Chapter I. 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 under- lies the Pythagorean tuning. As we shall see in the more detailed account of Greek tunings, the Pythagorean tuning perse 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 A". 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 TUNING AND TEMPERAMENT the observations of musicians themselves? His specific conten- tion was that the judgment of the ear with regard to intervals was superior to mathematical ratios. And so we find him talking about "parts'7 of an octave rather than about string-lengths „ One of Aristoxenus7 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. How- ever, 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 itused superparticular ratios, such as 5:4, 11:10, etc. All of the tuning varieties which he advocated himself are con- structed 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' dia- tonic 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 ver- HISTORY OF TUNING AND TEMPERAMENT sion, was unknown throughout the Middle Ages. Boethius dis- cussed 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 Pytha- gorean tuning easily held its own. But, later, thirds and sixths were freely used and were con- sidered 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 intui- tively 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 com- mon 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 inven- tion 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 !See J. Murray Barbour," The Persistence of the Pythagorean Tuning Sys- tem," Scripta Mathematica, I (1933), 286-304. TUNING AND TEMPERAMENT 4ths perfect, which indeed makes a great many Errors in the other Intervals of 3rd and 6th." After another half century we find Abbe 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 per- formers: Almost certainly some men did dislike the too-sharp major thirds and the too-flat minor thirds so much that they at- tempted 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. Al- though 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' con- temporaries 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 2lt occurs, for example, in such a general work as Sir James Jeans' Science and Music (New York, 1937). HISTORY OF TUNING AND TEMPERAMENT respect as the first of a long line of innovators and reformers in the field of tuning. As the words "tuning" and "temperament" aye 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 in- tegers. 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 mono- chords for temperaments the numbers given for certain (or all) string-lengths are only approximations, carried out to a partic- ular 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 tuner's method was to obtain unisons between pitches on a mono- chord 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 dimi- nution 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 Ga- furius characterized as "minimae ac latentis incertaeque quo- demmodo quantitatis" was actually so small that organs so tuned came closer to being in equal temperament than in just intonation 3Franchinus Gafurius, Practica musica (Milan, 1496), Book 2, Chapter 3. TUNING AND TEMPERAMENT or the meantone temperament. This belief is substantiated by two German methods of organ temperament which appeared in print a score of jgears 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. Shohe Tanaka's description of Schlick's method4 as the meantone 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 meantone 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 semi- tones equal and the other two smaller. The first precise math- ematical 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. "5 To facilitate con- structing 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 propor- tionals, 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 tempera- ment, then, is chiefly the history of its adoption upon keyboard instruments. 4aStudien im Gebiete der reinen Stimmung," Vierteljahrsschrift fiir Musik- wissenschaft, VI (1890), 62, 63. 5Francisco Salinas, De musica libri VH (Salamanca, 1577), p. 173. 6 HISTORY OF TUNING AND TEMPERAMENT 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 Tsai- yii'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 cen- turies, 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 Schbnberg and other advanced theo- rists 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 prac- tical 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 key- board 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 tuning that was then still considered as suitable for lutes and viols only. The first really practical approximation for equal tempera - Gjohann Philip Albrecht Fischer, Verhandlung van de Klokken en het Klokke- Spel (Utrecht, 1738), p. 19, gave a bell temperament, with C equal to 192.000. This was equal temperament, with a few minor errors. 7 TUNING AND TEMPERAMENT ment 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 chro- matic semitones is the same on all strings. Hence, in playing chords, C* might be sounded on one string and DD on another, and this will be a very false octave unless the instrument is in equal temperament. Vicentino had referred to a serious difficulty that arose from the common practice of having one kind of tuning (meantone) for keyboard instruments and another (equal) for fretted instruments. Since the pitches were so divergent, there was dissonance when- ever the two groups were used together. By 1600, theorists like Artusi and Bottrigari said that these different groups of instru- ments were not used simultaneously because of the pitch diffi- culties. That is why such large instrumental groups were needed as those employed in the Ballet Comique de la Reine or in Mon- teverdi's Orfeo — selected groups of like instruments sounded well, but the mixture of different tunings made tuttis impracti- cable. It would seem that this consideration would have brought about the universal adoption of equal temperament long before it did come. However, after the unfretted violins became the back- bone of the seventeenth century orchestra, their flexibility of in- tonation made this problem less pressing than when lutes and viols had been opposed to organs and claviers. Before we leave the sixteenth century, we should examine the contribution to tuning history for which Vicentino is especially known. His archicembalo was an instrument with six keyboards, with a total of thirty-one different pitches in the octave. He de- scribed its tuning as that of the "usage and tuning common to all the keyboard instruments, as organs, cembali, clavichords, and HISTORY OF TUNING AND TEMPERAMENT the like. "^ This would have been the ordinary meantone temper- ament, in which the fifths were tempered by 1/4 comma. Huy- ghens, a century and a half after Vicentino, showed that there was very close correspondence between a system in which the octave is divided into thirty-one logarithmically equal parts and the meantone system, similarly extended to thirty- one parts. A simpler type of multiple division was the cembalo with nineteen notes in the octave. Both Zarlino and Salinas intended their variants of the meantone temperament (with fifths tempered by 2/7 and by 1/3 comma respectively) for such an instrument, and the latter 's temperament would result in an almost precisely equal division. Praetorius described such an instrument also, and it has received favor with some twentieth century writers, especially Yasser. The best system of multiple division within the limits of prac- ticability divides the octave into fifty-three parts. This is lit- erally a scale of commas, and, as such, was suggested by the ancient Greek writers on the Pythagorean system. Mersenne and Kircher in the seventeenth century mentioned the system. Mercator realized its advantages for measuring intervals. But especial honor should be paid to the nineteenth century English- man Bosanquet for devising an harmonium with a "generalized keyboard" upon which the 53-system could be performed. Other varieties of equal multiple division will be discussed in Chapter VI, together with a number of unequal divisions, most of which are extensions of just intonation. Practical musicians have rejected all of them, chiefly because they are more difficult to play, as well as being more expensive, than our ordinary key- boards. Just intonation, as has already been mentioned, has had few devotees since the early seventeenth century. The history of the meantone temperament makes more interesting reading, since various theorists in addition to Zarlino and Salinas had conflict- ing ideas as to the amount by which the fifths should be tempered. Silbermann's temperament of 1/6 comma for the fifths is the most significant for us, because he represents the more con- ^Nicola Vicentino, L'antica musica ridotta alia moderna prattica (Rome, 1555), Book 5, Chapter 6. 9 TUNING AND TEMPERAMENT servative practice during the time of Bach and Handel. In his temperament the thirds are slightly sharp, but the wolves are almost as ravenous as in the Aron 1/4 comma system. To some extent the final adoption of equal temperament for an individual organ or clavier might have meant substituting this temperament for some type of meantone temperament. We are told that organs in England were still generally in meantone tem- perament until the middle of the nineteenth century. England must have lagged behind the Continent in this respect, and it is quite possible that the change, when it did come, was radical. But it is more likely that in most cases the change to equal temperament was made more smoothly than this. The importance of unequal systems of twelve notes to the octave has been gen- erally neglected by the casual historians of tuning, to whom only the Big Four (Pythagorean, just, meantone, and equal) are of moment. It is my opinion, however, that the unequal systems were of the greatest possible significance in bringing about the supremacy of our present tuning system. Reference has already been made to the early sixteenth century irregular systems of Schlick and Grammateus. The former resembled the meantone temperament; the latter was derived from the Pythagorean tuning. Bermudo repeated Grammateus' tuning, and his own second method was basically Pythagorean also. Ramis and Agricola crossed just intonation with the Pythagorean tuning, with fairly happy issue. Ganassi and Artusi treated just intonation and the meantone temperament much as Grammateus and Bermudo had treated the Pythagorean tuning. Only a few years later than Grammateus, Aron described for organs the meantone temperament, mentioned above. In it every fifth save one was tempered by such an amount (1/4 comma, or about 1/18 semitone) that four fifths less two octaves would pro- duce a pure major third. Thus arose the system that, with var- ious modifications, was to be the strongest opponent of equal temperament, so far as keyboard instruments were concerned, for two or three hundred years. In the meantone temperament a sharped note, as G% is lower in pitch than the equivalent flattened note, as AD, by the great diesis, which is almost half as large as a semitone. 10 HISTORY OF TUNING AND TEMPERAMENT After Aron's time the meantone temperament, or some similar system, was generally accepted for organ and clavier. But there were a few dissenting voices. One was that of his exact contem- porary Lanfranco, whose practical tuning rules for keyboard in- struments seem to agree with no system other than equal tem- perament. Another was that of Fogliano, who was apparently the first sixteenth century writer to follow Ramis' lead and use in a tuning system both the pure fifths and the pure thirds of just in- tonation. But there is a difference; for he realized that the triads on D and BD would be hopelessly out of tune in such a system, and therefore recommended that there be a mean D and Bb, each differing by half a comma from a pair of just pitches. These two mean pitches hint at Aron's meantone system. Otherwise this is what we ordinarily understand just intonation to be. Ironically enough, Fogliano's method, although containing more perfect thirds than Ramis' did, is far inferior to it if one goes beyond the ordinary bounds of two flats and three sharps. Beyond these bounds lay in wait the howling wolves, to muffle whose voices was the task of many a later worker in this field. Fogliano had no immediate followers as an advocate of just intonation, since the following generation was more concerned with temperament. But almost a century later, certain mathe- maticians — as Galileo, de Caus,and Kepler — proclaimed again the validity of pure thirds and fifths. Occasional lone figures, both mathematicians and music theorists, were to speak in favor of just intonation, even until our own day. But it is significant that the great music theorists, such as Zarlino, Mersenne, and Rameau, presented just intonation as the theoretical basis of the scale, but temperament as a practical necessity. Equally great mathematicians with some understanding of music, from Stevin to Max Planck, have hailed temperament. From the middle of the sixteenth century, all the theorists agreed that the fretted instruments, lutes and viols, were tuned in equal temperament. Vicentino made the first known reference to this fact, going so far as to state that both types of instrument had been so tuned from their invention. If we may believe pic- torial evidence, especially that of the Flemish painters, so me- ticulous about detail, frets were adjusted to equal temperament 11 TUNING AND TEMPERAMENT as early as 1500, although there is not complete agreement on this point. In the National Gallery in London, for example, there are several paintings in which the position of frets is shown plainly. A Concert, by Ercole de Roberti (1450-96), contains a nine- stringed lute and a small four-stringed viol, both apparently in equal temperament. Marco Marziale's Madonna and Child En- throned with Saints, painted between 1492 and 1507, has an eleven- stringed lute with intervals equally proportional. And The Am- bassadors, painted by Hans Holbein the Younger in 1533, has a six-stringed lute, again in equal temperament. Negative evidence is furnished by a painting by the early sixteenth century painter Ambrogio de Predis, whose Angel Playing on a Musical Instru- ment is playing a nine- stringed lute on which the semitones run large, small, small, large, and then three equal, as if the notes might have been C, C*, D, E , E, etc. Because of the ease of tuning perfect fifths, the Pythagorean tuning has been the foundation of many of the later irregular sys- tems, including that of Kirnberger. It also had some importance for such sophisticated writers as Werckmeister, Neidhardt, and Marpurg, whose systems with subtly divided commas were di- rected to the intellect rather than to the ear of the practical mu- sician. It becomes apparent, however, from the works of the men just mentioned that an instrument that was "well tempered" was not necessarily tempered equally. The title of Bach's famous "48" meant simply that the clavier was playable in all keys. Werck- meister and Neidhardt explained clearly that in their systems the key of C would be the best and D*3 the worst, with the conso- nance of the other keys somewhere between these extremes. Mersenne's and Rameau's modification of the 1/4 comma meantone temperament resembles somewhat the "good" temper- aments of Werckmeister and Neidhardt, and Gallimard, with the aid of logarithms, reached a very similar goal. Perhaps the best of these many irregular systems was Thomas Young's second method, in which six fifths are perfect, and the other six are tuned 12 HISTORY OF TUNING AND TEMPERAMENT flat by 1/6 Pythagorean comma, as in Silbermann's tuning. This would have been simpler to construct by ear than most of the systems, and does have an orderly progression from good to poor tuning as one departs from the most common keys. In almost all of these irregular systems, from Grammateus to Young, all the major thirds were sharp to some extent, thus differing from just intonation and the meantone temperament, in which the usable thirds were perfect and the others very harsh. For the practical musician it would have been an easy matter, as time went on, to tune the "common" thirds still sharper, so that all the thirds would be equally sharp, and his instrument would be substantially in equal temperament. Probably this is exactly what did happen. The recorded opposition to equal temperament on the part of such men as Werckmeister and even Sebastian Bach was to the rigorous mathematical treatment implied by the name "gleich- schwebend." Theirs was a practical approximation to equality, and, from the keyboard compositions of Bach, it is evident that his practice must have been as satisfactory as that of our present- day tuners, else the great majority of his compositions would have been unbearable. 13