Is a scale n-wise well formed?

Tests whether a scale has a generalized type of well formedness (pairwise or n-wise well formedness).

Usage

isgwf(set, setword = NULL, allow_de = FALSE, edo = 12, rounder = 10)

Arguments

set

Numeric vector of pitch-classes in the set

setword

A vector representing the ranked step sizes of a scale (e.g. c(2, 2, 1, 2, 2, 2, 1) for the diatonic). The distinct values of the setword should be consecutive integers. If you want to test a step word instead of a list of pitch classes, set must be entered as NULL.

allow_de

Should the function test for degenerate well-formed and distributionally even scales too? 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

Boolean: is the set n-wise well formed?

Details

David Clampitt's 1997 dissertation ("Pairwise Well-Formed Scales: Structural and Transformational Properties," SUNY Buffalo) offers a generalization of the notion of well-formedness from 1-dimensional structures with a single generator to 2-dimensional structures that mediate between two well-formed scales. Ongoing research suggests that this can be extended further to "n-wise" or "general" well-formedness, though n-wise well-formed scales are increasingly rare as n grows larger.

Examples

meantone_diatonic <- c(0, 2, 4, 5, 7, 9, 11)
just_diatonic <- j(dia)
some_weird_thing <- convert(c(0, 1, 3, 6, 8, 12, 14), 17, 12)
example_scales <- cbind(meantone_diatonic, just_diatonic, some_weird_thing)

apply(example_scales, 2, howfree)
#> meantone_diatonic     just_diatonic  some_weird_thing 
#>                 1                 2                 3 
apply(example_scales, 2, isgwf)
#> meantone_diatonic     just_diatonic  some_weird_thing 
#>              TRUE              TRUE              TRUE