Skip to contents

Voice-leading brightness relationships for a scale's modes

Source: R/brightness_comparisons.R
brightness_comparisons.Rd

The essential step in creating the brightness graph of a scale's modes is to compute the pairwise comparisons between all the modes. Which ones are strictly brighter than others according to "voice-leading brightness" (see "Modal Color Theory," 6-7)? This function makes those pairwise comparisons in a manner that's useful for more computation.

Usage

brightness_comparisons(set, goal = NULL, edo = 12, rounder = 10)

Arguments

set

Numeric vector of pitch-classes in the set

goal

Numeric vector of same length as set. Defaults to NULL.

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 goal=NULL, an n-by-n matrix where n is the size of the scale. Row i represents mode i of the scale in comparison to all n modes. If the entry in row i, column j is -1, then mode i is "voice-leading darker" than mode j. If 1, mode i is "voice-leading brighter". If 0, mode i is neither brighter nor darker, either because contrary motion is involved or because mode i is identical to mode j. (Entries on the principal diagonal are always 0.)

If goal is a set, the result is a 2n-by-2n matrix whose first n rows and columns represent the modes of set and whose last n rows and columns represent the modes of goal. (Thus the upper left n-by-n square is the same as if goal were NULL and the lower right n-by-n square is the result of entering goal as set with an empty goal parameter. The upper-right and lower-left quadrants of the matrix make comparisons between the modes of set and goal.) The meaning of entries -1, 0, and 1 are as above.

Details

Note that the returned value shows all voice-leading brightness comparisons, not just the transitive reduction of those comparisons. (That is, dorian is shown as darker than ionian even though mixolydian intervenes in the brightness graph.)

See also

brightnessgraph() for a human-readable presentation of the same information.

Examples

# Because the diatonic scale, sc7-35, is non-degenerate well-formed, the only
# 0 entries should be on its diagonal.
brightness_comparisons(sc(7, 35))
#>   1  2  3  4  5  6  7
#> 1 0 -1 -1 -1 -1 -1 -1
#> 2 1  0  1  1 -1  1  1
#> 3 1 -1  0  1 -1 -1  1
#> 4 1 -1 -1  0 -1 -1 -1
#> 5 1  1  1  1  0  1  1
#> 6 1 -1  1  1 -1  0  1
#> 7 1 -1 -1  1 -1 -1  0

mystic_chord <- sc(6,34)
colSums(sim(mystic_chord)) # The sum brightnesses of the mystic chord's 6 modes
#> [1] 25 31 31 31 31 31
brightness_comparisons(mystic_chord) 
#>   1  2  3  4  5  6
#> 1 0 -1 -1 -1 -1 -1
#> 2 1  0  0  0  0  0
#> 3 1  0  0  0  0  0
#> 4 1  0  0  0  0  0
#> 5 1  0  0  0  0  0
#> 6 1  0  0  0  0  0
# Almost all 0s because very few mode pairs are comparable.
# That's because nearly all modes have the same sum, which means they have sum-brightness
# ties, and voice-leading brightness can't break a sum-brightness tie.
# (See "Modal Color Theory," 7.)

major <- c(0, 4, 7)
minor <- c(0, 3, 7)
brightness_comparisons(major, minor)
#>    1  2  3 4  5  6
#> 1  0  0 -1 1 -1 -1
#> 2  0  0 -1 1 -1 -1
#> 3  1  1  0 1  1  1
#> 4 -1 -1 -1 0 -1 -1
#> 5  1  1 -1 1  0  0
#> 6  1  1 -1 1  0  0