musicMCT (version 0.1.2)

make_roth_ineqmat: Define hyperplanes for Rothenberg arrangements

Description

Although the Rothenberg propriety of a single scale can be computed directly with isproper(), propriety is a scalar feature (like modal "color") which is defined by a scale's position in the geometry of continuous pc-set space. That is, propriety, contradictions, and ambiguities are all determined by a scale's relationship to a hyperplane arrangement, but the arrangements which define these properties are different from the ones of Modal Color Theory. make_roth_ineqmat() creates the ineqmats needed to study those arrangements, similar to what makeineqmat() does for MCT arrangements. make_rosy_ineqmat() creates the combination of Rothenberg and MCT arrangements. (The name puns on the "Roth" of Rothenberg meaning "red," lending a reddish or rosy tint to the "colors" of the MCT arrangement.)

Usage

make_roth_ineqmat(card)
make_rosy_ineqmat(card)

Value

A matrix with card+1 columns and k rows, where k is the number of hyperplanes in the arrangement.

Arguments

card

The cardinality of the scale(s) to be studied

Details

Each row of a Rothenberg ineqmat represents a hyperplane, just like the rows produced by makeineqmat(). The rows are normalized so that their first non-zero entry is either 1 or -1, and their orientations are assigned so that a strictly proper set will return only -1s for its sign vector relative to the Rothenberg arrangement. A 0 in a Rothenberg sign vector represents an ambiguity. Note that the Rothenberg arrangements are never "central," which means that the hyperplanes do not all intersect at the perfectly even scale. (It is clear that they must not, because perfectly even scales have no ambiguities.) These arrangements also grow in complexity much faster than the MCT arrangements do: for tetrachords, MCT arrangements have 8 hyperplanes while Rothenberg arrangements have 22. For heptachords, those numbers increase to 42 and 259, respectively. Thus, this function runs slowly when called on cardinalities of only modest size (e.g. 12-24).

Examples

c_major <- c(0, 2, 4, 5, 7, 9, 11)
hepta_roth_ineqmat <- make_roth_ineqmat(7)
isproper(c_major)
cmaj_roth_sv <- signvector(c_major, ineqmat=hepta_roth_ineqmat)
table(cmaj_roth_sv)
hepta_roth_ineqmat[which(cmaj_roth_sv==0),] 
# This reveals that c_major has one ambiguity, which results from
# the interval from 4 to 7 being exactly half an octave.