Notices respecting New Boohs. 281 



z — \ 



Bv means of the simple transformation p — another series 



± 2 — z 



of harmonic numbers is obtained : 



c d e f g a b c 



i>= \ \ \ 1 2 7 oo, 



which bears an obvious resemblance to the third normal row. It, 

 however, lacks the components _£>=§"» 2"' an ^ ^' anc ^ contains 

 two with high symbols which do not occur in that row. The 

 latter notes, as the author plausibly explains, have really no part 

 in the scale of C, but belong to that of the dominant, being the 

 fifth and third respectively in that scale. Of the additional notes 

 required to complete the row, b]p ( p = 3) is very closely related to the 

 tonic, and, indeed, forms the ordinary modulation to the chord of 

 the sub-dominant. The remaining notes f% (p=^) and a\y (p =-|) 

 are both associated, though not so closely as the remaining notes 

 of the series, with the harmony of C major. The complete series 

 corresponding to the third normal row is finally 



« / ft 9 a \> a h c 



1 -L 2. -I 3. 9 o _ 



3 2 3 1 2 z ^ °° 



Since the harmonic numbers increase with the pitch, the author 

 terms this series "rising harmony." 



Another harmonic series is obtainable in the same way from the 



wave-lengths (I) by means of the transformation p — ~~ The 



£ — I 

 wave-lengths are inversely proportional to the vibrations, and we 



2 

 may for convenience assume that z= =-, and consequently pft = 2. 



Applying this transformation to the scale of A minor, we have 

 « 9 f e d c b . a 



~ _ 9 1 8 3 4 6 8 



z ~ Z 4 5 2 3 5" 7 1 



7 — 1 I- £ ± £ 5 7 



^ — L 7 4 3 2 3" 4 2 



P=0 i i I r 2 3 n 



The bar over the letters distinguishes the method of trans- 

 formation, and has no negative or minus significance. 



As in the previous case, the note with the high symbol is 

 omitted, and the missing members of the third normal row are 

 introduced, giving the complete series : 



a f e e\> d c$ c b a_ 



i>=0 1 f I T f 2 8 ». 



Pl.il. Mag. S. 6. Vol. 10. No. 56. Aug. 1905. U 



