musicMCT (version 0.1.2)

emb: How many instances of a subset-type exist within a scale? How many scales embed a subset?

Description

David Lewin's EMB and COV functions: see Lewin, Generalized Musical Intervals and Transformations (New Haven, CT: Yale University Press, 1987), 105-120. For EMB, given a group ("CANON") of transformations which are considered to preserve a set's type, find the number of instances of that type in a larger set (scale). Lewin characterizes this generally, but emb() only offers \(T_n\) and \(T_n / T_nI\) transformation groups as available canonical groups. Conversely, Lewin's COV function asks how many instances of a scale type include subset: this is implemented as cover() (not cov()!).

Usage

emb(subset, scale, canon = c("tni", "tn"), edo = 12, rounder = 10)
cover(subset, scale, canon = c("tni", "tn"), edo = 12, rounder = 10)

Value

Integer: count of subset or scale types satisfying the desired relation.

Arguments

subset

Numeric vector of pitch-classes in any representative of the subset type (Lewin's X)

scale

Numeric vector of pitch-classes in the larger set to embed into (Lewin's Y)

canon

What transformations should be considered equivalent? Defaults to "tni" (using standard set classes) but can be "tn" (using transposition classes)

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

emb(c(0, 4, 7), sc(7, 35))
emb(c(0, 4, 7), sc(7, 35), canon="tn")

# Works for continuous pc-space too:
emb(j(1, 3, 5), j(dia))
emb(j(1, 2, 3, 5, 6), j(dia))
emb(j(1, 2, 4, 5, 6), j(dia), canon="tn")

emb(c(0, 4, 7), c(0, 1, 3, 7))
emb(c(0, 4, 7), c(0, 1, 3, 7), canon="tn")

emb(c(0, 4), c(0, 4, 8))
cover(c(0, 4), c(0, 4, 8))

harmonic_minor <- c(0, 2, 3, 5, 7, 8, 11)
cover(c(0, 4, 8), harmonic_minor)
cover(c(0, 4, 8), harmonic_minor, canon="tn")