The note which is taken as the do or 1 of any scale is its keynote, and the tone it represents is the key tone or tonic of the scale. The scale already considered has c' for its keynote and is in the key of C. Suppose we form a major scale with g' for its keynote and compare the numbers of vibrations in its various tones with those in the key of C.
We see by this comparison that there are two tones each represented by a' and f", which differ in the number of their vibrations. The interval for the two a' s is 432: 426.6, or
81:80. This is called a comma. The f" s differ more widely, their interval being 135:128. This is sometimes called a semitone. In order to play music accurately in the key of G, two tones that are not found in the key of C would be required. One of these, the f" of the key of G, is introduced approximately by increasing the vibrations of f" in the key of C by multiplying the number by 
this new tone is called f" sharp or f"#. The number of vibrations for a' in the key of G differs so little from that of the same tone in the key of C, that in most instruments one tone serves for both.
Inasmuch as any tone in any scale may be taken for a new key tone, it is evident that to introduce two new tones for every new scale on such an instrument as the piano would
make the keyboard so large that it could not be used at all. Other complications come in also when the flat keys are used. In these (for instance, in the scale formed with f' as the keynote) the new tone required by the ratio of vibrations is secured by lowering the number of vibrations of the corresponding tone by multiplying it by 
The resulting tone is called the flat of the first, as b flat or bb. Between c' and d' there would be two tones, as follows: 
having for their respective vibrations, 256, 266.6, 276.5, 288.
