Articles producció científica> Enginyeria Informàtica i Matemàtiques

Tempered monoids of real numbers, the golden fractal monoid, and the well-tempered harmonic semigroup

  • Dades identificatives

    Identificador: imarina:6003787
  • Autors:

    Bras-Amoros, Maria
  • Altres:

    Autor segons l'article: Bras-Amoros, Maria;
    Departament: Enginyeria Informàtica i Matemàtiques
    Autor/s de la URV: Bras Amoros, Maria
    Paraules clau: Tempered monoid Numerical semigroup Musical harmonics Music Monoids Logarithm Increasing enumeration Golden ratio Fractal Equal temperament
    Resum: This paper deals with the algebraic structure of the sequence of harmonics when combined with equal temperaments. Fractals and the golden ratio appear surprisingly on the way. The sequence of physical harmonics is an increasingly enumerable sub-monoid of (R+,+) whose pairs of consecutive terms get arbitrarily close as they grow. These properties suggest the definition of a new mathematical object which we denote a tempered monoid. Mapping the elements of the tempered monoid of physical harmonics from R to N may be considered tantamount to defining equal temperaments. The number of equal parts of the octave in an equal temperament corresponds to the multiplicity of the related numerical semigroup. Analyzing the sequence of musical harmonics we derive two important properties that tempered monoids may have: that of being product-compatible and that of being fractal. We demonstrate that, up to normalization, there is only one product-compatible tempered monoid, which is the logarithmic monoid, and there is only one nonbisectional fractal monoid which is generated by the golden ratio. The example of half-closed cylindrical pipes imposes a third property to the sequence of musical harmonics, the so-called odd-filterability property. We prove that the maximum number of equal divisions of the octave such that the discretizations of the golden fractal monoid and the logarithmic monoid coincide, and such that the discretization is odd-filterable is 12. This is nothing else but the number of equal divisions of the octave in classical Western music.
    Àrees temàtiques: Mathematics Matemática / probabilidade e estatística Engenharias iv Algebra and number theory
    Accès a la llicència d'ús: https://creativecommons.org/licenses/by/3.0/es/
    ISSN: 00371912
    Adreça de correu electrònic de l'autor: maria.bras@urv.cat
    Identificador de l'autor: 0000-0002-3481-004X
    Data d'alta del registre: 2023-02-18
    Versió de l'article dipositat: info:eu-repo/semantics/acceptedVersion
    Enllaç font original: https://link.springer.com/article/10.1007/s00233-019-10059-4
    Referència a l'article segons font original: Semigroup Forum. 99 (2): 496-516
    Referència de l'ítem segons les normes APA: Bras-Amoros, Maria; (2019). Tempered monoids of real numbers, the golden fractal monoid, and the well-tempered harmonic semigroup. Semigroup Forum, 99(2), 496-516. DOI: 10.1007/s00233-019-10059-4
    URL Document de llicència: https://repositori.urv.cat/ca/proteccio-de-dades/
    DOI de l'article: 10.1007/s00233-019-10059-4
    Entitat: Universitat Rovira i Virgili
    Any de publicació de la revista: 2019
    Tipus de publicació: Journal Publications
  • Paraules clau:

    Algebra and Number Theory,Mathematics
    Tempered monoid
    Numerical semigroup
    Musical harmonics
    Music
    Monoids
    Logarithm
    Increasing enumeration
    Golden ratio
    Fractal
    Equal temperament
    Mathematics
    Matemática / probabilidade e estatística
    Engenharias iv
    Algebra and number theory
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