Density as a wave. The conclusive part of Atorène music course will make us grab that Canseliet omitted many details in his letters on the Last Cooking.
To reconstruct what was supposed to be the eighth sound, that would have eventually crowned the alchemists work. We will get why the third work is called Ars Musicae, or the art of music. Anyway, Atorène doesn’t go so far to also answer the implied question of what actually makes the ultimate product of the last cooking to whistle according to a musical scale. And implicitly why there is music from the Egg.
Canseliet dared to publish his secret correspondence to a friend on the whistles emitted by his Egg. Atorène, in “Le Laboratoire Alchimique” 1981, analyzes the musical proportions of the Egg densities comparing them to a musical organ. If you are experts in music theory you can directly go to the chapter on The Rhythms of the Universe.
You can find the first part, with the explanation of the Philosophical Week, at Atorène, Music Theory Course for Alchemists. Part 1. The readers educated in music theory can scroll down the general course to the chapter “The Rhythms of the Universe”.
The Musical Range of Zarlino
Conversely, a range still taught, called “natural”, is that of Giuseppe Zarlino (1517-1590). It is built from three perfect major chords, by means of three sounds that give then the major third (5/4) and the minor third (6/5), thirds defined by the Spaniard Bartolomeo Ramos de Pareja (a. 1440 – a.1521. In his treaty “Musica Pratica”, he shook the foundations of the then current theory, proposing inter alia, the simplification of the proportions of the intervals of major third (4/5) and minor (5/6) and the establishment of a twelve tones scale), based on the fa:
fa – la – do
do – mi – sol
sol – si – re
or in line:
fa la do mi sol si re
5/4 6/5 5/4 6/5 5/4 6/5
Which can be brought with no difficulties in the same octave, and one can get so:
from do to re:
5/4 x 6/5 x 5/4 x 6/5 = 9/4
and dividing by 2 = 9/8
from do to mi: 5/4 etc.
do re mi fa sol la si do
1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1
1 1.1250 1.2500 1.3333 1.500 1.6666 1.8750 2
From here the following intervals:
do re mi fa sol la si do
9/8 10/9 16/15 9/8 10/9 9/8 16/15
So one gets three fundamental intervals:
9/8 = 1.1250
10/9 = 1.1111
16/15 = 1.0666
The relationship between a major tone (1.1250) and a minor tone (1.1111) is the comma, general term designating a small range; in this case 1.0125, or 81/80.
For theorists, the scale of Zarlino however, is not more natural than Pythagoras. In fact, according to the terminology in use, a range is called natural when all the intervals are “natural”, that is, when the frequencies are to them as a set of integers. We due the authorship to Hermann von Helmholtz (1821 – 1894), well known for its resonators based on the analysis of Fourier. He said “the human soul is a particularly well-being in simple relationships, because it can grab and embrace them more easily.”
Likewise Pythagoras, the Zarlino armonics series can not be built but starting from a fifth below the tonic (fa for do range).
do re mi fa sol la si do
24/16 27/16 30/16 32/16 36/16 40/16 45/16 48/16
One can easily see that, in the harmonic “natural” series of do, there is no ratios 4/3 and 5/3.
The Temperated Range
To greatly simplify the problem, the octave was divided into 12 equal parts. We noted that the range of Pythagoras, or that of Zarlino, includes successively: 2 tone, half tone, 3 tone, half tone, this amounts to roughly 12 semitones.
Well, mathematicians have told the musicians that now they possessed twelve equal semitones.
Each semitone is therefore ¹²√2 , that’s to say 1.0595;
or, for a tone¹²√2² = 1.1225 ; from which, after transposition in diatonic major:
do re mi fa sol la si do
1.1225 1.1225 1.0595 1.1225 1.1225 1.1225 1.0595
1 ¹²√2² ¹²√24 ¹²√2 ¹²√2² ¹²√25 ¹²√27 ¹²√29 ¹²√2¹¹ 2
1 1.1225 1.2599 1.3348 1.4983 1.6818 1.8878 2
To modulate, is impeccable; unfortunately, the mathematical equality of semitones is only an idea of the spirit. Apart from the eighth, all intervals are unclean and expressionless. It should be noted, moreover, to compensate for this defect, instinctively pianists writhe.
Is this range, called tempered, that some western theorists have managed to impose.
There is also, to qualify ranges, a word that everyone knows: “chromatic”. In fact, in the language of music, is so called each scale processing for semitones, whether or not of temperate stairs. Nothing to do, then, with the color range of the Greeks (kröma = color).
The Harmony Problems
The reader would think perhaps that variations, for example with reports of Zarlino, appear very minimal. But, precisely the superiority of music lies in the rightness of his intonation intervals.
So the Eastern ranges, which already existed before the Buddha, including twenty-two unequal intervals per octave: the smallness of the intervals is not important in itself, but it allows to solve the many problems of harmony.
In the West, Christiaan Huygens (1629 – 1695) proposed a division of the octave in thirty-one parts. Moreover, examining the scores after this age, you can just see that there are thirty-one sounds:
the 7 notes (do, re, mi, fa , sol, la, si – C, D, E, F,G , A, B)
the 7 sharp ( one for each note)
the 7 flat (one for each note)
the 5 double sharp ( on: do, re, fa, sol, la)
the 5 double flat ( on: re, mi, sol, la, si)
The precision is necessary, because the music invites us to discover the sensitive soul of the world. If there is no absolutely perfect unison between some instrument, harmonics appear common disruptor beats.
In the contemporary treaties is said that the range of Pythagoras is no longer in use in our days. But, in fact, is practically what violinists use; and, on the other hand, all the string instruments, set for fifths, proceed from the cyclic scale. Wind instruments, in relation to how they are designed, generally give the harmonic range – not to be confused with a harmonic series – and don’t exactly sound like Zarlino’s do and la.
The tempered scale is not used as instruments to fixed sounds, such as piano, organ, cembalo; that’s to say after the eighteenth century.
The problems are not lacking in music theory; for example, the mi or fa sharp do not have exactly the same value for a singer and a violinist. Or again: the twelve classic fifths (= 129.7463) does not fit perfectly in the seven octaves (= 128); some gaps filled , restarting from a twelfth of the difference on every fifth.
In fact, each system has its advantages and its drawbacks. Each system, especially, it conforms better to a group of instruments, rather than to another; the skill of the performer neutralizes the effect discordant. And, indeed, there is no need to linger longer on this subject.