In musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system.
In just intonation, every interval between two pitches corresponds to a whole number ratio between their frequencies. Such just intervals have a stability, or purity to their sound. If one of those pitches is adjusted slightly, that stability decreases, and slow changes in the timbre of the interval's sound begin to appear—an effect known as beating. As the adjustment becomes more severe, the beating becomes faster. To intentionally choose an interval with beating as a substitute for a just interval is the act of tempering that interval. These adjustments can make different musical possibilities available to the musician that would be impractical in just intonation. The actual measure of these adjustments are usually called commas.
As early as 1496, church organists in Northern Italy engaged in the practice of pitch modification (tempering) by adjusting the lengths of organ pipes to accommodate certain intervals and pitch relations (especially the use of thirds) thus allowing for "different musical possibilities." As musicians sought more sophisticated and varied modes of expression the practice of temperament became a practical compromise. The practice of pitch modification in turn allowed for such compositional devices as modulation or intervallic variation to occur within changing melodic and harmonic contexts.
The implementation of temperament as it pertains to the evolution of tonality is a classic example of what Unification Principle refers to as Ih Bup, (reason-law) whereby the efficacy of acoustic principles (law) are preserved while allowing for greater expression (reason). The importance of this acoustic adaptation (choice) was to allow for music to explore and express a wider range of emotions. As tonality emerged as the prevalent syntax of Western music, this "key-centered" music exhibited new and highly evocative expressive dimensions.
The evolution toward the major-minor aspects of tonal music, of which temperament played a significant role, gave rise to the concept of harmonic polarity in which the complimentary opposite modes (major/minor) could be harmonized with the intent of creating greater expressive possibilities.
Before Meantone temperament became widely used in the Renaissance, the most commonly used tuning system was Pythagorean tuning. Pythagorean tuning was a system of just intonation which tuned every note in a scale from a progression of pure perfect fifths. This was quite suitable for much of the harmonic practice until then (See: Quartal harmony), but in the Renaissance, musicians wished to make much more use of Tertian harmony. The major third of Pythagorean tuning differed from a just major third by an amount known as Syntonic comma, which was deemed as being mathematically impure and thus to be avoided.
Their solution, laid out by Pietro Aron in the early sixteenth century, was to temper the interval of a perfect fifth slightly narrower than in just intonation, and then proceed much like Pythagorean tuning, but using this tempered fifth instead of the just one. With the correct amount of tempering, the Syntonic comma is removed from its major thirds, making them just. This compromise, however, leaves all fifths in this tuning system with a slight beating. However, because a sequence of four fifths makes up one third, this beating effect on the fifths is only one quarter as strong as the beating effect on the thirds of Pythagorean tuning, which is why it was considered a very acceptable compromise by Renaissance musicians.
Pythagorean tuning also had a second problem, which Meantone temperament does not solve, which is the problem of modulation (see below), which is restricted because both have a broken circle of fifths. A series of 12 just fifths as in Pythagorean tuning does not return to the original pitch, but rather differs by a Pythagorean comma, which makes that tonal area of the system more or less unusable. In meantone temperament, this effect is even more pronounced (the fifth over the break in the circle is known as the Wolf interval, as its intense beating was likened to a "howling"). 53 equal temperament provides a solution for the Pythagorean tuning, and 31 equal temperament for the Meantone.
Just intonation has the problem that it cannot modulate to a different key (a very common means of expression throughout the Common practice period of music) without discarding many of the tones used in the previous key, thus for every key the musician wishes to modulate to, his instrument must provide a few more strings, frets, or holes for him to use. When building an instrument, this can be very impractical.
Well temperament is the name given to a variety of different systems of temperament that were employed to solve this problem. 12 tone equal temperament (12-TET) is the modern standard version of it, and it can be seen as another modification of Pythagorean tuning. Unlike Meantone temperament, which alters the fifth to temper out the Syntonic comma, 12-TET tempers out the Pythagorean comma, thus creating a cycle of fifths that repeats itself exactly after 12 steps. This allowed the intervals of Tertian harmony, thirds and fifths, to be fairly close to their just counterpoints (the fifths almost imperceptibly beating, the thirds a little milder than the Syntonic beating of Pythagorean tuning), while permitting the freedom to modulate to any key and by various means (e.g. common-tone and enharmonic modulation, see modulation). This freedom of modulation also allowed substantial use of more distant harmonic relationships, such as the Neapolitan chord, which became very important to Romantic composers in the nineteenth century.
French composer and organist Jean-Phillipe Rameau (1683-1764) published his Traité de l'harmonie in 1722 and this theoretical discourse remains one of the most important documents on the subject of tonality. Unlike theoreticians before him, Rameau looked to science, specifically the overtone series, as a way to explain the nature of musical phenomena in relation to the theoretical properties of tonality. Influenced by the theories of Descartes and Sauveur, Rameau posited that there was a fundamental relationship between the harmonic principles in tonal music and the physics of sound (acoustics.) His theories were to influence musical thought for centuries and he became known as "the Newton of music."
He asserted that chords (triads) where the primary elements in music as opposed to melody or themes. His ideas regarding functional harmony, specifically the cadential relationship between the tonic, sub-dominant and dominant chords within a particular key center, became the underlying principles of what would become known as “the common practice” in musical composition in the Western music for three hundred years. The cadential relationship between tonic and dominant triads (as well as secondary dominants) is elemental to the tonal syntax.
Rameau's theories could not have been postulated had the practice of pitch modification been implimented since thirds had heretofore been avoided by composers. The evolution of music towards the use of Tertian harmony was a significant factor in establishing tonality.
Johann Sebastian Bach’s (1685-1750) seminal composition, The Well-Tempered Clavier, which was composed in the same year that Rameau published his Traité de l'harmoni, is the composition in which it could be said that the full establishment of tonal principles were initially manifested. In that composition Bach composed a set of works in all major and minor keys thereby exhibiting the veracity of tonality both theoretically and aesthetically. It should be noted that Equal Temperament did not become a fully accepted method of tuning until after World War I. Bach's tuning/temperament in 1722 was not the tuning that eventually came to be used in Equal Temperament in the early part of the twentieth century.
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