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Mathematics and Music: In Search of Harmony (Part 3)


The discovery of this relationship between the ratio of whole numbers and musical tones was significant at that time, so from this experiment Pythagoras established relationships between mathematics and music, respectively, associating the musical intervals for perfect consonances. These correspond to the fractions of a chord that give the highest notes of the mentioned ranges when the lowest chord is produced over the entire chord.

This discovery of consonant intervals is due to Pythagoras, although they were probably known long before in distinct ancient cultures.

The Samos thinker justified the underlining of small integers to the consonances by the fact that numbers 1, 2, 3, and 4 generated all perfection. The Pythagoreans considered the number four to be the origin of the entire universe, the material world, representing matter in its four elements: fire, air, earth, and water. The importance of this number for the Pythagoreans still emerges in the music scene when considering the tetrachord - the four-sound system, the extremes of which were within a fair quarter range.

Influenced by Eastern culture, Pythagorean doctrine held that "Everything is number and harmony." Thus the Pythagoreans believed that all knowledge would be reduced to numerical relations, placing them as the foundation of natural science.

The intervals are consonant, that is to say, of course, it was interesting to establish tunings that contained such pure intervals. Assuming that the octave was a fundamental range, the Pythagoreans take it as the universe of scale, the problem of establishing a scale was reduced to dividing the octave into sounds that determined the alphabet through which musical language could expressing, thus becoming natural from a note - determinant of the eighth universe along with its upper octave - walking in intervals of ascending and descending fifths, returning to the equivalent note - increased or decreased by an integer number of octaves-always that escaped the eighth universe.

The different octaves were reduced to only one, so each note is equivalent in all the other octaves, and particularly in that benchmark, when any score is achieved in the construction of scales, its significance is its position relative to the lowest note of the eighth. where you are.

2.2. The musical legacy of Arquitas de Tarento

One of the most important musicals of the classical Greek period, Arquitas de Tarento contributed significantly not only to the development of music, but to unraveling its rationale. Architas paid more attention believing that music should take a greater role than literature in children's education. For the Pythagoreans, music theory was divided into the study of the nature of sound properties, the establishment and calculation of musical intervals and musical proportions respectively. He wrote mainly related scientific works, while devoting himself to the first two, especially with regard to consonances. Among his contributions, he changed the old name of subcontract average to harmonic average, probably because the length relative to the fifth - whole string range - of great harmonic value for the Greeks, is the subcontract average between the length of the loose - whole string - and the one corresponding to the eighth - half of the string, fundamental consonant interval.

According to Arquitas, the consonances are produced by two or more simultaneous sounds perceived as only one, so the problem does not concern the phenomenon itself, but its perception, which led him to think of it as a subject problem.

With respect to theorizing about the nature of sound, Arquitas attributes - in the language of the time - tone differences to variations in the movements resulting from the flow that causes sound. Sounds differ according to their velocities in the middle, so the treble depended on a higher velocity than the bass. He illustrated the preceding statement with an example of a strong wind spread which produced a high pitched sound as opposed to a low pitched one.

Architas generalized such a process by analogously calculating the length of the chord corresponding to a third major interval above a given note as the harmonic average between the generating length of that note and that producing a fifth above.

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