# Tempo and Metrical Analysis by Tracking Multiple Metrical Levels Using Autocorrelation
## 2 Related Work
### Metrical Structure  
In the presence of a given pulsation in the musical excerpt that is being analyzed--let's say with a BPM of 120, i.e., with two pulses per second--the periodicity function will indicate a high periodicity score related to the period 0.5 s. But generally, as previously discussed, if there is a pulsation at a given tempo, multiples of the pulsation can also be found that are twice slower (1 s), three times slower, etc. For that reason, the periodicity function usually shows a series of peaks equally distant, corresponding to multiples of a given period. This can be seen for instance in Figure 4. This harmonicity of the rhythm periodicity has close connections with the notion of metrical structure in music, with a hierarchical ordering of rhythmical values such as whole notes, half notes, quarter notes, etc. One common approach to extract tempo from the periodicity function is to select the highest peak, within a range of beat periodicities considered to be most adequate, typically between 40 and 200 BPM, with a weighted emphasis on a range of best perceived periodicities. Studies have designed so-called \"resonance curves\" that weight the periodicity score depending on the period, so that periods around 120 BPMs would be preferred [13; 14]. This approach fails when tracking the temporal evolution of tempo over time, especially for pieces of music where different metrical levels are emphasized throughout the temporal development. For instance, if at a given moment of the piece of music, there is an accentuated quarter-note pulsation followed by an accentuated eighth-note pulsation, the tempo tracking will switch from one BPM value to another one twice faster, although the actual tempo might remain constant. And as we may imagine, such shift from one metrical level to another is very frequent in music. Figures 4 and 5 show an example of analysis using this simple method. Figure 4: Autocorrelogram for the first minutes of a performance of [PERSON]’s _Dance of the Goblins_. In each frame—i.e., column—, the highest peak of the autocorrelation function is highlighted. In [6], three particular metrical levels are considered to be core elements of the metrical structure:
* The _tactus_ is considered to be the most prominent level, also referred as the foot-tapping rate or the _beat_. The tempo is often identified with the tactus level. * The _tatum_--for \"temporal atom\"--is considered to be the fastest subdivision of the metrical hierarchy, such that all other metrical levels (in particular tactus and bar) are multiples of that tatum. * The _bar level_ or other metrical levels considered to be related to change of chords, melodic or rhythmic patterns, etc. [PERSON] et al. propose a method to track meter by modeling dependencies between successive instants [6]. In our view, one major interest of this approach is that it is based on a tracking of parallel metrical levels, namely the three main metrical levels discussed previously --tatum, tactus and bar level--which they propose to call respectively levels \(A\), \(B\) and \(C\). More precisely, the period associated with each of the three levels, \(\tau^{A}\), \(\tau^{B}\) and \(\tau^{C}\), are tracked in parallel along time, frame by frame. The proposed Hidden Markov Model is designed in such a way that the period at each level is conditioned by the period at the same level on the previous frame. Moreover, \(\tau^{C}\) is probabilistically constrained to be an integer multiple of \(\tau^{B}\) and \(\tau^{B}\) to be an integer multiple of \(\tau^{A}\). Finally, the values of the three periods \((\tau^{A},\tau^{B},\tau^{C})\) at a given frame are also conditioned by the periodicity function \(s\) at that frame, in the form of a state-conditional observation likelihoods \(p(s|\tau^{A},\tau^{B},\tau^{C})\), which is evaluated from a database of musical recordings where the meter has been hand-labeled.