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Finding Alternative Musical Scales

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Principles and Practice of Constraint Programming (CP 2016)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 9892))

Abstract

We search for alternative musical scales that share the main advantages of classical scales: pitch frequencies that bear simple ratios to each other, and multiple keys based on an underlying chromatic scale with tempered tuning. We conduct the search by formulating a constraint satisfaction problem that is well suited for solution by constraint programming. We find that certain 11-note scales on a 19-note chromatic stand out as superior to all others. These scales enjoy harmonic and structural possibilities that go significantly beyond what is available in classical scales and therefore provide a possible medium for innovative musical composition.

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Notes

  1. 1.

    Simple ratios also tend to produce intervals that are consonant in some sense, although consonance and dissonance involve other factors as well. One theory is that the perception of dissonance results from beats that are generated by upper harmonics that are close in frequency [18–21]. We will occasionally refer to simple ratios as resulting in “consonant” intervals, but this is not to deny the other factors involved.

  2. 2.

    We use the tempered pitch as a base for the percentage error because it is the same across all scales and so permits more direct comparison of errors. A cent is 1/1200 of an octave, or 1/100 of a semitone. Thus if two tones differ by c cents, the ratio of their frequencies is \(2^{c/1200}\). An error of +0.9 % is equivalent to +15.65 cents, and an error of \(-0.9\)% to \(-15.51\) cents.

  3. 3.

    This restriction excludes the classical harmonic minor scale, in which notes 6 and 7 are separated by three semitones, but the harmonic minor scale can be viewed as a variant of a natural minor scale in which note 7 is raised a semitone for cadences.

  4. 4.

    We follow the convention of numering the scales in the order of the tuples s treated as binary numbers.

  5. 5.

    For the Dorian, a solution with generators 3/2 and 5/3 is shown because it results in simpler ratios. The single generator 3/2 results in ratios 9/8, 32/27, 4/3, 3/2, 27/16, 16/9.

  6. 6.

    A complete list of all 50 solutions found for each of the 77 scales is available at web.tepper.cmu.edu/jnh/music/scales11notes19.pdf and as electronic supplementary material published online with this article.

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Author information

Authors and Affiliations

  1. Carnegie Mellon University, Pittsburgh, USA

    J. N. Hooker

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  1. J. N. Hooker
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Correspondence to J. N. Hooker .

Editor information

Editors and Affiliations

  1. University of Nice Sophia Antipolis , Sophia Antipolis, France

    Michel Rueher

Appendix

Appendix

A Chorale and Fugue for organ [8] uses scale 23 on 9 notes. The chorale cycles through the tonic (A) and the two most closely related keys (C\(\sharp \), F). The cadences illustrate that dominant seven chords need not play a role, even though they occur in the scale. Rather, the cadences use two leading tones and pivot on the tonic, often by moving from the lowered submediant. The chorale is followed by a double fugue that again cycles through the three keys A, C\(\sharp \), F. The first subject enters on these pitches but without a key change. The second subject (bar 96) illustrates the expanded possibilities for suspensions and pivots.

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Hooker, J.N. (2016). Finding Alternative Musical Scales. In: Rueher, M. (eds) Principles and Practice of Constraint Programming. CP 2016. Lecture Notes in Computer Science(), vol 9892. Springer, Cham. https://doi.org/10.1007/978-3-319-44953-1_47

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  • DOI: https://doi.org/10.1007/978-3-319-44953-1_47

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-44952-4

  • Online ISBN: 978-3-319-44953-1

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