Chapter IV. EQUAL TEMPERAMENT The first tuning rules that might be interpreted as equal tem- perament were given by Giovanni Maria Lanfranco. * As stated, these rules were for clavichords and organs (Monochordi & Organi), but Lanfranco extended them also to the common stringed instruments of his time. Thus there is none of the confusion that arose later when the keyboard instruments were tuned in one manner, the fretted instruments in another. Lanf ranco's essential rules concern the tempering of the fifths and the thirds: the fifths are to be tuned so flat "that the ear is not well pleased with them," and the thirds as sharp as can be endured. There seems to be a distinction here: for a fifth might be tuned only slightly flat and the ear would not then be wholly pleased with it; but the thirds are to be only a shade less harsh than those which cannot be endured at all. Most of Lanfranco 's contemporaries still knew no tuning but the Pythagorean, with its pure fifths and impossibly sharp thirds. Lanfranco's rules seem to represent a temperament of the Pytha- gorean tuning, rather than of just intonation. Equal temperament then fits his directions excellently. As further evidence, Lan- franco divided the notes to be tuned into two classes, sharps and flats. As with the meantone temperament, the sharps included F#, C#, and G#, "although most of these are also common to the flat class, if not in tuning, at least in playing." But, although the flats proper included only BD and ED, this class "occasionally needs in playing the black keys F* (Gb) and C# (D*3)." As Kin- keldey says, "the enlargement of the major third, the diminution of the minor third, the equivalence of the notes C# and DD, F* and GD— these are essential departures from his contempor- aries. "2 Aurelio Marinati^ honored Lanfranco by inserting in his "ex- IScintille de musica (Brescia, 1533), p. 132. 2(Xto Kinkeldey, Orgel und Klavier in der I 77 f. ^Somma di tutte le scienza (Rome, 1587), pp. 95-98. 2(Xto Kinkeldey, Orgel und Klavier in der Musik des 16. Jahrhunderts, pp. 77f. TUNING AND TEMPERAMENT ample of the tuning of clavichords and organs" a word-for-word account of Lanfranco's system, complete even to the title— with- out, however, giving him credit for it. Another plagiarist, Cerone, sufficiently appreciated Lanfranco to copy out his system for the benefit of organ-builders. 4 At the time when these men were writing, the meantone temperament was the recognized tuning norm for keyboard instruments. It is rather surprising that Cerone in particular, who had presented Zarlino's 2/7-comma system in detail, did not seem to realize that there was a con- flict between Zarlino's flat and Lanfranco's sharp major thirds. Lodovico Zacconi^ was more astute. He presented no tuning rules of his own, saying that it is "better that those who wish to know and to see should look to the source and to the original authors." For keyboard instruments he recommended Aron's meantone temperament. "As for the other instruments, such as the viole da braccio, viole da gamba, violins, and others, you can look at the end of Giovanni Maria Lanfranco's book, which indi- cates clearly how each one is to be tuned." In Zacconi's day and long before it, the fretted instruments were said to have equal semitones. To Zarlino, Salinas, and Galilei this meant equal temperament, with all semitones equal. To Grammateus and Bermudo, only ten semitones were equal, the others being smaller; to Artusi, and presumably also to Bot- trigari and Cerone, there were ten equal semitones, the other two being larger. But, of these three types of temperament- equal, modified Pythagorean, and modified meantone— only equal temperament had both flat fifths and sharp thirds in addition to equal semitones. Therefore, Zacconi, writing only sixty years after Lanfranco, is practically saying that the latter 's rules rep- resent equal temperament. In view of the excellent tuning methods of Lanfranco's immediate predecessors, Grammateus and Schlick, it is very likely that Lanfranco did intend equal tem- perament for all instruments, including clavichords and organs. Later writers who gave practical tuning rules for equal tem- perament were often no more precise than Lanfranco had been. 4See Kindeldey, op. cit., p. 80. 5Prattica di musica (Venice, 1592), Part I, p. 218. 46 EQUAL TEMPERAMENT Jean Denis, 6 for example, said nothing about the size of the thirds. But all the fifths are to be lowered a trifle (d'un poinct), "and all the fifths ought to be tempered equally." Denis may even have had some variety of meantone temperament in mind, for he directed that the tuning should begin with E*3 and end with G#. But if his "toutes" means what it says, his was equal tem- perament. Godfrey Keller's tuning rules for harpsichord or spinet were widely circulated, having been reprinted in the appendix to William Holder's Treatise . . . of Harmony (London, 1731), and in Part VI of Pierre Prelleur's long popular Modern Musick-Master.? Al- though they can refer to nothing but equal temperament, they are by no means accurate: "Observe all the Sharp Thirds must be as sharp as the Ear will permit; and all Fifths as flat as the Ear will permit. Now and then by way of Tryal touch Unison, Third, Fifth, and Eighth; and afterward Unison, Fourth, and Sixth." It is impossible for the thirds to be very sharp and the fifths simul- taneously very flat; for in the 1/5-comma variety of meantone temperament, in which the error of the fifths and the thirds is equal, the error is not large. Keller's rules would read better if he had said that the fifths were to be only slightly flat. Barthold Fritz** gave tuning rules for equal temperament that merited the approval of Emanuel Bach, to whom he had dedicated his little book. Bach said that "in my [Fritz's] few pages every- thing had been said that was necessary and possible, and that would satisfy far more needs than the sundry computations with which many a man has racked his brains; since the latter method ^Traite* de l'accord de l'espinette (Paris, 1650), pp. lOf. "*Keller'sbook had the title A Compleat Method . . . (London: Richard Meares). The British Museum has a copy dated 1707, but with a different printer. The Library of Congress copy does not contain the tuning rules; its copy of the Prelleur book is the 4th edition, dated 1738. The British Museum has an edition of the latter dated 1731. Part VI was printed separately with the title The Compleat Tutor for the Harpsichord or Spinet, and passed through several editions, with various printers, in the 1750's and '60's. °Anweisung wie man Claviere, Clavicins, und Orgeln, nach einer mechanis- chen Art, in alien zwblf Tonen gleich rein stimmen konne, . . . (3rd edition; Leipzig, 1780). 47 TUNING AND TEMPERAMENT of instruction was only for very few people, but mine was for everybody, the computers not excepted, because they depend upon the judgment of the ear as well as the others. "9 Fritz's rules were very simple. After going from F to A by four tempered fifths, he said, "I now have the already pure F as a major third to this A, and, by touching the A and by testing it with F, can hear whether it sounds sharp enough or so much up- wards that the beats are about the rapidity of eighth notes in common time. "10 Fritz began his tuning in the octave below middle C. From William Braid White's table, ** the tempered F- A in this octave will beat about 7 times per second, or over 400 times in a min- ute. Even allowing for the somewhat lower pitch of the eighteenth century, Fritz's eighth notes would be very fast, unless by "com- mon time" he meant alia breve. Mersennel2 also gave a practical tuning hint for equal tem- perament when he said, "Certain people believe that they can find the preceding accord of the equal semitones by beginning ut, re, mi, fa, etc. on each key of the spinet, or by the number of trem- blings or beats which the fifth and other tempered consonances make: for example, the fifth beats once in each second when it is tempered as it should be (as much for the organ as for the spinet); whereas when it is just it does not beat at all." From White's table, Mersenne's rule would apply best to the fifth D-A in the octave above middle C, and approximately to other fifths in that vicinity. Alexander Ellis' practical rules for the formation of equal temperament* 3 may be paraphrased as follows: If one tunes by upward fifths and downward fourths within the octave above mid- dle C, each fifth should beat once per second, and each fourth ^Ibid., Preface to 2nd edition. IQlbid., p. 14. 11 Piano Tuning and Allied Arts (4th edition; Boston, 1943), p. 68. l^Harmonie universelle (Paris, 1636-37), Nouvelles observations physiques & mathematiques, p. 20. 13h. L. F. Helmholtz, Sensations of Tone (2nd English edition, translated by Alexander J. Ellis; London, 1885), pp. 489 f. 48 EQUAL TEMPERAMENT three times in two seconds. Ellis stated that if this rule is fol- lowed accurately, the error for no pitch will be greater than two cents. Again using White's useful table, we find that the mean value of the beats of the tempered fifths in the C-C octave is 1.02 and of the tempered fourths, 1.47, proving that Ellis' rule is correct. White himself "lays the bearings" in the F-F octave, 14 just as Fritz did. Since the ratio of a tempered fifth is approximately 3:2, one might suppose that he would advocate beating rates that are 2/3 of Ellis' values: fourths once per second, and fifths twice in three seconds. However, he recommends that the fifths beat three times in five seconds, or 36 times per minute, and suggests setting a metronome at 72, with the bell ringing at every second tick. Since, from his own table, the mean value of the beats of his tempered fifths is .68 rather than .60, he would get better results from setting the metronome at 80. Bossier's methodic for achieving equal temperament is rem- iniscent of Aron's method for the meantone tuning. Aron, it may be remembered, first tuned his major third pure and then tuned equally flat the four fifths that were used in constructing the major third. Bossier first divided the octave by ear into three equal parts— C-E-G#-C. Then he tuned a group of four fifths, as C-G- D-A-E, slightly flat, so that the last would give the sharp major third already found. The method would be continued until the en- tire octave was tuned. Having these first three notes fixed gave him points of reference, so that he could never go far wrong. But he realized that the human ear is fallible, for he recommended that the tuner buy "steel forks from Frankfurt or Leipzig for all twelve notes." Geometrical and Mechanical Approximations One of the famous problems of antiquity was the duplication of the cube. It had been proved that the construction of the cube root of 2 could not be accomplished by Euclidean geometry, that 14Op. cit., p. 85. 15H. P. Bossier, Elementarbuch der Tonkunst (Speier, 1782), pp. xxiv-xxvi. 49 TUNING AND TEMPERAMENT is, by compass and ruler. This is the precise problem involved in the solution of equal temperament by geometry, if Bossier, for example, had desired to construct a monochord upon which would be located his C-E-G#-C. The first sixteenth century writer to suggest a geometrical or mechanical means of solving equal temperament was Fran- cisco Salinas. 16 Let him explain his method: "We judge this one thing must be observed by makers of viols, namely, that the oc- tave must be divided into 12 parts equally proportional, which 12 will be the equal semitones. And since they cannot accomplish this by the 9th of the 6th book [the mean proportional construc- tion] or by any other proposition of Euclid, it will be the task to use the instrument which we said was called the mesolabium, invented (as they believe) by Archimedes: by which they will be able to obtain aline divided into as many equal parts as they wish. We have not bothered to append the rule of its construction here, because mention is made of its principle by Vitruvius in his 9th book on architecture; from whom and from his expositors they will be able to obtain the method of constructing it: for it is to practical men for framing most matters not only useful, but well- nigh indispensible." The mesolabium had been previously advocated by Zar lino for constructing his 2/7- comma meantone temperament, and later Zarlino was to follow Salinas' lead in recommending it for equal temperament. Hutton defined the word as follows: "Mesolabe, or Mesolabium, a mathematical instrument invented by the an- cients, for finding two mean proportionals mechanically, which they could not perform geometrically. It consists of three paral- lelograms, moving in a groove to certain intersections. Its figure is described by Eutocius, in his Commentary on Archimedes. See also Pappius, Lib. 3. "17 With the aid of a clear diagram (Figure B) James Gow^S has explained the operation of the mesolabium as follows: "If AB, GH be the two lines between which it is required to find two mean 16De musica libri VH, p. 173. ^Charles Hutton, Mathematical Dictionary (new ed.; London, 1815). l^A Short History of Greek Mathematics (Cambridge, 1884; reprinted, New York, 1923), pp. 245 f. 50 EQUAL TEMPERAMENT proportionals, then slide the second frame under the firstandthe third under the second so that AG shall pass through the points C, E,at which the diameters of the second and third frames, re- spectively, cease to be visible. Then CD, EF are the required two mean proportionals." F H Fig. B. The Mesolabium (From James Gow, A Short History of Greek Mathematics [c. 1884]) Although Zarlino contended that the mesolabium might be used for finding any number of means, by increasing the number of parallelograms, his diagram is for two means only. Of course for equal temperament or for the 1/3-comma meantone temper- ament, two means would suffice. But Salinas also advocates it for an unlimited number of means, and Rossi would find the thirty means for Vicentino's division by its aid. Mersenne,19 however, in commenting upon Salinas' construction for equal temperament, said it was incorrect if he intended to use the mesolabium for more than two means, because the instrument mentioned by Vi- truvius "is of no use except for finding two means between two given lines." We shall not attempt to pass judgment upon these conflicting opinions, but it would seem that the difficulty of the process would be increased greatly with an increasing number of means. Zarlino^O has given three methods by which "to divide the octave directly into 12 equal and proportional parts or semi- tones." The first used the mesolabium, as already mentioned. The second used the method of Philo of Bysantium (second cen- tury, B.C.), which consisted of a circle and a variable secant ^Harmonie universelle, p. 224. 20Gioseffo Zarlino, Sopplimenti musicali (Venice, 1588), Chap. 30. 51 TUNING AND TEMPERAMENT through a point on its circumference. The third is a variation of the first, in that the string-length for one note is found by the mesolabium, and then the lengths for the other notes are found by similar proportions. Mersenne,21 too, has contributed non-Euclidean methods for finding two geometric means. The first, ascribed to Molthee, used straight lines only, in the form of intersecting triangles. The other method (Figure C) was furnished byRoberval and used Fig m 3 1 f A \\ |C ] — ^ v u \ y j \^ I . \ . C . Rob erval's Method for Finding Two Geometric Mean Pro- portionals (From Mersenne's HIarmonie universelle) Reproduce the Librai >d by courtesy of y of Congress a parabola and a circle. 22 Kircher23 combined the Euclidean method for finding one mean proportional with a mechanical method for finding two means. This latter is by still another method, consisting of two lines at right angles and two sliding 21Op. cit., p. 68. 22ibid, p. 408. 23Athanasius Kircher, Musurgia universalis (Rome, 1650), I, 207. 52 EQUAL TEMPERAMENT L -shaped pieces, like carpenters' squares (Figure D). Accord- ing to Rossi, 24 Kircher's is the method of Nicomedes, and Rossi considered it "more expeditious" than others that have been men- tioned. Marpurg25 ascribed Kircher's method to Plato, and added methods by Hero and by Newton, together with Descartes' method for finding any number of mean proportionals. Thus we have more than half a dozen geometrical and mechanical methods, proposed particularly for constructing a monochord in equal temperament. Fig. D. Nicomedes' Method for Finding Two Geometric Mean Pro- portionals (From Kircher's Musurgia universalis) Reproduced by courtesy of the Library of Congress Since these mechanical methods for finding two mean pro- portionals are rather awkward, the attempt has been made to use a satisfactory ratio for the major third or minor sixth, so that the remainder of the division could be made by the Euclidean construction for finding a single mean. Mersenne26 has given two such methods. In the second, which he said is "the easiest of all possible ways," the just value of the minor sixth (8:5) is used. By mean proportionals, eight equal semitones are found 24sistema musico, pp. 95 f. 25yersuch liber die musikalische Temperatur, 19. Abschnitt. 26Harmonieuniverselle, p. 69. 53 TUNING AND TEMPERAMENT between the fundamental and the minor sixth, and then, in like manner, the remaining four semitones between the minor sixth and the octave. As can be seen from Table 39, this method is not extremely close to correct equal temperament, because the just value of the minor sixth is about 14 cents higher than its value in the equal division. One might have expected the usually astute Mersenne to have chosen a tempered value in the first place. The equally tempered minor sixth is very nearly 100:63, as can be readily seen in Boulliau's table given by Mersenne, where it bears ex- actly this value. If this fraction is too difficult to work with, 27:17 will serve almost as well, and 19:12 comes rather close also. Any of these other ratios would have given a more satis- factory monochord than his. In Table 40, 19:12 is used for the minor sixth. Table 39. Mersenne's Second Geometrical Approximation Names CxDxEFxGxA x B C Cents 0 102 203 305 407 508 610 712 814 910 1007 1103 1200 M.D. 2.3; S. D. 2.5 Table 40. Geometrical Approximation (19:12 for Minor Sixth) Names C x D X E F X G Cents 0 99.5 198.9 298.4 397.8 497.3 596.7 696.2 Names X A X B C Cents 795.6 896.7 997.8 1098.9 1200.0 M.D. .76; S.D. .78 But we cannot be supercilious regarding Mersenne's other practical method for obtaining two mean proportionals. Mer- senne himself correctly said, "It serves for finding the mechan- ical duplication of the cube, to about 1/329 part. "27 By the fa- miliar Euclidean method he found the mean proportional between a line and its double, subtracted the original line from the mean, 27lbid., p. 68. 54 EQUAL TEMPERAMENT and then subtracted this difference from the doubled line. The length thus found was the larger of the desired means— that is, the string-length for the major third. In numbers, this ratio is (3 - 42): 2, or .79289, which represents 401.8 cents. The result is shown in Table 41, the remaining values being found by mean proportionals as in Mersenne's second approximation. This is an extremely fine geometrical way to approximate equal tem- perament. Table 41. Mersenne's First Geometrical Approximation Names C x D x E F X G Cents 0 100.4 200.9 301.3 401.8 501.6 601.3 701.1 Names X A x B C Cents 800.9 900.6 1000.4 1100.2 1200.0 M.D. .30; S.D. .32 Table 42. Ho Tchhe'ng-thyen's Approximation Lengths 900 849 802 758 715 677 638 601 570 536 509.5 479 450 Names C C* D D* E E# F# G G# A A* B C Cents 0 101 200 297 398 493 596 699 791 897 985 1091 1200 M.D. 4.8; S.D. 5.8 Numerical Approximations The earliest numerical approximation for equal temperament comes from China. About 400 A.D.,H6 Tchh^ng-thyen gave three monochords for the chromatic octave, with identical ratios, but with the fundamental taken as 9.00, 81.00, and 100.0 respec- tively.28 (string-lengths are given for the first of these tables only, since they illustrate the manner of its formation better than the other two.) Table 42 shows a remarkable temperament for the time when it was constructed, comparable to the brilliant solution of the ^"Maurice Courant, "Chine et Core'e," Encyclopedic de la musique et diction- naire du conservatoire (Paris, 1913), Part 1, Vol. I, p. 90. 55 TUNING AND TEMPERAMENT problem of equal temperament by Prince Tsai-yu over a thousand years later. At the time of Tchh^ng-thyenthe Pythagorean tuning was the accepted system in China. If we assume the calculation to begin with the higher C at 450 and proceed in strict Pythagorean manner to B# in the lower octave, the B^ will be at 888 instead of 900. This is 12 units too short. Let us, therefore, add 1 unit to 600, the value for G; 2 units to 800, the value for D; 3 units to 533, the value for A; and so forth, along a sequence of fifths, until we reach the correct value for C at 900. Tchh§ng-thyen's figures agree precisely with our hypothesis. A linear correction, such as Tchh§ng-thyen made, often pro- vides a good approximation, as we shall see elsewhere in this chapter. The difficulty with his correction is that if he had started with the lower C and had continued until he had reached the higher B^, the latter would have been only 6 units too short instead of 12. By adding 10 parts for A#, 8 for G^, etc., he obtained pitches that were much too low. If he had added 12 parts to 444 for the higher B^, the corrected length, 456, would have been at 1177, instead of 1200 cents, 23 cents flat! Let us consider the effect of adding precisely half the correction for each note. This would work well for the odd semitones, C D E F# G# A# B#, as might have been expected; but the lower three even semitones, C^ D# E^, are then as sharp as the higher odd semitones were flat before! We shall have better success if we continue the series of whole tones from G to Fx, the latter at 296 needing a correction of 4.2 to make a perfect octave to G, 600.5. Then the intermediate notes can be given a proportional linear correction, which would be doubled for the three notes C* D# E# when transposed to the lower octave. This improved temperament is shown in Table 43. The greatest error is at C*. Table 43. H6 Tchh£ng-thyen's Temperament, Improved Lengths 900 846.6 801 754.8 713 763 635 600.5 Names C X D x E F X G Cents 0 106 202 305 403 503 604 701 Lengths 566 534.1 504.5 475.7 450 Names X A x B C Cents 6 803 903 ] 1004 1101 M.D. 2.2; S.D. 2.7 1200 EQUAL TEMPERAMENT The arithmetical division of the 9:8 tone into 17:16 and 18:17 semitones was known to all sixteenth century writers through Ptolemy's demonstration that Aristoxenus could not have obtained equal semitones in this way. But Cardano (1501-76) may have been referring to some practical use of the 18:17 semitone when he wrote: "And there is another division of the tone into semi- tones, which is varied by putting the tone between 18 and 16; the middle voice is 17; the major semitone is between 17 and 16, but the minor between 18 and 17, the difference of which is 1/288. It is surprising how the minor semitone should be introduced so pleasingly in concerted music, but the major semitone never. "29 The simplest way to construct a monochord in equal tempera- ment is to choose a correct ratio for the semitone and then apply it twelve times, a construction that can be performed very easily by similar proportion. Vincenzo Galilei^O must be given the credit for explaining a practical, but highly effective, method of this type. For placing the frets on the lute he used the ratio 18:17 for the semitone, saying that the twelfth fret would be at the mid- point of the string. He went on to say that no other fraction would serve; for 17:16, etc., would give too few frets, and 19:18, etc., too many. Since 18:17 represents 99 cents, 17:16, 105 cents, and 19:18,94 cents, Galilei was correct in his contention. But he did not give a mathematical demonstration of his method. It remained Table 44. Galilei's Approximation Lengths 100000 94444 89197 84242 79562 75142 70967 Names C X D X E F X Cents 0 99 198 297 396 495 594 Lengths 67024 63301 59784 56463 53326 50000 Names G X A X B C Cents 693 792 891 990 1089 1200 M.D. 1.8: S.D. 3.3 29Girolamo Cardano, Opera omnia, ed. Sponius (Lyons, 1663), p. 549. 30Dialogo della musica antica e moderna (Florence, 1581), p. 49. 57 TUNING AND TEMPERAMENT for him a proof by intuition. The string- lengths in Table 44 were calculated by Kepler. 31 Mersenne32 testified that Galilei's method was favored by "many makers of instruments." The Portugese writer Domingos de S. Jose Varella33 gave a "way to divide the fingerboards of viols and guitars." This is precisely Galilei's method, and Varella told how the construction could be continued by similar proportion after the first 18:17 semitone had been formed. Like- wise Delezenne34 showed that 18:17 is very near the value for the correct equal semitone, and gave a geometrical construction for it used by Delannoy, the instrument maker, in placing the frets upon his guitars. Two other early nineteenth century references to what Gar- nault35 called the "secret compass" of the makers of fretted instruments were given in his tiny and not very trustworthy monograph on temperament. The first was from the Robet- Maugin Manuel du Luthier (1834), which stated that if the string is 2 feet in length, the first semitone will be at a distance of 16 lines from the end; this represents 16/2x12x12 = 1/18 the length of the string, thus giving 18:17 for the ratio of each semitone. Garnault's second reference was to the Bernard Romberg 'cello method (1839 ),36 which he said had been adopted byCheru- bini for use in the Paris Conservatoire. Romberg's directions "^Johannes Kepler, Harmonices mundi (Augsburg, 1619; edited by Ch. Frisch, Frankfort am Main, 1864), p. 164. ^^Harmonie universelle, p. 48. 33compendio de musica (Porto, 1806), p. 51. 34c E. J. Delezenne, "Memoire sur les valeurs numeriques des notes de la gamme," Recueil des travaux de la soci^te* des sciences, ... de Lille, 1826-27 p. 49, note (a), and p. 50. 35paul Garnault, Le temperament, son histoire, son application aux claviers, aux violes de gambe et guitares, son influence sur la musique du xviiie siecle (Nice, 1929), pp. 29 ff. 36in the German translation (original?), Violoncell Schull (Berlin, 1840 [?] ), the directions are given on page 17; in the English translation, A Complete Theoretical and Practical School for the Violoncello, they are omitted. 58 EQUAL TEMPERAMENT were much the same as those given previously. Although Gar- nault does not mention this, Romberg added that the directions given were for equal temperament, but the more advanced player would often make the sharped notes sharper and the flatted notes flatter than these pitches— another confirmation of the quasi-Py- thagorean tuning of instruments of the violin family. These references to the 18:17 semitone cover two and a half centuries. It is probable that they could be brought much nearer our own times if the makers of fretted instruments were, given a chance to express themselves. We must accept Galilei's method, therefore, as representing the contemporary practice. A player on a lute was not going to bother with the mesolabium or with a monochord on which were numbers representing the successive powers of the 12th root of 2. But he could place his frets by a simple numerical ratio such as 18:17, and we are glad that the frets thus placed served their purpose so well. Critics of Galilei were not slow to show that the 12th fret would not coincide precisely with the midpoint of the string. Passing by the inconveniently large numbers of Zarlino's ratios, we come to Kepler's result: if the entire string is 100,000 units in length, Galilei's 12th fret will be at 50,363 instead of 50,000. As we have already stated, his semitone has only 99 cents, so that the octave contains 1188 instead of 1200. There are various ways of correcting the octave distortion arising from the use of the 18:17 semitone. An obvious way is suggested by Mersenne's approximations: form only 4 semitones with the 18:17 ratio; then apply Mersenne's mean-proportional method to the remaining 8 semitones. The monochord thus con- structed (Table 45) is as good as Mersenne's first method. Table 45. Approximation a la Galilei and Mersenne Names C X D X E F X G Cents 0 99 198 297 396 496.5 597 697.5 Names X A X B C Cents 798 898.5 999 1099.5 1200 M.D. .67; S.D. .71 59 TUNING AND TEMPERAMENT An even simpler correction uses linear divisions only: since the length for the 12th fret is 363 units too great, divide 363 into 12 equal parts and subtract 30 units for the first fret, 61 for the second, 91 for the third, etc. As is always the case with this type of correction, there is a slight bulge in the middle of the octave, but the largest error is only 1.8 cents. The correction shown in Table 46 lends itself well to numer- ical computation, since the fundamental and its octave are in round numbers. But in practice, with a geometrical, not a numer- ical, construction, the following would be simpler and is even a trifle better: if 50,363 be considered the real middle of the string, the octave will be perfect. To make it the middle, shorten the entire string by twice the difference between 50,000 and 50,363, that is, by 726. Then everyone of the lengths as given by Kepler will be diminished by 726, and the 12th fret, 49,637, will be the exact middle of the string, 99,274. Note again the slight bulge in the middle of the division (Table 47), with the greatest distortion 1.0 cent. Table 46. Galilei's Temperament, with Linear Correction, No. 1 Lengths 100000 94414 89136 84151 79441 74991 70785 Names C X D X E F X Cents 0 99.5 199.1 298.8 398.5 498.3 598.2 Lengths 66812 63059 59512 56160 52993 50000 Names G X A X B C Cents 698.3 798.4 898.5 998.9 1099.4 1200 M.D. .26; S.D. .31 Table 47. Galilei's Temperament, with Linear Correction, No. 2 Lengths 100000 99274 88471 83516 78836 74416 70241 Names C X D X E F X Cents 0 99.7 199.4 299.3 399.1 499.0 599.0 Lengths 66298 62575 59058 55737 52600 49637 Names G X A X B C Cents 699.0 799.0 899.2 999.3 1099.7 1200 60 M.D. .17; S.D. .21 EQUAL TEMPERAMENT The improvements upon Galilei's tuning shown in Tables 46 and 47 could have been made by practical tuners. They are better divisions than many of the numerical expressions of equal tem- perament which will be shown later. They are better also than the temperament our contemporary tuners give our own pianos and organs. So there is nothing more that needs to be said, as far as practice is concerned. There are, however, several other and more subtle ways of improving Galilei's tuning which we should like to mention. These are of speculative interest solely. Letus return to the false octave generated by the 18:17 semi- tone. Mersenne suggested that "if the makers should increase slightly each 18:17 interval, they would arrive at the justness of the octave." The 11th fret is at 53326, leaving a ratio of 53326: 50000 for the remaining semitone. This, as its cents value in- dicates (111 cents), is about the size of the just 16:15 semitone. Let us pretend that the final digit in the antecedent is 5, and re- duce the ratio to 2133:2000. Now let us average this semitone with the eleven 18:17 semitones, using the arithmetical division generally followed by sixteenth century writers. Our desired semitone is 2000/2133 + 187/18 = 48319. In decimal form this 12 51192 is .9438779, as compared with the true equal semitone, .9438743. The successive powers of this decimal would deviate more and more from those of the 12th root of 2, but even then the octave would be only .1 cent flat. Another way of correcting Galilei's tuning is based upon the fact that his octave would be 12 cents, that is, half a Pythagorean comma, flat. A somewhat crude, but practical, manner of ad- justing the octave would be to form four 18:17 semitones, from C to E, then take the next five notes, F through A, as perfect fourths to the first five, and then the two remaining notes, B*3 and B, as perfect fourths to F and F#. A satisfactory monochord is shown in Table 48. Note particularly how much smaller its standard deviation is than that of Galilei's actual tuning. As an approach to a finer division using Pythagorean inter- vals, let us turn to Pablo Nassarre.37 37Escuela musica (Zaragoza, 1724), Part I, pp. 462 f. 61 TUNING AND TEMPERAMENT Table 48. Galilei's Temperament Combined with Pythagorean Names C X D X E F X G Cents 0 99 198 297 396 498 597 696 Names X A X B C Cents 795 894 996 1095 1200 M.D. 1.5; S.D. 1.6 He had discussed equal semitones upon fretted instruments, using much the same lan- guage as Praetorius,38 to the effect that a 16:15 diatonic semi- tone contains 5 commas and a 25:24 chromatic semitone 4 com- mas, but that these semitones have the peculiarity that they are all equal, containing 4 1/2 commas. They are obtained by a linear division of the 9:8 tone into 18:17 and 17:16 semitones. To place the frets, three or four 9:8 tones are constructed, and the distance between each pair of frets divided equally to form the semitones . Of course an arithmetical division of tones will not form precisely equal semitones. Furthermore, there is a fairly large distortion for the last semitone if the process is carried out through twelve semitones. Of course, as with Galilei's method, no single string would have had twelve frets. In Table 49 the division is made for the entire octave. The length for B was taken as the arithmetical mean between A^ and the middle of the string. Table 49. Nassarre's Equal Semitones Names C° X D° X E° (F) F#> (G) Cents 0 99 204 303 408 507 612 711 Names G#° (A) A*0 (B) C° Cents 816 915 1020 1107 1200 M.D. 4.2; S.D. 5.4 If Nassarre had divided each 9:8 tone into precisely equal semitones by a mean proportional, his errors would have been smaller. •^Syntagma musicum, Vol. 2, p. 66. 62 EQUAL TEMPERAMENT Table 50. Nassarre's Temperament Idealized Lengths 100000 94281 88889 83805 79012 74494 70233 Names c° X D° X E° (F) F#° Cents 0 102 204 306 408 510 612 Lengths 66216 62429 58859 55493 52319 50000 Names (G) G*° (A) A*> (B) C° Cents 714 816 918 1020 1110 1200 M.D. 3.7; S.D. 6.7 It is not particularly difficult to set down this tem- perament in figures, since the square root need be performed only for C*, after which a second series of 9:8 tones can be formed, starting with this note. If B is taken as the geometric mean between A^and C, its length is 52675, or 1110 cents, making the mean deviation 3.3, and the standard deviation 4.5. However, for the sake of an approximation to be made in Table 50, B is taken as the geometric mean between A^ and B^, with a relatively high standard deviation. If we now compare the cents values of the temperament shown in Table 50 with those of Galilei's tuning, we shall find that the error of the former is opposite to and twice as great as that of the latter. Therefore, for every pair of string- lengths, subtract the smaller (Nassarre) from the larger (Galilei), and then sub- tract 1/3 the difference from the larger number. The excellent monochord shown in Table 51 results. Table 51. Temperament a la Galilei and Nassarre Lengths Names Cents Lengths Names Cents 100000 C 0 66755 G 699.7 94390 x 99.9 63010 x 799.7 89094 D 199.9 59476 A 899.6 84096 x 299.9 56140 x 999.6 79379 E 399.8 52990 B 1099.6 74926 F 499.8 50000 C 1200 70722 x 599.7 M.D. .07; S.D. .13 63 TUNING AND TEMPERAMENT If the idealized Nassarre temperament had been extended one more semitone, the string-length for the octave would have been 49,328. When this number is adjusted with the 50,363 of Galilei's tuning, the octave proper to the above temperament becomes 50,018 or 1199.5 cents. Let us now make the same type of octave adjustment as with the original Galilei tuning, by subtracting 18 from the 12th semitone, and 1 or 2 less for each succeeding semitone. Then no length varies by more than 2 or 3 units from the correct value, that is, the maximum variation is less than .1 cent. This procedure sounds somewhat complicated. It is not nec- essary to go through the entire process three times, as shown above, in order to obtain the final monochord. The ratio for the semitone will be 17/9 + 2^2~/3 = 17 + 6/J2~ . Including the octave 3 27 correction, the formula for the string-length of the nth semitone is: 100,000 /l7 + 6^ \- 3(n-l) . Perhaps it woukfbe simpler 27/2 after all to stick to cube roots, especially when fortified with a table of logarithms ! Johann Philipp Kirnberger,39 however, used a very rounda- bout method of attaining equal temperament, believing it to be simpler in practice than tuning by beating fifths. He showed that the ratio 10935:8192 closely approaches the value of the fourth used in equal temperament. In practice this value would be ob- tained by tuning upward seven pure fifths and then a major third. In other words, if C° is the lower note, E^_1 is regarded to be the equivalent of F~^ , the tempered fourth. The basis for this equivalence lies in the fact that the schisma, the difference be- tween the syntonic and the ditonic commas, is almost exactly 1/12 ditonic comma, the amount by which the fourth must be tempered. The ratio given above becomes, in decimal form, .7491541 . . . , whereas the true tempered value is .7491535 .... The result is an extremely close approximation. 39Die Kunst des reinen Satzes in der Musik, 2nd part (Berlin, 1779), 3rd Di- vision, pp. 179 f. 64 EQUAL TEMPERAMENT Kirnberger spoke of Euler's approval of his method, and of Sulzer's and Lambert's publication of it. Marpurg^O showed that Lambert's method, when applied to an entire octave, will differ for no note by more than .00001. He praised it as a method that needs no monochord, and believed that the tuning of the just intervals used in it could be made more quickly and accurately than the estimation by ear of the tempering needed for the fourth or the fifth. However, the tuning of a pure major third is so dif- ficult that Alexander Ellis thought that better thirds can be ob- tained from four beating fifths than by tuning the thirds directly. If this be true, a type of tuning in which the essential feature is a pure major third could not be very accurate, without consider- ing the labor of tuning eight pure intervals in order to have only one tempered interval! Kirnberger 's approximation for equal temperament was next heard of in England, where John Farey^l seems to have dis- covered it independently. In Dr. Rees's New Cyclopedia^ 2 we are shown how Farey's method "differs only in an insensible degree" from correct equal temperament. Among the monochords shown by Marpurg is one by Daniel P . Strahle,43 allegedly in equal temperament, but actually unequal, as can be seen in Table 52. This is a geometric construction of a curious sort, for which Jacob Faggot computed the string- lengths by trigonometry (see Figure E). In brief, it went like this: upon the line QR, 12 units in length, erect an isosceles tri- angle, QOR, its equal legs being 24 units in length. Join O to the eleven points of division in the base. On QO locate P, 7 units from Q, and draw RP, extending it its own length to M. Then if RM represents the fundamental pitch and PM its octave, the ^Oyersuch liber die musikalische Temperatur, p. 148. 41uOn a New Mode of Equally Tempering the Musical Scale," Philosophical Magazine, XXVII (1807), pp. 65-66. 42ist American edition, Vol. 14, Part 1, article on Equal Temperament. 43«Nytt pafund, til at finna temperaturen, i stamningen for thonerne pa cla- veretock dylika instrumenter," Proceedings of the Swedish Academy, IV (1743), 281-291. .The second part of the article, "Trigonometriskutrakning," appears under Faggot's name. 65 TUNING AND TEMPERAMENT points of intersection of RP with the 11 rays from O will be the 11 semitones within the octave. Table 52. Faggot's Figures for Strahle's Temperament Lengths 10000 9379 8811 8290 7809 7365 6953 Names C X D X E F X Cents 0 111 219 325 428 529 629 Lengths 6570 6213 5881 5568 5274 5000 Names G X A X B C Cents 727 824 919 1014 1108 1200 M.D. 4.8; S.D. 5.7 Fig. E. Strahle's Geometrical Ap- proximation for Equal Temperament Reproduced by courtesy of the Library of the University of Michigan It is obvious from the construction that the distance between two consecutive points of division will be greater near R than near P, and hence that, superficially at least, the division will resemble a series of proportional lines, as in true equal tem- 66 EQUAL TEMPERAMENT perament. But, as Table 52 shows, there is a large bulge in the middle of the octave, and F*, which should be 5000^2 = 7071, is distorted very greatly. Now, if QR is given, the points of division are functions of QO (or RO), but they are also functions of QP. It is primarily the size of the angle QRP that determines the ratios of the string- lengths. Strahle's choice of 7 units for QP was unfortunate, or the distortion would not have been so great. To reduce the errors in this construction, let us attempt to find a value for the angle QRP for which the length for F# is correct, V2RM. Let A be the midpoint of QR and B the point 2 where OA cuts RM; so that BM is the length for F- . Then 1. RB = 42BP = a(2RP 1W2~ 2. OQR = cos-1 1/4 = 750 31' . By the sine law and from 1. and 2., 3. sin RPQ 12, or sin RPQ 12 sin PQR ~~ RP fl5/4 RB/1 + a/2" 4. cos QRP = 6/RB. From 3. and 4., 5. sin RPQ = ^30 cos QRP 1.1344 cos QRP 2(1+^2) " From 2., 6. QRP + RPQ = 104° 29' . As an approximate solution to 5. and 6., 7. QRP = 33° 36' and RPQ = 70° 53'. From 7., PQ = 7.028. But this is almost exactly Strahle's figure! A check reveals that Faggot made a serious error in computing the angles QRP and RPQ; so that his value for PQ was actually 8.605 rather than 7. Table 53 gives the correct figures for Strahle's temperament. 67 TUNING AND TEMPERAMENT Table 53. Correct Figures for Strahle's Temperament Lengths 100000 9432 8899 8400 7931 7490 7073 Names C X D X E F X Cents 0 101 202 302 401 500 600 Lengths 6676 6308 5955 5621 5303 5000 Names G X A X B C Cents 699 798 897 997 1098 1200 M.D. .83; S.D. 1.00 It is, therefore, possible to achieve superfine results by fol- lowing a method essentially the same as Strahle's. Although un- aware of the possibilities in Strahle's method, Marpurg has col- lected many unusual and interesting temperaments by other men. 44 Represented two monochords by Schrbter,both of which are excellent approximations to equal temperament constructed from tabular differences. In the first (Table 54), Schroter an- chored his column of differences upon the notes of the just minor triad, as C ED G C, with ratio 6:5:4:3. The intermediate notes were obtained by arithmetical divisions. This column of differ- ences is worth showing as a monochord in its own right, for the method of construction resembles that of Ganassi and Reinhard. The mean deviation is about the same as for the Pythagorean tuning, but the standard deviation is larger because the semitone B-C, with ratio 28:27, is much smaller than the others. Table 54. Schroter 's Column of Differences, No. 1 Lengths 54 51 48 45 42 40 38 36 Names C X D X E F X G Cents 0 99 204 317 435 520 608 702 Lengths 34 32 30 28 27 Names X A X B C Cents 804 906 1018 1137 1200 M.D. 11.9; S.D. 15.3 44yersuch uber die musikalische Temperatur, pp. 179 ff. 68 EQUAL TEMPERAMENT In Schrbter's monochord proper (Table 55) the upper funda- mental (451) is the sum of all the differences in the above table, save the first number to the left (54). Thus the lower fundamental (902) will be a true octave. This monochord is a highly satis- factory approximation to equal temperament. Table 55. Schroter's Approximation, No. 1 Lengths 902 851 803 758 716 676 638 Names C X D X E F X Cents 0 100.7 201.3 301.1 399.9 499.3 599.7 Lengths 602 568 536 506 478 451 Names G X A X B C Cents 700.0 800.7 901.1 1000.8 1099.4 1200 M.D. .52; S.D .59 Schroter's column of differences for the second approximation (Table 56), while also containing arithmetical divisions, is con- structed more carefully than the first. The minor thirds D-F and A-C have the unusual ratio 19:16 or 297 cents. All the notes in the tetrachord G-C are pure fifths above the notes inthetetra- chord C-F. Here the deviation is about the same as in Gram- mateus' tuning, thus ranking among the best of the irregular sys- tems .4 5 Table 56. Schrbter's Column of Differences, No. 2 Lengths 384 363 342 324 306 288 272 256 242 Names C X D X E F X G X Cents 0 97 201 294 393 498 597 702 799 Lengths 228 216 204 192 Names A X B C Cents 903 996 1095 1200 M.D. 3.8; S.D. 4.3 4^For Grammateus see the second part of Chapter VII. 69 TUNING AND TEMPERAMENT Schroter's second approximation (Table 57) is constructed from the above column of differences in the same manner as was his first. Its deviations, like those of the column of differences upon which it was based, are about 1/3 as large as those of the first monochord. 4843 600.3 Lengths 6850 6466 6103 5761 5437 5131 Names C X D X E F Cents 0 99.9 199.9 299.7 400.0 500.2 Lengths 4571 4315 4073 3845 3629 3425 Names G X A X B C Cents 700.3 800.1 900.0 M.D. . 15; 999.7 S.D. 18 1099.9 1200 Schroter's success in building up a monochord by using well- chosen tabular differences suggests that the same method be applied to Ganassi's tuning, which is rather similar to his first column of differences .46 The sum of the twelve numbers of Ganassi's monochord is 805, which is chosen, therefore, for the higher fundamental. As might have been expected, the mono- chord (Table 58) is very good. Table 58. Approximation Based on Ganassi's Monochord Lengths 1610 1520 1435 1355 1279 1207 1139 Names C X D X E F X Cents 0 99.6 199.3 298 6 398.5 498.8 599.2 Lengths 1075 1015 958 904 853 805 Names G X A X B C Cents 699.3 798.8 898.8 999.3 1099.9 1200 M.D. .42; S.D. .51 46see Chapter VII for Ganassi's tuning. 70 EQUAL TEMPERAMENT Table 59. Monochord from Difference Column, No. 1 Lengths 24 23 22 21 20 19 18 17 Names C X D X E F X G Cents 0 74 151 232 316 405 498 597 Lengths 15 14 13 12 Names A X B C Cents 815 933 1062 1200 M.D. 18.2; S.D. 19.7 16 702 These rather amusing improvements in poor or fair tuning systems suggest that the method be really put to the test by choosing for the original monochord an entirely unsatisfactory tuning. Accordingly, the thirteen numbers from 12 through 24 were chosen (Table 59). This is so perverted a tuning system that the major third (E), the fourth (F), and the fifth (G) are pre- cisely a semitone flat according to just intonation. However, a benighted anonymous writer in the Mercure de France in 1771 declared that if the entire string were divided into 24 parts, the numbers 12 through 24 would give all the semitones .47 Thanks to the regularity of its construction, the deviation of this system ranks it somewhere near the meantone tuning! In the next monochord (Table 60) the deviation is of the same class as that of Galilei's tuning. Its higher fundamental, 210, is the sum of the numbers 12 to 23 inclusive. Table 60. Monochord from Difference Column, No. 2 Lengths 420 397 375 354 334 315 297 Names C x D x E F x Cents 0 97.5 196.2 296.0 397.7 498.1 599.9 Lengths 280 264 249 235 222 210 Names G x A x B C Cents 702.0 803.9 905.2 1005.4 1103 9 1200 M.D. 1.6; S.D. 1.9 47Lionel de La Laurencie, Le violon de Lullya Viotti (Paris, 1924), Tome HI p. 74. 71 TUNING AND TEMPERAMENT For our third monochord (Table 61) we use the lengths of Table 60 as differences. Here the deviation is about the same as in Schrbter's second approximation. In the fourth and last approximation (Table 62) the errors have become too small to be recorded correctly when five-place logarithms are used. Apparently, however, the deviation is again about 1/10 that of the previous monochord. Table 61. Monochord from Difference Column, No. 3 Lengths 7064 6667 6292 5938 5614 5289 4992 Names C X D X E F X Cents 0 100.1 200.2 300.6 400.9 501.0 601.1 Lengths 4712 4448 4199 3964 3742 3532 Names G X A X B C Cents 701.0 800.9 900.6 M.D. .18; 1000.3 S. D. .21 1100.1 1200 Table 62. Monochord from Difference Column, No. 4. Lengths 118758 112091 105799 99861 94257 88968 83976 Names C X D X E F X Cents 0 100 200 300 400 500 600 Lengths 79264 74816 70617 66653 62911 59379 Names G X A X B C Cents 700 800 900 1000 1100 1200 Objection may be made to Schroter's approximations, and to ours as well, on the ground that the fundamentals are not round numbers such as most of the theorists used for the representa- tion of equal temperament. Let us see whether we can supply this lack. In our third monochord (Table 61) the length for F# is 4992. Let this be our higher fundamental. Add 8 to it, and 16 to its double, the lower fundamental. We could then make an arithmetical division to correct the intermediate numbers. It is little more trouble, however, to take the two left-hand digits of the numbers in this same monochord, starting with the value for 72 EQUAL TEMPERAMENT BD, 40. Multiply these and those for B, 37, by .4, as 16.0, 14.8, and all the pairs of digits to the left of BD by .2. Add these num- bers to the appropriate numbers in Monochord No. 3, and we have a corrected monochord, in which the maximum error is 4 units, or about 1 cent (see Table 63). Deviation is as in the original Monochord No. 3 (Table 61). Table 63. Monochord No. 3, Adjusted Lengths 10000 9439 8910 8411 7940 7496 7075 Names C X D X E F X Lengths 6678 6302 5947 5613 5297 5000 Names G X A X B C Fortunately, it is possible to make a similar adjustment of our five-digit monochord, No. 4 (Table 62). Here we shall take as our lower fundamental the length for ED, 99861. We need 139 to make a round number. This is about twice the length for G in Monochord No. 2. So we divide the numbers in the second mono- chord by 2 or by 4, and add to the appropriate numbers in Mono- chord No. 4. The maximum error is 6 units, or about 1/6 cent. A very useful approximation for equal temperament is to ex- press all its irrational ratios as comparatively small fractions. Alexander Ellis^S has made a table of about 150 intervals within the octave, which he has represented by logarithms, cents, and ratios, actual or approximate. Since all the intervals of equal temperament are contained in this table, it is easy to list them separately, as in Table 65. Table 64. Monochord No. 4, Adjusted Lengths 100,000 94,388 89,092 84,093 79,375 74,921 70,716 Names C x D x E F x Lengths 66,747 62,999 59,462 56,124 52,974 50,000 Names G x A x B C 48H. L. F. Helmholtz, Sensations of Tone, pp. 453-456. 73 TUNING AND TEMPERAMENT Table 65. Ellis' Fractional Approximations Ratios 1 89:84 449:400 44:37 63:50 303:227 140:99 433:289 Names Cx D xEF x G Ratios 100:63 37:22 98:55 168:89 2 Names x A x B C Charles Williamson^ has given the material in Table 65, wrongly ascribing it to Helmholtz rather than to Ellis. By con- tinued fractions he himself found that the majority of Ellis' ratios were correct. He objected to the ratio for the major second (449:400), stating that this interval can be represented more ac- curately as the inversion of a minor seventh. The ratios for the fourth (303:227) and fifth (4 33: 289) he thought were not sufficiently close either, and should likewise be paired. Ellis' ratio for the tritone (140:99) was good, but Williamson preferred to use the ratio for its inversion (99:70), which is no better. Williamson remarked that his ratio for the tone (55:49) oc- curs in Cahill's patent for the Telharmonium, and for the tritone (99:70) in Laurens Hammond's patent for the Hammond Electric Organ. He had not previously run across 295:221 or 442:295. It is interesting to note that here, as in many other instances, Pere Mersenne^O has anticipated the modern students of temperament. Mersenne stated that the minor third of equal temperament is approximately 6/5 x 112/113 = 672/565. Convergents to this ratio are 44:37 and 157:132, the first of these occurring in both tables above. Mersenne 's ratio for the major third was 5/4 x 127/126 = 635/504, convergents to which are 63:50 (as above) and 286:227. For the perfect fifth he gave the ratio 32 x 886/887 = 1329/887, the convergent to which is 442:295, used by Williamson. Williamson's reference to Hammond's patent^! suggests that the latter 's ratios be examined in their entirety. (It must be remembered that these ratios are based on the practical con- 49 "Frequency Ratios of the Tempered Scale," Journal of the Acoustical So- ciety of America, X (1938), 135. ^^Harmonie universelle, Nouvelles observations physiques & mathematiques, pTllL 51L. Hammond's Patent, 1,956,350, April 24, 1934, Sheet 18. 74 EQUAL TEMPERAMENT sideration of cutting teeth on gears.) The difficulty is that, al- though it is easy enough to reduce Hammond's frequencies to ratios with no more than two digits in numerator and denomin- ator, no one note appears as unity. (The ratios times 320 are the frequencies from middle C to its octave.) We cannot well compare this with Table 65. If either F or A, which have the simplest ratios in Table 66, is given the value of 1, more than half of the ratios will have three digits. Hence the composite table, Table 67, with decimal equivalents, gives a better idea of how the three systems compare. Table 66 . Hammond's Fractional Approximations Ratios 85:104 71:82 67:73 35:36 69:67 12:11 37:32 Names C X D x E F x Ratios 49:40 48:37 11:8 67:46 54:35 85.52 Names G X A x B C Table 67 . Compar ison of Three Approximations Ellis Williamson Hammond Equal Temperament C 200000 200000 200000 200000 B 188652 188652 188697 188775 x 178182 178182 178182 178180 A 168182 168182 168182 168179 x 158730 158730 158677 158740 G 149827 149831 149796 149831 x 141414 141429 141414 141421 F 133480 133484 133499 133484 E 126000 126000 125942 125992 x 118919 118919 118881 118921 D 112250 112245 112207 112246 x 105952 105952 105928 105946 C 100000 100000 100000 100000 Hammond has utilized some of the same ratios as Ellis and Williamson. His tone G-A is 55:49; his minor thirds F-Ab and F#-A are 44:37; his major third Eb-G is 63:50; his tritones ED-A and F-B are 99:70. He had another major third (Bb-D) 75 TUNING AND TEMPERAMENT with small ratio, 73:46, but this is a poorer approximation than 63:50. Note that many of Hammond's ratios are related in pairs, but not in the same way as Williamson's. The product of the ratios for F^ and G#, F and A, E and BD, and B and D^ is equal to 3:2. C and D are not so related. Of course the axis G is ap- proximately the square root of 3:2, and C*, the other axis, the square root of 3:4. Let us compare these three approximations with the true values for equal temperament to six places (see Table 67). For Ellis and Williamson these are the decimal equivalents of the fractions as given. For Hammond the note A was taken as the fundamental, and his frequencies as given in the patent have been divided by 1.1. In our absorption with quasi- equal temperaments that excel many presumably correct versions, we should not neglect the pioneers who first set down in figures the monochords constructed upon the 12th root of 2. The first European known to have formed such a monochord is Simon Stevin,52 about 1596, who said that since there are twelve proportional semitones in the octave, the problem is to "find 11 mean proportional parts between 2 and 1, which can be learned through the 45th proposition of my French arithmetic." There he had explained that mean proportionals can be found by extracting roots of the product of the extremes. He now applied this principle, by representing each semitone as the 12th root of some power of 2 (see Table 68). Table 68. Stevin's Monochord, No. 1 Lengths 10000 9440 8911 8408 7937 7493 7071 Names C X D X E F X Cents 0 99.7 199.6 300.2 400.0 499.6 600.0 Lengths 6675 6301 5945 5612 5298 5000 Names G X A X B C Cents 699.8 799.6 900.3 1000.1 1099.7 1200 52Van de Spiegeling der Singconst, pp. 26 ff. 76 EQUAL TEMPERAMENT In his actual calculations Stevin first computed notes 7, 4, and 5, that is, F^, Eb, and E. These involve no more difficult roots than cubic and quartic. There is now sufficient material to com- pute the remaining notes by proportion, "the rule of three." Thus the fifth note (7937), divided by the fourth (8408), gives the second (5440). This method is much easier than to extract the roots for each individual note, which runs into difficulties with the roots of prime powers, as for notes 2, 6, 8, and 12 (C*, F, G, B), where the 12th root itself must be extracted. But the method by pro- portion lacks in accuracy, for an error for any note is magnified in succeeding notes. Even so, the maximum error is only .4 cent. The deviation for Stevin 's monochord lies between those for Schrbter's two monochords. Stevin has worked out a second monochord for equal temper- ament upon the same principle as the first, but with a different order of notes. 53 Here the maximum error, for E, is 1 cent. The fact that the two monochords do differ indicates that pro- portion is not the ideal method (see Table 69). At the same time that Stevin was setting down the figures for equal temperament, or perhaps a few years earlier (1595), Prince Tsai-yii in China was making a much more elaborate and careful calculation of the same roots of 2.^4 We are not told how he performed his calculation, but, since it is correct to nine places, he must have extracted the appropriate root for each note sepa- rately—and without the aid of logarithms, which were to simplify Table 69. Stevin's Monochord, No. 2 Lengths 10000 9438 8908 8404 7936 7491 7071 Names E F X G X A X Lengths 6674 6298 5944 5611 5296 5000 Names B C X D X E 53Ibid., p. 72. 54pere Joseph Maria Amiot, De la musique des Chinois (Memoires concernant l'histoire, . . . des Chinois, " Vol. VI | Paris, 1780]), Part 2, Fig. 18, Plate 21. See also J. Murray Barbour, "A Sixteenth Century Approximation for IT," American Mathematical Monthly, XL (1933), 69-73. 77 TUNING AND TEMPERAMENT the problem so greatly for men who attempted it a few decades later. In some cases, since the tenth digit will be 5 or larger, modern computers would round off the number at the ninth digit by substituting the next higher digit. This is a convention of our mathematics, intended to reduce the error arising from rounding off a number. Tsai-yli never did this. Probably the first printed solution of equal temperament in numbers was made in Europe in 1630, a generation after Tsai- yii's time, when Johann Faulhaber solved a problem propounded by Dr. Johann Melder of Ulm.55 The problem was to divide a monochord 20000 units in length, so that all intervals of the same size should be equal. Faulhaber did not explain to his readers how he had arrived at his result (Table 71), presenting it rather as a riddle. His monochord was for equal temperament, but con- tained several errors of 1 in the unit's place. This is the sort of error likely to occur when logarithms are used, and we might suppose Faulhaber had made use of the logarithmic tables printed in his book. Table 70. Tsai-yii's Monochord c 500,000,000 F 749,153,538 B 529,731 ,547 E 793,700,525 X 561,231 ,024 X 840,896,415 A 594,603,557 D 890,898,718 X 629,960,524 X 943,874,312 G 667,419,927 C 1000,000,000 X 707,106,781 Table 71 . Faulhaber 's Monochord Lengths 20000 18877 17817 16817 15874 14982 14141 Names C X D X E F x Lengths 13347 12598 11891 11224 10594 10000 Names G X A X B C 55 Johann George Neidhardt, Sectio canonis harmonici (Ktinigsberg, 1724), p. 23. 78 EQUAL TEMPERAMENT Mersennehas given a number of different tables of equal tem- perament. The most characteristic, to six places, was furnished by Beaugrand, "very excellent geometer. "56 Mersenne also printed a table of first differences for the numbers in this mono- chord, to be used in connection with a method by Beaugrand for constructing the equal semitones. A comparison with Tsai-yu's table shows this one to be very inaccurate, the errors being much larger than if logarithms had been used. A much more ambitious table was contributed by Galle.57 In this table the lengths were given to eleven places. Beside it Mersenne printed a table with 144,000,000 as fundamental, so that the numbers might readily be compared with those of "the perfect clavier with 32 keys or steps to the octave," which had been presented in the book on the organ. This table will not be included here, for it seems likely that Mersenne himself com- puted these numbers from Galleys larger table, by multiplying them by .00144. Of the numbers in the table, the length for D is correct to only five places. The others agree fairly well with Tsai-yii to the ninth place, although there are some slight diver- gences. Beyond the ninth place no digits are correct. If Galle was using logarithms, he made some serious errors in interpo- lation. But if he was extracting roots, it is difficult to see how he failed to find correctly the middle number, the length for F#, which represents 1011 times the square root of 1/2. It should be ten units larger. The length for E*5 (1011 times the fourth root of 1/2) agrees neither with the correct value nor with the square root of the length for F#. Our final table from Mersenne^ Was supplied by Boulliau, "one of the most excellent astronomers of our age." In it he ex- pressed the string- lengths for equal temperament in degrees, minutes, and seconds. This is equivalent to having a fundamental of 14400 in decimal notation, and the errors should be no greater than for such a table. However, the errors are greater than in Stevin's four-place table, with a mean deviation of about 1 cent. We can only surmise how Boulliau computed his figures. Evi- ^"Mersenne, Harmonie universelle, p. 38. 5*7 Ibid., Nouvelles observations, p. 21. 58Ibid., pp. 384 f. 79 TUNING AND TEMPERAMENT dently the sexagesimal notation is somehow linked with his method of extracting the roots. Neidhardt printed six-place tables in equal temperament from Faulhaber, Mersenne, and Biimler, as well as several of his own. 59 His first original method was to divide the syntonic comma arithmetically, thus giving rise to a twofold error. The arithmetical division makes little difference, but the fact that the syntonic comma is about two cents smaller than the ditonic comma means that each fifth will be about .2 cent sharper than in correct equal temperament. Such a division is fairly easy to make, and, as the cents values indicate, the errors are small. The mean deviation is about 1 cent. Later, Neidhardt^O Was to divide the ditonic comma, both arithmetically and geometrically, the latter method being genuine equal temperament. He contended, however, that the differences between these two methods were negligible. Since the greatest variation is 5 units, in tables containing 6 digits, his contention was correct. Note that the numbers for the arithmetical division are the larger throughout the table. The true values come closer to his geometrical division, but in every instance lie between the two. Neidhardt's contemporary, Jakob Georg Meckenheuser,61 printed a table, "as computed in the first Societats-Frucht," evidently the proceedings of some learned society. From his figures, the syntonic comma is divided arithmetically, as in Neidhardt's first monochord. But evidently Meckenheuser's division ran to sharps, for seven of his notes were higher in pitch than the corresponding notes in Neidhardt's monochord. The higher C is not a true octave, but a B# tempered by a full syntonic comma, just as his F is really a tempered E#. The ratio of these pairs of enharmonic notes is the schisma, about 2 cents. Thus even when two temperaments are constructed upon the same hypothesis and both are intended for equal temperament, 59Neidhardt, Sectio canonis harmonici, p. 32. SOlbid., p. 19. "Ipie sogenannte allerneueste musicalische Temperatur (Quedlinburg, 1727), p. 51. 80 EQUAL TEMPERAMENT Table 72. Beaugrand's Monochord Lengths 200000 188770 178171 168178 158740 149829 141421 Names C x D x E F x Lengths 133480 125992 118920 112245 105945 100000 Names G x A x B C Table 73. Galle 's Monochord C 50,000,000,000 F 74,915,353,818 B 52,973,154,575 E 79,370,052,622 X 56,123,102,370 x 84,089,641,454 A 59,460,355,690 D 89,090,418,365 X 62,996,052,457 x 94,387,431,198 G 66,741,992,715 C 100,000,000,000 X 70,710,678,109 Table 74. Boulliau's Monochord Sexagesimal Notation Decimal Notation The Same, 20000 as Fundament 7200 10000 7632 10600 8092 11239 8573 11907 9072 12600 9605 13340 10179 14138 10772 14961 11405 15840 12110 16819 12823 17810 13580 18861 C 2° 0' 0" B 2 7 12 x 2 14 52 A 2 22 53 x 2 31 12 G 2 40 5 x 2 49 39 F 2 59 32 E 3 10 5 x 3 21 50 D 3 33 43 x 3 46 20 C 4 0 0 14400 20000 81 TUNING AND TEMPERAMENT Table 75. Neidhardt's Division of Syntonic Comma Lengths 200000 188867 178148 168229 158683 149845 141344 Names C Db D Eb E F F# Cents 0 99.1 200.3 299.5 400.6 499.9 601.0 Lengths 133472 126041 118888 112268 105898 100000 Names G Ab A B*> B c Cents 700.2 799.3 900.5 999.7 1100.8 1200 Table 76. Neidhardt's Division of Ditonic Comma • Arithmetical c 100000 B 105948 X 112247 A 118922 X 125994 G 133484 X 141424 F 149831 E 158743 X 168182 D 178182 X 188779 C 200000 Geometrical 100000 105945 112245 118920 125991 133483 141420 149830 158739 168178 178179 188774 200000 82 EQUAL TEMPERAMENT there may be a lack of agreement unless the process is followed through in exactly the same way for both. If it is true equal tem- perament, however, it does not matter in what order the notes are obtained, whether on the sharp or the flat side or mixed up in anyway whatever. In Table 77, Meckenheuser's numbers have been divided by 18. This tends to conceal his rather obvious arithmetical division of the comma: in the original, every num- ber except one (the length for D) ends in zero. There the value for G had been 240200000. This has been corrected to 240250000, since the number should be 240000000 tempered by 1/12 x 1/80 = 1/960. Since the syntonic comma is much easier to form than the ditonic, it is easy to see why it should have been preferred as the quantity to be divided. However, since the ratio of the two commas is about 11:12, an excellent approximation for equal temperament can be made by tempering the fifths by 1/11 syntonic comma. 62 This was done arithmetically by Sorge,with the results shown in Table 78. The mean tempering of his fifths is 1/886, whence the ratio of the fifth will be .667419962 . . . , instead of .667419927 .... However, there are larger errors for most notes, since the tem- perament is not built solely by fifths, and the temperament as a whole is comparable to Neidhardt's arithmetical division of the ditonic comma. Table 77. Meckenheuser's Division of Syntonic Comma Lengths 200,000,000 188,658,258 178,148,341 168,045,776 158,684,002 Names C C^ D D# E Cents 0 101.0 200.3 301 3 400.6 Lengths 149,685,380 141,346,458 133,472,222 125,903,184 118,889,159 Names E* F^ G G# A Cents 501.6 600.9 700.2 801.2 900.5 Lengths 112,147,215 105,899,532 99,894,201 Names A* B B^ Cents 1001.5 1100.8 1201.8 62Marpurg, Versuch iiber die musikalische Temperatur, p. 177. 83 TUNING AND TEMPERAMENT Table 78. Sorge's Division of Syntonic Comma Lengths 200000 188775 178182 168181 158743 149831 141422 Names C C* D D* E E* F# Lengths 133484 125994 118923 112247 105948 100000 Names G G* A A* B c The impression is likely to become quite strong as one reads the second half of this chapter that equal temperament is nothing but a mass of figures of astronomical size. Actually, as far as the ear is concerned, a wholly satisfactory monochord in equal temperament (or any other tuning system) would be obtained from the division of a string a meter long, marked off in millimeters. Mersenne63 gave such a table, considering it more practicable than the very complicated tables of Beaugrand and Galle. It could easily have been constructed from one of the more elaborate tables by rounding off the numbers at three places. Oddly, many of Mersenne's figures are one unit too large. The correct mono- chord is shown in Table 79. It is instructive to note that the de- viation for this monochord is larger than for one of Marpurg's irregular tunings, 64 and about the same as that for a couple of his other tunings. Thus, to three places, Marpurg's systems would have coincided with equal temperament. Table 79. Practical Equal Temperament, after Mersenne Lengths 1000 944 891 841 794 749 707 667 Names C X D X E F X G Cents 0 99.8 199.8 299.8 399.4 500.3 600.3 701.1 Lengths 630 595 561 530 500 Names X A X B C Cents 799.9 898.9 1000.7 1099.9 1200 M.D . .60; S.D. .81 ^Harmonie universelle, p. 339. ^Compare Marpurg's Temperaments E, B, and G in Chapter VH with the cents values of Table 79. 84 EQUAL TEMPERAMENT In 1706 young Neidhardt, full of importance as the author of a new book on temperament, Beste und leichteste Temperatur des Monochordi, held a tuning contest with Sebastian Bach's cousin, Johann Nikolaus Bach, in Jena. 6 5 Neidhardt tuned one set of pipes byamonochord he had computed by making an arith- metical division of the syntonic comma. Therefore, although he had worked out this division to six places, it was about as accurate as the practical monochord given above. Bach tuned another set of pipes entirely by ear, and won the contest handily, for a singer found it easier to sing a chorale in BD minor in Bach's tuning than in Neidhardt's. Perhaps part of Neidhardt's difficulty lay in the fact that it is difficult to tune a pipe to a string. Many years later, Adlung wrote that this same Johann Nikolaus Bach had what might be called a "monopipe"— a variable organ pipe with a sliding cyl- inder upon which the numbers of the monochord were inscribed. 66 Because of the end correction for a pipe, this method is likely to be faulty. However, forty years before the date of the historic tuning contest in Jena, Otto Gibelius67 described and pictured just such a pipe, intended for his meantone approximation dis- cussed in Chapter m. He also gave an end correction, amount- ing to 8/3 the width of the mouth of the pipe. In his accurately drawn copperplate (see Figure F) the width of the mouth is 11 millimeters, making the end correction about 30 millimeters. Since the internal depth is about 15 millimeters, his rule cor- responds very closely to our modern rule that the end correction for a rectangular pipe is twice the internal depth. The Dayton Miller Collection now at the Library of Congress contains several specimens of the "tuning pipe," most of them fairly small. Since the "tuning pipe" was not widely disseminated, organ- 65Philipp Spitta, Johann Sebastian Bach, trans. Clara Bell and J. A. Fuller- Maitland (2 vols.; London, 1884), I, 137 f. 66Jacob Adlung, Anleitung zu den musikalischen Gelahrtheit (Erfurt, 1758), p. 311. In addition to the Neidhardt- Bach test, he described a similar ex- perience that befell Meckenheuser in Riechenberg vor Goslar, where he tried for three days to tune the organ by his monochord, but in vain. See Jacob Adlung, Musica mechanica organoedi (Berlin, 1768), p. 56. 67propositiones mathematico-musicae, pp. 1-11. 85 TUNING AND TEMPERAMENT bC c o U (X T3 bD » c o C T3 3 O « o bp 86 EQUAL TEMPERAMENT ists tuning by the aid of the monochord probably had no more success than Neidhardt had. It is probable, however, that, like Johann Nikolaus— and Sebastian, too— the organists did not bother with a monochord but relied upon their ears. Hence the tuning rules given in the beginning of this chapter were of the greatest possible importance in practice. Some of them seem so vague that they would have needed to be supplemented by oral direc- tions. But if we could be sure that Mersenne's rule that a tem- pered fifth should beat once per second was to have been applied to the fifths in the vicinity of middle C, we would have as accurate a rule for equal temperament as that given by Alexander Ellis over two centuries later. Unfortunately, the more mathematically minded writers on equal temperament have given the impression that extreme ac- curacy in figures is the all- important thing in equal tempera- ment, even if it is patent that such accuracy cannot be obtained upon the longest feasible monochord. This is why Sebastian Bach and many others did not care for equal temperament. They were not opposed to the equal tuning itself , and their own tuning results were undoubtedly comparable to the best tuning accomplished today— upon the evidence of their compositions, as will be dis- cussed in the final chapter. But they needed a Mersenne to tell them that the complicated tables could well have had half their digits chopped off before using, and that, after all, a person who tunes accurately by beats gets results that the ear cannot dis- tinguish from the successive powers of the 12th root of 2. 87