Mathematics and Music: In Search of Harmony (part 8)

4.4. The music according to Descartes

French mathematician and philosopher, born in the small town of La Haye, René Descartes (1596-1650) wanted to systematize all knowledge according to structures analogous to those underlying the axiomatic model of Euclidean geometry in order to gain certainty.

In December 1618, the French philosopher completed his first work entitled Compendium Musicae. In an attempt to explain the basis of musical harmony and dissonance in mathematical terms, this work presents a large number of mathematical diagrams and tables illustrating the proportional relationships involved in various musical ranges. In order to organize his sensitive experience, making it compatible with his acoustic-mathematical-musical knowledge, Descartes established in the Compendium Musicae a generalized theory for the senses, through preliminaries in axiomatic form.

Observing such axioms, Descartes reveals a humanist side and in a certain sense little Cartesian, in the most common sense of the word, suggesting a reframing of the set of ideas and relationships that come to mind when we think of the French philosopher and thus symbolize his. dynamics / structure of thought.

The presence of analogies, mathematics, and Pythagoreanism in Descartes's work is manifest in the formulation of preliminary axioms, as well as in enlightening arguments for harmonic processes and rules of composition in music.

Concerning the idea of ​​the Harmonic Series, Descartes argued that no frequency could be heard without its higher octave in any way. Affirming that the octave was the only single interval produced by a divisive whole-string compromise, Descartes explained that no frequency consonant with one note of that range could be dissonant with the other. For the French thinker, just as there were only three concordant numbers, there were also only three major consonances - the fifth, third major, and third minor, from which the fourth and two sixths derived.

In the language of the French thinker, the lowest note was more powerful than the highest note, for the length of the chord that generates the first contains all those pertinent to the smallest, while the opposite does not occur.

Descartes also established the prohibition of the appearance of the tritone in the harmonic musical scenario, because it corresponds to the ratio of large and prime numbers to each other, as well as to be distant, as far as human auditory sensitivity is concerned, from any of the simple relations concerning the consonances. .

4.5. The Science-Music in Rameau

According to French composer and theorist Jean Philippe Rameau (1683-1764), music is the science of sounds, so sound is the main subject of music. Dividing this art / science into harmony and melody, the French theorist subordinated the latter to the former, admitting that knowledge of harmony is sufficient for a complete understanding of the properties of music.

Like Zarlino and Descartes, Rameau obtained the consonant intervals by dividing the chord into six parts, stating that the consonances underlie consecutive numbers, and that the order of such numbers determined the order and perfection of the consonances.

The French theorist pays particular attention to the argument for perfection of the octave range. Rameau stated that the upper note of an octave range is a replica of the lower one and that in the flute the emergence of such an interval depended only on the strength of the blow. He introduced in his work the idea of ​​equivalence of octaves by claiming that any number multiplied geometrically by some power of 2 - represented the same sound. In this sense, the single octave, double, triple octave, etc., were basically the same intervals as the fifth, twelfth, and so on.

The equivalence underlying the eighth still manifests itself when the French thinker states that the fundamental sound generated the octave and fifth intervals, but not the fourth, resulting from the difference between the eighth and fifth. It established a process for obtaining the mathematical relationship underlying a given inverted interval from that corresponding to the original interval by multiplying or dividing the number below or above the interval in question by 2 respectively.

With this, he presents himself as the first to define chords and their inversions, establishing numerical relationships underlying distinct dissonances, and also observing how the consonances conceived by Descartes distinguished in chords.

Concluded the first book of the Treaty of Harmony explaining how to relate fractions associated with the division of vibrations with multiplication of lengths.