] generators [1200., 390.155136790] bad 6547.764893 rms 157.873401 comp 9.346962 Hemiamity [10, 26, -34, -28, 18, -82, -79, -152, -155, 39] [[2, 1, -1, 13, 13], [0, 5, 13, -17, -14]] generators [600.000000000, 260.565078209] bad 6616.465091 rms .881985 comp 211.396985 Arnold [1, 4, -2, -1, 4, -6, -5, -16, -16, 4] [[1, 2, 4, 2, 3], [0, -1, -4, 2, 1]] generators [1200., 501.833702413] bad 6618.437371 rms 39.863722 comp 21.483576 [34, -24, 64, -28, -117, 6, -162, 216, 18, -300] [[2, -3, 9, -6, 12], [0, 17, -12, 32, -14]] generators [600.000000000, 217.771906631] bad 6646.001837 rms .339966 comp 375.555942 [2, 1, -4, 5, -3, -12, 1, -12, 8, 28] [[1, 1, 2, 4, 2], [0, 2, 1, -4, 5]] generators [1200., 353.356606337] bad 6651.131431 rms 27.107413 comp 27.157168 Pajarous [2, -4, -4, 10, -11, -12, 9, 2, 37, 42] [[2, 3, 5, 6, 6], [0, 1, -2, -2, 5]] generators [600.000000000, 109.882784796] bad 6667.906222 rms 12.267148 comp 43.767076 [6, 29, -2, -21, 32, -20, -54, -86, -149, -52] [[1, 4, 14, 2, -5], [0, -6, -29, 2, 21]] generators [1200., 483.287995700] bad 6718.809696 rms 1.555238 comp 151.808898 Ennealimmal [18, 27, 18, 144, 1, -22, 166, -34, 241, 342] [[9, 15, 22, 26, 37], [0, -2, -3, -2, -16]] generators [133.333333333, 48.8643746446] bad 6729.608260 rms .324647 comp 388.997739 [2, -6, 1, -2, -14, -4, -10, 19, 16, -9] [[1, 2, 1, 3, 3], [0, -2, 6, -1, 2]] generators [1200., 259.236678539] bad 6745.990413 rms 19.984071 comp 32.886454 [82, 28, 120, 155, -146, -40, -38, 200, 263, 20] [[1, -14, -3, -20, -26], [0, 82, 28, 120, 155]] generators [1200., 228.072122768] bad 6759.616416 rms .153762 comp 610.722159 [18, -14, 30, -20, -64, -3, -94, 109, 2, -160] [[2, 4, 4, 7, 6], [0, -9, 7, -15, 10]] generators [600.000000000, 55.2942867559] bad 6760.326292 rms 1.000617 comp 198.527159 Opossum [3, 5, 9, 4, 1, 6, -4, 7, -8, -20] [[1, 2, 3, 4, 4], [0, -3, -5, -9, -4]] generators [1200., 159.564330324] bad 6767.545993 rms 22.129858 comp 30.993567
Message: 6112 - Contents - Hide Contents Date: Mon, 20 Jan 2003 19:18:51 Subject: Re: An 11-limit linear temperament top 100 list From: Carl Lumma>If we define "epimericity" for p/q > 1 reduced to its lowest >form as log(p-q)/log(q),Now we're talking. Looks a lot like the "cent" heuristic...>then as I suggested in Message 4458,Apparently, *then* we were talking. I remember reading that, too, but I just didn't have a clue where it was coming from. Now, I think I'm catching on. -Carl
Message: 6113 - Contents - Hide Contents Date: Mon, 20 Jan 2003 20:16:28 Subject: Re: Calculating geometric complexity and badness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>> 5 limit >> >> In the 5 limit, if q = 2^u0 3^u1 5^u2 is a comma, >> then [u0, u1, u2] is the wedgie >> That's all the wedgie is for a single comma? I > take it this would be the chromatic uv in the > linear case?by COMMA, gene means COMMAtic unison vector.
Message: 6114 - Contents - Hide Contents Date: Mon, 20 Jan 2003 20:16:31 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> wrote:> I earlier referred to a "uselessnes" figure-of-demerit for notational > commas. This was obtained by multiplying the power of 3 in the comma > (number of fifths) by the size of the comma in cents. > > I have since realised that for the purpose of our notation (or any notation > based on a chain of best fifths) the ideal power of 3 in a comma is not > zero, but > > ideal_power_of_three = 7 * size_of_comma / size_of_apotome > > or > > 7 * comma_in_cents / 113.685 > > This ideal value would ensure that the comma always represented the same > fraction of an apotome irrespective of the size of the fifth. > > This can be used to show that a near-ideal half-apotome comma cannot exist. > It would need to have 3 to the power of 3.5, but the power of 3must be an> integer. The best we could do is find a near 57 cent comma with 3^3 or 3^4, > which is what we have in '|)) the 5:49 comma 392:405 (56.5 c). > > We can then define the "slope" of a comma as > > slope = actual_power_of_three - ideal_power_of_three > > For example (|( as the 5:11' comma 44:45 (38.9 c) has a slope of > > 2 - 7 * 38.9/113.7 = -0.4 > > which tells us that it should be extremely good at representing 2/7 of an > apotome across a wide range of fifth sizes. In fact it works for ETs > 49,56,63,70,77,84 and 91. > > I believe we should not use a comma for notating temperaments if it has an > absolute value of slope greater than 8. The following have slopes between 7 > and 8 and so I think they should also be avoided if possible > '| as 5' comma > ~|( as 7:13 comma > |~ as 23 comma > )|~ as 19' comma > |)) as 49' diesisEver since we started this notation project I've been basing my choice of commas primarily on musical considerations involving as little math as possible, and the results have been in general agreement with those arrived at when all of the mathematical analysis is brought in. I have two questions/comments in light of the above: 1) What are the most common symbols that we have proposed involving a slope greater than 8? 2) The choice of 23 vs. 23' comma and 19' vs. 19 comma involves respelling of notes using different nominals. For example for 16:19 with 16 as C, 19 will be Eb)| using the 19 (512:513) comma and D#)!~ using the 19' comma (19456:19683). Likewise, 23 will be F#|~ using the 23 comma (729:736) but Gb~|\ using the 23' comma (16384:16767). If one of these tones occurs in a heptatonic scale, we would certainly wish to notate the tones of the scale using all 7 nominals, so the choice of which comma applies should be automatic. I would imagine that a common use of the 23rd harmonic would be as a leading tone to the 24th harmonic (or dominant), in which case (with C as tonic) I would expect it to be spelled as F#|~ leading to G. And I would assume that a much less common usage would use it descending to the 22nd (11th) or 21st harmonics, in which case I would expect it to be spelled as Gb~|\. So I never expected that the status of 729:736 as the principal 23 comma would be brought into question, especially after all this time. What are the slopes for the five commas you listed above, taken to two decimal places?> This also says that we should equally allow both > '/|) as 7' diesis 57344:59049 > and its apotome complement > .(|\ as 7" diesis 27:28 > as you first suggested, since they have slopes of 6.9 and -6.9. In fact > true apotome complements always have the same absolute value of slope, but > opposite signs.This is good, since it wouldn't be very appropriate for a comma to be permitted but for its apotome complement to be rejected if both are notated with single-shaft symbols.> This suggests we modify "uselessness" to use slope instead ofnumber of fifths.> > uselessness = ABS(slope) * size_of_comma > > but in fact I find uselessness fairly useless now, and I'm happy to simply > limit the absolute slope to 8-- depending on your answers to my questions in points 1) and 2) above --> and the comma size to 70.17 cents, the > largest that could conceivably be notated as '((|I don't know how you got something that large. Two 7:11 commas plus a 5' comma are ~68.25c, and two 13:17 commas plus a 5' comma are ~69.19c. But I don't know what reason we would have to use anything with two convex left flags, so I don't think I would have any problem with any of these as an upper limit. --George
Message: 6115 - Contents - Hide Contents Date: Mon, 20 Jan 2003 20:27:45 Subject: Re: heuristic and straightness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>>>>> shortening the unison vectors makes the temperament worse, >>>>>> but in a given temperament, this would be counteracted by >>>>>> an increase in straighness, which makes the temperament >>>>>> better. >>>>>>>>>> You lost me. >>>>>>>> well, there are probably too many counterfactuals here. why >>>> don't we start again with any examples from the archives >>>> which came up in connection with straightness. your choice. >>>>>> Good idea. I did find some talk between you and Gene, >>> referring to results I couldn't find. :( At one point, you >>> mention low-badness blocks of one note... If you pick the >>> example, I'm happy to attempt to build the block and find the >>> alternate versions... no promises, though, since I'm new to >>> this. >>>> what are we talking about, anyway? (no offence) >> I'm trying to figure out what you meant in the top paragraph, > there...if you have a set of unison vectors A and a set of unison vectors B, where both sets have the same straightness, and vectors in A and B are about the same size intervals in JI, but the vectors in B are shorter than those in A, then the temperament defined by B will have larger error than the temperament defined by A. but if the straightness is different, and the vectors are the same length, then it is the straighter set that will have lower error. does the paragraph make more sense now?>>>>>> shortening the unison vectors makes the temperament worse, >> If we heuristically ignore the sizes, then yes. >>>>>>> but in a given temperament, this would be counteracted by >>>>>> an increase in straighness, which makes the temperament >>>>>> better. >> ??? If straightness is maximal when the uvs are maximally > orthogonal, how does this mean the uvs have gotten shorter?any basis for the temperament that uses much longer unison vectors will have to have much less straightness, because the area/volume/etc. enclosed by the UVs must remain constant. otherwise, you have torsion.
Message: 6116 - Contents - Hide Contents Date: Mon, 20 Jan 2003 20:30:42 Subject: Re: New file uploaded to tuning-math From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:> Paul, > > These are tha bomb! > > -Carljust one more to do . . . q, r, s, t, and u are each a factor of 10 zoom of the previous one, and i haven't done s yet (i'm away from the office). then i'm hoping we can get gene to produce an ultimate {2,3,7} list and an ultimate {2,5,7} list . . .
Message: 6117 - Contents - Hide Contents Date: Mon, 20 Jan 2003 20:37:07 Subject: Re: Superparticular temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>> In the 5-limit case, it is clearly not so, unless we >> think meantone is the last word. In the 7-limit case, >> we do miss some important systems, but the inclusion >> of some fairly obvious commas would catch them. > //>> There are 150 7-limit superparticular linear temperaments; >> I dunno Gene; looks like a bust to me. I wonder what > other easy stuff can be done to fractions to study commas. > There are jacks, but they're a subset of the above... > While we would expect superparticulars to be the smallest > intervals of a given complexity, there must be a cleaner > way of doing this... has anything been done on 'badness for > commas'?the heurstic. note that |n-d| is right in there, so if you define badness function that falls off instead of being log-flat, you can easily arrange for the superparticulars to be the "least bad". however, once a temperament is defined by more than one comma, this approach will miss, since straightness is being ignored. speaking of which, i was wondering -- in the ultimate 5-limit list, are there any commas which would make the list if one used heuristic log-flat badness? when i get back to work, i can calculate heuristic log-flat badness for each of them, and then i'm hoping gene can run a search to determine if anything's missing . . .
Message: 6118 - Contents - Hide Contents Date: Mon, 20 Jan 2003 03:42:34 Subject: Superparticular temperaments From: Gene Ward Smith Let us call a temperament "superparticular" if its kernel has a basis (not necessarily reduced) consisting of superparticular commas. Because these increase in number as the prime limit increases, and since commas of ever higher complexity appear, it seems likely that however we define "interesting" that sooner or later all of the linear temperaments of interest will be superparticular ones. This raises the question of when this happens for our purposes. In the 5-limit case, it is clearly not so, unless we think meantone is the last word. In the 7-limit case, we do miss some important systems, but the inclusion of some fairly obvious commas would catch them. This makes it seem plausible that we might not actually miss anything of significance in the 11-limit, and could probably insure that we do not merely by including some obvious additions. There are 150 7-limit superparticular linear temperaments; temperaments which are not superparticular include the following: Hexidecimal [36/35, 135/128] Double Wide [50/49, 875/864] Porcupine [64/63, 250/243] Superpythagorean [64/63, 245/243] Muggles [126/125, 525/512] Quartaminorthirds [126/125, 1029/1024] Supermajor Seconds [81/80, 1029/1024] Schismic [225/224, 3125/3087] Superkleismic [875/864, 1029/1024] Semififth [81/80, 6144/6125] Diaschismic [126/125, 2048/2025] Quartaminorthirds [126/125, 1029/1024] Octafiths [245/243, 2401/2400] Magic [225/224, 245/243] Orwell [225/224, 1728/1715] Tritonic [225/224, 50421/50000] Supersupermajor [245/243, 1029/1024] Shrutar [245/243, 2048/2025] Hemiwuerschmidt [2401/2400, 3136/3125] Hemikleismic [4000/3969, 6144/6125] Hemithird [1029/1024, 3136/3125] Wizard [225/224, 118098/117649] Duodecimal [225/224, 250047/250000] Slender [225/224, 589824/588245] Amity [4375/4374, 5120/5103] Hemififth [2401/2400, 5120/5103] If we toss out Hexidecimal, Slender, and Wizard, none of which are essential, we find that the TM reduced bases for the remaining temperaments employ superparticulars plus 250/243, 245/243, 525/512, 875/864, 1029/1024, 1728/1715, 2048/2025, 3125/3087, 3136/3125, 5120/5103, 4000/3969, 6144/6125 and 250047/250000, not all of which may be necessary (I didn't check.) Here are the superparticular 7-limit linear temperaments, in order of increasing geometric badness. Some absurdly funky systems are included. Ennealimmal [18, 27, 18, -34, 22, 1] [[9, 15, 22, 26], [0, -2, -3, -2]] generators [133.3333333, 48.99915090] comp 58.840778 rms .130449 bad 451.645972 Meantone [1, 4, 10, 12, -13, 4] [[1, 2, 4, 7], [0, -1, -4, -10]] generators [1200., 503.3520320] comp 15.101806 rms 3.665035 bad 835.864548 Miracle [6, -7, -2, 15, 20, -25] [[1, 1, 3, 3], [0, 6, -7, -2]] generators [1200., 116.5729472] comp 24.926629 rms 1.637405 bad 1017.380174 [0, 2, 2, -1, -3, 3] [[2, 3, 5, 6], [0, 0, -1, -1]] generators [600.0000000, 273.4076908] comp 4.275602 rms 59.723378 bad 1091.789279 [0, 0, 1, 2, -1, 0] [[1, 1, 2, 3], [0, 0, 0, -1]] generators [1200., 593.9303320] comp 1.661977 rms 405.957997 bad 1121.323723 [0, 1, 0, -3, 0, 2] [[1, 2, 3, 3], [0, 0, -1, 0]] generators [1200., 570.6132559] comp 2.050856 rms 287.792504 bad 1210.458384 [0, 0, 1, 3, -2, 0] [[1, 2, 3, 3], [0, 0, 0, 1]] generators [1200., 206.0696683] comp 1.661977 rms 473.700237 bad 1308.439092 [0, 0, 1, 2, -2, 0] [[1, 2, 2, 3], [0, 0, 0, -1]] generators [1200., 193.9303319] comp 1.661977 rms 511.941281 bad 1414.067236 [0, 0, 3, 7, -5, 0] [[3, 5, 7, 9], [0, 0, 0, -1]] generators [400.0000000, 193.9303319] comp 4.985930 rms 61.312549 bad 1524.199406 [2, 3, 1, -6, 4, 0] [[1, 2, 3, 3], [0, -2, -3, -1]] generators [1200., 270.7929767] comp 6.691597 rms 34.566097 bad 1547.782446 Pajara [2, -4, -4, 2, 12, -11] [[2, 3, 5, 6], [0, 1, -2, -2]] generators [600.0000000, 108.8143299] comp 11.925109 rms 10.903178 bad 1550.521642 Tripletone [3, 0, -6, -14, 18, -7] [[3, 5, 7, 8], [0, -1, 0, 2]] generators [400.0000000, 88.72066409] comp 14.168743 rms 8.100679 bad 1626.237917 [1, -1, -2, -2, 6, -4] [[1, 2, 2, 2], [0, -1, 1, 2]] generators [1200., 478.6926428] comp 4.972221 rms 65.953083 bad 1630.556687 Blackwood [0, 5, 0, -14, 0, 8] [[5, 8, 12, 14], [0, 0, -1, 0]] generators [240.0000000, 90.61325640] comp 10.254281 rms 15.815352 bad 1662.988586 [1, -1, 3, 10, -2, -4] [[1, 2, 2, 4], [0, -1, 1, -3]] generators [1200., 459.5087948] comp 6.535126 rms 43.659491 bad 1864.603865 Catakleismic [6, 5, 22, 37, -18, -6] [[1, 0, 1, -3], [0, 6, 5, 22]] generators [1200., 316.7238784] comp 34.269866 rms 1.610555 bad 1891.474472 [2, 1, 3, 4, 1, -3] [[1, 1, 2, 2], [0, 2, 1, 3]] generators [1200., 322.2119006] comp 6.245166 rms 48.926006 bad 1908.216790 Dominant seventh [1, 4, -2, -16, 6, 4] [[1, 2, 4, 2], [0, -1, -4, 2]] generators [1200., 497.7740225] comp 9.836560 rms 20.163282 bad 1950.956873 [0, 1, 1, 0, -2, 2] [[1, 2, 3, 3], [0, 0, -1, -1]] generators [1200., 273.4076908] comp 2.137801 rms 442.481365 bad 2022.224915 [0, 0, 2, 5, -3, 0] [[2, 3, 5, 6], [0, 0, 0, -1]] generators [600.0000000, 193.9303319] comp 3.323954 rms 186.002433 bad 2055.079021 [0, 1, 1, -1, -2, 2] [[1, 2, 2, 3], [0, 0, 1, 1]] generators [1200., 326.5923086] comp 2.137801 rms 458.017472 bad 2093.227909 [2, 1, -1, -5, 7, -3] [[1, 1, 2, 3], [0, 2, 1, -1]] generators [1200., 318.5700997] comp 6.245166 rms 53.747748 bad 2096.274853 [1, 1, 0, -3, 3, -1] [[1, 2, 3, 3], [0, -1, -1, 0]] generators [1200., 540.2785958] comp 3.054488 rms 225.884103 bad 2107.475631 [1, -1, 0, 3, 3, -4] [[1, 2, 2, 3], [0, -1, 1, 0]] generators [1200., 442.1793558] comp 3.917133 rms 142.097096 bad 2180.328488 [1, 1, 2, 2, 0, -1] [[1, 1, 2, 2], [0, 1, 1, 2]] generators [1200., 484.4129532] comp 3.529341 rms 188.646684 bad 2349.829555 [0, 2, -2, -10, 3, 3] [[2, 3, 5, 5], [0, 0, -1, 1]] generators [600.0000000, 291.2560966] comp 6.120838 rms 62.740535 bad 2350.552312 [0, 1, 0, -3, 0, 1] [[1, 1, 2, 3], [0, 0, 1, 0]] generators [1200., 229.3867439] comp 2.050856 rms 561.132227 bad 2360.128218 Augmented [3, 0, 6, 14, -1, -7] [[3, 5, 7, 9], [0, -1, 0, -2]] generators [400.0000000, 110.2596913] comp 12.019943 rms 16.598678 bad 2398.160778 [0, 1, 1, 0, -1, 1] [[1, 1, 2, 2], [0, 0, 1, 1]] generators [1200., 326.5923086] comp 2.137801 rms 526.643864 bad 2406.863715 [0, 0, 1, 3, -1, 0] [[1, 1, 3, 3], [0, 0, 0, -1]] generators [1200., 193.9303319] comp 1.661977 rms 875.848123 bad 2419.238652 [1, 2, 1, -3, 1, 1] [[1, 2, 3, 3], [0, -1, -2, -1]] generators [1200., 406.8431437] comp 3.917133 rms 157.889659 bad 2422.648543 [0, 1, 0, -2, 0, 1] [[1, 1, 2, 2], [0, 0, -1, 0]] generators [1200., 170.6132563] comp 2.050856 rms 577.847450 bad 2430.432628 Neutral thirds [2, 1, -4, -12, 12, -3] [[1, 1, 2, 4], [0, 2, 1, -4]] generators [1200., 358.0334333] comp 9.849244 rms 25.068246 bad 2431.810368 [0, 1, 1, -1, -1, 1] [[1, 1, 2, 3], [0, 0, -1, -1]] generators [1200., 273.4076908] comp 2.137801 rms 539.762916 bad 2466.820304 Beatles [2, -9, -4, 16, 12, -19] [[1, 1, 5, 4], [0, 2, -9, -4]] generators [1200., 356.3080304] comp 19.942653 rms 6.245316 bad 2483.820838 Diminished [4, 4, 4, -2, 5, -3] [[4, 6, 9, 11], [0, 1, 1, 1]] generators [300.0000000, 85.69820677] comp 11.405897 rms 19.136993 bad 2489.617178 [1, 2, 3, 1, -2, 1] [[1, 2, 3, 4], [0, -1, -2, -3]] generators [1200., 460.9163575] comp 4.972221 rms 100.967191 bad 2496.209750 Nonkleismic [10, 9, 7, -9, 17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]] generators [1200., 309.9514712] comp 27.531739 rms 3.320167 bad 2516.675801 [1, 0, 1, 2, 1, -2] [[1, 2, 2, 3], [0, -1, 0, -1]] generators [1200., 557.7664030] comp 3.054488 rms 271.083238 bad 2529.178945 Sharp [2, 1, 6, 11, -4, -3] [[1, 1, 2, 1], [0, 2, 1, 6]] generators [1200., 360.2272895] comp 9.849244 rms 26.099830 bad 2531.881842 [0, 0, 1, 1, -1, 0] [[1, 1, 1, 2], [0, 0, 0, 1]] generators [1200., 206.0696683] comp 1.661977 rms 917.874641 bad 2535.322905 [0, 0, 2, 4, -3, 0] [[2, 3, 4, 5], [0, 0, 0, 1]] generators [600.0000000, 206.0696683] comp 3.323954 rms 231.177173 bad 2554.199699 [1, 2, 0, -6, 3, 1] [[1, 2, 3, 3], [0, -1, -2, 0]] generators [1200., 352.0867411] comp 4.352116 rms 135.480508 bad 2566.124420 Tertiathirds [4, -3, 2, 13, 8, -14] [[1, 2, 2, 3], [0, -4, 3, -2]] generators [1200., 125.4687958] comp 14.729697 rms 12.188571 bad 2644.480844 Decimal [4, 2, 2, -1, 8, -6] [[2, 4, 5, 6], [0, -2, -1, -1]] generators [600.0000000, 249.0224992] comp 10.574200 rms 23.945252 bad 2677.407574 Decitone [2, 6, 6, -3, -4, 5] [[2, 3, 4, 5], [0, 1, 3, 3]] generators [600.0000000, 128.5123366] comp 11.925109 rms 18.863889 bad 2682.600333 [2, -1, 1, 5, 4, -6] [[1, 2, 2, 3], [0, -2, 1, -1]] generators [1200., 288.4250230] comp 6.691597 rms 59.930923 bad 2683.555226 [1, -3, -4, -1, 9, -7] [[1, 2, 1, 1], [0, -1, 3, 4]] generators [1200., 532.1557550] comp 8.756575 rms 35.154715 bad 2695.579144 [2, -2, 1, 8, 4, -8] [[1, 2, 2, 3], [0, -2, 2, -1]] generators [1200., 218.7617340] comp 8.112184 rms 41.524693 bad 2732.637321 [1, 1, 1, 0, 1, -1] [[1, 2, 3, 3], [0, -1, -1, -1]] generators [1200., 514.3017931] comp 2.851474 rms 336.706254 bad 2737.726652 [0, 0, 4, 9, -6, 0] [[4, 6, 9, 11], [0, 0, 0, 1]] generators [300.0000000, 6.069667048] comp 6.647907 rms 63.402645 bad 2802.058943 [1, 0, -1, -2, 4, -2] [[1, 1, 2, 3], [0, 1, 0, -1]] generators [1200., 466.5645474] comp 3.529341 rms 225.688309 bad 2811.229153 [1, 4, 3, -4, -2, 4] [[1, 2, 4, 4], [0, -1, -4, -3]] generators [1200., 496.0501669] comp 7.402240 rms 51.362639 bad 2814.321331 Injera [2, 8, 8, -4, -7, 8] [[2, 3, 4, 5], [0, 1, 4, 4]] generators [600.0000000, 93.65102578] comp 15.871133 rms 11.218941 bad 2825.971103 Squares [4, 16, 9, -24, -3, 16] [[1, 3, 8, 6], [0, -4, -16, -9]] generators [1200., 425.9591136] comp 28.922922 rms 3.443812 bad 2880.870808 [1, -1, 1, 5, 1, -4] [[1, 2, 2, 3], [0, -1, 1, -1]] generators [1200., 374.4750766] comp 4.352116 rms 154.263172 bad 2921.885210 [1, 0, 0, 0, 3, -2] [[1, 2, 2, 3], [0, -1, 0, 0]] generators [1200., 549.7582051] comp 2.851474 rms 360.237866 bad 2929.059955 Flattone [1, 4, -9, -32, 17, 4] [[1, 2, 4, -1], [0, -1, -4, 9]] generators [1200., 506.5439220] comp 19.685796 rms 7.652394 bad 2965.536695 Hemifourth [2, 8, 1, -20, 4, 8] [[1, 2, 4, 3], [0, -2, -8, -1]] generators [1200., 252.7423121] comp 15.298626 rms 12.690078 bad 2970.086942 [2, -2, -2, 1, 9, -8] [[2, 3, 5, 6], [0, 1, -1, -1]] generators [600.0000000, 167.6687407] comp 8.311748 rms 43.142169 bad 2980.483611 [1, -1, -5, -9, 11, -4] [[1, 2, 2, 1], [0, -1, 1, 5]] generators [1200., 441.3069864] comp 8.956788 rms 37.258834 bad 2989.054558 Semisixths [7, 9, 13, 5, -1, -2] [[1, -1, -1, -2], [0, 7, 9, 13]] generators [1200., 443.6203855] comp 24.364978 rms 5.052931 bad 2999.683372 [0, 0, 2, 5, -4, 0] [[2, 4, 5, 6], [0, 0, 0, 1]] generators [600.0000000, 6.069667048] comp 3.323954 rms 288.509783 bad 3187.648638 Kleismic [6, 5, 3, -7, 12, -6] [[1, 0, 1, 2], [0, 6, 5, 3]] generators [1200., 316.6640534] comp 16.383068 rms 12.273810 bad 3294.350652 [0, 2, 0, -6, 0, 3] [[2, 3, 5, 6], [0, 0, -1, 0]] generators [600.0000000, 170.6132563] comp 4.101712 rms 197.096298 bad 3315.956645 Pentadecimal [3, 5, 9, 7, -6, 1] [[1, 2, 3, 4], [0, -3, -5, -9]] generators [1200., 158.7318223] comp 14.382059 rms 16.143425 bad 3339.164305 [0, 0, 1, 4, -2, 0] [[1, 2, 4, 4], [0, 0, 0, -1]] generators [1200., 593.9303320] comp 1.661977 rms 1211.073885 bad 3345.188138 [1, -3, -2, 4, 6, -7] [[1, 2, 1, 2], [0, -1, 3, 2]] generators [1200., 513.7543330] comp 7.402240 rms 61.746458 bad 3383.283558 [0, 1, -1, -6, 2, 2] [[1, 2, 3, 3], [0, 0, -1, 1]] generators [1200., 291.2560966] comp 3.060419 rms 364.029464 bad 3409.559323 Heptadecal [0, 0, 7, 16, -11, 0] [[7, 11, 16, 20], [0, 0, 0, -1]] generators [171.4285715, 79.64461697] comp 11.633838 rms 25.354978 bad 3431.699329 [0, 1, 0, -4, 0, 2] [[1, 2, 3, 4], [0, 0, -1, 0]] generators [1200., 170.6132563] comp 2.050856 rms 838.919678 bad 3528.505242 [2, 3, 5, 3, -2, 0] [[1, 2, 3, 4], [0, -2, -3, -5]] generators [1200., 287.3658368] comp 8.353937 rms 51.525959 bad 3595.906950 [0, 1, 0, -2, 0, 2] [[1, 2, 2, 2], [0, 0, 1, 0]] generators [1200., 229.3867439] comp 2.050856 rms 861.314591 bad 3622.698490 [1, 2, 4, 4, -4, 1] [[1, 2, 3, 4], [0, -1, -2, -4]] generators [1200., 348.5748041] comp 6.139951 rms 97.801004 bad 3687.000243 [0, 2, 0, -5, 0, 3] [[2, 3, 4, 5], [0, 0, 1, 0]] generators [600.0000000, 229.3867439] comp 4.101712 rms 219.922942 bad 3699.993093 [1, 2, -2, -10, 6, 1] [[1, 2, 3, 2], [0, -1, -2, 2]] generators [1200., 454.5534905] comp 6.535126 rms 87.356149 bad 3730.795007 [1, 0, 2, 4, 0, -2] [[1, 1, 2, 2], [0, 1, 0, 2]] generators [1200., 398.8312176] comp 4.006648 rms 238.412445 bad 3827.288461 [2, 5, 3, -7, 1, 3] [[1, 1, 1, 2], [0, 2, 5, 3]] generators [1200., 315.3245950] comp 9.333509 rms 44.754977 bad 3898.802316 [0, 0, 5, 12, -8, 0] [[5, 8, 12, 14], [0, 0, 0, 1]] generators [240.0000000, 46.06966903] comp 8.309884 rms 57.395986 bad 3963.432270 [2, 4, 1, -9, 4, 2] [[1, 2, 3, 3], [0, -2, -4, -1]] generators [1200., 203.6485512] comp 8.112184 rms 60.827594 bad 4002.913457 [0, 4, 0, -11, 0, 6] [[4, 6, 9, 11], [0, 0, 1, 0]] generators [300.0000000, 29.38674378] comp 8.203424 rms 60.054354 bad 4041.428204 [2, 4, 4, -2, -1, 2] [[2, 3, 4, 5], [0, 1, 2, 2]] generators [600.0000000, 196.6785324] comp 8.311748 rms 61.297826 bad 4234.770049 [0, 2, 4, 4, -6, 3] [[2, 3, 5, 6], [0, 0, -1, -2]] generators [600.0000000, 152.3110691] comp 6.467985 rms 104.261749 bad 4361.772496 [1, 4, 5, 0, -5, 4] [[1, 2, 4, 5], [0, -1, -4, -5]] generators [1200., 520.6704216] comp 8.756575 rms 57.900863 bad 4439.699141 [1, 1, 1, -1, 2, -1] [[1, 1, 2, 3], [0, 1, 1, 1]] generators [1200., 285.6982066] comp 2.851474 rms 548.057280 bad 4456.201822 [0, 1, -1, -4, 1, 1] [[1, 1, 2, 2], [0, 0, -1, 1]] generators [1200., 291.2560966] comp 3.060419 rms 485.134310 bad 4543.847061 [2, 1, 0, -3, 6, -3] [[1, 1, 2, 3], [0, 2, 1, 0]] generators [1200., 417.2373637] comp 5.542165 rms 148.387948 bad 4557.824204 [0, 0, 1, 0, -1, 0] [[1, 1, 0, 2], [0, 0, 0, -1]] generators [1200., 193.9303319] comp 1.661977 rms 1673.375576 bad 4622.142544 [2, 1, 2, 2, 2, -3] [[1, 1, 2, 2], [0, 2, 1, 2]] generators [1200., 420.5480918] comp 5.542165 rms 155.619746 bad 4779.953169 [2, 0, 0, 0, 6, -5] [[2, 3, 5, 6], [0, 1, 0, 0]] generators [600.0000000, 250.2417952] comp 5.702948 rms 148.630126 bad 4833.989876 [0, 3, 0, -9, 0, 5] [[3, 5, 7, 9], [0, 0, 1, 0]] generators [400.0000000, 96.05341086] comp 6.152568 rms 133.979799 bad 5071.684361 [1, 1, 3, 5, -2, -1] [[1, 2, 3, 4], [0, -1, -1, -3]] generators [1200., 458.2361773] comp 4.722914 rms 231.090354 bad 5154.683175 [3, 5, 1, -12, 7, 1] [[1, 2, 3, 3], [0, -3, -5, -1]] generators [1200., 157.4375200] comp 10.922722 rms 43.469957 bad 5186.220813 [0, 2, 2, -1, -4, 4] [[2, 4, 5, 6], [0, 0, 1, 1]] generators [600.0000000, 26.59230899] comp 4.275602 rms 287.723620 bad 5259.809056 [0, 1, 0, -4, 0, 1] [[1, 1, 3, 4], [0, 0, -1, 0]] generators [1200., 570.6132559] comp 2.050856 rms 1255.321105 bad 5279.894151 [0, 1, 0, -1, 0, 1] [[1, 1, 2, 1], [0, 0, -1, 0]] generators [1200., 570.6132559] comp 2.050856 rms 1277.867746 bad 5374.725567 [6, 7, 5, -8, 9, -3] [[1, 3, 4, 4], [0, -6, -7, -5]] generators [1200., 286.0989745] comp 17.627870 rms 17.388827 bad 5403.435759 [0, 2, 6, 8, -9, 3] [[2, 3, 5, 7], [0, 0, -1, -3]] generators [600.0000000, 294.0505316] comp 9.353407 rms 62.423802 bad 5461.222835 [0, 1, 2, 1, -4, 2] [[1, 2, 2, 3], [0, 0, 1, 2]] generators [1200., 65.87074902] comp 3.233992 rms 522.430484 bad 5463.947575 [2, -3, -1, 6, 7, -9] [[1, 1, 3, 3], [0, 2, -3, -1]] generators [1200., 300.3607896] comp 9.333509 rms 63.538867 bad 5535.149429 [1, 2, 6, 8, -7, 1] [[1, 2, 3, 5], [0, -1, -2, -6]] generators [1200., 435.7072302] comp 8.956788 rms 70.186382 bad 5630.635755 [1, 1, -2, -8, 6, -1] [[1, 2, 3, 2], [0, -1, -1, 2]] generators [1200., 541.5638500] comp 5.319858 rms 199.354602 bad 5641.911563 [4, 9, -8, -44, 24, 5] [[1, 1, 1, 4], [0, 4, 9, -8]] generators [1200., 177.4647360] comp 27.604056 rms 7.456594 bad 5681.804332 [2, 11, 6, -17, -4, 13] [[1, 1, -1, 1], [0, 2, 11, 6]] generators [1200., 363.1818634] comp 19.942653 rms 14.296447 bad 5685.831463 [0, 3, 0, -8, 0, 5] [[3, 5, 7, 8], [0, 0, -1, 0]] generators [400.0000000, 37.27992253] comp 6.152568 rms 155.873944 bad 5900.467447 [2, 2, 2, -1, 2, -1] [[2, 3, 4, 5], [0, 1, 1, 1]] generators [600.0000000, 285.6982066] comp 5.702948 rms 184.020398 bad 5985.009686 [0, 2, 1, -3, -1, 2] [[1, 1, 3, 3], [0, 0, -2, -1]] generators [1200., 582.3155032] comp 3.845808 rms 405.273928 bad 5994.098393 [1, 0, -2, -4, 5, -2] [[1, 1, 2, 3], [0, 1, 0, -2]] generators [1200., 290.2267039] comp 4.722914 rms 272.846672 bad 6086.096297 [1, 0, -2, -4, 6, -2] [[1, 2, 2, 2], [0, -1, 0, 2]] generators [1200., 467.6680326] comp 4.722914 rms 274.917379 bad 6132.285324 [0, 0, 3, 8, -5, 0] [[3, 5, 8, 9], [0, 0, 0, -1]] generators [400.0000000, 60.59699869] comp 4.985930 rms 249.612405 bad 6205.240006 [2, 2, 1, -3, 4, -2] [[1, 2, 3, 3], [0, -2, -2, -1]] generators [1200., 336.7290694] comp 5.670929 rms 194.291227 bad 6248.296967 [1, 0, 2, 4, -1, -2] [[1, 2, 2, 3], [0, -1, 0, -2]] generators [1200., 255.7142370] comp 4.006648 rms 401.256919 bad 6441.467346 [0, 1, 3, 5, -6, 2] [[1, 2, 3, 4], [0, 0, -1, -3]] generators [1200., 394.0505317] comp 4.676704 rms 311.829091 bad 6820.187515 [0, 2, 0, -5, 0, 2] [[2, 2, 4, 5], [0, 0, 1, 0]] generators [600.0000000, 29.38674378] comp 4.101712 rms 405.448549 bad 6821.283937 [1, 1, 4, 7, -3, -1] [[1, 1, 2, 1], [0, 1, 1, 4]] generators [1200., 541.9922676] comp 6.138398 rms 182.241625 bad 6866.851562 [1, 0, 3, 6, -2, -2] [[1, 2, 2, 4], [0, -1, 0, -3]] generators [1200., 541.4436503] comp 5.319858 rms 246.900497 bad 6987.502433 [2, 2, 2, -1, 3, -2] [[2, 3, 5, 6], [0, -1, -1, -1]] generators [600.0000000, 114.3017931] comp 5.702948 rms 216.492360 bad 7041.115475 [2, 0, 1, 2, 4, -4] [[1, 2, 2, 3], [0, -2, 0, -1]] generators [1200., 352.7582023] comp 5.670929 rms 229.676703 bad 7386.274048 [0, 2, 2, -1, -2, 2] [[2, 2, 4, 5], [0, 0, 1, 1]] generators [600.0000000, 26.59230899] comp 4.275602 rms 405.399657 bad 7411.017522 [0, 1, 2, 3, -2, 1] [[1, 1, 3, 3], [0, 0, -1, -2]] generators [1200., 370.4928875] comp 3.233992 rms 730.932254 bad 7644.606586 [2, 1, 10, 20, -10, -3] [[1, 1, 2, 0], [0, 2, 1, 10]] generators [1200., 334.5486096] comp 15.864707 rms 30.441426 bad 7661.770385 [1, 2, -1, -9, 5, 1] [[1, 2, 3, 3], [0, -1, -2, 1]] generators [1200., 224.5581578] comp 5.297379 rms 278.453554 bad 7814.026023 [2, 0, 0, 0, 5, -4] [[2, 3, 4, 5], [0, -1, 0, 0]] generators [600.0000000, 149.7582050] comp 5.702948 rms 251.915620 bad 8193.208145 [2, -1, 0, 3, 6, -7] [[1, 1, 3, 3], [0, 2, -1, 0]] generators [1200., 484.9100284] comp 6.537864 rms 203.252537 bad 8687.758969 [4, 4, 12, 17, -8, -3] [[4, 6, 9, 10], [0, 1, 1, 3]] generators [300.0000000, 125.9743488] comp 18.891658 rms 24.610643 bad 8783.408917 [2, 3, 2, -3, 2, 0] [[1, 2, 3, 3], [0, -2, -3, -2]] generators [1200., 252.5079222] comp 6.537864 rms 209.085647 bad 8937.087487 [4, -4, -1, 9, 13, -16] [[1, 1, 3, 3], [0, 4, -4, -1]] generators [1200., 189.5576458] comp 15.650108 rms 37.374885 bad 9154.076279 [2, -1, 2, 7, 2, -6] [[1, 2, 2, 3], [0, -2, 1, -2]] generators [1200., 235.1959794] comp 7.234335 rms 178.776933 bad 9356.399761 [1, 4, 0, -12, 3, 4] [[1, 2, 4, 3], [0, -1, -4, 0]] generators [1200., 476.6658998] comp 7.962349 rms 148.388409 bad 9407.677587 [0, 1, 3, 3, -3, 1] [[1, 1, 2, 3], [0, 0, -1, -3]] generators [1200., 194.0505316] comp 4.676704 rms 436.527296 bad 9547.531318 [3, 2, 0, -6, 9, -4] [[1, 1, 2, 3], [0, 3, 2, 0]] generators [1200., 267.7220970] comp 8.482929 rms 136.530107 bad 9824.717681 [8, 13, 4, -27, 16, 2] [[1, 2, 3, 3], [0, -8, -13, -4]] generators [1200., 63.00613990] comp 28.070300 rms 12.646377 bad 9964.608569 [2, 3, 2, -4, 2, 1] [[1, 1, 1, 2], [0, 2, 3, 2]] generators [1200., 505.3868143] comp 6.537864 rms 234.327566 bad 10016.019720 [1, 4, 1, -8, 1, 4] [[1, 2, 4, 3], [0, -1, -4, -1]] generators [1200., 518.8451304] comp 7.416607 rms 187.506165 bad 10313.974820 [1, 1, 3, 6, -3, -1] [[1, 2, 3, 3], [0, -1, -1, -3]] generators [1200., 16.13091449] comp 4.722914 rms 494.772960 bad 11036.366530 [10, 14, 14, -7, 6, -1] [[2, 2, 3, 4], [0, 5, 7, 7]] generators [600.0000000, 139.6810044] comp 32.695891 rms 10.427607 bad 11147.334260 [3, 3, 3, -1, 3, -2] [[3, 5, 7, 8], [0, 1, 1, 1]] generators [400.0000000, 19.03154112] comp 8.554423 rms 157.513437 bad 11526.541520 [0, 8, 4, -13, -6, 12] [[4, 6, 9, 11], [0, 0, 2, 1]] generators [300.0000000, 17.68449710] comp 15.383232 rms 58.863881 bad 13929.774210 [1, -1, 11, 29, -15, -4] [[1, 2, 2, 7], [0, -1, 1, -11]] generators [1200., 453.7389106] comp 18.940672 rms 39.612986 bad 14211.121100 [5, 8, 3, -15, 9, 1] [[1, 2, 3, 3], [0, -5, -8, -3]] generators [1200., 103.5769428] comp 17.205721 rms 49.518880 bad 14659.412800 [4, 8, 5, -11, 3, 4] [[1, 1, 1, 2], [0, 4, 8, 5]] generators [1200., 198.4673924] comp 15.650108 rms 61.043960 bad 14951.244890 [0, 4, 2, -7, -4, 8] [[2, 4, 5, 6], [0, 0, 2, 1]] generators [600.0000000, 17.68449710] comp 7.691616 rms 287.546442 bad 17011.522780 [5, 6, 11, 8, -3, -2] [[1, 3, 4, 6], [0, -5, -6, -11]] generators [1200., 348.0209534] comp 18.827083 rms 52.250677 bad 18520.725480 [0, 2, -8, -23, 12, 3] [[2, 3, 4, 7], [0, 0, 1, -4]] generators [600.0000000, 234.1580374] comp 15.230577 rms 90.719918 bad 21044.343580 [4, 4, 3, -4, 7, -4] [[1, 3, 4, 4], [0, -4, -4, -3]] generators [1200., 464.8797794] comp 11.251842 rms 184.323319 bad 23336.061130 [0, 1, -4, -15, 8, 2] [[1, 2, 3, 3], [0, 0, -1, 4]] generators [1200., 70.92670834] comp 7.615289 rms 442.895836 bad 25684.690840 [4, 0, 1, 2, 9, -8] [[1, 3, 2, 3], [0, -4, 0, -1]] generators [1200., 473.9276617] comp 11.251842 rms 225.636070 bad 28566.418700 [1, 0, 7, 14, -7, -2] [[1, 1, 2, 0], [0, 1, 0, 7]] generators [1200., 452.0455031] comp 11.512379 rms 226.287912 bad 29991.037240 [4, 7, 0, -21, 12, 2] [[1, 2, 3, 3], [0, -4, -7, 0]] generators [1200., 99.13652417] comp 15.874272 rms 133.944397 bad 33752.984350 [0, 1, -4, -8, 4, 1] [[1, 1, 1, 4], [0, 0, 1, -4]] generators [1200., 539.2427832] comp 7.615289 rms 624.335655 bad 36206.861660
Message: 6119 - Contents - Hide Contents Date: Mon, 20 Jan 2003 08:23:14 Subject: Re: Superparticular temperaments From: Carl Lumma>In the 5-limit case, it is clearly not so, unless we >think meantone is the last word. In the 7-limit case, >we do miss some important systems, but the inclusion >of some fairly obvious commas would catch them. // >There are 150 7-limit superparticular linear temperaments;I dunno Gene; looks like a bust to me. I wonder what other easy stuff can be done to fractions to study commas. There are jacks, but they're a subset of the above... While we would expect superparticulars to be the smallest intervals of a given complexity, there must be a cleaner way of doing this... has anything been done on 'badness for commas'? -C.
Message: 6120 - Contents - Hide Contents Date: Mon, 20 Jan 2003 08:26:20 Subject: Re: Calculating geometric complexity and badness From: Carl Lumma I wrote..>> In the 5 limit, if q = 2^u0 3^u1 5^u2 is a comma, >> then [u0, u1, u2] is the wedgie >> That's all the wedgie is for a single comma? I > take it this would be the chromatic uv in the > linear case?D'oh, I meant commatic. -C.
Message: 6121 - Contents - Hide Contents Date: Tue, 21 Jan 2003 11:43:29 Subject: Re: Superparticular temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> If the badness can be calculated solely from the wedge product of the > commatic UVs, then it's naturally independent of the specific UVs > chosen. I'm not sure if that's what Gene's proposing.That's the idea. Badness can be calculated from the wedgie, or equivalently, from the mapping.
Message: 6122 - Contents - Hide Contents Date: Tue, 21 Jan 2003 12:15:17 Subject: Re: Superparticular temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> it should be a book of temperaments. for each, a notation, a > generalized keyboard, a halbestadt-imitation keyboard, a > horogram . . .A set of poptimal generators, a MOS list, a mapping, a wedgie, a TM reduced basis for the kernel, and a chord indexing function. I can never figure out these Erv Wilson things which consist of nothing but pictures, so if you can explain horograms that would be nice.
Message: 6123 - Contents - Hide Contents Date: Tue, 21 Jan 2003 12:51:49 Subject: A 13-limit comma list From: Gene Ward Smith Here is a list of 70 13-limit commas, with epimericity less than 0.25 and size less than 50 cents. There is a very good chance it is complete, since I took it from a similar list of 499 commas with epimericity less than 0.5 after I could add nothing further to it. [36/35, 77/75, 40/39, 128/125, 45/44, 143/140, 49/48, 50/49, 55/54, 56/55, 64/63, 65/64, 66/65, 78/77, 81/80, 245/242, 91/90, 99/98, 100/99, 105/104,121/120, 245/243, 126/125, 275/273, 144/143, 169/168, 176/175, 896/891, 196/195, 1029/1024, 640/637, 225/224, 1188/1183, 243/242, 1573/1568, 325/324, 351/350, 352/351, 364/363, 385/384, 847/845, 441/440, 1375/1372, 540/539, 4000/3993, 625/624, 676/675, 729/728, 2200/2197, 1575/1573, 5632/5625, 1001/1000, 4459/4455, 10985/10976, 1716/1715, 2080/2079, 2401/2400, 3025/3024, 4096/4095, 4225/4224, 4375/4374, 6656/6655, 140625/140608, 9801/9800, 10648/10647, 196625/196608, 151263/151250, 1990656/1990625, 123201/123200, 5767168/5767125]
Message: 6124 - Contents - Hide Contents Date: Tue, 21 Jan 2003 16:54:40 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> wrote [#5360:> > Perhaps we should ditch the (/| symbol entirely and use |)) for the 31' > comma since |)) is the more obvious symbol for the 49'-diesis.I had occasion to look at some old postings, and here's a sentence from one that you might be interested in: BEGIN QUOTE From: Manuel op de Coul, 18 Feb 2002 (#34402) Subject: Notation individualists Dave wrote:>... Some N-ET's should only be notated as every kth step of kN-ET.You probably write "should" because of your consistency requirement for the commas used, which I don't share. How would your system take care of for instance 105-tET? END QUOTE To answer the question, we would do it as a subset of 210-ET. But what symbols would we use for 10 and 11 degrees of 210 if not (/| and |\)? --George
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