Atorène, Music Theory Course for Alchemists. Part 1

By Iulia Millesima

The transition from the Greek to Roman letters, which are 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.

Is due to a Benedictine monk, Guido d’Arezzo (a. 990-1050 AD), the current name of the notes; he 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
Resonare fibris
Miira gestorum
Famuli tuorum
Solve polluti
labii reatum
Sancte Ioannes

To the faithful to sing loudly the wonders of your business, 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 fact is that 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? is called the tritone/three tones (the si united to fa),abnormal fifth interval. It is ignored the name of the debauched who invented it, 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), corresponding to our si flat:

Si bemolle (flat) = Si B molle (rounded, soft, flat)

The range of ut above is therefore sung with B quadratum. If B is flat, always solfeggiando 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 the method was simplified:

C D E F G A Bflat C

do re mi fa sol la si bemolle 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, which is the Tritus Plagalis (what amounts to as number three in the severe form), was misnamed Ipolidian, while 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 a proper behavior, and so even more for men …..

Despite all the cultural influences, it nevertheless remains a similar use in the Tritus Plagalis of the Middle Ages clergy. To escape the lure of pleasure, good fathers musicians had to invent endless subtleties.

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, and that was how Clytemnestra took his lover.

Sounds Features

The simple sounds, which don’t exist in nature, are caused by a sinusoidal vibration: they can appear as dull and monotonous. The complex sounds are formed by the superposition of many sinusoidal vibrations, and suddenly take life and develop depth.

As every sinusoidal vibration is multiple or submultiple of another, comes arbitrarily defined fundamental frequency, or fundamental sound. The other frequencies will then its harmonics which Boetius (1), in Middle Ages, did compared to subsequent circles originated by a stone thrown into the water.

Without mentioning the lasting, one’s ear detects the musical sound by 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. The la (A) of a violin 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 of them with their own amplitude, are produced by the same instrument in the same time.

We would need to add the transitory ( transitory phenomenon). Actually timbre characterizes the steady state of a sound; but between the start and steady state, the harmonics vary continuously. The transitory are equally produced at every modification of sound, in the same way music and speech very hardly present fix sounds, equally they are mostly composed of transitory, which precisely determinate the real timbre.

The musical effect of two simultaneous sounds depends on the ratio of their frequencies, not by 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 becomes the main source for the medieval theorists. The musical elements are defined according to the Pythagorean theory;

Texture; register of the human voice in the most favorable filed of singing;

The “la” is the central la of the piano keyboard;

Intervals

To determine intervals one can measure the sound pipes lengths ( and canes) and vibrating strings. The ancient din’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 – so 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 material, the frequency is inversely proportional to the density square. The same law is applied to the gases insufflated the musical pipes: if one increases the temperature, the gas is less dense and the sound Becomes sharper.

For instance,

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 the 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 our senses do detect the laws of harmony? This is a good mystery.

The octave interval 2/1
The frequency of a sound is a 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 so as 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 of notes within an octave.

The Musical Range of Pythagoras

We cannot deepen here such a rich history of the musical theory from antiquity to our days.

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, 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 distribution 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.

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