115 



mode, because the third, sixth, and seventh are smaller 

 intervals than in the ordinary mode of the scale, the 

 major. 



For convenience, an arbitrary number of vibrations may 

 be assumed for the tonic, and the other tones of the dia- 

 tonic scale calculated accordingly. 48 is a convenient 

 number for this purpose. The scale in C may be thus 

 represented : 



CDeFGabC 



48 54 60 64 72 80 90 96 



Suppose the scale to begin with the a below C : 



abCDeFGa 



40 45 48 54 60 64 72 80 



These vibration numbers show the following ratios to 

 the first : 



1 I f ft f f t 2, 



which are all simple except that belonging to D. If d 

 the major third be used in this scale instead of D the 

 key-note, there results a scale : 



abCdeFGa 



I I 



This is the pure minor scale, differing from the major 

 scale in the ratios of the third, sixth, and seventh. The 

 minor scales commonly taught are combinations of major 

 and minor intervals, especially by substituting the major 

 for the minor seventh. 



So far, no new tones are needed for harmonies in the 

 minor mode, as a complete key-board for major harmo- 

 nies supposes both thirds and key-notes in each letter. 

 But the same principle of related chords applies to the 

 minor mode as to the major ; if a is the tonic, e is the 

 dominant and d the subdominant. 



Prof. Poole's method of notation shows very clearly 



