The first song the station played used what Boston Globe columnist Marc Hirsh has dubbed the “Sensitive Female Chord Progression,” or I–V–vi–IV and its permutations (henceforth, “SFCP”). Now, pretty much everyone knows how ubiquitous this chord progression is in current popular music—and I know it's driven many to their wit's end—but except for brief excursions to the grocery store I'm pretty well insulated from the radio and popular music generally, so I wasn't too irritated. In fact, I took it as an opportunity to reflect on the phenomenon of stock melodic and harmonic patterns generally, from the familiar “doo-wop progression” (cf. “Stand By Me,” “Earth Angel”) to older specimens like the passamezzo antico and the folia. I was amused when the next song also used the SFCP; my musico-historical narrative was literally playing out in front of me. When the song after that also used the SFCP I found myself slightly more irritated than amused, and very quickly it became apparent that the station's playlist was cascading into an avalanche of harmonic sentimentality.
When they trotted out “Dark Horse” I thought it was going to be the coup de grâce. The song opens with the progression G♭M–D♭M–B♭m–A♭M, which—while not the SFCP—is nonetheless a close relative (just exchange B♭m and A♭M), and depends for its effect on the same sort of major-minor ambiguity (or more properly, major-Aeolian ambiguity).
Yet, despite the inherent ambiguity of this progression, the layering of a B♭m ostinato (D♭–[C–]B♭–F) in another voice clearly settles the matter of major and minor in favor of the latter. As such we should interpret the progression as VI–III–i–VII in B♭ minor rather than IV–I–vi–V in D♭ major. This isn't really that interesting, and at this point in the song I still felt like a victim of some musical equivalent of Chinese water torture (each cycle of chords like another drop on the forehead). By the end of the first verse, however, I was sold.
In terms of instrumental texture the verse is very sparse, comprising only a tonic pedal (B♭) in the bass, a rhythmically augmented variant of the B♭m ostinato, and various percussive sound effects (e.g., hand claps). Most of the interest lies in the vocal line, which is more elegant than Top 40 typically delivers. This melody is constructed modularly out of two fundamental phrases, which I have transcribed and labeled as A and B (along with the ostinato in the upper voice):
|Katy Perry feat. Juicy J, “Dark Horse,” verse phrases.|
Furthermore, the following is the structure that Perry erects out of these two modules:
|| A | – | A | – | A | – | A-2 | A-2 | B | – | B | – | A | – | A-2 | A-2' ||
Note that vertical bars indicate barlines and en dashes indicate the continuation of a phrase over the barline (i.e., since A and B are both two measures long, they spill over into the measure after the one in which they are first indicated). “A-2” refers to the second, one-measure motive in A, which in practice is varied slightly when it's stuttered. If we think in two-measure chunks we could reduce the above diagram down to A–A'–B–A', which of course is a tried-and-true design (it's the heart of sonata form). In this formal progression (both here and conventionally) the A material serves a normative function, while the B material serves as a point of contrast or digression. What I'll do now is characterize A and B in turn to demonstrate the ways in which they contrast (and connect).
Okay, looking at A first we can see that it is little more than an expansion of B♭ minor. The vocal line comprises—quite literally in the recording—two distinct motives: (1) an upper pentatonic neighbor on F, the dominant (i.e., SD4–5–4); and (2) an accented passing tone to the mediant (i.e., E♭–D♭), which then ‘steps’ down pentatonically to the tonic B♭. Taken together these motives outline the fifth B♭3–F4, which is harmonized by the ostinato. Overall we could characterize A as both pentatonic and scalar in its orientation (it proceeds completely by scalar step and implies no polyphony).
Turning our attention now to B, which follows after eight bars of A, we observe a striking contrast of melodic character. What was in A a stepwise pentatonic melody with static harmony becomes in B a diatonic compound melody with dynamic harmony. The first measure of B unfolds B♭–A♮ over a sustained F, implying (in this case) a i–V progression from the first beat to the second. In the second measure the lower unfolded line moves E♭–F, suggesting predominant-to-dominant motion (in fact when the upper line in the unfolding is provided explicitly as a vocal harmony in the second verse [~1:36], the harmony is iio6 [if we disregard the bass pedal]). Moreover, in terms of range B complements A by covering the octave's remaining fourth, F4–B♭4. Overall we could characterize B as diatonic and chordal in its orientation (it implies multiple lines moving by diatonic steps). B is, in other words, functionally tonal, proceeding as i–V–iio6–V (mostly in spite of the ostinato and of course the bass pedal).
Thus, the verse melody as a whole could be described as a dialectic between two paradigms of pitch organization, melodically-oriented pentatonicism and harmonically-oriented diatonicism, of which—for mostly formal reasons (A–A'–B–A')—the former registers as normative while the latter registers as deviant. Indeed, in relation to A, the leading tone and harmonic dynamism of B sound strikingly exotic, and I think that this contrast between the phrases is the most affective local manifestation of the “witch-y, spell-y kind of black magic-y” vibe that Perry claims to have been going for.
Yet, in addition to characterizing this contrast, it's also worth taking a look at how the competing paradigms of A and B are mediated. One strategy we have already noted is that of relating the phrases as complementary portions of the octave B♭3–4, particularly as lower fifth and upper fourth. This division of the octave into species of fourth and fifth has precedents in ancient theory and practice, and is a classic basis for structuring a melody. A more subtle connection between the A and B phrases is one of rhythmic symmetry. As one can see in my transcription, the second measure of A and the first measure of B feature the same rhythm in reverse order (4–4–88–4 || 4–88–4–4). Moreover, the first measure of A and the second measure of B both contrast with their subsequent and previous measures respectively by stressing longer durations. This is of course not a literal symmetry, and even the exact symmetry of the ‘inner’ measures is misleading in that the first beat of A-2 is continuously varied throughout the song. Nevertheless, in a broad sense there is a feeling of A as long–short (…88–4 ||) and B as short (|| 4–88…)–long.
The most elegant negotiation between the two phrases is somewhat abstract. In short, Perry connects A and B such that there are smooth ‘modulations’ between their respective pitch paradigms. To observe this, let us consider the progression from A to B and back again. First, A traces a complete B♭ minor pentatonic scale, opening with the upper neighbor figure SD4–5–4 and then descending SD(4–)3–2–1. All of this motion is by scalar step. It's worth noting that a feature of the minor pentatonic scale is that the fourths above SD1 and SD4 are of the same quality, m3 + M2 (e.g., B♭–D♭–E♭ and F–A♭–B♭). This relationship is emphasized somewhat in A in terms of the similar emphasis given to B♭–D♭ and F–A♭. But it plays a more important role in the ‘modulation’ to the functional tonality of B. The lead-in to B is a stuttering of A-2, which consists of the fourth over B♭ stuttered a couple times (E♭–D♭–B♭, or SD3–2–1). The first measure of B—in a manner that is supported by the rhythmic symmetry noted above—picks up this thread of melodic thought by outlining the upper fourth (F–A♭–B♭), but with the substitution of A♮ for A♭. The effect of this is—as we have seen in discussing B as a compound melody—is to obliterate the sense of the pitches as adjacent scale degrees in a pentatonic scheme, and instead cast them as members of a diatonic fourth, from which G♭/♮ is missing (i.e., F–A♮ is now a skip rather than a scalar step). Thus, by way of chromatic inflection the first measure of B reinterprets in a contrasting pitch paradigm the second measure of A. The return to the pentatonicism of A is also effected with some subtlety. Upon hearing the F–A♭–F that opens A, we can't help but hear this as a continuation of the two unfolded lines of B. That is, it registers initially not as SD4–5–4 in B♭ minor pentatonic, but as part of the lines B♭–A♮–A♭ (upper) and F–E♭–F (lower). By the time B♭ is reached at the end of the phrase though, it's clear that we're back in pentatonic territory, and that the F–A♭ dyad was functioning as something of a paradigmatic pivot. Overall, in moving between the phrases Perry highlights intervallic relationships that mean one thing in A and another in B.
There's probably some stuff to be said about the other parts of the song, but the verse melody is where I think the real art is in “Dark Horse.” One broad point that I'd like to bring up relates to the fact that the chorus revisits the SFCP-like chord progression of the introduction (or, considered the other way, the introduction was a foreshadowing of the chorus). While the verse is virtually monophonic (in that the instrumental texture scarcely achieves a more than drone-like significance) and harmonically static (B is harmonically dynamic, but only in brief contrast to the normative and pentatonic A), the chorus is homophonic, with a vocal line supported by explicit chord changes throughout. Thus, the contrast between static pentatonicism and functionally tonal (and dynamic) diatonicism that exists within the verse also exists at the higher hierarchical level of verse and chorus. Within the chorus we may also find this sort of fractalesque self-similarity, in that the progression VI–III–i–VI—as a cousin of the SFCP—is characterized by an ambiguity between functional tonality (when heard in the major) and pentatonic thinking (particularly in the root motion D♭–A♭–B♭).
Ultimately I think that the real appeal of this song is that it actively engages the pentatonic-tonal dialectic that sits at the heart of American popular music, rather than just passively accepting it. In this respect “Dark Horse” has an almost Mozartean quality. Sadly, after this song was over, it was back to the Salieris.
 Hirsh justifies the term, which at first glance may seem misogynistic, by claiming that he first noticed it being used in 1998 by “Lilith Fair types baring their souls for all to see” (which I guess could itself be taken as misogynistic, but whatever; it seems to be the only term in circulation).
 I'm not really sure who to credit as the “composer” of “Dark Horse” since Wikipedia lists multiple writers, but the impression I get from an MTV article on the song is that Perry and a friend wrote everything that I will be discussing. For this reason I'll just refer to Perry as the composer.
 Note that the phrases as transcribed—including A-2—are often varied superficially to suit the text. Also note that in the actual song the dotted half-note F in the A phrase in fact droops down to E♭ within the first measure, but I didn't transcribe this because it seemed to me more an expressive gesture than part of the melodic content in a Platonic sense.
 I'm using “diatonic” here to mean the seven-note diatonic scale, not more broadly in the sense of anything that can be derived from that scale.
 The Wikipedia article on “octave species” provides a pretty good crash course. I'm no expert on ancient music theory but there is an old tradition of dividing the octave into species of fourth (diatesseron) and fifth (diapente). Indeed, the species of fourth and fifth are the constructive modules in terms of which the eight “ecclesiastical” modes were (originally, I think) conceived, a lower fifth and upper fourth forming an authentic mode and a lower fourth and upper fifth forming a plagal mode. This way of thinking about the octave is reflected in practice as far back as the Seikilos epitaph, which spans an E octave, stressing mostly the upper fifth A–E and cadencing with the lower fourth E–A. A less ancient example may be observed in the opening line of Victoria's motet “O magnum mysterium” (c. 1570): lower fifth (“O magnum mysterium”), upper fourth (“et admirabile sacra[men]–”), lower fifth (“–mentum”).
 One could also hear a degree of symmetry within A: [88–4]+[sustain] | [sustain]+[88–4]. The pitches only heighten this hearing in that the 88–4's both emphasize minor thirds over the structural pitches F and B♭. Note that this interpretation hinges on the perception of the quarter E♭'s as equivalent on some level to a sustained pitch.