Methods and Results of the Analysis

The mean and standard deviation values were the primary focus of this analysis.  If either of these values had decreased over time, this would indicate a trend towards the given standard.  The opposite would be true if these values had increased over time.  Since the question to be answered relates to the existence of an evolution towards equal temperament, this tuning was the most important standard used.  The other two standards were included to see if a different type of evolution preceded the acceptance of equal temperament.

Since all of the data sets contained multiple outliers, different transformations were experimented with.  Square root transformations worked the best for most, but the just intonation standard still contained multiple outliers.  Linear regressions with the dates of publication as a predictor did not produce convincing results.  The best of all these regressions is the mean deviation with equal temperament as the standard (p-value of .001 w/DF 1).  This regression has an R-square value of (8.5%), which is not enough to prove any correlation.  The R-square value improved to (33%) after adding significant binary categorical variables to represent the countries in which each tuning was printed (found by stepwise regression).  The null hypothesis for the “date” coefficient was not refuted for this regression (p-value of .106 w/DF of 3).  In other words, adding the location of publications to the prediction made the date of publications statistically insignificant.  Regardless, the R-square value was still too low to prove any correlations. (see Appendix B for all tests and graphs mentioned in this paragraph)

A possible problem with the analysis could be the repetition of dates.  Some theorists and mathematicians published multiple methods in one publication, particularly in the seventeen hundreds.  They often included theories that would span a wide range of deviations.  This becomes evident when viewing a regression plot of the equal temperament mean deviation versus time (Appendix B).  To eliminate this factor as a possible “smokescreen” hiding trends, the data table was reduced by selecting only the first two theories from any publication.  There was no effort to select only “important” tunings.

With this data set, the residuals are more evenly spread out versus fits.  The R-square value of the “date” regression for this modified set is (2.2%).  The R-square value for stepwise regression with locations added is also lower than the first (17.6%).  The null hypothesis is not refuted for any of the coefficients except the binary variable for Germany (p-value of .001 w/DF 2).  No trends in the data could be identified with any regression models described so far.  Curve fittings were also experimented with, but with similar results (Appendix C).

Given the lack of any trends found by regression, classification methods were next experimented with.  Many divisions of the population were tried.  Dividing the time period into four groups (each about 92 years long) was the selected division as it produced the most significant results.   These quarters contain unequal numbers of cases, but the subgroup variances for equal temperament deviations (both mean and standard) are acceptable for an ANOVA.  The subgroup variances are too varied in size for the other standards (Just and Pythagorean), and non-parametric tests such as Mood’s median must be run when including these variables (Appendix D).

ANOVA’s and Mood’s median tests were preformed for each combination of standards and deviation by quarter.  The only two tests to show a distinction between quarters were both equal temperament standards.  Although the null hypothesis is refuted for these tests (p-value of .001 w/ DF of 92 for both tests), not all quarters are distinct from each other when considering the (95%) confidence intervals (the standard deviation test is shown below). 

 

 

 

 

 

 

 

 

The largest difference is between the second (1533-1623) and the third (1624-1715).  The third quarter has a dramatic shift closer to equal temperament with the fourth quarter returning closer to the second.  Unfortunately, the data could be skewed because of the previously mentioned problem of multiple tunings per publication.  Running the same test using only the first two tunings of each publication results in the following: 

 

 

 

 

 

 

 

 

With the p-value shifting from (.001) to (.379), it is obvious that the repeated values for dates have a significant impact on any analysis.  Removing cases from the repeated dates is not an appropriate answer to this problem.  It assumes an importance of the first two tunings of each publication and excludes numerous repeated values from important theorists such as Werkheimer and Neidhardt.  Therefore, tests were run on a new set of data containing the average of deviations from each publication (Appendix E).  This insures that no dated publication is given more weight than others and all tunings from each publication are represented.  Some theorists are counted two or three times because of multiple publications.  However, these publications have separate dates and can actually help in identifying trends.

The new data is more evenly spread than the first, but a square root transformation still improved the normality and brought the variances closer together in size.  The Just Intonation numbers contain multiple outliers for mean deviation regardless of transformations and the Pythagorean has one outlier for each of its graphs.  The dates are distributed in a more normal pattern, and are linear when presented in a series plot (Appendix F).

Once again, linear regressions were run on each of the standards. Using the date as a predictor, none of the standards produced a regression with coefficients disproving the null hypothesis (Appendix G). The only regression that contained significant coefficients was one predicting the date using a combination of the mean deviations of Pythagorean and Equal temperament standards. The R-square value for this regression is (16.9%). Adding the country data raises this R-square value. The most successful combination is below (full output can be seen in Appendix G).

 

 

 

 

 

 

Despite these stronger values, there are problems with the model.  Removing three samples with extreme residuals alters the regression to one with insignificant tuning variables.

 

 

Given the instability of this regression, classification methods were tried with the averaged data set.  Both transformed equal temperament variables grouped by quarters passed the Levene and Bartlett tests for equal variance.  The transformed mean deviation of just temperament barely passed these tests with p-values just above (.05) and nonparametric tests will be used for this variable.  The transformed standard deviation of just intonation deviation was more acceptable.  Both Pythagorean data sets were closer in variance with the untransformed data (Appendix H).

Based on the characteristics listed above, the appropriate tests (ANOVA or Mood’s median) were applied to the appropriate data (transformed or original) for each variable.  Every test failed to refute the null hypothesis (Appendix H).  Therefore, there is no distinction between any of the deviations when grouped into four quarters of the time period in question.

Multivariate tests were done with both data sets mentioned in this section.  These were generally used to explore the data before running the tests already described.  The first two tests were stepwise discriminant analyses run on the data as it was originally entered.  All transformed deviation standards were included for these tests.  The second test also included the binary country of publication data.  The first test produced two discriminant functions with the mean deviations of the equal tempered and Pythagorean data.  Only (46%) of cases were correctly predicted with this function.  The second stepwise discriminant analysis automatically removed all variables regarding pitch (Appendix I).  Similar discriminant analyses were attempted on the averaged data, but all variables (unaltered and transformed) were eliminated from any test (Appendix J).  Although redundant, these multivariate tests confirm the results of the other tests described in this section.