concords desired. Certainly the contrivance in its directness and efficiency is very clever. The scale therefore is, after casting out the alternatives not required in ascending, as follows. See how very Greek it is. ♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭ _b_—_c_‿_d_—_e_—_f_‿_g_—_a_ _b_—_c_‿_d_—_e_ \——————v—————/\—————v————/ \——————v—————/ And in the alternative:— ♭ ♭ ♭ x ♭v ♭ ♭ ♭ _b_—_c_‿_d_—_e_ _f_—_g_‿_a_—_b_—_c_‿_d_—_e_ \——————v——————/ \—————v————/\—————v————-/ Here the _f_x makes a perfect fourth to ♭_b_, but would not to _c_ below; and ♭_a_^v makes a perfect fourth to ♭_d_ above, but would not to ♭_e_ below. Each _c_ is to be taken as much nearer the _b_♭ than in our notation. The pentatonic is obtained by skipping over the half tones. These mysteries you can unravel if you care to take the trouble to cut strips of paper as I did of wood. Number them all at the bottom, and from the 9-7/8in. length you will get its fourth,—that is to say, three quarters of its original. Write on each the name of the note. And so on, getting octaves and fourths above or below, in the sequence I have given. As you go on, cut the strips to the lengths found and fold each strip in length into four; and then when you lay them out these curious tonal relations are made manifest. Thus you see why the sounds are what they are. The true lengths would prove in sounds perfect fourths if the diameters of the pipes had carried the geometrical law. Strangest of all remains the fact that my blind sticks proved true prophets, and led me in the way of evolution, the pitches of the pipes corroborating at every step. Reverting now to the details of the _Sheng_, there is one little hint too important to be omitted if any reader should happen to have the opportunity of measuring the actual pipes. He will find that the pipe that is longest in the speaking length—that is to say reckoning from the lower end of the slot—will be 10-1/8in. in length, instead of 9-7/8in. This excess of a quarter of an inch is common to all the pipes, and is that portion extended beyond the hollowed part of the foot which only reaches to the base of the metal tongue, and is therefore the real limit of the column of air. Consequently, this quarter should be allowed _off each pipe_ when measured, because if computed in the speaking length it would affect the accuracy of the half lengths. In my first analysis, I found difficulties arose when comparisons were instituted between the pipes themselves and the slips of wood of the lengths evolved as a problem; because, as I soon became aware, upon halving the total lengths as taken actually from the pipes, the half of this quarter inch was entering into every calculation, and was of course misrepresenting by an eighth of an inch the real speaking length to be credited to the half length and the three fourths length; and with the shortest of the pipes the discrepancy became serious.