or the Division of the Octave, 173 



The values of the perfect fifth and third eorrect to ten places 

 are : — 



Fifth = 7-01955 00086, 

 Third =4-- -13686 28614. 



II. To find the vibration-ratio of an interval given in E.T. 

 semitones. 



To the given number add —1— and * of itself; divide by 

 40. The quotient is the logarithm of the required ratio. 



The ratio of the E.T. third thus found is I- 125995 : 1. 



Intervals of Regular Systems. 



These intervals are formed by proceeding through a certain 

 number of fifths, disregarding octaves. Thus we have, repre- 

 senting a fifth by 7 + 8, 



Departure of 12 fifths = 12 (7 + 8) - 84 = 128, 

 2-fifthstone = 2(7 + 8) -12 = 1+28, 



7-fifths semitone = 7(7 + 8) -48 = 1 + 7'6 } 



5-fifths semitone = -5(7 + 8) + 36 = 1 -58, 



4-fifths third = 4(7 + 8) -24 = 4 + 48, 



8-fifths third = - 8 (7 + 8) + 60 = 4 - 88. 



r 

 Putting for 8 its value for any given system (8= - for any 



n 



cyclical system), we have at once the values of these intervals. 



Any others can be formed in a similar manner. The 4-fifths 



third is that of systems which Mr. Ellis calls " commatic \ 3i I 



reject it as useless except in negative systems. The 8-fifths third 



is that of systems which Mr. Ellis calls "skhismatic;" 1 employ 



it in positive systems, according to HelmhohVs proposal. 



Positive Systems. 



For the details relating to the treatment of these scales, which 

 are somewhat technical, I must refer to the ' Transactions ' of 

 the Musical Association ; I will here only allude to the double 

 second of the key. 



In any positive system the second of the key may be derived 

 in two ways : — first, as a fifth to the dominant (by 2 fifths from 

 the key-note) ; and, secondly, as a major sixth to the subdomi- 

 nant (by 10 fifths down from the key-note). Thus the first 

 second to c is d, the other \d. On account of the importance 

 of this double form of second, I will consider the derivation of 

 these two forms by the ordinary ratios, i. e. for just intonation ; 

 the positive systems furnish approximations to the same results. 



(3\ 2 1 9 

 2 ) X 2 = S' ^ ne 



