“Under the umbrella of seventeenth-century Lutheran theology, Bach’s musical discoveries – like Newton’s scientific advances, which Bach almost certainly did not know – took him to areas of the creative mind undreamed of before and ultimately pointed to the operations of God.” – Christoph Wolff

Introduction

Theorizing about proportional tempos in Bach’s music is nothing new and this book is certainly not the first publication to propose such a thing; however, this author is perhaps the first person to categorize these proportional tempos into a logical and coherent system of rows and columns consisting exclusively of integers (i.e., the mathematically ideal tempo matrix discussed in previous chapters and shown in this chapter in Table 1). Similarly, the related concept of proportionally related durations has been proposed by scholars long before this author’s “Bach tempo epiphany” in 1992. (Please refer to the author’s personal experience in Chapter 1.)

In 1959-60, musicologist Arthur Mendel (1905-1979) and organist/scholar Bernard Rose (1916-1996) exchanged views on proportional tempo relationships in Bach’s music.[i] As a by-product of this debate, an intriguing theory emerged, one of proportional relationships between performance times, referred to in this book as “duration ratios.” For example, Arthur Mendel argued for the same eighth-note speed in the Gloria and Et in terra pax sections of the Mass in B Minor, which results in a 1:2 duration ratio due to the virtually precise 1:2 ratio of eighth notes in each section, 302 and 606, respectively. In the Sanctus and Pleni sunt coeli sections of the Mass, which are notated in 4/4 with triplet eighths followed by 3/8, Bernard Rose argued for equal quarter-note speed, which the eminent Bach scholar Robert L. Marshall points out results in nearly precise equal durations due to the virtually equal number of eighth note in each movement, 376 and 363, respectively.[2]

Musicologist Don O. Franklin has addressed the issue of proportional duration ratios in a few other Bach works. Franklin explains that when the appropriate tempo giusto is assumed, a proportional duration ratio often results between adjacent movements.[3] For example, in the Prelude and Fugue in C Major from WTC I, 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] This 1:1 relationship, by the way, is shown in Diagram 1 below.

Aside from a few examples addressed by a handful of scholars, 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 Mass in B Minor and a few preludes and fugues? After all, Bach was arguably the most mathematical and systematic composer in the history of music, meaning that if such duration ratios occur in a few works, then similar ratios are bound to occur in more works. It seems hardly possible that Bach would have sought duration ratios only in the Gloria, Sanctus, and a few other works, but nowhere else. There needs to be an objective, scientific-based system that permits comparison and analysis of the tempos and durations of two or more movements simultaneously. Such a system was developed by this author in 1992, which is explained throughout the rest of this chapter.

Interpreting the Spreadsheets

Attempting to reconstruct Bach’s system of tempo (if he indeed operated with one) first requires the establishment of certain “axioms” or self-evident truths from which subsequent assumptions and conclusions can safely be based. The following three-step “if-then” logic, in which one statement, if true, leads to the next statement, provides a strong axiomatic base:

1. If Bach planned proportional duration ratios between one or more movements, then Bach must have planned specific tempos in beats per minute.

2. If Bach planned specific tempos in beats per minute, then they must have been whole numbers or integers.

3. If Bach planned specific tempos consisting of integers expressed in beats per minute and they all belonged to a mathematically proportional system, then Bach’s tempos must have been the same as those in the tempo matrix shown below (Table 1).

If step 3 is true, then Bach’s probable tempos can be determined by process of elimination, which is explained in Chapter 1. This makes it possible to apply tempos from the matrix, calculate durations, recognize relationships, and make conclusions. Otherwise, without applying absolute tempos in beats per minute, there would be no way to assess any possible relationships. In other words, the tempo must be *something*. The tempo application process often requires experimentation and always requires open ears and an open mind erased of the nineteenth-century biases discussed in Chapter 2. Let us now explain the spreadsheets. Please refer to Diagram 1 while reading the list “Interpreting the Spreadsheets.”

Interpreting the Spreadsheets

Each movement is presented in a spreadsheet consisting of one row and nine columns.
Movements are typically presented in duration pairs, such as a prelude and fugue, aria and chorus, two neighboring inventions or sinfonias, or two neighboring movements from a chamber work.
Column 1 – Lists the titles of the movements and tempo words if any.
(Column 2) – Lists the keys of the movements mostly in vocal and chamber works where the keys are different. This extra column is usually not necessary for preludes and fugues that are both in the same key.
Column 2 – Lists the meter of the movements. For sake of clarity, 4/4 is used instead of “C.” Likewise, 2/2 is used instead of C (with a line through it).
Column 3 – Lists the “actual” number of measures of the movement. For example, this prelude and fugue consist of 35 and 27 measures, respectively.
Column 4 – Lists the “ideal” number of measures of the movement, which is usually very close to and sometimes the same as the “actual” number of measures. For example, had Bach composed one more measure in the prelude to make a total of 36, the 1:1 duration ratio would be perfect with no discrepancy if played at Q = 63. (The popular early version of this prelude did indeed have 36 measures until it was reduced to 35 by modern scholars.)
Column 5 – Lists the tempo of the movement, which is determined by consideration of at least nine factors (discussed in Chapter 2) as well as process of elimination using the tempos in the appropriate column of the tempo matrix (discussed in Chapter 1).
Column 6 – Lists the “actual” duration, which results from the actual number of measures played at the assigned tempo in column 5. This is calculated by multiplying the number of measures by the beats per measure and dividing by the tempo. For example, for the prelude, 35 x 4 ÷ 63 = 2.222. The decimal is translated to seconds by multiplying it by 60, hence, 0.222 x 60 = 13.33 seconds, which makes the “actual” duration of the prelude 2:13.33 when played at the assigned tempo of Q = 63. All durations in this study are rounded off to the nearest hundredth of a second (i.e., two decimal digits).
Column 7 – Lists the “ideal” duration, which results from the “ideal” number of measures played at the assigned tempo in column 5. This is calculated in the same fashion as the previous example in column 6.
Column 8 – Lists the proportional relationship with the neighboring movement. For example, this prelude and fugue have virtually equal durations or a 1:1 duration ratio at 2:15-2:15.
Column 9 – Lists the “margin of error” or discrepancy between the actual and ideal durations, which is expressed in a percentage. This is calculated by dividing the difference of the two durations (in seconds) by the ideal duration (in minutes and seconds) and moving the decimal over by two places (or multiplying by 100). For example, in the prelude the difference between 21:13.33 and 2:15 is 1.67. Next, 2:15 translates to 135 seconds. Next, 1.67 ÷ 135 = .01237. Next, .01237 multiplied by 100 is 1.2%.

**Analysis of Gloria and et in terra pax from Mass in B Minor (BWV 232)**

Let us now put this spreadsheet system to the test by analyzing the *Gloria* and *et in terra pax* sections of the Mass in B Minor, movements 4 and 5, in which Arthur Mendel argued for equal eighth- and sixteenth-note speeds to achieve a 1:2 duration ratio (discussed above). To understand the complete picture, it is necessary to include the following movement in the equation, *Laudamus te*.
The first step in the analysis of these two movements is to experiment (by singing, conducting, etc.) with various tempos for each of the three sections whereby an appropriate tempo for each can be determined by process of elimination. The *Gloria* works well at the lively 3/8 Allegro of 56 bpm (one beat per measure), while the ensuing *et in terra pax* works well at the standard 4/4 Allegro of 84 bpm. These two tempos are taken from the “336” row and the “Sixes” and “Fours” columns of the matrix, respectively. Not only are these excellent Allegro-style tempos for these sections, which result in a virtually precise 1:2 duration ratio of 1:47-3:37 (as proposed by Mendel), but what is more, these sections add to 5:24 which is not far off from the duration of 5:10 for *Laudamus te* assuming a comfortable Andante-style tempo of 48 bpm.
This analysis, especially the comparison of the actual and ideal measures, is extremely revealing. More specifically, one less measure in *et in terra pax*, 75, results in a precise 4:3 measure ratio, 100:75, a precise 1:3 beat ratio, 100:300 (100 and 75 x 4), and a 2:3 tempo ratio, 56:84, resulting in a 1:2 duration ratio between the two sections, 1:47:3:34. In addition, two more measures in *Laudamus te*, 64, results in a duration of 5:20, which equals the total ideal duration of the two sections of the *Gloria*. But wait, there is more – the temporal organization of the *Confiteor* and *Sanctus* movements from the Mass are even more impressive than this!

**Analysis of Confiteor and Sanctus from Mass in B Minor (BWV 232)**

Let us now further test this spreadsheet system by analyzing the *Sanctus* and *Pleni sunt coeli* sections of the Mass in B Minor, movements 19 and 20, in which Bernard Rose argued for equal quarter- and eighth-note speeds to achieve equal durations (discussed above). To understand the complete picture, it is necessary to include the previous movement in the equation, the *Confiteor*, which consists of three sub-sections each with different tempos: “Normal” – *Adagio* – *Vivace ed allegro*.
The first step in the analysis of these two movements is to experiment (by singing, conducting, etc.) with various tempos for each of the five sections whereby an appropriate tempo for each can be determined by process of elimination. The *Confiteor* works well at a moderate tempo like 72 bpm, while the ensuing *Adagio* works well at a slower 48 bpm and the ensuing *Vivace ed allegro* works well at 96 bpm. These three tempos are taken from the “Fours” column from the matrix due to their binary beat subdivisions. Not only are these excellent tempos for these sections, but 72:48:96 relate or reduce to 3:2:4. In practical language, this means the *Adagio* is one-half slower than the *Confiteor* while the *Vivace ed allegro* is twice the speed of the *Adagio*.
Continuing with the assigning of tempos, the *Sanctus* works well at 84 bpm, taken from the “Fours” column, while the ensuing *Pleni sunt coeli* section works well at 56 bpm, taken from the “Sixes” column due to the six sixteenth notes for each beat (i.e., one beat per measure). These tempos indeed support Bernard Rose’s assertion that the two sections have equal quarter- and eighth-note speeds, since 84 and 56 belong to the same row of the matrix, in the “Fours” and “Sixes” columns, respectively, which results in similar durations of 2:14 and 2:09.
The most impressive aspect of the temporal organization of these two movements is that the total durations of theConfiteor and *Sanctus* share a precise 3:2 duration ratio of 6:36-4:24 when the five tempos 72:48:96 (which reduce to 3:2:4) and 84:56 (which reduce to 3:2) are assumed.

40 Examples of Duration Ratios in Bach’s Music

To subject this system to the preliminary testing, it is necessary to survey works in all major genres from a wide range of dates throughout Bach’s career. Thus, the following methodology of no fewer than ten steps is employed in this chapter and used throughout Part 2:

10 Steps to Reconstructing Bach’s Tempos and Duration Ratios

Duration ratios are categorized as 1:1, 1:2, 2:1, 2:3, 3:2.
Two examples of each ratio are listed and discussed for:
• Vocal works – mainly cantatas and other vocal works (BWV 1-200)
• Organ works (BWV 530-600s)
• Keyboard (Harpsichord) works (BWV 700s-900s)
• Instrumental and chamber works (BWV 1000 and over)
Only two movements are selected from each work. Complete analyses for the multi-movement works referenced here are found in Part 2 of this book.
Eight examples for each duration ratio are discussed, making a total of forty examples.
Choosing the most musically appropriate tempo for each movement is delicate and detailed, which includes the following process:
• For larger multi-movement works like chamber works and cantatas, which are analyzed in their entirety in Part 2, two movements are selected which share one of the five duration ratios.
• A few possible tempos from the tempo matrix are tried out for each movement and the most musically appropriate tempos are determined by process of elimination (as explained above).
• Professional recordings are consulted to determine the average sort of tempo taken by professionals, which either support or contradict the chosen tempo. Tempo adjustments are made at this stage depending on how different the average tempo is from the chosen tempo.

o For example, if the applied tempo happens to result in a duration of three minutes or very close to this, and the average tempo is within a reasonable range of three minutes (perhaps 2:45-3:15), then the applied tempo is confirmed.

o On the other hand, if the applied tempo happens to result in a duration of three minutes, and the average tempo is far away from this, then the applied tempo is changed to something closer to the average tempo unless there is good reason that indicates otherwise. Occasionally this happens (that is, when most performers completely misinterpret the correct tempo); however, none of the examples chosen in this chapter fall into this category.

o Thus, the applied tempos for each of the eighty movements selected in this chapter have been tested and confirmed by the most prominent world-class performers featured on YouTube. For example, according to the present theory, Bach’s monumental Passacaglia in C Minor minus the Fugue should last precisely eight minutes, 8:00. The average duration for this movement by most organists tends to be around 7:30-8:00, which confirms the applied tempo of 63 bpm.

6. Durations at the applied tempos are calculated and rounded off to the nearest hundredth of a second. Duration ratios between movements are indicated in the next to the last column in each spreadsheet with the numbers 1-2-3, as explained earlier in this chapter.

7. The discrepancy (i.e., margin of error), or percentage of difference between the actual and ideal duration, is listed in the last column in each spreadsheet. This indicates how close Bach was to achieving the number of measures that facilitates each duration and duration ratio. For example, in the first example below from Cantata No. 76, Bach apparently planned two more measures in the tenor aria, 96 instead of 94, which results in a duration of precisely three minutes, 3:00, at the applied tempo of 96 bpm. Likewise, the following alto aria in this cantata has 64 measures and a tempo of 64 bpm, that also facilitates a duration of three minutes, 3:00, indicating that Bach apparently planned these two arias to have equal durations of three minutes.

8. All percentages in the eighty discrepancy columns are added up, 111, and divided by 80 to determine the average discrepancy between the actual and ideal durations, 1.39, which translates to a mere 0.8 of a second (.0139 x 60 = 0.83). This indicates that, at least in these forty examples, Bach was less than one second off on average from achieving the desired duration and duration ratio precisely.

9. Such an infinitesimally small average discrepancy applied to forty examples in all genres from a wide range of years confirms and virtually proves the initial hypothesis that Bach consciously planned duration ratios in his music using the numbers 1-2-3. Moreover, this also confirms and virtually proves the initial hypothesis that Bach’s standard tempos were the same as the tempos in the matrix.

10. Bach’s system of tempo as well as his tempos and duration ratios are confirmed!

1:1 Ratios

The tenth and twelfth movements, a tenor and alto aria, are separated by a short recitative. The two arias have the same number of beats per measure, a close 3:2 beat ratio, 282:192 (94 x 3 and 64 x 3), and 3:2 tempo ratio, 96:64, resulting in similar durations, 2:56 and 3:00. Two more measures in the tenor aria, 96, results in a 3:2 beat ratio, 288:192, and equal durations of three minutes, 3:00. (equal beats per measure, 3:2 beat ratio, 3:2 tempo ratio, 1:1 duration ratio)

The first and fifth movements, a bass aria and alto-tenor duet, are the first and last major movements in this cantata, since the sixth movement is a short chorale. These outer movements have different meters, a close 1:2 beat ratio 220:444 (55 x 4 and 148 x 3), and 1:2 tempo ratio, 54:108, resulting in virtually equal durations, 4:04 and 4:06. One less measure for the first movement, 54, and four fewer measures for the fifth movement, 144, result in a 1:2 beat ratio, 216:432, and equal durations of four minutes, 4:00. (different meters, 1:2 beat ratio, 1:2 tempo ratio, 1:1 duration ratio)

The prelude and fugue have equal meters, 4/4, a virtually precise 7:8 beat ratio, 124:144 (31 x 4 and 36 x 4), a close 7:8 measure ratio, 32:36, and 7:8 tempo ratio, 63:72, resulting in equal durations of 2:17. (equal meters, 7:8 beat ratio, 7:8 tempo ratio, 1:1 duration ratio)

The first two movements, Vivace and Lento, have different meters, a 3:2 beat ratio, 360:240 (180 x 2 and 40 x 6), and virtually precise 3:2 tempo ratio, 96:63, resulting in virtually equal durations, 3:45 and 3:48. One-half less measure for the second movement, 39.5, results in equal durations of 3:45. (different meters, 3:2 beat ratio, 3:2 tempo ratio, 1:1 duration ratio)

Inventions 10 and 11 have different meters, a close 1:1 beat ratio, 96:92 (32 x 3 and 23 x 4), and 1:1 tempo ratio, 72:72, resulting in virtually equal durations, 1:20 and 1:18. One more measure for invention 11, 24, results in a 1:1 beat ratio, 96:96, and equal durations of 1:20. (different meters, 1:1 beat ratio, 1:1 tempo ratio, 1:1 duration ratio)

Sinfonias 1 and 15 have different meters, a close 3:4 beat ratio, 84:114 (21 x 4 and 38 x 4), and 3:4 tempo ratio, 54:72, resulting in virtually equal durations, 1:33 and 1:35. Bach achieved the ideal numbers of measures for the first and last sinfonias and, simultaneously, signed his name alphanumerically, 2138 = BACH. (different meters, 3:4 beat ratio, 3:4 tempo ratio, 1:1 duration ratio)

The first and second movements have different meters, a close 3:4 beat ratio, 216:282 (36 x 6 and 141 x 2), and 3:4 tempo ratio, 63:84, resulting in virtually equal durations, 3:25 and 3:21. Three more measures in the second movement, 144, results in a 3:4 beat ratio, 216:288, and equal durations of 3:25. (different meters, 3:4 beat ratio, 3:4 tempo ratio, 1:1 duration ratio)

The first and fifth movements have different meters, a close 3:4 measure ratio, 91:119, a close 3:2 beat ratio, 364:238 (91 x 4 and 119 x 2), and 3:2 tempo ratio, 84:56, resulting in virtually equal durations, 4:20 and 4:15. One less measure in the first movement, 90 and one more measure in the fifth movement, 120, result in a 3:4 measure ratio, 90:120, a 3:2 beat ratio, 360:240, and equal durations of 4:17. (different meters, 3:2 beat ratio, 3:2 tempo ratio, 1:1 duration ratio)

1:2 Ratios

The first and second movements, a chorus and tenor aria, have a close 1:3 measure ratio, 65:198, a close 2:3 beat ratio, 260:396 (65 x 4 and 198 x 2), and close 4:3 tempo ratio, 84:64, resulting in a virtually precise 1:2 duration ratio, 3:05-6:11. Two fewer measures for the chorus, 63, and six fewer measures for the aria, 192, results in a 1:2 duration ratio, 3:00-6:00. (different meters, 2:3 beat ratio, 4:3 tempo ratio, 1:2 duration ratio)

The sixth and seventh movements, *Gratias* and *Domine Deus*, both have four beats per measure, a close 1:2 measure ratio, 46:95, close 1:2 beat ratio, 184:380 (46 x 4 and 95 x 4), and 1:1 tempo ratio, 72:72, resulting in a close 1:2 duration ratio, 2:33-5:16. Two more measures for *Gratias*, 48, and one more measure for *Domine Deus*, 96, result in a 1:2 measure ratio, 48:96, a 1:2 beat ratio, 192:384, and 1:2 duration ratio, 2:40-5:20. (equal beats per measure, 1:2 beat ratio, 1:1 tempo ratio, 1:2 duration ratio)

The second and third movements, (Allemande) and Aria, have different meters, a close 1:3 measure ratio, 22:64, a close 7:16 beat ratio, 88:192 (22 x 4 and 64 x 3), and 7:8 tempo ratio, 63:72, resulting in a close 1:2 duration ratio, 1:23-2:40 (without repeats). One less measure for the second movement, 21, results in a 1:2 duration ratio, 1:20-2:40 (without repeats) or equal durations of 2:40 (with repeats). (different meters, 7:16 beat ratio, 7:8 tempo ratio, 1:2 duration ratio)

The sixth and seventh movements, *Gelobet seist du, Jesu Christ* and *Der Tag, der ist so freudenreich*, have equal meters, a close 7:16 beat ratio, 44:96 (11 x 4 and 24 x 4), and 7:8 tempo ratio, 42:48, resulting in a virtually precise 1:2 duration ratio, 1:02-2:00. One-half more measures in *Gelobet seist du*, 10.5, result in a 1:2 duration ratio, 1:00-2:00. (equal meters, 7:16 beat ratio, 7:8 tempo ratio, 1:2 duration ratio)

Inventions 8 and 9 have equal meters, 3/4, , equal measures, 34, equal beats, 102, and 2:1 tempo ratio, 96:48, resulting in a 1:2 duration ratio, 1:04-2:08. (equal meters, 1:1 beat ratio, 2:1 tempo ratio, 1:2 duration ratio)

The prelude and fugue have different meters, a close 1:3 measure ratio, 39:115, a close 1:1 beat ratio, 234:230 (39 x 4 and 115 x 2), and 2:1 tempo ratio, 96:48, resulting in a virtually precise 1:2 duration ratio, 2:26-4:52. Two more measures for the fugue, 117, results in a 1:3 measure ratio, 39:117, and a 1:2 duration ratio, 2:26-4:52. Also, one more measure for the prelude 40, and five more measures for the fugue, 120, result in a 1:2 duration ratio, 2:30-5:00. (different meters, 1:1 beat ratio, 2:1 tempo ratio, 1:2 duration ratio)

The first and second movements, *Grave* and *Fuga*, have different meters, a close 1:3 beat ratio 184:578 (23 x 8 and 289 x 2) and a 2:3 tempo ratio, 48:72, resulting in a close 1:2 duration ratio, 3:50-8:01. One more measure for the *Grave*, 24, and two fewer measures for the *Fuga*, 287, result in a 1:3 measure ratio, 192:574, and 1:2 duration ratio, 4:00-8:00. (different meters, 1:3 beat ratio, 2:3 tempo ratio, 1:2 duration ratio)

The second and fourth movements, Allegro and Allegro, have different meters, a close 2:3 beat ratio, 160:250 (80 x 2 and 125 x 2) and 9:7 tempo ratio, 72:56, resulting in a virtually precise 1:2 duration ratio, 2:13-4:27 (without repeats). One less measure for the fourth movement, 124, results in a 1:2 duration ratio, 2:13-4:26 (without repeats) or equal durations of 4:26 (with repeats). (equal beats per measure, 2:3 beat ratio, 9:7 tempo ratio, 1:2 duration ratio)

2:1 Ratios

The first and third movements, a soprano and alto aria, are separated by a short recitative. The soprano aria is organized symmetrically which is organized symmetrically with 29 measures of 12/8 for the outer sections and 25 measures of 4/4 for the inner section. At their assigned tempos, 84 and 72 bpm respectively, the total actual duration adds to 9:40 where the three sections share a 3:1:3 duration ratio of 4:09-1:23-4:09. The alto aria lasts 4:45 at its assigned tempo, 48 bpm, showing a 2:1 duration ratio between the two arias, 9:30-4:45.

The seventh and eighth movements, *Domine Deus* and *Qui tollis*, have different meters, a close 8:3 beat ratio 380:150 (95 x 4 and 50 x 3) and 4:3 tempo ratio, 72:54, resulting in a close 2:1 duration ratio, 5:16-2:46. One more measure for *Domine Deus*, 96, and four more measures for *Qui tollis*, 54, result in a 2:1 measure ratio, 96:48, an 8:3 beat ratio, 384:144, 2:1 duration ratio, 5:20-2:40. (different meters, 8:3 beat ratio, 4:3 tempo ratio, 2:1 duration ratio)

The second and third movements have different meters, a 7:4 beat ratio, 336:192 (28 x 12 and 64 x 3), and a 7:8 tempo ratio, 84:96, resulting in a 2:1 duration ratio, 4:00-2:00. (different meters, 7:4 beat ratio, 7:8 tempo ratio, 2:1 duration ratio)

Movements 11 and 12, two settings of *Dies sind die heiligen zehn Gebot*, have different meters, a close 8:3 beat ratio, 360:140 (60 x 6 and 35 x 4), and a close 4:3 tempo ratio, 84:64, resulting in a virtually precise 2:1 duration ratio, 4:17-2:11. Two-thirds more of measure for the first setting, 60.6, and three-tenths of a measure less for the second setting, 34.7, result in a 2:1 duration ratio, 4:20-2:10. (different meters, 8:3 beat ratio, 4:3 tempo ratio, 2:1 duration ratio)

The first and seventh (last) movements, Toccata and Gigue, both have four beats per measure, have a close 2:1 measure ratio, 108:52, a 2:1 beat ratio, 432:208 (108 x 4 and 52 x 4), and equal tempos, 54 bpm, resulting in a close 2:1 duration ratio, 8:00-3:51 (no repeats). Two more measures for the Gigue, 54, results in a 2:1 measure ratio, 108:54, a 2:1 beat ratio, 432:216, and a 2:1 duration ratio, 6:51-3:25 (no repeats), or equal durations of 8:00 (with repeats). (equal beats per measure, 2:1 beat ratio, 1:1 tempo ratio, 2:1 duration ratio)

The first and second movements, Aria and Var. 1, have equal meters, 3/4, equal measures, 32, a 1:1 beat ratio, 96:96, and a 1:2 tempo ratio, 48:96, resulting in a precise 2:1 duration ratio, 2:00-1:00. (equal meters, 1:1 beat ratio, 1:2 tempo ratio, 2:1 duration ratio)

The second and third movements, Adagio and Allegro assai, have different meters, a close 9:4 beat ratio, 342:160 (57 x 6 and 160), and 9:8 tempo ratio, 63:56, resulting in a close 2:1 duration ratio, 5:25-2:51. Three more measures for the first movement, 60, results in a 9:4 beat ratio, 360:160, a 2:1 duration ratio, 5:42:2:51. (different meters, 9:4 beat ratio, 9:8 tempo ratio, 2:1 duration ratio)

The first and third movements, (Allegro) and Presto, have different meters, a close 7:4 beat ratio, 472:278 (118 x 4 and 139 x 2), and a 7:8 tempo ratio, 84:96, resulting in a virtually precise 2:1 duration ratio, 5:40-2:50. One more measure for the first movement, 119, and three fewer measures for the third movement, 136, results in a 2:1 duration ratio, 5:40-2:50. (different meters, 7:4 beat ratio, 7:8 tempo ratio, 2:1 duration ratio)

2:3 Ratios

The first and second movements, a chorus and tenor aria, have equal beats per measure, a 7:6 beat ratio, 294:252 (147 x 2 and 126 x 2), and 7:4 tempo ratio, 84:48, resulting in a precise 1:2 duration ratio, 3:30-5:15. (equal beats per measure, 7:6 beat ratio, 7:4 tempo ratio, 2:3 duration ratio)

The fifteenth and sixteenth movements, both choruses, have equal beats per measure, a close 8:9 beat ratio, 147:159 (49 x 3 and 53 x 3), and 4:3 tempo ratio, 72:54, resulting in a virtually precise 2:3 duration ratio, 2:02-2:56. One less measure in the *Et incarnatus*, 48, and one more measure for the *Crucifixus*, 54, result in a 2:3 duration ratio, 2:00-3:00. (equal beats per measure, 8:9 beat ratio, 4:3 tempo ratio, 2:3 duration ratio)

The first and second movements have different meters, a close 2:1 measure ratio, 58:28, a close 2:3 beat ratio, 232:336 (58 x 4 and 28 x 12), and 1:1 tempo ratio, 84:84, resulting in a virtually precise 2:3 duration ratio, 2:45-4:00. Two fewer measures in the first movement, 56, results in a 2:1 measure ratio, 56:28, 2:3 beat ratio, 224:336, and a 2:3 duration ratio, 2:40-4:00. (different meters, 2:3 beat ratio, 1:1 tempo ratio, 2:3 duration ratio)

The toccata and fugue have different meters, a close 8:9 beat ratio, 396:444 (99 x 4 and 222 x 2), and 4:3 tempo ratio, 84:63, resulting in a virtually precise 2:3 duration ratio, 4:42-7:02. Two more measures in the toccata, 98, and one fewer measure in the fugue, 221, results in a 2:3 duration ratio, 4:40-7:00. (different meters, 8:9 beat ratio, 4:3 tempo ratio, 2:3 duration ratio)

Inventions 1 and 2 have equal meters, a close 7:9 measure ratio, 22:27, a close 7:9 beat ratio, 88:108 (22 x 4 and 27 x 4), and a 7:6 tempo ratio, 63:54, resulting in a close 2:3 duration ratio, 1:23-2:00. One less measure for invention 1, 21, results in a 2:3 duration ratio, 1:20-2:00. (equal meters, 7:9 beat ratio, 7:6 tempo ratio, 2:3 duration ratio)

The prelude and fugue have equal meters, a close 3:4 measure ratio, 36:46, a close 3:4 beat ratio, 144:184 (36 x 4 and 46 x 4), and 8:7 tempo ratio, 72:63, resulting in a close 2:3 duration ratio, 2:00-2:55 (without repeats). Two more measures for the fugue, 48, results in a 2:3 duration ratio, 2:00-3:00 (without repeats). (equal meters, 3:4 beat ratio, 8:7 tempo ratio, 2:3 duration ratio)

The first and second movements have different meters, a 1:2 beat ratio, 160:320 (20 x 8 and 160 x 2), and 3:4 tempo ratio, 63:84, resulting in a precise 2:3 duration ratio, 2:32-3:48 (without repeats), or 1:3 duration ratio, 5:04-3:48 (with repeats). (different meters, 1:2 beat ratio, 3:4 tempo ratio, 2:3 duration ratio)

The second and third movements have different meters, a close 9:20 beat ratio, 213:488 (71 x 3 and 244 x 2), and virtually precise 2:3 tempo ratio, 63:96, resulting in a 2:3 duration ratio, 3:22-5:05. One more measure in the second movement, 72, and four fewer measures in the third movement, 240, results in a 2:3 duration ratio, 3:20-5:00. (different meters, 4:9 beat ratio, 2:3 tempo ratio, 2:3 duration ratio)

3:2 Ratios

The second and fourth movements, an alto and tenor aria separated by a short recitative, have different meters, a close 2:3 measure ratio, 84:124, a close 2:1 beat ratio, 252:124 (84 x 3 and 124) and 4:3 tempo ratio, 56:42, resulting in a virtually precise 3:2 duration ratio, 4:30-2:57. Two more measures in the tenor aria, 126, results in a precise 3:2 duration ratio, 4:30-3:00. (different meters, 2:1 beat ratio, 4:3 tempo ratio, 3:2 duration ratio)

The third and fifth movements, a tenor aria and soprano-alto duet, have different meters, a close 3:1 measure ratio, 150:52, a close 3:2 beat ratio, 450:312 (150 x 3 and 52 x 6), and 1:1 tempo ratio, 126:126, resulting in a close 3:2 duration ratio, 3:34-2:28. Two measures less for the fifth movement, 50, results in a 3:1 measure ratio, 150:50, a 3:2 beat ratio, 150:100, and a 3:2 duration ratio, 3:34:2:28. (different meters, 3:2 beat ratio, 1:1 tempo ratio, 3:2 duration ratio)

The toccata and fugue have different meters, a close 4:3 beat ratio, 438:340 (438 and 170 x 2), and 8:9 tempo ratio, 56:63, resulting in a close 3:2 duration ratio, 7:49-5:23. Ten more measures in the toccata, 448, and two fewer measures in the fugue, 168, results in a 3:2 duration ratio, 8:00-5:20. (different meters, 4:3 beat ratio, 8:9 tempo ratio, 3:2 duration ratio)

The passacaglia and fugue have equal meters, a close 4:3 beat ratio, 504:372 (168 x 3 and 124 x 3), and 7:8 tempo ratio, 63:72, resulting in a close 3:2 duration ratio, 8:00-5:10. Four more measures for the fugue, 128, results in a 3:2 duration ratio, 8:00-5:20. Having fallen four measures short of the 128-measure ideal in the fugue may explain Bach's *adagio* indication in the penultimate measure. (equal meters, 4:3 beat ratio, 7:8 tempo ratio, 3:2 duration ratio)

The prelude and fugue have different meters, a close 6:7 beat ratio, 136:166 (34 x 4 and 83 x 2), and 4:7 tempo ratio, 48:84, resulting in a close 3:2 duration ratio, 2:58-1:58. Two more measures in the prelude, 36, and one more measure in the fugue, 84, results in a 3:2 duration ratio, 3:00-2:00. (different meters, 6:7 beat ratio, 4:7 tempo ratio, 3:2 duration ratio)

The prelude and fugue have different meters, a close 2:3 beat ratio, 186:284 (62 x 3 and 71 x 4), and 7:16 tempo ratio: 42:96, resulting in a precise 3:2 duration ratio, 4:25-2:57. In addition, one more measure for both, 63 and 72, results in a 3:2 duration ratio, 4:30-3:00. (different meters, 2:3 beat ratio, 7:16 tempo ratio, 3:2 duration ratio)

The third and fourth movements have different meters, a close 1:3 measure ratio, 29:61, a close 2:3 beat ratio, 116:183 (29 x 4 and 61 x 3), and a 7:16 tempo ratio, 42:96, resulting in a close 3:2 duration ratio, 2:45-1:54. One more measure for both movements, 30:60, results in a 1:2 measure ratio, 30:60, and a 3:2 duration ratio, 2:48-1:52. (different meters, 2:3 beat ratio, 7:16 tempo ratio, 3:2 duration ratio)

The first and third movements have different meters, a close 7:8 beat ratio, 426:488 (426 and 244 x 2), and 7:12 tempo ratio, 56:96, resulting in a precise 3:2 duration ratio, 7:36-5:04. Six fewer measures in the first movement, 420, and four more measures in the third movement, 240, result in a 3:2 duration ratio, 7:30-5:00. (different meters, 7:8 beat ratio, 7:12 tempo ratio, 3:2 duration ratio)

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Endnotes

[1] Arthur Mendel, “A Note on Proportional Relationships in Bach Tempi,” The Musical Times 100 (1959), 683-85. Also, Mendel, “Bach Tempi: A Rebuttal,” The Musical Times 101 (1960), 251. And, Bernard Rose, “A Further Note on Bach Tempi,” The Musical Times 101 (1960), 251.

[2] Robert L. Marshall, “Bach’s tempo ordinario: A Plaine and Easie Introduction to the System,” in Critica Musica: Essays in Honor of Paul Brainard, ed. J. Knowles (New York, 1996), 249-78.

[3] Don O. Franklin, “The Fermata as Notational Convention in the Music of J. S. Bach,” in Convention in Eighteenth- and Nineteenth-Century Music: Essays in Honor of Leonard G. Ratner (Stuvesant, NY: Pendragon Press, 1992), 345-381. Franklin refers to duration ratios as “dimensional relationships.”

[4] In comparing the A-minor Prelude and Fugue, WTC II, with the C-major Prelude and Fugue, WTC I, Franklin believes in the case of the latter, “. . . thirty-five measures for the prelude and twenty-seven for the fugue corresponds to the greater disparity of tempo.” He continues with, “To speak, however, of dimensional relationships between movements with the same meters but different tempos, on the basis of their actual duration in time (minutes) rather than on the duration of their beats, takes us beyond the scope of the present study.” 358-59.

Chapter 5: Reconstruction of Bach’s System of Tempo