Mathematics and Music: In Search of Harmony (part 7)

4.2. The science / music in Mersenne

Mathematician, philosopher and theoretical musician, Marin Marsenne (1588-1648) presents himself as one of the leading French thinkers of the 17th century, whose work - mostly devoted to the science, theory and practice of music - takes center stage in scientific and cultural movements. academics of the time. Mersenne moved into the confluence between the Renaissance and the Baroque in France as a character, whose work assumed an important role in the future developments of mathematics / music.

Believing that music was amenable to rational analysis and explanation, Mersenne attached importance to such science - as compared to other disciplines - as an area of ‚Äč‚Äčscientific research.

From a mathematical-acoustic-musical perspective, Mersenne raised crucial questions such as the apparent paradoxality of a note vibrating at various frequencies at the same time, suggesting more careful studies concerning harmonics. Although the French mathematician was a composer, he established a theory based on practice, for example, by defending and grounding an equal temperament in the construction of instruments and by rationally explaining tunings.

From 1630, its writers acquire new forms and interests, culminating in the elaboration of Harmonie Universelle in 1636, whose theoretical-practical approach encompasses reports of different ingenious experiments, studies of sound and reflections concerning the mathematical / music relationship that makes it. often be considered the father of acoustics. Varying lengths and stresses, Mersenne found that, for viewable frequencies, the vibration of a stretched wire was inversely proportional to the length of the string if its tension was constant; directly proportional to the square root of the tension if the string length were constant and inversely proportional to the square root of the mass per unit length for different wires of the same length and subject to the same tension.

For Mersenne, the arithmetic mean was superior to harmonic in character, since taking numbers proportional to vibrations - first causes of sound - the fifth in the lower position resulted from the arithmetic mean of the numbers that characterize the octave. Mersenne found, applying his principles, that the frequency ratio between a note and its octave was 1 to 2 respectively, further explaining the characteristics of vibrant air columns as well as echo and resonance phenomena. It was also the first to determine the frequency of an established musical note, as well as the speed of sound propagation in the air.

Mersenne considered the monochord as a fundamental support for the understanding not only of stringed instruments, but of all musical science, reveals a certain concern with Temperament when dividing the octave into 12 equals, obtaining in this last case the harmonic equality monochrome composed of 11 irrational numbers. resulting from proportional averages.

4.3. Kepler and the music of the planets

Weil-born mathematician, astronomer and philosopher Johannes Kepler (1571-1630) presented strong insights into music science. In 1601, Kepler assumed his post working on organizing calendars and predicting eclipses as a mathematician and astronomer at the court of Emperor Rudolf II in Prague until 1612, settling in Linz, where he completed and published his Harmonices Mundi in 1619. The main contribution of the German astronomer to music theory, this work consists of 5 books - the first two relate the origin of the 7 harmonies with inherent archetypes of geometry and God; book 3 presents a treatise on consonance and dissonance, intervals, modes, melody, and notation; book 4 discusses astrology while volume 5 addresses the Harmony of the Spheres.

Kepler found Pythagoras's experience with the monochord unsatisfactory for establishing consonant intervals. It is believed that such a posture might have led the Pythagoreans to disregard the third and sixth intervals with consonance, reproducing the monochord experiment with a greater number of chordal breakdowns.

He defended the existence, knew from the ancients, of musical scales peculiar to each planet, which sounded as if they were singing simple melodies, relating for this the velocities of the planets to the emitted frequencies. He considered the movements of the planets a song that reflected divine perfection. Thus he tried to explain the variation in velocity of a planet by a musical metaphor. Assuming that fast and slow movements were associated with high and low notes respectively in their imaginative construction, the German astronomer considered that the ratio of extreme velocities would determine a musical range representing the referred planet.

He also knew the laws of harmony concerning the relationship between musical intervals and string lengths, as well as the fundamental law of harmonics. Considered by Kepler, such law stated that in addition to emitting a fundamental sound, an oscillating string provided superior harmonics, corresponding to the fundamental sounds of strings twice, three times, etc. shorter than the initial string.

The German thinker also connects mathematics with music by establishing correspondences between the average distances of the planets to the sun and the frequency ratios on a diatonic musical scale relative to the first degree.