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

The formula of the metric weight $Wl,p(o)$ in equation 1 depends furthermore on the two parameters $l$ and $p$ , which can be varied by the user of the software. The parameter $l$ denotes the minimal length of local meters being considered in the calculation of the metric weight. By increasing the values of $l$ the user can exclude short local meters (shorter than $l$ ) from the calculation, whereas the parameter $p$ weights the contribution of the local meters to the metric weight depending on their length. Great values cause a greater contribution of longer local meters, small values cause a greater contribution of shorter local meters to the metric weight. By varying these parameters the user can obtain different metric perspectives on the same piece.

The metric weight of equation 1 describes the inner metric structure of a piece of music. Obviously it depends solely on the regularities caused by the notes of the piece without considering information given by the time signature. 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 introduced and discussed in Fleischer (2003), Fleischer (2002b) and Fleischer (2002a). Whenever a correspondence between inner and outer metric structure can be observed, metric coherence occurs.

Figure 1:

Metric weight $W2,2$ 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 1 shows an example regarding the time signature $2 4$ in Brahms’ Fourth Symphony. The metric weight $W2,2$ is characterized by different layers which 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 the second beats of all measures. The weights of the second and fourth eighths form a much lower layer, whereas the weak beats, such as the second and fourth sixteenths form the lowest layer. Hence metric coherence occurs.

Figure 2:

Excerpt (measures 1-35) from metric weight $W 2,2$ of the entire exposition of the 1st movement of the Third Symphony (time signature $6 4$ )

Figure 2 shows an example of metric coherence regarding the time signature $6 4$ in Brahms’ Third Symphony. One may distinguish the following layers: the beginnings of all measures (highest layer), the fourth beats of all measures, the second, third, fifth and sixth beats of all measures and the weak beats (lowest