Objective: The automatic discrimination between the coughing sounds produced by patients with tuberculosis (TB) and those produced by patients with other lung ailments. Approach: We present experiments based on a dataset of 1358 forced cough recordings obtained in a developing-world clinic from 16 patients with confirmed active pulmonary TB and 35 patients suffering from respiratory conditions suggestive of TB but confirmed to be TB negative. Using nested cross-validation, we have trained and evaluated five machine learning classifiers: logistic regression (LR), support vector machines (SVM), k-nearest neighbour (KNN), multilayer perceptrons (MLP) and convolutional neural networks (CNN). Main Results: Although classification is possible in all cases, the best performance is achieved using LR. In combination with feature selection by sequential forward selection (SFS), our best LR system achieves an area under the ROC curve (AUC) of 0.94 using 23 features selected from a set of 78 high-resolution mel-frequency cepstral coefficients (MFCCs). This system achieves a sensitivity of 93\% at a specificity of 95\% and thus exceeds the 90\% sensitivity at 70\% specificity specification considered by the World Health Organisation (WHO) as a minimal requirement for a community-based TB triage test. Significance: The automatic classification of cough audio sounds, when applied to symptomatic patients requiring investigation for TB, can meet the WHO triage specifications for the identification of patients who should undergo expensive molecular downstream testing. This makes it a promising and viable means of low cost, easily deployable frontline screening for TB, which can benefit especially developing countries with a heavy TB burden.