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Block Design for Piano

by Irritable Hedgehog



I call this piece Block Design for Piano, because that is the term used by mathematicians in combination theory for a particular configuration of combinations. The configuration here is technically a 4-(12,6,10) block design, which in my musical terms means that there are 12 notes, distributed into 6-note arpeggios. In fact, it is also true that every combination of 3 notes comes together 30 times in 30 different arpeggios, every pair of notes occurs 75 times in exactly 75 arpeggios, and each of the 12 notes occurs 165 times, in exactly half of the 330 arpeggios. In a way, the piece is a realization of Schoenberg's ideal, with exactly equal emphasis of each of the 12 notes. The music follows the numbers note for note, except that they occur two at a time when the difference between them is a major third.

The piece is dedicated to John McAlpine, who understands so well how to let music do what it "wants" to do, without forcing his will upon it. I must also acknowledge mathematician Patrick Solé (CNRS, France) whose correspondence guided me, and reassured me that my applied mathematics here is not really a distortion of the very sophisticated research that mathematicians do. This particular 4-(12,6,10) structure comes [from] page 47 of the CRC Handbook (CRC Press, 1996), where Donald L. Kreher gives the 30 "base blocks" from which one can generate an additional 300. A rather large literature concerning block design has evolved on several continents, mostly since the advent of the computer, but non-mathematicians like me must be prepared to take time for serious study if they wish to pursue this most interesting branch of combination theory.

Tom Johnson
Paris, August 2005


released December 12, 2015


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Irritable Hedgehog Kansas City, Missouri


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