¹ On the impact of Ohm's law, see Boring, Edwin G., Sensation and perception in the history of experimental psychology (New York, 1942), pp. 321–9Google Scholar; on the Ohm-Seebeck dispute, see Helmholtz, Hermann L. F., On the sensations of tone as a physiological basis for the theory of music, trans, by Ellis, Alexander J. (2nd edn., London, 1885), pp. 56–65.Google Scholar

² Also see Boring, , op. cit. (1), pp. 399–406.Google Scholar

³ See Aus Georg Simon Ohms handschriftlichem Nachlass, ed. Hartmann, Ludwig (Munich, 1927), pp. 137–87Google Scholar, and von Füchtbauer, Heinrich, Georg Simon Ohm. Ein Forscher wächst aus seiner Väter Art (Berlin, 1939), pp. 171–97.Google Scholar An excellent short account of Ohm's career is Caneva, Kenneth L., ‘Ohm, Georg Simon’, Dictionary of scientific biography (14 vols, to date, New York, 1970, in progress), x. 186–94.Google Scholar

⁴ Ohm, , ‘Bemerkungen über Combinationstöne und Stösse’, Annalen der Physik und Chemie, 2nd ser. xlvii (1839), 463–6.CrossRefGoogle Scholar The article is reprinted in Gesammelte Abhandlungen von Georg Simon Ohm, ed. Lommel, E. (Leipzig, 1892), pp. 573–6.Google Scholar All the following references to this article cite page numbers in Abhandlungen.

⁵ Annalen, 2nd ser. lix (1843), 513–65.Google Scholar This article is reprinted in Ohm, 's Abhandlungen, op. cit. (4), pp. 587–633.Google Scholar The following references to this article cite page numbers from the Abhandlungen. Sections of this article are translated in Lindsay, R. Bruce (ed.), Acoustics: historical and philosophical development (Stroundsburg, Penn., 1974), pp. 243–7.Google Scholar

⁶ On the siren, see Boring, , op. cit. (1), pp. 328–9Google Scholar, and Helmholtz, , Sensations of tone, op. cit. (1), pp. 11–13.Google Scholar A very brief account of Seebeck's career is ‘Seebeck, Ludwig Friedrich Wilhelm August’, Allgemeine Deutsche Biographie (mul. vols., Munich, 1891), xxxiii. 559–60.Google Scholar Seebeck's scientific publications are catalogued in Poggendorff, J. C. (ed.), Biographisch-literarisches Handwörterbuch zur Geschichte der exacten Wissenschaften (mul. vols., Leipzig, 1863), ii. 890–1.Google Scholar

⁷ See Dostrovsky, Sigalia, ‘The origins of vibration theory: the scientific revolution and the nature of music’ (Princeton University Ph.D. thesis, 1969)Google Scholar, and Lindsay, R. Bruce, ‘The story of acoustics’Google Scholar, in Lindsay, , Acoustics, op. cit. (5), pp. 8–10.Google Scholar

⁸ For example, see Young, Thomas, ‘Outlines of experiments and inquiries respecting sound and light’, The philosophical transactions of the Royal Society of London … abridged, xviii (1796–1800), 604–26, esp. pp. 616–26.Google Scholar

⁹ Ohm, , ‘Definition’, op. cit. (5), p. 588.Google Scholar Ohm is here quoting August Roeber from Repertorium der Physik, ed. Dove, H. W., iii (1839), 30–1.Google Scholar

¹⁰ Seebeck, ‘Boebachtungen über einige Bedingungen der Entstehung von Tönen’, Annalen, and ser. liii (1841), 423.Google Scholar

¹¹ Ohm, , ‘Definition’, op. cit. (5), pp. 591–2.Google Scholar Ohm also assumes that p remains constant. For brevity, I have followed Ohm, Seebeck, and Helmholtz in speaking of ‘a tone of frequency m’; what is meant, of course, is ‘a tone of such a pitch as is generated in the ear by a vibration of frequency m’.

¹² Ohm, , ‘Definition’, op. cit. (5), pp. 592–5.Google Scholar To facilitate the comparison of Ohm's and Seebeck's ideas, I have altered the original notation somewhat. Ohm's readiness to exploit the mathematical analysis of Fourier is the one theme of his galvanic researches still evident in the acoustical work.

¹³ These seem to have been characteristics of Ohm's scientific style; see Caneva's comments on Die galvanische Kette in ‘Ohm’, op. cit. (3), p. 190.Google Scholar

¹⁴ Ohm, , ‘Definition’, op. cit. (5), pp. 621–4.Google Scholar

¹⁵ Seebeck, ‘Ueber die Sirene’, Annalen, 2nd ser. lx (1843), 454–5.Google Scholar

¹⁶ Ibid. p. 456.

¹⁷ Ibid. pp. 466–7.

¹⁸ Ibid. p. 477.

¹⁹ Ibid. p. 474.

²⁰ Ibid. p. 480.

²¹ This letter is reprinted in Ohm, , Nachlass, op. cit. (3), pp. 188–90.Google Scholar Seebeck expressed to Ohm the hope that if he were ever in Dresden, ‘Sie sich nicht die Ohren vor den Gesängen meiner Sirene verstopfen werden’.

²² Ohm, , ‘Noch ein Paar Worte über die Definition des Tones’, Annalen, 2nd ser. lxii (1844), 1–18Google Scholar; all the following references are to the article as reprinted in Ohm, 's Abhandlungen, op. cit. (4), pp. 634–49.Google Scholar

²³ Ohm, , ‘Worte’, op. cit. (22), p. 639.Google Scholar

²⁴ Ibid., pp. 643–5. Seebeck's refutation appears in Seebeck, , ‘Ueber die Definition des Tones’, Annalen, 2nd ser. lxiii (1844), 357–8.Google Scholar

²⁵ Ohm, , ‘Worte’, op. cit. (22), pp. 645–6.Google Scholar

²⁶ Ibid., p. 646.

²⁷ Ibid., p. 647.

²⁸ Seebeck's later publications on this topic were as follows: (1) ‘Ueber die Definition des Tones’, op. cit. (24), pp. 353–67Google Scholar; (2) ‘Ueber die Erzeugung von Tönen durch getrennte Eindrücke, mit Beziehung auf die Definition des Tones’, Annalen, 2nd ser. lxiii (1844), 368–80.Google Scholar In this paper Seebeck examined the mathematical assumptions that must be made about the siren impulse if intense harmonics are to result, and he introduced new siren experiments to show that, no matter how the nature of the blast is altered, the harmonics remain almost inaudible; (3) ‘Akustik (1849)’, Repertorium der Physik, ed. Dove, W. H., viii (1849), 1–16.Google Scholar This article contains Seebeck's most mature statement of his position in the dispute.

²⁹ The issues did come more sharply into focus in one additional respect. By 1849 Seebeck had come to believe that the chief difference between him and Ohm involved the question of how the higher harmonics increase the intensity of the fundamental. On the other hand, Seebeck seemed to have been convinced by his own argument that the true harmonics are inaudible; in 1849 he returned to his original hypothesis that the harmonics actually heard arise through an interference effect. See Seebeck, , ‘Akustik (1849)’, op. cit. (28), pp. 14 and 10.Google Scholar

³⁰ Seebeck, , ‘Definition’, op. cit (24), pp. 364–5Google Scholar, and ‘Akustik (1849)’, op. cit. (28), pp. 14–15.Google Scholar In fact, Seebeck had argued that timbre is governed by the way in which the wave form departs from the pendular form even before Ohm's work appeared; see Seebeck, ‘Akustik (1842)’, Repertorium der Physik, ed. Dove, H. W., vi (1842), 6.Google Scholar

³¹ Seebeck, , ‘Definition’, op. cit. (24), pp. 360–1Google Scholar, and ‘Akustik (1849)’, op. cit. (28), pp. 13–14.Google Scholar

³² Seebeck, , ‘Definition’, op. cit. (24), p. 367Google Scholar, and ‘Akustik (1849)’, op. cit. (28), p. 16.Google ScholarOhm, 's charge cornes in ‘Worte’, op. cit. (22), p. 648.Google Scholar

³³ The standard source on Helmholtz's career is the biography by Koenigsberger, Leo, Hermann von Helmholtz (3 vols., Brunswick, 1902–1903)Google Scholar. An abridged translation by Welby, Frances A. is Hermann von Helmholtz (Oxford, 1906; New York, 1965)Google Scholar. For a shorter account, see Turner, R. Steven, ‘Helmholtz, Hermann von’, Dictionary of scientific biography, op. cit. (3), vi. 241–53.Google Scholar

³⁴ Koenigsberger, , op. cit. (33), i. 267–8.Google Scholar

³⁵ Annalen, 2nd ser. xcix (1856), 497–540.Google Scholar The article is reprinted in Wissenschaftliche Abhandlungen von Hermann Helmholtz (3 vols., Leipzig, 1882–1895), i. 263–302.Google Scholar The following references are to the page numbers in the Annalen. Short notes summarizing the contents of this paper were ‘Ueber die Combinationstöne oder Tartinischen Töne’, Niederrheinischen Sitzungsberichten. (Verhandlungen des naturhist. Vereins von Rheinland und Westphalen), xiii (1856), pp. lxxv–lxxviiGoogle Scholar, and ‘Ueber Combinationstöne’, Monatsbericht der königl. Akademie der Wissenschaften zu Berlin (22 05 1856), pp. 279–85.Google Scholar These articles are reprinted in Helmholtz, , Abhandlungen, i. 256–62, and iii. 7.Google Scholar

³⁶ The early history of investigations into combination tones seems never to have been adequately researched. See Boring, , op. cit. (1), pp. 352–6 and 392–3Google Scholar, and the historical notes in the primary sources cited below.

³⁷ This is an obvious mathematical result if the combination tone is regarded simply as the sum of the generators, i.e. as a1 sin 2π nft + a2 sin 2π mft. Experimentation on combination tones often supported this theoretical expectation of a combination tone of frequency f. Experimenters normally produced the generating frequencies on musical instruments and used common intervals like the fifth (2:3), the fourth (3:4), the major third (4:5), and the major whole tone (8:9). These did indeed give a combination tone equalling the greatest common divisor of the generating frequencies. This experimentation was extremely subjective, and anomalous intervals like the major sixth (3:5) were easily overlooked. This kind of combination tone effect is obviously that to which Seebeck referred in his hypothesis that harmonic overtones reinforce the intensity of the fundamental pitch.

³⁸ Hällström, , ‘Von den Combinationstönen’, Annalen, 2nd ser. xxiv (1832), 438–66CrossRefGoogle Scholar; also see the comments on this paper by Seebeck, in ‘Akustik (1842)’, op. cit. (30), pp. 90–3.Google Scholar

³⁹ For the more complex case of differing amplitudes and phase differences, see Wood, Alexander, Acoustics (London, 1940), pp. 195–6.Google Scholar As expression (5) shows, the beat theory strictly predicts a combination tone of frequency (ω1-ω2)/2, not one of frequency (ω1-ω2). This anomaly was much discussed but usually dismissed as resulting from different conventions about how to count vibrations. See Roeber, August, ‘Untersuchungen des Herrn. Scheibler in Crefeld über die sogenannten Schläge, Schwebungen oder Stösse’, Annalen, 2nd ser. xxxii (1834), 342–52Google Scholar, and Poggendorff, J. C., ‘Zusatz des Herausgebers’, Annalen, 2nd ser. xxxii (1834), 521–2.Google Scholar

⁴⁰ See the papers cited in notes 38 and 39; also Weber, Wilhelm, ‘Ueber die Tartinischen Töne’, Annalen, 2nd ser. xv (1829), 216–22CrossRefGoogle Scholar, and Roeber, , ‘Akustik (1839)’, op. cit. (9), pp. 6–54, 404–5.Google Scholar

⁴¹ Roeber, , ‘Scheibler’, op. cit. (39), pp. 492–528.Google Scholar

⁴² Roeber, , ‘Akustik (1839)’, op. cit. (9), pp. 30–1.Google Scholar

⁴³ Ohm, , ‘Bemerkungen’, op. cit. (4), pp. 573–6.Google Scholar Ohm may have omitted his calculations and assumptions because he planned to publish a longer paper on the topic that never materialized. Seebeck seemed to expect the appearance of such a paper in ‘Definition’, op. cit. (24), p. 367Google Scholar, and may have learned of Ohm's intention through private correspondence.

⁴⁴ Ohm, , ‘Bemerkungen’, op. cit. (4), pp. 574–5.Google Scholar In one short paper Helmholtz falsely attributes this idea to Poggendorff; see Helmholtz, , Abhandlungen, op. cit. (35), i. 256Google Scholar, and Poggendorff, , op. cit. (39), pp. 520–8.Google Scholar

⁴⁵ Evidence for this conclusion is found in the excerpts from letters in Koenigsberger, , op. cit. (33), i. 267–8Google Scholar, and in Helmholtz's procedure as described in ‘Combinationstöne’, op. cit. (35).

⁴⁶ Helmholtz, , ‘Combinationstöne’, op. cit. (35), pp. 502–10.Google Scholar

⁴⁷ This research also produced other results. Helmholtz confirmed Scheibler's result that the beats or higher-order combination tones can be heard even when the tones themselves cannot be. He discovered that, contrary to theory, the tuning fork possesses one harmonic overtone— the octave of its fundamental. He attributed this anomaly to a failure of the approximation implicit in the principle of undisturbed superposition, a result that anticipated the transformation theory. In a short note (but not in his main publication), Helmholtz made much of the fact that his results confirmed Hällström's formula for the frequency of first-order combination tones; but by 1856 this could hardly have been in doubt. See Helmholtz, , Abhandlungen, op. cit. (35), iii. 7.Google Scholar

⁴⁸ Helmholtz, , ‘Combinationstöne’, op. cit. (35), p. 523.Google Scholar

⁴⁹ Ibid., pp. 525–6.

⁵⁰ Helmholtz, , Sensations of tone, op. cit. (1), pp. 103–19, 179–97.Google Scholar

⁵¹ Helmholtz, , ‘Combinationstöne’, op. cit. (35), pp. 528–30.Google Scholar

⁵² Seebeck, , ‘Definition’, op. cit. (24), pp. 365–6Google Scholar; Seebeck attributes this objection to Wilhelm Weber.

⁵³ Helmholtz, , ‘Combinationstöne’, op. cit. (35), p. 528.Google Scholar

⁵⁴ Ibid., pp. 531–2.

⁵⁵ Ibid., pp. 518–21, 532–6.

⁵⁶ Ibid., pp. 536–40.

⁵⁷ See Helmholtz, , Sensations of tone, op. cit. (1), pp. 527–38Google Scholar (Addition by the translator [1885]: ‘Recent work on beats and combination tones’). Also see Strutt, John William, Rayleigh, Baron, The theory of sound (2nd edn., 2 vols, in 1, New York, 1945), ii. 432–78.Google Scholar

⁵⁸ On Helmholtz's philosophical ideas, see Erdmann, Benno, ‘Die philosophischen Grundlagen von Helmholtz' Wahrnehmungstheorie, kritisch erläutert’, Abhandlungen der preussischen Akademie der Wissenschaften. Philosophisch-historische Klasse, Jahrgang 1921, Nr. 1 (Berlin, 1921), 1–45.Google Scholar A short English discussion is Lenzen, Victor F., ‘Helmholtz's theory of knowledge’, Studies and essays in the history of science and learning offered in homage to George Sarton, ed. Montagu, M. F. Ashley (New York, 1947), pp. 299–320.Google Scholar

⁵⁹ The most important early lectures are ‘Ueber die Natur der menschlichen Sinnesempfindungen (Königsberg, 1852)’, in Helmholtz, , Abhandlugen, op. cit. (35), ii. 591–609Google Scholar; ‘Ueber Goethes naturwissenschaftliche Arbeiten (Königsberg, 1853)’, translated in Selected writings of Hermann von Helmholtz, ed. Kahl, Russell (Middletown, Connecticut, 1971), pp. 56–74Google Scholar; and ‘Ueber das Sehen des Menschen (Königsberg, 1855)’, Vorträge und Reden (2 vols., Braunschweig, 1884)Google Scholar. Helmholtz's most comprehensive Statement of his epistemology is ‘Die Thatsachen in der Wahrnehmung (Berlin, 1879)’, translated in Selected writings, ed. Kahl, pp. 366–408.

⁶⁰ Helmholtz, , ‘Ueber die Natur der menschlichen Sinnesempfindungen’, in Abhandlungen, op. cit. (35), pp. 607–9Google Scholar; ‘Recent progress in the theory of vision’, in Selected writings, op. cit. (59), pp. 216–20Google Scholar; ‘The physiological causes of harmony in music’, ibid., pp. 97–9.

⁶¹ Helmholtz, , ‘Combinationstöne’, op. cit. (35), p. 525.Google Scholar

⁶² Ibid., pp. 525–6. Cf. the discussion in Helmholtz, , Sensations of tone, op. cit. (1), pp. 58–65.Google Scholar

⁶³ See my forthcoming article, ‘Hermann von Helmholtz and the empiricist vision’, in Journal of the history of the behavioral sciences.