This investigation will seek to explain aspects of Stravinsky's twelve-tone music that have been neglected in the standard literature. In particular, this discussion will address the reasons Stravinsky implemented certain rows for particular formal events or to enhance the poetry in his twelve-tone works.
Some analyses of Igor Stravinsky's twelve-tone music have been carried out in the same manner as those analyses of twelve-tone music by Schoenberg or Webern, while others have focused on vertical pitch simultaneity that is analyzed harmonically, which, in light of the analysis introduced in this paper, might be explained as superfluous or coincidental. (Various types of Stravinsky analysis will be examined in the next chapter.) This paper will show that Stravinsky conceived all of his twelve-tone music in a fundamentally different way than the classical serialists, and that an Object-Oriented analysis is necessary for a thorough understanding of his serial works.
In the years between 1955 and 1966, when Stravinsky began composing with the twelve-tone method, his procedure evolved considerably. Despite this evolution, all of Stravinsky's twelve-tone works also exhibit many similarities in regard to formal structure and reinforcement of the poetry. Although the author will focus on Stravinsky's choral music in detail, this paper touches on each of the sixteen twelve-tone works that Stravinsky composed.
In his twelve-tone music, Stravinsky carefully chose tone rows designed to reinforce structural or poetic-textual events. Furthermore, the serial procedures of Transposition, Inversion, Retrogression, and Retrograde Inversion in Stravinsky's works enhanced the meaning of his poetic texts and his formal musical structures. In order easily to ascertain Stravinsky's row constructions, this paper will present a method of analyzing his twelve-tone works by projecting rows not into a standard matrix, appropriate for classical serialism, but rather onto three-dimensional objects. The objects are mathematical constructions formed out of arrangements of his twelve-tone rows by projecting them into three-dimensional space. In providing the analyst with a visual representation of the rows and transpositions that Stravinsky used, Object-Oriented analysis can aid in the understanding of Stravinsky's compositional choices that otherwise might appear arbitrary.
The majority of Stravinsky's twelve-tone compositions, and arguably the most substantial ones, are vocal music. In fact, of his sixteen twelve-tone works, eight include music for vocal soloists. Seven include choral parts. Only six are for instruments alone. Of those six instrumental works, three are brief, just over a minute long. Furthermore, Stravinsky's twelve-tone compositions with chorus are the longest of his later works. Without doubt, vocal music formed a cornerstone of Stravinsky's late compositional output.
In his vocal twelve-tone music, Stravinsky employed similar techniques for both voices and instruments in his choice of rows. Consequently, this paper plunges into in-depth analysis of both vocal and instrumental music. Nevertheless, with the addition of heretofore neglected poetic-textual analysis, this study enhances specifically Stravinsky's vocal music.
Throughout his years of twelve-tone composition, Stravinsky composed both short, simple works and long, complex ones. This paper begins with discussion of Stravinsky's shorter, simpler twelve-tone compositions, and then focuses on works of greater complexity. Since the large and more complex works from Stravinsky's last two decades build on techniques employed in his smaller works, this approach is more comprehensive than a chronological analysis of all of his twelve-tone music. The first works to be discussed are Epitaphium (1959), Anthem (1962), Elegy for J. F. K. (1964), Fanfare for a New Theater (1964), and The Owl and the Pussy-Cat (1965-66). Although these shorter works are analyzed using matrices, the analyses relate background concepts from which object-oriented analysis can be extrapolated. Next, applying such an extrapolation in an analyses of The Flood (1961-62), rows will be projected onto three-dimensional objects. In addition, the analysis of The Flood will discuss how object analysis can explain how Stravinsky embodied twelve-tone rows with structural and poetic-textual meaning. Following The Flood, Threni (1957-58), a work by Stravinsky that employs more complicated object types, will be examined. Finally, this paper will show how Object-Oriented analysis can enhance the preparation and performance of these works. An appendix will present tools to analyze Stravinsky's works of increased compositional complexity--including set rotation and verticals--in Canticum sacrum (1955), Introitus (1965), Abraham and Isaac (1962-63), Variations (1963-64), and Requiem Canticles (1965-66).
Table 1 lists chronologically all sixteen of Stravinsky's twelve-tone works. The first two works in the list, Canticum sacrum and Agon, are not entirely twelve-tone, as they contain some non-serial sections. Although he arranged and orchestrated the tonal (or modal) works of other composers, Stravinsky never composed any non-twelve-tone music after 1957.
Canticum sacrum ad honorem sancti Marci nominis (1955)
Threni: id est lamentationes Jeremiae prophetae (1957-58)
Epitaphium "für das Grabmal des Prinzen Max Egon zu Fürstenberg" (1959)
Double Canon "Raoul Dufy in Memoriam" (1959)
A Sermon, a Narrative, and a Prayer (1960-61)
Anthem "The dove descending breaks the air" (1962)
The Flood (1961-62)
Abraham and Isaac (1962-63)
Elegy for J. F. K. (1964)
Fanfare for a New Theater (1964)
Variations "Aldous Huxley in Memoriam" (1963-64)
Introitus "T. S. Eliot in Memoriam" (1965)
Requiem Canticles (1965-66)
The Owl and the Pussy-Cat (1965-66)
Below are described the typographical conventions regarding pitch, row names, and intervals used in this paper.
A pitch that is not fixed in register (i.e. Pitch Class) will be indicated with a capital letter: C or Ab refer to any C or Ab regardless of register. Specific pitch names in fixed register are consistent with the Acoustical Society: c' refers to middle C.
In order to increase clarity and ease in readability, pitch names are standardized in this paper to the following twelve pitch classes:
A Bb B C C# D Eb E F F# G Ab
Whenever possible in these analyses, enharmonic equivalents in Stravinsky's scores are replaced with the pitches listed above. (For example, B# is here called C, Gb is here called F#.)
The term "Prime form" is used in preference to "Original form" to indicate the basic, un-transposed, unaltered twelve-tone row. Although Stravinsky himself used the term "Original" in his manuscripts, that term has been replaced by "Prime," according to such analysts as Joseph Straus, George Perle, Claudio Spies, Stefan Kostka, and others who contribute to the standard twelve-tone literature of classical serial analysis. Furthermore, using the abbreviation "P" for Prime avoids the pitfalls of the letter "O" for Original, which can be confused with the number zero (0) in the course of analysis. In addition, following the standard serial literature "Retrograde" replaces "Crab" here.
Transpositions of rows are calculated from the first note of the Prime row, where P-0 is the original untransposed row beginning on a named pitch, and P-1 is the row transposed up one half-step. Hence, P-0 is the referential row, not necessarily the row that starts with pitch class 0 (or C).
When discussing hexachords and tetrachords, lower-case letters will be added to the names of the rows in the order and direction of the particular row discussed. For example, when Stravinsky uses hexachords, P-0a will refer to the first six notes in the Prime row, P-0b will refer to the last six notes. Note that the hexachords P-0b and R-0a contain the same six notes, but in reverse order. In the case of tetrachords, P-0a, P-0b, and P-0c will refer to the order positions 1-4, 5-8, and 9-12 of the Prime row.
When a twelve-tone matrix is shown, the intervals between each pitch (and their directions) are also listed horizontally above the names of each Inversion form of the row. The names of each Prime form of the row are listed in a column along the left side of the matrix.
Whenever intervals are mentioned where direction is not important, they will always be presented as the smallest interval class (ic). That is, all relations will be reduced to the smallest interval within the range of ic6 (a tritone). For example, the interval +10 will be reduced to ic2, and -7 will be ic5.
Stravinsky occasionally employs set rotation in his twelve-tone music. His row rotation most often involves the presentation of a row in the correct pitch order, but not beginning with the first pitch of the row. Instead, the first sounding pitch might be, for example, the third pitch of the row. The row proceeds until the twelfth pitch of the row is heard, after which the remaining pitches from the beginning of the row--in this case the first and second pitches of the row--are heard. A row that employs set rotation beginning with the second pitch of the row is designated rot1. Therefore, the row P-0 rot2 begins with the third pitch of the Prime row, and concludes with its second pitch.
The compositional process for Stravinsky's large-scale twelve-tone works changed somewhat after The Flood. Before Abraham and Isaac, Stravinsky's primarily building block was the row and its serially related forms. Subsequently, he exploited the relationships between various permutations of each row. In the later works he chopped up his rows into two or three parts (hexachords or tetrachords) and rotated each part.
Within the evolution of his twelve-tone compositional process, Stravinsky consistently fell back on his standard conception of the relationship between rows that can be elucidated by the technique of object analysis. For example, Introitus (1965) shows the combination of set rotation and row relationships by the first and last notes of the Prime row. Introitus is explained in an Appendix.
Topology (or analysis situs) is a branch of mathematics concerned with geometric configurations that are unaltered by elastic deformations such as stretching or twisting. Topology, a relatively new development in mathematics, was pioneered by A. F. Moebius (1790-1868), J. B. Listing (1802-1882), Bernhard Riemann (1826-1866). Topology concerns itself with problems such as the genus of a surface (how many holes are in a surface), tiling surfaces, knots, one-sided surfaces (such as a Moebius strip or Klein bottle), and coloring maps (the Five-Color theorem and the Four-Color theorem).
For the purposes of this paper, an object is something material that may be perceived by the senses, including abstract figures, which exist not in reality but in imaginary space. The abstract object on which this paper focuses specifically is the Torus, or doughnut-shaped topological object (a surface of genus 1). Other objects that appear in the course of analysis are the Sphere, the Cylinder, and Plane (surfaces of genus 0).
The Object-Oriented aspect of the analytical method presented in this paper is two-fold. First, the method employs objects in a mathematical or geometrical sense in that it uses graphical representations of collections of twelve-tone rows that form topological figures such as the Torus. Second, the method employs objects in the sense of Object-Oriented Programming, where data and the operations that work on that data are grouped together as an object, which can be called from and associated with other objects in the course of a computer program. Similarly, this paper will show that, depending on their construction, Stravinsky's rows can be grouped by topological figures, each type of which has properties effecting Stravinsky's compositional choices in the course of a piece.
Analysis using geometrical objects follows in the steps of Richard Cohn (Properties and Generability of Transpositionally Invariant Sets," Journal of Music Theory 35:1-2 (Spring/Fall 1991): 1-32), John Roeder ("A Geometric Representation of Pitch-Class Series," Perspectives of New Music 25:1-2 (Winter/Summer 1987): 362-409.), James Bennighof ("Set-Class Aggregate Structuring, Graph Theory, and Some Compositional Strategies," Journal of Music Theory 31:1 (Spring 1987): 51-98.), Walter O'Connell ("Tone Spaces," Die Reihe 8 (1968): 35-67.), and others.
Analysis with rows projected onto three-dimensional objects is only appropriate for works constructed with rows related to one another by the first and last notes of the Prime row. This study will show that Stravinsky conceived his formal and textual-poetic constructions using these kinds of rows.
Stravinsky's Topology. Doctoral Dissertation. Boulder, CO: University of Colordao, 2000. www.lulu.com/akuster