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POPULAR SCIENCE REVIEW. 
last flat. This (if there are more than one flat) will coincide 
with the second last flat. 
XVI. Writing the series of flat fifths in two alternate 
columns, as we did the sharp ones, they will form two sets of 
notes falling in natural succession from C and F alternately — 
thus : — 
0 C IF 
2 bB 3 bE 
4 bA 5 bD 
6 bG 7 bC 
The number of flats required by these key notes being placed 
beside them, it is evident that if a piece is to be transposed 
lower in pitch, we must add 2 flats in the signature for each 
note it is lowered, in order to get the right signature for the 
key of the new pitch it is to be put into. 
This, in the case of a piece already written in sharps, would 
be the same thing as taking off two of the sharps. 
Similarly in the case of raising a piece a note, which we 
spoke of in considering the sharps — instead of adding two 
sharps — if we had the piece originally written in a flat key, 
the same result would be obtained by taking off two flats. 
XVII. Another useful rule may be got by comparing the two 
successions of sharp and flat fifths. 
Series IX. 
(Sharp Keys) . . . C, G, D, A, E, B, # F, * C. 
(Flat Keys) . . . C, F, b B, b E, b A, b D, b G, b C. 
(Number in signature ) 0, 1, 2, 3, 4, 5, 6, 7. 
Examining this arrangement, it is seen that if any key-note 
has a certain number of sharps, the same note flattened has a 
number which is the difference between that number and 7, as 
its allowance of flats ; or, the signatures in sharp keys are the 
complements to 7 of the signatures in flat keys : — 
Thus/ G has 1 sharp. 
1 bG has 6 flats. 
„ / D has 2 sharps. 
1 bB has 4 flats. 
„ f A has 3 sharps. 
' bA has 4 flats. 
&c. &c. 
Now from this it results that, without ^altering the notes in 
which a piece is written, we may at once transpose it a semi- 
tone lower or higher by playing it inj the complementary 
number of flats or sharps. Thus, a piece written in 3 sharps 
may be played as if in 4 flats from the same written score 
(except any accidental sharp or flat it may contain), and this 
will lower its whole pitch one semitone. 
