Variations.

1. The rhetoric of variation.

The idea that models for variation form might be found in rhetoric arises not only from the explanatory vividness and sheer historical staying power of the latter but also from features common to both: their shared modes of display and their understanding of the persuasive power inherent in repetition and ornament. A specific correlation between rhetoric and variation can be grafted on to the common fund of rhetorical knowledge on which composers and theorists can be assumed to have drawn, and takes three forms: explicit connection, the existence of rhetorical models for the structure of variation form, and the idea of figures and figurations as flexible tools for analysing variations. As Abbé Vogler wrote in 1793: ‘Variations are a type of musical rhetoric, where the given meaning appears in different guises, with the distinction that the boundary lines are much more rigorously determined in music than in oratory’ (Verbesserung der Forkel'schen Veränderungen über … ‘God Save the King’, 1793, p.2).

Models for the structure of theme and variations come from the ars praedicandi, the medieval rhetoric of preaching, still alive in the 19th century. To construct a sermon, one was advised to choose a theme, a quotation from scripture, and then illuminate and amplify it in a series of divisions (in English, a term for variations). Each division could be a word from the quotation. In the second volume of Joseph Riepel's Anfangsgründe zur musicalischen Setzkunst (1755), the student and teacher argue over how much contrast to allow in a musical composition, with the student claiming that the composer, like the preacher quoting scripture, must stick to his theme, and the teacher claiming that digressions strengthen the sermon if the theme remains in memory. A second source for the linkage of variation form and rhetoric is the widely circulated 16th-century treatise by Erasmus on abundant language or copiousness (De copia, 1512), based on the necessity for developing the ability to say the same thing in different ways and drawing extensively on Quintilian. After providing the means of variety in a list of figures, Erasmus's demonstration included 150 variations of the sentence ‘Your letter pleased me mightily’ and 200 variations on ‘I will remember you as long as I live’. Variations of the first include:

Your epistle exhilarated me intensely.
Your brief note refreshed my spirit in no small measure.
Your pages engendered in me an unfamiliar delight.
Your communication poured vials of joy on my head.
Your letter promptly expelled all sorrow from my mind.
Good God, what a mighty joy proceeded from your epistle.
May I perish if I ever met with anything in my whole life more agreeable than your letter.

In pointing out the dangers inherent in the pursuit of copiousness, Erasmus echoed the concerns of the rhetoricians of antiquity while at the same time foreshadowing the cautions expressed in 18th-century treatises on variation technique, namely the tendency to fall into ‘glibness, which is both silly and offensive’ and to ‘pile up a meaningless heap of words and expressions without any discrimination, and thus obscure the subject they are talking about, as well as belabouring the ears of the unfortunate audience’. The principal advantage of copiousness is that it enables the speaker to avoid literal repetition (‘an ugly and offensive fault’) because ‘nature above all delights in variety’ (‘De duplici copia rerum ac verborum commentarii duo’, Collected Works of Erasmus, Literary and Educational Writings, ed. C.R. Thompson, xxiv: Copia: Foundations of the Abundant Style, ed. and trans. B.I. Knott, 1978).

Rhetorical variations were seen as a means of acquiring and polishing style because a display of copiousness, or varied repetition, might be called into action in many oratorical situations, especially those involved in showpieces known as epideictic or display orations. Cicero's description of display oratory describes the pleasure it gives, the neatness and symmetry of sentences, the ornamentation done ‘openly and avowedly, with no attempt at concealment, so that words correspond to words as if measured off in equal phrases, frequently things inconsistent are placed side by side, and things contrasted are paired, clauses are made to end in the same way and with similar sound’ (Cicero, v: ‘Orator’, ed. and trans. H.M. Hubbell, Loeb Classical Library, 1939, xi.37–xii.38). Cicero could as easily be describing the composer of variations. Indeed, as a means of acquiring and polishing style, variations were sometimes considered an early step in compositional training; Brahms urged his only composition student, Gustav Jenner, to begin with variations. Aristotle identified two other important features of epideictic rhetoric: its reliance on amplification, because the subject concerns ‘actions that are not disputed, so all that remains to be done is to attribute beauty and importance to them’, and its similarity to written prose, making it the only kind of display oratory meant to be read (Art of Rhetoric, I.ix.38–40; III.xii.6). Quintilian discussed amplification in detail, offering techniques for its purpose of revealing in ever stronger terms the importance of the subject (Institutio oratoria, VIII.iv). Thus both display oratory and variation straddle improvised and written forms, and both offer new ways to reclothe the subject.

The nature, application and control of ‘figures’ is a problem considered in almost identical terms by writers on rhetoric and writers on variation form. Both conclude that figures are natural and necessary but must not be overused. The tension between res and verba, or Gedanke and Ausdruck – how the thought is to be clothed in words – is greater in music than in verbal arts: in purely instrumental music, what is the res? Even the theme has its own figural ‘clothing’, as some early variation sets made clear when they called the first segment ‘Prima variatio’. Just as in verbal rhetoric, figures were the means necessary to adorn and make expressive a simpler musical entity, as well as the culprits in freighting it down unnecessarily. Overzealous labelling of motifs by late 16th- and early 17th-century theorists such as Burmeister and their 20th-century counterparts such as Schering, Unger and Gurlitt made the application of rhetorical figures to music appear problematic to more sober 20th-century scholars such as Brian Vickers and Peter Williams. But a figure is not necessarily co-extensive with a motif. Each of Erasmus's variations embodies a figure. As a means of showing mastery of style while elaborating on a theme, varied repetition may itself be seen as a kind of rhetorical figure that could include other figures. For example, ‘refining’ (expolitio), according to the [Rhetorical] Ad Herennium, ‘consists in dwelling on the same topic and yet seeming to say something ever new’ for ‘when we descant upon the same theme, we shall use a great many variations’; the example of a speech based on this figure includes the Theme expressed simply, the theme expressed in a new form, arguments from comparison, contrary and example, and a conclusion which restates the theme (IV.xlii.54–xliii.56). Puttenham renames this figure ‘the Gorgious’ because it has the same effect on speech and language as ‘rich and gorgious apparell’ has on the ‘bare and naked body’ in order to ‘attire it with copious and pleasant amplifications and much variety of sentences all running upon one point & to one intent’ (The Arte of English Poesie, 1589). Great effect may be produced by ‘Dwelling on the Point’ (commoratio), that is, ‘repeating the point several times in different words’ (Quintilian, Institutio oratoria, IX.i.27), or ‘synonymy’, which is described by Ad Herennium as a figure ‘which does not duplicate the same word by repeating it, but replaces the word that has been used by another of the same meaning, as follows: “You have overturned the republic from its roots; you have destroyed the state from its foundations”’; and by Peacham as ‘when by a variation and change of words that be of like signification, we iterat one thing diverse times’ (The Garden of Eloquence, 1577, 2/1593, p.149). Other figures that become central to variations are periphrasis (circumlocutio), the substitution of many words for one in order to amplify, and pleonasm, the addition of superfluous words for decoration and emphasis. It seems more than coincidental that the especially creative new thrust in Elizabethan rhetorical treatises accompanied the first flowering of variation form in England (see §5). The rhetorical underpinnings of variation form and variation technique reveal the aspects of persuasive (expressive) and pleasurable display common to both, and offer, as compensation for another layer of terminology, new tools for the analysis and valuation of variations even beyond those posited here. (See also Rhetoric and music.)

2. Terminology.

The roots of the word variatio in the adjective varius originally referred, in non-specialized antique usage, to an impression of mixed coloration in plants and animals, either in the sense of ‘colourful’ or the more negative connotation of ‘indeterminate’ or ‘fluctuating’. In his etymological analysis, Horst Weber (HMT, 1986) draws useful distinctions between the transitive and intransitive senses of variare (in German, verändern and sich ändern, respectively) with the connection of the first to process (varying, verändern) and the second to the result of that process (variation, Veränderung). Thus from the very beginning we see foreshadowed the twofold musical meanings of variation as technique and as form, and its connotations as positive and as problematic. Later associations of variation with colour can be seen in Zarlino's use of ‘Chromatico, quasi Colorato, o Variato’ for his chromatic genre (Istituto harmoniche, 1558, 3/1573, p.100). The idea of variety (varietas) played an important role in rhetoric, as the Latin writers drew on Aristotle, who himself had called on the authority of Euripides' Orestes: ‘Change also is pleasant, since change is in the order of nature; for perpetual sameness creates an excess of the normal condition; whence it was said: “Change [metabole] in all things is sweet”’ (Art of Rhetoric, 1371a, I.xi.20). Quintilian noted that ‘artistic structure [compositio] must be decorous, pleasing and varied’ (Institutio oratoria, IX.iv.146). Variety was a goal both in performance, especially in height and tempo of the voice (e.g. Ad Herennium, III.xii.22: ‘Relaxation from a continuous full tone conserves the voice, and the variety gives extreme pleasure to the hearer too’, trans. J.H. Freese) and in the realm of style (e.g. Ad Herennium, IV.xii.18: ‘To confer distinction [dignitas] upon style is to render it ornate, embellishing it by variety’, trans. H. Caplan).

Both variatio and its partial synonym mutatio are found in discussions of various kinds of ‘colourings’: the octave-related quality of different voices (Guido of Arezzo, Micrologus; Johannes Afflighemensis, De musica), hexachordal mutation and musica ficta (Marchetto da Padova, Lucidarium; Tinctoris, Diffinitorium; Finck, Practica musica; Zarlino, Le istitutioni harmoniche; Demantius, Isagoge artis musicae; and others). Varietas and variatio appear in discussions of the differentiae, the many possible ending-formulas for psalm tones used to link them to their antiphons. Whether the later Spanish term ‘diferencia’ for variation in the 16th century has any connection to this term is unclear; a similar question arises from the term divisiones for such endings (Regino of Prüm, c900) and the later English term ‘divisions’ for variations. The longstanding association of varius and variatio with rhythm, whether in a change of mode in mensural notation (from Franco on) or the rhythmically varied subdivisions of a final tone or cadence (from Guido on), makes the latter connection more plausible, especially in that the earliest variation sets of the 16th century subdivide the rhythm and may change the metre of the theme.

Rhetorical definitions of ‘figure’ as a schema, in which it is ‘altered from the simple and obvious method of expression’ (Quintilian, Institutio oratoria, IX.i.10–11, 13) thus make clear why variatio would become a figure to musical theorists of the 17th century: it takes a rhythmically plain (that is, ‘unfigured’) series of notes and recasts it as something rhythmically special, ‘ornate’ in the sense of ‘ornatus’, the soldier's armaments essential in the art of rhetoric. To Christoph Bernhard, variatio ‘occurs when an interval is altered through numerous shorter notes in such fashion that, instead of the longer note, numerous shorter notes rush to the following [principal] note through all kinds of runs and leaps’ (Tractatus compositionis augmentatus, p.73, Eng. trans. Musica poetica, 1997, p.434). It can thus include other figures and is itself part of other figures, like the transitus, one of Bernhard's figures of dissonance resolution. In writings by Bernhard, Prinz, Praetorius and Vogt, variatio is generally treated as synonymous with diminutio, coloratura and passaggio, and all of these are thought of both melodically, to fill in a large interval, and rhythmically, to subdivide a longer note. The related Spanish term glosa was discussed not only as the technique of diminution per se but also in the context of dissonance treatment (e.g. P. Nassarre, Fragmentos músicos, 2/1700). The sense of variatio as resolving a dissonance in small-note values carries through into the 18th century with Fux (Gradus ad Parnassum, 1725, p.217) and Scheibe (Compendium musices theoretico-practicum, ed. P. Benary in Die deutsche Kompositionslehre des 18. Jahrhunderts, 1961, p.62), and the much older view of it as a subdivided cadential note is recontextualized as the decoration of a cadence in an improvised cadenza (Riepel, Anfangsgründe zur musicalischen Setzkunst, iv: Erläuterung der betrüglichen Tonordnung, 1765, pp.89–90). A late echo of mutatio is found in Johann Walther's Musicalisches Lexicon of 1732, referring to each of its various changes (of accidentals, of mode, of manner, of register) as a Veränderung. The idea of variation as something intrinsic and essential, like ornatus, part of the artistic ‘tool-kit’ of the composer, might also make specific rules beyond those of part-writing impossible to prescribe: as Werckmeister put it, ‘one artist has a different Invention, a different Variation, and a different Genium from another’ (Harmonologia musica, 1702, p.84).

Variety and variation as the goal of art became an oft-repeated maxim during the 17th and 18th centuries, as a source of pleasure and as an approximation of the beautiful variety of nature. Simpson, Heinichen, Mattheson and Daube all asserted its primacy in music. Both florid counterpoint and elaborate figured-bass realizations were perceived to vary a simpler underlying model, and the art of ‘divisions upon a ground’ was predicated upon it: Christopher Simpson used ‘ground’, ‘subject’, ‘bass’ and ‘theme’ interchangeably (The Division-Violist, 1659, p.27; see Division). Whether the model was identified as a ‘given Moduli’ (Printz, Phrynis Mytilenaeus, 1696, pt 2, p.46), a ‘simple melody for singing or playing’ (Walther, Lexicon) or ‘certain bass notes’ (Mattheson's revision of Niedt's Handleitung zur Variation, 1721), the injunction that the original ought to be recognizable adds an important new strand to the ongoing evolution of the term. The possibilities of variations in fugues had been discussed from the early 17th century onwards, with subjects varied by inversion and by change of key and mode; in 1773, Daube described fugues with four-part invertible counterpoint, which with inversions ‘give rise to eight Veränderungen’ (Der musikalische Dilettant, 1773, p.330). Grassineau (translating Brossard, Dictionaire de musique, 1703) was specific about the result of the varied repetition, not merely the varied projection, of a simple model (Musical Dictionary, 1740): ‘Variation, is the different manner of playing or singing the same song, air, or tune, either by subdividing the notes into several of less value, or by adding of graces in such a manner, however, as one may still discern the ground [le simple] of the tune thro' all the enrichments’. The dictionaries of both Walther (1732) and Lacombe (1752) also adopted Brossard, but Lacombe was the first to use the plural Variations (Dictionnaire portatif des beaux-arts, 1752; subsequently used by Rousseau, 1768). Mattheson (1721) claimed that ‘what the French call double, we call variation, though this is not the best [term]. The name is entirely too general’.

Just as the term ‘variation’ could refer to different things, many different terms referred to variations. Double, originating in the pas doublé of court dance (Thoinot Arbeau, Orchésographie, 1588, p.33), appeared first in suites, such as an allemande or sarabande with one or more doubles. Like double but each with subtle semantic distinctions, the Spanish diferencia, Italian partita and English division all referred to a ‘partition’ – a segment, later called ‘theme’, that was repeated with alterations – as well as the sense of subdividing the original note values. In the realm of 17th-century dance music, however, it is sometimes unclear whether one double, glosa or variatio of an entire short piece is meant to be a varied repetition or an alternative version, especially since these terms refer to the practice of improvising diminutions as well as to sets of variations (e.g. Hernando de Cabezón's 1578 edition of his father Antonio's music, which included nine sets of diferencias, suggested that it be used as a model for glossing). The German Veränderung, like the Spanish mudanza and Italian mutanza, on the other hand, means change or alteration; German writings sometimes appear to distinguish between it and variatio, where the former is a broader category subsuming the latter as a figure (Horst Weber comments that all the different terms ‘make possible a latent distinction between the concept of a “figure” and that of a “segment”’). Mudanza and mutanza were choreographic terms of the 15th and 16th centuries, though both were applied to sets of variations (e.g. Antonio Valente's Intavolatura de cimbalo, 1576). Indeed, ‘variation’ as a term for a solo dance or ‘number’ persisted in the terminology of ballet, applied by Tchaikovsky and Stravinsky, among others, and perhaps connecting both with the sense of solo display often on view during performances of instrumental variations and with the possible origin of variations as varied repetitions of a piece of music to accompany the dancers. Certainly dancers from the 16th century onwards were given a vocabulary of steps to improvise in subdividing the basic steps, comparable to diminution practice. Such titles as ‘aria’ and ‘capriccio’ were at times given to pieces with variations or variation-based procedures, but more often not.

The echo of many terminological possibilities resonates in J.S. Bach's nomenclature (see §7): Aria variata all[a] man[iera] italiana, Partite diverse on chorale melodies, the unfinished Variationen of the Clavierbüchlein for Anna Magdalena Bach, the doubles in the suites, Aria mit verschiedenen Veränderungen (the Goldberg Variations) and Einige canonische Veränderungen über das Weynacht-Lied: Vom Himmel hoch da komm ich her. Whether the abandonment of ‘variation’ in favour of ‘Veränderung’ reflects a change in contemporary practice or in Bach's own sense of the relative valuation of his works is debatable. Certainly Brahms felt that Beethoven's use of that term for his Diabelli Variations reflected more intrinsic worth and greater ‘strictness’ (see §9). J.A.P. Schulz's rank-ordered list of variation types might lead to the same conclusion. In his article on variations for Sulzer's Allgemeine Theorie der schönen Künste, he placed the suite movements by Couperin and Bach lowest on the scale, then sonatas with varied reprises by C.P.E. Bach (not technically speaking variation form at all, although representing a type of ‘fully varied melody’), and finally, as ‘incontestably the highest type’, the contrapuntal variations with imitations and canons, as in Bach's Goldberg and Vom Himmel hoch sets. (Schulz also included Bach's Art of Fugue, fugues by D'Anglebert and even ‘the folie d'Espagne by the celebrated Corelli’ in this last category.)

Individual variations may or may not be labelled. Sweelinck's organ variations on Psalm cxl, as transmitted in the Fitzwilliam Virginal Book, have been identified as the earliest source to use the rubric ‘variatio’ for the heading of each segment (Weber, 1986). As with many other keyboard sets on sacred or secular themes (e.g. Sweelinck's Mein junges Leben hat ein End’), the first segment is not headed ‘theme’ but rather ‘Prima variatio’, showing that it already reflects a varied version of a familiar melody. Individual chorale variations were sometimes headed ‘versus’ (Scheidt's Tablatura nova, 1624) or ‘Verset’ (Titelouze, Hymnes de l'église, 1623) to distinguish them from secular variations, although this usage was not consistent and applied primarily to the first half of the 17th century. Many works of the late 16th century and the 17th use no title for the variations, merely numbers, while in variation movements from the 18th and 19th centuries composers often dispense altogether with any identifying title, number or other designator, especially in variation movements. In the 20th century, numbers make a comeback.

During the later 18th century, written discussion continued to treat variation as a technique, whether improvised or composed, and gave the first clear assessments of variation as a musical form (see §8). Writers rarely drew terminological distinctions between technique and form; Momigny was unusual in differentiating between ‘broderies’, varied repetitions of phrases and melodies in any form, and ‘variations’, or the ‘broderie’ of an ‘entire Air’ (Cours complet d'harmonie et de composition, ii, 1806, p.614). The latter type is more concerned with creating an overall ‘dessein’, and only in an Adagio variation do the frequently changing ‘broderie’-type figurations appear. After the term ‘theme and variations’ made its appearance (in Koch's Versuch einer Anleitung zur Composition, iii, 1793), all the earlier synonyms for variations except Veränderungen fell by the wayside; new terms were subsequently developed during the 19th century to create new hierarchies of value in variation (e.g. ‘formal’ versus ‘character’ variations; ‘decorative’ versus ‘contrapuntal’ versus ‘fantasy’ variations: see §9). In the 20th century, the term was applied to different sorts of process in addition to variation forms though with a tangential connection to the latter: Schoenberg's term ‘developing variation’ refers to the ‘endless reshaping of a basic shape’ by thematic regeneration; Fred Lerdahl's ‘expanding variation’ elaborates a simple model into cycles of increasing length and complexity, with stable events acting as points of departure for new growth (e.g. First String Quartet, 1978; Second String Quartet, 1982; Waves, 1988). And sometimes the term has been avoided entirely as composers seek to create different ‘views’ of a theme, as in Ruth Crawford Seeger's early work Kaleidoscopic Changes on an Original Theme ending with a Fugue (1924).

3. Variation types.

Every set of variations retains elements of the theme while altering or replacing others, and 20th-century typologies of variations (e.g. Nelson, 1948; von Fischer, 1956) developed terms that refer to the constant elements in a set, particularly constructive elements such as bass, melody, harmony and structure. While such terms have historical meaning primarily when all or most of the individual variations in a set meet that condition, variation sets also commonly mix variation types, especially after the 18th century. The nature of the theme, or given material – whether it is a melody (e.g. a song or hymn), a bass line, a harmonic progression (or ‘harmonic-metric scheme’, Esses, 1992) or a structural complex – exerts a certain predictive force on the type of variations to follow.

(a) Ostinato variations. Built upon a short pattern of notes, usually in the bass register, which functions as an Ostinato, §2 or Ground bass, this type includes continuous variations of late 16th- and 17th-century dance frameworks (e.g. Chaconne, Passacaglia), English grounds and their later progeny. It has two subgroups, ‘tonic-providing’ and ‘tonic-requiring’, depending on whether the pattern includes its own final cadence with a return to the tonic pitch at the end. The tonic-requiring type (e.g. Pachelbel's Canon in D major, on an expanded chaconne progression) creates a continuously regenerating series while tonic-providing ostinato variations (e.g. Bach's Passacaglia in C minor) may sound somewhat more sectional, although the presence of an upbeat generates continuity. Sometimes the bass line disappears or is submerged in the harmonies it generates, and sometimes it is transposed.

(b) Constant-melody or cantus firmus variations. The former is the broader category of which the latter is a historical instance. A melody, usually widely known, appears intact or with only slight embellishments in every variation, moving from voice to voice in the texture. Chorale variations by Sweelinck are perhaps the earliest instances (but see §4 for a discussion of the Western Wind masses). Composers as late as Weber (Variations on Schöne Minka op.40, variations 4 and 7) used the term ‘canto fermo’ to identify the melody when it appears in another voice. Until the mid-18th century, the cantus firmus is often set off from the other voices by its slower rhythmic values; after that time, when composers hardly ever used it for an entire set (Haydn's string quartets op.76 no.3, second movement, on Gott erhalte Franz den Kaiser, and op.76 no.6, first movement, are exceptions), it might appear in one or two variations, as in the introductory variations on the bass of the theme in Beethoven's op.35 and ‘Eroica’ Symphony finale.

(c) Constant-harmony variations. This broad category includes many variation sets of the 16th, 17th and 18th centuries in which the harmonic progression takes precedence in retentive power over the melody. The more sectional harmonic-metric schemes of Italian and Spanish dance frameworks, such as the Folia and the Romanesca, may be included here, as well as such topical, expressive and contrapuntal Baroque variation sets as Buxtehude's La capricciosa and Bach's Goldberg Variations. During the middle decades of the 18th century, a more restrictive subcategory of constant-harmony was popular: these constant-bass variations on a two-reprise (sectional) theme repeated the bass line of the theme in every variation, sometimes writing it out only once with ‘Repetetur [x] volte’. This subcategory may also include the variation suite, in which two or three suite movements (rarely an entire suite) maintain the same series of harmonies, generated from the same figured bass line, despite differences in metre, character and melody. F.E. Niedt's Handleitung zur Variation (1706) shows how to generate an entire suite from a single bass line, and Fischer (1957, p.118) cites a suite notated only by a figured bass (I-Fn XIX, 110).

(d) Melodic-outline variations. The theme's melody, or at least the ‘outline’ of its main notes, is recognizable despite figuration, simplification (unfigured variation) or rhythmic recasting. This much-maligned category (‘mere decoration’: Tovey correctly warns against ‘despising the embroidery variation on principle’) includes variations whose harmonies remain more or less unchanged (typical of the later 18th century) and those whose harmonies may change from variation to variation (more common in the first half of the 19th). It also favours periodic reiteration of the theme's melody more or less intact (melodic reprise) with figurations in another line. Types of figuration may be pleonastic (the addition of ‘superfluous’ notes within the melody or as a countermelody) or periphrastic (the original notes replaced by a more ornate line, though with sufficient resemblance to the original, especially at cadences). Many variation sets of the 18th and 19th centuries mix this type with constant-harmony variations.

(e) Formal-outline variations. Aspects of the theme's form and phrase structure are the only features to remain constant in this predominantly 19th-century type. Phrase lengths may expand or contract within the general outline, with harmonies usually referring to the theme at the beginning and end of a variation. Resemblance to the theme may be striking, subtle or ‘found only with the eyes’ (a criticism voiced by Brahms). Sets of this type may contain a mixture of (b) to (f), and include Beethoven's Diabelli Variations op.120 and Brahms's Handel Variations op.24.

(f) Characteristic variations. Individual numbers take on the character of different dance pieces, national styles or programmatic associations. Within such a set, types (c), (d) and (e) may be used, just as characteristic styles and topics may form individual variations within sets of those types. The historical staying power of this type is revealed in such examples as Poglietti, Rossignolo (1677); Herz, Variations caractéristiques sur un thème arabe op.137; Strauss, Don Quixote (1898; see also (g) below); Britten, Variations on a Theme of Frank Bridge op.10 (1937); and Ginastera, Variaciones concertantes (1953). By ‘characteristic’ is not meant the broad and by-now meaningless term ‘character variation’; from 18th-century discussions of character in variations, character emerges as a concomitant of figuration, metre and tempo, and is always present.

(g) Fantasy variations. In this 19th- and 20th-century type, occasionally used as a title, the variations allude to or develop elements of the theme, especially its melodic motifs, often departing from any clear structural similarity with it. Such pieces may also have a programmatic or characteristic element and may have either individual numbers (Elgar, ‘Enigma’ Variations) or a more continuous texture (Strauss calls his symphonic poem Don Quixote ‘fantastic variations on a theme of knightly character’). Indeed, while this type is on the one hand sometimes hard to distinguish from formal-outline variations, it has on the other hand even been called ‘free’ variation because of its melodic or motivic allusiveness and structural looseness. (Free variations were derided by Tovey because they violated his sole criterion of value, that the composer ‘know his theme’.) 20th-century fantasy variations have been written by Howard Hanson, George Perle, Donald Martino and others.

(h) Serial variations. Modification of a serial theme (a 12-note row or some slightly longer or shorter configuration) in which figuration and accompaniment are derived from the row. The structure of the theme usually remains constant. Because of this structural component, serial variations differ from other serial pieces in which the row itself, not the theme, is manipulated and varied. Examples include Schoenberg, Serenade op.24, third movement, and Webern, Symphony op.21, second movement.

Identifying variation types and even variation form itself is a much less straightforward matter than it might appear, because in both technique and overall shape variations are defined by boundaries and limits that are not always clear. Types (b) to (g) above may be mixed within a set, the vexed nature of melodic resemblance may complicate some determinations of identity, and varied repetition sometimes appears to be a surprisingly fragile principle such that a greatly contrasting segment may upset the whole. Writers as early as Volger were aware that ‘free’ variations might be inserted within a set (Betrachtungen der Mannheimer Tonschule, ii, 1779–80, pp.118–19; Momigny, Cours complet d'harmonie et de composition, ii, 1806, p.613). When variation forms take on elements of recurrence and return in addition to or perhaps instead of repetition, the resulting hybrid type raises questions: is it a rondo or a variation? a ternary form or a variation movement with freely varied minore? Then there are anomalous pieces with some strophic features – a structural unit that recurs, perhaps in different keys, perhaps with connective passages – to conjure a fantasy-like resemblance to variations, such as the slow movements of Mozart's Piano Concerto k449, Haydn's String Quartet op.76 no.6 (marked ‘Fantasia’) and Brahms's String Quintet op.111. Finally there are strophic vocal works with alterations in each stanza, whether to the accompaniment (as in Beethoven's Sehnsucht woo146 and An die ferne Geliebte op.98 or Brahms's Agnes op.59 no.5) or to the voice(s). A potent example is the quartet from Beethoven's Fidelio (‘Mir ist so wunderbar’), long erroneously referred to as a canon when the form is actually strophic constant-melody variation with each new voice entering with the original melody while the other voices continue with new counterpoints. Finally, a variation principle may also act as a structuring device in sonata form. Haydn favoured a paratactic organization of material that offsets the periodic (hypotactic) nature of the functional areas of sonata form (first group, bridge, second group, closing). In the first movement of the C major Sonata h XVI:50, figural, textural, registral and harmonic variations of the opening theme come to dominate the exposition and development. Varied repetitions dominate the first movements of Symphonies nos.85 and 87, but most paratactic is the first movement of Symphony no.88, in which each of the four areas in the exposition first presents a version of the opening quaver theme (bars 16, 44, 61, 77) and then accompanies that theme with a counterfigure in semiquavers (bars 24, 51, 71, 85). Each segment sounds like a melodic, rhythmic and structural variant and intensification of the one before. A different kind of intensification occurs when progressive diminution affects every functional area, as in the first movement of Mozart's D minor String Quartet k421/417b. The limits of variation, the points at which it spills over into development, transformation and fantasy, are not clearly drawn. Indeed, the structural integrity of the ‘given material’ can be violated even in works clearly identified as variations by their composers; the presence of paratactic and strophic structuring within fantasies and sonata-like forms suggests the power of the variation model.

Another direction such improvisations might take – the varied tone colours and figurations of different instruments – does come from a much earlier source, emerging from this advice in Guglielmo Ebreo's dancing tutor De pratica seu arte tripudii (appendix to the second edition, written within 12 years of the original 1463 edition, by which time he had converted and become Giovanni Ambrosio):

4. Origins.

Variation is a sufficiently broad procedure that finding the earliest instance of its systematic application, before the publication of Narváez's diferencias in 1538, has proved impossible. Those pieces that stand out as instances of varying in both secular and sacred vocal music are hard to categorize. In an anonymous 14th-century madrigal, E con chaval (PMFC, viii, 28), the cantus sets each line of text in its three-line strophe to a varied version of the first line (aa¹a²), possibly reflecting vocal improvisation. Another madrigal from the same volume, Quando i oselli canta (p.81), shows a similarly unusual variation principle in that the third line not only ornaments the first but repeats its text. Unfigured variation is found in a Gloria by Excetre from the Old Hall Manuscript, which paraphrases a Sarum chant in the upper voice ten times; the two phrases of every statement end on C, nearly every time with C in the lowest voice as well, lending a strikingly sectional air to the movement. The paraphrases differ in length and rhythm, but despite the occasional added notes do not resemble what later writers would term ‘diminutions’. If the original chant consists of a repeated melody, then the result may be a few ‘variations’, as in Olyver's Agnus Dei setting in the Old Hall Manuscript. Isorhythmic motets with ‘isomelic’ tendencies, in which the tenor melody is taken up by the upper voices especially at the beginnings of sections, might be considered a distant harbinger of such later genre variations as variation canzona or ricercare, but hardly of variations per se. Dalglish's categories ‘variation motet’, ‘hocket variation’ and ‘ostinato motet’ incorporate some techniques of variation, but only the last has some claim to embodying actual variations, and only when the upper voices dovetail with the ostinato, as in the eight ‘variations’ of Regina celi letare/Ave regina/Ave (ex.1: F-Pa 135, transcribed in H. Besseler, ‘Studien zur Musik des Mittelalters II’, AMw, viii, 1927, p.243) and the 28 ‘variations’ of Thomas gemma canturiae (transcribed in K. Levy, ‘New Material on the Early Motet in England’, JAMS, iv, 1951, pp.234–9). Certainly such celebrated ostinato-based pieces as the Sumer canon and Du Fay's Gloria ‘ad modum tubae’ are not variations. Successive polyphonic settings of the same plainchant, including most cantus firmus masses and even Magnificat cycles of the 16th century that go through all eight tones, thus do not necessarily refer to variations in any but the most general rhetorical sense.

Methods of organizing multi-sectional or multi-movement Renaissance works – masses and motets, for example – may include sophisticated repetition patterns, in which each statement of a cantus firmus changes one or more features of the original statement or accompanies other changes. Normally the continuity of the polyphonic web and the lack of an initially discrete structural entity works against both the intentional and perceptible nature of such works as variations. Josquin's three masses based on solmization syllables deserve mention, however, because of their ostinato treatment of these short and memorable formulae, though whether the result is variation is highly debatable. In some of the movements of his Missa ‘La sol fa re mi’ the five-note subject, which in some movements (especially beautifully in the Agnus) saturates the texture, is also heard in structural ‘units’: in the first Kyrie it occurs twice consecutively in each entering voice (STB) to form regular four-bar segments; in the Sanctus it announces a three-bar pattern with cadence, then the tenor turns it into an uninterrupted ostinato, just as it does on two different pitch levels in the Osanna. In the Benedictus the cantus firmus makes a six-bar segment successively in three voices (TSB) while the two-bar countersubject moves from voice to voice during the first two statements of the cantus firmus. The Osanna of the Missa ‘Faisant regretz’ turns its four-note subject (fa–re–mi–re) into a two-bar canon between tenor and bass, while the Benedictus turns it into a six-bar pattern similarly alternating between tenor and bass, with upper-voice imitation as well; each statement begins on a different pitch. In the Osanna of the Missa ‘Hercules Dux Ferrariae’, the threefold repetition of the cantus firmus is then repeated in a faster rhythm, suggesting the process of progressive diminution later to be a common feature in variations; the Benedictus offers three statements of the cantus firmus at different pitches paired with an ornate contrapuntal voice.

An exception to these strictures about cantus firmus masses are the Western Wind masses by Taverner, Tye and Sheppard, an unusual group of works constituting the first English masses based on a secular song and possibly the earliest English variations as well (c1535–42). The Taverner setting, usually considered the earliest of these, presents the melody 36 times, nine times in each of the four movements (21 treble, ten tenor, five bass statements), differing each time in texture, number of voice parts and metre, with occasional ornamental notes but without intervening material between statements. The requisite ‘entity’ concept is thus clear. Moreover, the arresting upward leaps of 5th and octave at the beginnings of the first and second phrases respectively and the meandering descents, largely in conjunct motion, make the tune instantly recognizable in whatever texture it is placed, although it is of course most evident in the treble. Tye puts the melody into the single voice not set by Taverner, the alto (mean), in 29 varied statements. Sheppard's total is 24, of which 13 omit the final phrase of the melody, and all are in the treble but no.10, the third phrase of 21, and 23, which are in the tenor. The Western Wind melody is neither more ornate than, nor moves at a different speed from, the other voices and thus differs appreciably from the melodies of other cantus firmus masses in its relationship to the texture of the mass as a whole. It makes sense to call these three works variation masses. (See Western Wind, exx.1 and ..\Frames/F010084.html2.)

In the realm of instrumental music, the most promising models are in dance music, especially (a) the possibility that short pieces for dancing were repeated with improvised variations rather than literally, and (b) the pairing of gliding and leaping dances, often with a proportional relationship in which the second, faster dance – the Nachtanz, Proportz or Tripla – would offer a varied version of the first. A variation principle informs even the original dance-pairs of the 14th century in GB-Lbl Add.29987, La Manfredina and Lamento di Tristano, each with its own Nachtanz called La Rotta, a quicker, compressed and somewhat ornamented version of the first dance. But varied dance pairs such as pavan–galliard and passamezzo–saltarello suggest that they were the forerunners not of sets of variations but rather of the varied allemande–courante pairs in longer sequences of dances in the early 17th century. That dance pieces were repeated, in part or as a whole, is clear not only from the length of the choreographic pattern compared to the notated music but also to explicit directions like ‘Questa Sonata farassi due volte’ (‘play this piece twice’, in the balletto Laura suave) and ‘Si torna à fare di nuovo detta Sciolta in Saltarello à quattro Tempi’ (‘Play the sciolta to this piece as a saltarello four more times’, in Furioso alla Spagnuola; Fabritio Caroso, Nobiltà di dame, 1600, Eng. trans., 1986, pp.10–11). It seems plausible to suggest that such repetitions were varied. Laura suave is itself a kind of variation suite, consisting of an opening pavan-like piece (including repetitions), the piece played as a galliard, then in triple proportion as a saltarello and finally as a canary (ex.2). As Thoinot Arbeau wrote in 1588 (Orchésographie, 1589, Eng. trans., 1967), ‘you can amplify this music to suit your pleasure and fancy’ (p.43) and ‘some dancers divide up the double that follows the two simples, and instead of the double comprising only four bars with four semi-breves, they introduce eight minims or sixteen crotchets, resulting in a great number of steps, passages and embellishments, all of which fit into the time and cadence of the music’ (p.66). This connects the variations of steps performed by dancers – called mutanze by Caroso as early as 1581, in Il ballarino – with musical variations, as does Antonio Valente's use of the term mutanze for sets of variations on dance themes such as the romanesca and the gagliarda napolitana (Naples, 1576). But the lateness of these sources, even if one were to speculate that they codify long-term existing practice, does not offer an actual precedent.

Get four or five kinds of instruments to play, such as shawms, organs, lute, harp, pipe and tabor, or whatever other instrument there is. Have them play one by one, and have them play a ballo, and [get] each one to play that [same] ballo, each one playing by itself. The [dancer] must dance to that air that the instruments play. For even though they are playing one [and the same] ballo, each one will play with his own air. [And] although they are playing the same ballo, the shawms will play in one air, the organ in one air, the harp in another air, the [pipe and tabor] in another air, but all will play one and the same ballo. Remember that the dancer must dance with that air and with that measure and with that rhythm that the said players are playing; that is, dancing each one on its own. And if the dancer always dances with one air, even though he dances with measure and in time but does not follow the air of the said players, his dancing will be imperfect and show little skill. (On the Practice or Art of Dancing, Eng. trans., 1993, p.235)

Although the meaning of ‘air’ is obscure, it plausibly refers to a kind of characteristic expressive style that may be related to tone quality or kind of figuration, especially since it has to affect the dancer's movements. What emerges from such a series of ‘intrinsic’ variations on a dance is a variation form. This sort of exercise may also lead to the kind of dancing described by Arbeau (p.16): ‘a kind of mute rhetoric by which the orator, without uttering a word, can make himself understood by his movements and persuade the spectators that he is gallant and worthy to be acclaimed, admired and loved’. Not only did Vogler make the same point about variations 200 years later, but it strongly reinforces the display element in both, which serves to convince the audience of the talents of the performer/composer.

Sets of variations appeared for the first time in the 16th century and, as their themes originating in dance and song indicate, they captured two forms of improvisation, the variations in repeated strains of dance music originating in its choreography and the varied settings and diminutions given successive stanzas of a song whose melody can be savoured as a cantus firmus or a springboard to figuration. Early 16th-century dances on short ostinato basses, such as Hugh Aston's Hornpype and the various forms of dump (e.g. My Lady Carey's Dompe, HAM, no.103), reveal a flourishing variation practice in England. The importance of diminutions in improvising embellished lines in polyphony and to a cantus firmus is attested by such treatises as Silvestro Ganassi, Fontegara (1535), Diego Ortiz, Trattado de glosas (1553) and Girolamo dalla Casa, Il vero modo di diminuir (1584). Intabulations of vocal polyphony and related dance-pairs such as pavan–galliard and passamezzo–saltarello in the Italian lute and keyboard repertories also participated in this development. As early as 1508, in J.A. Dalza's Intabulatura di lauto, two sorts of pavan herald the varied-reprise dance forms as well as the variation set per se: the pavana alla ferrarese with a series of open-ended phrases followed by varied repeats (AA'BB'CC' …) and freer material, and the pavana alla venetiana, with a somewhat longer phrase ending on the ‘tonic’ followed by a series of variations before a freer concluding passage (Horsley, 1959, calls these ‘multiple-strain’ and ‘single-strain’ variations respectively). ‘Suites’ of dances with related incipits, such as ‘Passa e mezzo antico’ (or ‘moderno’) and ‘Salltarello’, or ‘Passamezo–Padoana–Saltarel’ (all based on the same song) appear in Giacomo Gorzanis’s four books of Intabulatura de liuto (1561–79).

5. The 16th century.

The earliest published sets of variations, or diferencias, appeared in Spanish works for vihuela by Luis de Narváez (Delphín de música, 1538); especially influential were (1) the variations on O guárdame las vacas (ex.3), the Spanish version of the romanesca (identified as such by Francesco de Salinas, De musica libri septem, 1577), in which the first of three diferencias is much shorter than the theme while the third is followed by a coda; (2) Conde claros, on a two-bar theme; and (3) diferencias on a plainchant (O gloriosa domina, HAM, no.122). After the varied-reprise dances and ubiquitous vacas of Alonso de Mudarra (Tres libros de música, 1546), who used the terms manera and diferencia interchangeably, and Enríquez de Valderrábano (Silva de Sirenas, 1547, HAM, no.124), came the important diminution treatise of Ortiz (1553), which offered a series of recercadas on ‘plain songs [canto llano] which in Italy are commonly called tenors’ (bk.ii, f.47r). These recercadas resemble variations on the folia, romanesca and passamezzo antico and moderno, while Ortiz's recercadas on polyphonic compositions (e.g. Arcadelt's madrigal O felici occhi miei) have the cumulative effect of variations since diminution techniques are differently applied to each intabulation.

Antonio de Cabezón was the first master of the keyboard variation, with some published in Luis Venegas's Libro de cifra nueva (1557) and nine sets in his son Hernando's Obras de música (1578), including three on O guárdame las vacas and several calling attention to direct Italian influence, such as Diferencias sobre la pavana italiana and sobre la gallarda milanesa. Cabezón's four-voice polyphonic settings, while neither systematic nor progressive in figuration, often present diminutions first in the treble, then in the bass, then in both hands. Constant-melody technique is often present, sometimes with the theme melody lightly coloured. One wonders if such beautiful settings of plainchants as his Ave maris stella XI, which maintains the cantus firmus in the middle voice and surrounds it with precisely the kinds of figurations found in the diferencias, were spurs to further variation in performance, especially in pieces based on a strophic original. His sojourn in England with Philip II from 1554 to 1556 has spurred theories of reciprocal influence in variation.

The flowering of keyboard music in the later 16th and early 17th century in England among composers later known as the ‘English virginalists’ went hand in hand with a vogue for variations on dances and popular tunes. In general these combine constant-melody, constant-harmony and melodic-outline types. Sometimes the varied repeats in the pavan – AA¹BB¹CC¹ – were followed by yet another chain of varied repeats – A²A³B²B³C²C³ – before the metric variation of the galliard (perhaps with its own varied reprises). Occasionally an unnamed ‘alla venetiana’ model produced a one-strain pavan with a longer chain of variations, as in Bull's Spanish Pavan (eight-bar theme) with eight variations (Fitzwilliam Virginal Book, i, no.131); Bull's Quadran Pavan with eight variations followed by a ‘Variation of the Quadran Pavan’ with another eight; Byrd's Passamezzo Pavana (16-bar theme) with six variations followed by the ‘Galiardas Passamezzo’ with eight (Fitzwilliam Virginal Book, i, no.203); Peter Philips's Passamezzo pavana and Galiardas passamezzo with seven and ten respectively, the ninth in the latter labelled ‘Saltarella’. In most cases, however, variations on dances differentiated themselves from chains of variations by virtue of their emphasis on the varied reprise; thus Byrd's Monsieur's Alman has two parts each with a varied reprise (aa¹bb¹, followed by one variation in which each part has a varied reprise (a²a³b²b³), after which a ‘Variatio’ offers three further variations on the alman, each part with a varied reprise. Were all the varied reprises of Monsieur's Alman to be presented as a chain, it would be a theme with nine variations. While the vast majority of song variations are secular, the occasional plainchant appears, as in Bull's Salvator mundi, based on Veni creator and Miserere, and is treated as a cantus firmus in long notes moving from voice to voice in two- and three-part settings. Blitheman's six settings of Gloria tibi Trinitas and four of Aeterne rerum conditor in the Mulliner Book (MB, i, nos.91–6 and 49–52), if considered as variation sets rather than multiple settings of the same chant, may be the first chorale variations, challenging Sweelinck's position (see §6). Some ostinatos are lengthy and elaborate (e.g. Byrd, My Lady Nevell's Ground, 24 bars), while others consist of only two notes or chords (Byrd, The Bells, The woods so wild), perhaps the successors to the dump or older pes-like forms. Giles Farnaby's Rosasolis combines elements of both song and ostinato in that the theme is only four bars long and ends on a ‘half-cadence’, offering what would be an early example of the ‘tonic-requiring’ cycle, except that the piece ends on the ‘dominant’. (See also Passamezzo.)

The general pattern of many Elizabethan keyboard variations is progressive diminution in one or more groupings of several variations, broken up by dotted-rhythm variations (always seen as an intensification), and with triplets normally reserved for the last group of a cycle. A technique sometimes called ‘mirroring’ (by Mies, 1937, and Cavett-Dunsby, 1989), in which the treble figurative pattern in one variation goes into the bass in the next, helps to join variations as well. After a final build-up a quiet quasi-reprise, either reharmonized or simply a return to the theme's rhythm, leaves an elegiac sense of an ending. One of the best of these sets is Byrd's John come kiss me now, in which the melody is treated like a cantus firmus, except for its partial abandonment in variation 6; it appears in the treble in variations 1–9, 13 and 15, the tenor in 10–12, the bass in 14 and the alto in 16. Other groupings overcome the final bar-line to connect 5 and 6, so that the first variation without an immediately recognizable melody grows out of the previous one, and especially 11–14, in which both harmonically and rhythmically the variation is propelled into the next; this group also includes the triumphant triplet variation 12, which begins as a point of arrival and another rhythmic acceleration. The final variation disguises the theme melody in the alto and reharmonizes it while also crystallizing the upward and downward slopes of the theme into an articulation that grows out of the quasi-reprise variation 15 (ex.4). The tune Walsingham inspired some of these composers' longest sets – 30 virtuoso variations from Bull and 22 from Byrd – and most inventive figurations. Finally, dances with varied reprises and sets of variations for consort appear in Morley's First Book of Consort Lessons (1599) and other collections. Richard Alison's Goe from my window for broken consort treats the melody as a cantus firmus, moving from treble viol to flute to bass viol and to treble again, with the addition of florid figurations and counterpoints in the other instruments, unlike the keyboard set attributed to both Morley and Munday in the Fitzwilliam Virginal Book, in which the melody is much more variable than the bass and returns in a plain version only in variations 5 and 7. Byrd's 20 variations on Browning my dear (‘The leaves bee greene’) for viol consort retain the melody as a cantus firmus, beginning unobtrusively in the bass and working upwards, and are more contrapuntal than Alison's.

(i) The earlier 17th century.

6. The 17th century.

(ii) The later 17th century.

Variations, §6: The 17th century

(i) The earlier 17th century.

The beginning of the 17th century saw the remarkable Dutch organist J.P. Sweelinck create a north German organ ‘school’ by virtue of his exceptional teaching in which variations, both chorale-based and secular song-based, played a large part. In introducing the forms and figurations of the English virginalists to organ music, he helped to develop the new genre of chorale variations in which the chorale melody, always recognizable as a cantus firmus though occasionally slightly embellished, is embedded in an increasingly complex web of contrapuntal figurations in two, three or four voices. (His variations on Psalm cxl even appear in the Fitzwilliam Virginal Book, ii, no.151.) These figurations, often sequential repetition of short motifs, would change in the course of an individual variation, lending an exploratory aspect of the composer-performer at work to each piece; pieces begin with the first variation, perhaps because the melody was assumed to be familiar. In newly Protestant Amsterdam, these works would be performed during daily organ recitals in the church, rather than liturgically. In the rare cases where the chorale is itself highly figured, as in variation 6 of Erbarme Dich mein, o Herre Gott, the notes of the chorale remain at the metrically identical parts of the bar (ex.5). Sweelinck's variations on pavans and on songs like Est-ce Mars, Onder een linde groen and especially Mein junges Leben hat ein End’ also begin with the first variation, present written-out variants of each reprise, and change the type, placement and speed of figuration several times in a single variation. Unlike the chorale variations, his secular variations may also change within a variation the voice in which the melody appears. The variations of his principal students, Scheidt and Scheidemann, who took his pieces and techniques to Halle and Hamburg respectively, continued and extended these characteristics to include more brilliant figurations and longer sets of variations. Scheidt sometimes headed a variation with the location of the chorale or song melody (e.g. ‘Choralis in Alto’, ‘Variatio in Tenore’), and quick, slurred figures towards the end of a variation are marked ‘imitatio violistica’. Unlike Sweelinck, he usually began with the ‘theme’ as a simple chordal rendering before the ‘variations’.

With the rise of monody at the beginning of the 17th century, the repetitive impulse found outlet in two directions, both centred on retention of the bass: strophic arias with successive modifications to the melody or different melodies in each strophe, and vocal pieces, largely solos and duets, over an ostinato bass. The prototype of the former is perhaps Caccini's Ard'il mio petto misero, the only strophic aria in his Nuove musiche (1601) to be written out with some changes to the melody (fewer to the bass). (There are, however, strophic songs with varied lute parts and slight vocal changes already in Narváez, 1538, e.g. Sy tantos halcones.) A striking later example is Frescobaldi's ‘Aria di romanesca’ (Arie musicali, 1630), which also shows the incorporation of 16th-century dance frameworks into vocal music. The melody is different in each strophe except for the cadential note at the end of each three-bar phrase. On the other hand, Monteverdi's ciaccona for two voices and bass, Zefiro torna (1632), exemplifies a non-coincidence between bass and treble typical of his style but differing from many vocal ostinato-bass pieces in this era (Merula, Schütz). Inevitable cadencing in the two-bar bass line (a classic tonic-requiring pattern) is mitigated by shorter, longer and asymmetrical passages in the upper voices, while texturally the flat plane of the bass contrasts with the quasi-imitative and often off-beat give-and-take of the singers. Brief or prolonged absence of the ostinato may also have a profound programmatic effect, as in the final ‘tormented’ stanza of the poem (where the ‘sweet and joyous notes’ of Phyllida and Cloris give way to the pain of the narrator, ‘Sol io, per selve abbandonate e sole’) with its affective, chromatic and dissonant word-painting in recitative style before the final return of the ostinato. Monteverdi similarly frames and punctuates his later madrigal Lamento della ninfa (Book 8, 1638) with a group of three male commentators while setting the nymph's own words to the descending ‘lamento bass’ tetrachord, a powerful minor-mode evocation of despair that quickly spread to opera of the 1640s, 50s and 60s in works by Cavalli (including Egisto, Rosinda, Giasone and Eliogabalo), Cesti (Argia) and others. There is a difference, of course, between the endlessly recycling tetrachord that reaches only from i to V and the tetrachord that is extended to a final tonic-providing close; of the five elaborated lament basses by Cavalli given by Ellen Rosand (Opera in Seventeenth-Century Venice, Berkeley, 1991, p.649), only one ends on the dominant. Later examples include the justly celebrated lament from Purcell's Dido and Aeneas (1689), with its chromaticism and tonic-providing tail. The power of even the simple major-mode tetrachord may be seen in the opening and closing sections of the highly charged closing duet for Nero and Poppaea in Monteverdi's L'incoronazione di Poppea (1642), set to Benedetto Ferrari's text ‘Pur ti miro, pur ti godo’, perhaps the first operatic da capo (not knowing whether the music is by Ferrari, Sacrati, Cavalli, Laurenzi or a composer yet to be suggested does not detract from its ostinato-derived hold on the imagination). (See also Lamento; Ostinato, §§4–5; and Strophic variations.)

Whether the instrumental ostinato variations of the 17th and 18th centuries such as the Chaconne and Passacaglia owe anything to the vocal models is an open question, because they tend to take their points of articulation from the bass line, changing their figurations, textures and rhetoric at cadential points, and are thus true variation sets. The ‘set rhythm’ that builds throughout a span of continuous variations maintains the quality of overcoming its interior cadences at the same time that it uses the inevitability of cadences for energy and emotional power. Certainly variations on dance frameworks and progressions, or harmonic-metric ‘schemes’, constituted an important strand of variation writing up to the early 18th century, with occasional appearances thereafter. The first written variations of the chaconne, a dance imported from Latin America (chacona) in the late 16th century, appeared in Spanish guitar books of the early 17th century; the earliest set for keyboard is Frescobaldi's Partite sopra ciaccona (1627). Passacaglias originated in the early 17th century as a kind of ‘walking-around music’ for guitar that served as introductions, interpolated episodes and conclusions to songs and dances; these passages were also known as riprese or ritornelli, repeated several or many times with improvised variations. Frescobaldi's Partite sopra passacagli (1627) may have been the earliest for keyboard. Chaconnes tended to be major and passacaglias minor, and each had characteristic though similar bass lines and chord progressions, but early differences were often ignored. Indeed, Frescobaldi's Cento partite sopra passacagli (Secondo libro di toccate, 1637 edition) features sections headed ‘passacagli’ as well as ‘ciaccona’: both are in the minor, though with differing harmonic rhythms.

In addition to his vocal ‘partite’ on dance-bass patterns, Frescobaldi was thus the principal composer of variations and variation-inflected instrumental genres in the early 17th century in Italy. Of the established or evolving genres in which he was drawn to explore the concept of variation – suite, capriccio, canzona and ricercar – only the first had a natural connection with variation, developing as a stylized representation of dances with roots in both choreographic necessity and a form to which actual dancing occurred and to which repetition with improvised variations would have been common. Indeed, each of Lorenzo Allegri's eight balli (Il primo libro delle musiche, 1618) supposedly reflects the contents of actual dances performed as part of an intermedio at the Florentine court, and nearly all include pairs of dances in which the second is built on the bass and melody of the first (e.g. ‘Quinto ballo detto le Ninfe di Senna’, with opening pair: untitled ‘prima parte’ – ‘Canario seconda parte’; the following ‘Gavotta terza parte’ and ‘Corrente quarta & ultima parte’ are unrelated). Three German publications with longer series of linked dances later known as variation suites appeared in the second decade of the 17th century: Paul Peuerl's Newe Padouan, Intrada, Däntz unnd Galliarda (four-movement sequence, 1611), Isaac Posch's Musicalische Ehrenfreudt (three-movement sequence, 1618) and the most celebrated of these, Schein's Banchetto musicale (1617). Schein asserted that the dances are ‘arranged so that they correspond to one another in both mode and invention’, thus clearly referring to the rhetorical notion of the principal idea of each piece. The principal ideas are variations of each other, even if there is no literal bar-to-bar correspondence, with the most ornate version appearing first in the leisurely pavans (Padouana), followed by Gagliarda, Corente and finally the Allemande and its Tripla, a simplified metrical recasting of the allemande. As the dances progress from complex to simple, texturally and melodically, they also descend in style from higher to lower.

The canzona, ricercare and so-called capriccio (actually a type of ricercare) were contrapuntal single movements, often in multiple sections differentiated by metre and dominated by imitative texture; the term ‘variation’ has been used to describe the rhythmic transformations of the imitative subjects and countersubjects of successive sections. While such a composition may be considered cyclic, it is not a variation form in the sense of a common substratum underlying each section. On the other hand, the impulse to connect contrapuntal and variational forms suggests a relatively borderless connection between them. Several composers of this era cultivated what modern scholars call the variation canzona and variation ricercare: Andrea Gabrieli, Ercole Pasquini (1600), Ascanio Mayone (1603), G.M. Trabaci (1603) and Frescobaldi beginning with his Primo libro di toccate (1615). Ladewig (1987) has persuasively placed the origin of the variation canzona in late 16th-century Ferrara with Luzzaschi, Ercole Pasquini and ‘Giaches’ (de Wert or Brumel), who subsequently influenced Neapolitan musicians. Frescobaldi's capriccios are, in most cases, witty or clever ricercares, free contrapuntal investigations of a given subject (e.g. a solmization pattern), occasionally with an ostinato (Capriccio sopra il cucco, no.3 of the Primo libro di capricci, 1624), though in one case the term refers to a set of variations (on Or chè noi rimena in partite, a binary theme with varied repetitions). The Capriccio sopra l'aria di Ruggiero (no.12; Monumenti musicali italiani, viii, 1984, p.78) treats the four phrases of the Ruggiero bass (ABCD) in a way that shares elements of the imitative ricercares and canzonas, on the one hand, and sets of variations on the other. Apel (1972) calls it a ‘quadruple fugue’ and divides it into eight sections according to metre, principal cadences, and the presence of particular phrases of the bass. Table 1 amends his chart to reveal the two variation structures that emerge first from the sections in which all four theme phrases are present (I, III, VI, labelled variations 1–3) and second from alterations to the theme phrases themselves through metric and rhythmic changes (II, IV, V, labelled variations a–c), as well as the ‘finale’-grouping of sections VII and VIII (variation 4d) which combines both types by speeding up the figuration as well as presenting the B-phrase as a varied counter-figure (a chromatic 4th, both ascending and descending). The sections identified as variations, however, treat the phrases of the Ruggiero as separable entities, motivically and contrapuntally (ex.6); they are not structural variations like Frescobaldi's own Partite 8 sopra l'aria di Ruggiero (Toccate e partite … primo libro, 1615, expanded to 12 variations in the second edition), on the same popular subject already also set by Macque, Mayone and Trabaci. However, his Capriccio sopra soggetto scritto sopra l'aria di Ruggiero in the same volume adds the melody ‘Fra Jacopino’ to the harmonic pattern (Mw, xi). Dance-bass variations as well as variations on such melodies as ‘Tanto tempo hormai’ appeared in trio sonatas by Salomone Rossi (including 11 sets in his books 3 and 4, 1613–22), G.B. Buonamente (including eight of the 12 sonatas of his Book 4), Scarani and Merula.

Frescobaldi's Aria detta la Frescobalda (Secondo libro di toccate, 1627) is very possibly the first set of variations on an original theme and the first with variations labelled as dance-types: variations 3 and 5 are called Gagliarda and Corrente respectively. Apel (1972) considers this the earliest variation suite and suggests that the term ‘suite variations’ is more appropriate. Several other variation sets by Frescobaldi present the dance-flavoured alternation of duple and triple metre, such as the eight variations on the Aria detta balletto in the same collection, which alternates duple and triple metre for three pairs of variations, then for the last two increases the speed of figuration in duple metre, from triplets to semiquavers, while bringing back the melody in the ‘ultima parte’ for a theme reprise. Although the two triple-metre variations in the earlier Partite sopra l'aria di follia (variations 3 and 5, Primo libro di toccate, 1615) recall the corrente in rhythmic pattern, they are not labelled. A later, oddly titled type of variation suite is the Balletto corrente passacaglia (three in the 1637 edition of the Primo libro di toccate), with the corrente a triple-metre variant of the balletto, and the passacaglia a brief set of variations on a related bass line. Acknowledged as Frescobaldi's masterpiece in variations, the Partite sopra l'aria della romanesca (1615; HAM, no.192), offers both the bass and treble parts of the framework in the prima parte, though the treble is much less important than the bass in the variations. Several variations have a different structure from the first; the ripresa in particular is likely to be shortened. A single figurative pattern dominates in a few variations, a ‘middle section’ of slower variations (with notes of double value) right after the ‘proportio tripla’ variation that effectively speeds up the motion. The surface of both treble and bass is often so irregular, free and ornate that the underlying framework can scarcely be detected. A magical final variation strips all that away with simple off-beat chords and a newly emergent thematic essence never otherwise seen in this era.

Variations, §6: The 17th century

(ii) The later 17th century.

The later 17th century saw few innovations in variation writing. In Italy, composers continued to write dance-framework variations for keyboard (Michelangelo Rossi, Bernardo Storace, Gregorio Strozzi) or strings (Stradella, 24 variations for solo violin on the ‘Gran Duca theme’, G.B. Vitali's op.3 no.12 ‘sopra l'aria del pass' e mezzo’, Corelli's ciaccona forming the trio sonata op.2 no.12 and his celebrated sonata on the folia for violin and bass op.5 no.12 (1700), imitated or coincidentally joined within a few years by Vitali, Albicastro, Vivaldi and Reali); Alessandro Scarlatti's toccata concluding with 29 variations on the folia conflates the trends by following Corelli on the keyboard. G.B. Vitali carried the principles of the variation suite into the sonata. Bernardo Pasquini, the most prolific composer of keyboard variations in this period, sometimes made specific reference to works by Frescobaldi (Toccata con lo scherzo del cucco, with the same insistent pattern as his predecessor's Capriccio but without the counterpoint) and in variations used several of the same theme-types (Ruggiero, folia) as well as his own themes, in this way following Frescobaldi even into originality. Pasquini's original-theme sets, in most cases called ‘variationi’ to distinguish them from the ‘partite’ on pre-existing frameworks, include ‘Variationi capricciose’, with a theme that already sounds like a variation (the texture simplifies later), and ‘Variationi d'inventione’, with dance-types labelled (variations 5–7 ‘in corrente’, 11 ‘gagliarda’), allusions to several keyboard pieces by Frescobaldi and a final ‘alla zoppa’-type syncopated variation.

In England few composers followed the brilliant virginalists, and their variations consisted largely of grounds (John Blow, William Croft, Henry Purcell), usually with a tonic-providing pattern. The best-known of these, Purcell's Chacony in G minor, resembles Dido's lament in the force of its chromatic expressiveness. The important mid-century improvisation treatise by Christopher Simpson, The Division-Violist (1659, 2/1665), offered ways both to ‘break the ground’, in which the viol player plays divisions over the notes of the ground held by harpsichord or organ, and to create ‘descants to the ground’, in which the viol makes a ‘different-concording part unto the Ground’, resulting in a series of divisions, an improvised variation form (see Division). The torch of the most prolific variation composers had passed to Germany and Austria.

Froberger, a German musician who studied with Frescobaldi in Rome and travelled widely before joining the musical establishment of the imperial court in Vienna, wrote in several of the older genres, as can be seen in his Libro secondo (1649): variation canzona (e.g. fbwv305, on an attractive songlike subject, rhythmically transformed in successive contrapuntal sections in different metres), fantasia on a soggetto treated contrapuntally (e.g. fbwv201, ‘sopra ut re me fa sol la’, in which the hexachordal subject is diminished, treated with countersubjects of decreasing rhythmic value, changed metrically and finally offered chromatically), and variation suite (e.g. fbwv601–5, all in Allemand–Courant–Saraband format except 602, which adds a Gigue). Froberger's most celebrated variation set makes up the sixth suite, ‘auff Die Maÿerin’, fbwv606, also known as ‘Schweiget mir vom Frauen nehmen’ (the title of a poem by Georg Greflinger published in 1651; a variation set by Reincken with both titles made its way into the Bach family scriptorium). It consists of six partite with the melody coming out most clearly in the first and fifth (a simplified outline), and such interesting changes as 12/8 metre in the third (another way to write triplets), semiquaver mirroring in the fourth and fifth and pervasive chromaticism in the sixth (‘Grammatica’), followed by a ‘Courant sopra Maÿrin’ with its double and a ‘Sarabande sopra Maÿrin’. (The title ‘Grammatica’ has been interpreted as follows: as the first ‘a’ of the word was changed from an ‘o’, it seems that the artist who wrote the decorative titles of each partita on the autograph may have started to write ‘Cromatica’; ‘Grammatica’ may refer to the Pythagorean temperament explained in the early 16th-century treatise by Magister Henricus Grammateus of Vienna because it contains both D and E: see ‘Commentary on the Works’ in Froberger, Neue Ausgabe sämtlicher Clavier- und Orgelwerke, i, ed. S. Rampe, 1993.) It is odd to find suite movements labelled separately from the variations, especially because of Frescobaldi's practice in Aria detta la Frescobalda; this suggests that the partite are but one element in the larger suite and that dances and their doubles have a separate identity.

Alessandro Poglietti, who spent most of his career in Vienna, went considerably further in labelling his variations with characteristic and descriptive titles. His collection of pieces entitled Rossignolo (1677, written for Leopold I and his wife Eleanora) contains several series of pieces, among them a suite of dances (Allemande–Courente–Sarabande–Gigue) each with at least one double, and an ‘Aria Allemagna con alcuni variationi Sopra l'Età della M[aes]ta V[os]tra’, with 20 variations to correspond to her majesty's age. The designated characteristics include a ‘lyra’ (hurdy-gurdy, variation 5), ‘Bohemian bagpipe’ (8), ‘Dutch flageolet’ (9), ‘Bavarian shawm’ (11), ‘Acrobat's tightrope dance’ (16), ‘Polish swordplay’ (17), ‘Military fife’ (18) and ‘Hungarian violins’ (19) as well as such comedies of manners as ‘French hand-kissing’ and the satirically chromatic ‘old women's funeral procession’. The other variations are figural, and one alludes to the learned style with ‘soggetto rivoltato’ (4), but in this means only that the melody is in the left hand. Unfortunately, Poglietti's theme is pedestrian to the point of lameness, which somewhat undercuts the attractiveness or cleverness of his imitations and descriptions. The programmatic aspects, however, guarantee it a place in the history of variations.

Johann Pachelbel contributed excellent works to the variation repertory, nearly all for organ or harpsichord. He wrote seven sets of chorale partitas, a form credited to Georg Böhm, who adapted the older contrapuntal form of chorale variations into a newer, more homophonic type that drew on techniques of secular works and was intended for non-liturgical performance. The chorale melody dictates the structure and is surrounded by ‘accompaniments’ rather than ‘counterpoints’. The four in Pachelbel's Musicalische Sterbens-Gedancken (1683) possibly reveal a response to the recent death of his wife and son in the plague by incorporating one expressively chromatic variation in each set. He was also partial to the chaconne, writing six ciaccone for keyboard and the Canon in D, an ostinato variation set for three violins and bass. The Canon, with an eternally simple if utterly compelling bass line of root position triads, is a locus classicus of the tonic-requiring type, over which the canonic violins emerge, intensify and recede, towards the end revealing the power of the flattened seventh degree. In the F minor Ciaccona for organ, an intense series on the descending tetrachord, a segment of the piece modulates to the relative major and, as later in Bach’s Passacaglia in C minor, thins out the texture before a final series of returns. The Ciaccona in D, on the other hand, is a surprisingly sectional work, on a two-reprise theme with each part ending on the tonic. Although the first reprise uses the same chord progression as the Canon in D, its force is dramatically weakened by the tonic-providing close, the repeat and the second reprise. Pachelbel's ‘summa’ of variations is the Hexachordum Apollinis (1699), a collection of six arias with mixtures of constant-harmony and melodic-outline variations arranged in the keys of the hexachord (D, E, F, G, A, all minor except the one in F) until the final piece, which uses the key signature of B but is in F minor. The latter is in any case anomalous in the set: it is in triple metre, its first reprise remains in the tonic, and it has a title, ‘Aria Sebaldina’, referring though with unknown import to the Sebalduskirche in Nuremberg, where Pachelbel had worked since 1695. The collection as a whole appears to have had a serious purpose not often found in secular variations: with a preface attesting the ‘beliefs of many’ that music comes from the ‘Dreymal-Heilig’ sung by angels as well as from the harmony of the ‘heavenly bodies’ known by Pythagoras and Plato, it is dedicated to Buxtehude of Lübeck and F.T. Richter of Vienna, thus apparently attempting to unite north and south, Protestant and Catholic, religion and humanism.

Many composers of south Germany and Austria luxuriated in lengthy passacaglias and chaconnes, with notable contributions by Biber, Kerll, Georg Muffat and Fux; each composed other types of variation as well. Unlike his extraordinary Passacaglia for solo violin (65 repetitions of the descending tetrachord, perhaps originating for a religious purpose), however, Biber's ostinato movements are not always labelled more specifically than ‘Variatio’ (e.g. Sonata no.3 for violin, 1681, on the descending tetrachord, or Sonata no.1 on the bergamasca progression; the Mystery Sonatas also include many ostinato variations). Buxtehude, like Pachelbel an influence on J.S. Bach, imported the ostinato form into northern Germany, writing two ciaccone and a brilliant Passacaglia in D minor on four-bar themes as well as seven sonatas for violins and bass viol with ostinato movements. The Passacaglia presents a symmetrical structure of seven statements of the tonic-requiring bass in each of D minor, F, A minor and D minor. In the ciaccone the bass theme may be varied and moved from the pedals to the upper registers. In addition to chorale variations (one of which, ‘Auf meinem lieben Gott’, is actually a series of dance movements including a double) and suite doubles, he also composed six sets of secular variations, including one on a Lully ballet melody (‘Rofilis’) and his most celebrated, La capricciosa (buxwv 250). The latter uses as melody the song Kraut und Rüben, found in the quodlibet (variation 30) of Bach's Goldberg Variations, and as the bass I–IV–V–I pattern of the bergamasca (twice in each reprise). Also in G major and also with 32 sections (partite), La capricciosa has in common with the Goldberg Variations a multiplicity of topics and styles: quasi-imitative ‘canzona’ (2, 5, 15, 20 and the chromatic 12), brilliant ‘toccata’ (3, 6, 7, 11, 13, 16, 22, 26), lute-like style brisé (10, 17 and the sarabande-like 25), gigue (9, 19) and tonic-pedal ‘lyra’ (18) styles (ex.7). In several variations the theme’s harmonies are altered to I–IV–V–vi, leading in variation 25 to a striking expansion in phrase structure (giving 10 bars in each reprise instead of four or eight).

French variations of the 17th century had consisted primarily of doubles for lute (Denis Gaultier) with figural patterns of the style brisé, doubles and other variations for harpsichord (Louis Couperin, Chambonnières, D'Anglebert, variations on Les folies d'Espagne), which used both style brisé and a more idiomatic keyboard ‘division style’, and noël variations for organ (Lebègue, Gigault). Chaconnes and passecailles were usually en rondeau, with the theme serving as an unvaried refrain to punctuate the couplets, rather than as a source of continuous variations, although the latter appeared in rare instances (e.g. Lebègue, Chaconne in F). Composition in every one of these genres continued during the 18th century.

7. The early 18th century.

(i) Couperin and Rameau.

François Couperin's many ordres and pièces de clavecin contain relatively few variation movements. Most of his passacaglias and chaconnes for harpsichord are en rondeau. A notable example is the vivid and intense Passecaille from ordre no.8. But an extraordinary set of variations in ordre no.13 (Troisiéme livre de piéces de clavecin, 1722), entitled Les folies françoises, ou Les dominos (ex.8), appears to represent a meaningful French answer to the Folies d'Espagne with its increasing frenzy or virtuosity, the meaning of which, if it ever had one, had long before been lost. Here Couperin populates a masked ball with characters charting a colourful progress of love from Virginity (clear mask), Modesty (pink mask), Ardour (carnation), Hope (green), Fidelity (blue), Perseverance (grey linen), Boredom (violet) and Coquetry (diverse masks, represented musically by changing metres), to their dispiriting progeny: galant old men and faded beauties (purple mask and dead leaves), benevolent cuckolds (yellow), silent jealousy (Moorish grey) and frenzy or despair (black). While the first three numbers in the ordre appear unrelated to the Folies (‘Budding lilies’, ‘Reeds’, ‘Engaging one’), the concluding ‘L'âme-en-peine’ seems to be a doleful commentary on the foregoing. The eight-bar bass pattern whose twofold statement underlies each of the 12 couplets is reminiscent less of the folia than of Purcell's Dido's lament or Chacony in G minor (z730), or of the minor-key sections of Bach's Goldberg Variations, that is, a quasi-descending tetrachord (the third bar loosens the pattern so that it briefly resembles the i–V–i of the folia) followed by an ascending formula ending on the tonic. As in many 16th- and 17th-century sets, the first couplet is already a variation: the ‘pure form’ of the theme must wait until its presentation in the bass in semibreves in the seventh couplet.

Rameau's Gavotte with six doubles from the third collection of keyboard pieces (Nouvelles suites de pièces, c1728) is his longest variation set. Part of its popularity derives from its harmonies which, like the Couperin Folies, refer to several sequences of chords from earlier dance basses: in the first reprise, the opening i–IV–V–i from the chaconne and descending tetrachord from the passacaglia, and in the second, the progression from III to V is reminiscent of the folia, while the final melodic-minor inflected ascent recalls Purcell's Chacony. Mirroring techniques animate the first few variations, with the melody appearing complete or in part in either upper or middle voice. The final variation returns to an ornate version of the theme, rather than continuing an increasing rhythmic trajectory.

After 1780, Haydn began to infuse variation into most of his slow movements. Of the symphonies from the Paris set onwards, only the slow movements of nos.83, 98 and 99 are without significant variation. His last strophic variation movement, the first movement of the String Quartet op.76 no.6 (1797), explores the constant-melody variation type found in three other string quartet movements of the 1790s. While the beautiful ‘Kaiserhymn’ that Haydn had just composed as the new Austrian national anthem (‘Gott erhalte Franz den Kaiser’) seemed to call for this sort of treatment, the theme of op.76 no.6, on the other hand, is repetitive, circular and rhythmically static (ex.10a), so there is a touch of amusement in the exaggerated way in which it is repeated: in overture style in variation 2, with a mocking echo in variation 3. The movement even ends with a wonderful Allegro fugato (whose countersubject foreshadows the countersubject in the ‘Eroica’ finale). But this repetitive presentation has important consequences for the rest of the quartet. In the Fantasia slow movement, a series of repetitions of the same hymn-like theme in different keys connected by improvisatory transitions also gives way to a fugato, while in the Alternativo to the Minuet, the theme is an E major scale given the cantus firmus treatment: it is stated over and over again in successive instruments either ascending or descending, with new counterpoints and accompaniments every time (ex.10b). The latter piece has never been described as theme and variations, but it clearly derives from the textural constant-melody variations of the first movement; one feels that one is hearing a witty staging of the scale with different characters entering and exiting with their textured retinues. Finally, the last movement uses a scale-based theme metrically displaced to different parts of the bar. Repetition, variation, counterpoint and humour form the underlying ‘plot’ of the entire work. No better example could be found of the importance of variation principles to Haydn's musical mind.

Variations, §8: The Classical period

(iii) Mozart.

In contrast to Haydn, Mozart's variations were primarily strophic and largely for piano; out of 55 movements and sets, only seven are rondo-variations, one is ternary and only 15 are for genres without piano. As his performance and publication opportunities increased, so did the number of variation works: 16 were written between 1766 and 1779, mostly for Salzburg and Paris (half were independent sets for keyboard) and 32 in Vienna between 1781 and 1791. He wrote no symphonic variations. Whereas Haydn tended to write a few variations on an elaborately detailed theme, Mozart wrote larger numbers of variations on relatively simple themes, exceptions being the texturally complex minor-key themes of the finale of the String Quartet k421/417b (1783) and the second movement of the Piano Concerto k456 (1784), each with five variations and coda, and that of the second movement of the Piano Concerto k482 (1785), a rondo-variation. The themes of these last two are also rich in expressive rhetorical figures and establish the idyllic pastoral wind-serenade variation as an important part of the concerto oeuvre.

Mozart's variation sets and movements were widely known and admired during and after his lifetime, as evidenced by contemporary accounts adducing the marvels of his improvising variations at the keyboard and by multiple editions of both independent sets and of piano arrangements of variation movements up to 1817. One of the sets most highly valued by Mozart himself and his Viennese audiences has had a chequered reputation in the 20th century because its brilliantly rhetorical element has not been understood. This work, known as the Rondo in D major k382, was composed in March 1782 as a new finale to his earlier piano concerto k175 (1773). In sending a copy to his father (23 March 1782), Mozart noted that it was making a ‘big noise’ in Vienna and was always greeted with torrents of applause; he asked that Leopold ‘guard it like a jewel – and not give it to a soul to play … I composed it specially for myself – and no one else but my dear sister must play it’. Mozart continued to perform it for over a year, referring to it as a Variazion rondeau (Schmid, 1992, adduces the title as evidence of a vocal origin of the theme). It is a theme and variations in which the first reprise of the contredanse theme returns periodically like a ritornello or rondo refrain, as he was later to write in the piano variations k613. Persuasively arranged, the sectional format of the movement has the ritornello recurring first after a single variation (the solo foil to the tutti and thus the more personal signal of the genre), then after two variations (each increasing the speed of figuration), then after three (each with an expressive or characteristic reinterpretation: minore, scherzando with trill accompaniment, Adagio). The final ritornello is an Allegro in the triple metre flanking a real variation and a coda. No other movement by Mozart asserts first genre, then technical strategy, then expressive value: it is a rhetorical tour de force.

Several of Mozart's improvised keyboard sets were later published, such as the ones originating at a Viennese concert of 23 March 1783 (at the Burgtheater, with the emperor present) on themes by Paisiello (k398/416e) and Gluck (k455). (Other improvised sets that were mentioned in letters and contemporary accounts – on Figaro's ‘Non più andrai’ (two different occasions), on Michael Kelly's melody to Metastasio's canzonetta ‘Grazie agl'inganni tuoi’, one on a theme given by Archbishop Colloredo in Vienna – do not survive.) The Paisiello set appears closer to what must have been the virtuoso display of its original performance, given the presence of the melody in all the variations and of three improvisatory passages, two smaller and one fully fledged cadenza, between the last four variations. The minore of variation 4 triggers the first of these with a change in phrase structure when the melody returns to the opening in a new key (A major) rather than a return to the second element (the theme is abcb'd, in which b is a codetta to a, b', is on the dominant, and d is a short conclusion); here the short cadenza is Adagio and full of the minore's expressive values. The scherzando trill variation 6 consists only of a and b and their varied repetition with the hands exchanged (mirroring). Two altered bars and a short cadenza lead to a final variation in étude style – triplets in right and left hand – in which a and b are heard only once before the lengthy cadenza and coda based entirely on b. Thus the last two and then the last three segments of the theme are sheared off in the process of stripping the theme to its most memorable elements.

That k398/416e appeared in print so soon after its first performance suggests that Mozart may have changed it only little from the improvised form; all the sources before 1798 lack numbers for the variations and double bars between them, suggesting the through composition of an extempore performance. The Gluck variations, on the other hand, came out more than a year later and seem more highly crafted, especially in the ways in which Mozart departs from the strongly profiled unison descent from the tonic to the fifth degree of the scale in the first four bars, whether by substituting another melody completely (variation 2) or by finding substitutes for the dominant chord on which it lands (as in variation 4, where every version of the phrase has a different chord and resolution). A lengthy cadenza doubtless refers to its origin and perhaps increased the salability of such a work by enabling the performer to mimic extemporization.

Mozart alluded to Haydn's variations four times, and possibly used one as a model, during the 1780s in Vienna. The variation movement from the Wind Serenade k361/370a employs a theme remarkably like that in Symphony no.75; the minor variation of the same serenade plays with the opening of the minor theme in Haydn's Symphony no.63 (‘La Roxelane’); and the siciliana variation finale of Mozart's D minor Quartet k421/417b draws together the siciliana variation finale of Haydn's G major Quartet op.33 no.5 and Mozart's own earlier D minor variations in a piano and violin sonata (k377/374e, the sixth variation of which is a siciliana). Finally, one possible explanation for Mozart's revision to the theme of the slow movement of his Piano Concerto k450 after he had already begun composing the variations is that he realized that the material in the second reprise was too close to his model, Haydn's Symphony no.75, with its triple-metre hymn theme. One classic locus for the inventio of a theme is in imitation of the theme of another, or imitatio, so this kind of homage is especially appropriate to variations.

Mozart's methods of organizing sets of strophic variations differ in nearly every particular from Haydn's, but most obvious are cadenzas in keyboard sets and the way they chain variations together; opposite-mode variations, especially the two movements in C minor with two variations in two different keys (in the Wind Serenade k388/384a relative and parallel majors, in the Piano Concerto k491 with submediant and parallel majors; the rondo-variation movement in the Piano Concerto k482 also has relative and parallel major episodes); and the presence of an Adagio or an Adagio–Allegro pair. Of 29 movements, 20 have opposite-mode variations (half in fourth position out of five or six), seven have a penultimate Adagio variation, and 11 (including six of the sets with Adagios) conclude with a faster and often metrically altered variation. Mozart's minore variations tend to be imitative, chromatic and highly expressive, often with an air of mystery, as in the Piano Concerto k453 or the Divertimento for string trio k563.

The autograph of the String Quartet k464 supports the idea of a conscious plan for the ordering of variations. After completing five variations, Mozart decided that the coda should incorporate the cello figuration of the fourth variation, and thus reversed the order of the last two variations. The coda followed the original fifth variation in the manuscript. At that point the minore was added and given the number 4, and the original fourth variation was renumbered 6. The real question is why there was no minore movement originally: what was the rationale of the original plan? In fact, the most highly figured line in each variation descends through the parts from first violin to cello until, with every register sounded, a contrapuntal epitome offers a summation, obviously intended to be the final variation (ex.11). One other movement seeks to articulate its imminent close with a contrapuntal study, the slow movement of the Divertimento k563, which has four variations and a coda. Here the contrapuntal variation is not imitative, as in k464, but rather is a layered species exercise in which the chorale-like cantus firmus, played by the viola, itself includes two different note values (minims and crotchets) and represents the theme melody stripped down to a scale. The cello in semiquavers creates a largely third-species relationship to the viola, while the demisemiquavers maintain a constant second-species relationship to the cello. In format, the movement resembles the ubiquitous figure of refining (expolitio) as described in the Ad Herennium: the theme expressed simply, the theme expressed in a new form (three-part version in the varied repeats), arguments from comparison, contrary and example (variations 1–2, 3 and 4, respectively), and a conclusion that restates the theme.

Other questions of ordering arise from the striking finales to two piano concertos, k453 (1784) and k491 (1786). The former has a labelled ‘Finale’ after five variations and extension, in which an entirely new Presto theme (of practically the same length as the variation theme) forms the first of five quasi-paratactic segments, all of which contain repeated and sometimes varied material. That these five segments might be intended to match and sum up the theme and variations is suggested first by the resemblance of the finale theme and its piano reiteration to the tutti variation theme and solo first variation, and second by the reference of the mock-serious ombra passages and ascending imitative sequences with suspensions to the minore. The finale of k491 features alternation between piano-dominated and wind-band-dominated opening periods and at the same time a kind of three-part form in which the ‘middle section’ appears to develop aspects of the first, with a contrapuntal piano variation alluding to the march-like variation 3 and the maggiore variations (4 in A, 6 in C), developing the wind-band sonority of variation 2. The final section offers two variations without repeats and a coda: the first is a melodic reprise uniting all instrumental groups and the second has both a new time signature and a new auxiliary-note motif from which the coda develops.

Variations, §8: The Classical period

(iv) Beethoven.

In works by Haydn and Mozart, the decorum of a variation movement – its traditional and hence normative technical and expressive limits – depended upon position in the work, upon genre and upon the nature of the theme. In general, its implicit code included several different properties: ordering, in which simpler textures appeared early in a set but imitative polyphony never did; performance style, in which extremes of orchestration and dynamics would be introduced for local contrasts, rarely as the topic of an entire variation; and contrast and return, in which distantly related or contrasting material would be followed by returns of the theme melody. Finally, the theme itself would observe certain properties, in its (usually) two-reprise phrase structure with clearly delineated phrases, its degree of repetition, and contrast in melodic segments, rhythms and textures. All these properties devolve upon the concept of familiarity and recognition, without which, Koch said, ‘[the variations] give the impression of a group of arbitrarily related pieces which have nothing in common with each other, and for whose existence and ordering one can imagine no basis’ (Musikalisches Lexikon, 1802). Thus Beethoven issued the first serious challenge to the decorum of the classical variation in the very first variation of the String Quartet op.18 no.5 (1799), a gritty contrapuntal build-up starting in the cello which, by suddenly eradicating the harmony and registration of the theme, inserts a new level of difficulty into a previously more accessible form.

Beethoven's first published work was an ordinary set of variations on a minor-key march by Dressler (woo63), promoted by his teacher Neefe to attract attention and money to the 11-year-old composer. Only eight years later Beethoven wrote the exceptional set of 24 variations on Righini's ‘Venni amore’ (woo65, 1790–91), which featured an imaginative array of both melodic outline and constant-harmony techniques and a wide variety of piano figurations and sonorities sometimes reminiscent of the thicker and more contrast-orientated mode of writing for piano characteristic of such northern German composers as C.F. Fasch, J.A.P. Schulz and J.G. Müthel. The Viennese style in keyboard variations favoured thinner textures, as evidenced in the works of such composers as Mozart, Vanhal, Štepán and Kozeluch. For his early Viennese sets, Beethoven generally adopted many of the common features of Mozart's keyboard sets, especially the minore, Adagio and finale variations, though the last in his case are normally rondos with extended excursions into foreign keys and no cadenzas. His ‘peculiarities of style’, however, were copied by others and ‘palmed off with pride as their own’, as Beethoven wrote testily to Eleonore von Breuning in 1794, forcing him to write down and publish pieces that ‘I should never [otherwise] have written down’. The fine craftsmanship of the Righini set is once again in evidence in the set on Salieri's ‘La stessa, la stessissima’ (woo73, 1799); concluding with an ‘alla Austriaca’, Beethoven thus joins a popular trend to characteristic variations, one example of which is Vanhal's Theme avec VII variations caractéristiques (c1805), with its variations ‘alla Polonese’, ‘alla Ungarese’, ‘alla Francese’ and ‘alla Inglese’. In the set on Süssmayr's ‘Tändeln und Scherzen’ (woo76, 1799), Beethoven for the first time tried to link several variations tonally: variations 5, 6 and 7 in this F major set are in D major, B major and F major (Adagio), followed by an Allegro fugue in the eighth variation with a chain of third-related keys after it reaches the dominant. For all his efforts, however, it was reported (AMZ, 1799, col.607) that ‘he does not understand how to vary well’; in the same year Haydn's F minor Variations, just published, received the comment ‘varied as only a master can’.

That Beethoven revised his variation style is attested by the two sets op.34 and op.35, offered to Breitkopf in 1802 with the request that they be given the opus numbers of his ‘greater musical works, the more so as the themes have been composed by me’. On 18 October 1802 he claimed that these pieces were composed ‘in a completely new manner, and each in a separate and different way’. Considerable scholarly attention has been paid to these claims, mostly focussed on the demonstrably new elements in both sets: it has been suggested (Küthen, 1984) that the phrase ‘wirklich gantz neue Manier’, however, was intended as a parody of Antoine Reicha's ‘new kind of fugue’ in the 36 Fugues dedicated to Haydn in 1803, Beethoven having contemptuously interpreted the description to mean ‘the fugue is not a fugue’ (Reicha's L'art de varier, op.57, was however similarly exploratory and inventive). The F major set op.34, the first free-standing Adagio with variations, moves from F to variations in D, B, G, E and C minor before returning to F, also changing the time signature and tempo in every one but the first. The rather conventional techniques of melodic decoration are thus given new life in changes of character and in what sounds like a radical defamiliarization of the theme as early as the D major of the first variation, virtually a conjurer's trick. The E Variations op.35, later known as ‘Eroica’ or ‘Prometheus’ variations because the same contredanse theme was used in the finales of both those works, were called ‘grand’ by Beethoven in contrast to the ‘small’ F major set, probably because of their length (15 variations and fugue), enormous technical difficulty and large-scale contrapuntal pretensions – immediately shown by the ‘Introduzione col Basso del Tema’, with three constant-melody variations, the ‘canone all'ottava’ and the fugal finale. Beethoven first claimed to Breitkopf that this set had 30 variations, just like the Goldberg Variations, recently published in several editions starting in 1799. (The publication in 1802 of Handel's Chaconne and C.P.E. Bach's folia variations may have inspired Beethoven's passacaglia-like work on an eight bar theme, 32 Variations in C minor woo80, of 1806.) Like these older models, op.35 uses constant-harmony technique almost exclusively, in contrast to the melodic orientation of op.34. In variation movements of the middle period, however, melodic-outline technique prevails, but with very different cyclic shapes: the sublime D slow movement of the ‘Appassionata’ Sonata in F minor op.57 (1804–5) adumbrates the trajectory of its beautiful hymn theme by combining the gradatio of register with that of rhythm in an arc-like shape, while the finale of the E major String Quartet op.74 (1809), with a metrically ambiguous theme, offers an alternation in dynamics which effectively groups the variations into piano–forte doubles in which the first of each pair is melody-orientated, the second more motivic and figural.

Beginning with the ‘Eroica’, every one of Beethoven's odd-number symphonies has prominent conjunctions of variation and fugue, forecasting the central place these forms would have in his late style; moreover, every symphonic variation movement except that of the Ninth Symphony finale adopts some type of alternating-variation scheme, with extensions and transitions making a more or less continuous and fluid succession of variations such as also marked Haydn's symphonic movements and Mozart's piano concerto variations. The ‘Eroica’ finale modifies both the nature of the theme and the alternating plan itself in a virtual compendium of available variation techniques with a symmetrical harmonic plan, all within a progressive structure. The two themes are those familiar from op.35; the first theme (A) is the bass of the second (B), but they are always varied in turn (after the first two cantus firmus variations on the A theme), with the first always treated contrapuntally (cantus firmus, fugato, countermelody), the second with melodic reprise, melodic-outline and constant-harmony technique. The first fugato introduces a dynamic, asymmetrical ‘middle section’ in keys, largely minor, outside the tonic, and it is during the final truncated variation in a foreign key (B², C major, bar 258) that the two themes meet as equals, while in the second fugato B becomes a countersubject to A and helps to banish it entirely. During the final two Andante variations, the A theme is not present even as a bass line, and the return of the Presto introduction now leads to B; the predominance of A at the outset is more than compensated by the final emphasis on B.

The slow movements of the fifth and seventh symphonies draw on the alternating-variation model, each offering a different reading of it and resembling ‘Eroica’ more than any movement of Haydn's. Both suggest a three-part design together with their alternations. In the Fifth Symphony, a middle section is articulated by a series of shortened variations followed by a climactic reprise of the first theme; E.T.A. Hoffmann noted its resemblance to Haydn's variations separated by Zwischensätze (AMZ, x, 1810, col.641). The Seventh, on the other hand, has a more developmental middle section between the two maggiore sections: a variation with bass cantus firmus that creates a dominant pedal leads to a fugato and then to a shortened reprise-like variation. The maggiore sections also serve to defuse the climaxes generated by each group of variations on the first theme, including the extraordinary ever-expanding constant-melody variations of the first theme at the beginning. The Adagio of the Ninth treats its hymn theme to echoes and internal repetitions as in the Fifth, and its alternation of the themes in B and D major is initially unproblematic except for the deceptive cadence that ends each theme. When the second theme comes back in G major, however, ‘resolving’ D, future progressive developments are forecast: the following modulating episode on A begins in E major, develops the upbeat motif of the theme and generates the next round of alternations, in which the B theme disappears and variations on A now alternate with fanfare episodes both on A and on the upbeat, each time beginning in E and then modulating through darker-hued keys. The tonal palette is expanded and the alternating principle is extended to include the thematically derived episodic material, which is itself alternated to generate the movement's climax. In the finale of the Ninth, chains of strophic variations on the ‘Freude’ theme are used to generate climaxes as well, in the manner of the first section of the Allegretto of the Seventh, a kind of incrementum.

In chamber genres, Beethoven tried alternating variations twice, first in the Haydnesque Piano Trio op.70 no.2 (1808), where two consecutive variations of A interrupt the alternations before an unusual ending in the minor. In the late String Quartet in A minor op.132 (1825), the Heiliger Dankgesang apparently retreats from the progressive alternation of the symphonies in favour of the earlier five-part alternating model. Yet the movement is anything but Haydnesque: the extraordinary modal polarities of the otherworldly A (chorale) sections set the stage for the deceptive yet emphatic tonal resolutions at the beginnings of the virile B sections, while the tempo change is given a programmatic rationale (‘Neue Kraft fühlend’). The increasingly vigorous figurations of the already highly elaborated B sections become stile moderno foils to the gently accelerating contrapuntal figures of the stile antico variation (A¹) and fugue (A²) on the chorale.

Beethoven's late strophic variations are the Diabelli set and fully five of the slow movements in the piano sonatas (opp.109, 111) and string quartets (opp.127, 131, 135) in addition to the instrumental and vocal chains of variations on the ‘Freude’ theme amalgamated with the larger composite structure of the finale of the Ninth Symphony. The inner world of the hymn theme in the Adagio of the Ninth Symphony is previewed in the sonatas, which also examine the propriety of concluding a work with a slow variation movement. The solution is similar in conception but differently realized in each work: progressive diminution leading incrementally to extremes of registration, dynamics and speed of figuration (trills), followed by a theme reprise. In op.109 the diminutions occur within a single ‘sprung’ microcosm, variation 6, after the grouping of variations 3–5 covers three Baroque topics in turn: invention in invertible counterpoint, pastorale and canzona. In op.111, progressive diminution and registral expansion open out into a linking trill-laden modulatory episode with elements of a developmental minore between the ethereal fourth and retrospective fifth variations, the latter a conspectus of melodic reprise and two of the previous levels of rhythmic diminutions. A comparable episode appears with even larger-scale tonal resonances in op.127, because in that movement in A variation 3 is in E major, and the episode after variation 4 is in D/C minor, a connection reiterated in the coda. The highly detailed part-writing in variation 1, with its foretaste of Mahler's Adagietto, is unprecedented in variations, as is the suspended-animation introduction, a slowly unfolding dominant 7th chord out of which the theme emerges very tentatively (ex.12). In op.131, the central variation movement has a role to play in the integrated design of the quartet as a whole, beginning as the ‘aria’ following the 11-bar third movement with its recitative-like close. Perhaps these suggestions of an earlier style prompted Beethoven to use an Andante 2/4 theme ending with a cadence formula of the 1760s (bars 23–4 and 31–2) and even, during variation 3 with its texture of contrapuntal voice-pairs and false fugato, an ascending trio-sonata-like trill figure also found in the first movements of Haydn's quartets op.20 no.1 (copied out by Beethoven) and no.2. The compressed sonorities of variation 5 recast for quartet the technique already found in the Diabelli Variations op.120, variation 20 (ex.13) and later taken up by Schumann and Brahms. A lengthy coda brings in cadenza-like trills, several partial, decorated returns of the theme and final reiterations of the cadence formula. After the intricate part-writing of op.127 and 131, Beethoven's extremely slow last variation movement, op.135, the first to be marked Lento, is largely unfigured until the final variation, and also brings in melodic reprise after the minore.

Beethoven's two bouts of working on the Diabelli Variations, 1819 and 1822–3, surrounded the composition of the last piano sonatas. Like the Goldberg Variations, they represent a grandiose final statement, and like them had a strange reason for coming into existence: in 1819, Diabelli sent his waltz to composers important in Austria and asked each to contribute a single variation to a patriotic Vaterländischer Künstlerverein for the benefit of widows and orphans. The resulting cross-section of variation style was not published until 1824, the year after Diabelli brought out Beethoven's 33 variations, with the impressive total of 50 in alphabetical order from Assmayer to Worzischek (Voříšek), with a coda by Czerny. Included are a contrapuntal variation by Sechter, a minore by Schubert, one of brilliant virtuosity from Kalkbrenner (hastily commissioned during a visit to Vienna), variations by Czerny, Moscheles and Gelinek, an overture by Dreschler, a polonaise by Tomášek and, at opposite ends of the generational span, contributions from the elderly Förster and Schenk and the 11-year-old Liszt. As Kinderman's research has shown, Beethoven's 1819 draft already included 23 variations, to which he added variations 1–2, 15, 23–6 and 28–9, and revised the conclusion. The addition of the Alla Marcia maestoso as variation 1 is crucial because it instantly reveals contrast to be the primary aesthetic of the set. Just as Bach took the pensive Aria and turned it into an athletic two-part invention for the first Goldberg variation, so Beethoven announces a new topic, metre, tempo and texture immediately. Although characteristic variations do not remain at the fore during this formal-outline set, extremes of textures and of expressive types do. It is significant that after all the contrapuntal, brilliant, scherzando, fugal, epigrammatic, espressivo and comic variations, the finale should be a Tempo di Menuetto, grazioso e dolce, a tribute to Beethoven's and Diabelli's shared patrimony.

(iii) Symphonic variations after Beethoven.

9. The 19th century.

(i) Early 19th century.

(ii) Brahms.

Variations, §9: The 19th century

(i) Early 19th century.

Schubert was a master of the melodic reprise varied by means of beautiful new figurations or harmonies, in such pieces as the Andantes of the A minor String Quartet (a sonata-rondo), the C major String Quintet (an ABA variation) and late sonatas (alternating rondo-variations, d850, 894 and 958). Several of his variations, including the best-known variation movements, are based on his own lieder, and only four times did he vary themes not written by himself (d576, 624, 823 no.22, 908); the first of these gave a hint of Schubert's interest in linking the opposite-mode variation to one in a related foreign key. His first important set, on Die Forelle, was the Andantino of the Piano Quintet in A (1819), reflecting its vocal source with a constant melody during the first three variations (first violin, piano, second violin, cello). After the minore, a variation in the flat submediant – coloured by its own minor and flat mediant keys – simultaneously offers the most rhythmically layered variation in the entire movement. (This grouping of a minore with a different major key related to it is found also in the Piano Sonata in A minor d845, B Impromptu d935 no.3, and the C major variations on a theme from Hérold's Marie for piano four hands d908.) The final theme reprise returns to something outside the piece: the piano accompaniment of the lied itself. The Wanderer Fantasy (1822) not only links each movement by a head-motif related to the lied Der Wanderer, but clothes that eight-bar melody in the slow movement with different accompaniments, keys (C minor, E, C) and cadence patterns, finally dissolving the melody itself into periphrastic figurations. Virtuoso interludes and conclusions outside the variation chain assimilate the movement to the technically demanding cycle. The year 1824 saw the ornate Introduction and Variations on Trockne Blumen for flute and piano d802, the String Quartet in D minor with slow movement variations on Der Tod und das Mädchen, the Octet with variations on a duet from Die Freunde von Salamanka, and the A variations for piano four hands, with its nod in variation 5 to the Allegretto of Beethoven's Seventh Symphony. Of these, the quartet and piano sets are among Schubert's finest and most significant. Using the sombre chords of Der Tod for the theme, the quartet maintains a quasi-constant melody for variations 1 and 2 and the first reprise of 5, but breaks into a gallop for 3 based on a diminution of the theme's anapaestic rhythm. In the general shape of the set, including the ethereal maggiore of variation 5 and the major-mode coda, recalling the major ending of the lied, the movement appears to be modelled on the finale of Mozart's D minor Quartet k421/417d.

Of the composers making their name at the turn of the century who excelled at variations, especially in the eyes of contemporary reviewers, those who stand out are Spohr (for violin, clarinet, harp and string chamber music), Hummel (mostly for piano but some in chamber music; variations account for a quarter of his published piano works, some single opus numbers containing up to three sets), Prince Louis Ferdinand of Prussia (for piano and piano chamber music) and particularly Weber, who wrote a number of variations on original themes and in new genres like the concertinos for clarinet and for horn, as well as on such much-varied themes as Schöne Minka. These last pieces have elaborate introductions, which were becoming more and more common, and lengthy linked finales.

The 1820s and 30s saw a vast increase in the number of showy variation sets for piano on popular tunes tricked out with all manner of introductions, finales and virtuoso details in such a way that they became nearly indistinguishable from the numerous fantasies on popular melodies. Already under way in the 1790s with Daniel Steibelt and Abbé Gelinek, the trend accelerated with the growing concert scene in Paris, Vienna and Berlin and with such composers as Herz, Hünten, Kalkbrenner, Thalberg and Moscheles. That the 1830s exacerbated this trend may be seen in setting Diabelli's encyclopedic Vaterländische Künstlerverein of 1824 against the public display of the Hexaméron, a set of ‘grandes variations de bravoure’ on a theme from Bellini's I puritani commissioned for a benefit concert in Paris in 1837, with contributions by Liszt, Thalberg, Czerny, Pixis, Herz and Chopin. Indeed, the previous year saw the publication of Czerny's L'art d'improviser, in which Hummel, Moscheles, Kalkbrenner and Ries are upheld as models, although the student is also advised to study Mozart's k354/299a, Clementi's variations on the same theme in the finale of a B Sonata, and Beethoven's opp.34 and 35 and woo80. ‘The number of forms at the disposal of the artist is infinite’, Czerny proclaims, giving as examples figurations for either or both hands while retaining the melody or at least the harmonies of the first part; trills and ornaments of all kinds; a new cantabile theme on the original bass and harmony; ‘strict style’ with the theme placed in a higher or lower voice; canons or fugues on the theme; and changing the tempo, metre and key, as in an Adagio, Polonaise or Rondo, and in a finale with free development (p.92).

Schumann found most variations of this time irredeemably trite and vapid. Although he praised a few sets of variations by his contemporaries, especially Chopin's variations for piano and orchestra on Mozart's ‘Là ci darem la mano’ op.2 – which prompted his memorable ‘Hats off, gentlemen: a genius!’ – Schumann more often deplored their lack of meaning. He argued against using popular themes, notably ‘the most hackneyed Italian ones’, because the best sets have themes with personal resonances for the composer, and against the empty virtuosity and mechanical figurations of salon variations; in a scathing review of 1836 he asserted that ‘in no other genre of our art is more bungling incompetence displayed … variations should create a whole, whose centre is the theme. … The time is past when one can create astonishment with a sugary figure, a yearning suspension, an E major run over the keyboard. Now one strives for thoughts, for inner connections, with the whole bathed in fresh fantasy’ (Gesammelte Schriften, i, 219, 221, 223).

A more thoughtful typology from within the Schumann circle was published in 1860 by the Berlin composer Julius Schäffer:

The variation form, although cultivated by the masters with special partiality, is still so badly mistreated by bunglers and hacks that, when it appears, people avoid it or encounter it with mistrust, and as a consequence of its bad reputation noteworthy theorists and aestheticians scarcely want to grant it even a modest spot next to legitimate art forms. This appears to us unjust.
If we exclude the bravura variation, then the different forms of variation divide into three principal categories. In the first, which can appropriately be described as the decorative, all interest lies in the theme. In each variation, this is clothed, as it were, in a new attire, but it is not disguised. … It is usually a known melody and the goal of this genre is the ever-new charm of its differently turned-out repetitions.
In the second [category], which we call the contrapuntal, the centre of gravity lies in the variations themselves. … Here the theme is only the outline, on which different architectonic creations are built. … This category stands higher than the first … [and features] the creation of independent structures on the basis of the given harmonic relationships. …
In the third category, the centre of gravity lies neither in the theme alone nor in the variations alone, but rather in the psychological bond between the two. … That the theme is usually an invention of the composer's – a so-called original theme – is entirely in the nature of the thing. The individual variations will have to manifest a connection with the theme as well as with each other … ; in other respects, however, they will come into the world bringing with them their newborn motifs and new developmental laws, thus [each] to expand into autonomous art forms – often even as related movements not directly derived from the theme [but] like ‘intermezzi’ draw into their own realm. Just as the variation form in this genre achieves its highest significance, it reaches at the same time its outermost limits, striving to overcome them and to pass into the sphere of the free fantasy. It appears not inappropriate to give them the name Fantasy-Variations. (Echo, x, 1860, p.95; quoted in Puchelt, 142–3)

These remarks keep the same hierarchy as Schulz's rank-ordered list of 1774 while adding fantasy-variations. Brahms was later to argue that fantasy variations are not really variations at all.

Schumann's own variations chose a personally meaningful subject as early as the Abegg Variations op.1 (1830); he went on to vary themes by Clara Wieck (Impromptus op.5, based on her Romance variée op.3) and by Ernestine von Fricken's father (Symphonic études op.13), and to base another work, Carnaval op.9, on a cipher, A–S–C–H, in which the distinction between a motif to be varied and to be transformed breaks down. From Clara's theme he took an idealized bass line, which is initially presented alone, in the manner of Beethoven's op.35; its opening I–IV–V–I puts it in the much older tradition of the bergamasca and other dance schemes (ex.14). Von Fricken's theme, on the other hand, was a melody only, originally for flute. Schumann reassured von Fricken that he was very strict with the theme because the unfolding structure was based on it; in variations the object in view is ‘always before us’ but seen ‘as though through coloured glass’. Thus he sought to ‘break the pathetic [nature of von Fricken's C minor theme] into divers colours’ (Jugendbriefe, 251–4). In the second edition of the Symphonic études, now titled Etudes en forme de variations, Schumann also differentiated terminologically between ‘variations’ more closely related to the theme and structurally freer ‘études’. He followed Beethoven and Schubert in including a lengthy finale, here a rondo on a theme of Marschner. In addition to variations for piano, Schumann wrote variation movements in two string quartets, op.41 nos.2 and 3, as well as the finale of a sonata for piano and violin op.121. His compositional ‘last thought’ was a set of variations on a theme in E, on which he was working just before his suicide attempt in 1854.

Other responses to the superficiality of the form included two by Mendelssohn, his 17 Variations sérieuses op.54 (1841), which despite a fugato in variation 10, several scherzandos and an example of ‘Schumann shorthand’ in variation 11 (a curtailed outline of the theme's melody), stay close to the harmonically quixotic theme, and his chorale partita on Vater unser in Himmelreich (Six Sonatas, op.65 no.6), a set of cantus firmus variations with chorale fugue and finale. Formal seriousness and a striking set rhythm come to the fore in Liszt's two sets of ostinato variations on Bach's bass line from Cantata bwv12, Weinen, Klagen, Sorgen, Zagen. The first of these is a small-scale Prelude (1859) that is actually a passacaglia with 25 repetitions of the bass theme. The much larger Variations (1862) contain not only 43 variations but also a recitative, finale and chorale (‘Was Gott tut, dass ist wohlgetan’, the same one that ends Bach's cantata), and considerable chromaticism within its formal outlines. These two works mark the reappearance of the Baroque ostinato-variation as conscious archaism, although ostinato form had made an appearance in Chopin's Berceuse op.57, with a much simpler pattern. Liszt even included a series of variations on the folia within his Rhapsodie espagnole for piano, immediately after the introduction.

Variations, §9: The 19th century

(ii) Brahms.

Brahms was the first composer after Beethoven to whom variations were central. For over 40 years, in seven independent sets and nine variation movements from his op.1 to his op.120, he showed ‘what could still be done with the old forms’ (Kalbeck, quoting Wagner on op.24). He also used variation technique extensively in slow rondo movements, like the first movement of the Horn Trio op.40, slow movements of the Piano Quartet in A op.26, the String Quintet in F op.88 and the Violin Sonata in A op.100, and nearly every slow ABA movement. While the varying of theme returns is quite common in this period, Brahms exceeded contemporary practice by placing variations in sonata form movements: the second theme group in the first movement of the Piano Quartet in C minor op.60 consists of an eight-bar melody with three variations (the second in the minor), which is recapitulated with new variations, with the final one an actual recapitulation of the exposition's first. The development section of the Piano Quartet in A op.26 begins with three variations of the opening theme in B minor. Like Schumann, Brahms claimed that the theme was centrally important, but more particularly

In a theme for variations, it is almost only the bass that actually has any meaning for me. But this is sacred to me, it is the firm foundation on which I then build my stories. What I do with melody is only playing around. … If I vary only the melody, then I cannot easily be more than clever or graceful, or, indeed, [if] full of feeling, deepen a pretty thought. On a given bass, I discover new melodies in it, I create. (Letter of Adolf Schubring, Feb 1869; Briefwechsel, viii, 217–18)

Brahms went on to trace the ‘path made by the art of variation’, from Bach's Goldberg Variations to the melodic variations of Herz, to Beethoven and himself, and then to Schumann, asking wistfully ‘But could we not make a distinction between variations and fantasies on a melody, a motif?’ In two other letters Brahms seems eager to assess the history of the form and his place in it. To Joachim, with whom he was engaged in a ‘correspondence course’ in counterpoint in the summer of 1856 and who had just sent him some variations on an Irish folksong, he wrote:

From time to time I reflect on variation form and find that it should be kept stricter, purer. The Ancients were very strict about retaining the bass of the theme, their actual theme. With Beethoven, the melody, harmony and rhythm are so beautifully varied. I sometimes find, however, that the Moderns (both of us!) more often (I don't know the right expression) worry the theme. We anxiously retain the entire melody, but don't manipulate it freely. We don't really create anything new out of it; on the contrary we only burden it. The melody thus becomes scarcely recognizable.

Brahms's final verbal document on variations is the rather censorious letter he sent to Heinrich von Herzogenberg, who in 1876 was about to publish the first variations ever written on a theme of Brahms:

I wish people would distinguish between the title Variations and something else, possibly Fantasy-variations, or however we would want to call almost all the newer variation works. I have a singular affection for the variation form, and believe that this form still compels our talents and ability. Beethoven treats it so extraordinarily severely, he can even justly translate [the title variations as]: alterations [Veränderungen]. What comes after him, by Schumann, H[erzogenberg] or Nottebohm, is something else. I have, of course, as little against the method as against the music. But I wish people would distinguish by name what is different in the method.

What the variations Brahms labelled ‘fantasy-variation’ – by Nottebohm (on a theme of Bach, op.17), Herzogenberg (on a theme of Brahms, op.23) and Schumann – have in common are extensive alterations in the structure, metre and tempo of the theme while retaining its melody or motifs. Brahms's own variations, like Beethoven's, may depart from many details of the theme but normally retain its formal outline. Like Schumann, he often chose themes with personal significance and let the character and source of the theme play a role in the nature of the variations: a song theme led to melody-orientated variations, as in the sonatas opp.1 and 2 (with colourful harmonic substitutions) and op.21 no.2 on a Hungarian song, a Handel theme led to a stricter conception (as well as Baroque topics like siciliana and musette, canon and fugue), a Paganini theme led to virtuoso variations.

Brahms's independent variation sets up to 1862 show a tendency to pair stricter and freer conceptions. In the Schumann variations op.9 (1854), that contrast emerges from the attribution of most of the variations in the autograph manuscript either to ‘Brahms’ (variations 4, 7, 8, 14, 16) or to ‘Kreisler’ (5, 6, 9, 12, 13), referring to his Schumann-inspired alter ego from E.T.A. Hoffmann's character, and the piece becomes the embodiment of a dual persona. The ‘Brahms’ variations are nearly all slow, like the theme, and tend to have a lyrical melody which is sometimes treated in canon, while the ‘Kreislers’ are fast, feature melodic fragments embedded in figurations that consciously recall Schumann, contain codas, lack canons and depart more strikingly from the theme's structure, harmony and affect (ex.15). The theme itself, a poetic Albumblatt, was doubtless chosen by Brahms because Clara had written her op.20 variations on it the previous year, a colourful, pianistically rich set with a canon in variation 6 and melodic resemblance throughout; to make his own set still more Schumann-connected, Brahms quoted from Clara's Romance op.3, which Schumann had already varied in op.5 (variation 10), as well as from another Albumblatt (variation 9), and added his own Clara-cipher in variation 11. Brahms's next variation pair, op.21 nos.1 and 2 (1856–7), share several features: minore variations limited to a single large grouping, linking of most of the major variations by melody or speed of figuration, and a finale which includes a reworking of the first variation (the last also in Joachim's Variations in E for viola and piano op.10, 1854). The greater sophistication of op.21 no.1, written somewhat later than no.2, derives from its theme, Brahms's first written specially for a set of variations; the theme of no.2, an eight-bar Hungarian song, had been written down as early as 1853, and Brahms's interest in things Hungarian was rekindled with Clara's trip to Budapest in 1856. Both sets have Beethovenian elements. His final big pairing is the Handel Variations op.24 with the four-hand Schumann Variations op.23 of 1861; the theme of the latter, Schumann's ‘letzte Gedanke’, was chosen by Brahms for its ‘melancholy sound of farewell’ (letter to Joachim, 29 Dec 1862, Briefwechsel, v, 331). The stricter–freer paradigm can be seen in the conclusions of each set – the Handel set ends with a fugue, the Schumann with a funeral march – as well as in the telling differences between their two-part imitative minore variations: variation 6 in the Handel set is entirely canonic and resembles the theme's melody, while 4 in the Schumann set is more freely imitative and mysteriously evocative. Schubring's lengthy 1868 review of the latter piece includes an account of its emotional ‘meanings’ (Eng. trans. in The Compleat Brahms, ed. L. Botstein, 1999, p.200). Finally, the Handel set draws liberally on the Diabelli Variations, especially in variation 3 (Diabelli 11, 19), 7–8 (Diabelli 15), 14 (Diabelli 16) and 2 and 20 (Diabelli 3 and 12). The Handel ‘music-box’ variation 22 goes back to Couperin, and among the more contemporary styles are étude (2, 4, 21), introspective character-piece (5, 12), triumphal march (25) and Hungarian rhapsody (13). Of the two Paganini sets op.35, book 1 contains a preponderance of variations with melodic and harmonic resemblance to the theme, as well as older topics like the descending tetrachord bass (4) and quasi-musette (11), while the variations in book 2 immediately reinterpret the theme's harmonies, even at cadence points, and contain the variation most remote from the theme (12).

The variation movements up to 1864 also reflect pairing to a degree. The Sonata op.2, while close in variation technique to op.1, uses a more chromatic theme and recycles it as the scherzo theme. More profoundly connected are the two string sextets op.18 (1860) and op.36 (1864). The theme of the D minor Andante of op.18 suggests older models such as the folia and the Allegretto of Beethoven's Seventh Symphony, and its treatment includes expanding register while decreasing note values in the first three variations, a ‘heavenly’ maggiore musette (variation 5), a final quasi-reprise and a coda referring to Mozart's D minor quartet finale and the plagal major conclusion of Schubert's D minor Quartet ‘Der Tod und das Mädchen’. The slow movement of op.36, on the other hand, uses a floating, chromatic treble theme written much earlier (dedicated to Clara in a little pocket notebook of Christmas Eve 1854, and sent to her two months later) which fluctuates between minor and major, opens with a sequence on I and VII, and opposes rising 4ths and descending chromatic figures. In short, the theme is sufficiently ‘free’ for it to be varied strictly and still suggest fantasy, as in the first variation, which sounds like a continuation of the theme while actually incorporating augmentation, diminution and a new bass line. After two sets of paired variations (1–2, 3–4), the second resembling the aggressive counterpoint of Beethoven's Quartet op.18 no.5, the maggiore returns to melodic-outline technique using diminutions of the rising 4th radiated throughout the texture.

Brahms's later variation period, spanning years from the St Anthony Variations op.56 (1873) to the Clarinet Sonata in E op.120 no.2 (1894), reveals new preoccupations and older models of a different kind: finale variations which quote the theme of the first movement to bind the work together (String Quartet in B op.67, Clarinet Quintet op.115) or serve to conclude the work with a slow movement like Beethoven's late sonatas (op.120 no.2); the passacaglia (the tonic-requiring finale of the formal-outline variations in op.56, the bass subject deriving from the theme melody and the tonic-providing fourth movement of the Fourth Symphony op.98, the theme deriving from Bach); alternating variations based on a split theme, like the finale of the ‘Eroica’ (Andante con moto of the Piano Trio in C op.87); and fantasy-like reiterations of a theme in different keys, like Haydn's Fantasia in op.76 no.6 and Schubert's ‘Wanderer’ Fantasy slow movement (Adagio of the String Quintet in G op.111). The ‘St Anthony’ or ‘Haydn’ Variations (the chorale theme was not written by Haydn) was probably the first independent set written for orchestra and is widely viewed therefore as Brahms's ‘warm-up’ for a real symphony, yet he conceived of it in both orchestral and two-piano versions, each a fully autonomous work, and in an even more unusual move did not destroy the extensive sketches. The anapaest rhythm of Beethoven, Schubert and the Handel Variations returns in variation 6, and other evocations of op.24 appear here and there (e.g. 7, a combination of the Handel's 17 and musette 22). Characteristic variations, like the scherzando (5) and siciliano (7), are few. The embellished varied repeats of variation 3 contrast with the repeats in invertible counterpoint in 4 and with the added countermelody in 8. In the passacaglia finale, progressive diminution and expanding registers articulate the close of this novel form. In the finale of the Fourth Symphony, on the other hand, which bears a quite different kind of weight, Brahms drew on the formal articulations of Bach's Passacaglia for organ and Chaconne for solo violin, the second of which he had already arranged for piano left hand, while writing 30 variations and coda, the number (minus da capo) of the Goldberg Variations. There is a ‘middle section’ of slow variations, all but one in the major (12–15, the last two with ‘Wagnerian’ brass), after which the theme returns nearly unvaried (16); this return halfway through the set suggests the ‘Ouverture’ of the second half of the Goldberg set. A further return to the theme and to the rhetoric of the first two variations (23–5) immediately follows the ‘scherzo’ variation (22). Other variations deliberately recall the first movement of the symphony: 10, with its antiphonal echoes, strange harmonies, crescendos on a single chord; and especially the final group, 28–30, which brings back the descending 3rds of the opening theme.

Variations, §9: The 19th century

(iii) Symphonic variations after Beethoven.

The attractions of the ostinato variation as a format for conflations of old and new, as well as its appearance in composers as different as Liszt and Brahms, predicated the wholesale return of the passacaglia in the 20th century. However, within the post-Beethoven symphony, Brahms's Fourth was anomalous because so few composers wrote variation forms at all, preferring instead to vary returns and repeats in their slow movements. Those who did, notably Bruckner and Mahler, appropriated as models the Beethovenian alternating variation of the fifth, seventh and ninth symphonies. Most of Bruckner's symphonic adagios vary two themes in turn, usually by enriched orchestration, accompaniment patterns, countermelodies and different keys; like Beethoven's Ninth, they generate a substantial climax. Variations are important in the slow movements of Mahler's early symphonies, but only the Second and the Fourth can be said to have true variation forms, while the First and Third contain opening themes varied in inventive ways when they return in the course of more rondo-like structures. His Second Symphony alternates variations on themes in tonic major and minor (ABA¹B¹A², in A major and G minor) with lengthy mood-changing transitions and transparent techniques of variation; in early 1900 he told Natalie Bauer-Lechner that, unlike Brahms's strict variations, his own variations (in the Second and Third) are ‘more embellishments, playing around, and entwinements, than a careful following up and working through of the same grouping of notes’ (Gustav Mahler in den Erinnerungen von Natalie Bauer-Lechner, ed. H. Killian, 1980, p.153). The ‘entwinement’ doubtless includes the countermelody added in A¹. B¹ reaches a climactic level of scoring and dynamics. The Poco Adagio of his Fourth Symphony, on the other hand, draws more on the variation groupings of Beethoven's Seventh, even while referring to the pizzicato bass line of the Adagio of Beethoven's Ninth. Expansion of register, enrichment of orchestration and addition of countermelodies characterize the opening series of A variations, but the ascent leads to greater peacefulness rather than to a climactic restaging of the theme. Like the Seventh, Mahler's movement has bass elements of the A sections in the more episodic and increasingly dissonant B sections, which begin in the relative rather than tonic minor but move far afield. The second A group presents faster, more dance-like tempos and characters, while the third increases progressively in tempo and metre (Andante 3/4, Allegretto subito in 3/8, Allegro subito in 2/4, Allegro molto in 2/4) until the Anfangstempo der Variation returns, Andante subito in 3/4, and after it the Adagio. The ‘heavenly music’ of the finale appears in the coda, with an apotheosis of the bass figure in the timpani. In some ways the strophic form and progressive tonality of the finale are adumbrated in the variations of the Poco Adagio.

Brahms's St Anthony Variations launched the independent orchestral set as a genre. Closest to the model are sets on borrowed themes, like Dvořák's Symphonic Variations (1877), Parry's Symphonic Variations (1897) and the variations by Reger on themes by Hiller (op.100, 1907) and Mozart (op.132, 1914). Others, recalling such earlier works as Chopin's op.2, add a featured solo instrument, such as Tchaikovsky's Variations for Cello and Orchestra on a Rococo Theme (1876), Franck's Variations symphoniques (with piano, 1885) or Delius's Appalachia (1896), or adapt variations to a programme, as in Strauss's Don Quixote (also with prominent solo cello, 1896–7), d'Indy's Istar (1896) and Elgar's ‘Enigma’ Variations (1898–9). Dvořák begins in Brahmsian fashion without introduction; the theme, based on a Czech folksong the composer had recently set for male chorus (‘I'm a fiddler, poor as can be’), is in C with a raised fourth (the triumphant end of the coda repeats the opening four notes diatonically), stated at once with minimal accompaniment, then varied 27 times, concluding with a fugue. The first three variations maintain a constant melody, so that its three-strain (ABA) construction is firmly fixed in mind. After this point the departures in length, metre, tempo and key cannot obscure the essentially formal-outline status of the rest, with frequent returns of the melody, during which time characteristic topics of pastoral, scherzo, ‘mysterious’ adagio and martial exhortation make their appearance. The work was an enormous success. Franck's Variations symphoniques are a different matter: only one part of the lengthy piece is actually a theme plus strophic variations (bars 100–249), after a lengthy opening section on two different melodic ideas. The piano states the theme and the variations perform a kind of historical synopsis of the form, though out of chronological order: a dialogue in variation 1 emphasizes melodic-outline technique, constant-melody technique enters in 2 and 5, the étude-like 3 conceals just a few melody notes in its constant-harmony technique, while 4 is a centrally located expansive climax with a new structure. After the variations per se, the theme of the introduction returns at a slower tempo and is varied rather freely and developmentally, as it was in the introduction proper. An extended final section, Allegro non troppo (bar 285), brings back both themes in new guises, with the variation theme in the bass. The structure of the whole shares features with early 19th-century piano fantasias that might contain a set of variations within more improvisatory material (e.g. Beethoven's Fantasia op.77); while Nelson assigns the entire piece to a category of ‘free’ variations, it is clear that the variations themselves are not free but are embedded within a fantasia-like larger structure that alludes to a multi-movement work.

Strauss's Don Quixote, on the other hand, is subtitled ‘fantastische Variationen über ein Thema ritterlichen Charakters’, and is indeed a set of ‘fantasy-variations’ in an episodic form (Strauss's first notated idea for the work calls them ‘Verrückte, freien Variationen’). Like so many others, Strauss not only wrote down a few words on the history of variation but found a threefold historical typology:

I consider as the historical high points of the so-called variation form: the unbelievably brilliant Chaconne of J.S. Bach, in which the invention of figural elements appears to have sprung like Minerva from the head of Jupiter in a perfection since then no longer achieved. The paradisally beautiful Kaiservariations in the immortal Haydn's C major Quartet, as the ideal expression of the melodic clothing of a beautiful theme; and third, the metaphysical (I know no better expression), unearthly creation of the A Adagio from Beethoven's E Quartet op.127, with which I take the variation form as a purely musical creation, music-historically speaking, to have concluded. It found in the prelude to Rheingold and in Siegfried's Blacksmith's Song an application scenically full of meaning and dramatically important, after which it leads in my Don Quixote to representations of futile phantoms in the head of the Knight of the Sad Countenance – a kind of Satyrspiel ad absurdum. (Diary entry quoted in W. Werbeck, Die Tondichtungen von Richard Strauss, 1996, p.454)

Thus Strauss's term ‘fantastic variations’, like Brahms's ‘fantasy-variations’ category, consciously distances itself from the form, but Strauss goes further in appearing to mock it. There is no ‘ur’-structure to which the variations refer; the themes are melodies personified as Don Quixote (cello) and Sancho Panza (bass clarinet, bass tuba, viola), though the latter theme is marked simply maggiore and its status with respect to the first and primary theme is unclear; both are fully ‘thematized’ versions of melodies heard in the introduction (Dulcinea, also heard there, returns in variations 1, 3, 5 and 6) which subsequently appear varied by division and by other sorts of rhythmic and harmonic transformation as the programme requires. Unlike Till Eulenspiegel, a rondo in which the prankster's theme peeps out unvaried between episodes of mischief-making, in Don Quixote the themes are incorporated into the sometimes precisely delineated action, from the adventures with the windmills (variation 1) and sheep (2), ending with Quixote's death. More ruminative sections include the introduction, which details Quixote's process of derangement after reading knightly romances, and variation 3, a mad conversation between Don and squire.

The attention given to Elgar's Variations on an Original Theme op.36 has focussed primarily on the ‘enigma’ contained therein, a word authorized by the composer to be placed in the score, alluding possibly to an unwritten theme that goes ‘through and over the whole set’, as well as a ‘dark saying’ that ‘must be left unguessed’. Commentators on the works, while assessing Elgar's words and subsequent theories, have also made clear (as did Elgar himself, and Tovey) that the piece should be evaluated on its own musical terms (see Rushton, 1999). Each variation is a character portrait of someone close to Elgar, including himself, and the initials and other cryptic references that serve as headings to the variations have all been identified. The ternary structure of the G minor theme, with a B section in the parallel major, looks ahead both to the frequent appearance of variations in the major (3, 8, 10, 13, 14, as well as 6–7 in C major and 9 in E) and to the relatively greater importance of the melody of the A section as a virtual idée fixe. That melody, appearing in a variety of rhythmic transformations, in different registers, metres, tempos and instruments, has been the focus of intense scrutiny both because of the contrapuntal possibilities suggested by the enigma and because of its similarity to other melodies by Elgar and to the second theme of the slow movement of Mozart's Prague Symphony. Its prominent chains of 3rds and ascending bass line have also drawn comparisons with Brahms's Fourth Symphony, in the first and last movements respectively. As in Schumann's Etudes symphoniques, variations with less thematic resemblance are differently titled: 10, ‘Dorabella’, is an ‘Intermezzo’ and 13 ‘(***)’ a ‘Romanze’. Elgar's work appears to exemplify the ‘Fantasy-variation’ category of Schäffer, especially in the matter of the ‘psychological bond’ between theme and variations evidenced throughout but especially in variation 1 (C.A.E., his wife), which is expressively and texturally very complex, and 9 (‘Nimrod’, his friend Jaeger, who had saved him from recent depression with a conversation about Beethoven's slow movements), justly celebrated for its spacious and moving harmonies, and ostensible references to the ‘Pathétique’. The finale has been interpreted as two complete variations, the first quoting from ‘Nimrod’, the second from ‘C.A.E.’, with a peroration and coda (Rushton).

Because variation form seemed a good candidate either for relegation to the dustbin of tonality or for a final one-way flight into fantasy, its resilience in the 20th century is a surprising phenomenon. The serial variation emerged; the strophic form continued; the passacaglia flourished; diverse types of motivic fantasy left recognizable traces of variation form; and new forms of repetition promised a new paratactic emphasis, though without much in the way of reference to older forms. Moreover, composers seemed more willing to label movements with such titles as ‘Variationen’, ‘con variazioni’ and ‘Passacaglia’.

10. The 20th century.

Reger was perhaps the most dedicated composer of variations after Brahms and clearly upheld parts of his tradition, not only in writing large-scale independent sets for orchestra and for piano, often with a concluding fugue, but also in the evident care with which he connected type of theme and type of treatment. His two independent sets for orchestra (on themes by J.A. Hiller, op.100, 1907, and Mozart, op.132, 1914) joined piano sets on themes of Bach (op.81, 1904), Beethoven (op.86, 1904, two pianos, orchestrated 1915) and Telemann (op.134, 1914), a Chaconne in G minor for violin, two pieces called Introduction, Passacaglia and Fugue (op.96, 1906, two pianos; op.127, 1913, organ), Variations and Fugue on an Original Theme (op.73, 1903) for organ, as well as several choral fantasias for organ which include elements of the cantus-firmus-like chorale partita. In addition, his inclusion of many variation movements in chamber works – among them two violin sonatas (opp.84 and 139), two string quartets (opp.74 and 109), a string trio (op.141b), a clarinet quintet, a serenade – mark him as a successor to Brahms. Unlike Brahms, Reger keeps the early variations in a set quite close to the theme, perhaps because in some cases the themes themselves lack a memorable profile; these early variations are sometimes already in new keys (as in the Beethoven variations, on the Bagatelle op.119 no.11). The chamber variations are often transparent, with growing dissonance as the movements proceed, as in the String Trio op.141b. Typically dense, Reger's independent variations contain elaborate figurations marked by chromaticism, as well as thick textures dominated by a particular motif; the concluding fugues and double fugues are extremely long. The piano set on a theme from Bach's Cantata no.128 (op.81, 1904) is noticeably polyphonic and chromatic, while the figurations in the Telemann set (on a theme from the Musique de table) seem strikingly derivative, with Brahms's Handel Variations (also in B) as the main model and including echoes of Schubert's B Impromptu and Beethoven's op.109. In the Mozart set for orchestra, on the theme from the A major Sonata k331/300i, the first variation reiterates the theme melody against a rich and sparkling orchestral tutti texture, but this turns out to be the only variation in the tonic; while the melodic outline is nearly always apparent and is nearly reprised in variation 6, the keys are different in each variation, moving from F in variation 2 to A minor, E minor, A minor, D major and E major. The last variation is an expansive sostenuto of poetic breadth. Reger's variation sets, which combine melodic-outline and formal-outline types, are considered to be among his most important works.

Other notable variations of the first quarter of the 20th century include Glazunov's Thème et variations op.72 (1900) for piano, Rachmaninoff's Variations on a Theme of Chopin op.22 (1902–3) for piano, Dukas's Variations, interlude et final sur un thème de Rameau (1902) for piano, Szymanowski's Variations on a Polish Folk Theme op.10 (1904) for piano, Tovey's Brahmsian Elegiac Variations op.25 (1909) for piano and cello, the first movement of Joaquin Turina's Sonata romántica op.3 (1909) for piano and Dohnányi's Variations on a Nursery Theme op.25 (1914) for piano and orchestra. The Passecaille in Ravel's Piano Trio (1914) emerges from the depths, reaches a climax, and sinks down again. Sibelius's Fifth Symphony (1915), like Vaughan Williams's much later Fifth, has a variation-like slow movement that gradually loosens its structural components. Nielsen's Symphony no.6 (Sinfonia semplice, 1924–5) ends with a set of nine variations in which chaos and death intrude: variation 6 is a waltz that seems to function as a theme da capo, in 7 raucous noises break up the ball, 8 suggests the presence of death, and Nielsen compared 9 to a ‘grinning death's head’, finally dispelled by a ‘fanfare’. Vaughan Williams's Fantasia on a Theme by Thomas Tallis for string orchestra (1910), like other works with similar titles, presents ‘versions’ of a theme, rather than variations on a theme, and although a variational impulse is present it does not dominate. The work inherits the broadest strand of music ‘on’ another piece, whether capriccio, fantasia or variation, and becomes influential in new conceptions of variations later in the century (e.g. Britten's Lachrymae, ‘Reflections on a song of Dowland’ for viola and piano, 1950, or Foss's Baroque Variations, 1967; see below).

(i) Schoenberg, Berg, Webern.

(ii) Stravinsky, Hindemith, Britten.

(iii) Other mid-century approaches.

(iv) The later 20th century.

Schoenberg's contributions to the history of variation emerge from his writings and teachings as well as his compositions. He identified variation as an important ingredient in compositional technique generally (‘developing variation’), and variation form seemed congruent with the techniques of serialism. Indeed, his earliest set of variations, in the third movement of his String Quartet in D (1897), reveals a progressive texture already in the theme, which begins with a single line opening out into two-part counterpoint (bars 5–8). This texture, described as ‘unique in the variation literature’ (Frisch, The Early Works of Arnold Schoenberg, 1893–1908, 1993, p.43), appears to conjure both with the chromatic lament bass in its initial iambic semitone descent and the tetrachord span of its first incise and with the contrary-motion dynamic of the variation theme in the third movement of Brahms's Sextet no.2, op.36, while at the same time foreshadowing the solo serial theme of the Serenade op.24. As is often noted, Schoenberg's seven other variations sets plus the passacaglia (‘Nacht’) in Pierrot lunaire span a complete tonal arch from that early quartet, moving from the extended tonality of ‘Litanei’ in the Second String Quartet op.10 (1908) to the atonal ‘Nacht’, through the serial ‘Variationen’ of the Serenade op.24 (1923), ‘Thema mit Variationen’ from the Suite op.29 (1926) and Variationen für Orchester op.31 (1926–8), back to the tonality of the Variations on a Recitative for organ op.40 (1941) and Theme and Variations for wind band op.43a (1943). Schoenberg regarded variations as a ‘very strict form’, but allowed himself liberties of various kinds; one rationale might be his comment on Brahms's Handel Variations, that ‘Brahms fulfills his obligation to the theme in the first part of the measure, and is thus freed for the rest of the measure’ (see Nelson, 1964). In ‘Litanei’, the eight-bar structure of the theme remains constant while its four basic motifs, derived from the first and second movements, are manipulated and enriched in the presence of the soprano voice. Schoenberg identified these motifs (‘An Introduction to my Four Quartets’, Schoenberg, Berg, Webern: the String Quartets, a Documentary Study, ed. U. von Rauchhaupt, trans. E. Hartzel, 1971, pp.35–64), stating that his goal was their elaboration, omitted in the earlier movements; moreover the form would keep him from becoming ‘too dramatic’ for chamber music, a risk brought on by Stefan George's emotional poem. Indeed, the climax occurs not in the variations but in the overwrought coda, which has its own postlude. In this respect, the trajectory of the piece is not unlike that of the variations in Haydn's String Quartet op.20 no.4.

(i) Schoenberg, Berg, Webern.

In contrast to the strict structure of ‘Litanei’, in the passacaglia ‘Nacht’ the structural element of the theme in this normally pattern-based form disappears, with a motif of only three notes (e–g–e) radiating through the texture, first in minims, then in quavers, sometimes transposed and usually countered by chromatic scales. The singer has only one exposed statement of the motif, in minims (bar 10), ironically to the text ‘verschwiegen’ (silently); it also begins her final phrase (bar 23). Perhaps the note-based subject foreshadows the note-based serial structures to come, first in the ‘Variationen’ third movement of the Serenade op.24, with its 14-note theme. That the theme contains both the row and its retrograde immediately reveals the tension between the row and the theme. Schoenberg describes the variations as using ‘inversions and retrograde inversions, diminutions and augmentations, canons of various kinds, and rhythmic shifts to different beats – in other words, all the technical tools of the method are here, except the limitation to only twelve different tones’ (‘My Evolution’, Style and Idea, ed. L. Stein, 1984, p.91). The variations in the Suite op.29 exacerbate the row–theme issue by using a tonal melody, Silcher's Aennchen von Tharau.

During the composition of the Variationen für Orchester, op.31, Schoenberg wrote the Third String Quartet op.30 (1927), the Adagio of which still provokes controversy. After Erwin Stein's published analysis of it as a theme with variations and alternations (foreword to the Philharmonia score, 1927), Schoenberg asserted that the movement was really a rondo. Odegard (1966) supports the variation model, linking it to Haydn's alternating variations and the contrasts of rhythm and mood in Beethoven's Heiliger Dankgesang, while Dale (1993) claims that this view was ‘refuted’ by Schoenberg. But Schoenberg may have been hoist on the petard of his strict definition that variations require the ‘recurrence of one structural unit’, or he was disingenuously deflecting attention away from the Beethovenian echoes during 1927 centenary festivities. The orchestral Variations, Schoenberg's first independent set, take their place in the genre inaugurated by Brahms. Schoenberg himself gave a radio talk about the piece in 1931 and published an analysis in Style and Idea. The theme – which he describes as ‘simple’ and ‘characteristic’ – contains four forms of the row and generates a series of what he called both ‘formal’ or ‘developing’ variations, in which ‘everything develops from the theme and its individual features and there is, as usual, a general tendency toward quicker movement’, and ‘character’ variations, ‘in that each of them at the same time develops some particular character’ and contains a ‘characteristic motif’ (1960, p.36). In his radio talk he seemed to reiterate some of his general critique of variations, especially the aspect of ‘mere’ juxtaposition or ‘different views’ that makes inevitability and growth unlikely or impossible; he uses this idea to justify a ‘preparatory’ introduction and a ‘symphonic’ (that is, organic) finale. As in ‘Litanei’ and the Serenade, the theme is cantabile, focussing on shapes and rhythms that remain memorable. The cellos begin the theme (the prime, of five bars, is given in ex.16a; it is followed by a transposed retrograde inversion, seven bars; and a retrograde, five bars), and for the last phrase they are joined by the violins (transposed inversion, seven bars) while they return to the transposed prime, thus lending a three-part construction with a sense of return to the relatively symmetrical four-phrase theme. The focus is on finding motifs connected to the theme and a play of topics relating to genre, mood or dance-type (variation 2 is ‘chamber-music-like’, 3 ‘stormy’, 4 ‘idealized waltz’: ex.16b). The B–A–C–H motif is also quoted here and there. Neither of Schoenberg's last two variation sets, the independent works for organ and for band, approach the combination of inner complexity and surface sheen of op.31.

All Berg's variations, beyond the Brahms-inflected student piece Zwölf Variationen über ein eigenes Thema (1908), are in vocal works and concertos and show a remarkably dramatic and expressive organizing power. From the start he revealed a predilection for cantus firmus structures, in passacaglias (no.5 of the Altenberg lieder op.4; Wozzeck, Act 1 scene iv), and in chorale variations (Lulu, Act 3 scene i; Violin Concerto). His first set, ‘Hier ist Friede’ in the Altenberg lieder (1912), is based on three principal themes, stated one by one at the outset and varied nine times; five-bar segments are marked in the score. In an open dedicatory letter to Schoenberg, Berg wrote about the ‘motto’ of the first movement, ‘Thema scherzoso con variazioni’, of the Kammerkonzert for piano, violin and 13 wind instruments (1925): before the theme appears, three motifs give out the musical letters in the names Schoenberg (piano), Webern (violin) and Berg (muted horn). The motto is woven into the ‘Thema scherzoso’, and extends to 30 bars in three segments. Berg identified variations 1 and 5 as ‘reprise’ variations, using the prime form of the row, and 2–4 as ‘a kind of development section’ of increasing density and speed, featuring retrograde, inversion and retrograde inversion respectively. Often considered a less successful work because of its eclecticism, the variations are persuasive sonically because of their shape and recognizable motifs.

In his operas Berg used variation as he did other traditional forms, to structure scenes and acts; repetitive forms are particularly effective in charting the characters’ obsessions. The passacaglia in Wozzeck's scene with the Doctor (Act 1 scene iv) undergoes 21 variations in which the original serial theme, presented linearly as a bass line to the Doctor's initial manic admonishment, is fragmented, overlapped, turned into simultaneities, moved from part to part and, most importantly for the scenic movement, becomes more and more compressed as the exchanges between the characters become increasingly rapid. The variations at the beginning of Act 3, labelled ‘Inventions on a theme’, similarly intensify and speed up as Marie's soliloquy becomes more and more overwrought, but the theme, by contrast, is an eerily tonal and consciously archaic melody, in imitative texture and with a hollow 5th, to underscore the Bible's ancient wisdom before more chromatic motifs intrude with her present-day cries of anguish. Variations 1 and 5, which like the theme introduce a new Bible reading, are the most tonal; 2 and 6 are continuations. The fugue, a counterpart to and completion of the theme, also begins in a high register and works into climactic statements with the subject blared out in the bass. Whether the rest of the ‘inventions’ in Act 3 owe anything to at least a conception of variation form is moot, but the ‘Inventions on a rhythm’ in scene iii have a structural component that is suggestive. Lulu sees an even greater use of variations. The largest of these is the set-piece ‘Konzertante Chorale-Variationen’ during the dialogue between the Marquis and Lulu in Act 3 scene i, which is broken up by ‘intermezzi’ referring to earlier music (e.g. the English waltz from Act 1 scene iii). The chorale melody, introduced in Act 1 scene iii (bars 1113–22) and Act 2 scene i (bars 250–61), is presented in Act 3 in 12 variations (beginning in bar 83) that include a polonaise (variations 1 and 5), wind chorales (2 and 6), four-part canon (7), bass chorale linking 8, 9 and 10 and the chorale in diminution linking 11 and 12. Act 3 continues with a set of variations between scenes i and ii based on Wedekind's lute-song, already introduced as an intermezzo during the chorale variations (bars 103–18), and that returns later in the act as funeral music for Alwa. Like Schoenberg's Suite op.29 and Berg's later Violin Concerto, these variations use a tonal theme in non-tonal contexts.

The Adagio finale of Berg's Violin Concerto uses the chorale melody Es ist genug from Bach's Cantata no.60, granting it both serial and tonal treatments; the alternation between chorale phrases in dissonant counterpoint led by the solo violin and their repetition in ‘Bach’schen Harmonisierung’ by an ensemble of three clarinets, designed to sound like a small organ, is especially compelling. Traditional segmentation is altered in two ways. First, the layout of chorale and two variations gives way to a folksong (the Carinthian song with personal resonances heard originally in movement 1b and here played ‘wie aus der Ferne’) before the coda; second, in variation 1 (bar 164), Berg introduces a new melody, the ‘Klagegesang’ of Willi Reich's original programme note, which rises to a climax, bringing along the other violins. This and the chorale return in the coda, turning it into a kind of summary variation. Thus the clear structure and multiple rhythmic levels of the chorale variations are intercut with programmatic elements that fulfil a broader role for the finale of the work.

Like Berg's early piano variations of 1908, Webern's orchestral Passacaglia op.1 of the same year owes something to Brahms, but its debt to the finale of Brahms's Fourth Symphony is repaid in a mature and compelling work. (The passacaglia finale of Zemlinsky's Symphony no.2, 1892–3, is clearly also indebted to Brahms's Fourth, as is Reger's Suite ‘Den Manen J.S. Bachs’, op.16, for organ.) The chromatic crux in Brahms's theme is the raised fourth degree as the fifth of eight notes; here it is the lowered fifth degree as the fourth of eight. The eight-bar pizzicato theme, a bass line in D minor, exerts a hold on many of the variations, but during the continuous sweep of the overall three-part shape, with a ‘middle section’ of variations in the major (variations 12–15, like the Brahms and the Bach Chaconne), the theme tends to be lost during climactic points and at other times may be orchestrally redistributed note by note. As in the Brahms, a flute variation prepares the turn to major. After 20 years, Webern's next set of variations, the second movement of his Symphony op.21, turned its back on this world: 99 bars, lasting barely more than three minutes, in as compressed and lapidary a form imaginable. As in Schoenberg's Serenade, the clarinet gives out the theme but instead of a complete retrograde turns back on itself after only six pitches, in tritone transposition, and with its complete retrograde as accompaniment in harp and horns. Every section of the movement (theme, seven variations, coda) but variation 4 maintains a palindromic structure, and Webern claimed that it was ‘the midpoint of the whole movement, after which everything goes backward’ (The Path to the New Music; see Nelson, 1968–9); thus the entire movement is a palindrome (see Hitchcock, 1970, and Starr, 1970). The variations strive for contrast, each maintaining a distinctive scoring, texture, predominant motif and articulation: variation 1 is a double canon and 7 a quadruple one, 2 has a horn ostinato, and so on.

In Webern's last three variation works, questions central to the identity of any such work come to the fore: what is the subject for variation? what must remain constant? Indeed, in the first two of these the issue is more basic: where are the variations? The Variations for piano op.27 (1936) is in three movements, and analysts have disagreed about variation structures in the first two, whose overall designs are ABA (with three 18-bar sections of contrasting texture) and binary respectively, but with palindromic motifs and other mirror effects. Webern, however, composed the third movement first, writing to Hildegard Jone, ‘The completed part is a variations movement; the whole will be a kind of “Suite”’; Bailey (1973) takes this to mean that only the third movement contains true variations, with its six sections of 11 bars each. Here the nature of the form is increasingly abstract, with no identifiable theme beyond the level of the motif. Just as Webern had once described the last movement of his Cantata no.1 op.29 as a combination of variations, scherzo and fugue, so he wrote to Willi Reich that the ‘basic principle’ of the serial Variations for Orchestra op.30 (‘my overture’) is an ‘adagio’ form, which he clarified in a subsequent letter by relating each variation to a part in a sonata structure: theme = introduction, variation 1 = main theme, 2 = transition, 3 = second theme, 4 = reprise of main theme, ‘however in the manner of a development’ (‘for it is an andante form!’, i.e. slow-movement sonata form), 5 = ‘repeating the manner of the introduction and transition’, 6 = coda (see Bailey, 1973). (He similarly described the first movement of the String Quartet op.28 as variations in an adagio form.) Indeed, the ‘first theme’ is the closest thing in the piece to accompanied melody, while the ‘second theme’ after a chordal transition presents the tiny motifs slightly more lyrically than the theme. The row itself (ex.17) forms pitches 6–12 as the retrograde inversion of 1–6, and Webern identified the first six pitches (four in the double bass, two in the oboe, immediately retrograded so that two four-note motifs sound) as the source of all the rest of the material (the ‘Gestalt’ to be subjected to ‘metamorphoses’, that is, variations; but the theme itself is already a variation, itself a metamorphosis of the basic shape: letter to Jone, 26 May 1941). Yet despite this skein of motivic development in formally recognizable groupings, Webern described the variations as in ‘a quite different style’ with an ‘affinity with the type of presentation one finds in the Netherlanders’, thus forging a new path with elements of an archaic constructive principle.

Britten's exuberantly characteristic Variations on a Theme of Frank Bridge, op.10 (1937) runs a gamut of moods and topics, some of them parodistic, but arrestingly places its Adagio variation at the very beginning, contravening tradition, before turning to the March, Romance, Aria italiana (strumming violins and exaggerated melodic gestures), Bourrée classique and Wiener Walzer (full of discordant sighs and surface-skimming turns, some at an eerie distance). The mood abruptly changes during the Funeral March, Chant, and even the return of the theme as a cantus firmus during and after the fugue. Other early variations include the choral variations A Boy was Born (1933), the middle movement of the Sinfonietta op.1 (1932) and the Temporal Variations for oboe and piano (1936), the latter including an ‘Oration’ (marked ‘Lento quasi recitativo’), Commination (bearing the arresting marking ‘Adagio con fuoco’) and Chorale, in addition to the more common March, Waltz and Polka. His best-known set of variations, the Young Person's Guide to the Orchestra (Variations and Fugue on a Theme of Purcell) op.34, is, in the felicitous words of Paul Griffiths (‘Variations’, Grove6), ‘at once a tour of the forces and a tour de force’. Like the Bridge variations, it includes colourful character pieces while exploring orchestral colour and brings back the theme as a cantus firmus during the most frenzied moments of the fugal finale. Mood and character pieces and a concluding fugue appear again in a later work, Gemini Variations, ‘Twelve Variations and Fugue on an Epigram of Zoltán Kodály’ op.73, for flute, violin and piano four hands (1965); except for the ruminative final Romanza and lengthy fugue subject, the variations are very concise, including two ‘mirrors’ in contrary motion (variations 6 and 9 are Specchio 1 and 2 respectively). Like his contemporaries, Britten was attracted to the passacaglia, especially as a finale, notably in the Violin Concerto (1939), the second and third string quartets (1945, 1975), in Peter Grimes (1945, extracted as one of the Four Sea Interludes) and in The Turn of the Screw (1954). In this last work, the passacaglia is the culmination of a variation structure covering the entire opera, with a theme stated in the Prologue and 15 variations spread over the orchestral interludes that link the scenes. The theme rises in pitch in every variation during Act 1 from A, associated with the Governess, to A associated with the ghosts, then each time descending in pitch during Act 2 to the final confrontation, with all 12 tonal centres present during variation 15.

(ii) Stravinsky, Hindemith, Britten.

Stravinsky's eight variation movements on original themes are framed by the ‘Gavotte e due variazioni’ in Pulcinella and the orchestration of Bach's Vom Himmel hoch variations (1955–6), after which he wrote his single independent set, the Variations (Aldous Huxley in memoriam) (1963–4), which is of an entirely different stripe. Remarks he made to interviewers after the première of his Octet for wind (1922–3) about his affinity to earlier music, together with his comment that ‘in writing variations my method is to remain faithful to the theme as a melody’ (Nelson, 1962), have focussed attention on his tendencies towards constant-melody variation or use of at least parts of the theme melody as a cantus firmus. The layered rhythmic texture of many of his variations bears this out, and in the Octet's middle movement the theme melody returns in different keys in variations 1, 3 and 6, each time in the trombones with faster figurations in the other voices. (The Sonata for Two Pianos also returns to the melody of theme in variations 1, 2 and 6.) These are separated by dance-like and antic characteristic variations (march, waltz, cancan), sometimes with ostinato accompaniments. The lengthy legato theme melody also differs strikingly from the atomized melodies of the Viennese composers but is treated with octave displacement later in the set to reduce the degree of resemblance. Stravinsky described the ‘ribbons of scales’ he added to variation 1 that would return later, and pointed out the theme played in rotation by instrumental pairs in the final fugato variation. Similar elements appear in his ballet scores (Jeu de cartes, 1936; Danses concertantes, 1940–42) and other instrumental works (Concerto for Two Pianos, 1932–5; Sonata for Two Pianos, 1943–4; Ebony Concerto, 1945), even the serial Septet (1952–3), a relatively strict passacaglia on 16 notes in the second movement. The oscillating 3rds of the final movement of the Ebony Concerto appear unpromising material for a theme, yet always stand apart rhythmically from the surrounding texture; only in the third and final variation are they treated with diminution, and the repeated chords of the theme's accompaniment also become a subject for variation. Stravinsky's later variations increasingly feature disjunct themes, but from the beginning he treated his themes with octave displacement and instrument dispersal to recast the melody. Sometimes a fragment of the theme appears as an ostinato during a variation (Concerto, variation 4; Danses concertantes, variations 1 and 3; Sonata, variation 1) or becomes the subject of a fugal variation (Octet, variation 7; Sonata, variation 3). The Concerto for Two Pianos draws on Haydn's model of alternating variations in its third movement, though different melodic shapes obscure the relationships between the first theme and its variations; using the first theme as a subject for the fourth-movement fugue brings to mind Brahms's Piano Sonata op.2, in which the theme is used for the Scherzo, as well as his op.24, importing the closing fugue of variations into a multi-movement work (ex.18). Another procedure derives from Beethoven's op.34, placing every variation in a different key according to a pattern (Jeu de cartes, descending semitones; Danses concertantes, ascending semitones). In his last set, the serial Variations (Aldous Huxley in memoriam), the lack of a clear theme and degree of abstract concision resemble Webern, and the overall organization of the 12 sections suggests a refrain in the recurrence of dense polyphony (called ‘12-part variations’ by Stravinsky) in II (all violins), V (all strings) and XI (all wind) and the textures of I returning in XII. In these respects, as well as the appearance of a fugato (X), this set recalls at least the plans of his earlier ones, if not their style.

Important strands in the history of variations were contributed by Hindemith and Britten, beginning with their early works in the 1920s and 30s respectively. Hindemith was also drawn to the passacaglia, especially in finale position, in both instrumental music (String Quartet no.5, op.32, 1923) and vocal music (Das Marienleben, 1922–3; Cardillac, 1926; Die Harmonie der Welt, 1956–7, of which Act 5 is also a separate symphony). In the quartet finale the bass subject is sometimes restated intact, especially at the beginning, and sometimes participates rhythmically with the upper voices. The song cycle Das Marienleben includes a passacaglia in the ‘Darstellung Mariä in der Tempel’ as well as ostinato variations in the three pieces ‘Vom Tode Mariä’. Another early work, the Viola Sonata op.11 no.4 (1919), seems to have a surfeit of variations: the movements are Fantasie, Thema mit Variationen, Finale (mit Variationen). The Fantasie acts as an improvisatory introduction and the finale introduces a new theme which alternates with variations continuing from the second movement. Perhaps the connection of the fantasy with variations gave rise to Hindemith's more rhapsodic variations of the 1930s, such as the Philharmonisches Konzert (1932) and the finale of the viola concerto Der Schwanendreher (1935). A stricter strand of melodic-outline variation writing is evident in the variations on Mozart's lied Komm, lieber Mai that conclude the Sonata ‘Es ist so schönes Wetter draussen’ op.31 no.2 (1924) for violin. Hindemith's most important variation work, Die vier Temperamente for string orchestra and piano (1940, also well known with Balanchine's choreography, 1946), takes a rather extensive theme in three sections – a broad orchestral section, a faster, more scherzando section which introduces the piano, and a siciliano theme for strings, embellished by the piano (ex.19a) – and subjects it to four characteristic interpretations according to the ancient theory of humours: ‘Melancholic’, ‘Sanguinic’, ‘Phlegmatic’, and ‘Choleric’. Points of melodic contact with the theme are evident throughout, as are the theme's hollow 5th chords. The first, also slow–fast–slow, introduces a meditative solo violin into the first section; the second is entirely for strings; and the siciliano has been transformed into a ‘slow march’ (ex.19b). ‘Sanguine’ stays in ‘waltz’ time throughout, with the occasional dance-like repeated section; ‘Phlegmatic’, for five solo strings and piano, returns to the three-tempo structure; while ‘Choleric’ has the most improvisatory piano writing and give-and-take with the orchestra, and for the first time presents the final section of the theme, always previously dance-like, into a broad, even chorale-like, summary statement (ex.19c).

Other notable variations of the second quarter of the 20th century include Copland's astringent Piano Variations (1930, orchestrated in 1957), about which the composer wrote ‘it was not composed in the consecutive order of its finished state … I worked on the variations individually, not knowing exactly where or how they would eventually fit together. One fine day, when the time was right, the order of the variations fell into place’ (Copland and V. Perlis, Copland, i: 1900 through 1942, 1984); the sixth movement of Zemlinsky's String Quartet no.4, op.25 (1936), subtitled ‘Barcarole’; Kodály's Variations on a Hungarian Folksong, The Peacock, 1938–9, in which the 16 variations are grouped in threes, except for variation 10, which stands alone, and 11–14, the massive slow-movement group which reorganizes the whole into a three-part form; Poulenc's suite of character sketches Les soirées de Nazelles (1930–36), framed by a ‘Préambule’ and then a ‘Cadence’ and ‘Final’; three sets on Paganini's much-varied A minor Caprice: by Rachmaninoff (Rhapsody op.43, 1934) for piano and orchestra, by Lutosławski (for two pianos, 1941) and by Blacher (op.26 for orchestra, 1947); Rósza's Theme, Variations and Finale op.13 (1934), on a theme ‘in the manner of a Hungarian folksong’; Eisler's serial variations 14 Arten, den Regen zu beschreiben (1940) for flute, clarinet, piano and strings (written for a documentary film), notable for their lyricism, the ‘chorale-étude’ (no.3) and the initial ‘Anagramm’ on Schoenberg's name; and Dello Joio's ‘Variations, Chaconne and Finale’ for orchestra (1947), based on a Gregorian theme. The soaring lyricism of the second movement of Tippett's String Quartet no.3 (1945–6) hides the technique of constant-melody variation.

(iii) Other mid-century approaches.

Several composers during the middle of the century found variation techniques an essential part of their approach to composition, without writing many actual variations: Bartók, K.A. Hartmann and Shostakovich, among others. Apart from the second movement of his Violin Concerto (1937–8), some early pieces (Violin Sonata of 1903, piano variations) and two pieces in Mikrokosmos as true variation movements, Bartók tended to put into practice what he told the interviewer Denis Dille in 1937: ‘I never repeat [an idea] unvaried; this is connected to my love of variation, of thematic transformation’. He was partial to palindromic forms within a cycle, where a later movement may vary elements of an earlier one or be based on a varied version of the earlier movement's theme; this procedure he called a ‘Brückenform’, evident in the Violin Concerto, the Second Piano Concerto (1930–31) and the five-movement fourth and fifth quartets (1928, 1934). Whether the later movement may be considered ‘a variation’ of the earlier one is doubtful, however. His Music for Strings, Percussion and Celesta (1936) does reflect a four-movement-wide re-presentation of thematic material in varied forms. Bartók noted in his introduction to the 1937 score that in the development section of the second movement the theme of the first movement appears ‘in veränderter Gestalt’; that the theme of the fourth movement is alluded to there as well; that in the recapitulation of the same movement the 2/4 rhythm of the exposition is changed into 3/8; and that in the finale the theme of the first movement is ‘extended’ from its original chromatic form into a diatonic one (ex.20). Hartmann was similarly concerned with variation procedures and presented large-scale forms as variants of each other. His Symphony no.6 (1951–3), for example, concludes with a ‘Toccata variata’ of three fugues, in which the second two are variants of the first. Shostakovich's variation movements include the passacaglia slow movements of his string quartets nos.3 (1946, unusual in that the theme is a lyrical melody, rather than a framework), 6 (1956) and 10 (1964) as well as the Violin Concerto no.1 (1947–8). The finale of String Quartet no.2 (1944) is a variation movement beginning with a constant-melody build-up, which makes the remaining variations seem developmental; in fact, all the movements of the quartet are thematically connected and the variation movement alludes to the others. At the level of individual movements, however, it is unclear how to differentiate between cyclic procedures and variation techniques. If we have managed to avoid the temptation to see variations in the thematic transformation of developmental and cyclic forms of 19th-century music, then there is no reason to succumb to it for that of the 20th.

Stockhausen's critique of variation in 1952 began from the premise that musical ordering begins with the note, not the theme or figure, and thus took issue with Webern's identification of the entity to be varied as a ‘Gestalt’:

(iv) The later 20th century.

Musical variation assumes a pre-formed Gestalt that is varied. This Gestalt bears its own fixed ordering of tones. It is complete, not arising as the inevitable outcome of an idea for a specific arrangement of tones … In the ‘variation’, however, it is not a question of the to-be-varied [das Variierende], but rather of the varying [das Variieren]. (H. Weber, 1986, p.43)

Thus the composers involved in inventing total serialism generally eschewed variation, at least at the time; Stockhausen's later Inori for soloist and orchestra (1973–4) is related to variations. Boulez used the term ‘double’ in both Le marteau sans maître (1953–5) and Figures–Doubles–Prismes (1963) and ‘Variation’ for the first piece of Livre pour cordes for string orchestra (1968), the three parts of which he described respectively as ‘based on simple, static elements’, ‘a tangle of decorated lines’ and a combination of the two. But a composer could use the title ‘variation’ for nearly anything. Cage's Variations I–VIII (1958–78) are chance compositions: Variations I is for any number of players and any number and kind of instruments, the score consisting of clear plastic overlays with lines and dots, each player making an individual part; Variations V includes choreography by Merce Cunningham with film and video images. Cage's Hymns and Variations for 12 amplified voices (1979) takes two hymns by William Billings, subjects them to ‘subtraction’ (chance operations reducing their elements) and then follows with a series of ten ‘variations’ in which each has five such ‘subtractions’; the piece lasts nearly half an hour. Cage's Themes and Variations (1982) consisted of chance operations on text alone, designed to be read aloud; as in Hymns and Variations, the variations ‘succeed the theme as four more composite realizations of the original prose-poems’ (Radano, 1982). Electronic works are often only tangentially related to variations: Pierre Henri's Variations pour une porte et un soupir (1963) generates 25 variations from the musique concrète elements of a creaking door, breathing and a musical saw. Noah Creshevsky articulates the principle underlying his Variations (1987) as ‘perpetual variation’ while acknowledging that ‘sectional repetitions are interrelated solely through a few prominent motivic and rhythmic ’.

But the encroachments of the avant garde on the term ‘variation’ did not mean that composers ceased to produce remarkable ‘true’ variations during the 1950s and 60s. Ginastera's Variaciones concertantes for chamber orchestra (1953) combines characteristic and concertante approaches, as well as a new form: an interlude inserted between theme and variations, another interlude between the variations and a theme reprise, and a final variation ‘in modo di Rondo’. Berio's unusual Cinque variazioni for piano (1952–3) finds its theme (by Dallapiccola) in variation 5, only to edit it out in the 1966 revision (shades of Brahms editing Schumann and others out of his op.8). Elliott Carter's Variations for Orchestra (1954–5) invented two new modes of organization, in the first of which the variations progress from vivid contrast to the ‘misterioso’ variation 5, sonorous but without rhythmic propulsion (4 is ‘ritardando molto’, 6 is ‘accelerando molto’), then increase in textural and rhythmic complexity. The work also uses two ideas as ritornellos, the first a quick ascending one that gets slower at each restatement (variations 1, 3, 8, finale), the second a descending line (played by two violins during the theme) that gets faster as the work progresses (variations 2, 8, finale): the ritornellos meet at the end. Milton Babbitt's Semi-Simple Variations for piano (1956), a 36-bar serial piece, spreads its six-pitch theme (ex.21) over six bars, so that variation 1 contains the second hexachord which, as in the Webern Symphony, is a retrograde in tritone transposition. Moreover, the 12 notes are sustained, like a cantus firmus, and the set has been ‘registrally partitioned’ (Barkin, 1967); the composer noted in addition that ‘the sixteenth notes in the first six measures represent all 16 possible partitions of the quarter note in terms of the 16th note unit’. Charles Wuorinen wrote six inventive sets of variations, mostly for solo instruments (piano, flute, violin, cello, bassoon plus harp, and timpani) between 1963 and 1975. Peter Westergaard's serial Variations for Six Players (1963) features a disjunctive Webernian Klangfarbenmelodie: from the same year come Walton's Variations on a Theme by Hindemith (1962–3), which uses a meandering theme from the slow movement of Hindemith's Cello Concerto no.2, a source that generates the variations’ long-breathed melodies against vividly orchestrated faster-moving lines contrasting with more motivically organized variations.

The tradition of variations on a borrowed theme was extended in the 20th century to formal interactions with earlier music. There is perhaps a fruitful connection here with Picasso's 44 variations on Velázquez's Las Meninas (1957), in which the first painting is already a variation. 13 of the next 19 variations focus on the central figure, the Infanta in a white dress, and one can see her face progressively dissociating in 4–11. The vista broadens before a ‘theme reprise’ in 22, which inaugurates a series of four variations on the entire picture. The rest of the variations concern the secondary figures, with the Infanta returning only as part of a larger ensemble (36–8). This engagement with an earlier ‘text’ that creates varied ‘views’, and different perspectives of a model may be at work in Paul Lansky's Six Fantasies on a Poem by Thomas Campion (1978–9), an electronic piece in which each fantasy uses the text of the poem (‘Rose cheekt Lawra, come’) read in its entirety to very different effect; the whole has an extraordinary overall shape and impact. But Lukas Foss's Baroque Variations (1967), which the composer called ‘dreams about’ rather than ‘variations on’, uses three different pieces, each given in its entirety but realized in an entirely different way, without the necessary repetition and, in the case of no.3, with only a set of instructions for the performer. The latter situation is made especially vivid in Brian Eno's Variations on the Canon in D by Pachelbel (1975), which gives the players instructions on how to use selected parts of the (unplayed) theme so that each of the three variations unfolds as a process, a ‘self-regulating and self-generating system’. In the first (‘Fullness of Wind’), each player's tempo is decreased, with the rate of decrease governed by the instrument's register (bass = slow). In some respects this resembles Arvo Pärt's Cantus in memoriam Benjamin Britten (1977–80), in which all the strings inexorably descend the A minor scale with the upper strings moving much faster than the lower, and each progressively slowing. Rochberg's variations on Pachelbel's canon in his Quartet no.6 (1978), by contrast, maintain the structure of the theme even while moving far away from and then back towards its style. An entirely different approach to the historical sense of ‘Variations on …’ animates Mauricio Kagel's Variationen ohne Fuge for orchestra (1971–2), a peculiar hommage à Brahms that keeps the phrase structure and rhythm of the Handel Variations while changing their harmony and ordering (2, 19, 21, 3 etc.); the effect is rather like a smear of sound over a distanced but recognizable original. As in Alexander Goehr's Variations on the Sarabande from Bach's English Suite in E minor (a work with which it has nothing else in common), the Baroque theme appears only at the end. Kagel includes ad libitum parts for two actors to impersonate a silent Handel and a monologist Brahms. Other modes of calling up the past are rendered in Rochberg's Partita-Variations (1976), with its mix of 18th- to 20th-century styles in different characteristic variations; the theme, a ballade, is in the centre of the set.

The idea of repetition in combination with gradual change makes problematic the relationship between the minimalist musical style of the 1970s and 80s and variations. Very tiny ostinatos produce the feeling of pulsations rather than structures to be varied. Some pieces in a minimalist style or aesthetic do reveal a more structural approach, however, especially when governed by a text, as in Steve Reich's Tehillim (1981), which sets four psalms. The first presents strophic variations at each text repetition, with canons, elaborate vocal overlapping and interplay and increasing complexity, then dwells almost developmentally on two- or three-line segments, finally returning to the entire text in a set of recapitulatory variations. The last psalm uses incantatory canonic repetitions within each stanza while retaining the strophic form until the coda. Reich's Variations for Winds, Strings and Keyboards (1979) makes three increasingly complex statements of a long harmonic progression into a stretched chaconne.

Indeed, contemporary progeny not only of variations but of its older forms continue to exert fascination, as they did in the early 20th century. Mario Davidovsky's Chacona for violin, cello and piano (1971) is based on a regularly recurring pattern of durations; George Edwards's Draconian Measures (1976) and Czeched Swing (1994) maintain an element of harmonic structure in each variation, while in the latter, as in jazz, the bars may be variable in length but every variation has the same number of bars. Movements in larger works also adopt an updated ‘constant-harmony’ chaconne type: the slow movement of Tippett's Piano Sonata no.3 (1972–3) varies 17 chords; in John Harbison's Oboe Concerto (1991), the Passacaglia middle movement maintains a sarabande-like tread and harmonic structural elements especially in the tutti sections, while the solo sections are either connected to the substructure or more freely rhapsodic; in Thomas Adès's Concerto conciso for piano and orchestra (1997), a seven-bar chordal theme underlies the ciaconetta slow movement. Other composers interested in older forms of repetition include Birtwistle (‘varied ostinato’), Maxwell Davies (‘doubles’, a far cry from Theme and Variations: Mavis in Las Vegas, 1997), Schnittke (passacaglia) and Kernis (‘ground’). Ellen Zwilich's Prologue and Variations (1984) uses ‘prologue’ in place of ‘theme’ (ex.22a), in the sense of the introduction to the ‘characters’ in a drama, here represented by different musical motifs and textures over 69 bars of Andante misterioso; the four variations develop now one, now another of these aspects without maintaining a structural resemblance (102, 26, 86, 42 bars respectively, at Allegro (ex.22b), Lento, Presto, Tempo I). A look at these and other remarks composers make about their variations suggests a self-consciousness about claiming ‘true’ variation status, especially when the structure of the theme is changed, even when they use the title. This reveals the tenacity of the model both as a sense of limitation and as a testing-stone for the compositional imagination.

BIBLIOGRAPHY

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Grove6 (K. von Fischer and P. Griffith)

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b to 1600

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c baroque

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d classical

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K. von Fischer: ‘C.Ph.E. Bachs Variationenwerke’, RBM, vi (1952), 190–218

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E. Beuermann: ‘Die Reprisensonaten Carl Philipp Emanuel Bachs’, AMw, xiii (1956), 168–79

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A. Batta and S. Kovács: ‘Typbildung und Grossform in Beethovens frühen Klaviervariationen’, SMH, xx (1978), 125–56

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M. Zenck: ‘Rezeption von Geschichte in Beethoven’s “Diabelli-Variationen”’, AMw, xxxvii (1980), 61–75

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A. Münster: Studien zu Beethovens Diabelli-Variationen (Munich, 1982)

C. Reynolds: ‘Beethoven’s Sketches for the Variations in E, op.35’, Beethoven Studies 3, ed. A. Tyson (Cambridge, 1982), 47–84

E. Sisman: ‘Small and Expanded Forms: Koch’s Model and Haydn’s Music’, MQ, lxviii (1982), 444–75

D. Berg: ‘C.P.E. Bach’s “Variations” and “Embellishments” for his Keyboard Sonatas’, JM, ii (1983), 151–73

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E. Derr: ‘Beethoven’s Long-Term Memory of C.P.E. Bach’s Rondo in E flat, W. 61/1 (1787), Manifest in the Variations in E Flat for Piano, Opus 35 (1802)’, MQ, lxx (1984), 45–76

W. Kinderman: ‘Tonality and Form in the Variation Movements of Beethoven’s Late Quartets’, Beiträge zu Beethovens Kammermusik: Bonn 1984, 135–51

H.-W. Küthen: ‘Beethoven’s “wirklich gantz neue manier”: eine Persiflage’, Beiträge zu Beethoven’s Kammermusik: Bonn 1984, 216–24

C. Reynolds: ‘Ends and Means in the Second Finale to Beethoven’s op.30, no.1’, Beethoven Essays: Studies in Honor of Elliot Forbes, ed. L. Lockwood and P. Benjamin (Cambridge, MA, 1984), 127–45

M. Zenck: ‘“Bach, der Progressive”: Die “Goldberg-Variationen” in der Perspektive von Beethoven’s “Diabelli-Variationen”’, Johann Sebastian Bach: Goldberg-Variationen, Musik-Konzepte, no.42 (1985), 29–92

W. Kinderman: Beethoven’s Diabelli Variations (Oxford, 1987)

E. Helm: ‘C.P.E. Bach and the Great Chain of Variation’, Carl Philipp Emanuel Bach und die europäische Musikkultur: Hamburg 1988, 223–8

P. van Reijen: Vergleichende Studien zur Klaviervariationstechnik von Mozart und seinen Zeitgenossen (Buren, 1988)

H. Serwer: ‘C.P.E. Bach, J.C.F. Rellstab, and the Sonatas with Varied Reprises’, C.P.E. Bach Studies, ed. S. Clark (Oxford, 1988), 233–43

E. Cavett-Dunsby: Mozart’s Variations Reconsidered: Four Case Studies (K. 613, K. 501, and the finales of K.421/417b, and K. 491) (New York, 1989)

N. Marston: ‘Analysing Variations: the Finale of Beethoven’s String Quartet op.74’, MAn, viii (1989), 303–24

A. Brendel: ‘Must Classical Music be Entirely Serious? 2: Beethoven’s Diabelli Variations’, Music Sounded Out (London, 1990), 37–53

K. Komlós: ‘“Veränderte Reprise”: Aspects of an Idea’, MR, li (1990), 262–7

H. Lühning: ‘Der Variationensatz aus dem c-Moll-Trio op.1 Nr.3’, Beethovens Klaviertrios: Munich 1990, 45–64

E. Sisman: ‘Tradition and Transformation in the Alternating Variations of Haydn and Beethoven’, AcM, lxii (1990), 152–82

K. Komlós: ‘“Ich praeludirte und spielte Variazionen”: Mozart the Fortepianist’, Perspectives on Mozart Performance, ed. R.L. Todd and P. Williams (Cambridge, 1991), 27–54

W. Meredith: ‘Forming the New from the Old: Beethoven’s Use of Variation in the Fifth Symphony’, Beethoven’s Compositional Process, ed. W. Kinderman (Lincoln, NE, 1991), 102–21

S.M. Whiting: To the ‘New Manner’ Born: a Study of Beethoven’s Early Variations (diss., U. of Illinois, Urbana, 1991)

L. Plantinga and G.P. Johnson: ‘Haydn’s “Andante con variazioni”: Compositional Process, Text, and Genre’, The Creative Process, ed. R. Broude (New York, 1992), 129–67

M.H. Schmid: ‘Variation oder Rondo? zu Mozarts Wiener Finale KV 382 des Klavierkonzerts KV 175’, Mozart Studien, i (1992), 59–80

E. Sisman: Haydn and the Classical Variation (Cambridge, MA, 1993) [incl. discussion of the rhetoric of variation]

R.L. Marshall, ed.: Eighteenth-Century Keyboard Music (New York, 1994)

G. Stanley: ‘The “wirklich gantz neue Manier” and the Path to it: Beethoven’s Variations for Piano, 1783–1802’, Beethoven Forum, iii (1994), 53–79

M. Solomon: ‘Beethoven’s “Diabelli” Variations: the End of a Beginning’, Beethoven Forum, vii (1999), 137–51

e 19th century

C. Czerny: Fantasie-Schule, i: Systematische Anleitung zum Fantasieren auf dem Pianoforte, op.200 (Vienna, 1829/R; Eng. trans., 1983)

V. Luithlen: ‘Studie zu Johannes Brahms’ Werken in Variationenform’, SMw, xiv (1927), 286–320

M. Friedland: Zeitstil und Persönlichkeitsstil in den Variationenwerken der musikalischen Romantik: zur Geistesgeschichte und Schaffenspsychologie der Romantik (Leipzig, 1930)

W. Schwarz: Robert Schumann und die Variation, mit besonderer Berücksichtigung der Klavier-Werke (Kassel, 1939)

M.J.E. Brown: Schubert’s Variations (London, 1954)

B. Hansen: Variationen und Varianten in den musikalischen Werken Franz Liszts (Hamburg, 1959)

J. Moore: ‘An Approach to Elgar’s “Enigma”’, MR, xx (1959), 38–44

H. Hirsch: Rhythmisch-metrische Untersuchungen zur Variationstechnik bei Johannes Brahms (Freiburg, 1963)

G. Puchelt: Variationen für Klavier im 19. Jahrhundert (Darmstadt, 1973)

J. Brahms: Variations on a Theme of Haydn, ed. D.M. McCorkle (New York, 1976) [with analytical essays and book excerpts]

N.B. Reich: ‘Liszt’s Variations on the March from Rossini’s “Siège de Corinthe”’, FAM, xxiii (1976), 102–6

C. Floros: Brahms und Bruckner: Studien zur musikalischen Exegetik (Wiesbaden, 1980)

C.S. Becker: ‘A New Look at Schumann’s Impromptus’, MQ, lxvii (1981), 568–86

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G. Phipps: ‘The Logic of Tonality in Strauss’s Don Quixote: a Schoenbergian Evaluation’, 19CM, ix (1985–6), 189–205

M. Struck: ‘Dialog über die Variation präzisiert: Joseph Joachims “Variationen über ein irisches Elfenlied” und Johannes Brahms’ “Variationenpaar” op.21 im Licht der gemeinsamen gattungstheoretischen Diskussion’, Musikkulturgeschichte: Festschrift für Constantin Floros, ed. P. Petersen (Wiesbaden, 1990), 105–54

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P. Cook: ‘Vincent d’Indy’s Istar’, MR, lii (1991), 237–45

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