The Chord Dial: One Law for Every Chord the Ring Can Play, n Rings on a Necklace and the Interval Cycle 60/n

The cosine ruler of this series plays two notes at a time. Two equal rings crossing on the sixty-division wheel sound the pair of divisions where they meet. This paper closes the instrument into a necklace. Put n equal rings on the corners of a regular shape with n sides, any n that divides sixty, with each neighbouring pair set to the same dial setting k, and the whole necklace becomes one chord-playing instrument governed by a single law. Every crossing lands exactly on a marked division, every neighbouring pair plays the dial's interval 2k, and each pair's two notes step around the wheel by exactly 60/n divisions from the last. Read as music, the necklace of three rings plays the augmented family and, at the two golden settings, a triad of pure octaves, four rings play a pair of diminished sevenths, five play octave stacks, six the whole-tone scale, and twelve the full chromatic scale, stepping around the circle of fourths. One more law rides along at every setting. A crossing read on one ring and its partner read on the other always sit exactly thirty divisions apart, the tritone, built into the geometry itself. Every claim is verified by direct computation at all fourteen settings for eight necklace sizes. Paper 6 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.