# Tempo and Metrical Analysis by Tracking Multiple Metrical Levels Using Autocorrelation
## 2 Related Work
### Periodicity Analysis  
A pulsation corresponds to a periodicity in the succession of peaks in the accentuation curve. Classical signal-processing methods estimate periodicity using methods such as autocorrelation, YIN [10], bank of comb-filter resonators with a constant half-time [11] or phase-locking resonators [12]. (Cf. [6] for a detailed literature review.) Basically, a range of possible periodicity frequencies is considered, and for each frequency, periodicity is searched for.  
[PERSON] et al. consider that \"the key problems in meter analysis are in measuring the degree of musical accentuation and in modeling higher-level musical knowledge, not in finding exactly the correct period estimator\" [6]. They point out however a specific feature offered using a bank of comb-filter resonators, namely that high scores are given not only to the actual periods expressed in the input accentuation curve, but also to subharmonics of these periods. For instance, for a simple pulse of period 1 s--corresponding to a tempo of 60 beats per minute (BPM)--all methods (autocorrelation function, bank of resonators) give high scores to periods related to 1 s and all its multiples: 2 s (30 BPM), 3 s (20 BPM), etc.; but bank of resonators also give high scores to subharmonics such as 0.5 s (120 BPM), 0.33 s (180 BPM), 0.25 and 0.75 s, etc. In our view, the addition of hypothetical faster pulsations are not informative if they do not explicitly appear in the input sequence, and especially if they could contradict with the actual metrical structure.  
On the other hand, one main disadvantage of autocorrelation function is the fact that it does _not_ explicitly look for _periodic_ events in the input sequence of period \(T\), but only to the prominence of _pairs_ of high values in the input sequences separated by the duration \(T\). Periodic sequences of period \(T\) lead to high prominence of duration \(T\), but the reverse is not true: pairs of pulses of duration \(T\) can populate a sequence without being necessarily periodic. For instance, a 4-beat periodic pattern with accentuation of the first and fourth beat leads to a high score for the 4-beat period, but also for the 3-beat period, although the latter is not periodic. We propose in Section 3.2 a way to filter out these non-periodic high scores in the autocorrelation function.  
In the following, we will call _periodicity function_ the representation, such as autocorrelation function, showing the periodicity _score_ related to each possible _period_ (also called _lag_ in autocorrelation functions) expressed in seconds, or alternatively each possible frequency in Hz. Since the analysis is performed on a moving window, all periodicity functions can be grouped into one representation called _periodogram_. If the periodicity function is an autocorrelation function, this can be also called _autocorrelogram_. An example of autocorrelogram is shown in Figure 4, where each column displays an autocorrelation function computed on a time window of 5 s, which progressively scans the audio recording from beginning to end.  
It has been asserted that accentuation curves need to be estimated on several distinct registers--5 according to [PERSON] [11] and [PERSON] suggest even up to 40 different bands [6]--and that periodicity needs to be estimated on each register separately. This method would particularly work in the case of complex musical examples where the main pulsation can be extracted on a particular register frequency region. This multi-band strategy was possibly designed to counteract some limitations of the classical methods for accentuation curve: because they fail to produce clear accentuation curves for complex polyphony, a decomposition into frequency bands might produce clearer accentuation curves. However, for musical examples where the pulsation is scattered along various frequency bands, this method would not offer any particular advantage and could even fail to catch that periodicity.