[This article first appeared in Musikometrika, Vol. 5, 1993, pp. 69-89.
It may not be reproduced without permission.
Proof corrections were not implemented in the printed publication but are implemented here.
This is an early attempt which contains some elements of my later papers on composers' "fingerprints".
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The Pulse in German Folksong: A Statistical Investigation

Nigel Nettheim

Abstract The behaviour of music in relation to the metre has been thought to be related to a pulse-like feeling. This behaviour has previously been studied in respect of performance nuances, but is here studied instead in respect of musical scores. The data consists of a large collection of German folksongs. In this first study the metre is represented by the bar-line, and the musical notes by the direction of progression of their pitch and duration. Consistent characterizations are obtained, and a subclass is identified. Typical and atypical songs are extracted from the collection on the basis of their pulse.

1. Introduction

I will begin by proposing a definition:

Definition. The pulse in a given musical repertoire is the relationship between the notes and the metre.

This definition is sufficiently general to allow for the study of the pulse independently of any one theory of its nature and significance. One such theory, however, provides a convenient motivation for the definition and will now be briefly discussed.
The term 'pulse' was used by Clynes to refer to the character of music's conducting beat. On this view, the character of the beat is to be distinguished from that of the emotion expressed in the music. Thus joy can be expressed in a Beethovenian or a Mozartean manner, and it is then that manner which reflects the pulse. When these composers on another occasion express a different emotion, the underlying pulses remain unchanged. Thus the pulse normally operates in an essentially constant manner throughout a piece or repertoire, but differs between composers or repertoires. In its application to the works of a given composer it may reflect his individual personality; in its application to a given repertoire of folksongs, for example, it may reflect a general characteristic of that repertoire. Its character can be expressed in words such as "restrained" or "light", but a verbal description is never fully satisfactory. The pulse idea was perhaps first formulated by Becking (1928), who described with diagrams the associated gestures. It was studied in respect of musical performance by Clynes (1983);1 in the present paper it will be studied instead in respect of musical composition, that is, in its manifestation in musical scores, a task requiring entirely different methods and not requiring acceptance of Clynes' theory.
Returning now to my definition, the qualification "in a given musical repertoire" is included as the phenomenon is considered to appear not alone in any one bar of music but over a large enough number of bars adequately to represent the given pulse.2 The term "notes" covers all aspects of the music other than the metre: not only pitch and duration but also expression marks in a score or performance. Reference to "the metre" assumes a fixed metre, as free metre and mixed metre would require a different model.
The instructional value will be greatest when different pulses are compared but in the present paper just one repertoire, though containing sub-types, will be studied. It is intended in the future to conduct similar studies of other repertoires, according to the availability of suitable computer-readable data.

2. Some Earlier Statistical Studies

Before embarking on my own study, a brief survey of others may be in order.
Suchoff (1970) examined interval size, ignoring repeats, in a database of 1,306 Rumanian and Serbo-Croatian folksongs, after reducing the data to eight-note incipits. The intervals were related neither to note durations nor to the metre, but simply to the ordinal position in the incipit. As the study showed, the first melodic intervals tend to ascend. It was also shown that the Serbo-Croatian songs have a larger proportion of wide intervals (skips rather than steps). Further, on the basis of the comparison of the songs known to be old with the more modern ones the author suggested a concept of a "primitiveness indicator". Suchoff's was one of the earliest studies of its type, and was related to the still earlier work of the composer Bartok.
Pont (1990) studied interval direction (but not size) in three-note incipits, carried out in 10,859 Western pieces of the last few centuries, as well as in several much smaller repertoires. The first two intervals jointly tend to ascend in European repertoires, but the non-European repertoires instead preferred repetition.
Steinbeck (1976) analysed a database of two thousand German folksongs collected by Boehme (one of those used in the present paper), using cluster analysis to find similar musical fragments; he took into account pitch, duration and metre, the latter by adjusting bars of different metres to reduce them to conformity. His aim was to assist cataloguing.
Vos and Troost (1989) studied interval size and ordinal position in 469 incipits of classical pieces and 327 folksongs. This study revealed that large intervals tend to ascend and small ones to descend; that the rising fourth is a common beginning; and that the perfect fifth tends to ascend in folksong but to descend in classical pieces.
Detlovs (1991) took 33,800 intervals from Latvian folksongs and 32,000 from Western compositions, relating intervals to scale degrees. He confirmed the melodic tendencies of the scale degrees as embodied in conventional music theory, for instance the tendency of the seventh degree to rise by a semitone and the fourth and (presumably flat) sixth to fall. He also formulated other broader relationships from his data. including a comparison of earlier with more recent music from this point of view.3
Boroda (1991b) presented a detailed comparative analysis of metrical-rhythmical patterns ("rhythmic models"); these were derived from melodic motifs determined algorithmically. Folksongs of ten nationalities were sampled, yielding 5,388 motifs. Note durations and metrical positions were studied, but not pitch. Similarities were revealed in the rhythmic organization of folksongs within a given nationality in terms of common rhythmic models, together with significant differences between nationalities in terms of the frequency of occurrence of the models. It was shown that, while the set of all rhythmic patterns in a given region is usually very large, these can be considered as variations of a very small group of "basic rhythmic models". The study also drew attention to the relationship between such patterns in music and in spoken language. See also Boroda (1991a).

3. Data

A large computer database of German folksong has generously been placed in the public domain by the University of Essen through the courtesy of Professor Dr. Helmut Schaffrath. The data is provided in the form of melodic lines in the local ESAC code. Various items of commentary have been included, but not yet the songs' texts. Each song has been divided by the encoders into phrases, a valuable feature of the database even though there can of course be room for different opinions on the appropriate points of division. The data is grouped into eight collections; there is some overlap, but not a great deal, between and within collections.4 As far as one can judge by analytical checking, the accuracy of the data entry was quite high.5 Summary statistics are given in Table 1 where it is seen that there are over five thousand songs having on average, to the nearest integer, six phrases per song, two bars per phrase, and four notes per bar.

Table 1. Essen Database of German Folksong.

No. Collectiona Songsb Phrasesc Barsd Notese
1. Erk-Boehme 1,708  9,797 21,577  79,040
2. Boehme   703  4,826 10,378  40,288
3. Zuccalmaglio   673  3,984  9,255  35,166
4. Balladen   919  3,823  9,066  34,383
5. Fink   561  4,018  8,728  32,882
6. Altdeutscher   451  2,429  6,914  20,822
7. DeutscherVA   153    838  2,121   7,704
8. Kinderlieder   208  1,163  2,422   8,135
  Total 5,376 30,878 70,461 258,420

a Full source details are available from the author.
b The following 600 database songs have been excluded:
(A) the 181 songs in free metre (mostly ballads), for in these the pulse cannot be studied;
(B) the 418 songs with mixed metres, these requiring a different model;
(C) the two songs containing sprechgesang (indeterminate pitch), one of which also has mixed metres.
c As determined by the Essen encoders.
d Bars are counted by down-beats, rests at the beginning of a song having been removed.
e Tied notes and successive rests have been combined, and rests attached to a preceding note, but not across bars or phrases.

4. Analytical Approach

A beginning study of the pulse is all that can be attempted here, as the data consists only of the melodic pitches and durations and the metre, together with the phrase divisions. Relevant factors which are not available include the song texts,6 dynamic indications, implied harmony, the distinction between chord and non-chord tones, and any accompaniment which may belong to the music; also possibly relevant are the tempo and the dynamic level.
As the pulse is thought to characterise a given repertoire it may be likened to a fingerprint. It is not clear in advance whether this should be looked for in the common patterns of the music, in occasional significant departures from these, or in both ways. However that may be, the appropriate first step appears to be the determination of the most common patterns.
It should be mentioned that the author is not specially familiar with the present repertoire; whereas this would normally be a disadvantage to an analyst, it is here considered likely to assist in ruling out subjective influences. In any case, the aim is not to discover as much as possible about German folksongs in themselves, but rather to establish an analytical method applicable to this and other data.

5. Progressions at the Bar-line

As the main interest here lies in the pulse, we must first consider where this may profitably be looked for. The pulse is expected to be associated with a regularly recurring unit; the unit may be the beat as specified by the time signature, or it may be a multiple or submultiple of that beat. Notated time signatures and bar-lines do not necessarily correspond to the musical pulse as here defined, as they might be chosen partly according to orthographic considerations.7
We proceed by looking only at the progressions at the bar-lines of the melodies as coded in the database. The study of the significance of the other beats within each bar is postponed for future work in which particular metres will be examined separately. In the vicinity of each bar-line, then, three notes are directly relevant:

Na , the last note sounding before the bar-line;
Nb , the note immediately following the bar-line; and
Nc , the next note following Nb.

Note Nb functions as the conducting down-beat, and it is expected that in some cases Na functions to lead into the down-beat, and in some cases Nc functions as a completion or continuation of the pulse gesture. Thus sometimes the pulse is determined mainly by the two notes Na and Nb, sometimes mainly by Nb and Nc, and sometimes by all three notes Na, Nb and Nc. In the absence of slurs or other musical indications or the texts of the songs it seems impossible to distinguish algorithmically between these three possibilities, and accordingly we include all three notes in the following analysis. Other notes than these three may of course also have an influence on the gesture at the bar-line, but it seems natural in a first attempt to restrict attention to the immediate vicinity. In the present study attention is further restricted to within-phrase progressions, for although the degree of separation caused by a phrase division no doubt varies in different cases, a progression within a phrase has in general greater musical significance than one between phrases.8
We can now define the possible types of progressions in relation to the bar-line. The definitions will be chosen not according to mere book-keeping considerations but to reflect as far as possible the musical or pulse-like significance of each pattern. The music shown in Fig. 1 was constructed to illustrate concisely the various possible types of pulse-like configurations, the slurs indicating phrases.9 The nine bar-lines are numbered in the Figure and will now be described, using the notation 'T' and 'F' for progressions considered to be 'To' and 'From' the down-beat.

Fig. 1. Constructed examples of configurations at bar-lines. The slurs indicate phrases.

Musical examples of configurations at bar-lines

1: A normal configuration.
2: The rest is treated as a prolongation of the previous note for present purposes; it is thus considered to indicate short articulation of what would otherwise have been a quarter-note.10
3: The rest is treated as a prolongation of the previous note across the bar-line.
4: The tie across the bar-line is treated as if the tie were removed, so that the duration of the 'T' progression is an eighth; to a first approximation this seems most appropriate for the study of the pulse, though further work could of course treat such ties separately. Few such cases occurred in the present data.
5: No 'F' progression occurs because the following note begins beyond the next bar-line, by which time the previous pulse is finished and a new one is under way.11
6: No 'F' progression occurs, as at bar-line 5.
7: No 'T' progression occurs because the previous note is separated by a phrase division; although a down-beat cannot be felt without at least a brief preliminary up-beat feeling, no evidence is available in the score for the up-beat's character in such a case.
8: No 'T' progression occurs for the same reason as at bar-line 7; no 'F' progression occurs because the rest at the bar-line means that the pitch is undefined; this case occurs rarely in the present data.
9: No 'F' progression occurs because the piece ends with no further notes.
10: No bar-line is counted because the music as notated has ended, even if a following rest may be appropriate.

In summary, the types of joint bar-line progressions are:
(i) Both 'T' and 'F' present: normal (bar 1), rests treated as prolongations of their previous notes (bars 2,3), tie across the bar-line removed for present purposes (bar 4);
(ii) 'T' present, 'F' absent, as no following note occurs within the present pulse (bars 5,6,9);
(iii) 'T' absent, 'F' present, as a bar-line occurs at a phrase division (bar 7);
(iv) Neither 'T' nor 'F' present, as the phrase starts with a rest (bar 8).
On the basis of the above definitions we proceed with data analysis, using a custom computer program. The number of bar-line progressions of each type in the database is shown in Table 2.12 In order to maintain clarity in the presence of a multitude of possible variables, it was decided to attend only to bar-lines of type (i) above, in which the progressions are defined on both sides of the bar-line. This covers 72% of the total bar-lines. Note that case (iii) will be likely to occur when there is no up-beat to a song or phrase; although we have here taken the attitude that no evidence is then provided of the shape of the gesture into the down-beat, these cases could well be examined separately in a future study. They occurred relatively much more often in the Kinderlieder (25% of bar-lines) than elsewhere (10%).

Table 2. Number of Bar-line Progressions of each Type.

Type Number Percent
  (i)  (TF) 50,527  72
 (ii)  (T-) 12,568  18
(iii)  (-F)  7,239  10
 (iv)  (--)    127   0
Total 70,461 100

6. Marginal Distributions

The pitch and duration progressions to and from each included bar-line are now to be studied. Ultimately one would like to tabulate progressions involving particular degrees of the scale and particular durations, but a far simpler beginning is in order in which only the directions of these relationships are tabulated. These are shown in Table 3, where 'U' means an upward progression, 'R' a repeated, and 'D' a downward one, whether referring to pitch or to duration (so that a duration progression from a shorter to a longer note is coded 'U'), and each horizontal triple totals 100%.

Table 3. Pulses of Type (i).
Percent of Directions To and From the Bar-line.

Collection Pitch Duration Number
To From To From
U   R   D U   R   D U   R   D U   R   D
Erk-Boehme 46  19  35 27  26  47 50  43   7  3  57  40 15,480
Boehme 48  20  32 24  21  55 63  32   5  6  45  49  7,197
Zuccalmaglio 50  17  33 30  22  48 49  42   9  2  57  41  6,688
Balladen 47  20  33 28  27  45 53  40   7  3  52  45  7,111
Fink 47  20  33 26  23  51 61  33   6  3  48  49  5,946
Altdeutscher 45  19  36 26  27  47 49  40  11  4  53  43  5,095
Deutscher VA 47  20  33 31  25  44 55  37   8  7  53  40  1,457
Kinderlieder 44  14  42  9  44  47 41  53   6  1  75  24  1,553
Total 47  19  34 26  25  49 53  39   8  4  53  43 50,527

The first conclusion to be drawn from Table 3 is that the progressions to the bar-line are systematically different from those from the bar-line: tests of significance would be superfluous here, considering the rather large sample sizes indicated in the right column. In respect of duration this was to be expected; for example, the small percentages of progressions to longer notes from the bar-line correspond approximately to the small incidence of syncopation in these songs. But in respect of pitch this result was not previously known to the author and, crude though it is, it is encouraging as a first hint of pulse-like behaviour. In particular, there is an increased percentage of upward pitch progressions to the bar-line compared with those from the bar-line (47% vs 26%).
The second conclusion from Table 3 is that the eight song collections give similar results except for the Kinderlieder (children's songs).13 It was already known that Kinderlieder, on account of their greater simplicity or chant-like character, tend to have more repeated pitches than do other song types, but it can now be seen that this excess occurs not uniformly or randomly in the music but much more often from the bar-line than to the bar-line (44% vs 14%) and then at the expense only of the upward pitch progressions.14 The Kinderlieder also have a much greater percentage of repeated durations both to and from the bar-line. The only other noticeable departures from uniformity are seen in the duration behaviour in the Boehme and Fink songs, these departures being a good deal less extreme, however, than those of the Kinderlieder.

7. Bivariate Distributions

Our next task is the study of the joint behaviour of the four variables of Table 3 taken pair-wise, still considering only the direction of progression of each variable. This part of the study combines all eight collections. The two pitch variables are tabulated jointly on the left side of Table 4.

Table 4. Percent of Pitch Progressions Jointly To and From the Bar-line.
All Collections Combined (50,527 Pulses).

  Observations   Cross-products
To \ From U   R   D Total U   R   D
U 11   14   22  47 12   12   23
R  5    3   11  19  5    5    9
D 10    8   16  34  9    8   17
Total 26   25   49 100 26   25   49

It is now desired to test this joint distribution for independence of its two margins. Traditionally one reaches for a chi-square or likelihood ratio test which would in this case very strongly reject independence. However, such a test may be considered inappropriate here, for any slight departure from complete independence will be detected by a sufficiently large sample, and the sample here is indeed large.15 Further, the squared errors in the various cells might not be best represented as ratios to the expected cell values if small cells are less important in the application. More useful appears to be the direct comparison of the observed table with that obtained by multiplying the margins, as on the right side of Table 4. The general similarity between the cells on the two sides of the Table suggests that for present purposes the pitch directions to and from the bar-line may be considered approximately independent.
The corresponding observed percentages and calculated cross-products for durations are shown in Table 5. In this case there can be no question of independence, as the asterisked cross-product cells show large departures from the corresponding observation cells. For example, if a progression to the bar-line moves to a longer note, then the following note is much more likely to be shorter than if the two progressions were independent (38% vs 23%).

Table 5. Percent of Duration Progressions Jointly To and From the Bar-line.
All Collections Combined (50,527 Pulses).

  Observations   Cross-products
To \ From  U   R   D  Total  U   R   D 
U  2   13   38  53  2   28*   23*
R  2   34    3  39  1   21*   17*
D  0    6    2   8  1    4     3 
Total  4   53   43 100  4   53    43 

The next bivariate distribution, combining pitch and duration to the bar-line, is shown in Table 6. These variables proceeding from the bar-line are shown in Table 7. In both distributions approximate independence is apparent.

Table 6. Percent of Progressions To the Bar-line. Pitch vs Duration.
All Collections Combined (50,527 Pulses).

  Observations   Cross-products
Pitch \ Duration  U   R   D  Total  U   R   D 
U 26   18    3  47 25   19    3
R  9    8    2  19 10    7    2
D 18   13    3  34 18   13    3
Total 53   39    8 100 53   39    8

Table 7. Percent of Progressions From the Bar-line. Pitch vs Duration.
All Collections Combined (50,527 Pulses).

  Observations   Cross-products
Pitch \ Duration  U   R   D  Total  U   R   D 
U  1   14   11  26  1   14   11
R  0   14   11  25  1   13   11
D  3   25   21  49  2   26   21
Total  4   53   43 100  4   53   43

The remaining pairs of variables, "pitch to" vs "duration from", and "duration to" vs "pitch from", seem less natural for separate study.
Summarising, the joint durations to and from the bar-line are highly dependent, the other relevant bivariate distributions showing approximate independence.

8. Four-dimensional Pulse Distributions

The pulse, as here measured, will now be represented as a four-dimensional distribution arranged for convenience as a 2 x 2 table with the two pitch variables combined on the vertical axis and the two duration variables on the horizontal axis. These distributions are shown for all collections combined in Table 8 and for the Kinderlieder in Table 9. Once again, great uniformity was seen among all collections other than the Kinderlieder; in particular, the maximum cell in those seven matrices was the UD-UD one in each case, whereas in the Kinderlieder it was the UR-RR one.16 Instead of showing the matrices for the collections individually we use a summary measure of the differences between the various pairs of matrices, taking for this purpose the root-mean-square of all 81 cell differences—see Table 10, in which the symmetrical half is suppressed. While there is a general similarity among all the matrices, the main departing one is clearly that for the Kinderlieder.

Table 8. Pulse Matrix: Percent of Directions To and From the Bar-line.
All Collections Combined (50,527 Pulses).

Pitch \ Duration UU UR UD RU RR RD DU DR DD Total
UU  1  5  4  1  11
UR  2  6  6  1  15
UD  3 11  7  1  1  23
RU  1  2   3
RR  1  1   2
RD  1  1  4  5  1  12
DU  1  4  4  1  10
DR  1  3  3   7
DD  3  7  6  1  17
Total  1 12 42 38  2  5 100

Table 9. Pulse Matrix: Percent of Directions To and From the Bar-line.
Kinderlieder (1,553 Pulses).

Pitch \ Duration UU UR UD RU RR RD DU DR DD Total
UU  1  1  3   5
UR  4  3 19  2  28
UD  3  3  5  1  12
RU  1   1
RR  3   3
RD  3  3  3   9
DU  1  2  13
DR  2  1 10  1  14
DD  7 10  7  1  25
Total 20 22 53  5 100

Table 10. Root-mean-square Differences between Pulse Matrices.

Erk Boe Zuc Bal Fin Alt Dva Kin Total
Erk -
Boe 1.02 -
Zuc  .68  .94 -
Bal  .54  .98  .72 -
Fin 1.01  .44  .93  .97 -
Alt  .85 1.14 1.01  .89 1.14 -
Dva  .65  .97  .77  .72  .92  .99 -
Kin 1.96 2.73 2.46 2.14 2.73 2.28 2.28 -
Total  .38  .72  .63  .52  .72  .79  .65 2.19 -

9. Typical and Atypical Songs

It is often advantageous to be able to summarise a repertoire by citing typical representatives of it, and on the other hand to be able to draw attention to unusual examples.17 There are, of course, innumerable criteria which could be used for such purposes.18 The pulse character is only one aspect of a repertoire, and in using it as a criterion one must endeavour to concentrate attention on that aspect to the exclusion of other, possibly more immediately obvious, aspects. In any case, this approach provides a convenient illustration of the application of the results of this paper.
We accordingly assigned to each song of the present data a typicality rating, R, calculated according to the formula:

R  =  (100/M) [ (1/n) Σ ni=1 Σ nj=1 m2ij ] 1/2

where M is the maximum value among the cells in the pulse matrix {mij}, summation is carried out over the cells corresponding to the complete pulses occurring in the given song, and n is the number of such pulses. The higher the value of R for a given song, from 0% to 100%, the more typical are the song's pulses. The procedure was carried out using the overall pulse matrix of Table 8, and was repeated using the pulse matrix of Table 9 in order to distinguish typicality with respect to the differing Kinderlieder pulse. The resulting values of R were then surveyed, songs having only a few pulses being rejected for the present purpose.
A song very typical of the overall pulse is shown in Fig. 2.19 Here six of the eight pulses are of UD-UD type, the most common according to Table 8, while the first and fifth are of UR-UD type. The illustrative songs' values of R with respect to the two pulses are shown in Table 11 (given after Fig. 4).

Fig. 2. Song with pulse typical of non-Kinderlieder.

Score of 'Ritter und Magd'

The song most typical of the Kinderlieder pulse, shown in Fig. 3, happened to belong to the Ballad collection. All seven of its complete pulses are of type UR-RR, the most common according to Table 8, its last pulse being incomplete; hence its 100% rating with respect to the Kinderlieder pulse.

Fig. 3. Song with pulse typical of Kinderlieder.

Score of 'Die Rabenmutter'

Finally an atypical song is shown in Fig. 4. Three of its pulses are of type UU-RD, three DU-DR, the first and last being incomplete. These cells are empty in Table 9, accounting for the song's zero rating with respect to the Kinderlieder pulse.

Fig. 4. Song with pulse atypical of German folksong.

Score of 'Frohsinn'

Table 11. Percentage Typicality Ratings for Illustrative Songs.

Figure Title Overall Pulse Kinderlieder Pulse
2 Ritter und Magd 90  15
3 Die Rabenmutter 54 100
4 Frohsinn  6   0

By singing these songs with special attention to the music's gestures at the bar-lines one may feel that the pulse, crudely measured though it has so far been, has real musical significance.

10. Conclusions and Discussion

A first attempt has been made to examine the musical pulse in a large database. The results seem encouraging, both in the consistent statistical characterizations which have been realised and in their confirmation in illustrative songs. These characterizations may be interpreted as reflecting the shape of the gesture corresponding to the conducting beat. It might fairly be claimed that the statistical approach adopted here has succeeded in finding results of musical significance, and is capable of finding more such results, which would not likely have been found by an unaided connoisseur.
Nevertheless, the characterization of the pulse is seen as a long-term task to which the merest beginning has here been made. It is intended in further work to refine the measurement of pitch so as to consider not only its direction but its stepwise and skipping movements and eventually its degrees of the scale. Duration, similarly, may be refined into large and small differences and eventually into given note-lengths. For reliable estimates under these refinements the amount of data, which seemed large for the present paper's limited objectives, may come under pressure. A balance must be struck between the amount of data available and the amount of detail in the analysis.
A further refinement in which attention is restricted to individual metres will be needed in order to investigate the pulse behaviour at beats within the bar. Whereas averages have so far been taken over all the pulses of a repertoire, attention to the constitution of individual songs will show, for example, how much variety of pulse configuration is typically found within a song, perhaps leading eventually to an understanding of the pulse as a 'fingerprint'.
Finally, short folksongs can not show the full significance of the pulse. When data is available for other repertoires containing longer works, particularly those of the great composers of the period of common practice, the approach which has here been begun may become especially fruitful.


1 Clynes' proposed mechanism for the pulse in performance has been adversely criticised by Repp (1990) and others, but the notion of the pulse as felt is independent of its mechanism.
2 The term "pulse" is also used to refer to a particular instance in a given bar, the usage being clear from the context.
3 Detlovs apparently overlooked the possibility that a positive difference between successive intervals can occur even if both intervals have the same sign, the central note not then being a local minimum as depicted in his Fig. 3a, p.47; and similarly with his Fig. 3b.
4 The overlapping songs may constitute about eight cases per collection—personal communication from Prof. Dr. Schaffrath. No attempt has been made to eliminate variants, as no exact definition of a variant can be formulated.
5 A number of apparent errors have been found and are discussed in a separate paper not yet published. [See my subsequently published paper "On the Accuracy of Musical Data, with Examples from Gregorian Chant and German Folksong", Computers and the Humanities 27 (1993), 111-120.] These have not been corrected in the following work, partly in order to avoid faulty "corrections" and partly in order to allow others to reproduce the analyses if desired. The errors would not noticeably affect the present results.
6 The song text could have a special effect for example in the case of a song representing yodelling; such extreme cases are here assumed to be infrequent.
7 For example, the difference between two bars of 6/8 metre and one bar of 12/8 is in some circumstances known to be only orthographic, not musical.
8 Some statistics not reported here confirmed the noticeably different patterns occurring between phrases as compared with those within phrases so that, as expected, the phrase divisions indicated by the encoders were by no means random.
9 A few logical possibilities which would be unlikely to occur have been omitted.
10 Although not all rests following a note have this function, no reliable algorithm is available to determine the function. Such an algorithm might be more easily constructed when attention is restricted to a given metre.
11 It could be argued that the following note (C) can influence, even from this distance, the significance of the main note (B), but this view is not taken here.
12 Ties across the bar-line numbered only 310. As expected, no ties occurred between phrases.
13 Most of the collections have several component files which were also analysed separately, the results still indicating uniformity.
14 A chi-square test of independence of the three categories for "pitch from the bar-line" for Kinderlieder vs Others showed overwhelming significance, even considering that the data had been examined before formulating the hypothesis; see, however, the later comments on chi-square tests of independence.
15 Compare the hypothesis that the average height of men in a given population is 1.8 metres: if the true average is, say, 1.801, then a sufficiently large sample will almost certainly reject the hypothesis, although the rejection may not be appropriate in the application.
16 Explicitly, the cell UR-RR counts all bar-lines where the pitch ascended to the bar-line and was repeated from the bar-line, while the duration was repeated in both progressions.
17 The characterisation of repertoires has been well discussed by Meyer (1989, pp.61-62).
18 An unusual project led to an unusual criterion in Halperin (1979/80) p.17 where different results were obtained according as a phrase was processed forward from its beginning or backward from its end.
19 The songs in Figs. 2-4 have been transposed to C and each line shows a separate phrase; the identification tag in the Essen database and the song title are shown.


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