© Copyright 2005, John Charles Francis BSc (Hons.), MSc, PhD
CH3072
1 July 2005
A previous article by the author[1] published February 1^{st}, 2005, detailed the results of a systematic mathematical analysis of the diagram on the cover sheet of Das Wohltemperirte Clavier. Performed in terms of beats per second, the analysis considered diagram orientation (lefttoright, righttoleft), tuning direction (towards sharps, toward flats), starting position (12 positions) and beat rates for the interval closing the circle of fifths (0, 1, and 2). In total 144 possibilities were analysed, and for each, a system of twelve linear equations in twelve unknowns was constructed and solved to analytically obtain the pitches of the twelve semitones as both ratios and decimals, along with the corresponding temperament in cents. The automated analysis, performed using the symbolic equation solving software Mathematica, yielded the remarkable result that the diagram encoded two cammertoncornetton transpositions, confirming a tentative hypothesis that it represented two circles of fifths corresponding to the respective end points of the diagram. Moreover, both of the cammertoncornetton solutions conformed to our knowledge of historical pitch standards. It was further indicated that a specific marking by Bach, the ‘C’ of ‘Clavier’, showed the transpositions R21 (cammerton) and R122 (cornetton) as Bach’s preferred solution. R122 was derived as a righttoleft reading of the diagram and it will be shown that the mirror[2] form of R21 corresponds to a lefttoright reading. A statistical analysis was also performed according to the frequency of key signature usage in the BachWerkeVerzeichnis, considering both organ and other keyboard works. Strong correlations were found with the width of the major thirds for R122 and R21, and indeed the strongest correlation, not exceeded by any considered historic temperament, occurred at the midpoint of R122 and R21, i.e. the intervening key. The current paper presents the results in a more musicianfriendly manner, showing how the diagram relates to the circle of fifths. It also provides comparison with earlier interpretations proposed by Sparschuh, Zapf and a later one proposed by Lehman. In addition, it presents more details in terms of comparison with historical temperaments. The rationale behind the construction of the temperament is presented and the theoretical framework corresponding to the practical equalbeating tuning method is described.
The suggestion that the sinuous circles of J. S. Bach’s 1722 cover sheet for Das Wohltemperirte Clavier may depict a tuning scheme was made some years ago by Andreas Sparschuh, a mathematician from the
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The clavichord specialist Michael Zapf and the tuning theorist Herbert Anton Kellner, both independently met with Sparschuh. While Zapf was convinced that Bach’s diagram was indeed a tuning system, he was sceptical regarding Sparschuh’s initial interpretation, considering it at odds with historical tuning practice. Zapf subsequently made a proposal of his own, treating Sparschuh’s numbers as an equalbeating specification in terms of secondsperbeat[4]. Keith Briggs from BT Labs investigated the mathematics of Zapf’s proposal and noted that, by closing the circleoffifths, the pitch of Bach’s keyboard might be determined[5]. A proposal by the author, interpreted Sparschuh’s numbers as beatspersecond, and included both ends of Bach’s diagram in the analysis to close the circleoffifths[6]
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A subsequent proposal by Lehman[7] interpreted Sparschuh’s numbers as 1/12 and 1/6 of a Pythagorean comma, but without a consistent reading for the end points[8].
Lorenz Christoph Mizler’s main claim to fame today is as the founder of the society whose 14^{th} member was Johann Sebastian Bach. On the subject of tuning, he once noted that while Werckmeister’s temperament was the best in his day, it had nevertheless been improved upon since Neidhardt’s time[9]. However, we know that the best tuning practices of Mizler’s day did not satisfy Bach; for as his son Carl Philipp Emanuel noted, no one else could tune the harpsichord of his father to his satisfaction[10]. In this regard, Lindley proposes[11] that the elaborate theoretical tuning models expounded by the likes of Neidhardt and Sorge, were unable to capture the subtle nuances that Bach customarily achieved in tuning[12]. He, suggests, moreover, that the cause of the theoretical deficiency was that Sorge and Neidhardt would never split their basic unit of measurement for tempering, namely 1/12 of a Pythagorean comma. Lindley notes that this fraction is unable to satisfy the following three conditions simultaneously:
A gradual[13] variation of the major and minor thirds (or sixths)
The smallest major 3^{rd} CE beats larger than pure[14]
The most heavily tempered thirds are impure by less than a syntonic comma (i.e. Pythagorean Thirds are excluded)
We will see that the tuning for Das Wohltemperirte Clavier achieves all of these goals and that, as Lindley intimates, in so doing a finer division of the Pythagorean comma is needed.
Marpurg, while arguing the case for Equal Temperament, noted that there is only one ideal version of this temperament, whereas there are many unequal schemes[15]. He therefore suggested there would be chaos until Equal Temperament was universally achieved, as otherwise each would be inclined to use whatever they considered best. We will see that in Das Wohltemperirte Clavier, J. S. Bach offers a tuning system that is also an ideal of its kind.
J. S. Bach’s use of the generic term ‘Clavier’ (keyboard) leaves unspecified the precise instrument to be used. As noted by Richard Jones[16], this, in conjunction with the deliberately circumscribed keyboard compass, suggests that the work was intended to be universally accessible to keyboard players regardless of the particular type of instrument (harpsichord, clavichord, or organ[17]) that they might have at their disposal. An important issue arises here, however, in relation to the adoption of an unequal temperament for ‘welltempered’ instruments; namely coexistence at cammerton and cornetton pitch[18]. In the case of equal tempering, this is no concern as each key is identical. However, if unequally tempered instruments at cammerton and cornetton pitch are tuned identically to an identical temperament, then there will be inevitable intonation problems as the tuning of one instrument is two places removed from the other on the circleoffifths. It follows that there cannot be just one temperament for Das Wohltemperirte Clavier, given the generic term Bach uses, but rather there must be two variants: one for cammerton and another for cornetton. We will see that the temperament for Das Wohltemperirte Clavier consists of two transposed cammerton, cornetton variants, indicated by respective lefttoright and righttoleft readings of Bach’s diagram.
By virtue of certain telling features in the cammerton and cornetton tuning variants, Bach’s scheme can be shown to be an equalbeating one. Its transposition by two places on the circleoffifths requires adaptation to one of the beat rates, which Bach explicitly provides. While the equalbeating methods presented by Jorgensen have been viewed cautiously in some circles, they are nevertheless pertinent to Bach’s method. Jorgensen, in his impressive tome, Tuning, traces the historical understanding of beats, noting that in the Seventeenth Century there was no information published that they should increase in frequency when intervals are played higher up the scale[19]. To the musicians at that time, he notes, the quality of the fifths in a scale when played harmonically, seemed identical when their beat frequencies were the same; deceiving them into believing that tempered fifths were of the same size when their beat frequencies were identical[20]. Nor, it seems, had the situation changed by the end of the Eighteenth Century, for in the context of Thomas Young’s rules for well temperament of 1799, Jorgensen notes that it would be authentic practice to apply the much easier equalbeating methods[21].
C. P. E. Bach writing in 1753 concerning the tuning of the clavichord and harpsichord[22] refers to tempering most[23] of the fifths. He observed that the beats of fifths may be more easily heard by probing fourths[24]. In minimalist terms, he characterised unequal welltempered tuning[25]:
Both types of instrument must be tempered as follows: In tuning the fifths and fourths, testing minor and major and chords, take away from most of the fifths a barely noticeable amount of their absolute purity. All twentyfour tonalities will thus become usable. The beats of fifths can be more easily heard by probing fourths, an advantage that stems from the fact that the tones of the latter lie closer together than fifths.[26]
Clearly, this generic description does not indicate which fifths should be tempered, nor by how much. Moreover, although one is admonished to listen for the beats of fifths and fourths, there is no indication as to how fast those beats must be in order to take away from the fifth a barely noticeable amount of its absolute purity.
The diagram on the cover sheet of Das Wohltemperirte Clavier, read lefttoright, indicates a tuning scheme for cammertonpitched instruments. Bearings can be set by proceeding around the circleoffifths towards the flats using only fourths and fifths, so as to remain within the octave starting on Middle C[27]. Bach’s diagram indicates three types of interval: i) intervals beating once per second, ii) intervals beating twice per second and iii) pure intervals without beats. In the Eighteenth Century, such tempi could be readily determined from a pocket watch or pendulum clock; today, a metronome at 60 beats per minute will achieve the same result. The protrusion at the left of Bach’s diagram depicts the beatrate of the interval closing the circleoffifths, representing a check that the temperament has been set correctly. That interval arises as a side effect of the tuning scheme, and it beats exactly once per second. A realisation of the tuning procedure constrained to one octave, is illustrated in Figure 1, yielding a temperament denoted as R21[28]
Figure 1: Canonical form of the cammerton tuning procedure
See: Das Wohltemperirte Clavier  Frequencies for tuning instruments to the cammerton temperament
An alternative formulation, yielding the same temperament, is presented in Figure 2[29].
Figure 2: Optimised cammerton tuning procedure using single beat rate of 1 beat per second[31]
Instruments at cornetton pitch sound one tone higher than their cammerton counterparts and, to coexist harmoniously in an ensemble, their temperament must be transposed downwards by a whole tone. Read righttoleft, Bach’s diagram yields a transposed tuning scheme for cornettonpitched instruments. Bearings are set by proceeding around the circleoffifths towards the sharps, the protrusion at the right of Bach’s diagram giving the expected beat rate for the implicitly tuned interval closing the circleoffifths as two beats per second. A realisation of the tuning procedure within one octave is detailed in Figure 3, yielding a temperament denoted as R122[30].
Figure 3: Canonical form of the cornetton tuning procedure[32]
The pitch at which Bach’s tuning procedures were designed can be obtained by reverseengineering his diagram using the indicated beat rates (see appendices for details). The corresponding temperaments can then be derived from this pitch information. The pitch and temperament are summarised in Table 1 and Table 2, respectively, proving that the circles in Figure 1 and Figure 3 are exact transpositions.
R21 (Cammerton)  Frequency (Hz)  R122 (Cornetton) 
C  249.072 

D@  263.154 

D  279.331  C 
E@  296.049  D@ 
E  312.998  D 
F  332.43  E@ 
G@  350.873  E 
G  373.109  F 
A@  394.732  G@ 
A  417.997  G 
B@  443.573  A@ 
B  468.497  A 
 498.145  B@ 
 526.309  B 
Table 1: Pitch of Das Wohltemperirte Clavier[33]
See: Das Wohltemperirte Clavier  Frequencies for tuning instruments to the cammerton temperament
R21 (Cammerton)  R122 (Cornetton)  
Note  Temperament (Cents)  Deviation From Equal Temperament (Cents)  Note  Temperament (Cents)  Deviation From Equal Temperament (Cents) 
C  0  0 



D@  95.2136   4.8 



D  198.495   1.5  C  0  0 
E@  299.124   0.9  D@  100.628  0.6 
E  395.505   4.5  D  197.010   3.0 
F  499.782   0.2  E@  301.286  1.3 
G@  593.259   6.7  E  394.763   5.2 
G  699.637   0.4  F  501.141  1.1 
A@  797.169   2.8  G@  598.673   1.3 
A  896.314   3.7  G  697.818   2.2 
B@  999.128   0.9  A@  800.633  0.6 
B  1093.770   6.2  A  895.273   4.7 


 B@  1001.500  1.5 


 B  1096.720   3.3 
Table 2: Temperament of Das Wohltemperirte Clavier[34]
The temperament for Das Wohltemperirte Clavier arises naturally as an ideal providing an optimally smooth progression from worst to best thirds across the circleoffifths, so satisfying Lindley’s first condition. The thinking that led to its creation can be readily reconstructed from the well known consideration that four consecutive intervals on the circleoffifths determine the width of the corresponding major third. The Pythagorean third (~408 cents) is excluded, so that a sequence of four pure fifths does not occur, so satisfying Lindley’s second condition. The widest third must then consist of three consecutive pure intervals on the circleoffifths, followed by a tempered interval to narrow the third. This observation yields the sequence 0001 on the circleoffifths, corresponding to the worstcase third. Optimising for a gradual progression towards better thirds as Lindley suggests, leads to a progressive narrowing of the major thirds by one unit at a time. The results in Table 3 are now predicated.
Tempering of intervals on circleoffifths  Narrowing from Pythagorean Third 
(0, 0, 0, 1)  1 unit 
0, (0, 0, 1, 1)  2 units 
0, 0, (0, 1, 1, 1)  3 units 
0, 0, 0, (1, 1, 1, 1)  4 units 
0, 0, 0, 1, (1, 1, 1, 2)  5 units 
0, 0, 0, 1, 1, (1, 1, 2, 2)  6 units 
0, 0, 0, 1, 1, 1, (1, 2, 2, 2)  7 units 
0, 0, 0, 1, 1, 1, 1, (2, 2, 2, 2)  8 units 
0, 0, 0, 1, 1, 1, 1, 2, (2, 2, 2, 2)[35]  8 units 
Table 3: Derivation of Das Wohltemperirte Clavier tuning by progressive narrowing of thirds
Considering the resulting sequence 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, it can be seen that the sum of the digits is 14[36]. Accordingly, as the Pythagorean comma must be distributed over the circleoffifths, each tempering unit theoretically corresponds to 1/14^{th} part of a Pythagorean comma, the tempering fractions being 1/7[37] and 1/14. Adopting the historical precedent of placing the best thirds in the key of C, yields the theoretic temperament R1214P in Figure 4. The equalbeating method for tuning this temperament (R122) is depicted in Bach’s diagram, and realised in Figure 3[38].
Figure 4: R1214P  theoretic representation of R122 in terms of Pythagorean comma fractions
The cornetton temperament in Figure 4 must be transposed by a whole tone for cammertonpitched instruments to give R214P as shown in Figure 5.
Figure 5: R214P  theoretic representation of R21 in terms of Pythagorean comma fractions
The equalbeating method for tuning this temperament (R21) is depicted in Bach’s diagram, and has been realised in Figure 1 and Figure 2.
The difference in beat rates between cammerton (R21) and cornetton (R122) tuning procedures that occurs at the interval closing the circleoffifths, is explicitly represented in Bach’s diagram, and can easily be derived as indicated in Figure 6 from the following wellunderstood tuning principles:
Cammerton  C  D@  D  E@  E  F  G@  G  A@  A  B@  B  C  D@  Beat Rate 
C 



 F 







 1  




 F 



 B@ 


 1  


 E@ 





 B@ 


 1  


 E@ 



 A@ 




 0  
 D@ 





 A@ 




 0  
 D@ 



 G@ 






 0  





 G@ 



 B 

 2  



 E 





 B 

 2  



 E 



 A 



 2  

 D 





 A 



 2  

 D 



 G 





 2  
Left End  C 





 G 





 1 
Cornetton 

 C  D@  D  E@  E  F  G@  G  A@  A  B@  B 





 E@ 





 B@ 
 1  




 E@ 



 A@ 


 1  


 D@ 





 A@ 


 1  


 D@ 



 G@ 




 0  







 G@ 



 B  0  





 E 





 B  0  





 E 



 A 

 2  



 D 





 A 

 2  



 D 



 G 



 2  

 C 





 G 



 2  

 C 



 F 





 2  
Right End 






 F 



 B@ 
 2 
Figure 6: Comparison of the cammerton and cornetton tuning procedures.
Intervals in green cross the octave without changing beat rate (fourth is inverted upwards or pure interval).
However, when the interval in red crosses the octave it doubles beat rate since a fifth is inverted upwards.
The respective beat rates of 1 and 2 are captured by the protrusions at the left and right of Bach’s diagram.
The properties of the cornetton tuning scheme based on 1/14–comma fractions is now compared with the equal beating implementation derived from Bach’s diagram. Figure 7 gives the width of the major thirds, showing a smooth progression from best thirds CE and FA to the widest third EG#. The equal beating scheme tracks the theoretic one closely, generally matching it to within one cent. One exception is E@G, where the equal beating tuning is almost two cents narrower than theory.
Figure 8 shows the corresponding situation for minor thirds; the best being AC and DA, with a progressive narrowing until D@E. As before, the tracking of the equal beating tuning is generally within one cent, a minor exception being the third GB@.
Figure 9 shows the width of the fifths; the smallest occurring at C and F, while the largest are at E, B, and F#.
Figure 7: Width of major thirds in cents for R1214P (theoretic) and R122 (equal beating)
Figure 8: Width of minor thirds in cents for R1214P (theoretic) and R122 (equal beating)
Figure 9: Width of fifths in cents for R1214P (theoretic) and R122 (equal beating)
A comparison of R122 (cornetton) with other temperaments is given in terms of the Euclidian distance in cents in Figure 10 and in terms of the correlation distance in Figure 11. For the cammerton temperament R21, the comparison in terms of Euclidian distance in cents is given in Figure 12, while Figure 13 shows the correlation distance. In terms of correlation distance, the closest match to R122 is Zapf, followed by Lehman and Sorge[39]. For R21, the closest matches in terms of correlation distance are Neidhardt Circulating No. 1, Sparschuh, Young No. 1, Mercadier and Barnes.
Figure 10: Comparison of R122 with other temperaments (Euclidian distance in cents)[40]
Figure 11: Comparison of R122 with other temperaments (correlation distance)
Figure 12: Comparison of R21 with other temperaments (Euclidian distance in cents)
Figure 13: Comparison of R21 with other temperaments (correlation distance)
Noting Lindley’s remarks concerning the inadequacy of a 1/12comma Pythagorean tempering unit to capture the nuances of Bach’s tuning, it is instructive to consider what happens when Bach’s scheme is represented by other fractions of a comma. For the purpose of comparison and completeness, temperaments are presented for 1/11, 1/12, 1/13, 1/14, 1/15, 1/16, 1/17, 1/18 Pythagorean comma fractions, denoted as R1211P[41], R1212P[42], R1213P, R1214P, R1215P, R1216P, R1217P and R1218P, respectively. The results are shown in Table 4.
Cents  C  D@  D  E@  E  F  G@  G  A@  A  B@  B 
R122  0.0  100.6  197.0  301.3  394.8  501.1  598.7  697.8  800.6  895.3  1001.5  1096.7 
R1218P  0.0  103.3  198.7  304.6  397.4  500.7  601.3  699.3  803.9  898.0  1005.2  1099.3 
R1217P  0.0  102.6  198.4  303.8  396.8  500.8  600.7  699.2  803.2  897.6  1004.4  1098.7 
R1216P  0.0  102.0  198.0  302.9  396.1  501.0  600.0  699.0  802.4  897.1  1003.4  1098.0 
R1215P  0.0  101.2  197.7  302.0  395.3  501.2  599.2  698.8  801.6  896.5  1002.3  1097.3 
R1214P  0.0  100.3  197.2  300.8  394.4  501.4  598.3  698.6  800.6  895.8  1001.1  1096.4 
R1213P  0.0  99.2  196.7  299.5  393.4  501.7  597.3  698.3  799.4  895.0  999.7  1095.3 
R1212P  0.0  98.0  196.1  298.0  392.2  502.0  596.1  698.0  798.0  894.1  998.0  1094.1 
R1211P  0.0  96.6  195.4  296.3  390.8  502.3  594.7  697.7  796.4  893.1  996.1  1092.7 
Table 4: Cornetton temperaments in cents for different fractions of the Pythagorean comma,
theoretic ideal R1214P (green) and realisation based on equalbeating tuning R122 (yellow)
Table 5 shows the temperaments that occur when other fractions of a comma are used as approximations to Bach’s cammerton tuning. Temperaments are given for 1/11, 1/12, 1/13, 1/14, 1/15, 1/16, 1/17, 1/18 Pythagorean comma fractions, denoted as R211P, R212P, R213P, R214P, R215P, R216P, R217P and R218P, respectively.
Cents  C  D@  D  E@  E  F  G@  G  A@  A  B@  B 
R21  0.0  95.2  198.5  299.1  395.5  499.8  593.3  699.6  797.2  896.3  999.1  1093.8 
R218P  0.0  94.1  194.8  298.0  393.5  499.3  592.2  695.4  796.1  894.1  998.7  1092.8 
R217P  0.0  94.4  195.6  298.3  394.0  499.4  592.4  696.4  796.3  894.8  998.8  1093.2 
R216P  0.0  94.6  196.6  298.5  394.6  499.5  592.7  697.6  796.6  895.6  999.0  1093.6 
R215P  0.0  94.9  197.7  298.8  395.3  499.6  593.0  698.8  796.9  896.5  999.2  1094.1 
R214P  0.0  95.3  198.9  299.2  396.1  499.7  593.3  700.3  797.2  897.5  999.4  1094.7 
R213P  0.0  95.6  200.3  299.5  397.0  499.8  593.7  702.0  797.6  898.6  999.7  1095.3 
R212P[43]  0.0  96.1  202.0  300.0  398.0  500.0  594.1  703.9  798.0  900.0  1000.0  1096.1 
R211P  0.0  96.6  203.9  300.5  399.3  500.2  594.7  706.2  798.6  901.6  1000.4  1097.0 
Table 5: Cammerton temperaments in cents for different fractions of the Pythagorean comma,
theoretic ideal R214P (green) and realisation based on equalbeating tuning R21 (yellow)
The major and minor thirds and fifths can now be compared to see the impact of choosing different comma sizes. Only the cornetton case need be considered, as cammerton yields identical results under transposition. Moreover, the comparison is restricted to two of the more extreme variants, the 1/12comma Lehman solution on the one hand and the 1/18comma interpretation on the other. The progression in the quality of the major thirds from best to worst keys is shown in Figure 14; the minor thirds are given in Figure 15 and the fifths are presented in Figure 16. As can be seen, significant distortion occurs with both 1/12 and 1/18comma solutions, defeating the ideal of smooth, regular, transitions on the circleoffifths. In the case of R1212P, the major thirds on F, B@, E@, A@, D@ increase in size (reading right to left), but at G@ the third becomes smaller before increasing again at B and finally peaking at E. Comparison with R1214P shows that this imperfection arises because a 1/14comma design is being compressed due to a 1/12comma interpretation. In the minor third case, the behaviour of R1212P and R1218P is again arbitrary. In the case of R1212P, for example, there is a continuous increase in the width of the minor thirds from D@ to B@, but at the next step F there is an out of pattern four cents narrowing, and then the widening continues again until D. Again, comparison with R1214P shows that this is just a compression artefact due to a reading in terms of a course fractional unit. The 1/12comma reading in R1212P results in a widefifth, which, as Jorgensen notes, is undesirable harmonic waste (see Tuning). Comparison with R1214P shows that the imperfection only arises because a 1/14comma design is being compressed due to a 1/12comma reading.
Figure 14: Width of major thirds in R1214P (theoretic ideal), R1212P (Lehman) and R1218P
Figure 15: Width of minor thirds in R1214P (theoretic ideal), R1212P (Lehman) and R1218P
Figure 16: Width of fifths in R1214P (theoretic ideal), R1212P (Lehman) and R1218P
We have seen that a fourteenth part of a Pythagorean comma is the natural basic tempering unit to theoretically describe Bach’s temperament, and that larger fractions compromise the ideal of a progressive gradual change in the size of thirds on the circleoffifths, and lead to harmonic waste. The equalbeating methods R21 and R122, given by respective lefttoright and righttoleft readings of Bach’s diagram, were shown to yield cammerton and cornetton transpositions closely approximating the 1/14comma scheme. The 1/12comma reading proposed by Lehman was addressed as R1212P. To complete the picture, the interpretations of Andreas Sparschuh and Michael Zapf will also be considered.
The average minor third must necessarily have a width of 300 cents, the major third 400 cents, and the fifth 700 cents. Improving a third or fifth in one place, must necessarily degrade corresponding intervals elsewhere. The standard deviation, which is a measure of the inequality or colour of a temperament, is shown in Figure 17. R1214P and R214P have the least deviation of the thirds and fifths, while R1212P (Lehman) has the most. The deviation in the fifths, for example, is some 33% greater, perhaps, making it somewhat harder for singers to hit the right note.
The Zapf and Lehman proposals are essentially variants of the cornetton solution R1214P, while the proposal from Sparschuh is a variant of the cammerton solution R214P. The comparison in terms of major thirds is shown in Figure 18. Zapf has a best third FA comparable to R1214P and a peak AC#. R214P is simply a displaced version of R1214P, commencing on cammerton D rather than cornetton C. Lehman has a greater tempering of CE and FA with a widest third EG#. His reading further improves the already good thirds of FA, CA, GB and DA, at the expense of D@F, A@C, E@G and B@D. The temperament of Sparschuh is broadly comparable to R214P, although it wanders around somewhat.
Figure 19 shows the corresponding situation for minor thirds. R1214P has best minor thirds AC, EG, and DF, with a smooth progression toward the narrowest third D@E. Zapf has best minor thirds at AC, with a fairly smooth progression to the narrowest at F#A. Lehman improves the already good thirds AC, EG and DF at the expense of degrading FA@, CE@ and GB@. His temperament does not exhibit a monotonic rise, as there is an outofcharacter drop in width occurring at FA@. The temperament of Sparschuh is reasonably wellbehaved, although not entirely monotonic in ascent.
The sizes of fifths are shown in Figure 20. R1214 has three sizes of fifth, while Lehman has four including a wide fifth. Both Zapf and Sparschuh use more sizes of fifths.
Figure 17: Standard deviation of thirds and fifths (cents)
Figure 18: Comparison of major thirds (cents)
Figure 19: Comparison of minor thirds (cents)
Figure 20: Comparison of fifths (cents)
J. S. Bach’s specification for Das Wohltemperirte Clavier at the top of his 1722 manuscript is a prescriptive tuning method for an equalbeating temperament. The scheme uses an easytoset tempo of 1 beat per second. Read lefttoright, the diagram indicates the cammerton tuning procedure for the octave starting on Middle C. The beat rate of the implicitlytuned interval closing the circleoffifths is shown by the left protrusion of Bach’s diagram. Given that stringed instruments need to be regularly tuned, it makes eminent sense that the cammerton tuning should be read from lefttoright and not vice versa. The simple manner by which Bach could determine the relationship between cammerton and cornetton beat rates was presented. The cornetton tuning also takes place within the octave starting on Middle C, but commences on F, and is based on a righttoleft reading with the implicitlytuned interval closing the circleoffifths at the right. The manner in which cammerton and cornetton temperaments have been so combined into one diagram is indeed ingenious.
The rationale behind Bach’s temperament was shown to be an optimally smooth progression in the quality of thirds across the circleoffifths. It was shown that the tuning specification in terms of beats is closely linked to a 1/14comma theoretical scheme, and it was demonstrated that the principles behind the design are violated when a 1/12comma approximation is used. The 1/14comma scheme was also compared with the readings of Sparschuh and Zapf. Although all are workable musical solutions, only the 1/14comma reading achieves the ideal of the temperament design.
The pitches obtained by reverseengineering Bach’s diagram accord with our current knowledge of historical pitch[44]. The mathematics shows that the octave being tuned is the one based on Middle C, for both cammerton and cornetton cases, and that the respective lefttoright and righttoleft readings commence the tuning sequence on C and F, respectively. Optimisations can be made such that intervals that should beat twice per second can be dropped an octave where they beat once per second. Thus, a ‘2’ in Bach’s diagram can be interpreted to mean transpose down one octave and check for a one second beat rate.
An interesting question remains: should one tune in the theoretical manner of R1214P and R214P, or use the equalbeating temperaments R122 and R21? With choral music, the theoretic form may, perhaps, facilitate better intonation, as fewer intervals are used and all intervals are exact. On the other hand, it may be that certain works of Bach are more authentically rendered by using the equalbeating tuning procedure encoded in Das Wohltemperirte Clavier.
The preparation of this article was facilitated by Capella music software and by Yo Tomita’s Bach Musicological Font. Michael Zapf first drew my attention to the important discovery of Andreas Sparschuh, while my interest in Bach’s keyboard temperament was stimulated by the enthusiasm and kindness of the late Herbert Anton Kellner. Special thanks are due to Thomas Braatz for his invaluable assistance and I am also grateful to members of Michael Zapf’s Clavichord Discussion Group for comments received.
Reduce[{
(* Pitches f0…f11 form an octave, with relationship on
the circleoffifths according to Bach's WTC diagram read left
to right *)
3f0  2f7 == 1, (* left end: 5^{th} beating once per second *)
3f5  4f0 == 1, (* loop 1: 4^{th} beating once per second *)
3f10  4f5 == 1, (* loop 2: 4^{th} beating once per second *)
3f3  2f10 == 1, (* loop 3: 5^{th} beating once per second *)
3f8  4f3 == 0, (* loop 4: pure 4^{th} *)
3f1  2f8 == 0, (* loop 5: pure 5^{th} *)
3f6  4f1 == 0, (* loop 6: pure 4^{th} *)
3f11  4f6 == 2, (* loop 7: 4^{th} beating twice per second *)
3f4  2f11 == 2, (* loop 8: 5^{th} beating twice per second *)
3f9  4f4 == 2, (* loop 9: 4^{th} beating twice per second *)
3f2  2f9 == 2, (* loop 10: 5^{th} beating twice per second *)
3f7  4f2 == 2, (* loop 11: 4^{th} beating twice per second *)
(* Cents corresponding to pitch relations *)
c0 == 1200Log[2,1.0],
c1 == 1200Log[2,f1/f0],
c2 == 1200Log[2,f2/f0],
c3 == 1200Log[2,f3/f0],
c4 == 1200Log[2,f4/f0],
c5 == 1200Log[2,f5/f0],
c6 == 1200Log[2,f6/f0],
c7 == 1200Log[2,f7/f0],
c8 == 1200Log[2,f8/f0],
c9 == 1200Log[2,f9/f0],
c10 == 1200Log[2,f10/f0],
c11 == 1200Log[2,f11/f0]},{c1,c2,c3,c4,c5,c6,c7,c8,c9,c10,c11}]
(* Results c0,…,c11 in cents, and f0,…,f11 in Hz *)
c0==0.&&c1==95.2136&&c10==999.128&&c11==1093.77&&c2==198.495&&c3==299.124&&c4==395.505&&c5==499.782&&c6==593.259&&c7==699.637&&c8==797.169&&c9==896.314&&f0==249.072&&f1==263.154&&f10==443.573&&f11==468.497&&f2==279.331&&f3==296.049&&f4==312.998&&f5==332.43&&f6==350.873&&f7==373.109&&f8==394.732&&f9==417.997
Reduce[{
(* Pitches f0...f11 form an octave with relationship on the
Circleoffifths according to Bach's WTC diagram read right to
left *)
3f10  4f5 == 2, (* right end: 4^{th} beating twice per second *)
3f5  4f0 == 2, (* loop 11: 4^{th} beating twice per second *)
3f0  2f7 == 2, (* loop 10: 5^{th} beating twice per second *)
3f7  4f2 == 2, (* loop 9: 4^{th} beating twice per second *)
3f2  2f9 == 2, (* loop 8: 5^{th} beating twice per second *)
3f9  4f4 == 2, (* loop 7: 4^{th} beating twice per second *)
3f4  2f11 == 0, (* loop 6: pure 5^{th} *)
3f11  4f6 == 0, (* loop 5: pure 4^{th} *)
3f6  4f1 == 0, (* loop 4: pure 4^{th} *)
3f1  2f8 == 1, (* loop 3: 5^{th} beating once per second *)
3f8  4f3 == 1, (* loop 2: 4^{th} beating once per second *)
3f3  2f10 == 1, (* loop 1: 5^{th} beating once per second *)
(* Cents corresponding to pitch relations *)
c0 == 1200Log[2,1.0],
c1 == 1200Log[2,f1/f0],
c2 == 1200Log[2,f2/f0],
c3 == 1200Log[2,f3/f0],
c4 == 1200Log[2,f4/f0],
c5 == 1200Log[2,f5/f0],
c6 == 1200Log[2,f6/f0],
c7 == 1200Log[2,f7/f0],
c8 == 1200Log[2,f8/f0],
c9 == 1200Log[2,f9/f0],
c10 == 1200Log[2,f10/f0],
c11 == 1200Log[2,f11/f0]},{c1,c2,c3,c4,c5,c6,c7,c8,c9,c10,c11}]
(* Results c0,…,c11 in cents, and f0,…,f11 in Hz *)
c0==0.&&c1==100.628&&c10==1001.5&&c11==1096.72&&c2==197.01&&c3==301.286&&c4==394.763&&c5==501.141&&c6==598.673&&c7==697.818&&c8==800.633&&c9==895.273&&f0==279.331&&f1==296.049&&f10==498.145&&f11==526.309&&f2==312.998&&f3==332.43&&f4==350.873&&f5==373.109&&f6==394.732&&f7==417.997&&f8==443.573&&f9==468.49
[1] John Charles Francis, ‘The Esoteric Keyboard Temperaments of J. S. Bach’, Eunomios, Feb. 2005.
[2] The mirror forms had to be cut from the previous paper as the source file was too large for Acrobat.
[3] Andreas Sparschuh, Deutsche Mathematiker Vereinigung Jahrestagung, 1999.
[4] Zapf’s temperament derivation is available to members of his Clavichord Discussion Group at: http://launch.groups.yahoo.com/group/clavichord/files
[5] Keith Briggs, ‘Letter to the Editor’, Early Music Review, May 2003.
[6] In fact, two circles.
[7] Bradley Lehman, ‘Bach’s extraordinary temperament: our Rosetta Stone’, Vol. xxxiii, No. 1 & No. 2, Early Music, 2005.
[8] For his reading, Lehman turns Bach’s diagram upsidedown, explaining this curiosity as a facet of Bach’s pedagogic method.
[9] Lorenz Christoph Mizler, Neu eröffnete mus. Bibliothek oder gründliche Nachricht nebst unparteiischen Urteil v. mus. Schriften u. Büchern, vol. 1, Part 3, Leipzig 1737, p. 55.
[10] BD III, no. 801.
[11] Mark Lindley, ‘A Quest for Bach’s Ideal Style of Organ Temperament’, Stimmungen im 17. und 18. Jahrhundert : Vielfalt oder Konfusion? Stiftung Kloster Michaelstein, 1997, p. 4567 .
[12] Lindley notes in this context, C. P. E Bach’s remark that his father was not much given to theory.
[13] The emphasis is that of Lindley.
[14] In Bach’s case, we have testimony from Kirnberger as reported by Wilhelm Friederich Marpurg in his Versuch über die musikalische Temperatur,
[15] Marpurg, op. cit., p. 194.
[16] Richard Jones,
[17] Bach took issue with the temperament of at least one organ builder. For Andreas Sorge, commenting in 1748 on the tuning system of the organ builder Gottfried Silbermann, mentions Bach as having described four specific triads resulting from Silbermann's method as having a barbaric nature intolerable to a good ear (BachDokumente II, no. 575). Edward John Hopkins, also relates an anecdote whereby J. S. Bach as auditor of Silbermann's instruments said "You tune the organ in the manner you please, and I play the organ in the key I please"; following his remark with a Fantasy in Aflat major causing Silbermann to retire to avoid his own "wolf" (The Organ, London, 1855, p.143.).
[18] Bruce Haynes, A History of Performing Pitch, Scarecrow Press, 2002.
[19] Owen Jorgensen, Tuning,
[20] Jorgensen refers here to intervals within an octave.
[21] Jorgensen also cites Roger North’s tuning instructions of 1726, where he revealed some professional tuners were listening to the beats of fifths, which should beat at the same speed as slow quavers.
[22] Carl Philipp Emmanuel Bach, Versuch über die wahre Art das Clavier zu spielen, 1753, p. 10.
[23] Most, but not all, so excluding Equal Temperament.
[24] This remark reflects an understanding that a tempered fifth and its inversion downwards as a fourth must beat identically.
[25] «Beyde Arten von Instrumenten müssen gut temperirt seyn, indem man durch die Stimmung der Qvinten, Qvarten, Probirung der kleinen und grossen Tertien und gantzer Accorde, den meisten Qvinten besonders so viel von ihrer größten Reinigkeit abnimmt, daß es das Gehör kaum mercket und man alle vier und zwantzig Ton=Arten gut brauchen kan. Durch Probirung der Qvarten hat man den Vortheil, daß man die nöthige Schwebung der Qvinten deutlicher hören kan, weil die Qvarten ihrem Grund=Ton näher liegen als die Qvinten. Sind die Claviere so gestimmt, so kan man sie wegen der Ausübung mit Recht für die reinste Instrumente unter allen ausgeben, indem zwar einige reiner gestimmt aber nicht gespielet werden. Auf dem Claviere spielet man aus allen vier und zwantzig Ton=Arten gleich rein und welches wohl zu mercken vollstimmig, ohngeachtet die Harmonie wegen der Verhältnisse die geringste Unreinigkeit sogleich entdecket. Durch diese neue Art zu temperiren sind wir weiter gekommen als vor dem, obschon die alte Temperatur so beschaffen war, daß einige Ton=Arten reiner waren als man noch jetzo bey vielen Instrumenten antrift. Bey manchem andern Musico würde man vielleicht die Unreinigkeit eher vermercken, ohne einen KlangMesser dabey nöthig zu haben, wenn man die hervorgebrachten melodischen Töne harmonisch hören solte. Diese Melodie betrügt uns oft und läßt uns nicht eher ihre unreinen Töne verspüren, bis diese Unreinigkeit so groß ist, als kaum bey manchem schlecht gestimmten Claviere.»
[26] Jorgensen notes the erroneous historical belief that beats of fourths and fifths occurred between fundamentals.
[27] This is the octave referenced by the opening notes of the Praeludium in C BWV 846.
[28] The author’s previous article introduced this nomenclature.
[29] This procedure uses only one beat rate rather than two; that rate being exactly one beat per second (i.e., 60 beats per minute). Either the fourth or fifth may be used to hear the beats, as suggested by C. P. E. Bach. At each tuning step, the octave is set from a previously tuned note, and then the intervening note is tuned such that the fourth and/or fifth beat once per second. It is vital that such beating is achieved by narrowing the fifth, or equivalently by widening the fourth.
[30] David Griffel first suggested some years ago that the ‘C’ in ‘Clavier’ might be a pitch reference for a righttoleft reading of Bach’s diagram. This idea has been followedup in a previous paper by the author and most recently by Lehman. While the hook on the ‘C’ is a commonly found artefact in Bachrelated manuscripts, it is indeed possible that Bach intentionally placed the ‘C’ to show the righttoleft cornetton solution commences on F rather than the more usual C.
[31] All octaves, as well as fifths marked ‘0’, should be tuned pure. The fifths marked ‘1’ are narrowed by raising the pitch of the lower note until they beat once per second. The fourth can also be used to set or detect these beat rates, as the indicated fourths and fifths beat identically when the octave is pure.
[32] Optimisations can be made in like manner to Figure 2.
[33] Simple enharmonic spellings are used from now on.
[34] The cammerton and cornetton temperaments are exact transpositions.
[35] Including a 0 rather than the final 2 yields a Pythagorean third, while including a 1 rather than the final 2 breaks the pattern of grouping similar fifths, and requires a less convenient beat rate for tuning.
[36] Based on principles of gematria, “BACH” equates to 14.
[37] Historical precedents for 1/7comma usage include the Werckmeister Septenarium temperament.
[38] The equalbeating tuning method uses a rate of one beat per second to represent 1/14comma tempering and two beats per second to represent 1/7comma adjustments, where the requisite intervals are in the octave starting on Middle C. When transposed an octave lower, intervals beating twice per second will beat once per second, so 1/7comma adjustments can also be interpreted as an octave shift downwards, where the interval beats once per second.
[39] Georg Andreas Sorge (17031778) was the fifteenth member of the Mizler Society, joining immediately after Bach. The cornetton temperament may have been preserved on the
[40] The above were largely derived from J. Murray Barbour’s, Tuning and Temperament,
[41] The suffix nP indicates that the nth fraction of the Pythagorean comma is being considered as the smallest tempering unit.
[42] The 1/12^{th} comma approximation is the Lehman reading.
[43] This temperament has been mentioned by Lehman, not as a derivation from Bach’s diagram, but as a measure to accommodate modern transcriptions.
[44] See Bruce Haynes, A History of Performing Pitch.
[45] The computersupported analysis has been performed with the aid of the symbolic equation solving program, Mathematica (Version 4.2.1). The beat rate of each fourth is given by the difference in the frequency of the 4^{th} harmonic of the lower note and the 3^{rd} harmonic of the higher one, while the beat rate of each fifth is given by the difference in frequency between the 3^{rd} harmonic of the lower note and the 2^{nd} harmonic of the higher one. The tuning sequence starts at f0 moving towards the flats. Fourths and fifths are chosen appropriately so as to remain within the 12 contiguous semitones. The solution shown in the “results” section, shows that f0 is Middle C, implying that the cammerton octave starting on Middle C is tuned starting on C.
[46] The computersupported analysis has been performed with the aid of the symbolic equation solving program, Mathematica (Version 4.2.1). The beat rate of each fourth is given by the difference in the frequency of the 4^{th} harmonic of the lower note and the 3^{rd} harmonic of the higher one, while the beat rate of each fifth is given by the difference in frequency between the 3^{rd} harmonic of the lower note and the 2^{nd} harmonic of the higher one. The tuning sequence starts at f0 moving towards the sharps. Fourths and fifths are chosen appropriately so as to remain within the 12 contiguous semitones. The solution shown in the “results” section, shows that f0 is Middle C, implying that the cornetton octave starting on Middle C is tuned starting on F.
Last update: July 7, 2005