a metrical problem in webern's op. 27

A Metrical Problem in Webern's Op. 27Author(s): David LewinSource: Music Analysis, Vol. 12, No. 3 (Oct., 1993), pp. 343-354Published by: Blackwell PublishingStable URL: http://www.jstor.org/stable/854149 .Accessed: 04/09/2011 11:59

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DAVID LEWIN

A METRICAL PROBLEM IN WEBERN'S OP. 27

Some thirty years ago I published an article under the above title.' The point of departure was a question involving the second movement of the Piano Variations. Since one hears such a strong rhythmic pattern in groups of three quavers at the beginning of the movement, and since no other regular small groupings of quavers are anywhere nearly as audible, what is the sense of the written 2 time signature? In the earlier article, I used the question to introduce some heavy theoretical machinery. While the machinery still interests me, I no longer feel it gets us very far into the analysis of this particular piece. Here I shall propose another sort of response to the same question, a response which I now find much more to the analytical point.

I shall assume that the 2 time signature is meaningful in some sense. In particular, I shall assume that it is legitimate to articulate the piece metrically at moments of maximum density and dynamic that coincide with written barlines. I mean the barlines preceding bs 4, 9, 4', 9', 15, 20, 15' and 20'. At each of those moments one hears the lower of a pair of three-note chords.2 At each moment the dynamic is fortissimo, except for the chord in bs 4 and 4'. The forte at b.4, while not a maximum dynamic for the piece, is still louder than anything heard so far; the forte at b.4' is heard as a repeat of b.4. According to this view, the written 2 bar is to be heard as a formal mensural unit marking time in a regular way between the large-scale metric beats listed above. Using this unit of measure, one notices the regular metric pattern of Fig. 1:

Fig. 1

4 9 4' 9' 15 20 15' 20' ('3') 5 6 5 6 5 6 5 (3)

The top row of the figure gives the beats at the barlines preceding bs 4, 9, etc. The lower row counts the number of mensural units (written bars)

MUSIC ANALYSIS 12:3, 1993 343 ? Basil Blackwell Ltd. 1993. Published by Blackwell Publishers, 108 Cowley Road, Oxford OX4 1JF, UK and 238 Main Street, Cambridge, MA 02142, USA.

DAVID LEWIN

II

Sehr schnell J ca 160 1 2 4

10 11 12 13 14

P

f f j

T.

15 16 12 13 14

'0 0• f1

A 1516 17.18

6?"

Copyright 1937 by Universal Edition. Copyright renewed 1965. All rights reserved.

344 MUSIC ANALYSIS 12:3, 1993 ? Basil Blackwell Ltd. 1993


between successive beats. From b.4 to b.20', spans of 5 and 6 written bars alternate in that capacity. From the beat at the barline preceding b.20' to the end of the piece is 3 written bars; the number 3 accordingly appears in parentheses at the end of Fig. 1. I have also counted '3' written bars between the beginning of the piece and the barline preceding b.4. I have not counted the first half-bar of the music in this context, because the end of b.22' can be asserted to 'take care of it' in the global mensural context of Fig. 1. The ('3') and the (3) of Fig. 1 add up to another 6-bar span that alternates with 5-bar spans.

The durations of Fig. 1 can all be given as integral numbers (3, 5, 6) of 4 bars. The 5-spans could not be expressed as integral numbers of I bars, and that is one simple rationale for the written 2 time signature. Despite the locally experienced 3 groupings at the beginning of the piece, a i 'measuring rod' would not measure time in integral units between the asserted large beats of the music, while the 2 measuring rod does do so. That is as if all measurements came out evenly in integral inches, but not in integral centimetres; one would then prefer the inch as a formal mensural unit for the context. Peter Westergaard, in his magisterial analysis of the piece, points out this aspect of the 2 bar along with other similar duple mensural phenomena.3 He observes in particular that the repeated sections span eleven minims, and that the distances between successive low, loud three-note chords are 'either five or six minims', saying that 'this helps define ... the downbeat'. He does not, however, cite or investigate the regular alternation of 5 and 6-bar spans, displayed in Fig. 1.4

Nor does he explore an interesting quasi-isorhythmic structure which emerges from Fig. 1. Example 1 begins such an exploration. In the example, the span from b.4 to b.9 is compared with the span from b.9 to b.4'. Bars 9-11 (up to the repeat) match bs 4-6 very closely, with respect to the sketched attributes. The isorhythm is exact if grace notes are ignored. The dynamic patterns also match, if f and ff are counted as equivalent in the alternation of loud with soft. Dyadic contours match fairly well, as do attributes of accent/slur/staccato adhering to the various dyads - a notable exception being the staccato repeated A in b.9, which is a mismatch for the slur in b.4.

The quasi-isorhythm of the example can be continued so as to match b.7 with b. 1', and b.8 with b.2'. The isorhythm continues to be exact so far as attacks are concerned, if we ignore grace notes. Bar 3'is then an entire extra bar, extending the talea from 5 to 6 bars by echoing, about the barline preceding b.4', the rhythmic pattern already heard about the barline preceding b.3'. The purely rhythmic reading is plausible, but one is disturbed by the mismatch of dynamics and other textural features (tenuto, staccato, accent).

An alternative isorhythm arises if we match the last three quavers of b.7 with the first three quavers of b.2', the first three quavers of b.8 with the figure about the barline preceding b.3', and the figure about the barline

MUSIC ANALYSIS 12:3, 1993 345 ? Basil Blackwell Ltd. 1993

DAVID LEWIN

Ex. 1

w w

1g11 f p f p ift f P f

?> "? yf . ">

i" '

f p f P f P f p, f

preceding b.9 with the figure about the barline preceding b.4'. According to this reading, the rhythmically matching figures match dynamically and texturally as well. This seems particularly cogent as far as the three-note chords are concerned, the chords that mark the barlines preceding bs 9 and 4'. According to the new isorhythm, the talea of bs 4-9 is expanded during bs 9-4' by the repeated A in b. 1', which is inserted into the talea along with a quaver rest either before or after; then another quaver rest is inserted at the second or third quaver of b.3'. The disruptive role of the repeated-A figure seems much to the point in this connection; it is part of a joke. Of the essence is the insert of three quavers' duration into the talea, the same 'three quavers" duration that imposed a false metric grid on our rhythmic perception of bs 1-3.

Example 2 matches the span of bs 15-20 against the talea of bs 4-9. Here the fit is extremely close in all pertinent respects. The exact isorhythm is disrupted only by the displacement of the three-note chords in b.15; they come a quaver late, as was discussed earlier, so that the lower of the two chords can attack at the barline. Bars 15-20 are thus an isorhythmic reprise of the basic talea bs 4'-9', after the rhythmic complications of bs 9'-15.

Example 3 displays these complications. The span divides into three modules, each module presenting the same rhythm. As already noted (in Ex. 1), the rhythm of bs 9'-11'coincides with the rhythm of bs 4-6, the first half of the basic talea. Each module in Ex. 3 projects the first eight quavers of that rhythmic pattern. Example 3 thus prolongs and 'develops' the beginning of the talea. The eight-quaver rhythmic ostinato is another aspect of duple rhythm in the piece. The internal three-quaver groupings



Ex. 2

w w

41 11

p fp f fr pff f p f p ff f p f

15 f20

if p f p if f P if

Ex. 3

F, PI,

F313 15]

MUSIC ANALYSIS 12:3, 1993 347 C Basil Blackwell Ltd. 1993

DAVID LEWIN

within the eight-quaver module prevent the ostinato from becoming obtrusive; so does the variety in non-rhythmic aspects of the three modules. The extra quaver at the end of b.14, going beyond the eight- quaver module, becomes the missing quaver of b. 15, observed in Ex. 2.

The span of bs 20-15'is greatly more complicated rhythmically, but the large compositional gestures remain similar: after a rhythmic excursion (bs 20-15'), the basic talea returns (bs 15'-20').

The analytical work so far has responded to the opening question, finding a structural significance for the written 2 time signature. Another question now comes to the fore. If the piece really 'is' in 2, then what is the compositional sense of the strong 8 rhythmic groups that are so audible over the opening bars? A response to this question is furnished by analysing the phrase and thematic structure of bs 1-11, in conjunction with the rhythmic analysis above. I divide the section into three phrases.

Phrase 1 (opening-b.31/2): the phrase falls into regular, highly audible groups of three quavers. But all of this comes before the first structural downbeat of the piece, the barline preceding b.4. The phrase is highly thematic. In particular, as Wilbur Ogden has observed, the pitch classes BL-G?, A-A and C#-F are always presented in the same register throughout the piece.' Let us call the succession BV-G#-rest-A-A-rest-C -F, in the registers of the opening bars, the TUNE.

Phrase 2 (bs 3 3/4-8 1/2): the sonority attacked at the beginning of this phrase contains C and

F.,

the other pitch classes that remain fixed in register throughout the piece. The event destroys the 3 pseudometre of phrase 1, providing the first 'real' downbeat of the piece (according to my earlier analysis). The event also begins a talea. Because of all this, we can assign a particular thematic significance to the textural idea of the three- note chords. Let us call them CHORDS. We can think of them as a 'second idea', if not precisely a second theme.

Phrase 2 ends, at the first half of b.8, with the figure (EG)-(DB); the pitch classes and the grace notes recall the similar figure that ended phrase 1, (BD)-(GE) across the barline preceding b.3. These are the registrally mobile pitch classes of the composition; the end of phrase 2 in fact demonstrates such mobility for the first time. That is, up till b.8, all Bs, Ds, Gs and Es are in the registers of bs 2-3, the registers in which those pitch classes are first presented; at b.8 the pitch classes appear in different registers. Thus the end of phrase 1 introduces registrally mobile pitch classes for the first time; the rhyming end of phrase 2 demonstrates the mobility of those pitch classes for the first time. In fact, the end of phrase 2 demonstrates that some pitch classes are registrally mobile: up till this point, no pitch class - except for EWD# - has appeared in more than one register. The exception of EL/D# is easily rationalized by the requirements and characteristic features of the mirror canon.

Phrase 3 (bs 8 3/4-111/2): the TUNE of phrase 1 recurs at the opening of phrase 3, a sort of 'false reprise', which has a comic effect when the real



repeat supervenes, starting at b. 111/2. We can also hear phrase 3 as a sort of 'closing idea' because the opening of the TUNE, around the barline preceding b.9, is harmonized by CHORDS; phrase 3 thus combines thematic aspects of both phrase 1 and phrase 2. The CHORDS at the beginning of phrase 3 are the only other CHORDS in the first half of the piece, besides the CHORDS at the beginning of phrase 2.

Starting with the quaver upbeat to b.9, the succession B-G#-A-A- C#-F-C-F#, which spans phrase 3, runs through all the registrally fixed pitch classes of the piece, in their original order of appearance (bs 0-4). The succession consists of all the pitches of phrase 3 except for the notes that accompany B%-G# around the barline preceding b.9. Of these notes, F and C# are registrally fixed while B and G are registrally mobile; B and G are thus the unique registrally mobile pitch classes of the phrase. The B occurs in yet a third register; so does the G. As we shall see later, there is a sense in which these are the 'correct' registers for B and G.

The idea of a 'false reprise' bears further exploration. Its 'falsity' does not so much involve the quaver displacement in the written metre, which was not functional at the beginning of the piece anyway. Rather the sense of falsity attaches itself to the appearance of the TUNE at b.9 with a big structural downbeat on its second note. The large pulse is completely antithetical to the original metric character of the TUNE, which originally completely preceded the first structural downbeat of b.4. The large-scale metric mismatch is then corrected by the 'real' reprise of b. 111/2 etc.; this reprise places the beginning of the TUNE correctly, halfway through a talea rather than at the beginning of a talea.

The reprise of the TUNE in the second half of the piece (bs 18-20) is also metrically correct in the same sense: it begins halfway through a talea. The end of this reprise is compressed and jammed into the downbeat at b.20 because there are only 5 bars, not 6, in the talea of bs 15-20. The TUNE is heard clearly enough, despite the reversal of its C#-F into F-C# about the barline preceding b.20.

According to the above overview of phrase and thematic structure, the 8

pseudometre, heard at the opening of the piece, is indeed thematic. A specific thematic feature of this material is that it completely precedes the first structural downbeat of the music. Hence the 3 pseudometre never engages with the structural metric organization of the piece, and that lack of engagement is thematic; as it were, bs 0-3 are all played 'with the clutch down', and the transmission only engages at the barline preceding b.4. The idea is typically Brahmsian. One thinks of 'Immer leiser wird mein Schlummer', or the Horn Trio, or other pertinent pieces. They begin in the 'wrong' metre, or with the 'wrong' heard barlines, and only arrive at or settle into a 'real' structural downbeat and metre after the initial thematic idea has been fully presented.

Let us now explore further the characteristic alternation of '5' and '6' measurements displayed in Fig. 1. The scheme is abstractly suggestive


DAVID LEWIN

because measurements of 5 and 6 semitones characterize the three-note CHORDS, the chords which articulate the large metric beats. In fact, Catherine Nolan has argued compellingly that (5, 6) double cycles, in the pitch or pitch-class realm, govern the pitch structure of the composition.6 Her analysis relates the various CHORDS to other 5-plus-6-semitone structures in the music, registrally defined structures that are spread out over time rather than projected as simultaneities. In the remarks that follow, I shall appropriate her theoretical premise and incorporate a good deal of her analytical material. Her theoretical premise will be somewhat inflected by my own a priori arrangement of Fig. 2 below. Her analytical narrative will be abridged, and also somewhat modified by my desire to integrate a 'story' for the pitch-analysis with other 'stories' arising from my analysis so far.

Fig. 2

Segment 1: e3 bb2 f b' f ' Segment 2: d g # C' g' C2 order numbers: 1 2 3 4 5

Figure 2 displays two segments of abstract (5, 6) pitch cycles: Segment 1, specifically, projects alternating intervals of 5-semitones-down and 6- semitones-down; Segment 2 projects an analogous alternation of intervals 5 and 6 going up. The two Segments comprise all twelve pitch classes except for A and EIWD#. The beginnings of the Segments are articulated by the absent pitches d#4 and a3, to the left of Segment 1, and the absent pitches E? and A, to the left of Segment 2. The ends of the Segments are articulated by the pitch classes F# and C, taking us half a tritone away from A and EMID#, in either direction. The ends of the Segments are also articulated by the completion of the total chromatic, if one adjoins the missing A and EID#g to the pitch classes of the Segments. The abstract structure of Fig. 2 is further discussed below. At the bottom of Fig. 2, order numbers 1 to 5 identify the locations of various pitches within the Segments; f2 thus appears at order number 3 (or location 3) of Segment 1. The ordering of the Segments is to be considered as registral rather than temporal.

To the left of the first barline in Ex. 4a, the beamed trichords show how the notes in order-locations 1-2-3 of the two Segments are projected by the registral layout of phrase 1 (opening-b.31/2). The other notes of the phrase are a1, B and g3. a', the centre of the pitch mirror, is portrayed with an open notehead in Ex. 4a; the pitch class A does not participate in the Segmental structure of Fig. 2. The pitch classes B and G do participate in the Segments; the pitches B and g3 'ought to be' b' and g', according to the Segmental structure. Crossed-out noteheads appear in Ex. 4a to indicate that idea. The crossed-out b' in the treble clef would belong to Segment 1,



connected registrally with the beamed B%-F in a descending 5-6 interval structure; likewise, the crossed-out g' in the alto clef would belong to Segment 2, connecting registrally with the beamed G#-C# in an ascending 5-6 interval structure. The grace notes of the music, on the high G and the low B, are an interesting device for projecting the idea that these pitch classes 'do not really belong' where they are.

The first barline in Ex. 4a symbolizes the first big downbeat of the earlier rhythmic analysis, the pulse at the barline preceding b.4. fj' and f2, in the treble clef, are stemmed together; these pitches belong to Segment 1 along with the preceding beamed B%, F and E. Segment 1 would now be completely manifest in the piece, except that there is as yet no b' in the music to 'connect' f2 with f#'. The missing b' is indicated by a crossed-out notehead at the first barline of the example. Similarly, c2 and c#' in the alto clef of the example are stemmed together; these pitches belong to Segment 2 along with the preceding beamed G#, C# and D. Segment 2 would now be completely manifest in the piece, except that there is as yet no g' in the music to 'connect' c#' with c2. The missing g' is indicated by a crossed-out notehead at the barline.

Because the c2 of Segment 2 (alto clef) registrally overlaps the missing b' of Segment 1 (treble clef), we can hear the CHORD

[f'l, c2, f2] in the music at the end of b.3 as a 'substitute harmony' for the [fl', x b', f2] of Ex. 4a. In this analysis f#' and f2 remain, while the 'mediant' c2 is substituted for the 'mediant' b'. The principle of substitution here is formally analogous to major-for-minor, or harmonic-for-arithmetic, or perfect-for- plagal. Similarly, one can hear the CHORD [c ~', f~', c2] in the music at the beginning of b.4 as a 'substitute harmony' for [c~', x g', c2].

An interesting alternative analysis for the two CHORDS is possible as well. The chord in the treble clef of Ex. 4b symbolizes the last chord of b.3 in the music; the chord in the alto clef symbolizes the first chord of b.4. The treble-clef chord of the example is spelt with b~' instead of c2; the reading asserts that b#' is a ficta substitute for the crossed-out b' in Ex. 4a. Similarly, the alto-clef chord of Ex. 4b is spelt with gl' instead of g'; the reading asserts that g~' is a ficta substitute for the crossed-out g' in Ex. 4a. This analysis portrays an exciting enharmonic clash exactly at the big downbeat: Ex. 4b asserts that F# and B# are heard at the end of b.3 in the music, immediately followed by GC and C at the beginning of b.4.

Each of the alternative readings, for the CHORDS around the barline preceding b.4, portrays a structural tension about the CHORDS. In one reading, c2 knocks b' out of the picture, while f#' does the same for g'. In the other reading, b' is present, but only conceptually; it is represented acoustically by the ficta b'.; g' is similarly represented by the ficta gP'. Here, the ficta note of each CHORD clashes violently with its enharmonic equivalent in the other CHORD.

Example 4a shows how these structural tensions are definitively resolved by the CHORDS of b.20, marking another big structural downbeat of the


DAVID LEWIN

Ex. 4

a) W20 Lf

etc.

C)

b)1l5• - 1

APP RES

llel

AV

rhythmic analysis. In this example, the stemmed F0 and F of Segment 1 recur at b.20, and now they have with them their 'proper' mediant b'. b' is encircled on the example, to celebrate the event. Similarly, the stemmed Co and C of Segment 2 recur at b.20, and they now have with them their 'proper' mediant g', encircled on the example.

The encircled b' and g' appear for the first time in the piece at the big structural downbeat corresponding to the barline preceding b.9.7 At this moment, symbolized by the second barline of Ex. 4a, b' is grouped into a CHORD along with f2 and bV2; this projects order locations 2-3-4 of Segment 1. The reader may wish to review Fig. 2 in connection with the large beam-and-stem structure on the treble clef of Ex. 4a. The following story emerges. Phrase 1 (before b.4) projects order locations 1-2-3 of Segment 1; the downbeat at b.9 projects order locations 2-3-4; the downbeat at b.20 projects order locations 3-4-5 - an achievement that was thwarted at the b.4 downbeat. Every order-trichord of Segment 1 is represented in this narrative, and the narrative closes at the final (b.20) downbeat when this state of affairs has been achieved, solving the modal- or-enharmonic problem posed by the first (b.4) downbeat. A similar story can be told about the order-trichords of Segment 2, in connection with the beam-and-stem structure in the alto clef of Ex. 4a.

Example 4c brings the b.15 downbeat into the picture. It analyses the CHORDS of that downbeat as appoggiature to the beamed Segmental



structures that follow them in bs 17-18. The CHORDS of b. 15 thus inflect a return of the beamed order-locations 1-2-3 from the Segments. This heralds the final reprise of the TUNE, which commences with the B% and Go shown at the end of Ex. 4c.8

Figure 3 attempts to provide an etiology for Fig. 2, adapting Nolan's theoretical apparatus for the purpose:

Fig. 3

Segment 3a: E? A E BW F B Fo C G Co Go D A E? Segment 3b: E? A E B% F B Fo Segment 3c: C G Co Go D A E? Segment 3d: E? A D Go Co G C

Segment 3a is part of a (6, 7) pitch-class cycle. The segment begins with E?, the 'tonic' pitch class of the third movement in Op. 27. Pitch-class intervals of 6 and 7 then alternate until EF is reached once more.' The fourteen elements of Segment 3a state every one of the twelve pitch classes, repeating the boundaries E? and A. Segments 3b and 3c divide Segment 3a into two symmetrical halves. Segment 3d emphasizes the symmetry by aligning the retrograde of Segment 3c beneath Segment 3b; this displays an inversional relation between 3b and 3d. E? is one pitch-class centre of the inversion; A is the other pitch-class centre. Segments 1 and 2 are then easily derived from Segments 3b and 3d respectively, along with the idea of 'inversion about A'.

NOTES

1. David Lewin, 'A Metrical Problem in Webern's Op. 27', Journal of Music Theory, Vol. 6, No. 1 (Spring 1962), pp. 124-32.

2. The emphasis on the lower of the two chords accounts for the rhythmic displacement of the two-chord pair at the barline preceding b. 15.

3. Peter Westergaard, 'Webern and "Total Organization": An Analysis of the Second Movement of the Piano Variations, Op. 27', Perspectives of New Music, Vol. 1, No. 2 (Spring 1963), pp. 107-20.

4. There will be more discussion below regarding the alternation of 5s and 6s. 5. Wilbur Ogden, 'A Webern Analysis', Journal of Music Theory, Vol. 6, No. 1

(Spring 1962), pp.133-5. Ogden engages with matters of rhythm, phrase and theme; he also asserts the pertinence of isorhythm to the composition. His specific analysis of these matters proceeds from small rhythmic cells upwards, and thereby differs from mine, which proceeds from large hyperbeats downwards.

6. Catherine Nolan, 'Hierarchic Linear Structures in Webern's Twelve-Tone Music' (Diss.: Yale University, 1989). Nolan explores double-interval cycles


DAVID LEWIN

in general, and (5, 6) interval cycles in particular, as a basis for analysing Webern's twelve-note music. Much of her theoretical apparatus derives specifically from her assertion of 'the pivotal significance to Webern of two fundamental interval-classes, 5 and 6' (Abstract).

7. B and G thus find their rightful registers just after the end of phrase 2, where - as was discussed earlier - the registral mobility of the two pitch classes became manifest for the first time.

8. In the abstract, it would be possible to analyse the CHORDS of b.15 as arising from new, secondary, (5, 6) cycle-segments. The CHORDS certainly suggest such segments. But I have not found this suggestion carried through elsewhere in the music.

9. The cycle would not repeat there; it would go on with E?-Bb-E (the upper CHORD of b.20), etc.


a metrical problem in webern's op. 27

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