The Just Ring: Reading the Circle in Frequency Instead of Pitch, and Where the Just Intervals and the Shrutis Land

The dials of the preceding papers in this series divide the circle into equal steps of pitch, and so produce equal temperaments, the whole-tone scale on a 60-point ring, the chromatic scale on 120, quarter-tones on 240. Equal-pitch rings can never reach just intonation, because a just interval is a small whole-number frequency ratio, and the logarithm that turns ratio into pitch makes every such ratio (except the octave) an irrational number of steps. This paper changes the ring. If the circle's divisions are read as equal steps of frequency rather than of pitch, a linear-frequency ring, the geometry of a vibrating string, then a just ratio p/q lands on an exact division whenever the ring carries a multiple of q divisions across the octave. The just intervals reappear as exact points. The number of divisions a scale needs is the least common multiple of its ratios' denominators. The Pythagorean core and the seven-note just major scale need only 24 divisions, the full twelve-note 5-limit chromatic scale needs 480, and the twenty-two shrutis of Indian classical music, traced to their classical just ratios, need a ring whose size measures exactly how far that system reaches beyond simple ratios. The just intervals and the small-integer-ratio basis of just intonation are ancient, and the least-common-multiple counting is elementary, and these are cited as the established facts they are. What this paper introduces is the method that places them, the divided circle of the cosine ruler (Strassner 2026) read in frequency instead of pitch, the same circle whose pitch reading produced the equal temperaments of the earlier papers, so that just intonation and equal temperament become two readings of one construction. All results are verified by direct computation. Paper 3 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.