Infinite Continuum: Fractals in Ligetis Viola SonataBy: Blake Allen13 May 2013

Somewhere underneath, very deeply, theres a common place in our spirit where the beauty of mathematics and the beauty of music meet. But they dont meet on the level of algorithm or making music by calculation. Its much lower, much deeperor much higher, you could say. Gyrgy Ligeti[footnoteRef:1] [1: Steintz, 14]

What we talk about when we talk about fractals

Firstly, fractals are self-similar patterns and come from the nonexistent, the potential.[footnoteRef:2] Self-similarity is when an object is exactly, or approximately, the same as another part of itself (ie. a portion can be viewed as a replication of the whole.) This concept might be confusing, so it is best to demonstrate with an image of the Barnsley Fern[footnoteRef:3]: [2: Zukav, xxv] [3: Barnsley, 61]

The Barnsley fern is named after Michael F. Barnsley who first discussed its principles in his book Fractals Everywhere. A fern is the most basic, and accessible, version of a fractal because it is found in nature and most people can relate to having seen a similar leaf. Notice how at any amplification, the general pattern can be replicated and reproduced to make the same image.

Secondly, fractals are based upon attractors. There are three different types of attractors. The first are fixed attractors, which are a single point at which all motion tends. Imagine the center of a spiral. The second are periodic attractors, which are sets of points through which all motion oscillates but never settles at that given point. Imagine a pendulum and how it ticks back and forth, but never settles in the middle. One can envision where the middle of the arch is, but it never stops at that point even when turned off. Another very accessible example is the scientific and mathematic formulae of half-life (t1/2). If one divides something (typically radiation) in half and then that section in half etc, it will never completely erase. The third are strange attractors. Strange attractors are periodic cycles that are smeared out so that the orbits never repeat exactly [] Since they eventually return to the same general area but never exactly the same place, they are the essence of chaos.[footnoteRef:4] Imagine the same pendulum discussed earlier, but moving in a circle, swinging back and forth. Here are examples: [4: Madden, 7]

Rssler attractor[footnoteRef:5]Lorenz attractor[footnoteRef:6] [5: http://en.wikipedia.org/wiki/File:Roessler_attractor.png] [6: http://en.wikipedia.org/wiki/File:Lorenz.png]

Next, we need to talk about convergence, which is the point where all motion tends. In reality, the point of convergence is at infinity, but the human eye cannot sense it. This takes us to the Cauchy sequence, which is the point humans place the point of convergence. Imagine two mirrors across from each other or the lines of a train track creating infinity. We know that miles down the road the train tracks do not meet up, but our eye puts a point at where converge occurs. Below is a graph Augustin-Louis Cauchy created demonstrating this principle.[footnoteRef:7] [7: http://en.wikipedia.org/wiki/File:Cauchy_sequence_illustration.svg]

Now we need to talk about the Chaos Theorywhen the present determines the future, but the approximate present does not approximately determine the future.[footnoteRef:8] We as humans naturally think simple things behave simply and complex things are inconceivable. However, according to the chaos theory, simple things are complex, complex things are simple and the law of complexity is universal: complex systems breed turbulence and coherence simultaneously. Another example of the chaos theory is the butterfly effecta butterfly flaps its wings in LA and there is a tsunami in Japan. Refer back to the images of strange attractors. These are chaos. [8: Danforth]

Lets take a look at images of simple and complex fractals. Just like the Barnsley fern, the Sierpenski Triangle is a simple fractal.[footnoteRef:9] Just like the leaf, notice how at any given point of the triangle, one can see the exact image replicating. It is quite simple. [9: Duchesneau ,97]

Julia sets are where the true art of fractals appear, and they are quite complex. The same principle of replication applies to these two images.

What we talk about when we talk about fractals in music

With all the information, how is this applicable to music? There are things called music attractors. These can be key relationships, keynotes, poles of attractions, musical landmarks, meter, barlines and pitch. Though an orchestra typically tunes to 440 Hertz, it is impossible for every instrument to be exactly in tune. Each instruments pitch beats are wider or narrower, so we come as close to the exact maximal resonance as possible, but we accept a small amount of natural error as every instrument is different.

Are certain tones of the key naturally attractors? In C major, all motion leads to C. E and G are supporting tones[footnoteRef:10] to help establish C as the tonal center B is the leading tone which tends towards C while F always wants to lower to E, especially in a dominant seventh. [10: Madden, 45]

What about atonality? Stravinsky noted in his Poetics of Music:It is [] indispensable [] to recognize the existence of certain poles of attraction. Diatonic tonality is only one means of orienting music toward these poles. The function of tonality is completely subordinated to the force of attraction of the pole of sonority. All music is nothing more than a succession of impulses that converge towards a definite point of response. The general law of attraction is satisfied in only a limited way by traditional diatonic system [] The drawing together and separation of poles of attraction in a way determine the respiration of music.[footnoteRef:11] [11: Stravinksy, 35-37]

What about self-similarity in music? Below is a diagram of all the As on a piano given in Hz frequency. So, how does this create a fractal?

If you put all of the notes, and their given Hz on a straight line, you are given a self-similar spacing of the octave sequence[footnoteRef:12] [12: Madden, 28]

Lastly, the harmonic sequence is a fractal in itself. The harmonic sequence is not geometric, which as has been discussed earlier is the form of self-similarity. However, if we look at the sequence below (in C), we will notice that the notes get closer and closer together the farther up the sequence it gets. Convergence? Yes, indeed. It is a Cauchy Sequence. By transforming the nonlinear frequency sequence to a linear pitch notation, we have changed our point of view and revealed self-similarity. [footnoteRef:13] [13: Madden, 32]

What we talk about when we talk about Ligeti

Gyrgy Ligeti was born in Dicsszentmrton, Hungary (present day Trnveni, Romania) on May 28, 1923 and died June 12, 2006 in Vienna. He is known to the general public for Stanley Kubricks use of orchestral pieces in 2001: A Space Odyssey, The Shining and Eyes Wide Shut. However, in the classical realm, he is known for his micropolyphony and polyrhythmic compositions. Ligeti realized that the new theories which sought to explain the precarious balance between order and disorder, pattern and chaos, and the apparent origins of both conditions in measureable deterministic situations, had intriguing parallels with the way he composed [] Disorder, it was claimed, might seem disorderly, but in fact arose from basically very simple and universal principles.[footnoteRef:14] [14: Steintz, 15]

Ligeti often incorporates mathematical ideas, such as the chaos theory, calculations or proportions and Fibonacci numbers in his music. All of these mathematical ideas are determined by the question Is our world deterministic, or is it governed by chaos?[footnoteRef:15] [15: Steintz, 15]

In Ligetis Sonate for Viola Solo (1991 1994) he uses all of the principles of fractals that have been mentioned earlier. Only two movements will be discussed from here on. Firs off, why did Ligeti want to write a viola sonata this late in his career? He writes:The viola is seemingly a big violin tuned a fifth lower. In reality, the two instruments are worlds apart [] the low C-string gives the viola a unique acerbity, compact, somewhat coarse, with the aftertaste of wood, earth and tannic acid [the] C-string was the starting point for my fantasies of a viola sonata.[footnoteRef:16] [16: Ligeti, 5]

I. Hora lungIt evokes the spirit of Romanian folk music which [] Hora lung literally means slow dance but in the Romanian tradition this is not a dance but are sung folk melodies (in the northernmost province of the country, Maramures, in the centre of the Carpathian mountains), nostalgic and melancholy, richly ornamented [] This movement is played exclusively on the C-string and in it I make use of natural intervals (pure major third, pure major seventh and also the 11th harmonic).[footnoteRef:17] [17: Ligeti, 5]

This movement is all about the harmonic series, and even contains a convergence/Cauchy Sequence at the end when it plays the C harmonic series using harmonics on the C string with a note play even if the uppermost harmonics hardly sound.[footnoteRef:18] (see appendix) [18: Ligeti, 13]

Below is a table of the first movements expansion of range, and one notices the Cauchy sequence in the right column.[footnoteRef:19] [19: Duchesneau, 28]

t.o.1t.o.2t.o.3Interludet.o.4t.o.5t.o.6______________________________________________CbbagF#edcbbgCcbAbabgbgbgFfFfebebebebDddddccccccbbbbbaaaaagggggfffffCCCInterludeCCCIn the program notes of Hora lung, Ligeti writes:[footnoteRef:20] [20: Ligeti, 12]

Because the piece is based on the F series (except for the interlude mm. 15 16 and final measures), but played on the C string, Ligeti alters the third and fourth scale degrees. Ligetis use of this imaginary F harmonic series results in the dominant ending of a C harmonic series.

II. LoopThe title refers to the form; the same melodic figures are repeated, continually varied rhythmically and played progressively faster in tempo [much like the visual image of the fractal]. Double-stoppings are played throughout with one of the notes always being an open string. The performer is therefore compelled to carry out daring position changes which in the fast section of the movement creates a dangerous virtuosity. In addition, this movement music also be played in the spirit of jazz: elegant and relaxed.[footnoteRef:21] [21: Ligeti, 5]

Loop is a spiral that spins out of control into a self-similar Cauchy Sequence. It is comprised of 45 different notes, in various rhythms, that speed up, get faster and faster, until every note is a 16th, minus the fortissimo G# in m.90. The theme audibly is repeated eight times. The 45 notes are as follows:

The initial theme requires 148 semiquavers to be completed, the eighth and final [] requires only 48, down to 32%--or a reduction of 68%.[footnoteRef:22] The piece ends with three measures of rest, which indicates an unheard ninth repetition of the theme. [22: Duchesneau ,31]

Lastly, Ligeti was so fascinated by Benot Mandelbrots discovery of deterministic chaos and fractal geometry he invented his own Julia set. Even though he created musical fractals filled with self-similarity, chaos and Cauchy Sequences, he was fond of the mathematics behind it all. [footnoteRef:23] [23: Duchesneau ,95]

Bibliography

Barnsley, M. F., & Mathematics. (2012). Fractals Everywhere: New Edition (dover Books On Mathematics) (New ed.). Mineola, N.Y.: Dover Publications.

Danforth, C. M. (2013, April 4). Chaos in an Atmosphere Hanging on a Wall. Mathematics of Planet Earth 2013.

Duchesneau, L., & Marx, W. (Eds.). (2011). Gyrgy Ligeti: of Foreign Lands and Strange Sounds. Woodbridge, Suffolk.: Boydell Press.

Ligeti, G. (2001). Sonate for Viola Solo. London: Schott Music.

Madden, C. (2007). Fractals in Music: Introductory Mathematics for Musical Analysis Second Edition (2nd ed.). High Art Press.

Steintz, R. (1996, March). Music, Maths. The Musical Times.

Stravinsky, I. (2008). Poetics of Music in the Form of Six Lessons. New York: Hamlin Press.

Zukav, G. (2009). Dancing Wu Li Masters: an Overview of the New Physics. HarperOne.