Mathematics and Music: Seeking Harmony (Part 6)

The French mathematician realized that the organ builders had intuitively discovered the harmonics, mixing them through registers to obtain distinct tones. Organists mixed the organ's knobs in much the same way that painters mixed colors, imitating the harmony of nature observed in sound objects.

Contributors to the eighteenth-century mathematical / music relationship include Leonhard Euler (1707-1783), Jean Lê Rond d'Alembert (1717-1783) and Daniel Bernoulli (1700-1782). According to Euler, the ear tended to simplify the perceived reason, especially when jarring tones followed a harmonic progression.

The most precise connection between musical pitch and frequency - the speed of a vibratory movement - would occur in the eighteenth century with D'Alembert. He stated that a natural sound was not pure but complex, being obtained by overlapping several harmonics in a series. A harmonic was characterized by its amplitude, which assumes great importance in sound synthesis.

Daniel Bernoulli stated that the vibration of a sound body could be observed by superimposing its simple modes on distinct amplitudes, but there were no general principles upon which proof of such a statement could be experienced. This statement by Bernoulli was theoretically anchored in Fourier's experiments in areas seemingly distant from the musical universe. In the early decades of the nineteenth century, Fourier showed how to represent any periodic curve by superimposing sine waves corresponding to frequencies 1, 2, 3, 4 times the frequency of the original curve.

Not only did Fourier's theorem substantiate Daniel Bernoulli's point of view on acoustics, but it became the foundation for harmonic, consonance / dissonance, dissonant beats, as well as distinct apparently dissociated musical concepts from mathematics.

4.1. Gioseffo Zarlino's Mathematical-Musical Contribution

Italian theorist and composer of Chioggia - Zarlino integrated theory and practice as a model for his writings. Concerned about the reason for perfect consonance, Zarlino pioneered the recognition of the triad in harmonic terms while speaking only at intervals; in elaborating a rational explanation of the old rule that prohibited the use of fifth and eighth parallel as well as in the recognition of the minor / major antithesis.

Believing that music had the ability to provoke good and evil, the Italian composer agreed with Plato's concern for prudent use of this art. Although not always conceptually resonant with Zarlino's theoretical perspective, Pythagorean-Platonic and humanistic conceptions - most notably numerology - provided the Italian composer with a basis for establishing criteria for the use of his own inappropriate and generally related musical consonances and procedures, for example, consonant intervals with simple reasons.

According to Zarlino, the perfect harmony consisted of diversity, reluctant of elements distinct from each other, discordant and contrary having in their parts, proportions, movements and varying distances from the severe and acute regions. It compares music with nature and states that the truth and excellence of this admirable and useful warning are confirmed by the phenomena of nature, for by generating individuals of the same species it makes them similar to each other, but different in some particular respects, a difference or variety that bring greater pleasure to our senses.

Supported by the old doctrine of harmonic and arithmetic averages, Zarlino showed preference for the third largest over the smallest, since the latter was obtained by the arithmetic average of the lengths that produced the component notes of the fifth interval, while the third largest underlies the fraction. obtained by the harmonic mean of the inherent lengths of the component notes of the same range.

Unlike Pythagoras - which generated intervals by fifth overlaps - and like other theorists of his generation, Zarlino obtained intervals by dividing them by adding and subtracting them, but never inverting them. Zarlino explained the consonant property of Fridays as overlapping the intervals of Perfect Wednesday and Tuesday and not as inversion of Tuesdays, which would only occur from the early seventeenth century, when theorists explicitly used this procedure. The Italian composer and his predecessors understood dissonances as momentary cuts in the consonant process which, producing variety, emphasized the perfection of consonances.

Zarlino also argues that all harmony - composition and counterpoint - should consist mainly of consonance, using only dissonance secondarily and incidentally in the name of elegance and beauty.

At this point, the Italian composer established a strong link between poets, musicians and painters who, adapting or intending to paint stories or fables as they saw fit, adjusted the figures, arranging them in their composition. He therefore did not hesitate to place a consonance in a particular place as soon as he observed the original order of the story or fable he tried to represent. Moreover, countless painters have portrayed a simple subject in many ways. The musician should also look for variety in his counterpoint to the subject, and if he could invent many passages, he should choose the one that was best, best suited to his proposal, and best able to take his sonorous and orderly counterpoint.