Atorène’s course starts with the presentation of the Philosophical Week, which is the musical interval of the Last Cooking, then continues with the musical notation and Law of Attraction. Before getting to the Egg’s Densities Variations, and Degrees of Fire, the Canseliet’s apprentice provides the reader unfamiliar with the mathematical laws of the proportions of the sound with the construction of notes, intervals, and scales, and Pythagoras, as well as Zarlino’s, range.
The pages I have translated from “Le Laboratoire Alchimique”, 1981, are a real and comprehensive music theory course to provide solid foundations to figure out the variations of sounds from Canseliet’s Philosophical Egg during the famous seven whistles and the philosophical and mathematical implications of the resulting musical scale, through the ponderal (weight) accretions.
Readers unaware of music theory are recommended not to skip the theoretical parts to jump to the chapter on the Rhythms of the Universe, as the egg behaves just like a music resonator. And only by knowing the music theory might we understand what Canseliet did not say.
My translation from “Le Laboratoire Alchimique”, 1981:
Music Theory Course for Alchemists: The Week
Where do the days of the week come from?
From ancient Egypt, answers Dio Cassius Cocceianus (Nicaea in. 155 – in Bithynia. 230). He’s right, Egypt is the melting pot of the Week, and the passage of the witness up to our civilization was carried out mostly by Jews.
Monday is the day of the Moon, Lunae dies; Tuesday day of Mars, etc. Saturni dies is very deformed and was adopted by the Jews on the Sabbath (latin: sabbatu, greek: sabbaton, the Jewish sabbath, the day of rest, Spanish: sabado) and Sunday is the Lord’s day (latin: dominica, implicit diem: day of the Lord, Dominus). The English here is clear: for Saturday: Saturday, the day of Saturn; and on Sunday, the day of the Sun: Sunday; or, in German, sonntag (day: dies in latin, day in English; tag in German. Sole: solis, in latin; sun in English; sonne in German).
So we have the sequence of the celestial bodies Moon, Mars, Mercury, Jupiter, Venus, Saturn, and the Sun. A priori seems to be before a common disorder, as the progression gives heliocentric Mercury, Venus, Earth, Mars, Jupiter, Saturn, etc., while the classification using the ancient wisdom is: Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn.
A large number of investigators question this point. We will provide them with an explanation from the range of Pythagoras.
We have the planets in the order in which, according to the Tradition, they dispense their influences on the athanor and associate the sounds emitted by the philosophical Egg. To simplify, let’s call them according to the usual succession of notes in the solfeggio (without seeking a rigorous value).
Day of Moon 1 sound do C
Day of the Mercury 2 sound re D
Day of Venus 3 sounds mi E
Day of the Sun 4 sounds fa F
Day of Mars 5 sounds sol G
Day of Jupiter 6 sounds la A
Day 7 Saturn 7 sound si B
Let’s continue now, as the master of Samos, the great Pythagoras, by successive fifths, with a reduction of the octave. We get, starting from do:
do x 3/2 = sol
sol x 3/2 = re
re x 3/2 = la
la x 3/2 = mi
mi x 3/2 = si
si x 3/2 = fa ( being strict, fa sharp)
And ordering the planets that govern the area in relation to the succession of fifths, we immediately obtain the exact order of the days of the week. They are, in some way, the “chorded days” of the Hebdomas Hebdomadun, the week of the weeks:
do = Lunae dies = Monday
sol = Martis dies = Tuesday
re = Mercurii dies = Wednesday
la = Jovis dies = Thursday
mi = Friday dies = Veneris
si = Saturni dies = Saturday
fa = Solis dies = Sunday
We have seen that the seven notes indicated by Canseliet in his letters do not match what we have read above. So let’s go on to find out.
The Musical Notation and the Law of Attraction
Even if not making a complete history of musical notation, we say that for a specified period, following the principle inherited from the Greeks, some letters were used to describe sounds:
A B C D E F G.
The transition from the Greek to Roman letters, also currently used in Anglo-Saxon influence countries, is attributed to Boetius (480-524 AD). In his fundamental work De Institutione Musicae (500-507 AD), Boetius defines the elements of the music according to the Pythagorean tradition and adopts Alipius notation, replacing the Greek letters with the first 15 of the Latin alphabet.
Due to a Benedictine monk, Guido d’Arezzo (a. 990-1050 AD), the current name of the notes; made use of the first syllables of each verse of the St. John hymn to facilitate the study to his students when he taught at the Abbey of Pomposa.
This song, very well known at the time, and it seems foolproof against hoarseness, was written around 770 by Paul Deacon:
Ut queant laxis
To the faithful to sing the wonders of your business loudly, clears the error of the unworthy lip, oh St. John.
The name Ut became do only in the seventeenth century. It appeared only in the XVI, some 500 years after Guido d’Arezzo. The good fathers feared this note, whose relationship with fa generates a feeling of lasciviousness. Was not in the Middle Ages, the interval FB called diabolus in musica? The tritone/three tones (the si united to fa) is an abnormal fifth interval. The name of the debauched who invented it is ignored, the initials of Sancte Ioannes: maybe Anselm of Flanders, or François Lemaire.
In his method, the learned theorist Guido d’Arezzo adds syllables (ut, re, etc) to the traditional letters only to clarify different aspects of the range, for example, G. sol. re. ut. Likewise, mi-fa designated all semitones, hence the solfeggio range Ut:
C D E F G A B C
ut re mi fa sol re mi fa
But, before, they used two species of B, differentiated by St. Odo of Cluny (878-942) in two written representations. One, the A quadratum (angular, square), corresponds to our si natural; the other, the B rotundum (rounded, soft), corresponds to our si flat:
Si bemol (flat) = Si B molle (rounded, soft, flat)
The range of ut above is therefore sung with B quadratum. If B is flat, always solfeggio from C to C, one writes, moving semitones:
C D E F G A Bflat C
ut re mi fa re mi fa sol
We see that the notation is cumbersome: only much later was the method simplified:
C D E F G A Bflat C
do re mi fa sol la si flat do
This series is one of the eight modal ranges of cantus planus, as was called the real proper liturgic singing as opposed to the figurative and mensurato singing. The cantus planus has severe nature. The eight Gregorian chants are (next to the more proper name, there are put in brackets the traditional Greek names still in use): 1 Protus Authenticus (Doric); 2 Protus Plagalis (ipodorico); 3 Deuterus Authenticus (Phrygian) 4 Deuterus Plagalis (ipofrigio); Tritus Authenticus (Lydian); 6 Tritus Plagalis (Hypolydian); 7 Tetrardus Authenticus (Mixolydian); 8 Tetrardus Plagalis (Ipomixolidian).
This series in the late tenth century, to a regrettable mistake, was wanted to play with the Greek names of the great perfect system (“GPS”), after being Latinized. So our range, the Tritus Plagalis (which amounts to flat number three in the severe form), was misnamed Ipolidian. At the same time, its equivalent in (“GPS”) should be instead the Lydian (Tritus Authenticus).
On the other hand, we must consider that the Greeks of the fourth century b.C believed the Ionians and other ancestors as barbarians; their music, moreover, no longer possessed the exotic feature that at the time justified the severity of Plato (Republic, Book III):
… the only harmonies we need to preserve are the Dorian and Phrygian
… the Ionian is made for drunkards
… the Lydian is dangerous for women, whose duty requires proper behavior, and so even more for men …
Despite all the cultural influences, it remains a similar use in the Tritus Plagalis of the Middle Ages clergy. Good fathers musicians had to invent endless subtleties to escape the lure of pleasure.
In this strange range lies the Law of Attraction. Agamemnon was not ignorant of that. Before leaving for the siege of Troy had very recommended to its musicians not to play that in Doric or Phrygian. During his absence, Aegisthus hired them to play in Lydian, which was how Clytemnestra took his lover.
The simple sounds, which don’t exist in nature, are caused by a sinusoidal vibration: they can appear dull and monotonous. The superposition of many sinusoidal vibrations forms complex sounds and suddenly takes life and develops depth.
As every sinusoidal vibration is multiple or submultiple of another, it comes with arbitrarily defined fundamental frequency or fundamental sound. The other frequencies will then its harmonics which Boetius (1), in Middle Ages, did compare to subsequent circles originated by a stone thrown into the water.
Without mentioning the lasting, one’s ear detects the musical sound through three qualities:
– Intensity. The sound is barely audible (the amplitude of vibration is weak) or, on the contrary, so strong that it is necessary to close the ears ( large amplitude vibrations ).
–Pitch. The sound is sharp (high frequency) or grave ( low frequency. By convention was fixed in 1959, the “la” at 440 hertz. So to understand, we add that the human voice has an extension, approximately to the set of different textures (2), ranging from 80 to 1200 hertz.
–Timbre. A violin’s la (A) cannot be confused with the piano la (3). Timbre is caused by the harmonic frequencies of a fundamental sound; multiples and their fundamental sound, each with their amplitude, are produced by the same instrument simultaneously.
We would need to add the transitory ( transitory phenomenon). Timbre characterizes the steady state of a sound, but between the start and steady state, the harmonics vary continuously. The transitory is equally produced at every modification of sound; in the same way, music and speech hardly present fixed sounds. Equally, they are mostly composed of transitory, which precisely determines the real timbre.
The musical effect of two simultaneous sounds depends on the ratio of their frequencies, not on the absolute value of the frequencies: is this ratio to be called interval?
Manlius Severinus Boetius ( 480 around 526 A.D ) philosopher and lettered in the musical field. His fundamental work is De institutione Musicae, which became the main source for medieval theorists. The musical elements are defined according to the Pythagorean theory;
Texture; register of the human voice in the most favorable field of singing;
The “la” is the central la of the piano keyboard;
One can measure the sound pipes’ lengths ( and canes) and vibrating strings to determine intervals. The ancient didn’t talk of frequency but length. For a sound frequency to appear inversely proportional to the length of a canna or string, it is enough to pass from one to another to reverse the ratios.
Conversely, in one attaches various masses to the end of equal-length strings, the frequency is, this time, proportional to the square root of the mass.
And if the strings are of different materials, the frequency is inversely proportional to the density square. The same law applies to the gases insufflated in the musical pipes: if one increases the temperature, the gas becomes less dense, and the sound becomes sharper.
1) if a sound pipe emits a do (C) with gas at 300 °, at 450 ° C – taking a thermal expansion coefficient of 1/273 – we will have
2) if a string of density 16.5 emits a do, increasing the density to 20.6 we will have:
do x √² 16.5/20.6 = do x 1/1.117 = si flat,
( Octave down. Applying this law to cooking, the third sound would be lowered, just in theory, of a minor key).
It is time to define the intervals, which were empirically selected by the ear. It was the ear to have headed and set the music rules. Why do our senses detect the laws of harmony? This is a good mystery.
The octave interval 2/1
The frequency of a sound is double of the other b.
a = 2b
The fifth interval 3/2 = 1.50
It forms the arithmetic average of the octave:
The fourth interval 4/3 = 1.333
It forms the harmonic average of the octave:
The major third interval 5/4 = 1.250
The minor third interval 6/5 = 1.200
We will not enumerate them all, but it’s already possible to understand why the musicians use only certain sounds – notes -, chosen to present harmonious intervals between them.
Anyway, we note the beautiful numeric association:
2/1, 3/2, 4/3, 5/4, 6/5.
Theorists have tried to make a simple musical range compatible with the Ear’s needs. Building a musical scale means ordering notes within an octave.
To be continued at Atorène, Music Theory Course for Alchemists. Part 2