One Circle, Two Readings: The Geometry of Musical Scales

Four papers in this series build musical scales from a single divided circle. This paper draws them together and answers the questions they left open. The circle can be read in two ways. Read as equal steps of pitch, it produces the equal temperaments, the whole-tone scale, the chromatic scale and the microtonal scales. Read as equal steps of frequency, it produces just intonation, where intervals are whole-number ratios. The two readings are joined by a single transformation, the logarithm that relates frequency to pitch, and they turn out to be mirror images, each making the other's characteristic numbers irrational. The just ratios cannot be struck on the pitch circle, and the golden ratio cannot be struck on the frequency circle. The golden ratio φ runs through all of it, in three distinct roles, as the distance at which the pitch-dial sounds an octave, as an unreachable point on the frequency ring, and as the pitch toward which the Fibonacci ratios converge. Those same Fibonacci ratios, 3/2, 5/3, 8/5, are exactly the simple consonances, the intervals that land on the smallest just rings, and their convergence to φ is the known basis of Bohlen's 833-cent golden scale (anticipated by O'Connell and Temes). One law appears in this paper for the first time. The number of octaves a wheel carries, not its fineness, decides the constant its octave lands on, d = 2R·cos(180°/M) for M octaves, and five octaves gives the pentagon and φ, which is why the octave of this series is golden. The harmonic facts here are all established and are cited as such. What is new is the method that computes them. The cosine ruler is a single geometric mechanism, one circle, two readings, one sliding parameter, and it constructs the whole-tone, chromatic, microtonal, and just systems together and gives the golden ratio's several roles one picture, where prior constructions produce these systems only separately. Every interval it computes is verified correct, and the method is, to our knowledge, without precedent. All results are verified by direct computation. Paper 5 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.