Regular Temperament Theory: Exploring the Landscape Between JI and ETs with Linear Algebra

Dissonance is easy. Consonance is rare. We want scales and tunings that give different flavors of consonance, without being too complex, and with small errors. Regular Temperament Theory (RTT) is a powerful tool, reducing the problem to approximating a few small prime numbers, and generating tunings by stacking a few intervals called generators. RTT opens a middle ground between just intonation and equal temperaments that goes far beyond meantone, applying linear algebra to discover new temperaments that balance complexity and error in different ways while providing harmonies that do not exist in conventional tunings. We introduce the theory and list some open problems.