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

Wythoff Wisdom

International audienceSix authors tell their stories from their encounters with the famous combinatorial game Wythoff Nim and its sequences, including a short survey on exactly covering systems

Finite Countermodel Based Verification for Program Transformation (A Case Study)

Both automatic program verification and program transformation are based on program analysis. In the past decade a number of approaches using various automatic general-purpose program transformation techniques (partial deduction, specialization, supercompilation) for verification of unreachability properties of computing systems were introduced and demonstrated. On the other hand, the semantics based unfold-fold program transformation methods pose themselves diverse kinds of reachability tasks and try to solve them, aiming at improving the semantics tree of the program being transformed. That means some general-purpose verification methods may be used for strengthening program transformation techniques. This paper considers the question how finite countermodels for safety verification method might be used in Turchin's supercompilation method. We extract a number of supercompilation sub-algorithms trying to solve reachability problems and demonstrate use of an external countermodel finder for solving some of the problems.Comment: In Proceedings VPT 2015, arXiv:1512.0221

Numeration Systems: a Link between Number Theory and Formal Language Theory

We survey facts mostly emerging from the seminal results of Alan Cobham obtained in the late sixties and early seventies. We do not attempt to be exhaustive but try instead to give some personal interpretations and some research directions. We discuss the notion of numeration systems, recognizable sets of integers and automatic sequences. We briefly sketch some results about transcendence related to the representation of real numbers. We conclude with some applications to combinatorial game theory and verification of infinite-state systems and present a list of open problems.Comment: 21 pages, 3 figures, invited talk DLT'201

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

Dagstuhl Reports : Volume 1, Issue 2, February 2011

Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn

Resources Related to Mathematics, Art and Nature

The purpose of my sabbatical work was to learn about resources related to mathematics and art, nature, music and architecture. These subjects appeal to a wide audience, yet their mathematical connections are often on the periphery of a mathematics student\u27s studies. In this way I planned to increase my knowledge of mathematics but at the same time broaden my understanding of cross-disciplinary subjects