Measures of Consonances in a Goodness-of-fit Model for Equal-tempered Scales

In this paper a general model is described which measures the goodness of equal-tempered scales. To investigate the nature of this ’goodness’, the consonance measures developed by Euler and Helmholtz are discussed and applied to two different sets of intervals. Based on our model, the familiar 12-tone equal temperament does not have an extraordinary goodness. Others, such as the 19-tone equal temperament look as least as promising. A surprising outcome is that when intervals from the just minor scale are chosen to be approximated by an -tone equal temperament system, good values for are and , rather than the commonly used .