Instruments of Music, Instruments of Science: Hermann von Helmholtz's Musical Practices, his Classicism, and his Beethoven Sonata

Summary

The young Hermann Helmholtz, in an 1838 letter home, declared that he always appreciated music much more when he played it for himself. Though a frequent concert-goer, and celebrated for his highly influential 1863 work on the physiological basis of music theory, Die Lehre von den Tonempfindungen, it is likely that Helmholtz's enduring engagement with music began with his initial, personal experience of playing music for himself. I develop this idea, shifting the discussion of Helmholtz's work on sound sensation back to its origins, and examine the role of his material interaction with musical instruments and music itself. In his sound sensation studies, Helmholtz understood sound as an external, physical object. But Helmholtz also conceived of sound in musical terms. Further, Helmholtz's particular musical tastes as well as his deeply personal interaction with musical instruments allowed him to reconcile his conception of sound as physical object with his conception of sound as music. Helmholtz's physiological theory of sound sensation was both the product of and constitutive of how he heard and created sound. I argue that Helmholtz himself was the embodied reconciliation of his physiological theory of sound sensation and his belief that musical aesthetics were historically and culturally contingent.

Acknowledgements

I would like to thank M. Norton Wise, Ted Porter, Mitchell Morris, Myles Jackson, Julia Kursell, and David Cahan for comments on earlier versions of this paper. My research for this article was funded by the Deutscher Akademischer Austauschdienst (DAAD), supervised by Lorraine Daston at the Max Planck Institute for the History of Science in Berlin, Germany.

Notes

¹‘…ich habe immer weit mehr Vergnügen an der Musik, wenn ich sie selbst ausführe’. Hermann Helmholtz, November 5, 1838 letter (no. 6), Letters of Hermann von Helmholtz to his Parents: the Medical Education of a German Scientist, 1837–1846, edited by David Cahan (Franz Steiner Verlag, 1993), 51.

²The 1791 introduction of the single-piece cast iron frame marked the beginning of a century of innovation. The flurry of design innovation was fueled in part by a growing and robust piano market that mirrored economic growth and stability, especially for the middle classes.

³Hermann Helmholtz, November 5, 1838 letter (no. 6), Letters of Hermann von Helmholtz to his Parents, 51.

⁴Some very good historical work has explored the development and reception of Helmholtz's On the Sensations of Tone; Erwin and Elfrieda Hiebert present particularly useful analyses. Other scholars have traced the link between Helmholtz's effort to bridge the natural and moral sciences fueled by his belief in an inductive foundation rooted in empiricism. But all of these studies begin with Helmholtz's physics and physiology and then proceed to examine how his ideas related to and were received by contemporary musicians and music theorists. Current scholars have reduced his work with instruments to purely acoustic exercises and treated the piano largely as a machine. See Gary Hatfield, ‘Helmholtz and Classicism: The Science of Aesthetics and the Aesthetics of Science’, Hermann von Helmholtz and the Foundations of Nineteenth-Century Science, edited by David Cahan (California University Press, 1993); Elfrieda and Erwin Hiebert, ‘Musical Thought and Practice: Links to Helmholtz's Tonempfindungen’, Universalgenie Helmholtz: Rückblick nach 100 Jahren, edited by Lorenz Krüger, (Akademie Verlag, 1994); Elfrieda Hiebert, ‘Helmholtz's Musical Acoustics: Incentive for Practical Techniques in Pedaling and Touch at the Piano’, IMS Intercongressional Symposium, Budapest & Visegrad, (August 2004), 425–33; Edward Jurkowitz, ‘Helmholtz and the Liberal Unification of Science’, Historical Studies in the Physical Science, 32 (2003), 291–317.

⁵M. Norton Wise, ‘Making Visible’, Isis, 97 (2006), 75–82.

⁶Most recent scholarship has, however, focused on the visual sense only. Norton Wise argues that to make things visible is to make them real and therefore encourages us to recognize that visualization in science is not merely illustration but can be argument itself.¹ Pamela Smith presents the idea of fifteenth- and sixteenth-century artisanal knowledge in which image-making is knowledge-producing and further, in cases where images were imitations of the processes of nature, that visual art and science can be regarded as sharing the same goals. See M. Norton Wise, ‘Making Visible’, Isis, 97 (2006), 75–82; Peter Galison, Image and Logic: A Material Culture of Microphysics (The Chicago University Press, 1997); Pamela Smith, ‘Art, Science, and Visual Culture in Early Modern Europe’, Isis, 97 (2006), 83–100. See also Pamela Smith's text, The Body of the Artisan: Art and Experience in the Scientific Revolution (The Chicago University Press, 2004).

⁷Frederic Holmes and Kathryn Olesko, ‘The Images of Precision: Helmholtz and the Graphical Method in Physiology’, The Values of Precision, edited by M. Norton Wise (Princeton University Press, 1995).

⁸Hatfield also locates romantic themes in Helmholtz's work. Helmholtz acknowledged that imagination played the dominant role in artistic and aesthetic cognition. But Hatfield maintains that these romantic themes were secondary to the classicist ones. Imagination, for Helmholtz, was merely a filter for the law-like regularity of an ideal type. S ee Gary Hatfield, ‘Helmholtz and Classicism: The Science of Aesthetics and the Aesthetics of Science’, 524.

⁹M. Norton Wise, ‘A Spectacle for the Gods’, Bourgeois Berlin and Laboratory Science (in manuscript).

¹⁰Gary Hatfied, The Natural and the Normative: Theories of Spatial Perception from Kant to Helmholtz (MIT Press, 1990); Michael Heidelberger, ‘Force, Law, and Experiment: The Evolution of Helmholtz's Philosophy of Science’, Hermann von Helmholtz and the Foundations of Nineteenth-Century Science, edited by David Cahan (University of California Press, 1993); R. Steven Turner, In the Eye's Mind: Vision and the Helmholtz-Hering Controversy (Princeton University Press, 1994); R. Steven Turner, ‘Consensus and Controversy: Helmholtz on the Visual Perception of Space’, Hermann von Helmholtz and the Foundations of Nineteenth-Century Science, edited by David Cahan (University of California Press, 1993).

¹¹Timothy Lenoir, ‘The Politics of Vision: Optics, Painting, and Ideology in Germany, 1845–1895’, Instituting Science: The Cultural Production of Scientific Disciplines (Stanford University Press, 1997).

¹³Ferdinand Helmholtz, November 2, 1838 letter, Letters of Hermann von Helmholtz to his Parents, no. 2 in Cahan; English translation from translated edition of Leo Koenigsberger's Hermann von Helmholtz, by Leo Koenigsberger, translated by Frances A. Welby (Clarendon Press, 1906), 14. This belief in music as national cultural resource, as central to the moral development and discipline of the children of the Bildungsbürgertum was particularly prevalent throughout the first half of the nineteenth-century. See Myles Jackson, Harmonious Triads: Physicists, Musicians, and Instrument Makers in Nineteenth-Century Germany (The MIT Press, 2006), 245–8.

¹²Helmholtz's father, Ferdinand, at the time the subrector of the Potsdam Gymnasium, had published an article in 1837 in which he explained that it was through careful, practiced drawing that the power of the Idea was awakened and expressed, that the Gymnasium curriculum should be structured around this gradual acquisition and understanding of the beautiful. Ferdinand Helmholtz, ‘Die Wichtigkeit der allgemeine Erziehung für das Schöne’, Zu der öffentlichen Prüfung der Zöglinge des hiesigen Königlichen Gymnasiums den 21sten und 22sten März laden ganz ergebenst ein Director und Lehrercollegium (Decker'schen Geheimen Oberhofbuchdruckerei-Establissement, 1837), 1–44.

¹⁴Hermann Helmholtz, November 5, 1838 letter (no. 6), Letters of Hermann von Helmholtz to his Parents, 51.

¹⁵Hermann Helmholtz, December 1, 1838 letter (no. 7) Letters of Hermann von Helmholtz to his Parents, 53.

¹⁶‘An den beiden ersten Tagen musste ich mich mit Onkels Noten begnügen, die außer einer Mozartschen Sonate aus lauter für trefflichen Werken von Strauss, Lanner, Czerny, Hünten, Auber, Roß und Belllini etc. etc. etc. etc. bestanden; ich jagte sie alle hintereinander durch, wurde zuletzt aber ganz schlimm davon, dass ich immer wieder zu der Mozartschen Sonate und Cramers Etüden retirirte, um meinen geistigen Magen wieder etwas zu stärken. Am Mittwoch fand ich bei Schmidts noch eine zweite Mozartsche Sonate, und gestern traf mich Wiggers hier beim Spielen, ich erzählte ihm meine Noth, und er sagte ich sollte nur mit ihm kommen, er würde mir Noten geben. Da hat er mir denn einen Haufen aufgepackt unter dessen Last ich fast zusammengesunken bin (Poetische Hyperbel) aber das meiste sind Clavierconcerte und andre Künstlichkeiten von Weber, Moscheles, Kramer etc. nur ein Heft Mozartscher und eine Beethovensche Sonate sind darunter’. Hermann Helmholtz, August 23, 1839 letter (no. 13), Letters of Hermann von Helmholtz to his Parents, 68.

¹⁷Hermann Helmholtz, ‘On the Motion of the Strings of a Violin,’ Proceedings of the Philosophical Society of Glasgow, 1860.

¹⁸Hermann Helmholtz, On the Sensations of Tone as a Physiological Basis for the Theory of Music, 2^nd English edition, translated, revised and corrected from 4^th German edition, with additional notes and appendix, by Alexander Ellis (Dover Publications, 1954), 74.

¹⁹Hermann Helmholtz, ‘The Aim and Progress of Physical Science: An Opening Address delivered at the Naturforscher Versammlung, in Innsbrück’, 1869, republished in Popular Lectures on Scientific Subjects, translated by E. Atkinson (Longmans, Green, and Co., 1904), 321.

²⁰Hermann Helmholtz, Hermann von Helmholtz, By Leo Koenigsberger, 232–3. Helmholtz is referring to Beethoven's ballet Prometheus. The Mendelssohn Overture is likely the Opus 21 overture to his Midsummer's Night Dream. The concert also included a Bach chorus as well as Handel's Hallelujah Chorus.

²¹‘…aber diese erhöhnte theoretische Genauigkeit wäre practisch illusorisch, da schon jetzt der ganze Fehler von 1/885 bei den Quinten an der Grenze dessen liegt, was ein geübtes musikalisches Ohr unterscheiden kann’. Hermann Helmholtz, ‘Über musikalische Temperatur’, Naturh.-med. Verien in Heidelberg, (1860), 75.

²²Carl Dahlhaus, Nineteenth-Century Music, translated by J. Bradford Robinson (University of California Press, 1989), 9–15. Dahlhaus includes a very nice supportive analysis of Beethoven's Sonata in D minor.

²³Adolf Marx, ‘Die Form in Musik’, Die Wissenschaften in neunzehnten Jahrhundert, edited by J. A. Romberg, vol. 2 (Romberg's Verlag, 1856), republished and translated as ‘Form in Music’, Musical Form in the Age of Beethoven: Selected Writings on Theory and Method, ed. and tr. Scott Burnham (Cambridge University Press, 2006), 61.

²⁴Scott Burnham, ‘The role of sonata form in A. B. Marx's theory of form,’ Journal of Music Theory, 22, no. 2 (1989), 264–5.

²⁵Eduard Hanslick, Vom Musikalisch-Schönen: Ein Beitrag zur Revision der Asthetik der Tonkunst, 12th edition (Breitkopf Härtel, 1918), 59.

²⁶Eduard Hanslick, Vom Musikalisch-Schönen: Ein Beitrag zur Revision der Asthetik der Tonkunst, 12th edition (Breitkopf Härtel, 1918), 62.

²⁷Carl Dahlhaus, Nineteenth-Century Music, 50.

²⁸William Weber, Music and the Middle Class: The Social Structure of Concert Life in London, Paris and Vienna (Holmes and Meier Publishers, New York, 1975). Dahlhaus argues that the dichotomy between Italo-French opera and German classical instrumental music (reified in the contrasting of Rossini and Beethoven) was at the very roots of the nineteenth-century concept of music. Carl Dahlhaus, Nineteenth-Century Music, 9–11.

²⁹William Weber, Music and the Middle Class, 19–20.

³⁰Myles Jackson, Harmonious Triads, 232–3. See also Carl Dahlhaus, Nineteenth-Century Music, 50.

³¹Celia Applegate and Pamela Potter, ‘Germans as the ‘People of Music’: Genealogy of an Identity’, Music and German National Identity, edited by Celia Applegate and Pamela Potter (The University of Chicago Press, 2002). See also Celia Applegate's Bach in Berlin: Nation and Culture in Mendelssohn's Revival of the St. Matthew Passion, (Cornell University Press, 2005) in which she argues that the 1829 performance of St. Matthew Passion in Berlin was a moment of consolidation, transformation, and self-realization in collective life that suggested such moments were perhaps only possible through musical performance.

³²Sanna Pederson, ‘A. B. Marx, Berlin Concert Life and German National Identity’, 19 ^th -Century Music, xvii (1994–5), 88.

³³Believing that painting or sculpture borrowed material from human experience and usually attempted to describe or imitate nature, Helmholtz thought that they could also be investigated through scientific art-criticism in respect to their correctness and truth. Helmholtz's primacy of music over all other forms of art was not uncommon and was in fact the norm for the time period. Schopenhauer's widely read and embraced text, Die Welt als Wille und Vorstellung, for one, asserted that individual s could access their inner selves, their will, through introspection. This introspection was achieved through art. One could only know the world through art. Thus music, the purest form of art, was the world.

³⁴Hermann Helmholtz, ‘Ueber die physiologischen Ursachen der musikalischen Harmonie’, 1857 Bonn lecture, republished in Vorträge und Reden, 1, 199–155. English translation from ‘On the Physiological Causes of Harmony in Music’, Popular Lectures on Scientific Subjects (Longmans, Green, and Co., 1903), 53–4.

³⁵Hermann Helmholtz, On the Sensations of Tone, 182.

³⁶‘Just as in the rolling ocean, this movement, rhythmically repeated, and yet ever varying, rivets our attention and hurries us along. But whereas in the sea, blind physical forces alone are at work, and hence the final impression on the spectator's mind is nothing but solitude—in a musical work of art the movement follows the outflow of the artist's own emotions. Now gently gliding, now gracefully leaping, now violently stirred, penetrated or laboriously contending with the natural expression of passion, the stream of sound, in primitive vivacity, bears over into the hearer's soul unimagined moods which the artist has overheard from his own, and finally raises him up to that repose of everlasting beauty, of which God has allowed but few of his elect favorites to be the heralds’. Hermann Helmholtz, ‘On the Physiological Causes of Harmony in Music’, 93.

³⁷‘Viele, viele Grüsse nach Dahlem an mein kleines Fräulein, welches ich so sehr lieb habe, so sehr lieb, wie der Himmel hoch und das Meer tief ist, und welches ich gar nicht aufhören kann zu lieben, auch wenn Himmel und Meer längst untergegangen sein sollten; an mein liebes Fräulein Nachtigall mit den süßen und tiefen Gesängen aus der Alceste und den zierlichen weißen Händchen, mit denen es sich so hübsch spazieren promenieren lässt in großen und schönen Beethovenschen und anderen Hainen und Gärten. So viele Grüsse von dem glücklichen Menschen, dem der Himmel ein solches Bräutlein durch das ganze Leben zu behüten und zu lieben gegeben hat’. Hermann Helmholtz, June 20, 1848 letter (no. 16), Letters of Hermann von Helmholtz to his wife, 1847–1859, edited by Richard L. Kremer (Franz Steiner Verlag, 1990), 41. Alceste, Chrisoph Martin Wieland's text set to opera by Anton Scweitzer in 1774. Alceste is sometimes classified as Singspiel though the subject is serious and the work features only recitatives rather than spoken dialogue. It was the first serious opera to be set in German and was successful enough to be performed as a repertoire piece.

³⁸Hermann Helmholtz, Hermann von Helmholtz, by Leo Koenigsberger, 226.

³⁹Hermann Helmholtz, Hermann von Helmholtz, by Leo Koenigsberger, 226.

⁴⁰Helmholtz claimed that he did not usually like lieder at all, though he made an exception for Frau Richelot's performance of Beethoven's ‘Adelaide’, which he described as a ‘water cure’ (Brunnenkur): ‘Am Freitag Abend war ich noch bei Richelots, Frau Richelot sang Lieder von Beethoven zum Theil sehr gut. ‘Trocknet nicht Thränen der ewigen Liebe’ wäre gerade für Dich etwas nach der Brunnenkur zu singen; selbst Adelaide, die ich sonst nicht leiden mochte, trug sie so vor, dass es sich machte. Die mathematische Stelle in Berlin ist durch einen Breslauer besetzt, so dass Richelot hier bleibt’. Hermann Helmholtz, July 26, 1855 letter (no. 37), Letters of Hermann von Helmholtz to his wife, 1847–1859, 150.

⁴¹Carl Dahlhaus, Nineteenth-Century Music, 22. Dahlhaus explains that Palestrina was considered exemplary of Catholic church music, Bach for Protestant church music, Handel for the oratorio, Gluck for musical tragedy, Mozart for opera buffa, Haydn for the string quartet, Beethoven for the symphony, and Schubert for the lied.

⁴²See blocked quote on page 8. Hermann Helmholtz, Hermann von Helmholtz, By Leo Koenigsberger, 232–3.

⁴³‘Nachmittags wechselte ich Geld, hörte in einem Kaffeehause am Main ein Concert der östreichischen Regimentmusik, und ging dann in das Theater, wo ich erstens Dr. Robin sehr gut dargestellt sah, von einem neuen jungen Devrient, so gut, dass ich es aufgebe, die Rolle je zu spielen. Dann war nach einem meist langweiligen Concerte verschiedener Instrumental- und Vocalsolisten, worin zu meiner Verwunderung auch Fräulein Bywater aus Königsberg durch mässig schlecht gesungene Koloraturarien die Frankfurter sehr entzückte, ein höchst großartiges Fragment aus einer hinterlassenen Oper von Mendelssohn, Lorelei, was auch von Frau Anschütz gut dargestellt wurde’. Hermann Helmholtz, August 18, 1853 letter (no. 28), Letters of Hermann von Helmholtz to his wife, 1847–1859, 106.

⁴⁴See blocked quote on page 12. Hermann Helmholtz, Hermann von Helmholtz, by Leo Koenigsberger, 226.

⁴⁵‘Ihr kamt nicht—da war es denn auch mit meinem Hören schlecht bestellt. Es war mir als hätte bisher nur immer Deine Seele, mit ihrer tief musikalischen Innerlichkeit die Harmonien in mein Verständniss hinein geleitet. Meine Ohren hörten nur Musikalische Figuren und meine Seele hörte gar nichts. Natürlich war es die Mozart'sche Symphonie, bei der es mir so ging, eine der schönsten von ihm, über die alle um mich her in Entzücken schwammen. Ich, wie ich da war, vereinsamt, verlassen von der schöneren Hälfte meiner Seele, hätte ebenso gut können Skalen auf dem Clavier spielen hören’. Quoted in appended biography of Olga and Hermann Helmholtz, written circa 1902 by Betty Johannes [Olga Helmholtz's sister], likely in response to a request by Leo Koenigsberger, Letters of Hermann von Helmholtz to his wife, 1847–1859, 195.

⁴⁶Quoted in appended biography of Olga and Hermann Helmholtz, written circa 1902 by Betty Johannes [Olga Helmholtz's sister], likely in response to a request by Leo Koenigsberger, Letters of Hermann von Helmholtz to his wife, 1847–1859, 195.

⁴⁷See blocked quote on page 12. Hermann Helmholtz, Hermann von Helmholtz, by Leo Koenigsberger, 226.

⁴⁸‘Erst bei der Coriolan Overtüre (Beethoven's Overture to Collin's Coriolan) kam ich wieder zu mir—das ist ein Juwel, so kurz, bündig, so entschieden und stolz zwischen einer Menge von Unruhe und wirren Kämpfen und stirbt zuletzt so traurig in ein Paar melancholisten Tönen—ein Meisterwerk wie es nicht grösser sein kann’. Quoted in appended biography of Olga and Hermann Helmholtz, written in 1902 by Betty Helmholtz, Letters of Hermann von Helmholtz to his wife, 1847–1859, 195–6.

⁴⁹‘My room-fellow is the son of a Silesian engineer…. He has extraordinary execution on the piano, but only cares for florid pieces and for modern Italian music’. Hermann Helmholtz, Hermann von Helmholtz, by Leo Koenigsberger, 13–4.

⁵⁰‘We went to a smoking concert, i.e. to the rehearsal for the great orchestral concerts, where they give certain soli that are left out in the concert proper, and which the gentlemen of Utrecht listen to over their wine and cigars. I heard the Symphonic Preludes of Liszt, which are effective and extraordinary enough, but hardly beautiful; the Oberon Overture was very good, and as a piano solo in between we had the Variations Sérieuses of Mendelssohn, in the style of church music, which were very fine, and which I recommend you to study. Donders had been giving public lectures here on Acoustics, so that my book On the Sensations of Tone is known to everyone, even to the musicians. O. Jahn could not understand it, but hoped to study it with G., and told me he had had and enthusiastic letter about it from Claus Groth’. Hermann Helmholtz, March 14, 1863 letter to his wife, Hermann von Helmholtz, by Leo Koenigsberger, 222. Helmholtz was also critical of the ‘typical insensible style’ of Butera's Atala and a bell concert in a Scottish style that he believed could only be appreciated by its inventor. Hermann Helmholtz, letters (no. 24 and 25), Letters of Hermann von Helmholtz to his wife, 1847–1859, 86, 90.

⁵¹A harmonic interval is the distance between two pitches heard simultaneously. The number of steps between the pitches in a scale traditionally indicates this. From the pitch C up to E or down to A is a third. Further up to G or down to F is a fifth. From C to the C above or below is a perfect octave, an eighth.

⁵²When questioned whether an individual key had absolute character or only relative character in comparison to another, Helmholtz raised the possibility that the distinct character of keys was due, at least in part, to a particularity of the human ear. But, at least for pianos, bowed and wind instruments, the more likely cause of different characters of keys, according to Helmholtz, was the way in which a particular key was played on the instrument. Pianos keys, for instance, were struck differently depending on whether they were the black or white keys. For bowed and wind instruments the different lengths of the strings or wind chamber as a particular tone is sounded contributes to the supposed character of the key. Hermann Helmholtz, On the Sensations of Tone, 310–1.

⁵³Hermann Helmholtz, On the Sensations of Tone, 314.

⁵⁴Hermann Helmholtz, On the Sensations of Tone, 314.

⁵⁵Helmholtz promoted the cause of just-temperament in his 1857 popular lecture ‘On the Physiological Causes of Harmony in Music’, his article 1860 article ‘Über Musikalische Temperatur’, Tonempfindungen, and even referred to ‘preaching’ about unequal temperament to the organ-maker M. Cavallie in a letter to his wife. Hermann Helmholtz, Hermann von Helmholtz, by Leo Koenigsberger, 232.

⁵⁶The harmonium, a cheaper, smaller, and more rugged alternative to the pipe organ, was extremely popular in Europe and the United States through the nineteenth- and early-twentieth centuries. A pumping apparatus vibrates a bank of brass reeds. More sophisticated models (such as the one Helmholtz commissioned) had multiple reed banks and, correspondingly, multiple keyboards. Stops for drones allow the player a certain amount of control over the sound. The instrument's distinct timbre is the result of a double bellows system similar to that of the bagpipes, which allows the harmonium to create a sustained tone.

⁵⁷Hermann Helmholtz, On the Sensations of Tone, 315.

⁵⁸This very slight tempering of the fifths is perhaps the reason Helmholtz persisted in describing his commissioned harmonium as ‘just-tuned’ even though, by only keeping the thirds perfectly pure such a system would conventionally be considered a form of meantone temperament. Readers familiar with the complexities of tuning systems may find my use of ‘just temperament’ and ‘just tempered’ to be oxymoronic, but I do so in order to highlight that while Helmholtz claimed that his harmonium was just tuned, it was, because of the slight tempering of the fifths, not. Given Helmholtz's great care with terminology throughout On the Sensations of Tone, one can assume that his incorrect description of his harmonium as ‘just tuned’ was deliberate and likely part of his effort to promote just intonation. Alexander Ellis, the renowned English philologist and mathematician, began the English translation, with extensive comments and appendices, of Helmholtz's text in 1875. He too was quite interested in the musical systems of other cultures as well as the implementation of equal temperament (it is therefore of note and further indication of Helmholtz's deliberateness that Ellis was willing to translate Helmholtz's description of his harmonium as ‘just tuned’ without any comment). He claimed in a paper read before an 1880 Society of Arts meeting in Britain that though aimed at, equal temperament was only rarely attained. Alexander Ellis, ‘On the History of Musical Pitch’, Journal of the Society of Arts, April 2 (1880), 293–336. For an excellent discussion of the implementation of equal temperament see Ross W. Duffin's How Equal Temperament Ruined Harmony (and Why You Should Care) (W. W . Norton and Company, 2007). See also Mark Lindley, Lutes, Viols, and Temperaments (Cambridge University Press, 1984).

⁵⁹Hermann Helmholtz, On the Sensations of Tone, 319.

⁶⁰Hermann Helmholtz, On the Sensations of Tone, 319.

⁶¹Hermann Helmholtz, On the Sensations of Tone, 320. Myles Jackson examines the similar concerns of the first half of the nineteenth-century in his chapter, ‘Physics, Machines and Musical Pedagogy’, Harmonious Triads.

⁶²Hermann Helmholtz, On the Sensations of Tone, 322–3.

⁶³Hermann Helmholtz, On the Sensations of Tone, 322–3.

⁶⁴Hermann Helmholtz, On the Sensations of Tone, 322–3.

⁶⁵Hermann Helmholtz, On the Sensations of Tone, 324–5.

⁶⁶Hermann Helmholtz, On the Sensations of Tone, 324–5.

⁶⁷Hermann Helmholtz, On the Sensations of Tone, 320.

⁶⁸Hermann Helmholtz, On the Sensations of Tone, 327. Such statements can be read as perhaps alluding to the works of Richard Wagner, which were initially criticized as wild and jarring. It is not, however, clear what exactly Helmholtz's opinion of Wagner's music was though it certainly makes use of an aesthetic very different from those pieces and composers that Helmholtz openly admired. His second wife, Anna von Helmholtz, maintained a friendly correspondence with Cosima Wagner (Richard Wagner's wife, daughter of Franz Liszt) through the 1890s. Anna Helmholtz attended the Bayreuth festival in 1884 by herself then with Hermann in 1892, and then again alone (after his death) in 1899. See Petra Werner and Angelika Irmscher, Kunst und Liebe müssen sein: Breife von Anna von Helmholtz an Cosima Wagner 1889 bis 1899 (Druckhaus Bayreuth, 1993).

⁶⁹Hermann Helmholtz, On the Sensations of Tone, 327.

⁷⁰Hermann Helmholtz, On the Sensations of Tone, 327.

⁷¹David Pantalony, ‘Rudolph Koenig's Workshop of Sound: Instruments, Theories, and the Debate of Combination Tones’, Annals of Science, 62, no. 1 (2005), 57–82. See also Pantalony's Altered Sensations: Rudolph Koenig's Acoustical Workshop in Nineteenth-Century Paris (Springer: Archimedes Vol. 24, 2009).

⁷²In the fall of 1865, Helmholtz visited the workshop of the organ builder Cavaillé-Coll. He was given a tour of the workshop as well as taken to visit the Church of Saint-Sulpice, which housed an organ built by Cavaillé-Coll, then the largest organ in Europe. Helmholtz described the experience as highly interesting and commended the artisan for raising himself to the level of master, most intelligent and original. They then went to the house of the harmonium maker Mustel to see his latest invention, a tuning-fork piano with sustained tones. Helmholtz wrote to his (second) wife: ‘This confirmed my theoretical assumptions, and produced no special effect, which fact, however, is of some importance for my theory. The advance in the construction of the harmonium was very striking; it was like a very perfect and easily responding piano, with every kind of contrivance in the mechanism, for bringing out the treble parts’. Hermann Helmholtz, Hermann von Helmholtz, by Leo Koenigsberger, 232–3.

⁷³Hermann Helmholtz, ‘Über Musikalische Temperatur’, 74.

⁷⁴Hermann Helmholtz, ‘On the Motion of the Strings of a Violin’.

⁷⁵Hermann Helmholtz, On the Sensations of Tone, 19, 86.

⁷⁶Myles Jackson, ‘Physics, Machines and Musical Pedagogy’, Harmonious Triads.

⁷⁷Elfrieda Hiebert, ‘Helmholtz's Musical Acoustics: Incentive for Practical Techniques in Pedaling and Touch at the Piano’, 425–33.

⁷⁸Hermann Helmholtz, On the Sensations of Tone, 74–5.

⁷⁹Hermann Helmholtz, On the Sensations of Tone, 75.

⁸⁰Myles Jackson includes a nice discussion of Helmholtz's mechanical explanation of the piano and interest in piano makers in Harmonious Triads, 272–9.

⁸¹Hermann Helmholtz, On the Sensations of Tone, 75.

⁸³Hermann Helmholtz, On the Sensations of Tone, 76.

⁸²Hermann Helmholtz, On the Sensations of Tone, 76.

⁸⁴David Cahan offers a more thorough discussion of the relationship between Helmholtz and the Steinway family as well as Helmholtz's formal and informal exchange with American science more generally in his article, ‘Helmholtz in Gilded-Age America: The International Electrical Congress of 1893 and the Relations of Science and Technology’, Annals of Science 67 (2010), 1–38.

⁸⁵Hermann Helmholtz, June 9, 1871 letter to Theodore Steinway, Steinway and Sons Catalog, 1872. Cited in Richard K. Lieberman, Steinway and Sons (Yale University Press, 1995), 61.

⁸⁶D. W. Fostle, The Steinway Saga: An American Dynasty (Scribner, 1995), 286–7.

⁸⁷Richard K. Lieberman, Steinway and Sons, 61.

⁸⁸Erwin and Elfrieda Hiebert's discussion in, ‘Musical Thought and Practice: Links to Helmholtz's Tonempfindungen’, 307.

⁸⁹These letters were translated and maintained in the Steinway catalog for almost two decades along with endorsements from Lizst, Wagner, and others. One letter, from the Steinways to Helmholtz, is located at the Archiv der Berlin-Brandenburgischen Akademie der Wissenschaften. Two others are on loan to the La Guardia College Archive. Another is at the Steinway Archive and two others are published in the Steinway catalogs. See Elfrieda and Erwin Hiebert's article, ‘Musical Thought and Practice: Links to Helmholtz's Tonempfindungen’, for further discussion of these letters.

⁹⁰Elfrieda Hiebert, ‘Helmholtz's Musical Acoustics: Incentive for Practical Techniques in Pedaling and Touch at the Piano’, 427. Hiebert refers to Oscar Paul's text, Geschichte des Claviers vom Ursprunge bis zu den modernsten Formen dieses Instruments nebst einer Übersicht über Musikalische Abtheilung der Pariser Weltausstellung im Jahre 1867 (1868), 41.

⁹¹Hermann Helmholtz, On the Sensations of Tone, 6.

⁹²In 1842 Helmholtz wrote his father with instructions to tell his sister about the progress he was making on his Beethoven Sonata (Hermann Helmholtz, December 19, 1842 letter (no. 33), Letters of Hermann von Helmholtz to his Parents, 97). Twenty years later, he wrote to Carl Ludwig that he rarely played anything else (Hermann Helmholtz, Hermann von Helmholtz, by Leo Koenigsberger, 226).

⁹³Carl Dahlhaus, Nineteenth-Century Music, 81.

⁹⁴Carl Dahlhaus, Nineteenth-Century Music, 84–7.

⁹⁵Marx, not too surprisingly, cautioned composers against the arbitrary renunciation of form, lest they limit themselves. Further, to renounce form in and of itself, he explained, was to ‘return to spiritual chaos’, 90.

⁹⁶Adolf Marx, Die Lehre von der musikalischen Komposition, praktisch-theoretisch, 2nd edition (Breitkopf & Härtel, 1848), republished and translated, Musical Form in the Age of Beethoven: Selected Writings on Theory and Method, ed. and tr. Scott Burnham (Cambridge University Press, 2006), 138.

⁹⁷Marx described sonata form as normally consisting of a of three to four movements that making up a greater whole. The first would be lively (Allegro), followed by a peaceful Andante or Adagio. If there were four movements, then the third would be either a lively song from or a rondo, then, the finale in rondo or fugal form, 86.

⁹⁸Donald Francis Tovey, A Companion to Beethoven's Pianoforte Sonatas (The Associated Board of The R.A.M. and The R.C.M., 1931), 208.

⁹⁹Marx considered the same first seventeen measures of Figure 1 to be the opening of the sonata, but also included the repeated ‘consequent’ four-bar phrase that follow in this Satz, 110.

¹⁰⁰Readers well-versed in music theory will recognize that what Helmholtz referred to as ‘Doric’ is now termed ‘Phrygian’. Helmholtz went so far as to advocate for a new nomenclature based on the key signature of the interval of the diatonic scale beginning at C. So, the example Helmholtz gives: ‘the mode of Fourth of C’ has the signature of major scale of F. See Hermann Helmholtz, On the Sensations of Tone, 269.

¹⁰¹Hermann Helmholtz, On the Sensations of Tone, 308.

¹⁰²Hermann Helmholtz, On the Sensations of Tone, 327.

¹⁰³Marx pointed to Beethoven's use of C major and B-flat minor as a rejection of the normative rule that the relative major (G major) is requisite to avoid the ‘deeper gloom of minor piled onto minor’. Beethoven's contrarian choice allowed by ‘the rule and its foundation [to] fall away of themselves’, 138.

¹⁰⁴Julia Kursell ‘Sound Objects’, presented at ‘Sounds of Science, Schall im Labor (1800–1930)’ Workshop, Max-Planck-Institut für Wissenschaftgeschichte, Berlin, October 5–7, 2006.

¹⁰⁵It should be noted that Helmholtz was careful to distinguish between sensation and perception, the latter of which was the brain's processing of the former. When describing a study of sensation, Helmholtz explained that it was difficult to analyze sensations that could not be attached to corresponding differences in an external object, essentially that a psychophysical experiment based on the Fechner-Weber law would not work for a study of the ability to, say, distinguish harmonic overtones. The observer had to train himself to hear them—Helmholtz claimed that he could hear up to the sixteenth upper partial. See Hermann Helmholtz, On the Sensations of Tone, 49–51.

¹⁰⁶For example, Helmholtz mentioned neither psychophysics nor Fechner in On the Sensations of Tone. He was, however, aware of Fechner's work. The two men even had a brief correspondence in which Helmholtz clarified some points of his sound sensation theories for Fechner. A short letter from Helmholtz to Fechner, dated March 12, 1872, is included in the Fechner Nachlass at the Universitätbibliothek Leipzig.

¹⁰⁷Hermann Helmholtz, ‘Ueber Combinationstöne’, Poggendorff's Annalen der Physik und Chemie, 99 (1856), 497–540.

¹⁰⁸Gustav Hällstrom, ‘Von den Combinationstönen’, Annalen, 2nd series, 24 (1832), 438–66.

¹⁰⁹If the two original sounding tones are close together in frequency, the listener will hear beats. As the difference in the frequency of the sounding tones increases, the frequency of the beats also increases. When the difference between the frequencies of the sounding tones is high enough, the ear would hear the beats instead as an independent tone.

¹¹⁰R. Steven Turner, ‘The Ohm-Seebeck Dispute, Hermann von Helmholtz, and the Origins of Physiological Acoustics’, The British Journal for the History of Science, 10, no. 34 (1977), 13.

¹¹¹Hermann Helmholtz, ‘Ueber Combinationstöne’, 273–82.

¹¹²Helmholtz thought that the interference of the original sounding tones’ harmonic overtones occurred in ear canal between the eardrum and ossicles. But in this case the sound waves of the combination tones were still being created prior to simulating the auditory nerve and therefore still objectively existed independently of the listener. See Hermann Helmholtz, ‘Ueber Combinationstöne’, 300.

¹¹³R. Steven Turner, ‘The Ohm-Seebeck Dispute, Hermann von Helmholtz, and the Origins of Physiological Acoustics’, 14.

¹¹⁴‘Sie sehen, dass also auch das Clavier das Wellengewirre der Luft in seine einzelnen Bestandtheile auflöst. Was in unserem Ohr in demselben Falle geschieht, ist vielleicht dem eben beschriebenen Vorgange im Clavier sehr ähnlich…. Stellen wir weiter die Vermuthung auf, die freilich vorläufig nur Vermuthung bleibt, mir aber bei genauer Ueberlegung der physikalischen Leistungen des Ohres sehr wahrscheinlich erscheint, dass jedes solches Anhängselchen, ähnlich den Saiten des Claviers, auf einen Ton abgestimmt ist, so sehen Sie, nach dem Beispiel des Claviers, dass nur, wenn dieser Ton erklingt, das betreffende Gebilde schwingen und die zugehörige Nervenfaser empfinden kann, und dass die Gegenwart jedes einzelnen solchen Tones in einem Tongewirr auch stets durch die entsprechende Empfindung angezeigt werden muss. Das Ohr kann also, der Erfahrung nach, zusammengesetze Luftbewegungen in ihre Theile zerlegen’. Hermann Helmholtz, ‘Ueber die physiologischen Ursachen der musikalischen Harmonie’, 138–40.

¹¹⁵Hermann Helmholtz, ‘Ueber die physiologischen Ursachen der musikalischen Harmonie’, 146–7. In his 1869 address, ‘The Aim and Progress of Physical Science’, Helmholtz referred to this earlier discussion of the geistige ear: ‘nor need I remind you that the ear conveys to us sounds from without in no way in the ratio of their actual intensity, but strangely resolves them and modifies them, intensifying or weakening them in very different degrees, according to their varieties of pitch’. Hermann Helmholtz, ‘The Aim and Progress of Physical Science’, 342.

¹¹⁶See Gary Hatfield's text, The Natural and the Normative: Theories of Spatial Perception from Kant to Helmholtz and Timothy Lenoir's article, ‘The Eye as Mathematician: Clinical Practice, Instrumentation, and Helmholtz's Construction of an Empiricist Theory of Vision’.

¹¹⁷Hermann Helmholtz, ‘The Aim and Progress of Physical Science’, 343.

¹¹⁸Hermann Helmholtz, ‘Ueber Combinationstöne’, 289. There is here a resonance with Lamarckian evolutionary theory and the means by which an individual organism acquired physical traits, through repeated behaviors that over time resulted in a physiological change. Helmholtz certainly held a Lamarckian conception of evolution, describing the structural peculiarities acquired by parents and passed on to their offspring, and acknowledging that a single individual was capable of adapting itself. He understood Darwin's work as providing the law of transmission for this phenomenon. See Hermann Helmholtz, ‘The Aim and Progress of Physical Science’, 337–41.

¹¹⁹See Gary Hatfield, The Natural and the Normative: Theories of Spatial Perception from Kant to Helmholtz and ‘Helmholtz and Classicism: The Science of Aesthetics and the Aesthetics of Science’; Michael Heidelberger, ‘Force, Law, and Experiment: The Evolution of Helmholtz's Philosophy of Science’.

¹²⁰Hermann Helmholtz, ‘The Aim and Progress of Physical Science’, 325.

¹²¹Hermann Helmholtz, On the Sensations of Tone, 234–5.

¹²²Hermann Helmholtz, On the Sensations of Tone, 235.

¹²³Hermann Helmholtz, On the Sensations of Tone, 234–5.

¹²⁴Hermann Helmholtz, On the Sensations of Tone, 234.

¹²⁵Hermann Helmholtz, ‘Über die arabisch-persische Tonleiter’, presented on June 2, 1862 before the Naturh.-med. Verien in Heidelberg. See also Helmholtz's, On the Sensations of Tone, 279–85.

¹²⁶Hermann Helmholtz, On the Sensations of Tone, 282–3.

¹²⁷In his article, ‘Über die arabisch-persische Tonleiter’, Helmholtz suggested that the Alexandrian Greeks likely acquired fragments of the Persian music system after they had conquered them.

¹²⁸‘This reference to the history of music was necessitated by our inability in this case to appeal to observation and experiment for establishing our explanations… If our theory of the modern tonal system is correct it must also suffice to furnish the requisite explanation of the former less perfect stages of development’. Hermann Helmholtz, On the Sensations of Tone, 249.