The Math Behind the Music

The first chapter is about the connections, history, common abstract patterns, and the composers and compositions that used math. 

The second chapter is about the physical basis of harmony, pitch and timbre that make up music. Considerable attention is paid to the frequency relations of various harmonies, and it's a good refresher for those who don’t remember off the top that a fourth comes from any note by raising its frequency by 4/3, a fifth by raising its frequency by 1/2 and an octave by doubling. Sine curves are introduced to encapsulate the frequency patterns of various notes produced at different pitches by different instruments. Overtones are explained simply as the ratios of higher frequencies that a note produces that don’t block out the original frequencies and the relation between harmonies and overtones is shown. 

The third chapter discusses instrument tuning systems consistent with all the overtones and frequency relations between the notes of a scale. 

The fourth chapter is the most interesting in that it shows how themes and melodies can be varied with simple rules such as opposition, inversion, and transposition. The relation between these simple rules and group theory are examined, and various ways of notating and combining the rules are covered. 

The fifth chapter is about bell music, which is merely a variation of permutation and combination theory. 

The sixth chapter is about randomization in music, with many of the same methods used to construct music as we use for simple simulations in markets. 

The seventh chapter is about an attempt by one student to find the common basis, the patterns of harmony that make up the most popular songs. 

The eighth chapter is about how scores of music can be developed from visual cues, with rules to go from visual to music. The ninth and final chapter is about failed efforts to combine music and math, with particular reference to George Birkhoff's efforts to develop a complete theory of aesthetics by developing a scale of beauty based on the simplicity-to-complexity ratio of a composition. 

I found myself thinking many times of the relations between music and markets as I read the book. The combinations of opposites and inversions (where the intervals above a note and played the same intervals below, and transpositions (where the same theme is repeated a given number of intervals up) happens every day in the markets. The notation that musicians have developed to grapple with these techniques, including the summary of horizontal and vertical movements in visual sightings that the composer Villa-Lobos used to construct symphonies that depict buildings in a city, seems like a very fruitful field to augment technical analysis of markets. The book is full of anecdotes and charts and methods that will be right on the top of the page for market practitioners, and will spark many a fruitful extension by those who wish to take the pencil to paper, and systematize what they have been doing in markets or charting with the work of some great composers and mathematicians in this related field.