- 448 -Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory 
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Key Character and Key Statistics. Key character is determined by several factors. One is the complex aspect of keyboard tuning. The composers considered here strongly differ in their preference for Ab -Major. Exemplarily we will inspect the key character of this key and consider its impact on key preference. In mean tone tuning w.r.t. C, the wolf fifth g# - eb sounds very rough, since this fifth is 35.7 Cent higher than the just fifth (Meister (1991) p. 81 cited in Grönewald (2003)). Bach used Neidhardt’s tuning (Lindley2001). Even though in Neidhardt’s tuning the wolf is eliminated, Bach used Ab -Major and g# -minor only in WTC. Bach’s avoidance of these keys could be due to the reminiscence that these keys would have sounded odd in a tuning with the wolf on ab /g# .

On the other hand Ab -Major was the favored key of Chopin. What did make Chopin prefer this key? Did Ab -Major have a special key character for Chopin? Lindley (2003) points out »the tradition of tender affects« for this key which may have appealed to Chopin’s character. Chopin used equal temperament. So all intervals relative to the tonic were the same for all Major keys. For Chopin the key character may have been influenced by 18th century keyboard tuning, that preceded equal temperament. In tunings common at time when »Clementi was flourishing and Beethoven was not yet deaf«, Ab -Major had characteristic differences to tunings of keys with none or only few accidentals, e. g. C-Major. First the Major third ab -c in Ab -Major is stretched compared to c-e in C-Major, second the leading note - tonic step g-ab is squeezed compared to b-c. Therefore this tuning endows Ab -Major with a more nervous character and C-Major with a more solid one. 2

 
2  
The arguments in this subparagraph are from Lindley (2001, 2003) and from personal communication with Mark Lindley.


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Figure 10: Based on the key preference profiles (cf. Figures 8 and 9) for different composers the stylescape (upper) and the biplot of composers/keys (middle) with associated singular values are shown. Capital letters indicate Major keys, small ones minor keys, italics indicate composers. Upper: Haydn (JH) and Mozart (WAM) are very close. Beethoven (LvB) is nearby. Brahms (JB) and Bach (JSB) can be considered a group. Chopin (FC) is an outlier. Middle: In the biplot composers/keys we observe that the Viennese classic (JH, WAM, LvB) gather in the region of D-Major and G-Major. Chopin (FC) maintains his outlier position due to the distant position of his favored Ab -Major key. Lower: The explanatory values for the first (63.23 %) and the two most prominent factors (88 %) in correspondence analysis are high.



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- 448 -Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory