Editing Twelve-tone technique (section)

===Properties of transformations===

The tone row chosen as the basis of the piece is called the ''prime series'' (P). Untransposed, it is notated as P<sub>0</sub>. Given the twelve [[pitch class]]es of the chromatic scale, there are 12 [[factorial]]<ref>Loy 2007, 310.</ref> (479,001,600<ref name="Whittall 24"/>) tone rows, although this is far higher than the number of ''unique'' tone rows (after taking transformations into account). There are 9,985,920 classes of twelve-tone rows up to equivalence (where two rows are equivalent if one is a transformation of the other).<ref>Benson 2007, 348.</ref>

Appearances of P can be transformed from the original in three basic ways:
* [[Transposition (music)|transposition]] up or down, giving P<sub>χ</sub>.
* reversing the order of the pitches, giving the ''[[Permutation (music)|retrograde]]'' (R)
* turning each interval direction to its opposite, giving the ''[[Melodic inversion|inversion]]'' (I).

The various transformations can be combined. These give rise to a set-complex of forty-eight forms of the set, 12 transpositions of the ''four'' basic forms: P, R, I, RI. The combination of the retrograde and inversion transformations is known as the ''[[retrograde inversion]]'' (''RI'').

:{| class="wikitable"
|RI is:
|RI of P,
|R of I,
|and I of R.
|-
|R is:
|R of P,
|RI of I,
|and I of RI.
|-
|I is:
|I of P,
|RI of R,
|and R of RI.
|-
|P is:
|R of R,
|I of I,
|and RI of RI.
|}

thus, each cell in the following table lists the result of the transformations, a [[four-group]], in its row and column headers:

:{| class="wikitable"
|P:
|RI:
|R:
|I:
|-
|RI:
|P
|I
|R
|-
|R:
|I
|P
|RI
|-
|I:
|R
|RI
|P
|}

However, there are only a few numbers by which one may ''multiply'' a row and still end up with twelve tones. (Multiplication is in any case not interval-preserving.)