Examples of the equal division of the minor thirds of a scale are not wanting in other musical systems, and the explanation which Mr. Ellis has offered for these may prove to be the true one in this Chinese case. Celtic music, which is often compared with that of China on account of its pentatonic basis, would have another striking characteristic in common with our melodies were it played upon the Highland bagpipe with equally divided minor thirds whose scale is given by Mr. Ellis. The same writer reports also a modern Arabian scale which is practically identical with this. A minor third divided into two approximately equal parts is also found in the tetrachord called the equal diatonic, and ascribed to the Alexandrian musician Ptolemy (200 B.C.). The intervals of this order are 1211, 1110, 109, or 151, 165, and 182 cents; and they can be produced in a string by stopping it at a quarter of its length and at one-third and two-thirds of this quarter. Mr. Ellis regards this process of the tri-section of the quarter of a string as the ultimate source of all these equal divisions of the third. The intermediate pien-Koung of our scales may according to this have a mechanical and not a musical origin. Another hypothesis will, we shall find, suggest itself in the course of our examination of the Chinese system of modulation.
The non-diatonic intervals which result from the intermediate pien-Koung are the most striking feature of these Samien songs. The note is most commonly reached from or left for those immediately adjacent above and below, separated from it by about 150 cents, or three-quarters of a tone; the interval Tche—pien-Koung, or pien-Koung—Chang (350 cents), occurs nearly as often; and the progressions Kio—pien-Koung (350 cents) and pien-Tche—pien-Koung (650 cents) each occur once. To our ears trained in the diatonic scale all of these intervals have a very strange and half-barbaric sound; but the quality is most marked in that of about 350 cents, an interval between a minor and a major third (316-386 cents). This intermediate third neither charms the ear like the major nor touches the heart like minor, but stands between them with a character of gravity,
