the distance or space between two extremes, either in time or place. The word comes from the Latin intervallo, which, according to Isidore, signifies the space inter fossam & murum, "between the ditch and the wall;" others note, that the stakes or piles, driven into the ground in the ancient Roman bulwarks, were called vallia; and the interstices or vacancy between them, intervalla.
in music. The distance between any given sound and another, strictly speaking, is neither measured by any common standard of extension nor duration; but either by immediate sensation, or by computing the difference between the numbers of vibrations produced by two or more sonorous bodies, in the act of sounding, during the same given time. As the vibrations are slower and fewer during the same instant, for example, the sound is proportionally lower or graver; on the contrary, as during the same period the vibrations increase in number and velocity, the sounds are proportionally higher or more acute. An interval in music, therefore, is properly the difference between the number of vibrations produced by one sonorous body of a certain magnitude and texture, and of those produced by another of a different magnitude and texture in the same time.
Intervals are divided into consonant and dissonant. A consonant interval is that whole extremes, or whole highest and lowest sounds, when simultaneously heard, coalesce in the ear, and produce an agreeable sensation called called by Lord Kames a *tertium quid*. A dissonant interval, on the contrary, is that whose extremes, simultaneously heard, far from coalescing in the ear, and producing one agreeable sensation, are each of them plainly distinguished from the other, produce a grating effect upon the sense, and repel each other with an irreconcilable hostility. In proportion as the vibrations of different sonorous bodies, or of the same sonorous body in different modes, more or less frequently coincide during the same given time, the chords are more or less perfect, and consequently the intervals more or less consonant. When these vibrations never coincide at all in the same given time, the discord is consummate, and consequently the interval absolutely dissonant.
Intervals are not only divided according to their natures, but also with respect to their degrees. In this view, they are either enharmonic, chromatic, or diatonic. Of these therefore in their order, from the least to the greatest.
An enharmonic interval is what they call the eighth part of a tone, or the difference between a major and minor semitone generally distinguished by the name of a comma. Commas, however, are of three different kinds, as their quantities are more or less; but since these differences cannot be ascertained without long and intricate computations, it is not necessary for us to attempt an investigation, whose pursuit is so unpleasing, and whose result attended with so little utility. It has by musicians been generally called the eighth part of a tone; but they ought to have considered, that a comma is by no means the object of auricular perception, and that its estimate can only be formed by calculation. For a more minute disquisition of this matter, our readers may consult the article *Comma* in the Musical Dictionary, or the article *Music* in this Work, Notes, n s. A chromatic interval consists properly of a minor semitone, but may also admit the major. A diatonic interval consists of a semitone-major at least, but may consist of any number of tones within the octave. When an octave higher or lower is assumed, it is obvious that we enter into another scale which is either higher or lower, but still a repetition of the former degrees of sound.
Intervals again are either simple or compound. All the intervals within any one octave are simple; such as the second major or minor, the third, the fourth, the fifth, the sixth, the seventh, &c. of these afterwards. All intervals whose extremes are contained in different octaves, such as the ninth, the tenth, the eleventh, the twelfth, the thirteenth, the fourteenth, the fifteenth, &c. may be termed compound intervals.
The semitone either exactly or nearly divides the tone into two equal parts. In the theory of harmonical computation three kinds of semitones are recognized, viz. the greatest, the intermediate, and the smallest semitone. But in practice, to which these explanations are chiefly adapted, the semitone is only distinguished into major and minor. The semitone major is the difference between the third major and the fourth, as EF. Its ratio is as 15 to 16, and it forms the least of all diatonic intervals.
The semitone minor consists of the difference between the third major and minor; it may be marked in the same degree by a sharp or a flat, and it only forms a chromatic interval; its ratio is as 24 to 25.
Though some distinction is made between these semitones by the manner of marking them, yet on the organ and harpsichord no distinction can be made; nor is there any thing more common for us than to say, that D sharp in rising is E flat in descending; and so through the whole diapason above or below; besides, the semitone is sometimes major and sometimes minor, sometimes diatonic and sometimes chromatic, according to the different modes in which we compose or practise; yet in practice these are called semitones minor, which are marked by sharps or flats, without changing the degree; and semitones major are those which form the interval of a second.
With respect to the three semitones recognised in theory, the greatest semitone is the difference between a tone major and a semitone minor; and its ratio is as 25 to 27. The intermediate semitone is the difference between a semitone major and a tone major; and its ratio is as 128 to 135. In a word, the small semitone consists of the difference between the greatest and the intermediate semitone; and its ratio is as 125 to 128.
Of all these intervals, there is only the semitone major, which is sometimes admitted as a second in harmony.
The interval of a tone which characterizes the diatonic species of composition, is either major or minor. The former consists of the difference between the fourth and fifth; and its ratio is as 8 to 9; and the latter, whose ratio is as 9 to 10, results from the difference between the third minor and the fourth.
Seconds are distinguished into four kinds: two of which are not in practice sufficiently momentous to be mentioned. The second major is synonymous with the intervals of a tone; but as that tone may be either major or minor, its ratio may be either as 8 to 9, or as 9 to 10.
The second minor consists of the distance from B to C, or from EF; and its ratio is as 15 to 16.
The third is so called, because it consists of two gradations, or three diatonic sounds, as from G to B ascending, or from A to C, inclusive of the extremes; of which the first is a third major, composed of two full tones, and its ratio as 4 to 5; the second, a third minor consisting of a tone and a semitone major, and its ratio as 5 to 6.
The fourth has by some been reckoned an imperfect, but more justly by others a perfect, chord. It consists of three diatonic degrees, but takes its name from the four different sounds of which it is formed; or, in other words, the number by which it is denominated includes the extremes. It is composed of a tone major, a tone minor, and a semitone major, as from C to F ascending; its ratio as 3 to 4.
The fifth next to the octave, is, perhaps, the most perfect interval, as least susceptible of alteration. The number from whence it assumes its name likewise includes its extremes. It consists of two tones major, one minor, and a semitone major, as from A to E ascending; its ratio is as 2 to 3.
The sixth is not found among the natural order of consonances, but only admitted by combination. It is not here necessary to mention its various distinctions. and uses, as we only give an account of intervals in general.
The sixth major consists of four tones and a semitone major, as from G to E ascending; its ratio is as 3 to 5. The sixth minor contains three tones and two semitones major, as from E to C ascending; its ratio is as 5 to 8.
The seventh, as a reduplication of the second, is a dissonance. When major, it consists diatonically of five tones, three major, and two minor; and a major semitone, as from C to B ascending; its ratio is as 8 to 15.
When minor, it consists of four tones, three major and one minor, and two major semitones, as from E to D ascending; its ratio is as 5 to 9.
The octave is the most perfect of all chords, and in many cases hardly to be distinguished by the ear from an unison; that is to say, from that coincidence of sound produced by two musical strings, whose matter, lengths, diameters, and tensions, are the same. As the vibrations of two strings in unison during any given time, are precisely coincident; so whilst the lowest extreme of the octave vibrates once, the highest vibrates twice; and consequently its ratio is as 1 to 2, as from c to C ascending. It consists of six full tones and two semitones major. Its name is derived from the Latin octo, "eight;" because that number likewise includes its extremes. It may likewise be divided into twelve semitones. It contains the whole diatonic scale; and every series above or below consists only of the same returning sounds. From whence the natures, distances, and powers, of every interval greater than the octave, as the ninth, the tenth, the eleventh, the twelfth, the thirteenth, the fourteenth, the fifteenth, the triple octave, &c. may easily be computed.
During our past observations upon the term interval, we have either wholly neglected our faithful associate M. Rousseau, or only maintained a distant and momentary intercourse with him. We now propose to pay him a more permanent and familiar visit; but as he is engaged in the dispute between the Pythagoreans and Aristoxenians, we think it more advantageous to decline the controversy, and to follow him, after having escaped the fray, like a gentleman and a scholar. Having put the partizans of Aristoxenus to silence, let us, with him, forgo the lists of combat, nor stain his triumph by insulting the falling champions.
"We divide (says he) as did the ancients, intervals into consonant and dissonant. The consonances are perfect or imperfect*; dissonances are either such by nature, or become such by accident. There are only two intervals naturally dissonant, viz., the second and seventh, including their octaves or replications; nay, still these two may be reduced to one alone, as the seventh is properly no more than a replication of the second; for B, the seventh above the lowest C, where we have generally begun the scale, is really an octave above B, the note immediately below that C; and consequently the interval between these lower sounds is no more than that of a second major, to which all dissonances may therefore be ultimately reduced, whether considered as major or minor; but even all the consonances may become dissonant by accident. See Discord.
Besides, every interval is either simple or reduplicated. Simple intervals are such as the limits of a single octave comprehend. Every interval which surpasses this extent is reduplicated; that is to say, compounded of one or more octaves, and of the simple interval whose replication it is.
Simple intervals are likewise divided into direct and inverted. Take any simple interval whatever for a direct one; the quantity which, added to itself, is required to complete the octave, will be found an inverted interval; and the same observation holds reciprocally true of such as are inverted.
There are only six kinds of simple intervals; of which three contain such quantities, as, added to the other three, are required to complete the octave; and of consequence likewise the one must be inversions of the other. If you take at first the smallest intervals, you will have, in the order of direct intervals, the second, the third, and fourth; for inverted, the seventh, the fifth, and fifth. Suppose these to be direct, the others will be inverted; everything here is reciprocal.
To find the name of any interval whatever, it is only necessary to add the denomination of unity to the degree which it contains. Thus the interval of one degree shall give a second; of two, a third; of three, a fourth; of seven, an octave; of nine, a tenth, &c. But this is not sufficient to determine an interval with accuracy; for under the same name it may be either major or minor, true or false, diminished or redundant.
The consonances which are imperfect, and the two natural dissonances, may be major or minor; which, without changing their degree, occasions in the interval the difference of a semitone; so that if, from a minor interval, we still deduce a semitone, it becomes an interval diminished; if, by a semitone, we increase a major interval, it becomes an interval redundant.
The perfect consonances are by their nature invariable. When their intervals are such as they ought to be, we call them just, true; and if we dilate or contract this interval by a semitone, the consonance is termed false, and becomes a dissonance; redundant, if the semitone be added; diminished, if it be subtracted. We improperly give the name of a false fifth to the fifth diminished; this is taking the genus for the species; the fifth redundant is every jot as false as the diminished, it is even more so in every respect."
In the Musical Dictionary, plate C, fig. 2. may be seen a table of all the simple intervals practicable in music, with their names, their degrees, their values and their ratios.
Having ascertained the distinction between major and minor intervals, it is only necessary to add, that these may be natural or artificial. Of the natural we have already given some account, by ascertaining the distances and ratios of such as have been mentioned. Of the artificial, we may observe, that they are such as change their position from what it naturally is in the diatonic scale, to what the convenience of composition or transposition requires it to be. A note thus thus artificially heightened by a semitone, together with the character which expresses that elevation, is called a sharp; on the contrary, a note artificially depressed by a semitone, together with the character by which that depression is signified, is called a flat. The character which restores a note thus depressed or raised to its primary state, is called a natural. Major or minor intervals, as they prevail, characterize the major or minor mode. See Mode.