Pythagorean tuning

The Pythagorean tuning, also called quint just intonation, a tuning system, which is characterized in that the spacing of tones to each other (intervals ) are defined by a sequence of perfect fifths.

In the Middle Ages this was the general mood and atmosphere used. Beginning of the 16th century next octave and fifth, the major third were purely intoned in chord progressions and replaced the Pythagorean more and more by the meantone with keyboard instruments. At the present time the Pythagorean is reused in the context of reproduction, especially medieval music, but also in some cases in modern music.

The practical application of the Pythagorean in antiquity is not known. According to the legend of Pythagoras in the blacksmith whose music-theoretical description goes back to Pythagoras of Samos (c. 570-510 BC).

In the early and high Middle Ages, often contented himself with only the sounds of B - F - C - G - D - A - E - H - Fis to vote in perfect fifth intervals, the tones B and Fis were necessary mainly to the time when highly dissonant tritone perceived F - H by perfect fourths F # - B to bypass - H or F. With the expansion of Tonvorrats to 12 tones, the problem arises of the Pythagorean comma. One agrees with the pre-existing tones, the tones sharp and G sharp, and It and As in pure fifths, so give Gis and As not the same tone. You have to decide for Gis or As. Elaboration of the resulting impure fifth between G # - Eb or C # - As ( Pythagorean wolf fifth ) is about the Pythagorean comma too small and in most cases musically useless. For the music of the Middle Ages the position of the wolf fifth is between Gis - The least problematic. So, for example, is the music from the Robertsbridge Codex ( built around 1320) the position of the wolf fifth in Gis - It ahead.

A simple solution to the problem of the wolf fifth is obtained if one shrinks the fifths in the circle of fifths slightly by 1/ 12 of the Pythagorean comma. One then obtains the equal temperament, but - through continued due to the discovery and preference for pure third in the mood polyphony - only centuries later mainly used on the detour on the mean moods and well-tempered tunings than today.

For the first time tried the composer of the music of the Trecento (14th century) in Italy to establish the third as a consonant interval, but only in the second half of the 15th century, the musical transition from the Middle Ages to the Renaissance, began a fundamental shift in the listening habits where the third and the fourth to be consonant it were perceived as dissonant. For this kind of music Pythagorean was considered inadequate. A first workaround was to change the position of the wolf fifth. It was now placed between H and F sharp, since in this way good sounding, almost pure thirds ( D - F #, E - G #, A - C sharp and H - Dis) emerged. Actually, it involves diminished fourths ( D - Ges E - As, A - Des and H - Es), which were targeted in the music practice (eg in Buxheimer Organ Book, created 1460-1470 ). Mention is the location of the wolf fifth between H and F # for example by Ramis de Pareja in his Musica practica (Bologna 1482 ).

Example

To produce a diatonic scale on the root C is true - at a distance of perfect fifths - the following tones:

One arranges these diatonic to, leads to the following scale:

Thus, although the thirds are the fifths and fourths pure, but in comparison to the pure third (5: 4 = 80: 64), which results from the overtone series, too large and thus sharper sounding.

Interval table: See table on the Pythagorean scale.