Connections between mathematics and music have been recognized since the days of Ancient Greece. The Pythagoreans' association of musical intervals with integer ratios is so well known that it occludes the great variety of approaches to the music-mathematical relationship in Ancient Greece and Rome. The present article uncovers this diversity by examining how authors from Antiquity used one mathematical element – the number series 16, 17, 18 – to serve different ends. The number 17 provides the simplest way to divide the 9:8 whole tone into two parts, but the status of those two parts was debated, as was their relationship to the standard 256:243 semitone of Greek theory. I account for this diversity by appealing to the contexts in which the authors wrote – music treatises vs. commentaries on Plato's Timaeus – and to the importance placed on mathematics in Neoplatonist curricula. The article concludes by examining how confusion regarding the 16, 17, 18 series lingered even in the medieval period due to ambiguities in Boethius's De institutione musica, and how medieval authors eventually superseded the debate through a Euclid-inspired geometric division of the 9:8 ratio.

自古希腊时代以来，数学与音乐之间的联系就得到了认可。毕达哥拉斯人的音乐区间与整数比率的关联是众所周知的，以至于它掩盖了古希腊和罗马关于音乐与数学关系的多种方法。本文通过研究“上古”的作者如何使用一个数学元素（数字系列16、17、18）来服务于不同的目的，揭示了这种多样性。数字17提供了将9：8整个音调分为两个部分的最简单方法，但是这两个部分的状态以及它们与希腊理论的标准256：243半音的关系都引起了争议。我通过呼吁作者所写的上下文来解释这种多样性，其中包括音乐论文与柏拉图《提马乌斯》的评论–并且重视新柏拉图式课程中的数学。文章的结尾总结了16世纪，17世纪，18世纪系列的困惑，甚至在中世纪时期，由于Boethius的音乐作品的歧义如何依然存在，以及中世纪作家如何最终通过以欧几里得为灵感的9：8几何划分取代了这场辩论。比。