Abstract
In what follows an effort is made, first, to locate the essence of what Fokker understood and appropriated to himself from Huygens’s views on music theory, and then, to demonstrate how Fokker implemented and reconstructed Huygens’s ideas in connection with the objectives that he came to set for himself in his own research on musical acoustics.
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Notes
- 1.
Acoustically pure fifths (the fifths that belong to the harmonic series) are equal to 702 cents and are larger by 2 cents than equally tempered fifths (700 cents).
- 2.
A. D. Fokker, “Harmonische muziek,.” Archives du Musée Teyler, Serie III, vol. IX, Fascicule 5, (1942) 449–506.
- 3.
Fokker, “Harmonische muziek,” p. 451. “C’est donc en effet un plus haut degré de perfection auquel on a porté la Musique, y ayant introduit cette nouvelle espèce de consonances.”
- 4.
Historically, the diësis was introduced into Greek theory to represent an interval smaller than a semitone; in medieval treatises it is referred to as semitonium minus. In the Pythagorean scale, which derives all the tones in its scale from the interval of the pure fifth, the diësis was characterized as one of three categories of tuning. They are denoted as the diatonic, chromatic, and enharmonic genera. The diatonic scale is based on an octave divided into five tones and two semitones. The chromatic scale is based on an octave of 12 semitones in contrast to the 7-tone diatonic scale. The enharmonic scale is based on pair pitches that are not absolutely the same (such as G# and A♭) and is exploited for purposes of modulation. Enharmonic keyboards employ separate keys for some of the pairs of pitches that would otherwise be equivalent. Fokker and others who worked with keyboards having more than 12 keys per octave sometimes referred to the enharmonic keyboard as an instrument designed to produce different types of microtones. Microtones are smaller than a semitone and are distinguished from the several types of diesis by being some numerical fraction of a semitone rather than a tone in a particular collection of pitches that makes up a scale. Microtones have served both melodic and intonational functions in Western music since antiquity.
- 5.
A. D. Fokker, “Harmonische muziek,” 1942, p. 451.
- 6.
A. D. Fokker, “Harmonische muziek,” Inleiding p. 452.
- 7.
A. D. Fokker, “Christiaan Huygens’ oktaafverdeling in 31 gelijke diëzen,” Caecilia en de Muziek, 98 (1941) 149–152.
- 8.
Fokker, “Christiaan Huygens oktaafverdeling,” pp. 149–159. The Pythagorean comma as here used is the interval by which the sum of six whole tones, each designated by the ratio 9/8, exceeds the octave 2/1. A small interval, the comma, usually is taken to be about one-ninth of a whole tone.
- 9.
Aristoxenus (b. ca. 375–360, d. ? Athens) was a Greek musical theorist who challenged the Pythagorean tonal system of the musical scale based on mathematical ratios and rather placed the emphasis on aural reception.
- 10.
The term archicembalo was used by Vicentino in 1555 for a harpsichord or an organ with divided keys on a second manual that permitted the playing of intervals smaller than a semitone; that is, it permitted the playing of pure as against tempered intervals in a variety of keys. These instruments were conceived with the intention of making possible the performance of the diatonic, chromatic, and enharmonic genera of ancient Greek theory.
- 11.
Fokker, “Christiaan Huygens oktaafverdeling,” p. 151.
- 12.
Fokker, “Christiaan Huygens oktaafverdeling,” pp. 151–152/
- 13.
A. D. Fokker, “De keuze der muziektonen,” Caecilia en de Muziek, 99 (1942) 25–29; “Grundtoon, gidstoon en hun volledig akkord,” Ibid., 99 (1942) 72–75; “Vierklanken als getallenkwartetten,” 99 (1942) 116–118; “Intervalsnippers en de verdeeling van octaf en kwint in even groode deelen,” 99 (1942) 172–176; “Tartini en de zevende harmonische,” 100 (1943) 78–80.
References
Fokker, A.D. 1941. Christiaan Huygens’ oktaafverdeling in 31 gelijke diëzen. Caecilia en de Muziek 98: 149–152.
Fokker, A.D. 1942a. Harmonische muziek. Archives du Musée Teyler, Serie III, vol. IX, Fascicule 5: 449–506.
Fokker, A.D. 1942b. De keuze der muziektonen. Caecilia en de Muziek 99: 25–29.
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Hiebert, E. (2014). Temperament and the Circle of Fifths. In: The Helmholtz Legacy in Physiological Acoustics. Archimedes, vol 39. Springer, Cham. https://doi.org/10.1007/978-3-319-06602-8_22
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