# All About Audio Equalization: Solutions and Frontiers
## 4 Graphic Equalizers
### Bands in Graphic Equalizers  
The basic unit of the graphic equalizer is the band. A graphic equalizer typically has more bands than a parametric equalizer. The bands are usually distributed logarithmically in frequency, to match human perception. Let us denote the normalized lower and upper crossover frequencies of the \(m\)th band with \(\omega_{1,m}\) and \(\omega_{u,m}\) respectively. Bandwidth is the difference between the upper and lower crossover frequencies, \(B_{m}=\omega_{u,m}-\omega_{1,m}\). The frequency bands are adjacent [52], so the upper crossover of band \(m\) will be the lower crossover of band \(m+1\), \(\omega_{u,m}=\omega_{l,m+1}\). That is, input audio frequencies below this crossover will be primarily affected by the gain control for band \(m\), whereas input frequencies above it will be primarily affected by the gain control for band \(m+1\). The logarithmic distribution of the frequency bands can be specified using a fixed ratio \(R\) between each band, so \(\omega_{1,m+1}=R\omega_{1,m}\), \(\omega_{u,m+1}=R\omega_{u,m}\), or \(B_{m+1}=RB_{m}\). We also consider the geometric mean of the two crossover frequencies,  
\[\omega_{M,m}=\sqrt{\omega_{1,m}\omega_{u,m}}, \tag{34}\]  
where it can be seen that the same relationship holds for these values, \(\omega_{M,m+1}=R\omega_{M,m}\). Additionally, the bandwidth can be related to the geometric mean \(\omega_{M,m}\), because according to Equation (34), \(\omega_{M,m}=\sqrt{R}\omega_{1,m}=\omega_{1,m}/\sqrt{R}\):  
\[B_{m}=\omega_{u,m}-\omega_{1,m}=\left(\sqrt{R}-\frac{1}{\sqrt{R}}\right) \omega_{M,m}. \tag{35}\]  
Two common designs are octave and \(1/3\)-octave graphic equalizers. An octave is a musical interval defined by a doubling in frequency, so octave graphic equalizers will have the ratio \(R=2\) between each band. In a \(1/3\)-octave design, each octave contains three bands, which implies \(R^{3}=2\) or \(R\approx 1.26\). So starting at \(1000\,\mathrm{Hz}\), an octave spacing would have geometric mean frequencies at \(2000\,\mathrm{Hz}\), \(4000\,\mathrm{Hz}\), \(8000\,\mathrm{Hz}\) etc. and a \(1/3\)-octave spacing would have filters centered at \(1260\,\mathrm{Hz}\), \(1587\,\mathrm{Hz}\), \(2000\,\mathrm{Hz}\)_etc._  
The number of bands is determined by their spacing and the requirement to cover the entire audible spectrum. Octave graphic equalizers usually have \(10\) bands, ranging from about \(31\,\mathrm{Hz}\) at the lowest to \(16\,\mathrm{kHz}\) at the highest. Third-octave designs usually have \(30\) bands ranging from \(25\,\mathrm{Hz}\) to \(20\,\mathrm{kHz}\). These frequencies, shown in Tables 1 and 2, are standardized by the ISO (InternationalStandards Organization) [61]. Note that some frequencies have been rounded coarsely in the standard in order to have simpler numbers. The geometric mean of the crossover frequencies of a filter, \(\omega_{M,m}\), is not exactly the true center frequency where the filter reaches its maximum or minimum value, \(\omega_{\text{c,}m}\). We can find a relationship between the upper and lower crossover frequencies and the center frequency of a bandpass, bandstop, peaking or notch filter,  
\[\tan^{2}\left(\frac{\omega_{\text{c,}m}}{2}\right)=\tan\left(\frac{\omega_{ \text{u,}m}}{2}\right)\tan\left(\frac{\omega_{\text{l,}m}}{2}\right). \tag{36}\]  
However, the geometric mean is usually quite close to the center frequency. Thus, the bandwidth scales roughly proportionally with the center frequency and higher bands will have a larger bandwidth than lower ones. Since \(Q=\omega_{\text{c}}/B\), this is another way of saying that the Q factor is nearly constant for each band in a graphic equalizer (such constant \(Q\) graphic equalizers were first described in [62]).