in Anatomy, the lowest part of the intestinum rectum, called by anatomists the anus. See ANATOMY.

in general something that serves as a base or foundation for another.

in Music. A fundamental sound is that which forms the lowest note of the CHORD, and from whence are deduced the harmonical relations of the tonic; or, which serves for a key to the tone #. The fundamental bass is that which serves for a foundation to the harmony. A fundamental chord is that whose bass is fundamental, and in which the sounds are ranged in the same order as when they are generated, according to the experiment so often repeated by M. d'Alembert, in his Preliminary Discourse and Elements of Music. But as this order removes the parts to an extreme distance one from the other, they must be approximated by combinations or inversions; but if the bass remains the same, the chord does not for this reason cease to bear the name of fundamental. Such an example is this chord, ut mi sol, included in the interval of a fifth: whereas, in the order of its generation, ut sol mi, it includes a tenth, and even a seventeenth; since the fundamental ut is not the fifth of sol, but the octave of that fifth.

FUNDAMENTAL Bass. This part in music is, according to Rousseau, and indeed according to all authors who have proceeded upon M. Rameau's experiment, in its primary idea, that bass which is formed by the fundamental notes of every perfect chord that constitutes the harmony of the piece; so that under each chord it caules to be heard, or underflood, the fundamental sound of that particular chord; that is to say, the sound from whence it is derived by the rules of harmony. From whence we may see, that the fundamental bass can have no other contexture than that of a regular and fundamental succession, without which the procedure of the upper parts would be illegitimate.

To understand this well, it is necessary to be known, that, according to the system of Rameau, which Rousseau has followed in his Dictionary, every chord, though composed of several sounds, can only have one which is its fundamental, viz. that which produces this chord, and which is its bass according to the direct and natural order. Now, the bass which prevails under all the other parts, does not always express the fundamental sounds of the chords; for amongst all the sounds which form a chord, the composer is at liberty to transfer to the bass that which he thinks preferable; regard being had to the procedure of that bass, to the beauty of the melody, and above all to the expression, as may afterwards be explained. In this case the real fundamental sound, instead of retaining its natural station, which is in the bass, will either be transferred to some of the other parts, or perhaps even entirely suppressed, and Fundamental, such a chord is called an inverted chord.

In reality, says Rameau, a chord inverted does not differ from the chord in its direct and natural order from which it was produced: but as these sounds form different combinations, these combinations have long been taken for fundamental chords; different names have been given them, (which may be seen at the word ACCORD, in Rousseau's Dictionary). These names, by the persons who bestowed them, were thought to create and sanctify their distinctions; as if a difference in names could really produce a difference in the species.

M. Rameau in his Treatise of Harmony has shewn, and M. d'Alembert in his Elements of Music has still more clearly evinced, that many of these pretendedly different chords were no more than inversions of one single chord. Thus the chord of the sixth is no more than the perfect chord of the third transferred to the bass; by adding a fifth, we shall have the chord of the sixth and fourth. Here there are three combinations of a chord, which only consists of three sounds; those which contain four sounds are susceptible of four combinations, since each of these sounds may be transferred to the bass. But in adding beneath this another bass which, under all the combinations of one and the same chord, always presents the fundamental sound; it is evident, that consonant chords are reduced to the number three, and the number of dissonant chords to four. Add to this all the chords by supposition, which may likewise be reduced to the same fundamentals, and you will find harmony brought to a degree of simplicity in which no person could ever hope to see it whilst its rules remained in that state of confusion where M. Rameau found them. It is certainly, as that author observes, an astonishing occurrence, that the practice of this art could be carried so far as it really was, without knowing its foundation; and that all the rules were fo exactly found, without having discovered the principle on which they depended.

After having shewn what is the fundamental bass beneath the chords, let us now speak of its procedure, and of the manner in which it connects these chords among themselves. Upon this point the precepts of the art may be reduced to the fix following rules.

1. The fundamental bass ought never to found any other notes than those of the series or tone in which the composer finds himself, or at least those of the series or tone to which he chooses to make a transition. This of all the rules for the fundamental bass is the first and most indispensable.

2. By the second, its procedure ought to be so implicitly subjected to the laws of modulation, as never to suffer the idea of a former mode to be lost till that of a subsequent one can be legitimately affirmed; that is to say, that the fundamental bass ought never to be devious, or suffer us to be one moment at a loss in what mode we are.

3. By the third, it is subjected to the connexion of chords and the preparation of dissonances: a manoeuvre which, as we shall afterwards see, is nothing else but a method of producing this connexion, and which of consequence is only necessary when the connexion cannot subsist without it. See CONNEXION, PREPARATION.

4. By the fourth, it is necessitated, after every dis- fonance, to pursue that career which the resolution of the dissonance indispensably prescribes. See RESOLUTION.

5. By the fifth, which is nothing else but a consequence of the former, the fundamental bass ought only to move by consonant intervals; except alone in the operation of a broken cadence, or after a chord of the seventh diminished, where it rises diatonically. Every other motion of the fundamental bass is illegitimate.

6. By the sixth, in short, the fundamental bass or harmony ought not to be syncopated; but to distinguish the bars and the times which they contain, by changes of chords properly marked with cadences; in such a manner, for instance, that the dissonances which ought to be prepared may find their preparation in the imperfect time, but chiefly that all the repose may happen in the perfect time. This sixth rule admits of an infinite number of exceptions; but the composer ought however to be attentive to it, if he would form a music in which the movements are properly marked, and in which the bars may end gracefully.

Wherever these rules are observed, the harmony shall be regular and without fault: this, however, will not hinder the music from being detestable. See COMPOSITION.

A word of illustration on the fifth rule may not be useless. Whatever turn may be given to a fundamental bass, if it is properly formed, one of these alternatives must always be found; either perfect chords moving by consonant intervals, without which these chords would have no connexion; or dissonant chords in operations of cadence: in every other case, the dissonance can neither be properly placed nor properly resolved.

From thence it follows, that the fundamental bass cannot move regularly but in one of these three manners: 1st, To rise or descend by a third or by a sixth. 2dly, By a fourth or a fifth. 3dly, To rise diatonically by means of the dissonance which forms the connexion, or by a licent upon a perfect chord. With respect to a diatonic descent, it is a motion absolutely prohibited to the fundamental bass; or, at most, merely tolerated in cases where two perfect chords are in succession, divided by a close expressed or understood. This rule has no other exception: and it is from not discerning the foundation of certain transitions, that M. Rameau has caused the fundamental bass to descend diatonically under chords of the seventh; an operation which is impracticable in legitimate harmony. See CADENCE, DISSONANCE.

The fundamental bass, which they add for no other reason than to serve as a proof of the harmony, must be retrenched in execution, and often in practice it would have a very bad effect; for it is, as M. Rameau very properly observes, intended for the judgment, and not for the ear. It would at least produce a monotony extremely nauseous by frequent returns of the same chord, which they disguise and vary more agreeably by combining it in different manners upon the continued bass, without reckoning upon the different inversions of harmony, which furnish a thousand means of adding new beauties to the music and new energy to the expression. See CHORD, INVERSION.

But it will be objected, If the fundamental bass is not useful in composing good music, if it must even be Fundamen- tal. retrenched in practice, what good purpose, then, can it serve? We answer, that in the first place, it serves for a rule to scholars, upon which they may learn to form a regular harmony, and to give to all the parts such a diatonic and elementary procedure as is prescribed them by that fundamental bass. It does more, as we have already said: it proves whether a harmony already formed be just and regular; for all harmony which cannot be subjected to the test of a fundamental bass, must according to all rules be bad. Finally, it serves for the investigation of a continued bass under a given air: though, in reality, he who cannot directly form a continued bass will scarcely be able to form a fundamental bass, which is better; and much less still will he be able to transform that fundamental bass into a legitimate continued bass. These which follow are, however, the principal rules which M. Rameau prefers for finding the fundamental bass of a given air.

1. To ascertain with precision the mode in which the composer begins, and those through which he passes. There are also rules for investigating the modes; but so long, so vague, so incomplete, that with respect to this, the ear may be formed long before the rules are acquired; and the dunce who should try to use them would gain no improvement but the habit of proceeding always note by note, without even knowing where he is.

2. To try in succession under each note the principal chords of the mode, beginning by those which are most analogous, and passing even to the most remote, when the composer feels himself under a necessity of doing so.

3. To consider whether the chord chosen can suit the upper part in what precedes and in what follows, by a just fundamental succession; and when this is impracticable, to return the way he came.

4. Not to change the note of the fundamental bass till after having exhausted all the notes which are allowed in succession in the upper part, and which can enter into its chord; or till some syncopated note in the air may be susceptible of two or a greater number of notes in the bass, to prepare the dissonance which may be afterwards resolved according to rule.

5. To study the intertexture of the phrases; the possible succession of cadences, whether full or avoided; and above all, the pauses which for ordinary return at the end of every four, or of every two bars, so that they may always fall upon perfect and regular cadences.

6. In short, to observe all the rules formerly given for the composition of the fundamental bass.—These are the principal observations to be made for finding one under any given air; for there are sometimes several different ones which may be investigated. But, whatever may be said to the contrary, if the air has accent and character, there is only one just fundamental bass which can be adapted to it.

After having given a summary explication of the manner in which a fundamental bass should be composed, it should remain to suggest the means of transforming it into a continued bass; and this would be easy, if it were only necessary to regard the diatonic procedure and the agreeable air of this bass. But let andamen us not imagine that the bas, which is the guide and support of the harmony, the foul, and as it were the interpreter of the air, should be limited to rules so simple: there are others which depend upon principles more certain and more radical; fruitful, but latent principles, which have been felt by every artist of genius, without having been detected by any one. Rousseau hopes, that in his letter upon French music he infinuated this principle. For those who understand him, he imagines he has said enough concerning it, and can never say enough of it for those who do not. See Rousseau's Miscellanies, vol. ii. p. 1.

He does not here mention the ingenious system by M. Serre of Geneva, nor his double fundamental bas; because the principles which, with a facacity meritorious of praise, he had half detected, have afterwards been unfolded by M. Tartini, in a work of which Rousseau has given an account in his article SYSTEM.