Franz Schuberts "Wandererfantasie"

Franz Schubert’s D760 is entitled “fantasy”, although the four sections of the work recognisably reference the formal models of a four-movement sonata. Since those models appear in their traditional order, the “fantasy” elements have to manifest themselves differently, transgressing the norms of sonata in two ways: Schubert transforms and deconstructs the individual forms of the four-movement model, while suspending the autonomy of each movement. Both strategies are interrelated: by blurring the form of each movement, Schubert opens them up to the following sections. This is rendered plausible because the movements, which connect seamlessly, are derived from the same thematic material.The deconstruction of the formal models manifests itself in the elision of formal units, the interpolation of non-formal sections, and the startling curtailing of developmental procedures within the formal units. These formal licences generate ambiguous structures that do not lend themselves to definite formal interpretations. Thus formal ambiguity is a constituting element of the “fantastic” in D760.The thematic unity of the work is a result of the continuous transformation of a motif first presented in the main theme of the first movement; a process, in which new variants emerge from the synthesis of previous variations. Furthermore, the Presto, which stands in for the scherzo movement of the Fantasy, reverse engineers the sonata form of the first movement (which had been abandoned before the recapitulation) while completing and normalising the form of the first movement by aligning it with the scherzo form. Thus the Presto assumes the formal function of the missing recapitulation, whose “wrong” key of A flat major is “rectified” through the C-major finale.

The Rise of Linear Borders

Since roughly the late 19th century, international borders have generally been characterized by linearity, or the appearance as a series of one-dimensional points, connected by straight lines. Prior to this, various kinds of frontiers existed globally, some of them being more linear than others, but most included some kind of formal ambiguity. International relations (IR) often takes for granted the historical process which brought about the global linearization of borders, culminating in the late 19th century and still ongoing in ocean spaces and in outer space. But because cross-border relations are the main substance of inquiry in IR, many theories and areas of study in IR contain some perspective on that process, at least implicitly.

Theory of Response to Perturbations in Non-Hermitian Systems Using Five-Hilbert-Space Reformulation of Unitary Quantum Mechanics

Non-Hermitian quantum-Hamiltonian-candidate combination H λ of a non-Hermitian unperturbed operator H = H 0 with an arbitrary “small” non-Hermitian perturbation λ W is given a mathematically consistent unitary-evolution interpretation. The formalism generalizes the conventional constructive Rayleigh–Schrödinger perturbation expansion technique. It is sufficiently general to take into account the well known formal ambiguity of reconstruction of the correct physical Hilbert space of states. The possibility of removal of the ambiguity via a complete, irreducible set of observables is also discussed.

Herbie Hancock

Hancock’s postbop composition provided some of the signature sounds of the 1960s. “King Cobra” (My Point of View) relies on both major third and minor third axis progressions, often using a slash-chord vocabulary; several devices at the end of the form enhance its tonal and formal ambiguity. “Dolphin Dance” (Maiden Voyage) maintains an even heightened sense of tonal and formal ambiguity. It begins with major third axis progressions embellished through harmonic substitutions and elaborations; despite local harmonic cadences and goals, the composition suppresses a single-key orientation. “Jessica” (Fat Albert Rotunda) relies on a perfect fifth axis progression in its skeletal melody, and its horn scoring reveals significant details of Hancock’s harmonic vocabulary and harmonic substitution techniques.

Wayne Shorter

Wayne Shorter remains one of the most significant jazz composers of the 1960s, with challenging compositions noted for their harmonic and formal ambiguity. “El Toro” (Art Blakey, Freedom Rider) uses a major third axis progression in a 16-bar form, but the melodic structure cuts against an 8 + 8-bar regularity; “Virgo” (Night Dreamer) relies on a 29-bar ABAC form with irregular phrase lengths. A comparison of “Penelope” (Etc) and “El Gaucho” (Adam’s Apple) shows that they begin with a nearly identical melody, but Shorter subjects each to radically different harmonic environments. “Pinocchio” (Miles Davis, Nefertiti) is an 18-bar composition. Its major third axis uses postbop substitution strategies that echo more-conventional progressions. “Face of the Deep” (The All-Seeing Eye) is one of Shorter’s more exploratory compositions: it links to the jazz avant-garde, and its scoring for three horns reveals an expanded vocabulary of upper structure harmonies.