Chapter III. MEANTONE TEMPERAMENT It is not definitely known when temperament was first used. Vicentino stated that the fretted instruments had always been in equal temperament. As for the keyboard instruments, Zar- lino declared that temperament was as old as the complete chromatic keyboard. It may well be that some organs in the fifteenth century had had temperament of a sort, although the Pythagorean tuning continued to have too many advocates not to have been dominant in the earlier period. However that may be, Riemann discovered the first mention of temperament in a passage from Gafurius' Practica musica (1496). 1 There, among the eight rules of counterpoint, Gafurius said that organists as- sert that fifths undergo a small, indefinite amount of diminution called temperament (participata). Since he was reporting a con- temporary fact, rather than advocating an innovation, the practice may have begun decades earlier than his time. Notice that Gafurius stated that there was nothing regular about the temperament of his day, nor were the fifths diminished by any large amount. It seems reasonable to believe that when organists first became dissatisfied with the extremely sharp thirds of the Pythagorean tuning, they would go about any altera- tion of the fifths in a gingerly manner, lopping off a bit here and a bit there. Grammateus' division of Pythagorean tones into equal semitones came only twenty-two years after Gafurius' observation, 2 and ranks very high among irregular systems that approach equal temperament. It is easy to believe, therefore, that organs were tuned as well in 1500 as they generally are today. Dechales had no authority for stating that Guido of Arezzo was the father of temperament. «* The association of Ramis^ with ^ugo Riemann, Geschichte der Musiktheorie (Berlin, 1898), p. 327. 2See Chapter VII for Grammateus. 3R. P. Claudius Franciscus Milliet Dechales, Cursus seu mundus mathe- maticus (Lugduni, 1674), Tomus Tertius, pp. 15-17. ^See Preface and Chapter V. TUNING AND TEMPERAMENT temperament is one of the most common misconceptions in the history of tuning. And, although Schlick's system^ undoubtedly can properly be described as a temperament, it is just as surely of an irregular variety. It is well to mention these names, and discard each of them, before saying that full credit for describ- ing the meantone temperament must go to Pietro Aron. InAron's Toscanello^ there is a chapter entitled "Concerning the temperament (participation) and way of tuning the instrument. " The tuning is to be made in three successive stages (see Table 22). First, the major third, C-E, is to be made "sonorous and just." But the fifth C-G is to be made "a little flat." The fifth G-D is to be similarly flattened, and then A is to be tuned so that the fifths D-A and A-E are equal. The idea, of course, is to en- sure an equality of these four fifths, so far as it can be accom- plished by ear. Table 22. Aron's Meantone Temperament (1/4 Comma) 7 1.31+1 3 1 2 3+1 5 Names C° C#" D" Eb < E- F * F*" G-* G#" A" Bb * B" C° Cents 0 76 193 310 386 503 579 697 773 890 1007 1083 1200 M.D. 20.0; S.D. 20.2 In the second stage of tuning, the fifths F-C, B^-F, and E^-B0 are tempered exactly the same as the diatonic fifths had been. Finally, in the third stage, C^ and F# are tuned as pure thirds to A and D respectively. Aron says nothing about G^. With Kinkeldey we can say that this note "probably belongs to the third group, "7 and would be tuned as a pure third to E. The name "meantone" was applied to this temperament be- cause the tone, as C-D, is precisely half of the pure third, as 5See Chapter VII. ^Toscanello in musica (Venice, 1523); revised edition of 1529 was consulted.. ?Otto Kinkeldey, Orgel und Klavier in der Musik des 16. Jahrhunderts (Leip- zig, 1910), p. 76. 26 MEANTONE TEMPERAMENT C-E. Aron said nothing about the division of the comma. But since the pure E is a syntonic comma lower than the Pythagorean E,and each fifth is to be tempered by the same amount, the fifths will all be tempered by 1/4 comma. It is easy to calculate the ratio of the meantone fifth: the major third has the ratio 5:4; hence the ratio of the tone will be the square root of this, or /J5:2. The ratio of the major ninth will be twice the ratio of the tone, or ^5:1. The ratio of the fifth will be the square root of the ratio of the ninth, or 4V57l. If we consider the syntonic comma to be 21.5 cents, a fifth diminished by 1/4 comma will be 702.0- 5.4 = 696.6 cents. The deviation for the meantone temperament is nearly as large as for just intonation. That would seem to indicate that temperament makes for little improvement. Strangely enough, this is absolutely true, so far as the remote keys are concerned. However, if the deviation were to be measured only from E^ to G*, without allowing for the enharmonic uses of notes, the mean- tone temperament would be an easy victor over just intonation. That is, if we were computing the deviation of eleven fifths only, omitting the wolf fifth of 737 cents, the standard deviation for the meantone temperament would be much smaller than that for just intonation. But, since our ideal is equal temperament, the de- viation as computed shows accurately enough how very unsatis- factory this tuning is when its narrow bounds are overstepped. The meantone temperament was used from the beginning upon keyboard instruments only. It was the temperament that Vicen- tino intended for his Archicembalo when he said that it may be tuned "justly with the temperament of the flattened fifth, accord- ing to the usage and tuning common to all the keyboard instru- ments, as organs, cembali, clavichords, and the like."** Zarlino called the meantone temperament a "new temperament" and said that it is "very pleasing for all purposes" when used on key- board instruments. 9 To divide the major third into two mean tones, Zarlino advocated the Euclidean construction for a mean 8See Chapter VI. ^Gioseffo Zarlino, Dimostrationi armoniche (Venice, 1571), p. 267. 27 TUNING AND TEMPERAMENT proportional, and of course the fifth could be constructed from the major ninth by the same means. Verheijen's reply to Stevin's discussion of equal temperament explained the meantone temperament in detail 10 He even in- cluded amonochord for it (Table 23), and thus has the distinction of being the first person, so far as we know, to put its ratios into figures (cents values as in Aron, Table 22, beginning with F as 503). Table 23. Verheijen's Monochord for Meantone Temperament Lengths 10000 9750 8944 8560 8000 7477 7155 6687 6400 5961 7 19 13 1 3 Names F° F#" G" G#" A*"1 Bb+4 b"*~ C~* C#"2 D" 5590 5350 5000 1 5 Eb+- E" Fo In Spain, Sancta Maria described a practical tuning system that may have been the same as the meantone tuning. H He said that on the clavichord and thevihuela (the Spanish lute) each fifth is to be "a little flat." In fact, the diminution is to be "so small that it can scarcely be noticed." Since he did not say whether the thirds were to be pure or a little sharp, we cannot know whether his system was the real meantone or came nearer equal tem- perament. However, he held that a tone cannot be divided into two equal semitones, and consistently made the diatonic semi- tone larger than the chromatic semitone, as it would be in just intonation or the meantone temperament. The first German writer to describe the meantone tempera- ment was more explicit. This was Michael Praetor ius,* 2 m a lOsimon Stevin, Van de Spiegeling der Singconst, ed. D. Bierens de Haan (Amsterdam, 1884). Verheijen's letter is in Appendice A. Both discussion and reply remained in manuscript for almost three hundred years. ^Tom^s de Sancta Maria, Arte de taner fantasia (Valladolid, 1565), Chapter 53. 12syntagma musicum (Wolfenbuttel, 1618), Vol. II; new edition, 1884-94, pub- lished as Publikation alterer praktischen und theoretischen Musikwerke, Band 13, pp. 178 ff. 28 MEANTONE TEMPERAMENT chapter on the tuning of the "Regal, Clavicymbel, Symphonien und dergleichen Instrument." His was a practical system, with major thirds and octaves pure, and fifths flat. Praetor ius explained carefully how various intervals are altered by fractional parts of the comma. Otto Gibelius^ showed a method for obtaining an approxi- mately correct monochord for the meantone temperament. First he made a table in which were shown pairs of numbers differing by the syntonic comma for every note in a 14-note octave, ex- tending from A*3 toD#. Then he made an arithmetical division of each comma, with 3/4, 1/2, or 1/4 comma subtracted from the larger number, to obtain the tempered value. C, E, G^, and A*3 needed no temperament (see Table 24). His results check closely with numbers obtained by taking roots. ^ For example, his D is 193200; it should be 193196. His G is 144450 instead of 144447. Since the comma is small relative to the intervals of the scale and since as much as a quarter or a half of it is used, the error could not be great. An arithmetical division of the ditonic comma into twelfths in the construction of equal temper- ament would create greater errors than this for certain notes of the division. Table 24. Gibelius' Monochord for Meantone Temperament Lengths 216000 206720 193200 184896 180562.5 172800 161500 154560 Names C° C#" D" D*"* Eb+* E_1 F+4 F#" 144450 138240 135000 129200 120750 115560 108000 G"5 G#"2 Ab+1 A" Bb+2 B" C° Lemme Rossi,* ^ writing in the same year as Gibelius, would have approved the latter 's approximation for the meantone tem- 1,jPropositiones mathematico-musicae (Mlinden, 1666), copperplate opposite page 14 l^Wolffgang Caspar Printz, Phrynis Mytilenaeus oder der satyrische Com- ponist (Dresden and Leipzig, 1696), p. 73. l^sistema musico (Perugia, 1666), p. 59. 29 TUNING AND TEMPERAMENT perament, for he himself said that the arithmetical division of the comma differs "insensibly" from a geometrical division. In the example that he gave, the geometrical mean between the two numbers, 31104 and 30720, in the ratio of 81 to 80, is 30911, and the arithmetical mean is 30912, certainly a negligible difference. But, he said, the correct string- lengths for the meantone tem- perament can be obtained both "easily and quickly with the table of logarithms." Our final monochord for the meantone temperament proper will be Rossi's "Numeri del sistema participate "16 He has given it for a 19-note octave commencing on A (see Table 25). Since C itself is a tempered value here, we have transposed the system up a minor third from A to C, selecting those notes that would belong to the ordinary meantone scale. The number used for his fundamental had been previously used in a table of just intonation. Table 25. Rossi's Monochord for Meantone Temperament Lengths Names 41472 C° 24806 3 A" 39690 7 23184 37095 i d" 22187 5 B" 34668 Eb+^ 20736 C° 33178 -i E 31008 F+i 29676 3 F#~2 27734 i G" . 26542 G#"2 Another sort of approximation connected with the meantone temperament was given by Claas Douwes.l^ In describing the bonded clavichord he gave simple ratios (most of them super- particular) for various intervals that would occur on the same string. For example, the highest string has C, B, B*3, and A. C-A is 6:5; B-A, 19:17; Bb-A, 15:14. On the next string, G#-F is 7:6. Two octaves lower, the ninth string has only two notes, G# and G, with the ratio 24:23. Dou\ ^s had explained that his was a tempered system. His rational ratios are good approximations to the surds of the mean- 16lbid., p. 83. l^Grondig Ondersoek van de Toonen der Musijk (Franeker, 1699), pp. 98-104. 30 MEANTONE TEMPERAMENT tone temperament. His minor third, with ratio 6:5, is 316 cents; the meantone minor third is 310. His augmented second, 7:6, is 267 cents; the meantone augmented second is 270. His tone, 19:17, is almost 193 cents; the meantone tone is practically the same. His diatonic semitone, 15:14, is 119 cents; the meantone diatonic semitone, 117. His chromatic semitone, 24:23, is 74 cents; the meantone chromatic semitone, 76. His system agrees with itself as well as with the ordinary meantone system. For example, the tone should be the sum of the diatonic and the chro- matic semitones, or 15/14 x 24/23. This product is 3420:3059; his ratio for the tone, 19:17, equals 3420:3060, a close corre- spondence. In tracing the later history of the meantone temperament, it would be easy to name theorists in all the principal European countries who continued to favor an unequal tuning of keyboard instruments later than the first quarter of the eighteenth century. But, unless, like Galin in 1818, they specifically say that they favor the tuning in which the fifths are tempered by 1/4 syntonic comma or its equivalent (31-division)/^ we have no right to call their methods the meantone temperament. This is the fallacy of so much that has been written on this subject. Other Varieties of Meantone Temperament Strictly, there is only one meantone temperament. But theo- rists have been inclined to lump together under that head all sorts of systems intended for keyboard instruments. For ex- ample, the statement often appears in print that in England the meantone temperament was used for organs until the middle of the nineteenth century. William Crotch, 1* writing early inthat century, wrote: "As organs are at present tuned, (with unequal temperament), keys which have many flats or sharps will not have a good effect, especially if the time be slow." That state- ment is enough to cause a host of later English writers to say 18pierre Galin, Exposition d'une nouvelle me'thode pour l'enseignement de la musique (3rd edition, Bordeaux and Paris, 1862; 1st edition, 1818). ^Elements of Musical Composition (London, 1812), p. 112. 31 TUNING AND TEMPERAMENT that Crotch reported the meantone temperament to be in use in his age. But later in his book Crotch had this to say: "Unequal tem- perament is that wherein some of the fifths, and consequently some of the thirds, are made more perfect than on the equal temperament, which necessarily renders others less perfect. Of this there are many systems, which the student is now capable of examining for himself. "20 jn other words, Crotch is saying that there was a great diversity in the tuning of organs in his day. In Chapter VII, "Irregular Systems," twenty-odd men are mentioned who collectively have described fifty of the "many systems," none of which is the meantone temperament. In the present chapter we propose to describe still other systems of temperament, systems formed on the same general pattern as meantone temperament. Bosanquet called "regular" a tempera- ment constructed with one size of fifth. **■ The Pythagorean tun- ing, equal temperament, meantone temperament— all are regular systems. The systems that follow are also regular, with values for the fifth smaller than that of equal temperament and (usually) larger than that of the meantone temperament. Since their con- struction is similar, it is easy to describe them as varieties of the meantone temperament. In all of them, the tone is precisely half of the major third. No harm will be done by such a nomen- clature if we realize that these are regular temperaments which the earlier theorists themselves considered of the same type as the 1/4 -comma temperament and some of which they preferred to it. The first regular temperament to be advocated after the de- scription of the ordinary meantone temperament was that de- scribed by Zarlino in which "each fifth remains diminished and imperfect by 2/7 comma. "22 Although Zarlino showed a mono- chord with this tuning for the diatonic genus only, he intended it 20lbid., p. 135. 21R.H M. Bosanquet, An Elementary Treatise on Musical Intervals and Tem- perament (London, 1876), Chapter VTJI. 22cioseffo Zarlino, Istitutioni armoniche (Venice, 1558), pp. 126 ff. 32 MEANTONE TEMPERAMENT also for the chromatic genus— by which he meant the ordinary black keys. He also described an enharmonic genus, having 19 notes to the octave, as applied to a cembalo which Master Dom- enico Pesarese had made for him. This must have had the same tuning, although Zarlino did not clearly say so. Most of these varieties of the meantone temperament will have a smaller de- viation when applied to a keyboard with 19 or more notes to the octave than upon the usual keyboard. Zarlino's temperament corresponds to the 50-division, and, as such, will be discussed in the chapter on multiple division. In Table 26, we see the 2/7-comma temperament applied to a keyboard with 12 notes to the octave. Since the amount of tem- pering is greater than 1/4 comma, the deviation is greater than for Aron's system. It is, in fact, a very poor system, and Zar- lino later admitted it to be inferior to the 1/4- comma system. The only just interval in it is the chromatic semitone. Tanaka liked it "because all the imperfect consonances are impure alike, "23 that is, the major and minor thirds are 1/7 comma flat (3 cents), and the major and minor sixths are sharp by the same amount. To construct it on a monochord, Zarlino would use the questionable virtues of the mesolabium.24 Table 26. Zarlino's 2/7 - Comma Temperament 4 6 82 _12 2 _16 6 +4 10 Names C° C# D" Eb 7 E~7 F 7 F#" ' G~7 G#" 7 A~7 Bb * B- 7 C° Cents 0 70 191 313 383 504 574 696 817 887 1008 1078 1200 M.D. 25.0; S.D. 25.3 The next variety of meantone temperament is also highly un- satisfactory when applied to an octave of twelve semitones. This is the 1/3-comma temperament, the invention of Francisco Sali- nas, which he described as follows: "The first of them [the other two were the 2/7-comma and the 1/4-comma temperaments] has the comma divided into three parts equally proportional, of which 23Shoh^ Tanaka, "Studien im Gebiete der reinen Stimmung," Vierteljahrs- schrift fur Musikwissenschaft, VI (1890), 65. 24 For an account of the mesolabium, see the second part of Chapter IV. 33 TUNING AND TEMPERAMENT the minor tone is increased by one part and the major tone is decreased by two parts. "25 Salinas showed that his method re- sults in pure minor thirds, tritone, and major sixth. But the fifth is diminished by 1/3 comma, and so is the major third. On the whole this tuning does not compare favorably with the others, but Salinas added: "Although this imperfection is seen to be greater than that which is found in the other two temperaments, nevertheless it is endurable." Salinas intended his temperament for an octave containing 19 notes, divided into the three genera— diatonic, chromatic, and enharmonic. His special reason for advocating this tuning was the ease of realizing it upon the monochord. Seven of the notes can be obtained by a series of just minor thirds below and above the fundamental. Thus we obtain C, D#, E°, F#, Gb, A, and B#, and Salinas has given their string- lengths for the octave 22500 to 11250. To find the notes D and E, two mean proportionals must be inserted in the tritone, C— F^. This "will be very easy to those who know the use of a certain instrument invented by Archimedes, which is called mesolabium, from finding mean lines by it." The remainder of the notes can then be obtained by minor thirds from D and E. We agree with Salinas that the thirds and especially the fifths of the 1/3-comma temperament are less pleasing than those of the other two. But, in addition to its being capable of quicker tuning than the Zarlinian 2/7-comma method, it has an advantage possessed by neither of the other methods: it is practically a closed or cyclic system. Among its 19 notes there is no fifth containing a wolf; nor are there any discordant thirds. It is an equal temperament of 19 notes. In recent times the 19-division has had eloquent advocates, to whom reference is made in the chapter on multiple division. Let us see how well the 1/3-comma system is adapted to a 12- note keyboard. As Table 27 shows, this is the poorest tuning of all— like Zarlino's method, it is worse than just intonation. How- ever, too many theorists who have described these two systems have neglected to add that they are excellent foral9-note octave. 2&De musica libri VII, p. 143. 34 MEANTONE TEMPERAMENT Table 27. Salinas' 1 /3 - Comma Temperament 7 2 +1 44-L 2 l 8 +2 5 Names C° C^" D~3 Eb E" F s f#~ G" G#" a" Bb 3 B" C° Cents 0 64 190 316 379 505 569 695 758 884 1010 1074 1200 M.D. 30.3; S.D. 30.7 It would help us in portraying an orderly development of the 12-note temperaments if we could show that little by little the temperament of the fifth was reduced from the 1/4 comma of the meantone temperament to the 1/11 comma (1/12 ditonic comma equals 1/11 syntonic comma) of equal temperament. Probably there was such a tendency. But it is only a fortunate accident that Verheijen included the ratio of the fifth for the 1/5-comma temperament, together with the ratios for the three temperaments discussed by Zarlino and Salinas. 26 Verheijen's first ratio for the fifth is the cube root of 10:3 (1/3-comma temperament); then the fourth root of 5:1 (1/4-comma); the fifth root of 15:2 (1/5- comma); the seventh root of 50:3 (2/7-comma). Verheijen's casual reference to the 1/5-comma temperament indicates that even then some people were using it. Rossi, a couple of genera- tions later, also referred briefly to the 1/5-comma temperament, including it as one of the regular types then in use. ^ The temperament shown in Table 28 has in its favor, like the 1/3-comma temperament, the equal distortion of the fifths and the major thirds, the former being 1/5 comma flat, the latter sharp by the same amount. In it the diatonic semitone is pure. Table 28. 1/5 - Comma Temperament (Verheijen, Rossi) 7 2 3 4 1 6 1 8 3,2 Names C° C#'z D"5 Eb ' e" F ■ P*" G" G#" A" Bb » B_1 C° Cents 0 83 195 307 390 502 586 698 781 893 1005 1088 1200 M.D. 14.0; S.D. 14.2 ''"Simon Stevin, Van de Spiegeling der Singconst, Appendice D. 2'Sistema musico, p. 58. 35 TUNING AND TEMPERAMENT The deviation of this temperament is only about two-thirds that of the 1/4-comma system. There is an odd reference to the 1/5- comma temperament. Dechales^S gave a monochord which he called the "Diatonic scale of Guido of Arezzo." It is, however, a chromatic scale, and, so far as can be ascertained, has nothing in common with any of the ideas expressed by Guido. It seems evident that Dechaleshas intended the monochord in Table 29 for the 1/5-comma temperament. Its ninth note differs greatly from the cents value given in the previous table; but the note is A^ in Dechales' monochord and would naturally be more than a comma higher than the G^ more commonly used. Other divergences can be explained by the fact that Dechales has not expressed his numbers with great accuracy. However, the mean value for his diatonic semitone is 111.4, against 112.0 for the 1/5-comma temperament; for his chromatic semitone, 84.0 cents against 83.2. How he reached the conclusion that Guido favored such a temperament remains a mystery. Actually Dechales him- self ascribed the 1/4-comma temperament to Guido (rather than the 1/5-comma), contrary to the evidence of this monochord. Table 29. Dechales' "Guidonian" Temperament (1/5 - Comma) Lengths 60 57i 53| 50^ 47| 44^ 42| 40| 37| 35^ 33i 31§ 30 Names C C# D Eb E F F# G Ab A Bb B C Cents 0 85 194 312 395 502 587 696 808 893 1009 1090 1200 M.D. 13.3; S.D. 13.8 The 1/5-comma variety of meantone temperament comes close to the 43-division. As such, it is discussed briefly in Chapter VI, with the principal reference to Sauveur. Another temperament discussed by Rossi^ has its fifths flattened by 2/9 comma (see Table 30). He merely called it "another tempered system," without ascribing it to any theorist. Romieu identified this temperament with the 31 -division, and thus 28cursus seu mundus mathematicus, p. 20. 29sistema musico, p. 64. 36 MEANTONE TEMPERAMENT Table 30. Rossi's 2/9 - Comma Temperament 14 4 282 42 16 2 4 _10 Names C° C#"~ D" E°+S e" F+a F#" G" G#~ 9 A" Bb+9 B_ 9 C° Cents 0 79 194 308 389 503 582 697 777 892 1006 1085 1200 M.D. 17.0; S.D. 17.2 credited it to Huyghens.^O Actually, as we have already said, the 1/4-comma temperament comes closest to the 31-division. But perhaps other writers before Romieu confused these tem- peraments. For example, Printz^l spoke of a "still earlier" temperament that takes 2/9 comma from each fifth— earlier, perhaps, than Zarlino's 2/7- comma temperament, which he had been previously discussing. He also might have meant Vicentino's 31-division, since there are no early references to the 2/9- comma temperament. Since 2/9 is the harmonic mean between 1/4 and 1/5, the de- viation for this temperament is approximately the mean of the deviations of the other two temperaments. Like Zarlino's 2/7- eomma temperament, its third is altered half as much as its fifth, being 1/9 comma sharp. Its augmented second, as F-G#, is pure. The 74-division corresponds to the 2/9-comma tem- perament, and Drobisch liked this division best of all systems that form their major thirds regularly. Schneegass gave an interesting geometrical construction for what was much like the common meantone temperament, but more like the 2/9-comma temperament. His contention was that the diatonic semitone contains 3 1/4 "commas" and the chromatic semitone 2 1/4. (These commas of 35.3 cents have nothing in common with either the ditonic [23. 5J or the syntonic [2I.5] comma). Thus the tone contains 5 1/2 commas, and the octave 5x5 1/2 + 2x3 1/4 = 34 commas. As is shown in Chapter VI, the 34 -division has fifths that are almost 4 cents too large 30jean-Baptiste Romieu, "Memoire theorique & pratique sur les systemes tempe'res de musique," Me'moires de Tacademie royale des sciences, 1758, p. 837. 31Phrynis Mytilenaeus oder der satyrische Componist, p. 88. 32Cyriac Schneegass, Nova & exquisita monochordi dimensio (Erfurt, 1590), Chapter HI. 37 TUNING AND TEMPERAMENT and thirds that are 2 cents too large. But this was not what Schneegass had in mind. His theoretical fifth had the ratio 160:107, or 696.6 cents, which is precisely the size of the mean- tone fifth, and he directed that this ratio be used twice to form the tone. Then came the application of the doctrine about commas: A right triangle was to be constructed, with the space of the tone, G-A, as base, and thrice this length for the altitude (see Fig- ure A). Note that "space" here does not refer to the total length of a line, but rather to the distance from one point of division to another Since 3 1/4:2 1/4 = 13:9, the acute angle at thetopwas to be divided in the ratio of 13:9, with the larger angle toward A. The point where this line cut the base was to be G#. Now tan"1 1/3 = 18° 26', and 13/22 of this angle is 10° 53'. The space between G# and A, then, would be 3 tan 10° 53' = .57681 of the space between G and A. From the figures in his table, the divi- sion was made with extreme care. The ratio in the table of the space from G* to A to the space from G to A is 15/26 or .57692. By a series of lines parallel to the base, he cleverly divided the other tones (Bb-C, C-D, Eb-F, and F-G) into chromatic and diatonic semitones proportional to the division of G-A. Fig. A. Schneegass' Division of the Monochord Reproduced by courtesy of the Sibley Library* of the Eastman School of Music 38 MEANTONE TEMPERAMENT To examine the assumption that Schneegass made, let us des- ignate as a the angle 10° 53' and as ft the angle 18° 26 ', and as L the length for the note A. Then the length for G was L + tan a , and for G it was L + tan ft . His assumption: log/L + tanff ): log/ L + tan a \= ft : a In general this would be only a rough approximation. In this case, where ft : a = 22:13, it works very well indeed. Schneegass' actual fifth, G-D,of 698.1 cents is a little larger than his theoretical fifth of 696.6, and the mean of all 11 good fifths is 697.2 cents. This last figure is precisely the fifth of the 2/9 -comma temperament. The mean value of his tones is 194.0 cents, as compared with 194.4 cents of the 2/9- comma temperament, and his geometrical division of the tones yields semitones of 113.9 and 80.1 cents, compared with 114.0 and 80.4 cents. Schneegass' actual fifth has approximately the ratio 226:151, instead of his theoretical 160:107. It is idle to speculate why his figures fail to correspond with his theory, or why they agree so beautifully with the 2/9 -comma temperament. The significant thing is that they agree so well with themselves, which is an in- dication of the soundness of his mathematics! There is, how- ever, one puzzling clue to his division of the tone. Suppose the space of the tone G-A had been divided arithmetically in the ratio of 13:9, instead of the more complicated division of the angle actually used. Then Schneegass' G* would have been at 86.100 instead of at 85.967. This would have made the G* 3.3 cents lower than in the table, and his tone would have been divided into semitones of 117.7 and 76.0 cents. Nowthe semitones of the 1/4- comma temperament are of 117.1 and 76.0 cents respectively. Thus an arithmetical division of his tones would have come close to the temperament which is suggested by his theoretical fifth. However, his actual division (Table 31) with a 15:11 ratio, is very consistent with itself, as well as with the 2/9 -comma tem- perament. 39 TUNING AND TEMPERAMENT Table 31. Schneegass' Variety of Meantone Temperament Lengths 90.000 85.967 80.467 75.267 71.867 67.267 64.200 60.133 Names G G# A Bb B C C# D Cents 0 79 194 309 389 504 585 698 56.300 53.750 50.367 48.083 45.000 Eb E F F* G 812 892 1005 1085 1200 M.D. 16.7; S.D. 16.9 Robert Smith"^ is responsible for three wholly unsatisfactory varieties of the meantone temperament. He told first of a Mr. Harrison, who tuned his viol by "taking the interval of the major third to that of the octave, as the diameter of a circle to its cir- cumference— It follows from Mr. Harrison's assumption, that his 3rd major is tempered flat by a full fifth of a comma." If the ratio of the major third to the octave is l:tr , the third will have 382.0 cents, or be 1/5 comma flat, as Smith said. The fifth will then be tempered by 3/10 comma. Romieu^ barely mentioned 3/10- and 3/11 -comma temperaments, but did not discuss them on the ground that they were too like temperaments with unity in the numerator. Except for a few references to Smith and this tuning by rr, the 3/10- comma temperament has escaped further notice (see Table 32). Table 32. Harrison's 3/10 - Comma Temperament 21_ 3 _9_ 63 2 3_ 12 9 3 3 Names C° C#~10 D" Eb+l° E" F w F*" G"10 G#~~ A~a BD * B" C° Cents 0 69 191 314 382 504 573 696 764 887 1009 1078 1200 M.D. 26.2; S.D. 26.6 Since 3/10 is about the same as 2/7, the deviation for this temperament is approximately the same as for Zarlino's, both ■^Harmonics, or the Philosophy of Musical Sounds (Cambridge, 1749), pp. xi, xii. 3^In Memoires de l'acade'mie royale des sciences, 1758, p. 827. 40 MEANTONE TEMPERAMENT being inferior to just intonation. It has no special features to recommend it, since its one natural feature, the Tr ratio, is something to be determined by ear or by logarithms, and would not make the construction of a monochord any simpler. After referring to Harrison's system, as quoted above, Smith continued, "My third determined by theory, upon the principle of making all the concords within the extent of every three octaves as equally harmonious as possible, is tempered flat by one ninth of a comma; or almost one eighth, when no more concords are taken into the calculation than what are contained within one oc- tave." Later he showed that "to have all the concords in four octaves made equally harmonious," the thirds will be 1/10 comma flat. 35 With the third flat by 1/9 comma, the fifth will be tempered by 5/18 comma, a quantity impossible to judge by ear. In the second temperament, with the third 1/10 comma flat, the fifth will be 11/40 comma flat. The difference between these values of the fifth is only 1/360 comma! Therefore the temperaments would not vary for any note by as much as one cent. For this reason only the first of Smith's temperaments is shown in Table 33. Table 33. Smith's 5/18 - Comma Temperament 35 5 5 _10 5 5 5 20 5 +5 25 Names C° C#~18 D" Eb * E" 9 F '» F*" G"15 G#" 9 A" Bb s B"18 C° Cents 0 72 192 312 384 504 576 696 768 888 1008 1080 1200 M.D. 23.3; S.D. 23.7 Since 5/18 is also approximately the same as 2/7, Smith's temperament is only a little better than Zarlino's. We have pre- viously indicated that the 50-division has usually been considered the equivalent of the 2/7-comma temperament. Smith asserts, however, that his temperament corresponds to the 50-division, the error of the fifth in the latter being 41/148 comma. He is entirely correct in his claim. Smith did not suggest, however, that the octave be divided into fifty parts— merely that "a system of rational intervals deduced 35smith, Harmonics, p. 171. 41 TUNING AND TEMPERAMENT from dividing the octave into 50 equal parts, ...will differ insensi- bly from the system of equal harmony." His desire is more modest— to have at least 21 different pitches in the octave, pro- perly to differentiate the sharps, naturals, and flats. On the or- gan and harpsichord this could be done by adding extra pipes and strings. Performance would be facilitated by having "seven couples of secondary notes," governed by stops, so that the ap- propriate notes for a particular piece could be chosen. Of course, upon an instrument with 19 notes to the octave (the other two would be of little use), Smith's temperament, like Zarlino's and Salinas', would be far more acceptable than on the ordinary key- board. Smith himself considered that ordinary equal tempera- ment "far exceeds" both the 31- and 50-divisions, because of the cumbersomeness of the latter systems. The only other important variety of the meantone temperament was that practiced by Silbermann and his contemporaries. Ac- cording to Sorge, Silbermann tempered his fifths by 1/6 comma.36 Since Sorge himself made no distinction between the syntonic and ditonic commas, we might divide either. If we divide the ditonic comma, the deviation is precisely the same as for the Pythagorean tuning, M.D. 11.7, S.D. 11.8. But, for better com- parison with the other varieties of meantone temperament, let us divide the syntonic comma. Then the major third is 1/3 comma sharp, and the tritone is pure (see Table 34). Table 34. Silbermann's 1/6 - Comma Temperament 7 1 ,1 2,1 1 4 1 +1 5 Names C° C#" D" Eb a E" F « F*~ G~« G#"^ A" Bd ' B" C° Cents 0 89 197 305 394 502 590 698 787 895 1003 1092 1200 M.D. 9.3; S.D. 9.5 RomieuS? adopted the 1/6-comma temperament as his "tem- perament anacratique," showing its correspondence to the 55- division. A generation after Romieu, Barca called thistempera- 36Georg Andreas Sorge, Gesprach zwischen einem Musico theoretico und einem Studioso musices (Lobenstein, 1748), p. 20. 3?ln Memoires de l'academie royale des sciences, 1758, pp. 856 f. 42 MEANTONE TEMPERAMENT ment the "temperamento per coraune opinione perfettisimo,"38 and showed that it could be approximated by multiplying both terms of the ratio 81:80 by 6 and then tempering the fifth by the mean ratio 483:482, which gives 241:161 for the tempered fifth. (A better approximation is 220:147.) From additional references to the 55-division in Chapter VI, it would appear that this method of tuning was in use for well over a century. As a system upon which modulations might be made to any key, it was much better than the 1/4-comma meantone system, although inferior to most of the irregular systems discussed in Chapter VII. Romieu mentioned temperaments of 1/7, 1/8, 1/9, and 1/10 commas, but did not consider them sufficiently important to dis- cuss. The 1/10-comma temperament was included among Mar- purg's many temperaments. 39 Otherwise none of these tempera- ments has been advocated by any of our theorists. They should be presented, however, in order to complete our study of regular temperaments approaching equal temperament (see Tables 35-38). The syntonic comma has been divided in each case. Table 35. 1/7 - Comma Temperament 2 +3 4+l 6 18 3 ,2 5 Names C° C#_1 D" Eb ' E~~ F " F#" G" G#" a" Bb 7 B~~ C° Cents 0 92 198 303 396 501 593 699 791 897 1002 1095 1200 M.D. 6.3; S.D. 6.4 Table 36. 1/8 - Comma Temperament 7 l u+ 3 l + i 3 1 l 3 u+ 1 5 Names C° C#~a D" Eb ■ E" Fa F#" G" G#" A-"5 Bb * B_1 C° Cents 0 95 199 302 397 501 596 699 794 898 1001 1097 1200 M.D. 4.0; S.D. 4.1 38Alessandro Barca, "Introduzione a una nuova teoria di musica, memoria prima," Accademia di scienze, lettere ed arti in Padova. Saggi scientifici e lettari (Padova, 1786), pp. 365-418. 3^f.W. Marpurg, Versuch iiber die musikalische Temperatur (Breslau,1776), p. 163. 43 TUNING AND TEMPERAMENT With the exception of some of Marpurg's symmetrical versions of Neidhardt's unequal temperaments, the temperaments shown in Tables 37 and 38 come closer to equal temperament than any divisions that were not practical approximations to it. Table 37. 1/9 - Comma Temperament 7 2 14 1 21 8125 Names C° C#" D" Eb+3 E" F"*"5 F#" g" G#" A" Bb » B"5 C° Cents 0 97 199 301 398 500 598 700 797 899 1001 1098 1200 M.D. 2.3; S.D. 2.4 Table 38. 1/10 - Comma Temperament Names C° C#" D" Eb+" E" F+^ F#"s G"^ G#" A_T5 Bb+s B" C° Cents 0 99 200 301 399 500 599 700 798 899 1000 1099 1200 M.D. 1.2; S.D. 1.2 44