19. A flat is a character that causes the degree of the staff with which it is associated to represent a tone one half-step lower than it otherwise would. (See note under [Sec. 18] and apply the same discussion here.)
20. A double-sharp causes the staff degree on which it is placed to represent a pitch one whole-step higher than it would without any sharp. Similarly, a double-flat causes the staff degree on which it is placed to represent a pitch one whole-step lower than it would without any flat.
Double-sharps and double-flats are generally used on staff degrees that have already been sharped or flatted, therefore their practical effect is to cause staff degrees to represent pitches respectively a half-step higher and a half-step lower than would be represented by those same degrees in their diatonic condition. Thus in Fig. 10 (b) the first space in its diatonic condition[7] represents F-sharp, and the double-sharp on this degree would cause it to represent a pitch one-half step higher than F-sharp, i.e., F-double-sharp.
CHAPTER III
Symbols of Music Defined (Continued)
21. The natural[8] (sometimes called cancel) annuls the effect of previous sharps, flats, double-sharps, and double-flats, within the measure in which it occurs. After a double-sharp or double-flat the combination of a natural with a sharp, or a natural with a flat is often found: in this case only one sharp or flat is annulled. (Sometimes also the single sharp or flat will be found by itself, cancelling the double-sharp or double-flat). The natural is often used when a composition changes key, as in Fig. 11, where a change from E to G is shown.
22. The group of sharps or flats (or absence of them) at the beginning of a staff partially indicates the key in which the composition is written. They are called collectively the key-signature.