/ ON HARMONICS. 309 



,U*ose by whi^inve had left it. The simplest mode of arranging the twelve 

 ' s0un<|^ j *s*'to divide the octave into twelve equal intervals, all the notes 

 being in the same proportion to those which immediately follow them : 

 this is called the equal temperament, because the imperfection is equal in 

 all keys. In this system of temperament, the fifths, which consist of 

 seven semitones, are a little too flat, that is, the interval is a little too 

 small ; the minor thirds, consisting of three semitones, are also too flat, 

 and the major thirds too sharp. But it has generally been esteemed best 

 to preserve some keys more free from error ; partly for variety, and partly 

 because some are more frequently used than others : this cannot, however, 

 be done without making some of the scales more imperfect than in the equal 

 temperament. A good practical mode of performing it, is to make six 

 perfect fifths, in descending from the key note of the natural scale, and 

 six ascending fifths equally imperfect among themselves. We thus retain 

 a slight imperfection in the scales most commonly used, and make the keys 

 which are most remote from them considerably less perfect. Another 

 method, which is perhaps somewhat more easily executed, is to make the 

 fifth and third of the natural scale perfectly correct, to interpose between 

 their octaves, the second and sixth, so as to make three fifths equally tem- 

 pered, and to descend from the key note by seven perfect fifths, which will 

 complete the scale. Any of these modes of temperament may be actually 

 executed, either by the estimation of a good ear, or, still more accurately, 

 by counting the frequency of the beats which the notes make with each 

 other.* 



For denoting precisely the absolute as well as the relative frequency of 

 the sounds of the different octaves, we employ the first seven letters of the 

 alphabet ; A being the key note of the minor mode, in the scale of natural 

 notes, and C of the major. The peculiar characters used in music are 

 generally disposed on five or more lines, with their intervening spaces, 

 each implying a separate step in the scale, setting out from any line at 

 pleasure, which is marked with an ill formed G, a C, or an F : a sharp or 

 a flat implying that all the notes written on the line, or in the space, 

 to which it belongs, are to be raised or depressed a semitone, and a natural 

 restoring the note to its original value. The actual frequency of the vibra- 

 tion of any note, according to the pitch most usually employed, may be 

 found, if we recollect to call a noise, recurring every second, the first C, 

 then the C denoted by the mark of the tenor cliff will be the ninth, con- 

 sisting of 256 vibrations in a second. The fifth, consisting of sixteen 

 vibrations, will be nearly the lowest audible note, and the fourteenth the 

 highest note used in music, but the sixteenth, consisting of above 30,000 

 vibrations in a second, may perhaps be an audible sound. The frequency 

 of the vibrations of the other notes may easily be calculated from the 

 known relations which they bear to the note thus determined. (Plate 

 XXV. Fig. 355.) 



* Consult Marpurg's Anfangsgriinde der Theoretischen Musik, 4to, Leipz. 1757. 

 Versuch iiber die Temperatur, Bresl. 1776. Cavallo, Ph. Tr. 1788, p. 238. Robi- 

 son's Mech. Phil. 



