Again, let us regard the period of the theme not as an orbit but as diurnal rotation. We can then describe the codas of Brahms's Paganini-variations as produced by accelerating the spin till it breaks away for a while and then resumes for a few final catastrophic whirls; exactly like a dying top (though this, of course, does not accelerate its spin). Without acceleration Beethoven ended his wonderful C minor variations (most perfect of passacaglias) in this way. Brahms found in Haydn's Chorale St. Antoni the opportunity for another method. He took the first five bars as a ground-bass, within which narrow orbit the finale moves until its climax broadens out into the rest of the glorious theme, and so rounds off the whole work.
Bach poised the contrasts and climaxes of the Goldberg varia tions so accurately that the ending of the whole by a simple da capo of the theme is astonishingly effective. It is as if a charming old ancestress of a living line of great folk were to step from the frame of her Holbein portrait and bow to her assembled posterity.
To speak of the progress in variation-form since Beethoven is like speaking of the progress in reinforced concrete since the Parthenon. The classical variation-form is limited only by the composer's imagination and technique; and the removal of its foundations does not enlarge it at all. There is no reason to con demn other kinds of variation; and many great and beautiful works in non-classical variation-form exist, from Schumann's Etudes Symphoniques to Elgar's "Enigma" variations and Dohn any's Variations on a Nursery Song. But no "free" variation that breaks down the phrasing of its theme and follows its own dis cursive ways will ever achieve anything externally so unlike the theme as a strict harmonic and rhythmic variation on classical lines. (See Ex. 2b.) Nor will a series of such variations acquire anything like the classical momentum. On the contrary, in clumsy
hands the free variation becomes apologetic in the way in which it offers raw chunks of the original melody as evidence that it has not forgotten its duty, like Lewis Carroll's poetic Tema con V ariazioni, the preface to which is an unconscious epitome of modern misunderstandings of the form.
Variation writers may be scientifically classified into those who know their theme and those who do not. There is no reasonable doubt that many very clever composers, from Mendelssohn on wards, have completely misunderstood the nature of the deeper classical variations, and have thought that anything so unlike the original tune must be quite independent of it. Mendelssohn's Variations serieuses have a beautiful theme with a structure that might have given rise to splendid features; but Mendelssohn sim ply ignores this structure and replaces it by weaker things in almost every variation. Schumann shows more insight. He has no great grip of his theme, but he tries to distinguish by titles those variations which are true from those which are episodic ; thus in the Etudes Symphoniques the etudes are numbered separately from the variations; the andante of the F major quartet is called quasi variazioni; and the strictest set he ever wrote (on a theme by Clara Wieck) is called Impromptus.
Brahms stands alone in his grip of his theme. Reger is no nearer the classical form in his variations than in his other works. The present state of the form seems to indicate that if the composer does not aim at strict variations his most vital results will be on the line of melodic development, as in the above-mentioned works of Elgar and Dohnanyi, the Symphonic Variations of Dvofak, and those variations of Reger which are closest to this type. (D. F. T.) VARIATIONS, CALCULUS OF: see CALCULUS OF VARIATIONS.