Usually, it is most intuitive to music theorists to represent
a scale as a vector of the pitch-classes it contains. However, for certain
computations in the setting of Modal Color Theory, it is more convenient
to use a coordinate system with the "white" perfectly even scale as the
origin (because this is the point where all of the hyperplanes
in the arrangement defining scalar "colors" intersect). Therefore, these
two functions convert between the two coordinate systems: coord_to_edo
takes in a scale represented by its pitch classes and returns its
displacement vector from "white" and coord_from_edo does the reverse.
Details
It should be noted that the representative "white" scale used is not
necessarily the closest one to the scale in question. Instead, it is
the unique transposition of white that has 0 as its first coordinate.
This is natural in the context of Modal Color Theory, which essentially
always assumes transpositional equivalence with \(x_0 = 0\). The closest
transposition of "white" to set will be the one that has the same sum
class as set, guaranteeing that the voice leading between them is
"pure contrary" (Tymoczko 2011, 81ff; explored further in Straus 2018
doi:10.1215/00222909-7127694).
Examples
dominant_seventh_chord <- c(0, 2, 6, 9)
coord_to_edo(dominant_seventh_chord)
#> [1] 0 -1 0 0
ait1 <- c(0, 1, 4, 6)
ait2 <- c(0, 1, 3, 7)
coord_to_edo(ait1)
#> [1] 0 -2 -2 -3
coord_to_edo(ait2) # !
#> [1] 0 -2 -3 -2
weitzmann_pentachord <- coord_from_edo(c(0, -1, 0, 0, 0)) # See note 53 of "Modal Color Theory"
convert(weitzmann_pentachord, 12, 60)
#> [1] 0 7 24 36 48
coord_to_edo(weitzmann_pentachord)
#> [1] 0 -1 0 0 0