The Trigonometry of Music: Every Interval Is an Angle, and the Two Sliding Rings Read Its Cosine, Sine and Tangent at One Setting

The papers of this series build the scales of music on a circle with two sliding rings, and each of them uses the cosine of a setting as the distance between the two ring centers. This paper says what that use is. It is trigonometry, done on music, with the instrument of Two Sliding Rings. An interval is an angle, three degrees a semitone on every ring that carries five octaves, so the octave is 36 degrees, the fifth 21 degrees and the tritone 18 degrees. Set the two rings on that angle and the distance between the centers, halved, is the cosine, the distance between the two crossings, halved, is the sine, and the one divided by the other is the tangent, while the digits under the crossings check that the setting is right. The exact entries of the classical cosine table are the landmark intervals of the earlier papers, the fourth at 15 degrees, the tritone at 18, the minor seventh at 30 and the octave at 36. The sine of an interval is the cosine of the interval that makes two and a half octaves with it. Read in angles, the mirror of the companion paper One Circle, Two Readings becomes one line. A rational ratio is an irrational angle, and a rational angle is an irrational ratio, so the just fifth sits 0.0587 degrees past the 21 degree mark and on no mark of any ring. Every number in the paper is produced by the program in the appendix. Paper 10 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.