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

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)

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.

Value

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

Examples

emb(c(0, 4, 7), sc(7, 35))
#> [1] 6
emb(c(0, 4, 7), sc(7, 35), canon="tn")
#> [1] 3

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

emb(c(0, 4, 7), c(0, 1, 3, 7))
#> [1] 1
emb(c(0, 4, 7), c(0, 1, 3, 7), canon="tn")
#> [1] 0

emb(c(0, 4), c(0, 4, 8))
#> [1] 3
cover(c(0, 4), c(0, 4, 8))
#> [1] 1

harmonic_minor <- c(0, 2, 3, 5, 7, 8, 11)
cover(c(0, 4, 8), harmonic_minor)
#> [1] 6
cover(c(0, 4, 8), harmonic_minor, canon="tn")
#> [1] 3