Yuelü quanshu 樂律全書

Yuelü quanshu 樂律全書 "The whole book on musical tuning" is a book on music written during the Ming period 明 (1368-1644) by Zhu Zaiyu 朱載堉 (1536-c. 1610), courtesy name Boqin 伯勤, style Gouqu shanren 句曲山人. He was a son of Zhu Houwo 朱厚焥 (or Zhu Houwan 朱厚烷, d. 1591), Prince Gong of Zheng 鄭恭王, a member of the imperial house of the Ming, but was not allowed to dwell inside the palace because his father had been eliminated from the line of princely succession.

Zhu Zaiyu wrote quite a few books on music, including Lüxue xinshuo 律學新說, Yuexue xinshuo 樂學新說, Lülü jingyi 律呂精義 (part of the Yuelü quanshu), Yuelü zhenglun 律呂正論, Lülü zhiyi bianhuo 律呂質疑辨惑, Sepu 瑟譜, Zaoman guyue pu 操縵古樂譜, Shigong heyue pu 施官合樂譜, Xiangyin shiyue pu 鄉飲詩樂譜, Liudai xiaowu pu 六代小舞譜, Xiaowu yuepu 小舞鄉樂譜 and Lüli rongtong 律曆融通, as well as some books on mathematics, Jialiang suanjing 嘉量算經, and Suanxue xinshuo 算學新說.

Figure 1. Musical score of the air Guanju 關雎

The large words on the top are the words of the air (關關雎鳩，在河之洲). Below that, the basic tone of each syllable is written in terms of the pitch pipe designations (姑[洗], 蕤[賓], 應[鐘], 南[呂], etc.). Each syllable of the air is accompanied by a qin 琴 and a se 瑟 zither for which different notation systems are used, namely the gongche 工尺 system for the qin (the "strange" characters with 艹 tops and 乚 bottoms), and the pitch pipe designations for the se zither.

Figures 2-3. Images of dancers performing ceremonial dances

The 42-juan long book was finished in 1595 and submitted to the throne during the Wanli reign 萬曆 (1573-1619). The imperial bibliography in the official dynastic history Mingshi 明史 speaks of 40 juan of length. In fact, the Yuelü xinshu consists of eleven texts part of which is not divided into juan, so that it is not quite sure which statement is correct.

The chapter Lülü jingyi 律呂精義 "Essence of the meaning of the pitch pipes" is the most important part of the book, in which the basics of musical tuning are explained. Zhu Zaiyu argues that a line of 100 grains of millet in the long direction (hengshu 橫黍) has the same length as that of 81 grains of millet arranged side by side (zongshu 縱黍), and this length corresponds to 1 chi 尺 (the carpenter's foot, yingzao chi 營造尺, see weights and measures). He also explains that a pipe with half the length of the Yellow Bell pipe (huangzhong 黃鐘) does not harmonize (ying 應) with that of the full length, but half a Taicu pipe 太蔟 does. This theory was reproduced in the book Lülü zhengyi 律呂正義 from the Kangxi reign-period 康熙 (1662-1722) because it was the most exact explanation of the creation of a full musical scale out of the basic pipe (the Yellow Bell pipe).

Figures 4-5. The calculation of pitch pipes and their relation to measures

Diameters and lengths of the 16 pitch pipes. The Yellow Bell pipe (huangzhong 黃鐘) has a length of 8 cun 寸 or 80 fen 分, according to the length of the carpenter's foot (yingzao chi 營造尺), which in turn corresponds to the Shang foot (Shangchi 商尺). The diameter of the huangzhong pipe was 4 fen (0.4 cun), that of the notch 2.82 fen.

Set of 12 pitch pipes from the Han period 漢 (206 BCE-220 CE) in which the Yellow Bell pipe had a nominal length of 9 chi, corresponding to nine times the diameter of a standard coin (quanhuo 泉貨).

Figure 6. Deduction of a tempered scale from the largest pipe

Figure 7. Tuning the seven strings of a qin 琴 zither

The method of the deduction of a system of pitch pipes from the Yellow Bell pipe and the associated "main pipes" (zhenglü 正律) and "doubling pipes" (beilü 倍律) is based on two different mathematical operations, namely the Pythagorean method (gougu qiuxian 勾股求弦) and the hypotenuse (fangqiuxie 方求斜) method, but the former cannot any more be made use of when constructing the Nanlü pipe 南呂 from the Ruibin pipe 蕤賓 on.

The Lülü quanshu is included in the imperial series Siku quanshu 四庫全書.