434 Dr. 1{. Konig on the Simultaneous 



be sounded together. Suppose, for example, that in the dis- 

 turbed major third \n : 5fl + x there appeared, 



bn + x — An = n + x 



An — (n + ./•) = 3n — x 

 &n+x -(;hi-.r) = 2n + 2.r 



An — (2n-\-2x) = 2n — 2.r. 



and then 2n+2#,with 2n — 2#, allow 4# beats to be heard. By 

 this proceeding we always arrive at the true number of beats; 

 bnt we are obliged at the same time always to suppose the 

 existence of notes which are not only themselves unheard, but 

 are also often both the effect of some and the cause of other 

 note-, which are all equally inaudible. In the example here 

 given, ")// + ./• and -1// show the beat-note n + x at a certain in- 

 tensity : bin if we now sound a primary note n-f x of about 

 equal intensity and, at the same time, alone with the primary 

 note 4//, we shall only hear 4« beats, but under no circum- 

 stances a note 3n— x of such intensity that it could be in a 

 position to produce other notes again in fresh combinations. 

 This note, Sn—x, to judge by the analogy in other cases, 

 would not be strong enough for this, even if it were a beat- 

 note ; but it has been produced from n + x and An, and is 

 therefore only a difference note; and how far below the heat- 

 notes in intensity are the difference notes and summation 

 notes we shall see later, in the section that treats of these 

 notes. 



With how much caution we must regard an explanation of 

 the wider intervals of the beats by the combination notes 

 becomes still more obvious if, instead of an interval of the first 

 period, we examine a simultaneous sound of the second or 

 third. We have seen above that distinct secondary beats can 

 be heard in the ratio 2 : 7. If this is formed with the funda- 

 mental note c'" ', both the beat-notes m and m / = c // , and this 

 c" we can hear loudly and distinctly. If the interval 2 : 7 is 

 less out of tune, and m and m! consequently no longer in clear 

 unison, they sound together in precisely the same way as two 

 primary notes c" of the same intensity w T ould do if put out 

 of tune to the same extent, and we need no further inaudible 

 note for the explanation of this phenomenon ; but according 

 to the old view, 



7n + x — 2n (c'") = r dn + * (c iy + x) . 



5n + x — 2n — 3n + x(g' n + x) 

 In + x — (3n + x) = An (c iy ) 



hn + x — An = n + x (c /f + x) 



An — (n + x) = on — x (y'" - 



