Music and Mathematics–“Functional Melodies”

Interdisciplinary learning is a key strategy to bring students into a holistic awareness of their world, a systems-based awareness that fosters a path to unity consciousness and a greater sense of meaning in all learning and life itself.

Extensive training and experience in music performance and composition combined with  studies in engineering and mathematics ultimately led me to a most obvious interdisciplinary connection:  teaching music and mathematics in a fully integrated form.  This work, begun in my graduate studies at Stanford University led to the publication of by book, Functional Melodies by Key Curriculum Press in 2000.

Functional Melodies – The Book

(click on images below)

Read the introduction of Functional Melodies  for a strong summary and overview of the genesis and rationale for the work, a table of contents and activity outline.

Access Functional Melodies CD Audio Tracks here

The theoretical rationale for the value of interdisciplinary learning can be found in the case study (posted in “Evidence”) on this site:  “Teaching to Multiple Intelligences–Music and Mathematics.”

You will find photos of students engaged with this work under Student Work in this site.

Professional development workshops are offered to provide teachers with materials, confidence and expertise to conduct integrated music and mathematics lessons or units in their classroom.

Functional Melodies is no longer in print however copies are available through various online outlets, and directly from me.  An audio CD accompanies the book however the book copies now available online do not include the CD.  You access the  CD here.

It is important to note that over the years since it’s publication, a wide variety of variations, extensions and unique applications have evolved.  These applications are explored in teacher professional development workshops.

Sample Music and Mathematics Unit Sequence

Presented below is a highly summarized unit sequence combining various activities from Functional Melodies and others developed since.

  1.  Introduction–music and sound, physics, waves, simple harmonic motion, harmonics, is the major scale created by humans or by nature?
  2. The major scale as numbers–major scale tones are assigned number values, 1-8.  Melodies become number sequences and are graphed on a Cartesian Coordinate system.  Melodies are experienced and analyzed geometrically as well as sonically.
  3. Name That Graph! — A melody is played and three graphs are displayed, students determine which graph correlates to the melody played.
  4. Functional Composer — Scenario:  a composer has “writers block,” and can only come up with one 7-note motive.  As a mathematical means to solve the composer’s problem, students convert the notes to number, treat them as a function, graph the function and transform that function (translate, stretch, shrink, reflect) to generate variations while examining how the various math operations manifest in both the musical qualities of the variations as well as the geometric characteristics of the various function transformations.  The variations create a composition.  The cohesion of the group of melodies is attributed to their mathematical “DNA”.
  5. Inside OutVisual images beget a musical composition — To generate original motives, students trace photographic images (nature, sports players, dancers…) on a polar coordinate graph and use an interface to generate numbers that reflect the relationships in the shape of the image.  The numbers are converted to notes and played revealing that the essence of aesthetic beauty resides in structures that live with consistent integrity whether expressed in visual, sonic and/or numerical domains.
  6. Composition — Students choose “best” melodic material to generate original motives for a composition and generate extended musical material by experimenting with a wide variety of function transformations.
  7. Performance — Students perform (or the teacher performs if the student is not trained on an instrument) for the class.  Aesthetics are discussed.  Essence of musical “art” is discussed.  Mathematics–geometry and the nature of various operations are discussed.   Students write reflective journals and plan future work.