the art of forming any thing to certain proportion.

in reading, or speaking. See Reading.

in music, derived from the Latin modulari. This word in our language is susceptible of several different significations. It frequently means no more than an air, or a number of musical sounds properly connected and arranged. Thus it answers to what Mr Malcolm understands by the word tune, when he does not expressly treat concerning the tuning of instruments. Thus likewise it expresses the French word chant; for which reason, in the article Music, we have frequently expressed the one word by the other. But the precise and technical acceptation to which it ought to be confined, is the art of composing melody or harmony agreeably to the laws prescribed by any particular key, that of changing the key, or of regularly and legitimately passing from one key to another. In what remains to be said upon the subject we follow Rousseau.

Modulation (says he) is properly the manner of ascertaining and managing the modes; but at this time the word most frequently signifies the art of conducting the harmony, and the air successively through several modes, in a manner agreeable to the ear and conformed to rules.

If the different modes be produced by harmony, from thence likewise must spring the laws of modulation. These laws are simple in conception, but difficult in practice. We proceed therefore to show in what they consist.

To modulate properly in the same tone, it is necessary, 1. To run through all the sounds of it in an agreeable air, frequently repeating the sounds which are most essential to it, and dwelling upon these sounds with the most remarkable emphasis; that is to say, that the chord containing the sensible notes, and that of the tonic, should frequently be heard in it, but under different appearances, and obtained by different procedures to prevent monotony. 2. That repose or cadences should only be established upon these two chords: the greatest liberty, however, which ought to be taken with the rule is, that a cadence or repose may be established on the chord of the sub-dominant. 3. In short, that none of the sounds of the mode ought ever to be altered; for without quitting it we cannot introduce a sharp or a flat which does not belong to it, nor abstract any one which in reality does belong to it.

But passing from one mode to another, we must consult analogy, we must consider the relations which a key bears to the other notes in the series, and to the number of sounds common to both the modes, that from whence we pass, and that into which we enter.

If we pass from a mode major, whether we consider the fifth from the key as having the most simple relation with it except that of the octave, or whether we consider it as the first sound which enters into the harmonics of the same key, we shall always find, that this fifth, which is the dominant of the mode, is the chord upon which we may establish the modulation most analogous to that of the principal key.

This dominant, which constituted one of the harmonics of the first key, makes also one of its own peculiar key, of which it is the fundamental sound. There is then a connection between these two chords. Besides, that same dominant carrying, as well as the tonic, a perfect chord major upon the principle of resonance, these two chords are only different one from the other by the dissonance, which passing from the key to the dominant is the sixth superadded, and when ascending from the dominant to the key is the seventh. Now these two chords, thus distinguished by the dissonance which is suitable to each, by the sounds which compose them when ranged in order, form precisely the octave, or the diatonic scale, which we call a gamut, which determines the mode.

This same series of the key, altered only by a sharp, forms the scale belonging to the mode of the dominant; which shows how striking the analogy is between these two tones, and gives the easiest opportunity of passing from one to the other by means of one single alteration alone. The mode then of the dominant is the first which presents itself after that of the key in the order of modulations.

The same simplicity of relations which we find between a tonic and its dominant, is likewise found between the same tonic and its sub-dominant: for that fifth, in ascending, which is formed by the dominant with the tonic, is likewise formed by the sub-dominant in descending; but that sub-dominant does not form a fifth with the tonic, except by inversion; it is directly a fourth, if we take that tonic below, as it ought to be, and which fixes the degree of their relations: for in this sense the fourth, whose ratio is as 3 to 4, immediately follows the fifth, whose ratio is as 2 to 3. So that, if that sub-dominant does not enter into the chord of the tonic, in return the tonic enters into its perfect chord. For let ut mi sol be the chord of the tonic, that of the sub-dominant shall be fa la ut: thus it is the ut which here forms the connection, and the two other sounds of this new chord, are exactly the two dissonances of the preceding. Besides, we need not alter more sounds for this new mode than for that of the dominant; they are both in the one and the other quite the same chords of the principal mode, except one. Add a flat to the sensible note fa or B, and all the notes in the mode of ut or C will serve for that of fa or F. The mode of the sub-dominant then is scarcely less analogous to the principal mode than that of the dominant.

It ought likewise to be remarked, that after having made use of the first modulation in order to pass from a principal mode ut or C, to that of the dominant sol or G, we are obliged to make use of the second to return to the principal mode: for if sol or G be the dominant in the mode of ut or C, ut is the sub-dominant in the mode of sol: thus one of these modulations is no less necessary than the other.

The third sound which enters into the chord of the tonic is that of third formed by its mediant; and, after the preceding, it is likewise the most simple of relations. Here then is a new modulation which presents itself, and which is so much the more analogous, because two of the sounds of the principal tonic enter likewise into the minor chord of its mediant: for the former chord being ut mi sol, the latter must be mi sol fa, where it may be perceived that mi and sol are common. But what renders this modulation a little more remote, is the number of sounds which are necessary to be altered, even for the minor mode, which is most suitable to this mi. In the article Music (234.) will be found a table for all the modes; and Rouffeau, in his Musical Dictionary, has given the formula of a scale both for the major and minor: now, by applying this formula to the minor mode, we find nothing in reality, but the fourth sound fa heightened by a sharp in ascending; but in rising, we find two others which are altered, viz. the principal tonic ut, and its second re, which here becomes a sensible note: it is certain that the alteration of so many sounds, and particularly of the tonic, must remove the mode and weaken the analogy.

If we should invert the third as we have inverted the fifth, and take that third below the tonic on the fifth note la, which ought here to be called a sub-mediant, or the mediant below, we shall form upon this note la a modulation more analogous to the principal mode than that of mi; for as the perfect chord of this sub-mediant is la ut mi, there once more we find, as in that of the mediant, two of the sounds which enter into the chord of the tonic, viz. ut and mi: moreover, since the scale of this new key is composed, at least in descending, of the same sounds with that of the principal key; and since it has only two sounds altered in ascending, that is to say, one fewer than the series of the mediant; it follows that the modulation of this fifth note is preferable to that of the mediant; and by so much the more, that there the principal tonic forms one of the sounds essential to the mode; which is more proper for approximating the idea of the modulation.

The mi may afterwards follow.

Here then are four sounds, *mi fa sol la*, upon each of which we may modulate in passing from the major mode of *ut*. *Re* and *fa* remain, which are the two harmonics of the dominant. This last, as being a sensible note, cannot become a tonic by any proper modulation, at least it cannot immediately become one; this would be an abrupt application of ideas too much opposed to the same sounds, and would likewise be to give it a harmony too remote from the principal sound.

As to the second note *re*, we may likewise, by favour of a consonant procedure in the fundamental base, modulate upon it in a third minor; but this must be only continued for an instant, that the audience may not have time to forget the modulation of *ut*, which is itself altered in that place; otherwise, instead of returning immediately to *ut*, we must pass through intermediate modes, where we must run great hazard of deviation.

By following the same analogies, we may modulate in the following order, to make our exit from a minor mode; first upon the mediant, afterwards the dominant, next the sub-dominant, then the sub-mediant, or fifth note. The mode of each of these accessory keys is determined by its mediant taken from the principal sound. For instance, issuing from the major mode of *ut*, to modulate upon its mediant, we render the mode of that mediant minor; because *sol*, the dominant of the principal sound, forms a third minor with that mediant, which is *mi*. On the contrary, in our egress from the minor mode of *la*, we modulate upon its mediant *ut* in the major mode; because *mi*, the dominant of the tone from whence we issue, forms a third major with the key of that into which we enter, &c.

These rules, comprehended in one general formula, import, that the modes of the dominant and of the sub-dominant are like that of the tonic, and that the mediant and the fifth note require a mode opposed. We must, however, remark, that, by the right which we have of passing from the major to the minor, and vice versa, upon the same key, we may likewise change the order of modes from one key to another; but whilst we thus remove ourselves from the natural modulation, we must presently think of our return: for it is a general rule, that every piece of music ought to terminate in that key with which it began.

In his Musical Dictionary, plate B, fig. 6. and 7. Rousseau has collected in two examples, which are very short, all the modes to which we may immediately pass; the first, in passing from the major mode; and the second, from the minor. Each note indicates a particular modulation; and the value of the notes in each example likewise shows the relative duration suitable to each of these modes, according to its relation with the principal mode.

These immediate transitions from one mode to another, furnish us with the means of passing by the same rules to modes still more remote, and from thence to return to the principal mode, of which we never should lose sight. But it is not sufficient to know what course we ought to pursue; we must likewise be acquainted with the method of entering into it. A summary therefore of the precepts which are given in this department shall immediately follow.

In melody, in order to discover and introduce the modulation which we have chosen, nothing is necessary but to render perceptible the alterations which it causes in the sounds of that mode from whence we issue, to make them proper for the mode into which we enter. Are we now in the major mode of *ut*? there needs no more than to sound the note *fa* sharp, that we may discover the mode of the dominant; or a *fa* flat, that we may show the mode of the sub-dominant. Afterwards you may run over the sounds essential to the mode in which you enter; if it is well chosen, your modulation will always be just and regular.

In harmony, the difficulty is a little increased: for it is necessary that the change of modes should be made at the same time through all the parts, care must be taken of the harmony, and of the air, that we may avoid pursuing different modulations at the same time. Huygens has happily remarked, that the prohibition of two fifths in immediate succession proceeds upon this rule as its principle: in reality, between two parts it is scarcely possible to form a number of just fifths in uninterrupted succession without operating in two different modes.

To introduce a mode, a great many pretend that it is sufficient to form the perfect chord of its principal sound, and this is indispensible in order to produce the mode. But it is certain, that the mode cannot be exactly determined but by the chord containing the sensible note, or the dominant: we must then cause this chord to be heard when we enter into a new modulation. The most eligible rule would be, That in it the seventh, or minor dissonance, should always be prepared, at least the first time in which it is heard: but this method is not practicable in every admissible modulation; and provided that the fundamental basis proceeds by consonant intervals, that the connection of harmony be observed, the analogy of the mode pursued, and false relations avoided, the modulation will always be approved. Composers prescribe as another rule, That a mode should not be changed except after a perfect cadence: but this interdict is useless, and no person observes it.

All the possible methods of passing from one mode to another, are reducible to five with respect to the major mode, and to four with respect to the minor, which, in the Musical Dictionary, plate B, fig. 8. will be found implied in a fundamental basis intended for each modulation. If there be any other modulation which cannot be resolved into some one of these nine, unless that modulation be enharmonic, it must infallibly be illegitimate. See Enharmonic.