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June 24, 1999: print my music far and wide


Music theory geekiness today.

I've been reading a series of music theory treatises on the music of the Beatles, written by Alan W. Pollack. They're pretty interesting. Sometimes I feel like his analyses are a little over the top, though. I mean, I have no doubt that in the end, the music theory he's applying accurately reflects how he (and people with a similar musical background) experience the music and sense the motion within it. He's careful to disclaim the notion that the Beatles were conscious of it, though.

Sometimes he argues that the music goes a certain way because this underscores the lyrical focus of the song, and he often feels like he's reaching in these parts, but I can buy into the surface sheen, at least; the music is supposed to feel 'restless' because that reflects the lyrics, and the theoretical understanding of the music suggests that it's restless in the way of its progression, ok. But it's kind of odd to see the exact same musical move (say, heading to the V but avoiding the I) interepreted as reflecting two very different emotional attitudes. It seems more tempting to say "they wrote the music that way because it sounded cool; your analysis of why it sounded cool is fine. Your analysis of why they wrote it that way, I do not buy."

I've taken a little bit of music theory. Enough to be able to name the chords appropriately, but enough to really analyze a piece of music. If something I'm listening to goes through the chords "F-C-G", I call that a "IV-I-V" progression; in various songs and contexts, Mr. Pollack might refer to that progression as a "flat-VII - IV - I" progression, "IV of IV - IV - I" progression, or even "I - V - II".

Now, nobody else is ever going to analyze my music, so I figured I'd do it myself. In the process, I'm going to teach you Sean's Music Theory. This won't necessarily help you analyze other people's music. It might not be the best approach to explaining why my music sounds the way it does (that is, traditional music theory might offer better information). It definitely does however give you the best theory for understanding my compositional process.

For the sake of this discussion, we're going to look at one of the harmonically richest songs I've ever written, called "Seven Hours Ago". In the course of the song, 10 of the 12 possible major chords appear, as well as 6 of the 12 possible minor chords. Perhaps a studied classically-trained musician would hate the sound of it and the crazy chord changes. Or maybe not. I don't know.

The central conceit of the song is that at a particular moment in each verse stanza, a particular chord is struck, and the name of that chord is simultaneously sung (chosen, though, to fit in lyrically), and a beat is dropped for emphasis. In the course of the seven stanzas, all seven of the C-diatonically-rooted major chords will be played at that moment in the stanza.

The main theme of the song is a 4 measure sequence of eight chords which, according to Sean Music Theory, skip quickly through three key signatures, never gravitating far from the origin, which is F#. Some might argue it's eight measures of a very fast tempo, but the author knows that the emphasis is supposed to be a dropped beat, not a half-measure.


   chord       | F#   C#   | d#m  E   | B    A    | c#m   B   |
   role in F#    I    V       vi                          IV
   role in B                 iii  IV    I
   role in E                      I     V    IV     vi    V

I'm not sure how traditional music theory will explain the final resolve, which is "v - IV - I". An "IV - I" resolution is supposed to be notoriously weak, and indeed the sequence "c#m B F#" without any other context one would normally seem to imply the key of B (although I'm not saying music theory would argue that at all).

Now, let's go on a short excursion into an aspect of traditional music theory. This is something for which the facts are covered in even the simplest theory book, but the mechanics--the whys--are not, and so this constitutes guesswork on my part.

Central distinguishing elements of tonal music theory are the notion of a "home key", the notion of "modulation into other keys", and the notion of chords serving roles for one another.

So, a crucial issue is how to determine what key we're in. If someone wants to determine what key we're in, you might think, hey, whatever chord you get, that's the key. Well, that's the case if that's the only chord, but suppose there are lots of chords, and we're just looking at one of them at random, not necessarily the tonic?

Well, it turns out that in a strictly-in-key piece, there is one particularly special chord that uniquely determines the key: the dominant or V chord. It only does this by virtue of the version which includes the seventh, that is the V7 chord (e.g. G7 implies C). That is because there is no other major chord with a flattened-seventh in a given key. The other two majors have major sevenths, and the three minors all have minor sevenths.

The V7 chord is widely perceived as not merely resolving cleanly to the I (tonic), but wanting to resolve that way--implying a sort of motion. I don't know if this is really there or is simply a matter of us having this ingrained into us from childhood. (That may sound ludicrous, but I do think that's true about non-4 rhythms. Alan Pollack obviously knows what he's talking about, but he mentions being rhythmically bewildered by the middle instrumental section of Within You Without You. The Beatles Complete Score notates the passage in 5/4, but it's obviously in 5/8. It's worth keeping in mind as musician that what sounds perfectly obvious to you (e.g. strange rhythms that you've spent ten years learning) may make utterly no sense to the average listener.)

Thus, you get the extremely common "V7 - I" progression. This is a pair of chords that very unambiguously defines a key. (Yes, you can use the sequence without necessarily being in that key, but that's the basic situation implied by and common for it.)

Now, I basically never use seventh chords. They're just not part of the vocabulary I work with. But with "Seven Hours Ago", I explicitly decided that rather than writing a "random" chord progression that sounded good by ear, I was going to consciously think about modulations. So I needed a scheme by which I could modulate successfully--without knowing about existing music theory approaches.

The obvious idea is to use one or more chords which are harmonically ambiguous. The sequence "C - am" could be in the key of C, the key of F, or the key of G (not to mention the relative minors of each of those). This makes it easy to "set up" being in a new key. You can see this in the annotated chord progression above; each chord with more than one label "works" in one of several keys.

The question is how, without a V7-I cadence, to establish the new key.

To this, I turned away from traditional music theory and simply analyzed the situation myself. Just as there is only one possible flat-seventh major chord in a given scale, so there are "chord pairs" which uniquely imply a particular key. Much like a V7-I cadence, but ignoring the notion of cadence. I-V7 also uniquely implies the key, even if it's not necessarily a musically sensible move. So I'm just talking about pairs of chords like <I, V7>, rather than worrying about their order.

Without using sevenths, here are the ones which call out a unique key:

(There's 15 unique such pairs; only four of them have this property.)

Note that all it actually takes to spell out that it's the key of, say, C, is a chord with an F and a chord with a B. An F and a B alone are not sufficient (it could be C or Gb), but with any one other note, it's unambiguous. There are two choices of chords for each of these notes.

The second one of those is interesting, because "ii - V - I" is the most standard three-chord cadence, so it's worth noting that ii-V sets up the I in more ways than one. The remaining ones are all chords a (diatonic) step apart.

For completeness, we should reconsider this in light of the traditional music theory minor key, in which the V is made major instead of minor. However, Sean's Music Theory treats the relative minor as strictly identical to the major, so under SMT, the above is all there is. But, ok, I'll come back to this when we get to the refrain.

So, even if these aren't technically modulations by traditional music theory (since they're too brief), the goal was to construct these "key changes" via SMT in the hopes of making the chord changes make more sense. Another way of looking at it was that this was an authorial device, used to restrict my choices in the hopes of making it easier to find things that "sounded good".

So, here's the setup again, and now let's look at it in the light of those "unique chord pairs".


   chord       | F#   C#   | d#m  E   | B    A    | c#m   B   |
   role in F#    I    V       vi                          IV
   role in B                 iii  IV    I
   role in E                      I     V    IV     vi    V

Ok, I don't know enough music theory, so I'm going to call the final chord which is common to both keys the "pivot" chord. The first chord which doesn't fit the prior key is the "modulating" chord. The first chord which clearly establishes the new key is the "establishing" chord. Note that (in SMT) a modulating chord need not be an establishing chord. Ignoring extremes of a chord sequence like C-D-F, in which the chord movement to the modulating chord isn't diatonic, consider a sequence like G7-C-Em-D; if the D is part of a modulation, it could be from the key of D or G.

Use of a diatonic pair like <ii, V>--that is, a sequence like Em-A, would more strongly suggest a particular key. So, in the three SMT-style modulations in the brief passage above, consider what chords serve each of the roles:

The verse uses a somewhat stylized form in which two variant versions of the chordal theme are used (ignoring the internal variation caused by integrating the seven different chords at the appropriate spot). The first two verses both have the form AA'A, where the first A is instrumental. The final verse has the form AA'AA'.

Let's look at the first sung stanza, which is the A' form with a stressed chord of A major.

   | F#   C#   | E    B  |   A   c#m | g#m   A  |  c#m   B

Note that the modulation to B is on the surface removed. The jump from C# to E is unmotivated, but the iii-IV move from the previous version of the same progression motivates you to hear it the same way; things are just moving a little faster this time around.

This time around, the key of E is established more unambiguously (even if, perhaps, it seems that the key of B is omitted). The iii-IV in E is used to emphasize the "stressed" chord in this iteration of the verse.

Note that, given the conceit of using the sequence of chords A, B, C, D, E, F, G in each of the stanzas, the author chose to set the piece in the tonally distant key of F#. Keeps things challenging. By establishing right out from the beginning the tonally ambiguous nature of the piece (quick unexpected modulations and a lack of V-I resolutions), the later excursions far from the key and straight back will be unsurprising.

The final stanza of the first verse:

   | F#   C#   | d#m  E   | B    A    | c#m   B  |  g#m  F# |

This stanza is mostly noticeable because the selected chord actually is from the key. Thus, the choice of F# seems not so crazy, since the first verse doesn't need to go very far to integrate the chords. But it has to go a little ways to get to the A; thus the initial instrumental chord progression sets everything up.

The B can't drop straight into the F# in the fifth measure for rhythmic/vocal reasons, so instead it drops into g#m, perhaps hinting at a ii-V-I.

Instead, we get a refrain. The refrain was intentionally constructed around a sequence of chords which are non-diatonic: g#m-A#, acting as a sort of i-II. I picked these to be tonally ambiguous, and set out to find multiple ways to resolve them. While I wasn't conscious of the music theory underlying my decisions, it's worth pointing out where these chords to appear in music theory:

Most obviously, the chords g#m A# appear to be a iv-V setting up a I in a minor key, that is d#m. This is the standard "the V of a minor key is still major". I don't know if "iv-V-I" is reasonable in that context, though.

Second, the sequence could come from a D# major scale in which the IV has been altered to an iv, which is done from time to time; my Beatles notes refers to it as a 50's rock cliche, and in fact my friend dfan's band's has a song with an A-Dm-E progression in exactly this sense.

Let's look at the full refrain, then.

     |  g#m   A#   | d#m   F#    |
     |  g#m   A#   | b#m   C#    |
     |  g#m   A#   | E     B(7)  |

Only the second one of these used SMT; the first and last were done by ear.

The first time through makes perfect sense given the above analysis; the chords imply a modulation to the key of D# minor, the relative minor of the F# major of the verse.

The second sequence involved an explicit modulating chord, the b#m (that is, C minor), which is constructed to be diatonic with the previous A# chord. As noted before, there are two plausible key signatures this construction could be from (A# or D#); D# is plausible, since that makes the g#m seem to have been a simple iv chord. Nonetheless, the ear hears it as a slightly unexpected chord, despite its diatonic relation to the previous chord. A moment later, though, the C# chord comes out of nowhere, suddenly establishing via a iii-IV move an implied key of G#--which is not so outrageous, as it's the parallel major of the first chord of the refrain, and hence a natural candidate. This also makes return to g#m seem to be an IV-i cadence, nicely echoing the various IV-I cadences elsewhere.

The third solution I have no clue what's going on with it. I think the E is functioning as an IV-of-IV. The 7 in the B chord is added for the latter half of it as a "flavor", not as functionality. But it's noticeable that adding this to the E instead causes the progression to stagnate; the IV7-I E-B move has no further forward push. (The 7 in the IV7 resolves up a halfstep to the third, in parody of the V7 resolving down a halfstep.) This is interesting because of the obvious interrelationship of the E7 chord to the A#7 chord, but that seems irrelevent here. There is an obvious relationship between the g#m and the E (as well as the B); in one sense, we're clearly just modulating back to the key-leading-to-the-home-key. But A# doesn't seem to serve any purpose in either the key of B or E, the likely candidates. Maybe it doesn't really work, and it's just that the surprising change isn't very surprising after listening to the previous modulations in the refrain.

Here is the second verse.

  instrumental theme
      | F#   C#   | d#m  E   | B    A    | c#m   B   |
  first vocal
      | F#   C#   | E    B   | A    f#m  | bm    C   |  D   E   |
  second vocal
      | F#   C#   | d#m  E   | B    A    | c#m   D   |  E   F#  |

Some general comments: In the interest of sustaining an apparently repeating song form, note that the chord progressions remain rigid all the way up to the A. This has the effect of providing a regular form to the song, while also modulating temporarily from F# to E, making the sought-after chords a bit less remote. However, this leaves only one chord as a transition for the second vocal stanza in each verse; and while this works for A-?-B and A-?-D, it won't work for A-?-F, so the A isn't maintained that time around.

The first vocal line relies on a move down to f#m, a chord that's always been implied by the key of E, but was never voiced explicitly. This firmly establishes the key of E this time, but only for a moment, as we immediately jump to B minor. Note this is the first time we've seen the tonic move up by a fourth, that is, in the pattern of a V-I cadence. However, it's not that strong, since both chords are minor. This chord progression could be read as implying either A or D (ignoring minor keys, since this is SMT). A natural interpretation is A, since this makes the key changes a circle of fourths: F#, B, E, A.

However, this is again immediately usurped by a iii-IV progression out of nowhere to C. Given the number of iii-IV moves found so far, though, I imagine the listener is "getting used" to this device. The fact that it's used to bring the most remote chord from F# is crucial; it's all downhill from here (well, except F and G, but those involve cheating). After the requisite beat, this moves up in a IV-V, firmly grounding the key as G. If the f#m-bm move is read as implying BOTH keys A and D, then the circle of fourths is naturally extended by this: F#, B, E, A, D, G.

In something of a parody of rock cliche involving flat III-IV-V progressions or flat-VI - VII - I progressions, the progression steps up another whole step to E. While one could argue under SMT that the D-E pair is implying the key of A again, the way the E is used simply to lead back to F# implies that the sequence C-D-E-F# is simply an exaggerated extension of the stereotypical C-D-E progression.

The second go around is much simpler. Here, the move to A is explicit and involves the exact same sequence of steps as the move to G. It actually sounds different, however, because A is tonally connected to the previous key of E, so things are less forced. The extra F# at the end actually serves as a transition to the g#m of the refrain, and sounds much less over the top than the previous verse.

The second refrain is the same as the first, with an additional tag ending. I think of it as a variant refrain, not as an added-on bridge. Of course, one might argue that these are bridges not refrains in the first place anyway.

     | g#m       A#        | d#m      F#       |
     | g#m       A#        | b#m      C#       |
     | g#m       A#        | E        B(7)     |
     | g#m  F#   A    c#m  | F#   A   E     B  | d#m    C#   |

I'm not even going to try to analyze this coherently, since it was all done by ear. A few things of note: the F# and A in the first measure are related harmonically to the g#m, but not to each other. Their appearance in the first measure makes their appearance more tolerable in the second measure--where you can hear the F# as a substitution for F#m, I guess. This time around, the jump to E-B makes perfect sense, coming from the A, and thus returning to the key of E for the return to the verse. The sequence B-d#m-C# reiterates the end of the main verse progression, but a whole step higher; however, it is actually in the key of F# as well, and actually resolves unambiguously to F#, even using a V-I cadence. The last few chords are IV-vi-V-(I).

  instrumental theme
      | F#   C#   | d#m  E   | B    A    | c#m   B   |
  first vocal
      | F#   C#   | E    B   | A    c#m  | B     E   |  F#    B C# |
  second vocal
      | F#   C#   | d#m  E   | B    g#m  | F#    F   |  F#    B C# |
  third vocal
      | F#   C#   | E    B   | A    c#m  | g#m   G   |  F#   [end] |

The first time through the vocal line, the full progression is actually sustained all the way to the B for the first time. While the B-E could sound like a V-I in E--the apparent key signature--because of its location rhythmically, it doesn't feel that way, and the immediately following F# strongly implies a I-IV-V in B. However, with hints of a IV-of-IV - IV - I progression of E-B-F#, we instead immediately modulate strongly back to F# through B-C#, which in B sounds like V-I-II, but because of the expectation of F# as the home key is clearly itself, it's obviously a I-IV-V in F#, echoing the V-I cadence that led from the second refrain to this verse.

The second vocal line of this verse relies on a weak device to justify the use of the F. The vocal melody simply dips from F# to F at this point, singing the major 7th of the chord, and the motion of the chord following it is essentially just accidental, echoing the rock cliche of using such chords as pickups. More notable is the way we get to the F#--skipping getting to the key of E entirely, and immediately dropping back down to the F# by way of the g#m (which is common to all three of the keys F#, B, and E). Where the second vocal line normally drops to the refrain at this point, the same I-IV-V in F# is used to generate the expectation that we'll stay in F#. Thus the appearance of another verse fragment is less surprising than it might otherwise have been.

The final vocal line, whose text clearly deals with resolving the situation which has been described by the rest of the song, runs through the normal bits and ends with a ii-V-I cadence, with the G substituted for the C#7, a somewhat cliched stylization in certain kinds of popular music. Note, however, that not until the G chord is reached is it clear that a modulation away from E is occuring, since the c#m is not from F#, and the g#m is common to both E and F#. The G is the modulating chord, but lacking any context, it doesn't really establish anything, so it's not clear what's going on until the F# is hit--at which point the song has come to an end. On the surface, this is a measure early, but the previous two vocal stanzas also reached F# a measure early. In those cases, it didn't necessarily "feel" resolved, though, so the extra IV-V-I bits were thrown in to resolve it. This time around, it's all resolved by itself with no further effort.

There is one final unrelated aspect of all of the verse chord progressions which is too consistent for it to be a coincidence. As noted, the "natural" voicing of the main progression on the guitar is arranged so it moves up and down in a simple fashion (which is most notable in the top melody of the chords). Each of the remaining chord progressions continue the identical pattern--up for the first four chords, and down for the remainder of the chords--except the stressed chords. (Sometimes the chord doesn't move up or down, it stays in the same position, but it never goes down during the 'up' section or up during the 'down' section.)

Here's an analysis of the progressions in this light so you can see what's going on. The number listed is the fret number of the barre; this is always audible as the motion of the top voice in the guitar chords as well.

               ----- upward -----  - downward -
   verse 1:   | 2   4   | 6   7   | 7   5   | 4   2   |
              | 2   4   | 7   7   | 5   4   | 4   5   | 4   2   |
              | 2   4   | 6   7   | 7   5   | 4   2   | 4   2   |

   verse 2:   | 2   4   | 6   7   | 7   5   | 4   2   |
              | 2   4   | 7   7   | 5   2   | 2   3   | 5   7   |
              | 2   4   | 6   7   | 7   5   | 4   5   | 7   2   |

   verse 3:   | 2   4   | 6   7   | 7   5   | 4   2   |
              | 2   4   | 7   7   | 5   4   | 2   7   | 2   2 4 |
              | 2   4   | 6   7   | 7   4   | 2   1   | 2   2 4 |
              | 2   4   | 7   7   | 5   4   | 4   3   | 2       |

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