Index (click on a section, or read straight through):
Summary
1. The Theory of the Becking Curve
2. The Theory of Metre
3. A Schubert Fingerprint
4. The Theory of Tempo
5. Conclusions and Future Research
Appendix I. Some alternative implementations of the Schubert curve in different 'real' metres
Appendix II. Some non-essentially different notations for given 'real' metres
Appendix III. Fingerprint examples in score, with discussion
Appendix IV. Summary of terminology
Footnotes
References

     The conducting shape of music, taken between successive upbeats, is discussed according to the Becking curve. On that basis a distinction is made between 'real' and 'nominal' metre, reflecting the choice between essentially equivalent metrical notations in which a score may be written. A prepared appoggiatura pattern is proposed as a fingerprint characteristic of Schubert; 150 examples of it from his works are related to the Becking curve, helping to determine the number of 'nominal' bars per curve.
     A distinction is next made between 'real' and 'nominal' tempo, based upon the number of 'nominal' bars per curve, and it is shown that Schubert's verbal tempos indicated in the examples reflect this distinction, through being related generally to 'nominal' rather than to 'real' bars. The approach involving jointly the theories of curve, metre and tempo is thus considered to be supported by the observations, here just in the case of Schubert. Further work is foreshadowed, including comparisons between composers, when the fingerprint notion may be established more broadly.

Introduction
     The central aspect from which music is to be studied here is its internally felt conducting shape. According to this approach, associated with a piece or movement of music and running throughout it (with occasional possible exceptions) is a more or less exactly repeated cycle of pressure somewhat analogous to the human pulse. Each cycle typically takes about one to two seconds and corresponds to the shaping of the music from one upbeat through the downbeat and on to the next upbeat. It can be felt internally by any person who is in the presence of the music (whether by listening, performing, or silent score-reading) and who is sensitive to it. This notion has been thoroughly treated by Becking (1928), for which an English-language synopsis was provided by Nettheim (1996). ¹  Becking used a small baton in 'armchair' conducting, letting the music induce the motion; this is the opposite of the situation where a conductor influences an actual performance. The movement of Becking's baton is then supposed to reveal externally the internal pulse-like feeling associated with the music. Becking's purpose was to distinguish between the characters of the music of different composers, nationalities and historical times; that is an ultimate purpose of the present line of research too, but this paper deals just with the music of Schubert.

The Becking curve for Schubert
     Becking presented two-dimensional diagrams which became known as the 'Becking curves', indicating the way the baton moves in response to the music of each of a number of composers. His curve for Schubert is shown in Figure 1. The motion of the small baton starts near the top right (assuming a right-handed subject), proceeds to trace out the upbeat by moving further out to the pointed end (thus away from the subject's body), continues for the downbeat and follow-through by moving to the lower left via the lower path, and is completed by moving back for the re-gathering phase via the upper path to regain the starting point for the next upbeat. The thickness of the curve (which varies only a little in the case of Schubert) indicates the varying amount of force applied. The speed and quality of the motion are not indicated in the curves, the accompanying comments helping in those respects. An important feature, from the point of view of Becking's scheme, is that the Schubert curve belongs to the type that is pointed at one end and rounded at the other end. The motion therefore stops momentarily at the pointed end, whereas it continues without a break at the rounded end; that distinction can be felt upon reflection and can be embodied in performance in a way whose explanation is not part of the present purpose. ²

Figure 1. The Becking curve for Schubert (Becking, 1928)

Führen und schwingen
(Guide and swing)

     The exact shape the curve should have could of course be debated and, even though Becking's shapes may seem convincing, others are free to substitute their own. Becking's main movement for Schubert was towards the body ('führen', to guide/lead/conduct), the auxiliary one away ('schwingen', to swing). I find Schubert's curve to be similar to Becking's version but in the opposite direction—first moving away (with a slight touch of yearning at the beginning which diminishes in the course of the follow-through) and then easily back (see Figure 2). Further, I find the gesture to move forward in front of the body at an angle of about 20 degrees from the frontal toward the lateral plane. These modifications of the curve do not affect the essence of Becking's method or any of the results to be reported here.

Figure 2. The present author's version of the Becking curve for Schubert

     A term will be needed for the conducting shape of the music between one upbeat and the next. As no such term is in general use we will use, equivalently, the terms 'Becking curve', 'conducting shape', 'conducting curve' or simply 'curve', although strictly speaking these terms refer to a visual representation of the phenomenon rather than to the phenomenon itself. ³

Scope of the curve
     Becking occasionally indicated the number of printed bars per curve in his examples but often he did not indicate it. It turns out that that number can in practice be 1, ½ or 2 (or might perhaps very rarely take another value). A 'real' bar will be defined as the scope of one curve, by contrast with a 'nominal' bar as printed. (The unmodified term 'bar' will be taken to imply 'nominal' bar, when the emphasis provided by the word 'nominal' is not needed.) One of the main tasks of the present paper is to deal as objectively as possible with the problem of determining the number of 'nominal' bars per curve in a given example. Until that problem is solved, there is no satisfactory basis for comparing pieces or movements or for understanding and performing music in terms of the Becking curve. ⁴
     The determination of the number of 'nominal' bars per curve in a given example is not always easy. Several means may be considered, some more subjective, others more objective.

More subjective means
     (a) One might think through the music according to each of the three possible values in turn to see which seems most convincing. Each possibility should be worked through with full enthusiasm, but there is a risk that one will become convinced of the answer one first intuitively tries, after which it is difficult to allow fully the possibility of other answers. Thus the personal element might intrude too much.
     (b) Also subjective would be reliance on the opinion of master performers or other experts who, as is well known, often disagree with one another. More objective means are therefore to be sought.
More objective means
     (i) Given the musical form of a movement, its total length in 'nominal' bars can provide a clue, for if it is unusually short or long as thus measured, compared with the composer's other movements in that form, it might well have ½ or 2 'nominal' bars per curve, respectively, if other likely explanatory factors are absent. Similarly, an average length might suggest 1 bar per curve. Movements with 2 bars per curve naturally often have an even number of 'nominal' bars.
     (ii) A strong perfect cadence in the middle of a 'nominal' bar, not having the character of a feminine cadence, might suggest ½ bar per curve.
     (iii) Text underlay sometimes provides clear evidence, for a downbeat would not normally be occupied by a weak syllable (except perhaps in an emphatically broadened final cadence  ⁵  ), nor a weak beat by a strong syllable. ⁶
     (iv) The appearance in a given movement of a parallel passage shifted by half a 'nominal' bar might suggest ½ bar per curve. ⁷
     (v) Although the above fairly objective means may occasionally be clearly applicable, opportunities for their use are somewhat limited. I therefore propose considering here the evidence of 'fingerprints'—patterns characteristic of a given composer—which allows far more objective comparisons between pieces and movements; the intrusion of subjective determination is minimized as far as possible by carefully defining the fingerprints in terms of elements observable in the scores.

Introduction
     As the theory of metre to be presented here is based upon the notion of the Becking curve, it applies only to music which can be heard as yielding such a curve. The theory thus covers Western art music from about 1600 to 1900. The particular features referred to here are directed only to Schubert, but in some cases they might apply more widely within the period just mentioned.
     Metre, from the present point of view, is a hierarchical subdivision of the time occupied by one traversal of the curve of a given composer. The subdivision can be regarded as a grid enabling the listener to grasp readily the progress of the music through each curve. ⁸  Key spots in the grid may correspond directly, or more often indirectly, with key spots in the curve.
     The curve underlying music operates in continuous time: at every instant a certain degree of pressure or intensity is felt which in general varies continuously. Musical notation, on the other hand, operates for the most part only in discrete time, using the grid points of the metre and its subdivisions. The score therefore gives a discrete prescription from which the continuous curve arises when the notation is transformed into actual or imagined sound. ⁹

Essentially and non-essentially different metres
     Two given metres may be essentially different, as for instance 3/4 vs 4/4, or 3/4 vs 6/8. They may instead be non-essentially different, differing in manner of notation, as for instance 3/4 vs 3/8, or one bar of 6/8 vs two bars of 3/8. ¹⁰  More or less subtle (thus non-essential though not non-existent) differences of emphasis, tempo, etc. might well be implied by the mere manner of notation (to be discussed later). Whereas our terms 'essentially' and 'non-essentially' different metres refer to pairs of cases, we use 'real' and 'nominal' metre to refer to a single case.
     Clear evidence that two metres (or in this case three) may be non-essentially different appears in a letter of 20 June 1841 from Chopin to Julian Fontana (Hedley 1962, 196):

I am sending you a Tarantella [Op. 43]. Be kind enough to copy it out, but first of all go to Schlesinger's, or better still to Troupenas, and take a look at his edition of a collection of Rossini's songs which contains a Tarantella (in A). I don't know whether it is written in 6/8 time or 12/8. It can be written in both ways, but I should prefer it in the same way as Rossini. So if it is written in 12/8 or possibly common time with triplets, you should make two bars into one when you copy it out.

The first few bars of [Chopin's Nocturne op. 9] no. 2, in E flat major, were written on a card to Maria Wodzińska by Chopin, dated 22 September 1835... In this short extract the composer has barred the music as if it were in 6/8 [rather than 12/8 as published]! (State Collection, Warsaw).

Another well-known case occurs in Mozart's Bei Männern from Die Zauberflöte K620/7, written first in 3/8 and subsequently changed by Mozart to pairs of bars in 6/8; some significance is apparently intended here, though possibly not of the 'essential' type, for the duet begins not with a full bar of 6/8 but just a half bar. Examples of non-essentially different metres will be referred to as formal possibilities later in this section; examples from Schubert will be given in Section 4 (Tempo).

Alternative metres for a given curve
     The notion put forward here is that a given (continuous) underlying curve can be manifested in different (discrete) metres. That was to be expected in case the difference between such metres is non-essential. It is also true, however, even in the case of essentially different metres: although the corresponding conducting shapes might not then be strictly identical, they will be sufficiently similar for the practical purpose. This follows directly from Becking's notion of the curve, for a composer is by no means restricted to just one metre, but instead presents the given curve in different segments having various metres, 'real' as well as 'nominal'. (A 'segment' is here defined as a section of music having a fixed 'nominal' metre and tempo.) Becking, however, did not specifically deal with this aspect, that is, he did not compare various different discrete metrical grids or pillars in their relation to the flow of a single given continuous curve.
     The basis for subdivision, in Schubert's case as for most classical music, is the unit of two or three elements. These are the two smallest prime numbers; next would come five and seven. There are fourteen possible combinations up to the third level. Six of these have conventional names: 2 (simple duple), 3 (simple triple), 2x2 (simple quadruple), 2x3 (compound duple), 3x3 (compound triple), 2x2x3 (compound quadruple). Here we have represented first the highest level, having longest duration. Musical notations for the various metres will be seen in what follows. Whether quadruple metre is to be considered essentially different from duple metre is uncertain in advance—the outcome will depend upon an examination of the scores. ¹¹  Note also that the conventional terminology has no special term for the rather common cases 3x2 and 2x2x2 (which may be notated with 8th-note pairs in 3/4 and 4/4 respectively). The remaining six possibilities are: 2x3x2, 2x3x3, 3x2x2, 3x2x3, 3x3x2, 3x3x3. In subdivisions of more than three levels we are unlikely to observe any essentially new contribution to metrical behaviour, in so far as it delineates the conducting curve.
     It is natural that each composer may show some preference for the 'real' metre or metres in which he can best represent his conducting curve. He may also take his own attitude to the conventions of 'nominal' metre, for composers were not obliged to follow text-book usage in respect of metre any more than in respect of the significance of tempo indications, in respect of harmony, and so on. Indeed, in the case of many composers including Schubert, text-books appeared only some time after the works had been composed. Thus we must discard text-book approaches to metre, which do not distinguish between composers, and instead build up our understanding empirically for each composer separately.
     Only certain of the mathematically available subdivisions may be considered to have metrical or curve significance by a given composer. Thus Schubert often mixed pairs and triples at the lowest level of subdivision (even taking into account the notational question of triplet assimilation), which suggests that in such cases the significance of the lowest levels was not metrical for him. A case in point is Rastlose Liebe D138 bars 33-44, where a duple voice part is written against a triple accompaniment. Yet what is significant depends upon the composer's purpose, which may vary between cases. For the present purpose of systematically exploring manifestations of the conducting curve in Schubert, it turns out that most of the ground can be covered with just three categories: 2 (and possibly 2x2), 3, and 2x3—in conventional terminology simple duple (and possibly quadruple), simple triple, and compound duple.

Metrical implementations of the Schubert curve
     Implementations of the Schubert curve in different 'real' and 'nominal' metres will now be discussed in turn.

Different 'real' metres
     Implementations in different 'real' metres are shown in Figure 3. One bar as printed in the Figure corresponds to one curve (the number of 'nominal' bars per curve will be discussed later).

Figure 3. Implementations of the Schubert curve in the three main essentially different metres:

Figure 4. Regions in Figure 2:

Different 'nominal' metres
     Any metre appearing in a score can be regarded as resulting from three choices: (i) a 'real' metre, (ii) the number (1, ½ or 2) of notated or 'nominal' bars per curve and (iii) the note-value chosen as metrical unit. Given a 'real' metre, (ii) and (iii) determine the 'nominal' metre.
     (ii) The basis for the choice of the number of bars per curve is sometimes the purely practical matter of the number of bar lines desirable for visual clarity, but in other cases it may be the character of the music via the implied relative strength of the beats, a subtler difference than that between different 'real' metres (see e.g. Becking 1928, 41-42). (iii) The basis for the choice of the metrical unit is also sometimes a practical matter, the number of beams needed; in other cases it may be the music's perceived weight and speed, small units suggesting a lighter and quicker effect, larger ones a heavier and slower effect (see C.Brown 1998, for citations of early theorists), again a difference of degree rather than of kind; in still other cases the choice can probably be considered to be made just on conventional or arbitrary grounds.
     Table 1 will later be found helpful, showing Schubert's 'nominal' metres. The number of segments is shown, not the number of movements; reappearances of a given metre within a movement are counted as separate segments. Note-values or subdivisions used within the 'nominal' metres are also relevant but are not tabulated here. The 'nominal' metres provide no direct information on 'real' metres or on the conducting curve, so one should avoid drawing any conclusions from them concerning the essentials of Schubert's musical thought. ¹⁵

     We can now take into account also the number of bars per curve, showing a list of 'nominal' metres in Table 2. There is in principle an unlimited range of possible 'nominal' metres; we restrict the present list to those appearing in Schubert's works (thus, for example, 6/16 is not shown). However, some of Schubert's notations are not shown in Table 2, in particular those using triplet notation, as for instance 'real' compound duple metre notated as 2/4 in triplet 8th-notes with 1 bar per curve. In the Table '. . .' indicates metres which do not occur in Schubert, while '—' indicates metres which do not exist. A number of non-essentially different notations for given 'real' metres are shown in musical notation in Appendix II.

     The theory presented above will allow the systematic ¹⁶  comparison of examples from the musical literature and in particular from Schubert's music, which will be undertaken in the next section. The theory was derived initially from inspection of some examples and in turn led to the assimilation of further examples; thus theory and practice belong together.

Fingerprints as partial evidence of the curve in the score
     Schubert's extant scores constitute virtually our entire documentary evidence of his music. That is because, on the one hand, descriptions of performances at his time are quite inadequate for any detailed purposes and, on the other hand, the insights of later performers or commentators must be derived from the scores (there is clearly no possibility of a reliable unbroken line of emulation in performance). We adopt Becking's conducting curve as a governing notion, whether or not its specification is accepted in every detail, so we will search in the scores for evidence of the curve's scope and shape. As Becking (1928, 9) himself pointed out, that evidence can not sufficiently be seen in any one example considered by itself. The only remaining possibility is that the evidence resides in the scores as a whole oeuvre. We do not look into passages of simple scales or repeated notes, for instance, which any composer could write, but instead into the most characteristic places in the scores, for it is mainly from just those places that we derive our idea of Schubert's musical personality. To begin the search, an intuition is required as to what is or is not characteristic for the given composer; also required, if the results are to be convincing, is subsequent confirmation by the method of comparative analysis with explicitly defined score patterns, as will be seen later.
     Ideally, features of the prototypes and examples to be given would be found to match (directly or indirectly, specifically or on average) features of the conducting curve. That matching can be made explicit only to a limited extent at the present time, with the assistance once more of the rather broad regions of the conducting curve which were shown in Figure 4. The method to be used involves the study of many of Schubert's scores in an attempt to 'decode' them so as to reveal the evidence for the curve which lies hidden in them. This decoding can hardly be achieved completely, for the significance of the curve is that of a whole musical and human personality; the present paper, as a first attempt, therefore aims just to contribute a partial decoding.

Prototypes for the fingerprint S1
     The fingerprint to be described and exemplified will be called 'S1', where 'S' indicates Schubert and '1' indicates the first of several such Schubert fingerprints. The treatment of other Schubert fingerprints and fingerprints of other composers is intended for future work (see section 5). We will first present prototypes, then the definition, and finally the examples themselves.
     Prototypes are intended to assist in grasping the rather large collection of examples to be given. The most suitable scope for a prototype or other example appears to be two complete curves, beginning with the upbeat if present. ¹⁷  Prototypes for S1 are shown in Figure 5 for each of the main essentially different metres. The main ('downbeat') note of each curve falls just to the right of the dashed vertical lines. It is only the second curve in each case (with the heavier dashed lines) which is proposed as prototypical; the first curve might be taken to provide a 'warm-up' before the arrival of the main one which was aimed at. ¹⁸

Figure 5. Prototypes for the fingerprint S1.
Moment Musical D780/6 (compound duple), Das Wirtshaus D911/21 (simple duple), String Quartet in C major D046-2 (simple triple).

     Figure 5 shows not only the prototype figures to be defined shortly, but also illustrates, by means of vertical alignment of the notes, the relationship between the three main essentially different metres discussed in Section 2. The Figure also illustrates the idea of non-essentially different 'nominal' metres: the D780/6 example has 'nominal' metre 3/4 but 'real' metre 6/4 which is equivalent in turn to 6/8 if the note-values were halved, while the D911/2 example has 'nominal' metre 4/4 but 'real' metre 2/4. Thus four bars of the former music correspond to just one bar of the latter. In the final example, D046-2, the 'real' and 'nominal' metres are the same.
     A key feature of the present approach is that the proposed fingerprint can be matched convincingly in only one way to the curve shape; it is this matching which allows the determination of the number of 'nominal' bars per curve. An approximate matching of the score to the curve is again shown with circled letters.
     Note that an upbeat should be felt even if it does not appear in the score, as in D046/2. It should especially be noted that the turnaround occurs about half-way through the 'real' bar in duple (or quadruple) metre, but triple metre does not contain a half-way grid-point and in that case the turnaround occurs instead about two-thirds through the 'real' bar (see Becking 1928, 39-40). Thus although the vertical alignment in Figure 5 shows the progress through the printed score correctly, the three staves correspond differently with the phases of the conducting curve as it progresses at varying speeds in real time, and especially so for triple versus duple metre. The tempo of the prototypes and of the curve, as well as of the examples to follow, will be discussed in Section 4, but can be taken for the present to lie between 1 and 2 seconds per curve; the duration of a 'real' bar may differ between different prototypes and different examples. Both timing factors—the temporal shaping and the tempo itself—mean that the horizontal axis uniformly reflects note durations (through vertical alignment) but not chronometric time (except to some extent in collections of examples having the same tempo and 'real' metre). ¹⁹
     The partial matching of the curve to the scores just outlined is of course only as understood by the present author; although that might be considered a limitation, it is also an advantage in that it produces a consistent approach, whereas a combination of views or a consensus would produce an unworkable hybrid. The feeling of actually carrying out the conducting in the way Becking described, together with the convincing and appropriate character of the music thus thought or performed, is the only ultimate way of confirming these matchings. But beyond the matchings, the examples to be given may be invoked in support of the present approach.

Definition of S1
     We first attempt an explicit definition of S1 in its strictest manifestation. This can hardly result in a fully automated procedure, and some limited use of musical intuition and sensitivity will still be needed, especially in the recognition of close varieties. ²⁰  The definition will be formulated so as to be independent of both the 'real' metre and 'nominal' metre, the differences between the implementations in different metres being indicated where needed.

     'Preparatory' or 'upbeat' phase
u1. A single upbeat melody note is placed towards the end of the 'real' bar or curve: five- sixths through it in compound duple and simple triple metre, three-fourths in simple duple/quadruple metre.
u2. The upbeat follows fairly smoothly from the previous material, if any, so that the gesture initiated by it does not begin abruptly.
u3. The upbeat melody note may be consonant, or part of a dissonant chord leading smoothly into the harmony on the following downbeat.
u4. The upbeat texture is about equal to that of the following main note, or slightly lighter.

     'Main' or 'downbeat' phase
d1. The downbeat note is placed immediately after the following 'real' bar-line.
d2. The downbeat melody note has the same pitch as the previous one.
d3. The downbeat melody note is dissonant: an appoggiatura.

     'Follow-through' phase
f1. No note is struck in either the melody or the other parts after the downbeat and during its sounding.
f2. The downbeat notes last for one-half the 'real' bar (compound duple and simple duple/quadruple metre) or one-third (simple triple metre).

     'Resolution'
r1. The note resolving the appoggiatura occurs next, at a pitch one scale-degree lower. In compound duple and simple duple/quadruple metre it occurs at the 'return' after the turnaround, in simple triple metre before the turnaround.
r2. At the resolution the bass is sustained from that of the main note, with no new striking in other parts except for the minimum to resolve any other voices taking part in the dissonance.

     'Intermediate' or 'post-resolution' phase
i1. An intermediate phase may follow the resolution note. In duple/quadruple metre this phase begins shortly after the resolution, but in triple metre it occurs after the turnaround which takes place well after the resolution.
i2. The resolving note may last up to the next upbeat or, if other notes follow it, they are lighter-textured.

     General requirements
g1. A certain tempo range is required, about 1 to 2 seconds per 'real' bar (see section 4).
g2. Dynamics are about central (not violent, not timid). (In performance, the accompaniment is a little softer than central, the melody emphasized to give a singing quality, but that is only sometimes explicitly indicated in the score.)

Notes on the definition of S1
u1. The fractions refer to the literal score durations, not necessarily exactly to the durations in a performance with appropriate nuances.
d3. The accompaniment to the appoggiatura does not contain the resolution note (even in another octave, unless in the bass as a pedal-point). (In Beethoven it often does.)
f1. The fact that no note is struck during the sounding of the downbeat note allows a kind of 'soaring' which would have been impossible if intermediate notes had brought it 'down to earth'.
f1 and i2. The fact that after the dissonance the only conspicuous happening is the resolution corresponds to the fact that the tension felt in Schubert's conducting curve is very much confined to the initial phase, and thereafter dies away (Becking 1928, 192).
i1. The resolution's placement just after the turnaround (in duple/quadruple metre) can be understood as helping to make the shape of the gesture rounded, rather than pointed, at the end of the follow-through phase, for continuity is thus forced in that region. This observation does not apply, however, in triple metre.
Pattern of successive curves: S1 usually occurs as the second of a pair of 'real' bars constituting one half of a normal phrase (a 'phrase-member' in the terminology of Goetschius 1904, 22-25), but it can occur as the first such 'real' bar, or as both; this will be seen in some of the examples to be given.

Varieties of S1
     To proceed from prototypes to further comparable examples in Schubert's works may be considered as looking for more or less close varieties of the prototypes. A given conducting curve can of course be associated with very many different manifestations in the notes of the scores and, if we have chosen the prototypes well, there will be many manifestations differing from the prototypes to only a fairly small extent.
     We next list specifications of departures from the prototype definitions yielding varieties in two classes: close and more distant. Cases of the former class will be included in the collection of examples presented later, whereas those in the latter class will be excluded. Departures completely incompatible with Schubert's curve will, according to Becking's theory, not generally be found in Schubert's works. ²¹

     Close varieties
u1. The single 'preparation' or 'upbeat' note may be divided into two notes of half the length; typically the second of these has the same pitch as the main note, the first one degree higher (e.g. Moment musical D780/6 bars 15-16).
d2. The upbeat melody note might not have the same pitch as that of the main note but instead a nearby pitch ; typically it will then approach from a third below or a second above (e.g. Lied der Mignon IV D877/4 bars 14-15).
d3. As the appoggiatura is struck, the resolution note may occur elsewhere in the texture even though not as a pedal point (e.g. Gretchen am Spinnrade D118 bars 8-9), but this is not Schubert's most common case.
f1-1. The landing after the initial soaring can be modified in a smooth way by light anticipations of the resolution note (e.g. Deutsche Tänze D783/7 bars 1-2, where there is also a waltz accompaniment—see f1-2 below). In vocal music (e.g. Alfonso und Estrella D732/16 bars 100-101) this may hint at a kind of portamento.
f1-2. An accompanying figure may be added, if it does not conflict with the curve's shape, often running throughout (e.g. again Gretchen am Spinnrade D118).
r1. An upward resolution to the dissonance is also possible though less common (e.g. Moment musical D780/6 bars 7-8).
r2. At the resolution the bass may be struck again in the same octave (e.g. Am Meer D957/12 bars 19-20).

     More distant varieties
u1. The upbeat note may arrive later than specified in the definition (e.g. Der Neugierige D795/6 bars 3-4).
f2. The resolution may arrive early (e.g. Frühlingstraum D911/11 bars 1-2); see also 'Influence of vocal texts', below.
r2. At the resolution the bass may move more noticeably than just being struck again in the same octave (e.g. Der Lindenbaum D911/5 bars 81-82—the last two bars).
Other more distant varieties are occasionally mentioned under the excluded cases in the discussion of the examples to be given.

     Influence of vocal texts
The number of text syllables and their strength can naturally influence the implementation of an underlying musical figure (e.g. Erlkönig D328 bar 82, 1 bar per curve). The significance of the text can also play a part; for instance, shorter durations and more rests may be used to indicate breathlessness or urgency (e.g. again D328 bar 74 compared with bar 73). In such cases the resolution note arrives earlier than in the definition; for the sake of simplicity, however, such cases will be excluded from the evidence to be assembled here.

Fingerprint examples
     A collection of examples of the fingerprint S1 is given as score excerpts in Appendix III, together with a discussion of the individual examples. Here they are discussed in summary form.
     The examples are well distributed over time through Schubert's output. ²¹  They are also well distributed between vocal and instrumental works, bearing in mind the greater number of the former in his output. Even dances, which have their own characteristic rhythm, coexist with Schubert's personal rhythm (D041/17, D334, D783/7). But it is important to estimate to what extent counter-examples appear in Schubert's music, that is, prepared appoggiaturas whose behaviour differs essentially from that specified for S1. In respect of the resolution point of the appoggiatura, whether anticipated or approached from a nearby pitch, there seem to be few cases with a differing place in a given 'real' metre; cases that do occur appear mainly at broadened cadences. The significance of the relatively small group of examples with shorter approach to the appoggiatura, excluded here, is not yet determined with certainty.
     The collection of examples shown here numbers 150 (40 + 79 + 31 for the three main 'real' metres), all from different movements. The total number of works by Schubert is about 1000, of movements about 2000 and of segments having in a given 'nominal' metre and tempo about 4000. I have by no means studied all these works, movements and segments from the present point of view. Of course, S1 and its close varieties will occur in only a certain proportion of them; a formal estimate is not so far available for that proportion, but an informal estimate might be: more than half of vocal music and less than half of instrumental music. On the whole, the collection seems quite convincing to me.
     One can now estimate the proportion of Schubert's music written in each possible number of bars per curve, on the basis of the admittedly incomplete evidence provided by the examples of S1 so far found. These estimates have most meaning when made separately for each 'real' metre (see Table 3a). The segments having one bar per curve constitute about half (19/40) of those in 'real' compound duple metre, four-fifths (62/79) of those in 'real' simple duple/quadruple metre, and all of those in 'real' simple triple metre.

The distribution by 'nominal' metre is shown in Table 3b. Of interest there, in respect of Schubert's usage, are the facts that all cases of 'nominal' 6/8 metre have 1 bar per curve (though ½ bar per curve was possible) and that all cases of 12/8 'nominal' metre have ½ bar per curve (though 1 bar per curve was possible). Whether further data will bear out those observations remains to be seen.

Introduction
     The main tasks of the present paper have been attempted in the above sections, and we now briefly seek possible connections with musical tempo. The tempo of a fingerprint must be specified within a certain range if the character of the fingerprint, and of the music of which it is a part, is to be appropriately understood. The theories presented so far—of conducting curves, metres and fingerprints—do not provide a sufficient basis on which to deduce the absolute tempo of a given segment, for the curve can be implemented convincingly at a range of possible speeds. ²³  In the definition of the fingerprint S1 given earlier, an appropriate tempo was therefore taken for granted and not precisely specified.
     Tempo is a property of several different musical situations:
(a) Scores may indicate the tempo verbally or with a metronome mark. Either of these may be indicated at the beginning of a movement or segment without implying unchanged implementation throughout. A verbal indication is generally only a hint which may be modified according to the denominator of the notated metre and the prevailing note-values; other features of the music may also play a role. ²⁴
(b) Performances can be timed objectively by their whole duration or by a metronome mark (again the latter may not apply unchanged throughout).
(c) Listening (or silent score reading) involves, in addition to objective tempo, also perceived tempo. For instance, a performance with much (appropriate) nuance may seem faster than a performance at the same objective tempo but with less nuance (thus blander); articulation can influence perceived tempo too. When assigning a tempo indication the composer might have had in mind a certain manner of performance which, however, we can never fully know.
     Several units of measurement are available for tempo. It is not a priori clear whether a verbal indication (Allegro, etc.) should be considered a rate (and character) for individual notes of the metre denominator, for individual notes of the piece (of a suitable duration for counting), for 'nominal' bars, for 'real' bars, for the general psychological effect, or for some combination of those possibilities. It is quite possible for a slow (relatively long-lasting) musical unit to be formed from many fast (relatively short-lasting) notes within it or, conversely, a fast unit formed from a few slower notes. A central question can then be formulated as follows: in fast music, what is it that is fast? ²⁵
     For practical purposes of counting in performance and for metronome marks, a certain note-value (4th etc.) is generally taken as the unit of tempo. For theoretical purposes (and in some cases possibly also for practical purposes) it might be preferable to take the bar as unit. Thus a 'note-value' metronome mark can be converted to a 'nominal bar' metronome mark or to the corresponding 'real bar' metronome mark; this latter is the number of conducting curves per minute.
     Just as with metre, it turns out that tempo needs to be investigated separately for each composer, as the usage and conventions cannot be assumed to be the same for all composers. Here, naturally, we restrict attention to Schubert's music. ²⁶
     A detailed study of the metronome marks in Schubert's scores would require a separate paper. They appear only in the 20 songs of op. 1 to op. 7, the opera Alfonso und Estrella D732 (& D759A, D773), Drang in die Ferne D770, and the German Mass D872. Unfortunately their reliability seems too uncertain for use in the present investigation. ²⁷  We therefore consider Schubert's verbal tempo indications.

Schubert's verbal tempo indications
     The internal evidence of the verbal tempo indications in Schubert's scores also turns out to be somewhat inconsistent. Many factors no doubt affected his choice of tempo indication on different occasions, and a desire for literal consistency across his whole output was not necessarily prominent among them. Perhaps some were put down rather informally or casually while others responded to contemporary performances with a correctional hint: Nicht zu langsam, Nicht zu geschwind, etc.; this would make their interpretation today problematical. Such factors limit one's expectations in attempting a systematic study of his tempos.
     A survey of Schubert's verbal tempo indications is given in Table 4. In assigning the large number and variety of his indications to a small number of categories the following somewhat arbitrary decisions have been made:
Included categories (i) Nicht zu langsam, Nicht zu geschwind etc. are included as moderate tempos. (ii) Etwas langsam, Etwas geschwind are included together with Langsam, Geschwind.
Excluded categories (i) The tempos of the various segments of a movement are not always written out explicitly, but are sometimes instead implied from an earlier similar segment. (ii) Indications of the character of the music may to some extent imply a tempo, a matter subject to the judgement of the investigator. (iii) The tempos of dance movements are often not indicated explicitly but instead depend upon conventions well known at the time but not now known with certainty. (iv) Relative, rather than absolute, tempo indications (such as geschwinder) are not specific.
     These and other uncertainties arise, but for the present purpose a rough survey is adequate. The Table counts tempo segments, of which there may be more than one in a given movement. All versions of a given movement are included. Just as with Table 1 of Schubert's 'nominal' metres, it is emphasized that the full significance of this Table cannot be known until the number of bars per curve is taken into account (see later).

Verbal tempos for the fingerprint examples
     We now consider the question whether Schubert's verbal tempo indications correspond to 'nominal' or to 'real' bars. (It is also quite conceivable that they correspond in some cases to one of these and in other cases to the other, if Schubert did not pay particular attention to this matter.) That is, do they have practical ('nominal bar') significance or underlying ('real bar') significance? ²⁸  In this connection we survey the verbal tempos indicated by Schubert for the fingerprint examples given in Appendix III, where the indications are easily read down the columns of examples.
     In 'real' compound duple metre at 1 bar per curve, most of the verbal tempos are average, avoiding the extremes (with the possible exception of D613/2 and D616, Adagio and D713, Allegro). At 2 bars per curve they tend to be quicker than average. At ½ bar per curve they are generally slower than average. These are tendencies, not rules, and the categories are partly overlapping, not exclusive; thus Allegretto appears with both 1 and 2 bars per curve.
     In 'real' simple quadruple metre at 1 bar per curve, a rather full range of verbal tempos appears, whether the 'nominal' metre is 4/4, 2/2 or 2/4. Those written in 4/4 might seem generally faster than those in 2/2, however, with the exception of D103, Grave. The range extends from Grave or Sehr langsam to Schnell. At 2 bars per curve all three examples are marked Allegro. At ½ bar per curve they clearly tend to be slower than average.
     In 'real' simple triple metre, most of the tempos are slow or medium, and no very fast tempos appear. In all of these examples there is 1 bar per curve and they are all in 'nominal' 3/4 metre except for D911/09 which is in 3/8 (Sehr langsam).
     The above observations are summarised in Table 5. In compound duple metre the slower tempos at ½ bar per curve, if they apply to 'nominal' bars, correspond to a more nearly average rate per 'real' bar, as do also the faster tempos at 2 bars per curve. The same relationship applies in simple duple/quadruple metre, while simple triple metre does not contribute to this comparison. The Table therefore strongly suggests that Schubert's verbal tempos are generally related to 'nominal' rather than to 'real' bars, that is, to notated bars rather than to complete curves, or to the music as seen on the page rather than as heard and felt. ²⁹  What is Allegro or Largo is, then, a 'nominal' bar rather than a 'real' bar. This is the main conclusion of the present work on Schubert's tempos.

Schubert's versions
     Further evidence on Schubert's tempo usage in relation to the metre may be sought in those cases where he rewrote some music with changed metre. ³⁰  A range of such cases can arise, beginning with (a) small portions where the music is somewhat varied; these occupy one extreme and include the fingerprint examples. Remaining for consideration, therefore, are only (b) the rewriting of a whole movement or segment where the music is unchanged except for the metre and possibly the tempo, at the other extreme, and (c) the appearance of a fairly extended section of very similar music borrowed from one work for reuse in another, lying between the extremes.
     Versions where the number of bars per curve is unchanged, though the 'nominal' metre is changed, are not of interest for the present purpose. ³¹  Where the number of bars per curve is changed the 'real' metre will usually be unchanged. Eight such cases are known; they are shown in Table 6, in which the first seven cases are of type (b), the last of type (c). ³²

In the first case listed, D082-1, Adagio became Allegro vivace, suggesting that the former tempo can be considered approximately half the latter, while the last two cases, D944-1 and D732/11 vs D911/19, again suggest tempo conservation. The Ecossaises have no tempo indication. In the remaining three cases the verbal tempo has not been greatly changed. Thus the strong tendency noted in Table 5 above is only partly reflected in this small sample. On the other hand, the notion of conducting curve and of the number of bars per curve is supported: in D713 v1 bar 47 the S1 gesture is placed at the beginning of a 12/8 bar, whereas in the parallel passage in bar 52 it is placed at the middle of the bar, strongly suggesting that while using 12/8 notation Schubert was thinking of 6/8 music (Figure 6). ³³

Figure 6. Shift of ½ bar in Schubert D713 v1 bars 47, 52.

     Just one case is known where not only the number of bars per curve but also the 'real' metre is changed: D929-2 bars 1-21 (2/4 Andante con moto) & D929-4 bars 275-315 and 693-721 (6/8 Allegro moderato). Here the verbal tempo change seems to compensate for the number of bars per curve (see Newbould 1992, 218n).

     Carefully chosen implementations of the Schubert curve in different 'real' metres were shown in Figure 3. Some other conceivable implementations are shown in Figures 7a-c.
     The mathematically exact ones, simply duplicating the durations and attack points, are shown in Figure 7a; they have no musical value, because they are not appropriate to the effect of the metrical hierarchy on the human mental processing—that is, they fail because they ignore the character of the metre. Some other candidates, which are convincing to varying degrees, are shown in Figures 7b and 7c. In both Figures the staff immediately below the top was chosen for Figure 3; the other staves are possibly less convincing in realising the Schubert curve (they might conceivably be more appropriate to another composer). The differences involved might at first seem slight, but they are important in the present approach.

Figure 7a. Ineffective (mathematical) implementations of the 6/8 staff in C and 3/4 metre.

Figure 7b. Some alternative implementations of the 6/8 staff in C metre.

Figure 7c. Some alternative implementations of the 6/8 staff in 3/4 metre.

     Figures 8a-f show, for given 'real' metres, a number of non-essentially different notations, including those of Table 2 together with some others. In all six Figures staff A is the most common case. In Figure 8a, B & D have ½ bar per curve; they are less likely but theoretically possible.

Figure 8a. Some non-essentially different notations: 'real' simple duple.

In Figure 8b, B & D have ½ bar per curve.

Figure 8b. Some non-essentially different notations: 'real' simple triple.

In Figure 8c, C has 2 bars per curve. E is seen only in the Impromptu in Gb D899/3. The distinction between simple duple and simple quadruple metre is debatable.

Figure 8c. Some non-essentially different notations: 'real' simple quadruple.

In Figure 8d, B, D & I have 2 bars per curve. D was used by Schubert in D780/6. C, G & H have ½ bar per curve. F is quite common in Schubert.

Figure 8d. Some non-essentially different notations: 'real' compound duple.

In Figure 8e, B & D have ½ bar per curve. C occurs fairly often in Schubert.

Figure 8e. Some non-essentially different notations: 'real' compound triple.

In Figure 8f, B, C & E have 2 bars per curve. D is quite common in Schubert.

Figure 8f. Some non-essentially different notations: 'real' compound quadruple.

     The organization of a large collection of examples for mutual comparison could be handled in many different ways. Here the examples are classified according to the following hierarchy, the highest level being mentioned first: (i) by 'real' metre (thus in three main collections: compound duple, simple duple/quadruple, and simple triple), (ii) by number of bars per curve, (iii) by metrical unit and thus 'nominal' metre (in the case of simple duple/quadruple), and (iv) in order of their Deutsch catalogue D number (which is approximately chronological). ³⁷
     Many but not all of Schubert's scores were studied so, although the selection is quite large, it is not complete. In some segments several examples of S1 occur, but here at most one is given for each segment; this makes clearer the scope of the evidence. Some examples clearly satisfy the definition of S1, others are clearly 'close varieties' as discussed earlier, while in still other cases some judgement is needed. It is important to avoid bias in the selection procedure, not so much in establishing the fingerprint itself, which apparently can readily be done, but rather in connection with the theory of tempo (section 4). Every effort has been made to avoid such bias, yet it is only natural that the decision to include or exclude a candidate could not always be entirely a matter of algorithmic routine. Some cases of exclusion when there was room for doubt will be mentioned.
     In the following Figures each example covers two curves and all the music is aligned vertically. The dashed vertical lines between the examples function as for the prototypes in Figure 5; the heavier line (indicating S1) and the lighter one (possibly not S1) may appear in either order, the printed weight of those lines not necessarily reflecting the musical 'weight' of the curves themselves. ³⁸

Figure 9a. Fingerprint examples: compound duple (1 bar per curve).

Figure 9b. Fingerprint examples: compound duple (½ bar per curve).

Figure 9c. Fingerprint examples: compound duple (2 bars per curve).

     'Real' simple duple/quadruple, one bar per curve (Figures 10a(i), 10a(ii), 10a(iii))
     This type has the largest number of examples. They are shown in three groups according to the 'nominal' metre (4/4, 2/2, 2/4) to facilitate comparison between the three notations.
     Figure 10a(i) shows 24 cases in 'nominal' 4/4 metre. Most are straightforward and do not require special mention here. In D030, D409, D792, D795/12, D898-1, D922, D935/1 the upbeat is slightly ornamented; although the final upbeat note thus arrives later than in our definition, the upbeat group seems to belong to the specified position, the last 4th-note of 4/4. In D956-1 the upbeat duration is shortened by a rest, but the pattern is otherwise unaffected. Some of these examples have elements of 'real' compound quadruple metre rather than, or in addition to, simple duple/quadruple; perhaps only D898-1 and D935/1 properly belong to the former category, on account of their triplets appearing in the melody. The close similarity may be noted between the examples from D030 and D409, though these have very different tempos.
     Excluded were D419 bars 17-18, 21-22 (Geschwind), D540 bars 12-13 (Unruhig, schnell), D795/12 bars 7-8 (Ziemlich geschwind), D911/4 bars 62-63 (Ziemlich schnell) because the bass moves at the resolution.

Figure 10a. Fingerprint examples: simple duple/quadruple (1 bar per curve) (i) 4/4 (first page...).

Figure 10a. Fingerprint examples: simple duple/quadruple (1 bar per curve) (i) 4/4 (...concluded).

Figure 10a. Fingerprint examples: simple duple/quadruple (1 bar per curve) (ii) 2/2 (first page...).

Figure 10a. Fingerprint examples: simple duple/quadruple (1 bar per curve) (ii) 2/2 (...concluded).

Figure 10a. Fingerprint examples: simple duple/quadruple (1 bar per curve) (iii) 2/4).

     Comparing Figures 10a(i), 10a(ii) and 10a(iii), no reason appears for considering the different 'nominal' metres to result in essentially different music; comparison of the tempos will be considered later.

Figure 10b. Fingerprint examples: simple duple/quadruple (½ bar per curve).

     'Real' simple duple/quadruple, two bars per curve (Figure 10c)
     Three examples are shown, two in 2/2 'nominal' metre and one in 2/4.
     Excluded were D371 bars 1-4 (Langsam) and D399 bars 29-32 (Traurig, sehr langsam) because in both cases the anticipation arrives later than specified and the bass moves by an octave at the resolution—these might have only one bar per curve—as well as D810/4 bars 409-412 (Presto) because it has no upbeat to the appoggiatura.

Figure 10c. Fingerprint examples: simple duple/quadruple (2 bars per curve).

     'Real' simple triple (Figure 11)
     This case occurs only with 1 bar per curve, of which 29 examples are shown where S1 occurs as the second curve and two examples where it occurs as the first curve. Most of these need no special comment. D664/2 has no upbeat, although from the following bars an 8th-note may be inferred. D732/27 represents a 'real' compound triple metre (here 9/8) after rhythmic assimilation; its upbeat thus has, despite appearances, the duration of a triplet 8th-note. Close similarity is noted between the examples from D504 and D911/16.
     Excluded, because their upbeat 16th-note is shorter than the specified 8th-note, were: D112 bars 3-4 (Andante sostenuto), D296 bars 14-15 (Langsam), D756 bars 13-14 (Mässig), D794 bars 93-94 (Sehr langsam), D872/8 bars 1-2 (Nicht zu langsam), D957/13 bars 27-28 (Sehr langsam). They could well constitute an instructive group related very closely to S1 but, especially in view of anticipated comparisons with other composers in future work, they seem best excluded here. Excluded because the bass moves at the resolution were D046-2 bars 19-20 (Andante con moto), D857/1 bars 7-8 (Mässige Bewegung), D911/5 bars 81-82 (Mässig), D911/23 bars 31-32 (Nicht zu langsam).

Figure 11. Fingerprint examples: simple triple (1 bar per curve) (first page...).

Figure 11. Fingerprint examples: simple triple (1 bar per curve) (...continued...).

Figure 11. Fingerprint examples: simple triple (1 bar per curve) (...concluded).

Curve: The shape (in space, pressure, speed and character) of the armchair-conducting response to music, taken from one upbeat to the next. The term is used here to refer either to the response itself or to a diagrammatic representation of it. (Synonyms: Becking curve, conducting curve, conducting shape.)
'Real' metre: Metre applying to one complete curve.
'Nominal' metre: Metre as notated in a score.
'Real' bar: An amount of music corresponding to one complete curve (generally 1, ½, or 2 'nominal' bars).
'Nominal' bar: A bar of music as notated in a score.
Essentially different metres: Different 'real' metres.
Non-essentially different metres: Different 'nominal' metres for music having the same 'real' metre.
Segment: A portion of music with fixed 'nominal' metre and tempo indication.
S1: A fingerprint defined in the text (S = Schubert, 1 = the first such fingerprint).

     Die Gestalt der musikalischen Dirigierbewegung, zwischen aufeinanderfolgenden Auftakten genommen, wird nach der Becking-Kurve diskutiert. Auf dieser Basis wird ein Unterschied zwischen 'reelle' und 'nominelle' Metrik gemacht, der die Wahl zwischen grundsätzlich äquivalenten metrischen Notationen reflektiert, in denen eine Partitur geschrieben sein mag. Es wird ein Muster mit vorbereitete Appoggiatura als Kennzeichen Schuberts vorgeschlagen; 150 Beispiele aus seinen Werke werden der Becking Kurve verglichen, was die Anzahl der 'nominellen' Takte per Kurve festlegen hilft.
     Als nächstes wird dann ein Unterschied zwischen 'reellem' und 'nominellem' Tempo auf Grund der Anzahl der 'nominellen' Takte per Kurve aufgestellt und es wird gezeigt, daß die in den Beispielen angegebenen in Worten Tempi Schuberts diesen Unterschied reflektieren, indem sie im allgemeinen eher durch 'nominelle' als durch 'reelle' Takte verwandt sind. Der Ansatz, der die Theorien von Kurve, Metrik und Tempo gleichzeitig umfaßt, wird also als durch die Beobachtungen unterstützt betrachtet, hier lediglich im Falle Schuberts. Weitere Arbeit wird angedeutet, unter anderem Vergleiche zwischen Komponisten, wo der Begriff des Kennzeichens dann auf eine breitere Basis gestellt werden kann.

     La forme de la direction de la musique, prise entre des levés successifs, est examinée selon la courbe de Becking. Sur cette base, une distinction est établie entre le mètre "réel" et le mètre "nominal", reflétant le choix entre des notations métriques essentiellement équivalentes selon lesquelles une partition peut être écrite. Un modèle d'appoggiatura préparé est proposé comme empreinte distinctive caractéristique de Schubert; 150 exemples dans ses oeuvres sont rapportés à la courbe de Becking. Ceci aide à déterminer le nombre de mesures "nominales" par courbe.
     Une distinction basée sur le nombre de mesures "nominales" par courbe est ensuite établie entre le tempo "réel" et le tempo "nominal". Il est démontré que les tempos verbaux de Schubert indiqués dans les exemples reflètent cette distinction, par le fait d'être relié en général à des mesures "nominales" plutôt que "réelles". L'approche comportant conjointement les théories de la courbe, du mètre et du tempo est ainsi considérée comme étant appuyée par les observations, ici uniquement dans le cas de Schubert. Un travail plus approfondi comprenant des comparaisons entre compositeurs s'annonce, ce qui permettra détablir plus largement la notion d'empreinte distinctive.