Theory of the Beats of Mistuned Consonances. 271 



Smith's ' Harmonics/ prop. x. p. 81, 2nd edit. 



We will call a note having 100 vibrations in a certain time 

 a "unison;" and a note having 101 vibrations in the same 

 time will be said to make with the first an " imperfect unison." 

 Then the time in question, in which the notes make respec- 

 tively 100 and 101 vibrations, is what Smith calls a " simple 

 cycle," or " the period of the imperfect unisons." 



Consider notes making 2 and 3 vibrations respectively in 

 the time of 6 of the first note, or " unison.'''' Then the time 

 in which these notes make 2, 3, and 6 vibrations respectively 

 is what Smith calls the "short cycle" of the consonance. 



Suppose the consonance to be mistuned, so that 2 vibrations 

 of the one note = 6 of the unison (100), but three of the other 

 = 6 of the imperfect unison (101), then the 2 of the one note 

 still very nearly = 3 of the other, and each of these periods 

 or short cycles is still nearly 6 vibrations both of the unison 

 and the imperfect unison. Then Smith says: — 



" Take away the greatest equal numbers of short cycles 

 (of 6) that can be taken from both ends of the simple cycle, 

 or period of the imperfect unisons." 



(Take away two sets of 8 short cycles of 6 from 100 and 

 101 respectively.) 



" Then some part of another short cycle or two, as consist- 

 ing of unequal numbers of the quicker and slower vibrations 

 of the imperfect unisons, will always remain in the middle of 

 the cycle or period." 



(100-2x8x6 = 4.101-2.8.6 = 5. So that 4 and 5 

 vibrations respectively are the parts of the short cycles that 

 remain over.) 



" And this part, by interrupting the succession of the short 

 cycles " (of 6) "wherein the harmony of the consonance con- 

 sists, interrupts its harmony, and causes the noise which is 

 called a beat; especially as the interruption is made where 

 the short cycles on each side of it are the most imperfect and 

 inharmonious. Therefore the time between the successive 

 beats, made in the middle of each period or simple cycle of 

 the pulses of the imperfect unisons, or of the least imperfections 

 of the consonance, is equal to the time of the period/'' 



The term period of least imperfections is explained by noti- 

 cing that there would not be a real coincidence of the two 

 notes at the end of the period ; but there would very nearly 

 be one. 



This reasoning only points out the mode of calculating the 

 number of times that a certain form of grouping is repeated. 

 So far it is quite right ; and as forms of grouping will in this 

 subject always necessarily be a matter of interest and import- 



