Arturo Reghini’s third Chapter on Pythagorean Geometry and Acoustics and the fact, for a too easy septenary, of being impossible. My translation is from the Italian book “Sacred Numbers in Masonic Pythagorean Tradition”.
Chapter III:
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 therefore 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 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 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 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 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, and eighth chord has a frequency of 435 vibrations per second, and it is then easy to calculate the frequency of the other chords.
Now the third chord of the tetrachord of Philolaus, that’s to say G (sol), is the fifth of the eighth, and with the process now seen to extend the tetrachord of the scale we obtained only seven chords from the top chord to the second tetrachord (which begins with G sol) which is the fifth respects to the new octave which begins with G (sol); and so continuing with the development of this law of fifth it will determine all the seven notes. From the first three chords of the tetrachord of Philolaus, whose lengths are determined by the numbers of Tetractys and the Babylon proportion, by the law of the fifth, we will determine the seven chords. They are the odd prime numbers three, five, and seven contained within the decade, corresponding to the age of initiation of Blue Masonry, and for the frequency range, the seventh chord is also the latest and from that perfection of the number seven.
The seven chords, are written in order so that each chord is followed by its successive fifth
C G D A E B F C (do sol re la mi si fa do)
and, if one divides a circle into seven equal parts and at the points of division are writing the seven notes in that order, and then the three points are counted from three to three from G (do), one neatly gets the seven notes in the order of the musical scale; vice versa the seven notes written in the order of the musical scale in correspondence of the seven dividing points of the circumference and counting the points from five to five from C (do) , are neatly obtained the seven notes in the fifth order.
In the Pythagorean scale the intervals or proportions of the notes of the fundamental octave unit are expressed by
and it is easy to recognize the way in which it proceeded to the extension of the tetrachord all these reports contain the numerator and the denominator only powers of two and three. The maximum power of two is 128 = 2 to seventh, and the maximum power of three is 243 = 35. The same thing happens considering the relationship between any two notes of the octave. So while in the tetrachord appear only reports of the numbers 1, 2, 3, and 4 of Tetractys, the heptachord (five chords) displays only the ratios of the powers of the numbers of Tetractys, namely the first nine powers of two and the first six powers of the three, adding to the unit, that’s to say the numbers
1, 2, 4, 8, 16, 32, 64, 128, 256, 512
1, 3, 9, 27, 81, 243, 729
whose total sum is 2116 = 23 to the second.
Even in this way we obtain in the extension of the tetrachord the 5 and 7 because the two will appear at the seventh and not later and the three will appear at the fifth and no later. than. The natural scale differs from Pythagorean just because the length of the chords specified by the law of the fifth will replace the approximate values expressed by simple ratios and has:
C D E F G A B C (do re mi fa sol la si do)
In the Pythagorean scale the five “intervals” or tone between the G (do) and the D (re), between the D (re) and the E (mi), between the F (fa) and G (sol), between G (sol) and the A (la) and between the A (la) and the B (si) are exactly equal, and in the natural scale are quite equal; and in both scales, these intervals are greater than the two remaining intervals between the E (mi) and the F (fa) and between the B (si) and the C (do). To overcome this drawback the Pythagoreans inserted between the intervals the other five chords (which correspond to the piano black keys) so as to obtain twelve chords each of which differs from the previous of an interval substantially constant and equal to a semitone. In the tempered scale, introduced by Bach, these intervals are all absolutely equal and the lengths of the twelve chords form a geometric progression, but intervals are no longer expressed by simple ratios of numbers, that’s to say rational numbers, but irrational. In the case of stringed instruments, in which the length of the strings is determined by the player’s fingers and ear, Blaserna the physicist says that the great violin virtuosos tend to prefer the Pythagorean scale to the natural one, but is a bit ‘difficult to establish the correctness of this statement since only a very sensitive ear can tell the difference. Meanwhile, we note that the development of the fifth tetrachord has led us to number seven and, in connection, even to number twelve.
As a curiosity in the end we observe that, if at the seven points of the circle division, we write the names of the five planets known to the ancients and that of the Sun and the Moon in the order of their distance from the earth that is: Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn, and we proceed as we did for the chords of the range going from the first point (Sun) to fifth (Moon), and from this to the fifth (Mars) and so on, we obtain the days of the week in their order: Sunday (Sun-day), Monday, Tuesday, Wednesday, Thursday, Friday, Saturday (Satur-day). Removed the name of Sunday, which is Christian, and the Jewish Sabbath, these are the ancient pagan names of the days of the week still in use in almost every language and replaced with some exceptions, such as Russian and Portuguese. Starting the week with Sunday the fifth day is sacred to Jupiter and the sixth is the day of Venus, and we find the consecration of six to Aphrodite.
We observe, however, that the greek calendar did not know the week and that only some later Pythagoreans and some Christians may have resorted to these considerations or considerations equivalent to establish the consecration of the weekdays and the correspondence between the planets and the days of the week. We note finally that the week of our calendar is a conventional division, and that the planets are not seven so we can not establish the correspondence between the seven notes, the seven planets, the seven days of the week, etc. The only septenary which has a natural basis is the Pythagorean musical scale, and the distinction of the seven colors made by Newton, apparently by the analogy between optics and acoustics, is conventional because from an iris color we change to another through some thousand shades and not through a net gap like a musical note to another. A septenary law instead appears in the table of chemical elements of Mendelejeff.
We note in the end that the numbers three, five, and seven can also be obtained very simply by the Tetractys numbers, considering the ratio of 1: 2, 2: 3, 3: 4 which express the lengths of the last three strings of the tetrachord of Philolaus, and summing numerator and denominator. It is obtained in this way: 1 + 2 = 3, 2 + 3 = 5, 3 + 4 = 7.
In the Pythagorean literature, at least in that little which has come down to us, one does not find anything that confirms or excludes the path we’ve set out to reach the number five and number seven, starting from the tetrachord, although this path present similarities with the one held by Pythagoreans in order to divide the circumference into five and ten equal parts.
Previous Chapter Arturo Reghini Sacred Pythagorean Numbers 7.
Next Chapter Arturo Reghini Sacred Pythagorean Numbers 9.