Elementary voice leadings

Calculate elementary voice leadings which represent motion by a single arrow on a brightnessgraph(). vlsig() finds "voice-leading signature" of a set moving to transpositions of itself, as determined by vl_generators(). inter_vlsig() finds the elementary voice leadings from a set to some other set, i.e. where the goal parameter of brightnessgraph() is not NULL. By default, inter_vlsig() finds voice leadings for contextual inversions of a set.

Usage

vlsig(set, index = NULL, display_digits = 2, edo = 12, rounder = 10)

inter_vlsig(
  set,
  goal = NULL,
  index = NULL,
  type = c("ascending", "commontone", "obverse"),
  display_digits = 2,
  edo = 12,
  rounder = 10
)

Arguments

set: Numeric vector of pitch-classes in the set
index: Integer: which voice-leading generator should be displayed? Defaults to NULL, displaying all voice leadings.
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.
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.
type: For inter_vlsig() only. String: "ascending", "commontone", or "obverse". Defaults to "ascending", which makes the result prefer ascending voice leadings (as for vlsig()). The second makes the result prefer common tones (as might be expected for contextual inversions). The third option, "obverse", gives the obverse of a voice-leading in a sense that generalizes Morris (1998, doi:10.2307/746047 )'s concept for Neo-Riemannian PLR transformations. This option returns voice leadings that lead to set rather than away from it.

Value

List with three elements:

"vl" which shows the distance (in edo steps) that each voice moves,
"tn" which indicates the (chromatic) transposition achieved by the voice leading,
"rotation" which indicates the scalar transposition caused by the voice leading.

If index=NULL, returns instead a matrix whose rows are all the elementary voice leadings.

Details

Note that the voice leadings determined by vlsig() can be different from the corresponding ones at the same \(T_n\) level in vl_rolodex(). The latter function prioritizes minimal voice leadings, whereas vlsig() prioritizes elementary voice leadings derived from a set's brightnessgraph(). In particular, this means that vlsig() voice leadings will always be ascending, involve at least one common tone, and involve no contrary motion. See the odd_pentachord voice leadings in the Examples.

For vlsig() the value "rotation" in the result is non-arbitrary: if the rotation value is n, the voice leading takes set to the nth mode of set. For inter_vlsig(), there is no canonical correspondence between modes of set and goal, except to assume that the input modes are the 1st mode of each scale. If goal is NULL, finding contextual inversions of set, the first mode of the inversion is taken to be the one that holds the first and last pitches of set in common. These "rotation" values do not have a transparent relationship to the values of inter_vlsig()'s index parameter.

For inter_vlsig() results are not as symmetric between set and goal as you might expect. Since these voice-leading functions study ascending arrows on a brightness graph the possibilities for ascending from X to Y are in principle somewhat different from the possibilities for ascending from Y to X. See the examples for the "Tristan genus." Note that this is still true when type="commontone", which might lead to counterintuitive results.

Examples

# Hook's elementary signature transformation
major_scale <- c(0, 2, 4, 5, 7, 9, 11)
vlsig(major_scale, index=1)
#> $vl
#> [1] 0 0 0 1 0 0 0
#> 
#> $tn
#> [1] 7
#> 
#> $rotation
#> [1] 4
#> 

pure_major_triad <- j(1, 3, 5)
vlsig(pure_major_triad, index=1)
#> $vl
#> [1] 0.00 1.12 1.82
#> 
#> $tn
#> [1] 4.98045
#> 
#> $rotation
#> [1] 1
#> 
vlsig(pure_major_triad, index=2)
#> $vl
#> [1] 0.71 0.00 1.82
#> 
#> $tn
#> [1] 8.843587
#> 
#> $rotation
#> [1] 2
#> 

odd_pentachord <- c(0, 1, 4, 9, 11) # in 15-edo
vlsig(odd_pentachord, index=2, edo=15)
#> $vl
#> [1] 2 3 4 0 1
#> 
#> $tn
#> [1] 8
#> 
#> $rotation
#> [1] 2
#> 
vl_rolodex(odd_pentachord, edo=15)$"8" 
#> [1] -3  1  0 -1 -2

# Contextual inversions for Tristan genus:
dom7 <- c(0, 4, 7, 10)
halfdim7 <- c(0, 3, 6, 10)
inter_vlsig(dom7, halfdim7)
#>      [,1] [,2] [,3] [,4]
#> [1,]    1    0    0    1
#> [2,]    2    0    0    0
#> [3,]    1    1    0    0
#> [4,]    1    0    1    0
inter_vlsig(halfdim7, dom7)
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    1    1    0
#> [2,]    1    0    1    0
#> [3,]    1    1    0    0

# Elementary voice leadings between unrelated sets:
maj7 <- c(0, 4, 7, 11)
min7 <- c(0, 3, 7, 10)
inter_vlsig(min7, maj7)
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    1    0    1
#> [2,]    2    0    0    0
brightnessgraph(min7, maj7)


# Elementary inversional VL for just diatonic which is NOT a Q-relation:
inter_vlsig(j(dia), index=3)
#> $vl
#> [1] 0.71 0.71 0.00 0.71 0.71 0.00 0.00
#> 
#> $tni
#> [1] 7.726274
#> 
#> $rotation
#> [1] 5
#> 

# Obverse voice leadings:
# First we see the Parallel transformation which leads from minor to major:
minor <- c(0, 3, 7)
P <- inter_vlsig(minor, index=1)
print(P)
#> $vl
#> [1] 0 1 0
#> 
#> $tni
#> [1] 7
#> 
#> $rotation
#> [1] 0
#> 
# Compare to its obverse, Slide, leading *to* minor from major:
S <- inter_vlsig(minor, index=1, type="obverse")
print(S)
#> $vl
#> [1] 1 0 1
#> 
#> $tni
#> [1] 6
#> 
#> $rotation
#> [1] 0
#> 
# A voice-leading plus its obverse is a chromatic transposition:
P$vl + S$vl
#> [1] 1 1 1

Usage

Arguments

Value

Details

See also

Examples