DOCSLIB.ORG

Listening with Two Ears: Conflicting Perceptions of Space in Tonal Music

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Listening with Two Ears: Conflicting Perceptions of Space in Tonal Music Listening with Two Ears: Conflicting Perceptions of Space in Tonal Music Justin Hoffman Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2011 © 2011 Justin Hoffman All Rights Reserved ABSTRACT Listening with Two Ears: Conflicting Perceptions of Space in Tonal Music Justin Hoffman The Tonnetz is a spatial model of tonal pitch, constructed by placing fifths along the horizontal axis of a coordinate plane and thirds along the vertical axis. This dissertation examines the ways in which different conceptions of interval, including just- intonation ratios, diatonic scalar intervals, and pitch-class intervals, result in different Tonnetz geometries, representing different, and sometimes conflicting, modes of musical perception, and argues for treating conflicts between these often unexamined conceptions of interval as an explicit part of musical analysis. Chapter One considers relationships between a number of Tonnetz spaces, as well as the groups of intervals they model, using harmonic function theory. Chapter Two examines ways in which pitches may project multiple functions in Tonnetz spaces and uses these spaces to model some aspects of the harmonic theory of Jean-Philippe Rameau. Chapter Three considers the ways in which neo-Riemannian transformations, as ways of relating triads and seventh chords to one another, might be associated with changes of harmonic function in different Tonnetz spaces, and culminates in an analysis of Chopin’s E Minor Prelude. Chapter Four explores primary triads with chromatically altered roots and fifths and, in this context, analyzes an unusual modulation from A-flat major to E major in Hugo Wolf’s song “An den Schlaf.” Finally, Chapter Five considers harmonic function in nontriadic music, examining the beginning of the final movement of Bartók’s Fourth String Quartet. Contents List of Musical Examples ................................................................................................... ii List of Figures.................................................................................................................... iv Acknowledgements............................................................................................................ ix Introduction Listening with Two Ears.......................................................................... 1 Chapter One Tonal Spaces and Tonal Function.............................................................8 Chapter Two Plural Function in Letter-Class and Pitch-Class Spaces.........................52 Chapter Three Neo-Riemannian Transformations and Harmonic Function.................102 Chapter Four Functional Intensification of Bases and Associates..............................160 Chapter Five Inversion and Dualist Harmonic Function in the Last Movement of Bartók’s Fourth String Quartet .............................................................208 Works Cited .....................................................................................................................239 i Musical Examples Example 1-1 Schubert, String Quartet in G Major, Op. 161, I, mm. 15-24....................20 Example 1-2 Wagner, chromatic grail motif, from Parsifal (with three hearings).........28 Example 1-3 Wagner, diatonic grail motif, from Parsifal ..............................................34 Example 1-4 Wagner, chromatic grail motif, from Parsifal (two hearings)...................43 Example 1-5 Wagner, chromatic grail motif, from Parsifal (two hearings)...................44 Example 1-6 Wagner, chromatic grail motif, from Parsifal (two hearings)...................45 Example 1-7 Schubert, Sanctus, from Mass in E-flat Major, mm. 1-8 ...........................47 Example 2-1 Beethoven, Trio in D Major, Op. 70, no. 1 (Ghost), I, mm. 48-73............59 Example 2-2 Beethoven, Trio in D Major, Op. 70, no. 1 (Ghost), mm. 232-253 ...........62 Example 2-3 Beethoven, Trio in D Major, Op. 70, no. 1 (Ghost), mm. 8-21 .................64 Example 2-4 Beethoven, Trio in D Major, Op. 70, no. 1 (Ghost), mm. 1-15 .................67 Example 2-5 Rameau, Les Indes galantes, earthquake scene, mm. 21-36......................86 Example 2-6 Rameau, Hippolyte et Aricie, Act II, Second Trio of the Fates .................95 Example 2-7 Rameau, Les Indes galantes, earthquake scene, mm. 56-68......................99 Example 3-1 Schubert, Sanctus, from Mass in E-Flat Major, mm. 1-8 ........................108 Example 3-2 Liszt, “Un Sospiro,” from Trois Caprices Poetiques, mm. 66-70...........110 Example 3-3 Beethoven Piano Sonata in A-flat Major, Op. 26, III, mm. 1-29.............113 Example 3-4 Schumann, Novelette, Op. 21, no. 2, mm. 83-91 .....................................119 Example 3-5 Sanctus, from Schubert Mass in A-flat Major, mm. 1-20........................122 Example 3-6 Brahms, Concerto for Violin and Cello, I, mm. 270-278 ........................140 Example 3-7 Chopin, Prelude in E Minor, Op. 28, no. 4 ..............................................147 Example 4-1 Liszt, Organ Mass, Kyrie.........................................................................170 ii Example 4-2 Beethoven, Symphony No. 8 in F Major, IV, mm. 372-292 ...................175 Example 4-3 Chopin, Mazurka in F Minor, op. 68, no. 4 .............................................179 Example 4-4 Wolf, “An den Schlaf”.............................................................................187 iii Figures Figure 0-1 The just-intonation Tonnetz........................................................................... 2 Figure 0-2 The pitch-class Tonnetz ................................................................................. 2 Figure 1-1 Functional roles of each of the seven scale degrees .....................................11 Figure 1-2 Paradigmatic functional discharges ..............................................................12 Figure 1-3 Paradigmatic functional intensifications.......................................................13 Figure 1-4 Discharge and intensification in tonal space.................................................14 Figure 1-5 A segment of the just-intonation Tonnetz.....................................................16 Figure 1-6 Lewin’s just-intonation analysis of Schubert’s quartet.................................20 Figure 1-7 Chromatic and diatonic semitones................................................................22 Figure 1-8 Harmonic progressions illustrating diatonic and chromatic semitones ........22 Figure 1-9 A segment of note-class space ......................................................................23 Figure 1-10 The pitch-class Tonnetz ................................................................................26 Figure 1-11 Chromatic and diatonic semitones in the Tonnetz........................................27 Figure 1-12 The letter-class Tonnetz................................................................................31 Figure 1-13 The scale-degree Tonnetz .............................................................................32 Figure 1-14 Voice leadings in the letter-class Tonnetz ....................................................33 Figure 1-15 Harmonic functions in letter-class and pitch-class space .............................36 Figure 1-16 Mean distances between points in just-intonation and note-class space.......40 Figure 2-1 Dominant and subdominant seventh chords in scale-degree space ..............55 Figure 2-2 Secondary functional identities of V7 and II7 ...............................................56 Figure 2-3 Tonnetz inversions of seventh chords...........................................................57 Figure 2-4 Inversion of an E dominant seventh chord ...................................................63 iv Figure 2-5 Letter-class space centered on D and A triads..............................................65 Figure 2-6 Discharge patterns for dominant and subdominant harmonies.....................68 Figure 2-7 A dominant seventh chord transformed to a French sixth chord..................69 Figure 2-8 The augmented-sixth chord in the Tonnetz ..................................................69 Figure 2-9 Subdominant-to-dominant voice leading......................................................70 Figure 2-10 Revised discharge patterns............................................................................70 Figure 2-11 Rameau’s dominante and sous-dominante harmonies..................................72 Figure 2-12 The dominante and sous-dominante harmonies in Tonnetz space................73 Figure 2-13 Two different just-intonation contexts for a series of pitch classes..............76 Figure 2-14 Riemann’s illustration of connections between harmonies ..........................77 Figure 2-15 Tonnetz pathways associated with each of the six interval classes ..............80 Figure 2-16 Four harmonic contexts for interval class 1..................................................80 Figure 2-17 Harmonic progression with chromatic thirds................................................83 Figure 2-18 Two different just-intonation contexts for a series of pitch classes..............84 Figure 2-19 A Tonnetz representation of mm. 21-36 of the earthquake scene ................88 Figure 2-20 Tonnetz inversion of an A diminished seventh chord ..................................89 Figure 2-21 Rameau’s diatonique-enharmonique progression ........................................91
Load more

Recommended publications
  • Chapter 1 Geometric Generalizations of the Tonnetz and Their Relation To
    November 22, 2017 12:45 ws-rv9x6 Book Title yustTonnetzSub page 1 Chapter 1 Geometric Generalizations of the Tonnetz and their Relation to Fourier Phases Spaces Jason Yust Boston University School of Music, 855 Comm. Ave., Boston, MA, 02215 [email protected] 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 re- lationships and investigated the resulting geometries. This paper adopts the original common-tone based formulation and takes a geometrical ap- proach, 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 Ton- netze 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. 1. Formulations of the Tonnetz The Tonnetz originated in nineteenth-century German harmonic theory as a network of pitch classes connected by consonant intervals.
    [Show full text]
  • Riemann's Functional Framework for Extended Jazz Harmony James
    Riemann’s Functional Framework for Extended Jazz Harmony James McGowan The I or tonic chord is the only chord which gives the feeling of complete rest or relaxation. Since the I chord acts as the point of rest there is generated in the other chords a feeling of tension or restlessness. The other chords therefore must 1 eventually return to the tonic chord if a feeling of relaxation is desired. Invoking several musical metaphors, Ricigliano’s comment could apply equally well to the tension and release of any tonal music, not only jazz. Indeed, such metaphors serve as essential points of departure for some extended treatises in music theory.2 Andrew Jaffe further associates “tonic,” “stability,” and “consonance,” when he states: “Two terms used to refer to the extremes of harmonic stability and instability within an individual chord or a chord progression are dissonance and consonance.”3 One should acknowledge, however, that to the non-jazz reader, reference to “tonic chord” implicitly means triad. This is not the case for Ricigliano, Jaffe, or numerous other writers of pedagogical jazz theory.4 Rather, in complete indifference to, ignorance of, or reaction against the common-practice principle that only triads or 1 Ricigliano 1967, 21. 2 A prime example, Berry applies the metaphor of “motion” to explore “Formal processes and element-actions of growth and decline” within different musical domains, in diverse stylistic contexts. Berry 1976, 6 (also see 111–2). An important precedent for Berry’s work in the metaphoric dynamism of harmony and other parameters is found in the writings of Kurth – particularly in his conceptions of “sensuous” and “energetic” harmony.
    [Show full text]
  • Perceived Triad Distance: Evidence Supporting the Psychological Reality of Neo-Riemannian Transformations Author(S): Carol L
    Yale University Department of Music Perceived Triad Distance: Evidence Supporting the Psychological Reality of Neo-Riemannian Transformations Author(s): Carol L. Krumhansl Source: Journal of Music Theory, Vol. 42, No. 2, Neo-Riemannian Theory (Autumn, 1998), pp. 265-281 Published by: Duke University Press on behalf of the Yale University Department of Music Stable URL: http://www.jstor.org/stable/843878 . Accessed: 03/04/2013 14:34 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Duke University Press and Yale University Department of Music are collaborating with JSTOR to digitize, preserve and extend access to Journal of Music Theory. http://www.jstor.org This content downloaded from 128.84.127.82 on Wed, 3 Apr 2013 14:34:27 PM All use subject to JSTOR Terms and Conditions PERCEIVED TRIAD DISTANCE: EVIDENCE SUPPORTING THE PSYCHOLOGICAL REALITY OF NEO-RIEMANNIAN TRANSFORMATIONS CarolL. Krumhansl This articleexamines two sets of empiricaldata for the psychological reality of neo-Riemanniantransformations. Previous research (summa- rized, for example, in Krumhansl1990) has establishedthe influence of parallel, P, relative, R, and dominant, D, transformationson cognitive representationsof musical pitch. The present article considers whether empirical data also support the psychological reality of the Leitton- weschsel, L, transformation.Lewin (1982, 1987) began workingwith the D P R L family to which were added a few other diatonic operations.
    [Show full text]
  • Tonality and Transformation, by Steven Rings. Oxford Studies in Music Theory
    Tonality and Transformation, by Steven Rings. Oxford Studies in Music Theory. Richard Cohn, series editor. New York: Oxford University Press, 2011. xiv + 243 pp. ISBN 978-0-19-538427-7. $35 (Hardback).1 Reviewed by William O’Hara Harvard University teven Rings’ Tonality and Transformation is only one entry in a deluge of recent volumes1 in the Oxford Studies in Music Theory series,2 but its extension of the transformational ideas pioneered by S scholars like David Lewin, Brian Hyer, and Richard Cohn into the domain of tonal music sets it apart as a significant milestone in contemporary music theory. Through the “technologies” of transformational theory, Rings articulates his nuanced perspective on tonal hearing by mixing in generous helpings of Husserlian philosophy and mathematical graph theory, and situating the result in a broad historical context. The text is also sprinkled throughout with illuminating methodological comments, which weave a parallel narrative of analytical pluralism by bringing transformational ideas into contact with other modes of analysis (primarily Schenkerian theory) and arguing that they can complement rather than compete with one another. Rings manages to achieve all this in 222 concise pages that will prove both accessible to students and satisfying for experts, making the book simultaneously a substantial theoretical contribution, a valuable exposition of existing transformational ideas, and an exhilarating case study for creative, interdisciplinary music theorizing. [2] Tonality and Transformation is divided into two parts. The first three chapters constitute the theory proper—a reimagining of tonal music theory from the ground up, via the theoretical technologies mentioned above. The final 70 pages consist of four analytical essays, which begin to put Rings’ ideas into practice.
    [Show full text]
  • Accelerated Piano Technique and Music Theory Ii Course Syllabus
    ACCELERATED PIANO TECHNIQUE AND MUSIC THEORY II COURSE SYLLABUS Course: Accelerated Piano Technique and Music Theory II Credit: One Carnegie Unit Course Description Accelerated Piano Technique and Music Theory II is required for graduation as a vocal music major. It is for students who have completed the requirements of Accelerated Piano Technique and Music Theory I and completes the prerequisite for all other theory classes. It satisfies the Piano Lab II requirement for vocal majors and the Music Theory II requirement for both vocal and instrumental majors. This course covers the rudiments of music theory and emphasizes basic musicianship skills in the areas of sight singing, ear training, and dictation. Basic piano fundamentals are explored: familiarization with keyboard theory, hand coordination, grand staff note reading, and an introduction to the standard intermediate piano literature. Content Standards DCPS music content standards make up the core skills, concepts and knowledge for Music Theory II: 1. Perform a variety of repertoire. 2. Improvise, compose, and arrange. 3. Read and notate music. 4. Listen, analyze, and evaluate. These standards are incorporated in the course outline below. Course Outline 1. Perform all tasks covered in Accelerated Piano Technique and Music Theory I, with emphasis on reading and writing fluently in treble and bass clefs including identification, notation, reading and writing of all leger line notes above and below the staff. 2. Identify and write all major and minor key signatures; explain and construct a diagram of the circle of fifths. 3. Identify on the page and by ear, sing*, write, and play on the piano keyboard: a.
    [Show full text]
  • Melodies in Space: Neural Processing of Musical Features A
    Melodies in space: Neural processing of musical features A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Roger Edward Dumas IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Dr. Apostolos P. Georgopoulos, Adviser Dr. Scott D. Lipscomb, Co-adviser March 2013 © Roger Edward Dumas 2013 Table of Contents Table of Contents .......................................................................................................... i Abbreviations ........................................................................................................... xiv List of Tables ................................................................................................................. v List of Figures ............................................................................................................. vi List of Equations ...................................................................................................... xiii Chapter 1. Introduction ............................................................................................ 1 Melody & neuro-imaging .................................................................................................. 1 The MEG signal ................................................................................................................................. 3 Background ........................................................................................................................... 6 Melodic pitch
    [Show full text]
  • Andrián Pertout
    Andrián Pertout Three Microtonal Compositions: The Utilization of Tuning Systems in Modern Composition Volume 1 Submitted in partial fulfilment of the requirements of the degree of Doctor of Philosophy Produced on acid-free paper Faculty of Music The University of Melbourne March, 2007 Abstract Three Microtonal Compositions: The Utilization of Tuning Systems in Modern Composition encompasses the work undertaken by Lou Harrison (widely regarded as one of America’s most influential and original composers) with regards to just intonation, and tuning and scale systems from around the globe – also taking into account the influential work of Alain Daniélou (Introduction to the Study of Musical Scales), Harry Partch (Genesis of a Music), and Ben Johnston (Scalar Order as a Compositional Resource). The essence of the project being to reveal the compositional applications of a selection of Persian, Indonesian, and Japanese musical scales utilized in three very distinct systems: theory versus performance practice and the ‘Scale of Fifths’, or cyclic division of the octave; the equally-tempered division of the octave; and the ‘Scale of Proportions’, or harmonic division of the octave championed by Harrison, among others – outlining their theoretical and aesthetic rationale, as well as their historical foundations. The project begins with the creation of three new microtonal works tailored to address some of the compositional issues of each system, and ending with an articulated exposition; obtained via the investigation of written sources, disclosure
    [Show full text]
  • Boris's Bells, by Way of Schubert and Others
    Boris's Bells, By Way of Schubert and Others Mark DeVoto We define "bell chords" as different dominant-seventh chords whose roots are separated by multiples of interval 3, the minor third. The sobriquet derives from the most famous such pair of harmonies, the alternating D7 and AI? that constitute the entire harmonic substance of the first thirty-eight measures of scene 2 of the Prologue in Musorgsky's opera Boris Godunov (1874) (example O. Example 1: Paradigm of the Boris Godunov bell succession: AJ,7-D7. A~7 D7 '~~&gl n'IO D>: y 7 G: y7 The Boris bell chords are an early milestone in the history of nonfunctional harmony; yet the two harmonies, considered individually, are ofcourse abso­ lutely functional in classical contexts. This essay traces some ofthe historical antecedents of the bell chords as well as their developing descendants. Dominant Harmony The dominant-seventh chord is rightly recognized as the most unambiguous of the essential tonal resources in classical harmonic progression, and the V7-1 progression is the strongest means of moving harmony forward in immediate musical time. To put it another way, the expectation of tonic harmony to follow a dominant-seventh sonority is a principal component of forehearing; we assume, in our ordinary and long-tested experience oftonal music, that the tonic function will follow the dominant-seventh function and be fortified by it. So familiar is this everyday phenomenon that it hardly needs to be stated; we need mention it here only to assert the contrary case, namely, that the dominant-seventh function followed by something else introduces the element of the unexpected.
    [Show full text]
  • Generalized Interval System and Its Applications
    Generalized Interval System and Its Applications Minseon Song May 17, 2014 Abstract Transformational theory is a modern branch of music theory developed by David Lewin. This theory focuses on the transformation of musical objects rather than the objects them- selves to find meaningful patterns in both tonal and atonal music. A generalized interval system is an integral part of transformational theory. It takes the concept of an interval, most commonly used with pitches, and through the application of group theory, generalizes beyond pitches. In this paper we examine generalized interval systems, beginning with the definition, then exploring the ways they can be transformed, and finally explaining com- monly used musical transformation techniques with ideas from group theory. We then apply the the tools given to both tonal and atonal music. A basic understanding of group theory and post tonal music theory will be useful in fully understanding this paper. Contents 1 Introduction 2 2 A Crash Course in Music Theory 2 3 Introduction to the Generalized Interval System 8 4 Transforming GISs 11 5 Developmental Techniques in GIS 13 5.1 Transpositions . 14 5.2 Interval Preserving Functions . 16 5.3 Inversion Functions . 18 5.4 Interval Reversing Functions . 23 6 Rhythmic GIS 24 7 Application of GIS 28 7.1 Analysis of Atonal Music . 28 7.1.1 Luigi Dallapiccola: Quaderno Musicale di Annalibera, No. 3 . 29 7.1.2 Karlheinz Stockhausen: Kreuzspiel, Part 1 . 34 7.2 Analysis of Tonal Music: Der Spiegel Duet . 38 8 Conclusion 41 A Just Intonation 44 1 1 Introduction David Lewin(1933 - 2003) is an American music theorist.
    [Show full text]
  • Fourier Phase and Pitch-Class Sum
    Fourier Phase and Pitch-Class Sum Dmitri Tymoczko1 and Jason Yust2(B) Author Proof 1 Princeton University, Princeton, NJ 08544, USA [email protected] 2 Boston University, Boston, MA 02215, USA [email protected] AQ1 Abstract. 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. Keywords: Voice leading Fourier transform Tonal harmony Musical scales Chord geometry· · · · 1Introduction Early twenty-first century music theory explored a two-pronged generalization of traditional set theory. One prong situated sets and set-classes in continuous, non-Euclidean spaces whose paths represented voice leadings, or ways of mov- ing notes from one chord to another [4,13,16]. This endowed set theory with a contrapuntal aspect it had previously lacked, embedding its discrete entities in arobustlygeometricalcontext.AnotherpronginvolvedtheFouriertransform as applied to pitch-class distributions: this provided alternative coordinates for describing chords and set classes, coordinates that made manifest their harmonic content [1,3,8,10,19–21]. Harmonies could now be described in terms of their resemblance to various equal divisions of the octave, paradigmatic objects such as the augmented triad or diminished seventh chord. These coordinates also had a geometrical aspect, similar to yet distinct from voice-leading geometry. In this paper, we describe a new convergence between these two approaches. Specifically, we show that there exists a class of simple circular voice-leading spaces corresponding, in the case of n-note nearly even chords, to the nth Fourier “phase spaces.” An isomorphism of points exists for all chords regardless of struc- ture; when chords divide the octave evenly, we can extend the isomorphism to paths, which can then be interpreted as voice leadings.
    [Show full text]
  • 8.1.4 Intervals in the Equal Temperament The
    8.1 Tonal systems 8-13 8.1.4 Intervals in the equal temperament The interval (inter vallum = space in between) is the distance of two notes; expressed numerically by the relation (ratio) of the frequencies of the corresponding tones. The names of the intervals are derived from the place numbers within the scale – for the C-major-scale, this implies: C = prime, D = second, E = third, F = fourth, G = fifth, A = sixth, B = seventh, C' = octave. Between the 3rd and 4th notes, and between the 7th and 8th notes, we find a half- step, all other notes are a whole-step apart each. In the equal-temperament tuning, a whole- step consists of two equal-size half-step (HS). All intervals can be represented by multiples of a HS: Distance between notes (intervals) in the diatonic scale, represented by half-steps: C-C = 0, C-D = 2, C-E = 4, C-F = 5, C-G = 7, C-A = 9, C-B = 11, C-C' = 12. Intervals are not just definable as HS-multiples in their relation to the root note C of the C- scale, but also between all notes: e.g. D-E = 2 HS, G-H = 4 HS, F-A = 4 HS. By the subdivision of the whole-step into two half-steps, new notes are obtained; they are designated by the chromatic sign relative to their neighbors: C# = C-augmented-by-one-HS, and (in the equal-temperament tuning) identical to the Db = D-diminished-by-one-HS. Corresponding: D# = Eb, F# = Gb, G# = Ab, A# = Bb.
    [Show full text]
  • Hexatonic Cycles
    CHAPTER Two H e x a t o n i c C y c l e s Chapter 1 proposed that triads could be related by voice leading, independently of roots, diatonic collections, and other central premises of classical theory. Th is chapter pursues that proposal, considering two triads to be closely related if they share two common tones and their remaining tones are separated by semitone. Motion between them thus involves a single unit of work. Positioning each triad beside its closest relations produces a preliminary map of the triadic universe. Th e map serves some analytical purposes, which are explored in this chapter. Because it is not fully connected, it will be supplemented with other relations developed in chapters 4 and 5. Th e simplicity of the model is a pedagogical advantage, as it presents a circum- scribed environment in which to develop some central concepts, terms, and modes of representation that are used throughout the book. Th e model highlights the central role of what is traditionally called the chromatic major-third relation, although that relation is theorized here without reference to harmonic roots. It draws attention to the contrary-motion property that is inherent in and exclusive to triadic pairs in that relation. Th at property, I argue, underlies the association of chromatic major-third relations with supernatural phenomena and altered states of consciousness in the early nineteenth century. Finally, the model is suffi cient to provide preliminary support for the central theoretical claim of this study: that the capacity for minimal voice leading between chords of a single type is a special property of consonant triads, resulting from their status as minimal perturbations of perfectly even augmented triads.
    [Show full text]