feebled and rendered unpleasing to the car by the intemperate application of it.

The scale in common use is the diatonic; according to which, the system of the octave consists of a fifth and a fourth, or of twofourths, consisting of two tones and a semitone, and disjoined by the interval of a whole tone. The ancients supposed these two tetrachords (or fourths) to be similar; but they are not so, as will appear hereafter.

For the systematical divisions of a musical string, mathematicians have invented different canons, which agree in some of the main principles; as they needs must do, being founded in nature; but, in some particulars, the theory admits of variety.

When a musical string of any length is extended between two points, so as to vibrate freely, we call the sound given by the vibration of the whole string the unison, or fundamental tone; to which, one half of the string will sound an octave; two-thirds will found a fifth; three fourths will found a fourth. Hence the interval of an octave is expressed by the ratio of 2 to 1; a fifth by the ratio of 3 to 2, or 3; a fourth by 4 to 3, or 1.

The

The most simple Division of a String.

For a first experiment, a string may be divided (by a scale placed under it) into 12 equal parts. Six of these equal parts will sound an eighth to the whole string; eight of them will sound a fifth, and nine will found a fourth. The difference between the fourth and the fifth is a whole tone, arising naturally from this division of the scale, and is called the tone major, as distinguished from another which is somewhat lesser.

12)

or 3 to 4, sounds a fourth to the unison; or 2 to 3, sounds a fifth to the unison; and the difference between them, which is one of these twelve equal parts, gives us the tone major. Therefore the number 12, which was preferred by the ancients, is the most convenient number of equal parts into which a string can be divided, because it shews us the interval of a complete musical tone.

Composition and Resolution of Harmonic

Intervals.

Every greater interval divides itself into two lesser, of which one is always greater

than the other. Thus the octave, which is in the ratio of 2 to 1, or 4 to 2, is divisible into the two lesser ratios of 4 to 3, which is the fourth, and 3 to 2, which is the fifth: and these two being multiplied together, the greater term of the one by the greater term of the other, and the lesser term of the one by the lesser term of the other, produce 12 to 6; which have the same ratio to each other as 4 to 2 or 2 to 1; the whole ratio being compounded of the intermediate ratios into which it was divided.

Then again, the fifth, which is in the ratio of 3 to 2, or 6 to 4, divides itself into the ratios of 6 to 5, which is that of the lesser third, and 5 to 4 the greater third: and these being multiplied together as before, produce the ratio of 30 to 20, which is the fame as that of 3 to 2. The next step carries us to the division of the greater third, 5 to 4. If we take it as 10 to 8, it is resolvable into the two ratios of 10 to 9, which is the tone minor, and 9 to 8, which is the tone major. We double the numbers 3 to 2, and 5 to 4, taking in their stead the equivalents 6 to 4 and 10 to 8, because the ratio in the lesser terms being superparticular, that is, exceed

ing

ing only by unity, and admitting no other common measure, afford no medium.

The most ancient Greek writers seem to have been unacquainted with the difference of the two tones, major and minor; though Euclid, in his treatise on the section of the musical canon, has demonstrated that the octave, which contains five tones and two semitones, does not contain six tones: from whence the admission of the tone minor necessarily follows, and contributes greatly to the beauty and perfection of the modern scale. Ptolemy gives a major and a minor tone: but Didymus found out the minor tone long before him; only he placed the below the, Ptolemy above, as we have it However, neither of their divisions seem to have been generally adopted by theorists though it is scarcely to be doubted that the ear of practical musicians must have led them to the use of the true third 4 to 5, which does not contain two major tones, but 18 composed of the major and minor.

now.

From the divisions already mentioned, we have brought a semitone into the scale; which is the interval between the major third and the fourth, or, in other words, the com

plement

plement of the major third to the fourth. The fourth is 3 to 4, the third major is 4 to 5, which being subtracted by multiplying crosswise, (the antecedent of the first by the consequent of the second, and the antecedent of the second by the consequent of the first,) give us the ratio of 15 to 16: that is, if the interval on a musical string, which sounds a major third to the unison, is divided into 16 parts, the half note will be one of them, and this being added to the third will constitute the fourth. This half note therefore is the natural production of the scale itself. As the tone major arises from the division of the fourth and fifth, and is the interval between them; so is the semitone the interval between the third major and the fourth.

There are other things of lesser consideration which I shall forbear to speak of. These divisions which I have explained are the proper elements of musicians in the system of the octave; which, like the alphabet in a language, afford all that variety by their combinations, with which the ear is delighted. They are found by dividing the ratios of the octave, the fifth, and the greater third, into their