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Structural Musical Form
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 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." 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. 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.

According to Sisman, 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". YearWorkFormAB 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

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. as a set of double variations.

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

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

In the Baroque dance suite, a dance movement was sometimes immediately followed by a single variation, which was called the "double". 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.