Diatonic scale

Abstract: 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 the 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. The study of perceived pitch and of the perceived relations between tones differing in pitch has generally been approached from one of two quite different traditions: the psychoacoustic and the musical. The psychoacoustic approach has typically focused on simple, physically specifiable properties of tones isolated from any musical context— properties of frequency, separation in log frequency, or simplicity of integer ratios of frequencies. The results of such studies have provided some precise information about how the ear responds to isolated tones or tones in random sequences. We believe that they have been less informative with regard to how the listener perceives tones in organized musical sequences. Music theory suggests that the perception of such sequences may rely on the listener's sensitivity to different and structurally richer principles associated with tonal and diatonic organization. Such principles are useful in explaining the cognitive phenomena of reference point, motion, tension, and resolution that underlie the dynamic force of virtually all tonal music. They have, however, been subjected to relatively little systematic laboratory investigation or quantitative formulation.

Abstract: This paper aims to propose a hierarchical, generative account of diatonic harmonic progressions and suggest a set of phrase-structure grammar rules. It argues that the structure of harmonic progressions exceeds the simplicity of the Markovian transition tables and proposes a set of rules to account for harmonic progressions with respect to key structure, functional and scale degree features as well as modulations. Harmonic structure is argued to be at least one subsystem in which Western tonal music exhibits recursion and hierarchical organization that may provide a link to overarching linguistic generative grammar on a structural and potentially cognitive level.

Abstract: SUMMARY Cross-culturally, most music is tonal in the sense that one particular tone, called the tonic, provides a focus around which the other tones are organized. The specific orga- nizational structures around the tonic show considerable diversity. Previous studies of the perceptual response to Western tonal music have shown that listeners familiar with this musical tradition have internalized a great deal about its underlying organization. Krumhansl and Shepard (1979) developed a probe tone method for quantifying the perceived hierarchy of stability of tones. When applied to Western tonal contexts, the measured hierarchies were found to be consistent with music-theoretic accounts. In the present study, the probe tone method was used to quantify the perceived hierarchy of tones of North Indian music. Indian music is tonal and has many features in common with Western music. One of the most significant differences is that the primary means of expressing tonality in Indian music is through melody, whereas in Western music it is through harmony (the use of chords). Indian music is based on a standard set of melodic forms (called rags), which are themselves built on a large set of scales (thats). The tones within a rag are thought to be organized in a hierarchy of importance. Probe tone ratings were given by Indian and Western listeners in the context of 10 North Indian rags. These ratings confirmed the predicted hierarchical ordering. Both groups of listeners gave the highest ratings to the tonic and the fifth degree of the scale. These tones are considered by Indian music theorists to be structurally significant, as they are immovable tones around which the scale system is constructed, and they are sounded continuously in the drone. Relatively high ratings were also given to the vadi tone, which is designated for each rag and is given emphasis in the melody. The ratings of both groups of listeners generally reflected the pattern of tone durations in the musical contexts. This result suggests that the distribution of tones in music is a psychologically effective means of conveying the tonal hierarchy to listeners whether they are familiar with the musical tradition. Beyond this, only the Indian listeners were sensitive to the scales (thats) underlying the rags. For Indian listeners, multidimensional scaling of the correlations between the rating profiles recovered the theoretical representation of scales described by theorists of Indian music. Thus, the empirically measured tonal hierarchy induced by the rag contexts generates structure at the level of the underlying scales or thats, but its internalization apparently requires more extensive experience with music based on that scale system than that provided by the experimental context. There was little evidence that Western listeners assimilated the pitch materials to the major and minor diatonic system of Western music.

Abstract: THIS STUDY PRESENTS AND TESTS a theory of tonal tension (Lerdahl, 2001). The model has four components: prolongational structure, a pitch-space model, a surfacetension model, and an attraction model. These components combine to predict the rise and fall in tension in the course of listening to a tonal passage or piece. We first apply the theory to predict tension patterns in Classical diatonic music and then extend the theory to chromatic tonal music. In the experimental tasks, listeners record their experience of tension for the excerpts. Comparisons between predictions and data point to alternative analyses within the constraints of the theory. We conclude with a discussion of the underlying perceptual and cognitive principles engaged by the theory9s components.