Arturo Reghini’s third Chapter on the Pythagorean Geometry and Acoustics and the fact, for a too easy septenary, of being impossible. My translation from the italian book “Sacred Numbers in Masonic Pythagorean Tradition”.

The Odd Prime Numbers Triad within the Decade.

What do these numbers allude to? To the sacred numbers proposed to Apprentices, Companions and Masters meditation. And to the third degree Catechism.

Let’s start from 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)