March 30, 2021

The Mathematical Harmony in Music


“Music is the mathematics of one who does not know that he is counting.” Gottfried Wilhelm Leibniz (1646-1716)

 Tomoko realizes the preciseness and patterns of music. Not surprisingly then, math helps in reading music. The simplest task such as counting the beats uses math. Each time signature codes the number of beats per measure, and the notes represent fractions of a measure and beat – such as whole notes, quarter notes, eighth notes and so on. In turn, reading these rhythmic notations can help one read and solve math equations. 

In fact, mathematics lies underneath much musical composition and reflects the very nature of music itself. Even the concept of octaves is mathematical. An octave is the distance between a given note with a set sound frequency (that is, the string’s vibration) with another note with double that frequency. A perfect fifth is 1.5 times the frequency of the octave’s base note. Ratios help make music harmonious.

Music compositions reflect patterns, just as math does: symmetry, repetition, transposition, inversion.  The process for perceiving and generating those patterns mirrors mathematical processes. Johann Sebastian Bach very consciously incorporated mathematical principles into his keyboard compositions. His work “Musical Offering” is comprised of ten canons, in which each canon is a mathematical transformation of the principal musical line. In fact, a mathematical breakthrough enabled Bach to write “The Well-Tempered Clavier.” Keyboard instruments used to be tuned using a just-toned scale, which made shifting to keys other than the tonic sounded “off.” The equal (even)-tempered scale, popularized by Bach, evened out the frequency ratios between all 12 notes of the chromic scale so that shifts of harmonies to other keys would sound the same.

Tomoko rightly asserts that reading and playing music require good discipline, improve listening and collaborative skills, and strengthens mental and muscle memory.  Those practices can also build math skills and recall of math details. A harmonious blend!


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