Introduction
Background on Bartók's Compositional
Methods
The Fibonacci Series in the 1st Movements
Bassoon Solos
About the writer
One of the joys of playing music is achieving an understanding of how the music is organized. On many occasions I have been struck by the beauty of a melody or the logic with which a piece of music progresses. No doubt we have all experienced similar experiences, for it was moments like these that attracted us to become performers in the first place. Yet often we are totally unable to determine exactly what in the music attracts us. What are these mystical strands that a composer weaves into the fabric of his music? Is it possible for us to reach an understanding about how this music is constructed? This article is an attempt to search for answers to such questions. In particular, I have chosen the bassoon solos in opening of Bartók's Dance Suite because of the striking balance they contain and the beautiful, mysterious coherence they possess.
Background on Bartók's Compositional Methods
Erno Lendvai provides some useful insights into Bartók's unique method of composition. Lendvai takes note of Bartók's fascination with formations occurring in nature:
He was constantly augmenting his collection of plants, insects and mineral specimens. He called the sunflower his favourite plant, and was extremely happy whenever he found fircones placed on his desk. According to Bartók "also folk music is a phenomenon of nature. Its formations developed as spontaneously as other living natural organisms: the flowers, animals, etc." ("At the Sources of Folk Music": 1925).[1]
Fir cones, sunflowers, and certain sea shells are constructed with a spiral that has been described by the geometric series which is called the "Fibonacci series. " Starting with a "0, " the Fibonacci series progresses 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... Each succeeding number is obtained by adding the previous two numbers. (For instance 0 + 1 = 1; 1 + 1 = 2.) Lendvai has noticed that aspects of the Dance Suite are patterned after the Fibonacci series:
The first movement arises from major seconds (2); the second is built on minor thirds (3); the third summarises these former elements (2 + 3 + 2 + 3 + 2), presenting a pure pentatonic scale. The harmonies of this movement are based on 5 + 5. Finally, the melody of the fourth movement follows the pattern 8 = 5 + 3, where 5 = 3 + 2.[2]
This gives a segment of the Fibonacci series: 2, 3, 5, 8. It is Lendvai's contention that Bartók is using the Fibonacci series to provide an overall coherence to the Dance Suite.
The Fibonacci Series in the 1st Movement Bassoon Solos
Perhaps the most significant use of the Fibonacci series in the First Movement can be found in the tonal and rhythmic organization of the melody presented by the bassoons. The succession of intervals (in half-steps) in the melody in measures 2 through 9 is: 1 2 3 2 1 2 3 2 1 2 3 2 0 1 2 3 2 0 1 2 5 3 3 (Figure 2). There is an undulating pattern up and down the Fibonacci series with occasional skips. The climactic number " 5 " is presented towards the end of the phrase. Notice that the number " 4, " which is not a member of the Fibonacci series, is not present. Though the listener (or performer) may not recognize that the Fibonacci series is being used in this passage, nevertheless I believe that the movement from smaller to larger melodic intervals is heard. There is a logic that is pleasing to the ear. The phrase makes "musical sense" when we hear it.
Bartók also uses the Fibonacci series rhythmically in this passage. If we allow groups of notes to be initialized by the starts of ties or slurs the number of members in each group follow the Fibonacci series (Figure 3). 1 have ignored the grace note in measure 9 and counted the G in that measure as part of the group since it is an arrival point. The 1 + 4 groupings in measures 5 to 7 are a variation on the 5 grouping starting in measure 4 and therefore should also be considered a 5 grouping. Figure 4 is a continuation of this analysis of articulated groupings. The I + 5 groupings in measures 18 to 20 are another variation on the five grouping starting in measure 4. The last two groupings of 4 could be joined as components of a single group of 8, which is the next number in the Fibonacci series after 5.
Is all of this just playing with numbers? I think not. Listen to the music and keep in mind the Fibonacci series. Bartók has masterfully created these melodies in conformance with musical and numerical logic. The strength of its organic whole lies in the remarkable synthesis Bart6k achieves between the Fibonacci series and his musical ideas.
Terry Ewell, bassoonist, was the winner of the first annual Fernand Gillet performance competition. He was principal bassoon of the Hong Kong Philharmonic Orchestra for seven years, and has been a soloist under the direction of Jahja Ling, Gerard Schwarz, and Vilem Sokol. He received his Master of Arts in music theory in 1988. Currently, Mr. Ewell is on the faculty of Pacific Lutheran University. In addition, he is pursuing a Ph.D. in music theory at the University of Washington.
ENDNOTES
1. Erno Lendvai, Bela Bartók,
An Analysis of His Music (London: Kahn & Averill, 1971), p.
29.
2. Ibid., p. 49.