Chapter VIII. FROM THEORY TO PRACTICE In our intensive study of scores of tuning systems we have failed to note what may be learned from the music itself. Some of the theorists who have written on tuning were able composers as well. When they described with precision a particular division of the monochord, their theory may well have coincided with fact. But the tuning theories of the mere mathematicians do not carry so much weight. Nor do the rules of thumb the musicians more commonly presented. All of these theories may be put into neat little pigeonholes, but one can be sure that the practice itself, because of the limitations of the human ear, was even more var- ied than the extremely varied theories. It is not to be expected that a study of the music will provide a precise picture of tuning practice. It is to be used more by way of corroborating what the theorists have said. Let us con- sider first the contention of Vicentino that the fretted instruments were always in equal temperament. In general we can reach certain conclusions concerning tuning by examining the range of modulation. However, this is not definitive as regards the lutes and viols. Korte listed D#'s in lute music from 1508, an A* from 1523, and many D^'s from 1529.* But the mere presence of notes beyond the usual 12 -note compass proves little, because the lutes were not restricted to a total compass of 12 semitones. As shown in Table 180, the normal compass with the G tuning was Ct> to C# and for the A tuning from DD to D#. Ordinarily, lutes and viols had six strings, tuned by fourths, with a major third in the middle. Thus the open strings might be G C F A D G or A D G B E A. It is easy to see here the prototype of Sch'dnberg's chords built by fourths. Because of the perfect fourths, the fretted instruments might have inclined to- ward the Pythagorean tuning, as the later violins have done. Mersenne pointed out that the major third in the middle would iOskar Korte, "Laute und Lautenmusik bis zur Mitte des 16. Jahrhunderts," Internationale Musikgesellschaft, Beiheft 3 (1901). TUNING AND TEMPERAMENT then be sharp by a comma. But the strings of lutes and viols were tuned by forming uni- sons, fifths, or octaves with the proper frets on other strings, thus making the tuning uniform throughout the instrument. Vin- cenzo Galilei^ stated that if the tuning were not equal, semitones on the A string (mezzana) of the lute based on G would have the note names shown in Table 180. Since the frets were merely pieces of gut tied straight across the fingerboards at the correct places, the order of diatonic and chromatic semitones would have to be the same on all strings. Thus the chromatic compass of a lute with six strings and eight frets would be as shown in Table 180, if meantone temperament had been used. There might be some question for the G tuning regarding notes produced by the 6th fret, since B would be a better choice than C" on the 4th string. But the remaining notes for the 6th fret agree somewhat better with other notes in the compass than the equivalent sharped notes would have done. Galilei pointed out that Gt> (4th string, 1st fret) was not a pure fifth to C# (3rd string, 4th fret), nor was DD (5th string, 1st fret) a pure octave to the C#. He might have added that DD (1st string, 6th fret) was not a pure octave above the C# either. It is easy to multiply examples of unsatisfactory intervals on the unequally tuned lute in G. (Read them a tone higher for the A tuning.) Try building major triads upon the notes of the 6th string, starting with B13. C, D*3, and E*3 are satisfactory as roots also, but false triads are generated on B and D. On the 5th string, starting with D, the satisfactory triads are on E13, F, and A13; false triads on D, E, Gb, and G. On the 4th string, starting with G, the only unsatisfactory triad is on C13. On the 3rd string, starting with C, the other satisfactory triads are on D, E", and F, with false triads on C* and E. Thus, of 26 major triads in close position, only 17, about 2/3, are available. Some of the triads, those on G, D, and A, unsatisfactory in the lower octave, can be played correctly in the higher octave. But the complete E and B major triads are unavailable anywhere, because there are no G* and D# — unless, of course, the 6th fret runs to sharps rather than to flats. ^Fronimo (Venice 1581; revised edition, 1584), pp. 103 f. 186 FROM THEORY TO PRACTICE As illustrations of incongruous notes on particular frets, let us examine some of the Austrian lute music of the sixteenth cen- tury, as found in Volume 18 of the Austrian Denkm'aler. The first collection represented is Hans Judenkiinig's Ain schone kunstliche Underweisung (1523), His third Priamell is modal, but often suggests C minor. Like most of the German and Aus- trian composers, Judenkiinig used the A tuning of the lute. In bar 3 the note aD appears as the 4th fret on the 2nd string, indi- cating that this fret has a flat tuning (see Table 180). But in bar 4 there is a b and in bar 19 a c# , both of which belong to the sharp tuning for this fret. For Judenkiinig's fourth Priamell the editor has put the sig- nature of three sharps, as an indication of the prevailing sharp- ness. This even extends to the 6th fret, which would then include an e* . Actually there is an e# in the music, and no f '. There- fore it would have been possible to play this piece with an une- qual temperament, but not without changing the 6th fret from its normal flat tuning, Simon Gintzler's fifth Recercar (1547) used the Italian G tun- ing. Here the 6th fret has a flat tuning, as shown by aD and a very frequent eb . But in bar 10 there is a b instead of the c*5 belonging to the flat tuning. In Gintzler's setting of Senfl's song "Vita in ligno moritur," the 6th fret is again flat, but in bar 15 both ab and b occur. The a*3 and b also occur several times in Bakf ark's Fantas- ias (1565). More interesting is his setting of "Veni in hortum meum, soror mea" (1573). In bar 50, d* ' occurs as the third of the B major triad, indicating a sharp tuning for the 6th fret. This means that f' is not available on this fret; but f ' does occur in bar 56 and elsewhere. In bar 62 the complete C minor triad oc- curs: c' eb g' c", with the eb' the 4th fret on the 3rd string. But this fret must have had a sharp tuning, since the notes d#' g , and c occur on it with great frequency. It would be easy to multiply examples, from the music of Ital- ian, French, and Spanish composers. Those that have been given are sufficient to show that in the golden age of lute music the composers were indifferent to discords that would have arisen 187 TUNING AND TEMPERAMENT if an unequal temperament had been used. The example from Judenklinig occurs so early in the century (1523) that it seems very probable that lutes and viols did employ equal temperament from an early time, perhaps from the beginning of the sixteenth century. We need not be too much concerned with what the equal tem- perament for the fretted instruments was really like. It might have been the Grammateus-Bermudo tuning — Pythagorean with mean semitones for the chromatic notes* It might have been the Ganassi-Reinhard mean semitones applied to just intonation, or Artusi's more subtle system of mean semitones in meantone temperament. Or the frets might have been placed according to Galilei's 18:17 ratio, or (correctly) according to Salinas' ratio of the 12th root of 2. In any case, it would have been a good, workable temperament. Tuning of Keyboard Instruments In the early sixteenth century Schlick and Grammateus de- scribed systems for keyboard instruments that came close to equal temperament, and the correct application of Lanfranco's tuning rules must have resulted in equal temperament itself. But these systems were anomalous for a day when few acciden- tals were written. Examples of organ music from the late fif- teenth and the entire sixteenth century are found in numerous collections, such as Schering's Alte Meister aus der Friihzeit desOrgelspiels; Volume 1 of Bonnet's Historical Organ Recitals; Kinkeldey's Orgel und Clavier in der Musik des 16. Jahrhun- derts; Volume 1 of Margaret Glyn's Early English Organ Music; Volume 3 of Torchi's L'arte musicale in Italia; Wasielewski's Geschichte der Instrumentalmusik im 16. Jahrhundert; Volume 6 of the Italian Classics series. With the exception of the English composers, the compass used by all these composers was less than 12 notes — ED-F* or Bb-C . Both Tallis and Redford had D# in one piece and Eb in another, thus posing a problem with regard to the tuning. But except for them, there was no problem about performance: all 188 FROM THEORY TO PRACTICE of this organ music could have been played on an instrument in meantone temperament. Even 12 of Schlick's 14 little pieces (Monatshefte fur Musik- geschichte, 1869) lie within a compass of Eb-C#. One of the re- maining pieces has an A^; the other, G#. Since Schlick had di- rected that the wolf be divided equally between the fifths C*-G* and Ab-ED, these notes would have caused him no difficulty. Perhaps Tallis and Redford were dividing the error similarly. Much the same can be said for the clavier music of this pe- riod. Merian's Per Tanz in den deutschen Tabulaturbuchern (Leipzig, 1927) contains about 200 tiny keyboard pieces, and Vol- ume 2 of Bbhme's Geschichte des Tanzes about 20 more. None exceeds the Eb-G# compass. The famous English collection of virginal music, Parthenia, reveals nothing beyond the fact that Byrd preferred"!^ the younger composers Bull and Gibbons, D*. In Margaret Glyn's edition of Gibbons' Complete Keyboard Works, five of the 33 virginal pieces have a D#, but only two con- tain Eb's, one of these, a Pavan in G minor, having also an AD. But that does not necessarily mean that Gibbons did not use the meantone temperament. The virginals could have been set for an A13 at one time and for a D* at another — a point that will be discussed at some length later. More significant are the A^ and D# that occur in a G minor Fancy for organ by Gibbons. Unless Gibbons' tuning was appreciably better than the meantone tem- perament, this Fancy would have had some very rough places. This same Ab-D# was used in Tarquinio Merula's Sonata Cro- matica, a work having a modern ring because of its chromati- cism.3 Just a word about chromaticism. Other things being equal, a piece that contains many chromatic progressions is more likely to have an excessive tonal compass than one that is not chro- matic. But, since there are 12 different pitch names in the mean- tone compass, Eb-G#, it is entirely possible for a chromatic piece to lie within it. A Toccata by Michelangelo Rossi, for ex- ample, published in 1657, is very chromatic, but carefully re- 3Luigi Torchi, L/arte musicale in Italia (Milan, post 1897), III, 345-352. 189 TUNING AND TEMPERAMENT mains within the meantone bounds.4 The great English manuscript source of the early seventeenth century, the Fitzwilliam Virginal Book, is a monument to the boldness of the clavier composers of that time. Naylor^ has given a fascinating and exhaustive account of the music in this collection, and has shown that many of the progressions contain- ing accidentals resemble modulations to our major and minor keys more than they do modal cadences. Twenty-five of the 297 compositions contain D*'s, with Bull, Byrd, Farnaby, and Tom- kins in the lead0 Bull, Farnaby, Tisdall, and Oystermayre have A#'s also. With one exception, the largest compass in the entire collec- tion is that of Byrd's "Ut, re, mi, fa, sol, la," which extends from AD to D*0 That exception, of course, is John Bull's com- position on the hexachord, with the same title as Byrd's. It over- laps the circle of fifths by six notes, with the compass CD-A*. Bull states his Canto Fermo first on G and rises by tones through A, B, Db, Eb, and F. He then begins afresh with Ab, Bb, C, D, E, F% and G. An enharmonic modulation occurs at the begin- ning of Section 4, where the chord of F# is quitted as GD. The editors of the Fitzwilliam Virginal Book were so impressed with this passage that they correctly stated in a footnote, "This inter- esting experiment in enharmonic modulation is thus tentatively expressed in the MS.; the passage proves that some kind of 'equal temperament' must have been employed at this date." 6 This remarkable composition is not a mere juggling with sounds, as Nay lor has alleged. It has real musical interest, and because of its sustained style seems better adapted to the organ than to the clavier. But do not try to build up a theory of the use of equal temperament in England during Queen Elizabeth's reign on the basis of Dr. Bull's composition. Remember that it stands practically alone. It seems almost as if Bull had written a Fancy for four viols, and then, led by some mad whim, had transcribed 4 Ibid., p. 309. ^An Elizabethan Virginal Book (London, 1905). "J. A. Fuller-Maitland and W. Barclay Squire, The Fitzwilliam Virginal Book (Leipzig, 1899), I, 183. 190 FROM THEORY TO PRACTICE it for virginals and tuned his instrument to suit. One of the boldest of the keyboard composers of the early seventeenth century was Frescobaldi, an exact contemporary of Gibbons. Of his 31 works for organ and clavier, • three contain DD, three a D#, and one an A*. One of the most interesting of these is the Partite sopra Passacagli for organ, with a compass of D^-G*. The G* is the third of the dominant triad of A minor, and the D*3 the third of the subdominant triad of F minor. Hence the ordinary meantone temperament would be inadequate for Frescobaldi. In decided contrast to Frescobaldi are Sweelinck (German Denkm'aler, IV Band, 1. Folge) and Scheidt (German Denkmaler, I Band). Sweelinck' s Fantasia Cromatica, with E°-D# compass, was the only one of 36 pieces examined to exceed 12 scale de- grees, and Scheidt, although not averse to chromaticism and rather fond of D*'s, had no single composition, of 44 examined, with more than 12 degrees. As we reach the middle of the seventeenth century, we shall have to differentiate more carefully between music for organ and for clavier. The organ had a fixed compass, usually Eb-G#, but perhaps B^-D* or AD-C^. Even if the composer did not em- ploy AD and D#, for example, in the same composition, as Gib- bons and Merula had done, the presence of these notes in sepa- rate compositions was an indication that he was using at least a modified version of the meantone temperament." Not so for clavier. A study of the accidentals in clavier music suggests that tuning practice must have accommodated it- self to the music to be played. The performer would retune when changing from sharp to flat keys. Bach could tune his entire harpsichord in fifteen minutes; to change the pitches of only a couple of notes in each octave would have taken a much shorter time. Moreover, all the movements of the common dance suites 7I classici della musica italiana (Milan, 1919), Vol. XII; Torchi, op. cit., Vol. III. Q °The course of the argument and most of the examples in the remaining part of this section have been taken freely from my article "Bach and The Art of Temperament," Musical Quarterly, XXXIII (1947), 64-89. 191 TUNING AND TEMPERAMENT were in the same key, and this helped to restrict the compass to not more than twelve different pitch names, even if that compass was not the conventional ED-G^. The theorists give us little information about the variable tuning of claviers. Mersenne hinted at the practice^, He had given two keyboards in just intonation, the first with sharps only (except for BD) and the second with flats. Current practice, he said, was represented by either of these, but with tempered, not just, intervals^ Some eighty-five years later Kuhnau wrote to Mattheson that the strings of his Pantalonisches Cimbal (a large keyed dulcimer) vibrated so long he could not use equal temper- ament upon it, but had to "correct one key or another" when turn- ing from flats to sharps. More valuable evidence of the variable tuning practice for clavier comes from the music itself. Of Froberger's 67 clavier compositions (Austrian Denkmaler, VI, 2. Theil, and X, 2. Theil), 6 use 14 scale degrees, 10 use 13, and the remaining 51 use 12 or fewer. But only half (26) of the 51 lie wholly within the usual meantone compass. His accidentals range altogether from G^ to E#. Similarly, Johann Pachelbel's clavier music (Bavarian Denk- maler, 2. Jahrgang, 1. Band) suggests a variable tuning. Of 49 compositions examined, only 2 have more than 12 scale degrees. But of the remaining 47, only 21, or less than half, lie within the E^-G* compass, and the total range is from D*3 to B*. An ex- ception among Pachelbel's works, the Suite in AD (Suite ex Gis), beginning with anAllemand inAD minor, contains an enharmonic modulation at the point where the Fb major triad is treated as E major by resolving upon A minor, just before a cadence in E*3 major! With a range from D*30 to B for this single movement, it seems evident that for the moment Pachelbel was as reckless as Bach. Kuhnau' s works (German Denkmaler, IV Band, 1. Folge) give musical evidence of variability to buttress what he wrote to Mat- theson. Of his 28 clavier works, 3 of the 6 Biblical Sonatas have a compass of 14 scale degrees; the other 3 sonatas and 5 other works have 13. But of the remaining 17 works that have no more 192 FROM THEORY TO PRACTICE than 12 different pitches in the octave, only 2 lie wholly within the Eb-G* tuning. Actually Kuhnau preferred equal temperament upon the clavier. But most of these works would have been pass- able in meantone temperament if he had "corrected" some of the notes, just as he did on the Pantalon. Of Frangois Couperin's 27 charming suites for clavecin, only 6 have no more than 12 different scale degrees. They are all in the minor key, and in each the flattest note is a semitone higher than the keynote, as No. 8 in B minor has the compass C-E*. Twenty of the remaining 21 suites exceed the circle of fifths by one or two notes. But here again it is characteristic to have the flattest note a semitone above the tonic. For example, all five suites in D major-minor have the precise compass Eb-A#. Cou- perin leaves a strong impression that the dissonance inevitable in the slightly extended compass was a coolly calculated risk, and that a variable meantone tuning was used for these suites also. The one exception is No. 25, in Eb major and C major- minor. The compass here is 15 scale degrees, from Gb to D*. This would, perhaps, be carrying piquancy too far. There is ample evidence that in Italy during the first half of the eighteenth century equal temperament or its equivalent was being practiced. Three composers represented in the Italian Classics had, in a particular composition, a similar compass, 15 notes in the overlapping circle of fifths. They are Zipoli, Db-D#, Vol. 36; Serini, Cb-C#, Vol. 29; and Durante, Gb-G#, Vol. 11. Of 70 of Domenico Scarlatti's delightful little "sona- tas,"9 45, or more than half, overlap the circle. In one sonata he had a compass of 18 degrees, Db-B#; in another, 17, Gb-A#. All of these men upon occasion wrote notes so remote from the tonal center that meantone temperament seems wholly out of the question. Both Serini and Durante used Fx, and Scarlatti, Cx. At this time, in Germany, Telemann was advocating a form of multiple division with 55 notes in the octave, for a clavier with only 12 notes in the octave, which was practically the same as Silbermann's 1/6-comma variety of meantone temperament. We might expect, therefore, that his compositions for clavier 9Heinrich Barth, Klavierwerke von Domenico Scarlatti (4 vols.; Vienna, c. 1901). ~ 193 TUNING AND TEMPERAMENT would not exceed the bounds of the meantone temperament. How- ever, Telemann's 36 Clavier Fantasies have a total range of G"- B#, the same as for Couperin's suites. Only 8 of the fantasies overlap the circle, by one or two degrees. Of the remaining 28, only one lies within the ordinary meantone bounds, E"-G*. The others swing to the sharp side or the flat side, depending upon the key. Thus Telemann undoubtedly used the meantone temper- ament, but with variable intonation. It has been suggested in the preceding pages that composers such as Bull, Gibbons, Frescobaldi, and Domenico Scarlatti, whose works exceed the meantone bounds by several scale de- grees, were not using the meantone temperament. Were they, then, using equal temperament? That question is difficult to an- swer, especially since there was a type of tuning that would have been fairly satisfactory in many of these cases. The title of Bach's great collection of preludes and fugues, Das wohltem- perirte Clavier, has usually been taken to mean, as Parry called it, "The Clavichord Tuned in Equal Temperament." But even in Bach's day there was a good German phrase for equal tempera- ment — "die gleichschwebende Temperatur," "the equally beat- ing temperament." Bach's title might better be paraphrased, "The Well-Tuned Piano." Now, "well -tuned" had been used in a somewhat technical sense by the Flemish mathematician Simon Stevin, over a cen- tury before the first volume of the "48" was compiled in 1722, and by Bach's great French contemporary Rameau also, with a meaning nearly the same as Parry has given to it. To German theorists, however, there was a distinction. Andreas Werck- meister has erroneously been hailed as the father of equal tem- perament because of the title of one of his works on tuning, Musicalische Temperatur, and because of Mattheson's eulogy. Mattheson had said, "And thus the fame previously divided between Werckmeister and Neidhardt remains ineradicable — that they brought temperament to the point where all keys could be played without offense to the ear."^ (Underscoring is the present au- 10J. Mattheson. Critica musica. II, 162, 194 FROM THEORY TO PRACTICE thor's.) Werckmeister himself has used the phrase "wohl tem- perirt" as follows: "But if we have a well-tuned clavier, we can play both the major and minor modes on every note and transpose them at will. To one who is familiar with the entire range of keys, this affords variety upon the clavier and falls upon the ear very pleasantly." What did Werckmeister mean by these words? To use Neid- hardt's phrase, he meant a "completely circulating genus," that is, a tuning in which one could circumnavigate the circle of fifths without mal de son, Both men, as we have seen in Chapter VII, presented a number of different monochords, with the "foreign" thirds beating as much as a comma. Werckmeister said of them, "It would be very easy to let the thirds Db-F, Gb-Bb, Ab-C beat less than a full comma; but since thereby the other, more fre- quently used thirds obtain too much, it is better that the latter should remain purer, and the harshness be placed upon those that are used the least." Elsewhere Werckmeister described equal temperament with fair accuracy, but demurred, "I have hitherto not been able to approve this idea, because I would rather have the diatonic keys purer." And so to Werckmeister "well- tuned" meant "playable in all keys —but better in the keys more frequently used." If, then, a composer exceeded twelve different pitch names rarely and then only by a few scale degrees, his works could have been played to good advantage on a "well-tuned clavier." Com- posers like Bull and Pachelbel and Scarlatti, however, who ef- fected enharmonic modulations and used double sharps, would have been badly served even by Werckmeister' s best-known "correct" temperament, in which the key of Db had Pythagorean thirds for all its major triads. Equal temperament was needed for their works. -^ An equal temperament was needed for the keyboard works of] Bach, both for clavier and for organ. It is generally agreed that Bach tuned the clavier equally. Actually he was opposed to equal temperament, in the sense that there must be strict mathemati- cal ratios, which are first applied to the monochord and from there to the instrument to be tuned. Of course he was right. The best way to tune in equal temperament, as Ellis stated, is to 195 TUNING AND TEMPERAMENT count beats. Have you ever heard of a contemporary piano tuner who carried a monochordwith him? And yet the underlying the- ory must be correct or the result will be unsatisfactory: Ellis could not have given his practical tuning rule with assurance had he not been able to calculate accurately how far its use would fall short of the perfection implied by the term "equal tempera- ment." The organ works of Bach show as great a range of modulation as his clavier works do. Except for a dozen chorale preludes in the Orgelbiichlein, there are only 3 organ works of 148 examined that do not overstep the compass of the conventionally tuned or- gan. The compass of individual organ pieces is very frequently 13, 14, and 15 scale degrees, and even 18, 19, and 21 degrees have been observed. The compass of Bach's organ works as a whole is E°k-Cx, 25 degrees! In these works is a host of ex- amples of triads in remote keys that would have been dreadfully dissonant in any sort of tuning except equal temperament. For corroboration, if corroboration be necessary, we need but note the advice that Sorge gave to the instrument-maker Silbermann, two years before Bach's death. Sorge, a proponent of equal tem- perament, said: "In a word — Silbermann' s way of tempering cannot exist with modern practice. I call upon all impartial and experienced musicians — especially the world-famous Herr Bach in Leipzig — to witness that this is all the absolute truth. It is to be desired, therefore, that the excellent man [Silbermann] should alter his opinion regarding temperament. . . . * Just Intonation in Choral Music We have seen that just intonation exists in many different forms, and that the best version, if modulations are to be made to keys beyond B^ and A, comes near the Pythagorean tuning, as with Ramis. The contention has often been made that unaccom- panied voices sing in just intonation. Zarlino-^ listed instru- 11Georg Andreas Sorge, Gesprach zwischen einem Musico theoretico und einem Studioso musices, p. 21. 12Sopplimenti musicali, Chaps. 33-37. 196 FROM THEORY TO PRACTICE merits in three groups, each with a different tuning: keyboard instruments in meantone temperament; fretted instruments in equal temperament; voices, violins, and trombones in just in- tonation. His argument was that since intonation is free for these three last-named groups, they would use an intonation in which thirds and sixths are pure. Three hundred and forty-eight years later Lindsay Norden said, "As we shall show, no singer can sing a cappella in any temperament .... A cappella music, therefore, is always sung in just or untempered intonation. ^ Let us see what is implied by these statements. In the first place, singers must be able to sing the thirds and sixths purely.^ This may sound like a self-evident truth, too absurd to discuss. But scientific studies of intonation preferences show that the hu- man ear has no predilection for just intervals, not even the pure major third. ° Alexander Ellis declared that it was unreliable to tune the pure major thirds of meantone temperament directly, preferring results obtained by beating fifths. Hence the singers must be highly trained to be able to sing the primary triads of a key justly. In the second place, the singers must be able to differentiate intervals differing by the syntonic comma, 1/9 tone. We have seen that inPtolemy's version of the syntonic tuning theDminor triad, the supertonic triad of the key of C major, will be false. If, as Kornerup and others advocate, the Didymus tuning is used instead of Ptolemy's, the dominant triad will be false, which is a greater loss„ But a singer trained to niceties of intonation would have to vary his pitch by a comma in such critical places, and thus save the situation. Very good. But studies at the Uni- 13n. Lindsay Norden, "A New Theory of Untempered Music," Musical Quar- terly, XXII (1936), 218. 14Except for the reference to the Italian madrigalists, the remaining part of this section has been freely adapted from my article "Just Intonation Con- futed," Music and Letters, XIX (1938), 48-60 by permission of the editor of Music and Letters, 18 Great Marlborough Street, London, W. 1. 15Paul C. Greene, "Violin Intonation," Journal of the Acoustical Society of America, IX (1937-38), 43-44; Arnold M. Small, "Present-Day Preferences for Certain Melodic Intervals . . . ," Ibid., X (1938-39), 256; James F. Nick- erson, "Intonation of Solo and Ensemble Performance . . . ," Ibid., XXI (1949), 593-95. 197 TUNING AND TEMPERAMENT versity of Iowa1" have shown that there is no such thing as sta- bility of pitch among singers: scooping is found in almost half the attacks and averages a whole tone in extent; portamento is very common; the sustained part of the pitch varies from the true pitch by a comma or more in one-fourth of the notes ana- lyzed. If we add to these errors the omnipresent vibrato, with an average extent of a semitone, it would seem that the ambitious and optimistic director of an unaccompanied choir has an impos- sible task. Let us assume, for the moment, that it is possible for a choir to sing without these pitch fluctuations, that all its members can sing a note a comma higher or lower when necessary, and that the director has analyzed the music and marked the places where the comma shifts are to be made. What have we then? Strangely enough, if the harmony consists of simple diatonic progressions, typical of the seventeenth and eighteenth centuries, the pitch will probably falL With modal progressions, as in Palestrina, it is more likely to remain stationary. According to GustavEngel, if one were to consider possible comma shifts whenever a modu- lation occurs, most of the recitatives in Mozart's Don Giovanni would fall from one to four commas if sung unaccompanied, and the final pitch of the opera would be five or six semitones flatter than at the beginning, A or A*3 instead of D! If the music contains much chromaticism and remote modu- lations, even the best-trained choir would probably flounder. And yet there are choral compositions of the sixteenth and early seventeenth centuries that seem strikingly modern because of these very features. De Rore's madrigal "Calami sonum fer- entes" for four basses (c. 1555) begins with an ascending chro- matic scale passage treated in imitation. Later it has a re- markable faburden of inverted major triads a semitone apart — G F# G AD G. Caimo's madrigal "E ben raggion" (1585) contains a very smooth example of modulation in which the F* major triad is heard, and, 24 bars later, its enharmonic equivalent, the G" major triad. In just intonation the latter triad would be a large diesis (42 cents, or almost a quarter tone) higher than the former. 16Carl E„ Seashore, The Vibrato (Iowa City, Iowa, 1932). 198 FROM THEORY TO PRACTICE And what of Marenzio's madrigal "O voiche sospirate a mig- liornote,'' where there is a modulation around the circle of fifths from C to G", an enharmonic change from Gb to F#, and further modulation on the sharp side? According to Kroyer, from whom all these examples have been taken, this is the first time in mu- sic that the circle of fifths has been completed. *' Could Mar- enzio's madrigal have been sung in just intonation? Gesualdo has the respect of the moderns because of his har- monic freedom. The best known of his chromatic madrigals is the "Resta di darmi noia," in which he passes from G minor to E major, and then sequentially from A minor to F# major. Lis- ten to the recording of this madrigal by a group of unaccompa- nied singers in the album 2000 Years of Music and you will prob- ably agree that the attempt to record it was a noble experiment and nothing more. Of course the point that is missed by all these rabid expo- nents of just intonation in choral music is that this music was not ordinarily sung unaccompanied in the sixteenth century. A^ cappella meant simply the absence of independent accompani- ment, not of all accompaniment. If a choir usually sang motets accompanied by an organ in meantone temperament, it would quickly adapt itself to the intonation of the organ. If this choir were in the habit of singing madrigals accompanied by lutes or viols in equal temperament, its thirds would be as sharp as the thirds are today. Kroyer thought the pronounced chromaticism of the Italian madrigalists showed the influence of keyboard in- struments. On the contrary: it must have been the fretted in- struments, already in equal temperament, that influenced com- posers like de Rore, Caimo, Marenzio, and Gesualdo to write passages in madrigals that could not have been sung in tune with- out accompaniment. Present Practice What is tuning like today? A generation ago, Anglas made some excellent observations about the intonation of the symphony i n 1(Theodor Kroyer, "Die Anfange der Chromatik im italienischen Madrigal des XVI. Jahrhunderts," Internationale Musikgesellschaft, Beiheft 4 (1902). 199 TUNING AND TEMPERAMENT orchestra. 1° The pedals of the harp are constructed to produce the semitones of equal temperament; therefore, once the harp is put in tune with itself, it, and it alone of all the instruments, will be in equal temperament The violins show a tendency toward the Pythagorean tuning, both because of the way they are strung and because of the players' tendency to play sharps higher than enharmonic flats. Furthermore, in a high register both the vio- lins and the flutes are likely to play somewhat sharp for the sake of brilliance. He might have added that the brass instruments, making use of a more extended portion of the harmonic series than the woodwinds, have a natural inclination toward just into- nation in certain keys. The result is "a very great lack of pre- cision," with heterogeneous sounds that are a mixture of "just, Pythagorean, tempered, or simply false." Of course the ears of the audience, trained for years to endure such cacophony, ac- tually are pleased by what seems to be a good performance. LI. S. Lloyd has written an article with the frightening title "The Myth of Equal Temperament."1^ It would be pretty dis- couraging for the present author to have done extended research upon the history of equal temperament only to learn at last that his subject matter was in the class with the story of Cupid and Psyche! But Lloyd has not actually consigned equal temperament to the category of the tale of George Washington and the cherry tree. His argument is against rigidity of intonation, the rigidity that is inherent in any fixed system of tuning. He holds that the players in a string quartet or the singers in a madrigal group are likely to be guided by the music itself as to what intonation to use, sometimes approaching Pythagorean intervals when me- lodic considerations are paramount or just intervals when the harmony demands it. And undoubtedly this freedom of intonation, plus a well-defined vibrato, does increase the charm of these more intimate chamber ensembles. Not even the piano is exempt from the charge of inexactness. Three-quarters of a century ago Alexander Ellis showed that the 18J. P. L. Anglas, Precis d' acoustique physique, musical, physiologique (Paris, 1910), p. 206. 19Music and Letters, XXI (1940), 347-361. 200 FROM THEORY TO PRACTICE best British tuners of his day failed to tune pianos in equal tem- perament within desirable limits of error. There is no reason to believe that modern British tuners, or American ones either, are doing a better job than was done then. Schuck and Young even show that, because of the inharmonicity of the upper par- tials of the piano, a tuner is bound to tune the upper octaves pro- gressively sharper and the lowest octaves progressively flatter than those in the middle range. *® Their theoretical findings agree with measurements Railsback had already made of pianos tuned in equal temperament. However, the psychologists tell us that "stretched" octaves at top and bottom are a concomitant of normal hearing. Therefore the sharpness and the flatness re- spectively would probably be heard as correct intonation. Now all of this paints a dismal picture. Apparently nobody — not the pianist, nor the singer, nor the violinist, nor the wind- player — is able to perform in correct equal temperament. The harpist is left sitting alone, but no doubt he will be joined by the Hammond organist, whose instrument comes closest to the equal tuning. This contemporary dispute about tuning is perhaps a tempest in a teapoto It is probably true that all the singers and players are singing and playing false most of the time. But their errors are errors from equal temperament. No well-informed person today would suggest that these errors consistently resemble de- partures from just intonation or from any other tuning system described in these pages. Equal temperament does remain the standard, however imperfect the actual accomplishment may be. The trend of musical composition during the late nineteenth and the first half of the twentieth century has been to exploit the resources of equal temperament, of an octave divided into 12 equal parts, and hence also into 2, 3, 4, or 6 parts. To ascertain how far back this trend extends is not the purpose of this book. It would be foolish to deny that this modern trend is different in kind from the progressions of classic harmony, progressions that were almost as common in 1600 as in 1800. But it may be denied that these classic progressions were intimately connected 20O. H. Schuck and R. W. Young, "Observations on the Vibrations of Piano Strings," Journal of the Acoustical Society of America, XV (1943), 1-11. 201 TUNING AND TEMPERAMENT with the meantone temperament, as has often been alleged; for we have seen that the original 1/4 -comma meantone system did not even reign supreme in 1600, much less in 1700 or 1750. In 1600 there were half a dozen or more ways to tune the octave; in 1732 Neidhardt gave his readers a choice of twenty! Moreover, there is every reason to believe that in practice there were far greater departures from these extremely varied tuning methods of the seventeenth and eighteenth centuries than there are from equal temperament today. In the very nature of things, equal temperament has undergone vicissitudes during the last four hundred years, and will continue to do so. Perhaps the philosophical Neidhardt should be allowed to have the last word on the subject: "Thus equal temperament carries with itself its comfort and discomfort, like the holy es- tate of matrimony." 21 ^*Gantzlich erschopfte, mathematische Abtheilungen, p. 41. 202 LITERATURE CITED LITERATURE CITED Adlung, Jacob. Anleitung zu den musikalischenGelahrtheit. Er- furt, 1758. Musica mechanica organoedi. Berlin, 1768. Agricola, Martin. Musica instrumentalis deudsch. 4th ed., Wit- tenberg, 1545. Reprinted as Band 20 of Publikation alterer praktischer und theoretischer Musikwerke, 1896. Rudimenta musices. Wittenberg, 1539. Amiot, Pere Joseph Maria. De la musique des ChinoiSo (Me- moires concernant Thistoire, . . , des Chinois, Vol. VI.) Paris, 1780. Anglas, Jules Philippe Louis. Precis d'acoustique physique, musical, physiologique. Paris, 1910. Arieh Das Relativitatsprincip der musikalischen Harmonie, Band I. Leipzig, 1925. Aristoxenus. See H. S. Macran, The Harmonics of Aristoxenus. Oxford, 1902. Aron, PietrOo Toscanello in musica. Venice, 1523. Revised edition of 1529 was consulted. Artusi, Giovanni Maria. Seconda parte dell' Artusi overo della imperfettioni della moderna musica. Venice, 1603. Awraamoff, A. M. "Jenseits von Temperierung und Tonalitat," Melos, I (1920), 131-134, 160-166, 184-188. Bach, Johann Sebastian. For his keyboard works see particularly Vols. Ill, XIII, XIV, XV, XXV, and XXXVIII of the Bach-Gesellschaft Edition, Leipzig, 1851-1900; fac- simile edition, Ann Arbor, 1947. Bakfark, Valentin. (His lute music is reprinted in Denkmaler der Tonkunst in Osterreich, Vol. XVHI.) Barbour, J. Murray. "Bach and The Art of Temperament, " Mu- sical Quarterly, XXXIII (1947), 64-89. "Irregular Systems of Temperament," Journal of the American Musicological Society, I (1948), 20-26. 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Bottrigari, Ercole. II desiderio, owero de* concerti di varii stromenti musicali. Venice, 1594. New ed. byKathi Meyer, Berlin, 1924. Buttstett, Johann Heinrich. Kurze Anfuhrung zum General-Bass. 2nd ed. Leipzig, 1733. Caramuel (Juan Caramuel de Lobkowitz), Mathesis nova. Cam- pania, 1670. Cardano, Girolamo. Opera omnia, ed. Sponius. Lyons, 1663. Caus, Salomon de. Les raisons des forces mouvantes avec di- verses machines, Francfort, 1615. Cavazzoni, Girolamo. Composizioni. Reprinted in I classici della musica italiana, Vol. VI. Milan, 1919. Choquel, Henri Louis. La musique rendue sensible par la m4- chanique. New ed., Paris, 1762. Colonna, Fabio. La sambuca lincea. Naples, 1618. Couperin, Frangois. Pieces d'orgue. Paris, 1690. Reprinted as Tome VI, Oeuvres completes, Paris, 1932-33. Courant, Maurice. "Chine etCoree," Encyclopedic de la musique et dictionnaire du conservatoire (Paris, 1913), Parti, Vol. I, pp. 77-241. Crotch, William. Elements of Musical Composition. London, 1812. 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(Reference in Opelf s Allgemeine Theor ie der Musik.) Sabbatini, Galeazzo. Regola facile e breve per sonare sopra il basso continuo. Venice, 1628. (Reference in Kir- cher's Musurgia universalis.) Sachs, Curt (ed.). Two Thousand iears of Music (record album). London, 1931. Salinas, FranciscOo De musica libri VII. Salamanca, 1577. Sancta Maria, Tomas de. Arte de taner fantasia. Valladolid, 1565. 214 LITERATURE CITED Sauveur, Joseph. "Systeme general des intervalles des sons," Me'moires de Tacade'mie royale des sciences (1701), pp. 403-498. "Table geWrale des systemes tempe're's de musique," Me'moires de l'acade'mie royale des sciences (1711), pp. 406-417. Scarlatti, Domenico. Klavierwerke, ed. Heinrich Barth. 4 vols. Vienna, c. 1901. Scheidt, Samuel. Tabulatura nova. 3 vols. Hamburg, 1624. Re- printed in Denkm'aler deutscher Tonkunst, 1. Band. Schering, Arnold. Alte Meister aus der Frlihzeit des Orgelspiels. Leipzig, 1913. Schlick, Arnold. Tablaturen etlicher Lobgesang und Lidlein. 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GesprSch zwischen einem Musico the- oretico und einem Studioso musices. Lobenstein, 1748. 215 TUNING AND TEMPERAMENT Spitta, Philipp. Johann Sebastian Bach. 2 vols. Translated by Clara Bell and J. A. Fuller-Maitland. London, 1884. Vol. I. Stanhope, Charles, Earl. "Principles of the Science of Tuning In- struments with Fixed Tones," Philosophical Maga- zine, XXV (1806), 291-312. Stevin, Simon. Van de Spiegeling der Singconst. A manuscript work (c. 1600) edited by D. Bierens de Haan. Am- sterdam, 1884. Str'ahle, Daniel P. "Nytt pafund, til at finna temperaturen, i stamningen for thonerne pa claveretock dylika in- strumenter," Proceedings of the Swedish Academy, IV (1743), 281-291. Sulzer, Johann Georg. Allgemeine Theorie der schbnen Klinste. Leipzig, 1777-79. Suremain-Missery, Antoine. The'orie acoustico-musicale. Pa- ris, 1793. Sweelinck, Jan Pieters. (His compositions for organ and clavier are in Denkmaler deutscher Tonkunst, IV Band, 1. Folge.) Tanaka, Shohe\ "Studien im Gebiete der reinen Stimmung." Vierteljahrsschrift fur Musikwissenschaft, VI (1890), 1-90. Telemann, Georg Philipp. Drei Dutzend Klavier-Fantasien, ed. MaxSeiffert. 3rd ed., Kassel, 1935. Tempelhof, Georg Friedrich. Gedanken iiber die Temperatur des Herrn Kirnberger. Berlin and Leipzig, 1775. Thompson, Gen. Perronet. On the Principles and Practice of Just Intonation. 9th ed., 1866. (Reference in Helm- holtz.) Torchi, Luigi. L'arte musicale in Italia, Vol. III. Milan, post 1897. Varella, Domingos de S. Jose. Compendio de musica. Porto, 1806. Verheijen, Abraham. See Stevin' s Van de Spiegeling der Sing- const. 216 LITERATURE CITED Vicentino, Nicola. L'antica musica ridotta alia moderna prat- tica. Rome, 1555. Warren, Ambrose. The Tonometer. London, 1725. Wasielewski, Joseph Wilhelm von. Geschichte der Instrumental - musik im 16. Jahrhundert. Berlin, 1878. Werckmeister, Andreas. Hypomnemata musica. Quedlinburg, 1697. Musicalische Paradoxal-Discourse. Quedlinburg, 1707. Musicalische Temperatur. Frankfort and Leipzig, 1691. White, William Braid. Piano Tuning and Allied Arts. 4th ed., Boston, 1943. Wiese, Christian Ludwig Gustav, Baron von. Klangeintheilungs- , Stimmungs- und Temperatur-Lehre. Dresden, 1793. Williamson, Charles. " Frequency Ratios of the Tempered Scale," Journal of the Acoustical Society of America, X (1938), 135-136. Yasser, Joseph. A Theory of Evolving Tonality. New York, 1932. Young, Thomas. "Outlines of Experiments and Inquiries Re- specting Sound and Light," Philosophical Transac- tions, XC (1800), 106-150. Zacconi, LodovicOo Prattica di musica, Part I. Venice, 1592. Zarlino, Gioseffo. Dimostrationi armoniche. Venice, 1571. Istitutioni armoniche. Venice, 1558. Sopplimenti musicali. Venice, 1588. Zeising, Heinrich. Theatri machinarum. Altenburg, 1614. Zipoli, Domenico. Composizioni per organo e cembalo. Re- printed in I classici della musica italiana, Vol. XXXVI. Milan, 1919. 217 INDEX abacus Triharmonicus, 109 f. Abrege des regies de l'harmonie. See Levens. Adlung, J., 85. Agricola, M., 4, 10, 95, 149-151. Akustik. See C. E. Schumann. Alexander the Great, 122. Allgemeine Theorie der Musik. See F. W. Opelt. Allgemeine Theorie der Schonen Kiinste. See J. G. Sulzer. Amiot, J. M., 77. "Die Anfange der Chromatik im italienischen Madrigal des XVI. Jahrhunderts." See T. Kroyer. Anglas, J. P. L., 197 f. Anleitung zu den musikalischen Gelahrtheit. See J. Adlung. Anleitung zur Tonometrie. See J. D. Berlin. Anonymous, author of Exposition de quelques nouvelles vues mathematiques, 125. Anonymous, author of Pro clavichordiis fa- ciendis, 91 f. L'antica musica ridotta alia modernaprattica. See N. Vicentino. Anweisung wie man Claviere ... stimmen konne . See B. Fritz, approximations to equal temperament. See temperament, equal: geometrical and mechanical approximations, and numerical approximations . approximations to the meantone temperament. See temperament, meantone: approxima- tions. Appun, G., 119. Arabian scale. See multiple division: equal divisions: 17-division. Archicembalo, 27, 115 f, 152. Archicymbalam, 106. Archimedes, 34, 50. Archytas, 16 f, 19, 22 f, 143. Ariel, 113. Aristoxenus, 2, 16 f, 19, 22-24, 57. arithmetical division. See division, arithmet- ical. L'arithmetique des musiciens. See J. E. Gallimard. Aron, P., 10, 26, 49. L'art du facteur d'orgues. See F. Bedos. Arte de taner fantasia. See T. de Sancta Maria. L'arte musicale in Italia. See L. Torchi. Artusi, G. M., 8, 10, 46,_IT2, 144-148, 176, 186. Awraamoff, A. M., 24, 152. "Bach and The Art of Temperament." See J. M. Barbour. Bach, C. P. E., 47 f. Bach, J. N., 85-87. Bach, J. S., 10, 12 f, 85-87, 146, 189, 192-194. Bakfark, V., 185. Ballet comique de la reine, 8. Barbour, J. M., 3, 77, 112, 131, 161, 189-197. Barca, A., 42 f. Bartolus, A., 142 f. Beaugrand, J. de, 79, 81, 84. Bedos, F-, 125. Beer, J., 124. bells, 7, 142. Bendeler, J. P., 157-160. Berlin, J. D., 119. Bermudo, J., 3, 5, 46, 137, 162-164, 186. Bertelsen, N. P. J., 127. Beste und leichteste Temperatur des Mono- chordi. See J. G. Neidhardt. Blankenburg, Q. van, 105 f, 118. Bohme, F. M., 187. Boethius, A. M. S., 3, 121. bonded clavichord. See clavichord, bonded. Bonnet, J., 186. Bosanquet, R. H. M., 9, 32, 114, 117, 119-121, 123, 125-127, 129 f, 131. Bossier, H. P., 49 f. Bottrigari, E., 8, 46, 147 f. Boulliau, I., 54, 79-81. Bumler, G. H., 80. Bull, J., 187 f, 192 f. Buttstett, J. H., 105. Byrd, W., 188. Cahill, T., 74. Caimo, J., 196 f. Caramuel, J., 3. Cardano, G., 57. Caus, S. de, 11, 95 f, 101. cembalo. See keyboard instruments. cent, ii and passim. Cerone, P., 46. Cherubini, M. L-, 58. China, 7, 55 f, 77-79, 122, 150 f. "Chine et Coree." See M. Courant. Choquel, H. L., 152. choral music. See just intonation in choral music. chromatic genius. See Greek tunings. chromaticism, 187-189, 196 f. circle of fifths, 106, 188-194, 197. circulating temperaments. See irregular sys- tems: circulating temperaments. clavichord. See keyboard instruments. , bonded, 30 f, 145-147. clavier. See keyboard instruments. closed system. See temperament, regular. Cogitata physico-mathematica. See M. Mer- senne. Colonna, F., 23 f, 151-154. 219 TUNING AND TEMPERAMENT column of differences. See tabular differ- ences. comma, i and passim. See especially irregu- lar systems: divisions of ditonic comma. Compendio de musica. See D. Varella. Compendio del trattato de' generi, e de' modi- See G. B. Doni. A Compleat Method See G. Keller. The Compleat Tutor for the Harpsichord or Spinet. See P. Prelleur. continued fractions, 54, 74, 124, 128-130. correspondences between equal multiple divi- sions and varieties of meantone tempera- ment, 124. Couperin, F., 191 f. Courant, M., 55 f, 122, 150. Critica musica. See J. Mattheson. Crotch, W., 31 f. cube root. See duplication of the cube; loga- rithms; mesolabium. Cursus seu mundus mathematicus. See R. P. C. F. M. Dechales. De institutione musica. See A. M. S. Boethius. De la musique des Chinois. See J. M. Amiot. De musica libri VII. See F. Salinas, decameride, 120, 134. Dechales, R. P. C. F. M., 25, 36. Declaracion de instrumentos musicales. See J. Bermudo. Delannoy, 58. Delia imperfettioni della moderna musica. See G. M. Artusi. Delezenne, C. E. J., 58, 120. Denis, J., 47. Descartes, R., 53. U desiderio. See E. Bottrigari. deviation, iii and passim. Dialogo della musica antica e moderna. See V. Galilei, diaschismata, 121. diatonic genus. See Greek tunings. Dictionnaire de musique. See J. J. Rousseau. Didymus, 2, 18, 20 f, 23, 887109, 152, 195. diesis, 10, 108, 196. Dimostrationi armoniche. See G. Zarlino. Discorsi ... e due nuove scienze. See G. Gal- ilei, ditone, 21 f, 115. ditonic comma, i and passim. See especially irregular systems: divisions of ditonic comma, division, arithmetical, 21, 29 f, 56 f, 60-64, 68-73, 80, 83, 85, 99, 139-144, 150, 155, 162, 175. division, geometrical, 38 f, 80, 92 f, 150, 154. See also Euclidean construction. Don Giovanni. See W. A. Mozart. Doni, G. B., 109 f. Douwes, C, 30, 146. Dowland, J., 4, 24, 151, 153. Dowland, R., 151. Drobisch, M. W., 37, 114, 120, 123, 125, 127- 129. dulcimer, 112, 190. duplication of the cube, 49. Dupont, W., 91, 105. Durham Cathedral, 106. "Eine mathematisch-harmonische Analyse des Don Giovanni von Mozart." See G. Engel. Eitz, K. A. ii. Elementa musica. See Q. van Blankenburg. Elementa musicalis. See Faber Stapulensis. Elementarbuch der Tonkunst. See H. P. Bossier. An Elementary Treatise on Musical Intervals and Temperament. See R. H. M. Bosanquet. Elements of Musical Composition. See W. Crotch. An Elizabethan Virginal Book. See E. W. Naylor. Ellis, A. J., ii, 48 f, 65, 73-76, 87, 111, 131, 146, 193-195, 198 f. Elsasz, 113. Engel, G., 196. enharmonic genus. See Greek tunings. Enharmonic Organ, 110 f. Enharmonium, 111. eptameride, 120. equal temperament. See temperament, equal. Eratosthenes, 16, 18 f, 22 f. Erlangen University Library, 91. Escuela musica. See P. Nassarre. espinette, 47. An Essay upon Perfect Intonation. See H. Lis- ton. Euclidean construction, 27, 50, 52-55, 93, 137, 142, 144. Euharmonic Organ, 110. Euler, L., 65, 100 f, 109 f. Eutocius, 50. exponents, ii, 95, 102. Exposition de quelques nouvelles vues mathe- matiques dans la theorie de la musique, 125. Exposition d'une nouvelle methode pour l'en- seignement de la musique. See P. Galin. extensions of just intonation. See multiple di- vision: extensions of just intonation. Extract aus der neuen Musica oder Singkunst. See D. Hizler. Faber Stapulensis (Jacques le Febvre), 137, 144. Faggott, J. 65-67. 220 INDEX Farey, J., 65, 134. Farnaby, G., 188. Faulhaber, J., 78, 80. Febvre, J. le. See Faber Stapulensis. Fetis, F. J., 114. Fibonacci series, 115. Fischer, J. P. A., 7. Fitzwilliam Virginal Book, 188. Fludd, R., Frontispiece, 3. flute. See wind instruments. Fogliano, L., 11, 92-96, 104, 106. Foundling Hospital, 106. "Frequency Ratios of the Tempered Scale." See C. Williamson. Frescobaldi, G., 189, 192. fretted clavichord. See clavichord, bonded. fretted instruments, 6-8, 11, 25, 28, 40, 42, 45 f, 50, 57-59, 98 f, 103, 139-142, 144-149, 151, 162-164, 182-186, 188, 197. fretted instruments, in sixteenth century paint- ings, 12. Fritz, B., 47 f. Froberger, J. J., 190. Fronimo. See V. Galilei. Fuller-Maitland, J. A., 188. Gafurius, F., 3, 5, 25. Galilei, G., 11. Galilei, V., 8, 46, 57-64, 149, 184. Galin, P., 31. Galle (J. B. Gallet), 79, 81, 84. Gallimard, J. E., 12, 118, 134 f, 137 f. Ganassi, S., 10, 24, 68, 70, 139-143, 154, 176, 186. Gantzlich erschopfte, mathematische Abthei- lungen des diatonisch-chromatischen, tem- perirten Canonis Monochordi. See 3. G. Neidhardt. Garnault, P., 58 f. Gedanken iiber die Temperatur des Herrn Kirnberger. See G. F. Tempelhof. generalized keyboard, 9, 117, 130. G^ne'ration harmonique. See J. P. Rameau. geometrical approximations. See tempera- ment, equal: geometrical and mechanical approximations. geometrical division. See division, geometri- cal. Geschichte der Instrumentalmusik im 16. Jahrhundert. See J. W. Wasielewski. Geschichte der musicalischen Temperatur. See W. Dupont. Geschichte der Musiktheorie. See H. Riemann. Gesprach zwischen einem Musico theoretico und einem Studioso musices. See G. A. Sorge. Gesualdo, C., 197. Gibbons, O., 187, 189, 192. Gibelius, O., 29, 85. Gintzler, S., 185. Glyn, M. H., 186. Goetschius, P., 4. golden system, 127. Das goldene Tonsystem ... . See T. Kornerup. Gonzaga, 117. "good" temperaments. See irregular systems: circulating temperaments. Gow, J., 50 f. Grammateus, H., 3, 6, 10, 13, 25, 46, 69, 137- 139, 142, 144, 151, 157, 176, 186. Greek tunings, 15-24. diatonic, 15, 19-21. chromatic, 15, 17 f, 21. enharmonic, 15 f, 21, 109, 115. enharmonic, modern, 15,33 f, 188, 190, 193, 198. enharmonic of Salinas, 106 f. Greene, P. C., 195. Grondig Ondersoek van de Toonen der Musijk. See C. Douwes. Guido of Arezzo, 25, 36. guitar. See fretted instruments. hackebort. See dulcimer. Haser, A. fTTTOI. Hammond, L., 74-76. Hammond Electric Organ, 74-76, 199. Handel, G. F., 10, 106. Harmonices mundi. See J. Kepler. Harmonicorum libri xn. See M. Mersenne. Harmonicorum libri tres. See C. Ptolemy. Harmonics. See R. Smith. Harmonie universelle. See M. Mersenne. harmonium. See keyboard instruments. harp, 198 f. harpsichord. See keyboard instruments. Harrison, 40 f. Hawkes, W., 133 f, 137 f. Helmholtz, H. L. F., 48, 73 f , 89, 109 f. Henfling, K., 120. Hero (Heron) of Alexandria, 53. Herschel, J., 129. Hindoo scale. See multiple division: equal divisions: 22-division. Histoire des mathe'matiques. See J. E. Mon- tucla. Histoire generate de la musique. See F. J. Fetis. Hizler, D., 118. H& Tchheng-thyen, 55 f, 150. Holbein, H., 12. Holder, W., 47, 123. Hugo de Reutlingen, 88. Hutton, C., 50. Huyghens, C., 9, 37, 116-118. Hypomnemata musica. See A. Werckmeister. 221 TUNING AND TEMPERAMENT Ingenieurschul. See J. Faulhaber. inharmonic ity of upper partials, 199. instruments. See fretted instruments, key- board instruments, stringed instruments, wind instruments; also, bells, dulcimer, harp. "Intonation of Solo and Ensemble Perfor- mance." See J. F. Nickerson. "Introduzione a una nuova teoria di musica." See A. Barca. irregular systems, 4, 6, 10, 24, 32, 131-182, 200. irregular systems: circulating temperaments: 12 f , 155, 165, 168, 176-182, 192 f. irregular systems: circulating temperaments: Out-of-Tune Piano, 161. irregular systems: circulating temperaments: temperament by regularly varied fifths, iii, 179-182. irregular systems: divisions of ditonic com- ma, 12, 44, 84, 154-175. 1/2 comma, 138, 155-157; 1/3, 157-160; ~tj\, 159 f; 1/5, 159-161; 1767 161-164, 170, 179; 1/6, l/i, 171-173; 177, 164; 1/12, 1/4, 171 f; 1/12, 1/6, 136, 165-172, 178 f; 1/12, 1/6, 1/4", 173-175; 1/12, 3/16, 166; 1/12, 5/24, 170 f. irregular system: Metius' system, 175 f. irregular systems: modifications of regular temperaments, 24, 137-149. irregular systems: modifications of regular temperaments: Pythagorean, 46, 137 f, 186. irregular systems: modifications of regular temperaments: just, 139-144, 186. irregular systems: modifications of regular temperaments: meantone, 12, 46, 131-135, 144-149, 179, 186. irregular systems: temperaments largely Pythagorean, 12, 61, 149-154. "Irregular Systems of Temperament." See J. M. Barbour. Istitutioni armonische. See G. Zarlino. Jackson, W., 125. Jacobi, K. G. J., 129. Jank6, P. von, 120. Jeans, J., 4. "Jenseits von Temperierung und Tonalitat." See A. M. Awraamoff. Johann Sebastian Bach. See P. Spitta. Judenkunig, H., 185 f. just intonation, 2, 4, 9-11, 21, 88-104, 121, 131, 176, 199. just intonation, extensions of. See multiple division: extensions of just intonation. just intonation in choral music, 194-197. just intonation, modifications of. See irregu- lar systems: modifications of regular temperaments. just intonation, theory of, 101-104. "Just Intonation Confuted." See J. M. Barbour. Keller, G., 47. Kepler, J., 11, 58-60, 96 f, 99. keyboard instruments, 6-9, 25-48, 61, 89, 97-99, 103, 105-123, 130, 135 f, 139, 142, 147-149,186-195,197-199. See also abacus Triharmonicus, Archicembalo, Archicym- balam, Enharmonic Organ, Enharmonium, Euharmonic Organ, Hammond Electric Or- gan, Pantalonisches Cimbal, Telharmoni- um. King FSng; 122. Kinkeldey, O., iv, 26, 45, 186. Kircher, A., 9, 52 f, 108, 115, 122-124. Kirnberger, J. P., 12, 64 f, 155-157, 163. Kbrte, O., 183. Klangeintheilungs-, Stimmungs- und Temper- atur-Lehre. See C. L. G. von Wiese. Kornerup, T., 113, 121, 127 f, 195. Kroyer, T., 197. Kuhnau, J., 190 f. Die Kunst des reinen Satzes in der Musik. See J. P. Kirnberger. Kurze Anflihrung zum General- Bass. See J. H. Buttstett. La Laurencie, L. de, 71. Lambert, J. H., 65. Lanfranco, G. M., 11, 45 f, 186. "Laute und Lautenmusik bis zur Mitte des 16. Jahrhunderts." See O. Korte. Levens, 143-145. least squares, 127 f. linear correction. See division, arithmetical. Liston, H., 110 f. Lloyd, LI. S., 198. Lobkowitz. See J. Caramuel. logarithms, 3, 9, 30, 41, 64, 72 f, 77-79, 116, 118, 120, 128 f, 134, 154, 176. lute. See fretted instruments. madrigal, 144, 196 f. Malcolm, A., 3, 99 f, 141, 143. Manuel du luthier. See J. C. Robet-Maugin. Marchettus of Padua, 118. Marenzio, L., 144, 197. Marinati, A., 45 f. Marpurg, F. W., ii, 12, 43 f, 53, 65, 68, 84, 88, 98-100, 103 f, 139, 143, 154-182 (pas- sim). Mathematical Dictionary. See C. Hutton. Mathesis nova. See J. Caramuel. Mattheson, J., 124, 192. Marziale, M., 12. mean proportional. See Euclidean construc- tion. 222 INDEX mean-semitone temperament. See irregular systems: modifications of regular temper- aments. meantone temperament. See temperament, meantone. mechanical approximations. See tempera- ment, equal: geometrical and mechanical approximations. Meckenheuser, J. G., 80, 83. Melder, J., 78. "Memoire sur les valeurs numeriques des notes de la gamme." See C. E. J. Dele- zenne. "Memoire th£orique & pratique sur les sys- temes tempe're's de musique." See J. B. Romieu. Mercadier, J. B., 135, 166 f. Mercator, N., 9, 118, 123. Merian, W., 187. meride, 120. Mersenne, M., iii, 7, 9, 11 f, 48, 51-55, 58 f, 61, 74, 79 f, 84, 87, 97-99, 103, 106-108, 118 f, 122, 132-134, 137, 183, 190. Merula, T., 187, 189. mesolabium, 6, 33, 50 f, 59, 118 f, 144. Metius, A., 175 f, 181. Micrologus. See A. Ornithoparchus. Miller, D. C.,~85. Molth^e, 52. monochord, passim. Monochordum. See A. Reinhard. monopipe, 85-87. Monteverdi, C., 8. Montucla, J. E-, 89. Montvallon, A. B. de, 100 f . Morley, T., 151. Mozart, W. A., 196. multiple division, 105-130. See also split keys. multiple division: equal divisions, 111-130; 17-division, 112, 120, 123, 126, 128; 19-, 9, 34, 112-114, 120 f, 127, 129 f; 22-, 114 f, 119, 128-130; 24-, 115; 25-, 129; 28-, 129; 29-, 115, 120, 125 f, 128; 3L-, 9, 31, 36 f, 42, 51, 115-121, 127-129, 152; 34-, 37, 119, 129; 36-, 119; 41-, 119 f, 126, 128 f; 43-, 36, 118, 120, 123, 126, 128; 46-, 128; 50-, 33, 41 f, 120 f, 127; 51-, 128; 53-, 9, Tl8, 120-123, 125 f, 128-130; 55-, 42 f, 120, 122-126, 191; 56-, 119, 125 f; 58-, 125; . 65-, 125 f, 128; 67-, 122, 125; 70-, 126, 128; 74-, 37, 119, 125, 128; 77-, 126, 128; 79-, 125; 81-, 127; 84-, 125; 87^, 119, 125 f, 129; 89-, 126, 128; 9L-, 125; 94-, 126, 128; 98-, 125; 105-, 125; 112-, 125; 117-, 125; 118-, 125-129; 131-, 127; 142-, 128; 166-, 128; 212-, 127; 306-, 128; 343-, 127; 559-, 129; 612-, 129; 665-, 128 f; 817-, 129. multiple division: equal divisions: corres- pondences with varieties of meantone tem- perament, 124. multiple division: extensions of just intona- tion, 79, 106-112. multiple division, theory of, 126-130. "Music and Ternary Continued Fractions." See J. M. Barbour. Musica instrumentalis deudsch. See M. Agri- cola. Musica mathematica. See A. Bartolus. Musica mechanica organoedi. See J. Adlung. Musica practica Musica theorica See B. Ramis. See L. Fogliano. Musicae activae Micrologus. See A. Ornitho- parchus. "Musical Logarithma." See J. M. Barbour. Musicalische Paradoxal-Discourse. See A. Werckmeister. Musicalische Temperatur. See A. Werck- meister. La musique rendue sensible par la mechan- ique. See H. L. Choquel. Musurgia universalis. See A. Kircher. "The Myth of Equal Temperament." See LI. S. Lloyd. Nassarre, P., 61-64. National Gallery in London, 12. "Die natiirliche Stimmung in der modernen Vokalmusik." See M. Planck. Naylor, E. W., 188- negative system, 112-125 (passim). Neidhardt, J. G., ii, 12, 44, 78, 80, 82, 85-87, 115, 118, 154-182 (passim), 192 f, 200. New Cyclopedia. See A. Rees. Ayn new kunstlich Buech. See H. Grammateus. "A New Theory of Untempered Music." See N. L. Norden. Newton, I., 53. Nickerson, J. F., 195. Nicomedes, 53. Nierop, D. R. van, 175. "Nierop's Hackebort." See dulcimer. Norden, N. L., 195. Nouveau systeme de musique ... . See A. B. Montvallon. Nouveau systeme de musique theorique. See J. P. Rameau. Nouveau systeme de musique theorique et pratique. See J. B. Mercadier. Nova & exquisita Monochordi Dimensio. See C. Schneegass. "Novus cyclus harmonicus." See C.Huyghens. numerical approximations. See temperament, equal: numerical approximations; also, temperament, meantone: approximations. "Nytt Pifund, til at finna Temperaturen, i stamningen for thonerne pa Claveretock dylika Instrumenter." See D. P. Strahle. 223 TUNING AND TEMPERAMENT "Observations on the Vibrations of Piano Strings." See O. H. Schuck andR. W. Young. Odington, W., 3. Oettingen, A. von, 119. omega (o), 127. "On a New Mode of Equally Tempering the Musical Scale." See J. Farey. "On music." See J. Farey. "On Perfect Harmony in Music ... ." See H. W. Poole. "On Perfect Musical Intonation." See H. W. Poole. "On the History of Musical Pitch." See A. J. Ellis. "On the Musical Scales of Various Nations." See A. J. Ellis. On the Principles and Practice of Just Intona- tion. See P. Thompson. Opelt, F. W., 99, 114, 119 f. Orfeo. See C. Monteverdi. See keyboard instruments. See J. P. Bendeler. organ. Organopoeia. Orgelund Klavier in der Musik des 16. Jahr- hunderts. See O. Kinkeldey. Ornithoparchus, A., 3, 151. "Other Necessary Observations to Lute-Play- ing." See J. Dow land. "Outlines of Experiments and Inquiries Re- specting Sound and Light." See T. Young. Out-of-Tune Piano. See irregular systems: circulating temperaments: Out-of-Tune Piano. Oystermayre, J., 188. Pachelbel, J., 190, 193. paintings of the sixteenth century, 12. Palestrina, G. P. da, 196. Pantalonisches Cimbal, 190. Papius, A., 3. Pappius of Alexandria, 50. parfait diapason of Mersenne, 106-108. Parry, H., 192. Parthenia, 187. "The Persistence of the Pythagorean Tuning System." See J. M. Barbour. Pesarese, D., 33, 113. Philander, W., 118. Philo of Byzantium, 51. Philolaus, 121. The Philosophy of Musical Sounds. See R. Smith. Phrynis Mytilenaeus ... . See W.C. Printz. pi (it), 40 f, 77. piano. See keyboard instruments. Piano Tuning and Allied Arts. See W. B. White. A Plaine and Easie Introduction to Practicall Musicke. See T. Morley. Planck, M., 11. Plato, 53. Poole, H. W., 110. Populare Darstellung der Akustik. See H. Riemann. positive system, 112-125 (passim). Practica musica. See F. Gafurius. Praetorius, M., 9, 28 f, 62, 113. Prattica di musica. See L. Zacconi. Precis d'acoustique .... See J. P. L. Anglas. Predis, A. de, 12. Prelleur, P., 47. present practice of tuning, 197-200. "Present-Day Preferences for Certain Mel- odic Intervals." See A. M. Small. "Principles of the Science of Tuning Instru- ments with Fixed Tones." See C. Stanhope. Printz, W. C, 29, 37, 119, 142 f. Pro clavichordiis faciendis, 91 f. Propositiones mathematico-musicae. See O. Gibelius. Prout, E., 4. Ptolemy, C., 2, 16-23, 57, 88, 152, 195. Pythagoras, 1, 139. Pythagorean tuning, 1-4, 10, 21-23, 42, 45, 56, 59, 68, 88-91, 95, 101 f, 110-112, 121 f, 131, 147, 150 f, 176, 183, 194, 198. Pythagorean tuning, modifications of. See ir- regular systems: modifications of regular temperaments, and temperaments largely Pythagorean. Railsback, O. L., 199. Les raisons des forces mouvantes avec di- ver ses machines. See S. de Caus. Ramarin, 124. Rameau, J. P., 4, 11 f, 133, 137, 192. Ramis, B., 4, 10, 25, 88-92, 104 f, 151, 194. Redford, J., 186 f. Rees, A., 65. Regola facile e breve per sonare sopra il basso continuo. See G. Sabbatini. Regola Rubertina. See S. Ganassi. regular temperament. See temperament, reg- ular. Reinhard, A., 68, 141-143, 186. Das Relativitatsprincip der muslkalischen Harmonie. See Ariel. "Remarques sur les temperaments en mu- sique." See J. H. Lambert. Riemann, H., 25, 114, 116 f, 119 f, 135. Roberti, E. de, 12. Roberval, 52. Robet-Maugin, J. C, 58. Romberg, B., 58 f. Romieu, J. B., 37, 40, 42 f, 101, 114, 119 f, 123-126. Rore, C. da, 196 f. 224 INDEX Rossi, L., 29 f, 35 f, 51, 53, 115, 118-120. Rossi, M., 187 f. Rousseau, J., 105. Rousseau, J. J., 99 f, 104. Roussier, P. J., 4. Rudimenta musices. See M. Agricola. Ruscelli, G., 6. Sabbatini, G., 108. Sachs, C., iii, 197. St. Martin's Church in Lucca, 105 f. Salinas, F., 6, 9, 33-35, 42, 46, 50 f, 106 f, 113, 118, 186. Salmon, T., 143. Sambuca Lincea, 151-154. Sancta Maria, Tomas de, 28. Per satyrische Componist. See W.C. Printz. Sauveur, J., 112, 114, 118, 120, 123-126, 129. scale, Arabian. See multiple division: equal divisions: 17-. scale, Hindoo. See multiple division: equal divisions: 22-. scale, Siamese, 112. Scarlatti, D., 191-193. Scheidt, S., 189. A Scheme Demonstrating the Perfection and Harmony of Sounds. See W. Jackson. Schering, A., 186. schisma, 64, 80, 89, 92, 110 f, 154, 156. Schlick, A., 6, 10, 26, 46, 131, 135-139, 168, 181 f, 186 f. Schneegass, C., 37-40, 119. Schbnberg, A., 114, 183. Schola phonologica. See J. Beer. Schreyber, H. See Grammateus. Schroter, C. G., 68-73, 77. Schuck, O. H., 199. Schumann, K. E., 159. Science and Music. See J. Jeans. Scintille de musica. See G. M. Lanfranco. Scriabin, A., 113. Seashore, C. E., 196. Seconda parte dell' Artusi. See G. M. Artusi. Sectio Canonis harmonici. See J. G. Neidhardt. semi-meantone temperament. See irregular systems: modifications of regular tem- peraments: mean-semitone temperament. "Ein Sendschreiben liber Temperatur-Berech- nung." See C. G. Schroter. Senfl, L., 1857 Sensations of Tone. See H. L. F. Helmholtz; also, A. J. Ellis. Septenarium temperament, 164. Serini, G., 191. sesqui-. See superparticular ratio. Seu-ma Pyeou, 122. sexagesimal notation, 16, 79-81. A Short History of Greek Mathematics. See J. Gow. Siamese scale, 112. Silbermann, G., 9, 13, 42, 112, 124, 191, 194. Sistema musico. See L. Rossi. "A Sixteenth Century Approximation fortr." See J. M. Barbour. Small; A. M., iv, 195. Smith, R., 40-42. Societats-Frucht. See J. G. Meckenheuser. Die sogenannte allerneueste musicalisches Temperatur. See J. G. Meckenheuser. Somma de tutte le scienza. See A. Marinati. Sophiae cum moria certamen. See R. Fludd. Sopplementi musicali. See G. Zarlino. Sorge, G. A., 42, 83 f, 1247 159, 194. Spataro, G., 4. "Specimen de novo suo systemate musico." See K. Henfling. Spiegel der Orgelmacher und Organisten. See A. Schlick. spinet. See keyboard instruments. Spitta, P., 85. split keys, 33-35, 42, 97, 105 f. square root. See Euclidean construction. Squire, W. B.,~188. Stanhope, C, Earl, 157 f, 163. Steiner, J., 111. Stella, S., 117. Stevin, S., 7, 11, 28, 76 f, 79, 192. Str'ahle, D. P., 65-68. stretched octaves, 199. stringed instruments, 4, 8, 45 f, 58 f, 124, 195, 198 f. "Studien im Gebiete der reinen Stimmung." See S. Tanaka. Sulzer, J. G., 65. superparticular division, 2. See also Intervals with Superparticular Ratios, the table fol- lowing this index, superparticular division, of the tetrachord, 23 f. superparticular division, of the tone, 154. Suremain-Missery, A., 166. Sweelinck, J. P., 189. symmetry, 155-182 (passim). Syntagma musicum. See M. Praetorius. syntonic comma, i and passim. "Systeme general des intervalles des sons." See J. Sauveur. "Table general des syst^mes tempeVe's de musique. See J. Sauveur. tabular differences, 68-73. Tagore, S. M., 114. Tallis, T., 186 f. Tanaka, S., 6, 33, 109, 111, 117, 122 f, 135 f. Der Tanz in den deutschen Tabulaturblichern. See W. Merian. Telemann, G. P., 124, 191 f. Telharmonium, 74. 225 TUNING AND TEMPERAMENT temperament, 5. See also tunings. temperament, by regularly varied fifths. See irregular systems: temperament by regu- larly varied fifths. temperament, circulating. See irregular sys- tems: circulating temperaments. temperament, equal, 6-8, 10 f, 25, 29, 45-87, 90, 131, 142, 146 f, 164, 176-178, 183-186, 188, 191-195, 197-200. temperament, equal: geometrical and mechan- ical approximations, 49-55. temperament, equal: numerical approxima- tions, 55-87. temperament, history of, 1-14. temperament, meantone, 9-11, 25-44, 71, 106, 115, 124, 142, 176, 189-192, 197, 200. temperament, meantone: approximations, 29-31, 43. temperament, meantone: modifications. See irregular systems: modifications of regu- lar temperaments. temperament, meantone, varieties of, 31-44, 131; 1/3 comma, 9, 33-35, 51, 113; 1/5, iii, 35 f, 47, 120, 134; 1/6, 42 f, 112, 124, 135, 146, 168, 176, 191; 1/7, 43; 1/8, 43; 1/9, 44; 1/10, 44; 2/7, 9, 32 f, 35, 37, 46, 50, 119, 121; 2/9, 36 f, 119; 3/10, 40 f; 5/18, 41. temperament, meantone, varieties of: corres- pondences with equal multiple divisions, 124. temperament, "paper," 149, 154, 176. temperament, regular, 32t44, 91, 112-131. Le temperament. See P. Garnault. temperament anacritique, 124. "Temperament; or, the Division of the Octave." See R. H. M. Bosanquet. Tempelhof, G. F., 155. Temple Church in London, 106. Tentamen novae theoriae musicae. See L. Euler. Theatri machinarum. See H. Zeisung. Th^orie acoustico-musicale. See A. Sure- main-Missery. A Theory of Evolving Tonality. See J. Yasser. Thompson, P., 110. Tisdall, W., 188. Tomkins, T., 188. The Tonometer. See A. Warren. Das Tonsystem des Italieners Zarlino. See T. Kornerup. Torchi, L., 186. Toscanello in musica. See P. Aron. Trait6de l'accord de l'espinette. See J. Denis. Traits de l'harmonie. See J. P. Rameau. Traite de la viole. See J. Rousseau. Transponir-Harmonium, 111. Treatise ... of Harmony. See W. Holder. A Treatise of Musick. See A. Malcolm. trigonometry, 65-67. trombone. See wind instruments. Tsai-yii, Prince Chu, 7, 77-79. tuning, history of, 1-14. tuning pipe, 85-87. tuning today. See present practice of tuning. tunings. See Greek tunings, just intonation, Pythagorean tuning, etc. Two Thousand Years of Music. See C. Sachs. "liber mehr als zwolfstufigegleichschwebende Temperaturen." See P. von Jank6. "Uber musikalisches Tonbestimmung und Temperatur." See M. W. Drobisch. "liber wissenschafliche Begrundung derMusik durch Akustik." See A. F. Haser. unequal temperament. See temperament, meantone, and temperament, meantone, varieties of. See also irregular systems. Van de Spiegeling der Singconst. See S. Stevin. Varella, D., 58. Variety of Lute- Lessons. See R. Dowland. varieties of meantone temperament. See tem- perament, meantone, varieties of. Verhandlung van de Klokken en het Klokke- Spel. See J. P. A. Fischer. Verheijen, A., 28, 35, 120. Versuch liber die musikalische Temperatur. See F. W. Marpurg. The Vibrato. See C. E. Seashore. Vicentino, N., 8, 11, 25, 27, 37, 51, 115-119, 142, 152, 183. vihuela. See fretted instruments. viol. See fretted instruments. violin. See stringed instruments. "Violin Intonation." See P. C. Greene. Le violon de Lully~a~Viotti. See L. de La Laurencie. Violoncell Schull. See B. Romberg. violoncello. See stringed instruments. virginals. See keyboard instruments. Vitruvius, M., 50 f. voices, 124, 144, 194-197. Wang Pho, 150 f. Warren, A., 118. Wasieleswki, J. W. von, 186. Wedell, P. S., 127. Werckmeister, A., 12 f, 105, 154-182 (passim) 192 f. White, W. B., 48 f. Wiese, C. L. G., Baron von, 156 f. Williamson, C., 74-76. wind instruments, 7, 124, 195, 198. Wis-konstige Musyka. See D. R. van Nierop. wolf, 10 f, 27, 34, 91, 131 f, 134, 163. 226 INDEX Yasser, J., 9, 113 f. Young, R. W., 199. Young, T., 12 f, 135, 161, 165 f, 178 f, 181 f. Zacconi, L., 46. Zarlino, G., 6, 9, 11, 25, 27, 32, 35, 37, 42, 46, 50 f, 59, 113, 118, 121, 144, 147, 164, 194 f. Zeisung, H., 142. Zipoli, D., 191. Intervals with i Juperpartici ilar Ratios Ratios Intervals Cents Page References in Text 2:1 octave 1200 passim 3:2 perfect 5th 702 passim 4:3 perfect 4th 498 passim 5:4 major 3rd 386 passim 6:5 minor 3rd 316 passim 7:6 minor 3rd 267 18, 19 (Table 13), 22 f, 30 f. 8:7 maximum tone 234 19 (Table 13), 20, 23 f, 152. 9:8 major tone 204 passim 10:9 minor tone 182 passim 11:10 minimum tone 165 21 f, 154. 12:11 semitone 150 18, 21, 152-154. 13:12 • 139 23, 154. 14:13 15:14 approximation to meantone diatonic semitone 128 119 23. 17 (Table 5), 18, 23, 30 f, 152-1E 16:15 just diatonic semitone 112 passim 17:16 18:17 semitone approximation to semitone of equal temperament 105 99 57, 141, 153 f. 8, 57-64, 141, 153 f, 186. 19:18 semitone 93 17 (Table 7), 18, 57, 141, 154. 20:19 • 89 16 (Table 2), 17 (Table 7), 19 (Tables 13 and 14), 18, 141, 1 21:20 " 84 20, 22, 153 f. 22:21 • 81 18, 23, 151, 153. 227 TUNING AND TEMPERAMENT Ratios Intervals approximation to Cents Page References in Text 24:23 meantone chromatic semitone 74 16, 30 f. 25:24 just chromatic semitone 70 passim 26:25 quartertone 68 23. 27:26 - 65 154. 28:27 • 63 16, 18, 20, 22, 152. 31:30 ■ 57 21, 23, 109. 32:31 - 55 21, 23, 109. 33:32 ■ 53 152, 154. 36:35 - 49 16, 23. 39:38 ■ 45 16 (Table 2). 40:39 " 44 16 (Table 2), 23, 154. 45:44 " 39 154. 46:45 " 38 16, 23. 49:48 " 36 24, 152. 55:54 comma 32 152 f. 56:55 ■ 31 23. 64:63 approximation 27 23, 152. 74:73 to ditonic comma 24 passim 81:80 syntonic comma 22 passim 228 M 1/ DATE DUE ■ay a 9 1993 CAYLOHO PRINTED IN U.S.A. OCT 4 3 5002 00368 4748 Barbour, James Murray Tuning and temperament : a historical su ML 3809 B234 AUTHOR Barbour TITLE . -l~~ „^A \ r\ -v om/nT"* t Music ML 3809 B234 274194