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Aldous Huxley, one of the most preeminent intellectuals of his time, stated, “after silence, that which comes nearest to expressing the inexpressible is music.”(goodreads) Never a more verbally inexpressible art many of the world’s greatest minds have been drawn to this subject that bridges body and soul. Einstein stated, “If I were not a physicist, I would probably be a musician. I often think in music. I live my daydreams in music. I see my life in terms of music.” The logical reasoning of the head does not tend to understand the illogical reasoning of the heart. For many intellectuals, music is the bridge that leads to that understanding. Pythagoras saw music as the foundations of life as we know it. Every leaf, every planet, every person has a melody and rhythm of their own.

The life and mathematical achievements of Pythagoras of Samos, born around 570 BCE and died around 490 BCE, is a mystery to the mathematical world. Due to there being no surviving documents of Pythagoras it is unknown as to what the man actually did himself and what his followers did in his name. Though it is uncertain, still many innovations have been connected to the mathematician. Specifically, Pythagoras made music related mathematical achievements which are exemplified through his life, the music of the spheres, and the musical octave.

Said to be “the first pure mathematician”, Pythagoras of Samos is very important to the mathematics that we know and love today. Sadly, though he has been given this title, as stated above very little is known about his achievements in the field. There is nothing of Pythagoras’s writings that has survived to this day. This could be in part due to the secrecy of this influential man. “The society which he led, half religious and half scientific, followed a code of secrecy which certainly means that today Pythagoras is a mysterious figure. “ (O’Connor) Pythagoras spent some time in Egypt and it’s said that it was during this time that many of his beliefs were molded. “The accounts of Pythagoras’s time in Egypt suggest that he visited many of the temples and took part in many discussions with the priests. It is not difficult to relate many of Pythagoras’s beliefs, ones he would later impose on the society that he set up in Italy, to the customs that he came across in Egypt. (history)

To speak nothing on the fact that some of what is said of Pythagoras is pure myth would be a failure in giving an accurate summary of his life. The myths are said to be possible because of a trend started in the first centuries BCE in which Pythagoras was talked of as if he was a “semi-divine“ figure rather than just an ordinary man with an extraordinary mind. Writings of even plato and Aristotle began to be attributed to Pythagoras in an attempt to support this dissertation. “The Pythagorean question, then, is how to get behind this false glorification of Pythagoras in order to determine what the historical Pythagoras actually thought and did.“ (plato)

Pythagoras’s mother,Pythais, was a native of Samos. His father was named Mnersarchus and it is said that he brought corn to Samos as a traveling merchant from Tyre during a time of great famine. This gained him citizenship from the people as a show of gratitude. As a child Pythagoras spent his early years in Samos but travelled widely with his father.” (history) “There were, among his teachers, three philosophers who were to influence Pythagoras while he was a young man. One of the most important was Pherekydes who many describe as the teacher of Pythagoras. The other two philosophers who were to influence Pythagoras, and to introduce him to mathematical ideas, were Thales and his pupil Anaximander who both lived on Miletus. In [8] it is said that Pythagoras visited Thales in Miletus when he was between 18 and 20 years old.Thales was an old man by this time and had little practical knowledge to teach Pythagoras. He better served as a mentor in the time they shared, creating a strong impression on Pythagoras and inspiring him to pursue deeper knowledge of mathematics and astronomy in Egypt. Thales’s pupil, Anaximander, lectured on Miletus and Pythagoras attended these lectures. Anaximander certainly was interested in geometry and cosmology and many of his ideas would influence Pythagoras’s own views.” (O’Connor)

The path in which Pythagoras came to his “Musica Mundana”, Music of the Spheres, is centered around a story unknown as to if it is true or false. This should be of little surprise given what we know about Pythagoras. It is said that the man, as he walked by, took note of the sounds of blacksmiths’ hammers as they beat down on an object. The hammers were said to have different sounds based on their size. (Aboutscotland) This lead Pythagoras to go home and tie weights the same as each hammer to strings and pluck them to see what sound they made. He was said to have been an expert lyre player and as such would have had a working knowledge of strings based on that alone.(philclub.org)

This string theory is the beginning of both Musica Mundana as well as the octave as musicians know it today. Changing proportions on the strings changes frequencies. It was Pythagoras who noticed that the multiples of the same frequencies could be considered the same note.

Therefore, once a frequency is known, either halving or doubling the frequency will give the same note name below and above respectively for that specific note value; these are called octaves. As an example, an A1 on a piano has a frequency of 55.00Hz while A3 has a frequency of 220.00Hz. Since A1 is doubled to create A2 and A2 is doubled to create A3, the equation 2×2=4 is created. 55.00 multiplied by 4 is 220.00. To further the example, to get back to A2, A3 would need to be halved. Half of 220.00 Hz is 110.00 Hz. Upon looking at figure 1, it can be seen that A2 is in fact, 110.00 Hz. “If a taut string is merely touched at the center, so that the ratio of subdivided intervals is exactly 1:1, the string will emit a note that is an octave higher than the fundamental of the string. The note of the stopped string is also an octave higher. This stopped note is in a ratio of 1:2 in wavelength to open string. Thus its frequency will be twice as high, because it’s wavelength is twice as short. Here we see that Pythagoras was developing the beginnings of a wave theory.” (philclubcle) Different proportions will show the other notes within the modern harmonic progression. “By dividing the string into various other lengths, intervals of the fourth and fifth were produced, and so on. Pythagorasn and his followers conceived of the universe a vast lyre in which each planet, vibrating at a specific pitch, in relationships similar to the stopping of the monochord’s string, harmonized with other heavenly bodies to create a “music of the spheres.” (Richards)

Lamda, a “triangular figure of numbers, is the Tetrad of the Pythagoreans… It is a set of numbers whose relationships with each other seemed to summarize all the interdependent harmonies within the universe of space and time.” (aboutscotland) This set of numbers is an summary for everything including the planets. Pythagoras taught that each of the seven planets produced by its orbit a particular note according to its distance from the still center which was the earth.” (aboutscotland) Figure 3 shows the information on the notes for each of the planets in our solar system.

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