# Modelling the Microphone-Related Timbral Brightness of Record Signals
## 3 Brightness Modelling
### Correlates of Brightness  
In multiple application areas (e.g., comparisons between musical instruments, between variations of a single instrument, or between musical notes/chords), research has found that the spectral centroid (\(SC\)) correlates highly with the perception of brightness [3; 4; 5; 6]. The \(SC\) can be calculated from the discrete frequency representation of an audio signal as  
\[C=\frac{\sum_{n=1}^{N}f(n)x(n)}{\sum_{n=1}^{N}x(n)}, \tag{1}\]  
Figure 2: Listener ratings of brightness. Data for each source (from test part 1) have been scaled according to the cross-source ratings of the most and least bright recordings of that source (from test part 2).  
where \(C\) is the spectral centroid, \(f(n)\) is the frequency of the \(n\)th bin, and \(x(n)\) is the magnitude of the \(n\)th bin. For signals that have more high-frequency energy, the \(SC\) will be higher, and it will be lower for signals that contain more low-frequency energy.  
It has been noted (by a research team that included the second author) that the standard \(SC\) metric is calculated with a linearly spaced frequency scale, and suggested that use of a more perceptually meaningful scale, such as mel, Equivalent Rectangular Bandwidth (ERB), or cents, may relate better to perception [11].  
Brightness is commonly considered to relate to the high-frequency content of an audio signal [3]. Changes in low-frequency content are therefore unlikely to affect the perception of brightness, yet will affect the \(SC\). One potential improvement to the \(SC\) as a brightness predictor would therefore be to calculate it only for signal energy above a particular frequency, \(f_{c}\), as \(SC_{fc}\). This can be performed as shown in Equation (2), where \(C_{fc}\) is the frequency limited spectral centroid and \(n_{c}\) is the frequency bin that most closely matches \(f_{c}\).  
\[C_{fc}=\frac{\sum_{n_{c}}^{N}f(n)x(n)}{\sum_{n_{c}}^{N}x(n)} \tag{2}\]  
Two ratio-based metrics have been proposed to predict perceived brightness: the ratio of the total magnitude of the high-frequency portion of the signal to that of the low-frequency portion of the signal, as calculated from Equation (3) [7]; and the ratio of the total magnitude of the high-frequency portion of the signal to that of the full signal, as shown in Equation (4) [12]. Several different crossover frequencies, \(f_{c}\), have been suggested that range from 500 Hz to 3 kHz [7,8].  
\[\text{Ratio}_{1}=\frac{\sum_{n_{c}}^{N}x(n)}{\sum_{n=0}^{n_{c}-1}x(n)}, \tag{3}\]  
\[\text{Ratio}_{2}=\frac{\sum_{n_{c}}^{N}x(n)}{\sum_{n=0}^{N}x(n)}. \tag{4}\]  
For both the \(SC\) and ratio metrics, there are several additional manipulations that may make them more closely relate to the perception of brightness. Frequencies below the low-frequency limit of human hearing are unlikely to affect the perception of brightness but may affect the metrics; therefore, for this study, only frequencies greater than 20 Hz were considered (lower-frequency spectral bands were excluded). Similarly, human hearing is not equally sensitive to all frequencies within the audible range; this study therefore tested the application of A- and C-weighting filters prior to analysis. Finally, it has been shown that high-Q spectral peaks are less perceptible than low-Q peaks [13]; this study therefore trialed spectral smoothing to prevent less perceptible high-Q resonances from overly influencing the metrics.