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date: 21 November 2017

Tonic

Abstract and Keywords

This article proposes a general definition of tonic and explores its ramifications across repertories and intellectual traditions (cognitive, historical, ethnomusicological, music analytical, and phenomenological). The proposed definition admits of both wide and narrow applications: from broad conceptions that detect tonics in a wide range of world musics to a more narrow definition that limits the term (and its theoretical entailments) to bourgeois musical cultures in the West. These ideas are illustrated through discussions of diverse musical examples, from the “common practice” (Bach, Schubert) to the postwar avant-garde (Lutosławski), French house (Daft Punk), and rust-belt hard rock (Akron-based band Dia Pason).

Keywords: tonic, tonality, music cognition, ethnography, Lutosławski, Bach, Schubert, Daft Punk, Dia Pason

Criteria for Tonicity

Begin by listening to Audio Example 1, a passage for solo cello that lasts about one and a half minutes:

Much of the passage consists of the laconic, almost mechanical repetition of a single pitch: the D below middle C (i.e., D3). Its tic-toc reiterations are interrupted by unpredictable gestures that skitter and slide across registers, saturating chromatic space. Then, just as unpredictably, the tic-toc D3 returns. A question arises: Is the repeated D a tonic?

Few readers will likely answer with an unqualified “yes.” But it is worth noting that the sheer prevalence of the D—it persists unaccompanied for about one-third of the excerpt’s total duration—as well as its extreme salience compared to the other pitches in the excerpt, elevate it to at least potential tonic status in certain theories of tonal perception. Carol Krumhansl, for example, writes that

The means of emphasizing the tonic and organizing the other elements around it vary considerably across musical styles. In most cases, the tonic is emphasized both melodically and rhythmically; it is sounded with relative frequency and with longer duration; and it tends to appear near the beginning and end of major phrase boundaries and at points of rhythmic stress. (1990, 16)

The D3 in our excerpt emphatically satisfies Krumhansl’s second criterion: it “sounds with relative frequency and with longer duration” than any other pitch class in the passage.1 It is also emphasized rhythmically, via its insistent repetition. This repetition arguably lends it “rhythmic stress” as well, though whether it receives any metric stress (which Krumhansl might be implying) is an open question: it is not clear if the concept of meter, in any traditional sense, is relevant for this passage.2 The categories of melody and phrase are similarly problematic, making it difficult to say whether the D receives melodic emphasis or occurs “near the beginning and end of major phrase boundaries.” If anything, it is melodically emphasized through negation: its flat pitch contour stands in stark contrast to the extravagant profiles of the surrounding music (glissandi and all). As for phrase boundaries, the non-sequitur interruptions of the passage problematize the entire concept of phrase, to say nothing of its often-attendant notion of cadence. The returns of D seem to signal boundaries of some kind but they have no evident logic, or predictive relationship with what precedes and follows.

Indeed, the music’s many jump cuts throw into doubt the relationship between the flush-juxtaposed materials—between the laconically ticking D and the capricious gestures it abuts. The lack of conspicuous tonal threads between the reiterated D and its surroundings may well be the biggest hurdle for listeners who are asked to assign D tonic status.3 As if to highlight fact, the composer gives the repeated D3 the marking indifferente at each of its appearances in the score. Whatever the relationship of a tonic to the other pitches in a passage, surely it cannot be “indifferent.” Conversely, non-tonic pitches should not be “indifferent” to it. Rather the very idea of a tonic seems to assume some sort of heedful relationship between tonic and non-tonic. To be a tonic is to matter somehow to non-tonic pitches. Krumhansl suggests as much when she notes that the other pitch elements should be “organized around” the tonic; there is arguably no such organization at work in the passage in question, at least none that is aurally perspicuous upon listening without the score.

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Figure 1. Pitch-class distribution in the audio example. Values were determined by measuring pitch durations using Sonic Visualiser, in coordination with the score. Rests are included in the duration of the previous pitch. Glissandi are ignored and microtonal pitches are adjusted to the nearest equal-tempered pitch (per the performer’s intonation).

The music’s radically simplified pitch hierarchy further attenuates any residual tonic effects. D is clearly elevated hierarchically above the other pitch classes in the passage (as a result of its salience and duration), but there are no evident hierarchical distinctions among the remaining 11 pitch classes. As a result, the pitch hierarchy of the passage is like a highly distorted Chicago skyline: a field of low-rise buildings with one soaring Willis Tower (née Sears) in their midst. See Figure 1. By contrast, most theories of pitch distribution in tonal music posit pitch hierarchies that look more like the actual Chicago skyline, with the twelve pitch classes projecting various building heights based on their statistical prevalence. (We will see examples of such hierarchies later in Figure 4.) The Willis Tower in such tonal profiles is still the tallest, but the Trump and Aon Center are a close second and third, not unlike scale degrees 3 and 5 in traditional common-practice pitch hierarchies. Such a robustly textured hierarchy creates a multitiered sonic environment in which various characteristic relationships can emerge: some pitch classes may be perceived as “close” to the tonic and others “far” (to draw on a familiar distance metaphor); some may seem more “stable” or “at rest” than others (also familiar metaphors); and so on.4 In addition to the rhetorical disjunctions in the passage, the relative hierarchical “flatness” of the 11 non-D pitch classes5 frustrates attempts to hear these sorts of characteristic relationships between D and non-D as the passage unfolds. A hierarchical pitch profile would thus seem a necessary condition for the emergence of a tonic, as the absence of such a profile challenges our ability to relate the non-tonic pitch classes to the tonic in a variety of characteristic and meaningful ways.

Finally, some may object to assigning D tonic status on stylistic grounds, arguing that the music that falls between the repeated Ds is clearly coded as “atonal,” not only in its pitch content but in its gestural language. These readers may even recognize the piece: it is the opening of Witold Lutosławski’s Cello Concerto, composed in 1969–1970.6 Surely, the argument might go, it is inappropriate to listen for a tonic in such an atonal work. This objection differs from the previous ones. It is based not primarily on a phenomenological argument but on a stylistic one: this is music in which the concept of “tonic” is historically problematic or, to put it more dialectically, in which the idea of tonic continues to exert its influence, but negatively, through a strategic disavowal. Any local tonic effects that remain invite various interpretive responses, among them psychoanalytic readings.7 Less philosophically inclined interpreters—perhaps of a more music-theoretical bent—might simply assert that, given the surrounding “atonal” pitch structures, any emphasized pitches are not “true” tonics but fleeting epiphenomena.8 This view is often supported by theoretical commitments regarding what kinds of musical systems can support and sustain tonics in the first place. At one extreme, only art music of the common-practice era and closely related Western vernacular traditions—“tonal music” in the most conventional sense—can contain tonics properly so-called. Some ethnomusicologists may also wish to limit the concepts of “tonic” and “tonality” to these bourgeois European musical traditions but for different reasons: to avoid colonializing discourses and defer to more emic terminology for non-Western musics. At the other extreme—liberal in a different sense—the idea of tonic can be extended to a wide range of world musics in which a single pitch class predominates, from drones in Indian rāgs to concluding gongs in Javanese gamelan. Indeed, Krumhansl (1990, 16) adduces both of these as examples of world musics that exhibit “tonics,” broadly conceived.9 In such a view, genuine modernist atonality is a singular exception among the musics of the world.

A General Definition

We have seemingly begun at an angle to our main subject, with an oblique example from Lutosławski. But the discussion has in fact brought into focus several primitives about “tonicness.” We can even venture a basic definition:

Definition: A tonic is

  1. (a) a focal pitch class

  2. (b) with respect to which all remaining pitch classes in some musical passage are hierarchically arranged and perceived

  3. (c) even in its acoustic absence.

If the D in the Lutosławski fails as a tonic, it does so because of (b) and (c), however much it might momentarily satisfy (a).10

The adjective “focal” in (a) takes its inspiration from Donald Francis Tovey’s memorable simile that a tonic is “like the point of view, or the vanishing point, of a picture” (1949, 134). One often reads of the tonic as a “center,” or a “central” pitch class; Tovey’s simile captures the spirit of such spatial metaphors but has the additional advantage of indexing an action: the orienting of aural attention in a particular direction.11 Such an act of aural directing situates the non-tonic pitch classes with respect to the tonic focal point, just as a vanishing point orients peripheral elements in visual perspective. While D in the Lutosławski trivially acts as a focal pitch class during the passages of “indifferent” repetition, its all-but-complete disappearance as orienting presence in the rhapsodic sections vitiates whatever tonic potential it had when sounding alone. We never have an opportunity to fill out the sonic picture, so to speak—to experience the other pitch classes arrayed about the focal D. We instead hear an isolated singularity followed by stochastic scatter.

That the tonic is a pitch class might initially seem uncontroversial, but two clarifications are in order. First, the word tonic is also a common adjective for chords and keys: we refer to the “tonic chord” within a key, or to the “tonic key” within a musical work that modulates.12 But it should be clear that these uses depend on and extend the basic pitch-class definition. For one, the tonic pitch class is contained within the tonic chord (as root) and key (as first scale degree). We might say, after Daniel Harrison (1994, 50), that the tonic pitch class “transfers its franchise” to its embedding chord and key. Alternatively, we can observe that the tonic chord and key merit their labels by analogy to the tonic pitch class: they behave in their respective harmonic and modulatory environments more or less as the tonic pitch class does in its local diatonic world. Given the reciprocal part–whole logic of the relationship between tonic pitch class and its eponymous key and chord, we might refer to the latter as “tonics by synecdoche.”

Second, the tonic’s status as a pitch class, rather than a single pitch, should be clear on reflection. If one pitch D acts as a tonic, all Ds, in any register, should as well. Tonicness is thus a perceptual property that inheres in a pitch-class chroma, not in a single pitch.13 (Again, the pitch specificity of the reiterated D3 in the Lutosławski tells against its tonic potential.) By extension, non-tonics are also pitch classes. Note, however, that part (b) does not refer to the 11 remaining pitch classes, for in most notated tonal traditions (especially the common practice) pitch-class spelling matters. The number of non-tonic pitch classes is therefore without theoretical limit.14

The qualification “in some musical passage” in part (b) indicates an openness regarding the temporal extent of a tonic’s influence. While some theories of monotonality—for example those of Schenker, Schoenberg, and Lerdahl and Jackendoff—assert that a tonic applies for an entire composition, studies in music cognition have challenged the perceptibility of such long-range tonics.15 This will likely remain a contentious debate, if only because it involves so many variables. These include differences among listeners’ capacities (e.g., those with absolute pitch and those without), as well as differences in musical “occasion” (listening with or without a score, playing an instrument or listening to a recording, etc.). There may also be differences in scholarly purpose: a high-flown formal analysis of a tonal work, à la a Schenkerian sketch, need not be read as making the same kinds of perceptual assertions as an empirical study of listener cognition, though both might refer to “tonics.”16 We will return to such distinctions in the next section.

This also relates to part (c) of the definition. For a pitch class to act as a tonic, it must have the capacity to orient our hearing during moments when it is not acoustically present. The Lutosławski again motivates this condition: one reason D is so hard to accept as a tonic is its lack of orienting power when silent. Just how long a tonic can remain in effect in absentia is—again—a matter of scholarly debate. But most will agree that even at the most local, phrase level, the tonic typically will go silent at some point, at least in common-practice idioms that treat the dominant chord as the tonal lynchpin: the dominant is the harmony that most strongly evokes the tonic, in part because it is the only functional pillar that does not contain it as a pitch class. By contrast, consider musics anchored by a drone: the drone is a privileged pitch, to be sure, but is it a tonic? If we take (c) as a necessary condition for tonicness, drones will fall short, as they never have a chance to exert their influence in absentia.

Whether one wants to preserve the concept of tonic in non-Western drone-based musics is another matter of course. The definition remains strategically underdetermined in this sense: part (b) does not specify the ways in which the non-tonic pitch classes might be “hierarchically arranged and perceived” with respect to the tonic.17 As a result, it does not prescribe the range of musical systems within which tonics can operate. This leaves the door open for broad or narrow conceptions of tonality as discussed previously: from an encompassing view that treats tonality in some form as a property of nearly all world musics to a narrow one that considers it a historically contingent feature of bourgeois musical cultures in the West. As Brian Hyer (2002) notes, the concept of tonality in its broadest sense can be considered analogous to système musical.18 The present definition offers flexibility regarding just which systèmes one wishes to invoke as capable of sustaining tonics.

A Sarabande and Its Tonics (Empirical, Theoretical, Phenomenological)

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Figure 2. J. S. Bach, Sarabande from Cello Suite BWV 1011, annotated.

As noted earlier, our broad definition of tonic will likely receive subtly differing emphases based on scholars’ intellectual commitments. To illustrate the point, let us consider another example for solo cello: the Sarabande from J. S. Bach’s fifth cello suite (BWV 1011). Figure 2 shows a lightly annotated score of movement. Unlike the Lutosławski, this is a piece from the very heart of the canonical common practice—if any music is tonal, this surely is. Yet it is also an exceptional piece: its sparseness and severity set it apart even from Bach’s other movements for solo cello. In this highly reduced texture, local tonal effects speak with particular vividness, making it an especially effective site for exploring the various ways in which tonics can act as objects of perception, theoretical contemplation, and empirical research.

The annotations divide the Sarabande into four phrases. The first three phrases last four bars; the fourth is twice as long, though it is articulated into four-bar subphrases by the quasi-half cadence in m. 16. The contour inversion in mm. 17–18, which mirrors the recto statements in mm. 13–14, underscores the sense of subphrase parallelism, especially as a similar inversional relationship obtains between the beginnings of the first two phrases (compare mm. 1–2 and 5–6). Nevertheless, the continuous eighth-note motion in m. 16 attenuates the cadential effect, as does the linear-harmonic continuation: the G bass note of m. 16 can be heard to continue into the first two beats of m. 17, before moving to A♮ on the third beat, thus creating a continuous tonal motion that stitches across any putative cadential seam. The cadential endings in mm. 4, 8, 12, and 20, by contrast, are unmistakable. All are authentic cadences,19 all bring the eighth-note motion to momentary rest, and all exhibit conventional cadential rhetoric.

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Figure 3. The resolution of A♭ and B♮ in phrase one.

Phrases two through four traverse characteristically Bachian tonal trajectories: rather than staking out a key from beginning to end, each phrase traces a tonal motion that leads to cadential confirmation in a new key. No sooner is that goal confirmed than the first bar of the next phrase destabilizes it, setting off for a new tonal station. Only the first phrase exhibits a different logic…or does it? There can be little doubt that phrase one projects C minor throughout. The three-note gesture E♭–B♮–C in bar 1 (a highly thematic figure in the movement) points unambiguously to C minor. Moreover, a listener familiar with the tonal conventions of Bach’s suites, with their uniform keys across movements, will expect a movement in C minor, the key of the previous three movements. But in the Sarabande the opening and closing tonics in the first phrase differ considerably in effect. The cadential C of m. 4 secures the tonic via the most sonorous means available: the cello’s lowest pitch. The effect is almost like the confirmation of an initially tentative hypothesis. The movement begins in C minor and yet the first bar does not seem to sit squarely within the key: it lists to one side.20 This is due to the striking A♭ on beat three, which seems to nudge the C-minor tonic triad off center. The B♮ at the same location in bar 2 then partially rights the ship, aiming us toward the tonic resolution. Indeed, A♭ and B♮ both sound in this bar, repelling one another and pushing toward their respective resolutions: A♭ down to G, B♮ up to C. As Figure 3 illustrates, those resolutions arrive in bars 3 and 4, at the same metric position (beat three). These kinetics of initial destabilization and gradual resolution underlie the sense that the music, though clearly in C minor from the start, only truly stabilizes its tonic with the cadence of m. 4. We are reminded that one of the central roles of a tonic in the common practice is as a goal of tonal motion. The effect in the Sarabande is not unlike the “centripetal” tonality that Carl Dahlhaus ascribes to Brahms, though in microcosm: the music begins in a state of tonal imbalance and its “sole ambition is to reach its center” (1980, 74).

This, at least, is how it strikes me. Others’ ears may well hang on different details. For example, I suspect that a cellist’s experience of the music—when playing or listening—would exhibit all manner of somatically mediated texture that mine lacks. Enculturation, embodied practice, temperament, training, mode of musicking (playing, practicing, listening [closely or distractedly], analyzing)—all of these will mediate listeners’ phenomenological encounters with the music and its rhetorics of tonic (destabilized, confirmed, on the way, etc.). Scholars of different disciplinary and intellectual commitments nevertheless disagree on how great these experiential differences truly are. Some scholars in music perception and cognition, for example, have tended to downplay them. One such scholar, David Temperley, admits that

judgments about the kinds of [perceived musical] structure described [in his book] vary greatly among individuals—even among experts (and non-experts). Indeed, one might claim that there is so much subjectivity in these matters that the idea of pursuing a “formal theory of listener’s intuitions” is misguided.

But he then continues:

I do not deny that there are sometimes subjective differences about all the kinds of structure at issue here; however, I believe there is much more agreement than disagreement. The success of the computational tests I present here, where I rely on sources other than myself for “correct” analysis, offers some testimony to the general agreement that is found in these areas.21

Temperley’s “computational test” regarding the experience of tonics and keys draws on the work of Carol Krumhansl, Edward Kessler, Mark Schmuckler, and others who have sought to map the “tonal hierarchies” that underwrite listeners’ experiences of key.22 Krumhansl and her associates derived these hierarchies from “probe tone” experiments in which listeners were asked to rate the “fit” between the twelve chromatic pitch classes and a previously established key. Figure 4 shows Temperley’s slightly modified versions of the resulting tonal hierarchies in major and minor. As the comparison in 4(c) makes clear, Temperley’s major and minor profiles differ only in the values they assign scale degrees 3 and 6 (here, E♭ vs. E♮ and A♭ versus A♮).23 Not surprisingly, the tonic is the most hierarchically privileged pitch class in both key profiles.

Krumhansl, Temperley, and other scholars working in this tradition hypothesize that listeners internalize key profiles such as these through statistical exposure to a great deal of music in some tonal idiom.24 When hearing a musical passage, listeners seek the closest match between the pitch-class distribution in the music and the internalized profiles. What kind of thing is a tonic in this mode of hearing? It is often—but not always—the most statistically prevalent pitch class in the music. The exceptions arise because the pitch-class content of the entire musical texture must approximate one of the key profiles in toto: a statistically prevalent pitch class must be confirmed as tonic by the arrangement of the remaining pitch classes into the peaks and valleys characteristic of its major or minor key profile. In other words, the tonic is ennobled not (only) through its statistical prevalence but through the statistical distribution of the entire field of pitch-classes into a familiar, internalized hierarchy, over which it presides as head.

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Figure 5. Pitch distributions in the Bach Sarabande, compared with key profiles.

To illustrate this template-matching model, Figure 5(a) compares the pitch distribution of the entire Sarabande (the “input vector”) with the key profile for C minor.25 Even at sight it is clear that the match is remarkably close. Notably, tonic pitch-class C occurs more than any other in the piece.26 As Temperley has noted, however, the key profiles are most effective when applied not to pieces in their entirety but to seriated local segments, to better capture the effects of modulation. Figure 5(b) thus takes the pitch distribution of the first phrase as its input vector, comparing it with the profile for C minor. It is more difficult to judge the fit visually in this instance, as it is for the first bar alone, as shown in Figures 5(c) and (d): it is not clear at sight if its pitch distribution best fits, say, C minor or A♭ major.

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Figure 6. The scalar product of the input vector for m. 1 in the Sarabande with all 24 key profiles. Note the tie between C minor, A♭ major, and A♭ minor.

A computational comparison will clearly be more reliable. Temperley employs a “scalar product,” which multiplies the values in the input vector with the values for the 12 pitch classes in each of the 24 key profiles. These sums are then totaled, and the highest score reflects the best match between input vector and key profile.27 Figure 6 shows the results for bar 1, which register a tie between C minor, A♭ major, and A♭ minor. A look back at the music makes clear why this is: the bar contains pitches of the complete C minor, A♭ major, and A♭ minor triads and no others. Though pitch-class A♭ lasts twice as long as any other in the bar, and thus might seem to tip the scales in favor of A♭ as tonic, this is balanced out by the fact that either B or C will be chromatic in the A♭ key profiles, but both receive high values in the C-minor profile.

To calculate the modulating key structure of an extended passage, Temperley employs a key-finding algorithm that segments the music into measures and determines the best fit between each measure’s pitch classes and the key profiles via the scalar product.28 To reflect the “inertia” of tonal hearing—in which listeners presumably seek to maintain a key unless evidence for a modulation is strong enough to dislodge it—Temperley’s algorithm assesses a “penalty” for any change of key from bar to bar. Only when the new key is sufficiently strong to outweigh the persistence of the old does the algorithm register a modulation.

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Figure 7. Temperley’s key-finding algorithm applied to the Bach Sarabande.

Figure 7 shows the results of this algorithm for the entire Sarabande. Yellow and green cells indicate the best and second-best matches (respectively) between each measure’s input vector and the 24 key profiles; a heavy border indicates the preferred key based on the key-finding algorithm, with the modulation penalty set at –4.29 If the heavy border encloses a green or white cell, this means that the yellow cell or cells in that column have a score that is not sufficiently strong to motivate a change of key (viz., the score is less than four “points” greater than that in the heavily enclosed cell). A scan of the chart reveals that the algorithm’s selected keys match those of a traditional tonal analysis very closely, with modulations to E♭ major and F minor. The key of B♭ major in bars 13 and 14 is more questionable, but a quick glance at the score reveals it as the most plausible choice hereabouts, at least at this level of “resolution” (with single measures as the initial input). One also notes the increasing clarity of the algorithm’s results as each phrase comes to a close,30 a fact that recalls our earlier observations regarding the phrases’ Bachian trajectories from tonal destabilization to new tonic confirmation.

We have nevertheless come a long way from the fine-grained phenomenological distinctions with which this section began. Note, for example, that the pitches in the Sarabande could be scrambled within each bar and transposed to any register and the results of the key-finding algorithm would be unchanged. The method is thus insensitive both to syntax at the local, note-to-note level and to register. It moreover makes no distinction between bass notes, inner voices, and upper voices—distinctions that are essential in most tonal theories. Other empirical and computational methods address some of these issues. The key-finding approach of Brown, Butler, and Jones (1994; see also Brown and Butler 1984) attends not to pitch-class distributions but to rare intervals and intervallic cues that can signal a key. Such approaches would be sensitive to strong key signals such as the E♭–B♮–C in m. 1. Temperley himself (along with Elizabeth Marvin) has argued that “structural” models such as Brown, Butler, and Jones’s likely work in tandem with distributional ones such as his own in listeners’ processes of key finding (Marvin and Temperley 2008). Other researchers have developed key-finding models based on harmony rather than key profiles (Winograd 1968, Maxwell 1992), thus showing a sensitivity to chord rather than individual pitch class, while Jamshed Bharucha (1987, 1991) and Fred Lerdahl (2001) have proposed models that take into account the hierarchical interaction of pitches, chords, and keys. Huron (2006) and Aarden (2003) advance key-finding methods that are more sensitive to syntax, proposing that the key-profile model needs to be adjusted to reflect a distinction between mid-melody and phrase-final contexts.31 Finally, recent approaches to musical schemata (e.g., Gjerdingen 2007), though not primarily focused on tonal questions, generally model keys and tonics from a more ecological perspective, by tracking stock idioms and gestures that carry their tonal implications along with them, as a snail carries its shell.

For all of their differences, these psychological and empirical studies nevertheless have much in common. They share not only a commitment to certain experimental protocols but also a wariness of adopting any more traditional music theory than necessary (with the possible exception of Lerdahl 2001). If a key-finding model can succeed in identifying the keys in a piece of music through note counting and a few simple algorithmic steps, for example, why invoke sophisticated music-theoretical notions of key, tonic, and modulation? This strategic music-theoretical “thinness” is not merely a casualty of Occam’s razor; it also arises from the exigencies of experimental design. It should not be a surprise, then, that the concept of tonic looks quite different when we encounter it within the “thickest” of tonal theories: Schenkerian analysis.

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Figure 8. Generative Schenkerian reading of the Bach Sarabande.

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Figure 9. Eric Wen’s reading of the Sarabande’s first phrase (Wen 1999, 278, Example 3).

Figure 8 shows a generative Schenkerian analysis of the Sarabande, beginning from the minor tonic triad (a), which exfoliates into an Ursatz at (b). The middle-ground transformations of (c) and (d) show an initial ascent to the Kopfton scale degree 3, which is delayed to arrive over V (in a six-four configuration).32 Figure 8(e) shows further prolongations of the subsidiary harmonic stages in this trajectory—III, iv, and V—as the music moves closer to its phenomenal surface. Moving further into the foreground, Figure 9 presents one analysis, by Eric Wen (1999), of the prolongation of the tonic Stufe in the first phrase. Wen reads the striking A♭ bass note in bar 1 as a displaced inner voice. As a result, the opening harmony is implicitly bounded by tonic bass and tonic soprano, an exact reversal of the sounding music, in which tonic C3 falls registrally (and temporally) between the cello’s bounding G3 and A♭2, which outline a dissonant major seventh. The cello also traces a dissonant bounding interval in bar 2, now a minor ninth, which is again transformed in Wen’s reading to an underlying consonance: an implicit tenth between B♮2 and D4 (as shown in the bottom systems). While some may object to the reading’s relegation of the A♭ in bar 1 and G in bar 3 to inner voices,33 one can also argue that this transformation is partly responsible for the phrase’s expressive effect: the sounding music literally turns the underlying structure inside out.

The secure tonic embrace of the outer voices in bar 1 of Wen’s reading is emblematic of the role of the tonic in Schenkerian theory. Note the omnipresence of the tonic in Figure 8, beginning with the atemporal generative matrix of its triad. As the structure begins to unfurl, tonic manifests itself in two ways: as the harmonic Stufe that the composition prolongs and that forms the frame (Tonraum) for its temporal unfolding, and as scale degree 1, the telos of the Urlinie. The sketches visually reflect the tonic’s theoretical ubiquity: it seems to enfold the entire composition via the beamed outer voices of the Ursatz. Within this frame, all tonal phenomena ultimately derive from and are answerable to this encompassing tonic, including the putative local tonics of E♭ and F, which are subsumed into the encompassing key, as prolongations of III and iv. Indeed, these Stufen are less structurally prominent than the (non-tonicized) dominant (V) that begins in m. 16.

Schenker is well known for his criticism of theories of modulation and his insistence that all ostensible changes of key are in fact only tonicizations of greater or lesser strength. But as an analyst he was highly sensitive to the “illusory keys of the foreground.”34 Indeed, one might argue that tonic as a perceptual phenomenon is primarily a foreground effect for Schenker. It is part of what he calls the tonality (Tonalität) of the foreground, which is distinct from the austere diatony (Diatonie) of the background. The latter prolongs a conceptual tonic triad that may not be easily accessible to hearing, while the former harbors tonics that are the empirically sensible effects of late-middleground prolongation and composing-out.35 Schenker nevertheless states, in typically Delphic fashion,36 that “the tonal sparseness of diatony in the background and the fullness of tonality in the foreground are one and the same” (1979, 11). On one reading, Schenker may be asserting that that the process of composing-out mediates Diatonie and Tonalität, linking the empirically distant tonic of the former with the immediately sensible tonics of the latter via iterative prolongations (which have the potential, among other things, to transform Stufen into foreground keys). Another reading would have it that local and global tonics relate dialectically in Schenker’s theory: they are phenomenally divergent manifestations of a noumenal identity.

Tonic as Origin, Tonic as Goal

The Schenker of the 1906 Harmonielehre expresses a somewhat different view of tonics and tonicizations. He wonders at our propensity to take any consonant triad heard at the beginning of a piece as a tonic. Faced with examples in which this proves to be false—the finale of Beethoven’s Fourth Piano Concerto comes to mind (though Schenker does not cite it)—he wonders:

Should we conclude from these various possibilities that our assumption was erroneous to begin with? Or is our instinct rooted nevertheless in a natural cause?

It is the latter alternative that is correct. Our inclination to ascribe to any major or minor triad, first of all, the meaning of a tonic fully corresponds to the egotistic drive of the tone itself [dem Trieb und dem Egoismus des Tones], which…has to be evaluated from a biological point of view. This much is obvious: that the significance of the tonic exceeds that of the other scale steps, and these lose in value the farther they go from the tonic. Thus a scale-step does not aspire to the place of a VI or II in the system, but, on the contrary, it prefers to be a V at least, if not a I, a real tonic.37

The animistic language (the “egotistic drive of the tone”) is characteristic of Schenker, but here he employs it to describe the tonal effects of local chords rather than the generative processes of prolongation and composing-out, as in his later theories. The idea that all consonant chords aspire to tonic status suggests a view of tonal experience that is more volatile at the moment-to-moment level than his mature monotonal theories suggest.38

Schenker is of course not alone in asserting that listeners tend to take opening triads as tonics. Fred Lerdahl, whose cognitive theory is strongly based on Schenker’s, similarly argues that a chord or pitch heard in isolation will be taken as a tonic. But he roots the assertion in very different intellectual soil: the Egoismus des Tones gives way to the principle of cognitive economy, manifested by the “law of the shortest way”:

In this view, events are interpreted not only in the closest possible relation to one another but also in the closest proximity to a provisional tonic.

It follows that when a single note or chord sounds in isolation, the listener assumes that it is the tonic, for the shortest distance is from an event to itself. In terms of the algebraic pitch-space representation [Lerdahl’s “basic space,” which hierarchically arranges the 12 pitch classes in a manner resembling the Krumhansl-Kessler key profiles], the listener aligns the basic space to fit the pitch or chord in question, so that the pitch or chord is in the most stable position at the top of the hierarchy.

(Lerdahl 2001, 194)

Richard Cohn presents a similar view, though he is careful to specify its cultural and historical contingency:

For an acculturated listener, a major or minor triad, sounded in isolation and without prior context, signals the tonic status of its root by default. In a process first described by Gottfried Weber (1846 [1817–1821]), a listener spontaneously imagines an isolated triad housed within a diatonic collection, signifying a tonic that bears its name.

(Cohn 2012, 8)

Cohn is quick to note, however, that such provisional tonics

require confirmation, weakly through the remaining tones of its associated diatonic collection; more strongly by arranging those tones into a local cadence; more strongly yet by repeating that cadence, perhaps with supplementary rhetorical packaging, at the end of the movement or composition.

(Cohn 2012, 9)

Note that Cohn adduces closing gestures as the strongest means of confirming a hypothetical opening tonic. We thus arrive at one of the commonplaces of tonal theory: tonics often frame musical trajectories in time, acting as both origin and goal. Daniel Harrison elevates these to basic tenets in the “rhetoric of tonic”: “Tonic function ends a composition” and “Tonic begins compositional sections” (1994, 76–80).

That an opening tonic is more provisional than a closing one is also a commonplace of tonal theory. For this reason, some theorists have adopted skeptical positions about opening triads’ tonic status. For Schoenberg,

A triad standing alone is entirely indefinite in its harmonic meaning; it may be the tonic of one tonality or one degree of several others. The addition of one or more other triads can restrict its meaning to a lesser number of tonalities.

(Schoenberg 1954, 1)

Norman Cazden, writing in the same year, agrees:

[W]e may remark on the strategic logic by which a composition cannot begin on its tonic harmony. The work may be, let us say, in the key of C major, and it may begin with a simple C major chord, but there is no functional relationship as yet that makes us accept that chord as having a tonic role, and the further progress of the composition may easily demonstrate that it is really in another key.

(Cazden 1954, 25; quoted in Brown and Butler 1984, 9)

Few if any theorists have ever argued, however, that closing tonics are similarly provisional.

Though the key of a musical work is often derived by examining its opening and closing tonics, Cohn argues that these bounding tonics do not assure the tonal orientation of the music they bracket:

We can’t just go 〈B♭ major, Cough, Wheeze, Honk, B♭ major〉 and pretend that we have made coherent music in B♭ major (Straus 1987). If a tonal theory is to meet its claim of explanatory adequacy, it needs to be able to specify the role, with respect to the tonic, of the harmonies that separate the bounding tonics.

(Cohn 2012, 2).

TonicClick to view larger

Figure 10. Schubert, “Erster Verlust” (Goethe), D. 226.

Cohn is speaking here of enharmonically paradoxical chromatic progressions, but more or less diatonic passages and pieces can occasionally make the point too. Consider Schubert’s “Erster Verlust,” shown in Figure 10. The song opens and closes with an F-minor triad, and it has a key signature of four flats. Any young piano student who has learned a smattering of music theory will tell you that it is in F minor. Yet from the first measure the song wavers between F minor and A♭ major, the two keys taking on conventional semiotic roles: the former representing the desolate present, the latter a halcyon past. The keys infuse one another as the lyric subject broods on the (im)possibility of reanimating first love after it is lost. The singer’s tonal gestures point to self-deception: the opening vocal gesture sits more comfortably in A♭ major than in F minor, and the vocal line’s only two authentic cadences are in A♭ major (mm. 9 and 21). The piano has the last word, however, following the voice’s final cadence by drily reiterating the cadential gesture in F minor—an effect at once matter-of-fact and devastating. Rather than infusing one another, the two potential tonics now sit side by side, each represented by their most conventional generic signifiers of closure. The fluidly circulating and interpenetrating tonic effects of the song’s midsection have hardened and separated, as though we are sonic witness to a nascent dissociative disorder. To appreciate just how dissociative, note that one could remove the piano’s F-minor cadence, or replace it with some reiterated version of the A♭ cadence, without doing syntactical damage to the song. The meaning would change, to be sure—perhaps tilting toward the more pervasive delusion David Lewin (2006, 137–138) hears at the end of “Ihr Bild”—but the result would not violate tonal propriety. We would confidently analyze the recomposed song in A♭ major throughout, treating its opening F-minor triad not as tonic but as submediant (a role the song’s second chord seems to confirm). But now the tonal allegiance of all of the intervening music has shifted.

Of course, this is an exceptional song, perhaps even a limit case: the fact that it admits of such a richly bifocal tonal reading is a testament to Schubert’s often-celebrated psychological perspicacity. Yet the exercise should at least raise doubts about the relationship between opening and closing tonics and the music that falls between. This is in part a question of the perceptual status of monotonality, discussed previously. Many Schubert songs begin and end in different keys, after all, without offending listeners’ tonal sensibilities.39 As Brian Hyer has noted, “the dictum that pieces close on the original tonic was an aesthetic rather than a cognitive requirement” (2002, 742). But even in exceptional pieces that open and conclude with the same triad, like “Erster Verlust,” monotonality might not be assured.40 To adopt Cohn’s notation, a more accurate representation of the song might be: 〈F-minor triad, music poised between F minor and A♭ major, A♭-major cadence, F-minor cadence〉.

TonicClick to view larger

Figure 11. Schubert, String Quartet, D. 887, mvt. ii, mm. 52–60, reduced and annotated.

TonicClick to view larger

Figure 12. Two tonal strands in Schubert’s String Quartet, D. 887, mvt. ii, mm. 52–60.

It is nevertheless possible to construct a coherent and even compelling Schenkerian sketch of the song in F minor, as Carl Schachter does (1999b, 24, Ex. 1.7). In such a reading the F-minor tonic triad is prolonged for the entire song. But, as Schachter notes elsewhere (1987), prolongation and local sense of key can exhibit a certain independence from one another. In other words, to prolong a tonic does not mean to keep it always immediately before the listener’s ears. Another exceptional passage from Schubert makes the point vividly. The Sturm und Drang episode in mm. 43–60 of the slow movement of the composer’s final string quartet, D. 887, begins and ends in G minor. After confirming G minor via its dominant and subdominant in mm. 43–52, the music tumbles down a chain of minor thirds, touching on E minor (fleetingly), C♯ minor, and B♭ minor, before returning to G minor. Figure 11 illustrates. This equal-octave division, though tonally paradoxical (Cohn 1996, 2012), is hardly unprecedented: examples are legion not only in Schubert but throughout the nineteenth century. What is unprecedented is the alarming reiteration of the bounding G-minor triad’s root and third after each intermediate tonal station is reached. Though G-minor is the passage’s point of tonic origin and goal, and though Schenkerian analysts would likely read the progression as prolonging G minor,41 the prolonged chord’s phenomenal intrusion throughout the progression creates one of the most shocking effects in all of Schubert. Rather than unifying the music as tonic ambassadors, the insistent G–B♭ gestures seem to rip it into two tonal strands, as shown in Figure 12: sustained G minor (above) and descending-m3 (below). In mm. 54 and 56, local and bounding tonics enter into direct sonic conflict. While G minor’s role as point of origin and goal is beyond dispute, the passage makes palpable the gulf between tonic as theoretically prolonged and as immediately sensible.

Epilogue: Negotiating Popular Tonics

In March 2014 musician Owen Pallett published three articles on Slate.com analyzing songs by Katy Perry, Daft Punk, and Lady Gaga. In his discussion of Daft Punk’s 2013 single “Get Lucky,” he focuses in part on the four-chord loop that cycles through the entire song: Bm–D–F♯m–E.42 For Pallett, the loop is tonally ambiguous:

the song can be heard in two different keys. Most of the time it sounds as if it’s in the minor mode of F♯ Aeolian…

But the first chord of the progression isn’t F♯ minor, it’s B minor. The song slides smoothly back to it each time…The insistence of the B minor creates the aural illusion that the song could in fact be in the minor mode of B Dorian…

So, when the chord cycle comes back around to the beginning, the B minor, each time, the ear is tricked for a moment into thinking that the song is in a different key, a musical Tilt-a-Whirl. I am not going to lie: To my ears the song is clearly identifiable as F♯ minor, but on a Kinsey scale, I’d rate it a 3.

(Pallett 2014)

Pallett’s analysis generated considerable discussion in the article’s comments section, with many readers casting votes for various tonic candidates in his musical Tilt-a-Whirl. In addition to F♯ minor and B Dorian, several readers argued for A major, despite the A-major triad’s sonic absence in the song.43

The discussion is noteworthy in part for the various justifications the participants provide for one tonic hearing over another. Some appeal to music-theoretic principles: one poster (writing under the telling name Toccata) argues for an A-major hearing, in part because B minor lacks a leading tone and the quasi-Plagal cadence IV–i (E–Bm) is insufficient to establish a key. Poster Michael Curtis and others argue that Toccata’s criteria are too rooted in common-practice harmony: popular music in minor modes often eschews leading tones, and plagal progressions predominate in many pop idioms.44 As this suggests, the posters tend to base their judgments in their personal listening corpora: while Toccata seems to have a strong background in common-practice art music, others are clearly more literate in pop idioms, and one (Richard Worth) points to his jazz background as a possible reason for his preference to hear the song in B Dorian. Many also make more phenomenological appeals, urging their interlocutors to play the song at the piano, improvise over it, test certain pitches as tonics while listening to the changes, and so forth.

The participants’ widely varying subject positions are immediately evident. Some are musicians, some not; some are theoretically trained, others self-taught; some are literate in a wide range of vernacular musics, while others seem to have more parochial listening habits; and so on. The thickness of linguistic mediation is also striking: all of the negotiating and haggling over tonic effects is mediated through discourse, as participants deploy language (along with various illocutionary persuasion tactics) in an attempt to lead others’ ears toward their preferred hearings. The attempt to verbalize the progression’s sonic effect even seems to change some posters’ experience of its potential tonics. The interlocutors’ diverse backgrounds and competing discourses challenge any attempt to arrive at a consensus on the progression’s “true” tonic.

One might respond that this is clearly an ambiguous harmonic progression—we would not find such a wide variety of opinion in a less ambiguous example. While this is surely true, one final example will nevertheless suggest the extent to which fine-grained experiences of tonicness can vary even in ostensibly clear examples. For his 1999 book Metal, Rock, and Jazz: Perception and the Phenomenology of Musical Experience, Harris M. Berger conducted extensive interviews with two musicians about the tonal effects in their own songs, as part of his wide-ranging ethnographic fieldwork. Berger argues persuasively for the integrity, richness, and complexity of individuals’ situated musical experiences: “An entire dissertation could be written on one type of musical phenomenon experienced by one player” (1999, 174). As Berger notes, however, it is challenging to articulate that experience via a feedback interview:

At its best, the process [is] a collaboration, but such collaborations are asymmetrical and complex. The constant goal in the process is to share the participant’s experiences. On the one hand, this sharing can never be complete. Because both perceptual experiences and their interpretations are situational and historical, the participant’s experience is never the same twice, and its richness always exceeds even the participant’s best description. Further, the ethnographer constantly influences both the descriptions and the experiences themselves. On the other hand, partial sharing is possible; the same richness that makes experience exceed its descriptions enables the ethnographer to engage actively in the interpretation and even suggest lines of inquiry. Over time, the ethnographer can learn to engage with the world in ways similar to those of the participants.

(Berger 1999, 175)

The process of collaboration Berger describes is evident in his interviews with Chris Ozimek, guitarist and songwriter for the Akron-based, early-1990s commercial hard rock band Dia Pason. For example, Berger cannot begin to understand Ozimek’s experience of the song’s tonal character until he introduces some basic music-theoretic concepts and terminology that they can share. Specifically, he describes the tonic as feeling “at rest” and asks Ozimek which chords in the band’s song “Turn for the Worse” seem to him to act as tonics.

Ozimek states that the introduction—which is made up of syncopated riffs surrounding an A power chord—is centered on an A tonic. As Berger notes, “given the conventions of Western music theory, this was no great surprise” (1999, 187). But things become more interesting in the verse. When Ozimek sings the song’s first line, the harmony shifts to a D power chord. Ozimek also hears this as a tonic, despite Berger’s repeated attempts to suggest that it might sound like IV:

Pushing the point further and bringing it back to the question of tonality, I asked again if this D chord feels like the IV chord of a tonic A or if it feels like a I chord on its own. Chris said that on stage, when he has finished singing the first vocal phrase…, he jumps away from the mike to allow the audience’s focus to shift to the guitar; then, when the second vocal phrase of the verse comes up, he moves back to the mike to sing. As a result, he said, the vocal phrase with the D chord…and the guitar part…feel very distinct. Given all of this, we concluded that moving into the verse, D was the tonal center while it lasted.

(Berger 1999, 187–188)

Ozimek’s response to Berger’s prodding is fascinating: his tonal experience of the two chords is inseparable from his somatic experience as a performer. The two tonics correspond to his two positions on stage and to his shifting communicative stances with respect to the audience.

Theorists may well be incredulous reading such an account: perhaps Ozimek simply is not sufficiently attuned to the theoretical distinctions between a I chord and a IV chord to report on a phenomenological difference he must surely experience. Scholars in music cognition and perception might further argue that introspection in such matters is unreliable and should be replaced by more controlled experimentation, with linguistic mediation is kept to a minimum.45 But Berger’s ethical injunctions about taking seriously the situated complexity of participants’ musical experiences should give us pause, as should the ecological integrity of Ozimek’s account. Clearly, whatever these tonic sensations might be for Ozimek, they are inseparable from his somatic, performatively scripted experience as player and singer. We are reminded that our experience of any tonic, however unshakeable or inevitable it might feel, is as subject to the contingencies of culture, history, embodied practice, and musical occasion as is any other musical apperception. A more careful consideration of those contingencies might reveal ways in which they nourish rather than occlude tonal experience in our many daily acts of musicking.

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Notes:

(1) Other scholars in music cognition have explored the relationship between the statistical prevalence of pitch classes and their tonal status. Marvin and Temperley (2008) provide a valuable survey of this literature. See also Aarden (2003) and Huron (2006) for further discussion of the distributional model and its relationship to tonal hierarchy, syntax, and affect. We explore a distributional key-finding algorithm in Section III.

(2) The repeated D3 asserts a tactus, to be sure, but no clear metrical hierarchy.

(3) In my own listening, I hear such relationships only fleetingly. For example, the very first non-D event—a low tremolo F2—strikes my ear momentarily as a minor scale degree 3, though the effect is significantly attenuated by the rhetorical disjunction. And it is difficult for me to hear the scurrying figures immediately following the F2 in relation to the opening D. Only occasional later events carry a similar tonal charge for me; the most notable is the almost cadential tremolo figure on D4 at 1:18. Indeed, this is the closest moment to a phrase ending in the passage (at least to my ears), underscoring Krumhansl’s comments.

(4) On the richly metaphorical language that often arises in discussions of tonal effects, see Hyer (2002, 728–733). Such metaphors are surprisingly common in the empirical literature (both in experimental prompts and in definitions). David Huron (2006, 422), for example, first defines the tonic straightforwardly as “The first scale degree in the Western major and minor scales.” But when he reaches in the next sentence for the phenomenological effect of such tonics, metaphors take over: “The pitch in a scale that sounds most stable or closed.” Earlier in the book (145), Huron provides a fascinating account of the poetic and figurative language that theoretically knowledgeable listeners use to characterize the effect of different scale degrees. The metaphoric field for the tonic includes references to stability, pleasure, home, contentment, satisfaction, solidity, and strength.

(5) At best, we might note that C♯ and E♭ enjoy a slight edge, with E, F, and B tied for second and the remaining pitch classes trailing in a clump. Compared to the stratospheric D, however, these are negligible distinctions. Six-flat apartment buildings and single-family homes all that different in the shadow of a skyscraper.

(6) The recorded excerpt is performed by Heinrich Schiff (cello) and the Bavarian Radio Symphony, conducted by the composer (Philips 416 817–811).

(7) For examples of psychoanalytical hearings of “uncanny” tonal effects in late Schoenberg, see Cherlin (1993, 2007) and Kurth (2001). See also Straus’s (1990) Bloomian interpretation of such effects.

(8) For two classic examples of this line of argument, see Forte (1972, 1981). The latter—an analysis of Schoenberg’s op. 11, no. 1—appears alongside an explicitly tonal interpretation of the same piece (Ogdon 1981).

(9) This liberal extension of the concepts of tonic and tonality to non-Western music is relatively common in the music cognition literature. See, for example, Castellano, Bharucha, and Krumhansl (1984); Bharucha (1984); and Krumhansl (1990, 253–270, 2004). Such findings have led W. Jay Dowling (1984, 417) to assert that “the musical and psychological phenomenon of tonality as an organizing principle may be all but universal across cultures.”

(10) For some interpreters it might also fail for the broader historical/stylistic arguments presented previously, but for the moment we will bracket such extra-phenomenological considerations.

(11) For further exploration of this idea, which I refer to as “tonal intention,” see Rings (2011).

(12) Riemann (1990), with his emphasis on harmony and chord, stresses tonic-as-chord in his own definition of “Tonika” in the fifth edition of his Musik-Lexikon. After indicating that the word usually refers to “the pitch after which the key is named, that is, C in C major, G in G major, etc.,” he adds: “the newer harmonic theory [viz., his own] nevertheless understands by the word tonic the tonic triad, that is, the C-major chord in C major, the C-minor chord in C minor” (Riemann 1990, 1151, my translation).

(13) “Chroma” is the term music cognition scholars use for the perceptual property that all octave-related pitches share.

(14) David Temperley (2011) refers to such enharmonically distinct pitch classes as “tonal pitch classes,” while Julian Hook (2011) uses the term “spelled pitch classes.”

(15) The most-cited study on this topic is Cook (1987), though Robert Gjerdingen (1999, 164–166) has pointed out significant methodological flaws in Cook’s article. For a more experimentally sound study of long-range tonal audition, see Marvin and Brinkman (1999).

(16) On the distinction between analyses that seek to model prereflective experience and those that seek to stimulate new experiences, see Temperley (1999).

(17) The inclusion of both “arranged” and “perceived” points to both the compositional/performative (or poietic) aspects of tonal music making and their phenomenological (or esthesic) results. See Nattiez (1990).

(18) Hyer’s (2002) seminal article is a definitive account of tonality as a theoretical, historical, discursive, and ideological category. The essay contains much relevant material on tonic as a concept and a phenomenon. In order not to duplicate Hyer’s work, I have chosen not present a detailed historical survey of the term. For more on that history, see Gut (1976) and Beiche (1992).

(19) Only the final cadence, which curls up to the tonic with its last note, is an unambiguous perfect authentic cadence. The first three may be perfect or imperfect, depending on how one reads the implied upper voice; thus the neutral label “Cad” in the figure.

(20) Here I paraphrase Scott Burnham’s (1999) evocative description of Schubert’s tonally off-kilter secondary key areas.

(21) Temperley (2011, 7); for a similar argument, see Temperley (1999, 79–82). See also Huron (2006, 167): “One important complication of my account [of tonal qualia] is that people differ.” Huron then discusses differences in individual temperament but also cites psychological studies that argue for certain cross-cultural similarities in statistical learning and tonal experience (168–172).

(22) Seminal studies on the perception of key (sometimes called “tonality induction”) include Longuet-Higgins and Steedman (1971); Krumhansl and Kessler (1982); Krumhansl (1990, which discusses Schmuckler’s contribution); Brown and Butler (1984); and Brown, Butler, and Jones (1994). For a useful overview and literature survey see Vos (2000), along with the articles that follow in the summer 2000 issue (vol. 17, no. 4) of Music Perception.

(23) Temperley made his adjustments to Krumhansl’s tonal hierarchies in an effort to improve the performance of the Krumhansl-Schmuckler key-finding algorithm. For justification and an explanation of his changes, see Temperley (2011, 176–181). The Krumhansl-Kessler hierarchies and those derived from them (such as Temperley’s) do not recognize enharmonic differences in spelling, thus inhabiting a 12-pc universe. It is possible to add enharmonic distinctions later to the key-finding algorithm—as Temperley does—but I have chosen not to do so to for simplicity of demonstration.

(24) Huron (2006) presents a sustained argument for statistical learning as foundational to musical experience, though he critiques aspects of the Krumhansl-Kessler key-profile model, as we will see. Huron surveys a wide range of empirical studies that have explored cross-cultural differences and commonalities in the hearing of tonal phenomena (broadly construed) in world musics. He concludes (tentatively) that statistical learning underwrites such processes across cultures. See Huron (2006, 168–172).

(25) The values for the pitch profiles in Figure 5 were derived by tallying the duration of each pitch class in the Sarabande in eighth notes (the piece’s smallest rhythmic subdivision). The key profiles were then adjusted so that their maxima matched those of the input vectors.

(26) Some readers may find this surprising, as I did. The tonally off-center first bar, with its emphasized A♭ and fleeting C, can create such a vivid impression—standing as a seal over the entire Sarabande—that one can develop the (mistaken) impression that C is deemphasized throughout the piece. The statistical note-count in Figure 5(a) belies this, revealing that one of the means whereby Bach rights the tonal balance by movement’s end is by assuring C’s statistical dominance as the music progresses.

(27) This is a simplification of the formula in the Krumhansl-Schmuckler key-finding algorithm. See Temperley (2011, 175–176).

(28) On the justification for using measures as the smallest segments, see Temperley (2011, 189).

(29) Temperley (2011, 191) does not provide a method to determine the precise value of the modulation penalty, and he admits to experimenting with various values in his tests and then selecting the value “which seemed to yield the best performance.” I employed the same procedure in this study to arrive at the penalty of –4. For an alternative approach to modulation in a key-finding algorithm, see Huron and Parncutt (1993), which draws on echoic memory.

(30) The green cells in bar 8 seem to muddy the cadential clarity here, but this is visually misleading: there are four such cells because the key-finding readings are very coarse in this bar, given that it only includes one pitch. Moreover, the difference between the high scores (30) and the runners-up (27) in this column is greater than in any other bar.

(31) Aarden’s (2003) research also challenges the validity of the Krumhansl-Kessler profiles as models of statistical distribution, as the most prominent pitch class in many tonal corpora is not the tonic but the dominant. He thus proposes that the Krumhansl-Kessler profiles instead model phrase-final stability (or closure). See also the discussion in Huron (2006, 147–153).

(32) On the “controversial” but “characteristically Schenkerian” delay of the Kopfton scale degree 3 until its arrival over a cadential six-four, see Schmalfeldt (2011, 44). For an instance of such an analysis from Schenker, see Free Composition (Schenker 1979, Figure 40, 7).

(33) Among other things, one could adduce Bach’s arrangement of the Sarabande for lute (BWV 995). In Bach’s manuscript, these and the other bass pitches are placed on a separate lower staff, to be played on the lute’s ringing, open bass courses (or “bourdons”).

(34) See Schachter (1987) and Rings (2011, 158n12).

(35) But see Schachter (1999a) for a consideration of the ways that a background structure can at times impinge on (or manifest itself in) the foreground.

(36) Cf. Schachter (1999b, 184).

(37) Schenker (1906, 333). For the original German see Schenker (1954, 252).

(38) It also prefigures Schoenberg’s comments in his own Harmonielehre (written five years later) regarding the efforts of nontonic tonal regions to usurp the tonic’s control, though here political/military metaphors replace Schenker’s animism. See Schoenberg (1978, 150–153) and Hyer (2002, 731).

(39) This is also the case with popular songs that involve “truck-driver modulations”: wholesale modulations at the end of a song, often up by semitone (like gears shifting). Patrick McCreless (1996, 106) refers to this phenomenon as “Barry Manilow tonality.”

(40) “Erster Verlust” is arguably a better example of “tonal pairing” (Kinderman and Krebs 1996) than some often-cited examples that begin and end in different keys, as Schubert holds F minor and A♭ major in almost perfect equilibrium. For a similar reading of tonal pairing in a Schubert song that begins and ends with the same triad, see Harald Krebs’s (1996) discussion of “Meeres Stille.”

(41) See, e.g., Schenker (1979, Figure 114, 8); and Salzer and Schachter (1969, 215–218, esp. Example 7-71a).

(42) In the interest of accessibility, Pallett (2014) transposes the harmonies in his discussion to the white keys: Dm–F–Am–G (much to the consternation of some readers who posted comments). In quotes from Pallett I have substituted the actual, sounding chord names for his transposed ones.

(43) The scholarly literature on harmony, tonality, and modality in popular music has grown considerably in recent decades. Several studies are of relevance for the discussion that follows, including Biamonte (2010), de Clercq and Temperley (2011), Doll (2007), Everett (2004, 2009), Moore (2012), Nobile (2013), Tagg (2009), Temperley (2011), and Tymoczko (2014).

(44) A point supported by Tymoczko (2014).

(45) Some music cognition scholars nevertheless rely on introspection. David Temperley (1999, 78), for example, describes his working method for his own study of tonality in rock music thus: “I examine my intuitions as to what the tonal center is in many rock songs, assuming that these intuitions are the same as those of most other listeners.” Berger’s (1999) ethnography serves as a caution against such assumptions.