How Many Ultimate HB Pair Combinations Do K Pickups Have?
© 2016 Don Baker dba android originals LC
Notes on Humbucking Pair Placement showed how the ratios of in-phase to contra-phase humbucking pairs among 4 pickups could be changed by changing the number of the same poles up. On Changing the Personality of a Guitar – Part 1 suggested that pickups, plugged into fixed positions, with different poles could be moved from spot to spot for other pole arrangements. And that the pickup switching could be rearranged using a cross-point board or switch to maintain a switching progression from warm timbres (tones) to bright. On other words, just like a tone pot.
Applying Humbucking (HB) Pairs Analysis to the Music-Man St. Vincent Guitar and Applying Humbucking (HB) Pairs Analysis to the Paul Reed Smith 513 Guitar showed that using standard humbucker pickups in a humbucking pairs switching system can result in effectively equivalent tones for different HB pairs. How Many Useful Pickup Combinations Can You Get From 5 Coils? showed a number of different electrical circuit topologies for 2, 3, and 4 coil systems.
If you have K matched single-coil pickups in fixed positions, we know from combinatorial math and empirical testing that you get K*(K-1)/2 different combinations of two coils each, which can be either of two topologies, series- or parallel-connected, for a total of K*(K-1) different timbres. But if you start moving the poles around changing the ratio of N to S poles, how many different timbres are actually possible, and how many are equivalents? How many are in-phase? How many are contra-phase?
K = 2
For the lack of a better term, let’s call that the ultimate number of combinations for K pickups, UK. For the moment, let’s ignore the question of circuit topology. For HB pairs, you just take the number of unique combinations and multiply by 2. For K = 2, the answer almost trivial. You can have either two poles the same (contra-phase) or two poles different (in-phase).
So U1 = 2, and the total number of possible timbres is 2*U1 = 4. And because it doesn’t matter if you change the poles of both pickups at the same time, only one of them in one physical position needs to be changed to the opposite pole to change from in-phase to contra-phase and back. Note that this could be done by putting a standard humbucker in place of one of the single coils, and using either one pole or the other.
K = 3
K = 3 is a bit more complicated. The math may exist for predicting UK, but it isn’t immediately apparent. For K = 3, only one pole can be different, if any, and it doesn’t matter for this discussion if it is N or S. We know that for K = 3, we can have 3*2/(2*1) = 3 different HB pairs for each pole setup. We can have either 0 poles different or 1 pole different, since having either 2 N poles or 2 S poles still leaves just 1 pole different. The different pole an be in each of the 3 physical positions, so with that and 0 poles different, we have 4 different pole setups.
So we start by drawing out all the combinations, and striking out the ones that are effectively the same from the first pole setup in which they appear, to their appearance in the next setup(s). In the tables below, the black rectangle shapes represent pickups with N poles up, and the white, S poles up. Instead of the possibly confusing position designation N, M, B for neck, middle and bridge, we will use A, B and C, with AN meaning A position, N-up, BS meaning B position, S-up, etc. InØ means in-phase, and conØ means contra-phase. We take the pairs in the order, AB, AC, & BC, and placed the in-phase and contra-phase pairs in the proper column.
In humbucking pairs, AS,BN will produce the same tone as AN,BS, so the second is stricken from the tables, as are all the other similar tones. This leaves U3 = 6, with 3 in-phase, and 3 contra-phase. So with the series and parallel topologies, this means that the 4 pole setups can produce 2*U3 = 12 unique timbres with humbucking pairs.
Note that in the last figure, no unique humbucking pairs have been added. So if one were going to replace any single-coil pickups with dual-coil humbuckers, to switch the poles electronically, it would be sufficient to place them in the A and B positions only. Then they would add the 2 in-phase humbucking pairs in those positions, for a total of 8 unique pairs, or 16 humbucking pair timbres, consistent with the analysis in Applying Humbucking (HB) Pairs Analysis to the Paul Reed Smith 513 Guitar. Which also applies to the Music-Man Game Changer guitar with two humbuckers and a single.
K = 4
For K = 4, we use matched single-coil pickups in positions A, B, C and D. The pairs, numbering K*(K-1)/2 = 6, will be listed in the tables below in the order AB, AC, AD, BC, BD & CD. We start with the pole setup (S,S,S,S); then add one pole and shift it to each position: (N,S,S,S), (S,N,S,S), (S,S,N,S) & (S,S,S,N); then add another pole and shift the two poles to other positions: (N,N,S,S), (N,S,N,S) & (N,S,S,N); until no more unique pairs can be added. Note that these 8 setups suffice, as anything with 3 N poles up will merely mirror the setups with 1 N pole up.
Now we see the mathematical pattern for K number of matched single-coil pickups. The total number of possible in-phase pairs is K*(K-1)/2, and the total number of possible contra-phase pairs is K*(K-1)/2. So UK = K*(K-1) and the total number unique humbucking tones or timbres is 2* UK = 2* K*(K-1). So for K = 2,3,4,5,…, UK = 2, 6, 12, 20, …, and the number of possible unique timbres is 2* UK = 4, 12, 24, 40, …
It also appears that it takes only K sets of different pole configurations to obtain UK. Which K sets? It probably doesn’t matter. For K = 2, the possible different sets with unique pole configuration (excluding those mirrored in exchanging N and S), is (S,S) and (N,S), or 2 sets. For K = 3, the possible different sets with unique poles configurations is (S,S,S), (N,S,S), (S,N,S) & (S,S,N), or 4 sets. For K = 4, the possible unique pole configurations are (S,S,S,S), (N,S,S,S), (S,N,S,S), (S,S,N,S), (S,S,S,N), (N,N,S,S), (N,S,N,S) & (N,S,S,N), or 8 sets.
K = 5
For K = 5, the unique configurations are (S,S,S,S,S), (N,S,S,S,S), (S,N,S,S,S), (S,S,N,S,S), (S,S,S,N,S), (S,S,S,S,N), (N,N,S,S,S), (N,S,N,S,S), (N,S,S,N,S), (N,S,S,S,N), (S,N,N,S,S), (S,N,S,N,S), (S,N,S,S,N), (S,S,N,N,S), (S,S,N,S,N), (S,S,S,N,N), or 16 sets. For 3 N poles of 5, it only mirrors 3 S poles of 5. So the mathematical pattern appears to be for K coils, there are 2^(K-1) or 2 raised to the (K-1) power, K = 2, 3, 4, 5, … è 2, 4, 8, 16, … unique sets of pole configurations.
It is not immediately apparent how to prove that any K different pole configurations out of 2^(K-1) possible unique pole configurations can produce the maximum number of humbucking pairs and humbucking pair timbres. Other than by writing out all the tables and demonstrating that likelihood by induction, as was done here for UK. This exercise is left to the reader.
Purpose – a Whole New Universe of Customization
This article only means to show how many different timbres can be expected from K number of single-coil pickups that can each be switched out independently for opposite poles. If for example, you have a guitar with K = 4 pickups, switching between humbucking pairs and serial and parallel connections will give you 12 different timbres for each pole arrangement, and 24 different timbres across all the pole arrangements, of which you probably need only 4 different pole arrangements of the possible 8.
If the humbucking pair switching is set up to go from warm to bright, the 8 different pole arrangements will give you different progressions from warm to bright. As noted in Part 1, it is feasible to put a cross-connection board or cross-point switch in between the pickups and the switching to compensate for the pole arrangement to produce switching that goes progressively from warm to bright.
If you can switch out the poles in this manner, and the switching compensates to produce a progressive order from warm to bright, you have a 4-pickup guitar with 4 to 8 different personalities in timbre. You can go from all contra-phase outputs (with all 4 N poles up, or 4 S poles up), serial and parallel, to 2/3 in-phase and 1/3 contra-phase (with 2 N and 2 S poles up), as noted in Notes on Humbucking Pair Placement.
If you can further shade the timbres by moving the pickups physically, as in Patent U.S. 2016/0027422 A1, then that opens up a whole new universe in customizing the guitar to the guitarist. One that will require a great deal more study and experimentation.
Take, for example, a decision not to use plug-in pickups, but to place standard humbuckers in positions A, B, and C or D for K = 4, switching the poles electrically instead of physically. This adds a new wrinkle or two. First, that’s 7 coils in the space between the neck and bridge. That’s a lot of coils in a small space. Second, the humbuckers themselves add 6 new timbres (3 each with parallel and serial connections of their 2 poles) to the 24 they can produce in the analysis above for a total of 30.
Third, let’s say the humbuckers are all mounted with their N poles closer to the neck. That means the NN & SS (and to a lesser extent the NS and SN) connections of single poles between them now have slightly different physical positions than if they had be constructed of single-coil pickups. Are the differences tonally significant, or even audible enough, to nullify the previous strikeouts in the tables above for K =4?
Only experimentation and verification can tell.
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