王子的数学与声学——对朱载堉乐律学创新的思想资源和数学方法的考察

The Prince’s Mathematics and Acoustics:An Examination of the Sources of Zhu Zaiyu’s Innovation in Musical Temperament and His Mathematical Approach

明代皇室后裔、杰出学者朱载堉成功地解决了乐律学长期存在的问题:中国乐律中使用的“三分损益法”不能通过产生一个与起始音高成八度关系的音以达到“返宫”的理想目的。首先,朱载堉对此问题的解决方式反映出儒家经典中所描述的标准器“周鬴”在他的思维中占有重要地位。周鬴记录了长度、重量和体积的计量单位,并且以某种方式定义了标准音高。基于经典文献对周鬴的描述,朱载堉创造性地建立了一个方形及其外接圆的结构,并由此展开其律制的创新工作。其次,他使用了数术的方式去阐释此方和圆的周长数量关系,这一思想来自宋代学术。在这样的阐释中,他直接采用了一些数术理论中的数值,并且这些数值被进一步用于确定律管的尺寸。在朱载堉建立新律制所依赖的各种数学知识中,圆周长与其直径的关系,可以用现代数学方式表示为10×4/√9^(2)+9^(2),实际上是他极为自豪的重要发现。与这个周径关系相关联的数也被指称为“密率”,正与他的等程律中数据的名称相一致。朱载堉为新律制创建“密率”的数学方法的核心在于“勾股术”以及开平方和立方的操作。他完美地将寻找连续十二个律管长度问题转化为先求解音程中对称点音高对应长度的问题。以这样的方式,他所掌握的数学操作成功地处理了建立十二等程律过程中的数值。

The outstanding scholar Zhu Zaiyu朱載堉(1536-1611),a descendant of the Ming(1368-1644)royal family,successfully solved a long-standing problem in musical temperament posed by the sanfen sunyi method(三分損益):the method of“diminishing and augmenting[the length]by dividing[it]into three parts.”Used in musical temperament in China,this method had failed to achieve the ideal goal of returning to a starting note by producing a pitch that is in octave relationship to it.Zhu’s solution to this problem illustrates that the standard Zhou vessel described in Confucian classics was foremost in his mind.This ancient standard vessel recorded the measurement units of length,weight,and volume,and also defined the standard musical pitch in a particular way.On the basis of how the standard vessel was described in the classics,Zhu creatively established a configuration of a square and its circumscribed circle upon which he relied to unfold his temperament innovation.Secondly,he used a numerological method to interpret the numerical relation between the perimeter of the square and that of the circle,an interpretation that may have been influenced by Songdynasty scholarship.With this interpretation,Zhu directly brought numerological values into play,which he used in turn in his determination of the dimensions of pitch pipes.Among the mathematical knowledge upon which Zhu relied when establishing the new temperament,the relation between the circumference of a circle and its diameter,expressed in modern mathematical notation as 10×4/√9^(2)+9^(2),is an outstanding finding of which he was understandably proud.The numbers belonging to this relation are also designated as“precise lü,”which is the same as the designation of the data in his equal temperament.The core of Zhu’s mathematical approach to creating“precise lü”for the new temperament consists in the gougu procedure(i.e.,the rule corresponding to the Pythagoras’s theorem)and in the operations performed to determine square and cubic roots.Zhu perfectly transformed the question of finding the length of the twelve successive pipes into that of first finding the lengths that corresponded to key pitches in symmetric points of the musical scale.In this way,the mathematical operations that he had mastered could successfully deal with the values occurring in the process of establishing the twelve-tone equal temperament.