This Instructable demonstrates the procedure for composing twelve-tone melodies using a twelve-tone matrix.
This technique was developed by Arnold Schoenberg in 1921, and its purpose is to compose music in which each of the twelve pitches are heard equally. This technique prevents the emphasis of any one note, thereby avoiding any sense of key or tonality.
After learning this technique, you will be able to quickly write melodies for your compositions without emphasizing any particular tonality. With practice, creating a Twelve-Tone matrix is a breeze, taking less than five minutes to complete.
To complete the matrix, you will need to be able to add and subtract numbers between 1 and 12. Writing melodies from this matrix will require a basic understanding of music notation.
Items needed for this task include a 12 by 12 grid (as shown below) and a pen or pencil. To write melodies from the matrix, you will need staff paper or software for writing music.
To hear what this type of music can sound like, follow this link for an example of Schoenberg's twelve-tone music. www.youtube.com/watch
Step 1: Write Numbers in the Top Row
Write each of the whole numbers from 1 through 12 across the top row of the grid such that each number appears exactly once.
The order of the numbers can be either completely arbitrary or carefully planned. The intervals between these numbers will become the number of half-steps between the pitches of your melody. With this knowledge, you can place the numbers in the first row of this matrix at predetermined intervals for musical effect. I have placed numbers in the top row, as shown.
Step 2: Populate the First Column
While the notes in the first row could have been written in any order you chose, the first column depends entirely upon the first row, so you should not select numbers for this column in the same manner as you selected numbers for the first row.
Begin by determining the difference between the first two elements of the top row. In the example below, the difference between the first two elements is -2, since 1 - 3 = -2.
The opposite of the difference between the first two elements of the top row should be the difference between the first two elements of the first column. For example, since the difference between the first two elements of the first row is -2, the difference between the first two elements of the first column should be +2. This is achieved by adding 2 to the first element. Since 3 + 2 = 5, the second element is 5.
Follow the same procedure for each subsequent set of adjacent elements. Continuing with the example below, observe that the difference between 9 and 1 is +8. Therefore, the difference between the second and third elements of the first column should be -8. Subtracting 8 from 5 yields - 3. Note that -3 is not between 1 and 12. Whenever you encounter a result that is not between 1 and 12, add or subtract 12 to that number as needed to make the result between 1 and 12. In this case, -3 + 12 = 9, so a 9 appears as the third element of the first column.
An image of the completed first row and first column of our example also appears below.
Check your work. One way to verify that you have not made any mistakes is to make sure that each whole number from 1 through 12 appears exactly one time in the first column.
Step 3: Fill in the Second Row
Once the first row and column are complete, the cells in the second row can be populated. Like the first column, the remaining cells are dependent upon the first row.
Determine the difference between the first elements of the first and second rows. Continuing with our example, we can see that the difference between the first two elements is 2, since 5 - 3 = 2. Fill in the remaining elements of the second row such that the difference between each second row element and the first row element immediately above it is the same as the difference you have just determined. In the example below, this means that the second element of the second row should be 3, because 1 + 2 = 3, and the third element should be 11, since 9 + 2 = 11.
As in the previous step, if you encounter a number less than 1 or greater than 12, add or subtract 12 as needed so that the number you write in the matrix is in the range from 1 through 12.
An image of the completed second row of our example is shown.
Check your work. Make sure that each number from 1 through 12 appears only once in the row. If this is not the case, then you have made a mistake.
Step 4: Fill in the Remaining Rows
The remaining rows should be completed in the same manner as the second row was populated.
Determine the difference between first element of the lowest row that is not completed and the first element of the row immediately above it. This difference should be replicated throughout the rest of the row. Returning to our example, we can observe that the difference between 9 and 5 is 4, so each element of the third row should be 4 greater than the element in the cell above it.
Each subsequent row of our example has been completed in this manner, and the resulting completed matrix appears below.
You can check your work once again by verifying that each number from 1 through 12 appears only once in each row and each column.
One final way to verify that the matrix is properly completed is to look along the diagonal that runs from the top left corner of the matrix to the bottom right corner. The number in each of these cells should be the same. In our example, this is the case; the number 3 appears in each cell of the diagonal.
Step 5: Translate the Numbers to Pitches
Now that the matrix is complete, you can select a few rows or colums and "translate" them into music. Each number corresponds to a specific pitch according to this list.
C# / Db 2
D# / Eb 4
F# / Gb 7
G# / Ab 9
A# / Bb 11
Select one or more rows or columns from the matrix and translate them to pitches. You can read the rows from left to right or from right to left, and the columns can be read from top to bottom or from bottom to top.
Returning to our example, the seventh row, read from left to right, was chosen for the first half of a melody. The tenth column, read from bottom to top, was chosen for the second half of the melody. The row and column are translated as follows.
10 8 4 12 11 1 3 2 7 5 6 9
A G D# B Bb C D Db Gb E F Ab
11 2 3 1 6 5 7 9 8 4 12 10
Bb C# D C F E Gb Ab G Eb B A
Step 6: Write Music!
Make a melody from the pitches. Make sure that you don't change the orders of the pitches, as changing the order of the pitches defeats the purpose creating the matrix.
Recall that the pitches generated in step 5 were as follows:
A G D# B Bb C D Db Gb E F Ab
Bb C# D C F E Gb Ab G Eb B A
I have completed our example by writing music from these pitches.
Pick your time signature, rhythms, and dynamics in any way you like. Note that using a key signature is not necessary because your melodies were not derived with any sense of key or tonality. 20th Century composers are known for writing specific dynamic, articulation, and tempo markings, so you can imitate this style by specifically spelling out these factors as well. I have done this in the example below.
Your finished melodies will be atonal and consistent with the methodologies employed by Arnold Schoenberg and other 20th Century composers.