The Twelve-tone Equal Temperament
一种音乐定律方法，将一个八度音分成十二个音程相等的半音，任意相邻的两个半音的音程值均为2的12次方根。由明代音乐家朱载堉（1536—1611）最早提出。朱载堉是世界上最先确定十二平均律的学者，他不仅推动中国律学进入新的历史阶段，也在人类乐律史上树立了里程碑。早在周朝，古人就发现了十二律，但是历代学者用等差关系计算的十二律无法旋宫转调。而朱载堉认识到半音的音高关系其实是等比关系，经过巧妙计算，他创制了“新法密率”，与今天的十二平均律完全一致，其优点是可以实现旋宫转调。朱载堉还将十二平均律的原理应用到乐器中，设计制造了一些新乐器。在世界范围内，键盘类乐器均广泛使用十二平均律，钢琴就是据此原理发明并进行定音的乐器。
The twelve-tone equal temperament is a standard scale tuning in which an octave is divided into 12 equal semitones, with the interval value of any two adjacent semitones being the 12th root of 2. It was first proposed by Zhu Zaiyu (1536-1611), a musician of the Ming Dynasty. Zhu was the first scholar in the world to determine the twelve-tone equal temperament, and he not only created a new historical stage of Chinese temperament, but also set a milestone in the music history of the world. As early as the Zhou Dynasty, the ancient Chinese had discovered the twelve-tone temperament, but the temperament calculated by scholars over the ages in terms of equivalence could not enable modulation, concerning both key and mode changes. However, Zhu realized that the relationship between the pitches of the semitones was in fact an equiproportional relationship, and after clever calculations, he created a new method of dividing the temperament equally with geometric progression, which is exactly the same as the twelve-tone equal temperament of today, and has the advantage that it enables key and mode changes. Zhu also applied the principles of the twelve-tone equal temperament to musical instruments and designed and built some new instruments. The twelve-tone equal temperament is widely used in keyboard instruments all over the world and the piano is an instrument invented and tuned according to this principle.
引例 Citations：
◎昔黄帝令伶伦作为律。伶伦自大夏之西，乃之阮隃（昆仑）之阴，取竹于嶰谿（xièxī）之谷，以生空窍厚钧者，断两节间，其长三寸九分，而吹之，以为黄钟之宫，吹曰“舍少”。次制十二筒，以之阮隃（昆仑）之下，听凤皇之鸣，以别十二律。其雄鸣为六，雌鸣亦六，以比黄钟之宫，适合。（《吕氏春秋·古乐》）
以前黄帝命令一个叫伦的乐官创制乐律。伦从大夏山的西边，前往昆仑山的北边，从昆仑山山谷中取来竹子，选取中空而内壁厚度均匀的竹子，截取两个竹节的中间部分，长三寸九分，将吹此竹管发出的声音定为黄钟宫，命名为“舍少”。然后伦依次制作了十二根竹管，带往昆仑山下，聆听凤凰的鸣叫，用来区别十二律。雄凤的鸣叫有六律，雌凤的鸣叫也有六律，用它们来与黄钟宫的声音进行比较，都适度谐和。
In olden days, the Yellow Emperor commanded the court musician Lun to create a tuning system for music. Lun traveled westward beyond the Daxia mountains and onward to the north side of Mount Kunlun, where he took bamboo from the valley, selected a hollow bamboo with even thickness of the wall, cut a length from the middle part between two bamboo joints measuring three cun (寸) and nine fen (分), and designated the sound produced while blowing this bamboo pipe as Huangzhong Gong (黄钟宫), and named it Sheshao (舍少). After this, Lun made twelve bamboo pipes in turn which he took to the foot of Mount Kunlun to listen to the sound of the phoenixes, and used these to determine the temperament. The male phoenix's song was divided into six tones, likewise that of the female phoenix. He then compared these with the sound of Huangzhong Gong, and found them both moderate and harmonious. (Master Lv's Spring and Autumn Annals)
◎大抵不用三分损益，而用勾股之术及开立方之法求之所得也。是以隔八隔六，循环无端，上下相生，首尾一贯，以证往而不返之说为非。是盖二千余年所未有，实自我朝始耳。（朱载堉《律历融通·附录·音义》）
大体上不用三分损益的方法，而是用勾股定理和开立方计算可得到音律的高低。因此每相隔八个律或者六个律，就可以无限循环，前后相生，首尾连贯，这可以证明音律不能循环的说法是错误的。这大概是两千多年来从未有过的发现，确实是从我朝才有的理论。
In general, instead of using the method of finding a pitch by mathematical calculation based on the proportional relationship between the length of the wind or string and the pitch of the sound, the Pythagorean theorem and the extraction of a cubic root are used to calculate the level of the temperament. This way, every eight or six temperament interval could be cycled infinitely, with the last and first temperament intervals being produced in succession, proving that the saying that temperament intervals cannot be cycled is wrong. This is probably a discovery that had never been made in the past two thousand years and more, and is a theory conceived in our own dynasty. (Zhu Zaiyu: On the Relationship Between Temperament, Astronomy and Calendars)
推荐：教育部 国家语委
供稿：北京外国语大学 外语教学与研究出版社
责任编辑：钱耐安
