September 2, 1922] 



NA TURE 



3i7 



left to itself. Each such vibrator can accordingly be 

 set in vigorous vibration by very feeble forces, provided 

 they occur at or very nearly at the rate in question. 

 These sympathetic vibrations would quickly die away 

 when their stimulating cause ceased. Each such 

 vibrator is supposed to have nervous connexion with 

 the brain so that the fact and amplitude of its vibra- 

 tions may be transmitted thence and duly noted. 



The theory is not primarily concerned with the 

 exact nature or details of structure of these vibrators 

 provided only that they fulfil the foregoing conditions 

 for mathematical theory shows that the response 

 of one such vibrator to the forces exerted upon it by 

 another vibration obeys the same general laws quite 

 independently of the details and nature of the vibratory 

 responder under consideration. 



Facts of Audition. — For normal ears the following 

 may be regarded as the chief facts of audition with 

 which we are here concerned : 



(i.) The range of audition is limited at the upper 

 and lower ends, such limit varying with individuals, 

 but about eleven octaves are usually audible. 



(ii.) Before either limit of audition is reached the 

 notes may be recognised to be very high or low, but the 

 distinct location of pitch fails, so that only about 

 seven octaves are musically available. 



(iii.) At about the middle of the range the dis- 

 crimination of pitch between near notes when sounded 

 successively is, for a keen ear. about the twentieth of 

 an equal-tempered semitone or ^-} yth of an octave. 



(iv.) When two very near notes of almost equal 

 intensities are sounded simultaneously, the difference 

 of their frequencies may be recognised by any one 

 as the number of beats per second. This may serve 

 to discriminate a pitch-difference of the fortieth of a 

 semitone or half that just named. 



(v.) When two different notes at a considerable 

 interval (say C and G) are sounded together, both 

 notes can be heard and their interval estimated, they 

 are not mistaken for a single note of intermediate pitch 

 (E or Et>). (This deserves special notice as being the 

 direct opposite of colour vision for some parts of the 

 spectrum, and will be dealt with in another article.) 



(vi.) When several simple vibrations occur simulta- 

 neously, being produced in association from the same 

 vibrating source, string or wind, the resulting character 

 of the compound tone or note is recognised and spoken 

 of as its quality, quality of tone, or tone simply. 



(vii.) A musical shake of about ten notes per second 

 on a note of frequency about a hundred and ten per 

 second can be heard distinctly. 



Power of Theory to meet the Facts. — Having briefly 

 reviewed the chief facts of audition we may now 

 naturally ask what power the resonance theory has to 

 meet the demands thus made upon it. In other words, 

 can the physicist imagine a set of vibratory responders 

 the behaviour of which under vibratory stimuli would 

 give results which correspond to those of human 

 audition ? In trying to arrive at a right or possible 

 solution, obviously many variables are at our disposal. 

 They may be stated thus : 



(a) The total range of pitches of the set of responders. 



(b) The musical intervals between adjacent re- 



sponders. 



(c) The damping (or rate of dying away) of the 



NO. 2757, VOL. I 10] 



vibrations natural to these responders when 

 started and then left to themselves. 



(d) The constancy or otherwise of the intervals and 



of the dampings throughout the range. 



(e) The fineness of discrimination of relative ampli- 



tudes of vibrations of adjacent or other 

 responders by means of the nerves attached 

 to them. 

 These variables are more than are needed to make 

 a solution possible ; they leave a choice between a 

 variety of possibilities which may be imagined by 

 the physicist and suggested to the anatomist for 

 examination and rejection or acceptance. Thus, for 

 example, the less the damping natural to a set of 

 responders the easier is the location of pitch by them. 

 But the presence of objectionable damping could be 

 balanced by an enhanced fineness in the nervous 

 discrimination of relative amplitudes of adjacent 

 responders. A word or two of explanation may be 

 desirable as to the relation between the damping 

 natural to a vibratory responder and the nature of 

 its responses to various alternating forces of nearly 

 its own frequency. Without entering here into the 

 niceties of the mathematical theory it may be said 

 broadly that the best response follows only with the 

 best tuning between the frequency natural to a re- 

 sponder and that of the forces acting upon it. But 

 the actual value of this response and its falling off 

 consequent upon mistuning both depend on the 

 damping natural to the responder. If the damping 

 is very slight, then the response is very vigorous for 

 precise tuning, but for quite small mis tunings the 

 response is almost negligible. This is often summed 

 up by saying that for slightly damped responders 

 the resonance is sharp. On the other hand, for highly 

 damped responders the response is not so good for 

 best tuning as in the former case, but this response is 

 only slightly impaired by moderate mistuning of the 

 forces. In other words, for strongly damped re- 

 sponders the resonance is spread. 



Bearing these facts in mind we have to make a 

 choice among the possibilities open to tis so as best to 

 meet the facts of the case. The facts (iii.) and (v.) 

 show that the damping must not be too large, because 

 that would involve spread resonance instead of the 

 sharpness needed for the actual fineness of location 

 of pitch experienced. On the other hand, fact (vii.) 

 shows that the damping must not be too small, as in 

 that case the sound heard from one note of the shake 

 would run into that of the next and give a blurred 

 effect contrary to experience. We have thus found 

 limits between which the damping should lie. 



In view of these considerations we may submit the 

 following suggestions. Let it be supposed that in the 

 ear there is a set of vibratory responders which — 



1. Cover a range of seven or more octaves, 



2. Are about twelve to the octave in the middle of 



the range and have a suitable damping, 



3. Are in total number about a hundred. 



Simple Model and its Behaviour. — To test the 

 adequacy of the arrangement just postulated let the 

 following simple model be set up as shown in the 

 diagram, Fig. 1. 



This responsive model consists essentially of a 

 stout cord stretched across a room between the fixed 



