# Tempo and Metrical Analysis by Tracking Multiple Metrical Levels Using Autocorrelation
## 4 Experiments  
### Evaluation Campaigns Using Music with Constant Tempo  
The original algorithm was submitted to the Audio Tempo Extraction competition under the MIREX annual campaign [5; 15; 16]. This evaluation is made using 160 30-s excerpts of pieces of music of highly diverse music genres but with constant tempo. Listeners were asked to tap to the beat for each excerpt. From this, a distribution of perceived tempo was generated [17]. The two highest peaks in the perceived tempo distribution for each excerpt were selected, along with their respective heights, as the two tempo candidates for that particular excerpt. The height of a peak in the distribution is assumed to represent the perceptual salience of that tempo. Each algorithm participating to this MIREX task should also return two tempo candidates for each excerpt, with corresponding salience. This ground-truth data is then compared to the predicted tempo, using a tolerance of 8% in each tempo value in BPM. The two tempo candidates \(T1\) and \(T2\) associated with each excerpt of music correspond to the two main metrical levels, such as for instance 40 and 80 BPM. We should note however that other metrical levels can be accepted as well. For instance, the pulsation at 80 BPM might be further decomposed into two, leading to another metrical level of 160 BPM. If an algorithm returns the values 80 and 160 BPM while the ground-truth was 40 and 80 BPM, this does not necessarily mean that half of the results is incorrect. In Table 1, three values are given: the success rate to detect the two tempi correctly (\"both tempi\"), to detect one tempo correct out of the two (\"1 tempo\") and a \(P\)-score defined as follows:  
\[P=ST1*TT1+(1-ST1)*TT2 \tag{18}\]  
where \(ST1\) is the relative perceptual strength of \(T1\) (given by ground-truth data, varying from 0 to 1.0), \(TT1\) is the ability of the algorithm to identify \(T1\) within 8%, and \(TT2\) is the ability of the algorithm to identify \(T2\) within 8% [5; 16].  
In 2013 [18], our proposed model (OL) obtained the fourth highest \(P\)-score, compared to models from 2006 to 2013, as shown in Table 1. (The authors of model FW [19], submitted in 2015, already submitted a model in 2013 [20] of slighly lower P-score but still surpassing OL's score.) It can be noted that these three better models are applicable only to music with stable tempo. Since then, OL has been surpassed by the two aforementioned deep-learning models [1; 2].  
The current improved version of OL was submitted to the 2018 competition [21]. The frequency resolution of the spectrogram is decreased (1 Hz instead of 0.1 Hz) without damaging the results. This makes the computation significantly faster and less greedy in memory. To filter out non-relevant peaks, the first peak at the lowest lag in the autocorrelation function is constrained to be preceded by a valley with negative autocorrelation. As candidate metrical hierarchies can have various number of levels, comparing the summation of the score would penalize those with fewer number of levels.  
Consequently, when comparing pairs of metrical hierarchies, only the most dominant levels of each hierarchy are selected in such a way that we compare hierarchies with same number of levels. Finally, a periodicity that is higher than 140 BPM cannot belong to the two selected metrical levels, except if that fast pulsation is ternary, i.e., if the pulsation at the next level is three times lower. Apart from the optimization in time and space, these improvements were designed to improve the robustness of the algorithm. These improvements did not have an actual impact on the results of the MIREX task, though: OL 2018 does not offer any improvement in the results compared to the 2013 submission.