136 THE NATURE OF MUSIC 



at a). Chords whose components simultaneously rise 

 and fall have double cadences. Chords whose cadences 

 rise only or fall only have single cadences. Below at 

 b) and c) the double cadence of vii% is separated 

 into single cadences. At 6) sol is added to the two 

 lower tones of vii^^, thus forming the triad V 

 and becoming the bond-tone of the original rising 

 chord-cadence V — I, the authentic ending. At c) 

 do is added to the two upper tones of vii^^, thus 

 forming the triad IV and becoming the bond-tone 

 of the original falling cadence IV — I, the plagal 

 ending. 



7 



5 5 1 5 5 3 5 

 533533 13 



a) 3 1 ^)1 3 1 c)i 5 1 





vii^ I V V— I IV lY— I 



In this separation of the original cadence-tones in 

 the triads V and IV, re and ti retain their original 

 relations as 5 and 3 respectively while the relations 

 of la and /a are changed, la from 9 to 3, /a from 

 7 to 1. HarTnonic intervals are indicated by numbers 

 specifying the exact relation of a tone to its har- 

 monic root: such are the numbers over the above 

 chords. C/iord-intervals are computed from chord- 

 roots. Since any tone, that is, any component of a 

 harmony may be a chord-root it follows that chord- 

 roots and chord-intervals are sometimes harmonic 

 roots and harmonic intervals and sometimes not, 



