Tuning-Math Archive Section 11: 10726

Message: 10726 - Contents - Hide Contents Date: Wed, 31 Mar 2004 00:47:19 Subject: 11-limit Haluska linear temperaments From: Gene Ward Smith By these I mean linear temperaments defined by three successive Haluska commas. [33/32, 36/35, 45/44] [1, -3, 5, -1, -7, 5, -5, 20, 8, -20] [[1, 2, 1, 5, 3], [0, -1, 3, -5, 1]] [36/35, 45/44, 49/48] [2, 3, 1, 7, 0, -4, 4, -6, 6, 16] [[1, 2, 3, 3, 5], [0, -2, -3, -1, -7]] [45/44, 49/48, 50/49] [4, 2, 2, 10, -6, -8, 2, -1, 16, 21] [[2, 4, 5, 6, 9], [0, -2, -1, -1, -5]] [49/48, 50/49, 55/54] [4, 2, 2, 10, -6, -8, 2, -1, 16, 21] [[2, 4, 5, 6, 9], [0, -2, -1, -1, -5]] [50/49, 55/54, 56/55] [2, 6, 6, 0, 5, 4, -7, -3, -21, -21] [[2, 3, 4, 5, 7], [0, 1, 3, 3, 0]] [55/54, 56/55, 64/63] [0, 5, 0, -5, 8, 0, -8, -14, -29, -14] [[5, 8, 12, 14, 17], [0, 0, -1, 0, 1]] [56/55, 64/63, 81/80] [1, 4, -2, -6, 4, -6, -13, -16, -28, -10] [[1, 2, 4, 2, 1], [0, -1, -4, 2, 6]] [64/63, 81/80, 99/98] [1, 4, -2, -6, 4, -6, -13, -16, -28, -10] [[1, 2, 4, 2, 1], [0, -1, -4, 2, 6]] [81/80, 99/98, 100/99] [2, 8, 8, 12, 8, 7, 12, -4, 0, 6] [[2, 3, 4, 5, 6], [0, 1, 4, 4, 6]] [99/98, 100/99, 121/120] [6, 10, 10, 8, 2, -1, -8, -5, -16, -12] [[2, 4, 6, 7, 8], [0, -3, -5, -5, -4]] [100/99, 121/120, 126/125] [3, 5, 9, 4, 1, 6, -4, 7, -8, -20] [[1, 2, 3, 4, 4], [0, -3, -5, -9, -4]] [121/120, 126/125, 176/175] [9, 5, -3, 7, -13, -30, -20, -21, -1, 30] [[1, 1, 2, 3, 3], [0, 9, 5, -3, 7]] [126/125, 176/175, 225/224] [1, 4, 10, 18, 4, 13, 25, 12, 28, 16] [[1, 2, 4, 7, 11], [0, -1, -4, -10, -18]] [176/175, 225/224, 243/242] [8, 1, 18, 20, -17, 6, 4, 39, 43, -6] [[1, -1, 2, -3, -3], [0, 8, 1, 18, 20]] [225/224, 243/242, 385/384] [6, -7, -2, 15, -25, -20, 3, 15, 59, 49] [[1, 1, 3, 3, 2], [0, 6, -7, -2, 15]] [243/242, 385/384, 441/440] [6, -7, -2, 15, -25, -20, 3, 15, 59, 49] [[1, 1, 3, 3, 2], [0, 6, -7, -2, 15]] [385/384, 441/440, 540/539] [6, -7, -2, 15, -25, -20, 3, 15, 59, 49] [[1, 1, 3, 3, 2], [0, 6, -7, -2, 15]] [441/440, 540/539, 2401/2400] [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] [[0, 0, 0, 0, 0], [0, 0, 0, 0, 0]] [540/539, 2401/2400, 3025/3024] [6, -7, -2, 15, -25, -20, 3, 15, 59, 49] [[1, 1, 3, 3, 2], [0, 6, -7, -2, 15]] [2401/2400, 3025/3024, 4375/4374] [36, 54, 36, 18, 2, -44, -96, -68, -145, -74] [[18, 28, 41, 50, 62], [0, 2, 3, 2, 1]] [3025/3024, 4375/4374, 9801/9800] [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] [[0, 0, 0, 0, 0], [0, 0, 0, 0, 0]] [4375/4374, 9801/9800, 151263/151250] [36, 54, 36, 18, 2, -44, -96, -68, -145, -74] [[18, 28, 41, 50, 62], [0, 2, 3, 2, 1]] [9801/9800, 151263/151250, 1771561/1771470] [102, 210, 216, 222, 96, 56, -1, -88, -211, -124] [[6, 11, 17, 20, 24], [0, -17, -35, -36, -37]] [151263/151250, 1771561/1771470, 3294225/3294172] [102, 210, 216, 222, 96, 56, -1, -88, -211, -124] [[6, 11, 17, 20, 24], [0, -17, -35, -36, -37]] [1771561/1771470, 3294225/3294172, 781258401/781250000] [388, 658, 768, 821, 142, 128, -41, -64, -370, -352] [[1, -113, -192, -224, -239], [0, 388, 658, 768, 821]] [3294225/3294172, 781258401/781250000, 4882812500/4882786447] [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] [[0, 0, 0, 0, 0], [0, 0, 0, 0, 0]] [781258401/781250000, 4882812500/4882786447, 2541867610898/2541865828329] [842, 1472, 1716, 1846, 378, 356, 13, -148, -806, -754] [[2, -83, -146, -170, -182], [0, 421, 736, 858, 923]]

Message: 10728 - Contents - Hide Contents Date: Wed, 31 Mar 2004 00:56:33 Subject: 13-limit Haluska linear temperaments From: Gene Ward Smith [36/35, 40/39, 45/44, 49/48] [2, 3, 1, 7, 1, 0, -4, 4, -6, -6, 6, -9, 16, -1, -23] [[1, 2, 3, 3, 5, 4], [0, -2, -3, -1, -7, -1]] [40/39, 45/44, 49/48, 50/49] [4, 2, 2, 10, -2, -6, -8, 2, -18, -1, 16, -12, 21, -13, -44] [[2, 4, 5, 6, 9, 7], [0, -2, -1, -1, -5, 1]] [45/44, 49/48, 50/49, 55/54] [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] [[0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0]] [49/48, 50/49, 55/54, 56/55] [0, 0, 0, 0, 10, 0, 0, 0, 16, 0, 0, 23, 0, 28, 35] [[10, 16, 23, 28, 35, 37], [0, 0, 0, 0, 0, 1]] [50/49, 55/54, 56/55, 64/63] [0, 0, 0, 0, 10, 0, 0, 0, 16, 0, 0, 23, 0, 28, 35] [[10, 16, 23, 28, 35, 37], [0, 0, 0, 0, 0, 1]] [55/54, 56/55, 64/63, 65/64] [0, 5, 0, -5, -5, 8, 0, -8, -8, -14, -29, -30, -14, -14, 1] [[5, 8, 12, 14, 17, 18], [0, 0, -1, 0, 1, 1]] [56/55, 64/63, 65/64, 66/65] [1, -1, -2, -1, 1, -4, -6, -5, -2, -2, 1, 6, 4, 10, 7] [[1, 2, 2, 2, 3, 4], [0, -1, 1, 2, 1, -1]] [64/63, 65/64, 66/65, 78/77] [1, 4, -2, -1, -4, 4, -6, -5, -10, -16, -16, -24, 4, -4, -10] [[1, 2, 4, 2, 3, 2], [0, -1, -4, 2, 1, 4]] [65/64, 66/65, 78/77, 81/80] [1, 4, -2, -1, -4, 4, -6, -5, -10, -16, -16, -24, 4, -4, -10] [[1, 2, 4, 2, 3, 2], [0, -1, -4, 2, 1, 4]] [66/65, 78/77, 81/80, 91/90] [1, 4, -2, 11, 8, 4, -6, 14, 9, -16, 12, 4, 38, 30, -13] [[1, 2, 4, 2, 8, 7], [0, -1, -4, 2, -11, -8]] [78/77, 81/80, 91/90, 99/98] [4, 16, 9, 10, 15, 16, 3, 2, 9, -24, -32, -24, -3, 9, 15] [[1, 3, 8, 6, 7, 9], [0, -4, -16, -9, -10, -15]] [81/80, 91/90, 99/98, 100/99] [2, 8, 8, 12, 4, 8, 7, 12, -1, -4, 0, -20, 6, -18, -30] [[2, 3, 4, 5, 6, 7], [0, 1, 4, 4, 6, 2]] [91/90, 99/98, 100/99, 105/104] [4, 2, 2, -4, 8, -6, -8, -20, -2, -1, -16, 11, -18, 15, 42] [[2, 4, 5, 6, 6, 9], [0, -2, -1, -1, 2, -4]] [99/98, 100/99, 105/104, 121/120] [6, 10, 10, 8, 26, 2, -1, -8, 19, -5, -16, 23, -12, 36, 60] [[2, 4, 6, 7, 8, 11], [0, -3, -5, -5, -4, -13]] [100/99, 105/104, 121/120, 126/125] [3, 5, 9, 4, 17, 1, 6, -4, 16, 7, -8, 21, -20, 14, 44] [[1, 2, 3, 4, 4, 6], [0, -3, -5, -9, -4, -17]] [105/104, 121/120, 126/125, 144/143] [9, 5, -3, 7, 11, -13, -30, -20, -16, -21, -1, 7, 30, 42, 12] [[1, 1, 2, 3, 3, 3], [0, 9, 5, -3, 7, 11]] [121/120, 126/125, 144/143, 169/168] [8, 8, 8, 8, 8, -6, -10, -15, -17, -4, -9, -11, -5, -7, -2] [[8, 13, 19, 23, 28, 30], [0, -1, -1, -1, -1, -1]] [126/125, 144/143, 169/168, 176/175] [13, 9, 1, 19, 7, -16, -35, -15, -37, -23, 13, -17, 50, 16, -46] [[1, 4, 4, 3, 7, 5], [0, -13, -9, -1, -19, -7]] [144/143, 169/168, 176/175, 196/195] [9, 0, 9, 9, 9, -21, -11, -17, -19, 21, 21, 21, -6, -8, -2] [[9, 14, 21, 25, 31, 33], [0, 1, 0, 1, 1, 1]] [169/168, 176/175, 196/195, 225/224] [4, -3, 2, -4, 3, -14, -8, -20, -10, 13, 1, 18, -18, 1, 25] [[1, 2, 2, 3, 3, 4], [0, -4, 3, -2, 4, -3]] [176/175, 196/195, 225/224, 243/242] [8, 1, 18, 20, 27, -17, 6, 4, 13, 39, 43, 59, -6, 9, 19] [[1, -1, 2, -3, -3, -5], [0, 8, 1, 18, 20, 27]] [196/195, 225/224, 243/242, 325/324] [10, 2, 24, 25, 36, -20, 10, 5, 20, 50, 51, 76, -13, 12, 32] [[1, 5, 3, 11, 12, 16], [0, -10, -2, -24, -25, -36]] [225/224, 243/242, 325/324, 351/350] [12, 10, 44, 30, 28, -12, 36, 6, 0, 74, 35, 28, -68, -84, -14] [[2, 6, 7, 16, 14, 14], [0, -6, -5, -22, -15, -14]] [243/242, 325/324, 351/350, 352/351] [4, 9, -8, 10, -2, 5, -24, 2, -18, -44, -8, -38, 56, 24, -44] [[1, 1, 1, 4, 2, 4], [0, 4, 9, -8, 10, -2]] [325/324, 351/350, 352/351, 364/363] [2, -4, 30, 22, 16, -11, 42, 28, 18, 81, 65, 52, -42, -66, -26] [[2, 3, 5, 3, 5, 6], [0, 1, -2, 15, 11, 8]] [351/350, 352/351, 364/363, 385/384] [4, -8, 14, -2, -14, -22, 11, -17, -37, 55, 23, -3, -54, -91, -41] [[2, 3, 5, 5, 7, 8], [0, 2, -4, 7, -1, -7]] [352/351, 364/363, 385/384, 441/440] [3, 17, -1, -13, -22, 20, -10, -31, -46, -50, -89, -114, -33, -58, -28] [[1, 1, -1, 3, 6, 8], [0, 3, 17, -1, -13, -22]] [364/363, 385/384, 441/440, 540/539] [6, -7, -2, 15, 38, -25, -20, 3, 38, 15, 59, 114, 49, 114, 76] [[1, 1, 3, 3, 2, 0], [0, 6, -7, -2, 15, 38]] [385/384, 441/440, 540/539, 625/624] [6, -7, -2, 15, -34, -25, -20, 3, -76, 15, 59, -53, 49, -88, -173] [[1, 1, 3, 3, 2, 7], [0, 6, -7, -2, 15, -34]] [441/440, 540/539, 625/624, 676/675] [24, 20, 16, 60, 56, -24, -42, 12, 0, -19, 70, 56, 113, 98, -28] [[4, 6, 9, 11, 13, 14], [0, 6, 5, 4, 15, 14]] [540/539, 625/624, 676/675, 729/728] [6, 5, 22, -21, 14, -6, 18, -54, 0, 37, -66, 14, -135, -42, 126] [[1, 0, 1, -3, 9, 0], [0, 6, 5, 22, -21, 14]] [625/624, 676/675, 729/728, 1001/1000] [6, 5, 22, -21, 14, -6, 18, -54, 0, 37, -66, 14, -135, -42, 126] [[1, 0, 1, -3, 9, 0], [0, 6, 5, 22, -21, 14]] [676/675, 729/728, 1001/1000, 1716/1715] [12, 34, 20, 30, 52, 26, -2, 6, 38, -49, -48, -5, 15, 72, 69] [[2, 4, 7, 7, 9, 11], [0, -6, -17, -10, -15, -26]] [729/728, 1001/1000, 1716/1715, 2080/2079] [12, 34, 20, 30, 52, 26, -2, 6, 38, -49, -48, -5, 15, 72, 69] [[2, 4, 7, 7, 9, 11], [0, -6, -17, -10, -15, -26]] [1001/1000, 1716/1715, 2080/2079, 2401/2400] [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] [[0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0]] [1716/1715, 2080/2079, 2401/2400, 3025/3024] [36, 54, 36, 18, 108, 2, -44, -96, 38, -68, -145, 51, -74, 170, 307] [[18, 28, 41, 50, 62, 65], [0, 2, 3, 2, 1, 6]] [2080/2079, 2401/2400, 3025/3024, 4096/4095] [2, -57, -28, 46, 81, -95, -50, 66, 121, 95, 304, 399, 226, 331, 110] [[1, 1, 19, 11, -10, -20], [0, 2, -57, -28, 46, 81]] [2401/2400, 3025/3024, 4096/4095, 4225/4224] [2, -57, -28, 46, 81, -95, -50, 66, 121, 95, 304, 399, 226, 331, 110] [[1, 1, 19, 11, -10, -20], [0, 2, -57, -28, 46, 81]] [3025/3024, 4096/4095, 4225/4224, 4375/4374] [22, 48, -38, -34, -54, 25, -122, -130, -167, -223, -245, -303, 36, -11, -61] [[2, 7, 13, -1, 1, -2], [0, -11, -24, 19, 17, 27]] [4096/4095, 4225/4224, 4375/4374, 6656/6655] [22, 48, -38, -34, -54, 25, -122, -130, -167, -223, -245, -303, 36, -11, -61] [[2, 7, 13, -1, 1, -2], [0, -11, -24, 19, 17, 27]] [4225/4224, 4375/4374, 6656/6655, 9801/9800] [22, 48, -38, -34, -54, 25, -122, -130, -167, -223, -245, -303, 36, -11, -61] [[2, 7, 13, -1, 1, -2], [0, -11, -24, 19, 17, 27]] [4375/4374, 6656/6655, 9801/9800, 10648/10647] [22, 48, -38, -34, -54, 25, -122, -130, -167, -223, -245, -303, 36, -11, -61] [[2, 7, 13, -1, 1, -2], [0, -11, -24, 19, 17, 27]] [6656/6655, 9801/9800, 10648/10647, 123201/123200] [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] [[0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0]] [9801/9800, 10648/10647, 123201/123200, 5767168/5767125] [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] [[0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0]] [10648/10647, 123201/123200, 5767168/5767125, 192914176/192913083] [74, -108, 456, 200, -2, -343, 515, 61, -277, 1362, 838, 395, -1016, -1693, -747] [[2, 18, -17, 97, 47, 7], [0, -37, 54, -228, -100, 1]] [123201/123200, 5767168/5767125, 192914176/192913083, 4060088955/4060086272] [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] [[0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0]] [5767168/5767125, 192914176/192913083, 4060088955/4060086272, 14869765625/14869757952] [79, -348, 786, 1510, 563, -735, 1024, 2120, 600, 2802, 4710, 2595, 1520, -1328, -3640] [[1, -19, 93, -202, -390, -143], [0, 79, -348, 786, 1510, 563]] [192914176/192913083, 4060088955/4060086272, 14869765625/14869757952, 13051691536000/13051688172831] [173, 4210, 208, -6097, -3523, 6271, -156, -10262, -6224, -11336, -28721, -23759, -17836, -10660, 10374] [[1, -44, -1107, -52, 1610, 932], [0, 173, 4210, 208, -6097, -3523]] [4060088955/4060086272, 14869765625/14869757952, 13051691536000/13051688172831, 28344980104623/28344976000000] [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] [[0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0]]

Message: 10729 - Contents - Hide Contents Date: Wed, 31 Mar 2004 13:37:14 Subject: Re: 98 out of 99 From: Carl Lumma

>> >ctually, if you follow the thread, you'll see I'm still waiting >> for Gene to explain how he can get modulations of 9:5, 9:7, etc, >> by moving only distance 1 on the lattice. >

>You can't, but then you can't get any other modulations this way >either. Moving a distance of 1 always exchanges otonal and utonal >tetrads/quintads.

I call those modulations.

> I thought the whole

>> point of the observation that the dual of the 7-limit lattice is >> also a lattice was that you can represent all the possible >> modulations as a single step (there are 6 possible modulations >> in the 7-limit, and every point in the cubic lattice is connected >> to 6 others). >

>The six tetrads you get are the six tetrads sharing an interval.

Yes, that's what I meant.

>To do >the 9-limit more intrinsically, we would use a packing, not a lattice, >and we would have ten quintads, not six. We could take all points >[a,b,c] with [a,b,c] mod 5 [3,1,1] for major quintads, and [2,4,4] for >minor quintands, and link them when they share an interval. I don't >know what that looks like, but it's three-dimensional. Fascinating. -C.

Message: 10730 - Contents - Hide Contents Date: Wed, 31 Mar 2004 02:35:57 Subject: Musical harmony a fuzzy entropic characterization From: Gene Ward Smith Paul, I don't know if you are familiar with this article, but if not you will certainly want to be: Vidyamurthy and Murty, Musical harmony a fuzzy entropic characterization, Fuzzy Sets and Systems 48 (1992) #2, 195-200 The authors define a note by fuzzing it, as a function from R+ to [0,1], which attains the maximum of 1 (this suggests the function should be continuous or at least piecewise continuous to me.) They then define their version of harmonic entropy in general terms.

Message: 10731 - Contents - Hide Contents Date: Wed, 31 Mar 2004 22:29:19 Subject: Re: 98 out of 99 From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

>> To do >> the 9-limit more intrinsically, we would use a packing, not a lattice, >> and we would have ten quintads, not six. We could take all points >> [a,b,c] with [a,b,c] mod 5 [3,1,1] for major quintads, and [2,4,4] for >> minor quintands, and link them when they share an interval. I don't >> know what that looks like, but it's three-dimensional. > > Fascinating.

There's actually no reason not to simply add extra links to the lattice to do this, and you get three extra links, not four. To the six links [+-1,0,0], [0,+-1,0], [0,0,+-1] we add, if the quintad is otonal, links to quintads [0,2,1], [-1,2,2] and [0,1,2] away; and if the quintad is utonal, [0,-2,-1], [1,-2,-2] and [0,-1,-2] away. If we want to make a lattice for this, instead of a packing, we could link to [0,+-2,+-1], [-+1,+-2,+-2], and [0,+-1,+-2] around every point; now some of the links no longer involve a shared interval or even a shared note, however. For a Euclidean lattice, we would use [[1,-1,-1], [-1,5,4], [-1,4,5]] as the symmetric matrix of the bilinear form (inner product), and a^2+5b^2+5c^2-2ac-2ac+8bc as the corresponding quadratric form.

Message: 10732 - Contents - Hide Contents Date: Wed, 31 Mar 2004 20:42:18 Subject: Re: Questions for Carl From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:

> Paul G Hjelmstad wrote: > Yes, the column of the adjoint corresponding to the row of the original > matrix that represents the octave gives you a representative ET > mapping/val/constant structure. If you want to temper out all the > commas, that's the equal temperament you were looking for.

If the monzo matrix is unimodular, ie has determinant +-1, then all of the columns correspond to vals; if the monzos are comma monzos, then the vals will be +- equal temperament vals.

Message: 10734 - Contents - Hide Contents Date: Wed, 31 Mar 2004 13:38:44 Subject: Re: Questions for Carl From: Carl Lumma

>Thanks. I didn't word these questions that well. What I meant about >generators "lining up with temperament values" is the case where >a generator IS a step in the given temperament. (Perhaps 6 steps >out of 31-et, for example). I'll look into the Stern-Brocot tree.

Generator representations aren't unique. In 31-et, any interval will generate the tuning since 31 is prime. -Carl

Message: 10735 - Contents - Hide Contents Date: Wed, 31 Mar 2004 11:54:19 Subject: Re: Stoermer From: Manuel Op de Coul I calculated the 29- and 31-limit superparticulars. Pretty sure they are complete too, tested until 7*10^7. Only the denominators are shown. 29-limit: 28, 29, 57, 87, 115, 116, 144, 174, 203, 231, 260, 289, 319, 377, 405, 493, 550, 551, 608, 637, 725, 782, 783, 840, 1014, 1044, 1275, 1449, 1595, 1624, 1682, 2000, 2001, 2175, 2204, 2261, 2464, 2639, 2754, 2783, 3248, 3450, 3509, 4640, 4784, 4900, 5103, 5887, 5915, 6669, 7105, 7424, 7888, 8670, 9801, 10556, 11339, 12005, 12672, 13224, 13310, 13311, 13455, 19227, 20735, 23750, 24794, 25839, 26999, 30624, 30855, 35321, 47124, 53360, 72500, 83520, 87464, 136850, 158949, 166634, 168750, 176000, 176175, 184092, 240786, 244035, 303600, 410669, 418760, 613088, 949025, 1163799, 1235168, 1243839, 1625624, 1852200, 2697695, 4004000, 4090624, 8268799, 10556000, 18085704 31-limit: 30, 62, 92, 124, 154, 155, 186, 247, 279, 340, 341, 434, 464, 495, 527, 588, 620, 650, 713, 805, 836, 867, 899, 930, 960, 1053, 1209, 1364, 1425, 1518, 1519, 1767, 1859, 2015, 2232, 2944, 2975, 3564, 3750, 3875, 3968, 4185, 4959, 4991, 5642, 5796, 6075, 6137, 6292, 6324, 6479, 6727, 7656, 7904, 7935, 8091, 8463, 8525, 8959, 9424, 10880, 11780, 11934, 12121, 13299, 13454, 15624, 17576, 19250, 19343, 19964, 21141, 22815, 23374, 23715, 24024, 27404, 29791, 31464, 31899, 32798, 41261, 42687, 49010, 58310, 78336, 96875, 98735, 102486, 108375, 111320, 111475, 116280, 116963, 122264, 174096, 175769, 178125, 190463, 207575, 212381, 227447, 240064, 245024, 260337, 268800, 278783, 288144, 314432, 453375, 459172, 509795, 773604, 863939, 912950, 1147124, 1154439, 1255500, 1594175, 2307360, 2310399, 2345056, 3206268, 3301375, 3346109, 3897165, 14753024, 16092999 Manuel ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 10737 - Contents - Hide Contents Date: Thu, 01 Apr 2004 20:00:13 Subject: Re: a 3-Voci, doublestacked necklaces From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> Below are the interval vectors for I,ii,iii,IV,V,vi against > I,ii,iii,IV,V and vi. (I am working on all white-key trichords > and also the full set of trichords (double-stacked necklaces)) > > Here, there are only 8 unique ones. I've labelled the first appearance > of each one. (In this case it makes sense that, for example, I-IV and > I-V and ii-vi and iii-vi all have the same vectors. There are no Z- > relations) > > 3,0,0,2,2,2,0 I-I > 0,1,4,1,0,3,0 I-ii > 2,1,0,2,2,2,0 I-iii > 0,1,4,1,0,2,1 ii-iii > 1,1,2,1,1,3,0 I-IV > 0,2,3,0,1,2,1 iii-IV > 1,0,3,2,0,2,1 ii-V > 2,0,1,2,2,2,0 I-vi > > This is so easy I could have put it on "tuning", but since I am doing > the others here I put it here also. So these are the energy states > between chords an ordinary garage-band might use, (power chords!) for > what it's worth. > > It does show a little of the voice-leading potentialities, I guess. > I don't know if "energy-states between two necklaces" really > carries over from physics into music, that would be a question for > a psychoacoustician. EIS A045612 only considers cases where > the number of "positive and negative charges" are the same on the > two necklaces so I am already stretching things a bit. > > -- PHj

Hi Paul . . . I'm listening. Now, what is the matrix above? Easy as it may be, I'm afraid I don't understand it. -Paul

Message: 10739 - Contents - Hide Contents Date: Thu, 01 Apr 2004 00:05:11 Subject: Re: Questions for Carl From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

>> Thanks. I didn't word these questions that well. What I meant about >> generators "lining up with temperament values" is the case where >> a generator IS a step in the given temperament. (Perhaps 6 steps >> out of 31-et, for example). I'll look into the Stern-Brocot tree. >

> Generator representations aren't unique. In 31-et, any interval > will generate the tuning since 31 is prime. > > -Carl

Only if you assume that everything is cyclic around the octave. Gene's math never assumes that, so the only generator for 31-equal is (plus or minus) 1/31 of an octave. Possibly 1/31 of a tempered octave. But I don't think this is pertinent to the passage you were quoting.

Message: 10741 - Contents - Hide Contents Date: Thu, 01 Apr 2004 05:38:22 Subject: Re: Questions for Carl From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> Only if you assume that everything is cyclic around the octave. > Gene's math never assumes that, so the only generator for 31-equal is > (plus or minus) 1/31 of an octave. Possibly 1/31 of a tempered octave. > > But I don't think this is pertinent to the passage you were quoting.

To be even more impertinent, this made me consider the [1,1/31] linear temperament systems. In the five limit, |46 -10 -13>, in the seven limit <13 -10 6 -46 -27 42| with TM commas 225/224 and 589824/588245. TOP generators 1200.45 and 38.44 in the five limit, and 1200.34 and 38.43 in the seven limit. In the eleven limit, two systems have some plausibilty, but clearly the one to choose is the one with mapping [<1 2 2 3 4|, <0 -13 10 -6 -17|] and TM commas {225/224, 385/384, 1331/1323}, giving 1200.38 and 38.39 as generators. This is more interesting that one might suppose, since the improvement in the tuning over 31 equal or TOP 31 is considerable. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 10744 - Contents - Hide Contents Date: Sun, 04 Apr 2004 03:52:25 Subject: Why AMT (amity)? From: Paul Erlich Dear Gene, The acronym stands for "acute minor thirds". But the generator is c.340 cents. Isn't that an _obtuse_ minor third, not an acute one? Confused, Paul ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 10745 - Contents - Hide Contents Date: Sun, 04 Apr 2004 16:19:39 Subject: Re: Why AMT (amity)? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> Dear Gene, > > The acronym stands for "acute minor thirds". But the generator is > c.340 cents. Isn't that an _obtuse_ minor third, not an acute one?

The amity generator is a slightly sharp 243/200, and 243/200 is listed by Manuel as "acute minor third", the only interval which he calls by this name. I presume "acute" means "sharp".

Message: 10746 - Contents - Hide Contents Date: Sun, 04 Apr 2004 21:35:07 Subject: Re: Why AMT (amity)? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

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

>> The acronym stands for "acute minor thirds". But the generator is >> c.340 cents. Isn't that an _obtuse_ minor third, not an acute one? >

> The amity generator is a slightly sharp 243/200, and 243/200 is listed > by Manuel as "acute minor third", the only interval which he calls by > this name. I presume "acute" means "sharp".

Oh, right . . . presumably "acute" is the opposite of "grave" somehow (I don't know if I'd rather have acute pancreatitis or grave pancreatitis). Another question -- why "semisixths" but "hemifourths"? Why not "semifourths"? A hyphenated use of the latter word is found here: http://www.anaphoria.com/xen3a.PDF - Type Ok * [with cont.] (Wayb.)

Message: 10747 - Contents - Hide Contents Date: Sun, 04 Apr 2004 23:22:38 Subject: Re: 126 7-limit linears From: Paul Erlich Hi Gene, Would you be so kind as to produce a file like the one below, but instead of culling to 126 lines, leave all 32201 in there? That would be great. If that's too much, you could cut off the error and complexity wherever you see fit. The idea, though, is to produce a graph, and as most pieces of paper are rectangular, the data should fill a rectangular region. I'm *not* arguing for a rectangular badness function. Also could you provide the TM-reduced kernel bases -- at least for the 126 below? Thanks so much, Paul --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> I first made a candidate list by the kitchen sink method: > > (1) All pairs n,m<=200 of standard vals > > (2) All pairs n,m<=200 of TOP vals > > (3) All pairs 100<=n,m<400 of standard vals > > (4) All pairs 100<=n,m<=400 of TOP vals > > (5) Generators of standard vals up to 100 > > (6) Generators of certain nonstandard vals up to 100 > > (7) Pairs of commas from Paul's list of relative error < 0.06, > epimericity < 0.5 > > (8) Pairs of vals with consistent badness figure < 1.5 up to 5000 > > This lead to a list of 32201 candidate wedgies, most of which of > course were incredible garbage. I then accepted everything with a 2.8 > exponent badness less than 10000, where error is TOP error and > complexity is our mysterious L1 TOP complexity. I did not do any > cutting off for either error or complexity, figuring people could > decide how to do that for themselves. The first six systems are > macrotemperaments of dubious utility, number 7 is the {15/14, 25/24} > temperament, and 8 and 9 are the beep-ennealimmal pair, and number 13 > is father. After ennealimmal, we don't get back into the micros until > number 46; if we wanted to avoid going there we can cutoff at 4000. > Number 46, incidentally, has TM basis {2401/2400, 65625/65536} and is > covered by 140, 171, 202 and 311; the last is interesting because of > the peculiar talents of 311. > > > > 1 [0, 0, 2, 0, 3, 5] 662.236987 77.285947 2.153690 > 2 [1, 1, 0, -1, -3, -3] 806.955502 64.326132 2.467788 > 3 [0, 0, 3, 0, 5, 7] 829.171704 30.152577 3.266201 > 4 [0, 2, 2, 3, 3, -1] 870.690617 33.049025 3.216583 > 5 [1, 2, 1, 1, -1, -3] 888.831828 49.490949 2.805189 > 6 [1, 2, 3, 1, 2, 1] 1058.235145 33.404241 3.435525 > 7 [2, 1, 3, -3, -1, 4] 1076.506437 16.837898 4.414720 > 8 [2, 3, 1, 0, -4, -6] 1099.121425 14.176105 4.729524 > 9 [18, 27, 18, 1, -22, -34] 1099.959348 .036377 39.828719 > 10 [1, -1, 0, -4, -3, 3] 1110.471803 39.807123 3.282968 > 11 [0, 5, 0, 8, 0, -14] 1352.620311 7.239629 6.474937 > 12 [1, -1, -2, -4, -6, -2] 1414.400610 20.759083 4.516198 > 13 [1, -1, 3, -4, 2, 10] 1429.376082 14.130876 5.200719 > 14 [1, 4, -2, 4, -6, -16] 1586.917865 4.771049 7.955969 > 15 [1, 4, 10, 4, 13, 12] 1689.455290 1.698521 11.765178 > 16 [2, 1, -1, -3, -7, -5] 1710.030839 16.874108 5.204166 > 17 [1, 4, 3, 4, 2, -4] 1749.120722 14.253642 5.572288 > 18 [0, 0, 4, 0, 6, 9] 1781.787825 33.049025 4.153970 > 19 [1, -1, 1, -4, -1, 5] 1827.319456 54.908088 3.496512 > 20 [4, 4, 4, -3, -5, -2] 1926.265442 5.871540 7.916963 > 21 [2, -4, -4, -11, -12, 2] 2188.881053 3.106578 10.402108 > 22 [3, 0, 6, -7, 1, 14] 2201.891023 5.870879 8.304602 > 23 [0, 0, 5, 0, 8, 12] 2252.838883 19.840685 5.419891 > 24 [4, 2, 2, -6, -8, -1] 2306.678659 7.657798 7.679190 > 25 [2, 1, 6, -3, 4, 11] 2392.139586 9.396316 7.231437 > 26 [2, -1, 1, -6, -4, 5] 2452.275337 22.453717 5.345120 > 27 [0, 0, 7, 0, 11, 16] 2580.688285 9.431411 7.420171 > 28 [1, -3, -4, -7, -9, -1] 2669.323351 9.734056 7.425960 > 29 [5, 1, 12, -10, 5, 25] 2766.028555 1.276744 15.536039 > 30 [7, 9, 13, -2, 1, 5] 2852.991531 1.610469 14.458536 > 31 [2, -2, 1, -8, -4, 8] 3002.749158 14.130876 6.779481 > 32 [3, 0, -6, -7, -18, -14] 3181.791246 2.939961 12.125211 > 33 [2, 8, 1, 8, -4, -20] 3182.905310 3.668842 11.204461 > 34 [6, -7, -2, -25, -20, 15] 3222.094343 .631014 21.101881 > 35 [4, -3, 2, -14, -8, 13] 3448.998676 3.187309 12.124601 > 36 [1, -3, -2, -7, -6, 4] 3518.666155 18.633939 6.499551 > 37 [1, 4, 5, 4, 5, 0] 3526.975600 19.977396 6.345287 > 38 [2, 6, 6, 5, 4, -3] 3589.967809 8.400361 8.700992 > 39 [2, 1, -4, -3, -12, -12] 3625.480387 9.146173 8.470366 > 40 [2, -2, -2, -8, -9, 1] 3634.089963 14.531543 7.185526 > 41 [3, 2, 4, -4, -2, 4] 3638.704033 20.759083 6.329002 > 42 [6, 5, 3, -6, -12, -7] 3680.095702 3.187309 12.408714 > 43 [2, 8, 8, 8, 7, -4] 3694.344150 3.582707 11.917575 > 44 [2, 3, 6, 0, 4, 6] 3938.578264 20.759083 6.510560 > 45 [0, 0, 5, 0, 8, 11] 3983.263457 38.017335 5.266481 > 46 [22, -5, 3, -59, -57, 21] 4009.709706 .073527 49.166221 > 47 [3, 5, 9, 1, 6, 7] 4092.014696 6.584324 9.946084 > 48 [7, -3, 8, -21, -7, 27] 4145.427852 .946061 19.979719 > 49 [1, -8, -14, -15, -25, -10] 4177.550548 .912904 20.291786 > 50 [3, 5, 1, 1, -7, -12] 4203.022260 12.066285 8.088219 > 51 [1, 9, -2, 12, -6, -30] 4235.792998 2.403879 14.430906 > 52 [6, 10, 10, 2, -1, -5] 4255.362112 3.106578 13.189661 > 53 [2, 5, 3, 3, -1, -7] 4264.417050 21.655518 6.597656 > 54 [6, 5, 22, -6, 18, 37] 4465.462582 .536356 25.127403 > 55 [0, 0, 12, 0, 19, 28] 4519.315488 3.557008 12.840061 > 56 [1, -3, 3, -7, 2, 15] 4555.017089 15.315953 7.644302 > 57 [1, -1, -5, -4, -11, -9] 4624.441621 14.789095 7.782398 > 58 [16, 2, 5, -34, -37, 6] 4705.894319 .307997 31.211875 > 59 [4, -32, -15, -60, -35, 55] 4750.916876 .066120 54.255591 > 60 [1, -8, 39, -15, 59, 113] 4919.628715 .074518 52.639423 > 61 [3, 0, -3, -7, -13, -7] 4967.108742 11.051598 8.859010 > 62 [6, 0, 0, -14, -17, 0] 5045.450988 5.526647 11.410361 > 63 [37, 46, 75, -13, 15, 45] 5230.896745 .021640 83.678088 > 64 [1, 6, 5, 7, 5, -5] 5261.484667 11.970043 8.788871 > 65 [3, 2, -1, -4, -10, -8] 5276.949135 17.564918 7.671954 > 66 [1, 4, -9, 4, -17, -32] 5338.184867 2.536420 15.376139 > 67 [1, -3, 5, -7, 5, 20] 5338.971970 8.959294 9.797992 > 68 [10, 9, 7, -9, -17, -9] 5386.217633 1.171542 20.325677 > 69 [19, 19, 57, -14, 37, 79] 5420.385757 .046052 64.713343 > 70 [5, 3, 7, -7, -3, 8] 5753.932407 7.459874 10.743721 > 71 [3, 5, -6, 1, -18, -28] 5846.930660 3.094040 14.795975 > 72 [3, 12, -1, 12, -10, -36] 5952.918469 1.698521 18.448015 > 73 [6, 0, 3, -14, -12, 7] 6137.760804 5.291448 12.429144 > 74 [4, 4, 0, -3, -11, -11] 6227.282004 12.384652 9.221275 > 75 [3, 0, 9, -7, 6, 21] 6250.704457 6.584324 11.570803 > 76 [9, 5, -3, -13, -30, -21] 6333.111158 1.049791 22.396682 > 77 [0, 0, 8, 0, 13, 19] 6365.852053 14.967465 8.686091 > 78 [4, 2, 5, -6, -3, 6] 6370.380556 16.499269 8.391154 > 79 [1, -8, -2, -15, -6, 18] 6507.074340 4.974313 12.974488 > 80 [2, -6, 1, -14, -4, 19] 6598.741284 6.548265 11.820058 > 81 [2, 25, 13, 35, 15, -40] 6657.512727 .299647 35.677429 > 82 [6, -2, -2, -17, -20, 1] 6845.573750 3.740932 14.626943 > 83 [1, 7, 3, 9, 2, -13] 6852.061008 12.161876 9.603642 > 84 [0, 5, 5, 8, 8, -2] 7042.202107 19.368923 8.212986 > 85 [4, 2, 9, -6, 3, 15] 7074.478038 8.170435 11.196673 > 86 [8, 6, 6, -9, -13, -3] 7157.960980 3.268439 15.596153 > 87 [5, 8, 2, 1, -11, -18] 7162.155511 5.664628 12.817743 > 88 [3, 17, -1, 20, -10, -50] 7280.048554 .894655 24.922952 > 89 [4, 2, -1, -6, -13, -8] 7307.246603 13.289190 9.520562 > 90 [5, 13, -17, 9, -41, -76] 7388.593186 .276106 38.128083 > 91 [8, 18, 11, 10, -5, -25] 7423.457669 .968741 24.394122 > 92 [3, -2, 1, -10, -7, 8] 7553.291925 18.095699 8.628089 > 93 [3, 7, -1, 4, -10, -22] 7604.170165 7.279064 11.973078 > 94 [6, 10, 3, 2, -12, -21] 7658.950254 3.480440 15.622931 > 95 [14, 59, 33, 61, 13, -89] 7727.766150 .037361 79.148236 > 96 [3, -5, -6, -15, -18, 0] 7760.555544 4.513934 14.304666 > 97 [13, 14, 35, -8, 19, 42] 7785.862490 .261934 39.585940 > 98 [11, 13, 17, -5, -4, 3] 7797.739891 1.485250 21.312375 > 99 [2, -4, -16, -11, -31, -26] 7870.803242 1.267597 22.628529 > 100 [2, -9, -4, -19, -12, 16] 7910.552221 2.895855 16.877046 > 101 [0, 0, 9, 0, 14, 21] 7917.731843 14.176105 9.573860 > 102 [3, 12, 11, 12, 9, -8] 7922.981072 2.624742 17.489863 > 103 [1, -6, 3, -12, 2, 24] 8250.683192 8.474270 11.675656 > 104 [55, 73, 93, -12, -7, 11] 8282.844862 .017772 105.789216 > 105 [4, 7, 2, 2, -8, -15] 8338.658153 10.400103 10.893408 > 106 [0, 5, -5, 8, -8, -26] 8426.314560 8.215515 11.894828 > 107 [5, 8, 14, 1, 8, 10] 8428.707855 4.143252 15.190723 > 108 [6, 7, 5, -3, -9, -8] 8506.845926 6.986391 12.646486 > 109 [8, 13, 23, 2, 14, 17] 8538.660000 1.024522 25.136807 > 110 [0, 0, 10, 0, 16, 23] 8630.819015 11.358665 10.686371 > 111 [3, -7, -8, -18, -21, 1] 8799.551719 2.900537 17.521249 > 112 [0, 5, 10, 8, 16, 9] 8869.402675 6.941749 12.865826 > 113 [4, 16, 9, 16, 3, -24] 8931.184092 1.698521 21.324102 > 114 [6, 5, 7, -6, -6, 2] 8948.277847 9.097987 11.718042 > 115 [3, -3, 1, -12, -7, 11] 9072.759561 14.130876 10.062449 > 116 [0, 12, 24, 19, 38, 22] 9079.668325 .617051 30.795105 > 117 [33, 78, 90, 47, 50, -10] 9153.275887 .016734 112.014440 > 118 [5, 1, -7, -10, -25, -19] 9260.372155 3.148011 17.329377 > 119 [1, -6, -2, -12, -6, 12] 9290.939644 13.273963 10.377495 > 120 [2, -2, 4, -8, 1, 15] 9367.180611 25.460673 8.247748 > 121 [3, 5, 16, 1, 17, 23] 9529.360455 3.220227 17.366255 > 122 [6, 3, 5, -9, -9, 3] 9771.701969 9.773087 11.787090 > 123 [15, -2, -5, -38, -50, -6] 9772.798330 .479706 34.589494 > 124 [2, -6, -6, -14, -15, 3] 9810.819078 6.548265 13.618691 > 125 [1, 9, 3, 12, 2, -18] 9825.667878 9.244393 12.047225 > 126 [1, -13, -2, -23, -6, 32] 9884.172505 2.432212 19.449425

Message: 10748 - Contents - Hide Contents Date: Mon, 5 Apr 2004 17:10:39 Subject: Re: Comma names From: Manuel Op de Coul Gene wrote:

>15/14 major diatonic semitone (should be septimal diatonic semitone)

I wouldn't say "should be", it's another possibility. This list wasn't intended to be systematic but more as a list of historical usage.

>22/21 no name--unidecimal minor semitone?

You mean "undecimal", looks ok to me.

>33/32 unidecimal comma (bad name! Schoenberg's diesis?)

Why Schönberg's diesis and not undecimal 1/4-tone for example?

>540/539 Swets' comma (who is Swet?)

Wouter Swets is a Dutch ethnomusicologist. He mentioned this comma to me personally. I can't say more about it since he still has to publish his results. Manuel

Message: 10749 - Contents - Hide Contents Date: Mon, 05 Apr 2004 22:41:17 Subject: Re: 126 7-limit linears From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

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

>> Gene, you've done work of such breadth and depth. Is it going to >> remain unorganized, spread across thousands of posts, forever? >

> A little has been organized on my web site. I'm wondering again about > what and where to publish, after having seen what's gotten published > lately. There's enough material for a book, if I could get myself up > to speed and actually do it; but I started a book on Chebyshev > functions and never finished it, and in general have a problem > finishing projects.

So do I -- particularly the projects that mean most to me. This is why I have composed so little. You've done me better -- many, many better -- in this regard. I applaud you, Gene, and your compositions will receive due notice in my paper. Which website would you like me to reference?