a lower number of beats, and this introduces dis sonance. The following important laws may be here noted :— 1. The smaller the numbers which express the relative rates of vibration of two notes, the more agreeably do they strike the ear when they are sounded together. Simplicity, so to speak, is the law of harmony. 2. Dissonance is at a maximum when the beats number 33 per second; it lessens gradually, and disappears when the beats amount to 132 per second. To illustrate what has been said, attach points to the prongs of several tuning-forks of different pitch, cause them to vibrate, and, whilst doing so, allow the points slightly to touch a piece of smoked glass, draw the forks gently along, and a series of sinuous lines will be formed; by counting the number of indentations in a given length the pitch of the forks may be ascertained. Thus, supposing three forks sounding a fundamental note, a fifth, and an octave to be thus combined, and that an inch of the line formed by the fundamental fork contains sixteen undulations, an inch of that produced by the fork which sounds the fifth will contain twenty-four, while the fork that sounds the octave will produce thirty-two undulations. These numbers are in the ratio of 2: 3: 4, which, as we have already learned, express a note, its fifth, and its octave. A Musical Scale is a regular gradation of sounds occurring in the natural order of tones and semitones in groups of seven; each group constitutes a Gamut or Diatonic scale. These notes are distinguished by letters and names : Do, Re, Mi, Fa, Sol, La, Si, Do1, or C, D, E, F, G, A, B, C1. The first six of these names are the first syllables of C the first six verses of the hymn that is chanted at Rome on the feast of St. John. These notes may also be distinguished by numbers, which indicate the relative lengths of the strings and the relative numbers of vibrations necessary to produce the notes. If the length of the string producing the primary key-note be 32 inches, the length of the strings to produce the tones in the entire scale are― Do, Re, Mi, Fa, Sol, La, Si, Do1. 32, 30, 27, 24, 21, 20, 18, 16. Whatever be the number of vibrations per second necessary to produce the first note, Do, if we represent it by unity, then the numbers necessary to produce the other seven notes of the octave will be Do, Re, Mi, Fa, Sol, La, Si, Do1. And to whatever length this musical scale be extended, it will still be found a repetition of similar octaves. A column of air vibrating in a pipe obeys the same general law. The shorter the pipe, the higher the note. If the same note be produced on any musical instrument, that note is due to the same number of vibrations per second. The note of a piano produced by a string which vibrates 256 times in a second, is also produced by a flute in which a column of air vibrates the same number of times in a second. Vibrating Strings.-A stretched string executes its vibrations in equal times, consequently it produces musical notes. The string itself, being very slight, imparts but a small amount of motion to the air. It is connected with a surface more or less large, generally called a sound-board, which takes up the vibrations and transmits them to the air, as in the piano or violin. The force by which a string is stretched is called the tension of the string. The number of vibrations made by a string vary according to the following rules : : 1. The number of vibrations, or pitch of the note, increases with the square root of the tension. If two strings of the same length, material, or diameter be stretched by different weights, the rates of vibration are proportional to the square root of the stretching force. If one string be stretched by a weight of 9 lbs., and the other by a weight of 16 lbs., the rates of vibration of these strings will be as the square root of 9 is to the square root of 16, or as 3:4. The keypins of musical instruments are arranged to act as weights in giving tension to the string. 2. The number of vibrations is inversely proportional to the length of the string. That is, with the same tension, diameter, and material, a string 2 ft. long will vibrate twice as often as one 4 ft. long. Violin players vary the lengths of the vibrating parts of the strings of that instrument, by pressure of the fingers. Higher tones are in this manner produced. 3. The number of vibrations is in the inverse proportion of the diameter of the string. Other conditions being the same, a string one inch in diameter will vibrate twice as often as a string two inches in diameter. The strings for the lower notes of a piano are not only thick, but they are wrapped with wire to increase their diameter. 4. The pitch of a note is inversely proportional to the square root of the density of the substance composing the string. With tension, length, and diameter the same, let the number 4 represent the density or specific gravity of the substance of one string, and 16 that of another; then the vibrations will be as 4: 2-that is, the first string will vibrate twice as rapidly as the second. From these laws we obtain the following formula :Let n be the number of vibrations per second, the length of string, s the stretching force, d the diameter of the string : n = α × Td a is the number depending on the quality of the material of the string, and will vary if two different strings be compared, and it follows that a = by which the value of a is determined. nld, An instrument called a monochord may be used for illustration. For variation in length of string there is a moveable bridge D, Fig. 6. Other strings of varying thickness may be inserted for experiments with regard to diameters. For pressure the weight F may be removed and any other put in its place. For difference of material in class experiments arbitrary numbers may be chosen. (Calculations will be found in Exercise III.) Fig. 6. Harmonic Tones, their Generation and Function.-Let a rope or piece of india-rubber tubing be fixed at one end, and a jerk be given at the other end with the hand, the protuberance raised proceeds along the rope to the end and returns. A number of jerks given in rapid succession will produce a series of waves, which will be met by a return series; the direct and reflected waves so meet as in some parts to combine their forces, in others partially to neutralise each other, and in others, again, completely to destroy each other; this divides the rope into a number of vibrating parts, called ventral segments; and points at rest, named nodes, or nodal points. It is the same with a musical string or a vibrating column of air. The same string can vibrate as a whole, or it can divide itself into any number of vibrating parts, all of equal lengths. This division of a musical string may be rendered visible to the eye by placing little riders of paper on the string in the instrument called a monochord, before alluded to, at the ventral segments and at the nodes. When the string is sounded, the riders on the ventral segments are thrown off while those at the nodes are undisturbed. The nodes may be produced by placing small obstructions on the wire, pressing with the finger, for example, at the middle or at one-third from one end, or one-fourth, &c. The notes corresponding to the division of a string into its aliquot parts are called the harmonics of the string. In the Æolian harp a string emits its harmonic sounds, in addition to the fundamental or chief tone, being divided into vibrating segments by a current of air. These smaller vibrations are superposed upon the larger and mingle with it. The addition of these higher notes, which may be called overtones to the |