# Tempo and Metrical Analysis by Tracking Multiple Metrical Levels Using Autocorrelation
## 3 Proposed Method
### Metrical Structure  
The metrical grids constructed by the tracking method presented in the previous paragraph are so far made of a mere superposition of metrical periods. The ratio number associated with each metrical level should be considered relatively. For instance, the value 1 has no absolute meaning, it is arbitrarily given to the first level detected. Level 1.5 is 3 times slower than level 0.5. For each metrical grid, one or several of its metrical periods have been characterized as dominant because of their salience at particular instants of the temporal development of the metrical grid, and because such selection offers helpful guiding points throughout the temporal tracking of the metrical grid. Yet these selected dominant metrical periods simply highlight particular articulation of the surface and do not necessarily relate to the core metrical levels of the actual metrical structure. A metrical structure is composed of a certain number of metrical _levels_: they are particular periods of the metrical grid that are multiple of each other. For instance, in a typical meter of time signature 4/4, the main metrical level is the quarter note, the upper levels are the half note and the whole note, the lower levels are the eighth note, the sixteenth note, and any other subdivision by 2 of these levels. In the same example, dotted half note (corresponding to three quarter notes) is related to one metrical period in the metrical grid, because it is explicitly represented in the autocorrelation function as a possible periodicity, but it is not considered to be a metrical _level_. In the graphical representations shown in Figures 6-8, the metrical levels are shown in black while the other metrical layers are shown in gray. The metrical structure offers core information about meter. In particular, tempo corresponds to beat periodicity at one particular metrical level. In a typical meter, the main metrical level could be used as the tempo reference. In our example, with a typical time signature 4/4, the tempo could be inferred by reporting the period at the metrical level corresponding to the quarter note. However, in practice, there can be ambiguity related to the actual meter, and especially related to the choice of the main metrical level. For each metrical periodicity \(i\) can be associated a numerical score \(S_{i}\), computed as a summation across frames of the related periodicity score \(s_{i,n}\) for each frame \(n\). The metrical periodicities \(i\) are progressively considered in decreasing order of score \(S_{i}\) as potential metrical levels. In a first attempt, we integrate all possible periodicities as long as they form a coherent metrical structure. The metrical structure is initially made of one single metrical level corresponding to the strongest periodicity. Each remaining metrical period \(P\), from strongest to weakest, is progressively compared with the metrical levels of the metrical structure, in order to check that for each metrical level \(L\), \(P\) has a periodicity that is a multiple of \(L\), or the reverse. In such case, \(P\) is integrated into the metrical structure as a new metrical level. This method may infer incorrect metrical structures in the presence of a strong accentuated metrical period that is not considered to be a metrical level. This often happens in syncopated rhythm. For instance, a binary 4/4 meter with strong use of dotted quarter notes could lead to strongest periodicities at the eighth note (let's set this period to \(i=1\)), dotted quarter note (\(i=3\)) and whole note (\(i=8\)). One example is the rhythmical pattern 123-123-12, 123-123-12, etc. In such case, if the periodicities related to dotted quarter note (\(i=3\)) is stronger than the periodicities related to whole note (\(i=8\)), the first method would consider the meter to be ternary, of the form 6/8 for instance. Figure 7: Autocorrelation-based periodogram with tracking of the metrical structure for the first 160 s of a performance of the Scherzo of [PERSON]’s _Symphony No.