Rootedness in Conventional Tonality; Some Remarks on Historical

Let us next consider how the two above-discussed psychoacoustical factors of consonance, critical band and virtual pitch, pertain to the consonance system of conventional tonality.⁹⁵ One way to explain that system on the basis of these phenomena is the following. First, the intervals that produce the greatest critical-band effects (roughness), minor and major second and the tritone, are excluded, together with their octave generalizations. This leaves the major and minor triad as the only possible three-pitch-class collections within the diatonic set. Second, the registral position of the interval class most strongly determinative of virtual pitch, i. e, 5 (fifth/fourth), determines the stability of different triad inversions. If this interval class occurs as fb-interval 7 (fifth), in accordance with the harmonic series, the chord (5/3) is fully stable. If it occurs as fb-interval 5, whose “root-opposing” quality was discussed above, the chord (6/4) is unstable. If it occurs between upper voices, the stability of the resulting chord (6/3) is between these extremes.

Given that the concept of rootedness has its origins in conventional tonality, its pertinence to the traditional consonance system, as evident in the above explanation, may almost seem surprisingly slight. No root support besides the strongest—the fifth, or fb-interval 7—needs to be invoked in this explanation. However, while the position of interval class 5 is indeed decisive for the basic consonance system of conventional tonality, there are some more subtle indications

95 Explaining consonance systems on the basis of these two factors follows Terhardt's (1984) “two-component theory of musical consonance.” About other purported explanations of consonance and dissonance, see, for example, the discussion in Plomp and Levelt 1965.

that composers were also sensitive to the effects of weaker root supports. Let us first consider fb-interval 4 (major third). Since this interval is included in the major triad but not in the minor, the former is more strongly rooted than the latter. While this difference is not reflected in the basic consonance rules—major and minor triads are both stable—it is in several ways reflected in compositional practices, and also in the expressive connotations of major and minor. As for compositional practices, it may be first noted that altering minor chords to major by mixture is much more significant than the reverse procedure. The dominant occurs regularly as a major chord in minor, and the major tonic often occurs in endings (Picardy third). The reverse phenomena hardly ever occur. Modulations from minor to major keys (usually to the III) are also more significant than the reverse. All in all, as observed by numerous theorists, the major and minor triads are not entirely “equal” in compositional practices; there is a certain bias in favor of major. The conventional view that the expressive quality of the major triad is more

“happy” or “satisfied” is also well in accord with considerations of rootedness.

If the root-supporting effect of fb-interval 4 thus has some significance for the practices of tonal music, how about the weaker root supports? Fb-intervals 10 and 2 (minor seventh and major ninth) cannot occur in consonances under the principle that excludes interval class 2.

However, the minor seventh occurs alongside the stronger root supports in the most important dissonant chord in tonal music, the dominant seventh chord. The next conventional enlargement of this chord (in major) is the major ninth chord. Hence, these enlargements utilize root supports in the order of root-supporting weights. As is well known, Schenker vehemently renounced the connection between the seventh in the V⁷ and the seventh harmonic—to say nothing of the ninth—and sought to explain all dissonances on a purely linear basis.⁹⁶ However, linear functions and sonorous appeal do not rule each other out. While dominant sevenths and ninths require linear resolutions to triads in conventional tonality, it seems highly probable that the enormous popularity of such harmonies, and their tendency to govern extended stretches of time in Classic-Romantic music, had something to do with sonorous properties that stem from the connection with the harmonic series.

In this connection, one should also mention augmented-sixth chords, whose construction

96 Schenker 1906/1954 (§ 11) already declares that “no overtone beyond the fifth in the series has any application to our tonal system.” Similarly, Schenker 1935/1979 (§ 176) asserts that “where the seventh is not a suspension […] it is a passing tone. As such it has not the slightest relationship to the seventh overtone, which many textbooks hold to be identical with the seventh.” These quotes make evident that Schenker rejected the connection between the seventh and the seventh harmonic, but some other passages suggest that he went so far as to reject any kind of significance of the seventh as a vertical relationship. For example, Schenker 1926/1996 (9) states: “Therefore it contradicts the nature of the dissonant passing note to discriminate in any substantial way among the intervals of a fourth, a seventh and a ninth, to say nothing of positing an increasing scale of dissonance for these intervals: the vertical dimension is altogether excluded, everything hinges on the horizontal tension alone.” (My italics.) While Schenker actually discusses dissonant passing tones in strict counterpoint in this excerpt, the paradigmatic significance of strict counterpoint for his theories permits us to assume that more or less similar considerations apply to free composition. On the other hand, several scholars have observed the inconsistency between Schenker's theoretical statements and analytical practices concerning the position of the V⁷; see, for example, Ernst Oster's footnote in Schenker 1935/1979: 64, and Schachter 1999: 201–2.

resembles dominant sevenths (German sixth is enharmonically equivalent with V⁷). Like dominant sevenths and ninths, augmented-sixth chords certainly have a linear function but they, too, tend to assume much greater expressive and structural significance than would be characteristic of arbitrary dissonances produced by linear motions.

Whereas Schenker rightfully criticized conventional harmonic analysis for neglecting linear aspects of music, it is also unsatisfactory to go to the other extreme by renouncing the significance of dominant sevenths and ninths and augmented sixths as vertical constructions, just because their syntactic role depends on linear connections. That theorists have been inclined to give special labels to these chords in harmonic analysis reflects the fact that they “super-ficially” appear to be significant also as verticalities. Important as it is to see through such an appearance and recognize their eventual linear function, “superficial,” temporary appearances may also be vital for expressive qualities. Let the particularly eloquent Brahms passage in Example 16 suffice as illustration. Observe how the there is a pedaled pianissimo ninth chord on VI in mm. 19–24 appears as if temporarily ignorant of its linear origins or obligations—or, more hermeneutically, of the agony of the real world. A return to the latter is then brought about in m. 25 by the more or less normal resolution of the German sixth. For understanding the passage, it is essential to recognize both the conventional resolution and the preceding temporary suggestion of sonorous self-sufficiency.⁹⁷

EXAMPLE 16. A Brahms passage exploiting the sonorous qualities of a ninth chord

! examples of II and III, is not without prehistory in tonal music. Before being integrated into

97 For an extensive argument for the significance of different temporal perspectives in music analysis, see Lewin 1986.

unconventional referential harmonies, their sonorous features were tried out in non-structural (dissonant) harmonies in conventional tonality.⁹⁸ Such an evolutionary process is also evident in the stylistic evolution of individual composers, such as Scriabin. As observed by several scholars (e.g. Baker 1986), the II–V spans with “French sixth” type sonorities, characteristic of his “tonal” period, foreshadow similar structural frameworks in many of his “atonal”

works (cf. Example 7c). However, phases in such an evolutionary process are not always strictly chronological. As discussed in III, Debussy's mature work, such as the Préludes, contains cases in which non-triadic harmonies become referential for entire compositions (such as Voiles and Ce qu'a vu le vent d'ouest) alongside cases in which similar harmonies assume significance in more limited spans, being subordinate to triads in the overall organization.