INSTRUMENTS OF JUST INTONA TION. 57 



Non- co-ordinate or key-relationship symmetrical arrangements, 

 such as those of Mr. Poole and Mr. Brown, possess a similar pro- 

 perty of more limited extent. In these it is, for instance, possible 

 that a common chord may assume different forms to the finger in 

 cases where the key relationship is differently assumed: not so in 

 co-ordinate arrangements. 



I will only allude to one property of the division of the octave into 

 53 equal intervals, according to which the harmonium exhibited is 

 tuned. 



The mode in which the number 53 is arrived at has been ex- 

 plained by me, as part of a general theory. But we can verify its 

 properties independently by noticing, that if we take 31 units for 

 the fifth of the system, then 12 X 31 =372 and 7 X 53 = 371 ; so that 

 we see directly without formulae, that the departure of twelve fifths 

 = -sV of an octave = if of a semi-tone; and departure of one fifth = -^ 

 of a semitone. Now, the departure of a perfect fifth is ^rrsTj and the 

 difference is about atuu = TsVo of a semitone, which is the error of 

 the fifth of the system. Hence, we may say that the system of fifty- 

 three is sensibly identical with a system of perfect fifths. 



In the enharmonic organ recently constructed, I have applied to a 

 generalized key-board of forty-eight notes per octave, Helmholtz's 

 approximately just intonation, and also the mean tone system, which 

 is of historical interest. Each system is brought on to the key-board 

 separately by a draw-stop. In the same way all systems of interest 

 are accessible ; it is this employment of the key-board that I would 

 at present commend to those who inquire into its utility. 



Considering the facilities that we see about us for manipulating just 

 and approximately just systems, it is difficult to see why mere book 

 knowledge should continue to be alone regarded in the study of this 

 portion of the elements of music. When it is taught, for instance, 

 that certain vibration ratios correspond to certain musical effects, the 

 lesson should be taught experimentally; as it is, musicians for the 

 most part only know what consonances are from descriptions in 

 books. As illustrations, I may point out that we have, in the 

 harmonium now exhibited, the means of distinguishing three different 

 kinds of minor thirds, whose ratios are 6: 5, 32 -.27, 7 '&'> and these 

 sound quite different to the ear. Again, Pythagorean thirds can be 



