Tonal Hierarchy Research Articles

Tracing the dynamic changes in perceived tonal organization in a spatial representation of musical keys.

The cognitive representation of harmonic and tonal structure in Western music is investigated using a tone-profile technique. In this method listeners rate how well single tones (any one of the 12 tones of the chromatic scale) follow a musical element such as a scale, chord, or cadence. Very stable rating profiles reflecting the tonal hierarchies in major and minor keys are obtained, which, when intercorrelated and analyzed using multidimensional scaling, produce a four-dimensional spatial map of the distances between keys. The keys are located on the surface of a torus, in which the circle of fifths and the parallel and relative relations between major and minor keys are represented. In addition, single chords (major, minor, diminished, and dominant seventh) are found to be closely associated with the major and minor keys in which they play harmonic functions. The developing and changing sense of key during sequences of chords is traced by obtaining probe tone ratings following each chord in 10 different sequences, 8 of which contain modulations (changes) between keys. Modulations between closely related keys are found to be effected more immediately than are modulations between relatively distant keys. In all cases beyond the initial chord, the sense of the prevailing key is stronger than that produced by the last heard chord in isolation. Thus, listeners integrate harmonic functions over multiple chords, developing a sense of key that may need to be reevaluated as additional chords are sounded. It is suggested that the perceived relations between chords and keys and between different keys are mediated through an internal representation of the hierarchy of tonal functions of single tones in music. Music consists of tones varying in pitch, serve to highlight rhythmic patterns, further duration, loudness, and timbre, but the per- emphasize phrase structure, and distinguish ception of music extends well beyond the between tones constituting the primary meregistration of these physical attributes of lodic line and tones serving more ornamental the musical stimulus. Indeed, music contains or harmonic functions. Timbral characterconsiderable structure even in the relations istics may additionally provide important that obtain among the individual tones. For cues for the overall structure of the musical example, the durations are such that metri- composition.

Quantification of the hierarchy of tonal functions within a diatonic context.

Listeners rated test tones falling in the octave range from middle to high C according to how well each completed a diatonic C major scale played in an adjacent octave just before the final test tone. Ratings were well explained in terms of three factors. The factors were distance in pitch height from the context tones, octave equivalence, and the following hierarchy of tonal functions: tonic tone, other tones of the major triad chord, other tones of a diatonic scale, and the nondiatonic tones. In these ratings, pitch height was more prominent for less musical listeners or with less musical (sinusoidal) tones, whereas octave equivalence and the tonal hierarchy prevailed for musical listeners, especially with harmonically richer tones. Ratings for quarter tones interpolated halfway between the halftone steps of the standard chromatic scale were approximately the averages of ratings for adjacent chromatic tones, suggesting failure to discriminate tones at this fine level of division.

Stravinsky's ‘The Flood’

By now we are used to the tonal polarisation of Stravinsky's serial writing, but the resourcefulness with which he continues to throw fresh light on the method is amazing. In The Flood we have possibly the most overt example of his tonal twelve-note style so far, with what can be construed as an elaborate series of cadences on and around a final of C sharp. The five forms of the basic series shown in Ex. 1 illustrate what I mean, for they are used almost exclusively throughout the work. Taking the first and last notes in each case, we find a quasi tonal hierarchy outlined, with C sharp, its subdominant, dominant and supertonic. Significantly the form, O-F sharp and its retrograde which on these terms would have hinted at a flatter region are not so much in evidence. Even when cellular correspondences such as that which is obtained between notes 1–3 of O and I can be cited in justifying simultaneous presentation or continuation, we feel that considerations of cadence have really been uppermost in the composer's mind. Restriction to the five forms set out in Ex. 1 would of course have imposed too severe a discipline for a work lasting half an hour, and the way in which Stravinsky avails himself of variations is thoroughly typical. He does not utilise ‘enharmonic modulation’ by means of common cells : instead we find a standard permutation procedure in which notes 1 and 7 of a row become 6 and 12, or vice versa (Ex. 2). It is instructive that the particular forms chosen for permutation are frequently those which will yield the same cadential results as the forms shown in Ex. 1. Thus I-E as permutated in Ex. 2 is in its new form bounded by G sharp and C sharp, revealing affinities with O-G sharp as well as the more obvious I-G sharp (the shared final and almost identical shape). Ex. 3 shows one of the simpler uses of the permutation process for melodic variation and expansion. Here notes 6–12 of O-G sharp are followed by notes 6–12 of the substitute O-G sharp (permutated from O-C, with 6 and 12 becoming 1 and 7). He will also on occasion re-order to no particular plan beyond that of obtaining aural results which accord with the tonality of the stricter forms of this row.