Diatonic Set Research Articles

Unplayed Galant Melodies, the Ubiquity of the Rarest Interval, and the Heyday of the Major Mode

This article examines in a preliminary fashion the potential connections between the usage of Gjerdingen's (1988, 2007) skeletal galant schemata, the heyday of the major mode during the period 1750-1799 (Albrecht & Huron, 2014; Horn & Huron, 2015), and the rare intervals of the diatonic set (Browne, 1981). I discuss the relations between the rarity of the tritone and semitone in the diatonic template and in musical usage (Huron 2006, 2008; David Temperley, personal communication, 2017). I hypothesize that the skeletal usage of schemata emphasizes rare intervals (tritone and semitone) respective to their common counterparts. Though this is predominantly an armchair, speculative inquiry, a preliminary pilot analysis of a small expert-annotated corpus from Gjerdingen (2007) provides tentative support for the hypothesis that the skeletal usage of schemata overemphasizes vertical tritones, but not melodic semitones. The prevalence of skeletal tritones in the schemata abstracted by Gjerdingen suggests that the process of abstraction is associated with finding unambiguous cues for a local tonal context. While the present article relies on Gjerdingen's expert analytical annotations of a small corpus and extraction of a contrapuntal skeleton, I conclude by offering hypotheses for future testing regarding the increased prevalence and salience of tritones on the musical surface in the period 1750-1799, a subset of common-practice tonality.

Generic Sequences and the Generic Tonnetz

This article employs concepts from diatonic set theory, transformation theory, and neo-Riemannian theory in an investigation of sequences in generic (mod-7) pitch space. Sequences may be described using generic transposition operators, mod-7 analogs of the more familiar mod-12 chromatic transposition operators, and may be mapped in a generic Tonnetz analogous to the Tonnetz diagrams of neo-Riemannian theory. A complete classification of generic sequences with two-chord transposition blocks is derived from the paths described by the sequences in the Tonnetz. While it is tempting to regard generic space as “diatonic,” the examples demonstrate that generic structure governs many sequences that are not actually diatonic at all. The examples also show that the sequences found in the literature include many more types than are widely recognized and that generic sequence structure may apply in numerous other contexts besides patterns of chord roots.

From the Editor

From the Editor Ricardo D. Trimillos The previous issue of Asian Music, 51(1), foregrounded individual carriers of a genre as actors for aspects of continuity, innovation, and viability in contrast to framing a genre in terms of a single grand narrative. In a similar vein, the present number, 51(2), decenters the gaze on a musical genre as unified, homogeneous, and canonical. Rather, its four essays explore ways in which a specific locale or ethnie generates variation, innovation, and creativity, which in turn can inform the tradition writ large. The specific locale can be at a periphery, as Bond shows in the localization of Arabic texts in Gujarati South Asia through the performance of Sufi poetry, or even further afield, as Wisuttipat discusses in his study of Thai traditional music maintained in Chicago. A cultural center can constitute a locale: through its musical theater lkhaon bassac, Cambodia, as Billeri points out, reflects a history of appropriating and indigenizing features from other cultures, including Ramayana episodes from South Asia, theatrical conventions from China and Vietnam, and even the brass band from the Philippines. A locale and a cultural population can interact in contestations of identity and power, as Lai shows us in the case of the Tamil Indian minority in Singapore asserting its ethnic identity thorough drumming. A specific locale or cultural setting can function as a variable and as a nexus for negotiation between individual actors and the canonical tradition. Thus the four essays in this issue suggest distinctions between the category of tradition as a macro conceptualization referencing genre and the category of practice as a means for rationalizing diversity and variation within a specific genre. Maintenance is a common theme in the study of music and includes both continuity and change. Locale, as just discussed, is one variable for continuity and change. Maintenance, however, is a dynamic process rather than a stable state, and music engages with it in a number of ways. First, music performance can represent and reinforce a contested identity, as Tamil drumming attempts to do for the South Indian ethnic minority in Singapore. Second, it can serve as a vehicle for maintenance of other aspects of culture, including belief systems and cultural nationalism (that is, Merriam's functions of music). Social function is evident in the Gujarat, where a song genre using local languages reinforces Dar al-Islam (the Islamic world) and its adherents in regions in which Arabic is not the mother tongue and in Cambodia, where the so-called [End Page 1] operatic theater of lkhaon bassac has been consciously used by various regimes in the service of Khmer nationalism. Finally, maintenance is brought to bear upon the tradition itself as a means of its own preservation. Wisuttipat examines a specific project of maintenance for Central Thai music in Chicago and includes a critique of pedagogical methods developed in and for the diaspora. Specific locale and maintenance are general themes present in the major articles and also in a number of the print and media titles reviewed in 51(2). Furthermore, each article presents additional significant and provocative points relevant to our field. Nattapol Wisuttipat, in his essay "No Notation Needed: The Construction and Politics of Transnational Thai Identity through the Oral Transmission of Classical Music," examines a diasporic learning strategy that is aural and oral. The case speaks to a kind of soft decolonization, suggesting an alternative strategy for addressing critiques of current Southeast Asian institutional pedagogies that are notation based, such as gamelan. One fascinating act of decolonization is retuning a diatonic set of boomwhackers (plastic aerophonic tubes) to Thai tunings by cutting off part of the tube to change its vibrating length. Wisuttipat posits pedagogical strategies as a means of inculcating a Thai cultural sensibility (khwaampenthai) through music learning. Thus the Chicago project engages maintenance in terms of both cultural identity (music as the vehicle) and preservation of the genre itself (music as the object). A contribution to cultural theory is his concept of "resettling" a tradition in the diaspora as contrastive to an earlier concept of "unsettling" a tradition away from the homeland. The author's positionality is significant to the study. In a welcome reversal of...

Generic (Mod-7) Voice-Leading Spaces

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.

It’s only rock ‘n roll (but I like it): Chord perception and rock’s liberal harmonic palette

Both music-theoretic accounts and corpus analyses indicate that rock routinely employs chords that deviate from the norms of common-practice music. Yet we know little about how listeners experience the chords that make up rock’s liberal harmonic palette. In the present study, participants in two online experiments rated major chords that followed a short tonal sequence (a major scale + tonic major triad). Liking ratings obtained in Experiment 1 replicated earlier work showing that listeners prefer rock-typical targets—chords that are common in rock, but lie outside the basic diatonic set—to atypical, rarely used targets (Craton, Juergens, Michalak, & Poirier, 2016). Goodness of fit ratings with the same stimuli in Experiment 2 were similar to the liking ratings. High ratings for rock-typical target chords across the two experiments were not an artifact of their register and the pattern of responses was similar across four levels level of music training. In addition, the mean fitness ratings were approximated by simulations conducted with an auditory short-term memory (ASTM) model (Leman, 2000). Considered together, the findings provide evidence that listeners perceive rock-typical chords to be normal components of a key and that they may do so based on information in the auditory signal alone, without recourse to statistical learning mechanisms or representations of tonal knowledge. We speculate that bottom-up processes operating directly on the auditory signal create a perceptual ranking of chord fitness, which provides the harmonic palette from which composers/improvisers in different musical systems may conservatively (common-practice) or liberally (rock) choose. The mean data are available in tabular form for modelers in the online supplemental material.

Roll Over Beethoven? An Initial Investigation of Listeners’ Perception of Chords Used in Rock Music

Popular mainstream music (hereafter, rock) employs the basic set of chords built on the diatonic scale but also many others (de Clercq & Temperley, 2011; Stephenson, 2002). In an initial investigation of listeners’ perception of rock harmony, we adapted a rating task developed by Krumhansl (1990) in her pioneering studies of the common-practice harmonic hierarchy. Participants provided surprise and liking ratings for 35 (major, minor, dominant 7, and control) target chords that followed short key-establishing musical passages (a major scale + tonic major triad, or a major pentatonic scale + tonic major triad) or white noise. Liking ratings for targets that followed a musical passage indicated that listeners prefer some chords that are common in rock, but that lie outside the basic diatonic set, to atypical chords; this effect depended on chord quality and was clearest for major target chords. The findings provide an impetus for further research exploring the harmonic hierarchy in listeners with varying musical backgrounds; it appears to be more inclusive and less differentiated than is commonly thought.

The inequality factor: skewness and kurtosis as a measure of set-class cohesion

Building on current set-theoretic research, including similarity measures and ‘maximal evenness’, as well as findings in diatonic set theory and our own work on salience, we examine the interval-class content of set classes based on internal consistency (related to ‘cohesion’ and ‘regularity’) and inconsistency (‘skewness’). By systematically comparing one member tone of a set to the others, we posit abstract hierarchies between members based on salience, rather than the presence of specific interval classes or tonal function. Quantifying these factors allows us to hypothesize (1) what gives certain set classes the potential to function in a tonal way; (2) a ‘set-class-sensitive’ method to gauge the level of ‘dissonance’ in a composition (without presuppositions about the quality of specific interval classes); and (3) how different set classes might be deployed compositionally in order to achieve a variety of results.

FORM AND TRANSFORMATION IN THE ‘NOCTURNE’ FROM BRITTEN'S ‘SERENADE FOR TENOR, HORN AND STRINGS’

AbstractThe ‘Nocturne’ from Benjamin Britten's Serenade for Tenor, Horn and Strings (1943) presents a number of interesting melodic and motivic features effectively modelled by aspects of diatonic transformational theories. Following a brief review of important transformational operations in diatonic set theory (transposition within Mod-7 diatonic space, diatonic interval cycles, and ‘signature transformation’), this article presents an analysis of the ‘Nocturne’ drawing upon both traditional and recent developments in diatonic transformational theory. Doing so illustrates an intricate compositional technique, one that traces motivic associations in the vocal line, the horn part and the accompanying strings. A close reading reveals that these motivic techniques stem from the generic concept of echoing and reverberation at the heart of Tennyson's poem.

Enharmonic systems: A theory of key signatures, enharmonic equivalence and diatonicism

Key signatures and enharmonic equivalence are taken as points of departure for a study of the diatonic–chromatic relationship. Key signatures are modelled by signature vectors, i.e. seven-dimensional vectors with integer coordinates, each coordinate indicating the number of sharps (+) or flats (−) assigned to one of the seven letter names. A definition of standard signature vectors (those corresponding to the usual key signature of some major key) is readily formulated. These constructions do not depend on any convention regarding enharmonic equivalence of note names, but enharmonic equivalence (EE) conditions are intimately connected with key signatures, and in fact may also be formalized in terms of signature vectors, called in this context EE vectors. The canonical EE vector gives rise to a familiar twelve-note enharmonic system, but other systems are possible. Connections with maximal evenness and other properties familiar in diatonic set theory are investigated. The usual staff notation, including key signatures, may be realized within any enharmonic system, and various transformations (diatonic and chromatic transposition, and signature transformations that alter the key signature) may be applied to music thus notated. The interaction between the EE vector defining a system and the signature vector defining a seven-note subset thereof is subtle and sometimes unexpected. Several non-canonical enharmonic systems are explored, including those used in the Twelve Microtonal Etudes of Easley Blackwood.

Diatonic Trichords in Two Pieces from Kurtág's <i>Kafka-Fragmente</i>: A Neo-Riemannian Approach

The fundamentals of contextual transformation are explained, beginning with the idea of a repeated interval pair as found in Johannes Brahms and Gyorgy Kurtag, and proceeding to the transformations Parallel (P), Leittonwechsel (L),and Relative (R), as defined by Hugo Riemann, and slide (S),as defined by Lewin. The neo-Riemannian Tonnetz is introduced, with generalizations of P, L, R, and S to suit trichords other than the familiar consonant triads. Two Pieces from Kurtag's Kafka-Fragmente are analyzed by means of networks invoking the above transformations. In one case, pitch-class sets that are the union of two trichords related by P or S are shown to exist at two levels of structure. In the other case, generation of a complete diatonic set via two contextual inversions operating alternately on a set such as {C, D, F}, whose pairs form a major 2nd, minor 3rd, and perfect 4th, is shown to account for the progress of a melody.

Scales, Sets, and Interval Cycles: A Taxonomy

Recent studies in theory of by Agmon, Balzano, Carey and Clampitt, Clough and Douthett, Clough and Myerson, and Gamer have in common central role of interval cycle. Based on scale features defined in these studies, and an additional feature called distributional evenness defined here, a taxonomy is proposed for pitch-class sets (pcsets) correspond to interval cycles or to certain conjunctions thereof. Pairwise implicative relationships among features are explored. Of twenty sets of features that are consistent with these relationships, thirteen are found to be instantiated by actual pcsets and seven others are shown to be incapable of instantiation. Most instantiated feature-sets correspond to infinite classes of pcsets which are shown to be enumerable; one such feature-set is found to be uniquely realized (up to transposition) in usual diatonic pcset. Earlier versions of this paper, by Clough and Engebretsen, were presented at conferences of Society for Music Theory, New York, November 1995, and European Society for Cognitive Sciences of Music, Uppsala, Sweden, June 1997. WORKS CITED Agmon, Eytan. 1989. A mathematical model of diatonic system. Journal of Music Theory 33: 1-25. . 1991. Linear transformations between cyclically generated chords. Musikometrika 3: 15-41. . 1996. Coherent tone-systems: a study in theory of diatonicism. Journal of Music Theory 40: 39-59. Balzano, Gerald J. 1980. The group-theoretic description of 12fold and microtonal pitch systems. Computer Music Journal 4: 66-84. Browne, Richmond. 1981. Tonal implications of diatonic set. In Theory Only 5/6-7: 3-21. Carey, Norman, and David Clampitt. N.d. Two theorems concerning rational approximations: number theory and music theory. Typescript. -. 1989. Aspects of well-formed scales. Music Theory Spectrum 11: 187-206. Clampitt, David. 1997. Pairwise well-formed scales: structural and transformational properties. Ph.D. diss., State University of New York at Buffalo. Clough, John. 1994. Diatonic interval cycles and hierarchical structure. Perspectives of New Music 32/1: 228-53. Clough, John, and Gerald Myerson. 1985. Variety and multiplicity in diatonic systems. Journal of Music Theory 29: 249-70. Clough, John, and Gerald Myerson. 1986. Musical and generalized circle of fifths. American Mathematical Monthly 93: 695-701. Clough, John, and Jack Douthett. 1991. Maximally even sets. Journal of Music Theory 35: 93-173. Cohn, Richard. 1991. Properties and generability of transpositionally invariant sets. Journal of Music Theory 35: 1-32. This content downloaded from 207.46.13.33 on Wed, 12 Oct 2016 04:55:22 UTC All use subject to http://about.jstor.org/terms 104 Music Theory Spectrum . 1996. Maximally smooth cycles, hexatonic systems, and analysis of late-romantic triadic progressions. Music Analysis 15: 9-40. Gamer, Carlton. 1967. Some combinational resources of equaltempered systems. Journal of Music Theory 11: 32-59. Hanson, Howard. 1960. The harmonic materials of modern music: resources of tempered scale. New York: Appleton-Century-Crofts. Lewin, David. 1982. A formal theory of generalized tonal functions. Journal of Music Theory 26: 23-60. . 1996. Cohn functions. Journal of Music Theory 40: 181-216. Messiaen, Olivier. 1944. Technique de mon langage musicale. Paris: A. LeDuc. Morris, Robert D. 1987. Composition with pitch-classes. New Haven: Yale University Press. Rahn, Jay. 1996. Turning analysis around. Black Music Research Journal 16: 71-90. Soderberg, Stephen. 1995. Z-related sets as dual inversions. Journal of Music Theory 39: 77-100. Starr, Daniel, and Robert Morris. 1977-78. A general theory of combinatoriality and aggregate. Perspectives of New Music 16/1: 3-35, and 16/2: 50-84. Vieru, Anatol. 1995. Diatonicity and chromaticity. Muzica 21: 64-68. Wilson, Erv. 1975. Letter to John Chalmers re the basic structure of moments of symmetry. Winograd, Terry. N.d. An analysis of properties of deep scales in a T-tone system. Typescript. Wooldridge, Marc. 1992. Rhythmic implications of diatonic theory: a study of Scott Joplin's ragtime piano works. Ph.D. diss., State University of New York at Buffalo. This content downloaded from 207.46.13.33 on Wed, 12 Oct 2016 04:55:22 UTC All use subject to http://about.jstor.org/terms

Concert Music, Rock, and Jazz since 1945: Essays and Analytical Studies

Concert Music, Rock, and Jazz since 1945: Essays and Analytical Studies. Edited by Elizabeth West Marvin and Richard Hermann. (Eastman Studies in Music.) Rochester, N.Y.: University of Rochester Press, 1995. [x, 449 p. ISBN 1-878822-42-X. $95.00.] And so, with that discussion of the Bartok Concerto for Orchestra, we conclude our course in the analysis of music of the twentieth century. Presumably, this statement is now heard less often, but it is not difficult to understand why it has persisted. There been number of obstacles to the fruitful discussion of music since 1945. Foremost is the appearance of music for which the traditional values of coherence, motivic consistency, and organic process are either peripheral or irrelevant. Since these properties are the tenets upon which most traditional analysis and criticism are based, the usual means of discussing the music intelligently suddenly become frustrating, misleading, or inappropriate. Moreover, even when the music is meant to cohere, in this Beethovenian sense, it may be based on highly complex systems, which take time to learn and often do not seem terribly relevant to the perception of the music. Finally, the less the coherence of piece is dependent on pitch structure, the less applicable so much of our analytical arsenal seems to be. Concert Music, Rock, and Jazz since 1945 is collection of essays dedicated to the sensible analytical discussion of some of the music that is, in general, resistant to traditional analysis. In sense, it is snapshot of how we dealt with the problem of talking about recent music. It assures us it is neither history nor an exhaustive survey; rather, the editors have chosen essays that break new ground with respect to the compositions that they address, the perspectives from which they are written, or the analytical methodologies that they introduce or extend (p. 3). Although one might not call each essay groundbreaking, they comprise diverse sampling of what appears to be promising analysis these days, and they do present survey of the issues that seem to be pressing. The book is divided into three sections, but they are not those suggested by the title. The first, Compositional Poetics, is described by the editors as exploring current cultural contexts for compositional thought (p. 5). Part 2 is entitled Some Structuralist Approaches. These essays are concerned with demonstrating how the music exhibits comprehensible, organic structure in the traditional sense, and we shall begin here. In some cases, the music in question is dominated by familiar means of coherence, such as tonality or pitch structure. In others, the musical language is particularly nontraditional, and the essay must develop and present highly idiosyncratic analytical strategies in order to provide basis for fruitful discussion of the music. Fortunately, each author is remarkably lucid, and it is easy to imagine any of these essays as an introduction to new analytical methods for otherwise musically experienced readers. In his essay concerning the music of Elliott Carter, Andrew Mead shows how Carter's registral manipulation of pitch classes can be seen as an alternative to the temporal manipulations characterized by serialism, demonstrating a commonality with Schoenberg's practice (p. 99). In doing so, however, Mead takes the opportunity to review the relation between the total chromatic and the diatonic set; by consistently using relatively recent terminology such as class, he spares the novice reader the confusion usually brought about by older terminology (such as the term row signifying the [at most] forty-eight set forms). Therefore, this essay not only takes the reader through certain complex means of pitch organization, but presents worthwhile and clear account of the twelve-tone tradition as well. The subject of Walter Everett's essay, side two of the Beatles's Abbey Road album, also lends itself to somewhat traditional approach: here voice-leading graphs demonstrate the large-scale coherence of the song cycle. …

Analyzing Form and Measuring Perceptual Content in Mozart's Sonata K. 282: A New Theory of Parametric Analogues

Hierarchic analysis in music necessarily separates form from content. However, in active listening, the two are indivisible. To illustrate this, I first analyze in Part 1 the opening movement in Mozart's Sonata K. 282 from the top down, using traditional methods in music theory. Arriving at the manifest level, I then dissect the music from the bottom up, relying on the implication-realization model (Narmour, 1977,1989,1990,1991a, 1992). The contrasting perspectives reveal in great detail some of the movement's richly complex structuring. More generally, they confirm the inextricable feedback between parametric content and the meaning of form, specifically with respect to the contrary functions of closure and nonclosure. Following these analyses, Part 2 forges a synthesis by developing an implicative theory of analogical structures for melody, harmony, duration, and meter. Because, in terms of bottom-up processing, the analytical symbology for tracking structures is commensurable, we can, in all four primary parameters, weight similarity (aa), difference (ab), closure (stability), and nonclosure (implication) with comparable numbers. Further, by adding in some essential stylistic properties from the top down (scale step, diatonic pitch set, tonal cadential closure), we are able to represent the overall rhythmic shape of the first phrase in a single twodimensional graph. Thereby, we recapture from hierarchic analysis the perceptual sense that, in on-line listening, form and content are synthetically one.

Regions: A Theory of Tonal Spaces in Early Medieval Treatises

The discussion of harmonics in the ninth-century Scolica enchiriadis includes a diagram (reproduced as figure 1) that displays the relationships among the consonances and their correspondences with numerical proportions. This passage and diagram are included in Strunk's anthology (1950, 136).' The numbers in the diagram are the elements of the classical tetractys-6, 8, 9, 12---duplicated at the octave to form what we may refer to as the heptactys, the set {6,8,9,12,16,18,24}. The heptactys is our point of departure and the locus of intersection between ancient theoretical tradition and work of the last few decades that falls under the rubric of diatonic set theory, a relation already suggested in a general way in Gauldin (1983). In this paper we generate the elements of the Scolica diagram in a manner that may well be ahistorical, although it uses no concept that could not have been available in the ancient or medieval worlds. The heptactys will then be shown to belong to a class of pitch collections called regions, generated in the same way. Several other medieval constructions, including Pseudo-Odo's gamut, Guido's hexachord, the tetrachord of the finals, and the diamond-shaped diagrams in Guido's Micrologus and elsewhere, may be construed as regions, as we will see. Regions exist in pitch space. Taking a perspective that focuses exclu-

Harmonic Vocabulary in the Music of John Adams: A Hierarchical Approach

Following the minimalist tradition, much of John Adams's' music consists of long passages employing a single set of pitch classes (pcs) usually encompassed by one diatonic set.2 In many of these passages the pcs form a single diatonic triad or seventh chord with no additional pcs. In other passages textural and registral formations imply a single triad or seventh chord, but additional pcs obscure this chord to some degree. These phenomena suggest a hierarchical approach to the harmonic analysis of Adams's music that relates passages containing only a triad or seventh chord to passages incorporating additional pcs. For each individual passage I recognize three distinct pc levels arranged in an ordered triple of pc sets called a complex. A complex consists of (1) a chord-a strongly projected triad or seventh chord; (2) a sonority-all strongly presented pcs in the passage, encompassing the chord plus other strongly presented pcs, if any; and (3) a field-a complete diatonic collection plus strongly presented non-

Pitch salience and generative harmonic resolution of major triad inversions

Subjects from a broad range of musical experience were presented with inversions of the major triad, and asked to sing the most salient pitch, and the pitch that would best follow the chord. Responses were recorded by computer via voltage-to-music instrument digital interface conversion. Responses for salience did not differ significantly with musical experience, and correspond with predictions based on subharmonic matching [Tehrhardt, Tiefenstrukt ur der Musik (1982); Parncutt, Music Percept. 6, 65–94 (1988)], and with the results of other empirical measurements. Results for choice of resolution do not confirm the specific predictions of music theory, specifically the cadential resolution of the second inversion, except for musicians with highly developed skills in formal musical analysis. Responses do however show a significant increase in resolution to the tonic, to the dominant, and to the diatonic set, with experience. These results demonstrate the difference between psychoacoustic and cognitive levels of music perception, with the former less influenced by experiential variables, and the latter indicating a continual strengthening of the basic internalized schemata that underlie culturally based tonal systems throughout musical development.

A Study of Event Hierarchies in Tonal and Post-Tonal Music

It is paradoxical that although there is general agreemenrt that imlusici pitc1b perception is much more accurate when the listener is givetn a tonal cOl1tt for the stimulus, leading perceptual theories disagree in their definitiois o tonal context itself. Nearly all cognitive theories of so-called pitch istruct Li-., in music assume that relations among acoustical components of sound. or mathematical groups, or geometrical constructs, can serve as adequate bases on which rules of musical composition and comprehension of pitch relationl- ships may be formulated. These theories have an obvious intellectual iippecai but do not describe the temporal structure of pitch relations in actual music, and thus fail to account for significant differences in the compositiona treatment of pitch from one style period, or one culture, to another. I hc perceptual theory of event hierarchies, advanced most notably by Bharucha attempts to describe the perception of pitch relationships in musical timtc and thus is an important exception to the rule that perceptual theories oi musical pitch deal solely in timeless abstractions. The event-hierarchhx theory holds that as musical events (tones, chords) unfold in a specitic composition, they are interpreted hierarchically by the listener; this cxvent hierarchy activates in the listener a tonal hierarchy, a hierarchy of ccivt classes. The two hierarchies are thought to be perceptually co-generativec symbiotic partners in what Deutsch (1984) calls an "elaborate bootstrapping operation". No adequate description yet exists of this bootstrapping opera tion, in which relatively stable and unstable members of a totial hierarchx are said to generate a piece-specific event hierarchy as the event hicrarchx defines the tonal hierarchy. Beyond the critical chicken-egg character of such a theory, there is the likelihood that the theory is far more comuplex than the cognitive act it is supposed to describe. An inspection of represen- tative samples of Western tonal and post-tonal music shows that stvic- dependent differences exist in the time orders of intervals that occtur onlv rarely in the diatonic set. The paper includes the first report of the results of an experiment using temporal organising schemes extracted from actual tonal and post-tonal compositions. These results indicate that aural context conveyed by temporal collocation of rare intervals may be as influenti'il as pitch content is in eliciting a sense of tonal structure in the musical listener. and further that temporal organisation of rare intervals mav provide perceptual cues for metrical organisation in tonal music.

Aspects of Well-Formed Scales

Pentatonic, diatonic, and chromatic scales share the same underlying structure, that of the well-formed scale. Well-formedness is defined in terms of a relationship between the order in which a single interval generates the elements of a pitch-class set and the order in which those elements appear in a scale. Another characterization provides a recursive procedure for organizing all well-formed scales into hierarchies. Finally, well-formed scales are defined in terms of scale-step measure, and aspects of the diatonic set are examined. This content downloaded from 157.55.39.195 on Thu, 14 Apr 2016 05:35:50 UTC All use subject to http://about.jstor.org/terms

Describing the Perception of Tonality in Music: A Critique of the Tonal Hierarchy Theory and a Proposal for a Theory of Intervallic Rivalry

Strengths and limitations of the tonal hierarchy theory, and of the probetone testing procedure used to substantiate that theory, are discussed. The tonal hierarchy theory is characterized as an important contribution in that it begins to describe hierarchical relationships of tones in the diatonic set. The tonal hierarchy theory is, however, criticized because it does not describe the mental process or processes by which the tonal center of a piece of tonal music is recognized, nor does it account for the dynamic perception of tonality as it unfolds during actual musical listening. The probe-tone testing procedure most often used to substantiate the tonal hierarchy theory is criticized for the ambiguity of its response task, so that test results could be an artifact of effects of short-term memory. An alternative perceptual theory is proposed to describe the timedependent nature of pitch relationships in music. In this description, listeners are assumed to recognize the tonal center in tonal music on a bestevidence basis, and it is asserted that the clearest evidence is carried in the rarest-occurring intervals in the diatonic set. Evidence, gathered in a series of experiments, is cited to demonstrate that listeners both with and without extensive formal training in music form strong (and usually tacit) mental representations of unambiguous tonality when tones are arranged across time so as to form meaningful tonal referents.

The Interplay of Set Content and Temporal Context in a Functional Theory of Tonality Perception

The purpose of this study was to provide evidence for the perceptual component of an analysis of pitch relationships in tonal music that includes consideration of both formal analytic systems and musical listeners' responses to tonal relationships in musical contexts. It was hypothesized (1) that perception of tonal centers in music develops from listeners' interpretations of time-dependent contextual (functional) relationships among pitches, rather than primarily through knowledge of psychoacoustical or structural characteristics of the pitch content of sets or scales and (2) that critical perceptual cues to functional relationships among pitches are provided by the manner in which particular intervallic relationships are expressed in musical time. Excerpts of tonal music were chosen to represent familiar harmonic relationships across a spectrum of tonal ambiguity/specificity. The pitch-class sets derived from these excerpts were ordered: (1) to evoke the same tonic response as the corresponding musical excerpt, 2) to evoke another tonal center, and (3) to be tonally ambiguous. The effect of the intervallic contents of musical excerpts and strings of pitches in determining listeners' choices of tonic and the effect of contextual manipulations of tones in the strings in directing subjects' responses were measured and compared. Results showed that the musically trained listeners in the study were very sensitive to tonal implications of temporal orderings of pitches in determining tonal centers. Temporal manipulations of intervallic relationships in stimuli had significant effects on concurrences of tonic responses and on tonal clarity ratings reported by listeners. The interval rarest in the diatonic set, the tritone, was the interval most effective in guiding tonal choices. These data indicate that perception of tonality is too complex a phenomenon to be explained in the time-independent terms of psychoacoustics or pitch- class collections, that perceived tonal relationships are too flexible to be forced into static structural representations, and that a functional interpretation of rare intervals in optimal temporal orderings in musical contexts is a critical feature of tonal listening strategy.