© 2016 Don Baker dba android originals LC

Today, Aug 2, 2016, I filed a Provisional Patent Application for switching circuits to connect dual-coil humbucking pickups and matched single-coil pickups into humbucking quads, including digital controls. I’ve read that no one can patent a mathematical equation, so that is what this document discloses – the number of humbucking quads that can be made from KK number of dual-coil humbuckers and K number of matched single-coil electromagnetic pickups. The math shows that there are a surprising number of possible combinations.

**Dual-coil humbuckers**

We will assume that the individual coils in each dual-coil humbuckers can be connected together in either series or parallel. But always in-phase, unless the humbuckers is specially constructed with the same magnetic poles upwards towards the strings in each coil. And then they might as well be two single-coil pickups. So, series and parallel makes two different tones for each humbucker. From KK number of humbuckers, you can get 2*KK number of tones.

A pair of humbuckers makes a quad – four individual coils connected together. A pair of humbuckers can be connected together in series and parallel, in-phase and contra-phase (out of phase). That’s 4 different connections. The coils of each humbucker can be connected together in series or parallel. Over two humbuckers, that is 4 more choices, for a total of 16. Any number of KK humbuckers equal to 2 or greater can have KK*(KK-1)/2 different pairs of humbuckers (by combinatorial analysis). That makes a total of 16*KK*(KK-1)/2 number of tones.

So KK number of humbuckers can have:

KK = 2, 3, 4, 5 humbucking pairs

2*KK = 4, 6, 8, 10 humbucking pair tones

KK = 1, 3, 6, 10 humbucking quads

16*KK*(KK-1)/2 =16, 48, 96, 160 humbucking quad tones

Total Tones = 20, 54, 104, 170

**Matched single-coil pickups**

We already know that K number of single-coil pickups can have K*(K-1)/2 different humbucking pairs. Since the pairs can be connected in parallel or series, that makes K*(K-1) humbucking pair tones. But humbucking quads needs more discussion.

Start from the fact that no single can appear twice or more in a quad. It just wouldn’t be a humbucking quad. It is then easy to construct and see how many humbucking pairs from K = 4 singles, A, B, C & D, can be made into unique quads. They are (AB,CD), (AC,BD) and (AD, BC). There are no more than K-1.

Now add another coil, E, to get A, B, C, D, and E, for K = 5. All the combinations of A, B, C, and D are the same, plus those added by E. E can pair with the K-1 other coils, making K-1 pairs. To get the next pair, we take K-2 singles 2 at a time with combinatorial analysis, to get (K-2)*(K-3)/2 pairs for each of the K-1 pairs with E. So with coil E we add (K-1)*(K-2)*(K-3)/2 quads to the existing K-1 quads. The same analysis holds for K = 6 or more. The same factor of 16 tones per quad made of two humbuckers applies to these quads.

The number of single coil quads from K single coils can be expressed by:

Nqk = Summation of j = 4 to K: (K-1)*(K-2)*(K-3)/2, K > 3

And the number of quad tones is: 16*Nqk. This means that:

K = 2, 3, 4, 5, 6, 7, 8 singles

K*(K-1)/2 = 1, 3, 6, 10, 15, 21, 28 humbucking pairs

K*(K-1) = 2, 6, 12, 20, 30, 42, 56 humbucking pair tones

Nqk = 0, 0, 3, 15, 45, 105, 210 humbucking quads

16*Nqk = 0, 0, 48, 240, 720, 1680, 3360 humbucking quad tones

Total Tones = 2, 6, 60, 260, 750, 1722, 3416

**Note**: Some of *these* numbers were found to be inaccurate (too large) and were corrected 5 Aug 2016

**Is this real? **

Could you possibly get that many different tones out of that many pickups? Not exactly. First, as the pickups get closer together the tonal differences get more subtle. Eventually, the differences may disappear. Second, in-phase quad tones, by the very separation of the pickups, can be spread and averaged over longer distances of under the strings than pairs. Thus they can be expected to be warmer, losing the higher harmonics, and getting closer to the fundamental harmonic of the string. The closer to the fundamental they get, they less distinct they become. Contra-phase quad tones will likely be something else.

But still, most dual humbucker guitars, with a potential of 20 different tones, only have a 3-way switch. Three humbucker guitars have a potential of 54 different tones. Compared to a guitar with two humbuckers and a single, a guitar with five matched single-coil pickups, more spread out, has a potential of 260 tones.

How much have you been missing?

But check my math, please. I could have made a mistake.