Ordered Chaos: Understanding Musical Structure and Intention in Modernism

“That’s not real music.”

Any educator, performer, and advocate of new music has undoubtedly encountered this statement. Audiences like what is comfortable – recognizable structure (symphony having four movements, sonata form being in three primary sections, recurring melody, etc.), a consistent rhythm and beat (waltzes, common time, etc.) and of course, a westernized tonal center complying with the social construct of “major = happy, minor = sad”. All of these aspects are good, but anything beyond this often raises a few eyebrows and is cast into a box labeled as “noise”.

Yet, “noise” is simply how the ear categorizes the sounds it fails to comprehend.

In other words, what the ear cannot rationalize, it rejects.

Over the past few years, I’ve made a habit of asking my students, “What is music?” or “How do you define music?” These are some of the responses I have garnered:

“Music is a series of sounds that makes you feel something.”

“A bunch of notes that sound pretty.”

“It’s just noises… and silence. Anything is music.”

“That question hurts my brain.”

However, a recurring statement in most responses has been, “Music is a series of sounds that have some sort of form or meaning.”

But what about the sounds that are chaotic, disruptive, unpalatable? The sounds that feel unrelenting. Finales without resolution. Lack of cadential closure. Sounds that feel like an indiscernible swarm overwhelming the senses. Can those sounds be considered music, too?

Absolutely.

Although form is not a prerequisite for art, many works in the modernist sound world also have structure and intentionality. However, these structures often take on a different shape and require a different lens through which to analyze.

Historical Context

Let’s step back in time to post-WWI Europe. Regardless of national or political allegiance, countless countries, families, and livelihoods have been decimated. Composers who survived the military draft began to respond how they knew best – through writing music. However, the pre-war impetus that sparked the modernist movement – the premiere of Stravinsky’s The Rite of Spring – now seemed overly sensational to cope with complex themes of violence and trauma. As a response to grief and uncertainty, many composers reflected on older styles of music and thought, including Greco-Roman and medieval. Why is this important? Both schools had 1) unique views on how to define music (late Greco-Roman: Boethius’ (480-524 AD) philosophy of musica mundana, humana, and instrumentalis – in a nutshell, a broad spectrum on sound; Middle Ages: music as the metaphysical) and 2) both expected a mathematical component, even if not “pleasant” aurally.

In the Middle Ages, music education was divided into two departments: Mathematics (musicus) and rhetoric (the cantor). The former group involved the intellectual study of music (modern day ‘music theory’), the latter was the art of performance and composition. It was understood among all artists from this era that music must be a balance of both subjects. Yet, this concept seems to have largely dissipated in the 18th- early 20th century.

In modernism, these rediscovered principles, among others, guided many 20th century composers in their construction of new music. The abstract could become concrete, chaos become ordered, and the dissonant, a dialogue with divinity.

Practical Application

So, how is this applicable musically, or shall we say, in a “cantor” sense?

Let’s look at a well-known piano étude by composer György Ligeti titled L’escalier du diable, or “The Devil’s Staircase”. Take a listen to the piece here.

At first listen, you may find it cacophonous, vile, and distressing. And a cursory glance of the page elicits a similar response from many musicians. The bars are inconsistent, there are accents on weak beats, the score is split into three staves to be played simultaneously, and a dynamic direction of fortissississississimo marking the final motif seems physically impossible. It is an aurally and visually overwhelming work.

However, it is important to note that Ligeti was from a similar educational background as previously mentioned in the historical context. Aside from music, he was a well-rounded scholar in the subjects of art, architecture, and mathematics, with an especial interest in Benoit Mandelbrot’s theory of fractal geometry and concepts involving chaos theory. Using this lens, we can begin to pick apart L’escalier and gain an understanding.

The piece represents a staircase on which a prisoner continually climbs, thinking he has reached the surface, yet only finding himself to arrive both at the top and bottom simultaneously. This is a reference to the paradox known as Penrose Stairs or the impossible staircase (see figure).

One can hear the slow ascension, with a few “trips” on the stairs along the way (0:25 – 0:35), only to reach the surface and cascade back to the entry level (0:58 – 1:00). This cycle recurs throughout the piece, perpetually rising, then falling repeatedly. A new sound entry (2:40 – 5:15) introduces a vast, sustained theme which Ligeti marks as wildes Glockengeläute (wild ringing of bells). This theme is startingly interrupted (5:15) before recurring, combined with the initial “climbing theme”, as the étude’s finale (5:30 – end). This creates a structure of 1) Growth, 2) Plateau, 3) Growth, and 4) Plateau. However, it is unpredictable, and includes phases in which a lack of continuity is present.

Similarly, we can find this concept in mathematics. This pattern resembles the Cantor function, in which a variable is constant and increasing but not absolutely continuous. Not surprisingly, this function, when graphed, visually resembles a staircase (see figure). Ironically, this function is also referred to as the “Devil’s Staircase”!

This figure is a representation of the overall structure of the work. Yet, the “steps” of the staircase can be seen in even only the first four measures. Ligeti begins with a pattern in eighth notes of 2+2+3 followed by 2+2+2+3, which although may contain a few changing variables (influenced by Cantor function and chaos theory), maintain a relatively steady slope. (I’ve created a graph below to provide a visualization for this rhythm. Each entry of the first theme is marked with a purple “S” for subject, larger measures are marked in aqua, subdivisions are marked in fuchsia.)

Aside from mathematical equations, in the wildes Glockengeläute, one can almost discern a recognizable theme – the Dies Irae or “Day of judgment”, a sequence used in the Requiem Mass (see figure). An excerpt can be seen in measures 29 – 31 (3:08 – 3:23), and then again, rhythmically augmented and a bit distorted, in the finale (5:50-end).

Aside from his education in medieval music and the obvious reference to bells in this work, what is the significance of Ligeti’s choice to use the Dies Irae? Simply put, Ligeti was familiar with suffering. After surviving a labour camp during World War II, Ligeti discovered that nearly his entire family had been murdered in Auschwitz, leaving only his mother behind. His brother, Gábor, had been killed in Mauthausen-Gusen concentration camp, completely devastating György. Pictures from Mauthausen-Gusen show a familiar symbol – stairs. To enter the camp, prisoners had to climb a winding staircase known as “the stairs of death”. Once inside, a bell was rung – constant, but never continuous – summoning prisoners to work, and often leading to their death. A “devil’s staircase” with intermittent tolling, a somber reminder to the occupants of their impending “day of judgment” at the hands of the Third Reich.

If you have a moment, listen to L’escalier du diable again. Do you still hear it as noise? Or do you hear it as an autobiographical score – the music of Ligeti’s life? Although perhaps unpleasant, it is still music, nonetheless.

So the next time you encounter a bizarre piece that seems meaningless or aurally distressing, try approaching it from a different lens. Is it influenced by literature? Does it need a historical analysis? A philosophical debate? A mathematical equation? Or perhaps, is it a combination of all the above? In the words of composer Wolfgang Caspar Printz:

“Although we have two judges in music, ‘ratio’ or the intellect and ‘sensus’ or the ear, it is imperative that while they should agree, ratio retains the upper hand… for if the ear is given the upper hand and is allowed the freedom to judge independently, then absolutely nothing could be verified with certainty in music.”

by Brianna Jaeger, Catoctin School of Music

For Further Reading and Sources Used:

Application of Chaos Theory and Fractal Geometry in the Music of György Ligeti – Paul Merkley

The Cantor Function – O. Dovgoshey, O. Martio, V. Ryazanov, M. Vuorinen

The Dynamics of Disorder – Richard Steinitz

An Examination of Rhythmic Practices and Influences in the Keyboard Works of György Ligeti – Lauren Halsey

György Ligeti: Etudes pour piano, Deuxieme Livre (Schott Edition)

György Ligeti: Music and the Holocaust – ORT

Introduction to a Philosophy of Music – Peter Kivy

Music, Maths, Chaos – Richard Steinitz

Musica Poetica – Dietrich Bartel

Philosophy and Music – Theodore Gracyk & Andrew Kania

Rhetoric – Aristotle