Music, Math and Mind

Columbia University
Fall, 2019 -- AV4000
Dave Sulzer and Brad Garton

This course encompasses the physics and neuroscience of music. Topics will include the mathematics by which musical scales, rhythms, and harmonies are derived, a topic that spans the history of math from the monochord of Pythagoras, wave functions, calculus required for equal temperament, through fractal geometry. The perception of music encompasses the physics, anatomy, and neuroscience of the ear and auditory neural pathways, and synaptic mechanisms that occur within the midbrain and cerebral cortex, and there will be attention to what little is known about the physiology of emotion. Additional topics will include physics of sound and physiology and associated behaviors associated with animal sound including songbirds, cetaceans, insects, and bat echolocation. Previous coursework in math and physiology is not required, and it is intended to be useful from diverse backgrounds in science or the arts, including undergrad and grad students.

Some questions the class will address

• Sound waves are in three space dimensions (and time). What shape are they, how big and fast? how are sound waves different under water, through the earth?
• why do instruments sound different from each other? why do larger instruments play lower pitches?
• we have two eardrums and two ears, how can we distinguish so many different simultaneous sounds?
• neurons fire maybe as fast as 200 times a second (Hz). We can hear frequencies up to 20,000 Hz: how is that possible?
• do sound and anti-sound waves cancel each other out? (trick question)
• what is the Doppler effect on ambulance sirens?
• how can we tell the direction a sound is coming from?
• what sounds seem in tune and out of tune and how can a musical scale be made? Is it true that scales are really never in tune?
• related issue: what the heck are all those Pythagorean ratios like 3/2 for a perfect 5th: what does that really mean?
• what are sum and difference tones?
• what is a sin? what’s a log? what is a Fourier transform?
• what is the math definition of noise and different kinds of noise?
• how does the brain understand what it is listening too? warning; this does not yet have a really satisfying answer
• how are emotions carried by music?: ditto
• what kind of disorders are there for hearing sound and music?
• how do other animals hear and make sound differently?
• how do microphones and speakers work, and how is sound encoded on records and digitally?
• what are overtones and harmonic sounds and how are they different?
• how might a musician use math to come up with unusual forms and structures?

We don't have a textbook for the course, but my favorite classic on the topic, which is downloadable, is Helmholtz's On the Sensations of Tone. For those very interested in the math of musical scales, check out Harry Partch's Genesis of a Music, also freely downloadable. For inspiration on cortical processing of music, consider Wilder Penfield studies on epilepsy operations, one nice book is Speech and Brain-Mechanisms, but his papers on PubMed are useful. I will mention primary papers during the course of the lecture.