Markus Schmidmeier
Mathematical Sciences
Florida Atlantic University

Mathematics of the Tonal Systems

This course, offered in Spring 2012, can be taken as Honors Thesis in Mathematics, the course numbers are MAT 4970-001 and CRN 22098. We meet weekly on Wednesdays, 15:30 - 17:00 p.m., or by appointment.


In addition to the formal prerequisites for taking an honors thesis in mathematics, which can be found in the course catalog, familiarity with Introductory Number Theory MAS 3203 and Modern Algebra MAS 4301 is necessary. Most important is an eagerness to explore links between mathematics and music.


In this course we explore links between mathematics and music, in particular as they occur in the construction of scales. We review the physical properties of music (frequency, amplitude, duration, timbre) to understand the challenges in defining scales. The Pythagorean model is based on the octave (2:1) and the the fifth (3:2), and correspondingly, all admissible pitches are obtained from a base frequency and powers of 2 and 3. In the Western system the major third is introduced via the ratio (5:4), so the prime numbers 2, 3, and 5 define the system. In Bohlen-Pierce, those numbers are replaced by 3, 5 and 7.

How many numbers are there in a tune?

In order to define a scale, certain pitches need to be selected (the 7 white keys per octave on a piano tuned in the traditional Western system), and certain frequencies need to be identified. In the Pythagorean and the Western systems, for example, twelve fifths give seven octaves, so we "neglect" the comma which is (3/2)12 * (1/2)7. In music terms, F=G.


  1. Recognize that physical properties of sound give rise to mathematical models for traditional and modern musical scales.
  2. Express musical properties of scales in terms of number theory and group theory.
  3. Construct models (e.g. 3d-paper models) which mathematically explain features in music like: Which seven tones are selected to form each of the major and minor scales in traditional Western music?
  4. Define a new scale, and explore its properties.
  5. Give a seminar presentation.
  6. Write a short thesis.


Further Information

For the Code of Academic Integrity, Religious Accommodation and my grading scale please visit Infos for all my courses.

Contact Me

Office hours:   before and after the seminar

Telephone:  561-297-0275

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Last modified:  by Markus Schmidmeier