132 Dr. Young's Experiments and Inquiries 



2NM + 2 MS =KN + NM -f-NM + MO = KM -f NO, is 

 equal to the sum of the distances of the corresponding parts of 

 the simple vibrations. For instance, if the two sounds be as 

 80 : 81, the joint vibration will be as 80.5; the arithmetical 

 mean between the periods of the single vibrations. The greater 

 the difference in the pitch of two sounds, the more rapid the 

 beats, till at last, like the distinct puffs of air in the expe- 

 riments already related, they communicate the idea of a conti- 

 nued sound ; and this is the fundamental harmonic described 

 by Tartini. For instance, in Plate V. Fig. 34 — 37, the vibra- 

 tions of sounds related as 1 : 2, 4 : 5, 9 : 10, and 5 : 8, are 

 represented : where the beats, if the sounds be not taken too 

 grave, constitute a distinct sound, which corresponds with the 

 time elapsing between two successive coincidences, or near 

 approaches to coincidence : for, that such a tempered interval 

 still produces a harmonic, appears from Plate V. Fig. 38. But, 

 besides this primary harmonic, a secondary note is sometimes 

 heard, where the intermediate compound vibrations occur at a 

 certain interval, though interruptedly ; for instance, in the coa- 

 lescence of two sounds related to each other as 7 : 8, 5 : 7, or 

 4 : 5, there is a recurrence of a similar state of the joint motion, 

 nearly at the interval of -f^, -£-, or J- of the whole period: hence, 

 in the concord of a major third, the fourth below the key note 

 is heard as distinctly as the double octave, as is seen in some 

 degree in Plate V. Fig. 35 ; AB being nearly two-thirds of CD. 

 The same sound is sometimes produced by taking the minor 

 sixth below the key note; probably because this sixth, like every 

 other note, is almost always attended by an octave, as a harmo- 

 nic. If the angles of all the figures resulting from the motion 

 thus assumed be rounded off, they will approach more nearly 



