INSTRUMENTS OF JUST INTONATION. 55 



ON INSTRUMENTS OF JUST INTONATION. 





Mr. BOSANQUET : As the combinations which are to be dealt with 

 are complex, forms of arrangement have been employed which reduce 

 all the phenomena in practice and theory to a few simple types. 

 These I have called symmetrical arrangements. They were not 

 employed by General Thompson. 



There are two forms of symmetrical arrangements. Those which I 

 employ I shall distinguish co-ordinate symmetrical arrangements; 

 other forms employed by Mr. Poole and Mr. Brown may be called 

 key-relationship symmetrical arrangements. Into these latter I shall 

 not enter, but only remark that those which depend on co-ordinates 

 possess all symmetrical properties in a more extended form than the 

 others ; i.e., they include the key-relationship and more besides. 



Mr. Brown's position relations are exactly the same as Mr. Poole's, 

 with omission of the two series of sevenths. 



In my co-orclinate symmetrical arrangements a number of equal 

 temperament semi-tones, or twelfth parts of an octave, is taken as 

 abscissa, and deviation or departure from the note thus arrived at as 

 ordinate. Thus, the exact pitch of notes can be expressed by reference 

 to co-ordinates in a plane. 



To express a series of equal temperament (E.T.) fifths in this manner : 



The E.T. fifth is T V of an octave. Taking twelve fifths up, we have 

 seven octaves exactly. The result is expressed by abscissce only. 



To express a series of perfect fifths in this manner : 



The perfect fifth is seven semi-tones and a departure = '01955 E.T.S. 

 = TrrsT ' '* twelve fifths are seven octaves and a departure = '23460 

 E.T.S., the Pythagorean comma. The resulting notes have ordinates 

 which increase uniformly as we pass along the series of fifths ; and the 

 position arrived at after the twelve fifths has an ordinate which 

 represents the Pythagorean comma when the fifths are perfect, and 

 abscissa 7 X 12 = 84 or seven octaves. 



The form of symmetrical arrangement employed in the generalized 

 key-board, is arrived at by arranging a series of notes in the order of 

 the scale as far as the abscissas are concerned, and taking for ordinates 

 distances proportional to those thus arrived at. That is to say, the 



