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Concerning Serial Rotation in Stravinsky's Variations In Memoriam Aldous Huxley
The Variations In Memoriam Aldous Huxley is Stravinsky's last purely orchestral composition and was written from 1963 to 1964. During this time Stravinsky, and other composers such as Ernst Krenek, were interested in using serial rotation as a means for generating their pitch choices. The Variations contain several passages which demonstrate this technique remarkably well. A problem, however, concerning rotation is that, though many writers have dealt with the subject, there apparently has been taken no systematic approach regarding the classification and labeling of rotationally-related row-forms. This paper, then, will suggest such an approach by providing a conceptual model for rotation, and at the same time will try to shed light on certain aspects of Stravinsky's Variations.
For those to whom the concept is new, "rotation" is most easily thought of as simply beginning on some note other than the first note of a tone row. To illustrate this, refer to Example 1. The 12-tone series used within Stravinsky's Variations begins on D and ends with F. However, if one were to begin on the second note, C, and proceed to the last note of the row, F, finally attaching the first note, D, onto the end, then the result is the "first rotation" of the row. Likewise, beginning on the third note, proceeding to the last note, and attaching the D and C onto the end produces the second rotation of the row. The 12-tone row, then, has eleven rotational possibilities in addition to the original version, which would be the "zeroth rotation." See Example 1.
From this point on, pitch classes will be referred to by pitch class numbers. The "Fixed-Do" method will be used, so C will always equal "0," C-sharp and D-flat will equal "1," and so on, with B represented by "11." The classic serial operations of inversion and retrograde will be designated, as usual, with "I" and "R," respectively. The transposition of a row type will be labeled Tx ("T sub x"), where "x" equals (a) the first pitch class of a series, or (b) the last pitch class of the series if the series is a retrograde row-form. An individual row-form, then, will be labeled using the format (R)Tx(I), where the retrograde (R) component and the inversion (I) component are used only if applicable; all row-forms have as a minimum the label "Tx."
The so-called "forty-eight row-forms" of a 12-tone series are found by presenting all twelve transpositions of (1) the original row, or "prime-form" (Tx), (2) the inversion (TxI), (3) the retrograde (RTx), and (4) the retrograde inversion (RTxI) of a given series. These are typically represented by the familiar 12x12 matrix, as can be seen in Example 2a. However, if any of the twelve pitch classes contained within any of the 48 row-forms is allowed to constitute the initial pitch class of a series, then there results 576 possible rotationally-related row-forms. For example, all 48 row-forms could begin on their second note, or second order position, and a new 12x12 matrix would be generated. Example 2b shows the relationship between all 48 original forms and the 48 first rotations. See Example 2.
A definition for rotation can be shown as follows:
If a row (R)Tx(I) beginning on order position "n" (1 < n < 13) proceeds to the last note of the row,
The 576 possible row-forms can be represented by extending the 12x12 matrix so that it resembles as 12x12x12 cube. Each of the twelve 12x12 squares "sliced" from the cube represents all of the row-forms beginning on a given order position, n. Each square will be labeled Sp (where p = n-1); thus, the "zeroth rotation" of the row starts as usual with order position 1, the "first rotation" starts with order position 2, etc. The row-forms contained within Sp will be labeled (R)Tx(I)Sp. A "cube" constructed from the row of Stravinsky's Variations is shown in Example 3. It can be thought of as conveniently and graphically portraying all rotationally-related rows, many of which are actually found within the Variations. Indeed, it is not only graphically interesting, but it will be seen later that it also clearly demonstrates the basis for Stravinsky's choice of row-forms. See Example 3.
Before proceeding further, an example will be given to illustrate the usefulness of the cube as a labeling device for rotationally-related row-forms. To find, say, T8IS7, one first must locate square S7. This square, the 7th rotation, contains all 48 row-forms that begin with their 8th note. (Don't forget that S0 contains all row-forms that begin with their 1st note.) Within this square is to be found T8I, an inversion of the "original row" of S7 at a transposition such that its first note is pitch class 8. (The retrograde of this row, RT8IS7, would end on pitch-class 8.) This method is shown in Example 4. See Example 4.
The original row used by Stravinsky, according to Claudio Spies in Perspectives of New Music article entitled "Notes on Stravinsky's Variations," was first conceived as "a little melody" and is shown in Example 5. It is found in the composer's sketch book for the Variations, and Spies goes on to describe it as a "charming, spontaneous invention with a bit of 'Russia' in its rhythm." The hexachordal content of each half of the melody is the same (non-Z-related) and is of the type listed by Allen Forte as 6-9 (0, 1, 2, 3, 5, 7). This melody is never presented literally within the composition and Spies points out that its pitch content alone was used by Stravinsky to generate various serial charts, which the present writer believes can be summarized by the cube of Example 3. See Example 5.
One of the techniques employed by Stravinsky in his Variations is the utilization of rotationally-related row-forms that all either begin or end with the same pitch class. If the cube is drawn in such a manner that the upper left corner contains this "common tone," then the diagonal from top left to bottom right will conveniently represent the beginning or ending point of what will be termed here as all possible "common-tone" rotationally-related row-forms. In Example 6 it can be seen how the diagonal, or pitch class "2," generates all common-tone rotationally-related row-forms of transposition T2. See Example 6.
If, however, a pitch class other than that found in the upper left corner is the common tone, then the diagonal is useless for demonstrating the location of all 48 rotationally-related row-forms beginning with the given common tone. Instead, they are found distributed throughout the cube. In Example 7, primes, retrogrades, inversions and retrograde inversions that begin with the last note of the original row (namely, pitch class "5") are located within a cube having diagonal "2." One each, of types P, R, I and RI, is found within each square; or said differently, each order position is capable of generating a P, an R, an I, or an RI of the specified transposition T5. See Example 7.
Since only one of the twelve pitch classes can serve as the diagonal, it seems to be inefficient to construct charts solely based on these diagonals, as is so often done when discussing Stravinsky's transpositional rotational techniques. It would seem to be better to speak simply of the successive order positions of row-forms with a common initial or final tone. Or said differently, one should consider rows that begin or end with the same note (the same Tx) and taken from successive squares (i.e., S0, S1, S2, S3, etc.).
So, even though pitch class "5" does not represent the diagonal in the cube of Example 7, the prime forms beginning with p.c.5 (or retrogrades ending with p.c.5) can be taken from the cube and ordered by their successive order positions. See in Example 8a all rows of T5 with successive squares S0, S1, S2, etc.
As seen in Example 8b, the prime forms ending with "5" (or retrogrades beginning with "5") can be ordered by their successive order positions, though the numeral "5" is no longer reflected in the transposition number of the row-form (i.e., "5" is last, not first.) The inversion and retrograde inversion forms also, of course, can be similarly arranged. See Example 8.
Within the Variations, several passages will be examined in order to show the use of successive order-position row-forms with a common inital or final. However, first it is necessary to review the form of the Variations. Claudio Spies clearly shows the composition to be in twelve parts distinguished by textural differences or cessations of rhythmic activity. The sections are complete small forms in themselves, and the block form of the Variations is typical of Stravinsky's approach. The form is diagrammed in Example 9. The piece begins and ends with a chordal passage and contains three 12-part polyphonic sections and six sections based on various phrase structures. Following the opening chordal section is a monodic passage, and preceding the final 12-part polyphonic section is a fugato based on a rhythmic subject. See Example 9.
The present writer disagrees with Spies on two small points. "His" Section-I should be considered two separate sections because of the great difference of texture. The monodic section is complete unto itself, as will be seen shortly, and it should be treated separately in spite of the resulting brevity of the opening section (which, nevertheless, is also complete in itself). The second point concerns Spies' Section-III and Section-IV (mm. 33-46). These sections are no more differentiated or separated from each other than are the two contrasting phrases of Section VI. Therefore, this passage should be considered one section. These small changes in the formal diagram do not affect Spies' observation that the middle 12-part polyphonic section (Section-V), purposely placed "off-center" by Stravinsky, gives the composition an interesting asymmetrical structure.
Having established the form, various sections of the Variations will be considered in detail. Much attention will first be given to the monodic passage and this will be followed by an examination of the two contrasting phrases of Section-VI, the rhythmic fugato, and the first of the 12-part polyphonic sections. The final chordal section will be studied lastly.
An orchestral reduction of the monodic passage is provided in Excerpt 1. This passage (mm. 6-22) demonstrates Stravinsky's rotational method exceptionally well. The passage emphasizes pitch-class "2" (D), which serves as the final or initial note of each row-form used. (It will be remembered, of course, that D is the first note of the original melody.)
Beginning at m.6, the row-forms used in the monodic section are T0S6, T2S4, T2S3, RT2S2, RT2S1, T2S1, and T2S0. These rows can be found in the cube in the successive squares 6, 4, 3, 2, 1, 1, and 0, respectively. Another way of interpreting the cube is that each vertical column and each horizontal row represents an endless series repeated over and over. This type of endless series is to be thought of without regard for any particular 12x12 square and each will be referred to as a "Transposition Continuum" (T.C.). There are twelve vertical and twelve horizontal T.C.s, each presented in the cube twice--the top and bottom halves with respect to the horizontal T.C.s, and the left and right halves with respect to the vertical T.C.s. All 12-note rotational row-forms are segments "sliced out" of some T.C. This, of course, has an analogy in the study of the ancient "white-key" modes. Each of the modes is defined by an octave segment of the endless "Greater perfect System" and therefore all white-key modes are rotationally related and contained in the same transposition continuum. Thus, in rotationally-related 12-tone rows, the final and initial tones take on special importance.
Looking again at the monodic passage, it can be seen that the row-forms used can be found in successive T.C.s as well as successive squares. The appearance of pitch-class "2" (D) always indicates the ending and/or beginning of rows. It will be noticed, however, at the top of Excerpt 1, that T0S6 is indicated much lower in the cube than the others. This is because T0S6 ends with pitch-class "2" (D) while the others begin with D. Nevertheless, T0S6 is from T.C.5, which precedes the next row from T.C.4.
Interestingly, the introductory chordal section preceding the monodic passage concludes in m.5 with a chord containing pitch-classes (10, 4, 2, 7, and 9) or (B-flat, E, D, G, and A) and this forms a segment from T.C.6 which is the appropriate T.C. to use preceding the T.C.s of the monodic passage (T.C.s 5, 4, 3, 2, 1, 1, 0). This chord is shown at the top of Excerpt 1.
The row-forms of the monodic passage are connected by various types of elisions, and these points of elision generally involve repeated pitches.Footnote 1 Of special interest, at m.12, is the "elision" between T2S3 and RT2S2. T2S3 ends with (. . . 5, 3, and 0), but RT2S2 begins with pitch-class 5. Therefore, in order to make the elision, Stravinsky "backed up" after finishing T2S3 causing the succession of notes F - D-sharp - C - D-sharp - F, and thereby allowing pitch-class "5" (F) to serve as the note eliding the two rows. In this little reversal it will be noticed how the octave register of each of the twice-used pitches is kept constant.
Apparently there are two types of elision and because of this, sometimes it will be difficult to apply unambiguously an exact row label. The two types of elision will be called "actual elision" and "implied elision."
An actual elision occurs when the last note of one row is the same note (or a repeated note) as the first note of the following row. This can become more elaborate if the penultimate note(s) of the first row is the same as the second note(s) of the following row. This occurs between the first two rows of the monodic section and is shown in Example 10. See Example 10.
In m.7, the repeated note segment G-sharp-D both begins and ends on G-sharp. Seen in Example 11, each pairing of the two notes is merely a repetition; the final unpaired G-sharp indicates that the "middle" note, D (rather than G-sharp), is indeed the final note and the initial note, thereby eliding two rows. It will be seen that the structural importance of the "D" is undermined at this early point in the monodic passage by being assigned only to a grace note. See Example 11.
The repeated G-sharp seen above, the first repeated note of the composition, is quite important as it foreshadows the final note of the work--the bass clarinet's solitary G-sharp, which will be shown later to be quite significant.
In a "Double Elision" the penultimate note of the first row is the same as the first note of the second row, while the final note of the first row is the same as the second note of the second row. This is an intensification of the actual elision technique and is demonstrated in Example 12. See Example 12.
An implied elision occurs when the first note of the second row is the same as the first note of the first row. This could make the first row ambiguous, since it could be either of two rotations from the same Transposition Continuum. This occurs at m.10 between the second and third rows and is shown in Example 13. See Example 13.
In this case, Stravinsky musically makes it clear that T8S5 would not be the best label (because of the eighth-note rest separating T2S4 from T2S3).
An implied elision also takes place when the final note of the second row is the same as the final note of the first row. This could make the second row ambiguous for similar reasons.
Stravinsky makes use of implied elisions quite often throughout the Variations. The last four rows of the monodic passage indicate the use of implied elisions very well, as can be seen in Example 14. See Example 14.
As seen above, RT2S1 is followed simply by its own retrograde, T2S1, so in spite of the rests separating m.17 and m.18, the best label involves an actual elision (at the end of m.17). However, an implied elision is also possible if one begins the row with the new phrase on pitch-class 11 (B). If T2S1 is indeed chosen, then an implied elision occurs on pitch-class 2 (D) of m.20. However, if T11S2 is chosen, then an actual elision occurs on D. It must be remembered that the use of specific labels in ambiguous situations is not as important as simply appreciating the highly interesting elision techniques which purposely create the ambiguity in the first place.
In a sense, the monodic passage is theoretically brought to a close by descending through successive Transposition Continuums until Stravinsky's "original row" (T2S1) is obtained. The passage has the feel of closure in a more musical sense, as well. Of special interest are the fluctuating rhythmic subdivisions contained in each phrase. As the pitch structure proceeds from T.C. 5 through 4, 3, 2, 1, 1, and finally 0, the rhythmic structure proceeds with a series of written-out rhythmic accelerandos and ritardandos, giving a living, breathing quality to the passage. Stravinsky also used varying subdivisions seemingly to alter the pacing of the little motive of the opening of the bassoon melody of Le Sacre du Printemps, so this technique is not new to the composer. A diagram of the monodic section might look like that found in Example 15. See Example 15. (It is also shown in Excerpt 1.)
As seen above, the first note of a phrase is always in a new measure and separated from the previous phrase by a rest. The first phrase is a balance between accelerando and ritardando (triplet subdivisions--sixteenths--triplets). The middle four phrases all have a larger accelerando than ritardando. (The two center phrases have no ritardando.) The final phrase has, by contrast, a short accelerando, and a ritardando of a length much longer than any of the other accelerandos or ritardandos. The only phrase containing internal rests, this last phrase has a type of articulation slightly different than the previous phrases and it "haltingly" and "reluctantly" winds down the passage providing a sense of completion to the section. (The adverbs are those of Spies.)
Also, there is a symmetrical relationship between the rests that separate the phrases. The longest rest occurs at the beginning (m.7) and the second longest occurs at the end (mm.17-18). The shorter rests occur between the middle phrases. The entire monodic passage seems to be balanced around the solo violin statement of m.14. This septuplet is the fastest rhythm of the section and the only fragment played by a solo instrument. The constantly changing color doublings of all but m.14 are partly due to considerations of the ranges of each instrument, but more importantly, to a concern for maintaining a subtle variety of articulations and intensities for each individual note.
As attention is given to various other sections of the Variations, it is important to be remember the successive squares that the row-forms of the monodic passage came from. This technique is used repeatedly throughout the work.
Two Contrasting Phrases
The beginning of the two contrasting phrases that form Section VI (mm. 59-72) is shown in Excerpt 2. As the monodic section used rotationally-related row-forms beginning or ending on pitch-class 2 (D), which is the first note of the original row, the first six measures of Section VI use row-forms based on pitch-class 5 (F), which is the last note of the original row. Here, the three voices successively enter on F-final row-forms (though four of the six are presented as retrogrades causing F to be first) and move through the successive order-position squares S0, S11, S10, S9, S8, and S7.
Voice-1 begins playing S0, then voice-2 plays S11, and voice-3 plays S10; voice-1 plays S9, voice-2 plays S8, and voice-3 plays S7. (Voice-3 actually initiates S7 slightly before voice-2 initiates S8.) Voice-1 makes use of an implied elision and voice-2 utilizes an actual elision.
As shown in Excerpt 3, Section X consists of a rhythmic "fugato" in three voices (mm.101-117). Each voice of this section begins with the same rhythmic subject, though the row-forms of each voice are different.
Voice-1 uses prime forms that end with pitch-class "5" (or retrograde forms that begin with "5") and from the successive Sps: S7, S8, S9, S10, S11, and S0.
Voice-2 uses inverted forms that begin with pitch-class "2" (or retrograde inversion forms that end with "2") and from the successive Sps: S5, S4, S3, S2, S1, and S0.
Voice-3 uses inverted forms that end with pitch-class "5" (or retrograde inversion forms that begin with "5") and from the successive Sps: S7, S8, S9, S10, S11, and S0.
Of interest is a peculiarity contained in m.116. The viola's final F-sharp is not part of the viola's last row-form. Also, the double-bass had mostly been doubling the cello off and on; here it has as its penultimate note a pitch-class differing from that of the cello. The viola's final pitch-class 6 (F-sharp) and the bass' pitch-class 11 (B), and the cello's pitch-class 1 (C-sharp), all on the last sixteenth note of beat-2 of m.116, form a "vertical" out of a three-note segment near the end of the cello's final row. (RT8S0: 5, 4, 3, 7, 9, 2, 0, 6, 11, 1, 10, 8).
All three of the 12-part polyphonic sections (Sections III, V, and XI), though orchestrated slightly differently each time, contains two constants. The first constant is that there are twelve independent rhythms; each is assigned to one of the twelve voices in all three sections. In each section the pitch structure is based on different row-forms, but the pitch-classes are assigned to the same twelve rhythmic lines. The rhythmic lines of Section III will be labeled "1" through "12," and Sections V and XI are cross-referenced by Claudio Spies with Section III. This is shown in Example 16. See Example 16.
In Section V, only four rhythmic lines are "out of place" (voices 1, 4, 11, and 12 have been interchanged). The instrumentation of ten solo violas and two solo double basses is a darker version of, but similar to, the twelve solo violins of Section III.Footnote 2 In Section VI, there is a more decisive change. The ordering is "upside down" with rhythmic lines 1 and 12 interchanged, and 5 and 3 slightly out of place. The instrumentation now consists of wind instruments.
The rhythmic structure, then, of all three 12-part polyphonic sections moves toward an inverted ordering, while the orchestration seems to grow heavier. These three sections provide a successive "sagging" quality to the form as a whole.
The second constant in all three 12-part polyphonic sections is the use of row-forms that either begin with pitch-class 2 or end with pitch-class 5 (the first and last notes of the original row). The retrogrades of these rows are also used. May the idea be put forth that the structural emphasis of these two pitch-classes adds a slightly "pitch-centric" aspect to the 12-tone fabric. Now it can be seen, at least, that 2-initial and 5-final row-forms permeate the entire composition. Also, the various types of elisions mentioned earlier abound throughout these three 12-part polyphonic sections. Excerpt 4 provides the first of the 12-part polyphonic sections (section III) as well as a row analysis of the section.
All rows either begin or end with pitch-classes 2 or 5 (the first and last notes of T2S0). Bold-face type is used in the following to reveal when these pitch-classes are used to begin or end a row-form. The row-forms used within this passage are listed first below. Implied elisions are indicated by the parenthesis enclosing the implied row-form. Actual elisions are indicated by underlined pitch-classes.
All told, 6 Primes (from squares 7,8,9,10,11,0), 6 Retrogrades (from squares 7,8,9,10,11,0), 13 Inversions (from squares 0,0,0,1,2,3,4,5,7,8,9,10,11), and 12 Retrograde Inversions (from squares 0,0,1,2,3,4,5,7,8,9,10,11) are used within the passage. These 37 row-forms come from all of the squares except for S6. Violin-1 and Violin-8 employ four row-forms, while Violin-11 only employs two; the other nine parts each employ three. T8IS0 is the only row-form used twice.
For the most part, the melodic material of the Variations has been based on rotationally-related row-forms beginning on D or ending with F (the first and last notes of the original 12-note row). However, Stravinsky generates some of the harmonic material, especially in the last chordal section, from rotations of the individual hexachords. These hexachordal rotations form their own 6x6x6 cubes and are shown in Example 17. See Example 17.
The "diagonals" of these 6x6x6 cubes are then used as vertical collections of pitches throughout the final section. The diagonals of hexachord-1 are labeled A, B, C, D, E, and F, while the diagonals of hexachord-2 are labeled A', B', C', D', E', and F'. It should be pointed out that diagonals A and A', constituting the primary diagonal of each 6x6x6 cube, are simply unisons. The other "diagonal chords" contain doublings, and it is of great interest that the final six chords of the composition (mm.137-141) actually use the exact doublings of the diagonals. The final G-sharp of the bass clarinet, however, represents the complete diagonal, A', (8, 8, 8, 8, 8, 8). This progression of chords cycles through the diagonal chords exactly in a backward ordering: F', E', D', C', B', and the final A'.
Now, upon reflection, it becomes obvious that the contents of a diagonal change, depending on which pitch-class is used as the primary upper-left to lower-right diagonal. Diagonals are not based on serially derived operations. They are simply graphic and depend on how the cube is drawn. One cannot find transpositions or inversions of a diagonal within the same cube. So, to transpose a diagonal, one would need to redraw the entire cube, and this is clearly impractical.
The mathematically proper way of dealing with hexachordal diagonals, then, is to speak of (1) invariant order-positions, modulo hexachord, (2) relative to a common pitch-class, and (3) contained within successive order-position squares. For example, to find diagonal C' (based on the 6x6x6 cube made from hexachord-2, and having pitch-classes 3, 2, 11, 6, 10, 4) within the main 12x12x12 cube, one must proceed in the manner of Example 18. See Example 18.
The final chords in Stravinsky's Variations are, relative to T8S6-11, modulo hexachord-2, in terms of their order-positions:
The order positions remain invariant regardless of how the cube (or square) is drawn, and are based, therefore, on serially definable operations. Of course, it must be remembered that Stravinsky himself did construct charts with "diagonals" in mind. Indeed, the cube of Example 3 has a primary diagonal of pitch-class 2, one of the two structural pitch-classes (D and F) of the Variations. Let it be said, then that though the diagonals are more "intuitive" than are invariant order positions of successive Sp squares relative to a fixed pitch-class, the latter are more mathematically sound. Excerpt 5 shows the "diagonal chords" that Stravinsky used to conclude his Variations In Memoriam Aldous Huxley.
In conclusion, the cube is a convenient way of representing all possible rotations of the classic "48 forms" of a row. The basic concept of the cube could apply to most pieces that use rotation. Possibilities not exemplified in Stravinsky's Variations are abundant. For example, one may wish to consider potential combinatorial relationships between rotations of dissimilar order position, a possibility not compatible with the row used within the Variations. Another possibility would be to use the diagonal relationships of cubes generated from many, different-sized, arbitrary subsets of any of the 576 rotationally-related row-forms of some 12-tone row.
It is hoped that a consistent terminology, when discussing compositions that use rotation, will be of benefit to those who wish to study such pieces in the future.
1However, repeated pitches occur at one other exceptional spot. In Excerpt 1, at m.10, there is found the most curious "vertical" on the penultimate sixteenth note of m.10. This is followed in m.11 by a repetition of pitch-classes 10 and 9 (A-sharp and A-natural). This constitutes the only repetition of an interior segment of a row in the entire monodic passage.Return to main text
2Claudio Spies quite inexplicably says that Section V consists of four solo violins and six solo violas, as well as the two basses. The score actually specifies ten solo violas and two basses; however, four of the viola parts are notated in treble clef.Return to main text
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