The Development of the Equal Temperament Scale
Evolution or Radical Change?
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The Development of the Equal Temperament ScaleEvolution or Radical Change?
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Data Analyzed in this StudyThe main question this analysis seeks to answer is: Did the acceptance of the equal tempered scale fit into an evolutionary development or was it a radical departure from previously accepted methods? The country of origin for each tuning is included in the analysis because of the influence this variable might have. However, the study does not seek to answer any questions regarding the origin of each tuning method; the variable is only used to identify or account for any disparity in the data resulting from the variety of locations. The tunings used for this research come from various sources, with the majority from Tuning and Temperament by J. Murray Barbour.[i] Barbour used cents to analyze many historic temperaments. He also included references to the initial publications of these methods. Cross-referencing this data with tables found on the Internet, over ninety tunings with dates and countries of first publication are compiled for this research (Appendix A). Equal temperament attempts were not included for this study. There is no method to identify what tuning methods were prevalent for any given time or place. Debate is still ongoing concerning which scale Bach’s Well Tempered Clavier was designed for. If records are insufficient for a piece whose title alludes to a family of tuning systems, one can imagine the difficulty in accurately pinpointing the original temperament for other pieces. In this study, tunings are ordered and placed by their dates and locations of first publication. This is not an accurate portrayal of the music heard during those times, but theorists and mathematicians were probably influenced by the current practice and vice versa. Rather than skewing the results by trying to assess which tunings were important (based on modern notions), no documented twelve note tuning system from Europe was purposely excluded from this data set. The pitch systems were first analyzed by mean and standard deviation from three chosen standards: Equal Temperament, Just Intonation, and Pythagorean. These were chosen because they are all somewhat static and are historically significant. Total deviation values for a scale were calculated by adding the absolute value of differences between cent values of each note and the corresponding cent values from the standard. This sum was then divided by eleven to obtain the mean deviation (see example below). Eleven was used instead of thirteen because the first and last notes never change. Example
Calculation of Mean and Standard Deviation (using Silbermann w/ET as standard)
[i]
Barbour, J. Murry. Tuning
and Temperament – A Historical Survey, (East Lansing: Michigan State
College Press, 1951), | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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