What is the Difference between a Banjo and a Harley-Davidson Motorcycle? Answer: You Can Tune the Harley
Year: 1998 Authors: Daniel Fitzgerald
Core claim
Fretted tuning and diminished chords can be understood mathematically, using rational versus irrational tuning and a cyclic-group model of the twelve-tone scale.
Topics
music tuning, chord theory, cultural symbolism, twelve-tone scale
Domains
number theory, group theory, cyclic groups, music, visual symbolism, cross-cultural imagery
Methods
tuning comparison, group-theoretic model, symbolic interpretation
Media
fretted instrument, banjo, cover illustration
Paper text
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BRIDGES Mathematical Connections in Art, Music, and Science
What is the Difference between a Banjo and a Harley-Davidson Motorcycle? Answer: You Can Tune the Harley
Daniel Fitzgerald Mathematics Department Newman University Wichita, KS 67213 e-mail: fitzgeraldd@ksnewman.edu
It is a mathematical fact that a fretted instrument cannot be played in tune. Fretted instruments are designed according to the well-temperament theory of tuning. Differences between natural tunings and well-tempered tuning exist because one is based on rational numbers and the other on irrational numbers.
Another topic which requires mathematics to explain is that of chords. A peculiar property of diminished cords can be explained by elementary group theory. If a cyclic group is used to model a twelve-tone scale, it turns out that a diminished chord is a subgroup of order four and from that certain properties follow.
BRIDGES Mathematical Connections in Art, Music, and Science
The above picture is found on the cover of Sobre la muerte de la ciudad del valle germinará mi canto, written by Jorge Carrera Bolaños. The picture is a mixture of two cultures. First, you have the image of the dragon eating itself, that is a symbol of the infinite both in Old China and in Old Ireland. Instead of a dragon, a feathered snake is used. This is a symbol of Quetzalcoatl, one of the most important gods in all cultures of Mexico (before Columbus).