Well-formedness, Myhill's property, and/or moment of symmetry

Tests whether a scale has the property of "well-formedness" or "moment of symmetry."

Usage

iswellformed(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 answering "Is the scale MOS (with equivalence interval equal to the period)?" (if allow_de=FALSE) or "Is the scale well-formed in any sense?" (if allow_de=TRUE).

Details

The three concepts of "well-formedness," "Myhill's property," and "moment of symmetry" refer to nearly the same scalar property, generalizing one of the most important features of the familiar diatonic scale. See Clough, Engebretsen, and Kochavi (1999, 77) doi:10.2307/745921 for a useful discussion of their relationships. In short, except for a few edge cases, a scale possesses these properties if it is generated by copies of a single interval (as the Pythagorean diatonic is generated by the ratio 3:2) and all copies of the generator belong to the same generic interval (as the 3:2 generator of the diatonic always corresponds to a "fifth" within the scale). Such a structure typically means that all generic intervals come in 2 distinct sizes, which is the definition of "Myhill's property." An exception occurs if the generator manages to produce a perfectly even scale, e.g. when the whole tone scale is generated by 6 copies of 1/6 of the octave. Such a scale lacks Myhill's property and Carey & Clampitt (1989, 200) doi:10.2307/745935 call such cases "degenerate well-formed." Instead of Myhill's property, such scales have only 1 specific value in each intervalspectrum().

Clough, Engebretsen, and Kochavi define a related concept, distributionally even scales, which include the hexatonic and octatonic scales (Forte sc6-20 and sc8-28). Such scales are in some sense halfway between "degenerate" and "non-degenerate well-formed" because some of their interval spectra have 1 element while others have 2. From another perspective, distributionally even scales are non-degenerate well formed with a period smaller than the octave (e.g. as the hexatonic scales 1-3 step pattern repeats every third of an octave).

The term "moment of symmetry" refers to the non-degenerate well-formed scales and was coined by Erv Wilson 1975 (cited in Clough, Engebretsen, and Kochavi). It tends to be more widely used in microtonal music theory, e.g. https://en.xen.wiki/w/MOS_scale.

Scales with this property have considerably interesting voice-leading properties and are some of the most important landmarks in the geometry of MCT. See "Modal Color Theory," pp. 14, 17, 29, 33-34, and 36-37. A substantial portion of MCT amounts to an attempt to generalize ideas developed for MOS/NDWF scales to all scale structures.

Examples

iswellformed(sc(7, 35))
#> [1] TRUE
iswellformed(c(0, 2, 4, 6))
#> [1] TRUE
iswellformed(c(0, 1, 6, 7))
#> [1] FALSE
iswellformed(c(0, 1, 6, 7), allow_de=TRUE)
#> [1] TRUE
iswellformed(NULL, setword=c(2, 2, 1, 2, 1, 2, 1))
#> [1] TRUE