200 BULLETIN OP THE 



(abstract.) 



In the consideration and discussion of musical intervals from 

 a numerical standpoint, it is not unfrequeutly more convenient to 

 compare the logarithms of the ratios than the ratios themselves, 

 and the simplest form for the logarithmic series to be employed 

 is the number 2. Octaves, it is well known, progress, by integral 

 powers of 2. Intermediate tones progress hy fractional powers 

 of 2. 



In the modern major scale the seven intervals which comprise 

 the octave, or by which, proceeding from a fundamental the oc- 

 tave is reached, consist of but three different intervals, known 

 respectively as the major interval, minor interval, and semi- 

 interval — the entire octave comprising three of the major, two- 

 of .the minor, and two of the semi-intervals, so called. 



In the major interval there are eight vibrations of the lower 

 in the time of nine vibrations of the upper of the two tonea 

 which limit the interval. In the minor interval there are nine 

 vibrations of the lower to ten of the upper, of the two tones 

 which limit the interval. In the serai-interval, so called, there 

 are fifteen vibrations of the lower to sixteen of the upper of the 

 two tones which limit the interval. Thus making three distinct 

 ratios of progress, to wit, f , ^■^, ||. The ratio | is equivalent 

 to 2, raised to the power 0.170. The ratio y is equal to 2^ 

 raised to the power 0.152. The ratio || is equal to 2, raised ta 

 the power of 0.093. 



The advantage of employing the logarithms in place of the 

 ratios themselves in making comparisons is, that the operation 

 of comparison is thereby conducted by additions and subtractions 

 instead of by multiplications and divisions, the former processes 

 being obviously the simpler. 



The advantage of employing the number 2 as the base of the 

 system of logarithms, is the well-known fact that octaves progress 

 by integral powers of 2, hence it is convenient to have the inter- 

 mediate notes progress hy fractional powers o^ two. 



It may be observed that the semi-intervals, so called, are in 

 fact greater than the half of either the major or the minor inter- 

 vals. The sum of the logarithms — base 2 — of the three major 

 intervals, the two minor intervals, and the two semi-intervals 

 (comprising the octave), is necessarily unity: 



3 X .170 = 0.510 

 2 X .152 = 0.304 

 2 X .093 = 0.186 



1.000 

 Mr. A. ScHOTT presented an exposition on 



A NEW EYE-PIECE FOR OBSERVING PERSONAL EQUATIONS, 



