From Pythagoras to Johann Sebastian Bach: An Exploration in the Development of Temperament and Tuning

Temperament and tuning are at the core of all music—they are the embodiment of the correlation of music and mathematics in every musical work. In examining the shift from Pythagorean tuning to well temperament, compositional style and philosophy governs not only the evolution musical genres, but also the evolution of temperament and tuning systems. The association of mathematics and temperament defines ratios and pitch relations in every branch of temperament. An analysis of J.S. Bach\u27s Well-Tempered Clavier and his use of temperament display the contrast of melodious thirds of meantone temperament and emotional tension of chords in well temperament. The mathematical beauty and complexity of tuning governs the musical center of the work; a composition may focus upon minute intervallic relationships or larger harmonic structure, depending upon the primary focus of the utilized system of temperament. Arguably, mathematical relationships have more influence upon musical beauty than the composed music alone. J.S. Bach\u27s preludes and fugues of the Well-Tempered Clavier prove the existence of liveliness and color in every possible key

A Celebration of Faculty Scholarship - 2019

A listing of the scholarly and creative work produced by the faculty of the College of the Holy Cross. It includes articles, book chapters, productions, exhibitions, grants, reviews and other forms of scholarship reported for the 2018-2019 academic year.https://crossworks.holycross.edu/fac_bib/1009/thumbnail.jp

A Three-Part Study in the Connections Between Music and Mathematics

The idea for this thesis originated from my fascination with the studies of both music and mathematics throughout my entire life. As a triple major in Middle/Secondary Math Education, Mathematics, and Music, I have learned more than I thought possible of music and math. In proposing this thesis, I desired to use my knowledge of arithmetic and aesthetics to research how music and mathematics are intertwined. I am confident that the following three chapters have allowed me to develop as an academic in both music and mathematics. This thesis serves as a presentation of the connections of music and math and their application to my academic interests and studies as I conclude my undergraduate journey at Butler University

Faculty Promotion and Tenure Recipients, 2015

Event was sponsored by Chancellor and Provos

Scene Analysis under Variable Illumination using Gradient Domain Methods

The goal of this research is to develop algorithms for reconstruction and manipulation of gradient fields for scene analysis, from intensity images captured under variable illumination. These methods utilize gradients or differential measurements of intensity and depth for analyzing a scene, such as estimating shape and intrinsic images, and edge suppression under variable illumination. The differential measurements lead to robust reconstruction from gradient fields in the presence of outliers and avoid hard thresholds and smoothness assumptions in manipulating image gradient fields. Reconstruction from gradient fields is important in several applications including shape extraction using Photometric Stereo and Shape from Shading, image editing and matting, retinex, mesh smoothing and phase unwrapping. In these applications, a non-integrable gradient field is available, which needs to be integrated to obtain the final image or surface. Previous approaches for enforcing integrability have focused on least square solutions which do not work well in the presence of outliers and do not locally confine errors during reconstruction. I present a generalized equation to represent a continuum of surface reconstructions of a given non-integrable gradient field. This equation is used to derive new types of feature preserving surface reconstructions in the presence of noise and outliers. The range of solutions is related to the degree of anisotropy of the weights applied to the gradients in the integration process. Traditionally, image gradient fields have been manipulated using hard thresholds for recovering reflectance/illumination maps or to remove illumination effects such as shadows. Smoothness of reflectance/illumination maps is often assumed in such scenarios. By analyzing the direction of intensity gradient vectors in images captured under different illumination conditions, I present a framework for edge suppression which avoids hard thresholds and smoothness assumptions. This framework can be used to manipulate image gradient fields to synthesize computationally useful and visually pleasing images, and is based on two approaches: (a) gradient projection and (b) affine transformation of gradient fields using cross-projection tensors. These approaches are demonstrated in the context of several applications such as removing shadows and glass reflections, and recovering reflectance/illumination maps and foreground layers under varying illumination