The Digit Counts the Octave: How the Check Digits of the Fibonacci Ring Count Octaves, and How the Ring's Shapes Are the Chords of Music

Take a circle with sixty marks on it and write on each mark the last digit of a Fibonacci number, 0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9 and so on round. Those digits are the check digits of Neal's Net. Now treat the circle as a piano. Each mark is one key, one semitone, so twelve marks is an octave and the sixty marks hold five octaves. This paper asks what the digit does when you go up an octave. Number the marks 0 to 59. From a mark with an odd number, go up one octave at a time, twelve marks, and the digit changes by 2 each time, so from mark 1 the digits read 1, 3, 5, 7, 9. From a mark with an even number, go up two octaves at a time, twenty-four marks, and the digit again changes by 2 each time, so from mark 2 the digits read 1, 3, 5, 7, 9. Five steps bring the digit back where it started. So the digit counts octaves. This holds at all sixty marks. The paper then takes the ring's other regular shapes one at a time. They are the chords of music that cut the octave into equal parts, and each has a digit rule of its own. The square is the diminished seventh chord, and its digit is multiplied by 7 at each corner, from every mark alike. The triangle is the augmented triad, where two corners carry the same digit and the third carries that digit plus 5, and which corner is which depends on the mark's remainder when divided by 3, not on odd or even. The hexagon is the whole-tone scale, and digits next to each other on it add up to 0 or 5. Under all of these sits the one rule the ring was built from, that every digit is the sum of the two digits before it. John Coltrane's tone circle of 1961 turns out to be this same ring. Every number in the paper is checked by the short program in the appendix. Paper 9 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.