From Pythagoras’ scale to Bach’s well-tempered clavier

( Credit: the text below is translated from )

Music has been played since time immemorial. But it was in the 6th century BC. AD that Pythagoras had the idea of applying mathematics to Western classical music theory. To find out more, come and listen behind the scenes of the music.

Pythagorean music theory

From ancient times, liberal arts education (*i) defined mathematics as consisting of four disciplines: arithmetic, geometry, astronomy, and music. This mathematical conception of music has its source in the works of Pythagoras. He used a rudimentary instrument, the monochord, to highlight the fundamental properties that govern music. When the string of this instrument is played open, a sound of a definite pitch is produced. If the bridge is placed so that the length of the string is divided by half, the note produced will be the same as before but an octave (*ii) higher. By placing the bridge at a third of the length of the string, and by playing the shortest part of the string, we then fall on the fifth (*iii) higher than the octave of the initial note, that is to say whose the frequency is 3 times higher. These few examples led Pythagoras to find that the frequency of sound was inversely proportional to the length of the string (see fig. 1 and the box).

Illustration of a monochord. There are actually two strings. The first was used to produce a reference sound, while the second was equipped with a bridge to shorten its length. (Illustration: Athanasius Kircher/Wikimedia Commons, public domain)

The fifth is therefore an interval obtained by multiplying by 3 the frequency of the note produced by the string played open, but also by 1.5 because, as we have seen, multiplying or dividing a frequency by 2 amounts to producing the same note at different octaves. With the starting note and its fifth, we now have two different notes that “sound good” together. To build new notes, all you have to do is look for the next fifth from the note you just found. These two notes will also sound harmonious. Pythagoras, again, defines the starting note as being an F. Its fifth is do, whose frequency is 1.5 times that of fa. The fifth of C is G, and so on we find D, A, E and B. There are 7 notes, but the choice of this number is arbitrary, probably due to the symbolism of this figure, dear to the Pythagoreans. This process is called cycle of fifths and stops after 12 fifths (we add sharps # to the name of the notes already mentioned to define the notes from the eighth to the twelfth). We talk about a cycle because it happens that the fifth following the twelfth note (A#) results in a frequency ratio very close to that of the note from which we started the cycle (fa). Pythagoras considered, for lack of anything better, that the cycle was closed.

The approximation inherent in the cycle of fifths is the stumbling block of any musical theory. It is found in the construction of notes in the interval of an octave. Indeed, the ratio of 1.5 (3/2), raised to the power of 12, divided by a power of 2 to stay in the same octave, does not lead to an interval being exactly one octave (“Cycle des quintes” = circle of fifths. “quinte” = a fifth, “départ” = start) :

Illustration: Construction of notes following the cycle of fifths. We divide by 2, when necessary, to obtain values between 1 and 2 and thus remain in the same octave. The red C corresponds to the upper octave, the “starting” F is in the lower octave. (Illustration: Editorial

The interval separating the 12th note (A#) from the first (F) is not a perfect fifth, which makes it “ring out of tune”. It is called the wolf quint, for the impression of howling produced by this interval (see fig. 2).

modern music

From the Renaissance, this way of constructing music became truly problematic following the advent of polyphony, this technique consisting in playing several notes simultaneously in order to form harmonic intervals. The perfect major chord, which is the basis of harmony, superimposes a tonic (*iv), a major third (*v) (ratio to the tonic: 5/4 = 1.25) and a fifth (ratio to the tonic: 3/2 = 1.5). For example, the C major chord is made up of C, E, and G. The third is an interval which, like the fifth, sounds pleasant to the ear. However, the third does not appear in the cycle of fifths – or rather, the third is approximated by the succession of fifths as is the octave, in an imperfect way:

Translation: 1 Third (5/4 ratio) is approximately equal to 4 Fifths (3/2 ratio) => 5 is approximated as (3/2)4 (=5.0625)

This does not mean that the third is a larger interval than the fifth, but that the succession of 4 fifths (the gap between fa and la for example) is close to a third.

The impossibility of obtaining correct thirds with the Pythagorean scale had the consequence of relegating the third to the rank of “impure” intervals throughout the Middle Ages (see the video). To overcome this problem, a compromise was made by creating mid-tonic temperaments. This device consists of slightly reducing the 3/2 ratio of the natural fifth in order to fall exactly on a major third after 4 fifths. Only, by tuning the instruments in this way, the fifth of the wolf becomes even more pronounced. Moreover, the intervals between the different notes not being equal, one cannot transpose from one scale to another – for example shifting the C-E-G melody by a semitone towards the bass would give if- d#-fa# – without “twisting” the melody.

In the current musical system, another compromise has been chosen: equal temperament. It consists of dividing the octave into twelve equal intervals, called chromatic semitones. In this way, all transpositions are possible without altering the melody. But on the other side, the intervals are no longer pure – as in Pythagorean tuning, or to a lesser extent with mid-tone temperament. Each compromise has its advantages and disadvantages.

*Definition of specific words:

(i) Liberal arts: disciplines constituting a large part of the teaching of Latin letters and sciences from Antiquity to the Middle Ages.
(ii) Octave: interval separating two notes with the same name (eg do-do).
(iii) Fifth: interval between two notes separated by five degrees (eg C-G).
(iv) Tonic: base note on which a chord is built.
(v) Major third: interval of two tones (eg do-mi).

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