The second serious unitary form is the theme and variation. In this case the theme is itself a complete song form of perhaps two or three periods, and each variation is precisely of the same number of measures, and follows the same harmonic structure in many cases.
There are, however, in modern use, two types of the variation form. One of these, called formal variations, leaves the harmony entirely the same in all the variations, except, perhaps, to change the melody from major to minor of the same key and back again. In the best examples the harmony remains entirely unchanged, but the melody is diversified rhythmically in various ways. Good examples of this type of variations are to be found in the works of Mozart and in the second movement of Beethoven's Sonata in G major, opus 14, and in the second movement of the "Sonata Appassionata" of Beethoven.
The character variation pursues a different course. At times the key is changed and the harmony changes very much. In order to see how this can be accomplished without destroying the identity of the musical idea, it should be remembered that a musical idea consists essentially of three elements: it has a rhythm, a melodic figure, and a harmonic foundation. If the melodic figure is retained, and the harmonic figure, the rhythm can be diversified indefinitely, and, in fact, if any two of these elements are retained the third can be modified very much. In the latest practice of variation writing, two of these elements are changed at the same time, leaving only one element fixed, and in some instances it is difficult to find exactly where any of the original element of the theme remains. Beethoven began the development of character variation in his Sonata, opus 26, the first movement of which is in this form. He also did more or less in this direction in his famous "Thirty-two Variations." The variations of Schumann in the "Études Symphoniques" pass even beyond the bounds here defined. While remaining fast upon the original harmonic foundation, measure for measure, entirely new melodies come in and wholly different rhythms, so that in many instances only a few notes of the original theme are retained in any one variation. The student desiring to explore the most advanced variation writing will find examples ready to his hand in Brahms' variation on a theme of Händel and the two books of variations on a theme of Paganini. These may be considered as at present the ultima thule of variation-making art. The principle of the variation lies at the foundation of very much that meets us in the higher departments of music, even when the variation form is not heard of. All modifications or amplifications of a theme belong essentially to the variation type, and it is liberally applied to all long compositions where the same material is used a number of times.
When a piece of music consists of two fully developed melodic ideas, it is said to be binary in form, and these are all either song forms with trio or small rondo forms. Nearly all of the slow movements of Beethoven in the sonatas are binary forms, the dimensions of which may vary extremely. The student desiring to investigate this part of the subject more thoroughly is referred to the "Primer of Musical Forms," by W. S. B. Mathews (Arthur P. Schmidt & Co., Boston), where the principles are more fully unfolded.
There is one form in modern music which is the type of so large a proportion of extended instrumental movements that the student will do well to master its peculiarities at the earliest possible moment. This is the form sometimes called the sonata form or sonata-piece. The term sonata was originally used in two senses: in its larger sense it indicates an extended musical composition with three or four movements, all which taken together form the sonata. By the term sonata-piece, however, is meant the particular movement of the sonata which gives the name to the whole piece. This, as a rule, is the first movement, but sometimes it is the closing movement, and in some instances there are three of these movements in the same sonata, so arranged with reference to one another as to form the necessary contrasts. The sonata-piece is the form which contains within itself very much of the essence of all the smaller forms. It generally consists of three large chapters, beginning with the principal subject, which may be longer or shorter, according to the fancy of the composer, and may end on its own principal key or on the dominant, and may be followed by passage work or not, to any extent the composer chooses; then comes a second subject. According to Beethoven's almost invariable practice, the first subject of a sonata form is thematic in its character, and in developing this theme many of the principles of variation work are applied. The second subject is almost invariably a lyric melody, sometimes very charming, and always in a different key from that of the first subject, usually in the dominant; or, if the first subject is a minor, this will be in the relative major. Then follows a concluding paragraph of anywhere from six to thirty measures, and a double bar with a repeat sign. This forms the first chapter of the sonata-piece. After the double bar comes the second chapter, which is an elaboration or free fantasia on the material of the first part. At the end of this free fantasia, which may be longer or shorter at the fancy of the composer, comes the recapitulation, or the repetition of the entire first part, the only change being that the second subject is now in the principal key. In the elaboration of the sonata all sorts of musical fancies are liable to appear—queer juxtapositions of motives from the different parts of the first and second subjects, inversions, variations, and so on.
The sonata-piece is the type, not alone of the principal movement in symphonies and chamber quartets and trios, but it is also the type of all serious overtures, and therefore it has been well designated by German theorists as the Principal Form of modern music.
Whether longer or shorter, whether serious or lively, all musical forms have the same conditions to satisfy—viz., those of unity, or the preponderance of a single idea; symmetry, or just proportion of parts; and variety, the proper relief and introduction of new material. If the principal idea is repeated too much, monotony ensues; if there are too many accessory ideas, in place of variety we have looseness and want of unity. And in carrying out these principles in compositions of different lengths and in different styles, the composer has practically unlimited freedom.