Density as a wave. The conclusive part of the Atorène music course will make us grab that Canseliet omitted many details in his letters on the Last Cooking.

Reconstructing what was supposed to be the eighth sound that would have eventually crowned the alchemist’s work. We will get why the third work is called Ars Musicae, or the art of music. Anyway, Atorène doesn’t go so far as to answer the implied question of what 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 an expert 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. 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 employing three sounds that give them 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 a 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,* a 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. The authorship is due to Hermann von Helmholtz (1821 – 1894), well known for its resonators based on the analysis of Fourier. He said, “the human soul is particularly well-being in simple relationships because it can grab and embrace them more easily.”

Likewise, Pythagoras, the Zarlino harmonics 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 are no ratios between 4/3 and 5/3.

#### The Tempered Range

The octave was divided into 12 equal parts to simplify the problem greatly. We noted that the range of Pythagoras, or that of Zarlino, includes successively: 2-tone, halftone, 3-tone, and halftone; 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√2^{4 } ¹²√2 ¹²√2² ¹²√2^{5} ¹²√2^{7} ¹²√2^{9 }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, 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, include 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 into thirty-one parts. Moreover, examining the scores after this age, you can 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 flats (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 perfect unison between some instruments, harmonics appear as 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, this is what violinists use; 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 for fixed sounds, such as piano, organ, or 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) do not fit perfectly in the seven octaves (= 128); some gaps are filled, restarting from a twelfth of the difference on every fifth.

Each system has its advantages and its drawbacks. Each system, especially, conforms better to a group of instruments than another; the performer’s skill neutralizes the discordant effect. And, indeed, there is no need to linger longer on this subject.

#### The Rhythms of the Universe

Let’s return to our table of the Great, or Last, Cooking. To understand what we are talking about, get a glance at Brouaut’s Frontispiece, the Organ Pythagorean Proportions, and the article which originated the whole, Canseliet, the Art of Music & Weight.

the figures that you will find turn out to be very close to the range of Zarlino, but we find the same figures in the theories of the world:

– for semitones (1.066) we have 1.0670 and 1.0622;

– for minor tones (1.1111): 1.1144 and 1.1113;

– for major tones (1.125): 1.1217 and 1.1247; doubling the tonic weight to close the octave, we have also: 1.1292;

– for minor third (1.200): weight 5/weight 3 = 1.20097; w1 x 2/w6 = 1.19956;

– for major third (1.250): w6/w1 = 1.25015; w6/w4 = 1.24999;

– for fourth (1.333): w4/w1 = 1.33394; w6/w3 = 1.33657; w7/w4 = 1.32772; w1 x 2/w5 = 1.33306;

– for fifth (1.500: w5/w1 = 1.50030; w6/w2 = 1.48633; w1 x 2/w4 = 1. 49932;

– etc. You will get new third and other intervals by doubling the eighth of the seven weights. Admitting up to five centigrams of absolute measurement error, you get a relative error of 4 per 100,000 of each tone, up to 3 per 10,000 eighths.

Undoubtedly, the differences between the ratios of the cooking and the Western music intervals can not be due to inaccuracies in weighing. And that even if the differences derive from contingent influences, such as an atmospheric disturbance. In any event, the alteration remains extremely small.

If one tries to place the results in a harmonic series out of curiosity, it is necessary, even in this case, to start from a fifth below the tonic. Here is a very close series:

Weight 1 W2 W3 W4 W5 W6 W7 W8?

48/32 54/32 60/32 64/32 72/32 80/32 85/32 96/32;

series, which is derived from the following roundings:

48.0 53.842 60.005 72.014 80.025 85.013 –

rounding the group of third averages, fourth and fifth, mentioned above, and forming again the eighth, a for the accuracy is found the following values:

Weight 1 W2 W3 W4 W5 W6 W7 W8

164.7 185.3 205.9 219.6 247.05 274.05 292.8 329.4

As weight 1 as the basis – as already measured before cooking – one will thus come to the theoretical series in grams:

Weight 1 W2 W3 W4 W5 W6 W7 W8

164.7 185.3 205.9 219.6 247.05 274.5 292.8 329.4

One will get the same results with the harmonic scale, except the weight n. 7: 164.7 x 85/48 = 291.65. The equivalence with musical notes will then be:

Weight 1 W2 W3 W4 W5 W6 W7 W8

do re mi fa sol la si flat do

This range can be found all around the world: it is one of the ten Indian Thâta; or, among the Persian sects dastgâh, what is called rast-pandjgah; among the Arabs, between twelve main maquâmat, it bears the name of ouchaq.

#### Density Variations

When we have established comparisons, we were talking of weights: we have to distinguish between “measurements of weights” and “measurements of masses “, we focus on this point because if the mass increases, the volume does not vary.

In the chapter on the Transmutations ( Le Laboratoire Alchimique), we have seen that the transmutations are generally accompanied by a change in the mass of the order of ±20%.

Let’s call “V” the volume and “D” the density,” I “the index of the initial metal, and ” p ” that of the precious metal; according to Newtonian mechanics, one should expect the following relationship:

Vi Di = Vp Dp (mass conservation).

But, since the mass varies, we need to evaluate that VP – and this alone, at least as regards the gold – will be modified:

Vp = K V’ (where “K” is the coefficient of variation).

The variation thus will oscillate between the theoretical volume and the actual volume:

K = Vp / V’p

The association of the metal and the energy embodied in the Philosopher’s Stone seems to undergo a mysterious law of equilibrium. On the surface, everything happens as if the matter, in becoming royal, goes to achieve a certain “size”, like a vegetable.

In cooking, things go otherwise, and the phenomenon, always of relativistic order, appears even more strange if one can say so.

Remaining fixed on the volume observed, we say:

V x Di = V X Dn / In

being “i” the index of the first detection of weight; “n” that of detection of weight between 1 and 7; I is the interval of the degree “n”; “V” and “D” are, respectively, the volume and density.

It follows that:

In = Dn/ Di

In this case, it is the density that varies! It behaves like a wave.

#### The Degrees of Fire

The student will observe the time intervals in this area, even small variations, and for the first time that of 22 minutes on 24 hours, that’s to say in minutes: 1440/1418: he will rebuild the process without difficulty, even what concerns the last parameter which we will discuss, or rather talk about: the temperature. Most authors have shrouded it in mystery, but Philalethes is sometimes generous. It is necessary, he tells us in his Rules:

* “… the degree of heat which can be obtained from the lead (327 °) or by the tin in the merger (232 °) … So give start to your degree of heat, for the kingdom where nature left you* …”

The investigator who will not fail to read the works of Fulcanelli and Canseliet finds that the fourth degree indicates 340 for the second and 500 for the seventh.

It is only a question of scale, so the degrees of the Master of Savignies can be integrated into our system *degree-gram* by the relation:

T° G = T° Cans. x 2/3 * PA – 42.

His board of fire wheels can be inferred, even if you have a simple equation. For those who do not know how to count, here’s the development:

1° degree 310

2° degree 340

3° degree 370

4° degree 390

5° degree 435

6° degree 475

7 degree 500

8° degree 555

And now, child of the science, ora, lege, relege, labora et invenies ( pray, read, re-read, work and discover).

See also Brouaut’s Frontispiece, the Organ Pythagorean Proportions , Canseliet and the Art of Music and Weight , Hieronymus Bosch and the Concert in the Egg and Piero della Francesca and the Philosophical Pendent Egg , The Secret Night Chant of a Stradivarius Tree ;