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Alternatives to Semitones and Quartertones: Music-Theoretical Suggestions

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Figure 1

Notes

  1. An equal temperament is a self-similarity sequence. See my didactic video [20].

  2. 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.

  3. See Reference [21] for further examples of Bohlen’s non-octave tunings.

References

  1. Adriano Baratè and Luca Andrea Ludovico, “Generalizing Messiaen’s Modes of Limited Transposition to a n-tone Equal Temperament,” Proceedings of the 12th International Conference on Sound and Music Computing (SMC-15), Maynooth, Ireland (2015), pages 287–293.

  2. Heinz Bohlen, “13 Tonstufen in der Duodezime,” Acustica 39:2 (1978), pages 76–86.

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  3. 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.

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  4. Peter Burt, The Music of Tōru Takemitsu, Cambridge: Cambridge University Press, 2001 (pages 33–38).

  5. 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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  7. Ján Haluška, The Mathematical Theory of Tone Systems, New York: Marcel Dekker, 2004.

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  9. “List of Microtonal Software Plugins,” The Xenharmonic Alliance, 22 March 2017: http://xenharmonic.wikispaces.com/List+of+Microtonal+Software+Plugins.

  10. Olivier Messiaen, Technique de mon langage musical, vol. 1, Paris: Alphonse Leduc, 1944 (Chap. 16).

  11. Olivier Messiaen, The Technique of my Musical Language, vol. 1, John Satterfield (trans.), Paris: Alphonse Leduc, 1956 (Chap. 16).

  12. Athanase Papadopoulos, “Mathematics and Group Theory in Music,” Handbook of Group Actions, vol. 2, Lizhen Ji et al. (eds.), Somerville: International Press and Higher Education Press, 2015 (pages 525–572).

  13. Plato, The Republic, G. R. F. Ferrari (ed.), Tom Griffith (trans.), Cambridge: Cambridge University Press, 2000 (pages 7.531a–b).

  14. Rudolf Rasch, “Tuning and Temperament,” The Cambridge History of Western Music Theory, Thomas Christensen (ed.), Cambridge: Cambridge University Press, 2002 (pages 193–222).

  15. John Schneider, The Contemporary Guitar: Revised and Enlarged Edition, Lanham: Rowman & Littlefield, 2015 (page 60).

  16. Arnold Schoenberg, Theory of Harmony, Roy E. Carter (trans.), Berkeley: University of California Press, 1978 (page 425).

  17. George D. Secor, “Gift of the Gods,” Sagittal, 2 September 2015: http://sagittal.org/gift/GiftOfTheGods.htm.

  18. Reilly Smethurst, “Didactic Video: Bohlen’s 11 Non-Equal Divisions of 3:1,” The Mathematical Intelligencer, 4 November 2017 [Electronic supplementary material].

  19. Reilly Smethurst, “Didactic Video: Four-Note Harmonies from 18-Tone Equal Temperament,” The Mathematical Intelligencer, 4 November 2017 [Electronic supplementary material].

  20. Reilly Smethurst, “Didactic Video: Symmetrical Modes,” The Mathematical Intelligencer, 4 November 2017 [Electronic supplementary material].

  21. 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).

  22. Dinesh S. Thakur, “The Notion of Twenty-Two Shrutis,” Resonance 20:6 (2015), pages 515–531.

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  23. Daniel James Wolf, “Alternative Tunings, Alternative Tonalities,” Contemporary Music Review 22:1–2 (2003), pages 3–14.

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  24. Ozan Yarman, “A Comparative Evaluation of Pitch Notations in Turkish Makam Music,” Journal of Interdisciplinary Music Studies 1:2 (2007), pages 43–61.

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Correspondence to Reilly Smethurst.

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The Viewpoint column offers readers of The Mathematical Intelligencer the opportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, The publisher and editor-in-chief do not endorse them or accept responsibility for them. Articles for Viewpoint should be submitted to the Editor-in-Chief, Marjorie Senechal.

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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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