<The Musical Dimensions of Escher's Artistry: A Harmonious Illusion>
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“Ideas of chaos captivate us as they allow us to create order.” — Maurits Cornelis Escher (1898 — 1972)
As a teenager, I was captivated by a blacklight poster of M. C. Escher's "Other World." The combination of vibrant blacklight colors and Escher's intricate designs was unparalleled. For those unfamiliar with the artwork, here’s the original woodcut print:
<h2>Other World (Another World)</h2>
<div><h3>National Gallery of Art</h3></div>
<div><p>www.nga.gov</p></div>
It wasn't until I read Douglas Hofstadter's classic work, Gödel, Escher, Bach: an Eternal Golden Braid, that I began to truly appreciate the complex mathematics Escher illustrated with such finesse.
One of the remarkable concepts Escher brought to life is the idea of an infinite loop, as seen in his works "Ascending and Descending" and "Waterfall."
<h2>Ascending and Descending | The Art Institute of Chicago</h2>
<div><h3>Maurits Cornelis Escher, 1960</h3></div>
<div><p>www.artic.edu</p></div>
<h2>Waterfall | The Art Institute of Chicago</h2>
<div><h3>Maurits Cornelis Escher, 1961</h3></div>
<div><p>www.artic.edu</p></div>
These visual deceptions engage us in a unique way. They feel both correct and incorrect, prompting us to trace an impossible route in pursuit of a resolution that seems perpetually out of reach.
As a composer, I found myself questioning whether a musical representation of an endless staircase or a never-ending waterfall was possible. It turns out that with knowledge of the human auditory system and the clever application of mathematics, we can indeed create such experiences.
Psychoacoustics
Psychoacoustics examines the intricate relationship between sound's physical characteristics and the perceptions they create. This field advanced significantly due to the groundbreaking work of 19th-century scientist Hermann von Helmholtz.
Helmholtz's influential 1863 publication, "On the Sensations of Tone as a Physiological Basis for the Theory of Music," became a cornerstone reference in this area. He theorized that the ear functions as a resonator, with different cochlea sections vibrating at distinct frequencies. This idea, rooted in Joseph Fourier's work, illuminated how we perceive and differentiate pitch.
To clarify, pitch refers to our auditory perception of musical sounds and is typically associated with the fundamental frequency—the longest wavelength corresponding to a specific musical note.
The human ear can detect a broad range of sound pressure and frequency, approximately from 20 Hz to 20 kHz. Consequently, to maintain sensitivity where it's most needed, our sound perception has developed a logarithmic nature. This means we perceive the doubling of a fundamental frequency as the same note, merely higher in pitch, with the doubled frequency termed an octave above the original note.
In 1964, cognitive scientist Roger Shepard published "Circularity in Judgments of Relative Pitch" in the Journal of the Acoustical Society of America. He demonstrated how our logarithmic perception of pitch and loudness could be harnessed to create his famous auditory illusion known as Shepard tones. These tones create the illusion of endlessly ascending or descending pitch.
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Shepard tones achieve this effect through clever octave and volume manipulation, following these steps:
- Select a center tone, such as A440 (the A above middle C on the piano, vibrating at 440 Hz), which will be the loudest. Then, introduce tones above and below that extend across the audible range, spaced an octave apart.
- Adjust the volumes of the additional tones based on their distance from this central tone. The goal is to create a bell-shaped (Gaussian) envelope across the octaves that draws the listener's attention to the middle range while allowing the higher and lower notes to remain subtly audible.
- In a stepwise ascending scale, the highest tones gradually fade out as new tones are introduced at the lower end of the spectrum.
Because each note spans multiple octaves, this fading in and out across the octaves results in a seamless sensation of perpetual motion.
There is also a continuous version of Shepard tones, commonly referred to as a Risset scale or Shepard–Risset glissando. Here’s an animation demonstrating the illusion:
Some individuals find Shepard tones unsettling or disorienting. Researchers suggest that this type of scale can create an illusion of self-motion known as auditory vection, which may lead to feelings of imbalance or even nausea. A classic example is the video of Super Mario 64 from 1996, which might test your threshold.
Beyond Simple Scales
Given Escher's genius, it seems almost inadequate to compare the mind-bending elegance of his art to these intriguing yet formulaic scales. Nevertheless, Shepard tones have been integrated into various sophisticated musical compositions, from Hans Zimmer's score for Dunkirk to the Beatles' "I Am The Walrus."
Here’s a Vox video highlighting several instances of Shepard tones in films and popular music:
If you're feeling adventurous, you can create your own "Shepard Madness" at this fantastic site created by Stéphane Pigeon:
<h2>Binaural Shepard Tone Generator - The Audio Illusion</h2>
<div><h3>Shepard Madness can be very amusing, or quite unpleasant depending on the person who is listening. Make sure you can…</h3></div>
<div><p>mynoise.net</p></div>
Like Escher's creations, Shepard tones challenge us to question our senses, leading us to temporarily believe the implausible. In this way, both art forms gift us the opportunity to step back from certainty and engage with the unknown.
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