Alternatives to Semitones and Quartertones: Music-Theoretical Suggestions

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Figure 1

Notes

  1. 1.

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

  2. 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. 3.

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

References

  1. [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. [2]

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

    Google Scholar 

  3. [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.

    Google Scholar 

  4. [4]

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

  5. [5]

    Ricardo A. Godoy et al., “Indifference to Dissonance in Native Amazonians Reveals Cultural Variation in Music Perception,” Nature 535 (2016), pages 547–550.

    Article  Google Scholar 

  6. [6]

    Paul Griffiths, Olivier Messiaen and the Music of Time, Ithaca: Cornell University Press, 1985 (pages 96–97).

  7. [7]

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

    Google Scholar 

  8. [8]

    Ben Johnston, “Maximum Clarity” and Other Writings on Music, Bob Gilmore (ed.), Urbana: University of Illinois Press, 2006 (pages 254–255).

  9. [9]

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

  10. [10]

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

  11. [11]

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

  12. [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. [13]

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

  14. [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. [15]

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

  16. [16]

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

  17. [17]

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

  18. [18]

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

  19. [19]

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

  20. [20]

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

  21. [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. [22]

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

    Article  Google Scholar 

  23. [23]

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

    Article  Google Scholar 

  24. [24]

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

    MathSciNet  Google Scholar 

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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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