Brahms Metro

8/18/2019 Brahms Metro

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Metric Investigations in Brahms’ Symphonies

Anja Volk

Abstract

Rhythmic and metric ambiguities in Brahms’ compositions have been dis-cussed in various music theoretical approaches. Which compositional tech-niques cause the diverse forms of ambiguities, such as metric displacement?Themethod of inner metric analysis, as implemented in the RUBATO-Softwarefor Musical Analysis and Performance, explores the metric structureexpressed

by the onsets of notes without considering the information given by the timesignature. The report on the application of this method to the Second andThird Symphonies demonstrates surprising insights into the metric organiza-tion of these compositions. Furthermore it illustrates the contribution to the

precise description of metric peculiarities in the works of Brahms gained bythe inner metric analysis.

1 Introduction

»... Brahms, who had to extend so many of these inclinations totheir most complex reaches. In his recognition of ambiguity as acompositional value of extensive implications; ... and in his work-ing of rhythms, both local and more extensive, to produce syn-copes, metric displacements, and other avoidances of regularity ... .«1

The phenomenon of rhythmic and metric ambiguities observed in Brahms’Œuvre has often been discussed in music theory. Schönberg (1976) describes

the peculiarities by means of phrase lengths causing metrical displacements.According to him Brahms made use of irregular phrase lengths in such anextensive way, that under his influence the avoidance of regularity becamea common element of the syntax and grammar of musical structures (»zumfesten Bestandteil der Syntax und Grammatik vielleicht aller späteren mu-sikalischen Strukturen geworden.«2). Schönberg even declares Brahms’ avoid-ance of regularity as epoch-making to such a degree, that he appreciates him as

being a more inspiring innovator in music history than his antipodean Wagner.Frisch (1990) discusses metrical displacements as a fundamental characteristicof Brahms’ compositions as well, but does not agree with Schönberg concern-ing the phrase lengths as being the cause of the observed displacements:

»Phrases of irregular or variable length are, to be sure, a significantaspect of Brahms’s language ... But they differ fundamentally from

actual metrical displacement.«3

1 Epstein (1987, p. 157)2 Schönberg (1976, p. 61)


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Metric Investigations in Brahms’ Symphonies 2

Frisch instead regards notes’ placements as being responsible for metric dis-placements, as in the case of the piano quintet in F-Minor Op. 34, 1:

»Because the notated third beat is empty in bars 90-5 and the first beat is empty in bars 87-95 ... , they tend to sound like weak secondand fourth beats respectively. That is, notated beats 2 and 4 areperceived as 1 and 3. The bar line has thus drifted still further to

the right ... .«4

Epstein’s (1987) above mentioned quotation considers »metric displacement«as well. The following quotation suggests that Epstein might apply this termin a similar way as Schönberg and Frisch did:

»Rhythmic ambiguity also serves a broader design: by disguisingand in several ways deemphasizing rhythmic downbeats, the mu-sic in its longest spans is kept continually on a quasi-upbeat foot-ing, with few points of stability.«5

The mentioned deemphasizing of rhythmic downbeats might be achieved byplacing notes on weak beats and rests on strong beats, as Frisch argued con-cerning the piano quintet. However, since Epstein adumbrates »several ways«,one might detect other compositional methods being responsible for ambigu-

ities as well. In this article we want to apply the method of inner metric analysisto Brahms’ Symphonies in order to describe these several ways of producingambiguities and compare the results to the observations made by the abovementioned music theorists.

2 Inner metric analysis

The inner metric analysis, as implemented in the RUBATO-Software for Mu-sical Analysis and Performance, is based on the investigation of regularitieswithin the set of notes’ onsets of a given piece of music. It has been describedin detail in (Mazzola, 2002), (Fleischer, 2003), (Fleischer, 2002a) and (Fleischer,2002b), here we want to give just a brief overview. Inner metric analysis resultsin a metric weight for each note and is concerned with the metric structure ex-

pressed by the actual notes of a composition without taking into considerationthe information given by the time signature. The latter refers to the outer metricstructure, which is characterized by a metric hierarchy of accents depending onthe time signature and bar lines.

Inner metric analysis is based on the regularities within the set of all onsetsof the notes of a given piece. These regularities are described by means of localmeters. A local meter mk denotes a subset of equally distanced onsets. Theinner metric weight W l,p(o) of each onset o is the weighted sum of the length kof all local meters mk, which contain the onset o (the length k of a local metermk is defined as the number of onsets it consists of, decremented by one):

W l,p(o) =

∀mk ,k≥l:o∈mk

kp. (1)

3 Frisch (1990, p. 141)4 Frisch (1990, p. 147)5 Epstein (1987, p. 168)


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Figure 1: Metric weight W 2,2 of the measures 1-92 of the 3rd movement of the

Fourth Symphony (time signature 2

4 ): the higher the line, the greater the weight,grey lines in the background mark the bar lines

Figure 2: Excerpt (measures 1-35) from metric weight W 2,2 of the entire expositionof the 1st movement of the Third Symphony (time signature 6

4)

The formula of the metric weight W l,p(o) in equation 1 depends furthermore

on the two parameters l and p, which can be varied by the user of the soft-ware. The parameter l denotes the minimal length of local meters being con-sidered in the calculation of the metric weight. By increasing the values of lthe user can exclude short local meters (shorter than l) from the calculation,whereas the parameter p weights the contribution of the local meters to themetric weight depending on their length. Great values cause a greater con-tribution of longer local meters, small values cause a greater contribution of shorter local meters to the metric weight. By varying these parameters theuser can obtain different metric perspectives on the same piece.

The metric weight of equation 1 describes the inner metric structure of apiece of music. Obviously it depends solely on the regularities caused by thenotes of the piece without considering information given by the time signa-ture. The comparison of the results of the metric weights and the hierarchy of the outer metric structure led to a definition of metric coherence which we have

introducedand discussed in (Fleischer, 2003), (Fleischer, 2002b), and (Fleischer,2002a). Whenever a correspondence between inner and outer metric structurecan be observed, metric coherence occurs.

Figure 1 shows an example regarding the time signature 24

in Brahms’Fourth Symphony. The metric weight W 2,2 is characterized by different layerswhich correspond to layers of the outer metric hierarchy. The highest layer is

built upon the first beats of all measures, followed by the layer built upon thesecond beats of all measures. The weights of the second and fourth eighthsform a much lower layer, whereas the weak beats, such as the second andfourth sixteenths form the lowest layer. Hence metric coherence occurs.

Figure 2 shows an example of metric coherence regarding the time signa-ture 6

4 in Brahms’ Third Symphony. One may distinguish the following layers:

the beginnings of all measures (highest layer), the fourth beats of all meas-ures, the second, third, fifth and sixth beats of all measures and the weak beats(lowest layer). Hence four layers corresponding to the layers of the outer met-ric hierarchy can be detected within inner metric analysis.


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Figure 3: Metric weight W 2,2 of bassoon, violins, cello and bass (measures 127-155)of the first movement of the Second Symphony

Figure 3 illustrates another form of metric coherence, which is character-ized by a phase shift. A periodicity can be stated within the metric weight,which respects the layers of strong and weak beats concerning the outer hier-archy of 3

4, but the greatest metric weights are located on the second beats of

the measures instead of the first beats, as in the previous examples. We callthis phenomenon of a phase shift an upbeat of coherent character (for furtherdiscussion of this example see p. 12), since it occurs in those cases of a stablerelation between grouping and meter, where the beginnings of the groups do

not coincide with the beginnings of the bars. In this article we will use in themost cases the parameters l = 2 and p = 2, hence only cases with differentvalues for l and p will be indicated.6

Theauthor’s discussionof metric coherenceconcerning verydifferent stylesin music history in (Fleischer, 2003) proved the suitability of this music theor-etical term regarding the description of metricity of compositions. Metric co-herence very often occurs in those pieces which are typical representations of the important role of the metric hierarchy given by the bar lines (e.g. Renais-sance madrigals), whereas, for instance, in compositions by Bach coherenceoccurs more rarely. Hence inner metric analysis might serve as an appropriatemethod in order to describe metric ambiguities within Brahms’ compositionsas well.

3 Second Symphony»Perhaps no composer of the period so reveled in the structuralpossibilities of ambiguity as did Brahms. His Second Symphony isa case in point, ambiguous properties inherent in the basic ideas of the opening movement exerting pervasive effects upon the overallstructure of this and subsequent movements.«7

We have discussed metric analyses of all four Symphonies by Brahms in(Fleischer, 2003) in detail, now we want to focus on inner metric analyses of the Second Symphony.


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Figure 4: Excerpt from the metric weight of the entire exposition (measures 1-43)


Figure 6: Detail from figure 5


3.1 First movement

The metric weight of the entire exposition of the first movement Allegro nontroppo (time signature 3

4) is divided into three parts of very different regularit-

ies, as shown in figures 4, 5, and 7. The first part corresponds to the measures1-43, the second to measures 44-117 and the third to measures 118-186.

Figure 4 shows the first part of the metric weight which is characterized by metric coherence: the first onsets of the bars get the greatest metric weight.Metric coherence does not apply to the second part of the metric weight in fig-ure 5. This part is characterized by a completely different regularity (see also adetailed version in figure 6), which corresponds to the outer accent hierarchyof 4

4 instead of the notated 3

4, as shown in figure 9. Hence metric coherence

cannot be stated. The last part of the metric weight in figure 7 again reveals6 For a more detailed discussion of the influence of l and p see (Fleischer, 2002a).7 Epstein (1987, p. 162)


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Figure 8: Opening measures of the 1. movement

very different regularities. One may distinguish layers, but in contrast to thesecond part, within the highest layer no periodic differentiation can be stated.Therefore metric coherence cannot be found.

Inner metric analysis hence suggests a segmentation of the exposition intothree parts. These parts interestingly correspond to a segmentation of the ex-position based on a harmonic point of view, as suggested by Graham Phipps 8,which is a very special feature of this exposition according to him. Hence acoherence between inner metric and harmonic structure can be stated.

Now we go into more detail concerning these three parts of diverse innermetric structures within the metric weight of the entire exposition.

Concerning the first part of the exposition Epstein (1987) states an ambigu-ity regarding the mutual relationship of the first two measures of the move-ment (see figure 8): »Should they be heard as tonic or dominant oriented? Isthe first measure upbeat or downbeat?«9 This ambiguity is important not only

regarding the first two measures since it influences the following parts of themovement as well: »Both its harmonic and rhythmic properties are unclearand capable of producing various viewpoints, many of which are explored asthe music progresses.«10

According to Epstein, the harmonic and rhythmic properties do not allowan unequivocal decision concerning the relation of up- and downbeat withinthe first two measures and the corresponding measures over the course of thefirst 43 measures. But inner metric analysis (see figure 4) suggests a groupinginto groups of two measures within the first part of the exposition. Withinthese groups the second measure is the downbeat, since the metric weight of the first beat of each second measure is greater than those of the first measure.

8 Talk: »Die Überleitung in der Sonatenhauptsatzform. Auf den Spuren Martin Heideggers imersten Satz der II. Sinfonie von Brahms’« at the 1. congress of the German Society of Music Theory

in Dresden, 12. 09. 20019 Epstein (1987, p. 162)10 Epstein (1987, p. 162)


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Figure 9: Metric weight of figure 5, interpreted as 4

4

Figure 10: Detail from figure 9

Hence the inner metric structure in this case is unequivocal. The result of innermetric analysis on the other hand corresponds to the suggestion Epstein madeconcerning the performance of this piece: »From a performance viewpoint, themost efficacious decision is probably to consider the opening measure of thepiece an upbeat, leading to a (local) downbeat on the second measure with theentrance of the horns.«11

Furthermore his discussion of the horn’s motives corresponds to the men-tioned result of inner metric analysis: »By their attack and by the added tex-ture they create, the horns give an accent or emphasis to the second measure,thus lending it a downbeat quality. This pattern is carried out in the succeed-ing phrases: the entrances of the horn motive in measures 6, 10, 14 all givedownbeat qualities of emphasis to these measures ... .«12

The second part of the metric weight of the entire exposition (measures 44-117) in figure 5 reveals a highest layer built upon the weights of the onsets

1(measure 44), 3(measure 44), 2(measure 45), 1(measure 46), 3(measure 46), 2(measure 47), 1(measure 48) and so forthof successive measures. Hence the weight does not correspond to the outermetric structure of the time signature 3

4, but of 4

4 (see figure 9 and 10). In the

middle of figure 9 and at the end of figure 4 occur segments which are charac-terized by the rhythmic peculiarity of hemiolas. These segments correspondto the measures 42-43 and 78-81 (see figure 11). Hence the metric weight mightreflect the influence of the hemiolas causing a metric periodicity which doesnot correspond to the outer metric hierarchy.

Since the hemiolas are located at the edges of the segment of the meas-ures 42-81 the question arises, whether they are mainly responsible for the di-vergence between inner and outer metric structure within this great segment.This conjecture is tested by excluding these edges of the measures 42-43 and78-81, figure 12 shows the result of the corresponding analysis for the meas-

ures 44-77. Obviously this metric weight reveals a very different shape and is11 Epstein (1987, p. 177)12 Epstein (1987, p. 165)


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Figure 11: The hemiolas of measures 78 ff.

Figure 12: Metric weight of measures 44-77

Figure 13: Metric weight of figure 12 with shifted bar lines

not characterized by the periodicity corresponding to time signature 44

. Hence

the observed periodicity within the segment of measures 44-78 in figure 9 ismainly due to the influence of the hemiolas.

The metric weight in figure 12 concerning the measures 44-59 (the first partbefore the caesura in the middle of the figure) does not show layers correspond-ing to the ›strong‹ and ›weak‹ beats of the time signature 3

4, since the second

and fifth eighths gain the greatest metric weights in most cases. Hence byshifting the bar line (see figure 13), a correspondence with the time signature6

8 can be observed in the section before the caesura: first and fourth beats get

the greatest weights. The metric weight in figure 13 after the caesura (measures60-77) shows two different tendencies. In some measures one may distinguishlayers corresponding to the three strong beats and three weak beats of themeter 3

4, whereas in other measures great metric weights on the fourth eighths

prevent the occurrence of these layers. The evaluation of this segment in fig-

ure 12 results on the other hand in great metric weights on the third beats of the measures, as already observed in the part before the caesura.Hence the metric weight of the measures 44-77 is characterized on the one


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Figure 14: Measures 56 ff. of the first movement

Figure 15: Metric weight of cello and bass (measures 44-77)

hand by an ambiguity between 34

and 68

and on the other by a great metricweight on the third instead of the first beat.

Both characteristics correspond to observations in (Epstein, 1987). The fol-lowing quotation concerns the great metric weights on the third beats:

»As the big D major section after measure 44 develops to a climaxat measure 59, the rhythmic nature of the motive is again unclear.It appears at measure 59 with great downbeat force – the resolu-tion of a long dominant passage, further intensified by a crescendo.

However, its attack is on the third beat of the preceding measure,robbing this attack of a congruent emphasis with the local metricaldownbeat of measure 59 itself.«13

The development of this section results in a shifting of the downbeat to-wards the third beat of the measure. This is reflected by the metric weight of the segment of the measures 44-77, which was not detected within the analysisof the entire exposition. Hence by analyzing segments one may gain new in-formation about the piece. The ambiguity between the structures of 3

4 and 6

8

becomes even more evident in the analysis of the isolated instrumental part of cello and bass in figure 15. In this case the inner metric structure correspondsstrongly to the time signature of 6

8.

The confusion concerning these two time signatures Epstein (1987) statesas a further example regarding the ambiguities of this movement: »The suc-

ceeding passage at measures 63 ff. further confuses the rhythmic structure: the

13 Epstein (1987, p. 166)


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Figure 16: Measures 64 ff.

Figure 17: Metric weight of string instruments (measures 44-77)

Figure 18: Metric weight of wind instruments (measures 44-77)

motive combines with itself in diminution. Its first measure, in normal rhythm(measure 63), seems downbeat oriented until heard in the elision at measure64, where a new phrase apparently begins. This new phrase is also unclear... Is it to be heard as 6

8 or 3

4? ... The phrase is one of the most ambiguous

moments in the movement.«14

The rhythmic and melodic motives of the measures 64 ff. (see figure 16)confirm a 6

8-structure latent in the preceding passages as well, since the ob-

served correspondences in the mentioned metric weights can be stated in otherparts of the compositions. Another example is shown in figure 17 concerningthe analysis of the string instruments of this segment. In many measures thegreatest metric weights are located on the second and fifth eighths, hence wecan state a hidden 6

8-structure with a shift of an eighth as well.

On the other hand, metric coherence can be stated within the metric weightof the wind instruments in figure 18. The highest layer is built upon the three›strong‹ beats, the first beats get the greatest weights.

14 Epstein (1987, p. 166)


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Figure 19: Metric weight of the first violins (measures 44-77)

Figure 20: Measures 54 ff. in the first violins

A very distinct inner metric structure can be observed within the analysisof the first violins in figure 19. Great metric weights are located on the second,fourth and sixth eighths of the measures producing a kind of offbeat. Synco-pations (see figure 20) are responsible for this process.

Summing up the results of the analyses of different instrumental partswithin the second part of the exposition one can state the competing role of 3

4 and 6

8 , as well as very diverse periodicities of the weights depending onthe chosen segment or instrumental part, e.g. depending on the chosen con-text. As we have shown in (Fleischer, 2003), segmenting and voicing do notin any case produce such diverse periodicities of the metric weight. Examplesfrom Mozart’s Jupiter Symphony demonstrate15 that by selecting instrumentalparts or shorter segments one detects similar inner metric structures as exhib-ited within larger contexts. Hence these very diverse results of metric analysisdepending on the context in the case of the Second Symphony by Brahms may

be acknowledged as a characteristic of his music. This interpretation mightcorrespond to an observation of Epstein, that Brahms produces an unique mu-sical structure through anomalous contrasting and counter-balancing of ele-ments being responsible for the forward motion in music (»Brahms erreicht... eine einmalige musikalischen Struktur, wo das anomale Gegenüberstellenund Gegenbalancieren von Elementen ... die Mittel für die Vorwärtsbewegungin die Musik einbaut«16).

Within the third part of the exposition, where no periodic differentiationwithin the highest layer of the weight can be found, Epstein describes a pas-sage (see figure 21) as an ocean of rhythmic and metric uncertainty (»Nochwidersprüchlicher ist eine andere Passage im gleichen Satz, wo die Musikin einem Meer von rhythmischer und metrischer Ungewißheit schwimmt«17).The melodic motives (see the violins, cello and bass in figure 21) correspondto the time signature, but sound as they would begin on the first beat of themeasures (»Die melodischen Motive, die ihren Ursprung in den ersten Taktendes Satzes haben, passen in das 3

4-Metrum. Sie klingen, als wären ihre An-

fangstöne jeweils auf der ›Eins‹ des Taktes. So sind sie aber nicht notiert.«18).Moreover the syncopations of the horns (see viola in figure 21) disturb the

15 Fleischer (2003, p. 77)16 Epstein (1994, p. 12)17 Epstein (1994, p. 10)18 Epstein (1994, p. 10)


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Figure 23: Metric weight of clarinet, horn, and viola (measures 127-155)

emergence of any downbeats (» ... nivellieren die scheinbar nie endendenHörnersynkopen jegliches Gefühl für einen Schwerpunkt.«19). The same pas-sage Frisch (1990) discusses as a »massive canonic and metrically disorienting

episode«20.The metric weight of a greater segment including this passage (measures

127-155) in figure 22 allows no correspondence to any possible time signaturewhatsoever due to the great metric weights on the first, third, fourth and sixtheighths of the measures. Hence it confirms Epstein’s observation of metricuncertainty.

The analysis of the bassoon, violins, cello and bass already discussed inthe beginning (see figure 3), which form the melodic voice, confirms Epstein’sobservation regarding the second beats of the measures. The greatest metricweights are located on the second beats of the measures, the periodicity of the weight respects the layers of strong and weak beats of 3

4. Hence metric

coherence can be found, a phase shift takes place.The metric weight of the instrumental parts characterized by the synco-

pations (see figure 23) shows a periodicity which does not correspond to any19 Epstein (1994, p. 10)20 Frisch (1990, p. 155)


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Figure 24: Metric weight of measures 1-32 of the 2. movement ( 44

)

Figure 25: Metric weight of measures 62-91 of the 2. movement ( 44

)

Figure 26: Metric weight of measures 96-103 (time signature 44

)

time signature whatsoever. Great metric weights are located on the third andfourth sixteenths as well as on the seventh and eighth sixteenths and the el-eventh and twelfth sixteenths. It confirms Epstein’s observation that thesesyncopations are responsible for a lack of a clear metricity within this passage.

Inner metric analysis of the entire exposition of the first movement broughtto light three different parts of regularities. Furthermore we detected newcharacteristics of inner metric structure within these parts by choosing smallercontexts, such as segments or voicing. These characteristics were not inherentin the analysis of the entire exposition. Only in some cases metric coherencewas found, whereas very often a divergence between inner and outer metricstructure appeared, describing precisely the metric ambiguities observed in(Epstein, 1987), (Epstein, 1994) or (Frisch, 1990).

3.2 Second movement

The second movement Adagio non troppo is segmented into sections of differingtime signatures associated with different themes. Measures 1-32 are notatedas 4

4, measures 33-56 as 12

8 . The following passage (measures 57-61) is notated

as 44

in the string instruments, bassoon and trombone and as 128

in the windinstruments (despite bassoon and trombone). Measures 62-91 are notated as 4

4

for all instrumental parts, 92-96 as 128

and measures 96-103 again as 44

.Since measures 57-61 are transcribed as 12

8 in the used midi-file, e.g. a

quarter note of this segment corresponds to three eighth notes and is hence

different from a quarter note of the following segment notated as 4

4 , we havechosen for the analysis the segments of measures 33-61 ( 12

8 ) and 62-91 ( 4

4).

At first we want to discuss the results for the three segments notated as 44

.


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Figure 27: The beginning of the theme in the cello

Figure 28: Measures 6-9 in the accompanying parts


)

All metric weights in figures 24, 25, and 26 are characterized by great met-ric weights on the second and fourth beats of the measures. On the one handthis phase shift within the highest layer of the metric weight corresponds tothe upbeat of the theme (see figure 27), on the other we can observe a sim-ilar phenomenon which was mentioned by Frisch regarding the piano quintet(see page 2) in the accompanying voices. Rests are located on the first andthird beats whereas notes are placed on the weak second and fourth beats (seean example in figure 28), which according to Frisch results in a metric dis-placement. The very similarity of the weights regarding these three segmentsfurthermore demonstrates, that we obtained a common characteristic of thethematic material associated with the time signature of 4

4.

The analysis of the first segment notated as 128

reveals in figure 29 five sec-tions of different density within the flow of successive note events: quarternotes and eighth notes alternately shape the rhythmic flow in the first sec-tion (lowest metric weights), whereas the interplay of violins and violas in thesecond section results in a continuous sequence of eighth notes (measures 45-48). This sections follows a continuous sequence of sixteenth notes (measures49-52). The flow of events gets even more dense in the next segment (meas-

ures 53-57) since a continuous motion of thirty-second notes (segment of thegreatest metric weights in figure 29) appears. In measure 58 the first themefrom the very beginning starts its repetition.


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Figure 30: Detail of figure 29 for measures 53 ff.

Figure 31: Detail of figure 29 for measures 59 ff.

Within each segment of the metric weight up to the beginning of the firsttheme in measure 58, the highest layer is not differentiated, as the detailed

figure 30 illustrates. Layers can be distinguished, but the metric weights of the twelve beats of the measure are not differentiated at all within the highestlayer. This lack of differentiation corresponds to a characterization in (Epstein,1994) of this movement as being an example of a continuous motion with aminimum of structural accents (»... eines der atemberaubendsten Beispiele derTechnik der ununterbrochenen, geführten Bewegung und des Mindestmaßesan strukturellen Schwerpunkten.«21)

On the contrary, the repetition of the first theme at measure 58 results ina metric weight of the corresponding part, that assigns the greatest metricweights to the first, fourth, seventh and tenth beats within the highest layer(see figure 31). This differentiation enlightens the relationship between thetwo alternating time signatures of this movement: the 12

8 corresponds to 4

4

with quarter notes being divided into three eighth notes and does obviouslynot correspond to 3

2

(which requires great metric weights on the first, fifth andninth eighths).

The isolated analysis of the wind instruments of this section in figure 32reveals great metric weights on the first, fourth, seventh and tenth beats aswell as in the segment following measure 53 (see also figure 33). Furthermore,within the repetition of the first theme starting at measure 58 the greatest met-ric weights are located on the fourth and tenth beats, which corresponds to themetric characteristic of the first theme already detected within the segmentsof time signature 4

4 (since the fourth and tenth beats in 12

8 correspond to the

second and fourth beats of 44

). Hence this result confirms again the hypothesisthat the accentuation of the second and fourth beats is a characteristic of thefirst theme.

The analysis of the second segment notated as 128

in figure 34 shows (incontrast to the first 12

8 -segment) a differentiation within the layer of the twelve

beats. The greatest metric weights are located on the first, fourth, seventh and

21 Epstein (1994, p. 13)


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Figure 32: Metric weight of the wind instruments (measures 33-61)

Figure 33: Detail from figure 32 for measures 53 ff.


)

Figure 35: Metric weight of wind instruments (measures 92-96)

tenth beats, supporting the correspondence of 128

and 44

as previously men-tioned.

The isolated analysis for the wind instruments in figure 35 reveals thegreatest metric weights on the fourth and tenth beats, which originally wasdetected as the characteristic of the first theme associated with 4

4. Hence a

relationship between the different thematic materials associated with the twodistinct time signatures of 4

4 and 12

8 is discovered in inner metric analysis.

Inner metric analyses of the different segments of this movement enlightena characteristic of the thematic material notated as 4

4: greatest metric weights

are situated on the second and fourth beats. A similar observation is men-tioned in (Frisch, 1990) p. 154 regarding the piano quintet (see p. 1). Further-more a relationship between the two distinct themes associated with the twodifferenttime signatures could be stated within the analysis of the wind instru-

ments in measures 92-96. The lack of differentiation within the highest layerof the weight of the first segment notated as 12

8 furthermore corresponds to an

observation in (Epstein, 1994) concerning the minimum of structural accents.


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Figure 36: Metric weight of measures 1-32 of the third movement

Figure 37: The beginning of the third movement

Figure 38: Metric weight of measures 1-32, l = 76

3.3 Third Movement

The third movement Allegretto grazioso (Quasi Andantino) is segmented intosections of different time signatures as well. Measures 1-32, 107-125 and 190-240 are notated as 3

4, measures 33-106 as 2

4 and measures 126-189 as 3

8. Here

we discuss the first segment of each time signature (measures 1-32, 33-106 and126-189).

The greatest metric weights within the analysis of the first segment (meas-ures 1-32 in 3

4) in figure 36 are located on the first and third beats of the meas-

ures. The second layer is built upon the second beats of the measures, theweak beats (second, fourth and sixth eighth note) form a lower layer. Hencethe metric weight reflects the hierarchy of strong and weak beats of the outermetric structure, but the first and third beats compete within the highest layerof the strong beats. The great metric weights on the third beats correspond onthe one hand to the accents in the score (see figure 37) in the beginning.

The reason for the prominence of the weight of the third beats on the otherhand can be explored by incrementing the value of l . The metric weight cal-culated with l = 76 in figure 38 enables us to differ three local meters. Thelongest local meter of length k = 95 is built upon the first beats of all meas-ures, a second one of length k = 77 is built upon all eighth notes of measures

20-32, a third one of length k = 76 is built upon all eighth notes of measures7-19. Since the continuous sequence of eighth notes (which derives from themetric interplay of the wind instruments and the cello) in measures 7-19 stops


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Figure 39: Metric weight of measures 1-32, l = 63

Figure 40: Metric weight of measures 1-32 for the cello

Figure 41: Metric weight of measures 1-32 for the wind instruments

at the last eighth note of measure 19, we can observe a gap in the middle of figure 38.

The next differentiation of the metric weight by decreasing the values of loccurs in the metric weight for l = 63 (see figure 39) which clarifies the prom-inent role of the third beats of the measures. The first local meter of lengthk = 63 starts on the first beat of measure 1 and consists of the first and fourtheighth notes of all measures of this segment, the second local meter starts onthe second eighth note of the first measure and consists of the second and fiftheighth notes of all measures. Hence the first and third beats of the measuresparticipate in these local meters, whereas the second beats do not. Thereforethe highest layer within the metric weight is built upon the first and third beats

in the finest metric weight for l = 2 as well. The reason for the occurrence of the mentioned local meters is mainly due to the gaps in a quasi continuous mo-tion of eighth notes deriving from the interplay of the wind instruments andthe cello. One gap is located in measure 19 as already mentioned, two othersare located at the last eighths of measures 4 and 6 respectively (see figure 37).

The isolated analysis of the cello in figure 40 illustrates that the gaps withinthe motion of eighth notes even prevents the occurrence of layers correspond-ing to the outer metric structure within a large segment of this instrumentalpart. Within the last measures two layers can be separated, but within thehighest layer no differentiation can be observed. Hence metric coherence can-not be stated. On the other hand, the melodic contour of the cello in manycases is able to mediate the structure of the time signature 3

4.

The isolated analysis of the wind instruments in figure 41 shows metric co-herence. The greatest metric weights are situated on the second beats of the

measures, which is in contrast to the result of the analysis of all instruments infigure 36. Again an analysis with a higher value of l can reveal the reasons.

The metric weight with l = 31 in figure 42 shows a long local meter built


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Figure 42: Metric weight of measures 1-32 for the wind instruments, l = 31

Figure 43: Measures 11 ff. in the wind instruments


)

upon the second beats of all measures. This beat is the only one the windinstruments place continuously notes on, whereas the first beats lack an onsetin measures 15 and 27 due to syncopations.

The score in figure 43 illustrates the rhythmic analogy with the passage in

the first movement of measures 58/59 (see figure 14). In both cases a synco-pation takes place because of tying the third beat together with the first beatof the following measure. But whereas the syncopation in the first movementsupports great metric weights on the third beats, in this case the syncopationsupports great metric weights on the second beats. On the one hand the third

beat is empty in several measures, on the other hand long local meters cannot be constructed on the first beats due to the empty first beat in measures 15 and27. Hence the comparison of these passages of the first and third movementenlightens the different influence of syncopations on the inner metric structuredepending on the context.

Other score elements seem to prevent an accentuation of the third beat inthe case of the third movement as well, as for instance the decrescendo endingon the third beat of measure 14 (see figure 43). This is in contrast with the

mentioned passage of the first movement, where the third beat is pronounced by a sf (see figure 14). Furthermore, the accentuation of the second beats in theisolated metric analysis of the wind instruments corresponds to the rhythmic


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)

Figure 46: Metric weight of measures 1-59 of the fourth movement


Figure 48: Detail of figure 47

figures of measures 8-10 and 29-31 as well.The following segment Presto (measures 33-106) notated as 2

4 is character-

ized by a quasi continuous sequence of eighth notes which allows a differenti-ation into two layers within the metric weight (see figure 44) that distinguishesstrong and weak beats, but within the highest layer of the first and second

beats no differentiation takes place. Metric coherence cannot be stated.On the contrary the analysis of the second segment Presto (measures 126-

189) notated as 38

reveals metric coherence (see figure 45).The three analyzed segments of different outer metric structure of the third

movement were characterized by very diverse relations between inner andouter metric structure. Whereas the metric weight of the first segment showeda periodicity corresponding to the measures but without differentiating therole of the third and first beats, the metric weight of the second segment didnot show any differentiation within the layer of the strong beats. The lastsegment was characterized by metric coherence.

3.4 Fourth movement

Regarding the fourth movement Allegro con spirito (Alla breve) we will dis-cuss in the following the two segments associated with the two main themes,namely the measures 1-59 of the first theme and 78-154 of the second.


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Figure 49: Measures 38 ff. for the wind instruments

Figure 50: Metric weight of measures 1-59 for the first violins

The metric weight of the first theme in figure 46 shows only within thefirst 23 measures a differentiation which corresponds to the Alla breve, thecontinuous sequence of eighth notes in the following sections prevents anydifferentiation of the weight.

On the contrary, the metric weight of the wind instruments of the same seg-ment (figure 47) reveals a differentiation starting at measure 23, which accen-tuates the second beat, whereas in measures 37 ff. even the second and fourth

beats of the measures get great metric weights (see also the detailed figure 48).The rhythmic motives of measures 37-40 (see figure 49) supports the emer-gence of long local meters on these ›weak‹ beats, which has been observed

already in analyses of the second movement of this symphony. Indeed, ana-lyses of the other symphonies in (Fleischer, 2003) support these findings as acharacteristic of Brahms’ works in general, as mentioned as well in (Frisch,1990).

The isolated analysis of the first violins of this segment in contrast to thisshows in large parts great metric weights on the first and third beats, whereasthe third beat is the most prominent one (see figure 50). Hence the metricweight of the wind instruments and first violins are complementary and inter-act in such a way that the analysis of the entire composition results in a metricweight with a lack of any significant differentiation whatsoever. In (Fleischer,2002b) this phenomenon therefore was called mutual annihilation.

The analysis of the second part (measures78-154) in figures51 and 52 revealsgreat metric weights on the second and fourth beats as well. The isolated met-

ric analysis of the string instruments in figure 53 again shows a complementarystructure, since the highest layer is built upon the first and third beats of themeasures. In contrast to this the isolated analysis of the wind instruments in


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Figure 53: Metric weight for the string instruments (measures 78-154)


Figure 55: Metric weight of measures 78-154 for the first violins

figure 54 gains a less structured weight, nevertheless great metric weights onthe second beat often can be found.

The latter we can observe within the isolated analysis of the first violinsin figures 55 and 56 as well. The great metric weights on the second beatscorrespond to the rhythmic accentuation of this beat by placing half notes onit (see figure 57), the low metric weights on the first beats correspond to thetying together of the first beat of the measure with the last beat of the previousmeasure, such as in measures 84/85, 87/88 or 89/90.

3.5 Summary Second Symphony

Inner metric analysis of the first movement of Brahms’ Second Symphony sug-gests a segmentation of the exposition into three parts of verydifferent regular-ity of the inner metric structure, which furthermore corresponds to a segment-


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Figure 57: The second theme in the first violins (measures 78 ff.) of the 4. move-ment

ation of the exposition regarding harmonic structure. Detailed investigationsof the three segments enlightened the influence of hemiolas and syncopationson inner metric structure. Further peculiarities are concerned with regularit-ies of the metric weight, which correspond to a different time signature as thenotated one, such as the 4

4 instead of 3

4 (figure 5) or 6

8 instead of 3

4 (figures 12,

15). Another peculiarity concerns those metric weights, which exhibit period-icities that allow no correspondence to any time signature (figures 22, 3). Thecoherent first part of the metric weight of the exposition (figure 4) answers Ep-stein’s question concerning the relation of down- and upbeat of the first twomeasures.

Inner metric analysis of the second movement enlightens a typical featureof the thematic material notated as 4

4 in all three segments, which consists in

great metric weights on the second and fourth beats (figures 24, 25, 26) andcorresponds to an observation in (Frisch, 1990) regarding the piano quintet.Frisch considers this phenomenon a typical characteristic in Brahms’ Œuvre.The mentioned lack of structural accents observed in the second movement in(Epstein, 1994) corresponds to a metric weight without a differentiation of thehighest layer (figure 29). Furthermore a hidden relation between the two partsof different time signatures was detected (figure 35). The analysis of the third

movement demonstrates the very different influence of syncopes on the innermetric structure depending on the context in comparison to the first move-ment. The analysis of the fourth movement enlightens a complementary shapeof the inner metric structure between different instrumental parts causing mu-tual annihilation. Furthermore the phenomenon of great metric weights onsecond and fourth beats occurs (figure 52) as was already observed in thesecond movement.

4 Third Symphony

4.1 First movement

Frisch (1990) characterizes the first movement of the Third Symphony regard-ing metric displacements as »perhaps the most impressive – indeed, encyclo-pedic – work in this respect«22. In comparison to the first movement of the


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Figure 58: Metric weight of the entire exposition of the 1. movement of the ThirdSymphony, interpreted as 6

4

Figure 59: Detail from figure 58 for measures 36-72

Figure 60: Metric weight of the measures 1-35

Second Symphony we can observe the opposite tendency regarding segment-ation processes and inner metric structure. Measures 1-35 are notated as 6

4,

measures 36-72 as 94

and measures 49-72 as 64

. But surprisingly the metricweight of the entire exposition in figure 58 allows an interpretation regarding6

4 of the entire exposition despite the different time signatures (see also the de-

tailed versions in figures 2 and 59). Hence whereas the inner metric analysis of the entire exposition of the Second Symphony enlightened a three-partition of the inner metric weight, the exposition of the Third Symphony is segmentedregarding outer but not regarding inner metric structure.

Nevertheless we will now discuss analyses of the isolated segments asso-ciated with the different time signatures.

The metric weight of the isolated first segment in figure 60 shows a corres-

pondence to the outer metric structure of

6

4 as already observed withinthe ana-lysis of the entire exposition. Within the first seventeen measures the greatestmetric weights are located on the fourth beats, whereas in the following sec-tion on the first beats of the measures. Hence metric coherence can be stated.This result does not correspond to an observation in (Frisch, 1990) postulatinga conflict between the notated 6

4 and an »implicit« 3

2-meter: »The main theme,

entering in the violins in the third bar, begins to project a metrical profile, butone that fits more clearly into 3

2 than 6

4. Only in bar 7 is the duple division of

the bar firmly supported in all parts: the theme, the ‘motto’, and the harmonicvoices move every half bar.«23

The continuous motion in eighth notes in the accompanying voices mightprevent the emergence of this metric peculiarity in the metric weight of allinstrumental parts. Therefore we want to discuss the isolated analysis of the

first violins (see figure 62). The result in figure 61 confirms Frisch’s observation.

22 Frisch (1990, p. 155)23 Frisch (1990, p. 156)


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Figure 61: Metric weight of the first violins, measures 3-35

Figure 62: The theme in the first violins


Within the measures 3-6 the weight corresponds to 32

, whereas the prominenceof the fourth beat as the characteristic of 6

4 establishes a hierarchy correspond-

ing to outer metric hierarchy in the following part.The isolated analysis of the second segment in 9

4 (measures 36-48) in figure

63 reveals a regularity of the metric weight, which does not correspond tothe outer meter. The highest layer is built upon the weights of every secondquarter note. Hence metric coherence cannot be found. Furthermore the isol-ated analysis differs to a great extend to the corresponding part within theresult of the analysis of the entire exposition.

The score in figure 64 illustrates very distinct rhythmic structures of instru-mental parts. The accompanying parts (bass, cello and viola) may serve as

orientation concerning the 9

4 -structure, since the lowest notes in the bass markthe first beats of the measures whereas the notes of the cello mark the strongfirst, fourth and seventh beats. The leading voice in the clarinet is rhythmic-ally very distinct. The theme begins on the second quarter note of measure 36,its entrance motive is repeated in measure 37 in a slightly varied form. Butin measure 38 the motive starts on the third beat and is hence shifted. Thistechnique of shifting corresponding motives regarding their position in themeasure Frisch called metric displacement. Hence we want to discuss the ana-lysis of the theme (see clarinet in measures 36-39, the violas in measures 40-43,the oboe and clarinet in measures 44-46 and the violins in measures 47-48).

The metric weight of the theme in figure 65 shows a very distinct regularitycompared with the weight of all instrumental parts of this section. Within thefirst measure the greatest metric weights are located on the third and seventh

beats, in the second measure on the fourth and sixth beats and in the thirdmeasure on the first and ninth beats. Obviously this does not correspond tothe outer metric hierarchy.


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Figure 65: Metric weight of the theme in measures 36-48

Figure 66: Metric weight of figure 65 interpreted as 64

The interpretation of the same weight as 64

in figure 66 at least results in aregularity which fits to the bar lines. The greatest metric weights are situatedon the first and third beats of the measures. If great metric weights would

be located on the fifth beats as well we could state metric coherence to a 32

-meter. But in many cases the fifth beats get low metric weights. Hence metriccoherence cannot be stated. Interestingly, Brahms notated the theme withinthe development (measures 77 ff.) in a slightly varied version as 6

4. Hence

the result of the isolated analysis of the theme might enlighten a hidden struc-ture of the theme Brahms possibly was even aware of. This hidden structureis furthermore confirmed within the analysis of the entire exposition, whichrevealed a correspondence of this second segment to the 6

4-meter in figure 58.

The metric weight of the isolated third segment of the exposition in figure 67

is characterized by great metric weights on the fourth beats of the measures inthe first half of the picture, similar to the results of the isolated analysis of thefirst segment. Interestingly in the second half of the picture again a periodicity


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Figure 69: Metric weight of the theme

can be stated similar to the isolated analysis of the second segment: everysecond beat gets a greater metric weight.

4.2 Third movement

The third movement Poco Allegretto (time signature 38

) is segmented into threeparts of different key signatures: measures 1-53 in E sharp Major, measures 54-98 in A sharp Major, the last part in E sharp Major repeats the opening theme.Here we want to discuss the two segments in E sharp Major (for the discussionof the middle part see (Fleischer, 2003), p. 115).

The metric weight of the first segment in figure 68 shows a characteristic of this movement. The interplay of the instrumental parts without any caesuraresult in a continuous motion causing a metric weight without any clear dif-ferentiation within the highest layer24. The three strong beats form the highestlayer, but within this layer no differentiation can be observed. The isolatedanalysis of the theme in figure 69 on the other hand shows a differentiationwhich is divided into two parts. Within the first part of the picture the greatestmetric weights are located on the first beats of the measures, whereas withinthe second part on the third beats of the measures due to the syncopations (seefigure 70). Metric coherence can be stated.

The metric weight of the third part (measures 99-163) in figure 71 reveals incontrast to the corresponding weight of the first part a differentiation withinthe highest layer, due to the slowing motion towards the end of the movement.Whereas in the first part no caesura appears, we can find them in the second

part at the end. For instance, onsets of notes are placed solely on the second

24 Similar phenomena can be stated within the A sharp Major part.


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Figure 70: The theme in measures 26 ff.


Figure 72: Metric weight of the theme of measures 99-163

beats of the measures 151, 155 and 160, whereas the other beats are empty.These notes’ placements cause the great metric weights on the second beatswithin the whole segment.

The analysis of the melody of the third segment in figure 72 on the otherhand is very similar to the corresponding analysis of the first part of the move-ment.

4.3 Summary Third Symphony

The inner metric analysis of the first movement reveals the opposite relation between inner and outer metric structure as in the Second Symphony: despitesegments of different notated time signatures the inner metric structure allowshomogeneous interpretation. Concerning the theme the mentioned discrep-

ancy between 32

and 64

in (Frisch, 1990) is confirmed by inner metric analysis.The weight of the second theme notated as 9

4 fits 6

4 better, which Brahms used

in the development of the first movement. Within inner metric analysis of the second movement the continuous motion prevents a differentiation of theweight, whereas the isolated analysis of the theme shows metric coherence,which is characterized in the second part by a phase shift.

5 Conclusion

The metric structure of Brahms’ Second and Third Symphonies have beenstudied by means of notes’ onsets. Thereby discrepancies between inner andouter metric structure described precisely the often mentioned ambiguities

in Brahms’ Œuvre, such as the occurrence of regularities in the inner met-ric weight corresponding to the hierarchy of a different time signature thanthe notated one. Furthermore the results of inner metric analysis depend to


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a great extend on the chosen contexts (e.g. segments, instrumental parts),thus demonstrating the very diverse compositorial layers created by anom-alous contrasting and counter-balancing of elements.

Inner metric analysis considers the onsets of notes whereas melodic or har-monic features are neglected. As the discussion concerning the third move-ment of the Second Symphony (p. 18) illustrates, the melodic shape may me-diate metric accents in situations where inner metric analysis does not. Inso-

far melodic and harmonic information may contribute further insights into theanalysis of metric structure. Nevertheless the presented approach gains prom-ising results, whereas the consideration of great data bases, such as an entireexposition, are of great importance. The limits of punctual analysis are partof the critics Frisch (1990) mentioned regarding Schönberg’s approach: »... theanalyses never advance beyond the level of the individual theme, making noattempt to show how the shifting bar lines might affect the larger frameworkor dimension of a piece.«25 The analysis of large contexts permits the invest-igation of metric characteristics of themes and associated segments, as in thecase of the second movement of the Second Symphony. In (Fleischer, 2003) theanalysis of the entire first movement of the Fourth Symphony gained inter-esting results regarding the comparison of corresponding parts in exposition,development and reprise.

Brahms’ compositions in many cases are characterized by discrepancies between inner metric structure and the metric hierarchy given by the time sig-natures. Therefore metric coherence very often cannot be found. The observeddiscrepancies correspond in many cases to observations made by music the-orists (such as the phenomenon of metric displacement and great weights on›weak‹ beats) and hence may serve as precise descriptions with the method of inner metric analysis.

We have discussed some of the examples in detail concerning the questionwhich local meters are mainly responsible for the occurence or disturbance of metric coherence. This has been done by varying the parameter l. Further de-velopments of the software tool (a Java application JMetro has been designedin our research group KIT-MaMuTh and has been implemented by Chris Dyerand Monika Brand) now allow the listing of all local meters considered in themetric weight. Thus we detected some examples, for instance, revealing that

great metric weights on the beginnings of measures are not in each case due tolocal meters with a period26 of the measure (or multiples of the period). Evenafter excluding all local meters with a period of the measure, half-measure andmultiples of the measure the metric weight was characterized by metric coher-ence. This promises new insights into the complexity of the processes whichcause the emergence of metric coherence.

25 Frisch (1990, p. 149)26 the distance between consecutive onsets of the meter


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References

EPSTEIN, D AVID (1987). Beyond Orpheus. Oxford University Press.

EPSTEIN, DAVID (1994). Brahms und die Mechanismen der Bewegung: die Kom- position der Aufführung. In MEYER, MARTIN (ed.), Brahms-Studien, vol. 10,pp. 9–21. Johannes-Brahms-Gesellschaft, Hamburg.

FLEISCHER, ANJ A (2002a). A Model of Metric Coherence. In BUSIELLO, STEFANOET AL (ed.), Understanding and Creating Music. Caserta.

FLEISCHER, ANJ A (2003). Die analytische Interpretation. Schritte zur Erschließungeines Forschungsfeldes am Beispiel der Metrik . dissertation.de - Verlag im Inter-net Gmbh, Berlin.

FLEISCHER, ANJ A & THOMAS N OL L (2002b). Analytic Coherence and Perform-ance Control. In D E POL I & GUNNAR JOHANNSEN, GIOVANNI (ed.), Supervi-sion and Control in Engineering and Music, vol. 31, No.3, pp. 239–247. SpecialIssue of Journal of New Music Research.

FRISCH, WALTER (1990). The shifting barline: Metrical displacement in Brahms . InBOZARTH, GEORGE S. (ed.), Brahms Studies, pp. 139–163. Clarendon Press,

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MAZZOLA, GUERINO (2002). The Topos of Music. Birkhäuser, Basel.

SCHÖNBERG, A RNOLD (1976). Stil und Gedanke. In VOJTECH, I VAN (ed.), Ges-ammelte Schriften, vol. 1, pp. 35–71. S. Fischer, Frankfurt/Main.

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