A measure of the reasonableness of equal‐tempered musical scales

Using three criteria a desirability function is developed which allows comparison of the reasonableness of equal‐tempered musical scales to approximate just musical intervals. Unlike most other measurements, this scale is not dominated by very fine octave divisions. This feature allows for comparison of systems with few divisions to the octave with those with many divisions to the octave. Based on our measure, an equal‐tempered scale which must best represent the pure fifth requires 12, 41, or 53 notes to the octave. If the major third is to be well represented, equal‐tempered scales of 28 or 59 notes are required. Best representation of the minor third occurs for a 19‐note equal‐tempered scale. These results are shown to be in accordance with previously cited continued fraction results and other published evaluation criteria. The method is generalized to assess the reasonableness of equal‐tempered scales to approximate multiple just intervals simultaneously as well as to account for numerical weighting of intervals. Our results show that an equal‐tempered scale required to approximate the pure fifth, major third, and minor third equally requires 12, 19, 34, or 53 notes. This result is contrary to previously cited results in which 34 notes to the octave is relegated to a relatively minor role.