Aesthetics, Dynamics, and Musical Scales: A Golden Connection (original) (raw)
Related papers
Perfect balance: A novel principle for the construction of musical scales and meters
In T. Collins, D. Meredith, and A. Volk (Eds.), Mathematics and Computation in Music—MCM 2015, volume 9110 of LNAI, pages 97–108, Heidelberg. Springer.
We identify a class of periodic patterns in musical scales or meters that are perfectly balanced. Such patterns have elements that are distributed around the periodic circle such that their 'centre of gravity' is precisely at the circle's centre. Perfect balance is implied by the well established concept of perfect evenness (e.g., equal step scales or isochronous meters). However, we identify a less trivial class of perfectly balanced patterns that have no repetitions within the period. Such patterns can be distinctly uneven. We explore some heuristics for generating and parameterizing these patterns. We also introduce a theorem that any perfectly balanced pattern in a discrete universe can be expressed as a combination of regular polygons. We hope this framework may be useful for understanding our perception and production of aesthetically interesting and novel (microtonal) scales and meters, and help to dis-ambiguate between balance and evenness; two properties that are easily confused. Perfect balance: A novel principle for the construction of musical scales and meters. Available from: https://www.researchgate.net/publication/278683682\_Perfect\_balance\_A\_novel\_principle\_for\_the\_construction\_of\_musical\_scales\_and\_meters [accessed Jun 19, 2015].
Linking sonic aesthetics with mathematical theories
The Oxford Handbook of Algorithmic Music, 2018
Pure mathematics provides principles, procedures, and ways of thinking that can be fruitful starting points for music composition, performance, and algorithmic generation. In this chapter, a number of mathematical methods are suggested as useful ways to define and transform underlying musical structures such as meters and scales, and to realize these structures as finished pieces of music. The mathematical methods include the discrete Fourier transform, geometry, algebraic word theory, and tiling, and how these relate to musical features such as periodicity (or lack of periodicity), well-formedness, microtonality, canons, rhythmic hierarchies, and polyrhythms. The chapter closes with a detailed examination of a musical piece derived from the described processes.
The Sounds of Music : Science of Musical Scales I – Human Perception of Sound
2019
Both, human appreciation of music and musical genres, transcend time and space. The universality of musical genres and associated musical scales is intimately linked to the physics of sound and the special characteristics of human acoustic sensitivity. In this series of articles, we examine the science underlying the development of the heptatonic scale, one of the most prevalent scales of the modern musical genres, both western and Indian.
The Sounds of Music : Science of Musical Scales
Resonance, 2019
Sushan Konar works on stellar compact objects. She also writes popular science articles and maintains a weekly astrophysics-related blog called 'Monday Musings'. Both, human appreciation of music and musical genres, transcend time and space. The universality of musical genres and associated musical scales is intimately linked to the physics of sound and the special characteristics of human acoustic sensitivity. In this series of articles, we examine the science underlying the development of the heptatonic scale, one of the most prevalent scales of the modern musical genres, both western and Indian.
In search of universal properties of musical scales
Musical scales have both general and culture-specific properties. While most common scales use octave equivalence and discrete pitch relationships, there seem to be no other universal properties. This paper presents an additional property across the world’s musical scales that may qualify for universality. When the intervals of 998 (just intonation) scales from the Scala Archive are represented on an Euler lattice, 96.7% of them form star-convex structures. For the subset of traditional scales this percentage is even 100%. We present an attempted explanation for the star-convexity feature, suggesting that the mathematical search for universal musical properties has not yet reached its limits.
2020
González-Tisserand, senior researcher at the Institute for Microelectronics and Microsystems (IMM) of the National Research Council (CNR), Bologna Unit, in Italy, for their continuous support of my doctoral study and related research, for their patience, motivation, and immense knowledge. The altruistic collaboration of Ramón Arnau-Gómez, CEO of Arteco Consulting (Majorca, Spain), in the development of the plugin has been priceless. I also want to especially thank the help given by the guitarist and friendly colleague Neil Haverstick, who provided me with material from his forthcoming book, together with a great deal of information, and Dr. Julyan Cartwright (Spanish Research Council, CSIC) for his insightful comments and the encouragement to expand my research from the non-linear perspective. I am particularly grateful for the assistance given by Prof. Dr. Ozan Yarman from Istanbul University, for his insightful comments, and provision of material related to maqãm; and by my medical colleague, and Hindi musician expert, Dr. Vidyadhar Oke for his thoughts and criticism related to the shrutis, one of his areas of expertise. Two musicians also deserve individual recognition: Majorca based Cuban Pianist Ivon Frontela Rico, dear friend and outstanding critic of my work, and John Dodd for his friendly peer text review, comments, and suggestions. Besides them, and last but not least, many thanks to the those many other scholars who made contributions, either providing references, suggestions, comments, or images, to this dissertation. My gratitude is now for all of them and it will be particularly expressed in the corresponding chapters where they have contributed. Also, my gratitude to the many authors which have lent their artistic images in a copyright-free CC0 format.
Quantifying Harmony: The Mathematical Essence of Music
This research delves into the intricate correlation between music and mathematics, examining how mathematical principles play a fundamental role in shaping various aspects of musical theory and composition. Starting from Pythagoras' early insights on harmonic intervals to the complex patterns found in the Fibonacci sequence and fractal geometry, this study uncovers the deep connections between numerical concepts and musical notes. By extensively exploring historical perspectives, theoretical frameworks, and practical applications, we gain valuable insight into how mathematical ideas influence the melodies, harmonies, and rhythms that elevate our auditory experiences.
Nonlinear Dynamics, the Missing Fundamental, and Harmony
Communications in Computer and Information Science, 2009
We review the historical and current theories of musical pitch perception, and their relationship to the intriguing phenomenon of residue pitch. We discuss the nonlinear dynamics of forced oscillators, and the role played by the Fibonacci numbers and the golden mean in the organization of frequency locking in oscillators. We show how a model of the perception of musical pitch may be constructed from the dynamics of oscillators with three interacting frequencies. We then present a mathematical construction, based on the golden mean, that generates meaningful musical scales of different numbers of notes. We demonstrate that these numbers coincide with the number of notes that an equal-tempered scale must have in order to optimize its approximation to the currently used harmonic musical intervals. Scales with particular harmonic properties and with more notes than the twelve-note scale now used in Western music can be generated. These scales may be rooted in objective phenomena taking place in the nonlinearities of our perceptual and nervous systems. We conclude with a discussion of how residue pitch perception may be the basis of musical harmony.