project_onto: Closest point on a given flat

In musicMCT: Analyze the Structure of Musical Scales

Closest point on a given flat

Description

Projects a scale onto the nearest point that lies on a target flat of the hyperplane arrangement. project_onto() determines the target flat from a list of linearly independent rows in ineqmat which define the flat. match_flat() determines the target by extrapolating from a given scale on that flat. Note that while the projection lies on the desired flat (i.e. it will have all of the necessary 0s in its sign vector), it will not necessarily belong to any particular color. (That is, projection doesn't give you control over the 1s and -1s of the sign vector.)

Usage

project_onto(
  set,
  target_rows,
  ineqmat = NULL,
  start_zero = TRUE,
  edo = 12,
  rounder = 10
)

match_flat(
  set,
  target_scale,
  start_zero = TRUE,
  ineqmat = NULL,
  edo = 12,
  rounder = 10
)

Arguments

set	Numeric vector of pitch-classes in the set
target_rows	An integer vector: each integer specifies a row of ineqmat which helps to determine the target flat. The rows must be linearly independent.
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.
start_zero	Boolean: should the result be transposed so that its pitch initial is zero? Defaults to TRUE.
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.
target_scale	A numeric vector which represents a scale on the target flat.

Value

A numeric vector of same length as set, representing the projection of set onto the flat determined by target_rows or target_scale.

Examples

minor_triad <- c(0, 3, 7)
project_onto(minor_triad, 3)
project_onto(minor_triad, 1)
project_onto(minor_triad, c(1, 3))
# This last projection results in the perfectly even scale
# because that's the only scale on both hyperplanes 1 and 3.

major_scale <- c(0, 2, 4, 5, 7, 9, 11)
projected_just_dia <- match_flat(j(dia), major_scale)
print(projected_just_dia)

# This is very close to fifth-comma meantone:
fifth_comma_meantone <- sim(sort(((0:6) * meantone_fifth(1/5))%%12))[,5]
vl_dist(projected_just_dia, fifth_comma_meantone)

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