In addition, notes Tacchinardi (6), “it is notable that tetrachord contains the ranges most characteristic of the voice in declamation. In fact, questioning the voice goes up by a fourth; strengthening it grows by one degree, and in the end, ending, descends a fifth. “It should also be noted that (7) “Indo-European accent was an accent in height, the tonic vowel was characterized, not by a strengthening of the voice, as in German and English, but by an elevation. The greek tone consisted of an elevation of voice, the tonic vowel was a vowel toned sharper than unstressed vowels. The interval is given by Dionysius of Halicarnassus as an interval of a fifth. “And in the tetrachord of Philolaus the Sol (G) is the fifth of the Do (C) and the Do of the second octave is the fifth of Fa (F).

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 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 expering with 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 pythagorean: 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, important object of the 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 the Pythagorean geometry, which would take too long to treated, we limit ourselves to refer the reader to our work over the Pythagorean geometry (8).

We thus found that the radius of the circle circumscribed to an equilateral triangle of height h is equal to the 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 aknowledge (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 fundametal tetrachord of Philolaus to the scale or pythagoric range of the seven notes.

But before leaving this subject let’s 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 had 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.

Let us now turn to the tetrachord of Philolaus C, F, G, C (do, fa, sol, do) whose chords are respectively long 1, 3 : 4, 2 : 3, 1 : 2 such that

and where the second term is the arithmetic median of the extremes and the third is the harmonic median of the extremes, while the fourth is half of the first.

The last two terms can be considered as the first two terms of a new proportion in which the fourth term is, as in the case of the previous ratio, half of the first that’s to say 1 : 3 and the third term x should be accordingly calculated. Therefore it is

That is

G C X G ( sol do x sol)

and the new tetrachord G C D G ( sol do re sol)

The length of the third chord can be calculated in various ways, as a third of an unknown proportion, as the harmonic median of the extremes …

Is thus x = 4: 9, and, since this chord is less than 1: 2 it is external to the tetrachord, and takes instead its lower harmonic contained in the first tetrachord which will double length that’s to say 8: 9. This produces a new chord, contained within the extreme chords of the fundamental tetrachord, chord which we designate as D (re), and you have an equal chain ratio

and the new tetrachord

G C D G ( sol do re sol )

Operating again as before, for example, taking as first terms of a new ratio, or tetrachord, the last two terms of the proportion, or previous tetrachord and taking as before as fourth term half of the first, one obtains

and is obtained for the x the value x = 16 : 27 which is greater than 1: 2. The chord as long as that is therefore included within the extreme chords of essential tetrachord, and that is what we call A (la). We then have a third tetrachord

D G A D ( re sol la re)

and the ratio

Proceeding similarly, it has the proportion

which yields x = 32: 81, and since this fraction is less than half, one takes the chord which is its lower harmonic, that’s to say which has the length 64: 81. This corresponds to the E chord of the Pythagorean scale (although the natural scale E has a length 4 : 5 (slightly different). One has, therefore, the fourth tetrachord

A D E A ( la re mi la)

that is

Similarly considering the new proportion

E A X E ( mi la x mi)

That is

Which yields x = 128: 243 which is greater than 1 : 2, and then this is our chord, which is our B ( si), within the chords of the extreme fundamental tetrachord. It has therefore the fifth tetrachord E A B E ( mi la si mi.)

If we now consider the tetrachord B E X B ( si mi x si) that is

Which yields the value for x = 128 : 729, of which one should take the lower harmonic than the lower harmonic tha’s to say the chord of length 512 : 729 to get a chord within the tetrachord of Philolaus, but the interval between this chord and that of F ( = 3: 4 is too small because any ear can distinguish the two sounds, and therefore F fa) will replace this chord and makes it the sixth tetrachord

B E F B ( si mi fa si )

In the end considering the tetrachord F B X F ( fa si x fa) that’s to say the proportion

Which yields x = 1: 2 and is therefore the seventh tetrachord

F B C F (fa si do fa)

With this seventh tetrachord the loop is closed, because continuing to operate as we have done until now we would find the G ( sol), and so on.

Starting then from the three notes of the tetrachord of Philolaus C F G (do, fa, sol) and always working with the same law we have found four other notes and more. The Pythagorean range for this reason consists of seven notes which, written in decreasing order of lengths of the chords, are:

C D E F G A B C (do re mi fa sol la si do)

where the octave is the top of the first harmonic and the first of the upper octave. As is known, it is assumed by international convention as A (la) the third, eighth the chord that has the frequency of 435 vibrations per second, and it is then easy to calculate the frequency of the other chords.