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.

Harmonic Syntax

This chapter adapts traditional Schenkerian analytical methodology to form a theory of rock harmony rooted in the concept of prolongation. The chapter begins with the premise that focusing on small-scale chord-to-chord successions, as many existing theories do, tells us little about rock’s harmonic organization. After describing a new, syntactically based approach to harmonic function, the chapter defines the functional circuit: a large-scale harmonic trajectory spanning at least one complete song section and comprising the functional succession from tonic to pre-dominant to dominant and back to tonic. This trajectory is familiar from centuries of theoretical work on harmonic function, but its adaptation to the rock style is not trivial. In particular, it requires disentangling the notion that only certain chords can carry certain functions. For instance, dominant function can arise not only from the standard V chord but also from IV, ii, or sometimes even I chords.

Postbop Jazz in the 1960s

Innovations in postbop jazz compositions of the 1960s occurred in several dimensions, including harmony, form, and melody. Postbop jazz composers such as Wayne Shorter, Herbie Hancock, Chick Corea, along with others (Booker Little, Joe Henderson, Woody Shaw) broke with earlier tonal jazz traditions. Their compositions marked a departure from the techniques of jazz standards and original compositions that defined small-group repertory through the 1950s: single-key orientation, schematic 32-bar frameworks (in AABA or ABAC forms), and tonal harmonic progressions. The book develops analytical pathways through a number of compositions, including “El Gaucho,” “Penelope,” “Pinocchio,” “Face of the Deep” (Shorter); “King Cobra,” “Dolphin Dance,” “Jessica” (Hancock); “Windows,” “Inner Space,” “Song of the Wind” (Corea); as well as “We Speak” (Little); “Punjab” (Henderson); and “Beyond All Limits” (Shaw). These case studies offer ways to understand the works’ harmonic syntax, melodic and formal designs, and general principles of harmonic substitution. By locating points of contact among these postbop techniques—and by describing their evolution from previous tonal jazz practices—the book illustrates the syntactic changes that emerged during the 1960s.

Harmonic Syntax of the Twelve-Bar Blues Form

This paper describes the construction and analysis of a corpus of harmonic progressions from 12-bar blues forms included in the jazz repertoire collection The Real Book. A novel method of coding and analyzing such corpus data is developed, with a notion of “possible harmonic change” derived from the corpus and logit mixed-effects regression models that describe the difference between actually occurring harmonic events and possible but non-occurring ones in terms of various sets of theoretical constructs. Models using different sets of constructs are compared using the Bayesian Information Criterion, which assesses the accuracy and efficiency of each model. The principal results are that: (1) transitional probabilities are better modeled using root-motion and chord-frequency information than they are using pairs of individual chords; (2) transitional probabilities are better described using a mixture model intermediate in complexity between a bigram and full trigram model; and (3) the difference between occurring and non-occurring chords is more efficiently modeled with a hierarchical, recursive context-free grammar than it is as a Markov chain. The results have implications for theories of harmony, composition, and cognition more generally.

On “Crystal Growth” in Harmonic Space

James Tenney talks about his “crystal growth” algorithm, an idea that suggests a new “harmonic syntax” for harmonic space. As a quantitative model, it is both suggestively rich for future composition and plausible as a description of the history of tonal expansion. In this algorithm, sets of points are chosen, one by one, in some n-dimensional harmonic space, under the condition that each new point must have the smallest possible sum of harmonic distances to all points already in the set. That is, at each successive stage in the growth of the harmonic lattice, the next ratio added to the set is one whose sum of harmonic distances to each ratio already in the set is minimal. In relation to this algorithm, Tenney considers the Pythagorean pentatonic scale, which may be conceived as a pitch set that arises when extension into the 3,5-plane is just slightly delayed beyond the point where the algorithm would have begun that extension.

Corpus Studies of Harmony in Popular Music: A Response to Gauvin

This paper responds to the research presented in Gauvin’s paper on the evolution of harmonic syntax in popular music from the 1960s.  I begin by situating the findings from his second study (on flat-side harmonies) within the context of my corpus work with David Temperley on harmony in popular music.  Gauvin’s results are similar to ours, although some important differences are worth noting.  I also provide an interpretation of the results from his first study (on modulation), which Gauvin found to be inconsistent with his proposed hypothesis.  Specifically, I conjecture that modulation and harmonic palette may be in balance with one another, in that listeners may prefer songs where harmonic content is at a medium level of complexity.  Gauvin’s study also brings forth some basic issues with regard to harmonic encoding, in terms of both the subjectivity of the analytical process as well as the ease by which harmonic analyses are shared with and verified by other researchers.