Computes the magnitudes and phases of the DFT components for a given (multi)set
which can be input as either a vector of elements or as a distribution.
(See Amiot (2016) doi:10.1007/978-3-319-45581-5 for an overview of applications
of the DFT in this vein.) Entering a distribution takes priority over an entered set.
Arguments
- set
Numeric vector of pitch-classes in the set
- distro
Numeric vector representing a pitch-class distribution. Defaults to
NULLand overridessetandedoif entered.- edo
Number of unit steps in an octave. Defaults to
12.- rounder
Numeric (expected integer), defaults to
10: number of decimal places to round to when testing for equality.
Value
A 2-by-k real matrix, where k is the number of independent components. The ith column corresponds to the (i-1)th component (so that the first column gives the zeroth component). The first row gives the magnitudes of the components and the second row gives the phases. (See details regarding interpretation of the values: they are scaled by edo/(2*pi) from radians.)
Details
The scaling and orientation of phases corresponds to that used in Yust (2021) doi:10.1093/mts/mtaa0017: phases are reported as multiples of one kth of an octave (where the set is entered in k-edo), and oriented so that the \(\hat{f}_1\) component of a singleton points in the direction of the singleton (i.e. the phase of \(\hat{f}_1\) for pitch class 4 is 4). This differs from the phase values use in other publications, such as Yust (2015) doi:10.1215/00222909-2863409. Magnitudes are not squared, following Amiot (2016) rather than Yust (2021).
Examples
# Compare to Yust (2021), Example 10
reich_signature <- c(0, 1, 2, 4, 5, 7, 9, 10)
dft(reich_signature)
#> f0 f1 f2 f3 f4 f5 f6
#> magnitude 8 0.7320508 0 2 2 2.732051 0
#> phase 0 1.0000000 0 3 4 11.000000 0
# Magnitudes differ from Yust by squaring:
round(dft(reich_signature)[1, ]^2, digits=3)
#> f0 f1 f2 f3 f4 f5 f6
#> 64.000 0.536 0.000 4.000 4.000 7.464 0.000
reich_sig_distribution <- c(1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0)
dft(distro=reich_sig_distribution)
#> f0 f1 f2 f3 f4 f5 f6
#> magnitude 8 0.7320508 0 2 2 2.732051 0
#> phase 0 1.0000000 0 3 4 11.000000 0
# Z-related AITs differ in phase but not magnitude:
ait1 <- c(0, 1, 4, 6)
ait2 <- c(0, 1, 3, 7)
dft(ait1)
#> f0 f1 f2 f3 f4 f5 f6
#> magnitude 4 1.414214 2 1.414214 2 1.414214 2
#> phase 0 2.500000 0 1.500000 2 6.500000 0
dft(ait2)
#> f0 f1 f2 f3 f4 f5 f6
#> magnitude 4 1.414214 2 1.414214 2 1.414214 2
#> phase 0 1.500000 2 10.500000 2 1.500000 6