Scalar (and interscalar) interval matrix

As defined by Tymoczko 2008 ("Scale Theory, Serial Theory and Voice Leading") https://onlinelibrary.wiley.com/doi/10.1111/j.1468-2249.2008.00257.x, the scalar interval matrix represents the "rotations" of a set, transposed to begin on 0, in its columns. Its nth row represents the specific intervals which represent its generic interval of size n. If changed from its default (NULL), the parameter goal calculates Tymoczko's interscalar interval matrix from set to goal.

Usage

sim(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

Numeric n by n matrix where n is the number of notes in set

Examples

diatonic_modes <- sim(c(0, 2, 4, 5, 7, 9, 11))
print(diatonic_modes)
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,]    0    0    0    0    0    0    0
#> [2,]    2    2    1    2    2    2    1
#> [3,]    4    3    3    4    4    3    3
#> [4,]    5    5    5    6    5    5    5
#> [5,]    7    7    7    7    7    7    6
#> [6,]    9    9    8    9    9    8    8
#> [7,]   11   10   10   11   10   10   10

miyakobushi_modes <- sim(c(0, 1, 5, 7, 8)) # rows show trivalence
print(miyakobushi_modes)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    0    0    0    0    0
#> [2,]    1    4    2    1    4
#> [3,]    5    6    3    5    5
#> [4,]    7    7    7    6    9
#> [5,]    8   11    8   10   11

# Interscalar Interval Matrix:
sim(c(0, 3, 6, 10), c(0, 4, 7, 10))
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    1    1    0
#> [2,]    4    4    4    2
#> [3,]    7    7    6    6
#> [4,]   10    9   10    9

# Note that the interscalar matrices factor out transposition:
minor <- c(0, 3, 7)
major <- c(0, 4, 7)
sim(minor, major)
#>      [,1] [,2] [,3]
#> [1,]    0    1    0
#> [2,]    4    4    5
#> [3,]    7    9    9
sim(minor-1, major)
#>      [,1] [,2] [,3]
#> [1,]    0    1    0
#> [2,]    4    4    5
#> [3,]    7    9    9
sim(minor, major+2)
#>      [,1] [,2] [,3]
#> [1,]    0    1    0
#> [2,]    4    4    5
#> [3,]    7    9    9

# But not permutation:
major_64 <- c(0, 5, 9)
major_open <- c(0, 7, 4)
sim(minor, major_64)
#>      [,1] [,2] [,3]
#> [1,]    0    2    2
#> [2,]    5    6    5
#> [3,]    9    9   10
sim(minor, major_open)
#>      [,1] [,2] [,3]
#> [1,]    0    4   -3
#> [2,]    7    1    5
#> [3,]    4    9   12