16世纪有关音乐的数学理论--以马若利科、斯蒂文、程大位和朱载(土育)为例

文章对16世纪中国和欧洲与音乐有关的数学理论的不同传统进行了比较研究。 在欧洲,对诸如2~(1/2)这样的无理数的考虑对乐律的划分形成了一定的阻碍,因此后来的学者很难突破毕达哥拉斯的樊篱,马若利科的工作就是一个例子。另一方面,尽管古希腊的传统为多数学者所遵循,我们在16世纪还是看到了一些例外,例如西蒙·斯蒂文就提出了平均律的理论。 在中国,不存在那种对无限小数的禁忌,因此中国学者在发展音律学方面所遇到的困难要比其欧洲同行小一些。在16世纪,除了程大位遵循传统的损益方法,即利用整数和基于整数的有理数制定音律外,又出现了由朱载堉创立的平均律。 在欧洲,从古希腊发展而来的音律理论主要不是直接建立在音乐家的工作基础上,而是由一种称为独弦琴的弦乐器的理想模型所决定。 在中国,音乐家似乎比其欧洲同行更坚持理论。此外在中国,编钟的调音问题容易导致平均律的出现和对双调和谐问题的处理。无论如何,中国的音律理论主要基于管乐器,即称为十二律吕的一套律管。作者提到,在《汉书》中可以找到当时的中国人已经运用了律管旋宫转调的近代方法的证据。 在欧洲有关于星际球体运动发声的音乐理论;在中国则有关于空气的音乐理论,也就是所谓的候气说,它与寒暑干湿等季节性因素以及“地支”有关。

The Mathematics of Music During the 16th Century: The Cases of Francesco Maurolico, Simon Stevin, Cheng Dawei and Zhu Zaiyu

A comparison is made between the different traditions of China and Europe concerning the mathematics of music during the 16th century. In Europe, the considering of certain numbers ( square root of 2, ?) as irrational also impeded a division into tones of the musical scale, different from the Pythagorean one ( Francesco Maurolico). However, although this tenet was maintained by the majority of scholars, some exceptions occurred. Simon Stevin proposed an equable temperament.In China, there was no such prohibition on (infinite) numbers. Therefore, together with tones based on integers and on ratios between integers (Cheng Dawei) , the equable temperament was developed more easily than in Europe, as was done by Zhu Zaiyu.In Europe, musical theories derived from Greece were mainly based not directly on what musicians did, but rather on the idealization of string instruments called the monochord.In China, musicians appear to have insisted on theory more than in Europe. Moreover, in China, the problem of tuning sets of zhong (bells) favoured the equable temperament, as well as the making of two- tone asymmetrical shapes. However, the theory was mainly based on wind instruments which were tuned by means of the twelve liilii ( pitch pipes) . In the sequel, I will show that the Hanshu offers evidence that here someone already offered a modern solution to the problem of end-effect in pipes.In Europe, there was the music of spheres; in China, the music of the atmosphere. That is, music was based on the qi (breath) and linked to the hou ( seasonal terms) and the dizhi ( earthly branches, months ) .It is thus advisable to review the whole matter, mainly in view of the questions of cultural diversity that it poses, rather than its suggestions of a contest of priority.