GROUP TASK #17: The Key of Funeral Marches

Answer

Volume 17 of the New Grove Dictionary of Music and Musicians identifies seven classic funeral marches. As it turns out 5 of those 7 marches are in the key of F minor. Test the hypothesis that funeral marches tend to be composed in the key of F minor.

In order to calculate chi-square, we need to determine the probability of any given musical work being written in F minor.

The following table shows a distribution of keys from a convenience sample of 3,121 works from the common practice Western art music tradition.

Tonic	Major	Minor
C	358	6
C#	2	2
D	194	96
D#	0	2
Eb	125	0
E	24	35
F	757	10
F#	2	4
G	978	119
G#	0	2
Ab	11	0
A	126	61
Bb	198	2
B	2	5
Totals:	2,777	344

Notice that works in F minor account for just 0.32 percent of all works in the sample (10/3121). However, the sample includes both major- and minor-key works. As a proportion of minor-key works, works in F minor account for 2.9 percent of all works (10/344). As it stands, our hypothesis is perhaps poorly defined. One might reasonably argue that funeral marches would naturally all be written in the minor mode. So it seems unfair to include major key works in our calculation of what would be the “expected” distribution. A more conservative statistical test would limit our comparison to the distribution of works in minor keys. If we ignore the major mode, then the probability of a given minor-mode work being in F minor is 2.9 percent (i.e., 10/344). Therefore, for a sample of 7 works, we would normally expect 7 × 10/344 works in F minor. That is, we would expect 0.203489 of a piece.

Another question arises regarding the number of categories. Excluding the major keys, there are 14 categories in the above table, representing 14 possible minor-mode keys. Actually, since there were no instances of minor-mode works in A-flat minor, we might exclude that, reducing the number of minor-mode keys to 13. But are there 13 categories in our test? The hypothesis focuses specifically on F minor. So a better way to think of the data is between two categories: either in F minor, or in some other minor key. Accordingly, our chi-square test will be limited to two categories.

Is the tendency to write funeral marches in F minor statistically significant at the 0.01 level?

Once again, for a sample of 7 minor-key works, we would expect 0.2035 (2.9% of 7) works to be in the key of F minor.

Our chi-square value is 113.05. With one degree of freedom, at the 99% confidence level, the critical value of chi-square is 6.635. Since our calculated chi-square value exceeds the critical value, we can discard the null hypothesis, and accept that the results are consistent with the hypothesis. The results are statistical significant at p<0.01. The data are therefore consistent with the hypothesis that funeral marches tend to be written in the key of F minor.

Actually, our chi-square value is so big, that it would still have been statistically significant even if we had chosen the 99.99% confidence level.