Tempered Monoids of Real Numbers, the Golden Fractal Monoid, and the Well-Tempered Harmonic Semigroup

Abstract
We will analyze 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 (R, +) whose pairs of consecutive terms get arbitrarily close as they grow. These properties suggest to define 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 will see that, up to normalization, there is only one product-compatible tempered monoid, which is the logarithmic monoid, and there is only one non-bisectional fractal monoid which is generated by the golden ratio. The example of half-closed cylindrical pipes imposes to the sequence of musical harmonics one third property, the so-called odd-filterability property. We will see that the maximum number of equal divisions of the octave such that the discretization 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.