Ill 



in an instrument must be precise!}^ similar in arrangement 

 to that of every other octave of the same instrument. 



In a paper read before the Essex Institute on the even- 

 ing of the 3d of February last, a brief histor}^ of the 

 development of the modern scale was given, which it is 

 not necessary to repeat here, as the fjicts in relation to it 

 are to be found in the works of Helmholtz, Sedley Tay- 

 lor, and other writers on sound. It is enough to say here 

 that the ordinary scale of harmony, the diatonic major 

 scale, is now recos^uized as containino: a series of tones 

 whose viln-ation-numbers bear the following ratios to that 

 of the lowest, beginning with the lowest itself; 



1> ¥> I"? f? #1 "I? "V"' 2. 



It is to be remarked that a scale is merely a systemati- 

 cally arranged group of related tones, and there is no one 

 scale which includes all related tones in common use. 

 The diatonic major scale includes most of the simple 

 relations of tones, and is therefore the most common in 

 use. An analysis of this scale will show the simplicity 

 of its composition. 



In all harmony, some one tone is selected as a basis, to 

 which all the other tones of the harmony are related. 

 But when this tone, which is called the key-note, has 

 been selected, there are in reality used as the base or 

 foundation of harmonies made up from tones related to 

 that key-note, three tones ; the key-note itself, one whose 

 vibration number is f that of the key-note, and one to 

 which the key-note itself bears the relation f , and which 

 is therefore § of the key-note in its vibration number. 

 The interval indicated by the ratio f is called a fifth, be- 

 cause it is fifth in order in the diatonic scale. In every 

 key, then, there are used as the foundation of harmonies 

 belonging to that key, the fifth above, which is called the 

 dominant, the fifth below, called the subdomiuant, and 



