Applications of Math to Music

Mark Brautigam

(This are just a few math-music interactions that I'm aware of. I'm sure there are many more.)

Basic applications

Pitch. Math is the foundation of the 12-tone scale we use in western music today. The scale has evolved over the centuries from pitches based on integer ratios to our current system that uses a geometric progression of real numbers. This is the application I'm personally most familiar with.

Timbre. Different instruments sound different because they generate different overtones or harmonics. The presence and proportion of different overtones accounts for the differences in sound between, for example, a stringed instrument such as a violin vs. a reed instrument such as a saxophone.

Rhythm and Timing. It's sort of obvious that whole notes, half notes, quarter notes, etc. are based on math. (Also, they are a great way to teach fractions to students who play music but don't yet understand fractions.) Also, metronome markings such as ♩=60 or ♩=120 indicate speed in fractions of a minute.

Harmony. Chords and intervals can be described in mathematical terms. The differences between, for example, classical orchestral harmony and modern jazz harmony, can be described in mathematical terms. This is closely related to the math for pitch and timbre, which also involve overtones/harmonics. Consonance and dissonance can be explained mathematically.

Acoustics. The very creation of musical sounds depends on acoustic properties of the musical instrument(s) and the environment. This is related to the timbre of the instruments, but also to the way the environment absorbs and reflects the vibrations, and the way the ear responds to vibrations.

More complex interactions

Musical form. Forms such as sonata (ABABDAB'), rondo (ABACA), variations (AA'A"A'''), fugue, modern song form (ABABMB), etc. can be described in mathematical terms. Patterns within music, such as motifs, can be described mathematically.

Computer music generation. In aleatory music, elements of the composition are left to chance. Composers such as John Cage and Charles Ives used these techniques. 12-tone techniques such as used by Arnold Schoenberg, originally required precise record keeping to make sure the tones are all used in equal amounts, none predominating. Now all that record keeping and random number generation can be done by computer using mathematical algorithms.

Music imitation. Computers can be used to analyze music and create new compositions based on patterns found in various known sources. I myself was involved in a project at UC Santa Cruz in the 1990s to analyze the music of Bach and write new music in his style. The recent rise of AI techniques is going to make this even more feasible in the future (with varying degrees of success).

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