There is no theoretic limit to the subdivision of an organ-pipe, either stopped or open. In stopped pipes we begin with 1 semi-ventral segment, and pass on to 3, 5, 7, etc., semi-ventral segments, the number of vibrations of the successive notes augmenting in the same ratio. In open pipes we begin with 2 semi-ventral segments, and pass on to 4, 6, 8, 10, etc., the number of vibrations of the successive notes augmenting in the same ratio; that is to say, in the ratio 1:2:3:4:5, etc. When, therefore, we pass from the fundamental tone to the first overtone in an open pipe, we obtain the octave of the fundamental. When we make the same passage in a stopped pipe, we obtain a note a fifth above the octave. No intermediate modes of vibration are in either case possible. If the fundamental tone of a stopped pipe be produced by 100 vibrations a second, the first overtone will be produced by 300 vibrations, the second by 500, and so on. Such a pipe, for example, cannot execute 200 or 400 vibrations in a second. In like manner the open pipe, whose fundamental note is produced by 100 vibrations a second, cannot vibrate 150 times in a second, but passes, at a jump, to 200, 300, 400, and so on. [Illustration: FIG. 103.] In open pipes, as in stopped ones, the number of vibrations executed in the unit of time is inversely proportional to the length of the pipe. This follows from the fact, already dwelt upon so often, that the time of a vibration is determined by the distance which the sonorous pulse has to travel to complete a vibration. In Fig. 103, _a_ and _b_ (at the bottom) represent the division of an open pipe corresponding to its fundamental tone; _c_ and _d_ represent the division corresponding to its first, _e_ and _f_ the division corresponding to its second overtone, the dots marking the nodes. The distance _m n_ is one-half, _o p_ is one-fourth, and _s t_ is one-sixth of the whole length of the pipe. But the pitch of _a_ is that of a stopped pipe equal in length to _m n_; the pitch of _c_ is that of a stopped pipe of the length _o p_; while the pitch of _e_ is that of a stopped pipe of the length _s t_. Hence, as these lengths are in the ratio of 1/2:1/4:1/6, or as 1:1/2:1/3, the rates of vibration must be as the reciprocals of these, or as 3:2:1. From the mere inspection, therefore, of the respective modes of vibration, we can draw the inference that the succession of tones of an open pipe must correspond to the series of natural numbers. The pipe _a_, Fig. 103, has been purposely drawn twice the length of _a_, Fig. 93 (p. 215). It is perfectly manifest that to complete a vibration the pulse has to pass over the same distance in both pipes, and hence that the pitch of the two pipes must be the same. The open pipe, _a n_, consists virtually of two stopped ones, with the central nodal surface at _m_ as their common base. This shows the relation of a stopped pipe to an open one to be that which experiment establishes.