Abstract
When Hermann von Helmholtz died over a 100 years ago German science lost its preeminent physicist and physiologist. Over a time span of 50 years, as teacher, researcher, and science mediator in Königsberg, Bonn, Heidelberg, and Berlin, Helmholtz initiated substantial advances in the science of medicine, physiological optics, physiological acoustics, thermodynamics, hydrodynamics, electrodynamics, and mathematics. His reflections on science and its philosophical dimensions belong to an important genre of epistemological writings on the natural sciences. They represent a precedent-setting example in which the scientist becomes the intermediary who engages in reflections on the philosophy of science from the perspective not of philosopher but of scientist.
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Notes
- 1.
For a wide range of historical studies on music and mathematics: Günter Schnitzler, ed., Musik und Zahl. Interdisziplinäre Beiträge zum Grenzbereich zwischen Musik und Mathematik, Bonn-Bad Godesberg, 1976. See esp. Martin Vogel’s essay “Reine Stimmung und Temperierung.”
- 2.
“Musica est exercitium arithmeticae occultum nescientis se numerari anime.” This statement is made by Leibniz in a letter of 17 April 1712 addressed to the Russian mathematician Christian Goldbach. R. C. Archibald, “Mathematics and Music,” American Mathematical Monthly, 31 (1924), 1–2.
- 3.
The family of instruments with keyboard mechanisms, each of which has its own checkered history, includes: organs as the oldest keyboards that are sounded by air under pressure; the clavichord and harpsichord that are activated by struck or plucked mechanisms and that were in use from the fifteenth to the eighteenth centuries and revived at the end of the nineteenth; the pianoforte (piano) as an instrument central to musical life since the 3rd quarter of the eighteenth century; and the harmonium which is a pedal-operated organ with freely vibrating metal reeds – an instrument that developed during the first half of the nineteenth century.
- 4.
To a limited extent the woodwinds can adjust their pitches by lengthening or shortening an instrument’s vibrating air column at its upper extremity, by alternative fingerings, and by embouchure control. The same problem of pitch control maintains for the brasses. The French horn presents a unique case since a skillful player is able to sound the entire chromatic scale on the valveless natural horn by means of embouchure control and hand-in-bell techniques. Some of the tones on the natural horn will take on a unique muffled character – a phenomenon that on occasion has been said to elicit its own artistic appeal.
- 5.
Among orchestral instruments only the slide trombone and slide trumpet can claim true pitch flexibility. Except for its open strings the violin and the family of string instruments can claim maximum tuning independence.
References
Goldbach, Christian, and R.C. Archibald. 1924. Mathematics and music. American Mathematical Monthly 31: 1–2.
Schnitzler, Günter (ed.). 1976. Musik und Zahl. Interdisziplinäre Beiträge zum Grenzbereich zwischen Musik und Mathematik. Bonn: Verlag für Systematische Musikwissenschaft.
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Hiebert, E. (2014). The Helmholtz Setting in the Johannes Müller Circle in Berlin. In: The Helmholtz Legacy in Physiological Acoustics. Archimedes, vol 39. Springer, Cham. https://doi.org/10.1007/978-3-319-06602-8_1
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