Single-reed instruments can produce multiphonic sounds when they generate quasi-periodic oscillations. The aim of this article is to identify a minimal model of a single reed-instrument producing quasi-periodic oscillations. To better understand the influence of model parameters on the production of quasi-periodic regimes, the mapping between parameters and quasi-periodic regimes is explicitly identified using a support vector machine (SVM) classifier. SVMs enable the construction of boundaries between quasi-periodic and periodic regimes that are explicitly defined in terms of the parameters. Results and conclusions obtained from the numerical model are compared to published experiments related to t he the production of quasi-periodic oscillations with an alto saxophone. This qualitative comparison highlights the influence of key parameters on the production of multiphonic sounds.