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.

We are 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 Pythagoras

We cannot deepen here such a rich history of music theory from antiquity to today.

Aristossenus of Taranto (a. 370 – a. 300 BC), for instance, structured the musical legacy in a famous grid – the Great Perfect System – Which allowed to form palettes known from the main cultural currents: Dorian, Aeolian, Phrygian, and Lydian.

The range is a series of notes in a certain way, arranged in the order of increasing or decreasing frequency. The ranges differ for the different distributions of the intervals. While the range is, in principle, limited (it extends from the tonic to tonic), scale (échelle) is theoretically unlimited: the two terms are often used interchangeably. So let’s return two centuries to meet the great Pythagoras (a. 572-493. BC). He had an ear so fine that, legend has it, he heard singing the heavens. The construction of its range is entirely based on a progression of fifths: the octave is only an accessory and reduction. To simplify, we start from is:

fa – do – sol – re – la – mi – si – fa sharp. etc.

do re mi fa sol la si do

9/8 9/8 256/243 9/8 9/8 9/8 256/243

There are two fundamental intervals:

9/8 = 1.125

256/243 = 1.0535

for intervals with the tonic (the first note), we have:

do re mi fa sol la si do

1/1 9/8 81/64 4/3 3/2 27/16 243/128 2/1

1 1.1250 1.2656 1.3333 1.500 1.6875 1.8984 2

For how they are designed, the sounds in this range belong all to the same harmonic series based on fa:

do re mi fa sol la si do

384/256 486/256 586/256 729/256

432/256 512/256 648/256 768/256

Please note, to clarify the theory of harmonics, that if one starts from a sound “1” frequency “F”, all the integers that follow can be considered as harmonic, that is, multiple of the frequency 1F, so: 2F, 3F, 4F, etc. Joseph Fourier (1768-1839) did us this lovely service. In this series, some intervals are fatally musical and allow scholars to investigate sound ranges scientifically. With a base fa, the other fa of the octave will then be 1F, 2F, 4F, 8F, 16F, 32F, 64F, 128F, 256F, 512F, etc.

Thus, it is understandable that, according to this reasoning, the Pythagoras tonic corresponds to the harmonics n. 384 of the base fa (384 = 256 x 1.5).

This relative fa, of course, brings the number 512: it is sufficient to divide by the value of the tonic to restore the initial values.

However, apart from some details, this range said cyclic, is not considered a natural range! That would not be like the philosopher, based on the famous Tetraktys, whose sum is 10, the perfect number. The intervals of fourth (4/2), fifth (3/2), and octave (2/1), give a preferred series, allowing Pythagoreans to define all other intervals. It continues at Atorène, Music Theory Course for Alchemists. Part 2 with the application of these preliminary concepts on the philosopher’s egg.

#### 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 among other things, 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 this is what violinists use; on the other hand, all the string instruments, set for fifths, proceed from the cyclic scale. Concerning how they are designed, wind instruments 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.

A quite exhaustive essay on Pythagorean mathematics applied to music has been written by Arturo Reghini in *Numeri Sacri nella Tradizione Pitagorica Massonica*, or sacred numbers in the Pythagorean masonic tradition, and published in this site as *The Pythagorean Acoustics Based on Tetraktys* and *The Pythagorean Acoustic: Geometry & Music of Sirens*.

The previous part at Atorène, Music Theory Course for Alchemists. Part 1.

Now we are ready to read and appreciate Atorène: Fire and Weights in Canseliet’s Last Cooking