Semigroup Forum

, Volume 99, Issue 2, pp 496–516 | Cite as

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

  • Maria Bras-AmorósEmail author
Research Article


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 submonoid of \(({{\mathbb {R}}}^+,+)\) 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 \({{\mathbb {R}}}\) to \({{\mathbb {N}}}\) 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.


Musical harmonics Equal temperament Monoids Increasing enumeration Numerical semigroup Tempered monoid Logarithm Fractal Golden ratio 



The author would like to thank Julio Fernández, Pilar Bayer, Shalom Eliahou, Alfons Reverté, Pere Casulleras, Marly Cormar, and Felix Gotti for many stimulating and clarifying discussions. She would also like to thank the anonymous referees for their encouraging and helpful contributions. Finally she thanks Clifton Callender, Thomas Fiore, and Emmanuel Amiot. This work was supported by the Catalan Government under grant 2017 SGR 00705 and by the Spanish Ministry of Economy and Competitivity under grant TIN2016-80250-R.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Universitat Rovira i VirgiliTarragonaCatalonia

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