On Temperament, or the Division of the Octave. 507 



is to us, constituted as we are, unrealizable, unknowable, incon- 

 ceivable." The monads of Leibnitz and the demons of Max- 

 well express in words the personality implied in every manifes- 

 tation of force. 



LXVII. On Temperament, or the Division of the Octave. 

 By Mr. R. H. M. Bosanquet*. 



THE subjects of the paper were some points in the theory 

 of the division of the octave, and the employment of a 

 certain notation. 



The following definitions are fundamental : — 



" Regular Systems " are such that all their notes can be 

 arranged in a continuous series of fifths. 



u Regular Cyclical Systems " are not only regular, but return 

 into the same pitch after a certain number of fifths. Every 

 such system divides the octave into a certain number of ecmal 

 intervals. 



" Error " is deviation from a perfect interval. 



"Departure" is deviation from an equal -temperament in- 

 terval. 



E. T. is used as an abbreviation for "equal temperament/' 



Departure is called " positive " when intervals are sharper 

 than E. T., "negative " when flatter than E. T. 



Systems are called " positive " when the departure of twelve 

 fifths is positive, "negative" when the departure of twelve fifths 

 is negative. 



Systems are said to be " of the rth order,". positive or nega- 

 tive, when the departure of twelve fifths is +r units of the 

 system. 



Negative systems form their major thirds in accordance with 

 the ordinary notation of music ; for the departure of major 

 thirds is negative, i. e. they are flatter than E. T. thirds. And 

 if we take four negative fifths up, we have a third with negative 

 departure; thus, C# is either the third to A or 4 fifths up 

 from A. 



Positive systems form their thirds by 8 fifths down ; for 8 posi- 

 tive fifths down give a negative departure. Thus the third of A 

 should be D\). Hence positive systems require a separate notation. 

 Helmholtz proposes a notation for this purpose, which, however, is 

 unsuitable for use with musical symbols. The following notation 

 is adopted for positive systems. (Perfect and approximately per- 

 fect fifths are positive.) The fifths are arranged in series, each con- 



* Being an abstract of a paper read before the Musical Association, 

 November 2, 18/4. Communicated by the Author. 



