Alternatives to Semitones and Quartertones: Music-Theoretical Suggestions

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Figure 1


  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.


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Smethurst, R. Alternatives to Semitones and Quartertones: Music-Theoretical Suggestions. Math Intelligencer 40, 37–42 (2018).

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