Fuga in C Major
In our case, we have tied observations, i. Then, the average rank is assigned to each of the tied observations. Often p values are reported when the results of the tests are presented; a p value is the probability of obtaining an effect at least as extreme as the one observed in the sample. A p value less than some specified value usually 0. Calculations were made, using the routine kruskal.
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We find p values for the analysis of differences between categories in the cases of X 1 , X 2 and X 3 as 0. In other words, Kruskal—Wallis test reports no differences between the composer categories regarding X 1 or X 2 , while differences exist for X 3 not unreasonable, cf. To find between which groups there are differences, conventionally a suitable test for multiple comparisons also called post-hoc test is employed. For variables X 4 , X 5 , X 6 , for each variable no significant differences were found between categories, using the same methodology as above.
Fugue in C major
Note that in particular the variables X 4 and X 6 cf. The use of Kruskal—Wallis test might not be warranted. In this section, we investigate statistical relationships between the three variables considered by approaches from the field of multivariate statistics or, alternatively phrased, statistical learning. Within the latter paradigm, we here face a situation of so-called unsupervised learning, since we have, at least at this stage in the investigation, no specific input or output variables.
More on the techniques can be found e. New York: Springer-Verlag. By an analysis of principal components PCA , a low-dimensional representation of a data set is found, which captures as much information of the variation as possible. Overall aims are usually data reduction in the case of many variables and interpretation. In our case, the latter is in focus since we only consider six variables.
Mathematically speaking, linear combinations of the variables are studied. For observations, the first sample principal component is obtained, notated by Z 1. After the first principal component Z 1 has been deduced, the second principal component, Z 2 , is found. This is the linear combination which has the maximal variance out of all linear combinations uncorrelated with Z 1. In practice, software readily performs the optimisation in the PCA. There is also a connection to an eigenvalue-eigenvector problem for the sample covariance or correlation matrix of the involved variables, see e.
Johnson and Wichern Johnson, Richard A. Applied Multivariate Statistical Analysis. New Jersey: Pearson Education International. For our data, we find using the software R and the routine prcomp the following three first principal components:. Typically cumulative percentage of the total variance is reported. Moreover, usually an attempt is made of interpreting the coefficients, or loadings, as these are occasionally called.
Here, the first essentially is a weighted linear combination of the variables, with positive weights. Less weight is put on x 4 , initial interval in semitones. The second contrasts the x 1 and x 3 length and number of pitch classes, in a sense overall measures of the subject against x 4 and x 6 interval features, inner construction of subject. A visualisation is often made by plotting the observations in a plane spanned by the two principal components.
Observations plotted on the two first principal components categories shown.
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Figure 8. Figure 9. Observations plotted on the two first principal components indexed works, for reference. Upper-left Shorter themes, mostly diatonic nature. Shorter themes, mostly chromatic nature. Longer themes, often chromatic nature. Reubke fugue from organ sonata , Reger 2nd fugue subject, op. Dawes Dawes, Chris. Note that the descriptions of subjects as being of mostly diatonic or chromatic nature were made by the author reading and analysing the score, not by quantitative methodology.
Bach: obs. They have in common considerable lengths. It stands out somewhat from the other Bach works, and on stylistic grounds, this might be reasonable. In fact, for this work by Bach from his early production, quoting Williams Williams, Peter. The Organ Music of J. Cambridge: Cambridge University Press. Bach: C major fugue, BWV Clearly, the work is an early and imaginative response to the music of established masters, with marked similarities in figuration, texture, harmony and use of the organ, all of these implying a common genre. Reger: obs. This is a fugue in C major, op. This could be interpreted as a subject of type Spielfuge that its length does not show relatively many unique notes — a mostly diatonic theme.
Reger: C major fugue, op.
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These are located middle low z 1 , middle high z 2 , and are Reger fugues with opus , opus 7 D minor, second subject and opus These are quite early works in Reger's production. Reger subjects: op. Others: obs. For high z 2 , obs. Stanford: C minor fugue op. Finally, a remark on scaling of data. A scaling to unit variance is often strongly recommended, see e.
James et al. If not done, the magnitudes of the variables or units chosen may influence the result. The results given above were obtained after scaling. If not performing scaling, the first principal component would be dominated by X 1 due to its in general larger numbers The loading on variable X 1 is then as high as 0. We now consider regression models for the analysis. Preliminary investigations show that for all three composer categories, the correlations between X 1 , X 2 and X 3 are positive. In regression models, the response variable is of crucial importance and we consider two situations, where X 2 and X 3 , respectively, act as response variables.
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Of primary interest is then to see whether there is a difference between the three categories of composers. We here consider a regression model with X 2 as a response. As this is a count variable, we might choose Poisson regression as a first option. In our application, we might consider the covariate X 1 length entering per se, or in the form of a so-called offset term.
Probability and Risk Analysis. An Introduction for Engineers. Berlin: Springer-Verlag. Furthermore, the categories enter as so-called dummy variables. The three categories imply, using treatment coding, that two dummy variables need to be introduced. Goodness of fit is often checked by examining closer the residual deviance of the fitted model, see e.
Introduction to General and Generalized Linear Models. Hence, the model does not fit adequately, and another option must be taken. Often a negative-binomial response is considered in regression models for counts, when the simpler Poisson assumption fails Hilbe Hilbe, Joseph M. Negative Binomial Regression.
With X 1 as an offset, we then obtain, using the routine glm. The R environment requires that a baseline reference is given for the dummy variables.
Alchemy of Genius: A J.S. Bach and Chopin Musical Pairing
From the summary, we note further that X 4 initial interval , is clearly non-significant p value 0. Variables X 3 , X 5 and X 6 , on the other hand, are significant in the model. We now consider X 3 as the response variable and investigate its dependence on other variables. However, in practice X 3 cannot attain the value zero no music , and a fugue subject containing only one note would be peculiar; entirely rhythmic in nature and so far not encountered in music to the author's knowledge.