musicMCT (version 0.4.0)

tn: Transposition and Inversion

Description

Calculate the classic operations on pitch-class sets \(T_n\) and \(T_n I\). That is, tn adds a constant to all elements in a set modulo the octave, and tni essentially multiplies a set by -1 (modulo the octave) and then adds a constant (modulo the octave). If sorted is TRUE (as is default), the resulting set is listed in ascending order, but sometimes it can be useful to track transformational voice leadings, in which case you should set sorted to FALSE.

startzero transposes a set so that its first element is 0. (Note that this is different from tnprime() because it doesn't attempt to find the most compact form of the set. See examples for the contrast.)

Sometimes you just want to invert a set and you don't care what the index is. charm is a quick way to do this, giving a name to the transposition-class of \(T_0 I\) of the set. (The name charm is a reference to "strange" and "charm" quarks in particle physics: I like these as names for the "a" and "b" forms of a set class, i.e. the strange common triad is 3-11a = (0, 3, 7) and the charm common triad is 3-11b = (0, 4, 7). The name of the function charm means that if you input a strange set, you get out a charm set, but NB also vice versa.)

Usage

tn(
  set,
  n,
  sorted = TRUE,
  octave_equivalence = TRUE,
  optic = NULL,
  edo = 12,
  rounder = 10
)
tni(
  set,
  n = NULL,
  sorted = TRUE,
  octave_equivalence = TRUE,
  optic = NULL,
  edo = 12,
  rounder = 10
)
startzero(
  set,
  sorted = TRUE,
  octave_equivalence = TRUE,
  optic = NULL,
  edo = 12,
  rounder = 10
)
charm(set, edo = 12, rounder = 10)

Value

Numeric vector of same length as set

Arguments

set

Numeric vector of pitch-classes in the set

n

Numeric value (not necessarily an integer) representing the index of transposition or inversion. For tni() only, defaults to NULL, in which case n is chosen automatically to fix the first and last entries of set as common tones.

sorted

Do you want the result to be in ascending order? Boolean, defaults to TRUE.

octave_equivalence

Do you want to normalize the result so that all values are between 0 and edo? Boolean, defaults to TRUE.

optic

String: the OPTIC symmetries to apply. Defaults to NULL, applying symmetries most appropriate to the given function. If specified, overrides parameters sorted and octave_equivalence.

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.

Examples

c_major <- c(0, 4, 7)
tn(c_major, 2)
tn(c_major, -10)
tn(c_major, -10, optic="p") # Equivalent to tn(c_major, -10, octave_equivalence=FALSE)
tni(c_major, 4)
tni(c_major, 4, sorted=FALSE)
# If no index is supplied for tni, n is chosen to fix the first and last entries of the set:
tni(c_major)

tn(c(0, 1, 6, 7), 6)
tn(c(0, 1, 6, 7), 6, sorted=FALSE)

##### Difference between startzero and tnprime
e_maj7 <- c(4, 8, 11, 3)
startzero(e_maj7)
tnprime(e_maj7)
isTRUE(all.equal(tnprime(e_maj7), charm(e_maj7))) # True because inversionally symmetrical

##### Derive minimal voice leading from ionian to lydian
ionian <- c(0, 2, 4, 5, 7, 9, 11)
lydian <- rotate(tn(ionian, 7, sorted=FALSE), 3)
lydian - ionian

##### Easy to create a 12-tone matrix
row <- c(9, 10, 6, 8, 5, 7, 1, 2, 3, 11, 0, 4)
matrix_from_0 <- sapply(row, tni, set=row, optic="o")
matrix_from_9 <- tn(matrix_from_0, 9, optic="o")
print(matrix_from_0)
print(matrix_from_9)