VARIATIONS, in music, the term given to groups of progressively developed versions of a complete self-contained theme, retaining the form of that theme though not necessarily its melody. This at least is the classical sense of the term, though there are modern developments of the variation form to which this definition is at once too broad and too precise to apply. The aesthetic principle of variations appeared at very early stages of music; and it soon became something far more definite than the use of ornamental versions of a melodic phrase, a use which must have been natural almost as soon as music was articulate at all. During the 16th century principles aesthetically indistinguishable from some types of variation-form inevitably arose in the polyphonic treatment of Gregorian hymns verse by verse. Accordingly, the hymns and Magnificats of Palestrina might without great extravagance be described as contrapuntal sets of variations on ecclesiastical tunes, like very free examples of the type shown later in extreme simplicity and formality by Haydn's variations on his Austrian national anthem in the " Emperor " quartet (Op. 76, No. 3).
Already in the 16th century instrumental music was assuming such independence as it could attain by means of a primitive variation-form, growing partly out of the habit of playing vocal madrigals on the virginals or similar keyed instruments, or singing the top part as a solo to an instrumental accompaniment, with an overwhelming weight of ornaments beneath which the original madrigal was quite unrecognizable. (See, for example, the " diminutions " given in the 30th volume of Breitkopf & Hartel's complete edition of Palestrina's works.) A favourite plan, of which numerous examples may be found in the Fitzwilliam Virginal Book, was to put together several popular or original tunes, with an ornamental variation sandwiched between each. Sometimes, however, sets of variations on a single tune were produced, with essentially modern effect, as in Byrd's variations on " The Carman's Whistle." Such variations were naturally grouped in order of increasing complexity and brilliance. Some of the keyboard passages in which the early English variation-writers indulged are of extraordinary difficulty, even from the standpoint of modern pianoforte technique.
In the 17th century a highly artistic form of variation arose, very favourable to the earliest composers of the transition period, because of the simplicity of its principle, which relieved the composer of all the graver problems of formal organization. This was the ground-bass, a single phrase placed in the bass and repeating itself as long as the composer had fresh harmonies and superstructure with which to vary it. In typical examples the ground-bass was derived from the dance forms of the passacaglia and the chaconne, which in classical music resembled each other in being in slow time, and did not otherwise differ markedly, except that in the passacaglia the theme could be transferred now and then to the treble or to an inner part, a purely natural aesthetic resource which makes no radical difference to the art-form. The genius of, Purcell was cruelly hampered by the lack of possibilities for organizing large musical forms in his time, and nothing is more significant than the avidity with which he seizes upon the ground-bass as a means of giving coherence to his ideas.
By the time of Bach and Handel a lighter type of variationwork, less capable of high organization, and more like Byrd's variations on " The Carman's Whistle," had arisen. Bach's Aria variola alia maniera Italiana is an instance of this; and so is the air et doubles that appears now and then in Handel's instrumental works. The principle of this form is simply to take a symmetrical melody (generally in binary form) and embroider it. Such variations are called doubles whenever each variation divides the rhythm systematically into quicker notes than the one before. The most familiar example is that known as " The Harmonious Blacksmith " in Handel's E major suite. Sometimes the air itself was stated in a tangle of ornamentation, while the doubles made it float in a simplified form over an accompaniment of increasingly rapid flow. (See, for example, Handel's D minor suite and the little set in B flat on a theme afterwards varied in the noblest modern style by Brahms.)
But Bach had meanwhile applied the principle of the groundbass to variations on a complete symmetrical movement in binary form. His Air and 30 Variations, commonly known as the " Goldberg " variations, is (with the exception of Beethoven's 33 Veriinderungen on a waltz by Diabelli) not only the most gigantic set of variations in the world, but one of the three largest compositions in any form ever written for a single instrument. Of course in so large a work the conception of the ground-bass, as a clearly recognizable theme repeated with no more than slight ornament, would be inadequate whatever the variety of the superstructure: but so steady is the drift of Bach's bass that he is enabled to represent it by countless alternative harmonies and analogous chromatic progressions, without weakening its individuality. The grouping of the thirty variations is extremely subtle in balance and climax ; the more so because there are no means within the terms of Bach's art for making a free coda to the work, his ground-bass being both too long and too purely a bass to be taken as the theme of a fugue, like that in his great passacaglia for organ. Yet Bach contrives to round off the work perfectly by the simple direction aria da capo at the end. There is no question of retaining or varying the melody of the aria, which indeed is so ornamental as to be pointless and unrecognizable as a basis for variations; nor could it, like the above-mentioned Italian examples of Handel, be simplified, since most of its ornaments are integral parts of the phrases.
The next chapter in the history of the variation form is intimately connected with the sonata style. A set of variations used as a movement for a sonata inevitably tends to be variations on the melody. The sonata style implies the identification of themes by their melodies rather than by their texture, the very term " theme " being primarily used in a melodic connotation (see MELODY). Hence a set of exclusively harmonic variations would not be in the sonata style. Now, most of the best sets of variations by Mozart and Haydn are movements in their sonata works; and this should always be remembered in discussing the tendency of their treatment of the form. Few of their independent sets are of any importance, since most are very early works, or were written for pupils, or intended as encore pieces for concerts. Haydn shows a great fondness for a special form which, even if earlier specimens can be found, he may properly be said to have invented. It consists of alternating variations on two themes, the first a highly organized complete binary melody, and the other a shorter binary melody, often beginning with the same figure as the first, but clearly contrasted with it, inasmuch as, whichever theme is in the major, the other is in the minor. 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; but the effect is always that of a happy blend of a clearly marked variation form with a more contrasted scheme a little more highly organized than the roundand-round symmetry of a minuet and trio, but not so elaborate as a rondo. The only later example exactly corresponding to Haydn's form is the first allegretto of Beethoven's pianoforte trio in E flat, Op. 70, No. 2; although, with a wider range of key, a free application of the principle of alternating themes is magnificently illustrated by the slow movement of his C minor symphony.
Beethoven in his last works invented another variation-form on two themes, in which the first theme is very free in structure and the second theme is a more rigid melody in a different key and time. The examples of this are the slow movement of the gth Symphony and the Lydian figured chorale in the A minor quartet. A fine later development of this is the slow movement of Brahms's F major string quintet, Op. 88, in which the alternation of the two keys gives rise, in the last line of the movement, to one of the most astonishing and subtle dramatic strokes in all music.
In sonata works, Beethoven's examples of the normal variation form based on a single theme are as wonderful as may be expected from him; but nothing is more significant than his strict adherence in sonata works to the melodic principle of variation. He uses the form as an unsurpassable means of obtaining repose in slow movements. The extreme case of this is the slow movement of the sonata, Op. 57 (commonly called Appassionato), which is described in the article on SONATA FORMS. In this and in many other instances, his method is aesthetically that of the air et doubles, as being the simplest possible means of obtaining variety and climax without leaving the fundamental key. Until his latest works, such sets of variations are never finished. Their dramatic force is that of a repose which is too unearthly to last; and at the first sign of dramatic motion or change of key the sublime vision " fades into the light of common day," a light which Beethoven is far too great an idealist to despise. .(See the andante of the B flat trio, Op. 97 ; and the slow movement of the violin concerto, which contains two episodic themes in the same key.) In his later works Beethoven found means, by striking out into foreign keys or foreign rhythms, of organizing a coda which, as it were, finally spins down in fragmentary new variations, or even returns to the plain theme. Thus he was able to end his sonatas, Opp. 109 and in, with solemn slow movements in which, with the utmost richness of detail and novelty of idea, the melodic variation form is nevertheless paramount. Beethoven also found many ways of combining melodic variations with the principles of the rondo and other more highly organized continuous movements. Thus the finale of the Eroica Symphony has not only the theme but many ideas of the variations and fugue-passages in common with the brilliant set of variations for pianoforte on a theme from Prometheus, Op. 35; and the Fantasia for pianoforte, chorus and orchestra, and the choral finale of the pth Symphony, are sets of melodic variations with freely developed connecting links and episodes. In the case of the 9th Symphony, a second thematic idea eventually combines with the figures of the first theme in double fugue.
But Beethoven's highest art in variation-form is to be found in his independent sets of variations. In some of the earliest of these, notably in the 24 on a theme by Righini (which was his chief bravura performance as a young pianoforte player), he far transcends not only the earlier or sonata-form idea of melodic variations, but fuses their resources with those of the ground-bass, and adds to them his own unparalleled grasp of rhythmic organization. Beethoven is the first composer who can be said to have discovered that a theme consists not only of melody and harmony but of rhythm and form. With earlier composers the form of the theme was automatically preserved in consequence of the preservation of either its melody or its harmony; but Beethoven had an unerring judgment as to when the form of a theme might be definite enough to remain as a basis for a variation which departed radically from both the harmony and the melody. The climax in the history of variations dates from the moment when Beethoven was just about to begin his 9th Symphony, and received from A. Diabelli a waltz which that publisher was sending round to all the musicians in Austria so that each might contribute a variation to be published for the benefit of the sufferers in the late Napoleonic wars. Diabelli's theme was absurdly prosaic, but it happened to be perhaps the sturdiest piece of musical anatomy that Beethoven or any composer since has ever seen. Not only was its harmonic form exceptionally clear and firm, but its phrase-rhythm was as simple, recognizable and heterogeneous as its other qualities. Its melodic merit was nil, yet it had plenty of recognizable melodic figures. All these prosaic technicalities are far more likely to impress a great composer as good practical resources than those high poetic qualities which critics discuss incessantly, but which are to a great artist the air he breathes. Diabelli's waltz moved Beethoven to defer his work on the gth Symphony ! The shape of Diabelli's theme may be illustrated by a diagram Tonic. Dominant. Rising sequence. Close in dominant.
which represents its first sixteen bars; the upright strokes being the bars, and the brackets and dots (together with the names underneath) indicating the way in which the rhythm is grouped by correspondence of phrase and changes of harmony. The second part also consists of sixteen bars, moving harmonically back from the dominant to the tonic, and rhythmically of exactly the same structure as the first part. This harmonic and sequential plan, together with this straightforward square tapering rhythmic structure, is so formal in effect that Beethoven can substitute lor it almost anything equally familiar that corresponds in its proportions. Thus, the alternation of tonic and dominant in the first =ight bars may be represented by another familiar form in which three bars of tonic and a fourth of dominant are answered by three bars of dominant and a fourth of tonic; as in variation 14 (which must be reckoned in half-bars). Again, the antithesis of tonic and dominant is accompanied in Diabelli's theme by a part of the melodic figure being repeated a step higher at the change of harmony ; and th ; s naturally produces such devices as the answering of the tonic by the supertonic in variation 8, and, still more surprisingly, by the flat supertonic in variation 30. In so enormous and resourceful a work, occupying fifty minutes in performance, it is natural that some variations should drift rather farther from the anatomy of the theme than can be explained by any strict principle ; and so the jocular transformation of the beginning of Diabelli's bass into the theme of Mozart's Nolle e giorno faticar leads to a couple of extra bars at the end of its second part; otherwise the fughetta (variation 24) and variations 29 and 31 are the only cases in which any considerable part of the structure of the theme is lost, except the fugue (variation 32), which is simply an elaborate movement on a salient feature of what must by courtesy be called Diabelli's melody. A free fugue is a favourite solution of the difficult problem of the coda in a set of variations.
But for the works of Brahms, which invariably retain the classical conceptions while developing them in a thoroughly modern and living language, it can hardly be claimed that the art of variation-writing has advanced since Beethoven. The term is now used for a somewhat nondescript method of stringing together a series of short fantasias on a theme; a method which may be legitimate and artistic in individual cases, but hardly constitutes an art-form. There is this great disadvantage in variations that neglect the anatomy of the theme, that the only way in which, in the absence of other means of connexion, they can show any coherence at all is by more or less frequently harping on scraps of the melody. The effect is (except in unusually happy examples such as the Etudes symphoniques of Schumann and the Enigma Variations of Elgar) curiously apologetic; because no ambitious composer in the "free" modern variation style thinks a melodic variation quite worthy of his dignity, and so the melodic allusions become the more tiresome from their furtive manner. Many " advanced " specimens of variation-form undoubtedly owe their origin to a vague impulse of revolt from the unsound statements of unobservant writers of mid-19th century textbooks, who contented themselves with laying down crude rules such as that a variation might " either retain the melody and change the harmony, or retain the harmony and change the melody," etc., without any attempt to see how the classical composers really analysed their themes. It is very characteristic of Schumann's -modesty and grasp of facts that he, who was the first to produce serious art in a free non-anatomical variation style, did not call his experiments variations without qualification. He never wrote a set in which the anatomy of the theme was of real importance to the whole; and, with him, whenever at least the initial melodic figure of his theme is not traceable throughout a section, that section is simply an episode. But Schumann knows this perfectly well, and acknowledges it. The Etudes symphoniques are called variations only in those sections which are fairly strict variations. Elsewhere they are simply numbered as etudes. The slow movement of the F major string quartet (in which a second theme masquerades as the first variation, and some of the other variation-like sections are quite free) is called andante quasi variazione; and even the strictest of all his variation works is called Impromptus, on a theme by Clara Wieck, Op. 5. There is, no doubt, great scope for a variation-form which is neither melodic nor anatomic, and we have not a word to say against the legitimacy of many forms of effective modern fantasia-variations; but the fact remains that it is very hazardous to talk of an " advance " in the variation-form, when even the best fantasia-variations are not only unconnected with any classical type but evidently unable to get nearly as far from either the melody or the harmony of their theme as the 25th of Bach's " Goldberg " variations or many variations in the earliest sets by Beethoven. Indeed, the only sound classification of composers of modern variations, from the time of Mendelssohn onwards, is that which distinguishes the composers who seem to know their theme from those who do not. (D. F. T.)
Note - this article incorporates content from Encyclopaedia Britannica, Eleventh Edition, (1910-1911)