The music of Sirens, the harmony of spheres, Plato’s dodecahedron, and not only. The important role accorded to sacred geometric music in the Pythagorean school.
“In the beginning, you will come where the sirens are, who fascinate anyone who touches with his bow their shores” (Homer. Odyssey XII, 52-54). The Odyssey is a poem on the back home, very likely of a soul. Arturo Reghini’s “I Numeri Sacri nella Tradizione Pitagorica Massonica” or “ Sacred Numbers in Traditional Pythagorean Masonry” was posthumously and published in Roma in 1947. Fourth Chapter: “… the golden part, the division of the circumference into ten and five equal parts, to pentalpha, to the dodecahedron, and the harmonic median of the extreme segments of the two Tetractys formed with the elements of these two figures. We have seen that the catechism of Acousmatic places in the sanctuary of Delphi “the Tetractys wherein is the harmony in which Sirens are… “
Sirens, from Greek Σειρήν Seirḗn – pl.: Σειρῆνες Seirênes, were originally birds from religious characters featured by a seductive lure. Homer presents them as enchanting marine singers dwelling by Scylla and Charybdis. Legend has it that the mariners who surrendered to their charms and landed on the island then were dying.
Only Ulysses, warned by Circe, manages to escape having his companions plugged ears and him tied to the mast of the ship. Homer doesn’t describe them, perhaps thinking it had been made known by other myths, for instance, in Jason’s adventures and Argonauts, in which the sirens had the task of consoling the souls of the dead with their sweet songs and accompanying them in Hades.
But already, during the reign of Demetrius III Eucaerus (Seleucid kingdom first century b. C), we can find the first representation of fish-shaped Atargatis. This religious character will be later confused with that of the sirens. Only during the High Middle Ages did the ancient latin and Greek bird siren become the English mermaid or watery virgin. In the late Middle Ages, the myth of sirens mingled with that of Melusine and was used to decorate churches and monasteries’ capitals.
According to the Alchemic symbolism, the physical water and air belong to the element water, or that which flows. So it is very easy to take music, as we are used to knowing it, as the element water too. Now let’s read from Arturo Reghini:
To fully understand how important and meaningful should be for the Pythagoreans what we found about the dodecahedron, one should remember that for them and Plato, the dodecahedron was the symbol of the universe and that the five regular polyhedra, namely the cosmic figures, were the symbol of the four elements and universe. If we want to see why, there is only to read Plato’s Timaeus, the Pythagorean dialogue par excellence.
The regular tetrahedron, with its four triangular faces, four vertexes, and six edges, was the symbol of fire: and it may be that this correspondence has been determined by the shape of the solid whose vertex recalls the tip of the flame and has been confirmed by the erroneous etymology of the word “pyramid” used by the Greeks, instead of the tetrahedron, from the greek κυρ Fire (the right translation is Sun). Each face is divided by the three diameters of the circumscribed circumference leads to the vertexes of the face into six triangles rectangles equal to each other, and, considering the tetrahedron, which has for vertex the common center of the regular tetrahedron. For the basis of the 24 equal triangles in which the surface is divided, the tetrahedron consists of 24 equivalent tetrahedra. In this way, the octahedron has eight faces that are equilateral triangles, six vertexes, and 12 edges, so the surface of the octahedron is divided into 48 equal triangles, and correspondingly the polyhedron consists of 48 equivalent tetrahedra. Similarly, the icosahedron comprises twenty faces that are equilateral triangles, twelve vertexes, and thirty edges; its surface is divided into 120 equal right triangles. The icosahedron consists of 120 tetrahedra which, as a basis, has as a common vertex the center of the polyhedron. Each regular polyhedron has a polar polyhedron for which the numbers of faces and vertexes are exchanged while the number of edges remains unchanged.
The tetrahedron is self-polar; the octahedron’s polar polyhedron is a cube with six square faces, eight vertexes, and 12 edges. Philolaus saw the image of harmony in the cube because the number of its vertexes is the harmonic median of the numbers of the faces and edges, which of course, is also true of the octahedron. Each face of the cube is divided by the diameter of the circumscribed circumference passing through the vertexes in four equal isosceles triangles, so the surface of the cube is divided into 24 equal triangles, and the cube, or hexahedron, consists of 24 tetrahedra whose vertex is equivalent to the center of the cube. After having attributed to each of these four polyhedra the correspondence with the element fire, air, water, and earth, Plato silences Timaeus, who only says: “So it remains to us still a form of composition which is the fifth of what it has helped God for the design of the universe”. We observe that Plato and the Pythagoreans knew that the regular polyhedra are five and five only, as it is demonstrated simply, and we see that also through this way of the cosmic figures, one comes to the number five.
As for the sudden and unexpected silence of Plato that truncates the exposure of the subject, it also gives the eye to Robin (5), which simply says: “Au sujet du cinquième Polyedre Regulier, the dodécaedre … Platon est très mysterieux, when arrived to the fifth regular polyhedron, the dodecahedron… Plato is very mysterious”. But he does not investigate the reasons for Plato’s sudden silence.
Now the dodecahedron is the polar polyhedron of the icosahedron and thus has twelve faces that are regular pentagons, twenty vertexes, and thirty edges. Applying to it the previous subdivision procedure is that the diameters of the circumference circumscribed to a face, passing through the vertexes, divide it into ten equal right triangles.
Still, if in the face is inscribed the pentalpha, the pentagon is divided by the sides of pentalpha and by the diameters passing through the vertex of the pentalpha in thirty right triangles, which this time are neither isosceles, nor the beautiful right triangles dear to Timaeus (that’s to say with the double hypotenuse of minor cathetus ), nor are they all the same or equivalents. On the other hand, the surface of the dodecahedron is divided into 360 triangles. Correspondingly, the dodecahedron decomposes into 360 tetrahedra which have then as a basis and have as vertex the polyhedron’s center. Now 360 is the number of divisions of the twelve zodiac signs and the number of days of the Egyptian year.
This thing is fully confirmed by what two ancient writers say. Alcinous (1) after explaining the nature of the first four polyhedra, says that the fifth has twelve faces as the zodiac has twelve signs. And adds that each face is composed of five triangles (with the center of the face for the common vertex) each of which is composed of the other six (determined by the diameter and by two sides of pentalpha). A total of 360 triangles. Plutarch, in turn (2), has found that each of the twelve pentagonal faces of the dodecahedron consists of thirty right scalene triangles, adds that this shows that the dodecahedron represents both the zodiac and year because it is divided in the same number of shares of them. Plutarch alludes manifestly to the Egyptian year, composed of 12 months, each of thirty days, in which the five epagomenal days are not part of the Egyptian year.
To well understand the importance of Pythagoreans and Plato of these mathematical observations, it should be noted: 1 – that for them, the triangle is the atom (that’s to say, the last indivisible part) superficial because it is the polygon with the number of sides necessary and enough to delimit a plane portion, and that correspondingly the tetrahedron, or pyramid, is the solid atom because it is the polyhedron having the necessary number of faces and sufficient to delimit a portion of space. 2 ° -That, because of the exact definition of polygonal number, each polygonal number is always the sum of triangular, and for the definition of pyramidal number each pyramidal number is the sum of tetrahedral numbers. So we came to see that even the five cosmic figures, and in particular the symbol of the universe, were composed of tetrahedra; the entire universe was reduced to a sum of tetrahedral atoms.
The number twelve is the number of the faces of the dodecahedron and, consequently, is the number of the vertexes of the polar polyhedron, that’s to say, the icosahedron. Twelve is also the number of the cube’s edges and the polar polyhedron, the octahedron. Let’s consider the number twelve as consisting of the twelve vertexes of a dodecahedron. We develop this dodecahedral number within one of the angles by taking the vertex as the center of “omotetia” is obtained in the usual Pythagorean way the following dodecahedral numbers. The formulas of the regular polyhedral numbers (except for the tetrahedral number) were determined for the first time by Descartes, and are found in a manuscript of his that remained unpublished for over a century; in particular, the n ° dodecahedral number is given by the formula
but the dodecahedral n° can even be obtained through a relationship between the n ° pentagonal number and its gnomon. The pentagonal gnomons are the numbers of arithmetic series 1, 4, 7. 10 … so we have: pentagonal gnomons
pentagonal numbers
and it happens that adding to a pentagonal is a gnomon, you get the pentagonal following, and multiplying a pentagonal for the following gnomon one gets the corresponding dodecahedron number. Thus the sequence of dodecahedral numbers is :
dodecahedral
1 20 84 220 816… ;
relationship between pentagons and dodecahedra which arithmetically corresponds to the relationship between the number of sides of the pentagonal faces and the number of the faces of the dodecahedron. Even in the extension of the tetrachord to the eighth, we have seen a connection between five and twelve. Likewise, the Egyptian triangle of hypotenuse 5 has a perimeter given by 12. The number twelve on his own has traditionally already been sacred and universal. In addition to being the number of the months and signs of the zodiac, twelve was in Greece, Etruria, and Rome the number of the allowed gods, twelve was the number of members of some priestly colleges in archaic Rome, twelve was the number of Etruscan and Roman rods beam, and many surviving Celtic dodecahedra attest the importance that the ancient gave to this number and the dodecahedron. Facts and reasons confirm the dodecahedron’s choice as a symbol of the universe.