Generalized Tonnetze and Zeitnetze, and the topology of music concepts

The music-theoretic idea of a Tonnetz can be generalized at different levels: as a network of chords relating by maximal intersection, a simplicial complex in which vertices represent notes and simplices represent chords, and as a triangulation of a manifold or other geometrical space. The geometrical construct is of particular interest, in that allows us to represent inherently topological aspects to important musical concepts. Two kinds of music-theoretical geometry have been proposed that can house Tonnetze: geometrical duals of voice-leading spaces and Fourier phase spaces. Fourier phase spaces are particularly appropriate for Tonnetze in that their objects are pitch-class distributions (real-valued weightings of the 12 pitch classes) and proximity in these space relates to shared pitch-class content. They admit of a particularly general method of constructing a geometrical Tonnetz that allows for interval and chord duplications in a toroidal geometry. This article examines how these duplications can relate to important musical concepts such as key or pitch height, and details a method of removing such redundancies and the resulting changes to the homology of the space. The method also transfers to the rhythmic domain, defining Zeitnetze for cyclic rhythms. A number of possible Tonnetze are illustrated: on triads, seventh chords, ninth chords, scalar tetrachords, scales, etc., as well as Zeitnetze on common cyclic rhythms or timelines. Their different topologies – whether orientable, bounded, manifold, etc. – reveal some of the topological character of musical concepts.Accepted manuscrip

Geometric generalizations of the Tonnetz and their relation to Fourier phase space

Some recent work on generalized Tonnetze has examined the topologies resulting from Richard Cohn’s common-tone based formulation, while Tymoczko has reformulated the Tonnetz as a network of voice-leading relationships and investigated the resulting geometries. This paper adopts the original common-tone based formulation and takes a geometrical approach, showing that Tonnetze can always be realized in toroidal spaces,and that the resulting spaces always correspond to one of the possible Fourier phase spaces. We can therefore use the DFT to optimize the given Tonnetz to the space (or vice-versa). I interpret two-dimensional Tonnetze as triangulations of the 2-torus into regions associated with the representatives of a single trichord type. The natural generalization to three dimensions is therefore a triangulation of the 3-torus. This means that a three-dimensional Tonnetze is, in the general case, a network of three tetrachord-types related by shared trichordal subsets. Other Tonnetze that have been proposed with bounded or otherwise non-toroidal topologies, including Tymoczko’s voice-leading Tonnetze, can be under-stood as the embedding of the toroidal Tonnetze in other spaces, or as foldings of toroidal Tonnetze with duplicated interval types.Accepted manuscrip

A Method of Teaching English Speaking Learners to Produce Mandarin-Chinese Tones

Learning Mandarin Chinese tones is a big challenge for English speaking learners. The average tonal production accuracy is reported to be about 70 percent for intermediate-level learners and 40 percent for beginning-level Chinese learners. The Chinese tonal proficiency significantly influences the learners\u27 communicative effectiveness, including listening and speaking, but research often overlooks tonal production. This study proposed and tested a novel method of teaching English-speaking learners to pronounce Mandarin Chinese tones. This teaching method includes a Chinese tones bookmark, and a 30--50 minutes in-class training module. The research undertook five cycles of Design-Based Research (DBR) implementations with 31 public school students, adult learners, and Chinese teachers. Two audio recordings, one pre-training and one post-training, were collected and compared through the paired samples t-tests. Interviews, surveys, and class observations were adopted to determine the participants\u27 attitudes toward the training and the teaching model. The results revealed that the designed teaching method was effective to improve the tonal production accuracy of English speaking K-12 children and adult learners. In addition, the results indicated that the participants\u27 attitudes toward the designed method were positive. This study contributes to the current Chinese tonal teaching repertoire and presents a flexible, practical method for teachers to use when instructing students on Chinese tones

Fourier phase and pitch-class sum

Music theorists have proposed two very different geometric models of musical objects, one based on voice leading and the other based on the Fourier transform. On the surface these models are completely different, but they converge in special cases, including many geometries that are of particular analytical interest.Accepted manuscrip

Blueprints and Vignettes: Pitch-class sets, Serialism and Intervallicism, and the Integration of Systematic and Intuitive Music Making

Blueprints and Vignettes: Pitch-class sets, Serialism and Intervallicism, and the Integration of Systematic and Intuitive Music Making examines the interaction of systematic and intuitive elements in my compositional and improvisational practice, and the outcomes of my sustained investigation of pitch-class sets, serialism and intervallicism as creative tools. This submission consists of two parts: a portfolio of compositions, improvisations and recordings, and an accompanying analytical exegesis. The works in the composition portfolio are a product of three individual projects with three distinct instrumental and aesthetic settings – solo piano, avant-garde jazz quartet and chamber duo – that have occupied my practice over the past five years. Each project also focusses on a particular type of creative methodology, in order to pose answers to several key research questions. How can improvisation generate a composition? How can a composition facilitate improvisation? Does a composition treated in an improvisatory manner maintain its identity? What techniques can be used to assure that it does, or does not? What are the harmonic and melodic possibilities of pitch-class sets, serialism and intervallicism, particularly to musicians with other improvisational or compositional backgrounds? How to these structural devices relate to conventional tonal harmony? In the exegesis I examine these questions by analyzing the creative processes behind and improvisatory products of the compositions in the portfolio, and in so doing place the works on a theoretical continuum between improvisation and composition similar to the one proposed by Nettl (1974). I discuss my adoption of comprovisation (the use of pre-performed or recorded improvisations as compositional seeds or scaffolds) as an integral part of my creative process, and the various ways I incorporate improvisation into my composed works – through techniques such as open notation, aleatory, textual instructions, chord symbols and other systems of facilitating open-ended performance. This research positions my practice at a nexus of jazz, experimental improvisation and classical modernism, and offers a resource to those interested in systematic improvisation, intuitive composition or the uses of pitch-class sets, serialism and intervallicism

Symphony No. 1 and The Development of New Techniques in Contemporary Music Composition

The initial part of this dissertation is a symphony. Symphony No. 1 consists of three movements. Each movement begins with a monumental gesture designed to make a bold and unforgettable statement. Within each movement, there is an interesting array of harmonic and rhythmic schemes. In Symphony No. 1, the diametrically opposed concepts of “simple” and “complex” are featured prominently. These concepts are contrasted thematically, rhythmically, and harmonically. The next part of this dissertation establishes the existence of certain obstacles facing contemporary composers and the benefits of developing new techniques in composition. The music of György Ligeti and Arvo Pärt are examined in order to explore of the specific techniques used in their compositions. This involves a review of micropolyphony, including aspects of its origin and development. This also involves an examination of the tintinnabuli technique and its role in the music of Arvo Pärt. The final two sections of this dissertation include the presentation of a new compositional technique. Part of this presentation consists of a synopsis of some neo-Riemannian techniques, including aspects of harmonic dualism and parsimonious voice leading. This is followed by the comparative analysis of a new technique with methods used in established compositional works. The new technique is based on parsimonious voice leading extended to ninth chords and minor eleventh chords. It is also based on two distinct types of motives. The first motive is distinguished from the second in that it does not develop. Conversely, the second motive incorporates various aspects of motivic variation and development. The analyses presented in this section effectively demonstrate the uniqueness of the new technique as well as its legitimacy as a means of artistic expression and communication

Special collections: renewing set theory

The discrete Fourier transform on pitch-class sets, proposed by David Lewin and advanced by Ian Quinn, may provide a new lease on life for Allen Forte's idea of a general theory of harmony for the twentieth century based on the intervallic content of pitch-class collections. This article proposes the use of phase spaces and Quinn's harmonic qualities in analysis of a wide variety of twentieth-century styles. The main focus is on how these ideas relate to scale-theoretic concepts and the repertoires to which they are applied, such as the music of Debussy, Satie, Stravinsky, Ravel, and Shostakovich. Diatonicity, one of the harmonic qualities, is a basic concern for all of these composers. Phase spaces and harmonic qualities also help to explain the “scale-network wormhole” phenomenon in Debussy and Ravel and better pinpoint the role of octatonicism in Stravinsky's and Ravel's music.Accepted manuscrip

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

The aural skills acquisition process of undergraduate electroacoustic (EA) music majors in the context of a new aural learning method

Thesis (D.M.A.)--Boston UniversityElectroacoustic (EA) musicians require aural skills that exist beyond tonality and meter; however, specialized ear training courses for EA music are rare in university and college music programs that offer EA studies (EaSt) in their curricula. Since 2005, this researcher has been developing and teaching EA aural training at a Canadian university in that was inspired by concepts from Auditory Scene Analysis (ASA) studies, primarily integration and segregation. In the 2009/10 academic year, the researcher conducted an action study with his intact EA aural training class of 25 first year undergraduate students majoring in EaSt for the purposes of better understanding and improving the students' aural skill acquisition process. and of refining the teaching and learning sequence. The action study was organized into four cycles of observation, critical reflection, and action, and focused on optimizing and autonomizing the skill acquisition process within the large, varied group. Actions were designed in response to critical reflection on emerging problems, evaluations of students' views about the process, their moods and attitudes, and measurements of students' achievements-with specific attention to eight EA-oriented skills and seven tonal and metric skills. Qualitative and quantitative data gathered from questionnaires, in-class surveys and tests, homework, and competence tests provided evidence of skill acquisition, primarily in loudness discrimination, timbral discrimination, tonal awareness, interval discrimination, meter discrimination, and descriptive ability. The most notable emerging problems in the skill acquisition process were related to the group's variety of ability levels, including imbalances in difficulty levels, in students' level of interest in the activities, and in the all-inclusive effectiveness of the training. The main transformational aspects of the action study were autonomization of the skill acquisition process at home through weekly reflective practice reports and developing a cooperative learning environment in the classroom through regular in-class discussion