ON HARMONICS. SgS 



which are commonly considered as filling up the scale ; and on account of 

 its great resemblance to the fundamemtal note, it is described by the same 

 letter of the alphabet, or by the same syllable; so that all audible sounds are 

 considered as repetitions of a scries contained within the interval of an 

 octave. One third part of the string or pipe gives the fifth above the octave; 

 one fourth the double octave, and one fifth of the string its third. Thus 

 we obtain the common accord or chord, or the harmonic triad, consisting of 

 the fundamental note, with its third and fifth, which produces the most 

 perfect harmony, and which also contains the constituent parts of the most 

 simple and natural melodies. But we are still in want of intermediate steps 

 for the scale ; these are supplied by completing first, the triad of the fifth, 

 which gives us the second, and the seventh, of which Sand 15 vibrations 

 correspond respectively to 8 of the fundamental, and which may also be 

 found in the ascending series of natural harmonics; and in the" second place, 

 by adding the fourth and sixth in such proportions as to make up another 

 perfect triad with the octave; the respective notes consisting of 4 and 5 

 vibrations, while the fundamental note makes 3, and being no where found 

 among the natural harmonics. The complete scale is, therefore, formed by 

 these harmonic triads contiguous to and connected with each other; the 

 middle one being the triad of the key note, the superior one that of its 

 fifth, which is sometimes called the dominant or governing note, and the 

 inferior one that of the fourth, or subdominant. This scale is derived from 

 principles so simple, that it may properly be considered as a natuial arrange- 

 ment, and it appears to be found with little variation in barbarous as well as 

 in civilised countries. (Plate XXV. Fig. 354.) 



A long piece would, however, be too monotonous, unless the fundamental 

 note were sometimes changed; we may, therefore,^ take at pleasure one of 

 the auxiliary triads for the principal harmony, and we may continue the 

 modulation or progression, until every note of the scale becomes in succession a 

 key note. But, in order to fill up the intervals of these several scales in just 

 proportion, it becomes necessary to add several new notes to the first series , 

 for instance, if we take the seventh for a key note, we shall want five new 

 sounds within the octave, making twelve in the whole, which is the num- 

 ber usually employed in music The interval between any two adjoin- 

 ing sounds of tlie twelve is called a semitone or half note, two semitones 



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