by 16 vibrations in a second. If the number of vibrations is less, no sound is heard. The same physicist found that the highest sound which the ear can perceive corresponds to 48,000 vibrations in a second. Between these two limits it will be seen what an enormous quantity of sounds may be produced and perceived. Yet the sounds used in music, and more especially in singing, are comprised within much narrower limits. Thus the human voice has been compared with the sound produced by instruments, the number of whose vibrations could be ascertained; and it has been found that the lowest notes of a man's voice are made by 190 vibrations in a second, and the highest notes by 678. The lowest note of a woman's voice corresponds to 572 vibrations, and the highest to 1,606.

168. Musical scale. Gamut. The human ear can distinguish among several sounds not merely which is the highest, or the lowest, but it can also estimate the relations which exist between the numbers of vibrations corresponding to each of these sounds. Not, indeed, that we can say whether one sound produces two or three times as many vibrations as another; but whenever the number of vibrations of two successive or simultaneous sounds are in a simple ratio, these sounds excite in us an agreeable sensation, which varies with the ratio of the vibrations of the two sounds, and which the ear can readily estimate. Hence results a series of sounds characterised by relations, which have their origin in the nature of our organisation, and which constitute what is called the musical scale.

In this series the sounds are reproduced in the same order, in periods of seven, each period constituting a gamut; and the seven sounds or notes of each gamut by the names C, D, E, F, G, A, B, or by ut or do, re, mi, fa, sol, la, si. The first six of these letters are the first syllables of the lines of a hymn which was sung by the chorister children to St. John, their patron saint, when they prayed to be freed from hoarseness; and the word si is formed of the first letter of St John's name.

The word gamut is derived from gamma, the third letter of the Greek alphabet, because Guido d' Arezzo, who first (in the eleventh

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Intervals.

163

century) represented notes by points placed on parallel lines, denoted these lines by letters, and chose the letter gamma to designate the first line.

If we agree to represent by I the number of vibrations of the fundamental note C or do of the gamut, that is to say, of the deepest note; experiment shows that the numbers of vibrations of the other notes of the scale are those given in this table :

This table does not give the absolute numbers of the vibrations of the various notes, but only their relative numbers. Knowing the absolute number of vibrations of the fundamental C, we may deduce those of the other notes by multiplying them by,, or 2 respectively; and we thus find that at the octave (169), the number of vibrations is double that of the fundamental note.

4

...

The scale may be continued by taking the octaves of these notes namely, c, d, e, f, g, a, b, and again the octaves of these last, and so forth.

169. Intervals.—An interval is the ratio of one sound to another, that is, the relation between the numbers of vibrations which produce these sounds.

The interval between two consecutive notes of the gamut is called a second; such as the interval from do to re, from re to mi, from mi to fa, and so on.

If between any two notes which are compared, there are one, two, three, four, five, or six intermediate notes, these intervals are called respectively, a third, a fourth, fifth, sixth, seventh, and octave. Thus the interval from C to E is a third, that from C to F a fourth, from C to G a fifth, from C to A a sixth, and from C to B a seventh, and from C to c an octave.

Although two or more notes may be separately musical, it does not follow that, when sounded together, they produce a pleasant sensation. When the ear can distinguish without fatigue the ratio between two sounds, which is the case when the ratio is simple, the accord or co-existence of these two sounds forms a consonance; but if the number of vibrations is in a complicated ratio the ear is unpleasantly affected and we have dissonance.

The simplest concord is unison, in which the numbers of vibra

tions are equal; then comes the octave, in which the number of vibrations of one sound is double that of the other; then the fifth, where the ratio of the sounds is as 3 to 2; the fourth, of which the ratio is 4 to 3; and, lastly, the third, where the ratio is 5 to 4.

If three notes are sounded together they are concordant, when the numbers of their vibrations are as 4: 5: 6. Three such notes form a harmonic triad, and if sounded with a fourth note, which is the octave of the lowest, they constitute what is called a major chord. Thus C, E, G form a major triad, G, B, d form a major triad, and F, A, c form a major triad. C, G, and F have, for this reason, special names, being called respectively, the tonic, dominant, and sub-dominant, and the three triads the tonic, dominant, and sub-dominant triads or chords respectively.

If, however, the ratio of any three notes is as 10: 12: 15, the three sounds are slightly dissonant, but not so much so as to disqualify them from producing a pleasant sensation, at least under certain circumstances. When these three notes and the octave to the lower are sounded together they constitute what in music is called a minor chord.

The intervals between the notes in the scale are

9

It will be seen that there are here three kinds of intervals; the interval is called a major tone, and that of 10 a minor tone; the relation between the major and the minor tone is 2 : 10=81, and is called a comma. The interval 1 is called a major semitone. The major scale is formed of the following succession of intervals: a major tone, a minor tone, a major semitone, a major tone, a minor tone, a major tone, and a major semitone. This succession it is which constitutes the scale; the key note, or the tonic, may have any number of vibrations; but once its height is fixed, that of the other notes are always in the above ratio.

170. On semitones and on scales with different key notes.— It is found convenient for the purposes of music to introduce notes intermediate to the seven notes of the gamut; this is done by increasing or diminishing those notes by an interval of 25, which is called a minor semitone. When a note (say C) is increased by

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Musical Temperament.

165

this interval, it is said to be sharpened, and is denoted by the symbol C, called 'C sharp ;' that is C÷C=25. When it is decreased by the same interval, it is said to be flattened, and is represented thus—B?, called ' B flat;' that is, B÷B) = 25. If the effect of this be examined, it will be found that the number of notes in the scale from C up to c has been increased from seven to twenty-one notes, all of which can be easily distinguished by the Thus, reckoning C to equal 1, we have—

ear.

Hitherto we have made the note C the tonic or key note. Any other of the twenty-one distinct notes above-mentioned, e.g. G, or F, or Că, etc. may be made the key note, and a scale of notes constructed with reference to it. This will be found to give rise in each case to a series of notes, some of which are identical with those contained in the series of which C is the key note, but most of them different. And of course the same would be true for the minor scale as well as for the major scale, and indeed for other scales, which may be constructed by means of the fundamental triads.

171. On musical temperament. -The number of notes that arise from the construction of the scales described in the last article is enormous; so much so as to prove quite unmanageable in the practice of music; and particularly for music designed for instruments with fixed notes, such as the pianoforte. Accordingly it becomes practically important to reduce the number of notes, which is done by slightly altering their just proportions. This process is called temperament. By tempering the notes, however, more or less dissonance is introduced, and accordingly several different systems of temperament have been devised for rendering this dissonance as slight as possible. The system usually adopted -at least in intention-is called the system of equal temperament. It consists in the substitution between C and c of eleven notes at equal intervals, each interval being, of course, the twelfth root of 2, or 105946. By this means the distinction between the semitones is abolished, so that, for example, C and D✈ become the same note. The scale of twelve notes thus formed is called the chromatic scale. It of course follows that major triads become slightly dissonant. Thus, in the diatonic scale, if we reckon C to be 1, E is denoted by 125000, and G by 1.50000. On the system of equal

temperament if C is denoted by 1, E is denoted by 1·25992 and G by 1.49831.

172. The number of vibrations producing each note. The tuning fork.—Hitherto we have denoted the number of vibrations corresponding to the note C by m, and have not assigned any numerical value to that symbol. In the theory of music it is usual to assign 256 double vibrations to the middle C. This, however, is arbitrary. An instrument is in tune provided the intervals between the notes are correct, when C is yielded by any number of vibrations per second not differing much from 256. Moreover, two instruments are in tune with one another if, being separately in

Fig. 143.

tune, they have any one note, for instance, C, yielded by the same number of vibrations. Consequently, if two instruments have one note (say C) in common, they can then be brought into tune jointly, by having their remaining notes separately adjusted with reference to that fundamental note. A tuning fork is an instrument yielding a constant sound, and is used as a standard for tuning musical instruments. It consists of an elastic steel rod, bent as represented in fig. 143. It is made to vibrate either by drawing a bow across the ends, or by striking one of the legs against a hard body, or by