Contents

Introduction
Case study 3: Preludes and Fugues in C Major and D Major, WTC I
Case study 4: Prelude and Fugue in F Minor, WTC I
Case study 5: SDG-motive and symbolism in Prelude and Fugue in E-flat Major, WTC I
Summary and Conclusion

Introduction

The process Bach could have easily employed to achieve duration ratios involving integer durations was shown in Elaboration 4. It is a simple process that requires a little planning and just a few easy calculations. First, Bach would set a time constraint, such as two or four minutes. Second, Bach would multiply his ideal or assumed tempo by the number of beats and divide by the duration (in minutes) to achieve the ideal number of measures. Third, Bach would compose the movement with an important goal in mind—to attain the ideal number of measures.

Analysis shows integer durations like two, three, or four minutes—or durations very close to integer values, such as 1:58, 2:57, or 4:04—occur frequently in Bach’s works, probably about 40% of the time. About 60% of the time, however, analysis shows “imperfect” or non-integer durations like 1:42, 2:36, 3:50, or 4:15. In cases like these, the process of achieving duration ratios among movements is a little more complicated due to the presence of fractions or decimals, but nevertheless, can still be attained with a convenient method of calculation, known as cross-multiplication. Let us look at two preludes and fugues, which may illuminate this process: C Major and D Major, both from book 1 of The Well-Tempered Clavier.

Case study 3:  Preludes and Fugues in C Major and D Major, WTC I

The C-major Prelude and Fugue is one of the best examples of a 1:1 duration ratio with two-minute durations. Assuming a moderate tempo of quarter = 72 for the Prelude and a slightly slower tempo of quarter = 54 for the Fugue results in virtually equal durations of 1:57 and 2:00. Since both durations are two minutes or very close to two minutes, Bach could have achieved this simply by using the process outlined above.

The D-major pair consists of the same meter and number of measures as the C-major pair—both are in 4/4 with 35 and 27 measures, respectively. The tempos, however, are different. I assign a tempo of quarter = 84 to this Allegro-style Prelude, which makes the movement fast but at the same time not too fast to obscure the 32nd notes. It is significant that Bach did not mark the final measures Adagio (as it is often played), which implies a tempo for the whole Prelude that is no faster than the 32nds can be played cleanly. The Fugue resembles the style of a French overture, which I believe has the usual tempo of quarter = 63. Assuming a standard Allegro of quarter = 84 and standard overture tempo of quarter = 63 result this time not in virtually equal durations of two minutes, but 1:40 and 1:42 instead.

Equal durations of two minutes in the C-major Prelude and Fugue, WTC I

Prelude	4/4	35	Q = 72	1:56.66	2:00	1	2.8%
Fugue	4/4	27	Q = 54	2:00	2:00	1	0.0%

Equal durations of about 1:40 in the D-major Prelude and Fugue, WTC I

Prelude	4/4	35	Q = 84	1:40	1:40	1	0.0%
Fugue	4/4	27	Q = 63	1:42.85	1:40	1	2.9%

It is significant that the tempo relationship between the C-major and D-major pairs is exactly the same and both reduce to 4:3: 72/54 = 4/3 and 84/63 = 4/3. It is also significant that the measure counts of both pairs, 35 and 27, also reduce to a ratio very close to 4:3. In fact, had Bach added just one measure to each Prelude, 36, the measure ratios of 36/27 would equal 4/3 and the assumed tempos would result precisely in the respective 1:1 duration ratios of 2:00-2:00 and 1:42-1:42. Bach most likely was not aware of the actual durations of 1:42 and 1:40 for the D-major pair, but nevertheless, could have easily planned equal durations. After all, one need not know the precise duration of a movement in order to create a related duration for another movement. All that one needs is to set up a simple equation and solve for an unknown value. Let us retrace the way Bach could have achieved equal durations of 1:40 or 1:42 in the D-major pair.

Suppose Bach composed the Prelude first, chose a meter and a tempo to begin with (his “default” 4/4 Allegro of quarter = 84), and then composed the movement without much thought given to the number of measures, 35. Next, Bach chose a style, meter, and tempo for the Fugue before composing it, in this case apparently an overture style in 4/4 at its usual tempo of quarter = 63. Next, before composing the Fugue, Bach decided how its duration would relate to that of the Prelude. Let us assume in this case it was 1:1. Finally, before putting the Fugue down on paper Bach determined how many measures it should have to make its duration equal to that of the Prelude.

If both tempos and just one of the measure counts are known, among two movements with the same meter, the unknown measure count, “x”, can easily be determined by setting up one of two equations with ratios on either side. The first possibility is 84/63 = 35/x, which in prose translates to, “The ratio of the tempos, 84/63, equals the ratio of the measures, 35/x.” The second possibility is 84/35 = 63/x, which translates in prose to “The ratio of tempo to measures in the Prelude, 84/35, equals the ratio of tempo to measures in the Fugue, 63/x.” Next, “x” can easily be determined using cross-multiplication (multiplying the denominator of one side with the numerator of the other side, and vice versa). In this case, the unknown value “x” solves to 26 with a decimal (the decimal can be disregarded), meaning that the Fugue needs to have either 26 or 27 measures. Bach achieved 27 measures in this Fugue.

It is also possible that Bach composed the Fugue first, happened to end it by chance at 27 measures, and then determined 36 measures for the Prelude by setting up an opposite equation. It is also possible that Bach set up both measure counts at the same time, 36:27, so that their 4:3 ratio would equal the tempo ratio, 84:63 = 4:3, thus resulting in equal durations. Even though it is impossible to determine which of these three methods Bach would have used, it is apparent that he used at least one of them, since achieving equal durations of 1:40 by chance is highly unlikely. My analyses show hundreds or perhaps even thousands of accurate 1:1, 1:2, and 2:3 duration ratios of “imperfect” non-integer values like 1:42 or 2:36. (My final tally has yet to be determined.) Bach could have easily achieved any duration ratio he desired by simply setting up an equation and solving for an unknown value.

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Case study 4:  Prelude and Fugue in F Minor, WTC I

Let us look at another work, Prelude and Fugue in F Minor, also from book 1 of The Well-Tempered Clavier. The Prelude emulates the style of an Adagio suggesting the tempo of quarter = 48, which I find to be slow enough for this highly expressive movement, but at the same time, is not too fast. The Fugue is definitely not an Allegro but also not an Adagio, which suggests a tempo somewhere in between. I find an Andante of quarter = 63 to be ideal for this Fugue, since it seems fast enough for ample movement but also slow enough to allow expression. These tempos not only seem ideal from a theoretical and musical standpoint, but they result in a precise 1:2 duration ratio of two non-integer values: 1:50-3:40. If this observation is correct, then Bach achieved his goal precisely with absolutely no margin of error. Let us now retrace Bach’s compositional process.

Suppose Bach composed the Prelude first, chose a meter and a tempo to begin with (a 4/4 Adagio of quarter = 48), and then composed the movement without much thought given to the number of measures, 22. Next, Bach chose a style, meter, and tempo for the Fugue before composing it, in this case, apparently a 4/4 Andante of quarter = 63. Next, before composing the Fugue, Bach decided how its duration would relate to that of the Prelude. Let us assume Bach decided the Fugue would last twice as long as the Prelude, or a 1:2 duration ratio. Finally, before putting the Fugue down on paper, Bach determined how many measures it should have to make its duration equal to that of the Prelude.

Bach could have easily done this by setting up an equation with the ratio of the tempo to measures on either side, multiplying the right side by 2 (for the 1:2 duration ratio), and solving for an unknown value. Thus, the equation becomes 48/22 = 63/x(2), which in prose translates to “The ratio of tempo to measures in the Prelude equals two times the ratio of tempo to measures in the Fugue.” In this case, “x” solves to 57 with a decimal, meaning that the Fugue should have 57 or 58 measures. Bach achieved 58 measures in this Fugue. It is also possible that Bach composed the Fugue first, happened to end it by chance at 58 measures, and then determined 22 measures for the Prelude by setting up an opposite equation.

1:2 duration ratio of 1:50-3:40 in the F-minor Prelude and Fugue, WTC I

Prelude	4/4	22	Q = 48	1:50	1:50	1	0.0%
Fugue	4/4	58	Q = 63	3:40.95	3:40	2	0.4%

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Case study 5:  SDG-motive and symbolism in Prelude and Fugue in E-flat Major, WTC I

Perhaps the best example of all that may convince skeptics that Bach sought proportional duration ratios is the E-flat major Prelude and Fugue from volume 1 of the WTC. I assign a slow largo-style tempo of quarter = 54 for the Prelude and a standard allegro of quarter = 84 for the Fugue, which result in the durations 5:11 and 1:45. This shows a virtually precise 3:1 duration ratio in that had Bach given the Prelude just one more measure, 71, the 3:1 ratio would be precise at 5:15-1:45. The skeptic is bound to question my assumption of such a slow tempo for the Prelude, which is understandable, since it is usually performed considerably faster at closer to quarter = 72; however, further analysis shows some extremely revealing “secret symbolic codes” that corroborate this tempo choice and the 3:1 ratio.

This Prelude and Fugue is unique in that it is the only pair in the WTC to feature a prelude which is actually a fugue (in fact, a double fugue with two subjects). It is also unique in that it is the only piece in the WTC in which Bach prominently displayed his well-known motto of his Lutheran faith, Soli Deo Gloria (Glory to God alone). That is, the musical pitches in German for “SDG” are “Eb-D-G” whose three pitches with intervals of a descending half-step and ascending fourth Bach used as the basis for the slow, quarter-note fugue subject in the Prelude. (In German, “E-flat” is called “Es”, which also represents the letter of the alphabet.) The SDG-motive appears a total of 21 times in various transpositions, almost half of which include the motive in its original, untransposed form. This very important SDG-motive originates in the opening two bars of the Prelude, whose contour is constructed around the motive in the soprano line in m.1 (beats 3 and 4) to m.2 (beats 1-4) as Db-C-F.1

Just to show that the SDG-motive is no coincidence, one needs to look no further than The Art of Fugue where the motive appears in all its permutations (mostly in transposition and inversion) no fewer than a total of 87 times in Contrapuncti 8, 10, and 11. (Contrapuncti 8 and 11, the two triple fugues, use the same three subjects but inverted.) The first subject of Contrapunctus 8 and the second subject of Contrapunctus 11 contain the SDG-motive in transposition and inversion, respectively, while the first six pitches in Contrapunctus 10 present the motive in inversion and transposition. What is more, Bach ingeniously combined the SDG-motive with the well-known BACH-motive in Contrapuncti 8 and 11 whenever the corresponding subjects are presented simultaneously. (In German, “Bb” is referred to as “B” while “B” is referred to as “H”.) For example, Contrapunctus 8 presents SDG and BACH in their original, untransposed forms in m.44-45 and 159-160 with SDG in the bass and BACH in the soprano (in retrograde). In sum, Contrapuncti 8 and 11 present no fewer than a total of 20 simultaneous statements of SDG and BACH in various permutations and the complete The Art of Fugue contains well over one-hundred statements of SDG in all its permutations.

Extensive monographs and biographies have been written on J.S. Bach, which often include detailed analyses of his works, yet not one to my knowledge has even mentioned the SDG-motive. Moreover, highly detailed analyses and complete books devoted to The Art of Fugue exist, yet to my knowledge the SDG-motive has never even been mentioned even though it is clearly the most prevalent motive in the entire work (perhaps even more so than the opening subject). That a rather obvious motive like this has eluded even the most erudite and detail-oriented of scholars suggests that much more about Bach remains to be discovered, some things of which may be sitting right in front of our eyes. (For example, the SDG-motive does not get any more obvious than in m.3 of the extremely popular “St. Anne” Fugue for organ. Why has this never been mentioned?)

One of my unpublished studies (from 1994) investigates Bach's use of the SDG-motive, which shows that it was probably the most used motive in Bach’s musical vocabulary. This includes not just obvious instances like The Art of Fugue, the E-flat major Prelude from WTC I, the E-flat major Fugue from WTC II, and the E-flat major Fugue for organ (“St. Anne”), but more discrete and disguised instances in which Bach used the motive ingeniously in various guises.

For example, in the A-major Fugue from WTC I the first pitch (A) is followed by two eighth rests, which is then followed by two more pitches (G#-C#) that spell out a transposition of SDG. In the B-flat minor Fugue from WTC I the first three pitches, Bb-F-Gb, present the SDG-motive in retrograde, however ingeniously disguised, since the “Gb” is transposed up an octave. Even seemingly ordinary fugue subjects and melodies often contain SDG in one or more of its permutations, which Bach was certainly well aware of. For example, the famous subject of the C-major Fugue from WTC I states SDG in transposition in pitches 6-8 (F-E-A). But this is just the keyboard music. Analysis of Bach's complete ouevre shows a staggering number (probably in the thousands) of instances of the SDG-motive. For example, Cantata 171, “Gott, wie dein Name, so ist auch dein Ruhm”, translated as “God, as Thy Name is, so is Thy Praise”, opens very boldly with a choral fugue subject on the pitches D-C#-F#, which is SDG in transposition. And the words that accompany these pitches are “Gott, wie dein Name so” or “God, as thy Name is”. In this cantata, Bach very boldly and without hindrance accompanies “God as Thy Name is” with the musical pitches that symbolize “Glory be to God alone“. It is vital that one realizes how important the SDG-motive was to Bach in order to take seriously the following symbolic interpretation of the E-flat major Prelude and Fugue and to understand how this relates to tempo and duration.2

It is extremely revealing that not only does Bach thoroughly permeate the E-flat major Prelude with the SDG-motive, but that in order to make the motive possible in its untransposed form Bach had no other choice than choosing the key of E-flat major, and thus, three flats. (Also, look at two other works in E-flat major:  Fugue #7 from WTC II and the E-flat major Fugue for organ, “St. Anne”. The former states SDG no fewer than 12 times while the latter states SDG no fewer than 15 times, all either in original or transposed forms.) But the symbolic connotations go far beyond just the 3-note SDG-motive and a key signature with 3 flats. In fact, detailed analysis of this Prelude and Fugue reveals a literally jaw-dropping labyrinth of symbolic elements based exclusively on the numbers 3 and 7, which symbolize the 3-pitch SDG-motive (possibly also the Christian Trinity) and the number traditionally associated with the Christian ideal of “perfection” (i.e., the 7 days of God’s creation.)

I am convinced that no fewer than 95% of these elements can possibly be coincidental, but that Bach either deliberately planned each one or was well aware of virtually every one of them if they happened to work out coincidentally. In sum, the E-flat major Prelude and Fugue from WTC I is arguably the most symbolic and personalized of all pairs in the entire WTC and quite possibly in Bach’s entire ouevre. Are you ready? I will try to say everything in one paragraph, which I also summarize in the following list. Let me emphasize once again that virtually everything about this work is based exclusively on just two numbers, 3 and 7:

The E-flat major Prelude and Fugue from WTC I is the only pair in the entire first volume of the WTC to feature 3 fugue subjects, two in the Prelude and one in the Fugue, while the Prelude is the only piece in WTC I to feature the 3-note SDG-motive undisguised and in its original form. The work consists of 3 flats and is the 7th pair in the WTC. The SDG-motive in the Prelude appears a total of 21 times, which symbolizes the product of 3 and 7, while it appears in its original, untransposed form (Eb-D-G) a total of 10 times, which symbolizes the addition of 3 and 7. At the time of composition (in 1722) Bach was 37 years old. The Prelude begins with the 3rd finger playing the 7th letter of the alphabet (37), which in turn initiates the 7-note motive that is later combined with the 3-note SDG-motive (37). The number of measures of the Prelude, 70, symbolizes the number 7, while the number of measures in the Fugue, 37, symbolizes the unity of 3 and 7 as well as Bach’s age at this time. The difference between these measure counts, 33, symbolizes the number 3 doubled. The Fugue subject appears a total of 9 times, which symbolizes the product of 3 and 3. And now to the element of tempo and duration. Assuming a slow, deliberate largo-style tempo of quarter = 54 for the Prelude and a standard allegro of quarter = 84 for the Fugue results in a 3:1 duration ratio of 5:15-1:45, which symbolizes the number 3. (In other words, the Prelude lasts precisely three times longer than the Fugue.) But what is more, when added together these durations total precisely 7 minutes, while the difference between these tempos, 30, symbolizes the number 3. In sum, the 7th Prelude has a 7-note theme that begins on the 7th letter of the alphabet, has 70 measures, and lasts a total of 7 minutes with its Fugue to follow. Coincidence? I think not.

Symbolism with the numbers 3 and 7 in the E-flat major Prelude and Fugue, WTC I

1. This Prelude and Fugue is the only pair in the entire first volume of the WTC to feature a total of 3 fugue subjects—two in the Prelude (actually a double fugue) and one in the Fugue.
2. The Prelude is the only piece in the first volume of the WTC to symbolize with musical pitches, Eb-D-G, Bach's most important and sacred phrase, Soli Deo Gloria.
3. “Macro-unification” of the number 37 with respect to the beginning, ending, and the creator of the composition:  the 7-note theme in the Prelude begins with the 3rd finger playing the 7th letter of the alphabet (37); the Fugue consists of 37 measures; Bach was 37 years old at the time of publication (in 1722).
4. This Prelude and Fugue has 3 flats and is the 7th pair in the WTC, which symbolizes 37.
5. The Prelude consists of 70 measures, which symbolizes the number 7, and the Fugue consists of 37 measures, which symbolizes 3 and 7 combined. The difference between these two measure counts, 33, symbolizes the 3-note SDG-motive doubled.
6. The SDG-motive in the Prelude appears a total of 21 times (not including “tonal” entries), which symbolizes the product of 3 and 7.
7. The SDG-motive in the Prelude appears in its original, untransposed form (Eb-D-G) a total of 10 times, which symbolizes the addition of 3 and 7.
8. The two subjects in the Prelude consist of 3 and 7 notes so when combined (every time they appear) they symbolize the number 37, which represents the number of measures in the Fugue and Bach’s age at its composition. For example, in m.31-32 the 3-note SDG-motive appears in the bass while the 7-note motive introduced in m.1 appears simultaneously in the alto.
9. The respective tempos of quarter = 54 and 84 for the Prelude and Fugue result in a virtually precise 3:1 duration ratio of 5:15-1:45 with a discrepancy of merely one measure in the Prelude. That the Prelude lasts precisely three times longer than the Fugue symbolizes the 3-note SDG-motive upon which the entire work is based.
10. The durations of 5:15 and 1:45 not only relate 3:1 but add to precisely 7 minutes.
11. The difference between the two tempos, 30, symbolizes the 3-note SDG-motive upon which the entire work is based.
12. The difference between the two durations, 3:30, symbolizes the number 33 which in turn symbolizes the 3-note SDG-motive doubled. (See #5 above.)
13. The subject of the Fugue appears a total of 9 times, which symbolizes 3 x 3.
14. In sum, the 7th Prelude has a 7-note theme that begins on the 7th letter of the alphabet, has 70 measures, and lasts a total of 7 minutes with its Fugue to follow.

I have taken this seemingly whimsical and “speculative” diversion into alphanumeric symbolism for good reason, which is to corroborate the tempos of quarter = 54 and 84. If the present symbolic elements are not coincidental, meaning that Bach did indeed intend the 7th Prelude and Fugue to last 7 minutes which in turn have a 3:1 duration ratio to symbolize SDG—and all the other symbolic elements listed above—then the tempos of quarter = 54 and 84 stand proven. After all, these are the only tempos that produce a total duration of 7 minutes and a 3:1 duration ratio. If this is the case, then Bach intended the Prelude to be played in a slow and reverent fashion with one constant tempo throughout in order to translate into sound his most sacred and holy motto, Soli deo gloria. (Most performers choose two or three different tempos in this Prelude and play it considerably faster at about quarter = 72.)

What is also unique and special about this Prelude and Fugue is that it features arguably Bach's most “serious” and “holy” of Preludes in the entire WTC coupled with arguably the most “secular” and “humorous” of all Fugues in the entire WTC. To use an analogy, the Prelude is like a deep and meditative prayer, while the Fugue is like a day at the circus. Perhaps Bach chose this contrast to symbolize the whole of God’s creation, from the most holy and sacred to the most earthly and secular. Ever since discovering all these things, this pair has become by far my favorite in the entire WTC, since all the symbolic elements combined with the great musical contrast imbue the work with extra meaning in musical performance. Most performers ruin this Prelude with too fast of a tempo. The bottom line though, is that the tempos of quarter = 54 and 84 are the only tempos to produce a 3:1 duration ratio and a total duration of seven minutes. And considering that these tempos are proportionally unrelated, Bach could have achieved a 3:1 duration ratio here no other way than by employing the above process of cross-multiplication.

3:1 duration ratio and 7-minute duration in the E-flat major Prelude and Fugue, WTC I

Prelude	4/4	70	Q = 54	5:11.11	5:15	3	1.2%
Fugue	4/4	37	Q = 84	1:45.71	1:45	1	0.7%
(total)					7:00

Summary and Conclusion

To summarize, there exist only two ways Bach could have consciously achieved proportional duration ratios in his music. The first way was outlined in Part 1 of this series, Elaboration 4, where integer durations like two or four minutes were decided upon before putting pen to paper, in which case the presumed integer duration allowed Bach to determine the ideal number of measures. The second way, outlined on this page, is where duration ratios like 1:1 or 1:2 consisting of non-integer or “imperfect” durations like 1:42 or 3:50 were achieved by solving for an unknown value. These are two simple, foolproof methods that Bach could have easily employed. But just to be 100% certain that all this is no coincidence, let us analyze perhaps Bach’s grandest of works, the B-minor Mass. As we will see, Bach apparently used both these methods of cross-multiplication to achieve the unique and ingenious symmetrical architectural designs in his most monumental of works.

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