# CHAPTER 5 Twelve-Tone Rows and Geometrical Objects

## Analysis of the Relationships between Stravinsky's Rows

Because of Stravinsky's economical row choices, the full twelve-tone matrix is not necessary for the analysis of his works previously discussed. Rather, the relationships between the rows of those pieces can be observed in simpler charts that contain only the employed rows hinged by common pitches, which in the case of his smaller works are the first and last (twelfth) notes of the Prime row, and in the case of larger works like The Flood, include pitches at the beginning and ending of related rows as well.

As in most of his larger works, Stravinsky employed more rows in The Flood than in his smaller twelve-tone works. Nevertheless, Stravinsky most often maintained the important relationship between the first and last (twelfth) notes in his rows, i.e. pivot or hinge pitches. In these larger works, Stravinsky often used rows based on the first and last (twelfth) notes of other related rows, creating extensive patterns of related rows. For musical and poetic-textual meaning in many of his larger works, Stravinsky used rows that are related in ways other than those described thus far. In addition, in the larger works containing more rows, Stravinsky continued to use specific rows for definite musical-poetical or compositional effects. Analysts can easily become aware of these compositional devices and the relationships between rows when they project the rows onto the three-dimensional objects, as described below. When Stravinsky chose to exploit or contradict these relationships, he did so for a particular compositional effect.

The above analysis of The Flood illustrates how the patterns and characteristics of Stravinsky's related twelve-tone rows can be projected onto a three-dimensional object and facilitate analysis. What follows is a presentation of a method to obtain geometrical models useful in the analysis of The Flood and Stravinsky's other larger works.

## Deducing Objects from Analytical Devices

### The Matrix

The traditional analytical tool for twelve-tone analysis is the classic matrix, the format that shows the tone row in its four standard operations (P, R, I, RI), indicating all forty-eight potential row forms. Furthermore, the matrix is valuable for comparing rows for such compositional techniques as hexachordal invariance, combinatoriality, and (rarely) implications for tonal centricity or polarity. However, the matrix is less useful for the study of row succession and does not contribute to the analysis of large-scale formal organization. The above analysis of The Flood illustrates that often visual representations help to convey significant principles of row succession and other aspects of large-scale organization. Nevertheless, since the matrix shows all of the rows (albeit many that Stravinsky does not employ), it is a necessary starting point in the deduction of Object-Oriented analysis. Table 30, then, shows the basis for the rest of this chapter: the matrix for Stravinsky's The Flood.

 ``` int: -2 +1 +6 -3 +2 -1 -2 -4 -1 -2 +1 I-0 I-10 I-11 I-5 I-2 I-4 I-3 I-1 I-9 I-8 I-6 I-7 P-0 C# B C F# Eb F E D Bb A G Ab P-2 Eb C# D Ab F G F# E C B A Bb P-1 D C C# G E F# F Eb B Bb Ab A P-7 Ab F# G C# Bb C B A F E D Eb P-10 B A Bb E C# Eb D C Ab G F F# P-8 A G Ab D B C# C Bb F# F Eb E P-9 Bb Ab A Eb C D C# B G F# E F P-11 C Bb B F D E Eb C# A Ab F# G P-3 E D Eb A F# Ab G F C# C Bb B P-4 F Eb E Bb G A Ab F# D C# B C P-6 G F F# C A B Bb Ab E Eb C# D P-5 F# E F B Ab Bb A G Eb D C C# ```

### Rows Tied to Beginning and Ending Row Members

In the smaller works analyzed earlier, including Epitaphium, Anthem, Elegy for J. F. K., Fanfare, and the Owl and the Pussy-Cat, a graphical representation of the rows employed included only up to six rows. For these works, the graphical representations enhance understanding of row succession, form, and textual meaning. A similar graphical representation of the rows used in The Flood will leave out many important rows that are employed in the composition. Table 31 shows the same type of representation for The Flood as that used for the smaller works. It lists only the six basic rows related to the end notes of the Prime form (P-0, R-0, I-0, RI-0, I-2, RI-2), and leaves out many other rows that Stravinsky uses (P-7, RI-7, R-10, P-5, and R-5). (As shown in the previous chapter, these rows are the only ones sung by the duet of basses portraying God.) Because it is incomplete, this type of graphical representation must be extended in order to accommodate works that use more than the six related rows.

 ``` Eb F I-2 E | Bb v C# B ^ C | D RI-2 F# P-0 -> G A C# B C F# Eb F E D Bb A G Ab Eb D <- R-0 Ab I-0 B | A v Bb C ^ E | F RI-0 G F# ```

### The Extended Chain of Rows

An improved graphical representation of the important rows in The Flood shows an extended chain pattern of rows where the last note of each row becomes the first note of the following related row.

Table 32 shows a graphical representation of the chain of rows for The Flood. Only the beginning and ending notes of the rows are given, and the middle members of the rows are omitted.

The resulting row chains incorporate all of the rows employed by the voices and the Jacob's Ladder row in The Flood on two axes. The vertical axis contains Inverted rows in an upward direction and Retrograde Inversion rows in an downward direction. The horizontal axis contains Prime rows from left to right, and Retrograde rows from right to left.

If the representation continues through all of the possible rows in any direction, then the chain repeats indefinitely, or into infinity. In this case, the ends of the rows form a circle of fifths.

 ``` Eb I-2 | RI-2 Ab I-7 | RI-7 B P-10 -- R-10 F# P-5 -- R-5 C# P-0 -- R-0 Ab P-7 -- R-7 Eb I-0 | RI-0 F# ```

However, Table 32 is misleading because each important pitch from which rows are generated occurs in more than one location on the chart (except for the central pitch, C#). Therefore, the relationship between the rows starting or ending, for example, on Ab on the horizontal axis and those starting or ending on Ab on the vertical axis are not immediately apparent. One way to eliminate this drawback is to create a grid of rows where any pitch (that is, the end of any row) can be the center of the axes.

### The Grid of Rows

Table 33 shows a logical extension of the analytical method of the row chain graphical representation in Table 32. Whereas in the graph of row chains all of the rows center on the first note of the Prime row, in the grid any row that begins or ends with an important pitch can be traced in any direction. Again, as in the row chains above, each vertical axis contains inverted rows in an upward direction and Retrograde Inversion rows in an downward direction. Each horizontal axis contains Prime rows from left to right, and Retrograde rows from right to left.

 ``` C# P-0 - R-0 Ab P-7 - R-7 Eb - Bb - F I-0 I-7 I-2 | | | | | RI-0 RI-7 RI-2 F# P-5 - R-5 C# P-0 - R-0 Ab P-7 - R-7 Eb - Bb I-0 I-7 I-2 | | | | | RI-0 RI-7 RI-2 B P-10 - R-10 F# P-5 - R-5 C# P-0 - R-0 Ab P-7 - R-7 Eb I-0 I-7 I-2 | | | | | RI-0 RI-7 RI-2 E - B P-10 - R-10 F# P-5 - R-5 C# P-0 - R-0 Ab I-0 I-7 | | | | | RI-0 RI-7 A - E - B P-10 - R-10 F# P-5 - R-5 C# ```

Although this grid may prove more helpful in determining important relationships between rows, considerable repetition exists. In fact (again as in the case of the row chains above), if the grid is traced in any direction, then the same rows will recur an infinite number of times. Furthermore, along the diagonal from the upper left corner to the lower right corner, the identical rows repeat constantly. This redundancy, a fallout of two-dimensional cyclic organization, hinders the efficiency of the grid as a useful analytical tool.

### The Solution: Wrapped Repetitions form three-dimensional Objects

The goal for the best analytical tool to analyze Stravinsky's set succession, therefore, is one in which the pertinent information contained in the above grid can be displayed in a concise format that contains no duplication.

Table 34 shows a simplified version of the grid layout. For sake of clarity, this grid is limited to the pitch names for only the ends of the rows:

 ``` C# - Ab - Eb - Bb - F | | | | | F# - C# - Ab - Eb - Bb | | | | | B - F# - C# - Ab - Eb | | | | | E - B - F# - C# - Ab | | | | | A - E - B - F# - C# ```

Note that in Table 34 the rows along the diagonal of the grid from lower left to upper right corner. Unlike the rows along the diagonal from upper left to lower right, these rows contain information that is not immediately redundant. In Table 35 all rows that were superfluous have been eliminated. By eliminating all superfluous rows, the result is a chopped grid, shown in Table 35.

 ``` Bb - F | | Ab - Eb - Bb | | F# - C# - Ab | | E - B - F# | | A - E ```

Here, using any diagonal traced from the lower left to upper right of the grid of rows also eliminates any superfluous rows. Now the only evident redundancies are two occurrences of every other pitch letter: these are found by moving diagonally down from left to right. However, most important, no redundant rows exist.

For ease of illustration, Table 36 shows the chopped grid rotated 45 degrees counter-clockwise.

 ``` F / Bb Bb P Eb Ab Ab R / C# F# F# I \ B \ E E RI A ```

Now all the Prime rows occur diagonally from lower left to upper right, with Retrograde rows in the opposite direction. Likewise, all Inverted rows occur diagonally from upper left to lower right, with Retrograde Inversion forms in the opposite direction.

If Table 36 is regarded as a two-dimensional object or a plane, the left and right columns form the object's edges. Now, if the edge of the left column is "wrapped" so that it connects with the edge of the right column and so that the pitch names overlap, a three-dimensional cylindrical object is created.

Note that for this illustration, the edges of the object are wrapped "inwards," that is receding away from the perspective of the reader and into the page. For example, the typed pitch letters printed on this piece of paper would remain on the outside of the page if it were an inwardly wrapped object (see Table 37).

TABLE 37. Formation of the Cylindrical Object
Wrapping the edges away into the paper... forms a cylindrical object.

Since the pitch names that were redundant now overlap, they can be treated as the same pitch, so that no redundancy exists.

As a cylindrical object wrapped this way, all Prime rows will occur in an upward spiral from left to right, and all Retrograde rows will be along the same path, but in the opposite direction. Likewise, all Inverted rows will occur in a downward spiral from left to right, and all Retrograde Inversion forms will be along the same path, but in the opposite direction. Hence this cylindrical object contains all the rows, traceable in any direction, from any pitch, without immediate redundancy. (If the edges of the plane were wrapped "outwards" instead of "inwards" then the spirals of rows would occur in the opposite direction as those described here.)

With the internal pitches of the row of The Flood, an object-graph of the important rows that Stravinsky employs can be created, as shown in Table 38. Remember that the pitch names on the right and left column edges are "wrapped inwards" and are thus the same pitch. Note carefully the spiral patterns traced by the four basic rows: Prime, Retrograde, Inversion, and Retrograde Inversion.

 ``` Eb D F R-7 E E I-2 F Bb A C# B B C C Bb D P-7 C# F# RI-2 G G F# A Ab Ab Bb G A A I-7 Eb Bb R-0 F# D E E F F G Eb B F# RI-7 C C P-0 D B C# C Eb R-5 D D I-0 Eb Ab G B A A Bb Bb Ab C P-5 B E RI-0 F F E G F# F# Ab F G G I-5 C# Ab R-10 E C D D Eb Eb F C# A E RI-5 Bb Bb P-10 C A B ```

Only one type of redundancy still remains: if the rows continue to be traced upwards or downwards, they will eventually repeat infinitely. Table 39 shows that eventually the pitch F (and the entire pattern afterwards) returns.

 ``` F Bb Bb Eb Ab Ab C# F# F# B E E A D D G C C F ```

To eliminate this redundancy, the solution involves one final wrapping: the edges of the cylinder are connected to form a doughnut-shaped object known to mathematicians as a Torus, as shown in Table 40. Note that for this illustration the cylinder has been rotated 90 degrees. In addition, because of the shape of the Torus, for this example it does not matter whether the cylinder is wrapped inwards or outwards.

TABLE 40. The formation of a Torus
Wrapping the cylindrical object... forms a Torus.

This Torus-shaped three-dimensional object contains a projection of all of the rows on its surface that can be read in any direction with no redundancy. (For a related method of graphing representations of intervals in three-dimensional space, see John Roeder, "A Geometric Representation of Pitch-Class Series," in Perspectives of New Music 25:1-2 (Spring-Summer 1987): 366.)

Note that, because the object wraps left and right like a cylinder and up and down like a torus, the following two-dimensional representation of the object for The Flood in Table 41 shows the identical object depicted in Table 38 above.

 ``` Bb Eb Eb Ab C# C# F# B B E A A D G G C F F Bb ```

The unfolded, two-dimensional mapping of the three-dimensional Torus-shaped object in Table 41 is, perhaps, the most useful way to depict the object on two-dimensional paper.

## Various Intervals Create Several Different Object Types.

In the case of other twelve-tone rows, where the interval between the first and last (twelfth) pitch in the Prime row is not the same as that in The Flood (that is, other than ic5), the objects created by this analytical tool will be different than the Torus described above. For example, when a cylinder is created with a row with ic2 between the first and last pitch in the Prime row, the object will wrap into a Torus after rows beginning on only six pitches, and not all twelve as in the case of The Flood. In other words, to graphically represent all the rows of a row with ic2 between the first and last pitches using three-dimensional objects, two objects must be created. The first object will be based on the Prime form of the row, and will wrap after the Prime form is reiterated at six pitch levels. The second object will be based on the Prime form of the row transposed up (or down) one whole-step. This second object will contain rows with the six other pitch-class levels that were not mapped on the first object. (This will be further clarified with other illustrations below, but also refer back to Table 32.)

### Objects with ic2 Between First and Last Notes in the Row

Stravinsky employs the type of row that creates two objects in A Sermon, a Narrative, and a Prayer (again, the first and last notes of the row are ic2). In Table 42, an intervalic representation of each object is presented on the left next to each of the two objects that incorporate the first and last notes from the rows of A Sermon, a Narrative, and a Prayer. Remember that the left and right columns of each object are wrapped in the method described above to create a cylinder, and that the upper and lower edges of each cylinder are again wrapped to form two Torus-shaped objects.

Note that these same objects will be formed by ic2, but that the negative form (the interval -2) will be an inverted reflection of the positive form (the interval +2). Also, note that any works with rows so constructed will have the same starting and ending notes in their rows as those in A Sermon, a Narrative, and a Prayer, and hence will form identical objects, albeit with different internal row members.

The matrix and objects for A Sermon, a Narrative, and a Prayer are shown in an appendix. For now, please note the relationships created by the objects formed by the beginning and ending pitches of the row.

TABLE 42. Objects Generated by Rows with ic2 between 1 & 12
1st Object ic2 2nd Object ic2
```
0        0

-2

-4      -4

6

+4      +4

+2

0        0

```
```
Eb      Eb

C#

B       B

A

G       G

F

Eb      Eb

```
```
+1       +1

-1

-3       -3

-5

+5       +5

+3

+1       +1

```
```
E       E

D

C       C

Bb

Ab      Ab

F#

E       E

```

Other works by Stravinsky that use rows with a ic2 between their first and last members are the third and fourth movements of Canticum sacrum, Movements, Abraham and Isaac, and the second series of Requiem Canticles.

### Six Object Types

The different types and number of objects created by rows with different intervals between the first and last notes are listed in Table 43. (For pertinent information regarding cycles of invariant-transposition pitch-collections, see Richard Cohn, "Properties and Generability of Transpositionally Invariant Sets," Journal of Music Theory 35:1-2 (Spring-Fall 1991): 6.)

 Interval between 1 & 12 ic1 ic2 ic3 ic4 ic5 ic6 Number of Objects 1 2 3 4 1 6 Type of Object(s) Torus Torus Torus Torus Torus Sphere Number of Beginning and Ending Pitch Classes per Object 12 6 4 3 12 2

Table 43 shows that a pattern is formed with the beginning and ending interval classes ic1 to ic4 and the number of Torus-shaped objects. The projection of the row with ic5 creates a single Torus-shaped object.

When mapping rows made of ic6 (a tritone) between the first and last pitches in the row, six Spherical objects rather than Torus-shaped objects are formed. In this case, the object contains redundancies after only two row iterations in any direction (for example, P-0 and P-6 reiterate; as well as I-0 and I-6), and therefore it wraps immediately in any direction, creating a Sphere.

## Stravinsky's Twelve-tone Rows

Table 44 lists all of the Prime rows in all of Stravinsky's twelve-tone works, along with the interval between the first and last pitch in each row.

TABLE 44. Intervals between the first and last members of all of Stravinsky's Rows
Title of Stravinsky's Work Prime Row (P-0) Interval between 1 & 12
Canticum, II Ab G F D F# E Eb C# Bb C B A +1
Canticum, II & IV A Ab Bb C C# B E Eb F# D F G +2
Agon, Double Pas... Eb D E F C B C# Bb Ab A G F# +3
Agon, Pas... F F# A Ab G Bb B D C# E Eb C -5
Threni Eb Ab G Bb C# A D B E C F F# +3
Movements Eb E Bb Ab A D C B C# F# G F +2
Epitaphium C# Bb Eb E C B F# F D G Ab A -4
Double Canon F# F A Ab G D C Eb E C# B Bb +4
Sermon, Narr. Prayer Eb E C D C# Bb B F# G A Ab F +2
Anthem D F# E G A Bb C B C# Eb Ab F +3
The Flood C# B C F# Eb F E D Bb A G Ab -5
Abraham & Isaac G Ab Bb C C# A B Eb D E F# F -2
Elegy Ab D C Bb E F B A G F# Eb C# +5
Fanfare Bb A B C# D C Eb F E F# Ab G -3
Variations D C A B E Bb Ab C# Eb G F# F +3
Requiem, 1st Series F G Eb E F# C# B C D A Ab Bb +5
Requiem, 2nd Series F C B A Bb D C# Eb Ab F# E G +2
Owl & Pussy-Cat D E B C# Bb Ab G A C Eb F# F +3

From this table, it can be seen that Stravinsky favors rows whose beginning and ending pitches form ic2, ic3, and ic5. Rows of ic4 between the first and last pitches occur only twice, and ic1 rows occur only once. Stravinsky has no rows with ic6 between the first and last pitches.

### Objects with ic1 or ic5 Between First and Last Notes in the Row

The only composition in which Stravinsky uses a row with beginning and ending pitches of ic1 is the second movement of Canticum sacrum. Stravinsky uses rows with beginning and ending pitches of ic5 in the "Pas-de-Deux" of Agon, The Flood, Elegy, and the first series of Requiem Canticles.

Table 45 shows the objects created by rows with beginning and ending pitches of either ic1 or ic5. Either of these types of rows form a single Torus-shaped object with twelve distinct beginning and ending pitches.

TABLE 45. Objects Generated by Rows with ic1 or ic5 between 1 & 12
Object ic1 Canticum Sacrum, II Object ic5 The Flood
```
0

-1      -1

-2

-3      -3

-4

-5      -5

6

+5      +5

+4

+3      +3

+2

+1      +1

0

```
```
Ab

G        G

F#

F        F

E

Eb       Eb

D

C#       C#

C

B        B

Bb

A        A

Ab

```
```
0

+5       +5

-2

+3       +3

-4

+1       +1

6

-1       -1

+4

-3       -3

+2

-5       -5

0

```
```
C#

F#       F#

B

E        E

A

D        D

G

C        C

F

Bb       Bb

Eb

Ab       Ab

C#

```

### Objects with ic3 Between First and Last Notes in the Row

Table 46 shows the objects created by rows with ic3 between the first and last notes of the row. This type of row creates three Torus-shaped objects, each with four distinct beginning/ending pitches. Stravinsky uses this kind of row in the "Double Pas-de-Deux" of Agon, Threni, Anthem, Fanfare, Variations, and The Owl and the Pussy-Cat. The table shows the objects as used in Threni: id est lamentationes Jeremiae prophetae: the First Object contains the Prime row which begins with Eb and ends with F#.

TABLE 46. Objects Generated by Rows with ic3 between 1 & 12
1st Object ic3 2nd Object ic3 3rd Object ic3
```
0    0

-3

6    6

+3

0    0

```
```
Eb  Eb

C

A   A

F#

Eb  Eb

```
```
+1  +1

-2

-5  -5

+4

+1  +1

```
```
E   E

C#

Bb  Bb

G

E   E

```
```
+2  +2

-1

-4  -4

+5

+2  +2

```
```
F   F

D

B   B

Ab

F   F

```

### Objects with ic4 Between First and Last Notes in the Row

Table 47 shows the objects created by rows with ic4 between the first and last notes. When projected, this kind of row creates four Torus-shaped objects, each with three distinct beginning/ending pitches. The objects formed from this projection are particularly interesting because the cylinder must be twisted to accommodate the reiteration of rows, when the Torus is formed. The twisting takes place since at the bottom of the two-dimensional Torus representation the row beginning or ending pitches are on the outside of the cylindrical object, and at the top of the Torus the row beginning or ending pitches ar in the center of the cylinder. For the three-dimensional Torus, these all meet up at one point.

Stravinsky uses rows with intervals of ic4 between the first and last notes in only two short works: Epitaphium and Double Canon. Table 47 shows the objects for Double Canon: the First Object contains P-0, which begins with F# and ends with Bb.

TABLE 47. Objects Generated by Rows with ic4 between 1 & 12
1st Object ic4 2nd Object ic4
```
0

-4      -4

+4

0       0

```
```
F#

D       D

Bb

F#      F#

```
```
+1

-3      -3

+5

+1      +1

```
```
G

Eb      Eb

B

G       G

```
3rd Object ic4 4th Object ic4
```
+2

-2      -2

6

+2      +2

```
```
Ab

E       E

C

Ab      Ab

```
```
+3

-1      -1

-5

+3      +3

```
```
A

F        F

C#

A        A

```

### Objects with ic6 Between First and Last Notes in the Row

Although Stravinsky does not employ any rows with an interval of ic6 between the first and last notes of the row, the objects created from this type of row are given in Table 48.

TABLE 48. Objects Generated by Rows with ic6 between 1 & 12
1st Object ic6 2nd Object ic6 3rd Object ic6 4th Object ic6 5th Object ic6 6th Object ic6
```
0   0
6
0   0

```
```
+1  +1
-5
+1  +1

```
```
+2  +2
-4
+2  +2

```
```
+3  +3
-3
+3  +3

```
```
+4  +4
-2
+4  +4

```
```
+5  +5
-1
+5  +5

```

Thus far, these three-dimensional objects merely form theoretical models that are based upon row succession in Stravinsky's twelve-tone rows. The following chapter will show how these objects can be used to help determine formal structure and row succession, and to analyze textual-poetic meaning in Threni.

## This Document is Available for Purchase

Stravinsky's Topology. Doctoral Dissertation. Boulder, CO: University of Colordao, 2000. www.lulu.com/akuster