A Theory of Tonal Hierarchies in Music

musical structure of a culture or genre” (Bharucha 1984, p. 421). So, unlike tonal hierarchies that refer to cognitive representations of the structure of music across different pieces of music in the style, event hierarchies refer to a particular piece of music and the place of each event in that piece. The two hierarchies occupy complementary roles. In listening to music or music-like experimental materials (melodies and harmonic progressions), the listener responds both to the structure provided by the tonal hierarchy and the structure provided by the event hierarchy. Musical activity involves dynamic patterns of stability and instability to which both the tonal and event hierarchies contribute. Understanding the relations between them and their interaction in processing musical structure is a central issue, not yet extensively studied empirically.  The Basic Studies This section outlines the classic findings that illustrate tonal relationships and the methodologies used to establish these findings. 3.3.1 The Probe Tone Method Quantification is the first step in empirical studies because it makes possible the kinds of analytic techniques needed to understand complex human behaviors. An experimental method that has been used to quantify the tonal hierarchy is called the probe-tone method (Krumhansl and Shepard 1979). It was based on the observation that if you hear the incomplete ascending C major scale, C-D-E-F-G-A-B, you strongly expect that the next tone will be the high C. It is the next logical tone in the series, proximal to the last tone of the context, B, and it is the tonic of the key. When, in the experiment, incomplete ascending and descending scale contexts were followed by the tone C (the probe tone), listeners rated it highly as to how well it completed the scale (1 = very badly, 7 = very well). Other probe tones, however, also received fairly high ratings, and they were not necessarily those that are close in pitch to the last tone of the context. For example, the more musically trained listeners also gave high ratings to the dominant, G, and the mediant, E, which together with the C form the tonic triad. The tones of the scale received higher ratings than the nonscale tones, C# D# F# G# and A#. Less musically trained listeners were more influenced by how close the probe tone was to the tone sounded most recently at the end of the context, although their ratings also contained some of the tonal hierarchy pattern. A subsequent study used this method with a variety of contexts at the beginning of the trials (Krumhansl and Kessler 1982). Contexts were chosen because they are clear indicators of a key. They included the scale, the tonic triad chord, and chord 56 C.L. Krumhansl and L.L. Cuddy sequences strongly defining major and minor keys. These contexts were followed by all possible probe tones in the 12-tone chromatic scale, which musically trained listeners were instructed to judge in terms of how well they fit with the preceding context in a musical sense. The results for contexts of the same mode (major or minor) were similar when transposed to a common tonic. Also, the results were largely independent of which particular type of context was used (e.g., chord versus chord cadence). Consequently, the rating data were transposed to a common tonic and averaged over the context types. The resulting values are termed standardized key profiles. The values for the major key profile are 6.35, 2.23, 3.48, 2.33, 4.38, 4.09, 2.52, 5.19, 2.39, 3.66, 2.29, 2.88, where the first number corresponds to the mean rating for the tonic of the key, the second to the next of the 12 tones in the chromatic scale, and so on. The values for the minor key context are 6.33, 2.68, 3.52, 5.38, 2.60, 3.53, 2.54, 4.75, 3.98, 2.69, 3.34, 3.17. These are plotted in Fig. 3.1, in which C is assumed to be the tonic. Both major and minor contexts produce clear and musically interpretable hierarchies in the sense that tones are ordered or ranked according to music-theoretic descriptions. The results of these initial studies suggested that it is possible to obtain quantitative judgments of the degree to which different tones are perceived as stable reference tones in musical contexts. The task appeared to be accessible to listeners who differed considerably in their music training. This was important for further investigations of the responses of listeners without knowledge of specialized vocabularies for describing music, or who were unfamiliar with the musical style. Finally, the results in these and many subsequent studies were quite consistent over a variety of task instructions and musical contexts used to induce a sense of key.  Probe tone ratings for a C major context. (b) Probe tone ratings for a C minor context. Values from Krumhansl and Kessler (1982) 57 3 A Theory of Tonal Hierarchies in Music of the tonal hierarchies is an important first step in empirical research but, as seen later, a great deal of research has studied it from a variety of different perspectives. 3.3.2 Converging Evidence To substantiate any theoretical construct, such as the tonal hierarchy, it is important to have evidence from experiments using different methods. This strategy is known as “converging operations” (Garner et al. 1956). This section describes a number of other experimental measures that show influences of the tonal hierarchy. It has an effect on the degree to which tones are perceived as similar to one another (Krumhansl 1979), such that tones high in the hierarchy are perceived as relatively similar to one another. For example, in the key of C major, C and G are perceived as highly related, whereas C# and G# are perceived as distantly related, even though they are just as far apart objectively (in semitones). In addition, a pair of tones is heard as more related when the second is more stable in the tonal hierarchy than the first (compared to the reverse order). For example, the tones F#-G are perceived as more related to one another than are G-F# because G is higher in the tonal hierarchy than F#. Similar temporal-order asymmetries also appear in memory studies. For example, F# is more often confused with G than G is confused with F# (Krumhansl 1979). These data reflect the proposition that each tone is drawn toward, or expected to resolve to, a tone of greater stability in the tonal hierarchy. Janata and Reisberg (1988) showed that the tonal hierarchy also influenced reaction time measures in tasks requiring a categorical judgment about a tone’s key membership. For both scale and chord contexts, faster reaction times (in-key/outof-key) were obtained for tones higher in the hierarchy. In addition, a recency effect was found for the scale context as for the nonmusicians in the original probe tone study (Krumhansl and Shepard 1979). Miyazaki (1989) found that listeners with absolute pitch named tones highest in tonal hierarchy of C major faster and more accurately than other tones. This is remarkable because it suggests that musical training has a very specific effect on the acquisition of absolute pitch. Most of the early piano repertoire is written in the key of C major and closely related keys. All of these listeners began piano lessons as young as 3–5 years of age, and were believed to have acquired absolute pitch through exposure to piano tones. The tonal hierarchy also appears in judgments of what tone constitutes a good phrase ending (Palmer and Krumhansl 1987a, b; Boltz 1989a, b). A number of studies show that the tonal hierarchy is one of the factors that influences expectations for melodic continuations (Schmuckler 1989; Krumhansl 1991,  Cuddy and Lunney 1995; Krumhansl et al. 1999, 2000). Other factors include pitch proximity, interval size, and melodic direction. The influence of the tonal hierarchy has also been demonstrated in a study of expressive piano performance (Thompson and Cuddy 1997). Expression refers to 58 C.L. Krumhansl and L.L. Cuddy the changes in duration and dynamics (loudness) that performers add beyond the notated music. For the harmonized sequences used in their study, the performance was influenced by the tonal hierarchy. Tones that were tonally stable within a key (higher in the tonal hierarchy) tended to be played for longer duration in the melody than those less stable (lower in the tonal hierarchy). A method used more recently (Aarden 2003, described in Huron 2006) is a reaction-time task in which listeners had to judge whether unfamiliar melodies went up, down, or stayed the same (a tone was repeated). The underlying idea is that reaction times should be faster when the tone conforms to listeners’ expectations. His results confirmed this hypothesis, namely, that reaction times were faster for tones higher in the hierarchy. As described later, his data conformed to a very large statistical analysis he did of melodies in major and minor keys. Finally, tonal expectations result in event-related potentials (ERPs), changes in electrical potentials measured on the surface of the head (Besson and Faïta 1995; Besson et al. 1998). A larger P300 component, a positive change approximately 300 ms after the final tone, was found when a melody ended with a tone out of the scale of its key than a tone in the scale. This finding was especially true for musicians and familiar melodies, suggesting that learning plays some role in producing the effect; however, the effect was also present in nonmusicians, only to a lesser degree. This section has cited only a small proportion of the studies that have been conducted on tonal hierarchies. A closely related issue that has also been studied extensively is the existence of, and the effects of, a hierarchy of chords. The choice of the experiments reviewed here was to illustrate the variety of approaches that have been taken. Across the studies, consistent effects were found with many different kinds of experimental