Chapter VII. IRREGULAR SYSTEMS1 If we accept Bosanquet's definition that a "regular" tuning sys- tem is one in which every fifth, or every fifth save one, has the same value, this would include the Pythagorean tuning, equal tem- perament, and the several varieties of the meantone tempera- ment, as well as equal divisions with more than twelve notes in the octave. With the addition of just intonation, it would seem as if this covered the ground pretty thoroughly. There are, however, a great many tuning systems that do not fall into any of the above- mentioned classes. At first glance these irregular systems pre- sent a bewildering variety. But some of them have been offered by their sponsors as modifications of existing tuning systems, and others, although not so designated, are also closely related to regular systems. In fact, it is possible, by making the bounds sufficiently elastic, to fit every one of these irregular systems into one or another of certain subclasses. So that, unless we re- tain Bosanquet's strict definition, there is no such thing as an irregular system — one that is wholly a law unto itself! Our first group of irregular temperaments consists of modi- fications of the meantone temperament. The meantone wolf fifth is 35 cents sharp. The simplest modification of this tempera- ment is to divide this excess equally between the fifths C*-G^ and G*(Ab)-Eb (see Table 110). This is the modification gener- Table 110. Meantone Temperament with Two Sharp Fifths Names C C# D Eb E F F# G G# A Bb B C Cents 0 76 193 310 386 503 579 697 793 890 1007 1083 1200 MD. 17.2; S.D. 18.5 ally, but erroneously, ascribed to Schlick, and, according to Ellis, still in use in England in the early nineteenth century. The G* is now almost a comma sharper than in the pure 1/4-comma *Fora condensed version of the material in this chapter, see J. Murray Bar- bour, "Irregular Systems of Temperament," Journal of the American Mu- sicological Society, I (1948), 20-26. TUNING AND TEMPERAMENT temperament. The mean deviation is noticeably lower, but the standard deviation is affected less0 Mersenne has included a discussion of the meantone temper- ament with all his other tuning information. His account differs slightly in the different works where it occurs. In the Harmonie universelle (pp. 364 f.) he had made the fifths Eb-Bb-F perfect. In the Cogitata physico-mathematica (p. 338) he asked the reader to correct the "obvious errors" in the previous description. Here he indicated simply that the wolf fifth will be G#-Eb. Perhaps his real intent is to be found in Harmonicorum libri XH (p. 60), where these two fifths are to be sharp, but not so sharp as the wOlf fifth, which is still unusable. Mersenne said that the mean- tone fifth is tempered "1/136, which is about 1/4 comma." This is a gross misstatement, for the ratio given is larger than 1/2 comma. He probably meant 1/316, which is a reasonably close value. Mersenne' s improvements upon the regular meantone tem- perament are worth showing, even if the second will be only an approximation to what he had in mind. In the first temperament (Table 111) the fifths ED-Bb and Bb-F are pure. For the second (Table 112), note that the excess of the minor third G#(Ab)-F over the third of equal temperament is 30 cents. Let us divide this excess so that G#-ED bears only half of it, the other two fifths one -quarter each. Table 111. Mersenne's Improved Meantone Temperament, No. 1 Names C C# D Eb E F F# G G# A Bb B C Cents 0 76 193 299 386 503 579 697 773 890 1001 1083 1200 M.D. 17.2; S.D. 17.7 j Table 112. Mersenne's Improved Meantone Temperament, No. 2 Names C C* D Eb E F F# G G# A Bb B C Cents 0 76 193 288 386 503 579 697 773 890 996 1083 1200 M.D. 15.3; S.D. 16.9 134 IRREGULAR SYSTEMS In Mersenne's first improved meantone system, the mean de- viation is no lower than for the temperament previously shown; but the standard deviation is lower because more notes are in- volved in the change. Mersenne's second improvement was the pattern for a modification recommended by Rameau. Now Ra- meau is noted chiefly in tuning history for his advocacy of equal temperament. But he vacillated sufficiently in his adherence to it to follow Huyghens in acclaiming as "the most perfect of all" temperaments that in which "the fifth is diminished by the 1/4 part of a comma. ** But he was aware of the pitfalls of the mean- tone temperament; for he showed that, if the tuning is begun on C, G# will be a "minor comma," 2025/2048 too flat. The remain- ing fifths, therefore, should be tuned "more just," "to regain the minor comma that has been lost." It would be even better to be- gin with C*, in order to spread the discrepancy over more notes. This account sounds as if the excess should be divided equally among the last five fifths. But, in a later paragraph, Rameau declared that "the excess of the last two fifths and of the last four or five major thirds is tolerable, not only because it is al- most insensible, but also because it is found in modulations little used/ Apparently the first three of the five fifths are not to be so sharp as the final two fifths. Still later he recommended that "the division begin on B*3, and only those fifths that follow B-F* should be a little more just." These directions are as vague as Mersenne's. In Table 113 the division is begun on bP as Rameau suggested. The fifths from B to G* have been made pure, and the excess has been di- vided equally between G*-D* and E^-B^. Before considering a final, complicated modification of the 1/4-comma temperament, let us look at William Hawkes' im- Table 113. Rameau's Modified Meantone Temperament Names C C# D D# E F F# G G# A Bb B C Cents 0 87 193 298 386 503 585 697 789 890 1007 1083 1200 M.D. 12.5; S.D. 14.0 2J. P. Rameau, Nouveau systeme de musique th^orique (Paris, 1726), pp. 107 ff. 135 TUNING AND TEMPERAMENT provement upon the 1/5-comma temperament. This resembles Mersenne's first modification. In it, according to John Farey,^ "each ascending fifth is flattened by one-fifth of a comma as the instrument is tuned, except that the fifth above E*3 and the fifth below G^ are directed to be tuned perfect." Farey continued: u. . .but why these anomalies in the system are introduced I am at a loss to guess, especially as G* is thereby made 1/5 comma the worse by it," Hawkes' reason is perfectly valid — to dimin- ish the wolf fifth by 2/5 comma, although it will still be 16 cents sharp. The alteration results in a somewhat smaller deviation than for the pure 1/5-comma temperament. The most involved of all these temperaments was that of J. E. Gallimard,4 who brought a knowledge of logarithms to bear upon the problem, in order to obtain a subtly modified meantone tem- perament. He expressed intervals for all the principal tuning systems in Sauveur's Decamerides — four-place logarithms without the decimal point. The first of his original tempera- ments used the values of the 1/4-comma temperament for the eight notes from B^ to B, If Gallimard had continued in this fashion until the entire octave had been tuned, the final fifth (D -B") would have borne the usual wolf, amounting to 103 Deca. He split up this error by adding an ever-increasing amount to each logarithm for the five fifths from B to A*. Thus there would be a total of 1 + 2 + 3 + 4 + 5=15 parts to be divided into 103 Deca., or about 7 Deca. for each part. In cents, this means that the first seven fifths have a value of 696 or 697 cents each, the others 699, 702, 705, 708, 710 cents respectively. Gallimard has pure thirds in all the principal triads of the keys of Fand C, and the poorest thirds in the key of G^0 The third on G^ itself has 425 cents, practically a diesis sharp! In Gallimard' s second temperament, the first eight notes were tuned as in the previous temperament. But he distributed the error among the other five fifths, proportional to the series 1, 3, 6, 10, 15; that is, to the series n(n-l)/2. The cents values for 3"On Music," Philosophical Magazine, XXVI (1806), 171-176. ^L'arithm^tique des musiciens, p. 26. 136 IRREGULAR SYSTEMS these altered fifths are 698, 700, 704, 708, and 714» Here the worst fifths are worse than in his first temperament, and this error is reflected in a slightly higher deviation. His worst third, G^-B^, is still a diesis sharp. The deviations are still large for Gallimard's modification. Had he been willing to use a modification of the 1/6-comma tem- perament, with slightly sharp diatonic thirds, his system would have been better. Modifications of the latter temperament are to be found later in this chapter, by Young and Mercadier. Arnold Schlick' s temperament^ deserves special honor, for apparently he was the first writer in any country to describe a temperament for each note of the chromatic octave. Shohe Ta- naka and Hugo Riemann have broadcast the erroneous idea that Schlick founded the meantone system. The former spoke of the "exact instructions" that Schlick had given, and added, "In exact language this will mean that each fifth is to be flattened by 1/4 comma."" This reads well, but is utter nonsense with relation to what Schlick actually said. In place of "exact instructions" he gave very indefinite rules that create a problem for us. Beginning with F on the organ manual, the fifth F-C is to be somewhat flat. This same rule is to be followed in tuning the other "claves naturales" by fifths, making the octaves perfect. As to the major thirds, Schlick said that "although they will all be too high, it is necessary to make the three thirds C-E, F-A, and G-B better, ... as much as the said thirds are better, so much will G# be worse to E and B." The tuning of the black keys is to be made similarly, tuning upward by flat fifths from B to obtain F* and C#, and tuning down- ward from F to obtain Bb and ED. The semitone between G and A received special attention. As G* it was needed as the third above E; as A13 it was also needed as the third below C. So Schlick suggested a mean value for this note, directing that the fifth AD-ED is to be somewhat larger than a perfect fifth. Whatever Schlick' s system, it could not have been the mean- kSpiegel der Orgelmacher und Organisten, in Monatshefte fur Musikge- schichte, 1869, pp. 41 f. "Shohe Tanaka, in Vierteljahrsschrift fur Musikwissenschaft, VI (1890), 62, 137 TUNING AND TEMPERAMENT tone system as described so carefully by Tanaka; for it lacks pure thirds. Schlick said definitely that "all will be too high." Not even the diatonic thirds are to be pure, only made "better than the rest." What, then, was Schlick' s tuning method? All that can be said with assurance is that it was an irregular system, lying some- where between meantone and equal temperament. We cannot hope to reconstruct it exactly; but it will be worth while to give some idea, at least, of what it was like. Let us assume that Schlick used the same size of tempered fifth for each of the six diatonic fifths; a somewhat larger, but still flat, fifth for the four chro- matic fifths; and a sharp fifth for the two fifths A^-E^ and C*- G*. Call these temperaments x, y, and -z respectively. Then, since the ditonic comma must be absorbed in the course of the tuning, 6x + 4y - 2z = 24 cents. Now x is larger than y; let us assume that x = 2y. Since Schlick said that most of his fifths were to be "somewhat" flat and the other two fifths "somewhat" sharp, let us assume that x = z. Then 12y + 4y - 4y = 24 cents, y = 2 cents, x = z = 4 cents. Thus Schlick' s diatonic fifths, of 698 cents, will be tempered by 1/6 comma; his chromatic fifths, of 700 cents, will be the same size as those in equal temperament; his two sharp fifths will be of 706 cents. His diatonic thirds will be six cents sharp; his chromatic thirds, 8 or 10 cents; the thirds E-G* and A^-C, 18 cents (not unbearable); and the "foreign" thirds, B-D#, F#-A#, and D^-F, 26 cents, slightly more than a comma. The deviations for Schlick' s hypothetical temperament are less than half as large as those for the modified meantone tem- perament that Tanaka wrongly ascribed to him — the first tem- perament in this chapter. His is a good system, holding its own in comparison with systems that were proposed two or three centuries later. Of the irregular systems discussed in the first 138 IRREGULAR SYSTEMS section of this chapter, Schlick's is superior to Mersenne's, Ra- meau's, Hawkes', and Gallimard's. Even so, Schlick's system is not so good as that of Gram- mateus, next to be discussed. Therefore we must not assume that the present reconstruction has erred on the side of Schlick. As a temperament, it has far greater significance for us than if it had been the meantone temperament, with two sharp fifths. It is an indication that in the early sixteenth century organ temper- ament was nearer to equal temperament than it generally was for centuries after this time. Schlick's directions have the added weight that they represent the practice of an actual organist, un- concerned with mathematics or the theories of the ancient Greeks. Modifications of Regular Temperaments In the next main group of irregular temperaments the diatonic notes are tuned according to one of the well-known regular tem- peraments and then each tone is divided equally to form the chro- matic notes. The oldest and best of them was that of Henricus Grammateus, or Heinrich Schreyber of Erfurt. Grammateus tuned the diatonic notes of his monochord according to the Pyth- agorean ratios. But when it came to the black keys, the "minor semitones," he followed a different procedure. These were formed by dividing each tone into two equal semitones by the Eu- clidean method for finding a geometric mean proportional. Gram- mateus had a figure to illustrate the construction. Perhaps he obtained this method of halving intervals directly from Euclid. But he may have owed it to Faber Stapulensis^ (Jacques le Febvre), who had shown that it was impossible to divide a ton^ numerically into two equal parts, but that the halving of any in- terval could be accomplished by geometry. At any rate, Ber- mudo, whose one tuning method was identical with Grammateus', did depend upon Faber for the method of constructing the mean proportionals. Faber exerted great influence upon later writers 7"Arithmetica applicirt oder gezogen auff die edel Kunst musica," an appen- dix to his Ayn new kunstlich Buech (Niirnberg, 1518) a ^Elementa musicalia (Paris, 1496). 139 TUNING AND TEMPERAMENT Table 114 . Hawkes' Modified 1/5-Comma Temperament U 7 2 +£ 4 1 „6 1 7 3 2 Names C°C 'D" Eb 5 E~ = F+5 Fr* G" G#~5 A_1 Bb 5 B"1 C° Cents 0 83 195 303 390 502 586 698 785 893 1005 1088 1200 M.D. 12.7; S.D. 13.0 Table 115. Gallimard's Modified Meantone Temperament, No. 1 Names C C# D D# E F F# G G# A Bb B C Deca. 0 212 484 744 969 1263 1461 1747 1980 2232 2526 2716 3010 Cents 0 84 193 297 386 504 582 696 789 890 1007 1083 1200 M.D. 13.3; S.D. 14.9 Table 116. Gallimard's Modified Meantone Temperament, No. 2 Names C C# D D# E F F# G G# A Bb B C Deca. 0 204 484 734 969 1263 1457 1747 1969 2232 2526 2716 3010 Cents 0 81 193 293 386 504 581 696 785 890 1007 1083 1200 M.D. 14.0; S.D. 15.6 Table 117. Schlick's Temperament (Hypothetical) Names C° C#_1D" Eb+<5 E~~3 Fe F1"1^'^1"^" B^s B" C° Cents 0 90 196 302 392 502 590 698 796 894 1002 1090 1200 M.D. 8.0; S.D. 8.6 Table 118. Grammateus' Monochord (Pythagorean with Mean Semitones) Names C° Cr 2 D° D*^ (Eb+2) E° F° fHg° G#_2A° Bb+^ B? C° Cents 0 102 204 306 408 498 600 702 804 906 1008 1110 1200 M.D. 3.3; S.D. 4.5 140 IRREGULAR SYSTEMS who attempted to solve the tuning problem. Especially among mathematical writers who dabbled in this field, Faber's name was held in something of the same esteem as that of Boethius. This monochord division of Grammateus is seen to be of a subtle and theoretical nature. It is equivalent to dividing the Pythagorean comma equally between the fifths B-F# and B^-F. As such, it is identical with Marpurg's tuning K. This tuning may have been used in practice, but hardly by anyone who was ac- customed, like Schlick, to tune by ear. Note that it was presented as a method not for fretted instruments, but for organs. Gram- mateus said in his introduction: "There follows herewith an amusing reckoning which serves the art of song called music, and from such reckoning springs the division of the monochord, from which will then be taken the proportionate length and width of the organ pipes after the opinion of Pythagoras." So far as we know, Grammateus was the earliest writer with a method for finding equal semitones as applied to a tuning sys- tem. Of course only ten semitones will be equal, the other two being twelve cents smaller. Probably many men who later spoke about equal semitones on the lute may have had in mind some such division, perhaps made by dividing the tones arithmetically instead of geometrically. GanassF had a method for obtaining equal semitones on the lute and viol by linear divisions, using the ratios of just intona- tion for his basic scale. Although he described his procedure in more complicated terms, his monochord might have been tuned as follows: with A the fundamental, form the minor third C with the ratio 6:5; form F and G as perfect fourth and fifth to C with the respective ratios 4:3 and 3:2; divide the space between A and C into three equal parts for B^ and B; divide the space between C and F into five equal parts for C*, D, Eb, and E; F* will be half way between F and G, and G* halfway between G and the oc- tave A. The construction will be even easier if we start with C: form F and G as perfect fourth and fifth to C; divide the space between C and F into five equal parts, between F and G into two equal parts, and between G and the octave C into five equal parts. 9Sylvestro Ganassi, Regola Rubertina. Lettione seconda (1543); ed. Max Schneider (Leipzig, 1924), Chap. IV. 141 TUNING AND TEMPERAMENT In the monochord shown in Table 119, the lengths and ratios have been added according to Ganassi's directions. Actually, the above monochord does not quite represent Ga- nassi's ideas. His lute had only eight frets, so that the position of the notes above F is rather conjectural. However, he placed a dot where G, the tenth fret, would naturally fall, and it is rea- sonable to suppose that he would have made a linear division for the semitones on either side of G. A greater departure from his ideas lies in ignoring the tempering of the first and second frets: the second fret is to be placed higher than 8/9 by the width of the fret, and the first fret higher than 17/18 by half the width of the freto Similarly the sixth fret is to be placed lower than 17/24 by the width of the fret. His drawing for the monochord is made with unusual care (see Figure H). It appears as if the width of the fret were about 1/2 of 1 percent of the length of the string. This tempering would make B*3 and B sharper by about half a comma, and E*3 flatter by the same amount. The first two changes would not affect the tuning greatly, but the change in the position of the sixth fret would be harmful. Since Ganassi was not spe- cific as to the relative length and breadth of the string, we merely indicate here that he advocated these three tempered values. Fig. H. Ganassi's Method for Placing Frets on the Lute and Viol Reproduced by courtesy of the Library of Congress 142 IRREGULAR SYSTEMS Table 119. Ganassi's Monochord (Just with Mean Semitones) Lengths 120 114 108 102 96 90 85 Ratios 19/20 18/19 17/18 16/17 15/16 17/18 16/17 Names C° X D"1 X E"1 F° X Cents 0 88 182 281 386 498 597 Lengths 80 76 72 68 64 60 Ratios 19/20 18/19 17/18 16/17 15/16 Names G° x A"1 x B"1 C° Cents 702 790 884 983 1088 1200 M.D. 6.5; S.D. 7.8 Table 120. Reinhard's Monochord (Variant of Ganassi's) Lengths 60 56 2/3 53 1/3 50 2/3 48 45 42 1/2 40 38 Names C° X D° X E'1 F° X G° X Cents 0 99 204 292 386 498 597 702 790 Lengths 36 34 32 30 Names A"1 X B"1 C° Cents 884 983 1088 1200 M.D. 6.5; S.D. 7.8 Ratios Names C° Cents 0 Ratios Names G° Cents 702 Table 121. Malcolm's Monochord (Variant of Ganassi's) 16/17 17/18 18/19 19/20 15/16 16/17 17/18 x D° x E_1 F° 105 204 298 386 498 18/19 19/20 16/17 17/18 15/16 x A"1 x B-1 C° 796 884 989 1088 1200 M.D. 6.5; S.D. 7.8 x 603 143 TUNING AND TEMPERAMENT Except for the arithmetical divisions, Ganassi's tuning re- sembles Grammateus' treatment of the Pythagorean tuning, the difference being that the basic scale here is just intonation. It also resembles Artusi's treatment of the meantone temperament, shortly to be described. But even if Ganassi had used the Eu- clidean method to divide his tones, his monochord(M.D. 6.0; S.D. 7.3) would have been inferior to either of the other two, since the diatonic just scale varies more greatly from equal temperament than either the Pythagorean or meantone does. But this is a good division, and has the tremendous advantage that it is the easiest of all chromatic monochords to form. Ganassi's method was discovered independently by Andreas Reinhard,^ who described the syntonic tuning, and then gave a table in which the space of each tone, whether major or minor, is halved to obtain the chromatic note. His table gave string- lengths only, beginning with 45 for F. Since he used D° instead of D"1 , his intervals are in a slightly different order from Ga- nassi's. Ten years after Reinhard, his tuning method was taken over by Abraham Bartolus, * the sole difference being that the latter began with E (48) instead of F (45). Bartolus gave Reinhard as his source. At first he advocated the method for keyboard in- struments, and later prescribed it also for fretted instruments and bells. This general application of a tuning method is some- thing that is found in very few theorists of Bartolus' period, most of whom continued to say with Vicentinothat fretted instruments used equal temperament, and keyboard instruments, the mean- tone temperament. In one of the curious dialogs of Printz' s Phrynis Mytilenaeus-^ this same temperament is mentioned. "Charis" describes it and gives the string-lengths for the C octave, 360 to 180, thus avoid- ing the fractions that Reinhard had encountered. Very likely Printz intended this for Reinhard' s tuning, but his perplexing use ^"Monochordum (Leipzig, 1604). 11Musica mathematica: the 2nd part of Heinrich Zeising's Theatri machi- narum (Altenburg, 1614), pp. 151 f, 165 ff. 12Part 3, Chap. 6. 144 IRREGULAR SYSTEMS of anagrams effectively conceals Reinhard's name, if it is indeed hidden there. Alexander Malcolm** had a division very similar to those of Ganassi and Reinhard. In fact, it is the inversion of Ganassi's, with semitones paired in contrary motion. Although Malcolm said that the tones were to be divided arithmetically, as 16:17:18, his table of string-lengths (lengths of chords) represents a very unlikely division, difficult to make. Marpurg, who called the system ugly, has represented it by a series of increasing num- bers, as C, C% D are 48, 51, 54. This would mean that Mal- colm's ratios are to be taken as vibration numbers, improbable in view of his own terminology for them. Since Malcolm's scale contains the same ratios for semitones as Ganassi's and Reinhard's, although in a different order, the deviation for the three scales will be the same. But his chro- matic notes are all five or six cents higher than Reinhard's. It is very probable that Malcolm intended the same division as Reinhardo Malcolm stated that Thomas Salmon had written about this scale. But it is often referred to by Malcolm's name alone. Certainly these well-nigh equal semitones of Ganassi, Reinhard, Bartolus, Salmon, and Malcolm represent a long-lived (almost two centuries) and very good way to divide the octave with ease. Levens' "Sisteme"" also had linear divisions only, but was far less successful than those just described. His monochord had integer numbers starting with 48 for C. Ganassi's system had only five consecutive semitones formed by equal divisions of a larger interval, but Levens' had seven, from 42 for D to 28 for A. Thus Levens' consecutive semitones vary in size from 85 to 119 cents. Furthermore, his semitone A-B" is very small (63 cents), with the Archytas ratio, 28:27; whereas his semitone Bb-Cb, with the ratio 27:25, is more than twice as large (133 cents) . Levens' deviations are as great as for some varieties of just intonation. 13A Treatise of Musick (Edinburgh, 1721), p. 304. Abrege' des regies de l'harmonie (Bordeaux, 1743), p. 87. 145 TUNING AND TEMPERAMENT Since C is 48 in Levens' tuning, the monochord could easily be constructed with a foot rule. But it would not be so easy to construct a monochord of indefinite length for this tuning. A slight change in the values of A and B would greatly simplify the construction of the monochord, and at the same time would al- most cut the deviation in half. It would then be formed thus: Di- vide the entire string into 8 parts, putting D at the first point of division, F at the second, and AD at the third. Divide the space between C and D into two parts for C*. Divide the space between D and F into three parts, for E" and E, and apply EF twice from F toward A13, for F* and G. Divide the space from A*5 to the higher C (midpoint of the string) into four equal parts, for A, B", and B. The third distinct method of forming equal semitones upon the lute stems from Giovanni Maria Artusi. ** But, as with Gram- mateus' division, only ten of the semitones would be equal. In pointing out the "errors of certain modern composers," Artusi gave two examples of "intervals false for singing, but good for playing on the lute." Thus the diminished seventh, C*-B", in the beginning of Marenzio's madrigal "False Faith," is "false for voices and for modulation, but not false on the lute and the chitarone." On the lute, he continued, "the tone is divided into two equal semitones." So far Artusi had been speaking very much as had his predecessors. But he then stated that the tone in question is not the 9:8 tone, but the mean tone used on the lute and other in- struments. Later he called the tempered semitone "the just half of the mean tone." For constructing this temperament he men- tioned the mesolabium and the Euclidean construction for a mean proportional, with references to Zarlino and Faber, The meso- labium would have been useless for this purpose, unless Artusi had desired complete equal temperament. But Euclid's method would have served for constructing meantones from just major thirds, and then for constructing mean semitones from mean tones. Since Artusi did not give a detailed account of how his tem- perament was to be formed, we can only surmise that all the di- l^Seconda parte dell' Artusi overo della imperfettioni della moderna musica (Venice, 1603), pp. 30 ff. 146 IRREGULAR SYSTEMS Table 122. Levens' Monochord (Linear Divisions) Lengths 48 45 42 40 38 36 34 32 30 28 27 25 24 Names C° Db+1 D Eb+1 E F° x G° Ab+1 A Bbo Cb+2 C° Cents 0 112 231 316 404 498 597 702 814 933 996 1129 1200 M.D. 16.7; S.D. 19.9 Table 123. Levens' Monochord (Altered Form) Lengths 48 45 42 40 38 36 34 32 30 28 1/2 27 25 1/2 24 Names C° Db+1 D Eb+1 E F° x G° Ab+1 A Bb° B C° Cents 0 112 231 316 404 498 597 702 814 902 996 1095 1200 M.D. 8.8; S.D. 10.3 Table 124. Artusi's Monochord (Meantone with Mean Semitones) (Bonded Clavichord Tuning, No. 1) _ 1 j_l 1 3 5 Names C° x D f x E~ F 5 x G ? x A" x B~4 C° Cents 0 97 193 290 386 503 600 697 794 890 987 1083 1200 M.D. 5.7; S.D. 7.6 Table 125. Bonded Clavichord Tuning, No. 2 _i _2 .1 _1 _i _5 Names C° x D 3 x E 3 F 6 x G 6 x A 2 x B 6 C° Cents 0 97 197 294 394 502 599 698 795 895 992 1092 1200 M.D. 2.6; S.D. 3.8 147 TUNING AND TEMPERAMENT atonic notes were to be tuned as in the ordinary meantone tem- perament and the chromatic notes by dividing each of the tones in half. This is the "semi-meantone temperament" mentioned by Ellis, I® "in which the natural notes C, D, E, F, G, A, B were tuned in meantone temperament, and the chromatics were inter- polated at intervals of half a meantone." According to Ellis, it had been in use on "the old fretted or bonded clavichords." Un- fortunately, Ellis did not give the source of this information. If these bonded clavichords had had their notes paired CC* DD* E FF# GG# AA* B C, a fixed ratio could have existed be- tween the notes in each pair, so that C#, for example, would always be 96.5 cents higher than C. Of course, the two diatonic semi- tones, E-F and B-C, would be about a comma larger, at 117 cents each. Some writers have said that the bonded clavichords neces- sarily used the meantone temperament. But nothing would have prevented the performer from tuning his diatonic tones sharper than mean tones. Suppose, for example, it had become the fashion to diminish the fifth by 1/^ comma, as in Bach's day. Then the bonded clavichord would have had the scale shown in Table 125. In this tuning the standard deviation is fairly large because the semitones E-F and B-C have a deviation of eight cents,, If E and B are made four cents sharper, the mean deviation is un- changed, but the standard deviation is reduced to 3.0. This much can be done without changing the ratio of C to C*. But a bonded clavichord that was constructed at the time Douwes was writing (1699; see Chapter in) would have had the ratio of this pair of notes fixed according to the temperament then in use, perhaps the 1/6-comma meantone system, and the mean-semitone tuning would then have been even better than in Table 125. Furthermore, there is no valid reason why the ratio of the semitones on a single string could not have been -J2, if the bonded clavichord had been constructed at a time when equal tempera- ment was widely accepted. The only difficulty is that the free clavichords were more common then. But it is nonsense to think ^Alexander Ellis, "On the History of Musical Pitch," Journal of the Society of Arts, XXVIII (1880), 295. 148 IRREGULAR SYSTEMS that there was any connection between free clavichords and equal temperament, except where an old clavichord had retained sem- itonal ratios that belonged to a type of tuning that had been su- perseded. Even then, as we have shown, the open strings could have been tuned so that the instrument as a whole would have varied only slightly from equal temperament. The only troublesome situation would occur when the bonded clavichord had its ratios fixed so that, for example, the semitone between C° and D"^ was not a mean semitone, but C#~^. Re- member that Artusi was writing about equal semitones on the lute, not on the clavichord. And other theorists, advocating meantone temperament for keyboard instruments, made no dis- tinction between the clavichord, on one hand, and the organ and harpsichord, on the other. Let us see, in Table 126, what could be done when the fixed chromatic semitone has only 76 cents, the diatonic semitone, 117 cents. Here we assume that C-C#, F-F#,and G-G*are each 76 cents, and that D-E^andA-B^are each 117 cents. The other seven sem- itones are free. If we make them all equal, each will have 105.4 cents. That means that D and A are flatter than in the regular meantone temperament; E, F, G, and B sharper. After this somewhat eccentric tuning of the diatonic notes, the deviation is almost half that of the regular meantone temperament, but is still not quite so good as that of the old Pythagorean tuning, un- tempered. Therefore on a bonded clavichord that was built for the complete meantone temperament, even the most scientific tuning of the free strings would not make a very acceptable tem- perament. And such clavichords would certainly have delayed the acceptance of equal temperament. A corroboration of Artusi' s method of forming equal semi- tones on the lute came from Ercole Bottrigari.*' He had clas- sified instruments by their tuning, as Zarlino had done. He went on to show that the lute cannot play in tune with the cembalo. If the E string of the lute is tuned in unison with the E of the cem- *'I1 dcsiderio, ovvero de' concerti di varii stromenti musicali (Venice, 1594); new ed. by Kathi Meyer (Berlin, 1924). 149 TUNING AND TEMPERAMENT Table 126. Bonded Clavichord Tuning, No. 3 Names C C* D Eb E F F# G G# A Bb B C Cents 0 76 181 298 403 509 585 691 767 872 989 1094 1200 M.D. 12.0; S.D. 13.7 balo, the F's will be out of tune, the G's will again be in tune, and the G^'s out of tune. He explained that, since on the lute the tone was divided into two equal semitones, and on the cembalo into two unequal semitones, then the diatonic semitone E-F, with the ratio of 16:15 tempered, would be higher on the cembalo than on the lute; but the chromatic semitone G-G* (25:24 tempered) would be higher on the lute. This explanation would be true, even if the lute were in equal temperament. But the interesting question is why the G's were in tune if the E's were, and vice versa. If the lute were in equal temperament, it would have no pitches in unison with the cem- balo save the one that was tuned to a unison to begin with. Now, Bottrigari was referring to a tuning in which the order of strings was D, G, C, E, A, D. Of these the E string was called the "me- zanina," the middle string. On either D string or on the A string, the 2nd, 3rd, and 5th frets formed a diatonic sequence — A, B, C, D or D, E, F, G. Since the position of the frets was the same on all the strings, the succession on the E string would have been E, F*, G, A. Therefore, if the diatonic notes on the D and A strings were tuned in unison with those on the cembalo, as in Artusi's tuning, the notes E, F% G, and A on the E string will also be in unison. But E-F on the lute will behalf a mean tone and so will G-G^, whereas the E-F of the cembalo will be a tempered major semitone and the G-G* a tempered minor semitone. (F#-G, about which Bot- trigari said nothing, will be the ordinary major semitone of the meantone temperament on both instruments, and will be almost a comma larger than these other semitones on the lute.) This is the only reasonable explanation of Bottrigari' s statement, and, since it was made only nine years earlier than Artusi's account, we may surmise that this method of tuning was in common use about 1600. We should be careful, therefore, not to assume that 150 IRREGULAR SYSTEMS every statement about the use of equal semitones on the lute nec- essarily meant equal temperament, with the ratio of1 {2 for the semitone. Temperaments Largely Pythagorean A great many irregular temperaments are largely Pythagor- ean, that is, they contain many pure fifths. This is reasonable enough, since pure fifths are easy to tune and do not depart greatly from the fifths of equal temperament. As we shall see, many of these are typical "paper" temperaments, ill adapted either to tuning by ear or to setting upon a monochord. But first we shall examine several that used linear divisions only. 1ft Martin Agricola, ° who was responsible for a good version of just intonation, showed a monochord for the lute in which the diatonic notes, like those of Grammateus, were joined by pure fifths. To divide the tones into diatonic and chromatic semitones, Agricola applied the old doctrine that the tone is divisible into 9 commas, 5 for the chromatic semitone and 4 for the diatonic. He tuned a G string, marking off G* as 5/9 the distance from G to A. That means that G:G#:A as 81:76:72. Thus the diatonic semitone G*-A had the ratio 19:18, or almost 94 cents, instead of 256:243 or 90 cents, and the chromatic semitone 110 cents in- stead of 114. Agricola formed his A* and C* like the G*. As there were only seven frets on this string, he did not give values for D#, F, and F#. But F is of course a major tone below G, and he had previously shown ED (although he called it "dis") to be a tone be- low F. But there B^ had been shown to be a tone lower than C, 20 cents flatter than the A* on the other string. These incon- sistencies are bound to arise when any unequal tuning is used on a fretted instrument, as Galilei pointed out. For the sake of a logical construction, let us assume (see Table 127) that each of the five tones in the octave is divided into 5 + 4 commas. This may be slightly better than Agricola' s tuning would have been if 18Musica instrumentalis deudsch (4th ed.; Wittenberg, 1545). Reprinted as Band 20 of Publikation 'alterer praktischer und theoretischer Musikwerke, 1896. The reference here is to page 227 of the latter. 151 TUNING AND TEMPERAMENT Table 127. Agricola's Pythagorean- Type Monochord Names C°C#~5D° D#"6 E° F° F^G0 G^"5 A0 A#_1 B° C° Cents 0 110 204 314 408 498 608 702 812 906 1016 1110 1200 M.D. 8.3; S.D. 8.6 Table 128 WSng Pho's Pythagorean- Type Monochord Lengths 900 844 800 751 713 668 633 600 563 534 501 475 450 Names C C* D D* E E# F# G G* A A# B C Cents 0 111 204 313 403 516 609 702 812 904 1014 1107 1200 M.D. 8.9; S.D. 9.0 he had applied it to an entire octave. This system, if we can call it a system, is appreciably better than the ordinary Pythagorean tuning. It contains ten pure fifths; the fifth B-F# is four cents flat (1/6 comma), and A#-F is twenty cents flat. But none of the credit belongs to the inventor. Agric- ola, like many another good man, confused geometrical with arith- metical proportion The old statement about the sizes of semi- tones is very nearly correct when geometrical magnitudes are in question, but is less accurate when applied to linear divisions. Furthermore, it was a happy accident that led him to make his chromatic notes sharps. If he had divided the tone G-A into G- A^-Ainthis same manner, his diatonic semitone would have con- tained 88 cents, the chromatic, 116, thus diverging more widely from equality than the Pythagorean semitones do. An accidental improvement is the best we can say for this tuning of Agricola. Agricola's approximation for the Pythagorean tuning suggests the monochord of an early Chinese theorist, W&ng Ph5, who lived toward the end of the tenth century. ^ Perhaps he was familiar with the excellent temperament of Ho Tchheng-thyen, but, if so, was too timid to follow his example. Starting with the Pytha- gorean tuning for the octave 900-450, he has retained the purity ^Maurice Courant, in Encyclopedic de la musique et dictionnaire du conser- vatoire, Part 1, Vol. I, p. 90. 152 IRREGULAR SYSTEMS of G and D. He lowered the pitches of all the other notes by add- ing two units for C#, D*, E, and E#, and one unit for F#, G#, A, A*, and B. This was too small a correction for most of the notes, as can be seen from Table 128, which is comparable to that of Agricola. John Dowland is another writer whose tuning system, like those of Ramis, Grammateus, Agricola, and others, had a strong Pythagorean cast. In his account of "fretting the lute," C, D, F, G, and A have Pythagorean tuning.20 The chromatic semitone from C to C# is 33:31, or 108 cents, not far from the Pythago- rean of 114 cents. The diatonic semitone from D to E^ is 22:21, or 80 cents, considerably flatter than the Pythagorean of 90 cents. G* and B*3 form pure fifths to C# and E*5 respectively. An unu- sual feature of the tuning is F* taken as the arithmetical mean be- tween F and G, and E (!)asthe mean between E^° and F. The value for E thus obtained, 264:211, is 388 cents, almost a pure third above C, instead of the expected Pythagorean third. The third D-F#, of 393 cents, is likewise an improvement. Thus the deviation is somewhat less than that for the Pythagorean tuning, being almost the same as that of Agricola' s system. There is no B on this string, but we have made B a pure fifth above E. The trend of Dowland' s tuning resembles that of Ornithopar- chus, whose Micrologus was translated into English by Dowland. Ornithoparchus' division of the monochord was entirely Pytha- gorean, a ten-note system extending fromA^ to B by pure fifths. It was natural for Ornithoparchus to advocate the Pythagorean tuning, since most of his contemporaries had not yet departed from it„ But a century later, the Pythagorean tuning was becom- ing somewhat rare. And yet Dowland' s fellow countryman Thomas Morley, whose precepts have been quoted by everyone who writes about Elizabethan music, gave only a Pythagorean monochord. Unusual ratios are a feature of Colonna's tunings also, al- though he definitely included some ratios that belong to just in- tonation as well.21 He is noted in the field of multiple division 20Robert Dowland, Variety of Lute-Lessons (London, 1610). "Of Fretting the Lute" comes under "Other Necessary Observations to Lute-playing by John Dowland, Bachelor of Music." ^^Fabio Colonna, La sambuca lincea, p. 22. 153 TUNING AND TEMPERAMENT for having described an instrument, theSambucaLincea, similar to Vicentino's Archicembalo, upon which the division of the oc- tave into §1 parts could be accomplished. His mathematical the- ory of intervals is very ingenious, including superparticular pro- portions, but also more subtle fractions. He began with certain well-known consonant ratios: 1:1 (unison), 6:5 (minor third), 5:4 (major third), 4:3 (fourth), 3:2 (fifth), and 5:3 (major sixth). Then if a string of the monochord is divided to produce a certain in- terval, the sounding part of the string should produce with the other part (the Residuo) either one of the above intervals or a higher octave of it. This means that if any of the above ratios is called b:a, intervals derived from it have ratios of the form (2*b + a):2^b. For example, from 1:1 comes 17:16; from 6:5 comes 11:6; from 3:2 comes 25:24. Colonna's two chromatic monochords are shown in Tables 130 and 131. Each contains seven pure fifths and several pure thirds. The worst feature of both monochords is the 55:54 chromatic semitone of 30 cents (as G-G^ or B^-B) — not much larger than a comma. Almost as bad is the 12:11 diatonic semitone of 152 cents, as G*-A or B-C. * The 27:25 diatonic semitone of 134 cents, as F*~ -G or C* - D° , is not good either, but is a blemish found also in ordinary just intonation. A redeeming feature of the first monochord is the division of the 9:8 tone into 17:16 and 18:17 semitones. Colonna's division of the 10:9 tone into 12:11 and 55:54 "sem- itones" is reminiscent of the superparticular division of the 10:9 tone that Ptolemy used for his soft chromatic tetrachord, 5/f5 x 14/15 x 27/28, and of the common division of just intonation de- rived from Didymus' chromatic, 5/6 x 24/25 x 15/16. ^ Other 22Henri Louis Choquel used a 12:11 semitone between A and Bb and a 33:32 semitone between BD and B, in what was otherwise a monochord in ordinary just intonation. La musique rendue sensible par la me'chanique (New ed., Paris, 1762). 2*\A. m, Awraamoff in 1920 devised a tuning for the chromatic octave that out- does Colonna's. The natural seventh, 8:7, is exploited in this tuning, and such superparticular near-commatic intervals occur in it as 49:48 (36 cents) and 64:63 (27 cents)! "Jenseits von Temperierung und Tonalitat," Melos, Vol. I (1920). 154 IRREGULAR SYSTEMS Table 129. Dowland's Lute Tuning Ratios 1 33:31 9:8 33:28 264:211 4:3 24:17 3:2 Names C C* D Eb X F X G Cents 0 108 204 284 388 498 597 702 99:62 810 Ratios 27:16 99:56 [396:21l] 2:1 Names A Bb x C Cents 906 986 1090 1200 M.D. 8.2; S.D. 10.1 Table 130. Colonna's Irregular Just Intonation, No. 1 Lengths 50 48 45 [42 6/17] 40 371/2 36 Ratios 24/25 15/16 16/17 17/18 15/16 24/25 25/27 Names C° C¥~2 D"1 [Eb] E"1 F° F*-2 Cents 0 70 182 287 386 498 568 Lengths 331/3 32 8/11 30 28 4/17 26 2/3 25 Ratios 54/55 11/12 16/17 17/18 15/16 Names G° G* A-1 Bb B_1 C° Cents 702 732 884 989 1088 1200 M.D. 22.0; S.D. 30.3 Table 131. Colonna's Irregular Just Intonation, No. 2 Lengths 1920 2000 2160 2304 2400 2560 2688 Ratios 24/25 25/27 15/16 24/25 15/16 20/21 14/15 Names C° &* "2 D° Eb+1 E_1 F° F* Cents 0 70 204 316 386 498 618 Lengths 2880 3072 3200 3456 3520 3842 Ratios 15/16 24/25 25/27 54/55 11/12 Names G° Ab+1 A"1 Bb+1 B C° Cents 702 814 884 1018 1048 1200 M.D. 29.3; S.D. 33.8 155 TUNING AND TEMPERAMENT possible divisions of the 10:9 tone are 13:12 and 40:39, which is somewhat better than Colonna's division, and the linear division 19:18 and 20:19, as inGanassi. Divisions of the 9:8 tone include 17:16 and 18:17, as well as 15:14 and 21:20, both of which Co- lonna used. Other possible superparticular divisions of the 9:8 tone are 13:12 and 27:26; 12:11 and 33:32; 11:10 and 45:44; and 10:9 and 81:80, this last, of course, being the minor tone and comma. Divisions of the Ditonic Comma The Pythagorean-type temperaments in our second group are more difficult to construct, in that they contain unusual divisions of the ditonic comma. By ear, these temperaments would have been almost impossible in many cases, because there are no pure intervals to check by as in some varieties of the meantone tem- perament, nor are there even fairly definite tempered intervals, such as the C E G* C of equal temperament, which also provide a good check. For the division of the monochord, these temper- aments could have been set down readily with the aid of loga- rithms, and they can be expressed in our modern cents with the greatest of ease. Computers who did not use logarithms were able to achieve comparable results by a linear division of the comma, but had less success if they ignored the schisma which separates the syntonic from the ditonic comma. In most of our tables we shall assume, for the sake of convenience, that the di- tonic comma has been given a correct geometric division, and shall assign cents values to the intervals accordingly. The leading exponents of this sophisticated sort of comma- juggling were Werckmeister, Neidhardt, and Marpurg.^4 Each has expressed the alteration of his fifths and thirds in the 12th part of a comma, which, strictly, should be the ditonic comma. ^^See Johann George Neidhardt, Gantzlich erschopfte, mathematische Ab- theilungen des diatonisch-chromatischen, temperirten Canonis Monochordi (Konigsberg and Leipzig, 1732), pp. 29 (the Fifth-Circles) and 38 (Third- Circles). See also F.W.Marpurg, Versuch liber die musikalische Temper- atur, p. 158, for the lettered temperaments A through L. All other refer- ences will be indicated in footnotes. 156 IRREGULAR SYSTEMS Since the ditonic comma is approximately 24 cents, this means that 2 cents will be taken as the unit of tempering. Thus the oc- tave would contain 600 parts, or thereabouts. This is an inter- esting forerunner of the cents representation. In evaluating this group of temperaments, it should be pointed out that there are two opposing points of view. Since we are likely to regard most highly those irregular systems that come closest to equal temperament, there will be in each subclass a temperament by Mar pur g or Neidhardt that wins the award be- cause in it the altered fifths are symmetrically arranged among the entire 12 fifths of the temperament. In these temperaments all keys are pretty much alike, whether nearer to C major or F* major. But the whole intent of having a "circulating" temperament, of having the octave "well tempered," was to have greater con- sonance in the keys most used than in those more remote. This is made very clear in the writings of Werckmeister and Neid- hardt. We should fail in our duty, therefore, did we not refer at the end of this chapter to temperaments we have discussed that satisfy this ideal of graduated dissonance. Both Werckmeister and Neidhardt had a proper respect for equal temperament also, but a fanatic like Tempelhof, ^ writing fifty to seventy-five years later, could say that equal temperament was the worst possible temperament because one scale must differ from another in its tuning! The simplest alteration of the Pythagorean tuning is to divide the comma into two equal parts. If the altered fifths are consec- utive, there will be a temperament somewhat like the modifica- tion of the meantone temperament shown at the beginning of this chapter. This is Bamberger's tuning,2** except that he has di- vided the syntonic comma arithmetically between the fifths D-A and A-E, thus getting a slightly smaller deviation than if he had divided the ditonic comma (see Table 132). 2**Georg Friedrich Tempelhof, Gedanken iiber die Temperatur des Herrn Kirnberger (Berlin and Leipzig, 1775), pp. 10, 18. 2°J. P. Kirnberger, Die Kunst des reinen Satzes in der Musik, Part I, p. 13. 157 TUNING AND TEMPERAMENT Table 132. Kirnberger's Temperament (1/2-Comma) Ratios 1 256:243 9:8 32:27 5:4 4:3 45:32 3:2 128:81 Names C° Dbo D° Eb° E"1 F° F*-x G° Abo Cents 0 90 204 294 386 498 590 702 792 Ratios 270:161 16:9 15:8 2:1 Names ifl Bbo B-1 C° Cents 895 996 1088 1200 M.D. 9.0; S.D. 9 7 Baron von Wiese's second tuning was exactly the same as Kirnberger's. He was so confirmed a Pythagorean that he called E~\ F#"\ andB"1 by the respective names Fb°, Gb°, and Cb°, each of which would have been 2 cents (the schisma) flatter than the corresponding syntonic value. However, von Wiese's first temperament^' actually dic^ divide the ditonic comma, making his F* the mean between Db and B (Table 133). His ratio for F#, 5760:4073, is an excellent approximation for the square root of one-half. Von Wiese's other three temperaments are respectable enough, for in them the tempered fifths are separated by a minor or major third. Since the deviation is the same for all three, we show No. 3 only (Table 134). Von Wiese has indicated it as ex- tending from B" to D#; but from the construction it extends from Gb to B, with the fifths Eb-Bb and B-Gb each tempered by half the ditonic comma. The best arrangement of the tempered fifths is for them to be separated by a semitone or a tritone. The lat- Table 133. Von Wiese's Temperament, No. 1 (1/2-Comma) Names C° Db° D° Eb° E° F° F*~2 G° Ab° A° Bb ° B° C° Cents 0 90 204 294 408 498 600 702 792 906 996 1110 1200 M.D. 10.0; S.D. 10.8 ^Christian LudwigGustav, Baron von Wiese, Klangeintheilungs-, Stimmungs- und Temperatur-Lehre (Dresden, 1793), p. 9 (No. 1) and p. 12 (No. 3). 158 IRREGULAR SYSTEMS Table 134. Von Wiese's Temperament, No. 3 (1/2-Comma) Names C° Db+5 D° Eb+5 E° F° Gb+"2 G° Ab+5 A0 Bb° B° C° Cents 0 102 204 306 408 498 600 702 804 906 996 1110 1200 M.D. 5.0; S.D. 6.6 ter arrangement occurs in Grammateus' temperament, shown earlier in this chapter, which is identical with Marpurg's K. Note that von Wiese's No. 3 is the same as Grammateus' except for Bb. Next in order would be temperaments in which the ditonic comma is divided among three thirds. Charles, Earl Stanhope^o advocated such a division, but indicated that the syntonic comma should be divided among the fifths G-D, D-A, and A-E. This left the schisma, 2 cents, to be divided among the four fifths from Bb to G*3, the other five fifths being pure. Thus the four black keys are only one cent sharper than if the tuning were purely Pytha- gorean. He might better have divided the ditonic comma among his first three fifths, and not have had the approximate fifths to worry over. With the ditonic comma divided among three con- secutive fifths, the mean deviation is 9.0, the standard deviation 9.7. Stanhope's own temperament (Table 135) is slightly better than this, just as Kirnberger's was better than von Wiese's No. 1, because the former divided the syntonic comma. Werckmeister^ has shown a temperament in which the comma is divided into three parts. It is, however, even less satisfactory than Stanhope's, because it contains five fifths flat by 1/3 comma, two fifths sharp by 1/3 comma, and only five perfect fifths (see Table 136) . This is the poorest of the three temperaments Werck- meister called "correct." Bendeler has used the 1/3 -comma tempering in two of his 28"Principles of the Science of Tuning Instruments with Fixed Tones," Philo- sophical Magazine, XXV (1806), 291-312. 29Andreas Werckmeister, Musicalische Temperatur (Frankfort and Leipzig, 1691), Plate. 159 TUNING AND TEMPERAMENT Table 135. Stanhope's Temperament (1/3-Comma) Lengths 120 113.84 107.1 101.19 96 90 Names C° Dbo i D" 3 Ebo E"1 Fu Cents 0 91 197 295 386 498 Lengths 80 75.89 71.7 67.5 64 60 Names G° Abo 2 A"" Bbu B-1 C° Cents 702 793 892 996 1088 1200 85.38 nbo 589 M.D. 7.8; S.D. 8.7 three organ temperaments.^ In the first, the tempering is shared by the fifths C-G, G-D, and B-F* (Table 137) . Since these are not all consecutive fifths in the circle of fifths, his deviation is considerably less than Stanhope's. In Bendeler's second temperament (Table 138), the comma is divided among the three fifths C-G, D-A, and F*-C#. Since the Table 136. Werckmeister's Correct Temperament, No. 2 (1/3-Comma) Names C° C#~3D ho f'-'g" ,#■ .b+-i E- e 3 F° F" 'G3 G" 3A 3 B"'3 B C° Cents 0 82 196 294 392 498 588 694 786 890 1004 1086 1200 M.D. 9.2; S.D. 10.7 Table 137. Bendeler's Temperament, No. 1 (1/3-Comma) J-i Eb° E" F#_1 G" G#-iA-3 Bbo Names C° C" *D 3 E"" E 3 F° F' * G 3 G" *A 3 B"u B 3 C° Cents 0 90 188 294 392 498 588 694 792 890 996 1094 1200 M.D. 5.0; S.D. 5.8 Table 138. Bendeler's Temperament, No. 2 (1/3-Comma) bo Names C° C^D"3 E*30 E F° Fff 3 G G A B" B 3 C° Cents 0 90 196 294 392 498 596 694 792 890 996 1094 1200 M.D. 4.0; S.D. 4.8 30 J. P. Bendeler, Organopoeia (2nd ed.; Frankfurt and Leipzig, 1739), p. 40 (No. 1) and p. 42 (No. 2). 160 IRREGULAR SYSTEMS fifths are more widely separated than before, the deviation is less than for No. 1. The best arrangement of the three tempered fifths is to have them separated by major thirds, as in Marpurg's I, where E and G* are the same pitches as in equal temperament (see Table 139). The most famous of Werckmeister's irregular divisions has the comma divided equally among the four fifths C-G, G-D, D-A, and B-F#.31 since three of these fifths are consecutive, the de- viation is comparatively large (see Table 140). This is the only temperament that Sorge has ascribed to Werckmeister. The same division was accepted by Marpurg, and a modern acousti- cian, Karl Erich Schumann, ** has followed suit, without men- tioning any secondary source. In Werckmeister's third "correct" temperament (Table 141), five fifths (D-A, A-E, F#-C#, C#-G#, and F-C) are flattened by 1/4 comma, and one fifth, G*-D#, is raised by the same amount. Thanks, however, to the more nearly symmetrical arrangement of the tempered fifths, the deviation is slightly less than for his first temperamento In his third temperament, Bendeler,^^ unhampered by a sharp fifth and with a fairly symmetrical arrangement of the four flat- tened fifths (C-G, G-D, E-B, G*-D#), succeeded in achieving a very good division (Table 142). But, as usual, the best temperament for a particular division of the comma is completely symmetrical, and so Neidhardt, in his fourth Fifth-Circle (Table 143), gave Eb, F#, and A the same pitches they would have in equal temperament. (Marpurg's H is identical with this.) When the comma is divided into five parts and the tempered fifths are arranged as symmetrically as possible, the deviation begins to approach the vanishing point. (Paradoxically, this de- viation is lower than for a wholly symmetrical arrangement of six fifths tempered by 1/6 comma, shown in the next section.) In 31Werckmeister (see Table 1-iO), loc. cit. 32Akustik (Breslau, 1925), p. 31. 33Organopoeia, p. 42. 161 TUNING AND TEMPERAMENT Table 139. Marpurg's Temperament I (1/3-Comma) Names C° Cf^D° E°+* E~3 F+3 F*~3 G° G#" A0 Bb+3 B"5 C° Cents 0 106 204 302 400 506 604 702 800 906 1004 1102 1200 M.D. 3.0; S.D. 3.5 Table 140. Werckmeister's Correct Temperament, No. 1 (1/4-Comma) Names C° C#_1D~2 £b° E~4 F° F^G'4" G#~l A~4 Bb° B~4 C° Cents 0 90 192 294 390 498 588 696 792 888 996 1092 1200 M.D. 6.0; S.D. 7.5 Table 141. Werckmeister's Correct Temperament, No. 3 (1/4-Comma) Names C° C^D0 E 4 E 2 F * ¥w 2G° Gff A 4 B 4 B 2 C° Cents 0 96 204 300 396 504 600 702 792 900 1002 1098 1200 M.D. 5.0; S.D. 5.7 Table 142. Bendeler's Temperament, No. 3 (1/4-Comma) Names C° Cff *D 2 E E 2 F° Fff' 4 G 4 G* 4 A 2 Bu B~4 C° Cents 0 96 192 294 396 498 594 696 798 894 996 1092 1200 M.D. 3.3; S.D. 3.7 Table 143. Neidhardt's Fifth-Circle, No. 4 (1/4-Comma) U-2 _J h+-1 -i #--! -i H-2 _i ho -i Names C° C 4D 4 E 4E 2 F° F 2G 4 G 4A 4 B B 2 C° Cents 0 96 198 300 396 498 600 696 798 900 996 1098 1200 M.D. 2.7; S.D. 2.8 Table 144. Marpurg's Temperament G (1/5- Comma) Names C° Cff 5D 5 E 5E 5 F° F* 5 G° G * * A" 5 B 5 B~~5 C° Cents 0 100 199 299 398 498 602 702 802 901 1001 1100 1200 M.D. .7; S.D. 1.3 162 IRREGULAR SYSTEMS Marpurg'sG (Table 144) this near-symmetrical division is made. Marpurg called the amount of tempering 2^-/12 = 5/24 comma, which would be 5 cents, slightly larger than 1/5 comma or 4.8 cents. Although the difference between the two is wholly negli- gible, the latter amount of tempering has been used in making the table, with the values rounded off to even cents. The 1/6-comma temperament is recommended by Thomas Young, 34 as a simpler method than the irregular temperament described later in this chapter. In his own words, "In practice, nearly the same effect may be very simply produced, by tuning C to F, Bb, Eb, G#, C#, F# six perfect fourths; and C, G, D, A, E, B, F# six equally imperfect fifths." In other words, he had six consecutive fifths tempered by l/6ditonic comma (see Table 145). As a practical tuning method, this would not be difficult, Table 145. Young's Temperament No. 2 (1/6-Comma) Names C° Dbo D_3 Ebo E~3 F° Gbo G"* Ab° A~2 Bb° B~« C° Cents 0 90 196 294 392 498 588 698 792 894 996 1090 1200 M.D. 6.0; S.D. 6.8 and it certainly does differentiate between near and remote keys. This is the tuning of the Out-Of-Tune Piano, the sort of tuning into which a piano originally in equal temperament might fall if played upon by a beginner .35 Young's key of G is the best, that of Db the worst. If he had commenced his set of tempered fifths with F instead of C, the key of C would have been best. In Neidhardt's second Fifth-Circle (Table 146), all the fifths Table 146. Neidhardt's Fifth-Circle, No. 2 (1/6-Comma) Names C° Cff 2D° E 6 E 3 F « F 3G~? G 3A""e B 3 B "2 C° Cents 0 102 204 298 400 502 604 698 800 902 1004 1098 1200 M.D. 3.0; S.D. 3.4 34" Outlines of Experiments and Inquiries Respecting Sound and Light," Philo- sophical Transactions, XC (1800), 145. 35j. Murray Barbour, "Bach and The Art of Temperament," Musical Quar- terly, XXXm (1947), 66 f, 89. 163 TUNING AND TEMPERAMENT are altered by 1/6 comma, nine being lowered and three raised. Since the arrangement is completely symmetrical, the deviation is low. Of course, a symmetrical arrangement of fifths alternately pure and lowered by 1/6 comma comes closest to equal temper- ament. Both Neidhardt (Third Fifth-Circle) and Marpurg (F) have presented this temperament (Table 147). Observe that in it the consecutive notes are alternately the same as in equal tem- perament and 2 cents higher, so that the mean deviation and standard deviation both are equal to 2.0. More elaborate patterns of semitones either 2 cents higher or lower than in equal tem- perament could be obtained by having two pure fifths alternate Table 147. Neidhardt's Fifth-Circle, No. 3 (1/6-Comma) a-i --1 h+i --1 +-1 u-1 u-2 -1 hj- ' -J Names C° C* 2D 6 E 3E 3 F 6 F 2G° Gff 3A 6 B 6 B 3 C° Cents 0 102 200 302 400 502 600 702 800 902 1000 1102 1200 M.D. 2.0; S.D. 2.0 with two tempered fifths, or by having three pure fifths similarly alternate with three tempered ones. Bermudo,36 wno had also formed equal semitones on the lute by the method of Grammateus, made a real contribution to tuning theory in a chapter "concerning the seven-stringed vihuela upon which all the semitones can be played." This was a method in- tended for experienced players. His account of the division is necessarily lengthy and need not be given as a whole. G is the fundamental, and there are 10 frets, thus making no provision for F# on this string. The notes from E^ to G inclusive are formed by a succession of pure fifths. The thirds G-B and A-C* are each 2/3 syntonic comma sharper than pure thirds. The tone G-A is 1/6 comma less than a major tone. Then D and E form pure fourths with A and B, respectively, and G* is a fourth below C*. The geometry, which consists of linear divisions only, is easy to follow, especially with the aid of Bermudo's monochord dia- 3" J. Bermudo, Declaracion de instrumentos musicales (Ossuna, 1555), Book 4, Chap. 86. 164 IRREGULAR SYSTEMS gram (see Figure I). In ratios, as will be seen in Table 148, it becomes quite complicated, and, if these ratios were to be rep- r a b C D b f c b 1 v 1 ■ — !-■-, \ -i-H 1 — i — i i : ; , Fig. I. Bermudo's Method for Placing Frets on the Vihuela Reproduced by courtesy of the Library of Congress Table 148. Bermudo's Vihuela Temperament (1/6- 1/2-Comma) Names G° G#" A" Bb° B" c° Ratios 1 492075:463684 540:481 32:27 1215:964 4:3 Cents 0 102.9 200.3 294.2 400.6 498.0 Names C# i i 2 D"6 Eb° i F° k i G° Ratios 164025:115921 720:481 128:81 405:241 16:9 [218700:11592l] 2:1 Cents 600.9 698.3 792.1 898.6 996.1 [1098.9J 1200 M.D. 3.9; S.D. 4.2 resented by least integers, as was done in many of these systems, the fundamental note G would have to be 62,985,600! Let us as- sume that F*, the unused 11th fret, is a pure fourth above C*. The reason Bermudo's system is presented in connection with the use of fifths tempered by 1/6 comma is that that is precisely what he has. If the temperament of successive fifths is examined, it will be seen that the fifths on G, A, and B are each tempered by 1/6 comma, eight fifths are pure, and the usual wolf fifth, G#- E", is 1/2 comma flat. (It really should not be called a wolf fifth, since it is flat, not sharp, and the usual poor thirds of the mean- tone temperaments, on B through G#, are the best of all!) This is the first time, so far as is known, that any writer had suggested the formation of notes used in equal temperament by the proper division of the comma for those notes. Of course he was making an arithmetical division of the syntonic comma, and thus had small errors. But so did the late seventeenth and most of the eighteenth century comma-splitters from Werckmeister to Kirnberger and Stanhope. Bermudo's three tempered fifths 165 TUNING AND TEMPERAMENT are as symmetrically arranged as in the Neidhardt-Marpurg sys- tem shown before this. It is too bad he did not continue his proc- ess by tempering D# by 2/3 comma and E# by 5/6 comma. Then he would not have had the half-comma error concentrated on a single fifth, nor a Pythagorean third on E^. But this method of Bermudo is worthy of our respect as a very early approach to equal temperament, somewhat difficult, but not impracticable for a skilled performer to use. Werckmeister is the only later writer to temper his fifths by the 7th part of a comma, perhaps following the example of Zar- lino's 2/7-comma variety of meantone temperament.^' But his Septenarium temperament is a rather eccentric thing. In it the fifths C-G, BO-F, and B-F# are 1/7 comma flat; F#-C# is 2/7 comma flat; G-D is 4/7 comma flat; D-A and G*-D* are 1/7 comma sharp; the remaining five fifths are pure. (The cents Table 149. Werckmeister's Septenarium Temperament (1/7-Comma) Lengths 196 186 176 165 156 147 139 131 124 117 110 104 98 5 14 11 5 X it 4 t-» 4 Names C° Cri D~7 ED+"7 E_"7 F° F#_7G~7 G A~~7 B 7 B~7 C° Cents 0 91 186 298 395 498 595 698 793 893 1000 1097 1200 M.D. 4.7; S.D. 5.6 Table 150. Symmetric Septenarium Temperament (1/7-Comma) M-4 _J h+ 2 _2 +J U 4 _ 1 U 5 _2 h,J 3 Names C° C" 7D 7 E 7E "7 F 7 Fff""7G 7 Gff_7A 7 B 7 B" 7 C° Cents 0 100 201 301 401 501 598 699 799 899 999 1100 1200 M.D. 0.5; S.D. 1.0 values have been worked out from Werckmeister' s string-lengths , and are slightly inaccurate.) For the sake of a comparison with Werckmeister's temper- ament, a symmetric version of the 1/7 -comma temperament is shown in Table 150. It is even nearer equal temperament than Marpurg'sG, which had a symmetric distribution of the fifth part of the comma. 3'A. Werckmeister, Musicalische Temperatur, Plate. 166 IRREGULAR SYSTEMS Next we have a large group of temperaments in which some fifths are tempered by 1/5 comma and others by 1/12 comma, while the remaining fifths are pure. Since 1/12 comma is the temperament of the fifth of equal temperament, there will be as many pure fifths as there are fifths tempered by 1/5 comma. This group of temperaments might be considered, therefore, as variants of the previously described temperaments in which there are six pure fifths and six fifths tempered by 1/6 comma., Neidhardt was the great inventor of temperaments in which the comma was divided into both 6 parts and 12 parts. 38 All three "circulating" temperaments fall into this group. They hap- pen to be among the poorest ofthis type that he or the other the- orists have evolved — that is, when compared with equal temper- ament. But we shall see that they do satisfy Neidhardt' s purpose in creating them. The first circulating temperament (Table 151) has four fifths in each group — pure, tempered by 1/12 comma, and by 1/6 comma. Since four consecutive fifths in it are tem- pered by 1/6 comma, it may be considered a variant of the 1/5- comma meantone temperament. The first of Thomas Young's pair of temperaments is very like the Neidhardt temperament shown in Table 151. ^ Young said, "It appears to me, that every purpose maybe answered, by making C:E too sharp by a quarter of a comma, which will not offend the nicest ear; E:G# and At>:C equal; F*:A* too sharp by a comma; and the major thirds of all the intermediate keys more or less perfect as they approach more or less to C in the order of modulation." Table 151. Neidhardt's Circulating Temperament, No. 1 (1/12-, 1/6-Comma) _ J h° - -3 A 2 B B 4 C° 894 996 1092 1200 mes C° C#""6D~"3 Eb° -- #--5 --1 U- E a F° Fff 6 G 6 G* nts 0 94 196 296 392 498 592 698 796 M.D. 4.0; S.D. 4.6 "*°J. G. Neidhardt, Sectio canonis harmonici, pp. 16-18. 39Thomas Young, in Philosophical Transactions, XC (1800), 145 f. 167 TUNING AND TEMPERAMENT Young accomplished the first result by tempering the fifths on C, G, D, and A by 3/16 syntonic comma, and the other results by tempering the fifths on F, B", E, and B by approximately 1/12 syntonic comma, and leaving the other four fifths pure. The total amount of tempering would be 13/12 syntonic comma, this being sufficiently close to the ratio of the ditonic to the syntonic comma. Young has given numbers for his monochord, and they agree well with his theory. He has made a mistake, however, in calculating the length for ED (83810), which was intended as a pure fourth be- low gC The corrected length is given in Table 152. Table 152. Young's Temperament No. 1 (1/12- , 3/1C -Comma ) Lengths Names 100000 C° 94723 C 12 89304 3 D « 84197 79752 3 e""4 74921 i Y+ i2 71041 #-- Y i2 Cents 0 94 196 298 392 500 592 Lengths Names 66822 3 G 16 63148 G 12 59676 9 A"16 56131 53224 5 B"6 50000 C° Cents 698 796 894 1000 1092 1200 M.D. 5.3; S.D. 5.9 Now 3/16 syntonic comma is an awkward interval to deal with. If, instead, we take 1/6 ditonic comma as the temperament of Young's four diatonic fifths, and 1/12 ditonic comma for his sec- ond group of fifths, his monochord will be precisely of theNeid- hardt type. The differences from the monochord he did give are so small that the cents values do not differ. The arrangement of his second group of fifths is slightly different from Neidhardt's, and this accounts for the difference in deviation. Mercadier's temperament (Table 153) closely resembles Young's, even to the total amount of tempering — 13/12 syntonic comma. ^ He directed that the fifths from C to E should be flat by 1/6 syntonic comma, and those from E to G* flat by 1/12 comma. Then G# is taken as AD, the next three fifths are to be just, and the fifth F-C then turns out to be about 1/12 comma flat. 40Antoine Suremain-Missery, Theorie acoustico-musicale (Paris, 1793), p. 256. 168 IRREGULAR SYSTEMS Table 153. Mercadier's Temperament (1/12-, 1/6-Comma) Names C°C* 12D 3 E 12E 3 F 12 Fff 6 G 6 G* J A "2 B 12B~4 C° Cents 0 94 197 296 394 500 594 698 794 895 998 1094 1200 M.D. 4.1; S.D. 4.5 Table 154. Marpurg's Temperament D (1/12-, 1/6-Comma) #_ 2 _i h+i --£- '{- J ! # 3 1 l, l 1 Names C° C 3D 4E 4E 12 F° F 2 G 8 G 4A~4 B ~2B" C° Cents 0 98 198 300 398 498 600 698 798 900 998 1098 1200 M.D. 1.3; S.D. 1.6 Table 155. Neidhardt's Circulating Temperament, No. 2 (1/12-, 1/6-Comma) u-2 _i h+- -- +- #-^ --i ¥-- -- b+- -- Names C° Cff 4D 3 E 6E 12 F 12 F 3G 6 G 6A 2 B 6 B 12 C° Cents 0 96 196 298 394 500 596 698 796 894 1000 1096 1200 M.D. 3.3; S.D. 3.7 Table 156. Neidhardt's Circulating Temperament, No. 3 (1/12-, 1/6-Comma) #_3 _i h+- -— #--2 --1 #--5 --1 b+-i- -- Names C° C 4D 3E 6E 12 F° Fff 3 G 4 Gff 6 A 2 B 12 B 12 C° Cents 0 96 196 298 394 498 596 696 796 894 998 1096 1200 M.D. 2.7; S.D. 2.9 Table 157. Neidhardt's Third-Circle, No. 4 (1/12-, 1/6-Comma) Names C° Cff 4D 3 E E 2 F° Fff 3G 6 Gff 6A 2 B 6 B 3 C° Cents 0 96 196 296 396 498 596 698 796 894 1000 1094 1200, M.D. 2.7; S.D. 3.4 169 TUNING AND TEMPERAMENT As usual, Marpurg has presented the symmetric version (Ta- ble 154) of the above temperaments. It has negligible deviations. In the second and third of Neidhardt's "circulating" temper- aments, six fifths are tempered by 1/12 comma, and three each are pure or are tempered by 1/6 comma. These two tempera- ments (Tables 155 and 156) are quite similar, both containing three consecutive fifths tempered by 1/6 comma. Thus they pos- sibly represent the extreme case of modification of the 1/6-comma meantone temperament. Number 3 has a shade greater sym- metry and hence smaller deviation. Temperaments 4 and 3 of Neidhardt's Third-Circle have de- viations very similar to those of the temperaments shown in Ta- bles 155 and 156. In fact, their mean deviations are equal re- spectively to those of No. 2 and No. 3 in these tables, but their standard deviations are higher because they contain some sharp fifths. In No. 4 (Table 157), there are three fifths tempered by 1/12 comma and five by 1/6 comma; three fifths are pure, and one is 1/12 comma sharp. In No. 3 (Table 158), four fifths are 1/12 comma flat, six are 1/6 comma flat, and two are 1/6 comma sharp. (The same tempered fifths as in No. 3 appear in our hy- pothetical version of Schlick's temperament, but differently ar- ranged.) Once again Marpurg has given the symmetric version of Neid- hardt's temperaments, specifically of the second and third "cir- culating" temperaments. Logically we show next two temperaments (Tables 160 and 161) in which eight fifths are flat by 1/12 comma and two by 1/6 comma, while two are pure. Such a temperament is the fifth of Neidhardt's Third-Circle. The temperament shown in Table 160 comes so close to equal temperament that in practice it could not be improved upon. But the canny Marpurg has halved its deviation by using greater sym- metry (see Table 161). Another temperament of Neidhardt has the same deviations as those of his fifth Third-Circle (Table 160). This is the fifth tem- perament in his Fifth-Circle (Table 162), in which six fifths are 170 IRREGULAR SYSTEMS Table 158. Neidhardt's Third-Circle, No. 3 (1/12-, 1/6-Comma) Names C° C"*D""^ E^ e"^¥+T2 ¥f~T2G~~6 G*""6A"T2 Bb+* b" C° Cents 0 96 196 296 394 500 598 698 796 896 1002 1092 1200 MD. 3.3; S.D. 4.7 Table 159. Marpurg's Temperament C (1/12-, 1/6-Comma) Names C C#"3D--s Ebn E"i Fo FMG-*GHA-* Bb+° B~& C° Cents 0 98 200 300 400 498 600 700 800 898 1000 1100 1200 M.D. 1.0; S.D. 1.4 Table 160. Neidhardt's Third-Circle, No. 5 (1/12-, 1/6-Comma) U — 1 h+ 1 — 4- 1 U-— -— U 2 -- ho-1 -1 Names C° C ff~12D~« E * E »F • Fff 12G 12Gff~~3A 4 B " B_ 2 C° Cents 0 100 200 300 398 502 598 700 800 900 1000 1098 1200 M.D. 1.3; S.D. 2.0 Table 161. Marpurg's Temperament B (1/12- , 1/6-Comma) ■# 2 i hx1 1 -uJ- #-i -i #-.3 -i h + i - — Names C° C 3D~4 E° +4 E 3 F+12Fff 2G 6 G* 4A 3 B° 6 B 12 C° Cents 0 98 198 298 400 500 600 698 798 898 1000 1100 1200 M.D. .7; S.D. 1.1 Table 162. Neidhardt's Fifth-Circle, No. 5 (1/12- , 1/6-Comma) Names C° Cff 12D 6 E 6E * F 6 F* 2G 12G# 3A 3 B 4 B~3 C° Cents 0 100 200 298 402 502 600 700 800 898 1002 1102 1200 M.D. 1.3; S.D. 2.0 171 TUNING AND TEMPERAMENT flat by 1/12 comma and four by 1/6 comma, while two are sharp by 1/12 comma. The remaining temperaments in this group come from Mar- purg. The first (Table 163) of his temperaments in which some fifths are sharp contains six fifths flat by 1/6 comma, and three fifths each flat or sharp by 1/12 comma. ***■ Obviously, this is a variant upon the temperament in which six fifths are flat by 1/6 comma, the other six pure. The mean deviation, 2.0, is the same, but, as expected, the standard deviation is higher here. Other possible variants would contain, in addition to the six fifths tempered by 1/6 comma, two fifths each flat or sharp by 1/12 comma or pure; or four pure fifths and one each flat or sharp by 1/12 comma. The second temperament (Table 164) in this other set byMar- purg has fifths that do not differ greatly from those in the pre- vious temperament. Here the six fifths are tempered by the un- usual amount of 5/24 comma (shown as the same fraction that did duty as 1/5 in his Temperament G, but really 5/24 this time), and three each are pure or 1/12 comma sharp. In Marpurg's Temperament A (Table 165), ten fifths are flat by 1/12 comma, and one each is pure or 1/6 comma flat. This is as far as one can go in this direction, for the next step would be to have twelve fifths flat by 1/12 comma — that is, equal tem- perament. The other limit for this sequence of temperaments by Mar- purg is his own Temperament F, already shown as Neidhardt's Fifth-Circle, No. 3 (Table 147). In it there are no fifths tem- pered by 1/12 comma, and six fifths each pure or flat by 1/6 comma. Just before it in the set comes Temperament E (Ta- ble 166), which has two fifths flat by 1/12 comma, and five fifths each pure or flat by 1/6 comma. Marpurg's Temperament E, shown in Table 166, has the least deviation of the five temperaments in the set. Note the devia- tions again: A, 1.7,1.8; B, 0.7,1.1; C, 1.0, 1.4; D, 1.3, 1.6; E, 0.3,0.8. From the table for E it is easy to see why its deviation is low: there are seven consecutive notes with cents values end- ^Marpurg, Versuch liber die musikalische Temperatur, p. 163. 172 IRREGULAR SYSTEMS Table 163. Marpurg's Temperament, No. 1 (1/12- , 1/6-Comma) Names C° C 2D 12 E 12E 3 F 6 F 12G 12G 3A 6 B 4 B 4 C° Cents 0 102 202 304 400 502 602 704 800 902 1002 1104 1200 M.D. 2.0; S.D. 2.4 Table 164. Marpurg's Temperament, No. 2 (1/12-, 5/24-Comma) a_3 _± b+i _i __1 #_3 __5 #_2 __5. b_JL _i? Names C° C# 4D 12E 8 E 3 F 12 Fw 4G 24 G 3A 12 B 12B 24 C° Cents 0 96 194 297 400 496 594 697 800 896 994 1097 1200 M.D. 3.0; S.D. 3.1 Table 165. Marpurg's Temperament A (1/12- , 1/6-Comma) Names C° C# 2 D 6 E 4E 4 F 12 Fff 12G 12Gff 12A 6 B • B 3 C° Cents 0 102 200 300 402 500 602 700 802 902 1000 1102 1200 M.D. 1.7; S.D. 1.8 Table 166. Marpurg's Temperament E (1/12-, 1/6-Comma) Names C° C#_T2D" 12Eb+-5E~4 F+1 F#"^G_^G#~ 3 A-^ B°+* B_" C° Cents 0 100 202 302 402 502 602 700 800 900 1000 1100 1200 M.D. 0.3; S.n0.8 Table 167. Neidhardt's Fifth-Circle, No. 6 (1/12-, 1/4-Comma) M-— _J K+i -- i U -i -_L #5,1 hj^ 1 7 Names C° Cff 12D 3 E 4E 3 F 12 F 2G 12G 6A 4 B 6 B-" C° Cents 0 100 196 300 400 496 600 700 796 900 1000 1096 1200 M.D. 2.7; S.D. 3.3 Table 168. Neidhardt's Fifth-Circle, No. 9 (1/12-, 1/4-Comma) ji.i _i h+i -i ik 2 J. a 2 > hn ± Names C° C# ^ 3E 4 E 3 F° Fff 3 G 12 G 3A"3 B B" 12 C° Cents 0 98 196 300 400 498 596 700 800 898 996 1100 1200 M.D. 2.0; S.D. 2.4 173 TUNING AND TEMPERAMENT ing in 00, and five ending in 02. Therefore the total deviation will be only 4 cents, or a mean deviation of 0.3. In the other temperaments of the set, some values end in 00 and others in 98 or 02. But in no other temperament do all the 00' s come together as they do in E. Therefore the deviation is higher in the other temperamentSo But it need not have been higher. If in A the pure fifth is followed directly by the fifth flat by 1/6 comma, there will be only one note with an 02 ending, and eleven notes with 00. The fifths in B, C, and D can be so arranged that there will be respectively 2, 3, and 4 consecutive notes with an 02 (or 98) end- ing, the other endings being 00. Thus the minimum deviation (M.D. 0.3; S.D. 0.8) will be the same for all five temperaments, but this will not always involve the most symmetrical version of the fifths. The remaining nine temperaments are all by Neidhardt, and each contains some fifths tempered by 1/4 comma. His Fifth- Circle, No. 6 (Table 167) has four fifths each flat by 1/4 comma or flat or sharp by 1/12 comma. His arrangement is symmetric. In Temperament No. 9 of this same set (Table 168), Neidhardt has three fifths flat by 1/4 comma, three flat by 1/12 comma, and six pure. Again the arrangement is symmetric. The deviation is lower than for the previous temperament. In Temperaments 7 and 10 (Table 169 and 170), Neidhardt di- vides the comma into 4 or 6 parts. No. 7 is especially compli- cated, having eight fifths flat by 1/6 comma and two sharp by 1/6 comma, and one each flat or sharp by 1/4 comma. It would be difficult to construct a symmetric arrangement from such an ar- ray, and Neidhardt has not attempted to do so. Table 169. Neidhardt's Fifth-Circle, No. 7 (1/6-, 1/4-Comma) Names C° C# 6D 12E 6 E 3 F 6 F* 3 G 4 G* 3A 12B 3 2 C° Cents 0 94 194 298 400 494 596 696 800 892 996 1098 1200 M.D. 3.3; S.D. 4.1 174 IRREGULAR SYSTEMS Table 170. Neidhardt's Fifth-Circle, No. 10 (1/6-, 1/4-Comma) U-- -- h-l-i -2 a 2 1 i, 5 1 Un 2 Names C° C* 6D 4 E 6 E 3 F° F f 3G * G « A~~2 B B~3 C° Cents 0 94 198 298 392 498 596 696 796 894 996 1094 1200 M.D. 3.0; S.D. 3.8 Table 171. Neidhardt's Fifth-Circle, No. 10, Idealized ji_J _i h+i --Z- u-J. _i u-- --5. hn _-l Names C° C# 6D 4 E 6 E 12 F° F* 12G 4 G* 6 A 12B B 12 C° Cents 0 94 198 298 398 498 598 696 796 896 996 1096 1200 M.D. 1.3; S.D. 2.4 Table 172. Neidhardt's Sample Temperament, No. 2 (1/12-, 1/6-, 1/4-Comma) Names C° Cw D l2 E E 12 F 12 F 12G 6 G A 3 B 12 B 12 C° Cents 0 90 194 294 386 496 590 698 792 890 994 1088 1200 M.D. 6.3; S.D. 7.2 Temperament 10 (Table 170) is considerably simpler, with two fifths flat by 1/4 comma, three by 1/6 comma, and the remaining seven pure. The deviation is slightly lower than for No. 7. But in No. 10 also the arrangement is far from symmetric. Let us see what would result from an approach to symmetry. Al- though the deviation is about halved in Table 171, it is possible that, as in the alphabetically named temperaments by Marpurg, the least deviation for all four of these Neidhardt temperaments will not occur with the most nearly symmetric arrangement of the fifths. In the remaining five temperaments in this group, Neidhardt has tempered his fifths by 1/4, 1/5, and 1/12 comma. His second and third "sample" temperaments (the first was just intonation) have relatively high deviations. 42 No. 2 (Table 172) has three fifths flat by 1/4 comma, one by 1/5, two by 1/12, five pure, and one 1/12 comma sharp. 42 J. Go Neidhardt, Gantzlich erschopfte mathematische Abtheilung, p. 34. 175 TUNING AND TEMPERAMENT Neidhardt's No. 3 (Table 173) is somewhat less erratic than No. 2, with six pure fifths, and two each flat by 1/4, 1/6, or 1/12 comma. It also has a lower deviation than No. 2. Rather similar to the above sample temperaments is his Third-Circle, No. 1 (Table 174), in which five fifths are pure, two flat by 1/4 comma, one by 1/6, and four by 1/2. Two temperaments from the Fifth-Circle are considerably better than the three just mentioned. In No. 11 (Table 175) there are no pure fifths; two fifths are flat by 1/4 comma, two by 1/6, five by 1/12, while three are 1/12 comma sharp. Table 173. Neidhardt's Sample Temperament, No. 3 (1/12-, 1/6-, 1/4-Comma) » U _i ho— i. 5 U 2 x H H -1 ho 5 Names C°Cff"12D 3 E 12E-'5 F° Fff "6 G"5 Gff " 12A" 12B B 6 C° Cents 0 92 196 296 388 498 592 698 794 892 996 1090 1200 M.D. 5.7; S.D. 6.4 Table 174. Neidhardt's Third-Circle, No. 1 (1/12-, 1/6-, 1/4-Comma) Names C° C 6D 4 E 12E 4 F° F 8G 12 G 12A 2 B 12 B 4 C° Cents 0 94 198 296 390 498 592 700 794 894 998 1092 1200 M.D. 5.3; S.D. 5.9 Table 175. Neidhardt's Fifth-Circle, No. 11 (1/12- , 1/6-, 1/4-Comma) u-2 -J h+— -— +— 4--1 -X u-3 _J ho -- Names C° C* 4D 4 E 12E 12 F 12 Fff 12G 12Gff 3A 2 B B 2 C° Cents 0 96 198 296 394 500 598 700 800 894 996 1098 1200 M.D. 2.7; S.D. 3.2 Table 176. Neidhardt's Fifth-Circle, No. 12 (1/12- , 1/6- , 1/4-Comma) Names C° C^ 12D~4 E 4 E_2 F° Fff"2 G" 12G 4 A~4 B B"2 C° Cents 0 100 198 300 396 498 600 700 798 900 996 1098 1200 M.D. 2.0; S.D. 2.3 176 IRREGULAR SYSTEMS In No. 12 (Table 176) there are six pure fifths, and two each flat by 1/4, 1/6, or 1/12 comma. This has precisely the same number of each size of fifth as the third sample temperament, in which the deviation was almost three times as great. The reason, of course, is to be found in the symmetry of No. 12. Metius' System At the beginning of this chapter it was said that "by making the bounds sufficiently elastic" all irregular systems could be classified. That statement is severely tested by the final tuning method listed in this part of the chapter, one presented by Ad- rian Metius. It was not possible to see Metius' own description, and Nierop, who gave the monochord, seemed to have been puz- zled by it himself. * Nierop has shown this monochord in two forms, one from 1000 to 500 and the other from 11520 to 5760, with E the fundamental. It is evident from the context that the second monochord was given simply to show how its lengths have been increased or diminished by arithmetic divisions of the syn- tonic comma, and that only the first table comes from Metius directly., By using Metius' lengths, it is possible to reconstruct the tempering, indicated by the exponents. Apparently there is only one pure fifth, C-G. The fifths on BD and A are 1/12 comma flat, those on F and E 1/6 comma flat, on B and C* 1/2 comma flat, and on G 3/4 comma flat! The fifths on D and F* are 1/6 comma sharp, that on D* 1/3 comma sharp, and on G* 1/2 comma sharp. Metius' system does not seem to follow any known system of temperament or modification thereof. Specifically, it does not resemble the meantone temperament, for only the thirds on B*3 and E are pure, the other thirds varying in size up to 417 cents for GD-BD and 419 cents for AD-C. But there is no pattern ap- parent in the alterations, no planned shift from good to poor keys. The fifth G-D, 3/4 comma flat, is almost as unsatisfactory as this same fifth would be in just intonation. There is no good rea- son for both of the fifths B-F# and C#-G# to be half a comma flat 4**D,, r. van Nierop, Wis-konstige Musyka (Amsterdam, 1650). The reference here is to page 60 of the 2nd edition (1659). 177 TUNING AND TEMPERAMENT and then to have the fifth G#-D# half a comma sharp. All in all, Metius has been just about as erratic as he could be. And yet the system, despite its irregularities, is much bet- ter than the ordinary 1/4-comma meantone temperament and is slightly better than the Pythagorean or the 1/6 -comma mean- tone. That much we must grudgingly admit. Metius' tempera- ment contains eight different sizes of fifth. But that is not much less regular than many of the fairly good temperaments we have shown that had four sizes of fifth, while Werckmeister's Septe- nariumand Neidhardt's second sample temperament had five dif- ferent sizes. And so let us label it highly irregular, but not really unworkable. "Good" Temperaments With Metius' enigmatic temperament we have described the last of our irregular tuning systems, and are in a position to try to formulate a judgment upon them. It is easy to see how the modifications of the Pythagorean, just, or meantone system by the halving of tones, as in the systems of Grammateus, Ganassi, or Artusi, would make these systems much more like equal tem- perament. But it is more difficult to see what Werckmeister, Neidhardt, and Marpurg were driving at in their multifarious at- tempts to distribute the comma unequally among the twelve fifths. If, as was pointed out at the beginning of an earlier section of this chapter, our ideal is equal temperament, we shall praise highly some of the beautifully symmetric systems of Marpurg and Neidhardt. But the trouble is that they are too good! The deviations for most of them are lower than for a piano allegedly tuned in equal temperament by the most skillful tuner. In some cases these temperaments might have been successfully trans- ferred from paper to practice by calculating the number of beats for each of the beating fifths. Since most of the fifths were to be tuned pure, such a method might have been easier than that pur- sued today. These same temperaments might have been reduced to distances on a monochord with slightly greater ease than equal temperament could be, although it must be remembered that us- ually even the most innocent set of cents values needs logarith- 178 IRREGULAR SYSTEMS mic computation before yielding figures for a monochord. But it will be safe to dismiss most of these oversubtle systems as use- less, even for the age when they were devised. What do we have left? It will be of interest to consider which of his twenty systems Neidhardt considered the best. In the Sec- tio canonis he had said, "In my opinion, the first [of the circu- lating temperaments] is, for the most part, suitable for a vil- lage, the second for a town, the third for a city, and the fourth for the court." The fourth was equal temperament; the mean deviations of the other temperaments had been 4.0, 3.3, and 2.7 cents, respectively. In the much later Mathematische Abtheilungen Neidhardt pre- sented eighteen different irregular temperaments, together with just intonation and equal temperament. He then attempted to choose the best of these twenty tunings. He chose equal temper- ament, of course, and the two temperaments (Third-Circle, No. 2, and Fifth-Circle, No. 8) that were identical with the first and second circulating temperaments above. Now half of the rejected temperaments had deviations lower than that of the second cir- culating temperament (3.3), and a couple of others were just about as good. But none of these was considered worthy in the final appraisal, Neidhardt had, incidentally, changed his ideas somewhat as to the relative position of the best temperaments: the Circulating Temperament, No. 2 (Fifth-Circle, No. 8) is now considered best for a large city; No. 1 (Third-Circle, No, 2) for a small city; and Third-Circle, No. 1, not included before, for a village. If we examine the deviations of the major thirds in the three temperaments Neidhardt himself considered superior, we quickly find why he liked them. In the second circulating temperament (Table 155) the thirds on Cand Fare 8 cents sharper than a pure third, and the sharpness gradually increases in both directions around the circle of fifths until the three worst thirds are 18 cents sharp. In the first circulating temperament (Table 151) the third on C is only 6 cents sharp, and there is the same grad- ual increase until the five poorest thirds are all 18 cents sharp. In the Third-Circle, No. 1 (Table 174), the third on C is 4 cents sharp, and the six poorest thirds are either 18 or 20 cents sharp. 179 TUNING AND TEMPERAMENT. Werckmeister' s third temperament, the first of the three he has labeled "correct" (Table 140), is much like the Neidhardt temperament just mentioned. Its thirds on C and F are only 4 cents sharp, but the thirds of the principal triads in the key of D*3 are all a syntonic comma, 22 cents, sharp. Werckmeister himself said that some people who advocated equal temperament held that "in the future ... it will be just the same to play an air in D*3 as in C. ^ But he held consistently "that one should let the diatonic thirds be somewhat purer than the others that are seldom used." 45 A good comparison can be made between two temperaments of Neidhardt, already mentioned as having fifths of four different sizes and the same number of each size, but with a different ar- rangement. The Fifth-Circle, No. 12 (Table 176) has a sym- metric arrangement and a low mean deviation, 2.0. Its thirds show no trend whatever from near to far keys, but are sufficiently irregular to make this seem a poor attempt at equal tempera- ment. Not so its companion, the third sample temperament (Ta- ble 173), in which the third on C is only 2 cents sharp, whereas four of the five poorest thirds are 20 cents sharp. To be sure, the deviation for this temperament, 5.7, is almost three times as great as for the other one, and there is a painful lack of sym- metry. But the unsymmetric temperament is "circulating," and therefore deserves an honored place among the "good" temper- aments,, Thomas Young's temperaments also deserve mention for their circulating nature. His first temperament (Table 152) is equiva- lent to a temperament with four pure fifths and four fifths each tempered by 1/6 or 1/12 comma. It is constructed with scien- tific accuracy so that the thirds range in sharpness from 6 cents forC-E to 22 cents, a syntonic comma, forF#-A#. Its mean de- viation is 5.3. On the other hand, there is the symmetric form of this temperament, Marpurg's D (Table 154), with a mean de- viation of 1.3. And the even better, nonsymmetric form, with a "A. Werckmeister, Hypomnemata musica (Quedlinburg, 1697), p. 36. 45Werckmeister, Musicalische Paradoxal-Discourse (Quedlinburg, 1707), p. 113. 180 IRREGULAR SYSTEMS mean deviation of 0.3! But these last-mentioned temperaments are curiosities only, whereas Young's differentiated admirably between near and far keys. However, Young's first temperament was too difficult to con- struct, as he had described it with fifths tempered by 3/16 and "approximately" 1/12 syntonic comma. Therefore he substituted his second method (Table 145), which was of the utmost simplic- ity, with six consecutive perfect fifths and six consecutive fifths tempered by 1/6 ditonic comma. Its mean deviation was 6.0. In it the thirds on C, G, and Dare each 6 cents sharp, whereas those on F% C#, and G* are each 22 cents sharp. Neidhardt's Fifth- Circle, No. 3 (Marpurg's F) is the symmetric version of this temperament (Table 147), with a mean deviation of 2.0. Again we may well say that Young's version is an excellent irregular temperament, while the symmetrical version represents having fun with figures. So many versions of good circulating temperaments have ap- peared on these pages, each with its points of excellence, that we cannot resist the temptation to close this chapter with an irreg- ular temperament to end irregular temperaments! Gallimard's modification of the ordinary meantone temperament, by a sys- tematic variation in the size of the chromatic fifths, was good enough in principle, but could not have been too successful be- cause of the large number of other fifths tempered by 1/4 comma. What is really needed, in order to have a more orderly change in the size of the thirds, is to have the variable tempering ap- plied to all the fifths, instead of to only five of them. Let the fifth D- A be the flattest, and let each succeeding fifth in both di- rections around the circle of fifths be a little sharper until the fifth on A^ is the sharpest. Then the total parts to be added will be 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36 parts. Since these parts are to be added to 12 fifths, it is evident that D-A, the flat- test fifth, will be flatter than the fifth of equal temperament by three of these parts; the fifths B-F# and F-C will be precisely the size of the equal fifth; and the sharpest fifth, AD-E°, will be larger than the equal fifth by three parts. The thirds will vary as follows (the error being expressed as the number of parts be- low or above the third of equal temperament): C-E, -8; G-B, -8; 181 TUNING AND TEMPERAMENT D-F#, -6; A-C#, -2: E-G#, 2; B-D#, 6; Gb-Bb, 8; Db-F, 8; Ab-C, 6; Eb-G, 2; Bb-D, -2; F-A, -6. We can choose the value for one part that will give the de- sired size of thirds. If the part is one cent, the fifth D-A is 697 cents, practically a meantone fifth, and the fifth Ab-Eb is 703, practically perfect; the best thirds, C-E and G-B, are 392, 1/4 comma sharp; the poorest thirds, Gb-Bb and Db-F, are 408, precisely a Pythagorean third. Table 178 should have satisfied the desire of Werckmeister and his contemporaries for a circulating temperament in which all the thirds are sharp, but none more than a comma, and all the fifths are flat or pure. As the size of the part is reduced, the tuning approaches equal temperament. When the part is in- creased to 1 3/4 cents, the best thirds are pure. But the poor- est thirds are now 414 cents, about 5/4 comma sharp. Thus Ta- ble 178 probably represents the limit of a tolerable temperament in the extreme keys. Since the mean deviation for the entire se- ries of temperaments formed in this manner is precisely pro- portional to the size of the part, it would be easy to devise a sys- tem with the deviation of any of the systems in this chapter, but with a more orderly distribution of the errors, as regards com- mon keys and less-used keys. The Temperament by Regularly Varied Fifths may be re- garded as the ideal form of Werckmeister' s "correct" temper- aments and of Neidhardt's "circulating" temperaments and of all "good" temperaments that practical tuners have devised by rule of thumb. Let us see, therefore, how closely it is approached by these other temperaments. In Table 179, the deviations have been computed, not only from equal temperament, but also from our temperament with variable fifths. The table shows clearly that the temperaments with greatest symmetry do not fit so well into the desired pattern as do those that are much less regular in their construction. In general, the temperaments with lowest deviation from the one ideal temperament will have a high devi- ation from the other. Neidhardt's second circulating tempera- ment has the unique position of ranking the same with regard to both. 182 IRREGULAR SYSTEMS Table 177. Metius' Irregular Temperament Lengths 1000 940 896 837 800 749 704 668 628 596 563 530 500 "3 G#_1 A+~& Bb+T5B M.D. 9.5; S.D. 11.6 Names E° F 6 F# 3 G 3 Gff A 12 B 12B 6 C 3 C# 2D 12D#~2 E° Table 178. Temperament by Regularly Varied Fifths Names CxDxEFxGxAxB C Cents 0 92 197 297 392 500 591 699 794 894 999 1091 1200 M.D. 5.8; S.D. 6.6 Table 179. Deviations of Certain Temperaments From Equal Temperament From Varied Fifths M.D. S.D. M.D. S.D. Neidhardt's Circulating, No. 1 4.0 4.6 2.1 2.3 No. 2 3.3 3.7 3.3 3.7 No. 3 2.7 2.9 4.2 4.7 Third- Circle, No. 1 5.3 5.9 1.2 1.5 Wer ckmeister 's Correct, No. 1 6.0 7.5 1.9 2.3 No. 2 9.2 10.7 4.7 5.7 No. 3 5.0 5.7 3.8 4.2 Neidhardt's Fifth-Circle, No. 12 2.0 2.3 6.2 6.7 Sample, No. 3 5.7 6.4 1.5 1.8 Young's No. 1 5.3 5.9 1.7 1.9 Marpurg's Letter D 1.3 1.6 6.7 7.1 Young's No. 2 6.0 6.8 1.9 2.0 Neidhardt's Fifth-Circle, No. 3 2.0 2.0 5.0 5.8 Schlick's (Hypothetical) 8.0 8.6 2.7 3.1 Neidhardt's Third- Circle, No. 3 3.3 4.7 3.0 3.8 Our hypothetical reconstruction of Arnold Schlick's tempera- ment had the same size of fifths as Neidhardt's Third-Circle, No. 3, but differently arranged, and with a fairly high deviation. Ob- serve that, with this other standard of varied dissonance, Schlick's temperament is even a little better than Neidhardt's. Of all the temperaments shown in our table, Neidhardt's Third-Circle, No. 183 TUNING AND TEMPERAMENT 1 seems to be the best, with our new standard, although Neid- hardt himself said it was best for a village! But it would have been difficult to tune, and therefore Thomas Young's Tempera- ment, No. 2 probably cannot be surpassed from the practical point of view. Even so, the highest honor must be paid to old Arnold Schlick, writing so long before these other men, but stat- ing as clearly as need be for his very practical purpose, "Al- though they will all be too high, it is necessary to make the three thirds C-E, F-A, and G-B better, „ . .as much as the said thirds are better, so much will G be worse to E and B." Table 180. Compass of the Lute G Tuning A Tuning 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 1. G Ab A Bb B C Db(C#) D Eb A Bb B C C#(Db) D Eb E F 2. D Eb E F F# G Ab(G#) A Bb E F F# G G#(Ab) A Bb B C 3. A Bb B C C#D Eb(D#) E F B C C# D D#(Eb) E F F# G 4. F Gb G Ab A Bb Cb(B) C Db G Ab A BbB (Cb) C Db D Eb 5. C Db D Eb E F Gb(F#) G Ab D Eb E F F#(Gb) G Ab A Bb 6. G Ab A Bb B C Db(C#) D Eb A Bb B C C#(Db) D Eb E F 184