## 4 Construction of a Timbre Perception Feature Model
### Calculation Method  
The acoustic parameters were calculated as follows. The spectral centroid for the magnitude spectrum of the STFT [60] is given by:  
\[C_{t}=\frac{\sum\limits_{n=1}^{N}M_{t}[n]\times n}{\sum\limits_{n=1}^{N}M_{t}[n]}, \tag{1}\]  
where \(M_{t}[n]\) is the magnitude of the Fourier transform at frame \(t\) and frequency \(n\). This centroid is a measure of the spectral shape, where higher centroid values indicate \"brighter\" sounds. Spectral slope was calculated using a linear regression over spectral amplitude values. It should be noted that spectral slope is linearly dependent on the spectral centroid as follows [62]:  
\[\text{slope}(t_{m})=\frac{1}{\sum\limits_{k=1}^{K}a_{k}(t_{m})}\times\frac{ \sum\limits_{k=1}^{K}f_{k}\cdot a_{k}(t_{m})-\sum\limits_{k=1}^{K}f_{k}\cdot \sum\limits_{k=1}^{K}a_{k}(t_{m})}{K\sum\limits_{k=1}^{K}f_{k}^{2}-(\sum \limits_{k=1}^{K}f_{k})^{2}}, \tag{2}\]  
where \(\text{slope}(t_{m})\) is the spectral slope at time \(t_{m}\), \(a_{k}\) is the spectral amplitude at \(k\), and \(f_{k}\) is the frequency at \(k\). Tristimulus values were introduced by [PERSON] and [PERSON] as a timbral equivalent to color attributes in vision. The tristimulus comprises three different energy ratios, providing a description of the first harmonics in a spectrum [63]:  
\[ \begin{split} T1(t_{m})=\frac{a_{1}(t_{m})}{\sum\limits_{k=1}^{H} a_{1}(t_{m})},\\ T2(t_{m})=\frac{a_{2}(t_{m})a_{3}(t_{m})+a_{4}(t_{m})}{\sum\limits_{k=1 }^{H}a_{1}(t_{m})},\\ T3(t_{m})=\frac{\sum\limits_{k=1}^{H}a_{1}(t_{m})}{\sum\limits_{k=1 }^{H}a_{1}(t_{m})},\end{split}  \tag{3}\]  
where \(H\) is the total number of partials and \(a_{h}\) is the amplitude of partial \(h\). Spectral flux is a time-varying descriptor calculated using STFT magnitudes. It represents the degree of variation in a spectrum over time, defined as unity minus the normalized correlation between successive \(a_{k}\) terms [64]:  
\[\text{spectral flux}=1-\frac{\sum\limits_{k=1}^{K}a_{k}(t_{m-1})a_{k}(t_{m}) }{\sqrt{\sum\limits_{k=1}^{K}a_{k}(t_{m-1})^{2}}\sqrt{\sum\limits_{k=1}^{K}a _{k}(t_{m})^{2}}}. \tag{4}\]  
Inharmonicity measures the departure of partial frequencies \(f_{h}\) from purely harmonic frequencies \(hf_{0}\). It is calculated as a weighted sum of deviations from harmonicity for each individual partial [62]:  
\[\text{inharmo}(t_{m})=\frac{2}{f_{0}(t_{m})}\frac{\sum\limits_{h=1}^{H}(f_{h }(t_{m})-hf_{0}(t_{m}))a_{h}^{2}(t_{m})}{\sum\limits_{h=1}^{H}a_{h}^{2}(t_{m})}, \tag{5}\]where \(f_{0}\) is the fundamental frequency and \(f_{h}\) is the frequency of partial \(h\). Spectral roll-off was proposed by [PERSON] and [PERSON] [65]. It is defined as the frequency \(f_{c}(t_{m})\) below which 95% of the signal energy is contained:  
\[\sum_{f=0}^{f_{c}(t_{m})}a_{f}^{2}(t_{m})=0.95\sum_{f=0}^{sr/2}a_{f}^{2}(t_{m}), \tag{6}\]  
where _sr_/2 is the Nyquist frequency and \(a_{f}\) is the spectral amplitude at frequency \(f\). In the case of harmonic sounds, it can be shown experimentally that spectral roll-off is related to the harmonic or noise cutoff frequency. The spectral roll-off also reveals an aspect of spectral shape as it is related to the brightness of a sound. The odd-to-even harmonic energy ratio distinguishes sounds with a predominant energy at odd harmonics (such as the Guan) from other sounds with smoother spectral envelopes (such as the Suona). It is defined as:  
\[\text{OER}(t_{m})=\frac{\sum\limits_{h=1}^{H/2}a_{2h-1}^{2}(t_{m})}{\sum \limits_{h=1}^{H/2}a_{2h}^{2}(t_{m})}. \tag{7}\]  
Twelve time-varying statistics were calculated for the 54 parameters, including the maximum, minimum, mean, variance, standard deviation, interquartile range, skewness coefficient, and kurtosis coefficient, producing an objective acoustic parameter set containing 166 dimensions (see Table 6). In this paper, Timbre Toolbox [62] and MIRtoolbox [66] were used for feature extraction.