If there is any moral to be drawn from the history of this theory, it is that considering how near Sauveur's errors were to the truth, it behooves us to exercise some caution also with regard to the new theory. And in reality there seems to be reason for doing so. The fact that a musician will never confound a more perfectly consonant chord on a poorly tuned piano with a less perfectly consonant chord on a well tuned piano, although the roughness in the two cases may be the same, is sufficient indication that the degree of roughness is not the only characteristic of a harmony. As the musician knows, even the harmonic beauties of a Beethoven sonata are not easily effaced on a poorly tuned piano; they scarcely suffer more than a Raphael drawing executed in rough unfinished strokes. The _positive physiologico-psychological_ characteristic which distinguishes one harmony from another is not given by the beats. Nor is this characteristic to be found in the fact that, for example, in sounding a major third the fifth partial tone of the lower note coincides with the fourth of the higher note. This characteristic comes into consideration only for the investigating and abstracting reason. If we should regard it also as characteristic of the sensation, we should lapse into a fundamental error which would be quite analogous to that cited in (1). The _positive physiological_ characteristics of the intervals would doubtless be speedily revealed if it were possible to conduct aperiodic, for example galvanic, stimuli to the single sound-sensing organs, in which case the beats would be totally eliminated. Unfortunately such an experiment can hardly be regarded as practicable. The employment of acoustic stimuli of short duration and consequently also free from beats, involves the additional difficulty of a pitch not precisely determinable. FOOTNOTES: [Footnote 134: This article, which appeared in the Proceedings of the German Mathematical Society of Prague for the year 1892, is printed as a supplement to the article on "The Causes of Harmony," at page 32.] [Footnote 135: The present exposition is taken from the volumes for 1700 (published in 1703) and for 1701 (published in 1704), and partly also from the _Histoire de l'Académie_ and partly from the _Mémoires_. Sauveur's later works enter less into consideration here.] [Footnote 136: Euler, _Tentamen novae theoriae musicae_, Petropoli, 1739.] [Footnote 137: In attempting to perform his experiment of beats before the Academy, Sauveur was not quite successful. _Histoire de l'Académie_, Année 1700, p. 136.] [Footnote 138: _Histoire de l'Académie_, Année 1701, p. 134.] [Footnote 139: _Ibid._, p. 298.] [Footnote 140: _Histoire de l'Académie_, Année 1702, p. 91.] [Footnote 141: From the _Histoire de l'Académie_, Année 1700, p. 139.] [Footnote 142: Because all octaves in use in music offer too great differences of rates of vibration.]