For many years I taught a college course entitled “Form and Analysis,” and one of my favorite lectures was one in which the students and I discussed the organization of the ostinato principle in the Crucifixus from Bach’s Mass in B-Minor, long before I knew about the earlier version of the work from Cantata BWV 12. It was my chance to introduce the students to the meaning of the work through Bach’s use of number symbolism, dissonance, texture change from polyphony to homophony, and finding where B-flat + A + C + B-natural, his well-known musical signature, is hidden. We then listened to two different recorded performances of the work, and discussed the pros and cons of each performance.
When completing his B-Minor Mass in 1748–49 why did Bach reuse a 35-year-old composition from the opening choral movement of cantata BWV 12, Weinen, Klagen, Sorgen, Zagen (Weeping, Lamenting, Worrying, Quaking), as the basis (parody) for his recomposed setting for the Crucifixus?1 Undoubtedly, he realized that the cantata text demonstrated an equivalency of meaning with the text of the Crucifixus. “Indeed, the basic theme of the Crucifixus is identical to that of the cantata movement, since it also concerns the suffering inflicted upon a man, Christ, because of humanity’s sins.”2 An examination of the cantata text and that of the Crucifixus demonstrates this equivalency.
Weinen, Klagen,
Sorgen, Zagen,
Angst und Not,
Sind der Christen Tränenbrot
die das Zeichen tragen.
Weeping, lamenting,
Worrying, quaking,
Anxiety and distress:
[These] are the bread of affliction for Christians
Who bear the mark of Christ.
Crucifixus
Crucifixus etiam pro nobis
sub Pontio Pilato passus
et sepultus est.
He was crucified also for us
suffered under Pontius Pilate
and was buried.3
The cantata movement, composed in 1714, is a large ABA form in F minor. Both the original and the recomposed Crucifixus are based on the ostinato/passacaglia bass of section A, marked Lente (hereafter, simply called ostinato): the discarded B section, marked un poc’allegro, is not based on an ostinato principle. The bass line of section A is notated in half notes that state the chromatic ostinato 12 times.
Let us now look closely at the symbols and numbers in the first movement of cantata BWV 12. The key signature of three flats may be a symbol of the Trinitarian God, or the three days Christ spent in the tomb. (The key of F minor now has four flats, but minor keys in Bach’s day were derived from the Dorian mode, which requires the sixth scale step to be lowered when needed.) According to eighteenth-century writers on key characteristics, the key of F minor was “the most pathetic of all, lugubrious and despairing.”4 The meter is 3/2, which might refer to Christ as the second person of the Trinity, or if 3 and 2 are added to total five, a stronger symbol for Christ results because 5 symbolizes the wounds of Christ on the cross, as well as the cross of Christ with its 5 points, and man with his 5 senses and 5 appendages.5 (I am not suggesting that whenever Bach used 3/2 meter he was referring to the Trinity and Christ; I merely suggest the possibility here because of the subject matter.) The ostinato bass line has a total of 13 notes—a symbol of Christ and the apostles or the Last Supper, and also a universal symbol of misfortune.
The halfway point
For proportional or symbolic reasons, Bach often marked the halfway point in his compositions.6 Techniques that he used to indicate these points include: 1) a change of texture; 2) introducing new material; 3) using a previously introduced motive in a way that had not been used before, and will not be used again in the same composition; and 4) introducing his own musical signature (B-flat+A+C+B-natural) at this halfway point. In BWV 12 this point occurs on the first beat of measure 25, where the chorus comes to a stop with the ostinato at the first full cadence; then a sudden change of texture demands attention with the first homophonic choral phrase in the movement so far. (Example 1)
Other symbols
On the first beat of measure 33, two remarkable and unmistakable symbols occur: the first is a sudden silence brought about by the only notated rest for the chorus at the moment that the ninth ostinato statement begins. Of course, the number 33 represents Christ’s age, and the number 9 the hour of his death. (Example 2) In light of the preceding picture, the 12 statements of the ostinato can be seen now as a reference to the Church as founded by the 12 Apostles.
Crucifixus from Mass in B Minor, BWV 232/17
An examination of the Crucifixus shows several alterations from the earlier cantata movement. (Orchestration changes and modernizations are not relevant to this study of numbers and symbols.) The first noticeable change is the four-measure introductory statement of the ostinato; this changes the total number of statements from 12 to 13, and the number of measures from 49 to 53; might the number 53 be a symbol of the atonement of Christ as found in the 53rd chapter of Isaiah? This thirteenth ostinato statement may be a reference to the Last Supper with Jesus and the 12 apostles, as well as to misfortune/death often associated with this number; and if one counts from the first entrance of the chorus in measure five, ignoring the introductory statement, then the numbers 33 and 9, symbols for the death of Jesus at the ninth hour as found in the cantata, are still valid.
The original half notes of the ostinato bass have been changed to quarter notes, creating an effect of quiet excitement and movement, as in the beating of an anxious human heart. The 24 notes of the ostinato might refer to rebirth or regeneration, since 2 x 4 is 8, and Christ rose on the 8th day after his entry into Jerusalem; the eighth day of the week is a new beginning, just as the eighth note of the octave renews the musical series. Another important change is the change of key to E minor, a key that is described as “pensive, profound, sad and expressive of grief; in such a way, however, that some chance of consolation remains” (emphasis mine). The letter “E” is the fifth letter of the alphabet, and another symbol of the cross and the wounds of Christ.
The cantata movement and the Crucifixus both make use of many dissonant intervals, harmonically and melodically; clearly these dissonances symbolize the pain of the crucifixion. Some of these intervals are called “false intervals;” take for example, the notes G up to A#, an augmented second on the page, but a minor third to the ear. Bach introduced additional “false” intervals to the soprano and alto voices in measures 13 and 14 of the Crucifixus, changing the original major seconds to augmented seconds, undoubtedly to symbolize the falsely accused Christ.
As previously stated, the number two refers to Christ as the second person of the Trinity, and number five refers to Christ and the cross. Let us now look at the first entrance of the chorus in measure five. The soprano enters on the second beat with a five-note motive of two different pitches, C and B; in the German scale B is actually H; therefore the pitches spell the first two letters in Christ’s name. (Example 3)
In the concluding ostinato statement, a modulation is required to connect with the next movement, the “Et resurrexit” in D major. Bach ends the Crucifixus a third higher in G major (Christ rose on the third day), and then as if to confirm his belief that Christ died for him, he introduced his musical signature (B-flat+A+C+B-natural) hidden between alto and soprano voices, beginning on the third beat of measure 51. (Example 4)
Numerically equivalent numbers
Basic rules or techniques of number symbolism are found in treatises from the Middle Ages that explain how to find numerically equivalent numbers. The procedure is a simple one—they can be found through cross-multiplication or cross-addition. An example using the number 33 will make this clear: this number represents 9 as 3 x 3; therefore 33 and 9 are symbolically equivalent numbers. On the other hand, if cross-addition is applied as in 3 + 3, then 6 and 33 are also symbolically equivalent numbers.7
Equating letters with numbers is an ancient tradition known as gematria. Descriptions of it are found in the Jewish mystical practice of cabala, and there is some evidence to indicate that the technique was known in ancient Greece. The alphabet in Bach’s day consisted of only 24 letters, in which the letter “I” and the modern “J” were counted as one, as were the modern “U” and the Latin “V.” To demonstrate gematria and numerical equivalency, let us begin with BACH: the corresponding alphabet numbers are 2138; by adding these numbers (2+1+3+8) we arrive at the number 14; if we multiply them (2 x 1 x 3 x 8) we arrive at 48; therefore, 14 and 48 are also numerically equivalent numbers.
One may wonder where I am going with this, and why it might be important. The reason is that the gematria for the title Weinen, Klagen, Sorgen, Zagen is 237; when multiplied, 2 x 3 x 7 equals 42, an equivalent BACH number as in 3 x 14. The gematria for Crucifixus is 127, and 1 x 2 x 7 equals 14, or BACH; thus Bach’s signature as composer can be found in the very titles of these two compositions—accidental perhaps, but surely deserving of further study.
Is it just a coincidence, or did Bach really think it important to use numbers and other symbols in his music? Did he do it to challenge his own artistic abilities to create effortlessly sounding music that was secretly confined by numerical constraints? Did he do it to give pleasure to future analytically minded generations, or did he believe that numerical associations strengthened artistic perfection? The only answer to these questions is that we can never know the true answer to any of them. Yet who can deny that the discovery of these numbers gives us an unexpected pleasure that is separate from the pleasure we receive from hearing or playing the music itself? If we cannot agree or believe that employing numbers in his music was a decision Bach consciously made, then perhaps we can at least admit the truth of the great German philosopher/mathematician Leibnitz who wrote in 1712 that ‘Music is a secret exercise in the arithmetic of the soul, unaware of its act of counting’.8
