# Music Theory/Serialism

In general, serialism in music is the compositional technique that uses series of musical elements such as pitches, durations, and dynamics, often a series containing every type of that element. Big historical names in serialism are:
* Original members of the Second Viennese School:
** Arnold Schoenberg, who originally invented the twelve-tone technique
** Anton von Webern, who finalized and popularized the twelve-tone technique
** Alban Berg, who combined the technique with elements of Romanticism
* Pierre Boulez, known for serializing almost all musical elements instead of only pitch

## Twelve-tone Technique

The twelve-tone technique (Zwölftontechnik in German), also known as dodecaphony or twelve-tone serialism, is the most well-known form of musical serialism. It aims to emphasize each of the twelve notes in the chromatic scale equally; thus, this kind of music is void of any key (in the traditional sense).

### Pitch Class Philosophy

Many composers and theorists who work with the twelve-tone technique prefer to use the name "pitch class" rather than "note".
The twelve-tone technique aims to emphasize each of the twelve notes in the chromatic scale equally, regardless of the octave the note is in. A "note" C can be specific to an octave: you can have a middle C, or a low C, or a high C. A pitch class C, however, represents all the notes from all octaves that are called C. This means that, of course, there exists only twelve pitch classes.
The twelve-tone technique also disregards how the notes themselves are presented, all different spellings of the same pitch are considered identical. The pitch class C♯ is identical to the pitch class D♭. For this reason, many theorists prefer to replace the diatonic-favoring letter names with numbers 0 through 11. This also allows theorists to apply principles from set theory.
For brevity, the terms "note" and "pitch class" will largely be used interchangeably.

### Tone Row

The twelve-tone technique starts with a tone row, which is a unique ordering of the twelve notes in the chromatic scale, or the twelve pitch classes. The twelve notes can be in any order, so long as no note occurs more than once in the order.
For instance, here's a tone row we'll call X: C, B, F, D, E, F♯, A, G♯, D♯, C♯, A♯, G.
The tone row can be subjected to several transformations:
* The row can remain unchanged, in which case it is called the row's prime form (P).
* The row can be reversed, with the last note first and the first note last, creating the row's retrograde (R). So, the retrograde of row X would be G, A♯, C♯, D♯, G♯, A, F♯, E, D, F, B, C.
* The row can have its intervals inverted, so all ascending intervals become descending intervals and vice versa, creating the row's inversion (I). For instance, if a row Y started with the notes E, B, the inversion of Y would start with the notes E, A; since the first interval in row Y was an ascending perfect fifth (or a descending perfect fourth), the first interval in the inversion of Y is a descending perfect fifth (or an ascending perfect fourth). So, the inversion of row X would be C, C♯, G, A♯, G♯, F♯, D♯, E, A, B, D, F.
* The row can be both inverted and reversed, creating the row's retrograde inversion (RI). So, the retrograde inversion of row X would be F, D, B, A, E, D♯, F♯, G♯, A♯, G, C♯, C.
* The row, its retrograde, its inversion, or its retrograde inversion can also be transposed to any note in the chromatic scale. So, for instance, row X can be transposed up a perfect fifth (or down a perfect fourth) to become G, F♯, C, A, B, C♯, E, D♯, A♯, G♯, F, D.
All transformations of a tone row are considered different "versions" of the same base tone row. Thus, including the prime form, there are 48 unique "versions" of any tone row, except for some rows in which some transformations give the same row (like the ascending chromatic scale, which is identical to the retrograde inversion transposed up a half step).

### Using Tone Rows

Because of the many different transformations of a single tone row, pieces of considerable length can be composed with merely the transformations of one row.
Some rules must be met in order to constitute a valid usage of a tone row:
* All of the notes in the row must be played in the order they appear in the row.
* Any instance of a note can appear in any octave.
* A single note within the row can be repeated any number of times.
* Several rows may be played simultaneously, synchronously or asynchronously.
* Any row may be repeated or switched out for another row, but all notes within the row must be played before doing so.
* Some composers also allow the ability to, within a row, alternate between the current note and the note directly before in the row before going to the next note in the series.
All else is up to the composer's discretion.
Recalling our definition of row X as C, B, F, D, E, F♯, A, G♯, D♯, C♯, A♯, G, here is an example of a valid usage of row X's prime form according to our rules:

### Tone Matrices

Composers who use the twelve-tone technique often use what are called tone matrices. A tone matrix is a twelve-by-twelve grid of pitch classes that visually lays out all of the possible transformations of a tone row.
Along the top row is the prime form of the row, starting from the top-left corner and ending at the top-right corner. Along the left-most column is the row's inversion, starting on the top-left corner (the same place the prime row started) and ending on the bottom-left corner. Along the other rows are the transpositions of the prime rows, starting with the left-most pitch.
Recalling our definition of row X as C, B, F, D, E, F♯, A, G♯, D♯, C♯, A♯, G, here is the tone matrix of row X:
| | I$_0$ | I$_{11}$ | I$_5$ | I$_2$ | I$_4$ | I$_6$ | I$_9$ | I$_8$ | I$_3$ | I$_1$ | I$_{10}$ | I$_7$ | |
| P$_0$ | C | B | F | D | E | F♯ | A | G♯ | D♯ | C♯ | A♯ | G | R$_0$ |
| P$_1$ | C♯ | C | F♯ | D♯ | F | G | A♯ | A | E | D | B | G♯ | R$_1$ |
| P$_7$ | G | F♯ | C | A | B | C♯ | E | D♯ | A♯ | G♯ | F | D | R$_7$ |
| P$_{10}$ | A♯ | A | D♯ | C | D | E | G | F♯ | C♯ | B | G♯ | F | R$_{10}$ |
| P$_8$ | G♯ | G | C♯ | A♯ | C | D | F | E | B | A | F♯ | D♯ | R$_8$ |
| P$_6$ | F♯ | F | B | G♯ | A♯ | C | D♯ | D | A | G | E | C♯ | R$_6$ |
| P$_3$ | D♯ | D | G♯ | F | G | A | C | B | F♯ | E | C♯ | A♯ | R$_3$ |
| P$_4$ | E | D♯ | A | F♯ | G♯ | A♯ | C♯ | C | G | F | D | B | R$_4$ |
| P$_9$ | A | G♯ | D | B | C♯ | D♯ | F♯ | F | C | A♯ | G | E | R$_9$ |
| P$_{11}$ | B | A♯ | E | C♯ | D♯ | F | G♯ | G | D | C | A | F♯ | R$_{11}$ |
| P$_2$ | D | C♯ | G | E | F♯ | G♯ | B | A♯ | F | D♯ | C | A | R$_2$ |
| P$_5$ | F | E | A♯ | G | A | B | D | C♯ | G♯ | F♯ | D♯ | C | R$_5$ |
| | RI$_0$ | RI$_{11}$ | RI$_5$ | RI$_2$ | RI$_4$ | RI$_6$ | RI$_9$ | RI$_8$ | RI$_3$ | RI$_1$ | RI$_{10}$ | RI$_7$ | |
Reading along any row or column, forwards or backwards, gives some transformation of row X.
Each row is also given a label: P is a prime row, I is an inversion, R is a retrograde, RI is a retrograde inversion; numbers at the end indicate a transposition by that number of semitones. Thus, R$_7$ indicates the retrograde transposed up a perfect fifth (or down a perfect fourth).