Harmonytimbre's Structures in the Works of Vladimir Tarnopolski: On the Problem of Harmonic Analysis of New Music

Композиторская техника Владимира Тарнопольского основывается на создании индивидуальных детализированных звуковысотных структур, которые трактуются как единые звучности. Композитор называет такие структуры «гармониетембрами». Можно не только рассматривать их с точки зрения тембровой краски, но и анализировать их звуковысотную структуру, преобразование которой происходит по законам гармонии. В статье проанализирована гармоническая сторона гармониетембровых структур в музыке Тарнопольского на примере сочинений разных лет: «Кассандра» (1991), «Маятник Фуко» (2004), «Be@thoven. Invokation» (2017). В результате анализа выявлены основные приемы работы с гармониетембрами и сформулированы закономерности построения звуковысотных структур в музыке Тарнопольского. Vladimir Tarnopolski's compositional technique is based on the individual detailed pitch structures, which are interpreted as a one sound. The composer calls such structures “harmonytimbres.” We can consider them not only from the point of view of timbre character, but also analyze their pitch structure, which develops according to the laws of harmony. The article is devoted to the harmonic side of harmonytimbre's structures in Tarnopolski's music on the example of his works from different years: “Cassandra” (1991), “Foucault's Pendulum” (2004), “Be@thoven. Invokation” (2017). As a result of the analysis, the main methods of working with harmonytimbres are identified and the regularities of constructing pitch structures in Tarnopolski's music are formulated.

Literary contexts of Józef Hołard’s works

The aim of the article is to demonstrate the relationship between the work of Józef Hołard and literature that inspired his work. A valuable document explaining this issue is found in the unpublished text of the artist entitled Credo, written in connection with the conduct of proceedings at the university. Associated with Bielsko-Biała, Professor Józef Hołard died in January 2015, but his painting, drawing and design works have not been sufficiently explored. They contain many symbolic elements derived from the esoteric and occult sciences. However, they are used and understood very freely by the artist, who gives priority to artistic creation. For many years he drew inspiration from literature dealing with similar subjects, especially Foucault’s Pendulum by Umberto Eco. The series of drawings and paintings According to Umberto Eco is best known in Hołard’s oeuvre.

The Formulations of Classical Mechanics with Foucault’s Pendulum

Since the pioneering works of Newton (1643–1727), mechanics has been constantly reinventing itself: reformulated in particular by Lagrange (1736–1813) then Hamilton (1805–1865), it now offers powerful conceptual and mathematical tools for the exploration of dynamical systems, essentially via the action-angle variables formulation and more generally through the theory of canonical transformations. We propose to the (graduate) reader an overview of these different formulations through the well-known example of Foucault’s pendulum, a device created by Foucault (1819–1868) and first installed in the Panthéon (Paris, France) in 1851 to display the Earth’s rotation. The apparent simplicity of Foucault’s pendulum is indeed an open door to the most contemporary ramifications of classical mechanics. We stress that adopting the formalism of action-angle variables is not necessary to understand the dynamics of Foucault’s pendulum. The latter is simply taken as well-known and simple dynamical system used to exemplify and illustrate modern concepts that are crucial in order to understand more complicated dynamical systems. The Foucault’s pendulum first installed in 2005 in the collegiate church of Sainte-Waudru (Mons, Belgium) will allow us to numerically estimate the different quantities introduced.