The term ‘Ohm's law’ traditionally denotes the formula of Georg Simon Ohm relating voltage, current, and resistance in metallic conductors. But to students of sensory physiology and its history, ‘Ohm's law’ also denotes another relationship: the fundamental principle of auditory perception that Ohm announced in 1843. This aspect of Ohm's science has attracted very little attention, partly because his galvanic researches so thoroughly eclipsed it in success and importance, and partly because Ohm's work in physiological acoustics had so little immediate impact on the science of his time. On announcing his hypothesis in 1843, Ohm found himself drawn into a bitter dispute with the physicist August Seebeck, who successfully discredited the hypothesis and forced Ohm to withdraw from the field.
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. 321–9; 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.
2 Also see Boring, , op. cit. (1), pp. 399–406.
3 See Aus Georg Simon Ohms handschriftlichem Nachlass, ed. Hartmann, Ludwig (Munich, 1927), pp. 137–87, and von Füchtbauer, Heinrich, Georg Simon Ohm. Ein Forscher wächst aus seiner Väter Art (Berlin, 1939), pp. 171–97. 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.
4 Ohm, , ‘Bemerkungen über Combinationstöne und Stösse’, Annalen der Physik und Chemie, 2nd ser. xlvii (1839), 463–6. The article is reprinted in Gesammelte Abhandlungen von Georg Simon Ohm, ed. Lommel, E. (Leipzig, 1892), pp. 573–6. All the following references to this article cite page numbers in Abhandlungen.
5 Annalen, 2nd ser. lix (1843), 513–65. This article is reprinted in Ohm, 's Abhandlungen, op. cit. (4), pp. 587–633. 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.
6 On the siren, see Boring, , op. cit. (1), pp. 328–9, and Helmholtz, , Sensations of tone, op. cit. (1), pp. 11–13. A very brief account of Seebeck's career is ‘Seebeck, Ludwig Friedrich Wilhelm August’, Allgemeine Deutsche Biographie (mul. vols., Munich, 1891), xxxiii. 559–60. 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.
7 See Dostrovsky, Sigalia, ‘The origins of vibration theory: the scientific revolution and the nature of music’ (Princeton University Ph.D. thesis, 1969), and Lindsay, R. Bruce, ‘The story of acoustics’, in Lindsay, , Acoustics, op. cit. (5), pp. 8–10.
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.
9 Ohm, , ‘Definition’, op. cit. (5), p. 588. Ohm is here quoting August Roeber from Repertorium der Physik, ed. Dove, H. W., iii (1839), 30–1.
10 Seebeck, ‘Boebachtungen über einige Bedingungen der Entstehung von Tönen’, Annalen, and ser. liii (1841), 423.
11 Ohm, , ‘Definition’, op. cit. (5), pp. 591–2. 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. 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.
14 Ohm, , ‘Definition’, op. cit. (5), pp. 621–4.
15 Seebeck, ‘Ueber die Sirene’, Annalen, 2nd ser. lx (1843), 454–5.
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. 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), 1–18; all the following references are to the article as reprinted in Ohm, 's Abhandlungen, op. cit. (4), pp. 634–49.
23 Ohm, , ‘Worte’, op. cit. (22), p. 639.
24 Ibid., pp. 643–5. Seebeck's refutation appears in Seebeck, , ‘Ueber die Definition des Tones’, Annalen, 2nd ser. lxiii (1844), 357–8.
25 Ohm, , ‘Worte’, op. cit. (22), pp. 645–6.
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. 353–67; (2) ‘Ueber die Erzeugung von Tönen durch getrennte Eindrücke, mit Beziehung auf die Definition des Tones’, Annalen, 2nd ser. lxiii (1844), 368–80. 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. 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.
30 Seebeck, , ‘Definition’, op. cit (24), pp. 364–5, and ‘Akustik (1849)’, op. cit. (28), pp. 14–15. 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.
31 Seebeck, , ‘Definition’, op. cit. (24), pp. 360–1, and ‘Akustik (1849)’, op. cit. (28), pp. 13–14.
32 Seebeck, , ‘Definition’, op. cit. (24), p. 367, and ‘Akustik (1849)’, op. cit. (28), p. 16.Ohm, 's charge cornes in ‘Worte’, op. cit. (22), p. 648.
33 The standard source on Helmholtz's career is the biography by Koenigsberger, Leo, Hermann von Helmholtz (3 vols., Brunswick, 1902–1903). An abridged translation by Welby, Frances A. is Hermann von Helmholtz (Oxford, 1906; New York, 1965). For a shorter account, see Turner, R. Steven, ‘Helmholtz, Hermann von’, Dictionary of scientific biography, op. cit. (3), vi. 241–53.
34 Koenigsberger, , op. cit. (33), i. 267–8.
35 Annalen, 2nd ser. xcix (1856), 497–540. The article is reprinted in Wissenschaftliche Abhandlungen von Hermann Helmholtz (3 vols., Leipzig, 1882–1895), i. 263–302. 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–lxxvii, and ‘Ueber Combinationstöne’, Monatsbericht der königl. Akademie der Wissenschaften zu Berlin (22 05 1856), pp. 279–85. These articles are reprinted in Helmholtz, , Abhandlungen, i. 256–62, and iii. 7.
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 392–3, 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), 438–66; also see the comments on this paper by Seebeck, in ‘Akustik (1842)’, op. cit. (30), pp. 90–3.
39 For the more complex case of differing amplitudes and phase differences, see Wood, Alexander, Acoustics (London, 1940), pp. 195–6. 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), 342–52, and Poggendorff, J. C., ‘Zusatz des Herausgebers’, Annalen, 2nd ser. xxxii (1834), 521–2.
40 See the papers cited in notes 38 and 39; also Weber, Wilhelm, ‘Ueber die Tartinischen Töne’, Annalen, 2nd ser. xv (1829), 216–22, and Roeber, , ‘Akustik (1839)’, op. cit. (9), pp. 6–54, 404–5.
41 Roeber, , ‘Scheibler’, op. cit. (39), pp. 492–528.
42 Roeber, , ‘Akustik (1839)’, op. cit. (9), pp. 30–1.
43 Ohm, , ‘Bemerkungen’, op. cit. (4), pp. 573–6. 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. 367, and may have learned of Ohm's intention through private correspondence.
44 Ohm, , ‘Bemerkungen’, op. cit. (4), pp. 574–5. In one short paper Helmholtz falsely attributes this idea to Poggendorff; see Helmholtz, , Abhandlungen, op. cit. (35), i. 256, and Poggendorff, , op. cit. (39), pp. 520–8.
45 Evidence for this conclusion is found in the excerpts from letters in Koenigsberger, , op. cit. (33), i. 267–8, and in Helmholtz's procedure as described in ‘Combinationstöne’, op. cit. (35).
46 Helmholtz, , ‘Combinationstöne’, op. cit. (35), pp. 502–10.
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.
48 Helmholtz, , ‘Combinationstöne’, op. cit. (35), p. 523.
49 Ibid., pp. 525–6.
50 Helmholtz, , Sensations of tone, op. cit. (1), pp. 103–19, 179–97.
51 Helmholtz, , ‘Combinationstöne’, op. cit. (35), pp. 528–30.
52 Seebeck, , ‘Definition’, op. cit. (24), pp. 365–6; Seebeck attributes this objection to Wilhelm Weber.
53 Helmholtz, , ‘Combinationstöne’, op. cit. (35), p. 528.
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. 527–38 (Addition by the translator : ‘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.
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. 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.
59 The most important early lectures are ‘Ueber die Natur der menschlichen Sinnesempfindungen (Königsberg, 1852)’, in Helmholtz, , Abhandlugen, op. cit. (35), ii. 591–609; ‘Ueber Goethes naturwissenschaftliche Arbeiten (Königsberg, 1853)’, translated in Selected writings of Hermann von Helmholtz, ed. Kahl, Russell (Middletown, Connecticut, 1971), pp. 56–74; and ‘Ueber das Sehen des Menschen (Königsberg, 1855)’, Vorträge und Reden (2 vols., Braunschweig, 1884). 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. 607–9; ‘Recent progress in the theory of vision’, in Selected writings, op. cit. (59), pp. 216–20; ‘The physiological causes of harmony in music’, ibid., pp. 97–9.
61 Helmholtz, , ‘Combinationstöne’, op. cit. (35), p. 525.
62 Ibid., pp. 525–6. Cf. the discussion in Helmholtz, , Sensations of tone, op. cit. (1), pp. 58–65.
63 See my forthcoming article, ‘Hermann von Helmholtz and the empiricist vision’, in Journal of the history of the behavioral sciences.
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