Double variation

The double variation (also known as alternating variations) is a musical form used in classical music. It is a type of theme and variations that employs two themes. In a double variation set, a first theme (to be called A here) is followed by a second theme (B), followed by a variation on A, then a variation on B, and so on with alternating A and B variations. Often there is a coda at the end.

The double variation is strongly associated with the composer Joseph Haydn, who wrote many such movements during his career.

The double variation in Haydn

The double variation first appears in Haydn's work of the 1770s. Haydn may have been inspired by an earlier example of Carl Philipp Emanuel Bach, the sixth of that composer's Sonatas with Varied Reprises, (W. 50/6, H. 140), in C minor (1760). Elaine Sisman, an authority on variations, notes "This set of sonatas was advertised in Vienna several times in the period in which Haydn wrote his first [double] variations."^[1]

While Haydn's double variations show considerable diversity, there are some general patterns.

• Both themes have the same tonic, but in opposite modes, so that if A is major, B is minor, and vice versa.
• The second theme is usually thematically reminiscent of the first, though not so close as to be an actual variation of it.^[2]
• The total number of variations is small, often just one or two for each theme.
• The number of variations is (with just one exception) arranged to place the major theme last. Thus, if the first theme is major, Haydn generally uses ABABA form, but if the first theme is minor, Haydn uses ABABAB.

As Haydn's career proceeded, he moved toward a very particular type of double variations, having the following additional specific characteristics.

• The tempo is moderate, typically andante.
• The minor theme is placed first.
• Each theme is divided into two sections, and each section is repeated.
• The internal arrangement of both themes is often that of sonata form, with the music moving to the dominant or relative major key in the first part, to remote keys in the first half of the second part, and then to a recapitulation of the opening material in the tonic key. This observation is made by Charles Rosen (in The Classical Style) concerning the double variations in the Drumroll Symphony; it holds true in several other cases as well.
• While assessments of emotional content are necessarily subjective, it is reasonable to claim that the minor themes sound tense and the major themes blissful. Jean-Yves Bras, writing in program notes for a performance of the Piano Trio H:23 (Harmonia Mundi 901400), describes the minor and major themes as "somber" and "radiant", respectively. Charles Rosen, writing of the major theme from the Piano Trio H. 13, says that in it Haydn created "an emotion that was completely his own and that no other composer, not even Mozart, could duplicate – a feeling of ecstasy that is completely unsensual, almost amiable." Rosen's remark could be applied to several of the other major double variation themes.

List of works by Haydn written in double variation form

According to Sisman,^[1] Haydn wrote 21 double variation movements. Sisman's list is restated below in chronological order. Where different authorities provide different dates, both are given; NG = the New Grove (used by Sisman), MH = Maurice Hinson's edition of the piano sonatas.^[3] For the keys of the A and B sections, lower case designates minor; upper case major. The structural synopses are taken from Sisman with minor corrections; in Sisman's notation an asterisk means "altered".

Year	Work	Form	A	B
1770–75? (ES); before 1780 (MH)	Piano sonata H. XVI:36. 2: Scherzando	ABA1B1A2coda	A	a
1771-3 (ES); ca. 1768–1770? (MH)	Piano sonata H. XVI:44. 2: Allegretto	ABA1B1; a minuet	g	G
before 1778 (ES); 1771–1773? (MH)	Piano sonata H. XVI:33. Finale: Tempo di Minuet	ABA1B1A2; a minuet	D	d
before 1778 (ES); 1773 (MH)	Piano sonata H. XVI:22. Finale: Tempo di Minuet	ABA1B1A2; a minuet	E	e
1778/79	Symphony No. 53, "L'Impériale". 2: Andante	ABA1B1A2A3	A	a
1778/79	Symphony No. 70, 2: Specie d'un canone in contrapunto doppio: Andante	ABA1B1A2.	d	D
1779	Symphony No. 63, "La Roxelane". 2: Allegretto (O piu tosto allegro)	ABA1B1A2*B2*	c	C
1781	String quartet Op. 33, No. 6. 4: Allegretto	ABA1B1A2	D	d
before 1784	Piano sonata H. XVI:34. 3: Vivace molto	ABA1*B1A2. First variation in A is lengthened by a reprise of the initial section.	e	E
1784	Piano sonata H. XVI:40. 1: Allegro innocente	ABA1B1A2. In the following movement, in ternary (ABA) form, the A sections form yet two more variations of the A theme of the opening movement.	G	g
1789	Piano sonata H. XVI:48. 1: Andante con espressione	ABA1B1A2	C	c
1786	Symphony No. 82. 2: Allegretto	ABA1B1A2coda	F	f
1787	String quartet Op. 50, No. 4. 2: Andante	ABA1B1A2	A	a
1788	Symphony No. 90. 2: Andante	ABA1B1A2coda	F	f
1788	String quartet Op. 55, No. 2, "The Razor". 1: Andante più tosto Allegretto	ABA1B1A2B2	f	F
1789	Piano trio H. XV:13 in C minor. 1: Andante	ABA1B1A2B2	c	C
1793	Variations for solo piano in F minor, H. XVII:6. Andante	ABA1B1A2B2A* with extensive coda. This work is widely admired by commentators; Sisman calls it the "most profound" of all of Haydn's alternating variations.	f	F
1793	String quartet Op. 71, No. 3. 2: Andante con moto	ABAA1B1A2coda	B♭	b♭
1794	Piano trio H. XV:19 in G minor. 1: Andante	ABA1B1 followed by a second quasi-variation on B in Presto tempo, expanded to full sonata form. For discussion of this expansion, see Rosen (1997:83–88).	g	G
1795	Piano trio H. XV:23 in D minor. 1: Andante molto	ABA1B1A2B2 with coda	d	D
1795	Symphony No. 103, "The Drumroll". 2: Andante più tosto Allegretto	ABA1B1A2B2 form, with a long coda based on B. The themes are said to be based on Croatian folk tunes.	c	C

The double variation in Beethoven

Although the double variation is associated strongly with Haydn, Elaine Sisman has pointed out that, provided we adopt a somewhat looser definition of the form, Ludwig van Beethoven also emerges as a major composer of double variations.^[1] With the partial exception of the Piano Trio in E flat major, Op. 70 No. 2, which Sisman sees as an homage to Haydn, Beethoven's double variations have a rather different character. For instance, sometimes only the A theme is strongly varied, with B remaining relatively constant. Beethoven also likes to interrupt or truncate one or both themes, producing a less regular structure than Haydn's, seen in the often-complex structural formulae given below.

Thus flexibly construed, the double variation emerges as the musical form for some of the most famous of Beethoven's works. Here is a list of movements for which Sisman argues that a double-variation structure is present.

Year	Work	Form	A	B
1802	Third Symphony. 4: Allegro molto	Intro-A-A1-A2-B-Ax-B1-A3-B2-Ax1-B3-B4-coda	various, centered on E♭	various, centered on E♭
1808	Fifth Symphony. 2: Andante con moto	A-B-A1-B1-A2-cadenza on A-B2-A3-A4-coda based on A	A♭/once in A♭ minor	A♭-C, A♭-C, C
1808	Piano Trio Op. 70, No. 2. 2: Allegretto	A-A1-B-A2-B1 with coda	C	c
1812	Seventh Symphony. 2: Allegretto	Intro(1 chord)-A-A1-A2-A3-B-A4-A5-B1-coda	a	A to C, A
1824	Ninth Symphony. 3: Adagio molto e cantabile	A-B-A1-B1-A2-A3-coda	B♭/once in E♭	D, then G
1825	String Quartet No. 15, Op. 132. 3: Molto Adagio — Andante ("Heiliger Dankgesang")	A-B-A1B1A2	F Lydian, notated C	D

As Sisman notes, Beethoven placed his double variations in the same genres as Haydn: the piano trio, the string quartet, and the symphony.

Later double variations

After Beethoven, the double variation appears to have been only seldom employed. The following list is ordered chronologically.

Brahms

The second movement of Johannes Brahms' String Quintet No. 1 (1882) is described by Joanna Wyld^[4] as a set of double variations.

Bruckner

The second movement of Anton Bruckner's Seventh Symphony (1883/1885) is described by A. Peter Brown^[5] as a set of double variations.

Dvořák

The Larghetto movement of Antonín Dvořák's String Quintet Op. 97 (1893) is described by Colin Lawson ^[6] as a set of double variations.

Other senses of the term "double variation"

Distinct variations for repeated sections

Occasionally, authors on music use the term "double variation" with a different meaning. This definition presupposes that the theme consists of two parts, each one repeated (that is, AABB). In a double variations of this kind, each repeat gets its own variation, as shown below:

AABB A₁A₂B₁B₂ A₃A₄B₃B₄ ...

Alternatively, some of the variations can be single (A_xA_xB_xB_x) and others double.

An example of this usage is found in Cedric T. Davie's discussion ^[7] of the last movement of Beethoven's Piano Sonata Opus 109, in which some but not all of the variations are double in the intended sense. The full formula for this movement (adapting Davie's verbal description) is:

Theme: AABB

I. A₁A₁B₁B₁

II. A₂A₃B₂B₃

III. A₃A₄B₃B₄

IV. A₅A₅B₅B₅

V. A₆A₇B₆B₇

VI. A₈A₉B₈B₉

Coda, incorporating the original AABB

The two kinds of "double variation" are not mutually exclusive. In Haydn's Piano Trio H:13, the first movement is a double variations in the first sense given in this article (that is, it takes the form ABA₁B₁A₂B₂), and the last variation of the B theme (B₂) is a double variation in the second sense, with different treatment of the repeats in each half of the theme. There appears to be no standard nomenclature for keeping the two senses distinct.

"Double" as designating a single variation

In the Baroque dance suite, a dance movement was sometimes immediately followed by a single variation, which was called the "double".^[8] A widely known example is the first partita from Johann Sebastian Bach's sonatas and partitas for solo violin where each of the four dance movements is followed by a double that elaborates on the bass-line of the previous piece.

Notes

1. Sisman (1990)
2. Charles Rosen writes, "Haydn's double variations are almost never intended to sound as if they contain two distinct themes; the second melody appears as a free variation of the first, and the form is that of a monothematic rondo." Rosen (1997:331)
3. Hinson, Maurice, ed. (1991) Haydn: The Complete Piano Sonatas. In three volumes. Van Nuys, CA: Alfred Publishing.
4. Wyld, Joanna (2009). "Sleeve Notes: Nash Ensemble – Brahms String Quintets". Archived from the original on 25 July 2011. Retrieved 13 April 2010.
5. Brown, A. Peter (2003) The Symphonic Repertoire. Volume 4. the Second Golden Age of the Viennese Symphony: Brahms, Bruckner, Dvořák, Mahler, and Contemporaries. Bloomington, Indiana: Indiana University Press.
6. Lawson, Colin (2003) "The string quartet as a foundation for larger ensembles," in Robin Stowell and Jonathan Cross, eds, The Cambridge Companion to the String Quartet, pp. 310–327. Cambridge: Cambridge University Press.
7. Davie, Cedric T. (1953) Musical Structure and Design. London: Dennis Dobson. A more recent reprint is available from Dover Publications.
8. Geiringer, Karl and Irene Geiringer (1966) Johann Sebastian Bach: The Culmination of an Era. Oxford: Oxford University Press. P. 125.

References

• Rosen, Charles (1997) The Classical Style, 2nd. ed. New York: Norton.
• Sisman, Elaine (2001). "Variation". In Sadie, Stanley; Tyrrell, John (eds.). The New Grove Dictionary of Music and Musicians (2nd ed.). London: Macmillan Publishers. ISBN 978-1-56159-239-5. ‎
• Sisman, Elaine R. (1990) "Tradition and transformation in the alternating variations of Haydn and Beethoven," Acta Musicologica 62:152–182.