Timothy J. Buell
A Note on the Compositional Background
When Stravinsky, speaking of Movements (1958), announced that he had "discovered new (to me) serial combinations," it is likely that he was referring to his unique system of hexachordal transposition-rotation. Figure 1 is a reproduction of a page from Stravinsky's sketchbook for Movements. Several of the compositional procedures which pervade the late twelve-tone works are evidenced in this sketch. The physical layout of the sketch, and the inscribed hexachordal blocks, indicate Stravinsky's predilection for regarding the twelve-note series in terms of its discrete hexachordal components. The "construction of twelve verticals,"  which Stravinsky stated he employed in the fifth movement, can also be seen. The music for "4 v.c. soli," derived from this vertical alignment of the row forms, appears at m. 170 of Movements. Finally, Stravinsky's comment that "five orders are rotated instead of four,  "refers to his addition of a fifth form of the twelve-note row to the conventional four orders. This fifth form, which Stravinsky has clearly labelled "RIV-INV," or the inversion of the retrograde inverted, is a melodic inversion around the first pitch of the RI form. It is apparent that this so-called IRI form duplicates a transposition of RI - in the case of the series in Movements the IRI form is a duplication of RI4. However, as Spies points out, the distinction between the two set forms should be made because Stravinsky consistently regarded the IRI form as a separate, untransposed form, often in preference to the RI form. This IRI form represents one of Stravinsky's more idiosyncratic twelve-tone procedures, and this form of the set figures prominently in all of Stravinsky's later works, from Movements through to the Requiem Canticles.
Stravinsky's tendency to regard the twelve-note set in terms of distinct hexachordal components, in conjunction with the separate treatment of each hexachord in the application of transposition-rotation, pervades the compositional design of the majority of his twelve-tone works. Although hexachordal transposition-rotation makes an appearance as early as Movements, it does not emerge as the primary compositional procedure until The Flood (1962). In earlier works, such as Movements and A Sermon, A Narrative and a Prayer, hexachordal transposition-rotation appears only tentatively, and the procedure is clearly subservient to a more traditional twelve-tone system. Following The Flood, the exclusive use of transposition-rotation was to continue in Abraham and Isaac (1963), and the Requiem Canticles (1966). The unique exception is the Variations (1964), in which the entire twelve-note set is treated as the rotated unit.In all of the other late twelve-tone works it is the hexachordal component of the twelve-note row which is the rotated unit. Of these works, perhaps none is as exclusively hexachordal in design as Abraham and Isaac. The pre-eminence of the hexachord was acknowledged by Stravinsky when commenting on the work: "A twelve-note series is employed, but hexachordal and small units are stressed rather than full orders.
Figure 1: Page from Stravinsky's sketchbook for Movements (1958) 
Figure 2: Rotations of the five set-forms of Abraham and Isaac
Figure 3: Abraham and Isaac, mm. 17-36
Copyright 1965 by Boosey & Hawkes Publications Ltd. All rights reserved. Reprinted by permission of Boosey & Hawkes, Inc.
Figure 4: Abraham and Isaac, mm. 41-47
Copyright 1965 by Boosey & Hawkes Publications Ltd. All rights reserved. Reprinted by permission of Boosey & Hawkes, Inc.
Figure 5: Abraham and Isaac, mm. 220-239
Copyright 1965 by Boosey & Hawkes Publications Ltd. All rights reserved. Reprinted by permission of Boosey & Hawkes, Inc.
Stravinsky's transposition-rotation procedure in Abraham and Isaac can be briefly introduced as follows: taking as its unit the hexachordal segment of the twelve-note row, the operation of rotation permutes the interval order of the hexachord (in the manner indicated by the diagonal lines in Figure 2), while maintaining the first pitch of the original, untransposed hexachord as the common tone initiating each rotation. And so the first rotation begins with the original hexachord's second interval (though beginning on the common tone, that is, the initial pitch-class of the original, untransposed hexachord) and ends with the original hexachord's first interval. The second rotation begins with the original hexachord's third interval (again beginning with that common pitch-class), and ends with its second interval. The cycle concludes with the fifth rotation, which begins with the last interval of the original, and ends with the first. Stravinsky's preferred procedure is to treat a given "block" of hexachordal rotations as a single pitch-complex, reading either from left to right or right to left, as indicated by the arrows in Figure 3.
The hexachordal statements are predominantly linear in design, a compositional technique habitual to Stravinsky throughout the twelve-tone works. Linear separation of hexachordal statements is most apparent in the distinction between the baritone and the accompaniment: seldom do they share in a common hexachordal unfolding. Moreover, hexachordal statements (as illustrated in Figure 3) are usually confined to the particular instrument or combination of instruments which initiated them. For example, the accompaniment at mm. 41-46 consists of a shared partial articulation of Ib/2, Ib/3 and Ia/3 by the horn and tenor trombone (see Figure 4). At the same time, the baritone is independently articulating hexachords RIb/2. Ib/4, Ia/0 and Ib/0. The new series of hexachordal statements beginning at m.47 enjoins a change in instrumentation: Sb/2 - oboe; Sb/5 - shared by clarinet and bass clarinet; IRIa/0 - first violins, later to be joined by the second violins and violas.
Hexachordal transposition-rotation creates a second species of hexachord, obtained through a vertical reading of the hexachordal block. These "verticals" as they were called by Stravinsky, are formed by taking a single order-element of the hexachordal block of five rotations plus the original (see Figure 2). One of the most striking deployments of these verticals occurs in the chords beginning at m.229, where the hexachordal verticals associated with Ib are stated. Observe the independence of the baritone's line, separately articulating complete statements of the hexachords of RI and I (Figure 5).
Although Abraham and Isaac is based on a twelve-note row, it is, as we are discovering, almost exclusively hexachordal in design. The pre-eminence of the hexachord as the primary compositional element, in conjunction with Stravinsky's stated preference for "hexachordal and smaller units" of the complete set, makes the inference of the twelve-note set somewhat problematic. Spies cites the opening four measures, played by the violas and one bassoon, as the only occurrence of the ordered twelve-note set in the entire work (see Figure 6 below.)
Figure 6: Abraham and Isaac, m. 1-5
This simple priority would appear to be justification enough for the labelling of this particular twelve-note row as the basic set, for, according to Spies, the appearance of this row at the outset "bespeaks Stravinsky's aversion to obscurantism for its own, or for any other, sake. "Yet the independence of the hexachordal unit in Abraham and Isaac has been acknowledged; moreover - as we will see - pitch order within the rotated hexachords is frequently unstable, giving way in some instances to reordered or incomplete statements of a rotation. And so the analytic usefulness of placing a priority on a specific ordering of a particular set of twelve pitch-classes is rather limited, since, in Stravinsky's transposition-rotation universe, the very operation of rotation ensures that the interval created between the 6th and 1st order-elements of any given pair of hexachords will be in a constant state of flux. Further, since Stravinsky treats each hexachord independently (that is, the operation of rotation is performed separately on each hexachordal component of the twelve-note set) the interval between order-elements 0 and 6 of each of the five transposed-rotated versions of the twelve-note set will be different for each rotation. The result is an intervallic spanning of the hexachordal divider which is different for each rotation from that of the basic set (see Figure 2). Since - as we will see - the rotated versions of the row appear as frequently as the original forms, no single ordering of a twelve-note row attains any degree of priority, making it somewhat arbitrary to label the initial twelve-pitch classes as the basic set simply by virtue of their initial appearance. Analytically, the label has limited usefulness since that ordered twelve-note row never again appears as a referential entity.
Milton Babbitt has described the structural function of the set as follows:
May it not be then that the hierarchically derived concept of a set can replace the structural differentiation provided by tonal harmonic "structure" in that the twelve pitch-class set, like a "tonic," is distinguished from other aggregates not simply by internal content or even interval relations (since any aggregate, in itself, is compositionally as definitely ordered in time as well as in space as what is adjudged to be a set need be) just as a "tonic" is determined not sufficiently by its triadic structure (or representation thereof) but by its relation to other such structures within and over a composition?" 
Babbit's assertion that the twelve-note set can be distinguished from other aggregates is surely dependant upon one's hearing of the set as referential, in order that the set may be heard in relation to "other such structures within and over a composition." In a work so exclusively hexachordal as Abraham and Issac, it is the hexachord, and not the twelve-note statement, which is audibly referential. In fact, it would appear that the twelve-note set functions in this music more on a pre-compositional level. And if this is the case, then one must look beyond simple priority in determining the basic set.
Speaking of Abraham and Isaac, Stravinsky's admission that "key gravitations will be found to exist" was doubtless an acknowledgment of the two pitch-classes which assume a high degree of centricity throughout: F and C#. The baritone's part is essential to the casting of F and C# in their centric roles. Of the baritone's nineteen verses, sixteen either begin or end with F or with C#, and in many cases these instances are emphasized by pitch duplications in the accompaniment. The opening vocal phrase, for example (mm.12-22), begins on F (doubled an octave lower by the trombone) and ends on F. In the rare instances where a verse does not begin (or end) with F or C# their influence is, nonetheless, pervasive. Verse 2, for example, begins with a B in the baritone's part (m.26), however, C# emerges as the centric pitch in the following measure: the C# is starkly isolated as it is sung without any accompaniment, it is emphasized as the baritone's longest-held note and highlighted registrally as the highest note in the phrase, preceded by a melodic leap of a seventh. The C# is immediately echoed in m. 28 by the bass trombone two octaves below; the baritone simultaneously sings an F.
Similarly, although the third verse (m.52) does not begin with F or C# in the baritone's part, C# is registrally emphasized as the highest note in the baritone's line, and it is reiterated as the highest pitch in the bassoon's accompanying ostinato-like figure (mm.53-55). Three of the work's canons begin on F: the first canon at m.56 (F played by the bassoon, tuba and flute): the second canon at m.73 (F sung by the baritone and played by the bassoon): and the fifth canon at m.197 (F sung by the baritone and played by the horn). Of the other two canons, canon 4 (m.163-166, between tuba and flute) begins on C# and canon 3 (mm.129-135, between baritone and trumpet) is accompanied by a C#-A ostinato in the tuba. The trumpet's final pitch in the canon is C# (m.135), as is the baritone's accented penultimate pitch (C# mm.133-134). The centricity of F and C# is manifested throughout the work by frequent octave doublings, pitch duplications, and repetitions. See for example, the unison C#s in the orchestra at m.205, followed by the baritone's repeated C#s in the orchestra at m.205, followed by the baritone's repeated C#s at the beginning of verse 15 (m. 207). Of these two centric pitch-classes, F will emerge as the more prominent, most particularly by virtue of its pervasive function as a point of initiating and ending phrases. Indeed, throughout the work, F and C# often appear in a cadence-like fashion. Already noted is the frequent consecutive articulation of these pitches as the final note of one verse and the initial note of the next. The orchestra's last three notes are F (E#) D# and C#. In the fourth canon, beginning at m.163, the tuba's initial entry is on C# simultaneous with the baritone's F. The flute follows with an inversion in diminution, beginning on C#. The vocal line of the section beginning at the Meno mosso at m. 105 concludes with a cadential F (E#-C#) at m.111, punctuated with a sfz chord in the strings.
Stravinsky's underlining of the centricity of F and C# often results in the re-ordering of a hexachord. While complete hexachordal statements occur frequently enough, this is by no means the rule, and hexachordal unfoldings are often re-ordered or truncated precisely in order to ensure the centricity of a particular pitch-class. And though the statements of hexachordal rotations are predominately linear, the resulting vertical sonorities are by no means simply fortuitous. At m.22, for example, the concomitant statements of Sa/3 by the baritone and RIb/0 by the English horn and bassoon both end with the octave duplication of the centric pitch-class F; the ensuing instrumental statement of Ib/0 begins with its second order-element - C#- instead of the first, effecting an F-C#-F progression (see Figure 3). A similar hexachordal re-ordering occurs at m.117, where the clarinet's statement of Ib/0 (T5) is re-ordered to (5,6,1,2,3,4), so that the initial pitch will duplicate the flute's C# two octaves above (see Figure 7 below).
Figure 7: Abraham and Isaac, mm. 115-118
Hexachordal re-orderings to emphasize pitch-class centricity are often achieved at the expense of complementation. Hexachordal complementation occurs frequently enough; see, for example, the consecutive statements of Ia/0 - Ib/0 at mm. 43-45 by the baritone (Figure 4). But hexachordal complementation does by no means predominate, and articulations of the aggregate are often made subservient to other compositional needs. In Figure 4, for example, observe the linear, but not the vertical integrity of the aggregate, as several of the baritone's pitches are duplicated in the accompaniment. The result in this instance is a profusion of the "fifths and doubled intervals" spoken of by Stravinsky with regard to Abraham and Isaac:  observe the fifths created between the brass accompaniment and the baritone in mm.42-45.
If one follows the course of pitch-class F and C# throughout Abraham and Isaac, it becomes apparent that these centric pitches are associated in the majority of their appearances with the particular hexachordal blocks created by the unfolding of the two hexachords (and their rotations) designated as Sa and Sb in Figure 2. Articulations of these two hexachordal regions are most frequently associated with the baritone part. The solo baritone, or "cantor-baritone," as Stravinsky referred to the role, is the most highly visible element in Abraham and Isaac. Except for a few brief instrumental interludes, the baritone is singing throughout; all of the dramatic roles are entrusted to this single performer. As documented by Spies,  Stravinsky took a great deal of care over the setting of the Hebrew text. Although it was the physical sound of the Hebrew words which was initially attractive to Stravinsky, the subsequent attention he paid to matters of scansion and syntax - which amounted virtually to learning Hebrew at the age of eighty-two - indicates the intensity of Stravinsky's very personal response to the text. The baritone is never upstaged by orchestral sound or texture; seldom is he accompanied by more than four or five instruments, and there are many brief solo passages for the baritone.
Table 1: Summary of Hexachords and Rotated Hexachords from Hexachordal Blocks
Table 1, above, reveals several important features with regard to hexachordal usage in Abraham and Isaac. Perhaps most significant is the overwhelming preponderance of statements of untransposed hexachords culled from the Sa and Sb hexahcordal blocks, with the balance slightly in favor of Sb. This preponderance holds even when the total statements of both transposed and untransposed hexachordal statements from the hexachordal blocks are compared. Though articulations of I hexachords gain some weight, there are still about twice as many statements of the S hexachords. There is a total of 170 statements of untransposed hexachords culled from the hexachordal blocks of all five set-forms, while there are only 54 statements of transposed forms of these hexachords. This is in keeping with Stravinsky's habitual practice of treating the untransposed form of the row as the norm, while transpositions are the exception. Van den Toorn provides this description of the procedure:
The four untransposed orders of the set - Prime (P) and its standard affiliates, Retrograde (R), Inversion (I) and Retrograde Inversion (RI) - are apt to acquire a privileged status, if only by virtue of their predominance, and may therefore assume the character of a "home base, "with respect to which transpositions are perceived as departure or auxiliaries. 
Table 1 reveals that the two component hexachords of S - Sa and Sb - are almost never stated in a transposed form. The reason appears obvious: hexachords Sa and Sb begin, respectively, with the centric pitch-classes F and C#. Most telling is the manner in which transpositions of the hexachords are selected. Almost without exception, transposition levels are chosen in order that one or both of the centric pitch-classes F and C# will appear at an opportune moment, most frequently at the beginning or end of a phrase. As occurrences of this procedure are too numerous to cite in full, the following notable example will perhaps suffice:
- The selection of RIb3 (T2) at m. 171, in order that God's phrase "Do not lay they hand upon they boy" should begin on C#. Observe also the change in dramatic character, achieved through dynamic contrast, from p (narrator) to f (God). See Figure 8.
Figure 8: Abraham and Isaac, mm. 169-172
-The selection of Ib/5(t4) in the baritone's solo at m. 109 provides a common-tone linkage (C#) with the previous hexachord. In the next measure, the selection of RIa/0(t4) for the baritone's music effects a cadential E-E#-C# at the end of the verse (see Figure 9, below):
Figure 9: Abraham and Isaac, mm. 108-111
The baritone solo, in its initial entry at m.12, articulates the first occurrence of the "home base," hexachords Sa and Sb, with a reordering of the order-elements of Sb to (1,6,5,4,3,2). Stravinsky often plays with the initial order-elements of a hexachord, this sort of "doubling back" following the statement of the first order-element is one of the more common hexachordal re-ordering games played by Stravinsky. It is this form of the set, then, which I have designated as the "basic set," or" S," for the following reasons:
1) Unfoldings of the component hexachords (and their associated rotations) of this form of the set are preponderant throughout Abraham and Isaac.
2) The highly prominent solo baritone is most frequently associated with articulations of S (of which the aforementioned opening phrase is but the first of many occurrences).
3) The two hexachordal components of S, along with each of their associated rotations begin, respectively, with the centric pitch-classes F and C#.
In this reading, therefore, the opening twelve measures, instead of being designated as "S," become a statement of "RI" (see Figure 10); accordingly, the set form which Spies labels as the basic set becomes the RI form in the chart in Figure 2. The introductory twelve measures will thus articulate RI, followed by I at m.5, ushering in the initial statement of the basic set by the baritone. Since the final pitch-class of RI is F, common-tone linkage is established between the last note of the instrumental introduction and the first note of the baritone solo. Contextually, then, there is a solid basis for designating the set articulated by the baritone's opening statement as the basic set. The mechanics of Stravinsky's scheme of transposition-rotation ensure that each hexachordal rotation of Sa begins with F; each Sb rotation begins with C# Further, Stravinsky can alter the linear direction of the statements of the rotated hexachords (i.e. reading the rotations of the hexachordal blocks in Figure 2 from left to right, or from right to left), with the result that phrases will frequently begin or end with these centric pitch-classes. The first appearance of S in the baritone is also the first instance of the unfolding of an entire hexachordal block.
Having duly noted the difficulty inherent in the inference of the twelve-note row in a work so exclusively hexachordal, the following characteristics exhibited by our "S" should be mentioned:
- the second trichord of Sa appears in retrograde (at T2) as the first trichord of Sb.
- the set inverts at I2 into the retrogrades of the five-note cells contained within each hexachord (order-elements 5 and 6 exchange their positions on either side of the hexachordal divider) See Figure 10. Since Abraham and Isaac is so resolutely hexachordal in design, this second feature, which in order to be audible would require complete statements of the twelve-note row, does not emerge as a compositional determinant.
The set is non-combinatorial. The two component hexachords are, however, Z-related. There is little doubt that Stravinsky was keenly aware of the intervallic equivalence of the hexachords' contents, for, as it will be seen, it is the interval content of the hexachords, and not any single linear ordering of the interval succession of a particular hexachord, which is at the crux of Stravinsky's transposition-rotation system.
Figure 10: Abraham and Isaac, Row Properties
The use of the dual criteria of hexachordal preponderance and F-C# centricity in determining which form of the row should be designated as "S" rests upon the assumption that Stravinsky did indeed intend that this form of the set be heard as a kind of referential "home base." And, since the rotations of Sa and Sb figure just as prominently as the unrotated hexachords, the audibility of Sa and Sb as referential entities must extend to their rotations as well. In other words, the Sa and Sb hexachordal blocks must be audible as referential hexachordal regions. What follows in a summary of the methods by which Stravinsky has endowed the Sa and Sb hexachordal regions with just such a referential quality.
The highly prominent baritone part is invariably associated with complete unfoldings of the Sa and Sb hexachordal regions. Of the 89 untransposed statements of hexachords from the Sa and Sb hexachordal blocks (see table 1) the solo baritone sings 42 of them. Moreover, of the total of 107 hexachordal statements sung by the baritone, statements of hexachords culled from the Sa and Sb regions account for by the largest proportion (see table 2 below.
Table 2: Derivation of Hexachordal Statements of Baritone (Transposed and Untransposed).
A further expression of the pre-eminence of Sa and Sb is the fact that complete traversals of the hexachordal block, in the manner indicated in Figure 3, are exclusively the domain of the baritone, and are invariably associated with complete unfoldings of the Sa and Sb hexachordal blocks. The single exception is the articulation of the IRIb/0 hexachordal block at mm.65-72 (see Figure 1). Yet even this passage is framed by complete statements of the Sb hexachordal block on the one side (mm.56-64) and by Sa/0 on the other (beginning of verse 4: baritone m.73) Moreover, Ia0 and IRIb0 form the accompaniment for both the Sa and Sb unfoldings by the baritone. The appearance of IRIb0 in the flute at m.62, which carries through into the baritone's ensuing traversal of IRIb, functions as a kind of hexachordal pivot. All this endows the entire passage with a kind of tonic/modulatory section/tonic feeling, very much in keeping with the notion of the Sa and Sb hexachordal regions as a "home base". Further, complete statements of a hexachordal block are always coincident with the enunciation of a complete textual unit: in many instances an entire verse is set to the unfolding of a complete chain of hexachordal rotations. In the passage just under discussion, for example, the baritone's music for verse 4 (m.73) is set to a complete unfolding of the Sa hexachordal rotations, moving down the hexachordal "ladder" from Sa0 to Sa5, with interval ordering within each hexachord moving left-to-right or right-to-left, according to the arrows in Figure 11. Parenthetically, the aforementioned low priority given to hexachordal complementation is again evidenced here: Sa1 duplicates three pitches from Sa0. The sparse accompaniment, provided by a single bassoon and violin which independently state isolated hexachordal rotations, similarly duplicate several pitches.
Stravinsky's transposition-rotation scheme ensures that each of the five rotations of the hexachord will result in a distinct transposition of that hexachord. The mechanics of this procedure are described by Babbitt in this manner:
Each such collection (i.e. a hexachordal "block") is composed of six hexachords, derived by successive order transposition (rotation) and pitch transposition (by the transpositions number equal to the mod. 12 complement of the pitch-class number of the element which, as the result of the rotation, occupies the initial order position in the so-derived hexachord) of the elements of - respectively - the first hexachord of the set, its inversion, the final hexachord of the set, its inversion. 
Table 3: Comparison of transposition levels obtained from the rotation of hexachord Sa with the twelve possible transpositions.
Figure 11: Abraham and Isaac, mm. 56-82
This is simply to say that Stravinsky wanted a common pitch-class to initiate each rotated hexachord. This procedure ensures that the six hexachords (the original plus the five rotated hexachords) represent all the possible transpositions of that hexachord in which that particular initiating pitch-class occurs as a common element (see table 3 above). The resultant transpositions are therefore re-ordered, since the interval order of the original hexachord is permuted in each rotation.
Although interval-order is permuted, there remains that common pitch-class which initiates each rotation. As can be observed in table 3, no other pitch class will occur more than three times in the series of six hexachords which comprise the hexachordal block. And so, it would appear that issues of pitch-class centricity are very much at the crux of transposition-rotation, since the traversal of a hexachordal block ensures the articulation of all available transpositions of the hexachord in which the initial order-element occurs as a common pitch-class.
Since, in terms of Abraham and Issac, the complete traversal of a hexachordal block is almost exclusively confined to articulations of Sa and Sb hexachordal regions, the pitch-class centricity enforced by transposition-rotation is specifically that of F and C#. Most telling is the fact that of all the unfolding of the pervasive Sa and Sb hexachords and their rotations, transposed statements of these hexachords appear in only five instances (see table 1), and these are confined to isolated hexachordal statements (as opposed to the complete unfolding of a hexachordal block). Never is there a complete statementof a hexahcordal block in a transposed form. See, for example, the baritone's statements of Sb5(t3) at m.201 (Figure 12). Yet even in this instance Stravinsky is intent upon avoidng any kind of ambiguity in terms of our perception of the untransposed Sa and Sb hexachordal blocks as centric regions: this statement of Sb5(T3) is preceded by the baritone's statement of Sa0; untransposed hexachords also appear prominently in the accompaniment (Sa3 in the horn, and Sa0 and Sa2 in the tuba); finally, the entire passage is framed by statements of Sb verticals.
Along with the wealth of hexachordal variety and independence already encountered, Abraham and Isaac seems to exhibit yet another perspective from which Stravinsky has viewed the hexachordal block: that of a "diagonal" reading. This is certainly one of the more fascinating aspects of the hexachordal apparatus in Abraham and Isaac; and the procedure may well be unique to this work. It has not been remarked upon in other commentaries on the Variations, The Flood or the Requiem Canticles, though I suspect that this procedure may very well account for what Spies has called the "arbitrary, unrelatable fragments" in his commentary on the Requiem Canticles. Measures 184-194 (see Figure 13) exhibit a diagonal reading of the hexachordal block (henceforth abbreviated as a "Δ" rotation).
Figure 12: Abraham and Isaac, m. 199-206
While the pitch material in the accompaniment can be seen to have been culled from various Sa and Sb hexachordal rotations, much of the vocal line cannot be accounted for so readily. While the (F-C#-D-C-A#-B) hexachord at m.184 ("And he lifted Abraham") can be labeled as a statement of Rb0(T6), with order positions permuted to (2,3,4,5,6,1), or as an untransposed statement of either Sa5 or Rb5 (again, with order permutations); many of the ensuing pitches in the vocal line cannot be culled from complete or even partial linear statements of any of the ten hexachordal blocks. For example, the baritone's final three pitches in this section, F#-G-C (m.194), cannot be accounted for in this particular ordering as partial linear statements - untransposed or transposed - of any single hexachord in the system. Further, the consistent association of textual phrases or entire verses with complete unfoldings of chains of hexachordal rotations, or at least subsets culled from a particular hexachordal block's pitch-complex, has been seen to be an essential compositional determinant in Abraham and Isaac. And indeed, the final phrase of the passage under discussion: ("...and brought him up for a burnt offering...") does articulate a hexachord: (6,5 10,1,2,7), with F=0. In normal order this hexachord can be expressed as (0,1,2,5,6,9), the second trichord of which is not contained in either of the two Z-related hexachords - (1,2,3,4,5,6) and (0,1,2,3,4,7), with F=0. In normal order this hexachord can be expressed as (0,1,2,5,6,9), the second trichord of which is not contained in either of the two Z-related hexachords - (0,12,3,5,6) and (0,1,2,3,4,7) - which constitute the twelve-note set of Abraham and Isaac. But the hexachord (6,5,10,1,2,7) does occur, in this ordering, as the "Δ" rotation of RI, with the constituent pitch-classes being designated as follows:
RIb: (0-1, 1-2, 2-3, 3-4, 4-5, 5-6)
- Where the first number in each of the pairs represents the rotation,(0 being the unrotated hexachord), and the second number represents the pitch's order position within the rotation. For example, in Δ RIb, above, the "2-3" points to the D# in the second rotation, the third order element in that rotation.
Further, the (0,8,9,7,5,6) hexachord, which began this passage (m.184), articulates the other Δ rotations of RI (see Figure 13 and also table 4). But what exactly are the musical results obtained by Stravinsky when the rotation of a hexachordal block is articulated linearly? In the case of Δ RIb (0,8,9,7,5,6) - the "upper-right-to-lower-left" Δ rotation (see Figure 13) - the Δ rotation charts the progress of the initial order-element, in this case pitch-class F, through the various transposition levels arising from the rotations. Thus in the case RIb, the final element in the unrotated hexachord, RIb0, is F, (again, note the pervasiveness of this centric pitch-class), which is the initial pitch of Δ RIb (0,8,9,7,5,6). The Δ rotation is therefore a linear statement of all the transposition levels associated with a particular hexachordal block, or, in other words, a statement of the intervals necessary to transpose each order-element of a given unrotated hexachord to its initial order-element. In terms of hexachord RIb, this operation can be expressed by the following pitch-class sum:
6 10 9 11 1 0
+ 0 8 9 7 5 6
6 6 6 6 6 6
rotations also represent the unfolding of consecutive order-numbers while moving down through the hexachordal block, with the result that each order-number of a Δ rotation points to a discrete rotation.
The left-to-right Δ rotation demonstrates some different characteristics. Most significantly, this Δ rotation represents the transposition of subsets of the hexachord by the transposition levels associated with each rotation complex. In terms of RIb, for example, Δ RIb (6,5,10,1,2,7) represents the transposition of subset trichord (6,9,1) or RIb (6,10,9,11,1,0) by the six transposition levels associated with RIb (see Table 4).
6 9 1 6 9 1
+ 0 8 9 1 2 7
6 5 10 1 2 7
Sb verticals: (1) [ ] unison Fs. (2) [0,1,4,6] (3) [0,1,3,6,8]
(4) [0,2,6,8] (5) [0,1,3,,6,8] (6) [0,1,4,6]
Table 4: Rotations of the five set forms expressed in pitch-class notation.
The transposition level resulting from each hexachordal rotation is also indicated (Pitch-class F=0)
Figure 13: Abraham and Isaac, m. 182-189
The next Δ rotation in this series will be RI (0-2, 1-3, 2-4, 3-5, 4-6, 1-1), with the final order-element in this hexachord being the "wrap-around." This represents the transpositions of the subset (10 11 0) of RIb by the six transposition levels:
10 11 0 10 11 0
+ 0 8 9 7 5 6
10 7 9 5 4 6
It will be observed that the two subsets employed as examples thus far have been culled from alternate order-elements of the hexachord) i.e. order-elements 1 3 5; 0 2 4; etc.). Transposition of subsets consisting of consecutive order-elements will yield segments of alternate verticals:
6 10 9 6 10 9
+ 0 8 9 7 5 6
6 6 6 1 2 3
10 9 11 10 8 11
+ 0 8 9 8 5 6
10 5 8 5 2 5
Measures 91-98 exhibit just such a deployment of partial verticals (see Figure 14 below); however, their derivation is still open to question. Are the incomplete statements of verticals the result of a compositional operation such as the one described above, or simply an intuitive decision on Stravinsky's part?
And so, to the more conventional perspectives from which the rotated hexachordal block can be viewed: the linear statements of hexachordal chains and the statement of the hexachordal verticals; we can now add a third way of "looking" at the hexachordal block - the diagonal reading or Δ rotation.
Assuming, then, that the baritone's music in mm.184-194 (see Figure 13) is bounded at both ends by statements of Δ RIb, how should we interpret the intervening pitches? Given Stravinsky's preferred procedure of staying within the confines of a particular hexachordal block until some, if not all, of the rotated hexachords have been articulated, it would appear justified to try to account for those pitches of the baritone's part framed on either side by statements of Δ RIb within that particular hexachordal region. The phrase at m.183 which follows the initial Δ RIb statement ("... his eyes, and saw behold...") articulates a (6 8 10 9 7 5) collection. In terms of RIb this points to order-positions (1-2-3) of RIb2 and order-positions (4-3-2) of RIb1, with the apparent truncation of the linear statement and the accompanying movement up or down the hexachordal ladder defined by the diagonal boundary formed by the initial Δ rotation (see Figure 13). Observe how the common pitch class 9 (the D natural) effects common-tone linkage up the hexachordal ladder. The baritone's ensuing 11-note pattern ("...a ram (was) behind (him) caught in a thicket by his horns...") weaves diagonally through the hexachordal block on the other side of the diagonal boundary (at pitch-classes 4,3,2,5,7,8), before arriving at the boundary at pitch-class 8, following which the melodic line "crosses" the diagonal boundary, finally ending with the ubiquitous (0,1,2) trichord, and the centric F. Although the next phrase ("...And (he) went Abraham and took the ram...") could be culled from further diagonal weavings within the RIb hexachordal block, the phrase can also be accounted for as Ra0, with permuted order-positions. Without access to Stravinsky's sketches or "charts" for Abraham and Isaac one can only point to the analytical possibilities that do exist in this situation, and assert that while the more straight-forward linear occurrence of this hexachord as Ra0 presents itself as the more obvious solution, Stravinsky's idiosyncratic manner of "looking" at the hexachordal block (and portions thereof) from the diagonal as well as the linear standpoint, combined with his displayed preference for treating a particular hexachordal block as a referential region during an extended linear statement, could well make a case for extracting the (C#-D#-D-C-A#-A) melodic statement from partial Δ rotations of RI. Moreover, the unambiguous statement of the hexachord (6,5,10,1,2,7) at the conclusion of verse 13 ("...and brought him up for a burnt offering in place of his son"), which states the other RI diagonal, clearly suggests that the vocal line is still very much within the RIb hexachordal region.
Figure 14: Abraham and Isaac, mm. 91-92
Consideration of the dramatic exigencies of the text at this point suggest another - though admittedly more tenuous - reason for Stravinsky's use of Δ rotations. Claudio Spies has pointed out several instances where musical punctuations serve as a general means for underlining the text. At mm.207-217 (the beginning of Section 9) the accompaniment immediately prior to the angel's speech consists of a single line shared by the horn and tuba (there is no overlapping or doubling), yet just after the angel's proclamation ("And he said...") the horn and tuba each play their independent pitches together, and so depict musically the dramatic change in narrative. A further example occurs at m.221, at Abraham's words "and multiply I will the seed: (see Figure 15). Note the unique appearance of the pizzicato quintuplet figure ("multiplication" of the sixteenths?), the repeated notes in the clarinet, and the displaced rhythmic accent in the vocal line. This passage is also notable for the unique appearance of shared verticals between baritone and accompaniment. Now, to return to mm.185-188 (Figure 13): are the circuitous diagonal unfoldings of the rotations, "caught" around the diagonal boundary, musically symbolic of the text?  As Spies suggests, the use of such terms as "word-painting" or "programmatic" invites "the risk of possible semantic vagueness;  "but there is little doubt that the subtle yet pervasive degree of musical expressionism endows Abraham and Isaac with much of its dramatic intensity. To this it may be added that in a work of the unique format of Abraham and Isaac, in which the dramatic roles of God, the angel of the Lord, Abraham and Isaac, and the narrator are all entrusted to a single performer, a means must found to replace the musico-dramatic interaction between performers that would have occurred naturally in a work with more than one performer. That Abraham and Isaac maintains the dramatic tension of the narrative despite the limited and compressed nature of the work demonstrates just how well Stravinsky was able to compensate for the lack of dramatic action with devices purely musical.
FIgure 15: Abraham and Isaac, mm.221-222
Finally, what effect does the presence of the Δ RIb hexachords have on our perception of centric Sa and Sb hexachordal regions as centric? Most telling is the manner in which the baritone's wandering into this remote hexachordal region (see Figure 13) is continually brought to earth by the pervasiveness of the Sa and Sb hexachordal regions. The entire passage is symmetrically framed by S verticals articulated by the orchestral accompaniment; the second series of verticals is a non-literal retrograde of the first: Sa verticals 1 through 6 from mm.182-184, and Sb verticals from mm.195-197. The symmetrical, arch-like deployment of the verticals is enhanced by the similar orchestration (flutes, low brass and strings) of both passages, while in contrast, the intervening vocal passage (with its Δ RIb wanderings) is accompanied only by a clarinet, bassoon and bass clarinet. Yet observe the pervasiveness of Sa in the accompaniment, which consists of successive entries of Sa rotations, which often overlap. Instrumental independence of hexachordal statements is preserved until the final three measures of the passage (m.192-195), when the only statement of Sa4 in the passage is shared by the clarinet and bass clarinet, below which the bassoon unfolds consecutive statements of Sa2 and Sa3. Observe how the order-positions in the initial statement of Sa2, played by the clarinet at m.184, permute order-positions (1-5; 2 3 4 5 6; 1-5), which not only reinforces the centricity of pitch-class F (and the association of the of F with Sa), but also establishes common-tone linkage with the ensuing entry of the bassoon, which also begins with that E-F semitone.
Stravinsky's well-known remarks in his Poetics of Music on "polarity" have elicited much discussion. Arthur Berger provides the following translation of the relevant passage: 
What preoccupies us, then, is less tonality, properly so called, than what might be described as the polarity of a sound, of an interval, or even of a sonic complex (complexe sonore).
In discussing the possible involvement of the octatonic scale in Petroushka, Berger has proposed, as a necessary condition of polarity, the "denial of priority to a single pitch-class precisely for the purpose of not deflecting from the priority of a whole complexe sonore." The pitch class universe created by hexachordal transposition-rotation might at first glance appear rather removed from the octatonic complexe sonore of Stravinsky's earlier works, yet I would suggest that the notion of a complexe sonore can be equally useful in our discussion of the hexadhordal regions created by transposition-rotation.
Linear considerations ...
What distinguishes Stravinsky's scheme of hexachordal transposition-rotation from conventional operations of set transposition is that it allows non-adjacent interval classes contained within a given set to be expressed as a melodic adjacency. For example, hexachord Sa0 contains one tritone, which occurs between order-elements 1 and 6 (F-B) of the unrotated hexachord (see Figure 2). In the fifth rotation, Sa5, this tritone occurs as the initial melodic adjacency. Similarly, the tritone is Sb0 occurs as a non-adjacency between order-elements 2 and 6(A-D#), but sounds as a melodic adjacency in Sb3. Articulation of all rotated set forms will, therefore, achieve a complete serialization of all possible melodic adjacencies as defined by the original hexachord. The result is that no one ordered succession of intervals in a hexachord attains any degree of priority over any other such succession during the unfolding of a hexachordal block (i.e. the interval succession represented by Sa does not figure any more or less prominently than the interval successions represented by the associated rotations of Sa). Berger's statement could, then, be amended to "... the denial of priority to a single hexachordal ordering precisely for the purpose of not deflecting from the priority of a whole hexachordal region (read "complexe sonore.") Polarity is thus created between linearly expressed hexachordal regions; and recognition of these hexachordal regions is very much dependent upon pitch-class centricity. As has been seen, the "home base" attribute of the Sa and Sb hexachordal regions is largely dependent upon the association of each hexachordal region with, respectively, F and C#. Indeed, pitch-class centricity is inherent in the mechanics of transposition-rotation: the common initiating pitch-class will occur more frequently than any other pitch-class when a complete hexachordal block is unfolded. In Abraham and Isaac, this "naturally occurring" centricity is reinforced by "artificially" emphasizing these pitches (F and C#) by such musical devices as dynamics, registral placement, repetition and duplication. The identification of the hexachordal region is also largely dependent upon the one-pitchclass verticals, which means, plainly, the first verticals of Sa and Sb: the F and C# verticals.
It is not surprising, then, to discover that the preponderance of linear unfoldings of the Sa and Sb hexachordal regions is matched by the statements of Sa and Sb verticals: of the twelves instances in which are stated partial or complete hexachordal verticals, statements of Sa and Sb verticals form the majority - eight statements in all. Moreover, only in statements of the Sa and Sb verticals are the one-pitch-class verticals included. For example, the statement of the Ib verticals which accompanies verse 18 (mm.229-239, see Figure 5) ends with the second vertical: the one-pitch-class vertical (i.e. pitch-class A) is not stated. Observe also how the centric C# from vertical 2 of Ib is regularly emphasized by the tuba, reinforcing the baritone's concluding C# an octave above. Finally, the fact that verticals are never stated in a transposed form further supports the contention that we are meant to hear those one-pitch-class verticals - and by this is meant the Fs and C#s of the first verticals of Sa and Sb - as pointing to the centric hexachordal regions of Sa and Sb.
As has been noted, linear hexachordal statements are almost exclusively confined to the independent voices which initiated them. The baritone never shares in a hexachordal statement with the accompaniment; when two or more instruments share in the articulation of a hexachord it is invariably in a linear fashion. The precedent is set in the opening measures, and a predominant characteristic of Abraham and Isaac is that of a sinewy interweaving of contrapuntal lines, punctuated by vertical simultaneities. The frequent sparseness of the instrumental accompaniment as a means of highlighting the baritone solo has been remarked upon. While the instrumental texture is often thin, the variety of instrumental combinations is endless. Changes in instrumental combination invariably reflect the succession of hexachordal unfoldings: see, for example, at mm.42-47 (Figure 4) the statement of Ib3, shared by horn, tenor and bass trombone, followed (after a one measure rest) by a simultaneous articulation of Sb2 (oboe),Sb5 (clarinet, and bass clarinet), and IRa0 (violin I). Indeed, the "combinatorial" treatment of the orchestra, remarked upon by Milton Babbitt in his discussion of Movements, is surely at its most insistent in Abraham and Isaac. Not once in the entire work is a full orchestral tutti achieved
Table 5: Abraham and Isaac, mm. 221-222, Instrumental combinations
Changes in instrumental combinations occur almost per measure - or per hexachord. The opening verse, for example (mm.1-25), features no less than sixteen different instrumental combinations, none of which will be exactly duplicated later in the work (see table 5). When instrumental combinations are repeated, it is invariably with a subtle variation. For example, an attractive formal symmetry is created by the domination of the opening and closing measures by the violas and low woodwinds; yet in mm.1-12 the instrumentation consists of viola, oboe, English horn, clarinet, bass clarinet and bassoon; in mm.248-254 the accompaniment consists of viola, clarinet and bass clarinet.
.... and Vertical Considerations
While the "horizontal" music in Abraham and Isaac is invariably associated with contrapuntal, linear hexachordal unfoldings, the passages of vertical simultaneities are analogously confined to articulations of the hexachordal verticals. In a work so heavily contrapuntal, the instances of simultaneously-sounded vertical textures are particularly striking. Yet the contrast between the linearly deployed hexachords and their rotations and the deployment of the hexachordal verticals goes far beyond this surface level distinction. Recall at this point Stravinsky's comment, speaking of Abraham and Isaac: "Octaves occur frequently, and doubled intervals are everywhere, and I suppose that `key' gravitations will be found to exist"  The fifths spoken of by Stravinsky are very much vertical sonorities in Abraham and Isaac. Stravinsky's habitual fondness for using the fifth as a means of "acoustic support" is manifested throughout; and, as we might expect, F and C# are invariably the pitch-classes which receive this support. Next to octave or unison duplications, the fifth figures the most prominently in reinforcing the centricity of F and C# While instances of octave duplications and fifths occur throughout Abraham and Isaac, this feature by itself is not unique to this work, for, as Spies points out: "fifths abound, in like fashion, in comparatively as many measures of any of Stravinsky's works written since 1954." Spies goes on to suggest that fifths do not "perform a sufficiently exclusive, tangible `function'" nor do they "arise from any remarkably peculiar set properties." In fact, the fifth (in addition to the intervals of the fourth and major third) is the one interval most conspicuously absent as a linear melodic adjacency in the unfolding of a hexachordal block. It is necessary to articulate a twelve-note row - and melodically span the hexachordal divider - in order for a fifth to occur as a melodic adjacency. For example, the consecutive statements of Ia2 with Ib2 as twelve-note series will produce a fifth as a melodic adjacency across the hexachordal divider (i.e. order-positions 5 and 6). Yet the compositional design of Abraham and Isaac is so exclusively hexachordal that twelve-note rows are almost never deployed in this manner, and, on the infrequent occasions that they are so deployed  it is an interval other than the fifth which will span the hexachordal divider. That Stravinsky chose not to deploy his rotated set forms in this manner is evidence that the fifth (and octave) are very much vertical, as opposed to linear, sonorities. Certainly, Stravinsky was very much concerned that the fifth would not appear in the hexachordal segments of the twelve-note row.
This tells us something about the possible compositional process of Abraham and Isaac. To briefly reiterate some earlier observations: complete unfoldings of hexachordal blocks - which are confined almost without exception to statements of the Sa and Sb hexachordal regions - are the exclusive domain of the baritone, while the instrumental accompaniment to these unfoldings is usually comprised of isolated or fragmented rotations. The choice of the transposition levels of the single (transposed) hexachords and hexachordal rotations articulated by the baritone often appears to be governed by the capability of a particular transposition level to produce one or both of the centric pitch-classes F and C# at an opportune moment, particularly at the beginning and ending of phrases. Moreover, the transposition levels of the hexachords which appear in the accompaniment so frequently produce vertical fifths (and fourths) that their selection was obviously governed by the vertical sonorities which would arise during simultaneous linear hexachordal statements. See, for example,m.151-152, where the vertical coincidence of Sb0(T10) and Rb5(T7) in the accompaniment with the baritone's statement of RIa3(T8) creates the fifth-abundant vertical sonorities E-F#-B, and C#-G# (see Figure 16 below).
It is therefore likely that Stravinsky composed the baritone part first, according to this preferred procedure of patterned chains of hexachordal unfoldings - and with a preponderance of complete unfoldings of the "home" hexachordal regions of Sa and Sb - and then "harmonized" the vocal part with contrapuntal lines culled form other hexachords and their rotations (both transposed and untransposed). In doing so, Stravinsky would have chosen those rotations and transposition levels which achieved the desired vertical sonorities (fifths and octaves, emphasis of the centric F and C# and so on) as a result of their simultaneous articulation with the vocal line.
Figure 16: Abraham and Isaac, mm. 149-155
To return to our discussion of the serial verticals, we find that the vertical sonorities so carefully orchestrated by Stravinsky in his contrapuntal deployment of hexachords - fifths, fourths and octaves - also occur in abundance in the serial verticals. The occurrence of these intervals in statements of the verticals is invariably highlighted. We see a notable instances of this in the statement of the Ib verticals in mm.229-239 (see Figure 5) where the fourths, fifths and octave-unisons contained in these verticals are starkly isolated in the highest register by the flute and trumpet doubling - a unique and particularly beautiful orchestration. Note the tuba's continual reaffirmation of these intervals in the lowest register, frequently creating fifths or octaves with the highest notes of the chords.
A Final Word
And so there emerge a distinct contrast between the "vertical" and the "horizontal" music of Abraham and Isaac. This is evidenced by the intervallic content of the hexachords generated by the serial verticals, which differs greatly from that of the linear, unrotated hexachords. For example, the five verticals of Sb (excluding the first, one-pitch-class verticals), when placed in normal order, create three distinct chord types, none of which is contained in the linear hexachordal region of Sb (see table 4). Vertical 4 from the Sa hexachordal block represents a (0,2,4,6) whole-tone tetrachord, a sonority similarly unobtainable from any linear hexachordal unfolding. The hexachordal regions can, therefore, generate two distinct "sonic complexes," dependent upon a horizontal or vertical reading of the hexachordal block. Ultimately, the key to the aural identification of a hexachordal region is the centric pitch-class, which is articulated linearly through the successive unfoldings of the hexachordal rotations (thereby ensuring the repetition of the initiating, common pitch-class more frequently than any other constituent pitch-class), or vertically, by virtue of the one-pitch-class vertical. The preponderance of complete unfoldings of the Sa and Sb hexachordal blocks, and most telling, the insistence upon the untransposed forms of their associated hexachords, place these hexachords in their centric position, and so cast the hexachords belonging to the other set-forms (which appear most frequently in an isolated or transposed form, in order to support the centricity of F or C#) in a subordinate role.
Thus to the polarity of the two "sonic complexes" represented by the linear unfoldings of the Sa and Sb hexachordal regions we must consider a second, and perhaps more profound, kind of polarity: that of the vertical and horizontal sonorities, each derived from a different yet - by their mutual origins in the Sa and Sb hexachordal regions -ultimately related complexe sonore.
Babbit, Milton. "Since Schoenberg." In Perspectives of New Music, no.12 (1973-74), pp.3-28.
_______________ "Remarks on the Recent Stravinsky." In Perspectives on Schoenberg and Stravinsky, edited by Benjamin Boretz and Edward T.Cone. New York: W.W. Norton, 1972, pp.165-185.
Berger, Arthur. "Problems of Pitch Organization in Stravinsky." In Persepctives on Schoenberg and Stravinsky, edited by Benjamin Boretz and Edward T. Cone, New York: W.W.Norton, 1972, pp.123-154.
Kohl, Jerome. "Exposition in Stravinsky's Orchestral Variations." In Perspectives of New Music, no 18 (1979-80). pp. 391-405. Spies, Claudio. (a) "Notes on Stravinsky's Abraham and Isaac." In Perspectives on Schoenberg and Stravinsky. edited by Benjamin Boretz and Edward T. Cone, New York: W.W. Norton 1972, pp.186-209.
_______________ (b) "Notes on Stravinsky's Variations." In Perspectives on Schoenberg and Stravinsky, edited by Benjamin Boretz and Edward T. Cone, New York: W.W. Norton, 1972, pp. 210-222.
_______________ (c) "Some Notes on Stravinsky's Requiem Settings." In Perspectives on Schoenberg and Stravinsky, edited by Benjamin Boretz and Edward T. Cone,
New York: W.W. Norton, 1972, pp.223-249. Stravinsky, Igor and Robert Craft. Dialogues and a Diary. New York:Doubleday, 1963. _________________________________ Memories and Commentaries. New York:Doubleday, 1963.
_________________________________ Themes and Episodes. New York, Knopf,1967. van den Toorn, Pieter. The Music of Igor Stravinsky. New Haven: Yale University Press, 1983. Whittal, Arnold. "Thematicism in Stravinsky's Abraham and Isaac." in Tempo, no.89 (1969), pp.12-16.
Young, Douglas. "1964: Abraham and Isaac." in Tempo, no.97 (1971),pp.27-37.
 Stravinsky (1959) p.99.
 Ibid, p.100.
 Ibid, p.100.
 Spies (1972a) p.200.
 See Kohl (1979) also Spies (1972b).
 Stravinsky and Craft (1966) p.55.
 Stravinsky, Vera, and Robert Craft. Stravinsky in Pictures and Documents New York: Simon and Shuster, 1979, 1978. (The caption in this edition actually misidentifies plate 20 as a sketch for the Variations (1964)!) This example is reproduced with the kind permission of the Stravinsky estate.
 See van den Toorn (1983) p.389.
 Spies op. cit. p.202. n.18.
 Or "alternates", as Stravinsky intuitively referred to his rotations. See Memoires and Commentaries, p.100.
 Sa or Sb, for example, do not appear any more or less prominently than their rotated versions.
 Babbit (1973-74) pp.7-8.
 For a concise account of the various contrapuntal devices employed by Stravinsky in these canons, see Spies op.cit.
 Stravinsky and Craft (1967) p.55.
 Spies, op.cit. pp.186-187.
 van den Toorn op.cit. p.389.
 I have borrowed the term "hexachordal ladder" from van den Toorn's discussion of Stravinsky's serial works. See chapters 13 and 14 of The Music of Igor Stravinsky.
 Babbit (1972) p.183.
 Spies (1972c) p.244.
 Spies (1972a) p.194.
 Ibid, p.195.
 Duly noted are Stravinsky's protestations to the contrary: "...I do not wish the listener any luck in discovering musical descriptions or illustrations..."
 Spies, op.cit. p.196.
 There are some puzzling anomalous pitches here: the third chord in m.182 contains a G natural, as opposed to the G# of Sa vertical 3; similarly, chord 4 contains F and G, which are foreigh to Sa vertical 4. However, it seems contextually indisputable that Stravinsky intended the chords as a deployment of Sa verticals, and so these "wrong" notes can perhaps be considered as errors in the published score.
 Berger (1972) p.137.
 Ibid, p.137.
 Babbit (1972) p.182.
 The single exception to this procedure occurs at m.16 which features a vertical articulation of hexachord Ib0 by the strings.
 Stravinsky and Craft (1967) pp.55-56.
 Spies (1972a) p.207.
 See, for example, the viola's statement of RIa0 and RIb0, mm.1-2, and the baritone's statement of Ia0 and Ib0 at mm.233-239.