Bach Scholar™
Elaboration 6 in
a series of 8
(It is recommended that the elaborations be read in their proper sequence.)
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Bach, the “musical scientist”
accompanied by a complete analysis of the B-minor Mass
by Bach Scholar, the “musical detective”
Contents
- Essay: Bach the “musical scientist”
- The first half of the B-minor Mass (Missa)
- Outer parameters: Kyrie and Cum sancto spiritu
- Large-scale plan from the Christe to the Quoniam
- Christe and Kyrie II
- Gloria and Laudamus
- Gratias, Domine deus
- Domine deus, Qui tollis
- Gloria to Qui tollis
- Qui sedes, Quoniam
- Summary of the first half
- The second half of the B-minor Mass
- Credo, Patrem, Et in unum deum, Et incarnatus, Crucifixus
- Et resurrexit, Et in spiritum sanctum
- Confiteor, Sanctus
- Osanna, Benedictus, Agnus dei, Dona nobis pacem
- Summary of the second half
- Summary of the complete B-minor Mass
- Conclusion
Bach, the “musical scientist”
I know what you may be thinking:
“This theory is crazy and totally outlandish. Why in the world would Bach straightjacket his compositions into strict, inflexible, and idealistic duration molds that seem directly antithetical to the human element of making music?”
In order to better understand the veracity of my theory and to answer the question above, it is necessary to realize Bach’s uniqueness as a composer as well as the period in history in which he lived. Bach grew up during the end of the historical period known as the “Scientific Revolution,” a period like no other in history whose greatest scientist and icon, Isaac Newton (1642-1726), was at the pinnacle of his career when Bach was born in 1685. In his book Bach: The Learned Musician, the eminent Bach scholar and biographer, Christoph Wolff, eloquently summarizes the major achievements of Newton, who has been claimed by some as having been “the most intelligent person in history.”
Isaac Newton, it is still fair to say today, played the most critical role in the foundation of modern science. In addition to his most spectacular accomplishments—inventing calculus, discovering the laws of motion and the principles of optics, and explaining the concept of universal gravitation—his fundamental contributions covered an astounding range of fields. He studied space, time, heat, and the chemistry and theory of matter; he formulated the basic concepts of mass and dynamics; he invented the gravitational theory of tides; and he helped design such scientific instruments as the reflecting telescope. Toward the end of his career, he turned to alchemy, history, chronology, biblical exegesis, and theological issues. Newton’s theoretical and experimental works exemplified a new kind of scientific method, characterized by a kind of “contrapuntal alternation between mathematical constructs and comparisons with the real world.” And a traditional element, typical of a pre-Enlightenment outlook, was Newton’s firm belief that his discoveries “pointed to the operations of God.” Unlike later science, which focused solely on understanding nature, the Newtonian search for truth encompassed both natural and divine principles. Trying to grasp the relationship between God and nature led Newton to explore the boundaries between them, where he ultimately saw the fusion of natural and divine principles (Wolff, 6-7).
Based on what some of Bach’s contemporaries said about the “Honorary Capellmeister” as well as how Bach described his own work in his own words, Wolff emphasizes that Bach considered himself to be not merely an organist, music maker, or even a composer, but rather a “musical scientist” who produced works of “musical science”:
Not coincidentally, Bach found himself at work in 1733 on a special project that would occupy him for some time to come. On July 27 of that year, he dedicated to the electoral court in Dresden the Missa in B minor—the Kyrie and Gloria of what would become the B-minor Mass [Bach’s own words]: “To your Royal Highness I submit in deepest devotion the present small work of that science which I have attained in musique.” Stripping the phrase of the conventional formalities and courtly protocol, the statement reveals what this work was to someone who held the directorship of music in Leipzig’s two principle churches; he considered himself a musical scholar producing works of musical science (Wolff, 3-4).
C. F. Daniel Schubart said it all in just a few words: “What Newton was as philosopher, Sebastian Bach was as musician” (Wolff, 1). Wolff summarizes Bach’s major achievements in “musical science”:
- Fugue and canon (The Art of Fugue)
- Major-minor tonality (The Well-Tempered Clavier)
- Harmonic expansion (the Chromatic Fantasy and Fugue)
- Extended polyphony (the unaccompanied violin, cello, and flute pieces)
- Instrumentation (the Brandenburg Concertos)
- Instrumental and vocal genres (Bach employed virtually all contemporary models and types—from aria, cantata burlesque, and canzona to oratorio, scherzo, and sinfonia)
- Small-scale form (the Orgelbüchlein) and large-scale form (the St. Matthew Passion)
- Style and compositional technique, from retrospective to modern (the B-minor Mass)
- Musical affect and meaning (the church cantatas)
Considering that Bach was not merely an organist, composer, and Capellmeister, but a “musical scientist,” as well as considering the unique time in history in which Bach lived, it becomes very easy to understand why Bach would have sought idealistic formulas and molds with which to unify his works. As shown in analyses in previous elaborations and in virtually all other analyses not revealed to the public, Bach was apparently preoccupied, and I would even go so far as to say “obsessed,” with achieving special duration ratios consisting exclusively of the whole numbers 1, 2, and 3. For this reason, I would add to Wolff’s list above, the scientific achievement, “musical architecture as a means of compositional perfection (the majority of works unified by the proportional duration ratios of 1:1, 1:2, 2:1, 2:3, and 3:2).”
Analysis shows Bach sought only seven duration ratios in all his music, which consist of all the possible combinations of the first three numbers: 1:1, 1:2, 2:1, 2:3, 3:2, 1:3, 3:1. (My analyses show that 1:3 and 3:1 duration ratios are rare, but Bach did seek them occasionally.) For some reason, Bach stopped here and did not try to achieve 3:4 or 4:3 duration ratios or any other ratio with fives or any other higher number. Perhaps Bach sought to unify his temporal proportions with all combinations of the numbers 1, 2, and 3 in order to unify the “relationship between God and nature” similar to what Newton was presumably doing in the realm of science. But rather than speculating why Bach would have done all this, let us discover what Bach did and how he did this by turning to the work of “science” to which Bach himself referred, the B-minor Missa, as well as the second half of the complete Mass, which includes all movements from the Credo to the Dona nobis pacem.
First half of the B-minor Mass (Missa)
The first half of the B-minor Mass, the Missa, consists of two large sections: (1) the Kyrie, which consists of the first Kyrie, Christe, and second Kyrie, and; (2) the Gloria, a succession of eight movements from the Gloria to Cum sancto spiritu. The Missa to which Bach referred as his work of “musical science” is represented by the first eleven movements of the complete B-minor Mass. Analysis of the measure counts and resulting durations based on tempo assignments reveals, indeed, a tightly constructed work of musical science or architecture. Analysis of Bach’s multi-movement cycles almost always reveals that Bach sought duration ratios mostly in pairs and in groups of two or more movements. (For example, my study “Tempo and Architecture in the Inventions and Sinfonias” shows how Bach organized the 15 Inventions into several duration pairs just like in preludes and fugues.)
Outer parameters: Kyrie and Cum sancto spiritu
Analysis of Bach's works shows that when movements have no duration partners, they almost always have integer durations. For example, a non-integer duration like 6:36 almost never stands alone, but usually relates in some way to either one or both durations to either side. (See the discussion of the Confiteor-Sanctus pair.) Analysis also shows movements that stand alone with no duration partners almost always have integer durations, which Bach most likely did to “perfect” the otherwise isolated “imperfect” duration.
The Missa is certainly no exception to these rules. After a brief four-measure introduction, the Kyrie proper consists of 122 measures of 4/4, whose Largo indication suggests a tempo of quarter = 54, resulting in a duration of nine minutes, or 9:02 to be precise. Perhaps Bach intended the Missa to open with a nine-minute movement because 9 equals 3 x 3, which symbolizes the Trinity squared? What better way would there be to begin a large-scale work of “musical science” than this?
Nine-minute duration of the opening Kyrie of the Missa
| 1. Kyrie, Largo ed un poco piano (SATB) |
b | 4/4 | 122 | Q = 54 | 9:02.22 | 9:00 | 0.4% |
As Bach opened the Missa with an integer duration, he also ended it with an integer duration. The Cum sancto spiritu consists of 128 measures of 4/4, whose Vivace indication suggests a lively tempo of quarter = 96, resulting in a duration of precisely four minutes. Analysis shows four minutes was one of the most common integer durations in Bach’s music. (For example, please refer to the discussion of the Italian Concerto in Elaboration 4.)
Four-minute duration of the final Cum sancto spiritu of the Missa
| 11. Cum sancto spiritu, Vivace (SATB) |
D | 3/4 | 128 | Q = 96 | 4:00 | 4:00 | 0.0% |
Large-scale plan from the Christe to the Quoniam
Between the Kyrie and Cum sancto spiritu stand nine movements (once again, the number 9) whose durations relate in pairs and groups. The measure counts themselves are extremely revealing, indicating that something suspicious was going on in Bach’s mind as he planned the length of these movements. We need not even assign tempos to discover that the measure ratios are always very close to the ratios of 2:3, 3:2, 2:1, or 4:3. (Although Bach apparently did not seek 3:4 or 4:3 duration ratios, they are quite common among measure counts.) Here is Bach’s measure ratio plan from the Christe to the Quoniam
Table 1. Measure counts and their ratios of all movements from the Christe to the Quoniam
| Christe 85 (84) |
Kyrie II 59 (56) |
84:56 = 3:2 |
| Gloria 100 (100) |
et in terra 76 (75) |
100:75 = 4:3 |
| Laudamus 62 (64) |
Gratias 46 (48) |
64:48 = 4:3 |
| Domine deus 95 (96) |
Qui tollis 50 (48) |
96:48 = 2:1 |
| Qui sedes 86 (84) |
Quoniam 127 (126) |
84:126 = 2:3 |
Furthermore, the measure ratio from the Laudamus to Qui tollis, 62:46:95:50, approaches a 3:2:4:2 ratio, showing that Bach very likely intended the durations of all four of these movements—which function as two aria-chorus pairs—to have proportionally related durations.
Table 2. 3:2:4:2 measure ratio from the Laudamus to the Qui tollis
Laudamus |
Gratias |
Domine deus |
Qui tollis |
| 62 | 46 | 95 | 50 |
| (64) | (48) | (96) | (48) |
| 3 | 2 | 4 | 2 |
Let us now assign tempos and see what these measure ratios produce.
Christe, second Kyrie
The Allegro style and bright D-major tonality of the Christe suggest quarter = 84, whose 85 measures of 4/4 result in a duration just over four minutes, 4:02. The “Alla breve” indication and 4/2 meter in the second Kyrie suggest that the quarter-note pulse in the Christe becomes the half-note pulse in the second Kyrie, whose 59 measures of 4/2 result in a duration of 2:48. This indicates that Bach, indeed, was aiming for the 3:2 measure ratio of 84:56 (shown in Table 1) in order to achieve a precise 3:2 duration ratio of 4:00-2:40. Once again, we see the ubiquitous duration of four minutes.
3:2 ratio of 4:00-2:40 between the Christe and second Kyrie
| 2. Christe (S, S) | D | 4/4 | 85 | Q = 84 | 4:02.85 | 4:00 | 3 | 1.9% |
| 3. Kyrie, Alla breve (SATB) |
f# | 4/2 | 59 | H = 84 | 2:48.57 | 2:40 | 2 | 5.1% |
Gloria, Laudamus
The first duration pair, the Christe and second Kyrie, consists of an aria followed by a chorus. However, the second duration pair, the Gloria and Laudamus, consists of a chorus followed by an aria. Assuming an equal eighth-note pulse for both sections of the Gloria, 3/8 followed by 4/4, naturally results in a 2:1 duration ratio. The Allegro style of the 3/8 section suggests eighth = 168 (dotted-quarter = 56), which becomes quarter = 84 in the et in terra pax section, in 4/4. This results in the durations of 1:47 and 3:37, which show a virtually precise 1:2 ratio and add to a total duration of 5:24. This indicates that Bach, indeed, was aiming for the 4:3 measure ratio of 100:75 (shown in Table 1), because this naturally results in a precise 1:2 duration ratio between the two sections (assuming an equal eighth-note speed).
1:2 ratio of 1:46-3:33 between the two sections of the Gloria
| 4. Gloria (SATB) | D | 3/8 | 100 | DQ = 56 | 1:47.14 | 1:46.66 | 1 | 0.5% |
| et in terra pax | D | 4/4 | 76 | Q = 84 | 3:37.14 | 3:33.33 | 2 | 1.8% |
| (total) | 5:24.28 | 5:20 |
Although Bach was most likely not conscious of the actual durations of the two sections of the Gloria—1:47 and 3:37, which relate 1:2—he did apparently plan the total duration to be 5:20. This can be seen after everything is put into perspective through analysis of the next four movements. The frequent thirty-seconds and Andante character in the following soprano aria, the Laudamus, suggest a tempo of quarter = 48, whose 62 measures of 4/4 result in a duration of 5:10. This indicates that Bach, indeed, was aiming for one less measure in the et in terra pax, 75, and two more measures in the Laudamus, 64 (shown in Table 1), since this results in equal durations of 5:20 when the correct tempos are assumed.
Equal durations averaging 5:20 between the Gloria and Laudamus
| 4. Gloria (SATB) | D | 3/8 | 100 | DQ = 56 | 1:47.14 | 1:46.66 | 1 | 0.5% |
| et in terra pax | D | 4/4 | 76 | Q = 84 | 3:37.14 | 3:33.33 | 2 | 1.8% |
| (total) | 5:24.28 | 5:20 | 1 | 1.3% | ||||
| 5. Laudamus (S) | A | 4/4 | 62 | Q = 48 | 5:10 | 5:20 | 1 | 3.1% |
Gratias, Domine deus
The rather infrequent appearance of eighth notes in the Gratias and its bright tonality of D major suggest a rather lively tempo. A normal Allegro of half = 84 seems a little too fast, which is why I assign a slightly slower, more moderate tempo of half = 72. At this tempo, its 46 measures of 4/2 result in a duration of 2:33. I assign the following aria, the Domine deus (a soprano and tenor duet), a moderate tempo of quarter = 72 because quarter = 63 seems a little too slow and quarter = 84 seems a little too fast. Moreover, the rhythmic character seems to resemble that of a gavotte, whose usual tempo I believe is half = 72. This indicates that Bach, indeed, was aiming for two more measures in the Gratias, 48, and one more measure in the Domine deus, 96 (shown in Table 1), since this 1:2 measure ratio of 48:96 results in a 2:1 duration ratio of 5:20-2:40 when the correct tempos are assumed.
1:2 ratio of 2:40-5:20 between the Gratias and Domine deus
| 6. Gratias (SATB) | D | 4/2 | 46 | H = 72 | 2:33.33 | 2:40 | 1 | 4.2% |
| 7. Domine deus (S, T) | G | 4/4 | 95 | Q = 72 | 5:16.66 | 5:20 | 2 | 1.0% |
Domine deus, Qui tollis
As previously explained, the 95 measures of 4/4 at quarter = 72 for the Domine deus results in a duration of 5:17. The “Lente” indication and sarabande style of the following chorus, the Qui tollis, suggest a slow tempo of quarter = 54 (the most common sarabande tempo), whose 50 measures of 3/4 result in a duration of 2:47. This indicates that Bach, indeed, was aiming for one more measure in the Domine deus, 96, and two fewer measures in the Qui tollis, 48 (shown in Table 1), since this 2:1 measure ratio of 96:48 results in a 2:1 duration ratio of 5:20-2:40 when the correct tempos are assumed.
2:1 ratio of 5:20-2:40 between the Domine deus and Qui tollis
| 7. Domine deus (S, T) | G | 4/4 | 95 | Q = 72 | 5:16.66 | 5:20 | 2 | 1.0% |
| 8. Qui tollis, Lente (SATB) |
b | 3/4 | 50 | Q = 54 | 2:46.66 | 2:40 | 1 | 4.2% |
Gloria to Qui tollis
We can now see that the measure counts and their ratios from the Gloria to the Qui tollis (shown in Table 2) are highly significant and that Bach’s plan was one of achieving a 2:2:1:2:1 duration ratio between these five movements. This is undoubtedly some of the “musical science” to which Bach referred when mentioning the Missa. This “science” is clear and unequivocal. Bach simply sought to unify durations by choosing special numbers of measures combined with specific, absolute tempos. In sum, these five movements consist of just two durations, 2:40 and its double, 5:40.
2:2:1:2:1 ratio of 5:20-5:20-2:40-5:20-2:40 from the Gloria to the Qui tollis
| 4. Gloria (SATB) | D | 3/8 | 100 | DQ = 56 | 1:47.14 | 1:46.66 | 1 | 0.5% |
| et in terra pax | D | 4/4 | 76 | Q = 84 | 3:37.14 | 3:33.33 | 2 | 1.8% |
| (total) | 5:24.28 | 5:20 | 2 | 1.3% | ||||
| 5. Laudamus (S) | A | 4/4 | 62 | Q = 48 | 5:10 | 5:20 | 2 | 3.1% |
| 6. Gratias (SATB) | D | 4/2 | 46 | H = 72 | 2:33.33 | 2:40 | 1 | 4.2% |
| 7. Domine deus (S, T) | G | 4/4 | 95 | Q = 72 | 5:16.66 | 5:20 | 2 | 1.0% |
| 8. Qui tollis, Lente (SATB) |
b | 3/4 | 50 | Q = 54 | 2:46.66 | 2:40 | 1 | 4.2% |
Qui sedes, Quoniam
The measure ratio between the Qui sedes and Quoniam, 86:127, virtually equals 2:3, which stands apart from all the measure ratios encountered from the Gloria to the Qui tollis. This suggests Bach intended these two arias (alto followed by bass) to function as a duration pair whose durations are not necessarily related to those of the Gloria to Qui tollis.
The Qui sedes is notated in 6/8 and Bach supplied staccato marks on many of the eighth notes. I have found that when Bach indicates staccato marks it usually indicates a rather fast tempo. The gigue-like 6/8 meter, staccato marks, and general gigue-like character of the Qui sedes suggest the most common 6/8 gigue tempo, which I believe is eighth = 168 (dotted-quarter = 56). This is substantially faster than it is often performed, as most performers and conductors tend to choose a slower tempo closer to the 6/8 siciliano or pastorale styles. Another reason that suggests a lively tempo of dotted-quarter = 56 for the Qui sedes is that the previous movement, the Qui tollis, is the slowest movement since the opening Kyrie, and as a general rule, Bach usually chose faster movements to follow slower movements, and vice versa, in order to create musical variety. All this evidence suggests it is highly unlikely that Bach would have intended yet another slow tempo after the already slow Qui tollis. With 86 measures of 6/8 at dotted-quarter = 56, the Qui sedes has a duration just over three minutes, 3:04.
The Quoniam consists of 127 measures of 3/4. The 3/4 meter and distinctive rhythmic patterns—for example, in the first two measures of the bass instruments—unmistakably show the style of a polonaise, and thus, suggest the usual polonaise tempo of quarter = 84. (For more on the polonaise tempo, please refer to Elaboration 3.) With 127 measures of 3/4 at quarter = 84, the Quoniam has a duration of 4:32.
Observation of the durations of the Qui sedes, 3:04, and Quoniam, 4:32, shows a virtually precise 2:3 ratio of 3:00-4:30. This suggests Bach was aiming for two fewer measures in the Qui sedes, 84, and one less measure in the Quoniam, 126, to achieve these durations and this ratio. Moreover, the ratio of the ideal measure counts, 84:126, equals 2:3 which also happens to equal the usual tempos of the polonaise and minuet.
2:3 ratio of 3:00-4:30 between the Qui sedes and Quoniam
| 9. Qui sedes (A) | b | 6/8 | 86 | DQ = 56 | 3:04.28 | 3:00 | 2 | 2.4% |
| 10. Quoniam (B) | D | 3/4 | 127 | Q = 84 | 4:32.14 | 4:30 | 3 | 0.8% |
Summary of the first half
We can now view the complete eleven-movement Missa, showing that “Bach the scientist” organized the work in a symmetrical fashion according to duration ratios. The Kyrie and Cum Sancto spiritu stand alone at the outer parameters, whose durations are whole-number or integer values, 9:00 and 4:00. My analyses show Bach was preoccupied very much with symmetry; it is common to see movements stand alone at the outer parameters, which may symbolize outer “walls” or “pillars” using an analogy from architecture. (For more on symmetry, please refer to my study, “Discovering Bach’s Secret Technique of Outer-Parameter Symmetry”. Also, my study “Tempo and Architecture in the Inventions and Sinfonias” shows how Bach’s large-scale plan for the 15 Inventions was governed exclusively by symmetry.)
On the inside of the outer parameters of the Missa stand two duration pairs, Christe-Kyrie and Qui sedes-Quoniam, whose symmetrical duration ratios of 3:2 and 2:3 add further to the overall symmetry. In the middle of the Missa stands a five-movement group that consists exclusively of just two durations, 2:40 and its double of 5:20. All this is achieved within a mere 1.9% average discrepancy.
Large-scale symmetrical duration plan for the Missa
Kyrie |
Christe-Kyrie II |
Gloria-Laudamus-Gratias- |
Qui sedes-Quoniam |
Cum sancto |
| 9:00 | 4:00-2:40 | 5:20-5:20-2:40-5:20-2:40 | 3:00-4:30 | 4:00 |
| integer | 3:2 | 2:2:1:2:1 | 2:3 | integer |
| outer | inner | middle | inner | outer |
The second half of the B-minor Mass
Like the first half (the Missa), the second half of the B-minor Mass also consists of two large sections: (1) the Credo (Symbolum Nicenum), which consists of a succession of nine movements from the Credo to Sanctus, and; (2) a succession of five movements from the Osanna to Dona nobis pacem. Analysis of the second half of the B-minor Mass reveals once again the impeccable work of a “musical scientist”; however, for some reason almost none of the measure counts show proportional ratios as in the Missa. Nevertheless, even without relying on proportional measure ratios, Bach was somehow able to achieve astonishingly accurate 2:3 and 3:2 duration ratios. In fact, the second half of the Mass consists almost exclusively of 2:3 and 3:2 duration ratios, which is realized after assigning the proper tempos to each movement.
Credo-Patrem-Et in unum-Et incarnatus-Crucifixus
Analysis shows a perfectly symmetrical duration scheme for the five-movement group from the Credo to Crucifixus. Moreover, the instrumentation is symmetrical, in that two choruses appear at the outer parameters while the only aria, Et in unum Dominum, appears alone in the middle.
The appropriate tempo for the Credo can be determined by comparison with the B-minor Prelude from book 1 of The Well-Tempered Clavier. These two works are strikingly similar in style, which is defined by a continual walking bass line and slow-moving contrapuntal lines above this, the only difference is that the note values are doubled in the Credo. Comparing a piano reduction of the Credo to the B-minor Prelude indicates that Bach most likely conceived both at the same tempo due to similarity of style. (This type of comparison is what I refer to as “cross-referencing.” Please refer to Elaboration 1.)
Bach marked the B-minor Prelude “Andante,” which indicates a tempo slower than the more normal default Allegro of quarter = 84. Therefore, I believe the tempo of this Prelude and Credo to be the most normal or usual Andante of quarter/half = 63. This is much slower than how most modern “historically informed” conductors take the Credo, which for some reason has become outrageously fast at an Allegro or Vivace of about half = 96. This is clearly much too fast for the Credo.
Kirnberger provides more evidence for a slow tempo for the Credo, judging by his association of the Credo's meter, 4/2, with “emphatic and weighty” musical character. The popular tempo of about half = 96 for the Credo is very far from being emphatic and weighty. The tempo of half = 63, however, is emphatic and weighty indeed. There is absolutely no good reason that supports a fast Allegro-style tempo for the Credo, which makes “historically-informed performance ” in this case an oxymoron.
2/1 meter (also 4/2), which is also called large alla breve by some, consists of two whole notes or semibreves per measure. However, as in the case with the 6/2 meter of two triple beats that are derived from it, it is no longer used because of the confusion caused by the rests, since the same rest has a value of half a measure at one time and a whole measure at another. In place of these, it is better to use 2/2 and 6/4 with the adjective Grave to indicate the emphatic and weighty performance required by these meters. I know of only one Credo by the elder Bach in the large alla breve of two beats, which he designated, however, with C to show that the rests have the same value as in ordinary alla brevetime (386).
With 45 measures of 4/2 at half = 63, the Credo has a duration of just under three minutes, 2:51. Bach followed the slow Credo with a fast Patrem, whose 84 measures of 2/2 and Allegro-style tempo of half = 84 result in a duration of precisely two minutes. This offers yet another reason for a slow tempo for the Credo. From a purely subjective, musical standpoint, choosing an Allegro for the Credo and another Allegro for the Patrem seems to be redundant. Bach usually followed slower movements by faster movements, and vice versa, in order to achieve musical variety and contrast. Choosing two similar Allegro tempos for both movements ruins such interesting musical contrast. (Bach even inscribed the number “84” after the final measure in the Patrem in his autograph copy, which may be one of the few cases that unequivocally proves Bach was conscious of the numbers of measures he achieved.)
Bach marked Et in unum Dominum “Andante,” which suggests a normal Andante of quarter = 63. With 80 measures of 4/4 at quarter = 63, it has a duration just over five minutes, 5:05.
Et incarnatus is clearly a slower movement in 3/4, although its lack of sixteenths suggests a faster tempo than the other slow movement in 3/4, the sarabande-style Qui tollis. I suggest quarter = 72 for the Et incarnatus, whose 49 measures of 3/4 result in a duration of just over two minutes, 2:03.
Bach undoubtedly chose 3/2 for the Crucifixus in order to indicate its “emphatic and weighty” character as Kirnberger describes. I assign the tempo half = 54 because this is the usual tempo for most of Bach’s slower pieces in 3/2, which is also the usual tempo for the 3/2 French courante, one of the slowest and majestic of all court dances. (Please refer to my study, “Of Bach and Courante Tempos”.) With 53 measures of 3/2 at half = 54, the Crucifixus has a duration just under three minutes, 2:57.
(The Crucifixus is clearly a slow movement, in which case Nicholas Harnoncourt’s well-known comment of it being a “dance,” and thus has a lively tempo, is a mistake. Although he is correct by referring to the Crucifixus as a “dance” (passacaglia/chaconne), assuming all dances to have lively tempos and bouncy characters is incorrect. After all, not all dances are fast and some are even slow. Hence, the popular phrase “dance-like” for Bach's music is almost always a misnomer, since it implies a fast tempo and lively character even for slower and more expressive movements. Please refer to my favorite article on this subject, a brilliantly written essay by Jan Swafford: “Speed Freaks Do Bach.”)
We can now view Bach’s plan from the Credo through the Crucifixus. Bach apparently planned a symmetrical duration scheme that reflects the symmetrical nature of the instrumentation. The choruses Credo and Crucifixus appear at the outer parameters, whose virtually equal durations suggest Bach was aiming for two more measures in the Credo, 47, and one more measure in the Crucifixus, 54, to achieve precisely three minutes. Inside these outer parameters appear two more choruses, the Patrem and Et incarnatus, whose virtually equal durations suggest Bach was aiming for one fewer measure in the Et incarnatus, 48, to achieve precisely two minutes. In the middle of the five-movement group appears Et in unum Dominum, whose duration of 5:05 suggests Bach was aiming for one less measure, 79, to achieve precisely five minutes.
Spreadsheet analysis of the Credo to Crucifixus (3:00-2:00-5:00-2:00-3:00)
| 12. Credo (SATB) | A | 4/2 | 45 | H = 63 | 2:51.42 | 3:00 | 3 | 4.8% |
| 13. Patrem (SATB) | D | 2/2 | 84 | H = 84 | 2:00 | 2:00 | 2 | 0.0% |
| 14. Et in unum Dominum, Andante (S, A) |
G | 4/4 | 80 | Q = 63 | 5:04.76 | 5:00 | 5 | 1.6% |
| 15. Et incarnatus (SATB) | b | 3/4 | 49 | Q = 72 | 2:02.5 | 2:00 | 2 | 2.0% |
| 16. Crucifixus (SATB) | e | 3/2 | 53 | H = 54 | 2:56.66 | 3:00 | 3 | 1.9% |
For some unexplainable but fascinating reason, Bach’s plan here is based exclusively on fives. First, there are a total of five movements in this group. Second, the Credo and Patrem add to five minutes (3:00 + 2:00). Third, the Et incarnatus and Crucifixus add to five minutes (2:00 + 3:00). And finally, the Et in unum Dominum itself lasts five minutes.
Symmetrical duration plan from the Credo to Crucifixus (based on the number 5)
Credo |
Patrem |
Et in unum Dominum |
Et incarnatus |
Crucifixus |
| 45 (47) | 84 (84) | 80 (79) | 49 (48) | 53 (54) |
| H = 63 | H = 84 | Q = 63 | Q = 72 | H = 54 |
| 3:00 | 2:00 | 5:00 | 2:00 | 3:00 |
| outer | inner | middle | inner | outer |
Bach may have even intended some alphanumeric symbolism here, in that the Credo and Crucifixus both begin with the letter “C,” which symbolizes the number 3 and their durations of 3:00, while the Et in unum Dominum in the middle begins with the letter “E” which symbolizes the number 5 and its duration of 5:00. Furthermore, the outer movements have letter counts that are divisible by 5—Credo has 5 letters, Crucifixus has 10 letters, while Et in unum Dominum has 15 letters. Of course, these alphanumeric interpretations are only speculation, although I do have a strong feeling (and evidence in other works) that Bach did employ such cryptic modes of alphanumeric symbolism from time to time in his music.
Possible symbolic (alphanumeric) interpretation of the Credo to Crucifixus
Credo |
Et in unum Dominum |
Crucifixus |
| 5 letters | 15 letters | 10 letters |
| C = 3 | E = 5 | C = 3 |
| 3:00 | 5:00 | 3:00 |
| outer | middle | outer |
Et resurrexit-Et in spiritum
The Et resurrexit is unambiguously a fast Allegro or Vivace-style chorus, whose 131 measures of 3/4 at quarter = 96 results in a duration of just over four minutes, 4:06. The bass aria, Et in spiritum sanctum, resembles the style of a pastorale, suggesting a slow to moderate tempo. I assign an Andante-style tempo of dotted-quarter = 48, whose 144 measures of 6/8 results in a duration of precisely six minutes. This suggests Bach was aiming for three fewer measures in the Et resurrexit, 128, in order to achieve a precise 2:3 duration ratio of 4:00-6:00 with the Et in spiritum sanctum.
2:3 ratio of 4:00-6:00 between the Et resurrexit and Et in spiritum
| 17. Et resurrexit (SATB) | D | 3/4 | 131 | Q = 96 | 4:05.62 | 4:00 | 2 | 2.3% |
| 18. Et in spiritum (B) | A | 6/8 | 144 | DQ = 48 | 6:00 | 6:00 | 3 | 0.0% |
Confiteor-Sanctus
The Confiteor and Sanctus pair is one of the most impressive and baffling of cases of duration ratios in all of Bach's works, which clearly shows the profound side of “Bach the scientist” more than any other example in the B-minor Mass. I still cannot figure out how Bach did it, but he somehow created a virtually precise 3:2 duration ratio in spite of the fact that the Confiteor consists of three sections each with different tempos, while the Sanctus consists of a 4/4 and 3/8 section. The Confiteor consists of a total of 251 measures subdivided into three sections: a 2/2 of 120 measures, an Adagio transitional section in 2/2 of 25.5 measures, and a final Vivace section in 2/2 of 105.5 measures. The Adagio indication after 120 measures implies that the first section probably does not have a slower 2/2 tempo like slower “stile antico” movements—for example, Contrapunctus 1 from The Art of Fugue, whose tempo is a rather slow half = 54—since an Adagio indication in this case would seem redundant. For this reason, I assign a moderate tempo of half = 72 for the first section. Assuming an Adagio of half = 48 for the next 25.5 measures and Vivace or Allegro tempo of half = 96 for the final 105.5 measures result in a total duration of 6:36. What is baffling about this is that these three tempos, half = 72-48-96, result in an impressively accurate 3:1:2 duration ratio among the three sections, the only discrepancy being about three measures.
But this is just the beginning. The Sanctus is divided into a 4/4 section with triplets (similar to a 12/8) followed by an Allegro-style 3/8 section. I assume the tempo of quarter = 84 for the 4/4 section, because this is the usual tempo I believe Bach intended for his 12/8 gigues that have few or no sixteenths. (For more on Bach’s gigue tempos, please refer to my study “Of Bach and Gigue Tempos”.) With 47 measures of 4/4 at quarter = 84, the first section has a duration of 2:14. With 121 measures of 3/8 at an Allegro of dotted-quarter = 56, the second section has a duration of 2:10. Thus, the two sections of the Sanctus are virtually equal at 2:14 and 2:10, suggesting Bach intended perhaps one less measure for the Sanctus, 46, and three more measures for the Pleni sunt coeli, 123, in order to achieve equal durations averaging about 2:12.
But are you ready for more? When added together, the three sections of the Confiteor have a total duration of 6:36 (which themselves relate 3:1:2), while the two sections of the Sanctus have a total duration of 4:24 (which themselves relate 1:1). And 6:36 and 4:24 happen to be at a precise 3:2 ratio! I have known this for a long time and still cannot figure out how Bach achieved such precision. Perhaps the total measure counts have something to do with it, since the total measures of the Confiteor, 251, and Sanctus, 168, show a precise 3:2 ratio (251:168 = 3:2). Perhaps the durations of the Vivace, Sanctus, and Pleni sections also have something to do with it, since these three successive movements have virtually equal durations of 2:12, 2:14, and 2:10, showing a 1:1:1 ratio in addition to the 3:1:2 and 3:2 ratios. This example represents one of the most profound accomplishments of “Bach the scientist” or “Bach the mathematician,” but I know it is not a coincidence because similar instances like it happen in other works.
3:2 ratio of 6:36-4:24 between the Confiteor (3:1:2) and Sanctus (1:1)
| 19. Confiteor (SATB) | f# | 2/2 | 120 | H = 72 | 3:20 | 3:15 | 3 | 2.6% |
| Adagio | 2/2 | 25.5 | H = 48 | 1:03.75 | 1:05 | 1 | 1.9% | |
| Vivace ed allegro | D | 2/2 | 105.5 | H = 96 | 2:11.87 | 2:10 | 2 1 | 1.4% |
| (total) | 6:35.62 | 6:36 | 3 | 0.1% | ||||
| 20. Sanctus (SATB) | D | 4/4 | 47 | Q = 84 | 2:14.28 | 2:12 | 1 | 1.7% |
| Pleni sunt coeli | D | 3/8 | 121 | DQ = 56 | 2:09.64 | 2:12 | 1 | 1.8% |
| (total) | 4:23.92 | 4:24 | 2 | 0.0% |
(back to discussion of Kyrie and Cum sancto spiritu)
Osanna-Benedictus-Osanna da capo-Agnus dei-Dona nobis pacem
Just like the five-movement group from the Gloria to the Qui tollis in the first half, the final five-movement section of the second half also consists of just two durations, although here they relate 2:3 rather than 1:2 as in the former five-movement group. The Osanna chorus is clearly an Allegro-style movement, whose 149 measures of 3/8 at dotted-quarter = 56 results in a duration of 2:40. The following tenor aria, Benedictus, resembles the style of a certain type of sarabande, very much like the sarabande from the third keyboard Partita. This is a good example of cross-referencing, as both pieces are in 3/4 and feature triplet sixteenths and thirty-seconds. My research shows this type of sarabande has a slow Adagio-style tempo of quarter = 42, since this speed allows for the ornate figurations and note values but at the same time is not too fast. With 57 measures of 3/4 at quarter = 42, the Benedictus has a duration of just over four minutes, 4:04.
Following this is the Osanna da capo, whose duration we have already determined to be 2:40, indicating that Bach intended a symmetrical 2:3:2 duration ratio of 2:40-4:00-2:40 between the Osanna-Benedictus-Osanna da capo. This suggests that Bach intended one less measure for the Benedictus, 56, in order to achieve precisely four minutes.
Symmetrical 2:3:2 duration plan among the Osanna, Benedictus, and Osanna da capo
Osanna |
Benedictus |
Osanna da capo |
| 3/8, 149, DQ = 56 | 3/4, 57, Q = 42 | 3/8, 149, DQ = 56 |
| 2:40 | 4:04 (4:00) | 2:40 |
| 2 | 3 | 2 |
There is a big problem with the tempo most performers choose for the Agnus dei, an alto aria. Traditionally, it is taken very slowly and lugubriously at around quarter = 42 or even slower. Analysis, however, suggests a much faster tempo. The first bit of evidence for a faster tempo lies in the fact that Bach notated the bass line as eighths with eighth rests, which persists throughout the entire aria. This constant pizzicato style is a very good indication that Bach conceived this movement not at a slow tempo, but at a rather lively tempo. For example, the D-major Prelude from volume 1 of The Well-Tempered Clavier has eighths with eighth rests in the bass line, and this most definitely is a lively piece.
The second bit of evidence for a faster tempo in the Agnus is that sixteenth notes are almost totally absent and function more on the embellishing rather than structural level. A slow tempo for the Agnus would perhaps be justifiable were it to have many sixteenth notes, but this is clearly not the case. Moreover, a very slow tempo like quarter = 42 makes it virtually impossible for the singer to take the first four-measure phrase in one breath, but rather, the phrase must be broken up into smaller fragments destroying the musical line.
The text for the Agnus does not even suggest a slow tempo, since if anything, one should be happy and optimistic about having one’s sins taken away by the Lamb of God. After all, why should one be sad and lugubrious about being cleansed of one’s sins? Finally, had Bach really intended such a slow tempo in the Agnus, he would have had no other choice but to modify the otherwise lively looking pizzicato style with a tempo word such as “Adagio.” And this Bach certainly did not do. Therefore, all the evidence clearly suggests a faster rather than slower tempo for the Agnus.
I believe a moderate tempo of quarter = 72 to be the best tempo for the Agnus, since quarter = 84 seems a little too fast and quarter = 63 seems a little too slow. Moreover, a tempo of quarter = 72 allows the singer to easily sing in four-measure phrases. Perhaps the strongest bit of evidence for a moderately-fast tempo for the Agnus is that with 49 measures of 4/4 at quarter = 72 its duration is 2:43, which is virtually equal to that of both the Osanna and Dona nobis pacem. This suggests Bach intended one less measure, 48, to give it the same measure count and duration as the Dona nobis pacem. (Also, these two movements share a common pulse of M.M. 72.) The Agnus is by far the shortest, most umimposing, and lightest aria in the entire Mass; however, singers and conductors routinely turn this two-and-a-half to three-minute jaunt into a six-minute saga of epic proportions. The Agnus dei is without question the most misunderstood and misinterpreted movement in all of Bach’s works, often being performed at less than half speed!
The Dona nobis pacem is simply a restatement of the Gratias from the Missa, the only difference is that Bach inserted a new text. Assuming the same tempo and duration as before shows the Dona nobis pacem to have a duration of 2:33. As concluded before, this suggests Bach intended two more measures, 48, to achieve a duration of 2:40.
It now becomes clear that Bach planned the final five-movement group in the second half of the Mass to have a 2:3:2:2:2 duration scheme at 2:40-4:00-2:40-2:40-2:40, suggesting that he was aiming for one less measure in the Agnus, 48, to achieve 2:40. Our analysis of the B-minor Mass is now complete.
2:3:2:2:2 ratio of 2:40-4:00-2:40-2:40-2:40 from the Osanna to the Dona nobis pacem
| 21. Osanna (SATB) | D | 3/8 | 149 | DQ = 56 | 2:39.64 | 2:40 | 2 | 0.2% |
| 22. Benedictus (T) | b | 3/4 | 57 | Q = 42 | 4:04.28 | 4:00 | 3 | 1.8% |
| Osanna da capo | D | 3/8 | 149 | DQ = 56 | 2:39.64 | 2:40 | 2 | 0.2% |
| 23. Agnus dei (A) | g | 4/4 | 49 | Q = 72 | 2:43.33 | 2:40 | 2 | 1.2% |
| 24. Dona nobis pacem (SATB) |
D | 4/2 | 46 | H = 72 | 2:33.33 | 2:40 | 2 | 2.4% |
Summary of the second half
We can now view the complete thirteen-movement second half, showing that “Bach the scientist” organized the work in a symmetrical fashion according to duration ratios similarly although a little differently than in the first half. The Credo to Crucixus and Osanna to Dona nobis pacem represent two five-movement duration groups at the outer parameters, whose durations are the symmetrical 2:00-3:00-5:00-3:00-2:00 and the non-symmetrical 2:40-4:00-2:40-2:40-2:40 (2:3:2:2:2), respectively.
On the inside of these five-movement groups stand two duration pairs, Et resurrexit-Et in spiritum and Confiteor-Sanctus, whose symmetrical duration ratios of 2:3 (4:00-6:00) and 3:2 (6:36-4:24) add further to the overall symmetry. All this is achieved within a mere 1.5% average discrepancy.
Large-scale symmetrical duration plan for the second half
Credo-Patrem-Et in unum Deum- |
Et resurrexit- |
Confiteor- |
Osanna-Benedictus-Osanna da capo- |
| 3:00-2:00-5:00-2:00-3:00 | 4:00-6:00 | 6:36-4:24 | 2:40-4:00-2:40-2:40-2:40 |
| five movements | 2:3 | 3:2 | five movements |
| outer | inner | inner | outer |
Summary of the complete B-minor Mass
Let us now summarize “Bach the scientist’s” temporal organization of the B-minor Mass. The first half, the Missa, is organized symmetrically in which the outer parameters, the Kyrie and Cum sancto spiritu, stand alone with no duration partners but have integer durations instead, 9:00 and 4:00. Inside these outer parameters stand two duration pairs, Christe-Kyrie and Qui sedes-Quoniam, which have 3:2 and 2:3 duration ratios of 4:00-2:40 and 3:00-4:30. In the middle stands a five-movement group, the Gloria to Qui tollis, which has a 2:2:1:2:1 duration scheme of 5:20-5:20-2:40-5:20-2:40. It is significant that the measure ratios Bach intended in all movements from the Christe to the Quoniam are clearly recognizable, since they consist exclusively of integer values: 1:2, 2:1, 2:3, 3:2, 4:3.
The complete first half of the Mass (large-scale symmetrical duration plan)
| 1. Kyrie, Adagio (SATB) | b | 4/4 | 4 | Q = 42 | 0:22.85 | |||
| Largo ed un poco piano | b | 4/4 | 122 | Q = 54 | 9:02.22 | 9:00 | 0.4% | |
| (total) | 9:25.07 | |||||||
This marks the separation between movements 1 and 2-3. |
||||||||
| 2. Christe (S, S) | D | 4/4 | 85 | Q = 84 | 4:02.85 | 4:00 | 3 | 1.9% |
| 3. Kyrie, Alla breve (SATB) | f# | 4/2 | 59 | H = 84 | 2:48.57 | 2:40 | 2 | 5.1% |
This marks the separation between movements 2-3 and 4-8. |
||||||||
| 4. Gloria (SATB) | D | 3/8 | 100 | DQ = 56 | 1:47.14 | 1:46.66 | 1 | 0.5% |
| et in terra pax | D | 4/4 | 76 | Q = 84 | 3:37.14 | 3:33.33 | 2 | 1.8% |
| (total) | 5:24.28 | 5:20 | 2 | 1.3% | ||||
| 5. Laudamus (S) | A | 4/4 | 62 | Q = 48 | 5:10 | 5:20 | 2 | 3.1% |
| 6. Gratias (SATB) | D | 4/2 | 46 | H = 72 | 2:33.33 | 2:40 | 1 | 4.2% |
| 7. Domine deus (S, T) | G | 4/4 | 95 | Q = 72 | 5:16.66 | 5:20 | 2 | 1.0% |
| 8. Qui tollis, Lente (SATB) | b | 3/4 | 50 | Q = 54 | 2:46.66 | 2:40 | 1 | 4.2% |
This marks the separation between movements 4-8 and 9-10. |
||||||||
| 9. Qui sedes (A) | b | 6/8 | 86 | DQ = 56 | 3:04.28 | 3:00 | 2 | 2.4% |
| 10. Quoniam (B) | D | 3/4 | 127 | Q = 84 | 4:32.14 | 4:30 | 3 | 0.8% |
This marks the separation between movements 9-10 and 11. |
||||||||
| 11. Cum sancto spiritu, Vivace (SATB) |
D | 3/4 | 128 | Q = 96 | 4:00 | 4:00 | 0.0% | |
The second half of the Mass begins with a symmetrically arranged five-movement group from the Credo to the Crucifixus consisting exclusively of durations and other elements (see above) that relate to the number 5: 3:00-2:00-5:00-2:00-3:00. The second half also ends with a five-movement duration group, the Osanna to the Dona nobis pacem, which shows a 2:3:2:2:2 duration scheme of 2:40-4:00-2:40-2:40-2:40. Between these two five-movement groups stand the Et resurrexit and Et in spiritum sanctum, which have a 2:3 duration ratio of 4:00-6:00, as well as the Confiteor and Sanctus, which have a 3:2 duration ratio of 6:36-4:24 (and whose sections relate 3:2 and 1:1).
The complete second half of the Mass (large-scale symmetrical duration plan)
| 12. Credo (SATB) | A | 4/2 | 45 | H = 63 | 2:51.42 | 3:00 | 3 | 4.8% |
| 13. Patrem (SATB) | D | 2/2 | 84 | H = 84 | 2:00 | 2:00 | 2 | 0.0% |
| 14. Et in unum Dominum, Andante (S, A) |
G | 4/4 | 80 | Q = 63 | 5:04.76 | 5:00 | 5 | 1.6% |
| 15. Et incarnatus (SATB) | b | 3/4 | 49 | Q = 72 | 2:02.5 | 2:00 | 2 | 2.0% |
| 16. Crucifixus (SATB) | e | 3/2 | 53 | H = 54 | 2:56.66 | 3:00 | 3 | 1.9% |
This marks the separation between movements 12-16 and 17-18. |
||||||||
| 17. Et resurrexit (SATB) | D | 3/4 | 131 | Q = 96 | 4:05.62 | 4:00 | 2 | 2.3% |
| 18. Et in spiritum (B) | A | 6/8 | 144 | DQ = 48 | 6:00 | 6:00 | 3 | 0.0% |
This marks the separation between movements 17-18 and 19-20. |
||||||||
| 19. Confiteor (SATB) | f# | 2/2 | 120 | H = 72 | 3:20 | 3:15 | 3 | 2.6% |
| Adagio | 2/2 | 25.5 | H = 48 | 1:03.75 | 1:05 | 1 | 1.9% | |
| Vivace ed allegro | D | 2/2 | 105.5 | H = 96 | 2:11.87 | 2:10 | 2 1 | 1.4% |
| (total) | 6:35.62 | 6:36 | 3 | 0.1% | ||||
| 20. Sanctus (SATB) | D | 4/4 | 47 | Q = 84 | 2:14.28 | 2:12 | 1 | 1.7% |
| Pleni sunt coeli | D | 3/8 | 121 | DQ = 56 | 2:09.64 | 2:12 | 1 | 1.8% |
| (total) | 4:23.92 | 4:24 | 2 | 0.0% | ||||
This marks the separation between movements 19-20 and 21-24. |
||||||||
| 21. Osanna (SATB) | D | 3/8 | 149 | DQ = 56 | 2:39.64 | 2:40 | 2 | 0.2% |
| 22. Benedictus (T) | b | 3/4 | 57 | Q = 42 | 4:04.28 | 4:00 | 3 | 1.8% |
| Osanna da capo | D | 3/8 | 149 | DQ = 56 | 2:39.64 | 2:40 | 2 | 0.2% |
| 23. Agnus dei (A) | g | 4/4 | 49 | Q = 72 | 2:43.33 | 2:40 | 2 | 1.2% |
| 24. Dona nobis pacem (SATB) |
D | 4/2 | 46 | H = 72 | 2:33.33 | 2:40 | 2 | 2.4% |
To summarize, Bach organized the duration ratios in each half of the B-minor Mass symmetrically, although with entirely different plans, 1—2—5—2—1 and 5—2—2—1, respectively (including the Osanna da capo).
Symmetrical duration groupings in the first half
| 1 | 2 | 5 | 2 | 1 |
Symmetrical duration groupings in the second half
| 5 | 2 | 2 | 5 |
Bach achieved his duration ratios and symmetrical designs with astonishing accuracy, at an average discrepancy of just 1.4%, which translates to a discrepancy of less than one second for every minute of music. This is calculated by taking the total of 2491 measures (not including the 4-measure introduction), taking the total discrepancy between the “actual” and “ideal” measure counts, 35, and then dividing the total measures by the total discrepancy, 35/2491 = .014 = 1.4%, which translates to .84 of a second.
Conclusion
This has been a long journey and I commend anyone for having read all the Elaborations from the beginning up to this point. The purpose of presenting this long and detailed analysis of the B-minor Mass is not necessarily to provide insight into the fascinating and cryptic workings of “Bach the scientist” through analysis of perhaps his greatest of all works, but simultaneously, to prove the theory introduced on my home page and explained further in Elaborations 1-5. Let us now summarize the key points from the beginning up to now:
1. A mathematically ideal system of tempos is derived that anyone could derive simply by setting up a grid and using elementary arithmetic (Elaboration 2).2. Some reasonable assumptions are made and important axioms are formulated, which are based on some of the key points made by Bach’s most famous student and disciple, Johann Philip Kirnberger (Elaboration 3).
3. It is shown how Bach could have easily achieved any duration or duration ratio he so desired by using elementary arithmetic and solving for unknown values (Elaborations 4 and 5).
4. Bach’s esteemed biographer, Christoph Wolff, explains how Bach was not just an organist, musician, or even a composer, but a “musical scientist” who did not merely compose nice music, but more importantly, produced “works of musical science” (this Elaboration). Bach’s self-proclaimed work of “musical science” was the Missa he dedicated in 1733 to the electoral court in Dresden.
5. By a highly detailed and objective-based analysis of the complete B-minor Mass, it is shown that Bach sought proportional duration ratios in his Mass, which could have only been attained by employing the methods hypothesized in Elaborations 4 and 5.
6. The final statistics in the B-minor Mass overwhelmingly corroborate the theory of durations and duration ratios even further. Half of the entire Mass is dominated by integer durations within a very small margin of error. That is, out of 24 movements, 12 have integer durations: two of 2:00, three of 3:00, four of 4:00, one of 5:00, one of 6:00, and one of 9:00. Moreover, there exist a total of two 1:1 ratios, one 1:2 ratio, three 2:3 ratios, three 3:2 ratios, one 1:1:1 ratio, one 3:1:2 ratio, one 2:2:1:2:1 ratio, and one 2:3:2:2:2 ratio. The only section to have neither an integer duration nor be a part of a duration ratio is the brief four-measure introduction. Other than this, the entire Mass is unified exclusively by integer durations and duration ratios. Furthermore, the Mass is divided into two halves that are each organized in perfectly symmetrical fashions according to duration groups. The average of all the margin of error figures (the percentages in the far right columns) comes to just 1.7%, meaning that Bach achieved all these things—integer durations, duration ratios, symmetry—within a mere one-second discrepancy for every minute of music. On top of this, the durations for each movement in each half of the Mass add up to virtually equal figures of 49:04 and 50:41 which total 99:45, showing that Bach achieved a total duration of 100 minutes divided equally as 50:00 + 50:00 within a mere 0.9% average margin of error. Could we have expected anything less from the greatest “musical scientist” in the history of music?
Summary of statistics in the B-minor Mass
| number of movements surveyed | 24 |
| number of integer durations | 12 |
| movements lasting 2:00 | 2 |
| movements lasting 3:00 | 3 |
| movements lasting 4:00 | 4 |
| movements lasting 5:00 | 1 |
| movements lasting 6:00 | 1 |
| movements lasting 9:00 | 1 |
| 1:1 duration ratios | 2 |
| 1:2 duration ratios | 1 |
| 2:3 duration ratios | 3 |
| 3:2 duration ratios | 3 |
| 1:1:1 duration ratios | 1 |
| 3:1:2 duration ratios | 1 |
| 2:2:1:2:1 duration ratios | 1 |
| 2:3:2:2:2 duration ratios | 1 |
| average margin of error | 1.7% |
Large-scale 1:1 duration ratio of 50:00 + 50:00 in both halves
| total duration of first half | 49:04 | 50:00 | 1 | 1.9% |
| total duration of second half | 50:41 | 50:00 | 1 | 0.6% |
| total duration of Mass | 99:45 | 100:00 | 0.3% |
Considering all the six steps above, it can be concluded that Bach’s tempos were nothing other than those assumed in the matrix in Elaboration 2. It can also be concluded that Bach indeed planned a style, meter, and tempo first, which then enabled him to calculate the ideal numbers of measures to compose when finally putting pen to paper. And finally, unless the durations and their ratios and the large-scale symmetrical designs within a mere 1.7% discrepancy in the B-minor Mass can be shown to be coincidental, then my theory as well as Bach’s tempos and durations can be considered proven. Let us now return to the initial statements posed at the beginning of this essay:
“This theory is crazy and totally outlandish. Why in the world would Bach straightjacket his compositions into strict, inflexible, and idealistic duration molds that seem directly antithetical to the human element of making music?”
I think the answer to this question is clear now.
Continue to next Elaboration -->
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(Postscript: At this writing, I have analyzed Bach's other major works in the same fashion as the B-minor Mass and have found similar results and statistics. I have not converted these analyses to HTML—which is very time consuming—but rather, they exist in Microsoft Word format. Unfortunately, they are not available to the public at this time. For details on these analyses—which include virtually all of Bach's complete works—please refer to the summary of my book in progress, “Breaking the Bach Tempo Code”.)