From Pythagoras to Johann Sebastian Bach: An Exploration in the Development of Temperament and Tuning

Temperament and tuning are at the core of all music—they are the embodiment of the correlation of music and mathematics in every musical work. In examining the shift from Pythagorean tuning to well temperament, compositional style and philosophy governs not only the evolution musical genres, but also the evolution of temperament and tuning systems. The association of mathematics and temperament defines ratios and pitch relations in every branch of temperament. An analysis of J.S. Bach\u27s Well-Tempered Clavier and his use of temperament display the contrast of melodious thirds of meantone temperament and emotional tension of chords in well temperament. The mathematical beauty and complexity of tuning governs the musical center of the work; a composition may focus upon minute intervallic relationships or larger harmonic structure, depending upon the primary focus of the utilized system of temperament. Arguably, mathematical relationships have more influence upon musical beauty than the composed music alone. J.S. Bach\u27s preludes and fugues of the Well-Tempered Clavier prove the existence of liveliness and color in every possible key