Vanya Dolav
History of Theory, December 2018
The Great Equal Temperament Debate
1) Tuning and Temperament – Summary
In the chapter on “Tuning and Temperament” in the Cambridge History of Western Music Theory, Rudolf Rasch discusses the different tuning systems and temperaments, their development against the background of musical history and the principal musical theorists who promoted each of the systems.
A tuning system is a system of pitching using only just intervals as opposed to a temperament, which slightly alters (or ‘tempers’) just intervals making them wider or narrower. In mathematical terms all intervals of a tuning system can be expressed in rational numbers, while some or all of the intervals of temperament cannot be expressed in rational numbers.
The major tuning systems are the Pythagorean system and just intonation. The major temperaments as discussed by Rasch are the meantone temperament, concentric tuning and equal temperament.
The Pythagorean system is based on the octave (2:1) and the pure fifth (3:2). The fifths are just and the major thirds are formed as the sum of four fifths minus two octaves. In this system the circle of fifths cannot be closed; if eleven fifths are just, the twelfth one is too small (by a ditonic or Pythagorean comma).
The system of just intonation is based on the octave (2:1), the pure fifth (3:2) and the pure major third (5:4). The just fifths and just thirds are in one scale.
The meantone temperament uses the pure major third (5:4), by diminishing the fifths by one quarter of a syntonic comma. The whole tone is half of the pure major third, being the mean (average) of the two tones, which explains the name of the system.
In concentric tuning, the system is concentrated on the central fifths. A number of fifths are left in their just form; others are tempered by certain amounts. The total tempering sums up to the diatonic comma.
In equal temperament the octave is equally divided into twelve notes and thus the comma is divided equally among all (twelve) fifths, mathematically each the 12th root of 2.
Certain advances in science, in particular mathematics, helped the study of tuning and temperament. During the sixteenth century mathematicians developed root extraction methods and these methods were used to calculate string lengths for notes to divide an interval in equal parts. In this way tempered intervals could be calculated. Later measurement of intervals took place in cents as the unit of intervals, applying logarithms using logarithm tables. Christian Huygens (1629-1695) devised tables that could be used in musical calculations and he wrote extensively about the algebraic expression of meantone temperament.
Other people whose ideas have led the way in promoting a given tuning system are the following: Heinrich Glarean (1488-1563) promoted the Pythagorean system. In his book Dodecachordon (1547) he used octaves with string-length ratios 1:2 and fifths with string-length ratios 2:3, adhering to Pythagorean principles of interval theory. Francisco Salinas (1513-1590) promoted just intonation, discussing scales in which as many just intervals are to be realized as possible. Gioseffo Zarlino (1517-1590) wrote extensively about the system of just intonation and worked out a meantone temperament. Andreas Werckmeister (1645-1705) developed ideas about concentric tuning. Simon Stevin (1548-1620) and Friedrich Marpurg (1718-1795) were very influential in promoting equal temperament.
The Pythagorean system was prominent in the Middle Ages because the fifths were the predominant consonant intervals in medieval music and the thirds were of secondary importance so that their poor tuning could be accepted. During the Renaissance, the system of just intonation was appealing because of their mix of ‘natural’ perfect and imperfect consonances. From the 16th to the 18th centuries the meantone system became popular and was widely used for the tuning of keyboards. As of the eighteenth century equal temperament became widespread. From the nineteenth century the high sharps and low flats of the Pythagorean system became the underlying principle in melodic intonation, since it strengthens the leading note and stresses the major-minor opposition in nineteenth century harmony. Since the late nineteenth century equal temperament is the most commonly used system in Western classical music.
Developments in musical practice led to development of tuning systems. For example in the meantone system the single use of the raised keys (either as sharp or as flat) limited possibilities. The free use of all 24 major or minor keys became possible in equal temperament.
2) Views of Stuart Isacoff and Ross Duffin in the Great Equal Temperament Debate and my view
Stuart Isacoff and Ross Duffin have diametrically-opposed views towards the history of equal temperament: Isacoff believes equal temperament is the natural and inevitable consequence of the evolution and progress of tuning, while Duffin does not favor it over other systems and is very critical of equal temperament.
My own point of view is that the music itself can guide the players what intonation to use. For example in a string quartet Pythagorean intervals or just intervals can be used depending on the harmony.
The principal advantage of equal temperament appears to be that it can be used in every key with identical musical effect. Therefore, equal temperament is a good starting point, because all notes appear in a simple structure and it is possible to perform the intervals, chords and tonalities without a particular coloring.
Isacoff compares temperament to the pruning of a tree to balance the asymmetry of nature. Even tunings did not resolve the problem of the ‘wolf’, the dissonant fifth (usually G♯ – E♭), while temperament does. However, in equal temperament the uniform intervals can lack expression. Another approach is that the leading note determines the tuning.
Over time Pythagoras’ formulas that were suitable for medieval music did not meet the demands of harmonies with major thirds, minor thirds and sixths, reason why Zalino devised a new system. His contemporary Galileo Galilei (1654-1642) felt that music must be freed entirely from the tyranny of the pure ratios that were seen at the time as the natural and divine numbers. He argued that just intonation with its pure fifths and pure thirds may be an ideal, but in practice it was a fantasy. The admonition of Giordano Bruno (1548-1600) to poets “Throw off the yoke of authority; there are no rules other than the ones you make” allows for the freedom that composers and musicians also needed.
Rasch showed that each period in the history of music has had its own theory of tuning in order to meet its own musical needs. From this perspective different tuning systems or temperament can be applied depending on the musical piece and the time in which it was composed. Historical performers argue for the use of historical temperaments over equal temperament, but it can be difficult to determine which one to use because there are many historical systems and different theories which ones were used at times.
The discussion of tuning and temperament is mostly centered on the tuning of keyboard instruments. This shows that the choice also depends on the instrument. Non-keyboard performers already were not compromising the position of the sharps and flats, as a violin board that Peter Prelleur made in 1731 shows. Keyboard temperament is a compromise to enable the keyboard player to play in different keys, but some intervals do not sound good. Non-keyboard players are not limited to twelve unchangeable notes to the octave. They can continually adjust while they play, altering the intervals.
From this perspective, temperament is a too rigid concept to apply to players of stringed instruments and singers who do not have instruments with immovable pitch (like the organ and harpsichord). However, from a theoretical perspective tuning and temperament followed the development of music and therefore are very important.
I think it is important to be flexible; different tuning systems and temperaments have advantages and form an important theoretical foundation.