Rothenberg propriety

Rothenberg (1978) doi:10.1007/BF01768477 identifies a potentially desirable trait for scales which he calls "propriety." Loosely speaking, a scale is proper if its specific intervals are well sorted in terms of the generic intervals they belong to. A scale is strictly proper if, given two generic sizes g and h such that g < h, every specific size corresponding to g is smaller than every specific size corresponding to h. A scale if improper if any specific size of g is larger than any specific size of h. An ambiguity occurs if any size of g equals any size of h: scales with ambiguities are weakly but not strictly proper.

Usage

isproper(set, strict = FALSE, edo = 12, rounder = 10)

has_contradiction(set, edo = 12, rounder = 10)

strictly_proper(set, edo = 12, rounder = 10)

Arguments

set

Numeric vector of pitch-classes in the set

strict

Boolean: should only strictly proper scales pass?

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

Boolean which answers whether the input satisfies the property named by the function

See also

make_roth_ineqmat() creates an ineqmat for a hyperplane arrangement that lets you explore propriety-related issues in finer detail.

Examples

c_major <- c(0, 2, 4, 5, 7, 9, 11)
has_contradiction(c_major)
#> [1] FALSE
strictly_proper(c_major)
#> [1] FALSE
isproper(c_major)
#> [1] TRUE
isproper(c_major, strict=TRUE)
#> [1] FALSE

isproper(j(dia), strict=TRUE)
#> [1] TRUE

pythagorean_diatonic <- sort(((0:6)*z(3/2))%%12)
isproper(pythagorean_diatonic)
#> [1] FALSE
has_contradiction(pythagorean_diatonic)
#> [1] TRUE