LAWS OF SONOROUS VIBRATIONS. 741 



est, and the last, the highest. The half may be divided again, producing a 

 second octave, and so on, within the limits of appreciation of musical sounds. 

 If the string be divided so that f of its length will vibrate, there are 72 vibra- 

 tions in a second, and this note is the 5th in the scale. If the string be 

 divided again, so as to leave $ of its length, there are 60 vibrations, which 

 give the 3d note in the scale. These are the most natural subdivisions of 

 the note ; and the 1st, 3d, 5th and 8th, when sounded together, make Avhat 

 is known as the common major chord. Three-fourths of the length of the 

 original string make 64 vibrations, and give the 4th note in the scale. 

 With of the string, there are 54 vibrations, and the note is the 2d in the scale. 

 With | of the string, there are 80 vibrations, or the 6th note in the scale. 

 With ^ of the string, there are 90 vibrations, or the 7th note in the scale. 

 The original note, which may be called C, is the key-note, or the tonic. In 

 this scale, which is oalled the natural, or diatonic, there is a regular mathe- 

 matical progression from the 1st to the 8th. This is called the major key. 

 Melody consists in an agreeable succession of notes, which may be assumed, 

 for the sake of simplicity, to be pure. In a simple melody every note must 

 be one of those in the scale. When a different note is sounded, the melody 

 passes into a key which has a different fundamental note, or tonic, with a 

 different succession of 3ds, 5ths etc. Every key, therefore, has its 1st, 3d, 

 5th and 8th, as well as the intermediate notes. If a note formed by a string 

 the length of the tonic instead of , be substituted for the major 3d, the key 

 is converted into the minor. The minor chord, consisting of the 1st, the 

 diminished 3d, the 5th and the 8th, is perfectly harmonious, but it has a 

 quality quite different from that of the major chord. The "notes of a melody 

 may progress in the minor key as well as in the major. Taking the small 

 numbers of vibrations merely for convenience, the following is the mode -of 

 progression in the natural scale, which may be assumed to be the scale of 

 major : 



1st. 2d. 3d. 4th. 5th. 6th. 7th. 8th. 



Note CDEFGAB C 



Lengths of the string 1 f f Vk i 



Number of vibrations 48 54 60 64 72 80 90 96 



The intervals between the notes of the scale, it is seen, are not equal. 

 The smallest, between the 3d and 4th and the 7th and 8th, are called semi- 

 tones. The other intervals are either full perfect tones or small perfect 

 tones. Although there are semitones, not belonging to the key of C, between 

 C and D, D and E, F and G, G and A, and A and B, these intervals are not 

 all composed of exactly the same number of vibrations ; so that, taking the 

 notes on a piano, with D as the tonic, the 5th would be A. It is assumed 

 that D has 54 vibrations, and A, 80, giving a difference of 26. With C as 

 the tonic and G as the fifth, there is a difference of 24. It is on account of 

 these differences in the intervals, that each key in music has a more or less 

 peculiar and distinctive character. 



Even in melody, and still more in harmony, in long compositions, the ear 

 becomes fatigued by a single key, and it is necessary, in order to produce the 



