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BAIN MUSC 525 Post-Tonal Music Theory

MUSC 525 Straus Ch. 6 Overview: of 4

Straus Chapter 6 More Twelve-Tone Topics

“Twelve-tone composers share a premise—that interesting and expressive music

can be written with reference to a pre-composed ordering of the twelve pitch classes.”

“Twelve-tone composition is a world of musical possibilities, and within that world, each composer has discovered or created a new country or province with its own distinctive landscape.”

Joseph Straus, Introduction to Post-Tonal Theory, 3rd ed. (Upper Saddle River, NJ: Prentice Hall, 2005).

TERMS & CONCEPTS

Webern and Derivation (pp. 217-221) Trichordal derivation Tetrachordal derivation Generator Order operations that determine

the derivation (see Fig. 6-2) Three Levels of Structural

Unfolding - Note to note - Chord to chord - Series form to series form Transformation networks created

series form statements (see Fig. 6-4)

Toward total organization (see Ex. 6-3)

Pointillism Webern, Concerto, Op. 24

(1934) and String Quartet, Op. 28 (1938)

Schoenberg and Hexachordal Combinatoriality

(pp. 222-30) Aggregate completion in the

horizontal and vertical dimensions

Complementary hexachords have the same ic content: Hexchords are either Z-related or

self-complementary (map onto their complement under Tn or TnI)

Four Types of Hex. Comb. - P-comb. (7/50 or 14%) - I-comb. (19/50 or 38%) - R-comb. (4/50 or 8%), at a level other than R0 - RI-comb. (13/50 or 26%) P0 is always R-comb. with R0 All-combinatorial hexachords

(see Fig. 6-11) Hex. comb. areas – Ax (see Fig. 6-13) Small-scale succession Large scale organization “Modulation” between areas Schoenberg, Variations for

Orchestra, Op. 31 (1928); String Quartet No. 4, Op. 37 (1936); Piano Piece, Op. 33a

Stravinsky and Rotational

Arrays (pp. 231-34) Rotation Rotational arrays Untransposed forms of the

series: P, I, R & IR Stravinsky hexachordal array (see Fig. 6-16): - Rows → melodies - Verticals → harmonies - Diagonals Four-part array (Straus 1999) Twelve verticals (Straus 1999) Stravinsky, Requiem Canticles

(1966)

Ruth Crawford Seeger (pp. 234-35) Short series Rotation and transposition Multi-level series projection Crawford, Diaphonic Suite, No.

1, III (1930) Integral Serialism (pp. 235-36) Total serialization Pitch series Duration series Dynamic series Articulation series Boulez, Structures 1a (1952) Boulez and Multiplication (pp. 237-40) Multiplicands Multipliers Subset embedding Le Marteau sans Maître (1955),

III. "L'Artisanant furieux" Babbitt and Trichordal Arrays

(p. 240-45) A combination of Schoenbergian

combinatoriality and Webernian derivation

Trichordal array Duration rows Time point series (Wuorinen

1994) Babbitt, String Quartet No. 2

(1954); Semi-Simple Variations (1956)



HEXACHORDAL COMBINATORIALITY

“The aggregate—a collection consisting of all twelve pitch classes—is the basic harmonic unit in twelve-tone music….”

“Combinatoriality is the general term for combining a collection with one or more transposed or inverted forms of itself (or its complement) to create an aggregate.”

“Schoenberg, in his mature twelve-tone music, always constructs series in

which the two hexachords are related by inversion.”

Joseph Straus, Introduction to Post-Tonal Theory

Series from Schoenberg, Fourth String Quartet, Op. 37 (1936)

1 2 3 4 5 6 7 8 9 10 11 12 P2 D Cƒ A Bß F Eß E C Aß G F# B I7 G Aß C B E Fƒ F A Cƒ D Eß Bß

Notice that the first hexachord of P2 (blue) is equivalent to the second hexachord of I7 (red). The same relation obviously holds for R2, and RI7. Straus calls this group of four closely-related series forms an area. Areas are designated by the symbol Ax, for example, the members of A2 are P2, I7, R2, and RI7.As shown below, when two hexachordally-combinatorialy series forms are combined in counterpoint, the aggregate if formed both horizontally and vertically as shown below:

aggregate → (D,Cƒ,A,Bß,F,Eß) (E,C,Aß,G,Fƒ,B) aggregate → (G,Aß,C,B,E,Fƒ) (F,A,Cƒ,D,Eß,Bß) ↑ aggregate ↑ aggregate

THE FOUR TYPES OF HEXACHORDAL COMBINATORIALITY

P-comb. I-comb. R-comb. RI-comb. P0: H1 H2 P0: H1 H2 P0: H1 H2 P0: H1 H2

Px H2 H1 Ix H2 H1 Rx H2 H1 RIx H2 H1

Tn(H1) = H2 TnI(H1) = H2 Rn(H1) = H1 TnI(H1) = H1 Hexachordal mapping relations

Another way to look at hexachordal combinatoriality is to determine if H2 may be obtained by applying Tn or TnI to H1. For example, can Tn or TnI be applied to the set (D,Cƒ,A,Bß,F,Eß) to obtain (E,C,Aß,G,Fƒ,B)? Straus describes the combinatorial properties of all fifty hexachordal set classes on p. 264. He indicates the number of levels at which each set class is P-comb., R-comb., I-comb., and RI-comb. Notice that every series is R-comb. R0, The values in the table are obtained by examining the ic vector (for Tn) and index vectors (for TnI) of the hexachords for the following properties:

All P-comb hexachords have a 0 entry in their ic vector.

All R-comb hexachords have a 6 (or 3 for ic6) entry in their ic vector. All I-comb hexachords have a 0 entry in their index vector.

All RI-comb hexachords have a 6 (or 3 for ic6) entry in their index vector.

For more information, see pp. 222-30.

THE SIX ALL COMBINATORIAL HEXACHORDS Of the 50 hexachordal set classes, only 6 are P-comb., I-comb., R-comb. and RI-comb.

Set Class P I R RI

1st Order 6-1 (012345) 6-8 (023457) 6-32 (024579)

1 1 1

1 1 1

1 1 1

1 1 1

2nd Order 6-7 (012678) 6-20 (014589)

2 3

2 3

2 3

2 3

3rd Order 6-35 (02468T) 6 6 6 6



DERIVED SERIES

“Webern’s music is highly concentrated motivically. It tends to make intensive use of just a few intervals or sets.”

“Webern often guaranteed a high degree of motivic concentation by employing a derived series.”

Joseph Straus, Introduction to Post-Tonal Theory

In a derived series, all of the discrete trichords or tetrachords belong to the same set class.1 In the following series, all of the discrete trichords are members of 3-3 (014):

Webern, Concerto for Nine Instruments, Op. 24, I (1934) B–Bß–D Eß–G–Fƒ Gƒ–E–F C–Cƒ–A

P11 RI6 R5 I0 In this series all of the discrete tetrachords belong to 4-1 (0123):2

Webern, String Quartet, Op. 28 (1938)

G–Fƒ–A–Gƒ C–Dß–Bß–B Eß–D–F–E P7 I0 P3

To calculate the specific order relations involved, make small chart of the P, I, R and RI forms of the generator as follows: P7 G–Fƒ–A–Gƒ R7 I7 G–Aß–F–Fƒ RI7 For more information, see pp. 217-221.

INTEGRAL SERIALISM

Pitch series from Boulez, Structures 1a (1952)

Eß–D–A–Aß–G–Fƒ–E–Cƒ–C-Bß–F–B

Boulez borrowed his pitch series from Messiaen's Mode de valeurs et d'intensités (Brindle 1987, 23-25). The duration series is given in Fig. 6-17 (p. 236). Boulez also employs a dynamic series and articulation series in this work. For more information see Brindle 1987, 26-28 & DeYoung 1978.

1 Of the 12 trichordal set classes, only 3-10 (036) cannot act as the generator of a trichordally derived series. 2 Any tetrachord that excludes ic4 (4-1, 4-6, 4-9, 4-10, 4-13. 4-23, and 4-28) can act as the generator of a tetrachordally derived series.



BOULEZ AND MULTIPLICATION

Series from Boulez, Le Marteau sans Maître (1955)

Eß–F–D–Cƒ–Bß–B–A–C–Gƒ–E–G–Fƒ

Divided into 5 segments: A B C D E [Eß,F] [Bß,B,Cƒ,D] [A,C] [Gƒ] [E,Fƒ,G] Multiplication of segments:

BA = [Eß,F] * [Bß,B,Cƒ,D] ] = [Bß,C] + [C,Cƒ] + [Cƒ,Dƒ] + [D,E] = [Bß,B,C,Cƒ,D,Dƒ,E]

with transposition by a predetermined interval derived from a constant pitch class. For example, if pc F is the constant, then Tx for segment A would be T10 because the first pitch class of segment A is Eß, and the opci from F-Eß is 10, and T10

[Bß,B,C,Cƒ,D,Dƒ,E] = [Aß,A,Bß,C,Cƒ,D]. As Straus (2005) puts it, in Boulez's Le Marteau sans Maître the "series is no longer an explicit part of the music–it's never presented as a tune. Rather, it exerts its influence on the music from a structural distance." For more information, see pp. 237-40 and Koblyakov 1990. STRAVINKSY AND ROTATIONAL ARRAYS See pp. 231-34 and Straus 1999. References Bain, Reginald. Twelve-Tone Assistant. Available online at: <http://www.reginaldbain.com>. Brindle, Reginald Smith. The New Music: The Avant Garde Since 1945. New York: Oxford University Press, 1987. DeYoung, Lynden. 1978. "Pitch Order and Duration Order in Boulez Structure Ia", Perspectives of New Music 16, no. 2:27–

34. Koblyakov, Lev. Pierre Boulez: A World of Harmony. Chur, Switzerland: Harwood, 1990. Straus, Joseph N. "Stravinsky's 'Construction of Twelve Verticals': An Aspect of Harmony in the Serial Music." Music

Theory Spectrum, Vol. 21, No. 1 (Spring, 1999), 43-73. Wuorinen, Charles. Simple Composition. New York: C.F. Peters, 1994.

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