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Saturday, January 25, 2014

Pythagoras ipse dixit!

Pythagoras : The Mathematics of harmony

At Capo Colonna, near Crotone, the Greeks had a temple dedicated to Hera Lacinia - i.e.  48 columns aligned in the direction of the rays of the rising sun. On a starry night in the month of May, the women of the city went in procession to the temple to ask for the gift of the goddess of fertility. In Byzantine devotion to the goddess moved to Mary Theotokos, the Mother of God, but the tradition of the procession remained unchanged, and continues today.

The town of Crotone is most proud, however, without a doubt, the school that Pythagoras of Samos, the great mathematician and philosopher, founded when he moved there from Greece around 530 BC. It prospered for thirty years, until the Pythagoreans get involved in the political affairs of the city, supporting the wrong party . They were persecuted and driven out, the school was burned, and Pythagoras fled to Metapontum, where he died shortly after.
To commemorate these disparate historical memories, the Province of Crotone combines the traditional Marian procession initiatives of a singular " May the Pythagorean ." The event alternates lectures on mathematical and musical concerts, and culminates May 24 with a "Concert dawn " held at sunrise , at four in the morning, at Cape Colonna.

The music is not involved in the program in a purely casuality. It was in fact just a musical intuition that allowed Pythagoras to make the link between mathematics and nature, which is probably the most profound discovery and fruitful history of the whole of human thought.

According to Iamblichus, the episode is as follows. Pythagoras spent a day in front of a blacksmith workshop and realized that the sound of hammers on anvils was sometimes consonant, sometimes dissonant. Intrigued, he entered the workshop, where did show the hammers, and found that those who rang in consonance had a specific weight ratio. For example, if one of the hammers weighed twice of the other, they produced sounds an octave away. If one of the hammers weighed one and a half the other, distant sounds they produced a fifth ( the interval between the C and the G).

Back home, Pythagoras did some experiments with nerves of ox, to see if any similar rule was worth and applicable to the sounds generated by string instruments such as the lyre. Surprisingly, the rule was even the same! For example, if one of the strings had twice the length of the other, they produced sounds an octave away 

If one of the strings was a long one and half the other, they produced sounds a distant fifth.
In perfect scientific style of observation and experiment, Pythagoras deduced a theory: the coincidence of music, mathematics and nature. More specifically, he argued that there were three types of music: instrumental music itself, played the human body, and the socialite played by cosmos. The substantial coincidence of the three music was responsible for an emotional side effect product for literal resonance, the melody man, and on the other the possibility to deduce the mathematical laws of the universe to the music .

Because the laws of harmony discovered by Pythagoras concerned only fractional numbers, also called rational numbers and harmonic ratios perfectly matched to numerical ratios, Pythagoras enunciated his discovery in the famous maxim: everything is (number ) rational. It encodes the belief in mathematical intelligibility of nature, and it is the metaphysical presupposition of the entire scientific enterprise , of which Pythagoras was indeed the founding father .

More precisely, "reason" was nothing more than the ability to express concepts through a "relationship " number, as evidenced by the use of the same word for both terms , both in greek (logos) which in Latin ( ratio). Since then , for the Greeks, logos also meant the "word " itself, the word came to express a triple coincidence of language, rationality and mathematics. Although this coincidence is still alive and well, and the Treaty of Wittgenstein is but the latest reformulation revised. A discovery so profound that it could not be regarded Pythagoras or a real divinity, or at least a depository of divine wisdom. His teaching could not be questioned, and he applied himself for the first time the expression "ipse dixit". His school took on the character of a religious brotherhood, and their followers were divided into two categories: acousmatic, or listeners, and mathematicians, or apprentices. The first were taught in an esoteric and superficial way, while the latter were initiated to esoteric and deep teachings.

A typical example of the dichotomy is the Pythagorean cosmological theory, the esoteric aspect which has been handed down by Plato in the uneasy, complex Timaeus' dialogue. Through mysterious constructions based on the numbers 1 , 2 and 3, which correspond to the numerical ratios of the octave and the fifth, we arrive at determination of relations: harmonics that govern the motion of the planets. The solar system is therefore seen as a seven-stringed lyre played by Apollo , in which the planets produce the sounds that correspond to them, and which together make up the music of the spheres.

The Pythagorean esoteric aspect of the model was for centuries the reference point for cosmology, so that even in 1619, Kepler used it in his amazing book "The harmony of the world". In it, he described the musical laws that govern the motion of the planets, specifying that the celestial symphony Mercury sings soprano, tenor is Mars, Saturn and Jupiter as basso, and the Earth and Venus from high. And in the third of the three famous laws of Kepler reappears, miraculously, the fifth report of the square of the rotational period of a planet around the Sun is proportional to the cube of its distance from it. In fact, one of the conferences in May Pythagorean is therefore dedicated to " The ultimate dream of Kepler ."

The key breakthrough of modern physics, made ​​by Newton in the Principia , corresponds instead to an explicit attempt to discover the esoteric aspect of the Pythagorean cosmology, hidden under the " vulgar speech " as Music of the SpheresLike many of his contemporaries, Newton took the view that the fundamental knowledge of the world, the so-called prisca sapientia, had already been revealed by God to the first men, engraved on two pillars: they were rediscovered after the Flood of Pythagoras and Hermes Trismegistus, which covered the truth in their esoteric philosophies. Be that as it may, the fact is that on this basis showed that Newton's law of universal gravitation was implicit in the laws of harmony Pythagorean, and declared that it had therefore already be known to Pythagoras himself.
Pythagorism remains very much alive even in modern physics, and not just as generic mathematization of nature. First of all, if classical physics had been formulated as the Pythagorean motto : "Everything is (number ) real" or "everything is (number ) imaginary " atomic physics seems to be returned to the original version, which are precisely the integers to determine the characteristics of nature at microscopic level, through the quantization of the amount that is supposed continuous, first of all the energy. In addition, the most recent attempt to arrive at a unified theory of nature, the so-called string theory, Witten, breaking the constituents of matter are no longer thought of as points (im) material, but as pieces of string that vibrate in a multidimensional space, and whose modes of vibration ( or sounds ) are the elementary particles. Which justifies the title " From Pythagoras to Witten", a conference in May Pythagorean.

Even the history of music, as that of physics, has implemented and developed in a profound way the Pythagorean belief. Pythagoras himself had already discovered that his music theory had some problems: in fact, the numerical ratios corresponding, respectively, in a tone and two semitones did not coincide, and differ by a small amount but perceptible to the ear, which was called the Pythagorean commaThe mathematical solution of temperament, that is to divide the octave into two equal semitones, was found only in the eighteenth century and required the assignment of a value to irrational semitone.

Not surprisingly, the solution, which initially fathered resistance lively, was polarized by the 48 preludes and fugues of the Well-Tempered Clavier . Bach was in fact sensitive to the mathematical structure of the music, and works such as the Goldberg Variations, the Musical Offering and the Art of Fugue in a systematic way using geometric transformations that reverse, overturn and expand the musical themes. The same transformations, basic for the whole polyphony, were then explicitly formulated at the beginning of the century as the rules of the twelve-tone technique. Which explains the presence of many works of Bach, "L'Offerta musicale" in particular, as well as Webern and Berg, concerts in May Pythagorean .
In conclusion, it remains to note that the Pythagorean thought has now become the basic metaphysics of planetary culture. The science and technology, whether we like it or not, have surpassed all geographical boundaries and pervaded the entire globe , are in fact based on precisely the coincidence between nature and mathematics that Pythagoras was the first to perceive and pursue known, revealing more universal and deeper than any other prophet or thinker , from Buddha to Christ, from Plato to Marx.

The planet, now unified by science and technology, continues to remain divided by religions. Perhaps in this field Pythagoras, who believed that God was simply the harmony of the universe, and that the religious purification is got through contemplation mathematics, may one day hold up the right way. Which is, moreover, already contained in the esoteric version of the Pythagorean beginning of the Gospel according to John: "In the beginning was Reason, and Reason was with God, and the name was God."

... not a perfect translation, yet hinting to the genius of Pythagoras...

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