Point clouds are effective data structures for the rep- resentation of three-dimensional media and hence adopted in a wide range of practical applications. In many cases, the portrayed data is expected to be visualized by humans. After acquisition, point clouds may undergo different processing operations such as compression or denoising, potentially affecting their perceived quality. Although subjective experiments are still the most re- liable form of assessing the intensity of degradation, they are expensive and time-consuming, pushing many systems to depend on objective metrics. Such algorithms are used to model the human visual system, and their performance is usually assessed through their correlation with subjective visual quality scores. In this paper, an objective quality metric capable of evaluating distortions between a reference and a distorted point cloud at multiple scales is presented. The proposed metric is based on the point cloud structural similarity metric (PointSSIM), which computes a score based on the difference between statistical estimators obtained on the distribution of the luminance attribute over local neighborhoods. A collection of PointSSIM scores is produced for multiple scales obtained through the voxelization of both models at different bit depth precisions. These scores are then pooled through a weighted sum, with the importance of each scale being defined through logistic fitting to subjective mean opinion scores, producing one MS-PointSSIM score. Three datasets were employed for fitting and performance assessment, demonstrating a clear advantage of the proposed metric when compared to the single-scale baseline. Moreover, the presented MS-PointSSIM is shown to be the best predictor according to the average Pearson correlation coefficient across the three datasets when compared to state-of-the-art metrics.