334 OUR PHYSICAL WORLD 



It is evident that there is a discrepancy between the rates of 

 vibration needed for the notes of the C scale and the numbers 

 needed for the D and E scales. The E note will serve for the 

 third note of the C scale and reasonably well for the second note 

 of the D scale, but F will not do both for the fourth note of the 

 C scale and the third note of the D settle, so an additional key 

 has been put into the piano at this point as a black key, which 

 we call F sharp. Similarly, the upper C will not do for the 

 seventh note of the D scale, and so another black key is intro- 

 duced as C sharp. So it will be evident from the requirements 

 of the E scale and others that additional black keys are required, 

 and it has been found that we can reasonably well meet all 

 requirements by putting in five additional black keys in each 

 octave, and, of course, the corresponding strings. Even then 

 you will find we have to put up with notes that do not exactly 

 meet requirements. 



The same thing is true in all instruments in which the number 

 of vibrations is fixed by the mechanical limitations of its manu- 

 facturers. That is not true for the violin, for the length of the 

 string and consequently the pitch of the note is determined by 

 the pressure of the finger of the player, and this can be applied 

 to the string anywhere. So the skilful violinist can render his 

 notes exactly true where the pianist must be satisfied with 

 approximately correct ones. A hundred years ago the piano 

 used to be tuned so that certain of the notes met the requirements 

 exactly, others only approximately. For instance, we may 

 tune the E above middle C so it will vibrate at the rate of 320 

 per second and meet exactly the requirements of the C scale. 

 Then, however, it will not meet the requirements of the second 

 note of the D scale. Or we may tune E to 322 vibrations, 

 when it will meet the requirements for both these scales more 

 nearly, but for neither exactly. In the old method of tuning, 

 certain notes sounded just right, others were distinctly unharmoni- 

 ous and were known as "wolves" because they howled so badly. 

 Now the piano is tuned so that the twelve intervals in the octave 



