828 SPECIAL SENSES. 



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 of major: 



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



Note CDEFGABC 



Lengths of the string 1 f f I 5 W 2 



Number of vibrations 24 27 30 32 36 40 45 48 



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 semitones. The other intervals 

 are either full perfect tones or small perfect tones. Although there are semitones, not 

 belonging to the key of 0, between 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, if we have D as the tonic, its 5th would be A. We 

 assume that D has 27 vibrations, and A, 40, giving a difference of 13. With C as the 

 tonic and G as the 5th, we have a difference of 12. It is on account of these differences 

 in the intervals, that each key in music has a peculiar and an individual character. 



In tuning a piano, which is the single instrument most commonly used for accompani- 

 ment and the general interpretation of musical compositions, the ordinary method is by 

 the 5ths. We bring the 5th of in exact accord with the tonic ; then the 5th of D ; 

 then the 5th of E, and finally the 5th of F. The 5th of E should be the octave of C, 

 but, by progressing in this way, the last note (C) is too sharp and is not the octave of 

 the lower 0. If this progression were continued higher and higher, the octaves would 

 become more and more out of tune ; and, to avoid this, the octaves are made perfect and 

 the 5ths and 3ds are tuned down, so that the inequality is distributed throughout the 

 scale. This is called tempering the scale, and, with this " temperament," the notes are 

 not exactly true ; still, musicians are accustomed to this, and they fail to recognize the 

 mathematical defect. 



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 most pleasing effects, 

 to change the tonic, by what is called modulation, returning afterward to the original key. 



Quality of Musical Sounds. By appropriate means, we can analyze or decompose 

 white light into prismatic colors ; and, in the same way, nearly all musical sounds, which 

 seem at first to be simple, can be resolved into certain well-defined constituents. There 

 are few absolutely simple sounds used in music. We may take an example, however, in 

 the notes of great stopped-pipes in the organ. These are simple, but are of an unsatis- 

 factory quality and wanting in richness. Almost all other musical sounds, however, 

 have a fundamental tone, which we recognize at once ; but this tone is accompanied 

 by harmonics caused by secondary vibrations of subdivisions of the sonorous body. 

 The number, pitch, and intensity of these harmonic, or aliquot vibrations affect what 

 is called the quality, or timbre of musical notes, by modifying the form of the sonorous 

 waves. This fact, which we shall discuss more elaborately farther on, requires little 

 argument for its support. If we suppose a string vibrating a certain number of times in 

 a second, the vibrations being perfectly simple, we should have, according to the laws 

 of vibrating bodies, a simple musical tone; but, if we suppose that the string subdivides 

 itself into different segments, one of which gives the 3d, another, the 5th, and so on, of the 

 fundamental tone, it is evident that the form of the vibrations must be considerably 

 modified. This is the fact ; and, with these modifications in form, the quality, or timbre 

 of the note is changed. We can illustrate this roughly on the piano. If we strike the 

 note 0, we have a certain quality of sound. We may assume, for sake of argument, 

 that this is a simple tone, although in reality it is complex. We now strike simultaneously 

 the fundamental note, its 3d, 5th, and 8th, making the common chord of major. The 

 predominant note is still C, but the addition of the harmonious notes modifies its quality. 



