In the Middle Ages this mood was the generally accepted and used mood. At the beginning of the 16th century, in addition to the octave and fifth, the major third was also purely intoned in chordal combinations, and the Pythagorean tuning was increasingly being replaced by the mid-tone tuning on keyboard instruments . Nowadays the Pythagorean tuning is used again in connection with the reproduction of mainly medieval music, but also in some cases with modern music.
Nothing is known about the practical application of the Pythagorean tuning in antiquity . According to the legend of Pythagoras in the forge , its music-theoretical description goes back to Pythagoras of Samos (around 570 to 510 BC).
In the early and high Middle Ages, people were often content with tuning only the nine notes B - F - C - G - D - A - E - B - F sharp in intervals of a fifth, with the notes B and F sharp mainly used for the time to avoid the tritone F - B felt as strongly dissonant by the pure fourths F sharp - B or F - Bb. With the expansion of the tone supply to twelve tones, the problem of the Pythagorean comma arose . If you tune the tones C sharp and G sharp as well as E flat and A flat to the already existing tones, G sharp and A flat do not result in the same tone (the G sharp would be approx. 23.5 cents higher). You have to choose G sharp or A flat. The resulting impure fifth between G sharp - E flat or C sharp - A flat ( Pythagorean wolf fifth ) is too small by the Pythagorean comma and in most cases musically useless. For the music of the Middle Ages, the position of the wolf fifth between G sharp - Es is the least problematic. For example B. the music from the Robertsbridge Codex (originated around 1320) the position of the wolf fifth in G sharp - Es ahead.
|Example of a perfect fifth a′-e ″ / pyth. Wolf fifth gis′-es ″|
A simple solution to the problem of the wolf fifth is obtained by reducing the fifths in the circle of fifths slightly by 1 ⁄ 12 of the Pythagorean comma (approx. 1.955 cents). One then obtains the equal tuning , which however - because of the discovery and preference for the pure third in polyphony - only prevailed centuries later via the detour of the medium-tone tunings and well-tempered tunings as the predominantly used tuning today.
For the first time the composers of the music of the Trecento (14th century) tried to establish the third as a consonant interval in Italy, but it was not until the second half of the 15th century, in the musical transition from the Middle Ages to the Renaissance , that a fundamental change in listening habits began , in which the third was perceived as a consonant and the fourth as dissonant. For this type of music, the Pythagorean tuning with its impure-sounding Pythagorean thirds (approx. 408 cents) was viewed as inadequate. Together with the wolf fifth, however, a tuning with perfect fifths also produces four almost perfect thirds (approx. 384 cents B-Eb, F-sharp-B, C sharp-F and G sharp-C). Therefore, a first remedy was to change the position of the wolf fifth. It has now been placed between B and F sharp (actually G flat), since this is how the good-sounding, almost pure thirds D - F sharp, E - G sharp, A - C sharp and B - D flat emerged. Actually, these are diminished fourths (D - Ges, E - As, A - Des and B - Es), which were specifically used in music practice (e.g. in the Buxheim organ book , written between 1460 and 1470). The position of the wolf fifth between H and F sharp z. B. by Bartolomé Ramos de Pareja in his Musica practica (Bologna 1482).
To create a diatonic scale on the root C one tunes in the following tones - spaced by perfect fifths:
Arranging these diatonic results in the following scale:
|Whole tone||Whole tone||Limma||Whole tone||Whole tone||Whole tone||Limma|
|9 : 8||9 : 8||256 : 243||9 : 8||9 : 8||9 : 8||256 : 243|
|203.9 cents||203.9 cents||90.2 cents||203.9 cents||203.9 cents||203.9 cents||90.2 cents|
This means that the fifths and fourths are pure, the thirds (frequency ratio 81 : 64 = approx. 407.8 cents), however, in comparison to the pure third (frequency ratio 5 : 4 = 80 : 64 = approx. 386.5 cents) from the overtone series results in the syntonic comma (81 : 80 = approx. 21.5 cents) being too large and therefore sounding sharper.
- Roland Eberlein : History of the organ tuning. I. The Pythagorean mood. In: Website of the Walcker Foundation. Retrieved July 8, 2020 .