Given any scale, attempts to find a scale defined as integers mod k
which belongs to the same hue as the input (i.e. would return TRUE
when same_hue() is applied). This function thus is similar in spirit to
quantize_color() but seeks a more precise structural match between
input and quantization. Note, though, that while quantize_color() should always
be able to find a suitable quantization (if nmax is set high enough),
this is not necessarily true for quantize_hue(). There are lines in
\(\mathbb{R}^n\) which pass through no rational points but the origin, so some hues
(including ones of musical interest like the 5-limit just diatonic scale)
may not have any quantization.
Arguments
- set
Numeric vector of pitch-classes in the set
- nmax
Integer, essentially a limit to how far the function should search before giving up. Although every real color should have a rational representation in some mod k universe, for some colors that k must be very high. Increasing nmax makes the function run longer but might be necessary if small chromatic universes don't produce a result. Defaults to
12.- reconvert
Boolean. Should the scale be converted to 12edo? Defaults to
FALSE.- 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
If reconvert=FALSE, a list of two elements: element 1 is set with a vector of integers
representing the quantized scale; element 2 is edo representing the number k of unit steps in the
mod k universe. If reconvert=TRUE, returns a single numeric vector measured relative
to the unit step size input as edo: these generally will not be integers. May also return a vector
of NAs the same length as set if no suitable quantization was found beneath the limit given by
nmax.
Examples
meantone_diatonic <- sort(((0:6)*meantone_fifth())%%12)
quantize_hue(meantone_diatonic) # Succeeds
#> $set
#> [1] 0 2 4 6 7 9 11
#>
#> $edo
#> [1] 12
#>
quantize_hue(j(dia), nmax=15) # Fails no matter how high you set nmax.
#> [1] NA NA NA NA NA NA NA
quasi_guido <- convert(c(0, 2, 4, 5, 7, 9), 13, 12)
quantize_color(quasi_guido)
#> $set
#> [1] 0 2 4 5 7 9
#>
#> $edo
#> [1] 12
#>
quantize_hue(quasi_guido)
#> $set
#> [1] 0 2 4 5 7 9
#>
#> $edo
#> [1] 13
#>