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The Search for Pure Intonation

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The Search for Pure Intonation Intonation in Performance Practice: An Historical, Mathematical and Aural Comparison of Four Tunings Intonation in Performance Practice: An Historical, Mathematical and Aural Comparison of Four Tunings Nick Nyderek, Music Dr. Matthew Faerber,Ph.D. Department of Music ABSTRACT This paper explores several different scale intonations and temperaments, including Pythagorean intonation, limit-5 just intonation, mean-tone temperament, and equal temperament. Historical context is provided as well as possible reasons for the development of new temperaments and intonations. These temperaments and intonations are compared mathematically and musically with pure intonation. A primary focus of the provision, for aural analysis and comparison, is a computer generated audio recording of each of the specified intonations and temperaments. There are also graphical representations comparing different temperaments and intonations. Introduction Since the time of Pythagoras, scale intonations have been an ongoing subject of study and refinement for musicians. Evolving musical styles, development of new instruments, and individual taste have all been components in the development of different intonations. For modern musicians and music educators, the importance of understanding different intonations and temperaments lies largely within performance practice. Tuning is a primary element of performance practice. For example, a vocal ensemble may not present a historically informed performance if the equal tempered tuning of a piano is the basis for the chosen intonation: equal temperament was not implemented in practice until a century or more after the Renaissance. Four prominent intonations and temperaments include Pythagorean intonation, just intonation, mean-tone temperament, and equal temperament. Each of these intonations represents a significant development in the history of intonation in western music. When understood mathematically and aurally, the differences in these intonations allow the modern performer to make informed decisions regarding the treatment of pitch and performance. 183 Intonation in Performance Practice: An Historical, Mathematical and Aural Comparison of Four Tunings Review of Literature Irregular Systems of Temperament by J. Murray Barbour is an article that opens with a basic history and explanation of several intonations and temperaments, including Pythagorean, just, mean-tone, and equal. This introduction is useful because it concisely explains some aspects of the scales not mentioned in other sources. Beyond Temperament: Non-Keyboard Intonation in the 17th and 18th Centuries by Bruce Haynes is a beneficial article for this research. It defines the difference between intonation and temperament, and provides further explanation of Pythagorean tuning, equal temperament, mean-tone temperament, and just intonation. It focuses on the intonations more than on the temperaments, describing their composition in great detail. It also includes an appendix containing several quotations about intonation and temperaments from historical musicians. A Venerable Temperament Rediscovered by Douglas Leedy focuses exclusively on mean-tone temperament, one of the more mysterious temperaments. The several varieties of mean-tone temperament derive from rather complex equations. Since mean-tone held a prominent place for a large span of musical history, this is a helpful resource. Perfect harmony: A mathematical analysis of four historical tunings by Michael F. Page delves into the raw mathematics of just intonation, quarter-comma mean-tone temperament, well temperament, and equal temperament. Since the mathematics involved in each of the intonations and temperaments studied are a main focus of this research, this article is indispensable. The Technique of the Monochord by Cecil Adkins defines and explains the application of the monochord. The monochord is a tool used by musicians throughout history which lays out the foundations of intonation and temperament. Comprehending the monochord is an important component for understanding the history of intonation and temperament, and provides evidence for the origin of tunings. Illustrations of Just and Tempered Intonation by Alexander John Ellis takes a somewhat different approach to the formation of just intervals by describing them in terms of the harmonic series. It is an older publication, and perhaps the harmonic series basis for just intervals has since become assumed knowledge in the modern world. This definition of just intervals also serves as the definition used in this paper. 184 Intonation in Performance Practice: An Historical, Mathematical and Aural Comparison of Four Tunings History of Intonations and Temperaments The Pythagorean intonation system dates back to the sixth century B.C. and was the primary tuning used through the Middle Ages.1 Pythagoras was the first to designate mathematical ratios to produce musical intervals. He based his system on the division of a string into halves and thirds, producing pure2 octaves and fifths. These intervals are the foundation of the Pythagorean intonation system,3 which is also characterized by wide thirds. This system was used extensively in practice until the Renaissance. While Middle Age organum was based on fifths, fourths (inverted fifths), and octaves, the Renaissance was characterized in part by the use of thirds. This shift created a desire for purer thirds rather than the wide thirds of Pythagorean intonation. In the mid-1500’s, when the interval of a third became more important in practice, just intonation was introduced. Suggested as early as the second century A.D. by Ptolemy, just intonation is based on the harmonic series.4 The characteristics of just intonation include pure thirds (with respect to the harmonic series) which sound narrow to an ear accustomed to equal temperament5, and sevenths which also sound narrow when compared to equal temperament. Another characteristic of just intonation is the idea that in order to maintain such pure intervals, flexibility of pitch was required when changing tonal centers. Just intonation was readily acceptable to instruments with flexible pitches such as the sackbut and the voice, but the development and more widespread use of instruments of fixed pitch such as organs and harpsichords created a need for a tuning system more accessible throughout a wider range of keys. One solution to this tuning problem was the introduction of fixed- note “temperaments,” or “closed systems designed to help make the intonation of instruments with immovable pitch (like the organ and 1 J. Murray Barbour. "Irregular Systems of Temperament." Journal of the American Musicological Society 1, no. 3 (Autumn 1948): 20. 2 Pure intonation refers to intervals in which no beats are heard and is based on the harmonic series. ‘Beats’ in an interval are the aural sensation experienced when the sound waves differ by an amount such that they amplify each other (in phase) and cancel each other out (out of phase) at a rate that is audible. A visual example of beats is given in Appendix C. 3 Andre Barbera: 'Pythagoras', Grove Music Online ed. L. Macy (Accessed 9 July 2005), <http://www.grovemusic.com> 4 J. Murray Barbour. "Irregular Systems of Temperament." Journal of the American Musicological Society 1, no. 3 (Autumn 1948): 20. 5 Equal Temperament has been the standard tuning system in western music since the nineteenth century. 185 Intonation in Performance Practice: An Historical, Mathematical and Aural Comparison of Four Tunings harpsichord) convincing.”6 The most common of these temperaments was mean-tone. Of the various forms of mean-tone temperament, the most frequently used was quarter-comma mean-tone.7 Introduced as early as the fifteenth century, mean-tone became the principal temperament in the sixteenth and seventeenth centuries. The term “quarter-comma” signifies how much the fifths are tempered to accommodate purer thirds. The philosophy behind mean-tone was to create a fixed temperament that consisted of pure thirds and allowed the performer to move through different keys without compromising the purity of the intervals. This was achieved to a degree, but upon modulation to a key far from that in which the instrument was tuned, there would be heightened dissonance (see Appendix E). An extreme example of this was the “wolf” interval (most commonly Eb-G#), which was distastefully wide.8 The desire for more flexibility in the choice of keys caused musicians and theorists to turn toward equal temperament. Equal temperament was suggested by Aristoxenus as early as the third century B.C.9 However, it was not implemented in practice until the sixteenth century by fretted instruments and not until the nineteenth century by keyboard instruments. One reason for this reluctance to shift to equal temperament is its mathematical nature. This temperament is based on exactly equal semitones rather than ratios that relate to the harmonic series. Thus, every key is as in tune as the next, but pure intervals do not exist in any key. Not only was it difficult to tune instruments to these exact mathematical ratios due to limited technology, but many musicians balked at the concept of a temperament that contained no pure intervals. Eventually, the desire for key flexibility outweighed the desire for pure intervals within a temperament, and by the mid-nineteenth century equal temperament became the standard temperament in western music. Mathematics of Intonations and Temperaments There are solid mathematical formulas that
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