In this paper, we study the possibility of defining pseudo-distances between musical chords of different cardinality, from the distance defined on the generalized voice-leading space by Callender, Quinn, and Tymoczko [“Generalized Voice-Leading Spaces.” Science 320 (5874): 346–348]. This latter distance measures the size of the minimal voice leading between two chords. A pseudo-distance is a weaker notion than distance, which allows more flexibility with fewer mathematical constraints. We define one distance and two pseudo-distances. The distance is the Hausdorff metric between sets. The two pseudo-distances, based on a similar approach, use the projection/immersion of chords into lower/higher-dimensional voice-leading chord spaces. Finally, we propose some musical applications.

在本文中，我们研究了由Callender，Quinn和Tymoczko在“广义语音引导空间”（“广义语音引导空间”）上定义的距离，来定义不同基数的音乐和弦之间的伪距离。科学320（5874）：346–348]。后一个距离测量两个和弦之间的最小声音大小。伪距离是比距离弱的概念，它可以在较少的数学约束的情况下提供更大的灵活性。我们定义一个距离和两个伪距离。距离是集合之间的Hausdorff度量。基于类似的方法，这两个伪距使用和弦的投影/沉入较低/较高维的语音引导和弦空间。最后，我们提出一些音乐应用程序。