Filling the Whole-Tone Gap: How a 120-Point Ring Completes the Chromatic Scale

A sliding two-circle dial built on a 60-point ring sounds only even intervals, the whole-tone scale, because its two crossing points are always an even number of semitones apart. The perfect fifth and perfect fourth, the foundation of tonal music, never appear at any setting. This paper shows the limitation is not fundamental but a matter of resolution. Doubling the ring to 120 points, so that one semitone spans two divisions, halves the interval step. The two crossing points now lie k semitones apart instead of 2k. As the single setting k advances, the dial walks the entire chromatic scale, all twelve intervals, one per step, and the perfect fifth appears cleanly at k = 7. The even settings reproduce the 60-ring's whole tones. The odd settings fill the gap, and bring new exact distances onto the dial with them, √2 itself among them, at k = 15. A short general law, interval = 120k/N semitones, places both rings on one footing and shows that 120 is the smallest ring on which the dial completes the chromatic scale. Throughout, the golden ratio keeps its special role unchanged. The octave occurs only where the center distance equals R·φ and R/φ, a fact that holds on every five-octave ring, however finely it is divided. All results are verified by direct computation. Paper 1 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.