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The demands of tuning (attaining the perfect scale) and temperament (the compromises necessary for composing in every key) have challenged musicians from the earliest civilizations onward. This guide surveys these longstanding problems, devoting a chapter to each principal theory and offering a running account of the complete history of tuning and temperament. Organized chronologically, the book features a helpful glossary and numerous illustrative tables, and it requires minimal background in music theory. This new reissue is currently the only edition in print of a much-quoted classic. 9 figures. 180 tables.
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Tuning and Temperament
A Historical Survey
By J. Murray Barbour
Dover Publications, Inc.Copyright © 2004 Dover Publications, Inc.
All rights reserved.
HISTORY OF TUNING AND TEMPERAMENT
The tuning of musical instruments is as ancient as the musical scale. In fact, it is much older than the scale as we ordinarily understand it. If primitive man played upon an equally primitive instrument only two different pitches, these would represent an interval of some sort – a major, minor, or neutral third; some variety of fourth or fifth; a pure or impure octave. Perhaps his concern was not with interval as such, but with the spacing of soundholes on a flute or oboe, the varied lengths of the strings on a lyre or harp. Sufficient studies have been made of extant specimens of the wind instruments of the ancients, and of all types of instruments used by primitive peoples of today, for scholars to come forward with interesting hypotheses regarding scale systems anterior to our own. So far there has been no general agreement as to whether primitive man arrived at an instrumental scale by following one or another principle, several principles simultaneously, or no principle at all. Since this is the case, there is little to be gained by starting our study prior to the time of Pythagoras, whose system of tuning has had so profound an influence upon both the ancient and the modern world.
The Pythagorean system is based upon the octave and the fifth, the first two intervals of the harmonic series. Using the ratios of 2:1 for the octave and 3:2 for the fifth, it is possible to tune all the notes of the diatonic scale in a succession of fifths and octaves, or, for that matter, all the notes of the chromatic scale. Thus a simple, but rigid, mathematical principle underlies the Pythagorean tuning. As we shall see in the more detailed account of Greek tunings, the Pythagorean tuning per se was used only for the diatonic genus, and was modified in the chromatic and enharmonic genera. In this tuning the major thirds are a ditonic comma (about 1/9 tone) sharper than the pure thirds of the harmonic series. When the Pythagorean tuning is extended to more than twelve notes in the octave, a sharped note, as G#, is higher than the synonymous flatted note, as Ab.
The next great figure in tuning history was Aristoxenus, whose dispute with the disciples of Pythagoras raised a question that is eternally new: are the cogitations of theorists as important as the observations of musicians themselves? His specific contention was that the judgment of the ear with regard to intervals was superior to mathematical ratios. And so we find him talking about "parts" of an octave rather than about string-lengths. One of Aristoxenus' scales was composed of equal tones and equal halves of tones. Therefore Aristoxenus was hailed by sixteenth century theorists as the inventor of equal temperament. However, he may have intended this for the Pythagorean tuning, for most of the other scales he has expressed in this unusual way correspond closely to the tunings of his contemporaries. From this we gather that his protest was not against current practice, but rather against the rigidity of the mathematical theories.
Claudius Ptolemy, the geographer, is the third great figure in early tuning history. To him we are in debt for an excellent principle in tuning lore: that tuning is best for which ear and ratio are in agreement. He has made the assumption here that it is possible to reach an agreement. The many bitter arguments between the mathematicians and the plain musicians, even to our own day, are evidence that this agreement is not easily obtained, but may actually be the result of compromise on both sides. To Ptolemy the matter was much simpler. For him a tuning was correct if it used superparticular ratios, such as 5:4, 11:10, etc. All of the tuning varieties which he advocated himself are constructed exclusively with such ratios. To us, nearly 2000 years later, his tunings seem as arbitrary as was that of Pythagoras.
Ptolemy's syntonic diatonic has especial importance to the modern world because it coincides with just intonation, a tuning system founded on the first five intervals of the harmonic series – octave, fifth, fourth, major third, minor third. Didymus' diatonic used the same intervals, but in slightly different order. If it could be shown that Ptolemy favored his syntonic tuning above any of the others which he has presented, the adherents of just intonation from the sixteenth century to the twentieth century would be on more solid ground in hailing him as their patron saint. Actually he approved the syntonic tuning because its ratios are superparticular; but so are the ratios of three of the four other diatonic scales he has given.
Just intonation, in either the Ptolemy or the Didymus version, was unknown throughout the Middle Ages. Boethius discussed all three of the above-mentioned authorities on tuning, but gave in mathematical detail only the system of Pythagoras. It was satisfactory for the unisonal Gregorian chant, for its small semitones are excellent for melody and its sharp major thirds are no drawback. Even when the first crude attempt at harmony resulted in the parallel fourths and fifths of organum, the Pythagorean tuning easily held its own.
But, later, thirds and sixths were freely used and were considered imperfect consonances rather than dissonances. It has been questioned whether these thirds and sixths were as rough as they would have been in the strict Pythagorean tuning, or whether a process of softening (tempering) had not already begun. At least one man, the Englishman Walter Odington, had stated that consonant thirds had ratios of 5:4 and 6:5, and that singers intuitively used these ratios instead of those given by the Pythagorean monochord. In reply one might note that some theorists continued to advocate the Pythagorean tuning for centuries after the common practice had become something quite different. If it was good enough for them, surrounded as they were by other, less harsh, tuning methods, it must have sufficed for most of those who lived in an age when no other definite system of tuning was known.
The later history of the Pythagorean tuning makes interesting reading. It was still strongly advocated in the early sixteenth century by such men as Gafurius and Ornithoparchus, and formed the basis for the excellent modification made by Grammateus and Bermudo. At the end of the century Papius spoke in its favor, and so, forty years later, did Robert Fludd. In the second half of the seventeenth century Bishop Caramuel, who has the invention of "musical logarithms" to his credit, said that "very many" (plurimi) of his contemporaries still followed in the footsteps of Pythagoras. Like testimony was given half a century later from England, where Malcolm wrote that "some and even the Generality ... tune not only their Octaves, but also their 5ths as perfectly ... Concordant as their Ear can judge, and consequently make their 4ths perfect, which indeed makes a great many Errors in the other Intervals of 3rd and 6th." After another half century we find Abbé Roussier extolling "triple progression," as he called the Pythagorean tuning, and praising the Chinese for continuing to tune by perfect fifths.
Like the systems of Agricola in the sixteenth century and of Dowland in the early seventeenth century, many of the numerous irregular systems of the eighteenth century contained more pure than impure fifths. The instruments of the violin family, tuned by fifths, have a strong tendency toward the Pythagorean tuning. And a succession of roots moving by fifths is the basis of our classic system of harmony from Rameau to Prout and Goetschius. Truly the Pythagorean tuning system has been long-lived, and is still hale and hearty!
To return to the fifteenth century and the dissatisfied performers: Almost certainly some men did dislike the too-sharp major thirds and the too-flat minor thirds so much that they attempted to improve them. But history has preserved no record of their experiments. And the vast majority must have still been using the Pythagorean system, with all its imperfections, when Ramis de Pareja presented his tuning system to the world.
To be sure, Ramis did not present himself as the champion of a tremendous innovation. He was not a Luther nailing his ninety-five theses to the church door. His tuning was offered as a method which would be easier to work out on the monochord, and thus would be of greater utilitarian value to the singer, than was the Pythagorean tuning, with its cumbersome ratios. Although Ramis' monochord contained four pure thirds, with ratio 5:4, it was not the usual form of just intonation applied to the chromatic octave, in which eight thirds will be pure. It is rather to be considered an irregular tuning, combining features of both the Pythagorean tuning and just intonation. Some of Ramis' contemporaries assailed his tuning method, but his pupil Spataro explained it as a sort of temperament of the Pythagorean tuning. From these polemics arose the entirely false notion that Ramis was an advocate of equal temperament. But he is worthy of our respect as the first of a long line of innovators and reformers in the field of tuning.
As the words "tuning" and "temperament" are used today, the former is applied to such systems as the Pythagorean and just, in which all intervals may be expressed as the ratio of two integers. Thus for any tuning it is possible to obtain a monochord in which every string-length is an integer. A temperament is a modification of a tuning, and needs radical numbers to express the ratios of some or all of its intervals. Therefore, in monochords for temperaments the numbers given for certain (or all) string-lengths are only approximations, carried out to a particular degree of accuracy. Actually it is difficult in extreme cases to distinguish between tunings and temperaments. For example, Bermudo constructed a monochord in which the tritone G-C# has the ratio 164025:115921. This differs by only 1/7 per cent from the tritone of equal temperament, and in practice could not have been differentiated from it. But his system, which consists solely of linear divisions, should be called a tuning rather than a temperament.
It is not definitely known when the practice of temperament first arose in connection with instruments of fixed pitch, such as organs and claviers. Even in tuning an organ by Pythagorean fifths and octaves, the result would not be wholly accurate if the timer's method was to obtain unisons between pitches on a monochord and the organ pipes. This would be a sort of unconscious temperament. More consciously he may have tried to improve some of the harsh Pythagorean thirds by lopping a bit off one note or another. Undoubtedly this was being done during the fifteenth century, for we find Gafurius, at the end of that century, mentioning that organists assert that fifths undergo a small diminution called temperament (participata).
We have no way of knowing what temperament was like in Gafurius' age; but it is my belief that this diminution which Gafurius characterized as "minimae ac latentis incertaeque quodemmodo quantitatis" was actually so small that organs so tuned came closer to being in equal temperament than in just intonation or the mean-tone temperament. This belief is substantiated by two German methods of organ temperament which appeared in print a score of years later than Gafurius' tome. The earlier of the two was Arnold Schlick's temperament, an irregular method for which his directions were somewhat vague, but in which there were ten flattened and two raised fifths, as well as twelve raised thirds. Shohé Tanaka's description of Schlick's method as the mean-tone temperament is wholly false; for in the latter the eight usable thirds are pure. Actually, from Schlick's own account, the method lay somewhere between the mean-tone temperament and the equal temperament. More definite and certainly very near to equal temperament was Grammateus' method, in which the white keys were in the Pythagorean tuning and the black keys were precisely halfway between the pairs of adjoining white keys.
Just what the players themselves at this time understood by equal semitones is not known. Perhaps they would have been satisfied with a tuning like that of Grammateus, with ten semitones equal and the other two smaller. The first precise mathematical definition of equal temperament was given by Salinas: "We judge this one thing must be observed by makers of viols, so that the placing of the frets may be made regular, namely that the octave must be divided into twelve parts equally proportional, which twelve will be the equal semitones." To facilitate constructing this temperament on the monochord, Salinas advised the use of the mesolabium, a mechanical method for finding two mean proportionals between two given lines. Zarlino also gave mechanical and geometric methods for finding the mean proportionals, intended primarily for the lute. (Zarlino did include, however, Ruscelli's enthusiastic plea that all instruments, even organs, should be tuned equally.) The history of equal temperament, then, is chiefly the history of its adoption upon keyboard instruments.
Neither Salinas nor Zarlino gave monochord lengths for equal temperament, although the problem was not extremely difficult: to obtain the 12th root of 2, take the square root twice and then the cube root. The first known appearance in print of the correct figures for equal temperament was in China, where Prince Tsaiyü's brilliant solution remains an enigma, since the music of China had no need for any sort of temperament. More significant for European music, but buried in manuscript for nearly three centuries, was Stevin's solution. As important as this achievement was his contention that equal temperament was the only logical system for tuning instruments, including keyboard instruments. His contemporaries apologetically presented the equal system as a practical necessity, but Stevin held that its ratios, irrational though they may be, were "true" and that the simple, rational values such as 3:2 for the fifth were the approximations! In his day only a mathematician (and perhaps only a mathematician not fully cognizant of contemporary musical practice) could have made such a statement. It is refreshingly modern, agreeing completely with the views of Schönberg and other advanced theorists and composers of our day.
The most complete and important discussion of tuning and temperament occurs in the works of Mersenne. There, in addition to his valuable contributions to acoustics and his descriptions of instruments, Mersenne ran the whole gamut of tuning theory. He expressed equal temperament in numbers, indicated geometrical and mechanical solutions for it, and finally put it upon the practical basis of tuning by beats as used today. Fully as catholic is his list of instrumental groups for which this temperament should be used: all fretted instruments, all wind instruments, all keyboard instruments, and even percussion instruments (bells). The widespread influence of Mersenne's greatest work, Harmonie universelle (Paris, 1636 – 37), undoubtedly helped greatly to popularize a timing that was then still considered as suitable for lutes and viols only.
The first really practical approximation for equal temperament had been presented by Vincenzo Galilei half a century before Mersenne. He showed that the ratio of 18:17 was convenient in fretting the lute. Since references to this size of semitone cover two and a half centuries, it is probable that it has been used even longer by makers of lutes, guitars, and the like. Of course the repeated use of the 18:17 ratio would not give an absolutely pure octave, but a slight adjustment in the intervals would correct the error. Galilei's explanation of the reason for equal semitones on the lute is logical and correct: Since the frets are placed straight across the six strings, the order of diatonic and chromatic semitones is the same on all strings. Hence, in playing chords, C# might be sounded on one string and Db on another, and this will be a very false octave unless the instrument is in equal temperament.
Excerpted from Tuning and Temperament by J. Murray Barbour. Copyright © 2004 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
|I.||History of Tuning and Temperament||1|
|Other Varieties of Meantone Temperament||31|
|Geometrical and Mechanical Approximations||49|
|Theory of Just Intonation||102|
|Theory of Multiple Division||128|
|Modifications of Regular Temperaments||139|
|Temperaments Largely Pythagorean||151|
|Divisions of Ditonic Comma||156|
|VIII.||From Theory to Practice||185|
|Tuning of Keyboard Instruments||188|
|Just Intonation in Choral Music||196|
|Intervals with Superparticular Ratios||226|