LIGHT AND SOUND. 



the three the highest of the third triad. The 

 numbers representing the vibrations of the second 

 triad will be 6, 1\, 9 ; and of the third triad, 2$, 3$, 

 4 ; these sets of numbers having the proportion of 

 4, 5, and 6. In addition to our original note C, 

 and that derived from it by doubling its vibrations, 

 we have other six notes. And if we place the six 

 notes between the first two, by doubling the vibra- 

 tions of those below C, and halving the vibrations 

 of the one above double C, we have the following 

 series : 



4 4* 5 Si 6 6f l\ 8; 



C 



or, dividing each number by 4, and naming the 

 notes by the first seven letters of the alphabet, we 

 have : 



CDE F G 



if 

 A 



B 



We have now eight notes in all, forming what is 

 called the diatonic scale, and c is called the octave 

 above C ; G is a fifth above it, and E a third. The 

 scale may be extended both upwards and down- 

 wards, by doubling the numbers for the right-hand 

 extension, and halving them for that on the left, 

 and thus a common pianoforte has a compass of 

 seven octaves. The lowest notes of each of the 

 three triads formed above have received special 

 names, C being called the tonic, G the dominant, 

 and F, the octave above the lowest of the third 

 triad, the subdominant. 



If, instead of making our triad consist of notes 



whose vibrations are as i, \\, ij, we make the 

 ratios I, i\, \\, the three notes form a minor triad, 

 and when sounded with the octave of the lowest 

 note, produce a pleasing, though somewhat melan- 

 choly effect. A scale formed from three minor 

 triads would be a minor scale, though the minor 

 scale in practical use is formed differently. In the 

 ascending scale, the tonic has a minor triad, while 

 the dominant and subdominant have major triads ; 

 but in the descending scale, all the triads are 

 minor. The ratio of a major to a minor third is 

 25 to 24, and this ratio is called an interval of a 

 semitone. In music, it is convenient to sub- 

 divide the interval between two consecutive notes, 

 and this is done by raising the one note, or lower- 

 ing the other, by half a tone ; that is, multiplying 

 the number of vibrations of the one, or dividing 

 that of the other by -f-J. The note formed by rais- 

 ing C one half-tone, or by sharpening C, is called C 

 sharp, and is written C$ ; and that by lowering or 

 flattening D, is D flat, written Db. The vibrations 

 per second of these four notes are as the numbers 



C cl Db D J and th Se f C ft and D ^ are S nearly 

 equal that only a practised ear can recognise the 

 difference of pitch. In the pianoforte, the same 

 note serves for both, and the new notes introduced 

 correspond to the black keys. As the interval 

 between E and F, and between B and c, is very 

 nearly a half-tone, it is not necessary to sharpen 

 E or B ; and consequently the chromatic scale, as 

 it is termed, is as follows : 



VIBRATIONS OF STRINGS AND COLUMNS OF AIR. 



If a stretched string or wire, AB, be sharply 

 struck, or pulled aside, or agitated by a resined 

 bow, it is thrown into to-and-fro vibrations, which 

 .give rise to a musical note. By placing the 



bridge, B, at 

 different dis- 

 tances from 

 A, we can 

 vary the 

 length of the 

 sounding- 

 string ; and 



Fig- 33- 



to increase the intensity of the sound, a sounding- 

 board, CD, which can vibrate in unison with the 

 string, is placed below it. The blows given 

 directly to the air by the string are too small to 

 produce a loud sound; but the vibrations of the 

 sound-board being continually reinforced by those 

 of the string, soon become appreciable, and a 

 comparatively large mass of air is set in motion. 

 If the bridge be successively placed at points 

 which give vibrating lengths proportional to the 

 numbers 



all the notes of the diatonic scale are produced. 

 Therefore, we conclude that : 



(i.) When the length of the string alone varies ; 

 the number of vibrations in a given time is 

 inversely as the length. After a given note has 

 been sounded, let the stretching-weight be in- 

 creased four times, and it will be found that the 

 new note is the octave above the last, or that the 

 number of vibrations is doubled. If the two 

 weights are as 16 to 25, the two notes form a 

 major third, or the vibrations are as 4 to 5. The 

 second law is therefore : 



(2.) When the stretching-weight alone varies, 

 the number of vibrations in a given time is as the 

 square root of the weight. Let two strings of the 

 same material, length, and tension, be placed side 

 by side, but let the diameter of the second be half 

 as much again as that of the first, or the double of 

 it, and it will be found that the note of the second 

 is a major fifth, or an octave below that of the 

 first, or that the number of vibrations in a given 

 time are as 3 to 2, and 2 to i. So that the third 

 law is : 



(3.) When the diameter of the string alone 

 varies, the number of vibrations in a given time is 

 inversely as the diameter. And the fourth law 

 is : 



(4.) When the density of the string alone varies, 

 the number of vibrations per second is inversely 

 as the square root of the density. When the 

 string AB is vibrating, and giving out its primary 

 note, if it be lightly touched in the middle, it now 



365 



