Perfect harmony
Title: Perfect harmony: A mathematical analysis of four historical tunings
Author: Michael F. Page.
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. © 2004 Acoustical Society of America.
The PDF file can be found from the website http://lit.gfax.ch//tunings/MathAnalysisOf4HistoricalTunings.pdf.
The table below are measures of central tendency and dispersion for absolute deviations from perfect harmony. Two measures of central tendency (the mean and median) and three measures of dispersion (the variance, range, and interquartile range) are shown. Tunings include those mentioned above as well as other selected tunings. Some tunings include links explaining the appropriate tunings. The table is ordered by mean (lowest to highest).
Title: Perfect harmony: A mathematical analysis of four historical tunings
Author: Michael F. Page.
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. © 2004 Acoustical Society of America.
The PDF file can be found from the website http://lit.gfax.ch//tunings/MathAnalysisOf4HistoricalTunings.pdf.
The table below are measures of central tendency and dispersion for absolute deviations from perfect harmony. Two measures of central tendency (the mean and median) and three measures of dispersion (the variance, range, and interquartile range) are shown. Tunings include those mentioned above as well as other selected tunings. Some tunings include links explaining the appropriate tunings. The table is ordered by mean (lowest to highest).