The Development of the Equal Temperament Scale
Evolution or Radical Change?
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The Development of the Equal Temperament ScaleEvolution or Radical Change?
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Mean-Tone TemperamentIn previous sections of this paper, tunings have been described by string length ratios. By the Renaissance, frequency ratios were the common way to define tunings. The conversion from one to the other is simple; the reciprocal of one equals the other. For example, a perfect fifth has a string length ratio of (2:3) and a frequency ratio of (3:2). In keeping with the writings of those who developed the following tuning methods, frequency ratios will be used to describe the remaining tuning systems. Mean-tone temperament is a modification of the Pythagorean scale. When tuning using the Pythagorean method, the fourth note tuned from “C” is “E,” which should be a major third. As mentioned before, music during the Renaissance favored major thirds over perfect fifths as harmonies. The aim of mean-tone temperament was to “purify” the major third. The Pythagorean “E” had a frequency ratio of (81:64). The “just” ratio for “E” is (5:4). The difference between these ratios became known as the syntonic comma (also known as the Ptolemaic comma) with a ratio of (81:80). This comma is large enough to be audible to most listeners. If perfect fifths are tuned flatter or sharper than (3:2),
other intervals in the Pythagorean method can be made closer to “just”
intervals. If fifths are tuned
one-fourth of a syntonic comma flat:
Quarter
Comma Meantone Temperament vs. Just Intonation
Click here to listen to the 1/4 comma mean-tone scale. The name “mean-tone” comes from the major second (D) being the geometric mean of the root and the major third (C and E).[i] This method does not resolve the problem of having two ratios per chromatic. Identifying which combination divides the octave most evenly is the deciding factor for which sharp and flat ratios are use for chromatics. In the above chart, the note names identify which combination was chosen for this version. The combination also makes the “wolf” interval spread out between multiple pairs of notes. This first documented mean-tone tuning was known as “Quarter Comma Mean-tone Temperament” and was first published by Italian music theorist Pietro Aaron (1490 – 1545 CE) in the early fifteen hundreds.[ii] Musicians using this scale would not stray too far from the key of “C” (i.e. keys with too many flats or sharps) because of the more arbitrary way accidentals were tuned.Click here to listen to mean-tone applied to Mozart's Sonata no. 11 in A major (3 sharps). Mathematical theory is not the sole basis of mean-tone temperament. Compromising the ratio of a perfect fifth in order to make a “just” third is a decision guided by aesthetics. This decision overcorrected the fifth causing problems with harmonies other than major thirds. Many other mean-tone temperaments were developed with different divisions of the syntonic comma. These will be included for analysis in this research.
[i] Helmholtz, Hermann. On
the Sensations of Tone as a Physiological Basis for the Theory of Music (2nd
English edition), (New York: Dover Publications, Inc., 1954), 546 [ii]
Frazer, Peter. “The Development of Musical Tuning Systems,” Midicode,
http://www.midicode.com/tunings/temperament.shtml#5.2. (accessed August 13,
2005) |
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, the note “E” will have a ratio of (5:4).
