On The Musicality Of M.C. Escher

“Order is repetition of units.  Chaos is multiplicity without rhythm.”

“My work is a game, a very serious game.”

“Are you really sure that a floor can’t also be a ceiling?”

– M.C. Escher

I’ve long been curious about M.C. Escher’s (1898-1972) drawings and woodcuts because of their precision, their order and symmetry, their use of repetition and optical illusions, and the way they seem to point towards what could be called the infinite. Lately I’ve been thinking about what these qualities in Escher’s art have to offer those of us working in music (whether making it or writing about it). Let’s take a look.

First, Escher incorporated tessellations into his work, a technique he picked up in his study of tile mosaics while visiting Alhambra, a Moorish palace in Spain in the early 1920s. (Which reminds me of an article on the advanced geometry of 12-century Islamic art.) Seeing the tile mosaics inspired Escher to use geometric grids as the basis for his art as a way of gaining precision. Tessellations, by the way, are the composite result of geometric shapes that are repeated without overlaps or gaps. Honeycombs and interlocking pavement tiles are examples of tessellations. We see tessellations in Escher works such as these:

Second, Escher depicted in his work transformation/transmutations where we see one shape becoming another. These transformations appear most clearly in Escher’s tessellation pieces. In his woodcut Sky and Water, for example, we see birds becoming fish/fish becoming birds.

Or in this piece, Day and Night, a whole landscape shifting:

Third, Escher was fascinated by so-called “impossible constructions” or visual illusions such as the Necker cube and the Penrose triangle that take advantage of quirks of perception and perspective. You can see impossible constructions depicted in Escher’s famous “Relativity” piece that depicts people simultaneously ascending and descending stairs in an infinite loop. Are the figures moving up or down, sideways this way or that way? I like to rotate this piece onto its different sides to see how it holds up. Miraculously, Escher makes the work cohere no matter what viewing perspective we try to bring to it:

Fourth, and speaking of infinite loops, Escher’s works illustrate the idea of recursiveness—that is, something feeding back upon itself in a never-ending cycle. Relativity, above, depicts such infinite loops, as does the work Drawing Hands:

And this one that depicts lizards crawling to life/becoming tessellations:

These works and others present the viewer with a visual chicken/egg dilemma: Where does it all start and end? I like that.

Fifth, it’s been said that Escher’s art demonstrated an “intuitive” understanding of mathematical order and symmetry and perhaps this is the reason why his works are so pleasing to look at? What’s remarkable is that this intuitive understanding was so accurate that in the late 1950s the Canadian mathematician H.S.M. Coxeter said of Escher’s hyperbolic tessellations (regular tilings of a hyperbolic plane): “Escher got it absolutely right to the millimeter.” Here is his Circle Limit III:

This notion of Escher’s intuitive mathematical understanding reminds me of a quote from the philosopher-mathematician Gottfried Lebniz (1646-1716) that always made intuitive sense to me: “Music is a hidden arithmetic exercise of the soul, which does not know that it is counting.”

Finally, there’s an intangible quality to Escher’s work that some critics have described as an interest in exploring infinity. The repetition, the tessellations depicting nature’s transformations and evolution, the impossible constructions playing with our perceptions, the infinite loops feeding back upon themselves—all of these characteristics of Escher’s art suggest an artist trying to represent that which can’t be represented, a reality beyond, a time-space outside our everyday experience of space-time. You even see it in tiny details, like when Escher draws a reflection of himself. In his work The Eye, for example, the reflection is twofold: there’s the mirror-image close up of his face where we see the folds around his eye, and there’s also that next level reflection deep in his eye’s pupil where we see Escher post-Escher–he’s already a corpse! It’s these kinds of little details that suggest that Escher was always somehow thinking beyond the Now even as he had intricate, and serious fun (“My work is a game, a very serious game”) constructing its beguiling representations:


For me, Escher’s work has musical resonances and looking at his pieces reminds me of the work of various composers, especially that of the American minimalists such as Steve Reich and Philip Glass. Over the years I’ve spent much time thinking through their work (you can read more about their work here; and you can view a Ventrilo-Dialogue with Reich here). Escher’s tessellations remind me of minimalist music that is similarly built out of small repeating units of sound strung together to make long rhythmic tapestries. (Now that I think of it, a lot of electronic dance music fits this bill as well.) Escher’s transformations/transmutations remind me of how minimalist music changes over time through subtle additive or subtractive procedures—adding or taking away a note here and there to transform one motive into another before our ears. (Ditto for electronic dance music.) And Escher’s impossible constructions remind me of minimal music’s perceptual artifacts—where as a listener you’re not sure if you’re listening to three groups of four beats or four groups of three beats, for example. Like Escher’s Relativity, a piece like Reich’s Drumming allows the listener to hear both perspectives at once. As for recursiveness, a lot of classic minimal music really does have an endless quality about it: a sense that it could, and just might, go on forever—or at least long enough for the listener to stop worrying about where it’s “going.” (It’s not going anywhere, just being something for a time.) Finally, to return to Escher’s intuitive understanding of math: Aren’t composers kinds of mathematicians too in that in one way or another they’re concerned with numbers and quantity, structure, space, and change? Like Escher, most composers frame what they do not in clinical terms (“I spend a lot of time exploring e-minor…” or “I do most of my compositional work in 5/4 time…”) but in intuitive and emotional terms (“In this song I was trying to capture the sadness of my break-up with a girlfriend…”)  And isn’t music a good example of a kind of equation in sound that presents not an argument or a “proof” but rather shares the results of a procedure, solving itself and bringing us along for the ride?

On Perception And Playing A Polyrhythm

A polyrhythm is the simultaneous sounding of more than one rhythm. I find polyrhythms endlessly interesting, mainly because they play with our perceptions, especially our sense of what is foreground and what is background. In this way, polyrhythms are the aural equivalent of those optical illusions you may remember from Psychology 101, such as the faces/vase illusion

and the young woman/old woman illusion.

These optical illusions come to life to the degree that you can bend your perception through them. If there is a “trick” to seeing them the “right” way, it’s to allow yourself to perceptually move among multiple viewing perspectives. Similarly, you can learn to hear the aural illusions of polyrhythms with the same perceptual flexibility. So if you hear a so-called “two against three” beat polyrhythm that superimposes a three beat pattern over a two beat pattern, in your mind’s ear you can foreground the two or the three, or even hear both of them–their rhythmic gestalt as it were–at the same time.  And the best way to learn how to hear something is to learn to play it.


To play a two against three beat polyrhythm here’s what you do. Place your hands on a table top or your thighs and play the following six beat rhythm (“T” = hands together; “R” = right hand only; “L” = left hand only; “-” = a silent count or rest):

Hands Play:  T  –  R   L   R   –
Beat:              1  2  3   4   5   6

Another way to think about the rhythm is:

Long (T) – short (R) short (L) long (R) – [repeat]

If it helps you stay in time, you can count the beats out loud as your hands play the gestalt T – R L R – pattern.

Play this rhythm very slowly over and over again until it feels comfortable in your hands. Next, speed it up, but make sure you keep the six beat structure intact by observing the rests (on beats 2 and 6 of the pattern). You’re playing a polyrhythm! It’s a polyrhythm with a three beat pattern in your right hand and a two beat pattern in your left.

Once you’re comfortable playing the pattern continuously, you can start playing with your perception of it. One way to induce a perceptual shift is to change the loudness of your hand tapping. While playing the pattern, try making your right hand’s dynamic very soft. As you do this, you’ll notice that the “three” of your two against three pattern fades to the background while your left hand’s “two” pattern is foregrounded because it’s louder. Now bring the right hand’s dynamic back up and then try diminishing the volume of your left hand. As you do this, you’ll notice that the “two” of your pattern is now background, foregrounding your right hand’s “three” pattern. The tricky part of this is keeping your hands steady while you play with changing their dynamics. It’s tricky because what your ear hears conflicts with what your hands feel.


You can read more about aural illusions in music here.

On Minimalism and Aural Illusions

One of the enduring contributions of the so-called American “minimalist” composers–particularly Steve Reich and Philip Glass–to global music culture was to re-introduce shape-shifting, metamorphosing aural illusions to our listening experience through intense repetition, polyrhythm and additive rhythms. These rhythmic devices are not new in music–you can certainly hear them in some African and Indonesian musics–but they were newly foregrounded in the concert hall back in the 1960s and 70s when minimalism burst onto the scene.

“Foregrounding” is an apt term in that the use of these musical devices reminds us of those perceptual puzzles from Psychology 101–like the picture of the two faces/vase that foregrounds one or the other depending on your interpretive listening stance:

A good percentage of the bliss in a vintage Reich or Glass piece derives from how the music plays with our senses, inviting the transformation of our (mis)perception to become part and parcel of the music’s affect. Reich’s early piece Drumming (1971), for instance, features perceptual artifacts the composer calls “resultant patterns” that arise out of the music’s polyrhythmic web. Reich found inspiration for this concept from his study of West African drumming.  (A similar concept, “inherent patterns” was discussed by ethnomusicologist Gerhard Kubik in the early 1960s.) Musicians performing Reich’s music foreground these patterns by playing or singing them to help us along in our listening. Moreover, the careful design of the music supports our multiple and shifting interpretations: Drumming is in a 12/8 meter which can be rhythmically perceived in a variety of ways (3 groups of 4 beats, 4 groups of 3, 6 groups of 2, 2 groups of 6)–often simultaneously.

Here are excerpts from a recent performance of Drumming (and you can forward the clip to 2:00 to hear the singers’ “resultant patterns”):

Glass’s early piece Music In Twelve Parts (1971-1974) works its perceptual magic not through polyrhythms but through additive rhythms. The composer structures his piece around short rhythmic units that repeat at a steady tempo but also grow in length incrementally. Glass found inspiration for this technique from his study of Indian music with Ravi Shankar. After sufficient repetition, these repeating rhythmic blocks induce subtle perceptual shifts–playing especially with our sense of time. The music can make you feel like it’s foregrounding a slower time dimension behind its frantic surface.

Here is Music In Twelve Parts:

In both cases, the composers use minimal techniques to yield maximal perceptual results.

On The Neuroscience Of Magic And The Magic Of Aural Illusions In Music

In their engaging book Sleights Of Mind, Stephen Macknik and Susana Martinez-Conde explore the neurobiology of attention in the context of the magician’s art.  They argue that illusions, slights of hand and other tricks have much to offer the study of how we think because magicians specialize in playing with our perceptions and (mis)guiding our attention.  Magic manipulates the reality of our here and now (242) and even “the nature of our conscious experience” (255).  For instance, our visual systems have a spotlight of attention beyond which we are almost blind.  Magicians capitalize on this by directing what we see in controlled ways, creating “frames” or windows of space to localize where and what we attend to (66).  Magic works its wonders over and over again by hacking into and manipulating our awareness and understanding of how we think the world should work.  Simply put, a magic trick succeeds when it manages to violate our expectations (159).

Magicians are not the first artistic community to have made important discoveries about our cognitive processes.  European painters in the 15th-century discovered the rules of perspective so that–presto!–paintings quickly moved from looking flat and oddly proportioned like this

to having a depth of field like this

and much more recently, toying with our sense of perception through impossible figures like this

What kinds of discoveries have music composers and performers made over the centuries?  After all, music, like magic, is also an art of holding the listener’s attention and awareness through time.  (Music may well be, as a friend once said to me, one of the provinces of magic.)  So what about aural illusions in music?  How does music create phantom presences out of such seemingly simple sonic materials?  Is it fair to say that music is an elaborate art of aural illusionry?

There are certainly many musics around the world that make use of different kinds of aural illusions to achieve their affect.  A few examples come to mind.  Consider, for instance, akadinda music from Uganda.  The akadinda is a large wooden xylophone played by several musicians at once.  The music consists of very fast interlocking patterns that whiz by at upwards of 300 beats per minute.  But what’s even more remarkable about this music is that its rapid-fire textures give rise to an auditory illusion called “inherent patterns.” First described by ethnomusicologist Gerhard Kubik about fifty years ago (Kubik has spent much of his life in the field studying and documenting music making in Sub-Saharan Africa), inherent patterns are resultant or composite melodic or rhythmic patterns that seem to (magically) rise to the surface of what is otherwise just a very dense musical texture.  Inherent patterns can be considered perceptual artifacts–patterns that arrive at our senses because they seem inherent or embedded in what we’re hearing.  In the case of akadinda music, if you listen through the fast interlocking rhythms you can sometimes hear a simple two note melody that no single musician is playing on their own.  Rather, the melody is the sum of several sets of hands playing adjacent akadinda keys.  Your brain notices this inherent melody because the fast interlocking pattern repeats insistently to the point that your brain can focus in and detect it.

Inherent melodies can be heard in other African musical traditions as well, such as mbira music from Zimbabwe.  In mbira music, you’ll sometimes hear performers sing along with the inherent patterns as they perceive them in the music they’re playing.  And the American composer Steve Reich has written numerous pieces (such as Drumming) that exploit this perceptual phenomenon as well.

Another musical example that taps our propensity for noticing sounds beyond sounds is throat singing from Tuva and Mongolia.  Skilled throat singers from these regions have evolved techniques that involve constricting their throats and shaping their lips in such a way that they can produce not one but two pitches at once.  The low-pitched droning note is called a fundamental and the higher pitched note an overtone or harmonic. With one (long) breath, a singer produces two distinct frequencies.  To our ears it sounds like two people are voicing sound, the higher pitched voice making an ethereal whistle.

A final example of musical illusions can be heard in the fugues of the German Baroque composer J.S. Bach (1685-1750).  A fugue is a type of intricate composition for several musical voices (or musical lines) in which each voice has its own independent melody that coexists and interacts with those of the other voices.  In a fugue, a theme is introduced by one voice, then picked up by the other voices as they enter, one by one, in different pitch registers.  Once all the voices are in and chattering away, the fugal texture can sound a little chaotic but it all holds together through careful musical design.

If you listen to Bach’s (unfinished) Art Of Fugue (composed circa 1740s), you can hear the four distinct voices enter one at a time and make a multi-part musical conversation.  What is interesting in terms of aural illusions, however, is how your ear, upon hearing a voice entering the texture (for instance, voice two enters at 0:12), tracks that voice for a while as it articulates the fugue’s theme.  This narrowing of your attention on one fugal voice–would Macknik and Martinez-Conde call it auditory “framing”?–continues until you are distracted by a new voice making its entrance (at 0:20) and then yet another one (at 0:30).  The magic of a great fugue composer like Bach is to design a musical texture that can guide your attention in a way that is satisfyingly challenging.

As with the magician’s art, engaging music presents us–with or without our consent–with a set of perceptual problems to figure out, whether they be inherent patterns, phantom overtone melodies, or fugal counterpoint equations.