Chapter 4: The Alpha & The Omega and the Imitation of Nature
“Symmetry, as wide or narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.” – Hermann Weyl
The Quest for Symmetry
The letters Alpha and Omega, Α and Ω, are the first and last letters of the Classical Greek alphabet, which symbolize the unity and oneness of “God, the Father” and “God, the Son” in Christian theology. Since Christ and God are understood as being equal, incorporating such symbolism into multi-movement musical works requires making the first and last “elements” equal, which may refer to the first and last notes of a melody or fugue subject. For example, the last fugue Bach would ever pen, Contrapunctus 14 from The Art of Fugue, has an opening subject consisting of seven notes that are the same played backwards as forwards. In this case, the seven notes symbolize “perfection” while their symmetrical arrangement and the identical first and last notes symbolize The Alpha & The Omega.” (See Diagram 1.)
Diagram 1. Opening subject of Contrapunctus 14 (1749)
The “elements” that define The Alpha & The Omega may also refer to the first and last movements of multi-movement works which are equal in performance time creating large-scale outer parameter symmetry. Since achieving perfect symmetry in melodies and fugue subjects (like in Contrapunctus 14) is impractical and impossible to achieve more than just a few times, Bach apparently preferred unifying the outer parameters by making the first and last movements equal in duration. The easiest way to do this is simply to repeat the first movement at the end. For example, with a quick flourish of his pen that took a few seconds to write “Aria da capo,” Bach created perfect outer parameter symmetry in the Goldberg Variations. (See Diagram 2.)
Diagram 2: Outer parameter symmetry in the Goldberg Variations, BWV 988 (1741)
Most performers and listeners take it for granted that the exact same music appears at the end of the Goldberg Variations, as if it there were a rule among composers that one must end a theme and variations the same way as it begins. Surprisingly, however, ending a set of theme and variations with the identical opening theme almost never occurs in the classical music repertory. In fact, this author is not aware of the existence of any such theme and variations, except the Goldberg Variations. Seen in this light, outer parameter symmetry in the Goldberg Variations must be understood as an anomaly.
About twenty-five years before Bach composed the Goldberg Variations (BWV 988), he composed his only other set of variations, Aria variata (BWV 989), which consists of a theme and 10 variations, each 12 measures long. The Theme and Var. 10 both feature stately melodies with the eighth note as the fastest non-ornamental note value, requiring the same tempo. This suggests that seeking equal first and last movements was not a new technique the mature Bach used in just the Goldberg Variations, but a technique the young Bach had started using as early as 1709 as a 24-year-old. If true, then Bach’s “musical science” of achieving proportional duration ratios began early in the composer’s career. (See Diagram 3.)
Diagram 3. Outer parameter symmetry in the Aria variata, BWV 989 (1709)
The Goldberg Variations is not the only work to consist of identical first and last movements, as Bach did this occasionally in other multi-movement works. There may be more examples, but this author has counted a total of six occurrences of identical first and last movements in Bach’s works:
Multi-movement works with identical first and last movements
Cantata No. 30 (BWV 30) – 160 measures of 2/4
Cantata No. 117 (BWV 117) – 100 measures of 6/8
BWV 228 – 77 measures of 4/4
BWV 248III – 96 measures of 3/8
Goldberg Variations (BWV 988) – 32 measures of 3/4
Violin and Harpsichord Sonata No. 6 (earlier two versions, BWV 1019a) – 91 measures of 4/4
The last work on this list, BWV 1019, is the one work (other than sinfonias 1 and 15, discussed below) that comes closest to proving the hypothesis that Bach sought equal durations for the first and last movements. This sixth and final Sonata for Violin and Harpsichord went through three stages. In the two early versions, BWV 1019a, Bach simply repeated the first movement at the end thereby creating outer parameter symmetry the easiest possible way as he would almost twenty years later in the Goldberg Variations. (See Diagram 4.)
Diagram 4. Outer parameter symmetry in BWV 1019a (1717-1723)
Upon his final revision, however, Bach opted for a completely new last movement by replacing 91 measures of 4/4 with 119 measures of 6/8 in which both share the sixteenth note as the fastest non-ornamental note value as well as the same tempo indication, Allegro. Assuming the same “Allegro” sixteenth-note speed for both results in virtually equal durations due to the similar number of sixteenth notes in each, 1456 and 1428, respectively. This is calculated by multiplying the number of measures by the number of sixteenth notes per measure (91 x 16 = 1456, 119 x 12 = 1428). In other words, had Bach achieved merely two to three measures more for the newly composed last movement in 6/8, it would have had the same number of sixteenth notes, and hence, precisely the same duration as the first movement in 4/4 assuming an equal sixteenth-note speed. Bach’s revision of BWV 1019 proves his striving for outer parameter symmetry to symbolize The Alpha & The Omega. (See Diagram 5.)
Diagram 5. Outer parameter symmetry in BWV 1019 (1717-1723)
Bach used the same technique as in BWV 1019 to achieve outer parameter symmetry in the 15 Sinfonias; however, this instance represents one of the most impressive and mind-boggling of Bach’s mathematical feats due to the presence of another symbolic element, Bach’s own personal signature. Sinfonia 1 is in 4/4 time and consists of 21 measures, while sinfonia 15 is in 9/16 time and consists of 38 measures. The number of measures combined symbolize Bach’s name alphanumerically, 2138 = BACH, showing that Bach inscribed his signature into the first and last sinfonias by way of number of measures. However, this is just the beginning.
Not only do these two bookends to the sinfonias symbolize Bach’s name, but both share the sixteenth note as the fastest non-ornamental note value, like the first and last movements of BWV 1019, and both consist of virtually the same number of sixteenth notes, also like BWV 1019. More specifically, sinfonias 1 and 15 consist of 336 and 342 sixteenth notes, respectively, which is calculated by multiplying the number of measures with the number of sixteenth notes per measure: 21 x 16 = 336, 38 x 9 = 342. This indicates that Bach was a mere six sixteenth notes off from achieving precisely the same number of sixteenth notes in both sinfonias, which implies equal sixteenth-note speed. And like adding a cherry on top of a dessert, Bach topped off this impressive mathematical feat with his own personal signature. (See Diagram 6.)
Diagram 6. Outer parameter symmetry in the 15 Sinfonias (1723)
The accompanying 15 Inventions published with the 15 Sinfonias also contain a similar outer parameter plan minus the personal signature. Inventions 1 and 15 are the only two inventions in 4/4 with 22 measures, which likewise imply equal tempos and durations. (See Diagram 7.)
Diagram 7. Outer parameter symmetry in the 15 Inventions (1723)
Bach seldom composed single-movement works. For example, Bach rarely composed “a fugue” or “a prelude” but rather preludes always belong to fugues, and vice versa. Hence, a “prelude and fugue” is one entity and not just two single, unrelated pieces randomly paired together. This is demonstrated most clearly in Clavierübung III (also known as the “German Organ Mass”) in which Bach placed the E-flat Major Prelude and Fugue (BWV 552) each at the book ends of the collection. Considering Bach’s symmetrical plan in all the previous examples, it becomes likely that Bach intended this prelude and fugue to have equal durations to unify the opus with The Alpha & The Omega. This work is discussed further in the next chapter as well as Part 2, but for now, let us contemplate on the same performance duration for the great Prelude and Fugue in E-flat Major that begins and ends Clavierübung III. (See Diagram 8.)
Diagram 8. Outer parameter symmetry in Clavierübung III (1739)
