Author: John Benton
Title: "An Archaeological Dig into the Mathematical Foundations of Western Music"

This paper began as an attempt to gain an understanding of why the black keys on the piano are where they are. I started with a matrix of powers of four-thirds and three-halves and converted it into a column of ascending frequencies. I identified first the seven white keys, followed by the five black keys and then the black keys separated into flats and sharps. I converted the matrix into a graph which effectively rotated the matrix forty-five degrees. All twelve of the musical scales could then be read from left to right on adjacent (but not connected) nodes of the graph. I then converted the graph into a table-based checkerboard pattern and expanded it to show the Pythagorean Commas. I developed an equation to compute the ratio of frequencies between any two notes in the original matrix. Having started using powers of four-thirds as well as three-halves, and from my perspective as an electronics engineer and computer scientist, I developed proceedure for exploring triads, Ptolemaic, Just and Equal Temperament Tuning, including a root-sum-square based analysis of the different tuning systems. This paper follows my explorations and of how I felt I had caught glimpses of the original development of the mathematical foundation of music. Thus the title of the paper. I have the hope that the reader will share some of my sense of excitement and awe at the beauty of how the simple beginnings of Pythagorean tuning lead to magnificent structures which still form the framework of Western music, both classical and popular.