Related papers: On the mathematical structure of Tonal Harmony
Kepler's thinking is highly original and the inspiration for discovering his famous third law is based on his rather curious geometric approach in his Harmonices mundi for explaining consonances. In this article we try to use a modern…
After briefly revising the concepts of consonance/dissonance, a respective mathematic-computational model is described, based on Helmholtz's consonance theory and also considering the partials intensity. It is then applied to characterize…
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…
We introduce a new notion of Almgren's frequency which is adapted to solutions of a sub-Laplacian (harmonic functions) on a Carnot group of arbitrary step $\bG$. With this notion we investigate some new functionals associated with the…
Recently, Engel et al. discussed phonon broadening as observed in 3D quasicrystals on the basis of calculations on the Fibonacci chain. We show that the paper contains several statements and assumptions that are contradicted by factual…
Fourier Series is the second of monographs we present on harmonic analysis. Harmonic analysis is one of the most fascinating areas of research in mathematics. Its centrality in the development of many areas of mathematics such as partial…
We address certain structural innovations in the music of Claudio Monteverdi, which defined the pivotal transition from the Renaissance prima pratica to the Baroque seconda pratica. To formalize this analysis, we employ Mazzola's…
The origins of consonance in human music has long been contested, and today there are three primary hypotheses: aversion to roughness, preference for harmonicity, and learned preferences from cultural exposure. While the evidence is…
In this sequel to arXiv:0905.3327, we continue to study the congruence properties of the alternating version of multiple harmonic sums. As contrast to the study of multiple harmonic sums where Bernoulli numbers and Bernoulli polynomials…
The purpose of this article is to survey certain aspects of multilinear harmonic analysis related to notions of transversality. Particular emphasis will be placed on the multilinear restriction theory for the euclidean Fourier transform,…
In this article, we analyze the structure and relationships between magnitude homology and Eulerian magnitude homology of finite graphs. Building on the work of Kaneta and Yoshinaga, Sazdanovic and Summers, and Asao and Izumihara, we…
The purpose of this paper is to show through particular examples how group theory is used in music. The examples are chosen from the theoretical work and from the compositions of Olivier Messiaen (1908-1992), one of the most influential…
A phenomenologically successful two-component hadronic duality picture led to Veneziano's amplitude, the fundamental first step to string theory. This picture is briefly recalled and its two components are identified as the open strings…
In this paper, the theory of harmonic maps is extended. The soliton or traveling wave solutions of Euler's equations of the extended harmonic maps are studied. In certain cases, the chaotic behaviors of these partial equations can be found…
Harmonic theory provides a mathematical framework to describe the structure, behavior, evolution and emergence of harmonic systems. A harmonic system is context aware, contains elements that manifest characteristics either collaboratively…
We go further on the study of harmonicity for almost contact metric structures already initiated by Vergara-Diaz and Wood. By using the intrinsic torsion, we characterise harmonic almost contact metric structures in several equivalent ways…
We study the harmonic polytope, which arose in Ardila, Denham, and Huh's work on the Lagrangian geometry of matroids. We describe its combinatorial structure, showing that it is a $(2n-2)$-dimensional polytope with…
A musical score provides the essential instructions for its performance while containing indications - at times implicit - regarding the composer's intentions. The presence of authorial variants, and even more so complex series of revisions…
The divergent part of the generating functional of the Resonance Chiral Theory is evaluated up to one loop when one multiplet of scalar an pseudoscalar resonances are included and interaction terms which couple up to two resonances are…
In the framework of Galilei classical mechanics (i.e., general relativistic classical mechanics on a spacetime with absolute time) developed by Jadczyk and Modugno, we analyse systematically the relations between symmetries of the geometric…