Given an input scale, identify which adjacent colors represent good approximations of it, in a sense consistent with "Modal Color Theory," pp. 31-32.
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, ...)
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).
Numeric vector of pitch-classes in the set
Boolean: should the result be transposed so that its pitch
initial is zero? Defaults to TRUE.
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.
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.
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.
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.
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".
Integer: how many digits to display when naming any non-integral interval sizes. Defaults to 2.
Number of unit steps in an octave. Defaults to 12.
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().
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.