Nabokov, Nicolas [Nikolay]

(iii) The division of time.

Vestiges of the system of proportional mensuration signs persisted to the 18th century in some places. After 1500 the more complex proportions are found only in theoretical works, and in a few 16th-century polyphonic works to illustrate the text (e.g. the pieces by Renaldi and Striggio quoted by Morley, 1597). Nevertheless proportional signatures, in the form of fractions like those of the 15th century and, as then, cancelled by the reciprocals of the fractions, were used in the 17th and the early 18th centuries in Italy and Germany for pieces with short sections in different metres (e.g. 12/8 is cancelled by 8/12, 6/4 by 4/6). Apel’s suggestion (Notation of Polyphonic Music, 5/1961, pp.163, 442) that these mensuration changes might have carried connotations of tempo change seems only speculative. Examples of this notation occur in Frescobaldi (fig.83), Corelli’s op.5, Georg Muffat (Apparatus musico-organisticus, 1690), F.X. Murschhauser (Prototypon longo-breve organicum, 1707) and the manuscript CS-KRa II/133. On the more complicated question of C and C as proportional signatures, see below.

A fundamental change in tactus notation occurred, however, from the 17th century; it eroded the significance of proportional signatures for tempo, and makes unambiguous determination of tempo very difficult in a good deal of later music. In medieval notation there was a progressive slowing down of note values (see §3 above), and this continued during the 16th century, partly no doubt owing to the proliferation of short note values. By the second half of the century the minim had become the normal beat in polyphony. But this slowing down did not continue uniformly throughout the polyphonic repertory: even though the crotchet became the main time unit for much music in the 17th century, the minim continued to be the normal beat in much music produced for popular consumption, such as the metrical psalters and hymnals, and in church music in the stile antico. Indeed it still survives as the normal beat in hymn tunes and Anglican chants; the crotchet has only recently taken its place in some hymn tunes, and then usually only when settings are complex or connotations of a modern style are sought.

The change, then, lies largely in the increasing readiness of notators arbitrarily to adopt different note lengths as the main beat in different contexts (for an example of this seeMadrigal, fig.2). This variety in tactus notation presumably had its roots in the 15th-century notation of augmentation and diminution, which continued to be expounded as the basis for theoretical distinctions such as those between C and C. By the time of Beethoven any note value between the semiquaver and the dotted minim was capable of functioning as the main beat (compare parts of the Arietta of the Sonata op.111, fig.89 below, with his scherzos: that of the Ninth Symphony includes specific recognition of the dotted minim as the beat – ‘ritmo di tre battute’, etc.).

With this increasing variety of augmentation and diminution, especially from the 18th century, any note value could theoretically function as the beat, independent of considerations of tempo, through a novel explanation of fractional time signature (found at least as early as G.M. Bononcini’s Musico prattico, 1673). Here, as in modern theory, the denominator of the fraction representing the time signature indicates the note value on which the metre is based (usually the beat, with the figure 1 for the semibreve, 2 for the minim, 4 for the crotchet etc.), and the numerator indicates the number of such note values to the bar.

Parallel with the partial emancipation of the time signature from tempo, there were two developments tending to make the determination of tempo and time easier: the increased use of bar-lines and of verbal specifications of tempo. Vertical lines had been used through staves in medieval score notation, not in their modern sense as bar-lines but to divide sections from one another; but some 15th- and 16th-century keyboard and lute sources include the visual separation of units of one or more bars, either by a space left between them or by a bar-line. Such bar-lines are used with varying degrees of consistency and frequency (for bar-lines marking off single beats, see Apel, 1942, p.67). The bar-line was used in vocal music also (mainly in scores) from the late 16th century, but was not adopted generally in mensural notation until the 18th century. Even later than that some notation lacked it, as do some 19th- and 20th-century editions of old music. In 20th-century music dotted bar-lines were used to clarify the subdivisions of larger bars, as in Debussy’s Les sons et les parfums tournent dans l’air du soir (from Préludes, i, 1910). (For the use of conflicting bar-lines, see below.)

The increasing unreliability of time signatures as indicators of tempo is also reflected in the adoption of Italian (and later German, English and other) terms for this purpose. These tempo indications (in Donington’s terminology, ‘time-words’: see Interpretation of Early Music, 3/1974, pp.386ff), like bar-lines, seem to have appeared first in polyphonic music for soloists, perhaps because of the complexity of this music from the individual performer’s point of view when compared with vocal music. Tempo indications occur in Luis de Milán’s El maestro (1536) but were not generally adopted before the 17th century: the terms adagio, allegro, grave, largo, lento and presto are attested between 1596 and 1619, with others during the 17th century. These tempo indications have not always carried their present connotations, nor has their significance always been precise. From the early 19th century (Beethoven) they were supplemented still further by metronome indications, and in the 20th century (Bartók) by precise indications of the duration of a piece in minutes and seconds (see Tempo and expression marks and Metronome (i)).

Despite these developments, time signatures never completely lost their associations with tempo, although the associations of numerical (fractional) time signatures, taken in isolation, are seldom unambiguous. No consistency exists in the music of the last three centuries even in the relationship between the note values chosen to function as beats and tempo.

With and C, the circular and semicircular signatures inherited from the Middle Ages, practice was even more confused. Theorists continued to expound the significance of vertical bars in time signatures (as in C), and of the reversal of symbols (e.g. in ) as signifying diminution; but even when distinctions can be drawn between C meaning 4/4 and (‘alla breve’) meaning 2/2 or 4/2 (the latter, for example, in the Credo of Bach’s B minor Mass), it is by no means clear that the tempos of the beats were intended to be equivalent. In the 16th century the sign C became uncommon, the basic duple metre of most polyphony being signified by C; precisely the reverse convention became common at the beginning of the 17th century, and the reason for the change is obscure, for no change in meaning seems necessarily to have been intended. This change occurred in English music printing quite suddenly about 1594 in madrigal partbooks and about 1621 in psalm books: see fig.84. These signs may enclose a dot without changing in meaning. Similarly, is sometimes used as a synonym for , C 3, C 3/2, C 3/2, C 3/2, 3, 3/1 and so on, for quick triple time; from the 17th century and seem to have been dropped until was later revived with a new meaning (see below). These time signatures were used without total consistency well into the 19th century at least; Schubert often used C adagio movements in 4/4 and C for fast movements, whereas Bruckner used C for fast 4/4 movements and C for slow movements. (In the first movement of Bruckner’s Seventh Symphony, moreover, it is hard to see a real tempo difference being intended between the allegro moderato C of the opening and the ruhig C at bar 51.) Even books of rudiments may imply that C and C are synonyms (Dibdin, Music Epitomized, 1808 edn., p.37).

The most that can be said is that vertical bars in time signatures, and reversed (‘retorted’) signs, indicate relatively fast tempos, but not always reliably; this is particularly likely when C is used for music in the stile antico, or when the time signature changes during the course of a piece. (For relevant passages from a wide variety of theorists, see Donington, Interpretation of Early Music, 3/1974, pp.405ff.) There has been no thorough investigation of time signatures used since the 16th century; the degree of ambiguity at different times and places can scarcely be assessed, and the reasons for it are unknown.

Signs sometimes used in the 18th and 19th centuries as equivalents of 4/2 (‘great alla breve’) rather than C (2/2 or 4/4) include 2/1, , revived in a new sense (e.g. Bach, Gigue from Partita no.6, bwv830), CC (Rossini), C C and C C (Schubert, Impromptu d899 no.3, altered to C by Schubert’s publisher). was sometimes written as C | (i.e. with its elements spaced out).

Even when the tempo implied by a time signature is clear, divisions of the time within individual bars are not always literally those written, even apart from considerations of agogics, in terms of which most notators intended the written note values to be interpreted with some degree of rhythmic freedom. Unwritten rhythmic conventions cannot generally be guessed in European music before the 16th century. At that period Tomás de Santa María wrote of pairs of notes written equal but intended to be played unequally (Arte de tañer fantasia, 1565); this may have been widespread, and a similar practice is well attested in 17th- and 18th-century France (see Notes inégales). In some cases of unequal performance of note pairs, the first note of each pair was to be lengthened; in others it was the second note, though this was generally indicated in some other way. Even written-out ornaments must have been subject to some rhythmic alteration in practice: some trills written as successions of even rapid notes were intended to begin slowly and then speed up. Such practice was in certain contexts so much a matter of course that notational devices were used to indicate the absence of inequality: either a verbal direction (e.g. ‘notes égales’) or dots written over individual notes (Marin Marais). A comparable unwritten rhythmic convention, attested in some 19th-century keyboard music, is the introduction of unwritten rhetorical pauses (see Franklin Taylor’s preface to his Clementi edition, concerning the Didone abbandonata Sonata), though Lussy (1874) sought to restrict this practice to salon pieces (Eng. trans., pp.207–8).

Discrepancies between written and intended rhythms are particularly likely when different strands of the texture are notated in different time signatures; these multiple time signatures (in Read’s terminology, ‘polymeters’) occur fairly frequently in the 18th century (as in Bach and Mozart). In such cases it is usually at least possible that the conflicting rhythms which result are to be accommodated to one another (for exceptions, see below). Dotted-note figures with duple subdivisions against triplets are cases in point: particularly in late Baroque music it often seems likely that dotted rhythms are to be relaxed into triplet rhythms: the use of a crotchet and quaver under the numeral 3 (and often slurred), for the more accurate notation of triplet rhythms in cases such as these, is attested in the mid-18th century (Arne, 1756; see fig.85, where it is immediately adjacent to the older convention) but not generally used before the 19th. In some 19th-century notation, for example chorus parts in Verdi operas, the accommodation of dotted rhythms to triplets also seems highly probable. There is even some evidence in the 19th century of duplets in a melody and triplets in an accompaniment being accommodated to each other (Bochsa, New and Improved Method of Instruction for the Harp, c1818–19, p.60); although theorists explain triplets as three notes in the time of two, they do not always state that all the notes involved are equal in length. Some piano notation, with polyphonic figuration and notes occurring in more than one polyphonic voice (fig.86), absolutely requires accommodation of the rhythms, though its evidence ought not, perhaps, to be pressed into proving rhythmic accommodation elsewhere. The correct practice in Schubert’s music, where dotted and triplet rhythms often appear simultaneously, is particularly difficult to establish (see MT, civ (1963), 626, 713, 797, 873).

On the other hand, even an apparently straightforward case of dotted notes against triplets in Paisiello was cited as early as 1806 by Callcott (fig.87, from A Musical Grammar, p.236, repeated in many subsequent editions) as an ambiguous case: ‘There is some doubt whether this Melody should be played as written, or as if it were compound; that is, one dotted Crotchet, one Crotchet, and one Quaver, in the first Measure’. The possibility of maintaining conflicting rhythms in certain contexts had been raised by some in the second half of the 18th century (Quantz, Eng. trans., 1966, p.82; Türk, 1789). A general tendency in cases of conflict to accommodate all rhythms to the most relaxed within a texture ultimately lacks logic, and a cautious approach combined with aesthetic judgment seems advisable in the present lack of detailed studies based on a large cross-section of notational evidence.

A distinction between intended and written rhythms, literally interpreted, is likely, on the grounds of the category to which a piece belongs, until the 19th century at least and even later in popular music. Double-dotting, though attested in Marais as early as 1701, is, like the precise notation of triplets, uncommon in mensural notation before the 19th century; yet the French overture is a well-known example of a category with a well-defined tradition of tempo and double-dotting in performance, which was required but not normally spelt out in the notation. There were conventions concerning the tempo and rhythmic features of various categories, particularly dances, such as the minuet, gavotte, chaconne and pastorale; these conventions varied to some extent according to period and country, and were sometimes ambiguous even to contemporaries. Nevertheless the notation used for them might for the sake of convenience vary in literal detail from the intended effect, if there was no danger of misunderstanding at the time – for example in the choice of C as the time signature of a gigue, with quick triple time intended throughout (see Ferguson, 1975, pp.92–3).

The tradition-bound approach to notation implicit in practice such as this led some to retain traditional ‘category’ notation even when the notation was no longer suitable to the category. Chopin, for example, retained the 3/4 time signature of the scherzo (descended from the 18th-century minuet) even though the tempo had so greatly increased by his time that the bar contains only one beat, and single phrases frequently contain notes tied over three or four bars, as well as rests of two bars or more (fig.88). According to later 19th-century rhythmic theory, this traditional notation seemed inadequate (for criticisms of category notation based ultimately on this theory, see S. Macpherson, 1908, rev. 1915, pp.39ff, and 1911, rev. 1932, p.17). Nevertheless, category (or rather style) notation remained alive in 20th-century popular music notation: notated almost universally in C, it very often requires to be interpreted on grounds of style as if notated with unequal pairs of notes closer to 12/8; moreover, certain other rhythmic characteristics (syncopation etc.) are frequently required by the style though not spelt out in the notation.

In addition to these complexities, there are many contexts in 18th-century music where triple subdivisions of notes are extensively tolerated. The lack of theoretical provision for such practice at times renders passages difficult to read: a notable example is the Arietta of Beethoven’s sonata op.111 (fig.89) where the demisemiquavers are to be taken as ‘perfect’ in the medieval sense unless followed by a hemidemisemiquaver and thus rendered ‘imperfect’ (this detail of Beethoven’s notation survives in modern editions of the sonatas).

The confusion in the use of time signatures and subdivision of beats, especially in relation to tempo, was recognized by Riemann (Musikalische Dynamik und Agogik, 1884). He recognized that the choice of note length for the beat was often arbitrary, particularly since 6/8, for example, could represent either two or six beats to the bar. Accordingly, and to facilitate the representation of beats with triple subdivisions, he proposed a system of time signatures based on the beat, ignoring differences that had long been artificial, such as that between 2/4 and 2/8. In his reformed system, time signatures were to comprise simple integers (e.g. 2, for two beats to the bar, whatever their nominal value), integers separated by dots (e.g. 2.3, meaning six beats in two groups of three each) or fractions (e.g. 2/3, meaning two beats with triple subdivisions, or 3.2/3 meaning six beats with triple subdivisions in three groups of two). However soundly based and economical, this ingenious system did not win general acceptance, though the use of the simple integer (found with a more restricted meaning earlier in the French Baroque), omitting the denominator of the conventional fractional time signature, is fairly widespread in 20th-century mensural notation.

Another abortive attempt to increase the variety of the subdivisions of the beat, in this case well beyond the capacity even of 14th-century Ars Subtilior notation though not matching the theoretical potential of 15th-century proportional notation, was the system proposed by Henry Cowell. He sought to supplement the traditional vocabulary of note lengths with ‘two-third notes’, ‘four-fifth notes’, ‘four-seventh notes’ (i.e. as fractions of a semibreve) and further submultiples of the semibreve, which were to be represented with void notes of various shapes without stems (triangles, squares, lozenges etc.). Some of these shapes were borrowed from traditional American shape-note notation (see §5(iv) below). These notes would then generate others by the addition of stems and flags, in the same way as the semibreve: with a stem, for example, ‘third-notes’, ‘two-fifth notes’, ‘two-seventh notes’ and so forth. The system is exemplified in Cowell’s Fabric of 1917, published in 1922 (fig.90: see also Read, 1964, 2/1969, pp.76–7; Stone, 1963, p.19). A further system of durational proportions, based on ratios and not requiring novel note shapes – and thus more flexible than Cowell’s system – has been adopted by Stockhausen (Klavierstück I).

Multiple time signatures occasionally occur from the 18th century in pieces where there is clearly no question of accommodating the rhythms in one part to those in another: Fux’s Concentus musico-instrumentalis (DTÖ, xlvii, Jg.xxiii/2) has a movement with a simultaneous ‘Aria italiana’ in 6/8 and an ‘Aire française’ in C; and the ‘Fanfare’ from La triomphante in Couperin’s tenth ordre (fig.91) has conflicting signatures, with the explanatory note ‘Quoy que les valeurs du dessus ne semblent pas se raporter avec celles de la basse; il est d’usage de le marquer ainsi’. In some 20th-century music and editions of older music, both conflicting signatures and conflicting bar-lines may be found (figs.92 and 93: examples from Bartók, String Quartet no.3, 1929, and an edition of Monteverdi madrigals by Leichtentritt).

Time signatures representing additive metres (those in which the beats within a bar cannot be subdivided into groups of equal size) are found in the 18th century: Handel included a few orchestral bars in 5/8 in Orlando (1733) to represent madness; Burney (History) termed this ‘a division of time which can only be borne in such a situation’. (The principle of additive rhythm occurs much earlier than this in vers mesurés à l’antique.) Additive time signatures were used by Reicha (‘3/8 et 2/8’), Chopin, Tchaikovsky, Bartók (3 + 3 + 2 over 8) and others, but Moritz Hauptmann’s rhythmic theory regarded them as ‘inorganic’ and thus to be condemned (1853, Eng. trans., 1888, pp.196ff). A list of 20th-century works with additive and other unusual signatures may be found in Read, 1964, 2/1969, pp.159ff; some are based (not necessarily consistently) on small rapid note values, rather than beats, as the unit for the numerator of the fraction. A variant of this occurs in Stockhausen’s Klavierstück I, with the smallest note value, not necessarily directly represented in the changing time signatures, regarded as the basic time division. As changes of time signature within a piece became increasingly common in the 20th century, various notational details were simplified: the double bar previously usual before a change was often reduced to an ordinary single bar-line, time signatures were enlarged and written between the staves, or above the staff (Debussy).

For music partly or entirely outside a system of regular beats, notational practice has varied. The oldest sign representing an indefinite prolonging of a note’s duration is the pause sign or fermata (a semicircle over a dot), inherited from the Middle Ages and still universally used in Western notation (in the 20th century it was sometimes modified to provide various degrees of extra length, or inverted to signify shortening rather than lengthening). Relatively lengthy passages of music without a regular metric beat occur from the 17th century onwards in recitative; Italian recitative was normally divided arbitrarily into bars of four beats, with C as a time signature, whereas French recitative was notated with frequent changes of time signature more closely reflecting the declamation of the text. (The arbitrary use of a time signature in Italian recitative is paralleled by the occasional 16th- and 17th-century practice of notating pieces in duple metre even though their musical sense is triple: see Apel, Notation, pp.66–7.) An experimental notation is found in some French harpsichord pieces of the last quarter of the 17th century and the early 18th century, where conventional time divisions are quite abandoned and the music notated either entirely in ‘semibreves’ grouped with slurs, or in a mixture of ‘semibreves’ joined with slurs and ‘short note values’ joined with beams (fig.94; for details of this notation and its interpretation see Prélude non mesuré, and Moroney, 1976).

Various methods were used from the 18th century to notate irregular expressive melodies in free rhythm in instrumental music as for example in written-out cadenzas or in keyboard fantasias (Mozart, Beethoven). They may be notated, like Italian recitative, by using a regular time signature and bar-lines and by dividing the passage arbitrarily into beats, as in Beethoven’s Ninth Symphony (fourth movement) or in his Piano Sonata op.31 no.2 (first movement). This notation has been maintained by later composers, including Wagner, for the shepherd’s piping in Act 3 scene i of Tristan und Isolde. Alternatively the passages may be written without a time signature, either in notes of conventional size, or in small notes, or a mixture of the two, grouped with beams as required to clarify the accentuation, and with any bar-lines irregularly placed as an auxiliary means of grouping: this method was used by Mozart and many later composers (fig.95).

In notation of this sort the note values carry only imprecise connotations of duration. A more systematic use of imprecise durations was required in some 20th-century works: notes of indeterminate length were notated in various novel ways, sometimes with imprecise distinctions between longer and shorter notes (see Risatti, 1975, pp.1ff). In some 20th-century music, great rhythmic variety within the texture had to be represented without the aid of regular beats in any strand, and increased precision was sought in the notation of accelerandos and decelerandos, previously indicated only with verbal directions in the score. This was sometimes achieved through the modification of aspects of notation not formerly used to represent the division of time: converging and diverging multiple beams used for groups of short note values notated in an otherwise traditional fashion may denote increasing and decreasing tempos; or staves may be slanted upwards from the horizontal to denote an increase in tempo and downwards to denote a decrease in tempo (Bussotti). Other devices include the multiplication of metronome indications at short intervals, the use of ‘+’ and ‘−’ signs for (imprecise) tempo changes and so on (see Risatti, 1975, pp.21ff). Rhythmic proportions more complex than those available with the traditional numerals (3 = three in the time of two, 5 = five in the time of four, etc.) have been represented by a precise spacing of notes to a specific scale (e.g. one second to 2·5 cm of score); this notation may be supplemented with auxiliary signs specifying the duration of individual ‘bars’ (short subdivisions) within the music, resembling the specifications of precise lengths in an architectural scale-drawing. In 20th-century music generally, whatever the style, vertical alignment of different parts of the texture became a generally reliable indication of the order in which notes are to be sounded, and whether or not notes are simultaneous; and spacing of notes tends generally to be proportional to their durations.

Notation, §III, 4: Mensural notation from 1500