The Development of the Equal Temperament Scale
Evolution or Radical Change?
|
The Development of the Equal Temperament ScaleEvolution or Radical Change?
|
|
|
Equal TemperamentMany mathematicians and music theorists considered the equal temperament scale well before the first accurate table of string lengths for it was published. Those working on other temperaments saw the advantage of spreading imperfections over multiple fifths rather than having one interval completely out of tune, so the next logical step was extending this to all fifths. Galileo’s father, Vincenzo Galilei (~1525-1591 CE), wrote about the placement of frets on a lute in 1581. Fretted instruments have more complications in temperament because the parallel strings (usually tuned a perfect fourth apart) have identical string length ratios throughout their ranges due to the perpendicular frets on the fingerboard. Galilei proposed that the only temperament that will work with this setup is one that divides the octave equally. He gave an approximation of this division with each semitone frequency ratio of (18:17).[i] This ratio does not divide the octave equally, but it proved to be very close to the solution described below. Until the seventeenth century, no practical and accurate application of equal temperament was published in Europe. This was not due to lack of inspiration on the part of those trying to develop a tuning method; the barrier was directly related to an unsolved mathematics problem. The classical problem of doubling a cube is one that Greek mathematicians had tried to solve for centuries. In the nineteenth century, French mathematician Pierre Wantzel (1814-1848 CE) proved that the geometric solution they were attempting was impossible using only a ruler and compass. The cube root of two is more than just irrational; it cannot
be constructed with a ruler and compass.[ii]
Spreading the error over twelve fifths of a Pythagorean tuning involved
splitting the error into twelve equal parts.
Since these are proportional relationships, this meant taking the twelfth
root. Three is a factor of twelve,
and the cube root of a Pythagorean comma is part of the calculation of values of
an equal tempered scale. Since the
denominator of
A simpler way of illustrating this problem is to ignore the
Pythagorean comma and create a scale based on the desired result.
All semitones should have the same proportion because all intervals are
consistent in an equal tempered scale. Twelve
of these semitones will equal one octave with the frequency ratio of (2:1). Therefore, a semitone must have a ratio of
(
Equal
Temperament
Click here to listen to this scale. The advantage to this is clear; all keys signatures sound exactly the same. If people can accept the sound of one major key in equal temperament, all major keys will sound satisfactory to their ears. The same could be said for other modes, such as those used in Gregorian chants. Click here to listen to this scale applied to Mozart's Sonata no. 11 in A major. Unfortunately, this could not be tested until the values for each chromatic were known. Since the tuning contains the cube root of two, the scale had to wait until mathematicians were capable of accurately estimating this value. Dutch mathematician Simon Stevin (1548-1620) attempted to calculate the equal tempered scale during the 1580’s. In his writing, he did not refer to it as temperament. He believed equal proportions of the octave produced the ideal scale, and all other tunings were misconceptions. To make his calculations, he first established a value for “E” by estimating the cubed root of two. He used this value and the properties of proportions to calculate values for the remaining notes. Although he had many errors due to improper rounding of decimals, his table of values produces a tuning that resembles the equal temperament used today.[iii] Because Stevin’s work was never published, equal temperament was not experimented with until after 1636 when Marin Mersenne published various calculations of equal temperament in his Harmonie Universelle. The most accurate of these calculations was given to Mersenne by French engineer Jean Gallé (1580-? CE). With the exception of one (probably typographical) error, these calculations are indistinguishable from the current version. Mersenne is often referred to as the inventor of equal temperament with Stevin or Gallé hardly mentioned. Although Stevin was the first Western European known to tune using equal temperament, it is possible that Mersenne and Stevin found this method from China. Chinese prince Chu Tsai-Yu published an equal temperament scale in 1584. His calculations were as accurate as those done by Gallé fifty years later. It would seem unlikely to have influenced Europe so quickly; however, Italian Jesuit Matteo Ricci (1552-1610 CE) went to China to start a mission in 1583. Although religion was part of his mission, his main focus was sharing Western mathematical and scientific methods with Chinese scholars. He became a well-respected figure in their country. Trying to gain the respect of Chinese royalty, Ricci was most likely aware of Chu Tsai-Yu’s publication. There is no evidence of Ricci sending the publication to Europe. Whether or not he directly influenced Western European music, Chu Tsai-Yu is the first person known to have made an accurate table for tuning in equal temperament.[iv] Another mathematical advancement contributing to equal temperament was the logarithm. Scottish mathematician John Napier (1550-1617 CE) created a calculation table. His table was later updated by Henry Briggs (1561-1631 CE) to reflect changes planned by Napier shortly before his death. Briggs’ update (published in 1628 CE) became the form of logarithms used today.[v] Thanks to this new tool, calculations for frequencies could be made more accurately and quickly without the time-consuming extraction of roots. Taking logarithms of frequency ratios (base two) and multiplying them by the number (1200) later became the standard for measuring pitch called cents. This measure is beneficial because it is small enough to measure important intervals such as the Pythagorean comma (approximately 23.46 cents), and one hundred cents represent a semitone in equal temperament. Alexander Ellis (1814-1890 CE) first popularized this measure in his English translation of Hermann Helmholtz’s (1821-1894 CE) On the Senations of Tone.[vi] The following chart shows each of the values for an equal tempered scale, followed by the logarithm and cents for each of these values. Equal
Temperament with Logarithms (base 2) and Cents (log *1200)
Although this new standard of measure was obviously inspired by equal temperament, logarithms advanced the study of all temperament methods. As can be seen by the dates for methods mentioned so far, there were overlaps due to conflicting opinions about which method was better. Even Mersenne invented his own modification of mean tone temperament after publishing the equal tempered scale.[vii] Equal temperament did not become the dominant tuning method until the end of the eighteenth century.[viii] Logarithms (and later cents) became a convenient way to calculate and express frequency values for the many versions of mean-tone and well temperament. The analysis in the next section will use cents to express the frequencies of all tuning methods including the Pythagorean. By the nineteenth century, Western European music had become so complex that equal temperament was the only practical method for tuning fixed pitch instruments. Contrary to Stevin’s view of the scale, most saw this as a necessary compromise. Other temperaments favored particular key signatures. Musical compositions were created in all twenty-four major and minor keys by this time. As well as changing keys mid-piece, harmonic progressions often mixed chords from other key signatures with the diatonic chords. Without equal temperament, fixed pitch instruments would have needed retuning constantly (even in the middle of a performance) to facilitate these new compositional techniques. In the next section, selected common tunings leading up to this acceptance of equal temperament will be statistically analyzed.
[i]
Barbour, J. Murry. Tuning
and Temperament – A Historical Survey, (East Lansing: Michigan State
College Press, 1951), 49 [ii]
Katz, Victor. A History of
Mathematics – An Introduction (2nd edition), (Massachussetts:
Addison-Wesley Longman, Inc., 1998), 664 [iii]
Christensen, Thomas. The Cambridge History of Western Music Theory,
(New York: Cambridge University Press, 2002) , 205 [iv]
Temple, Robert. The Genius
of China – 3000 Years of Science, Discovery, and Invention, (New York:
Simon and Schuster, 1987), 209 [v]
Katz, Victor. A History of
Mathematics – An Introduction (2nd edition), (Massachussetts:
Addison-Wesley Longman, Inc., 1998), 420 [vi]
White, John D. Theories of Musical Texture in Western History, (New
York: Garland Publishing Inc., 1995), 30 [vii]
Barbour, J. Murry. Tuning
and Temperament – A Historical Survey, (East Lansing: Michigan State
College Press, 1951), 134 [viii]
Christensen, Thomas. The Cambridge History of Western Music Theory,
(New York: Cambridge University Press, 2002), 220 |
|
Send mail to mgrenfell@charter.net with
questions or comments about this web site.
|
).
