A CLASSIFICATION OF GROUPS 279 



beating nor upon the duration except that the periodicity of any 

 given group was to remain constant. 3 



This procedure gave as its data (1) an alternate series of 

 groups and rests, and (2) a series of discrete but contiguous dura- 

 tions, i.e., durations of groups, and durations of rest periods. 

 Generalized the data would appear as follows: 



(1) Group, Rest, Group, Rest, G, R, G, R, G, R, etc. 



(2) G-Dur., R-Dur., G-Dur., R-Dur., G-D, R-D, G-D, R-D, etc. 



Quantitatively expressed, the above two series may be ex- 

 pressed : 



(3) a-G, R, b-G, R, c-G, R, w-G,R, x-G, R, y-G, R, z-G 



(4) a-sec., b-sec., c-sec., d-sec., e-sec., f-sec., g-sec., etc. 



where a, 6, c, d, etc., in (3) stand for simple numbers, each de- 

 scribing and defining the group to which it has reference from 

 the point of view of the number of elements contained in it; 

 and where the corresponding coefficients in (4) qualify the dura- 

 tions of both groups and rests in seconds. Inasmuch as the rest 

 periods are not otherwise quantitatively described in series (1) 

 and (3), they may be entirely disregarded in these equations or 

 series, and the latter may be written as follows: 



(5) a-G, b-G, c-G, d-G, w-G, x-G, y-G, z-G 



The above series only describes groups as to the number of 

 elements or beats contained in them; it says nothing regarding 

 the other characteristic properties of groups, such as their peri- 

 odicities, or their durations; it disregards entirely the rests and 

 their durations, but it is the fundamental series of this paper, 

 which seeks to answer the following questions: 



(a) What groups are beat by any observer? 



(b) Is a general classification possible? 



Our observers were seated in one of the standard tablet chairs, 

 the right fore-arm resting on the tablet, the index finger crooked 



3 It goes without saying that observers were not permitted to count either 

 silently or aloud while beating. 



