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Here the fx makes a perfect fourth to ♭b, but would not to c below; and ♭av 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.
Time also, I found, had occasioned a little variation, as the bamboos in drying lengthen a little; but it is a mere trifle.
One or two points I must not forget to direct attention to. Notice that the reeds in the Sheng have their faces turned to the wall of the bowl, and in this way a reflecting surface acts to the advantage of the reed; the air also acts less wildly than might be the case if the reeds were turned toward the centre of the bowl. The reed tongues are very thin, and are not lifted from the level of the plates; consequently they may be caused to sound both by drawing with the breath and by blowing, although the latter is prohibited in practice, as the moisture from blowing condensing on the reed alters the pitch, and corrodes the metal. Any excessive forcing of the tone the reeds are not liable to, because the air is passing at the same time through all the pipes, those that are sounding and those that are not.
Fairly, then, I think that I may claim to have transformed myself into an early Chinaman, and to have shown that I possess a sympathetic, inquisitive, barbarian sort of a mind, and ought to have lived years ago. The plan that I hit upon in a wild, instinctive way appears to be identical with the plan upon which the Sheng was evolved; for no other seems so easy and natural as this, alike in regard to the origin of the instrument and to the development of the music.