## 1 Introduction  
Multi-pitch estimation or multiple fundamental frequency estimation is the process of extracting musical notation (pitches) from a given acoustic signal. Multi-pitch estimation is a task that belongs to content-based music information retrieval. Music information retrieval (MIR) has drawn more attention due to the exponential growth of digital music [1]. Howbeit, there is a significant gap between high-level human perception and low-level signal features. Some music features such as rhythm and pitch have an important role in helping bridge this gap since they are more closely related to the human perception of music. Multiple fundamental frequency estimation was introduced by [PERSON] [2] in his research on separating co-channel speech signals. Later, multiple-F0 estimation was extended to polyphonic pitch estimation or multi-pitch estimation (MPE) in the context of automatic music transcription (AMT) for polyphonic music signals [3, 4]. Among all musical instruments, the piano is one of the most popular instruments worldwide and one of the most complex in what concerns pitch variety and number of simultaneous notes [5]. These are the main reasons that motivate us to research the multi-pitch estimation of piano sounds. A single piano note is a monophonic sound comprising a quasi-harmonic spectrum [6], where the lowest frequency of the harmonic series corresponds with the perceived pitch or fundamental frequency (F0). The combination of multiple piano notes (monophonic sounds) results in a polyphonic sound with multiple pitches. MPE refers to determination of the underlying pitches of an obtained polyphonic sound. Unlike mono-pitch estimation, multi-pitch estimation deals with issues such as the source number ambiguity and the octave ambiguity (Figure 1) (Figure extracted from [7], with the author's consent). Therefore, multi-pitch estimation is a challenging problem. Although there has been a lot of research devoted to it, it still remains completely unsolved. In general, multi-pitch estimation approaches and algorithms assume that there might be more than one harmonic source in the same short-time signal. As explained by [PERSON] et al. [8], that signal may also be expressed as a sum of harmonic sources plus a residual (The residual, \(z[n]\), comes from components that are not explained by sinusoids, for instance, the background noise, spurious components, or nonharmonic partials. ):  
\[y[n]=\sum_{m=1}^{M}y_{m}[n]+z[n],M>0\text{ with }y_{m}[n]\approx y_{m}[n+N_{m}], \tag{1}\]  
where \(M\) is the number of harmonic sources, \(n\) represents the discrete time, \(y_{m}[n]\) is the quasi-periodic part of the \(m\)th source signal, \(N_{m}\) represents the period of the \(m\)th source, and \(z[n]\) is the residual. According to Fourier series and the signal quasi periodicity, \(y[n]\) can be written as follows:  
\[y[n] = \sum_{m=1}^{M}\Big{\{}\sum_{h=1}^{\infty}A_{m,h}\cos(h\omega_{m}n +\phi_{m,h})\Big{\}}+z[n] \tag{2}\] \[\approx \sum_{m=1}^{M}\sum_{h=1}^{H_{m}}A_{m,h}\cos(h\omega_{m}n+\phi_{m,h})+z[n].\]  
Figure 1: Example of source number and octave ambiguity - The spectrum of a single piano note C4, Musical Instrument Digital Interface (MIDI) note number 60, is almost identical to the spectrum of the two-key combination of C4 and C5. C5 is one octave apart from C4. (a) Spectrum of C4. (b) Spectrum of the mixture of C4 and C5. (c) Signal difference between (a,b). Last-step approximation is common for practical usage: a finite and small number of sinusoids \(H\) is commonly used to approximate a quasi-periodic signal. The main difficulty associated with MPE is dealing with the modeling of \(y[n]\). This task implies estimating the number of harmonic sources and their related F0s. Decomposition of the observed signal into an unknown number of model sources not only is a problem of pattern matching but also can be seen as a classification problem, i.e., identifying the most likely combination of F0s for the modeling of \(y[n]\).