IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2024i11p1658-d1401823.html
   My bibliography  Save this article

Partition Entropy as a Measure of Regularity of Music Scales

Author

Listed:
  • Rafael Cubarsi

    (Departament de Matemàtiques, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain)

Abstract

The entropy of the partition generated by an n -tone music scale is proposed to quantify its regularity. The normalized entropy relative to a regular partition and its complementary, here referred to as the bias, allow us to analyze various conditions of similarity between an arbitrary scale and a regular scale. Interesting particular cases are scales with limited bias because their tones are distributed along specific interval fractions of a regular partition. The most typical case in music concerns partitions associated with well-formed scales generated by a single tone h . These scales are maximal even sets that combine two elementary intervals. Then, the normalized entropy depends on each number of intervals as well as their relative size. When well-formed scales are refined, several nested families stand out with increasing regularity. It is proven that a scale of minimal bias, i.e., with less bias than those with fewer tones, is always a best rational approximation of l o g 2 h .

Suggested Citation

  • Rafael Cubarsi, 2024. "Partition Entropy as a Measure of Regularity of Music Scales," Mathematics, MDPI, vol. 12(11), pages 1-23, May.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:11:p:1658-:d:1401823
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/11/1658/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/11/1658/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Jack Douthett & Richard Krantz, 2007. "Maximally even sets and configurations: common threads in mathematics, physics, and music," Journal of Combinatorial Optimization, Springer, vol. 14(4), pages 385-410, November.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.

      Corrections

      All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:12:y:2024:i:11:p:1658-:d:1401823. See general information about how to correct material in RePEc.

      If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

      If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

      If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

      For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

      Please note that corrections may take a couple of weeks to filter through the various RePEc services.

      IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.