Fabrications for Winds and Percussion: an Analysis Douglas G
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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. -
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. -
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. -
On Chords Generating Scales; Three Compositions for Orchestra
INFORMATION TO USERS This reproduction was made from a copy of a document sent to us for microfilming. While the most advanced technology has been used to photograph and reproduce this document, the quality of the reproduction is heavily dependent upon the quality of the material submitted. The following explanation of techniques is provided to help clarify markings or notations which may appear on this reproduction. 1. The sign or “target” for pages apparently lacking from the document photographed is “Missing Page(s)”. If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting through an image and duplicating adjacent pages to assure complete continuity. 2. When an image on the film is obliterated with a round black mark, it is an indication of either blurred copy because of movement during exposure, duplicate copy, or copyrighted materials that should not have been filmed. For blurred pages, a good image of the page can be found in the adjacent frame. If copyrighted materials were deleted, a target note will appear listing the pages in the adjacent frame. 3. When a map, drawing or chart, etc., is part of the material being photographed, a definite method of “sectioning” the material has been followed. It is customary to begin filming at the upper left hand comer of a large sheet and to continue from left to right in equal sections with small overlaps. If necessary, sectioning is continued again—beginning below the first row and continuing on until complete. -
MTO 16.3: Jenkins, After the Harvest
Volume 16, Number 3, August 2010 Copyright © 2010 Society for Music Theory After the Harvest: Carter’s Fifth String Quartet and the Late Late Style J. Daniel Jenkins NOTE: The examples for the (text-only) PDF version of this item are available online at: http://www.mtosmt.org/issues/mto.10.16.3/mto.10.16.3.jenkins.php Received September 2009 I. Introduction (1) [1] While reflecting on Elliott Carter’s Night Fantasies (1980) and those pieces that followed, David Schiff writes, “after years of ploughing through rocky soil it was now time for the harvest” ( Schiff 1988 , 2). As various scholars have demonstrated, in the “harvest period,” Carter cultivated two main structural elements in his music: a harmonic language that focused on all-interval twelve-note chords and a rhythmic language that relied on long-range polyrhythms. (2) Almost all of Carter’s works composed between Night Fantasies and the Fifth String Quartet (1995) include either an all-interval twelve-note chord or a structural polyrhythm, and many contain both. Exceptions are either occasional pieces, or compositions for solo instruments that often do not have the range to accommodate the registral space that all-interval rows require. [2] Because of the constancy of these elements in Carter’s music over a period of fifteen years, it is remarkable that in the Fifth String Quartet he eschews all-interval twelve-note chords while simultaneously removing the rhythmic scaffolding provided by a structural polyrhythm. (3) Carter has incorporated all-interval twelve-note chords in some of the compositions that postdate the Fifth String Quartet, but global long-range polyrhythms seem to have disappeared as a structural force in his music. -
Lecture Notes on Pitch-Class Set Theory Topic 1
Lecture Notes on Pitch-Class Set Theory Topic 1: Set Classes John Paul Ito General Overview Pitch-class set theory is not well named. It is not a theory about music in any common sense – that is, it is not some set of ideas about music that may or may not be true. It is first and foremost a labeling system. It makes no claims about music itself, but it does make some claims about the basic materials of music, and those claims are objectively, mathematically true, like geometry. In what follows we first deal with fundamental concepts of pitch-class and interval class, and then we work toward the definition of the set class. Pitch vs. Pitch Class We begin with a distinction introduced in the first term of freshman theory, between pitch and pitch class. A pitch is a note-name in a specific register: for example, F4, the F above middle C. The idea of “an F” is the idea of pitch-class, of note-name leaving aside register. Thus F is a pitch class, F4 is a pitch. Technically, a pitch class is the group (class) of all pitches related by octave equivalence – that is, it is not so much the abstract idea of F’s in general, but the group of F’s in all audible registers, F0, F1, F2, etc. For the purposes of pitch-class set theory, we will not distinguish among enharmonically- equivalent spellings of the same pitch or pitch class; F#4 and Gb4 are considered to the same pitch, and F# and Gb are interchangeable names for the same pitch class. -
Sets and Nonsets in Schoenberg's Atonal Music Author(S): Allen Forte Reviewed Work(S): Source: Perspectives of New Music, Vol
Sets and Nonsets in Schoenberg's Atonal Music Author(s): Allen Forte Reviewed work(s): Source: Perspectives of New Music, Vol. 11, No. 1, Tenth Anniversary Issue (Autumn - Winter, 1972), pp. 43-64 Published by: Perspectives of New Music Stable URL: http://www.jstor.org/stable/832462 . Accessed: 30/01/2012 02:31 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]. Perspectives of New Music is collaborating with JSTOR to digitize, preserve and extend access to Perspectives of New Music. http://www.jstor.org SETS AND NONSETS IN SCHOENBERG'S ATONAL MUSIC ALLEN FORTE A consideration of Schoenberg's atonal music ought to refer to a theoretical framework for atonal music in general. The present article, however, will not set forth such a framework, but will fo- cus, instead, on an elementary problem in Schoenberg's atonal oeuvre,1that of distinguishing compositional units or sets from sec- ondary formations or nonsets.2 In order to concentrate on this aspect of Schoenberg's music and to avoid digressions into other interesting aspects, no effort will be made to show the function of sets in the music. No com- plete analyses will be presented, although every musical example quoted is extracted from a complete analysis-and, indeed, could not be convincingly and confidently presented to the reader other- wise. -
Mathematical Perspectives on Music-Theoretical Knowledge
Mathematical Perspectives on Music-theoretical Knowledge Thomas Noll 1 1 Dear Colleagues, it is a great pleasure to be invited to this excellent institute and Iʼm very glad that you are interested in the application of mathematics to music theory. My talk will focus on very basic music-theoretical concepts which are inspired by some properties of the traditional notation system. The links which I will draw to actual music are mostly concerned with the bass line. This is somewhat complementary to the focus of the joint project which Tom Fiore and I are currently working on. That one deals with voice leadings in harmonic progressions. The mathematical trajectory of my talk will take us from linear maps on Zn to linear maps on the free Z-module of rank 2 and turther on to associated automorphisms of the free group on two generators. All this is closely linked the the field of algebraic combinatorics on words. Elementary Observations about Musical Notation and the Diatonic System (Almost) every generic interval comes in two species. “Myhill's Property” Exceptions: P1 P8 P15 2 2 In traditional musical notation note heads are placed on parallel horizontal staff lines and occasionally accidentals are attached to them. This system is well- adopted to the notation of music, which - in the broadest sense - can be called diatonic. Before going into sophisticated aspects of this concept, let us start from a simplified perspective: Diatonic tone relations are those which can be notated with no use sharps or flats. There is a generic way to measure intervals, namely to count the height differences between two notes on the staff. -
II Maniates, M
.1111 IAN BENT 602 I Lester, J. "The Fux-Mattheson Correspondence: An Annotated Translation," Current 1 Musicology 24 (1977), pp. 37-62 · 19. Between Modes and Keys: German Theory 1592-1802, Stuyvesant, NY, Pendragon Press, 11 1989 Twelve-tone theory Maniates, M. R. "Bottrigari versus Sigoni: On Vicentino and his Ancient Music Adapted to II Modern Practice," in Musical Humanism and Its Legacy: Essays in Honor of Claude V. ii JOHN COVACH 111 Palisca, ed. N. Baker and B. Hanning, Stuyvesant, NY, Pendragon, 1992, pp. 79-107 "The Cavalier Ercole Bottrigari and his Brickbats: Prolegomena to the Defense of Don 111•, .. Nicola Vicentino against Messer Gandolfo Sigonio," in Music Theory and the Exploration 111 of the Past, ed. C. Hatch and D. Bernstein, University of Chicago Press, 1993, pp. 111 1111 139-89 As scholars begin to gain a sense ofhistorical perspective on art music in the twentieth 11 Mann, A. "Beethoven's Contrapuntal Studies with Haydn," Musical ()Jf;arterly 66 (1970), pp. century, it seems clear that the introduction and development of twelve-tone compo- 111 1,il 711 -26; reprinted in The Creative World ofBeethoven, ed. P. H. Lang, New York, Norton, sitional procedures will remain one ofthe cardinal markers ofmusical modernism. The 1971,pp.209-24 careers of Schoenberg, Berg, Webern, Boulez, Stockhausen, Babbitt, and even !:llli "Haydn and Mozart,'' in Theory and Practice: The Great Composer as Student and Teacher, !,I Stravinsky (among many others) are all at some point intimately bound up with dodec- i:: New York, Norton, 1987, pp. 41-54 aphonic concerns, as is the course ofavant-garde music generally.