Which voice leadings are irreducible? — monochrome

Often, the elementary voice leadings of a set (given by vlsig()) can be broken into two intermediate voice leadings through a different set (i.e. ones given by inter_vlsig() with some suitable choice of goal). A classic exmaple is the voice leading (0, 1, 2) that takes C major (C, E, G) to F major (C, F, A). This voice leading is elementary for major triads, but it can be decomposed into the succession of Neo-Riemannian voice leadings R-then-L by passing through a minor triad. Such decompositions are not always possible, though: given a choice of set and goal classes, sometimes the elementary path from one mode of set to another does not pass through any mode of goal. Such a voice leading is "monochrome" in the sense that it uses the restricted palette of the modes of a single set.

Usage

monochrome_vl(
  set,
  goal = NULL,
  bool = FALSE,
  display_digits = 2,
  edo = 12,
  rounder = 10
)

Arguments

set: Numeric vector of pitch-classes in the set
goal: For inter_vlsig() only, vector of the transposition type to voice lead to. Defaults to NULL, producing voice leadings to the inversion of set.
bool: Should the result be a Boolean TRUE/FALSE value? Defaults to FALSE.
display_digits: Integer: how many digits to display when naming any non-integral interval sizes. Defaults to 2.
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

If bool=FALSE, a voice-leading matrix formatted after inter_vlsig(). If bool=TRUE, a single Boolean value indicating whether any monochrome voice leadings exist for set and goal.

Examples

maj7 <- c(0, 4, 7, 11)
mM7 <- c(0, 3, 7, 11)

# Just a few basic transformations lead between these seventh chords:
inter_vlsig(maj7, mM7)
#>      [,1] [,2] [,3] [,4]
#> [1,]    1    0    1    1
#> [2,]    3    0    0    0
inter_vlsig(mM7, maj7)
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    1    0    0

# But we can see from their brightness graph that modes III and I of maj7
# have no intermediate voice leading that involves the minor-major seventh:
brightnessgraph(maj7, mM7)


# monochrome_vl detects this voice leading:
monochrome_vl(maj7, mM7)
#>      [,1] [,2] [,3] [,4]
#> [1,]    2    2    0    0

# Note that the equivalent does not apply to minor-major seventh, which always
# has some mode of the major 7th chord decomposing its elementary voice leadings:
monochrome_vl(mM7, maj7)
#>      [,1] [,2] [,3] [,4]

# Finally, note that the presence of monochrome voice leadings is dependent on 
# the pair of chord types you choose, not simply the "set." For instance, we can define
# a chord that will decompose the voice leading from mode III to mode I of the major 7th:
dom7 <- c(0, 4, 7, 10)
monochrome_vl(maj7, dom7)
#>      [,1] [,2] [,3] [,4]
brightnessgraph(maj7, dom7)