A tradition reported by Diogenes Laertius tells how Pythagoras, listening to the sounds emitted by the hammer of a blacksmith pounding on the anvil, observed that the height of those sounds depended on the thickness of the hammers, and then experimenting with strings equally stretched drawn from the same rope, found that with the decreasing length of the string sound was rising, and that were obtained some sounds that the human ear perceives the chord when the lengths ratios of the strings were expressed by simple numerical ratios. If the tradition reported by Diogenes Laertius is true this would be the first example of a scientific discovery obtained by the orthodox method of scientific observation, followed by the experiment, and, as the simplest possible ratios are the three ratios: 1: 2, 2 : 3, 3: 4, Pythagoras would have experimentally acknowledged that, taking a single rope and three single strings as long as with the previous ratio, we precisely obtained the lyre of Orpheus or tetrachord of Philolaus. In addition, when the strings were arranged in the descending order of their lengths 1, 3: 4, 2: 3, 1: 2, was immediate the realization that they form a geometrical ratio, that the second string is as long as the length the arithmetic average of the lengths of the extreme strings, and that the third rope is the harmonic average. And, if we accept the tradition reported by Iamblichus, it may be that the knowledge of Babylonian proportions has led Pythagoras to explore ropes having those lengths, and to find by ear the chord of the sounds emitted by them and their identification with the sounds from the strings of the lyre of Orpheus and the tetrachord of Philolaus. However, one can imagine the admiration that this discovery must have aroused within Pythagoreans: by means of the Tetractys numbers one can get the Tetractys of the strings of the Philolaus tetrachord, and the lengths of these strings are nothing more than the simplest case of the Babylonian proportion.

In the end, it is worth noting how these measures may also be suggested by the linear view of the linear, polygonal, and pyramidal numbers, an important object of Pythagorean arithmetic. In fact, if in a long segment h one takes its midpoint, the segment is divided into two segments each of length 1: of 2 h. If then one considers the fourth triangular number which is the Tetractys and supposes that the form is that of an equilateral triangle, is easy to intuitively recognize that there are points located on the boundary of the triangle and only a single central point, that the three height s of the triangle meet at this point, and it is equidistant from the three vertices as well as from the three sides, and that it divides the three heights into two parts, which the lesser is the half of the major and the third part of the whole height h, and that the major is the 2: 3 of h. The rigorous recognition of this property requires the development of Pythagorean geometry, which would take too long to treat, we limit ourselves to referring the reader to our work over Pythagorean geometry (10).

We thus found that the radius of the circle circumscribed to an equilateral triangle of height h is equal to two-thirds of this height. In a similar way and taking advantage of the isotropy of the regular tetrahedron it is recognized that, if the points making up the fifth tetrahedral number are arranged so that the bases are regular triangles, they may be disposed at five equidistant planes, of which the first meeting through the vertex of the tetrahedron, the second containing with three points, the third one six, the fourth the ten points forming the Tetractys, and the fifth the triangular base of the tetrahedron. The center of Tetractys also belongs to the tetrahedral number, and we intuitively acknowledge (but it can be proved) that the four heights of the tetrahedron are equal, that they meet at a point that belongs to the four Tetractys located above the four bases, and that the center of the tetrahedron divides every height into two parts of which is the lesser is 1 : 4 of the height, while the major the 3 : 4 of the height. Thus the radius of the sphere circumscribed to the regular tetrahedron is three times the radius of the inscribed sphere and the 3 : 4 of the height of the tetrahedron. The property can be stated by saying that, given a segment h, the Tetractys of height h and the tetrahedron of height h, the entire segment h and its half are the extremes of a geometrical ratio which other terms are the radius of the circumcircle circumscribed to the Tetractys and the radius of the sphere circumscribed to the tetrahedron. Thus considering the tetraktys of height h and the tetrahedron of the same height, it happens that the radius of the circumcircle circumscribed to the tetraktys is the harmonic average of the height and its half, and the radius of the sphere circumscribed to tetrahedron is the arithmetic average of the height and its half.

Let’s see now how to pass from the fundamental tetrachord of Philolaus to the scale or Pythagorean range of the seven notes.

But before leaving this subject lets us introduce another notice, still in connection to the law of the fifth, that’s to say the ratio 2 : 3. Cicero when looking at the tomb of Archimedes in Syracuse was able to find and identify it because above it there was the figure of the cylinder and the equilateral cone circumscribed to the sphere. Archimedes discovered that the total area of the circumscribed cylinder (6π r2) was a proportional average between the surface of the sphere (6π r2) and the circumscribed equilateral cone (9π r2), having the diameter of the base equal to the apothem; and the same he showed that the volume of the cylinder (2π r3) was a proportional average between that one of the sphere

and that one of the circumscribed equilateral cone (3π r3). This discovery and property should be considered important and worthy of appearing on the tomb of the great Geometer. It can be easily deduced that the four relations between the surface of the sphere and the whole surface of the circumscribed cylinder, between the volumes of the two solids, between the surface of the cylinder and the total area of the circumscribed equilateral cone, and between the volumes of two solids, they are all four equal to the ratio 2 : 3, that’s to say the ratio of the fifth, the ratio of Do (C) : Sol (G) basic of the tetrachord of Philolaus, the typical interval of the elevation in the spoken language so appreciated by Dionysius of Halicarnassus.

Next Arturo Reghini Sacred Numbers 8.

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- See Delatte, Etudes, 255;

Fram. 2 reported by Mieli, Le scuole eleatica, jonica e pitagorica, Florence 1916, p.. 251; JAMBLICHI, Nicomachi arithm. introduced., ed. Teubner, p.. 100; - See A. Reghini, Per la restituzione della geometria pitagorica, Rome 1935;
- See G. LORIA, etc. Le scienze esatte., 36;

The lyre and harp (from which the guitar) differ slightly, were the instrument of Orpheus, Amphion, and Apollo. Amphion, with the sound of the lyre, is said to have built the walls of Thebes, with the sound of the lyre Orpheus exerted an action on animals and plants; - Tacchinardi, Acustica musicale, Milan, Hoepli, 1912, p.. 175;

See A. Meillet, Aperçu d’une histoire de la langue grecque. Paris, 1912, p.. 22, see also page 296;

A. Reghini, Per la restituzione della geometria pitagorica;