Beyond the Semitone: Microtonal Rings and the Resolution Ladder

A two-circle dial on a 60-point ring plays whole tones, and doubled to 120 points it plays the full chromatic scale. This paper takes the next step on the same ladder. Doubling once more, to a 240-point ring, makes one semitone span four divisions and the dial's interval step shrink to a quarter-tone. The dial then walks all twenty-four notes of the quarter-tone scale (24-tone equal temperament), one per setting, reaching the microtonal intervals that lie between the twelve familiar notes, intervals no coarser ring can produce. Finer rings continue the pattern. A ring of 360 points gives third-tones, 480 gives eighth-tones, and in general a ring of N points has interval step 120/N semitones. The 60-, 120-, and 240-point rings are the first three rungs of a single resolution ladder, whole tone, semitone, quarter-tone, each a finer reading of the same crossing geometry. The golden ratio is unmoved by all of this. On every five-octave ring, including the microtonal ones, the octave occurs only where the center distance equals R·φ and R/φ, because the octave is a fixed angle on the circle. All results are verified by direct computation. Paper 2 of 10 in the music series of The Strassner Series: Fibonacci Geometry and Harmony. The program that checks every number in the paper is printed in its appendix.