I find myself happily trudging through the explorative analysis of the greater body of a genre of music from the years roughly 1950-2015 as I sit firmly on 22,583 words of my treatise, ‘Theory of Pop Music.’

Much confusion has ensued in the course of discussions with close friends, colleagues, and professional musicians, both Pop and Classical as to what it is and what it is not.

Understandably, my concept of Pop is broad. It references Popular Music, as distinct from Western Art Music. In the same way that I am uneasy when someone references me as a classical musician because of my false assumption, they might lump my writing into the same category as ‘easy classics with meditative beach noise to relax to’ on sale exclusively at Target, I understand that the Scandinavian-post-grunge-dragon-math-core-metal guitarist desires autonomy from the likes of Taylor Swift. Every musician hates being asked for a list of influences as a prerequisite resume to your art; I get it. But that is exactly what I have done.

Similarly, the term “Theory” is somewhat of a gross misnomer. At least, it is misleading; but with good purpose. The majority of pop musicians think of theory as ‘how to write music’ which is why so many oppose the notion of learning theory. They dislike rules to be imposed on their sense of creativity. And that makes sense.

More accurately, a theory seeks to make logical and coherent connections of the properties, and materials that make the essence of particular noumena. A theory must always proceed a phenomenon, not introduce it. It must describe what happened, not inform us what will happen. It is methodical science, not prophetic oligarchy.

The irony is, however, that most pop musicians ARE looking for ways to bring coherency to their music. They are judging their contemporaries and studying their spiritual progeny. They’re curious. They want to be taught. But they are doing this without a formulated technical language and in many cases, without a notational system to describe it.

One of the cruxes of conventional music theory as most understand it is that is focused on the music primarily of three composers. Haydn, Mozart, and Beethoven. The logical reason for this is two-fold:

  1. It is this literature, as well as the literature of the previous 100 years that shares enough similarity to it, including the literature of the next 100 years that uses the classical era as a model, that will make up 80% of the music that pays the bills of the majority of college music majors, after they earn their accolades for the rest of their life. Hence, it is important to know how this music works.
  2. For those musicians who are composers, the classical era stands as a model, as it did to the 19th and 20th century, despite their disparate stylistic departures. It is crucial for a composer to learn these normative structures before they can employ logical deviations, even when those deviations make the music so radically different in style that comparison to the classical era seems distasteful; Yet, the same engine lies under the hood of both. It is theory that makes the similarities readily available and obvious.

For all its complaint’s against it, however, Pop music is no less fundamentally rooted in theory. Some sad attempts of the past have tried to make a correlation, with embarrassing results. If I had to guess(which is more of a postulate than a guess), it is that while the 19th-century music of Brahms, Chopin, and Schubert share ancestry with the classical composers, Popular music shares ancestry with only music as early as Brahms and Chopin. It is the illegitimate offspring to a throne, no less epically vivacious, but the bloodline was transmitted, not inherited.

To use a more visual analogy, when Walt Disney first conspired to make the first ‘moving paintings’, he employed classical artists and painters to help him revolutionize art in a way which we now know as film. However, the generation that ascended in the wake of the retirement of those initial artists were trained as animators, not classical artists; a vocation that didn’t exist in the day that Disney first assembled his ‘9 old men.’ The animators were trained to be animators. The painters were trained to be painters, who later invented animation. That subtle difference is key to understanding why Classical theory fails with Pop Music, and why Pop musicians repudiate theory after a year of college courses.

If, theory, however started with the extended chromatic era of the 19th century, and the shift from the ‘circle of fifths’ to relationships by thirds, many of those musicians who initially rejected theory at the local community college, would, with a bit of work, begin to see an immediately emerging coherence between the music that they’re constantly exposed to, the music they hear in their head, and the theory being presented as to what they’re hearing, and why they might be perceiving it that way.

That is to say, when one is learning classical theory, it is easy to see how Mozart is utilizing harmony. But it is quite non-existent, except in perhaps very basic examples(which generally need no explanation to begin with). I can’t think of many popular musicians who sound like Mozart, unless on purpose, like Queen’s Bohemian Rhapsody. But there are plenty of damn fine good examples of people sounding like Schubert. Is there any wonder why Chopin and the Tonnetz works so convincingly without sounding out of place in the music of Muse, while this…

… does not.

Unfortunately, the theory that concerns the 19th-century romantics was quickly abandoned at the onset of the 20th century when the likes of Schoenberg and Stravinsky opened the floodgates of harmonic experimentation. Theorists, taking a nod from Noah, were busy trying to build an Ark mid-float to safeguard tonality as it drifted off in different directions to various theoretical shores.

One vestigial form, however, survived and slowly matured; Film music. And alongside it, Popular music emerged. Jazz, and Filmscore, and Popular music all intermarried in their harmonic exile from the theoretical observance of scholarly musicology for the greater whole of the 20th century.

 

Hence, a book on the theory of popular music is both parts observation and a treatise of creative tools for the active musician today. It does not attempt to fit the 19th-century concepts on to the 20th and 21st century, but it assumes it as a basic model. A parental guardian, of which, the circle of fifths can be considered a distant relative, inheriting particular traits, but sharing more in common with the firstborn generation of the new country.

 

As an example, I would like to discuss Radiohead’s Karma police. An example that I toyed with, but ultimately omitted from the text, because its harmonic ingenuity undermines it’s usefulness as a model in the course of study. Never the less, a guided analysis will show some fascinating features working under the hood of late-century popular music.

 

The song begins with the progression A minor, D/F#, E minor and G major. Assuming that the song is in A minor, the roman analysis would be i  iv6  v  bVII. A minor, hence, is problematic. The E minor should be an E major, especially if the F# is present in the chord before, which adheres to the melodic minor scale. Furthermore, the G major, should be G#, as it is the leading tone. Such a progression would sound like this:

Figure 1.

 

 

Assuming the song is in A minor is wrong. Because of the lack of the G#, and the fact that the dominant chord, E, is minor, traditional theory condemns this interpretation. The mantra “always raise the leading tone in minor” is one of the first true principles stressed in classical theory. Furthermore, the song has a strong pull toward G major. When a minor tonality leads to a major, however, it is almost always from bVII to III, not VI to bVII.  Such a progression sounds below:

Figure 2.

 

So, we can rule A minor out. If you’re beginning to disagree with me… Great. You’re right. Let go of that thought for now.

Back to the drawing board, if we take an inventory of the chord classes, we can see that the key is best expressed as either G major or E minor. E minor is highly unlikely because listening to the progression E minor(the third chord) sounds deceptive. A failed goal. The D/F#(the second chord) both leads to E minor and then away from E back to G. Put another way, The F# attempts to move to E minor from above, thinks to itself ‘no, that’s not right’ and then moves up instead, where it does sound ‘complete.’

This is shown in the image below:

Figure 3.

 

The first F# draws an arrow to the E, where the red circle here indicates a failed goal, the F# then moves up to G, where it does sound like a goal has been met. Furthermore to add to the sense of G being more integral than A minor or E minor… The concept of a tonic is that it acts both as the starting place, a home base, and the goal, the end of either a progression or an entire piece.

Sure enough, the song starts in A minor, and we can assume then, as with the overwhelming majority of music, that it is the tonic. However, in the last measure, the final A minor chord sounds absolutely nothing like a goal or a tonic. It sounds like an intervening harmony, that is to say, a harmony that is acting as a movement between one goal and the next. How, in such a short time did the tonic transform into a subordinate harmony? This dualism is part of the alluring intoxication of the song.

Listen here to hear the progression in isolation:

It should be blatantly obvious that the last chord does not sound like a goal has been reached. It is a thought mid-statement. It is an adjective rather than a noun. However, at the beginning of the song, it most certainly was a noun. Wasn’t it? If you’re having trouble deciding, Compare this audio example with the first two examples that are in the ‘classical’ style.

 

If you did, you most certainly heard the overwhelming sense of ‘goal’ in the former examples, compared to the open-ended-ness of the later. This is a further case for G major. Isn’t it? But how then do we explain that all of the verses start on A minor, and lead to A minor? Let’s compare a rather unorthodox idea presented by Beethoven in the Opus 31 no. 3. Don’t worry it’s only the first 30 seconds, or so.

 

Here, Beethoven deviates from the classical norm with an almost extreme prejudice by replacing the opening tonic with a predominant function. The II65 furthers it’s intensity by transforming into the predominant function of a diminished 7th belonging to V(dominant),  which then arrives at the dominant  in the form of a Cadential 64 where it coyishly spreads its feathers like a peacock, so to speak, before its transformation into a V7 position flirtatiously makes itself aware to its tonic where, at 0:15 seconds there is no mistaking the arrival of the tonic, which with all buoyancy, lightly leaping back upwards toward the II65, repeating the progression. Here, there is no doubt what Beethoven is doing. We retrospectively interpret that the composer has avoided the opening tonic. He spends much effort slowly emphasizing the functions through dynamic changes, and when he arrives at the tonic, he lets all the energy prance out of the gate.

 

Using this as a model, The Radiohead example starts with A minor and moves to G major as a goal. A minor is the II(predominant) of G major. Analogous to the Beethoven. It also leads to a type of dominant movement prior to G major. Hence, this might be a good interpretation. Therefore, the analysis is as follows:

Figure 4.

 

Now it is starting to make sense. The parentheses indicate that I consider the E minor a neighbor chord in this interpretation, rather than a deceptive goal, according to the previous interpretation. That is, it embellishes D major, but ultimately is not functional. Disagree with me? Good, you’re right. What we can see then, is the emergence of a dualism.

The progression is best described as both A minor and G major at the same time. They are functioning at the same moment in time, both confirming and obscuring each other as tonal centers. The E minor is the dominant goal of the A minor, while the A minor also acts as the predominant, to the dominant D major, to the tonic of G:

Figure 5.

 

At a deep structural level, if we were to untangle the two progressions we should be hearing both of these progressions:

In the audio example above, The first three chords project the A minor progression represented by red arrows in the illustration above. The second progression, G major, is represented by blue arrows. The final 5 chord progression represents a hypothetical superimposition of the progressions, in which one can begin to conceptualize how the dualism of the piece is conceived.

There is, however, one other special interpretation that I have saved until the last, and it is probably by far the most accepted analysis of this piece. That is, that the piece is in the Dorian mode. While I assume most musicians understand modes, I will never the less define it as follows: A mode is a cyclic permutation of a diatonic scale. Before diatonic tonality was codified somewhere in the middle of the 17th century, music was often thought about in terms of modes, rather than keys. That is to say, in Dorian, or Mixolydian, rather than in A major. As vertical structures began to be explored, and particularly those of triadic structures, one mode in particular gained prominence; The Ionian mode.

This mode is equivalent to our standard Major scale. The Minor scale is equivalent to the Aeolian mode, however, theorists argued about its adoption presenting Dorian as a more likely candidate. Ultimately, Aeolian won out, most likely due to the fact that it represents a minor compliment to major in its primary functional chords; Tonic, Predominate, and Dominant, where as Dorian does not.

The Ionian mode gained its prominence for 2 reasons:

1. The primary chords, tonic(I), subdominant(IV), and dominant(V) are all Major. These primary chords stand in relation to the circle of fifths. The Dominant lies 5 steps above the tonic, while the subdomaint lies 5 steps below. All subordinate chords are minor, the supertonic(II) mediant(III) and submediant(VI). This is true of every major key.

2. The remaining chord, the leading tone chord, VII is diminished. So strong is the pull of a diminished chord to the chord directly to the right of it, that its authority to stand as a consonant triad on its own is diminished, hence the name. That is to say, it is a special chord that’s only use is to highlight the chord next to it, rather than be heard as a chord in itself. This feature is of utmost importance to tonal music. It is by this triad, either through its formation of a diminshed 7th, or when subsumed into the dominant 7, that modulations are able to confirm so strongly a new key. Listen here:

Figure 6.

Every held chord(the beginning chord) has a diminished formation in it. You can hear it’s instability. The first two are fully diminished 7th chords; They compromise of two superimposed diminished triads. The last 3 are Dominant 7ths. You can hear that the Dominant 7 is considerably softer in its dissonance because of the support of the major triad which roots the upper diminished triad. It is easy to hear why this diminished triad is not suitable as a tonic, a goal, the period of a grammatical sentence. Notice, however, how each tonic sounds convincingly as a goal. This would not be the case if we removed the diminished triads:

While the chords do make sense functionally, they no longer sound like ‘goals’ rather, they all come under the influence of the gravitational pull of the opening and ending chord. Their function has changed from goals in themselves to steps in progressing to the goal.

Therefore, because of the diminished triad’s ability to highlight the chord directly to the right of it, the permutation whose tonic is positioned such that the diminished chord comes right before its reappearance at the end of the mode is emphasized to such dramatic exaggeration, that whatever the next chord to the right of it is takes on a highly elevated hierarchical position, as can be observed from the example above. Hence the name ‘Major’ i.e., have a major advantage to all other modes.

Stated another way, the Ionian mode became the Major scale because the penultimate chord, before the progression reached the tonic again, was a chord that highlighted the final chord, which was also the beginning chord. In a literary context, it placed the climax at the end and emphasized the character at the beginning as the character at the end. Without this diminished triad, it would be largely impossible to ‘terminate’ the progression and have a sense of end; This is a concept we will explore further in the Radiohead example.

You can imagine then, that this diminished chord wreaked havoc on coherency if it appeared anywhere in the middle of a scale rather than the penultimate chord.

 

For instance, if you were in G Mixolydian, and your progression was I, III, IV, V, I, you encountered the problem that your IV was emphasized to such a degree, that your sense of G as a tonic was lost. This is because the III that precedes it is the diminished chord.

 

Figure 7.

The chord that starts out the progression starts out sounding like a tonic, but by the end of the progression, it no longer sounds like the goal. Compare to the same progression in Ionian:

 

Figure 8.

 

Or, if you were to write in Lydian, the diminished chord falls on the subdominant IV, making the dominant V, a chord who’s function is to strongly insist on resolving to the tonic, instead, sound like the new tonic.

 

Figure 9.

Again, the F major starts out sounding like a tonic, but by the end of the progression it’s function has changed as a result of the placement of the diminished triad.

In Phrygian, the problem is more compounded by the fact that the dominant position, the function that is so strongly associated with the tonic, a function that can so convincingly compliment, contrast, and confirm the tonic, instead is the diminished triad itself. Here, the dominant moves to the tonic triad, bypassing the chord of resolution(VI) because of its dominant status as a structural pillar, and thus the progression sounds like a complete mistake rather than a transfer of function as in the previous examples.

 

Figure 10.

 

How the minor was determined from Aeolian was not as straight forward as the Ionian. As can be deduced from the previous sampling of modes, each one has a Major aroma or a Minor aroma. However, the preferred Minor aroma that composers gravitated toward was Dorian, not Aeolian. Aeolian has its supertonic(II) as a diminished triad, which is why minor overwhelmingly modulates to III, as we explored in the second audio example of Karma Police. In Dorian, the diminished triad falls on VI, where, it highlights the VII. Dorian tends to contrast with its subdominant, IV, where it is major. This is different than the Ionian, where we described that the primary chords I, IV, and V are all Major. Here, instead, we have two different qualities of chords, Major and Minor. Furthermore, Dorian tends to use VII, a Major chord in this case, as its dominant. This mode was in use so frequently that it retains its high sensitivity to the medieval aesthetic. Even today, it is the sound of fantasy worlds, of the Renaissance, Game of Thrones, Catholic choirs, the Crusades, Middle-Earth, etc.

 

Figure 11.

You can hear, however, that the ‘sadness’, the ‘sinister’ is considerably lightened. It is less ‘dark’ than the minor scale we know today

 

Figure 12.

 

Above, you will note the absence of the dominant V in favor of the more prominent Major IV and Major VII the latter of which acts as the dominant and gives Dorian it’s characteristic sound.

 

However, Dorian has a more lively, jovial sound as well. While outside of the scope of this article, Dorian is also the sound of Carlos Santana’s Oye Como Va

 

As well as Mark Ronson and Bruno Mar’s Uptown Funk:

 

Why was such a versatile mode dropped in favor of the Aeolian? As we observed before, the Ionian mode was chosen because the diminished triad precedes the tonic. It also carries the beneficial feature of having all primary chords as Major. While the Dorian had mixed quality primary chords, Aeolian featured the mirror image, as it were, of the Ionian primary chords. All of its primary chords are minor. Its secondary chords, III, VI, and VII are all Major. Its diminished triad is supertonic II, which as we’ve stated multiple times thus far is the reason minor progressions naturally levitate toward the Major III. In essence, Aeolian was the perfect structural complement to Ionian. The primary functions in Major were all major chords, and the secondary chords were all minor. Aeolian reverses this. Dorian, could not.

So then, Aeolian won a debate on theoretical criteria, rather than the practical. We should not roll our eyes at this, just yet. As a diatonic field on its own, it lacks the harmonic appeal in comparison to Dorian, but the implications of its connection as a ‘mirror’ of Major carry with it vital consequences, of which were utilized by 19th-century composers much more than their Classical ancestors. This brings to light a poignant observation in terms of the philosophy of harmony as the Classical Masters thought about it in contrast to their Romantic progeny of the next century did.

One caveat is available for discussion. The composers that predate the Baroque period thought of Major and Minor, not as happy and sad, but as structurally more and less stable. Hence, again the qualifications of Major and Minor. Indeed as can be viewed as early as the works of Bach, when a composer wished to destabilize a key and give ‘looser’ connections between harmonies, the minor mode softened the expectation of chord stability in such a way as to have more harmonic freedom. I propose that this is still a reason most musicians say they prefer Minor over Major. It is cognitively less about the actual sound and more about it’s the ability to encompass so many convincing harmonic progressions without sounding harsh.

Moving along, in the classical era Bach, Mozart, and Haydn viewed Major and Minor as separate fields. To help facilitate the minor key as vitally important on its own as a tonal area, they followed a model of composers that made a chromatic alteration, so that it would further resemble an equivalent quality of its Major counterpart. This practice largely suppressed the diminished triad of II and created a new leading tone to the tonic, so that a new diminished chord was created on VII, which as discussed prior, in its natural state is a Major chord. By doing so they also affected the dominant chord to transform it to Major. Hence:

 

Figure 13.

Which can be heard here:

The raised G# turns the progression from Aeolian into the more familiar A minor that we know today. It is effectively darker because of the intensity that G# creates in its desire to resolve to the tonic. This alteration was made to ‘correct’ the harmony of a minor key.

In order to create a smoother melodic line, a second chromatic alteration was made. the sixth scale degree was raise as to not create the awkward augmented second interval. The problem proposed was that, while the iconography on the page looked like a diatonic step, aurally it is the equivalent of a minor third. It was, therefore,  problematic for instruments that were not fixed in their pitch, such as the voice and strings to execute with any assured frequency. Musically speaking, the augmented second in performance is reminiscent of picking up an empty soda can when you think it to be full. It throws you off and makes you reevaluate for a second.

 

Figure 14.

 

The romantic composers lead by Beethoven, by contrast, exhibited a philosophy that saw Major and Minor as a codified system. A Yin and Yang, Dark and Light, Spiritual and Physical. It all belonged to the same field, and they were more easily ready to move to the minor or major variants of chord degrees without preparation. Not only that, but Romantic composers obscured functions by juxtaposing relationships in such a way as to confuse the tonic, the predominant and the dominant.

It is from this latter philosophy that modern pop musicians originate, albeit often at an intuitive level. It is also this dualism that is at play in the Radiohead example discussed in this article.

Therefore, we should summarize to this point what is happening with the progression of Karma Police.

We determined that it is both in G major and A minor interacting at the same time. This is true. It is also in A Dorian. That is why, as we observed with the modal examples, the A minor starts out sounding like a tonic, but the function is changed when it is arrived at again, proceeding after the G major.

Again:

 

We need to now quickly discuss the relationship between A minor and G major. We compared it to the Beethoven Opus 31. 3, where Beethoven starts out on supertonic II harmony. But we determined that where Beethoven unequivocally makes clear the functions, the Radiohead song does not, by merit that the function of the A minor(tonic at the beginning) changes functions to predominate at its next arrival, whereas in the Beethoven, the opening chord(F minor 7 in first inversion) doesn’t sound anything like a tonic at all. Therefore the Beethoven example never trades functions.

Furthermore, when a Major key relates to a Minor it does so in two ways. Either by emphasizing its submediant effectively changing the mode from Ionian to Aeolien of the same scale, or by chromatically altering the key, where it keeps the same tonic, but projects a minor shift. The difference in sound is demonstrated below:

Relative Minor(different mode, same scale):

 

Figure 15.

Parallel Minor(chromatic alteration, different scale, same tonic):

Figure 16.

 

Note how the first example, the relative minor smoothly melts into A minor in comparison to the harsh aberration of the transformation of the tonic from Major to Minor.  And the 6th chord, we expect to hear C major, but instead the minor chord sounds and adds an abrupt drama to the cadence. Notice also from the two examples: In relative minors(first example) functions change while scales do not. In Parallel minors, as harsh as the contrast is, functions remain the same while scales change from C major to C minor.

However, the Radiohead example uses A minor and G major, neither of which are related to these previous two observations. We have already concluded that these two are not related in a predominant and tonic relationship as is the case with Beethoven’s Opus 31 no. 3. At least not outright.  We can conclude that the keys are not related by relative or parallel minor as well, expressed in the examples above.

Here is what an example of A minor modulating to G major might sound like:

Figure 17.

 

It does not sound like a terribly large tonal distance has been spanned. That is because, A minor is the relative minor of C major, and C major relates to G major by fifth, according to the circle of fifths. Furthermore, as discussed with the melodic minor: F, the 6th scale degree of A minor is chromatically altered to become F#. The F# CAN be shared then, by both A minor and G major when chromatically altered in A minor. But then that leaves us with the curious G-G# relationship, which is a huge tonal leap and sounds incredibly foreign to the key.

 

One last point about modes before we are able to clarify the progression in question. We said that a mode is a cyclic permutation of a scale. Below we see C major and D Dorian. They are related by being the same scale. They differ in that Dorian simply starts on the second note of the scale and ends on the same note an octave higher, hence the sense of tonic is shifted. Notice that even though the scale is C major, by starting on the second step, D, the characteristic flavor is drastically altered.

Figure 18.

If you are having trouble hearing Dorian as an autonomous entity, this example compares C major against C Dorian, where the character of Dorian is emphatically highlighted in the same way that Parallel minor and majors are. This parallel connection with Dorian is not required for further analysis in the article. This is simply to capture the sound of Dorian.

Figure 19.

This connection between C and D is important at a deep structural level for Karma Police. It expresses a similar relationship between A and G.

Figure 20.

 

Another relationship is analogous to this. The E minor, as we remember is flanked by F# as seen in Figure 3.3 The relationship E to F# is the same thematic interval as described above.

 

Figure 21.

You can see that each major second fragment, identified by the barline splits, is identical intervalically to each other. In set theory, these three dyads are all considered equivalent by transposition.

Furthermore, the scale that you see in this example underlies Karma police, as we will see in the chorus.

So we know that the progression plays with dualism: A minor and G major occurring simultaneously. The entire progression can be heard as A Dorian with one exception: The second appearance of A minor changes function from tonic to an intermediate harmony. As seen from the example above the emphasis of tonal areas are also expressed in a ‘modal’ shift from one harmony to another. In this last analytical interpretation, whereas I had changed the function of E minor from a deceptive goal at Figure 3. to an intervening chord subsumed into dominant harmony at Figure 4., now I assume both based on the theme of dualistic functions applied at the same time. This is in contrast to reassigned functions at two temporally unique points as in the case of the change of function between the tonic A minor and the second appearance A minor.

To sum it up, the Dualism can be expressed as:

  1. G major vs. A minor vying for dominance while weaved through Dorian
  2. Functional changes to A minor
  3. A thematic modal shift between one tone and the tone a major second above.
  4. simultaneous and differing functions with the same chord.

These postulates will be referred to as D1., D2., D3., and D4. for ease of communication in future reference

 

We can now discuss Karma Police’s secret battery. What makes the whole thing work:

It exists in the F# and the transformations which it will adopt. It is at first, D major with the F# in the bass. While this might be controversial, and many including myself would normally disagree, for the purposes of the article I reappropriate the term “pitch class” to encompass any note designated by a letter, rather than a particular frequency. Hence, Pitch Class F can express F# or Fb. Pitch class G, G# and Gb. This is for ease of understanding the dualism, and assigning both F,F# and Fb equal status as will hopefully be made clear shortly.

TAKE NOTE GUITARISTS: If you’re playing the D major in root position, you commit violence to the symmetry of the voice-leading and destroy an integral, if not the integral artifact of the entire piece.

As mentioned in Figure 4. The F# both descends to E minor invoking D3. and ascends to F# where it functions as the leading tone of G major. Note that D3. is a separate function than the leading tone just described.  Also keep in mind that the distance between E and F# is a major second, while F# to G is a minor second. This is important because Major and Minor second relations from the same tone is another dualist theme.

Let’s study the second part of the phrase:

Figure 22.

 

In this example, we see that where D/F# first appeared in the first half of the phrase, it has been replaced by F major. That is to say, F# has moved down by a minor second to create an entirely new harmony and a rather contrasting harmony at that. None the less, here, it’s function remains the same.

 

Figure 23.

But the transformation from an F# root chord to an F major chord is beyond striking. Whereas, the F# moved to G major in Figure 3. and made it sound like a goal, the F major acts as an upper leading tone that leads the opposite way, to E minor and now E minor sounds like the goal.  Whereas E and F# were in relation by a major second descending, thus making E minor weaker compared to G major, F major ascends to G major by major second, making G major considerably weaker than E minor. Although as we can see, the chord that directly precedes the last G major is again, F#,  it’s function has been demoted from a dominant to simply a passing chord. That is, a chord that fills in a space. You can easily hear the function stripped from it. So strong is the upper leading tone(F), that the lower leading tone(F#) cannot override its consequences. In effect, the E minor, where at its first appearance was a deceptive goal, is now inarguably the goal:

Figure 24.

 

As a goal it does not sound like a tonic, but it does have a strong secondary goal orientation. Said another way, when we listen to the example, we hear that it is indeed emphasized, but that it can not end the phrase; we expect more music to come. In this way, it has become a Dominant. That is because it is the dominant of A minor(albeit a minor dominant, which is congruent with a A Dorian interpretation). As illustrated in Figure 5. the two underlying progressions that are interweaving the harmonic texture are vying for prominence as the tonal key. Although we can see it theoretically in Figure 5., here we actually hear it. The F# and F, remember, we said do not change function. But how can that be if F# is the dominant of G, and F major acts as the dominant of the dominant of A minor?

The pitch class ‘F’ in this piece acts as a switch. When chords are modified that contain F in them, harmonic function changes, tonal goals shift, whether that is F, or F#(traditionally F and F# would be considered separate pitch classes, but for the remainder of the article, it is assumed that any note that is designated as F, F#, or Fb). In this piece that is F major containing F, D major containing F#, F# major containing F#, B minor containing F#. When the switch is flipped one way, G major is active, when flipped the other way, A minor becomes active. In this sense, while the chords containing F do interact with the surface progression, they can be considered ‘outside’ the progressions. Like supernatural forces divinely intervening in the physical realm.

We can now focus on the last section of the verse in isolation:

Figure 25.

 

 

I have included the first measure of the next phrase, which begins the repeat from the beginning of the song, to show how A minor has regained its tonic status.

 

At the beginning of this phrase, we hear A minor again. This time again, sounding like the tonic. This is because, as discussed in the previous example, E minor gained dominant function due to F major’s influence on it. The switch flipped, bringing A minor back as tonic. So it would only make sense that if E was the goal dominant, that the tonic of that dominant should follow shortly, as here it does. However, simultaneously it is a predominant harmony of G major. The next three chords are exactly the same functions as the Beethoven Opus 31. no 3. At the onset of the G major in first inversion,  with B in the bass, justifying the progression to be in G major is troublesome. A I6 moving to a II, to a III to a V is not a logical progression by classical or functional standards. One might argue that it doesn’t need to be functional because it sounds good. But that analysis, which is often adopted by a majority of musicians does nothing to satisfy the curiosity as to why it sounds good. The reason for it working so convincingly is because up to this point, the first two phrases have worked to destabilize our sense of anyone unified key. It is the expression of two simultaneous keys that has brought coherence to the progressions up to this point. On it’s own, I6, a tonic function, moves to II, a predominant function moves to III, a tonic substitute, to V, a dominant. Once a predominant is introduced it must move to dominant to be considered predominant. If it moves back to tonic, as it does here, the supertonic II has to be considered a contrapuntal expansion of tonic. That is, the II fills in a gap between I6 and III. The reason that I and III can both be considered a tonic function is because I, here, GBD, shares two common tones with III, here BDF#. Further evidence that supports this is that the I is in first inversion so that B is in the bass, which is the same for III; B is in the bass. If we look at the voice leading, the G leads to F#, which can be considered a suspension. Listen here to get the sense of I and III creating a tonic expansion:

Therefore I6, II, III is summed up into I6. The next chord in the progression is V. I and V could be argued to be somewhat logical, but if a predominant does not intervene, and then move to the dominant, then I and V are considered to express a prolongation function. There are many reasons this analysis is faulty, but the most convincing reason is that the Dominant, D major leads to A minor. The progression I, V, II, is a backwards progression.

Illustrated another way, its like taking a sentence, “The cat drank his milk.” and arranging in such a way that it read “drank cat the milk his.”

This is one major reason why musicians who are not versed in theory struggle to ‘fit on’ the next chord in a progression. If they’re lucky, a series of trial and error will give them another suitable chord in the sequence that sounds good. But more often than not, writing music in this way, by adding on chords as you go, tends to force the progression into the most basic chord functions. A great irony exists in this: Often a musician who states that Theory will force him/her to write like everyone else ends up writing like everyone else because they are like a person trying to draw a picture in the dark. Only the most basic illustrations will come off as convincing because one has to rely on what is familiar shapes. Any illustration of complexity requires the ability for the sight of perspective, proportion, and detail. Each time the pencil is forced to leave the paper to start from another angle, it is anyone’s guess what the end result will be. In the same way, the more complex the relationships of the harmonies are, the more difficult it is to plan its long-term coherence. That is, to be able to deviate into strange harmonic areas(like the F major of Figure 23.) and be able to convincingly rotate back to the original functions, (like the A minor at the start of Figure 25.), rather than spindling off further and further into tonal areas, that tend to lose all sense of coherence.

 

There is, however, one interesting aspect of the last analysis: I, V, II. This is a backward progression. These chords are related by ascending fifths. And this is yet another prominent feature of the dualism of the piece. This feature will be discussed shortly.

 

Getting back to the progression at hand. You will see in the example that A minor is simultaneously expressed as functioning at the same time as G major. While it is undoubtedly clear that the opening three chords, A minor, D major, G major represent a fully functional predominant, dominant function, the A minor suggests I, IV, (VII), III, VII, I, II, IV, I. You will notice that that the first VII is in parenthesis and that is simply because at that moment, it is not functional to A minor. We discussed already that it is clear that here, G major is switched on. But I to IV expresses the beautiful progression that gives Dorian it’s the quintessential flavor. Listen again to the flute and piano example at And we already established that the progression at Figure 12. This is the same progression, except in A minor. We stated that IV can stand in for predominant and VII often stands in for the dominant in a Dorian progression and that III can substitute I. Here, what we have is I, IV, VII, III. That is: tonic, predominant, dominant, tonic.

 

A small interjection: We see also in the figure, C major with the chords I, V6, VI. This is simply to highlight the relation expressed before about the backward progression, and what we will be discussing next. Furthermore, we talked about how Minor tends to modulate to III, it’s relative Major. C major is the relative major of A minor, and here, one can hear a very familiar gesture in tonal music. C also relates to G by fifth, backward. That is the opposite of the circle of fifths. This again, is a dualist procedure that will be shown clearly next.

As the progression moves on, we have I, II, IV, I. Tonic, predominant, predominant, tonic?

We just mentioned that the backward progression of ascending fifths, the opposite of the circle of fifths , was a feature in play here.

The way that tonal music works is that it revolves strongly around the concept of a tonic relating to its dominant. In the key of C, C is the tonic and G is the dominant. In the key of G, G is the tonic and D is the dominant. In the key of C, C is the tonic, D is the supertonic, and G is the dominant. That is, I, II, V. That is expressed functionally, tonic, predominant, dominant. But if ordered in the way where C is tonic, G is the dominant, and D is the dominant of the dominant, we get the backward progression C G D. The reason it is backward is that if a dominant precedes the tonic, tonality has been established to the tonic. But if we ascend to the dominant, the sense is that we have jumped out of the key into a new key. Listen to these two progressions:

Figure 26.

You will no doubt hear how ‘closed in’ the first progression sounds. It sounds like it is exploring around the room before coming back to its spot. It stays within the family as it were. By comparison the second progression, the ‘backward’ progression seems to be as if it were walking through a series of doors, lost until the last chord. The harmonic progression wanders, whereas the first harmonic progression is focused toward its goal. However, the end chord of the second progression does sound goal oriented, but in a strangely different way. The first progressions movement is called “authentic” in classical theory. The second, “Plagal.” A plagal cadence is a beautiful alternative to the authentic cadence. Essentially the subdominant IV chord changes functions to dominant. That in itself is a dualist procedure. The plagal cadence furthermore requires more careful attention, because as a predominant, it has the ability to tend to sound like a predominant and not a dominant. Furthermore, in the key of C major, F major, the subdominant, is the predominant. However, in F Major, C major is the dominant. Depending on where these chords fall, one can easily make the mistake of turning a tonic, your goal, into the dominant.

In a similar point I made earlier, if someone with no interest in theory, leading chords one after the other in an additive fashion without any sense of a projected goal, that person could involuntarily end up with a chord that theoretically should seem to work, but that practically sounds ‘wrong’ without much clue as to why the discrepancy,  inevitably  abandoning an otherwise perfect opportunity for a beautifully expressed progression. Spotting your opportunities and having the math preemptively worked out, so to speak, is what songwriting(vs. experimenting) should be about. A contractor does not start building a house brick by brick, seeing how it ‘just goes’, without much trial and error, the expense of time, and more than likely settling on a house that is ‘useable’ rather than what his original vision was. By contrast, if after the contractor had a vision of how he would like the house to be, the intricate and stunning features of the house, he then sat down and thought ‘how would I do that?’ He would probably not scoff at theoretical understandings as a means of ‘stunting his creativity.’ His creativity envisioned a second story double-panel flooring of glass, with fish and water contained within that cast shadows as they swim above heads of his guests while during a dinner evening. Yet, it was still substantial ‘theory’ that brought his creativity to life.

However, when dealing with the physical world, such an observation is almost comedic. It is all too apparent when a structure fails in the real world(the ceiling leaks, the fish die, the walls cave in). By contrast, the elusively abstract nature of music is not readily apparent. None the less, it is remarkable how even the untrained ear is perilous in its search for coherence. It has an uncanny ability to rightfully perceive those harmonic mistakes that are ‘corrected’ and so, retrospectively accepted as interesting and those harmonic mistakes which repulse the ear. This is not to say dissonant loud music as opposed to beautiful pleasant music. It has to do with style and logic. If in the middle of a Jay Z album one song ended and the next began and it was a Dixie Chicks song(Let’s just say Wide Open Spaces), the effect is awkward. If it then began to deviate from the original, and/or was superimposed with characteristics associated with Hip-hop(metric polyphony, rhythmic complexity, repetitious fragmentation) and Jay Z began to verse over the rearrangement of textures, the effect would be ingenious. If, however, It was just Wide Open Spaces, with Jay Z simply covering the song, the effect is most likely repulsive.

Knowing the science behind the magic trick is what makes the magic trick convincing; getting your audience to say “what just happened?” is the reward of a successful ‘controlling of harmonic tensions.’ It signifies craft, rather than luck(of the latter, luck, is more aptly referred to as inspiration, and waiting for inspiration to strike, or luck to manifest, is not a feasible way to conduct one’s productive career.) Stated a slightly different way; One assumes that experimentation.

 

To illustrate this susceptibility of confusing the tonic by virtue of Plagal movement sounding accidentally authentic, let us listen to these very similar progressions alone:

Figure 27.

 

In example:

1. (:00 seconds)We hear a properly executed ‘chord progression’ as defined by classical theory. A tonic leads to a predominant and then a Dominant that is held for two chords(the one labeled V64-7) leads back to the Tonic. There is absolutely no ambiguity to the progression.

2. (:10 seconds) Whereas example 1. sustained a dominant chord over two chords, example 2, by contrast, moves the dominant chord back to tonic, and then to F major. This has two effects. First, the F major does not sound predominant but sounds like a goal that has been arrived at. Secondly, because the Dominant was terminated earlier than the previous example, the ear perceives that the F is ‘added on’ to the progression and left hanging. That is to say, even though both examples share 5 chords spanning over 3 measures with each ending on the downbeat of the third measure, the second example sounds like it has gone past that downbeat. It sounds like another chord should come after it, which, would realistically then place the two phrases at asymmetrical juxtaposition. This is an auditory illusion and a fascinating one for the discourse of composition. That is to say, that 4+5 can=8 to the ear, based on where the harmony is placed. Never the less, the F major does not sound convincing, either as a goal or as a chord that leads to a goal, simply on grounds that it’s the final chord and has nowhere to go. One wants to ‘write more out”, which again, as we spoke about earlier exhibits a type of writing that has not taken its goal into account. One might inquire as to why such a goal needs to be taken into account. If we bring it into a simple real-composition situation, let’s say that the first chord progression of example 1 highlight’s ‘twinkle twinkle little star.’ The second ‘how I wonder what you are.” By having to add more chords to sound like an ‘end’ what do you with the lyrics? “How I wonder what you are what you you are.”

Figure 28.

 

Granted, it worked for Taylor Swift, but look at what it made her do, OOH, look at what it just made her do.

As the Taylor Swift example demonstrates at 1:05 into the song, like all things in theory, deviations can be remarkably refreshing if the deviation is under strict control in order emphasize itself against the normative rather than the ex nihilo, arbitrary deviation that sounds like a mistake.

3. (:20 seconds) Example 3 fixes example 2 by bringing in the B flat from F major and applying it to the tonic of C.  This removes the obscurity from example 2, where one cannot decide if the tonic is the dominant of the predominant of if the tonic comes early and the predominant hands on to the end of the progression arbitrarily. That is to say that in the same way that Tonic C is related to Dominant F, Predominant F relates to Tonic C. Another way of saying this is, C is 5 diatonic steps below G. F is also 5 diatonic steps below C. This is the reason this procedure of the plagal cadence requires special attention. Here, however, F does most certainly sound like the goal.

4. (:30 seconds) This is the correct plagal movement. Unlike any of the previous examples, F major has been able to switch its function from Predominant to a convincing dominant. The C that comes at the end sounds like the tonic, not the dominant of the predominant, and the sense of the downbeat has not been obscured.

 

One other feature of the plagal movement is the ability to give a simultaneous effect of both ‘cutting off’ the progression early, and also convincingly giving it an end. Listen to this example for comparison. The second version is exactly the same to the fourth audio example above. This invokes D2., the postulate concerning simultaneous functions.

Figure 29.

Both progressions are equally convincing and well balanced. However, the second progression sounds like it might move on, but when it ends on the tonic, we retrospectively assign a different function to the F major from predominant to dominant, effectively minimizing the temporal span to a goal. You can imagine how this fusing of predominant and dominant functions allows for even further deviations to come back into a normative whole by giving the sense of an expansion that is then cut short through a plagal cadence. It is like having 8 as the goal in mind. Phrase 1 is 2+2=4. Phrase 2 is 3+1= 4. We are building an expectation of a phrase being extended which is then cut short by the plagal cadences ability to ‘make it quick.’

You can hear the first phrase ending around :06 seconds, setting up the expectation for a rhythmically symmetrical phrase. around 0:10-0:11 seconds we get a sense that we are going to need to extend the progression, and thus affect the symmetry of both lines if we are to get back to the tonic, but the proceeding chord cancels out all expectation to extend by using a plagal motion into the tonic. Again, the effect is abrupt, but the goal has been convincingly established.

 

What does all this mean for Karma Police? Going back to Figure 25. we stated that the progression was clearly a II V I progression in G, but it simultaneously was operating at a dualistic level with A minor.  As you can see from the Figure, A minor is confirmed by its plagal cadence, D major to A minor. D major is the last chord before the entire verse is repeated with the A minor coming on the downbeat. However, while D major is the IV of A minor, here, acting as a dominant, it is also the V of G major, the actual dominant. Which means it has a certain binaural openness. It can relate to either A minor or G major. Another dualist feature of the piece.

Figure 30.

 

In the above example, the only alteration that has been made is swapping out the A minor chord at the beginning of the restatement of the verse, with a G major chord. This passage, as was discussed in Figure 25., can be both G major or A minor. In the unfolding of the progression, we sense that it starts in G major, but ends in A minor. The actual song moves back to A minor. Here, it moves instead to G major, making the D major the Dominant of G. This is the ONLY chord that has been altered. However, the harmonically androgynous progression sets up an expectation that can convincingly switch either way. Again, this switch is initiated from a chord that contains pitch class F. In this example, F#(in the D major) has flipped the switch to G major, but The F# convincingly switches the progression to A minor as well, which is how Radiohead utilizes it, in the form of a plagal movement. Listen to the above example again and pay attention to the chords that come after the G major(0:14 seconds). The chords are identical to those found in Figure 1. Yet, how drastically different they sound when the G major, rather than the A minor proceeds them. In Figure 30. all of the functions proceeding the G major have changed considerably. It is as if we get a glimpse of ‘what could have been’ by using G major. And yet, it is so convincing, that one must acknowledge the equal validity that G major possesses over the progression. We can remind ourselves of the fact that G major and A minor are a major second apart and share no common tones. Again, this invokes D2., referring to simultaenous functions at a larger structural level.

 

We stated that a III could substitute a I, as is the case with the G major and B minor in Figure 25. where both chords contain two common tones: B and D. Also, VI can substitute I, G major and E minor, as in the interpretation in Figure 4. and 5. where we interpreted it like the Beethoven Opus 31. no 3. example. In A minor, C major can substitute as III as in Figures 22.-24., or F major, as VI, as in Figure 25. Where we suggested that C major might be implied by its familiar progression from classical theory. Again, this is because A minor and F major share A and C, whereas A minor and C major share C and E as common tones. This relation applies to all chords that are in relation where roots are apart by a third. This is indeed a common and overdetermined technique to prolong a particular harmony.

But that is clearly not the case between two chords that are a second apart. If we listen back to Figure 17. we are reminded that the modulation between both chords is quite pronounced. The only reason why these two progressions can work without disrupting the structure of the entire verse progression of Karma Police is because of its underlying simultaneous progressions that are ‘switched’ by pitch class F.

Again, lets remember that the progression is contained within an A Dorian mode and that the relationship of Dorian is what binds G major to A minor. That is to say, a shift upwards by a second takes G major into A Dorian territory. This is essentially happening on two different levels. Like the cogs of a clock that contain clogs within them. The entire progression can be thought of as being in Dorian, while simultaneously being considered opposing keys, G major and A minor connected by a Dorian relation. This last feature isn’t true about all, nor the majority of Dorian progressions. It is inherently unique to this song. This invokes D1 and D3.

We can now consider one more observation as to why this dualism is so effective. In a major scale the seventh scale degree lies a half-step below the tonic. We discussed this in relation to Figure 3.

In G major, that leading tone is F#, and has the ability to confirm G as the tonic.

However, in minor modes, the half-step leading tone exists between the fifth and sixth scale degrees, and it leads downward to the fifth, because the fifth is part of the tonic triad. This in itself is a dualist observation.  But more importantly, in A minor, the sixth scale degree is F natural.

Figure 31.

This relationship is partially responsible for the ‘switch.’ and even more important for why the functions switch between these two key areas. Again, if the progression was simply ‘dorian’ then the sense of functions would not switch, but the juxtaposition of F# belonging to a melodic minor, and F natural belonging to the natural minor, makes the interpretation of the progression simply being in Dorian problematic.

 

We can now move on to the Chorus, which encompasses all of these techniques into one of the most unorthodoxicaly beautiful chord progressions to ever be presented in Popular music.

The verse is exactly the same leading up to the chorus, but this time, instead of D major moving to A minor, or G major, it moves to C major. C major is both the predominant IV of G major and III substitution of A minor. It is also the parallel major of A minor which means it can stand on its own as a key.

In the key of G, the progression is IV, V, I. An entirely plausible chord progression. In A minor, it is III, IV, VII. The tonic does not even appear, therefore, we can assume that this is in the key of G. Everything is smooth until the introduction of an F# major chord. Here, for the first time we have both A# and C# present. This chord does not exist in either of the harmonic keys G major or A minor. It belongs somewhere completely different.

Going back, we remember that we talked about a feature in Figure 3. which was expanded upon in Figure 23. involving the dualism of F# and F. In the first phrase of Karma police, F# stood a major second above E minor and a minor second below G major(because F# is the leading tone to G major). In the second phrase F natural stood a minor second above E minor, and a major second below G major, and now we can say that it is because the F natural is the upper leading tone to E minor confirmed by the discussion at Figure 31. 

In the same way that we said the G major relates to A minor by an ascending major second, invoking D4, what would happen if we reversed this relationship to a Major chord(contrasting against A minor) by a descending minor 2nd(contrasting against an ascending major second)?

The F# major chord just mentioned. As should be obvious without a doubt at this point, the F#major belongs to that pitch class which we said stands outside of the progression that ‘triggers’ it into a new function.

 

And here… It is this pivotal moment in the piece that all the other underlying features that we have spoken about up to this point are briefly exposed for a second. This chord, unlike any of the others up hitherto, cannot be explained as being part of A minor, G major, or A Dorian. Radiohead draws attention to the man behind the curtain, as it were. The great and powerful Oz in all its billowing gloom.

 

THIS IS WHAT YOU GET WHEN YOU MESS WITH DIATONIC FUNCTIONS.

 

Here I suppose you can say that the underlying structure of a major movement upwards, G major to A minor, which are the first two chords in the key of G major, have been reversed. This would create a Phrygian connection. I will not go into an example of how that sounds because it is both F# major and G major.

But, another reversal has happened. F# is supposed to lead to G. But in this case, G is leading to F#. So, to make harmonic sense, we should probably invoke their enharmonic equivalents, that is, two notes that sound the same but are different in name. These are the ‘black keys’ on the piano. Between D and E is a black key. It can be D# but it is also E flat. If you were to read F# Eb F# Eb you would play the same note on a piano 4 times.

In the case here then, F# becomes G flat. G becomes A double flat. Upholding this last interpretation, We have another dualist feature, where, some pitch class G that normally leads to some pitch class A by ascending second(D3), become inverted to lead both downward and by minor 2nd. While this is a fascinating feature of the music that is of considerable note(If not paramount), for the context of this analysis it will become needlessly difficult to continue to analyze in these enharmonic equivalents. Especially because now the Gb/F# needs to ‘loop’ back to C major, D major, and G, all of which prefer an F# major. It will suffice to realize that underneath these chords is the theme ‘pitch class G to A via F.” And here again, for the first time F# major, rather than D major of F major has forced G and A into a ‘new’ identity, namely G flat and A double flat. Also, for the first time, it’s reversed the direction of these two pitch classes.

Let’s just listen to get a sense of what we are dealing with in this progression:

 

Figure 32.

 

The progression is both stable, goal oriented to the G, and then, with the inclusion of F# is made endless, in a loop. This type of progression, by and large, represents an overwhelming majority of pop ‘progressions.’ It is this type of writing that is often misrepresented by musicians who claim that pop music ‘simply uses the same 4 chords.’ It is stated as a negative attribute. I argue that it is, instead, a fascinating quality that popular music possesses. One that is not often found in classical music. Two examples in the classical literature that do make use of such a constant ‘looping’ progression for the entire movement is Ravel’s Bolero and Beethoven’s 7th symphony, II.

In both the Ravel and Beethoven, textures, orchestration, and countermelodies develop, but the underlying harmony stays the same. Of course, in both examples, There is a harmonically contrasting section that helps alleviate the repetition, as is often the case in popular music as well, in such cases where the verses often utilize different chords than the chorus. However, it is just as likely that pop music will use the same 4 chords throughout the entire song. Other features, such as the chorus melody, are considerably different and therefore alter the perception of those harmonies not unfamiliar to techniques we have discussed up to now.

The features of such a ‘looping progression’ are poignantly displayed in this chorus by Radiohead. C major starts out the progression, but it was preceded by D major. It also proceeds to D major. This makes an interpretation of C major highly problematic. In Figure 3. we discussed the probability of A minor acting as II of G major resolving to the deceptive cadence E minor through a D major in first inversion, where it then lead back to G, creating a good case for G major which was only disrupted by the fact that the next phrase insisted on an A minor interpretation, thus keeping it in a state of ambiguity. Furthermore, we concluded that these progressions were encased solemnly in A Dorian. None the less, it is a gesture that repeats itself at the third phrase seen in Figure 25. where this time, the same progression was undoubtedly referring to a II, V, I in G major. That is to say, the same progression featured in the first phrase ‘fails’ to become a cadential gesture due to the way the chords function after it, whereas in the third phrase they are confirmed to be cadential leaning toward G major because of the material coming after it. The D major at the end of the third phrase is reinterpreted as a plagal cadence retrospectively due to the sudden shift to A minor at the start of the next verse.

 

We also discussed that A minor and C major can substitute for each other in the same way that G major and E minor can, on the principle that they share 2 notes in common, while the third note is different only by one step.

 

In Figure 32. , C major stands in place of A minor, which changes the quality from minor to Major but can support the same function as A minor. Hence, the chorus achieves a smooth transition to a different harmonic area. Whereas each verse landed on A minor up to this point, now it moves to C major. No other chord, not even A major(due to it’s C#) can both contrast and affirm similarities. Therefore, this chord progression is equivalent to the same progression found in both the first and third phrases of the song discussed at Figure 3. and Figure 25. This is wholly supported by classical theory where both II, V, I and IV, V, I are considered interchangeable without obscuring predominant function.

But, in staying with the theme of this analysis, a dualistic operation can be perceived by interpreting this progression in another way. The D major that precedes the C does not make syntactical sense. It assumes that the Dominant initiated a cadential function instead of a tonic. And here, in the chorus, much instrumental texture has been stripped away. The drums cease, and the dynamic level suddenly diminishes rather than increases as is common for most chorus sections. This places emphatic stress on the chord progression.

We talked about how difficult it can be to express function with chords that lie a fifth away from each other(thus being a considerable reason why plagal cadences work.) This was discussed at Figure 27. 

We also talked about the modal encasing of such progressions being able to give a more cohesive sense of progression and goal, while the key areas were then ‘free to mix’ from G major and A minor.

Here, I propose that we look at this chord progression is being in C Lydian. In the Lydian mode, the characteristic scale degree is 4. Thus, it affects all chords built on II, IV, and VII. If this is the case then the D major that precedes the C major at the onset of the chorus acts as a II. To be sure, the B minor that precedes the D major that precedes the C major at the chorus acts as VII of C Lydian. Both VII and II flank C major on either side, which gives it a purely modal characteristic and, also establishes it as a convincing key area. Indeed, C major sounds strangely stable as well.

Figure 33.

Isolated in Figure 33. we can hear how convincingly the B minor and D major emphasize and establish C major as a goal. We can justify this even further by virtue that D major has always lead back to A minor(a substitute of C major) and at the end of 2 verses, B minor -> D major has lead back to A minor, where the D major acted as a plagal confirmation. Furthermore, we start to see D4. taking effect in that, functions are needing to be simultaneously bifunctional for one progression to make sense in relation to another. The B minor at Figure 25. stands in for a G major tonality(by substitution) while it also acts as II, IV, I(in contrast to II, V, I) in A minor with a plagal cadence described also at Figure 25. And now, it is acting as the VII, II, I in a C Lydian progression. It is this, and for only this reason, that C major can sound like a convincing goal.

The D major that proceeds the C and moves to G major, does again make us reinterpret the functions. But the F#(Gb) major chord disrupts the sense of G major.

First, it is another pitch class F chord, this time F# major, that flicks a ‘switch.’ Secondly, This F# is placed at the end of each bar which is in the strongest metrical position to announce itself as a chord that leads, rather than a chord that is lead to. Lastly, the chord that it leads to is back to C major. This is all highly implicative that C is a goal. In our discussion of modes starting at Figure 7. we talked about the problematic diminished chord that causes whatever chord that is to the right of the diminished chord to sound as the goal, rather than the tonic of the mode. Briefly for recap, if III is the diminished chord(Mixolydian) then IV, will sound like I.

In the case of Lydian, the diminished chord falls on the IV chord. In C Lydian, as you may have guessed, that is F#. That would mean that V, the dominant, would sound like the tonic, a compounded problem when we consider the influence discussed at Figure 27. in regards to chords that lie a 5th away from each other. The diminished chord does not even need to lead to the tonic for the reinterpretation of dominant as tonic to be expressed. It is part of what is inherent to a chord with leading function; we know instinctively where it is supposed to go. For our example, it is compounded further by the fact that G major is treated cadentially(IV, V, I.) To understand what the F# major is, we have to briefly venture again into modal systems.

Each mode has a characteristic flavor that makes it distinctive from Major and Minor tonality, and at the same time, allows it to exist within those tonalities as well. Both Bach and Chopin, for instance, were highly sensitive to their cultivation of modal flirtations within a tonal context. In eras before them, modality was not contrasted against tonality yet, as tonality had not been theorized. Remember, the Ionian mode won out on account that the diminished chord was placed in such a position that all modes eventually gravitated to the Ionian as the ultimate mode. It was from this mode that ‘functions’ arose, and so to assume functions back on to modal systems is problematic. In more recent times, the modality has been utilized and thought about quite differently than the composers of the Renaissance, and that of the classical, and the extended era of the romantics. In all of these prior systems, whether cognizant in the case of Chopin and Bach, or intuitive in Palestrina and Josquin Des Prez, the modes have been under the influence of Major tonality, not minor(Again, because Major had an overwhelmingly gravitational force between tonic, predominant and dominant, whereas minor did not, hence the reason these two were not named Happy and Sad keys, as we think of them today.)

Composers of the early 20th century, starting most visibly in the works of Debussy, favored modes as alternatives to tonal systems; Tonalities in themselves, rather than subsections of common era tonality. It is this favoring of modal systems that is still highly prevalent in use today with pop musicians, and one major reason that classical theory is flawed in attempting to present a model for artists of today. How, then, does one ensure that the diminished chord does not negate the effect of the mode? It is a simple solution. They alter the structure of the chord so that where a diminished chord would appear turns instead,  into a consonant triad; Either major or minor. This for the first time opened up a practical usage for the Locrian mode. Where the tonic would otherwise be a diminished chord, if one note was altered it became a minor chord, and thus, would be able to function like the other modes. Another development came from the utilization of modes in this way.

In a post-tonal world, the concept of a tonic, or ‘a key’ has become embedded in us, such that if we used many random chords that did not seem to be connected in any logical sense, the perception by most is that of nonsense. Even in the Radiohead example while functions change and chords from other key areas are used, we still have a sense of a goal; either G major, A minor, C major, etc. Other chords are of less importance, but still important in their context to support and emphasize goals. This brings us to another observation about pop music in general.

If we can assume that Pop music is more easily driven by modality, than tonality, we can then begin to understand the dualist nature that operates at the surface level of much pop music.

If we start with C major, and then we move to D Dorian, it is still part of C major. Said another way, D Dorian is C major starting on the second scale degree of C major. If we start on E, we have E Phrygian which belongs under the C major tonality. F Lydian is the C major scale starting from the fourth scale degree of C major, etc.

The above example is much like the composers of old thought about it. Here, it is easy to flow in and out of modal inflections; Here a bit of Dorian, there a bit of Lydian, back to Ionian, all while maintaining the same ‘key.’

However, if we take C, and chromatically alter notes so that it becomes another mode then the sense of tonality stays with C, while the ‘modal inflection’ affects the entire key. C major and C minor vs C major and A minor is case and point. This was all discussed at Figures 15., 16. and 19.

As we chromatically alter these tones to create different modes on the same tonic, we introduce a series of new functions as well. Remember, because of the placement of the diminished chord in any mode, and because of the characteristic ‘cadences’ of each mode, functions change under different modes. IV-I, or, the plagal cadence, is derived from Dorian, where composers of the renaissance favored it over the minor IV, and V, of the natural minor(Aeolian.) It can also cadence convincingly at VII-I. Phyrgian had a characteristic cadence of II to I, where their distance of a minor second apart became characteristic of much middle-eastern, Spanish, and Russian folk music. Mixolydian had a cadence of VII to I, or minor V to I. Mixolydian is much the sound of robin hood, a merry variant of Dorian. Since all of this chromatic alteration is free to happen and still be contained within the same key, it only makes sense that the diminished chord has a higher responsibility to be altered to avoid tonal ambiguity. Indeed, this seems paramount in the case of the more formally androgynous system of modality that utilizes a central tonic.

 

It is for the reasons discussed above that again I yield to the perspective that traditional theory is useless in explaining what is intuitive in artists of the past 75 years, and therefore why theory as a whole has been rejected. The ‘rules’, even as observations, do not observe what is particularly fascinating for most artists who are busy creating from their intuition.

 

Coming back to the example, What is F# major, the dominant of B major, doing in the middle of the chorus, without any sense of modulation, for one metric beat, and then leads straight back to C? More pertinent, why does it work?

Because F# major is the altered IV of C Lydian, and, in keeping with Radiohead’s insistence of using Plagal cadences to simultaneously confuse the preceding tonality and lead convincingly to another tonality, #IV acts as a type of ‘modal-plagal’ cadence. C sounds like a goal, then does not when D major enters and leads to G, which then sounds like the goal, then does not, when F# enters and leads to C, which sounds like the goal, then does not when D major enters and leads to G, which then sounds like the goal, then does not… Again, this ‘switch’ of functions is caused by a chord with some Pitch class F. But instead of it being F natural and F#, it is two chords that contain F#. One leads to G major, the other to C major. This is what catches us in the dualistic ‘loop’ that is so provocatively on display for the entire chorus. It could indeed go on like this indefinitely and there is just enough contrast and resolve for it to be aesthetically pleasing. I think it would be rare for anyone to characterize this chorus as harmonically ‘dissonant’ and yet classical theory tells us that these two chords are quite dissonant, as the outline a diminished fifth. What else outlines a diminished fifth? The eponymous diminished chord. And why are diminished chords problematic? Because they are a chord that leads, not a chord that is lead to. In a normal classical cadence, that is, when we want to clearly establish a key, we present chords that highlight the three major functions. Tonic, Predominant, Dominant, Tonic. That is what a cadence does after all; it confirms the key by presenting the functions uninterrupted(this is why in our earlier analysis, the interruption of chords outside of cadential areas made the interpretation of a particular key over another problematic, and this is a feature that makes the music interesting. Again, when I say problematic, I don’t mean ‘confusing’ to our perceptions, but to simply say, It sounds great, tells us nothing as to why it sounds great.) Here then we can state that a cadence starts with the tonic and ends with the tonic. The penultimate chord that confirms it is the dominant chord.

By contrast, the Radiohead progression at the chorus uses two Major chords that are a diminished fifth away. If you were to flip around a diminished fifth so that it became an argument fourth, that is to say C-F# becomes F#- C, they are still the same distance away. 6 semitones apart. This is in contrast to C- E, where they are 4 semitones apart, and E- C where they are 8 semitones apart. They are inversionally invariant. They can’t be altered to change their sound.

Figure 34.

 

C and F# outline the diminished chord of G major, which has been one of the tonalities vying for tonic in the piece. This would assume that this progression acts as a harmonic progression equivalent to one of the ‘switch’ chords of the entire piece. But C major substituting A minor in this progression can be said to be part of the diminished fifth phenomenon happening here. In C major, It’s dominant, G major is present, and the F# major, acts, as we said, a Lydian Plagal cadence that sounds like it belongs to C more than it belongs to F# major. Even still, we said that F#major is really Gb major, while G major is A double flat major, which would mean that the chord of resolution in G major, turns into the pitch class for its opposing harmony, some pitch class A, which then moves down by minor second, instead of up by major second to Gb, so that in reality what is happening is an enharmonic dance; A gesture that starts out the progression as C, representing A minor, is reinterpreted by the next chord to belong to G major, which represents A double flat minor be re-interpretation, that begins to move down to Gb, which is the reinterpreted as F# major belonging to C major Lydian.

This last interpretation is problematic for several reasons, but it highlights the reason why the progression works as it does. We have a peripatetic progression that seems to be on an infinite loop, much like the symbol for infinity itself.

 

The two chords that move upward C and D point to the goal of G, but G and F#major point to the goal of C. C and D represent a Dorian shift upward(D3) while G and F# represent its inverse, a minor second, or Phrygian Shift downward. This play between the Major second and minor second was present when the D/F# of the first phrase of the verse became the F major of the second verse of the phrase as highlighted in Figure 23. In that progression, F# in the bass moved down to F natural. Here, G moves to F# in the same way. The difference between these two chord successions is that D/F# has a root a minor third away from F major, while G major and F# major are a minor second. If we look back at Figure 21. we see a rough outline of the Dorian subsections that were emphasized by some of these ‘switch’ functions of the chords. Each one created a major second fragment.

Figure 35.

 

Notice, that to the left of the figure, C and D are present. Notice to the right of E F# and G are present. If we build major chords on all of these notes, we would have the same progression as the chorus; But what tones are left out from Figure 21? The A and the E. Remember that we said that C major and F# major were like harmonic bookends to a G major tonality. But viewing the progression with this insight of Figure 21., it would seem that the progression really wants to highlight a chord built on E. And, all the chords of the progression are major chords, which might suggest that the chord it wants to rotate around is E major. Conspicuously, E major is the dominant of A minor, which has been missing from the entire piece, which would establish it as A minor. After all, no one would doubt the obvious overwhelming A minor flavor of the entire piece. The piece starts in A minor, all the verses pick up A minor again, and the chorus modulates to C major, which we already stated was the most probable of choices to modulate to when in a minor mode. Indeed, outside of this analysis you would be hard pressed for any musician familiar with the song to conclude that the piece is in G major, or anything other than A minor.

So, how can it be that a song in A minor can be missing the only other chord that can confirm it’s tonality? The dominant? First, let’s test our hypothesis.

Figure 36.

 

Assuming, as we said that E major is the dominant of A minor, I have lead the E major to A minor in the progression. It is convincing, and furthermore the E major terminates the loop and sends it toward a goal; A minor. But this E major never manfiests.

Taking a slightly more traditional theoretical view, The F# major can be explained another way, giving way, to yet again another dualistic feature. F# major is the dominant of B. Here, it could be explained as a secondary dominant to B minor. B minor, as we have discussed previously is a chord that contains the pitch class F trigger tone, in this case, F#. B minor also acts as a chord that exists in both A minor and G major. in A minor, the F# comes from an alteration to the diminished triad on II, turning it instead into a predominant harmony in that key. In the key of G major, it acts as substitute harmony to the tonic of G major. If we replace the first E major with a B minor that moves to an E major, it is a logical and syntactical progression in A minor. Here, the B minor also terminates the infinite loop progression and directs it to a goal: A minor.

Figure 37.

 

However, by altering the E dominant 7 to D dominant 7, the progression is equally convincing as G major as well.

Figure 38.

 

As it turns out, the progression indeed uses an altered version of this last progression to terminate the infinte loop. It does so, however, by emphasizing C major. The longer duration of C major at the end of the progression gives a sense that C major is a type of tonic, if even an agitated, restless tonic. But it is then immediately reinterpreted as unstable against the Bm – D that ultimately moves back to A minor for the onset of the subsequent verse progressions. This last moves reaffirms the plagal movement that has usurped G major as a tonality throughout the entire piece. Notice that we showed at Figure 33. that Bm to D flanked C major and acted as a type of cadential figure to C major. Here, B minor and D major proceed C major, where it reinterprets the C major is unstable. This is entirely analogous to our discussion that outlined the phenomenon of plagal movements.  Under Figure 25. I discussed backward progressions. We can view the B minor and D major leading into the progression, and the same two chords leading out of the progression as gateways into the infinite loop progression. One leads to C major while the other leads away from C major.

 

The final progression for observation is the outro/bridge which highlights the text “For a minute there, I lost myself.” The end of the chorus leads into this progression. The beginning of this progression is exact to the end of the last progression. B minor and D major is thus repeated, but instead of leading to C major, as it did in the chorus, or to A minor as it did in the verse, it leads instead to G major. The progression of B minor, D major to G major is what we postulated as a ‘missing’ progression in Figure 38. Thus, Radiohead is confirming at least one of the goal-oriented progressions absent from the chorus,  establishing G major as the tonality. The Chord progression then toggles between D major, a ‘switch chord’ and G major 2 times. A third interest of D however, moves to E major.

 

AND FOR A MINUTE THERE, I LOST THE DOMINANT…

 

Figure 39.

 

The E major chord seems to come suddenly, from the depths of nowhere. It is the only chord that contains a chromatic alteration where also Pitch Class F is not present. Or does it? But as was stated in Figure 37., it is the missing chord needed to confirm A minor as a key. Here it is given special emphasis by lasting twice as long as the harmonies that precede it. Furthermore, The E major is quite smooth in this context because it acts as the plagal dominant of B minor, and serves as both an obstructed goal(wanting to resolve to A minor) and being able to loop the progression through B minor to G major(as a plagal movement). The E major is quite striking, though not as much as the F# major, but nonetheless, these two chords share a dramatic and functional similarity. Indeed, these two chords, E major and F# major are related at a deeper level. By an ascending major second, invoking D3., hey are the missing middle section of the three aligned  ‘ascending-major-second-steps’ of D3.  This relates to that discussed in Figure 21. and Figure 35. 

 

But there is one last interpretation that throws us completely for a loop. At the onset of describing the Pitch Class “F” chords as ‘switches, I mentioned chords affected by F, F# and Fb. But there has been no chord expressing an Fb. However, an Fb major chord is an enharmonic E major chord. They sound the same but are spelled differently. And here, on grounds that pop music does not think in proper notational expression, this chord again assumes a dualistic function, and at the same time completes the pitch class aggregate of F, F# and Fb. Its unusual placement at the end and its emphasis is enough to warrant the classification. Assuming this interpretation, that would bind F flat major to F# major, in that the A# contained within F# major stands in relation analogous to C and F# major. Furthermore, any chord that contains an E natural, can be thought of as being under the influence again of Pitch Class F; That would represent the triads of C major, A minor, and E minor, renaming them as D double flat major, B# major and F flat major respectively. This is analogous to our interpretation of G flat and A double flat major. Hence Pitch class F stands as a dominant, if not the ultimate feature of the piece.

Thom Yorke’s deceptively important lyrics ‘For a minute there, I lost myself” at such a crucial moment of the piece stands as a poignant description to a higher level structure of the piece; Enharmonic spelling, tonality, and functionality all at once. Perhaps a stroke of intuitive brilliance.

Lastly, the E major chord is technically the last chord of the end of the functional boundary of this progression and is in an optimal place to lead back to the A minor verse. As a bridge that proceeds the chorus, it should be able to lead with utmost conviction back to a verse. This is then, how it would progress back to the verse:

However, the bridge is also acting as an outro progression to lead the song to a close. This simultaneous function invokes D4, except it affects an entire formal unit rather than just simply a chord. If the E major did indeed move back to another verse section, it would be easy to confirm that the piece is in A minor. However, that analysis is destroyed by the fact that the E major is acting as a plagal movement to B minor, a chord that we said belongs both to G major as a substitute tonic, and A minor as a predominant II, and the more obvious point that the E major must proceed to A minor in order for the E major to be bestowed the dominant function. Radiohead evades this analysis by virtue that it has given a dualistic function to the bridge, which also acts simultaneously as an outro. Yet, indeed the peculiarity of the placement of the E major at the end of the piece, and the overwhelming sense of the entire piece being in A minor is sufficient enough for us to associate E major as dominant to A minor, and consider the E major to also be a failed cadential motion to A minor.

 

So what key is the piece in then? The piece started in A minor, and all of the verses begin with A minor. Furthermore, the chorus starts at a C major tonality, which is the common mediant that minor keys modulate to, and here at a chorus, its modulation would be most appropriate. The chorus itself terminates back into A minor tonality. Within the structure of the strong pillars of A minor tonality, G major is also emphasized greatly. First of all, the overwhelming presence of F# assumes G major, not A minor. Most of the D major chords lead to G major as tonic functions. E minor, the dominant of A minor, is more appropriate to a G major deceptive goal rather than it is the minor dominant of A minor, and it can easily be analyzed this way ignoring its also dual function to the dominant in a modal inflection of A minor. The C major which is the goal of the A minor can also be analyzed as the IV, a predominant function of G major(Figure 36.), or, as an outgrowth of G major tonality in terms of prolongation(Figure 25.) of G major tonality. Both types of function are present and both are convincing. The F# major chord upon which the entire dualistic structure is revealed is both the dominant of a Tonic substitute in G major(III) and the dominant of a predominant(II) in A minor. Classical theory would assume that whatever the first and last chords of a piece are is more than likely what key the piece is in. So what is the last chord of this piece?

 

Unfortunately, Radiohead ends the piece by adding a colorful distorted synth sound that overtakes the bridge/outro and then begins a modulatory deflation that abandons any sense of a final chord. As if a balloon has been inflating over a prolonged period, inside of which contains the answer. When it pops the answer will be revealed. Instead, at the height of its inflation, it begins to simply deflate instead without revealing it’s secret.

One then can make a case for G major as the last progression is so heavily emphasizing that harmonic area. This would interpret most of the A minors as a sort of predominant function to G major(of which the Beethoven Opus 31. 3 serves as a good model.)

One can also make the case for A minor, as it starts in A minor, and the last functional chord is an E major; A chord that is not present until the very end, and therefore seems to suggest that A minor, and its dominant E major, are containing all of the other subordinate surface level harmonies.

 

In summary what we can observe as definite are these qualities.

  1. The Dorian mode acts as a harmonic setting, in which two tonalities vie for importance. It is like the setting of a play where two characters will engage in dramatic conflict and both will give convincing arguments for their positions.
  2. The concept of Dorian has a structural role as well as the surface level, or a major ascending second. Dorian displaces a tonality from its first degree to it’s second. In C major this means C moves upward to D(minor). In G major, this means G moves upward to A(minor); The two keys in opposition in the piece. the two massively important climactic chords that are chromatically outside of the key, E major and F# major are also in this relation to each other, which was also seen at the surface level wherein the first verse F# leads to E of the E minor chord, and that E leads back to F#. In the same way, the C and D major that starts out the chorus are also reflecting this relationship of a ‘Dorian shift’ where major chords are built on those scale degrees.
  3. The opposite inverse is integral as well. That is, a Phrygian structure, or a minor second descending. We see this with the A double flat leading to G flat, the second two chords of the chorus. We see this also in the second verse surface level where F major now leads to E minor, instead of F# to E. We see this again at the end of the Chorus where, C leads to D in the upward progression(expressed in point 2 here) now leads to B minor, downward by half step to terminate the infinite loop.
  4. The chorus itself is split in half. The ascending portion moves up by ascending major second(C-D), the last two chords move by descending minor second(G-F#) All four chords are major, which is impossible in any diatonic collection; That is to say, that there are only three major chords in any given key. Hence this split emphasizes a dualistic operation prominently wherein the verses it is hidden.
  5. Any chord in the piece that contains Pitch Class “F” takes on the class of a higher level functioning chord. One interpretation could make a case for its ‘tonic’ quality by insisting that it is from these chords, D major, F major, F# major, and Fb major that carry the most prominent weight of the piece. Furthermore, the opposing classes, A minor, C major, and E minor are ‘brought into the fold’ of the Pitch class F aggregate, as secondary harmonic areas, hence creating a cohesive field to contain all of the chords of the piece.
  6. When we composed out the various ways that the verses, choruses, and bridge/outro could have played out to be more normative, we were able then to better understand the dualistic choices of the composition. In essence, the material that Radiohead is omitting(that which was not composed) is just as important as the material that is included(that which is composed.) In this view, it opens up an interpretation to the concept of ‘background dialogues’ of which we have normative expectations that are either deferred, abandoned, or confirmed. It is like a ‘choose your own adventure’ storybook. There are many outcomes, but they work within a matrix of dialogue choices. They have the ability to create entirely different stories for one musician and another. However, the choices are not arbitrary, even when those choices are intuitive. As one chooses one harmonic gesture over another, a new series of dialogue options open up. While that leads to tremendous variety, it also speaks of the background matrix that is controlling those options, assuming we are desiring to write cohesive, aesthetic and formally pleasing music.

This last point is one of many reasons for  a ‘Theory of Pop Music.’

The article is a sort of thesis as to why I propose a different mode of analysis. It overdetermines that position by showing how difficult it is to balance classical theory with what’s actually happening in the compositional reality of artists.

Another model of theory can illustrate these concepts in a more concise manner if we are willing to approach it differently than the classical ‘Roman numeral analysis’ that only serves an explanation where functions and scale degrees are behaving nicely within tonal key centers, and still maintain its complications.

It’s not enough to for the theorist to simply say, “It doesn’t follow theoretical models, and hence the reason for the deviation’ as much as it is not enough for the artist to say, “it’s a cool song because it doesn’t follow theory.” That tells us nothing about the phenomenon of the composing. It turns the music into a religious mythos separate from a scientific measurement. And yet where we do see common relations to classical era tonality, sometimes it’s superficial; In appearance only. Sometimes it is borrowing heavily from classical theory, but such cases are exceptional and overtly pronounced in its model as operatic, or ‘classical.’

 While my treatise does not go as in depth as this analysis, it takes into account the very many articulating procedures that are discussed here. They are, of course, presented in such a way that the musician who is not interested in theory can easily grasp. Essentially because it aims to amplify and make known what a musician is already doing intuitively so that the choices that a musician makes are more in line with their own inspiration and mental visualization.

It is a tool so that when a musician says “that doesn’t sound right? Why?” they can hopefully find an answer quickly. It is not meant as a series of rules for ‘good’ music but rather observations and ways of looking music that make the music of the ethereal realm more ripe for picking into the physical.