LAWS OF SONOROUS VIBRATIONS. 827 



employed in music, those of from 40 to 4,000 vibrations in a second, embracing about 

 seven octaves. In an orchestra, the double bass gives the lowest note, which has 40-25 

 vibrations in a second, and the highest note, given by the small flute, has 4,752 vibrations. 

 In grand organs, there is a pipe which gives a note of 16 -5 vibrations, and the deepest 

 note of modern pianos has 27'5 vibrations ; but delicate shades of pitch in these low notes 

 are not appreciable to most persons. Sounds above the limits just indicated are painfully 

 sharp, and their pitch cannot be exactly appreciated by the ear. The physiological inter- 

 est connected with these facts is, that the limits of the appreciation of musical sounds are 

 probably due to the anatomical arrangement of the auditory apparatus, as we have a 

 limit to the acuteness of vision, which can be explained by the structure of the eye. This 

 fact is the basis of the accepted theories of the appreciation of musical sounds. 



Musical Scale. We have thus far considered musical sounds, without any reference 

 to the relations of different notes to each other. A knowledge of these relations lies at 

 the foundation of the science of music ; and, without a clear idea of certain of the funda- 

 mental laws of music, we cannot thoroughly comprehend the mechanism of audition. 



It requires very little cultivation of the ear to enable us to comprehend the fact, that 

 the successions and combinations of tones must obey certain fixed laws; and, long before 

 these laws were the subject of mathematical demonstration, the relations of the different 

 notes of the scale were established, merely because certain successions and combinations 

 were agreeable to the ear, while others were discordant and apparently unnatural. Now 

 that we are pretty thoroughly acquainted with the laws of vibrations, we can study the 

 scale from a scientific, as well as from an esthetic point of view. 



The most convenient notes for our study are those produced by vibrating strings, and 

 the phenomena here observed are essentially the same for all musical sounds; for it is 

 by means of vibrations communicated to the air that the waves of sound find their 

 way to the auditory apparatus. Let us take, to begin with, a string vibrating 24 

 times in a second. If this string be divided into two equal parts, each part will vibrate 

 48 times in a second. The note thus produced is the octave, or the 8th of the primary 

 note, called the 8th, because the natural scale, as we Shall see, contains eight notes, of 

 which the first is the lowest and the last, the highest. We may divide the half again, 

 producing a second octave, and so on, within the limits of our appreciation of musical 

 sounds. If we divide the string so that f of its length will vibrate, we have 36 vibrations 

 in a second, and this note is the 5th in the scale. If we divide the string again, so as to 

 leave f of its length, we have 30 vibrations, which gives the 3d note in the scale. These 

 are the most natural subdivisions of the note; and the 1st, 3d, 5th, and 8th, when sound- 

 ed together, make what is known as the common major chord. Three-fourths of the 

 length of the original string makes 32 vibrations, and gives the 4th note in the scale. If 

 we take f of the string, we have 27 vibrations, and the note is the 2d in the scale. Witli 

 | of the string, we have 40 vibrations in a second, or the 6th note in the scale. With -fa 

 of the string, we have 45 vibrations in a second, or the 7th note in the scale. 



It will be observed that we have started with a note, which we may call C. This is 

 the key-note, or the tonic. In this scale, which is called the natural, or diatonic key, we 

 have a regular mathematical progression from the 1st to the 8th. This is called the 

 major key of 0. Melody consists in an agreeable succession of notes, which we may 

 assume, for sake of simplicity, to be pure. We cannot, in a simple im-lotly. sound any 

 note but one of those in the scale. When a different note is sounded, we pa.-s into a k-y 

 which has a different fundamental note, or tonic, with a different succession of 3<K r>rhs, 

 etc. Every key, therefore, has its 1st, 3d, 5th, and 8th, as well as tlie inti-nneclijite 

 notes. If we substitute for the 3d a note formed by a string | the length <f tin- tonic 

 instead of f, we have the key converted into the minor. The minor chord, consisting of the 

 1st, the diminished 3d, the 5th, and the 8th, is perfectly harmonious, but it lm> a quality 

 quite different from that of the major chord. The notes of a melody may progress in the 



