Suomeksi | In English

Meantone temperament

Meantone temperament represents an attempt to avoid the problems of Pythagorean tuning or just intonation. In fact, several meantone temperament systems exist, but they all share the tendency to narrow the fifths in order to maintain the purity of the “common thirds”.

The system is called meantone temperament because the size of each whole tone is identical, 192 C – the mean between the “weak” (182 C or 10/9) and “strong” (203 C, or 9/8) whole tone. This also results in pure major thirds.

A major whole tone consists of two pure fifths: 702+702=204(+1200). If the whole tone is narrowed by half of the syntonic comma (to generate the mean), it follows that each fifth must be narrowed with a fourth of the comma. This is why the system is sometimes called one-quarter comma meantone tuning. This first meantone tuning is attributed to Pietro Aron (1523).

Meantone tuning results in nine keys (B flat, F, C, G, D, A, g, d and a) that sound relatively good, but the tempering also reduces the fifths to 696.6 cents. The impurity of the incomplete fifth, however, only becomes disturbing when it occurs as an open interval. The difference of the final fifth to be tempered to an equally tempered fifth is, nevertheless, 38 C (11 · 3.4 C). This oversize fifth (738 C) is usually placed between the E flat and the G sharp (enharmonic fourth).

Gottfried Silbermann (1683–1753) is known to have narrowed the fifths only by one-sixth of the comma, resulting in slightly imperfect thirds, but also reducing the size of the oversize fifth and slightly increasing the purity of the other fifths.

Silbermann's tuning and several other meantone tuning systems were used in pipe organs until the 19th century, and some meantone tunings are being utilised in keyboard instruments again today. The meantone tuning system is an example of a so-called regular temperament, where each fifth is narrowed by an equal amount.

In irregular temperaments some fifths are narrowed, but some are not. The temperament known as Werckmeister III (invented by Andreas Werckmeister, 1645–1706) features a considerable number of pure fifths, but only four Pythagorean thirds (F sharp – A sharp, D flat – F, E – G sharp and A flat – C); all the remaining thirds are at least equal-tempered. A temperament with pure natural major thirds (C–E, F–E, G–B and B flat – D) that still features pure fifths in H, F sharp and C sharp is attributed to J.-P. Rameau.

The notion of the so-called enharmonic comma, formed by stacking three major thirds (A flat – C – E – G sharp, for example) on top of each other, deserves a mention in connection with tuning systems. If each third was pure (386 C), the “octave” from A flat to G sharp would be 41 cents (a fifth of a whole tone!) short. In other words: the thirds cannot be stacked using only pure thirds. The problem of “balancing the major thirds” has been one of the key questions in creating usable temperaments. The much-used Vallott temperament has five purer, two equal and five inferior thirds than equally tempered tuning.