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Table 1: tuning of five consecutive perfect fifths on the circle of fifths The assumption will be that Wilhelm Friedemann was taught to tune his fifths justly (i.e. pure); this assumption being motivated by several factors: the ease of tuning in just fifths [6], the rapidity with which this can be done [7], and the historical precedent of employing just fifths. The resulting tuning is summarised in Table 1. Returning now to Figure 2 and keeping in mind the hypothesis that the trills embody a meaning for tuning purposes, it is observed that the trills are of two different kinds: the first four are mordants, while the latter is a, so-called, doppelt cadence u. mordant. For the time being, the focus will be on the four mordents; these being defined in the introductory table of ornaments that J. S. Bach included at the beginning of Wilhelm Friedemann’s Klavierbüchlein as indicated in Figure 3.
Figure 3: a mordant relates tonic and leading tone From this, it follows that the four mordents occurring in the first three bars of BWV 924 (Figure 2), identify four relationships between tonic and leading tone, as shown in Table 2.
Table 2: relationships implied by mordents
From the notes already tuned (Table 1), the leading notes in Table 2 must now be tuned. Since semitone relations cannot be tuned directly by ear with any useful accuracy, there is but one practical possibility: to tune by thirds. For the time being, the assumption will be to tune these thirds justly [8]; and under this condition the tuning shown in Table 3 can be derived. Note, it is possible, and indeed easier, to tune only one leading note from the third below and then proceed by tuning in perfect fifths as shown in Table 4 [9]. Using either of these equivalent procedures, the original circle of fifths in Table 1 is extended as shown in Table 5.
Table 3: tuning of the notes Bach marked by mordents as just thirds
Table 4: alternative method for tuning the notes Bach marked by mordents as just thirds
Table 5: extended circle of fifths
It can be observed that the second step of the tuning procedure dovetails nicely with the first, extending the circle of fifths by four steps. A further remark concerns the interval B - F# , which was not tuned explicitly, but rather defined implicitly as a result of other tuning operations: this interval is a “wolf”, which needs to be dealt with. This is optimally achieved by spreading the wolf equally across the intervals: A - E, E - B, B - F# .Note, that the tuning steps shown in Table 1 form part of the method of Pythagorean tuning, and so suffer from a perceived “defect”, namely, that the major thirds C1 - E1 and G1 - B1 are wide in comparison to the ideal of justly tuned thirds [10]. Accordingly, a prior, we might expect such notes to be tempered (narrowed) to remedy this problem; conveniently, distributing the wolf equally, also serves to narrow the thirds. Returning now to the score of BWV 924 (Figure 1), and once again keeping in mind the hypothesis that the trills embody tuning instructions, it is observed that the notes B1 and E1 in bars 3 and 5, respectively, are marked by trills which Bach called doppelt cadence u. mordant. In view of the placement of this type of trill on precisely those notes that a priori need tempering to distribute the wolf and “improve” thirds, the doppelt cadence u. mordant on E1 and B1 is taken to imply a tempering operation. In this regard, note the shape of these trills: both of which point downwards, the direction in which the E1 and B1 must be tempered to distribute the wolf and improve the thirds. No guesswork is needed concerning the amount of this tempering, since the fundamental rationale for the operation is to distribute the wolf in Table 5 evenly, i.e., to adjust these two notes such that the error in the wolf interval B1- F2# is spread evenly over the three [11] intervals A0- E1, E1- B1, B1- F2# . This requirement precisely determines the tempering needed for E1 and B1, and after the relevant calculations are performed, the results indicated in Table 6 and Table 7 are obtained. Returning again to the score of BWV 924 (Figure 1), it will be seen that the G1# in bar 4 is also marked by a doppelt cadence u. mordant, but that it points upwards, not downwards. Earlier, the doppelt cadence u. mordant on E1 and B1 were taken to imply a narrowing of the thirds C1 - E1 and G1 - B1, and by analogy the inverted doppelt cadence u. mordant on G1# can be interpreted as a widening of the interval E1 - G1# . There are two ways such widening can be achieved:
Two semitones still remain to be specified, B@ and F respectively, there being no specific information from any trills in the score as to how the intervals should be tuned. Assuming tuning by just fifths, then starting at D# /E@ the intervals E@ - B@ and B@ - F can be tuned. Alternatively, starting at C, the intervals F - C and B@ - F can be tuned in just fifths. In fact, it turns out that with the tuning steps considered so far, the difference is virtually imperceptible [13].
Table 6: optimum tempering to distribute the wolf
Table 7: the fifths resulting from tempering in Error! Reference source not found.
Table 8: method for tuning the notes Bach marked by mordents as sharp thirds
It is now shown how the tuning method above can be modified to tune the thirds in Table 4 wide, rather than justly. The procedure is as follows:
The tuning systems considered so far are summarised in Table 10. The derived temperaments are referred to as Temperament I, Temperament II, Temperament III and Temperament IV, respectively. Their corresponding deviations from 12-tone Equal Temperament are given in Table 11, while a comparison between the fifths of all the derived temperaments is provided in Table 12. Thereafter, the characteristics of specific temperaments are described by tables of intervals, tables of thirds and fifths, and tables of the major and minor tetrachords.
Discussion The structure of the sequence of fifths within the derived temperaments can be represented as shown in Table 9, where J, denotes a just fifth, N, a fifth narrowed by 1/3 syntonic comma, W, a fifth widened by 1/3 syntonic comma, E, an Equal Tempered fifth and X, the fifth that is 1/3 syntonic comma smaller than an Equal Tempered fifth. Considering the major thirds of Temperament I (Table 14) and Temperament II (Table 18), it can be observed that the two narrowest major thirds are D - F# and A - C# , and that these are tuned justly. The widest occur at F# - A# and C# - F, and are Pythagorean thirds, while the remaining thirds in Table 14 fall between these extremes in a progressive manner according to the circle of fifths. Looking now to the major (Table 15) and minor (Table 16) tetrachords for Temperament I, it will be noticed that duplication occurs: specifically, the major tetrachords starting on A@ and E@ are the same; likewise those starting on B@ and F. With regard to the minor tetrachords, the ones starting on B@ and F are identical. Comparing now with Temperament II, where, G# /A@ has been sharpened, it can be observed that all tetrachords are unique (Table 19 and Table 20). This provides an excellent rationale for sharpening G# /A@ , namely, to add explicit key colour and variety; a further rationale is the creation of a wide fifth which adds further colour to the temperament. A similar observation can be made regarding Temperament III: namely, that the major tetrachords on A@ and E@ are identical, as are those on B@ and F (Table 23). In the case of the minor tetrachords, there are also two duplications: those rooted in A@ and E@ are the same, and likewise those rooted in B@ and F (Table 24). Comparing now with Temperament IV, where explicit sharpening of G# /A@ has occurred, it is important to note that all major (Table 27) and minor (Table 28) tetrachords have been rendered unique as a result of this tempering operation. That the tempering corresponding to the upward pointing doppelt cadence u. mordant should be exactly that needed to render all the tetrachords unique, is hardly a coincidence. Accordingly, the prescriptive interpretation of the trill on G# /A@ in Figure 1 can be assumed. This eliminates Temperaments I and III from consideration. By the above assumption, Temperaments II and IV remain as viable options. If the third-hand account by Friedrich Wilhelm Marpurg [7] is taken at face value: namely, that Kirnberger was expressly required by Bach to tune all the thirds sharp, then both Temperaments I and II must be rejected as invalid options. That would leave uniquely Temperament IV as the implied temperament. However, the possibility remains that Marpurg’s statement was merely a form of words to express Well Temperament. It is also possible that Marpurg misconstrued, or even misrepresented, Kirnberger’s remarks by implying that each and every fifth is tuned sharp. Alternatively, Bach may have taught Kirnberger a different tuning system to his own. Moreover, it is possible, that Wilhelm Friedemann may have used the simpler procedure of tuning the thirds justly, while his father adopted the expedient of tuning the thirds wide. It likewise conceivable, that J. S. Bach may have used different tuning variants at different stages of his career and in different situations. Moreover, if J. S. Bach is assumed to have derived the system embedded in BWV 924 himself, that suggests an experimental disposition. and accordingly, he may have used both of these variants at one time or another.
Table 9: structure of derived temperaments in terms of fifths One final area of uncertainty to be addressed concerns the tuning of B@ and F, and whether they should these be tuned from E@ or C (or both). In the case, of Temperaments I and II, the choice is of little import, with a resulting difference in tuning of two cents. However, in the case of Temperaments III and IV, the difference is significant. The following options can be considered:
Temperaments I, II, III and IV have been compared with other historic temperaments and the results are shown in Figure 4, Figure 5, Figure 6, and Figure 7, respectively. Each figure indicates the calculated distance [19] of the historic temperament from the temperament derived from BWV 924, providing a measure of how far the historic temperament differs, on average, from those derived from BWV 924. The following observations apply to all the derived temperaments:
Conclusions This paper has reported the discovery by the author of several tuning systems derived from an analysis of BWV 924. These systems were determined under the assumption that BWV 924 was explicitly constructed by Johann Sebastian Bach to provide a pedagogic tuning aid for his young son, Wilhelm Friedemann; the purpose being to serve as a reminder of lessons already taught. Lack of information with regard to the exact teaching of J. S. Bach led to the range of candidate solutions [21], being considered.
Two possible interpretations with regard to tempering A@ /G# were possible: one pedagogic (and arguably contrived), and the other prescriptive: the former corresponded to Temperaments I and III, the latter to Temperaments II and IV. It was found that explicitly sharpening G# /A@ renders all major and minor tetrachords unique, and also offers the benefit of one wide fifth, adding colour and variety to the keys. The author notes, that in the case of Temperament IV, for example, the key of A@ , alluded to by Edward John Hopkins [6], is most charming. In general, the variety and colour introduced by sharpening G# /A@ is very pleasing. The choice between Temperaments II and IV, depends on the credence given to the remark of Friedrich Wilhelm Marpurg [7]. Accepting Marpurg’s proposition, implies accepting Temperament IV. The implications of this analysis will be relevant for musicians concerned with the historical informed performance practice of Bach’s music. The results obtained have shown that several historic temperaments, including Werckmeister and Kirnberger, are unlike those derived from BWV 924; the Kellner temperament is likewise very different. Equal Temperament is a viable performance option, but lacks the benefit of justly tuned fifths and key colour. Accordingly, musicians are invited to experiment with the temperaments presented in this paper, and Temperament 4 is recommended as the starting point.
Table 10: derived temperaments based on tempering by 1/3 syntonic comma [24]
Table 11: deviation of temperaments from Equal Temperament [25]
Table 12: comparison of fifths
Table 13: Temperament I interval analysis
Table 14: Temperament I thirds and fifths
Table 15: Temperament I major tetrachords
Table 16: Temperament I minor tetrachords
Table 17: Temperament II interval analysis
Table 18: Temperament II thirds and fifths
Table 19: Temperament II major tetrachords
Table 20: Temperament II minor tetrachords
Table 21: Temperament III interval analysis
Table 22: Temperament III thirds and fifths
Table 23: Temperament III major tetrachords
Table 24: Temperament III minor tetrachords
Table 25: Temperament IV interval analysis
Table 26: Temperament IV thirds and fifths
Table 27: Temperament IV major tetrachords
Table 28: Temperament IV minor tetrachords
Table 29: sizes of intervals occurring in each temperament [34] Figure 4: Temperament I comparison with historic temperaments [35]
Figure 5: Temperament II comparison with historic temperaments
Figure 6: Temperament III comparison with historic temperaments
Figure 7: Temperament IV comparison with historic temperaments Music Examples See: Keyboard Temepramernt of J.S. Bach – Music Examples
References
Footnotes [1] „In denen vier schlimmen Triadibus aber ist ein rauhes, wildes, oder, wie Herr Kapellmeister Bach in Leipzig redet, ein barbarisches Wesen enthalten, welches einem guten Gehör unerträglich fällt“ [2] In the terminology of tuning, a “wolf” is an interval that is out of tune and metaphorically howls. [3] „alle grossen Terzen scharf“ [4] The fourteen trills in BWV 924 are perhaps indicative of Bach’s personal signature, since according to the principles of Gematria, the name Bach is represented by the number 14 (B A C H = 2 + 1 + 3 + 8 = 14). [5] The tenor octave is, incidentally, the area where the tension of the strings of a harpsichord is greatest; the practical corollary being that this octave should be tuned before others. [6] It is relatively straightforward to tune two notes so that they are a just fifth apart, the procedure being to approximate the fifth and then tune to eliminates beats between the notes. Next to tuning in unisons and octaves, this is the most elementary of tuning procedures. [7] Forkel’s observation regarding the maximum 15 minute keyboard tuning period needed by Bach motivates consideration of simple, rapid, procedures. [8] The alternative hypothesis that the thirds were tuned wide is considered later in the paper. [9] This simpler method is possible because the reference notes for tuning purposes ( Table 1) are separated by justly tuned fifths. It is a matter of conjecture which method was taught to Wilhelm Friedemann. In fact, all the notes needed to tune each third individually are already tuned, and to explicitly tune each interval in thirds would certainly have provided needed practice with the method, while avoiding the requirement for the young Wilhelm Friedemann to tune a circle of fifths in remote keys. However, if the alternative hypothesis that the thirds should be tuned sharp is entertained, then almost certainly but one third would be explicitly tuned, and the remaining notes tuned by fifths.[10] While the Pythagorean third was considered consonant by Bach’s time, it still lay at the limit of acceptability and, accordingly, was often banished to remote keys. [11] When the wolf is distributed over A0 : E1, E1 : B1, B1 : F2#., each of the intervals contains only 1/3 of the error that was originally present in the interval B1 : F2#, and this small amount is no longer a problem for the ear. [12] Equivalently, the interval G1# - D1# should be narrowed by 1/3 syntonic comma. [14] The syntonic comma is the interval between two notes of frequency ratio 81:80, or around 21 cents (1200 cents is an octave). It is defined as the difference between the interval of four contiguous just perfect fifths and the interval of two octaves and a just major third. A just perfect fifth has notes with a frequency ratio of 3:2 which is approximately 702 cents, and four of them are roughly 2807 cents. A just major third has notes with a frequency ratio of 5:4, which is equal to approximately 386 cents, implying that two octaves and a major third are about 2786 cents. The difference is around 21 cents. [15] Alternatively, tune F1# such that the interval F1# - B1 is a just fourth. In either case, the effect is to widen the third D1 - F1# by 1/3 syntonic comma. [16] If the first option of step 6 is assumed, continue tuning a just circle of fifths until F. [17] Equivalently, narrow the interval G1# - D1# by 1/3 syntonic comma (i.e., similar tempering to other tempered fifths). [18] These tuning choices yield significantly different results. Later in the paper, it is determined which option is the correct one. [19] For the purposes of this comparison, all temperaments were normalised to C and a Euclidian metric defined by taking the square root of the sum of the squares of the deltas for each note. [20] Barnes statistical method does not claim to exactly reproduce Bach’s temperament, but rather to approximate it. [21] Alternatives, based on reversing the sequence of tuning operations derived from BWV 924 have not been presented in this paper. Such options have been analysed, however, but led to less satisfactory results. [22] This interval is tuned implicitly, of course. [23] About 2 cents more in fact. Table 11 shows alternative tunings for F and Bb.[25] The values in curved brackets are obtained by tuning F justly from C and Bb justly from F. [26] Diatonic semitone of ratio 256 : 243 in Pythagorean tuning (Limma). [27] Chromatic semitone of ratio 135 : 128 in Just Intonation and interval C# - D in Pythagorean tuning. [28] Diatonic semitone F# - G in Pythagorean tuning. [29] Chromatic semitone F - F# in Pythagorean tuning. [30] Diatonic semitone of ratio 16:15 in Just Intonation. [31] Major third of ratio 5 : 4 in Just Intonation. [32] Major third of ratio 18/55. [34] The comments provided on this table are not intended to be complete! [35] The centre of the figure denotes the temperament derived from BWV 924, and the distance from the centre is a measure of the fit of the historic temperament from that derived from BW924.
Written by Charles Francis (June 2004) © |
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