Scale of the Sounds of the Sheng.
These numbers indicate the sequence in evolution of pipe lengths by the process described.
The scale really comprises one octave and a fourth and the master pipe is the e♭, it being so marked on every instrument I have handled, as shown in the illustration at pipe 14. This is the pipe giving the note corresponding in pitch to the imperial standard pipe, yet it is one fourth less than that in length, because, though both are cylindrical, the one is whistle or flute blown and the other reed blown—such is the law of these reed pipes—whilst the real standard length standing beside it, No. 15, gives a sound a fourth lower, and is the lowest in sound in the scale.
Yet b♭ is not the tonic; the Chinese have not in their music our kind of reckoning; but their e♭, at the junction of the two tetrachords, corresponds to the mese or middle note of the Greek scale. And in passing let me say that in the middle tetrachord you leave out in descending the notes 10 and 4, and in ascending leave out 12 and 13, according as the conjunct tetrachords are formed in the upper or in the lower part of the scale; and thus the conditions required by the tetrachord are maintained. Although, to make exposition easy, the notes are here presented in our modern notation, you should still bear in mind that the relations of note to note are not the same, are not exact in ratios; most of the notes are flatter or sharper than indicated, for the simple reason that there is no other ratio of interval than the fourth taken in relation to intervening upper or lower octaves; and since two fourths will not comprise an octave, each successive step in fourths that are perfect takes us away from diatonic accuracy. Thus the g given as a fourth above d♭ looks odd; yet it is from that actual pitch length, as one may say, that the c above is derived. The c is a flat note not expressed by our notation, but we have to signify the notes in the nearest terms we can for convenience, none being quite accurate. A very curious puzzle, you will answer; but very clear I can assure you when you have once found your way through the labyrinth.
Writers upon the Sheng all say that the pipes in the range numbered 2 and 6 are mere duplicates, and also 4 and 8. But they are altogether mistaken; they give not any intimation whatever why they exist. If it had been so then speaking lengths would have been in duplicate, which they are not. But I can demonstrate why they are there; and that they are not duplicates either as regards length or in pitch, but are necessary in the evolution. There is nothing fantastic in the arrangement; all the notes come naturally from one to the other; they are necessary; not one too many to complete the idea, not one left out; and, in truth, that last one in the sequence given of evolution—which I have marked ♭v_{a}, to indicate an extra flatness—has every suggestion of being an afterthought. For the pipe No. 2 in the order exists for no other reason than to make an A♭ that shall be a true fourth to the high D♭; a sounding pipe, for which a place is found where otherwise a second mute pipe would have been, corresponding with that on the opposite side. Why are there two pipes with the ventage hole turned inwards to be closed by a finger of the right hand? Because the thumb ranges over several pipes, but could not properly close more than one at a time; and to meet the difficulty, pipes 3 and 4 have the closure operating behind. So that when required for making fourths or thirds with 2 or 5 or 6 or 7, in the order that comes under the thumb of the right hand, then the finger comes to aid in producing the simple 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.
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And in the alternative:—