Stylistic information in pitch-class distributions

ABSTRACT

This study examines pitch-class distributions in a large body of tonal music from the seventeenth, eighteenth and nineteenth centuries using the DFT on pitch-class sets. The DFT, applied over the pitch-class domain rather than a temporal domain, is able to isolate significant and salient qualities characteristic of tonal pitch-class distributions, such as diatonicity and triadicity. The data reveal distinct historical trends in tonal distributions, the most significant of these is a marked decrease in diatonicity in the eighteenth and nineteenth centuries. Comparing distributions for beginnings, endings, and whole pieces reveals a strong similarity between beginnings and whole pieces. Endings, by contrast, are more distinct in the properties of their distributions overall and show some historical trends not shared by beginnings and whole pieces, whose differences do not appear to interact with composer date.

KEYWORDS:

• Pitch-class distribution
• tonality
• mode
• harmony
• corpus analysis
• discrete Fourier transform

Acknowledgements

Many thanks to Matthew Chiu, who wrote code to calculate distributions from the YCA data and devoted much time and effort to processing the large body of data used in this study, which could not have been completed without his efforts.

Notes

1 Pieces by Byrd listed as ending in a different mode were retained because they appear to be Picardy-third endings rather than genuine key changes.

2 Previous studies, especially Albrecht and Shanahan (2013), show that the chorales are idiosyncratic due especially to their often short length and rapid harmonic rhythm, which is particularly important when isolating beginnings and endings (see below). There are also a large number of them in the corpus, so they would disproportionately affect the results for Bach if included.

3 One might wonder how all qualities can decrease in size, given that data are normalised. The DFT conserves total power, the sum of squared magnitudes. But data are normalised by cardinality (f₀) rather than total power. A distribution that is more evenly spread out between the pitch classes will have a greater share of its power in f₀, which means it will normalise to a lower-power distribution, and all components will tend to be lower. Therefore, an overall decrease of DFT magnitudes indicates a more chromatic distribution in the sense of more evenly spread between all pitch classes.

4 This appears not to be the case for f₄, but this is an artifact of the conversion to polar coordinates for small-magnitude components. The trendlines by date are parallel in complex space, but because the one for beginnings passes by on a different side of, and much further from, the origin, the trend in magnitudes looks different.

5 This is not a perfect one-to-one correspondence, however, and the numerical correspondence of category 5 and f₅ is coincidental. Category 2 corresponds roughly to f₆ and category 6 imperfectly to f₂, while category 3 tends to correspond to f₄ and category 4 to f₃. In general, such correspondences need not exist at all – see Quinn (2006–2007).