Abstract
The spectral information of a pitch-class set or distribution relates to its interval content and what Ian Quinn calls its harmonic qualities, the magnitudes of a discrete Fourier transform of a pitch-class vector. The spectrum is invariant with respect to transposition and inversion, but the existence of Z-related sets, which have equivalent spectra but are not related by transposition or inversion, means that the spectrum is not a complete description of a set class. We show how to isolate transposition-invariant phase information using products of Fourier coefficients. We describe some of the mathematical features of these coefficient products and show how they encode aspects of tonality, and can be useful for analyzing non-tonal music with an example from Takemitsu’s “Air” for solo flute.
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Notes
- 1.
For 80% of pc-sets, the error is larger than 10%.
- 2.
Not necessarily distinct.
- 3.
The qualification “regular” distinguishes these from an arbitrary product, but since non-regular products are of no evident interest, we will typically omit the qualifier.
- 4.
For example, \(\hat{a}_5 \hat{a}_5 \hat{a}_2\) or \(\hat{a}_4 \hat{a}_4 \hat{a}_4\). These are also regular coefficient products and can have interesting applications, but we focus instead on products of three unique coefficients here.
- 5.
A complication here is that there are two contributors to \(\hat{a}_2\) in the quadratic term, \(\hat{a}_3\hat{a}_7\) and \(\hat{a}_5\hat{a}_5\), and the latter is larger in the distribution derived using the clipping filter. In the original distribution, the phase of \(\hat{a}_2\) is closer to that of \(\hat{a}_3\hat{a}_7\) than \(\hat{a}_5\hat{a}_5\).
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Yust, J., Amiot, E. (2022). Non-spectral Transposition-Invariant Information in Pitch-Class Sets and Distributions. In: Montiel, M., Agustín-Aquino, O.A., Gómez, F., Kastine, J., Lluis-Puebla, E., Milam, B. (eds) Mathematics and Computation in Music. MCM 2022. Lecture Notes in Computer Science(), vol 13267. Springer, Cham. https://doi.org/10.1007/978-3-031-07015-0_23
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