No doubt part of the enduring attraction of the variation principle lay in its very starkness. Whereas such dialectical procedures as fugue or dramatic forms as sonata tended to focus interest upon a composer's command of longer-term continuity, the constant repetition in variation form of at least some aspect of an initial theme provided an exact measure of a composer's inventiveness from bar to bar. And, of course, deciding which aspect of a theme to retain could itself have interesting consequences. Taking the top line, the tune, as a vehicle for progressively more florid decoration was only ever the most obvious option. It was favoured, though, for equally obvious reasons, by instrumental virtuosos all the way from the frantic scurryings of the 17th-century English division-violists by way of the young Chopin pianistically bedizening Mozart's La ci darem la mano ('Hats off, gentlemen] a genius,' responded Schumann) to the swoonings and rampages of Rachmaninov's Rhapsody on a Theme of Paganini - which, despite its title, amounts to quite a tight set of variations.
But most tunes imply an accompaniment, even if they do not come with one, offering the alternative possibility of keeping the melody more or less constant and progressively changing the harmony. For that matter, both tune and harmony may be varied over a regularly repeating bass line - a form of continuous variations which the Baroque evolved under the title of ground, chaconne, or passacaglia. Not least, the tune, harmony and bass whose inter-relations comprise the structure of a theme, can all be variously mutated according to their own characteristics, provided the overall shape of the structure - its proportions, phrasing or bar-count - is itself kept intact.
Most of these techniques are already to be found in sets by that earliest great master of variation form, William Byrd. But as so often, it was Bach who was to provide the most compendious synthesis of possibilities up to his time in his Aria mit verschiedenen Veranderungen ('Aria with different Alterations'), reputedly composed for the harpsichordist, Johann Goldberg, to beguile an insomniac patron.
As published in 1742, the 30 so- called Goldberg Variations are laid out with a grand symmetry; every third variation developing a different kind of canon over the harmonic bass of the source Aria, which frames the sequence as a whole. Yet it is probable that Bach still considered it more a collection to be selected from than an immutable sequence of inventions to be played complete at a sitting. Only later in the 18th century did composers start trying to invest such sequences with at least a latently symphonic overall sweep.
Haydn's frequent recourse was to so-called double, or alternating variations on two, often highly contrasting themes; a procedure which, in his plangent Piano Variations in F minor of 1793, generated such tension that he found himself having to add an extended finale to resolve it. Yet Beethoven was able to wrest such extreme contrasts from the single theme of his Diabelli Variations for piano of 1823 as to demand a kind of double finale comprising a tumultuous fugue balanced by an ineffably elegant minuet. The sets of variations that meanwhile found their way into many a classical concerto and symphony - epitomised by the double-variation slow movement of Beethoven's Ninth Symphony - seemed, by contrast, to serve a relatively stabilising function between the more volatile procedures of sonata and rondo form. And it was not until the appearance of Brahms's Variations on a Theme of Haydn (sic) in 1873 that orchestral variations emerged as a self-sufficient form in their own right.
No one to this day is quite certain who really composed the 18th-century tune Brahms used. But, intrigued by its subtle asymmetries, he took it as a pretext for reasserting the ancient techniques of variation writing at their strictest, culminating in a finale on a repeating bass. His particular target was the altogether looser early Romantic development of 'fantasy variations' in which mere fragments of an initial theme might be used to generate a semi-independent sequence of character pieces. Yet so successful was Brahms in disguising the scholastic ingenuities of his Haydn Variations that its initial impact was rather to encourage the composing of orchestral 'fantasy variations' too - the most successful being Richard Strauss's picturesque Don Quixote of 1898 and Elgar's friendly 'Enigma' Variations of the following year.
All the same, the austere rigours of Webern's Variations for Orchestra (1940), Copland's Orchestral Variations (1957) and Stravinsky's late Variations: Aldous Huxley In Memoriam (1964) - to say nothing of Elliott Carter's grand Variations for Orchestra (1955) based effectively on three themes - proclaim pretty conclusively Brahms's achievement in establishing orchestral variations as a kind of demonstration form. And never more so than in the instance of Schoenberg.
After the post-Wagnerian flux of his early Romanticism, after the Expressionist delirium of his middle period, Schoenberg thought he had at last struck upon a new way of ordering music through the forming of the 12 tones of the basic scale into a specific configuration for each work. This series of notes, variously combined with itself, could be used as a source for all the themes and harmonies. Having tried a few preliminary 12-tone piano pieces and a more extended Wind Quintet, he felt by 1926 it was time for a large-scale orchestral form - significantly choosing not concerto or symphony, but variations.
Doubtless there are listeners who continue to find that sectional structure of variation form itself frustrating to their sense of flow. But frequent stops and starts may seem a positive help in coming to terms with a manner so intense and concentrated as Schoenberg's. And in the Variations for Orchestra, Op 31 he takes every care to clarify his procedures - poetically assembling the pitches of his 12-tone row one by one in the introduction, and throwing in a counter-subject on the notes BACH to signal his aim of continuing a great tradition. The long cello theme itself, though entirely serial in content and accompaniment, preserves a late Romantic profile. And though some of the ensuing variations, to say nothing of the last pages of the finale, are as abrasively constructivist as anything he wrote, they are interspersed with variations of enchanting delicacy, ashimmer with sounds of harp, mandolin and celesta.
If the work still reaches the concert hall fairly infrequently, this may be at least partly because of its fearsomely virtuoso demands on sections of the orchestra. Yet Furtwangler was proud to conduct the premiere in 1928 with the Berlin Philharmonic, and post- war advocates have included not only Solti and Boulez but, more surprisingly, Karajan. The determination and real insight with which Simon Rattle has promoted Schoenberg since the outset of his career has proved one of his most admirable if least marketable traits. His readings of Op 31 in his 'Towards the Millennium' programme with the CBSO in Symphony Hall, Birmingham, next Wednesday; St David's Hall, Cardiff, on Thursday; and the Royal Festival Hall, next Saturday, deserve all support.