I. One object to be aimed at is, to make the sum of the temperaments of all the concords the least possible. Since experience teaches us that the harshness of a given concord increases with its temperament, it is obvious that of two systems which agree in other respects, the best is that in which the sum of the temperaments is least. II. When other things are equal, the best adjustment of the imperfections of the scale is that which diminishes the harmoniousness of all the different concords proportionally. The succession of a worse to a better harmony, is justly regarded by several of the best writers on this subject, as one of the principal causes of offence to the ear, in instruments imperfectly tuned. III. When different chords of the same kind are of unequally frequent occurrence, there is an advantage, _cæteris paribus_, in giving the greatest temperament to that which occurs most seldom. This important consideration has indeed been neglected by Dr. Smith, in the systems which he recommends, both for his changeable and the common fixed scale; as it is, also, by the numerous advocates of the system of equal semitones. But many authors on temperament, and most instrument-makers, pay a vague regard to it. Their aim has been, although in a loose and conjectural manner, to make the prominent chords of the simplest keys the nearest to perfection, whilst a greater temperament is thrown upon those which occur only in the more complex keys. Thus Dr. Young, in the Philos. Trans. for 1800, recommends a scheme which increases the temperament of the IIIds, on the key note of the successive keys, as we modulate by fifths from C, nearly in arithmetical progression. Earl Stanhope assigns as a reason for the small temperament which is given to several of the IIIds in his system, that they are on the tonic of the simpler keys. The irregularities in Mr. Hawkes's scheme may be traced to the same cause. And, with the instrument-makers, it is a favourite maxim to lay the wolf, as they term it, where it will be most seldom heard. But if the above consideration deserves any weight at all, it deserves to be accurately investigated. Not only ought the relative frequency of different chords to be ascertained with the greatest accuracy, of which the nature of the subject is susceptible, but the degree of weight which this consideration ought to have, when compared with the two others above-mentioned, should be determined: for it is plain that neither of them ought to be ever left out of view. Accordingly, the principal design of the following propositions will be to investigate the actual frequency of occurrence of different chords in practice; and from this and the two other above-mentioned considerations united, to deduce the best system of temperament for a scale, containing any given number of sounds to the octave, and particularly for the common Douzeave, or scale of twelve degrees. PROPOSITION I.