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  • From actantial model to conceptual graph: Thematized action in John Cage's 0′00′(4′33′′No. 2)
    J. Math. Music (IF 0.381) Pub Date : 2020-05-29
    Michael D. Fowler

    In this article, I build an actantial model, M, of John Cage's 1962 indeterminate work 0′00′′ (4 ′33 ′′ No. 2). To further investigate Greimas' actantial axes of desire, knowledge and power, I generate an ontology, O, that records the facts of the model through hierarchies of relations and concepts. This allows for a conceptual graph (CG), C, that describes the score's instructions and its actants

  • Musical pitch quantization as an eigenvalue problem
    J. Math. Music (IF 0.381) Pub Date : 2020-05-24
    Peter beim Graben; Maria Mannone

    How can discrete pitches and chords emerge from the continuum of sound? Using a quantum cognition model of tonal music, we prove that the associated Schrödinger equation in Fourier space is invariant under continuous pitch transpositions. However, this symmetry is broken in the case of transpositions of chords, entailing a discrete cyclic group as transposition symmetry. Our research relates quantum

  • Mathematical approaches to defining the semitone in antiquity
    J. Math. Music (IF 0.381) Pub Date : 2020-04-30
    Caleb Mutch

    Connections between mathematics and music have been recognized since the days of Ancient Greece. The Pythagoreans' association of musical intervals with integer ratios is so well known that it occludes the great variety of approaches to the music-mathematical relationship in Ancient Greece and Rome. The present article uncovers this diversity by examining how authors from Antiquity used one mathematical

  • Symbolic structures in music theory and composition, binary keyboards, and the Thue–Morse shift
    J. Math. Music (IF 0.381) Pub Date : 2020-03-17
    Ricardo Gómez; Luis Nasser

    We address the broad idea of using mathematical models to inform music theory and composition by implementing them directly in the process of music creation. The mathematical model we will use is the Thue–Morse dynamical system. We briefly survey previously published works that were similarly motivated, and then discuss a new piece of music we composed, inspired by this symbolic dynamical system. In

  • Quantum GestART: identifying and applying correlations between mathematics, art, and perceptual organization
    J. Math. Music (IF 0.381) Pub Date : 2020-03-11
    Maria Mannone; Federico Favali; Balandino Di Donato; Luca Turchet

    Mathematics can help analyze the arts and inspire new artwork. Mathematics can also help make transformations from one artistic medium to another, considering exceptions and choices, as well as artists' individual and unique contributions. We propose a method based on diagrammatic thinking and quantum formalism. We exploit decompositions of complex forms into a set of simple shapes, discretization

  • An alternative approach to generalized Pythagorean scales. Generation and properties derived in the frequency domain
    J. Math. Music (IF 0.381) Pub Date : 2020-03-05
    Rafael Cubarsi

    Abstract scales are formalized as a cyclic group of classes of projection functions related to iterations of the scale generator. Their representatives in the frequency domain are used to built cyclic sequences of tone iterates satisfying the closure condition. The refinement of cyclic sequences with regard to the best closure provides a constructive algorithm that allows to determine cyclic scales

  • Generalized Tonnetze and Zeitnetze, and the topology of music concepts
    J. Math. Music (IF 0.381) Pub Date : 2020-03-02
    Jason Yust

    The music-theoretic idea of a Tonnetz can be generalized at different levels: as a network of chords relating by maximal intersection, a simplicial complex in which vertices represent notes and simplices represent chords, and as a triangulation of a manifold or other geometrical space. The geometrical construct is of particular interest, in that allows us to represent inherently topological aspects

  • Some remarks on hypergestural homology of spaces and its relation to classical homology
    J. Math. Music (IF 0.381) Pub Date : 2020-02-27
    Juan Sebastián Arias-Valero; Emilio Lluis-Puebla

    Classical homology of a topological space provides invariants of the space by means of triangulation or squaring made up from singular simplices (simplicial homology) or singular cubes (cubical homology) in the space. In much the same way, Mazzola's hypergestural homology intends to associate invariants to topological categories and, in particular, topological spaces by means of approximation with

  • Gauge models of musical forces
    J. Math. Music (IF 0.381) Pub Date : 2020-02-17
    Reinhard Blutner; Peter beim Graben

    Metaphors involving motion and forces are a source of inspiration for understanding tonal music and tonal harmonies since ancient times. Starting with the rise of quantum cognition, the modern interactional conception of forces as developed in gauge theory has recently entered the field of theoretical musicology. We develop a gauge model of tonal attraction based on SU(2) symmetry. This model comprises

  • On the topological characterization of gestures in a convenient category of spaces
    J. Math. Music (IF 0.381) Pub Date : 2020-02-13
    Timothy L. Clark

    In this paper, we reconsider the topological characterization of gestures in a convenient category of spaces mentioned by Mazzola in 2009, recovering Arias's 2018 result that the relevant equivalence is a homeomorphism. We also show the topological characterization of gestures extends to an adjunction between the category of gestures and the category of continuous maps whose domain is a one-dimensional

  • Difference tones in “ non-Pythagorean” scales based on logarithms
    J. Math. Music (IF 0.381) Pub Date : 2020-01-21
    Thomas Morrill

    In order to explore tonality outside of the “Pythagorean” paradigm of integer ratios, Robert Schneider introduced a musical scale based on the logarithm function. We seek to refine Schneider's scale so that the difference tones generated by different degrees of the scale are themselves octave equivalents of notes in the scale. In doing so, we prove that a scale which contains all its difference tones

  • Axiomatic scale theory
    J. Math. Music (IF 0.381) Pub Date : 2020-01-10
    Daniel Harasim; Stefan E. Schmidt; Martin Rohrmeier

    Scales are a fundamental concept of musical practice around the world. They commonly exhibit symmetry properties that are formally studied using cyclic groups in the field of mathematical scale theory. This paper proposes an axiomatic framework for mathematical scale theory, embeds previous research, and presents the theory of maximally even scales and well-formed scales in a uniform and compact manner

  • Functorial semiotics for creativity
    J. Math. Music (IF 0.381) Pub Date : 2019-11-27
    Guerino Mazzola

    In this paper, we develop a mathematically conceived semiotic theory. This project seems essential for a future computational creativity science since the outcome of the process of creativity must add new signs to given semiotic contexts. The mathematical framework is built upon categories of functors, in particular linearized categories deduced from path categories of digraphs and the Gabriel–Zisman

  • Modulation in tetradic harmony and its role in jazz
    J. Math. Music (IF 0.381) Pub Date : 2019-09-05
    Octavio A. Agustín-Aquino; Guerino Mazzola

    After a quick exposition of Mazzola's quantum modulation model for the so-called triadic interpretation of the major scale within the equal temperament, we study the model for the tetradic interpretation of the same scale. It is known that tetrads are fundamental for jazz music, and some classical objects for this kind of music are recovered.

  • An analysis of pitch-class segmentation in John Cage's Ryoanji for oboe using morphological image analysis and formal concept analysis
    J. Math. Music (IF 0.381) Pub Date : 2019-09-05
    Michael D. Fowler

    In 1983, John Cage used the traditional stone garden, or karesansui at the Zen temple, Ryōan-ji in Kyoto as a model to generate a series of visual and musical works that utilized tracings of a collection of his own rocks. In this article, I analyze the first of the musical works, Ryoanji for oboe, using mixed methods drawn from morphological image analysis and formal concept analysis (FCA). I introduce

  • On features of fugue subjects. A comparison of J.S. Bach and later composers
    J. Math. Music (IF 0.381) Pub Date : 2019-05-17
    Jesper Rydén

    The musical form fugue has inspired many composers, in particular writing for the organ. By quantifying a fugue subject, comparisons can be made on a statistical basis between J.S. Bach and composers from later epochs, a priori dividing works into three categories. The quantification is made by studying the following features: length, expressed in number of notes written; range (in semitones); number

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