TO THE COMPARISON OF SEVERAL MAGNITUDES. 65 



and again we have it in the value of A n , which is A 10 .p xo + A 9 . q Q + A 8 . r 8 = 

 220 + 70 + 28. 



The series for B is formed exactly in the same way, only that its commence- 

 ment is delayed one step ; in other words, B x is held as zero, and B 2 is made unit. 



Similarly for C, C 1? C 2 are held as zeroes, and C s is made unit. We may 

 in the same way find series for D, E, F, and so on. 



The eighth set of values give 28, 19, 12 as nearly proportional to the 



logarithms of 5, 3, and 2. Assuming these as sufficiently near for the purposes 



of musicians, we must divide the interval corresponding to the ratio 1 : 2, called 



by them an octave, into 12 parts, to which the name semitone is given. In this 



way, counting in semitones, we have log 2 = 12, log 3 = 19, log 5 = 28 ; 



3 5 



whence log 9 = 7, log ^ = 4 , and so on ; whence the arrangement of the 



gamut is at once obtained. According to these values, we should have 



9 10 



log g = 2, and log Q = 2 , wherefore the degree of precision obtained by these 



numbers is not sufficient to discriminate between the tone major and the tone 



minor. 



In order to obtain a closer approximation, we must proceed further along 



the series. Now it is important to keep to the nomenclature and arrangement 



of semitones ; wherefore we search among the series C for some member 



divisible by 12 ; no one of those above given is so divisible, and therefore we 



look for some compound of two of them which may be a multiple of 12. Thus 



362 

 C 10 + C n = 156, so that, still counting in semitones, we have log 5 = -r^ = 



11 247 156 



27 j3 , log 3 = jcr = 19 , log 2 = -J3 =12. From these values the logarithm 



9 

 of the tone major represented by the ratio g is still 2, but that of the tone minor 



represented by q- is 1 jo ; in the same way the corrected values of the other 



musical intervals may be obtained. 



By putting A, B, C to represent the periodic times of three astronomical 

 phenomena, we may ascertain the intervals between their simultaneous recur- 

 rence. Thus, if we put A for the time of revolution of the moon's node, B and 

 C for the earth's and moon's periodic times, we shall obtain directly the law of 

 recurrence of eclipses. 



If we take three contiguous sets of values, and thence compute the succeed- 

 ing set, we obtain 



A n _|_ 3 = p „ + 2 . Aj + 2 + 9\ n + 1 • A n +1 + r n A» 



B„ + 3 = p n + 2 • B„ _|_ 2 + q n + 1 . B„ _|_ i + T n B n 



^n + 3 = P n + 2 • ^n + 2 + ?» + l • Ui+1 + *'n ^n I 



K 



A„ + i 



Ai + 2 



B„ 



B„ + i 



B„ + 2 



c„ 



C„ + i 



^n+2 



