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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Interpretation of the Results
It is harder to prove that something does not exist than to prove that it does. This analysis did not find any proof of an evolution leading to equal temperament. If one existed, it would likely be reflected in the writing of theorists and mathematicians. Even the tests that came close to identifying a trend did so in a manner that is counterintuitive to what was being searched for. For example, the regression based on averaged data predicted that equal temperament would occur earliest in Spain in the year 2168 CE. Even if this were a valid test, equal temperament becoming the standard would have been a radical change in the nineteenth century. As well as not finding any trends, dividing the tunings into groups of time failed to find a statistically significant difference between any two quarters. The first and last quarters were not distinct even though they were separated by almost two hundred years. This is not to say tuning methods did not change. A dot plot of the deviations from equal temperament over time shows an oscillating pattern, occasionally coming close to equal temperament (Appendix K). Many of the theorists and mathematicians mentioned in this study were familiar with the concept of equal temperament. Many of them were also capable of the root extraction or logarithm methods for accurately finding equal tempered measurements. Those who favored well temperaments sometimes came close to our current tuning. Time series plots featuring individual well temperament advocates (who published more than two tunings) show that they often approached equal temperament and then strayed away from it in later tunings (Appendix K). For example, Neidhardt’s Third Circle Number Five came very close to equal temperament (mean deviation of .7 cents and standard deviation of 1.3 cents). Twenty-six years later, Neidhardt published his Sample Number Two tuning, which was a drastic shift away from equal temperament (mean deviation of 8.0 cents with a standard deviation of 9.1 cents). This could be due to a conflict between mathematical perfection and aesthetics. It is possible that authors presenting multiple scales per publication used most of them as examples to refute, while promoting one scale. The references used for this research presented each of these scales as equally important, but reading and analyzing each of the original publications is the only verification that all scales included for analysis represent the true intents of each author. Therefore, future research is needed to substantiate the findings of this thesis. The standards and analysis outlined in chapters eleven and twelve could be reapplied to a filtered version of the data set if any of the included scales were in fact “straw men.” Given the obscure nature of the publications in question, this could prove to be a long and arduous process.
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