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Distances between continuous transpositions of a set

Source: R/minimize_VL.R
tndists.Rd

One way to think about the voice-leading potential of a set is to consider the minimal voice-leadings by which it can move to transpositions of itself (or another set). For instance, the major triad's closest transpositions are \(T_4\) and \(T_8\) while its most distant transposition is \(T_6\), and potentially also \(T_{\pm 2}\) depending on the distance metric you use. For the major triad restricted to 12-tone equal temperament, this set of relationships is well modeled by Richard Cohn's discussion of Douthett & Steinbach's "Cube Dance" in Audacious Euphony (102-106). The behavior of other sets is not always what you might expect extrapolating from the case of tertian sonorities. For instance, the trichord (027) has different minimal neighbors depending on the metric chosen: its nearest neighbors are \(T_{\pm 4}\) under the Euclidean metric but \(T_{\pm 5}\) under the taxicab metric.

This function allows us to visualize such relationships by plotting the minimal voice leading distance from a set to transpositions of its goal in continuous pc-space. (In spirit, it is like a continuous version of vl_rolodex() except that it visualizes a voice-leading distance rather than reporting the specific motions of the set's individual voices.) The main intended use of the function is the plot that it produces, which represents many discrete \(T_n\)s of the set (for a sampling of each edo step divided into subdivide amounts) on the x axis and voice-leading distance on the y axis. Secondarily, tndists() invisibly returns the distance values that it plots, named according to the \(T_n\) they correspond to.

Usage

tndists(
  set,
  goal = NULL,
  method = c("taxicab", "euclidean", "chebyshev", "hamming"),
  subdivide = 100,
  edo = 12,
  rounder = 10
)

Arguments

set

Numeric vector of pitch-classes in the set

goal

Numeric vector like set: what is the tn-type of the voice leading's destination? Defaults to NULL, in which case the function uses set as the tn-type.

method

What distance metric should be used? Defaults to "taxicab" but can be "euclidean", "chebyshev", or "hamming".

subdivide

Numeric: how many small amounts should each edo step be divided into? Defaults to 100.

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

Numeric vector of length edo * subdivide representing distances of the transpositions. Names indicate the transposition index that corresponds to each distance.

Examples

major_triad <- c(0, 4, 7)
taxicab_dists <- tndists(major_triad)

euclidean_dists <- tndists(major_triad, method="euclidean")

tns_to_display <- c("1.9", "1.92", "1.95", "2", "2.05", "2.08", "2.1")
taxicab_dists[tns_to_display]
#>  1.9 1.92 1.95    2 2.05 2.08  2.1 
#> 5.70 5.76 5.85 6.00 5.85 5.76 5.70 
euclidean_dists[tns_to_display]
#>      1.9     1.92     1.95        2     2.05     2.08      2.1 
#> 3.290897 3.325538 3.377499 3.464102 3.550704 3.602666 3.581899