Best ways to regularize a scale — simplify

Given an input scale, identify which adjacent colors represent good approximations of it, in a sense consistent with "Modal Color Theory," pp. 31-32.

Usage

simplify_scale(
  set,
  start_zero = TRUE,
  ineqmat = NULL,
  scales = NULL,
  signvector_list = NULL,
  adjlist = NULL,
  method = c("euclidean", "taxicab", "chebyshev", "hamming"),
  display_digits = 2,
  edo = 12,
  rounder = 10
)

best_simplification(set, ...)

Arguments

set: Numeric vector of pitch-classes in the set
start_zero: Boolean: should the result be transposed so that its pitch initial is zero? Defaults to TRUE.
ineqmat: Specifies which hyperplane arrangement to consider. By default (or by explicitly entering "mct") it supplies the standard "Modal Color Theory" arrangements of getineqmat(), but can be set to "white," "roth", "pastel," or "rosy", giving the ineqmats of make_white_ineqmat(), make_roth_ineqmat(), make_pastel_ineqmat(), and make_rosy_ineqmat(). For other arrangements, the desired inequality matrix can be entered directly.
scales: List of scales representing the faces of your hyperplane arrangement. Defaults to NULL in which case the function looks for representative_scales in the global environment.
signvector_list: A list of signvectors to use as the reference by which colornum assigns a value. Defaults to NULL and will attempt to use representative_signvectors, which needs to be downloaded and assigned separately from the package musicMCT.
adjlist: Adjacency list structured in the same way as color_adjacencies. Defaults to NULL in which case the function looks for color_adjacencies in the global environment.
method: What distance metric should be used? Defaults to "euclidean" (unlike most functions with a method parameter in musicMCT) but can be "taxicab", "chebyshev", or "hamming".
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.
...: Other arguments to be passed from best_simplification() to simplify_scale().

Value

A matrix with n+6 rows, where n is the number of notes in the scale. Each column represents a scale which is a potential simplification of the input set, together with details about that simplified scale. The first n entries of the column represent the pitches of the scale itself:

The n+1th row indicates the color number of the simplification.
The n+2th row shows how many degrees of freedom the simplification has (always between 0 and d-1 where d is set's degree of freedom).
The n+3th row calculates the voice-leading distance from set to the simplified scale (according to the chosen method, for which Euclidean distance is the default because it corresponds to the assumption that orthogonal projection finds the closest point on a neighboring flat).
The n+4th row counts how many more hyperplanes the simplified scale lies on compared to set.
The n+5th row is a quotient of the previous two rows (distance divided by number of new regularities).
The n+6th row calculates a final "score" which is used to order the columns from best (first) to worst (last) simplifications. This score is the inverse of the previous row divided by the total number of hyperplanes in the arrangement. (Without this normalization, scores for higher cardinalities quickly become much larger than scores for low cardinalities.)

If display_digits is a value other than NULL, the function prints to console a suitably rounded representation of the data, while invisibly returning the unrounded information.

best_simplification() returns simply a numeric vector with the scale judged optimal by simplify_scale() (i.e. the first n entries of its first column, without all the other information).

Details

Suppose that you've gathered data on how a particular instrument is tuned. Two intervals in its scale differ by about 12 cents: does it make sense to consider those intervals to be essentially the same, up to some combination of measurement error and the permissiveness of cognitive categories? simplify_scale() helps to answer such a question by considering whether eliding a precisely measured difference results in a significant simplification of the overall scale structure.

It accomplishes this by starting from two premises:

Any simplification should move to an adjacent color with fewer degrees of freedom.
There's a tradeoff between moving farther (i.e. requiring more measurement fuzziness) and achieving greater regularity. Therefore it starts by projecting the input scale onto all neighboring flats with fewer degrees of freedom. Some projections can be rejected immediately because the closest point on the flat isn't actually an adjacent color. The non-rejected projections can therefore be ranked by calculating the "cost" of each additional regularity: for every 1 or -1 in the sign vector that is converted to a 0, how far does one have to move in voice leading space?

To answer this question, simplify_signvector needs access to data about the hyperplane arrangement in question. For the basic "Modal Color Theory" arrangements, this is the data in representative_scales.rds, representative_signvectors.rds, and color_adjacencies.rds. The function assumes that, if you don't specify other data, you have those three files loaded into your workspace. It can't function without them.

Examples

if (FALSE) { # exists("representative_scales") && exists("representative_signvectors") && exists("color_adjacencies")
# For this example to run, you need the necessary data files loaded.
# Let's see what happens if we try to simplify the 5-limit just diatonic:

simplify_scale(j(dia))

# So the best option is color number 942659, which is the "well-formed"
# structure of the familiar diatonic scale. The particular saturation of
# that meantone structure is very close to 1/5-comma meantone:

simplified_jdia <- best_simplification(j(dia))
fifth_comma_dia <- sim(sort((meantone_fifth(1/5)*(0:6))%%12))[,5]
vl_dist(simplified_jdia, fifth_comma_dia)
}