"The riddles of God are more satisfying than the solutions of man."
Variation V from Elgar's Enigma Variations is dedicated to Richard Penrose Arnold (1856 - 1908), the son of poet Matthew Arnold. His friendship with Elgar was driven chiefly by their mutual interest in golf. Ein feste Burg plays "through and over" Variation V as shown in Figure 14.1. The missing Principal Theme plays in the upper staff while the variation appears in the lower two staves in the form of a piano reduction. The notes of the Principal theme are based on Bach’s rendering of Ein feste Burg from the final chorale of cantata BWV80. An audio file of this melodic mapping supports the efficacy of this contrapuntal solution.
Figure 14.2 illustrates precisely how Ein feste Burg was carefully mapped over Variation V based on melodic interval mirroring and the principles of counterpoint. Melodic interval mirroring occurs when note intervals from Ein feste Burg are reflected in the variation over comparable or identical distances between notes. These notes do not necessarily appear in the melody line of the variation. The contrapuntal devices of similar and contrary motion were also considered in this analysis. Similar motion is when both voices move in the same direction, but not necessarily by the same degree. Contrary motion takes place when Ein feste Burg moves in the opposite direction than the variation, again not necessarily by the same interval. Similar motion is indicated by SM, and contrary motion by CM. For the purposes of this analysis, similar motion includes any instances of parallel motion, and contrary motion any instances of oblique motion. In some cases, the upper voice of the variation moves parallel with Ein feste Burg while the bass line moves in a contrary manner. An effective counterpoint typically employs a fairly balanced mix of contrary and similar motion, something clearly evident with this mapping.
In Figure 14.2 a melodic conjunction is represented by a diamond-shaped note head, and a harmonic conjunction by a triangle-shaped note head. A melodic conjunction is defined as any matching melody note between Ein feste Burg and the movement's melody line. A harmonic conjunction is defined as match between a melody note from the covert Principal Theme and any non-melodic note from the movement. Both melodic and harmonic conjunctions must sound together to be considered a match.
Table 14.1 summarizes 68 melodic conjunctions between Ein feste Burg and Variation V. A melodic conjunction is any shared note between both melodies. Share melody notes are dispersed over 20 out of 24 measures in Variation V. Ein feste Burg was deemed to be dormant in 2 of the 24 measures, namely measures 172 and 173. This means that Ein feste Burg plays in 22 out of 24 measures, or 91.7 % of the time. In measures where the covert Principal Theme is active, shared melody notes are present in 20 of those 22 measures, or almost 91% of all active measures. A total of 68 melodic conjunctions is truly remarkable because these note matches appear sequentially. The odds of such a feat occurring by chance are infinitesimally small.
Table 14.2 breaks down melodic conjunctions between Ein feste Burg and Variation V by note type. There are 8 shared melody notes with frequencies range from 1 to 21.
Table 14.3 gives a complete breakdown of all shared notes between Ein feste Burg and a piano reduction of Variation V. There are 166 shared notes spread out over 28 measures. There are 10 shared note types with frequencies ranging from 1 to 66. Of these shared notes, 9 are shared by both melodies. These shared notes are A flat, A, B flat, B, C, D, E, F, and G.