Math 1:\u00a0\u00a0\u00a0\u00a0\u00a0 n! = n*(n-1)*(n-2)* \u2026 2*1,\u00a0 i.e.,\u00a0 4! = 4*3*2*1 = 24<\/p>\n
Summation series<\/strong>: the sum of a set of expressions with the same formula, differing according to the incremental change (usually an integer change) of one or more terms.\u00a0 Such as, the summation of n! from 1 to 4, by increments of 1:<\/p>\n Math 2:\u00a0 Product series<\/strong>: the product of a series of expressions, which change according an increment of one of the terms, such as the product of i as i decreases from n to 1:<\/p>\n Math 3:\u00a0 Permutations<\/strong>: the number of unique<\/em> ways to separate k things out of n things, if the order of which thing comes first matters:<\/p>\n Math 4:\u00a0 For example, the number of ways can 8 Olympians be awarded gold, silver and bronze is 8!\/5! = 40320\/120 = 336.\u00a0 Note the two notations (of several) shown above for permutations.\u00a0 We are not really going to use permutations, but it helps to understand combinations.<\/p>\n Conbinations<\/strong>: the number of unique<\/em> ways n things can be taken k at a time without regard to order:<\/p>\n Math 5:\u00a0 For example, the number of ways that 8 Olympians can stand on the podium, regardless of who wins what is 336\/3! = 56.\u00a0 Another way to think of it is to put 3 factors in the top and bottom of a fraction:<\/p>\n Math 6:\u00a0 Combinations of pickups don\u2019t tell you how to connect them together.\u00a0 They only tell you how many you have to work with.\u00a0 In this case, the formula is simple:<\/p>\n Math 7:\u00a0 For example, if you have 3 humbuckers, and you want to know how many pairs of humbuckers (the equivalent of a humbucking quad of matched single-coil pickups) you can make out of them, the answer is:<\/p>\n Math 8:\u00a0 The total number of all possible different unique combinations, from MM = 1 to KK is:<\/p>\n Math 9:\u00a0 In other words, for 3 humbuckers, the number of single, pairs and triples (equivalent to pairs, quads, and hexes of matched single-coil pickups) is:<\/p>\n Math 10:\u00a0 Table 1 shows how the number of combinations fall out for 1 to 6 dual-coil humbuckers.\u00a0 Note that the columns in the table are binomial coefficients, and the bottom row follows a progression, 1, 1+2, 3+4, 7+8, 15+16, 31+32.\u00a0 So the table could be extended indefinitely in that manner.<\/p>\n Although the number of PAF-sized humbuckers that can fit between the neck and bridge of a guitar is probably no more than 4, a piano would have a lot more room.<\/p>\n This is a bit more tricky.\u00a0 At the humbucking pair level, dual-coil humbuckers are already fixed.\u00a0 If you have humbuckers AB, CD & EF, for the inner coils A, B, C, D, E & F, coil A is always connected to coil B.\u00a0 But for the matched single-coil pickups, A, B, C, D, E & F, any coil can associate with any other coil.<\/p>\n For the coil set A & B, we have a single pair, AB, or the number of the combination of 2 things taken 2 at a time.\u00a0 For the coil set A, B & C, we have the pairs AB, AC & BC, or the number C(3,2) = 3.\u00a0 For the coil set A, B, C & D, we have the pairs, AB, AC, AD, BC, BD, and CD, or the number C(4,2) = 6.\u00a0 We also have the unique quad sets, (AB,CD), (AC,BD) & (AD, BC), the number of 3.\u00a0 This is obiously not the same as 6 pairs taken 2 at a time, C(6,2) = 15.\u00a0 It can\u2019t be because if we had a combination like (AB,AC), it would not be a quad, and there is no way it could be humbucking.<\/p>\n So what happens if we add a 5th<\/sup> coil, E?\u00a0 First, all the combinations from the first four still exist.\u00a0 To those we add the combinations derived from adding E.\u00a0 E can combine uniquely with each of the original 4 pickups for form K-1 pairs additional pairs, for a total of 10.\u00a0 To form quads, the remaining 3 pickups can be taken 2 at a time to form the second pair, for a total of 3. So E adds 4*3 = 12 humbucking quads for a total of 15.<\/p>\n If we add a 6th<\/sup> coil, F, We have not only more pairs and quads, but a humbucking hex, (AB, CD, EF).\u00a0 If we add a 7th<\/sup> coil, G, we have more pairs, quads, and hexes.\u00a0 If we add an 8th<\/sup> coil, H, we have more pairs, quads, hexes, and a humbucking oct.<\/p>\n Without going through the entire development, the equation that results for the number, SNK,M<\/sub>, of unique combinations of K matched single-coil pickups organized as M humbucking pairs is:<\/p>\n Math 11:\u00a0 Note that in the last form, (j-2M)! need not be calculated, but simply removed from the sequence of the product (j-1)!.<\/p>\n As before with dual-coil humbuckers, the pairs, quads, etc., for a given K can be added together to get the total number of humbucking combinations of matched single-coil pickups, TSNK<\/sub>.<\/p>\n Math 12:\u00a0 <\/p>\n Again, this does not tell us how to connect the pickups together, only how many humbucking pairs, quad, hexes, octs, etc., of electromagnetic pickups there are to work with.<\/p>\n Published Aug 5, 2016.\u00a0 Correction<\/strong>: Table 2 was calculated in a spreadsheet, and indicated errors in the previously posted page<\/a> in HumbuckingPairs.com.\u00a0 Those have been corrected.<\/p>\n Aug 6, 2016 Note<\/strong>: It’s possible for this math to contain errors.\u00a0 Math 11 has been reworked.\u00a0 If you find any, please let me know, mailto:\u00a0 axe1 [a] TulsaSoundGuitars [d] c*m\u00a0 No solicitations for web site development or mobile apps, please.\u00a0 I do my own work.<\/p>\n <\/p>\n","protected":false},"excerpt":{"rendered":" \u00a9 2016 Don Baker dba android originals, TulsaSoundGuitars.com, HumbuckingPairs.com Mathematical Definitions First we need some definitions of mathematical terms to bring everyone onto the same page. Factorial n: the product of all numbers from n down to 1 Math 1:\u00a0\u00a0\u00a0\u00a0\u00a0 … Continue reading ![\"BEMath02\"](\"http:\/\/android-originals.com\/tulsasound\/wp-content\/uploads\/2016\/08\/BEMath02.png\")
<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\nUnique combinations of KK dual-coil humbucking pickups taken MM at a time, NKK,MM<\/sub><\/h2>\n
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\n<\/em><\/strong><\/p>\nUnique combinations of K match single-coil pickups taken M pairs at a time, SNK,M<\/sub><\/h2>\n
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