Dynamical systems theory for music dynamics

We show that, when music pieces are cast in the form of time series of pitch variations, the concepts and tools of dynamical systems theory can be applied to the analysis of {\it temporal dynamics} in music. (i) Phase space portraits are constructed from the time series wherefrom the dimensionality is evaluated as a measure of the {\pit global} dynamics of each piece. (ii) Spectral analysis of the time series yields power spectra ($\sim f^{-\nu}$) close to {\pit red noise} ($\nu \sim 2$) in the low frequency range. (iii) We define an information entropy which provides a measure of the {\pit local} dynamics in the musical piece; the entropy can be interpreted as an evaluation of the degree of {\it complexity} in the music, but there is no evidence of an analytical relation between local and global dynamics. These findings are based on computations performed on eighty sequences sampled in the music literature from the 18th to the 20th century.Comment: To appear in CHAOS. Figures and Tables (not included) can be obtained from [email protected]

Beyond pitch/duration scoring: Towards a system dynamics model of electroacoustic music

Based on a hierarchy of discrete pitches and metrically sub-divisible duration, Western tonal art music is usually modelled through printed music scores. Scoring acoustic musical events beyond this paradigm has resulted in non-standard graphs in two dimensions. New digitally generated ‘soundscape’ forms are often not conceived or understandable within traditional musical paradigms or notation models, and often explore attributes of music such as spatial processing that fall outside two- dimensional graphic scoring. To date there is not a commonly accepted model that approximates the structural dynamics of electroacoustic music; providing a conceptual framework independent of the music to the degree of standard music notation. Based on recent work in spectro-morphology as a way of explaining sound shapes, a systems dynamics model is proposed through mapping a dynamic taxonomy for structural listening as an aid to composition. This approach captures formal but not semiotic discourse

Tonal prisms: iterated quantization in chromatic tonality and Ravel's 'Ondine'

The mathematics of second-order maximal evenness has far-reaching potential for application in music analysis. One of its assets is its foundation in an inherently continuous conception of pitch, a feature it shares with voice-leading geometries. This paper reformulates second-order maximal evenness as iterated quantization in voice-leading spaces, discusses the implications of viewing diatonic triads as second-order maximally even sets for the understanding of nineteenth-century modulatory schemes, and applies a second-order maximally even derivation of acoustic collections in an in-depth analysis of Ravel's ‘Ondine’. In the interaction between these two very different applications, the paper generalizes the concepts and analytical methods associated with iterated quantization and also pursues a broader argument about the mutual dependence of mathematical music theory and music analysis.Accepted manuscrip

A space for inflections: following up on JMM's special issue on mathematical theories of voice leading

Journal of Mathematics and Music's recent special issue 7(2) reveals substantial common ground between mathematical theories of harmony advanced by Tymoczko, Hook, Plotkin, and Douthett. This paper develops a theory of scalar inflection as a kind of voice-leading distance using quantization in voice-leading geometries, which combines the best features of different approaches represented in the special issue: it is grounded in the concrete sense of voice-leading distance promoted by Tymoczko, invokes scalar contexts in a similar way as filtered point-symmetry, and abstracts the circle of fifths like Hook's signature transformations. The paper expands upon Tymoczko's ‘generalized signature transform’ showing the deep significance of generalized circles of fifths to voice-leading properties of all collections. Analysis of Schubert's Notturno for Piano Trio and ‘Nacht und Träume’ demonstrate the musical significance of inflection as a kind of voice leading, and the value of a robust geometrical understanding of it.Accepted manuscrip

Negative Harmony: The Shadow Of Harmonic Polarity On Contemporary Composition Techniques

If we are to blame any composer for the puzzling question ‘What is Negative Harmony?’ that caused a worldwide itch on the creative minds of many young composers in this century, it would be Jacob Collier, who introduced the world with prodigious, colorful harmonies upon releasing his first album “In My Room” in 2016. Provided with brief answers during interviews with June Lee that explained the concept of tones and chords being mirrored on an axis center in the opposite direction, the surface of Negative Harmony was exposed but led to even more confusion. Prompted by even more questions, the internet exploded with articles and blogs about Collier’s interpretations of Negative Harmony. Unfortunately, they all varied in their approach and results, which led curious fans such as myself back to the original question, ‘What is Negative Harmony?’ Even though Collier shows innovation in his compositions, the musical principles applied are built on a solid foundation laid by composers and music theorists that was built centuries before him. For this reason, we can best learn about Negative Harmony if we evaluate it within its historical context

George Perle’s Twelve–Tone Tonality: some developments for CAC using PWGL

This paper presents a description and some developments on Perle’s theory and compositional system known as Twelve-Tone Tonality, a system that, because of its characteristics and fundamentals, is currently associated with Schoenberg dodecaphonic system. Some research has been made in the last few decades in order to develop his model in a Computer Assisted Composition (CAC) environment. After some efforts in order to analyse these prototypes, we realize that in general they were discontinued or outdated. A three-scope proposal is so outlined: Firstly, to simplify the grasp of a system that presents an easily understandable starting premise but afterwards enters a world of unending lists and arrays of letters and numbers; Secondly, to present the implementation process already started using PWGL [1] (see Laurson, 1996; Laurson, 2003; Laurson, 2009). Finally, the model is applied in a short original compositional work, and it is presented and analysed emphasizing the standpoints properties of the system. Some further considerations were made regarding the continuity of this project where the construction of a dedicated PWGL library of Perle’s model reveals a pre-compositional necessary tool. PWGL software was selected due to its specific fitting features: it is based on Common Lisp - perfectly powerful and suitable to process lists of integers — and it is specialized in CAC

Upholding a Modernist Mentality: Experimentalism and Neo-tonality in the Symphonies of Einojuhani Rautavaara

Einojuhani Rautavaara (1928-2016) is one of the most significant Finnish composers to emerge after Sibelius, and his music has been highly acclaimed for its originality and accessibility. After experimenting with modernism in the 1960s, his more approachable recent style has often been misunderstood. This thesis will consider this music, and re-consider terms describing it, such as “mystical”, “neo-Romantic” and “postmodern”, which are used in a variety of music criticism and in some biographical accounts. As such, it offers a new understanding of Rautavaara’s idiom, its larger significance, and its relationship with broader, historical trends within the Long Twentieth Century. This thesis addresses the larger, integrated role of modernism in Rautavaara’s music within the multi-stylistic context of contemporary composition. Modernism is examined both in technical, stylistic terms as well as a more general mentality of progress, arguing that both definitions have informed Rautavaara’s recent music more than has been acknowledged. After considering modernism in its broadest terms and in the more specific Finnish context, the thesis draws on the concepts of “reactive modernism” (J.P.E. Harper-Scott) and the “moderate mainstream” (Arnold Whittall) to argue that, in the late-twentieth century, Rautavaara continues an individualistic, critical approach that did not reject either “modern” or “anachronistic” techniques. Musical analyses focussing on the eight symphonies, several of which have received no previous detailed discussion, support a new contextualisation of Rautavaara’s entire symphonic cycle as a major pillar of his output. These analytical chapters work inwards from differing experiences of global form, to surface-level thematic processes, to melodic and harmonic processes, in particular how Rautavaara reconciles dodecaphonic and tonal thinking. Issues of symmetry and duality occur throughout. From these close readings, contextual discussion assesses Rautavaara’s legacy, focusing on his mentorship of younger Finnish composers, as well as determining common and recurring compositional features

Reconsidering Pitch Centricity

Analysts commonly describe the musical focus upon a particular pitch class above centricity. But this seemingly simple concept is complicated by a range of factors. can be understood variously as a compositional feature, a perceptual effect arising from or listening strategies, or some complex combination thereof. Second, the relation the theoretical construct of tonality (in any of its myriad conceptions) is often not consistently theorized. Finally, various musical contexts manifest or evoke pitch centricity in seemingly and to differing degrees. This essay examines a range of compositions by Ligeti, Carter, and others to arrive at a more nuanced perspective of pitch centricity - one that takes perceptual foundations, recognizes its many forms and intensities, and addresses its tonal structure in a given composition