Time Dilation and Homecoming in Hans Zimmer’s Inception Score

It’s difficult to think of any music that’s been so popularly over-analyzed as Hans Zimmer’s soundtrack to Inception. There are clear reasons why: Zimmer’s soundtrack music seems to do a good job of participating in the movie’s dramatic elements. I’d like to take a look at a few of those forms of participation—dilation and homecoming. 

We all know by now the delightful mechanic of Inception (and if you don’t know it and don’t want it spoiled, this is your final warning). When you dream, you experience time more slowly than it is progressing while you are asleep in the real world. But certain elements of the dreamer’s outside world will seep through into the dream (this is true of dreams, in my experience—alarm clocks, honking trucks, shouting toddlers, a cat scratching your face). Consequently, a noise heard in the dream goes through some kind of time dilation. As its frequencies get distended, the pitch gets lower and the rhythm gets slower.

And this is where Edith Piaf comes in. Throughout the film, one character plays the song “Non, je ne Regrette Rien” into the sleepers’ dreams, to signal to them that the dream is ending. They hear the noise and know they have limited time left. But, crucially, when they hear it, rather than hearing Piaf’s original track, they hear a slower and lower version of it. This is what one YouTuber noticed (well, I think probably quite a few people noticed it; he just went viral). Zimmer confirmed this was the case and he acted a little bemused people didn’t get it earlier. He called it a “signpost”, as if the audience, not just the characters, will hear the dilated, ominous sound and understand (consciously or not) that the dream is ending. We’ll discuss this more later.

But I am here to discuss how Zimmer goes further still. Here is the relevant part of Edith Piaf’s song. The orchestration is pretty arresting, since the trombones swell to a dynamic fervor that grates against a modern sensibility: piaf trombones

Joseph Gordon-Levitt is a DJ and Edith Piaf is his sample and that is why he breaks it down in that Matrix scene


Now, whenever Arthur (Joseph Gordon-Levitt, for those of you who like me can never remember character’s names in movies you’ve seen fourteen times) plays this song to someone who’s asleep, they hear it but they hear it in the dream, which means time is slowed down.

nn-ts nn-ts nn-ts DJ KING ARTHUR

(And it’s just so very Hans Zimmer to leave you hanging with that upbeat that’s fortissimo, just like his soundtrack to The Crown.) Now, all of this may be obvious. But let me insult your intelligence and summarize where we are so far: two things seem to happen to music as it travels from waking reality to dream (or dream to dream). First, its rhythm gets slower, i.e. diminution. Second, its pitch goes down, i.e. transposition. (It’s a little convenient that the diminution and transposition are always to speeds and intervals that are tonally just perfect, whereas actual audio slowdown is always pretty messy.  (Like, where does that middle trombone note go?) But this is Hollywood!) This can be visualized thus:

So, what does Hans Zimmer do that goes beyond even this? So far, we have talked about diegetic music. That is, music within the story itself, music that can be heard by the characters as they experience it. Musical scores to films, however, are usually nondiegetic, especially the big orchestral ones. Non-diegetic is music that cannot be heard by the characters. When the fellowship of the Ring mounts some heroic peak in Peter Jackson’s film, the characters are not hearing Howard Shore’s soaring strings, nor do we assume they are. That music is just for us. (Some clever films try to exist in an ambiguous middle realm, like the phenomenal movie Baby Driver.) In Inception, the Edith Piaf song is usually functioning diegetically, that is, it is inside the story. What is creative about what Zimmer does is that the diegetic elements of the film have leaked into the structure of the non-diegetic film music. Edith Piaf’s song is actually everywhere in the score: frequently during dream sequences you’ll hear the trombones doing this annoying braying rhythm (the technical term is “bramm BRAMM”, sadly I am not joking), but in situations where the characters clearly aren’t intended to be hearing it. For instance, in Zimmer’s track “Dream Within a Dream.”

notice the bramms, if you would, as well as oscillating violins, which will come in later, like Chekhov’s gun

Now, isn’t that funny? The bramms are now non-diegetic. It’s not unusual for a diegetic sound (i.e. characters can hear it) to slip into the non-diegetic music (i.e. character’s can’t hear it), such as when Anne of Avonlea is suddenly accompanied by a full imaginary orchestra when all she’s literally hearing is some Cape Breton sap pathetically scratching his fiddle. But what is unusual is to have a diegetic element, like the Piaf trombones, become a motivic element of the entire non-diegetic score, even when it is not directly relevant. In other words, the Piaf trombones have become structural to the music. They appear like some incessant prophetic reminder: you are in a dream. The dream dilates time. Every sound, every experience, every moment of your above-life (or above-dream) is being stretched out. (This whole time business really seems to bother Christopher Nolan. It shows up in a lot of his movies. Get over it, man! You’re gonna get old some day.) Even if the characters are not always aware of this haunting aural reminder, we are—because we can hear these dratted trombones in the non-diegetic score.

But Hans Zimmer goes still further. This time dilation has invaded every aspect of his orchestration. For instance, listen to those violins in the above clip. Play it again for a minute. Do you hear how they’re oscillating between two different notes? They’re doing it at this plodding 8th-note level. But here, earlier in the same song, is that same chord progression in a higher key. And look at what the violins are doing.

there, the gun went off. happy, anton?

There is a clear aural and visual relationship here: the higher passage had its motivic materials presented faster and the lower one presented slower. It’s our chart again. At the higher level of the dream musically, it’s 16th-note violin oscillation. At the lower level, it’s slowed down to the 8th-note level. Pretty cool.*

Of course, dreams and time dilation aren’t the only themes of Inception. Homecoming, or nostos as Homer might say, is the whole point of the story. Cob has to get back to his kids. And I think we can hear the traces of homecoming make their way through Zimmer’s score, this time at the level of harmony.

Take a gander at this chart (and if it doesn’t mean anything to you, then just keep scrolling to get to the conclusion below, I will never know). It’s an analysis of the chords we just heard, which I will dwell on ever so briefly. Roman numerals, despite what many people seem to think, are nothing more than a clumsy description of how theorists think common practice chords function. That is, why does one chord lead to another? Why do some chords not lead to others? If you can label their function, you can capture some of their logic. Here is my analysis of the chords from “Dream Within a Dream.” (That’s the music from above that was tense-sounding, feel free to listen again.)

In essence, the Roman numeral system assumes a tonal framework, which is fair for Inception, since it uses good old triads and gives us some apple-pie tonic-dominant progressions. But these chords are certainly hard to fit into a G minor context, as my analysis avers. (Gm, you see, is “i”.) E-flat is easy enough, since it can be constructed with a G minor scale, but G-flat chords and C-flat chords? These are far afield. Our ears enjoy hearing chords that “fit” with our sense of key, and these do not. So why does Zimmer write them this way—and why do we (well, some of us) like them? 

As the notation indicates, there is some “borrowing” going on in these chords, indicated by the slash. The numerator “borrows” from the denominator. In order to move from G minor to G-flat major, G minor has to borrow a chord from its close relative, E-flat. E-flat doesn’t actually have any native relationship with G-flat either, but it turns out that E-flat’s own cousin, E-flat minor, does. Thus, in order to explain the motion from the first to the second chord, one goes G minor –> E-flat major –> E-flat minor –> G-flat. Dizzying. This same logic gets us the C-flat major chord, as well.

According to this analysis, then, Zimmer is making you hear in two different keys, and which one you hear depends on which chord you happen to be hearing. During the odd-numbered chords (G minor and E-flat), you seem to be operating in a G minor world. But during the even-numbered chords (G-flat and C-flat), you are hearing some chords belonging to E-flat minor, whose presence is felt but never made explicit. These relationships are largely governed by the famous chromatic mediant, used by Medievals and Wagner and Star Wars for all sorts of magical effects. The overall sense is one of harmonic alienation and upheaval. It’s not far-fetched to describe this chord progression as symbolically portraying Cob’s distance from home. After all, I initially described the chords as “far afield”—it is the normal metaphor theorists would use, because there is some truth to it.

But now let’s look at the chord progression of arrival-at-home. These are those chords you listened to for 10-hours straight after the movie came out, from “Time.” 


maudlin nonsense, you all came here just to listen to this clip didn’t you

These chords are the opposite of above. All these Roman numerals are happy, straight-forward, even a little boring. But that’s okay: after the upheaval and alienation of before, nothing can more profoundly give us a sense of peace. It’s worth noting that I assumed this piece was in G major, but it could be analyzed in any number of different keys fairly plausibly (A dorian, D mixolydian). Every chord is so related and fits so consonantly with the other, it could happily be considered any of those. It’s a remarkably bland bit of harmony, but—for that very reason—beautiful music for a homecoming.

So, the implications of what I’m arguing for should start to be clear. Zimmer is portraying at-home-ness through these near-by harmonic relations, through diatonicism. And he is portraying distant-from-home-ness by using those far-flung harmonic relations, by using chromaticism. It is a modified sort of leitmotif, but one that formally organizes the music and refers to something abstract. The real clincher, the one that makes Zimmer’s soundtrack work so well, is the false home that he portrays. It’s in those moments of the film where Cob is caught between staying in his dreams of home and the promise of real home in his waking life. It is this chord progression, which is a combination of the previous two:

music theory is just mystification so that you’ll take me more seriously when I make ridiculous claims


As you listen to this, you’ll notice your emotions seem to get tugged both directions at once. The first two chords have that serenity (diatonicism) of “Time” but the next two darken the color (chromaticism). This can be explained by my Roman numerals again. In every major key, certain chord degrees (I, IV and V) can be expected to be major and certain minor (ii, iii, vi). Now, here’s what cool. Without ever explicitly having a G major chord, Hans Zimmer can imply “G major” by using the Am (ii) and Em (vi), because we’re used to hearing those accompany G major. But when he gives us that G minor chord, we realize we’ve been duped! What fools we were to think it was major. We are now sad. We’re back to our old friend mode mixture from “Dream Within a Dream” above: G major is reaching into G minor in order to access some chromatic sonorities. We even get good old chromatic mediants in the second half of the phrase. And the whole time, you’ll notice the chord progression perfectly mirrors the “Time” progression, but filtered through some demented glass.

One of my favorite moments is when Zimmer combines our two elements: the false homecoming chords and the bramm BRAMMs of the trombones. (Oh, and also, if you can listen closely, the oscillating violins are there too! See, it all connects.)

him who has ears let him hear Nolan/Zimmer illuminati unite


This is a pretty nifty moment. There is that aching pull from the chord progression, presenting false home, embodied in whatever-her-name-is-whom-Marion-Cotillard-plays. But then there’s that rude braying from the trombones, embodied in Ariadne reminding Cob that he is still dreaming. It is a bit like Mercury upbraiding Aeneas, “Building her gorgeous city, doting on your wife… Wasting time in Libya—what hope misleads you so?” Carthage is not his real home.

So where does this leave us? We started with alienating harmonies, we moved through a kind of “false” sense of home, and then at the end, the last track “Time” (the one with the falling (?) top) we finally get our true (??) home. All this is convincing only if Zimmer gives us the original “home” progression near the beginning, so we have a sense of where we’re headed, which he does. He gives us the barest outline of the eventual homecoming progression in the film’s first moments, along with some bramm BRAMMs for good measure, which nicely sums up this post.

I promised a word about authorial intention. I have argued that Zimmer portrayed time dilation in his score (even in those oscillating strings) and that he’s portrayed homecoming and false homecomings in his chromaticism. Am I arguing, therefore, that Zimmer meant to do all this? That this was running through his mind while he composed? That he expected us to pick up on it? Well, if the answer to the last one is “yes”, then either it was a very subliminal sense in which we picked up on it, or else it was a disastrous attempt, because most people probably did not pick up on these themes. I do happen to think that it’s quite possible soundtracks operate in some subliminal ways, but I am not willing to bet on it. I think it’s probably fairer to say this: the screenplay and the music are both richly allusive and imbue each other with meaning. 

Music and screenplay work together as a single text; the better the movie, the better they work together. (Sure, maybe Inception is not a Great Film, but it’s a pretty good flick.) Whether all the things I wrote about went through Zimmer’s head is ultimately unknowable and irrelevant. But I can say “Zimmer portrays these things in his music” not because he thought so, nor because you sensed it, but because the music is just so good at portraying it. 

P.S. I am sure there are other films where diegetic elements (especially foley, I’m guessing?) become structural elements of the film’s music. But I can’t think of any off the top of my head. If you can, let me know!

* Yes, I’m aware the oscillation isn’t quite the same rate, since it switches off every two notes in the one and every five notes in the other. But if you try the other alternative, which would be switching off every two notes on 16th notes, it would be rapidly tedious-sounding when played so fast. It was a good choice for Zimmer, since it preserves the sound of a relationship between the two passages without being literal to the point of aural tedium.

I used The Aeneid translated by Robert Fagles.

For other thoughts on ambiguity, equivocality, and authorial intention in music, see Kofi Agawu’s “Ambiguity in Tonal Music: A Preliminary Study”.


The Chiastic Tragedy of La La Land

Is it possible to hear one melody with two different harmonies at the same time? It seems it is in “La La Land,” if by “hearing” we mean that you yourself are supplying the harmonies in your mind, and no one is playing them in reality.

I admit this is a bit dizzying, but I can demonstrate that this is happening. Take the famous “Mia & Sebastian’s Theme” by Justin Hurwitz, as you first hear it being played by Ryan Gosling’s character in the restaurant.

mia and sebastian's theme with chords

Mia & Sebastian’s wistful but happy melody

As my annotation says, the theme is wistful, yes, but it’s happy all the same. We become accustomed to hearing this theme with these harmonies underneath it, because it is used with daring regularity throughout the movie, and without fail it’s associated with Mia & Sebastian’s romance. In the following clip, for instance.

But here is the curious thing. Hurwitz, either by luck or ingenuity, has embedded in this theme a potential to be heard as both tragic and romantic, as is demonstrated in the ending scene where Sebastian plays the love theme without his left hand.

Mia and Sebastian's theme, right hand alone
Mia & Sebastian’s theme, but now sad

When we hear Hurwitz’s melody without its accompanying harmonies, it becomes unambiguously tragic. This tragedy is made poignant because of the memory in our minds of a happier harmonic interpretation. Why is the melody so different in the absence of any harmony?

When the melody is heard by itself, it has an implied harmony of F# minor. And, yes, for the average audience of “La La Land”, minor means sad. This can be seen if we do a generative analysis of melody, peeling back some of the harmonic underlay. (a) below is the original melody, (b) highlights the notes of the melody that outline an F# minor chord (on accented beats for the most part), (c) shows those notes isolated, and (d) makes them into a chord progression.

Yes, Schenker is rolling over in his grave, I don't care, be quiet
Yes, Schenker is rolling over in his grave, I don’t care, be quiet

This is a canny move on Hurwitz’s part. He has introduced this melody to you many times with A major harmonies underneath it. When your ear hears the melody without any harmonies, it will assume the ones in (d). But Hurwitz has put this melody, without chords, at the end of the movie, so we have built up enough context that we can hear it with (d) harmonies but remember it with the old ones. They are both there. There is a powerful sense of the sad, minor (d) harmony, because that is the salient interpretation. But there is a wistful memory of the earlier harmonies, because we have become so used to hearing it with the major tonality.

And what could be a better musical representation of nostalgia? What was a source of sweetness, removed from its context, becomes tragic; yet it is the very fact that we can remember how it sounded as a happier melody that makes it so tragic. It is a musical twisting-of-the-knife.

And it is a metaphor for the rest of this essay, which is about the story of “La La Land”, Damien Chazelle’s upbeat and mass-appealing veneer over the same story he told in his darker “Whiplash.” Both stories are about jazz, about selling out, about the celibacy of the artist, about the struggle of artists to justify themselves to society and their families.

Damien Chazelle’s worldview is a grim fork in the road for the artist: you can live in solitude, an abused victim of your artistic muses, hoping to achieve greatness, or you and your dreams can sell out in pursuit of wealth and commercial success. Just for the sake of ease, I’ll refer to Chazelle’s categories as romanticism (follow your dreams, don’t sell out the art) and capitalism (art needs mass appeal to survive, you have to make a living).

Mia and Sebastian are twin chiasms in mirror image. Sebastian flirts with commercial success but returns to his artistic ideals, whereas Mia flirts with artistic ideals but returns to her realization of commercial success. Their relationship can exist only in the moments of intersection when they are crossing between romanticism and capitalism. 

Their lines are color coordinated with their onscreen outfits; these details make my blog great
Their lines are color coordinated with their onscreen outfits; these details make my blog great

This is illustrated in two of the film’s narratives. Keith, who wants to pop-ify Sebastian’s pure jazz sound, makes what seems to me to be a sound argument against Sebastian’s aesthetic philosophy: 

But you say you want to save jazz. How are you going to save jazz if no one’s listening? Jazz wouldn’t exist if people hadn’t gotten tired of what they were listening to before. I mean, do you really think a bunch of ninety-year-olds in a basement is the future of the form? Traditionalists whined when Kenny Clarke started dropping bombs. If traditionalists had their way, we’d still be playing Dixieland.

But at this point Mia has swapped places with Sebastian and is now the play-writing acolyte at the altar of pure art. She refreshes Sebastian on the exacting demands made on them by Personal Authenticity and Autonomous Art:

Mia. It matters if you’re going to give up your dream to be on the road for years. Sebastian. Do you like the music I’m playing? Mia. Yes. I do. I just didn’t think you did. …Sebastian. This is what you wanted from me. Mia. To be in this band? Sebastian. To have a steady job. Mia. Yes, I wanted you to have a job so you could take care of yourself and start your club.

The other illustration is the converse moment, Mia’s audition and acting breakthrough. It’s her turn to sell out. How she does it is the most insightful part of Chazelle’s storytelling. She sings about her aunt, a woman who followed her gut, lived life with abandon, threw caution to the wind, seized the day with a little madness. Yet these are the very things that Mia will refuse to do with Sebastian once her career gets off the ground. This is what the kids these days call a “performative contradiction,” and it’s a glorious one. By feigning authenticity, by narrating, acting, performing the artistic lifestyle, Mia obviates her need to commit to art the way Sebastian does. There is a bit of Chazelle himself in Mia: obsessed with jazz, fascinated by the lifestyle of the artist committed to no commercial gain beyond the art itself—and using that narrative to make lots of money in Hollywood.

Of course I don’t mean to imply that I agree with any of this drivel about commitment to art versus the wickedness of commercial gain and so forth. I agree with this LA Times critic that such thinking is wrong-headed, but I don’t agree with him that it is outmoded or old-fashioned. The opposite is true. Chazelle has put his finger on something. I struggle to bring to mind musicians who don’t evince Sebastian’s attitude or music consumers who don’t think along Mia’s lines—off the top of my head, I am reminded of Lady Gaga’s recent Netflix documentary when she talks about her relationships with boyfriends and producers. Or similar themes in the Tchaikovsky Competition documentaries from the 90s, those glorious panegyrics to the abusive tendencies of Classical music.

The historian in me is compelled to ask, where does this all come from? No doubt the scholarly literature on that is longer than my entire Goodreads. But the attitude reminds me of Beethoven’s Heiligenstadt Testament. It is an early and concise exploration of many of these themes, and its author is the archetypal suffering artist. Found among Beethoven’s effects upon his death, it’s a personal expression of grief and despair about his calling as an artist and his affliction of deafness. His contemporaries might have accused Beethoven, like we might Sebastian, of being a douche-bag—like that moment when Sebastian bumps past Mia—of being “hostile stubborn or Misanthropic” in Beethoven’s words. But he insists that that’s not a fair portrayal.

Born with a fiery Lively Temperament susceptible even to the Diversions of Society, I soon had to keep to myself, pass my life in solitude, if I attempted from time to time to rise above all this, o how harshly then was I repulsed by the doubly sad Experience of my bad Hearing… Such Happenings brought me close to Despair; I was not far from ending my own life—only Art, only art held me back. …I am resigned—to be forced already in my 28th year to become a Philosopher is not easy, and harder for an Artist than for anyone else.

Chazelle’s stories about art are stories that end in isolation and loneliness. When compared with “Whiplash”, “La La Land” is made the more effective because there is that tantalizing hope, the dream of la la land in Mia’s final fantasy, that the artist is not in solitude, that the melody can have the harmony along with it.

Things I mentioned:

Ludwig van Beethoven, “Heiligenstadt Testament”, in Piero Weiss and Richard Taruskin, Music In the Western World: A History In Documents.


Different Angles on a Musical Object

Josquin, Gestalt, and Aphex Twin

I know how annoying those blog post titles are that try to connect three random things, just to hook you and make you think, “Wait, how is he going to do that?” But just so you know, I had to take things out of this blog title. Boards of Canada and African polyrhythm, for instance.

Ethnomusicologist Simha Arom notes that much of the Central African Republic’s polyphony, amongst the Banda-Linda horn ensembles he researched, is based on a “polyphonic cell,” or a “cyclic rhythmic figure” that constitutes the formal makeup of the music. It’s a result of 16-or-so horn playing a rhythmic pattern that repeats indefinitely. The music varies itself by having individuals come in or out of the texture. But each individual player always repeats the same short groove. When Arom speaks of a “polyphonic cell”, he is speaking of the sound that would occur if they were all playing at once. This polyphonic cell can be presented as a paradigm, in that the whole piece is derived from its pattern, but it may not appear as such at any given time in the performance of a piece. It is, however, the blueprint upon which different realizations and variations are based. Now, what he says next is pretty interesting, quoted at length:


Since the piece is based on the varied reiteration of that combinatory formula, the latter also behaves as paradigm. In consequence, if we were to place on the paradigmatic axis all of the combinatory formulas as they appear during a performance, we would obtain a “meta-paradigm” encompassing the totality of the piece, i.e., the entirety of the individual realizations as well as of the combinations performed therein.

 So, not only is there a polyphonic cell that exists outside of any specific moment in the music—a paradigm of a single iteration of the groove—but there is also a “meta-paradigm” for how that polyphonic cell unfolds, often combinatorially, throughout time.

It’s not a new observation that much of modern electronic music owes its procedures to African and Afrodiasporic music. But in the spirit of what Kofi Agawu calls “contrapuntal readings” between different cultures (he considers this a healthy practice), I’d like to look at several different manifestations of this phenomenon Arom is noting of the “polyphonic cell” and compare it to the music of Josquin. I think this impulse to look at music as a “cell” or “block” or object that can be prismatically understood from several different angles is transhistoric and transcultural.

Gestalt psychology’s principle of invariance
from Wikimedia Commons

In Gestalt psychology, there is a “principle of invariance” whereby humans tend to be able to recognize an object that they’ve never seen before as the same as another object, even if it is rotated, stretched, drawn differently or disproportionately. So, given that Gestalt psychology is the investigation of how the human mind tends to group disparate objects, we tend to group objects on basis of what we assume to be their 3D shape, even if we have never seen or experienced the object from multiple sides. We just guess in our minds what the rest of the object is like. (For more on Gestalt psychology applied to music, see Leonard Meyer below.)

I want to suggest that this approach to the polyphonic cell is a way of treating music like an object, one that demands to be seen from different angles. Polyphony is complex: lots of different actors (or, in electronic music, a single actor’s multiple tracks, “actants” in Bruno Latour’s language) are producing melodies that vie for our aural attention. What do we listen for? It’s probably impossible to really get the complexity of polyphonic music the first try.

This is why electronic music often gives us multiple tries at it. Polyphonic EDM repeats, and repeats, and repeats, and each time it gives us the sound “object”, or Arom’s “polyphonic cell” with a different voice removed or another voice added. This is the akin to the combinatorial aspect that Arom notes in Banda-Linda horn music.

How would we visually represent this? Arom speaks of a “paradigm” and a “meta-paradigm.” I am not positive I have interpreted him correctly, but I am going to assume that this maps onto the categories “synchronic” and “diachronic” (in a particular moment in time of the music vs. across the temporal span of the music). If we combine these, we get the paradigmatic axis that makes up the “totality” of the piece. So, in other words, we might be able to represent the piece analytically in such a way that there would be no information loss in our portrayal of the piece, even though we didn’t do a score-format transcription. Somebody, just looking at our paradigmatic analysis, might be able to reproduce an entire song, if the paradigm + meta-paradigm really did constitute the “totality” of the piece.

Let’s take Aphex Twin’s “We Are the Music Makers” from Selected Ambient Works 85-92. The song is an 8-minute process of muting and unmuting various tracks, which, heard together, are our polyphonic cell. Richard James has come up with a pretty complex polyphonic cell, which is the musical material on the left below. You never do end up hearing all the elements simultaneously. The closest you get is something like the “climax” of this song at 6:00, but it is missing the bass. Overall, you end up hearing all sorts of different combinations of tracks (“voices”, as we might call them in the fifteenth century). This allows you a curious insight into the nature of the cell. You hear all sorts of things you wouldn’t otherwise have noticed; for instance, you might have assumed that Richard James’s bass track, if you had just heard it as 3a+3b, was all a single “voice”, but he has cleverly split it up into two separate tracks and effectively gets two melodies out of one. Sometimes you hear 3a, sometimes 3b. (I am sorry my figure doesn’t notate when. I’ll make it better someday.)

This starts to feel a lot like the Gestalt principle of invariance. Although we are hearing something different each iteration of the cell, we can recognize that something is being repeated—even though each constituent part is absent at one point or another. So it’s a bit like looking at a musical object from several different angles, yet recognizing it to be the same object. One obvious caveat is: there has been information loss, since I just put “drum” and didn’t actually notate out the drum (because it was too hard and didn’t repeat as cleanly as the notes). It’s not coincidental that drums are the hardest aspect of EDM to be formally reductive about.

Another example is Boards of Canada, “Roygbiv“, a just really lovely song. (Another one my wife introduced me to.) Here I have analyzed it slightly differently. Since it operates in clear 4-bar phrases (in some ways, it’s strangely Mozartian), I just outlined the “cell” as I would a normal score. This cell gets repeated five times and then fades on the sixth. Here you can really sense how your brain is getting an education in how to listen to this music: you hear the bass first, then some supplementary background lines, then the melody, then just melody and accompaniment, and then everyone altogether. Had you heard it together from the outset, you likely would not have been able to comprehend it. If we treat the polyphonic cell as an object and apply the principle of invariance across the form of the song, it suddenly gets that Boards of Canada magic.

“Roygbiv” polyphonic cell
“Roygbiv” diachronic paradigm

Now the moment you’ve been waiting for: Josquin, this blog’s namesake. His situation is a bit more complicated, because he is going to introduce two variables into the situation. First, unlike Aphex Twin or BoC, Josquin is going to decouple musical material from track. That is, if you equate “track” with “voice”, the superius of Josquin might sing material that the tenor sings and vice versa. However, you can still make a paradigm of the music and track its existence through the tracks/voices. Second, Josquin is also going to allow for minor transformations of his constituent elements, up or down by a fifth usually. (Here I am not speaking historiographically, since Josquin is not responding to EDM. Hoping that’s obvious.) These two changes means that, although there is a similarity in the way you can analyze Josquin’s music, its approach to the “musical object” is more complicated.

Take “Sanctus” from Missa Hercules Dux Ferrarie. Here you have a very few number of musical elements on display. Josquin is reusing and modifying a lot of his material cyclically. I have outlined the constituent parts all at the top of this colorful figure, except for a tiny bit in perfection 14 which doesn’t fit with anything else (noted in gray). Then I’ve put their disposition across time below. This time, I have to use colors, since the equivalent of Josquin’s “tracks” no longer corresponds to the constituent parts one-to-one. There is some information loss, obviously, since I haven’t noted when something has been transposed up or down (although, if I had spent more time in Inkscape, how hard would that have been?). However, the music is operating in a similar way, reducible to this sort of paradigmatic understanding.

“Sanctus”, Missa Hercules Dux Ferrarie

But this raises an interesting question: what is the cell in Josquin’s music? Because he has unmoored the musical motives from specific “tracks”, it is unclear what object it is he is giving us different angles on. It is something like the cantus firmus, the long red lines in alto and then tenor. This is not too far from the historical reality, since one of the central questions in mass repertory of the fifteenth and sixteenth centuries is how composers treat cantus firmi in musical form and how the surrounding voices react (see Sparks and Wegman below). Like the others, Josquin is turning the musical object around, seeing what different angles reveal, but the object has become more internally complex, fluid, reacting to the very act of being examined, like a smartphone’s accelerometer changing its display depending on how you angle it. (Sorry, that metaphor was not poetic, but at least it’s relatable.) Josquin’s music, besides all the surface-level differences, has a similar feel to BoC and Aphex Twin (and, of course, to African polyphonies, although people said so long before me, like Arom and England), owing to Josquin’s love of repetition, his “obsessive compositional personality” (see Rodin).

So I find myself again thinking that musical repetition in our own time is less about inducing trance, revisiting trauma, or numbing ourselves with consumerist titillation, as I railed about in the last post. It seems more, in these cases, about cultivating a kind of contrapuntal intelligence. Repetition of this kind is almost a discipline, an education in attending to that which would be apparent to none of us on a first hearing.

Things I mentioned:

Simha Arom, “The Music of Banda-Linda Horn Ensembles: Form and Structure” and African Polyrhythm and Polyphony

Kofi Agawu, The African Imagination in Music and Representing Africa

Leonard Meyer, Emotion and Meaning in Music

Edgar Sparks, Cantus Firmus in Mass and Motet 1420-1520

Rob Wegman, Born for the Muses: The Life and Masses of Jacob Obrecht

Nicholas England, “Bushman Counterpoint”

Jesse Rodin, Josquin’s Rome

Big thanks to Victoria Chang for starting me thinking about this.