Abstract
Having for several years devoted himself exclusively to investigations focused on what can be achieved in regard to acoustically pure intervals using the Huygens 31-tone system, Fokker chose in 1949, by way of comparison and contrast, to redirect his attention to examining the limited extent to which just intonation is achievable for music structured on the ordinary equal temperament scale of 12 notes to the octave using instruments that produced tones with fixed pitch. While the pianoforte and the organ gain the advantage of complete freedom of modulation in this system, they sacrifice the purity and beauty of perfect concords. It is for this reason, as Fokker remarked, “that stress must be laid on a special training to acquire a just intonation of perfect intervals and chords.”
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Notes
- 1.
Fokker, Just Intonation and the Combination of Harmonic Diatonic Melodic Groups, the Hague, 1949, pp. 1–50. Quotation p. 1.
- 2.
Fokker, Just Intonation, p. 2.
- 3.
Tonic sol-fa is a type of musical notation and is associated with a method of sight-singing developed in England in the nineteenth century by John C. Curwen (1816–1880). He published music in the sol-fa system with the aim of teaching beginners to sing accurately and to foster proficiency in sight-reading.
- 4.
Fokker, Just Intonation (1949), p. 34.
- 5.
Fokker, “The Qualities of the Equal Temperament by 31 Fifths of a Tone in the Octave,” International Society for Musical Research Fifth Congress Report, Amsterdam, 1953, pp. 191–192.
- 6.
Adriaan Fokker, “Equal Temperament and the Thirty-one-keyed Organ,” Scientific Monthly (Published by the American Association for the Advancement of Science, Washington, D.C.) 83 (1955).
- 7.
Fokker, “Equal Temperament,” pp. 162/3.
- 8.
Fokker, “Equal Temperament,” loc. cit. pp. 164–165.
- 9.
In 1855 the French physicist J.A. Lissajous (1822–1880) invented a way to study acoustical vibrations by reflecting a light beam from the vibrating object onto a screen thus to provide visual demonstration of the wave form and obtain precise tuning of musical instruments without using the ear. In Tonempfindungen, in the section on musical tones of bowed instruments, Helmholtz gave a detailed description of the use of Lissajous figures to calculate the whole motion of a string and the intensity of its upper partials.
References
Fokker, Adriaan. 1949. Just intonation and the combination of harmonic diatonic melodic groups, 1–50. The Hague: M. Nijhoff.
Fokker, Adriaan. 1953. The Qualities of the equal temperament by 31 fifths of a tone in the octave. International Society for Musical Research. Fifth Congress Report, 191–192. Amsterdam.
Fokker, Adriaan. 1955. Equal temperament and the thirty-one-keyed organ. Scientific Monthly 83. Washington, DC: Published by the American Association for the Advancement of Science.
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Hiebert, E. (2014). Just Intonation and the 12-tone System (1949). In: The Helmholtz Legacy in Physiological Acoustics. Archimedes, vol 39. Springer, Cham. https://doi.org/10.1007/978-3-319-06602-8_24
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