The Pythagoreans linked musical intervals with integer ratios, cosmic order, and the human soul. The empirical approach of Aristoxenus, based on real musicians making real music, was neglected. Today, many music scholars and researchers still conceptualize intervals as ratios. We argue that this idea is fundamentally incorrect and present convergent evidence against it. There is no internally consistent “Just” scale: a 6th scale degree that is 5:3 above the 1st is not a perfect 5th (3:2) above the 2nd (9:8). Pythagorean tuning solves this problem, but creates another: ratios of psychologically implausible large numbers. Performers do not switch between two ratios of one interval (e.g., 5:4 and 81:64 for the major third), modern studies of performance intonation show no consistent preferences for specific ratios, and no known brain mechanism is sensitive to ratios in musical contexts. Moreover, physical frequency and perceived pitch are not the same. Rameau and Helmholtz derived musical intervals from the harmonic series, which is audible in everyday sounds including voiced speech; but those intervals, like musical intervals, are perceived categorically. Musical intervals and scales, although they depend in part on acoustic factors, are primarily psychocultural entities—not mathematical or physical. Intervals are historically and culturally variable distances that are learned from oral traditions. There is no perfect tuning for any interval; even octaves are stretched relative to 2:1. Twelve-tone equal temperament is not intrinsically better or worse than Just or Pythagorean. Ratio theory is an important chapter in the history Western musical thought, but it is inconsistent with a modern evidence-based understanding of musical structure, perception and cognition.

中文翻译：

---

音乐间隔的心理文化理论

毕达哥拉斯人将音乐区间与整数比率，宇宙秩序和人类灵魂联系在一起。忽略了以真实音乐家创作真实音乐为基础的Aristoxenus的经验方法。时至今日，许多音乐学者和研究人员仍将音程概念化为比率。我们认为这个想法从根本上是错误的，并提出了反对它的趋同证据。内部没有一致的“正义”标度：比第一标度高5：3的第六标度度并不比第二标度（9：8）完美的第五标度（3：2）。毕达哥拉斯调律解决了这个问题，但又产生了另一个问题：心理上难以置信的大量数字的比率。表演者不会在一个间隔的两个比率之间切换（例如，大三分之二是5：4和81:64），现代的语调研究表明，对于特定比率没有一致的偏好，而且没有已知的大脑机制对音乐环境中的比例敏感。而且，物理频率和感知的音调是不同的。Rameau和Helmholtz从泛音系列中得出音乐间隔，在包括浊音在内的日常声音中可以听到这种间隔。但是这些间隔，例如音乐间隔，是可以分类的。音乐间隔和音阶虽然部分取决于声学因素，但主要是心理文化实体，而不是数学或物理上的实体。间隔是从口头传统中学到的历史和文化上可变的距离。任何间隔都没有完美的调整；甚至八度音阶都相对于2：1拉伸。十二音律相同的气质本质上并不比Just或Pythagorean好或坏。比率理论是西方音乐思想史上的重要一章，