to be approached in an analytic form and restrict to formal aspects of art. However, it does not pretend to deny the transcendental importance of the connotative aspect of every creative art.

Pythagoras (550 BC) explained music as the expression of that universal harmony which is also realized in arithmetic and astronomy. Plato recognized the importance of the mathematical element. He said that if to any art, the arithmetic, measure and weight are omitted, what remains is not much. He also expressed that through measure and proportion one reaches beauty and excellence. Aristotle mantained that the ones who claim that mathematics has nothing to say about beauty and goodness are wrong, and that some of the elements of beauty are the order and the symmetry, and that these are the properties to which mathematics pay attention. The point of view of the Greek philosophy was inclined to select the form and the proportion as typical elements of beauty.

The mathematician Luca Pacioli in his »De Divina Proporzione« of 1509 considers the golden section, which was used by his friend Michelangelo and which we will deal with it later. During the 17th century and the beginning of the 18th century the concepts of »ingenious« and »good taste« prevailed. To this last one is implicit an effort of attention, then an intuitive aesthetic judgement depending on the good taste, and finally the analysis. Leibniz could accept perceptions and aesthetic judgements as part of knowledge and defined music as counting without knowing that we are counting. This last statement coincides with Birkhoff’s concept in the sense that the density of certain ordered relations between the notes considered intuitively yield the measure of the aesthetic effect. De Crousaz writes that good taste makes us appreciate at the beginning by sensations that which reason would have approved.

Rameau observed that a fundamental sound and several partials constitute a musical note and that the notes that differ by an octave are similar according to their aesthetic effect and can be considered almost identical. These facts lead to the understanding of Western music. It was d’Alembert who gave a precise presentation of the work of Rameau (which is qualitative, while Birkhoff’s is quantitative). So, the degree of harmonicity is different from the degree of pleasure or aesthetic measure. For example, the unison and the octave are the most harmonious of the intervals but certainly not the most pleasurable according to Birkhoff.

In 1739, Euler developed a consonance theory based on the Pythagorean law. The smaller the numbers that express the frequency relation between two notes, the more consonant they would be. In this way, Euler established harmonic criteria of any interval or chord which coincide with the observed facts. It is interesting that Euler established a quantitative law for the measure of harmonicity. So Euler’s general concept about the nature of aesthetic pleasure coincides with that of Birkhoff, which in Helmholtz’ words, years later, established that the easiest form in which we perceive the order that characterize the observed objects, these would seem more simple and perfect, and easily and joyfully we will recognize them. An order that is difficult to discover, even if it certainly pleases us, will carry or associate a certain degree of tiredness and sadness. Birkhoff points out that his theory is a simple enumeration that lacks of any mathematics and that it is a mere essay. In his work, he develops the psychological basis of his formula, applies it to