The Lülü xinshu 律呂新書 "New Book on the Pitchpipes" is a book on music compiled by the Song period 宋 (960-1279) scholar Cai Yuanding 蔡元定 (1135-1198), courtesy name Cai Jitong 蔡季通, who came from Jianyang 建陽 (modern Jianyang, Fujian). The Neo-Confucian master Zhu Xi 朱熹 called his method of tuning very exact. The 2 juan "scrolls" long book is divided into two parts with 23 chapters. The first part (Lülü benyuan 律呂本原) describes the basics of musical tuning, while the second (Lülü zhengbian 律呂證辨) discusses particular problems of the production of pitchpipes (lülü 律呂).
Cai Yuanding describes the method of the Han period 漢 (206 BCE-220 CE) scholar Jing Fang 京房 to derive the pitchpipes of a whole gamut from the basic Yellow Bell Pipe 黃鐘 with determined measures by alternatively subtracting and adding a third of the dimension of a pipe to produce the next in the series. Cai adds to this method the possibility to deviate from this method and to create "changed pipe tones" (bianlü 變律) with tone steps lesser than a full tone, but only for six of the twelve pipes, so that his gamut consists of eighteen tones. He stressed that it was important to check that the diameter of the pipes was always equal, which is not very easy in a natural product like bamboo tubes. For the calculation of the length of the pipes he made use of an old method of the "progessing nine" (jiujin zhi 九進制), meaning that it was necessary to deduct one tenth of each digit to exactly meet the correct tone. He therefore also commented on the treatise on the pitchpipes, Lüshu 律書, in the history Shiji 史記. His chapter on the musical modes is particularly interesting to understand the concept of harmony and musical mode during the Song period 宋 (960-1279).
There were several studies on Cai Yuanding's book, like Xu Zhen's 許珍 Lülü xinshu fenzhu tuzuan 律呂新書分注圖纂, Li Wencha's 李文察 Lülü xinshu buzhu 律呂新書補注, Zhou Mo's 周模 Lülü xinshu zhu 律呂新書注, Han Bangqi's 韓邦奇 Lülü xinshu zhijie 律呂新書直解, Wang Chaoxi's 王朝璽 Lülü xinshu sijie 律呂新書私解 or Wen Zaoxiang's 文藻翔 Lülü xinshu jianshi 律呂新書淺釋.
Unfortunately the book is full of errors in the quotations of older texts and has therefore much been criticized. It is included in the imperial reprint series Siku quanshu 四庫全書.
Gao Liushui 高流水 (1996). "Lülü xinshu 律呂新書", in: Zhuzi baijia da cidian 諸子百家大辭典, Shenyang: Liaoning renmin chubanshe, p. 433.
Guo Shuqun 郭樹群 (1996). "Lülü xinshu 律呂新書", in: Zhongguo xueshu mingzhu tiyao 中國學術名著提要, Yishu 藝術卷, Shanghai: Fudan daxue chubanshe, p. 86.
Li Xueqin 李學勤, Lü Wenyu 呂文鬰 (1996). Siku da cidian 四庫大辭典, Changchun: Jilin daxue chubanshe, vol. 1, p. 622.
律呂本原 Lülü benyuan The origin and development of the pitchpipes
1 黃鐘 Huangzhong The Yellow Bell Pipe
2 黃鐘之實 Huangzhong zhi shi The volume of the Yellow Bell Pipe
3 黃鐘生十一律 Huangzhong sheng shiyi lü Creating the other eleven pitchpipes from the Yellow Bell Pipe
4 十二律之實 Shier lü zhi shi The volumes of the twelve pitchpipes
5 變律 Bianlü The alteration of pipes
6 律生五聲圖 Lü sheng wu sheng tu Chart of producing the five tones by the pipes
7 變聲 Biansheng The alteration of tones
8 八十四聲圖 Bashisi sheng tu Chart of the 84 tones
9 六十調圖 Liushi diao tu Chart of the 60 keys
10 候氣 Houqi Relations of the tones to the months
11 審度 Shendu Diligence for length measures
12 嘉量 Jialiang Excellence in volume measures
13 謹權衡 Qin quanhengCarefulness in weight measrues
律呂證辨 Lülü bianzheng Discussion of the pitchpipes
1 造律 Zaolü The production of pipes
2 律長短圍徑之數 Lü changduan weijing zhi shu The dimensions of diameters, circumferences and lengths of the pipes
3 黃鐘之實 Huangzhong zhi shi The volume of the Yellow Bell pipe
4 三分損益上下相生 Sanfen sunyi shangxia xiansheng The mutual production of pipes by adding and subtracting a third
5 和聲 Hesheng Harmonzing sounds
6 五聲大小之次 Wusheng daxiao zhi ci The order of the higher and lower of the five tones
7 變宮變徵 Biangong bianzheng Alteration of the first and the fourth tone
8 六十調 Liushi diao The 60 keys
9 候氣 Houqi Relations of the tones to the months
10 度量權衡 Duliang quanheng Length, volume and weight measures