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[1.1] Theories of harmony rest on two fundamental questions: 1) What are the objects (i.e., the harmonies) and; 2) What are the relationships among the objects (i.e., harmonic progressions/successions)? The success of recent work in neo-Riemannian theory addresses situations for which the answer to the first question involves standard or syntactic harmonies, but the answer to the second question involves non-standard or non-syntactic progressions (Childs 1998, Cohn 1998a and 1998b, Douthett and Steinbach 1998, Gollin 1998, Hyer 1995). A roughly analogous situation arises for addressing jazz harmonic progressions, particularly for compositions after 1960. Compositions of this period often use standard jazz harmonies, but in progressions that suppress or ignore harmonic function. Yet the analogy is rough because, in general, jazz harmonies employ chordal extensions of sevenths, ninths, elevenths, and/or thirteenths. Further, harmonies in a number of jazz compositions after 1960 also make use of non-syntactic (or non-standard) chords, many of which are derived from the hexatonic collection and use pcset [0148] as a subset.(2)[1.2] In this paper I continue to examine these issues. I first delimit the harmonies in this repertory as subsets of one of four collections: diatonic, acoustic, octatonic, and hexatonic. I use recent work in neo-Riemannian theory as a springboard to suggest methods to model harmonic succession, and ultimately supply a three-dimensional Tonnetz in order to consider ways in which ninth-chord harmonies from these four underlying collections interact. I use the opening progression of Wayne Shorter's composition "Vonetta" (recorded in 1967 by the Miles Davis Quintet on the album Sorcerer) as a test case in order to examine progressions of chords derived from different collections. Throughout, the essay uses interactive graphics for modeling these two- and three-dimensional pitch spaces.Chord QualityExample 1. Jazz harmonies from Diatonic, Acoustic, Octatonic, and Hexatonic Collections(click to enlarge and see the rest)[2.1] Jazz compositions of the 1960s draw primarily from harmonies that are subsets of four fundamental collections: the diatonic, the acoustic (also considered a mode of the ascending melodic minor scale), the octatonic, and the hexatonic.(3) Example 1 indicates some representative jazz harmonies from these four underlying collections. Example 1a shows harmonies derived from the C Diatonic Collection. I refer here to diatonic collections by the letter D and a number that corresponds to the pitch class of the major form of the collection: the label D0 refers to the C major collection, D1 the C/D major collection, etc. The diatonic harmonies include major seventh harmonies (and their extensions), minor seventh harmonies (and their extensions), dominant seventh harmonies (and their diatonic extensions), and others. Example 1b reveals harmonies derived from the F acoustic collection. I refer to acoustic collections by the letter A and a number that corresponds to the pitch class of the final of the acoustic collection: A5 is the F acoustic collection, A6 is the F acoustic collection, etc. (The F acoustic collection may also be regarded as a mode of the C ascending melodic minor scale.) Common jazz harmonies from this A5 collection include F13(11), CminM7 with extensions, Amin7/9(5), B7(altered), and Emaj7(5). Many of these acoustic harmonies include pc-set class [0148]: with the A5 collection [0148] consists of the pitch classes of either C, E, G, B, or E, G, B, D.[2.2] Example 1c contains chords from the D/E octatonic collection (Oct 2,3), indicating ic3-related dominant and diminished chords. Example 1d contains ic4-related hexatonic chords, primarily minM7 and maj7(5), which form set-class [0148]. Note that [0148] harmonies also appear as subsets of acoustic collection harmonies. In jazz compositional contexts, however, hexatonic harmonies appear as supersets of [0148], typically [01458]. …

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Modeling Diatonic, Acoustic, Hexatonic, and Octatonic Harmonies and Progressions in Two- and Three-Dimensional Pitch Spaces; or Jazz Harmony after 1960