Category Archives: Theory

Answering the Dirty Projectors’ “Impregnable Question”

T he first seconds of the Dirty Projectors’ newest album speak volumes about what’s to follow. Dave Longstreth’s trembly hums at the open of “Offspring Are Blank” lay out the first of many “challenging harmonies” to follow (as various reviews delicately put it). The rhythmic puzzler “About to Die” follows, and next is the incomparably weird “Gun Has No Trigger”; it’s not until the fourth track that there’s some relief.

So Swing Lo Magellan isn’t lacking for things to talk about here. But for this first post in a long while (sorry!!!), I’ve just got a (comparatively) quick note about one of the simplest, most accessible songs on the whole record: “Impregnable Question.”

This album’s got a bit of a positive correlation between track length and musical complexity – and “Impregnable Question” is the second shortest tune. It’s very conventional, with none of the metric or harmonic twists that permeate the other songs. Instead, what’s interesting lies on a more subtle level: the relationship between the harmonies and the lyrics.

The structure of “Impregnable Question” is very simple, with just a twelve-measure progression repeated four times. That’s plenty of time, though, to establish one major musical theme: the denial of any traditional authentic cadence — that is, a sequence where the V chord resolves to the tonic. That’s the cornerstone of traditional “functional harmony,” the theory that each chord in any given key has a purpose, a “function.” The idea is that there are three basic major chords in every key: the tonic, or I, which is the starting and ending point for progressions; the sub- or predominant, or IV, which usually leads to the dominant; and the dominant, or V, which resolves back to I. V-I is the most common and traditional cadence there is.

But there’s no V-I motion anywhere in this song. Every V (here, E-flat major) instead steps down to D-flat, the IV. This thematic denial of any expected resolution is taken even further in the melody. Longstreth never even sings the leading tone, the seventh note of the scale that so yearns to resolve upward to the tonic. It’s the one scale degree that calls that strongly for resolution — but we never hear it.

Take a listen:

The chords:

   Ab                               | Db         Cm
01 If there is                      | ever the impregnable ques-
                             
   Ab             Bbm               | Db            Eb
03 -tion of "Why?    What           | did or didn't pass?"
                             
   Db            Ab                 | (Ab)
05 It would help to seek            |
                             
   Db         Ab                    | (Ab)
07 comfort in destiny,              |
                             
   Db        Ab        Bbm          | Dø7       Db
09 but I really don't. We don't see | eye to eye.    But I
                             
   Ab        Eb7add4                | Db           Ab
11 need you,   and you're           | always on my mind

This expected-resolution-denial theme is driven home at the most dramatic point in the melody, right at the start of the tenth measure. Take another listen to the video – the first time is right at 0:29 (then again at 1:07, 1:42, and 2:16).

Longstreth sets it up by systematically lulling us with two slow-moving D-flat/A-flat pairings. “It would help to seek…” [pause] “…comfort in destiny…” then another pause.

But then he finishes that sentence with “but I really don’t,” and the melody suddenly moves on through a minor chord and down to the crucial moment. There’s no pause this time as he reaches down to the low F and swings back up to the high A-flat for “We don’t see eye to eye.”

The chord that hits on the first “eye” is called a “secondary dominant” – in functional harmony terms, it’s a dominant chord poised to resolve to something besides the tonic (most often the V or vi). Here, it’s a half-diminished seven chord rooted on D, the tritone, one of the most most conspicuous notes he could choose in the key of A-flat major (noted above as Dø7). In most worlds, you could expect this chord would resolve to E-flat. But in “Impregnable Question,” nothing resolves the way it “should.” Just like all the other dominant chords, this one moves to IV instead.

I should note here that it is not at all uncommon for a V chord to resolve to a IV, or even for a V or vii/V (a secondary dominant of V) to resolve to IV. It’s such a common device that there’s a term for it, based on the whole functional harmony theory mentioned earlier, and the standard progression I-IV-V-I. That progression’s reversal — I-V-IV-I, or a retrogression — appears in probably thousands of pop songs.

In this case, however, it’s notable because of a great marriage of music to lyrics. The opening lines are a direct call to disregarding the seemingly inevitable: “If there is ever the impregnable question… it would help to seek comfort in destiny, but I really don’t.” And immediately after that, Longstreth sets up the song’s central tension: “We don’t see eye to eye, but I need you, and you’re always on my mind.” The second verse repeats that line, driving the refrain home.

But in the third and final verse, Longstreth answers the tension: Rather than the conflicted “we don’t see eye to eye,” instead he sings “What is mine is yours in happiness and in strife. You’re my love, and I want you in my life.” In other words, there may always be tension, be it in music or relationships. But Longstreth’s narrator refuses to let the tension control him; his dominant chords obstinately resolve to IV, and he’ll dedicate himself to his love regardless of any discord. Most rewarding to us, the listeners — the music cued us in to that response well before Longstreth states it outright.

Just one enjoyable moment of many on this excellent album. More to come!

Becky Sullivan, with special thanks to Nick Curry


Songs cited: The Dirty Projectors, “Impregnable Question,”  Swing Lo Magellan, © 2012 by Domino Records, WIGLP272.

Expectation and surprise: a primer from the Mountain Goats and the Flaming Lips

T he way we hear music is defined entirely by what we expect to hear.  These expectations can come from anywhere, including before music even begins – if it’s a late Romantic opera, my ears are primed for something a lot more harmonically complex than a mid-80s pop song.  That’s a no-brainer; Richard Wagner’s disparate tonalities would be way more surprising in a Journey tune than they are in Tristan und Isolde.

More simply, though, our ears create expectations based on what they’re already hearing.  So, when a song starts in one key, you can reasonably expect it to continue in that key.  When you hear a certain four-chord progression a few times, you expect that progression to continue.  When your brain hears a downbeat every four notes, it expects the song to stay in 4/4.

But when one of those expectations goes bust – say an F-sharp appears in the key of C major, or the chord progression abruptly changes – well, we notice it.  That’s the reason it sounds weird to hear that F-sharp, or for a measure to suddenly add a beat.  There’s some research out there that posits that this expectative setup and subsequent surprise could be a reason why some music seems more interesting than others.  In other words, our brains are pros at being efficient; they pay less attention to things they expected to happen, so that when something unexpected occurs, it sticks out much more.  And because of that, we actually pay more attention to challenging music than we do extremely simple songs.1

I n other words, the setup-and-surprise shtick is rewarding to hear.  And it appears often in pop music, much to my enjoyment, and hopefully yours too.  Let’s start with the chorus of the Mountain Goats’ “No Children” – but be sure to listen to the first 1:10 or so, so that you’re properly primed when the good stuff hits.

A chord/lyric transcription:

01 (Dbm)   (Gb)        
02 (Dbm)   (Gb)
03 (Gb)    (Ab)        
04 (Db)
05 (Db) I hope that our few re- (Ab) maining friends
06 (Gb) give up on trying to (Db) save us.  I
07 (Db) hope we come out with a (Ab) failsafe plot to piss
08 (Gb) off the dumb few that for- (Db) gave us
09 (Db) I hope the fences we (Ab) mended
10 (Gb) fall down beneath their own (Ab) weight... and I
11 (Db) hope we hang on past the (Bbm) last (Ab) exit.  I
12 (Gb) hope it's already too (Ab) late... and
13 (Db) I hope the junkyard a few blocks from here
14 (Bbm) someday burns down... and
15 (Gb) I hope the rising black smoke carries me far a-
16 (Gb) way, and I never come (Ab) back to this town a-
17 (Gb) gain. In my
18 (Ab) life, I hope I
19 (Db) lie, and
20 (Bbm) tell everyone you were a good wife, and I hope you die,
21 (Db) I hope we
22 (Ab) both die.
23 (Db)
24 (Dbm)   (Gb), etc., return to verse..

Hopefully you had a moment of surprise at the end of that chorus.  It’s really a clever bit of songwriting on John Darnielle’s part.  The lyric, obviously, is a bit shocking.  “I hope I lie and tell everyone you were a good wife” is frank enough, but “I hope you die, I hope we both die” is, well, a very strongly-worded statement.  That alone would probably grab a listener’s attention.

But a major part of what makes that line work so well is the unexpected way the chorus is phrased – that is, the way its melody and harmony are structured.  Up until this point in the song, everything has been in groups of two.  Each line in the verse is two measures long, demarcated by the initializing “I hope”s.  And even the chorus opens up like this – the G-flat measure is paired with an A-flat measure, followed by D-flat/B-flat minor, and D-flat/A-flat.  Each line of the chorus comes in a pair: “In my life / I hope I lie,” “and tell everyone you were a good wife / and I hope you die.”

Until, that is, the stunning “I hope we both die,” which Darnielle declines to pair with a second line.  Instead, it simmers – and that simmering takes place over a single measure, not a pair.  Both antecedents (the lyric, the final chorus measure) come without a satisfactory conclusion, and so the line lingers in your mind for an awkward moment before the song returns to its opening chords.2  And of course, I probably don’t have to draw you the connection between the unsatisfactory conclusions of the music and what we can probably assume was an analogous experience in Darnielle’s life.  [Edit.: John Darnielle tweeted this to me a few hours after I posted this: “No analogous life events tho – all fiction in that song!”]

O n a happier (and less uncomfortably candid) note: “It’s Summertime” by the Flaming Lips.  I don’t want to spoil anything, so without further comment, listen to just the opening just-under-five seconds:

This intro is a great example of how good our brains are at building expectations.  It’s only four seconds, but that’s enough to establish a meter in the mind of the listener.  We hear a group of sixteen notes, the first of which is clearly accented, and then a second group of notes, apparently identical, begins.  There are some fancy terms (namely, “metric projection”) for how our mind hears this.  In effect, the idea is that when our brain hears 1) a unit of time, followed by 2) the beginning of a second unit, it expects the second interval to equal the first, and for repeating identical intervals to continue indefinitely.3  On top of that, sixteen is a very easy number of beats for our Western ears to understand; it’s four groups of four – so usual, in fact, that they even call it the “common time.”  So when the second group begins, there’s little doubt in our minds what the meter is, and more importantly, where the downbeat is (at the beginning of the song, and then there’s another one sixteen sixteenth notes later, of course).

But Wayne Coyne ain’t gonna let us off so easy.  Play on!  The first ten seconds or so is plenty:

Wait, what?  Pause and listen again, if you need to.  Here’s what happened (pretend the Os are the notes):

|Oooooooooooooooo|Oooooooooooooooo|
         band should join here -->|X
   
|Oooooooooooooooo|Oooooooooooooooo|
band actually joins here ->|X (???)
                            
in retrospect, it was actually this:
Oooooooooo|ooooooOooooooooo|oooooo
  (band joins on downbeat) |X        

WTF, Wayne.  After taking just enough seconds for the bass guitar to create the expectation of a certain meter, the rest of the band crashes the party a beat and a half early.  It’s disorienting!  And the bass even sticks with the misaligned downbeats until the lyrics come in.  It rights itself temporarily at about 0:46, but the misalignment – a form of metrical dissonance called “displacement dissonance” – comes back for good at 2:05.4

This song actually asks a philosophical question about expectation and music that I love to think about.  Which is more important: what you’re experiencing at a specific moment, or what you know about that moment afterward?  Tell me what you think.  I have my own thoughts, of course, but I’ll expound on them later.

Becky Sullivan


1 On the other hand, there’s the fact that your brain gives itself a dopaminergic pat on the back every time its predictions come true, which is why it can be rewarding to hear a song you already know.  In reality, there’s probably a sweet spot between total randomness and bland repetition.  NPR’s blog 13.7 speculated a bit back in February about this “Goldilocks zone,” though it doesn’t delve in very far.

2 The word “antecedent” carries some extra implications when it comes to music theory, but those can largely be ignored in this instance. I’m referring simply to the fact that this is a unit that inherently carries the expectation of a second, complementary unit.

3 This philosophy (and the term “metric projection”) courtesy of Christopher Hasty and his book Meter As Rhythm. A thick but fantastic read.

4 The term “displacement dissonance” comes from an unusual monograph on meter by Harald Krebs called Fantasy Pieces: Metrical Dissonance in the Music of Robert Schumann. Instead of the usual dry music theory fodder, Krebs imagines a lively conversation between Schumann and a friend at a coffee shop where in between cups of joe and mid-afternoon desserts, the two posit a slew of interesting ideas about meter and rhythm.

Songs cited:
  • The Mountain Goats, “No Children,” Tallahassee, © 2002 by 4AD Records, CAD 2215CD. 
  • The Flaming Lips, “It’s Summertime,” Yoshimi Battles the Pink Robots, © 2002 by Warner Bros. Records, 9 48141-2. 

The apoplexy over appoggiatura: an explanation

T his weekend, the Wall Street Journal tried its hand at music theory, and it wasn’t pretty.  (NPR ran a similar story on Monday’s episode of All Things Considered.)  In the article, WSJ writer Michaeleen Doucleff claims that “science has found the formula” to why Adele’s ballad “Someone Like You” makes everyone cry, and it’s not that it’s sad or Adele’s a badass or anything like that.  No, Doucleff says, it’s because of a marvelous musical device called an appoggiatura!  The appoggiatura, she claims, is that little ornament of the held-out “you” in Adele’s chorus “never mind, I’ll find someone like you,” where it dips down for a split second mid-note.  Take a listen:

Well, there’s something there, but it ain’t an appoggiatura.

And it wasn’t just me who noticed this error.  I’m not sure about the Wall Street Journal, but NPR received an onslaught of emails from outraged theory nerds like myself.  Yesterday, they aired a correction.  While it’s a step in the right direction, their composer-theorist guest (for various reasons) avoids a precise definition of appoggiatura and confusingly implies that other notes in “Someone Like You” are appoggiaturas when they are, in fact, not.1

An appoggiatura – forgive the cumbersome term – is one very specific kind of “non-chord tone,” or a note in the melody which doesn’t belong to the chord being played behind it.  Yes, Adele’s little dip is a non-chord tone, but it is not an appoggiatura, which has these requirements:2

  1. It must occur on an accented beat, usually the chord change itself.
  2. It must be approached by an interval larger than a step.  (The dip fails on this requirement; it’s approached by only a step.)
  3. It must resolve to a chord tone by a step.
  4. The resolution should be in the opposite direction of the approach – in other words, if the melody leaps up to the appoggiatura, it should resolve down.  And if the melody leaps down to the appoggiatura, it should then resolve up.

The so-called “appoggiatura” in “Someone Like You” fails requirement #2, and is therefore what you could call an accented neighboring tone – that is, a neighbor tone (where the melody goes up/down a step to the “neighboring” non-chord tone, and then back to the chord tone) that occurs on the chord change.  Or, because the note is so short and insignificant, you could even just slap the noncommittal title “ornament” on it and move along.

Later in the NPR follow-up, composer/theorist Rob Kapilow goes on to suggest that some of the melodies in the verse are appoggiaturas.  There are at least two, he says.  First, the song’s third and fourth words: “that you’re,” then the “found a girl.”  (Listen above to the beginning of the song.)  Both of these are also not appoggiaturas.  “That” isn’t approached by a leap (it’s an accented passing tone), and “found a” isn’t even a non-chord tone.

T rue appoggiaturas have a very distinctive sound.  Dissonant notes on the downbeat are often noticeable anyway, but the sound of an appoggiatura is extra accentuated by its arrival by a leap, because the melody leading up to it isn’t smooth.

There’s a great example in one of my favorite songs ever: the Muppets’ “Rainbow Connection.”  In fact, an NPR listener fingered this very tune as an example of the elusive appoggiatura.3  It occurs on the “nec” syllable of “connection” in the line “someday we’ll find it: the rainbow connection.”  Check it out:

Here’s a transcription to help you out – the notes in parentheses are the chords, while above the words are the notes of the melody.

      D   F#  A            G#   A   F#
(Bm7) Someday we'll  | (E) find it, the
       E   G#  B            G# A#    D
(C#m7) rainbow con-  | (F#) nection, the
    D F#    A              C#        B
(D) lovers, the      | (E) dreamers, and
    A
(A) me.4

The bona fide appoggiatura is that G-sharp in the fourth measure, right on “con-NEC-tion.”  It’s clearly a non-chord tone; F-sharp major contains only the notes F-, A-, and C-sharp, so G-sharp doesn’t belong.  Second, the note is accented by its location on the downbeat and chord change.  Third, it’s approached by a leap (an interval of more than one step) – even though it’s only 1.5 steps, it still counts.  And finally, it resolves up to the A-sharp, a chord tone of F-sharp major, in the opposite direction of its approach.  It’s a very weirdly placed note, especially when followed by an accidental.  In effect, it draws the listener’s attention just before an out-of-key A-sharp.

Grizzly Bear’s 2009 hit “Two Weeks” is another example.  This appoggiatura happens in the opposite direction – the melody leaps up to the appoggiatura, then resolves down.  It happens throughout the chorus, on the first syllables of each “always,” “sometimes,” and “easy.”  Take a listen for yourself:

“Two Weeks” is in the key of F major.  The syllables in question are Gs over a B-flat major chord; they’re all approached by an upward fourth and resolve down to the F.  Here’s a transcription:

                               C     D
                 | (Am)    (C) would you
     G F                       C  D
(Bb) always      | (F)     (C) maybe
     G   F                     C    D
(Bb) sometime    | (F)     (C) make it
     G F                       C    D 
(Bb) easy        | (F)     (C) take your
     A  G C
(Bb) ti-i-ime...

This is a prime example of why it’s called the “leaning tone.”  There’s no doubt that the G will resolve to the F.  And in an analogous move, the entire B-flat major chord is similarly expected to resolve down to the F major, and sure enough, it’s the next fall after the G-F move in the melody.  It’s sort of a double lean that starts the downward momentum that carries through each line of the chorus.

On the surface, though, the usage of appoggiatura in “Two Weeks” and “Rainbow Connection” is similar.  They’re both used to delay the arrival of the chord change to the melody.  It’s an effective move; the unexpected non-chord tone draws the listener’s attention, which makes it extra satisfying when it resolves.  That said, the appoggiatura is still pretty rare.  In order to identify even two or three examples beyond “Rainbow Connection,” which I learned in school as an example of appoggiatura, I had to listen to nearly 100 songs.  And even then, a couple were too weak too include.5

R egardless, the major takeaway here should not about the magical power of the elusive appoggiatura.  It’s simply the name given to a very specific form of non-chord tone.  Yes, non-chord tones are effective melodic devices.  By moving off the chord, you create dissonance; dissonance begets tension; tension begets emotional response from the listener.  But they also occur very naturally in songwriting and don’t really warrant this kind of laser-focused analysis in a discussion about a topic as broad as “why music makes me cry.”

It’s true that Wall Street Journal and NPR got their definitions wrong.  But it’s much more important to understand that songs like “Someone Like You” don’t derive their emotional punch from something as simple as a single appoggiatura in the chorus.  What actually makes “Someone Like You” so powerful is a combination of more salient things: solid songwriting, tried-and-true chord progressions (here, I-iii-vi-IV and I-V-vi-IV), the contrast between the vocal ranges of the verses and the chorus, etc.  And of course, it always helps to have a very, very talented performer behind the microphone.

Becky Sullivan


1 All Things Considered only has two hours of airtime a day to devote to all of the news, and though they do an admirable job of considering all things, it’s asking a lot to spare more than a few minutes for extended music theory pedantry about the exact definition of a very specific musical term (no offense to self). So instead, Rob Kapilow provides a correct example from the classical literature, then the discussion moves on to other moments in “Someone Like You” (though, unfortunately, they are also incorrectly labeled appoggiaturas). 

2 This definition is the current standard for music theory classes and comes from my experience/textbooks in music theory classes in both high school and college. There is one major alternative view that calls an appoggiatura any accented non-chord tone that resolves by a step. This old-fashioned definition is not taught or widely used because it erases the (useful) distinctions between several different kinds of non-chord tones.

3 The other song this listener indicates, the Beatles’ “Try to See It My Way,” is unfortunately not an example of appoggiatura. The non-chord tones in question this time are accented neighboring tones again.

4 Man, that song is really nice. I’ll even admit to tearing up a bit in the new Muppets movie when Animal broke out with the drums.

5 Most of my failed appoggiaturas were similar to those misidentified by NPR – accented passing or neighbor tones (which fail because their approach is by step, not leap), or they were chord tones that sounded like non-chord tones on first listen.  Other times, the non-chord tone was unaccented (it fell on an offbeat), or it appeared at the start of a phrase (meaning it had no approach at all). 

Songs cited:
  • Adele Adkins and Dan Wilson, “Someone like You,” performed by Adele on 21, © 2011 by XL Records, XLCD 520. 
  • Paul Williams and Kenneth Ascher, “Rainbow Connection,” performed by Kermit the Frog on the Rainbow Connection / I Hope That Somethin’ Better Comes Along 7″, © 1979 by Atlantic Records, STA 37283. 
  • Grizzly Bear, “Two Weeks,” Veckatimest, © 2009 by Warp Records, WARPCD182.