Nigel Nettheim

[This article was first published in 2000 in Systematische Musikwissenschaft Vol VI/4, 1998, pp. 363-413.
It may not be reproduced without permission.
A few minor corrections have been made here.]

Index (click on a section, or read straight through):
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

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(Summary in German or French)

     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.


     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)

Becking curve for Schubert
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

Becking curve for Schubert, Nettheim's version

     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.


     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.

A similar case is seen in M.J.E.Brown (1972, 59):

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:

Implementations of Schubert curve in the 3 main metres

Figure 4. Regions in Figure 2:

Regions in Figure 2, with letters in circles

     The top staff, labelled A, is taken from the beginning of his Moment musical in Ab D780/6 and shows, for the sake of clarity, just the melody. 12  The tempo can provisionally be taken as about two seconds per curve. This material is typical of Schubert and corresponds well with his curve shape, as will be discussed more fully later. When the music is heard or thought (together with the harmonizing notes in the score), the pressure of the curve can be felt internally and its conducting shape followed externally. Several distinguishable phases are indicated along the curve in Figure 4, and the matching of the curve to the score between Figures 3 and 4 is indicated by means of circled initial letters in each Figure. These phases are: the ‹P›reparatory phase, or upbeat (the initial 8th-note); the Tip (at the end of the upbeat note); the ‹M›ain phase, or downbeat (just after the bar line); the Follow-through (on the long second note); the Turnaround (the end of the follow-through and change of the direction of the conducting gesture a little before the centre of the bar); and the ‹R›eturn phase, beginning a little after the turnaround (the third note and the following rest). 13  The return phase has somewhat less strength than the main thrust. This matching of the score to the curve is only approximate, the labels in Figures 3 and 4 referring to regions rather than to precise points of application; in any case, fine detail of the matching is not needed for the present purpose. The approximate matching was possible in this case only because two conditions were satisfied: (i) the number of 'nominal' bars per curve was assumed known and (ii) the excerpt was both typical and simple; for many excerpts the matching would be less clear.
     Figure 3B shows my attempt to write similar music yielding the same conducting curve at the same tempo but in an essentially different metre. (The metre is shown as 4/4 but might as well, for the present purpose, have been shown as 2/2.) To make such an attempt the first requirement is a change of mental attitude to install the new metre with full conviction. One then tries to find suitable places in the score for the same distinguishable phases as were observed above. The upbeat note cannot occur at the same time as in Figure 3A, for the new metre of common time has no natural division at exactly that point. It has been placed a little earlier as a quarter-note (the staves in the Figure are accurately aligned vertically); it could conceivably have been placed a little later as an 8th-note, but Schubert's music (to be discussed later) indicates instead a preference for the quarter-note upbeat in this metre. The downbeat still occurs at the bar line. The slight change in upbeat duration might have caused a change in the duration of the follow-through, but in fact it has not done so: the change of direction still takes place just before the middle of the bar. A following rest seems not needed here, the slur perhaps indicating the true gesture more accurately than would a rest. As some notes had to be slightly shifted by comparison with the top staff, the implied curve is not quite unchanged, but has been maintained as nearly as could be managed and retains its character quite well. The Becking curve for a given composer should in any case be understood to represent not a single curve but a family of such curves having similar shape and character.
     Figure 3C shows my attempt to find a corresponding rendition in triple metre. Again the new metre must be installed mentally. The upbeat occurs at the same time as in the top staff, though having a different metrical significance. The down-beat again occurs just after the bar line. The downbeat note has been shortened by comparison with those in the other metres (a lengthening would not correspond to Schubert's typical writing, as will be seen later) and the follow-through continues into the second note of the bar. The turnaround could not occur naturally before the middle of the bar in this case; it occurs instead just before the rest and the return begins at about the beginning of the rest. Thus the second half of the conducting gesture in duple and quadruple metres is speeded up in triple metre (as was mentioned by Becking 1928, 39). The rest functions similarly to that in Figure 3A.
     Figure 3D shows twelve equal divisions of the bar. Those divisions are not assigned metrical significance in themselves, but provide reference points for the placement of the notes and rests in the different metres in the higher staves.
     We have now shown how a given curve can be represented in music having essentially different metres. The idea is that Schubert himself could conceivably have written such music with little or no violation to his personal curve. Taking the 6/8 version as a starting point, we showed in Figure 3, for the C and 3/4 versions, a particular implementation or transcription chosen from a variety of possible ones; it will later be seen that the ones chosen for that Figure are of the kind that Schubert often used, but some other (presumably less likely) candidates for such implementations are shown and discussed in Appendix I.
     We now briefly consider several further metres, whose behaviour remains to be confirmed later (as indeed does that of the metres discussed above). The metre 12/8 is normally used very much like 4/4 with triplets, for one does not easily look more closely into four beats per curve. Twelve beats is a large number in this context for, although they can no doubt be performed and listened to as constituting one curve, it is not so easy for each beat to be appreciated for its individual place in the metrical hierarchy. For the same reason 9/8 is very much like 3/4 with triplets. On the other hand, 6/8 is often not very much like 2/4 with triplets; the reason appears to be that six is a small enough number for humans to deal with in the context of metre (cf. Miller 1956). In the comparison between 2/4 and 4/4, it seems that the two divisions of 2/4 are often not enough to delineate a conducting shape of much sophistication, unless there is further metrical division into two or three parts, producing essentially 4/4 or 6/8 respectively. The 2/4 metre without further subdivision may however be suited to simpler music, such as folksong or Schubert's Ecossaises. Summing up, 6/8, C and 3/4 can be considered Schubert's main 'real' metres, with 2/4, 9/8 and 12/8 less commonly used as essentially different from one of those. 14

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

Table 1. Schubert's 'Nominal' Metres (All Works):

Nominal Metre Segments Works
No. %
2/2 648  15.60   
2/4 510  12.28   
3/2 14  .34   
3/4 1036  24.95   
3/8 59  1.42   
4/2 .02    D899/3
4/4 1427  34.36    Includes almost all recitative
6/4 .02    D015A
6/8 393  9.46   
9/8 .14    D684, D738, D770, D780/2, D896B, D957/3v1
12/8 43  1.04   
18/8 .02    D605A
2/2 & 6/8 .19    D929v1, v2
4/4 & 12/8 .02    D097
Unknown .14    Fragments: D072/1, D084/13, D453, D469/2
Total 4154  100.00   

     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.

Table 2. 'Nominal' Metres by Bars per Curve and 'Real' Metre (All Works):

Bars/curve 'Real' Metre 'Nominal' Metre
1 Simple duple 2/2 2/4 . . .
"     triple 3/2 3/4 3/8
"     quadruple 4/2 4/4 4/8
Compound duple . . . 6/4 6/8
"     triple . . . . . . 9/8
"     quadruple . . . . . . 12/8
½ Simple duple 4/2 4/4 4/8
"     triple . . . 6/4 6/8
"     quadruple    
Compound duple . . . . . . 12/8
"     triple . . . . . . 18/8
"     quadruple    
2 Simple duple    
"     triple — (hemiola)
"     quadruple 2/2 2/4 . . .
Compound duple 3/2 3/4 3/8
"     triple    
"     quadruple . . . 6/4 6/8

     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).

Prototypes for the fingerprint S1, in 3 metres

     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).

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.

Table 3a. Bars per Curve by 'Real' Metre (Fingerprint Examples).

'Real' Metre    Bars per Curve Total
Compound Duple 10 19 11 40
Simple Duple/Quadruple 14 62 3 79
Simple Triple 31 31
Total 24 112 14 150

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.

Table 3b. Bars per Curve by 'Nominal' Metre (Fingerprint Examples).

'Nominal' Metre    Bars per Curve Total
2/2 4 22 2 28
2/4 2 16 1 19
3/4 31 9 40
4/4 13 24 37
3/8 1 2 3
6/8 18 18
12/8 5 5
Total 24 112 14 150


     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).

Table 4. Schubert's 'Nominal' Verbal Tempo Indications (All Works).

Tempo Indication Category Segments
No.  %
Sehr langsam, etc. 75 2.9
Langsam, Adagio, etc. 505 19.4
Andante, Andantino, Mässig, Moderato, etc. 808 31.0
Allegretto, etc. 176 6.8
Allegro, Geschwind, Schnell, etc. 952 36.6
Allegro molto, Presto, etc. 86 3.3
Total 2602 100.0
Other (including no indication) 1560  

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.

Table 5. Schubert's Verbal Tempo Markings by 'Real' Metre and Bars per Curve (Fingerprint Examples).

Bars per Curve 'Real' Metre
½ Slower Slower
1 Average Full range Slow to medium
2 Faster Faster

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

Table 6. Schubert's Versions with Changed Number of Bars per Curve but Unchanged 'Real' Metre.

Title Deutsch
Bars per
Symphony in D D082-1 1-20
Allegro vivace
Ecossaise (No.8/9) D145/37 v1

Nähe des Geliebten D162 v1
Sehr langsam
Frühlingsgesang D709
Etwas lebhaft
Etwas geschwind
Der Unglückliche D713 v1
Ecossaise (No.6/8) D735/7

Symphony in C D944-1 1-16
[Allegro ...] Piu moto
Alfonso und Estrella
Andante con moto
Etwas geschwind

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.

Half-bar shift in D713

     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).


     The theory of the conducting curve and of metre presented here has enabled the establishing of a carefully defined fingerprint by systematic comparison, while at the same time the comparison of numerous examples has corroborated the theory. The two approaches— theory and practice—thus support one another. The theory and practice of tempo yielded some further, more qualified, support for this approach. 34
     The 'essential nature' of an individual piece or movement (or, more colloquially, 'how it goes') has often been determined on the basis of the external evidence of favourite performers, possibly unreliable metronome marks, historical anecdotes, or personal opinions. Here, however, it is taken here to be the resultant of three elements: the 'real' metre, the approximate tempo, and the conducting curve behaviour, and is studied by reference to the internal evidence of the notes in the scores. The present work can therefore be regarded as not merely of theoretical significance but as bearing directly upon the practical tasks of musical activity, whether in listening, performing, or silent score-reading.
     The following directions are anticipated for future research:
     (i) For a complete study of Schubert's music it is necessary to determine the 'real' metre for every movement or segment of all his works, in so far as it can be determined with sufficient certainty. That is clearly a large task, and in some cases the correct determination will probably remain uncertain; the present paper has provided only a first attempt with still incomplete coverage. This task can be divided into three parts. (a) A more complete survey of examples of S1 is to be compiled. (b) Further fingerprints (S2,...) are to be defined, for S1 does not appear in all the segments by any means. 35  Note that when one fingerprint has been found in a given segment, the 'real' metre is thereby known for the remainder of that segment, facilitating the search for different fingerprints in that segment. (c) Other means of determining the number of bars per curve, mentioned at the end of Section 1, are to be explored more fully and coordinated with the fingerprints.
     (ii) More composers are to be included, compared and distinguished—this may be seen as an ultimate task of analytical musicology, bearing on a question often informally raised: how do we recognize a newly heard piece as being the work of a certain composer, as when we say, for instance, "That must be Mozart!"? 36
     (iii) Implications for performance are to be developed, taking into account (i) and (ii) above. Other writers have approached composer-specific performance nuances directly, whereas it might be preferable to build up to an attempt at that difficult task with supporting theory and evidence such as that presented in the present paper.



Some alternative implementations of the Schubert curve in different 'real' metres

     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.

Implementations of Schubert curve in the 3 main metres

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

Implementations of Schubert curve in the 3 main metres

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

Implementations of Schubert curve in the 3 main metres



Some non-essentially different notations for given 'real' metres

     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.

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.

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.

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.

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.

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.

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



Fingerprint examples in score, with discussion

     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

     'Real' compound duple, one bar per curve (Figure 9a)
     We begin with 19 examples of compound duple 'real' metre having one bar per curve. Fully satisfying our definition of S1 are D051, D537/1, D613/2, D713; these have an anticipation of the appoggiatura (on the same pitch). Close varieties, approaching the appoggiatura from a nearby pitch (by step or small skip), are D121, D157/2, D795/3, D872/7, D877/4. The resolution occurs in the accompaniment as the appoggiatura is sung in D118 and D515. D084/21 has a light anticipation of the resolution in the accompaniment; the triplets in the voice part indicate compound duple time for S1—this is the only one of these cases where the 'nominal' metre is not 6/8. Further close varieties are D081, D616, D911/13, D911/19, D957/10 having a graceful anticipation of the resolution note in the manner of portamento singing, which even seems to enhance the S1 effect. D795/9 has an upper dominant note in the melody which does not disturb the S1 effect. In D795/10 the inner part (top note of the left hand) and the melody combine to produce the S1 effect. Some of the above examples have several of the features mentioned here.
     Excluded were four examples ('near misses', not printed here). In D892/2 bars 74-75 (Andante un poco mosso), the bass moves on resolution. In D946/2 (Allegretto) the left-hand activity interferes slightly with the smooth effect of a strict S1, reflecting a possible influence of Beethoven in this work. In D107 bars 6-7 (Allegretto) and D911/11 bars 1-2 (Etwas bewegt) the resolution arrives slightly early.

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

Fingerprint examples: compound duple (1 bar per curve). (741 x 1188 pixels)
'Real' compound duple, half bar per curve (Figure 9b)
     Ten examples of this type are shown. Different 'nominal' metres appear: 2/2, 2/4 and 4/4, all with triplet notation, and 12/8, in which the triplets are incorporated into the time signature. There seems to be no essential difference between the effects of these notations.
     All ten examples seem straightforward cases of S1. In D191, D222, D238, D418 the appoggiatura is anticipated, while in D100, D516, D541, D894-1, D908, D911/17 it is approached from a nearby pitch. In D191 and D541 the dotted notation evidently requires assimilation to the prevailing triplets (but little is lost for present purposes for any readers who do not favour assimilation). In D911/17 the appoggiatura is written as a grace-note and the S1 gesture arrives at the beginning rather than at the centre of the notated bar (on the metrical shifting in this song see Feil 1986); this bar location appears also in D541.
     Excluded was D894-1 bar 64 (Molto moderato e cantabile) because the bass moves on resolution.

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

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

     'Real' compound duple, two bars per curve (Figure 9c)
     Ten cases of this type are shown where S1 occurs as the second curve, and one case (D783/7) where it occurs as the first curve. The 'nominal' metres are 3/4 and 3/8, no essential difference appearing between these notations. In all cases the dissonance is either anticipated or approached from a nearby pitch. Slightly ornamented resolutions occur in D007, D180, D625/2, D783/7.
     Excluded was D782-5 bars 1-4 (Sehr langsam) where the bass moves strongly at the resolution; this seems to be not a case of S1, though some features happen to be shared with it.

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

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...).

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

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

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

     Figure 10a(ii) shows 22 examples in 'nominal' 2/2 metre. Most have no new features beyond those mentioned above. In D106 the resolution takes place upward in the voice part but downward in the middle staff (the bassoon part). In D285 the S1 melodic pattern appears in the right hand of the piano part in its middle voice.

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

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).

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

     Figure 10a(iii) shows 16 examples in 'nominal' 2/4 metre. 'Real' compound quadruple metre is seen in D252 and D301. No other special features arise.
     Excluded cases were: D911/11 bars 29-30 (Langsam) because the bass moves; D414 bars 13-14 (Ruhig, zart), D537/2 bars 3-4 (Allegretto quasi Andantino), D795/6 bars 3-4 (Langsam) and D795/17 bars 9-10 (Ziemlich geschwind) because the upbeat arrives later than specified.

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

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.

     'Real' simple duple/quadruple, half bar per curve (Figure 10b)
     Fourteen examples are shown, all in 'nominal' 4/4 metre except for D760/2 and D935-3 in 2/2. Again no new features arise. Close similarity is noted between D760/2, D768 and D911/21.
     Excluded was D289 bar 21 (Langsam, feierlich) where the dissonance and resolution take place in the left hand of the piano part while the upbeat character occurs in the voice part; the combined effect approximates S1 but does not satisfy our definition.

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

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).

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...).

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

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

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

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

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


Summary of terminology

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).



     1 Shove and Repp (1995) also presented a synopsis of Becking (1928) but from a different point of view, that of experimental psychology.
     2 Traversing a Becking curve is somewhat analogous to riding a fair-ground roller-coaster. To understand the curve more fully may require reference to one of the synopses mentioned above or to the extensive discussion in the original, Becking (1928).
     3 The terms 'pulse' or 'beat' have generally been used to refer to a smaller time scale, as in the undifferentiated ticking of a watch or the four beats of common time, respectively (e.g. Cooper and Meyer 1960, 3-4, Randel 1986, 489). Although Becking often used the term 'beat' or 'beating' (Schlag, schlagen) in reference to the conducting of the whole shape, it could then be misunderstood to refer to just a part of it. Terms such as 'gesture' lack the connotation of repeated similar application. The term 'sweep' could be considered, meaning, as a noun, one complete traversal of a conducting curve and, as a verb, to carry out such a traversal, but it might falsely suggest motion of a particular character.
     4 The notion of the number of bars per curve has appeared in the literature with various terminologies. The case of ½ bar per curve has been referred to as 'zusammengesetzten' or 'joined' metres (in literature cited by C.Brown 1998, 4, who did not discuss the possibility of 2 bars per curve, presumably 'divided' metres). The determination of the number of bars per curve, and its importance for enlightened performance, was discussed by Lussy (1909, 70ff), who mentioned the case where "a mental process is necessary to halve, as it were, each bar; every bar of four beats becoming two bars of two beats" (ibid., 71). Lussy also gave specific examples from Mendelssohn and Adam in score (ibid., 24-25). Another example is seen in Georgiades (1986, 87) concerning Schubert's Wandrers Nachtlied D768: "it is not the whole but rather the half bar that represents the basic unit of the composition".
     5 An example of a broadened cadence ending a section occurs in Memnon D541, where the cadence in bar 29 has the pattern of bars 20 and 26 in every respect except for its broadened rhythm (it will later be confirmed that the number of bars per curve for that section is ½).
     6 Thus in Die Forelle D550, bar 2, the second syllable of 'hel-le' can hardly occupy the strongest part of a curve, to that extent ruling out the possibility of ½ bar per curve.
     7 See later Figure 6 and its discussion.
     8 Other organization occurs at higher levels, as in the grouping of curves into phrases, periods, etc. In that case the appropriate unit is the 'real' bar which, as will be seen, is not always equivalent to the 'nominal' bar. Strictly speaking, one should therefore preferably refer not to a four-bar phrase, but to a four-curve phrase, or a phrase of four 'real' bars.
     9 This notion is supported by many if not all musicians, e.g. Dubal (1991, 84):
Menuhin: ... once you ... realize that [music is] a living substance like a river flowing —and that it can be guided within limits, navigated, it is not flowing at a metronomic pace—then it suddenly comes to life.
Dubal: This is not in the notation...
Menuhin: No, but it must be deduced from the notation, no matter how limited.
Becking (1928, 11) expressed it in terms of a continuous flow over pillars. Expressing it in almost mathematical terms, the musical score provides a discrete approximation to a continuous curve; the conducting curve has, however, not only spacial form but also varying intensity and a certain character.
     10 Whether 4/4 with triplets and 12/8 are essentially equivalent may sometimes be debatable, but when they are used simultaneously they are clearly equivalent. This is seen in Trost an Elisa D097 bars 19-20, Gruppe aus dem Tartarus D583 bars 1-20, and the Four-hand Variations D624 bars 151-165. An even less usual case occurs in the Piano Trio D929/4 bars 337-357, and compare the first version: 2/2 and 6/8 used simultaneously.
     11 "The recurrence of groups of four pulses, as in 4/4, may be termed quadruple meter but is also a special case of duple meter." (Randel 1986, 489).
     12 For the present purpose the metre is shown as 6/8, though the original is notated in pairs of bars of 3/4.
     13 The terms 'upbeat' and 'downbeat' are not strictly accurate here, for in the case of Schubert and some other composers the beats as rendered in the 'armchair conducting' of the Becking curves have a horizontal component rather than the real-world conductor's vertical component. The conventional terms will be used here nevertheless.
     14 The conventional metrical notation has an inconsistency (at least in respect of Schubert's music): whereas in 6/8 the upbeat note has the duration of the metrical denominator, in 3/4 it does not. That is, the role of the final 8th-note of the bar is recognized in the conventional metrical notation 6/8 but not in 3/4. This presumably evolved in the interests of indicating the essential difference between the two metres as simply as possible for the score-reader, at the expense of strict logic. The two metres could, for instance, have been named 6A/8 and 6B/8.
     15 For that purpose a complete tabulation of Schubert's 'real' metres would be needed, but that is not yet available.
     16 Here and throughout, 'systematic' is used in the sense of' 'thorough and orderly' rather than 'pertaining to a quasi-organic system', but the latter sense also plays a role in this work via the Becking curve.
     17 Although each excerpt shows just a few isolated bars, one should ideally be aware of the whole musical situation from which it is taken.
     18 The study of the relationship between the successive curves in a movement belongs to an area of higher musical forms than is dealt with here.
     19 Possible implications for performance lie beyond the scope of the present paper (see Section 5).
     20 But see Nettheim (1993a, 1998) for indications of some ways in which automation can assist, given sufficient data. The benefits of explicit definitions are likely to increase when comparisons between composers are later considered (see Section 5). The present method of searching for the fingerprint differs from that of Cope (1991), whose purpose was also rather different.
     21 Thus a preparation, appoggiatura and resolution with proportions other than defined above will typically be found only in other composers' works; the conducting curve is then qualitatively different.
     22 This incidentally provides some confirmation of Becking's notion that each composer has an inborn attitude which operates throughout his life. The present writer obtained a familiarity with Schubert's little-known earliest works when preparing his theses (Nettheim 1990, 1999?).
     23 It would be desirable to estimate the average and range of the tolerable number of conducting curves per minute for a given composer; such estimates are presently unavailable.
     24 Cf. Leopold Mozart (1756, 33): " It is true that at the beginning of every piece special words are written which are designed to characterize it, such as 'Allegro' (merry), 'Adagio' (slow) and so on. But both slow and quick have their degrees... So one has to deduce it [the tempo] from the piece itself... Every melodious piece has at least one phrase from which one can recognize quite surely what sort of speed the piece demands".
     25 Radcliffe (1976, 104) compares an example from Mendelssohn's 'Reformation' Symphony with one from Haydn's Symphony No. 104. A four-bar phrase in 2/2 Allegro con fuoco in the former corresponds to a one-bar figure in 4/4 Adagio with notes of one quarter the length in the latter. Radcliffe comments: "quite apart from this resemblance, so much of Mendelssohn's movement, in spite of its indication of tempo, seems really to be slow music in disguise, and the rapid quaver passages achieve fussiness rather than real energy".
     26 Systematic attempts for other composers include Kolisch (1943), Rothschild (1961), Zaslaw (1974) and Marty (1988).
     27 The present writer has undertaken a fairly close unpublished study of the internal evidence, which suggests a definite lack of consistency in the metronome marks. The external evidence deals with the question to what extent Schubert himself was responsible for the marks, by what means the numerical values may have been assigned, and the reliability of their transference into publication. Also to be taken into account is the virtual unplayability of some of them. At that early time in the metronome's history the reliability of its physical functioning, including the operation of its winding mechanism, can presumably not be known.
     A recent study by C.Brown (1998) seeks to establish the reliability of the marks. The internal evidence considered there was restricted to the Opera D732 on the ground that it is the only case with Italian verbal indications, a slender ground in my opinion. Brown's external evidence that Schubert himself was responsible for the marks in op. 1 to op. 7 is based on the observation that he was "directly involved in the process of publication" (ibid., 3), hardly sufficient proof. A method of assigning values is supported only by the claim that it "seems probable" (ibid., 3); Brown's formulation mirrors that of Stadlen (1993). The transference of the values to publication included a clear error in Breitkopf & Härtel's and Bärenreiter's D732/8 correctly reported by Brown (ibid., 8), as well as an apparent error in those editions at the beginning of the Overture to D732 (8th = 92 instead of dotted 8th = 92) perhaps not so far reported; these particular cases reduce one's confidence in the marks in general. The remaining marks in the Overture seem to be good examples of virtually unplayable ones, therefore requiring explanation. Altogether, it may fairly be concluded that the question is not yet resolved.
     28 The question of absolute tempos is a separate one specially related to performance and is not investigated here.
     29 The 'near misses' excluded from the examples in Appendix III, whose tempos are shown in the discussion of those examples, do not seem to contradict this conclusion.
     30 Cases where only the tempo indication is changed, as for instance the five versions of Die Forelle D550, will not be surveyed here.
     31 A list of known cases is available from the author.
     32 Readers' help in finding any more such cases will be welcome. In another possible case, the Variations D908, the 'nominal' metre and probably the number of bars per curve is changed in variation 7, but the music is also considerably varied.
     33 The same phenomenon is seen in Der Blinde Knabe D833 bars 31 & 36, there comparing 'nominal' 4/4 with 'real' 2/4 metre. Examples in Chopin include his Ballade Op.52 (see Nettheim 1993b, 96-7) and Nocturne Op.62/1.
     34 The collection of examples could of course be considered instructive even in the absence of the present theory; consideration of the examples might by itself lead to a better understanding of Schubert's music.
     35 A different Schubert fingerprint was discussed by Misch (1962). He defined a Lieblingsmotiv (favourite motif) with some care and found about 20 examples, most in songs and most having 1 bar per curve.
     36 If, for instance, it turns out that Schubert's prepared appoggiaturas are typically longer than Mozart's, this might lend explicit support to the view that there is, literally and possibly also figuratively, an element of "long"-ing in Schubert's musical character that does not belong to Mozart's. The analytical method for distinguishing Schubert from Mozart involves demonstrating that fingerprints S1, S2... are typical of Schubert but not of Mozart, while M1, M2... are typical of Mozart but not of Schubert. When more composers are included, the number of required demonstrations increases quadratically, and fingerprints defined earlier might need to be partly redefined so that the scheme can work as a whole.
     37 If instead we combined all the examples into a single list ordered by D number, the important points of the various curves would not quite line up vertically, especially in the case of duple versus triple metre. Then it would not be so easy to compare the examples and to consider the evidence for the theories presented in this paper.
     In the present classification the 'real' metre and the number of bars per curve are determined according to the behaviour of the fingerprint pattern in the music. Yet the existence of the large collection of examples so classified is needed to confirm the hypothesis of the fingerprint itself. This is not circular reasoning, however, for the examples, as collected and classified, provide the evidence for the whole theory; if the collection of examples were not convincing, one would need to reconsider the theory from the beginning.
     38 Vertical alignment was achieved with the aid of the SCORE commercial music printing program; in cases where the tempos of the examples differ, the horizontal axis shows notational rather than chronometric time. To save space, excerpts are identified only by their D numbers without titles. The number of the first complete notated bar is given at the left side of each example. Because of the necessarily small size of reproduction of each example, open note-heads might sometimes appear closed.


Becking, Gustav
1928 Der Musikalische Rhythmus als Erkenntisquelle. (Musical rhythm as a source of insight.) Augsberg: Benno Filser.
Brown, Clive
1998 "Schubert's tempo conventions". In: Schubert Studies, ed. Brian Newbould, Aldershot: Ashgate (1-15).
Brown, Maurice J.E.
1972 Chopin. An Index of his Works in Chronological Order. Second, Revised Edition. New York: Da Capo. (First edition 1960.)
Cooper, Grosvenor and Leonard B. Meyer
1960 The Rhythmic Structure of Music. Chicago: University of Chicago Press.
Cope, David
1991 Computers and Musical Style. Madison: A-R Editions.
Dubal, David.
1991 Conversations with Menuhin. London: Heinemann.
Feil, Arnold
1986 "Two Analyses". In: Schubert—Critical and Analytical Studies, edited by Walter Frisch. Nebraska: University of Nebraska Press. (104-125.)
Georgiades, Thrasybulos
1986 "Lyric as Musical Structure: Schubert's Wandrers Nachtlied ('Über allen Gipfeln.', D768)". In: Schubert—Critical and Analytical Studies, edited by Walter Frisch. Lincoln: Univ. of Nebraska Press. (84-103.) Translated by Marie Louise Göllner.
Goetschius, Percy
1904 Lessons in Music Form. Philadelphia: Oliver Ditson. (Reprinted Westport, Connecticut: Greenwood Press, 1970.)
Hedley, Arthur (translator and editor)
1962 Selected Correspondence of Frederyk Chopin. London: Heinemann. (Original: Korespondencja Frederyka Chopina, Warsaw: Panstwowy Instytut Wydawniczy, 1955.)
Kolisch, Rudolf
1943 "Tempo and Character in Beethoven's Music". Musical Quarterly 29 (1943): 169-187, 291-312. Translated by Arthur Mendel.
Lussy, Mathis
1909 A short treatise on rhythm. Translated by Ernest Fowles. London: The Vincent Music Company. (Original: Le Rhythme Musical, Place? Publisher? Date?)
Marty, Jean-Pierre
1988 The Tempo Indications of Mozart. New Haven: Yale University Press.
Miller, George A.
1956 "The magical number seven, plus or minus two: some limits on our capacity for processing information." The Psychological Review 63: 81-97.
Misch, Ludwig
1962 "Ein Lieblingsmotiv Schuberts". (A favourite motif of Schubert.) Die Musikforschung 15: 146-152.
Mozart, Leopold
1756 Versuch einer gründlichen Violinschule. Augsburg. English translation by Editha Knocker, London, O.U.P., 1948.
Nettheim, Nigel [Some of these items may be read on the author's web site.]
1990 Schubert's Earliest Works: an Analytical Study. M.Litt thesis, UNE.
1993a "The pulse in German folksong: a statistical investigation". Musikometrika, 5: 69-89. [Proof corrections were not implemented by the editor; errata are available from the author.]
1993b "The Derivation of Chopin's Fourth Ballade from Bach and Beethoven". The Music Review, 54/2: 95-111 (published May 1996).
1996 "How Musical Rhythm Reveals Human Attitudes: Gustav Becking's Theory". International Review of the Aesthetics and Sociology of Music, 27: 101-122. [Erratum: 107 line -4: Schütz -> Schulz.]
1998 "Melodic Pattern Detection using MuSearch in Schubert's Die Schöne Müllerin". In: Melodic Similarity, edited by Walter B. Hewlett and Eleanor Selfridge-Field. Cambridge: MIT Press.
1999? Schubert's Early Progress: On the Internal Evidence of his Compositions up to Gretchen am Spinnrade. PhD Thesis, UNSW (in progress).
Newbould, Brian.
1992 Schubert and the Symphony. Surrey, UK: Toccata Press.
Radcliffe, Philip
1976 Mendelssohn. London: Dent. (Original edition 1954).
Randel, Don, editor
1986 The New Harvard Dictionary of Music. Cambridge, Massachusetts: Harvard University Press.
Rothschild, Fritz
1961 Musical Performance in the Times of Mozart and Beethoven. London: Adam and Charles Black.
Shove, Patrick and Bruno H. Repp
1995 "Musical motion and performance: theoretical and empirical perspectives". In: The Practice of Performance: Studies in Musical Interpretation, edited by John Rink. Cambridge: Cambridge University Press.
Stadlen, Peter
1993 "Können die Metronomisierungen in Schuberts Werken als authentisch gelten? Weiterer Bemerkungen". (Are the metronome marks in Schubert's works authentic? Further remarks) Schubert durch die Brille 10:93-94. (English version in SIUK Newsletter, no.5, 4-5.)
Zaslaw, Neal.
1974 "Mozart's Tempo Conventions". In: International Musicological Society: Report of the Eleventh Congress, Copenhagen 1972. Copenhagen: Wilhelm Hansen (720-733).


Ein Kennzeichen Schuberts,
mit Bezug auf die Theorie von Metrik, Tempo, und der Becking-Kurve.

     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.


Une empreinte distinctive de Schubert
en relation avec la théorie du mètre, du tempo et de la courbe de Becking.

     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.

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