Chapter V. JUST INTONATION The seeds of just intonation had been sown early in the Christian era, when Didymus and Ptolemy presented monochords that con- tained pure fifths and major thirds (see Chapter II). But they remained dormant during the Middle Ages. Even after the seeds had sprouted near the beginning of the modern era, the plants were to bear fruit only occasionally and haphazardly. Enough of our metaphor. We shall consider in this chapter all 12-note systems that contain some arrangement of pure fifths and major thirds. The Pythagorean tuning may be thought of as the limiting form of just intonation, since it has a great many pure fifths, but no pure major thirds. As the various chromatic notes were added to the scale during the latter Middle Ages, they were tuned by pure fifths or fourths to notes already present in the scale. Finally, fifteenth century writers were describing the formation of a complete chromatic monochord, using the Pythagorean intervals. Such a writer was Hugo de Reutlingen, whose altered notes consisted of two sharps and three flats. Since the more typical tuning has G# instead of AD, that is shown in Table 80. Of course the deviation would be the same as for Hugo's tuning. The ratio for each diatonic semitone is 256:243, and for the chromatic semitone 2187:2048. Compare with these ratios the relative simplicity of the ratios for Marpurg's first tuning, the model form of just intonation. (The lengths are very much simpler also.) The first known European writer to break away from the Pythagorean tuning for the tuning of the chromatic monochord was Bartolomeus Ramis de Pareja.2 Ramisgave specific direc- tions for tuning the monochord that resulted in a system in which the six notes A^° -G° are joined by perfect fifths, as in the Pyth- agorean tuning, and the remaining six notes, D^-F*-1 , also joined by fifths, lie a comma higher than the corresponding notes ^Flores musicae omnis cantus Gregoriani (Strassburg, 1488), Chapter EL ^Musica practica (Bologna, 1482); new edition, by Johannes Wolf (1901), pub- lished as Beiheft der Internationale Musikgesellschaft. TUNING AND TEMPERAMENT Table 80. Pythagorean Tuning Lengths 629856 589824 559872 531441 497664 472392 442368 Names C° C^° D° EbP E° F° F*o Cents 0 114 204 294 408 498 612 Lengths 419904 393216 373248 354294 321776 314928 Names G° G#o A0 Bbo B° C° Cents 702 816 906 M.D. 11.7; 996 S.D. 11.8 1110 1200 Names C Cents 0 C* D Table 81. Ramis' Monochord bO -1 0 -W~1_0 h° Bt B 92 182 294 386 498 590 702 792 884 996 1088 1200 M.D. 10.0; S.D. 10.1 in the Pythagorean tuning (see Table 81). Thus there are pure major thirds to only the four notes BD-G. Montucla,^ writing a "history of music," gave string-lengths for a 17 -note tuning, in which twelve notes are the same as in Ramis. The other five extend the scale to A*-1 and to GD'°. This is a wholly useless extension because such enharmonic pairs as D^0 and C*"1 differ by the schisma, 2 cents. Helmholtz was more astute in constructing his 24 -note harmonium in just intonation, in which the eight notes from C° through C#° are joined by fifths; the next eight, E"1 through E#_1 , furnish major thirds to notes in the first series; and the remaining eight, AD+1 through A+1, are considered (by disregarding the schisma) as equivalent to the thirds above the notes in the second series, i. e., G#~2- Gx"2.4 Ramis' monochord does not differ perceptibly from the Pyth- agorean tuning. If he had substituted DD° and all the other Pyth- agorean enharmonic equivalents of the syntonic notes, he would 3jean Etlenne Montucla, Histoire des mathe'matiques (New ed.; Paris, 1802), IV, 650. 4H. L. F. Helmholtz, Sensations of Tone, pp. 316 f. 90 JUST INTONATION have had a monochord from E^b0 through G°, in Pythagorean tuning. His reason for making the new division was solely to simplify the construction of the monochord. In his own words, the Pythagorean tuning, as given by Boethius, is "useful and pleasing for theorists, but tiresome for singers and irksome to the mind. But because we have promised to satisfy both [singers and theorists], we shall simplify the division of the monochord." Later he expressed the same idea in these words: "So therefore we have made all our divisions very easy, because the fractions are common and are not difficult." Undoubtedly Ramis' method is easier. But if he had desired to obtain the equivalent of the Pythagorean tuning from AD to C#, he would have commenced his tuning with F# instead of with C, having notes with zero exponents from D° to C*° and with -1 from G#-1 to Fx-1 . On such a monochord, however, as on the usual Pythagorean monochord, the eight most common thirds would have been very sharp and the four useless thirds, E-A", B-E13, F^-B^, and C*-F, would have been pure. The monochord, as Ramis actually tuned it, has as its four pure thirds, B^-D, F-A, C-E, and G-B. Thus, although Ramis professed to be mak- ing his division of the octave solely for the sake of simplicity, the accidental result was that several pure triads were available in keys frequently used. The bitter critics of Ramis in his own day failed to realize that his tuning was just what he had described: a simplified equivalent of the Pythagorean tuning — shifted, however, by six scale degrees to the flat side. To them, any tampering with the old intervals was sacrilege. Many later writers, misled by Ramis' announced intentions, have stated, without examining his monochord, that he had advocated temperament. As we have de- fined temperament and as the word is usually understood, this is a serious misconception. It has even been stated that Ramis ad- vocated equal temperament! Since Ramis' book is accessible in a modern edition, there is no longer any excuse for repeating such myths. It must be said, somewhat sadly, that Ramis was not aware himself of the peculiar properties of the monochord he had 91 TUNING AND TEMPERAMENT fathered. For example, he explained that although Eb does not form a major third to B, D# is not really needed, for the minor triad B D F* can be used in making a Phrygian cadence on E. But his interval B^-E^0 is slightly better than the Pythagorean thirds, Ab°-C° and Eb°-G°, that were acceptable to him! Ramis must have been a good practical musician. Although his system would not now be called a temperament, we might do well to take him at his own evaluation and hail him as the first of modern tuning reformers. Corroboration of Ramis' tuning system is found in an inter- esting anonymous German manuscript of the second half of the fifteenth century, Pro clavichordiisfaciendis, which Dupont^ ran across in the Erlangen University Library. Starting with the note B, C is to be a just semitone (16:15) higher, E a perfect fourth, G a just minor sixth (8:5), etc. A succession of pure fifths on the flat side extends to Gb, below which there is a just major third (5:4), E^b, and the monochord is completed by add- ing BbD, the fifth above Ebb! The complete monochord is shown in Table 82. The deviation for this tuning is almost precisely the same as for that of Ramis, and it too contains many pure fifths and sev- eral pure thirds. However, it has one peculiar feature as Du- pont has presented it. In every other tuning system we have ex- amined, there has been an uninterrupted succession of notes connected by fifths from the flattest to the sharpest. In the Pyth- agorean and other regular tuning systems, such as the meantone, the wolf fifth would be very flat or sharp, and in the irregular systems there would be other divergences. But the note names persisted, usually from Eb to G* inclusive. Table 82. The Erlangen Monochord Names C° Db° Ebb+1Ebo E"1 F° Gb° G° Ab° Bbb+1Bb° B_1 C° Cents 0 90 202 294 386 498 588 702 792 904 996 1088 1200 M.D. 10.3; S.D. 10.5 ^Wilhelm Dupont, Geshichte der musicalischen Temperatur (Erlangen, 1935), pp. 20-22. 92 JUST INTONATION But in the Erlangen monochord there is no D or A, and the notes that Dupont has given as their enharmonic equivalents, E^b and B^b, are not in a fifth -relation with any other notes in the monochord. Therefore it seems obvious that the anonymous writer intended these notes to be D° and A0, each of which is higher by the schisma than E^b*1 and Bbb+1 respectively. Then the notes that are pure thirds above D° and A0 will be F*-1 and C^"1 , notes that continue the fifth-series from B"1. It would then be immaterial whether to call the semitone between G and A by the name AD° or G* * , since either would complete the scale correctly. The original writer, by the way, had not named the black keys, merely designating the semitone between C and D as the first, between D and E as the second, between F and G as the third, and between G and A as the fourth. In renaming some of the black keys, therefore, we are not violating his in- tent, but rather confirming it. The revised monochord, with schismatic alterations, is shown in Table 83. These two pre-sixteenth-century tunings, the one in Spain and the other in Germany, are sufficient indication of the trend of men's thinking with regard to consonant thirds. Lodovico Fogliano," half a century later than Ramis, offered no apologies for using the 5:4 ratio for the major third. But he was not con- tent to present ordinary just intonation. Realizing that D° formed an imperfect fifth below A"1 , he advocated D-1 as a consonant fifth. This in turn led him to BD° as a pure major third below D-1, as well as the Bb+1 as third below D°. But he said the "practical musicians" used only one key each for D and B*3, "neither right nor left, but the mean between both." "Such a mean D or BD, moreover, is nothing else than a point dividing the proportion of the comma into two halves." Table 83. Erlangen Monochord, Revised Names C° C*"1 D° E* E_1 F° F^G0 G*" A° Bb° B_1 C° Cents 0 92 204 294 386 498 590 702 794 906 996 1088 1200 M.D. 10.0; S.D. 10.1 6Musica theorica (Venice, 1529), fol. 36. 93 TUNING AND TEMPERAMENT To obtain the mean proportional by geometry, Fogliano used the familiar Euclidean construction, and appended a figure to show how the division was to be made. This alteration of pure values, he said is "what they [the practical musicians] call temperament." Here is the germ of the meantone temperament, which his countryman Aron had described in its complete format aboutthis same time. For the sake of showing monochords in just intonation from the early sixteenth century, there are set down here three mon- ochords after Fogliano, first with his one pair of D's and B^'s, then with the second pair, and finally with the mean D and B*3. The first monochord (Table 84) is the best, having two groups of four notes each with like exponents. The second monochord (Table 85) would have had the same deviation as the first if it had had F#_1 (in place of F#"2) as third above D°. (This is Mar- purg's first monochord, Table 96.) The monochord with the two meantones (Table 86) ranks between the first two. If Fogliano had formed three meantones, including one on F#, the deviation would be slightly less than for the first monochord. The result is given in Table 87. Table 84. Fogliano's Monochord, No. 1 Lengths 3600 3456 3240 3000 2880 2700 2592 2400 Names C° c*-2 D"1 Eb+i E"1 F° F#-2 G° Cents 0 70 182 316 386 498 568 702 Lengths 2304 2160 2025 1920 1800 Names G,_2 A"1 Bbo B_I C° Cents 772 884 996 1088 1200 M.D. 21.3; S.D. 23.6 94 JUST INTONATION Table 85. Fogliano's Monochord, No. 2 Lengths 3600 3456 3200 3000 2880 2700 2592 2400 Names C° c#-2 D° Eb+1 E"1 F° F*-2 G° Cents 0 70 204 316 386 498 568 702 Lengths 2304 2160 2000 1920 1800 Names Gfr. A-1 Bb+i B"1 C° Cents 772 884 1018 1088 1200 M.D. 25.0; S.D. 26.7 Table 86. Fogliano's Tempered Just Intonation Lengths 3600 3456 |_3220j 3000 2880 2700 2592 2400 Names C° c*-2 D-i Eb+i E"1 F° F*« G° Cents 0 70 193 316 386 498 568 702 Lengths 2304 2160 [2012.5] 1920 1800 Names G#-2 A"1 Bb+I B"1 C° Cents 772 884 1007 1088 1200 M.D. 23.2; S.D. 24.7 Table 87. Fogliano's Tempered Just Intonation, Revised Lengths 3600 3456 [3220] 3000 2880 2700 [2576] Names C° c#-2 1 D 2 Eb+1 E"1 F° F# 2 Cents 0 70 193 316 386 498 579 Lengths 2400 2304 2160 [2012.5] 1920 1800 Names G° G#-l A'1 Bb+i B"1 C° Cents 702 772 884 M.D. 21.3; 1007 S.D. 22.3 1088 1200 95 TUNING AND TEMPERAMENT Martin Agricola' resembled Ramis in his tuning ideas. He gave a monochord in which the eight diatonic notes, including B*3, were joined by pure fifths, as in the Pythagorean tuning. Then he directed that the interval from B to the end of the string be divided into ten parts, with C* at the first point of division, D# at the second, and G* at the fourth. Then F* was to be a pure fourth to C*. Thus these black keys were given syntonic values, and the whole monochord is made up of notes with 0 and -1 ex- ponents (see Table 88). Ramis' monochord is slightly better than Agricola' s, with a ratio of 6:6 for the number of fifths in each group, in place of 8:4. Table 88. Agricola 's Monochord Names C° C^D0 D*"1 E° F° F*"1 G° G*-1 A0 Bb° B° C° Cents 0 92 204 296 408 498 590 702 794 906 996 1110 1200 M.D. 10.3; S.D. 10.5 It will be observed that the better of Fogliano's untempered monochords has more than twice the deviation of Ramis'. Thus it might be thought that Fogliano had been unfortunate in his choice of intervals. Quite the contrary. The most symmetric form of just intonation for the series ED-G* has four notes with the same exponent, followed by four more with exponents that are one less. Of the remaining four notes, two would have +2 and two would have -2 as exponents. This is precisely Fogliano's second monochord, if we should substitute F^"1 in it. Fogliano's first monochord has the exponential pattern 1,4,4,3, which is just as satisfactory. (That is, the tuning contains one note with ex- ponent + 1, 4 with 0 and -1 exponents, and 3 with -2.) The diffi- culty, therefore, is inherent in just intonation itself, as will be discussed further a bit later. Salomon de Caus^ was one of several mathematicians of the early seventeenth century who were interested in just intonation. '"De monochorea' dimensione," in Rudimenta musices (Wittemberg, 1539) °Les raisons des forces mouvantes avecdiverses machines (Francfort, 1615) Book 3, Problem III. 96 JUST INTONATION If we follow his directions, we obtain the monochord shown in Table 89. Here there are three groups of four notes each with the same exponent — the most symmetric arrangement of all. The deviation is appreciably less than in Fogliano' s arrangement. Johannes Kepler^ gave some genuine tuning lore together with an elaborate discussion of the harmony of the spheres. His two monochords in just intonation (Tables 90 and 91) are identi- cal except that the second has a G* in place ofanAD. Since Kep- ler had five notes with zero exponents in both monochords, the deviation for his systems is lower than most that have been pre- sented in this chapter. Table 89. De Caus's Monochord Names C° C^D"1 D#_2 E" ?#-2 no 1#"« A-l B bo B_1 Cents 0 70 182 274 386 498 568 702 772 884 996 1088 1200 M.D. 17.7: S.D. 20.1 Table 90. Kepler's Monochord, No. 1 Lengths 1620 1536 1440 1350 1296 1215 1152 Names C° c"-» D° Eb+i E_1 F° jr#-i Cents 0 92 204 316 386 498 590 Lengths 1080 1024 960 900 864 810 Names G° G*+i A0 Bb+i B"1 C° Cents 702 794, 906 1018 1088 1200 M.D. 14.0; S.D. 15.8 ^Harmonices mundi, p. 163. 97 TUNING AND TEMPERAMENT Although Marin Mersenne was a zealous advocate of equal temperament in practice, he took pains to present literally doz- ens of tables in just intonation. He repeated, among others, Kepler's two monochords shown in Tables 90 and 91, together with tables for keyboards with split keys. Four of his monochords (Tables 92-95) are worth including here, as evidence of the va- riety that is possible in a type of tuning that is ordinarily thought to be fixed and uniform.^ None is as good as either of Kep- ler's two. Table 91. Kepler's Monochord, No. 2 Lengths 100000 93750 88889 833333 80000 75000 71111 Names C° c#-i D° Eb+i E_1 F° f'-' Cents 0 92 204 316 386 498 590 Lengths 66667 62500 60000 56250 53333 50000 Names G° Ab+i A0 Bb+i B"1 C° Cents 702 814 906 M.D. 14.0; 1018 S.D. 15.8 1088 1200 Table 92. Mersenne's Spinet Tuning, No. 1 Lengths 3600 3375 3240 3000 2880 2700 Names C° Db+1 D"1 Eb+1 E_1 F° Cents 0 112 182 316 386 498 Lengths 2400 2250 2160 2025 1920 1800 Names G° Ab+1 A-1 Bbo B*1 C° Cents 702 814 884 M.D. 17.7 996 S.D. 1088 20.1 1200 lOMersenne, Harmonie universelle, pp. 54, 117 f. 98 2531 1/4 nb+i 610 JUST INTONATION Table 93. Mersenne's Spinet Tuning, No. 2 Lengths 3600 3456 3200 3072 2880 2700 2592 2400 Names C° C#-2 D° D*-2 E"1 F° F#"2 G° Cents 0 70 204 274 386 498 568 702 Lengths 2304 2160 2025 1920 1800 Names Gf-2 A"1 Bb B"1 Co Cents 772 884 996 1088 1200 M.D. 21.3; S.D. 23.6 Table 94. Mersenne's Lute Tuning, No. 1 Names C° Db+1 D"1 Eb+1 E"1 F° Gb+1 G° Ab+1A_1 Bb+1 B"1 C° Cents 0 112 182 316 386 498 610 702 814 884 1018 1088 1200 M.D. 21.3; S.D. 23.6 Table 95. Mersenne's Lute Tuning, No. 2 Names C° Db+1 D° Eb+1 E"1 F° Gb+1 G° Ab+1 A"1 Bb+1 B_1 C° Cents 0 112 204 316 386 498 610 702 814 884 1018 1088 1200 M.D. 17.7; S.D. 20.1 Table 96. Marpurg's Monochord, No. 1 Lengths 900 864 800 750 720 675 640 Ratios 24/25 25/27 15/16 24/25 15/16 128/135 15/16 Names C° c*-2 D° Eb+1 E"1 F° F*'1 Cents 0 70 204 316 386 498 590 Lengths 600 576 540 500 480 450 Ratios 24/25 15/16 25/27 24/25 15/16 Names G° G*-2 A"1 Bb B"1 C° Cents 702 772 884 1018 1088 1200 M.D. 21.3; S.D. 23.6 99 TUNING AND TEMPERAMENT Table 97. Marpurg's Monochord, No. 3 Names C° C#_2D° Eb * E^1 F° F#_1 G° G^2 A0 Bb° B"1 C° Cents 0 70 204 306 386 498 590 702 772 906 996 1088 1200 MD. 19.3; S.D. 22.0 Table 98. Marpurg's Monochord, No. 4 Names C° C^D-1 Eb+1 E _1 F° F#~2G0 G^A"1 Bb +1 B_1 C° Cents 0 70 182 316 386 498 568 702 772 884 1018 1088 1200 M.D. 25.0; S.D. 26.7 Note that Mersenne's first spinet tuning (Table 92) has flats for its black keys and the second tuning (Table 93) has sharps except for BO. The first tuning is constructed exactly the same as deCaus's tuning (Table 89), except that it begins a major third lower, with Gb instead of B°. Mersenne's first lute tuning (Ta- ble 94) differs from his first spinet tuning (Table 92) at only one pitch (BD+1 instead of B130), but that is enough to increase its de- viation to that of the second spinet tuning (Table 93). The second lute tuning (Table 95), although differing from the first spinet tuning (Table 92) at two places, has the same deviation. Friedrich Wilhelm Marpurg, *■ who wrote brilliantly about temperament 140 years after Mersenne, included four mono- chords in just intonation. The second of these was Kepler's first, and need not be repeated here. The other three are shown in Tables 96-98. In each of them the notes, according to their exponents, are grouped into four classes. The first may be con- sidered the model form of just intonation, the ideal form of Fog- liano's second monochord (Table 85). Opelt has shown two monochords in just intonation from Rous- seau's Dictionary. *•* The first (Table 99) was by Alexander Malcolm, whose linear improvement upon just intonation is to be found in Chapter VII. This is the same as Kepler's second mon- ochord (Table 91), transposed a fifth lower. 11Versuch liber die musikalische Temperatur, pp. 118, 123. 12F 100 12F. W. Opelt, Allgemeine Theorie der Musik (Leipzig, 1852), p. 46. JUST INTONATION Rousseau tried to "improve" upon this tuning by substituting other just pitches in place of D*3"1"1, F*"1 , and B"0, with very un- satisfactory results, since his division of the major tone of 204 cents was into semitones of 70 and 134 cents! This monochord (Table 100) is the reverse of Marpurg's fourth (Table 98), with semitones paired in contrary motion, when Rousseau's A*3*1 is made to coincide with Marpurg's G*~2. Table 99. Malcolm's Monochord Names C° Db+1 D ° Eb+1 E"1 F° f""1 G° Ab+1 A"1 Bb° B"1 C° Cents 0 112 204 316 386 498 590 702 814 884 996 1088 1200 M.D. 14.0; S.D. 15.8 Table 100. Rousseau's Monochord Names Cu C#"2 D° Eb+1 E"1 F° F^"2 G° Ab+l A"1 Bb+1 B"1 C° Cents 0 70 204 316 386 498 568 702 814 884 954 1088 1200 M.D. 25.0; S.D. 26.7 Table 101. Euler's Monochord Names C° C^2 D° D#_2 E_1 F° F*"1 G° G*~2A_1 A*"2 B_1 C° Cents 0 70 204 274 386 498 590 702 772 884 976 1088 1200 M.D. 17.1; S.D. 20.1 Table 102. Montvallon's Monochord Names C° C#_1 D° E^1 E"1 F° F#_1 G° G#_1 A"1 Bb° B"1 C° Cents 0 92 204 316 386 498 590 702 794 884 996 1088 1200 M.D. 12.0; S.D. 13.3 101 TUNING AND TEMPERAMENT Table 103. Romieu's Mono-chord Names C° C#"2D° Eb+1 E"1 F° F^G0 G^A"1 B°° B"1 C° Cents 0 70 204 316 386 498 590 702 772 884 996 1088 1200 M.D. 17.7; S.D. 20.1 Euler's monochord ran entirely to sharps. ** However, it has the same symmetric grouping of its notes as de Caus's (Table 89), only transposed a fifth higher. Montvallon's monochord, given by Romieu,14 follows a more familiar order in the selection of notes than Euler's did (see Ta- ble 102). Romieu himself contributed an example (Table 103) of a "syst^me juste." 15 it has a somewhat more complicated pattern than Euler's (Table 101), but the same deviation. Theory of Just Intonation In the foregoing pages there have been presented more than twenty different monochords in authentic just intonation, i. e., with pure fifths and major thirds. Their mean deviations have varied from 10.0 to 25.0. And yet each has a right to be called just intonation! This great divergence can be explained by math- ematics. Let us consider first a monochord in the Pythagorean tuning. Its mean deviation is 11.7. A Pythagorean chromatic semitone, as C°-C*°, is 114 cents; the diatonic semitone, as C#° -D° , 90. Hence the deviation for the pair of semitones is 24 cents. When the just semitones are used, the chromatic semi- tone, C-C*"1, is 92 cents; the diatonic, C#_1-D°, 112. The deviation for the pair of just semitones is 20 cents, or 4 cents less than for the pair of Pythagorean semitones. Therefore the substitution of each just note reduces the deviation by 4/12 or .3 cent. **A. F. H'aser, "Uber wissenschaftliche Begriindung der Musik durch Akustik," Allgemeine musikalische Zeitung, 1829, col. 145. 14"Me"moire theorique & pratique sur les systemes tempe're's de musique," Memoires de l'acade'mie royale des sciences, 1758, p. 867. 15Ibid., p. 865. 102 JUST INTONATION But the sixth note to be altered around the circle of fifths is adjacent to the first note to have been altered, and therefore the total deviation is unchanged. The same is true for the seventh note. The eighth note lies between two notes, each sharper by the syntonic comma. Therefore, when it too is raised, the syn- tonic semitones already present are changed to Pythagorean semitones, and the deviation is increased by .3 cent. This proc- ess continues until all twelve notes have been raised by a comma, and the monochord is again in Pythagorean tuning. If we call the number of notes with -1 exponent nx , and with 0 exponent n2 , the following formula gives the mean deviation: 3D, = 29 + n , - 6 + 6 - n, - 6 The minimum deviation of 10.0 cents occurs when (nx ,n2) = (5,7), (6,6), or (7,5). Thus Ramis' monochord (Table 81) with 6,6 is one of the three best possible. When there are notes with three different exponents, the change of a single note may cause a greater change in the devi- ation than was possible with two exponents only. Suppose a mon- ochord contains the notes C° C#_1 D"1, the total deviation being 18 cents for the two semitones. When C* "2 is used, the devia- tion becomes 42 cents, an increase of 24 cents. But if the notes had originally been C C#_1 D°, the change to C*"2 would in- crease the deviation from 20 cents to 64 cents, that is, by 44 cents, or two commas. Again, the deviation of the two semitones C#_1 D° Eb+1 is 24 cents; with D'1 it is 44 cents, an increase of 20 cents. Thus when a note is changed by a comma, the change in the mean deviation may be 1/3 (as before) or 6/3 or 11/3 or 5/3 . A much more complicated formula, therefore, is needed to express the deviation with the three exponents. If we call the number of notes with -1 exponent nx , with 0, n2 , and with +1, n3 , the mean deviation is given by the formula: 103 TUNING AND TEMPERAMENT 3 D3 = 23 + ru - 6 + n3 - 6 + 6 - rii - 6 6 - n3 - 6 7(k2 - kx) + 5(k4 - k3), where kx = the larger of n2 and (7 - nx), k2 = the smaller of 7 and (12 - ni), k3 = the larger of n2 and (5 - ni), and k* = the smaller of 5 and (12 - m). The terms con- taining the k's are zero whenever k2