Exploring Fractal Structures in Bach's Cello Suites
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The Fractal Nature of Bach's Music
Bach's iconic cello suites serve as a fascinating example of fractal structures in music. The intricate patterns present in these compositions have intrigued musicians and mathematicians alike.
"It’s simple to play any musical instrument: all one must do is touch the correct key at the appropriate time, and the instrument will play itself."
— Johann Sebastian Bach (1685–1750)
Like many music enthusiasts throughout history, I hold a deep admiration for Bach's work. My true appreciation blossomed during my counterpoint studies at Berklee College of Music. However, it wasn't until I participated in the NSF-supported Yale Fractal Geometry Workshops, guided by Benoit Mandelbrot, that I began to grasp the depths of Bach's genius, particularly in relation to the concept of music as a fractal.
As a composer, Bach is celebrated for his extraordinary ability to fuse aesthetic beauty with mathematically inspired musical transformations. While his six cello suites may not be as universally recognized as pieces like The Brandenburg Concertos or The Goldberg Variations, they have firmly secured their place in the classical music repertoire.
Remarkably, these suites were largely overlooked for a century following Bach's death. They re-emerged sporadically in the 19th century, but it was not until the renowned cellist Pablo Casals recorded and performed them extensively in the late 1930s that their captivating beauty gained widespread recognition. As Casals poignantly noted, "They are the very essence of Bach, and Bach is the essence of music."
Analyzing Fractal Scaling in Cello Suite No. 3
In a previous article, we examined structural scaling within Cello Suite No. 3. Here, we will delve into findings from a paper I published in the journal Fractals, which introduced the concept of intervallic scaling.
Understanding Pitch-Related Scaling
Pitch, along with rhythm, is a fundamental element of music. It represents the perceived frequency of a note. When two pitches are played simultaneously, they create a harmonic interval, while playing them in succession forms a melodic interval. The changes in a series of melodic intervals can be viewed as the second derivative of pitch, which we will refer to as melodic moment.
Typically, the spectrum of musical pitch is not continuous. In Western music, the minimum distance between two pitches is known as a semitone, corresponding to the distance between adjacent piano keys. The modern equal temperament tuning system relates note frequencies logarithmically as follows:
F_p = F_ref * 2^(k/12)
where ( F_p ) is the frequency of the target pitch, ( F_ref ) is the frequency of a reference pitch, and ( k ) is the intervallic distance in semitones. An interval of 12 semitones corresponds to an octave, which is perceived as the same note at a higher or lower pitch.
Depending on various factors such as musical style and the composer’s intent, only a select subset of available frequencies is deemed "appropriate" for use in a composition.
Binning Data for Analysis
To analyze pitch-related distributions, it can be beneficial to "bin" the data, also referred to as "coarse graining." For our analysis, we will group data into equal-size bins, each covering the same range of values. For a dataset ( D = {y_1, y_2, y_3, ldots} ) with a range ( r = D_{max} - D_{min} ), we partition ( D ) into ( n ) bins of size ( r/n ). Each bin ( B_1, B_2, ldots, B_n ) will then contain closely related elements.
Assigning a single value to a group of neighboring values helps reveal trends that may not be immediately apparent. However, this method cannot fabricate a power-law where none exists.
Fractal Dimensions in Bach's Suites
There are six cello suites, each with multiple movements. For simplicity, we will reference them by their suite and movement numbers. Our focus will be on movements with minimal harmonic intervals to ensure a clear analysis of melodic intervals.
The individual movements consist of approximately 200 to 1400 notes, providing a relatively small sample size. We will assume a strong correlation between the melodic characteristics and the underlying intervallic distribution.
Applying this approach to Suite 1.2, the allemande serves as an excellent example of fractal scaling. After excluding 24 notes that provide harmonic support, we can examine the distribution of melodic intervals.
The Enchanting Prelude
In the video below, enjoy Yo-Yo Ma's rendition of the Prelude from Cello Suite No. 1, which beautifully showcases Bach's melodic brilliance.
The Significance of Fractal Analysis
The analysis of intervallic scaling reveals the dynamics of melody, suggesting an underlying consistency in how melodies evolve. Lower fractal dimension values indicate a balance between small and large melodic jumps, while higher values reflect a tendency for smaller jumps to occur more frequently, punctuated by larger leaps.
An intriguing question remains: Can listeners perceive differences in fractal dimensions between compositions? While our analysis focuses on melody, rhythm and tempo also significantly influence our perception of musical dynamics.
Bach's work stands out for its transparent quality and timeless nature, allowing listeners to connect with the fractal patterns inherent in the world he experienced. If you're interested in exploring the original research paper "Intervallic Scaling in the Bach Cello Suites," feel free to reach out!
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