This appendix presents matrices and objects for works by Stravinsky that were mentioned parenthetically in the main body of this paper. To supplement the discussion in chapters two and five, this appendix includes Matrices for all of Stravinsky's twelve-tone works not examined in detail. Furthermore, this appendix includes brief analytical discussion of the three works Canticum sacrum, Double Canon, and Introitus. The matrices, objects, and analyses in this appendix are ordered by date of compositional inception.
Agon, a ballet for twelve dancers with orchestra, was started in 1953 and finished in 1957. The parts of Agon composed last incorporate twelve-tone rows. Table 1 and Table 2 show rows that Stravinsky used in Agon.
int: -1 +2 +1 -5 -1 +2 -3 -2 +1 -2 -1
I-0 I-11 I-1 I-2 I-9 I-8 I-10 I-7 I-5 I-6 I-4 I-3
P-0 Eb D E F C B C# Bb Ab A G F#
P-1 E Eb F F# C# C D B A Bb Ab G
P-11 D C# Eb E B Bb C A G Ab F# F
P-10 C# C D Eb Bb A B Ab F# G F E
P-3 F# F G Ab Eb D E C# B C Bb A
P-4 G F# Ab A E Eb F D C C# B Bb
P-2 F E F# G D C# Eb C Bb B A Ab
P-5 Ab G A Bb F E F# Eb C# D C B
P-7 Bb A B C G F# Ab F Eb E D C#
P-6 A Ab Bb B F# F G E D Eb C# C
P-8 B Bb C C# Ab G A F# E F Eb D
P-9 C B C# D A Ab Bb G F F# E Eb
|
int: +1 +3 -1 -1 +3 +1 +3 -1 +3 -1 -3
I-0 I-1 I-4 I-3 I-2 I-5 I-6 I-9 I-8 I-11 I-10 I-7
P-0 F F# A Ab G Bb B D C# E Eb C
P-11 E F Ab G F# A Bb C# C Eb D B
P-8 C# D F E Eb F# G Bb A C B Ab
P-9 D Eb F# F E G Ab B Bb C# C A
P-10 Eb E G F# F Ab A C B D C# Bb
P-7 C C# E Eb D F F# A Ab B Bb G
P-6 B C Eb D C# E F Ab G Bb A F#
P-3 Ab A C B Bb C# D F E G F# Eb
P-4 A Bb C# C B D Eb F# F Ab G E
P-1 F# G Bb A Ab B C Eb D F E C#
P-2 G Ab B Bb A C C# E Eb F# F D
P-5 Bb B D C# C Eb E G F# A Ab F
|
One of the two works by Stravinsky before Threni, and his first work to employ any twelve-tone technique, Canticum sacrum is an exploratory work both structurally and sonically. It was composed in 1955 in response to a commission by Alessandro Piovesan and the Venice Biennale International Festival of Contemporary Music. The work was designed to be performed in St. Mark's Basilica, and its five movements are said to be analogous with the five domes of the cathedral. The first performance was conducted by the composer on 13 September 1956 at St. Mark's.
Table 3 shows the matrix for the second movement of Canticum sacrum, and Table 4 shows the rows employed.
int: -1 -2 -3 +4 -2 -1 -2 -3 +2 -1 -2
I-0 I-11 I-9 I-6 I-10 I-8 I-7 I-5 I-2 I-4 I-3 I-1
P-0 Ab G F D F# E Eb C# Bb C B A
P-1 A Ab F# Eb G F E D B C# C Bb
P-3 B Bb Ab F A G F# E C# Eb D C
P-6 D C# B Ab C Bb A G E F# F Eb
P-2 Bb A G E Ab F# F Eb C D C# B
P-4 C B A F# Bb Ab G F D E Eb C#
P-5 C# C Bb G B A Ab F# Eb F E D
P-7 Eb D C A C# B Bb Ab F G F# E
P-10 F# F Eb C E D C# B Ab Bb A G
P-8 E Eb C# Bb D C B A F# Ab G F
P-9 F E D B Eb C# C Bb G A Ab F#
P-11 G F# E C# F Eb D C A B Bb Ab
|
| Measures | Text | Tenor Rows | Orchestral Rows |
|---|---|---|---|
| 46 | R-0 | ||
| 47-50 | Surge aquilo | P-0 | RI-2 |
| 50-52 | Perfla | R-0 (12-7) RI-8 (12-8) |
R-3 |
| 52-55 | Hortum meum | R-0 | R-0 |
| 56-57 | RI-2 | ||
| 58-62 | Veniat | I-8 | RI-2 |
| 63-65 | Veni | R-0 | Harp: R-0 Fl: I-0 |
| 66-68 | Soror | RI-8 | |
| 68-73 | Sponsa | R-0 | |
| 73-81 | I-2P-0 | ||
| 82-85 | P-0 | ||
| 86-87 | Comedite | I-1 | Fl: I-9Hp & CI: I-5 |
| 88-93 | Et bibite | P-0 |
Table 5 shows the matrix for the third and fourth movements of Canticum sacrum, and Table 6 and Table 7 show the rows employed.
int: -1 +2 +2 +1 -2 +5 -1 +3 -4 +3 +2
I-0 I-11 I-1 I-3 I-4 I-2 I-7 I-6 I-9 I-5 I-8 I-10
P-0 A Ab Bb C C# B E Eb F# D F G
P-1 Bb A B C# D C F E G Eb F# Ab
P-11 Ab G A B C Bb Eb D F C# E F#
P-9 F# F G A Bb Ab C# C Eb B D E
P-8 F E F# Ab A G C B D Bb C# Eb
P-10 G F# Ab Bb B A D C# E C Eb F
P-5 D C# Eb F F# E A Ab B G Bb C
P-6 Eb D E F# G F Bb A C Ab B C#
P-3 C B C# Eb E D G F# A F Ab Bb
P-7 E Eb F G Ab F# B Bb C# A C D
P-4 C# C D E F Eb Ab G Bb F# A B
P-2 B Bb C D Eb C# F# F Ab E G A
|
| Measures | Text | Rows | Form |
|---|---|---|---|
| 94-106 | Organ: P-0, R-5 Tbs: R-10; winds: RI-1 (alt?) |
Caritas | |
| 107-115 | WW & trp: I-1 Tbs: R-0; Str: R-4 |
||
| 116-129 (repeated) | Diliges | Sop: RI-4 Alto: R-7 Ten: RI-2, I-2 Tr: RI-1, I-1 |
|
| 130-147 | Organ & Cbn: P-3, RI-1 Tr: RI-9; Bn: RI-1 Ob & CI: I-10 |
Spes | |
| 148-153 | Qui confidunt | Ten: I-0 Bar: P-10 Orchestra: P-5,? |
|
| 154-162 | Sustinuit | Sop: RI-1, R-4 Alto: I-1, I-7 Orchestra: I-3 |
|
| 163-168 | Non | Ten: I-0 Bar: P-10 Orchestra: P-5,? |
|
| 169-177 | Speravit | Sop: R-4, P-3 Alto: I-7, I-11 Orchestra: I-3 |
|
| 178-183 | Qui habitat | Ten: I-0 Bar: R-10 Orchestra: P-5,? |
|
| 184-203 | Organ: P-2 Tbs: RI-10 Cb & cbn: R-1 |
Fides | |
| 204-218 | Credidi | Coro: P-1, P-1 Tr: I-11; Orch:? |
|
| 219-236 | Ego autem | Sop: I-6, RI-8; Alto: I-11, RI-1; Ten: I-4, R-2; Bass: I-9; Tr 1 & cbtb: I-11; Ob 1 & btb: P-6 |
|
| 237-249 | Tr 1: P-2, Tr 2: I-5 Organ: P-4; Str: P-2 |
| Measures | Text | Vocal Rows | Orchestral Rows |
|---|---|---|---|
| 250-253 | RI-0 | ||
| 253-265 | Jesus | Bar: P-0 Coro: P-0 |
Tr doubling vocal notes Orchestra: R-0 |
| 265-273 | Si potes | Bar: R-0 Coro: R-0 |
Tr doubling vocal notes |
| 274-280 | Et continuo | Sop: P-5; Alto: I-7, P-9 Ten: RI-10, I-10; Bass: R-5 |
|
| 281 | Cum lacrimis | Bar: RI-2 | |
| 282-294 | Credo | Bar: P-10 | R-5, RI-5 |
| 295-306 | Adjuva | Bar: I-10, RI-10 | P-10 |
Stravinsky composed Movements, a ten-minute-long work for piano and orchestra, in 1958-59. Table 8 shows the matrix for Movements.
int: +1 -6 -2 +1 +5 -2 -1 +2 +5 +1 -2
I-0 I-1 I-7 I-5 I-6 I-11 I-9 I-8 I-10 I-3 I-4 I-2
P-0 Eb E Bb Ab A D C B C# F# G F
P-11 D Eb A G Ab C# B Bb C F F# E
P-5 Ab A Eb C# D G F E F# B C Bb
P-7 Bb B F Eb E A G F# Ab C# D C
P-6 A Bb E D Eb Ab F# F G C C# B
P-1 E F B A Bb Eb C# C D G Ab F#
P-3 F# G C# B C F Eb D E A Bb Ab
P-4 G Ab D C C# F# E Eb F Bb B A
P-2 F F# C Bb B E D C# Eb Ab A G
P-9 C C# G F F# B A Ab Bb Eb E D
P-8 B C F# E F Bb Ab G A D Eb C#
P-10 C# D Ab F# G C Bb A B E F Eb
|
The analysis of The Flood shows that Stravinsky employs techniques whose use can be explained by way of analysis through projecting rows onto an object. The projection of rows for The Flood resulted in a single object. In addition, only a small section of the object was appropriate for that analysis.
In other works, like Threni and A Sermon, a Narrative, and a Prayer Stravinsky employs rows that create multiple objects. A brief look at Double Canon, a smaller work that employs multiple objects, can help to further illustrate the techniques used in the larger works.
Stravinsky's Double Canon "Raoul Dufy in Memoriam" is a short work for String Quartet written in 1959. The canon uses Prime and Retrograde, and Retrograde Inversion (never Inversion) forms at various degrees of transposition. The rhythm of the Retrograde and Retrograde Inversion forms is a retrograde of the rhythm of the Prime forms. In addition, the dynamic contour is retained in the Retrograde versions as well.
The Prime form of the row of Double Canon is taken from the first aggregate in the first violin part. The standard twelve-tone matrix is given in Table 9.
int: -1 +4 -1 -1 -5 -2 +3 +1 -3 -2 -1
I-0 I-11 I-3 I-2 I-1 I-8 I-6 I-9 I-10 I-7 I-5 I-4
P-0 F# F A Ab G D C Eb E C# B Bb
P-1 G F# Bb A Ab Eb C# E F D C B
P-9 Eb D F# F E B A C C# Bb Ab G
P-10 E Eb G F# F C Bb C# D B A Ab
P-11 F E Ab G F# C# B D Eb C Bb A
P-4 Bb A C# C B F# E G Ab F Eb D
P-6 C B Eb D C# Ab F# A Bb G F E
P-3 A Ab C B Bb F Eb F# G E D C#
P-2 Ab G B Bb A E D F F# Eb C# C
P-5 B Bb D C# C G F Ab A F# E Eb
P-7 C# C E Eb D A G Bb B Ab F# F
P-8 D C# F E Eb Bb Ab B C A G F#
|
All of the rows that Stravinsky uses in Double Canon are indicated in TAble 10. Each instrumental part is barred differently, therefore the chart below indicates the system in which each row begins.
| System | Violin 1 Rows | Violin 2 Rows | Viola Rows | Cello Rows |
|---|---|---|---|---|
| 1 | P-0 (1st Object) | |||
| P-10 (3rd Object) | ||||
| R-0 (1st Object) | ||||
| 2 | P-0 (1st Object) | R-2 (3rd Object) | ||
| P-0 (1st Object) | ||||
| R-0 (1st Object) | ||||
| RI-8 (1st Object) | ||||
| 3 | R-0 (1st Object) | |||
| RI-10 (3rd Object) | ||||
| RI-8 (1st Object) | ||||
| 4 | RI-8 (1st Object) |
The entire work employs six rows: P-0, R-0, RI-8, P-10, RI-10, and R-2. Again, note that no Inversion row is employed. When projected onto objects, the rows Stravinsky used (with an interval between the first and last notes in the row of ic4) create four objects, two of which are exploited in the work. These objects are shown in the table above.
From the four possible objects formed by ic4 rows, Stravinsky uses rows from the First and Third Objects in Double Canon. The rows P-0, R-0, and RI-8 are from the First Object. The rows P-10, RI-10, and R-2 are from the Third Object. These two objects are shown in Table 11 and Table 12.
The relationship of the First and Third Objects is important for the construction of the Double Canon. The First Object includes rows beginning or ending with two of the following three pitches: F#, Bb, and D. The Third Object includes rows beginning or ending with pitches a +2, +/-6, and -2 away from those in the First Object: C, E, and Ab.
The Third Object is identical with the First Object, but entirely transposed +2, +/-6, or -2. The ic2 relationships are important in Double Canon.
D
Eb Eb
R-4 F B I-8
Ab C
G C#
E F# Violin I (2x)
F# Ab Violin II
B F
P-4 C E RI-8
C# G
A A
Bb Bb
B B
G C#
I-4 Ab E R-0
A Eb
D C Viola (2x)
E D Cello
C# G
C Ab
RI-4 Eb A P-0 Violin I (2x)
F F Violin II
F#
G G
R-8 A Eb I-0
C E
B F
Ab Bb
Bb C
Eb A
P-8 E Ab RI-0
F B
C# C#
D D
|
E
F F
R-6 G C# I-10
Bb D
A Eb
F# Ab
Ab Bb
C# G Violin II
P-6 D F# RI-10
Eb A
B B
C C
C# C#
A Eb
I-6 Bb F# R-2
B F Cello
E D
F# E
Eb A
D Bb
RI-6 F B P-2
G G
Ab
A A
R-10 B F I-2
D F#
C# G
Bb C
Violin II C D
F B
P-10 F# Bb RI-2
G C#
Eb Eb
E E
|
The instruments in Double Canon are paired together: the two violins play one canon, while the viola and the cello play another. The first violin and viola are the leading parts, the second violin and cello are the following parts. The first violin and viola play rows from the first object two times each (first violin P-0, P-0, RI-8, RI-8; viola R-0, R-0). The second violin and cello follow with the same row as the leading part transposed by ic2, immediately followed by the row that the leading instrument played. In other words, the second violin plays P-10 (the leading first violin row transposed -2) followed by P-0, the row that the first violin is playing, and then RI-10 (first violin row +2) followed by RI-8. Likewise, the cello plays R-2 (viola row +2) followed by R-0, the form that the viola is playing. The transposed rows (ic2) that occur in the second violin and the cello parts are from the Third Object.
The term "double" in the title Double Canon refers not only to the canons between the first and second violins (two Prime forms, followed by two Retrograde Inversion forms), and between the viola and cello (two Retrograde forms). The canon is also "double" in that Stravinsky's row choices are from two objects related by ic2; both +2 and -2 are in the Third Object. Of all the rows from the matrix, Stravinsky specifically chose rows that are related by ic2 in the First and Third Objects formed by the projection of the rows.
Composed in 1960-61, Stravinsky's A Sermon, a Narrative, and a Prayer is a cantata for alto and tenor soloists, speaker, chorus, and orchestra. Table 13 shows the work's matrix, and Table 14 and Table 15 show the objects obtained from the row.
int: +1 -4 +2 -1 -3 +1 -5 +1 +2 -1 -3
I-0 I-1 I-9 I-11 I-10 I-7 I-8 I-3 I-4 I-6 I-5 I-2
P-0 Eb E C D C# Bb B F# G A Ab F
P-11 D Eb B C# C A Bb F F# Ab G E
P-3 F# G Eb F E C# D A Bb C B Ab
P-1 E F C# Eb D B C G Ab Bb A F#
P-2 F F# D E Eb C C# Ab A B Bb G
P-5 Ab A F G F# Eb E B C D C# Bb
P-4 G Ab E F# F D Eb Bb B C# C A
P-9 C C# A B Bb G Ab Eb E F# F D
P-8 B C Ab Bb A F# G D Eb F E C#
P-6 A Bb F# Ab G E F C C# Eb D B
P-7 Bb B G A Ab F F# C# D E Eb C
P-10 C# D Bb C B Ab A E F G F# Eb
|
A A
Ab C
C C#
I-6 Bb B R-4
B Bb
D Eb
C# D
F# F
F F#
RI-6 Eb E P-4
E Ab
G
Bb F#
B Bb
R-2 A Ab I-4
Ab A
C# C
C B
Eb E
P-2 E Eb RI-4
D C#
F# D
F F
E Ab
Ab A
I-2 F# G R-0
G F#
Bb B
A Bb
D C#
C# D
RI-2 B C P-0
C E
Eb
F# D
G F#
R-10 F E I-0
E F
A Ab
Ab G
B C
P-10 C B RI-0
Bb A
D Bb
C# C#
C E
E F
I-10 D Eb R-8
Eb D
F# G
F F#
Bb A
A Bb
RI-10 G Ab P-8
Ab C
B
D Bb
Eb D
R-6 C# C I-8
C C#
F E
E Eb
G Ab
P-6 Ab G RI-8
F# F
Bb F#
A A
|
Bb Bb
A C#
C# D
I-7 B C R-5
C B
Eb E
D Eb
G F#
F# G
RI-7 E F P-5
F A
Ab
B G
C B
R-3 Bb A I-5
A Bb
D C#
C# C
E F
P-3 F E RI-5
Eb D
G Eb
F# F#
F A
A Bb
I-3 G Ab R-1
Ab G
B C
Bb B
Eb D
D Eb
RI-3 C C# P-1
C# F
E
G Eb
Ab G
R-11 F# F I-1
F F#
Bb A
A Ab
C C#
P-11 C# C RI-1
B Bb
Eb B
D D
C# F
F F#
I-11 Eb E R-9
E Eb
G Ab
F# G
B Bb
Bb B
RI-11 Ab A P-9
A C#
C
Eb B
E Eb
R-7 D C# I-9
C# D
F# F
F E
Ab A
P-7 A Ab RI-9
G F#
B G
Bb Bb
|
Stravinsky's Abraham and Isaac, for baritone and chamber orchestra, was composed in 1962-63. Table 16 shows the work's matrix.
int: +1 +2 +2 +1 -4 +2 +4 -1 +2 +2 -1
I-0 I-1 I-3 I-5 I-6 I-2 I-4 I-8 I-7 I-9 I-11 I-10
P-0 G Ab Bb C C# A B Eb D E F# F
P-11 F# G A B C Ab Bb D C# Eb F E
P-9 E F G A Bb F# Ab C B C# Eb D
P-7 D Eb F G Ab E F# Bb A B C# C
P-6 C# D E F# G Eb F A Ab Bb C B
P-10 F F# Ab Bb B G A C# C D E Eb
P-8 Eb E F# Ab A F G B Bb C D C#
P-4 B C D E F C# Eb G F# Ab Bb A
P-5 C C# Eb F F# D E Ab G A B Bb
P-3 Bb B C# Eb E C D F# F G A Ab
P-1 Ab A B C# D Bb C E Eb F G F#
P-2 A Bb C D Eb B C# F E F# Ab G
|
Stravinsky composed the five-minute-long Variations in 1963-64 for orchestra. Table 17 shows the matrix for Variations.
int: -2 -3 +2 +5 -6 -2 +5 +2 +4 -1 -1
I-0 I-10 I-7 I-9 I-2 I-8 I-6 I-11 I-1 I-5 I-4 I-3
P-0 D C A B E Bb Ab C# Eb G F# F
P-2 E D B C# F# C Bb Eb F A Ab G
P-5 G F D E A Eb C# F# Ab C B Bb
P-3 F Eb C D G C# B E F# Bb A Ab
P-10 C Bb G A D Ab F# B C# F E Eb
P-4 F# E C# Eb Ab D C F G B Bb A
P-6 Ab F# Eb F Bb E D G A C# C B
P-1 Eb C# Bb C F B A D E Ab G F#
P-11 C# B Ab Bb Eb A G C D F# F E
P-7 A G E F# B F Eb Ab Bb D C# C
P-8 Bb Ab F G C F# E A B Eb D C#
P-9 B A F# Ab C# G F Bb C E Eb D
|
Stravinsky's third-to-last composition, Introitus is dedicated to the memory of his friend T. S. Eliot. The work is under four minutes long, and calls for two-part men's chorus (TB), harp, piano, tam-tams, timpani, solo viola, and solo contrabass.
Stravinsky uses rows horizontally (melodically) in the vocal parts, and uses set rotation in the instrumental parts. The Prime form of the row of Introitus is taken from the first aggregate sung by the tenors beginning in M. 3. The standard twelve-tone matrix for the work is given in Table 18.
int: +5 +1 -2 +4 +1 -6 -1 -1 -3 +1 -4
I-0 I-5 I-6 I-4 I-8 I-9 I-3 I-2 I-1 I-10 I-11 I-7
P-0 Ab C# D C E F B Bb A F# G Eb
P-7 Eb Ab A G B C F# F E C# D Bb
P-6 D G Ab F# Bb B F E Eb C C# A
P-8 E A Bb Ab C C# G F# F D Eb B
P-4 C F F# E Ab A Eb D C# Bb B G
P-3 B E F Eb G Ab D C# C A Bb F#
P-9 F Bb B A C# D Ab G F# Eb E C
P-10 F# B C Bb D Eb A Ab G E F C#
P-11 G C C# B Eb E Bb A Ab F F# D
P-2 Bb Eb E D F# G C# C B Ab A F
P-1 A D Eb C# F F# C B Bb G Ab E
P-5 C# F# G F A Bb E Eb D B C Ab
|
The rows employed for the entire work are given in Table 19. Characteristic of Stravinsky's other twelve-tone works, the standard four rows are sung by the voices: Prime, Retrograde, Inversion, and Retrograde Inversion beginning with the last note of the Prime form.
| Measures | Tenor Rows | Bass Rows | Instrumental Rows |
|---|---|---|---|
| 1-2 | |||
| 3-9 | P-0 | R-0 | |
| 9-10 | |||
| 10-15 | R-0 | P-0 | |
| 16-18 | Speaking | Speaking | |
| 19-25 | R-0 | ||
| 24-30 | I-0 | RI-2 | |
| 31-33 | |||
| 33-39 | RI-2 | R-0 | |
| 40-41 | Speaking | Speaking | |
| 42-45 | R-0 (1-10) | RI-2 (1-8) | P-0 |
| 46-47 | |||
| 48-49 | R-0 (9-12) | RI-2 (8-12) | P-0? |
| 50-53 |
The object formed by this row has the same first and last row notes as the object of The Flood. However, because Stravinsky only employs a few rows, and these forms are generated from the first and last notes of the Prime form of the row, projecting the object is not useful for the present analysis. Rather, the analyst simply needs a small graphical representation of the rows used, as shown in Table 20.
Bb
F
E
F#
D
C#
^ G
| Ab
RI-2 A
P-0 -> C
B
Ab C# D C E F B Bb A F# G Eb
Eb
D <- R-0
E I-0
C |
B v
F
F#
G
Bb
A
C#
|
Requiem Canticles, for alto and bass soloists, chorus, and orchestra, was Stravinsky's penultimate composition, written in 1965-66. The work uses two rows, the matrices for which are shown in Table 21 and Table 22.
int: +2 -4 +1 +2 -5 -2 +1 +2 -5 -1 +2
I-0 I-2 I-10 I-11 I-1 I-8 I-6 I-7 I-9 I-4 I-3 I-5
P-0 F G Eb E F# C# B C D A Ab Bb
P-10 Eb F C# D E B A Bb C G F# Ab
P-2 G A F F# Ab Eb C# D E B Bb C
P-1 F# Ab E F G D C C# Eb Bb A B
P-11 E F# D Eb F C Bb B C# Ab G A
P-4 A B G Ab Bb F Eb E F# C# C D
P-6 B C# A Bb C G F F# Ab Eb D E
P-5 Bb C Ab A B F# E F G D C# Eb
P-3 Ab Bb F# G A E D Eb F C B C#
P-8 C# Eb B C D A G Ab Bb F E F#
P-9 D E C C# Eb Bb Ab A B F# F G
P-7 C D Bb B C# Ab F# G A E Eb F
|
int: -5 -1 -2 +1 +4 -1 +2 +5 -2 -2 +3
I-0 I-7 I-6 I-4 I-5 I-9 I-8 I-10 I-3 I-1 I-11 I-2
P-0 F C B A Bb D C# Eb Ab F# E G
P-5 Bb F E D Eb G F# Ab C# B A C
P-6 B F# F Eb E Ab G A D C Bb C#
P-8 C# Ab G F F# Bb A B E D C Eb
P-7 C G F# E F A Ab Bb Eb C# B D
P-3 Ab Eb D C C# F E F# B A G Bb
P-4 A E Eb C# D F# F G C Bb Ab B
P-2 G D C# B C E Eb F Bb Ab F# A
P-9 D A Ab F# G B Bb C F Eb C# E
P-11 E B Bb Ab A C# C D G F Eb F#
P-1 F# C# C Bb B Eb D E A G F Ab
P-10 Eb Bb A G Ab C B C# F# E D F
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Stravinsky's Topology. Doctoral Dissertation. Boulder, CO: University of Colordao, 2000. www.lulu.com/akuster
(C) Copyright 2000 Andrew Kuster. All Rights Reserved.