Contents

The idea of proportional duration ratios
Factors in determining tempo
Musical comparisons and cross-referencing
Description of the analytical system, spreadsheets, and statistics
Explanation of my methodology and the scientific method
Endnotes

The idea of proportional duration ratios

In 1959-60, Arthur Mendel and Bernard Rose exchanged views on proportional tempo relationships in Bach’s music.1  As a by-product of this debate, an intriguing theory emerged, one of proportional relationships between performance times, which I refer to as “duration ratios.” For example, Arthur Mendel argued for the same eighth-note pulse in the Gloria and Et in terra pax sections of the B-minor Mass, which results in a virtually precise 1:2 duration ratio since the sections consist of 302 and 606 eighth notes, respectively. In the Sanctus and Pleni sunt coeli sections of the Mass, which appear in 4/4 with triplet eighths followed by 3/8, Bernard Rose argued for equal quarter-note speed, which Robert L. Marshall points out results in a nearly precise 1:1 duration ratio due to the 376 binary eighth notes and 363 eighth notes, respectively.2

Don O. Franklin has addressed the issue of proportional duration ratios in several other works. Franklin explains that when the appropriate tempo giusto (i.e., the most natural tempo) is assumed, a proportional duration ratio often results between adjacent movements.3  For example, Franklin points out that in the C-major Prelude and Fugue from book 1 of The Well-Tempered Clavier, a slightly slower tempo for the 27-measure Fugue, in 4/4, than for the 35-measure Prelude, also in 4/4, results in roughly the same duration for both pieces.4

Aside from a few examples discussed by Mendel, Rose, Marshall, and Franklin, the topic of proportional duration ratios remains largely unexplored. (Proportional tempo relationships, however, have been much more widely discussed.) But why stop at a few movements from the B-minor Mass and some other isolated pieces? After all, Bach was arguably the most methodical and systematic composer in the history of music—a “musical scientist” as Christoph Wolff so amply puts it (see Elaboration 6)—making it likely that if such duration ratios occur in a few works, then similar ratios are bound to occur in more works. Being the methodical “musical scientist” he was, it seems hardly possible that Bach would have sought duration ratios only in the Gloria, Sanctus, and a handful of other works, but nowhere else.

If the above examples of duration ratios are just rare exceptions—meaning that they are coincidental and Bach did not consciously plan them—then four venerable Bach scholars have merely theorized in vain and the matter need not be investigated further. On the other hand, if these examples are not coincidental and Bach did indeed plan them, then it becomes our obligation as performers and scholars to analyze the temporal dimensions of all of Bach’s works in order to determine how often Bach planned duration ratios and what kind of operating procedures he employed. This is what I have concerned myself with as a scholar since 1992, which has required me to develop a special and unique analytical system and work within my own terms.

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Factors in determining tempo

It would not be enough merely to count measures and beats and form conclusions based on these factors. For example, theorizing about the C-major Prelude and Fugue having tempos that result in equal performance durations may appeal to abstract-minded theorists; however, offering no absolute tempos or concrete durations in minutes and seconds makes this theory highly impractical and of hardly any value to performers. As a performer, I am interested in specifics and absolute values. I want numbers. I want to know the precise tempos and durations Bach may have planned for the C-major Prelude and Fugue, if he indeed planned such things. In order to bring absolute tempos into the equation, I have assumed a complete mathematically ideal system of tempos from the outset, which I introduce in Elaboration 2. For now though, I would like to explain the factors involved with making tempo choices.

Assigning tempos to works seems like a subjective process, which is why most objective and skeptically-oriented scholars have a difficult time accepting absolute tempos. Of course, some degree of subjectivity is impossible to avoid when assigning tempos; however, it is possible to subdue this subjectivity by making informed and unbiased decisions based on a variety of objective factors. In sum, at least ten factors contribute to the determination of tempo for any one composition:

Ten factors in the determination of tempo

1. style of composition (dance or other style)
2. tempo words (if any)
3. meter
4. density or sparsity of note values
5. rhythmic motives
6. melodic motives
7. rhythmic phraseology
8. harmonic rhythm
9. text (if vocal work)
10. articulation marks (staccato, legato)

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Musical comparisons and cross-referencing

Perhaps one of the most overlooked and under-researched topics in Bach studies is that to which I refer as “cross-referencing,” which allows the formation of broad stylistic categories or “families.” Cross-referencing occurs when similarities in two or more works justify the same or a related tempo. If done correctly, cross-referencing is probably the most effective and reliable method on which to rely when assigning tempos. Unfortunately, it is very difficult to apply because it requires that one have nearly encyclopedic knowledge of Bach’s entire output. In essence, cross-referencing allows one to see “all the trees in the forest” before coming to an informed decision about tempo even in seemingly simple and unimposing works, say, a two-part Invention, or a chorale from the Orgelbüchlein.

I do not claim to have encyclopedic knowledge of Bach’s oeuvre, but I try to employ musical comparisons and cross-referencing as much as possible in order to gain an overview of “Bach’s forest,” and thus, to make the best and most logical decisions in my tempo choices. Here is a list of ten good examples of musical cross-referencing and the conclusions that can be drawn from them:

Ten examples of musical cross-referencing:

1. The Sinfonia from Cantata 29 and Invention 8 have much in common. Both are in 3/4, both are exuberant and lively pieces, and both feature strikingly similar rhythmic patterns. This is the 3/4 Italian corrente style, exemplified in the lively 3/4 Courante (Corrente) from French Suite 5. Since Bach marked the Sinfonia “Presto,” it can be assumed that he associated “Presto” with the 3/4 corrente style. Therefore, all we have to do now is identify all the works that fit into this stylistic category, determine what Bach's “Presto” was, and assign all these works this tempo.

2. The bass aria from Cantata 130, “The ancient serpent burns with spite,” and the D-major Prelude from WTC II, both have the same meter, 12/8[4/4], employ the same note values, and have strikingly similar rhythmic characteristics. This suggests the same tempo for both as well as for all other works in this category.

3. The second movement from the F-minor Sonata for Violin and Harpsichord, BWV 1018, and the A-minor Invention feature identical figurations (i.e., the Invention subject) that define the movements' similar affects. Since Bach marked the sonata movement “Allegro,” it can be assumed he also conceived the Invention at “Allegro.” This makes it desirable to choose a tempo for the Invention that is no faster than one would choose for the sonata movement, and vice versa.

4. Comparing the Credo from the B-minor Mass to the B-minor Prelude from WTC I shows strikingly similar styles and texture. (I have a piano reduction of the Credo that I compare alongside the Prelude.) Both feature walking bass lines with slow-moving polyphonic melodies above, the only difference is that the note values in the Credo are doubled. Bach marked this Prelude “Andante,” making it logical that he intended the Credo to have this same kind of “Andante.” This clearly shows that the popular Allegro tempo for the Credo is incorrect. Ultimately, one should perform the Credo no faster than one would perform this Prelude, since the musical evidence suggests Bach conceived them at the same tempo. (For more on the Credo, please refer to Elaboration 6.)

5. The Courante from French Suite 1 and the C-sharp minor Prelude from WTC I have much in common. First, the meters of 3/2 and 6/4 are directly related, since they both consist of six quarter notes per measure and both are in “large” meters that Kirnberger explains have “emphatic and heavy” tempos and characters. Second, both feature identical opening six-note motives (not including the pick-up in the Courante). Third, they are in minor keys with subdued affects. In essence, this Prelude emulates the French courante style, which is well-known to be “slow and majestic.”5  This makes it desirable to choose a tempo for the Prelude that is no slower than one would choose for this Courante, and vice versa.

6. Just like the two pieces discussed in #5, the F-sharp minor Fugue from WTC I, in 6/4, and the second section of the E-flat major Fugue for organ, in 6/4, also fall into this slow and majestic French courante category. There is no theoretical justification for the popular Molto Adagio tempo for this Prelude and the popular Allegro tempo for this Fugue. Both are in the rare meter of 6/4, which indicates a “slow” and “broad” tempo, and both feature the same note values (eighths as the fastest non-embellishing note values). This suggests the same tempo for both, Bach's usual courante tempo of Andante, which is about midway between Adagio and Allegro.

7. The Quia fecit bass aria from the Magnificat (the 5th movement) and the Gavotte from French Suite 5 are defined by identical opening rhythmic formulas, the only difference is the note values are doubled in the latter. In essence, the Quia fecit emulates the gavotte style, making it desirable to choose the same tempo for both pieces. This also applies to all other works in this category, which can be assumed as having “moderate” gavotte-style tempos.

8. The Gigue from French Suite 1 and the A-minor Fugue from WTC II have much in common. Both are in 4/4 and both feature the same number of 32nd notes that occur in the same types of groups and patterns making the rhythmic phraseology strikingly similar. Bach apparently conceived both as belonging to the same style, making it desirable to choose the same tempo for both. This can also be extended to include all other works in this category.

9. The B-flat minor Prelude from WTC I and Prelude from English Suite #4 have much in common rhythmically, in that both are defined by a prominent rhythmic formula consisting of two “shorts” followed by three “longs”:  -- _ _ _ -- _ _ _. These pieces obviously do not have the same tempo, since the former is clearly in an Adagio style and the latter is clearly in an Allegro style. But what this identical rhythmic phraseology suggests is a tempo for the B-flat minor Prelude that is precisely half that of the F-major Prelude, and vice versa. This observation helps in choosing a tempo for the B-flat minor Prelude that is not too slow, and at the same time, choosing a tempo for the F-major Prelude that is not too fast. These rhythmic similarities suggest Bach's usual Adagio and Allegro were most likely at 1:2 ratios.

10. The Loure from French Suite 5 and Gigue from Overture in the French Manner have much in common, although like the comparison in #9 these two pieces obviously do not have the same tempo. Traditionally, loures were referred to as “slow gigues” since the rhythmic figurations are identical to one style of gigue, known sometimes as a “dotted-note gigue” or “French gigue” (also known as a “canary” in the Renaissance). Comparing this Loure with this Gigue (a “French” dotted-note gigue) shows a 2:1 ratio in meters, 6/4 and 6/8, as well as identical rhythmic patterns. This makes it desirable to assign a tempo for the Loure that is precisely half that of the tempo of this particular Gigue. To put it simply in generic terminology, “fast” is precisely two times “slow,” and vice versa. In essence, this Gigue could be transformed into a Loure by playing it at half speed, and conversely, this Loure could be transformed into a Gigue by playing it at twice its speed.

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Description of analytical system, spreadsheets, and statistics

The methodology employed in my study of tempo and duration in Bach’s music consists of six steps:

1. Tempo and metric categories are established, using the main tempo words and dance styles as points of reference. Each movement is then assigned one tempo from about a dozen standard tempo categories from Adagio to Presto (i.e., tempos in the ideal tempo matrix). All categories are listed in an appendix (unfortunately, not made public on this web site). Cross-referencing among movements is used in order to ensure consistency of tempo assignments. For example, a cantata aria may emulate the style of a courante from a dance suite, thus making it desirable to assign the same courante-style tempo and to establish a cross-reference between the two movements. Likewise, a movement from a chamber work, marked Andante, may be similar in style to a keyboard work with no tempo indication, making it desirable to assign the same Andante tempo to the latter and to establish a cross-reference between the two works. The far right column of each category in the appendix lists cross-references.

2. After assigning each movement to a tempo and metric category, the works as complete entities are analyzed—preludes and fugues, cantatas, chamber works, etc. The duration of each movement is then calculated with a calculator. In order to ensure scientific accuracy, the durations are listed in minutes and seconds. For example, the 35-measure C-major Prelude in 4/4 at Q = 72 results in a duration of 1:56.66, which is calculated by multiplying the number of measures, 35, by the number of beats, 4, dividing by the tempo, 72, keeping the integer as the number of minutes, 1, and multiplying the decimal, .9444444, by 60 to attain the seconds, 56.66. All durations are expressed to the hundredth of a second, not rounded off.

3. After calculating the duration of each movement, the durations are then compared and proportional relationships are sought. For example, the 35-measure Prelude in 4/4 at a moderate tempo of Q = 72 lasts 1:56.66 while the ensuing 27-measure Fugue in 4/4 at a slower Q = 54 lasts 2:00, showing that the durations are virtually equal, the discrepancy being merely three seconds. In other words, had Bach given the Prelude just one more measure, 36 instead of 35, its duration would be precisely two minutes like the Fugue.6

4. After uncovering almost or precisely equal duration ratios such as 1:1, 1:2, or 2:3, the margins of error or discrepancies between “actual” and “ideal” durations are then calculated. The margin of error is calculated by dividing the difference of the “actual” and “ideal” duration by the “ideal” duration in seconds, and multiplying the decimal by 100 to attain a percentage figure. For example, the “actual” duration of the C-major Prelude is 1:56.66 and the “ideal” duration is 2:00, in which case accounting for the difference between the two durations, 3.34 seconds, and dividing this by 120 seconds (two minutes) results in .0278333, which becomes 2.8% when multiplied by 100. In this case, the margin of error is just 2.8%, which translates to a discrepancy of merely 1.68 seconds for every minute of music. All margin of error figures are rounded off to the nearest tenth of a second. The computation of margin of error figures is absolutely vital to this study, since it is the only way to calculate Bach’s overall accuracy in attaining precise duration ratios.

5. All of the data are presented in spreadsheets consisting of a row for each movement and a column for each of the criteria. The far left column lists the title and tempo indication, if any. The next column to the right usually lists the key or tonality. The next column to the right lists the meter. The next column to the right lists the number of measures. The next column to the right lists the assigned tempo from one of several standard tempo categories.7 The next column to the right lists the actual duration, which results from the meter, number of measures, and tempo. The next column to the right lists the ideal duration, which is often a rounded-off duration close to the actual duration. The next column to the right lists vertically aligned numbers that indicate the ratio between the durations—for example, a vertically aligned 1 and 1 indicates equal durations, 1:1, a vertically aligned 1 and 2 indicates that the latter duration is two times longer than the former duration, 1:2, and a vertically aligned 2 and 3 indicates that latter duration is one-half longer than the former duration, 2:3. The column to the far right lists in percentage form the margin of error or discrepancy between the two durations in the same column. The following spreadsheet uses the C-major Prelude and Fugue as an example.

Spreadsheet analysis of 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%

6. After completing the spreadsheet analyses, the total number of duration ratio are tallied up and the margins of error are averaged. This makes it possible to assess whether duration ratios in Bach’s music are coincidental or not. Considering that out of about two thousand cases, 1:1, 1:2, and 2:3 ratios occur about 99.5% of the time at a 1.7% average margin of error—just a one-second discrepancy for every minute of music—it can be concluded that it is impossible such ratios are coincidental. Otherwise, the ratios would probably occur no more than about 40% of the time and with a much larger average margin of error, such as 7%. The final statistics confirm beyond a reasonable doubt all hypotheses or axioms assumed from the outset, including the ideal tempo matrix, which proves with at least 99% certainty the precise tempo and duration that Bach planned for each movement.

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Explanation of my methodology and the scientific method

The methodology employed in this study departs from traditional musicological methodology in that it does not rely on the usual sources in the form of treatises, books, and articles, but rather obtains its information from more practically-oriented empirical data. Literally, the word “empiricism” means “observation” or “experience” and has been used since the time of Aristotle, who was probably the first to base a philosophical system on data obtained by direct human observation or experience. For example, a concept such as “the sun is hot” need not be proven mathematically or supported by anyone else’s research, simply because “the sun is hot” is true based on human feeling and observation. “The sun is hot” is, thus, an empirically-based statement, and for this reason requires no footnotes or references because it is common knowledge.

Similarly, when I make a statement like “Bach’s Allegro was 84 beats per minute,” I have based it exclusively on my own research and findings, or on my own observations and empirical evidence as a performer and listener. Traditionally, musicologists would expect a statement like this to be supported by previous research from other scholars; however, since such a statement was derived exclusively from my own research, I make no apologies for not conforming to traditional musicological methodology and supplying citations and references.

Rather than relying on the research of other modern scholars, my empirical data and scientifically-based reasoning have been based on the following sources: (1) some of the writings of Bach’s student, Johann Philip Kirnberger; (2) various editions of Bach’s music; (3) my own skill as a performer on piano, harpsichord, and organ; (4) my experience as a Renaissance and Baroque dance accompanist; (5) numerous CD recordings; (6) my skill and patience working with numbers and calculations.

The “scientific method” commonly refers to the system of reasoning that eventually culminated in the European “Scientific Revolution” of the 1500-1600s. The scientific method is usually described as a three-step process that includes: (1) the formation of a hypothesis; (2) the use of “pure reasoning” (i.e., mathematics, logic, statistics) in order to test, and hopefully, confirm the hypothesis; (3) proof of the hypothesis based on the merit of step #2. Thus, the scientific method can be summarized as:

hypothesis —> experimentation and testing —> proof of hypothesis

The “innovative scientific method” to which I refer on my home page has been based on this three-step process. My main set of interrelated hypotheses is that Bach sought proportional duration ratios between movements (such as 2:00-4:00, which reduces to a 1:2 ratio), and could have done so only by planning “ideal” tempos and measure counts for each movement. After all, in order to plan a movement to last, say, precisely four minutes, one has to assume a tempo and number of measures from the outset. This simply cannot be done any other way.

Hence, in order to make this a workable and practical hypothesis, I assume a hypothetical table of tempos from the outset, in order to have a set of absolute values that can be used as “hypothetical tools” and applied to Bach’s music. (Otherwise, we would have to work with imaginary variables like x, y, or z, which would only result in confusion.) Therefore, the reader should understand that when I assign a tempo like quarter = 96 to a movement, it is really just an “assumed” tempo. It is only through the application of this tempo in many works over the course of many years that it has become not just an assumed or hypothetical tempo, but a “proven” tempo.

My main means of testing or experimentation (step #2 in the scientific method) has been applying the assumed or hypothetical tempos to most of Bach’s major works, which makes it possible to calculate precise durations and proportional duration ratios. For example, assuming quarter = 72 for a movement in 4/4 with 35 measures makes it possible to determine its duration by simply multiplying the number of beats per measure by the number of measures and dividing by the tempo, meaning that this movement would have a duration of 1:57 (as explained above). Once tempos have been assigned and durations have been calculated, it becomes possible to compare durations and discover duration ratios. Doing this just for a few works would prove nothing; however, the fact that accurate duration ratios occur in at least 99% of Bach movements, and complete multi-movement works are regularly unified by logical duration schemes, prove the initial hypothesis.

I realize that the skeptical reader may have a difficult time accepting my claims of having proven Bach’s tempos and discovered some of Bach’s cryptic secrets. For this reason, I ask the reader to please keep in mind that most of my research and data have never been publicly revealed and that my conclusions have been based on sound, scientifically-based reasoning applied over a long period of time. That is, my hypotheses have been tested and continually re-tested in order to ensure accuracy and validity. I try to approach Bach’s works and my analyses in the same fashion as a scientist would approach an experiment—with as much objectivity as possible. Assigning tempos is, indeed, a subjective process. There is simply no getting around this fact. However, my interpretation of all other facets (tempo ratios, measure ratios, duration ratios) is just as objective as a scientist’s interpretation of, say, physical data and natural laws.

In sum, my scientific method is one of formulating a set of interrelated hypotheses, assuming a set of tempos from the outset, applying these tempos to all of Bach’s major works, discovering relationships, and thus, proving the initial hypotheses:

1. Hypotheses

1. Bach planned ideal tempos in beats per minute. For example (Bach thinking to himself), “I want this movement to have a tempo of 84 beats per minute.”

2. Bach planned ideal numbers of measures for his movements. For example, “I want this movement to consist of 84 measures.”

3. Bach planned movements to share proportional duration ratios of 1:1, 1:2, and 2:3. For example, “I want the first movement to last two minutes and the second movement to last four minutes so that the durations relate at a 1:2 ratio.”

Sub-hypothesis: IF Bach planned ideal tempos, THEN they were most likely members of a logical hierarchy of tempos consisting exclusively of integers (i.e., the assumed tempo matrix, see Elaboration 2).

2. Experimentation or Application

The assumed or hypothetical tempos have been applied to all of Bach’s major works over the course of sixteen years, which result in extremely accurate 1:1, 1:2, and 2:3 duration ratios at least 99% of the time. Furthermore, the analysis of Bach’s measure counts often show proportional relationships. For example, one movement might have 50 measures while the next movement has 100 measures, showing a 1:2 measure ratio. The combination of special ratios in measure counts and the natural tempos for the style (tempo giusto) result in integer durations about 40% of the time and proportional duration ratios at least 99% of the time.

3. Proof

Extremely accurate duration ratios at least 99% of the time (in about one-thousand or more cases) with a discrepancy of only about 1.7% (which translates to an average of a little over one second per every minute of music) could not have possibly been the result of coincidence. The chances of these accurate duration ratios being coincidental this frequently are null, meaning that Bach must have planned them. For this reason, all hypotheses have been proven using the following logic:

SINCE the hypothetical tempos result in extremely accurate duration ratios over 99% of the time, THEN this confirms that Bach did indeed use the assumed tempo matrix assumed, and, did indeed plan special measure counts to achieve proportional duration ratios.

In short, the scientific method employed in this study can be summarized as:

Formulate hypotheses, assume tempos —> apply these tempos to Bach’s music, special relationships result —> hypotheses confirmed, assumed tempos and all hypotheses proven

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Endnotes

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