Notes
An equal temperament is a self-similarity sequence. See my didactic video [20].
Musical intervals are frequency-ratios (or approximations of frequency-ratios) that are measured in relation to a reference pitch, 1:1. Intervals are not measured in relation to the preceding note of a scale; steps are measured in relation to the preceding note of a scale. “Error” is the difference between a pure frequency-ratio and its tempered approximation.
See Reference [21] for further examples of Bohlen’s non-octave tunings.
References
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Heinz Bohlen, “13 Tonstufen in der Duodezime,” Acustica 39:2 (1978), pages 76–86.
Heinz Bohlen, “13 Tone Steps in the Twelfth,” Heinz Bohlen and Brad Stockwell (trans.), Acta Acustica united with Acustica 87:5 (2001), pages 617–624.
Peter Burt, The Music of Tōru Takemitsu, Cambridge: Cambridge University Press, 2001 (pages 33–38).
Ricardo A. Godoy et al., “Indifference to Dissonance in Native Amazonians Reveals Cultural Variation in Music Perception,” Nature 535 (2016), pages 547–550.
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George D. Secor, “Gift of the Gods,” Sagittal, 2 September 2015: http://sagittal.org/gift/GiftOfTheGods.htm.
Reilly Smethurst, “Didactic Video: Bohlen’s 11 Non-Equal Divisions of 3:1,” The Mathematical Intelligencer, 4 November 2017 [Electronic supplementary material].
Reilly Smethurst, “Didactic Video: Four-Note Harmonies from 18-Tone Equal Temperament,” The Mathematical Intelligencer, 4 November 2017 [Electronic supplementary material].
Reilly Smethurst, “Didactic Video: Symmetrical Modes,” The Mathematical Intelligencer, 4 November 2017 [Electronic supplementary material].
Reilly Smethurst, “Two Non-Octave Tunings by Heinz Bohlen: A Practical Proposal,” Bridges Finland Conference Proceedings, Kristóf Fenyvesi et al. (eds.), Phoenix: Tessellations, 2016 (pages 519–522).
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Smethurst, R. Alternatives to Semitones and Quartertones: Music-Theoretical Suggestions. Math Intelligencer 40, 37–42 (2018). https://doi.org/10.1007/s00283-018-9800-z
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DOI: https://doi.org/10.1007/s00283-018-9800-z