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VARIATIONS, in music, the term given to groups of pro gressively developed versions of a complete self-contained theme, retaining the form of that theme though not necessarily its melody. This is the classical sense of the term, but there are modern developments of the variation form to which this defini tion is at once too broad and too precise to apply. The aesthetic principle of variations appeared at very early stages of music. During the i6th century an artistically mature variation-form automatically rose in the polyphonic treatment of Gregorian hymns verse by verse. Accordingly, the hymns and Magnificats of Palestrina might be described as contrapuntal sets of varia tions on ecclesiastical tunes, like rich and free examples on the simple.plan shown later by Haydn's variations on his Austrian national anthem in the "Emperor" quartet (op. 76, No. 3). Already in the i6th century instrumental music was climbing up the trellis of a primitive variation-form. A favourite plan (see the Fitzwilliam Virginal Book, passim) was to put together sev eral popular or original tunes, with an ornamental variation sandwiched between. Sometimes sets of variations on a single tune were produced, with excellent effect, as in Byrd's variations on "The Carman's Whistle." Such variations were naturally grouped in order of increasing brilliance, and they often include passages that would catch the greatest pianoforte players.

In the 17th century a highly artistic form of variation solved with great simplicity the problem of expanding instrumental pieces to a length admitting of growth to a big climax. This was the ground-bass, a single phrase placed in the bass and repeating itself ad infinitum. It originated in the dance forms of the passacaglia and the chaconne. Both were in slow triple time, the chaconne having a strong accent on the second beat, while the passacaglia, by some chance, developed the liberty to transfer its theme to other parts than the bass. The genius of Purcell was cruelly hampered by the non-existence of large musical forms in his time, and he seized upon the ground-bass with avidity. By the time of Bach and Handel lighter sets of variations, consisting essentially of embroidery on a melody, haci come into vogue. Bach's Aria variata ally maniera I taliana tells us where this fashion began ; and in France the air et doubles was taken over from early English virginal music. Doubles are variations each of which divides the rhythm into quicker notes than the one before. The most familiar example is that known as "The Harmonious Black smith" in Handel's E major suite. Sometimes the air itself was stated in a tangle of ornamentation, while the doubles simplified the melody and varied the accompaniment. But Bach had mean while applied the principle of the ground-bass to variations on a complete symmetrical movement in binary form. His Air and 3o Variations, commonly known as the "Goldberg" variations, is (with the exception of Beethoven's 33 V eriinderungen on a waltz by Diabelli) the most gigantic set of variations in the world.

A melodically interesting ground-bass could not be maintained on so large a scale; but the 32 bars of Bach's theme are so many clear harmonic steps which can be represented by many analogous progressions, without loss of identity. (Ex. Ia.) There is no question of retaining or varying the melody of the aria, which is a tissue of ornaments that will bear neither development nor simplification.

The rise of the sonata style again brought the melodic em broidery variation into prominence ; for in sonata forms we iden tify themes entirely by their melodies. Now, with not more than three or four exceptions, the best sets of variations by Mozart and Haydn are movements in their sonata works; and their inde pendent sets are either early or perfunctory exercises and encore pieces. Two common mistakes of professional and amateur criti cism are, first, the judging of Haydn's and Mozart's variations by these parerga, and secondly, the much graver error of despis ing the embroidery variation on principle. It is either vulgar or sublime. And it is handled lovingly by precisely the greatest masters of deep harmonic and rhythmic variation, Beethoven and Brahms. Haydn is fond of a special form first known in Philipp Emanuel Bach. It consists of alternating variations on two themes, alternately major and minor; the first a rich and complete binary melody, and the other a shorter binary melody, often beginning with the same figure as the first. The first theme usually returns as if it were going to be unvaried, but its first repeat is an ornamental variation. The form is rarely worked out far enough to include more than one variation of the second theme; and sometimes (as in the famous "Gypsy" trio) there are new episodes instead of variations of the second theme, so that the form becomes a sectional rondo. The only strict example of Haydn's type of alternating variations in later music is the first allegretto of Beethoven's pianoforte trio in E flat (op. 7o, No. 2) ; but a magnificent application of it, without change of mode, though with a wide range of key, is shown in the slow movement of his C minor symphony.

Beethoven, in his last works, invented another variation-form on two themes, of which the second is in a different key and time.

The examples of this are the slow movement of the 9th symphony and the Lydian figured chorale in the A minor quartet. In the slow movement of Brahms's F major string quintet (op. 88), the alternation of the two keys gives rise, in the last line of the move ment, to one of the most astonishing dramatic strokes in all music. Beethoven uses embroidery variations as means of obtain ing extraordinary repose in slow movements. The extreme case of this is the slow movement of the sonata op. 57 (commonly called "Appassionata"), which is described in the article on