signvector: Detect a scale's location relative to a hyperplane...

In musicMCT: Analyze the Structure of Musical Scales

Detect a scale's location relative to a hyperplane arrangement

Description

As "Modal Color Theory" describes (pp. 25-26), each distinct scalar "color" is determined by its relationships to the hyperplanes that define the space. For any scale, this function calculates a sign vector that compares the scale to each hyperplane and returns a vector summarizing the results. If the scale lies on hyperplane 1, then the first entry of its sign vector is 0. If it lies below hyperplane 2, then the second entry of its sign vector is -1. If it lies above hyperplane 3, then the third entry of its sign vector is 1. Two scales with identical sign vectors belong to the same "color".

vl_signvector() is a convenience wrapper that computes the sign vector of a voice leading rather than a set. It applies coord_from_edo() to its first argument and defaults to "infrared" for its ineqmat argument.

Usage

signvector(set, ineqmat = NULL, edo = 12, rounder = 10)

vl_signvector(vl, ineqmat = "infrared", edo = 12, rounder = 10)

Arguments

set	Numeric vector of pitch-classes in the set
ineqmat	Specifies which hyperplane arrangement to consider. By default (or by explicitly entering "mct") it supplies the standard "Modal Color Theory" arrangements of getineqmat(), but can be set to strings "white," "black", "gray", "roth", "infrared", "pastel", "rosy", "infrared", or "anaglyph", giving the ineqmats of make_white_ineqmat(), make_black_ineqmat(), make_gray_ineqmat(), make_roth_ineqmat(), make_infrared_ineqmat(), make_pastel_ineqmat(), make_rosy_ineqmat(), make_infrared_ineqmat(), or make_anaglyph_ineqmat(). For other arrangements, this parameter accepts explicit matrices.
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.
vl	Numeric vector representing a voice leading, whose entries should represent the motions of voices in registral order from low to high. (That is, the first entry should represent the motion of the lowest voice, and the last entry should represent the motion of the highest voice.) Only used by vl_signvector(), where it substitutes for signvector()'s set argument.

Value

A vector whose entries are 0, -1, or 1. Length of vector equals the number of hyperplanes in ineqmat.

Examples

# 037 and 016 have identical sign vectors because they belong to the same trichordal color
signvector(c(0, 3, 7))
signvector(c(0, 1, 6))

# Just and equal-tempered diatonic scales have different sign vectors because they have 
# different internal structures (e.g. 12edo dia is generated but just dia is not). 
dia_12edo <- c(0, 2, 4, 5, 7, 9, 11)
just_dia <- j(dia)
isTRUE( all.equal( signvector(dia_12edo), signvector(just_dia) ) )

# For voice leadings, consider the opening of *Tristan und Isolde*:
tristan_chord <- c(5, 11, 3, 8)
v7_of_a <- c(4, 8, 2, 11)
tristan_vl <- v7_of_a - tristan_chord
vl_signvector(tristan_vl)
# which is identical to
signvector(coord_from_edo(tristan_vl), ineqmat="infrared")

musicMCT documentation built on June 21, 2026, 9:06 a.m.