Chapter VI. MULTIPLE DIVISION If a keyboard instrument is not in equal temperament, its intona- tion can be improved by a judicious increase in the number of notes in the octave. The first reference to split keys came from Italy, where before 1484 the organ of St. Martin's at Lucca had separate keys for E^ and D* and also for G* and A*3.* At this same time, Ramis^ noted that split keys were being used in Spain, but objected to having separate keys for A*3 and G* and for F* and G*3, on the ground that this would be mixing the chromatic with the diatonic genus. From Germany came further evidence of the divided keyboard from Arnold Schlick,^ who referred to an organ constructed at the turn of the sixteenth century "that had double semitones on manual and pedal . . . which were called half semitones or 'ignoten.'" There are frequent references to multiple division during the sixteenth and seventeenth centuries, chiefly by Italian theorists. Jean Rousseau^ in 1687 deplored the fact that the French clave- cins did not have the "doubles feintes" common inltaly, and con- sequently had "mauvais effets dans les Tons transposez." But the split keys must have been very common in Germany during the latter part of the seventeenth and beginning of the eighteenth centuries, if we may judge by the copious references to "sub- semitonia" by Werckmeister and his successors. Buttstett, it is true, said in 1733 that the sub- and supersemitonia were "mehr curieux als practicabel."5 But six years later, in Holland, van Blankenburg was to show u't Gesnede Clavier" with three extra Iwilhelm Dupont, Geschichte der musicalischen Temperatur, p. 45. ^Musica practica, Tract. 2, Cap. 4. ^Spiegel der Orgelmacher und Organisten (Maintz, 1511), Chap. 8. Reprinted in Monatshefte fiir Musikgeschichte, 1869. 4Traite~ de la viole (Paris, 1687), p. 50. ^Johann Heinrich Buttstett, Kurze Anfiihrung zum General-Bass (2nd edition; Leipzig, 1733), p. 20. TUNING AND TEMPERAMENT keys, as well as an "Archicymbalam" with eighteen notes in the octave. 6 Handel played on English organs with split keys.' Father Smith's Temple Church organ in London, constructed in 1682-83, had the same pairs of divided keys as the Lucca organ, G#-Ab and D#-Eb, and so did Durham Cathedral. The organ of the Foundling Hospital (1759) had an ingenious mechanism by which Db and Ab could be substituted for C# and D#, or D# and A# for Eb and Bb, thus increasingthe compass to sixteen notes, without increasing the number of keys. Many of the sources said nothing about the tuning of the extra notes, and we can freely assume that whatever variety of mean- tone temperament was used for the twelve regular notes was ex- tended both clockwise and counterclockwise around the circle (or, rather, spiral) of fifths. More interesting to us are the sys- tems that represent just intonation, as extended to the enharmonic scale. We have already noted that Fogliano (1529) had felt the need for two D's and two Bb's, to ensure just triads, but was willing to settle for a mean D and a mean B*3. But van Blanken- burg, mentioned above, included both pairs of notes inhisArchi- cymbalam, and so did almost all of the men whose systems will be described below. The "enharmonic genus" of Salinas" was one of the earliest and best of these systems. Although it contained twenty-four notes, it had nothing in common with a real enharmonic scale composed of quarter tones. It is just intonation extended to seven sharps and six flats. In tabular form it would appear as shown in Table 104. Observe that all the notes in the right diagonal are duplicated on the left, a comma lower. Thus it is possible to play all major triads from Gb through G*, and all minor triads from Eb through E*. Mersenne's "parfait diapason" ^ is based upon Salinas' sys- tem, with the addition of seven more notes, or thirty-one in all "Quirinus van Blankenburg, Elementa musica (The Hague, 1739), p. 112. 'Helmholtz, Sensations of Tone, p. 434. 8De musica libri VII, p. 122. ^Harmonie universelle, p. 338. 108 MULTIPLE DIVISION (see Figure G). These would be joined to Table 104 on the left side, as shown in Table 105. Table 104. Salinas' Enharmonic Genus A#-2 E#-2 B#-2 ,#-i n¥-i n#-i »#-! yfl-1 Cf-1 Qff-1 DP-1 A D° A° E° B° F*° Bb+i F+i c+i G+i D+1 Gb+2 Db+2 Ab+2 Eb+2 Bb+2 Gb+3 f-1 Fig. G. Mersenne's Keyboard with Thirty-One Notes in the Octave (From Mersenne's Harmonie universelle) Reproduced by courtesy of the Library of Congress Table 105. Mersenne's Addition to Salinas' System A" Fu E"1 B"1 G° Ab+i Eb+i This is not a particularly clever addition. Note that Mersenne did not have a C°. Furthermore, for the sake of symmetry, there should have been Db+1 in the lowest line of Mersenne's additional notes, Bbb+2 , Fb+2 , and Cb+2 in the line below it, and C*~2, G*"2, and D*~2 in the line above the highest line, or a total of thirty-nine notes. 109 TUNING AND TEMPERAMENT The praiseworthy thing about Mersenne's addition is that it recognized the need for having more pairs of notes differing by a comma. Imperfect as his scheme was, it would be much more useful than the 34-note keyboard of Galeazzo Sabbatini, given by Kircher. 10 There were, as usual with Kircher, many errors in the figures, and an erratic manner of naming the notes. The ac- tual notes of Sabbatini' s keyboard are shown in Table 106. Table 106. Sabbatini's Keyboard Cx-3 Gx-3 Dx-3 Ax A*"2 E#_2 B#-2 [F#-2] F#-i C*"1 G*-1 D#_1 D° A0 E° Bu Bb* F+1 C+1 G+1 |Gb+2] Db+2 Ab+2 Eb+2 3 Bbb+3 Fb+3 cb+3 gbb+3 cbb^ Gbb^ Dbb^ Abb Ebbb+5 Bbbb+5 [pbb+5] Dbbb-^ Except for the three notes in brackets which have been sup- plied, this is a beautifully symmetric scheme. But how different from that of Salinas! Here there are no notes differing by the syntonic comma, with the result that no major triad based on a note in the diagonal on the right will have a pure fifth, and there will be a similar series of defective minor triads. With this in- tonation it is not even possible to supply a missing note by its enharmonic equivalent, because no pair of notes differs by the ditonic comma either. The most characteristic small interval in it is the great diesis of 42 cents, as between A#~2 and BD+1, whereas A^"1, needed as the fifth of the D* triad, lies almost half way between these two notes, 22 cents higher than A*"2 and 20 cents lower than B^*1. Other small intervals of little use contain 28, 14, and 8 cents. This, then, is an example of just l^Athanasius Kircher, Musurgia universalis, I, 460. 110 MULTIPLE DIVISION intonation carried to an absurd end. Doni's three-manual organ keyboard* * (abacus Triharmon- icus) was more elaborate than any system previously described, with sixty keys in the octave, but with only thirty -nine distinct pitches. The lowest keyboard was the Dorian, then the Phrygian, and finally the Lydian. The arrangement of the notes on each keyboard was identical, and the keyboards were tuned a major third apart, so that the Dorian E, the Phrygian C, and the Lydian A^ were the same pitch. The tuning was largely just, as can be seen from Table 107, which represents seventeen of the twenty notes on one keyboard. Table 107. Doni's Keyboard 2 JT#-2 C#-2 Q#-2 D^2 D-1 A"1 E"1 B"1 *b° C° G° Gb+i Ab+i Eb+i This arrangement is somewhat lacking in symmetry, and the additional three notes, which were real quarter tones, were of no use except to illustrate the scales of the Greeks, this being one of the uses of the organ. The enharmonic notes were formed, as Didymus formed his, by an arithmetical division of the syntonic semitone, 16:15, into 32:31 and 31:30 quarter tones. 2 The nineteenth century was particularly rife with proposals to increase greatly the number of notes in the octave. Many of the instruments upon which the inventors practiced their ingen- uity were harmoniums, intended for experimental purposes only. One of the more modest was Helmholtz's, already mentioned in Chapter V, with only twenty-four notes in the octave. 13 it fol- lowed a suggestion by Euler in 1739 that each manual be in the 11GiovanniBattistaDoni, Compendiodel trattato de' generi, e de' modi (Rome, 1635), Chap. 13. 12Shohe' Tanaka (in Vierteljahrsschrift fur Musikwissenschaft, VI [1890] , 85) was in error in showing these notes of Doni as only a comma higher than the lower note of the pair forming the semitone. 13Helmholtz, Sensations of Tone, p. 316 f. Ill TUNING AND TEMPERAMENT Pythagorean tuning, the one manual a comma higher than the other. General Thompson followed Doni's lead by having three manuals on his Enharmonic Organ, with forty different pitches in the octave. Henry Poole's Euharmonic Organ had only two black keys on the keyboard; but through a series of eleven ped- als all the notes could be transposed into five sharp and five flat keys, giving fifty distinct pitches in the octave. Liston's organ also relied upon pedals to obtain a great vari- ety of notes with the minimum number of keys. 14 With only twelve keys to the octave, tuned in just intonation, he was able by means of six pedals to add their enharmonic equivalents, thus having twenty-four notes in his normal scale. These are shown in Table 108. Then by three acute pedals all these notes could be raised in pitch by a comma. Two grave pedals similarly lowered nine or eleven of the normal notes by a comma. Thus Liston had a total of fifty -nine pitches available. Of Liston's fifty-nine notes, there were ten pairs, such as DId0-C*~2, which differed by the schisma, 2 cents. Further- more, Cx-3 and E#~3 differed by only six cents from D*3 +1 and F""1"3 respectively, and could be considered equivalent pairs also. Thus there were essentially only 47 separate pitches. These in- cluded four larger intervals: between C and C*~2 and between Cx"4 and D"1 there were two commas; between E#_1 and F#~' and between A#~ and B there were three. If these larger in- tervals had been divided, the octave would have contained 43 + 2x2 + 2x3 = 53 commas, which is the number one might have Table 108. Liston's Enharmonic Organ B#-3 Fx-3 Cx-3 G*-2 D*-2 A#-2 E*-2 A-1 E-1 B"1 F#_1 C#_1 Bb° Fo co QO Do ^b-t-1 qD+i Db+i ^b+i e^1 nbb +2 Tpb +2 14Henry Liston, An Essay upon Perfect Intonation (Edinburgh, 1812), pp. 3-7, 33-40. 112 MULTIPLE DIVISION anticipated. These "commas" are not all the same size. The ditonic comma does not occur at all except as the sum of the syntonic comma and the schisma. The syntonic comma is, as is evident from the scheme of pedals, the most common interval. But intervals of 20 cents, as D*~2 -E*30 , and of 26 cents, as G+1 -G*~3 , also occur. More ambitious was Steiner's system. ° For the key of C he used 12 notes in just intonation, symmetrically arranged in three groups of 4 notes each. But these could be transposed me- chanically into any of 12 different keys, the keynotes being tuned by perfect fifths. Thus there were 144 notes, but only 45 distinct pitches. Shohe Tanaka adopted Steiner's idea of having 12 key- notes in Pythagorean tuning, for mechanical transposition. But he extended his keyboard to 26 different notes, as shown in Ta- ble 10S. Of the 312 notes to the octave of Tanaka's "Transponir- Harmonium" or "Enharmonium," there were only 70 unduplicated pitches, no more than on an organ described by Ellis which had a total of 14 x 11 or 154 notes to the octave, with 70 separate pitches. Table 109. Tanaka's Enharmonium F#-2 c#-2 G#-2 D*-2 A#-2 E#-2 G"1 D"1 A-1 E-1 B"1 F#_1 C*"1 Bb° Fo co Go Do Ao £o Qb+i dd+1 Ab+1 Eb+1 Bb+1 F+1 Equal Divisions With Tanaka's Enharmonium we may safely drop the subject of just intonation extended. The theory is simple enough: pro- vide at least four sets of notes, each set being in Pythagorean tuning and forming just major thirds with the notes in another set; construct a keyboard upon which these notes may be played with the minimum of inconvenience. Only in the design of the keyboards did the inventors show their ingenuity, an ingenuity that might better have been devoted to something more practical. 15Tanaka, op. cit., pp. 18 f. and 23 ff. 113 TUNING AND TEMPERAMENT The other direction in which multiple division developed had far greater possibilities. This was the division of the octave into more than twelve acoustically equal parts. " Any regular sys- tem of tuning — a system constructed on a fixed value of the fifth — will eventually reach a point where its "comma," the er- ror for the enharmonic equivalent of the keynote, is small enough to be disregarded. Thus we have closed systems that agree more or less closely with the various types of meantone temper- ament, etc. If the Pythagorean tuning is extended to 17 notes, an interval of 66 cents is formed — a doubly diminished third, as Ax-C. Di- vided among 17 notes, the deficit is about 4 cents, the amount by which each fifth must be raised to have a closed system. The fifth (now taken as 10/17 octave) contains 706 cents, being raised by about the same amount that it is lowered in the Silbermann variety of meantone temperament. The major third (6/17 octave) contains 423 cents, being more than twice as sharp as it is in equal temperament, and the minor third is correspondingly very flat. If we take 5 parts for the third, this becomes a neutral third of 353 cents, such as the thirds found in some scales of the Orient. In the 17 -division, the tone is composed of 3 equal parts, of which the diatonic semitone comprises 1 part and the chromatic semitone 2 parts. Since the diatonic semitone, 70 cents, is even smaller than in the Pythagorean tuning, this system is well adapted to melody. It is, of course, wholly unacceptable for harmony because of its outsize thirds. It is notatedwith 5 sharps and 5 flats only, D* and A* being considered the equivalent of Fb and Cb, and Gb and Db the equivalent of E# and B#» The 17- division is the well-known Arabian scale of third-tones. *-1 A much more popular system is the 19-division. It arises in much the same way as the 17 -division, except that, as in just in - l"For the sake of completeness two smaller divisions should be mentioned: the Javanese equal pentatonic and the Siamese equal heptatonic. For a strange reference to the latter see J. Murray Barbour, "Nierop's Hacke- bort," Musical Quarterly, XX (1934), 312-319. * 'Joseph Sauveur ("Systeme general des intervalles des sons," M^moires de l'academie royale des sciences, 1701, pp. 445 f.) made an early reference to this scale, and of course it is discussed in all modern accounts of Arabian theory. 114 MULTIPLE DIVISION tonation, the diatonic semitone is considered the larger, with 2 parts to 1 for the chromatic semitone. Since the octave contains 5 tones and 2 semitones, it will have 5x3 + 2x2= 19 parts. The history of the 19-division goes back to the middle of the six- teenth century, whenZarlino and Salinas discussed, among types of meantone tuning, one in which the fifth was tempered by 1/3 comma. Like the other two types (1/4 and 2/7 comma) it was in- tended for a cembalo with 19 notes to the octave. ° Salinas' claim as inventor has not been disputed. He was rather apolo- getic concerning it, because of its greater deviation from pure intervals than the other two. He apparently did not realize that this could not be distinguished from an equal division into 19 parts, and that thus, as a closed system, it possessed a great advantage. It can be notated with 6 sharps and 6 flats, Cb being the equivalent of B* and E* of Fb. We have plenty of evidence from past centuries of cembali with 19 notes in the octave, for which this division would have been the ideal tuning. Zarlino19 described such a cembalo that Master Domenico Pesarese had made for him. Elsasz is fre- quently but erroneously called the inventor of the 19-note cem- balo, because his instrument is described in Praetorius' Syn- tagma. After having been neglected during the nineteenth century for the more elaborate systems such as have been described in the previous section of this chapter, the 19-division was revived in the second quarter of the twentieth century. It has had eloquent contemporary advocates in Ariel, Kornerup, and Yasser. Of all these enthusiasts, Yasser has gone to the greatest pains to show the construction of the system and its possibilities. He differs radically from its other adherents, who have proposed it partly for the sake of differentiating enharmonic pairs of notes, but chiefly because its triads are more consonant than those of equal temperament. Yasser holds that the harmony of Scriabin and the *°See Chapter III for further discussion of the various equivalents of the cy- clic multiple systems. ^Institutioni armoniche, p. 140. ^Joseph Yasser, A Theory of Evolving Tonality (New York, 1932). 115 TUNING AND TEMPERAMENT tone-rows of Schonberg show an intuitive striving toward the 19- division, since a scale as used should contain unequal divisions, being a selection from an equal division of more parts. Thus the Siamese scale of 7 equal parts is suitable for pentatonic melodies; the ordinary 12-note chromatic scale, for heptatonic melodies; and the 19-division for melodies built upon the 12- note scale. Yasser's attempt to give a historical foundation is so defective that his case emerges considerably weaker than if he had presented his system simply from the speculative point of view. There does not seem to be much chance of the 19-division coming into use in our day. Its thirds and fifths have been dis- cussed in Chapter III. To modern ears, accustomed to the sharp major thirds of equal temperament, the thirds of 379 cents, 1/3 comma flat, would sound insipid in the extreme. There would seem to be a better chance for the acceptance of a system that does not differ so markedly in its intervals from our own. The 22 -division belongs next in our study of equal divisions. It was not discussed by Sauveur, Romieu, or Drobisch. In fact, Bosanquet did not even mention it in his comprehensive book on temperament, although Opelt had treated it carefully twenty -five years before. * But the following year Bosanquet contributed an article to the Royal Society, "On the Hindoo Division of the Octave." In it he referred to S. M. Tagore's Hindu Music and an article in Fetis' Histoire generate. There the Hindoo scale was said to consist of 22 small intervals called "S'rutis." If these are considered equal, a new system arises with "practi- cally perfect" major thirds (actually, being 381.5 cents, they are almost 5 cents flat) and very sharp fifths (709 cents, or 7 cents sharp). Riemann later was to include the 22-division in his dis- cussion of various systems, and it is frequently mentioned today. Unfortunately, the Hindoo theory does not make the S'rutis all equal, but that does not prevent the division from finding an hon- ored place among these others. The thirds of the 22-division are better than those of the 19- division, and its fifths are no worse. However, it is not so good 21F. W. Opelt, Allgemeine Theorie der Musik, Chap. IV. 116 MULTIPLE DIVISION a system for the performance of European music. The difficulty lies in the formation of the major third. The fifth is taken as 13/22 octave, whence the tone has 4 parts and the ditone, 8. But 8/22 octave is 436 cents, an impossibly high value. Hence the major third must be only 7 parts, or 381.5 cents. This means that D# is taken as the major third above C, and Fb (or Cx) as the third above B. This is an awkward feature, but one that we shall run into with most of these equal divisions. It is not or- dinarily possible to retain our ideas of tone relations while mak- ing a division of the octave that will provide good fifths and thirds. The 24 -division has the same good fifths and sharp thirds as the 12-division, and the deviations for the 29-division are very similar, but with plus and minus signs reversed. Both the 25- and the 28-divisions have good thirds and quite poor fifths. So none of these four divisions is of great import. The 24 -division does have its place, as a possible realization of Aristoxenus' theory that the enharmonic diesis is a true quarter tone, the half of the equal semitone. Kircher22 presented it as such, together with a geometrical method of obtaining the quarter tones on the monochordo Rossi2** later gave the string- lengths for equal quarter tones, and Neidhardt offered a similar table many years afterwards. 24 The 29-division has its place as a member in the series that contains the 17 -division, but that fact does not im- prove the quality of its thirds. The next system of importance is the 31-division. It is the most ancient of them all and well worth the attention that has been given to it. Observe that 31 logically follows 19 in the Fi- bonacci series: 5, 7, 12, 19, 31, 50, 81, This system was first described by Vicentino25 in 1555, as the method of tuning his Archicembalo . In theory this was constructed in an attempt to reconcile the ideas of the ancient Greeks with those of six- teenth century practice. In reality it was a clever method for extending the usual meantone temperament of 1/4 comma until 22Musurgia universalis, I, p. 208. 2^Sistema musico, p. 102. 24J. G. Neidhardt, Sectio canonis harmonici, p. 31. 25L'antica musica, Book 5, Chaps. 3-5. 117 TUNING AND TEMPERAMENT it formed practically a closed system. The Archicembalo contained six ranks of keys, of which the first two represented the ordinary harpsichord keyboard with 7 natural keys, 3 sharps, and 2 flats. The third "order" contained 4 more sharps and 3 flats. The fourth order continued the flat succession with 7 more keys, and the fifth added 5 more sharps. (The sixth order is in tune with the first.) Thus all the notes would lie in a succession of fifths from G*30 to Ax, and the cir- cle would be completed by taking Ex as equivalent to G^b or C^b to Ax. (Vicentino himself gave a second tuning to the fourth or- der that showed that he considered the above to be equivalent pitches.) Vicentino specified that the first three orders of the Archi- cembalo should be tuned "justly with the temperament of the flattened fifth, according to the usage and tuning common to all the keyboard instruments, as organs, cembali, clavichords, and the like." But the other three orders may be tuned "with the perfect fifth" to the first three orders. For example, the G of the fourth order (that is, Abb) is to be a perfect fifth above the C of the first order. It must be admitted that this part of Vicen- tino' s scheme does not seem to make sense. If we ignore this puzzling doctrine of the perfect fifth, we have a logical system, formed by a complete sequence of 31 tem- pered fifths. The amount of tempering is not specified, but was to be the same as that of common practice. The common prac- tice was the ordinary meantone temperament, in which major thirds are perfect. This is undoubtedly what Vicentino used. By logarithms Christian Huyghens^" showed that the 31-di- vision does not differ perceptibly from the 1/4-comma tempera- ment. More specifically he said: "The fifth of our division is no more than 1/110 comma higher than the tempered fifths, which difference is entirely imperceptible; but which would render that consonance so much the more perfect." Riemann^' was con- fused by this remark, not realizing that Huyghens meant that this fifth was 1/110 comma higher than a fifth tempered by 1/4 comma. 26"Novus cyclus harmonious," Opera varia (Leyden, 1724), pp. 747-754. 2'Geschichte der Musiktheorie, p. 359. 118 MULTIPLE DIVISION The difference between the logarithm of the meantone fifth, .174725011, and that of 21B/3\ .1757916100, is .0000491089, which is quite close to 1/110 of the logarithm of the syntonic comma, .0053951317. Tanaka^S and Riemann have described Gonzaga's harpsichord intheMuseoCivico inBologna, dated 1606. Essentially the same as Vicentino's instrument, its arrangement of notes is somewhat different, the second row, for example, consisting solely of sharped notes, instead of 3 sharps and 2 flats. Father Scipione Stella's eight-manual harpsichord also resembled Vicentino's, but had a couple of manuals duplicated to facilitate the execu- tion.29 An improved version of Vicentino's Archicembalo was Colon - na's 6 -manual Sambuca Lincea.30 The difficulty with Vicentino's system was the unsystematic arrangement of the second and third orders. Both C# and Eb, for example, were in the second order, while Db and D* were in the third. If the instrument was to be considered merely an extension of an ordinary cembalo with twelve notes in the octave, such an arrangement was no doubt good enough. But, for its complete possibilities to be available, any such instrument needs what Bosanquet called a "generalized keyboard." Colonna came close to supplying this lack. Each of his or- ders contained seven notes, and was 1/5 tone above the preced- ing order. In our notation, the notes between C in the first or- der and D in the sixth would be DDb, C*, Db, and Cx„ Colonna' s notation for them was Cx, C#, DD, and C*, respectively. This is very clumsy; but his idea of the division was entirely correct, as can be seen from the scales he listed as examples of the ca- pabilities of the instrument. He included such remote major keys as Cb, A#, Ebb, and G# - all of course with his peculiar notation. 28Shoh£ Tanaka, in Vierteljahrsschrift fur Musikwissenschaft, VI (1890), pp. 74 f. ~ 29Fabio Colonna, La sambuca lincea (Naples, 1618), p. 6. 3Qlbid., passim. 119 TUNING AND TEMPERAMENT The germ of the 31-division lay in the contention of Marchet- tus of Padua that a tone could be divided into five parts. After Vicentino, Salinas and Mersenne discussed the system without realizing its value. Hizler31 referred to a 31-note octave, but used in practice only 13 notes, having both a D# and an E". Rossi3^ anticipated Huyghens in obtaining by logarithms the string- lengths for the 31-division, but did not call attention to the fact that its pitches were so close to those of the meantone temperament which he also presented. (With A at 41472, his meantone E was 27734, the 31-division E, 27730.) Gallimard33 was to follow Huyghen's lead in comparing the logarithms of the two temperaments. Van Blankenburg3"* was to use the 31-divi- sion as a sort of tuning measure, much as Sauveur used the 43- division and Mercator the 53-division. According to van Blank - enburg, Neidhardt's equal temperament was full of "young wolves, each 1/3 of the large wolf," because the major third of equal temperament contains 10 1/3 parts instead of the 10 parts of the 31-division. The string-lengths for the 31-division were also given by Ambrose Warren,3^ for the octave 8000.0 to 4000.0. Warren showed how this temperament could be applied to the fingerboard of the violin, for a string 13 inches long. For obtaining the 31-division mechanically, Rossi recom- mended the mesolabium. Salinas, Zarlino, and Philander have stated that the mesolabium could be used for finding an unlimited number of geometrical means between two lines, provided the number of parallelograms was increased correspondingly. Per- haps so, but Rossi3** was undoubtedly correct in saying that "in dividing the octave into 31 parts you will experience greater dif- 31Daniel Hizler, Extract aus der neuen Musica Oder Singkunst (Niirnberg, 1623), p. 31. ^^Sistema musico, pp. 86, 64. 33J. E. Gallimard, L'arithmetique des musiciens (Paris, 1754), Table XVI, p. 25. 3**Elementa musica, p. 115. 35The Tonometer (London, 1725), table at end of book. 3"Sistema musico, p. 111. 120 MULTIPLE DIVISION ficulty because of the great number of rectangles," and Mer- senne37 said flatly that it "is of no use except for finding two means between two given lines." Romieu^° included the 31 -division among those for which he had obtained correspondences, calling it a temperament of 2/9 comma. This is not very close, for 1/4 - 1/110 = 53/220. (Dro- bisch's 74-division is the real 2/9-comma temperament.) It is possible that writers before Romieu had this tuning in mind when they wrote about the 2/9-comma temperament. Printzy^ for ex- ample, spoke of a "still earlier" temperament that took 2/9 comma from each fifth. Earlier, perhaps, than Zarlino's 2/7 comma, which he had been discussing previously. But Lemme Rossi, who gave a detailed treatment to the 2/9-comma tuning, did not identify it with the 31 -division. The 34-division is a positive system, like the 22-division. That is, its fifth of 706 cents is larger than the perfect fifth, be- ing the same size as for the 17 -division. Its third is about 2 cents sharp. Thus it provides slightly greater consonance than the 31 -division. But, like the 22-division, it has remained one of the stepchildren of multiple division, largely because it is in a series for which ordinary notation cannot be used. There is a surprising mention of the 34-division by Cyriac Schneegass in 1591 (see Chapter III), but his own monochord came closer to the 2/9-comma division. Bosanquet had indicated the relation between the 22- and 34-divisions, and had praised the 56- and 87-divisions also as similar systems. Opelt, too, has included it in his fairly short list. The 36 -division has little to recommend it, although its string- lengths were worked out by Berlin,'*" and Appun and Oettingen both found it worth describing.'*-'- The 41-division has excellent fifths (702.4 cents), but thirds 3'Harmonie universelle, p. 224. JOIn Memoires de 1 academie royale des sciences, 1758, p. 837. ^"phrynis Mytilenaeus oder der satyrische Componist, p. 88. 40Johann Daniel Berlin, Anleitung zur Tonometrie (Copenhagen and Leipzig, 1767), pp. 26-27. 41Hugo Riemann, Populare Darstellung der Akustik (Berlin, 1896), p. 138. 121 TUNING AND TEMPERAMENT (380.5) that are almost six cents flat, being in this latter respect inferior to the 31- and 34 -divisions. It occurs in a worthy se- ries: 12, 17, 29, 41, 53, ... . This system was not singled out by any of the earlier writers, but received considerable atten- tion from such nineteenth century theorists as Delezenne, Dro- bisch, andBosanquet. Paul von Janko"*^ set himself the task of as- certaining the best system between 12 and 53 divisions, and chose the 41 -division. Rather naively, he concluded he had discovered this system, since Riemann had not mentioned it! The 43-division is associated with the name of Sauveur, 4^ who used its intervals (Merides) as a unit of musical measure. The Merides were divided into seven parts called Eptamerides. For more subtle distinctions, Sauveur suggested using Decam- erides, 10 of which comprised one Eptameride. But he did not use the Decamerides in practice. Thus there were 43 x 7 = 301 Eptamerides in the octave, or 3010 Decamerides. Since .30103 is the common logarithm of 2, it is possible to convert directly from logarithms to Eptamerides by dropping the decimal point and all but the first three digits of the logarithm. The 43-division is a closed system approximating the 1/5- comma variety of meantone temperament, which, as we saw in Chapter III, had been mentioned by Verheijen and Rossi. Its thirds and fifths have an equal and opposite error of slightly over four cents, thus making it somewhat inferior to the 34 -division, although the equality of the error may have some weight in rank- ing the two systems. Since 43 is a number occurring in a useful series for multiple division — 12, 19, 31, 43, 55, ... — this divi- sion was treated by Romieu, Opelt, Drobisch, and Bosanquet. The 50-division need not detain us long. It may be thought of as an octave composed of ditonic commas, since 1200 r 24 = 50. It was advocated by Henfling in 1710 and criticized by Sauveur44 the following year. A century later Opelt was to mention it. 42"Uber mehr als zwolfstufige gleichschwebende Temperaturen," Beitrage zur Akustik und Musikwissenschaft, 1901, pp. 6-12. 43joseph Sauveur, in Memoires de l'academie royale des sciences, 1701, pp. 403-498. 44joseph Sauveur, "Table generate des systemes temp£re's de musique," Mlmoires de l'academie royale des sciences, 1711, p. 406 f. 122 MULTIPLE DIVISION Bosanquet has included it as a member of the series: 12, 19, 31, 50, .... This division shows no improvement over the 31- division. Its fifths have about the same value as those of the latter, and its thirds are flatter than the latter' s were sharp. Kornerup^ has waxed lyrical in its praise, as a closed system corresponding to Zarlino's 2/7 -comma meantone temperament. He showed that the value for Zarlino's chromatic semitone (70.6724 cents) came very close to the mean of the chromatic semitones for the 19- and 31 -divisions (70.2886), and might have added that this similarity extends throughout, since all three are regular systems. He found that the greatest deviation of the 2/7 -comma tuning from the 50-division is a little over three cents, and is much less for most notes. We shall have more to say later about the special part of Kornerup's theory that has caused him to overvalue this system. The most important system after the 31- is the 53-division. In theory it is also the most ancient. According to Boethius, " Pythagoras' disciple Philolaus held that, since the tone is divis- ible into minor semitones and a comma, and since the semitone is divisible into two diaschismata, the tone is then divisible into four diaschismata plus a comma. If, now, the diaschisma is taken as two commas exactly, the tone is divided into nine commas. (Note what was said about the ditonic comma in connection with the 50-division.) This dictum about the number of commas in a tone was one of the most persistent parts of the Pythagorean system. Writers in the early sixteenth century sometimes mentioned the fact that there are nine commas in a tone, without giving any other tuning lore. They probably included, however, the statement that the diatonic semitone contains four commas, the chromatic semitone, five. Amusingly enough, after just intonation became the ideal, writers continued to talk about commas; but now it was the chro- matic semitone that contained four commas, the diatonic semi- tone, five. Since the Pythagorean diatonic semitone contains 90 cents, 45Thorvald Kornerup, Das Tonsystem des Italieners Zarlino (Copenhagen, 1930). 46A. M. S. Boethius, De institutione musica, Book 3, Chap. 8. 123 TUNING AND TEMPERAMENT and the chromatic, 114, their ratio is 3 3/4:4 3/4, or approxi- mately 4:5. Similarly, if we choose the larger just chromatic semitone of 92 cents and the smaller just diatonic semitone of 112 cents, the ratio will be 4 1/2:5 1/2, or, again, 4:5. But the ratio might be taken as 5:6, giving rise to the 67-division dis- cussed below. The comma, taken as 1/9 Pythagorean tone, would have a mean value of 22.7 cents, lying between the syntonic and the ditonic commas. If there are 9 commas in a tone, the octave contains 5x9 + 2 x 4 = 53 commas — provided we are thinking in terms of the Pythagorean tuning. If we are thinking in terms of just intona- tion, with a large diatonic semitone, there will be 5x9 + 2x5 = 55 commas. Thus the 55-division has received attention also. There are several advantages to the 53-division. Its fifths are practically perfect (.1 cent flat), so that it is unnecessary to use a monochord for tuning. Its thirds are very slightly flat (1.4 cents). However, since it is a positive system, with fifths sharper than those of equal temperament, the pure major third above C is F", with 17 parts, whereas C-E represents the Pyth- agorean third, with 18 parts. This would be confusing to the performer. After the time of the Greeks, the history of the 53-division takes us to China, where the Pythagorean tuning had been known for many centuries, probably since the invasion of Alexander the Great. In 1713 it was confirmed as the official scale, however widely instrumental tunings may have differed from it in practice. One of the most remarkable of the early Chinese theorists was King FSng, who, according to Courant, "calculated ex- actly the proportional numbers to 60 111," that is, he extended the Pythagorean system to 60 notes. These results were published by Seu-ma Pyeou, who died in 306 A. D. King FSng observed that the 54th note was almost identical with the first note. Cou- rant's figures are 177,147 for the first; 176,777 for the 54th. Seventeenth century European theorists who referred defin- itely to this system include Mersenne and Kircher. Tanaka ^Maurice Courant, "Chine et Core"e," Encyclopedic de la musique et dic- tionnaire du conservatoire (Paris, 1913), Part I, Vol. I, p. 88. 124 MULTIPLE DIVISION mentioned Kircher's name in this connection, thus differing from the majority of his contemporaries, who ascribed the system to Mercator. According to Holder,48 Nicholas Mercator had "de- duced an ingenious Invention of finding and applying a least Com- mon Measure to all Harmonic Intervals, not precisely perfect, but very near it." This was the division into 53 commas. There is no evidence, in Holder's account, that Mercator intended this system to be used on an instrument. It was to be merely a "Common Measure." Of 25 systems that Sauveur discussed, only two, the 17- and 53-divisions, were positive. He was unable to appreciate the splendid value of the thirds of the latter, since, according to his theory, its thirds would have to be as large as Pythagorean thirds. Romieu did not even mention this system. Drobisch, too, did not at first (1853) appreciate the 53-division, discarding it because of its sharp thirds. But two years later he re-evaluated both the 41- and the 53-divisions, showing that a just major scale could be obtained with them by using C D Fb G Bbb Cb C.49 The stage was thus set for Bosanquet' s detailed study of mul- tiple division, which culminated in his invention of the "gener- alized keyboard" for regular systems. In his article in the Royal Society's Proceedings, 1874-75, Bosanquet gave a clear and com- prehensive treatment of regular systems, both positive and neg- ative, with a possible notation for them. He showed how various systems could be applied to his keyboard, especially the 53- and 118 -divisions. In his symmetrical arrangement, 84 keys were needed for the 53 different notes in the octave. Obviously, then, Bosanquet' s name should be singled out for especial mention, since he applied the system to an enharmonic harmonium and did not simply discuss it as his predecessors had done. As has been noted above, the 55-division is the negative coun- terpart of the 53-division, thus having the advantage that ordi- nary notation can be used. That is its only advantage, for its fifths (698.2 cents) are no better than those of the 43-division, 48william Holder, Treatise . . .of Harmony, p. 79. 49m, W. Drobisch, "Uber musikalische Tonbestimmung und Temperatur," Abhandlungen der mathematisch-physischen Classe der koniglich s'achsis- chen Gesellschaft der Wissenschaften, IV (1855), 82-86. 125 TUNING AND TEMPERAMENT and its thirds (392.7 cents) are inferior to the latter' s. Sauveur devoted considerable space to this system, saying it was "fol- lowed by the musicians." This is a reasonable statement, for this system corresponds closely to the 1/6 -comma variety of meantone temperament favored by Silbermann. Thus we have confirmation from France of the spread of this method. Romieu showed the correspondence between the 55-division and the 1/6-comma tuning, and adopted the latter for his "tem- perament anacratique. " He referred to Sauveur, and also to Ramarin's system as given in Kircher. Mattheson^l presented this division from Johann Beer's Schola phonologica, saying that it required "that an octave should have 55 commas, but no ma- jor or minor tones." Sorge, after disapproving of the ordinary 1/4 -comma mean- tone, continued: "I am better pleased by the famous Capell- meister Telemann's system of intervals, in which the octave is divided into 55 geometrical parts (commas), that grow smaller from step to step." 52 Sorge explained that in its complete state it could not be used on the clavier; but it might be applied to the violin and to certain wind instruments, and was easiest for singers. ^Correspondences between multiple divisions and temperaments by fractional parts of the syntonic comma can be worked out by continued fractions. When the temperament of the fifth is 1/2 comma, the octave contains 26 parts. If d is the denominator of the fractional part of the comma (21.5 cents), the following formula gives the parts in the octave for 2 , is 7/12 oc- tave; that is, the fifth of equal temperament. The third, similarly, approaches 1/3 octave. Therefore, the farther the series goes, the better become its fifths, the poorer its thirds. This would seem, then, to be an inferior theory. In other divisions listed by Sauveur the difference between the two sizes of semitone was two, three, or even four parts. Here, again, the fifth eventually comes close to 7/12 octave and the third to 3/12 octave. Romieu followed Sauveur' s theory. To an extent so did Bosanquet. But the latter added the theory of positive systems. The primary positive system is 17, 29, 41, 53, 65, 77, 89, Here the fifth can be expressed as (7n + 3)/ (12n + 5) octave. Just as in the negative systems above, the limit of this ratio is 7/12 octave,, For the 53- and 65-divisions the fifths are practically perfect; the thirds of these divisions have approximately equal, but opposite, deviations. This suggests a 128 MULTIPLE DIVISION secondary positive system, the mean between the former two: 118, with both fifths and thirds well-nigh perfect. But there is nothing in these series themselves to facilitate choosing the best division or the two best. That had to be ascertained by compar- ing the intervals in the various divisions after they had been chosen. Again it would seem as if there were an arbitrary fac- tor present. We have already spoken of Kornerup and his fondness for the 50-division.56 His "golden" system of music was suggested by a study made by P.S. Wedell and N. P. J. Bertelsen in 1915. By the method of least squares they obtained the following octave series in which both the major third (5:4) and the augmented sixth (that is, the minor seventh, 7:4) approach their pure val- ues: 3, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, These of course are "golden" numbers, the law of the series being Sn = sn-i + sn-2- As n ><*>, s >a[5-1 =.61803398, a s~n 2 an+ i ratio which Kornerup called go . It is this ratio which is used in the golden section of a line, where (1 - co )/co = 6U, and which Kornerup used as the basis of his tuning system. By rather simple arithmetic we find that the golden fifth is (15 - /\[5)/22 oc- tave, or 696.2144738 cents. The golden third is 384.8579 cents, only a fair approximation, since the pure third is 386.3137 cents. Therefore, even if the series is continued indefinitely, the fifth will never be less than about 6 cents, nor the third than 1.5 cents flat. Since we have already observed several systems with bet- ter thirds and fifths than this, it would seem as if the golden system is an ignis fatuus. Drobisch57 gave an interesting formula which combined Bo- sanquet's primary and secondary positive systems. The fifth of these systems will be: (7n - l)/2(6n - 1). For odd values of n, the octave contains 6n - 1 parts; for even values, twice as many. "'"See Thorvald Kornerup, Das goldene Tonsystem als Fundament des theo- retischen Akustik (Copenhagen, 1935). "* 'In Abhandlungen der mathematisch-physischen Classe der koniglich sach- sischen Gesellschaft der Wissenschaften, IV (1855), 79 f. 129 TUNING AND TEMPERAMENT Hence he obtained the series (with n ranging from 4 through 15) : 46, 29, 70, 41, 94, 53, 118, 65, 142, 77, 166, 89. Somewhat more general was Drobisch's attempt to find a di- vision of the octave that would insure a good value for the fifth. He expressed the ratio of the fifth (log 3/2) to the octave (log 2) as a decimal, .5849625, or as a fraction, 46797/80000. From this ratio, by binary continued fractions, he obtained the series 2, 5, 12, 41, 53, 306, 665, [15601] , Next he found all the powers of 3/2 from the 13th to the 53rd, in order to ascertain which approach a pure octave. This should have checked closely with his previous list, to which 17 and 29 would be semi-conver- gents. This, however, is his complete list: 17, 19, 22, 29, 31, 41, 43, 46, 51, 53. Having eliminated all positive divisions (those with raised fifths), he still had 19, 31, and 43 to add to his prev- ious list. Although the 50-division did not appear on either list, Dro- bisch anticipated Kornerup by showing that its fifth lies almost exactly between the fifths of the 19- and 31 -divisions. After these promising beginnings, he went off at a tangent by trying to find, by least squares, the value of the fifth that would produce the best values for five different intervals. Then, again using continued fractions, he found that successive approximations to this value (.5810541) form the series: 2, 5, 7, 12, 31, 74, This is why the 74-division had an especial appeal for him. Drobisch's continued fractions were the first really scien- tific method of dividing the octave with regard to the principal consonances, the thirds and the fifths. The difficulty with it is that there are three magnitudes to be compared (third, fifth, and octave), but only one ratio (third to octave, fifth to octave, pos- sibly third to fifth) can be approximated by binary continued frac- tions. If we must choose a single ratio, it is better to use that of the fifth to the octave, as Drobisch did, since the third may be expressed in terms of the fifth. But the usual formula, T = 4F - 20, is valid only through O = 12. We have already noted that as fine a musical theorist as Sauveur failed to appreciate the 53- division, since he used the above formula and obtained a third that was one part large. Since the syntonic comma is about 1/56 130 MULTIPLE DIVISION octave, this formula will fail to give a correct number of parts for the third for any octave division greater than 28. Thus if O = 41, and F = 24, the formula makes T = 4 x 24 - 2 x 41 = 14, whereas the correct value is 13. If O = 665, and F = 389, T = 4 x 389 - 2 x 665 = 226, instead of 214. Knowing the value of the comma, we can correct our formula to read: T = 4F - 20 - TO"]. [_56j But even this would only by accident give a value for the third with as small a deviation as that for the fifth in the same divi- sion., What is needed is a method that will approach the just val- ues for third and fifth simultaneously. The desired solution can be obtained only by ternary con- tinued fractions, which are a means by which the ratios of three numbers may be approximated simultaneously, just as the ra- tios of two numbers may be approximated by binary continued fractions. When the ordinary or Jacobi ternary continued frac- tions are applied to the logarithms of the major third (5:4), per- fect fifth (3:2), and octave (2:1), the octave divisions will be: 3, 25, 28, 31, 87, 817, There are two serious faults in these results. In the first place, the expansion converges too rapidly, and we are interested chiefly in small values, those for which the octave has fewer than 100 parts. In the second place, the first few terms are foreign to every other proposed solution, such as those by Sauveur and Drobisch on previous pages. To insure slow convergence, a mixed expansion was evolved, which yields the following excellent series of octave divisions: 3, 5, 7, 12, 19, 31, 34, 53, 87, 118, 559, 612, 58 The only serious omission is the Hindoo division, with 22 parts in the oc- tave o The last term shown above (612) was said by Bosanquet to have been considered very good by Captain J. Herschel. There is no record that Captain Herschel ever constructed an experimental instrument with 612 separate pitches in the oc- tave. Even if he had done so, it would have been a mechanical monster, incapable of producing genuine music at the hands of a ^8J. Murray Barbour, "Music and Ternary Continued Fractions," American Mathematical Monthly, LV (1948), 545-555. 131 TUNING AND TEMPERAMENT performer. With the possible exception of the 19- and 22-divi- sions, the same can be said of all these attempts at multiple di- vision. Bosanquet's 53-division apparently was a success on the harmonium he constructed with the "generalized keyboard." But it, too, was cumbersome to play, and would have been very ex- pensive if applied to a pipe organ or piano. Thus the mathemat- ical theory, worked out laboriously by ternary continued frac- tions, remains theory and nothing more. The practice for the past five hundred years has favored almost exclusively systems with only twelve different pitches in the octave. There seems no immediate prospect of that practice being discarded in favor of any system of multiple division. 132