musicMCT (version 0.1.2)

coord_to_edo: Coordinate systems for scale representation

Description

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.

Usage

coord_to_edo(set, edo = 12)
coord_from_edo(set, edo = 12)

Value

Numeric vector of same length as set. Same scale, new coordinate system.

Arguments

set

Numeric vector of pitch-classes in the set

edo

Number of unit steps in an octave. Defaults to 12.

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)

ait1 <- c(0, 1, 4, 6)
ait2 <- c(0, 1, 3, 7)
coord_to_edo(ait1)
coord_to_edo(ait2) # !

weitzmann_pentachord <- coord_from_edo(c(0, -1, 0, 0, 0)) # See note 53 of "Modal Color Theory"
convert(weitzmann_pentachord, 12, 60)
coord_to_edo(weitzmann_pentachord)