The Mathematics of Tuning Musical Instruments — a Simple Toolkit for Experiments

  • Erich Neuwirth
Part of the Mathematics and Visualization book series (MATHVISUAL)


This paper gives an overview of the (rather simple) mathematics underlying the theory of tuning musical instruments. Besides demonstrating the fundamental problems and discussing the different solutions (only on an introductory level), we also give Mathematica code that makes it possible to listen to the constructed scales and chords. To really get a “feeling” for the contents of this paper it is very important to hear the tones and intervals that are mentioned. The paper also has the purpose of giving the reader a Mathematica toolkit to experiment with different tunings.


Frequency Ratio Base Frequency Musical Instrument Lower Tone Base Tone 
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  1. 1.
    Blackwood E. (1999) The Structure of Recognizable Diatonic Tunings, Princeton University PressGoogle Scholar
  2. 2.
    Lindley M. Turner-Smith R., Mathematical Models of Musical Scales, Verlag für systematische MusikwissenschaftGoogle Scholar
  3. 3.
    Neuwirth E. (1997) Musical Temperaments, Springer VerlagGoogle Scholar
  4. 4.
    Neuwirth E. (2001) Designing a Pleasing Sound Mathematically, in Mathematics MagazineGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Erich Neuwirth
    • 1
  1. 1.University of ViennaAustria

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