NOTE TO THE FOLLOWING ARTICLE. 



The following paragraphs were removed from the body of the article at 

 page 511, as not necessary to its design, they are here prefixed as a conve- 

 nience to those who may not have seen the more elaborate article on Beats, 

 in " Smith's Harmonics." 



When two sounds are heard nearly harmonizing, there are heard 

 at the same time irregularities of sounds, or Beatb ; the frequency 

 of which depends on the nature of the sounds. When a true Do, 

 (32 vibrations per second,) is accompanied by a sound of 31 or 33 vi- 

 brations, one beat per second is heard. Sounds of 31 and 33 vibra- 

 tions would produce 2 beats ; 256 and 259 would produce 3 beats, &c, 

 as in the following diagram. Where the sounds are nearly 5ths, 

 3rds, or any concords, their vibrations per second must be multiplied 

 by the ratio of the interval, so as to produce nearly equal numbers, 

 and then the difference is the number of beats per second. Thus the 

 beats of the imperfect fifth between La 1 27 and Mi 3 40 are found to 

 be one per second ; by subtracting 80 (twice 40) from 81 (3 times 

 27.) In the same way, the beats of the major third, recommended in 

 the lamented Prof. Fisher's Table for Tuning (Sill. Journal, Vol. 1, 

 p. 195,) between Fan 5 (325.68 vibrations,) and Lan 5 (42S.92 vibra- 

 tions per second,) will be found by multiplying these numbers by 4 

 and 5 respectively, to be 37.28 per second. 



In the following figure the points represent the vibrations of im- 

 perfect unison, as of Sol 5 ; the commas only, of imperfect fifths, as Do„ 

 and SoK The beats, which are the same in both cases, are denoted 

 by" b 



b b b 



These beats furnish us with the most ready way, though the least 

 satisfactory way, of ascertaining the number of vibrations in any pitch. 

 It is easy to tune two tubes so that they shall differ in pitch precisely 

 comma. It lias long been known that their vibrations then are 80:8 1, 

 but it is more difficult to ascertain the vibrations of either. Let the 

 sounds be Si 4 and Si 4 . If they beat 3 limes per second, we know 

 that Si 4 vibrates 240, and 81* 243 times p'T second. Again, by means 

 of beats the most perfect tuning can be executed, by the aid of an un- 

 practised ear. Even the most perfect interval, (he octave, can be tuned 

 more accurately by making use of their beats with an intermediate 

 sound, than in any other way. This process is, however, too slow 

 for the practical tuner, unless it be in tuning setts of tuning forks. 



