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Mathematics and Music: In Search of Harmony (Part 4)


It is noteworthy that the third major interval obtained by Arquitas agrees with that present in the Harmonic Series. Such a phenomenon leads us to imagine that Architas had a sensitive ear when he realized that the third corresponding to (4/5) - lower than Pythagorean, (64/81) - sounded more natural, since it merged exactly within natural harmonics. of a note. While Pythagoras calculates underlying fractions of scale using only farm paths, Architas strongly considers calculations of arithmetic and harmonic averages in the generation of his musical system. Redistributes the length relationships underlying the Pythagorean scale, obtaining different fractions such as (4/5) corresponding to the Tuesday interval, formerly associated with (64/81) by Pythagoras.

As mentioned earlier, Arquitas developed a theory for the nature of sound with immediate implications for music, in which he related strength and speed to musical height - the stronger and faster a movement, the sharper the sound produced. Considering air as a fundamental element evidenced the centralization of interest in the sound propagation medium by the Tarentine, who claimed that there was greater compression of a stronger sound. Such statement deprived the inherent qualities of the instrument and its acoustic possibilities, from the focus of attention in the study of what today is called acoustics, towards the understanding of how the sound was transported.

Induced by the observation that the same note was produced by different values ​​of different lengths, tensions or capacities, Arquitas was forced to abandon the traditional Pythagorean theory, embarking on the theoretical search for a principle that explained the essence of sound, regardless of the resonant body structure.

Representing a significant contribution of Pythagoreanism, the acoustic studies of the Tarentine thinker unfortunately did not prosper, being resumed only in Modernity, when the development of modern acoustics began.

3. Brief Overview of Music in the Middle Ages

During this period, there was the strong contribution of the Roman citizen and writer Boetius (480-524 AD) to the systematization of Western music. The Roman thinker published in five volumes the De Institutione Musica, in which he considers music to be a force that pervaded the entire universe and a unifying principle of both man's body and soul and his body parts. His treatise is based on the Pythagorean doctrine of consonances, and makes use of mathematics to rationalize musical consonances and the principle of division of monochord.

Developed in the 8th and 9th century AD, the Cantochon features one of the oldest recognized musical forms, consisting of a single melody limited by an octave range. Although documents and facts prior to the ninth century reveal some rudiments of harmony, man sang and played in unison for centuries. Around the ninth century, what began to be called the first polyphonic songs, called in their earliest form Parallel Organum, began to appear.

Over the next two centuries, singing evolved into the Free Organum. At this time, it is important to emphasize the importance of the pedagogue and musical theorist Guido d'Arezzo (955-1050 AD) who, using new methods of notation and teaching, played a decisive role in the constitution of our music theory. He wrote Micrologus - the first complete treatise on musical practice - which, including a discussion of Polyphonic music and Cantochon, developed, among others, a first-time singing technique based on the syllables ut, re, mi, fa, sun, there - name of the musical notes currently employed in much of the world.

In Free Organum, the organal voice frees itself from the parallel path to navigate in opposite, oblique, and direct movements. This form lasts until the beginning of the twelfth century, when the note-to-note style is abandoned to take over the Melismatic Organum, where the main voice is held, supporting long notes, while another voice was freely developed with less valuable notes. It also means rhythmic diversification.

In the thirteenth century, clausulae notes began to use words independent of the text, giving rise to a type of popular music called moteto, where a third voice - triplum - comes into play with completely independent words, sometimes in other languages.

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