## 1 Introduction  
Sinusoidal models are widely used in the analysis [1; 2], synthesis [2; 3], and transformation [4; 5] of musical instrument sounds. The musical instrument sound is modeled by a waveform consisting of a sum of time-varying sinusoids parameterized by their amplitudes, frequencies, and phases [1; 2; 3]. Sinusoidal analysis consists of the estimation of parameters, synthesis comprises techniques to retrieve a waveform from the analysis parameters, and transformations are performed as changes of the parameter values. The time-varying sinusoids, called partials, represent how the oscillatory modes of the musical instrument change with time, resulting in a flexible representation with perceptually meaningful parameters. The parameters completely describe each partial, which can be manipulated independently. Several important features can be directly estimated from the analysis parameters, such as fundamental frequency, spectral centroid, inharmonicity, spectral flux, onset asynchrony, among many others [2]. The model parameters can also be used in musical instrument classification, recognition, and identification [6], vibrato detection [7], onset detection [8], source separation [9], audio restoration [10], and audio coding [11]. Typical transformations are pitch shifting, time scaling [12], and musical instrument sound morphing [5]. Additionally, the parameters from sinusoidal models can be used to estimate alternative representations of musical instrument sounds, such as spectral envelopes [13] and the source-filter model [14; 15]. The quality of the representation is critical and can impact the results for the above applications. In general, sinusoidal models render a close representation of musical instrument sounds because most pitched musical instruments are designed to present very clear modes of vibration [16]. However, sinusoidal models do not result in perfect reconstruction upon resynthesis, leaving a modeling residual that contains whatever was not captured by the sinusoids [17]. Musical instrument sounds have particularly challenging features to represent with sinusoids, such as sharp attacks, transients, inharmonicity, and instrumental noise [16]. Percussive sounds produced by plucking strings (such as harpsichords, harps, and the _pizzicato_ playing technique) or striking percussion instruments (such as drums, idiophones, or the piano) feature sharp onsets with highly nonstationary oscillations that die out very quickly, called transients [18]. Flute sounds characteristically comprise partials on top of breathing noise [16]. The need in woodwind instruments presents a highly nonlinear behavior that also results in attack transients [19], while the stiffness of piano strings results in a slightly inharmonic spectrum [18]. The residual from most sinusoidal representations of musical instrument sounds contains perceptually important information [17]. However, the extent of this information ultimately depends on what the sinusoids are able to capture. The standard sinusoidal model (SM) [1; 20] was developed as a parametric extension of the short-time Fourier transform (STFT) so both analysis and synthesis present the same time-frequency limitations as the Discrete Fourier Transform (DFT) [21]. The parameters are estimated with well-known techniques, such as peak-picking and parabolic interpolation [20; 22], and then connected across overlapping frames (partial tracking [23]). Peak-picking is known to bias the estimation of parameters because errors in the estimation of frequencies can bias the estimation of amplitudes [22; 24]. Additionally, the inherent time-frequency uncertainty of the DFT further limits the estimation because long analysis windows blur the temporal resolution to improve the frequency resolution and _vice-versa_[21]. The SM uses quasi-stationary sinusoids (QSS) under the assuption that the partials are relatively stable inside each frame. QSS can accurately capture the lower frequencies because these have fewer periods inside each frame and thus less temporal variation.