The investigations which form the subject of the present communication had their origin in a practical source. It is conceived important that this should be understood, as many musicians regard the problem as a purely theoretical one, undeserving of practical attention. It was in taking part in the tuning of an organ that the effect of the ordinary equal temperament was first realised by the writer as a matter affecting musical sounds in practice, as distinguished from theory. It is the writer's experience that after the ear has once been attracted to this effect, it never fails to perceive it in the tones of instruments tuned in the ordinary way. Some tones show the effect more, some show it less. Among keyed instruments the worst effects are produced by the ordinary harmonium; next to this comes the full-toned modern grand piano, which is often unpleasant in slow harmony, especially if, as is generally the case, the temperament is not very uniform;∗ and the best tones, which show the effect least, are those of soft-toned pianos with little power, and the ordinary organ diapason-stops—especially those old-fashioned, sweet-toned diapasons, which are rapidly disappearing before the organ-builders of the present day. The constant perception of these effects served as an inducement to a study of the subject, pursued in the first instance with the practical object of the improvement of instruments, and afterwards also for the sake of the interest attaching to the theory developed.