INTRODUCTION 



V d' J f tf a' V d' d" e" f" g" a" b" 

 264 297 330 352 396 440 495 528 594 660 704 792 880 990 



Underneath the ordinary musical symbols we have placed 

 the convenient literal notation employed by German writers. 

 This may be continued upwards by means of additional accents 

 (c'", c*, ...), and downwards by suffixes (C,, C,,, ...). 



If in the construction of the scale we had used, instead of 

 the major, the minor chord, which consists of a Minor and a 

 Major Third in ascending order, the frequencies being as 

 10 : 12 : 15, we should have required three notes not included 

 in the above scheme. And if, starting from any note already 

 obtained (other than C) as a new key-note, we proceed to 

 construct a major or a minor scale, further additional notes are 

 required. In the case of the violin, or of the human voice, or of 

 some other wind-instruments which allow of continuous varia- 

 tion of pitch, this presents no difficulty. But in instruments 

 like the piano or organ the multiplication of fixed notes beyond 

 a moderate limit is impracticable. It is found, however, that 

 by a slight tampering with the correct numerical relations the 

 requirements of most keys can be fairly well met by a system 

 of twelve notes in each octave, which are known as 

 C C* D D* E F F* G GS A A3 B. 



This process of adjustment, or compromise, is called "tempera- 

 ment"; on the usual system of " equal " temperament the 

 intervals between the successive notes are made equal, the 

 octave being accordingly divided into twelve steps for each of 



which the vibration-ratio is 2 T X Thus the ratio of G to C is 

 made to be 2 T *= T4983 instead of 1-5. 



