Contents

Case study 1: Prelude and Fugue in F Major, WTC II
Case study 2: Italian Concerto
Summary and Conclusion

Case study 1:  Prelude and Fugue in F Major, WTC II

My analytical system serves as a tool in the uncovering of durations and relationships between durations, which I refer to as “duration ratios.” In my essay, Bach, the “musical scientist,” and in the following analysis of the B-minor Mass, I give reasons why Bach would have sought duration ratios in his music. This elaboration, however, represents the first part of the inquiry into how Bach achieved proportional duration ratios in his music.

I understand that the idea of Bach deliberately planning a movement to be a certain number of measures and length of time in minutes and seconds seems farfetched to a lot of people. Once one realizes how easy this is to achieve though, one may in fact realize that this is not such a farfetched idea after all.

Let us assume the six axioms formulated in Elaboration 3. Suppose Bach were planning to compose a prelude and fugue. First he would choose the styles and meters of each piece. Let us use a concrete example, Prelude and Fugue in F Major from book 2 of The Well-Tempered Clavier. In this work, Bach decided that the Prelude would be in 3/2 and the Fugue would be in 6/16. It is clear that this Prelude has a “majestic” character and a slower rather than faster tempo, which Bach indicated by choosing 3/2 instead of 3/4. Had Bach wanted this Prelude to have a less majestic character and a faster tempo, he would have had no other choice but to notate it in 3/4.

Let us now suppose that Bach not only intended this Prelude to be slow and majestic, but even had an absolute tempo in mind, say, half = 54 (which I believe was Bach's usual 3/2 courante tempo). Let us now suppose that Bach decided this Prelude to last, say, four minutes. He would then have to calculate the number of measures in 3/2 to compose in order to achieve four minutes at his assumed tempo. This is very easy to do and is not “rocket science” at all. Simply multiply the tempo by 4 (54 x 4 = 216) and divide this by the number of beats per measure, 3 (216/3 = 72), which would have told Bach that he should aim for 72 measures in order to achieve a “perfect” four-minute Prelude. It is really that simple.

Let us now turn to the Fugue. Bach chose 6/16 for this Fugue, which indicates a rather lively tempo in a gigue style. The thirty-second notes in the final section, however, limit the tempo substantially, as Kirnberger makes clear: “Dance pieces involving 16th and 32nd notes have a slower tempo than those that tolerate only 8th and at the most 16th notes as the fastest note values in the same meter.” Like the Prelude, let us suppose Bach chose not only the 6/16 and gigue style, but an absolute tempo as well, say, dotted-eighth = 96 (which I believe was one of Bach's usual triplet gigue tempos).

Let us now suppose Bach wanted the Fugue to last exactly half as long as the Prelude, that is, intending a 2:1 duration ratio. Thus, Bach would have to calculate the number of measures in 6/16 to compose in order to achieve two minutes. Once again, this is very easy to do. Simply multiply the tempo by 2 (96 x 2 = 192) and divide this by the number of beats per measure, 2 (192/2 = 96), which would have told Bach that he should aim for 96 measures in order to achieve a “perfect” two-minute Fugue.

Let us now examine this Prelude and Fugue as it stands. The Prelude consists of 72 measures of 3/2, and the Fugue consists of 99 measures of 6/16. The point here is not to prove the tempos half = 54 and dotted-eighth = 96, but merely to show how easy it would have been for Bach to achieve idealized, designated durations. From a realistic, practical standpoint these are, nevertheless, very good tempos and I doubt that any better ones can be found. Assuming these to be the “definite” or “ideal” tempos referred to by Kirnberger, spreadsheet analysis indicates that Bach achieved the ideal number of measures for the Prelude, 72, but went over the ideal in the Fugue by just three measures, 99 instead of 96. This amounts to about four seconds actual discrepancy in playing time, or just 3.1% of the total duration.

2:1 duration ratio of 4:00-2:00 in the F-major Prelude and Fugue, WTC II

Prelude	3/2	72	H = 54	4:00	4:00	2	0.0%
Fugue	6/16	99	DE = 96	2:03.75	2:00	1	3.1%

Bach was only human, and if he had indeed planned special measure counts to achieve special durations, he most likely would not have been able to achieve it precisely in every case. This is simply very difficult to do. Planning special measure counts is easy, however, attaining this, even allowing for a small margin of error, requires the precision and tenacity of a scientist. And Bach was, indeed, something of a “musical scientist.”

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Case study 2:  Italian Concerto

Let us now turn to a very popular composition, the “Italian Concerto.” I assign the first movement quarter = 96, the second movement (Andante) eighth = 72, and the third movement (Presto) half = 108. These tempos are arrived at through process of elimination and simply by using good judgment. I choose a slightly faster tempo for the third movement than for the first movement, because Bach’s indication of “Presto” implies this and, from a purely subjective musical point of view, this seems like the correct choice. Most performers do play the third movement faster than the first movement anyway, so there is nothing radical about my tempo choices.

I choose eighth = 72 for the second movement, because the ornate writing and frequent 32nd notes suggest subdivision of the primary note value, which is the quarter note. I also choose eighth = 72 because Bach’s indication of “Andante” implies a moving tempo rather than a more stagnant one, which would be the case had Bach chosen “Adagio” instead; (which is the way most performers play it).

I believe these are quite simply the best tempos that can be found for the three movements of the Italian Concerto. Not only are they excellent tempos in actual practice, but calculating the resulting durations shows three movements very close to four minutes. The first movement consists of 192 measures of 2/4, whose tempo of quarter = 96 results in a duration of precisely four minutes. The second movement consists of 49 measures of 3/4, whose tempo of eighth = 72 results in a duration just over four minutes, 4:05. The third movement consists of 210 measures of 2/2, whose tempo of half = 108 results in a duration just under four minutes, 3:53. This suggests in his initial blueprint Bach was aiming for one less measure in the second movement, 48, and six more measures in the third movement, 216, to achieve precisely four minutes.

Three equal durations of four minutes in the Italian Concerto (F Major)

1. (Allegro)	2/4	192	Q = 96	4:00	4:00	1	0.0%
2. Andante	3/4	49	E = 72	4:05	4:00	1	2.0%
3. Presto	2/2	210	H = 108	3:53.33	4:00	1	2.9%

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Summary and Conclusion

As seen in the F-major Prelude and Fugue and the Italian Concerto, as well as hundreds of more examples not yet released to the public, integer durations like two or four minutes occur quite frequently in Bach’s works. The chances of this being coincidental are virtually zero, since integer durations occur so frequently within such small margins of error. In the Italian Concerto, Bach could have attained this four-minute ideal only by knowing his tempos in beats per minute beforehand, which would have allowed him to calculate the number of measures for which to aim. After all, three durations of four minutes at an average discrepancy of merely 1.6% do not just happen by themselves. And as we can see, the process Bach would have used to achieve such a plan is not a problem of “rocket science” at all, but is really quite simple.

First, Bach planned to compose three movements and decided that they would all be equal at four minutes. This is analogous to an architect planning three rooms of a house to be equal in dimensions. Second, Bach decided on the meters and tempos he would employ (i.e., those given in the spreadsheet above). Third, Bach calculated the numbers of measures that result in four minutes at the assumed tempos, which as shown earlier, is very easy to do. Finally, Bach put his pen to paper and composed the three movements with an important goal in mind—to achieve 192 measures in the first movement, 48 measures in the second movement, and 216 measures in the third movement. And this Bach certainly did with remarkable accuracy. Let us now continue by investigating the method Bach employed to achieve non-integer durations.

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