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Theory
  • Precomposition in Stravinsky’s 12-tone Music

Stravinsky’s Precomposition from Movements On
Paul Lombardi


These charts differ from Stravinsky’s actual precomposition charts in that they do not contain serial mistakes. Also, the alpha and beta hexachord labels are reversed for the R and IR rotational arrays. The series and their transpositions employed in the pieces are given in the form of an S-chart.[1]



Movements (1958-59)[2]

0 1 2 3 4 5 6 7 8 9 A B
I 3 2 8 A 9 4 6 7 5 0 B 1 RI
T6I
P 3 4 A 8 9 2 0 B 1 6 7 5 R
TAI
RIR 7 6 0 2 1 8 A B 9 4 3 5 IR


Pα 3 4 A 8 9 2 Pβ 0 B 1 6 7 5
3 9 7 8 1 2 0 2 7 8 6 1
3 1 2 7 8 9 0 5 6 4 B A
3 4 9 A B 5 0 1 B 6 5 7
3 8 9 A 4 2 0 A 5 4 6 B
3 4 5 B 9 A 0 7 6 8 1 2
Iα 3 2 8 A 9 4 Iβ 6 7 5 0 B 1
3 9 B A 5 4 6 4 B A 0 5
3 5 4 B A 9 6 1 0 2 7 8
3 2 9 8 7 1 6 5 7 0 1 B
3 A 9 8 2 4 6 8 1 2 0 7
3 2 1 7 9 8 6 B 0 A 5 4
Rα 2 9 8 A 4 3 Rβ 5 7 6 1 B 0
2 1 3 9 8 7 5 4 B 9 A 3
2 4 A 9 8 3 5 0 A B 4 6
2 8 7 6 1 0 5 3 4 9 B A
2 1 0 7 6 8 5 6 B 1 0 7
2 1 8 7 9 3 5 A 0 B 6 4
IRα 8 1 2 0 6 7 IRβ 5 3 4 9 B A
8 9 7 1 2 3 5 6 B 1 0 7
8 6 0 1 2 7 5 A 0 B 6 4
8 2 3 4 9 A 5 7 6 1 B 0
8 9 A 3 4 2 5 4 B 9 A 3
8 9 2 3 1 7 5 0 A B 4 6



Epitaphium (1959)[3]

0 1 2 3 4 5 6 7 8 9 A B
I 1 4 B A 2 3 8 9 0 7 6 5 RI
T2I
P 1 A 3 4 0 B 6 5 2 7 8 9 R
T6I
RIR 5 8 3 2 6 7 0 1 4 B A 9 IR



Double Canon (1959)
This work uses the highlighted portions of the matrix, which are P, R and IR, and their transpositions of up or down a whole step.


P 6 5 9 8 7 2 0 3 4 1 B A R
7 6 A 9 8 3 1 4 5 2 0 B
3 2 6 5 4 B 9 0 1 A 8 7
TA(P) 4 3 7 6 5 0 A 1 2 B 9 8
5 4 8 7 6 1 B 2 3 0 A 9
A 9 1 0 B 6 4 7 8 5 3 2
0 B 3 2 1 8 6 9 A 7 5 4
9 8 0 B A 5 3 6 7 4 2 1
8 7 B A 9 4 2 5 6 3 1 0 T2(R)
B A 2 1 0 7 5 8 9 6 4 3
1 0 4 3 2 9 7 A B 8 6 5
2 1 5 4 3 A 8 B 0 9 7 6
IR T2(IR)



A Sermon, a Narrative and a Prayer (1960-61)[4]

0 1 2 3 4 5 6 7 8 9 A B
I 3 2 6 4 5 8 7 0 B 9 A 1 RI
T6I
P 3 4 0 2 1 A B 6 7 9 8 5 R
TAI
RIR 7 6 A 8 9 0 B 4 3 1 2 5 IR


Pα 3 4 0 2 1 A Pβ B 6 7 9 8 5
3 B 1 0 9 2 B 0 2 1 A 4
3 5 4 1 6 7 B 1 0 9 3 A
3 2 B 4 5 1 B A 7 1 8 9
3 0 5 6 2 4 B 8 2 9 A 0
3 8 9 5 7 6 B 5 0 1 3 2
Iα 3 2 6 4 5 8 Iβ 7 0 B 9 A 1
3 7 5 6 9 4 7 6 4 5 8 2
3 1 2 5 0 B 7 5 6 9 3 8
3 4 7 2 1 5 7 8 B 5 A 9
3 6 1 0 4 2 7 A 4 9 8 6
3 A 9 1 B 0 7 1 6 5 3 4
Rα A 1 2 0 4 3 Rβ 5 8 9 7 6 B
A B 9 1 0 7 5 6 4 3 8 2
A 8 0 B 6 9 5 3 2 7 1 4
A 2 1 8 B 0 5 4 9 3 6 7
A 9 4 7 8 6 5 A 4 7 8 6
A 5 8 9 7 B 5 B 2 3 1 0
IRα 0 9 8 A 6 7 IRβ 5 2 1 3 4 B
0 B 1 9 A 3 5 4 6 7 2 8
0 2 A B 4 1 5 7 8 3 9 6
0 8 9 2 B A 5 6 1 7 4 3
0 1 6 3 2 4 5 0 6 3 2 4
0 5 2 1 3 B 5 B 8 7 9 A



The Flood (1961-62)[5]

0 1 2 3 4 5 6 7 8 9 A B
I 1 3 2 8 B 9 A 0 4 5 7 6 RI
T2I
P 1 B 0 6 3 5 4 2 A 9 7 8 R
T4I
RIR 3 5 4 A 1 B 0 2 6 7 9 8 IR


Pα 1 B 0 6 3 5 Pβ 4 2 A 9 7 8
1 2 8 5 7 3 4 0 B 9 A 6
1 7 4 6 2 0 4 3 1 2 A 8
1 A 0 8 6 7 4 2 3 B 9 5
1 3 B 9 A 4 4 5 1 B 7 6
1 9 7 8 2 B 4 0 A 6 5 3
Iα 1 3 2 8 B 9 Iβ A 0 4 5 7 6
1 0 6 9 7 B A 2 3 5 4 8
1 7 A 8 0 2 A B 1 0 4 6
1 4 2 6 8 7 A 0 B 3 5 9
1 B 3 5 4 A A 9 1 3 7 8
1 5 7 6 0 3 A 2 4 8 9 B
Rα 5 3 6 0 B 1 Rβ 8 7 9 A 2 4
5 8 2 1 3 7 8 A B 3 5 9
5 B A 0 4 2 8 9 1 3 7 6
5 4 6 A 8 B 8 0 2 6 5 7
5 7 B 9 0 6 8 A 2 1 3 4
5 9 7 A 4 3 8 0 B 1 2 6
IRα B 1 A 4 5 3 IRβ 8 9 7 6 2 0
B 8 2 3 1 9 8 6 5 1 B 7
B 5 6 4 0 2 8 7 3 1 9 A
B 0 A 6 8 5 8 4 2 A B 9
B 9 5 7 4 A 8 6 2 3 1 0
B 7 9 6 0 1 8 4 5 3 2 A



Anthem (1962)[6]

0 1 2 3 4 5 6 7 8 9 A B
I 2 A 0 9 7 6 4 5 3 1 8 B RI
T4I
P 2 6 4 7 9 A 0 B 1 3 8 5 R
TAI
RIR 8 4 6 3 1 0 A B 9 7 2 5 IR



Abraham and Isaac (1962-63)[7]

0 1 2 3 4 5 6 7 8 9 A B
I 5 4 6 8 7 B 1 3 2 0 A 9 RI
T2I
P 5 6 4 2 3 B 9 7 8 A 0 1 R
TAI
RIR 9 8 A 0 B 3 5 7 6 4 2 1 IR


Pα 5 6 4 2 3 B Pβ 9 7 8 A 0 1
5 3 1 2 A 4 9 A 0 2 3 B
5 3 4 0 6 7 9 B 1 2 A 8
5 6 2 8 9 7 9 B 0 8 6 7
5 1 7 8 6 4 9 A 6 4 5 7
5 B 0 A 8 9 9 5 3 4 6 8
Iα 5 4 6 8 7 B Iβ 1 3 2 0 A 9
5 7 9 8 0 6 1 0 A 8 7 B
5 7 6 A 4 3 1 B 9 8 0 2
5 4 8 2 1 3 1 B A 2 4 3
5 9 3 2 4 6 1 0 4 6 5 3
5 B A 0 2 1 1 5 7 6 4 2
Rα B 3 2 4 6 5 Rβ 1 0 A 8 7 9
B A 0 2 1 7 1 B 9 8 A 2
B 1 3 2 8 0 1 B A 0 4 3
B 1 0 6 A 9 1 0 2 6 5 3
B A 4 8 7 9 1 3 7 6 4 2
B 5 9 8 A 0 1 5 4 2 0 B
IRα 3 B 0 A 8 9 IRβ 1 2 4 6 7 5
3 4 2 0 1 7 1 3 5 6 4 0
3 1 B 0 6 2 1 3 4 2 A B
3 1 2 8 4 5 1 2 0 8 9 B
3 4 A 6 7 5 1 B 7 8 A 0
3 9 5 6 4 2 1 9 A 0 2 3



Variations (1963-64)[8]

0 1 2 3 4 5 6 7 8 9 A B
I 2 4 7 5 0 6 8 3 1 9 A B RI
T4I
P 2 0 9 B 4 A 8 1 3 7 6 5 R
TAI
RIR 8 A 1 B 6 0 2 9 7 3 4 5 IR


Pα 2 0 9 B 4 A Pβ 8 1 3 7 6 5
2 B 1 6 0 4 8 A 2 1 0 3
2 4 9 3 7 5 8 0 B A 1 6
2 7 1 5 3 0 8 7 6 9 2 4
2 8 0 A 7 9 8 7 A 3 5 9
2 6 4 1 3 8 8 B 4 6 A 9
Iα 2 4 7 5 0 6 Iβ 8 3 1 9 A B
2 5 3 A 4 0 8 6 2 3 4 1
2 0 7 1 9 B 8 4 5 6 3 A
2 9 3 B 1 4 8 9 A 7 2 0
2 8 4 6 9 7 8 9 6 1 B 7
2 A 0 3 1 8 8 5 0 A 6 7
Rα A 4 B 9 0 2 Rβ 5 6 7 3 1 8
A 5 3 6 8 4 5 6 2 0 7 4
A 8 B 1 9 3 5 1 B 6 3 4
A 1 3 B 5 0 5 3 A 7 8 9
A 0 8 2 9 7 5 0 9 A B 7
A 6 0 7 5 8 5 2 3 4 0 A
IRα 0 6 B 1 A 8 IRβ 5 4 3 7 9 2
0 5 7 4 2 6 5 4 8 A 3 6
0 2 B 9 1 7 5 9 B 4 7 6
0 9 7 B 5 4 5 7 0 3 2 1
0 A 2 8 1 3 5 A 1 0 B 3
0 4 A 3 5 2 5 8 7 6 A 0



Elegy for J.F.K. (1964)[9]

0 1 2 3 4 5 6 7 8 9 A B
I 8 2 4 6 0 B 5 7 9 A 1 3 RI
T6I
P 8 2 0 A 4 5 B 9 7 6 3 1 R



Fanfare for a New Theatre (1964)[10]

0 1 2 3 4 5 6 7 8 9 A B
I A B 9 7 6 8 5 3 4 2 0 1 RI
T8I
P A 9 B 1 2 0 3 5 4 6 8 7 R
T2I
RIR 4 5 3 1 0 2 B 9 A 8 6 7 IR



Introitus (1965)[11]

0 1 2 3 4 5 6 7 8 9 A B
I 8 3 2 4 0 B 5 6 7 A 9 1 RI
T4I
P 8 1 2 0 4 5 B A 9 6 7 3 R
T6I
RIR A 5 4 6 2 1 7 8 9 0 B 3 IR


Pα 8 1 2 0 Pβ 4 5 B A Pγ 9 6 7 3
8 9 7 3 4 A 9 3 9 A 6 0
8 6 2 7 4 3 9 A 9 5 B 8
8 4 9 A 4 A B 5 9 3 0 1
Iα 8 3 2 4 Iβ 0 B 5 6 Iγ 7 A 9 1
8 7 9 1 0 6 7 1 7 6 A 4
8 A 2 9 0 1 7 6 7 B 5 8
8 0 7 6 0 6 5 B 7 1 4 3
Rα 0 2 1 8 Rβ A B 5 4 Rγ 3 7 6 9
0 B 6 A 4 A 3 9 3 2 5 B
0 7 B 1 A 9 3 4 3 6 0 4
0 4 6 5 A 4 5 B 3 9 1 0
IRα 6 4 5 A IRβ 8 7 1 2 IRγ 3 B 0 9
6 7 0 8 8 2 3 9 3 4 1 7
6 B 7 5 8 9 3 2 3 0 6 2
6 2 0 1 8 2 1 7 3 9 5 6



Requiem Canticles (1965-66), Series 1[12]

0 1 2 3 4 5 6 7 8 9 A B
I 5 3 7 6 4 9 B A 8 1 2 0 RI
TAI
P 5 7 3 4 6 1 B 0 2 9 8 A R
T8I
RIR 3 1 5 4 2 7 9 8 6 B 0 A IR


Pα 5 7 3 4 6 1 Pβ B 0 2 9 8 A
5 1 2 4 B 3 B 1 8 7 9 A
5 6 8 3 7 9 B 6 5 7 8 9
5 7 2 6 8 4 B A 0 1 2 4
5 0 4 6 2 3 B 1 2 3 5 0
5 9 B 7 8 A B 0 1 3 A 9
Iα 5 3 7 6 4 9 Iβ B A 8 1 2 0
5 9 8 6 B 7 B 9 2 3 1 0
5 4 2 7 3 1 B 4 5 3 2 1
5 3 8 4 2 6 B 0 A 9 8 6
5 A 6 4 8 7 B 9 8 7 5 A
5 1 B 3 2 0 B A 9 7 0 1
Rα 1 6 4 3 7 5 Rβ A 8 9 2 0 B
1 B A 2 0 8 A B 4 2 1 0
1 0 4 2 A 3 4 3 1 0 B 9
1 5 3 B 4 2 A 8 7 6 4 5
1 B 7 0 A 9 A 9 8 6 7 0
1 9 2 0 B 3 A 9 7 8 1 B
IRα 7 2 4 5 1 3 IRβ A 0 B 6 8 9
7 9 A 6 8 0 A 9 4 6 7 8
7 8 4 6 A 5 A 5 7 8 9 B
7 3 5 9 4 6 A 0 1 2 4 3
7 9 1 8 A B A B 0 2 1 8
7 B 6 8 9 5 A B 1 0 7 9



Requiem Canticles (1965-66), Series 2[13]

0 1 2 3 4 5 6 7 8 9 A B
I 5 A B 1 0 8 9 7 2 4 6 3 RI
TAI
P 5 0 B 9 A 2 1 3 8 6 4 7 R
T2I
RIR 9 2 3 5 4 0 1 B 6 8 A 7 IR


Pα 5 0 B 9 A 2 Pβ 1 3 8 6 4 7
5 4 2 3 7 A 1 6 4 2 5 B
5 3 4 8 B 6 1 B 9 0 6 8
5 6 A 1 8 7 1 B 2 8 A 3
5 9 0 7 6 4 1 4 A 0 5 3
5 8 3 2 0 1 1 7 9 2 0 A
Iα 5 A B 1 0 8 Iβ 9 7 2 4 6 3
5 6 8 7 3 0 9 4 6 8 5 B
5 7 6 2 B 4 9 B 1 A 4 2
5 4 0 9 2 3 9 B 8 2 0 7
5 1 A 3 4 6 9 6 0 A 5 7
5 2 7 8 A 9 9 3 1 8 A 0
Rα 2 A 9 B 0 5 Rβ 7 4 6 8 3 1
2 1 3 4 9 6 7 9 B 6 4 A
2 4 5 A 7 3 7 9 4 2 8 5
2 3 8 5 1 0 7 2 0 6 3 5
2 7 4 0 B 1 7 5 B 8 A 0
2 B 7 6 8 9 7 1 A 0 2 9
IRα 0 4 5 3 2 9 IRβ 7 A 8 6 B 1
0 1 B A 5 8 7 5 3 8 A 4
0 A 9 4 7 B 7 5 A 0 6 9
0 B 6 9 1 2 7 0 2 8 B 9
0 7 A 2 3 1 7 9 3 6 4 2
0 3 7 8 6 5 7 1 4 2 0 5



The Owl and the Pussycat (1966)

0 1 2 3 4 5 6 7 8 9 A B
I 2 0 5 3 6 8 9 7 4 1 A B RI
T4I
P 2 4 B 1 A 8 7 9 0 3 6 5 R
TAI
RIR 8 6 B 9 0 2 3 1 A 7 4 5 IR


© 2005 Paul Lombardi



[1] Paul Lombardi, “A Symmetrical Property of Rotational Arrays in Stravinsky’s Late Music” (Music Theory Society of New York State. Baruch College, CUNY at New York, April 10, 2005).

[2] Milton Babbitt, “Order, Symmetry, and Centricity in Late Stravinsky,” Confronting Stravinsky, ed. Jann Pasler (Berkeley, CA: University of California Press, 1986) 252; “Remarks on the Recent Stravinsky,” The Collected Essays of Milton Babbitt, ed. Stephen Peles (Princeton, NJ: Princeton University Press, 2003) 165; “Stravinsky’s Verticals and Schoenberg’s Diagonals: A Twist of Fate,” The Collected Essays of Milton Babbitt, ed. Stephen Peles (Princeton, NJ: Princeton University Press, 2003) 412. Joseph Straus, Stravinsky’s Late Music (New York, NY: Cambridge UP, 2001) 30-31, 66-68, 129, 182; “Stravinsky’s ‘Construction of Twelve Verticals’: An Aspect of Harmony in the Serial Music,” Music Theory Spectrum 21/1 (1999): 71.

[3] Straus, Stravinsky’s Late Music 62.

[4] Robert Morris, “Generalizing Rotational Arrays,” Journal of Music Theory 2/1 (1988): 76. Joseph Straus, “A Strategy of Large-Scale Organization in Stravinsky’s Late Music,” Integral 11 (1997): 27. Straus, “Stravinskys Construction” 52; Stravinsky’s Late Music 104, 168-69.

[5] Straus, “A Strategy” 30-32; “ Stravinsky’s Construction” 65; Stravinsky’s Late Music 151, 154, 174.

[6] Straus, Stravinsky’s Late Music 142.

[7] Straus, Stravinsky’s Late Music 87n, 89; “A Strategy of” 35.

[8] Jerome Kohl, “Exposition in Stravinsky’s Orchestral Variations,” Perspectives of New Music 18/1-2 (1979): 392. Claudio Spies, “Notes on Stravinsky’s Variations,” Perspectives of New Music 4/1 (1965): 66. Straus, Stravinsky’s Late Music 201.

[9] Claudio Spies, “Some Notes on Stravinsky’s Requiem Settings,” Perspectives of New Music 5/2 (1967): 108. Straus, Stravinsky’s Late Music 229.

[10] Straus, Stravinsky’s Late Music 91.

[11] Spies 104, 108. Straus, Stravinsky’s Late Music 162-64.

[12] Spies 108-11. Straus, “Stravinsky’s Construction” 47, 63, 68; Stravinsky’s Late Music 76, 172, 219, 242, 245.

[13] Straus, Stravinsky’s Late Music 71, 242.



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