174 Mr. R. H. M. Bosanquet on Temperament, 



fifth down gives the subdominant, and a sixth up the depressed 



second, or - x ^ = -^- ; and the ratio of the two notes in ques- 



9 10 8] 

 tion is - -h — = «^r, an ordinary comma. 



The following example illustrates the difficulty thus intro- 

 duced into ordinary music, in which the depressed second is not 

 recognized. 



It is impossible to assign a position for the note marked -x- 

 which does not involve a fourth or fifth faulty to the extent of 

 a comma ; and, according to my experience, this cannot be tole- 

 rated. It is necessary therefore to avoid such combinations in 

 dealing with positive systems. 



The little circle (o) is used to neutralize the sigus of elevation 

 and depression in exactly the same way as (&) neutralizes sharps 

 or flats in ordinary music. 



Harmonic Seventh. 



If we compute (rule I.) the interval of the harmonic seventh 

 (7: 4), we find 10 — *31174; or it is less than ten semitones by 

 about one third of a semitone. In fact it is well known that 

 if we flatten a minor seventh by some such quantity, we obtain 

 a smooth combination free from beats. We can obtain a pretty 

 close representation of this by taking 14 fifths down in positive 

 systems according to an approximation pointed out by Helm- 

 holtz, or by 10 fifths up in negative systems (Mr. Ellis). 



Rule. The harmonic seventh on the dominant must not be 

 suspended, so as to form a fourth with the key-note. 



In ratios this stands as follows: 



Tonic : dominant : : 4 : 3, 

 Dominant : h. seventh : : 4 : 7; 

 .-. tonic : h. seventh of dominant : : 16 : 21 : : 48 : 63. 

 But 



Fourth : tonic : : 4 : 3 : : 64 : 48 ; 

 .'. fourth : h. seventh of dominant : : 64 : 63 ; 



and the. error of the h. seventh of the dominant, if treated as 

 a fourth to the tonic, is more than a comma. This implies a 

 Great restriction on the use of the harmonic seventh. 



