Prime numbers and the law of Tetraktys. The first was Pythagoras’ limit, and the second was an enigma that continues today. The basis of Pythagoras school musical theory.

Arturo Reghini (1878-1946) was an Italian mason, writer of renowned essays on esotericism, and, almost all, a mathematician. His book “ I Numeri Sacri nella Tradizione Pitagorica Massonica” or “ Sacred Numbers in Traditional Pythagorean Masonry” posthumous published in Roma 1947, soon became a reference text for the understanding of the remaining enigmas on the Pythagorean mathematics: the sacred parts, from a masonic and sapiential point of view. Among them are the Pythagorean Laws of Acoustics.

To restore the sapiential implications in Pythagorean mathematics and an explanation of Tetraktys from the beginning, you can go to Arturo Reghini, Sacred Pythagorean Numbers. Part 1.

My translation from “I Numeri Sacri nella Tradizione Pitagorica Massonica”, Chap 3:

In addition to the essential Tetractys, ancients considered other Tetractys, quadruple, or quaternion, and Plutarch distinguishes several of them (1). He called Pythagorean the one composed of the first four odd numbers and the first four even numbers, that’s to say (1 + 2) + (3 + 4) + (5 + 6) + (7 + 8) = 36

where 36 is the square of the first perfect number. We observe that 36 is the first triangle that is also a square: and that six is the only triangle whose square is still triangular. It is formed by the sum of two Tetractys, one formed by the odd numbers 1 + 3 + 5 + 7 = 16, which is the square of 4, and the one formed by the even numbers 2 + 4 + 6 + 8, which of course is equal to the twice of the decade. Plato Plutarch calls platonic Tetractys the one formed by the numbers which made the soul of the world whose creation is exhibited in the Timaeus; the Platonic quadruplet is formed by the sum of the two quaternions that have both completed the first unit and then are made with the first three powers of two and three, that are: 1, 2, 4, 8 and 1, 3, 9, 27. The first has 15 for sum; the second has 40 for sum: there is a total of 55, which is the tenth triangular number. Therefore, seeing the generation of numbers compounds contained within the decade, it remains to be seen how to arrive at the prime numbers 5 and 7, the first of which appears as a factor of ten because 2 · 5 = 10.

In contrast, the second is not generated by the multiplication by any number of the decade and does not generate any number of the decade. For this reason, the seven were assimilated to Minerva because the goddess Athena, the Minerva of the Latins and the Etruscans, was a virgin, was not generated, but had jumped out of the brain of Jupiter armed to the teeth. Observing the thing and the consecration of the number seven to Minerva confirms that the number generation was done in a Pythagorean way by multiplication, utilizing the way we have held to the composed numbers.

We must now follow another way. And there are two ways, independent of one another, leading to the numbers five and seven, one starting from the consideration of a tetrachord of Philolaus, the other starting from the consideration of the polygonal and pyramidal numbers.

According to Archytas (1), a Pythagorean living a little later than Pythagoras, there are three progressions: arithmetic, geometric and harmonic, and Iamblichus stated (2) that in the school of Pythagoras, there were considered the three arithmetic, geometric and harmonic averages. We need to remember that according to the Pythagoreans, there is arithmetical proportion within four numbers a, b, c, d when a – b = c – d, and in the particular case of the continuous proportion, that’s to say if both the media b and c are equal, that’s to say if the proportion is a – b = b – c, the average is called the arithmetic median or the arithmetic median of c, and is equal to half their sum, that’s to say

If you have numbers instead of segments a, b, c, and d, the definition of arithmetical ratio and the arithmetic average are the same. Similarly, it has a geometrical ratio between four numbers when a : b = c : d, and in the case of a continuous proportion, that is, if b = c and the proportion becomes a: b = b: c, and b is the proportional average between a and c, and has b = √ a c

If you have numbers instead of segments, the definition of geometric proportion is the same, and the average segment proportion between two segments, a and c are also called the geometric average. This segment can always be built in Pythagorean geometry, even if the given segments are not commensurable, employing an application of the Pythagorean theorem, and this building and the demonstration of the Pythagorean theorem are independent of the postulate of parallels or Euclid postulate, as we have shown in another study (3).

It is said in the end that the four numbers a, b, c, and d are in harmonic proportion when their inverse is in an arithmetic proportion, that’s to say when

and in particular, one has continuous harmonic proportions when

Thus the number b is the harmonious average between a and c when b is equal to the arithmetic average of the inverse of a and c. This definition differs from the definition handed down by the Tarentine Pythagorean Archytas in his piece. Still, it is equal to it, and we derive the same consequences from the two definitions. From the definition of harmonic average, from the relation

and here above it is obtained:

which one can also write

and for the fundamental property of proportions, one has also

and therefore

an important relationship that is found in Nicomachus. This ratio allows the construction of the harmonious average segment between two segments, a and c assigned.

According to Iamblichus (4), Pythagoras would have learned this important proportion in Babylon and was the first to deliver it in Greece. In this Babylonian proportion, the extremes are two numbers or segments a and c, and their average is their arithmetic average and harmonic average. Since then, the rectangle sides a and c are equal to the square of side √ as

there is also the proportion

which contains the three arithmetic, geometric and harmonic averages. The property expressed by this relation can be stated by saying that the geometric average between two numbers or segments a and c is the geometric average between their arithmetic average and their harmonic average. Given the two segments a and c Pythagorean geometry teaches to build their arithmetic, geometric and harmonic average even if the segments are immeasurable, and all that apart from the theory of parallels and the related assumption. If it’s numbered, you can determine in which cases the three arithmetic, geometric and harmonic averages of two integers a and c are integer numbers, but we abstain from this digression.

In the special case where both a = 2c the Babylonian proportion is:

and if a = 1

This quaternion contains the numbers that are the respective measures of the length of four and the tetrachord of Philolaus. It is nothing more than the lyre of Orpheus (5), that’s to say, the instrument he used to accompany his acting and even singing. If the first string gives the sound of our Do (C), the fourth string, having half-length, gives the double sound frequency, that’s to say, the first harmonic of Do (C), or the Do (C) upper octave, while the sounds emitted by the other two strings are respectively those of Fa (F)and Sol (G). The first harmonic of Sol (G) is also the second harmonic of Do (C), and proportionality also the first harmonic of the second Do (C) coincides with the second harmonic of Fa (F). The ear perceives and likes these chords and concordances. In addition, notes Tacchinardi (6), “it is notable that tetrachord contains the ranges most characteristic of the voice in declamation. 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 toned sharper than unstressed vowels. Dionysius of Halicarnassus gives the interval 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 string, the sound was rising, and that were obtained some sounds that the human ear perceives the chord when simple numerical ratios expressed the lengths ratios of the strings. 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: utilizing 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 essential object of Pythagorean arithmetic. 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, it 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. It is equidistant from the three vertices as well as from the three sides, and 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 (8).

We thus found that the radius of the circle circumscribed to an equilateral triangle of height h equals 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. 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 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. 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, the third is the harmonic median of the extremes, and 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, therefore, x = 4: 9, and, since this chord is less than 1: 2 it is external to the tetrachord and takes instead the lower harmonic contained in the first tetrachord, which will double the 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 the 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 the 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 the 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, our B ( si), within the chords of the extreme fundamental tetrachord. It has 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

This yields the value for x = 128 : 729, of which one should take the lower harmonic than the lower harmonic, that’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 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, and the eighth chord has a frequency of 435 vibrations per second. It is then easy to calculate the frequency of the other chords.