Mathematics and Music: In Search of Harmony (part 5)

4. Contributions to Science / Music from the Renaissance

From the musical point of view, the Renaissance is characterized by the evolution of polyphony - superposition of melodies - and the consequent development of harmony. Performing mathematical speculations concerning this area in his Theoretical Music, Ludovico Fogliani (1470-1539) provided strong insights for Gioseffe Zarlino (1517-1590) - one of the greatest musical theorists of the time - to organize in his work Inztituzioni Armonique (1558) a basis of scientific-cultural education throughout Europe for two centuries.

Continuing the aforementioned theorist is the Spanish Francisco Salinas (1513-1590), as well as the French priest and mathematician Marin Mersenne (1588-1648) who, still devoting himself to acoustics, presents himself as the first theorist to ground the study of harmony in the phenomenon of resonance. Maintaining frequent correspondence with René Descartes (1596-1650), Mersenne discussed problems and unclear aspects of the French philosopher Compedium Musicae in 1618. Descartes established in the compendium these aesthetic concepts of marked influence on the Treatise of Harmony, written by Jean Philippe Rameau ( 1683-1764) one hundred years later.

Substantially modifying the Pythagorean conception, Galileo Galilei wrote in 1638 that neither length nor tension nor linear density of strings was the direct and immediate reason underlying musical intervals, but ratios of the number of vibrations and impacts of sound waves. that hit the eardrum. Considering the sound that reached the ear rather than the vibrating object that produced it, Galileo found that musical pitch was directly related to frequency by recording scratches drawn on a metal plate from a vibrating rod in solidarity with a membrane that received sound vibrations.

Galileo's perception in the seventeenth century that the sense of musical height is directly related to the concept of frequency marks the beginning of the physics of music in its present conception. This idea prompted efforts to understand musical harmonics, since during this century it seemed paradoxical - at first from Mersenne - that a single object could vibrate simultaneously at different frequencies. The foundations of this idea by means of demonstrative mathematical formulas later materialized through Newton, Laplace, and Euler. The passage from the concept of vibration to the wave occurred in association with the propagation velocity, generating progressive waves.

Classified as a wave, sound gained a new dimension that made it possible to study it in the light of the wave theory developed by Huygens (1629-1695), a significant strategic point in the interaction of mathematics with music in that, understanding the nature of sound, makes it sound. It is possible to better understand, represent and manipulate musical phenomena.

The skepticism of Pythagorean arithmetic dogmatism in music sparked interest in the physical determinants of musical height in the late sixteenth and early seventeenth centuries. Such a stance manifests itself when Vicenzo Galilei raised the then paradox that various fractions could be associated with a certain interval.

The resolution of ambiguity presented by Vincenzo Galilei is found in his son Galileo (1564-1642) who modified the sound reference of the time, considering in his analysis the sound that reached the ear instead of the vibrant object that produced it. Galileo explained the consonance / dissonance of some intervals by writing in 1638 that the direct and immediate explanation underlying the musical intervals was not the length of the string, nor the tension to which it was subjected, but the reason for the number. vibrations and impacts of air waves that hit the ear directly.

He stated that the degree of consonance produced by two tones was associated with the proportion of impacts of the high pitched sound that coincided with those resulting from the low pitched sound. That the frequency produced by an intended rope was inversely proportional to the square root of the linear density of such a rope, a fact corroborated and generalized by Mersenne in determining, through experiments involving dense ropes over 30 meters, other parameters of which the frequency of vibration of a rope still depended.

One of the biggest questions that had been accompanying musical acoustics so far referred to the mystery of sound harmonics. The French mathematician realized the importance of harmonics, requesting his numerous correspondents to seek an explanation for such a phenomenon.

Concerning harmonics, Fotenelle referred to the idea underlying the principle of superposition when she stated that each half, third and quarter of an instrument string performed its partial vibrations at the same time as the entire string vibrated. The science / music relationship still resonates with the genius of the French mathematician Joseph Saveur (1653-1716) who, despite being deaf and dumb, first discovered a means of calculating the absolute number of vibrations of a sound. Often considered the father of acoustics, he was the first to calculate the frequency of beats produced by two notes.