J ON HARMONICS. 307 



dti a bud*5irtJrn, must consist of the harmonics of a single sound only : and 

 when teiaccompaniment is performed by a French horn, the length of the 

 instrument is first adjusted to the principal note, and all the sounds which 

 it is to produce are selected from this natural series. But the notes consti- 

 tuting the most natural scale are not, without exception, comprehended 

 among the harmonics ; they are, however, all immediately dependent on a 

 similar relation. A sound of which the vibrations are of equal frequency 

 with those of another, is called a unison ; if two vibrations occur for every 

 one of the fundamental note, the sound is called its superior octave, being 

 the eighth of those which are commonly considered as filling up the scale ; 

 and on account of its great resemblance to the fundamental note, it is de- 

 scribed by the same letter of the alphabet, or by the same syllable ; so that 

 all audible sounds are considered as repetitions of a series 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 har- 

 monic 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 9 and 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 with the octave to make up another perfect 

 triad ; the respective notes consisting of 4 and 5 vibrations, while the fun- 

 damental note makes 3, and being no where found among the natural har- 

 monics. 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 natural arrangement, and it 

 appears to be found with little variation in barbarous as well as in civil- 

 ized countries. (Plate XXV. Fig. 354.) 



A long piece would, however, be too monotonous, unless the funda- 

 mental 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 suc- 

 cession 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 number usually employed in music. The interval between 

 any two adjoining sounds of the twelve is called a semitone or half note, 

 two semitones making a tone or note ; these terms are, however, sometimes 

 employed with various subordinate distinctions and limitations. The five 

 additional sounds have no separate names, but they are denominated from 



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