$6 SECTION PHYSICS. 



ordinate of any note is proportional to its distance from a fixed point 

 in the series of fifths. 



The series of fifths is selected for the application of the conditions, 

 because it is the most convenient ; but the variations of all the concords 

 in any system are linked together in such a manner that it is indifferent 

 which is taken as independent variable, so to speak ; the results would 

 be always the same. 



The generalized key-board, of which the harmonium exhibited offers 

 an example, may be conveniently described with reference to abscissae 

 from left to right, and horizontal ordinates on plan from back to front. 

 The vertical ordinates are one-third of the horizontal ones. 



In the abscissae, half an inch corresponds to an E.T. semi-tone. 

 Twelve semi-tones make an octave. The octave measures 6 inches in 

 abscissa, and nothing in ordinate. 



In the ordinates on plan, 3 inches correspond to the Pythagorean 

 comma or departure of twelve fifths. Thus the difference between the 

 ordinates of two notes on same abscissa, between which one series of 

 twelve fifths lies, is 3 inches. 



The ordinates of the intermediate fifths are increased by inch at 

 each step upwards in the series of fifths, so that twelve steps upwards 

 in the series correspond to 3 inches. 



I have described the key-board as connected with the system of 

 perfect fifths ; and it is so in this harmonium to all intents and pur- 

 poses. But it is clear that if each fifth have any departure from E.T. 

 whatever, this may be equally represented by the ordinates in ques- 

 tion, as no use has been made of the amount of the departure ; and we 

 can say that a key-board, constructed in the form of a co-ordinate sym- 

 metrical arrangement, forms a graphical representation of the interval 

 relations of any set of notes belonging to a regular succession of fifths. 



Thirds can always be referred to fifths. In systems such as that of 

 perfect fifths which we are dealing with here by means of a theorem 

 brought into notice by Helmholtz : in other cases, in other ways. 



The most important property of key arrangements which form 

 graphical representations of their intervals is, that any combination 

 of intervals has the same form to the finger on whatever notes or in 

 whatever key it may be taken. Thus a common chord always has 

 the same form. 



