Filtered Point-Symmetry (FiPS) is a tool for modeling relationships between iterated maximally even sets. Common musical relationships can be studied by using FiPS to model chords contained within a specific scalar context (such as C major or the 01-octatonic collection), and by capturing those relationships in a FiPS configuration space. In this work, many FiPS configuration spaces are presented; some are isomorphic to commonly referenced voice-leading spaces like the neo-Riemannian Tonnetz , and others show tonal networks that have not previously been explored. A displacement operation is introduced to codify the traversal of a configuration space, and a short music analysis is provided to demonstrate some benefits of the approach.

Mathematical Music Theory, pp. 167-182 (2018) No AccessChapter 8: Harmonious OppositionRichard PlotkinRichard PlotkinUniversity at Buffalo, The State University of New York, New York, USAhttps://doi.org/10.1142/9789813235311_0008Cited by:0 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: The following sections are included: Introduction Parsimonious Transformations in Scalar Contexts Scalar Polarity Conclusions Bibliography FiguresReferencesRelatedDetails Mathematical Music TheoryMetrics History PDF download

In this paper, we examine relationships between signature transformations, Filtered Point-Symmetry (FiPS), and voice-leading spaces, with a strong emphasis on the role of scalar context in each model. While these models overlap, the differences between them are of substantial analytical importance. We determine how to expand the signature group using FiPS, and map familiar transformational graphs to maximally even coordinate spaces, thereby separating transformational groups from the scalar contexts in which they are most often explained. We also look at differences in the definition and usage of maximal evenness in continuous space, and examine the impact of scalar context on voice-leading distance.

It is exciting to have this opportunity to read and react to three other substantial papers. Marek Žabka’s idea to advocate this format deserves substantial praise, as does Rick Cohn’s organization of the most recent Clough Conference. In order to offer our reactions, we need to briefly expand our discussion of two relevant concepts in Filtered Point-Symmetry, which were left out of ‘Scalar Context in Musical Models’.

This paper introduces a system in which parsimonious and continuous transformations occur seamlessly between triads and tetrachords. Such fluidity is abundant in common practice music, but unprecedented in theoretical literature, largely because there has been no consistent way to approach transformations independent of cardinality. Neo-Riemannian theory elegantly unites harmonic change and voice-leading efficiency, but deals exclusively with set class [037] in a 12-gamut pcset space. Attempts to extend the neo-Riemannian approach to tetrachords in 12-gamut space often fall short; the elegant characteristics of the triadic theory do not carry over. However, when a scalar context arbitrates the parsimoniousness of transformations, triads and tetrachords can be treated in a consistent manner. Within this consistently modeled space, cardinality itself can be transformed. In this paper, we see that filtered point-symmetry is an essential tool for working through the iterated maximally even sets that establish scalar contexts. To understand cardinality transformations, we also extend filtered point-symmetry to model partially symmetric distributions and relatively even sets.

Richard Plotkin Department of Music University of Chicago Chicago, IL 60637 rplotkin@uchicago.edu Developments in music theory, musical acoustics, and psychoacoustics over the last 15 to 20 years have resulted in a single structural basis that provides common ground for analysis of musical systems that include non-standardlnon-octavesystems, microtonal systems, and systems that use unusual tunings common in non-western music. These developments make it likely that a confluence of understanding in musical acoustics, psychoacoustics, and music synthesis is about to take place. This common structural basis will lead to generalized rules of composition, unheard of since the classic work Treatise On Harmony by Rameau (1722), based on generalizations of the well-known properties of our familiar 12tone scales. Indeed, some of these generalizations have already been articulated. As participants in various aspects of this work for the past 14 years [1-7], we are in a unique position to collect, assess, and distill the important contributions to this, truly, multidisciplinary field. It is our purpose, in this paper, to articulate a 21st century approach to music composition using the latest results from mathematical music theory and musical acoustics. In our presentation we will produce audio and visual examples applying these techniques.