The basics of the Pythagorean acoustics and geometry in this end of the second chapter of Arturo Reghini’s “ Sacred numbers in masonic Pythagorean Tradition” .

The quaternion of composed and synthetic numbers. Arturo Reghini: In addition to the essential Tetractys ancients considered other Tetractys or quadruple or quaternion, and Plutarch distinguishes several of them (1). He called Pythagorean the one composed by 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 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 so seen 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 while 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 was 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. The observation of the thing and the consecration of the number seven to Minerva confirms that the numbers generation was done in a Pythagorean way by multiplication, that by means of the way we have held to the composed numbers.

We must now follow another way. And there are two ways, independent of one another, both leading to the numbers five 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 (2), a Pythagorean living a little later Pythagoras, there are three progressions: arithmetic, geometric and harmonic, and Iamblichus stated (3) 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 arithmetic average is the same. Similarly, it has geometrical ratio between four numbers when a: b = c: d, and in the case of 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 proportional between two segments a and c is also called the geometric average. This segment can always been build in Pythagorean geometry, even if the given segments are not commensurable, by means of 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 shown in another study (4).

It is said in the end that the four numbers a, b, c, d are in harmonic proportion when their inverse are in 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 but is equal to it, and from the two definitions we derive the same consequences. From the definition of harmonic average that 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

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

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 also the geometric average between their arithmetic average and their harmonic average. Given the two segments a and c the 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 numbers you can determine in which cases the three arithmetic, geometric and harmonic average 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: