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Message: 6250 - Contents - Hide Contents Date: Tue, 28 Jan 2003 03:28:57 Subject: Re: Calculating geometric complexity II From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Gene Ward Smith wrote: >>> I'm not sure if we are speaking the same language, but I'm using lexicographical order; that is, z[0,1], z[0,2] .... z[0,n] would >> be followed by z[1,2]...z[1,n] and so forth. This gives a linear temperament wedgie as the product of two vals, and puts the 2-part, which is related to the generators column of the period-generator matrix, at the beginning. >> Oh, that's good. It should be the same as my invariant. But are > 7-limit wedge products taken from vectors or vals?Either, but I follow the val ordering.> I get 7-limit meantone as 21.97, 11-limit meantone as 31.72 and > h12^h19^h22 in the 11-limit as 29.52. The planar temperament with > 441:440 and 225:224 is 34.44.I'm afraid I don't know what these numbers mean. I have 7-limit meantone: h50^h31 = 126/125^81/80 = [1, 4, 10, 12, -13, 4] h12^h19^h22 = 100/99^225/224 = [-1,2,-2,2,2,-8,-5,-2,14,-6] 225/224^441/440 = h41^h31^h12 = [1,-2,3,-2,6,-6,5,-13,11,-4]>> I'm simply being unsophisticated about it--I store the wedgies as lists, and reverse the ordering when I compute from commas, etc. in order to get the lists to be the same. >> That sounds like taking the complement. I thought you said you didn't > have to because you were using duality.I said I was using duality to identify compliments. I started out trying to do things the right way, as you seem to be doing, but it gave me trouble, so I settled for a fast, simple-minded approach, which means I have a separate program for each kind of wedge product I want to take. And how can you be sure that> reversing the list will do the trick? Some of the coefficients should > be negated if you aren't using a special ordering.Sometimes I do. I simply make the wedge product of "vectors", or what I would call intervals, correspond to the wedge product for vals, which I take as the basis.
Message: 6251 - Contents - Hide Contents Date: Tue, 28 Jan 2003 08:31:32 Subject: Top 135 11-limit planar temperaments From: Gene Ward Smith I took the same set of commas used for the 11-limit linear search and did a planar search. Below is everything which came of it with a badness less than 7000. Commas: [27/25, 15/14, 16/15, 35/33, 81/77, 21/20, 22/21, 25/24, 126/121, 80/77, 28/27, 125/121, 33/32, 36/35, 77/75, 128/125, 45/44, 49/48, 50/49, 55/54,56/55, 64/63, 625/616, 81/80, 245/242, 99/98, 100/99, 121/120, 245/243, 126/125, 1728/1715, 1331/1323, 2200/2187, 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, 41503/41472, 43923/43904, 2401/2400, 3025/3024, 117649/117612, 4375/4374, 250047/250000, 9801/9800, 151263/151250, 1771561/1771470, 3294225/3294172] Temperaments: Odin [6, -12, 18, 12, -24, 12, -8, 17, -10, 2] [[6, 0, 0, 8, 17], [0, 1, 0, -2, -4], [0, 0, 1, 2, 3]] [200., 1901.938862, 2786.350249] [9801/9800, 151263/151250] bad 1016.28998247272 comp 60.8872902832050 rms .035131850660382 Thor [2, 8, -6, -14, 10, -2, -2, 5, 14, -25] [[2, 0, 0, 2, 5], [0, 1, 0, 7, 5], [0, 0, 1, -4, -3]] [600., 1901.942580, 2786.188554] [3025/3024, 4375/4374] bad 1035.92638329421 comp 38.5752608313015 rms .112087379468910 Baldur [8, -4, 12, -2, -14, 10, -10, 22, 4, -23] [[2, 0, 1, 3, 7], [0, 2, 1, 1, -2], [0, 0, 2, 1, 3]] [600., 950.9556220, 617.7106490] [2401/2400, 9801/9800] bad 1449.76377668198 comp 43.7518714674570 rms .114500045202118 Portent [3, 0, -3, 1, 4, 1, -10, 11, -10, 17] [[1, 1, 0, 3, 5], [0, 3, 0, -1, 4], [0, 0, 1, 0, -1]] [1200., 233.6891150, 2784.876116] [385/384, 441/440] bad 1764.18257049630 comp 18.3842363308401 rms 1.21738947132896 Marvel [1, -2, -3, -2, -1, -4, 5, 12, -9, -19] [[1, 0, 0, -5, 12], [0, 1, 0, 2, -1], [0, 0, 1, 2, -3]] [1200., 1900.698768, 2783.222880] [225/224, 385/384] bad 1870.58824531347 comp 16.9370609095528 rms 1.58446655877984 Freya [8, -4, -6, -2, 13, -8, -10, 21, -18, 11] [[1, 1, 3, 3, 2], [0, 2, 3, 2, 1], [0, 0, 4, 2, -3]] [1200., 350.9629470, -466.6890570] [2401/2400, 3025/3024] bad 2010.98776597205 comp 41.0390934677348 rms .186386748151713 Prodigy [1, -2, 3, -2, 6, -6, 5, -13, 11, -4] [[1, 0, 0, -5, -13], [0, 1, 0, 2, 6], [0, 0, 1, 2, 3]] [1200., 1900.058168, 2783.119618] [225/224, 441/440] bad 2285.58316112181 comp 17.1265830917959 rms 1.88286946716198 Thrush [1, -3, 5, 2, -2, -4, 1, -5, 10, -8] [[1, 0, 0, -1, -5], [0, 1, 0, -2, -2], [0, 0, 1, 3, 5]] [1200., 1902.348537, 2791.393074] [126/125, 176/175] bad 2547.25063891524 comp 15.3842837174898 rms 2.74396443686221 Tyr [6, -12, 0, 12, 3, -6, -8, 16, -32, 36] [[3, 0, 0, 4, 8], [0, 2, 0, -4, 1], [0, 0, 1, 2, 0]] [400., 950.9401100, 2786.210081] [3025/3024, 102487/102400] bad 2857.01276885075 comp 47.1866974264405 rms .186794130466611 [0, 0, 1, 0, 1, -2, 0, -4, 6, -2] [[1, 0, 4, 6, 0], [0, 1, -1, -2, 0], [0, 0, 0, 0, 1]] [1200., 1918.862743, 4179.117018] [16/15, 21/20] bad 2921.33279504550 comp 4.97222074067480 rms 52.9913559112739 [1, -1, 0, 1, -1, 1, -2, 5, -5, 3] [[1, 0, 0, 2, 5], [0, 1, 0, -1, -1], [0, 0, 1, 1, 0]] [1200., 1879.053186, 2792.327260] bad 2996.69876379486 comp 5.91311172315335 rms 35.2454351574855 Ennealimmic [0, 0, 18, 0, -27, 18, 0, 1, 22, -34] [[9, 1, 1, 12, 0], [0, 2, 3, 2, 0], [0, 0, 0, 0, 1]] [133.3333330, 884.3105210, 4151.266830] [2401/2400, 4375/4374] bad 3075.58283502960 comp 58.8407776477704 rms .115806095418123 Indra [5, -4, -3, -3, 9, -9, 0, 10, -8, -6] [[1, 0, 0, 0, 2], [0, 1, 3, 3, 0], [0, 0, 5, 4, -3]] [1200., 1901.732842, -583.8771670] [540/549, 1375/1372] bad 3085.38152624593 comp 27.8143651821687 rms .756198817751678 Loki [2, -6, 8, -8, 4, 20, 21, -18, -30, 30] [[2, 0, 0, -21, -18], [0, 1, 0, 4, 2], [0, 0, 1, 3, 4]] [600., 1902.157169, 2786.807115] [5632/5625, 9801/9800] bad 3246.38438780793 comp 44.4356991774615 rms .246643625789512 [0, 1, -1, 1, -1, 1, -4, 4, -1, 3] [[1, 0, 4, 0, -1], [0, 1, -1, 0, 1], [0, 0, 0, 1, 1]] [1200., 1940.929772, 3437.027245] bad 3283.51818866258 comp 5.61567074804458 rms 43.9374991940989 [1, 0, -1, 2, -2, 2, -6, 9, -6, 6] [[1, 0, 0, 6, 9], [0, 1, 0, -2, -2], [0, 0, 1, 0, -1]] [1200., 1911.208075, 2806.289811] [56/55, 64/63] bad 3287.79356404259 comp 9.57116278552758 rms 11.6009294089245 Wonder [4, -2, 0, -1, 10, -5, -5, -2, 1, 13] [[1, 1, 1, 2, 2], [0, 2, 1, 1, 5], [0, 0, 2, 1, 0]] [1200., 350.4014210, 617.6277210] [243/242, 441/440] bad 3323.60893011696 comp 22.0117554169689 rms 1.46208701772333 Skadi [6, -14, 1, -4, 11, -25, 22, 1, -6, -41] [[1, 0, 5, 8, 1], [0, 1, 1, 3, 2], [0, 0, 6, 14, 1]] [1200., 1901.910634, -852.6376980] [3025/3024, 703125/702464] bad 3329.41757716090 comp 59.1250020573940 rms .123862643866053 Heimdall [2, 8, -24, -14, 37, -20, -2, 4, -8, 9] [[1, 0, 0, 1, 2], [0, 2, 0, 14, 37], [0, 0, 1, -4, -12]] [1200., 950.9853460, 2786.261233] [4375/4374, 117649/117612] bad 3374.25698568491 comp 78.7460888054938 rms .0613204982504788 Minerva [1, -2, 4, -2, 2, 4, 5, -9, -2, 8] [[1, 0, 0, -5, -9], [0, 1, 0, 2, 2], [0, 0, 1, 2, 4]] [1200., 1900.667337, 2786.672949] [99/98, 176/175] bad 3418.39320450227 comp 13.8705775701070 rms 4.77073894768755 [0, 0, 0, 0, 2, -2, 0, -3, 3, 1] [[2, 3, 0, 1, 0], [0, 0, 1, 1, 0], [0, 0, 0, 0, 1]] [600., 2675.614809, 4049.362941] bad 3429.80427625012 comp 4.27560160479231 rms 90.7352957149063 Ares [1, 0, 2, 2, -2, -4, -6, 2, 12, -8] [[1, 0, 0, 6, 2], [0, 1, 0, -2, -2], [0, 0, 1, 0, 2]] [1200., 1909.527660, 2790.324322] [64/63, 100/99] bad 3446.14554928860 comp 11.4062212681811 rms 7.84294438996711 [0, 0, 1, 0, 1, 0, 0, -4, 3, 3] [[1, 0, 4, 3, 0], [0, 1, -1, 0, 0], [0, 0, 0, 0, 1]] [1200., 1951.404028, 4239.678629] bad 3449.23559856868 comp 3.91713341569379 rms 113.580022537560 [12, 10, 2, -46, 22, 26, 16, 2, -1, -37] [[2, 0, 4, -6, 1], [0, 1, 1, 3, 2], [0, 0, 6, -5, 1]] [600., 1901.944923, -252.6019720] [9801/9800, 1771561/1771470] bad 3599.72148039770 comp 108.199696824479 rms .0295599564509325 Varuna [2, -4, 6, 8, -12, 0, -9, 12, 3, -6] [[2, 0, 0, 9, 12], [0, 1, 0, -4, -6], [0, 0, 1, 2, 3]] [600., 1901.338331, 2786.378135] [441/440, 8019/8000] bad 3604.16117147728 comp 27.4049690301862 rms .916707774439380 Agni [4, -2, -6, -1, 3, -3, -5, 23, -19, -2] [[1, 1, 1, 2, 5], [0, 2, 1, 1, 0], [0, 0, 2, 1, -3]] [1200., 350.6699170, 616.8206970] [385/384, 1375/1372] bad 3661.03469122519 comp 25.8506216717486 rms 1.07752254294854 [1, -10, 4, -1, 2, -16, 22, -9, 2, -35] [[1, 0, 0, -22, -9], [0, 1, 0, 1, 2], [0, 0, 1, 10, 4]] [1200., 1902.216951, 2786.696695] bad 3689.07864872765 comp 40.9627801052499 rms .343513930669761 Apollo [1, -2, 2, -2, -2, 8, 5, 2, -14, 6] [[1, 0, 0, -5, 2], [0, 1, 0, 2, -2], [0, 0, 1, 2, 2]] [1200., 1903.626101, 2781.116799] [100/99, 225/224] bad 3783.90063671516 comp 14.5345146867970 rms 4.69827448567886 Zeus [2, 3, 1, -1, 1, 2, -11, 3, 10, 4] [[1, 0, 1, 4, 2], [0, 1, 1, -1, 1], [0, 0, 2, -3, 1]] [1200., 1902.084928, -156.7887080] [121/120, 176/175] bad 4038.86681626742 comp 15.7875289629287 rms 4.07825269244272 [1, 1, -2, -2, 2, -2, -2, 5, 1, -6] [[1, 0, 0, 2, 5], [0, 1, 0, 2, 2], [0, 0, 1, -1, -2]] [1200., 1905.077356, 2819.545578] bad 4120.83748370374 comp 8.61226945529442 rms 18.9317806545970 [4, -2, 16, -1, 8, 0, -5, -36, 38, 19] [[1, 1, 1, 2, -3], [0, 2, 1, 1, 8], [0, 0, 2, 1, 8]] [1200., 351.0551660, 617.8770160] bad 4150.05970855924 comp 48.4863010792248 rms .253516780100018 [0, 0, 1, 0, 1, 3, 0, -4, -2, 10] [[1, 0, 4, -2, 0], [0, 1, -1, 3, 0], [0, 0, 0, 0, 1]] [1200., 1937.485616, 4196.602633] bad 4263.57694300046 comp 6.53512582974509 rms 39.0516348452538 [1, -1, 1, 1, 0, -1, -2, 1, 1, 1] [[1, 0, 0, 2, 1], [0, 1, 0, -1, 0], [0, 0, 1, 1, 1]] [1200., 1901.430980, 2868.772160] bad 4297.46887769516 comp 5.01275383685932 rms 76.3874117536148 [1, -3, 2, 2, -2, 2, 1, 2, -8, 6] [[1, 0, 0, -1, 2], [0, 1, 0, -2, -2], [0, 0, 1, 3, 2]] [1200., 1909.065474, 2795.586753] bad 4354.57924454404 comp 10.4836397276885 rms 12.2367469308100 [4, 4, 0, -6, -2, -2, -11, 17, 17, -31] [[2, 1, 0, 7, 8], [0, 2, 0, 3, -1], [0, 0, 1, -1, 0]] [600., 650.6253940, 2784.681994] bad 4434.31807040888 comp 28.2069024337545 rms 1.04939345373849 [42, -16, 7, -56, 77, -20, 8, 7, -4, -24] [[1, 0, 11, 4, 2], [0, 1, 7, 4, 3], [0, 0, 42, 16, 7]] [1200., 1901.959384, -564.9380620] bad 4442.28465166623 comp 206.699026701649 rms .00723203142559257 [2, -5, 8, 0, 4, -10, 6, -18, 21, -12] [[1, 0, 0, -3, -9], [0, 1, 0, 0, 2], [0, 0, 2, 5, 8]] [1200., 1901.674053, 1393.524953] bad 4449.40081377987 comp 29.0694348417321 rms .976585021148923 [0, 0, 2, 0, -3, 1, 0, 0, 4, -6] [[1, 0, 0, 2, 0], [0, 2, 3, 1, 0], [0, 0, 0, 0, 1]] [1200., 941.9274440, 4142.971906] bad 4496.66454378065 comp 6.69159712103067 rms 38.8209381002883 [10, 4, -12, -16, 23, -10, -12, 26, -4, -14] [[1, 0, 3, 0, -1], [0, 2, 3, 2, 1], [0, 0, 5, -2, -6]] [1200., 950.9827840, -733.3644130] bad 4498.84469293319 comp 64.1711108498080 rms .136380446391062 [10, 4, 6, -16, -4, 8, -12, 27, 18, -48] [[2, 0, 1, 2, 6], [0, 1, 4, 0, 2], [0, 0, 5, -2, 3]] [600., 1901.887933, -1084.267089] bad 4510.72411538062 comp 55.6533187885891 rms .195217432227642 [3, 1, 5, -3, 3, 6, -6, -6, 8, 12] [[1, 0, 0, 2, -2], [0, 1, 0, 1, 1], [0, 0, 3, -1, 5]] [1200., 1902.851683, 929.9710570] bad 4520.22999251532 comp 20.7925558036751 rms 2.29293024678861 [0, 2, -2, 16, -16, -20, -30, 30, 33, -36] [[2, 0, 30, 0, 33], [0, 1, -8, 0, -10], [0, 0, 0, 1, 1]] [600., 1901.747174, 3368.579148] bad 4528.71193061989 comp 56.2415835909200 rms .190910952661430 [1, -1, 1, -1, 2, -1, 1, -2, 1, 0] [[1, 0, 0, -1, -2], [0, 1, 0, 1, 2], [0, 0, 1, 1, 1]] [1200., 1891.805669, 2733.530853] bad 4550.06595854289 comp 5.85389490599201 rms 54.8788988885822 [0, 2, -1, 16, -8, 19, -30, 15, -26, 77] [[1, 0, 15, 0, -13], [0, 1, -8, 1, 10], [0, 0, 0, 2, 1]] [1200., 1901.756442, 733.4287120] bad 4635.27790843236 comp 54.8983800591757 rms .207575911526645 [0, 2, -2, 4, -4, -8, -11, 11, 14, -16] [[2, 0, 11, 0, 14], [0, 1, -2, 0, -4], [0, 0, 0, 1, 1]] [600., 1904.449829, 3372.039360] bad 4636.38347760921 comp 20.0726293671069 rms 2.56843562735841 [0, 2, 0, -1, 0, -2, -3, 0, 10, -8] [[1, 1, 2, 0, 4], [0, 2, 1, 0, -2], [0, 0, 0, 1, 0]] [1200., 342.7110090, 3342.837584] bad 4646.90101665519 comp 8.41342500821798 rms 22.6324386363937 [2, -2, 4, 0, -4, 4, -1, 4, -2, -2] [[2, 0, 0, 1, 4], [0, 1, 0, 0, -2], [0, 0, 1, 1, 2]] [600., 1901.942580, 2777.587751] bad 4659.00557393774 comp 12.7064245264378 rms 8.09533503160310 [1, 3, 2, -3, -2, 0, -5, 2, 16, -16] [[1, 0, 0, 5, 2], [0, 1, 0, 3, -2], [0, 0, 1, -3, 2]] [1200., 1903.304567, 2780.387946] bad 4668.05450884960 comp 15.7836276214392 rms 4.71648929658626 [0, 0, 2, 0, 4, -4, 0, -11, 12, 2] [[2, 0, 11, 12, 0], [0, 1, -2, -2, 0], [0, 0, 0, 0, 1]] [600., 1907.232521, 4157.670513] bad 4688.04945823466 comp 11.9251094548197 rms 9.54632575563360 [2, -4, 0, -4, 5, -10, 10, -1, 2, -23] [[1, 1, 0, -3, 2], [0, 2, 0, 4, 5], [0, 0, 1, 2, 0]] [1200., 350.1386570, 2783.638547] bad 4699.80013684326 comp 23.2170394162774 rms 1.80951543867521 [1, -1, 1, 6, -10, 4, -10, 17, -7, 2] [[1, 0, 0, 10, 17], [0, 1, 0, -6, -10], [0, 0, 1, 1, 1]] [1200., 1903.401073, 2788.404313] bad 4725.98288138721 comp 21.2327511023451 rms 2.27497418074208 [6, 0, 6, -10, -2, 10, -1, 10, 1, -17] [[2, 1, 0, 2, 3], [0, 3, 0, 5, -1], [0, 0, 1, 0, 1]] [600., 433.9278370, 2785.854451] bad 4742.23673530433 comp 32.9477999681190 rms .761056989457215 [0, 3, -3, 0, 0, 0, -7, 7, 2, 0] [[3, 0, 7, 0, 2], [0, 1, 0, 0, 0], [0, 0, 0, 1, 1]] [400., 1905.928439, 3366.473574] bad 4749.03332977513 comp 11.5777906862646 rms 10.4121635984525 [0, 1, 0, 1, 0, -1, -4, 0, 5, 1] [[1, 0, 4, 0, 5], [0, 1, -1, 0, -1], [0, 0, 0, 1, 0]] [1200., 1912.530379, 3382.632651] bad 4770.90052914812 comp 5.20540008245681 rms 77.1729452821527 [0, 0, 2, 0, -1, -1, 0, -3, 7, -5] [[1, 1, 2, 3, 0], [0, 2, 1, -1, 0], [0, 0, 0, 0, 1]] [1200., 331.8288160, 4097.311597] bad 4777.27735548368 comp 6.24516582199003 rms 49.0140421889222 [3, -12, 3, 11, -1, -7, 2, 5, -22, 19] [[1, 2, 0, -8, 1], [0, 3, 0, -11, -1], [0, 0, 1, 4, 1]] [1200., -165.9852710, 2785.769277] bad 4809.19708363336 comp 40.0595447124733 rms .473486446948210 [1, -1, 0, -1, -1, 1, 1, 5, -5, -4] [[1, 0, 0, -1, 5], [0, 1, 0, 1, -1], [0, 0, 1, 1, 0]] [1200., 1875.774222, 2713.274074] bad 4822.64578053677 comp 6.26868584035112 rms 49.0167035588588 [2, -3, 4, 4, -4, -2, -5, 4, 4, -2] [[1, 0, 1, 4, 4], [0, 1, 0, -2, -2], [0, 0, 2, 3, 4]] [1200., 1902.349260, 790.0166980] bad 4827.10298605382 comp 15.2804909068804 rms 5.28863138105167 [2, 0, -2, -1, 1, -1, -4, 10, -4, -3] [[1, 0, 0, 2, 5], [0, 2, 0, 1, 1], [0, 0, 1, 0, -1]] [1200., 951.4862280, 2793.241000] bad 4834.44470918493 comp 10.6309677091032 rms 13.1194189272698 [1, -3, -4, 2, 3, -1, 1, 8, -20, 13] [[1, 0, 0, -1, 8], [0, 1, 0, -2, 3], [0, 0, 1, 3, -4]] [1200., 1900.444933, 2788.289556] bad 4894.18134365504 comp 17.9776359054058 rms 3.57148315218545 [4, -8, 12, 4, -12, 12, 1, 5, -13, 8] [[4, 0, 0, -1, 5], [0, 1, 0, -1, -3], [0, 0, 1, 2, 3]] [300., 1901.870393, 2785.537589] bad 4922.46283598179 comp 37.8832370980165 rms .557268815348865 [14, -16, 12, 10, -11, 4, -18, 38, -28, 13] [[1, 0, 5, 7, 7], [0, 2, 3, 2, 1], [0, 0, 7, 8, 6]] [1200., 950.9433560, -866.6405490] bad 4984.07511702271 comp 67.6904137910735 rms .132210715868233 [1, -2, 1, -2, 2, -2, 5, -2, -1, -6] [[1, 0, 0, -5, -2], [0, 1, 0, 2, 2], [0, 0, 1, 2, 1]] [1200., 1894.806413, 2782.197963] bad 4989.55528617158 comp 9.29436892650194 rms 18.9457617330096 [0, 0, 1, 0, -4, 10, 0, 4, -13, 12] [[1, 0, -4, -13, 0], [0, 1, 4, 10, 0], [0, 0, 0, 0, 1]] [1200., 1896.439144, 4147.010107] bad 4999.23585680722 comp 15.1018056341298 rms 5.64067964887982 [2, -3, 1, -1, 1, -1, 3, 3, -6, -3] [[1, 0, 1, 0, 2], [0, 1, 1, 2, 1], [0, 0, 2, 3, 1]] [1200., 1907.306700, -149.5641090] bad 5051.05976228133 comp 10.9393279472889 rms 12.7616227340794 [30, -26, 5, -10, 55, -46, -8, 5, -3, 13] [[1, 0, 17, 15, 3], [0, 1, 25, 22, 6], [0, 0, 30, 26, 5]] [1200., 1901.959890, -2172.088780] bad 5054.14609645120 comp 150.891809580502 rms .0180710598561816 [0, 0, 2, 0, -1, 3, 0, -3, 1, 4] [[1, 1, 2, 2, 0], [0, 2, 1, 3, 0], [0, 0, 0, 0, 1]] [1200., 338.1800570, 4135.477131] bad 5071.90931761187 comp 6.24516582199003 rms 52.0369153334717 [3, -2, -2, -15, 12, -18, 20, -4, 16, -60] [[1, 0, 1, -6, -2], [0, 1, 0, 5, 4], [0, 0, 3, 2, -2]] [1200., 1902.268388, 528.8302800] bad 5082.65078152533 comp 48.4939242527045 rms .310364433347883 [0, 0, 1, 0, -2, 3, 0, 1, -2, 1] [[1, 0, -1, -2, 0], [0, 1, 2, 3, 0], [0, 0, 0, 0, 1]] [1200., 1933.962964, 4153.457761] bad 5108.71749544833 comp 4.97222074067480 rms 92.6693006392783 [1, 0, 0, 2, -6, 0, -6, 13, 0, -10] [[1, 0, 0, 6, 13], [0, 1, 0, -2, -6], [0, 0, 1, 0, 0]] [1200., 1907.277788, 2793.834156] bad 5115.16292740936 comp 13.1462917409038 rms 8.16302838574596 [1, -1, 1, 1, 2, -3, -2, -2, 4, 2] [[1, 0, 0, 2, -2], [0, 1, 0, -1, 2], [0, 0, 1, 1, 1]] [1200., 1874.652558, 2783.299225] bad 5166.27216976288 comp 7.24905888052384 rms 36.5151604789725 [0, 1, 0, 3, 0, -1, -7, 0, 5, 8] [[1, 0, 7, 0, 5], [0, 1, -3, 0, -1], [0, 0, 0, 1, 0]] [1200., 1881.404744, 3343.845626] bad 5185.30545361514 comp 9.25520386685862 rms 19.8979974517799 [16, -8, 6, -4, -1, 2, -20, 43, -14, -12] [[1, 1, 1, 2, 3], [0, 2, 3, 2, 1], [0, 0, 8, 4, 3]] [1200., 350.9363160, 66.68503200] bad 5260.22319737066 comp 61.1116174300952 rms .180175081218233 [0, 1, -2, -4, 8, -2, 4, -8, 1, 4] [[1, 0, -4, 0, 1], [0, 1, 4, 0, -2], [0, 0, 0, 1, 2]] [1200., 1896.257650, 3368.850068] bad 5286.08883875188 comp 14.6615151257675 rms 6.42225175032022 [2, 8, 12, -14, -17, 16, -2, 6, 36, -59] [[1, 0, 0, 1, 3], [0, 2, 0, 14, -17], [0, 0, 1, -4, 6]] [1200., 950.9551580, 2786.215616] bad 5289.70487905852 comp 60.8245774554324 rms .183330064505816 [0, 2, 0, -1, 0, 5, -3, 0, -1, 8] [[1, 1, 2, 0, 2], [0, 2, 1, 0, 5], [0, 0, 0, 1, 0]] [1200., 347.4239790, 3353.944124] bad 5302.18434287586 comp 9.49320221652931 rms 19.0951464272005 [2, 2, 0, 3, 5, 5, -15, -1, -1, 36] [[1, 1, 0, 6, 2], [0, 2, 0, -3, 5], [0, 0, 1, -1, 0]] [1200., 350.0353330, 2783.690904] bad 5307.34511186710 comp 24.1129366653730 rms 1.85888304024675 [0, 0, 1, 0, -4, -2, 0, 4, 6, -16] [[1, 0, -4, 6, 0], [0, 1, 4, -2, 0], [0, 0, 0, 0, 1]] [1200., 1902.825929, 4161.942947] bad 5312.54032759458 comp 9.83655960404403 rms 17.5062920810931 [1, -1, 2, 1, -2, 0, -2, 2, 2, -2] [[1, 0, 0, 2, 2], [0, 1, 0, -1, -2], [0, 0, 1, 1, 2]] [1200., 1883.854888, 2760.650470] bad 5327.23844444991 comp 6.45343693589445 rms 50.3528967349848 [2, -4, 6, -4, 0, 12, 10, -7, -16, 14] [[2, 0, 0, -10, -7], [0, 1, 0, 2, 0], [0, 0, 1, 2, 3]] [600., 1900.740942, 2783.292216] bad 5330.07264966956 comp 25.7891807374471 rms 1.57811723901127 [3, 6, 3, -4, -1, 2, -16, 5, 26, -12] [[1, 2, 0, 8, 1], [0, 3, 0, 4, -1], [0, 0, 1, -2, 1]] [1200., -166.1887900, 2784.046789] bad 5356.07287304441 comp 29.1232408162276 rms 1.17016532276278 [1, 0, 0, 0, 1, 0, -3, 2, 0, 3] [[1, 0, 0, 3, 2], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0]] [1200., 1960.744316, 2909.707396] bad 5425.30706477891 comp 4.48428118953036 rms 127.406689126362 [0, 0, 6, 0, 7, -2, 0, -25, 20, 15] [[1, 1, 3, 3, 0], [0, 6, -7, -2, 0], [0, 0, 0, 0, 1]] [1200., 116.7221960, 4149.216970] bad 5429.45146744141 comp 24.9266291754662 rms 1.75023785683065 [0, 1, 0, -4, 0, 6, 4, 0, -6, 0] [[1, 0, -4, 0, -6], [0, 1, 4, 0, 6], [0, 0, 0, 1, 0]] [1200., 1891.730262, 3358.625256] bad 5437.41784395894 comp 11.3766284463437 rms 12.4554313320885 [1, 0, 0, 2, -1, 0, -6, 5, 0, 4] [[1, 0, 0, 6, 5], [0, 1, 0, -2, -1], [0, 0, 1, 0, 0]] [1200., 1902.082652, 2781.111944] bad 5444.65984256136 comp 7.25309732593763 rms 38.4292571476626 [0, 0, 0, 0, 0, 3, 0, 0, -5, 7] [[3, 5, 7, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]] [400., 3445.781227, 4228.273263] bad 5506.43666495546 comp 4.98593037019576 rms 99.1985046234926 [0, 0, 0, 0, 0, 3, 0, 0, -5, 7] [[3, 5, 7, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]] [400., 3445.781227, 4228.273263] bad 5506.43666495546 comp 4.98593037019576 rms 99.1985046234926 [6, -6, 6, 7, -2, -5, -14, 10, 4, 7] [[1, 2, 0, 0, 1], [0, 6, 0, -7, -2], [0, 0, 1, 1, 1]] [1200., -83.08195100, 2785.793290] bad 5534.35366902164 comp 30.8076809807602 rms 1.05055643641137 [0, 2, -2, -1, 1, -1, -3, 3, 3, -3] [[1, 1, 2, 0, 1], [0, 2, 1, 0, -1], [0, 0, 0, 1, 1]] [1200., 351.7654310, 3328.499465] bad 5548.06687345962 comp 7.44298259357962 rms 36.7091373473631 [5, -5, 5, 1, 8, -9, -4, -7, 11, 5] [[1, 4, 0, 0, 5], [0, 5, 0, -1, 8], [0, 0, 1, 1, 1]] [1200., -579.5988980, 2787.933793] bad 5708.26336277119 comp 28.1192151191875 rms 1.36143219469305 [1, 0, -1, -3, 3, -3, 2, 1, 2, -9] [[1, 0, 0, -2, 1], [0, 1, 0, 3, 3], [0, 0, 1, 0, -1]] [1200., 1921.494592, 2799.739773] bad 5772.35111379124 comp 9.28313739507226 rms 21.9844595794731 [0, 0, 3, 0, 0, -6, 0, -7, 18, -14] [[3, 0, 7, 18, 0], [0, 1, 0, -2, 0], [0, 0, 0, 0, 1]] [400., 1909.219664, 4160.960951] bad 5845.63240156657 comp 14.1687433668534 rms 7.73576318461447 [1, 1, -1, -2, 3, 1, -2, 1, -1, 4] [[1, 0, 0, 2, 1], [0, 1, 0, 2, 3], [0, 0, 1, -1, -1]] [1200., 1917.964536, 2812.033033] bad 5850.40705217659 comp 7.86842099338344 rms 33.6873704842274 [3, 1, -2, -3, 0, -2, -6, 15, 1, -15] [[1, 0, 0, 2, 5], [0, 1, 0, 1, 0], [0, 0, 3, -1, -2]] [1200., 1899.268919, 927.6248760] bad 5856.84502927373 comp 16.8439337516371 rms 5.02984869721676 [3, -6, 9, 6, -6, -6, -4, -1, 14, -10] [[3, 0, 0, 4, -1], [0, 1, 0, -2, -2], [0, 0, 1, 2, 3]] [400., 1901.316048, 2784.680152] bad 5924.26808915630 comp 29.1330056575679 rms 1.29321719456868 [2, 3, -8, -1, 6, 5, -11, 16, -20, 25] [[1, 0, 1, 4, 4], [0, 1, 1, -1, -1], [0, 0, 2, -3, -8]] [1200., 1901.869607, -156.7846250] bad 5932.17437291159 comp 28.6390579838659 rms 1.35150322545767 [0, 1, 0, 1, 0, 1, -4, 0, 2, 6] [[1, 0, 4, 0, 2], [0, 1, -1, 0, 1], [0, 0, 0, 1, 0]] [1200., 1925.501074, 3435.427219] bad 5950.68546075884 comp 5.36337697012359 rms 89.3246221913220 [0, 1, -1, -4, 4, -4, 4, -4, 7, -12] [[1, 0, -4, 0, 7], [0, 1, 4, 0, -4], [0, 0, 0, 1, 1]] [1200., 1900.205459, 3362.797718] bad 5962.88164251638 comp 11.9317492431541 rms 12.1253949917086 [1, 1, 0, -2, -1, -1, -2, 5, 5, -12] [[1, 0, 0, 2, 5], [0, 1, 0, 2, -1], [0, 0, 1, -1, 0]] [1200., 1881.623704, 2790.367608] bad 6029.09414756640 comp 8.52092102046404 rms 28.4469504258604 [2, 1, -1, -5, -3, -4, 0, 14, 7, -35] [[1, 0, 0, 0, 7], [0, 1, 1, 2, -2], [0, 0, 2, -1, -1]] [1200., 1903.657388, 440.8614670] bad 6032.12596742398 comp 21.1891738616514 rms 2.91867233128147 [1, 6, -4, -3, 3, 6, -12, 8, 0, 12] [[1, 0, 0, 12, 8], [0, 1, 0, 3, 3], [0, 0, 1, -6, -4]] [1200., 1903.200558, 2789.604546] bad 6033.71083981913 comp 23.6613689462420 rms 2.21556667685763 [0, 6, 6, -5, -5, 1, -6, -6, 36, -29] [[1, 0, 1, 0, 6], [0, 6, 5, 0, 1], [0, 0, 0, 1, -1]] [1200., 316.8989880, 3367.203476] bad 6060.66704142351 comp 32.1188280117315 rms 1.03662386026966 [1, 0, -1, 2, 3, 2, -6, 1, -6, 20] [[1, 0, 0, 6, 1], [0, 1, 0, -2, 3], [0, 0, 1, 0, -1]] [1200., 1911.252794, 2784.377076] bad 6097.24566869775 comp 12.1891257953617 rms 11.7544495429905 [1, 4, 3, -7, -6, -3, -1, 6, 27, -48] [[1, 0, 0, 1, 6], [0, 1, 0, 7, -6], [0, 0, 1, -4, 3]] [1200., 1900.938869, 2784.928920] bad 6097.32420980217 comp 30.5888025104082 rms 1.17823820056869 [4, -3, 2, 3, 2, -3, -9, 6, 0, 9] [[1, 0, 1, 3, 2], [0, 1, 1, 0, 1], [0, 0, 4, 3, 2]] [1200., 1901.051086, -78.22830700] bad 6100.54845391948 comp 18.1415826138066 rms 4.35192050616832 [2, 3, 8, -1, 4, 10, -11, -18, 17, 31] [[1, 0, 1, 4, -5], [0, 1, 1, -1, 6], [0, 0, 2, -3, 8]] [1200., 1902.087581, -157.5264690] bad 6107.17540768098 comp 33.7422267279707 rms .923434392736174 [1, -1, 0, 1, 1, -1, -2, 2, -2, 4] [[1, 0, 0, 2, 2], [0, 1, 0, -1, 1], [0, 0, 1, 1, 0]] [1200., 1883.854888, 2839.056406] bad 6125.81003163150 comp 5.35837602749742 rms 92.1680834046060 [1, -1, 0, -1, 1, -1, 1, 2, -2, -3] [[1, 0, 0, -1, 2], [0, 1, 0, 1, 1], [0, 0, 1, 1, 0]] [1200., 1876.360113, 2751.454131] bad 6135.69616331766 comp 5.22191053930940 rms 98.4669026674191 [4, 16, -30, -28, 47, -22, -4, 9, 6, -16] [[1, 1, 0, 8, 14], [0, 2, 1, 10, 16], [0, 0, 2, -8, -15]] [1200., 350.9851650, 1217.630414] bad 6170.82598478571 comp 109.159321116142 rms .0495668402293985 [0, 1, 1, 1, 1, -1, -4, -4, 8, 4] [[1, 0, 4, 0, 8], [0, 1, -1, 0, -1], [0, 0, 0, 1, -1]] [1200., 1942.325909, 3437.591028] bad 6171.98251941751 comp 7.30373127100998 rms 42.8117246192801 [1, -1, 2, -1, 0, 2, 1, -1, -1, 1] [[1, 0, 0, -1, -1], [0, 1, 0, 1, 0], [0, 0, 1, 1, 2]] [1200., 1876.360113, 2673.048195] bad 6187.81876691039 comp 6.34058390955111 rms 61.1243775595360 [1, 0, 1, 2, 2, -2, -6, -2, 6, 8] [[1, 0, 0, 6, -2], [0, 1, 0, -2, 2], [0, 0, 1, 0, 1]] [1200., 1904.075168, 2772.086725] bad 6199.56512459235 comp 9.33257230919805 rms 23.3001017536660 [0, 0, 1, 0, -4, 3, 0, 4, -2, -4] [[1, 0, -4, -2, 0], [0, 1, 4, 3, 0], [0, 0, 0, 0, 1]] [1200., 1903.983644, 4147.084306] bad 6204.98656488581 comp 7.40224022206568 rms 41.6229548624566 [2, 0, 0, -1, -2, 0, -4, 10, 0, -9] [[1, 0, 0, 2, 5], [0, 2, 0, 1, -2], [0, 0, 1, 0, 0]] [1200., 942.7110090, 2763.822953] bad 6215.29346167813 comp 10.1816360114984 rms 18.7896080667385 [5, -4, 3, 9, -8, 1, -19, 23, -7, 11] [[1, 0, 4, 7, 7], [0, 1, 1, -1, -1], [0, 0, 5, 4, 3]] [1200., 1900.951561, -783.0042210] bad 6255.21219785550 comp 31.5942420963977 rms 1.11486477241930 [2, -4, 1, -4, 1, 0, 10, 3, -11, -11] [[1, 0, 1, -3, 2], [0, 1, 1, 4, 1], [0, 0, 2, 4, 1]] [1200., 1899.901252, -158.2562060] bad 6316.82337427400 comp 17.6721673176495 rms 4.81143027826092 [2, -2, 0, 0, 0, 0, -1, 7, -7, 0] [[2, 0, 0, 1, 7], [0, 1, 0, 0, 0], [0, 0, 1, 1, 0]] [600., 1916.088501, 2800.340450] bad 6415.79165308161 comp 9.25518561364888 rms 24.6199643485286 [1, 1, 0, 0, -1, -1, -5, 5, 5, -5] [[1, 0, 0, 5, 5], [0, 1, 0, 0, -1], [0, 0, 1, -1, 0]] [1200., 1878.198137, 2708.743904] bad 6564.31085516441 comp 7.34075273450734 rms 44.9611747617408 [0, 1, -1, -2, 2, 1, 1, -1, -1, 1] [[1, 0, -1, 0, -1], [0, 1, 2, 0, 1], [0, 0, 0, 1, 1]] [1200., 1959.383531, 3406.564894] bad 6568.76050609645 comp 5.61567074804458 rms 87.8980693450637 [1, 4, -9, -7, 16, 1, -1, -1, -13, 23] [[1, 0, 0, 1, -1], [0, 1, 0, 7, 16], [0, 0, 1, -4, -9]] [1200., 1901.481512, 2785.832097] bad 6584.31898000100 comp 34.3486362521353 rms .952219596767383 [2, -2, 2, 0, 4, -4, -1, -4, 5, 2] [[2, 0, 0, 1, -4], [0, 1, 0, 0, 2], [0, 0, 1, 1, 1]] [600., 1893.018151, 2764.536904] bad 6637.68370205685 comp 12.0778528195730 rms 13.0930920487608 [0, 0, 2, 0, 1, 1, 0, -6, 4, 5] [[1, 0, 3, 2, 0], [0, 2, -1, 1, 0], [0, 0, 0, 0, 1]] [1200., 929.3541260, 4094.341969] bad 6673.59061219533 comp 6.69159712103067 rms 57.6149378145246 [1, -1, 1, -1, 0, 1, 1, 1, -2, -1] [[1, 0, 0, -1, 1], [0, 1, 0, 1, 0], [0, 0, 1, 1, 1]] [1200., 1848.103022, 2765.599534] bad 6746.13329653979 comp 5.01275383685932 rms 119.912366216817 [1, -8, 3, 15, -6, 3, -8, 6, -24, 42] [[1, 0, 0, 8, 6], [0, 1, 0, -15, -6], [0, 0, 1, 8, 3]] [1200., 1901.393195, 2786.166712] bad 6782.60738207617 comp 39.2081611778529 rms .704620851966020 [1, 1, 2, -2, -2, 2, -2, 2, 6, -8] [[1, 0, 0, 2, 2], [0, 1, 0, 2, -2], [0, 0, 1, -1, 2]] [1200., 1910.170892, 2803.852328] bad 6793.09143112948 comp 9.28137732639499 rms 25.8842977989784 [4, 6, -7, -2, 7, 7, -22, 19, -10, 29] [[1, 0, 1, 4, 3], [0, 1, 1, -1, 0], [0, 0, 4, -6, -7]] [1200., 1902.497058, -78.79599600] bad 6833.17735874029 comp 37.5154722826377 rms .792677713212396 [0, 1, -1, 1, -1, -2, -4, 4, 4, -4] [[1, 0, 4, 0, 4], [0, 1, -1, 0, -2], [0, 0, 0, 1, 1]] [1200., 1957.739619, 3373.636674] bad 6840.44251062967 comp 6.31048797658235 rms 68.3796444468896 [4, -1, 2, -5, 2, 2, -1, 6, -1, -7] [[1, 0, 3, 1, 3], [0, 1, 3, 2, 2], [0, 0, 4, 1, 2]] [1200., 1900.622770, -1628.696407] bad 6866.68096580402 comp 17.3162049434602 rms 5.50319721956654 [2, -5, 9, 0, 0, 0, 6, -14, 8, 0] [[1, 0, 0, -3, -7], [0, 1, 0, 0, 0], [0, 0, 2, 5, 9]] [1200., 1903.038127, 1394.604900] bad 6875.79407457772 comp 24.1492767266702 rms 2.39917839064525 [3, -6, 3, -6, -1, 8, 15, 5, -25, -5] [[1, 2, 0, -1, 1], [0, 3, 0, 6, -1], [0, 0, 1, 2, 1]] [1200., -166.4319540, 2783.336505] bad 6876.40374165867 comp 28.5566325724214 rms 1.57795234732042 [0, 0, 1, 0, 8, -14, 0, -15, 25, -10] [[1, 0, 15, 25, 0], [0, 1, -8, -14, 0], [0, 0, 0, 0, 1]] [1200., 1902.113831, 4151.084459] bad 6883.00626940537 comp 24.4144735481044 rms 2.33700561136825 [0, 1, -1, -4, 4, 8, 4, -4, -12, 16] [[1, 0, -4, 0, -12], [0, 1, 4, 0, 8], [0, 0, 0, 1, 1]] [1200., 1897.495509, 3367.662350] bad 6922.52880455597 comp 16.5816358914270 rms 6.18295855540234 [0, 1, -2, 8, -16, 10, -15, 30, -18, 6] [[1, 0, 15, 0, -18], [0, 1, -8, 0, 10], [0, 0, 0, 1, 2]] [1200., 1901.667571, 3367.462292] bad 6970.21197579567 comp 35.4232371924615 rms .933308815670537 [0, 3, 0, -5, 0, 4, 1, 0, 4, -8] [[1, 2, 3, 0, 4], [0, 3, 5, 0, 4], [0, 0, 0, 1, 0]] [1200., -164.5380770, 3367.789497] bad 6971.43219652988 comp 13.4226284511329 rms 10.5615603736245 [1, 5, 4, 1, 2, 6, -16, -9, 19, 23] [[1, 0, 0, 16, -9], [0, 1, 0, -1, 2], [0, 0, 1, -5, 4]] [1200., 1901.931251, 2786.246421] bad 6995.89835564255 comp 28.0802795888660 rms 1.67432561646129
Message: 6252 - Contents - Hide Contents Date: Tue, 28 Jan 2003 01:10:43 Subject: dummy math question From: monz hi all, here's what i think is probably a dummy math question, meaning that i don't know how to do it 'cause i'm a dummy ... if x = y^z, and i know x and y, how do i find z? thanks. -monz
Message: 6253 - Contents - Hide Contents
Date: Tue, 28 Jan 2003 10:18:44
Subject: Re: dummy math question
From: manuel.op.de.coul@xxxxxxxxxxx.xxx
Assuming there is a solution:
log x
z = -----
log y
Manuel
Message: 6254 - Contents - Hide Contents Date: Tue, 28 Jan 2003 01:28:36 Subject: Re: dummy math question From: monz thanks, Manuel! it was just what i needed. -monz> From: <manuel.op.de.coul@xxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, January 28, 2003 1:18 AM > Subject: Re: [tuning-math] dummy math question > > > Assuming there is a solution: > > log x > z = ----- > log y
Message: 6255 - Contents - Hide Contents Date: Wed, 29 Jan 2003 11:45:15 Subject: Re: A common notation for JI and ETs From: David C Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" <gdsecor@y...> wrote: Hi George, Thanks for updating the quick reference. It's great.>> Incidentally, I think we should point out that the 5'-comma symbol > should>> stay to the left of any arrow symbol, even in text, so they're > always>> treated as a single compound symbol. e.g. >> Score: ./| # C-notehead >> Text: C#./ >> >> This will also reduce the problem of . being taken as punctuation. > > Yes, absolutely!I note that in the quick reference where you've given a combination of a shorthand ASCII sagittal with a conventional sharp, you've put the sagittal leftmost. You need a note to say that this is how it would appear on a score (but of course the ASCII characters should never appear on a score), but in text the sagittal should be rightmost. On second thoughts, since the ASCII characters never appear on a score, wouldn't it be better to show them in text order and add a note to say that on a score the sagittal should be leftmost (i.e. furthest from the letter name or note head in both cases). This is in contrast to the 5' accent marks which remain the leftmost component of a compound sagittal whether in text or on a score.>> but any other >> combinations of these shorthand symbols would represent multiple > sagittal>> symbols in the obvious way (we may yet find a use for this). >> For anything else, I would suggest going to the sagittal ascii that > we've previously been using, which has characters to indicate each > component of the actual symbol.You may have misunderstood me. What I'm saying is that, for example, /f can be used as equivalent to the pair of symbols /| |) (5-comma up, 7-comma up) but never the single symbol /|) for which n is available. The only exceptions to this are the pairs // \\ ff tt which are taken as equivalent to the single symbols //| \\! |)) !)) [or possibly (/| (\! ] unless the author explicitly states otherwise. There are two possible usages I have in mind for multiple sagittals against a single note. 1. A one-symbol-per-prime notation. Some may yet prefer it. 2. Linear-temperament-specific notations where more than 7 pseudo-nominals are produced by using one lot of sagittals (with the usual 7 nominals) to notate a small proper MOS. And then using a second lot of sagittals to represent the chroma (chromatic unison vector) and its multiples, that offset the other notes from that MOS. For example, in miracle temperament the 10 pseudo-nominals could be (as a chain of secors): F#\ G Ab/ Af Bv Ct C#\ D Eb/ Ef The above combinations of sagittals with sharps and flats would be better notated by double-shaft sagittals. The chroma is t (the 7-comma down) and so the next row could be notated F#\t Gt Ab/t Aft Bvt Ctt C#\t Dt Eb/t Eft where tt is to be taken as two separate sagittals. Whether or not this is a good idea, I see no reason to forbid using the shorthand symbols in this way.> Since the two versions differ > according to whether or not a symbol contains either |, !, X, or Y, > then there's no problem in > determining which ascii version of the notation is being used.Yes. A good feature.> By all means let's use x *only* for the double-sharp. I initially > suggested Y for this purpose (I like its lateral symmetry, > particularly for the upward-pointing legs), which we can compare with > k for appearance: > > up down > /|\ \!/ > ||\ !!/ > |||) !!!) > X\ Y/ k/ > X) Y) k) > /X\ \Y/ \k/ > > I don't see any conflict between Y and y, because they won't ever > occur together, or even in two different symbols in the same ascii > version (as with X and x).The conflict still exists in figuring out which of X and Y are up and down when no straight flags are present. Nothing about the X suggests any direction and if the user happens to have learnt the shorthand ASCII, she may well assume that the Y points up in the same way that lowercase y does. However, the asymmetry of the k is ugly and I can't think of anything better so I'll go with the Y.>> By the way George, I hope you realise I still think there are > serious>> problems with the triple shafts and X shafts. It's only the > availability of>> the dual-symbol version of the notation that allows me to ignore > them. >> The problem that you had with single-symbol notation way back when > included double-shaft symbols, but you didn't mention those in the > above statement, so you need to explain what you mean by that.This problem is not with the existence of single sagittal symbols between sharp and double-sharp or flat and double-flat, but specifically with the appearance of their shafts. There is a Ted Mook type objection (sight-reading in bad light) to the triple-shafts, which I refer to as 2-3-confusability. We addressed that partially by agreeing not to pack the triple-shafts into the same width as the double-shafts, but I believe the problem still remains. There is also the objection that the X-shaft up symbols look like they are _adding_ something to a conventional double sharp whereas they are intended to represent something smaller than a double sharp (or equal in the final case). e.g. /X looks like it ought to mean a double sharp _plus_ a 5-comma, whereas you have it meaning a double sharp _minus_ a 55-comma. Or looking at it another way, you want the X-shaft itself to mean sesqui-sharp so that /X means sesqui-sharp plus a 5-comma, but this is a very problematic redefinition of the X from double-sharp to sesqui-sharp. There is no similar problem with the double-shaft || representing semi-sharp because it does not look like a complete sharp symbol, in fact it contains exactly half the strokes of a conventional sharp. Only when it attains the two straight flags as /||\ does it resemble (and have the same number of strokes as) a conventional sharp symbol. This is very good. Ideally the sesqui-sharp shaft would only look like an X with the addition of the two straight flags. But this seems impossible to me.> I continue to have serious problems with the double-symbol notation > (especially when it results in an occasional _de facto_ triple symbol > whenever a double-flat is modified) which only the availability of > the single-symbol version allows me to ignore.OK. I believe you suggested that the font should contain a single glyph for conventional double-flat which looks like two conventional flats touching each other (and maybe even squashed sideways a bit). I'll go along with that now.> The only problems that I see with the single-symbol version are that: > 1) The performance notation has a steeper learning curve; > 2) The ascii simulation is rather cumbersome, particularly for three- > shaft symbols; > 3) An ascii shorthand does not seem to be feasible; > 4) More symbols are required in a font.Yeah. But it's still worth having. I just think the other problems I mentioned above might yet be ameliorated.> And the only problems that I have with the double-symbol notation > involve the performance version is actually a single problem that has > dual consequences: > 1) Lower efficiency (in contrast with the the single-symbol version, > in which every line segment conveys information);Redundancy in communication is often an asset in preventing errors.> a) Less legibility, i.e., a more cluttered appearance on the > printed page; > b) Less clarity, i.e., in a polyphonic part or score, it is not > always obvious which symbols modify which notes;I agree with those two.> c) Less intuitive, i.e., more symbols preceding a note-head often > symbolize a smaller amount of alteration (e.g., \!# is a smaller > alteration than #), and down-arrow symbols frequently appear when the > pitch is actually being altered upward (e.g., \!#).I suspect that many musicians/composers have more of a 12-pseudo-nominals orientation, rather than 7-nominals. For example Joseph Pehrson. Such a person is likely to find the double-symbol notation _more_ intuitive.> But I would not want to abandon either version, because having both > available immediately puts off criticism from anyone else who might > have problems accepting one version or the other. Indeed.-- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
Message: 6256 - Contents - Hide Contents Date: Wed, 29 Jan 2003 14:38:49 Subject: Re: A common notation for JI and ETs From: David C Keenan Of course this last-ditch effort to improve on the triple and X-shafts is motivated by the fact that I've agreed to draw the outline versions of these symbols for the font. Another problem with the X shafts (these problems have all been mentioned before) is that it is unclear when looking at it, whether the note being modified is one aligned with the arrowhead or one aligned with the point where the shafts cross. Attention is unavoidably drawn to that crossing point, no matter how much one might know that we must look at the head, to be consistent with the other saggitals. This is partly intrinsic to the crossing itself and partly a matter of long habit from reading conventional double-sharp x symbols. This kind of distraction caused by shaft features was the main reason you gave for rejecting a shortened middle shaft for the triple shaft symbols. Any suggestions for better sharp and sesqui-sharp valued shafts. By shaft value I mean currently we have shaft values: | natural 1:1 || semi-sharp 704:729 ||| sharp 2048:2187 X sesqui-sharp 2048:2187 * 704:729
Message: 6257 - Contents - Hide Contents Date: Wed, 29 Jan 2003 09:34:31 Subject: Re: Calculating geometric complexity II From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote: So> what other surprises do you have up your sleeve?Hopefully, none. I changed my 7-limit wedgie (a routine I wrote before the Browne book arrived on the scene) to conform to standard order. My 11-limit linear wedgie already tracks with yours. In neither case do the definitions of geometric complexity need changing.>> h12^h19^h22 = 100/99^225/224 = [-1,2,-2,2,2,-8,-5,-2,14,-6] >> (0, 1, 2): 1 * > (0, 1, 3): 2 > (0, 1, 4): 2 * > (0, 2, 3): -2 * > (0, 2, 4): 2 > (0, 3, 4): 8 * > (1, 2, 3): -5 > (1, 2, 4): 2 * > (1, 3, 4): 14 > (2, 3, 4): 6 * > > Here, it's the right order but the signs are wrong. It's also different > to the invariant I defined, which always converts to the smaller bases.I simply reversed the ordering for the product of two vals and used it for the planar wedgie determined from the product of two intervals, but we could change that.> I can change that to always have the dual flag set and I don't think > there'll be any repercussions. And the signs don't matter for the > complexity calculation, so this one's okay.Actually, the signs do matter in this case--this is the only time your calculations should be of from mine.
Message: 6260 - Contents - Hide Contents Date: Thu, 30 Jan 2003 16:50:13 Subject: Seven limit temperament graphs From: Gene Ward Smith These show relationships between temperaments, with a line drawn between them if the bilinear form on 7-limit linear temperaments is zero. This means the temperaments have a common comma.
Message: 6261 - Contents - Hide Contents Date: Thu, 30 Jan 2003 18:06:50 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" <gdsecor@y...> > wrote: > Hi George, > > Thanks for updating the quick reference. It's great.I've updated it further, per your comments, plus a couple of corrections and extensive additions. Have another look.>>> Incidentally, I think we should point out that the 5'-comma symbol should >>> stay to the left of any arrow symbol, even in text, so they're always >>> treated as a single compound symbol. e.g. >>> Score: ./| # C-notehead >>> Text: C#./There's now a section on order of symbol components.> I note that in the quick reference where you've given a combination of a > shorthand ASCII sagittal with a conventional sharp, you've put the sagittal > leftmost. You need a note to say that this is how it would appear on a > score (but of course the ASCII characters should never appear on a score), > but in text the sagittal should be rightmost. > > On second thoughts, since the ASCII characters never appear on a score, > wouldn't it be better to show them in text order and add a note to say that > on a score the sagittal should be leftmost (i.e. furthest from the letter > name or note head in both cases).I've addressed both of these -- see what you think.> ... > You may have misunderstood me. What I'm saying is that, forexample, /f can> be used as equivalent to the pair of symbols /| |) (5-comma up, 7- comma > up) but never the single symbol /|) for which n is available.That makes the distinction of a tridecimal schisma and thus a departure from one of the distinguishing characteristics of our notation. (It remains to be seen whether anyone will actually do that, but the possibility is there.) Are you expecting perhaps that others will use these as one-character-per-prime symbols as a sort of bridge to sagittal notation (as I see in the next statement below).> ... > There are two possible usages I have in mind for multiple sagittals against > a single note. > 1. A one-symbol-per-prime notation. Some may yet prefer it.They might then want shorthand characters to represent the 19 and 23 commas, which we haven't covered. At the very least I think that the 19 comma should be available, since I recall that, when we started on this project, you thought that our notation would need to be 19-limit. Why don't we do the smallest commas this way: '| ' 5'-comma sharp 32768:32805 .! . 5'-comma flat )| " 19-comma sharp 512:513 )! ; 19-comma flat |( ( 5:7-comma sharp 5103:5120 !( c 5:7-comma flat Since the 19-comma is about twice as large as the 5' comma, it would be appropriate if it were to use characters indicating an approximate double. (If you think that the colon would be better than the semicolon for 19-comma-down, that would be okay with me.) And the 5:7 comma gets a better deal in the process. This would also allow us to notate three more ETs with single characters: 80, 104, and 152 (which I'm sure Paul would appreciate).> 2. Linear-temperament-specific notations where more than 7 pseudo- nominals > are produced by using one lot of sagittals (with the usual 7 nominals) to > notate a small proper MOS. And then using a second lot of sagittals to > represent the chroma (chromatic unison vector) and its multiples, that > offset the other notes from that MOS. > > For example, in miracle temperament the 10 pseudo-nominals could be (as a > chain of secors): > > F#\ G Ab/ Af Bv Ct C#\ D Eb/ Ef > > The above combinations of sagittals with sharps and flats would be better > notated by double-shaft sagittals. > > The chroma is t (the 7-comma down) and so the next row could be notated > F#\t Gt Ab/t Aft Bvt Ctt C#\t Dt Eb/t Eft > where tt is to be taken as two separate sagittals. > > Whether or not this is a good idea, I see no reason to forbid using the > shorthand symbols in this way.I tried playing around with something like this a few weeks ago (when linear temperament notation was being discussed), but I came to the conclusion that it's much simpler just to use 72-ET notation for Miracle. This is all so very specialized that I don't think very many are going to want to bother with it.> ...>>> By the way George, I hope you realise I still think there are serious >>> problems with the triple shafts and X shafts. It's only the availability of >>> the dual-symbol version of the notation that allows me to ignore them. >>>> The problem that you had with single-symbol notation way back when >> included double-shaft symbols, but you didn't mention those in the >> above statement, so you need to explain what you mean by that. >> This problem is not with the existence of single sagittal symbols between > sharp and double-sharp or flat and double-flat, but specifically with the > appearance of their shafts. There is a Ted Mook type objection > (sight-reading in bad light) to the triple-shafts, which I refer to as > 2-3-confusability. We addressed that partially by agreeing not to pack the > triple-shafts into the same width as the double-shafts, but I believe the > problem still remains.When I did some legibility testing with a few subjects last year, I found that the distance at which the number of shafts could be easily distinguished was greater than that at which wavy flags could easily be distinguished from concave ones. Given that, I don't believe that there is a problem such as you describe.> There is also the objection that the X-shaft up symbols look like they are > _adding_ something to a conventional double sharp whereas they are intended > to represent something smaller than a double sharp (or equal in the final > case). e.g. /X looks like it ought to mean a double sharp _plus_ a 5-comma, > whereas you have it meaning a double sharp _minus_ a 55-comma. Or looking > at it another way, you want the X-shaft itself to mean sesqui-sharp so that > /X means sesqui-sharp plus a 5-comma, but this is a very problematic > redefinition of the X from double-sharp to sesqui-sharp.Then we need to campaign for the elimination of these antiquated sharp and flat symbols (and especially their doubles) with all deliberate speed so as to eliminate the possibility of any confusion as soon as possible! ;-)> There is no similar problem with the double-shaft || representing > semi-sharp because it does not look like a complete sharp symbol, in fact > it contains exactly half the strokes of a conventional sharp. Only when it > attains the two straight flags as /||\ does it resemble (and have the same > number of strokes as) a conventional sharp symbol. This is very good.And the conventional double-sharp symbol has half as many lines as (and is a lot smaller than) a sharp symbol. This is not very good. More justification for our campaign! (:You are with me on this, aren't you?:)> Ideally the sesqui-sharp shaft would only look like an X with the addition > of the two straight flags. But this seems impossible to me.Hey, you're just nitpicking, now. Anyone can see that it's logical enough once they are told that the shafts by themselves count for n-1 semi-apotomes. It's the old symbol that makes little sense.>> I continue to have serious problems with the double-symbol notation >> (especially when it results in an occasional _de facto_ triple symbol >> whenever a double-flat is modified) which only the availability of >> the single-symbol version allows me to ignore. >> OK. I believe you suggested that the font should contain a single glyph for > conventional double-flat which looks like two conventional flats touching > each other (and maybe even squashed sideways a bit). I'll go along with > that now.I really wasn't trying to suggest that, but it doesn't sound like a bad idea. I don't think that anyone would be confused by it.>> The only problems that I see with the single-symbol version are that: >> 1) The performance notation has a steeper learning curve; >> 2) The ascii simulation is rather cumbersome, particularly for three- >> shaft symbols; >> 3) An ascii shorthand does not seem to be feasible; >> 4) More symbols are required in a font. >> Yeah. But it's still worth having. I just think the other problems I > mentioned above might yet be ameliorated.I would say fix it only if we find that it doesn't work. We're going to need to make the staves a little larger than usual in order to be able to read the flags. Oh, speaking of flags, I forgot to mention that single symbol notation results in larger (and thus more- readable) flags whenever a symbol has more than one shaft. (Do we have to go through all this again? I think we are still agreeing to disagree. But now, after having written everything else in this message, I've come back to this point, because I think I've figured out why you've brought all of this up. You're testing me to see if I still feel the same way about all of this, because you don't want to do a lot of work on the font, only to have me change my mind.)>> And the only problems that I have with the double-symbol notation >> involve the performance version is actually a single problem that has >> dual consequences: >> 1) Lower efficiency (in contrast with the the single-symbol version, >> in which every line segment conveys information); >> Redundancy in communication is often an asset in preventing errors. >>> a) Less legibility, i.e., a more cluttered appearance on the >> printed page; >> b) Less clarity, i.e., in a polyphonic part or score, it is not >> always obvious which symbols modify which notes; >> I agree with those two. >>> c) Less intuitive, i.e., more symbols preceding a note-head often >> symbolize a smaller amount of alteration (e.g., \!# is a smaller >> alteration than #), and down-arrow symbols frequently appear when the >> pitch is actually being altered upward (e.g., \!#). >> I suspect that many musicians/composers have more of a 12-pseudo- nominals > orientation, rather than 7-nominals. For example Joseph Pehrson. Such a > person is likely to find the double-symbol notation _more_ intuitive.You don't have to specular about that. He's already using a double- symbol notation and has therefore gotten very comfortable with it. By contrast, I've always used a single-symbol notation for *everything* (having never needed to notate anything more complicated than 41 tones in the octave), and my early exposure to Pythagorean tuning, meantone temperament, and 17, 19, 22, 31, and 41-ET (and avoidance of 24-ET) made it very easy for me to accept the idea that so-called enharmonic sharps and flats are distinctly different pitches. (And I *still* consider every one of those tunings as more practical than 72-ET! If you don't believe me, then tell me where you can get a 72-ET keyboard or 72-ET guitar.) The idea of 12 nominals in a notation is foreign to all of my microtonal experience, so it should be no surprise that I view the current popularity of 72- ET on the tuning list with mixed feelings, inasmuch as it perpetuates a 12-ET mentality, which dispenses with the meantone diesis. For me the joyful discovery that sharps and flats could be different was an important step in getting into microtonality via 19 or 31-ET, the meantone temperament, and the enharmonic genus, with the 5 comma being the troublesome adversary to be banished (or vanished), at least initially. With 72-ET these links to the past are disregarded. So I cannot help feeling that a double-symbol notation is just one more thing to perpetuate an excessive dependency on 12-ET. Repeating a quote from one of my postings around a year ago, #33068, would be appropriate here: << Should I start handing out bumper stickers that say "Kick the #/b habit -- go [single-symbol] saggital!" >> It is also interesting to see how relevant the rest of that posting is to the issue of sharp and flat symbols and their relationship (or lack thereof) to 12 nominals, all of which came up because of my proposal to replace the conventional sharp and flat symbols with others that mean exactly the same thing (and still do). (: That thread has finally come full circle -- keep those cards and letters coming, folks. :)>> But I would not want to abandon either version, because having both >> available immediately puts off criticism from anyone else who might >> have problems accepting one version or the other. > > Indeed.Aaargh! After all that idealism of a year ago, I've ended up heading off any criticism with a compromise! --George
Message: 6262 - Contents - Hide Contents Date: Thu, 30 Jan 2003 18:09:13 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> wrote:> Of course this last-ditch effort to improve on the triple and X- shafts is > motivated by the fact that I've agreed to draw the outline versions of > these symbols for the font. > > Another problem with the X shafts (these problems have all been mentioned > before) is that it is unclear when looking at it, whether the note being > modified is one aligned with the arrowhead or one aligned with the point > where the shafts cross. Attention is unavoidably drawn to that crossing > point, no matter how much one might know that we must look at the head, to > be consistent with the other saggitals. This is partly intrinsic to the > crossing itself and partly a matter of long habit from reading conventional > double-sharp x symbols.This is alleviated by the fact that X symbols will not occur very often, so that: 1) The alleged problem should not occur very often; and 2) The habit of getting the alignment from the flags should be well established from the much more frequent occurrence of the other symbols.> This kind of distraction caused by shaft features was the main reason you > gave for rejecting a shortened middle shaft for the triple shaft symbols.This was something that I found much more distracting than an X. If anything, this made it more difficult to distinguish the ||| from the X, since the symbols have the same width and both have two lines sticking out one end.> Any suggestions for better sharp and sesqui-sharp valued shafts. By shaft > value I mean currently we have shaft values: > > | natural 1:1 > || semi-sharp 704:729 > ||| sharp 2048:2187 > X sesqui-sharp 2048:2187 * 704:729None of the symbols as you show them above will ever occur in any piece of music, ever. These were originally conceived as: /|\ semi-sharp /||\ sharp /|||\ sesqui-sharp /X\ double-sharp inasmuch as you never see the shaft(s) without any flags, and the symbols are grouped (in one's mind) by rounding upward to the half- apotome, approximately. The values you give above apply strictly only to the | and ||| cases. The || and X symbols are defined as apotome-complements or (double-apotome complements) of single-shaft symbols. For example, ||) is defined as 2048:2187 / 63:64, not 704:729 * 63:64. So it is not very meaningful to give exact values for || and X, because their symbols are not defined that way. --George
Message: 6263 - Contents - Hide Contents Date: Thu, 30 Jan 2003 19:39:53 Subject: Re: Seven limit temperament graphs From: Carl Lumma>These show relationships between temperaments, with a line >drawn between them if the bilinear form on 7-limit linear >temperaments is zero. This means the temperaments have a >common comma.Cool. These could come in handy for "transferring" (I think that was Ivor's term) tunings mid-piece. Have you considered: () Labeling the lines by their commas? (Wait, I suppose the Paul's dualzoom thing already shows this...) () Making the lines longer in temp3da? -C.
Message: 6264 - Contents - Hide Contents Date: Thu, 30 Jan 2003 20:46:59 Subject: Re: Seven limit temperament graphs From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>> These show relationships between temperaments, with a line >> drawn between them if the bilinear form on 7-limit linear >> temperaments is zero. This means the temperaments have a >> common comma. >> Cool. These could come in handy for "transferring" (I think > that was Ivor's term) tunings mid-piece. Have you considered: > > () Labeling the lines by their commas? > (Wait, I suppose the Paul's dualzoom thing already shows this...)no, that thing is only 5-limit.
Message: 6265 - Contents - Hide Contents Date: Fri, 31 Jan 2003 19:22:26 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> David C Keenan wrote: >>> This brings up the point that what applies "in the development" need not >> apply in the dissemination. I think there is an advantage incalling it the>> 5-schisma. We could invent another term for the sub-cent ones like >> "schismina" (pron. skizmeena) literally "a small schisma". >> I'd certainly expect the 5-comma to be 81:80. So if you're talking > about schismas, it'd be much more straightforward if you called them > schismas. > > > GrahamDave's concern here is that we really need two different terms for all of the things that everybody else gives the label "schisma." He believes (and I agree) that the first interval to be given that name (32678:32805) should continue to be called that. However, our problem is that, although it was considered to be a very small interval at the time it was first named, it does not vanish in our rational notation -- rather, we have devised a symbol for it. What we do need, however, is a term that will apply to the sub-cent intervals that *do* vanish in our rational notation, and I think that "schismina" should do very nicely. But we will need to clarify whether we should consider a schismina a subclass of schisma (and if so, whether the term should apply only to those particular schisminas that vanish in our rational notation), or whether there should be a boundary that distinguishes a schismina from a schisma (and if so, exactly what size, and why exactly that size). --George
Message: 6266 - Contents - Hide Contents Date: Fri, 31 Jan 2003 04:33:22 Subject: Re: Seven limit temperament graphs From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:> () Making the lines longer in temp3da?Did you see temp3db?
Message: 6267 - Contents - Hide Contents Date: Fri, 31 Jan 2003 05:56:58 Subject: Re: Seven limit temperament graphs From: Carl Lumma>> >) Making the lines longer in temp3da? >> Did you see temp3db?I missed it; it needs longer lines too. :) -C.
Message: 6270 - Contents - Hide Contents Date: Fri, 31 Jan 2003 06:43:18 Subject: Files archive From: Gene Ward Smith One of these days we'll need to clean house. Does anyone see anyhthing now we could axe?
Message: 6271 - Contents - Hide Contents Date: Fri, 31 Jan 2003 14:31:42 Subject: Re: A common notation for JI and ETs From: David C Keenan>--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" ><gdsecor@y...> wrote: >--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> >wrote:>> Thanks for updating the quick reference. It's great. >>I've updated it further, per your comments, plus a couple of >corrections and extensive additions. Have another look. Good work!To the end of the first paragraph you might add the words "They are all smaller than a cent." There's typo in the paragraph immediately under "ASCII SHORTHAND FOR SINGLE-SHAFT SYMBOLS" "These single characters may also be used in combination a 5'-comma accent mark" should be: "These single characters may also be used in combination with a 5'-comma" ^^^^ The paragraph: The 5' comma has traditionally been known as the "schisma", but in the development of the sagittal notation we have used that term to indicate the difference between two intervals that are represented by the same symbol, i.e., one that vanishes in this notation. I suggest: The 5' comma has traditionally been known as the "schisma", but in the development of the sagittal notation we have used that term to indicate the the tiny difference between two commas that are represented ^^^^ ^^^^^^ by the same symbol, i.e., one that vanishes in this notation. Or you could say "two commas or dieses" if you prefer. This brings up the point that what applies "in the development" need not apply in the dissemination. I think there is an advantage in calling it the 5-schisma. We could invent another term for the sub-cent ones like "schismina" (pron. skizmeena) literally "a small schisma". The advantage relates to a wider consideration - how the symbols should be pronounced when reading them out loud. You wisely replaced all my "sharp" and "flat" with "up" and "down" and it is natural to want the whole name to be as short as possible to say. This leads me to prefer "55-comma" to "11-5 comma". And those "prime"s (as in "five prime comma") sound silly and don't really tell you anything about the size. It would be good if, when there are two notational commas available for a given prime number N (or combination of primes), the smaller is called the N-comma and the larger the N-diesis. This already occurs in many cases, so one can drop the "prime" for them. But in the case of N = 5 we can't do that so it would be good to call them 5-comma and 5-schisma. This will also work for N = 17 and 19 although in the 17 case it would be better to call the small one the 17-kleisma. Incidentally the traditional kleisma is the 5^6-kleisma and the "septimal kleisma" is the 7:25-kleisma. Both of these are notated '|( so commas notated as |( (5:7) should probably be called kleismas too. If we set the cutoff between a kleisma and comma at exactly half of a Pythagorean comma or 11.73 cents, this will work in the maximum number of cases. The cutoff between schisma and kleisma doesn't matter too much for this purpose since no combination of primes ever has two useful commas in this range. So I go to Manuel's collected interval names in the file intnam.par that comes with Scala. Having hauled one into a spreadsheet some time ago and sorted it by size, I find that a 3.80 cent interval is the largest referred to as a schisma (33554432:33480783, septimal) and the smallest called a kleisma is 4.50 cents (384:385). Halfway between the 19-schisma )| and the 5:7-kleisma |( would be 4.57 cents, so I propose that the cutoff should be infinitesimally below 384:385 or at 4.50 cents. The best cutoff between comma and diesis for this purpose would be exactly half a pythagorean limma or 45.11 cents. However this would omit the 25-diesis and THE diesis (125:128) so I propose placing the cutoff infinitesimally below 125:128 or at 41.05 cents. We then need an easy-to-say way to distinguish large dieses from small for things like the 11 and 13 dieses. This includes the 35, 5:49, 625 and 13:19 dieses. The cutoff between these is obviously at exactly a half apotome or 56.84 cents. Any suggestions? Is there a common suffix meaning "big" that we can tack on to "diesis"? I suppose we could just use "diesis" and "big diesis". The upper limit for a big diesis would be 70.17 cents for our purposes. The boundary between schisma and schismina (or whatever) would be half a 5-schisma or 0.98 cents. So here's the summary: 0 c schismina 0.98 c schisma 4.50 c kleisma 11.73 c comma 41.05 c diesis 56.84 c big diesis 70.17 c semitones, limmas, apotome to about 135 c>There's now a section on order of symbol components. >>> I note that in the quick reference where you've given a combination >of a>> shorthand ASCII sagittal with a conventional sharp, you've put the >sagittal>> leftmost. You need a note to say that this is how it would appear >on a>> score (but of course the ASCII characters should never appear on a >score),>> but in text the sagittal should be rightmost. >> >> On second thoughts, since the ASCII characters never appear on a >score,>> wouldn't it be better to show them in text order and add a note to >say that>> on a score the sagittal should be leftmost (i.e. furthest from the >letter>> name or note head in both cases). >>I've addressed both of these -- see what you think.I'm afraid I find it too complicated. I figure folks eyes will just glaze over. I think that when they are writing text they shouldn't have to worry about whether they are using it "abstractly" or not. They should just type the sharp or flat before the other accidental. That's how it's always been done on this list. This gets us down from three categories to two. Text is text and staff is staff ...>> 1. A one-symbol-per-prime notation. Some may yet prefer it. >>They might then want shorthand characters to represent the 19 and 23 >commas, which we haven't covered. At the very least I think that the >19 comma should be available, since I recall that, when we started on >this project, you thought that our notation would need to be 19-limit. > >Why don't we do the smallest commas this way: > >'| ' 5'-comma sharp 32768:32805 >.! . 5'-comma flat >)| " 19-comma sharp 512:513 >)! ; 19-comma flat > |( ( 5:7-comma sharp 5103:5120 > !( c 5:7-comma flat > >Since the 19-comma is about twice as large as the 5' comma, it would >be appropriate if it were to use characters indicating an approximate >double. (If you think that the colon would be better than the >semicolon for 19-comma-down, that would be okay with me.) And the >5:7 comma gets a better deal in the process. This would also allow >us to notate three more ETs with single characters: 80, 104, and 152 >(which I'm sure Paul would appreciate).I'd like to have the 19-schisma in the single-ASCII, and if so ; and " would be the obvious choice (semicolon, not colon for reasons I gave earlier). But I really don't like using ( for 5:7-kleisma up. 1. It will get missed in text (i.e. parsed as an opening parenthesis). 2. Folks are already used to thinking of ([<{ as meaning down and )]>] as up. Scala uses ( for diesis down. I thought we already agreed not to use () purely for reason 1. As you say, we're scraping the bottom of the barrel. It can't be an uppercase character. In approximate keyboard order: It can't be `,~!|@#%^&()+-{}{}\/'.";?<>. It can't be qwtyuosdfhjxcvbnm. Already used or rejected for any use. That only leaves $*_=:eripagklz. A lowercase character shouldn't be used unless it has a descender, or no-ascender and is open at the bottom. That eliminates eaklz leaving ripg. p and g are too big to represent something that small. Cant use $ because it is wavy not concave. I want to reserve colon for placing between notes to form chords. _ is obviously down, not up. = suggests no direction and is utterly unlike an arrow. That leaves *ri. I note that k isn't a bad looking down symbol and might be paired with p for some use, for lack of anything else to pair it with and because p's obvious partner b is already taken. I also note that e and a or g and a might make a pair, and possibly $ and z. But none of these suit a small right concave flag. r is more like |) or )|). i looks like an inverted ! which should at least make it an up symbol, but I'm inclined to go with * because of its smallness and upwardness and because it seems better to use special characters rather than letters when possible. Do we want to consider something other than c for its partner? k bears a vague resemblance to *, but it seems a bit too big. What do you think?>I tried playing around with something like this a few weeks ago (when >linear temperament notation was being discussed), but I came to the >conclusion that it's much simpler just to use 72-ET notation for >Miracle. This is all so very specialized that I don't think very >many are going to want to bother with it.Yeah. It's pretty ugly. I guess if folks use combinations of these single-ASCIIs they just have to spell out what they mean by it. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
Message: 6272 - Contents - Hide Contents Date: Fri, 31 Jan 2003 18:22:05 Subject: Re: A common notation for JI and ETs From: David C Keenan Hi George,>--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" ><gdsecor@y...> wrote: >--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> >wrote:>> This problem is not with the existence of single sagittal symbols >between>> sharp and double-sharp or flat and double-flat, but specifically >with the>> appearance of their shafts. There is a Ted Mook type objection >> (sight-reading in bad light) to the triple-shafts, which I refer to >as>> 2-3-confusability. We addressed that partially by agreeing not to >pack the>> triple-shafts into the same width as the double-shafts, but I >believe the>> problem still remains. >>When I did some legibility testing with a few subjects last year, I >found that the distance at which the number of shafts could be easily >distinguished was greater than that at which wavy flags could easily >be distinguished from concave ones. Given that, I don't believe that >there is a problem such as you describe.I understood that these subjects were only involved in deciding whether they preferred the middle shaft shortened or not, and that the distance recognition tests were only conducted with yourself as subject. Is this correct? Can you tell me something about your subjects and the test. How many? What were their musical backgrounds? Were they likely to be impartial or were they friends or relatives who might pick up subliminal cues as to which you preferred? What instructions were they given? How was the test conducted? We both know the answers to these questions in the case of the subject Ted Mook. He has been involved in trying out various microtonal notations as a performer. He only knows me from one previous email exchange in which I asked him why he didn't like the tartini symbols. I expect you still have the email exchange relating to the test. The test was conducted by email, so subliminal cues would be difficult. Ted preferred the shortened middle shaft. But I acknowledge that a result from a single subject means almost nothing. Even if we assume your results are valid above, your argument from them is a non-sequitur. It might only mean that we have a bigger problem with distinguishing wavy flags than we do with distinguishing triple shafts. But even this is not the case, because the consequences of mistaking a wavy flag for a straight one are not very serious musically (about 15 cents), while mistaking a triple shaft for a double is very serious (about 100 cents). A useful test would provide random sagittal symbols, in dim light at a large enough distance for a significant error rate and ask the subject whether she thinks they are 1, 2, 3 or X shaft and whether they are up or down (having first allowed her to examine them closely to learn these categories). Then the same test would be repeated with the shortened middle shaft (again being allowed to examine them closely). Subjects would alternate according to which set they did first. At best I only expect the shortened middle shaft to provide a minor improvement. And I have another suggestion (later) so I'll drop it.>> There is also the objection that the X-shaft up symbols look like >they are>> _adding_ something to a conventional double sharp whereas they are >intended>> to represent something smaller than a double sharp (or equal in the >final>> case). e.g. /X looks like it ought to mean a double sharp _plus_ a >5-comma,>> whereas you have it meaning a double sharp _minus_ a 55-comma. Or >looking>> at it another way, you want the X-shaft itself to mean sesqui-sharp >so that>> /X means sesqui-sharp plus a 5-comma, but this is a very >problematic>> redefinition of the X from double-sharp to sesqui-sharp. >>Then we need to campaign for the elimination of these antiquated >sharp and flat symbols (and especially their doubles) with all >deliberate speed so as to eliminate the possibility of any confusion >as soon as possible! ;-)Hee hee. And pigs might fly. :-)>> There is no similar problem with the double-shaft || representing >> semi-sharp because it does not look like a complete sharp symbol, >in fact>> it contains exactly half the strokes of a conventional sharp. Only >when it>> attains the two straight flags as /||\ does it resemble (and have >the same>> number of strokes as) a conventional sharp symbol. This is very >good. >>And the conventional double-sharp symbol has half as many lines as >(and is a lot smaller than) a sharp symbol. This is not very good. >More justification for our campaign! (:You are with me on this, >aren't you?:)Yes. I agree it would be nice if the sagittals replaced the existing sharp and flat symbols but it isn't going to happen overnight. So you can't just ignore the problems that will occur during the (possibly decades-long) transition. In fact if you set up the sagittal notation for a head-on clash like the one I've described above, it is much less likely that a transition will ever be made. This is as much a political decision as it is an aesthetic or logical one (rather like much ancient-greek thought ;-).>> Ideally the sesqui-sharp shaft would only look like an X with the >addition>> of the two straight flags. But this seems impossible to me. >>Hey, you're just nitpicking, now.I don't think so. I get the feeling, from this and earlier exchanges on this topic, that the triple and X shafts are somewhat sacred cows to you. But that's the wrong pantheon, Hermes. :-)>Anyone can see that it's logical >enough once they are told that the shafts by themselves count for n-1 >semi-apotomes.Huh? To me, your three-semi-apotome symbol has always had two shafts, not four. The general n-1 idea is fine because conceptually a line has zero width and we're really looking at the overall width of the tail to roughly correspond to its value. I just thought the reason for the X was simply that four shafts is just too wide (and 3-4 confusability is much worse than 2-3) so we just find something completely different. Never mind how many strokes it's got.> It's the old symbol that makes little sense.Whether the old symbols make sense or not is all but irrelevant given their ubiquitousness. But one way in which they do "make sense" is that they are very very different from each other and therefore very difficult to confuse.>> ... I just think the other problems I >> mentioned above might yet be ameliorated. >>I would say fix it only if we find that it doesn't work.I find that X shafts don't work.> We're going >to need to make the staves a little larger than usual in order to be >able to read the flags. Sure. > Oh, speaking of flags, I forgot to mention >that single symbol notation results in larger (and thus more- >readable) flags whenever a symbol has more than one shaft.Sure. That helps with the 2-3 confusability but has no bearing on the two problems I've cited with the X shaft. 1. Clash of meanings with existing double-shaft. 2. Distraction caused by the crossing point.>(Do we >have to go through all this again? I think we are still agreeing to >disagree. But now, after having written everything else in this >message, I've come back to this point, because I think I've figured >out why you've brought all of this up. You're testing me to see if I >still feel the same way about all of this, because you don't want to >do a lot of work on the font, only to have me change my mind.)It's the font work, yes - as I already said in a subsequent post. But I'm not testing you to see if you still feel the same way. I'm trying to brutally force you to have the rational discusion that I gave up on previously when I got the "sacred cow" feeling. Or maybe its not so much a sacred cow, but it's just that you have been using them yourself for a long time and would find in very inconvenient to change. But (and I'm always saying this to someone in these standardisation efforts) you are only one person. What is your inconvenience compared to that of all those who may come after you?>--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" ><gdsecor@y...> wrote: >--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> >wrote:>> Of course this last-ditch effort to improve on the triple and X- >shafts is>> motivated by the fact that I've agreed to draw the outline versions >of>> these symbols for the font. >> >> Another problem with the X shafts (these problems have all been >mentioned>> before) is that it is unclear when looking at it, whether the note >being>> modified is one aligned with the arrowhead or one aligned with the >point>> where the shafts cross. Attention is unavoidably drawn to that >crossing>> point, no matter how much one might know that we must look at the >head, to>> be consistent with the other saggitals. This is partly intrinsic to >the>> crossing itself and partly a matter of long habit from reading >conventional>> double-sharp x symbols. >>This is alleviated by the fact that X symbols will not occur very >often, so that: >1) The alleged problem should not occur very often; and >2) The habit of getting the alignment from the flags should be well >established from the much more frequent occurrence of the other >symbols.I hope so. But if you could eliminate this distraction without significant bad side-effects, why wouldn't you? I earlier wrote that it would be good if the sesqui-apotome-valued shaft only came to look like an X when topped by two straight shafts, i.e when it becomes a double-apotome symbol. It seems from your draft XH article, that you are in favour of resemblances to existing symbols. You describe the correspondences between sagittal and four other systems that are otherwise unrelated to each other: the quartertone arrows, the Bosanquet slashes, the Tartini symbols and the Couper symbols (and hence the conventional symbols). You specifically claim that a resemblance to the conventional double-sharp symbol is an advantage, which indeed it might be, if there was only the double-apotome symbol itself, and not all the other X-shaft symbols _smaller_ than it. A resemblance where the same feature or sub-symbol means different things is clearly not an advantage. So I now have a proposal to replace the X shaft. It solves _both_ of the problems I mentioned. It has no features to distract from the arrowhead and it doesn't look like an X until it gets to be a double-apotome (but even then the resemblance is a bit strained). Take a conventional double-sharp X symbol. Lose the square blobs on the ends of the strokes. Grab hold of the two top ends and bend them apart. Keep bending them down past horizontal until they become a pair of upward pointing straight flags on top of a V shaft (inverted V in this case).>> This kind of distraction caused by shaft features was the main >reason you>> gave for rejecting a shortened middle shaft for the triple shaft >symbols. >>This was something that I found much more distracting than an X.It seems very odd to me that anyone would find the convergence of a pair of lines that don't actually appear, to be more distracting than ones that do.>If >anything, this made it more difficult to distinguish the ||| from the >X, since the symbols have the same width and both have two lines >sticking out one end.Yes I expect it would increase confusability between 3 and X, but I figure there's a lot of room to play with there, and not much between 2 and 3. Anyway, we can probably forget about short middle shafts.>> Any suggestions for better sharp and sesqui-sharp valued shafts? By >shaft>> value I mean currently we have shaft values: >> >> | natural 1:1 >> || semi-sharp 704:729 >> ||| sharp 2048:2187 >> X sesqui-sharp 2048:2187 * 704:729 >>None of the symbols as you show them above will ever occur in any >piece of music, ever.No. That's why I refer to "shaft values", not symbol values, in the same way we talk about "flag values". Of course the bare single shaft may occur with a 5-schisma accent but that isn't relevant to this discussion.>These were originally conceived as: > /|\ semi-sharp > /||\ sharp > /|||\ sesqui-sharp > /X\ double-sharpYes I understand that.>inasmuch as you never see the shaft(s) without any flags, and the >symbols are grouped (in one's mind) by rounding upward to the half- >apotome, approximately.In your mind perhaps, but not mine. In my mind, and I suspect many others, the flag values are _added_to_ the shaft value, approximately. I believe you have talked in those terms extensively yourself, during our long cooperative effort.>The values you give above apply strictly only to the | and ||| >cases. The || and X symbols are defined as apotome-complements or >(double-apotome complements) of single-shaft symbols. For example, >||) is defined as 2048:2187 / 63:64, not 704:729 * 63:64. So it is >not very meaningful to give exact values for || and X, because their >symbols are not defined that way.Yes. Good point. Consider them deleted. Although we could argue whether the approximation (or offset) is in the flags or in the shafts, or partly in both, this is a meaningless distinction and is not relevant to the current discussion. Here are some more alternative shaft ideas. I can see that one objection to the V shaft might be that it is too narrow to have a value as large as a sesqui-apotome. So I propose moving the two shafts apart until they are as far apart at the head, as are the two parallel shafts of the existing double-shaft symbols. This opens up the possibility of using the actual V shaft in place of the triple shaft. There is a certain order to that. The tail area increases steadily with shaft value. | 0/2 apotome || 1/2 apotome \/ 2/2 apotome (note these are shafts not straight flags) \ / 3/2 apotome But note that the > 2/2 apotome symbols can have the same size flags as the > 0/2 apotome symbols, and the > 3/2 the same size as the > 1/2. This would make my job easier with the outline font, but of course I won't use that as an argument for accepting it. These shafts could be represented in ASCII as V A W M. Regards, -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
Message: 6273 - Contents - Hide Contents Date: Fri, 31 Jan 2003 09:36:07 Subject: Re: A common notation for JI and ETs From: Graham Breed David C Keenan wrote:> This brings up the point that what applies "in the development" need not > apply in the dissemination. I think there is an advantage in calling it the > 5-schisma. We could invent another term for the sub-cent ones like > "schismina" (pron. skizmeena) literally "a small schisma".I'd certainly expect the 5-comma to be 81:80. So if you're talking about schismas, it'd be much more straightforward if you called them schismas. Graham
Message: 6274 - Contents - Hide Contents Date: Sat, 01 Feb 2003 09:16:23 Subject: Re: A common notation for JI and ETs From: David C Keenan Some corrections. I wrote: "Sure. That helps with the 2-3 confusability but has no bearing on the two problems I've cited with the X shaft. 1. Clash of meanings with existing double-shaft. 2. Distraction caused by the crossing point." That should have been 1. Clash of meanings with existing double-sharp. ^^^^^ I wrote: "There is a certain order to that. The tail area increases steadily with shaft value. | 0/2 apotome || 1/2 apotome \/ 2/2 apotome (note these are shafts not straight flags) \ / 3/2 apotome" Since it would be natural to assume that I was giving the tails for up symbols, that should have been | 0/2 apotome || 1/2 apotome /\ 2/2 apotome (note these are shafts not straight flags) / \ 3/2 apotome Hi Graham, Good to hear from you in this thread. I assume you are saying that you greatly prefer the terms 5-comma and 5-schisma to 5-comma and 5'-comma for 80:81 and 32768:32805 respectively. If so, good. It seems we're all agreed on that now. I also assume that you were not commenting on the suggested distinction between schismas and schisminas. Is that correct? I think that these should be distinguished purely on the basis of size, not whether they vanish in sagittal. They would be more generally useful that way and we would still have the advantage, when talking about the development of sagittal, that only schisminas vanish.
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