It is well known to those who have attended to the subject of musical ratios, that a fixed scale of eight degrees to the octave, which shall render all its concords perfect, is impossible. It has been demonstrated by Dr. Smith, from an investigation of all the positions which the major, the minor, and the half-tone can assume, that the most perfect scales possible, of which there are two equally so, differing only in the position of the major and the minor tone above the key note, must have one Vth and one 3d too flat, and consequently the supplementary 4th and VIth too sharp, by a comma. In vocal music, and in that of perfect instruments, this defect in the scale is not perceived, because a small change may be made in the key, whenever the occurrence of either of those naturally imperfect intervals renders such a change necessary to perfect harmony. But in instruments with fixed scales, such as the guitar, the piano-forte, and the organ, if we begin with tuning as many concords as possible perfect, the resulting chords above-mentioned will be necessarily false in an offensive degree. Hence it is an important problem in practical harmonics, to distribute these imperfections in the scale among the different chords, in such a manner as to occasion the least possible injury to harmony. But this is not the only nor the principal difficulty which the tuner of imperfect instruments has to encounter. In order that these instruments may form a proper accompaniment for the voice, and be used in conjunction with perfect instruments, it is necessary that music should be capable of being executed on them, in all the different keys in common use; and especially that they should be capable of those occasional modulations which often occur in the course of the same piece. Now only five additional sounds to the octave are usually inserted for this purpose, between those of the natural scale, which, of course, furnish it with only three sharps and two flats. Hence, when a greater number of flats or sharps is introduced, the music can be executed only by striking, in the former case, the sharp of the note next below; and, in the latter, the flat of the note next above. But as the diatonic semitone is more than half the major, and much more than half the minor tone, if the additional sounds in the common artificial scale be made perfect for one of the above employments, they must be extremely harsh for the other. Hence arises the necessity of adjusting the position of these five inserted sounds so that they may make tolerable harmony, whichever way employed. A change in these will require corresponding changes in the position of the several degrees of the natural scale; so that it is highly probable that the best scheme of temperament will leave no concord, either of the natural or artificial scale, absolutely perfect. In adjusting the imperfections of the scale, the three following considerations have been usually taken into view.