Z-Related Hexachords Explained by Transpositional Combination and the Complement Union Property
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James M. Baker, David W. Beach, and Jonathan Ber- Nard, Eds. Music
James M. Baker, David W. Beach, and Jonathan Ber nard, eds. Music Theory in Concept and Practice. Uni versity of Rochester Press, 1997.529 pp. Reviewed by Marilyn Nonken Music Theory In Concept and Practice is a seventieth-birthday tribute to Allen Forte, a theorist of unprecedented influence. Its editors claim that "in the modern rise of music theory . no single individual has been more central" (6). If this is an exaggeration, it is only a slight one. In addition to authoring landmark works (including Tonal Theory in Concept and Practice and The Structure of Atonal Music), Forte has also been an active and motivating presence in the field: as the editor who ushered the Jour nal of Music Theory through its transition from a fledgling publication to one of prominence; as a catalyst behind the foundation of the Society for Music Theory; and as a guiding force whose efforts resulted in the estab lishment of the doctoral program in music theory at Yale, the first of its kind at a major American institution. As depicted by Jonathan Bernard in the first of nineteen essays in this collection, Forte stands as heir to a tradition of musical inquiry that extends to the earliest decades of the century. With contributions by scholars on a wide variety of topics, this volume cannily evidences the transforming effect of one scholar's work on the discipline. Music Theory in Concept and Practice is divided into two broad categories ("Historical and Theoretical Essays" and "Analytical Studies"), then fur ther divided into four subsections. The essays are categorized according to their musicological orientfltion ("Historical Perspectives" and "Theoretical Perspectives"), and the analyses to the repertoire they treat ("The Tonal Repertoire" and "Twentieth-Century Music"). -
Surviving Set Theory: a Pedagogical Game and Cooperative Learning Approach to Undergraduate Post-Tonal Music Theory
Surviving Set Theory: A Pedagogical Game and Cooperative Learning Approach to Undergraduate Post-Tonal Music Theory DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Angela N. Ripley, M.M. Graduate Program in Music The Ohio State University 2015 Dissertation Committee: David Clampitt, Advisor Anna Gawboy Johanna Devaney Copyright by Angela N. Ripley 2015 Abstract Undergraduate music students often experience a high learning curve when they first encounter pitch-class set theory, an analytical system very different from those they have studied previously. Students sometimes find the abstractions of integer notation and the mathematical orientation of set theory foreign or even frightening (Kleppinger 2010), and the dissonance of the atonal repertoire studied often engenders their resistance (Root 2010). Pedagogical games can help mitigate student resistance and trepidation. Table games like Bingo (Gillespie 2000) and Poker (Gingerich 1991) have been adapted to suit college-level classes in music theory. Familiar television shows provide another source of pedagogical games; for example, Berry (2008; 2015) adapts the show Survivor to frame a unit on theory fundamentals. However, none of these pedagogical games engage pitch- class set theory during a multi-week unit of study. In my dissertation, I adapt the show Survivor to frame a four-week unit on pitch- class set theory (introducing topics ranging from pitch-class sets to twelve-tone rows) during a sophomore-level theory course. As on the show, students of different achievement levels work together in small groups, or “tribes,” to complete worksheets called “challenges”; however, in an important modification to the structure of the show, no students are voted out of their tribes. -
Pitch-Class Set Theory: an Overture
Chapter One Pitch-Class Set Theory: An Overture A Tale of Two Continents In the late afternoon of October 24, 1999, about one hundred people were gathered in a large rehearsal room of the Rotterdam Conservatory. They were listening to a discussion between representatives of nine European countries about the teaching of music theory and music analysis. It was the third day of the Fourth European Music Analysis Conference.1 Most participants in the conference (which included a number of music theorists from Canada and the United States) had been looking forward to this session: meetings about the various analytical traditions and pedagogical practices in Europe were rare, and a broad survey of teaching methods was lacking. Most felt a need for information from beyond their country’s borders. This need was reinforced by the mobility of music students and the resulting hodgepodge of nationalities at renowned conservatories and music schools. Moreover, the European systems of higher education were on the threshold of a harmoni- zation operation. Earlier that year, on June 19, the governments of 29 coun- tries had ratifi ed the “Bologna Declaration,” a document that envisaged a unifi ed European area for higher education. Its enforcement added to the urgency of the meeting in Rotterdam. However, this meeting would not be remembered for the unusually broad rep- resentation of nationalities or for its political timeliness. What would be remem- bered was an incident which took place shortly after the audience had been invited to join in the discussion. Somebody had raised a question about classroom analysis of twentieth-century music, a recurring topic among music theory teach- ers: whereas the music of the eighteenth and nineteenth centuries lent itself to general analytical methodologies, the extremely diverse repertoire of the twen- tieth century seemed only to invite ad hoc approaches; how could the analysis of 1. -
Music in the 20Th Century Serialism Background Information to Five Orchestral Pieces
AOS 2: Music in the 20th Century Serialism Background Information to Five Orchestral Pieces Five Orchestral Pieces 1909 by Arnold Schoenberg is a set of atonal pieces for full orchestra. Each piece lasts between one to five minutes and they are not connected to one another by the use of any musical material. Richard Strauss requested that Schoenberg send him some short orchestral pieces with the possibility of them gaining some attention from important musical figures. Schoenberg hadn’t written any orchestral pieces since 1903 as he had been experimenting with and developing his ideas of atonality in small scale works. He had had a series of disappointments as some of his works for both chamber orchestra and larger ensembles had been dismissed by important musical figures as they did not understand his music. As a result of this Schoenberg withdrew further into the small group of like minded composers e.g. The Second Viennese School. Schoenberg uses pitches and harmonies for effect not because they are related to one another. He is very concerned with the combinations of different instrumental timbres rather than melody and harmony as we understand and use it. Schoenberg enjoyed concealing things in his music; he believed that the intelligent and attentive listener would be able to decipher the deeper meanings. Schoenberg believed that music could express so much more than words and that words detract from the music. As a result of this belief titles for the Five Orchestral Pieces did not appear on the scores until 13 years after they had been composed. -
The Generalised Hexachord Theorem
THEOREM OF THE DAY The Generalised Hexachord Theorem Suppose that S is a subset of the set of pitch classes comprising Zn, n even, with |S | = s; then the interval class vectors of S and its complement Zn \ S differ component- wise by |n − 2s|, except for the last component, for which the difference is |n/2 − s|. The interval class vector of S is the vector of length n/2 whose i-th component is the number of pairs a, b ∈ S , a < b, for which min(b − a, n − b + a) = i. For Z12, the integers modulo 12, this is illustrated above, using the opening bars of Chopin’s op. 10, no. 5, for i = 5 (the upper, red, curves in bars 2–3) and i = 2 (the lower, green, curves). The word ‘class’ is used to indicate that no distinction is made between the interval from, say, B♭ down to E♭ (the first red curve) or from B♭ up to E♭; both intervals have value 5 = min(10 − 3, 12 − 10 + 3). The complete interval class vectors have been calculated here using David Walters’ PC Set Calculator, which is attached to Gary Tucker’s excel- lent Introduction to Pitch Class Analysis. The right-hand part of Chopin’s ´etude uses only the five black-key notes of the piano, the complement of this set being all the white keys. The Generalised Hexachord Theorem then tells us that components 1–5 of the corresponding interval classes differ uniformly by 12 − 2 × 5 = 2. The original Hexachord Theorem, due to the composer Milton Babbitt and theorist David Lewin in the 1950s, applies to subsets of exactly six notes out of the twelve note scale, the interval class vectors of this ‘hexachord’ or ‘hexad’ and its complement then being identical. -
To Allen Forte from His Former Advisees: Tributes and Reminiscences
Gamut: Online Journal of the Music Theory Society of the Mid-Atlantic Volume 6 Issue 2 Article 9 December 2013 To Allen Forte from His Former Advisees: Tributes and Reminiscences David Carson Berry University of Cincinnati, College-Conservatory of Music Follow this and additional works at: https://trace.tennessee.edu/gamut Recommended Citation Berry, David Carson (2013) "To Allen Forte from His Former Advisees: Tributes and Reminiscences," Gamut: Online Journal of the Music Theory Society of the Mid-Atlantic: Vol. 6 : Iss. 2 , Article 9. Available at: https://trace.tennessee.edu/gamut/vol6/iss2/9 This A Music-Theoretical Matrix: Essays in Honor of Allen Forte (Part V), edited by David Carson Berry is brought to you for free and open access by Volunteer, Open Access, Library Journals (VOL Journals), published in partnership with The University of Tennessee (UT) University Libraries. This article has been accepted for inclusion in Gamut: Online Journal of the Music Theory Society of the Mid-Atlantic by an authorized editor. For more information, please visit https://trace.tennessee.edu/gamut. TO ALLEN FORTE FROM HIS FORMER ADVISEES: TRIBUTES AND REMINISCENCES Festschriften have often included a tabula gratulatoria, that is, a listing of names of colleagues, former students, and friends who are sending compliments and felicitations to the honoree. Less often, the gratulationes have transcended the tabula to become full-fledged trib- utes and reminiscences. Perhaps the latter are not as common because Festschriften (of the particular kind that circulates in academe) are viewed as repositories of scholarship and not personal reflections—that is, a place for contributions to the broader field and not remembrances pertinent to the few. -
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. -
The Consecutive-Semitone Constraint on Scalar Structure: a Link Between Impressionism and Jazz1
The Consecutive-Semitone Constraint on Scalar Structure: A Link Between Impressionism and Jazz1 Dmitri Tymoczko The diatonic scale, considered as a subset of the twelve chromatic pitch classes, possesses some remarkable mathematical properties. It is, for example, a "deep scale," containing each of the six diatonic intervals a unique number of times; it represents a "maximally even" division of the octave into seven nearly-equal parts; it is capable of participating in a "maximally smooth" cycle of transpositions that differ only by the shift of a single pitch by a single semitone; and it has "Myhill's property," in the sense that every distinct two-note diatonic interval (e.g., a third) comes in exactly two distinct chromatic varieties (e.g., major and minor). Many theorists have used these properties to describe and even explain the role of the diatonic scale in traditional tonal music.2 Tonal music, however, is not exclusively diatonic, and the two nondiatonic minor scales possess none of the properties mentioned above. Thus, to the extent that we emphasize the mathematical uniqueness of the diatonic scale, we must downplay the musical significance of the other scales, for example by treating the melodic and harmonic minor scales merely as modifications of the natural minor. The difficulty is compounded when we consider the music of the late-nineteenth and twentieth centuries, in which composers expanded their musical vocabularies to include new scales (for instance, the whole-tone and the octatonic) which again shared few of the diatonic scale's interesting characteristics. This suggests that many of the features *I would like to thank David Lewin, John Thow, and Robert Wason for their assistance in preparing this article. -
Geometrical Music Theory
Clifton Callender College of Music, Florida State University. Ian Quinn Music Department, Yale University. Dmitri Tymoczko Music Department, Princeton University. To whom correspondence should be addressed. All three authors contributed equally to this report. GENERALIZED VOICE-LEADING SPACES SUMMARY: We translate basic concepts of music theory into the language of contemporary geometry. ABSTRACT: Western musicians traditionally classify pitch sequences by disregarding the effects of five musical transformations: octave shift, permutation, transposition, inversion, and cardinality change. Here 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. To interpret music is to ignore information. A capable musician can understand the sequence of notes (C4, E4, G4) in various ways: as an ordered pitch sequence (an “ascending C-major arpeggio starting on middle C”), an unordered collection of octave- free note-types (a “C major chord”), an unordered collection of octave-free note-types modulo transposition (a “major chord”), and so on. Musicians commonly abstract away from five types of information: the octave in which notes appear, their order, their specific pitch level, whether a sequence appears “rightside up” or “upside down” (inverted), and the number of times a note appears. Different purposes require different information; consequently, there is no one optimal degree of abstraction. Here we model this process. -
An Explanation of Anomalous Hexachords in Four Serial Works by Igor Stravinsky
University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Masters Theses Graduate School 8-2011 An Explanation of Anomalous Hexachords in Four Serial Works by Igor Stravinsky Robert Sivy [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes Part of the Musicology Commons, and the Music Theory Commons Recommended Citation Sivy, Robert, "An Explanation of Anomalous Hexachords in Four Serial Works by Igor Stravinsky. " Master's Thesis, University of Tennessee, 2011. https://trace.tennessee.edu/utk_gradthes/1025 This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a thesis written by Robert Sivy entitled "An Explanation of Anomalous Hexachords in Four Serial Works by Igor Stravinsky." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Music, with a major in Music. Brendan P. McConville, Major Professor We have read this thesis and recommend its acceptance: Barbara Murphy, Donald Pederson Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) An Explanation of Anomalous Hexachords in Four Serial Works by Igor Stravinsky A Thesis Presented for the Master of Music Degree The University of Tennessee, Knoxville Robert Jacob Sivy August 2011 Copyright © 2011 by Robert Jacob Sivy All rights reserved. -
Guido of Arezzo and His Influence on Music Learning
Musical Offerings Volume 3 Number 1 Spring 2012 Article 4 2012 Guido of Arezzo and His Influence on Music Learning Anna J. Reisenweaver Cedarville University, [email protected] Follow this and additional works at: https://digitalcommons.cedarville.edu/musicalofferings Part of the Musicology Commons, Music Pedagogy Commons, and the Music Practice Commons DigitalCommons@Cedarville provides a publication platform for fully open access journals, which means that all articles are available on the Internet to all users immediately upon publication. However, the opinions and sentiments expressed by the authors of articles published in our journals do not necessarily indicate the endorsement or reflect the views of DigitalCommons@Cedarville, the Centennial Library, or Cedarville University and its employees. The authors are solely responsible for the content of their work. Please address questions to [email protected]. Recommended Citation Reisenweaver, Anna J. (2012) "Guido of Arezzo and His Influence on Music Learning," Musical Offerings: Vol. 3 : No. 1 , Article 4. DOI: 10.15385/jmo.2012.3.1.4 Available at: https://digitalcommons.cedarville.edu/musicalofferings/vol3/iss1/4 Guido of Arezzo and His Influence on Music Learning Document Type Article Abstract Throughout the history of Western music, Guido of Arezzo stands out as one of the most influential theorists and pedagogues of the Middle Ages. His developments of the hexachord system, solmization syllables, and music notation revolutionized the teaching and learning of music during his time and laid the foundation for our modern system of music. While previous theorists were interested in the philosophical and mathematical nature of music, Guido’s desire to aid singers in the learning process was practical. -
Belatedness, and Sonata Structure in Rochberg's (Serial) Second
Trauma, Anxiety (of Influence), Belatedness, and Sonata Structure in Rochberg’s (Serial) Second Symphony Richard Lee University of Georgia Serialism is special to theorists and composers alike. For composer George Rochberg, doubly so: “I needed a language expressive and expansive enough to say what I had to. My war experience had etched itself deep into my soul.1 This essay is an exploration of Rochberg’s Second Symphony (1955–56)—“the first twelve-tone symphony composed by an American2— analyzed as a narrative of trauma, anxiety, and belatedness that emerges from the composer’s biography, his reliance on tradition (form), and his theorizing/deployment of serialism within a mid-20th-century compositional trend. Throughout this analysis, serialism acquires agency: it drives the following interpretation and has a capacity to act on (behalf of) Rochberg. Symphony No. 2 contains an array of thematic content that signifies trauma. The work stands as a response to World War II, therefore it makes sense to pin a biographical account of musical narrative to it. Rochberg was drafted in 1942 and his composition teacher, Hans Weisse, was driven out of Europe by the Nazi regime. In 1950, Rochberg went to Rome to study with Luigi Dallapiccola (known for his lyrical twelve-tone compositions), later telling Richard Dufallo that “one of the most powerful impulses toward twelve-tone, serialism, whatever you want to call it, was my reaction to my war experience which began to take over after the war.”3 1 George Rochberg, Five Lines, Four Spaces: The World of My Music, ed. Gene Rochberg and Richard Griscom (Chicago: University of Illinois Press, 2009), 14.