Western musicians traditionally classify pitch sequences by disregarding the effects of five musical transformations: octave shift, permutation, transposition, inversion, and cardinality change. We model this process mathematically, showing that it produces 32 equivalence relations on chords, 243 equivalence relations on chord sequences, and 32 families of geometrical quotient spaces, in which both chords and chord sequences are represented. This model reveals connections between music-theoretical concepts, yields new analytical tools, unifies existing geometrical representations, and suggests a way to understand similarity between chord types.

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Generalized voice-leading spaces.

How shall we define the musical stimulus? Traditionally, music theorists interested in pitch phenomena have sought the definition in terms of ratios of whole numbers. Such ratios can provide a description of most, if not all, of the musical intervals in use today, and can be translated readily into ratios of physically realizable tone frequencies. Psychoacousticians, following the dictates of reductionism, have sought a finer grain of analysis than this, pointing out that the essential constituent of an interval is a tone, and that the study of music perception must ultimately refer to the perception of single tones and their frequency components. Accordingly, a great deal of scientific effort has gone into studying the perception of isolated tones.

The Pitch Set as a Level of Description for Studying Musical Pitch Perception

Hermeneutics is defined by the Shorter Oxford English Dictionary as "the art or science of interpretation, esp. of Scripture." It was toward the end of the last century that Hermann Kretschmar applied the word to the verbal elucidation of musical meaning, but musical hermeneutics was an art (or science) that, under one rubric or another, had long been practised. Music, it was generally agreed, had meaning. It was an art of expression; therefore it should be possible to determine what it expressed and how. Kretschmar, with his detailed attempts at exact explanation, thus represented what might be called the dogmatic climax of a long tradition. At the very time he was writing, however, a more formalistic view, expounded by critics like Gurney and Hanslick and furthered by theorists like Schenker, was gaining support. Indeed, for a time the purists were successful in discrediting hermeneutical methods and results--although their victories may have been due less to the force of their own logic than to a widespread reaction against the excesses of ultra-realistic program music on the one hand and of literalistic "interpretations" on the other. As a result, even those who still defended the possibility and relevance of musical expression were loath to define its nature in any specific terms. Today, however, when it is once more acceptable to admit an interest in the subjectmatter and iconography of a painting, it has become increasingly feasible to discuss the putative meaning of a musical composition without evoking immediate derision. There have been a number of recent books that try, from various points of view, to come to grips with the problems involved.'

Schubert's Promissory Note: An Exercise in Musical Hermeneutics

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 listeners9 responses to tonal relationships in musical contexts. It was hypothesized (1) that perception of tonal centers in music develops from listeners9 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 listeners9 choices of tonic and the effect of contextual manipulations of tones in the strings in directing subjects9 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.

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

The Early History of Music Writing in the West

This new series of Norton books, devoted to informed discussion of contemporary music, draws principally upon articles first published in Perspectives of New Music, which Richard Kostelanetz has described as among the most consistently interesting magazines in America. The Perspectives books will comprise a repository of the clearest thinking and most serious writing about twentieth-century music, forming an essential addition to the libraries of both professionals and amateurs concerned with understanding recent developments."

Perspectives on contemporary music theory

4 General Equal-Tempered Harmony: Par ts 2 and 3 IAN QUINN 64 On Tempo Relations ANDREW MEAD 110 Symmetrical Permutation, the Twelve Tones, and Messiaen’s Catalogue d’oiseaux WAI-LING CHEONG 138 Vignettes of Old Masters IV: Jim Randall’s benfest (for Jim Randall at 77) BENJAMIN BORETZ 154 The History of Set Theory from a European Point of View LUIGI VERDI 184 A Model of Melodic Expectation for Some Neo-Romantic Music of Penderecki SCOTT MURPHY 224 Interpreting a Composition, Composing an Interpretation: Schnebel’s Schubert-Phantasie (1978, rev. 1989) PIETER BERGÉ 236 Formant Synthesis, Granular Synthesis, and Waveshaping in Halo, Bells and Voices, Bessell DAVID BESSELL

Perspectives of New Music

The analyses are almost entirely limited to pitch relationships, but are outstanding of their type. Interestingly, Sessions, in his cracking-good and still fresh essay, \"To the Editor,\" warns against \"the pitfalls inherent in the present vogue of 'analysis' as an independent and self-contained musical discipline\" (page 119). This is especially true for pitch analysis, which is to music what sentence parsing is to drama, and is of value chiefly to composers in their role as craftsmen and to other analysts; for the performer its message is faint, and for the listener, fainter still. The conversations and confessions, on the other hand, are of value in many directions; they are full of the glancing revelations that abound in the discourse of lively minds and that seem to apply to any lively recipient. As a group they would be of particular benefit as background reading for the large number of university personnel now in the throes of curriculum revi-

Benjamin Boretz

Papers

Perspectives on contemporary music theory

Perspectives of New Music

Perspectives on American composers

Language, as a Music

Being About Music Textworks 1960-2003