# Enhancement of Conventional Beat Tracking System Using Teager-Kaiser Energy Operator
## 2 Dataset and Methods
### Tempo Representation  
To create a tempo structure of given recordings, we need a representation of tempo--how the density of onsets, or more precisely repetitiveness of significant onsets, is distributed. This can be done by several techniques, in this study we focused on the method of dynamic beat tracking system proposed in [12]. This system estimates beat positions in an onset envelope and uses them to pick the right peaks within a given interval (default tempo). The default tempo is set up before the calculation (or it is calculated automatically based on autocorrelation function with respect to the standard 120 BPM) and therefore it has to be estimated by listening to the particular audio track or estimated from the sheet music to work as we want. The calculated peak positions can deviate from the default tempo in adjustable boundaries (depends on settings, e.g., [PERSON] reports approximately 10% [12]). The parameter \"tightness\", which corresponds to the detection tolerance (from the default tempo), was set to the number 50 in all cases. At first, this looks like an inappropriate method for the varying tempo of string quartet music (second database), but with good parameterization and segmentation of particular motifs, it fits our need.  
Beat detectors are based on a calculation of beats in an audio signal and therefore the metric structure from an elementary point of view. Usually, there is not enough information to consider dividing beats into bars without manual correction, but with proper segmentation, midi reference and dynamic time warping (DTW) techniques, this is possible [25]. However, one does not need such a method to calculate the GT of a given track. In this case, we only focused on the GT and ABD. Figure 8 shows how this system picks onset candidates from the onset curve and creates the beat positions by using periodicity information.  
Figure 9 shows the estimated time positions of beats at the beginning of a string quartet segment. As we can see, the system is using periodicity information to calculate beat positions even at places where no onsets are detected--in this specific part, second violin and viola are playing very quietly (and no onset is detected) and then a violin solo begins. Between the 6 th and the 10 th second of  
Figure 7: Spectrogram and onset detection function for a solo violin using TKEO.  
this track, there are strong onsets in the calculated onset curve. Their periodicity information is then used to fill the gap in the silent part of this recording, which is one of the advantages of the dynamic programming search system.  
The disadvantage of such a beat tracking system is the adjustable default tempo--the algorithm searches for beat positions within a given interval, but there is no guarantee that true beat positions exist within specified limits (also concerning the tolerance parameter). The reference global tempo can be misleading if the recording is rhythmically unstable or the tempo changes significantly over time. A similar problem exists in the metric pulse. If the system detected 100 BPM as the GT and the reference is 50 BPM, it does not mean that the system is completely wrong. That is why we also calculated the ABD.  
Figure 8: Comparison of the onset and beat positions.  
Figure 9: Estimated beat positions by the dynamic programming system.