- 303 -Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory 
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layer). Hence four layers corresponding to the layers of the outer metric hierarchy can be detected within inner metric analysis.


PIC
Figure 3: Metric weight W2,2 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 characterized 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 hierarchy 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 call this phenomenon of a phase shift an upbeat of coherent character (for further discussion of this example see p. 653), since it occurs in those cases of a stable relation 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 the most cases the parameters l = 2 and p = 2 , hence only cases with different values for l and p will be indicated.6

 
6  
For a more detailed discussion of the influence of l and p see (Fleischer, 2002a).

The author’s discussion of metric coherence concerning very different styles in music history in Fleischer (2003) proved the suitability of this music theoretical term regarding the description of metricity of compositions. Metric coherence 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. Renaissance madrigals), whereas, for instance, in compositions by Bach coherence occurs more rarely. Hence inner metric analysis might serve as an appropriate method in order to describe metric ambiguities within Brahms’ compositions as well.

3 Second Symphony

»Perhaps no composer of the period so reveled in the structural possibilities of ambiguity as did Brahms. His Second Symphony is a case in point, ambiguous properties inherent in the basic ideas of the opening movement


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- 303 -Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory