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For an n-TET tuning system, we propose a formalism to study the transformations of k-chords over a generalized non-degenerate Tonnetz generated by a given interval structure. Root and mode are the two components of a directed chord on which the algebra operates, so that chord transformations within one chord cell or towards other cells, and paths or simple circuits over the chord network can be determined without resorting to computational algorithms or geometrical representations. The one-step transformations over the edges of the chord network associated with the $ k{-}1 $ k − 1 drift operators generalize the basic operators P, R and L of the Neo-Riemmanian triadic progressions and the maximally smooth cycles of the 12-TET system to any higher dimensional space. [ABSTRACT FROM AUTHOR]

This article draws on the mathematical approaches of Tymoczko (2011) and Hook (2008) to construct a transformational space of closed-position diatonic triads. The group acts on a set of twenty-one objects: the seven diatonic triads differentiated by the three closed-position inversions. After constructing the transformation group as an abstract voice-leading space, I interpret the system as an instrumental space (De Souza 2017) that captures the chord shapes required to play triadic progressions on a keyboard. This emphasis on the space's ability to capture such chord shapes leads to generalizations of the system that explore interactions between voice leading and chord spacing using geometric and diatonic theory. [ABSTRACT FROM AUTHOR]

The revival of Formenlehre studies has initiated a renewed interest in formal functions in atonal music. Schoenberg, who pioneered the study of formal functions, did not extensively analyze his atonal music from a formal point of view. Recent scholars, however, have drawn parallels between the formal construction of his atonal music and common-practice tonal music. One impediment to such analysis is the lack of a theory of counterpoint and harmony upon which formal-functional analysis rests. I propose a normative contrapuntal-harmonic structure for Schoenberg's atonal music I call the "chromatic cadence." I trace its origins to voice-leading techniques used by Schoenberg in his earlier tonal music and develop fundamental principles that guide the analysis and identification of the cadence in his atonal music. I conclude with a detailed analysis of Schoenberg's piano piece, op. 19, no. 1, in which I examine the interrelationship of its motivic construction, contrapuntal-harmonic structure, and its formal functions. [ABSTRACT FROM AUTHOR]

Musicians often operate with a hierarchy of scale-like collections, each embedded within the next, and with transposition and inversion available at every level. A particularly common technique is to counteract a transformation at one level with an analogous transformation in the intrinsic scale consisting of a chord's own notes. [ABSTRACT FROM AUTHOR]

In this paper, we propose a model for idealized diatonic and chromatic voice leadings between seventh chords and study its computational aspects. The model provides a mathematical formalization of the concept of tonal seventh chord realizations including augmented sixth chords. Certain contrapuntal aspects, such as preparation and resolution of certain dissonant intervals, are taken into account. Possible applications of the model are discussed using a graph-theoretic approach. In particular, we present an algorithm for generating concrete voicings from sequences of seventh-chord symbols. [ABSTRACT FROM AUTHOR]

While previous scholarship has focused on the close relationship between John Adams's pieces, China Gates and Phrygian Gates, this paper offers an alternative interpretation of China Gates based on Adams's recent revelation that the piece was inspired by an endless winter rainstorm that beat down on his cottage. This paper utilizes a two-part methodology: musical elements are analyzed first using theoretical tools, after which I adopt a semiotic framework based on cognitive theories of cross-domain conceptual blending to demonstrate how these musical elements signify rain through symmetry, phasing, parsimonious voice-leading, as well as pitch-class and set-class repetition. In this paper, I unveil the analogical processes by which the physical phenomenon of rain is represented by musical features and illustrate the action of cross-domain mapping in constructing a conceptual blend. I argue that in producing a sonic analog for rain, China Gates suggests a narrative tension between consistency and dynamism. [ABSTRACT FROM AUTHOR]

Prevalent neuroscientific theories are combined with acoustic observations from various studies to create a consistent geometric model for music perception in order to rationalize, explain and predict psycho-acoustic phenomena. The space of all chords is shown to be a Whitney stratified space. Each stratum is a Riemannian manifold which naturally yields a geodesic distance across strata. The resulting metric is compatible with voice-leading satisfying the triangle inequality. The geometric model allows for rigorous studies of psychoacoustic quantities such as roughness and harmonicity as height functions. In order to show how to use the geometric framework in psychoacoustic studies, concepts for the perception of chord resolutions are introduced and analyzed. [ABSTRACT FROM AUTHOR]

Most previous analytical studies of The Rite of Spring have focused on its harmony and rhythm. This article shifts attention to its melodies—mostly short fragments that move repetitively within a narrow registral frame—and shows that they take on expressive meanings closely linked with the work’s dramatic scenario. In addition to operating in the manner of Wagnerian leitmotivs in relation to characters or dramatic situations, they are often bound together in expressive families through the manipulation of their dyads. [ABSTRACT FROM AUTHOR]

This article contains scientific reflections on the ways of performing vocal parts in a contemporary opera on the example of the opera by a living Polish composer Elżbieta Sikora entitled Madame Curie, telling the story of life and work of an outstanding Polish scientist and Nobel Prize winner, Maria Curie-Skłodowska. Based on her own experiences in performing one of the roles in the opera - an American journalist, Missy Meloney - the author analyses the vocal part in terms of melodic, intonation, rhythmic, articulation and acting difficulties and forms conclusions about the specifics of voice leading.

Jack Douthett wrote a number of letters to John Clough and Richard Cohn concerning Cohn's "P-Relations," single-semitone voice-leading relationships. The ideas in these letters led to graph-theoretic and geometric models. The following selection has been edited and prepared for publication by Richard Cohn and Dani Zanuttini-Frank. [ABSTRACT FROM AUTHOR]

Jack Douthett's work over three decades was central to defining an era in mathematical theory. The present special issue attests to his abiding influence over the field, as well as the energy he brought to research in all areas of mathematical music theory through his collaborations, correspondence, and relationships. [ABSTRACT FROM AUTHOR]

We generalize orbifold models for chords and voice leading to incorporate loudness, allowing for the modeling of resting voices, which are used frequently by composers and arrangers across genres. In our generalized setting (strictly speaking, that of orbispaces rather than an orbifolds), passages with resting voices, passages with two or more voices in unison, and fully harmonized passages occupy distinct subspaces that interact in mathematically precise and musically interesting ways. In particular, our setting includes previous orbifold models by way of constant-loudness subspaces, and provides a way to model voice leading between chords of different cardinalities. We model voice leading in this general setting by morphisms in the orbispace path groupoid, a category for which we give a formal definition. We demonstrate how to visualize such morphisms as singular braids, and explore how our approach relates to (and is consistent with) selected previous work. [ABSTRACT FROM AUTHOR]

This article examines the transformation of the period in the early Romantic era. While in Classical periods the end of the antecedent phrase is harmonically closed off from the beginning of the consequent phrase, Romantic periods often join the end of the antecedent to the beginning of the consequent, creating an undivided progression. Voice-leading analyses illustrate three different means of joining the antecedent to the consequent: 1) by overlapping the two phrases, 2) by prolonging and resolving the antecedent's closing dominant, and 3) by ending the antecedent on a predominant. All three procedures involve converting a dividing dominant into a progressive dominant. [ABSTRACT FROM AUTHOR]

Prevalent neuroscientific theories are combined with acoustic observations from various studies to create a consistent geometric model for music perception in order to rationalize, explain and predict psycho-acoustic phenomena. The space of all chords is shown to be a Whitney stratified space. Each stratum is a Riemannian manifold which naturally yields a geodesic distance across strata. The resulting metric is compatible with voice-leading satisfying the triangle inequality. The geometric model allows for rigorous studies of psychoacoustic quantities such as roughness and harmonicity as height functions. In order to show how to use the geometric framework in psychoacoustic studies, concepts for the perception of chord resolutions are introduced and analyzed.

Parsimonious transformations are common patterns in different musical styles and eras. In some cases, they can be represented on the Tonnetz, Cube Dance, Power Towers, or the central region of an orbifold, mainly when they only include the most even trichords and tetrachords. In this paper, two novel graphs, called Cyclopes, are presented, which include more than double the number of chord types in previously published graphs, thus allowing to represent a larger musical repertoire in a practical way. Apart from parsimonious transformations, they are also especially suitable for representing trichords a major third apart, tetrachords a minor third apart, and the cadences V7–I(m) and II Ø –V7–I(m) with major or minor tonic chords. Therefore, they allow to clearly visualize the relationship among the corresponding chords and better understand those patterns, as well as being efficient mnemonic resources, all of which make them useful tools both for music analysis and composition. [ABSTRACT FROM AUTHOR]

Auditory scene analysis is an elementary aspect of music perception, yet only little research has scrutinized auditory scene analysis under realistic musical conditions with diverse samples of listeners. This study probed the ability of younger normal-hearing listeners and older hearing-aid users in tracking individual musical voices or lines in JS Bach's The Art of the Fugue. Five-second excerpts with homogeneous or heterogenous instrumentation of 2–4 musical voices were presented from spatially separated loudspeakers and preceded by a short cue for signaling the target voice. Listeners tracked the cued voice and detected whether an amplitude modulation was imposed on the cued voice or a distractor voice. Results indicated superior performance of young normal-hearing listeners compared to older hearing-aid users. Performance was generally better in conditions with fewer voices. For young normal-hearing listeners, there was interaction between the number of voices and the instrumentation: performance degraded less drastically with an increase in the number of voices for timbrally heterogeneous mixtures compared to homogeneous mixtures. Older hearing-aid users generally showed smaller effects of the number of voices and instrumentation, but no interaction between the two factors. Moreover, tracking performance of older hearing aid users did not differ when these participants did or did not wear hearing aids. These results shed light on the role of timbral differentiation in musical scene analysis and suggest reduced musical scene analysis abilities of older hearing-impaired listeners in a realistic musical scenario.

Auditory scene analysis is an elementary aspect of music perception, yet only little research has scrutinized auditory scene analysis under realistic musical conditions with diverse samples of listeners. This study probed the ability of younger normal-hearing listeners and older hearing-aid users in tracking individual musical voices or lines in JS Bach's The Art of the Fugue. Five-second excerpts with homogeneous or heterogenous instrumentation of 2–4 musical voices were presented from spatially separated loudspeakers and preceded by a short cue for signaling the target voice. Listeners tracked the cued voice and detected whether an amplitude modulation was imposed on the cued voice or a distractor voice. Results indicated superior performance of young normal-hearing listeners compared to older hearing-aid users. Performance was generally better in conditions with fewer voices. For young normal-hearing listeners, there was interaction between the number of voices and the instrumentation: performance degraded less drastically with an increase in the number of voices for timbrally heterogeneous mixtures compared to homogeneous mixtures. Older hearing-aid users generally showed smaller effects of the number of voices and instrumentation, but no interaction between the two factors. Moreover, tracking performance of older hearing aid users did not differ when these participants did or did not wear hearing aids. These results shed light on the role of timbral differentiation in musical scene analysis and suggest reduced musical scene analysis abilities of older hearing-impaired listeners in a realistic musical scenario. [ABSTRACT FROM AUTHOR]

We generalize orbifold models for chords and voice leading to incorporate loudness, allowing for the modeling of resting voices, which are used frequently by composers and arrangers across genres. In our generalized setting (strictly speaking, that of orbispaces rather than an orbifolds), passages with resting voices, passages with two or more voices in unison, and fully harmonized passages occupy distinct subspaces that interact in mathematically precise and musically interesting ways. In particular, our setting includes previous orbifold models by way of constant-loudness subspaces, and provides a way to model voice leading between chords of different cardinalities. We model voice leading in this general setting by morphisms in the orbispace path groupoid, a category for which we give a formal definition. We demonstrate how to visualize such morphisms as singular braids, and explore how our approach relates to (and is consistent with) selected previous work.

This paper details a corpus study examining Renaissance voice-leading practices. Palestrina's masses are searched for progressions matching contrapuntal 'rules' taken from Vicentino (1555). Vicentino's treatise provides a quasi-systematic organization of contrapuntal rules according to the minimum size of the vocal texture in which they ought to be set. Palestrina's realizations of these progressions illustrate the exact size of vocal texture employed, enabling a direct comparison of theory and practice. The analysis reveals a general agreement, but suggests that Vicentino's taxonomy is too strict. The results are examined in the context of uncovering guiding theoretical and perceptual principles. [ABSTRACT FROM AUTHOR]

In a representative historical database of musical scores, a sonority was defined at every onset in any voice. For eight trichords—major (047), minor (037), suspended (027), diminished (036), 015, 045, 025, 035—immediately preceding and following sonorities were analysed. Chroma prevalence profiles depended surprisingly little on century or temporal position—especially for major/minor trichords (cf. Krumhansl's key profiles). Profiles for nine non-chord chromas per trichord were influenced more strongly by diatonic scales and 5th relationships than missing fundamentals (roots) or completion tones (tones that complete familiar tetrachords) according to predictions of simple models. [ABSTRACT FROM AUTHOR]

Music theorists have modeled voice leadings as paths through higher-dimensional configuration spaces. This paper uses topological techniques to construct two-dimensional diagrams capturing these spaces' most important features. The goal is to enrich set theory's contrapuntal power by simplifying the description of its geometry. Along the way, I connect homotopy theory to "transformational theory," show how set-class space generalizes the neo-Riemannian transformations, extend the Tonnetz to arbitrary chords, and develop a simple contrapuntal "alphabet" for describing voice leadings. I mention several compositional applications and analyze short excerpts from Gesualdo, Mozart, Wagner, Stravinsky, Schoenberg, Schnittke, and Mahanthappa. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we study the possibility of defining pseudo-distances between musical chords of different cardinality, from the distance defined on the generalized voice-leading space by Callender, Quinn, and Tymoczko ["Generalized Voice-Leading Spaces." Science 320 (5874): 346–348]. This latter distance measures the size of the minimal voice leading between two chords. A pseudo-distance is a weaker notion than distance, which allows more flexibility with fewer mathematical constraints. We define one distance and two pseudo-distances. The distance is the Hausdorff metric between sets. The two pseudo-distances, based on a similar approach, use the projection/immersion of chords into lower/higher-dimensional voice-leading chord spaces. Finally, we propose some musical applications. [ABSTRACT FROM AUTHOR]

This article constructs generic voice-leading spaces by combining geometric approaches to voice leading with diatonic set theory. Unlike the continuous mod-12 spaces developed by Callender, Quinn, and Tymoczko, these mod-7 spaces are fundamentally discrete. The mathematical properties of these spaces derive from the properties of diatonic pitch-class sets and generic pitch spaces developed by Clough and Hook. After presenting the construction of these voice-leading spaces and defining the OPTIC relations in mod-7 space, this article presents the mod-7 OPTIC-, OPTI-, OPT-, and OP-spaces of two- and three-note chords. The final section of the study shows that, although the discrete mod-7 versions of these lattices appear quite different from their continuous mod-12 counterparts, the topological space underlying each of these graphs depends solely on the number of notes in the chords and the particular OPTIC relations applied. [ABSTRACT FROM AUTHOR]

The theory of harmonic vectors (THV) postulates that, of the six possible root progressions in a given tonality, those up a fourth, down a third, and up a second (+4, -3, +2) are present in significantly greater numbers than the complementary root motions down a fourth, up a third, and down a second (-4, +3, -2) (Meeùs 1988; 1989; 2000). This imbalance sheds light on a specific aspect of the tonal system (Meeùs 2001, 63). The progressions +4, -3, +2 arise almost accidentally in pre-tonal polyphony through the constraints of contrapuntal rules, but in later repertoires they become a decisive syntactical feature that actively constrains tonality. A model combining voice leading and harmonic progression is tested against a body of madrigal cycles by Verdelot, Arcadelt, Lassus, Rore, Wert, and Monteverdi. These cycles, published between c. 1530 and 1638, contain about 50,000 chord progressions, permitting a close examination of the phenomena that reflect the changing status of the prevalent root progressions, the technical aspects that may have fostered it, and the compositional possibilities that result from the evolution outlined. [ABSTRACT FROM AUTHOR]

Albert Simon's analytical approach, or Tonfeldtheorie, is applicable to most compositions of the late nineteenth and early to mid-twentieth centuries. Pitch fields are not based on voices or voice leading; instead, individual pitches are understood as elements of a particular pitch field. The identity of pitches is partly defined systematically, partly through the sound material itself, that is, by material aspects that determine respective concrete, sensually experienceable properties. Compositions, then, are analyzed as aggregates of timbrally defined individual notes whose material identity reveals their membership in structures. [ABSTRACT FROM AUTHOR]

An integral feature in many of Scriabin's late musical narratives is the presence of an atonal problem--a musical event that threatens a harmony which the piece is based on. I offer a new interpretation of Scriabin's late music, in which the idea of an atonal problem becomes a defining feature of his style (after op. 58). This atonal problem is defined as a non-chord tone, which disrupts the balance of the collection (octatonic, whole-tone, or Mystic) which the work is based on. Drawing from Schoenberg's concept of a tonal problem and from Straus's expansion of this concept in Disability Studies in music, I use Scriabin's piano miniatures to show that, within each work, a single pitch class always stands out registrally, dynamically, and/or rhythmically, and becomes an important staple of Scriabin's late style. Thus, the accommodation of this "wrong" note no longer represents that pitch class as a disruptive note, but rather it adds to the unique aspect of that work. [ABSTRACT FROM AUTHOR]

Chromatic, major-third root movement comprises a special class of triadic progression. The contrary motion of half steps, described as PL voice leading, produces a perceptual paradox that simultaneously destroys any sense of background diatonic collection and forces irreconcilable interpretations of consonance and dissonance. Richard Cohn, Richard Taruskin, Ma&hew Bribi;er- Stull, and Sco& Murphy identify examples from 19th-century music and film scores that connect this type of progression with descriptions of uncanny or supernatural phenomena. This article explores the same association in contemporary pop music. Surveying a wide variety of songs from 1956-2016, I examine different tonal and formal contexts in which this phenomenon occurs and provide hermeneutic readings where the association between lyrics and harmony is less obvious. While not every PL/LP transformation evokes an uncanny experience, when the transformations are used conspicuously between adjacent or framing harmonies, the pervasive associations with the uncanny, across sub-genre and generational lines, are hard to ignore. [ABSTRACT FROM AUTHOR]

In this paper, I explore how voice-leading analysis can illuminate text-music relations at both structural and foreground levels in three songs by David Bowie: "Life on Mars," "Five Years," and "We Are the Dead." The analyses provide a way of relating seemingly disparate stylistic examples through text-music relations applicable to a wide variety of rock music. [ABSTRACT FROM AUTHOR]

A venerable rule of traditional Western part writing is the so-called Direct Octaves Rule (also known as Hidden or Exposed octaves), whereby similar pitch motion (i.e., two or more voices moving in the same direction) to a perfect octave should be avoided unless step motion is used. A number of interpretations have been offered as to why musicians might follow this rule. A traditional account (Fux, 1725/1966) exhibits several inconsistencies. A modern interpretation based on auditory scene analysis appears to have merit. However, this interpretation has yet to be tested empirically. Three experiments test the scene-analysis account using numerosity judgments for complex chords as the dependent measure. In Experiment 1, musician listeners show decreasing accuracy in numerosity judgments when an octave is present in the sonority – as predicted. In experiments 2 and 3, chords were preceded by a single neighboring or distant tone. It was hypothesized that neighboring primes would increase the accuracy of numerosity judgments for octave-containing chords more than distant primes – consistent with the Direct Octaves rule. However, no such facilitation was observed. Nevertheless, a post-hoc test showed improved accuracy when the octave was approached by step motion in highest voice compared with step-approach to the lowest voice. This latter finding is consistent with the most restrictive formulations of the direct octaves rule. [ABSTRACT FROM AUTHOR]