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