simplify_scale {musicMCT} | R Documentation |
Best ways to regularize a scale
Description
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 |
ineqmat |
Specifies which hyperplane arrangement to consider. By default (or by
explicitly entering "mct") it supplies the standard "Modal Color Theory" arrangements
of |
scales |
List of scales representing the faces of your hyperplane
arrangement. Defaults to |
signvector_list |
A list of signvectors to use as the reference by
which |
adjlist |
Adjacency list structured in the same way as |
method |
What distance metric should be used? Defaults to |
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 |
rounder |
Numeric (expected integer), defaults to |
... |
Other arguments to be passed from |
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 a0
, 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.
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+1
th row indicates the color number of the simplification.The
n+2
th row shows how many degrees of freedom the simplification has (always between0
andd-1
whered
isset
's degree of freedom).The
n+3
th row calculates the voice-leading distance fromset
to the simplified scale (according to the chosenmethod
, 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+4
th row counts how many more hyperplanes the simplified scale lies on compared toset
.The
n+5
th row is a quotient of the previous two rows (distance divided by number of new regularities).The
n+6
th 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).
Examples
# 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)