musicMCT (version 0.1.2)

isym: Test for inversional symmetry

Description

Is the pc-set inversionally symmetrical? That is, does it map onto itself under \(T_n I\) for some appropriate \(n\)? isym() can return either TRUE/FALSE or an index of symmetry but defaults to the former. isym_index() is a simple wrapper for isym() that returns the latter. isym_degree() counts the total number of inversional symmetries (i.e. the number of distinct inversional axes of symmetry).

Usage

isym(set, return_index = FALSE, edo = 12, rounder = 10)
isym_index(set, ...)
isym_degree(set, ...)

Value

isym() returns the Boolean value from testing for symmetry, unless return_index=TRUE, in which case isym() and isym_index()

return a numeric value for one index of inversion at which the set is symmetrical. If the set is not inversionally symmetrical, they will return NA. isym_degree() gives the degree of inversional symmetry.

Arguments

set

Numeric vector of pitch-classes in the set

return_index

Should the function return a specific index at which the set is symmetrical? 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.

...

Arguments to be passed to isym()

Details

isym() is evaluated by asking whether, for some appropriate rotation, the step-interval series of the given set is equal to the step-interval series of the set's inversion. This is designed to work for sets in continuous pc-space, not just integers mod k. Note also that this calculates abstract pitch-class symmetry, not potential symmetry in pitch space. (See the second example.)

Examples

#### Mod 12
isym(c(0, 1, 5, 8))
isym(c(0, 2, 4, 8))

#### Continuous Values
qcm_fifth <- meantone_fifth()
qcm_dia <- sort(((0:6)*qcm_fifth)%%12)
just_dia <- j(dia)
isym(qcm_dia)
isym(just_dia)

#### Rounding matters:
isym(qcm_dia, rounder=15)

### Index and Degree
hexatonic_scale <- c(0, 1, 4, 5, 8, 9)
isym_index(hexatonic_scale) # Only returns one suitable index
isym_degree(hexatonic_scale)