musicMCT (version 0.4.0)

move_to_hyperplane: Intersection of a line with a hyperplane

Description

Navigate scale space by finding the point where a given line intersects with a given hyperplane. Hyperplanes are specified by the row and ineqmat parameters. That is, the hyperplane is given by the rowth row of the specified ineqmat. An arbitrary hyperplane can be specified by entering row=1 with the desired hyperplane as a 1-row matrix as the input to ineqmat.

For the line, two different use cases are available. In the first, set is specified. In this case, the line in question is the given set's "hue" (i.e., the line which runs from edoo() to set). This is useful when exploring non-central arrangements such as the Rothenberg arrangements (make_roth_ineqmat()); for an arrangement centered on edoo(), it will simply return edoo(). In the second, set is left NULL and both point and direction must be specified. Here, the given line is the one which runs through point parallel to the vector specified by direction.

Usage

move_to_hyperplane(
  row,
  set = NULL,
  point = NULL,
  direction = NULL,
  ineqmat = NULL,
  method = c("taxicab", "euclidean", "chebyshev", "hamming"),
  edo = 12,
  rounder = 10
)

Value

A list with three entries: set, dist, and sign. set is a numeric vector with the intersection point of the given line and hyperplane. dist gives the voice-leading distance (as measured using method) from input set or point to the result set. sign indicates whether the intersection point lies along the given direction or opposite it. (For instance, when a hue is specified by input set, a positive sign indicates that the intersection belongs to the same color as input set, whereas a negative sign indicates that the intersection is a scalar involution of the input.)

If the line lies entirely on the given hyperplane, the returned set simply matches the input, while dist and sign are 0. If the line and hyperplane do not intersect, the result set is a vector of NAs the same length as the input set or point; dist is Inf and sign is NA. In both of these cases, a warning is given.

Arguments

row

Integer specifying the hyperplane to be intersected as a row of ineqmat.

set

Numeric vector representing a scale. Defaults to NULL; if specified, will give the intersecting line as the scale's "hue."

point

Numeric vector representing a point on the line. Overridden by set.

direction

Numeric vector representing the direction of the line. Overridden by 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.

method

What distance metric should be used? Defaults to "taxicab" but can be "euclidean", "chebyshev", or "hamming".

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.

Examples

major_triad <- c(0, 4, 7)
# Let's find where the major triad's hue first intersects a Rothenberg hyperplane:
move_to_hyperplane(3, set=major_triad, ineqmat="roth")
same_hue(major_triad, c(0, 4, 6))
strictly_proper(major_triad)
strictly_proper(c(0, 4, 6))

# But the major triad's hue intersects every MCT hyperplane at the center of the space:
move_to_hyperplane(3, set=major_triad, ineqmat="mct")

# Let's move away from the major triad in other directions than its hue:
lower_third <- c(0, -1, 0)
move_to_hyperplane(1, point=major_triad, direction=lower_third)
move_to_hyperplane(2, point=major_triad, direction=lower_third)
move_to_hyperplane(3, point=major_triad, direction=lower_third)