Tone-row music and serialism

September 23, 24, & 25 1997

Tone-row music. Arnold Schoenberg (1974-1951) and his sudents/friends Alban Berg (1885-1935) and Anton Webern (1883-1945). The second Vienese school. Terminology:

row, tone-row
set
series

Set forms: P (prime), I (inversion), R (retrograde), RI (retrograde inversion).

tone-row music
12-tone music
serial music
dodecaphony

Assigning numbers to pitches. Octave equivalence as a congruence (equivalence) relation modulo (mod.) 12. The matrix of a tone-row.

Tone-row matrix à la 104

RECIPE

Find the tone-row
Assign the number 0 to the first sound of the row
Write a chromatic scale starting with the first sound of the row
Number each step of the chromatic scale, in order, starting with 0 until you reach the upper octave. (If D# = 0 => E = 1, F = 2, ...., D = 11).
Assign a number to each sound of the tone-row according to the numbering system described above. If the row starts with the sequence D#, F, D, C#... the numbers will be: 0, 2, 11, 10...
Write the inversion of the tone-row and number all sounds as described in 4. You can also take the numbers representing the row and subtract each of them from 12 (the octave): 12-0=0, 12-2=10, 12-11=1, 12-10=2, etc.
Write the tone-row (using numbers) on a line; this is the top line of your matrix:
```
	    0  2  11  10 ...
```
Write the inversion (using numbers) as the leftmost column of your matrix:
```
       	    0
       	   10
       	    1
	    2
	    .
	    .
```
Transpose the tone-row on each sound of the inversion, by adding the first number of each line (i.e. the sound of the inversion on which you transpose the tone-row) to each number of the top line:
```
	    0  2  11 10 ...
	   10 ..........
	    1  3   0 11 ...       (0+1=1; 2+1=3; 11+1=12=0; 10+1=11)
	    2  4   1 ....
	    .
	    .
	
```
If the result of your addition is 12 or a number larger than 12, subtract 12 from the result and enter only numbers between 0 and 11. These are pitch classes; in this system the octave (or register) does not count. An equivalence relation modulo m is at work. It can be notated as:
where s is a sound number, m = 12 (twelve sounds in every octave), k is the octave number and n is the pitch class or the letter name of the sound in question. Using again the same example, your matrix should look like this:
```
	    0  2  11  10  ...
	   10  0   9   8  ...    (10+2=12=0; 11+10=21 21-12=9; 20-12=8)
            1  3   0  11  ...
	    2  4   1   0  ...
           .. ..  ..  ..  ...  
```
In the end, you'll have a concentrated picture of everything you wanted to know about your tone-row: the prime form and its 12 transpositions (P) on lines (read left-to-right); the retrograde and its transpositions (R) on lines (read right-to-left); the inversion and its transpositions (I) on columns (read top-to-bottom); the retrograde inversion and its transpositions (RI) on columns (read bottom-to-top).
Transpositions are notated with the initial of the set-form (P, I, R, RI) and an index showing the number of the sound on which it starts.

A. Schoenberg - Variations for Orchestra Op. 31 Note the BACH motive appearing a few times in the Introduction. The Theme has the cello line presenting the tone-row:
The matrix of this tone-row is:
```
	 0   6   8   5   7  11   4   3   9  10   1   2

	 6   0   2  11   1   5  10   9   3   4   7   8

	 4  10   0   9  11   3   8   7   1   2   5   6

	 7   1   3   0   2   6  11  10   4   5   8   9

	 5  11   1  10   0   4   9   8   2   3   6   7

	 1   7   9   6   8   0   5   4  10  11   2   3

	 8   2   4   1   3   7   0  11   5   6   9  10

	 9   3   5   2   4   8   1   0   6   7  10  11

	 3   9  11   8  10   2   7   6   0   1   4   5

	 2   8  10   7   9   1   6   5  11   0   3   4

	11   5   7   4   6  10   3   2   8   9   0   1

	10   4   6   3   5   9   2   1   7   8  11   0
```
The rest of the music (the accompaniment) is strictly derived from the same tone-row. Using the same numbering system to represent the sounds, we find:
or
Since sounds in chords are attacked simultaneously, it is not possible to determine an unique order in which they appear. However, looking at the matrix, we find the Inversion starting with 9:
The notation for set-forms (P, R, I, RI) consists of the letter(s) with the number showing the first sound in that set-form as a subscript: P0; I9; etc.

In Schoenberg's Variations for Orchestra, the Theme , in the cello, uses:
while the accompaniment is its retrograde:
Disrupting this symmetry, the cello introduces at the end P3.

Note that the set-froms of the Theme are in an ascending 5ths succession (0, 7, 2, and 9). Also, the set-froms of the cello/violin line and those of the accompaniment are in a hexachordal combinatorial relationship:
Combinatoriality is the simultaneous presentation of two different forms of a single tone-row so constructed that new 12-tone aggregates are created by the combination of their hexachords. Although hexachordal combinatoriality is the most frequently used form of combinatoriality, it is possible to apply the same principle to other divisions of the row.

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