Voice leadings between inversions with maximal common tones

Clampitt (2007, 467; doi:10.1007/978-3-642-04579-0_46 ) defines two \(n\)-note sets to be Q-related if they:

Have all but one tone in common
Are related by tni()
Have a strictly crossing-free voice leading which preserves all \(n-1\) common tones This function finds all sets which are Q-related to an input set in this sense. The relation is defined to generalize the smooth voice leadings between consonant triads and diatonic scales to other sets, in particular demonstrating that non-singular pairwise well-formed scales (see isgwf()) demonstrate similarly nice voice leading properties.

(Strictly speaking, Clampitt includes tn() in the second part of the definition. However, the first criterion is only possible under tn() if the set is generated and therefore inversionally symmetrical. Therefore if a set satisfies Clampitt's definition by tn(), it also satisfies the tni() requirement.)

If the third part of the definition is relaxed, allowing the voice leading to involve voice crossing, Clampitt (1997, 121) identifies this as the Q*-relation. The Q*-relation can be computed with this function by setting method="hamming". (All other methods provided by vl_dist() give equivalent results in this context.)

Usage

clampitt_q(
  set,
  index = NULL,
  method = c("taxicab", "euclidean", "chebyshev", "hamming"),
  edo = 12,
  rounder = 10
)

Arguments

set: Numeric vector of pitch-classes in the set
index: Integer: which Q-related set and voice leading should be returned? Defaults to NULL, in which case all options are returned.
method: What distance metric should be used? Defaults to "taxicab" but can be "euclidean", "chebyshev", or "hamming".
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

A list with two entries, "sets" and "vls". The former is a matrix whose columns are the sets which are Q-related to the input set, in OPC-normal form. The latter is a matrix whose rows represent the voice-leading motions which transform set into its goals. (This follows the general practice of musicMCT of representing scales as columns and voice leadings as rows.) The rows of "vls" correspond to the columns of "sets", but the columns of "vls" correspond to the order of the input set, which may not match the normal form of the output sets. (See the last example.)

Examples

# The Neo-Riemannian P, L, and R transformations on triads are all Q-relations:
major_triad <- c(0, 4, 7)
clampitt_q(major_triad)
#> $sets
#>      [,1] [,2] [,3]
#> [1,]    9    0    4
#> [2,]    0    3    7
#> [3,]    4    7   11
#> 
#> $vls
#>      [,1] [,2] [,3]
#> [1,]    0    0    2
#> [2,]    0   -1    0
#> [3,]   -1    0    0
#> 

# A well-formed scale like the diatonic has two Q-relations given by its signature transformations:
major_scale <- c(0, 2, 4, 5, 7, 9, 11)
clampitt_q(major_scale)
#> $sets
#>      [,1] [,2]
#> [1,]    4    6
#> [2,]    5    7
#> [3,]    7    9
#> [4,]    9   11
#> [5,]   10    0
#> [6,]    0    2
#> [7,]    2    4
#> 
#> $vls
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,]    0    0    0    0    0    0   -1
#> [2,]    0    0    0    1    0    0    0
#> 

# A non-singular pairwise well-formed scale also has Q-relations:
clampitt_q(j(dia))
#> $sets
#>           [,1]      [,2]
#> [1,]  5.902237 10.882687
#> [2,]  7.019550  0.000000
#> [3,]  8.843587  1.824037
#> [4,] 10.882687  3.863137
#> [5,]  0.000000  4.980450
#> [6,]  2.039100  7.019550
#> [7,]  3.863137  8.843587
#> 
#> $vls
#>      [,1]       [,2] [,3]      [,4] [,5] [,6] [,7]
#> [1,]    0  0.0000000    0 0.9217872    0    0    0
#> [2,]    0 -0.2150629    0 0.0000000    0    0    0
#> 

# Set-class 7-31 is pairwise well-formed:
clampitt_q(sc(7, 31))
#> $sets
#>      [,1] [,2]
#> [1,]   10    1
#> [2,]    0    3
#> [3,]    1    4
#> [4,]    3    6
#> [5,]    4    7
#> [6,]    6    9
#> [7,]    7   10
#> 
#> $vls
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,]    0    0    0    0    0    0    1
#> [2,]   -2    0    0    0    0    0    0
#> 
# It also has two additional Q*-related sets:
clampitt_q(sc(7, 31), method="hamming")
#> $sets
#>      [,1] [,2] [,3] [,4]
#> [1,]   10    7    4    1
#> [2,]    0    9    6    3
#> [3,]    1   10    7    4
#> [4,]    3    0    9    6
#> [5,]    4    1   10    7
#> [6,]    6    3    0    9
#> [7,]    7    4    1   10
#> 
#> $vls
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,]    0    0    0    0    0    0    1
#> [2,]    0    0    0    0    4    0    0
#> [3,]    0    0   -5    0    0    0    0
#> [4,]   -2    0    0    0    0    0    0
#> 

# Most other types of scales have at most one Q-relation:
dominant_seventh <- c(0, 4, 7, 10)
clampitt_q(dominant_seventh)
#> $sets
#>      [,1]
#> [1,]    2
#> [2,]    4
#> [3,]    7
#> [4,]   10
#> 
#> $vls
#>      [,1] [,2] [,3] [,4]
#> [1,]    2    0    0    0
#> 

# The order of "sets" may not match the order of "vls":
clampitt_q(c(0, 1, 4, 7))
#> $sets
#>      [,1]
#> [1,]    1
#> [2,]    4
#> [3,]    7
#> [4,]    8
#> 
#> $vls
#>      [,1] [,2] [,3] [,4]
#> [1,]   -4    0    0    0
#>

Voice leadings between inversions with maximal common tones

Usage

Arguments

Value

See also

Examples