This work proposes graphic equalizer designs with third-octave and Bark frequency divisions using symmetric band filters with a prescribed Nyquist gain to reduce approximation errors. Both designs utilize an iterative weighted least-squares method to optimize the filter gains, accounting for the interaction between the different band filters, to ensure excellent accuracy. A third-octave graphic equalizer with a maximum magnitude-response error of 0.81 dB is obtained, which outperforms the previous state-of-the-art design. The corresponding error for the Bark equalizer, which is the first of its kind, is 1.26 dB. This paper also applies a recently proposed neural gain control in which the filter gains are predicted with a multilayer perceptron having two hidden layers. After the training, the resulting network quickly and accurately calculates the filter gains for third-order and Bark graphic equalizers with maximum errors of 0.86 dB and 1.32 dB, respectively, which are not much more than those of the corresponding weighted least-squares designs. Computing the filter gains is about 100 times faster with the neural network than with the original optimization method. The proposed designs are easy to apply and may thus lead to widespread use of accurate auditory graphic equalizers.Appl. Sci. 2020, 10, 1222 2 of 22 using a neural network, which handled the optimization calculation [19]. The second paper extended the neurally-controlled equalizer (EQ) method to the widely used third-octave GEQ design [20], where the 31 EQ bands also loosely approximated the bandwidths of human auditory filters [21].Compared to our previous conference paper [20], this article presents an improved design method using second-order band filters with a symmetric shape on the logarithmic frequency scale, which was recently proposed for an octave GEQ [22]. The proposed design allows the parametric EQ used as band filters in the GEQ to have a controllable gain at the Nyquist limit, in contrast to the earlier designs that forced the Nyquist gain to one (i.e., 0 dB) [3]. Furthermore, this paper extends both the weighted least-squares (WLS) GEQ design method and the neurally-controlled EQ method to operate in Bark bands, which are a more accurate approximation of the human auditory resolution than the third-octave bands [23].Currently, two important metrics in GEQ design are accuracy and computational complexity. Both aspects have an impact when a GEQ is controlled by a computer in automated tasks without a human listener fine-tuning the gains, e.g., when equalizing music to be above a masking threshold in time varying noise [24,25]. Previous GEQs that are not very accurate (see the measurements, e.g., in [10]) have typically required a human operator to listen to the effect of the command gains as he/she controls them, since the accuracy of different command gain settings, as well as the visual feedback of the command gain sliders can be highly inaccurate. One of the goals of this work was to get the proposed neurally controlled GEQs to provide ...