Chapter 27

The New World

By

Robert Fleisher, Northern Illinois University

After studying tonal harmony for nearly two years, students are sometimes shocked to learn that composers have already been writing music without keys or key signatures for nearly a century.¹ While some wonder why they've worked so hard to grasp the seemingly sacred "rules" of tonality only to discover how many composers have systematically abandoned these, others are intrigued to discover a tool—pitch-class set theory—that offers a new way to think about the familiar sounds of tonal music and, more importantly, a useful method to identify and describe pitch relationships in atonal music. Just as with tonal analysis, much (at first) unfamiliar terminology and procedure must first be learned.² The principles of set theory can be made more comprehensible by applying them to familiar sonorities (even if these are more often heard in tonal rather than in atonal music).

A familiar pair of complementary sets can be seen on the black keys and white keys of the piano.³ The black-key pentatonic scale is a transposition down a half step of the "major pentatonic" in Ex. 26-5 (p. 455). The complement of this pentachord (5-note) set is the septachord (7-note) set comprising all (white key) pitches of the C-major scale. If you follow the procedures outlined on pp. 496-503, you should come up with the prime forms [0,2,4,7,9] and [0,1,3,5,6,8,10] for these sets.⁴ Complements have remarkably similar interval content: the interval class vector of the larger one is essentially an expanded version of the smaller: 254361 vs. 032140.⁵ The most prominent interval class in both sets is ic5; the least prominent in both sets are ic1 and ic6 (both of which are absent in the pentachord).

Saying a bit more about symmetry will clarify what is meant by distinct forms (DF). A non-symmetrical set has 24 distinct forms (12 transpositions and 12 inversions), no two of which contain all the same pitch classes. Because most sets are non-symmetrical, "most set classes have 24 distinct forms" (p. 548). A symmetrical set will duplicate all of its pitch classes in one or more transpositions or inversions. All sets possessing any degree of symmetry consequently have fewer than 24 distinct forms. The more symmetrical the set, the fewer distinct forms it possesses. Our discussion will conclude with a comparison of two very different trichord (3-note) sets.

The prime form of the set representing the minor triad is [0,3,7].⁶ We can play different minor triads on every note of the chromatic scale, so this set has 12 transpositions. By reversing the interval order in this set (minor 3rd, major 3rd), we discover that its inversion is the major triad (major 3rd, minor 3rd). The principle of inversional equivalence tells us that the 12 transpositions of the major triad are members of the same set class represented by the minor triad (the best normal order). The 12 transpositions of the minor triad and its 12 inversions represent the 24 distinct forms comprising this trichord set class. Would you say this set [0,3,7] is or isn't symmetrical?

The trichord set corresponding to the augmented triad [0,4,8] is transpositionally and inversionally symmetrical, which you can easily confirm at the piano.⁷ Because there are three transpositions that all duplicate the same pitch classes—e.g., (C,E,G#), (E,G#,C), (G#,C,E) and three inversions that each duplicate one of the transpositions (see contents of previous parentheses), this set class has only four distinct forms.⁸ The number of locations in which a set reproduces its original pitch classes by transposition and inversion are termed symmetry indices (or indexes)—3,3 for the set [0,4,8]. These indices are directly linked to the number of distinct forms.⁸ Divide 24 (the maximum number of distinct forms) by the sum of the two symmetry indices for any set; the number of distinct forms will result. This formula for [0,4,8]: 3+3 = 6; 24/6 = 4.⁹

Notes
¹ The emergence of atonality is generally associated with Arnold Schoenberg (1874-1951) and his students, Anton Webern (1883-1945) and Alban Berg (1885-1935). Ex. 27-1 (pp. 494-495) is from an early atonal composition by Schoenberg; his Second String Quartet, Op. 10 (1907-08) is often cited as the first work to abandon the use of a key signature (in the final movement). Since the earliest atonal compositions preceded the emergence of atonal theory by more than a half century, we might wonder how these and many other composers since managed to write so much music without the "net" provided for nearly 300 years by tonality and functional harmony. The view expressed by Schoenberg provides one possible answer: "For me, it is certain that the laws of the old art are also those of the new art." In Arnold Schoenberg, Style and Idea (Berkeley and Los Angeles: University of California Press, 1984), p. 375.
² This chapter introduces the essential concepts and methodology of atonal theory, but the discussion of Ex. 27-1 (pp. 494-495) and related questions in Self-Test 27-1-C and 27-1-D (pp. 504-505) also reflect the kind of potent observations this approach can facilitate concerning pitch organization in atonal music. Another important concept in atonal music is the aggregate, which describes any complete statement of all 12 chromatic pitch classes. Curiously, in the 11 measures (and 61 notes!) of Ex. 27-1, no aggregate has been completed, since Schoenberg uses all pitch classes except for one. (Which one is missing?) But in the next measure (p. 421 in the Burkhart Anthology, 6^th ed.)—where the tempo, texture, rhythm, register, and dynamics all undergo sudden and drastic change—the first aggregate in this movement is completed by the lowest sounding pitch heard up to this point. (Aggregate completion in atonal music isn't always so dramatic!) A second aggregate is rapidly completed, by the middle of m. 12! Play through Ex. 26-17 and Ex. 26-37 at the piano; what pitch completes the first aggregate in each of these excerpts? 
³ Here's a succinct definition of complement relations: "For any set, the pitch classes it excludes constitute its complement." In Joseph N. Straus, Introduction to Post-Tonal Theory, 2nd ed. (Upper Saddle River: Prentice-Hall, 2000), p. 81. Chapter 1 provides clear explanations of many essential concepts of atonal theory; for further discussion of complement relations also see Straus, pp. 81-82.
⁴ Notice that the prime form actually represents the B-Locrian mode, the best normal order of all pitches in the C-major scale-and in D-Dorian, E-Phrygian, F-Lydian, G-Mixolydian, and A natural minor (see pp. 453-455)!
⁵ Interval class vectors (or "interval vectors") summarize the number of times each interval class (ic) appears in a set. Play and compare the major and Hirajoshi pentatonic scales in Ex. 26-5 (p. 455). The "major pentatonic" sounds quite different from the "Hirajoshi" because these sets contain very different distributions of the six interval classes. The vector of the "major pentatonic" set-032140-shows that it contains no half steps (ic1), 3 whole steps (ic2), 2 minor 3rds (ic3), 1 major 3rd (ic4), 4 perfect 4ths or 5ths (ic5), and no tritones (ic6). For more about interval vectors, see Stefan Kostka, Materials and Techniques of Twentieth Century Music, 4th ed. (Upper Saddle River: Prentice-Hall, 2012), pp. 177-179.
⁶ In atonal theory, the trichord (like the triad in tonal music) is considered the smallest meaningful set. Since the 220 set classes of atonal music far outnumber the basic sonorities in tonal music, they are identified numerically. Appendix C (pp. 548-551) lists the Forte names (followed by the number of distinct forms if other than 24), prime forms, and interval class vectors for the 220 set classes of three to nine pitches, all arranged in complementary pairs.
⁷ There are several dozen inversionally symmetrical sets and only 12 transpositionally symmetrical sets (all but one of which is also inversionally symmetrical). See Straus, pp. 74-79. 
⁸ Since complements share the same degree of symmetry (seen in their indices), they also share the same number of distinct forms: e.g. 4 for both the trichord [0,4,8] and its complementary nonachord [0,1,2,4,5,6,8,9,10]. 
⁹ The indices represent (in order) transpositional and inversional symmetry: for all non-symmetrical sets, the indices are 1,0; the "1" here simply means that all sets are capable of "mapping" onto themselves, thereby reproducing their original pitch classes at a "zero" level of transposition. The formula for all such sets (including [0,3,7]) is simply: 24/1 = 24 DF.