The Logic of Six-Based Minor for Harmonic Analyses of Popular Music

Harmonic analyses of popular music typically take the minor tonic to be Roman numeral “one.” By nature, this “one-based” approach requires a new numbering scheme when songs shift between relative key centers. Recent scholarship has argued, however, that popular music often involves ambiguity between relative tonalities, as exemplified in the “Axis” progression, if not sometimes a tonal fusion of two relative keys. I thus argue for the utility of a “six-based” approach to the minor tonic, where the minor tonic is taken to be scale-degree 6. This six-based approach, common among practitioners of popular music as seen in the Nashville number system, avoids the forced choice of a single tonic, and it thus offers a consistent way to track chord function and behavior across shifts between relative key centers. After considering these shifts in a diatonic context on the levels of both phrase and song form, I posit that popular music involves three possible tonalities, together which form a “triple-tonic complex” akin to Stephenson’s three harmonic palettes: a major system, a parallel-minor system, and a relative-minor system. I conclude by considering how chromatic chords common in a major key, such as II and ".fn_flat('')."VII, correspond to their counterparts in the relative minor, IV and ".fn_flat('')."II, thereby collapsing the landscape of diatonic modes into three modal complexes. Overall, the paper serves to reveal the logic of six-based minor—why it is useful, what issues it resolves, and what insights it can afford us about harmonic syntax in popular music.

Schenkerian Theory, Neo-Riemannian Theory and Late Schubert: A Lesson from Tovey

Current analytical studies on Schubert’s tonality have tended to favour either Schenkerian theory or, more recently, neo-Riemannian theory to explain the composer’s signature harmonic progressions. What remains unclear with respect to these two prevailing analytical purviews is the extent to which one may relate to the other.This article offers a new way of understanding how Schenkerian and neo-Riemannian views of Schubert’s late tonal practices may be complementary, using Tovey’s concept of key-relations from his article ‘Tonality in Schubert’ of 1928. It suggests that Tovey’s key-relations can function as a bridge between these two theories because they approximate parsimonious voice-leading operations while preserving chord function within a tonal hierarchy. In forming an intermediate pathway between Schenkerian diatony and neo-Riemannian theory’s parsimonious voice-leading operations, Tovey’s key-relations highlight the important contributions that Schenkerian theory and neo-Riemannian theory offer to our understanding of Schubert’s tonality.

Nonlinear semiclassical dynamics of open systems

A semiclassical approximation for an evolving density operator, driven by a ‘closed’ Hamiltonian and ‘open’ Markovian Lindblad operators, is reviewed. The theory is based on the chord function, i.e. the Fourier transform of the Wigner function. It reduces to an exact solution of the Lindblad master equation if the Hamiltonian is a quadratic function and the Lindblad operators are linear functions of positions and momenta. The semiclassical formulae are interpreted within a (real) double phase space, generated by an appropriate classical double Hamiltonian. An extra ‘open’ term in the double Hamiltonian is generated by the non-Hermitian part of the Lindblad operators in the general case of dissipative Markovian evolution. Decoherence narrows the relevant region of double phase space to the neighbourhood of a caustic for both the Wigner function and the chord function. This difficulty is avoided by the definition of a propagator, here developed in both representations. Generalized asymptotic equilibrium solutions are thus presented for the first time.

Differences in Electric Brain Responses to Melodies and Chords

The music we usually listen to in everyday life consists of either single melodies or harmonized melodies (i.e., of melodies “accompanied” by chords). However, differences in the neural mechanisms underlying melodic and harmonic processing have remained largely unknown. Using EEG, this study compared effects of music-syntactic processing between chords and melodies. In melody blocks, sequences consisted of five tones, the final tone being either regular or irregular (p = .5). Analogously, in chord blocks, sequences consisted of five chords, the final chord function being either regular or irregular. Melodies were derived from the top voice of chord sequences, allowing a proper comparison between melodic and harmonic processing. Music-syntactic incongruities elicited an early anterior negativity with a latency of approximately 125 msec in both the melody and the chord conditions. This effect was followed in the chord condition, but not in the melody condition, by an additional negative effect that was maximal at approximately 180 msec. Both effects were maximal at frontal electrodes, but the later effect was more broadly distributed over the scalp than the earlier effect. These findings indicate that melodic information (which is also contained in the top voice of chords) is processed earlier and with partly different neural mechanisms than harmonic information of chords.

Harmonic Stasis and Oscillation in Björk’s Medúlla

Björk Guðmundsdóttir’s 2004 all-vocal album Medúlla provides an opportunity to explore processes of harmonic oscillation, the repetition of two or more alternating harmonies, and stasis, the continuous repetition of a single harmony. Qualities of timelessness are evoked through harmonic stasis, which allows for intricate vocal textures to come to the fore in songs such as “Komið” and “Öll Birtan.” “Who Is It” illustrates how local-level and global processes of harmonic oscillation interact and how traditional chord function may be negated through oscillation. To account for harmonic progressions that suggest tonal centricity, concepts of repose and tension are applied as interpretive models to passages from “Triumph of a Heart.” This paper offers a model to describe oscillation and combines models developed by Tim Hughes to explore other uses of harmonic oscillation and stasis. Overarching oscillations support extra-musical narratives suggested by song lyrics and music videos. These models of analysis can be applied to other songs by Björk, as well as other popular music, as an alternative to traditional methods of understanding tonal harmony, which break down for such songs.

Computer-Based Recognition of Perceptual Patterns in Harmonic Dictation Exercises

During the 1975–1976 academic year, student response data was saved for a group of 17 freshman music majors as they worked through 15 units of harmonic dictation exercises delivered on the University of Delaware's GUIDO system. Analysis of the student data base led to the identification of seven confusion tendencies that affect the perception of harmonies: bass line confusions, confusions of inversion, confusions of chord function, confusions of chord quality, unperceived sevenths, unperceived roots, and favorite response confusions. The level of student achievement on individual harmonies was found to be highly correlated with the percentage of times these harmonies were asked in the curriculum.