On Temperament, or the Division of the Octave. 165 



branches of experimental science, the history of which invariably 

 indicates a procedure by successive approximations, that I re- 

 duce the discussion to that of corrections to, or departures from, 

 equal-temperament positions. I am obliged to mention this, as 

 there is a tendency on the part of some, especially pure mathe- 

 maticians, to object to the treatment by approximations. But 

 we must remember that there is no instance of any theory being 

 reduced to observation or experiment except by means of ap- 

 proximation, or with a limited accuracy of some kind ; and astro- 

 nomy itself, the greatest example of modern science, is the greatest 

 example of the procedure of approximations. 



We can easily imagine the octave divided into any number of 

 parts. And we can easily imagine that a of these parts will 

 be the best approximation to the fifth, b to the third, and so on ; 

 and if we tabulated all the values for all possible modes of divi- 

 ding the octave, we should no doubt be able to find out all the 

 properties of every division. But this would be a very clumsy pro- 

 ceeding ; and accordingly I have preferred to start from a quasi- 

 analytical point of view, and to define a number having certain re- 

 lations with any system, which' I call its order. This admits readily 

 of doing two things : — first, determining the best approximation 

 to the fifth ; and secondly, arranging the whole system into scales 

 corresponding to that approximation. Thirds and other inter- 

 vals are made to depend upon the fifths. Now the definition of 

 the order of a system is really simple. Since the equal-tempe- 

 rament fifth is 7 semitones and the octave is 12, it is clear that 

 12 equal-temperament fifths = 7 octaves. Hence the deviation 

 of 12 fifths of any system from 7 octaves is an important pro- 

 perty of the system ; and if we take some approximate fifth 

 of any system and multiply its number of units by 12, the 

 number of units by which the result exceeds or falls short of 7 

 octaves is what I call the order of the system for that approxi- 

 mate fifth. With ordinary systems there is only one approxi- 

 mation that merits discussion at all, and consequently only one 

 order practically for each system. The importance of the order 

 is not only that systems are arranged into scales in different 

 manners, according to their orders, but also, since the depar- 



ture of the fifth of the system n of order r is -, it is only neces- 

 sary to compare this fraction with the departure of the perfect 

 fifth (-01955), or of any other fifth, to ascertain with facility all 

 that is to be known about it. 



Having said so much as to the general method, I pass on to 

 notice a few points in the history of the subject. For general 

 references, see Mr. A. J. Ellis, F.R.S.,"On the Temperament 

 of Instruments with Fixed Tones" (Proc. Roy. Soc. 1864). 



