title: Perfect harmony: a mathematical analysis of four historical tunings.
abstract: In Western music, a musical interval defined by the frequency ratio of two notes is generally considered consonant when the ratio is composed of small integers. Perfect harmony or an "ideal just scale," which has no exact solution, would require the division of an octave into 12 notes, each of which would be used to create six other consonant intervals. The purpose of this study is to analyze four well-known historical tunings to evaluate how well each one approximates perfect harmony. The analysis consists of a general evaluation in which all consonant intervals are given equal weighting and a specific evaluation for three preludes from Bach's "Well-Tempered Clavier," for which intervals are weighted in proportion to the duration of their occurrence. The four tunings, 5-limit just intonation, quarter-comma meantone temperament, well temperament (Werckmeister III), and equal temperament, are evaluated by measures of centrality, dispersion, distance, and dissonance. When all keys and consonant intervals are equally weighted, equal temperament demonstrates the strongest performance across a variety of measures, although it is not always the best tuning. Given C as the starting note for each tuning, equal temperament and well temperament perform strongly for the three "Well-Tempered Clavier" preludes examined.
authors.affiliation: The Pingry School, Martinsville, New Jersey 08836, USA. michaelfpage@att.net
authors.forename: Michael F
authors.initials: MF
authors.lastname: Page
mesh_terms: D049668:History, 15th Century; D049669:History, 16th Century; D049670:History, 17th Century; D049671:History, 18th Century; D049672:History, 19th Century; D049690:History, Ancient; D049691:History, Medieval; D006801:Humans; D009146:Music / Q000266:history*; D010897:Pitch Discrimination / Q000502:physiology*; D011571:Psychoacoustics