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A Model of Metric Coherence

Anja Fleischer

Abstract

This paper discusses a notion of metric coherence, based upon a mathematical model ofthe inner metric structure of a piece of music as implemented in the ’RUBATO Worksta-tion for Musical Analysis and Performance’. The inner metric analysis studies the metricstructure expressed by the notes of a given piece without considering the time signatureand barlines and results in a metric weight for each note of the score. The given timesignature defines a certain regular accent structure upon the possible notes or onsets ofa bar which we call outer metric structure. The notion of metric coherence describes thecorrespondences of varying degrees between the outer and inner metric structure. As aresult of the explorative work with the model, a higher degree of coherence was detectedwithin those works, which are typical representations of the accent scheme given by thetime signature. Furthermore, metric ambiguities within pieces of different composers, suchas the works of Johannes Brahms, can be described as divergence between inner and outermetric structure. Since RUBATO enables the transformation of analytical results into var-ious performance parameters, the definition of metric coherence was tested furthermore byusing metric weights to shape a performance of the analyzed piece and to evaluate theseperformances within an empirical experiment. As the outcome a relationship was detectedbetween metric coherence and the understanding of the corresponding interpretation by thelisteners.

1 Introduction

The time signature and barlines define a regular accent structure upon the notes withinthe bars. Or do they merely assist for orientation by dividing the sequence of notes ofa piece of music into units of equal duration? The example of figure 1 is described in[de la Motte 1981] p. 32 as a typical example of the varietas technique which avoids anykind of repetition. Hence it provokes the question, whether it would be proper to assign aregular accent structure to the notes which expresses the metricity of the piece accordingto the barlines or not. A very distinct example is given in figure 2. The piece by Hasslerbelongs to those Renaissance madrigals which are obviously influenced by dances of a clearmetricity.

A Model of Metric Coherence 2

Fig. 1: First 10 measures of the soprano of the Kyrie II of Dufay’s mess “Se la

face ay pale” (according [de la Motte 1981], p. 32)

Fig. 2: Beginning of the madrigal “Tanzen und Springen” by Hans Leo Hassler

These two examples may illustrate that the role of the time signature and barlines con-cerning the metricity of a piece of music differ to a high degree between various musicalstyles.

Hence the question arises, as to how far the different influence of the accent hierarchygiven by the time signature can be described precisely. In the following we want to introducea mathematical model realized by the MetroRubette1 of the software RUBATO whichdescribes the metric structure of the notes without considering the time signature andbarlines and which therefore can help us answering the above mentioned questions.

2 Outer and inner metric analysis

Before getting into details concerning the metric analysis realized by the MetroRubette wewant to sketch the idea of “periodicity” which seems to be of important role for the notionof metricity, as the following quotations suggest.

The elements that make up a metrical pattern are beats. ... metrical structureis inherently periodic. We therefore assert, as a first approximation, that beatsmust be equally spaced. ... Fundamental to the idea of meter is the notion ofperiodic alternation of strong and weak beats;2

Lerdahl and Jackendoff describe metric structures on the basis of beats which are equallyspaced. In a similar way Parncutt describes a pulse as a set of equally spaced elements:

1name for the tool which is concerned with metric analysis2[Lerdahl/Jackendoff 1983], p. 18/19


A pulse is a chain of events, roughly equally spaced in time. A pulse may becompletely specified by two pieces of information: its period and its phase. Theperiod is the time interval between successive events. The phase may be specifiedby the actual time at which any of the events occurs, relative to some referencetime.3

The model of the MetroRubette is based upon the detection of “metrical patterns” or“pulses” in a similar way. All onsets of notes which are equally distanced form an innerlocal meter. Since the model looks for metric regularities inside the given piece we call thecorresponding analytic structure inner metric structure which is opposed to the outer metricstructure given by the time signature and barlines. (In [Fleischer/Noll 2002] outer metrichierarchy is defined as a hierarchy of a metric scale {p1|p2| . . . |pn} of mutually dividingperiods and a common phase where all metric layers meet. Period and phase are defined asin the quotation from [Parncutt 1987].)

The model looks for all inner local meters of the given piece containing at least threeonsets. Hence inner metric analysis treats a composition as if it were written for an ensembleof metronomes. They vary in period and phase and are switched on and off whenever theycan click along the onsets of the notes of the piece.

Fig. 3: A short example with all inner local meters

Figure 3 gives an example showing all local meters of a short piece of music, which aredenoted by a, b, c, d and e. The first row below the notes illustrates the projection ofthe notes onto the set of onsets starting with 0. In mathematical terms, a local meter isa subset of onsets. The set of all onsets of this short piece of music we denote by X ={0, 2, 3, 4, 6, 10, 12}, the local meter a is then defined by a = {0, 2, 4, 6}. It requires a furtherrefinement of the term inner local meter. Only maximal local meters are considered withinthe analysis, e.g. those local meters, which are not subsets of other local meters. In the

3[Parncutt 1987], p. 132


example the local meters a′ = {0, 2, 4} and a′′ = {2, 4, 6} are subsets of a, therefore theyare not considered.

2.1 The inner metric weight

The outcome of the inner metric analysis is the inner metric weight, defined for each onset,which represents the numerical codification of the metric meaning of each note within itscontext based upon the detection of the inner local meters. One possible way of codificationof metric meaning of an onset o is the number of local meters of the piece which contain o.But the inner metric weight takes a further information into account, which is the lengthof the local meters.

Let ms,d,k = {s + id, i = 0, . . . , k} denote a local meter with a first onset in s, periodd and length k. The calculation of the metric weight takes only the length k into account,therefore we denote it by mk. The weight wp(mk) of the meter mk is defined by

wp(mk) = kp,

where p denotes a specific exponent which can be varied by the user. The weight Wp(o) ofan onset o of the piece is calculated as the sum of the weights wp(mk) of all local metersmk which contain o:

Wp(o) =X

∀mk:o∈mk

kp. (1)

Equation 1 enlightens the role of p. The higher the value of p the higher the contribution oflong local meters to the metric weight, the shorter the value of p the higher the contributionof short local meters. The MetroRubette provides another parameter for the user whichinfluences the outcome of the analysis. The parameter l controls the minimum of the lengthk of the local meters being considered in equation 1. All local meters m with k(m) < l arenot considered which results in the following equation:

Wl,p(o) =X

∀mk:k≥l,o∈mk

kp. (2)

Since Wl,p(o) is based on the inner local meters we call it the inner metric weight. Recallingthe idea of outer metric structure we define the outer metric weight of an onset o as thenumber of layers to which it belongs.

Figure 3 shows the metric weight Wl,p = W2,2 for each onset above the notes. Thelength of the black lines represents the numerical value of the metric weight: the higher theline the greater the weight. The choice l = 2 results in the consideration of all local metersfor the calculation of the weight because of the restriction in the definition of local meter(a local meter consists of at least three onsets). The choice p = 2 results in the quadraticinfluence of the length of each local meter on the weight. As figure 3 shows, the onset o = 6get the greatest weight, which is contained in the four local meters a, c, d and e which isthe greatest number of meters in this example (see the following table).


onset o 0 2 3 4 6 10 12

local meterscontaining o a, c, e a, b, d b, c a, b a, c, d, e d a

W2,2 17 17 8 13 21 4 4

W2,0 3 3 2 2 4 1 1

The values of W2,0 of this short examples demonstrate, that the choice p = 0 results inmetric weights, which count the number of considered local meters depending on l. In thecase of l = 2 it represents the number of all local meters. The influence of the length of thelocal meters in W2,2 can be studied by comparing, for instance, the weights W2,2(3) andW2,2(4). Both onsets are contained in two local meters, but since onset 3 is contained intwo local meters of length 3 and onset 4 is contained in one local meter of length 4 and inone of length 3 it gets a greater value W2,2(4).

The metric weight of this example demonstrates that the inner metric analysis yieldsa very different information about the piece in comparison to the outer metric analysis.It provokes the question, in how far the above described model of inner metric structurecan enlighten important metric characteristics of pieces of music. This question can onlybe answered within an intensive explorative work with the model. The results of such anintensive work are documented in [Fleischer 2002], in the following we want to give a briefoverview.

2.2 Some results of the application of the inner metric weight

A surprising finding in [Fleischer 2002] of the application of the model concerns the greatamount of pieces, which show an obvious correspondence between inner and outer metricstructure.

Fig. 4: Excerpt from the metric weight of the entire exposition of the 1. movement

of Brahms’ Third symphony (measures 1-35), W2,2,64, gray lines in the background

mark the barlines

Figure 4 shows an excerpt from the metric weight of the entire exposition of the firstmovement of Brahms’ Third Symphony (time signature 6

4). The first beat of all measures

get the greatest metric weight, followed by the metric weight of the fourth beat. The second,third, fifth and sixth beat form a lower layer, whereas the ’weak beats’ get the lowest weights.Obviously there exists a strong correspondence between the inner metric weight and thehierarchy of the accents of the outer metric structure.

Another example in figure 5 shows an excerpt from the first movement of Brahms’Second Symphony. Epstein argues in [Epstein 1994] p. 10, that the melodic motives (cello,


Fig. 5: Measures 136 ff. in string instruments (first movement of Brahms’ Second

Symphony)

bass, supplemented by the bassoon) in this passage sound as if their beginnings wouldcoincide with the first beat of the measures – but actually they begin on the second beat.

Fig. 6: Metric weight W2,2 (measures 127-155) of bassoon, cello and bass

The metric weight of this part in figure 6 shows a periodicity which corresponds to theperiodicity of the noted time signature 3

4, the phase is shifted by one beat. The greatest

metric weight is situated on the second beat of the measure which according to Epstein“sounds” as if it would be the first beat. This case is a typical conflicting situation betweengrouping and meter: the beginnings of groups do not coincide with beginnings of measures.Whenever the relation between grouping and meter is quite stable, as in this case, one canfind traces in the metric weight. The upbeat gets greater metric weight than the first beatof the measure. We call this phenomenon an upbeat of coherent character.

Fig. 7: Metric weight W2,2 for “Adieu sweet Amarillis” by John Wilbye, 44

Figures 7 and 8 show two examples for the time signature 44. In both cases the greatest

metric weights are situated on the first beat of the measure, followed by the weights of thethird beat, the second and fourth beat form a much lower layer.


Fig. 8: Excerpt from the metric weight of the entire exposition of the 1. movement

of Mozarts symphony K 551 (measures 1-55), p = 2 and l = 2, 44

Last but not least we want to return to the both examples mentioned in the beginningin section 1 (see figures 1 and 2).

Fig. 9: Metric weight W2,2 for “Tanzen und Springen” by Hassler, 34

The metric weight of the Renaissance madrigal by Hassler in figure 9 shows a clear corre-spondence to the accent hierarchy of the 3

4time signature.

Fig. 10: Metric weight W2,2 for the soprano of the Kyrie II of the mess “Se la face

ay pale” by Guillaume Dufay

This correspondence is not the case concerning the piece of Dufay (see figure 10). Themetric weight shows no periodicity which reminds us of the varietas technique underlyingthis piece – repetitions are avoided, hence periodicities within the set of onsets are veryrare. The different role of metric accent hierarchies in the Renaissance madrigal and thepiece of Dufay is therefore properly described by the inner metric weight.

Furthermore a correspondence between inner and outer metric structure can very oftenbe stated in those works, which are typical representations of the accent scheme given bythe time signature, as the above given examples illustrate. This leads us to the definitionof metric coherence.


2.3 Metric coherence

Whenever a correspondence between inner and outer metric structure can be stated, metriccoherence occurs. Therefore a presupposition for coherence is the occurrence of regularityor periodicity in the inner metric weight, i.e. the existence of weight layers correspondingto specific periods. Metric coherence is hence concerned with a metric subscale of the outermetric hierarchy (see page p.2) being significant in the inner metric weight. Furthermoreone may distinguish phase-coincidence from phase-displacement on each layer. The latteris the case in the example of an upbeat of a coherent character in figure 6. An examplewhere no coherence occurs is given in figure 10. Phase-coincidence can be observed in theexamples of figure 4, 7, 8, and 9.

3 The influence of l and p

The model of inner metric structure does not provide a classification of “useful” choicesfor the two parameters l and p a priori. The weights discussed above were calculated withthe parameters l = 2 and p = 2. The following examples concerning pieces of Handel andSchubert illustrate in how far the variation of the parameters can enlighten the differentcontribution of long and short local meters to the constitution of metric coherence.

3.1 Handel: Sarabande D-Minor

Fig. 11: George Frederic Handel: Sarabande from Suite D-Minor

Handel’s Sarabande (see figure 11) is a stylized version from the Baroque era of a folk dancewhich probably came from Mexico. Hence the question arises as to how far the originalmetric characteristics of the dance Sarabande could be recovered within this stylized formof piano music. In the following we discuss the analysis of the theme including the repetition.

At first we choose the fixed parameter p = 2 and vary the parameter l.

Fig. 12: Metric weight W31,2


The longest local meter m of length k = 31 is built upon the first onset of the measures.Therefore the first differentiation of the metric weight can be stated by choosing l = 31.All other onsets get the weight 0 (see figure 12).


Within the theme there exist no local meters of length 30, 29, 28, 27, and 26. Hence thenext differentiation of the weight can be observed within the analysis with the parameterl = 25, as shown in figure 13. There are two local meters with length 25 built upon thesecond semibreve and the last minim of each measure (one local meter within the measures1-13, the other within the measures 17-29). The first onset of each measure get the greatestweight.


There exist no local meters of length 24, 23, 22, and 21, hence the next differentiationcan be stated within the metric weight W20,2 (see figure 14). An inner local meter m withk(m) = 20 is built upon the onsets of every second semibreve of each second measure andthe last minim of each second measure. The greatest weights within each measure are nowlocated on the second semibreve, the last minim or first semibreve alternating. The firstonset of each measure are contained only in the longest local meter and do not contributeto shorter local meters.


The next differentiation of the weight by using l = 15 is shown in figure 15. We can observea differentiation within the highest layer of the weight (the second semibreve and the last


minim of each measure), the first onsets of each measure again do not contribute to a shorterlocal meter.


The metric weight for l = 14 in figure 16 shows a further differentiation within the layersbuilt upon the second semibreve and the last minim of each measure.


The beginnings of each measure contribute to a local meter of length 13, which can beobserved in figure 17: the weights of the beginnings are now differentiated into varyingvalues.


Figure 18 shows the finest differentation W2,2. The greatest metric weights of the measuresare situated on the first semibreve in the following measures: 4, 6, 8, 10, 12, 14, 16, 18,24, 26, 30, and 32. In the measures 1, 3, 7, 9, 11, 13, 15, 17, 19, 21, 22, 23, 25, 27, 28,29, and 31 the greatest metric weights are located on the second semibreve. Interestingly,the metric characteristic of the original Sarabande is a strong accent on the second beatof the measure. The competing role of the first and second semibreve within the innermetric hierarchy establishes at least a trace of this original characteristic, but prevents theemergence of metric coherence.

The choice of a greater value for the parameter p enlightens the contribution of the longlocal meters to the metrical architecture of the piece.

The choice of the parameter p = 3 in figure 19 results in growing weights of the firstsemibreve of each measure.




The analysis using p = 4 in figure 20 shows a regular metric weight with greatest values onthe beginning of each measure.

The incrementing of the values of p results in a correspondence between the inner andouter metric structure regarding the noted 3

2time signature: the greatest weights are located

on the first beat of each measure. This correspondence was not observable in the metricweight W2,2, where the competing role of the first and second semibreve did not allowa regular pattern of metrical hierarchy. The analyses produced with varying values of pand l elucidate the different role of long and short local meters within the constitution ofinner metric hierarchy. Shorter local meters prevent the emergence of periodicity withinthe metric weight but enlightens the prominent role of the second beat of the Sarabandewhereas longer local meters are responsible for the emergence of a correspondence betweenthe inner and outer metric structure. Hence the influence of the original dance is verydifferent in comparison to the Renaissance madrigals discussed in [Fleischer 2002]. Thelatter are characterized by metric coherence, whereas the stylized example from the Baroqueera shows only a trace of the metricity of the original dance.

3.2 Schubert: Moment Musical op. 94 No. 4

The B-part in D flat Major of Schubert’s Moment Musical op. 94 No. 4 provides anotherinteresting example regarding the study of the distinct role of long and short local meters.Both melody, harmony and rhythm seem to be shaped in a way, that the quarter note onthe second eighth in figure 21 (see the accent mark) might be interpreted as the ’actual’beginning of the bar. By shifting the barline the grouping structure of the piece would bein accordance with an anacrousis or upbeat.

If we exclude all local meters with a length shorter than 79, as shown in figure 22, thefirst three sixteenth of each bar get the same weight. By excluding the shorter local meters


Fig. 21: Beginning of part B of the Moment Musical op. 94 No. 4

Fig. 22: Excerpt from the metric weight with l = 79

we therefore do not get an answer concerning the question of the ‘right’ placement of thebarlines.

Fig. 23: Excerpt from the metric weight with l = 38

Within the further refinement of the metric weight by decrementing the value of l the firstinteresting metric weight concerning the question of the ‘proper‘ location of the barlinesoccurs for l = 38 (see figure 23). The greatest metric weight is situated on the second eighthof the bar, which is the case within all further refinements of the metric weight betweenl = 38 and l = 3 as well. Therefore we can state a phase-shift within the highest layer ofthe metric weight. Another example of phase-shift is given in example 6 which we calledan upbeat of coherent character, since the greatest metric weight is situated on the upbeatinstead of the first onset of the bar.

The latter does not apply to the Moment Musical. The greatest metric weight is locatedon the second eighth and does not coincide with the beginning of the grouping. Instead,the greatest metric weight coincides with the accent mark which might be interpreted asthe ’actual’ beginning of the bar. Obviously, the role of inner and outer metric structure isreversed in comparison to the situation of an upbeat of coherent character as in example 6.Figures 24 and 25 illustrate this reversion by using the accent mark in order to identify theaccent of the inner metric structure and the V-shaped sign in order to identify the accentof the outer metric structure.

The number of inner local meters, which are considered within an analysis for l = 3,amounts to 576. Within the finest differentiation of the metric weight by chosing l = 2further 6391 local meters of length k = 2 are included in the calculation. By enlarging the


Fig. 24: Accents of inner and outer metric structure as they apply in the Moment

Musical

Fig. 25: Accents of inner and outer metric structure in the case of the upbeat of

coherent character

contribution of these short local meters to the weight by chosing p = −2 one gets a metricweight of a very different shape (see figure 26).

Fig. 26: Metric weight for l = 2, p = −2

The metric weights of the first and second sixteenth of each bar are very small at thebeginning, but they increase up to the middle of the piece and decrease towards the endof the piece. This process is responsible for the two ’pyramids’ which can be observed infigure 26. The metric weight of the first onset towers above the second eighth in bar 74 forthe first time, which is the first bar in this part B of the Moment Musical, where the accentmarks on the second eighth disappear in the score. The highest metric weight of the firstonset coincides with the beginning of the reprise in bar 89.

The great amount of short local meters results from the repetitions of the rhythmicmotive of the first three notes within the bars and end in an ambiguity, which arises fromthis process of repetitions. Short local meters are in this case responsible for the ambiguityas well as for the disturbance of metric coherence.


4 Schumann’s Third Symphony

In the following section an example of Schumann’s Third Symphony will demonstrate aspecial relationship between inner and outer metric structure.

Fig. 27: Measures 1-20 of the first violins, first movement of Robert Schumann’s

Third Symphony

Epstein describes a peculiarity concerning the metric organization of the first movement:

... that the first six measures of the opening theme, in their metric organizationand in its reinforcement through orchestration, make unclear whether the musicis to be perceived as 3

4... or as 3

2... The two meters possible through this

ambiguity are played upon throughout the movement, at times one version orthe other explicit, at other times both possibilities implicit, ... creating situationsincapable of a definitive singular interpretation.4

He observes an ambiguity between the noted and perceived meter or time signature in theopening: the noted meter of 3

4(see figure 27) is perceived as 3

2.

Fig. 28: Metric weight W2,2 of measures 1- 20 of the first violins, 1. movement of

Schumann’s Third Symphony

The metric weight of the opening in the first violins in figure 28 shows a significant regularity.However, this regularity does not correspond to the noted 3

4time signature, since it expresses

a periodicity of two. The reinterpretion of this metric weight according a 22

time signaturein figure 29 demonstrates this periodicity even more evidently.

The inner metric weight enlightens in this case a discrepancy between the noted (outer)metric structure and the inner metric structure of the notes. The regularity of the innermetric weight shows a different periodicity than the outer metric regularity. Furthermorethe reinterpretation of the weight suggests a corresponding with a 2

2time signature, not

with a 32

time signature, as proposed by Epstein.

4[Epstein 1987], p.154


Fig. 29: Metric weight W2,2 of measures 1- 20 of the first violins, interpreted as

2/2

Fig. 30: Part of the metric weight W2,2 for the first violins of the entire exposition

(measures 1-92)

Since he stated this ambiguity between the two meters not only concerning the opening, butwithin the entire movement, the question arises, how this ambiguity is constituted withinlarger parts of the movement. Figure 30 shows an excerpt of the metric weight for the firstviolins of the entire exposition regarding the first 92 measures. In the first instance one canobserve a grouping into units of two measures due to the greater metric weight of the firstbeat of each second measure. Furthermore the weight of the first 20 measures differs to agreat extend to the weight of the separate analysis of these measures in figure 28 as thedetailed representation in figure 31 illustrates.

Fig. 31: Part of the metric weight of figure 30 (the first 20 measures), interpreted

in the original time signature

The periodicity of the weight of the isolated first 20 measures in figure 28 caused in anaccent hierarchy, which contradicts the accent hierarchy of the noted time signature, as forexample in the measures 3, 7, 8, 11 or 15 of the theme. In these measures the greatest metricweight is not situated on the first beat. The periodicity of the weight in figure 31 contradictsthe accent hierarchy of the noted time signature first in measure 16, where the second beatgets a higher weight than the first beat. The reeinterpretation of the weight according to a32

meter illustrates a more convincing interpretation of the weight’s periodicity (see figure32).

The evaluation of the weight according to this interpretation shows an avoidance of anydiscrepancy between inner and outer metric structure of the 3

2meter: in each measure the


Fig. 32: Metric weight of figure 31, interpreted as 32

greatest weight is situated on the first beat of the measure. Therefore this interpretation isthe most plausible one. Analyzing the greater context of the entire exposition results in aperiodicity of the weight within the first 20 measures, which best corresponds to the timesignature 3

2, as suggested by Epstein.

Fig. 33: Part of the metric weight W2,2 for the first violins of the entire exposition

(measures 1-92), interpreted as 32

Fig. 34: Part of the metric weight for the first violins of the entire exposition

(measures 93-184), interpreted as 32

Furthermore the evaluation of the weight of the entire exposition (see figures 33 and 34)detects a clear correspondence with the 3

2meter throughout this entire part.

Epstein stated an ambiguity between noted and perceived meter within this momeventof Schumann’s symphony. By comparing the inner and outer metric structure we can de-scribe this ambiguity as a discrepancy between the periodicities of the inner and outermetric weights. Furthermore the choice of different contexts, such as the first 20 measuresor the entire expositions, resulted in different periodicites of the weights, demonstrating theimportant influence of the context on the inner metric structure. Hence beside the variationof the parameters l and p the variation of the context can enlighten interesting principlesof metric organization.


5 Brahms

Metric ambiguities are a characteristic of the works of Brahms as well, as mentioned in[Epstein 1987]: “Schumann’s Third Symphony ... suggests an interesting kinship in musicalthought between Schumann and his younger protege, Brahms, who had to extend so manyof these inclinations to their most complex reaches. In his recognition of ambiguity as acompositional value of extensive implications”5. These ambiguities are discussed as wellin [Frisch 1990] or [Schonberg 1976]. In [Fleischer 2002] the four symphonies by Brahmsare discussed very detailed, here we want to give ate least two examples from the secondsymphony concerning metric peculiarities.

A phenomenon, which very often can be observed within inner metric analysis (e.g.[Frisch 1990] p. 147) is shown in figure 35: “weak” beats concerning the outer metric hi-erarchy get great inner metric weights, as in this case the second and fourth beats of themeasures.

Fig. 35: Brahms’ Second symphony, 2. movement, metric weight W2,2 of bars 1-32

( 44)

Frisch characterizes the already mentioned passage from the first movement in figure 5as “massive canonic and metrically disorienting episode” ([Frisch 1990], p. 155), Epsteindiscusses this passage as well in [Epstein 1994].

Fig. 36: Metric weight of the bars 127-155 for all instrumental parts

Within the metric weight of all instrumental parts in figure 36 the great metric weights onthe first and third beat as well as on the fourth and sixth beat prevent a correspondenceto any time signature.

Fig. 37: Metric weight of the bars 127-155 for violins, bassoon, cello and bass

5[Epstein 1987], S. 157


The metric weight of the violins, bassoon, cello and bass in figure 37 as well as the alreadydiscussed weight of the bassoon, cello and bass in shown in figure 6 however are characterizedby metric coherence.

Fig. 38: Metric weight of the bars 127-155 for clarinet, horn and violas

On the contrary, the metric weight of the syncopations in the clarinet, horn and violas infigure 38 shows a very distinct periodicity which cannot be interpreted in accordance toany time signature: great metric weights are located on the third and fourth sixteenth, aswell as on the seventh and eighth sixteenth and eleventh and twelfth. The analyses of theseparate voices of this passage in figures 6, 37 and 38 enlightens the very diverse rhythmicvoices of this passage which contribute to a metric structure (see figure 36) that is lackinga correspondence to the given time signature.

6 Listening experiments

Since RUBATO enables the transformation of analytical results into various performanceparameters that affect the musical performance, realized as a midi-file output, the notionof metric coherence was tested concerning the question, whether the analytic informationcan help to shape a performance of the piece in such a way, that it expresses the structureof the piece to the listeners in a meaningful way.

Listening experiments have been performed by means of RUBATO-performances (formore details see [Fleischer 2002]. Drum rhythms were played with various structures ofaccentuation, arising from metric weights of different degrees of coherence. Figures 39, 40,and 41 give examples for different metric weights in the case of the vocal parts of theCredo-fugue from Bach’s B Minor Mess which were used for listening experiments.

Fig. 39: Metric weight of the complete Credo-Fugue for all 5 voices, p = 2 and

l = 2

The metric weight of the complete Credo-Fugue in figure 39 demonstrates a regularity of ahigh degree. The layer of the weights of the first and third beat can be separated from thelayer which is built upon the weights of the second and fourth beat. The weak beats forma much lower third layer.


Fig. 40: Metric weight of the first soprano of the Credo movement, p = 2 and

l = 2

Within the metric weight of the first soprano in figure 40 accents can be observed of thefourth beat of each measure in the second half of the picture. In the first half of the figuregreat metric weight are sometimes located on the first and third beat, but on the secondor fourth beat as well. Therefore such a clear regularity as in figure 39 cannot be stated.

Fig. 41: Metric weight of the second soprano of of the Credo movement, p = 2

and l = 2

The metric weight of the second soprano in figure 41 is characterized by great weights onthe first and third beats of each measure, but they cannot be separated by layers as in thecase of the analysis of all vocal parts as in figure 39. The weights of the other vocal partsare similar: the regularity which can be observed in these weights is of much lower degreein comparison to the regularity of the analysis of all vocal parts.

By comparing these weights for the same piece the question arises, whether a perfor-mance based upon the metric weight of figure 39 would lead to a more convincing interpre-tation regarding the question in how far the metric structure was expressed properly thana performance based upon the metric weights of the separate voices.

Another question being considered in the listening experiments concerns the descriptionof adequacy or suitability of the chosen structures of accentuation within the performance.In how far is it important, that the structures of accentuation “fit” with the inner metricstructure? In order to test this question a metric weight stemming from another piece whichis characterized by coherence of a high degree was used to perform the piece. Listenersevaluated the performances by comparing pairs of different performances and assigning thebetter one.

As the outcome of the experiment conducted with 46 persons, a relationship was de-tected between metric coherence and the understanding of the corresponding interpretationby the listeners. Metric weights of higher degree of coherence led to a more convincing in-terpretation regarding the question in how far the metric structure was expressed properly.This result may be taken as an indication for a relationship between analytical structuresof the score and the understanding of the performed music by listeners through suitableexpression of these structures within a performance.


References

[de la Motte 1981] de la Motte, Diether: Kontrapunkt. Barenreiter 1981.

[Epstein 1987] Epstein, David: Beyond Orpheus. Oxford University Press, 1987.

[Epstein 1994] Epstein, David: Brahms und die Mechanismen der Bewegung: die Kom-position der Auffuhrung. In: Brahms-Studien, Bd. 10, Hrsg.: Martin Meyer im Auftragder Johannes-Brahms-Gesellschaft, Hamburg 1994, 9-21.

[Fleischer 2002] Fleischer, Anja: Die analytische Interpretation. Schritte zur Er-schließung eines Forschungsfeldes am Beispiel der Metrik. Dissertation at HumboldtUniversity Berlin.

[Fleischer/Noll 2002] Fleischer, Anja & Thomas Noll: Analytical Coherence and Per-formance Regulation. In: Johannsen, G. and G. De Poli (Guest Editors): Human Su-pervision and Control in Engineering and Music. Special Issue, Journal of New MusicResearch, Vol. 31, No. 1 (in print).

[Frisch 1990] Frisch, Walter: The shifting barline: Metrical displacement in Brahms. In:Brahms Studies, Hrsg.: George S. Bozarth, Clarendon Press, Oxford 1990, 139-163.

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[Parncutt 1987] Parncutt, Richard: The Perception of Pulse in Musical Rhythm. In:Action and Perception in Rhythm and Music, Hrsg.: Alf Gabrielsson, Royal SwedishAcademy of Music No. 55, 1987, S. 127-138.

[Schonberg 1976] Schonberg, Arnold: Stil und Gedanke. In: Gesammelte Schriften Bd.1, Hrsg.: Ivan Vojtech, Frankfurt/Main, S. Fischer, 1976.

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