Define hyperplanes for transposition-sensitive arrangements

The "black" hyperplane arrangement compares a set's scale degrees individually to the pitches of edoo(card) (where card is the number of notes in set). This primarily has the purpose of attending to the overall transposition level of a set. Most applications of Modal Color Theory assume transpositional equivalence, but occasionally it is useful to relax that assumption. Sum class (Straus 2018, doi:10.1215/00222909-7127694) is a natural precise way to track this information, but the "black" arrangements do so qualitatively in the spirit of modal color theory. make_black_ineqmat() returns only the inequality matrix for the "black" arrangement, while make_gray_ineqmat() for convenience combines the results of make_white_ineqmat() and make_black_ineqmat().

Usage

make_black_ineqmat(card)

make_gray_ineqmat(card)

Arguments

card

The cardinality of the scale(s) to be studied

Value

A card by card+1 inequality matrix (for make_black_ineqmat()) or the result of combining white and black inequality matrices (in that order) for make_gray_ineqmat().

See also

make_white_ineqmat()

Examples

# The set (1, 4, 7)'s elements are respectively below, equal to, and
# above the pitches of edoo(3).
test_set <- c(1, 4, 7)
signvector(test_set, ineqmat=make_black_ineqmat(3))
#> [1]  1  0 -1

# The result changes if you transpose test_set down a semitone:
signvector(test_set - 1, ineqmat=make_black_ineqmat(3))
#> [1]  0 -1 -1

# The results from signvector(..., ineqmat=make_black_ineqmat) can
# also be calculated with coord_to_edo():
sign(coord_to_edo(test_set))
#> [1]  1  0 -1
sign(coord_to_edo(test_set - 1))
#> [1]  0 -1 -1