ABSTRACT: To the three categories of hemiolas I identified in an earlier study I add a fourth, overlapping hemiolas. Particularly in the Baroque repertoire, and especially in the works of Handel, there are frequent instances of two successive two-bar hemiolas that overlap because they are collapsed onto three bars. I examine the larger context in which their preparation, elaboration, and peroration may represent central compositional issues, especially in light of the brevity of most Baroque compositions.
 Among the difficulties of observing the hemiola and approaching it from a theoretical or analytical perspective is the high degree of uncertainty (not to say ambiguity) that surrounds its articulation. The compositional environment will often suggest hemiolas by emphasizing the third beat of a measure (or the second beat in the following measure) through thematic, textural, harmonic, or registral stresses of varying impact, but it will not necessarily follow it up with an explicitly marked hemiola formation. In other words, it will stop just short of defining three contiguous long, hemiolic beats (each divided into two short internal beats, one strong, the other weak) over the span of two measures in triple meter. Compounding the consequent uncertainty is the well-established absence of a requirement for the hemiola to be articulated by both outer voices: The bass may not necessarily support the suggestion of a hemiola by the upper voice, and the upper voice will not necessarily corroborate its realization by the bass. Especially in such cases, the emphases articulating the hemiola will accrue largely through a series of design stresses--rhythmic accents that (as just intimated) are due to melodic, textural, harmonic, or registral intensification of purely local origin, at the very surface. In borderline instances, it is up to such design accents to try and supplant the metrical accents of the notated meter, whose supporting grid extends to deeper levels of durational structure. And indeed, there are many cases where it is in fact the notated meter that wins out. (On the other hand, the presence of several hemiolas may foster parallelisms that will reinforce the design stresses in potentially ambiguous passages.) Since not all voices necessarily participate in the hemiolic activity, the hemiola's effect on the composition's harmonic rhythm and other underlying paces will vary considerably from one hemiola to another.
 The formation I call overlapping hemiolas, which may be defined as a set of two hemiolas collapsed onto three measures, makes use of just this possibility of articulating the hemiola incompletely; see Example 1, which also depicts the possibility of a third overlapping hemiola, sometimes found in Bach and occasionally in Handel.
The formation often begins, somewhat unexpectedly and perhaps furtively, at a distance of three, rather than two, bars from the closing cadential progression of a phrase or subphrase, namely at the beginning of the group. Its location lends it a thematic weight different in nature, if not in emphasis or impact, from that of hemiolas that appear closer to the cadential chords of the group in question; its length allows it to shape the entire group in a way no single hemiola could. (Single hemiolas may, of course, also appear at the beginning of a group without being followed by an overlapping hemiola.)
 In the first bar of the formation, two-beat patterns in the soprano or bass (or in both) briefly contradict the notated meter and establish the first hemiola. In the second bar, the hemiola's articulation becomes uncertain, as the broken stems in Example 1 show. On the one hand, the metrical accent on the first beat is not attenuated to the same degree it usually is in hemiola formations; on the other, the expected stress on the second beat is elided or contradicted by adjacent stresses. The first two beats of the second measure may be regarded as a grey area, a durational "pivot zone," that plays host to two events: the premature peroration of the first hemiola, and the sly entrance of the second hemiola. For the second hemiola enters, however surreptitiously, at the downbeat of this second measure; it now begins to make its presence known. As a result, the first hemiola's division of long hemiolic beats into strong and weak internal beats undergoes metrical reinterpretation, as the diagram of accentual conflicts at the bottom of Example 1 demonstrates. The first hemiola's conclusion thus assumes a jagged and unsettled quality, a sense of tentative and incomplete realization that may have significant repercussions for the metrical design in the phrases and sections that follow. Once the third measure has been reached, the outline of the second hemiola emerges quite clearly. Because it is usually the second hemiola that enlists the cadential support of the bass, it brings the formation to a decisive close and at the same time sends out the signal that it has neutralized and absorbed the first hemiola.
 To illustrate the foregoing explanation and its more extended ramifications, I should like to offer several observations on the appearance of overlapping hemiolas in the Sarabande from Handel's G-minor Suite of 1720. The Sarabande opens with a four-bar subphrase that strikes an immediately memorable impression by replacing the expected, largely stepwise outlines of the typical sarabande melody with a series of bold leaps; see the braces in Example 2.
While these help underscore the Sarabande's four-bar grouping they also prevent it from assuming the predictable melodic contours of the genre. The overlapping hemiolas occupy bars 5-7, much of the second four-bar subphrase (Example 3).
 The two-chord patterns in bars 5-6 begin to establish a hemiola and continue through to the downbeat of bar 6 (see the brackets atop Example 3). The melodic line's two pairs of descending thirds, F- natural5-D5 and E-flat5-C5, are supported by rising fourths in the bass, D3-G3 and C3-F3. Together, melody and bass articulate an S-W, S-W formation that reaches into the beginning of bar 6. The progression stands out--and can superimpose a hemiolic interpretation over the notated meter--because it comprises the Sarabande's first genuinely harmonic progressions spanning only two internal beats each, and because it suggests a durational contraction (as well as a harmonic realization) of the longer, contrapuntally conceived progressions that straddle the barline at the turn of bars 2, 3, and 4.
 At the downbeat of bar 6, then, two events overlap: The statement
of the hemiola's fourth, weak internal beat, and the beginning of an
unexpectedly sustained F-major chord, whose continuation through
the second beat of the measure supports the notated meter as
established in bars 1-4 and generates what Joel Lester called a
 Because the overlapping hemiolas span three bars and occupy the greater part of a four-bar subphrase, the temporary metrical uncertainty they portray in their central measure has the potential for creating a genuine disturbance in the rhythmic flux of the piece. To foster a compositional and rhetorical milieu in which such disturbance will realize its assigned effect without appearing out of line, Handel often prepares the onset of overlapping hemiolas in some way during the preceding passages and then only gradually allows them to dissipate in the phrase or phrases that follow. By the time their peroration is completed, then, their very presence in the piece, as such, will likely have promoted additional rhythmic dialectics, and these might have in turn become intimately linked to the compositional essence of the piece.
 The two-chord patterns that launch the first hemiola in bars 5-6 are prepared by the falling third F-sharp4-D4 in bar 4. Their movement forward begins with the quasi-improvisatory reappearance of this falling third in the two-line octave, F-natural5 conspicuously replacing F-sharp5 at the turn of bar 5. The transfer of F-sharp to the high register and its alteration to F-natural represents yet another way of sidestepping excessive thematic predictability within the Sarabande's periodic framework.
 Let us look now at several later allusions to the overlapping hemiolas; these, too, help prevent the remaining three eight-bar phrases from outlining an excessively mechanical profile. Although the four-bar group that follows the double bar brings back the rhythmic outlines of bars 1-4, the unexpected entrance of a two-beat G major chord on the second beat of bar 12, at the close of the group (and in the manner of a written-out sforzato), nearly usurps the cadential status of F, the subtonic that entered on the downbeat of the measure (see the arrow in Example 4).
It is as if Handel insisted on
affirming, following the upheaval of bars 5-7, that we are indeed
listening to a Sarabande, but that further upheaval is in store for us as
 The uncertainties and tensions of the Sarabande's overlapping hemiolas find their complete peroration only in its remaining two phrases (bars 17-24 and 25-32). The subphrase in bars 17-20 restores the outlines of bars 1-4 but forgoes the striking leaps of the first subphrase in favor of a more traditionally cantabile, conjunct setting of the characteristic incipit. The following subphrase, in bars 21-24, shows a comparatively mechanical quality (a quality underscored by the repetition of the same subphrase in bars 29-32). The literal transposition of bars 21 and 29 in bars 22 and 30 (a third higher) underscores the insistence of the subphrase's repeated notes (A4, C5), as well as the closing impulse of its plainly cadential hemiolas (bars 22- 23, 30-31). This is Handel's response to the accentual irregularities occasioned by the overlapping hemiolas: Though they play their role well, the two subphrases in question count among the less inspired passages in the 1720 collection of suites (the Sarabande dates from about 1703-06, but it was revised for the 1720 publication). Their slight awkwardness exemplifies the difficulties Baroque composers faced when, in the course of very short pieces, they felt need to resolve complex issues they had raised moments earlier but did not quite have the needed durational space in which to work out the resolution.
 It is only in the course of bars 25-28, the first subphrase of the closing phrase, that a vestige of genuine tension reemerges. At bar 26, Handel sidesteps the opportunity to present generic repeated-chord patterns (in imitation of bars 17ff. and bar 25) and sustains the second-beat D chord through the end of the measure instead. Handel can thereby suggest a "hidden hemiola" in bars 25-26 without fully realizing it. The seventh chord on G, sustained in different positions through the second and third beats of bar 25, militates against the realization of the hemiola (Example 5).
Along the same lines, by
beginning bar 27 with two quarter notes, Handel evokes the preceding
cadential hemiola (that of bars 22-23) without really articulating it (cf.
the brackets in Example 5). As a group, bars 25-27 thus portray an
attenuated rhythmic reference to the Sarabande's temporal
peculiarities (the suggestion of hemiolic closure in bars 26 and 27 not
being matched by that of hemiolic opening in bars 25 or 26), but at the
same time they steer clear of the peculiarities' actual portrayal--a
signal indication that the turbulence of the Sarabande's temporality
has finally reached the point of complete peroration.
 I should like to take the opportunity of writing again about the
hemiola to modify my account of the hemiola formations in two of the
pieces I presented in "The Two-Length Bar Revisited": Those in the
Courante from Handel's F minor Suite, and those in the Musette from
the G minor Concerto Grosso, Op. 6 No. 6 (Examples 3 and 4 in that
article). Even though the hemiolas in each piece appear at the close of
a phrase, one could make a plausible case for reading them as
constellations of overlapping hemiolas.
 In bars 17-19 of the Courante (Example 6 in the present essay), the upper voice's first hemiola (bars 17-18) is self-evident; that an additional, overlapping hemiola spans bars 18-19 in the upper voice is suggested by the entrance of the bass G at the turn of bar 18, which underscores the downbeat of the measure, by the subsequent emphasis on the supertonic 6/5 chord on the third beat of bar 18, and by the articulation of the dominant's 6/4-5/3 resolution on the first two beats of bar 19, which places substantial motivic weight on the second beat of that measure.
The succession of these events suggests a metric reinterpretation similar to that displayed in the present Example 1.
 Though it is the second, rather than the first, hemiola that is relatively tentative here--its interpretation is supported by the grouping of the upper voice but contradicted by the return of the bass tone G at the beginning of bar 19--the durationally angular setting of the three-bar segment and its collection of conflicting accents does evoke the thematic and rhythmic contraction typical of overlapping hemiolas and as such assumes growing significance in the course of the piece. First introduced as the climax to a set of imitative entrances of the Courante's short theme over the dominant (bars 10- 14-17), it reappears in the form of seemingly incidental allusions to the tentative hemiola of bars 18-19 during the cadential progression of bars 27-29 and, more concretely, in the progression of bars 30-32 (and, transposed but complete, in bars 41-43).
 The overlapping hemiolas in the Musette from the G minor Concerto Grosso (bars 11-13) stand, by contrast, on nearly equal footing (bars 1-15 are quoted in Example 7 of the present article; my new analytical observations are illustrated in Examples 8-10).
The first hemiola is outlined by three pairs of chords whose articulation of
strong and weak internal beats--SW, SW, SW--supports their
contrapuntal and harmonic priorities. The first chord of each pair
acts as an accented appoggiatura to the second. The metrical
reinterpretation that takes place in the course of bar 12 is made
possible by this contrapuntal and harmonic function: As bar 12
unfolds, the second chord of each pair is reinterpreted accentually as
the principal chord of the pair. The primacy of the rising fourths in
the bass in bars 11-12 is gradually replaced by that of the falling thirds
in bars 12-13, which are supported by the melody's movement in
parallel tenths with the bass beginning at the downbeat of bar 12.
Both the B-flat and C of the bass in bar 12 gain in accentual
importance because they help extend A-flat through an ascent to the
supertonic's inner- voice tone C, a third above, and in so doing
gradually define the contour of the expanded cadential progression
that spans bars 11-13 (closing in bar 14). By the time the B-flat
dominant enters on the second beat of bar 13, it has gained its
requisite hemiolic strong accent.
 The effect of a hemiola depends in large measure on its relation to the prevailing rhythmic and thematic design. From this perspective, the Musette's overlapping hemiolas are particularly noteworthy for they introduce the third change in design within the six- and eight-bar groups in which they occur (bars 1-6 and 7-14; the hemiolas are first suggested, in germinal form, by the chordal pairs in bars 5-6). The Musette's incipit, which moves in quarter notes (bars 1-2, 7-8), is followed by an extraordinary rising slide to a sustained E-flat (bars 3, 7) and a responding, falling slide to a sustained B-flat; the slides represent displaced contractions of the quarter-note incipit (Example 9).
The hemiolas (incipiently in bars 5-6, fully in bars 11-13) reintroduce the movement in quarter notes but contradict their earlier grouping in threes through the hemiolas' characteristic articulation in pairs. The expansive nature of the dialogue between the concertino and ripieno groups that follows--and many of the ensuing tonal and durational enlargements--throughout the movement arise from the need to work out and then dissipate the tensions resulting from this uncommonly close succession of changes in design and, in particular, from the effects of contraction occasioned by the slides and the chordal pairs.
 To my original remarks regarding the influence the passage exerts on the rest of the Musette I should like to add also the observation that durational groupings in the Musette's extended developmental passages are remarkably regular. Like the brief spurt of periodicity that follows the double bar in the F-minor Courante and the growing accentual conformity of the G-minor Sarabande, they mitigate the accentual uncertainties of the earlier, thematic hemiolas. More important, later allusions to the chordal pairs that originally introduced the overlapping hemiolas present them in an expanded state, their ascending bass fourths altered to ascending fifths (bars 49- 53, Example 10 in the present article).
The bass line carrying the ascending fifths rises, like the bass line that carried the hemiolas' original ascending fourths, from E-flat to F and G; instead of continuing up to A-flat, though, it quickly falls back to F and E-flat. By allowing each chord to occupy a full measure rather than just a quarter note, Handel sidesteps the questions that marked the original progression and paves the way for the introduction of altogether new tensions in the stormy C minor section that is to follow. It is perhaps a mere coincidence--but what a wonderful coincidence--that the C minor section (bars 81ff.) opens, as well, with a rising fifth in the bass, which ascends from tonic to dominant; however common and, in this instance, isolated from a sequential setting, the progression nonetheless carries a recollection of the earlier sequence (Example 10).
 The hemiola, to be sure, occupies a special place in the Handelian repertoire, but its cultivation as a durational and tonal resource with consequences ranging well beyond its immediate appearance was hardly within Handel's exclusive compositional domain. Overlapping hemiolas, in particular, appear with great frequency in the music of J.S. Bach, and especially in Bach's Courantes, Sarabandes, and Minuets. Particularly worthy of study are such pieces as the Sarabande from the G-major Partita for Clavier, in which almost every four-bar group contains overlapping hemiolas that are expressed in different ways by different constellations of inner and outer voices.
 Although the literature on the hemiola remains modest in scope, several important studies that shed considerable light on its durational properties (and offer some intriguing further references) have appeared in the recent and relatively recent past; I should like to close by calling them to the reader's attention.
2. Joel Lester, The Rhythms of Tonal Music (Carbondale and
Edwardsville, Ill.: Southern Illinois University Press, 1986), pp. 3-4
and 18-21. This durational accent is considerably intensified by the
trill embellishing C5, but the trill, which appears in brackets or
parentheses in the most reliable editions, is implicit in Handel's
melodic outline (note the two eighths by which it is followed) rather
than explicit in his text.
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3. It is possible to find several hemiolic implications in bars 9-12,
especially across bars 10-11 and 11-12. These, I believe, represent an
allusion to bars 5-7 that is superseded by the more obvious
parallelisms among the second-beat emphases throughout the passage.
The emphases win out (as we shall presently see) on account of the
stronger need to evoke the durational design of bars 1-4.
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4. Floyd K. Grave (private communication) has suggested that the
Sarabande rhythm's second-beat emphasis comes into its own in bars
13-15 and persists as an unnotated expression of 2/4 time through the
second beat of bar 15 (the tied E-flat5 standing for a quarter-note rest
at the beginning of bar 15); and that the sense of an event is
underscored by the entrance of A-flat5 in bar 13. The melodic peak of
the Sarabande, it also contradicts the inner-voice A-natural4 of bar 12
in much the same way that the F-natural5 of bar 5 contradicted the F-
sharp4 of bar 4. These wonderful observations are by no means
incompatible with mine and reflect the extraordinary richness of
meaning with which Handel invests the surface--particularly
remarkable in view of the substantial rests and static F-minor
harmony (also pointed out by Grave) that pervade the passage.
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5. The hemiolic implications of bars 25-27 echo those of bars 10-12
(see fn. 3, above) but they are more explicitly articulated owing to the
intensified activity on the third beat of bar 25.
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6. Like most of Handel's Courantes, the piece under discussion is really
a Corrente, but Handelians--unlike Bach scholars--persist in using the
French term even when the Italian is clearly preferable. To maintain
Handelian consistency, I retain the French form here.
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7. Bar 12 includes the overlapping hemiolas' grey area, their "pivot
zone," spanning beats 1 and 2. Our retrospective interpretation of the
area's metrical disposition allows the chords on B-flat and C (beats 1
and 3) to assert their accentual primacy over the appoggiatura-like
chords by which they are preceded. The parallel tenths between the
outer voices in bars 12-13 allow both performers and listeners to
absorb the accentual reinterpretation in a fluent, unforced way.
Paradoxically, both appearances of the underlying A-flat (bar 11, beat
2; bar 13, beat 1) are metrically weaker than those of B-flat and C,
which prolong A-flat; but the metrical weakness of structural chords
following their introduction by sequential appoggiatura chords and
preceding the appearance of still more structural chords (such as the
dominant on beats 2 and 3 of bar 13) is quite common.
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8. Similar pairs of hemiolic chords appear as an idiom under more
straightforward circumstances throughout the Classical era. In their
later formation, they extend for four--rather than three--bars, and
therefore divide into two sets of complete hemiolas. See, for instance,
the second movement from Haydn's Keyboard Sonata in D, No. 61
[Hoboken XVI: 51], bars 31-34, for a particularly dramatic example;
or the Menuet from Haydn's Symphony No. 99 in E-flat, bars 52-55.
Though they don't occur very frequently, such groups of successive
hemiolas are by no means uncommon in the Classical style (see also
Mozart's Little Gigue in G, K. 574, which incorporates successive
hemiolas in its principal thematic material as a well-known borrowing
of similar hemiolas from the Gigue of Handel's F-minor Suite). For
a set of genuine late 18th-century overlapping hemiolas see Mozart's
Piano Concerto in B-flat, K. 595, third movement, bars 51-52; the
hemiolas appear in the bass line, extending from the middle of bar 51
through the end of bar 52 (the movement is in 6/8 time, and the
hemiolas span the duration of what would be three bars in 3/8 time).
Canonic hemiolas in the Menuetto from Mozart's G-minor Symphony,
K. 550, that in some ways resemble overlapping hemiolas are discussed
by Richard Cohn in "Metric and Hypermetric Dissonance in the
'Menuetto' of Mozart's Symphony in G minor, K. 550," Integral
6 (1992), pp. 22-26, Example 7 (bars 28-36). Canonic hemiolas
occur in different voices; overlapping hemiolas are usually
collapsed onto a single voice or onto several voices moving
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Corrigan, Vincent. "Hemiola in the Eighteenth Century," in Seymour L. Benstock, ed., Johann Sebastian: A Tercentenary Celebration (Westport, CT: Greenwood Press, 1992), pp. 23-32.
Grave, Floyd K. "Metrical Dissonance in Haydn," Journal of Musicology 13.2 (Spring, 1995), pp. 168-202.
Hlawicka, Karl. "Die rhythmische Verwechslung," Die Musikforschung 11.1 (1958), pp. 33-49; and "Musikalischer Rhythmus und Metrum," Die Musikforschung 24.4 (1971), pp. 385-406.
Jackson, Timothy L. "Mozart's `Little Gigue' in G major K. 574: A Study in Rhythmic Shift--A Reminiscence of the Competition with Haessler," Mitteilungen der Internationalen Stiftung Mozarteum 37.1-4 (July 1989), pp. 70-80.
Krebs, Harald. "Some Extensions of the Concepts of Metrical Consonance and Dissonance," Journal of Music Theory 31.1 (Spring, 1987), pp. 99-120.
Little, Meredith Ellis, and Natalie Jenne. Dance and the Music of J. S.Bach. Bloomington: Indiana University Press, 1992.
Schneider, Herbert. "Structure metriques du menuet au XVIIe et au debut du XVIIIe siecle," Revue de Musicologie 78.1 (1992), pp. 27-65.
Sutton, Julia. "The Minuet: An Elegant Phoenix," Dance Chronicle 8.3-4 (1985), pp. 119-52.
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