A Tutorial on Pitch Class Sets
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Perceptual Interactions of Pitch and Timbre: an Experimental Study on Pitch-Interval Recognition with Analytical Applications
Perceptual interactions of pitch and timbre: An experimental study on pitch-interval recognition with analytical applications SARAH GATES Music Theory Area Department of Music Research Schulich School of Music McGill University Montréal • Quebec • Canada August 2015 A thesis submitted to McGill University in partial fulfillment of the requirements of the degree of Master of Arts. Copyright © 2015 • Sarah Gates Contents List of Figures v List of Tables vi List of Examples vii Abstract ix Résumé xi Acknowledgements xiii Author Contributions xiv Introduction 1 Pitch, Timbre and their Interaction • Klangfarbenmelodie • Goals of the Current Project 1 Literature Review 7 Pitch-Timbre Interactions • Unanswered Questions • Resulting Goals and Hypotheses • Pitch-Interval Recognition 2 Experimental Investigation 19 2.1 Aims and Hypotheses of Current Experiment 19 2.2 Experiment 1: Timbre Selection on the Basis of Dissimilarity 20 A. Rationale 20 B. Methods 21 Participants • Stimuli • Apparatus • Procedure C. Results 23 2.3 Experiment 2: Interval Identification 26 A. Rationale 26 i B. Method 26 Participants • Stimuli • Apparatus • Procedure • Evaluation of Trials • Speech Errors and Evaluation Method C. Results 37 Accuracy • Response Time D. Discussion 51 2.4 Conclusions and Future Directions 55 3 Theoretical Investigation 58 3.1 Introduction 58 3.2 Auditory Scene Analysis 59 3.3 Carter Duets and Klangfarbenmelodie 62 Esprit Rude/Esprit Doux • Carter and Klangfarbenmelodie: Examples with Timbral Dissimilarity • Conclusions about Carter 3.4 Webern and Klangfarbenmelodie in Quartet op. 22 and Concerto op 24 83 Quartet op. 22 • Klangfarbenmelodie in Webern’s Concerto op. 24, mvt II: Timbre’s effect on Motivic and Formal Boundaries 3.5 Closing Remarks 110 4 Conclusions and Future Directions 112 Appendix 117 A.1,3,5,7,9,11,13 Confusion Matrices for each Timbre Pair A.2,4,6,8,10,12,14 Confusion Matrices by Direction for each Timbre Pair B.1 Response Times for Unisons by Timbre Pair References 122 ii List of Figures Fig. -
Unified Music Theories for General Equal-Temperament Systems
Unified Music Theories for General Equal-Temperament Systems Brandon Tingyeh Wu Research Assistant, Research Center for Information Technology Innovation, Academia Sinica, Taipei, Taiwan ABSTRACT Why are white and black piano keys in an octave arranged as they are today? This article examines the relations between abstract algebra and key signature, scales, degrees, and keyboard configurations in general equal-temperament systems. Without confining the study to the twelve-tone equal-temperament (12-TET) system, we propose a set of basic axioms based on musical observations. The axioms may lead to scales that are reasonable both mathematically and musically in any equal- temperament system. We reexamine the mathematical understandings and interpretations of ideas in classical music theory, such as the circle of fifths, enharmonic equivalent, degrees such as the dominant and the subdominant, and the leading tone, and endow them with meaning outside of the 12-TET system. In the process of deriving scales, we create various kinds of sequences to describe facts in music theory, and we name these sequences systematically and unambiguously with the aim to facilitate future research. - 1 - 1. INTRODUCTION Keyboard configuration and combinatorics The concept of key signatures is based on keyboard-like instruments, such as the piano. If all twelve keys in an octave were white, accidentals and key signatures would be meaningless. Therefore, the arrangement of black and white keys is of crucial importance, and keyboard configuration directly affects scales, degrees, key signatures, and even music theory. To debate the key configuration of the twelve- tone equal-temperament (12-TET) system is of little value because the piano keyboard arrangement is considered the foundation of almost all classical music theories. -
MTO 19.3: Nobile, Interval Permutations
Volume 19, Number 3, September 2013 Copyright © 2013 Society for Music Theory Interval Permutations Drew F. Nobile NOTE: The examples for the (text-only) PDF version of this item are available online at: http://www.mtosmt.org/issues/mto.13.19.3/mto.13.19.3.nobile.php KEYWORDS: Interval permutations, intervals, post-tonal analysis, mathematical modeling, set-class, space, Schoenberg, The Book of the Hanging Gardens ABSTRACT: This paper presents a framework for analyzing the interval structure of pitch-class segments (ordered pitch-class sets). An “interval permutation” is a reordering of the intervals that arise between adjacent members of these pitch-class segments. Because pitch-class segments related by interval permutation are not necessarily members of the same set-class, this theory has the capability to demonstrate aurally significant relationships between sets that are not related by transposition or inversion. I begin with a theoretical investigation of interval permutations followed by a discussion of the relationship of interval permutations to traditional pitch-class set theory, specifically focusing on how various set-classes may be related by interval permutation. A final section applies these theories to analyses of several songs from Schoenberg’s op. 15 song cycle The Book of the Hanging Gardens. Received May 2013 [1] Example 1 shows the score to the opening measures of “Sprich nicht immer von dem Laub,” the fourteenth song from Schoenberg’s Book of the Hanging Gardens, op. 15. Upon first listen, one is drawn to the relationship between the piano’s gesture in the first measure and the vocal line in the second. -
The Social Economic and Environmental Impacts of Trade
Journal of Modern Education Review, ISSN 2155-7993, USA August 2020, Volume 10, No. 8, pp. 597–603 Doi: 10.15341/jmer(2155-7993)/08.10.2020/007 © Academic Star Publishing Company, 2020 http://www.academicstar.us Musical Vectors and Spaces Candace Carroll, J. X. Carteret (1. Department of Mathematics, Computer Science, and Engineering, Gordon State College, USA; 2. Department of Fine and Performing Arts, Gordon State College, USA) Abstract: A vector is a quantity which has both magnitude and direction. In music, since an interval has both magnitude and direction, an interval is a vector. In his seminal work Generalized Musical Intervals and Transformations, David Lewin depicts an interval i as an arrow or vector from a point s to a point t in musical space. Using Lewin’s text as a point of departure, this article discusses the notion of musical vectors in musical spaces. Key words: Pitch space, pitch class space, chord space, vector space, affine space 1. Introduction A vector is a quantity which has both magnitude and direction. In music, since an interval has both magnitude and direction, an interval is a vector. In his seminal work Generalized Musical Intervals and Transformations, David Lewin (2012) depicts an interval i as an arrow or vector from a point s to a point t in musical space (p. xxix). Using Lewin’s text as a point of departure, this article further discusses the notion of musical vectors in musical spaces. t i s Figure 1 David Lewin’s Depiction of an Interval i as a Vector Throughout the discussion, enharmonic equivalence will be assumed. -
The Computational Attitude in Music Theory
The Computational Attitude in Music Theory Eamonn Bell Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2019 © 2019 Eamonn Bell All rights reserved ABSTRACT The Computational Attitude in Music Theory Eamonn Bell Music studies’s turn to computation during the twentieth century has engendered particular habits of thought about music, habits that remain in operation long after the music scholar has stepped away from the computer. The computational attitude is a way of thinking about music that is learned at the computer but can be applied away from it. It may be manifest in actual computer use, or in invocations of computationalism, a theory of mind whose influence on twentieth-century music theory is palpable. It may also be manifest in more informal discussions about music, which make liberal use of computational metaphors. In Chapter 1, I describe this attitude, the stakes for considering the computer as one of its instruments, and the kinds of historical sources and methodologies we might draw on to chart its ascendance. The remainder of this dissertation considers distinct and varied cases from the mid-twentieth century in which computers or computationalist musical ideas were used to pursue new musical objects, to quantify and classify musical scores as data, and to instantiate a generally music-structuralist mode of analysis. I present an account of the decades-long effort to prepare an exhaustive and accurate catalog of the all-interval twelve-tone series (Chapter 2). This problem was first posed in the 1920s but was not solved until 1959, when the composer Hanns Jelinek collaborated with the computer engineer Heinz Zemanek to jointly develop and run a computer program. -
Interval Cycles and the Emergence of Major-Minor Tonality
Empirical Musicology Review Vol. 5, No. 3, 2010 Modes on the Move: Interval Cycles and the Emergence of Major-Minor Tonality MATTHEW WOOLHOUSE Centre for Music and Science, Faculty of Music, University of Cambridge, United Kingdom ABSTRACT: The issue of the emergence of major-minor tonality is addressed by recourse to a novel pitch grouping process, referred to as interval cycle proximity (ICP). An interval cycle is the minimum number of (additive) iterations of an interval that are required for octave-related pitches to be re-stated, a property conjectured to be responsible for tonal attraction. It is hypothesised that the actuation of ICP in cognition, possibly in the latter part of the sixteenth century, led to a hierarchy of tonal attraction which favoured certain pitches over others, ostensibly the tonics of the modern major and minor system. An ICP model is described that calculates the level of tonal attraction between adjacent musical elements. The predictions of the model are shown to be consistent with music-theoretic accounts of common practice period tonality, including Piston’s Table of Usual Root Progressions. The development of tonality is illustrated with the historical quotations of commentators from the sixteenth to the eighteenth centuries, and can be characterised as follows. At the beginning of the seventeenth century multiple ‘finals’ were possible, each associated with a different interval configuration (mode). By the end of the seventeenth century, however, only two interval configurations were in regular use: those pertaining to the modern major- minor key system. The implications of this development are discussed with respect interval cycles and their hypothesised effect within music. -
Motivic Similarity and Form in Boulez's Anthèmes
Motivic similarity and form in Boulez’s Anthèmes Cecilia Taher Music Theory Area, Department of Music Research Schulich School of Music, McGill University Montreal, Canada June 2016 A thesis submitted to McGill University in partial fulfillment of the requirements for the degree of Doctor of Philosophy. © 2016 Cecilia Taher i To the memory of the musical genius Pierre Boulez (1925-2016), whose music and words inspired the very genesis of this research project. ii iii Abstract This dissertation combines theoretical, analytical, and empirical methods to study aspects of motivic similarity and form in Boulez’s Anthèmes. The concept of similarity is central to the definition of motives and musical form. Nevertheless, little is known about the perception of motivic relationships and form while listening to post-tonal music. The most traditional theoretical methods used for post-tonal analysis (such as pitch-class set theory) have tended to emphasize abstract similarity relations based on local pitch and interval structures, disregarding features other than pitch as well as larger formal levels. However, experimental research has demonstrated that the perception of these pitch relationships in post-tonal music is secondary and extremely difficult. According to the empirical evidence, the kinds of structures that are particularly salient during music listening (including post-tonal music listening) are defined by both local- and global-level relationships involving surface features such as articulation and rhythm. This marks a gap between theoretical and perceptual research agendas. Boulez seemed to be aware of this disconnect, as he explicitly intended to reconcile perception with compositional practice. He described the form of Anthèmes in terms of constantly changing yet clearly recognizable motives, with surface features acting as large-scale formal markers. -
Set Theory: a Gentle Introduction Dr
Mu2108 Set Theory: A Gentle Introduction Dr. Clark Ross Consider (and play) the opening to Schoenberg’s Three Piano Pieces, Op. 11, no. 1 (1909): If we wish to understand how it is organized, we could begin by looking at the melody, which seems to naturally break into two three-note cells: , and . We can see right away that the two cells are similar in contour, but not identical; the first descends m3rd - m2nd, but the second descends M3rd - m2nd. We can use the same method to compare the chords that accompany the melody. They too are similar (both span a M7th in the L. H.), but not exactly the same (the “alto” (lower voice in the R. H.) only moves up a diminished 3rd (=M2nd enh.) from B to Db, while the L. H. moves up a M3rd). Let’s use a different method of analysis to examine the same excerpt, called SET THEORY. WHAT? • SET THEORY is a method of musical analysis in which PITCH CLASSES are represented by numbers, and any grouping of these pitch classes is called a SET. • A PITCH CLASS (pc) is the class (or set) of pitches with the same letter (or solfège) name that are octave duplications of one another. “Middle C” is a pitch, but “C” is a pitch class that includes middle C, and all other octave duplications of C. WHY? Atonal music is often organized in pitch groups that form cells (both horizontal and vertical), many of which relate to one another. Set Theory provides a shorthand method to label these cells, just like Roman numerals and inversion figures do (i.e., ii 6 ) in tonal music. -
Pitch-Class Sets and Microtonalism Hudson Lacerda (2010)
Pitch-class sets and microtonalism Hudson Lacerda (2010) Introduction This text addresses some aspects of pitch-class sets, in an attempt to apply the concept to equal-tempered scales other than 12-EDO. Several questions arise, since there are limitations of application of certain pitch-class sets properties and relations in other scales. This is a draft which sketches some observations and reflections. Pitch classes The pitch class concept itself, assuming octave equivalence and alternative spellings to notate the same pitch, is not problematic provided that the interval between near pitches is enough large so that different pitches can be perceived (usually not a problem for a small number of equal divisions of the octave). That is also limited by the context and other possible factors. The use of another equivalence interval than the octave imposes revere restrictions to the perception and requires specially designed timbres, for example: strong partials 3 and 9 for the Bohlen-Pierce scale (13 equal divisions of 3:1), or stretched/shrinked harmonic spectra for tempered octaves. This text refers to the equivalence interval as "octave". Sometimes, the number of divisions will be referred as "module". Interval classes There are different reduction degrees: 1) fit the interval inside an octave (13rd = 6th); 2) group complementar (congruent) intervals (6th = 3rd); In large numbers of divisions of the octave, the listener may not perceive near intervals as "classes" but as "intonations" of a same interval category (major 3rd, perfect 4th, etc.). Intervals may be ambiguous relative to usual categorization: 19-EDO contains, for example, an interval between a M3 and a P4; 17-EDO contains "neutral" thirds and so on. -
Reducing Pessimism in Interval Analysis Using Bsplines Properties: Application to Robotics∗
Reducing Pessimism in Interval Analysis using BSplines Properties: Application to Robotics∗ S´ebastienLengagne, Universit´eClermont Auvergne, CNRS, SIGMA Cler- mont, Institut Pascal, F-63000 Clermont-Ferrand, France. [email protected] Rawan Kalawouny Universit´eClermont Auvergne, CNRS, SIGMA Cler- mont, Institut Pascal, F-63000 Clermont-Ferrand, France. [email protected] Fran¸coisBouchon CNRS UMR 6620 and Clermont-Auvergne University, Math´ematiquesBlaise Pascal, Campus des C´ezeaux, 3 place Vasarely, TSA 60026, CS 60026, 63178 Aubi`ere Cedex France [email protected] Youcef Mezouar Universit´eClermont Auvergne, CNRS, SIGMA Cler- mont, Institut Pascal, F-63000 Clermont-Ferrand, France. [email protected] Abstract Interval Analysis is interesting to solve optimization and constraint satisfaction problems. It makes possible to ensure the lack of the solu- tion or the global optimal solution taking into account some uncertain- ties. However, it suffers from an over-estimation of the function called pessimism. In this paper, we propose to take part of the BSplines prop- erties and of the Kronecker product to have a less pessimistic evaluation of mathematical functions. We prove that this method reduces the pes- simism, hence the number of iterations when solving optimization or con- straint satisfaction problems. We assess the effectiveness of our method ∗Submitted: December 12, 2018; Revised: March 12, 2020; Accepted: July 23, 2020. yR. Kalawoun was supported by the French Government through the FUI Program (20th call with the project AEROSTRIP). 63 64 Lengagne et al, Reducing Pessimism in Interval Analysis on planar robots with 2-to-9 degrees of freedom and to 3D-robots with 4 and 6 degrees of freedom. -
Euromac 9 Extended Abstract Template
9th EUROPEAN MUSIC ANALYSIS CONFERENC E — E U R O M A C 9 Scott Brickman University of Maine at Fort Kent, USA [email protected] Collection Vectors and the Octatonic Scale (013467), 6z23 (023568), 6-27 (013469), 6-30 (013679), ABSTRACT 6z49 (013479) and 6z50 (014679). Background 6z13 transforms to 6z50 under the M7 operation. 6z23 and 6z49 Allen Forte, writing in The Structure of Atonal Music (1973), map unto themselves under the M7 operation, as do 6-27 and tabulated the prime forms and vectors of pitch-class sets 6-30 whose compliments are themselves. (PCsets). Additionally, he presented the idea of a set-complex, a set of sets associated by virtue of the inclusion relation. So, a way to perceive these hexachords is that they provide five Post-Tonal Theorists such as John Rahn, Robert Morris and areas. Joseph Straus, allude to Forte’s set-complex, but do not treat the concept at length. The interval vector, developed by Donald Martino, tabulates the intervals (dyads) in a collection. In order to comprehend the The interval vector, developed by Donald Martino (“The Source constitution of the octatonic hexachords, I decided to create a Set and Its Aggregate Formations,” Journal of Music Theory 5, "collection vector" which would tabulated the trichordal, no. 2, 1961) is a means by which the intervals, and more tetrachordal and pentachordal subsets of each octatonic specifically the dyads contained in a collection, can be identified hexachord. and tabulated. I began exploring the properties of the OH by tabulating their There exists no such tool for the identification and tabulation of trichordal subsets. -
Composing with Pitch-Class Sets Using Pitch-Class Sets As a Compositional Tool
Composing with Pitch-Class Sets Using Pitch-Class Sets as a Compositional Tool 0 1 2 3 4 5 6 7 8 9 10 11 Pitches are labeled with numbers, which are enharmonically equivalent (e.g., pc 6 = G flat, F sharp, A double-flat, or E double-sharp), thus allowing for tonal neutrality. Reducing pitches to number sets allows you to explore pitch relationships not readily apparent through musical notation or even by ear: i.e., similarities to other sets, intervallic content, internal symmetries. The ability to generate an entire work from specific pc sets increases the potential for creating an organically unified composition. Because specific pitch orderings, transpositions, and permutations are not dictated as in dodecaphonic music, this technique is not as rigid or restrictive as serialism. As with integral serialism, pc numbers may be easily applied to other musical parameters: e.g., rhythms, dynamics, phrase structure, sectional divisions. Two Representations of the Matrix for Schönberg’s Variations for Orchestra a. Traditional method, using pitch names b. Using pitch class nomenclature (T=10, E=11) Terminology pitch class — a particular pitch, identified by a name or number (e.g., D = pc 2) regardless of registral placement (octave equivalence). interval class (ic) — the distance between two pitches expressed numerically, without regard for spelling, octave compounding, or inversion (e.g., interval class 3 = minor third or major sixth). pitch-class set — collection of pitches expressed numerically, without regard for order or pitch duplication; e.g., “dominant 7th” chord = [0,4,7,10]. operations — permutations of the pc set: • transposition: [0,4,7,10] — [1,5,8,11] — [2,6,9,0] — etc.