arXiv subjects

James S. Walker

Publications and source records attributed to James S. Walker.

3 recordsLinked to original sources

arXiv2020

A Geometric Analysis Of The Harmonic Structure of "In My Life"

After our book---Mathematics and Music: Composition, Perception, and Performance, 2nd edition, CRC Press, 2020---was published, we found a striking example of the importance of the Tonnetz for analyzing the harmonic structure of The Beatles' song, "In My Life." Our Tonnetz analysis will illustrate the highly structured geometric logic underlying the numerous chord progressions in the song. Spectrograms provide a way for us to visualize chordal harmonics and their connection with voice leading. We shall also describe the interesting harmonic rhythms of the song's chord progressions. A lot of this harmonic rhythm lends itself well to a geometric description.

math.HO↗

arXiv2020

Symmetry and Art

We use some fundamental ideas from complex analysis to create symmetric images and animations. Using a domain coloring algorithm, we generate mappings to the entire complex plane or the hyperbolic upper half-plane. The resulting designs can have rotational, translational, or mirror symmetry according to our chosen mapping functions. An appealing feature of these designs is how they reveal important properties of Euclidean and non-Euclidean geometries. We can also generate animations of our designs. Our goal is to create designs and animations having significant artistic content.

math.MG↗

arXiv2018

Generating Bézout trees for Pythagorean pairs

Relatively prime pairs of integers can be represented as nodes in three way branching trees. We construct trees of Bézout coefficients which correspond to the relatively prime pairs in the aforementioned trees. As one application, we compare the Bézout coefficients in these trees to those returned by the gcd function in Matlab. As another application, we use these trees to decrease the computation time required to create computer generated hyperbolic wallpaper designs.

math.NT↗