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- W2594300838 abstract "R-moulds of numerical semigroups are defined as increasing sequences of real numbers whose discretizations may give numerical semigroups. The ideal sequence of musical harmonics is an R-mould and discretizing it is equivalent 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 two important properties of moulds are derived, those of being Pythagorean and fractal. It is demonstrated that, up to normalization, there is only one Pythagorean mould and one non-bisectional fractal mould. Furthermore, it is shown that the unique non-bisectional fractal mould is given by the golden ratio. The case of half-closed cylindrical pipes imposes to the sequence of musical harmonics one third property, the so-called even-filterability property. The paper proves that the maximum number of equal divisions of the octave such that the discretizations of the Pythagorean mould and the non-bisectional fractal mould coincide, and such that the discretization is half-closed-pipe admissible is 12. This result gives a mathematical justification for the choice of 12 as the maximum number of equal divisions of the octave in equal temperaments." @default.
- W2594300838 created "2017-03-16" @default.
- W2594300838 creator A5021954232 @default.
- W2594300838 date "2017-03-03" @default.
- W2594300838 modified "2023-09-27" @default.
- W2594300838 title "Metric molds, fractal molds on the golden ratio, half-closed-pipe admissible molds, and the mathematical emergence of the well tempered harmonic semigroup" @default.
- W2594300838 hasPublicationYear "2017" @default.
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