The Twelve-Tone Music

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Points to remember when writing a piece of 12-tone music

 How to fill in a matrix - The HARD WAY!

The easy way is to use an online calculator or my spreadsheet

The quickest way to write down all guises of a row is to use a matrix. In such a matrix the rows are read as follows.  (You can download some blank matrices as a Word file

Prime Rows (P): 0-11

The series as it is constructed is read from left to right as are all the transpositions of it.

Inversions (I): 0-11

These are read downwards.

Retrogrades (R):

The 12 prime series in reverse order so you just read from right to left.

Retrograde Inversions (RI):

Each inverted row is sounded in reverse order so you read the inverted rows (columns) series upwards.

 

Stage 1

Write P0 in the leftmost box of the second row and then write in your tone row across the same row of the matrix, using the numbers 0-11 to represent the chromatic scale c c# d d# e f f# g g# a a# b. Leave the rightmost box blank for now.

Stage 2

Fill in the second column from the left working downwards. This row is, in fact the Inversion of the original tone row. All moves up become moves down and vice versa. Calculate the difference between the 1st two numbers of the top row. We have 3 and 2 so the difference is -1. Thus, we add +1 [i.e. the opposite of -1] to 3 to give our next number in the column.

After this we go from 2 to 4, a difference of +2. We need a difference of -2 [the opposite of +2], to give 2. Now, we meet 4 and 9, a difference of +5. If we subtract 5 from 2 we get -3. By the rules of "Clock Arithmetic" (Modulo 12) the number we actually have is 9. Briefly (using a line of numbers rather than a 'clock' you start on 2 and move backwards (minus) 5 places. Note that we have 0-11 not 1-12.

<etc. 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 etc.>

Click here to find  out more about Clock Arithmetic  or here

We can now finish this column. Here are stages 1 and 2. Leave the bottom box blank for now.

                           
P0 3 2 4 9 1 5 8 11 10 0 6 7  
  4                        
  2                        
  9                         
  5                        
  1                        
  10                        
  7                        
  8                        
  6                        
  0                        
  11                        
                           
 

Stage 3

We need to label the left most column. P0 starts with the number 3. P1 is one semitone higher, which is the row which starts with 4. Label this P1. Then find 5 and label this P3 and so on. Use clock arithmetic to find where P9 is!

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Stage 4

The Inversions are labelled across the top. Simply match up the numbers in the 'P' column but replace the 'P' with an 'I'. Clearly, the first column of notes is I0. Likewise, because the next column has 2 at the top, this must be I11. You may as well also label the R and RI boxes as the numbers of the rows and columns stays the same.

  I0 I11 I1 I6 I10 I2 I5 I8 I7 I9 I3 I4  
P0 3 2 4 9 1 5 8 11 10 0 6 7 R0
P1 4

 

  

  

  

  

  

  

  

  

  

  

R1
P11 2                       R11
P6 9                       R6
P2 5                       R2
P10 1                       R10
P7 10                       R7
P4 7                       R4
P5 8                       R5
P3 6                        R3
P9 0                       R9
P8 11                       R8
  RI0 RI11 RI1 RI6 RI10 RI2 RI5 RI8 RI7 RI9 RI3 RI4  

Stage 5

All that remains is to fill in the rows. Start with P1. We have already added 1 to the 3 at the top of the column. Work along the row adding 1 to all the numbers. Then do the same for row P2. You can either work from row P1 which you have just done and continue to add 1 or, you can always work from P0 and add the number implied by the row label. Don't forget to use clock arithmetic when you add 1 to 11 to get 12. (Subtract 12 to leave zero.)

  I0 I11 I1 I6 I10 I2 I5 I8 I7 I9 I3 I4  
P0 3 2 4 9 1 5 8 11 10 0 6 7 R0
P1 4 3 5 10 2 6 9 0 11 1 7 8 R1
P11 2                       R11
P6 9                       R6
P2 5 4 6 11 3 7 10 1 0 2 8 9 R2
P10 1                       R10
P7 10                       R7
P4 7                       R4
P5 8                       R5
P3 6                       R3
P9 0                        R9
P8 11                       R8
  RI0 RI11 RI1 RI6 RI10 RI2 RI5 RI8 RI7 RI9 RI3 RI4  

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Here is the completed matrix.

  I 0 I 11 I 1 I 6 I 10 I 2 I 5 I 8 I 7 I 9 I 3 I 4  
  \/ \/ \/ \/ \/ \/ \/ \/ \/ \/ \/ \/  
P 0 > 3 2 4 9 1 5 8 11 10 0 6 7 < R 0
P 1 > 4 3 5 10 2 6 9 0 11 1 7 8 < R 1
P 11 > 2 1 3 8 0 4 7 10 9 11 5 6 < R 11
P 6 > 9 8 10 3 7 11 2 5 4 6 0 1 < R 6
P 2 > 5 4 6 11 3 7 10 1 0 2 8 9 < R 2
P 10 > 1 0 2 7 11 3 6 9 8 10 4 5 < R 10
P 7 > 10 9 11 4 8 0 3 6 5 7 1 2 < R 7
P 4 > 7 6 8 1 5 9 0 3 2 4 10 11 < R 4
P 5 > 8 7 9 2 6 10 1 4 3 5 11 0 < R 5
P 3 > 6 5 7 0 4 8 11 2 1 3 9 10 < R 3
P 9 > 0 11 1 6 10 2 5 8 7 9 3 4 < R 9
P 8 > 11 10 0 5 9 1 4 7 6 8 2 3 < R 8
  /\ /\ /\ /\ /\ /\ /\ /\ /\ /\ /\ /\  
  RI 0 RI 11 RI 1 RI 6 RI 10 RI 2 RI 5 RI 8 RI 7 RI 9 RI 3 RI 4  

You then need to convert the numbers back into notes to give the following.  

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  I 0 I 11 I 1 I 6 I 10 I 2 I 5 I 8 I 7 I 9 I 3 I 4  
  \/ \/ \/ \/ \/ \/ \/ \/ \/ \/ \/ \/  
P 0> D# D E A C# F G# B A# C F# G < R 0
P 1> E D# F A# D F# A C B C# G G# < R 1
P 11> D C# D# G# C E G A# A B F F# < R 11
P 6> A G# A# D# G B D F E F# C C# < R 6
P 2> F E F# B D# G A# C# C D G# A < R 2
P 10> C# C D G B D# F# A G# A# E F < R 10
P 7> A# A B E G# C D# F# F G C# D < R 7
P 4> G F# G# C# F A C D# D E A# B < R 4
P 5> G# G A D F# A# C# E D# F B C < R 5
P 3> F# F G C E G# B D C# D# A A# < R 3
P 9> C B C# F# A# D F G# G A D# E < R 9
P 8> B A# C F A C# E G F# G# D D# < R 8
  /\ /\ /\ /\ /\ /\ /\ /\ /\ /\ /\ /\  
  RI 0 RI 11 RI 1 RI 6 RI 10 RI 2 RI 5 RI 8 RI 7 RI 9 RI 3 RI 4  

The whole matrix was derived from the original tone-row series. But using this matrix for musical composition is not as easy as you might think.

MUSICAL EXAMPLE TWO - "Lex-x-ede".

Examine the piece "Lec-x-ede" which uses some of the rows from the matrix above. "Lec-x-ede" is not a very representative 12-tone piece; in fact it is rather contrived because it was composed in an afternoon. You need to study a few real examples and work more steadily. Read the notes and see which 'hints' "Lec-x-ede" has ignored.

Also consider the following.

  1. Does "Lec-x-ede" have any interesting ideas?
  2. Was P0 a 'good' row?
  3. Are the instruments used idiomatically?
  4. Comment on the plan.
  5. How many ideas are there?

 

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Points to think about when composing a 12-tone piece.

 
1

Avoid rows which include recognisable tonal scales or arpeggios.

2

Avoid using too many melodic intervals of the same or similar size because these may lead to melodic monotony.

3

You may wish to avoid pairs of notes a semitone apart if you feel that they will sound like moving from the leading note to the tonic in a major or minor scale.

4

Unless you actually want your row to contain several examples of a particular interval (say, 3rds of various sizes) then aim for balanced number of each kind of interval size.

5

One way of creating a row is to compose a fragment of melody first and then adapt it. Details of this methods are elsewhere.

6

Any pitch of the series may be written in any octave and can be spelled enharmonically (F sharp = G flat).

7

The order of the notes must remain the same. If you want a different order, use a different transformation; that's why they are there!

8

Try to avoid using a rhythmic pattern in consecutive or nearby measures. However, this may happen if imitation is being used.

9

Quite a few serial pieces call for unusual combinations of instruments in Webern's music. The instruments, used at the extremes of their registers, frequently play one at a time and very little at a time. This is known as pointillism.

10

Webern went further than Schoenberg by avoiding repetition of pitch colour. Sometimes, each note of a melody is played by a different instrument.

11

Serial pieces can have very precise instructions about the way the notes are to be played - pizzicato, muted, unmuted, what dynamic [There are no such details in "Lec-x-ede" because it was merely composed as an exercise.]

12

Serial music often uses canonic principles.

 

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