Calculating Stop Combinations, Part I
As soon as the first primitive slider and pallet wind chests were introduced into 15th century organs in Europe so that 2 independent registers in the instrument controlled by draw knobs called "stops" ( because pushing them "in" stopped the sound ) were supplied to play either singly or in combination, the discovery was quickly made that those same 2 stops (A, B) could be used in 3 different ways: A alone, B alone, or A + B.
In organs provided with 3 independent stops (A, B, C) it was found that 7 different ways were possible (A alone, B alone, C alone, A + B, A + C, B + C, or A + B + C).
As more independent stops were provided it was learned that 4 stops (A, B, C, D) could be used in 15 different ways, 5 stops (A, B, D, C, E) in 31 different ways, and so on. This progression proceeds predictably according to a simple, mathematical formula.
Early performers would notice another aspect of this mathematical relationship: it applied to the practicing of polyphonic pieces having multiple voices in general, and fugues in particular. When the 1st voice entered there was only one way, obviously, to practice that single line. When the 2nd voice entered they found there were 3 different ways to practice these 2 lines (i.e. right hand, left hand, both hands), and it became 3 times harder for them because if broken down separately each of the 3 different ways had to be coordinated enough to sound clearly. The entrance of the 3rd voice in trio texture made things 7 times harder because now there were 7 different ways to practice it (right hand, left hand, both hands, pedal, right hand and pedal, left hand and pedal, both hands and pedal). They also found that the addition of the 4th voice made the texture 15 times harder, and so on, according to formula.
We therefore are well-advised to adopt the stance that, unlike most other music, fugues require a special approach to practicing [See blog, How To Learn A Fugue].
(con't in Part II)