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8 CHINESE MUSIC. sound rendered was the perfect fifth, which in our Western music is also expressed by the ratio of 3 to 2. The second bamboo being treated on the ~ame principle, produced a third tube measuring exactly two-thirds of the length, and giving a note a perfect fifth higher than that of the second t ube. This new sound seeminbo· too far distant from the first or fundamental note the lenoth ' b of the producing t ube was doubled (that is, fom-thirds of the second tube's whole length was t aken instead of two-thirds), and the note became an octave lower. All the tubes were cut on the same principle,1 the relation of 3 to 2 representing the harmony existing between heaven and ear th. They engendered one another and always measured two-thirds or four-thirds of the whole length of their generator. The lus were therefore divided into two classes, the ~ fft (yang lu s) and the ~ g (yin ·tu s), or males and females, positives and negatives, perfect and imperfect. According to the ~ ~ ( I King), chaos was divided into two parts, yang answering to male energy, and yi n corresponding to the female principle. All that is strong and superior is yang; yin indicates dependence, inferiority. Everything in Nature belongs to one of these two grand categories, from whose combinations and reciprocal action results all that exists or takes place in the universe. The lus 1, 3, 5, 7, 9, and 1 1 were considered as yang ; the even numbers were classed as yin ; but it is w,ell to remark that these distinctions did not at all affect the tones, and were made simply to please the Chinese ideas of the time. Other comparisons were drawn between the 12 lus, the 12 moons, the 1 2 Chinese hours, etc. 2 The fiTS t tube, which was considered as the basis, the generator of all the others, received the name '.ft~ (hucmg-chung). The sound produced by it was named ~ (lcung), and became the tonic or key-note of a kind of semi-diatonic scale of 12 degrees, nearly i dentical with our ch1·omatic gamut, the only difference being that our scale is tempered, while that of the Chinese is left untouched. Temperament deno tes a small, and to the ear almost imperceptible, deviation from the absolute purity of intervals which compose our scale. It is well known that 12 pe1·fect fifths employed within the space of an octave (like the 1 2 Chinese sounds) exceed the ratio of the octave, or that of 2 to 1, by the ditonio cornmia, a small interval expressed by the ratio· of 531,441 to 524,288. Our ear is so constructed that it cannot endure the excess or deficiency of a whole comma in any interval without being offended, and therefore it has been found expedient to diminish each fifth by one-twelfth of the ditonic comma, instead of diminishing only one :fifth by the entire comma. Tha.t is what we call temperament in Western music, and it is the absence of it that causes some of the Grunese intervals to appear to us either too high or too flat. We will prove mathematically the difference when speaking of the diatonic scale. The following diagram will illustrate the lils, giving their names, the moons, hours, etc., to which they correspond, the musical sounds they emit (supposing huang-chung to give 0m C), their corresponding notes in our music, etc. The lus follow each other at the interval of half a tone. I.:: ~ m ~ J: "f 5 $.• ' lie Ji fr WI JI½ A ~ Im f/1\ + = A't·